diff --git a/changes-set.txt b/changes-set.txt index 1b1a7bfa7c..e5ee8391cb 100644 --- a/changes-set.txt +++ b/changes-set.txt @@ -22,8 +22,6 @@ proposed cbviing cbviin proposed cbviung cbviun proposed nfiing nfiin proposed nfiung nfiun -proposed nfrexg nfrex -proposed nfrexdg nfrexd proposed nfaba1g nfaba1 proposed nfabg nfab proposed hblemg hblem @@ -41,8 +39,6 @@ proposed cbviin cbviinv proposed cbviun cbviunv proposed nfiin nfiinv proposed nfiun nfiunv -proposed nfrex nfrexv -proposed nfrexd nfrexdv proposed nfaba1 nfaba1v proposed nfab nfabv proposed hblem hblemv @@ -74,7 +70,6 @@ proposed syl5com imtridcom proposed syl5 imtrid alternate proposal: sylid proposed syl56 imtridi proposed syl5d imtridd -underway syl5bi biimtrid proposed syl5bir biimtrrid proposed syl5ib imbitrid proposed syl5ibcom imbitridcom @@ -92,8 +87,67 @@ make a github issue.) DONE: Date Old New Notes +19-Jan-25 --- --- Moved surreal zero and one theorems + from SF's mathbox to main set.mm +19-Jan-25 fnssintima [same] Moved from SF's mathbox to main set.mm +18-Jan-25 elintabg [same] Moved from RP's mathbox to main set.mm +17-Jan-25 --- --- Moved Conway cut theorems + from SF's mathbox to main set.mm +17-Jan-25 rexrab [same] Moved from SF's mathbox to main set.mm +17-Jan-25 riotarab [same] Moved from SF's mathbox to main set.mm +16-Jan-25 --- --- Moved surreal ordering and birthday theorems + from SF's mathbox to main set.mm +16-Jan-25 --- --- Moved surreal full-eta theorems + from SF's mathbox to main set.mm +16-Jan-25 snres0 [same] Moved from SF's mathbox to main set.mm +15-Jan-25 eqfunressuc [same] Moved from SF's mathbox to main set.mm +15-Jan-25 eqfunresadj [same] Moved from SF's mathbox to main set.mm +15-Jan-25 sotr3 [same] Moved from SF's mathbox to main set.mm +15-Jan-25 dford5 [same] Moved from SF's mathbox to main set.mm +13-Jan-25 elunnel2 [same] moved from GS's mathbox to main set.mm +12-Jan-25 rimrhm [same] revised - eliminated unnecessary hypotheses +12-Jan-25 isrim [same] revised - eliminated unnecessary antecedent +12-Jan-25 isrim0 [same] revised - eliminated unnecessary antecedent +12-Jan-25 elrnust --- obsolete - use elfvunirn instead +12-Jan-25 elunirn2 --- obsolete - use elfvunirn instead +12-Jan-25 --- --- Moved surreal birthday and density theorems + from SF's mathbox to main set.mm +12-Jan-25 sotrine [same] Moved from SF's mathbox to main set.mm +12-Jan-25 funeldmb [same] Moved from SF's mathbox to main set.mm +11-Jan-25 fldcrng fldcrngo +11-Jan-25 recrng resrng +11-Jan-25 --- --- Moved surreal ordering theorems from + SF's mathbox to main set.mm +11-Jan-25 --- --- Moved ordinal sequence theorems from + SF's mathbox to main set.mm +11-Jan-25 tfisg [same] Moved from SF's mathbox to main set.mm +11-Jan-25 3orel13 [same] Moved from SF's mathbox to main set.mm +11-Jan-25 3pm3.2ni [same] Moved from SF's mathbox to main set.mm +10-Jan-25 syl5bi biimtrid +10-Jan-25 fldcrngd [same] Moved from SN's mathbox to main set.mm +10-Jan-25 drngringd [same] Moved from SN's mathbox to main set.mm +10-Jan-25 drnggrpd [same] Moved from SN's mathbox to main set.mm + 9-Jan-25 fvresval [same] Moved from SF's mathbox to main set.mm + 9-Jan-25 brtp [same] Moved from SF's mathbox to main set.mm + 9-Jan-25 --- --- Moved initial surreal definitions from + SF's mathbox to main set.mm + 9-Jan-25 nfrexdg nfrexd consistent with nfrald + 9-Jan-25 nfrexg nfrex consistent with nfral + 7-Jan-25 rhmdvdsr [same] Moved from TA's mathbox to main set.mm + 7-Jan-25 rhmopp [same] Moved from TA's mathbox to main set.mm + 7-Jan-25 elrhmunit [same] Moved from TA's mathbox to main set.mm + 7-Jan-25 rhmunitinv [same] Moved from TA's mathbox to main set.mm + 7-Jan-25 ringi ringdilem + 5-Jan-25 nfrexd nfrexdw consistent with nfraldw + 5-Jan-25 nfrex nfrexw consistent with nfralw + 5-Jan-25 rabss3d [same] moved from TA's mathbox to main set.mm + 4-Jan-25 smorndom smocdmdom 4-Jan-25 ralimda --- obsolete - use ralimdaa instead 3-Jan-25 srgi srgdilem + 2-Jan-25 frnnn0fsuppg fcdmnn0fsuppg + 2-Jan-25 frnnn0suppg fcdmnn0suppg + 2-Jan-25 frnnn0fsupp fcdmnn0fsupp + 2-Jan-25 frnnn0supp fcdmnn0supp 2-Jan-25 pr2nelem enpr2 28-Dec-24 fovrnd fovcdmd 28-Dec-24 fovrnda fovcdmda @@ -127,7 +181,7 @@ Date Old New Notes 13-Dec-24 wksonproplem [same] revised - eliminated hypothesis 13-Dec-24 mptmpoopabovd [same] revised - eliminated hypotheses 13-Dec-24 mptmpoopabbrd [same] revised - eliminated hypotheses -13-Dec-24 wlkRes --- obsolete - use obapresex2 instead +13-Dec-24 wlkRes --- obsolete - use opabresex2 instead 13-Dec-24 opabresex2d --- obsolete - use opabresex2 instead 27-Nov-24 eldifsucnn [same] moved from SF's mathbox to main set.mm 27-Nov-24 nnasmo [same] moved from SF's mathbox to main set.mm @@ -273,7 +327,7 @@ Date Old New Notes 26-Jun-24 rp-imass resssxp moved from RP's mathbox to main set.mm 26-Jun-24 sscon34b [same] moved from RP's mathbox to main set.mm 26-Jun-24 rcompleq [same] moved from RP's mathbox to main set.mm -26-Jun-24 funresd [same] moved from GS's mathbox to main set.mm +26-Jun-24 funresd [same] moved from GS's mathbox to main set.mmisCyclic_of_subgroup_isDomain 26-Jun-24 cmnmndd [same] moved from SN's mathbox to main set.mm 26-Jun-24 syl6req eqtr2di compare to eqtr2i or eqtr2d 16-Jun-24 syl6eqr eqtr4di compare to eqtr4i or eqtr4d diff --git a/discouraged b/discouraged index 0557785111..fc3d0bc39a 100644 --- a/discouraged +++ b/discouraged @@ -1768,6 +1768,7 @@ "bj-inftyexpidisj" is used by "bj-minftynrr". "bj-inftyexpidisj" is used by "bj-pinftynrr". "bj-inftyexpiinv" is used by "bj-inftyexpiinj". +"bj-velpwALT" is used by "bj-elpwgALT". "blo3i" is used by "ipblnfi". "blocn" is used by "blocn2". "blocn2" is used by "ubthlem1". @@ -2441,8 +2442,6 @@ "bnj518" is used by "bnj546". "bnj519" is used by "bnj535". "bnj519" is used by "bnj97". -"bnj521" is used by "bnj535". -"bnj521" is used by "bnj927". "bnj523" is used by "bnj600". "bnj523" is used by "bnj908". "bnj523" is used by "bnj934". @@ -3267,8 +3266,6 @@ "cbvsbc" is used by "cbvcsb". "cbvsbc" is used by "cbvsbcv". "ccat2s1fvwOLD" is used by "ccat2s1fstOLD". -"ccatlenOLD" is used by "ccat2s1lenOLD". -"ccatlenOLD" is used by "wlklenvclwlkOLD". "ccatval1OLD" is used by "ccat2s1p1OLD". "ccatval1OLD" is used by "ccats1val1OLD". "ccatw2s1assOLD" is used by "ccat2s1fvwOLD". @@ -4785,7 +4782,7 @@ "df-erq" is used by "nqerid". "df-erq" is used by "nqerrel". "df-exid" is used by "isexid". -"df-fld" is used by "fldcrng". +"df-fld" is used by "fldcrngo". "df-fld" is used by "flddivrng". "df-fld" is used by "isfld2". "df-gcdOLD" is used by "ee7.2aOLD". @@ -5138,7 +5135,6 @@ "dfsb1" is used by "bj-dfsb2". "dfsb1" is used by "drsb1". "dfsb1" is used by "frege55b". -"dfsb1" is used by "subsym1". "dfsb2" is used by "dfsb3". "dfvd1imp" is used by "gen11". "dfvd1impr" is used by "gen11". @@ -5292,6 +5288,9 @@ "dicdmN" is used by "dicvalrelN". "dicelvalN" is used by "dicelval2N". "dicfnN" is used by "dicdmN". +"dif1enOLD" is used by "findcard2OLD". +"dif1enlemOLD" is used by "dif1enOLD". +"dif1enlemOLD" is used by "rexdif1enOLD". "dihglbcN" is used by "dihmeetcN". "dihglbcpreN" is used by "dihglbcN". "dihglblem2N" is used by "dihglblem3N". @@ -5563,7 +5562,7 @@ "drnfc1" is used by "nfcvb". "drnfc1" is used by "nfriotad". "drngoi" is used by "dvrunz". -"drngoi" is used by "fldcrng". +"drngoi" is used by "fldcrngo". "drsb1" is used by "iotaeq". "drsb1" is used by "sb2ae". "drsb1" is used by "sbco3". @@ -9063,6 +9062,7 @@ "ispsubclN" is used by "pmapsubclN". "ispsubclN" is used by "psubcli2N". "ispsubclN" is used by "psubcliN". +"isrim0OLD" is used by "isrimOLD". "isrngo" is used by "isrngod". "isrngo" is used by "rngoi". "isrngod" is used by "iscringd". @@ -10378,11 +10378,11 @@ "nfral" is used by "nfra2". "nfral" is used by "opreu2reuALT". "nfrald" is used by "nfral". -"nfrald" is used by "nfrexdg". +"nfrald" is used by "nfrexd". "nfreud" is used by "nfreu". -"nfrexdg" is used by "nfiundg". -"nfrexdg" is used by "nfrexg". -"nfrexg" is used by "nfiung". +"nfrex" is used by "nfiung". +"nfrexd" is used by "nfiundg". +"nfrexd" is used by "nfrex". "nfs1" is used by "sb8". "nfs1" is used by "sb8e". "nfsabg" is used by "nfabg". @@ -12597,6 +12597,7 @@ "ringcvalALTV" is used by "ringccofvalALTV". "ringcvalALTV" is used by "ringchomfvalALTV". "riotaocN" is used by "glbconN". +"riotaocN" is used by "glbconNOLD". "rnbra" is used by "bra11". "rnbra" is used by "cnvbraval". "rnelshi" is used by "hmopidmchi". @@ -12907,6 +12908,7 @@ "scandx" is used by "zlmdsOLD". "scandx" is used by "zlmlemOLD". "scandx" is used by "zlmtsetOLD". +"selsALT" is used by "elALT". "setrec1lem1" is used by "setrec1lem2". "setrec1lem1" is used by "setrec1lem4". "setrec1lem1" is used by "setrec2fun". @@ -14280,7 +14282,7 @@ New usage of "0psubN" is discouraged (0 uses). New usage of "0psubclN" is discouraged (1 uses). New usage of "0r" is discouraged (11 uses). New usage of "0reALT" is discouraged (0 uses). -New usage of "0sdom1domOLD" is discouraged (0 uses). +New usage of "0sdom1domALT" is discouraged (0 uses). New usage of "0sdomgOLD" is discouraged (0 uses). New usage of "0vfval" is discouraged (9 uses). New usage of "139prmALT" is discouraged (0 uses). @@ -14307,7 +14309,7 @@ New usage of "1p2e3ALT" is discouraged (0 uses). New usage of "1pi" is discouraged (11 uses). New usage of "1pr" is discouraged (16 uses). New usage of "1psubclN" is discouraged (0 uses). -New usage of "1sdom2OLD" is discouraged (0 uses). +New usage of "1sdom2ALT" is discouraged (0 uses). New usage of "1sdomOLD" is discouraged (0 uses). New usage of "1sr" is discouraged (8 uses). New usage of "1strbasOLD" is discouraged (0 uses). @@ -14414,7 +14416,6 @@ New usage of "aaanOLD" is discouraged (0 uses). New usage of "ab0ALT" is discouraged (0 uses). New usage of "ab0OLD" is discouraged (0 uses). New usage of "ab0orvALT" is discouraged (0 uses). -New usage of "abbiOLD" is discouraged (0 uses). New usage of "abfOLD" is discouraged (0 uses). New usage of "ablo32" is discouraged (4 uses). New usage of "ablo4" is discouraged (3 uses). @@ -14872,9 +14873,9 @@ New usage of "bj-csbsnlem" is discouraged (1 uses). New usage of "bj-currypeirce" is discouraged (0 uses). New usage of "bj-denot" is discouraged (0 uses). New usage of "bj-dfid2ALT" is discouraged (0 uses). -New usage of "bj-dtru" is discouraged (0 uses). New usage of "bj-elabd2ALT" is discouraged (0 uses). New usage of "bj-elissetALT" is discouraged (0 uses). +New usage of "bj-elpwgALT" is discouraged (0 uses). New usage of "bj-equsalhv" is discouraged (0 uses). New usage of "bj-eximALT" is discouraged (1 uses). New usage of "bj-gl4" is discouraged (0 uses). @@ -14896,6 +14897,7 @@ New usage of "bj-sbeqALT" is discouraged (0 uses). New usage of "bj-sbidmOLD" is discouraged (0 uses). New usage of "bj-ssbid1ALT" is discouraged (0 uses). New usage of "bj-ssbid2ALT" is discouraged (0 uses). +New usage of "bj-velpwALT" is discouraged (1 uses). New usage of "bj-vtoclgfALT" is discouraged (0 uses). New usage of "bj-xpima1snALT" is discouraged (0 uses). New usage of "blo3i" is discouraged (1 uses). @@ -15156,7 +15158,6 @@ New usage of "bnj446" is discouraged (3 uses). New usage of "bnj517" is discouraged (1 uses). New usage of "bnj518" is discouraged (2 uses). New usage of "bnj519" is discouraged (2 uses). -New usage of "bnj521" is discouraged (2 uses). New usage of "bnj523" is discouraged (3 uses). New usage of "bnj524" is discouraged (0 uses). New usage of "bnj525" is discouraged (7 uses). @@ -15328,7 +15329,6 @@ New usage of "cbvab" is discouraged (4 uses). New usage of "cbvabwOLD" is discouraged (0 uses). New usage of "cbval" is discouraged (4 uses). New usage of "cbval2" is discouraged (1 uses). -New usage of "cbval2vOLD" is discouraged (0 uses). New usage of "cbval2vv" is discouraged (0 uses). New usage of "cbvald" is discouraged (16 uses). New usage of "cbvaldva" is discouraged (1 uses). @@ -15381,6 +15381,7 @@ New usage of "cbvreuwOLD" is discouraged (0 uses). New usage of "cbvrex" is discouraged (4 uses). New usage of "cbvrex2v" is discouraged (0 uses). New usage of "cbvrexcsf" is discouraged (1 uses). +New usage of "cbvrexdva2OLD" is discouraged (0 uses). New usage of "cbvrexf" is discouraged (1 uses). New usage of "cbvrexsv" is discouraged (1 uses). New usage of "cbvrexv" is discouraged (3 uses). @@ -15397,10 +15398,8 @@ New usage of "ccat2s1fstOLD" is discouraged (0 uses). New usage of "ccat2s1fvwALT" is discouraged (0 uses). New usage of "ccat2s1fvwALTOLD" is discouraged (0 uses). New usage of "ccat2s1fvwOLD" is discouraged (1 uses). -New usage of "ccat2s1lenOLD" is discouraged (0 uses). New usage of "ccat2s1p1OLD" is discouraged (0 uses). New usage of "ccat2s1p2OLD" is discouraged (0 uses). -New usage of "ccatlenOLD" is discouraged (2 uses). New usage of "ccats1val1OLD" is discouraged (0 uses). New usage of "ccatval1OLD" is discouraged (2 uses). New usage of "ccatw2s1assOLD" is discouraged (2 uses). @@ -15821,6 +15820,7 @@ New usage of "csbrngVD" is discouraged (0 uses). New usage of "csbsngVD" is discouraged (0 uses). New usage of "csbunigVD" is discouraged (0 uses). New usage of "csbxpgVD" is discouraged (0 uses). +New usage of "cshwsexaOLD" is discouraged (0 uses). New usage of "csmdsymi" is discouraged (0 uses). New usage of "currysetALT" is discouraged (0 uses). New usage of "cvati" is discouraged (1 uses). @@ -16052,7 +16052,7 @@ New usage of "dfnul3OLD" is discouraged (0 uses). New usage of "dfnul4OLD" is discouraged (0 uses). New usage of "dfpjop" is discouraged (4 uses). New usage of "dfrecs3OLD" is discouraged (0 uses). -New usage of "dfsb1" is discouraged (4 uses). +New usage of "dfsb1" is discouraged (3 uses). New usage of "dfsb2" is discouraged (1 uses). New usage of "dfsb3" is discouraged (0 uses). New usage of "dfsn2ALT" is discouraged (0 uses). @@ -16114,7 +16114,9 @@ New usage of "dicelval2N" is discouraged (0 uses). New usage of "dicelvalN" is discouraged (1 uses). New usage of "dicfnN" is discouraged (1 uses). New usage of "dicvalrelN" is discouraged (0 uses). -New usage of "dif1enALT" is discouraged (0 uses). +New usage of "dif1enOLD" is discouraged (1 uses). +New usage of "dif1enlemOLD" is discouraged (2 uses). +New usage of "dif1ennnALT" is discouraged (0 uses). New usage of "difidALT" is discouraged (0 uses). New usage of "difopabOLD" is discouraged (0 uses). New usage of "dih0bN" is discouraged (0 uses). @@ -16258,6 +16260,7 @@ New usage of "drsb1" is discouraged (3 uses). New usage of "dsndx" is discouraged (20 uses). New usage of "dtruALT" is discouraged (0 uses). New usage of "dtruALT2" is discouraged (6 uses). +New usage of "dtruOLD" is discouraged (0 uses). New usage of "dtrucor2" is discouraged (0 uses). New usage of "dummylink" is discouraged (0 uses). New usage of "dvadiaN" is discouraged (1 uses). @@ -16510,6 +16513,7 @@ New usage of "elimnv" is discouraged (1 uses). New usage of "elimnvu" is discouraged (5 uses). New usage of "elimph" is discouraged (3 uses). New usage of "elimphu" is discouraged (3 uses). +New usage of "elintabOLD" is discouraged (0 uses). New usage of "ellnfn" is discouraged (5 uses). New usage of "ellnop" is discouraged (9 uses). New usage of "elnanel" is discouraged (1 uses). @@ -16542,6 +16546,7 @@ New usage of "elreal" is discouraged (7 uses). New usage of "elreal2" is discouraged (3 uses). New usage of "elringchomALTV" is discouraged (1 uses). New usage of "elrngchomALTV" is discouraged (1 uses). +New usage of "elrnustOLD" is discouraged (0 uses). New usage of "elsetrecslem" is discouraged (1 uses). New usage of "elspancl" is discouraged (1 uses). New usage of "elspani" is discouraged (2 uses). @@ -16553,6 +16558,7 @@ New usage of "elspansn5" is discouraged (2 uses). New usage of "elspansncl" is discouraged (1 uses). New usage of "elspansni" is discouraged (2 uses). New usage of "eltpsgOLD" is discouraged (0 uses). +New usage of "elunirn2OLD" is discouraged (0 uses). New usage of "elunirnALT" is discouraged (0 uses). New usage of "elunop" is discouraged (7 uses). New usage of "elunop2" is discouraged (0 uses). @@ -16560,6 +16566,7 @@ New usage of "en0ALT" is discouraged (0 uses). New usage of "en0OLD" is discouraged (0 uses). New usage of "en1OLD" is discouraged (0 uses). New usage of "en1bOLD" is discouraged (0 uses). +New usage of "en1eqsnOLD" is discouraged (0 uses). New usage of "en1unielOLD" is discouraged (0 uses). New usage of "en2snOLD" is discouraged (0 uses). New usage of "en2snOLDOLD" is discouraged (0 uses). @@ -16568,6 +16575,7 @@ New usage of "en3lplem1VD" is discouraged (1 uses). New usage of "en3lplem2VD" is discouraged (1 uses). New usage of "enfiALT" is discouraged (0 uses). New usage of "enfiiOLD" is discouraged (0 uses). +New usage of "enp1iOLD" is discouraged (0 uses). New usage of "enpr2dOLD" is discouraged (0 uses). New usage of "enqbreq" is discouraged (5 uses). New usage of "enqbreq2" is discouraged (4 uses). @@ -16580,7 +16588,7 @@ New usage of "enrer" is discouraged (8 uses). New usage of "enrex" is discouraged (9 uses). New usage of "ensn1OLD" is discouraged (0 uses). New usage of "ensucne0OLD" is discouraged (0 uses). -New usage of "epweonOLD" is discouraged (0 uses). +New usage of "epweonALT" is discouraged (0 uses). New usage of "eq0ALT" is discouraged (0 uses). New usage of "eq0OLDOLD" is discouraged (0 uses). New usage of "eq0rdvALT" is discouraged (0 uses). @@ -16676,6 +16684,7 @@ New usage of "f1cofveqaeqALT" is discouraged (0 uses). New usage of "f1dmOLD" is discouraged (0 uses). New usage of "f1dom2gOLD" is discouraged (0 uses). New usage of "f1eqcocnvOLD" is discouraged (0 uses). +New usage of "f1finf1oOLD" is discouraged (0 uses). New usage of "f1omoALT" is discouraged (0 uses). New usage of "f1oweALT" is discouraged (0 uses). New usage of "f1rhm0to0ALT" is discouraged (0 uses). @@ -16695,6 +16704,7 @@ New usage of "fimacnvOLD" is discouraged (0 uses). New usage of "fimadmfoALT" is discouraged (0 uses). New usage of "findOLD" is discouraged (0 uses). New usage of "findcard2OLD" is discouraged (0 uses). +New usage of "findcard3OLD" is discouraged (0 uses). New usage of "fineqvacALT" is discouraged (0 uses). New usage of "fldcALTV" is discouraged (0 uses). New usage of "fldcatALTV" is discouraged (1 uses). @@ -16740,13 +16750,13 @@ New usage of "funopg" is discouraged (0 uses). New usage of "funopsn" is discouraged (2 uses). New usage of "fvimacnvALT" is discouraged (0 uses). New usage of "fvmptopabOLD" is discouraged (0 uses). +New usage of "fvn0fvelrnOLD" is discouraged (0 uses). New usage of "fvpr1OLD" is discouraged (0 uses). New usage of "fvpr2OLD" is discouraged (0 uses). New usage of "fvpr2gOLD" is discouraged (0 uses). New usage of "fvprcALT" is discouraged (0 uses). +New usage of "fvssunirnOLD" is discouraged (0 uses). New usage of "gcdabsOLD" is discouraged (0 uses). -New usage of "gcdmultipleOLD" is discouraged (0 uses). -New usage of "gcdmultiplezOLD" is discouraged (0 uses). New usage of "gen11" is discouraged (19 uses). New usage of "gen11nv" is discouraged (5 uses). New usage of "gen12" is discouraged (7 uses). @@ -16773,6 +16783,7 @@ New usage of "gidsn" is discouraged (1 uses). New usage of "gidval" is discouraged (3 uses). New usage of "glb0N" is discouraged (2 uses). New usage of "glbconN" is discouraged (1 uses). +New usage of "glbconNOLD" is discouraged (0 uses). New usage of "glbconxN" is discouraged (1 uses). New usage of "goeqi" is discouraged (0 uses). New usage of "golem1" is discouraged (1 uses). @@ -16835,7 +16846,6 @@ New usage of "grpstr" is discouraged (0 uses). New usage of "grpsubfvalALT" is discouraged (0 uses). New usage of "gsumbagdiagOLD" is discouraged (1 uses). New usage of "gsumbagdiaglemOLD" is discouraged (2 uses). -New usage of "gsumccatOLD" is discouraged (0 uses). New usage of "gt-lt" is discouraged (0 uses). New usage of "gt-lth" is discouraged (1 uses). New usage of "gt0srpr" is discouraged (2 uses). @@ -17344,9 +17354,12 @@ New usage of "indistpsALTOLD" is discouraged (0 uses). New usage of "indistpsx" is discouraged (0 uses). New usage of "indpi" is discouraged (1 uses). New usage of "indthincALT" is discouraged (0 uses). +New usage of "infn0ALT" is discouraged (0 uses). New usage of "infpssALT" is discouraged (0 uses). +New usage of "infsdomnnOLD" is discouraged (0 uses). New usage of "int2" is discouraged (3 uses). New usage of "int3" is discouraged (1 uses). +New usage of "intidOLD" is discouraged (0 uses). New usage of "intnatN" is discouraged (0 uses). New usage of "intprOLD" is discouraged (0 uses). New usage of "intprgOLD" is discouraged (0 uses). @@ -17389,6 +17402,7 @@ New usage of "ipval3" is discouraged (1 uses). New usage of "ipz" is discouraged (1 uses). New usage of "isablo" is discouraged (3 uses). New usage of "isabloi" is discouraged (3 uses). +New usage of "isanmbfmOLD" is discouraged (0 uses). New usage of "isarep1OLD" is discouraged (0 uses). New usage of "isass" is discouraged (1 uses). New usage of "isblo" is discouraged (5 uses). @@ -17416,6 +17430,7 @@ New usage of "ishlatiN" is discouraged (0 uses). New usage of "ishlo" is discouraged (4 uses). New usage of "ishmo" is discouraged (0 uses). New usage of "ishst" is discouraged (4 uses). +New usage of "isinfOLD" is discouraged (0 uses). New usage of "isline4N" is discouraged (0 uses). New usage of "islno" is discouraged (5 uses). New usage of "islpolN" is discouraged (6 uses). @@ -17441,6 +17456,8 @@ New usage of "ispointN" is discouraged (2 uses). New usage of "isposixOLD" is discouraged (0 uses). New usage of "ispsubcl2N" is discouraged (0 uses). New usage of "ispsubclN" is discouraged (10 uses). +New usage of "isrim0OLD" is discouraged (1 uses). +New usage of "isrimOLD" is discouraged (0 uses). New usage of "isrngo" is discouraged (2 uses). New usage of "isrngod" is discouraged (1 uses). New usage of "issgrpALT" is discouraged (1 uses). @@ -17461,6 +17478,7 @@ New usage of "itg1addlem4OLD" is discouraged (0 uses). New usage of "itvndx" is discouraged (6 uses). New usage of "iunconnALT" is discouraged (0 uses). New usage of "iunconnlem2" is discouraged (1 uses). +New usage of "iunidOLD" is discouraged (0 uses). New usage of "iunopabOLD" is discouraged (0 uses). New usage of "ivthALT" is discouraged (0 uses). New usage of "jaoded" is discouraged (1 uses). @@ -17623,7 +17641,6 @@ New usage of "lsatfixedN" is discouraged (1 uses). New usage of "lshpinN" is discouraged (0 uses). New usage of "lshpnel2N" is discouraged (0 uses). New usage of "lshpset2N" is discouraged (1 uses). -New usage of "lsmidmOLD" is discouraged (0 uses). New usage of "ltaddnq" is discouraged (7 uses). New usage of "ltaddpr" is discouraged (5 uses). New usage of "ltaddpr2" is discouraged (1 uses). @@ -17738,6 +17755,7 @@ New usage of "matvscaOLD" is discouraged (0 uses). New usage of "max1ALT" is discouraged (0 uses). New usage of "mayete3i" is discouraged (1 uses). New usage of "mayetes3i" is discouraged (0 uses). +New usage of "mbfmbfmOLD" is discouraged (0 uses). New usage of "mdbr" is discouraged (6 uses). New usage of "mdbr2" is discouraged (7 uses). New usage of "mdbr3" is discouraged (0 uses). @@ -17982,8 +18000,8 @@ New usage of "nfralwOLD" is discouraged (0 uses). New usage of "nfreu" is discouraged (0 uses). New usage of "nfreud" is discouraged (1 uses). New usage of "nfreuwOLD" is discouraged (0 uses). -New usage of "nfrexdg" is discouraged (2 uses). -New usage of "nfrexg" is discouraged (1 uses). +New usage of "nfrex" is discouraged (1 uses). +New usage of "nfrexd" is discouraged (2 uses). New usage of "nfriotad" is discouraged (0 uses). New usage of "nfrmo" is discouraged (0 uses). New usage of "nfrmod" is discouraged (0 uses). @@ -18131,6 +18149,7 @@ New usage of "nnexALT" is discouraged (3 uses). New usage of "nnfiOLD" is discouraged (0 uses). New usage of "nnindALT" is discouraged (0 uses). New usage of "nnne0ALT" is discouraged (0 uses). +New usage of "nnsdomgOLD" is discouraged (0 uses). New usage of "noelOLD" is discouraged (0 uses). New usage of "nonbooli" is discouraged (0 uses). New usage of "norcomOLD" is discouraged (0 uses). @@ -18296,6 +18315,7 @@ New usage of "odubasOLD" is discouraged (0 uses). New usage of "oldmm3N" is discouraged (2 uses). New usage of "olposN" is discouraged (0 uses). New usage of "om0x" is discouraged (0 uses). +New usage of "ominfOLD" is discouraged (0 uses). New usage of "omlfh1N" is discouraged (2 uses). New usage of "omlfh3N" is discouraged (0 uses). New usage of "omllaw2N" is discouraged (3 uses). @@ -18324,6 +18344,8 @@ New usage of "onomeneqOLD" is discouraged (0 uses). New usage of "onsetreclem1" is discouraged (2 uses). New usage of "onsetreclem2" is discouraged (1 uses). New usage of "onsetreclem3" is discouraged (1 uses). +New usage of "ontrciOLD" is discouraged (0 uses). +New usage of "onuniorsuciOLD" is discouraged (0 uses). New usage of "opabid" is discouraged (2 uses). New usage of "opabresex2d" is discouraged (1 uses). New usage of "opelcn" is discouraged (1 uses). @@ -18362,6 +18384,7 @@ New usage of "ordelordALT" is discouraged (0 uses). New usage of "ordelordALTVD" is discouraged (0 uses). New usage of "ordpinq" is discouraged (6 uses). New usage of "ordpipq" is discouraged (5 uses). +New usage of "ordsucOLD" is discouraged (0 uses). New usage of "orim12dALT" is discouraged (0 uses). New usage of "orthcom" is discouraged (6 uses). New usage of "orthin" is discouraged (2 uses). @@ -18725,6 +18748,8 @@ New usage of "r19.30OLD" is discouraged (0 uses). New usage of "r19.35OLD" is discouraged (0 uses). New usage of "r1omALT" is discouraged (0 uses). New usage of "r1pwALT" is discouraged (0 uses). +New usage of "rabbiiaOLD" is discouraged (0 uses). +New usage of "rabeqcOLD" is discouraged (0 uses). New usage of "rabeqiOLD" is discouraged (0 uses). New usage of "rabid2OLD" is discouraged (0 uses). New usage of "rabrabiOLD" is discouraged (0 uses). @@ -18814,12 +18839,14 @@ New usage of "retbwax1" is discouraged (0 uses). New usage of "retbwax2" is discouraged (3 uses). New usage of "retbwax3" is discouraged (0 uses). New usage of "retbwax4" is discouraged (0 uses). +New usage of "reuanidOLD" is discouraged (0 uses). New usage of "reutruALT" is discouraged (0 uses). New usage of "rexabOLD" is discouraged (0 uses). New usage of "rexbiOLD" is discouraged (0 uses). New usage of "rexbidvALT" is discouraged (0 uses). New usage of "rexbidvaALT" is discouraged (0 uses). New usage of "rexcomOLD" is discouraged (0 uses). +New usage of "rexdif1enOLD" is discouraged (0 uses). New usage of "reximdvaiOLD" is discouraged (0 uses). New usage of "reximiaOLD" is discouraged (0 uses). New usage of "rexlimddvcbv" is discouraged (0 uses). @@ -18842,6 +18869,7 @@ New usage of "riesz2" is discouraged (0 uses). New usage of "riesz3i" is discouraged (2 uses). New usage of "riesz4" is discouraged (4 uses). New usage of "riesz4i" is discouraged (1 uses). +New usage of "rimrhmOLD" is discouraged (0 uses). New usage of "ringcbasALTV" is discouraged (7 uses). New usage of "ringcbasbasALTV" is discouraged (3 uses). New usage of "ringccatALTV" is discouraged (5 uses). @@ -18855,9 +18883,10 @@ New usage of "ringcinvALTV" is discouraged (1 uses). New usage of "ringcisoALTV" is discouraged (0 uses). New usage of "ringcsectALTV" is discouraged (1 uses). New usage of "ringcvalALTV" is discouraged (3 uses). -New usage of "riotaocN" is discouraged (1 uses). +New usage of "riotaocN" is discouraged (2 uses). New usage of "rlimaddOLD" is discouraged (0 uses). New usage of "rlimmulOLD" is discouraged (0 uses). +New usage of "rmoanidOLD" is discouraged (0 uses). New usage of "rmoanimALT" is discouraged (0 uses). New usage of "rmobidvaOLD" is discouraged (0 uses). New usage of "rmodislmodOLD" is discouraged (0 uses). @@ -19019,6 +19048,7 @@ New usage of "scandx" is discouraged (20 uses). New usage of "scmateALT" is discouraged (0 uses). New usage of "sdom0OLD" is discouraged (0 uses). New usage of "sdom1OLD" is discouraged (0 uses). +New usage of "selsALT" is discouraged (1 uses). New usage of "seq1hcau" is discouraged (0 uses). New usage of "seqp1iOLD" is discouraged (0 uses). New usage of "setrec1lem1" is discouraged (3 uses). @@ -19131,9 +19161,10 @@ New usage of "smfval" is discouraged (17 uses). New usage of "smgrpassOLD" is discouraged (1 uses). New usage of "smgrpismgmOLD" is discouraged (1 uses). New usage of "smgrpmgm" is discouraged (1 uses). -New usage of "sn-elALT2" is discouraged (0 uses). +New usage of "sn-exelALT" is discouraged (0 uses). New usage of "sn-wcdeq" is discouraged (0 uses). New usage of "snatpsubN" is discouraged (3 uses). +New usage of "snelpwiOLD" is discouraged (0 uses). New usage of "snelpwrVD" is discouraged (0 uses). New usage of "snexALT" is discouraged (1 uses). New usage of "snnen2oOLD" is discouraged (0 uses). @@ -19296,6 +19327,7 @@ New usage of "strndxid" is discouraged (2 uses). New usage of "sucdom2OLD" is discouraged (0 uses). New usage of "sucdomOLD" is discouraged (0 uses). New usage of "suceloniOLD" is discouraged (0 uses). +New usage of "sucexeloniOLD" is discouraged (0 uses). New usage of "sucidALT" is discouraged (0 uses). New usage of "sucidALTVD" is discouraged (0 uses). New usage of "sucidVD" is discouraged (0 uses). @@ -19578,7 +19610,6 @@ New usage of "wl-section-prop" is discouraged (0 uses). New usage of "wl-syls1" is discouraged (0 uses). New usage of "wl-syls2" is discouraged (0 uses). New usage of "wlkResOLD" is discouraged (0 uses). -New usage of "wlklenvclwlkOLD" is discouraged (0 uses). New usage of "wrecseq123OLD" is discouraged (0 uses). New usage of "wunfuncOLD" is discouraged (0 uses). New usage of "wunnatOLD" is discouraged (0 uses). @@ -19591,6 +19622,7 @@ New usage of "xihopellsmN" is discouraged (0 uses). New usage of "xorbi12iOLD" is discouraged (0 uses). New usage of "xorcomOLD" is discouraged (0 uses). New usage of "xpexgALT" is discouraged (0 uses). +New usage of "xpfiOLD" is discouraged (0 uses). New usage of "xrge0tmdALT" is discouraged (0 uses). New usage of "zaddablx" is discouraged (0 uses). New usage of "zexALT" is discouraged (1 uses). @@ -19626,7 +19658,7 @@ Proof modification of "0nelopabOLD" is discouraged (91 steps). Proof modification of "0nnnALT" is discouraged (11 steps). Proof modification of "0posOLD" is discouraged (66 steps). Proof modification of "0reALT" is discouraged (15 steps). -Proof modification of "0sdom1domOLD" is discouraged (31 steps). +Proof modification of "0sdom1domALT" is discouraged (31 steps). Proof modification of "0sdomgOLD" is discouraged (53 steps). Proof modification of "139prmALT" is discouraged (758 steps). Proof modification of "19.21a3con13vVD" is discouraged (107 steps). @@ -19640,7 +19672,7 @@ Proof modification of "1onOLD" is discouraged (9 steps). Proof modification of "1onnALT" is discouraged (16 steps). Proof modification of "1p1e2apr1" is discouraged (7 steps). Proof modification of "1p2e3ALT" is discouraged (7 steps). -Proof modification of "1sdom2OLD" is discouraged (18 steps). +Proof modification of "1sdom2ALT" is discouraged (18 steps). Proof modification of "1sdomOLD" is discouraged (294 steps). Proof modification of "1strbasOLD" is discouraged (22 steps). Proof modification of "1strwunOLD" is discouraged (20 steps). @@ -19686,7 +19718,6 @@ Proof modification of "aaanOLD" is discouraged (38 steps). Proof modification of "ab0ALT" is discouraged (33 steps). Proof modification of "ab0OLD" is discouraged (146 steps). Proof modification of "ab0orvALT" is discouraged (19 steps). -Proof modification of "abbiOLD" is discouraged (51 steps). Proof modification of "abfOLD" is discouraged (13 steps). Proof modification of "abn0OLD" is discouraged (28 steps). Proof modification of "abrexexgOLD" is discouraged (43 steps). @@ -19828,6 +19859,7 @@ Proof modification of "bj-19.21t0" is discouraged (39 steps). Proof modification of "bj-19.41al" is discouraged (51 steps). Proof modification of "bj-a1k" is discouraged (10 steps). Proof modification of "bj-ab0" is discouraged (39 steps). +Proof modification of "bj-abex" is discouraged (35 steps). Proof modification of "bj-abf" is discouraged (13 steps). Proof modification of "bj-ablsscmn" is discouraged (10 steps). Proof modification of "bj-ablsscmnel" is discouraged (5 steps). @@ -19835,6 +19867,8 @@ Proof modification of "bj-ablssgrp" is discouraged (10 steps). Proof modification of "bj-ablssgrpel" is discouraged (5 steps). Proof modification of "bj-abv" is discouraged (42 steps). Proof modification of "bj-abvALT" is discouraged (29 steps). +Proof modification of "bj-adjfrombun" is discouraged (28 steps). +Proof modification of "bj-adjg1" is discouraged (63 steps). Proof modification of "bj-aecomsv" is discouraged (18 steps). Proof modification of "bj-alcomexcom" is discouraged (45 steps). Proof modification of "bj-aleximiALT" is discouraged (25 steps). @@ -19846,6 +19880,8 @@ Proof modification of "bj-ax12v3ALT" is discouraged (44 steps). Proof modification of "bj-ax6e" is discouraged (43 steps). Proof modification of "bj-ax6elem1" is discouraged (27 steps). Proof modification of "bj-ax6elem2" is discouraged (23 steps). +Proof modification of "bj-axadj" is discouraged (38 steps). +Proof modification of "bj-axbun" is discouraged (20 steps). Proof modification of "bj-axc10" is discouraged (30 steps). Proof modification of "bj-axc10v" is discouraged (37 steps). Proof modification of "bj-axc11nv" is discouraged (5 steps). @@ -19856,6 +19892,7 @@ Proof modification of "bj-axc16g16" is discouraged (25 steps). Proof modification of "bj-axc4" is discouraged (15 steps). Proof modification of "bj-axd2d" is discouraged (16 steps). Proof modification of "bj-axdd2" is discouraged (28 steps). +Proof modification of "bj-axsn" is discouraged (12 steps). Proof modification of "bj-axtd" is discouraged (26 steps). Proof modification of "bj-bibibi" is discouraged (34 steps). Proof modification of "bj-bijust0ALT" is discouraged (6 steps). @@ -19893,6 +19930,7 @@ Proof modification of "bj-ceqsaltv" is discouraged (45 steps). Proof modification of "bj-ceqsalv" is discouraged (10 steps). Proof modification of "bj-clel3gALT" is discouraged (59 steps). Proof modification of "bj-cleljusti" is discouraged (20 steps). +Proof modification of "bj-clex" is discouraged (50 steps). Proof modification of "bj-cmnssmnd" is discouraged (30 steps). Proof modification of "bj-cmnssmndel" is discouraged (5 steps). Proof modification of "bj-consensusALT" is discouraged (30 steps). @@ -19907,7 +19945,6 @@ Proof modification of "bj-dfnnf3" is discouraged (34 steps). Proof modification of "bj-disj2r" is discouraged (88 steps). Proof modification of "bj-disjsn01" is discouraged (18 steps). Proof modification of "bj-drnf2v" is discouraged (10 steps). -Proof modification of "bj-dtru" is discouraged (146 steps). Proof modification of "bj-dtrucor2v" is discouraged (31 steps). Proof modification of "bj-dvelimdv" is discouraged (64 steps). Proof modification of "bj-dvelimdv1" is discouraged (63 steps). @@ -19916,6 +19953,7 @@ Proof modification of "bj-eeanvw" is discouraged (32 steps). Proof modification of "bj-elabd2ALT" is discouraged (100 steps). Proof modification of "bj-elabtru" is discouraged (25 steps). Proof modification of "bj-elissetALT" is discouraged (24 steps). +Proof modification of "bj-elpwgALT" is discouraged (29 steps). Proof modification of "bj-endbase" is discouraged (81 steps). Proof modification of "bj-endcomp" is discouraged (81 steps). Proof modification of "bj-endmnd" is discouraged (234 steps). @@ -19998,6 +20036,9 @@ Proof modification of "bj-opelidb1ALT" is discouraged (46 steps). Proof modification of "bj-orim2" is discouraged (31 steps). Proof modification of "bj-peircecurry" is discouraged (58 steps). Proof modification of "bj-peircestab" is discouraged (65 steps). +Proof modification of "bj-prex" is discouraged (32 steps). +Proof modification of "bj-prexg" is discouraged (43 steps). +Proof modification of "bj-prfromadj" is discouraged (28 steps). Proof modification of "bj-pw0ALT" is discouraged (28 steps). Proof modification of "bj-rababw" is discouraged (34 steps). Proof modification of "bj-rabtrALT" is discouraged (36 steps). @@ -20016,6 +20057,9 @@ Proof modification of "bj-sbidmOLD" is discouraged (41 steps). Proof modification of "bj-sbievv" is discouraged (25 steps). Proof modification of "bj-smgrpssmgm" is discouraged (51 steps). Proof modification of "bj-smgrpssmgmel" is discouraged (5 steps). +Proof modification of "bj-snex" is discouraged (25 steps). +Proof modification of "bj-snexg" is discouraged (46 steps). +Proof modification of "bj-snfromadj" is discouraged (22 steps). Proof modification of "bj-spcimdv" is discouraged (56 steps). Proof modification of "bj-spcimdvv" is discouraged (56 steps). Proof modification of "bj-spimtv" is discouraged (52 steps). @@ -20030,8 +20074,10 @@ Proof modification of "bj-stdpc5" is discouraged (20 steps). Proof modification of "bj-subst" is discouraged (50 steps). Proof modification of "bj-substax12" is discouraged (89 steps). Proof modification of "bj-substw" is discouraged (49 steps). +Proof modification of "bj-unexg" is discouraged (106 steps). Proof modification of "bj-vecssmod" is discouraged (18 steps). Proof modification of "bj-vecssmodel" is discouraged (5 steps). +Proof modification of "bj-velpwALT" is discouraged (24 steps). Proof modification of "bj-vjust" is discouraged (22 steps). Proof modification of "bj-vtocl" is discouraged (12 steps). Proof modification of "bj-vtoclf" is discouraged (29 steps). @@ -20058,7 +20104,6 @@ Proof modification of "catcoppcclOLD" is discouraged (402 steps). Proof modification of "catcxpcclOLD" is discouraged (864 steps). Proof modification of "cayleyhamiltonALT" is discouraged (657 steps). Proof modification of "cbvabwOLD" is discouraged (60 steps). -Proof modification of "cbval2vOLD" is discouraged (85 steps). Proof modification of "cbveuALT" is discouraged (48 steps). Proof modification of "cbveuwOLD" is discouraged (29 steps). Proof modification of "cbvexsv" is discouraged (29 steps). @@ -20070,16 +20115,15 @@ Proof modification of "cbvopabvOLD" is discouraged (20 steps). Proof modification of "cbvralfwOLD" is discouraged (139 steps). Proof modification of "cbvreuvwOLD" is discouraged (13 steps). Proof modification of "cbvreuwOLD" is discouraged (145 steps). +Proof modification of "cbvrexdva2OLD" is discouraged (109 steps). Proof modification of "cbvriotavwOLD" is discouraged (13 steps). Proof modification of "cbvrmowOLD" is discouraged (58 steps). Proof modification of "ccat2s1fstOLD" is discouraged (43 steps). Proof modification of "ccat2s1fvwALT" is discouraged (116 steps). Proof modification of "ccat2s1fvwALTOLD" is discouraged (150 steps). Proof modification of "ccat2s1fvwOLD" is discouraged (203 steps). -Proof modification of "ccat2s1lenOLD" is discouraged (66 steps). Proof modification of "ccat2s1p1OLD" is discouraged (93 steps). Proof modification of "ccat2s1p2OLD" is discouraged (162 steps). -Proof modification of "ccatlenOLD" is discouraged (119 steps). Proof modification of "ccats1val1OLD" is discouraged (38 steps). Proof modification of "ccatval1OLD" is discouraged (145 steps). Proof modification of "ccatw2s1assOLD" is discouraged (48 steps). @@ -20140,6 +20184,7 @@ Proof modification of "csbrngVD" is discouraged (266 steps). Proof modification of "csbsngVD" is discouraged (204 steps). Proof modification of "csbunigVD" is discouraged (302 steps). Proof modification of "csbxpgVD" is discouraged (538 steps). +Proof modification of "cshwsexaOLD" is discouraged (168 steps). Proof modification of "curryset" is discouraged (72 steps). Proof modification of "currysetALT" is discouraged (13 steps). Proof modification of "currysetlem" is discouraged (49 steps). @@ -20185,7 +20230,9 @@ Proof modification of "dfvd3anir" is discouraged (19 steps). Proof modification of "dfvd3i" is discouraged (19 steps). Proof modification of "dfvd3ir" is discouraged (19 steps). Proof modification of "dfwrecsOLD" is discouraged (619 steps). -Proof modification of "dif1enALT" is discouraged (335 steps). +Proof modification of "dif1enOLD" is discouraged (313 steps). +Proof modification of "dif1enlemOLD" is discouraged (285 steps). +Proof modification of "dif1ennnALT" is discouraged (335 steps). Proof modification of "difidALT" is discouraged (14 steps). Proof modification of "difopabOLD" is discouraged (171 steps). Proof modification of "dih2dimbALTN" is discouraged (450 steps). @@ -20206,6 +20253,7 @@ Proof modification of "drnfc2" is discouraged (59 steps). Proof modification of "drnfc2OLD" is discouraged (52 steps). Proof modification of "dtruALT" is discouraged (79 steps). Proof modification of "dtruALT2" is discouraged (161 steps). +Proof modification of "dtruOLD" is discouraged (157 steps). Proof modification of "dummylink" is discouraged (1 steps). Proof modification of "dveel2ALT" is discouraged (22 steps). Proof modification of "dveeq1-o" is discouraged (20 steps). @@ -20377,7 +20425,7 @@ Proof modification of "el1" is discouraged (12 steps). Proof modification of "el12" is discouraged (19 steps). Proof modification of "el123" is discouraged (26 steps). Proof modification of "el2122old" is discouraged (25 steps). -Proof modification of "elALT" is discouraged (27 steps). +Proof modification of "elALT" is discouraged (16 steps). Proof modification of "elALT2" is discouraged (48 steps). Proof modification of "elabOLD" is discouraged (10 steps). Proof modification of "elabgOLD" is discouraged (47 steps). @@ -20403,6 +20451,7 @@ Proof modification of "eliminable2b" is discouraged (7 steps). Proof modification of "eliminable2c" is discouraged (8 steps). Proof modification of "eliminable3a" is discouraged (7 steps). Proof modification of "eliminable3b" is discouraged (8 steps). +Proof modification of "elintabOLD" is discouraged (75 steps). Proof modification of "elopabrOLD" is discouraged (58 steps). Proof modification of "elopaelxpOLD" is discouraged (47 steps). Proof modification of "eloprabgaOLD" is discouraged (269 steps). @@ -20412,12 +20461,15 @@ Proof modification of "elpwgdedVD" is discouraged (23 steps). Proof modification of "elpwi2OLD" is discouraged (21 steps). Proof modification of "elrabiOLD" is discouraged (55 steps). Proof modification of "elrefsymrels3" is discouraged (65 steps). +Proof modification of "elrnustOLD" is discouraged (4 steps). Proof modification of "eltpsgOLD" is discouraged (73 steps). +Proof modification of "elunirn2OLD" is discouraged (65 steps). Proof modification of "elunirnALT" is discouraged (38 steps). Proof modification of "en0ALT" is discouraged (62 steps). Proof modification of "en0OLD" is discouraged (86 steps). Proof modification of "en1OLD" is discouraged (164 steps). Proof modification of "en1bOLD" is discouraged (83 steps). +Proof modification of "en1eqsnOLD" is discouraged (86 steps). Proof modification of "en1unielOLD" is discouraged (39 steps). Proof modification of "en2snOLD" is discouraged (56 steps). Proof modification of "en2snOLDOLD" is discouraged (35 steps). @@ -20426,10 +20478,11 @@ Proof modification of "en3lplem1VD" is discouraged (95 steps). Proof modification of "en3lplem2VD" is discouraged (267 steps). Proof modification of "enfiALT" is discouraged (42 steps). Proof modification of "enfiiOLD" is discouraged (14 steps). +Proof modification of "enp1iOLD" is discouraged (132 steps). Proof modification of "enpr2dOLD" is discouraged (114 steps). Proof modification of "ensn1OLD" is discouraged (47 steps). Proof modification of "ensucne0OLD" is discouraged (82 steps). -Proof modification of "epweonOLD" is discouraged (86 steps). +Proof modification of "epweonALT" is discouraged (86 steps). Proof modification of "eq0ALT" is discouraged (31 steps). Proof modification of "eq0OLDOLD" is discouraged (6 steps). Proof modification of "eq0rdvALT" is discouraged (25 steps). @@ -20485,6 +20538,7 @@ Proof modification of "f1cofveqaeqALT" is discouraged (124 steps). Proof modification of "f1dmOLD" is discouraged (19 steps). Proof modification of "f1dom2gOLD" is discouraged (77 steps). Proof modification of "f1eqcocnvOLD" is discouraged (361 steps). +Proof modification of "f1finf1oOLD" is discouraged (211 steps). Proof modification of "f1omoALT" is discouraged (39 steps). Proof modification of "f1oweALT" is discouraged (805 steps). Proof modification of "f1rhm0to0ALT" is discouraged (161 steps). @@ -20500,6 +20554,7 @@ Proof modification of "fimacnvOLD" is discouraged (78 steps). Proof modification of "fimadmfoALT" is discouraged (108 steps). Proof modification of "findOLD" is discouraged (70 steps). Proof modification of "findcard2OLD" is discouraged (535 steps). +Proof modification of "findcard3OLD" is discouraged (488 steps). Proof modification of "fineqvacALT" is discouraged (40 steps). Proof modification of "fldidomOLD" is discouraged (34 steps). Proof modification of "fncoOLD" is discouraged (95 steps). @@ -20682,13 +20737,13 @@ Proof modification of "funmoOLD" is discouraged (115 steps). Proof modification of "fvilbdRP" is discouraged (27 steps). Proof modification of "fvimacnvALT" is discouraged (102 steps). Proof modification of "fvmptopabOLD" is discouraged (185 steps). +Proof modification of "fvn0fvelrnOLD" is discouraged (94 steps). Proof modification of "fvpr1OLD" is discouraged (56 steps). Proof modification of "fvpr2OLD" is discouraged (44 steps). Proof modification of "fvpr2gOLD" is discouraged (75 steps). Proof modification of "fvprcALT" is discouraged (26 steps). +Proof modification of "fvssunirnOLD" is discouraged (47 steps). Proof modification of "gcdabsOLD" is discouraged (170 steps). -Proof modification of "gcdmultipleOLD" is discouraged (348 steps). -Proof modification of "gcdmultiplezOLD" is discouraged (230 steps). Proof modification of "gen11" is discouraged (27 steps). Proof modification of "gen11nv" is discouraged (14 steps). Proof modification of "gen12" is discouraged (16 steps). @@ -20700,6 +20755,7 @@ Proof modification of "ggen22" is discouraged (13 steps). Proof modification of "ggen31" is discouraged (22 steps). Proof modification of "ghomidOLD" is discouraged (178 steps). Proof modification of "ghomlinOLD" is discouraged (161 steps). +Proof modification of "glbconNOLD" is discouraged (904 steps). Proof modification of "grpbaseOLD" is discouraged (11 steps). Proof modification of "grpinvfvalALT" is discouraged (161 steps). Proof modification of "grposnOLD" is discouraged (291 steps). @@ -20707,7 +20763,6 @@ Proof modification of "grpplusgOLD" is discouraged (11 steps). Proof modification of "grpsubfvalALT" is discouraged (176 steps). Proof modification of "gsumbagdiagOLD" is discouraged (209 steps). Proof modification of "gsumbagdiaglemOLD" is discouraged (571 steps). -Proof modification of "gsumccatOLD" is discouraged (996 steps). Proof modification of "hashf1lem1OLD" is discouraged (886 steps). Proof modification of "hashfacenOLD" is discouraged (753 steps). Proof modification of "hashge3el3dif" is discouraged (229 steps). @@ -20781,23 +20836,30 @@ Proof modification of "indistps2ALT" is discouraged (43 steps). Proof modification of "indistpsALT" is discouraged (64 steps). Proof modification of "indistpsALTOLD" is discouraged (64 steps). Proof modification of "indthincALT" is discouraged (202 steps). +Proof modification of "infn0ALT" is discouraged (38 steps). Proof modification of "infpssALT" is discouraged (52 steps). Proof modification of "infregelb" is discouraged (185 steps). +Proof modification of "infsdomnnOLD" is discouraged (39 steps). Proof modification of "int2" is discouraged (14 steps). Proof modification of "int3" is discouraged (17 steps). +Proof modification of "intidOLD" is discouraged (57 steps). Proof modification of "intprOLD" is discouraged (112 steps). Proof modification of "intprgOLD" is discouraged (80 steps). Proof modification of "iotaexOLD" is discouraged (57 steps). Proof modification of "iotassuniOLD" is discouraged (35 steps). Proof modification of "iotavalOLD" is discouraged (101 steps). +Proof modification of "isanmbfmOLD" is discouraged (15 steps). Proof modification of "isarep1OLD" is discouraged (106 steps). Proof modification of "iscmgmALT" is discouraged (57 steps). Proof modification of "iscsgrpALT" is discouraged (57 steps). Proof modification of "iseriALT" is discouraged (54 steps). +Proof modification of "isinfOLD" is discouraged (588 steps). Proof modification of "ismgmALT" is discouraged (53 steps). Proof modification of "ismgmOLD" is discouraged (292 steps). Proof modification of "isosctrlem1ALT" is discouraged (363 steps). Proof modification of "isposixOLD" is discouraged (72 steps). +Proof modification of "isrim0OLD" is discouraged (177 steps). +Proof modification of "isrimOLD" is discouraged (64 steps). Proof modification of "issgrpALT" is discouraged (53 steps). Proof modification of "issmgrpOLD" is discouraged (85 steps). Proof modification of "istrkg2d" is discouraged (439 steps). @@ -20806,6 +20868,7 @@ Proof modification of "isvciOLD" is discouraged (178 steps). Proof modification of "itg1addlem4OLD" is discouraged (1803 steps). Proof modification of "iunconnALT" is discouraged (56 steps). Proof modification of "iunconnlem2" is discouraged (580 steps). +Proof modification of "iunidOLD" is discouraged (77 steps). Proof modification of "iunopabOLD" is discouraged (117 steps). Proof modification of "ivthALT" is discouraged (1080 steps). Proof modification of "jaoded" is discouraged (26 steps). @@ -20816,7 +20879,6 @@ Proof modification of "joincomALT" is discouraged (83 steps). Proof modification of "latmassOLD" is discouraged (309 steps). Proof modification of "lcmftp" is discouraged (1057 steps). Proof modification of "lediv2aALT" is discouraged (264 steps). -Proof modification of "lsmidmOLD" is discouraged (116 steps). Proof modification of "luk-1" is discouraged (73 steps). Proof modification of "luk-2" is discouraged (52 steps). Proof modification of "luk-3" is discouraged (20 steps). @@ -20835,6 +20897,7 @@ Proof modification of "mathbox" is discouraged (1 steps). Proof modification of "matscaOLD" is discouraged (73 steps). Proof modification of "matvscaOLD" is discouraged (69 steps). Proof modification of "max1ALT" is discouraged (48 steps). +Proof modification of "mbfmbfmOLD" is discouraged (9 steps). Proof modification of "meetcomALT" is discouraged (83 steps). Proof modification of "merco1" is discouraged (57 steps). Proof modification of "merco1lem1" is discouraged (191 steps). @@ -20899,6 +20962,7 @@ Proof modification of "mnringvscadOLD" is discouraged (30 steps). Proof modification of "mobidvALT" is discouraged (48 steps). Proof modification of "moelOLD" is discouraged (101 steps). Proof modification of "mof0ALT" is discouraged (67 steps). +Proof modification of "mopickr" is discouraged (77 steps). Proof modification of "mpjao3danOLD" is discouraged (29 steps). Proof modification of "mpofunOLD" is discouraged (95 steps). Proof modification of "mpteq12daOLD" is discouraged (41 steps). @@ -20988,6 +21052,7 @@ Proof modification of "nnexALT" is discouraged (34 steps). Proof modification of "nnfiOLD" is discouraged (14 steps). Proof modification of "nnindALT" is discouraged (15 steps). Proof modification of "nnne0ALT" is discouraged (19 steps). +Proof modification of "nnsdomgOLD" is discouraged (97 steps). Proof modification of "noelOLD" is discouraged (77 steps). Proof modification of "norcomOLD" is discouraged (27 steps). Proof modification of "normlem7tALT" is discouraged (177 steps). @@ -20999,6 +21064,7 @@ Proof modification of "nsnlpligALT" is discouraged (110 steps). Proof modification of "o2p2e4OLD" is discouraged (67 steps). Proof modification of "odfvalALT" is discouraged (181 steps). Proof modification of "odubasOLD" is discouraged (62 steps). +Proof modification of "ominfOLD" is discouraged (79 steps). Proof modification of "omsindsOLD" is discouraged (89 steps). Proof modification of "ondomon" is discouraged (287 steps). Proof modification of "onfrALT" is discouraged (88 steps). @@ -21015,6 +21081,8 @@ Proof modification of "onfrALTlem5" is discouraged (320 steps). Proof modification of "onfrALTlem5VD" is discouraged (320 steps). Proof modification of "onnevOLD" is discouraged (26 steps). Proof modification of "onomeneqOLD" is discouraged (251 steps). +Proof modification of "ontrciOLD" is discouraged (9 steps). +Proof modification of "onuniorsuciOLD" is discouraged (16 steps). Proof modification of "opabresex2d" is discouraged (48 steps). Proof modification of "opelopab4" is discouraged (69 steps). Proof modification of "opidon2OLD" is discouraged (80 steps). @@ -21041,6 +21109,7 @@ Proof modification of "orbi1rVD" is discouraged (101 steps). Proof modification of "orcomdd" is discouraged (13 steps). Proof modification of "ordelordALT" is discouraged (128 steps). Proof modification of "ordelordALTVD" is discouraged (202 steps). +Proof modification of "ordsucOLD" is discouraged (67 steps). Proof modification of "orim12dALT" is discouraged (34 steps). Proof modification of "p0exALT" is discouraged (2 steps). Proof modification of "peano1OLD" is discouraged (9 steps). @@ -21110,6 +21179,8 @@ Proof modification of "r19.30OLD" is discouraged (66 steps). Proof modification of "r19.35OLD" is discouraged (72 steps). Proof modification of "r1omALT" is discouraged (13 steps). Proof modification of "r1pwALT" is discouraged (151 steps). +Proof modification of "rabbiiaOLD" is discouraged (39 steps). +Proof modification of "rabeqcOLD" is discouraged (37 steps). Proof modification of "rabeqiOLD" is discouraged (59 steps). Proof modification of "rabid2OLD" is discouraged (57 steps). Proof modification of "rabrabiOLD" is discouraged (38 steps). @@ -21179,12 +21250,14 @@ Proof modification of "retbwax1" is discouraged (195 steps). Proof modification of "retbwax2" is discouraged (127 steps). Proof modification of "retbwax3" is discouraged (20 steps). Proof modification of "retbwax4" is discouraged (13 steps). +Proof modification of "reuanidOLD" is discouraged (37 steps). Proof modification of "reutruALT" is discouraged (45 steps). Proof modification of "rexabOLD" is discouraged (40 steps). Proof modification of "rexbiOLD" is discouraged (65 steps). Proof modification of "rexbidvALT" is discouraged (10 steps). Proof modification of "rexbidvaALT" is discouraged (10 steps). Proof modification of "rexcomOLD" is discouraged (67 steps). +Proof modification of "rexdif1enOLD" is discouraged (120 steps). Proof modification of "reximdvaiOLD" is discouraged (28 steps). Proof modification of "reximiaOLD" is discouraged (21 steps). Proof modification of "rexlimivOLD" is discouraged (23 steps). @@ -21194,8 +21267,10 @@ Proof modification of "rexn0OLD" is discouraged (17 steps). Proof modification of "rexprgOLD" is discouraged (17 steps). Proof modification of "rexsngOLD" is discouraged (10 steps). Proof modification of "rgen2a" is discouraged (81 steps). +Proof modification of "rimrhmOLD" is discouraged (25 steps). Proof modification of "rlimaddOLD" is discouraged (159 steps). Proof modification of "rlimmulOLD" is discouraged (159 steps). +Proof modification of "rmoanidOLD" is discouraged (37 steps). Proof modification of "rmoanimALT" is discouraged (93 steps). Proof modification of "rmobidvaOLD" is discouraged (10 steps). Proof modification of "rmodislmodOLD" is discouraged (1641 steps). @@ -21270,6 +21345,7 @@ Proof modification of "sbtT" is discouraged (7 steps). Proof modification of "scmateALT" is discouraged (236 steps). Proof modification of "sdom0OLD" is discouraged (29 steps). Proof modification of "sdom1OLD" is discouraged (73 steps). +Proof modification of "selsALT" is discouraged (34 steps). Proof modification of "seqp1iOLD" is discouraged (20 steps). Proof modification of "setsidvaldOLD" is discouraged (93 steps). Proof modification of "setsmsbasOLD" is discouraged (55 steps). @@ -21284,9 +21360,9 @@ Proof modification of "slotsbhcdifOLD" is discouraged (96 steps). Proof modification of "smgrpassOLD" is discouraged (54 steps). Proof modification of "smgrpismgmOLD" is discouraged (22 steps). Proof modification of "sn-00id" is discouraged (50 steps). -Proof modification of "sn-elALT2" is discouraged (52 steps). +Proof modification of "sn-exelALT" is discouraged (52 steps). +Proof modification of "snelpwiOLD" is discouraged (20 steps). Proof modification of "snelpwrVD" is discouraged (37 steps). -Proof modification of "snex" is discouraged (64 steps). Proof modification of "snexALT" is discouraged (48 steps). Proof modification of "snnen2oOLD" is discouraged (116 steps). Proof modification of "snssgOLD" is discouraged (58 steps). @@ -21333,6 +21409,7 @@ Proof modification of "stdpc5t" is discouraged (23 steps). Proof modification of "sucdom2OLD" is discouraged (343 steps). Proof modification of "sucdomOLD" is discouraged (35 steps). Proof modification of "suceloniOLD" is discouraged (160 steps). +Proof modification of "sucexeloniOLD" is discouraged (163 steps). Proof modification of "sucidALT" is discouraged (28 steps). Proof modification of "sucidALTVD" is discouraged (28 steps). Proof modification of "sucidVD" is discouraged (24 steps). @@ -21564,7 +21641,6 @@ Proof modification of "wl-section-prop" is discouraged (1 steps). Proof modification of "wl-syls1" is discouraged (12 steps). Proof modification of "wl-syls2" is discouraged (14 steps). Proof modification of "wlkResOLD" is discouraged (50 steps). -Proof modification of "wlklenvclwlkOLD" is discouraged (197 steps). Proof modification of "wrecseq123OLD" is discouraged (211 steps). Proof modification of "wunfuncOLD" is discouraged (338 steps). Proof modification of "wunnatOLD" is discouraged (341 steps). @@ -21572,6 +21648,7 @@ Proof modification of "wunressOLD" is discouraged (145 steps). Proof modification of "xorbi12iOLD" is discouraged (31 steps). Proof modification of "xorcomOLD" is discouraged (27 steps). Proof modification of "xpexgALT" is discouraged (84 steps). +Proof modification of "xpfiOLD" is discouraged (373 steps). Proof modification of "xrge0tmdALT" is discouraged (166 steps). Proof modification of "zexALT" is discouraged (34 steps). Proof modification of "zfcndac" is discouraged (256 steps). diff --git a/iset.mm b/iset.mm index 41507a23cf..7154811fa2 100644 --- a/iset.mm +++ b/iset.mm @@ -1872,13 +1872,6 @@ statements not containing the new symbol (or new combination) should ( wb wi biimp syl ) ABCEBCFDBCGH $. $} - $( The antecedent of one side of a biconditional can be moved out of the - biconditional to become the antecedent of the remaining biconditional. - (Contributed by BJ, 1-Jan-2025.) $) - imbibi $p |- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) ) $= - ( wi wb biimp jarr a1d syl biimpr com23 impbidd ) ABDZCEZABCNMCDZABCDZDMCFO - PAABCGHINCABMCJKL $. - ${ mpbi.min $e |- ph $. mpbi.maj $e |- ( ph <-> ps ) $. @@ -1942,12 +1935,12 @@ statements not containing the new symbol (or new combination) should $} ${ - syl5bi.1 $e |- ( ph <-> ps ) $. - syl5bi.2 $e |- ( ch -> ( ps -> th ) ) $. + biimtrid.1 $e |- ( ph <-> ps ) $. + biimtrid.2 $e |- ( ch -> ( ps -> th ) ) $. $( A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 5-Aug-1993.) $) - syl5bi $p |- ( ch -> ( ph -> th ) ) $= + biimtrid $p |- ( ch -> ( ph -> th ) ) $= ( biimpi syl5 ) ABCDABEGFH $. $} @@ -2395,7 +2388,7 @@ statements not containing the new symbol (or new combination) should in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) $) 3imtr4g $p |- ( ph -> ( th -> ta ) ) $= - ( syl5bi syl6ibr ) ADCEDBACGFIHJ $. + ( biimtrid syl6ibr ) ADCEDBACGFIHJ $. $} ${ @@ -2726,6 +2719,13 @@ statements not containing the new symbol (or new combination) should ( ph -> ( ps -> ch ) ) ) $= ( wi imdi bicomi ) ABCDDABDACDDABCEF $. + $( The antecedent of one side of a biconditional can be moved out of the + biconditional to become the antecedent of the remaining biconditional. + (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, + 5-Jan-2025.) $) + imbibi $p |- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) ) $= + ( wi wb pm5.4 imbi2 bitr3id pm5.74rd ) ABDZCEZABCJAJDKACDABFJCAGHI $. + $( Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. @@ -2829,7 +2829,7 @@ statements not containing the new symbol (or new combination) should $( Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.) $) ancomsd $p |- ( ph -> ( ( ch /\ ps ) -> th ) ) $= - ( wa ancom syl5bi ) CBFBCFADCBGEH $. + ( wa ancom biimtrid ) CBFBCFADCBGEH $. $} ${ @@ -7232,8 +7232,8 @@ Many theorems of this section actually hold for stable propositions (see [WhiteheadRussell] p. 101 with decidability condition added. (Contributed by Jim Kingdon, 13-May-2018.) $) pm2.13dc $p |- ( DECID ph -> ( ph \/ -. -. -. ph ) ) $= - ( wdc wn wo df-dc notnotrdc con3d orim2d syl5bi pm2.43i ) ABZAACZCZCZDZKALD - KOAEKLNAKMAAFGHIJ $. + ( wdc wn wo df-dc notnotrdc con3d orim2d biimtrid pm2.43i ) ABZAACZCZCZDZKA + LDKOAEKLNAKMAAFGHIJ $. $( Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) $) @@ -9831,8 +9831,8 @@ converse also holds (see ~ pm5.6dc ). (Contributed by Jim Kingdon, $( Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.) $) 3jao $p |- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) -> ( ( ph \/ ch \/ th ) -> ps ) ) $= - ( w3o wo wi w3a df-3or jao syl6 3imp syl5bi ) ACDEACFZDFZABGZCBGZDBGZHBACDI - PQROBGZPQNBGRSGABCJNBDJKLM $. + ( w3o wo wi w3a df-3or jao syl6 3imp biimtrid ) ACDEACFZDFZABGZCBGZDBGZHBAC + DIPQROBGZPQNBGRSGABCJNBDJKLM $. $( Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.) $) 3jaob $p |- ( ( ( ph \/ ch \/ th ) -> ps ) <-> @@ -12799,10 +12799,10 @@ holds for classical logic but not (for all propositions) in intuitionistic (Contributed by Mario Carneiro, 2-Feb-2015.) $) 19.33b2 $p |- ( ( -. E. x ph \/ -. E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) $= - ( wex wn wal orcom alnex orbi12i bitr4i pm2.53 orcoms al2imi orim12d syl5bi - wo wi com12 19.33 impbid1 ) ACDEZBCDEZPZABPZCFZACFZBCFZPZUEUCUHUCBEZCFZAEZC - FZPZUEUHUCUBUAPUMUAUBGUJUBULUABCHACHIJUEUJUFULUGUDUIACBAUIAQBAKLMUDUKBCABKM - NORABCST $. + ( wex wn wo wal orcom alnex orbi12i bitr4i wi pm2.53 orcoms al2imi biimtrid + orim12d com12 19.33 impbid1 ) ACDEZBCDEZFZABFZCGZACGZBCGZFZUEUCUHUCBEZCGZAE + ZCGZFZUEUHUCUBUAFUMUAUBHUJUBULUABCIACIJKUEUJUFULUGUDUIACBAUIALBAMNOUDUKBCAB + MOQPRABCST $. $( Converse of ~ 19.33 given ` -. ( E. x ph /\ E. x ps ) ` and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version @@ -14034,8 +14034,8 @@ definition has the properties we expect of proper substitution (see $( An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) $) sbequ2 $p |- ( x = y -> ( [ y / x ] ph -> ph ) ) $= - ( wsb weq wi wa wex df-sb simpl com12 syl5bi ) ABCDBCEZAFZMAGBHZGZMAABCIPMA - NOJKL $. + ( wsb weq wi wa wex df-sb simpl com12 biimtrid ) ABCDBCEZAFZMAGBHZGZMAABCIP + MANOJKL $. $( One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is ~ stdpc6 .) Translated to @@ -15152,8 +15152,8 @@ Predicate calculus with distinct variables (cont.) 25-Dec-2017.) $) sbi1v $p |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) $= - ( wsb weq wi wal sb6 ax-2 al2imi sb2 syl6 sylbi syl5bi ) ACDECDFZAGZCHZAB - GZCDEZBCDEZACDITPSGZCHZRUAGSCDIUCRPBGZCHUAUBQUDCPABJKBCDLMNO $. + ( wsb weq wi wal sb6 ax-2 al2imi sb2 syl6 sylbi biimtrid ) ACDECDFZAGZCHZ + ABGZCDEZBCDEZACDITPSGZCHZRUAGSCDIUCRPBGZCHUAUBQUDCPABJKBCDLMNO $. $( Reverse direction of ~ sbimv . (Contributed by Jim Kingdon, 18-Jan-2018.) $) @@ -16948,10 +16948,10 @@ Predicate calculus with distinct variables (cont.) ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E. y A. x ( ph -> x = y ) ) ) $= ( wex wsb wa weq wi sb8e impexp bi2.04 bitri 2albii nfs1v nfri eximi alim - wal alimi a7s exim syl syl5com syl5bi sylbi ) ABEABCFZCEZAUGGBCHZIZCSBSZA - UIIZBSZCEZIABCDJUKUGULIZCSBSZUHUNUJUOBCUJAUGUIIIUOAUGUIKAUGUILMNUHUGBSZCE - ZUPUNUGUQCUGBABCOPQUPUQUMIZCSZURUNIUOUTCBUOBSUSCUGULBRTUAUQUMCUBUCUDUEUF - $. + wal alimi a7s exim syl syl5com biimtrid sylbi ) ABEABCFZCEZAUGGBCHZIZCSBS + ZAUIIZBSZCEZIABCDJUKUGULIZCSBSZUHUNUJUOBCUJAUGUIIIUOAUGUIKAUGUILMNUHUGBSZ + CEZUPUNUGUQCUGBABCOPQUPUQUMIZCSZURUNIUOUTCBUOBSUSCUGULBRTUAUQUMCUBUCUDUEU + F $. $} ${ @@ -17343,9 +17343,9 @@ Predicate calculus with distinct variables (cont.) $( Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) $) 2euex $p |- ( E! x E. y ph -> E. y E! x ph ) $= - ( wex weu wmo wa eu5 excom hbe1 hbmo 19.8a moimi df-mo eximdh syl5bi impcom - wi sylib sylbi ) ACDZBEUABDZUABFZGABEZCDZUABHUCUBUEUBABDZCDUCUEABCIUCUFUDCU - ACBACJKUCABFUFUDRAUABACLMABNSOPQT $. + ( wex weu wmo wa excom hbe1 hbmo wi 19.8a moimi df-mo sylib eximdh biimtrid + eu5 impcom sylbi ) ACDZBEUABDZUABFZGABEZCDZUABRUCUBUEUBABDZCDUCUEABCHUCUFUD + CUACBACIJUCABFUFUDKAUABACLMABNOPQST $. $( Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) $) @@ -20339,7 +20339,7 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo $( A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) $) eqneqall $p |- ( A = B -> ( A =/= B -> ph ) ) $= - ( wne wceq wn df-ne pm2.24 syl5bi ) BCDBCEZFJABCGJAHI $. + ( wne wceq wn df-ne pm2.24 biimtrid ) BCDBCEZFJABCGJAHI $. $( Decidable equality expressed in terms of ` =/= ` . Basically the same as ~ df-dc . (Contributed by Jim Kingdon, 14-Mar-2020.) $) @@ -20581,7 +20581,7 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo $( Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) $) necon3ad $p |- ( ph -> ( A =/= B -> -. ps ) ) $= - ( wne wceq wn df-ne con3d syl5bi ) CDFCDGZHABHCDIABLEJK $. + ( wne wceq wn df-ne con3d biimtrid ) CDFCDGZHABHCDIABLEJK $. $} ${ @@ -20648,8 +20648,8 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo $( Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.) $) necon1aidc $p |- ( DECID ph -> ( A =/= B -> ph ) ) $= - ( wne wceq wn wdc df-ne wi con1dc mpd syl5bi ) BCEBCFZGZAHZABCIPAGNJOAJDA - NKLM $. + ( wne wceq wn wdc df-ne wi con1dc mpd biimtrid ) BCEBCFZGZAHZABCIPAGNJOAJ + DANKLM $. $} ${ @@ -20808,8 +20808,8 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo $( Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.) $) necon4addc $p |- ( ph -> ( DECID A = B -> ( ps -> A = B ) ) ) $= - ( wceq wdc wne wn wi df-ne imbi1i condc syl5bi sylcom ) ACDFZGZCDHZBIZJZB - PJZETPIZSJQUARUBSCDKLPBMNO $. + ( wceq wdc wne wn wi df-ne imbi1i condc biimtrid sylcom ) ACDFZGZCDHZBIZJ + ZBPJZETPIZSJQUARUBSCDKLPBMNO $. $} ${ @@ -21096,7 +21096,7 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo $( A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) $) elnelall $p |- ( A e. B -> ( A e/ B -> ph ) ) $= - ( wnel wcel wn df-nel pm2.24 syl5bi ) BCDBCEZFJABCGJAHI $. + ( wnel wcel wn df-nel pm2.24 biimtrid ) BCDBCEZFJABCGJAHI $. $( @@ -24496,8 +24496,9 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo 4-Jan-2017.) $) spcimgft $p |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) ) $= - ( wcel cvv cv wceq wi wal elex wex issetf exim syl5bi 19.36-1 syl6 syl5 ) - DEHDIHZCJDKZABLZLCMZACMBLZDENUEUBUDCOZUFUBUCCOUEUGCDGPUCUDCQRABCFSTUA $. + ( wcel cvv cv wceq wi wal elex wex issetf exim biimtrid 19.36-1 syl6 syl5 + ) DEHDIHZCJDKZABLZLCMZACMBLZDENUEUBUDCOZUFUBUCCOUEUGCDGPUCUDCQRABCFSTUA + $. $( A closed version of ~ spcgf . (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.) $) @@ -24510,8 +24511,9 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo 4-Jan-2017.) $) spcimegft $p |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. B -> ( ps -> E. x ph ) ) ) $= - ( wcel cvv cv wceq wi wal wex elex issetf exim syl5bi 19.37-1 syl6 syl5 ) - DEHDIHZCJDKZBALZLCMZBACNLZDEOUEUBUDCNZUFUBUCCNUEUGCDGPUCUDCQRBACFSTUA $. + ( wcel cvv cv wceq wi wal wex elex issetf exim biimtrid 19.37-1 syl6 syl5 + ) DEHDIHZCJDKZBALZLCMZBACNLZDEOUEUBUDCNZUFUBUCCNUEUGCDGPUCUDCQRBACFSTUA + $. $( A closed version of ~ spcegf . (Contributed by Jim Kingdon, 22-Jun-2018.) $) @@ -24699,8 +24701,8 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) $) rspc $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $= ( wral cv wcel wi wal df-ral nfcv nfv nfim wceq eleq1 imbi12d spcgf - pm2.43a syl5bi ) ACEHCIZEJZAKZCLZDEJZBACEMUFUGBUEUGBKCDECDNUGBCUGCOFPUCDQ - UDUGABUCDERGSTUAUB $. + pm2.43a biimtrid ) ACEHCIZEJZAKZCLZDEJZBACEMUFUGBUEUGBKCDECDNUGBCUGCOFPUC + DQUDUGABUCDERGSTUAUB $. $( Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, @@ -24749,8 +24751,8 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo by Mario Carneiro, 4-Jan-2017.) $) rspcimdv $p |- ( ph -> ( A. x e. B ps -> ch ) ) $= ( wral cv wcel wi wal df-ral wceq wa simpr eleq1d biimprd imim12d mpid - spcimdv syl5bi ) BDFIDJZFKZBLZDMZACBDFNAUGEFKZCGAUFUHCLDEFGAUDEOZPZUHUE - BCUJUEUHUJUDEFAUIQRSHTUBUAUC $. + spcimdv biimtrid ) BDFIDJZFKZBLZDMZACBDFNAUGEFKZCGAUFUHCLDEFGAUDEOZPZUH + UEBCUJUEUHUJUDEFAUIQRSHTUBUAUC $. $} rspcimedv.2 $e |- ( ( ph /\ x = A ) -> ( ch -> ps ) ) $. @@ -24866,11 +24868,11 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo substitution. (Contributed by BJ, 2-Dec-2021.) $) rspc2gv $p |- ( ( A e. V /\ B e. W ) -> ( A. x e. V A. y e. W ph -> ps ) ) $= - ( wral cv wcel wi wal wa df-ral albii wceq eleq1 syl5bi bi2anan9 imbi12d - imbi2i 19.21v bicomi impexp bitr3id spc2gv pm2.43a ) ADHJZCGJCKZGLZUJMZCN - ZEGLZFHLZOZBUJCGPUNULDKZHLZAMZDNZMZCNZUQBUMVBCUJVAULADHPUCQVCULUTMZDNZCNZ - UQBVBVECVEVBULUTDUDUEQVFUQBVDUQBMZCDEFGHVDULUSOZAMUKERZURFRZOZVGULUSAUFVK - VHUQABVIULUOVJUSUPUKEGSURFHSUAIUBUGUHUITTT $. + ( wral cv wcel wi wal wa df-ral albii wceq eleq1 biimtrid imbi2i bi2anan9 + 19.21v bicomi impexp imbi12d bitr3id spc2gv pm2.43a ) ADHJZCGJCKZGLZUJMZC + NZEGLZFHLZOZBUJCGPUNULDKZHLZAMZDNZMZCNZUQBUMVBCUJVAULADHPUAQVCULUTMZDNZCN + ZUQBVBVECVEVBULUTDUCUDQVFUQBVDUQBMZCDEFGHVDULUSOZAMUKERZURFRZOZVGULUSAUEV + KVHUQABVIULUOVJUSUPUKEGSURFHSUBIUFUGUHUITTT $. $} ${ @@ -25626,15 +25628,15 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo -> E! z A. x ( ph -> z = A ) ) $= ( vw wa wceq wi wal wex cv exbii bitri imim2i weu cbvexv biantrur 3bitr2i isseti 19.41v excom wb eqeq2 biimpr an31 imbi1i impexp 3bitr3i syl 2alimi - sylib 19.23v albii 19.21v eximdv syl5bi pm4.24 biimpi eqtr3 syl56 alanimi - imp anim12 com12 syl5 alrimivv adantl eqeq1 imbi2d albidv eu4 sylanbrc ) - ABLZFGMZNZDOCOZACPZLAEQZFMZNZCOZEPZWGAKQZFMZNZCOZLZWDWIMZNZKOEOZWGEUAWBWC - WHWCWDGMZBLZDPZEPZWBWHWCBDPZWTABCDIUBXAWQEPZBLZDPWREPZDPWTBXCDXBBEGHUEUCR - XDXCDWQBEUFRWRDEUGUDSWBWSWGEWBWRWFNZDOZCOZWSWGNZWAXECDWAVSWEWQUHZNZXEVTXI - VSFGWDUITXJVSWQWENZNZXEXIXKVSWEWQUJTVSWQLZWENWRALZWENXLXEXMXNWEABWQUKULVS - WQWEUMWRAWEUMUNUQUOUPXGWSWFNZCOXHXFXOCWRWFDURUSWSWFCUTSUQVAVBVHWCWPWBWCWO - EKWMAWNNZCOZWCWNWFWKXPCAAALZWFWKLWEWJLWNAXRAVCVDAWEAWJVIWDWIFVEVFVGXQWCWN - XQWCWNNAWNCURVDVJVKVLVMWGWLEKWNWFWKCWNWEWJAWDWIFVNVOVPVQVR $. + sylib 19.23v albii 19.21v eximdv biimtrid imp pm4.24 biimpi eqtr3 alanimi + anim12 syl56 com12 syl5 alrimivv adantl eqeq1 imbi2d albidv eu4 sylanbrc + ) ABLZFGMZNZDOCOZACPZLAEQZFMZNZCOZEPZWGAKQZFMZNZCOZLZWDWIMZNZKOEOZWGEUAWB + WCWHWCWDGMZBLZDPZEPZWBWHWCBDPZWTABCDIUBXAWQEPZBLZDPWREPZDPWTBXCDXBBEGHUEU + CRXDXCDWQBEUFRWRDEUGUDSWBWSWGEWBWRWFNZDOZCOZWSWGNZWAXECDWAVSWEWQUHZNZXEVT + XIVSFGWDUITXJVSWQWENZNZXEXIXKVSWEWQUJTVSWQLZWENWRALZWENXLXEXMXNWEABWQUKUL + VSWQWEUMWRAWEUMUNUQUOUPXGWSWFNZCOXHXFXOCWRWFDURUSWSWFCUTSUQVAVBVCWCWPWBWC + WOEKWMAWNNZCOZWCWNWFWKXPCAAALZWFWKLWEWJLWNAXRAVDVEAWEAWJVHWDWIFVFVIVGXQWC + WNXQWCWNNAWNCURVEVJVKVLVMWGWLEKWNWFWKCWNWEWJAWDWIFVNVOVPVQVR $. $} ${ @@ -25824,11 +25826,11 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo ( E! x e. A E. y e. B ph -> E! y e. B E. x e. A ph ) ) ) $= ( wral cv wcel wa wmo wex wrex wreu wal weu df-reu r19.42v df-rex bitri wi wrmo df-rmo ralbii df-ral moanimv albii bitr4i bitr3i an12 exbii eubii - wdc 2euswapdc syl7bi imbi2i syl6ibr syl5bi ) ACEUAZBDFCGEHZAIZCJZBDFZBGDH - ZUTIZCKZBKULZACELZBDMZABDLZCEMZTZURVABDACEUBUCVBVDCJZBNZVFVKVBVCVATZBNVMV - ABDUDVLVNBVCUTCUEUFUGVFVMVHVDBKZCOZTVKVHVEBOZVFVMVPVHVCVGIZBOVQVGBDPVRVEB - VRUSVCAIZIZCKZVEVRVSCELWAVCACEQVSCERUHVTVDCUSVCAUIUJSUKSVDBCUMUNVJVPVHVJU - SVIIZCOVPVICEPWBVOCWBUTBDLVOUSABDQUTBDRUHUKSUOUPUQUQ $. + wdc 2euswapdc syl7bi imbi2i syl6ibr biimtrid ) ACEUAZBDFCGEHZAIZCJZBDFZBG + DHZUTIZCKZBKULZACELZBDMZABDLZCEMZTZURVABDACEUBUCVBVDCJZBNZVFVKVBVCVATZBNV + MVABDUDVLVNBVCUTCUEUFUGVFVMVHVDBKZCOZTVKVHVEBOZVFVMVPVHVCVGIZBOVQVGBDPVRV + EBVRUSVCAIZIZCKZVEVRVSCELWAVCACEQVSCERUHVTVDCUSVCAUIUJSUKSVDBCUMUNVJVPVHV + JUSVIIZCOVPVICEPWBVOCWBUTBDLVOUSABDQUTBDRUHUKSUOUPUQUQ $. $} ${ @@ -25843,18 +25845,18 @@ atomic formula (class version of ~ elsb1 ). (Contributed by Rodolfo ( vw wcel wa wceq wi wal wex cv wrex bitri wral wreu eleq1d anbi12d exbii cbvexv r19.41v rexcom4 risset anbi1i 3bitr4ri wb eqeq2 imim2i biimpr an31 imbi1i impexp 3bitr3i sylib syl 2alimi 19.23v an12 adantr pm5.32ri bitr4i - eleq1 19.42v albii 19.21v reximdvai syl5bi imp pm4.24 biimpi anim12 eqtr3 - expd syl56 alanimi com12 syl5 a1d ralrimivv adantl imbi2d albidv sylanbrc - eqeq1 reu4 ) FHLZAMZGHLZBMZMZFGNZOZDPCPZWMCQZMWMERZFNZOZCPZEHSZXDWMKRZFNZ - OZCPZMZXAXFNZOZKHUAEHUAZXDEHUBWSWTXEWTXAGNZBMZDQZEHSZWSXEWTWODQZXQWMWOCDC - RDRNZWLWNABXSFGHJUCIUDUFXOEHSZDQXNEHSZBMZDQXQXRXTYBDXNBEHUGUEXOEDHUHWOYBD - WNYABEGHUIUJUEUKTWSXPXDEHWSXAHLZXPXDWSXNWOMZXCOZDPZCPZYCXPMZXDOZWRYECDWRW - PXBXNULZOZYEWQYJWPFGXAUMUNYKWPXNXBOZOZYEYJYLWPXBXNUOUNWPXNMZXBOYDWMMZXBOY - MYEYNYOXBWMWOXNUPUQWPXNXBURYDWMXBURUSUTVAVBYGYHXCOZCPYIYFYPCYFYDDQZXCOYPY - DXCDVCYQYHXCYQYCXOMZDQYHYDYRDYDWNXOMYRXNWNBVDXOYCWNXNYCWNULBXAGHVHVEVFVGU - EYCXODVITUQTVJYHXCCVKTUTVSVLVMVNWTXMWSWTXLEKHHWTXLYCXFHLMXJWMXKOZCPZWTXKX - CXHYSCWMWMWMMZXCXHMXBXGMXKWMUUAWMVOVPWMXBWMXGVQXAXFFVRVTWAYTWTXKYTWTXKOWM - XKCVCVPWBWCWDWEWFXDXIEKHXKXCXHCXKXBXGWMXAXFFWJWGWHWKWI $. + eleq1 19.42v albii 19.21v expd reximdvai biimtrid imp pm4.24 biimpi eqtr3 + anim12 syl56 alanimi com12 syl5 ralrimivv adantl eqeq1 imbi2d albidv reu4 + a1d sylanbrc ) FHLZAMZGHLZBMZMZFGNZOZDPCPZWMCQZMWMERZFNZOZCPZEHSZXDWMKRZF + NZOZCPZMZXAXFNZOZKHUAEHUAZXDEHUBWSWTXEWTXAGNZBMZDQZEHSZWSXEWTWODQZXQWMWOC + DCRDRNZWLWNABXSFGHJUCIUDUFXOEHSZDQXNEHSZBMZDQXQXRXTYBDXNBEHUGUEXOEDHUHWOY + BDWNYABEGHUIUJUEUKTWSXPXDEHWSXAHLZXPXDWSXNWOMZXCOZDPZCPZYCXPMZXDOZWRYECDW + RWPXBXNULZOZYEWQYJWPFGXAUMUNYKWPXNXBOZOZYEYJYLWPXBXNUOUNWPXNMZXBOYDWMMZXB + OYMYEYNYOXBWMWOXNUPUQWPXNXBURYDWMXBURUSUTVAVBYGYHXCOZCPYIYFYPCYFYDDQZXCOY + PYDXCDVCYQYHXCYQYCXOMZDQYHYDYRDYDWNXOMYRXNWNBVDXOYCWNXNYCWNULBXAGHVHVEVFV + GUEYCXODVITUQTVJYHXCCVKTUTVLVMVNVOWTXMWSWTXLEKHHWTXLYCXFHLMXJWMXKOZCPZWTX + KXCXHYSCWMWMWMMZXCXHMXBXGMXKWMUUAWMVPVQWMXBWMXGVSXAXFFVRVTWAYTWTXKYTWTXKO + WMXKCVCVQWBWCWJWDWEXDXIEKHXKXCXHCXKXBXGWMXAXFFWFWGWHWIWK $. $} ${ @@ -26417,11 +26419,11 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use sbciegft $p |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) ) $= ( wcel wnf cv wceq wb wi wal w3a wsbc imim2i alimi biimpa sylan2 3adant1 - wa wex sbc5 biimp impd 19.23t syl5bi biimpr com23 19.21t 3ad2ant1 sylibrd - sbc6g impbid ) DEFZBCGZCHDIZABJZKZCLZMZACDNZBVAUPATZCUAZUTBACDUBUOUSVCBKZ - UNUSUOVBBKZCLZVDURVECURUPABUQABKUPABUCOUDPUOVFVDVBBCUEQRSUFUTBUPAKZCLZVAU - OUSBVHKZUNUSUOBVGKZCLZVIURVJCURUPBAUQBAKUPABUGOUHPUOVKVIBVGCUIQRSUNUOVAVH - JUSACDEULUJUKUM $. + wa wex sbc5 biimp impd 19.23t biimtrid biimpr com23 19.21t sbc6g 3ad2ant1 + sylibrd impbid ) DEFZBCGZCHDIZABJZKZCLZMZACDNZBVAUPATZCUAZUTBACDUBUOUSVCB + KZUNUSUOVBBKZCLZVDURVECURUPABUQABKUPABUCOUDPUOVFVDVBBCUEQRSUFUTBUPAKZCLZV + AUOUSBVHKZUNUSUOBVGKZCLZVIURVJCURUPBAUQBAKUPABUGOUHPUOVKVIBVGCUIQRSUNUOVA + VHJUSACDEUJUKULUM $. $} ${ @@ -26925,8 +26927,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use also ~ rspsbca and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) $) rspsbc $p |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) ) $= - ( vy wral wsb wcel wsbc cbvralsv dfsbcq2 rspcv syl5bi ) ABDFABEGZEDFCDHAB - CIZABEDJNOECDABECKLM $. + ( vy wral wsb wcel wsbc cbvralsv dfsbcq2 rspcv biimtrid ) ABDFABEGZEDFCDH + ABCIZABEDJNOECDABECKLM $. $( Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.) $) @@ -32914,8 +32916,8 @@ we define it to be empty in this case (see ~ opprc1 and ~ opprc2 ). For ~ preqr2 . (Contributed by Jim Kingdon, 21-Sep-2018.) $) preqr2g $p |- ( ( A e. _V /\ B e. _V ) -> ( { C , A } = { C , B } -> A = B ) ) $= - ( cpr wceq cvv wcel wa prcom eqeq12i preqr1g syl5bi ) CADZCBDZEACDZBCDZEAFG - BFGHABEMONPCAICBIJABCKL $. + ( cpr wceq cvv wcel wa prcom eqeq12i preqr1g biimtrid ) CADZCBDZEACDZBCDZEA + FGBFGHABEMONPCAICBIJABCKL $. ${ preqr1.1 $e |- A e. _V $. @@ -32961,11 +32963,11 @@ we define it to be empty in this case (see ~ opprc1 and ~ opprc2 ). For ( A e. { C , D } /\ B e. { C , D } ) ) ) $= ( wceq wn cpr wcel wa eleq2 mpbii wo wi elpr eqeq2 notbid prid1 prid2 jca orel2 syl6bi com3l imp ancrd orel1 orim12d orcom bitri preq12b 3imtr4g ex - syl5bi impd impbid2 ) ABIZJZABKZCDKZIZAVBLZBVBLZMVCVDVEVCAVALVDABEUAVAVBA - NOVCBVALVEABFUBVAVBBNOUCUTVDVEVCVDACIZADIZPZUTVEVCQZACDERUTVHVIUTVHMZBDIZ - BCIZPZVFVKMZVGVLMZPVEVCVJVKVNVLVOVJVKVFUTVHVKVFQVKUTVHVFVKUTVGJVHVFQVKUSV - GBDASTVGVFUDUEUFUGUHVJVLVGUTVHVLVGQVLUTVHVGVLUTVFJVHVGQVLUSVFBCASTVFVGUIU - EUFUGUHUJVEVLVKPVMBCDFRVLVKUKULABCDEFGHUMUNUOUPUQUR $. + biimtrid impd impbid2 ) ABIZJZABKZCDKZIZAVBLZBVBLZMVCVDVEVCAVALVDABEUAVAV + BANOVCBVALVEABFUBVAVBBNOUCUTVDVEVCVDACIZADIZPZUTVEVCQZACDERUTVHVIUTVHMZBD + IZBCIZPZVFVKMZVGVLMZPVEVCVJVKVNVLVOVJVKVFUTVHVKVFQVKUTVHVFVKUTVGJVHVFQVKU + SVGBDASTVGVFUDUEUFUGUHVJVLVGUTVHVLVGQVLUTVHVGVLUTVFJVHVGQVLUSVFBCASTVFVGU + IUEUFUGUHUJVEVLVKPVMBCDFRVLVKUKULABCDEFGHUMUNUOUPUQUR $. $( A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.) $) @@ -33778,8 +33780,8 @@ we define it to be empty in this case (see ~ opprc1 and ~ opprc2 ). For (Contributed by NM, 18-Nov-1995.) $) intss1 $p |- ( A e. B -> |^| B C_ A ) $= ( vx vy wcel cint cv wal vex elint wceq eleq1 eleq2 imbi12d spcgv pm2.43a - wi syl5bi ssrdv ) ABEZCBFZACGZUAEDGZBEZUBUCEZQZDHZTUBAEZDUBBCIJUGTUHUFTUH - QDABUCAKUDTUEUHUCABLUCAUBMNOPRS $. + wi biimtrid ssrdv ) ABEZCBFZACGZUAEDGZBEZUBUCEZQZDHZTUBAEZDUBBCIJUGTUHUFT + UHQDABUCAKUDTUEUHUCABLUCAUBMNOPRS $. $( Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.) $) @@ -33810,9 +33812,9 @@ we define it to be empty in this case (see ~ opprc1 and ~ opprc2 ). For 9-Jul-2011.) $) intmin $p |- ( A e. B -> |^| { x e. B | A C_ x } = A ) $= ( vy wcel cv wss crab cint wi wral elintrab ssid wceq sseq2 eleq2 imbi12d - vex rspcv mpii syl5bi ssrdv ssintub a1i eqssd ) BCEZBAFZGZACHIZBUFDUIBDFZ - UIEUHUJUGEZJZACKZUFUJBEZUHAUJCDRLUFUMBBGZUNBMULUOUNJABCUGBNUHUOUKUNUGBBOU - GBUJPQSTUAUBBUIGUFABCUCUDUE $. + vex rspcv mpii biimtrid ssrdv ssintub a1i eqssd ) BCEZBAFZGZACHIZBUFDUIBD + FZUIEUHUJUGEZJZACKZUFUJBEZUHAUJCDRLUFUMBBGZUNBMULUOUNJABCUGBNUHUOUKUNUGBB + OUGBUJPQSTUAUBBUIGUFABCUCUDUE $. $( Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) $) intss $p |- ( A C_ B -> |^| B C_ |^| A ) $= @@ -34505,8 +34507,8 @@ same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $) iinss $p |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C ) $= ( vy wss wrex ciin cv wcel wral cvv wb vex eliin ax-mp wi reximi r19.36av - ssel syl syl5bi ssrdv ) CDFZABGZEABCHZDEIZUFJZUGCJZABKZUEUGDJZUGLJUHUJMEN - AUGBCLOPUEUIUKQZABGUJUKQUDULABCDUGTRUIUKABSUAUBUC $. + ssel syl biimtrid ssrdv ) CDFZABGZEABCHZDEIZUFJZUGCJZABKZUEUGDJZUGLJUHUJM + ENAUGBCLOPUEUIUKQZABGUJUKQUDULABCDUGTRUIUKABSUAUBUC $. $} ${ @@ -35092,19 +35094,19 @@ same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and bitri nfcri csbeq1a eleq2d cbvrex simplrr simplrl ad3antrrr sseldd simprr sylib jca simp-4l sbie sylibr nfs1v nfan nfim anbi1d equequ1 sbequ anbi2d imbi12d equequ2 rspc2 syl3c eqeltrd inelcm syl2anc rexlimddv exp31 impcom - sylbi syl5bi necon2bd impancom 3impa alimdv eq0 ) ABCUAZDBJZEBJZDEKZLUBZU - CZMFNZADCUDZAECUDZKZOZUEZFUFZXFLUBXBWQXIWQXCCOZABUGZFUFXBXIAFBCUHXBXKXHFW - RWSXAXKXHPWRWSMZXKXAXHXLXKMZXGWTLXGXJADQZXJAEQZMZXMWTLUIZXGXCXDOZXCXEOZMX - PXCXDXEUJXRXNXSXOAXCDCUKAXCECUKULURXKXLXPXQPZXKXJXJAGUMZMZAGRZPZGBUNABUNZ - XLXTPXJAGBXJGSUOYEXLXPXQYEXLMZXPMZXCAHNZCUPZOZXQHDYGXNYJHDQYFXNXOTXJYJAHD - XJHSAFYIAYHCUQUSZAHRZCYIXCAYHCUTVAZVBVHYGYHDOZYJMZMZXCAINZCUPZOZXQIEYPXOY - SIEQYFXNXOYOVCXJYSAIEXJISAFYRAYQCUQUSZAIRCYRXCAYQCUTVAZVBVHYPYQEOZYSMZMZY - NYHEOXQYGYNYJUUCVDZUUDYHYQEUUDYHBOZYQBOZMYEYJXJAIUMZMZHIRZUUDUUFUUGUUDDBY - HYFWRXPYOUUCYEWRWSTVEUUEVFUUDEBYQYFWSXPYOUUCYEWRWSVGVEYPUUBYSTZVFVIYEXLXP - YOUUCVJUUDYJUUHYGYNYJUUCVCUUDYSUUHYPUUBYSVGXJYSAIYTUUAVKVLVIYDUUIUUJPZYJY - AMZHGRZPAGYHYQBBUUMUUNAYJYAAYKXJAGVMVNUUNASVOUULGSYLYBUUMYCUUNYLXJYJYAYMV - PAHGVQVTGIRZUUMUUIUUNUUJUUOYAUUHYJXJGIAVRVSGIHWAVTWBWCUUKWDYHDEWEWFWGWGWH - WJWIWKWLWMWNWOWKWIFXFWPVL $. + sylbi biimtrid necon2bd impancom 3impa alimdv eq0 ) ABCUAZDBJZEBJZDEKZLUB + ZUCZMFNZADCUDZAECUDZKZOZUEZFUFZXFLUBXBWQXIWQXCCOZABUGZFUFXBXIAFBCUHXBXKXH + FWRWSXAXKXHPWRWSMZXKXAXHXLXKMZXGWTLXGXJADQZXJAEQZMZXMWTLUIZXGXCXDOZXCXEOZ + MXPXCXDXEUJXRXNXSXOAXCDCUKAXCECUKULURXKXLXPXQPZXKXJXJAGUMZMZAGRZPZGBUNABU + NZXLXTPXJAGBXJGSUOYEXLXPXQYEXLMZXPMZXCAHNZCUPZOZXQHDYGXNYJHDQYFXNXOTXJYJA + HDXJHSAFYIAYHCUQUSZAHRZCYIXCAYHCUTVAZVBVHYGYHDOZYJMZMZXCAINZCUPZOZXQIEYPX + OYSIEQYFXNXOYOVCXJYSAIEXJISAFYRAYQCUQUSZAIRCYRXCAYQCUTVAZVBVHYPYQEOZYSMZM + ZYNYHEOXQYGYNYJUUCVDZUUDYHYQEUUDYHBOZYQBOZMYEYJXJAIUMZMZHIRZUUDUUFUUGUUDD + BYHYFWRXPYOUUCYEWRWSTVEUUEVFUUDEBYQYFWSXPYOUUCYEWRWSVGVEYPUUBYSTZVFVIYEXL + XPYOUUCVJUUDYJUUHYGYNYJUUCVCUUDYSUUHYPUUBYSVGXJYSAIYTUUAVKVLVIYDUUIUUJPZY + JYAMZHGRZPAGYHYQBBUUMUUNAYJYAAYKXJAGVMVNUUNASVOUULGSYLYBUUMYCUUNYLXJYJYAY + MVPAHGVQVTGIRZUUMUUIUUNUUJUUOYAUUHYJXJGIAVRVSGIHWAVTWBWCUUKWDYHDEWEWFWGWG + WHWJWIWKWLWMWNWOWKWIFXFWPVL $. $} ${ @@ -36124,8 +36126,8 @@ same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and $( In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) $) trel3 $p |- ( Tr A -> ( ( B e. C /\ C e. D /\ D e. A ) -> B e. A ) ) $= - ( wtr wcel w3a wa 3anass trel anim2d syl5bi syld ) AEZBCFZCDFZDAFZGZOCAFZHZ - BAFROPQHZHNTOPQINUASOACDJKLABCJM $. + ( wtr wcel w3a wa 3anass trel anim2d biimtrid syld ) AEZBCFZCDFZDAFZGZOCAFZ + HZBAFROPQHZHNTOPQINUASOACDJKLABCJM $. ${ $d x A $. $d x B $. @@ -36142,9 +36144,9 @@ same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and $( The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) $) trin $p |- ( ( Tr A /\ Tr B ) -> Tr ( A i^i B ) ) $= - ( vx wtr cin wss wral wcel elin trss im2anan9 syl5bi ssin syl6ib ralrimiv - wa cv dftr3 sylibr ) ADZBDZPZCQZABEZFZCUDGUDDUBUECUDUBUCUDHZUCAFZUCBFZPZU - EUFUCAHZUCBHZPUBUIUCABITUJUGUAUKUHAUCJBUCJKLUCABMNOCUDRS $. + ( vx wtr wa cin wss wral wcel elin trss im2anan9 biimtrid syl6ib ralrimiv + cv ssin dftr3 sylibr ) ADZBDZEZCPZABFZGZCUDHUDDUBUECUDUBUCUDIZUCAGZUCBGZE + ZUEUFUCAIZUCBIZEUBUIUCABJTUJUGUAUKUHAUCKBUCKLMUCABQNOCUDRS $. $} $( The empty set is transitive. (Contributed by NM, 16-Sep-1993.) $) @@ -36674,11 +36676,11 @@ conditioning it (with ` x e. z ` ) so that it asserts the existence of a repizf2 $p |- ( A. x e. w E* y ph -> E. z A. x e. { x e. w | E. y ph } E. y e. z ph ) $= ( vv wmo cv wral wrex wex crab wi vex rabex wceq weu repizf2lem raleqf wa - nfcv nfrab1 repizf syl6bir syl5bi wel cab df-rab nfv nfex nfan nfab nfeq2 - nfcxfr exbid sylibd vtocle ) ACHBEIZJZACDIKZBACLZBUSMZJZDLZNGVCVBBUSEOPGI - ZVCQZUTVABVFJZDLZVEUTACRZBVCJZVGVIABCESVGVKVJBVFJVIVJBVFVCBVFUBZVBBUSUCZT - ABCGDFUDUEUFVGVHVDDDVFVCDVCBEUGZVBUAZBUHVBBUSUIVODBVNVBDVNDUJADCFUKULUMUO - UNVABVFVCVLVMTUPUQUR $. + nfcv nfrab1 repizf syl6bir biimtrid wel cab df-rab nfex nfan nfcxfr nfeq2 + nfv nfab exbid sylibd vtocle ) ACHBEIZJZACDIKZBACLZBUSMZJZDLZNGVCVBBUSEOP + GIZVCQZUTVABVFJZDLZVEUTACRZBVCJZVGVIABCESVGVKVJBVFJVIVJBVFVCBVFUBZVBBUSUC + ZTABCGDFUDUEUFVGVHVDDDVFVCDVCBEUGZVBUAZBUHVBBUSUIVODBVNVBDVNDUNADCFUJUKUO + ULUMVABVFVCVLVMTUPUQUR $. $} @@ -37142,19 +37144,19 @@ This definition and how we use it is easiest to understand (and most 29-Jul-2023.) $) exmidsssnc $p |- ( B e. V -> ( EXMID <-> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) ) ) $= - ( vu vy vz vw wcel wem cv csn wss c0 wceq wo wi wal wb spcgv cvv sneq wdc - exmidsssn sseq2d eqeq2d orbi2d bibi12d alimdv syl5bi biimp syl6 wn ssrab2 - alimi crab snexg rabexg sseq1 eqeq1 orbi12d imbi12d 3syl mpii wral rabeq0 - wex snmg r19.3rmv syl bitr4id biimpd wa snidg adantr simpr eleqtrrd biidd - elrab simprbi ex orim12d syld orcom syl6ib df-dc syl6ibr alrimdv df-exmid - a1dd impbid ) BCHZIAJZBKZLZWLMNZWLWMNZOZPZAQZWKIWNWQRZAQZWSIWLDJZKZLZWOWL - XCNZOZRZDQZAQWKXAADUCWKXHWTAXGWTDBCXBBNZXDWNXFWQXIXCWMWLXBBUAZUDXIXEWPWOX - IXCWMWLXJUEUFUGSUHUIWTWRAWNWQUJUNUKWKWSEJZMKLZMXKHZUBZPZEQIWKWSXOEWKWSXNX - LWKWSXMXMULZOZXNWKWSXPXMOZXQWKWSXMFWMUOZMNZXSWMNZOZXRWKWSXSWMLZYBXMFWMUMW - KWMTHXSTHWSYCYBPZPBCUPXMFWMTUQWRYDAXSTWLXSNZWNYCWQYBWLXSWMURYEWOXTWPYAWLX - SMUSWLXSWMUSUTVASVBVCWKXTXPYAXMWKXTXPWKXTXPFWMVDZXPXMFWMVEWKGJWMHGVFXPYFR - GBCVGXPFGWMVHVIVJVKWKYAXMWKYAVLZBXSHZXMYGBWMXSWKBWMHZYABCVMVNWKYAVOVPYHYI - XMXMXMFBWMFJBNXMVQVRVSVIVTWAWBXPXMWCWDXMWEWFWIWGEWHWFWJ $. + ( vu vy vz vw wcel wem cv csn wss c0 wceq wo wi wal wb spcgv cvv biimtrid + exmidsssn sneq sseq2d eqeq2d orbi2d bibi12d alimdv biimp syl6 wdc wn crab + alimi ssrab2 snexg rabexg sseq1 orbi12d imbi12d 3syl mpii wral rabeq0 wex + eqeq1 r19.3rmv syl bitr4id biimpd snidg adantr simpr eleqtrrd biidd elrab + snmg wa simprbi ex orim12d syld orcom df-dc syl6ibr a1dd alrimdv df-exmid + syl6ib impbid ) BCHZIAJZBKZLZWLMNZWLWMNZOZPZAQZWKIWNWQRZAQZWSIWLDJZKZLZWO + WLXCNZOZRZDQZAQWKXAADUBWKXHWTAXGWTDBCXBBNZXDWNXFWQXIXCWMWLXBBUCZUDXIXEWPW + OXIXCWMWLXJUEUFUGSUHUAWTWRAWNWQUIUNUJWKWSEJZMKLZMXKHZUKZPZEQIWKWSXOEWKWSX + NXLWKWSXMXMULZOZXNWKWSXPXMOZXQWKWSXMFWMUMZMNZXSWMNZOZXRWKWSXSWMLZYBXMFWMU + OWKWMTHXSTHWSYCYBPZPBCUPXMFWMTUQWRYDAXSTWLXSNZWNYCWQYBWLXSWMURYEWOXTWPYAW + LXSMVFWLXSWMVFUSUTSVAVBWKXTXPYAXMWKXTXPWKXTXPFWMVCZXPXMFWMVDWKGJWMHGVEXPY + FRGBCVQXPFGWMVGVHVIVJWKYAXMWKYAVRZBXSHZXMYGBWMXSWKBWMHZYABCVKVLWKYAVMVNYH + YIXMXMXMFBWMFJBNXMVOVPVSVHVTWAWBXPXMWCWIXMWDWEWFWGEWHWEWJ $. $} ${ @@ -37576,20 +37578,20 @@ This definition and how we use it is easiest to understand (and most 11-Jul-2011.) $) copsexg $p |- ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) $= - ( vz vw cv cop wceq wa wex wb wi vex 19.8a weq syl5bi syl5 weu euequ1 wal - eqvinop 19.23bi ex opth anbi1i 2exbii nfe1 wnf wo dveeq2or nfae anass a1i + ( vz vw cv cop wceq wa wex wb wi vex 19.8a weq biimtrid syl5 weu euequ1 + eqvinop 19.23bi ex opth anbi1i 2exbii nfe1 wal wnf wo dveeq2or nfae anass anim2d eximd biidd drex1 sylibd exbii 19.40 19.9t biimpd anim1d syl6 jaoi - ax-mp exlimi equcom eubii eupick com12 sylan9 sylbi impbid anbi1d 2exbidv - mpbi mpan eqeq1 bibi2d imbi12d mpbiri adantr exlimivv pm2.43i ) DBGZCGZHZ - IZAWJAJZCKBKZLZWJDEGZFGZHZIZWPWIIZJZFKEKWJWMMZEFDWGWHBNCNUBWSWTEFWQWTWRWQ - WTWRAWRAJZCKZBKZLZMWRAXCWRAXCXAXCCXBBOUCUDWREBPZFCPZJZXCAMWNWOWGWHENFNUEZ - XCXGAJZCKZBKZXGAXAXIBCWRXGAXHUFUGXKXEXFAJZCKZJZBKZXGAXJXOBXNBUHCBPCUAZXEC - UIZUJXJXOMZCBEUKXPXRXQXPXJXNCKXOXPXIXNCCBCULXIXEXLJZXPXNXEXFAUMZXPXLXMXEX - LXMMXPXLCOUNUOQUPXNXNCBXPXNUQURUSXQXJXNXOXJXSCKZXQXNXIXSCXTUTYAXECKZXMJXQ - XNXEXLCVAXQYBXEXMXQYBXEXECVBVCVDRQXNBOVEVFVGVHXEXOXMXFAXOXEXMXEBSZXOXEXMM - BEPZBSYCBETYDXEBBEVIVJVRXEXMBVKVSVLXMXFAXFCSZXMXFAMCFPZCSYECFTYFXFCCFVIVJ - VRXFACVKVSVLVMRQVNVOWQWJWRWMXDDWPWIVTZWQWLXCAWQWKXABCWQWJWRAYGVPVQWAWBWCW - DWEVNWF $. + a1i ax-mp exlimi equcom eubii mpbi eupick com12 sylan9 sylbi impbid eqeq1 + mpan anbi1d 2exbidv bibi2d imbi12d mpbiri adantr exlimivv pm2.43i ) DBGZC + GZHZIZAWJAJZCKBKZLZWJDEGZFGZHZIZWPWIIZJZFKEKWJWMMZEFDWGWHBNCNUAWSWTEFWQWT + WRWQWTWRAWRAJZCKZBKZLZMWRAXCWRAXCXAXCCXBBOUBUCWREBPZFCPZJZXCAMWNWOWGWHENF + NUDZXCXGAJZCKZBKZXGAXAXIBCWRXGAXHUEUFXKXEXFAJZCKZJZBKZXGAXJXOBXNBUGCBPCUH + ZXECUIZUJXJXOMZCBEUKXPXRXQXPXJXNCKXOXPXIXNCCBCULXIXEXLJZXPXNXEXFAUMZXPXLX + MXEXLXMMXPXLCOVFUNQUOXNXNCBXPXNUPUQURXQXJXNXOXJXSCKZXQXNXIXSCXTUSYAXECKZX + MJXQXNXEXLCUTXQYBXEXMXQYBXEXECVAVBVCRQXNBOVDVEVGVHXEXOXMXFAXOXEXMXEBSZXOX + EXMMBEPZBSYCBETYDXEBBEVIVJVKXEXMBVLVRVMXMXFAXFCSZXMXFAMCFPZCSYECFTYFXFCCF + VIVJVKXFACVLVRVMVNRQVOVPWQWJWRWMXDDWPWIVQZWQWLXCAWQWKXABCWQWJWRAYGVSVTWAW + BWCWDWEVOWF $. $} ${ @@ -38334,9 +38336,9 @@ We have not yet defined relations ( ~ df-rel ), but here we introduce a few 27-Mar-1997.) $) po3nr $p |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B R C /\ C R D /\ D R B ) ) $= - ( wpo wcel w3a wa wbr wn po2nr 3adantr2 df-3an potr anim1d syl5bi mtod ) AE - FZBAGZCAGZDAGZHIZBCEJZCDEJZDBEJZHZBDEJZUFIZSTUBUIKUAABDELMUGUDUEIZUFIUCUIUD - UEUFNUCUJUHUFABCDEOPQR $. + ( wpo wcel w3a wa wbr wn po2nr 3adantr2 df-3an potr anim1d biimtrid mtod ) + AEFZBAGZCAGZDAGZHIZBCEJZCDEJZDBEJZHZBDEJZUFIZSTUBUIKUAABDELMUGUDUEIZUFIUCUI + UDUEUFNUCUJUHUFABCDEOPQR $. ${ $d x y z R $. @@ -38452,13 +38454,13 @@ We have not yet defined relations ( ~ df-rel ), but here we introduce a few sowlin $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( B R C -> ( B R D \/ D R C ) ) ) $= ( vx vy vz wcel w3a wbr wo wi cv wceq breq1 imbi2d breq2 wral wa wor rsp2 - orbi1d imbi12d orbi2d orbi12d df-iso 3anass syl6 impd syl5bi adantl sylbi - wpo rsp com12 vtocl3ga impcom ) BAICAIDAIJAEUAZBCEKZBDEKZDCEKZLZMZUSFNZGN - ZEKZVEHNZEKZVHVFEKZLZMZMUSBVFEKZBVHEKZVJLZMZMUSUTVNVHCEKZLZMZMUSVDMFGHBCD - AAAVEBOZVLVPUSVTVGVMVKVOVEBVFEPVTVIVNVJVEBVHEPUCUDQVFCOZVPVSUSWAVMUTVOVRV - FCBERWAVJVQVNVFCVHERUEUDQVHDOZVSVDUSWBVRVCUTWBVNVAVQVBVHDBERVHDCEPUFQQUSV - EAIZVFAIZVHAIZJZVLUSAEUNZVLHASGASZFASZTWFVLMZFGHAEUGWIWJWGWFWCWDWETZTWIVL - WCWDWEUHWIWCWKVLWIWCWHWKVLMWHFAUOVLGHAAUBUIUJUKULUMUPUQUR $. + orbi1d imbi12d orbi2d orbi12d wpo df-iso 3anass syl6 impd biimtrid adantl + rsp sylbi com12 vtocl3ga impcom ) BAICAIDAIJAEUAZBCEKZBDEKZDCEKZLZMZUSFNZ + GNZEKZVEHNZEKZVHVFEKZLZMZMUSBVFEKZBVHEKZVJLZMZMUSUTVNVHCEKZLZMZMUSVDMFGHB + CDAAAVEBOZVLVPUSVTVGVMVKVOVEBVFEPVTVIVNVJVEBVHEPUCUDQVFCOZVPVSUSWAVMUTVOV + RVFCBERWAVJVQVNVFCVHERUEUDQVHDOZVSVDUSWBVRVCUTWBVNVAVQVBVHDBERVHDCEPUFQQU + SVEAIZVFAIZVHAIZJZVLUSAEUGZVLHASGASZFASZTWFVLMZFGHAEUHWIWJWGWFWCWDWETZTWI + VLWCWDWEUIWIWCWKVLWIWCWHWKVLMWHFAUNVLGHAAUBUJUKULUMUOUPUQUR $. $} $( A strict order relation has no 2-cycle loops. (Contributed by NM, @@ -39609,7 +39611,7 @@ rather than an implication (but the two are equivalent by ~ bm1.3ii ), depend on ` x ` . (Contributed by Mario Carneiro, 18-Nov-2016.) $) eusvnfb $p |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) ) $= ( cv wceq wal weu wnfc cvv wcel eusvnf wex euex vex eqeltrrdi sps exlimiv - wa id syl jca isset nfcvd nfeqd nfrd eximdv syl5bi eusv1 sylibr impbii + wa id syl jca isset nfcvd nfeqd nfrd eximdv biimtrid eusv1 sylibr impbii imp ) BDZCEZAFZBGZACHZCIJZRZUOUPUQABCKUOUNBLZUQUNBMUNUQBUMUQAUMCULIUMSBNO PQTUAURUSUOUPUQUSUQUMBLUPUSBCUBUPUMUNBUPUMAUPAULCUPAULUCUPSUDUEUFUGUKABCU HUIUJ $. @@ -39681,13 +39683,13 @@ rather than an implication (but the two are equivalent by ~ bm1.3ii ), <-> E. x e. A A. y e. B ( ph -> x = C ) ) ) $= ( wcel wa wrex wceq wi wral cv bitri ralbii eleq1d anbi12d cbvrexv risset nfra2xy nfv nfim ralcom impexp bi2.04 r19.21v rsp sylbi com3l imp31 eqeq1 - eqcom bitrdi imbi2d ralbidv syl5ibrcom reximdv ex expimpd rexlimi reusv3i - com23 syl5bi impbid1 ) AHFLZMZDGNZABMHIOZPZEGQDGQZACRZHOZPZDGQZCFNZVLBIFL - ZMZEGNVOVTPZVKWBDEGDRERZOZABVJWAJWEHIFKUAUBUCWBWCEGVOVTEVNDEGGUEVTEUFUGWD - GLZBWAWCWAVPIOZCFNZWFBMZWCCIFUDWIVOWHVTWIVOWHVTPWIVOMZWGVSCFWJVSWGAVMPZDG - QZWFBVOWLVOWFBWLVOBWLPZEGQZWFWMPVOVNDGQZEGQWNVNDEGGUHWOWMEGWOBWKPZDGQWMVN - WPDGVNABVMPPWPABVMUIABVMUJSTBWKDGUKSTSWMEGULUMUNUOWGVRWKDGWGVQVMAWGVQIHOV - MVPIHUPIHUQURUSUTVAVBVCVGVHVDVEUMABCDEFGHIJKVFVI $. + eqcom bitrdi ralbidv syl5ibrcom reximdv ex com23 biimtrid expimpd rexlimi + imbi2d reusv3i impbid1 ) AHFLZMZDGNZABMHIOZPZEGQDGQZACRZHOZPZDGQZCFNZVLBI + FLZMZEGNVOVTPZVKWBDEGDRERZOZABVJWAJWEHIFKUAUBUCWBWCEGVOVTEVNDEGGUEVTEUFUG + WDGLZBWAWCWAVPIOZCFNZWFBMZWCCIFUDWIVOWHVTWIVOWHVTPWIVOMZWGVSCFWJVSWGAVMPZ + DGQZWFBVOWLVOWFBWLVOBWLPZEGQZWFWMPVOVNDGQZEGQWNVNDEGGUHWOWMEGWOBWKPZDGQWM + VNWPDGVNABVMPPWPABVMUIABVMUJSTBWKDGUKSTSWMEGULUMUNUOWGVRWKDGWGVQVMAWGVQIH + OVMVPIHUPIHUQURVGUSUTVAVBVCVDVEVFUMABCDEFGHIJKVHVI $. $} ${ @@ -39970,12 +39972,12 @@ rather than an implication (but the two are equivalent by ~ bm1.3ii ), shortened by Andrew Salmon, 12-Aug-2011.) $) ssorduni $p |- ( A C_ On -> Ord U. A ) $= ( vx vy con0 wss cuni wtr word cv wral wcel wrex eluni2 wa wi ssel onelss - syl6 rexlimdv syl5bi anc2r syl ssuni syl8 ralrimiv dftr3 sylibr onelon ex - ssrdv ordon trssord 3exp mpii sylc ) ADEZAFZGZUQDEZUQHZUPBIZUQEZBUQJURUPV - BBUQVAUQKZVACIZKZCALZUPVBCVAAMZUPVEVBCAUPVDAKZVEVAVDEZVHNZVBUPVHVEVIOZOVH - VEVJOOUPVHVDDKZVKADVDPZVDVAQRVHVEVIUAUBVAVDAUCUDSTUEBUQUFUGUPBUQDVCVFUPVA - DKZVGUPVEVNCAUPVHVLVEVNOVMVLVEVNVDVAUHUIRSTUJURUSDHZUTUKURUSVOUTUQDULUMUN - UO $. + syl6 rexlimdv biimtrid anc2r syl ssuni ralrimiv dftr3 sylibr onelon ssrdv + syl8 ex ordon trssord 3exp mpii sylc ) ADEZAFZGZUQDEZUQHZUPBIZUQEZBUQJURU + PVBBUQVAUQKZVACIZKZCALZUPVBCVAAMZUPVEVBCAUPVDAKZVEVAVDEZVHNZVBUPVHVEVIOZO + VHVEVJOOUPVHVDDKZVKADVDPZVDVAQRVHVEVIUAUBVAVDAUCUISTUDBUQUEUFUPBUQDVCVFUP + VADKZVGUPVEVNCAUPVHVLVEVNOVMVLVEVNVDVAUGUJRSTUHURUSDHZUTUKURUSVOUTUQDULUM + UNUO $. $} $( The union of a set of ordinal numbers is an ordinal number. Theorem 9 of @@ -40068,10 +40070,10 @@ rather than an implication (but the two are equivalent by ~ bm1.3ii ), ordsucim $p |- ( Ord A -> Ord suc A ) $= ( vx word csuc wtr cv wral ordtr suctr syl wcel wceq wo csn df-suc eleq2i cun elun velsn dford3 orbi2i 3bitri wal simprbi df-ral sylib 19.21bi treq - wi syl5ibrcom jaod syl5bi ralrimiv sylanbrc ) ACZADZEZBFZEZBUPGUPCUOAEZUQ - AHZAIJUOUSBUPURUPKZURAKZURALZMZUOUSVBURAANZQZKVCURVFKZMVEUPVGURAOPURAVFRV - HVDVCBASUAUBUOVCUSVDUOVCUSUIZBUOUSBAGZVIBUCUOUTVJBATUDUSBAUEUFUGUOUSVDUTV - AURAUHUJUKULUMBUPTUN $. + wi syl5ibrcom jaod biimtrid ralrimiv sylanbrc ) ACZADZEZBFZEZBUPGUPCUOAEZ + UQAHZAIJUOUSBUPURUPKZURAKZURALZMZUOUSVBURAANZQZKVCURVFKZMVEUPVGURAOPURAVF + RVHVDVCBASUAUBUOVCUSVDUOVCUSUIZBUOUSBAGZVIBUCUOUTVJBATUDUSBAUEUFUGUOUSVDU + TVAURAUHUJUKULUMBUPTUN $. $} $( The successor of an ordinal number is an ordinal number. Proposition 7.24 @@ -41159,8 +41161,8 @@ numbers have the property (conclusion). Exercise 25 of [Enderton] $( Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) $) tfis2f $p |- ( x e. On -> ph ) $= - ( wsb cv wral con0 wcel sbie ralbii syl5bi tfis ) ACDACDHZDCIZJBDRJRKLAQB - DRABCDEFMNGOP $. + ( wsb cv wral con0 wcel sbie ralbii biimtrid tfis ) ACDACDHZDCIZJBDRJRKLA + QBDRABCDEFMNGOP $. $} ${ @@ -42668,12 +42670,12 @@ of the Cartesian product represent the top (integer) and bottom A. x A. y A. z ( <. <. x , y >. , z >. e. A -> <. <. x , y >. , z >. e. B ) ) ) $= ( vw cvv cxp wss cv cop wcel wi wal ssel alrimiv wex eleq1 sylib dfss2 - alrimivv wceq elvvv imbi12d biimprcd alimi 19.23v 2alimi syl5bi com23 a2d - 19.23vv alimdv 3imtr4g com12 impbid2 ) DGGHGHZIZDEIZAJBJKCJKZDLZUTELZMZCN - ZBNANZUSVDABUSVCCDEUTOPUAVEURUSVEFJZDLZVFUQLZMZFNVGVFELZMZFNURUSVEVIVKFVE - VGVHVJVEVHVGVJVHVFUTUBZCQZBQAQZVEVKABCVFUCVEVMVKMZBNANVNVKMVDVOABVDVLVKMZ - CNVOVCVPCVLVKVCVLVGVAVJVBVFUTDRVFUTERUDUEUFVLVKCUGSUHVMVKABULSUIUJUKUMFDU - QTFDETUNUOUP $. + alrimivv wceq elvvv imbi12d biimprcd alimi 19.23v 2alimi 19.23vv biimtrid + com23 a2d alimdv 3imtr4g com12 impbid2 ) DGGHGHZIZDEIZAJBJKCJKZDLZUTELZMZ + CNZBNANZUSVDABUSVCCDEUTOPUAVEURUSVEFJZDLZVFUQLZMZFNVGVFELZMZFNURUSVEVIVKF + VEVGVHVJVEVHVGVJVHVFUTUBZCQZBQAQZVEVKABCVFUCVEVMVKMZBNANVNVKMVDVOABVDVLVK + MZCNVOVCVPCVLVKVCVLVGVAVJVBVFUTDRVFUTERUDUEUFVLVKCUGSUHVMVKABUISUJUKULUMF + DUQTFDETUNUOUP $. $( Extensionality principle for ordered triples, analogous to ~ eqrel . Use ~ relrelss to express the antecedent in terms of the relation @@ -43123,27 +43125,27 @@ of the Cartesian product represent the top (integer) and bottom wo snex opeqsn bitrdi 2exbidv imbi12d spcv sneq cbvexv a9ev equcom 19.41v exbii mpbi mpbiran eqid a1bi 3bitr2ri sylib prexg mp2an mpi opeqpr preqsn idd eqtr2 simplbi syl dfsn2 preq2 eqtr2id eqtrid biimpd expd com12 adantr - anbi12d mpd expcom impd jaod syl5bi 2eximdv exlimiv imp sylbi simpr equid - syl2an jctl sylibr eqtr4d opeq12 spc2ev adantlr preq12 biimpa dfop jaodan - eqtr4di ex 3imtr4g ssrdv exlimivv impbii ) CDJZUDYCKKUAZUBZCALZUEZMZDYFBL - ZUFZMZNZBOAOZYCUCYEYMYEGLZCUEZMZYNYFYIJZMZBOAOZPZGUGZYNCDUFZMZYSPZGUGZNZY - MYEYNYCQZYNYDQZPZGUGZUUFGYCYDUHUUJYTUUDNZGUGUUFUUIUUKGUUIYPUUCURZYSPUUKUU - GUULUUHYSYNCDGUIEFUJZABYNUKULYPYSUUCUMRUNYTUUDGUORRUUACHLZUEZMZHOZUUBYQMZ - BOAOZYMUUEUUAYOYOMZABSZYHNZBOZAOZPZUUQYTUVEGYOCEUSYPYPUUTYSUVDYNYOYOUPYPY - RUVBABYPYRYOYQMZUVBYNYOYQUPUVFYQYOMUVBYOYQUQYFYICAUIZBUIZEUTRVAVBVCVDUUQY - HAOUVDUVEUUPYHHAHASZUUOYGCUUNYFVETVFUVCYHAUVCUVABOZYHBASZBOUVJBAVGUVKUVAB - BAVHVJVKUVAYHBVIVLVJUUTUVDYOVMVNVOVPUUEUUBUUBMZUUSUUBVMUUDUVLUUSPGUUBCKQD - KQUUBKQEFCDKKVQVRUUCUUCUVLYSUUSYNUUBUUBUPUUCYRUURABYNUUBYQUPVBVCVDVSUUQUU - SYMUUPUUSYMPHUUPUURYLABUURYLCYJMZDYGMZNZURZUUPYLUURYQUUBMUVPUUBYQUQYFYICD - UVGUVHEFVTRUUPYLYLUVOUUPYLWBUUPUVMUVNYLUVMUUPUVNYLPZUVMUUPNZUVAUVQUVRYJUU - OMZUVACYJUUOWCUVSUVABHSYFYIUUNUVGUVHHUIWAWDWEUVMUVAUVQPUUPUVAUVMUVQUVAUVM - UVNYLUVAUVOYLUVAUVMYHUVNYKUVAYJYGCUVAYGYFYFUFZYJYFWFZYFYIYFWGZWHTUVAYGYJD - UVAYGUVTYJUWAUWBWITWNWJWKWLWMWOWPWQWRWSWTXAXBXFXCYLYEABYLGYCYDYLUULYNUUNI - LZJZMZIOHOZUUGUUHYLUULUWFYLYPUWFUUCYHYPUWFYKYHYPNZYNYFYFJZMZUWFUWGYNYOUWH - YHYPXDYHUWHYOMZYPYHAASZYHNUWJYHUWKAXEXGYFYFCUVGUVGEUTXHWMXIUWEUWIHIYFYFUV - GUVGUVIIASNUWDUWHYNUUNUWCYFYFXJTXKWEXLYLUUCNZYRUWFUWLYNYGYJUFZYQYLUUCYNUW - MMYLUUBUWMYNCDYGYJXMTXNYFYIUVGUVHXOXQUWEYRHIYFYIUVGUVHUVIIBSNUWDYQYNUUNUW - CYFYIXJTXKWEXPXRUUMHIYNUKXSXTYAYBR $. + anbi12d mpd expcom impd biimtrid 2eximdv exlimiv syl2an sylbi simpr equid + jaod imp sylibr eqtr4d opeq12 spc2ev adantlr preq12 biimpa eqtr4di jaodan + jctl dfop ex 3imtr4g ssrdv exlimivv impbii ) CDJZUDYCKKUAZUBZCALZUEZMZDYF + BLZUFZMZNZBOAOZYCUCYEYMYEGLZCUEZMZYNYFYIJZMZBOAOZPZGUGZYNCDUFZMZYSPZGUGZN + ZYMYEYNYCQZYNYDQZPZGUGZUUFGYCYDUHUUJYTUUDNZGUGUUFUUIUUKGUUIYPUUCURZYSPUUK + UUGUULUUHYSYNCDGUIEFUJZABYNUKULYPYSUUCUMRUNYTUUDGUORRUUACHLZUEZMZHOZUUBYQ + MZBOAOZYMUUEUUAYOYOMZABSZYHNZBOZAOZPZUUQYTUVEGYOCEUSYPYPUUTYSUVDYNYOYOUPY + PYRUVBABYPYRYOYQMZUVBYNYOYQUPUVFYQYOMUVBYOYQUQYFYICAUIZBUIZEUTRVAVBVCVDUU + QYHAOUVDUVEUUPYHHAHASZUUOYGCUUNYFVETVFUVCYHAUVCUVABOZYHBASZBOUVJBAVGUVKUV + ABBAVHVJVKUVAYHBVIVLVJUUTUVDYOVMVNVOVPUUEUUBUUBMZUUSUUBVMUUDUVLUUSPGUUBCK + QDKQUUBKQEFCDKKVQVRUUCUUCUVLYSUUSYNUUBUUBUPUUCYRUURABYNUUBYQUPVBVCVDVSUUQ + UUSYMUUPUUSYMPHUUPUURYLABUURYLCYJMZDYGMZNZURZUUPYLUURYQUUBMUVPUUBYQUQYFYI + CDUVGUVHEFVTRUUPYLYLUVOUUPYLWBUUPUVMUVNYLUVMUUPUVNYLPZUVMUUPNZUVAUVQUVRYJ + UUOMZUVACYJUUOWCUVSUVABHSYFYIUUNUVGUVHHUIWAWDWEUVMUVAUVQPUUPUVAUVMUVQUVAU + VMUVNYLUVAUVOYLUVAUVMYHUVNYKUVAYJYGCUVAYGYFYFUFZYJYFWFZYFYIYFWGZWHTUVAYGY + JDUVAYGUVTYJUWAUWBWITWNWJWKWLWMWOWPWQXEWRWSWTXFXAXBYLYEABYLGYCYDYLUULYNUU + NILZJZMZIOHOZUUGUUHYLUULUWFYLYPUWFUUCYHYPUWFYKYHYPNZYNYFYFJZMZUWFUWGYNYOU + WHYHYPXCYHUWHYOMZYPYHAASZYHNUWJYHUWKAXDXPYFYFCUVGUVGEUTXGWMXHUWEUWIHIYFYF + UVGUVGUVIIASNUWDUWHYNUUNUWCYFYFXITXJWEXKYLUUCNZYRUWFUWLYNYGYJUFZYQYLUUCYN + UWMMYLUUBUWMYNCDYGYJXLTXMYFYIUVGUVHXQXNUWEYRHIYFYIUVGUVHUVIIBSNUWDYQYNUUN + UWCYFYIXITXJWEXOXRUUMHIYNUKXSXTYAYBR $. $} ${ @@ -44536,15 +44538,15 @@ of the Cartesian product represent the top (integer) and bottom to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) $) iss $p |- ( A C_ _I <-> A = ( _I |` dom A ) ) $= - ( vx vy cid wss cdm cres wceq cv cop wcel wb wal wa ssel wi opeldm syl5bi - vex wrel a1i jcad wbr df-br bitr3i wex eldm2 opeq2 eleq1d biimprcd sylcom - ideq exlimdv imbi2d syl5ibcom impd impbid opelres alrimivv reli relss mpi - bitr4di relres eqrel sylancl mpbird resss sseq1 mpbiri impbii ) ADEZADAFZ - GZHZVLVOBIZCIZJZAKZVRVNKZLZCMBMZVLWABCVLVSVRDKZVPVMKZNZVTVLVSWEVLVSWCWDAD - VROZVSWDPVLVPVQABSZCSZQUAUBVLWCWDVSWCVPVQHZVLWDVSPZWCVPVQDUCWIVPVQDUDVPVQ - WHULUEZVLWDVPVPJZAKZPWIWJWDVSCUFVLWMCVPAWGUGVLVSWMCVLVSWCWMWFWCWIVSWMWKWI - WMVSWIWLVRAVPVQVPUHUIZUJRUKUMRWIWMVSWDWNUNUORUPUQVPVQDVMWHURVCUSVLATZVNTV - OWBLVLDTWOUTADVAVBDVMVDBCAVNVEVFVGVOVLVNDEDVMVHAVNDVIVJVK $. + ( vx vy cid wss cdm cres wceq cv cop wcel wb wal ssel vex opeldm biimtrid + wa wi wrel a1i jcad wbr df-br ideq bitr3i wex eldm2 opeq2 eleq1d biimprcd + sylcom exlimdv imbi2d syl5ibcom impd impbid opelres bitr4di alrimivv reli + relss mpi relres eqrel sylancl mpbird resss sseq1 mpbiri impbii ) ADEZADA + FZGZHZVLVOBIZCIZJZAKZVRVNKZLZCMBMZVLWABCVLVSVRDKZVPVMKZRZVTVLVSWEVLVSWCWD + ADVRNZVSWDSVLVPVQABOZCOZPUAUBVLWCWDVSWCVPVQHZVLWDVSSZWCVPVQDUCWIVPVQDUDVP + VQWHUEUFZVLWDVPVPJZAKZSWIWJWDVSCUGVLWMCVPAWGUHVLVSWMCVLVSWCWMWFWCWIVSWMWK + WIWMVSWIWLVRAVPVQVPUIUJZUKQULUMQWIWMVSWDWNUNUOQUPUQVPVQDVMWHURUSUTVLATZVN + TVOWBLVLDTWOVAADVBVCDVMVDBCAVNVEVFVGVOVLVNDEDVMVHAVNDVIVJVK $. $} ${ @@ -45130,12 +45132,12 @@ of the Cartesian product represent the top (integer) and bottom Carneiro, 2-Nov-2015.) $) poirr2 $p |- ( R Po A -> ( R i^i ( _I |` A ) ) = (/) ) $= ( vx vy wpo cid cres cin c0 wss wceq wrel relres relin2 cv wcel wbr wa wn - syl5bi mp1i cop df-br brin bitr3i wi vex brres poirr wb ideq breq2 notbid - sylbi syl5ibcom expimpd ancomsd con2d imnan sylib pm2.21d relssdv ss0 syl - ) ABEZBFAGZHZIJVGIKVECDVGIVFLVGLVEFAMBVFNUACOZDOZUBZVGPZVHVIBQZVHVIVFQZRZ - VEVJIPZVKVHVIVGQVNVHVIVGUCVHVIBVFUDUEVEVNVOVEVLVMSUFVNSVEVMVLVMVHVIFQZVHA - PZRVEVLSZVHVIFADUGZUHVEVQVPVRVEVQVPVRVEVQRVHVHBQZSVPVRAVHBUIVPVTVLVPVHVIK - VTVLUJVHVIVSUKVHVIVHBULUNUMUOUPUQTURVLVMUSUTVATVBVGVCVD $. + biimtrid mp1i cop df-br brin bitr3i wi vex brres poirr breq2 sylbi notbid + ideq syl5ibcom expimpd ancomsd con2d imnan sylib pm2.21d relssdv ss0 syl + wb ) ABEZBFAGZHZIJVGIKVECDVGIVFLVGLVEFAMBVFNUACOZDOZUBZVGPZVHVIBQZVHVIVFQ + ZRZVEVJIPZVKVHVIVGQVNVHVIVGUCVHVIBVFUDUEVEVNVOVEVLVMSUFVNSVEVMVLVMVHVIFQZ + VHAPZRVEVLSZVHVIFADUGZUHVEVQVPVRVEVQVPVRVEVQRVHVHBQZSVPVRAVHBUIVPVTVLVPVH + VIKVTVLVDVHVIVSUMVHVIVHBUJUKULUNUOUPTUQVLVMURUSUTTVAVGVBVC $. $} $( The relation induced by a transitive relation on a part of its field is @@ -47204,20 +47206,20 @@ We use their notation ("onto" under the arrow). (Contributed by NM, wfun opeq1 eqeq1 opeq2 eqeq2 csn wa wex wrel funrel relop sylib opth opid preq1i dfop preq2i snex zfpair2 3eqtr4ri eqeq2i bitr3i wal dffun4 simprbi prid1 eleq2 mpbiri prid2 jca w3a opeq12 3adant3 eleq1d 3adant2 anbi12d wb - opex eqeq12 3adant1 spc3gv mp3an syl2im syl5bi dfsn2 preq2 eqtr2id eqeq2d - eqtr3 expcom syl6bi com13 imp sylcom exlimdvv mpd vtocl2g 3impia ) ACLBDL - ABMZUCZABNZEOZFOZMZUCZEFPZQAXEMZUCZAXENZQXBXCQEFABCDXDANZXGXJXHXKXLXFXIXD - AXEUDUAXDAXEUEUBXEBNZXJXBXKXCXMXIXAXEBAUFUAXEBAUGUBXGXDGOZUHZNZXEXNHOZRZN - ZUIZHUJGUJZXHXGXFUKZYAXFULGHXDXEESZFSZUMUNXGXTXHGHXGXTGHPZXHXTXFXNXNMZXNX - QMZRZNZXGYEXTXFXOXRMZNYIXDXEXOXRYCYDUOYJYHXFYFXOXRRZRXOUHZYKRYHYJYFYLYKXN - GSZUPUQYGYKYFXNXQYMHSZURUSXOXRXNYMUTGHVAURVBVCVDXGIOZJOZMZXFLZYOKOZMZXFLZ - UIZJKPZQZKVEJVEIVEZYIYFXFLZYGXFLZUIZYEXGYBUUEIJKXFVFVGYIUUFUUGYIUUFYFYHLY - FYGXNXNYMYMVTVHXFYHYFVIVJYIUUGYGYHLYFYGXNXQYMYNVTVKXFYHYGVIVJVLXNTLZUUIXQ - TLUUEUUHYEQZQYMYMYNUUDUUJIJKXNXNXQTTTIGPZJGPZKHPZVMZUUBUUHUUCYEUUNYRUUFUU - AUUGUUNYQYFXFUUKUULYQYFNUUMYOYPXNXNVNVOVPUUNYTYGXFUUKUUMYTYGNUULYOYSXNXQV - NVQVPVRUULUUMUUCYEVSUUKYPXNYSXQWAWBUBWCWDWEWFXPXSYEXHQYEXSXPXHYEXSXEXONZX - PXHQYEXRXOXEYEXOXNXNRXRXNWGXNXQXNWHWIWJXPUUOXHXDXEXOWKWLWMWNWOWPWQWRWSWT - $. + opex eqeq12 3adant1 spc3gv mp3an syl2im biimtrid dfsn2 preq2 eqeq2d eqtr3 + eqtr2id expcom syl6bi com13 imp sylcom exlimdvv mpd vtocl2g 3impia ) ACLB + DLABMZUCZABNZEOZFOZMZUCZEFPZQAXEMZUCZAXENZQXBXCQEFABCDXDANZXGXJXHXKXLXFXI + XDAXEUDUAXDAXEUEUBXEBNZXJXBXKXCXMXIXAXEBAUFUAXEBAUGUBXGXDGOZUHZNZXEXNHOZR + ZNZUIZHUJGUJZXHXGXFUKZYAXFULGHXDXEESZFSZUMUNXGXTXHGHXGXTGHPZXHXTXFXNXNMZX + NXQMZRZNZXGYEXTXFXOXRMZNYIXDXEXOXRYCYDUOYJYHXFYFXOXRRZRXOUHZYKRYHYJYFYLYK + XNGSZUPUQYGYKYFXNXQYMHSZURUSXOXRXNYMUTGHVAURVBVCVDXGIOZJOZMZXFLZYOKOZMZXF + LZUIZJKPZQZKVEJVEIVEZYIYFXFLZYGXFLZUIZYEXGYBUUEIJKXFVFVGYIUUFUUGYIUUFYFYH + LYFYGXNXNYMYMVTVHXFYHYFVIVJYIUUGYGYHLYFYGXNXQYMYNVTVKXFYHYGVIVJVLXNTLZUUI + XQTLUUEUUHYEQZQYMYMYNUUDUUJIJKXNXNXQTTTIGPZJGPZKHPZVMZUUBUUHUUCYEUUNYRUUF + UUAUUGUUNYQYFXFUUKUULYQYFNUUMYOYPXNXNVNVOVPUUNYTYGXFUUKUUMYTYGNUULYOYSXNX + QVNVQVPVRUULUUMUUCYEVSUUKYPXNYSXQWAWBUBWCWDWEWFXPXSYEXHQYEXSXPXHYEXSXEXON + ZXPXHQYEXRXOXEYEXOXNXNRXRXNWGXNXQXNWHWKWIXPUUOXHXDXEXOWJWLWMWNWOWPWQWRWSW + T $. $} ${ @@ -47288,14 +47290,14 @@ We use their notation ("onto" under the arrow). (Contributed by NM, subclass. (Contributed by NM, 15-Aug-1994.) $) funssres $p |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G ) $= ( vx vy wfun wss wa cdm cres wceq cv cop wcel wb wal vex wi wex imp wrel - opelres ssel opeldm a1i jcad adantl weu funeu2 eldm2 eximdv syl5bi eupick - ancrd syl2an exp43 com23 com34 pm2.43d impd impbid alrimivv relres funrel - bitr4id relss mpan9 eqrel sylancr mpbird ) AEZBAFZGZABHZIZBJZCKZDKZLZVNMZ - VRBMZNZDOCOZVLWACDVLVSVRAMZVPVMMZGZVTVPVQAVMDPZUAVLVTWEVKVTWEQVJVKVTWCWDB - AVRUBZVTWDQVKVPVQBCPZWFUCUDUEUFVLWCWDVTVLWCWDVTQVLWCWDWCVTVJVKWCWDWCVTQZQ - ZQVJWCVKWJVJWCVKWDWIVJWCGWCDUGWCVTGZDRZWIVKWDGDVPVQAUHVKWDWLWDVTDRVKWLDVP - BWHUIVKVTWKDVKVTWCWGUMUJUKSWCVTDULUNUOUPSUQURUSUTVDVAVLVNTBTZVOWBNAVMVBVJ - ATVKWMAVCBAVEVFCDVNBVGVHVI $. + opelres ssel opeldm jcad adantl funeu2 eldm2 ancrd eximdv biimtrid eupick + a1i syl2an exp43 com23 pm2.43d impd impbid bitr4id alrimivv relres funrel + weu com34 relss mpan9 eqrel sylancr mpbird ) AEZBAFZGZABHZIZBJZCKZDKZLZVN + MZVRBMZNZDOCOZVLWACDVLVSVRAMZVPVMMZGZVTVPVQAVMDPZUAVLVTWEVKVTWEQVJVKVTWCW + DBAVRUBZVTWDQVKVPVQBCPZWFUCULUDUEVLWCWDVTVLWCWDVTQVLWCWDWCVTVJVKWCWDWCVTQ + ZQZQVJWCVKWJVJWCVKWDWIVJWCGWCDVCWCVTGZDRZWIVKWDGDVPVQAUFVKWDWLWDVTDRVKWLD + VPBWHUGVKVTWKDVKVTWCWGUHUIUJSWCVTDUKUMUNUOSVDUPUQURUSUTVLVNTBTZVOWBNAVMVA + VJATVKWMAVBBAVEVFCDVNBVGVHVI $. $} $( Equality of restrictions of a function and a subclass. (Contributed by @@ -47314,15 +47316,15 @@ We use their notation ("onto" under the arrow). (Contributed by NM, ( vx vy vz wfun wa cdm wrel cv wi wal anim12i wo wn 19.21bi opeldm dffun4 wcel vex cin c0 wceq cun cop funrel relun sylibr elun anbi12i anddi bitri adantr disj1 biimpi imnan sylib nsyl orel2 syl con2d orel1 orim12d adantl - syl5bi simprbi 19.21bbi jaao syld alrimiv alrimivv sylanbrc ) AFZBFZGZAHZ - BHZUAUBUCZGZABUDZIZCJZDJZUEZVTSZWBEJZUEZVTSZGZWCWFUCZKZELZDLCLVTFVOWAVRVO - AIZBIZGWAVMWMVNWNAUFBUFMABUGUHUMVSWLCDVSWKEVSWIWDASZWGASZGZWDBSZWGBSZGZNZ - WJWIWQWOWSGZNZWRWPGZWTNZNZVSXAWIWOWRNZWPWSNZGXFWEXGWHXHWDABUIWGABUIUJWOWR - WPWSUKULVRXFXAKVOVRXCWQXEWTVRXBOXCWQKVRWBVPSZWBVQSZGZXBVRXIXJOKZXKOVRXLCV - RXLCLCVPVQUNUOPZXIXJUPUQWOXIWSXJWBWCACTZDTZQWBWFBXNETZQMURXBWQUSUTVRXDOXE - WTKVRXJXIGZXDVRXJXIOKXQOVRXIXJXMVAXJXIUPUQWRXJWPXIWBWCBXNXOQWBWFAXNXPQMUR - XDWTVBUTVCVDVEVOXAWJKVRVMWQWJVNWTVMWQWJKZDEVMXRELDLZCVMWMXSCLCDEARVFPVGVN - WTWJKZDEVNXTELDLZCVNWNYACLCDEBRVFPVGVHUMVIVJVKCDEVTRVL $. + biimtrid simprbi 19.21bbi jaao syld alrimiv alrimivv sylanbrc ) AFZBFZGZA + HZBHZUAUBUCZGZABUDZIZCJZDJZUEZVTSZWBEJZUEZVTSZGZWCWFUCZKZELZDLCLVTFVOWAVR + VOAIZBIZGWAVMWMVNWNAUFBUFMABUGUHUMVSWLCDVSWKEVSWIWDASZWGASZGZWDBSZWGBSZGZ + NZWJWIWQWOWSGZNZWRWPGZWTNZNZVSXAWIWOWRNZWPWSNZGXFWEXGWHXHWDABUIWGABUIUJWO + WRWPWSUKULVRXFXAKVOVRXCWQXEWTVRXBOXCWQKVRWBVPSZWBVQSZGZXBVRXIXJOKZXKOVRXL + CVRXLCLCVPVQUNUOPZXIXJUPUQWOXIWSXJWBWCACTZDTZQWBWFBXNETZQMURXBWQUSUTVRXDO + XEWTKVRXJXIGZXDVRXJXIOKXQOVRXIXJXMVAXJXIUPUQWRXJWPXIWBWCBXNXOQWBWFAXNXPQM + URXDWTVBUTVCVDVEVOXAWJKVRVMWQWJVNWTVMWQWJKZDEVMXRELDLZCVMWMXSCLCDEARVFPVG + VNWTWJKZDEVNXTELDLZCVNWNYACLCDEBRVFPVGVHUMVIVJVKCDEVTRVL $. $} ${ @@ -47616,19 +47618,19 @@ We use their notation ("onto" under the arrow). (Contributed by NM, ( vy vx vz vw vv cv ccnv wfun wss wo wral wa wceq wrex wi wal weq cbvrexv cab cuni cnveq eqeq2d wcel funeqd sseq1 sseq2 orbi12d ralbidv rspcv funeq anbi12d biimprcd cnvss orim12i wb sseq12 syl5ibrcom expd syl6com rexlimdv - ancoms com23 alrimdv anim12ii syl5bi alrimiv vex eqeq1 rexbidv elab ralab - df-ral anbi2i imbi12i albii bitr2i sylib fununi cnvuni cnvex dfiun2 eqtri - syl ciun funeqi sylibr ) BIZJZKZWJCIZLZWMWJLZMZCANZOZBANZDIZEIZJZPZEAQZDU - BZUCZKZAUCJZKWSFIZKZXIGIZLZXKXILZMZGXENZOZFXENZXGWSXIXBPZEAQZXJXKXBPZEAQZ - XNRZGSZOZRZFSZXQWSYEFXSXIHIZJZPZHAQWSYDXRYIEHAEHTXBYHXIXAYGUDUEUAWSYIYDHA - YGAUFWSYHKZYGWMLZWMYGLZMZCANZOZYIYDRWRYOBYGABHTZWLYJWQYNYPWKYHWJYGUDUGYPW - PYMCAYPWNYKWOYLWJYGWMUHWJYGWMUIUJUKUNULYJYIXJYNYCYIXJYJXIYHUMUOYNYIYBGYNY - AYIXNYNXTYIXNRZEAXAAUFYNYGXALZXAYGLZMZXTYQRYMYTCXAACETYKYRYLYSWMXAYGUIWMX - AYGUHUJULYTXTYIXNYTXNXTYIOZYHXBLZXBYHLZMYRUUBYSUUCYGXAUPXAYGUPUQUUAXLUUBX - MUUCYIXTXLUUBURXIYHXKXBUSVDXKXBXIYHUSUJUTVAVBVCVEVFVGVBVCVHVIXQXIXEUFZXPR - ZFSYFXPFXEVOUUEYEFUUDXSXPYDXDXSDXIFVJDFTXCXREAWTXIXBVKVLVMXOYCXJXDYAXNGDD - GTXCXTEAWTXKXBVKVLVNVPVQVRVSVTXEFGWAWFXHXFXHEAXBWGXFEAWBEDAXBXAEVJWCWDWEW - HWI $. + ancoms com23 alrimdv anim12ii biimtrid alrimiv df-ral eqeq1 rexbidv ralab + vex elab anbi2i imbi12i albii bitr2i sylib fununi syl cnvuni cnvex dfiun2 + ciun eqtri funeqi sylibr ) BIZJZKZWJCIZLZWMWJLZMZCANZOZBANZDIZEIZJZPZEAQZ + DUBZUCZKZAUCJZKWSFIZKZXIGIZLZXKXILZMZGXENZOZFXENZXGWSXIXBPZEAQZXJXKXBPZEA + QZXNRZGSZOZRZFSZXQWSYEFXSXIHIZJZPZHAQWSYDXRYIEHAEHTXBYHXIXAYGUDUEUAWSYIYD + HAYGAUFWSYHKZYGWMLZWMYGLZMZCANZOZYIYDRWRYOBYGABHTZWLYJWQYNYPWKYHWJYGUDUGY + PWPYMCAYPWNYKWOYLWJYGWMUHWJYGWMUIUJUKUNULYJYIXJYNYCYIXJYJXIYHUMUOYNYIYBGY + NYAYIXNYNXTYIXNRZEAXAAUFYNYGXALZXAYGLZMZXTYQRYMYTCXAACETYKYRYLYSWMXAYGUIW + MXAYGUHUJULYTXTYIXNYTXNXTYIOZYHXBLZXBYHLZMYRUUBYSUUCYGXAUPXAYGUPUQUUAXLUU + BXMUUCYIXTXLUUBURXIYHXKXBUSVDXKXBXIYHUSUJUTVAVBVCVEVFVGVBVCVHVIXQXIXEUFZX + PRZFSYFXPFXEVJUUEYEFUUDXSXPYDXDXSDXIFVNDFTXCXREAWTXIXBVKVLVOXOYCXJXDYAXNG + DDGTXCXTEAWTXKXBVKVLVMVPVQVRVSVTXEFGWAWBXHXFXHEAXBWFXFEAWCEDAXBXAEVNWDWEW + GWHWI $. $( The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.) $) @@ -48444,8 +48446,8 @@ We use their notation ("onto" under the arrow). (Contributed by NM, frn sstrd ) ABCDZCCEZCFZGZABGZTCHCUCIABCJCKLTUAAIZUBBIUCUDITUAAMUEABCNUAAOL ABCRUAAUBBPQS $. - $( A function with bounded domain and range is a set. This version is proven - without the Axiom of Replacement. (Contributed by Mario Carneiro, + $( A function with bounded domain and codomain is a set. This version is + proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) $) fex2 $p |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V ) $= ( wf wcel w3a cxp cvv xpexg 3adant1 wss fssxp 3ad2ant1 ssexd ) ABCFZADGZBEG @@ -49106,8 +49108,8 @@ We use their notation ("onto" under the arrow). (Contributed by NM, EACIUGUDCJZUAUEQCKUHUEUAAUDCLMNROPABCSBAUBST $. $( A relation is a one-to-one onto function iff its converse is a one-to-one - onto function with domain and range interchanged. (Contributed by NM, - 8-Dec-2003.) $) + onto function with domain and codomain/range interchanged. (Contributed + by NM, 8-Dec-2003.) $) f1ocnvb $p |- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) ) $= ( wrel wf1o ccnv f1ocnv wceq wb dfrel2 f1oeq1 sylbi syl5ib impbid2 ) CDZABC @@ -49698,12 +49700,12 @@ We use their notation ("onto" under the arrow). (Contributed by NM, ( vz cv wcel wbr wa wex weu cfv weq wb wal elfv wi biimpr breq2 sylib vex alimi ceqsalv anim2i eximi elequ2 anbi12d cbvexv exsimpr df-eu sylibr jca nfeu1 nfa1 nfan nfex nfim biimp ax-14 syl6 com23 impd anc2ri com12 eximdv - nfv sps syl5bi exlimi imp impbii bitri abbi2i ) AFZBFZGZCVODHZIZBJZVQBKZI - ZACDLZVNWBGVNEFZGZVQBEMZNZBOZIZEJZWAEBVNCDPWIWAWIVSVTWIWDCWCDHZIZEJVSWHWK - EWGWJWDWGWEVQQZBOWJWFWLBVQWERUBVQWJBWCEUAVOWCCDSUCTUDUEWKVREBEBMWDVPWJVQE - BAUFWCVOCDSUGUHTWIWGEJZVTWDWGEUIVQBEUJZUKULVSVTWIVRVTWIQBVTWIBVQBUMWHBEWD - WGBWDBVFWFBUNUOUPUQVTWMVRWIWNVRWGWHEWGVRWHWGVRWDWFVRWDQBWFVPVQWDWFVQVPWDW - FVQWEVPWDQVQWEURBEAUSUTVAVBVGVCVDVEVHVIVJVKVLVM $. + nfv sps biimtrid exlimi imp impbii bitri abbi2i ) AFZBFZGZCVODHZIZBJZVQBK + ZIZACDLZVNWBGVNEFZGZVQBEMZNZBOZIZEJZWAEBVNCDPWIWAWIVSVTWIWDCWCDHZIZEJVSWH + WKEWGWJWDWGWEVQQZBOWJWFWLBVQWERUBVQWJBWCEUAVOWCCDSUCTUDUEWKVREBEBMWDVPWJV + QEBAUFWCVOCDSUGUHTWIWGEJZVTWDWGEUIVQBEUJZUKULVSVTWIVRVTWIQBVTWIBVQBUMWHBE + WDWGBWDBVFWFBUNUOUPUQVTWMVRWIWNVRWGWHEWGVRWHWGVRWDWFVRWDQBWFVPVQWDWFVQVPW + DWFVQWEVPWDQVQWEURBEAUSUTVAVBVGVCVDVEVHVIVJVKVLVM $. $} ${ @@ -49995,7 +49997,7 @@ We use their notation ("onto" under the arrow). (Contributed by NM, $( A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.) $) - foelrni $p |- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y ) + foelcdmi $p |- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y ) $= ( wfo wcel cv cfv wceq wrex crn forn eleq2d wfn fofn fvelrnb syl bitr3d wb biimpa ) BCDFZECGZAHDIEJABKZUBEDLZGZUCUDUBUECEBCDMNUBDBOUFUDTBCDPABEDQ @@ -50457,12 +50459,12 @@ We use their notation ("onto" under the arrow). (Contributed by NM, F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C ) $= ( cv wceq wi wal cmpt wcel wa w3a cfv simp2 fveq1d wrex cvv elex nfa1 nfv risset nfeq1 nfim simprl simplr simprr eqeltrd eqid fvmpt2 syl2anc simpll - nffvmpt1 fveq2d 3eqtr3d exp43 a2i com23 rexlimd syl7 syl5bi imp32 3adant2 - sps eqtrd ) AHZBIZCDIZJZAKZFAECLZIZBEMZDGMZNZOZBFPBVMPZDVRBFVMVLVNVQQRVLV - QVSDIZVNVLVOVPVTVOVIAESZVLVPVTJABEUDVPDTMZVLWAVTDGUAVLVIWBVTJZAEVKAUBWBVT - AWBAUCAVSDAECBUOUEUFVKVHEMZVIWCJJAVKVIWDWCVIVJWDWCJVIVJWDWBVTVIVJNZWDWBNZ - NZVHVMPZCVSDWGWDCTMWHCIWEWDWBUGWGCDTVIVJWFUHZWEWDWBUIUJAECTVMVMUKULUMWGVH - BVMVIVJWFUNUPWIUQURUSUTVFVAVBVCVDVEVG $. + nffvmpt1 fveq2d 3eqtr3d exp43 a2i com23 sps rexlimd syl7 biimtrid 3adant2 + imp32 eqtrd ) AHZBIZCDIZJZAKZFAECLZIZBEMZDGMZNZOZBFPBVMPZDVRBFVMVLVNVQQRV + LVQVSDIZVNVLVOVPVTVOVIAESZVLVPVTJABEUDVPDTMZVLWAVTDGUAVLVIWBVTJZAEVKAUBWB + VTAWBAUCAVSDAECBUOUEUFVKVHEMZVIWCJJAVKVIWDWCVIVJWDWCJVIVJWDWBVTVIVJNZWDWB + NZNZVHVMPZCVSDWGWDCTMWHCIWEWDWBUGWGCDTVIVJWFUHZWEWDWBUIUJAECTVMVMUKULUMWG + VHBVMVIVJWFUNUPWIUQURUSUTVAVBVCVDVFVEVG $. $} ${ @@ -50935,10 +50937,10 @@ We use their notation ("onto" under the arrow). (Contributed by NM, ffvelcdmda $p |- ( ( ph /\ C e. A ) -> ( F ` C ) e. B ) $= ( wf wcel cfv ffvelcdm sylan ) ABCEGDBHDEICHFBCDEJK $. - ffvelrnd.2 $e |- ( ph -> C e. A ) $. + ffvelcdmd.2 $e |- ( ph -> C e. A ) $. $( A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) $) - ffvelrnd $p |- ( ph -> ( F ` C ) e. B ) $= + ffvelcdmd $p |- ( ph -> ( F ` C ) e. B ) $= ( wcel cfv ffvelcdmda mpdan ) ADBHDEICHGABCDEFJK $. $} @@ -51048,10 +51050,10 @@ We use their notation ("onto" under the arrow). (Contributed by NM, dff4im $p |- ( F : A --> B -> ( F C_ ( A X. B ) /\ A. x e. A E! y e. B x F y ) ) $= ( wf cxp wss cv wbr weu wral wa wreu dff3im wcel cop df-br ssel opelxp2 - syl6 syl5bi pm4.71rd eubidv df-reu bitr4di ralbidv pm5.32i sylib ) CDEFEC - DGZHZAIZBIZEJZBKZACLZMUKUNBDNZACLZMABCDEOUKUPURUKUOUQACUKUOUMDPZUNMZBKUQU - KUNUTBUKUNUSUNULUMQZEPZUKUSULUMERUKVBVAUJPUSEUJVASULUMCDTUAUBUCUDUNBDUEUF - UGUHUI $. + syl6 biimtrid pm4.71rd eubidv df-reu bitr4di ralbidv pm5.32i sylib ) CDEF + ECDGZHZAIZBIZEJZBKZACLZMUKUNBDNZACLZMABCDEOUKUPURUKUOUQACUKUOUMDPZUNMZBKU + QUKUNUTBUKUNUSUNULUMQZEPZUKUSULUMERUKVBVAUJPUSEUJVASULUMCDTUAUBUCUDUNBDUE + UFUGUHUI $. $( An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.) $) @@ -51128,10 +51130,10 @@ We use their notation ("onto" under the arrow). (Contributed by NM, ${ $d A x $. $d C x $. - fvmptelrn.1 $e |- ( ph -> ( x e. A |-> B ) : A --> C ) $. + fvmptelcdm.1 $e |- ( ph -> ( x e. A |-> B ) : A --> C ) $. $( The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) $) - fvmptelrn $p |- ( ( ph /\ x e. A ) -> B e. C ) $= + fvmptelcdm $p |- ( ( ph /\ x e. A ) -> B e. C ) $= ( wcel cmpt wf wral eqid fmpt sylibr r19.21bi ) ADEGZBCACEBCDHZIOBCJFBCED PPKLMN $. $} @@ -52594,19 +52596,19 @@ is the image of an element of its domain (see ~ foima ). (Contributed ( vx vy wf1 wa wceq ccom wi adantr wfn f1fn adantl cv wcel wbr wb syl cid ccnv cres f1cocnv1 coeq2 eqeq1d syl5ibcom equid resieq mpbiri anidms breq cfv ad2antlr mpbird wex vex brco cop wfun cdm fnfun fndm biimpar funopfvb - eleq2d syl2anc bicomd df-br eqcom 3bitr4g biimpd bitr4id brcnv eximdv cvv - anim12d syl5bi anim1i adantll funfvex funfni eqvincg 3syl sylibrd adantlr - mpd eqfnfvd eqcomd ex impbid ) ABCGZABDGZHZCDIZCUBZDJZUAAUCZIZWLWOWSKWMWL - WPCJZWRIWOWSABCUDWOWTWQWRCDWPUEUFUGLWNWSWOWNWSHZDCXAEADCWNDAMZWSWMXBWLABD - NZOZLWNCAMZWSWLXEWMABCNLZLXAEPZAQZHZXGXGWQRZXGDUMZXGCUMZIZXIXJXGXGWRRZXHX - NXAXHXNXHXHHXNXGXGIEUHAXGXGUIUJUKOWSXJXNSWNXHXGXGWQWRULUNUOWNXHXJXMKWSWNX - HHZXJFPZXKIZXPXLIZHZFUPZXMXJXGXPDRZXPXGWPRZHZFUPXOXTFXGXGWPDEUQZYDURXOYCX - SFXOYAXQYBXRXOYAXQXOXGXPUSZDQZXKXPIZYAXQXOYGYFXODUTZXGDVAZQZYGYFSWNYHXHWN - XBYHXDADVBTLWNYJXHWNYIAXGWNXBYIAIXDADVCTVFVDXGXPDVEVGVHXGXPDVIXPXKVJVKVLX - OYBXRXOXGXPCRZXLXPIZYBXRXOYKYECQZYLXGXPCVIXOCUTZXGCVAZQZYLYMSWNYNXHWNXEYN - XFACVBTLWNYPXHWNYOAXGWNXEYOAIXFACVCTVFVDXGXPCVEVGVMXPXGCFUQYDVNXPXLVJVKVL - VQVOVRXOXBXHHZXKVPQZXMXTSWMXHYQWLWMXBXHXCVSVTYRAXGDXGDWAWBFXKXLVPWCWDWEWF - WGWHWIWJWK $. + eleq2d syl2anc bicomd df-br eqcom 3bitr4g biimpd bitr4id anim12d biimtrid + eximdv cvv anim1i adantll funfvex funfni eqvincg 3syl sylibrd adantlr mpd + brcnv eqfnfvd eqcomd ex impbid ) ABCGZABDGZHZCDIZCUBZDJZUAAUCZIZWLWOWSKWM + WLWPCJZWRIWOWSABCUDWOWTWQWRCDWPUEUFUGLWNWSWOWNWSHZDCXAEADCWNDAMZWSWMXBWLA + BDNZOZLWNCAMZWSWLXEWMABCNLZLXAEPZAQZHZXGXGWQRZXGDUMZXGCUMZIZXIXJXGXGWRRZX + HXNXAXHXNXHXHHXNXGXGIEUHAXGXGUIUJUKOWSXJXNSWNXHXGXGWQWRULUNUOWNXHXJXMKWSW + NXHHZXJFPZXKIZXPXLIZHZFUPZXMXJXGXPDRZXPXGWPRZHZFUPXOXTFXGXGWPDEUQZYDURXOY + CXSFXOYAXQYBXRXOYAXQXOXGXPUSZDQZXKXPIZYAXQXOYGYFXODUTZXGDVAZQZYGYFSWNYHXH + WNXBYHXDADVBTLWNYJXHWNYIAXGWNXBYIAIXDADVCTVFVDXGXPDVEVGVHXGXPDVIXPXKVJVKV + LXOYBXRXOXGXPCRZXLXPIZYBXRXOYKYECQZYLXGXPCVIXOCUTZXGCVAZQZYLYMSWNYNXHWNXE + YNXFACVBTLWNYPXHWNYOAXGWNXEYOAIXFACVCTVFVDXGXPCVEVGVMXPXGCFUQYDWGXPXLVJVK + VLVNVPVOXOXBXHHZXKVQQZXMXTSWMXHYQWLWMXBXHXCVRVSYRAXGDXGDVTWAFXKXLVQWBWCWD + WEWFWHWIWJWK $. $} ${ @@ -52912,17 +52914,17 @@ H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) $= isotr $p |- ( ( H Isom R , S ( A , B ) /\ G Isom S , T ( B , C ) ) -> ( G o. H ) Isom R , T ( A , C ) ) $= ( vx vy vz vw cv wbr cfv wb wral wa wcel wceq wf1o simpl f1oco syl2anr wf - ccom wiso f1of ad2antrr simprl ffvelrnd simprr simplrr breq1 fveq2 breq1d - bibi12d breq2 breq2d rspc2va syl21anc fvco3 syl2anc breq12d bitr4d bibi2d - 2ralbidva biimpd impancom imp jca df-isom anbi12i 3imtr4i ) ABHUAZIMZJMZD - NZVPHOZVQHOZENZPZJAQIAQZRZBCGUAZKMZLMZENZWFGOZWGGOZFNZPZLBQKBQZRZRZACGHUF - ZUAZVRVPWPOZVQWPOZFNZPZJAQIAQZRABDEHUGZBCEFGUGZRACDFWPUGWOWQXBWNWEVOWQWDW - EWMUBVOWCUBABCGHUCUDWDWNXBVOWNWCXBVOWNRZWCXBXEWBXAIJAAXEVPASZVQASZRZRZWAW - TVRXIWAVSGOZVTGOZFNZWTXIVSBSVTBSWMWAXLPZXIABVPHVOABHUEZWNXHABHUHUIZXEXFXG - UJZUKXIABVQHXOXEXFXGULZUKVOWEWMXHUMWLXMVSWGENZXJWJFNZPKLVSVTBBWFVSTZWHXRW - KXSWFVSWGEUNXTWIXJWJFWFVSGUOUPUQWGVTTZXRWAXSXLWGVTVSEURYAWJXKXJFWGVTGUOUS - UQUTVAXIWRXJWSXKFXIXNXFWRXJTXOXPABVPGHVBVCXIXNXGWSXKTXOXQABVQGHVBVCVDVEVF - VGVHVIVJVKXCWDXDWNIJABDEHVLKLBCEFGVLVMIJACDFWPVLVN $. + ccom wiso f1of simprl ffvelcdmd simprr simplrr breq1 fveq2 breq1d bibi12d + ad2antrr breq2 breq2d rspc2va syl21anc fvco3 syl2anc bitr4d bibi2d biimpd + breq12d 2ralbidva impancom imp jca df-isom anbi12i 3imtr4i ) ABHUAZIMZJMZ + DNZVPHOZVQHOZENZPZJAQIAQZRZBCGUAZKMZLMZENZWFGOZWGGOZFNZPZLBQKBQZRZRZACGHU + FZUAZVRVPWPOZVQWPOZFNZPZJAQIAQZRABDEHUGZBCEFGUGZRACDFWPUGWOWQXBWNWEVOWQWD + WEWMUBVOWCUBABCGHUCUDWDWNXBVOWNWCXBVOWNRZWCXBXEWBXAIJAAXEVPASZVQASZRZRZWA + WTVRXIWAVSGOZVTGOZFNZWTXIVSBSVTBSWMWAXLPZXIABVPHVOABHUEZWNXHABHUHUQZXEXFX + GUIZUJXIABVQHXOXEXFXGUKZUJVOWEWMXHULWLXMVSWGENZXJWJFNZPKLVSVTBBWFVSTZWHXR + WKXSWFVSWGEUMXTWIXJWJFWFVSGUNUOUPWGVTTZXRWAXSXLWGVTVSEURYAWJXKXJFWGVTGUNU + SUPUTVAXIWRXJWSXKFXIXNXFWRXJTXOXPABVPGHVBVCXIXNXGWSXKTXOXQABVQGHVBVCVGVDV + EVHVFVIVJVKXCWDXDWNIJABDEHVLKLBCEFGVLVMIJACDFWPVLVN $. $} ${ @@ -53985,21 +53987,21 @@ ordered pairs (for use in defining operations). This is a special case ( vw va vt vr vs cv cop wceq wa wex cvv wcel vex wi oprabidlem weu coprab opex opexg mp2an eqvinop biimpi eqeq1 opth1 syl6bi opeq1 eqeq2d w3a otth2 df-3an bitri anbi1i anass 3bitri 3exbii eximi excom 3imtr4i anim2i euequ1 - 3syl sylbi eupick mpan syl6 3impd syl5bi com12 syl5 bitrdi anbi1d 3exbidv - eqcom imbi1d imbi12d mpbiri adantr exlimivv com3l mpdd mpcom 19.8a impbid - ex df-oprab elab2 ) EJZBJZCJZKZDJZKZLZAMZDNZCNZBNZAEWPABCDUAWNOPWOOPWPOPW - LWMBQZCQZUBZDQZWNWOOOUCUDWQXAAWKFJZGJZKZLZXHWPLZMZGNFNZWQXAARZWQXLFGWKWNW - OXDXEUEUFXKWQXMRZFGXIXNXJXIWQXFWNLZXMXIWQXJXOWKXHWPUGXFXGWNWOFQGQUHUIXOXI - WQXMXOXFHJZIJZKZLZXRWNLZMZINHNXIXNRZHIXFWLWMXBXCUEYAYBHIXSYBXTXSXIWKXRXGK - ZLZXNXSXHYCWKXFXRXGUJUKYDXNWPYCLZYEAMZDNCNBNZARZRYGWLXPLZWMXQLZWOXGLZAMZD - NZMCNZMZBNZYEAYGYIYJYLMZMZDNZCNBNZYPYFYRBCDYFYIYJMZYKMZAMUUAYLMYRYEUUBAYE - YIYJYKULZUUBWLWMXPXQWOXGXBXCXEUMZYIYJYKUNUOUPUUAYKAUQYIYJYLUQURUSYTYIYQDN - ZMZCNBNZYIUUECNZMZBNYPYSBNZCNUUFBNZCNYTUUGUUJUUKCYQBDHSUTYSBCVAUUFBCVAVBU - UEBCHSUUIYOBUUHYNYIYLCDISVCUTVEVFYPYEAYEUUCYPAUUDYPYIYJYKAYPYIYNYJYKARZRY - IBTYPYIYNRBHVDYIYNBVGVHYNYJYMUULYJCTYNYJYMRCIVDYJYMCVGVHYKDTYMUULDGVDYKAD - VGVHVIVIVJVKVLVMYDWQYEXMYHYDWQYCWPLYEWKYCWPUGYCWPVQVNZYDXAYGAYDWRYFBCDYDW - QYEAUUMVOVPVRVSVTUIWAWBVFWCWDWAWBWEWQAXAWRWSWTXAWRDWFWSCWFWTBWFVEWHWGABCD - EWIWJ $. + 3syl sylbi eupick mpan syl6 3impd biimtrid com12 syl5 eqcom bitrdi anbi1d + 3exbidv imbi1d imbi12d mpbiri adantr exlimivv com3l mpcom 19.8a ex impbid + mpdd df-oprab elab2 ) EJZBJZCJZKZDJZKZLZAMZDNZCNZBNZAEWPABCDUAWNOPWOOPWPO + PWLWMBQZCQZUBZDQZWNWOOOUCUDWQXAAWKFJZGJZKZLZXHWPLZMZGNFNZWQXAARZWQXLFGWKW + NWOXDXEUEUFXKWQXMRZFGXIXNXJXIWQXFWNLZXMXIWQXJXOWKXHWPUGXFXGWNWOFQGQUHUIXO + XIWQXMXOXFHJZIJZKZLZXRWNLZMZINHNXIXNRZHIXFWLWMXBXCUEYAYBHIXSYBXTXSXIWKXRX + GKZLZXNXSXHYCWKXFXRXGUJUKYDXNWPYCLZYEAMZDNCNBNZARZRYGWLXPLZWMXQLZWOXGLZAM + ZDNZMCNZMZBNZYEAYGYIYJYLMZMZDNZCNBNZYPYFYRBCDYFYIYJMZYKMZAMUUAYLMYRYEUUBA + YEYIYJYKULZUUBWLWMXPXQWOXGXBXCXEUMZYIYJYKUNUOUPUUAYKAUQYIYJYLUQURUSYTYIYQ + DNZMZCNBNZYIUUECNZMZBNYPYSBNZCNUUFBNZCNYTUUGUUJUUKCYQBDHSUTYSBCVAUUFBCVAV + BUUEBCHSUUIYOBUUHYNYIYLCDISVCUTVEVFYPYEAYEUUCYPAUUDYPYIYJYKAYPYIYNYJYKARZ + RYIBTYPYIYNRBHVDYIYNBVGVHYNYJYMUULYJCTYNYJYMRCIVDYJYMCVGVHYKDTYMUULDGVDYK + ADVGVHVIVIVJVKVLVMYDWQYEXMYHYDWQYCWPLYEWKYCWPUGYCWPVNVOZYDXAYGAYDWRYFBCDY + DWQYEAUUMVPVQVRVSVTUIWAWBVFWCWHWAWBWDWQAXAWRWSWTXAWRDWEWSCWEWTBWEVEWFWGAB + CDEWIWJ $. $} $( The result of an operation is a set. (Contributed by Jim Kingdon, @@ -55370,25 +55372,25 @@ ordered pairs (for use in defining operations). This is a special case $( An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.) $) - fovrn $p |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> + fovcdm $p |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C ) $= ( cxp wf wcel co wa cop opelxpi cfv df-ov ffvelcdm eqeltrid sylan2 3impb ) DEGZCFHZADIZBEIZABFJZCIZUBUCKUAABLZTIZUEABDEMUAUGKUDUFFNCABFOTCUFFPQRS $. ${ - fovrnd.1 $e |- ( ph -> F : ( R X. S ) --> C ) $. + fovcdmd.1 $e |- ( ph -> F : ( R X. S ) --> C ) $. $( An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) $) - fovrnda $p |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C ) $= - ( wcel co cxp wf fovrn syl3an1 3expb ) ABEIZCFIZBCGJDIZAEFKDGLPQRHBCDEFGM - NO $. + fovcdmda $p |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C ) $= + ( wcel co cxp wf fovcdm syl3an1 3expb ) ABEIZCFIZBCGJDIZAEFKDGLPQRHBCDEFG + MNO $. - fovrnd.2 $e |- ( ph -> A e. R ) $. - fovrnd.3 $e |- ( ph -> B e. S ) $. + fovcdmd.2 $e |- ( ph -> A e. R ) $. + fovcdmd.3 $e |- ( ph -> B e. S ) $. $( An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) $) - fovrnd $p |- ( ph -> ( A F B ) e. C ) $= - ( cxp wf wcel co fovrn syl3anc ) AEFKDGLBEMCFMBCGNDMHIJBCDEFGOP $. + fovcdmd $p |- ( ph -> ( A F B ) e. C ) $= + ( cxp wf wcel co fovcdm syl3anc ) AEFKDGLBEMCFMBCGNDMHIJBCDEFGOP $. $} ${ @@ -56492,18 +56494,18 @@ ordered pairs (for use in defining operations). This is a special case $( Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.) $) caofinvl $p |- ( ph -> ( G oF R F ) = ( A X. { B } ) ) $= - ( vw cfv cof co cmpt csn cxp cv wf wcel wa adantr ffvelcdmda ffvelrnd - eqid fmptd feq1d mpbird wfn wceq wral ralrimiva fnmpt dffn5im feqmptd - syl fneq1d offval2 fveq1d simpr fveq2 fveq2d fvmptg syl2anc oveq1d id - eqtrd oveq12d eqeq1d rspcdva mpteq2dva fconstmpt eqtr4di ) AIHFUAUBZS - DEUCZDEUDUEAWBSDSUFZITZWDHTZFUBZUCWCASDWEWFFIHKGGMADGWDIADGIUGDGCDCUF - ZHTZJTZUCZUGACDWJGWKAWHDUHZUIGGWIJAGGJUGZWLPUJADGWHHNUKULZWKUMZUNADGI - WKQUOUPUKADGWDHNUKZAIDUQZISDWEUCURAWQWKDUQZAWJGUHZCDUSWRAWSCDWNUTCDWJ - WKGWOVAVDADIWKQVEUPSDIVBVDASDGHNVCVFASDWGEAWDDUHZUIZWGWFJTZWFFUBZEXAW - EXBWFFXAWEWDWKTZXBAWEXDURWTAWDIWKQVGUJXAWTXBGUHXDXBURAWTVHXAGGWFJAWMW - TPUJWPULCWDWJXBDGWKWHWDURWIWFJWHWDHVIVJWOVKVLVOVMXABUFZJTZXEFUBZEURZX - CEURBGWFXEWFURZXGXCEXIXFXBXEWFFXEWFJVIXIVNVPVQAXHBGUSWTAXHBGRUTUJWPVR - VOVSVOSDEVTWA $. + ( vw cfv cof co cmpt csn cxp cv wcel adantr ffvelcdmda ffvelcdmd eqid + wf wa fmptd feq1d mpbird wfn wceq wral ralrimiva fnmpt fneq1d dffn5im + feqmptd offval2 fveq1d simpr fveq2 fveq2d fvmptg syl2anc eqtrd oveq1d + syl id oveq12d eqeq1d rspcdva mpteq2dva fconstmpt eqtr4di ) AIHFUAUBZ + SDEUCZDEUDUEAWBSDSUFZITZWDHTZFUBZUCWCASDWEWFFIHKGGMADGWDIADGIULDGCDCU + FZHTZJTZUCZULACDWJGWKAWHDUGZUMGGWIJAGGJULZWLPUHADGWHHNUIUJZWKUKZUNADG + IWKQUOUPUIADGWDHNUIZAIDUQZISDWEUCURAWQWKDUQZAWJGUGZCDUSWRAWSCDWNUTCDW + JWKGWOVAVNADIWKQVBUPSDIVCVNASDGHNVDVEASDWGEAWDDUGZUMZWGWFJTZWFFUBZEXA + WEXBWFFXAWEWDWKTZXBAWEXDURWTAWDIWKQVFUHXAWTXBGUGXDXBURAWTVGXAGGWFJAWM + WTPUHWPUJCWDWJXBDGWKWHWDURWIWFJWHWDHVHVIWOVJVKVLVMXABUFZJTZXEFUBZEURZ + XCEURBGWFXEWFURZXGXCEXIXFXBXEWFFXEWFJVHXIVOVPVQAXHBGUSWTAXHBGRUTUHWPV + RVLVSVLSDEVTWA $. $} $} @@ -56640,7 +56642,7 @@ ordered pairs (for use in defining operations). This is a special case $( If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.) $) - fornex $p |- ( A e. C -> ( F : A -onto-> B -> B e. _V ) ) $= + focdmex $p |- ( A e. C -> ( F : A -onto-> B -> B e. _V ) ) $= ( wfo wcel cvv cdm crn wfun funrnex syl5com wf wceq fof fdm syl eleq1d forn fofun 3imtr3d com12 ) ABDEZACFZBGFZUCDHZCFZDIZGFZUDUEUCDJUGUIABDTCDKLUCUFAC UCABDMUFANABDOABDPQRUCUHBGABDSRUAUB $. @@ -56649,9 +56651,9 @@ ordered pairs (for use in defining operations). This is a special case can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.) $) f1dmex $p |- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V ) $= - ( wf1 wcel cvv crn wss f1rn ssexg sylan ex ccnv wfo wf1o f1cnv f1ofo fornex - syl syl5com syld imp ) ABDEZBCFZAGFZUDUEDHZGFZUFUDUEUHUDUGBIUEUHABDJUGBCKLM - UDUGADNZOZUHUFUDUGAUIPUJABDQUGAUIRTUGAGUISUAUBUC $. + ( wf1 wcel cvv crn wss f1rn ssexg sylan ex ccnv wfo f1cnv f1ofo syl focdmex + wf1o syl5com syld imp ) ABDEZBCFZAGFZUDUEDHZGFZUFUDUEUHUDUGBIUEUHABDJUGBCKL + MUDUGADNZOZUHUFUDUGAUITUJABDPUGAUIQRUGAGUISUAUBUC $. ${ $d x y A $. $d y B $. @@ -57669,8 +57671,8 @@ ordered pairs (for use in defining operations). This is a special case 27-Jan-2020.) $) ovmpoelrn $p |- ( ( A. x e. A A. y e. B C e. M /\ X e. A /\ Y e. B ) -> ( X O Y ) e. M ) $= - ( wcel wral cxp wf co fmpo fovrn syl3an1b ) EFKBDLACLCDMFGNHCKIDKHIGOFKAB - CDEFGJPHIFCDGQR $. + ( wcel wral cxp wf co fmpo fovcdm syl3an1b ) EFKBDLACLCDMFGNHCKIDKHIGOFKA + BCDEFGJPHIFCDGQR $. $} ${ @@ -57963,28 +57965,28 @@ ordered pairs (for use in defining operations). This is a special case cres wfo ccnv wfun wf1o f1f fo2ndf weq wral f2ndf cxp wss fssxp cop ssel2 elxp2 sylib anim12dan fvres adantr ad2antlr eqeq12d op2nd eqeq12i funopfv wf vex f1fun anim12d eqcom biimpi eqeqan12d simpl anim12i f1veqaeq sylan2 - opeq12 syl6 com14 syl6bi pm2.43i syld impcom syl5bi sylbid adantl adantlr - com13 com12 wb eleq1 bi2anan9 anbi2d fveq2 simpllr imbi12d imbi2d 3imtr4d - simpr rexlimdvva rexlimivv imp mpcom ralrimivv dff13 df-f1 simprbi dff1o3 - sylanbrc ) ABCUAZCCUBZJCUCZUDZXNUEUFZCXMXNUGXLABCVHZXOABCUHZABCUIKXLCBXNU - AZXPXLCBXNVHZDLZXNMZELZXNMZNZDEUJZOZECUKDCUKXSXLXQXTXRABCULKXLYGDECCCABUM - ZUNZXLYACPZYCCPZQZYGOXLXQYIXRABCUOKYIYLXLYGYIYLXLYGOZYAFLZGLZUPZNZGBRFARZ - YCHLZILZUPZNZIBRHARZQYIYLQZYMYIYJYRYKUUCYIYJQYAYHPYRCYHYAUQFGYAABURUSYIYK - QYCYHPUUCCYHYCUQHIYCABURUSUTYRUUCUUDYMOZYQUUCUUEOZFGABYNAPZYOBPZQZYQUUFUU - IYQQZUUBUUEHIABUUJYSAPZYTBPZQZQZUUBUUEUUNUUBQZYIYPCPZUUACPZQZQZXLYPXNMZUU - AXNMZNZYPUUANZOZOZUUDYMUUNUUSUVEOZUUBUUIUUMUVFYQUUSUUIUUMQZUVEUURUVGUVEOY - IUURUVGUVEUURUVGQZUVBXLUVCUVHUVBYPJMZUUAJMZNZXLUVCOZUVHUUTUVIUVAUVJUURUUT - UVINZUVGUUPUVMUUQYPCJVAVBVBUUQUVAUVJNUUPUVGUUACJVAVCVDUVKGIUJZUVHUVLUVIYO - UVJYTYNYOFVIGVIVEYSYTHVIIVIVEVFUVHXLUVNUVCUVGUURXLUVNUVCOZOXLUURUVGUVOXLU - URYNCMZYONZYSCMZYTNZQZUVGUVOOXLCUFZUURUVTOABCVJUWAUUPUVQUUQUVSYNYOCVGYSYT - CVGVKKXLUVGUVTUVOUVNUVGUVTXLUVCUVNUVGUVTUVLOOUVTUVNUVGUVNUVLUVTUVNUVPUVRN - ZUVGUVNUVLOOUVQUVSYOUVPYTUVRUVQYOUVPNUVPYOVLVMUVSYTUVRNUVRYTVLVMVNXLUVGUV - NUWBUVCXLUVGUVNUWBUVCOOXLUVGQZUWBUVNUVCUWCUWBFHUJZUVOUVGXLUUGUUKQUWBUWDOU - UIUUGUUMUUKUUGUUHVOUUKUULVOVPABYNYSCVQVRUWDUVNUVCYNYOYSYTVSSVTTSWAWBWAWCW - ATWDWJWETWFWGTSWHWKWIVBUUOYLUURYIUUNYJUUPUUBYKUUQYQYJUUPWLUUIUUMYAYPCWMVC - YCUUACWMWNWOUUOYGUVDXLUUOYEUVBYFUVCUUNUUBYBUUTYDUVAYQYBUUTNUUIUUMYAYPXNWP - VCYCUUAXNWPVNUUOYAYPYCUUAUUIYQUUMUUBWQUUNUUBXAVDWRWSWTSXBSXCXDXESTXEXFDEC - BXNXGXKXSXTXPCBXNXHXIKCXMXNXJXK $. + opeq12 syl6 com14 syl6bi pm2.43i syld com13 impcom biimtrid sylbid adantl + com12 adantlr wb eleq1 bi2anan9 anbi2d fveq2 simpllr simpr imbi12d imbi2d + 3imtr4d rexlimdvva rexlimivv mpcom ralrimivv dff13 sylanbrc df-f1 simprbi + imp dff1o3 ) ABCUAZCCUBZJCUCZUDZXNUEUFZCXMXNUGXLABCVHZXOABCUHZABCUIKXLCBX + NUAZXPXLCBXNVHZDLZXNMZELZXNMZNZDEUJZOZECUKDCUKXSXLXQXTXRABCULKXLYGDECCCAB + UMZUNZXLYACPZYCCPZQZYGOXLXQYIXRABCUOKYIYLXLYGYIYLXLYGOZYAFLZGLZUPZNZGBRFA + RZYCHLZILZUPZNZIBRHARZQYIYLQZYMYIYJYRYKUUCYIYJQYAYHPYRCYHYAUQFGYAABURUSYI + YKQYCYHPUUCCYHYCUQHIYCABURUSUTYRUUCUUDYMOZYQUUCUUEOZFGABYNAPZYOBPZQZYQUUF + UUIYQQZUUBUUEHIABUUJYSAPZYTBPZQZQZUUBUUEUUNUUBQZYIYPCPZUUACPZQZQZXLYPXNMZ + UUAXNMZNZYPUUANZOZOZUUDYMUUNUUSUVEOZUUBUUIUUMUVFYQUUSUUIUUMQZUVEUURUVGUVE + OYIUURUVGUVEUURUVGQZUVBXLUVCUVHUVBYPJMZUUAJMZNZXLUVCOZUVHUUTUVIUVAUVJUURU + UTUVINZUVGUUPUVMUUQYPCJVAVBVBUUQUVAUVJNUUPUVGUUACJVAVCVDUVKGIUJZUVHUVLUVI + YOUVJYTYNYOFVIGVIVEYSYTHVIIVIVEVFUVHXLUVNUVCUVGUURXLUVNUVCOZOXLUURUVGUVOX + LUURYNCMZYONZYSCMZYTNZQZUVGUVOOXLCUFZUURUVTOABCVJUWAUUPUVQUUQUVSYNYOCVGYS + YTCVGVKKXLUVGUVTUVOUVNUVGUVTXLUVCUVNUVGUVTUVLOOUVTUVNUVGUVNUVLUVTUVNUVPUV + RNZUVGUVNUVLOOUVQUVSYOUVPYTUVRUVQYOUVPNUVPYOVLVMUVSYTUVRNUVRYTVLVMVNXLUVG + UVNUWBUVCXLUVGUVNUWBUVCOOXLUVGQZUWBUVNUVCUWCUWBFHUJZUVOUVGXLUUGUUKQUWBUWD + OUUIUUGUUMUUKUUGUUHVOUUKUULVOVPABYNYSCVQVRUWDUVNUVCYNYOYSYTVSSVTTSWAWBWAW + CWATWDWEWFTWGWHTSWIWJWKVBUUOYLUURYIUUNYJUUPUUBYKUUQYQYJUUPWLUUIUUMYAYPCWM + VCYCUUACWMWNWOUUOYGUVDXLUUOYEUVBYFUVCUUNUUBYBUUTYDUVAYQYBUUTNUUIUUMYAYPXN + WPVCYCUUAXNWPVNUUOYAYPYCUUAUUIYQUUMUUBWQUUNUUBWRVDWSWTXASXBSXCXJXDSTXDXED + ECBXNXFXGXSXTXPCBXNXHXIKCXMXNXKXG $. $} ${ @@ -58046,35 +58048,35 @@ ordered pairs (for use in defining operations). This is a special case vv wex w3a 3an6 weq wo poirr intnand im2anan9 ioran syl6ibr imp 3ad2antr1 ex an4 potr 3impia orcd 3expia expdimp breq2 biimpa expcom adantrd adantl jaod anim2d orim2d breq1 equequ1 anbi1d orbi12d imbi2d syl5ibr expd com12 - impd jaao an4s sylan2b biimpi 3adant2 jctild adantld syl5bi jca wb breq12 - anidms notbid 3ad2ant1 3adant3 3adant1 anbi12d imbi12d xporderlem anbi12i - notbii imbi12i bitrdi expcomd sylbi com3l exlimivv syl3anb com3r ralrimiv - 3exp 3imp ralrimivva df-po sylibr ) CEUCZDFUCZOZUAPZYDGQZRZYDUBPZGQZYGUIP - ZGQZOZYDYIGQZSZOZUICDUDZUEZUBYOUEUAYOUEYOGUCYCYPUAUBYOYOYCYDYOTZYGYOTZOZO - YNUIYOYCYSYIYOTZYNSYSYTYCYNYQYRYTYCYNSZYQYDIPZJPZUFZUGZUUBCTZUUCDTZOZOZJU - JIUJZYRYGKPZLPZUFZUGZUUKCTZUULDTZOZOZLUJKUJZYTYIMPZNPZUFZUGZUUTCTZUVADTZO - ZOZNUJMUJZUUAIJYDCDUHKLYGCDUHMNYICDUHUUJUUSUVHUUAUUIUUSUVHUUASSIJUVHUUIUU - SUUAUVGUUIUUSUUASSMNUUSUVGUUIUUAUURUVGUUIUUASSKLUUIUURUVGUUAUUIUURUVGUUAU - UIUURUVGUKUUEUUNUVCUKZUUHUUQUVFUKZOUUAUUEUUHUUNUUQUVCUVFULUVIUVJUUAUVIYCU - VJYNYCUVJOZYNUVIUUFUUFOUUGUUGOOZUUBUUBEQZIIUMZUUCUUCFQZOZUNZOZRZUUFUUOOUU - GUUPOOZUUBUUKEQZIKUMZUUCUULFQZOZUNZOZUUOUVDOUUPUVEOOZUUKUUTEQZKMUMZUULUVA - FQZOZUNZOZOZUUFUVDOUUGUVEOOZUUBUUTEQZIMUMZUUCUVAFQZOZUNZOZSZOZUVKUVSUXBYC - UUQUUHUVSUVFYCUUHOUVQUVLYCUUHUVQRZYCUUHUVMRZUVPRZOUXDYAUUFUXEYBUUGUXFYAUU - FUXECUUBEUOVBYBUUGUXFYBUUGOUVOUVNDUUCFUOUPVBUQUVMUVPURUSUTUPVAUWNUVTUWGOZ - UWEUWLOZOUVKUXAUVTUWEUWGUWLVCUVKUXHUXAUXGUVKUXHUWTUWOUVJYCUUFUUOUVDUKZUUG - UUPUVEUKZOUXHUWTSZUUFUUGUUOUUPUVDUVEULYAUXIYBUXJUXKYAUXIOZYBUXJOZOUWEUWLU - WTUXLUWAUWLUWTSZUXMUWDUXLUWAUXNUXLUWAOUWHUWTUWKUXLUWAUWHUWTYAUXIUWAUWHOZU - WTYAUXIUXOUKUWPUWSYAUXIUXOUWPCUUBUUKUUTEVDVEVFVGVHUWAUWKUWTSUXLUWAUWIUWTU - WJUWIUWAUWTUWIUWAOUWPUWSUWIUWAUWPUUKUUTUUBEVIVJVFVKVLVMVNVBUXMUWBUWCUXNUW - BUXMUWCUXNSUWBUXMUWCUXNUXMUWCOZUXNUWBUWLUWHUWIUWROZUNZSUXPUWKUXQUWHUXPUWJ - UWRUWIUXMUWCUWJUWRDUUCUULUVAFVDVHVOVPUWBUWTUXRUWLUWBUWPUWHUWSUXQUUBUUKUUT - EVQUWBUWQUWIUWRIKMVRVSVTWAWBWCWDWEWFWEWGWHUVJUWOYCUUHUVFUWOUUQUUHUVFOUWOU - UFUUGUVDUVEVCWIWJVMWKWLWMWNUVIYNUUDUUDGQZRZUUDUUMGQZUUMUVBGQZOZUUDUVBGQZS - 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impcom wo cuni brtpos2 simplbi elun sylib adantl mpjaod ex exlimdv syl5bi - cun ssrdv sylibr impbii ) AUAZDZEZFADZGZHZWEWCWEFBIZAJZBKWCHZBFALMWHWIBWH - FWGWAJZWIWGNGZWJWHOBPZWGANUBQZWCFWBGZWJWCWNFNNUCZGZNNUDWCWBWOTZWNWPRWBUEZ - WBWOFUFSUGFWGWALWLUHUIUJUKSULWFWQWCWFCWBWOCIZWBGWSWGWAJZBKWFWSWOGZBWSWACP - MWFWTXABWFWTXAWFWTUMZWSWDUNZGZXAWSFUOZGZXDXARXBXCWOWSXCEXCWOTWDUPXCUEUQUR - USWTWFXFXARXFWTWFXAXFWTWJWFXARZXFWSFWGWAWSFUTVAWJWHXGWMWHWEXAFWGALWLUHVBS - VCVDVEWTXDXFVFZWFWTWSXCXEVQGZXHWTXIWSUOUNVGWGAJZWKWTXIXJUMOWLWSWGANVHQVIW - SXCXEVJVKVLVMVNVOVPVRWRVSVT $. + impcom cun cuni brtpos2 simplbi elun sylib adantl mpjaod exlimdv biimtrid + wo ex ssrdv sylibr impbii ) AUAZDZEZFADZGZHZWEWCWEFBIZAJZBKWCHZBFALMWHWIB + WHFWGWAJZWIWGNGZWJWHOBPZWGANUBQZWCFWBGZWJWCWNFNNUCZGZNNUDWCWBWOTZWNWPRWBU + EZWBWOFUFSUGFWGWALWLUHUIUJUKSULWFWQWCWFCWBWOCIZWBGWSWGWAJZBKWFWSWOGZBWSWA + CPMWFWTXABWFWTXAWFWTUMZWSWDUNZGZXAWSFUOZGZXDXARXBXCWOWSXCEXCWOTWDUPXCUEUQ + URUSWTWFXFXARXFWTWFXAXFWTWJWFXARZXFWSFWGWAWSFUTVAWJWHXGWMWHWEXAFWGALWLUHV + BSVCVDVEWTXDXFVPZWFWTWSXCXEVFGZXHWTXIWSUOUNVGWGAJZWKWTXIXJUMOWLWSWGANVHQV + IWSXCXEVJVKVLVMVQVNVOVRWRVSVT $. $( The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) $) @@ -58438,14 +58440,14 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and rntpos $p |- ( Rel dom F -> ran tpos F = ran F ) $= ( vx vy vw vz cdm wrel crn cv wcel wbr wex vex elrn cop wceq breldm elrel wi cvv ctpos ccnv dmtpos eleq2d syl5ib relcnv mpan breq1 wb brtposg mp3an - syl6 bitrdi opex brelrn syl6bi syli exlimdv syl5bi ex syl5 bitr4di impbid - exlimivv eqrdv ) AFZGZBAUAZHZAHZVGBIZVIJZVKVJJZVLCIZVKVHKZCLVGVMCVKVHBMZN - VGVOVMCVOVGVNDIZEIZOZPZELDLZVMVGVOVNVFUBZJZWAVOVNVHFZJVGWCVNVKVHCMZVPQVGW - DWBVNAUCUDUEWBGWCWAVFUFDEVNWBRUGULVTVOVMSDEVTVOVRVQOZVKAKZVMVTVOVSVKVHKZW - GVNVSVKVHUHVQTJVRTJVKTJWHWGUIDMZEMZVPVQVRVKATTTUJUKZUMWFVKAVRVQWJWIUNVPUO - UPVDUQURUSVMVNVKAKZCLVGVLCVKAVPNVGWLVLCWLVGVNWFPZDLELZVLWLVNVFJZVGWNVNVKA - WEVPQVGWOWNEDVNVFRUTVAWMWLVLSEDWMWLWHVLWMWLWGWHVNWFVKAUHWKVBVSVKVHVQVRWIW - JUNVPUOUPVDUQURUSVCVE $. + syl6 bitrdi opex brelrn syl6bi exlimivv syli exlimdv biimtrid syl5 impbid + ex bitr4di eqrdv ) AFZGZBAUAZHZAHZVGBIZVIJZVKVJJZVLCIZVKVHKZCLVGVMCVKVHBM + ZNVGVOVMCVOVGVNDIZEIZOZPZELDLZVMVGVOVNVFUBZJZWAVOVNVHFZJVGWCVNVKVHCMZVPQV + GWDWBVNAUCUDUEWBGWCWAVFUFDEVNWBRUGULVTVOVMSDEVTVOVRVQOZVKAKZVMVTVOVSVKVHK + ZWGVNVSVKVHUHVQTJVRTJVKTJWHWGUIDMZEMZVPVQVRVKATTTUJUKZUMWFVKAVRVQWJWIUNVP + UOUPUQURUSUTVMVNVKAKZCLVGVLCVKAVPNVGWLVLCWLVGVNWFPZDLELZVLWLVNVFJZVGWNVNV + KAWEVPQVGWOWNEDVNVFRVCVAWMWLVLSEDWMWLWHVLWMWLWGWHVNWFVKAUHWKVDVSVKVHVQVRW + IWJUNVPUOUPUQURUSUTVBVE $. $( The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) $) @@ -58576,7 +58578,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and HZCIZAJZEZUKFZBGZHABCKULBUKKUFUJUMUOUFUGUMUIACLMUFUGUIUOUFUGHZUOUIUPUNUHB UPCNZDZUNUHGUGURUFUGUQAACQOPCRSUAUBUCUDABCTULBUKTUE $. - $( The domain and range of a transposition. (Contributed by NM, + $( The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) $) tposf2 $p |- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) ) $= ( wrel wf ccnv ctpos crn wss wfo wfn ffn dffn4 sylib tposfo2 syl5 imp fof @@ -58602,14 +58604,14 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( wrel wf1 wfo wa ccnv ctpos wf1o tposf12 tposfo2 anim12d df-f1o 3imtr4g ) ADZABCEZABCFZGAHZBCIZEZSBTFZGABCJSBTJPQUARUBABCKABCLMABCNSBTNO $. - $( The domain and range of a transposition. (Contributed by NM, + $( The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.) $) tposfo $p |- ( F : ( A X. B ) -onto-> C -> tpos F : ( B X. A ) -onto-> C ) $= ( cxp wfo ccnv ctpos wrel wi relxp tposfo2 ax-mp wceq cnvxp foeq2 sylib wb ) ABEZCDFZSGZCDHZFZBAEZCUBFZSITUCJABKSCDLMUAUDNUCUERABOUAUDCUBPMQ $. - $( The domain and range of a transposition. (Contributed by NM, + $( The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) $) tposf $p |- ( F : ( A X. B ) --> C -> tpos F : ( B X. A ) --> C ) $= ( cxp wf ccnv ctpos wrel wi relxp tposf2 ax-mp cnvxp feq2i sylib ) ABEZCD @@ -58921,9 +58923,9 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and smoiun $p |- ( ( Smo B /\ A e. dom B ) -> U_ x e. A ( B ` x ) C_ ( B ` A ) ) $= ( vy wsmo cdm wcel wa cv cfv ciun wrex eliun con0 wi smofvon smoel 3expia - ontr1 expcomd sylsyld rexlimdv syl5bi ssrdv ) CEZBCFGZHZDABAIZCJZKZBCJZDI - ZUJGULUIGZABLUGULUKGZAULBUIMUGUMUNABUGUKNGZUHBGZUIUKGZUMUNOBCPUEUFUPUQBCU - HQRUOUMUQUNULUIUKSTUAUBUCUD $. + ontr1 expcomd sylsyld rexlimdv biimtrid ssrdv ) CEZBCFGZHZDABAIZCJZKZBCJZ + DIZUJGULUIGZABLUGULUKGZAULBUIMUGUMUNABUGUKNGZUHBGZUIUKGZUMUNOBCPUEUFUPUQB + CUHQRUOUMUQUNULUIUKSTUAUBUCUD $. $( If ` F ` is an isomorphism from an ordinal ` A ` onto ` B ` , which is a subset of the ordinals, then ` F ` is a strictly monotonic function. @@ -59016,22 +59018,22 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and raleq imbi2d r19.21v cres cdm wfun simplll adantr simpld funfn sylib word wfn simpllr eloni ordelss sylan simplr sstrd simprd fnssres syl2anc fveq2 eqeq12d ad2antrr rspcv syl3c rspcdva adantl 3eqtr4d eqfnfvd fveq2d reseq2 - simpr fvres sselda ralrimiva cbvralv sylibr exp31 expcom a2d syl5bi tfis2 - vtoclga mpcom mpi ) ACCMZBNZEOZWQFOZPZBCQZCUDCUEUFAWPXARZGAUANZCMZWTBXCQZ - RZRZAXBRUACUEXCCPZXFXBAXHXDWPXEXAXCCCUGWTBXCCUIUHUJXGALNZCMZWTBXIQZRZRZUA - LXCXIPZXFXLAXNXDXJXEXKXCXICUGWTBXCXIUIUHUJXMLXCQAXLLXCQZRXCUEUFZXGAXLLXCU - KXPAXOXFAXPXOXFRAXPSZXOXDXEXQXOSZXDSZUBNZEOZXTFOZPZUBXCQXEXSYCUBXCXSXTXCU - FZSZEXTULZDOZFXTULZDOZYAYBYEYFYHDYEUCXTYFYHYEEEUMZVAZXTYJMYFXTVAYEEUNZYKY - EYLCYJMZYEAYLYMSXSAYDAXPXOXDUOUPZHTZUQEURUSYEXTCYJYEXTXCCXSXCUTZYDXTXCMXS - XPYPAXPXOXDVBXCVCTXCXTVDVEXRXDYDVFVGZYEYLYMYOVHVGYJXTEVIVJYEFFUMZVAZXTYRM - YHXTVAYEFUNZYSYEYTCYRMZYEAYTUUASYNITZUQFURUSYEXTCYRYQYEYTUUAUUBVHVGYRXTFV - IVJYEUCNZXTUFZSZUUCEOZUUCFOZUUCYFOZUUCYHOZUUEWTUUFUUGPBXTUUCWQUUCPWRUUFWS - UUGWQUUCEVKWQUUCFVKVLUUEYDXOXTCMZWTBXTQZXSYDUUDVFXSXOYDUUDXQXOXDVFVMYEUUJ - UUDYQUPXLUUJUUKRLXTXCXIXTPXJUUJXKUUKXIXTCUGWTBXIXTUIUHVNVOYEUUDWBVPUUDUUH - UUFPYEUUCXTEWCVQUUDUUIUUGPYEUUCXTFWCVQVRVSVTYEWREWQULZDOZPZYAYGPBCXTWQXTP - ZWRYAUUMYGWQXTEVKZUUOUULYFDWQXTEWAVTVLYEAUUNBCQYNJTXSXCCXTXRXDWBWDZVPYEWS - FWQULZDOZPZYBYIPBCXTUUOWSYBUUSYIWQXTFVKZUUOUURYHDWQXTFWAVTVLYEAUUTBCQYNKT - UUQVPVRWEWTYCBUBXCUUOWRYAWSYBUUPUVAVLWFWGWHWIWJWKWLWMWNWO $. + simpr fvres sselda ralrimiva cbvralv sylibr exp31 expcom biimtrid vtoclga + a2d tfis2 mpcom mpi ) ACCMZBNZEOZWQFOZPZBCQZCUDCUEUFAWPXARZGAUANZCMZWTBXC + QZRZRZAXBRUACUEXCCPZXFXBAXHXDWPXEXAXCCCUGWTBXCCUIUHUJXGALNZCMZWTBXIQZRZRZ + UALXCXIPZXFXLAXNXDXJXEXKXCXICUGWTBXCXIUIUHUJXMLXCQAXLLXCQZRXCUEUFZXGAXLLX + CUKXPAXOXFAXPXOXFRAXPSZXOXDXEXQXOSZXDSZUBNZEOZXTFOZPZUBXCQXEXSYCUBXCXSXTX + CUFZSZEXTULZDOZFXTULZDOZYAYBYEYFYHDYEUCXTYFYHYEEEUMZVAZXTYJMYFXTVAYEEUNZY + KYEYLCYJMZYEAYLYMSXSAYDAXPXOXDUOUPZHTZUQEURUSYEXTCYJYEXTXCCXSXCUTZYDXTXCM + XSXPYPAXPXOXDVBXCVCTXCXTVDVEXRXDYDVFVGZYEYLYMYOVHVGYJXTEVIVJYEFFUMZVAZXTY + RMYHXTVAYEFUNZYSYEYTCYRMZYEAYTUUASYNITZUQFURUSYEXTCYRYQYEYTUUAUUBVHVGYRXT + FVIVJYEUCNZXTUFZSZUUCEOZUUCFOZUUCYFOZUUCYHOZUUEWTUUFUUGPBXTUUCWQUUCPWRUUF + WSUUGWQUUCEVKWQUUCFVKVLUUEYDXOXTCMZWTBXTQZXSYDUUDVFXSXOYDUUDXQXOXDVFVMYEU + UJUUDYQUPXLUUJUUKRLXTXCXIXTPXJUUJXKUUKXIXTCUGWTBXIXTUIUHVNVOYEUUDWBVPUUDU + UHUUFPYEUUCXTEWCVQUUDUUIUUGPYEUUCXTFWCVQVRVSVTYEWREWQULZDOZPZYAYGPBCXTWQX + TPZWRYAUUMYGWQXTEVKZUUOUULYFDWQXTEWAVTVLYEAUUNBCQYNJTXSXCCXTXRXDWBWDZVPYE + WSFWQULZDOZPZYBYIPBCXTUUOWSYBUUSYIWQXTFVKZUUOUURYHDWQXTFWAVTVLYEAUUTBCQYN + KTUUQVPVRWEWTYCBUBXCUUOWRYAWSYBUUPUVAVLWFWGWHWIWLWJWMWKWNWO $. $} ${ @@ -59384,22 +59386,22 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and cbvexdva fneq2 raleq exbidv r19.21v cop csn cun w3a wrex cab tfrlem3 wfun cvv fveq2 eleq1d anbi2d cbvalv sylib adantr simplr simpll opeq12d uneq12d wal 3anbi123d cbvrexdva cbvabv adantl fveq12d reseq12d cbvraldva2 cbvralv - sneqd tfrlemiex expr expcom a2d syl5bi tfis3 impcom ) FUDUEAHPZFUFZDPZXBQ - ZXBXDUGZIQZUKZDFRZSZHUHZAXBLPZUFZXHDXLRZSZHUHZUIZAMPZNPZUFZXDXRQZXRXDUGZI - QZUKZDXSRZSZMUHZUIZAXKUILNFLNTZXPYGAYIXOYFHMYIHMTZSZXMXTXNYEYKXLXSXBXRYIY - JUJZYIYJULZUMYKXHYDDXLXSYMYKXEYAXGYCYKXDXBXRYLUNYKXFYBIYKXBXRXDYLUOUPUQUR - USVAUTXLFUKZXPXKAYNXOXJHYNXMXCXNXIXLFXBVBXHDXLFVCUSVDUTYHNXLRAYGNXLRZUIXL - UDUEZXQAYGNXLVEYPAYOXPAYPYOXPUIAYPYOXPAYPYOSZSLUAOCDEXTXREUEZUBPZXRXSXRIQ - ZVFZVGZVHZUKZVIZMUHZNXLVJZUBVKHGUCIBCLUAEGHIJVLAIVMZXLIQZVNUEZSZLWEZYQAUU - HBPZIQZVNUEZSZBWEUULKUUPUUKBLBLTZUUOUUJUUHUUQUUNUUIVNUUMXLIVOVPVQVRVSVTUU - GGPZOPZUFZUUREUEZUCPZUURUUSUURIQZVFZVGZVHZUKZVIZGUHZOXLVJUBUCUBUCTZUUFUVI - NOXLUVJNOTZSZUUEUVHMGUVLMGTZSZXTUUTYRUVAUUDUVGUVNXSUUSXRUURUVLUVMUJZUVJUV - KUVMWAZUMUVNXRUUREUVOVPUVNYSUVBUUCUVFUVJUVKUVMWBUVNXRUURUUBUVEUVOUVNUUAUV - DUVNXSUUSYTUVCUVPUVNXRUURIUVOUPWCWNWDUQWFVAWGWHYQYPAYPYOULWIYQUUTCPZUURQZ - UURUVQUGZIQZUKZCUUSRZSZGUHZOXLRZAYQYOUWEYPYOUJYGUWDNOXLUVKYFUWCMGUVKUVMSZ - XTUUTYEUWBUWFXSUUSXRUURUVKUVMUJUVKUVMULUMUWFYDUWADCXSUUSUWFDCTZSZYAUVRYCU - VTUWHXDUVQXRUURUVKUVMUWGWAZUWFUWGUJZWJUWHYBUVSIUWHXRUURXDUVQUWIUWJWKUPUQU - VKUVMUWGWBWLUSVAWMVSWIWOWPWQWRWSWTXA $. + sneqd tfrlemiex expr expcom a2d biimtrid tfis3 impcom ) FUDUEAHPZFUFZDPZX + BQZXBXDUGZIQZUKZDFRZSZHUHZAXBLPZUFZXHDXLRZSZHUHZUIZAMPZNPZUFZXDXRQZXRXDUG + ZIQZUKZDXSRZSZMUHZUIZAXKUILNFLNTZXPYGAYIXOYFHMYIHMTZSZXMXTXNYEYKXLXSXBXRY + IYJUJZYIYJULZUMYKXHYDDXLXSYMYKXEYAXGYCYKXDXBXRYLUNYKXFYBIYKXBXRXDYLUOUPUQ + URUSVAUTXLFUKZXPXKAYNXOXJHYNXMXCXNXIXLFXBVBXHDXLFVCUSVDUTYHNXLRAYGNXLRZUI + XLUDUEZXQAYGNXLVEYPAYOXPAYPYOXPUIAYPYOXPAYPYOSZSLUAOCDEXTXREUEZUBPZXRXSXR + IQZVFZVGZVHZUKZVIZMUHZNXLVJZUBVKHGUCIBCLUAEGHIJVLAIVMZXLIQZVNUEZSZLWEZYQA + UUHBPZIQZVNUEZSZBWEUULKUUPUUKBLBLTZUUOUUJUUHUUQUUNUUIVNUUMXLIVOVPVQVRVSVT + UUGGPZOPZUFZUUREUEZUCPZUURUUSUURIQZVFZVGZVHZUKZVIZGUHZOXLVJUBUCUBUCTZUUFU + VINOXLUVJNOTZSZUUEUVHMGUVLMGTZSZXTUUTYRUVAUUDUVGUVNXSUUSXRUURUVLUVMUJZUVJ + UVKUVMWAZUMUVNXRUUREUVOVPUVNYSUVBUUCUVFUVJUVKUVMWBUVNXRUURUUBUVEUVOUVNUUA + UVDUVNXSUUSYTUVCUVPUVNXRUURIUVOUPWCWNWDUQWFVAWGWHYQYPAYPYOULWIYQUUTCPZUUR + QZUURUVQUGZIQZUKZCUUSRZSZGUHZOXLRZAYQYOUWEYPYOUJYGUWDNOXLUVKYFUWCMGUVKUVM + SZXTUUTYEUWBUWFXSUUSXRUURUVKUVMUJUVKUVMULUMUWFYDUWADCXSUUSUWFDCTZSZYAUVRY + CUVTUWHXDUVQXRUURUVKUVMUWGWAZUWFUWGUJZWJUWHYBUVSIUWHXRUURXDUVQUWIUWJWKUPU + QUVKUVMUWGWBWLUSVAWMVSWIWOWPWQWRWSWTXA $. $} ${ @@ -59442,27 +59444,27 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and cdm suceloni onss df-ss unieqd wtr eloni ordtr 3syl unisuc eleqtrrid fndm eqtrd ad2antrr eleqtrrd fneq2 raleq anbi12d rspcev sylibr simplrr simplrl eldm mpan tfrlem5 imp syl22anc breqtrd exlimddv brelrn elssuni ex exlimdv - syl5bi alrimiv fvss rnex ssex exlimiv vtoclg impcom ) EHMAEGUGZNZOMZAKPZY - SNZOMZUHAUUAUHKEHUUBEQZUUDUUAAUUEUUCYTOUUBEYSUIUJUKAUAPZUUBULZULZRUMZUNZU - OZUBPZUUFNUUFUULUPGNQZUBUUJUQZSZUAURZUUDAUUJRMZUUPUUQUUJUSZUUJOMUUIRUTUUR - UUHRVAUUIVBVDUUIUUHRUUGUUBKVCZVEZVEVFVGUUJVHVIZAKUCUBDUUJLUAGBCKUCDFLGIVJ - ZAGVOZBPZGNZOMZSZBVKUVCUUBGNZOMZSZKVKJUVGUVJBKUVDUUBQZUVFUVIUVCUVKUVEUVHO - UVDUUBGUIUJVLVMVPVNVQUUOUUDUAUUOUUCUUFVRZUNZUTZUUDUUOUUBCPZYSVSZUVOUVMUTZ - UHZCVKUVNUUOUVRCUVPUUBUVOVTZUDPZMZUVTDMZSZUDURZUUOUVQUVPUUBUVODUNZVSUVSUW - EMUWDUUBUVOYSUWEKUCDLGUVBWAWBUUBUVOUWEWCUDUVSDWDWEUUOUWCUVQUDUUOUWCUVQUUO - UWCSZUVOUVLMZUVQUWFUUBUVOUUFVSZUWGUWFUUBUVDUUFVSZUWHBUWFUUBUUFWRZMUWIBURU - WFUUBUUJUWJUWFUUBUUGUUJUUBUUSWFUWFUUJUUHUNZUUGUWFUUIUUHUWFUUHRUTZUUIUUHQU - WFUUHRMZUWLUWFUUGRMZUWMUWFUUBRMZUWNUWCUWOUUOUWCUVTUEPZUOZUFPZUVTNUVTUWRUP - GNQUFUWPUQZSZUWOUERUWCUWBUWTUERWGUWAUWBWHBCUEUFDFGUVTIUDVCWIVPUWCUWPRMZUW - TSZSZUXAUUBUWPMZUWOUWCUXAUWTWJUXCUWQUWAUXDUWCUXAUWQUWSWKUWAUWBUXBWLUWPUUB - UVOUVTWMWNUWPUUBWOWNWPWQUUBWSTZUUGWSTUUHWTTUUHRXAVPXBUWFUUGXCZUWKUUGQUWFU - WNUUGUSUXFUXEUUGXDUUGXEXFUUGUUTXGVPXJXHUUKUWJUUJQUUNUWCUUJUUFXIXKXLBUUBUU - FUUSXTVPUWFUWISZUUBUVDUVOUUFUWFUWIWHZUXGUUFDMZUWBUWIUUBUVOUVTVSZUVDUVOQZU - UOUXIUWCUWIUUOUUFLPZUOZUUMUBUXLUQZSZLRWGZUXIUUQUUOUXPUVAUXOUUOLUUJRUXLUUJ - QUXMUUKUXNUUNUXLUUJUUFXMUUMUBUXLUUJXNXOXPYABCLUBDFGUUFIUAVCZWIXQXKUUOUWAU - WBUWIXRUXHUXGUWAUXJUUOUWAUWBUWIXSUUBUVOUVTWCXQUXIUWBSUWIUXJSUXKKUCCBDLUAU - DGUVBYBYCYDYEYFUUBUVOUUFUUSCVCYGTUVOUVLYHTYIYJYKYLCUUBUVMYSYMTUUCUVMUVLUU - FUXQYNVGYOTYPTYQYR $. + biimtrid alrimiv fvss rnex ssex exlimiv vtoclg impcom ) EHMAEGUGZNZOMZAKP + ZYSNZOMZUHAUUAUHKEHUUBEQZUUDUUAAUUEUUCYTOUUBEYSUIUJUKAUAPZUUBULZULZRUMZUN + ZUOZUBPZUUFNUUFUULUPGNQZUBUUJUQZSZUAURZUUDAUUJRMZUUPUUQUUJUSZUUJOMUUIRUTU + URUUHRVAUUIVBVDUUIUUHRUUGUUBKVCZVEZVEVFVGUUJVHVIZAKUCUBDUUJLUAGBCKUCDFLGI + VJZAGVOZBPZGNZOMZSZBVKUVCUUBGNZOMZSZKVKJUVGUVJBKUVDUUBQZUVFUVIUVCUVKUVEUV + HOUVDUUBGUIUJVLVMVPVNVQUUOUUDUAUUOUUCUUFVRZUNZUTZUUDUUOUUBCPZYSVSZUVOUVMU + TZUHZCVKUVNUUOUVRCUVPUUBUVOVTZUDPZMZUVTDMZSZUDURZUUOUVQUVPUUBUVODUNZVSUVS + UWEMUWDUUBUVOYSUWEKUCDLGUVBWAWBUUBUVOUWEWCUDUVSDWDWEUUOUWCUVQUDUUOUWCUVQU + UOUWCSZUVOUVLMZUVQUWFUUBUVOUUFVSZUWGUWFUUBUVDUUFVSZUWHBUWFUUBUUFWRZMUWIBU + RUWFUUBUUJUWJUWFUUBUUGUUJUUBUUSWFUWFUUJUUHUNZUUGUWFUUIUUHUWFUUHRUTZUUIUUH + QUWFUUHRMZUWLUWFUUGRMZUWMUWFUUBRMZUWNUWCUWOUUOUWCUVTUEPZUOZUFPZUVTNUVTUWR + UPGNQUFUWPUQZSZUWOUERUWCUWBUWTUERWGUWAUWBWHBCUEUFDFGUVTIUDVCWIVPUWCUWPRMZ + UWTSZSZUXAUUBUWPMZUWOUWCUXAUWTWJUXCUWQUWAUXDUWCUXAUWQUWSWKUWAUWBUXBWLUWPU + UBUVOUVTWMWNUWPUUBWOWNWPWQUUBWSTZUUGWSTUUHWTTUUHRXAVPXBUWFUUGXCZUWKUUGQUW + FUWNUUGUSUXFUXEUUGXDUUGXEXFUUGUUTXGVPXJXHUUKUWJUUJQUUNUWCUUJUUFXIXKXLBUUB + UUFUUSXTVPUWFUWISZUUBUVDUVOUUFUWFUWIWHZUXGUUFDMZUWBUWIUUBUVOUVTVSZUVDUVOQ + ZUUOUXIUWCUWIUUOUUFLPZUOZUUMUBUXLUQZSZLRWGZUXIUUQUUOUXPUVAUXOUUOLUUJRUXLU + UJQUXMUUKUXNUUNUXLUUJUUFXMUUMUBUXLUUJXNXOXPYABCLUBDFGUUFIUAVCZWIXQXKUUOUW + AUWBUWIXRUXHUXGUWAUXJUUOUWAUWBUWIXSUUBUVOUVTWCXQUXIUWBSUWIUXJSUXKKUCCBDLU + AUDGUVBYBYCYDYEYFUUBUVOUUFUUSCVCYGTUVOUVLYHTYIYJYKYLCUUBUVMYSYMTUUCUVMUVL + UUFUXQYNVGYOTYPTYQYR $. $} ${ @@ -60363,15 +60365,15 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( vy con0 wfn cv cfv cres wceq wral wa nfv wi fveq2 wb imp nfan wcel nfim nfra1 eqeq12d imbi2d r19.21v rsp onss tfri1 fvreseq mpanl2 syl6bir sylan2 wss ancoms adantr tfri2 jctr jcab sylibr eqeq12 adantl mpbird exp43 com4t - syl6 exp4a pm2.43d syl com3l a2d syl5bi tfis2f com12 ralrimi eqfnfv mpan2 - impd biimpar syldan ) BHIZAJZBKZBWCLZDKZMZAHNZWDWCCKZMZAHNZBCMZWBWHOZWJAH - WBWHAWBAPWGAHUDUAZWCHUBZWMWJWMWJQZWMGJZBKZWQCKZMZQZAGWMWTAWNWTAPUCWCWQMZW - JWTWMXBWDWRWIWSWCWQBRWCWQCRUEUFXAGWCNWMWTGWCNZQWOWPWMWTGWCUGWOWMXCWJWOWBW - HXCWJQZWHWOWBXDWHWOWGQZWOWBXDQZQWGAHUHXEWOXFXEWOWOWBXDWOWBOZXCXEWOWJXGXCX - EWOWJXGXCOZXEWOOZOWJWFCWCLZDKZMZXHXLXIXGXCXLWBWOXCXLQZWOWBWCHUOZXMWCUIWBX - NOXCWEXJMZXLWBCHIZXNXOXCSACDEFUJZGHWCBCUKULWEXJDRUMUNUPTUQXIWJXLSZXHXEWOX - RXEWOWGWIXKMZOZXRXEXEWOXSQZOWOXTQXEYAAWCCDEFURUSWOWGXSUTVAWDWFWIXKVBVGTVC - VDVEVFVHVIVJVKVSVLVMVNVOVPWBWLWKWBXPWLWKSXQAHBCVQVRVTWA $. + syl6 exp4a pm2.43d syl com3l impd a2d biimtrid tfis2f com12 ralrimi mpan2 + eqfnfv biimpar syldan ) BHIZAJZBKZBWCLZDKZMZAHNZWDWCCKZMZAHNZBCMZWBWHOZWJ + AHWBWHAWBAPWGAHUDUAZWCHUBZWMWJWMWJQZWMGJZBKZWQCKZMZQZAGWMWTAWNWTAPUCWCWQM + ZWJWTWMXBWDWRWIWSWCWQBRWCWQCRUEUFXAGWCNWMWTGWCNZQWOWPWMWTGWCUGWOWMXCWJWOW + BWHXCWJQZWHWOWBXDWHWOWGQZWOWBXDQZQWGAHUHXEWOXFXEWOWOWBXDWOWBOZXCXEWOWJXGX + CXEWOWJXGXCOZXEWOOZOWJWFCWCLZDKZMZXHXLXIXGXCXLWBWOXCXLQZWOWBWCHUOZXMWCUIW + BXNOXCWEXJMZXLWBCHIZXNXOXCSACDEFUJZGHWCBCUKULWEXJDRUMUNUPTUQXIWJXLSZXHXEW + OXRXEWOWGWIXKMZOZXRXEXEWOXSQZOWOXTQXEYAAWCCDEFURUSWOWGXSUTVAWDWFWIXKVBVGT + VCVDVEVFVHVIVJVKVLVMVNVOVPVQWBWLWKWBXPWLWKSXQAHBCVSVRVTWA $. $} ${ @@ -60662,17 +60664,17 @@ defined for all sets (being defined for all ordinals might be enough if rdgon $p |- ( ( ph /\ B e. On ) -> ( rec ( F , A ) ` B ) e. On ) $= ( vy vf vg vz con0 wcel wa cvv cv cfv wceq adantr wral crdg cdm ciun cmpt cun df-irdg wfun funmpt a1i word ordon w3a vex 3ad2ant1 dmex fveq2 eleq1d - wf simpl3 simpr wb fdm eleq2d syl mpbid ffvelrnd rspcdva ralrimiva fveq2d - cbvralv sylibr iunon sylancr onun2 dmeq fveq1 iuneq12d uneq2d eqid fvmptg - syl2anc eqeltrd cuni csuc eleq2i biimpi adantl suceloni biimpri tfrcl + wf simpl3 simpr fdm eleq2d syl ffvelcdmd rspcdva ralrimiva fveq2d cbvralv + wb mpbid sylibr iunon sylancr onun2 syl2anc dmeq fveq1 uneq2d eqid fvmptg + iuneq12d eqeltrd cuni csuc eleq2i biimpi adantl suceloni biimpri tfrcl unon ) ADLMZNZHLIECUAJOCBJPZUBZBPZWNQZEQZUCZUEZUDZLDBJECUFXAUGWMJOWTUHUIL UJWMUKUIWMHPZLMZXBLIPZURZULZXDXAQZCBXDUBZWPXDQZEQZUCZUEZLXFXDOMXLLMZXGXLR IUMZXFCLMZXKLMZXMWMXCXOXEAXOWLFSUNXFXHOMXJLMZBXHTZXPXDXNUOXFKPZXDQZEQZLMZ KXHTXRXFYBKXHXFXSXHMZNZWPEQZLMZYBBLXTWPXTRYEYALWPXTEUPUQXFYFBLTZYCWMXCYGX - EAYGWLGSUNSYDXBLXSXDWMXCXEYCUSZYDYCXSXBMZXFYCUTYDXEYCYIVAYHXEXHXBXSXBLXDV - BVCVDVEVFVGVHXQYBBKXHWPXSRZXJYALYJXIXTEWPXSXDUPVIUQVJVKBXHXJOVLVMCXKVNWAZ - 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B \/ A = B \/ B e. A ) ) $= ( vx vy com wcel wceq cv wi eleq2 eqeq2 eleq1 3orbi123d c0 csuc wo 3orass w3o sylibr syl imbi2d 0elnn olc wa df-3or elex elsuc2g syl6bir nnsucelsuc - cvv 3mix1 elsuci syl6bi orbi2i biimpi orcomd olcd syl6 jaao syl5bi finds2 - eqcom ex vtoclga impcom ) BEFAEFZABFZABGZBAFZRZVFACHZFZAVKGZVKAFZRZIVFVJI - CBEVKBGZVOVJVFVPVLVGVMVHVNVIVKBAJVKBAKVKBALMUAVOANFZANGZNAFZRZADHZFZAWAGZ - WAAFZRZAWAOZFZAWFGZWFAFZRZVFCDVKNGVLVQVMVRVNVSVKNAJVKNAKVKNALMVKWAGVLWBVM - WCVNWDVKWAAJVKWAAKVKWAALMVKWFGVLWGVMWHVNWIVKWFAJVKWFAKVKWFALMVFVRVSPZVTAU - BWKVQWKPVTWKVQUCVQVRVSQSTWAEFZVFWEWJIWEWBWCPZWDPWLVFUDWJWBWCWDUEWLWMWJVFW - DWLWAUJFZWMWJIWAEUFWNWMWGWJAWAUJUGWGWHWIUKUHTVFWDWIWFAGZPZWJVFWDWFAOFWPWA - AUIWFAULUMWPWGWHWIPZPWJWPWQWGWPWIWHWPWIWHPWOWHWIWFAVBUNUOUPUQWGWHWIQSURUS - UTVCVAVDVE $. + 3mix1 elsuci syl6bi eqcom orbi2i biimpi orcomd olcd syl6 jaao biimtrid ex + cvv finds2 vtoclga impcom ) BEFAEFZABFZABGZBAFZRZVFACHZFZAVKGZVKAFZRZIVFV + JICBEVKBGZVOVJVFVPVLVGVMVHVNVIVKBAJVKBAKVKBALMUAVOANFZANGZNAFZRZADHZFZAWA + GZWAAFZRZAWAOZFZAWFGZWFAFZRZVFCDVKNGVLVQVMVRVNVSVKNAJVKNAKVKNALMVKWAGVLWB + VMWCVNWDVKWAAJVKWAAKVKWAALMVKWFGVLWGVMWHVNWIVKWFAJVKWFAKVKWFALMVFVRVSPZVT + AUBWKVQWKPVTWKVQUCVQVRVSQSTWAEFZVFWEWJIWEWBWCPZWDPWLVFUDWJWBWCWDUEWLWMWJV + FWDWLWAVBFZWMWJIWAEUFWNWMWGWJAWAVBUGWGWHWIUJUHTVFWDWIWFAGZPZWJVFWDWFAOFWP + WAAUIWFAUKULWPWGWHWIPZPWJWPWQWGWPWIWHWPWIWHPWOWHWIWFAUMUNUOUPUQWGWHWIQSUR + USUTVAVCVDVE $. $( A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.) $) @@ -62867,8 +62869,8 @@ ordinals implies excluded middle ( ~ ordsucunielexmid ). (Contributed 12-Aug-2015.) $) qsss $p |- ( ph -> ( A /. R ) C_ ~P A ) $= ( vx vy cqs cpw cv wcel cec wceq wrex vex elqs wss sseq1 syl5ibrcom velpw - ecss syl6ibr rexlimdvw syl5bi ssrdv ) AEBCGZBHZEIZUEJUGFIZCKZLZFBMAUGUFJZ - FBUGCENOAUJUKFBAUJUGBPZUKAULUJUIBPAUHCBDTUGUIBQREBSUAUBUCUD $. + ecss syl6ibr rexlimdvw biimtrid ssrdv ) AEBCGZBHZEIZUEJUGFIZCKZLZFBMAUGUF + JZFBUGCENOAUJUKFBAUJUGBPZUKAULUJUIBPAUHCBDTUGUIBQREBSUAUBUCUD $. qsss.2 $e |- ( ph -> R e. V ) $. $( The union of a quotient set. (Contributed by Mario Carneiro, @@ -63145,7 +63147,7 @@ ordinals implies excluded middle ( ~ ordsucunielexmid ). (Contributed BVFUTBVFUABUSEUBZUCBCUDZVCVFEUTVBUSFUEBUSEUFZUHUGUIUJAEITZBHUKUSHTUTITZAV LBHKULVLVMBUSHBUTIVIUMVJEUTIVKUNURUOLMBUSVAEUPUQ $. - $( The domain and range of the function ` F ` . (Contributed by Mario + $( The domain and codomain of the function ` F ` . (Contributed by Mario Carneiro, 23-Dec-2016.) $) qliftf $p |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) ) $= ( vy wfun cv cec cmpt crn wf cqs wceq qliftlem fliftf wrex cab df-qs eqid @@ -63273,33 +63275,33 @@ ordinals implies excluded middle ( ~ ordsucunielexmid ). (Contributed wi adantl reeanv sylibr adantr ecexg elisset 3syl biantrud 2rexbidv mpbid 19.42v bicomi rexbii rexcom4 bitri sylib eceq1 eqeq2d anbi1d oveq1 eceq1d cvv anbi12d anbi2d oveq2 cbvrex2v anbi12i bitr4i wbr wer wss simprll erth - sseldd simprrl cxp wf fovrnd 3imtr3d eqeq2 biimprcd syl6 impd wb bi2anan9 - eqeq1 imbi12d syl5ibrcom anassrs rexlimdvva syl5bir alrimivv eu4 sylanbrc - ) AQMUPZRNUPZUQZUQZQUCURZIUSZUTZRUBURZJUSZUTZUQZBURZYJYMHVAZKUSZUTZUQZUBF - VBZUCEVBZBVCZUUCYPUOURZYSUTZUQZUBFVBUCEVBZUQZBUOVDZVKZUOVEBVEUUCBVFYIYPYT - BVCZUQZUBFVBZUCEVBZUUDYIYPUBFVBUCEVBZUUOYIYLUCEVBZYOUBFVBZUQZUUPYHUUSAYFU - UQYGUURUUQQEIVGMUCEQIVHUDVIUURRFJVGNUBFRJVHUEVIVJVLYLYOUCUBEFVMVNYIYPUUMU - CUBEFYIUULYPYIKSUPZYSWMUPUULAUUTYHUFVOYRSKVPBYSWMVQVRVSVTWAUUOUUBBVCZUCEV - BUUDUUNUVAUCEUUNUUABVCZUBFVBUVAUUMUVBUBFUVBUUMYPYTBWBWCWDUUAUBBFWEWFWDUUB - UCBEWEWFWGYIUUKBUOAUUKYHUUIQUAURZIUSZUTZRDURZJUSZUTZUQZYQUVCUVFHVAZKUSZUT - ZUQZDFVBZQTURZIUSZUTZRCURZJUSZUTZUQZUUEUVOUVRHVAZKUSZUTZUQZCFVBZUQZTEVBUA - EVBZAUUJUWHUVNUAEVBZUWFTEVBZUQUUIUVNUWFUATEEVMUUCUWIUUHUWJUUAUVMUVEYOUQZY - QUVCYMHVAZKUSZUTZUQUCUBUADEFUCUAVDZYPUWKYTUWNUWOYLUVEYOUWOYKUVDQYJUVCIWHW - IWJUWOYSUWMYQUWOYRUWLKYJUVCYMHWKWLWIWNUBDVDZUWKUVIUWNUVLUWPYOUVHUVEUWPYNU - VGRYMUVFJWHWIWOUWPUWMUVKYQUWPUWLUVJKYMUVFUVCHWPWLWIWNWQUUGUWEUVQYOUQZUUEU - VOYMHVAZKUSZUTZUQUCUBTCEFUCTVDZYPUWQUUFUWTUXAYLUVQYOUXAYKUVPQYJUVOIWHWIWJ - UXAYSUWSUUEUXAYRUWRKYJUVOYMHWKWLWIWNUBCVDZUWQUWAUWTUWDUXBYOUVTUVQUXBYNUVS - RYMUVRJWHWIWOUXBUWSUWCUUEUXBUWRUWBKYMUVRUVOHWPWLWIWNWQWRWSAUWGUUJUATEEUWG - UVMUWEUQZCFVBDFVBAUVCEUPZUVOEUPZUQZUQZUUJUVMUWEDCFFVMUXGUXCUUJDCFFAUXFUVF - FUPZUVRFUPZUQZUXCUUJVKAUXFUXJUQZUQZUVMUWEUUJUXLUWEUUJVKUVMUVDUVPUTZUVGUVS - UTZUQZUWDUQZUVKUUEUTZVKUXLUXOUWDUXQUXLUXOUVKUWCUTZUWDUXQVKUXLUVCUVOIWTZUV - FUVRJWTZUQUVJUWBKWTUXOUXRUNUXLUXSUXMUXTUXNUXLUVCUVOILALIXAUXKUGVOUXLELUVC - AELXBUXKUJVOAUXDUXEUXJXCZXEXDUXLUVFUVRJOAOJXAUXKUHVOUXLFOUVFAFOXBUXKUKVOA - UXFUXHUXIXFZXEXDWNUXLUVJUWBKPAPKXAUXKUIVOUXLGPUVJAGPXBUXKULVOUXLUVCUVFGEF - HAEFXGGHXHUXKUMVOUYAUYBXIXEXDXJUWDUXQUXRUUEUWCUVKXKXLXMXNUVMUWEUXPUUJUXQU - VIUWEUXPXOUVLUVIUWAUXOUWDUVEUVQUXMUVHUVTUXNQUVDUVPXQRUVGUVSXQXPWJVOUVLUUJ - UXQXOUVIYQUVKUUEXQVLXRXSXNXTYAYBYAYBVOYCUUCUUHBUOUUJUUAUUGUCUBEFUUJYTUUFY - PYQUUEYSXQWOVTYDYE $. + sseldd simprrl wf fovcdmd 3imtr3d eqeq2 biimprcd syl6 impd eqeq1 bi2anan9 + cxp imbi12d syl5ibrcom anassrs rexlimdvva syl5bir alrimivv eu4 sylanbrc + wb ) AQMUPZRNUPZUQZUQZQUCURZIUSZUTZRUBURZJUSZUTZUQZBURZYJYMHVAZKUSZUTZUQZ + UBFVBZUCEVBZBVCZUUCYPUOURZYSUTZUQZUBFVBUCEVBZUQZBUOVDZVKZUOVEBVEUUCBVFYIY + PYTBVCZUQZUBFVBZUCEVBZUUDYIYPUBFVBUCEVBZUUOYIYLUCEVBZYOUBFVBZUQZUUPYHUUSA + YFUUQYGUURUUQQEIVGMUCEQIVHUDVIUURRFJVGNUBFRJVHUEVIVJVLYLYOUCUBEFVMVNYIYPU + UMUCUBEFYIUULYPYIKSUPZYSWMUPUULAUUTYHUFVOYRSKVPBYSWMVQVRVSVTWAUUOUUBBVCZU + CEVBUUDUUNUVAUCEUUNUUABVCZUBFVBUVAUUMUVBUBFUVBUUMYPYTBWBWCWDUUAUBBFWEWFWD + UUBUCBEWEWFWGYIUUKBUOAUUKYHUUIQUAURZIUSZUTZRDURZJUSZUTZUQZYQUVCUVFHVAZKUS + ZUTZUQZDFVBZQTURZIUSZUTZRCURZJUSZUTZUQZUUEUVOUVRHVAZKUSZUTZUQZCFVBZUQZTEV + BUAEVBZAUUJUWHUVNUAEVBZUWFTEVBZUQUUIUVNUWFUATEEVMUUCUWIUUHUWJUUAUVMUVEYOU + QZYQUVCYMHVAZKUSZUTZUQUCUBUADEFUCUAVDZYPUWKYTUWNUWOYLUVEYOUWOYKUVDQYJUVCI + WHWIWJUWOYSUWMYQUWOYRUWLKYJUVCYMHWKWLWIWNUBDVDZUWKUVIUWNUVLUWPYOUVHUVEUWP + YNUVGRYMUVFJWHWIWOUWPUWMUVKYQUWPUWLUVJKYMUVFUVCHWPWLWIWNWQUUGUWEUVQYOUQZU + UEUVOYMHVAZKUSZUTZUQUCUBTCEFUCTVDZYPUWQUUFUWTUXAYLUVQYOUXAYKUVPQYJUVOIWHW + IWJUXAYSUWSUUEUXAYRUWRKYJUVOYMHWKWLWIWNUBCVDZUWQUWAUWTUWDUXBYOUVTUVQUXBYN + UVSRYMUVRJWHWIWOUXBUWSUWCUUEUXBUWRUWBKYMUVRUVOHWPWLWIWNWQWRWSAUWGUUJUATEE + UWGUVMUWEUQZCFVBDFVBAUVCEUPZUVOEUPZUQZUQZUUJUVMUWEDCFFVMUXGUXCUUJDCFFAUXF + UVFFUPZUVRFUPZUQZUXCUUJVKAUXFUXJUQZUQZUVMUWEUUJUXLUWEUUJVKUVMUVDUVPUTZUVG + UVSUTZUQZUWDUQZUVKUUEUTZVKUXLUXOUWDUXQUXLUXOUVKUWCUTZUWDUXQVKUXLUVCUVOIWT + ZUVFUVRJWTZUQUVJUWBKWTUXOUXRUNUXLUXSUXMUXTUXNUXLUVCUVOILALIXAUXKUGVOUXLEL + UVCAELXBUXKUJVOAUXDUXEUXJXCZXEXDUXLUVFUVRJOAOJXAUXKUHVOUXLFOUVFAFOXBUXKUK + VOAUXFUXHUXIXFZXEXDWNUXLUVJUWBKPAPKXAUXKUIVOUXLGPUVJAGPXBUXKULVOUXLUVCUVF + GEFHAEFXPGHXGUXKUMVOUYAUYBXHXEXDXIUWDUXQUXRUUEUWCUVKXJXKXLXMUVMUWEUXPUUJU + XQUVIUWEUXPYEUVLUVIUWAUXOUWDUVEUVQUXMUVHUVTUXNQUVDUVPXNRUVGUVSXNXOWJVOUVL + UUJUXQYEUVIYQUVKUUEXNVLXQXRXMXSXTYAXTYAVOYBUUCUUHBUOUUJUUAUUGUCUBEFUUJYTU + UFYPYQUUEYSXNWOVTYCYD $. eropr.12 $e |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) } $. @@ -63328,15 +63330,15 @@ ordinals implies excluded middle ( ~ ordsucunielexmid ). (Contributed 30-Dec-2014.) $) eroprf $p |- ( ph -> .+^ : ( J X. K ) --> L ) $= ( cxp wf cv cec wceq wa co wrex cio cmpo wcel wral cab wi ad2antrr adantr - cqs fovrnda ecelqsg syl2anc eleqtrrdi eleq1a syl rexlimdvva abssdv eroveu - adantld weu iotacl sseldd ralrimivva eqid fmpo sylib erovlem feq1d mpbird - ) APQVCZRKVDWTRBCPQBVEZUGVEZLVFVGCVEZUFVEZMVFVGVHZDVEZXBXDJVIZNVFZVGZVHZU - FHVJUGGVJZDVKZVLZVDZAXLRVMZCQVNBPVNXNAXOBCPQAXAPVMXCQVMVHZVHZXKDVOZRXLXQX - KDRXQXJXFRVMZUGUFGHXQXBGVMXDHVMVHZVHZXIXSXEYAXHRVMXIXSVPYAXHINVSZRYANUCVM - ZXGIVMXHYBVMAYCXPXTUJVQXQXBXDIGHJAGHVCIJVDXPUQVRVTIXGNUCWAWBVBWCXHRXFWDWE - WIWFWGXQXKDWJXLXRVMADEFGHIJLMNOPQSTXAXCUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURWH - XKDWKWEWLWMBCPQXLRXMXMWNWOWPAWTRKXMABCDEFGHIJKLMNOPQSTUCUDUEUFUGUHUIUJUKU - LUMUNUOUPUQURUSWQWRWS $. + cqs ecelqsg syl2anc eleqtrrdi eleq1a syl adantld rexlimdvva abssdv eroveu + fovcdmda iotacl sseldd ralrimivva eqid fmpo sylib erovlem feq1d mpbird + weu ) APQVCZRKVDWTRBCPQBVEZUGVEZLVFVGCVEZUFVEZMVFVGVHZDVEZXBXDJVIZNVFZVGZ + VHZUFHVJUGGVJZDVKZVLZVDZAXLRVMZCQVNBPVNXNAXOBCPQAXAPVMXCQVMVHZVHZXKDVOZRX + LXQXKDRXQXJXFRVMZUGUFGHXQXBGVMXDHVMVHZVHZXIXSXEYAXHRVMXIXSVPYAXHINVSZRYAN + UCVMZXGIVMXHYBVMAYCXPXTUJVQXQXBXDIGHJAGHVCIJVDXPUQVRWIIXGNUCVTWAVBWBXHRXF + WCWDWEWFWGXQXKDWSXLXRVMADEFGHIJLMNOPQSTXAXCUCUDUEUFUGUHUIUJUKULUMUNUOUPUQ + URWHXKDWJWDWKWLBCPQXLRXMXMWMWNWOAWTRKXMABCDEFGHIJKLMNOPQSTUCUDUEUFUGUHUIU + JUKULUMUNUOUPUQURUSWPWQWR $. $} ${ @@ -63552,21 +63554,21 @@ the cancellation property (fifth hypothesis), show that the relation ( vu wcel wa cv cec wceq wex ecelqsdm cqs co wi wal ee4anv eleq1 bi2anan9 wmo an4 wb adantr biimpac anim12i an4s adantl wer a1i wbr simprl cdm erdm eqtr2 ax-mp simpll sylancr mpbird simprr simplr eqeltrrd syl22anc syl2anc - erth mp2and erthi eqeq12 syl5ibrcom expimpd syl5bi exlimdvv syl5bir eqeq1 - alrimivv anbi2d 2exbidv eceq1 eqeq2d oveq12 eceq1d anbi12d cbvex2v bitrdi - mo4 sylibr ) FJIUAZNZGWNNZOZFBPZIQZRZGCPZIQZRZOZAPZWRXAHUBZIQZRZOZCSBSZFD - PZIQZRZGEPZIQZRZOZMPZXKXNHUBZIQZRZOZESDSZOZXEXRRZUCZMUDAUDXJAUHWQYFAMYDXI - YBOZESDSZCSBSWQYEXIYBBCDEUEWQYHYEBCWQYGYEDEYGXDXQOZXHYAOZOWQYEXDXHXQYAUIW - QYIYJYEWQYIOZYEYJXGXTRZYKWSWNNZXBWNNZOZWSXLRZXBXORZOZYLYIWQYOXDWQYOUJXQWT - WOYMXCWPYNFWSWNUFGXBWNUFUGUKULYIYRWQWTXMXCXPYRWTXMOYPXCXPOYQFWSXLVBGXBXOV - BUMUNUOYOYROZXFXSIJJIUPZYSKUQZYSWRXKIURZXAXNIURZXFXSIURZYSUUBYPYOYPYQUSZY - SWRXKIJUUAYSIUTJRZYMWRJNZYTUUFKJIVAVCZYMYNYRVDZJWRITVEZVLVFYSUUCYQYOYPYQV - GZYSXAXNIJUUAYSUUFYNXAJNZUUHYMYNYRVHZJXAITVEZVLVFYSUUGXKJNZUULXNJNZUUBUUC - OUUDUCUUJYSUUFXLWNNUUOUUHYSWSXLWNUUEUUIVIJXKITVEUUNYSUUFXOWNNUUPUUHYSXBXO - WNUUKUUMVIJXNITVELVJVMVNVKXEXGXRXTVOVPVQVRVSVSVTWBXJYCAMYEXJXDXRXGRZOZCSB - SYCYEXIUURBCYEXHUUQXDXEXRXGWAWCWDUURYBBCDEWRXKRZXAXNRZOZXDXQUUQYAUUSWTXMU - UTXCXPUUSWSXLFWRXKIWEWFUUTXBXOGXAXNIWEWFUGUVAXGXTXRUVAXFXSIWRXKXAXNHWGWHW - FWIWJWKWLWM $. + mp2and erthi eqeq12 syl5ibrcom expimpd biimtrid exlimdvv syl5bir alrimivv + erth eqeq1 anbi2d 2exbidv eqeq2d oveq12 eceq1d anbi12d cbvex2v bitrdi mo4 + eceq1 sylibr ) FJIUAZNZGWNNZOZFBPZIQZRZGCPZIQZRZOZAPZWRXAHUBZIQZRZOZCSBSZ + FDPZIQZRZGEPZIQZRZOZMPZXKXNHUBZIQZRZOZESDSZOZXEXRRZUCZMUDAUDXJAUHWQYFAMYD + XIYBOZESDSZCSBSWQYEXIYBBCDEUEWQYHYEBCWQYGYEDEYGXDXQOZXHYAOZOWQYEXDXHXQYAU + IWQYIYJYEWQYIOZYEYJXGXTRZYKWSWNNZXBWNNZOZWSXLRZXBXORZOZYLYIWQYOXDWQYOUJXQ + WTWOYMXCWPYNFWSWNUFGXBWNUFUGUKULYIYRWQWTXMXCXPYRWTXMOYPXCXPOYQFWSXLVBGXBX + OVBUMUNUOYOYROZXFXSIJJIUPZYSKUQZYSWRXKIURZXAXNIURZXFXSIURZYSUUBYPYOYPYQUS + ZYSWRXKIJUUAYSIUTJRZYMWRJNZYTUUFKJIVAVCZYMYNYRVDZJWRITVEZWAVFYSUUCYQYOYPY + QVGZYSXAXNIJUUAYSUUFYNXAJNZUUHYMYNYRVHZJXAITVEZWAVFYSUUGXKJNZUULXNJNZUUBU + UCOUUDUCUUJYSUUFXLWNNUUOUUHYSWSXLWNUUEUUIVIJXKITVEUUNYSUUFXOWNNUUPUUHYSXB + XOWNUUKUUMVIJXNITVELVJVLVMVKXEXGXRXTVNVOVPVQVRVRVSVTXJYCAMYEXJXDXRXGRZOZC + SBSYCYEXIUURBCYEXHUUQXDXEXRXGWBWCWDUURYBBCDEWRXKRZXAXNRZOZXDXQUUQYAUUSWTX + MUUTXCXPUUSWSXLFWRXKIWLWEUUTXBXOGXAXNIWLWEUGUVAXGXTXRUVAXFXSIWRXKXAXNHWFW + GWEWHWIWJWKWM $. $} ${ @@ -64833,17 +64835,17 @@ the first case of his notation (simple exponentiation) and subscript it cxp snex xpexg adantr rnex a1i cop wrex sneq xpeq1d anbi2d wb elixpsn elv uniex ixpeq1d eleq2d bitr3id anbi1d xpsn eqeq2i eqid opeq2 sneqd rspceeqv anbi2i mpan2 op2nda eqcomi jctir eqeq1 rexbidv rneq unieqd syl5ibrcom imp - anbi12d wi eqidd ancli eleq1w syl5bi imbi12d rexlimiv impbii vtoclbg f1od - bitri ) EFKZAHCBELZCUAZWMAMZLZUDZHMZNZOZDPPGWLWQPKZWOCKZWLWMPKWPPKXAEFUBW - OAQZUEWMWPPPUFUCUGWTPKWLWRWNKZRWSWRHQUHURUIXBWRIMZLZWPUDZSZRZWRXEJMZUJZLZ - SZJCUKZWOWTSZRZXBWRWQSZRXDXORIEFXEESZXHXQXBXRXGWQWRXRXFWMWPXEEULZUMTUNXRX - NXDXOXNWRBXFCUAZKZXRXDYAXNUOIBJXECWRPUPUQXRXTWNWRXRBXFWMCXSUSUTVAVBXIXBWR - XEWOUJZLZSZRZXPXHYDXBXGYCWRXEWOIQZXCVCVDVIYEXPXBYDXPXBXPYDYCXLSZJCUKZWOYC - NZOZSZRXBYHYKXBYCYCSYHYCVEJWOCXLYCYCXJWOSXKYBXJWOXEVFVGVHVJYJWOXEWOYFXCVK - VLVMYDXNYHXOYKYDXMYGJCWRYCXLVNVOYDWTYJWOYDWSYIWRYCVPVQTVTVRVSXNXOYEXMXOYE - WAZJCXJCKZYLXMWOXLNZOZSZXBXLYCSZRZWAYPWOXJSZYMYRYOXJWOXEXJYFJQVKVDYMYRYSY - MXLXLSZRYMYTYMXLWBWCYSXBYMYQYTAJCWDYSYCXLXLYSYBXKWOXJXEVFVGTVTVRWEXMXOYPY - EYRXMWTYOWOXMWSYNWRXLVPVQTXMYDYQXBWRXLYCVNUNWFVRWGVSWHWKWIWJ $. + anbi12d eqidd ancli eleq1w biimtrid imbi12d rexlimiv impbii bitri vtoclbg + wi f1od ) EFKZAHCBELZCUAZWMAMZLZUDZHMZNZOZDPPGWLWQPKZWOCKZWLWMPKWPPKXAEFU + BWOAQZUEWMWPPPUFUCUGWTPKWLWRWNKZRWSWRHQUHURUIXBWRIMZLZWPUDZSZRZWRXEJMZUJZ + LZSZJCUKZWOWTSZRZXBWRWQSZRXDXORIEFXEESZXHXQXBXRXGWQWRXRXFWMWPXEEULZUMTUNX + RXNXDXOXNWRBXFCUAZKZXRXDYAXNUOIBJXECWRPUPUQXRXTWNWRXRBXFWMCXSUSUTVAVBXIXB + WRXEWOUJZLZSZRZXPXHYDXBXGYCWRXEWOIQZXCVCVDVIYEXPXBYDXPXBXPYDYCXLSZJCUKZWO + YCNZOZSZRXBYHYKXBYCYCSYHYCVEJWOCXLYCYCXJWOSXKYBXJWOXEVFVGVHVJYJWOXEWOYFXC + VKVLVMYDXNYHXOYKYDXMYGJCWRYCXLVNVOYDWTYJWOYDWSYIWRYCVPVQTVTVRVSXNXOYEXMXO + YEWJZJCXJCKZYLXMWOXLNZOZSZXBXLYCSZRZWJYPWOXJSZYMYRYOXJWOXEXJYFJQVKVDYMYRY + SYMXLXLSZRYMYTYMXLWAWBYSXBYMYQYTAJCWCYSYCXLXLYSYBXKWOXJXEVFVGTVTVRWDXMXOY + PYEYRXMWTYOWOXMWSYNWRXLVPVQTXMYDYQXBWRXLYCVNUNWEVRWFVSWGWHWIWK $. $( A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.) $) @@ -65005,7 +65007,7 @@ the first case of his notation (simple exponentiation) and subscript it $( The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) $) f1oeng $p |- ( ( A e. C /\ F : A -1-1-onto-> B ) -> A ~~ B ) $= - ( wcel wf1o cvv cen wbr wfo f1ofo fornex syl5 imp f1oen2g 3com23 mpd3an3 + ( wcel wf1o cvv cen wbr wfo f1ofo focdmex syl5 imp f1oen2g 3com23 mpd3an3 ) ACEZABDFZBGEZABHIZRSTSABDJRTABDKABCDLMNRTSUAABDCGOPQ $. $( The domain of a one-to-one function is dominated by its codomain. @@ -66035,37 +66037,37 @@ the first case of his notation (simple exponentiation) and subscript it ( ( A ^m B ) ^m C ) ~~ ( A ^m ( B X. C ) ) ) $= ( vx vy wcel cmap co cv cfv cmpt cvv wf wral wa eqid wceq vf w3a cxp cmpo vg wfn fnmap elex 3ad2ant1 3ad2ant2 fnovex mp3an2i 3ad2ant3 xpexg 3adant1 - elmapi ffvelcdmda an32s ralrimiva sylib simp1 elmapd syl5ibr adantl fovrn - syl fmpo 3expa sylanl1 fmptd wb elmapg 3adant3 ad2antrr mpbird ex sylibrd - simp3 elmapfn ad2antll fnovim 3adant2 simp1l2 simp1l1 fex2 syl3anc fvmpt2 - adantlrl syl2anc fveq1d simp2 vex ovexg mp3an eqtrd mpoeq3dva eqtr4d nfcv - sylancl nfmpt1 nfmpt nfeq2 fveq1 ralrimi jctil mpoeq123 eqeq2d syl5ibrcom - a1d sylancr ad2antrl feqmptd simprl sylan mpteq2dva nfmpo2 eqidd nfv fvex - nfmpo1 ovmpt4g mp3an3 oveq eqeq1d expcomd ralrimd mpteq12 syl6an impbid + elmapi ffvelcdmda an32s ralrimiva fmpo sylib elmapd syl5ibr adantl fovcdm + simp1 3expa sylanl1 fmptd wb elmapg 3adant3 ad2antrr mpbird simp3 sylibrd + syl elmapfn ad2antll fnovim adantlrl 3adant2 simp1l2 simp1l1 fex2 syl3anc + ex fvmpt2 syl2anc fveq1d simp2 ovexg mp3an sylancl eqtrd mpoeq3dva eqtr4d + vex nfcv nfmpt1 nfmpt nfeq2 fveq1 a1d ralrimi mpoeq123 sylancr syl5ibrcom + jctil eqeq2d ad2antrl simprl sylan mpteq2dva nfmpo2 eqidd nfmpo1 nfv fvex + feqmptd ovmpt4g mp3an3 oveq eqeq1d expcomd ralrimd mpteq12 syl6an impbid en3d ) ADIZBEIZCFIZUBZUAUEABJKZCJKZABCUCZJKZGHBCGLZHLZUALZMZMZUDZHCGBYSYT UELZKZNZNZJOOUCUFZYNYOOIZCOIZYPOIUGUUIYNAOIZBOIZUUJUGYKYLUULYMADUHUIZYLYK UUMYMBEUHUJABOOJUKULZYMYKUUKYLCFUHUMYOCOOJUKULUUIYNUULYQOIZYROIUGUUNYLYMU UPYKBCEFUNUOZAYQOOJUKULUUAYPIZUUDYRIYNYQAUUDPZUURUUCAIZHCQZGBQUUSUURUVAGB UURYSBIZRUUTHCUURYTCIZUVBUUTUURUVCRZBAYSUUBUVDUUBYOIBAUUBPZUURCYOYTUUAUUA - 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entr syl2an anandirs ex exlimdv syl5bi ) AEZBEZFGWNWOSUAZUBZSUCAHIZBHIZTZ - ABFGZWNWOSUDWTWQXASWTWQXAWRWSWQXAWRWQTZAWOAWPUEZJZKZFGXEBFGXAWSWQTZXBAWPA - LZXEFXBXGAXBWNWOWPUFZWOMIZXGAFGZWQXHWRWNWOWPUGUHBDUMXHXITAWNUIAMIXJAUJCWN - WOAWPMUKULNOWRWQXGWPWNAJZKZLZXEWRAXLWPWRAUNAXLPAUOAUPQRWQWPWNLZXDKXNWPXKL - ZKZXEXMWQXDXOXNWQWPWNUQZAWNIZXDXOPWNWOWPURZACUSZWNAWPUTNVAWQWOXNXDWQWPVBZ - WPWPVEZLZWOXNYCYAWPVCVDWQWNWOWPVFZYAWOPWNWOWPVGZWNWOWPVHZQWQYBWNWPWNWOWPV - IRVJVKWQWPVLVMZXMXPPWQYDYGWNWOWPVNVQWNXKWPVOQVPVRVSXFBXEWQWSXCWOIZBXEFGWQ - XCYAIZYHWQXQXRYIXSXTWNAWPVTNWQYDYIYHWAYEYDYAWOXCYFWBQWCBXCDAWPMMSWDCWEWFW - GOAXEBWHWIWJWKWLWM $. + entr syl2an anandirs ex exlimdv biimtrid ) AEZBEZFGWNWOSUAZUBZSUCAHIZBHIZ + TZABFGZWNWOSUDWTWQXASWTWQXAWRWSWQXAWRWQTZAWOAWPUEZJZKZFGXEBFGXAWSWQTZXBAW + PALZXEFXBXGAXBWNWOWPUFZWOMIZXGAFGZWQXHWRWNWOWPUGUHBDUMXHXITAWNUIAMIXJAUJC + WNWOAWPMUKULNOWRWQXGWPWNAJZKZLZXEWRAXLWPWRAUNAXLPAUOAUPQRWQWPWNLZXDKXNWPX + KLZKZXEXMWQXDXOXNWQWPWNUQZAWNIZXDXOPWNWOWPURZACUSZWNAWPUTNVAWQWOXNXDWQWPV + BZWPWPVEZLZWOXNYCYAWPVCVDWQWNWOWPVFZYAWOPWNWOWPVGZWNWOWPVHZQWQYBWNWPWNWOW + PVIRVJVKWQWPVLVMZXMXPPWQYDYGWNWOWPVNVQWNXKWPVOQVPVRVSXFBXEWQWSXCWOIZBXEFG + WQXCYAIZYHWQXQXRYIXSXTWNAWPVTNWQYDYIYHWAYEYDYAWOXCYFWBQWCBXCDAWPMMSWDCWEW + FWGOAXEBWHWIWJWKWLWM $. $} ${ @@ -66303,16 +66305,16 @@ the first case of his notation (simple exponentiation) and subscript it vf wa syl2anc cv wf1 wex wb peano2 brdomg biimpa cfv ad2antrr sssucid a1i simpr simplll f1imaen2g syl22anc difexg word nnord orddif wfn f1fn sucidg imaeq2d fnsnfv difeq2d ccnv wfun wf simprbi imadif eqtr4d f1f crn imassrn - df-f1 sstrid ssdifd eqsstrrd eqsstrd ssdomg sylc endomtr simpllr ffvelrnd - frn phplem3g domentr exlimddv ex ) ACDZBCDZSZAEZBEZFGZABFGZWLWOSZWMWNRUAZ - UBZWPRWLWOWSRUCZWLWNCDZWOWTUDWKXAWJBUEHZWMWNCRUFIUGWQWSSZAWNAWRUHZJZKZFGZ - XFBLGWPXCAWRAMZLGXHXFFGZXGXCXHAXCWSXAAWMNZWJXHALGWQWSULWLXAWOWSXBUIZXJXCA - UJUKWJWKWOWSUMZWMWNAWRCUNUOOXCXFPDZXHXFNXIXCXAXMXKWNXECUPIXCXHWRWMAJZKZMZ - XFXCWJXHXPQXLWJAXOWRWJAUQAXOQAURAUSIVCIXCXPWRWMMZXEKZXFXCXRXQWRXNMZKZXPXC - XEXSXQXCWRWMUTZAWMDZXEXSQWSYAWQWMWNWRVAHXCWJYBXLACVBIZWMAWRVDTVEWSXPXTQZW - QWSWRVFVGZYDWSWMWNWRVHZYEWMWNWRVOVIWMXNWRVJIHVKXCXQWNXEXCYFXQWNNWSYFWQWMW - NWRVLHZYFXQWRVMWNWRWMVNWMWNWRWEVPIVQVRVSXHXFPVTWAAXHXFWBTXCBXFXCWKXDWNDBX - FLGWJWKWOWSWCXCWMWNAWRYGYCWDBXDWFTOAXFBWGTWHWI $. + df-f1 frn sstrid ssdifd eqsstrrd eqsstrd ssdomg endomtr simpllr ffvelcdmd + sylc phplem3g domentr exlimddv ex ) ACDZBCDZSZAEZBEZFGZABFGZWLWOSZWMWNRUA + ZUBZWPRWLWOWSRUCZWLWNCDZWOWTUDWKXAWJBUEHZWMWNCRUFIUGWQWSSZAWNAWRUHZJZKZFG + ZXFBLGWPXCAWRAMZLGXHXFFGZXGXCXHAXCWSXAAWMNZWJXHALGWQWSULWLXAWOWSXBUIZXJXC + AUJUKWJWKWOWSUMZWMWNAWRCUNUOOXCXFPDZXHXFNXIXCXAXMXKWNXECUPIXCXHWRWMAJZKZM + ZXFXCWJXHXPQXLWJAXOWRWJAUQAXOQAURAUSIVCIXCXPWRWMMZXEKZXFXCXRXQWRXNMZKZXPX + CXEXSXQXCWRWMUTZAWMDZXEXSQWSYAWQWMWNWRVAHXCWJYBXLACVBIZWMAWRVDTVEWSXPXTQZ + WQWSWRVFVGZYDWSWMWNWRVHZYEWMWNWRVOVIWMXNWRVJIHVKXCXQWNXEXCYFXQWNNWSYFWQWM + WNWRVLHZYFXQWRVMWNWRWMVNWMWNWRVPVQIVRVSVTXHXFPWAWEAXHXFWBTXCBXFXCWKXDWNDB + XFLGWJWKWOWSWCXCWMWNAWRYGYCWDBXDWFTOAXFBWGTWHWI $. $} ${ @@ -66429,16 +66431,16 @@ the first case of his notation (simple exponentiation) and subscript it need to know ` B = C ` or ` -. B = C ` , respectively. (Contributed by Jim Kingdon, 5-Sep-2021.) $) fidceq $p |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C ) $= - ( vx vf wcel cv wceq wdc com biimpi wa wn wo adantl ffvelrnd elnn syl2anc - cfv wi cfn w3a cen wbr wrex isfi 3ad2ant1 wf1o bren ad2antll f1of simpll2 - wex wf simplrl simpll3 nndceq exmiddc f1of1 f1veqaeq syl12anc fveq2 con3i - syl wf1 a1i orim12d mpd df-dc sylibr exlimddv rexlimddv ) AUAFZBAFZCAFZUB - ZADGZUCUDZBCHZIZDJVMVNVRDJUEZVOVMWADAUFKUGVPVQJFZVRLZLZAVQEGZUHZVTEVRWFEU - MZVPWBVRWGAVQEUIKUJWDWFLZVSVSMZNZVTWHBWESZCWESZHZWMMZNZWJWHWMIZWOWHWKJFZW - LJFZWPWHWKVQFWBWQWHAVQBWEWFAVQWEUNWDAVQWEUKOZVMVNVOWCWFULZPVPWBVRWFUOZWKV - QQRWHWLVQFWBWRWHAVQCWEWSVMVNVOWCWFUPZPXAWLVQQRWKWLUQRWMURVDWHWMVSWNWIWHAV - QWEVEZVNVOWMVSTWFXCWDAVQWEUSOWTXBAVQBCWEUTVAWNWITWHVSWMBCWEVBVCVFVGVHVSVI - VJVKVL $. + ( vx vf wcel cv wceq wdc com biimpi wa wn wo cfv adantl ffvelcdmd syl2anc + elnn wi cfn w3a cen wbr wrex isfi 3ad2ant1 wf1o wex bren ad2antll wf f1of + simpll2 simplrl simpll3 nndceq exmiddc syl wf1 f1of1 f1veqaeq fveq2 con3i + syl12anc a1i orim12d mpd df-dc sylibr exlimddv rexlimddv ) AUAFZBAFZCAFZU + BZADGZUCUDZBCHZIZDJVMVNVRDJUEZVOVMWADAUFKUGVPVQJFZVRLZLZAVQEGZUHZVTEVRWFE + UIZVPWBVRWGAVQEUJKUKWDWFLZVSVSMZNZVTWHBWEOZCWEOZHZWMMZNZWJWHWMIZWOWHWKJFZ + WLJFZWPWHWKVQFWBWQWHAVQBWEWFAVQWEULWDAVQWEUMPZVMVNVOWCWFUNZQVPWBVRWFUOZWK + VQSRWHWLVQFWBWRWHAVQCWEWSVMVNVOWCWFUPZQXAWLVQSRWKWLUQRWMURUSWHWMVSWNWIWHA + VQWEUTZVNVOWMVSTWFXCWDAVQWEVAPWTXBAVQBCWEVBVEWNWITWHVSWMBCWEVCVDVFVGVHVSV + IVJVKVL $. $} ${ @@ -66580,20 +66582,20 @@ the first case of his notation (simple exponentiation) and subscript it ( A \ { X } ) ~~ M ) $= ( vf com wcel cen wbr wf1o csn cdif ensymd 3ad2ant1 syl2anc adantr adantl syl wceq cres wfo csuc w3a cv wex simp2 bren sylib wa cfv cfn peano2 nnfi - enfii simpl3 wf f1of sucidg ffvelrnd fidifsnen syl3anc word orddif eleq1d - nnord ibi ccnv wfun wfn crn dff1o2 simp2bi f1ofo wb cdm wrel f1orel resdm - f1odm reseq2d eqtr3d foeq1 mpbid simpl1 f1osng f1ofn mpbird resdif f1oeng - cop fnressn eqbrtrd entr exlimddv ) BEFZABUAZGHZCAFZUBZWOADUCZIZACJKZBGHZ - DWRWOAGHWTDUDWRAWOWNWPWQUEZLWOADUFUGWRWTUHZXAABWSUIZJZKZGHZXGBGHXBXDAUJFZ - WQXEAFZXHWRXIWTWRWOUJFZWPXIWNWPXKWQWNWOEFXKBUKWOULQMXCAWOUMNOWNWPWQWTUNXD - WOABWSWTWOAWSUOWRWOAWSUPPWRBWOFZWTWNWPXLWQBEUQMOZURZCXEAUSUTXDBXGXDBWOBJZ - KZXGGWRBXPRZWTWNWPXQWQWNBVAXQBVDBVBQZMOXDXPEFZXPXGWSXPSZIZXPXGGHWRXSWTWNW - PXSWQWNXSWNBXPEXRVCVEMOXDWSVFVGZWOAWSWOSZTZXOXFWSXOSZTZYAWTYBWRWTWSWOVHZY - BWSVIARWOAWSVJVKPXDWOAWSTZYDWTYHWRWOAWSVLPXDWSYCRYHYDVMXDWSWSVNZSZWSYCXDW - SVOZYJWSRWTYKWRWOAWSVPPWSVQQWTYJYCRWRWTYIWOWSWOAWSVRVSPVTWOAWSYCWAQWBXDYF - XOXFBXEWIJZTZXDXOXFYLIZYMXDWNXJYNWNWPWQWTWCXNBXEEAWDNXOXFYLVLQXDYEYLRZYFY - MVMXDYGXLYOWTYGWRWOAWSWEPXMWOBWSWJNXOXFYEYLWAQWFWOXOAXFWSWGUTXPXGEXTWHNWK - LXAXGBWLNWM $. + enfii simpl3 wf f1of sucidg ffvelcdmd fidifsnen syl3anc word nnord orddif + eleq1d ibi ccnv wfun wfn crn dff1o2 simp2bi f1ofo wb cdm wrel resdm f1odm + f1orel reseq2d eqtr3d foeq1 mpbid cop simpl1 f1osng fnressn mpbird resdif + f1ofn f1oeng eqbrtrd entr exlimddv ) BEFZABUAZGHZCAFZUBZWOADUCZIZACJKZBGH + ZDWRWOAGHWTDUDWRAWOWNWPWQUEZLWOADUFUGWRWTUHZXAABWSUIZJZKZGHZXGBGHXBXDAUJF + ZWQXEAFZXHWRXIWTWRWOUJFZWPXIWNWPXKWQWNWOEFXKBUKWOULQMXCAWOUMNOWNWPWQWTUNX + DWOABWSWTWOAWSUOWRWOAWSUPPWRBWOFZWTWNWPXLWQBEUQMOZURZCXEAUSUTXDBXGXDBWOBJ + ZKZXGGWRBXPRZWTWNWPXQWQWNBVAXQBVBBVCQZMOXDXPEFZXPXGWSXPSZIZXPXGGHWRXSWTWN + WPXSWQWNXSWNBXPEXRVDVEMOXDWSVFVGZWOAWSWOSZTZXOXFWSXOSZTZYAWTYBWRWTWSWOVHZ + YBWSVIARWOAWSVJVKPXDWOAWSTZYDWTYHWRWOAWSVLPXDWSYCRYHYDVMXDWSWSVNZSZWSYCXD + WSVOZYJWSRWTYKWRWOAWSVRPWSVPQWTYJYCRWRWTYIWOWSWOAWSVQVSPVTWOAWSYCWAQWBXDY + FXOXFBXEWCJZTZXDXOXFYLIZYMXDWNXJYNWNWPWQWTWDXNBXEEAWENXOXFYLVLQXDYEYLRZYF + YMVMXDYGXLYOWTYGWRWOAWSWIPXMWOBWSWFNXOXFYEYLWAQWGWOXOAXFWSWHUTXPXGEXTWJNW + KLXAXGBWLNWM $. $} ${ @@ -66669,14 +66671,14 @@ the first case of his notation (simple exponentiation) and subscript it fin0 $p |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) ) $= ( vn vm vf cfn wcel cv cen wbr c0 wex com wrex wa wceq simplrr sylib syl wn wne wb isfi biimpi csuc simpr breqtrd en0 nner necon2bi 2falsed adantr - n0r wf1o ensymd bren cfv wf f1of adantl sucidg ad3antlr eleqtrrd ffvelrnd - simplr elex2 exlimddv ex rexlimdva imp nn0suc ad2antrl mpjaodan rexlimddv - 2thd wo ) BFGZBCHZIJZBKUAZAHBGALZUBZCMVQVSCMNCBUCUDVQVRMGZVSOOZVRKPZWBVRD - HZUEZPZDMNZWDWEOZVTWAWJBKPZVTTWJBKIJWKWJBVRKIVQWCVSWEQWDWEUFUGBUHRZBKUISW - JWKWATWLWABKABUMZUJSUKWDWIWBWDWHWBDMWDWFMGZOZWHWBWOWHOZVRBEHZUNZWBEWPVRBI - JWRELWPBVRWOVSWHVQWCVSWNQULUOVRBEUPRWPWROZVTWAWSWAVTWSWFWQUQZBGWAWSVRBWFW - QWRVRBWQURWPVRBWQUSUTWSWFWGVRWNWFWGGWDWHWRWFMVAVBWOWHWRVEVCVDAWTBVFSZWMSX - AVOVGVHVIVJWCWEWIVPVQVSDVRVKVLVMVN $. + n0r wf1o ensymd bren cfv adantl sucidg ad3antlr simplr eleqtrrd ffvelcdmd + wf f1of elex2 2thd exlimddv ex rexlimdva imp wo nn0suc ad2antrl rexlimddv + mpjaodan ) BFGZBCHZIJZBKUAZAHBGALZUBZCMVQVSCMNCBUCUDVQVRMGZVSOOZVRKPZWBVR + DHZUEZPZDMNZWDWEOZVTWAWJBKPZVTTWJBKIJWKWJBVRKIVQWCVSWEQWDWEUFUGBUHRZBKUIS + WJWKWATWLWABKABUMZUJSUKWDWIWBWDWHWBDMWDWFMGZOZWHWBWOWHOZVRBEHZUNZWBEWPVRB + IJWRELWPBVRWOVSWHVQWCVSWNQULUOVRBEUPRWPWROZVTWAWSWAVTWSWFWQUQZBGWAWSVRBWF + WQWRVRBWQVDWPVRBWQVEURWSWFWGVRWNWFWGGWDWHWRWFMUSUTWOWHWRVAVBVCAWTBVFSZWMS + XAVGVHVIVJVKWCWEWIVLVQVSDVRVMVNVPVO $. $} ${ @@ -66686,13 +66688,13 @@ the first case of his notation (simple exponentiation) and subscript it fin0or $p |- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) ) $= ( vn vm vf cfn wcel cv cen wbr c0 wceq wex wo com wrex wa simplrr sylib ex isfi biimpi csuc nn0suc ad2antrl simpr breqtrd wf1o adantr ensymd bren - en0 cfv wf f1of adantl sucidg ad3antlr simplr eleqtrrd ffvelrnd elex2 syl - exlimddv rexlimdva orim12d mpd rexlimddv ) BFGZBCHZIJZBKLZAHBGAMZNZCOVIVK - COPCBUAUBVIVJOGZVKQQZVJKLZVJDHZUCZLZDOPZNZVNVOWBVIVKDVJUDUEVPVQVLWAVMVPVQ - VLVPVQQZBKIJVLWCBVJKIVIVOVKVQRVPVQUFUGBULSTVPVTVMDOVPVROGZQZVTVMWEVTQZVJB - EHZUHZVMEWFVJBIJWHEMWFBVJWEVKVTVIVOVKWDRUIUJVJBEUKSWFWHQZVRWGUMZBGVMWIVJB - VRWGWHVJBWGUNWFVJBWGUOUPWIVRVSVJWDVRVSGVPVTWHVROUQURWEVTWHUSUTVAAWJBVBVCV - DTVEVFVGVH $. + en0 cfv wf f1of adantl sucidg ad3antlr simplr eleqtrrd ffvelcdmd exlimddv + elex2 syl rexlimdva orim12d mpd rexlimddv ) BFGZBCHZIJZBKLZAHBGAMZNZCOVIV + KCOPCBUAUBVIVJOGZVKQQZVJKLZVJDHZUCZLZDOPZNZVNVOWBVIVKDVJUDUEVPVQVLWAVMVPV + QVLVPVQQZBKIJVLWCBVJKIVIVOVKVQRVPVQUFUGBULSTVPVTVMDOVPVROGZQZVTVMWEVTQZVJ + BEHZUHZVMEWFVJBIJWHEMWFBVJWEVKVTVIVOVKWDRUIUJVJBEUKSWFWHQZVRWGUMZBGVMWIVJ + BVRWGWHVJBWGUNWFVJBWGUOUPWIVRVSVJWDVRVSGVPVTWHVROUQURWEVTWHUSUTVAAWJBVCVD + VBTVEVFVGVH $. $} ${ @@ -67069,15 +67071,15 @@ the first case of his notation (simple exponentiation) and subscript it ( vz cv wral wrex c0 wceq raleq rexbidv wcel wa ad4antr vw vu vv wbr wn csn cun wex wne cfn wb fin0 syl mpbid biantru exbii sylib df-rex sylibr ral0 wss cdif notbid ralbidv cbvrexv wpo simp-4r simprl ad2antrr simplr - breq1 w3o simprr eldifad eldifbd simpr fimax2gtrilemstep ex findcard2sd - rexlimdva syl5bi ) ABKZCKZEUDZUEZCUAKZLZBDMWECNLZBDMZWECUBKZLZBDMZWECWJ - UCKZUFUGZLZBDMZWECDLZBDMUAUBUCDWFNOWGWHBDWECWFNPQWFWJOWGWKBDWECWFWJPQWF - WNOWGWOBDWECWFWNPQWFDOWGWQBDWECWFDPQAWBDRZWHSZBUHZWIAWRBUHZWTADNUIZXAIA - DUJRZXBXAUKHBDULUMUNWRWSBWHWRWECUTUOUPUQWHBDURUSWLJKZWCEUDZUEZCWJLZJDMA - WJUJRZSZWJDVAZWMDWJVBRZSZSZWPWKXGBJDWBXDOZWEXFCWJXNWDXEWBXDWCEVKVCVDVEX - MXGWPJDXMXDDRZSZXGWPXPXGSZBCDEWJWMXDADEVFXHXLXOXGFTAWDWBWCOWCWBEUDVLCDL - BDLXHXLXOXGGTAXCXHXLXOXGHTAXBXHXLXOXGITAXHXLXOXGVGXMXJXOXGXIXJXKVHVIXMX - OXGVJXQWMDWJXMXKXOXGXIXJXKVMVIZVNXQWMDWJXRVOXPXGVPVQVRVTWAHVS $. + breq1 simprr eldifad eldifbd simpr fimax2gtrilemstep rexlimdva biimtrid + w3o ex findcard2sd ) ABKZCKZEUDZUEZCUAKZLZBDMWECNLZBDMZWECUBKZLZBDMZWEC + WJUCKZUFUGZLZBDMZWECDLZBDMUAUBUCDWFNOWGWHBDWECWFNPQWFWJOWGWKBDWECWFWJPQ + WFWNOWGWOBDWECWFWNPQWFDOWGWQBDWECWFDPQAWBDRZWHSZBUHZWIAWRBUHZWTADNUIZXA + IADUJRZXBXAUKHBDULUMUNWRWSBWHWRWECUTUOUPUQWHBDURUSWLJKZWCEUDZUEZCWJLZJD + MAWJUJRZSZWJDVAZWMDWJVBRZSZSZWPWKXGBJDWBXDOZWEXFCWJXNWDXEWBXDWCEVKVCVDV + EXMXGWPJDXMXDDRZSZXGWPXPXGSZBCDEWJWMXDADEVFXHXLXOXGFTAWDWBWCOWCWBEUDVSC + DLBDLXHXLXOXGGTAXCXHXLXOXGHTAXBXHXLXOXGITAXHXLXOXGVGXMXJXOXGXIXJXKVHVIX + MXOXGVJXQWMDWJXMXKXOXGXIXJXKVLVIZVMXQWMDWJXRVNXPXGVOVPVTVQVRHWA $. $} $} @@ -67120,9 +67122,9 @@ the first case of his notation (simple exponentiation) and subscript it $( An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.) $) infm $p |- ( _om ~<_ A -> E. x x e. A ) $= - ( vf com cdom wbr cv wf1 wcel wex brdomi wa c0 cfv wf f1f adantl ffvelrnd - peano1 a1i elex2 syl exlimddv ) DBEFZDBCGZHZAGBIAJZCDBCKUDUFLZMUENZBIUGUH - DBMUEUFDBUEOUDDBUEPQMDIUHSTRAUIBUAUBUC $. + ( vf com cdom wbr cv wf1 wcel wex brdomi wa c0 wf adantl peano1 ffvelcdmd + cfv f1f a1i elex2 syl exlimddv ) DBEFZDBCGZHZAGBIAJZCDBCKUDUFLZMUERZBIUGU + HDBMUEUFDBUENUDDBUESOMDIUHPTQAUIBUAUBUC $. $( An infinite set is not empty. (Contributed by NM, 23-Oct-2004.) $) infn0 $p |- ( _om ~<_ A -> A =/= (/) ) $= @@ -67154,23 +67156,23 @@ the first case of his notation (simple exponentiation) and subscript it ( vf c2o wcel wa adantr c0 c1o simplr simpr eqtr4d ad2antrr mpbid eqeltrd wceq syl ad3antrrr mpjaodan vx cen wbr w3a wne cpr wf1o wex bren 3ad2ant1 cv biimpi cfv wb wf1 f1of1 adantl simpll3 f1fveq syl12anc simpllr simpll2 - prid2g prid1g wn neneqd mtbird ad4antr pm2.21dd wo wf f1of ffvelrnd elpri - df2o3 eleq2s ffvelcdmda ex ssrdv wss prssi syl2anc eqssd exlimddv ) CEUBU - CZACFZBCFZUDZABUEZCABUFZQZWHWIGZCEDUKZUGZWKDWHWNDUHZWIWEWFWOWGWEWOCEDUIUL - UJHWLWNGZCWJWPUACWJWPUAUKZCFZWQWJFZWPWRGZWQWMUMZIQZWSXAJQZWTXBGZBWMUMZIQZ - WSXEJQZXDXFGZWQBWJXHXAXEQZWQBQZXHXAIXEWTXBXFKXDXFLMWTXIXJUNZXBXFWTCEWMUOZ - WRWGXKWPXLWRWNXLWLCEWMUPUQZHZWPWRLZWPWGWRWEWFWGWIWNURZHZCEWQBWMUSUTZNOWTB - WJFZXBXFWTWGXSXQABCVCRZNPXDXGGZAWMUMZIQZWSYBJQZYAYCGZWQAWJYEXAYBQZWQAQZYE - XAIYBWTXBXGYCVAYAYCLMWTYFYGUNZXBXGYCWTXLWRWFYHXNXOWPWFWRWEWFWGWIWNVBZHZCE - 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sylanbrc elun sylnibr olcd wdc cv wral fof ffvelrnd eqeq1 dcbid rspc2va - wf eqeq2 syl21anc exmiddc adantr mpjaodan ) AHEFOZPZHEFUAZOZPZWOQZUBZWL - QZAWLRZWOWPWSHWKEFUCZOZUDZPZWOWSWLXCAWLUEHWKXAUISWNXBHWNEFWTUDZOXBWMXDE - FUFUGEFWTUHUJUKZULUMAWRRZHFEUNZTZWQXHQZXFXHRZWOWPXJXCWOXJHXAPZXCXJXHXKX - FXHUEAXHXKVBZWRXHAHXGUCZPZXHXKAHDPZXNXHVBNHXGDUOSAXMXAHAEGUPZFGPXMXATAG - DEUQZXPJGDEURSAWMGFLAFUSPFWMPKFUSUTSVAZGFEVCVDVEVFZVGVHHXAWKVISXEULUMXF - XIRZWPWOXTXCWOXTWLXKUBZXCXTWRXKQYAQAWRXIVJXTXHXKXFXIUEAXLWRXIXSVGVKWLXK - VLVMHWKXAVNVOXEVOVPAXHXIUBZWRAXHVQZYBAXOXGDPBVRZCVRZTZVQZCDVSBDVSYCNAGD - FEAXQGDEWEJGDEVTSXRWAIYGYCHYETZVQBCHXGDDYDHTYFYHYDHYEWBWCYEXGTYHXHYEXGH - WFWCWDWGXHWHSWIWJMWJ $. + sylanbrc elun sylnibr olcd wdc cv wral wf ffvelcdmd eqeq1 dcbid rspc2va + fof eqeq2 syl21anc exmiddc adantr mpjaodan ) AHEFOZPZHEFUAZOZPZWOQZUBZW + LQZAWLRZWOWPWSHWKEFUCZOZUDZPZWOWSWLXCAWLUEHWKXAUISWNXBHWNEFWTUDZOXBWMXD + EFUFUGEFWTUHUJUKZULUMAWRRZHFEUNZTZWQXHQZXFXHRZWOWPXJXCWOXJHXAPZXCXJXHXK + XFXHUEAXHXKVBZWRXHAHXGUCZPZXHXKAHDPZXNXHVBNHXGDUOSAXMXAHAEGUPZFGPXMXATA + GDEUQZXPJGDEURSAWMGFLAFUSPFWMPKFUSUTSVAZGFEVCVDVEVFZVGVHHXAWKVISXEULUMX + FXIRZWPWOXTXCWOXTWLXKUBZXCXTWRXKQYAQAWRXIVJXTXHXKXFXIUEAXLWRXIXSVGVKWLX + KVLVMHWKXAVNVOXEVOVPAXHXIUBZWRAXHVQZYBAXOXGDPBVRZCVRZTZVQZCDVSBDVSYCNAG + DFEAXQGDEVTJGDEWESXRWAIYGYCHYETZVQBCHXGDDYDHTYFYHYDHYEWBWCYEXGTYHXHYEXG + HWFWCWDWGXHWHSWIWJMWJ $. $} ${ @@ -67873,8 +67875,8 @@ texts assume excluded middle (in which case the two intersection foima anbi2d imaeq2 eleq1d imbi12d ima0 0fin eqeltri a1i cfv wn csn cun sseq1 wfn simprl fofn simprr sucid sseldd fnsnfv syl2anc uneq2d imaeq2i vex df-suc imaundi eqtri eqtr4di simpr snssd ssequn2 sylib sssucid sstr - eqtr3d mpan ad2antll simplr mp2and eqeltrd fof ffvelrnd unsnfi eqeltrrd - syl3anc wdc wral simpll fidcenumlemrk mpjaodan exp31 finds mpcom mpan2 + eqtr3d mpan ad2antll simplr mp2and eqeltrd fof ffvelcdmd unsnfi syl3anc + eqeltrrd wdc wral simpll fidcenumlemrk mpjaodan exp31 finds mpcom mpan2 wf ) AEFJZDKAFDEUCZXCDLHFDEUGMAFFNZXCKOZFUDFUEOZAXEPZXFAXGXEIQAUARZFNZP ZEXIJZKOZSATFNZPZETJZKOZSAUBRZFNZPZEXRJZKOZSZAXRUFZFNZPZEYDJZKOZSXHXFSU AUBFXITLZXKXOXMXQYIXJXNAXITFUTUHYIXLXPKXITEUIUJUKXIXRLZXKXTXMYBYJXJXSAX @@ -67886,8 +67888,8 @@ texts assume excluded middle (in which case the two intersection LVJEXRUUCVMVNVOZQYSYTYANUUAYALYSYPYAYOYQVPVQYTYAVRVSWBYOYBYQYOAXSYBUUGY EXSYNAXRYDNYEXSXRVTXRYDFWAWCWDZYMYCYFWEWFZQWGYOYRPZUUAYGKYOUUBYRUUJQUUM YBYPDOZYRUUAKOYOYBYRUULQYOUUNYRYOFDXREYOXDFDEXBYOAXDUUGHMZFDEWHMUUHWIZQ - YOYRVPYAYPDWJWLWKYOBCDEXRFYPYOABRCRLWMCDWNBDWNUUGGMUUOYMYCYFWOUUKUUPWPW - QWRWSWTXAWK $. + YOYRVPYAYPDWJWKWLYOBCDEXRFYPYOABRCRLWMCDWNBDWNUUGGMUUOYMYCYFWOUUKUUPWPW + QWRWSWTXAWL $. $} $} @@ -68039,10 +68041,10 @@ texts assume excluded middle (in which case the two intersection sbthlemi10 $p |- ( ( EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B ) $= ( wem cdom wbr wa cv wf1 wex brdom cvv anbi12i eeanv bitr4i wcel wf1o cen w3a cuni cres ccnv cdif cun resex cnvex eqeltri sbthlemi9 f1oen3g - vex unex sylancr 3expib exlimdvv syl5bi imp ) LBCMNZCBMNZOZBCUFNZVGBC - EPZQZCBFPZQZOZFRERZLVHVGVJERZVLFRZOVNVEVOVFVPBCEKSCBFHSUAVJVLEFUBUCLV - MVHEFLVJVLVHLVJVLUGGTUDBCGUEVHGVIDUHZUIZVKUJZBVQUKZUIZULTJVRWAVIVQEUR - UMVSVTVKFURUNUMUSUOABCDEFGHIJUPBCGTUQUTVAVBVCVD $. + vex unex sylancr 3expib exlimdvv biimtrid imp ) LBCMNZCBMNZOZBCUFNZVG + BCEPZQZCBFPZQZOZFRERZLVHVGVJERZVLFRZOVNVEVOVFVPBCEKSCBFHSUAVJVLEFUBUC + LVMVHEFLVJVLVHLVJVLUGGTUDBCGUEVHGVIDUHZUIZVKUJZBVQUKZUIZULTJVRWAVIVQE + URUMVSVTVKFURUNUMUSUOABCDEFGHIJUPBCGTUQUTVAVBVCVD $. $} $} $} @@ -68382,21 +68384,21 @@ all reals (greatest real number) for which all positive reals are greater. supmoti $p |- ( ph -> E* x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) ) $= ( vw cv wbr wn wral wrex wi wa weq breq1 wrmo wcel anbi2ci 3bitr4i ralnex - ancom rexbidv imbi12d rspcva breq2 cbvrexv syl6ibr con3d expimpd ad2antrl - an42 syl5bi ad2antll anim12d wb ralrimivva equequ1 notbid anbi12d bibi12d - equequ2 rspc2v mpan9 sylibrd ralbidv imbi1d rmo4 sylibr ) ABLZCLZIMZNZCHO - ZVOVNIMZVODLZIMZDHPZQZCGOZRZKLZVOIMZNZCHOZVOWFIMZWBQZCGOZRZRZBKSZQZKGOBGO - WEBGUAAWPBKGGAVNGUBZWFGUBZRZRZWNVNWFIMZNZWFVNIMZNZRZWOWNWLVRRZWDWIRZRZWTX - EVRWIRZWLWDRZRXJWIVRRZRWNXHXIXKXJVRWIUFUCVRWDWIWLUPWLVRWDWIUPUDWTXFXBXGXD - WQXFXBQAWRWQWLVRXBVRVPCHPZNWQWLRZXBVPCHUEXMXAXLXMXAVNVTIMZDHPZXLWKXAXOQCV - NGCBSZWJXAWBXOVOVNWFITXPWAXNDHVOVNVTITUGUHUIVPXNCDHVOVTVNIUJUKULUMUQUNUOW - RXGXDQAWQWRWDWIXDWIWGCHPZNWRWDRZXDWGCHUEXRXCXQXRXCWFVTIMZDHPZXQWCXCXTQCWF - GCKSZVSXCWBXTVOWFVNITYAWAXSDHVOWFVTITUGUHUIWGXSCDHVOVTWFIUJUKULUMUQUNURUS - UQAFESZFLZELZIMZNZYDYCIMZNZRZUTZEGOFGOWSWOXEUTZAYJFEGGJVAYJYKBESZVNYDIMZN - ZYDVNIMZNZRZUTFEVNWFGGFBSZYBYLYIYQFBEVBYRYFYNYHYPYRYEYMYCVNYDITVCYRYGYOYC - VNYDIUJVCVDVEEKSZYLWOYQXEEKBVFYSYNXBYPXDYSYMXAYDWFVNIUJVCYSYOXCYDWFVNITVC - VDVEVGVHVIVAWEWMBKGWOVRWIWDWLWOVQWHCHWOVPWGVNWFVOITVCVJWOWCWKCGWOVSWJWBVN - WFVOIUJVKVJVDVLVM $. + ancom rexbidv imbi12d rspcva breq2 cbvrexv syl6ibr con3d biimtrid expimpd + an42 ad2antrl ad2antll anim12d ralrimivva equequ1 anbi12d bibi12d equequ2 + wb notbid rspc2v mpan9 sylibrd ralbidv imbi1d rmo4 sylibr ) ABLZCLZIMZNZC + HOZVOVNIMZVODLZIMZDHPZQZCGOZRZKLZVOIMZNZCHOZVOWFIMZWBQZCGOZRZRZBKSZQZKGOB + GOWEBGUAAWPBKGGAVNGUBZWFGUBZRZRZWNVNWFIMZNZWFVNIMZNZRZWOWNWLVRRZWDWIRZRZW + TXEVRWIRZWLWDRZRXJWIVRRZRWNXHXIXKXJVRWIUFUCVRWDWIWLUPWLVRWDWIUPUDWTXFXBXG + XDWQXFXBQAWRWQWLVRXBVRVPCHPZNWQWLRZXBVPCHUEXMXAXLXMXAVNVTIMZDHPZXLWKXAXOQ + CVNGCBSZWJXAWBXOVOVNWFITXPWAXNDHVOVNVTITUGUHUIVPXNCDHVOVTVNIUJUKULUMUNUOU + QWRXGXDQAWQWRWDWIXDWIWGCHPZNWRWDRZXDWGCHUEXRXCXQXRXCWFVTIMZDHPZXQWCXCXTQC + WFGCKSZVSXCWBXTVOWFVNITYAWAXSDHVOWFVTITUGUHUIWGXSCDHVOVTWFIUJUKULUMUNUOUR + USUNAFESZFLZELZIMZNZYDYCIMZNZRZVEZEGOFGOWSWOXEVEZAYJFEGGJUTYJYKBESZVNYDIM + ZNZYDVNIMZNZRZVEFEVNWFGGFBSZYBYLYIYQFBEVAYRYFYNYHYPYRYEYMYCVNYDITVFYRYGYO + YCVNYDIUJVFVBVCEKSZYLWOYQXEEKBVDYSYNXBYPXDYSYMXAYDWFVNIUJVFYSYOXCYDWFVNIT + VFVBVCVGVHVIUTWEWMBKGWOVRWIWDWLWOVQWHCHWOVPWGVNWFVOITVFVJWOWCWKCGWOVSWJWB + VNWFVOIUJVKVJVBVLVM $. ${ supeuti.2 $e |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ @@ -68507,11 +68509,11 @@ all reals (greatest real number) for which all positive reals are greater. ( C R sup ( B , A , R ) <-> E. z e. B C R z ) ) $= ( vw wcel wa wbr cv ad3antrrr suplubti expdimp breq2 cbvrexv wn simplll csup wrex simplr supubti sylc wo simpr wor wi simpllr wss sseldd sowlin - supclti syl13anc mpd ecased ex rexlimdva syl5bi impbid ) AIGPZQZIHGJUGZ - JRZIDSZJRZDHUHZAVHVKVNABCDEFGHIJKLUAUBVNIOSZJRZOHUHVIVKVMVPDOHVLVOIJUCU - DVIVPVKOHVIVOHPZQZVPVKVRVPQZVKVJVOJRZVSAVQVTUEAVHVQVPUFVIVQVPUIZABCDEFG - HVOJKLUJUKVSVPVKVTULZVRVPUMVSGJUNZVHVOGPVJGPZVPWBUOAWCVHVQVPMTAVHVQVPUP - VSHGVOAHGUQVHVQVPNTWAURAWDVHVQVPABCDEFGHJKLUTTGIVOVJJUSVAVBVCVDVEVFVG + supclti syl13anc mpd ecased ex rexlimdva biimtrid impbid ) AIGPZQZIHGJU + GZJRZIDSZJRZDHUHZAVHVKVNABCDEFGHIJKLUAUBVNIOSZJRZOHUHVIVKVMVPDOHVLVOIJU + CUDVIVPVKOHVIVOHPZQZVPVKVRVPQZVKVJVOJRZVSAVQVTUEAVHVQVPUFVIVQVPUIZABCDE + FGHVOJKLUJUKVSVPVKVTULZVRVPUMVSGJUNZVHVOGPVJGPZVPWBUOAWCVHVQVPMTAVHVQVP + UPVSHGVOAHGUQVHVQVPNTWAURAWDVHVQVPABCDEFGHJKLUTTGIVOVJJUSVAVBVCVDVEVFVG $. $} $} @@ -68666,15 +68668,15 @@ all reals (greatest real number) for which all positive reals are greater. supisoti $p |- ( ph -> sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) ) $= ( vw vj wbr wral vk cima csup cfv cv wcel weq wn wa ralrimivva wiso isoti - wb syl mpbid r19.21bi anasss wf1o wf isof1o f1of 3syl supclti ffvelrnd wi - wrex supubti ralrimiv suplubti expd supisolem mpbi2and simpld simprd impr - mpdan eqsuptid ) AQUAEFHLIUBZIGJUCZLUDZKAFUEZHUFZEUEZHUFFEUGZWAWCKSUHWCWA - KSUHUIUMZAWBUIWEEHAWEEHTZFHAWDWAWCJSUHWCWAJSUHUIUMZEGTFGTZWFFHTZAWGFEGGPU - JAGHJKLUKZWHWIUMMEFGHJKLULUNUOUPUPUQAGHVSLAWJGHLURGHLUSMGHJKLUTGHLVAVBABC - DEFGIJPOVCZVDAVTQUEZKSUHZQVRAWMQVRTZWLVTKSZWLUAUEKSUAVRVFZVEZQHTZAVSRUEZJ - SUHZRITZWSVSJSZWSDUEJSDIVFZVEZRGTZWNWRUIZAWTRIABCDEFGIWSJPOVGVHAXDRGAWSGU - FXBXCABCDEFGIWSJPOVIVJVHAVSGUFXAXEUIXFUMWKARDQUAGHIVSJKLMNVKVPVLZVMUPAWLH - UFWOWPAWQQHAWNWRXGVNUPVOVQ $. + wb syl mpbid r19.21bi anasss wf1o isof1o f1of 3syl supclti ffvelcdmd wrex + wf wi supubti ralrimiv suplubti expd supisolem mpdan mpbi2and simpld impr + simprd eqsuptid ) AQUAEFHLIUBZIGJUCZLUDZKAFUEZHUFZEUEZHUFFEUGZWAWCKSUHWCW + AKSUHUIUMZAWBUIWEEHAWEEHTZFHAWDWAWCJSUHWCWAJSUHUIUMZEGTFGTZWFFHTZAWGFEGGP + UJAGHJKLUKZWHWIUMMEFGHJKLULUNUOUPUPUQAGHVSLAWJGHLURGHLVEMGHJKLUSGHLUTVAAB + CDEFGIJPOVBZVCAVTQUEZKSUHZQVRAWMQVRTZWLVTKSZWLUAUEKSUAVRVDZVFZQHTZAVSRUEZ + JSUHZRITZWSVSJSZWSDUEJSDIVDZVFZRGTZWNWRUIZAWTRIABCDEFGIWSJPOVGVHAXDRGAWSG + UFXBXCABCDEFGIWSJPOVIVJVHAVSGUFXAXEUIXFUMWKARDQUAGHIVSJKLMNVKVLVMZVNUPAWL + HUFWOWPAWQQHAWNWRXGVPUPVOVQ $. $} $( Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) $) @@ -68951,41 +68953,41 @@ all reals (greatest real number) for which all positive reals are greater. ( vx vy vz vw cep cv cfv wceq wcel wi weq fveq2 id eqeq12d wa wb syl wiso word w3a wral con0 wss ordsson sseld eleq1 imbi12d imbi2d r19.21v ordelss 3ad2ant2 3ad2antl2 sselda pm5.5 ralbidva isof1o 3ad2ant1 ad2antrr simpll3 - ccnv wf1o simpr wf f1of simplrl ffvelrnd ordtr1 f1ocnvfv2 syl2anc eqeltrd - jca sylc wbr simpll1 f1ocnv isorel syl12anc vex epelc a1i cvv wfn funfvex - 3syl f1ofn funfni sylan epelg 3bitr3d mpbird simplrr rspcv eqtr3d rspccva - simprr epel biimpri adantl simpl2 simprl mpbid eqeltrrd eqrdv expr sylbid - impbida ex com23 a2i syl5bi tfis2 com3l mpdd adantll ffvelcdmda 3ad2antl1 - ralrimiv adantlr wrex crn simpl1 f1ofo eleq2d adantr fvelrnb bitr3d simpl - wfo forn simplr exp43 syldd imp rexlimdv impbid mpdan ) ABHHCUAZAUBZBUBZU - CZDIZCJZUUDKZDAUDZABKUUCUUFDAUUCUUDALZUUDUELZUUFUUCAUEUUDUUAYTAUEUFUUBAUG - UNUHUUIUUCUUHUUFUUCUUHUUFMZMZUUCEIZALZUULCJZUULKZMZMZDEDENZUUJUUPUUCUURUU - HUUMUUFUUOUUDUULAUIUURUUEUUNUUDUULUUDUULCOUURPQUJUKUUQEUUDUDUUCUUPEUUDUDZ - MZUUIUUKUUCUUPEUUDULUUTUUKMUUIUUCUUSUUJUUCUUHUUSUUFUUCUUHUUSUUFMUUCUUHRZU - 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(Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, @@ -69439,15 +69441,15 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler ( cinl wfn wf cfv c2nd wceq adantl fveq2d c1st va cres crn wss updjudhf ccom cdju ffn syl wf1 inlresf1 f1fn mp1i f1f ax-mp fnco syl3anc cv wcel frn wa fvco2 sylan fvres c0 cif cmpt a1i eqeq1d ifbieq12d 1stinl adantr - fveq2 iftrued eqtrd djulcl 2ndinl simpr eqeltrd ffvelrnd fvmptd eqfnfvd - 3eqtrd ) AUACHLCUBZUFZFAHCDUGZMZWDCMZWDUCWFUDZWECMAWFEHNWGABCDEFGHIJKUE - WFEHUHUICWFWDUJZWHACDUKZCWFWDULUMZCWFWDNZWIAWJWMWKCWFWDUNUOCWFWDUTUMWFC - HWDUPUQACEFNZFCMICEFUHUIAUAURZCUSZVAZWOWEOZWOWDOZHOZWOLOZPOZFOZWOFOAWHW - PWRWTQWLCHWDWOVBVCWQWTXAHOXCWQWSXAHWPWSXAQAWOCLVDRSWQBXABURZTOZVEQZXDPO - ZFOZXGGOZVFZXCWFHEHBWFXJVGQWQKVHWQXDXAQZVAZXJXATOZVEQZXCXBGOZVFZXCXKXJX - PQWQXKXFXNXHXIXCXOXKXEXMVEXDXATVMVIXKXGXBFXDXAPVMZSXKXGXBGXQSVJRXLXNXCX - OWQXNXKWPXNACWOVKRVLVNVOWPXAWFUSACDWOVPRWQCEXBFAWNWPIVLWQXBWOCWPXBWOQAC - WOVQRZAWPVRVSVTWAVOWQXBWOFXRSWCWB $. + fveq2 iftrued eqtrd djulcl 2ndinl simpr eqeltrd ffvelcdmd fvmptd 3eqtrd + eqfnfvd ) AUACHLCUBZUFZFAHCDUGZMZWDCMZWDUCWFUDZWECMAWFEHNWGABCDEFGHIJKU + EWFEHUHUICWFWDUJZWHACDUKZCWFWDULUMZCWFWDNZWIAWJWMWKCWFWDUNUOCWFWDUTUMWF + CHWDUPUQACEFNZFCMICEFUHUIAUAURZCUSZVAZWOWEOZWOWDOZHOZWOLOZPOZFOZWOFOAWH + WPWRWTQWLCHWDWOVBVCWQWTXAHOXCWQWSXAHWPWSXAQAWOCLVDRSWQBXABURZTOZVEQZXDP + OZFOZXGGOZVFZXCWFHEHBWFXJVGQWQKVHWQXDXAQZVAZXJXATOZVEQZXCXBGOZVFZXCXKXJ + XPQWQXKXFXNXHXIXCXOXKXEXMVEXDXATVMVIXKXGXBFXDXAPVMZSXKXGXBGXQSVJRXLXNXC + XOWQXNXKWPXNACWOVKRVLVNVOWPXAWFUSACDWOVPRWQCEXBFAWNWPIVLWQXBWOCWPXBWOQA + CWOVQRZAWPVRVSVTWAVOWQXBWOFXRSWBWC $. $d B b x $. $d G b x $. $d H b $. $d ph b $. $( The composition of the mapping of an element of the disjoint union to @@ -69458,15 +69460,15 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler vb wss ffn syl wf1 inrresf1 f1fn mp1i f1f ax-mp fnco syl3anc cv wcel wa frn c2nd fvco2 sylan fvres c1st cif cmpt a1i eqeq1d ifbieq12d wn 1stinr fveq2 1n0 neii eqeq1 mtbiri adantr iffalsed eqtrd djurcl 2ndinr eqeltrd - simpr ffvelrnd fvmptd 3eqtrd eqfnfvd ) AUFDHLDUAZUBZGAHCDUCZMZWJDMZWJUD - WLUGZWKDMAWLEHNWMABCDEFGHIJKUEWLEHUHUIDWLWJUJZWNACDUKZDWLWJULUMZDWLWJNZ - WOAWPWSWQDWLWJUNUODWLWJVAUMWLDHWJUPUQADEGNZGDMJDEGUHUIAUFURZDUSZUTZXAWK - OZXAWJOZHOZXALOZVBOZGOZXAGOAWNXBXDXFPWRDHWJXAVCVDXCXFXGHOXIXCXEXGHXBXEX - GPAXADLVEQRXCBXGBURZVFOZSPZXJVBOZFOZXMGOZVGZXIWLHEHBWLXPVHPXCKVIXCXJXGP - ZUTZXPXGVFOZSPZXHFOZXIVGZXIXQXPYBPXCXQXLXTXNXOYAXIXQXKXSSXJXGVFVNVJXQXM - XHFXJXGVBVNZRXQXMXHGYCRVKQXRXTYAXIXCXTVLZXQXBYDAXBXSTPZYDDXAVMYEXTTSPTS - VOVPXSTSVQVRUIQVSVTWAXBXGWLUSACDXAWBQXCDEXHGAWTXBJVSXCXHXADXBXHXAPADXAW - CQZAXBWEWDWFWGWAXCXHXAGYFRWHWI $. + simpr ffvelcdmd fvmptd 3eqtrd eqfnfvd ) AUFDHLDUAZUBZGAHCDUCZMZWJDMZWJU + DWLUGZWKDMAWLEHNWMABCDEFGHIJKUEWLEHUHUIDWLWJUJZWNACDUKZDWLWJULUMZDWLWJN + ZWOAWPWSWQDWLWJUNUODWLWJVAUMWLDHWJUPUQADEGNZGDMJDEGUHUIAUFURZDUSZUTZXAW + KOZXAWJOZHOZXALOZVBOZGOZXAGOAWNXBXDXFPWRDHWJXAVCVDXCXFXGHOXIXCXEXGHXBXE + XGPAXADLVEQRXCBXGBURZVFOZSPZXJVBOZFOZXMGOZVGZXIWLHEHBWLXPVHPXCKVIXCXJXG + PZUTZXPXGVFOZSPZXHFOZXIVGZXIXQXPYBPXCXQXLXTXNXOYAXIXQXKXSSXJXGVFVNVJXQX + MXHFXJXGVBVNZRXQXMXHGYCRVKQXRXTYAXIXCXTVLZXQXBYDAXBXSTPZYDDXAVMYEXTTSPT + SVOVPXSTSVQVRUIQVSVTWAXBXGWLUSACDXAWBQXCDEXHGAWTXBJVSXCXHXADXBXHXAPADXA + WCQZAXBWEWDWFWGWAXCXHXAGYFRWHWI $. $} $d A h k x $. $d A k x y z $. $d B h k x $. $d B k x y z $. @@ -69622,16 +69624,16 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler cdjucase ccom fveq1i cin c0 fnfun syl csn cxp djulf1o f1ocnv f1ofun funco wf1o sylancl dmco wrel cop df-inl funmpt2 funrel dfrel2 mpbi a1i imaeq12d cv eqtrid df-fn sylanbrc c1o djurf1o imacnvcnv eqtri djuin wa elexd f1odm - fndm eleqtrrdi jctil funfvima sylc fvun1 syl112anc f1ofn wf f1of ffvelrnd - fvco2 sylancr f1ocnvfv1 fveq2d 3eqtrd ) ABIJZDEUDZJZWQDIKZUEZJZWQWTJZDJZB - DJAWSWQXAELKZUEZUBZJZXBWQWRXGDEUCUFAXAICMZNZXFLEOZMZNZXIXLUGUHPZWQXIQZXHX - BPAXARZXAOZXIPXJADRZWTRZXPADCNZXRFCDUIUJUHUKSULZSWTUQZXSSYAIUQZYBUMSYAIUN - TZYASWTUOTDWTUPURAXQWTKZDOZMXIDWTUSAYEIYFCYEIPZAIUTZYGIRZYHUASUHUAVIVAIUA - VBVCZIVDTIVEVFVGAXTYFCPFCDWAUJVHVJXAXIVKVLAXFRZXFOZXLPZXMAERXERZYKGVMUKSU - LZSXEUQZYNSYOLUQYPVNSYOLUNTYOSXEUOTEXEUPURYMAYLXEKXKMXLEXEUSLXKVOVPVGXFXL - VKVLXNACXKVQVGAYIBIOZQZVRBCQXOAYRYIABSYQABCHVSZYCYQSPUMSYAIVTTWBYJWCHCBIW - DWEXIXLXAXFWQWFWGVJAWTYANZWQYAQXBXDPYBYTYDYASWTWHTASYABISYAIWIZAYCUUAUMSY - AIWJTVGYSWKYADWTWQWLWMAXCBDAYCBSQXCBPUMYSSYABIWNWMWOWP $. + fndm eleqtrrdi jctil funfvima sylc fvun1 syl112anc f1of ffvelcdmd sylancr + f1ofn wf fvco2 f1ocnvfv1 fveq2d 3eqtrd ) ABIJZDEUDZJZWQDIKZUEZJZWQWTJZDJZ + BDJAWSWQXAELKZUEZUBZJZXBWQWRXGDEUCUFAXAICMZNZXFLEOZMZNZXIXLUGUHPZWQXIQZXH + XBPAXARZXAOZXIPXJADRZWTRZXPADCNZXRFCDUIUJUHUKSULZSWTUQZXSSYAIUQZYBUMSYAIU + NTZYASWTUOTDWTUPURAXQWTKZDOZMXIDWTUSAYEIYFCYEIPZAIUTZYGIRZYHUASUHUAVIVAIU + AVBVCZIVDTIVEVFVGAXTYFCPFCDWAUJVHVJXAXIVKVLAXFRZXFOZXLPZXMAERXERZYKGVMUKS + ULZSXEUQZYNSYOLUQYPVNSYOLUNTYOSXEUOTEXEUPURYMAYLXEKXKMXLEXEUSLXKVOVPVGXFX + LVKVLXNACXKVQVGAYIBIOZQZVRBCQXOAYRYIABSYQABCHVSZYCYQSPUMSYAIVTTWBYJWCHCBI + WDWEXIXLXAXFWQWFWGVJAWTYANZWQYAQXBXDPYBYTYDYASWTWKTASYABISYAIWLZAYCUUAUMS + YAIWHTVGYSWIYADWTWQWMWJAXCBDAYCBSQXCBPUMYSSYABIWNWJWOWP $. $} ${ @@ -69645,16 +69647,16 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler cdjucase ccom fveq1i cin csn cxp wf1o djulf1o f1ocnv f1ofun funco sylancl c0 dmco imacnvcnv eqtri a1i df-fn sylanbrc fnfun syl c1o djurf1o wrel cop cv df-inr funmpt2 funrel dfrel2 mpbi imaeq12d eqtrid djuin wa elexd f1odm - fndm eleqtrrdi jctil funfvima sylc fvun2 syl112anc f1ofn wf f1of ffvelrnd - fvco2 sylancr f1ocnvfv1 fveq2d 3eqtrd ) ABIJZDEUDZJZWQEIKZUEZJZWQWTJZEJZB - EJAWSWQDLKZUEZXAUBZJZXBWQWRXGDEUCUFAXFLDMZNZOZXAICNZOZXJXLUGUPPZWQXLQZXHX - BPAXFRZXFMZXJPZXKADRXERZXPFUPUHSUIZSXEUJZXSSXTLUJYAUKSXTLULTXTSXEUMTDXEUN - UOXRAXQXEKXINXJDXEUQLXIURUSUTXFXJVAVBAXARZXAMZXLPXMAERZWTRZYBAECOZYDGCEVC - VDVEUHSUIZSWTUJZYESYGIUJZYHVFSYGIULTZYGSWTUMTEWTUNUOAYCWTKZEMZNXLEWTUQAYK - IYLCYKIPZAIVGZYMIRZYNUASVEUAVIVHIUAVJVKZIVLTIVMVNUTAYFYLCPGCEWAVDVOVPXAXL - VAVBXNAXICVQUTAYOBIMZQZVRBCQXOAYRYOABSYQABCHVSZYIYQSPVFSYGIVTTWBYPWCHCBIW - DWEXJXLXFXAWQWFWGVPAWTYGOZWQYGQXBXDPYHYTYJYGSWTWHTASYGBISYGIWIZAYIUUAVFSY - GIWJTUTYSWKYGEWTWQWLWMAXCBEAYIBSQXCBPVFYSSYGBIWNWMWOWP $. + fndm eleqtrrdi jctil funfvima sylc fvun2 syl112anc f1of ffvelcdmd sylancr + f1ofn wf fvco2 f1ocnvfv1 fveq2d 3eqtrd ) ABIJZDEUDZJZWQEIKZUEZJZWQWTJZEJZ + BEJAWSWQDLKZUEZXAUBZJZXBWQWRXGDEUCUFAXFLDMZNZOZXAICNZOZXJXLUGUPPZWQXLQZXH + XBPAXFRZXFMZXJPZXKADRXERZXPFUPUHSUIZSXEUJZXSSXTLUJYAUKSXTLULTXTSXEUMTDXEU + NUOXRAXQXEKXINXJDXEUQLXIURUSUTXFXJVAVBAXARZXAMZXLPXMAERZWTRZYBAECOZYDGCEV + CVDVEUHSUIZSWTUJZYESYGIUJZYHVFSYGIULTZYGSWTUMTEWTUNUOAYCWTKZEMZNXLEWTUQAY + KIYLCYKIPZAIVGZYMIRZYNUASVEUAVIVHIUAVJVKZIVLTIVMVNUTAYFYLCPGCEWAVDVOVPXAX + LVAVBXNAXICVQUTAYOBIMZQZVRBCQXOAYRYOABSYQABCHVSZYIYQSPVFSYGIVTTWBYPWCHCBI + WDWEXJXLXFXAWQWFWGVPAWTYGOZWQYGQXBXDPYHYTYJYGSWTWKTASYGBISYGIWLZAYIUUAVFS + YGIWHTUTYSWIYGEWTWQWMWJAXCBEAYIBSQXCBPVFYSSYGBIWNWJWOWP $. $} @@ -69790,34 +69792,34 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler Kingdon, 9-Aug-2023.) $) difinfsnlem $p |- ( ph -> G : _om -1-1-> ( A \ { B } ) ) $= ( wceq wcel com wa c1o ad2antrr adantr vm cinl cfv cinr cif csn cdif wral - cv c0 wi wf1 cdju wf f1f syl 0lt1o djurcl mp1i ffvelrnd wn elsni necon3ai - wne eldifd djulcl adantl con3i wdc eqeq12 dcbid rspc2gv syl2anc ralrimiva - mpd ifcldadc simplr eqtr4d ad3antrrr ad2antrl simprr f1veqaeq syl12anc wb - simpr inl11 ad3antlr mpbid a1d djune necomd sylancl mtod iftrued iffalsed - neneqd eqeq12d mtbird pm2.21d wo mpjaodan simprl sylbid sylibd ralrimivva - exmiddc adantrr 2fveq3 eqeq1d ifbieq2d f1mpt sylanbrc ) AFUIZUBUCZGUCZENZ - UJUDUCZGUCZXOUEZDEUFZUGZOZFPUHXSUAUIZUBUCZGUCZENZXRYEUEZNZXMYCNZUKZUAPUHF - PUHPYAHULAYBFPAXMPOZQZXPXRXOYAAXRYAOYKXPAXRDXTAPRUMZDXQGAYMDGULZYMDGUNZKY - MDGUOUPZUJROZXQYMOZAUQPRUJURZUSUTAXREVDXRXTOZVALYTXREXREVBVCUPVESYLXPVAZQ - XODXTYLXODOZUUAYLYMDXNGAYOYKYPTYKXNYMOZAPRXMVFZVGUTZTUUAXOXTOZVAYLUUFXPXO - EVBVHVGVEYLBUIZCUIZNZVIZCDUHBDUHZXPVIZAUUKYKITYLUUBEDOZUUKUULUKUUEAUUMYKJ - TUUJUULBCXOEDDUUGXONUUHENZQUUIXPUUGXOUUHEVJVKVLVMVOZVPVNAYJFUAPPAYKYCPOZQ - ZQZXPYJUUAUURXPQZYFYJYFVAZUUSYFQZYIYHUVAXNYDNZYIUVAXOYENZUVBUVAXOEYEUURXP - YFVQUUSYFWEVRUVAYNUUCYDYMOZUVCUVBUKZAYNUUQXPYFKVSUURUUCXPYFYKUUCAUUPUUDVT - 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simpl3 neqned eqid difinfsnlem mptex spcev wo ffvelrnd jca eqeq12 rspc2gv - dcbid exmiddc mpjaodan exlimddv reldom brrelex2i difexg 3ad2ant2 brdomg + simpl3 neqned eqid difinfsnlem mptex spcev wo ffvelcdmd jca dcbid rspc2gv + eqeq12 exmiddc mpjaodan exlimddv reldom brrelex2i difexg 3ad2ant2 brdomg sylc ) AJZBJZKZUCZBCUAACUAZLCUBUDZDCMZUEZLCDUFZUGZUBUDZLUUHEJZNZEUHZUUFLO UIZCFJZNZUULFUUFUUMCUBUDZUUOFUHUUFUUMLUJUDUUDUUPUKUUCUUDUUEULUUMLCUMUNUUM CFUOPUUFUUOQZUPUQRZUUNRZDKZUULUUTURZUUQUUTQZLUUHUUNUSLUTZVAZNZUULUVBLCUVD @@ -69854,7 +69856,7 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler QZSUUQUUEUVAUUCUUDUUEUUOXRZSUUQUUOUVAUUFUUOVMZSUWSUUSDUUQUVAVMXSUXBXTYAUU KUXCEUXBILUXAXJYBLUUHUUJUXBXLYCPUUQUUTUCZUUTUVAYDUUQUUSCMZUUEQUUCUXGUUQUX HUUEUUQUUMCUURUUNUUQUUOUUMCUUNVHUXFUUMCUUNVIPUWOUUQUWQWDYEUXEYFUXDUUBUXGA - BUUSDCCYSUUSKYTDKQUUAUUTYSUUSYTDYGYIYHYRUUTYJPYKYLUUFUUHTMZUUIUULWPUUDUUC + BUUSDCCYSUUSKYTDKQUUAUUTYSUUSYTDYIYGYHYRUUTYJPYKYLUUFUUHTMZUUIUULWPUUDUUC UXIUUEUUDCTMUXILCUBYMYNCUUGTYOPYPLUUHTEYQPXA $. $} @@ -69977,30 +69979,30 @@ property of disjoint unions (see ~ updjud in the case of functions). -> E. f f : _om -onto-> ( A |_| 1o ) ) ) $= ( vg vn vy vz vw vu cv wcel com c1o wa c0 wceq cfv cinl fveq2d eqtr4d wex wfo cdju cinr cuni cif cmpt wf wrex 0lt1o djurcl ax-mp a1i wn simpllr fof - wral syl nnpredcl ad2antlr ffvelrnd djulcl nndceq0 adantl ifcldadc fmpttd - wdc simprl foelrn syl2anc csuc ad2antrl simplrr wtr word nnord ordtr 3syl - peano2 wb unisucg mpbid simprr peano3 neneqd iffalsed eqid eqeq1 ifbieq2d - unieq eqeltrrd fvmptd3 fveq2 rexlimddv rexlimdvaa peano1 el1o sylib eqtrd - rspceeqv iftruei eqtr4di eqeltri fvmptg sylancr wo biimpi ralrimiva dffo3 - djur mpjaod sylanbrc omex mptex foeq1 spcev ex exlimdv cbvexv syl6ibr ) A - JBKAUAZLBCJZUBZCUALBMUCZDJZUBZDUAZLYDYBUBZCUAYAYCYGCYAYCYGYAYCNZLYDELEJZO - PZOUDQZYJUEZYBQZRQZUFZUGZUBZYGYILYDYQUHFJZGJZYQQZPGLUIZFYDUQYRYIELYPYDYIY - JLKZNZYKYLYOYDYLYDKZUUDYKNOMKUUEUJBMOUKULZUMUUDYKUNZNZYNBKYOYDKUUHLBYMYBU - UHYCLBYBUHYAYCUUCUUGUOLBYBUPURUUCYMLKYIUUGYJUSUTVABMYNVBURUUCYKVGYIYJVCVD - VEVFYIUUBFYDYIYSYDKZNZYSHJZRQZPZHBUIZUUBYSUUKUDQZPZHMUIZUUJUUMUUBHBUUJUUK - BKZUUMNZNZUUKIJZYBQZPZUUBILUUTYCUURUVCILUIYAYCUUIUUSUOUUJUURUUMVHILBUUKYB - VIVJUUTUVALKZUVCNZNZUVAVKZLKZYSUVGYQQZPUUBUVDUVHUUTUVCUVAVSVLZUVFYSUVGOPZ - YLUVGUEZYBQZRQZUFZUVIUVFYSUVNUVOUVFYSUULUVNUUJUURUUMUVEVMUVFUVMUUKRUVFUVM - 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Kingdon, 16-Aug-2023.) $) ctssdclemn0 $p |- ( ph -> E. g g : _om -onto-> ( A |_| 1o ) ) $= ( vm vy com c1o wcel cfv cinl c0 wceq wa vx vz cdju cinr cif cmpt wfo wex - cv wf wrex wral ad2antrr fof syl simpr ffvelrnd djulcl 0lt1o djurcl ax-mp - wn a1i eleq1 dcbid adantr rspcdva ifcldadc fmpttd ad3antrrr simplr foelrn - wdc syl2anc iftrued 2fveq3 ifbieq1d ad5antr sseldd eqeltrd fvmptd3 fveq2d - eqid wss simpllr eqtrd 3eqtr4rd ex reximdva wi ssrexv rexlimdva2 iffalsed - peano1 eqeltrdi el1o sylib fveq2 rspceeqv wo djur biimpi adantl ralrimiva - mpd mpjaod dffo3 sylanbrc omex mptex foeq1 spcev ) AMBNUCZKMKUIZCOZXNFPZQ - PZRUDPZUEZUFZUGZMXMDUIZUGZDUHAMXMXTUJUAUIZLUIZXTPZSZLMUKZUAXMULYAAKMXSXMA - XNMOZTZXOXQXRXMYJXOTZXPBOXQXMOYKCBXNFYKCBFUGZCBFUJZAYLYIXOIUMCBFUNZUOYJXO - UPUQBNXPURUOXRXMOZYJXOVBTRNOYOUSBNRUTVAZVCYJEUIZCOZVMZXOVMEMXNYQXNSYRXOYQ - XNCVDVEAYSEMULYIHVFAYIUPVGVHVIAYHUAXMAYDXMOZTZYDUBUIZQPZSZUBBUKZYHYDUUBUD - PZSZUBNUKZUUAUUDYHUBBUUAUUBBOZTZUUDTZYGLCUKZYHUUKUUBYEFPZSZLCUKZUULUUKYLU - 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vy wss w3a c1o cdju wn cfv cif cmpt wfn crn wf simp3 fof ffvelrnd simpllr - elequ1 dcbid simpll2 rspcdva ifcldadc fmpttd ffnd wrex wb ad2antrr foelrn - fvelrnb sylancom simpll1 fveq2 ifbieq1d sselda ad4antr ffvelcdmda fvmptd3 - eqid simp2 iftrued eqtrd adantr eqtr4d ex reximdva ssrexv sylsyld eqeltrd - mpd eqeltrrd rexlimdva2 impbid bitr4d eqrdv df-fo mptex foeq1 spcev elex2 - sylanbrc omex ctm mpbird simpl1 ctssdclemn0 eleq1 peano1 exmiddc mpjaodan - a1i 3expia exlimdv 3impia 3com23 exlimiv cbvexv sylib ctssdclemr impbii - wo ) DIZJUCZYAABIZKZBLZCIZYAMZNZCJUAZUDZDLZJAUEUFZYCKZBLZYKJYLEIZKZELZYNY - JYQDYBYIYEYQYBYIYEYQYBYIOYDYQBYBYIYDYQYBYIYDUDZPYAMZYQYSUGZYRYSOZYQJAYOKZ - ELZUUAJAFJFIZYAMZUUDYCUHZPYCUHZUIZUJZKZUUCUUAUUIJUKZUUIULZAQUUJUUAJAUUIUU - AFJUUHAUUAUUDJMZOZUUEUUFUUGAUUNUUEOYAAUUDYCYRYAAYCUMZYSUUMUUEYRYDUUOYBYIY - DUNZYAAYCUORZSUUNUUETUPUUNUUEUGZOYAAPYCYRUUOYSUUMUURUUQSYRYSUUMUURUQUPUUN - YHUUENCJUUDYFUUDQYGUUECFDURUSYBYIYDYSUUMUTUUAUUMTVAVBVCVDZUUAUBUULAUUAUBI - ZUULMZGIZUUIUHZUUTQZGJVEZUUTAMZUUAUUKUVAUVEVFUUSGJUUTUUIVIRUUAUVFUVEUUAUV - 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(Contributed by Jim Kingdon, 13-Mar-2023.) $) enumctlemm $p |- ( ph -> G : _om -onto-> A ) $= ( vy vx com cv cfv wceq wrex wcel wa adantr wf wral wfo c0 cif ffvelcdmda - fof syl adantlr ffvelrnd ad2antrr wdc simpr nndcel syl2anc ifcldadc fmptd - wn foelrn sylan eleq1w fveq2 ifbieq1d elnn iftrued eqeltrd fvmptd3 eqeq2d - wb eqtrd rexbidva mpbird wi wss omelon onelssi ssrexv mpd ralrimiva dffo3 - 3syl sylanbrc ) AMBEUAKNZLNZEOZPZLMQZKBUBMBEUCACMCNZFRZWHDOZUDDOZUEZBEAWH - MRZSZWIWJWKBAWIWJBRWMAFBWHDAFBDUCZFBDUAGFBDUGUHZUFUIAWKBRWMWIURAFBUDDWPIU - JUKWNWMFMRZWIULAWMUMAWQWMHTWHFUNUOUPJUQAWGKBAWCBRZSZWFLFQZWGWSWTWCWDDOZPZ - LFQZAWOWRXCGLFBWCDUSUTAWTXCVIWRAWFXBLFAWDFRZSZWEXAWCXEWEXDXAWKUEZXAXECWDW - LXFMEBJWHWDPWIXDWJXAWKCLFVAWHWDDVBVCXEXDWQWDMRAXDUMZAWQXDHTWDFVDUOXEXFXAB - XEXDXAWKXGVEZAFBWDDWPUFVFVGXHVJVHVKTVLAWTWGVMZWRAWQFMVNXIHMFVOVPWFLFMVQWA - TVRVSLKMBEVTWB $. + fof syl adantlr wn ffvelcdmd ad2antrr simpr nndcel syl2anc ifcldadc fmptd + wdc foelrn sylan eleq1w fveq2 ifbieq1d elnn iftrued eqeltrd fvmptd3 eqtrd + wb eqeq2d rexbidva mpbird wi wss omelon onelssi ssrexv 3syl mpd ralrimiva + dffo3 sylanbrc ) AMBEUAKNZLNZEOZPZLMQZKBUBMBEUCACMCNZFRZWHDOZUDDOZUEZBEAW + HMRZSZWIWJWKBAWIWJBRWMAFBWHDAFBDUCZFBDUAGFBDUGUHZUFUIAWKBRWMWIUJAFBUDDWPI + UKULWNWMFMRZWIURAWMUMAWQWMHTWHFUNUOUPJUQAWGKBAWCBRZSZWFLFQZWGWSWTWCWDDOZP + ZLFQZAWOWRXCGLFBWCDUSUTAWTXCVIWRAWFXBLFAWDFRZSZWEXAWCXEWEXDXAWKUEZXAXECWD + WLXFMEBJWHWDPWIXDWJXAWKCLFVAWHWDDVBVCXEXDWQWDMRAXDUMZAWQXDHTWDFVDUOXEXFXA + BXEXDXAWKXGVEZAFBWDDWPUFVFVGXHVHVJVKTVLAWTWGVMZWRAWQFMVNXIHMFVOVPWFLFMVQV + RTVSVTLKMBEWAWB $. $} ${ @@ -70211,14 +70213,14 @@ property of disjoint unions (see ~ updjud in the case of functions). ( vk vx cv wfo wex com c1o cdju wcel wa c0 wceq simpll wb adantl syl fo00 wi foeq2 mpbid sylib 0ct djueq1 foeq3 exbidv mpbiri simpl2im cfv cif cmpt cvv omex mptex simplr simpr enumctlemm foeq1 spcegv mpsyl wf fof ad2antrr - eqid ffvelrnd eleq1 sylc ctm mpbird wo 0elnn mpjaodan ex impcom rexlimiva - exlimiv ) DGZABGZHZBIZJAKLZCGZHZCIZDJWCVTJMZWGWBWHWGUBBWBWHWGWBWHNZVTOPZW - GOVTMZWIWJNZWAOPZAOPZWGWLOAWAHZWMWNNWLWBWOWBWHWJQWJWBWORWIVTOAWAUCSUDAWAU - AUEWNWGJOKLZWEHZCICUFWNWFWQCWNWDWPPWFWQRAOKUGWDWPJWEUHTUIUJUKWIWKNZWGJAWE - HZCIZEJEGZVTMXAWAULOWAULZUMZUNZUOMWRJAXDHZWTEJXCUPUQWRAEWAXDVTWBWHWKQWBWH - WKURWIWKUSZXDVGUTWSXECXDUOJAWEXDVAVBVCWRFGZAMZFIZWGWTRWRXBAMZXJXIWRVTAOWA - WBVTAWAVDWHWKVTAWAVEVFXFVHZXKXHXJFXBAXGXBAVIVBVJFACVKTVLWHWJWKVMWBVTVNSVO - VPVSVQVR $. + eqid ffvelcdmd eleq1 sylc ctm mpbird wo mpjaodan exlimiv impcom rexlimiva + 0elnn ex ) DGZABGZHZBIZJAKLZCGZHZCIZDJWCVTJMZWGWBWHWGUBBWBWHWGWBWHNZVTOPZ + WGOVTMZWIWJNZWAOPZAOPZWGWLOAWAHZWMWNNWLWBWOWBWHWJQWJWBWORWIVTOAWAUCSUDAWA + UAUEWNWGJOKLZWEHZCICUFWNWFWQCWNWDWPPWFWQRAOKUGWDWPJWEUHTUIUJUKWIWKNZWGJAW + EHZCIZEJEGZVTMXAWAULOWAULZUMZUNZUOMWRJAXDHZWTEJXCUPUQWRAEWAXDVTWBWHWKQWBW + HWKURWIWKUSZXDVGUTWSXECXDUOJAWEXDVAVBVCWRFGZAMZFIZWGWTRWRXBAMZXJXIWRVTAOW + AWBVTAWAVDWHWKVTAWAVEVFXFVHZXKXHXJFXBAXGXBAVIVBVJFACVKTVLWHWJWKVMWBVTVRSV + NVSVOVPVQ $. $} ${ @@ -70245,8 +70247,8 @@ property of disjoint unions (see ~ updjud in the case of functions). $( A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.) $) ctfoex $p |- ( E. f f : _om -onto-> ( A |_| 1o ) -> A e. _V ) $= - ( com c1o cdju cv wfo cvv wcel wa omex fornex ax-mp djuexb sylibr exlimiv - wi simpld ) CADEZBFZGZAHIZBUAUBDHIZUASHIZUBUCJCHIUAUDQKCSHTLMADNORP $. + ( com c1o cdju cv wfo cvv wcel wa omex focdmex ax-mp djuexb sylibr simpld + wi exlimiv ) CADEZBFZGZAHIZBUAUBDHIZUASHIZUBUCJCHIUAUDQKCSHTLMADNOPR $. $} @@ -70433,30 +70435,30 @@ property of disjoint unions (see ~ updjud in the case of functions). 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N , 1o , (/) ) ) = X ) $= ( com wcel wa cfv c0 wceq c1o cif wdc wn simpllr simpr adantr wo mpjaodan c2o syl xnninf cmpt simplr simplll nninfisollem0 cuni simp-4r simp-4l wne - cv neqned nninfisollemne nninfisollemeq wf nninff nnpredcl ffvelrnd df2o3 - cpr adantl eleqtrdi elpri ad2antrr nndceq0 exmiddc neii fveq1d eqid eleq1 - 1n0 ifbid id word nnord ordirr iffalsed peano1 eqeltrdi fvmptd3 ad3antrrr - eqtrd 3eqtr3rd ex mtoi olcd df-dc sylibr simpl ) BDEZCUAEZFZBCGZHIZADAUJZ - BEZJHKZUBZCIZLZWLJIZWKWMFZBHIZWSXBMZXAXBFABCWIWJWMXBNWKWMXBUCWIWJWMXBUDXA - XBOUEXAXCFZBUFZCGZHIZWSXFJIZXDXGFABCWIWJWMXCXGUGWKWMXCXGNWIWJWMXCXGUHXDBH - UIZXGXDBHXAXCOUKZPXDXGOULXDXHFABCWIWJWMXCXHUGWKWMXCXHNWIWJWMXCXHUHXDXIXHX - JPXDXHOUMWKXGXHQZWMXCWKXFHJUSZEXKWKXFSXLWKDSXECWJDSCUNWICUOUTZWIXEDEWJBUP - PUQURVAXFHJVBTVCRWIXBXCQZWJWMWIXBLXNBVDXBVETVCRWKWTFZWRWRMZQWSXOXPWRXOWRJ - HIZJHVJVFXOWRXQXOWRFZBWQGZWLHJXRBWQCXOWROVGWIXSHIWJWTWRWIXSBBEZJHKZHWIABW - PYADWQDWQVHWNBIWOXTJHWNBBVIVKWIVLWIYAHDWIXTJHWIBVMXTMBVNBVOTVPZVQVRVSYBWA - VTWKWTWRUCWBWCWDWEWRWFWGWKWLXLEWMWTQWKWLSXLWKDSBCXMWIWJWHUQURVAWLHJVBTR + neqned nninfisollemne nninfisollemeq cpr nninff adantl nnpredcl ffvelcdmd + cv wf df2o3 eleqtrdi elpri ad2antrr nndceq0 exmiddc 1n0 neii fveq1d eleq1 + eqid ifbid id word nnord ordirr iffalsed eqeltrdi fvmptd3 eqtrd ad3antrrr + peano1 3eqtr3rd ex mtoi olcd df-dc sylibr simpl ) BDEZCUAEZFZBCGZHIZADAUR + ZBEZJHKZUBZCIZLZWLJIZWKWMFZBHIZWSXBMZXAXBFABCWIWJWMXBNWKWMXBUCWIWJWMXBUDX + AXBOUEXAXCFZBUFZCGZHIZWSXFJIZXDXGFABCWIWJWMXCXGUGWKWMXCXGNWIWJWMXCXGUHXDB + HUIZXGXDBHXAXCOUJZPXDXGOUKXDXHFABCWIWJWMXCXHUGWKWMXCXHNWIWJWMXCXHUHXDXIXH + XJPXDXHOULWKXGXHQZWMXCWKXFHJUMZEXKWKXFSXLWKDSXECWJDSCUSWICUNUOZWIXEDEWJBU + PPUQUTVAXFHJVBTVCRWIXBXCQZWJWMWIXBLXNBVDXBVETVCRWKWTFZWRWRMZQWSXOXPWRXOWR + JHIZJHVFVGXOWRXQXOWRFZBWQGZWLHJXRBWQCXOWROVHWIXSHIWJWTWRWIXSBBEZJHKZHWIAB + WPYADWQDWQVJWNBIWOXTJHWNBBVIVKWIVLWIYAHDWIXTJHWIBVMXTMBVNBVOTVPZWAVQVRYBV + SVTWKWTWRUCWBWCWDWEWRWFWGWKWLXLEWMWTQWKWLSXLWKDSBCXMWIWJWHUQUTVAWLHJVBTR $. $} @@ -70596,21 +70598,21 @@ elements or fails to hold (is equal to ` (/) ` ) for some element. wex fveq1 isomnimap ibi ad3antlr simpr wb com 2onn relen brrelex2i elmapg cvv sylancr mpbid adantr f1of adantl syl2anc simpllr mpbird rspcdva f1ofn fco wfn ad2antlr sylancom ffvelcdmda fveq2d rspcedv sylbid rexlimdva ccnv - fvco2 f1ocnv syl ffvelrnd simplr f1ocnvfv2 3eqtr3rd ralrimiva orim12d mpd - fveq2 ex exlimddv ) ABUAUBZAHIZBHIZWTXAJZXBCKZDKZLZMNZCBOZXFPNZCBQZUCZDRB - UDUEZQZXCXKDXLXCXEXLIZJZABEKZUFZXKEWTXQEUNZXAXNWTXRABEUGUHUIXOXQJZFKZXEXP - UJZLZMNZFAOZYBPNZFAQZUCZXKXSXTGKZLZMNZFAOZYIPNZFAQZUCZYGGRAUDUEZYAYHYANZY - KYDYMYFYPYJYCFAYPYIYBMXTYHYAUOZSUKYPYLYEFAYPYIYBPYQSULUMXAYNGYOQZWTXNXQXA - YRFAGHUPUQURXSYAYOIZARYATZXSBRXETZABXPTZYTXOUUAXQXOXNUUAXCXNUSWTXNUUAUTZX - AXNWTRVAIZBVFIZUUCVBABUAVCVDZRBXEVAVFVEVGUIVHVIXQUUBXOABXPVJVKZABRXEXPVQV - LXSUUDXAYSYTUTVBWTXAXNXQVMRAYAVAHVEVGVNVOXSYDXHYFXJXSYCXHFAXSXTAIZJZYCXTX - PLZXELZMNZXHUUIYBUUKMXSUUHXPAVRZYBUUKNXQUUMXOUUHABXPVPZVSAXEXPXTWGVTSUUIX - GUULCUUJBXSABXTXPUUGWAUUIXDUUJNZJZXFUUKMUUPXDUUJXEUUIUUOUSWBSWCWDWEXSYFXJ - XSYFJZXICBUUQXDBIZJZXDXPWFZLZYALZUVAXPLZXELZPXFUUSUUMUVAAIUVBUVDNXQUUMXOY - FUURUUNURUUSBAXDUUTXQBAUUTTZXOYFUURXQBAUUTUFUVEABXPWHBAUUTVJWIURUUQUURUSW - JZAXEXPUVAWGVLUUSYEUVBPNFAUVAXTUVANYBUVBPXTUVAYAWQSXSYFUURWKUVFVOUUQUURXQ - UVDXFNXOXQYFUURVMXQUURJUVCXDXEABXDXPWLWBVTWMWNWRWOWPWSWNWTXBXMUTZXAWTUUEU - VGUUFCBDVFUPWIVIVNWR $. + fvco2 f1ocnv syl ffvelcdmd simplr f1ocnvfv2 3eqtr3rd ralrimiva ex orim12d + fveq2 mpd exlimddv ) ABUAUBZAHIZBHIZWTXAJZXBCKZDKZLZMNZCBOZXFPNZCBQZUCZDR + BUDUEZQZXCXKDXLXCXEXLIZJZABEKZUFZXKEWTXQEUNZXAXNWTXRABEUGUHUIXOXQJZFKZXEX + PUJZLZMNZFAOZYBPNZFAQZUCZXKXSXTGKZLZMNZFAOZYIPNZFAQZUCZYGGRAUDUEZYAYHYANZ + YKYDYMYFYPYJYCFAYPYIYBMXTYHYAUOZSUKYPYLYEFAYPYIYBPYQSULUMXAYNGYOQZWTXNXQX + AYRFAGHUPUQURXSYAYOIZARYATZXSBRXETZABXPTZYTXOUUAXQXOXNUUAXCXNUSWTXNUUAUTZ + XAXNWTRVAIZBVFIZUUCVBABUAVCVDZRBXEVAVFVEVGUIVHVIXQUUBXOABXPVJVKZABRXEXPVQ + VLXSUUDXAYSYTUTVBWTXAXNXQVMRAYAVAHVEVGVNVOXSYDXHYFXJXSYCXHFAXSXTAIZJZYCXT + XPLZXELZMNZXHUUIYBUUKMXSUUHXPAVRZYBUUKNXQUUMXOUUHABXPVPZVSAXEXPXTWGVTSUUI + XGUULCUUJBXSABXTXPUUGWAUUIXDUUJNZJZXFUUKMUUPXDUUJXEUUIUUOUSWBSWCWDWEXSYFX + JXSYFJZXICBUUQXDBIZJZXDXPWFZLZYALZUVAXPLZXELZPXFUUSUUMUVAAIUVBUVDNXQUUMXO + YFUURUUNURUUSBAXDUUTXQBAUUTTZXOYFUURXQBAUUTUFUVEABXPWHBAUUTVJWIURUUQUURUS + WJZAXEXPUVAWGVLUUSYEUVBPNFAUVAXTUVANYBUVBPXTUVAYAWQSXSYFUURWKUVFVOUUQUURX + QUVDXFNXOXQYFUURVMXQUURJUVCXDXEABXDXPWLWBVTWMWNWOWPWRWSWNWTXBXMUTZXAWTUUE + UVGUUFCBDVFUPWIVIVNWO $. $} $( Omniscience is invariant with respect to equinumerosity. 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Markov -> B e. Markov ) ) $= ( vy vg vh vx vf cmarkov wcel wa cv cfv c1o wceq wral wn c0 wrex c2o wf cen wbr wi cmap co wf1o wex bren biimpi ad2antrr ccom ccnv f1ofn ad3antlr - wfn f1ocnv f1of syl simpr ffvelrnd syl2anc fveqeq2 simplr rspcdva simpllr - fvco2 f1ocnvfv2 sylancom fveq2d 3eqtr3rd ralrimiva ex con3d fveq1 ralbidv - eqeq1d notbid rexbidv imbi12d ismkvmap ibi com cvv relen brrelex2i elmapg - 2onn sylancr mpbid adantr adantl fco ad2antlr ffvelcdmda fveqeq2d rspcedv - wb mpbird sylbid rexlimdva 3syld exlimddv ) ABUAUBZAHIZBHIZXCXDJZXECKZDKZ - LZMNZCBOZPZXIQNZCBRZUCZDSBUDUEZOZXFXODXPXFXHXPIZJZABEKZUFZXOEXCYAEUGZXDXR - XCYBABEUHUIUJXSYAJZXLFKZXHXTUKZLZMNZFAOZPZYFQNZFARZXNYCYHXKYCYHXKYCYHJZXJ - CBYLXGBIZJZXGXTULZLZYELZYPXTLZXHLZMXIYNXTAUOZYPAIYQYSNYAYTXSYHYMABXTUMZUN - YNBAXGYOYNBAYOUFZBAYOTYAUUBXSYHYMABXTUPUNBAYOUQURYLYMUSUTZAXHXTYPVFVAYNYG - YQMNFAYPYDYPMYEVBYCYHYMVCUUCVDYNYRXGXHYLYMYAYRXGNXSYAYHYMVEABXGXTVGVHVIVJ - VKVLVMYCYDGKZLZMNZFAOZPZUUEQNZFARZUCZYIYKUCGSAUDUEZYEUUDYENZUUHYIUUJYKUUM - UUGYHUUMUUFYGFAUUMUUEYFMYDUUDYEVNZVPVOVQUUMUUIYJFAUUMUUEYFQUUNVPVRVSXDUUK - GUULOZXCXRYAXDUUOFAGHVTWAUNYCYEUULIZASYETZYCBSXHTZABXTTZUUQXSUURYAXSXRUUR - XFXRUSXCXRUURWQZXDXRXCSWBIZBWCIZUUTWGABUAWDWEZSBXHWBWCWFWHUJWIWJYAUUSXSAB - XTUQWKZABSXHXTWLVAYCUVAXDUUPUUQWQWGXCXDXRYAVESAYEWBHWFWHWRVDYCYJXNFAYCYDA - IZJZYJYDXTLZXHLZQNZXNUVFYFUVHQYCUVEYTYFUVHNYAYTXSUVEUUAWMAXHXTYDVFVHVPUVF - XMUVICUVGBYCABYDXTUVDWNUVFXGUVGNZJXGUVGQXHUVFUVJUSWOWPWSWTXAXBVKXCXEXQWQZ - XDXCUVBUVKUVCCBDWCVTURWJWRVL $. + wfn f1ocnv f1of syl simpr ffvelcdmd fvco2 syl2anc fveqeq2 rspcdva simpllr + simplr f1ocnvfv2 sylancom fveq2d 3eqtr3rd ralrimiva ex con3d fveq1 eqeq1d + ralbidv notbid rexbidv imbi12d ismkvmap ibi com cvv 2onn brrelex2i elmapg + relen sylancr mpbid adantr adantl fco mpbird ad2antlr ffvelcdmda fveqeq2d + wb rspcedv sylbid rexlimdva 3syld exlimddv ) ABUAUBZAHIZBHIZXCXDJZXECKZDK + ZLZMNZCBOZPZXIQNZCBRZUCZDSBUDUEZOZXFXODXPXFXHXPIZJZABEKZUFZXOEXCYAEUGZXDX + RXCYBABEUHUIUJXSYAJZXLFKZXHXTUKZLZMNZFAOZPZYFQNZFARZXNYCYHXKYCYHXKYCYHJZX + JCBYLXGBIZJZXGXTULZLZYELZYPXTLZXHLZMXIYNXTAUOZYPAIYQYSNYAYTXSYHYMABXTUMZU + NYNBAXGYOYNBAYOUFZBAYOTYAUUBXSYHYMABXTUPUNBAYOUQURYLYMUSUTZAXHXTYPVAVBYNY + GYQMNFAYPYDYPMYEVCYCYHYMVFUUCVDYNYRXGXHYLYMYAYRXGNXSYAYHYMVEABXGXTVGVHVIV + JVKVLVMYCYDGKZLZMNZFAOZPZUUEQNZFARZUCZYIYKUCGSAUDUEZYEUUDYENZUUHYIUUJYKUU + MUUGYHUUMUUFYGFAUUMUUEYFMYDUUDYEVNZVOVPVQUUMUUIYJFAUUMUUEYFQUUNVOVRVSXDUU + KGUULOZXCXRYAXDUUOFAGHVTWAUNYCYEUULIZASYETZYCBSXHTZABXTTZUUQXSUURYAXSXRUU + RXFXRUSXCXRUURWQZXDXRXCSWBIZBWCIZUUTWDABUAWGWEZSBXHWBWCWFWHUJWIWJYAUUSXSA + BXTUQWKZABSXHXTWLVBYCUVAXDUUPUUQWQWDXCXDXRYAVESAYEWBHWFWHWMVDYCYJXNFAYCYD + AIZJZYJYDXTLZXHLZQNZXNUVFYFUVHQYCUVEYTYFUVHNYAYTXSUVEUUAWNAXHXTYDVAVHVOUV + FXMUVICUVGBYCABYDXTUVDWOUVFXGUVGNZJXGUVGQXHUVFUVJUSWPWRWSWTXAXBVKXCXEXQWQ + ZXDXCUVBUVKUVCCBDWCVTURWJWMVL $. $} $( Being Markov is invariant with respect to equinumerosity. For example, @@ -71142,20 +71144,20 @@ of Omniscience (WLPO). wral cen wbr cmap wf1o wex bren biimpi ad2antrr ccom fveq1 eqeq1d ralbidv co dcbid iswomnimap ibi com cvv 2onn relen brrelex2i elmapg sylancr mpbid wb adantr f1of adantl fco syl2anc simpllr mpbird rspcdva wfn f1ofn f1ocnv - ccnv syl ffvelrnd fvco2 fveqeq2 simplr f1ocnvfv2 fveq2d ad4ant24 3eqtr3rd - ralrimiva ffvelcdmda adantlr eqtrd impbida exlimddv ex ) ABUAUBZAHIZBHIZW - NWOJZWPCKZDKZLZMNZCBTZOZDPBUCUMZTZWQXCDXDWQWSXDIZJZABEKZUDZXCEWNXIEUEZWOX - FWNXJABEUFUGUHXGXIJZFKZWSXHUIZLZMNZFATZOZXCXKXLGKZLZMNZFATZOZXQGPAUCUMZXM - XRXMNZYAXPYDXTXOFAYDXSXNMXLXRXMUJUKULUNWOYBGYCTZWNXFXIWOYEFAGHUOUPQXKXMYC - IZAPXMRZXKBPWSRZABXHRZYGXGYHXIXGXFYHWQXFSWNXFYHVEZWOXFWNPUQIZBURIZYJUSABU - AUTVAZPBWSUQURVBVCUHVDVFXIYIXGABXHVGVHZABPWSXHVIVJXKYKWOYFYGVEUSWNWOXFXIV - KPAXMUQHVBVCVLVMXKXPXBXKXPXBXKXPJZXACBYOWRBIZJZWRXHVQZLZXMLZYSXHLZWSLZMWT - YQXHAVNZYSAIYTUUBNXIUUCXGXPYPABXHVOZQYQBAWRYRXIBAYRRZXGXPYPXIBAYRUDUUEABX - HVPBAYRVGVRQYOYPSVSZAWSXHYSVTVJYQXOYTMNFAYSXLYSMXMWAXKXPYPWBUUFVMXIYPUUBW - TNXGXPXIYPJUUAWRWSABWRXHWCWDWEWFWGXKXBJZXOFAUUGXLAIZJZXNXLXHLZWSLZMUUIUUC - UUHXNUUKNXIUUCXGXBUUHUUDQUUGUUHSAWSXHXLVTVJUUIXAUUKMNCBUUJWRUUJMWSWAXKXBU - UHWBXKUUHUUJBIXBXKABXLXHYNWHWIVMWJWGWKUNVDWLWGWNWPXEVEZWOWNYLUULYMCBDURUO - VRVFVLWM $. + ccnv ffvelcdmd fvco2 fveqeq2 simplr f1ocnvfv2 ad4ant24 3eqtr3rd ralrimiva + syl fveq2d ffvelcdmda adantlr eqtrd impbida exlimddv ex ) ABUAUBZAHIZBHIZ + WNWOJZWPCKZDKZLZMNZCBTZOZDPBUCUMZTZWQXCDXDWQWSXDIZJZABEKZUDZXCEWNXIEUEZWO + XFWNXJABEUFUGUHXGXIJZFKZWSXHUIZLZMNZFATZOZXCXKXLGKZLZMNZFATZOZXQGPAUCUMZX + MXRXMNZYAXPYDXTXOFAYDXSXNMXLXRXMUJUKULUNWOYBGYCTZWNXFXIWOYEFAGHUOUPQXKXMY + CIZAPXMRZXKBPWSRZABXHRZYGXGYHXIXGXFYHWQXFSWNXFYHVEZWOXFWNPUQIZBURIZYJUSAB + UAUTVAZPBWSUQURVBVCUHVDVFXIYIXGABXHVGVHZABPWSXHVIVJXKYKWOYFYGVEUSWNWOXFXI + VKPAXMUQHVBVCVLVMXKXPXBXKXPXBXKXPJZXACBYOWRBIZJZWRXHVQZLZXMLZYSXHLZWSLZMW + TYQXHAVNZYSAIYTUUBNXIUUCXGXPYPABXHVOZQYQBAWRYRXIBAYRRZXGXPYPXIBAYRUDUUEAB + XHVPBAYRVGWFQYOYPSVRZAWSXHYSVSVJYQXOYTMNFAYSXLYSMXMVTXKXPYPWAUUFVMXIYPUUB + WTNXGXPXIYPJUUAWRWSABWRXHWBWGWCWDWEXKXBJZXOFAUUGXLAIZJZXNXLXHLZWSLZMUUIUU + CUUHXNUUKNXIUUCXGXBUUHUUDQUUGUUHSAWSXHXLVSVJUUIXAUUKMNCBUUJWRUUJMWSVTXKXB + UUHWAXKUUHUUJBIXBXKABXLXHYNWHWIVMWJWEWKUNVDWLWEWNWPXEVEZWOWNYLUULYMCBDURU + OWFVFVLWM $. $} $( Weak omniscience is invariant with respect to equinumerosity. For @@ -71258,30 +71260,30 @@ of Omniscience (WLPO). nninfwlpoimlemg $p |- ( ph -> G e. NN+oo ) $= ( vj c2o com wcel c0 wceq c1o cif wa a1i syl adantr syl2anc vf cmap cfv co cv csuc wss wral xnninf wrex 0lt2o 1lt2o cfn wdc peano2 adantl 2ssom - wf nnfi ad2antrr simpr ffvelrnd sselid peano1 ralrimiva finexdc ifcldcd - elnn nndceq fmptd 2onn elexi omex elmap sylibr wn iftrued suceq rexeqdv - csn wo ifbid fvmptd3 wb cun df-suc rexeqi rexun bitri ifbi ax-mp eqtrdi - ifordc eqtrd 3eqtr4d eqimss eqeltrd el2oss1o iffalsed sseqtrrd mpjaodan - exmiddc fveq1 sseq12d ralbidv df-nninf elrab2 sylanbrc ) AEIJUBUDZKZHUE - ZUFZEUCZXKEUCZUGZHJUHZEUIKAJIEURXJACJBUEZDUCZLMZBCUEZUFZUJZLNOZIEAXTJKZ - PZYBLNILIKZYEUKQNIKZYEULQYEYAUMKZXSUNZBYAUHYBUNYEYAJKZYHYDYJAXTUOUPZYAU - SRYEYIBYAYEXQYAKZPZXRJKZLJKZYIYMIJXRUQYMJIXQDAJIDURZYDYLFUTYMYLYJXQJKZY - EYLVAYEYJYLYKSXQYAVHTVBVCYOYMVDQXRLVIZTVEXSBYAVFTVGGVJIJEIJVKVLVMVNVOAX - OHJAXKJKZPZXSBXLUJZXOUUAVPZYTUUAPZXMXNMXOUUCUUALXSBXLVTZUJZLNOZOZLXMXNU - UCUUALUUFYTUUAVAZVQYTXMUUGMUUAYTXMUUAUUEWAZLNOZUUGYTXMXSBXLUFZUJZLNOZUU - JYTCXLYCUUMJEIGXTXLMZYBUULLNUUNXSBYAUUKXTXLVRVSWBYSXLJKZAXKUOUPZYTUULLN - IYFYTUKQZYGYTULQZYTUUKUMKZYIBUUKUHUULUNYTUUKJKZUUSYTUUOUUTUUPXLUORZUUKU - SRYTYIBUUKYTXQUUKKZPZYNYOYIUVCIJXRUQUVCJIXQDAYPYSUVBFUTUVCUVBUUTYQYTUVB - VAYTUUTUVBUVASXQUUKVHTVBVCYOUVCVDQYRTVEXSBUUKVFTVGZWCZUULUUIWDUUMUUJMUU - LXSBXLUUDWEZUJUUIXSBUUKUVFXLWFWGXSBXLUUDWHWIUULUUILNWJWKWLYTUUAUNZUUJUU - GMYTXLUMKZYIBXLUHUVGYTUUOUVHUUPXLUSRYTYIBXLYTXQXLKZPZYNYOYIUVJIJXRUQUVJ - JIXQDAYPYSUVIFUTUVJUVIUUOYQYTUVIVAYTUUOUVIUUPSXQXLVHTVBVCYOUVJVDQYRTVEX - SBXLVFTZUUAUUELNWMRWNSUUCXNUUALNOZLYTXNUVLMZUUAYTCXKYCUVLJEIGXTXKMZYBUU - ALNUVNXSBYAXLXTXKVRVSWBAYSVAYTUUALNIUUQUURUVKVGWCZSUUCUUALNUUHVQWNWOXMX - NWPRYTUUBPZXMNXNYTXMNUGZUUBYTXMIKUVQYTXMUUMIUVEUVDWQXMWRRSUVPXNUVLNYTUV - MUUBUVOSUVPUUALNYTUUBVAWSWNWTYTUVGUUAUUBWAUVKUUAXBRXAVEXLUAUEZUCZXKUVRU - CZUGZHJUHXPUAEXIUIUVREMZUWAXOHJUWBUVSXMUVTXNXLUVREXCXKUVREXCXDXEUAHXFXG - XH $. + wf nnfi ad2antrr simpr ffvelcdmd sselid peano1 nndceq ralrimiva finexdc + elnn ifcldcd fmptd 2onn elexi omex elmap sylibr wn csn iftrued wo suceq + rexeqdv ifbid fvmptd3 cun df-suc rexeqi rexun bitri ax-mp eqtrdi ifordc + wb ifbi eqtrd 3eqtr4d eqimss eqeltrd el2oss1o iffalsed sseqtrrd exmiddc + mpjaodan fveq1 sseq12d ralbidv df-nninf elrab2 sylanbrc ) AEIJUBUDZKZHU + EZUFZEUCZXKEUCZUGZHJUHZEUIKAJIEURXJACJBUEZDUCZLMZBCUEZUFZUJZLNOZIEAXTJK + ZPZYBLNILIKZYEUKQNIKZYEULQYEYAUMKZXSUNZBYAUHYBUNYEYAJKZYHYDYJAXTUOUPZYA + USRYEYIBYAYEXQYAKZPZXRJKZLJKZYIYMIJXRUQYMJIXQDAJIDURZYDYLFUTYMYLYJXQJKZ + YEYLVAYEYJYLYKSXQYAVHTVBVCYOYMVDQXRLVEZTVFXSBYAVGTVIGVJIJEIJVKVLVMVNVOA + XOHJAXKJKZPZXSBXLUJZXOUUAVPZYTUUAPZXMXNMXOUUCUUALXSBXLVQZUJZLNOZOZLXMXN + UUCUUALUUFYTUUAVAZVRYTXMUUGMUUAYTXMUUAUUEVSZLNOZUUGYTXMXSBXLUFZUJZLNOZU + UJYTCXLYCUUMJEIGXTXLMZYBUULLNUUNXSBYAUUKXTXLVTWAWBYSXLJKZAXKUOUPZYTUULL + NIYFYTUKQZYGYTULQZYTUUKUMKZYIBUUKUHUULUNYTUUKJKZUUSYTUUOUUTUUPXLUORZUUK + USRYTYIBUUKYTXQUUKKZPZYNYOYIUVCIJXRUQUVCJIXQDAYPYSUVBFUTUVCUVBUUTYQYTUV + BVAYTUUTUVBUVASXQUUKVHTVBVCYOUVCVDQYRTVFXSBUUKVGTVIZWCZUULUUIWLUUMUUJMU + ULXSBXLUUDWDZUJUUIXSBUUKUVFXLWEWFXSBXLUUDWGWHUULUUILNWMWIWJYTUUAUNZUUJU + UGMYTXLUMKZYIBXLUHUVGYTUUOUVHUUPXLUSRYTYIBXLYTXQXLKZPZYNYOYIUVJIJXRUQUV + JJIXQDAYPYSUVIFUTUVJUVIUUOYQYTUVIVAYTUUOUVIUUPSXQXLVHTVBVCYOUVJVDQYRTVF + XSBXLVGTZUUAUUELNWKRWNSUUCXNUUALNOZLYTXNUVLMZUUAYTCXKYCUVLJEIGXTXKMZYBU + UALNUVNXSBYAXLXTXKVTWAWBAYSVAYTUUALNIUUQUURUVKVIWCZSUUCUUALNUUHVRWNWOXM + XNWPRYTUUBPZXMNXNYTXMNUGZUUBYTXMIKUVQYTXMUUMIUVEUVDWQXMWRRSUVPXNUVLNYTU + VMUUBUVOSUVPUUALNYTUUBVAWSWNWTYTUVGUUAUUBVSUVKUUAXARXBVFXLUAUEZUCZXKUVR + UCZUGZHJUHXPUAEXIUIUVREMZUWAXOHJUWBUVSXMUVTXNXLUVREXCXKUVREXCXDXEUAHXFX + GXH $. $} ${ @@ -71292,22 +71294,22 @@ of Omniscience (WLPO). G = ( i e. _om |-> 1o ) <-> A. n e. _om ( F ` n ) = 1o ) ) $= ( com c1o wceq cfv wa wcel c0 wrex c2o a1i adantr syl2anc cmpt wral cif cv csuc suceq rexeqdv ifbid simpr 0lt2o 1lt2o cfn wdc peano2 adantl syl - nnfi 2ssom wf ad3antrrr ffvelrnd sselid peano1 nndceq ralrimiva finexdc - elnn ifcldcd fvmptd3 vex sucid fveqeq2 rspcev iftrued eqtrd 1n0 simpllr - neii fveq1d eqid eqidd eqeq1d mtbiri pm2.65da wo ffvelcdmda elpri df2o3 - cpr eleq2s orcomd ecased wn eqeq1 ralimi ralnex cbvrexv sylnib ad2antlr - sylib wss elomssom ssrexv 3syl mtod iffalsed mpteq2dva eqtrid impbida + nnfi wf ad3antrrr elnn ffvelcdmd sselid peano1 nndceq ralrimiva finexdc + 2ssom ifcldcd fvmptd3 vex sucid fveqeq2 rspcev iftrued 1n0 neii simpllr + eqtrd fveq1d eqidd eqeq1d mtbiri pm2.65da wo ffvelcdmda cpr elpri df2o3 + eqid eleq2s orcomd ecased wn eqeq1 ralimi ralnex sylib cbvrexv ad2antlr + sylnib wss elomssom ssrexv 3syl mtod iffalsed mpteq2dva eqtrid impbida wi ) AFCIJUAZKZDUDZELZJKZDIUBZAXLMZXODIXQXMINZMZXOXNOKZXSXTXMFLZOKZXSXT MZYABUDZELZOKZBXMUEZPZOJUCZOXSYAYIKXTXSCXMYFBCUDZUEZPZOJUCZYIIFQHYJXMKZ YLYHOJYNYFBYKYGYJXMUFUGUHXQXRUIZXSYHOJQOQNXSUJRJQNZXSUKRXSYGULNZYFUMZBY - GUBYHUMXSYGINZYQXRYSXQXMUNUOZYGUQUPXSYRBYGXSYDYGNZMZYEINOINZYRUUBQIYEUR - UUBIQYDEAIQEUSZXLXRUUAGUTUUBUUAYSYDINXSUUAUIXSYSUUAYTSYDYGVGTVAVBUUCUUB + GUBYHUMXSYGINZYQXRYSXQXMUNUOZYGUQUPXSYRBYGXSYDYGNZMZYEINOINZYRUUBQIYEVG + UUBIQYDEAIQEURZXLXRUUAGUSUUBUUAYSYDINXSUUAUIXSYSUUAYTSYDYGUTTVAVBUUCUUB VCRYEOVDTVEYFBYGVFTVHVISYCYHOJYCXMYGNZXTYHUUEYCXMDVJVKRXSXTUIYFXTBXMYGY - DXMOEVLVMTVNVOYCYBJOKZJOVPVRZYCYAJOYCYAXMXKLJYCXMFXKAXLXRXTVQVSYCCXMJJI - XKQXKVTYNJWAXSXRXTYOSYPYCUKRVIVOWBWCWDXSXTXOXSXNQNXTXOWEZXQIQXMEAUUDXLG - SWFUUHXNOJWIQXNOJWGWHWJUPWKWLVEAXPMZFCIYMUAXKHUUICIYMJUUIYJINZMZYLOJUUK - YLYFBIPZXPUULWMAUUJXPXTDIPZUULXPXTWMZDIUBUUMWMXOUUNDIXOXTUUFUUGXNJOWNWC - WOXTDIWPWTXTYFDBIXMYDOEVLWQWRWSUUKYKINZYKIXAYLUULXJUUJUUOUUIYJUNUOYKXBY + DXMOEVLVMTVNVRYCYBJOKZJOVOVPZYCYAJOYCYAXMXKLJYCXMFXKAXLXRXTVQVSYCCXMJJI + XKQXKWIYNJVTXSXRXTYOSYPYCUKRVIVRWAWBWCXSXTXOXSXNQNXTXOWDZXQIQXMEAUUDXLG + SWEUUHXNOJWFQXNOJWGWHWJUPWKWLVEAXPMZFCIYMUAXKHUUICIYMJUUIYJINZMZYLOJUUK + YLYFBIPZXPUULWMAUUJXPXTDIPZUULXPXTWMZDIUBUUMWMXOUUNDIXOXTUUFUUGXNJOWNWB + WOXTDIWPWQXTYFDBIXMYDOEVLWRWTWSUUKYKINZYKIXAYLUULXJUUJUUOUUIYJUNUOYKXBY FBYKIXCXDXEXFXGXHXI $. $} @@ -71666,32 +71668,32 @@ Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, p0ex breqtrrdi 1onn domrefg djudom sylancl dju1p1e2 adantl cinr cfv 0lt1o domentr djurcl elex2 jctil vex djuex mp2an 2onn breq2 anbi2d foeq2 exbidv imbi12d albidv spcgv eleq2 breq1 anbi12d foeq3 mpsyl sylc wn wo simpr fof - cinl csuc elelsuc df-2o eleqtrri a1i ffvelrnd adantr eqeltrrd 0ex djulclb - wf wb orcd df-dc cpr 1oex prid2 df2o3 wrex simp-4r djulcl syl2anc simplrr - foelrn fveq2d simp-5r 3eqtrd elpri eleq2s ad2antrl mpjaodan rexlimddv wne - djune neii pm2.65da olcd sseqtrrdi exmidfodomrlemeldju exlimdd ex alrimiv - simplr df-exmid ) CFZBFZGZCHZYTAFZIJZKZUUCYTDFZUCZDHZLZBUDZAUDZEFZMUEZUFZ - MUULGZUGZLZEUDUHUUKUUQEUUKUUNUUPUUKUUNKZNUULOUAZUUFUCZUUPDUUKUUNDUUJDAUUI - DBUUEUUHDUUEDUIUUGDUJUKULULUUNDUIUMUUPDUIUURUUKYSUUSGZCHZUUSNIJZKZUUTDHZU - UKUUNUNUURUVCUVBUUNUVCUUKUUNUUSOOUAZIJZUVFNUOJUVCUUNUULOIJOOIJZUVGUUNUULU - UMOIUUMPGUUNUULUUMIJLURUULUUMPUPQUQUSORGZUVHUTORVAQUULOOOVBVCVDUUSUVFNVIV - CVEMVFVGZUUSGZUVBMOGZUVKVHUULOMVJQCUVJUUSVKQVLUUSPGZUUKUUBYTNIJZKZNYTUUFU - CZDHZLZBUDZUVDUVELZUULPGUVIUVMEVMUTUULOPRVNVONRGUUKUVSLVPUUJUVSANRUUCNSZU - UIUVRBUWAUUEUVOUUHUVQUWAUUDUVNUUBUUCNYTIVQVRUWAUUGUVPDUUCNYTUUFVSVTWAWBWC - QUVRUVTBUUSPYTUUSSZUVOUVDUVQUVEUWBUUBUVBUVNUVCUWBUUAUVACYTUUSYSWDVTYTUUSN - IWEWFUWBUVPUUTDYTUUSNUUFWGVTWAWCWHWIUURUUTKZMUUFVGZMWNVGZSZUUPUWDUVJSZUWC - UWFKZUUOUUOWJZWKZUUPUWHUUOUWIUWHUWEUUSGZUUOUWHUWDUWEUUSUWCUWFWLUWCUWDUUSG - UWFUWCNUUSMUUFUUTNUUSUUFXEZUURNUUSUUFWMVEZMNGUWCMOWOZNUVLMUWNGVHMOWPQWQWR - WSWTZXAXBMPGZUUOUWKXFXCUULOMPXDQZTXGUUOXHZTUWCUWGKZOUUFVGZUWESZUUPUWTUVJS - ZUWSUXAKZUWJUUPUXCUUOUWIUXCUWKUUOUXCUWTUWEUUSUWSUXAWLUWSUWTUUSGUXAUWSNUUS - OUUFUWCUWLUWGUWMXAONGUWSOMOXIZNMOXJXKXLWRWSWTZXAXBUWQTXGUWRTUWSUXBKZUWJUU - PUXFUWIUUOUXFUUOUWEUVJSZUXFUUOKZUWEUBFZUUFVGZSZUXGUBNUXHUUTUWKUXKUBNXMUUR - UUTUWGUXBUUOXNUUOUWKUXFUULOMXOVEUBNUUSUWEUUFXRXPUXHUXINGZUXKKZKZUXIMSZUXG - UXIOSZUXNUXOKZUWEUXJUWDUVJUXHUXLUXKUXOXQUXQUXIMUUFUXNUXOWLXSUWCUWGUXBUUOU - XMUXOXTYAUXNUXPKZUWEUXJUWTUVJUXHUXLUXKUXPXQUXRUXIOUUFUXNUXPWLXSUWSUXBUUOU - XMUXPXNYAUXLUXOUXPWKZUXHUXKUXSUXIUXDNUXIMOYBXLYCYDYEYFUXGWJUXHUWEUVJUWPUW - PUWEUVJYGXCXCMMPPYHVOYIWSYJYKUWRTUWSUULUWTUWCUULOUFUWGUWCUULUUMOUUKUUNUUT - YQUQYLZXAUXEYMYEUWCUULUWDUXTUWOYMYEYNYOYPEYRT $. + cinl wf csuc elelsuc df-2o eleqtrri a1i ffvelcdmd adantr eqeltrrd djulclb + wb 0ex orcd df-dc cpr 1oex prid2 df2o3 wrex simp-4r djulcl foelrn syl2anc + simplrr fveq2d simp-5r 3eqtrd elpri ad2antrl mpjaodan rexlimddv wne djune + eleq2s neii pm2.65da olcd simplr sseqtrrdi exmidfodomrlemeldju exlimdd ex + alrimiv df-exmid ) CFZBFZGZCHZYTAFZIJZKZUUCYTDFZUCZDHZLZBUDZAUDZEFZMUEZUF + ZMUULGZUGZLZEUDUHUUKUUQEUUKUUNUUPUUKUUNKZNUULOUAZUUFUCZUUPDUUKUUNDUUJDAUU + IDBUUEUUHDUUEDUIUUGDUJUKULULUUNDUIUMUUPDUIUURUUKYSUUSGZCHZUUSNIJZKZUUTDHZ + UUKUUNUNUURUVCUVBUUNUVCUUKUUNUUSOOUAZIJZUVFNUOJUVCUUNUULOIJOOIJZUVGUUNUUL + UUMOIUUMPGUUNUULUUMIJLURUULUUMPUPQUQUSORGZUVHUTORVAQUULOOOVBVCVDUUSUVFNVI + VCVEMVFVGZUUSGZUVBMOGZUVKVHUULOMVJQCUVJUUSVKQVLUUSPGZUUKUUBYTNIJZKZNYTUUF + UCZDHZLZBUDZUVDUVELZUULPGUVIUVMEVMUTUULOPRVNVONRGUUKUVSLVPUUJUVSANRUUCNSZ + UUIUVRBUWAUUEUVOUUHUVQUWAUUDUVNUUBUUCNYTIVQVRUWAUUGUVPDUUCNYTUUFVSVTWAWBW + CQUVRUVTBUUSPYTUUSSZUVOUVDUVQUVEUWBUUBUVBUVNUVCUWBUUAUVACYTUUSYSWDVTYTUUS + NIWEWFUWBUVPUUTDYTUUSNUUFWGVTWAWCWHWIUURUUTKZMUUFVGZMWNVGZSZUUPUWDUVJSZUW + CUWFKZUUOUUOWJZWKZUUPUWHUUOUWIUWHUWEUUSGZUUOUWHUWDUWEUUSUWCUWFWLUWCUWDUUS + GUWFUWCNUUSMUUFUUTNUUSUUFWOZUURNUUSUUFWMVEZMNGUWCMOWPZNUVLMUWNGVHMOWQQWRW + SWTXAZXBXCMPGZUUOUWKXEXFUULOMPXDQZTXGUUOXHZTUWCUWGKZOUUFVGZUWESZUUPUWTUVJ + SZUWSUXAKZUWJUUPUXCUUOUWIUXCUWKUUOUXCUWTUWEUUSUWSUXAWLUWSUWTUUSGUXAUWSNUU + SOUUFUWCUWLUWGUWMXBONGUWSOMOXIZNMOXJXKXLWSWTXAZXBXCUWQTXGUWRTUWSUXBKZUWJU + UPUXFUWIUUOUXFUUOUWEUVJSZUXFUUOKZUWEUBFZUUFVGZSZUXGUBNUXHUUTUWKUXKUBNXMUU + RUUTUWGUXBUUOXNUUOUWKUXFUULOMXOVEUBNUUSUWEUUFXPXQUXHUXINGZUXKKZKZUXIMSZUX + GUXIOSZUXNUXOKZUWEUXJUWDUVJUXHUXLUXKUXOXRUXQUXIMUUFUXNUXOWLXSUWCUWGUXBUUO + UXMUXOXTYAUXNUXPKZUWEUXJUWTUVJUXHUXLUXKUXPXRUXRUXIOUUFUXNUXPWLXSUWSUXBUUO + UXMUXPXNYAUXLUXOUXPWKZUXHUXKUXSUXIUXDNUXIMOYBXLYHYCYDYEUXGWJUXHUWEUVJUWPU + WPUWEUVJYFXFXFMMPPYGVOYIWTYJYKUWRTUWSUULUWTUWCUULOUFUWGUWCUULUUMOUUKUUNUU + TYLUQYMZXBUXEYNYDUWCUULUWDUXTUWOYNYDYOYPYQEYRT $. $} ${ @@ -71713,31 +71715,31 @@ Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, orcd df-dc sylibr simp-4r djulcl foelrn syl2anc simprr wb eqeq1d ad2antlr wrex fvres mpbird adantr simpr fveq2d simp-5r 3eqtrd elpri df2o3 ad2antrl cpr eleq2s mpjaodan rexlimddv wne 0ex djune neii a1i mtbiri pm2.65da olcd - eqeq12d simplr sseqtrrdi wf 1oex eleqtrri ffvelrnd exmidfodomrlemreseldju - fof prid2 csuc elelsuc df-2o exlimdd ex alrimiv df-exmid ) CFZBFZGZCHZUUF - AFZIJZKZUUIUUFDFZUAZDHZLZBUBZAUBZEFZMUDZUEZMUURGZUFZLZEUBUGUUQUVCEUUQUUTU - VBUUQUUTKZNUUROUHZUULUAZUVBDUUQUUTDUUPDAUUODBUUKUUNDUUKDUIUUMDUJUKULULUUT - DUIUMUVBDUIUVDUUQUUEUVEGZCHZUVENIJZKZUVFDHZUUQUUTURUVDUVIUVHUUTUVIUUQUUTU - VEOOUHZIJZUVLNUNJUVIUUTUUROIJOOIJZUVMUUTUURUUSOIUUSPGUUTUURUUSIJLUOUURUUS - PUPQUSUQORGZUVNUTORVAQUUROOOVBVCVDUVEUVLNVIVCVEMVFSZUVEGZUVHMOGZUVQVGUURO - MVHQCUVPUVEVJQVKUVEPGZUUQUUHUUFNIJZKZNUUFUULUAZDHZLZBUBZUVJUVKLZUURPGUVOU - VSEVLUTUUROPRVMVNNRGUUQUWELVOUUPUWEANRUUINTZUUOUWDBUWGUUKUWAUUNUWCUWGUUJU - VTUUHUUINUUFIVPVQUWGUUMUWBDUUINUUFUULVRVSWAVTWBQUWDUWFBUVEPUUFUVETZUWAUVJ - UWCUVKUWHUUHUVHUVTUVIUWHUUGUVGCUUFUVEUUEWCVSUUFUVENIWDWEUWHUWBUVFDUUFUVEN - UULWFVSWAWBWGWHUVDUVFKZUVAMUULSZMWIUURWJSZTZKZUVBUWJMVFOWJSZTZUWIUWMKZUVA - UVAWKZWLZUVBUWPUVAUWQUWIUVAUWLWMWNUVAWOZWPUWIUWOKZUVAOUULSZUWKTZKZUVBUXAU - WNTZUWTUXCKZUWRUVBUXEUVAUWQUWTUVAUXBWMWNUWSWPUWTUXDKZUWRUVBUXFUWQUVAUXFUV - AUWKUWNTZUXFUVAKZMWISZUCFZUULSZTZUXGUCNUXHUVFUXIUVEGZUXLUCNXEUVDUVFUWOUXD - UVAWQUVAUXMUXFUUROMWRVEUCNUVEUXIUULWSWTUXHUXJNGZUXLKZKZUXJMTZUXGUXJOTZUXP - UXQKZUWKUXKUWJUWNUXPUWKUXKTZUXQUXPUXTUXLUXHUXNUXLXAUVAUXTUXLXBUXFUXOUVAUW - KUXIUXKMUURWIXFZXCXDXGZXHUXSUXJMUULUXPUXQXIXJUWIUWOUXDUVAUXOUXQXKXLUXPUXR - KZUWKUXKUXAUWNUXPUXTUXRUYBXHUYCUXJOUULUXPUXRXIXJUWTUXDUVAUXOUXRWQXLUXNUXQ - UXRWLZUXHUXLUYDUXJMOXPZNUXJMOXMXNXQXOXRXSUVAUXGWKUXFUVAUXGUXIUVPTUXIUVPMP - GZUYFUXIUVPXTYAYAMMPPYBVNYCUVAUWKUXIUWNUVPUYAUWNUVPTZUVAUVRUYGVGMOVFXFQYD - YHYEVEYFYGUWSWPUWTUURUXAUWIUUROUEUWOUWIUURUUSOUUQUUTUVFYIUSYJZXHUWTNUVEOU - ULUWINUVEUULYKZUWOUVFUYIUVDNUVEUULYPVEZXHONGUWTOUYENMOYLYQXNYMYDYNYOXRUWI - UURUWJUYHUWINUVEMUULUYJMNGUWIMOYRZNUVRMUYKGVGMOYSQYTYMYDYNYOXRUUAUUBUUCEU - UDWP $. + eqeq12d simplr sseqtrrdi wf fof eleqtrri ffvelcdmd exmidfodomrlemreseldju + 1oex prid2 csuc elelsuc df-2o exlimdd ex alrimiv df-exmid ) CFZBFZGZCHZUU + FAFZIJZKZUUIUUFDFZUAZDHZLZBUBZAUBZEFZMUDZUEZMUURGZUFZLZEUBUGUUQUVCEUUQUUT + UVBUUQUUTKZNUUROUHZUULUAZUVBDUUQUUTDUUPDAUUODBUUKUUNDUUKDUIUUMDUJUKULULUU + TDUIUMUVBDUIUVDUUQUUEUVEGZCHZUVENIJZKZUVFDHZUUQUUTURUVDUVIUVHUUTUVIUUQUUT + UVEOOUHZIJZUVLNUNJUVIUUTUUROIJOOIJZUVMUUTUURUUSOIUUSPGUUTUURUUSIJLUOUURUU + SPUPQUSUQORGZUVNUTORVAQUUROOOVBVCVDUVEUVLNVIVCVEMVFSZUVEGZUVHMOGZUVQVGUUR + OMVHQCUVPUVEVJQVKUVEPGZUUQUUHUUFNIJZKZNUUFUULUAZDHZLZBUBZUVJUVKLZUURPGUVO + UVSEVLUTUUROPRVMVNNRGUUQUWELVOUUPUWEANRUUINTZUUOUWDBUWGUUKUWAUUNUWCUWGUUJ + UVTUUHUUINUUFIVPVQUWGUUMUWBDUUINUUFUULVRVSWAVTWBQUWDUWFBUVEPUUFUVETZUWAUV + JUWCUVKUWHUUHUVHUVTUVIUWHUUGUVGCUUFUVEUUEWCVSUUFUVENIWDWEUWHUWBUVFDUUFUVE + NUULWFVSWAWBWGWHUVDUVFKZUVAMUULSZMWIUURWJSZTZKZUVBUWJMVFOWJSZTZUWIUWMKZUV + AUVAWKZWLZUVBUWPUVAUWQUWIUVAUWLWMWNUVAWOZWPUWIUWOKZUVAOUULSZUWKTZKZUVBUXA + UWNTZUWTUXCKZUWRUVBUXEUVAUWQUWTUVAUXBWMWNUWSWPUWTUXDKZUWRUVBUXFUWQUVAUXFU + VAUWKUWNTZUXFUVAKZMWISZUCFZUULSZTZUXGUCNUXHUVFUXIUVEGZUXLUCNXEUVDUVFUWOUX + DUVAWQUVAUXMUXFUUROMWRVEUCNUVEUXIUULWSWTUXHUXJNGZUXLKZKZUXJMTZUXGUXJOTZUX + PUXQKZUWKUXKUWJUWNUXPUWKUXKTZUXQUXPUXTUXLUXHUXNUXLXAUVAUXTUXLXBUXFUXOUVAU + WKUXIUXKMUURWIXFZXCXDXGZXHUXSUXJMUULUXPUXQXIXJUWIUWOUXDUVAUXOUXQXKXLUXPUX + RKZUWKUXKUXAUWNUXPUXTUXRUYBXHUYCUXJOUULUXPUXRXIXJUWTUXDUVAUXOUXRWQXLUXNUX + QUXRWLZUXHUXLUYDUXJMOXPZNUXJMOXMXNXQXOXRXSUVAUXGWKUXFUVAUXGUXIUVPTUXIUVPM + PGZUYFUXIUVPXTYAYAMMPPYBVNYCUVAUWKUXIUWNUVPUYAUWNUVPTZUVAUVRUYGVGMOVFXFQY + DYHYEVEYFYGUWSWPUWTUURUXAUWIUUROUEUWOUWIUURUUSOUUQUUTUVFYIUSYJZXHUWTNUVEO + UULUWINUVEUULYKZUWOUVFUYIUVDNUVEUULYLVEZXHONGUWTOUYENMOYPYQXNYMYDYNYOXRUW + IUURUWJUYHUWINUVEMUULUYJMNGUWIMOYRZNUVRMUYKGVGMOYSQYTYMYDYNYOXRUUAUUBUUCE + UUDWP $. $} ${ @@ -73002,12 +73004,12 @@ from there to positive integers ( ~ df-ni ) and then positive rational ltmpig $p |- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A ( C .N A ) ( s e. ( 1st ` L ) \/ r e. ( 2nd ` L ) ) ) ) $= ( cltq wbr wcel cnq cplq co vy vx vf vg vh cv c1st c2nd wo wi wa wceq wrex ltexnqi adantl subhalfnqq ad2antrl simprr simplrl adantr syl2anc - cfv ltanqi simplrr breqtrd simprl addclnq wf ad4antr ffvelrnd wor w3a - ltsonq sowlin mpan syl3anc simpr addassnqg breq1d mpbird wb addcomnqg - ltanqg caovord2d oveq2 fveq2 breq12d rspcev oveq1 rexbidv crab fveq2i - mpd cop nqex rabex op1st eqtri elrab2 sylanbrc ex oveq12d sylan breq2 - id op2nd orim12d rexlimddv ralrimivva ) AFUFZGUFZOPZXJEUGVBZQZXKEUHVB - ZQZUIZUJFGRRAXJRQZXKRQZUKZUKZXLXQYAXLUKZXJUAUFZSTZXKULZXQUARXLYEUARUM - YAUAXJXKUNUOYBYCRQZYEUKZUKZUBUFZYISTZYCOPZXQUBRYFYKUBRUMYBYEUBYCUPUQY - HYIRQZYKUKZUKZXJYJSTZYIDVBZYISTZOPZYQXKOPZUIZXQYNYOXKOPZYTYNYOYDXKOYN - YKXRYOYDOPYHYLYKURYHXRYMYBXRYGAXRXSXLUSUTUTZYJYCXJVCVAYBYFYEYMVDVEYNY - ORQZXSYQRQZUUAYTUJZYNXRYJRQZUUCUUBYNYLYLUUFYHYLYKVFZUUGYIYIVGVAXJYJVG - VAYHXSYMYBXSYGAXRXSXLVDUTUTZYNYPRQZYLUUDYNRRYIDARRDVHXTXLYGYMKVIUUGVJ - ZUUGYPYIVGVAROVKUUCXSUUDVLUUEVMRYOXKYQOVNVOVPWMYNYRXNYSXPYNYRXNYNYRUK - ZXRXJHUFZSTZUULDVBZOPZHRUMZXNYNXRYRUUBUTZUUKYLXJYISTZYPOPZUUPYHYLYKYR - USZUUKUUSUURYISTZYQOPZUUKUVBYRYNYRVQUUKUVAYOYQOUUKXRYLYLUVAYOULUUQUUT - UUTXJYIYIVRVPVSVTUUKUCUDUEUURYPYIORSUCUFZRQZUDUFZRQZUEUFZRQVLUVCUVEOP - UVGUVCSTUVGUVESTOPWAUUKUVCUVEUVGWCUOUUKXRYLUURRQUUQUUTXJYIVGVAYNUUIYR - UUJUTUUTUVDUVFUKUVCUVESTUVEUVCSTULUUKUVCUVEWBUOWDVTUUOUUSHYIRUULYIULZ - UUMUURUUNYPOUULYIXJSWEUULYIDWFZWGWHVAJUFZUULSTZUUNOPZHRUMZUUPJXJRXMUV - JXJULZUVLUUOHRUVNUVKUUMUUNOUVJXJUULSWIVSWJXMUVMJRWKZUUNUULSTZBUFZOPZH - RUMZBRWKZWNZUGVBUVOEUWAUGNWLUVOUVTUVMJRWOWPZUVSBRWOWPZWQWRWSWTXAYNYSX - PYNYSUKXSUVPXKOPZHRUMZXPYNXSYSUUHUTYNYLYSUWEUUGUWDYSHYIRUVHUVPYQXKOUV - HUUNYPUULYISUVIUVHXEXBVSWHXCUVSUWEBXKRXOUVQXKULUVRUWDHRUVQXKUVPOXDWJX - OUWAUHVBUVTEUWAUHNWLUVOUVTUWBUWCXFWRWSWTXAXGWMXHXHXAXI $. + cfv ltanqi simplrr breqtrd simprl addclnq wf ad4antr ffvelcdmd ltsonq + wor w3a sowlin syl3anc mpd simpr addassnqg breq1d mpbird wb addcomnqg + mpan ltanqg caovord2d oveq2 fveq2 breq12d rspcev oveq1 rexbidv fveq2i + crab nqex rabex op1st eqtri elrab2 sylanbrc ex id oveq12d sylan breq2 + cop op2nd orim12d rexlimddv ralrimivva ) AFUFZGUFZOPZXJEUGVBZQZXKEUHV + BZQZUIZUJFGRRAXJRQZXKRQZUKZUKZXLXQYAXLUKZXJUAUFZSTZXKULZXQUARXLYEUARU + MYAUAXJXKUNUOYBYCRQZYEUKZUKZUBUFZYISTZYCOPZXQUBRYFYKUBRUMYBYEUBYCUPUQ + YHYIRQZYKUKZUKZXJYJSTZYIDVBZYISTZOPZYQXKOPZUIZXQYNYOXKOPZYTYNYOYDXKOY + NYKXRYOYDOPYHYLYKURYHXRYMYBXRYGAXRXSXLUSUTUTZYJYCXJVCVAYBYFYEYMVDVEYN + YORQZXSYQRQZUUAYTUJZYNXRYJRQZUUCUUBYNYLYLUUFYHYLYKVFZUUGYIYIVGVAXJYJV + GVAYHXSYMYBXSYGAXRXSXLVDUTUTZYNYPRQZYLUUDYNRRYIDARRDVHXTXLYGYMKVIUUGV + JZUUGYPYIVGVAROVLUUCXSUUDVMUUEVKRYOXKYQOVNWCVOVPYNYRXNYSXPYNYRXNYNYRU + KZXRXJHUFZSTZUULDVBZOPZHRUMZXNYNXRYRUUBUTZUUKYLXJYISTZYPOPZUUPYHYLYKY + RUSZUUKUUSUURYISTZYQOPZUUKUVBYRYNYRVQUUKUVAYOYQOUUKXRYLYLUVAYOULUUQUU + TUUTXJYIYIVRVOVSVTUUKUCUDUEUURYPYIORSUCUFZRQZUDUFZRQZUEUFZRQVMUVCUVEO + PUVGUVCSTUVGUVESTOPWAUUKUVCUVEUVGWDUOUUKXRYLUURRQUUQUUTXJYIVGVAYNUUIY + RUUJUTUUTUVDUVFUKUVCUVESTUVEUVCSTULUUKUVCUVEWBUOWEVTUUOUUSHYIRUULYIUL + ZUUMUURUUNYPOUULYIXJSWFUULYIDWGZWHWIVAJUFZUULSTZUUNOPZHRUMZUUPJXJRXMU + VJXJULZUVLUUOHRUVNUVKUUMUUNOUVJXJUULSWJVSWKXMUVMJRWMZUUNUULSTZBUFZOPZ + HRUMZBRWMZXEZUGVBUVOEUWAUGNWLUVOUVTUVMJRWNWOZUVSBRWNWOZWPWQWRWSWTYNYS + XPYNYSUKXSUVPXKOPZHRUMZXPYNXSYSUUHUTYNYLYSUWEUUGUWDYSHYIRUVHUVPYQXKOU + VHUUNYPUULYISUVIUVHXAXBVSWIXCUVSUWEBXKRXOUVQXKULUVRUWDHRUVQXKUVPOXDWK + XOUWAUHVBUVTEUWAUHNWLUVOUVTUWBUWCXFWQWRWSWTXGVPXHXHWTXI $. $( Lemma for ~ cauappcvgpr . The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.) $) @@ -78604,28 +78606,28 @@ in HoTT although some of the conditions are redundant (for example, vy cv wrex crab wa cnp wb cauappcvgprlemcl nqprlu syl df-iplp addclnq genpelvu syl2anc biimpa breq2 rexbidv fveq2i rabex op2nd eqtri elrab2 wceq nqex biimpi adantr adantl simpld vex ltnqex gtnqex elab2 ltrelnq - brel sylbi simprd ad2antll mpbird ad2antrr ad5antr wf simplr ffvelrnd - eleq1 ltanqg syl3anc mpbid simpr w3a addcomnqg caovord2d ltsonq sotri - simpllr breqtrrd ex reximdva mpd sylanbrc rexlimdvva ssrdv ) AUAFIUNZ - DOPIUHZDBUNZOPZBUHZUIZUJQUKULZXNGUNZRQZYAEULZDRQOPGSUOZISUPZYCYARQZDR - QZXPOPZGSUOZBSUPZUIUKULZAUAUNZXTTZYLYKTZAYMUQZYLUBUNZUCUNZRQZVOZUCXSU - KULZUOUBFUKULZUOZYNAYMUUBAFURTXSURTZYMUUBUSABCEFGHIJKLMUTADSTZUUCNBDI - VAVBUDUEUFUGUMFXSYLUBUCUJRUGUMUFUEUDVCUEUNZUFUNZVDVEVFVGYOYSYNUBUCUUA - YTYOYPUUATZYQYTTZUQZUQZYSYNUUJYSUQZYLSTZYGYLOPZGSUOZYNUUKUULYRSTZUUKY - PSTZYQSTZUUOUUKUUPYFYPOPZGSUOZUUJUUPUUSUQZYSUUIUUTYOUUGUUTUUHUUGUUTYF - XPOPZGSUOZUUSBYPSUUAXPYPVOUVAUURGSXPYPYFOVHVIUUAYBYCOPGSUOZISUPZUVBBS - UPZUIZUKULUVEFUVFUKMVJUVDUVEUVCISVPVKUVBBSVPVKVLVMVNVQVRVSVRZVTZUUJUU - QYSUUHUUQYOUUGUUHUUDUUQUUHDYQOPZUUDUUQUQXQUVIBYQYTUCWAXPYQDOVHXOXRIDW - BBDWCVLWDZDYQSSOWEWFWGWHWIVRZYPYQVDVFYSUULUUOUSUUJYLYRSWPVSWJUUKUUSUU - NUUKUUPUUSUVGWHUUKUURUUMGSUUKYASTZUQZUURUUMUVMUURUQZYGYRYLOUVNYGYFYQR - QZOPZUVOYROPZYGYROPUVNUVIUVPUUKUVIUVLUURUUJUVIYSUUHUVIYOUUGUUHUVIUVJV - QWIVRWKUVNUUDUUQYFSTZUVIUVPUSAUUDYMUUIYSUVLUURNWLUUKUUQUVLUURUVKWKZUV - NYCSTUVLUVRUVNSSYAEASSEWMYMUUIYSUVLUURJWLUUKUVLUURWNZWOUVTYCYAVDVFZDY - QYFWQWRWSUVNUURUVQUVMUURWTUVNUDUEUFYFYPYQOSRUDUNZSTZUUESTZUUFSTXAUWBU - UEOPUUFUWBRQUUFUUERQOPUSUVNUWBUUEUUFWQVSUWAUUKUUPUVLUURUVHWKUVSUWCUWD - UQUWBUUERQUUEUWBRQVOUVNUWBUUEXBVSXCWSYGUVOYROSXDWEXEVFUUJYSUVLUURXFXG - XHXIXJYIUUNBYLSYKXPYLVOYHUUMGSXPYLYGOVHVIYEYJYDISVPVKYIBSVPVKVLVNXKXH - XLXJXHXM $. + brel sylbi simprd ad2antll eleq1 mpbird ad2antrr ad5antr wf ffvelcdmd + simplr ltanqg syl3anc mpbid simpr addcomnqg caovord2d ltsonq breqtrrd + w3a sotri simpllr ex reximdva mpd sylanbrc rexlimdvva ssrdv ) AUAFIUN + ZDOPIUHZDBUNZOPZBUHZUIZUJQUKULZXNGUNZRQZYAEULZDRQOPGSUOZISUPZYCYARQZD + RQZXPOPZGSUOZBSUPZUIUKULZAUAUNZXTTZYLYKTZAYMUQZYLUBUNZUCUNZRQZVOZUCXS + UKULZUOUBFUKULZUOZYNAYMUUBAFURTXSURTZYMUUBUSABCEFGHIJKLMUTADSTZUUCNBD + IVAVBUDUEUFUGUMFXSYLUBUCUJRUGUMUFUEUDVCUEUNZUFUNZVDVEVFVGYOYSYNUBUCUU + AYTYOYPUUATZYQYTTZUQZUQZYSYNUUJYSUQZYLSTZYGYLOPZGSUOZYNUUKUULYRSTZUUK + YPSTZYQSTZUUOUUKUUPYFYPOPZGSUOZUUJUUPUUSUQZYSUUIUUTYOUUGUUTUUHUUGUUTY + FXPOPZGSUOZUUSBYPSUUAXPYPVOUVAUURGSXPYPYFOVHVIUUAYBYCOPGSUOZISUPZUVBB + SUPZUIZUKULUVEFUVFUKMVJUVDUVEUVCISVPVKUVBBSVPVKVLVMVNVQVRVSVRZVTZUUJU + UQYSUUHUUQYOUUGUUHUUDUUQUUHDYQOPZUUDUUQUQXQUVIBYQYTUCWAXPYQDOVHXOXRID + WBBDWCVLWDZDYQSSOWEWFWGWHWIVRZYPYQVDVFYSUULUUOUSUUJYLYRSWJVSWKUUKUUSU + UNUUKUUPUUSUVGWHUUKUURUUMGSUUKYASTZUQZUURUUMUVMUURUQZYGYRYLOUVNYGYFYQ + RQZOPZUVOYROPZYGYROPUVNUVIUVPUUKUVIUVLUURUUJUVIYSUUHUVIYOUUGUUHUVIUVJ + VQWIVRWLUVNUUDUUQYFSTZUVIUVPUSAUUDYMUUIYSUVLUURNWMUUKUUQUVLUURUVKWLZU + VNYCSTUVLUVRUVNSSYAEASSEWNYMUUIYSUVLUURJWMUUKUVLUURWPZWOUVTYCYAVDVFZD + YQYFWQWRWSUVNUURUVQUVMUURWTUVNUDUEUFYFYPYQOSRUDUNZSTZUUESTZUUFSTXEUWB + UUEOPUUFUWBRQUUFUUERQOPUSUVNUWBUUEUUFWQVSUWAUUKUUPUVLUURUVHWLUVSUWCUW + DUQUWBUUERQUUEUWBRQVOUVNUWBUUEXAVSXBWSYGUVOYROSXCWEXFVFUUJYSUVLUURXGX + DXHXIXJYIUUNBYLSYKXPYLVOYHUUMGSXPYLYGOVHVIYEYJYDISVPVKYIBSVPVKVLVNXKX + HXLXJXHXM $. $( Lemma for ~ cauappcvgprlemladd . The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) $) @@ -78672,43 +78674,43 @@ in HoTT although some of the conditions are redundant (for example, ( cplq co cltq wbr cnq wcel vr vv vf vg vh cfv wrex crab cop c2nd cab cv cpp weq breq2 rexbidv nqex rabex op2nd elrab2 cauappcvgprlemladdfl wa c1st wn wss oveq2 fveq2 oveq1d breq12d cbvrexv a1i rabbiia oveq12d - wb id breq1d opeq12i fveq2i sseqtrdi adantr wceq wf ad2antrr ffvelrnd - simplr addassnqg syl3anc addclnq syl2anc sylancom eqeltrrd wor ltsonq - simpr so2nr mpan addcomnqg adantl w3a caov32d ltanqg caovord2d biimpd - 3bitr2d wral oveq1 oveq2d breq2d anbi12d ralbidv rspcv mpd rsp simpld - wi sylc breqtrd jctird mtod nrexdv ffvelcdmda elrab3 syl mtbird op1st - eleq2i sylnibr ssneldd adantlr wo cnp cauappcvgprlemcl nqprlu addclpr - prop prloc sylan orcomd ecased ex rexlimdva expimpd syl5bi ssrdv ) AU - AIULZGULZOPZUUFEUFZDOPZQRZGSUGZISUHZUUHUUFOPZDOPZBULZQRZGSUGZBSUHZUIZ - UJUFZFUUEDQRIUKDUUOQRBUKUIZUMPZUJUFZUAULZUUTTUVDSTZUUNUVDQRZGSUGZVBAU - 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Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) $) @@ -78843,20 +78845,20 @@ in HoTT although some of the conditions are redundant (for example, L

. ) $= ( cltq wbr cnq wcel vx cfv cplq co cv wa cab cop cltp ltaddnq syl2anc - wrex ffvelrnd ltanqi ltbtwnnqq sylib c2nd simprl adantr simprrl fveq2 - c1st wceq oveq12d breq1d rspcev breq2 rexbidv crab fveq2i rabex op2nd - id nqex eqtri elrab2 sylanbrc simprrr breq1 elab sylibr ltnqex gtnqex - vex op1st eleqtrrdi rspe syl12anc cnp cauappcvgprlemcl addclnq nqprlu - wb syl ltdfpr mpbird rexlimddv ) ADFUBZDUCUDZUAUEZQRZWTWRDEUCUDZUCUDZ - QRZUFZGJUEZXCQRZJUGZXCBUEZQRBUGZUHZUIRZUASAWSXCQRZXEUASULADXBQRZWRSTZ - XMADSTZESTZXNOPDEUJUKASSDFKOUMZDXBWRUNUKUAWSXCUOUPAWTSTZXEUFZUFZXLWTG - UQUBZTZWTXKVBUBZTZUFZUASULZYAXSYCYEYGAXSXEURZYAXSHUEZFUBZYIUCUDZWTQRZ - HSULZYCYHYAXPXAYMAXPXTOUSAXSXAXDUTYLXAHDSYIDVCZYKWSWTQYNYJWRYIDUCYIDF - VAYNVMVDVEVFUKYKXIQRZHSULZYMBWTSYBXIWTVCYOYLHSXIWTYKQVGVHYBXFYIUCUDYJ - QRHSULZJSVIZYPBSVIZUHZUQUBYSGYTUQNVJYRYSYQJSVNVKYPBSVNVKVLVOVPVQYAWTX - HYDYAXDWTXHTAXSXAXDVRXGXDJWTUAWDXFWTXCQVSVTWAXHXJJXCWBBXCWCWEWFYFUASW - GWHYAGWITZXKWITZXLYGWMAUUAXTABCFGHIJKLMNWJUSAUUBXTAXCSTZUUBAXOXBSTZUU - CXRAXPXQUUDOPDEWKUKWRXBWKUKBXCJWLWNUSGXKUAWOUKWPWQ $. + wrex ffvelcdmd ltanqi ltbtwnnqq sylib c2nd c1st simprl adantr simprrl + wceq fveq2 oveq12d breq1d rspcev breq2 rexbidv crab fveq2i nqex rabex + id op2nd eqtri elrab2 sylanbrc simprrr vex breq1 sylibr ltnqex gtnqex + elab op1st eleqtrrdi rspe syl12anc wb cauappcvgprlemcl addclnq nqprlu + cnp syl ltdfpr mpbird rexlimddv ) ADFUBZDUCUDZUAUEZQRZWTWRDEUCUDZUCUD + ZQRZUFZGJUEZXCQRZJUGZXCBUEZQRBUGZUHZUIRZUASAWSXCQRZXEUASULADXBQRZWRST + ZXMADSTZESTZXNOPDEUJUKASSDFKOUMZDXBWRUNUKUAWSXCUOUPAWTSTZXEUFZUFZXLWT + GUQUBZTZWTXKURUBZTZUFZUASULZYAXSYCYEYGAXSXEUSZYAXSHUEZFUBZYIUCUDZWTQR + ZHSULZYCYHYAXPXAYMAXPXTOUTAXSXAXDVAYLXAHDSYIDVBZYKWSWTQYNYJWRYIDUCYID + FVCYNVMVDVEVFUKYKXIQRZHSULZYMBWTSYBXIWTVBYOYLHSXIWTYKQVGVHYBXFYIUCUDY + JQRHSULZJSVIZYPBSVIZUHZUQUBYSGYTUQNVJYRYSYQJSVKVLYPBSVKVLVNVOVPVQYAWT + XHYDYAXDWTXHTAXSXAXDVRXGXDJWTUAVSXFWTXCQVTWDWAXHXJJXCWBBXCWCWEWFYFUAS + WGWHYAGWMTZXKWMTZXLYGWIAUUAXTABCFGHIJKLMNWJUTAUUBXTAXCSTZUUBAXOXBSTZU + UCXRAXPXQUUDOPDEWKUKWRXBWKUKBXCJWLWNUTGXKUAWOUKWPWQ $. $} ${ @@ -78998,41 +79000,41 @@ This proof (including its lemmas) is similar to the proofs of vh wn wceq ltsonq ltrelnq son2lpi simprl cv wi breq1 fveq2 opeq1 eceq1d wral fveq2d oveq2d breq12d oveq12d breq2d anbi12d imbi12d oveq1d breq1d breq2 cbvral2v sylib wcel rspc2v syl2anc mpd imp simpld adantr sotri wb - 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N. A .

. ) ) $= ( cltq cfv wbr cab cop cpp c1st wcel cltp c1o ceq cec crq cplq cnpi - cv co wrex clti wa ltrelpi brel simprd caucvgprlemk ffvelrnd ltanqi - syl cnq syl2anc wceq opeq1 eceq1d fveq2d oveq2d fveq2 oveq1d rspcev - breq12d crab wb oveq1 breq1d rexbidv elrab3 caucvgprlemladdrl sseld - sylbird mpd cnp caucvgprlemcl nqprlu addclpr nqprl mpbid ) AJHUAZKL - UOZDTUBLUCDBUOZTUBBUCUDZUEUPZUFUAZUGZWOWNTUBLUCWNWPTUBBUCUDWRUHUBZA - WNEUOZUIUDZUJUKZULUAZUMUPZXBHUAZDUMUPZTUBZEUNUQZWTAJUNUGZWNJUIUDZUJ - UKZULUAZUMUPZWNDUMUPZTUBZXJAIUNUGZXKAIJURUBXRXKUSRIJUNUNURUTVAVFVBZ - AXNDTUBWNVGUGZXQADIJRSVCAUNVGJHMXSVDZXNDWNVEVHXIXQEJUNXBJVIZXFXOXHX - PTYBXEXNWNUMYBXDXMULYBXCXLUJXBJUIVJVKVLVMYBXGWNDUMXBJHVNVOVQVPVHAXJ - WNWOXEUMUPZXHTUBZEUNUQZLVGVRZUGZWTAXTYGXJVSYAYEXJLWNVGWOWNVIZYDXIEU - NYHYCXFXHTWOWNXEUMVTWAWBWCVFAYFWSWNABCDEFGHKLMNOPQWDWEWFWGAXTWRWHUG - ZWTXAVSYAAKWHUGWQWHUGZYIABCEFGHKLMNOPWIADVGUGYJQBDLWJVFKWQWKVHBWNWR - LWLVHWM $. + cv co wrex clti wa ltrelpi syl simprd caucvgprlemk ffvelcdmd ltanqi + brel syl2anc opeq1 eceq1d fveq2d oveq2d fveq2 oveq1d breq12d rspcev + cnq wceq crab oveq1 breq1d rexbidv elrab3 caucvgprlemladdrl sylbird + wb sseld mpd cnp caucvgprlemcl nqprlu addclpr nqprl mpbid ) AJHUAZK + LUOZDTUBLUCDBUOZTUBBUCUDZUEUPZUFUAZUGZWOWNTUBLUCWNWPTUBBUCUDWRUHUBZ + AWNEUOZUIUDZUJUKZULUAZUMUPZXBHUAZDUMUPZTUBZEUNUQZWTAJUNUGZWNJUIUDZU + JUKZULUAZUMUPZWNDUMUPZTUBZXJAIUNUGZXKAIJURUBXRXKUSRIJUNUNURUTVFVAVB + ZAXNDTUBWNVPUGZXQADIJRSVCAUNVPJHMXSVDZXNDWNVEVGXIXQEJUNXBJVQZXFXOXH + XPTYBXEXNWNUMYBXDXMULYBXCXLUJXBJUIVHVIVJVKYBXGWNDUMXBJHVLVMVNVOVGAX + JWNWOXEUMUPZXHTUBZEUNUQZLVPVRZUGZWTAXTYGXJWEYAYEXJLWNVPWOWNVQZYDXIE + UNYHYCXFXHTWOWNXEUMVSVTWAWBVAAYFWSWNABCDEFGHKLMNOPQWCWFWDWGAXTWRWHU + GZWTXAWEYAAKWHUGWQWHUGZYIABCEFGHKLMNOPWIADVPUGYJQBDLWJVAKWQWKVGBWNW + RLWLVGWM $. $} ${ @@ -79455,24 +79457,24 @@ This proof (including its lemmas) is similar to the proofs of L

. ) $= ( cnq vx cfv c1o cop ceq cec crq cplq co cv cltq wbr cltp wrex wcel - cab caucvgprlemk cnpi clti ltrelpi brel syl simprd ffvelrnd syl2anc - wa ltanqi ltbtwnnqq sylib c2nd c1st simprl simprrl wceq fveq2 opeq1 - adantr eceq1d fveq2d oveq12d breq1d rspcev rexbidv crab fveq2i nqex - breq2 rabex op2nd eqtri elrab2 sylanbrc simprrr breq1 sylibr ltnqex - vex elab gtnqex op1st eleqtrrdi rspe syl12anc caucvgprlemcl addclnq - cnp wb nqprlu ltdfpr mpbird rexlimddv ) AJHUBZJUCUDZUEUFZUGUBZUHUIZ - UAUJZUKULZXQXLDUHUIZUKULZVFZKLUJZXSUKULZLUPZXSBUJZUKULBUPZUDZUMULZU - ATAXPXSUKULZYAUATUNAXODUKULXLTUOZYIADIJRSUQAURTJHMAIURUOZJURUOZAIJU - SULYKYLVFRIJURURUSUTVAVBVCZVDZXODXLVGVEUAXPXSVHVIAXQTUOZYAVFZVFZYHX - QKVJUBZUOZXQYGVKUBZUOZVFZUATUNZYQYOYSUUAUUCAYOYAVLZYQYOEUJZHUBZUUEU - CUDZUEUFZUGUBZUHUIZXQUKULZEURUNZYSUUDYQYLXRUULAYLYPYMVQAYOXRXTVMUUK - XREJURUUEJVNZUUJXPXQUKUUMUUFXLUUIXOUHUUEJHVOUUMUUHXNUGUUMUUGXMUEUUE - JUCVPVRVSVTWAWBVEUUJYEUKULZEURUNZUULBXQTYRYEXQVNUUNUUKEURYEXQUUJUKW - GWCYRYBUUIUHUIUUFUKULEURUNZLTWDZUUOBTWDZUDZVJUBUURKUUSVJPWEUUQUURUU - PLTWFWHUUOBTWFWHWIWJWKWLYQXQYDYTYQXTXQYDUOAYOXRXTWMYCXTLXQUAWQYBXQX - SUKWNWRWOYDYFLXSWPBXSWSWTXAUUBUATXBXCYQKXFUOZYGXFUOZYHUUCXGAUUTYPAB - CEFGHKLMNOPXDVQAUVAYPAXSTUOZUVAAYJDTUOUVBYNQXLDXEVEBXSLXHVBVQKYGUAX - IVEXJXK $. + cab caucvgprlemk cnpi clti ltrelpi brel syl simprd ffvelcdmd ltanqi + syl2anc ltbtwnnqq sylib c2nd c1st simprl adantr simprrl fveq2 opeq1 + wa wceq eceq1d fveq2d oveq12d breq1d rspcev breq2 rexbidv crab nqex + fveq2i rabex op2nd eqtri elrab2 sylanbrc simprrr elab sylibr ltnqex + vex breq1 gtnqex op1st eleqtrrdi rspe syl12anc cnp wb caucvgprlemcl + addclnq nqprlu ltdfpr mpbird rexlimddv ) AJHUBZJUCUDZUEUFZUGUBZUHUI + ZUAUJZUKULZXQXLDUHUIZUKULZVPZKLUJZXSUKULZLUPZXSBUJZUKULBUPZUDZUMULZ + UATAXPXSUKULZYAUATUNAXODUKULXLTUOZYIADIJRSUQAURTJHMAIURUOZJURUOZAIJ + USULYKYLVPRIJURURUSUTVAVBVCZVDZXODXLVEVFUAXPXSVGVHAXQTUOZYAVPZVPZYH + XQKVIUBZUOZXQYGVJUBZUOZVPZUATUNZYQYOYSUUAUUCAYOYAVKZYQYOEUJZHUBZUUE + UCUDZUEUFZUGUBZUHUIZXQUKULZEURUNZYSUUDYQYLXRUULAYLYPYMVLAYOXRXTVMUU + KXREJURUUEJVQZUUJXPXQUKUUMUUFXLUUIXOUHUUEJHVNUUMUUHXNUGUUMUUGXMUEUU + EJUCVOVRVSVTWAWBVFUUJYEUKULZEURUNZUULBXQTYRYEXQVQUUNUUKEURYEXQUUJUK + WCWDYRYBUUIUHUIUUFUKULEURUNZLTWEZUUOBTWEZUDZVIUBUURKUUSVIPWGUUQUURU + UPLTWFWHUUOBTWFWHWIWJWKWLYQXQYDYTYQXTXQYDUOAYOXRXTWMYCXTLXQUAWQYBXQ + XSUKWRWNWOYDYFLXSWPBXSWSWTXAUUBUATXBXCYQKXDUOZYGXDUOZYHUUCXEAUUTYPA + BCEFGHKLMNOPXFVLAUVAYPAXSTUOZUVAAYJDTUOUVBYNQXLDXGVFBXSLXHVBVLKYGUA + XIVFXJXK $. $} $} @@ -79739,18 +79741,18 @@ This proof (including its lemmas) is similar to the proofs of ( wbr cltp cnp wcel vf vg vh clti wa cv c1o cop ceq cec crq cplq cltq cfv co cab ltsopr ltrelpr simprl caucvgprprlemval simpld adantr sotri cpp son2lpi syl2anc w3a wb ltaprg adantl cnq ad2antrr nqprlu syl cnpi - ffvelrnd recnnpr wceq addcomprg caovord2d mpbird simprr jca mtoi nnnq - ltaddpr ex recclnq 3syl addnqpr breq1d anbi1d mtbird ) AHGUDQZUEZJUFZ - CHUGUHUIUJZUKUNZULUOZUMQJUPWSIUFZUMQIUPUHZHFUNZRQZGFUNZWPGUGUHUIUJUKU - NZUMQJUPXEWTUMQIUPUHZVDUOZWPCUMQJUPCWTUMQIUPUHZRQZUEXHWPWRUMQJUPWRWTU - MQIUPUHZVDUOZXBRQZXIUEZWOXMXHXDRQZXDXHRQZUEZXHXDRSUQURVEWOXMXPWOXMUEZ - XNXOXQXNXKXDXJVDUOZRQZXQXLXBXRRQZXSWOXLXIUSWOXTXMWOXTXDXBXJVDUORQABHG - DEFIJKLMUTVAVBXKXBXRRSUQURVCVFXQUAUBUCXHXDXJRSVDUAUFZSTZUBUFZSTZUCUFZ - STVGYAYCRQYEYAVDUOYEYCVDUORQVHXQYAYCYEVIVJXQCVKTZXHSTAYFWNXMPVLICJVMV - NAXDSTZWNXMAVOSGFLOVPVLZAXJSTZWNXMAHVOTZYINIHJVQVNVLYBYDUEYAYCVDUOYCY - AVDUOVRXQYAYCVSVJVTWAXQXDXGRQZXIXOXQYGXFSTZYKYHAYLWNXMAGVOTYLOIGJVQVN - VLXDXFWFVFWOXLXIWBXDXGXHRSUQURVCVFWCWGWDWOXCXLXIWOXAXKXBRWOYFWRVKTZXA - XKVRAYFWNPVBAYMWNAYJWQVKTYMNHWEWQWHWIVBICWRJWJVFWKWLWM $. + ffvelcdmd recnnpr wceq addcomprg caovord2d mpbird ltaddpr simprr mtoi + jca ex nnnq recclnq 3syl addnqpr breq1d anbi1d mtbird ) AHGUDQZUEZJUF + ZCHUGUHUIUJZUKUNZULUOZUMQJUPWSIUFZUMQIUPUHZHFUNZRQZGFUNZWPGUGUHUIUJUK + UNZUMQJUPXEWTUMQIUPUHZVDUOZWPCUMQJUPCWTUMQIUPUHZRQZUEXHWPWRUMQJUPWRWT + UMQIUPUHZVDUOZXBRQZXIUEZWOXMXHXDRQZXDXHRQZUEZXHXDRSUQURVEWOXMXPWOXMUE + ZXNXOXQXNXKXDXJVDUOZRQZXQXLXBXRRQZXSWOXLXIUSWOXTXMWOXTXDXBXJVDUORQABH + GDEFIJKLMUTVAVBXKXBXRRSUQURVCVFXQUAUBUCXHXDXJRSVDUAUFZSTZUBUFZSTZUCUF + ZSTVGYAYCRQYEYAVDUOYEYCVDUORQVHXQYAYCYEVIVJXQCVKTZXHSTAYFWNXMPVLICJVM + VNAXDSTZWNXMAVOSGFLOVPVLZAXJSTZWNXMAHVOTZYINIHJVQVNVLYBYDUEYAYCVDUOYC + YAVDUOVRXQYAYCVSVJVTWAXQXDXGRQZXIXOXQYGXFSTZYKYHAYLWNXMAGVOTYLOIGJVQV + NVLXDXFWBVFWOXLXIWCXDXGXHRSUQURVCVFWEWFWDWOXCXLXIWOXAXKXBRWOYFWRVKTZX + AXKVRAYFWNPVBAYMWNAYJWQVKTYMNHWGWQWHWIVBICWRJWJVFWKWLWM $. $} ${ @@ -79765,20 +79767,20 @@ This proof (including its lemmas) is similar to the proofs of { q | ( *Q ` [ <. J , 1o >. ] ~Q ) . )

. ) ) $= ( cltq wbr cab cltp wceq wa cv c1o cop ceq cec crq cfv cplq co ltsopr - cpp cnp ltrelpr son2lpi wcel ffvelrnd ad2antrr cnq adantr syl recclnq - cnpi nnnq nqprlu ltaddpr syl2anc simprr sotri simprl wb addnqpr mpbid - breq1d ex mtoi opeq1 eceq1d fveq2d oveq2d breq2d abbidv opeq12d fveq2 - jca breq12d anbi1d adantl mtbird ) AHGUAZUBZJUCZCHUDUEZUFUGZUHUIZUJUK - ZQRZJSZWQIUCZQRZISZUEZHFUIZTRZGFUIZWMGUDUEZUFUGZUHUIZQRJSXIWTQRISUEZU - MUKZWMCQRJSCWTQRISUEZTRZUBZWMCXIUJUKZQRZJSZXOWTQRZISZUEZXFTRZXMUBZWLY - BXFXLTRZXLXFTRZUBZXFXLTUNULUOUPWLYBYEWLYBUBZYCYDYFXFXKTRZXMYCYFXFUNUQ - ZXJUNUQZYGAYHWKYBAVDUNGFLOURUSWLYIYBWLXIUTUQZYIWLXHUTUQZYJWLGVDUQZYKA - YLWKOVAGVEVBXHVCVBZIXIJVFVBZVAXFXJVGVHWLYAXMVIXFXKXLTUNULUOVJVHYFXLXL - XJUMUKZTRZYOXFTRZYDWLYPYBWLXLUNUQZYIYPWLCUTUQZYRAYSWKPVAZICJVFVBYNXLX - JVGVHVAYFYAYQWLYAXMVKWLYAYQVLYBWLXTYOXFTWLYSYJXTYOUAYTYMICXIJVMVHVOVA - VNXLYOXFTUNULUOVJVHWFVPVQWKXNYBVLAWKXEYAXMWKXCXTXDXFTWKWSXQXBXSWKWRXP - JWKWQXOWMQWKWPXICUJWKWOXHUHWKWNXGUFHGUDVRVSVTWAZWBWCWKXAXRIWKWQXOWTQU - UAVOWCWDHGFWEWGWHWIWJ $. + cpp cnp ltrelpr son2lpi wcel cnpi ffvelcdmd ad2antrr cnq nnnq recclnq + adantr syl nqprlu ltaddpr syl2anc simprr sotri simprl wb breq1d mpbid + addnqpr jca ex opeq1 eceq1d fveq2d oveq2d breq2d abbidv opeq12d fveq2 + mtoi breq12d anbi1d adantl mtbird ) AHGUAZUBZJUCZCHUDUEZUFUGZUHUIZUJU + KZQRZJSZWQIUCZQRZISZUEZHFUIZTRZGFUIZWMGUDUEZUFUGZUHUIZQRJSXIWTQRISUEZ + UMUKZWMCQRJSCWTQRISUEZTRZUBZWMCXIUJUKZQRZJSZXOWTQRZISZUEZXFTRZXMUBZWL + YBXFXLTRZXLXFTRZUBZXFXLTUNULUOUPWLYBYEWLYBUBZYCYDYFXFXKTRZXMYCYFXFUNU + QZXJUNUQZYGAYHWKYBAURUNGFLOUSUTWLYIYBWLXIVAUQZYIWLXHVAUQZYJWLGURUQZYK + AYLWKOVDGVBVEXHVCVEZIXIJVFVEZVDXFXJVGVHWLYAXMVIXFXKXLTUNULUOVJVHYFXLX + LXJUMUKZTRZYOXFTRZYDWLYPYBWLXLUNUQZYIYPWLCVAUQZYRAYSWKPVDZICJVFVEYNXL + XJVGVHVDYFYAYQWLYAXMVKWLYAYQVLYBWLXTYOXFTWLYSYJXTYOUAYTYMICXIJVOVHVMV + DVNXLYOXFTUNULUOVJVHVPVQWFWKXNYBVLAWKXEYAXMWKXCXTXDXFTWKWSXQXBXSWKWRX + PJWKWQXOWMQWKWPXICUJWKWOXHUHWKWNXGUFHGUDVRVSVTWAZWBWCWKXAXRIWKWQXOWTQ + UUAVMWCWDHGFWEWGWHWIWJ $. $} ${ @@ -79842,24 +79844,24 @@ This proof (including its lemmas) is similar to the proofs of { u | ( *Q ` [ <. J , 1o >. ] ~Q ) . )

E. t e. Q. t e. ( 2nd ` L ) ) $= ( wcel cltq wbr vx vf vg vh cv c1o cfv c2nd cnq wrex cnp c1st cop 1pi - cnpi a1i ffvelrnd prop prmu 3syl wa c1q co simprl 1nq addclnq sylancl - cplq ceq cec crq cab cltp simprr wb adantr nqpru syl2anc mpbid ltaprg - cpp w3a adantl nqprlu syl mp1i wceq addcomprg caovord2d fveq2i rec1nq + cnpi a1i ffvelcdmd prop prmu 3syl c1q cplq simprl 1nq addclnq sylancl + wa co ceq cec crq cab cpp simprr wb adantr nqpru syl2anc mpbid ltaprg + cltp w3a adantl nqprlu syl mp1i wceq addcomprg df-1nqqs fveq2i rec1nq eqtr3i breq2i abbii breq1i opeq12i oveq2i addnqpr 3brtr4d fveq2 opeq1 eceq1d fveq2d breq2d abbidv breq1d opeq12d oveq12d rspcev breq2 breq1 - df-1nqqs rexbidv crab nqex rabex op2nd eqtri sylanbrc eleq1 rexlimddv - elrab2 ) AUAUEZUFHUGZUHUGZRZCUEZIUHUGZRZCUIUJZUAUIAYDUKRZYDULUGZYEUMU - KRYFUAUIUJAUOUKUFHNUFUORZAUNUPUQZYDURUAYEYLUSUTAYCUIRZYFVAZVAZYCVBVHV - CZUIRZYRYHRZYJYQYOVBUIRZYSAYOYFVDZVEYCVBVFVGZYQYSJUEZHUGZLUEZUUDUFUMZ - VIVJZVKUGZSTZLVLZUUIKUEZSTZKVLZUMZWAVCZUUFYRSTZLVLZYRUULSTZKVLZUMZVMT - ZJUOUJZYTUUCYQYMYDUUFUFUFUMZVIVJZVKUGZSTZLVLZUVFUULSTZKVLZUMZWAVCZUVA - VMTZUVCYMYQUNUPYQYDUUFVBSTZLVLZVBUULSTZKVLZUMZWAVCZUUFYCSTLVLYCUULSTK - VLUMZUVRWAVCZUVLUVAVMYQYDUVTVMTZUVSUWAVMTYQYFUWBAYOYFVNYQYOYKYFUWBVOU - UBAYKYPYNVPZKYCYDLVQVRVSYQUBUCUDYDUVTUVRVMUKWAUBUEZUKRZUCUEZUKRZUDUEZ - UKRWBUWDUWFVMTUWHUWDWAVCUWHUWFWAVCVMTVOYQUWDUWFUWHVTWCUWCYQYOUVTUKRUU - BKYCLWDWEUUAUVRUKRYQVEKVBLWDWFUWEUWGVAUWDUWFWAVCUWFUWDWAVCWGYQUWDUWFW - HWCWIVSUVLUVSWGYQUVKUVRYDWAUVHUVOUVJUVQUVGUVNLUVFVBUUFSVBVKUGUVFVBVBU - VEVKXLWJWKWLZWMWNUVIUVPKUVFVBUULSUWIWOWNWPWQUPYQYOUUAUVAUWAWGUUBVEKYC - VBLWRVGWSUVBUVMJUFUOUUDUFWGZUUPUVLUVAVMUWJUUEYDUUOUVKWAUUDUFHWTUWJUUK - UVHUUNUVJUWJUUJUVGLUWJUUIUVFUUFSUWJUUHUVEVKUWJUUGUVDVIUUDUFUFXAXBXCZX - DXEUWJUUMUVIKUWJUUIUVFUULSUWKXFXEXGXHXFXIVRUUPUUFBUEZSTZLVLZUWLUULSTZ - KVLZUMZVMTZJUOUJZUVCBYRUIYHUWLYRWGZUWRUVBJUOUWTUWQUVAUUPVMUWTUWNUURUW - PUUTUWTUWMUUQLUWLYRUUFSXJXEUWTUWOUUSKUWLYRUULSXKXEXGXDXMYHUUFMUEUUIVH - VCZSTLVLUXAUULSTKVLUMUUEVMTJUOUJZMUIXNZUWSBUIXNZUMZUHUGUXDIUXEUHQWJUX - CUXDUXBMUIXOXPUWSBUIXOXPXQXRYBXSYIYTCYRUIYGYRYHXTXIVRYA $. + caovord2d crab nqex rabex op2nd eqtri elrab2 sylanbrc eleq1 rexlimddv + rexbidv ) AUAUEZUFHUGZUHUGZRZCUEZIUHUGZRZCUIUJZUAUIAYDUKRZYDULUGZYEUM + UKRYFUAUIUJAUOUKUFHNUFUORZAUNUPUQZYDURUAYEYLUSUTAYCUIRZYFVGZVGZYCVAVB + VHZUIRZYRYHRZYJYQYOVAUIRZYSAYOYFVCZVDYCVAVEVFZYQYSJUEZHUGZLUEZUUDUFUM + ZVIVJZVKUGZSTZLVLZUUIKUEZSTZKVLZUMZVMVHZUUFYRSTZLVLZYRUULSTZKVLZUMZWA + TZJUOUJZYTUUCYQYMYDUUFUFUFUMZVIVJZVKUGZSTZLVLZUVFUULSTZKVLZUMZVMVHZUV + AWATZUVCYMYQUNUPYQYDUUFVASTZLVLZVAUULSTZKVLZUMZVMVHZUUFYCSTLVLYCUULST + KVLUMZUVRVMVHZUVLUVAWAYQYDUVTWATZUVSUWAWATYQYFUWBAYOYFVNYQYOYKYFUWBVO + UUBAYKYPYNVPZKYCYDLVQVRVSYQUBUCUDYDUVTUVRWAUKVMUBUEZUKRZUCUEZUKRZUDUE + ZUKRWBUWDUWFWATUWHUWDVMVHUWHUWFVMVHWATVOYQUWDUWFUWHVTWCUWCYQYOUVTUKRU + UBKYCLWDWEUUAUVRUKRYQVDKVALWDWFUWEUWGVGUWDUWFVMVHUWFUWDVMVHWGYQUWDUWF + WHWCXLVSUVLUVSWGYQUVKUVRYDVMUVHUVOUVJUVQUVGUVNLUVFVAUUFSVAVKUGUVFVAVA + UVEVKWIWJWKWLZWMWNUVIUVPKUVFVAUULSUWIWOWNWPWQUPYQYOUUAUVAUWAWGUUBVDKY + CVALWRVFWSUVBUVMJUFUOUUDUFWGZUUPUVLUVAWAUWJUUEYDUUOUVKVMUUDUFHWTUWJUU + KUVHUUNUVJUWJUUJUVGLUWJUUIUVFUUFSUWJUUHUVEVKUWJUUGUVDVIUUDUFUFXAXBXCZ + XDXEUWJUUMUVIKUWJUUIUVFUULSUWKXFXEXGXHXFXIVRUUPUUFBUEZSTZLVLZUWLUULST + ZKVLZUMZWATZJUOUJZUVCBYRUIYHUWLYRWGZUWRUVBJUOUWTUWQUVAUUPWAUWTUWNUURU + WPUUTUWTUWMUUQLUWLYRUUFSXJXEUWTUWOUUSKUWLYRUULSXKXEXGXDYBYHUUFMUEUUIV + BVHZSTLVLUXAUULSTKVLUMUUEWATJUOUJZMUIXMZUWSBUIXMZUMZUHUGUXDIUXEUHQWJU + XCUXDUXBMUIXNXOUWSBUIXNXOXPXQXRXSYIYTCYRUIYGYRYHXTXIVRYA $. $} ${ @@ -79966,34 +79968,34 @@ This proof (including its lemmas) is similar to the proofs of E. t e. Q. ( s . ] ~Q ) . ) )

X

( F ` K )

L

. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) . ) ) ) ) $= ( wbr cop co cnpi c1p cnp cv clti cfv c1o ceq cec crq cltq cab cltp - cpp wa wi wral cer cplr cltr wcel wb caucvgsrlemf ad2antrr ffvelrnd - wf simplr adantr ffvelcdmda recnnpr syl addclpr syl2anc prsrlt wceq - caucvgsrlemfv prsradd adantlr oveq1d eqtrd breq12d anbi12d ralbidva - bitrd imbi2d mpbird ) AGUAZEUAZUBOZWDIUCZWEIUCZJUAWDUDPUEUFUGUCZUHO - JUIWIDUAUHODUIPZUKQZUJOZWHWGWJUKQZUJOZULZUMZERUNZGRUNWFWDHUCZWEHUCZ - WJSUKQSPUOUFZUPQZUQOZWSWRWTUPQZUQOZULZUMZERUNZGRUNLAWQXGGRAWDRURZUL - ZWPXFERXIWERURZULZWOXEWFXKWLXBWNXDXKWLWGSUKQSPUOUFZWKSUKQSPUOUFZUQO - ZXBXKWGTURZWKTURZWLXNUSXKRTWDIARTIVCZXHXJABCDEFGHIJKLMNUTZVAAXHXJVD - ZVBZXKWHTURZWJTURZXPXIRTWEIAXQXHXRVEVFZXKXHYBXSDWDJVGVHZWHWJVIVJWGW - KVKVJXKXLWRXMXAUQXIXLWRVLXJABCDWDEFGHIJKLMNVMVEZXKXMWHSUKQSPUOUFZWT - UPQZXAXKYAYBXMYGVLYCYDWHWJVNVJXKYFWSWTUPAXJYFWSVLXHABCDWEEFGHIJKLMN - VMVOZVPVQVRWAXKWNYFWMSUKQSPUOUFZUQOZXDXKYAWMTURZWNYJUSYCXKXOYBYKXTY - DWGWJVIVJWHWMVKVJXKYFWSYIXCUQYHXKYIXLWTUPQZXCXKXOYBYIYLVLXTYDWGWJVN - VJXKXLWRWTUPYEVPVQVRWAVSWBVTVTWC $. + cpp wa wi wral cplr cltr wcel wb wf caucvgsrlemf ad2antrr ffvelcdmd + cer simplr adantr ffvelcdmda recnnpr syl addclpr wceq caucvgsrlemfv + syl2anc prsrlt prsradd adantlr eqtrd breq12d bitrd anbi12d ralbidva + oveq1d imbi2d mpbird ) AGUAZEUAZUBOZWDIUCZWEIUCZJUAWDUDPUEUFUGUCZUH + OJUIWIDUAUHODUIPZUKQZUJOZWHWGWJUKQZUJOZULZUMZERUNZGRUNWFWDHUCZWEHUC + ZWJSUKQSPVCUFZUOQZUPOZWSWRWTUOQZUPOZULZUMZERUNZGRUNLAWQXGGRAWDRUQZU + LZWPXFERXIWERUQZULZWOXEWFXKWLXBWNXDXKWLWGSUKQSPVCUFZWKSUKQSPVCUFZUP + OZXBXKWGTUQZWKTUQZWLXNURXKRTWDIARTIUSZXHXJABCDEFGHIJKLMNUTZVAAXHXJV + DZVBZXKWHTUQZWJTUQZXPXIRTWEIAXQXHXRVEVFZXKXHYBXSDWDJVGVHZWHWJVIVLWG + WKVMVLXKXLWRXMXAUPXIXLWRVJXJABCDWDEFGHIJKLMNVKVEZXKXMWHSUKQSPVCUFZW + TUOQZXAXKYAYBXMYGVJYCYDWHWJVNVLXKYFWSWTUOAXJYFWSVJXHABCDWEEFGHIJKLM + NVKVOZWAVPVQVRXKWNYFWMSUKQSPVCUFZUPOZXDXKYAWMTUQZWNYJURYCXKXOYBYKXT + YDWGWJVIVLWHWMVMVLXKYFWSYIXCUPYHXKYIXLWTUOQZXCXKXOYBYIYLVJXTYDWGWJV + NVLXKXLWRWTUOYEWAVPVQVRVSWBVTVTWC $. $} ${ @@ -81989,38 +81991,38 @@ specified by (11.2.9) in HoTT whereas this theorem fixes the rate of wi eqid caucvgsrlembound caucvgprpr wcel prsrcl ad2antrl oveq2 breq2d cnr anbi12d imbi2d rexralbidv simplrr adantr wreu srpospr riotacl syl adantll rspcdva nfv nfcv nfre1 nfralya nfan nfra1 nfrexya wb wf simpr - ad4antr ffvelrnd simplrl addclpr syl2anc prsrlt caucvgsrlemfv adantlr - prsradd oveq2d eqtrd breq12d bitrd oveq12d 3bitrd ralbida mpbid breq2 - prsrriota rexbid fveq2 breq1d oveq1d imbi12d cbvralv rexbii ralrimiva - sylib ex oveq1 breq1 ralbidv rspcev rexlimddv ) AFUFZGUFZUGOZYJUAPUAU - FJUHUBUFQUIRQUJUKULUMUBSUNUOZUHZUCUFZUDUFZUIRZUPOZYNYMYOUIRZUPOZUQZVD - ZGPURZFPUSZUDSURZUTBUFZTOZYIEUFZUGOZUUGJUHZCUFZUUEVARZTOZUUJUUIUUEVAR - ZTOZUQZVDZEPURFPUSZVDZBVMURZCVMUSZUCSAUDUCDQFGHIYLKAUAUBDGHIJYLKLMNYL - VEZVBZAUAUBDGHIJYLKLMNUVAVCAUAUBDGHIJYLKLMNUVAVFVGAYNSVHZUUDUQZUQZYNQ - UIRQUJUKULZVMVHZUUFUUHUUIUVFUUEVARZTOZUVFUUMTOZUQZVDZEPURZFPUSZVDZBVM - URZUUTUVCUVGAUUDYNVIVJUVEUVOBVMUVEUUEVMVHZUQZUUFUVNUVRUUFUQZYKYJJUHZU - VHTOZUVFUVTUUEVARZTOZUQZVDZGPURZFPUSZUVNUVSYKYMYNUEUFQUIRQUJUKULUUEUM - ZUESUNZUIRZUPOZYNYMUWIUIRZUPOZUQZVDZGPURZFPUSZUWGUVSUUCUWQUDSUWIYOUWI - UMZUUAUWOFGPPUWRYTUWNYKUWRYQUWKYSUWMUWRYPUWJYMUPYOUWIYNUIVKVLUWRYRUWL - YNUPYOUWIYMUIVKVLVNVOVPUVRUUDUUFAUVCUUDUVQVQVRUVQUUFUWISVHZUVEUVQUUFU - 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c1r eqcomd wf ad2antrr simpr ffvelrnd cnp simplr recnnpr prsrcl 1sr - 3syl a1i addcomsrg adantl addasssrg caov32d adantlr caucvgsrlemofff - w3a oveq1d caucvgsrlemasr 3eqtr2d breq12d wb ltasrg addclsr syl2anc - caovord2d 3bitr4d anbi12d biimpd imim2d ralimdva mpd ) AFULZDULZUCU - DZWSGUEZWTGUEZJULWSUFUGUHUIUJUEZUMUDJUNXDBULUMUDBUNUGZUOUPOUOUGUQUI - ZPOZQUDZXCXBXFPOZQUDZURZUSZDRUTZFRUTXAWSHUEZWTHUEZXFPOZQUDZXOXNXFPO - ZQUDZURZUSZDRUTZFRUTLAXMYBFRAWSRSZURZXLYADRYDWTRSZURZXKXTXAYFXKXTYF - XHXQXJXSYFXBVDPOZXGVDPOZQUDXNCPOZXPCPOZQUDXHXQYFYGYIYHYJQYFYIYGYDYI - YGVAYEABCDEFGHWSIJKLMNVBVCZVEYFYHXCVDPOZXFPOXOCPOZXFPOYJYFUAUBUKXCX - FVDTPYFRTWTGARTGVFYCYEKVGZYDYEVHZVIZYFYCXEVJSXFTSZAYCYEVKZBWSJVLXEV - MVOZVDTSYFVNVPZUAULZTSZUBULZTSZURUUAUUCPOZUUCUUAPOVAYFUUAUUCVQVRZUU - BUUDUKULZTSWCZUUEUUGPOUUAUUCUUGPOPOVAYFUUAUUCUUGVSVRZVTYFYMYLXFPAYE - YMYLVAYCABCDEFGHWTIJKLMNVBWAZWDYFUAUBUKXOCXFTPYFRTWTHARTHVFYCYEABCD - EFGHIJKLMNWBVGZYOVIZACTSYCYEACEGMWEVGZYSUUFUUIVTWFWGYFUAUBUKXBXGVDQ - TPUUHUUAUUCQUDUUGUUAPOUUGUUCPOQUDWHYFUUAUUCUUGWIVRZYFRTWSGYNYRVIZYF - 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sylancl c1r w3a ltasrg adantl ad3antrrr simpr ffvelrnd simpllr wceq - simplr addcomsrg caovord2d caucvgsrlemoffval breq1d bitrd addasssrg - wf adantlr caov32d breq2d ad2antrr ffvelcdmda 1sr a1i syl3anc mp2an - m1p1sr eqtri oveq2i 0idsr eqtrid eqtrd breq12d 3bitrd biimpd oveq1d - eqtrdi anim12d imim2d ralimdva breq2 anbi12d imbi12d cbvralv syl6ib - syl fveq2 reximdv impr oveq1 breq1 imbi2d rexralbidv ralbidv rspcev - rexlimddv ) AUFBUGZUHUIZFUGZUAUGZUJUIZYJKUOZUBUGZYGRSZUHUIZYMYLYGRS - ZUHUIZUKZUPZUAULUMZFULUNZUPZBTUMZYHYIGUGZUJUIZUUDJUOZCUGZYGRSZUHUIZ - UUGUUFYGRSZUHUIZUKZUPZGULUMFULUNZUPZBTUMZCTUNZUBTABUBDUAFGHIKMADEGH - IJKLMNOPQVBZADEGHIJKLMNOPQUQADEGHIJKLMNOPQURUSAYMTUTZUUCUKZUKZYMERS - ZVARSZTUTZYHUUEUUFUVCYGRSZUHUIZUVCUUJUHUIZUKZUPZGULUMZFULUNZUPZBTUM - ZUUQUVAUVBTUTZVATUTZUVDUVAUUSETUTZUVNAUUSUUCVCAUVPUUTAEHJPVDZVEYMEV - FZVGVHUVBVAVFVJAUUSUUCUVMAUUSUKZUUBUVLBTUVSYGTUTZUKZUUAUVKYHUWAYTUV - JFULUWAYTYKYJJUOZUVEUHUIZUVCUWBYGRSZUHUIZUKZUPZUAULUMUVJUWAYSUWGUAU - LUWAYJULUTZUKZYRUWFYKUWIYOUWCYQUWEUWIYOUWCUWIYOUWBVKRSZYNERSZUHUIZU - 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adantr ffvelrnd ltadd1sr adantl breqtrd wn nlt1pig pm2.21d w3o pitri3or - mpan mpjao3dan w3a wb ltasrg ffvelcdmda 1sr addclsr addcomsrg caovord2d - sylancl m1r mpbid addasssrg syl3anc c0r mp2an m1p1sr eqtri 0idsr eqtrid - eqtrd ralrimiva caucvgsrlembnd ) ABCDLHUEZUFMNZEFUAGHIJKAUUDUAUGZHUEZOP - UAQAUUEQUHZUIZUUDUUFRMNZUFMNZUUFOUUHUUCUUIOPZUUDUUJOPUUHLUUEUJPZUUKLUUE - UKZUUELUJPZUUHUULUUKUUHUULUUCUUFIUGZLLULZUMUNZUOUEZUPPZIUQZUURDUGZUPPZD - UQZULZSURNZSULZVEUNZMNZOPZUUKALFUGZUJPZUUCUVJHUEZUVGMNZOPZUSZFQUTZUUGUU - LUVIUSZAUVKUVNUVLUUCUVGMNZOPZUIZUSZFQUTZUVPAGUGZUVJUJPZUWCHUEZUVLUUOUWC - LULZUMUNZUOUEZUPPZIUQZUWHUVAUPPZDUQZULZSURNZSULZVEUNZMNZOPZUVLUWEUWPMNZ - OPZUIZUSZFQUTUWBGQLUWCLUKZUXBUWAFQUXCUWDUVKUXAUVTUWCLUVJUJVAUXCUWRUVNUW - TUVSUXCUWEUUCUWQUVMOUWCLHVBZUXCUWPUVGUVLMUXCUWOUVFVEUXCUWNUVESUXCUWMUVD - SURUXCUWJUUTUWLUVCUXCUWIUUSIUXCUWHUURUUOUPUXCUWGUUQUOUXCUWFUUPUMUWCLLVC - VDVFZVGVHUXCUWKUVBDUXCUWHUURUVAUPUXEVIVHVJVOVPVDZVQVKUXCUWSUVRUVLOUXCUW - EUUCUWPUVGMUXDUXFVLVGVMVNVRKLQUHZAVSWFVTUWAUVOFQUVTUVNUVKUVNUVSWAWBWGWH - 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(Contributed by wa cv wi wex elreal df-rex breq2 oveq1 eqeq1d anbi2d rexbidv imbi12d cltr df-0 breq1i ltresr cmr recexgt0sr opelreal anbi1i a1i mulresr df-1 eqeq2i wb eqid 1sr 0r opthg2 mp2an mpbiran2 bitrdi anbi12d pm5.32da bitrid oveq2 - rspcev syl6bir expd rexlimdv syl5 syl5bi gencl imp ) BEFZGBHIZGAUAZHIZBWF - JKZLMZTZAENZGCUAZOPZHIZWGWMWFJKZLMZTZAENZUBWEWKUBWLQFZWDCWMBWDWMBMZCQNWSW - TTCUCCBUDWTCQUERWTWNWEWRWKWMBGHUFWTWQWJAEWTWPWIWGWTWOWHLWMBWFJUGUHUIUJUKW - NOWLULIZWSWRWNOOPZWMHIXAGXBWMHUMUNOWLUORXAODUAZULIZWLXCUPKZSMZTZDQNWSWRDW - LUQWSXGWRDQWSXCQFZXGWRWSXHXGTZXCOPZEFZGXJHIZWMXJJKZLMZTZTZWRXPXHXOTWSXIXK - XHXOXCURUSWSXHXOXGWSXHTZXLXDXNXFXLXDVDXQXLXBXJHIXDGXBXJHUMUNOXCUORUTXQXNX - EOPZLMZXFXQXMXRLWLXCVAUHXSXRSOPZMZXFLXTXRVBVCYAXFOOMZOVESQFOQFYAXFYBTVDVF - VGXEOSOQQVHVIVJRVKVLVMVNWQXOAXJEWFXJMZWGXLWPXNWFXJGHUFYCWOXMLWFXJWMJVOUHV - LVPVQVRVSVTWAWBWC $. + rspcev syl6bir expd rexlimdv syl5 biimtrid gencl imp ) BEFZGBHIZGAUAZHIZB + WFJKZLMZTZAENZGCUAZOPZHIZWGWMWFJKZLMZTZAENZUBWEWKUBWLQFZWDCWMBWDWMBMZCQNW + SWTTCUCCBUDWTCQUERWTWNWEWRWKWMBGHUFWTWQWJAEWTWPWIWGWTWOWHLWMBWFJUGUHUIUJU + KWNOWLULIZWSWRWNOOPZWMHIXAGXBWMHUMUNOWLUORXAODUAZULIZWLXCUPKZSMZTZDQNWSWR + DWLUQWSXGWRDQWSXCQFZXGWRWSXHXGTZXCOPZEFZGXJHIZWMXJJKZLMZTZTZWRXPXHXOTWSXI + XKXHXOXCURUSWSXHXOXGWSXHTZXLXDXNXFXLXDVDXQXLXBXJHIXDGXBXJHUMUNOXCUORUTXQX + NXEOPZLMZXFXQXMXRLWLXCVAUHXSXRSOPZMZXFLXTXRVBVCYAXFOOMZOVESQFOQFYAXFYBTVD + VFVGXEOSOQQVHVIVJRVKVLVMVNWQXOAXJEWFXJMZWGXLWPXNWFXJGHUFYCWOXMLWFXJWMJVOU + HVLVPVQVRVSVTWAWBWC $. $} ${ @@ -83800,17 +83802,17 @@ natural numbers (notably, ~ axcaucvg ). (Contributed by Jim Kingdon, ( cop cnr cnpi wcel wa cv c1o ceq cec cltq wbr cab c1p cpp co cer c0r cfv wceq crio opeq1 eceq1d breq2d abbidv breq1d opeq12d oveq1d opeq1d cmpt fveq2d eqeq1d riotabidv adantl simpr axcaucvglemcl fvmptd eqcomd - a1i wreu wb eqeltrd cr wf adantr caddc wral pitonn eleqtrrdi ffvelrnd - c1 cint elrealeu sylib eqcom reubii eqeq2d riota2 syl2anc mpbird ) AK - UAUBZUCZNUDZKUESZUFUGZUHUIZNUJZXBEUDZUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZI - UPZKJUPZUOSZUQZXMDUDZUOSZUQZDTURZXNUQZWSXNXTWSFKWTFUDZUESZUFUGZUHUIZN - UJZYDXEUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZIUPZXRUQZDTURZXTUAJTJFUAYPVGUQW - SRVPYBKUQZYPXTUQWSYQYOXSDTYQYNXMXRYQYMXLIYQYLXKUOYQYKXJUNYQYJXIUKYQYI - XHUKULYQYFXDYHXGYQYEXCNYQYDXBWTUHYQYCXAUFYBKUEUSUTZVAVBYQYGXFEYQYDXBX - EUHYRVCVBVDVEVFUTVFVHVIVJVKAWRVLABCDEIKLNOPVMZVNZVOWSXNTUBXSDTVQZXPYA - VRWSXNXTTYTYSVSWSXRXMUQZDTVQZUUAWSXMVTUBUUCWSLVTXLIALVTIWAWRPWBWRXLLU - BAWRXLWHBUDZUBCUDWHWCUMUUDUBCUUDWDUCBUJWILBCEKNWEOWFVKWGDXMWJWKUUBXSD - TXRXMWLWMWKXSXPDTXNXQXNUQXRXOXMXQXNUOUSWNWOWPWQ $. + wreu wb eqeltrd cr wf adantr c1 caddc wral pitonn eleqtrrdi ffvelcdmd + a1i cint elrealeu sylib eqcom reubii eqeq2d riota2 syl2anc mpbird ) A + KUAUBZUCZNUDZKUESZUFUGZUHUIZNUJZXBEUDZUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZ + IUPZKJUPZUOSZUQZXMDUDZUOSZUQZDTURZXNUQZWSXNXTWSFKWTFUDZUESZUFUGZUHUIZ + NUJZYDXEUHUIZEUJZSZUKULUMZUKSZUNUGZUOSZIUPZXRUQZDTURZXTUAJTJFUAYPVGUQ + WSRWHYBKUQZYPXTUQWSYQYOXSDTYQYNXMXRYQYMXLIYQYLXKUOYQYKXJUNYQYJXIUKYQY + IXHUKULYQYFXDYHXGYQYEXCNYQYDXBWTUHYQYCXAUFYBKUEUSUTZVAVBYQYGXFEYQYDXB + XEUHYRVCVBVDVEVFUTVFVHVIVJVKAWRVLABCDEIKLNOPVMZVNZVOWSXNTUBXSDTVPZXPY + AVQWSXNXTTYTYSVRWSXRXMUQZDTVPZUUAWSXMVSUBUUCWSLVSXLIALVSIVTWRPWAWRXLL + UBAWRXLWBBUDZUBCUDWBWCUMUUDUBCUUDWDUCBUJWILBCEKNWEOWFVKWGDXMWJWKUUBXS + DTXRXMWLWMWKXSXPDTXNXQXNUQXRXOXMXQXNUOUSWNWOWPWQ $. $} ${ @@ -83835,36 +83837,36 @@ natural numbers (notably, ~ axcaucvg ). 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XDYTWOYQ $. + bitr3id ralrimiva rspcev ad5antr oveq2d breq12d syl ffvelcdmd oveq12d + wf 3imtr3d ex sylan2br oveq1 ) AUHUAUIZUJRZUBUIZGUIZUKRZUUOJUPZUCUIZU + ULULUMZUJRZUURUUQUULULUMZUJRZSZVFZGTUNZUBTUOZVFZUAUQUNZURBUIZUSRZFUIZ + UUOUSRZUUOIUPZCUIZUVIUTUMZUSRZUVNUVMUVIUTUMZUSRZSZVFZGKUNFKUOZVFZBVGU + NZCVGUOZUCUQAUAUCEUBGHJMABCDEFGHIJKLMNOPQVAZABCDEFGHIJKLMNOPQVBVCAUUR + UQVDZUVHSZSUURUHVEZVGVDZUVJUVLUVMUWHUVIUTUMZUSRZUWHUVQUSRZSZVFZGKUNZF + KUOZVFZBVGUNZUWDUWFUWIAUVHUWIUWFUURVHVIVJUWGAUWFUUMUUNUDUIZUKRZUWSJUP + ZUUSUJRZUURUXAUULULUMZUJRZSZVFZUDTUNZUBTUOZVFZUAUQUNZSZUWRUXJUVHUWFUX + IUVGUAUQUXHUVFUUMUXGUVEUBTUXFUVDUDGTUDGVKZUWTUUPUXEUVCUWSUUOUUNUKVLUX + LUXBUUTUXDUVBUXLUXAUUQUUSUJUWSUUOJVMZVNUXLUXCUVAUURUJUXLUXAUUQUULULUX + MVOVQVPVRVSVTWAWBWCAUXKSZUWQBVGUXNUVIVGVDZSZUVJUWPUXPUVJSZUEUIZUHVEZU + VIWDZUWPUEUQUXOUXTUEUQUOZUXNUVJUXOUYAUEUVIWEWFWGUXQUXRUQVDZUXTSZSZUFU + IZUWSUKRZUXAUURUXRULUMZUJRZUURUXAUXRULUMZUJRZSZVFZUDTUNZUWPUFTUYDUWTU + YKVFZUDTUNZUBTUOZUYMUFTUOUYDUXJUHUXRUJRZUYPUXPUXJUVJUYCAUWFUXJUXOWHWI + UYDUXTUVJUYQUXQUYBUXTWJZUXPUVJUYCWKUXTUVJUYQUXTURUXSUSRZUVJUYQUXSUVIU + RUSVLUYSUHUHVEZUXSUSRUYQURUYTUXSUSWPWLUHUXRWMWQWNWRWOUYBUXJUYQUYPVFZV + FUXQUXTUXIVUAUAUXRUQUAUEVKZUUMUYQUXHUYPUULUXRUHUJVLVUBUXFUYNUBUDTTVUB + UXEUYKUWTVUBUXBUYHUXDUYJVUBUUSUYGUXAUJUULUXRUURULWSVQVUBUXCUYIUURUJUU + LUXRUXAULWSVQVPWTXAVRXBVJXFUYOUYMUBUFTUBUFVKZUYNUYLUDTVUCUWTUYFUYKUUN + UYEUWSUKXCXGXDXEXTUYDUYETVDZUYMSZSZMUIZUYEXHVEXIXJZXKRMXLVUHEUIZXKREX + LVEXMXNUMXMVEXOXJUHVEZKVDZVUJUUOUSRZUWMVFZGKUNZUWPVUDVUKUYDUYMVUDVUJX + PUVIVDUVNXPUTUMUVIVDCUVIUNSBXLXQKBCEUYEMXRNXSVJVUFVUMGKVUFUUOKVDZSZVU + GUGUIZXHVEXIXJZXKRMXLVURVUIXKREXLVEXMXNUMXMVEXOXJUHVEZUUOWDZVUMUGTVUO + VUTUGTUOVUFBCUGEUUOKMNYAYBVUPVUQTVDZVUTSZSZUYEVUQUKRZVUQJUPZUYGUJRZUU + RVVEUXRULUMZUJRZSZVULUWMVVCUYLVVDVVIVFUDTVUQUDUGVKZUYFVVDUYKVVIUWSVUQ + UYEUKVLVVJUYHVVFUYJVVHVVJUXAVVEUYGUJUWSVUQJVMZVNVVJUYIVVGUURUJVVJUXAV + VEUXRULVVKVOVQVPVRVUPUYMVVBUYDVUDUYMVUOWHYCVUPVVAVUTYJZYDVVCVVDVUJVUS + USRZVULVVCVUDVVAVVDVVMYEVUPVUDVVBUYDVUDUYMVUOYFYCVVLEUYEVUQMYGWOVVCVU + SUUOVUJUSVUPVVAVUTWJZVQYKVVCVVFUWKVVHUWLVVFVVEUHVEZUYGUHVEZUSRVVCUWKV + VEUYGWMVVCVVOUVMVVPUWJUSVVCVUSIUPZVVOUVMVVCAVVAVVQVVOWDUXQAUYCVUEVUOV + VBAUXKUXOUVJYHYIZVVLABCDEFGHIJVUQKLMNOPQYLWOVVCVUSUUOIVVNYMYNZVVCUWHU + XSUTUMZVVPUWJVVCUWFUYBVVTVVPWDUXPUWFUVJUYCVUEVUOVVBAUWFUXJUXOYFUUAVUF + UYBVUOVVBUXQUYBUXTVUEYFWIZUURUXRYOWOUYDVVTUWJWDVUEVUOVVBUYDUXSUVIUWHU + TUYRUUBYPYNUUCYRVVHUWHVVGUHVEZUSRVVCUWLUURVVGWMVVCVWBUVQUWHUSVVCVVOUX + SUTUMZVWBUVQVVCVVEUQVDUYBVWCVWBWDVVCTUQVUQJVVCATUQJUUGVVRUWEUUDVVLUUE + VWAVVEUXRYOWOVVCVVOUVMUXSUVIUTVVSUYDUXTVUEVUOVVBUYRYPUUFYNVQYRVPUUHYQ + YSUWOVUNFVUJKUVKVUJWDZUWNVUMGKVWDUVLVULUWMUVKVUJUUOUSXCXGXDYTWOYQYQUU + IYSUUJUWCUWRCUWHVGUVNUWHWDZUWBUWQBVGVWEUWAUWPUVJVWEUVTUWNFGKKVWEUVSUW + MUVLVWEUVPUWKUVRUWLVWEUVOUWJUVMUSUVNUWHUVIUTUUKVQUVNUWHUVQUSXCVPWTXAW + TXDYTWOYQ $. $} $} @@ -84920,15 +84922,15 @@ product of extended reals (common case). (Contributed by David A. ( vx vy cr wcel clt wbr cltrr cv w3a cmnf csn cun cpnf cxp wo brun wceq wi wa copab df-ltxr breqi bitri eleq1 breq1 3anbi13d breq2 3anbi23d brabg eqid simp3 syl6bi brxp simprbi elsni syl a1i renepnf neneqd pm2.24 syl6ci - adantl simplbi renemnf adantr jaod syl5bi 3adant3 ibir orcd sylibr 3expia - wn wb impbid ) AEFZBEFZUAZABGHZABIHZWAABCJZEFZDJZEFZWCWEIHZKZCDUBZHZABELM - ZNZOMZPZWKEPZNZHZQZVTWBWAABWIWPNZHWRABGWSCDUCUDABWIWPRUEZVTWJWBWQVTWJVRVS - WBKZWBWHVRWFAWEIHZKXACDABEEWIWCASWDVRWGXBWFWCAEUFWCAWEIUGUHWEBSWFVSXBWBVR - WEBEUFWEBAIUIUJWIULUKZVRVSWBUMUNWQABWNHZABWOHZQVTWBABWNWORVTXDWBXEVSXDWBT - VRVSXDBOSZXFVOWBXDXFTVSXDBWMFZXFXDAWLFXGABWLWMUOUPBOUQURUSVSBOBUTVAXFWBVB - VCVDVRXEWBTVSVRXEALSZXHVOWBXEXHTVRXEAWKFZXHXEXIVSABWKEUOVEALUQURUSVRALAVF - VAXHWBVBVCVGVHVIVHVIVRVSWBWAXAWRWAXAWJWQXAWJVRVSWJXAVPWBXCVJVKVLWTVMVNVQ - $. + wn adantl simplbi renemnf adantr jaod biimtrid 3adant3 ibir sylibr 3expia + wb orcd impbid ) AEFZBEFZUAZABGHZABIHZWAABCJZEFZDJZEFZWCWEIHZKZCDUBZHZABE + LMZNZOMZPZWKEPZNZHZQZVTWBWAABWIWPNZHWRABGWSCDUCUDABWIWPRUEZVTWJWBWQVTWJVR + VSWBKZWBWHVRWFAWEIHZKXACDABEEWIWCASWDVRWGXBWFWCAEUFWCAWEIUGUHWEBSWFVSXBWB + VRWEBEUFWEBAIUIUJWIULUKZVRVSWBUMUNWQABWNHZABWOHZQVTWBABWNWORVTXDWBXEVSXDW + BTVRVSXDBOSZXFVDWBXDXFTVSXDBWMFZXFXDAWLFXGABWLWMUOUPBOUQURUSVSBOBUTVAXFWB + VBVCVEVRXEWBTVSVRXEALSZXHVDWBXEXHTVRXEAWKFZXHXEXIVSABWKEUOVFALUQURUSVRALA + VGVAXHWBVBVCVHVIVJVIVJVRVSWBWAXAWRWAXAWJWQXAWJVRVSWJXAVOWBXCVKVLVPWTVMVNV + Q $. $} @@ -87265,9 +87267,9 @@ it could represent the (meaningless) operation of identity of the complex numbers. (Contributed by AV, 17-Jan-2021.) $) addid0 $p |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) ) $= ( cc wcel wa caddc co wceq cc0 simpl simpr subaddd eqcom subid adantr eqtrd - cmin wi ex syl5bi sylbird oveq2 addid1 sylan9eqr impbid ) ACDZBCDZEZABFGZAH - ZBIHZUHUJAAQGZBHZUKUHAABUFUGJZUNUFUGKLUFUMUKRUGUMBULHZUFUKULBMUFUOUKUFUOEBU - LIUFUOKUFULIHUOANOPSTOUAUFUKUJRUGUFUKUJUKUFUIAIFGABIAFUBAUCUDSOUE $. + cmin wi ex biimtrid sylbird oveq2 addid1 sylan9eqr impbid ) ACDZBCDZEZABFGZ + AHZBIHZUHUJAAQGZBHZUKUHAABUFUGJZUNUFUGKLUFUMUKRUGUMBULHZUFUKULBMUFUOUKUFUOE + BULIUFUOKUFULIHUOANOPSTOUAUFUKUJRUGUFUKUJUKUFUIAIFGABIAFUBAUCUDSOUE $. $( Adding a nonzero number to a complex number does not yield the complex number. (Contributed by AV, 17-Jan-2021.) $) @@ -87337,14 +87339,14 @@ it could represent the (meaningless) operation of negf1o $p |- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } ) $= ( vy cr cv cneg wcel wceq wa imp wi cc recn eleq1d adantl negeq elrab wss crab wf1o ccnv cmpt ssel renegcl syl6 negneg syl biimpcd mpd sylanbrc weq - eqcomd simpr a1i syl5bi wb syl6com ad3antrrr negcon2 syl2anc exp31 impcom - sylbi f1ocnv2d simpld ) BGUAZBCHZIZBJZCGUBZDUCDUDFVMFHZIZUEKVIAFBVMAHZIZV - ODEVIVPBJZLZVQGJZVQIZBJZVQVMJVIVRVTVIVRVPGJZVTBGVPUFZVPUGUHMVSWCWBVIVRWCW - DMVRWCWBNVIWCVRWBWCVPWABWCVPOJZVPWAKVPPZWEWAVPVPUIUOUJQUKRULVLWBCVQGVJVQK - VKWABVJVQSQTUMVIVNVMJZVOBJZWGVNGJZWHLZVIWHVLWHCVNGCFUNVKVOBVJVNSQTZWJWHNV - IWIWHUPUQURMVRWGLVIVPVOKVNVQKUSZWGVRVIWLNZWGWJVRWMNWKWJVRVIWLWJVRLZVILWEV - NOJZWLWNVIWEVRVIWENWJVIVRWCWEWDWFUTRMWIWOWHVRVIVNPVAVPVNVBVCVDVFVEVEVGVH - $. + eqcomd simpr a1i biimtrid wb syl6com ad3antrrr negcon2 exp31 sylbi impcom + syl2anc f1ocnv2d simpld ) BGUAZBCHZIZBJZCGUBZDUCDUDFVMFHZIZUEKVIAFBVMAHZI + ZVODEVIVPBJZLZVQGJZVQIZBJZVQVMJVIVRVTVIVRVPGJZVTBGVPUFZVPUGUHMVSWCWBVIVRW + CWDMVRWCWBNVIWCVRWBWCVPWABWCVPOJZVPWAKVPPZWEWAVPVPUIUOUJQUKRULVLWBCVQGVJV + QKVKWABVJVQSQTUMVIVNVMJZVOBJZWGVNGJZWHLZVIWHVLWHCVNGCFUNVKVOBVJVNSQTZWJWH + NVIWIWHUPUQURMVRWGLVIVPVOKVNVQKUSZWGVRVIWLNZWGWJVRWMNWKWJVRVIWLWJVRLZVILW + EVNOJZWLWNVIWEVRVIWENWJVIVRWCWEWDWFUTRMWIWOWHVRVIVNPVAVPVNVBVFVCVDVEVEVGV + H $. $} @@ -89986,6 +89988,17 @@ would be equivalent to negated equality and this would follow readily (for FUQVGRVLUQUPVHAVIUQUPUKVJSTUIUJULPUM $. $} + ${ + $d A x $. + $( A complex number has a multiplicative inverse if and only if it is apart + from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from + real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.) $) + recapb $p |- ( A e. CC -> ( A =//= 0 <-> E. x e. CC ( A x. x ) = 1 ) ) $= + ( cc wcel cc0 cap wbr cv cmul co c1 wceq wrex recexap ex wa simprl simprr + simpl 1ap0 eqbrtrdi mulap0bad rexlimdvaa impbid ) BCDZBEFGZBAHZIJZKLZACMZ + UEUFUJABNOUEUIUFACUEUGCDZUIPZPZBUGUEULSUEUKUIQUMUHKEFUEUKUIRTUAUBUCUD $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -91359,6 +91372,19 @@ would be equivalent to negated equality and this would follow readily (for DBKABCIJPDABCEFMHNON $. $} + ${ + $d A x $. + $( A real number has a multiplicative inverse if and only if it is apart + from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim + Kingdon, 18-Jan-2025.) $) + rerecapb $p |- ( A e. RR -> ( A =//= 0 <-> E. x e. RR ( A x. x ) = 1 ) ) $= + ( cr wcel cc0 cap wbr cv cmul co c1 wceq wrex cdiv rerecclap recn recidap + wa cc sylan oveq2 eqeq1d rspcev syl2anc ex wss wi ax-resscn ssrexv recapb + ax-mp biimprd syl2im impbid ) BCDZBEFGZBAHZIJZKLZACMZUOUPUTUOUPRKBNJZCDBV + AIJZKLZUTBOUOBSDZUPVCBPZBQTUSVCAVACUQVALURVBKUQVABIUAUBUCUDUEUOVDUTUSASMZ + UPVECSUFUTVFUGUHUSACSUIUKVDUPVFABUJULUMUN $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -92535,23 +92561,23 @@ Ordering on reals (cont.) mp3an12 syl cneg ixi 1re renegcli eqeltri readdcld remulcl anbi12d rspcev simpr syl12anc rexbidv syl5ibrcom rexlimivv an4 cc0 resubcl pnpcan syl5ib ancoms adantrl adantrr subdid nnncan1 3com23 eqtr3d anim12d subeq0 biimpd - oveq1 3expb rimul 3syld syl5bi ralrimivva reu4 sylanbrc ) BEFZBAGZHIZJFZK - BXCLIZMIZJFZSZAENZXIBCGZHIZJFZKBXKLIZMIZJFZSZSZXCXKOZUAZCEUBAEUBXIAEUCXBB - XKKDGZMIZHIZOZDJNCJNXJCDBUDYDXJCDJJXKJFZYAJFZSZXJYDYCXCHIZJFZKYCXCLIZMIZJ - FZSZAENZYGXKYBLIZEFZYCYOHIZJFZKYCYOLIZMIZJFZYNYEXKEFZYBEFZYPYFXKUEZYFKEFZ - YAEFZUUCPYAUEZKYAUFUGZXKYBUHUIYGYQXKXKHIZJYGXKYBXKYEUUBYFUUDUJZYFUUCYEUUH - UKZUUJULYEUUIJFZYFYEUULXKXKUMUNUJUOYGYTKKMIZYAYAHIZMIZJYGYTKKUUNMIZMIZUUO - YGYSUUPKMYGYSYBYBHIUUPYGXKYBYBUUJUUKUUKUPYGKYAYAUUEYGPUQYFUUFYEUUGUKZUURU - RUSQYGUUNEFZUUOUUQOZYGYAYAUURUURUTUUEUUEUUSUUTPPKKUUNVAVCVDUSYGUUMJFUUNJF - UUOJFUUMVBVEJVFVBVGVHVIYGYAYAYEYFVNZUVAVJUUMUUNVKUGUOYMYRUUASAYOEXCYOOZYI - YRYLUUAUVBYHYQJXCYOYCHRTUVBYKYTJUVBYJYSKMXCYOYCLRQTVLVMVOYDXIYMAEYDXEYIXH - YLYDXDYHJBYCXCHWNTYDXGYKJYDXFYJKMBYCXCLWNQTVLVPVQVRVDXBXTACEEXRXEXMSZXHXP - SZSZXBXCEFZUUBSZSZXSXEXHXMXPVSUVHUVEXCXKLIZJFZKUVIMIZJFZSZUVIVTOZXSUVHUVC - UVJUVDUVLUVCXDXLLIZJFUVHUVJXDXLWAUVHUVOUVIJXBUVFUUBUVOUVIOBXCXKWBWOTWCUVD - XOXGLIZJFZUVHUVLXPXHUVQXOXGWAWDUVHUVPUVKJUVHKXNXFLIZMIUVPUVKUVHKXNXFUUEUV - HPUQXBUUBXNEFUVFBXKUHWEXBUVFXFEFUUBBXCUHWFWGUVHUVRUVIKMXBUVFUUBUVRUVIOZXB - UUBUVFUVSBXKXCWHWIWOQWJTWCWKUVMUVNUAUVHUVIWPUQUVGUVNXSUAXBUVGUVNXSXCXKWLW - MUKWQWRWSXIXQACEXSXEXMXHXPXSXDXLJXCXKBHRTXSXGXOJXSXFXNKMXCXKBLRQTVLWTXA + oveq1 3expb rimul 3syld biimtrid ralrimivva reu4 sylanbrc ) BEFZBAGZHIZJF + ZKBXCLIZMIZJFZSZAENZXIBCGZHIZJFZKBXKLIZMIZJFZSZSZXCXKOZUAZCEUBAEUBXIAEUCX + BBXKKDGZMIZHIZOZDJNCJNXJCDBUDYDXJCDJJXKJFZYAJFZSZXJYDYCXCHIZJFZKYCXCLIZMI + ZJFZSZAENZYGXKYBLIZEFZYCYOHIZJFZKYCYOLIZMIZJFZYNYEXKEFZYBEFZYPYFXKUEZYFKE + FZYAEFZUUCPYAUEZKYAUFUGZXKYBUHUIYGYQXKXKHIZJYGXKYBXKYEUUBYFUUDUJZYFUUCYEU + UHUKZUUJULYEUUIJFZYFYEUULXKXKUMUNUJUOYGYTKKMIZYAYAHIZMIZJYGYTKKUUNMIZMIZU + UOYGYSUUPKMYGYSYBYBHIUUPYGXKYBYBUUJUUKUUKUPYGKYAYAUUEYGPUQYFUUFYEUUGUKZUU + RURUSQYGUUNEFZUUOUUQOZYGYAYAUURUURUTUUEUUEUUSUUTPPKKUUNVAVCVDUSYGUUMJFUUN + JFUUOJFUUMVBVEJVFVBVGVHVIYGYAYAYEYFVNZUVAVJUUMUUNVKUGUOYMYRUUASAYOEXCYOOZ + YIYRYLUUAUVBYHYQJXCYOYCHRTUVBYKYTJUVBYJYSKMXCYOYCLRQTVLVMVOYDXIYMAEYDXEYI + XHYLYDXDYHJBYCXCHWNTYDXGYKJYDXFYJKMBYCXCLWNQTVLVPVQVRVDXBXTACEEXRXEXMSZXH + XPSZSZXBXCEFZUUBSZSZXSXEXHXMXPVSUVHUVEXCXKLIZJFZKUVIMIZJFZSZUVIVTOZXSUVHU + VCUVJUVDUVLUVCXDXLLIZJFUVHUVJXDXLWAUVHUVOUVIJXBUVFUUBUVOUVIOBXCXKWBWOTWCU + VDXOXGLIZJFZUVHUVLXPXHUVQXOXGWAWDUVHUVPUVKJUVHKXNXFLIZMIUVPUVKUVHKXNXFUUE + UVHPUQXBUUBXNEFUVFBXKUHWEXBUVFXFEFUUBBXCUHWFWGUVHUVRUVIKMXBUVFUUBUVRUVIOZ + XBUUBUVFUVSBXKXCWHWIWOQWJTWCWKUVMUVNUAUVHUVIWPUQUVGUVNXSUAXBUVGUVNXSXCXKW + LWMUKWQWRWSXIXQACEXSXEXMXHXPXSXDXLJXCXKBHRTXSXGXOJXSXFXNKMXCXKBLRQTVLWTXA $. $} @@ -92913,8 +92939,8 @@ Positive integers (as a subset of complex numbers) $( A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) $) nngt1ne1 $p |- ( A e. NN -> ( 1 < A <-> A =/= 1 ) ) $= - ( cn wcel c1 clt wbr wne cr 1re ltne mpan wceq wn nn1gt1 ord syl5bi impbid2 - df-ne ) ABCZDAEFZADGZDHCTUAIDAJKUAADLZMSTADRSUBTANOPQ $. + ( cn wcel c1 clt wbr wne cr 1re ltne mpan wceq wn df-ne nn1gt1 ord biimtrid + impbid2 ) ABCZDAEFZADGZDHCTUAIDAJKUAADLZMSTADNSUBTAOPQR $. $( The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) $) @@ -92943,20 +92969,20 @@ Positive integers (as a subset of complex numbers) c1 oveq1 cr vy wa cv caddc nnnlt1 pm2.21d rgen breq1 oveq2 cbvralv adantr cc nncn ax-1cn pncan sylancl simpl eqeltrd syl5ibrcom a1dd rspcv nnre 1re ltsubadd mp3an2 syl2anr subsub3 mp3an3 syl2an biimpd syl9r nn1m1nn adantl - wb wo mpjaod ralrimdva syl5bi nnind rspcva sylan2 cc0 nngt0 posdif impbid - syl5ibr ) AEFZBEFZUBZABGHZBAIJZEFZWHWGCUCZBGHZBWMIJZEFZKZCELZWJWLKZWMDUCZ - GHZWTWMIJZEFZKZCELWMRGHZRWMIJZEFZKZCELWMUAUCZGHZXIWMIJZEFZKZCELZWMXIRUDJZ - GHZXOWMIJZEFZKZCELZWRDUABWTRMZXDXHCEYAXAXEXCXGWTRWMGNYAXBXFEWTRWMISOPQWTX - IMZXDXMCEYBXAXJXCXLWTXIWMGNYBXBXKEWTXIWMISOPQWTXOMZXDXSCEYCXAXPXCXRWTXOWM - GNYCXBXQEWTXOWMISOPQWTBMZXDWQCEYDXAWNXCWPWTBWMGNYDXBWOEWTBWMISOPQXHCEWMEF - ZXEXGWMUEUFUGXNWTXIGHZXIWTIJZEFZKZDELZXIEFZXTXMYICDEWMWTMZXJYFXLYHWMWTXIG - UHYLXKYGEWMWTXIIUIOPUJYKYJXSCEYKYEUBZWMRMZYJXSKWMRIJZEFZYMYNXSYJYMYNXRXPY - MXRYNXORIJZEFYMYQXIEYMXIULFZRULFZYQXIMYKYRYEXIUMZUKUNXIRUOUPYKYEUQURYNXQY - QEWMRXOIUIOUSUTUTYPYJYOXIGHZXIYOIJZEFZKZYMXSYIUUDDYOEWTYOMZYFUUAYHUUCWTYO - XIGUHUUEYGUUBEWTYOXIIUIOPVAYMUUDXSYMUUAXPUUCXRYEWMTFZXITFZUUAXPVNZYKWMVBX - IVBUUFRTFUUGUUHVCWMRXIVDVEVFYMUUBXQEYKYRWMULFZUUBXQMZYEYTWMUMYRUUIYSUUJUN - XIWMRVGVHVIOPVJVKYEYNYPVOYKWMVLVMVPVQVRVSWQWSCAEWMAMZWNWJWPWLWMABGUHUUKWO - WKEWMABIUIOPVTWAWLWJWIWBWKGHZWKWCWGATFBTFWJUULVNWHAVBBVBABWDVIWFWE $. + wb mpjaod ralrimdva biimtrid nnind rspcva sylan2 cc0 nngt0 posdif syl5ibr + wo impbid ) AEFZBEFZUBZABGHZBAIJZEFZWHWGCUCZBGHZBWMIJZEFZKZCELZWJWLKZWMDU + CZGHZWTWMIJZEFZKZCELWMRGHZRWMIJZEFZKZCELWMUAUCZGHZXIWMIJZEFZKZCELZWMXIRUD + JZGHZXOWMIJZEFZKZCELZWRDUABWTRMZXDXHCEYAXAXEXCXGWTRWMGNYAXBXFEWTRWMISOPQW + TXIMZXDXMCEYBXAXJXCXLWTXIWMGNYBXBXKEWTXIWMISOPQWTXOMZXDXSCEYCXAXPXCXRWTXO + WMGNYCXBXQEWTXOWMISOPQWTBMZXDWQCEYDXAWNXCWPWTBWMGNYDXBWOEWTBWMISOPQXHCEWM + EFZXEXGWMUEUFUGXNWTXIGHZXIWTIJZEFZKZDELZXIEFZXTXMYICDEWMWTMZXJYFXLYHWMWTX + IGUHYLXKYGEWMWTXIIUIOPUJYKYJXSCEYKYEUBZWMRMZYJXSKWMRIJZEFZYMYNXSYJYMYNXRX + PYMXRYNXORIJZEFYMYQXIEYMXIULFZRULFZYQXIMYKYRYEXIUMZUKUNXIRUOUPYKYEUQURYNX + QYQEWMRXOIUIOUSUTUTYPYJYOXIGHZXIYOIJZEFZKZYMXSYIUUDDYOEWTYOMZYFUUAYHUUCWT + YOXIGUHUUEYGUUBEWTYOXIIUIOPVAYMUUDXSYMUUAXPUUCXRYEWMTFZXITFZUUAXPVNZYKWMV + BXIVBUUFRTFUUGUUHVCWMRXIVDVEVFYMUUBXQEYKYRWMULFZUUBXQMZYEYTWMUMYRUUIYSUUJ + UNXIWMRVGVHVIOPVJVKYEYNYPWEYKWMVLVMVOVPVQVRWQWSCAEWMAMZWNWJWPWLWMABGUHUUK + WOWKEWMABIUIOPVSVTWLWJWIWAWKGHZWKWBWGATFBTFWJUULVNWHAVBBVBABWCVIWDWF $. nnsub.1 $e |- A e. NN $. nnsub.2 $e |- B e. NN $. @@ -94394,10 +94420,10 @@ Nonnegative integers (as a subset of complex numbers) integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) $) 0mnnnnn0 $p |- ( N e. NN -> ( 0 - N ) e/ NN0 ) $= ( cn wcel cc0 cmin co cn0 wnel 0re cneg df-neg eqcomi eleq1i cle wbr nn0ge0 - wn cr nnre le0neg1d clt nngt0 0red lenltd pm2.21 syl6bi mpid sylbird syl5bi - wi syl5 mt2i df-nel sylibr ) ABCZDAEFZGCZQUPGHUOUQDRCZIUQAJZGCZUOURQZUPUSGU - SUPAKLMUTDUSNOZUOVAUSPUOVBADNOZVAUOAASZTUOVCDAUAOZVAAUBUOVCVEQVEVAUJUOADVDU - OUCUDVEVAUEUFUGUHUKUIULUPGUMUN $. + wn cr nnre le0neg1d clt nngt0 wi 0red lenltd pm2.21 syl6bi sylbird biimtrid + mpid syl5 mt2i df-nel sylibr ) ABCZDAEFZGCZQUPGHUOUQDRCZIUQAJZGCZUOURQZUPUS + GUSUPAKLMUTDUSNOZUOVAUSPUOVBADNOZVAUOAASZTUOVCDAUAOZVAAUBUOVCVEQVEVAUCUOADV + DUOUDUEVEVAUFUGUJUHUKUIULUPGUMUN $. ${ un0addcl.1 $e |- ( ph -> S C_ CC ) $. @@ -94410,11 +94436,11 @@ Nonnegative integers (as a subset of complex numbers) ( wcel caddc co cc0 wo wa eleq2i elun bitri cc sselda eqeltrd csn ssun1 cun sseqtrri sselid expr addid2d wss a1i elsni oveq1d eleq1d syl5ibrcom wi impancom jaodan sylan2b 0cnd snssd unssd eqsstrid addid1d simpr jaod - oveq2d syl5bi impr ) ADCIZECIZDEJKZCIZVIEBIZELUAZIZMZAVHNZVKVIEBVMUCZIV - OCVQEGOEBVMPQVPVLVKVNVHADBIZDVMIZMZVLVKUNZVHDVQIVTCVQDGODBVMPQAVRWAVSAV - RVLVKAVRVLNNBCVJBVQCBVMUBGUDZHUEUFAVLVSVKAVLNZVKVSLEJKZCIWCWDECWCEABREF - SUGABCEBCUHAWBUISTVSVJWDCVSDLEJDLUJUKULUMUOUPUQVPVKVNDLJKZCIVPWEDCVPDAC - RDACVQRGABVMRFALRAURUSUTVASVBAVHVCTVNVJWECVNELDJELUJVEULUMVDVFVG $. + oveq2d biimtrid impr ) ADCIZECIZDEJKZCIZVIEBIZELUAZIZMZAVHNZVKVIEBVMUCZ + IVOCVQEGOEBVMPQVPVLVKVNVHADBIZDVMIZMZVLVKUNZVHDVQIVTCVQDGODBVMPQAVRWAVS + AVRVLVKAVRVLNNBCVJBVQCBVMUBGUDZHUEUFAVLVSVKAVLNZVKVSLEJKZCIWCWDECWCEABR + EFSUGABCEBCUHAWBUISTVSVJWDCVSDLEJDLUJUKULUMUOUPUQVPVKVNDLJKZCIVPWEDCVPD + ACRDACVQRGABVMRFALRAURUSUTVASVBAVHVCTVNVJWECVNELDJELUJVEULUMVDVFVG $. $} un0mulcl.3 $e |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M x. N ) e. S ) $. @@ -94424,12 +94450,12 @@ Nonnegative integers (as a subset of complex numbers) ( wcel cmul co cc0 wo wa eleq2i elun bitri sseqtrri cc sselda csn wi expr cun ssun1 sselid mul02d wss ssun2 c0ex mpbir eqeltrdi elsni oveq1d eleq1d snss syl5ibrcom impancom jaodan sylan2b 0cnd snssd eqsstrid mul01d oveq2d - unssd jaod syl5bi impr ) ADCIZECIZDEJKZCIZVKEBIZELUAZIZMZAVJNZVMVKEBVOUDZ - IVQCVSEGOEBVOPQVRVNVMVPVJADBIZDVOIZMZVNVMUBZVJDVSIWBCVSDGODBVOPQAVTWCWAAV - TVNVMAVTVNNNBCVLBVSCBVOUEGRHUFUCAVNWAVMAVNNZVMWALEJKZCIWDWELCWDEABSEFTUGL - CIVOCUHVOVSCVOBUIGRLCUJUPUKZULWAVLWECWADLEJDLUMUNUOUQURUSUTVRVMVPDLJKZCIV - RWGLCVRDACSDACVSSGABVOSFALSAVAVBVFVCTVDWFULVPVLWGCVPELDJELUMVEUOUQVGVHVI - $. + unssd jaod biimtrid impr ) ADCIZECIZDEJKZCIZVKEBIZELUAZIZMZAVJNZVMVKEBVOU + DZIVQCVSEGOEBVOPQVRVNVMVPVJADBIZDVOIZMZVNVMUBZVJDVSIWBCVSDGODBVOPQAVTWCWA + AVTVNVMAVTVNNNBCVLBVSCBVOUEGRHUFUCAVNWAVMAVNNZVMWALEJKZCIWDWELCWDEABSEFTU + GLCIVOCUHVOVSCVOBUIGRLCUJUPUKZULWAVLWECWADLEJDLUMUNUOUQURUSUTVRVMVPDLJKZC + IVRWGLCVRDACSDACVSSGABVOSFALSAVAVBVFVCTVDWFULVPVLWGCVPELDJELUMVEUOUQVGVHV + I $. $} $( Closure of addition of nonnegative integers. (Contributed by Raph Levien, @@ -95299,10 +95325,10 @@ nonnegative integers (cont.)". $) Kingdon, 14-Mar-2020.) $) zapne $p |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M =//= N <-> M =/= N ) ) $= ( cz wcel wa cap wbr wne cc wi zcn apne syl2an wceq wn clt wo w3o cr zre wb - df-ne ztri3or 3orrot 3orass bitri sylib reaplt bitrdi sylibrd syl5bi impbid - ord orcom ) ACDZBCDZEZABFGZABHZUOAIDBIDURUSJUPAKBKABLMUSABNZOZUQURABUBUQVAB - APGZABPGZQZURUQUTVDUQVCUTVBRZUTVDQZABUCVEUTVBVCRVFVCUTVBUDUTVBVCUEUFUGUMUOA - SDZBSDZURVDUAUPATBTVGVHEURVCVBQVDABUHVCVBUNUIMUJUKUL $. + df-ne ztri3or 3orrot 3orass bitri sylib reaplt orcom bitrdi biimtrid impbid + ord sylibrd ) ACDZBCDZEZABFGZABHZUOAIDBIDURUSJUPAKBKABLMUSABNZOZUQURABUBUQV + ABAPGZABPGZQZURUQUTVDUQVCUTVBRZUTVDQZABUCVEUTVBVCRVFVCUTVBUDUTVBVCUEUFUGUMU + OASDZBSDZURVDUAUPATBTVGVHEURVCVBQVDABUHVCVBUIUJMUNUKUL $. $( Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.) $) @@ -97225,7 +97251,7 @@ nonnegative integers (cont.)". $) ( vj cv cle wbr cz crab cuz wceq breq1 rabbidv df-uz zex rabex fvmpt ) CB CDZADZEFZAGHBREFZAGHGIQBJSTAGQBREKLCAMTAGNOP $. - $( The domain and range of the upper integers function. (Contributed by + $( The domain and codomain of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.) $) uzf $p |- ZZ>= : ZZ --> ~P ZZ $= ( vj vk cv cle wbr cz crab cpw wcel wral cuz wss ssrab2 elpw2 mpbir rgenw @@ -97269,10 +97295,10 @@ nonnegative integers (cont.)". $) eluzuzle $p |- ( ( B e. ZZ /\ B <_ A ) -> ( C e. ( ZZ>= ` A ) -> C e. ( ZZ>= ` B ) ) ) $= ( cuz cfv wcel cz cle wbr w3a wa eluz2 simpll simpr2 cr zre ad2antrr adantl - 3ad2ant1 3ad2ant2 simplr simpr3 letrd syl3anbrc ex syl5bi ) CADEFAGFZCGFZAC - HIZJZBGFZBAHIZKZCBDEFZACLUMUJUNUMUJKZUKUHBCHIUNUKULUJMUMUGUHUINUOBACUKBOFUL - UJBPQUJAOFZUMUGUHUPUIAPSRUJCOFZUMUHUGUQUICPTRUKULUJUAUMUGUHUIUBUCBCLUDUEUF - $. + 3ad2ant1 3ad2ant2 simplr simpr3 letrd syl3anbrc ex biimtrid ) CADEFAGFZCGFZ + ACHIZJZBGFZBAHIZKZCBDEFZACLUMUJUNUMUJKZUKUHBCHIUNUKULUJMUMUGUHUINUOBACUKBOF + ULUJBPQUJAOFZUMUGUHUPUIAPSRUJCOFZUMUHUGUQUICPTRUKULUJUAUMUGUHUIUBUCBCLUDUEU + F $. $( A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.) $) @@ -97792,16 +97818,16 @@ nonnegative integers (cont.)". $) caddc nnnlt1 pm2.21d rgen wex wb 1nn elex2 nfra1 r19.3rm mp2b wa wo com12 rsp adantl wsbc nfv nfsbc1v nfim sbceq1a imbi12d cbvral mpbi rspec dfsbcq vex sbcie bitr3id biimprcd syl6 adantr jcad jaob syl6ibr cle nnleltp1 nnz - cz zleloe syl2an bitr3d ancoms sylibrd ralimdva syl5bi nnind mpd ) CJZKLD - JZWLMNZBOZDKPZAWMGJZMNZBOZDKPWMQMNZBOZDKPWMHJZMNZBOZDKPZWMXBQUDUAZMNZBOZD - KPZWPGHWLWQQUBZWSXADKXJWRWTBWQQWMMRSTGHUCZWSXDDKXKWRXCBWQXBWMMRSTWQXFUBZW - SXHDKXLWRXGBWQXFWMMRSTGCUCZWSWODKXMWRWNBWQWLWMMRSTXADKWMKLZWTBWMUEUFUGXEX - EDKPZXBKLZXIQKLIJKLIUHXEXOUIUJIQKUKXEDIKXDDKULUMUNXPXEXHDKXPXNUOZXEXCDHUC - ZUPZBOZXHXQXEXDXRBOZUOXTXQXEXDYAXNXEXDOXPXEXNXDXDDKURUQUSXPXEYAOXNXPXEACX - BUTZYAXEYBOZHKWPAOZCKPYCHKPYDCKFUGYDYCCHKYDHVAXEYBCXECVAACXBVBVCCHUCZWPXE - AYBYEWOXDDKYEWNXCBWLXBWMMRSTACXBVDVEVFVGVHXRBYBBACWMUTXRYBABCWMDVJEVKACWM - XBVIVLVMVNVOVPXCBXRVQVRXQXGXSBXNXPXGXSUIXNXPUOWMXBVSNZXGXSWMXBVTXNWMWBLXB - WBLYFXSUIXPWMWAXBWAWMXBWCWDWEWFSWGWHWIWJFWK $. + cz zleloe syl2an bitr3d ancoms sylibrd ralimdva biimtrid nnind mpd ) CJZK + LDJZWLMNZBOZDKPZAWMGJZMNZBOZDKPWMQMNZBOZDKPWMHJZMNZBOZDKPZWMXBQUDUAZMNZBO + ZDKPZWPGHWLWQQUBZWSXADKXJWRWTBWQQWMMRSTGHUCZWSXDDKXKWRXCBWQXBWMMRSTWQXFUB + ZWSXHDKXLWRXGBWQXFWMMRSTGCUCZWSWODKXMWRWNBWQWLWMMRSTXADKWMKLZWTBWMUEUFUGX + EXEDKPZXBKLZXIQKLIJKLIUHXEXOUIUJIQKUKXEDIKXDDKULUMUNXPXEXHDKXPXNUOZXEXCDH + UCZUPZBOZXHXQXEXDXRBOZUOXTXQXEXDYAXNXEXDOXPXEXNXDXDDKURUQUSXPXEYAOXNXPXEA + CXBUTZYAXEYBOZHKWPAOZCKPYCHKPYDCKFUGYDYCCHKYDHVAXEYBCXECVAACXBVBVCCHUCZWP + XEAYBYEWOXDDKYEWNXCBWLXBWMMRSTACXBVDVEVFVGVHXRBYBBACWMUTXRYBABCWMDVJEVKAC + WMXBVIVLVMVNVOVPXCBXRVQVRXQXGXSBXNXPXGXSUIXNXPUOWMXBVSNZXGXSWMXBVTXNWMWBL + XBWBLYFXSUIXPWMWAXBWAWMXBWCWDWEWFSWGWHWIWJFWK $. $} ${ @@ -98458,13 +98484,13 @@ Rational numbers (as a subset of complex numbers) ( vz wcel cc0 clt wbr wa cv cdiv co wceq cn wrex cz rexcom wi cr adantl cq elq bitri breq2 wb zre nnre adantr nngt0 gt0div syl3anc sylan9bb elnnz bicomd simplbi2 imp weq oveq1 eqeq2d simpll rspcedvd ex sylbid com13 impl - rexlimdva reximdva syl5bi impcom sylibr ) CUAEZFCGHZICAJZBJZKLZMZANOZBNOZ - VPBNOANOVLVKVRVKCDJZVNKLZMZDPOZBNOZVLVRVKWABNODPOWCDBCUBWADBPNQUCVLWBVQBN - VLVNNEZIWAVQDPVLWDVSPEZWAVQRWAWDWEIZVLVQWAWFVLVQRWAWFIZVLFVSGHZVQWAVLFVTG - HZWFWHCVTFGUDWFWHWIWFVSSEZVNSEZFVNGHZWHWIUEWEWJWDVSUFTWDWKWEVNUGUHWDWLWEV - NUIUHVSVNUJUKUNULWGWHVQWGWHIZVPWAAVSNWGWHVSNEZWFWHWNRZWAWEWOWDWNWEWHVSUMU - OTTUPADUQZVPWAUEWMWPVOVTCVMVSVNKURUSTWAWFWHUTVAVBVCVBVDVEVFVGVHVIVPABNNQV - J $. + rexlimdva reximdva biimtrid impcom sylibr ) CUAEZFCGHZICAJZBJZKLZMZANOZBN + OZVPBNOANOVLVKVRVKCDJZVNKLZMZDPOZBNOZVLVRVKWABNODPOWCDBCUBWADBPNQUCVLWBVQ + BNVLVNNEZIWAVQDPVLWDVSPEZWAVQRWAWDWEIZVLVQWAWFVLVQRWAWFIZVLFVSGHZVQWAVLFV + TGHZWFWHCVTFGUDWFWHWIWFVSSEZVNSEZFVNGHZWHWIUEWEWJWDVSUFTWDWKWEVNUGUHWDWLW + EVNUIUHVSVNUJUKUNULWGWHVQWGWHIZVPWAAVSNWGWHVSNEZWFWHWNRZWAWEWOWDWNWEWHVSU + MUOTTUPADUQZVPWAUEWMWPVOVTCVMVSVNKURUSTWAWFWHUTVAVBVCVBVDVEVFVGVHVIVPABNN + QVJ $. $( A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.) $) @@ -100497,26 +100523,26 @@ Infinity and the extended real number system (cont.) xaddpnf1 breqtrrid 2thd 3jaoi sylbi ltpnf simpr breqtrrd oveq1d cneg rexneg renegcl eqeltrd rexrd renemnfd xaddpnf2 syl2anc pnfxr xrltnr breq12 oveq12d 0re ltnri pnfaddmnf mnfltpnf mpbiri oveq1 sylan9eq pnfnemnf mp2an syl eleq1 - xaddmnf2 mnfxr mnfnepnf neeq1 mnfaddpnf 3jaod syl5bi imp ) ACDZBCDZABEFZGBA - UAZHIZEFZUBZXPBUCDZBJKZBLKZUDXOYABUEXOYBYAYCYDXOAUCDZAJKZALKZUDZYBYAUMZAUEZ - YEYIYFYGYEYBYAYEYBMZXQGBAUFIZEFXTABUGYKXSYLGEYBYEXSYLKBAUHUINUJOYFYBYAYFYBM - ZXQXTYMXQJBEFZYBYFXPYNPZBUKZXPYOYFBULUNUOYMAJBEYFYBUPUQURYMXTGLEFZGCDYQPUSG - UTVAZYMXSLGEYMXSBLHIZLYMXRLBHYMXRJUAZLYFXRYTKYBAJVBZQVCRVDYBXPYFBJSZYSLKZYP - YBUUBYFBVEUNBVFZVGVHNTVIOYGYBYAYGYBMZXQXTUUEALBEYGYBUPYBLBEFYGBVJUNVKUUEGJX - SEVLUUEXSBJHIZJYGXSUUFKYBYGXRJBHYGXRLUAJALVBVMRVDZQYBXPYGBLSZUUFJKYPYBUUHYG - BVNUNBVOVGVHVPVQOVRVSXOYHYCYAUMZYJYEUUIYFYGYEYCYAYEYCMZXQXTUUJAJBEYEAJEFYCA - VTQYEYCWAZWBUUJGJXSEVLUUJXSJXRHIZJUUJBJXRHUUKWCUUJXRCDZXRLSZUULJKYEUUMYCYEX - RYEXRAWDUCAWEAWFWGZWHQYEUUNYCYEXRUUOWIQXRWJWKVHVPVQOYFYCYAYFYCMZXQXTUUPXQJJ - EFZJCDZUUQPWLJWMVAAJBJEWNTUUPXTGGEFZGWPWQZUUPXSGGEUUPXSJLHIGUUPBJXRLHYFYCWA - YFXRLKYCYFXRYTLUUAVCRZQWOWRRNTVIOYGYCYAYGYCMZXQXTUVBXQLJEFWSALBJEWNWTUVBGJX - SEVLUVBXSJJHIZJYGYCXSUUFUVCUUGBJJHXAXBUURJLSUVCJKWLXCJVOXDRVPVQOVRVSXOYHYDY - AUMZYJYEUVDYFYGYEYDYAYEYDMZXQXTUVEXQALEFZUVEXOUVFPZYEXOYDAUKQAUTZXEUVEBLAEY - EYDWAZNURUVEXTYQYRUVEXSLGEUVEXSLXRHIZLUVEBLXRHUVIWCYEUVJLKZYDYEXRUCDZUVKUUO - UVLUUMXRJSUVKXRUKXRVEXRXGWKXEQVHNTVIOYFYDYAYFYDMZXQXTUVMXQUVFUVMXOUVGYFXOYD - YFXOUURWLAJCXFWTQUVHXEUVMBLAEYFYDWANURUVMXTYQYRUVMXSLGEUVMXSYSLYFXSYSKYDYFX - RLBHUVAVDQYDXPYFUUBUUCYDXPLCDZXHBLCXFWTYDUUBYFYDUUBLJSXIBLJXJWTUNUUDVGVHNTV - IOYGYDYAYGYDMZXQXTUVOXQLLEFZUVNUVPPXHLWMVAALBLEWNTUVOXTUUSUUTUVOXSGGEUVOXSL - JHIZGYGYDXSUUFUVQUUGBLJHXAXBXKRNTVIOVRVSXLXMXN $. + xaddmnf2 mnfxr mnfnepnf neeq1 mnfaddpnf 3jaod biimtrid imp ) ACDZBCDZABEFZG + BAUAZHIZEFZUBZXPBUCDZBJKZBLKZUDXOYABUEXOYBYAYCYDXOAUCDZAJKZALKZUDZYBYAUMZAU + EZYEYIYFYGYEYBYAYEYBMZXQGBAUFIZEFXTABUGYKXSYLGEYBYEXSYLKBAUHUINUJOYFYBYAYFY + BMZXQXTYMXQJBEFZYBYFXPYNPZBUKZXPYOYFBULUNUOYMAJBEYFYBUPUQURYMXTGLEFZGCDYQPU + SGUTVAZYMXSLGEYMXSBLHIZLYMXRLBHYMXRJUAZLYFXRYTKYBAJVBZQVCRVDYBXPYFBJSZYSLKZ + YPYBUUBYFBVEUNBVFZVGVHNTVIOYGYBYAYGYBMZXQXTUUEALBEYGYBUPYBLBEFYGBVJUNVKUUEG + JXSEVLUUEXSBJHIZJYGXSUUFKYBYGXRJBHYGXRLUAJALVBVMRVDZQYBXPYGBLSZUUFJKYPYBUUH + YGBVNUNBVOVGVHVPVQOVRVSXOYHYCYAUMZYJYEUUIYFYGYEYCYAYEYCMZXQXTUUJAJBEYEAJEFY + CAVTQYEYCWAZWBUUJGJXSEVLUUJXSJXRHIZJUUJBJXRHUUKWCUUJXRCDZXRLSZUULJKYEUUMYCY + EXRYEXRAWDUCAWEAWFWGZWHQYEUUNYCYEXRUUOWIQXRWJWKVHVPVQOYFYCYAYFYCMZXQXTUUPXQ + JJEFZJCDZUUQPWLJWMVAAJBJEWNTUUPXTGGEFZGWPWQZUUPXSGGEUUPXSJLHIGUUPBJXRLHYFYC + WAYFXRLKYCYFXRYTLUUAVCRZQWOWRRNTVIOYGYCYAYGYCMZXQXTUVBXQLJEFWSALBJEWNWTUVBG + JXSEVLUVBXSJJHIZJYGYCXSUUFUVCUUGBJJHXAXBUURJLSUVCJKWLXCJVOXDRVPVQOVRVSXOYHY + DYAUMZYJYEUVDYFYGYEYDYAYEYDMZXQXTUVEXQALEFZUVEXOUVFPZYEXOYDAUKQAUTZXEUVEBLA + EYEYDWAZNURUVEXTYQYRUVEXSLGEUVEXSLXRHIZLUVEBLXRHUVIWCYEUVJLKZYDYEXRUCDZUVKU + UOUVLUUMXRJSUVKXRUKXRVEXRXGWKXEQVHNTVIOYFYDYAYFYDMZXQXTUVMXQUVFUVMXOUVGYFXO + YDYFXOUURWLAJCXFWTQUVHXEUVMBLAEYFYDWANURUVMXTYQYRUVMXSLGEUVMXSYSLYFXSYSKYDY + FXRLBHUVAVDQYDXPYFUUBUUCYDXPLCDZXHBLCXFWTYDUUBYFYDUUBLJSXIBLJXJWTUNUUDVGVHN + TVIOYGYDYAYGYDMZXQXTUVOXQLLEFZUVNUVPPXHLWMVAALBLEWNTUVOXTUUSUUTUVOXSGGEUVOX + SLJHIZGYGYDXSUUFUVQUUGBLJHXAXBXKRNTVIOVRVSXLXMXN $. $( Under certain conditions, the conclusion of ~ lesubadd is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.) $) @@ -100763,11 +100789,11 @@ Infinity and the extended real number system (cont.) ( ( A O B ) i^i ( B P C ) ) = (/) ) $= ( cxr wcel c0 wbr w3a co cin wss wceq elin elixx1 3adant3 biimpa simp3d cv wa wb adantrr wn 3adant1 simp2d simpl2 simp1d syl2anc mpbid pm2.65da - adantrl pm2.21d syl5bi ssrdv ss0 syl ) EQRZFQRZGQRZUAZEFMUBZFGHUBZUCZSU - DVOSUEVLDVOSDUKZVORVPVMRZVPVNRZULZVLVPSRZVPVMVNUFVLVSVTVLVSVPFJTZVLVQWA - VRVLVQULVPQRZEVPITZWAVLVQWBWCWAUAZVIVJVQWDUMVKABCEFVPIJMNUGUHUIUJUNVLVR - WAUOZVQVLVRULZFVPKTZWEWFWBWGVPGLTZVLVRWBWGWHUAZVJVKVRWIUMVIABCFGVPKLHOU - GUPUIZUQWFVJWBWGWEUMVIVJVKVRURWFWBWGWHWJUSPUTVAVCVBVDVEVFVOVGVH $. + adantrl pm2.21d biimtrid ssrdv ss0 syl ) EQRZFQRZGQRZUAZEFMUBZFGHUBZUCZ + SUDVOSUEVLDVOSDUKZVORVPVMRZVPVNRZULZVLVPSRZVPVMVNUFVLVSVTVLVSVPFJTZVLVQ + WAVRVLVQULVPQRZEVPITZWAVLVQWBWCWAUAZVIVJVQWDUMVKABCEFVPIJMNUGUHUIUJUNVL + VRWAUOZVQVLVRULZFVPKTZWEWFWBWGVPGLTZVLVRWBWGWHUAZVJVKVRWIUMVIABCFGVPKLH + OUGUPUIZUQWFVJWBWGWEUMVIVJVKVRURWFWBWGWHWJUSPUTVAVCVBVDVEVFVOVGVH $. $} ${ @@ -101531,11 +101557,11 @@ Infinity and the extended real number system (cont.) ( ( A [,) B ) i^i ( B [,) C ) ) = (/) ) $= ( vx cxr wcel w3a cico co cin c0 wss wceq wa clt wbr cle wb elico1 biimpa cv 3adant3 simp3d adantrr wn 3adant1 simp2d simpl2 simp1d xrlenlt syl2anc - elin mpbid adantrl pm2.65da pm2.21d syl5bi ssrdv ss0 syl ) AEFZBEFZCEFZGZ - ABHIZBCHIZJZKLVGKMVDDVGKDUAZVGFVHVEFZVHVFFZNZVDVHKFZVHVEVFULVDVKVLVDVKVHB - OPZVDVIVMVJVDVINVHEFZAVHQPZVMVDVIVNVOVMGZVAVBVIVPRVCABVHSUBTUCUDVDVJVMUEZ - VIVDVJNZBVHQPZVQVRVNVSVHCOPZVDVJVNVSVTGZVBVCVJWARVABCVHSUFTZUGVRVBVNVSVQR - VAVBVCVJUHVRVNVSVTWBUIBVHUJUKUMUNUOUPUQURVGUSUT $. + elin mpbid adantrl pm2.65da pm2.21d biimtrid ssrdv ss0 syl ) AEFZBEFZCEFZ + GZABHIZBCHIZJZKLVGKMVDDVGKDUAZVGFVHVEFZVHVFFZNZVDVHKFZVHVEVFULVDVKVLVDVKV + HBOPZVDVIVMVJVDVINVHEFZAVHQPZVMVDVIVNVOVMGZVAVBVIVPRVCABVHSUBTUCUDVDVJVMU + EZVIVDVJNZBVHQPZVQVRVNVSVHCOPZVDVJVNVSVTGZVBVCVJWARVABCVHSUFTZUGVRVBVNVSV + QRVAVBVCVJUHVRVNVSVTWBUIBVHUJUKUMUNUOUPUQURVGUSUT $. $( If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by @@ -102127,16 +102153,16 @@ Infinity and the extended real number system (cont.) ( cuz wcel cmin co cz cle wbr w3a wi wa adantr cr zre adantl syl2anr cc zcn cfv cfz eluz2 simpr zsubcl adantlr zsubcld 3jca 3adant3 com12 imp caddc cc0 subge0d exbiri com23 3impia impcom resubcl addge02d mpbid 3ad2ant2 3ad2ant1 - ex subsubd breqtrrd subge0 imp31 subge02d jca elfz2 sylanbrc 3adant2 syl5bi - wb sylbi ) ABDUAEZCADUAEZCABFGZFGZBCUBGEZVQBHEZAHEZBAIJZKZVRWALBAUCVRWCCHEZ - ACIJZKZWEWAACUCWBWDWHWALWCWBWDMZWHWAWIWHMZWBWFVTHEZKZBVTIJZVTCIJZMWAWIWHWLW - BWHWLLWDWHWBWLWCWFWBWLLWGWCWFMZWBWLWOWBMZWBWFWKWOWBUDWOWFWBWCWFUDNZWPCVSWQW - CWBVSHEWFABUEUFUGUHVDUIUJNUKWJWMWNWJBCAFGZBULGZVTIWJUMWRIJZBWSIJWHWIWTWCWFW - GWIWTLWOWIWGWTWOWIWTWGWOWIMCAWOCOEZWIWFXAWCCPZQNWOAOEZWIWCXCWFAPZNNUNUOUPUQ - URWJBWRWIBOEZWHWBXEWDBPZNZNWHWROEZWIWCWFXHWGWFXAXCXHWCXBXDCAUSRUIQUTVAWJCAB - WHCSEZWIWFWCXIWGCTVBQWHASEZWIWCWFXJWGATVCQWIBSEZWHWBXKWDBTNNVEVFWJUMVSIJZWN - WBWDWHXLWBWHWDXLWBWHXLWDWHXCXEXLWDVOWBWCWFXCWGXDVCZXFABVGRUOUPVHWJCVSWHXAWI - WFWCXAWGXBVBQWHXCXEVSOEWIXMXGABUSRVIVAVJVTBCVKVLVDVMVNVPUK $. + ex subsubd breqtrrd subge0 imp31 subge02d jca elfz2 sylanbrc biimtrid sylbi + wb 3adant2 ) ABDUAEZCADUAEZCABFGZFGZBCUBGEZVQBHEZAHEZBAIJZKZVRWALBAUCVRWCCH + EZACIJZKZWEWAACUCWBWDWHWALWCWBWDMZWHWAWIWHMZWBWFVTHEZKZBVTIJZVTCIJZMWAWIWHW + LWBWHWLLWDWHWBWLWCWFWBWLLWGWCWFMZWBWLWOWBMZWBWFWKWOWBUDWOWFWBWCWFUDNZWPCVSW + QWCWBVSHEWFABUEUFUGUHVDUIUJNUKWJWMWNWJBCAFGZBULGZVTIWJUMWRIJZBWSIJWHWIWTWCW + FWGWIWTLWOWIWGWTWOWIWTWGWOWIMCAWOCOEZWIWFXAWCCPZQNWOAOEZWIWCXCWFAPZNNUNUOUP + UQURWJBWRWIBOEZWHWBXEWDBPZNZNWHWROEZWIWCWFXHWGWFXAXCXHWCXBXDCAUSRUIQUTVAWJC + ABWHCSEZWIWFWCXIWGCTVBQWHASEZWIWCWFXJWGATVCQWIBSEZWHWBXKWDBTNNVEVFWJUMVSIJZ + WNWBWDWHXLWBWHWDXLWBWHXLWDWHXCXEXLWDVOWBWCWFXCWGXDVCZXFABVGRUOUPVHWJCVSWHXA + WIWFWCXAWGXBVBQWHXCXEVSOEWIXMXGABUSRVIVAVJVTBCVKVLVDVPVMVNUK $. $( Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van @@ -102640,8 +102666,8 @@ Infinity and the extended real number system (cont.) 1z fzsuc2 sylancr eqcomd feq2d mpbid simpr3 feq1d mpbird wfn wb fnresdisj reseq1d ffn uneq1d resundir uncom un0 eqtr2i 3eqtr4g fnresdm 3eqtrd nn0zd fveq1d peano2zd snidg fvres fveq1i fvsng eqtrid 3eqtr3d incom uneq2d 3jca - eqcomi 3eqtrrd fzssp1 fssres sylancl nnuz eluzfz2 ffvelrnd eqeltrrd opeq2 - wss fnressn sneqd eqtrd eqtr4id uneq12d reseq2d resundi eqtr2di impbida + eqcomi 3eqtrrd wss fzssp1 sylancl nnuz eluzfz2 ffvelcdmd eqeltrrd fnressn + fssres opeq2 sneqd eqtrd eqtr4id uneq12d reseq2d resundi eqtr2di impbida syl ) FHIZJFUAKZACLZBAIZDCEMZNZUBZJFJUCKZUAKZADLZYLDOZBNZCDYFPZNZUBZYEYKU DZYNYPYRYTYNYMAYILZYTYFYLQZMZAYILZUUAYTYGUUBAELZYFUUBUEZRNZUUDYEYGYHYJUFZ YTUUBBQZAEYTYLSIZYHUUBUUIELZYEUUJYKFUGZUHZYEYGYHYJUIZUUJYHUDZUUKEYLBUQZQZ @@ -102654,11 +102680,11 @@ Infinity and the extended real number system (cont.) PTYTUUJYHUVLBNUUMUUNUUOUVLYLUUQOBYLEUUQGWQYLBSAWRWSULWTYTYQYIYFPZCYFPZCYT DYIYFUVIWBYTUWBEYFPZMUWBRMZUWAUWBYTUWCRUWBYTUUBYFUEZRNZUWCRNZYTUWEUUFRUUB YFXAUUTWSYTUUKUVTUWFUWGVTUURUUBUUIEWCUUBYFEWATVOXBCEYFWEUWDUWBUWBWGXDWIYT - YGUVRUWBCNUUHUVSYFCWJTXEXCYEYSUDZYGYHYJUWHYGYFAYQLZUWHYNYFYMXNUWIYEYNYPYR - UFZJFXFYMAYFDXGXHUWHYFACYQYEYNYPYRVPZVQVRUWHYOBAYEYNYPYRUIZUWHYMAYLDUWJUW + YGUVRUWBCNUUHUVSYFCWJTXEXCYEYSUDZYGYHYJUWHYGYFAYQLZUWHYNYFYMXFUWIYEYNYPYR + UFZJFXGYMAYFDXNXHUWHYFACYQYEYNYPYRVPZVQVRUWHYOBAYEYNYPYRUIZUWHYMAYLDUWJUW HYLJVBOZIYLYMIZUWHYLSUWMYEUUJYSUULUHXIVIJYLXJYDZXKXLUWHYIYQUVJMZDYMPZDUWH CYQEUVJUWKUWHEUUQUVJGUWHUVJYLYOUQZQZUUQUWHDYMVSZUWNUVJUWSNUWHYNUWTUWJYMAD - WCZYDUWOYMYLDXOULUWHYPUWSUUQNUWLYPUWRUUPYOBYLXMXPYDXQXRXSUWHUWQDUUCPUWPUW + WCZYDUWOYMYLDXMULUWHYPUWSUUQNUWLYPUWRUUPYOBYLXOXPYDXQXRXSUWHUWQDUUCPUWPUW HYMUUCDUWHUVAUVDUVEVJUWHFHUVCYEYSVCUVGVIUVHVLXTDYFUUBYAYBUWHYNUWTUWQDNUWJ UXAYMDWJTXEXCYC $. $} @@ -103079,14 +103105,14 @@ Finite intervals of nonnegative integers (or "finite sets of sequential ( cz wcel co cfz wa cn0 cle wbr w3a cc0 wi wb adantr cr zre adantl imp cmin caddc elfz2 znn0sub biimpcd impcom zaddcl adantlr zred letr syl3anc addge01 syl2an elnn0z simplbi2 sylbird syld df-3an bitr3i 3anim123i sylbi lesubadd2 - 3ancoma syl biimprcd 3jca exp31 com23 3adant2 com12 syl5bi elfz2nn0 sylibr - ) CDEZABBCUBFZGFEZHABUAFZIEZCIEZVQCJKZLZVQMCGFEVNVPWAVPBDEZVODEZADEZLZBAJKZ - AVOJKZHZHZVNWAABVOUCWIVNWAWEWHVNWANZWBWDWHWJNWCWBWDHZVNWHWAWKVNWHWAWKVNHZWH - HVRVSVTWHWLVRWFWLVRNWGWLWFVRWKWFVROVNBAUDPUEPUFWLWHVSWLWHBVOJKZVSWLBQEZAQEZ - VOQEWHWMNWKWNVNWBWNWDBRZPZPWKWOVNWDWOWBARZSPWLVOWBVNWCWDBCUGUHUIBAVOUJUKWLW - MMCJKZVSWKWNCQEZWSWMOVNWQCRZBCULUMVNWSVSNWKVSVNWSCUNUOSUPUQTWHWLVTWGWLVTNWF - WLVTWGWLWOWNWTLZVTWGOWLWDWBVNLZXBWLWBWDVNLXCWBWDVNURWBWDVNVCUSWDWOWBWNVNWTW - RWPXAUTVAABCVBVDVESUFVFVGVHVITVJVKTVQCVLVM $. + 3ancoma biimprcd 3jca exp31 com23 3adant2 com12 biimtrid elfz2nn0 sylibr + syl ) CDEZABBCUBFZGFEZHABUAFZIEZCIEZVQCJKZLZVQMCGFEVNVPWAVPBDEZVODEZADEZLZB + AJKZAVOJKZHZHZVNWAABVOUCWIVNWAWEWHVNWANZWBWDWHWJNWCWBWDHZVNWHWAWKVNWHWAWKVN + HZWHHVRVSVTWHWLVRWFWLVRNWGWLWFVRWKWFVROVNBAUDPUEPUFWLWHVSWLWHBVOJKZVSWLBQEZ + AQEZVOQEWHWMNWKWNVNWBWNWDBRZPZPWKWOVNWDWOWBARZSPWLVOWBVNWCWDBCUGUHUIBAVOUJU + KWLWMMCJKZVSWKWNCQEZWSWMOVNWQCRZBCULUMVNWSVSNWKVSVNWSCUNUOSUPUQTWHWLVTWGWLV + TNWFWLVTWGWLWOWNWTLZVTWGOWLWDWBVNLZXBWLWBWDVNLXCWBWDVNURWBWDVNVCUSWDWOWBWNV + NWTWRWPXAUTVAABCVBVMVDSUFVEVFVGVHTVIVJTVQCVKVL $. $( Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, @@ -103190,21 +103216,21 @@ Finite intervals of nonnegative integers (or "finite sets of sequential ( ( ( P ` 0 ) = a /\ ( P ` 1 ) = b ) /\ ( ( P ` 2 ) = c /\ ( P ` 3 ) = d ) ) ) $= ( c3 cuz cfv wcel cc0 wa cv wceq wrex c1 c2 rexbii bitri cfz co simpr cn0 - wf 0nn0 elnn0uz mpbi 3nn0 uzss ax-mp sseli eluzfz sylancr adantr ffvelrnd - wss risset eqcom sylib 1eluzge0 cz cle 1z 3z 1le3 eluz2 mpbir3an 2eluzge0 - wbr jca uzuzle23 mpan r19.42v anbi2i 2rexbii r19.41v anbi1i 3bitri sylibr - jca32 ) BHIJZKZLBUAUBZCAUEZMZLAJZDNZOZDCPZQAJZENZOZECPZMZRAJZFNZOZFCPZHAJ - ZGNZOZGCPZMZMZWIWMMZWRXBMZMGCPZFCPZECPDCPZWFWOWSXCWFWJWNWFWGCKZWJWFWDCLAW - CWEUCZWCLWDKZWEWCLLIJZKZBXNKXMLUDKXOUFLUGUHWBXNBHXNKZWBXNUQHUDKXPUIHUGUHZ - LHUJUKULLLBUMUNUOUPXKWHWGOZDCPWJDWGCURXRWIDCWHWGUSSTUTWFWKCKZWNWFWDCQAXLW - CQWDKZWEWCQXNKBQIJZKXTVAWBYABHYAKZWBYAUQYBQVBKHVBKQHVCVJVDVEVFQHVGVHQHUJU - KULQLBUMUNUOUPXSWLWKOZECPWNEWKCURYCWMECWLWKUSSTUTVKWFWPCKZWSWFWDCRAXLWCRW - DKZWEWCRXNKBRIJKYEVIBVLRLBUMUNUOUPYDWQWPOZFCPWSFWPCURYFWRFCWQWPUSSTUTWFWT - 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3impia biimpd exp4b 3imp syl5bi 3jca ex impcom elfzo0 sylibr ) ABCZADBUMEFZ - GAHFZBUAFZABIJZKZADBUBEFVPVOVTVPVQBHFZABLJZKZVOVTMABUCWCVOVTWCVOGVQVRVSVQWA - WBVOUDWCVOVRVQWAWBVOVRMVOBACZVQWAGZWBVRABUEZWEWBWDVRWEWBWDGZVSVRWEVSWGVQANF - ZBNFZVSWGUFWAAOBOZABUGUHUIZVQWAVSVRMZVQWHDALJZGWAWLMAUJWHWAWMWLWHWAGZWMVSVR - WNWMVSGZDBIJZVRWNDTFATFZBTFZWOWPMWNUKWHWQWAAUNULWAWRWHBUOPDABUPUQWAWPVRMZWH - WAWIWSWJVRWIWPBURUSUTPVAQVBRSVCQVDVESWCVOVSVOWDWCVSWFVQWAWBWDVSMVQWAWBWDVSW - EWGVSWKVFVGVHVISVJVKRVLABVMVN $. + 3impia biimpd exp4b 3imp biimtrid 3jca ex impcom elfzo0 sylibr ) ABCZADBUME + FZGAHFZBUAFZABIJZKZADBUBEFVPVOVTVPVQBHFZABLJZKZVOVTMABUCWCVOVTWCVOGVQVRVSVQ + WAWBVOUDWCVOVRVQWAWBVOVRMVOBACZVQWAGZWBVRABUEZWEWBWDVRWEWBWDGZVSVRWEVSWGVQA + NFZBNFZVSWGUFWAAOBOZABUGUHUIZVQWAVSVRMZVQWHDALJZGWAWLMAUJWHWAWMWLWHWAGZWMVS + VRWNWMVSGZDBIJZVRWNDTFATFZBTFZWOWPMWNUKWHWQWAAUNULWAWRWHBUOPDABUPUQWAWPVRMZ + WHWAWIWSWJVRWIWPBURUSUTPVAQVBRSVCQVDVESWCVOVSVOWDWCVSWFVQWAWBWDVSMVQWAWBWDV + SWEWGVSWKVFVGVHVISVJVKRVLABVMVN $. ${ $d k N $. @@ -104212,14 +104238,14 @@ Finite intervals of nonnegative integers (or "finite sets of sequential /\ ( F ` K ) e/ ( F " ( 1 ..^ K ) ) ) ) ) $= ( vv cc0 co cn0 wcel wa cpr cima cin c0 wceq cfv wnel cle wbr adantl wn cfz wf c1 cfzo wfn ffn adantr 0nn0 a1i simpr nn0ge0 elfz2nn0 syl3anbrc id - nn0re leidd fnimapr syl3anc ineq1d eqeq1d cv wral wb simpl ffvelrnd eleq1 - disj notbid df-nel bitr4di ralprg syl2anc bitrid bitrd ) EBUAFZCAUBZBGHZI - ZAEBJKZAUCBUDFKZLZMNEAOZBAOZJZVTLZMNZWBVTPZWCVTPZIZVRWAWEMVRVSWDVTVRAVOUE - ZEVOHZBVOHZVSWDNVPWJVQVOCAUFUGVREGHZVQEBQRZWKWMVRUHUIVPVQUJVQWNVPBUKSEBUL - UMZVQWLVPVQVQVQBBQRWLVQUNZWPVQBBUOUPBBULUMSZVOEBAUQURUSUTWFDVAZVTHZTZDWDV - BZVRWIDWDVTVGVRWBCHWCCHXAWIVCVRVOCEAVPVQVDZWOVEVRVOCBAXBWQVEWTWGWHDWBWCCC - WRWBNZWTWBVTHZTWGXCWSXDWRWBVTVFVHWBVTVIVJWRWCNZWTWCVTHZTWHXEWSXFWRWCVTVFV - HWCVTVIVJVKVLVMVN $. + nn0re leidd fnimapr syl3anc ineq1d eqeq1d cv wral disj wb simpl ffvelcdmd + eleq1 notbid df-nel bitr4di ralprg syl2anc bitrid bitrd ) EBUAFZCAUBZBGHZ + IZAEBJKZAUCBUDFKZLZMNEAOZBAOZJZVTLZMNZWBVTPZWCVTPZIZVRWAWEMVRVSWDVTVRAVOU + EZEVOHZBVOHZVSWDNVPWJVQVOCAUFUGVREGHZVQEBQRZWKWMVRUHUIVPVQUJVQWNVPBUKSEBU + LUMZVQWLVPVQVQVQBBQRWLVQUNZWPVQBBUOUPBBULUMSZVOEBAUQURUSUTWFDVAZVTHZTZDWD + VBZVRWIDWDVTVCVRWBCHWCCHXAWIVDVRVOCEAVPVQVEZWOVFVRVOCBAXBWQVFWTWGWHDWBWCC + CWRWBNZWTWBVTHZTWGXCWSXDWRWBVTVGVHWBVTVIVJWRWCNZWTWCVTHZTWHXEWSXFWRWCVTVG + VHWCVTVIVJVKVLVMVN $. $} $( The difference between two elements in a half-open range of nonnegative @@ -104231,16 +104257,16 @@ Finite intervals of nonnegative integers (or "finite sets of sequential 3ad2ant1 adantl cfzo cneg cn0 cn wi elfzo0 nn0re nnre resubcl ancoms nn0ge0 w3a wb posdif biimp3a addgegt0 syl2anc nn0cn nncn 3ad2ant2 breqtrrd possumd subadd23d mpbid readdcl addge02 syl lelttrdi impancom imp ltsubadd2d mpbird - 3jca jca ex syl5bi 3adant2 sylbi ) ADCUAEZFZBVSFZCUBABGEZHIZWBCHIZJZVTAUCFZ - CUDFZACHIZULWAWEUEZACUFWFWHWIWGWABUCFZWGBCHIZULZWFWHJZWEBCUFWMWLWEWMWLJZWCW - DWNDWBCKEZHIWCWNDACBGEZKEZWOHWNALFZWPLFZJDAMIZDWPHIZJDWQHIWMWRWLWSWFWRWHAUG - ZNZWJWGWSWKWGWJWSWGCLFZBLFZWSWJCUHZBUGZCBUIOUJPQWMWTWLXAWFWTWHAUKNWJWGWKXAW - JXEXDWKXAUMWGXGXFBCUNOUOQAWPUPUQWNABCWMARFZWLWFXHWHAURNNWLBRFZWMWJWGXIWKBUR - STWLCRFZWMWGWJXJWKCUSUTTVCVAWNWBCWMWRXEWBLFWLXCWJWGXEWKXGSZABUIOWLXDWMWGWJX - DWKXFUTZTZVBVDWNWDABCKEZHIZWMWLXOWFWLWHXOWFWLJZACXNXPWRXDXNLFZWFWRWLXBNWLXD - WFXLTWLXQWFWJWGXQWKWJXEXDXQWGXGXFBCVEOPTVMXPDBMIZCXNMIZWLXRWFWJWGXRWKBUKSTX - PXDXEJZXRXSUMWLXTWFWJWGXTWKWGWJXTWGXDWJXEXFXGQUJPTCBVFVGVDVHVIVJWNABCWMWRWL - XCNWLXEWMXKTXMVKVLVNVOVPVQVRVJ $. + 3jca jca ex biimtrid 3adant2 sylbi ) ADCUAEZFZBVSFZCUBABGEZHIZWBCHIZJZVTAUC + FZCUDFZACHIZULWAWEUEZACUFWFWHWIWGWABUCFZWGBCHIZULZWFWHJZWEBCUFWMWLWEWMWLJZW + CWDWNDWBCKEZHIWCWNDACBGEZKEZWOHWNALFZWPLFZJDAMIZDWPHIZJDWQHIWMWRWLWSWFWRWHA + UGZNZWJWGWSWKWGWJWSWGCLFZBLFZWSWJCUHZBUGZCBUIOUJPQWMWTWLXAWFWTWHAUKNWJWGWKX + AWJXEXDWKXAUMWGXGXFBCUNOUOQAWPUPUQWNABCWMARFZWLWFXHWHAURNNWLBRFZWMWJWGXIWKB + URSTWLCRFZWMWGWJXJWKCUSUTTVCVAWNWBCWMWRXEWBLFWLXCWJWGXEWKXGSZABUIOWLXDWMWGW + JXDWKXFUTZTZVBVDWNWDABCKEZHIZWMWLXOWFWLWHXOWFWLJZACXNXPWRXDXNLFZWFWRWLXBNWL + XDWFXLTWLXQWFWJWGXQWKWJXEXDXQWGXGXFBCVEOPTVMXPDBMIZCXNMIZWLXRWFWJWGXRWKBUKS + TXPXDXEJZXRXSUMWLXTWFWJWGXTWKWGWJXTWGXDWJXEXFXGQUJPTCBVFVGVDVHVIVJWNABCWMWR + WLXCNWLXEWMXKTXMVKVLVNVOVPVQVRVJ $. $( @@ -106214,46 +106240,46 @@ more restricted (for example, specify rational arguments). 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frecsuc fveq1i fveq2i 3eqtr4g frecuzrdgrclt ffvelrnd adantlr frecuzrdgg - wf1o f1ocnvfv2 opeq1d simpr caovcld opelxp sylanbrc crn peano2 eleqtrrd - cdm fdm fvelrn eqeltrrd funopfv sylc oveq2d eqtr4d ) AEFUBUCZTZUDZEUMUE - UNZGUCZEELUFUCZHUCZUGUCZKUNZEEGUCZKUNYFGUHZYGYLUIZGTYHYLUJYFYDIGUOZYNAY - PYEABCDFGHIJKMNOPQSUKULYDIGUPUQZYFYIURZHUCZYOGYFYSEYKBCYDJBUSZUMUEUNZYT - CUSZKUNZUIZUTZUNZYOYFYSEYKUIZUUEUCZUUFYFYSYJUUEUCZUUHYFYRUUEFDUIZVAZUCZ - YIUUKUCZUUEUCZYSUUIYFUAUSZUUEUCZYDIVBZTZUAUUQVCUUJUUQTZYIVDTZUULUUNUJYF - UURUAUUQYFUUOUUQTZUDZUUPUUOVHUCZUMUEUNZUVCUUOUGUCZKUNZUIZUUQUVBUUPUVCUV - EUUEUNZUVGUVBUUPUVCUVEUIZUUEUCUVHUVBUUOUVIUUEUVAUUOUVIUJYFUUOYDIVEVFVGU - VCUVEUUEVIVJUVBUVCYDTZUVEJTUVGUUQTZUVHUVGUJUVAUVJYFUUOYDIVKVFZUVBIJUVEA - IJVLZYEUVAOVMUVAUVEITYFUUOYDIVNVFZVOUVBUVDYDTZUVFITZUVKUVBUVJUVOUVLFUVC - VPUQUVBUVCUUBKUNZITZUVPCIUVEUUBUVEUJZUVQUVFIUUBUVEUVCKVQZVSUVBUUCITZCIV - CZUVRCIVCBYDUVCYTUVCUJZUWAUVRCIUWCUUCUVQIYTUVCUUBKVRZVSVTAUWBBYDVCYEUVA - AUWABCYDIPWAVMUVLWBUVNWBUVDUVFYDIWCWDZBCUVCUVEYDJUUDUVGUUEUVDUVQUIUUQUW - CUUAUVDUUCUVQYTUVCUMUEVRUWDWEUVSUVQUVFUVDUVTWFUUEWHZWGWIWJUWEWKWLAUUSYE - AFYDTZDITZUUSAFWMTZUWGMFWNUQNFDYDIWCWDULAVDYDLXFZYEUUTABFLMRWOZVDYDELWP - WQZUAUUJYIUUQUUEWRWIYRHUUKQWSYJUUMUUEYIHUUKQWSWTXAYFYJUUGUUEYFYJYJVHUCZ - YKUIZUUGYFYJUUQTZYJUWNUJYFVDUUQYIHAVDUUQHUOZYEABCDFHIJKMNOPQXBULZUWLXCZ - YJYDIVEUQYFUWMEYKYFUWMYILUCZEYFBCDFHIJKLYIAUWIYEMULAUWHYENULAUVMYEOULZA - YTYDTUUBITUDUWAYEPXDZQUWLRXEAUWJYEUWSEUJUWKVDYDELXGWQWJXHWJZVGWJEYKUUEV - IVJYFYEYKJTYOUUQTZUUFYOUJAYEXIZYFIJYKUWTYFUWOYKITUWRYJYDIVNUQZVOYFYGYDT - ZYLITUXCYEUXFAFEVPVFYFBCEYKYDIIKUXAUXDUXEXJYGYLYDIXKXLBCEYKYDJUUDYOUUEY - GEUUBKUNZUIUUQYTEUJUUAYGUUCUXGYTEUMUEVRYTEUUBKVRWEUUBYKUJUXGYLYGUUBYKEK - VQWFUWFWGWIWJYFYSHXMZGYFHUHZYRHXPZTYSUXHTYFUWPUXIUWQVDUUQHUPUQZYFYRVDUX - JYFUUTYRVDTUWLYIXNUQYFUWPUXJVDUJUWQVDUUQHXQUQZXOYRHXRWDAGUXHUJYESULZXOX - SYGYLGXTYAYFYMYKEKYFYNUUGGTYMYKUJYQYFYJUUGGUXBYFYJUXHGYFUXIYIUXJTYJUXHT - UXKYFYIVDUXJUWLUXLXOYIHXRWDUXMXOXSEYKGXTYAYBYC $. + frecsuc fveq1i fveq2i 3eqtr4g frecuzrdgrclt ffvelcdmd adantlr f1ocnvfv2 + wf1o frecuzrdgg opeq1d simpr caovcld opelxp sylanbrc crn cdm peano2 fdm + eleqtrrd fvelrn eqeltrrd funopfv sylc oveq2d eqtr4d ) AEFUBUCZTZUDZEUMU + EUNZGUCZEELUFUCZHUCZUGUCZKUNZEEGUCZKUNYFGUHZYGYLUIZGTYHYLUJYFYDIGUOZYNA + YPYEABCDFGHIJKMNOPQSUKULYDIGUPUQZYFYIURZHUCZYOGYFYSEYKBCYDJBUSZUMUEUNZY + TCUSZKUNZUIZUTZUNZYOYFYSEYKUIZUUEUCZUUFYFYSYJUUEUCZUUHYFYRUUEFDUIZVAZUC + ZYIUUKUCZUUEUCZYSUUIYFUAUSZUUEUCZYDIVBZTZUAUUQVCUUJUUQTZYIVDTZUULUUNUJY + FUURUAUUQYFUUOUUQTZUDZUUPUUOVHUCZUMUEUNZUVCUUOUGUCZKUNZUIZUUQUVBUUPUVCU + VEUUEUNZUVGUVBUUPUVCUVEUIZUUEUCUVHUVBUUOUVIUUEUVAUUOUVIUJYFUUOYDIVEVFVG + UVCUVEUUEVIVJUVBUVCYDTZUVEJTUVGUUQTZUVHUVGUJUVAUVJYFUUOYDIVKVFZUVBIJUVE + AIJVLZYEUVAOVMUVAUVEITYFUUOYDIVNVFZVOUVBUVDYDTZUVFITZUVKUVBUVJUVOUVLFUV + CVPUQUVBUVCUUBKUNZITZUVPCIUVEUUBUVEUJZUVQUVFIUUBUVEUVCKVQZVSUVBUUCITZCI + VCZUVRCIVCBYDUVCYTUVCUJZUWAUVRCIUWCUUCUVQIYTUVCUUBKVRZVSVTAUWBBYDVCYEUV + AAUWABCYDIPWAVMUVLWBUVNWBUVDUVFYDIWCWDZBCUVCUVEYDJUUDUVGUUEUVDUVQUIUUQU + WCUUAUVDUUCUVQYTUVCUMUEVRUWDWEUVSUVQUVFUVDUVTWFUUEWHZWGWIWJUWEWKWLAUUSY + EAFYDTZDITZUUSAFWMTZUWGMFWNUQNFDYDIWCWDULAVDYDLXFZYEUUTABFLMRWOZVDYDELW + PWQZUAUUJYIUUQUUEWRWIYRHUUKQWSYJUUMUUEYIHUUKQWSWTXAYFYJUUGUUEYFYJYJVHUC + ZYKUIZUUGYFYJUUQTZYJUWNUJYFVDUUQYIHAVDUUQHUOZYEABCDFHIJKMNOPQXBULZUWLXC + ZYJYDIVEUQYFUWMEYKYFUWMYILUCZEYFBCDFHIJKLYIAUWIYEMULAUWHYENULAUVMYEOULZ + AYTYDTUUBITUDUWAYEPXDZQUWLRXGAUWJYEUWSEUJUWKVDYDELXEWQWJXHWJZVGWJEYKUUE + VIVJYFYEYKJTYOUUQTZUUFYOUJAYEXIZYFIJYKUWTYFUWOYKITUWRYJYDIVNUQZVOYFYGYD + TZYLITUXCYEUXFAFEVPVFYFBCEYKYDIIKUXAUXDUXEXJYGYLYDIXKXLBCEYKYDJUUDYOUUE + YGEUUBKUNZUIUUQYTEUJUUAYGUUCUXGYTEUMUEVRYTEUUBKVRWEUUBYKUJUXGYLYGUUBYKE + KVQWFUWFWGWIWJYFYSHXMZGYFHUHZYRHXNZTYSUXHTYFUWPUXIUWQVDUUQHUPUQZYFYRVDU + XJYFUUTYRVDTUWLYIXOUQYFUWPUXJVDUJUWQVDUUQHXPUQZXQYRHXRWDAGUXHUJYESULZXQ + XSYGYLGXTYAYFYMYKEKYFYNUUGGTYMYKUJYQYFYJUUGGUXBYFYJUXHGYFUXIYIUXJTYJUXH + TUXKYFYIVDUXJUWLUXLXQYIHXRWDUXMXQXSEYKGXTYAYBYC $. $} ${ @@ -107356,8 +107382,8 @@ the previous value and the value of the input sequence (second operand). fveq2d df-ov eqtr4di xp1st xp2nd peano2uz caovclg adantlr caovcld opelxpi elexd adantr syl2anc fvoveq1 oveq2d opeq12d opeq2d ovmpog syl3anc eqeltrd oveq1 eqid eqtrd jca frecfcl 3syl wi c0 csuc eqeq12d imbi2d fveq1i frec0g - eqtrid eqtr4d ad2antlr simpll ffvelrnd adantll sylib eqeltrrd simpr oveq2 - cbvralv frecsuc fveq2i 3eqtr4rd exp31 finds impcom eqfnfvd rneqd eqtr4id + eqtrid eqtr4d ad2antlr simpll ffvelcdmd adantll cbvralv sylib simpr oveq2 + eqeltrrd frecsuc fveq2i 3eqtr4rd exp31 finds impcom eqfnfvd rneqd eqtr4id a2d ) AFIJUGBCJUHOZUIBUJZUMUKPZCUJZYOIOZFPZQZULZJJIOZQZUNZUOGUOBCFIJUPAGU UCAUAUQGUUCAUQYMHURZGVFZGUQUSABCUUAJGHUIDEYMHEUJZDUJZUMUKPIOZFPZULZKAYNIO ZHRZUUAHRZBYMJYNJSUUKUUAHYNJITUTAUULBYMMVAZAJVGRJYMRZKJVBVCZVDZHUIVEAHVHV @@ -107382,9 +107408,9 @@ the previous value and the value of the input sequence (second operand). UYLHUYJUYTUYLHRZVUBUXLYMHWCVCZWIZUYJUYNUYKUYLUUJPZQZUYQUUDUYJVUGUYPUYNUYJ UYRVUDUYPHRVUGUYPSVUCVUEUYJUCUDUYLUYOHHHFUYIUWGUWHUXNAUWGUWHUYHUWIXQWFVUE UYJUWEIOZHRZUYOHRUCYMUYNUWEUYNSVUIUYOHUWEUYNITUTAVUJUCYMVMZUYHUXNAUWJVUKU - UNUULVUJBUCYMYNUWESUUKVUIHYNUWEITUTYBXRXNUYJUYRUYNYMRZVUCJUYKWDVCZVDWGZDE + UNUULVUJBUCYMYNUWESUUKVUIHYNUWEITUTXRXSXNUYJUYRUYNYMRZVUCJUYKWDVCZVDWGZDE UYKUYLYMHUUIUYPUUJUUFUYOFPHUUGUYKSUUHUYOUUFFUUGUYKUMIUKWLWMUUFUYLUYOFWSUU - JWTZWPWQZWOZUYJVULVUGHRVUHUUDRZVUMUYJVUGUYPHVUPVUNWRUYNVUGYMHWHWKZXSBCUYK + JWTZWPWQZWOZUYJVULVUGHRVUHUUDRZVUMUYJVUGUYPHVUPVUNWRUYNVUGYMHWHWKZYBBCUYK UYLYMUIYSUYQYTUYNYPUYOFPZQUUDYNUYKSZYOUYNYRVUTYNUYKUMUKWSZVVAYQUYOYPFYNUY KUMIUKWLWMWNYPUYLSZVUTUYPUYNYPUYLUYOFWSWOUWQWPWQUYJUXQUXLYTOZUYMUYJUXQUXM YTOZVVDUYJUVBUVCUYHUXQVVESAUVBUYHUXNUWRXNAUVCUYHUXNUWSXNZVUAUBUUBUXKUUDYT @@ -107432,8 +107458,8 @@ the previous value and the value of the input sequence (second operand). ovmpod frecuzrdgrclt ffnd wf wfn c1st 1st2nd2 adantl fveq2d df-ov eqtr4di c2nd xp1st xp2nd elexd peano2uz opelxpd fvoveq1 oveq2d opeq2d eqid ovmpog opeq12d syl3anc eqtrd cz uzid jca frecfcl ffn 3syl c0 csuc eqeq12d imbi2d - fveq1i frec0g eqtrid eqtr4d ad2antlr simpll ffvelrnd frecsuc simpr fveq2i - 3eqtr2d 3eqtr4g 3eqtrd 3eqtr4rd exp31 a2d finds impcom eqfnfvd eqtr4id + fveq1i frec0g eqtrid eqtr4d simpll ffvelcdmd frecsuc simpr 3eqtr2d fveq2i + ad2antlr 3eqtr4g 3eqtrd 3eqtr4rd exp31 a2d finds impcom eqfnfvd eqtr4id rneqd ) AHJKUGBCKUHQZUIBUJZUOUKRZCUJZUUHJQZHRZULZUMZKKJQZULZUNZUPIUPBCHJK UQAIUUPAUAURIUUPAURUUFFUSZIABCUUNKIFUIDEUUFFEUJZDUJZUOUKRJQZHRZUMZLNFUIUT AFVAVBAUUGUUFSZUUIFSZVCZVCZUUGUUIUVBRZUUKFUVFDEUUGUUIUUFFUVAUUKUVBFUVFUVB @@ -107462,24 +107488,24 @@ the previous value and the value of the input sequence (second operand). UIUUHUVGULZUMZUUOUNZQZUUOYAIVUPMYEAUXAVUQUUOTUYJUUOVUOUUQYFWCYGAUXAUYTUUO TUYJUUOUUMUUQYFWCYHVUBURSZAVUEVUIVURAVUEVUIVURAVCZVUEVCZVUCWOQZVUCXAQZUUM RZVVAUOUKRZVVBVVDJQZHRZULZVUHVUGVUTVVAUUFSZVVBUISZVVGUUQSVVCVVGTVUTVUCUUQ - SZVVHVUTURUUQVUBIAURUUQIWMVURVUEUWPYIVURAVUEYJZYKZVUCUUFFXBWCZVUTVVBFVUTV + SZVVHVUTURUUQVUBIAURUUQIWMVURVUEUWPYOVURAVUEYIZYJZVUCUUFFXBWCZVUTVVBFVUTV VJVVBFSZVVLVUCUUFFXCWCZXDZVUTVVDVVFUUFFVUTVVHVVDUUFSVVMKVVAXEWCZVUTUWAVVF - FSZAUWAVURVUEUWBYIVUTVVNVVEGSZUWAVVRVRVVOVUTUWGVVSUBUWHVVDUVMVVDTUWFVVEGU - VMVVDJVSVMAUWIVURVUEUWNYIVUTVVHVVDUWHSVVMKVVAWBWCWDUVTVVRBCVVBVVEFGUUGVVB + FSZAUWAVURVUEUWBYOVUTVVNVVEGSZUWAVVRVRVVOVUTUWGVVSUBUWHVVDUVMVVDTUWFVVEGU + VMVVDJVSVMAUWIVURVUEUWNYOVUTVVHVVDUWHSVVMKVVAWBWCWDUVTVVRBCVVBVVEFGUUGVVB TUUIVVETVCUVSVVFFUUGVVBUUIVVEHWEVMWFWGWHZXFBCVVAVVBUUFUIUULVVGUUMVVDUUIVV EHRZULUUQUUGVVATZUUHVVDUUKVWAUUGVVAUOUKVLZVWBUUJVVEUUIHUUGVVAUOJUKXGXHXLU UIVVBTZVWAVVFVVDUUIVVBVVEHVLXIUYHXKXMVUTVUHVUDUUMQZVUCUUMQZVVCVUTUWTUXAVU - RVUHVWETAUWTVURVUEUYIYIAUXAVURVUEUYJYIZVVKUDUUOVUBUUQUUMYLXMVUTVUCVUDUUMV - USVUEYMWRVUTVWFVVAVVBULZUUMQVVCVUTVUCVWHUUMVUTVVJVUCVWHTVVLVUCUUFFWPWCZWR - VVAVVBUUMWSWTYOVUTVUGVVAVVBVUORZVVDVVAVVBUVBRZULZVVGVUTVUGVWHVUOQZVWJVUTV + RVUHVWETAUWTVURVUEUYIYOAUXAVURVUEUYJYOZVVKUDUUOVUBUUQUUMYKXMVUTVUCVUDUUMV + USVUEYLWRVUTVWFVVAVVBULZUUMQVVCVUTVUCVWHUUMVUTVVJVUCVWHTVVLVUCUUFFWPWCZWR + VVAVVBUUMWSWTYMVUTVUGVVAVVBVUORZVVDVVAVVBUVBRZULZVVGVUTVUGVWHVUOQZVWJVUTV UGVUCVUOQZVWMVUTVUFVUPQZVUBVUPQZVUOQZVUGVWNVUTUWQVUOQZUUQSZUDUUQVJZUXAVUR VWOVWQTAVWTVURVUEAVWSUDUUQUXCVWRUXEUXDUXFUVBRZULZUUQUXCVWRUXDUXFVUORZVXBU XCVWRUXKVUOQVXCUXCUWQUXKVUOUXLWRUXDUXFVUOWSWTUXCUXMUXNVXBUUQSVXCVXBTUXOUX RUXCUXEVXAUUFFUXSUXCVXAUXHFUXCUXMUXPUXTVXAUXHTUXOUXQUYBDEUXDUXFUUFFUVAUXH UVBUURUXGHRFUUSUXDTUUTUXGUURHUUSUXDUOJUKXGXHUURUXFUXGHVLUVBXJZXKXMUYBWIXF ZBCUXDUXFUUFUIVUNVXBVUOUXEUXDUUIUVBRZULUUQUYEUUHUXEUVGVXFUYFUUGUXDUUIUVBV - LXLUYGVXFVXAUXEUUIUXFUXDUVBVNXIVUOXJZXKXMXNVXEWIVTYIVWGVVKUDUUOVUBUUQVUOY - LXMVUFIVUPMYEVUCVWPVUOVUBIVUPMYEYNYPVUTVUCVWHVUOVWIWRXNVVAVVBVUOWSWTVUTVV + LXLUYGVXFVXAUXEUUIUXFUXDUVBVNXIVUOXJZXKXMXNVXEWIVTYOVWGVVKUDUUOVUBUUQVUOY + KXMVUFIVUPMYEVUCVWPVUOVUBIVUPMYEYNYPVUTVUCVWHVUOVWIWRXNVVAVVBVUOWSWTVUTVV HVVIVWLUUQSVWJVWLTVVMVVPVUTVVDVWKUUFFVVQVUTVWKVVFFVUTVVHVVNVVRVWKVVFTVVMV VOVVTDEVVAVVBUUFFUVAVVFUVBUURVVEHRFUUSVVATUUTVVEUURHUUSVVAUOJUKXGXHUURVVB VVEHVLVXDXKXMZVVTWIXFBCVVAVVBUUFUIVUNVWLVUOVVDVVAUUIUVBRZULUUQVWBUUHVVDUV @@ -107547,14 +107573,14 @@ the previous value and the value of the input sequence (second operand). ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) ) $= ( vz vw c1 caddc co cfv cv wcel wceq va vb vc cseq cuz cmpo cvv cop cfrec vd eluzel2 syl fveq2 eleq1d ralrimiva uzid rspcdva wss ssv a1i iseqvalcbv - cz iseqovex seq3val frecuzrdgsuct mpdan ffvelrnd peano2uz caovcld fvoveq1 - eqid seqf oveq2d oveq1 ovmpog syl3anc eqtrd ) AHNOPZDFGUDZQZHHVSQZLMGUEQZ - EMRZLRZNOPFQZDPZUFZPZWAVRFQZDPZAHWBSZVTWHTIABCGFQZHGVSUAUBWBUGUARZNOPWMUB - RUCUJWBEUJRUCRNOPFQDPUFPUHUFGWLUHUIZEUGWGAWKGVBSZIGHUKULZABRZFQZESZWLESBW - BGWQGTWRWLEWQGFUMUNAWSBWBJUOZAWOGWBSWPGUPULUQEUGURAEUSUTABCLMDEFGJKVCUAUB - UCUJDEUGFGBCLMVAZABCLMDWNEFGWPXAJKVDVEVFAWKWAESWJESWHWJTIAWBEHVSABCDEFGWB - WBVKWPJKVLIVGZABCWAWIEEEDKXBAWSWIESBWBVRWQVRTWRWIEWQVRFUMUNWTAWKVRWBSIGHV - HULUQVILMHWAWBEWFWJWGWCWIDPEWDHTWEWIWCDWDHNFOVJVMWCWAWIDVNWGVKVOVPVQ $. + iseqovex seq3val frecuzrdgsuct mpdan eqid seqf ffvelcdmd peano2uz caovcld + cz fvoveq1 oveq2d oveq1 ovmpog syl3anc eqtrd ) AHNOPZDFGUDZQZHHVSQZLMGUEQ + ZEMRZLRZNOPFQZDPZUFZPZWAVRFQZDPZAHWBSZVTWHTIABCGFQZHGVSUAUBWBUGUARZNOPWMU + BRUCUJWBEUJRUCRNOPFQDPUFPUHUFGWLUHUIZEUGWGAWKGVKSZIGHUKULZABRZFQZESZWLESB + WBGWQGTWRWLEWQGFUMUNAWSBWBJUOZAWOGWBSWPGUPULUQEUGURAEUSUTABCLMDEFGJKVBUAU + BUCUJDEUGFGBCLMVAZABCLMDWNEFGWPXAJKVCVDVEAWKWAESWJESWHWJTIAWBEHVSABCDEFGW + BWBVFWPJKVGIVHZABCWAWIEEEDKXBAWSWIESBWBVRWQVRTWRWIEWQVRFUMUNWTAWKVRWBSIGH + VIULUQVJLMHWAWBEWFWJWGWCWIDPEWDHTWEWIWCDWDHNFOVLVMWCWAWIDVNWGVFVOVPVQ $. $} ${ @@ -107641,14 +107667,14 @@ the previous value and the value of the input sequence (second operand). ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) ) $= ( vz vw c1 co cfv cv wcel va vb vc caddc cseq cuz cmpo wceq cvv cop cfrec vd cz eluzel2 syl wss ssv seqovcd iseqvalcbv seqvalcd frecuzrdgsuct mpdan - a1i eqid seqf2 ffvelrnd eleq1d ralrimiva eluzp1p1 rspcdva caovcld fvoveq1 - fveq2 oveq2d oveq1 ovmpog syl3anc eqtrd ) AIPUDQZFGHUEZRZIIVTRZNOHUFRZDOS - ZNSZPUDQGRZFQZUGZQZWBVSGRZFQZAIWCTZWAWIUHJABCHGRZIHVTUAUBWCUIUASZPUDQWNUB - SUCULWCDULSUCSPUDQGRFQUGQUJUGHWMUJUKZDUIWHAWLHUMTJHIUNUOZKDUIUPADUQVCABCN - ODEFGHMLURUAUBUCULFDUIGHBCNOUSZABCNODEFWOGHWPWQKLMUTVAVBAWLWBDTWKDTWIWKUH - JAWCDIVTABCDEFGHWCKLWCVDWPMVEJVFZABCWBWJDEDFLWRABSZGRZETZWJETBHPUDQUFRZVS - WSVSUHWTWJEWSVSGVMVGAXABXBMVHAWLVSXBTJHIVIUOVJVKNOIWBWCDWGWKWHWDWJFQDWEIU - HWFWJWDFWEIPGUDVLVNWDWBWJFVOWHVDVPVQVR $. + a1i eqid seqf2 ffvelcdmd fveq2 ralrimiva eluzp1p1 rspcdva caovcld fvoveq1 + eleq1d oveq2d oveq1 ovmpog syl3anc eqtrd ) AIPUDQZFGHUEZRZIIVTRZNOHUFRZDO + SZNSZPUDQGRZFQZUGZQZWBVSGRZFQZAIWCTZWAWIUHJABCHGRZIHVTUAUBWCUIUASZPUDQWNU + BSUCULWCDULSUCSPUDQGRFQUGQUJUGHWMUJUKZDUIWHAWLHUMTJHIUNUOZKDUIUPADUQVCABC + NODEFGHMLURUAUBUCULFDUIGHBCNOUSZABCNODEFWOGHWPWQKLMUTVAVBAWLWBDTWKDTWIWKU + HJAWCDIVTABCDEFGHWCKLWCVDWPMVEJVFZABCWBWJDEDFLWRABSZGRZETZWJETBHPUDQUFRZV + SWSVSUHWTWJEWSVSGVGVMAXABXBMVHAWLVSXBTJHIVIUOVJVKNOIWBWCDWGWKWHWDWJFQDWEI + UHWFWJWDFWEIPGUDVLVNWDWBWJFVOWHVDVPVQVR $. $} ${ @@ -107920,21 +107946,21 @@ seq K ( .+ , G ) ) $= ser3mono $p |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) ) $= ( caddc cfz co wcel cfv cr cz syl adantr c1 vk vy cseq cv wa eqid eluzel2 - cuz adantlr serfre elfzuz uztrn syl2anr ffvelrnd cmin cle cc0 wceq breq2d - wbr fveq2 wral ralrimiva simpr wb eluzelz peano2zm elfzelz adantl fzaddel - 1zzd syl22anc mpbid cc ax-1cn npcan sylancl oveq2d eleqtrd rspcdva fzelp1 - zcn syldan eleq1d fzss1 fzp1elp1 sseldd addge01d readdcl breqtrrd monoord - wss seq3p1 ) AUAKCEUCZDFHAUAUDZDFLMZNZUEZEUHOZPWOWNWRBCEWSWSUFAEQNZWQADWS - NZWTGEDUGRSABUDZWSNZXBCOZPNZWQIUIUJWQWODUHOZNXAWOWSNZAWODFUKGDWOEULUMZUNZ - AWODFTUOMZLMNZUEZWOWNOZXMWOTKMZCOZKMZXNWNOUPXLUQXOUPUTZXMXPUPUTXLUQXDUPUT - ZXQBDTKMZFLMZXNXBXNURZXDXOUQUPXBXNCVAZUSAXRBXTVBXKAXRBXTJVCSXLXNXSXJTKMZL - MZXTXLXKXNYDNZAXKVDXLDQNZXJQNZWOQNZTQNXKYEVEXLXAYFAXAXKGSZEDVFRXLFQNZYGXL - FXFNZYJAYKXKHSDFVFRZFVGRXKYHAWODXJVHVIXLVKWOTDXJVJVLVMXLYCFXSLXLYJYCFURZY - LYJFVNNTVNNYMFWBVOFTVPVQRZVRVSVTXLXMXOAXKWQXMPNXLWODYCLMZWPXKWOYONAWODXJW - AVIXLYCFDLYNVRZVSZXIWCXLXEXOPNBWSXNYAXDXOPYBWDAXEBWSVBXKAXEBWSIVCSXLXNEFL - MZNXNWSNXLWPYRXNXLXAWPYRWLYIDEFWERXLXNYOWPXKXNYONAWODXJWFVIYPVSWGXNEFUKRV - TWHVMXLBUBKPCEWOAXKWQXGYQXHWCAXCXEXKIUIXBPNUBUDZPNUEXBYSKMPNXLXBYSWIVIWMW - JWK $. + cuz adantlr serfre elfzuz uztrn syl2anr ffvelcdmd cmin cle cc0 wceq fveq2 + wbr breq2d wral ralrimiva wb eluzelz peano2zm elfzelz adantl 1zzd fzaddel + simpr syl22anc mpbid cc ax-1cn npcan sylancl oveq2d eleqtrd fzelp1 syldan + zcn rspcdva eleq1d fzss1 fzp1elp1 sseldd addge01d readdcl seq3p1 breqtrrd + wss monoord ) AUAKCEUCZDFHAUAUDZDFLMZNZUEZEUHOZPWOWNWRBCEWSWSUFAEQNZWQADW + SNZWTGEDUGRSABUDZWSNZXBCOZPNZWQIUIUJWQWODUHOZNXAWOWSNZAWODFUKGDWOEULUMZUN + ZAWODFTUOMZLMNZUEZWOWNOZXMWOTKMZCOZKMZXNWNOUPXLUQXOUPUTZXMXPUPUTXLUQXDUPU + TZXQBDTKMZFLMZXNXBXNURZXDXOUQUPXBXNCUSZVAAXRBXTVBXKAXRBXTJVCSXLXNXSXJTKMZ + LMZXTXLXKXNYDNZAXKVKXLDQNZXJQNZWOQNZTQNXKYEVDXLXAYFAXAXKGSZEDVERXLFQNZYGX + LFXFNZYJAYKXKHSDFVERZFVFRXKYHAWODXJVGVHXLVIWOTDXJVJVLVMXLYCFXSLXLYJYCFURZ + YLYJFVNNTVNNYMFWBVOFTVPVQRZVRVSWCXLXMXOAXKWQXMPNXLWODYCLMZWPXKWOYONAWODXJ + VTVHXLYCFDLYNVRZVSZXIWAXLXEXOPNBWSXNYAXDXOPYBWDAXEBWSVBXKAXEBWSIVCSXLXNEF + LMZNXNWSNXLWPYRXNXLXAWPYRWLYIDEFWERXLXNYOWPXKXNYONAWODXJWFVHYPVSWGXNEFUKR + WCWHVMXLBUBKPCEWOAXKWQXGYQXHWAAXCXEXKIUIXBPNUBUDZPNUEXBYSKMPNXLXBYSWIVHWJ + WKWM $. $} ${ @@ -107958,29 +107984,29 @@ seq K ( .+ , G ) ) $= cv fveq2 eqeq12d imbi12d imbi2d cz seq3p1 eluzel2 simpl cle wbr eluzelz wa adantr adantl cr eluzle peano2re lep1d letrd eluz2 syl3anbrc syl2anc zred seq3-1 eqtr4d a1i13 peano2fzr imim1d oveq1 simprl peano2uz adantlr - expr uztrn eqid seqf ffvelrnd eleq1d wral ralrimiva fzssuz sstrid simpr - wss ssel2 syl2an rspcdva caovassg syl13anc syl5ibr animpimp2impd uzind4 - uzss mpcom mpd ) AJIUBUCPZJUDPZQZJEGHUEZRZIXJRZJEGXGUEZRZEPZSZAJXGUFRZQ - ZXIMXGJUGUHXRAXIXPTZMABUKZXHQZXTXJRZXLXTXMRZEPZSZTZTAXGXHQZXGXJRZXLXGXM - RZEPZSZTZTAUAUKZXHQZYMXJRZXLYMXMRZEPZSZTZTAYMUBUCPZXHQZYTXJRZXLYTXMRZEP - ZSZTZTAXSTBUAXGJXTXGSZYFYLAUUGYAYGYEYKXTXGXHUIUUGYBYHYDYJXTXGXJULUUGYCY - IXLEXTXGXMULUJUMUNUOXTYMSZYFYSAUUHYAYNYEYRXTYMXHUIUUHYBYOYDYQXTYMXJULUU - HYCYPXLEXTYMXMULUJUMUNUOXTYTSZYFUUFAUUIYAUUAYEUUEXTYTXHUIUUIYBUUBYDUUDX - TYTXJULUUIYCUUCXLEXTYTXMULUJUMUNUOXTJSZYFXSAUUJYAXIYEXPXTJXHUIUUJYBXKYD - XOXTJXJULUUJYCXNXLEXTJXMULUJUMUNUOXGUPQZAYGYKAYHXLXGGRZEPYJABCEFGHINOKU - QAYIUULXLEABCEFGXGAXRUUKMXGJURUHZAXTXQQZVCZAXTHUFRZQZXTGRZFQZAUUNUSUUOH - UPQZXTUPQZHXTUTVAUUQAUUTUUNAIUUPQZUUTNHIURUHZVDZUUNUVAAXGXTVBVEZUUOHIXT - UUOHUVDVNAIVFQZUUNAIAUVBIUPQNHIVBUHVNVDZUUOXTUVEVNZAHIUTVAZUUNAUVBUVINH - IVGUHVDUUOIXGXTUVGUUOUVFXGVFQUVGIVHUHUVHUUOIUVGVIUUNXGXTUTVAAXGXTVGVEVJ - VJHXTVKVLOVMZKVOUJVPVQYMXQQZAYSUUAUUEYRAUVKVCUUAYNYRAUVKUUAYNUVKUUAVCZY - NAYMXGJVRVEWDVSYRUUEAUVLVCZYOYTGRZEPZYQUVNEPZSYOYQUVNEVTUVMUUBUVOUUDUVP - UVMBCEFGHYMUVMUVKXGUUPQZYMUUPQAUVKUUAWAZAUVQUVLAUVBUVQNHIWBUHZVDXGYMHWE - VMAUUQUUSUVLOWCAXTFQCUKZFQVCXTUVTEPFQUVLKWCZUQUVMUUDXLYPUVNEPZEPZUVPUVM - UUCUWBXLEUVMBCEFGXGYMUVRAUUNUUSUVLUVJWCZUWAUQUJUVMAXLFQZYPFQUVNFQZUVPUW - CSAUVLUSAUWEUVLAUUPFIXJABCEFGHUUPUUPWFUVCOKWGNWHVDUVMXQFYMXMUVMBCEFGXGX - QXQWFAUUKUVLUUMVDUWDUWAWGUVRWHUVMUUSUWFBUUPYTUUIUURUVNFXTYTGULWIAUUSBUU - PWJUVLAUUSBUUPOWKVDAXHUUPWOUUAYTUUPQUVLAXHXQUUPXGJWLAUVQXQUUPWOUVSHXGXD - UHWMUVKUUAWNXHUUPYTWPWQWRABCDXLYPUVNFELWSWTVPUMXAXBXCXEXF $. + expr uztrn eqid seqf ffvelcdmd eleq1d wral ralrimiva fzssuz uzss sstrid + wss simpr syl2an rspcdva caovassg syl13anc syl5ibr animpimp2impd uzind4 + ssel2 mpcom mpd ) AJIUBUCPZJUDPZQZJEGHUEZRZIXJRZJEGXGUEZRZEPZSZAJXGUFRZ + QZXIMXGJUGUHXRAXIXPTZMABUKZXHQZXTXJRZXLXTXMRZEPZSZTZTAXGXHQZXGXJRZXLXGX + MRZEPZSZTZTAUAUKZXHQZYMXJRZXLYMXMRZEPZSZTZTAYMUBUCPZXHQZYTXJRZXLYTXMRZE + PZSZTZTAXSTBUAXGJXTXGSZYFYLAUUGYAYGYEYKXTXGXHUIUUGYBYHYDYJXTXGXJULUUGYC + YIXLEXTXGXMULUJUMUNUOXTYMSZYFYSAUUHYAYNYEYRXTYMXHUIUUHYBYOYDYQXTYMXJULU + UHYCYPXLEXTYMXMULUJUMUNUOXTYTSZYFUUFAUUIYAUUAYEUUEXTYTXHUIUUIYBUUBYDUUD + XTYTXJULUUIYCUUCXLEXTYTXMULUJUMUNUOXTJSZYFXSAUUJYAXIYEXPXTJXHUIUUJYBXKY + DXOXTJXJULUUJYCXNXLEXTJXMULUJUMUNUOXGUPQZAYGYKAYHXLXGGRZEPYJABCEFGHINOK + UQAYIUULXLEABCEFGXGAXRUUKMXGJURUHZAXTXQQZVCZAXTHUFRZQZXTGRZFQZAUUNUSUUO + HUPQZXTUPQZHXTUTVAUUQAUUTUUNAIUUPQZUUTNHIURUHZVDZUUNUVAAXGXTVBVEZUUOHIX + TUUOHUVDVNAIVFQZUUNAIAUVBIUPQNHIVBUHVNVDZUUOXTUVEVNZAHIUTVAZUUNAUVBUVIN + HIVGUHVDUUOIXGXTUVGUUOUVFXGVFQUVGIVHUHUVHUUOIUVGVIUUNXGXTUTVAAXGXTVGVEV + JVJHXTVKVLOVMZKVOUJVPVQYMXQQZAYSUUAUUEYRAUVKVCUUAYNYRAUVKUUAYNUVKUUAVCZ + YNAYMXGJVRVEWDVSYRUUEAUVLVCZYOYTGRZEPZYQUVNEPZSYOYQUVNEVTUVMUUBUVOUUDUV + PUVMBCEFGHYMUVMUVKXGUUPQZYMUUPQAUVKUUAWAZAUVQUVLAUVBUVQNHIWBUHZVDXGYMHW + EVMAUUQUUSUVLOWCAXTFQCUKZFQVCXTUVTEPFQUVLKWCZUQUVMUUDXLYPUVNEPZEPZUVPUV + MUUCUWBXLEUVMBCEFGXGYMUVRAUUNUUSUVLUVJWCZUWAUQUJUVMAXLFQZYPFQUVNFQZUVPU + WCSAUVLUSAUWEUVLAUUPFIXJABCEFGHUUPUUPWFUVCOKWGNWHVDUVMXQFYMXMUVMBCEFGXG + XQXQWFAUUKUVLUUMVDUWDUWAWGUVRWHUVMUUSUWFBUUPYTUUIUURUVNFXTYTGULWIAUUSBU + UPWJUVLAUUSBUUPOWKVDAXHUUPWOUUAYTUUPQUVLAXHXQUUPXGJWLAUVQXQUUPWOUVSHXGW + MUHWNUVKUUAWPXHUUPYTXDWQWRABCDXLYPUVNFELWSWTVPUMXAXBXCXEXF $. $} $} @@ -108071,23 +108097,23 @@ seq K ( .+ , G ) ) $= ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) ) $= ( vn cv cfzo co wcel wa cseq cfv c1 caddc wceq wral cuz cz eluzel2 adantr eqid syl ralrimiva fveq2 eleq1d rspccva sylan adantlr seqf elfzouz adantl - ffvelrnd cfz fzssuz fzofzp1 sselid syl2an anassrs ralrimivva oveq1 oveq1d - eqeq12d 2ralbidv oveq2d oveq2 rspc2va syl21anc seq3caopr3 ) ABCFGHIUBJKLM - NOPRSTUAAUBUCZMNUDUEUFZUGZWFFKMUHZUIZHUFWFUJUKUEZKUIZHUFZWFFJMUHZUIZDUCZG - UEZWKJUIZEUCZGUEZFUEZWOWRFUEZWPWSFUEZGUEZULZEHUMDHUMZWOWJGUEZWRWLGUEZFUEZ - XBWJWLFUEZGUEZULZWHMUNUIZHWFWIWHBCFHKMXMXMURZAMUOUFZWGANXMUFXORMNUPUSUQZW - HIUCZKUIZHUFZIXMUMZBUCZXMUFZYAKUIZHUFZAXTWGAXSIXMTUTZUQXSYDIYAXMXQYAULZXR - YCHXQYAKVAVBVCVDAYAHUFCUCZHUFUGZYAYGFUEZHUFWGOVEZVFWGWFXMUFAWFMNVGVHZVIAX - TWKXMUFZWMWGYEWGMNVJUEXMWKMNVKMNWFVLVMZXSWMIWKXMXQWKULZXRWLHXQWKKVAVBVCVN - WHWOHUFWRHUFZYAWPGUEZYGWSGUEZFUEZYIXCGUEZULZEHUMDHUMZCHUMBHUMZXFWHXMHWFWN - WHBCFHJMXMXNXPAYBYAJUIZHUFZWGAXQJUIZHUFZIXMUMZYBUUDAUUFIXMSUTZUUFUUDIYAXM - YFUUEUUCHXQYAJVAVBVCVDVEYJVFYKVIAUUGYLYOWGUUHYMUUFYOIWKXMYNUUEWRHXQWKJVAV - BVCVNAUUBWGAUUABCHHAYHUGYTDEHHAYHWPHUFWSHUFUGYTQVOVPVPUQUUAXFWQYQFUEZWOYG - FUEZXCGUEZULZEHUMDHUMBCWOWRHHYAWOULZYTUULDEHHUUMYRUUIYSUUKUUMYPWQYQFYAWOW - PGVQVRUUMYIUUJXCGYAWOYGFVQVRVSVTYGWRULZUULXEDEHHUUNUUIXAUUKXDUUNYQWTWQFYG - WRWSGVQWAUUNUUJXBXCGYGWRWOFWBVRVSVTWCWDXEXLXGWTFUEZXBWJWSFUEZGUEZULDEWJWL - HHWPWJULZXAUUOXDUUQUURWQXGWTFWPWJWOGWBVRUURXCUUPXBGWPWJWSFVQWAVSWSWLULZUU - OXIUUQXKUUSWTXHXGFWSWLWRGWBWAUUSUUPXJXBGWSWLWJFWBWAVSWCWDWE $. + ffvelcdmd fzssuz fzofzp1 sselid syl2an anassrs ralrimivva oveq1d 2ralbidv + cfz oveq1 eqeq12d oveq2d oveq2 rspc2va syl21anc seq3caopr3 ) ABCFGHIUBJKL + MNOPRSTUAAUBUCZMNUDUEUFZUGZWFFKMUHZUIZHUFWFUJUKUEZKUIZHUFZWFFJMUHZUIZDUCZ + GUEZWKJUIZEUCZGUEZFUEZWOWRFUEZWPWSFUEZGUEZULZEHUMDHUMZWOWJGUEZWRWLGUEZFUE + ZXBWJWLFUEZGUEZULZWHMUNUIZHWFWIWHBCFHKMXMXMURZAMUOUFZWGANXMUFXORMNUPUSUQZ + WHIUCZKUIZHUFZIXMUMZBUCZXMUFZYAKUIZHUFZAXTWGAXSIXMTUTZUQXSYDIYAXMXQYAULZX + RYCHXQYAKVAVBVCVDAYAHUFCUCZHUFUGZYAYGFUEZHUFWGOVEZVFWGWFXMUFAWFMNVGVHZVIA + XTWKXMUFZWMWGYEWGMNVRUEXMWKMNVJMNWFVKVLZXSWMIWKXMXQWKULZXRWLHXQWKKVAVBVCV + MWHWOHUFWRHUFZYAWPGUEZYGWSGUEZFUEZYIXCGUEZULZEHUMDHUMZCHUMBHUMZXFWHXMHWFW + NWHBCFHJMXMXNXPAYBYAJUIZHUFZWGAXQJUIZHUFZIXMUMZYBUUDAUUFIXMSUTZUUFUUDIYAX + MYFUUEUUCHXQYAJVAVBVCVDVEYJVFYKVIAUUGYLYOWGUUHYMUUFYOIWKXMYNUUEWRHXQWKJVA + VBVCVMAUUBWGAUUABCHHAYHUGYTDEHHAYHWPHUFWSHUFUGYTQVNVOVOUQUUAXFWQYQFUEZWOY + GFUEZXCGUEZULZEHUMDHUMBCWOWRHHYAWOULZYTUULDEHHUUMYRUUIYSUUKUUMYPWQYQFYAWO + WPGVSVPUUMYIUUJXCGYAWOYGFVSVPVTVQYGWRULZUULXEDEHHUUNUUIXAUUKXDUUNYQWTWQFY + GWRWSGVSWAUUNUUJXBXCGYGWRWOFWBVPVTVQWCWDXEXLXGWTFUEZXBWJWSFUEZGUEZULDEWJW + LHHWPWJULZXAUUOXDUUQUURWQXGWTFWPWJWOGWBVPUURXCUUPXBGWPWJWSFVSWAVTWSWLULZU + UOXIUUQXKUUSWTXHXGFWSWLWRGWBWAUUSUUPXJXBGWSWLWJFWBWAVTWCWDWE $. $} ${ @@ -108131,14 +108157,14 @@ seq K ( .+ , G ) ) $= $( Lemma for ~ seq3f1o . (Contributed by Jim Kingdon, 21-Aug-2022.) $) iseqf1olemkle $p |- ( ph -> K <_ ( `' J ` K ) ) $= ( cfv clt wbr wceq wa cz wcel syl adantr zred ccnv cle co elfzelz wf1o wf - cfz f1ocnv f1of ffvelrnd simpr ltled cr eqle sylan f1ocnvfv2 syl2anc cfzo - cv fveq2 id eqeq12d cuz elfzuz elfzo2 syl3anbrc rspcdva eqtr3d syldan w3o - wral ztri3or mpjao3dan ) ADDCUAZKZLMZDVOUBMZDVONZVODLMZAVPOZDVOVTDADPQZVP - ADEFUGUCZQZWAHDEFUDRZSTVTVOAVOPQZVPAVOWBQZWEAWBWBDVNAWBWBVNUEZWBWBVNUFAWB - WBCUEZWGIWBWBCUHRWBWBVNUIRHUJZVOEFUDRZSTAVPUKULADUMQVRVQADWDTDVOUNUOZAVSV - RVQAVSOZVOCKZDVOWLWHWCWMDNAWHVSISAWCVSHSWBWBDCUPUQWLBUSZCKZWNNZWMVONBEDUR - UCZVOWNVONZWOWMWNVOWNVOCUTWRVAVBAWPBWQVKVSJSWLVOEVCKQZWAVSVOWQQAWSVSAWFWS - WIVOEFVDRSAWAVSWDSAVSUKVOEDVEVFVGVHWKVIAWAWEVPVRVSVJWDWJDVOVLUQVM $. + f1ocnv f1of ffvelcdmd simpr ltled cr eqle sylan f1ocnvfv2 syl2anc cv cfzo + cfz id eqeq12d wral cuz elfzuz elfzo2 syl3anbrc rspcdva eqtr3d syldan w3o + fveq2 ztri3or mpjao3dan ) ADDCUAZKZLMZDVOUBMZDVONZVODLMZAVPOZDVOVTDADPQZV + PADEFUSUCZQZWAHDEFUDRZSTVTVOAVOPQZVPAVOWBQZWEAWBWBDVNAWBWBVNUEZWBWBVNUFAW + BWBCUEZWGIWBWBCUGRWBWBVNUHRHUIZVOEFUDRZSTAVPUJUKADULQVRVQADWDTDVOUMUNZAVS + VRVQAVSOZVOCKZDVOWLWHWCWMDNAWHVSISAWCVSHSWBWBDCUOUPWLBUQZCKZWNNZWMVONBEDU + RUCZVOWNVONZWOWMWNVOWNVOCVKWRUTVAAWPBWQVBVSJSWLVOEVCKQZWAVSVOWQQAWSVSAWFW + SWIVOEFVDRSAWAVSWDSAVSUJVOEDVEVFVGVHWKVIAWAWEVPVRVSVJWDWJDVOVLUPVM $. $} ${ @@ -108153,13 +108179,13 @@ seq K ( .+ , G ) ) $= $( Lemma for ~ seq3f1o . (Contributed by Jim Kingdon, 21-Aug-2022.) $) iseqf1olemklt $p |- ( ph -> K < ( `' J ` K ) ) $= ( cfv clt wbr wceq co wf1o wcel adantr syl neneqd wa f1ocnvfv2 syl2anc cv - ccnv cfz cfzo fveq2 id eqeq12d wral cuz cz wf f1ocnv f1of ffvelrnd elfzuz - elfzelz simpr elfzo2 syl3anbrc rspcdva eqtr3d mtand w3o ztri3or ecase23d + ccnv cfz cfzo fveq2 id eqeq12d wral cz wf f1ocnv ffvelcdmd elfzuz elfzelz + cuz f1of simpr elfzo2 syl3anbrc rspcdva eqtr3d mtand w3o ztri3or ecase23d ) ADDCUFZLZMNZDVKOZVKDMNZADVKKUAZAVNVMVOAVNUBZVKCLZDVKVPEFUGPZVRCQZDVRRZV QDOAVSVNISAVTVNHSVRVRDCUCUDVPBUEZCLZWAOZVQVKOBEDUHPZVKWAVKOZWBVQWAVKWAVKC - UIWEUJUKAWCBWDULVNJSVPVKEUMLRZDUNRZVNVKWDRAWFVNAVKVRRZWFAVRVRDVJAVRVRVJQZ - VRVRVJUOAVSWIIVRVRCUPTVRVRVJUQTHURZVKEFUSTSAWGVNAVTWGHDEFUTTZSAVNVAVKEDVB - VCVDVEVFAWGVKUNRZVLVMVNVGWKAWHWLWJVKEFUTTDVKVHUDVI $. + UIWEUJUKAWCBWDULVNJSVPVKEUSLRZDUMRZVNVKWDRAWFVNAVKVRRZWFAVRVRDVJAVRVRVJQZ + VRVRVJUNAVSWIIVRVRCUOTVRVRVJUTTHUPZVKEFUQTSAWGVNAVTWGHDEFURTZSAVNVAVKEDVB + VCVDVEVFAWGVKUMRZVLVMVNVGWKAWHWLWJVKEFURTDVKVHUDVI $. $} ${ @@ -108174,17 +108200,17 @@ seq K ( .+ , G ) ) $= ( cfv co wcel wa ad2antrr syl cz cle wbr elfzelz zred ccnv cfz wceq c1 wn cmin cif wf wf1o f1of w3a elfzel1 elfzel2 peano2zm 3jca elfzle1 clt simpr eqcom sylnib wo ad2antlr zleloe syl2anc mpbid zltlem1 letrd lem1d elfzle2 - wb ecased jca elfz2 sylanbrc ffvelrnd zdceq adantr ifcldadc f1ocnv fzdcel - wdc 3syl syl3anc ) ABDDCUAZJZUBKLZBDUCZDBUDUFKZCJZUGBCJZEFUBKZAWFMZWGDWIW - KADWKLZWFWGGNWLWGUEZMZWKWKWHCAWKWKCUHZWFWNAWKWKCUIZWPHWKWKCUJOZNWOEPLZFPL - ZWHPLZUKEWHQRZWHFQRZMWHWKLWOWSWTXAWOWMWSAWMWFWNGNZDEFULOZWOWMWTXDDEFUMOZW - OBPLZXAAXGWFWNABWKLZXGIBEFSOZNZBUNOZUOWOXBXCWOEDWHWOEXETWODWOWMDPLZXDDEFS - ZOZTWOWHXKTZWOWMEDQRXDDEFUPOWODBUQRZDWHQRZWOXPDBUCZWOWGXRWLWNURBDUSUTWODB - QRZXPXRVAZWFXSAWNBDWEUPVBWOXLXGXSXTVJXNXJDBVCVDVEVKWOXLXGXPXQVJXNXJDBVFVD - VEVGWOWHBFXOWOBXJTZWOFXFTWOBYAVHWOXHBFQRAXHWFWNINBEFVIOVGVLWHEFVMVNVOAWGW - AZWFAXGXLYBXIAWMXLGXMOZBDVPVDVQVRAWJWKLWFUEAWKWKBCWRIVOVQAXGXLWEPLZWFWAXI - YCAWEWKLYDAWKWKDWDAWQWKWKWDUIWKWKWDUHHWKWKCVSWKWKWDUJWBGVOWEEFSOBDWEVTWCV - R $. + ecased jca elfz2 sylanbrc ffvelcdmd wdc zdceq adantr ifcldadc f1ocnv 3syl + wb fzdcel syl3anc ) ABDDCUAZJZUBKLZBDUCZDBUDUFKZCJZUGBCJZEFUBKZAWFMZWGDWI + WKADWKLZWFWGGNWLWGUEZMZWKWKWHCAWKWKCUHZWFWNAWKWKCUIZWPHWKWKCUJOZNWOEPLZFP + LZWHPLZUKEWHQRZWHFQRZMWHWKLWOWSWTXAWOWMWSAWMWFWNGNZDEFULOZWOWMWTXDDEFUMOZ + WOBPLZXAAXGWFWNABWKLZXGIBEFSOZNZBUNOZUOWOXBXCWOEDWHWOEXETWODWOWMDPLZXDDEF + SZOZTWOWHXKTZWOWMEDQRXDDEFUPOWODBUQRZDWHQRZWOXPDBUCZWOWGXRWLWNURBDUSUTWOD + BQRZXPXRVAZWFXSAWNBDWEUPVBWOXLXGXSXTWAXNXJDBVCVDVEVJWOXLXGXPXQWAXNXJDBVFV + DVEVGWOWHBFXOWOBXJTZWOFXFTWOBYAVHWOXHBFQRAXHWFWNINBEFVIOVGVKWHEFVLVMVNAWG + VOZWFAXGXLYBXIAWMXLGXMOZBDVPVDVQVRAWJWKLWFUEAWKWKBCWRIVNVQAXGXLWEPLZWFVOX + IYCAWEWKLYDAWKWKDWDAWQWKWKWDUIWKWKWDUHHWKWKCVSWKWKWDUJVTGVNWEEFSOBDWEWBWC + VR $. ${ $d A u $. $d J u $. $d K u $. $d M u $. $d N u $. @@ -108219,25 +108245,25 @@ seq K ( .+ , G ) ) $= /\ -. B e. ( K ... ( `' J ` K ) ) ) ) $= ( cfv wcel wceq adantr syl ccnv cfz co wn wa cima cmin iseqf1olemqval c1 cif simprl iftrued eqtrd wf1o f1ocnvfv2 syl2anc ad2antrr wfn f1ofn - wss cuz elfzuz fzss1 3syl wf f1ocnv f1of ffvelrnd elfzuz3 fzss2 sstrd - elfzubelfz ad2antlr fnfvima syl3anc eqeltrrd w3a cle elfzelz peano2zm - cz wbr 3jca clt simpr eqcom sylnib wo elfzle1 wb zleloe mpbid zltlem1 - ecased zred lem1d elfzle2 letrd jca elfz2 sylanbrc wdc zdceq ifcldadc - eqeltrd simprr iffalsed wf1 f1of1 f1elima mtbird eqneltrd pm2.65da ) + wss cuz elfzuz fzss1 3syl wf f1ocnv f1of ffvelcdmd elfzuz3 elfzubelfz + fzss2 sstrd ad2antlr fnfvima syl3anc eqeltrrd cz w3a cle wbr peano2zm + elfzelz 3jca clt simpr eqcom sylnib wo elfzle1 wb zleloe mpbid ecased + zltlem1 zred lem1d elfzle2 letrd jca elfz2 sylanbrc wdc zdceq eqeltrd + ifcldadc simprr iffalsed wf1 f1of1 f1elima mtbird eqneltrd pm2.65da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seq K ( .+ , G ) ) $= by Jim Kingdon, 27-Aug-2022.) $) iseqf1olemmo $p |- ( ph -> A = B ) $= ( wcel wn wa ad2antrr syl ccnv cfv cfz co wceq wf1o simplr iseqf1olemab - simpr jca iseqf1olemnab pm2.21dd wo wdc cz elfzelz f1ocnv f1of ffvelrnd - wf 3syl fzdcel syl3anc exmiddc adantr mpjaodan eqcomd iseqf1olemnanb ) + simpr jca iseqf1olemnab pm2.21dd wo cz elfzelz wf f1ocnv f1of ffvelcdmd + wdc 3syl fzdcel syl3anc exmiddc adantr mpjaodan eqcomd iseqf1olemnanb ) ACGGFUAZUBZUCUDZPZCDUEZVLQZAVLRZDVKPZVMVPQZVOVPRBCDEFGHIAGHIUCUDZPZVLVP JSAVRVRFUFZVLVPKSACVRPZVLVPMSADVRPZVLVPNSACEUBZDEUBZUEZVLVPOSLAVLVPUGVO VPUIUHVOVQRZVLVQRZVMWFVLVQAVLVQUGVOVQUIUJAWGQVLVQABCDEFGHIJKMNOLUKSULAV - PVQUMZVLAVPUNZWHADUOPZGUOPZVJUOPZWIAWBWJNDHIUPTAVSWKJGHIUPTZAVJVRPWLAVR - VRGVIAVTVRVRVIUFVRVRVIUTKVRVRFUQVRVRVIURVAJUSVJHIUPTZDGVJVBVCVPVDTZVEVF + PVQUMZVLAVPUTZWHADUNPZGUNPZVJUNPZWIAWBWJNDHIUOTAVSWKJGHIUOTZAVJVRPWLAVR + VRGVIAVTVRVRVIUFVRVRVIUPKVRVRFUQVRVRVIURVAJUSVJHIUOTZDGVJVBVCVPVDTZVEVF AVNRZVPVMVQWPVPRZVPVNRZVMWQVPVNWPVPUIAVNVPUGUJAWRQVNVPABDCEFGHIJKNMAWCW DOVGLUKSULWPVQRBCDEFGHIAVSVNVQJSAVTVNVQKSAWAVNVQMSAWBVNVQNSAWEVNVQOSLAV - NVQUGWPVQUIVHAWHVNWOVEVFAVLUNZVLVNUMACUOPZWKWLWSAWAWTMCHIUPTWMWNCGVJVBV + NVQUGWPVQUIVHAWHVNWOVEVFAVLUTZVLVNUMACUNPZWKWLWSAWAWTMCHIUOTWMWNCGVJVBV CVLVDTVF $. $} @@ -108373,26 +108399,26 @@ seq K ( .+ , G ) ) $= ( cfv wceq co wcel cif cle wbr syl cv cfz wral cfzo csn wa ccnv c1 cmin cun elfzole1 adantl elfzle2 elfzolt2 anim12ci cr elfzoelz zred elfzoel2 clt wi cz elfzel2 adantr ltleletr syl3anc mpd wb elfzel1 elfz mpbir2and - wn zltnle syl2anc mpbid intnanrd wf1o f1ocnv f1of 3syl ffvelrnd elfzelz - mtbird iffalsed r19.21bi eqtrd simpr eleq1w eqeq1 oveq1 fveq2d ifbieq2d - eqeltrd ifbieq12d fvmptg ralrimiva iseqf1olemqval elfzuz2 iseqf1olemkle - wf fveq2 eluz2 syl3anbrc eluzfz1 iftrued eqid iftruei eqtrdi id eqeq12d - cuz ralsng mpbird ralun elfzuz fzisfzounsn raleqdv ) ABUAZDMZXRNZBGFUBO - ZUCXTBGFUDOZFUEZUJZUCZAXTBYBUCXTBYCUCZYEAXTBYBAXRYBPZUFZXSXRFFEUGZMZUBO - ZPZXRFNZFXRUHUIOZEMZQZXREMZQZXRYHXRGHUBOZPZYRYBPXSYRNYHYTGXRRSZXRHRSZYG - UUAAXRGFUKULYHXRFUTSZFHRSZUFZUUBAUUDYGUUCAFYSPZUUDIFGHUMTXRGFUNZUOYHXRU - PPFUPPHUPPUUEUUBVAYHXRYGXRVBPZAXRGFUQULZURYHFYGFVBPZAXRGFUSULZURYHHAHVB - PZYGAUUFUULIFGHVCTVDZURXRFHVEVFVGYHUUHGVBPZUULYTUUAUUBUFVHUUIAUUNYGAUUF - UUNIFGHVITVDUUMXRGHVJVFVKYHYRXRYBYHYRYQXRYHYLYPYQYHYLFXRRSZXRYJRSZUFZYH - UUOUUPYHUUCUUOVLZYGUUCAUUGULYHUUHUUJUUCUURVHUUIUUKXRFVMVNVOVPYHUUHUUJYJ - VBPZYLUUQVHUUIUUKAUUSYGAYJYSPUUSAYSYSFYIAYSYSEVQYSYSYIVQYSYSYIWTJYSYSEV - RYSYSYIVSVTIWAYJGHWBTZVDXRFYJVJVFWCWDAYQXRNBYBLWEWFZAYGWGWMCXRCUAZYKPZU - VBFNZFUVBUHUIOZEMZQZUVBEMZQYRYSYBDUVBXRNZUVCYLUVGUVHYPYQCBYKWHUVIUVDYMU - VFYOFUVBXRFWIUVIUVEYNEUVBXRUHUIWJWKWLUVBXREXAWNKWOVNUVAWFWPAYFFDMZFNZAU - VJFYKPZFFNZFFUHUIOEMZQZFEMZQZFACFDEFGHIJIKWQAUVQUVOFAUVLUVOUVPAYJFXKMPZ - UVLAUUJUUSFYJRSUVRAUUFUUJIFGHWBZTUUTABEFGHAUUFHGXKMZPIFGHWRTIJLWSFYJXBX - CFYJXDTXEUVMFUVNFXFXGXHWFAUUFUUJYFUVKVHIUVSXTUVKBFVBYMXSUVJXRFXRFDXAYMX - IXJXLVTXMXTBYBYCXNVNAXTBYAYDAUUFFUVTPYAYDNIFGHXOGFXPVTXQXM $. + wn zltnle syl2anc mpbid intnanrd wf1o f1ocnv f1of 3syl ffvelcdmd mtbird + elfzelz iffalsed r19.21bi eqtrd simpr eqeltrd eleq1w eqeq1 oveq1 fveq2d + wf ifbieq2d fveq2 ifbieq12d fvmptg ralrimiva iseqf1olemqval cuz elfzuz2 + iseqf1olemkle eluz2 syl3anbrc eluzfz1 iftrued iftruei eqtrdi id eqeq12d + eqid ralsng mpbird ralun elfzuz fzisfzounsn raleqdv ) ABUAZDMZXRNZBGFUB + OZUCXTBGFUDOZFUEZUJZUCZAXTBYBUCXTBYCUCZYEAXTBYBAXRYBPZUFZXSXRFFEUGZMZUB + OZPZXRFNZFXRUHUIOZEMZQZXREMZQZXRYHXRGHUBOZPZYRYBPXSYRNYHYTGXRRSZXRHRSZY + GUUAAXRGFUKULYHXRFUTSZFHRSZUFZUUBAUUDYGUUCAFYSPZUUDIFGHUMTXRGFUNZUOYHXR + UPPFUPPHUPPUUEUUBVAYHXRYGXRVBPZAXRGFUQULZURYHFYGFVBPZAXRGFUSULZURYHHAHV + BPZYGAUUFUULIFGHVCTVDZURXRFHVEVFVGYHUUHGVBPZUULYTUUAUUBUFVHUUIAUUNYGAUU + FUUNIFGHVITVDUUMXRGHVJVFVKYHYRXRYBYHYRYQXRYHYLYPYQYHYLFXRRSZXRYJRSZUFZY + HUUOUUPYHUUCUUOVLZYGUUCAUUGULYHUUHUUJUUCUURVHUUIUUKXRFVMVNVOVPYHUUHUUJY + JVBPZYLUUQVHUUIUUKAUUSYGAYJYSPUUSAYSYSFYIAYSYSEVQYSYSYIVQYSYSYIWMJYSYSE + VRYSYSYIVSVTIWAYJGHWCTZVDXRFYJVJVFWBWDAYQXRNBYBLWEWFZAYGWGWHCXRCUAZYKPZ + UVBFNZFUVBUHUIOZEMZQZUVBEMZQYRYSYBDUVBXRNZUVCYLUVGUVHYPYQCBYKWIUVIUVDYM + UVFYOFUVBXRFWJUVIUVEYNEUVBXRUHUIWKWLWNUVBXREWOWPKWQVNUVAWFWRAYFFDMZFNZA + UVJFYKPZFFNZFFUHUIOEMZQZFEMZQZFACFDEFGHIJIKWSAUVQUVOFAUVLUVOUVPAYJFWTMP + ZUVLAUUJUUSFYJRSUVRAUUFUUJIFGHWCZTUUTABEFGHAUUFHGWTMZPIFGHXATIJLXBFYJXC + XDFYJXETXFUVMFUVNFXKXGXHWFAUUFUUJYFUVKVHIUVSXTUVKBFVBYMXSUVJXRFXRFDWOYM + XIXJXLVTXMXTBYBYCXNVNAXTBYAYDAUUFFUVTPYAYDNIFGHXOGFXPVTXQXM $. $} ${ @@ -108413,17 +108439,17 @@ seq K ( .+ , G ) ) $= ( cfv wcel syl va cv cuz wa csb cle wbr cif cmpt csbeq2i cvv wceq cfz co wf cfn wf1o f1of cz elfzel1 elfzel2 fzfigd fex syl2anc nfcvd fveq1 fveq2d ifeq1d mpteq2dv csbiegf eqtrid fveq2 eleq1d wral cbvralv sylib - ralrimiva ad2antrr simpr wb simplr elfz5 mpbird ffvelrnd rspcdva uzid - elfzuz wn wdc eluzelz zdcle syl2anr ifcldadc fvmpt2d eqeltrd ) ABUBZK - UCRZSZUDZWPGIDUEZRWPLUFUGZWPIRZHRZKHRZUHZFABWQXEWTFAWTGIBWQXAWPGUBZRZ - HRZXDUHZUIZUEZBWQXEUIZGIDXJQUJAIUKSZXKXLULAKLUMUNZXNIUOZXNUPSXMAXNXNI - UQXONXNXNIURTZAKLAJXNSZKUSSZMJKLUTTZAXQLUSSZMJKLVATZVBXNXNUPIVCVDGIXJ - XLUKXMGXLVEXFIULZBWQXIXEYBXAXHXCXDYBXGXBHWPXFIVFVGVHVIVJTVKWSXAXCXDFW - SXAUDZUAUBZHRZFSZXCFSUAWQXBYDXBULYEXCFYDXBHVLVMAYFUAWQVNZWRXAAWPHRZFS - ZBWQVNYGAYIBWQPVQYIYFBUAWQWPYDULYHYEFWPYDHVLVMVOVPZVRYCXBXNSXBWQSYCXN - XNWPIAXOWRXAXPVRYCWPXNSZXAWSXAVSYCWRXTYKXAVTAWRXAWAAXTWRXAYAVRWPKLWBV - DWCWDXBKLWGTWEWSXAWHZUDZYFXDFSUAWQKYDKULYEXDFYDKHVLVMAYGWRYLYJVRYMXRK - WQSAXRWRYLXSVRKWFTWEWRWPUSSXTXAWIAKWPWJYAWPLWKWLWMZWNYNWO $. + ralrimiva ad2antrr simpr simplr elfz5 mpbird ffvelcdmd elfzuz rspcdva + wb wn uzid wdc eluzelz zdcle syl2anr ifcldadc fvmpt2d eqeltrd ) ABUBZ + KUCRZSZUDZWPGIDUEZRWPLUFUGZWPIRZHRZKHRZUHZFABWQXEWTFAWTGIBWQXAWPGUBZR + ZHRZXDUHZUIZUEZBWQXEUIZGIDXJQUJAIUKSZXKXLULAKLUMUNZXNIUOZXNUPSXMAXNXN + IUQXONXNXNIURTZAKLAJXNSZKUSSZMJKLUTTZAXQLUSSZMJKLVATZVBXNXNUPIVCVDGIX + JXLUKXMGXLVEXFIULZBWQXIXEYBXAXHXCXDYBXGXBHWPXFIVFVGVHVIVJTVKWSXAXCXDF + WSXAUDZUAUBZHRZFSZXCFSUAWQXBYDXBULYEXCFYDXBHVLVMAYFUAWQVNZWRXAAWPHRZF + SZBWQVNYGAYIBWQPVQYIYFBUAWQWPYDULYHYEFWPYDHVLVMVOVPZVRYCXBXNSXBWQSYCX + NXNWPIAXOWRXAXPVRYCWPXNSZXAWSXAVSYCWRXTYKXAWFAWRXAVTAXTWRXAYAVRWPKLWA + VDWBWCXBKLWDTWEWSXAWGZUDZYFXDFSUAWQKYDKULYEXDFYDKHVLVMAYGWRYLYJVRYMXR + KWQSAXRWRYLXSVRKWHTWEWRWPUSSXTXAWIAKWPWJYAWPLWKWLWMZWNYNWO $. $} ${ @@ -108439,17 +108465,17 @@ seq K ( .+ , G ) ) $= cfz c1 cmin cfn cz elfzel1 elfzel2 fzfigd mptexg eqeltrid nfcvd fveq1 syl fveq2d ifeq1d mpteq2dv csbiegf eqtrid fveq2 eleq1d wral ralrimiva cbvralv sylib ad2antrr wf iseqf1olemqf simpr wb simplr syl2anc mpbird - elfz5 ffvelrnd elfzuz rspcdva uzid wdc eluzelz zdcle syl2anr ifcldadc - wn fvmpt2d eqeltrd ) ABUBZKUCRZSZUDZWSGEDUEZRWSLUFUGZWSERZHRZKHRZUHZF - ABWTXHXCFAXCGEBWTXDWSGUBZRZHRZXGUHZUIZUEZBWTXHUIZGEDXMQUJAEUKSZXNXOTA - ECKLUNULZCUBZJJIUMRUNULSXRJTJXRUOUPULIRUHXRIRUHZUIZUKOAXQUQSXTUKSAKLA - JXQSZKURSZMJKLUSVFZAYALURSZMJKLUTVFZVACXQXSUQVBVFVCGEXMXOUKXPGXOVDXIE - TZBWTXLXHYFXDXKXFXGYFXJXEHWSXIEVEVGVHVIVJVFVKXBXDXFXGFXBXDUDZUAUBZHRZ - FSZXFFSUAWTXEYHXETYIXFFYHXEHVLVMAYJUAWTVNZXAXDAWSHRZFSZBWTVNYKAYMBWTP - VOYMYJBUAWTWSYHTYLYIFWSYHHVLVMVPVQZVRYGXEXQSXEWTSYGXQXQWSEAXQXQEVSXAX - DACEIJKLMNOVTVRYGWSXQSZXDXBXDWAYGXAYDYOXDWBAXAXDWCAYDXAXDYEVRWSKLWFWD - WEWGXEKLWHVFWIXBXDWPZUDZYJXGFSUAWTKYHKTYIXGFYHKHVLVMAYKXAYPYNVRYQYBKW - TSAYBXAYPYCVRKWJVFWIXAWSURSYDXDWKAKWSWLYEWSLWMWNWOZWQYRWR $. + elfz5 ffvelcdmd elfzuz rspcdva wn uzid eluzelz zdcle syl2anr ifcldadc + wdc fvmpt2d eqeltrd ) ABUBZKUCRZSZUDZWSGEDUEZRWSLUFUGZWSERZHRZKHRZUHZ + FABWTXHXCFAXCGEBWTXDWSGUBZRZHRZXGUHZUIZUEZBWTXHUIZGEDXMQUJAEUKSZXNXOT + AECKLUNULZCUBZJJIUMRUNULSXRJTJXRUOUPULIRUHXRIRUHZUIZUKOAXQUQSXTUKSAKL + AJXQSZKURSZMJKLUSVFZAYALURSZMJKLUTVFZVACXQXSUQVBVFVCGEXMXOUKXPGXOVDXI + ETZBWTXLXHYFXDXKXFXGYFXJXEHWSXIEVEVGVHVIVJVFVKXBXDXFXGFXBXDUDZUAUBZHR + ZFSZXFFSUAWTXEYHXETYIXFFYHXEHVLVMAYJUAWTVNZXAXDAWSHRZFSZBWTVNYKAYMBWT + PVOYMYJBUAWTWSYHTYLYIFWSYHHVLVMVPVQZVRYGXEXQSXEWTSYGXQXQWSEAXQXQEVSXA + XDACEIJKLMNOVTVRYGWSXQSZXDXBXDWAYGXAYDYOXDWBAXAXDWCAYDXAXDYEVRWSKLWFW + DWEWGXEKLWHVFWIXBXDWJZUDZYJXGFSUAWTKYHKTYIXGFYHKHVLVMAYKXAYPYNVRYQYBK + WTSAYBXAYPYCVRKWKVFWIXAWSURSYDXDWPAKWSWLYEWSLWMWNWOZWQYRWR $. $} $} $} @@ -108469,14 +108495,14 @@ seq K ( .+ , G ) ) $= ( cfv wcel wceq syl csb cle wbr cif cv cuz cmpt csbeq2i cvv cfz co wf cfn wf1o f1of cz elfzel1 elfzel2 fzfigd fex syl2anc nfcvd fveq1 fveq2d ifeq1d mpteq2dv csbiegf eqtrid simpr breq1d ifbieq1d elfzuz elfzle2 fveq2 eleq1d - wa iftrued ralrimiva ffvelrnd rspcdva eqeltrd fvmptd eqtrd ) ACGFDUAZQCKU - BUCZCFQZHQZJHQZUDZWGABCBUEZKUBUCZWJFQZHQZWHUDZWIJUFQZWDEAWDGFBWOWKWJGUEZQ - ZHQZWHUDZUGZUAZBWOWNUGZGFDWTPUHAFUIRZXAXBSAJKUJUKZXDFULZXDUMRXCAXDXDFUNXE - MXDXDFUOTZAJKAIXDRZJUPRLIJKUQTAXGKUPRLIJKURTUSXDXDUMFUTVAGFWTXBUIXCGXBVBW - PFSZBWOWSWNXHWKWRWMWHXHWQWLHWJWPFVCVDVEVFVGTVHAWJCSZVPZWKWEWMWGWHXJWJCKUB - AXIVIZVJXJWLWFHXJWJCFXKVDVDVKACXDRZCWORNCJKVLTAWIWGEAWEWGWHAXLWENCJKVMTVQ - ZAWJHQZERZWGERBWOWFWJWFSXNWGEWJWFHVNVOAXOBWOOVRAWFXDRWFWORAXDXDCFXFNVSWFJ - KVLTVTWAWBXMWC $. + wa iftrued ralrimiva ffvelcdmd rspcdva eqeltrd fvmptd eqtrd ) ACGFDUAZQCK + UBUCZCFQZHQZJHQZUDZWGABCBUEZKUBUCZWJFQZHQZWHUDZWIJUFQZWDEAWDGFBWOWKWJGUEZ + QZHQZWHUDZUGZUAZBWOWNUGZGFDWTPUHAFUIRZXAXBSAJKUJUKZXDFULZXDUMRXCAXDXDFUNX + EMXDXDFUOTZAJKAIXDRZJUPRLIJKUQTAXGKUPRLIJKURTUSXDXDUMFUTVAGFWTXBUIXCGXBVB + WPFSZBWOWSWNXHWKWRWMWHXHWQWLHWJWPFVCVDVEVFVGTVHAWJCSZVPZWKWEWMWGWHXJWJCKU + BAXIVIZVJXJWLWFHXJWJCFXKVDVDVKACXDRZCWORNCJKVLTAWIWGEAWEWGWHAXLWENCJKVMTV + QZAWJHQZERZWGERBWOWFWJWFSXNWGEWJWFHVNVOAXOBWOOVRAWFXDRWFWORAXDXDCFXFNVSWF + JKVLTVTWAWBXMWC $. $} ${ @@ -108519,54 +108545,54 @@ seq K ( .+ , G ) ) $= -> ( seq K ( .+ , [_ J / f ]_ P ) ` ( `' J ` K ) ) = ( seq K ( .+ , [_ Q / f ]_ P ) ` ( `' J ` K ) ) ) $= ( vv ccnv cfv c1 cmin co csb cseq caddc cz wcel cle wbr cuz elfzelz - cfz syl wf1o wf f1ocnv f1of ffvelrnd peano2zm iseqf1olemklt zltlem1 - clt wb syl2anc mpbid eluz2 syl3anbrc 1zzd cv wa wceq cif adantr w3a - elfzel1 elfzel2 adantl peano2zd 3jca cr elfzle1 letrd lep1d elfzle2 - zred 1red leaddsub syl3anc mpbird jca elfz2 sylanbrc iseqf1olemqval - iftrued eqtrd zleltp1 gtned neneqd iffalsed cc pncan1 fveq2d 3eqtrd - zcnd iseqf1olemqf1o adantlr iseqf1olemfvp zsubcld lem1d simpr uztrn - elfzuz iseqf1olemjpcl syldan iseqf1olemqpcl seq3shft2 iseqf1olemkle - npcan1 eluzfz1 eqidd 3eqtr4d oveq12d zltp1le seq3m1 seq3-1p eqeltrd - 3eqtr4rd f1ocnvfv2 fveq2 eleq1d ralrimiva rspcdva eqid caovcomd + cfz syl wf1o wf f1ocnv f1of ffvelcdmd peano2zm clt iseqf1olemklt wb + zltlem1 syl2anc mpbid eluz2 syl3anbrc 1zzd cv wa cif adantr elfzel1 + wceq w3a elfzel2 adantl peano2zd 3jca zred cr elfzle1 letrd elfzle2 + lep1d 1red leaddsub syl3anc mpbird jca elfz2 iseqf1olemqval iftrued + sylanbrc eqtrd zleltp1 neneqd iffalsed cc zcnd pncan1 fveq2d 3eqtrd + gtned iseqf1olemqf1o adantlr iseqf1olemfvp lem1d simpr elfzuz uztrn + zsubcld 3eqtr4rd iseqf1olemjpcl syldan iseqf1olemqpcl iseqf1olemkle + seq3shft2 f1ocnvfv2 eluzfz1 3eqtr4d oveq12d zltp1le seq3-1p eqeltrd + npcan1 eqidd seq3m1 fveq2 eleq1d ralrimiva rspcdva eqid caovcomd seqf ) ANMUJZUKZULUMUNZGJMFUOZNUPZUKZUUIUUKUKZGUNUUIGJHFUOZNULUQUNZ UPZUKZNUUOUKZGUNZUUIUULUKUUIGUUONUPUKZAUUMUURUUNUUSGAUUMUUJULUQUNZU UQUKUURABCGIUIUUKUUOULNUUJANURUSZUUJURUSZNUUJUTVAZUUJNVBUKZUSANOPVD UNZUSZUVCUCNOPVCZVEZAUUIURUSZUVDAUUIUVGUSZUVKAUVGUVGNUUHAUVGUVGUUHV FZUVGUVGUUHVGAUVGUVGMVFZUVMUDUVGUVGMVHVEUVGUVGUUHVIVEUCVJZUUIOPVCVE - ZUUIVKVEANUUIVNVAZUVEABMNOPTUCUDUEUFVLZAUVCUVKUVQUVEVOUVJUVPNUUIVMV + ZUUIVKVEANUUIVLVAZUVEABMNOPTUCUDUEUFVMZAUVCUVKUVQUVEVNUVJUVPNUUIVOV PVQNUUJVRVSAVTZAUIWAZNUUJVDUNUSZWBZUVTULUQUNZHUKZLUKUVTMUKZLUKUWCUU - OUKUVTUUKUKUWBUWDUWELUWBUWDUWCNWCZNUWCULUMUNZMUKZWDZUWHUWEUWBUWDUWC - NUUIVDUNZUSZUWIUWCMUKZWDUWIUWBEUWCHMNOPAUVHUWAUCWEZAUVNUWAUDWEZUWBO - URUSZPURUSZUWCURUSZWFOUWCUTVAZUWCPUTVAZWBUWCUVGUSUWBUWOUWPUWQAUWOUW - AAUVHUWOUCNOPWGVEWEZAUWPUWAAUVHUWPUCNOPWHVEWEZUWBUVTUWAUVTURUSZAUVT - NUUJVCWIZWJZWKUWBUWRUWSUWBOUVTUWCUWBOUWTWQZUWBUVTUXCWQZUWBUWCUXDWQZ - UWBONUVTUXEANWLUSUWAANUVJWQWEZUXFAONUTVAZUWAAUVHUXIUCNOPWMVEWEUWANU - VTUTVAZAUVTNUUJWMWIZWNZUWBUVTUXFWOZWNUWBUWCUUIPUXGAUUIWLUSZUWAAUUIU - VPWQWEZUWBPUXAWQZUWBUWCUUIUTVAZUVTUUJUTVAZUWAUXRAUVTNUUJWPWIZUWBUVT - WLUSULWLUSUXNUXQUXRVOUXFUWBWRUXOUVTULUUIWSWTXAZAUUIPUTVAZUWAAUVLUYA - UVOUUIOPWPVEWEZWNXBUWCOPXCXDZUGXEUWBUWKUWIUWLUWBUVCUVKUWQWFNUWCUTVA - ZUXQWBUWKUWBUVCUVKUWQUWBUVHUVCUWMUVIVEZAUVKUWAUVPWEUXDWKUWBUYDUXQUW - BNUVTUWCUXHUXFUXGUXKUXMWNUXTXBUWCNUUIXCXDXFXGUWBUWFNUWHUWBUWCNUWBNU - WCUXHUWBUXJNUWCVNVAZUXKUWBUVCUXBUXJUYFVOUYEUXCNUVTXHVPVQXIXJXKUWBUW - GUVTMUWBUVTXLUSUWGUVTWCUWBUVTUXCXPUVTXMVEXNXOXNUWBBUWCFIHJLNOPUWMAU - VGUVGHVFUWAAEHMNOPUCUDUGXQZWEUYCABWAZOVBUKZUSZUYHLUKZIUSZUWAUBXRZUH - XSUWBBUVTFIMJLUVTOPUWBUWOUWPUXBWFOUVTUTVAZUVTPUTVAZWBUVTUVGUSUWBUWO - UWPUXBUWTUXAUXCWKUWBUYNUYOUXLUWBUVTUUJPUXFAUUJWLUSUWAAUUJAUUIULUVPU - VSXTWQWEZUXPUXSUWBUUJUUIPUYPUXOUXPUWBUUIUXOYAUYBWNWNXBUVTOPXCXDZUWN - UYQUYMUHXSYSAUYHUVFUSZUYJUYHUUKUKIUSAUYRWBUYRNUYIUSZUYJAUYRYBAUYSUY - RAUVHUYSUCNOPYDVEZWENUYHOYCVPZABEFHIJLMNOPUCUDUGUBUHYEYFZAUYHUUPVBU - KZUSZUYRUYHUUOUKIUSZAVUDWBZVUDUUPUVFUSZUYRAVUDYBVUFUVCUUPURUSZNUUPU - TVAVUGAUVCVUDUVJWEZVUFNVUIWJVUFNVUFNVUIWQWONUUPVRVSUUPUYHNYCVPAUYRU - YJVUEVUAABEFHIJLMNOPUCUDUGUBUHYGYFZYFZQYHAUVBUUIUUQAUUIXLUSUVBUUIWC - AUUIUVPXPUUIYJVEXNXGAUUIMUKZLUKNLUKZUUNUUSAVULNLAUVNUVHVULNWCUDUCUV - GUVGNMYTVPXNABUUIFIMJLNOPUCUDUVOUBUHXSAUUSNHUKZLUKZVUMABNFIHJLNOPUC - UYGUCUBUHXSZAVUNNLAVUNNUWJUSZNNWCZNNULUMUNMUKZWDZNMUKZWDVUTNAENHMNO - PUCUDUCUGXEAVUQVUTVVAAUUIUVFUSZVUQAUVCUVKNUUIUTVAVVBUVJUVPABMNOPTUC - UDUEYINUUIVRVSNUUIYKVEXFAVURNVUSANYLXFXOZXNXGYMYNABCGIUUKNUUIUVJAVU - HUVKUUPUUIUTVAZUUIVUCUSANUVJWJZUVPAUVQVVDUVRAUVCUVKUVQVVDVOUVJUVPNU - UIYOVPVQUUPUUIVRVSZVUBQYPAUVAUUSUURGUNUUTABCDGIUUONUUIQSVVFUVJVUJYQ - ABCUUSUURIGRAUUSVUOIVUPAUYLVUOIUSBUYIVUNUYHVUNWCUYKVUOIUYHVUNLUUAUU - BAUYLBUYIUBUUCAVUNNUYIVVCUYTYRUUDYRAVUCIUUIUUQABCGIUUOUUPVUCVUCUUEV + OUKUVTUUKUKUWBUWDUWELUWBUWDUWCNWFZNUWCULUMUNZMUKZWCZUWHUWEUWBUWDUWC + NUUIVDUNZUSZUWIUWCMUKZWCUWIUWBEUWCHMNOPAUVHUWAUCWDZAUVNUWAUDWDZUWBO + URUSZPURUSZUWCURUSZWGOUWCUTVAZUWCPUTVAZWBUWCUVGUSUWBUWOUWPUWQAUWOUW + AAUVHUWOUCNOPWEVEWDZAUWPUWAAUVHUWPUCNOPWHVEWDZUWBUVTUWAUVTURUSZAUVT + NUUJVCWIZWJZWKUWBUWRUWSUWBOUVTUWCUWBOUWTWLZUWBUVTUXCWLZUWBUWCUXDWLZ + UWBONUVTUXEANWMUSUWAANUVJWLWDZUXFAONUTVAZUWAAUVHUXIUCNOPWNVEWDUWANU + VTUTVAZAUVTNUUJWNWIZWOZUWBUVTUXFWQZWOUWBUWCUUIPUXGAUUIWMUSZUWAAUUIU + VPWLWDZUWBPUXAWLZUWBUWCUUIUTVAZUVTUUJUTVAZUWAUXRAUVTNUUJWPWIZUWBUVT + WMUSULWMUSUXNUXQUXRVNUXFUWBWRUXOUVTULUUIWSWTXAZAUUIPUTVAZUWAAUVLUYA + UVOUUIOPWPVEWDZWOXBUWCOPXCXFZUGXDUWBUWKUWIUWLUWBUVCUVKUWQWGNUWCUTVA + ZUXQWBUWKUWBUVCUVKUWQUWBUVHUVCUWMUVIVEZAUVKUWAUVPWDUXDWKUWBUYDUXQUW + BNUVTUWCUXHUXFUXGUXKUXMWOUXTXBUWCNUUIXCXFXEXGUWBUWFNUWHUWBUWCNUWBNU + WCUXHUWBUXJNUWCVLVAZUXKUWBUVCUXBUXJUYFVNUYEUXCNUVTXHVPVQXPXIXJUWBUW + GUVTMUWBUVTXKUSUWGUVTWFUWBUVTUXCXLUVTXMVEXNXOXNUWBBUWCFIHJLNOPUWMAU + VGUVGHVFUWAAEHMNOPUCUDUGXQZWDUYCABWAZOVBUKZUSZUYHLUKZIUSZUWAUBXRZUH + XSUWBBUVTFIMJLUVTOPUWBUWOUWPUXBWGOUVTUTVAZUVTPUTVAZWBUVTUVGUSUWBUWO + UWPUXBUWTUXAUXCWKUWBUYNUYOUXLUWBUVTUUJPUXFAUUJWMUSUWAAUUJAUUIULUVPU + VSYDWLWDZUXPUXSUWBUUJUUIPUYPUXOUXPUWBUUIUXOXTUYBWOWOXBUVTOPXCXFZUWN + UYQUYMUHXSYEAUYHUVFUSZUYJUYHUUKUKIUSAUYRWBUYRNUYIUSZUYJAUYRYAAUYSUY + RAUVHUYSUCNOPYBVEZWDNUYHOYCVPZABEFHIJLMNOPUCUDUGUBUHYFYGZAUYHUUPVBU + KZUSZUYRUYHUUOUKIUSZAVUDWBZVUDUUPUVFUSZUYRAVUDYAVUFUVCUUPURUSZNUUPU + TVAVUGAUVCVUDUVJWDZVUFNVUIWJVUFNVUFNVUIWLWQNUUPVRVSUUPUYHNYCVPAUYRU + YJVUEVUAABEFHIJLMNOPUCUDUGUBUHYHYGZYGZQYJAUVBUUIUUQAUUIXKUSUVBUUIWF + AUUIUVPXLUUIYRVEXNXGAUUIMUKZLUKNLUKZUUNUUSAVULNLAUVNUVHVULNWFUDUCUV + GUVGNMYKVPXNABUUIFIMJLNOPUCUDUVOUBUHXSAUUSNHUKZLUKZVUMABNFIHJLNOPUC + UYGUCUBUHXSZAVUNNLAVUNNUWJUSZNNWFZNNULUMUNMUKZWCZNMUKZWCVUTNAENHMNO + PUCUDUCUGXDAVUQVUTVVAAUUIUVFUSZVUQAUVCUVKNUUIUTVAVVBUVJUVPABMNOPTUC + UDUEYINUUIVRVSNUUIYLVEXEAVURNVUSANYSXEXOZXNXGYMYNABCGIUUKNUUIUVJAVU + HUVKUUPUUIUTVAZUUIVUCUSANUVJWJZUVPAUVQVVDUVRAUVCUVKUVQVVDVNUVJUVPNU + UIYOVPVQUUPUUIVRVSZVUBQYTAUVAUUSUURGUNUUTABCDGIUUONUUIQSVVFUVJVUJYP + ABCUUSUURIGRAUUSVUOIVUPAUYLVUOIUSBUYIVUNUYHVUNWFUYKVUOIUYHVUNLUUAUU + BAUYLBUYIUBUUCAVUNNUYIVVCUYTYQUUDYQAVUCIUUIUUQABCGIUUOUUPVUCVUCUUEV VEVUKQUUGVVFVJUUFXGYM $. $} @@ -108584,46 +108610,46 @@ seq K ( .+ , G ) ) $= -> ( seq K ( .+ , [_ J / f ]_ P ) ` N ) = ( seq K ( .+ , [_ Q / f ]_ P ) ` N ) ) $= ( vv ccnv cfv clt wbr csb cseq wceq wa c1 caddc co seq3f1olemqsumkj - adantr cz wcel cle cuz cfz wf1o wf f1ocnv syl f1of ffvelrnd elfzelz - peano2zd elfzel2 simpr wb zltp1le syl2anc mpbid eluz2 syl3anbrc cif - cv cmin ad2antrr elfzel1 adantl 3jca cr zred peano2re iseqf1olemkle - elfzle1 letrd lep1d elfzle2 elfz2 sylanbrc iseqf1olemqval ad3antrrr - w3a jca ltp1d ltletrd pm2.65da iffalsed eqtrd fveq2d iseqf1olemqf1o - lensymd wral r19.21bi iseqf1olemfvp 3eqtr4rd eluzelz eluzle adantlr - ralrimiva iseqf1olemjpcl syldan iseqf1olemqpcl oveq12d elfzuz uztrn - seq3fveq seq3split 3eqtr4d 3eqtr3d wo zleloe mpjaodan ) ANMUJZUKZPU - LUMZPGJMFUNZNUOZUKZPGJHFUNZNUOZUKZUPYOPUPZAYPUQZYOYRUKZPGYQYOURUSUT - ZUOUKZGUTYOUUAUKZPGYTUUFUOUKZGUTYSUUBUUDUUEUUHUUGUUIGAUUEUUHUPZYPAB - CDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHVAZVBUUDBCGIUIYQYTUUFPUUDUUFVCVDP - VCVDZUUFPVEUMZPUUFVFUKZVDUUDYOAYOVCVDZYPAYOOPVGUTZVDZUUOAUUPUUPNYNA - UUPUUPYNVHZUUPUUPYNVIAUUPUUPMVHZUURUDUUPUUPMVJVKUUPUUPYNVLVKUCVMZYO - OPVNVKZVBZVOAUULYPANUUPVDZUULUCNOPVPZVKZVBZUUDYPUUMAYPVQUUDUUOUULYP - UUMVRUVBUVFYOPVSVTWAUUFPWBWCZUUDUIWEZUUFPVGUTVDZUQZUVHHUKZLUKUVHMUK - ZLUKUVHYTUKUVHYQUKUVJUVKUVLLUVJUVKUVHNYOVGUTVDZUVHNUPNUVHURWFUTMUKW - DZUVLWDUVLUVJEUVHHMNOPAUVCYPUVIUCWGZAUUSYPUVIUDWGZUVJOVCVDZUULUVHVC - VDZXCOUVHVEUMZUVHPVEUMZUQUVHUUPVDUVJUVQUULUVRAUVQYPUVIAUVCUVQUCNOPW - HVKZWGUVJUVCUULUVOUVDVKUVIUVRUUDUVHUUFPVNWIZWJUVJUVSUVTUVJOUUFUVHAO - WKVDZYPUVIAOUWAWLZWGAUUFWKVDZYPUVIAYOWKVDZUWEAYOUVAWLZYOWMZVKZWGUVJ - UVHUWBWLZAOUUFVEUMZYPUVIAOYOUUFUWDUWGUWIAONYOUWDANAUVCNVCVDZUCNOPVN - VKZWLUWGAUVCONVEUMUCNOPWOVKABMNOPTUCUDUEWNZWPAYOUWGWQWPZWGUVIUUFUVH - VEUMZUUDUVHUUFPWOWIZWPUVIUVTUUDUVHUUFPWRWIXDUVHOPWSWTZUGXAUVJUVMUVN - UVLUVJUVMYOUVHULUMUVJUVMUQZYOUUFUVHAUWFYPUVIUVMUWGXBZUWSUWFUWEUWTUW - HVKUVJUVHWKVDUVMUWJVBZUWSYOUWTXEUVJUWPUVMUWQVBXFUWSUVHYOUXAUWTUVMUV - HYOVEUMUVJUVHNYOWRWIXLXGXHXIXJUVJBUVHFIHJLNOPUVOUVJEHMNOPUVOUVPUGXK - UWRUVJBWEZLUKIVDZBOVFUKZAUXCBUXDXMYPUVIAUXCBUXDUBXTWGXNZUHXOUVJBUVH - FIMJLNOPUVOUVPUWRUXEUHXOXPUUDUXBUUNVDZUXBUXDVDZUXBYQUKIVDZUUDUXFUQZ - UVQUXBVCVDZOUXBVEUMUXGAUVQYPUXFUWAWGUXFUXJUUDUUFUXBXQWIZUXIOUUFUXBA - UWCYPUXFUWDWGAUWEYPUXFUWIWGUXIUXBUXKWLAUWKYPUXFUWOWGUXFUUFUXBVEUMUU - 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mpbird wf1o f1of ffvelrnd eqeltrd eleq1w eqeq1 fvoveq1 ifbieq2d fveq2 - weq ifbieq12d fvmptg eqtrd fveq2d cmpt csbeq2i fzfigd mptexg eqeltrid - cvv nfcvd fveq1 ifeq1d mpteq2dv csbiegf eqtrid 2fveq3 ifbieq1d elfzuz - cfn iftrued eleq1d wral rspcdva fvmptd adantlr seqeq1d fveq1d 3eqtr4d - breq1 seq3split seqeq1 ralrimiva iseqf1olemqf 3eqtr4rd iseqf1olemjpcl - cbvralv sylib fex iseqf1olemqpcl seq3fveq seq3f1olemqsumk zcnd npcan1 - cc oveq12d w3a elfzuz3 eleqtrrd eqeq12d wo zleloe mpjaodan ) AONUKULZ - PGJMFUMZOUNZUOZPGJHFUMZOUNZUOZUPZONUPZAUVBUQZNURUSUTZUVDUOZPGUVCUVLUR - VAUTZUNZUOZGUTUVLUVGUOZPGUVFUVNUNZUOZGUTUVEUVHUVKUVMUVQUVPUVSGUVKBCGI - UIUVCUVFOUVLUVKOVBVCZUVLVBVCZOUVLVDULZUVLOVEUOZVCAUVTUVBANOPVFUTZVCZU - VTUCNOPVGVPZVHZUVKNVBVCZUWAAUWHUVBAUWEUWHUCNOPVIVPZVHZNVJVPUVKUVBUWBA - UVBVKUVKUVTUWHUVBUWBWFUWGUWJONVLVMVNOUVLVOVQZUVKUIWGZOUVLVFUTVCZUQZUW - LHUOZLUOZUWLMUOZLUOZUWLUVFUOZUWLUVCUOZUWNUWOUWQLUWNUWOUWLNNMVRUOZVFUT - ZVCZUWLNUPZNUWLURUSUTMUOZVSZUWQVSZUWQUWNUWLUWDVCZUXGUWDVCUWOUXGUPUWNU - 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neqne wo f1ocnv ffvelrnd zdceq exmiddc mpjaodan ) ALLKUHZUIZUJZNOUKUL - ZYSHUMZUNZBUMZYTUIZUUBUJZBNLUKULZUOZOFENUPZUIZOFMNUPUIZUJZUQZHURZYRUS - ZAYRUTZKVAVBZYSYSKUNZUUBKUIZUUBUJZBUUEUOZOFHKEVDZNUPZUIZUUIUJZUQZUULA - UUOYRAYSYSKVCZYSVEVBZUUOAUUPUVEUCYSYSKVFVGANOALYSVBZNVHVBZUBLNOVIZVGA - UVGOVHVBZUBLNOVJZVGVKYSYSVEKVLVMVNUUNUUPUUSUVCAUUPYRUCVNUUNUUSUURBNLV - QULZLVOZVPZUOZUUNUURBUVLUOZUURBUVMUOZUVOAUVPYRUDVNUUNUVQLKUIZLUJZUUNU - VSYQLUJZYRUVTAYRUVTLYQVRVSVTAUVSUVTWAZYRAUUPUVGUVGUWAUCUBUBYSYSLLKWBW - CVNWDUUNLVHVBZUVQUVSWAAUWBYRAUVGUWBUBLNOWEVGZVNUURUVSBLVHUUBLUJZUUQUV - RUUBLUUBLKWFUWDWGWHWIVGWDUURBUVLUVMWJVMAUUSUVOWAYRAUURBUUEUVNAUVGLNWK - UIZVBUUEUVNUJUBLNOWLNLWMWNWOVNWDAUVCYRUEVNWPUUKUVDHKVAHKWRUUPUUSUVCHU - UPHWSUUSHWSHUVBUUIHOUVAHFUUTNHNWRZHFWRZHKEWQWTHOWRZXAXBXCYTKUJZUUAUUP - UUFUUSUUJUVCYSYSYTKXDUWIUUDUURBUUEUWIUUCUUQUUBUUBYTKXIXEXFUWIUUHUVBUU - IUWIOUUGUVAUWIEUUTFNHKEXGXHXJXEXKXLXMAUUMUTZYSYSUGYSUGUMZLYQUKULVBUWK - LUJLUWKXNXOULKUIXPUWKKUIXPZXQZUNZUUBUWMUIZUUBUJZBUUEUOZOFHUWMEVDZNUPZ - 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ad2antrr ad2antlr simpr2 sylibr elfzoelz fzval3 simpr3 exlimdv syl5bi - wb mpbid ex expcom a2d fzind2 mpcom ) MLMUFUGZUHZAUUCUUCHURZUIZBURZUU - EUJZUUGUKZBUUCULZMFELUMZUJZMFKLUMZUJZUKZUNZHUOZAMLUPUJZUHZUUDQLMUQUSA - UUFUUIBLUBURZUFUGZULZUUOUNZHUOZUTAUUFUUIBLLUFUGZULZUUOUNZHUOZUTZAUUFU - UIBLUCURZUFUGZULZUUOUNZHUOZUTAUUFUUIBLUVJVAVBUGZUFUGZULZUUOUNZHUOZUTA - UUQUTUBUCMLMUUTLUKZUVDUVHAUVTUVCUVGHUVTUVBUVFUUFUUOUVTUUIBUVAUVEUUTLL - UFVCVDVEVFVGUBUCVHZUVDUVNAUWAUVCUVMHUWAUVBUVLUUFUUOUWAUUIBUVAUVKUUTUV - JLUFVCVDVEVFVGUUTUVOUKZUVDUVSAUWBUVCUVRHUWBUVBUVQUUFUUOUWBUUIBUVAUVPU - UTUVOLUFVCVDVEVFVGUUTMUKZUVDUUQAUWCUVCUUPHUWCUVBUUJUUFUUOUWCUUIBUVAUU - CUUTMLUFVCVDVEVFVGUVIUUSABCDEFGHIJILKLMNOPQRSAUUSLUUCUHQLMVIUSRUUGIUJ - ZUUGUKZBLLVKUGZULZAUWGUWEBVJULUWEBVLUWEBUWFVJLVMVNVOVPAMFHIEVQZLUMUUM - AUWHKFLAHIEKVRAUUCUUCIVSZUUCVTUHIVRUHAUUCUUCIUIZUWIRUUCUUCIWAUSALMAUU - SLWBUHQLMWCUSAUUSMWBUHQLMWEUSWDUUCUUCVTIWFWGUUEIUKZEKUKAUWKBUURUUGMWH - WIZUUHJUJZLJUJZWPZWJBUURUWLUWDJUJZUWNWPZWJEKUWKBUURUWOUWQUWKUWLUWMUWP - 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mpbird ffvelrnd wn eluzel2 wdc zdcle ifcldadc adantlr seq3fveq eqtr3d - uzid sylan2br exlimddv ) AJKUEUFZXKUAUGZUHZBUGZXLSZXNUIZBXKUJZKEUBJUK - SZUBUGZKULUMZXSXLSHSZJHSZUPZUNZJUOSZKEIJUOSZUIZUQZYFKEHJUOSZUIZUAABCD - YDEFUAGHIJKLMNOPQRUBBXRYCXNKULUMZXOHSZYBUPZUBBURZXTYKYAYLYBXSXNKULUSX - SXNHXLUTVAZVBVCYHAXMUCUGZXLSZYPUIZUCXKUJZYGUQZYJYSXQXMYGYRXPUCBXKUCBU - RZYQXOYPXNYPXNXLVFUUAVGVDVEVHAYTVIZYEYFYIAXMYSYGVJUUBBCEFUDYDHJKAKXRT - ZYTOVKUUBUDUGZXKTZVIZUUDYDSZUUDKULUMZUUDXLSZHSZYBUPZUUJUUDHSZUUFUUDXR - TZUUKFTUUGUUKUIUUEUUMUUBUUDJKVLVMZUUFUUKUUJFUUFUUHUUJYBUUEUUHUUBUUDJK - VNVMVOZUUFUUJUULFUUFUUIUUDHUUFYRUUIUUDUIUCXKUUDUCUDURZYQUUIYPUUDYPUUD - XLVFUUPVGVDXMYSYGAUUEVQUUBUUEVPVRVSZUUFXNHSZFTZUULFTBXRUUDBUDURUURUUL - FXNUUDHVFVTAUUSBXRUJZYTUUEAUUSBXRQWAZWBUUNVRWCWCUBUUDYCUUKXRFYDUBUDUR - XTUUHYAUUJYBXSUUDKULUSXSUUDHXLUTVAYDWDZWFWEUUOUUQWGUUBXNXRTZVIZXNYDSZ - YMFUVDUVCYMFTUVEYMUIUUBUVCVPZUVDYKYLYBFUVDYKVIZXSHSZFTZYLFTUBXRXOXSXO - UIUVHYLFXSXOHVFVTUVGUUTUVIUBXRUJZAUUTYTUVCYKUVAWHUVIUUSUBBXRYNUVHUURF - XSXNHVFVTVEZWIUVGXOXKTXOXRTUVGXKXKXNXLUVGXMXKXKXLWJUUBXMUVCYKAXMYSYGW - KWBXKXKXLWLWMUVGXNXKTZYKUVDYKVPUVGUVCKWNTZUVLYKWOUVDUVCYKUVFVKAUVMYTU - VCYKAUUCUVMOJKWPWMZWHXNJKWQWEWRWSXOJKVLWMVRUVDYKWTZVIZUVIYBFTUBXRJXSJ - UIUVHYBFXSJHVFVTAUVJYTUVCUVOAUUTUVJUVAUVKWIWHUVPJWNTZJXRTAUVQYTUVCUVO - AUUCUVQOJKXAWMWHJXHWMVRUVDXNWNTZUVMYKXBUVCUVRUUBJXNWPVMAUVMYTUVCUVNWB - XNKXCWEXDZUBXNYCYMXRFYDYOUVBWFWEUVSWCAUVCUUSYTQXEAXNFTCUGZFTVIXNUVTEU - FFTYTLXEXFXGXIXJ $. + mpbird ffvelcdmd wn eluzel2 uzid wdc ifcldadc adantlr seq3fveq eqtr3d + zdcle sylan2br exlimddv ) AJKUEUFZXKUAUGZUHZBUGZXLSZXNUIZBXKUJZKEUBJU + KSZUBUGZKULUMZXSXLSHSZJHSZUPZUNZJUOSZKEIJUOSZUIZUQZYFKEHJUOSZUIZUAABC + DYDEFUAGHIJKLMNOPQRUBBXRYCXNKULUMZXOHSZYBUPZUBBURZXTYKYAYLYBXSXNKULUS + XSXNHXLUTVAZVBVCYHAXMUCUGZXLSZYPUIZUCXKUJZYGUQZYJYSXQXMYGYRXPUCBXKUCB + URZYQXOYPXNYPXNXLVFUUAVGVDVEVHAYTVIZYEYFYIAXMYSYGVJUUBBCEFUDYDHJKAKXR + TZYTOVKUUBUDUGZXKTZVIZUUDYDSZUUDKULUMZUUDXLSZHSZYBUPZUUJUUDHSZUUFUUDX + RTZUUKFTUUGUUKUIUUEUUMUUBUUDJKVLVMZUUFUUKUUJFUUFUUHUUJYBUUEUUHUUBUUDJ + KVNVMVOZUUFUUJUULFUUFUUIUUDHUUFYRUUIUUDUIUCXKUUDUCUDURZYQUUIYPUUDYPUU + DXLVFUUPVGVDXMYSYGAUUEVQUUBUUEVPVRVSZUUFXNHSZFTZUULFTBXRUUDBUDURUURUU + LFXNUUDHVFVTAUUSBXRUJZYTUUEAUUSBXRQWAZWBUUNVRWCWCUBUUDYCUUKXRFYDUBUDU + RXTUUHYAUUJYBXSUUDKULUSXSUUDHXLUTVAYDWDZWFWEUUOUUQWGUUBXNXRTZVIZXNYDS + ZYMFUVDUVCYMFTUVEYMUIUUBUVCVPZUVDYKYLYBFUVDYKVIZXSHSZFTZYLFTUBXRXOXSX + OUIUVHYLFXSXOHVFVTUVGUUTUVIUBXRUJZAUUTYTUVCYKUVAWHUVIUUSUBBXRYNUVHUUR + FXSXNHVFVTVEZWIUVGXOXKTXOXRTUVGXKXKXNXLUVGXMXKXKXLWJUUBXMUVCYKAXMYSYG + WKWBXKXKXLWLWMUVGXNXKTZYKUVDYKVPUVGUVCKWNTZUVLYKWOUVDUVCYKUVFVKAUVMYT + UVCYKAUUCUVMOJKWPWMZWHXNJKWQWEWRWSXOJKVLWMVRUVDYKWTZVIZUVIYBFTUBXRJXS + JUIUVHYBFXSJHVFVTAUVJYTUVCUVOAUUTUVJUVAUVKWIWHUVPJWNTZJXRTAUVQYTUVCUV + OAUUCUVQOJKXAWMWHJXBWMVRUVDXNWNTZUVMYKXCUVCUVRUUBJXNWPVMAUVMYTUVCUVNW + BXNKXHWEXDZUBXNYCYMXRFYDYOUVBWFWEUVSWCAUVCUUSYTQXEAXNFTCUGZFTVIXNUVTE + UFFTYTLXEXFXGXIXJ $. $} $} @@ -108854,21 +108880,21 @@ seq K ( .+ , G ) ) $= ( va vb cseq cfv cuz cv cle wbr cif cmpt cfz co wcel wceq elfzle2 iftrued adantl elfzuz fveq2 eleq1d wral ralrimiva adantr wf1o f1of syl ffvelcdmda wa wf rspcdva eqeltrd breq1 2fveq3 ifbieq1d fvmptg syl2an2 3eqtr4rd simpr - eqid cbvralv sylib ad2antrr cz eluzel2 eluzelz ad2antlr syl32anc ffvelrnd - eluzle elfz4 uzid wdc zdcle ifcldadc syl2anc seq3fveq cbvmptv seq3f1oleml - wn eqtrd ) ALEJKUCUDLEUAKUEUDZUAUFZLUGUHZXBHUDIUDZKIUDZUIZUJZKUCUDLEIKUCU - DABCEFGJXGKLPAGUFZKLUKULZUMZVHZXHLUGUHZXHHUDZIUDZXEUIZXNXHXGUDZXHJUDXJXOX - NUNAXJXLXNXEXHKLUOUPUQZXJXHXAUMAXOFUMXPXOUNXHKLURXKXOXNFXQXKBUFZIUDZFUMZX - NFUMBXAXMXRXMUNXSXNFXRXMIUSUTAXTBXAVAZXJAXTBXARVBZVCXKXMXIUMXMXAUMAXIXIXH - HAXIXIHVDXIXIHVIZQXIXIHVEVFZVGXMKLURVFVJVKUAXHXFXOXAFXGXBXHUNXCXLXDXNXEXB - XHLUGVLXBXHIHVMVNXGVSZVOVPTVQSAXRXAUMZVHZXRXGUDZXRLUGUHZXRHUDZIUDZXEUIZFY - GYFYLFUMYHYLUNAYFVRYGYIYKXEFYGYIVHZUBUFZIUDZFUMZYKFUMUBXAYJYNYJUNYOYKFYNY - JIUSUTAYPUBXAVAZYFYIAYAYQYBXTYPBUBXAXRYNUNXSYOFXRYNIUSUTVTWAWBYMYJXIUMYJX - AUMYMXIXIXRHAYCYFYIYDWBYMKWCUMZLWCUMZXRWCUMZKXRUGUHZYIXRXIUMAYRYFYIALXAUM - ZYRPKLWDVFZWBAYSYFYIAUUBYSPKLWEVFZWBYFYTAYIKXRWEZWFYFUUAAYIKXRWIWFYGYIVRX - RKLWJWGWHYJKLURVFVJAXEFUMZYFYIWSAXTUUFBXAKXRKUNXSXEFXRKIUSUTYBAYRKXAUMUUC - KWKVFVJWBYFYTAYSYIWLUUEAYSYFUUDVCXRLWMVPWNZUAXRXFYLXAFXGXBXRUNXCYIXDYKXEX - BXRLUGVLXBXRIHVMVNZYEVOWOUUGVKMWPABCDEFHIXGKLMNOPQRUABXAXFYLUUHWQWRWT $. + cbvralv sylib ad2antrr cz eluzel2 eluzelz ad2antlr eluzle elfz4 ffvelcdmd + eqid syl32anc wn uzid zdcle ifcldadc syl2anc seq3fveq cbvmptv seq3f1oleml + wdc eqtrd ) ALEJKUCUDLEUAKUEUDZUAUFZLUGUHZXBHUDIUDZKIUDZUIZUJZKUCUDLEIKUC + UDABCEFGJXGKLPAGUFZKLUKULZUMZVHZXHLUGUHZXHHUDZIUDZXEUIZXNXHXGUDZXHJUDXJXO + XNUNAXJXLXNXEXHKLUOUPUQZXJXHXAUMAXOFUMXPXOUNXHKLURXKXOXNFXQXKBUFZIUDZFUMZ + XNFUMBXAXMXRXMUNXSXNFXRXMIUSUTAXTBXAVAZXJAXTBXARVBZVCXKXMXIUMXMXAUMAXIXIX + HHAXIXIHVDXIXIHVIZQXIXIHVEVFZVGXMKLURVFVJVKUAXHXFXOXAFXGXBXHUNXCXLXDXNXEX + BXHLUGVLXBXHIHVMVNXGWIZVOVPTVQSAXRXAUMZVHZXRXGUDZXRLUGUHZXRHUDZIUDZXEUIZF + YGYFYLFUMYHYLUNAYFVRYGYIYKXEFYGYIVHZUBUFZIUDZFUMZYKFUMUBXAYJYNYJUNYOYKFYN + YJIUSUTAYPUBXAVAZYFYIAYAYQYBXTYPBUBXAXRYNUNXSYOFXRYNIUSUTVSVTWAYMYJXIUMYJ + XAUMYMXIXIXRHAYCYFYIYDWAYMKWBUMZLWBUMZXRWBUMZKXRUGUHZYIXRXIUMAYRYFYIALXAU + MZYRPKLWCVFZWAAYSYFYIAUUBYSPKLWDVFZWAYFYTAYIKXRWDZWEYFUUAAYIKXRWFWEYGYIVR + XRKLWGWJWHYJKLURVFVJAXEFUMZYFYIWKAXTUUFBXAKXRKUNXSXEFXRKIUSUTYBAYRKXAUMUU + CKWLVFVJWAYFYTAYSYIWSUUEAYSYFUUDVCXRLWMVPWNZUAXRXFYLXAFXGXBXRUNXCYIXDYKXE + XBXRLUGVLXBXRIHVMVNZYEVOWOUUGVKMWPABCDEFHIXGKLMNOPQRUABXAXFYLUUHWQWRWT $. $} ${ @@ -109056,22 +109082,22 @@ seq K ( .+ , G ) ) $= ( cfv wcel wceq co vw vk cuz cseq cfz elfzuz3 syl cv wi c1 fveqeq2 imbi2d caddc cz elfzuz eluzelz wa simpr adantr uztrn syl2anc syldan seq3-1 eqtrd seqeq1 fveq1d eqeq1d syl5ibcom cmin eluzel2 seq3m1 oveq2d oveq1 ralrimiva - adantlr wral eqid seqf eluzp1m1 sylan ffvelrnd rspcdva 3eqtrd uzp1 mpjaod - ex a1i seq3p1 oveq1d oveq2 fveq2 eleq1d peano2uz expcom a2d uzind4 mpcom - wo ) IGUCQZRZAIDFHUDZQJSZAGHIUETRZWTOGHIUFUGAUAUHZXAQJSZUIAGXAQZJSZUIZAUB - UHZXAQZJSZUIAXIUJUMTZXAQZJSZUIAXBUIUAUBGIXDGSXEXGAXDGJXAUKULXDXISXEXKAXDX - IJXAUKULXDXLSXEXNAXDXLJXAUKULXDISXEXBAXDIJXAUKULXHGUNRZAGHSZXGGHUJUMTUCQR - ZAGDFGUDZQZJSXPXGAXSGFQZJABCDEFGAGHUCQZRZXOAXCYBOGHIUOUGZHGUPUGABUHZWSRZY - DYARZYDFQZERZAYEUQYEYBYFAYEURAYBYEYCUSGYDHUTVALVBKVCPVDXPXSXFJXPGXRXADFGH - VEVFVGVHAXQXGAXQUQZXFGUJVITZXAQZXTDTYKJDTZJYIBCDEFHGAHUNRZXQAYBYMYCHGVJUG - ZUSZAXQURAYFYHXQLVOZAYDERCUHZERUQZYDYQDTERZXQKVOZVKYIXTJYKDAXTJSXQPUSVLYI - YDJDTZJSZYLJSBEYKYDYKSUUAYLJYDYKJDVMVGAUUBBEVPXQAUUBBENVNUSYIYAEYJXAYIBCD - EFHYAYAVQYOYPYTVRAYMXQYJYARYNHGVSVTWAWBWCWFAYBXPXQWRYCHGWDUGWEWGXIWSRZAXK - XNAUUCXKXNUIAUUCUQZXKXNUUDXKUQZXMXJXLFQZDTZJUUFDTZJUUDXMUUGSXKUUDBCDEFHXI - UUDUUCYBXIYARZAUUCURAYBUUCYCUSGXIHUTVAZAYFYHUUCLVOAYRYSUUCKVOWHUSUUEXJJUU - FDUUDXKURWIUUDUUHJSZXKUUDJYDDTZJSZUUKBEUUFYDUUFSUULUUHJYDUUFJDWJVGAUUMBEV - PUUCAUUMBEMVNUSUUDYHUUFERBYAXLYDXLSYGUUFEYDXLFWKWLAYHBYAVPUUCAYHBYALVNUSU - UDUUIXLYARUUJHXIWMUGWBWBUSWCWFWNWOWPWQ $. + adantlr wral eqid seqf eluzp1m1 sylan ffvelcdmd rspcdva 3eqtrd ex wo uzp1 + mpjaod a1i seq3p1 oveq1d oveq2 fveq2 eleq1d peano2uz expcom uzind4 mpcom + a2d ) IGUCQZRZAIDFHUDZQJSZAGHIUETRZWTOGHIUFUGAUAUHZXAQJSZUIAGXAQZJSZUIZAU + BUHZXAQZJSZUIAXIUJUMTZXAQZJSZUIAXBUIUAUBGIXDGSXEXGAXDGJXAUKULXDXISXEXKAXD + XIJXAUKULXDXLSXEXNAXDXLJXAUKULXDISXEXBAXDIJXAUKULXHGUNRZAGHSZXGGHUJUMTUCQ + RZAGDFGUDZQZJSXPXGAXSGFQZJABCDEFGAGHUCQZRZXOAXCYBOGHIUOUGZHGUPUGABUHZWSRZ + YDYARZYDFQZERZAYEUQYEYBYFAYEURAYBYEYCUSGYDHUTVALVBKVCPVDXPXSXFJXPGXRXADFG + HVEVFVGVHAXQXGAXQUQZXFGUJVITZXAQZXTDTYKJDTZJYIBCDEFHGAHUNRZXQAYBYMYCHGVJU + GZUSZAXQURAYFYHXQLVOZAYDERCUHZERUQZYDYQDTERZXQKVOZVKYIXTJYKDAXTJSXQPUSVLY + IYDJDTZJSZYLJSBEYKYDYKSUUAYLJYDYKJDVMVGAUUBBEVPXQAUUBBENVNUSYIYAEYJXAYIBC + DEFHYAYAVQYOYPYTVRAYMXQYJYARYNHGVSVTWAWBWCWDAYBXPXQWEYCHGWFUGWGWHXIWSRZAX + KXNAUUCXKXNUIAUUCUQZXKXNUUDXKUQZXMXJXLFQZDTZJUUFDTZJUUDXMUUGSXKUUDBCDEFHX + IUUDUUCYBXIYARZAUUCURAYBUUCYCUSGXIHUTVAZAYFYHUUCLVOAYRYSUUCKVOWIUSUUEXJJU + UFDUUDXKURWJUUDUUHJSZXKUUDJYDDTZJSZUUKBEUUFYDUUFSUULUUHJYDUUFJDWKVGAUUMBE + VPUUCAUUMBEMVNUSUUDYHUUFERBYAXLYDXLSYGUUFEYDXLFWLWMAYHBYAVPUUCAYHBYALVNUS + UUDUUIXLYARUUJHXIWNUGWBWBUSWCWDWOWRWPWQ $. $} ${ @@ -109115,20 +109141,20 @@ seq K ( .+ , G ) ) $= ( C T ( seq M ( .+ , G ) ` N ) ) ) $= ( co wcel vz va vb cseq cfv cv cmpt wceq wral adantr ralrimivva weq oveq1 wa eleq1d oveq2 cbvral2v sylib rspc2va syl21anc eqid fvmptg simprl simprr - syl2anc oveq12d 3eqtr4d cuz eqtr4d eqeltrd seq3homo eluzel2 seqf ffvelrnd - cz syl caovcld eqtr3d ) AKEIJUDZUEZUAFDUAUFZGSZUGZUEZKEHJUDUEDVTGSZABCEEF - IHWCJKLONABUFZFTZCUFZFTZUNZUNZDWFWHESZGSZDWFGSZDWHGSZESWLWCUEZWFWCUEZWHWC - UEZESMWKWLFTZWMFTZWPWMUHLWKDFTZWSUBUFZUCUFZGSZFTZUCFUIUBFUIZWTAXAWJRUJZLA - XFWJAWFWHGSZFTZCFUIBFUIXFAXIBCFFQUKXIXEXBWHGSZFTBCUBUCFFBUBULXHXJFWFXBWHG - UMUOCUCULXJXDFWHXCXBGUPUOUQURZUJZXEWTDXCGSZFTZUBUCDWLFFXBDUHXDXMFXBDXCGUM - UOZXCWLUHXMWMFXCWLDGUPUOUSUTUAWLWBWMFFWCWAWLDGUPWCVAZVBVEWKWQWNWRWOEWKWGW - NFTZWQWNUHAWGWIVCZWKXAWGXFXQXGXRXLXEXQXNUBUCDWFFFXOUCBULXMWNFXCWFDGUPUOUS - UTUAWFWBWNFFWCWAWFDGUPXPVBVEWKWIWOFTZWRWOUHAWGWIVDZWKXAWIXFXSXGXTXLXEXSXN - UBUCDWHFFXOUCCULXMWOFXCWHDGUPUOUSUTUAWHWBWOFFWCWAWHDGUPXPVBVEVFVGAWFJVHUE - ZTZUNZWFIUEZWCUEZDYDGSZWFHUEZYCYDFTZYFFTZYEYFUHOYCXAYHXFYIAXAYBRUJOAXFYBX - KUJXEYIXNUBUCDYDFFXOXCYDUHXMYFFXCYDDGUPUOUSUTZUAYDWBYFFFWCWAYDDGUPXPVBVEP - VIYCYGYFFPYJVJLVKAVTFTWEFTWDWEUHAYAFKVSABCEFIJYAYAVAAKYATJVOTNJKVLVPOLVMN - VNZABCDVTFFFGQRYKVQUAVTWBWEFFWCWAVTDGUPXPVBVEVR $. + syl2anc oveq12d 3eqtr4d cuz eqtr4d eqeltrd seq3homo eluzel2 syl ffvelcdmd + cz seqf caovcld eqtr3d ) AKEIJUDZUEZUAFDUAUFZGSZUGZUEZKEHJUDUEDVTGSZABCEE + FIHWCJKLONABUFZFTZCUFZFTZUNZUNZDWFWHESZGSZDWFGSZDWHGSZESWLWCUEZWFWCUEZWHW + CUEZESMWKWLFTZWMFTZWPWMUHLWKDFTZWSUBUFZUCUFZGSZFTZUCFUIUBFUIZWTAXAWJRUJZL + AXFWJAWFWHGSZFTZCFUIBFUIXFAXIBCFFQUKXIXEXBWHGSZFTBCUBUCFFBUBULXHXJFWFXBWH + GUMUOCUCULXJXDFWHXCXBGUPUOUQURZUJZXEWTDXCGSZFTZUBUCDWLFFXBDUHXDXMFXBDXCGU + MUOZXCWLUHXMWMFXCWLDGUPUOUSUTUAWLWBWMFFWCWAWLDGUPWCVAZVBVEWKWQWNWRWOEWKWG + WNFTZWQWNUHAWGWIVCZWKXAWGXFXQXGXRXLXEXQXNUBUCDWFFFXOUCBULXMWNFXCWFDGUPUOU + SUTUAWFWBWNFFWCWAWFDGUPXPVBVEWKWIWOFTZWRWOUHAWGWIVDZWKXAWIXFXSXGXTXLXEXSX + NUBUCDWHFFXOUCCULXMWOFXCWHDGUPUOUSUTUAWHWBWOFFWCWAWHDGUPXPVBVEVFVGAWFJVHU + EZTZUNZWFIUEZWCUEZDYDGSZWFHUEZYCYDFTZYFFTZYEYFUHOYCXAYHXFYIAXAYBRUJOAXFYB + XKUJXEYIXNUBUCDYDFFXOXCYDUHXMYFFXCYDDGUPUOUSUTZUAYDWBYFFFWCWAYDDGUPXPVBVE + PVIYCYGYFFPYJVJLVKAVTFTWEFTWDWEUHAYAFKVSABCEFIJYAYAVAAKYATJVOTNJKVLVMOLVP + NVNZABCDVTFFFGQRYKVQUAVTWBWEFFWCWAVTDGUPXPVBVEVR $. $} ${ @@ -109184,18 +109210,18 @@ seq K ( .+ , G ) ) $= ( co wcel cc0 caddc cfv cle wbr wi wceq fveq2 breq2d cr vw cfz cseq cuz vj vv eluzfz2 cv c1 imbi2d ralrimiva eluzfz1 rspcdva cz eluzel2 readdcl syl adantl seq3-1 breqtrrd a1i cfzo eqid seqf ad2antrr elfzouz ad2antlr - wa wf ffvelrnd eleq1d wral adantr peano2uz simpr fzofzp1 addge0d seq3p1 - adantlr ex expcom a2d fzind2 mpcom ) EDEUBIZJZAKELCDUCZMZNOZAEDUDMZJZWF - FDEUGUQAKUAUHZWGMZNOZPAKDWGMZNOZPZAKUEUHZWGMZNOZPAKWRUILIZWGMZNOZPAWIPU - AUEEDEWLDQZWNWPAXDWMWOKNWLDWGRSUJWLWRQZWNWTAXEWMWSKNWLWRWGRSUJWLXAQZWNX - CAXFWMXBKNWLXAWGRSUJWLEQZWNWIAXGWMWHKNWLEWGRSUJWQWKAKDCMZWONAKBUHZCMZNO - ZKXHNOBWEDXIDQXJXHKNXIDCRSAXKBWEHUKZAWKDWEJFDEULUQUMABUFLTCDAWKDUNJFDEU - OUQZGXITJUFUHZTJVHZXIXNLITJZAXIXNUPZURZUSUTVAWRDEVBIJZAWTXCAXSWTXCPAXSV - HZWTXCXTWTVHZKWSXACMZLIXBNYAWSYBYAWJTWRWGAWJTWGVIXSWTABUFLTCDWJWJVCXMGX - RVDVEXSWRWJJZAWTWRDEVFZVGZVJXTYBTJZWTXTXJTJZYFBWJXAXIXAQZXJYBTXIXACRZVK - AYGBWJVLXSAYGBWJGUKVMXSXAWJJZAXSYCYJYDDWRVNUQURUMVMXTWTVOYAXKKYBNOBWEXA - YHXJYBKNYISAXKBWEVLXSWTXLVEXSXAWEJAWTDEWRVPVGUMVQYABUFLTCDWRYEXTXIWJJZY - GWTAYKYGXSGVSVSXOXPYAXQURVRUTVTWAWBWCWD $. + wa ffvelcdmd eleq1d wral adantr peano2uz fzofzp1 addge0d adantlr seq3p1 + wf simpr ex expcom a2d fzind2 mpcom ) EDEUBIZJZAKELCDUCZMZNOZAEDUDMZJZW + FFDEUGUQAKUAUHZWGMZNOZPAKDWGMZNOZPZAKUEUHZWGMZNOZPAKWRUILIZWGMZNOZPAWIP + UAUEEDEWLDQZWNWPAXDWMWOKNWLDWGRSUJWLWRQZWNWTAXEWMWSKNWLWRWGRSUJWLXAQZWN + XCAXFWMXBKNWLXAWGRSUJWLEQZWNWIAXGWMWHKNWLEWGRSUJWQWKAKDCMZWONAKBUHZCMZN + OZKXHNOBWEDXIDQXJXHKNXIDCRSAXKBWEHUKZAWKDWEJFDEULUQUMABUFLTCDAWKDUNJFDE + UOUQZGXITJUFUHZTJVHZXIXNLITJZAXIXNUPZURZUSUTVAWRDEVBIJZAWTXCAXSWTXCPAXS + VHZWTXCXTWTVHZKWSXACMZLIXBNYAWSYBYAWJTWRWGAWJTWGVRXSWTABUFLTCDWJWJVCXMG + XRVDVEXSWRWJJZAWTWRDEVFZVGZVIXTYBTJZWTXTXJTJZYFBWJXAXIXAQZXJYBTXIXACRZV + JAYGBWJVKXSAYGBWJGUKVLXSXAWJJZAXSYCYJYDDWRVMUQURUMVLXTWTVSYAXKKYBNOBWEX + AYHXJYBKNYISAXKBWEVKXSWTXLVEXSXAWEJAWTDEWRVNVGUMVOYABUFLTCDWRYEXTXIWJJZ + YGWTAYKYGXSGVPVPXOXPYAXQURVQUTVTWAWBWCWD $. $} ser3le.3 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR ) $. @@ -109207,13 +109233,13 @@ seq K ( .+ , G ) ) $= ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) ) $= ( vx cc0 caddc cfv cmin co cle wbr wcel cr cseq cvv cv cmpt wceq resubcld cuz wa vex fveq2 oveq12d eqid fvmptg sylancr eqeltrd cfz elfzuz sylan2 wb - subge0d mpbird breqtrrd ser3ge0 recnd ser3sub breqtrd cz eluzel2 ffvelrnd - syl serfre mpbid ) ALFMDEUAZNZFMCEUAZNZOPZQRVPVNQRALFMKUBKUCZDNZVRCNZOPZU - DZEUANVQQABWBEFGABUCZEUGNZSZUHZWCWBNZWCDNZWCCNZOPZTWFWCUBSWJTSWGWJUEZBUIW - FWHWIIHUFZKWCWAWJUBTWBVRWCUEVSWHVTWIOVRWCDUJVRWCCUJUKWBULUMUNZWLUOAWCEFUP - PSZUHZLWJWGQWOLWJQRZWIWHQRZWNAWEWQWCEFUQZJURWNAWEWPWQUSWRWFWHWIIHUTURVAWN - AWEWKWRWMURVBVCABDCWBEFGWFWHIVDWFWIHVDWMVEVFAVNVPAWDTFVMABDEWDWDULZAFWDSE - VGSGEFVHVJZIVKGVIAWDTFVOABCEWDWSWTHVKGVIUTVL $. + subge0d mpbird breqtrrd ser3ge0 recnd ser3sub breqtrd cz serfre ffvelcdmd + eluzel2 syl mpbid ) ALFMDEUAZNZFMCEUAZNZOPZQRVPVNQRALFMKUBKUCZDNZVRCNZOPZ + UDZEUANVQQABWBEFGABUCZEUGNZSZUHZWCWBNZWCDNZWCCNZOPZTWFWCUBSWJTSWGWJUEZBUI + WFWHWIIHUFZKWCWAWJUBTWBVRWCUEVSWHVTWIOVRWCDUJVRWCCUJUKWBULUMUNZWLUOAWCEFU + PPSZUHZLWJWGQWOLWJQRZWIWHQRZWNAWEWQWCEFUQZJURWNAWEWPWQUSWRWFWHWIIHUTURVAW + NAWEWKWRWMURVBVCABDCWBEFGWFWHIVDWFWIHVDWMVEVFAVNVPAWDTFVMABDEWDWDULZAFWDS + EVGSGEFVJVKZIVHGVIAWDTFVOABCEWDWSWTHVHGVIUTVL $. $} @@ -109267,17 +109293,17 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( vx vy cn wcel cmul c1 cfv cc0 cap wbr wi wceq breq1d cc vw csn cxp cseq vk cv caddc co fveq2 imbi2d cuz wa elnnuz biimpri fvconst2g syl2an adantr 1zzd eqeltrd mulcl adantl seq3-1 1nn sylancl eqbrtrd wf nnuz sylan2b seqf - eqtrd simpl ffvelrnd ad2antlr simpr mulap0d biimpi adantll seq3p1 syl2an2 - wb peano2nnd oveq2d mpbird exp31 a2d nnind mpcom ) CIJACKIBUBUCZLUDZMZNOP - ZFAUAUFZWIMZNOPZQALWIMZNOPZQAUEUFZWIMZNOPZQAWQLUGUHZWIMZNOPZQAWKQUAUECWLL - RZWNWPAXCWMWONOWLLWIUISUJWLWQRZWNWSAXDWMWRNOWLWQWIUISUJWLWTRZWNXBAXEWMXAN - OWLWTWIUISUJWLCRZWNWKAXFWMWJNOWLCWIUISUJAWOBNOAWOLWHMZBAGHKTWHLAURZAGUFZL - UKMZJZULXIWHMZBTABTJZXIIJZXLBRXKDXNXKXIUMZUNIBXITUOUPAXMXKDUQUSZXITJHUFZT - JULZXIXQKUHTJZAXIXQUTZVAZVBAXMLIJXGBRDVCIBLTUOVDVJEVEWQIJZAWSXBYBAWSXBYBA - ULZWSULZXBWRBKUHZNOPZYDWRBYCWRTJWSYCITWQWIAITWIVFYBAGHKTWHLIVGXHXNAXKXLTJ - ZXOXPVHYAVIVAYBAVKZVLUQAXMYBWSDVMYCWSVNABNOPYBWSEVMVOYCXBYFVTWSYCXAYENOYC - XAWRWTWHMZKUHYEYCGHKTWHLWQYBWQXJJZAYBYJWQUMVPUQAXKYGYBXPVQXRXSYCXTVAVRYCY - IBWRKAXMYBWTIJYIBRDYCWQYHWAIBWTTUOVSWBVJSUQWCWDWEWFWG $. + eqtrd simpl ffvelcdmd ad2antlr mulap0d wb biimpi adantll seq3p1 peano2nnd + simpr syl2an2 oveq2d mpbird exp31 a2d nnind mpcom ) CIJACKIBUBUCZLUDZMZNO + PZFAUAUFZWIMZNOPZQALWIMZNOPZQAUEUFZWIMZNOPZQAWQLUGUHZWIMZNOPZQAWKQUAUECWL + LRZWNWPAXCWMWONOWLLWIUISUJWLWQRZWNWSAXDWMWRNOWLWQWIUISUJWLWTRZWNXBAXEWMXA + NOWLWTWIUISUJWLCRZWNWKAXFWMWJNOWLCWIUISUJAWOBNOAWOLWHMZBAGHKTWHLAURZAGUFZ + LUKMZJZULXIWHMZBTABTJZXIIJZXLBRXKDXNXKXIUMZUNIBXITUOUPAXMXKDUQUSZXITJHUFZ + TJULZXIXQKUHTJZAXIXQUTZVAZVBAXMLIJXGBRDVCIBLTUOVDVJEVEWQIJZAWSXBYBAWSXBYB + AULZWSULZXBWRBKUHZNOPZYDWRBYCWRTJWSYCITWQWIAITWIVFYBAGHKTWHLIVGXHXNAXKXLT + JZXOXPVHYAVIVAYBAVKZVLUQAXMYBWSDVMYCWSVTABNOPYBWSEVMVNYCXBYFVOWSYCXAYENOY + CXAWRWTWHMZKUHYEYCGHKTWHLWQYBWQXJJZAYBYJWQUMVPUQAXKYGYBXPVQXRXSYCXTVAVRYC + YIBWRKAXMYBWTIJYIBRDYCWQYHVSIBWTTUOWAWBVJSUQWCWDWEWFWG $. $} ${ @@ -109291,24 +109317,24 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) ) $= ( vx vy cc wcel cc0 wbr wceq c1 clt cmul cn cfv cdiv co cif wa ad2antrr wn cz cap cle wo csn cxp cseq cneg cexp 1cnd wf simp1 nnuz 1zzd fvconst2g - w3a cv simpl eqeltrd mulcl adantl syl simp2 simpr elnnz sylanbrc ffvelrnd - seqf znegcld simplr eqcom sylnib ioran wb 0zd zleloe mtbird zltnle mpbird - syl2anc zred lt0neg1d mpbid simpll3 ecased exp3vallem recclapd wdc simpl2 - zdclt ifcldadc zdceq sneq xpeq2d seqeq3d fveq1d oveq2d ifeq2d eqeq1 breq2 - ifeq12d fveq2 negeq fveq2d ifbieq12d ifbieq2d df-exp ovmpog syld3an3 ) AE - FZBUAFZAGUBHZGBUCHZUDZBGIZJGBKHZBLMAUEZUFZJUGZNZJBUHZXSNZOPZQZQZEFABUIPYE - IXJXKXNUPZXOJYDEYFXORUJYFXOTZRZXPXTYCEYHXPRZMEBXSYFMEXSUKZYGXPYFXJYJXJXKX - NULZXJCDLEXRJMUMXJUNXJCUQZMFZRYLXRNAEMAYLEUOXJYMURUSYLEFDUQZEFRYLYNLPEFXJ - YLYNUTVAVHVBZSYIXKXPBMFYFXKYGXPXJXKXNVCZSYHXPVDBVEVFVGYHXPTZRZYBYRMEYAXSY - FYJYGYQYOSYRYAUAFGYAKHZYAMFYRBYFXKYGYQYPSZVIYRBGKHZYSYRUUAXMTZYRXMXPGBIZU - DZYRYQUUCTUUDTYHYQVDYRXOUUCYFYGYQVJBGVKVLXPUUCVMVFYRGUAFZXKXMUUDVNYRVOZYT - GBVPVTVQZYRXKUUEUUAUUBVNYTUUFBGVRVTVSYRBYRBYTWAWBWCYAVEVFZVGYRAYAYFXJYGYQ - YKSYRXLXMUUGXJXKXNYGYQWDWEUUHWFWGYHUUEXKXPWHYHVOXJXKXNYGWIGBWJVTWKYFXKUUE - XOWHYPYFVOBGWLVTWKCDABEUAYNGIZJGYNKHZYNLMYLUEZUFZJUGZNZJYNUHZUUMNZOPZQZQY - 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(Contributed by Mario @@ -109429,17 +109455,17 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ cn0 wo wi expcllem ex adantr cdiv simpll sselid simplr simprl recnd nnnn0 wceq ad2antll expineg2 syl22anc crab ssrab2 simpl sylibr sstri cmul sseli cv syl2an anim1i sylbi mulap0 sylanbrc 1ap0 mpbir2an syl2anc sylib simprd - oveq2 eleq1d imbi12d vtoclga sylc eqeltrd jaod syl5bi 3impia ) CEKZCLMNZD - UAKZCDUBOZEKZWMDUGKZDUCKZDUDZUEKZPZUHWKWLPZWODUFXAWPWOWTWKWPWOUIWLWKWPWOA - BCDEFGHUJUKULXAWTWOXAWTPZWNQCWRUBOZUMOZEXBCRKWLDRKWRUGKZWNXDUTXBERCFWKWLW - TUNUOWKWLWTUPXBDXAWQWSUQURWSXEXAWQWRUSVAZCDVBVCXBXCEKZXCLMNZXDEKZXBJVKZLM - NZJEVDZEXCXKJEVEZXBCXLKZXEXCXLKZXBXAXNXAWTVFXKWLJCEXJCLMSTVGXFABCWRXLXLER - XMFVHAVKZXLKZBVKZXLKZPXPXRVIOZEKZXTLMNZXTXLKXQXPEKZXREKZYAXSXLEXPXMVJXLEX - RXMVJGVLXQXPRKZXPLMNZPZXRRKZXRLMNZPZYBXSXQYCYFPYGXKYFJXPEXJXPLMSTYCYEYFER - XPFVJVMVNXSYDYIPYJXKYIJXREXJXRLMSTYDYHYIERXRFVJVMVNXPXRVOVLXKYBJXTEXJXTLM - STVPQXLKQEKQLMNZHVQXKYKJQEXJQLMSTVRUJVSZUOXBXGXHXBXOXGXHPYLXKXHJXCEXJXCLM - STVTWAYFQXPUMOZEKZUIXHXIUIAXCEXPXCUTZYFXHYNXIXPXCLMSYOYMXDEXPXCQUMWBWCWDY - CYFYNIUKWEWFWGUKWHWIWJ $. + oveq2 eleq1d imbi12d vtoclga sylc eqeltrd jaod biimtrid 3impia ) CEKZCLMN + ZDUAKZCDUBOZEKZWMDUGKZDUCKZDUDZUEKZPZUHWKWLPZWODUFXAWPWOWTWKWPWOUIWLWKWPW + OABCDEFGHUJUKULXAWTWOXAWTPZWNQCWRUBOZUMOZEXBCRKWLDRKWRUGKZWNXDUTXBERCFWKW + LWTUNUOWKWLWTUPXBDXAWQWSUQURWSXEXAWQWRUSVAZCDVBVCXBXCEKZXCLMNZXDEKZXBJVKZ + LMNZJEVDZEXCXKJEVEZXBCXLKZXEXCXLKZXBXAXNXAWTVFXKWLJCEXJCLMSTVGXFABCWRXLXL + ERXMFVHAVKZXLKZBVKZXLKZPXPXRVIOZEKZXTLMNZXTXLKXQXPEKZXREKZYAXSXLEXPXMVJXL + EXRXMVJGVLXQXPRKZXPLMNZPZXRRKZXRLMNZPZYBXSXQYCYFPYGXKYFJXPEXJXPLMSTYCYEYF + ERXPFVJVMVNXSYDYIPYJXKYIJXREXJXRLMSTYDYHYIERXRFVJVMVNXPXRVOVLXKYBJXTEXJXT + LMSTVPQXLKQEKQLMNZHVQXKYKJQEXJQLMSTVRUJVSZUOXBXGXHXBXOXGXHPYLXKXHJXCEXJXC + LMSTVTWAYFQXPUMOZEKZUIXHXIUIAXCEXPXCUTZYFXHYNXIXPXCLMSYOYMXDEXPXCQUMWBWCW + DYCYFYNIUKWEWFWGUKWHWIWJ $. $} ${ @@ -109631,11 +109657,11 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ( cc wcel cc0 cap wbr cz cneg cexp co c1 cdiv wceq cr cn0 wo wa elznn0 wi expnegap0 3expia adantr simpl simprl recnd simprr expineg2 oveq2d ad2ant2rl syl12anc expcl simpll simplr nn0zd expap0i syl3anc recrecapd eqtr2d expimpd - expr jaod syl5bi 3impia ) ACDZAEFGZBHDZABIZJKZLABJKZMKZNZVGBODZBPDZVHPDZQZR - VEVFRZVLBSVQVMVPVLVQVMRVNVLVOVQVNVLTVMVEVFVNVLABUAUBUCVQVMVOVLVQVMVORZRZVKL - LVIMKZMKVIVSVJVTLMVSVQBCDVOVJVTNVQVRUDVSBVQVMVOUEUFVQVMVOUGZABUHUKUIVSVIVEV - OVICDVFVMAVHULUJVSVEVFVHHDVIEFGVEVFVRUMVEVFVRUNVSVHWAUOAVHUPUQURUSVAVBUTVCV - D $. + expr jaod biimtrid 3impia ) ACDZAEFGZBHDZABIZJKZLABJKZMKZNZVGBODZBPDZVHPDZQ + ZRVEVFRZVLBSVQVMVPVLVQVMRVNVLVOVQVNVLTVMVEVFVNVLABUAUBUCVQVMVOVLVQVMVORZRZV + KLLVIMKZMKVIVSVJVTLMVSVQBCDVOVJVTNVQVRUDVSBVQVMVOUEUFVQVMVOUGZABUHUKUIVSVIV + EVOVICDVFVMAVHULUJVSVEVFVHHDVIEFGVEVFVRUMVEVFVRUNVSVHWAUOAVHUPUQURUSVAVBUTV + CVD $. $( Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.) $) @@ -109795,21 +109821,21 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ negnegd simp2r nnnn0d nn0negz syl eqeltrrd expclzap mulcomd simp3l negdid 3eqtr4d impancom simp3r eqtrd 1t1e1 oveq1i eqtr4di expap0i ax-1cn mpanl12 divmuldivap eqtr4d addcld nn0addcld eqeltrd expineg2 oveq12d jaod sylan2b - nn0zd syl5bi impr ) ADEZAFGUAZHZBIEZCIEZABCUBJZKJZABKJZACKJZLJZMZWSCNEZCU - CEZCOZUDEZHZUEZWQWRHXECUFWRWQBNEZBUCEZBOZUDEZHZUEZXKXEUGBUFWQXQHXFXEXJWQX - LXFXEUGZXPWOXLXRWPWOXLXFXEABCUHPUIWQXPXFXEABCUJPULWQXLXJXEUGXPWQXJXLXEWQX - JXLXEWQXJXLUMZACBUBJZKJXCXBLJXAXDACBUJXSWTXTAKXSBCXSBWQXJXLUKZUNXSCWQXGXI - XLUOUPZUQURXSXBXCXSWOXLXBDEWOWPXJXLUSZYAABUTVAXSWOWPWSXCDEYCWOWPXJXLVBXSX - HOZCIXSCYBVCXSXHNEZYDIEXSXHWQXGXIXLVDVEXHVFVGVHACVIQVJVMPVNWQXPXJXEWQXPXJ - UMZRAWTOZKJZSJZRAXNKJZSJZRAXHKJZSJZLJZXAXDYFYIRRLJZYJYLLJZSJZYNYFYIRYPSJY - QYFYHYPRSYFYHAXNXHUBJZKJZYPYFYGYRAKYFBCYFBWQXMXOXJUOUPZYFCWQXPXGXIVKUPZVL - ZURYFWOXNNEZYEYSYPMWOWPXPXJUSZYFXNWQXMXOXJVDVEZYFXHWQXPXGXIVOVEZAXNXHUHQV - PURYORYPSVQVRVSYFYJDEZYJFGUAZYLDEZYLFGUAZYNYQMZYFWOUUCUUGUUDUUEAXNUTVAYFW - OWPXNIEUUHUUDWOWPXPXJVBZYFXNUUEWLAXNVTQYFWOYEUUIUUDUUFAXHUTVAYFWOWPXHIEUU - JUUDUULYFXHUUFWLAXHVTQRDEZUUMUUGUUHHUUIUUJHHUUKWAWARRYJYLWCWBTWDYFWOWPWTD - EYGNEXAYIMUUDUULYFBCYTUUAWEYFYGYRNUUBYFXNXHUUEUUFWFWGAWTWHTYFXBYKXCYMLYFW - OWPBDEUUCXBYKMUUDUULYTUUEABWHTYFWOWPCDEYEXCYMMUUDUULUUAUUFACWHTWIVMPULWJW - KWMWN $. + nn0zd biimtrid impr ) ADEZAFGUAZHZBIEZCIEZABCUBJZKJZABKJZACKJZLJZMZWSCNEZ + CUCEZCOZUDEZHZUEZWQWRHXECUFWRWQBNEZBUCEZBOZUDEZHZUEZXKXEUGBUFWQXQHXFXEXJW + QXLXFXEUGZXPWOXLXRWPWOXLXFXEABCUHPUIWQXPXFXEABCUJPULWQXLXJXEUGXPWQXJXLXEW + QXJXLXEWQXJXLUMZACBUBJZKJXCXBLJXAXDACBUJXSWTXTAKXSBCXSBWQXJXLUKZUNXSCWQXG + XIXLUOUPZUQURXSXBXCXSWOXLXBDEWOWPXJXLUSZYAABUTVAXSWOWPWSXCDEYCWOWPXJXLVBX + SXHOZCIXSCYBVCXSXHNEZYDIEXSXHWQXGXIXLVDVEXHVFVGVHACVIQVJVMPVNWQXPXJXEWQXP + XJUMZRAWTOZKJZSJZRAXNKJZSJZRAXHKJZSJZLJZXAXDYFYIRRLJZYJYLLJZSJZYNYFYIRYPS + JYQYFYHYPRSYFYHAXNXHUBJZKJZYPYFYGYRAKYFBCYFBWQXMXOXJUOUPZYFCWQXPXGXIVKUPZ + VLZURYFWOXNNEZYEYSYPMWOWPXPXJUSZYFXNWQXMXOXJVDVEZYFXHWQXPXGXIVOVEZAXNXHUH + QVPURYORYPSVQVRVSYFYJDEZYJFGUAZYLDEZYLFGUAZYNYQMZYFWOUUCUUGUUDUUEAXNUTVAY + FWOWPXNIEUUHUUDWOWPXPXJVBZYFXNUUEWLAXNVTQYFWOYEUUIUUDUUFAXHUTVAYFWOWPXHIE + UUJUUDUULYFXHUUFWLAXHVTQRDEZUUMUUGUUHHUUIUUJHHUUKWAWARRYJYLWCWBTWDYFWOWPW + TDEYGNEXAYIMUUDUULYFBCYTUUAWEYFYGYRNUUBYFXNXHUUEUUFWFWGAWTWHTYFXBYKXCYMLY + FWOWPBDEUUCXBYKMUUDUULYTUUEABWHTYFWOWPCDEYEXCYMMUUDUULUUAUUFACWHTWIVMPULW + JWKWMWN $. $( Product of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative @@ -109840,27 +109866,27 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ cneg nn0cnd simp1l simp2r nnnn0d eqtr3d expcl syl2anc simp1r nnzd expap0i mulneg1d nn0zd exprecap eqtr4d mulcld nn0mulcld eqeltrrd expineg2 3eqtr4d oveq1d jaodan simp3l mulneg2d recclapd recap0d recrecapd mul2negd 3eqtrrd - simp2 simp3r 3eqtrd jaod sylan2b syl5bi impr ) ADEZAFGHZIZBJEZCJEZABCKLZM - LZABMLZCMLZNZWMCOEZCUAEZCUMZUBEZIZUCZWKWLIWRCUDWLWKBOEZBUAEZBUMZUBEZIZUCZ - XDWRUEBUDWKXJIWSWRXCWKXEWSWRUEZXIWIXEXKWJWIXEWSWRABCUFUGUHWKXIWSWRWKXIWSU - IZPAWNUMZMLZQLZPAXGMLZQLZCMLZWOWQXLXOPXPCMLZQLZXRXLXNXSPQXLAXGCKLZMLZXNXS - XLYAXMAMXLBCXLBWKXFXHWSUJUKZXLCWKXIWSULZUNZVDZRXLWIXGOEZWSYBXSNWIWJXIWSUO - ZXLXGWKXFXHWSUPZUQZYDAXGCUFTURRXLXPDEZXPFGHZWMXRXTNXLWIYGYKYHYJAXGUSZUTXL - WIWJXGJEZYLYHWIWJXIWSVAZXLXGYIVBAXGVCZTXLCYDVEXPCVFTVGXLWIWJWNDEZXMOEZWOX - ONZYHYOXLBCYCYEVHXLYAXMOYFXLXGCYJYDVIVJAWNVKZSXLWPXQCMXLWIWJBDEZYGWPXQNZY - HYOYCYJABVKZSVMVLUGVNWKXEXCWRUEXIWKXEXCWRWKXEXCUIZXOPWPXAMLZQLZWOWQUUDXNU - UEPQUUDABXAKLZMLZXNUUEUUDUUGXMAMUUDBCUUDBWKXEXCWBZUNZUUDCWKXEWTXBVOUKZVPZ - RUUDWIXEXAOEZUUHUUENWIWJXEXCUOZUUIUUDXAWKXEWTXBWCUQZABXAUFTURRUUDWIWJYQYR - YSUUNWIWJXEXCVAZUUDBCUUJUUKVHUUDUUGXMOUULUUDBXAUUIUUOVIVJYTSUUDWPDEZWPFGH - ZCDEZUUMWQUUFNUUDWIXEUUQUUNUUIABUSUTUUDWIWJWLUURUUNUUPUUDBUUIVEABVCTUUKUU - OWPCVKSVLUGWKXIXCWRWKXIXCUIZWQXRPXQXAMLZQLZWOUUTWPXQCMUUTWIWJUUAYGUUBWIWJ - XIXCUOZWIWJXIXCVAZUUTBWKXFXHXCUJUKZUUTXGWKXFXHXCUPZUQZUUCSVMUUTXQDEXQFGHU - USUUMXRUVBNUUTXPUUTWIYGYKUVCUVGYMUTZUUTWIWJYNYLUVCUVDUUTXGUVFVBYPTZVQUUTX - PUVHUVIVRUUTCWKXIWTXBVOUKZUUTXAWKXIWTXBWCZUQZXQCVKSUUTUVBPPXPXAMLZQLZQLUV - MWOUUTUVAUVNPQUUTYKYLXAJEZUVAUVNNUVHUVIUUTXAUVKVBZXPXAVFTRUUTUVMUUTYKUUMU - VMDEUVHUVLXPXAUSUTUUTYKYLUVOUVMFGHUVHUVIUVPXPXAVCTVSUUTAXGXAKLZMLZUVMWOUU - TWIYGUUMUVRUVMNUVCUVGUVLAXGXAUFTUUTUVQWNAMUUTBCUVEUVJVTRURWDWAUGVNWEWFWGW - H $. + simp2 simp3r 3eqtrd jaod sylan2b biimtrid impr ) ADEZAFGHZIZBJEZCJEZABCKL + ZMLZABMLZCMLZNZWMCOEZCUAEZCUMZUBEZIZUCZWKWLIWRCUDWLWKBOEZBUAEZBUMZUBEZIZU + CZXDWRUEBUDWKXJIWSWRXCWKXEWSWRUEZXIWIXEXKWJWIXEWSWRABCUFUGUHWKXIWSWRWKXIW + SUIZPAWNUMZMLZQLZPAXGMLZQLZCMLZWOWQXLXOPXPCMLZQLZXRXLXNXSPQXLAXGCKLZMLZXN + XSXLYAXMAMXLBCXLBWKXFXHWSUJUKZXLCWKXIWSULZUNZVDZRXLWIXGOEZWSYBXSNWIWJXIWS + UOZXLXGWKXFXHWSUPZUQZYDAXGCUFTURRXLXPDEZXPFGHZWMXRXTNXLWIYGYKYHYJAXGUSZUT + XLWIWJXGJEZYLYHWIWJXIWSVAZXLXGYIVBAXGVCZTXLCYDVEXPCVFTVGXLWIWJWNDEZXMOEZW + OXONZYHYOXLBCYCYEVHXLYAXMOYFXLXGCYJYDVIVJAWNVKZSXLWPXQCMXLWIWJBDEZYGWPXQN + ZYHYOYCYJABVKZSVMVLUGVNWKXEXCWRUEXIWKXEXCWRWKXEXCUIZXOPWPXAMLZQLZWOWQUUDX + NUUEPQUUDABXAKLZMLZXNUUEUUDUUGXMAMUUDBCUUDBWKXEXCWBZUNZUUDCWKXEWTXBVOUKZV + PZRUUDWIXEXAOEZUUHUUENWIWJXEXCUOZUUIUUDXAWKXEWTXBWCUQZABXAUFTURRUUDWIWJYQ + YRYSUUNWIWJXEXCVAZUUDBCUUJUUKVHUUDUUGXMOUULUUDBXAUUIUUOVIVJYTSUUDWPDEZWPF + GHZCDEZUUMWQUUFNUUDWIXEUUQUUNUUIABUSUTUUDWIWJWLUURUUNUUPUUDBUUIVEABVCTUUK + UUOWPCVKSVLUGWKXIXCWRWKXIXCUIZWQXRPXQXAMLZQLZWOUUTWPXQCMUUTWIWJUUAYGUUBWI + WJXIXCUOZWIWJXIXCVAZUUTBWKXFXHXCUJUKZUUTXGWKXFXHXCUPZUQZUUCSVMUUTXQDEXQFG + HUUSUUMXRUVBNUUTXPUUTWIYGYKUVCUVGYMUTZUUTWIWJYNYLUVCUVDUUTXGUVFVBYPTZVQUU + TXPUVHUVIVRUUTCWKXIWTXBVOUKZUUTXAWKXIWTXBWCZUQZXQCVKSUUTUVBPPXPXAMLZQLZQL + UVMWOUUTUVAUVNPQUUTYKYLXAJEZUVAUVNNUVHUVIUUTXAUVKVBZXPXAVFTRUUTUVMUUTYKUU + MUVMDEUVHUVLXPXAUSUTUUTYKYLUVOUVMFGHUVHUVIUVPXPXAVCTVSUUTAXGXAKLZMLZUVMWO + UUTWIYGUUMUVRUVMNUVCUVGUVLAXGXAUFTUUTUVQWNAMUUTBCUVEUVJVTRURWDWAUGVNWEWFW + GWH $. $} $( Exponentiation of negative one to an even power. (Contributed by Scott @@ -111393,20 +111419,20 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ c1 co leidi eqbrtrdi impexp wo cz wb simpl nn0zd peano2nn0 zleloe syl2anc clt nn0leltp1 cmul faccl nn0re peano2re syl nnnn0d nn0ge0d nn0p1nn nnge1d lemulge11d facp1 breqtrrd adantr faccld letr syl3anc mpan2d com23 sylbird - imim2d leidd syl5ibcom syl5 a1dd jaod sylbid com13 com4l a2d imp4a syl5bi - ex nn0ind 3impib 3com12 ) BCDZACDZABEFZAGHZBGHZEFZXAXBXCXFXBAUAUCZEFZIZXD - XGGHZEFZJXBAKEFZIZXDKGHZEFZJXBAUBUCZEFZIZXDXPGHZEFZJZXBAXPUKUDULZEFZIZXDY - BGHZEFZJZXBXCIZXFJUAUBBXGKRZXIXMXKXOYIXHXLXBXGKAELMYIXJXNXDEXGKGNOPXGXPRZ - XIXRXKXTYJXHXQXBXGXPAELMYJXJXSXDEXGXPGNOPXGYBRZXIYDXKYFYKXHYCXBXGYBAELMYK - XJYEXDEXGYBGNOPXGBRZXIYHXKXFYLXHXCXBXGBAELMYLXJXEXDEXGBGNOPXMXDXNXNEXMAKG - XBXLAKRAUEUFUGXNXNUKQUHUIUJUMUNYAXBXQXTJZJZXPCDZYGXBXQXTUOYOYNXBYCYFYOXBY - MYCYFJYCYOXBYMYFXBYOYCYMYFJZXBYOYCYPJXBYOIZYCAYBVDFZAYBRZUPZYPYQAUQDYBUQD - YCYTURYQAXBYOUSUTYQYBYOYBCDXBXPVAZSUTAYBVBVCYQYRYPYSYQYRXQYPAXPVEYQYMXQYF - YQXTYFXQYQXTXSYEEFZYFYOUUBXBYOXSXSYBVFULYEEYOXSYBYOXSXPVGZTZYOXPQDYBQDXPV - HXPVIVJYOXSYOXSUUCVKVLYOYBXPVMVNVOXPVPVQSYQXDQDZXSQDZYEQDZXTUUBIYFJXBUUEY - OXBXDAVGTZVRYOUUFXBUUDSYOUUGXBYOYEYOYBUUAVSTSXDXSYEVTWAWBWEWCWDYQYSYFYMXB - YSYFJYOYSXDYERZXBYFAYBGNXBXDXDEFUUIYFXBXDUUHWFXDYEXDELWGWHVRWIWJWKWQWLWMW - NWOWPWRWSWT $. + imim2d leidd syl5ibcom syl5 a1dd sylbid ex com13 com4l a2d imp4a biimtrid + jaod nn0ind 3impib 3com12 ) BCDZACDZABEFZAGHZBGHZEFZXAXBXCXFXBAUAUCZEFZIZ + XDXGGHZEFZJXBAKEFZIZXDKGHZEFZJXBAUBUCZEFZIZXDXPGHZEFZJZXBAXPUKUDULZEFZIZX + DYBGHZEFZJZXBXCIZXFJUAUBBXGKRZXIXMXKXOYIXHXLXBXGKAELMYIXJXNXDEXGKGNOPXGXP + RZXIXRXKXTYJXHXQXBXGXPAELMYJXJXSXDEXGXPGNOPXGYBRZXIYDXKYFYKXHYCXBXGYBAELM + YKXJYEXDEXGYBGNOPXGBRZXIYHXKXFYLXHXCXBXGBAELMYLXJXEXDEXGBGNOPXMXDXNXNEXMA + KGXBXLAKRAUEUFUGXNXNUKQUHUIUJUMUNYAXBXQXTJZJZXPCDZYGXBXQXTUOYOYNXBYCYFYOX + BYMYCYFJYCYOXBYMYFXBYOYCYMYFJZXBYOYCYPJXBYOIZYCAYBVDFZAYBRZUPZYPYQAUQDYBU + QDYCYTURYQAXBYOUSUTYQYBYOYBCDXBXPVAZSUTAYBVBVCYQYRYPYSYQYRXQYPAXPVEYQYMXQ + YFYQXTYFXQYQXTXSYEEFZYFYOUUBXBYOXSXSYBVFULYEEYOXSYBYOXSXPVGZTZYOXPQDYBQDX + PVHXPVIVJYOXSYOXSUUCVKVLYOYBXPVMVNVOXPVPVQSYQXDQDZXSQDZYEQDZXTUUBIYFJXBUU + EYOXBXDAVGTZVRYOUUFXBUUDSYOUUGXBYOYEYOYBUUAVSTSXDXSYEVTWAWBWEWCWDYQYSYFYM + XBYSYFJYOYSXDYERZXBYFAYBGNXBXDXDEFUUIYFXBXDUUHWFXDYEXDELWGWHVRWIWQWJWKWLW + MWNWOWPWRWSWT $. $( A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) $) @@ -111782,38 +111808,38 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ eluz2 syl3anbrc adantrr nnuz eleqtrdi cvv fvi eluzelcn eqeltrid seq3split elv elfzuz3 eluznn syl2anc facnn oveq1d 3eqtr4d expr faccld nncnd mulid2d oveq2d eqtr3d fveq2 fac0 eqtrdi oveq1 0p1e1 seqeq1d fveq1d oveq12d eqeq2d - syl5ibrcom wo fznn0sub elnn0 sylib mpjaod eqid mulcl seqf ffvelrnd nnnn0d - nnap0d divcanap5d 3eqtrd div0apd simpr mul02d mul01d nn0uz elfz5 ad2antrr - syl clt nn0re nnre ad2antlr 0z sylancl 0zd bitr4d mtbid simpl adantr elfz - subge0d zltnle zltp1le mpbid nn0ge0 mpbir2and 0cn mp1i nnz bcval3 syl3an2 - seq3z 3expa 3eqtr4rd wdc fzdcel exmiddc mpjaodan ) BUFCZAUGCZDZAEBUHFCZBA - UIFZBGHBAUJFZIUKFZULZJZAKJZLFZMUVGUQZUVFUVGDZUVHBKJZUVIKJZUVMGFZLFZUVRUVL - GFZUVSLFUVNUVGUVHUVTMUVFABUMNUVPUVQUWAUVSLUVPUVIUGCZUVQUWAMZUVIEMZUVFUVGU - WBUWCUVFUVGUWBDDZBGHIULZJZUVIUWFJZUVLGFUVQUWAUWEUAUBUCGOHIUVIBUWEUAPZOCZU - BPZOCZDZDUWIUWKUWEUWJUWLUNUWEUWJUWLUOUPUWEUWJUWLUCPZOCZURDUWIUWKUWNUWEUWJ - 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ralrimiva ex syl5bi nn0ind sylan ) BEFBCGZHIZEFZCJKZAJFBAHIZEFZDGZVSHIZEF - ZCJKLVSHIZEFZCJKUAGZVSHIZEFZCJKZWJMUBIZVSHIZEFZCJKZWBDUABWELNZWGWICJWRWFW - HEWELVSHOPQWEWJNZWGWLCJWSWFWKEWEWJVSHOPQWEWNNZWGWPCJWTWFWOEWEWNVSHOPQWEBN - ZWGWACJXAWFVTEWEBVSHOPQWICJVSJFZVSLLUCIFZWIXCUDZXBXCSVSLNZWIXCXEXBVSLUEUF - XEWHLLHIZEVSLLHRXFMELEFZXFMNTLUGUHULUIUJUMXBXDSWHLEXGXBXDWHLNTVSLUKUNTUJX - BLJFZXHXCXDVDZXBUOZXJXBXHXHUPXCUQXIVSLLURXCUSUMUTVAVBWMWJWEHIZEFZDJKZWJEF - ZWQWLXLCDJVSWENWKXKEVSWEWJHRPVCXNXMWQXNXMSZWPCJXOXBSWKWJVSMVEIZHIZUBIZWOE - XNXBXRWONXMVSWJVFVGXMXBXREFXNXMXBSWKXQXLWLDVSJWEVSNXKWKEWEVSWJHRPVHXBXMXP - JFXQEFZVSVIXLXSDXPJWEXPNXKXQEWEXPWJHRPVHVJVKVLVMVNVOVPVQWAWDCAJVSANVTWCEV - SABHRPVHVR $. + ralrimiva ex biimtrid nn0ind sylan ) BEFBCGZHIZEFZCJKZAJFBAHIZEFZDGZVSHIZ + EFZCJKLVSHIZEFZCJKUAGZVSHIZEFZCJKZWJMUBIZVSHIZEFZCJKZWBDUABWELNZWGWICJWRW + FWHEWELVSHOPQWEWJNZWGWLCJWSWFWKEWEWJVSHOPQWEWNNZWGWPCJWTWFWOEWEWNVSHOPQWE + BNZWGWACJXAWFVTEWEBVSHOPQWICJVSJFZVSLLUCIFZWIXCUDZXBXCSVSLNZWIXCXEXBVSLUE + UFXEWHLLHIZEVSLLHRXFMELEFZXFMNTLUGUHULUIUJUMXBXDSWHLEXGXBXDWHLNTVSLUKUNTU + JXBLJFZXHXCXDVDZXBUOZXJXBXHXHUPXCUQXIVSLLURXCUSUMUTVAVBWMWJWEHIZEFZDJKZWJ + EFZWQWLXLCDJVSWENWKXKEVSWEWJHRPVCXNXMWQXNXMSZWPCJXOXBSWKWJVSMVEIZHIZUBIZW + OEXNXBXRWONXMVSWJVFVGXMXBXREFXNXMXBSWKXQXLWLDVSJWEVSNXKWKEWEVSWJHRPVHXBXM + XPJFXQEFZVSVIXLXSDXPJWEXPNXKXQEWEXPWJHRPVHVJVKVLVMVNVOVPVQWAWDCAJVSANVTWC + EVSABHRPVHVR $. $( A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, @@ -112121,12 +112147,12 @@ This definition (in terms of ` U. ` and ` ~<_ ` ) is not taken directly hashfz1 $p |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N ) $= ( vx vy cn0 wcel c1 cfz co chash cfv cz cv caddc cmpt cc0 cfrec wceq wf1o com syl2anc ccnv cen wbr cuz wf eqid frec2uzf1od f1ocnv f1of 3syl elnn0uz - biimpi ffvelrnd frecfzennn ensymd hashennn wa oveq1 cbvmptv freceq1 ax-mp - 0zd fveq1i f1ocnvfv2 eqtr3id eqtrd ) ADEZFAGHZIJZABKBLZFMHZNZOPZUAZJZCKCL - ZFMHZNZOPZJZAVGVOSEVOVHUBUCVIVTQVGOUDJZSAVNVGSWAVMRZWASVNRWASVNUEVGBOVMVG - VBVMUFZUGZSWAVMUHWASVNUIUJVGAWAEZAUKULZUMVGVHVOBVMAWCUNUOCVHVOUPTVGWBWEVT - AQWDWFWBWEUQVTVOVMJAVOVMVSVLVRQVMVSQBCKVKVQVJVPFMURUSOVLVRUTVAVCSWAAVMVDV - ETVF $. + 0zd biimpi ffvelcdmd frecfzennn ensymd hashennn wa cbvmptv freceq1 fveq1i + oveq1 ax-mp f1ocnvfv2 eqtr3id eqtrd ) ADEZFAGHZIJZABKBLZFMHZNZOPZUAZJZCKC + LZFMHZNZOPZJZAVGVOSEVOVHUBUCVIVTQVGOUDJZSAVNVGSWAVMRZWASVNRWASVNUEVGBOVMV + GULVMUFZUGZSWAVMUHWASVNUIUJVGAWAEZAUKUMZUNVGVHVOBVMAWCUOUPCVHVOUQTVGWBWEV + TAQWDWFWBWEURVTVOVMJAVOVMVSVLVRQVMVSQBCKVKVQVJVPFMVBUSOVLVRUTVCVASWAAVMVD + VETVF $. $( Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by @@ -112165,13 +112191,6 @@ This definition (in terms of ` U. ` and ` ~<_ ` ) is not taken directly OPABNIJKL $. $} - $( The codomain of an onto function is a set if its domain is a set. - (Contributed by AV, 4-May-2021.) $) - focdmex $p |- ( ( A e. V /\ F : A -onto-> B ) -> B e. _V ) $= - ( wcel wfo wa crn cvv wf fof anim2i ancomd rnexg 3syl wb forn eleq1d adantl - fex mpbid ) ADEZABCFZGZCHZIEZBIEZUDABCJZUBGCIEUFUDUBUHUCUHUBABCKLMABDCTCINO - UCUFUGPUBUCUEBIABCQRSUA $. - ${ $d A f $. $d B f $. $d F f $. $( The size of two finite sets is equal if there is a bijection mapping one @@ -112706,7 +112725,7 @@ are used instead of sets because their representation is shorter (and more u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) ) $= ( cc0 co cima cres cfzo cfv csn cun wcel wceq syl sylib wfn cfz cop chash imaundi cuz cn0 elfzonn0 elnn0uz fzisfzounsn imaeq2d ffnd syl2anc 3eqtr4a - fnsnfv uneq2d reseq2d resundi eqtrdi wfun funfn ffvelrnd fnressn eqtrd + fnsnfv uneq2d reseq2d resundi eqtrdi wfun funfn ffvelcdmd fnressn eqtrd cdm ) ACBHDUAIZJZKZCBHDLIZJZKZCDBMZNZKZOZVJVKVKCMUBNZOAVGCVIVLOZKVNAVFVPC ABVHDNZOZJVIBVQJZOVFVPBVHVQUDAVEVRBADHUEMPZVEVRQADUFPZVTADHBUCMZLIZPZWAGD WBUGRDUHSHDUIRUJAVLVSVIABWCTWDVLVSQAWCCVDZBFUKGWCDBUNULUOUMUPCVIVLUQURAVM @@ -112757,29 +112776,29 @@ are used instead of sets because their representation is shorter (and more adantll f1ocnv ad2antrr syl2anc ex f1oeq1 elab coex cnvex ad2antlr ancoms 3imtr4g adantlr anbi12i coass cid cres f1ococnv1 adantrr fcoi1 3syl eqtrd coeq2d eqtr2id f1ococnv2 coeq1d adantrl fcoi2 eqtr3id eqeq12d eqcom f1of1 - bitrdi wf1 ad2antrl cocan1 syl3anc f1ofn ad2antll cocan2 3bitr3d exlimivv - syl5bi en3d sylbir syl2anb ) ABUAUBABFHZIZFJZCDGHZIZGJZACEHZIZEUCZBDYCIZE - UCZUAUBZCDUAUBABFUFCDGUFXSYBKXRYAKZGJFJYHXRYAFGUGYIYHFGYIUDUEYEYGXTUDHZLZ - XQUHZLZXTUHZUEHZXQLZLZYIYECAMUIZNYACNOZANOZYRNOZXRYACXTUJNCDXTUKXTGPZULUM - ZXRAXQUJNABXQUKXQFPZULUMZMNNUNURZYSYTUUAUOCANNMUPUQUSYIYDEYRYDYCYROZYIACY - CQZACYCRYAYSYTUUGUUHSXRUUCUUECAYCNNUTUSVAVBVCYIYGDBMUIZNYADNOZBNOZUUINOZX - RYADXTVDZNYACDXTVEUUMDTCDXTVFCDXTVGVHXTUUBVIUMZXRBXQVDZNXRABXQVEZUUOBTABX - QVFZABXQVGVHXQUUDVIUMZUUFUUJUUKUULUODBNNMUPUQUSYIYFEUUIYFYCUUIOZYIBDYCQZB - DYCRYAUUJUUKUUSUUTSXRUUNUURDBYCNNUTUSVAVBVCYIACYJIZBDYMIZYJYEOZYMYGOYIUVA - UVBYIUVAKADYKIZBAYLIZUVBYAUVAUVDXRACDXTYJVJVKZXRUVEYAUVAABXQVLVMBADYKYLVJ - VNZVOYDUVAEYJUDPZACYCYJVPVQZYFUVBEYMYKYLXTYJUUBUVHVRXQUUDVSVRBDYCYMVPVQWB - YIBDYOIZACYQIZYOYGOZYQYEOYIUVJUVKYIUVJKDCYNIZADYPIZUVKYAUVMXRUVJCDXTVLVTX - RUVJUVNYAUVJXRUVNABDYOXQVJWAWCZADCYNYPVJVNZVOYFUVJEYOUEPZBDYCYOVPVQZYDUVK - EYQYNYPXTUUBVSYOXQUVQUUDVRVRACYCYQVPVQWBUVCUVLKUVAUVJKZYIYJYQTZYOYMTZSZUV - CUVAUVLUVJUVIUVRWDYIUVSUWBYIUVSKZYKXTYQLZTZYPYMXQLZTZUVTUWAUWCUWEUWFYPTUW - GUWCYKUWFUWDYPUWCUWFYKYLXQLZLZYKYKYLXQWEUWCUWIYKWFAWGZLZYKUWCUWHUWJYKXRUW - HUWJTYAUVSABXQWHVMWMUWCUVDADYKQUWKYKTYIUVAUVDUVJUVFWIADYKRADYKWJWKWLWNUWC - UWDXTYNLZYPLZYPXTYNYPWEUWCUWMWFDWGZYPLZYPUWCUWLUWNYPYAUWLUWNTXRUVSCDXTWOV - TWPUWCUVNADYPQUWOYPTYIUVJUVNUVAUVOWQADYPRADYPWRWKWLWSWTUWFYPXAXCUWCCDXTXD - ZACYJQZACYQQZUWEUVTSYAUWPXRUVSCDXTXBVTUVAUWQYIUVJACYJRXEUWCUVKUWRYIUVJUVK - UVAUVPWQACYQRVHACDXTYJYQXFXGUWCUUPYOBURZYMBURZUWGUWASXRUUPYAUVSUUQVMUVJUW - SYIUVABDYOXHXIUWCUVBUWTYIUVAUVBUVJUVGWIBDYMXHVHABXQYOYMXJXGXKVOXMXNXLXOXP - $. + bitrdi wf1 ad2antrl cocan1 syl3anc f1ofn ad2antll cocan2 3bitr3d biimtrid + en3d exlimivv sylbir syl2anb ) ABUAUBABFHZIZFJZCDGHZIZGJZACEHZIZEUCZBDYCI + ZEUCZUAUBZCDUAUBABFUFCDGUFXSYBKXRYAKZGJFJYHXRYAFGUGYIYHFGYIUDUEYEYGXTUDHZ + LZXQUHZLZXTUHZUEHZXQLZLZYIYECAMUIZNYACNOZANOZYRNOZXRYACXTUJNCDXTUKXTGPZUL + UMZXRAXQUJNABXQUKXQFPZULUMZMNNUNURZYSYTUUAUOCANNMUPUQUSYIYDEYRYDYCYROZYIA + CYCQZACYCRYAYSYTUUGUUHSXRUUCUUECAYCNNUTUSVAVBVCYIYGDBMUIZNYADNOZBNOZUUINO + ZXRYADXTVDZNYACDXTVEUUMDTCDXTVFCDXTVGVHXTUUBVIUMZXRBXQVDZNXRABXQVEZUUOBTA + BXQVFZABXQVGVHXQUUDVIUMZUUFUUJUUKUULUODBNNMUPUQUSYIYFEUUIYFYCUUIOZYIBDYCQ + ZBDYCRYAUUJUUKUUSUUTSXRUUNUURDBYCNNUTUSVAVBVCYIACYJIZBDYMIZYJYEOZYMYGOYIU + VAUVBYIUVAKADYKIZBAYLIZUVBYAUVAUVDXRACDXTYJVJVKZXRUVEYAUVAABXQVLVMBADYKYL + VJVNZVOYDUVAEYJUDPZACYCYJVPVQZYFUVBEYMYKYLXTYJUUBUVHVRXQUUDVSVRBDYCYMVPVQ + WBYIBDYOIZACYQIZYOYGOZYQYEOYIUVJUVKYIUVJKDCYNIZADYPIZUVKYAUVMXRUVJCDXTVLV + TXRUVJUVNYAUVJXRUVNABDYOXQVJWAWCZADCYNYPVJVNZVOYFUVJEYOUEPZBDYCYOVPVQZYDU + VKEYQYNYPXTUUBVSYOXQUVQUUDVRVRACYCYQVPVQWBUVCUVLKUVAUVJKZYIYJYQTZYOYMTZSZ + UVCUVAUVLUVJUVIUVRWDYIUVSUWBYIUVSKZYKXTYQLZTZYPYMXQLZTZUVTUWAUWCUWEUWFYPT + UWGUWCYKUWFUWDYPUWCUWFYKYLXQLZLZYKYKYLXQWEUWCUWIYKWFAWGZLZYKUWCUWHUWJYKXR + UWHUWJTYAUVSABXQWHVMWMUWCUVDADYKQUWKYKTYIUVAUVDUVJUVFWIADYKRADYKWJWKWLWNU + WCUWDXTYNLZYPLZYPXTYNYPWEUWCUWMWFDWGZYPLZYPUWCUWLUWNYPYAUWLUWNTXRUVSCDXTW + OVTWPUWCUVNADYPQUWOYPTYIUVJUVNUVAUVOWQADYPRADYPWRWKWLWSWTUWFYPXAXCUWCCDXT + XDZACYJQZACYQQZUWEUVTSYAUWPXRUVSCDXTXBVTUVAUWQYIUVJACYJRXEUWCUVKUWRYIUVJU + VKUVAUVPWQACYQRVHACDXTYJYQXFXGUWCUUPYOBURZYMBURZUWGUWASXRUUPYAUVSUUQVMUVJ + UWSYIUVABDYOXHXIUWCUVBUWTYIUVAUVBUVJUVGWIBDYMXHVHABXQYOYMXJXGXKVOXLXMXNXO + XP $. $} $( Version of ~ isorel for strictly increasing functions on the reals. @@ -112787,7 +112806,7 @@ are used instead of sets because their representation is shorter (and more 9-Sep-2015.) $) leisorel $p |- ( ( F Isom < , < ( A , B ) /\ ( A C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> ( F ` C ) <_ ( F ` D ) ) ) $= - ( clt wiso cxr wss wa wcel wbr wn cfv cle sseldd xrlenlt syl2anc ffvelrnd + ( clt wiso cxr wss wa wcel wbr wn cfv cle sseldd xrlenlt syl2anc ffvelcdmd wb w3a simp1 simp3r simp3l isorel notbid syl12anc simp2l simp2r wf1o isof1o wf f1of 3syl 3bitr4d ) ABFFEGZAHIZBHIZJZCAKZDAKZJZUAZDCFLZMZDENZCENZFLZMZCD OLZVGVFOLZVCUPVAUTVEVITUPUSVBUBZUPUSUTVAUCZUPUSUTVAUDZUPVAUTJJVDVHABDCFFEUE @@ -112833,36 +112852,36 @@ are used instead of sets because their representation is shorter (and more wf isof1o f1of eldifbd elsn2g mtbid wo cv wral eldifad rspcdva wss sseldd breq1 zleloe mpbid ecased cdm wn nn0zd peano2zm zltnle breq2d mtbird nsyl fdm neleqtrrd fsnunfv syl3anc 3brtr4d fveq2d 2thd zfz1isolemsplit eleqtrd - elun sylib mpjaodan ad2antlr lensymd ffvelrnd eqtrd 2falsed eqtr4d breq1d - ltnrd ) ACUAGFUBZUCZUDUEZUFUGZOZCDPQZCEGUDUEZFUHUBUKZUEZDYQUEZPQZUIZCYPUB - ZOZAYNUJZDYMOZUUADUUBOZUUDUUEUJZYOCEUEZDEUEZPQZYTUUGYMYKPPEULZYNUUEYOUUJU - 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V /\ A e. CC ) -> ( F shift A ) e. _V ) $= ( vz vw vu wcel cc wa cshi co cv cmin wbr copab cvv wceq shftfvalg ancoms cnex a1i cab crn rnexg ad2antrr wss breq2 elab simpr simpl subcld brelrng - vex mp3an2 sylan ex syl5bi ssrdv adantll ssexd opabex3d eqeltrd ) BCGZAHG - ZIZBAJKZDLZHGZVGAMKZELZBNZIDEOZPVDVCVFVLQDEABCRSVEVKDEHHPGVETUAVEVHIVKEUB - ZBUCZPVCVNPGVDVHBCUDUEVDVHVMVNUFVCVDVHIZFVMVNFLZVMGVIVPBNZVOVPVNGZVKVQEVP - FUMZVJVPVIBUGUHVOVQVRVOVIHGZVQVRVOVGAVDVHUIVDVHUJUKVTVPPGVQVRVSVIVPBHPULU - NUOUPUQURUSUTVAVB $. + vex mp3an2 sylan ex biimtrid ssrdv adantll ssexd opabex3d eqeltrd ) BCGZA + HGZIZBAJKZDLZHGZVGAMKZELZBNZIDEOZPVDVCVFVLQDEABCRSVEVKDEHHPGVETUAVEVHIVKE + UBZBUCZPVCVNPGVDVHBCUDUEVDVHVMVNUFVCVDVHIZFVMVNFLZVMGVIVPBNZVOVPVNGZVKVQE + VPFUMZVJVPVIBUGUHVOVQVRVOVIHGZVQVRVOVGAVDVHUIVDVHUJUKVTVPPGVQVRVSVIVPBHPU + LUNUOUPUQURUSUTVAVB $. $} ${ @@ -114562,17 +114581,17 @@ a function (ordinarily on a subset of ` CC ` ) and produces a new ( cv cltrr wbr cfv co cr caddc wa cn wcel clt syl2anc wb cmul wceq crio c1 wi wral cdiv cle simplr nnred simpr ltle cuz eluznn ex nnz eluz1 syl syl6bi jcad anim1i syl5ibr impbid adantl biimpar r19.21bi syldan ltxrlt - cz expr syld ad2antrr ffvelrnd nnrecred readdcld cc0 cap caucvgrelemrec - wf nnap0 oveq2d breq2d bitr4d anbi12d 3imtr3d ralrimiva ) ACHZBHZIJZWGD - KZWHDKZWGEHUALUDUBEMUCZNLZIJZWKWJWLNLZIJZOZUEZBPUFCPAWGPQZOZWRBPWTWHPQZ - OZWGWHRJZWJWKUDWGUGLZNLZRJZWKWJXDNLZRJZOZWIWQXBXCWGWHUHJZXIXBWGMQZWHMQZ - XCXJUEXBWGAWSXAUIZUJZXBWHWTXAUKZUJZWGWHULSWTXAXJXIWTXAXJOZWHWGUMKZQZXIW - TXSXQWSXSXQTAWSXSXQWSXSXAXJWSXSXAWHWGUNUOWSXSWHVIQZXJOZXJWSWGVIQXSYATWG - UPWGWHUQURZXTXJUKUSUTXQXSWSYAXAXTXJWHUPVAYBVBVCVDVEWTXIBXRAXIBXRUFCPGVF - VFVGVJVKXBXKXLXCWITXNXPWGWHVHSXBXFWNXHWPXBXFWJXEIJZWNXBWJMQXEMQXFYCTXBP - MWGDAPMDVSWSXAFVLZXMVMZXBWKXDXBPMWHDYDXOVMZXBWGXMVNZVOWJXEVHSXBWMXEWJIX - BWLXDWKNXBXKWGVPVQJZWLXDUBXNXBWSYHXMWGVTURWGEVRSZWAWBWCXBXHWKXGIJZWPXBW - KMQXGMQXHYJTYFXBWJXDYEYGVOWKXGVHSXBWOXGWKIXBWLXDWJNYIWAWBWCWDWEWFWF $. + cz expr syld wf ad2antrr ffvelcdmd nnrecred readdcld cc0 caucvgrelemrec + cap nnap0 oveq2d breq2d bitr4d anbi12d 3imtr3d ralrimiva ) ACHZBHZIJZWG + DKZWHDKZWGEHUALUDUBEMUCZNLZIJZWKWJWLNLZIJZOZUEZBPUFCPAWGPQZOZWRBPWTWHPQ + ZOZWGWHRJZWJWKUDWGUGLZNLZRJZWKWJXDNLZRJZOZWIWQXBXCWGWHUHJZXIXBWGMQZWHMQ + ZXCXJUEXBWGAWSXAUIZUJZXBWHWTXAUKZUJZWGWHULSWTXAXJXIWTXAXJOZWHWGUMKZQZXI + WTXSXQWSXSXQTAWSXSXQWSXSXAXJWSXSXAWHWGUNUOWSXSWHVIQZXJOZXJWSWGVIQXSYATW + GUPWGWHUQURZXTXJUKUSUTXQXSWSYAXAXTXJWHUPVAYBVBVCVDVEWTXIBXRAXIBXRUFCPGV + FVFVGVJVKXBXKXLXCWITXNXPWGWHVHSXBXFWNXHWPXBXFWJXEIJZWNXBWJMQXEMQXFYCTXB + PMWGDAPMDVLWSXAFVMZXMVNZXBWKXDXBPMWHDYDXOVNZXBWGXMVOZVPWJXEVHSXBWMXEWJI + XBWLXDWKNXBXKWGVQVSJZWLXDUBXNXBWSYHXMWGVTURWGEVRSZWAWBWCXBXHWKXGIJZWPXB + WKMQXGMQXHYJTYFXBWJXDYEYGVPWKXGVHSXBWOXGWKIXBWLXDWJNYIWAWBWCWDWEWFWF $. $} $( Convergence of real sequences. @@ -114591,26 +114610,26 @@ A Cauchy sequence (as defined here, which has a rate of convergence bitri rexbii sylibr c1 wss simpr peano2nnd uznnssnn ssralv 3syl cz eluznn cle sylan simplr nnzd eluz1 syl biimpd impancom mpd simprd ad2antlr nnred 1re ltadd1 mp3an3 syl2anc nnleltp1 bitr4d syldan mpbird syl2an adantll wf - 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The modified sequence ` G ` is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.) $) cvg1nlemf $p |- ( ph -> G : NN --> RR ) $= - ( cn cv cmul cr wcel adantr co cfv wa simpr nnmulcld ffvelrnd fmptd + ( cn cv cmul cr wcel adantr co cfv wa simpr nnmulcld ffvelcdmd fmptd wf ) ACOCPZHQUAZFUBRGAUIOSZUCZORUJFAORFUHUKITULUIHAUKUDAHOSUKMTUEUFLU G $. $} @@ -114667,38 +114686,38 @@ A Cauchy sequence (as defined here, which has a rate of convergence ( ( G ` n ) < ( ( G ` k ) + ( 1 / n ) ) /\ ( G ` k ) < ( ( G ` n ) + ( 1 / n ) ) ) ) $= ( cfv co caddc clt wbr cn vb va cv c1 cdiv wa cuz wral wcel cmul wceq - cr simplr wf ad2antrr nnmulcld ffvelrnd fveq2d fvmptg syl2anc eqeltrd - oveq1 eluznn adantll nndivred readdcld nnrecred oveq1d breq2d anbi12d - rpred fveq2 breq1d oveq2 oveq2d breq12d oveq12d raleqbidv bitri sylib - cbvralv ralbii rspcdva cle eluzle adantl nnred nnrpd lemul1d mpbid cz - wb nnzd eluz mpbird simpld ltmul1dd nncnd mulid2d lt2mul2divd mulcomd - 1red 3bitr3d ltadd2dd lttrd simprd jca ralrimiva ) AEUCZGOZDUCZGOZUDX - IUEPZQPZRSZXLXJXMQPZRSZUFZDXIUGOZUHETAXITUIZUFZXRDXSYAXKXSUIZUFZXOXQY - CXJXLBXIHUJPZUEPZQPZXNYCXJYDFOZULYCXTYGULUIXJYGUKAXTYBUMZYCTULYDFATUL - FUNXTYBIUOZYCXIHYHAHTUIXTYBMUOZUPZUQZCXICUCZHUJPZFOZYGTULGYMXIUKYNYDF - YMXIHUJVBURLUSUTZYLVAZYCXLYEYCXLXKHUJPZFOZULYCXKTUIZYSULUIXLYSUKXTYBY - TAXKXIVCVDZYCTULYRFYIYCXKHUUAYJUPZUQZCXKYOYSTULGYMXKUKYNYRFYMXKHUJVBU - RLUSUTZUUCVAZYCBYDABULUIXTYBABJVKUOZYKVEZVFYCXLXMUUEYCXIYHVGZVFYCXJYF - RSZXLXJYEQPZRSZYCUUIUUKUFYGYFRSZXLYGYEQPZRSZUFZYCUUOYGYSYEQPZRSZYSUUM - RSZUFZYCYGUAUCZFOZYEQPZRSZUVAUUMRSZUFZUUSUAYDUGOZYRUUTYRUKZUVCUUQUVDU - URUVGUVBUUPYGRUVGUVAYSYEQUUTYRFVLZVHVIUVGUVAYSUUMRUVHVMVJYCUBUCZFOZUV - ABUVIUEPZQPZRSZUVAUVJUVKQPZRSZUFZUAUVIUGOZUHZUVEUAUVFUHUBTYDUVIYDUKZU - VPUVEUAUVQUVFUVIYDUGVLUVSUVMUVCUVOUVDUVSUVJYGUVLUVBRUVIYDFVLZUVSUVKYE - UVAQUVIYDBUEVNZVOVPUVSUVNUUMUVARUVSUVJYGUVKYEQUVTUWAVQVIVJVRAUVRUBTUH - ZXTYBAXIFOZXKFOZBXIUEPZQPZRSZUWDUWCUWEQPZRSZUFZDXSUHZETUHZUWBKUWLUWCU - VAUWEQPZRSZUVAUWHRSZUFZUAXSUHZETUHUWBUWKUWQETUWJUWPDUAXSXKUUTUKZUWGUW - NUWIUWOUWRUWFUWMUWCRUWRUWDUVAUWEQXKUUTFVLZVHVIUWRUWDUVAUWHRUWSVMVJWAW - BUWQUVREUBTXIUVIUKZUWPUVPUAXSUVQXIUVIUGVLUWTUWNUVMUWOUVOUWTUWCUVJUWMU - VLRXIUVIFVLZUWTUWEUVKUVAQXIUVIBUEVNZVOVPUWTUWHUVNUVARUWTUWCUVJUWEUVKQ - UXAUXBVQVIVJVRWAVSVTUOYKWCYCYRUVFUIZYDYRWDSZYCXIXKWDSZUXDYBUXEYAXIXKW - EWFYCXIXKHYCXIYHWGYCXKUUAWGYCHYJWHWIWJYCYDWKUIYRWKUIUXCUXDWLYCYDYKWMY - CYRUUBWMYDYRWNUTWOWCYCUULUUQUUNUURYCYFUUPYGRYCXLYSYEQUUDVHVIYCXLYSUUM - RUUDVMVJWOYCUUIUULUUKUUNYCXJYGYFRYPVMYCUUJUUMXLRYCXJYGYEQYPVHVIVJWOZW - PYCYEXMXLUUGUUHUUEYCYEXMRSZBXIUJPZHXIUJPZRSZYCBHXIUUFYCHYJWGYCXIYHWHZ - ABHRSXTYBNUOWQYCUXHUDYDUJPZRSUXHYDRSUXGUXJYCUXLYDUXHRYCYDYCYDYKWRWSVI - YCBXIUDYDUUFUXKYCXBYCYDYKWHWTYCYDUXIUXHRYCXIHYCXIYHWRYCHYJWRXAVIXCWOZ - XDXEYCXLUUJXPUUEYCXJYEYQUUGVFYCXJXMYQUUHVFYCUUIUUKUXFXFYCYEXMXJUUGUUH - YQUXMXDXEXGXHXH $. + cr simplr wf ad2antrr nnmulcld ffvelcdmd oveq1 fveq2d syl2anc eqeltrd + fvmptg eluznn adantll nndivred readdcld nnrecred oveq1d breq2d breq1d + rpred fveq2 anbi12d oveq2 oveq2d breq12d oveq12d cbvralv ralbii bitri + raleqbidv sylib rspcdva eluzle adantl nnred nnrpd lemul1d mpbid cz wb + cle nnzd eluz mpbird simpld ltmul1dd mulid2d 1red lt2mul2divd mulcomd + nncnd 3bitr3d ltadd2dd lttrd simprd jca ralrimiva ) AEUCZGOZDUCZGOZUD + XIUEPZQPZRSZXLXJXMQPZRSZUFZDXIUGOZUHETAXITUIZUFZXRDXSYAXKXSUIZUFZXOXQ + YCXJXLBXIHUJPZUEPZQPZXNYCXJYDFOZULYCXTYGULUIXJYGUKAXTYBUMZYCTULYDFATU + LFUNXTYBIUOZYCXIHYHAHTUIXTYBMUOZUPZUQZCXICUCZHUJPZFOZYGTULGYMXIUKYNYD + FYMXIHUJURUSLVBUTZYLVAZYCXLYEYCXLXKHUJPZFOZULYCXKTUIZYSULUIXLYSUKXTYB + YTAXKXIVCVDZYCTULYRFYIYCXKHUUAYJUPZUQZCXKYOYSTULGYMXKUKYNYRFYMXKHUJUR + USLVBUTZUUCVAZYCBYDABULUIXTYBABJVKUOZYKVEZVFYCXLXMUUEYCXIYHVGZVFYCXJY + FRSZXLXJYEQPZRSZYCUUIUUKUFYGYFRSZXLYGYEQPZRSZUFZYCUUOYGYSYEQPZRSZYSUU + MRSZUFZYCYGUAUCZFOZYEQPZRSZUVAUUMRSZUFZUUSUAYDUGOZYRUUTYRUKZUVCUUQUVD + UURUVGUVBUUPYGRUVGUVAYSYEQUUTYRFVLZVHVIUVGUVAYSUUMRUVHVJVMYCUBUCZFOZU + VABUVIUEPZQPZRSZUVAUVJUVKQPZRSZUFZUAUVIUGOZUHZUVEUAUVFUHUBTYDUVIYDUKZ + UVPUVEUAUVQUVFUVIYDUGVLUVSUVMUVCUVOUVDUVSUVJYGUVLUVBRUVIYDFVLZUVSUVKY + EUVAQUVIYDBUEVNZVOVPUVSUVNUUMUVARUVSUVJYGUVKYEQUVTUWAVQVIVMWAAUVRUBTU + HZXTYBAXIFOZXKFOZBXIUEPZQPZRSZUWDUWCUWEQPZRSZUFZDXSUHZETUHZUWBKUWLUWC + UVAUWEQPZRSZUVAUWHRSZUFZUAXSUHZETUHUWBUWKUWQETUWJUWPDUAXSXKUUTUKZUWGU + WNUWIUWOUWRUWFUWMUWCRUWRUWDUVAUWEQXKUUTFVLZVHVIUWRUWDUVAUWHRUWSVJVMVR + VSUWQUVREUBTXIUVIUKZUWPUVPUAXSUVQXIUVIUGVLUWTUWNUVMUWOUVOUWTUWCUVJUWM + UVLRXIUVIFVLZUWTUWEUVKUVAQXIUVIBUEVNZVOVPUWTUWHUVNUVARUWTUWCUVJUWEUVK + QUXAUXBVQVIVMWAVRVTWBUOYKWCYCYRUVFUIZYDYRWLSZYCXIXKWLSZUXDYBUXEYAXIXK + WDWEYCXIXKHYCXIYHWFYCXKUUAWFYCHYJWGWHWIYCYDWJUIYRWJUIUXCUXDWKYCYDYKWM + YCYRUUBWMYDYRWNUTWOWCYCUULUUQUUNUURYCYFUUPYGRYCXLYSYEQUUDVHVIYCXLYSUU + MRUUDVJVMWOYCUUIUULUUKUUNYCXJYGYFRYPVJYCUUJUUMXLRYCXJYGYEQYPVHVIVMWOZ + WPYCYEXMXLUUGUUHUUEYCYEXMRSZBXIUJPZHXIUJPZRSZYCBHXIUUFYCHYJWFYCXIYHWG + ZABHRSXTYBNUOWQYCUXHUDYDUJPZRSUXHYDRSUXGUXJYCUXLYDUXHRYCYDYCYDYKXBWRV + IYCBXIUDYDUUFUXKYCWSYCYDYKWGWTYCYDUXIUXHRYCXIHYCXIYHXBYCHYJXBXAVIXCWO + ZXDXEYCXLUUJXPUUEYCXJYEYQUUGVFYCXJXMYQUUHVFYCUUIUUKUXFXFYCYEXMXJUUGUU + HYQUXMXDXEXGXHXH $. $} ${ @@ -114717,55 +114736,55 @@ A Cauchy sequence (as defined here, which has a rate of convergence weq anbi1i cdiv wceq oveq2 breq2d anbi12d rexralbidv simplr rphalfcld c2 simpr rspcdva sylbir cmul ad4antr 2re remulcld rerpdivcld nndivred rpred a1i simprl readdcld arch syl adantr nnmulcld wf ad6antr simplrl - nnred eluznn sylancom ffvelrnd rehalfcld simpllr oveq1d breq1d oveq2d - ad3antrrr breq12d oveq12d raleqbidv simprd simplrr cvg1nlemcxze ltled - ad2antrr leadd2dd ltletrd cle rpmulcld rpdivcld nnrpd ltaddrp2d lttrd - simprr 2rp cz wb nnzd syl2anc mpbird oveq1 fveq2d fvmptg mpbid simpld - eluz ltadd1dd recnd addassd breqtrd ralrimiva rexlimddv jca rspcev ex - rpcnd 2halvesd sylbi reximdva mpd ) AUAUFZJUGZCUFZUBUFZRSZTUHZUUKUUJU - ULRSZTUHZUIZUAUCUFZUJUGZUKZUCULUOZUBUMUKZCUPUOEUFZIUGZUUKBUFZRSZTUHZU - UKUVDUVERSZTUHZUIZEFUFZUJUGZUKZFULUOZBUMUKZCUPUOAUBCUAUCGHJADFGHIJKLM - NOPQUNADFGHIJKLMNOPQUQURAUVBUVOCUPAUUKUPUSZUIZUVBUVOUVQUVBUIZUVNBUMUV - RUVEUMUSZUIZUVQUUQUAUDUFZUJUGZUKZUDULUOZUBUMUKZUIZUVSUIZUVNUVRUWFUVSU - VBUWEUVQUVAUWDUBUMUUTUWCUCUDULUCUDVEUUQUAUUSUWBUURUWAUJUTVAVBVCVDVFZU - WGUUJUUKUVEVOVGSZRSZTUHZUUKUUJUWIRSZTUHZUIZUAUUSUKZUVNUCULUWGUVTUWOUC - ULUOZUWHUVTUVAUWPUBUMUWIUULUWIVHZUUQUWNUCUAULUUSUWQUUNUWKUUPUWMUWQUUM - UWJUUJTUULUWIUUKRVIVJUWQUUOUWLUUKTUULUWIUUJRVIVJVKVLUVQUVBUVSVMUVTUVE - 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Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.) $) recvguniqlem $p |- ( ph -> F. ) $= - ( clt wbr cfv cmin co c2 caddc readdcld recnd ffvelrnd resubcld rehalfcld - cn cr ltadd1dd addassd 2halvesd oveq2d pncan3d 3eqtrd breqtrd lttrd ltnrd - cdiv pm2.21fal ) ABBLMABEDNZBCOPZQUOPZRPZBGAUQUSAUDUEEDFIUAZAURABCGHUBZUC - ZSGJAUTCUSRPZUSRPZBLAUQVDUSVAACUSHVCSVCKUFAVECUSUSRPZRPCURRPBACUSUSACHTZA - USVCTZVHUGAVFURCRAURAURVBTUHUIACBVGABGTUJUKULUMABGUNUP $. + ( clt wbr cfv cmin co c2 caddc readdcld recnd cdiv cn ffvelcdmd rehalfcld + cr resubcld ltadd1dd addassd 2halvesd oveq2d pncan3d 3eqtrd breqtrd lttrd + ltnrd pm2.21fal ) ABBLMABEDNZBCOPZQUAPZRPZBGAUQUSAUBUEEDFIUCZAURABCGHUFZU + DZSGJAUTCUSRPZUSRPZBLAUQVDUSVAACUSHVCSVCKUGAVECUSUSRPZRPCURRPBACUSUSACHTZ + AUSVCTZVHUHAVFURCRAURAURVBTUIUJACBVGABGTUKULUMUNABGUOUP $. $} ${ @@ -115208,18 +115227,18 @@ A Cauchy sequence (as defined here, which has a rate of convergence ( cfv clt wbr c1 wcel wi cn vw vk caddc co cz nnzd peano2zd wb nnltp1le cle syl2anc mpbid cv fveq2 breq1d imbi2d resqrexlemdec mpdan a1i w3a wa wceq crp wf resqrexlemf ad2antrr simplr2 1red zred cc0 nnred nngt0d 0re - cr ltle mpan sylc addge02d simplr3 letrd elnnz1 sylanbrc ffvelrnd rpred - peano2nnd simpll simpr lttrd ex expcom a2d uzind syl3anc pm2.43i ) AFEN - ZGENZOPZAGQUCUDZUERZFUERWRFUJPZAWQSZAGAGKUFUGZAFLUFAGFOPZWTMAGTRZFTRXCW - TUHKLGFUIUKULAUAUMZENZWPOPZSAWRENZWPOPZSZAUBUMZENZWPOPZSAXKQUCUDZENZWPO - PZSXAUAUBWRFXEWRVBZXGXIAXQXFXHWPOXEWREUNUOUPXEXKVBZXGXMAXRXFXLWPOXEXKEU - NUOUPXEXNVBZXGXPAXSXFXOWPOXEXNEUNUOUPXEFVBZXGWQAXTXFWOWPOXEFEUNUOUPXJWS - AXDXIKABCDEGHIJUQURUSWSXKUERZWRXKUJPZUTZAXMXPAYCXMXPSAYCVAZXMXPYDXMVAZX - OXLWPYEXOYETVCXNEATVCEVDYCXMABCDEHIJVEVFZYEXKYEYAQXKUJPXKTRZWSYAYBAXMVG - ZYEQWRXKYEVHYEWRAWSYCXMXBVFVIYEXKYHVIAQWRUJPZYCXMAVJGUJPZYIAGVNRZVJGOPZ - YJAGKVKZAGKVLVJVNRYKYLYJSVMVJGVOVPVQAQGAVHYMVRULVFWSYAYBAXMVSVTXKWAWBZW - EWCWDYEXLYETVCXKEYFYNWCWDYEWPYETVCGEYFAXDYCXMKVFWCWDYEAYGXOXLOPAYCXMWFY - NABCDEXKHIJUQUKYDXMWGWHWIWJWKWLWMWN $. + cr ltle mpan addge02d simplr3 letrd elnnz1 sylanbrc peano2nnd ffvelcdmd + sylc rpred simpll simpr lttrd ex expcom a2d uzind syl3anc pm2.43i ) AFE + NZGENZOPZAGQUCUDZUERZFUERWRFUJPZAWQSZAGAGKUFUGZAFLUFAGFOPZWTMAGTRZFTRXC + WTUHKLGFUIUKULAUAUMZENZWPOPZSAWRENZWPOPZSZAUBUMZENZWPOPZSAXKQUCUDZENZWP + OPZSXAUAUBWRFXEWRVBZXGXIAXQXFXHWPOXEWREUNUOUPXEXKVBZXGXMAXRXFXLWPOXEXKE + UNUOUPXEXNVBZXGXPAXSXFXOWPOXEXNEUNUOUPXEFVBZXGWQAXTXFWOWPOXEFEUNUOUPXJW + SAXDXIKABCDEGHIJUQURUSWSXKUERZWRXKUJPZUTZAXMXPAYCXMXPSAYCVAZXMXPYDXMVAZ + XOXLWPYEXOYETVCXNEATVCEVDYCXMABCDEHIJVEVFZYEXKYEYAQXKUJPXKTRZWSYAYBAXMV + GZYEQWRXKYEVHYEWRAWSYCXMXBVFVIYEXKYHVIAQWRUJPZYCXMAVJGUJPZYIAGVNRZVJGOP + ZYJAGKVKZAGKVLVJVNRYKYLYJSVMVJGVOVPWDAQGAVHYMVQULVFWSYAYBAXMVRVSXKVTWAZ + WBWCWEYEXLYETVCXKEYFYNWCWEYEWPYETVCGEYFAXDYCXMKVFWCWEYEAYGXOXLOPAYCXMWF + YNABCDEXKHIJUQUKYDXMWGWHWIWJWKWLWMWN $. $} ${ @@ -115322,33 +115341,33 @@ A Cauchy sequence (as defined here, which has a rate of convergence ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) ) $= ( wcel cexp co cmin c1 c4 cdiv cle wbr oveq2d adantr vw vk cn cfv c2 cv caddc wceq fveq2 oveq1d oveq1 breq12d imbi2d cneg cc0 renegcld 0red crp - wi resqrexlemf 1nn ffvelrnd rpred resqcld le0neg2d mpbid leadd2dd recnd - a1i negsubd addid1d 3brtr3d 1m1e0 oveq2i cc 4cn exp0 ax-mp eqtri eqtrid - div1d breqtrrd wa cr peano2nn adantl resubcld ffvelcdmda 4re 4pos elrpd - clt rerpdivcld nnz peano2zm syl rpexpcld resqrexlemcalc2 lediv1d biimpa - wf cz letrd ad2antrr rpcnd rpap0d divdivap1d simpr pncan1 simplr expm1t - cmul nncnd sylancr eqtrd eqtr4d breqtrd ex expcom a2d nnind impcom ) FU - CJAFEUDZUEKLZDMLZNEUDZUEKLZOFNMLZKLZPLZQRZAUAUFZEUDZUEKLZDMLZYGOYLNMLZK - LZPLZQRZUSAYGDMLZYGONNMLZKLZPLZQRZUSAUBUFZEUDZUEKLZDMLZYGOUUENMLZKLZPLZ - QRZUSAUUENUGLZEUDZUEKLZDMLZYGOUUMNMLZKLZPLZQRZUSAYKUSUAUBFYLNUHZYSUUDAU - VAYOYTYRUUCQUVAYNYGDMUVAYMYFUEKYLNEUIUJUJUVAYQUUBYGPUVAYPUUAOKYLNNMUKSS - ULUMYLUUEUHZYSUULAUVBYOUUHYRUUKQUVBYNUUGDMUVBYMUUFUEKYLUUEEUIUJUJUVBYQU - UJYGPUVBYPUUIOKYLUUENMUKSSULUMYLUUMUHZYSUUTAUVCYOUUPYRUUSQUVCYNUUODMUVC - 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TUVMUVJUVKUWDUWBUHVPAUVKUULXJOUUEXKXNXOSXPXQXRXSXTYAYB $. + resqrexlemf 1nn ffvelcdmd rpred resqcld le0neg2d mpbid leadd2dd negsubd + wi a1i recnd addid1d 3brtr3d 1m1e0 oveq2i cc 4cn exp0 ax-mp eqtri div1d + eqtrid breqtrrd wa cr peano2nn adantl resubcld ffvelcdmda 4re clt elrpd + wf 4pos rerpdivcld cz nnz peano2zm syl rpexpcld resqrexlemcalc2 lediv1d + biimpa letrd ad2antrr rpcnd rpap0d divdivap1d simpr nncnd pncan1 simplr + cmul expm1t sylancr eqtrd eqtr4d breqtrd ex expcom a2d nnind impcom ) F + UCJAFEUDZUEKLZDMLZNEUDZUEKLZOFNMLZKLZPLZQRZAUAUFZEUDZUEKLZDMLZYGOYLNMLZ + KLZPLZQRZVHAYGDMLZYGONNMLZKLZPLZQRZVHAUBUFZEUDZUEKLZDMLZYGOUUENMLZKLZPL + ZQRZVHAUUENUGLZEUDZUEKLZDMLZYGOUUMNMLZKLZPLZQRZVHAYKVHUAUBFYLNUHZYSUUDA + UVAYOYTYRUUCQUVAYNYGDMUVAYMYFUEKYLNEUIUJUJUVAYQUUBYGPUVAYPUUAOKYLNNMUKS + SULUMYLUUEUHZYSUULAUVBYOUUHYRUUKQUVBYNUUGDMUVBYMUUFUEKYLUUEEUIUJUJUVBYQ + UUJYGPUVBYPUUIOKYLUUENMUKSSULUMYLUUMUHZYSUUTAUVCYOUUPYRUUSQUVCYNUUODMUV + CYMUUNUEKYLUUMEUIUJUJUVCYQUURYGPUVCYPUUQOKYLUUMNMUKSSULUMYLFUHZYSYKAUVD + YOYEYRYJQUVDYNYDDMUVDYMYCUEKYLFEUIUJUJUVDYQYIYGPUVDYPYHOKYLFNMUKSSULUMA + YTYGUUCQAYGDUNZUGLYGUOUGLYTYGQAUVEUOYGADHUPAUQAYFAYFAUCURNEABCDEGHIUSZN + UCJAUTVIVAVBVCZAUODQRUVEUOQRIADHVDVEVFAYGDAYGUVGVJZADHVJVGAYGUVHVKVLAUU + CYGNPLYGUUBNYGPUUBOUOKLZNUUAUOOKVMVNOVOJZUVINUHVPOVQVRVSVNAYGUVHVTWAWBU + UEUCJZAUULUUTAUVKUULUUTVHAUVKWCZUULUUTUVLUULWCZUUPUUKOPLZUUSQUVMUUPUUHO + PLZUVNUVLUUPWDJUULUVLUUODUVLUUNUVLUUNUVLUCURUUMEAUCUREWLUVKUVFTUVKUUMUC + JAUUEWEWFVAVBVCADWDJUVKHTZWGTUVLUVOWDJUULUVLUUHOUVLUUGDUVLUUFUVLUUFAUCU + RUUEEUVFWHVBVCUVPWGZUVLOOWDJUVLWIVIUOOWJRUVLWMVIWKZWNTUVLUVNWDJUULUVLUU + KOUVLYGUUJAYGWDJUVKUVGTUVLOUUIUVRUVKUUIWOJZAUVKUUEWOJUVSUUEWPUUEWQWRWFW + SZWNZUVRWNTUVLUUPUVOQRUULABCDEUUEGHIWTTUVLUULUVOUVNQRUVLUUHUUKOUVQUWAUV + RXAXBXCUVMUVNYGUUJOXLLZPLUUSUVMYGUUJOAYGVOJUVKUULUVHXDUVMUUJUVLUUJURJUU + LUVTTZXEUVJUVMVPVIUVMUUJUWCXFUVMOUVLOURJUULUVRTXFXGUVMUURUWBYGPUVMUUROU + UEKLZUWBUVLUURUWDUHUULUVLUUQUUEOKUVLUUEVOJUUQUUEUHUVLUUEAUVKXHXIUUEXJWR + STUVMUVJUVKUWDUWBUHVPAUVKUULXKOUUEXMXNXOSXPXQXRXSXTYAYB $. $} ${ @@ -115362,53 +115381,53 @@ A Cauchy sequence (as defined here, which has a rate of convergence resqrexlemnmsq $p |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) ) $= - ( c2 cexp co cmin c1 cn wcel cfv c4 cdiv clt resqrexlemf ffvelrnd rpred - crp resqcld recnd nnncan2d resubcld 1nn a1i cz rpexpcld nnrpd nnzd 1zzd - 2z 4nn zsubcld rpdivcld wbr resqrexlemover mpdan cr difrp syl2anc mpbid - wb ltsubrpd cle resqrexlemcalc3 ltletrd eqbrtrrd ) AGEUAZNOPZDQPZFEUAZN - OPZDQPZQPZVRWAQPREUAZNOPZUBGRQPZOPZUCPZUDAVRWADAVRAVQAVQASUHGEABCDEHIJU - EZKUFUGUIZUJAWAAVTAVTASUHFEWILUFUGUIZUJADIUJUKAWCVSWHAVSWBAVRDWJIULZAWA - DWKIULULWLAWHAWEWGAWDNASUHREWIRSTAUMUNUFNUOTAUTUNUPAUBWFAUBUBSTAVAUNUQA - GRAGKURAUSVBUPVCUGAVSWBWLADWAUDVDZWBUHTZAFSTWMLABCDEFHIJVEVFADVGTWAVGTW - MWNVKIWKDWAVHVIVJVLAGSTVSWHVMVDKABCDEGHIJVNVFVOVP $. + ( c2 cexp co cmin c1 cn wcel cfv c4 clt crp resqrexlemf ffvelcdmd rpred + cdiv resqcld recnd nnncan2d resubcld 1nn a1i cz rpexpcld 4nn nnrpd nnzd + 2z 1zzd zsubcld rpdivcld resqrexlemover mpdan cr wb difrp syl2anc mpbid + wbr ltsubrpd cle resqrexlemcalc3 ltletrd eqbrtrrd ) AGEUAZNOPZDQPZFEUAZ + NOPZDQPZQPZVRWAQPREUAZNOPZUBGRQPZOPZUHPZUCAVRWADAVRAVQAVQASUDGEABCDEHIJ + UEZKUFUGUIZUJAWAAVTAVTASUDFEWILUFUGUIZUJADIUJUKAWCVSWHAVSWBAVRDWJIULZAW + ADWKIULULWLAWHAWEWGAWDNASUDREWIRSTAUMUNUFNUOTAUTUNUPAUBWFAUBUBSTAUQUNUR + AGRAGKUSAVAVBUPVCUGAVSWBWLADWAUCVKZWBUDTZAFSTWMLABCDEFHIJVDVEADVFTWAVFT + WMWNVGIWKDWAVHVIVJVLAGSTVSWHVMVKKABCDEGHIJVNVEVOVP $. $( Lemma for ~ resqrex . The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.) $) resqrexlemnm $p |- ( ph -> ( ( F ` N ) - ( F ` M ) ) < ( ( ( ( F ` 1 ) ^ 2 ) x. 2 ) / ( 2 ^ ( N - 1 ) ) ) ) $= - ( co c1 c2 cexp cdiv cmul wcel cfv cmin c4 clt crp resqrexlemf ffvelrnd - cn rpred resubcld resqcld cc 2cn expm1t sylancr 2nn a1i nnnn0d nnexpcld - wceq nnrpd eqeltrrd remulcld 1nn nnzd rpexpcld 4re 4pos elrpii peano2zm - cz syl rpdivcld caddc cle rpaddcld rpmulcld wa cr adantr resqrexlemdecn - wbr cc0 simpr wb difrp mpbid rpge0d recnd subidd fveq2 oveq2d sylan9req - syl2anc 0re eqlei wo zleloe mpjaodan 1red nnrecred addid1d resqrexlemlo - 0red mpdan rpgt0d lt2addd eqbrtrrd readdcld ltdivmul2d ltled lemulge11d - breqtrd rpcnd mulassd mulcomd subsq eqtr4d oveq1d eqtr3d resqrexlemnmsq - ltmul1dd lelttrd rpap0d div32apd 4d2e2 oveq1i nnm1nn0 expdivapd eqtr3id - 2cnd cn0 recdivapd eqtrd recclapd mul32d mulcld divrecapd ) AGEUAZFEUAZ - UBNZOEUAZPQNZUCGOUBNZQNZRNZPUUDQNZPSNZSNZUUCPSNZUUGRNZUDAUUAYSPQNZYTPQN - ZUBNZUUHSNZUUIAYSYTAYSAUHUEGEABCDEHIJUFZKUGZUIZAYTAUHUEFEUUPLUGZUIZUJZA - UUNUUHAUULUUMAYSUURUKAYTUUTUKUJZAUUHAPGQNZUUHUEAPULTGUHTZUVCUUHUTUMKPGU - NUOZAUVCAPGPUHTAUPUQZAGKURUSZVAZVBZUIZVCAUUFUUHAUUFAUUCUUEAUUBPAUHUEOEU - UPOUHTAVDUQUGAPUVFVEVFZAUCUUDUCUETAUCVGVHVIUQZAGVKTZUUDVKTAGKVEZGVJVLZV - FZVMUIZUVJVCAUUAUUAYSYTVNNZUUHSNZSNZUUOVOAUUAUVSUVAAUVSAUVRUUHAYSYTUUQU - USVPUVIVQUIZAGFUDWBZWCUUAVOWBZGFUTZAUWBVRZUUAUWEYTYSUDWBZUUAUETZUWEBCDE - FGHADVSTUWBIVTAWCDVOWBUWBJVTAUVDUWBKVTAFUHTUWBLVTAUWBWDWAUWEYTVSTZYSVST - ZUWFUWGWEAUWHUWBUUTVTAUWIUWBUURVTYTYSWFWNWGWHAUWDVRWCUUAUTUWCAUWDWCYSYS - UBNUUAAYSAYSUURWIZWJUWDYSYTYSUBGFEWKWLWMWCUUAWOWPVLAGFVOWBZUWBUWDWQZMAU - VMFVKTUWKUWLWEUVNAFLVEGFWRWNWGWSAOUVSAWTZUWAAOUVRUVCSNZUVSUDAOUVCRNZUVR - UDWBOUWNUDWBAUWOWCVNNUWOUVRUDAUWOAUWOAUVCUVGXAZWIXBAUWOWCYSYTUWPAXDUURU - UTAUVDUWOYSUDWBKABCDEGHIJXCXEAYTUUSXFXGXHAOUVRUVCUWMAYSYTUURUUTXIZUVHXJ - WGAUVCUUHUVRSUVEWLXMXKXLAUUAUVRSNZUUHSNUVTUUOAUUAUVRUUHAUUAUVAWIZAUVRUW - QWIZAUUHUVIXNXOAUWRUUNUUHSAUWRUVRUUASNZUUNAUUAUVRUWSUWTXPAYSULTYTULTUUN - UXAUTUWJAYTUUTWIYSYTXQWNXRXSXTXMAUUNUUFUUHUVBUVQUVIABCDEFGHIJKLMYAYBYCA - UUIUUJOUUGRNZSNZUUKAUUIUUCUXBSNZPSNZUXCAUUFUUGSNZPSNUUIUXEAUUFUUGPAUUFU - VQWIAUUGAPUUDAPUVFVAZUVOVFZXNZAYKZXOAUXFUXDPSAUXFUUCUUGUUERNZSNUXDAUUCU - UEUUGAUUCUVKXNZAUUEUVPXNZUXIAUUEUVPYDZYEAUXBUXKUUCSAUXBOUUEUUGRNZRNUXKA - UUGUXOORAUUGUCPRNZUUDQNUXOUXPPUUDQYFYGAUCPUUDAUCUVLXNUXJAPUXGYDAUVDUUDY - LTKGYHVLYIYJWLAUUEUUGUXMUXIUXNAUUGUXHYDZYMYNWLXRXSXTAUUCUXBPUXLAUUGUXIU - XQYOUXJYPYNAUUJUUGAUUCPUXLUXJYQUXIUXQYRXRXM $. + ( co c1 c2 cexp cdiv cmul wcel cfv cmin c4 clt cn resqrexlemf ffvelcdmd + crp resubcld resqcld cc wceq 2cn expm1t sylancr 2nn a1i nnnn0d nnexpcld + rpred nnrpd eqeltrrd remulcld 1nn nnzd rpexpcld 4pos elrpii cz peano2zm + 4re syl rpdivcld caddc cle rpaddcld rpmulcld wbr cc0 wa cr adantr simpr + resqrexlemdecn difrp syl2anc mpbid rpge0d recnd subidd oveq2d sylan9req + fveq2 0re eqlei zleloe mpjaodan 1red nnrecred addid1d 0red resqrexlemlo + wb mpdan rpgt0d lt2addd eqbrtrrd readdcld ltdivmul2d breqtrd lemulge11d + ltled rpcnd mulassd mulcomd subsq eqtr4d oveq1d resqrexlemnmsq ltmul1dd + wo eqtr3d lelttrd 2cnd rpap0d div32apd 4d2e2 oveq1i cn0 nnm1nn0 eqtr3id + expdivapd recdivapd eqtrd recclapd mul32d mulcld divrecapd ) AGEUAZFEUA + ZUBNZOEUAZPQNZUCGOUBNZQNZRNZPUUDQNZPSNZSNZUUCPSNZUUGRNZUDAUUAYSPQNZYTPQ + NZUBNZUUHSNZUUIAYSYTAYSAUEUHGEABCDEHIJUFZKUGZUTZAYTAUEUHFEUUPLUGZUTZUIZ + AUUNUUHAUULUUMAYSUURUJAYTUUTUJUIZAUUHAPGQNZUUHUHAPUKTGUETZUVCUUHULUMKPG + UNUOZAUVCAPGPUETAUPUQZAGKURUSZVAZVBZUTZVCAUUFUUHAUUFAUUCUUEAUUBPAUEUHOE + UUPOUETAVDUQUGAPUVFVEVFZAUCUUDUCUHTAUCVKVGVHUQZAGVITZUUDVITAGKVEZGVJVLZ + VFZVMUTZUVJVCAUUAUUAYSYTVNNZUUHSNZSNZUUOVOAUUAUVSUVAAUVSAUVRUUHAYSYTUUQ + UUSVPUVIVQUTZAGFUDVRZVSUUAVOVRZGFULZAUWBVTZUUAUWEYTYSUDVRZUUAUHTZUWEBCD + EFGHADWATUWBIWBAVSDVOVRUWBJWBAUVDUWBKWBAFUETUWBLWBAUWBWCWDUWEYTWATZYSWA + TZUWFUWGXCAUWHUWBUUTWBAUWIUWBUURWBYTYSWEWFWGWHAUWDVTVSUUAULUWCAUWDVSYSY + SUBNUUAAYSAYSUURWIZWJUWDYSYTYSUBGFEWMWKWLVSUUAWNWOVLAGFVOVRZUWBUWDYAZMA + UVMFVITUWKUWLXCUVNAFLVEGFWPWFWGWQAOUVSAWRZUWAAOUVRUVCSNZUVSUDAOUVCRNZUV + RUDVROUWNUDVRAUWOVSVNNUWOUVRUDAUWOAUWOAUVCUVGWSZWIWTAUWOVSYSYTUWPAXAUUR + UUTAUVDUWOYSUDVRKABCDEGHIJXBXDAYTUUSXEXFXGAOUVRUVCUWMAYSYTUURUUTXHZUVHX + IWGAUVCUUHUVRSUVEWKXJXLXKAUUAUVRSNZUUHSNUVTUUOAUUAUVRUUHAUUAUVAWIZAUVRU + WQWIZAUUHUVIXMXNAUWRUUNUUHSAUWRUVRUUASNZUUNAUUAUVRUWSUWTXOAYSUKTYTUKTUU + NUXAULUWJAYTUUTWIYSYTXPWFXQXRYBXJAUUNUUFUUHUVBUVQUVIABCDEFGHIJKLMXSXTYC + AUUIUUJOUUGRNZSNZUUKAUUIUUCUXBSNZPSNZUXCAUUFUUGSNZPSNUUIUXEAUUFUUGPAUUF + UVQWIAUUGAPUUDAPUVFVAZUVOVFZXMZAYDZXNAUXFUXDPSAUXFUUCUUGUUERNZSNUXDAUUC + UUEUUGAUUCUVKXMZAUUEUVPXMZUXIAUUEUVPYEZYFAUXBUXKUUCSAUXBOUUEUUGRNZRNUXK + AUUGUXOORAUUGUCPRNZUUDQNUXOUXPPUUDQYGYHAUCPUUDAUCUVLXMUXJAPUXGYEAUVDUUD + YITKGYJVLYLYKWKAUUEUUGUXMUXIUXNAUUGUXHYEZYMYNWKXQXRYBAUUCUXBPUXLAUUGUXI + UXQYOUXJYPYNAUUJUUGAUUCPUXLUXJYQUXIUXQYRXQXJ $. $} ${ @@ -115420,30 +115439,30 @@ A Cauchy sequence (as defined here, which has a rate of convergence E. r e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( r + x ) /\ r < ( ( F ` i ) + x ) ) ) $= ( c2 co cn crp a1i wcel clt wbr vk vn c1 cfv cexp cr resqrexlemf rpssre - cmul wss fssd 1nn ffvelrnd cz 2z rpexpcld 2rp rpmulcld cv cdiv caddc wa - cuz wral cmin wf ad2antrr simplr rpred eluznn adantll resubcld rpdivcld - nnzd nnrpd cc0 cle eluzle adantl resqrexlemnm cc wceq 2cn expm1t oveq2d - sylancr rpcnd cn0 nnm1nn0 syl cap 2ap0 1zzd zsubcld expap0d divcanap5rd - expcld eqtrd breqtrrd uzid ax-mp nnnn0d bernneq3 mpbid lttrd ltsubadd2d - ltdiv2d readdcld adantr simpr resqrexlemdecn ltled fveq2 eqcomd eqle wo - syl2an wb zleloe syl2anc mpjaodan ltaddrpd lelttrd jca ralrimiva cvg1n + cmul wss fssd 1nn ffvelcdmd cz 2z rpexpcld 2rp rpmulcld cv caddc wa cuz + cdiv wral cmin wf ad2antrr simplr eluznn adantll resubcld nnzd rpdivcld + rpred nnrpd cc0 cle eluzle adantl resqrexlemnm cc expm1t sylancr oveq2d + wceq 2cn rpcnd cn0 nnm1nn0 syl expcld cap 2ap0 1zzd zsubcld divcanap5rd + expap0d eqtrd breqtrrd uzid ax-mp nnnn0d ltdiv2d mpbid lttrd ltsubadd2d + bernneq3 readdcld adantr simpr resqrexlemdecn ltled fveq2 eqcomd syl2an + eqle wo wb zleloe syl2anc mpjaodan ltaddrpd lelttrd jca ralrimiva cvg1n ) ABIUCHUDZMUENZMUINZMUINZFGUAUBHAOPUFHACDEHJKLUGZPUFUJAUHQUKAYIMAYHMAY GMAOPUCHYKUCORZAULQUMMUNRZAUOQUPMPRZAUQQZURYOURZAUBUSZHUDZUAUSZHUDZYJYQ - UTNZVANSTZYTYRUUAVANZSTZVBZUAYQVCUDZVDUBOAYQORZVBZUUEUAUUFUUHYSUUFRZVBZ - UUBUUDUUJYRYTVENZUUASTUUBUUJUUKYJMYQUENZUTNZUUAUUJYRYTUUJYRUUJOPYQHAOPH - VFUUGUUIYKVGZAUUGUUIVHZUMVIZUUJYTUUJOPYSHUUNUUGUUIYSORZAYSYQVJVKZUMVIZV - LUUJUUMUUJYJUULAYJPRUUGUUIYPVGZUUJMYQYNUUJUQQZUUJYQUUOVNZUPZVMVIUUJUUAU - UJYJYQUUTUUJYQUUOVOZVMZVIZUUJUUKYIMYQUCVENZUENZUTNZUUMSUUJCDEHYSYQJAEUF + VCNZUTNSTZYTYRUUAUTNZSTZVAZUAYQVBUDZVDUBOAYQORZVAZUUEUAUUFUUHYSUUFRZVAZ + UUBUUDUUJYRYTVENZUUASTUUBUUJUUKYJMYQUENZVCNZUUAUUJYRYTUUJYRUUJOPYQHAOPH + VFUUGUUIYKVGZAUUGUUIVHZUMVNZUUJYTUUJOPYSHUUNUUGUUIYSORZAYSYQVIVJZUMVNZV + KUUJUUMUUJYJUULAYJPRUUGUUIYPVGZUUJMYQYNUUJUQQZUUJYQUUOVLZUPZVMVNUUJUUAU + UJYJYQUUTUUJYQUUOVOZVMZVNZUUJUUKYIMYQUCVENZUENZVCNZUUMSUUJCDEHYSYQJAEUF RZUUGUUIKVGZAVPEVQTZUUGUUILVGZUUOUURUUIYQYSVQTZUUHYQYSVRVSZVTUUJUUMYJUV - HMUINZUTNUVIUUJUULUVPYJUTUUJMWARZUUGUULUVPWBWCUUOMYQWDWFWEUUJYIUVHMUUJY - IUUJYHMUUJYGMUUJOPUCHUUNYLUUJULQUMYMUUJUOQUPUVAURWGUUJMUVGUVQUUJWCQZUUJ - UUGUVGWHRUUOYQWIWJWQUVRUUJMUVGUVRMVPWKTUUJWLQZUUJYQUCUVBUUJWMWNWOUVSWPW - 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wrex wceq oveq2 breq2d anbi12d rexralbidv adantr resqrexlemf ffvelrnd - cr wb rpred difrp syl2anc biimpa rspcdva fveq2 raleqdv cbvrexv breq1d - sylib oveq1d simprr simprl nnzd ad2antrr simpr eluz2 syl3anbrc simprd - cz rpcnd recnd pncan3d breqtrd ad3antrrr pm2.21fal cc0 resqrexlemdecn - ltnrd uzid syl addsubassd breqtrrd readdcld ltaddsub2d mpbird ltadd1d - cc wf ltnsymd wo zlelttric mpjaodan rexlimddv inegd lenltd ) AJIHUBZU - CQXKJRQZUDAXLAXLUEZFUFZHUBZJJXKUGUHZSUHZRQZJXOXPSUHZRQZUEZFUAUFZUIUBZ - UJZUKUAULXMYAFGUFZUIUBZUJZGULUNZYDUAULUNXMXOJEUFZSUHZRQZJXOYISUHZRQZU - EZFYFUJGULUNZYHEUMXPYIXPUOZYNYAGFULYFYPYKXRYMXTYPYJXQXORYIXPJSUPUQYPY - LXSJRYIXPXOSUPUQURUSAYOEUMUJXLOUTAXLXPUMTZAXKVCTZJVCTZXLYQVDAXKAULUMI - HABCDHKLMVAZPVBZVEZNXKJVFVGVHVIYGYDGUAULYEYBUOYAFYFYCYEYBUIVJVKVLVNXM - YBULTZYDUEZUEZYBIUCQZUKIYBRQZUUEUUFUEZJJRQUUHJXKXPSUHZJRUUHXKXQRQZJUU - IRQZUUHYAUUJUUKUEFYCIXNIUOZXRUUJXTUUKUULXOXKXQRXNIHVJZVMUULXSUUIJRUUL - XOXKXPSUUMVOUQURUUEYDUUFXMUUCYDVPZUTUUHYBWDTZIWDTZUUFIYCTUUEUUOUUFUUE - 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RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( G ` k ) < ( A + e ) /\ A < ( ( G ` k ) + e ) ) ) $= ( wbr wcel cv cfv caddc co clt wa cuz wral cn wrex crp c1 cexp cdiv - c2 cr wf resqrexlemf adantr 1nn ffvelrnd cz rpexpcld simpr rpdivcld - a1i 2z rpred 1red readdcld arch wceq simpllr eluznn syl2anc simplll - syl fveq2 oveq1d fvmptg cmin resubcld ad3antrrr 4re 4pos elrpii cn0 - c4 nnm1nn0 nn0zd rerpdivcld cle resqrexlemcalc3 nnred simplr eluzle - reexpcld adantl ltletrd ltaddsubd mpbid 4z 2re 2lt4 ltleii mpbir3an - bernneq3 sylancr ltdiv23d lelttrd ltsubadd2d eqbrtrd resqrexlemover - eluz2 lttrd eqeltrd breqtrrd ltaddrpd jca ralrimiva ex reximdva mpd - ) AIUAZKUBZEFUAZUCUDZUESZEYEYFUCUDZUESZUFZIHUAZUGUBZUHZHUIUJZFUKAYF - UKTZUFZULJUBZUOUMUDZYFUNUDZULUCUDZYLUESZHUIUJZYOYQUUAUPTZUUCYQYTULY - QYTYQYSYFYQYRUOYQUIUKULJAUIUKJUQZYPACDEJMNOURZUSULUITYQUTVFVAUOVBTZ - YQVGVFVCZAYPVDZVEVHZYQVIVJZUUAHVKVQYQUUBYNHUIYQYLUITZUFZUUBYNUUMUUB - UFZYKIYMUUNYDYMTZUFZYHYJUUPYEYDJUBZUOUMUDZYGUEUUPYDUITZUURUKTYEUURV - LUUPUULUUOUUSYQUULUUBUUOVMZUUNUUOVDYDYLVNVOZUUPUUQUOUUPUIUKYDJUUPAU - 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reximdva syld ralrimdva syl5bi raleqbidv ad2antlr oveq2d anim2i w3a ralbi - 3expia sylc bitrid sylibd impbid ) AEIUBZKUCZHUBZKUCZGUDZLUCZFUBZUEUFZUGZ - IYKUHUCZUIZHMUJZFUKUIZEYJJUBZKUCZGUDZLUCZYOUEUFZJYIUHUCZUIZUGIYRUIZHMUJZF - UKUIZUUAEYNUAUBZUEUFZUGZIYRUIHMUJZUAUKUIZAUUKYTUUOFUAUKFUAULZYQUUNHIMYRUU - QYPUUMEYOUULYNUEUPUMUNUOAUUPUUJFUKAYOUKUQZUGZUUPEYNYOURUSUDZUEUFZUGZIYRUI - ZHMUJZUUJUURUUPUVDUTZAUURUUTUKUQUVEYOVAUUOUVDUAUUTUKUULUUTVBZUUNUVBHIMYRU - VFUUMUVAEUULUUTYNUEUPUMUNVCVDVEUUSUVCUUIHMUUSYKMUQZUGZEIYRUIZUVAIYRUIZUGU - VIUUHIYRUIZUGZUVCUUIUVHUVIUVJUVKUVIUVHBIYRUIZUVJUVKUTEBIYROVFZUVHUVMUGZUV - JCDUUCYLGUDZLUCZUUTUEUFZUGZJYRUIZUGZUVKUVOUVJUVTCUVHUVMUVJUVTUVMUVJUGZBUV - AUGZIYRUIZUVHUVTBUVAIYRVGZUWDUVTUTUVHUWDUVTUWCUVSIJYRIJULZBDUVAUVRUWFYJUU - CVBBDVIYIUUBKVJZQVDUWFYNUVQUUTUEUWFYMUVPLUWFYJUUCYLGUWGVHVKVLVMUOVNVOVPVQ - UVGUVMCUUSUVGYKYRUQZUVMCUVGYKVRUQUWHMVRYKNVSYKWAVDZBCIYKYRIHULZYJYLVBBCVI - YIYKKVJZPVDVTZWBWCWLUVHUVMUVJUWAUVKUTZUWBUWDUVHUWMUWEUUSUWDUWMUTUVGUUSUWA - UWDUVKUUSUWAUWDUVKUTUUSUWAUGZUWCUUHIYRUWNYIYRUQZUGBUVAUUHUWNBUWOUVAUUHUTU - 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elpr oveq1d breqtrd syl5bi ralrimiv prid1g ad4antr rspcev sylancom prid2g - jaod ad4antlr ad2antrr simplr maxabslemlub mpjaodan ex ralrimiva 3jca ) C - EFZDEFZGZCDHIZCDJIZKLZHIZMNIZEFWJAOZPQZRZACDUBZSWKWJPQZWKBOZPQZBWNUAZUCZA - ESWEWIWEWFWHCDUDWEWGWECDWECWCWDUEZUFZWEDWCWDUGZUFZUHUIUJUKZWEWMAWNWKWNFWK - CULZWKDULZUMWEWMWKCDAUNVDWEXEWMXFWEWMXEWJCPQZRWECWJWTXDCDUOUPXEWLXGWKCWJP - TUQURWEWMXFWJDPQZRWEDWJXBXDWEDDCHIZDCJIKLZHIZMNIZWJUSWDWCDXLUSQDCUOUTWEXK - WIMNWEXIWFXJWHHWEDCXCXAVAWEDCXCXAVBVCVEVFUPXFWLXHWKDWJPTUQURVNVGVHWEWSAEW - EWKEFZGZWOWRXNWOGZWKCPQZWRWKDPQZXOXPCWNFZWRWCXRWDXMWOXPCDEVIVJWQXPBCWNWPC - WKPTVKVLXOXQDWNFZWRWDXSWCXMWOXQCDEVMVOWQXQBDWNWPDWKPTVKVLXOCDWKWEWCXMWOWT - VPWEWDXMWOXBVPWEXMWOVQXNWOUGVRVSVTWAWB $. + elpr oveq1d breqtrd jaod biimtrid ralrimiv prid1g ad4antr rspcev sylancom + prid2g ad4antlr ad2antrr simplr maxabslemlub mpjaodan ex ralrimiva 3jca ) + CEFZDEFZGZCDHIZCDJIZKLZHIZMNIZEFWJAOZPQZRZACDUBZSWKWJPQZWKBOZPQZBWNUAZUCZ + AESWEWIWEWFWHCDUDWEWGWECDWECWCWDUEZUFZWEDWCWDUGZUFZUHUIUJUKZWEWMAWNWKWNFW + KCULZWKDULZUMWEWMWKCDAUNVDWEXEWMXFWEWMXEWJCPQZRWECWJWTXDCDUOUPXEWLXGWKCWJ + PTUQURWEWMXFWJDPQZRWEDWJXBXDWEDDCHIZDCJIKLZHIZMNIZWJUSWDWCDXLUSQDCUOUTWEX + KWIMNWEXIWFXJWHHWEDCXCXAVAWEDCXCXAVBVCVEVFUPXFWLXHWKDWJPTUQURVGVHVIWEWSAE + WEWKEFZGZWOWRXNWOGZWKCPQZWRWKDPQZXOXPCWNFZWRWCXRWDXMWOXPCDEVJVKWQXPBCWNWP + CWKPTVLVMXOXQDWNFZWRWDXSWCXMWOXQCDEVNVOWQXQBDWNWPDWKPTVLVMXOCDWKWEWCXMWOW + TVPWEWDXMWOXBVPWEXMWOVQXNWOUGVRVSVTWAWB $. $} ${ @@ -117479,15 +117498,15 @@ A Cauchy sequence (as defined here, which has a rate of convergence ( vx vy cr cv cle wbr wa wi wral wrex wceq breq1 imbi1d ralbidv wcel csup wss simpl imim2i ralimi reximi simpr jca cbvrexv anbi12i reeanv cpr maxcl bitr4i clt adantl r19.26 anim12 wb simplrl simplrr sselda syl3anc syl5ibr - maxleastb ralimdva syl5bir rspcev syl6an rexlimdvva syl5bi impbid2 ) CHUB - ZDIZEIZJKZABLZMZECNZDHOZVPAMZECNZDHOZVPBMZECNZDHOZLZVTWCWFVSWBDHVRWAECVQA - VPABUCUDUEUFVSWEDHVRWDECVQBVPABUGUDUEUFUHWGFIZVOJKZAMZECNZGIZVOJKZBMZECNZ - LZGHOFHOZVMVTWGWKFHOZWOGHOZLWQWCWRWFWSWBWKDFHVNWHPZWAWJECWTVPWIAVNWHVOJQR - SUIWEWODGHVNWLPZWDWNECXAVPWMBVNWLVOJQRSUIUJWKWOFGHHUKUNVMWPVTFGHHVMWHHTZW - LHTZLZLZWHWLULHUOUAZHTZWPXFVOJKZVQMZECNZVTXDXGVMWHWLUMUPWPWJWNLZECNXEXJWJ - WNECUQXEXKXIECXKXIXEVOCTZLZWIWMLZVQMWIAWMBURXMXHXNVQXMXBXCVOHTXHXNUSVMXBX - CXLUTVMXBXCXLVAXECHVOVMXDUCVBWHWLVOVEVCRVDVFVGVSXJDXFHVNXFPZVRXIECXOVPXHV - QVNXFVOJQRSVHVIVJVKVL $. + maxleastb ralimdva syl5bir rspcev syl6an rexlimdvva biimtrid impbid2 ) CH + UBZDIZEIZJKZABLZMZECNZDHOZVPAMZECNZDHOZVPBMZECNZDHOZLZVTWCWFVSWBDHVRWAECV + QAVPABUCUDUEUFVSWEDHVRWDECVQBVPABUGUDUEUFUHWGFIZVOJKZAMZECNZGIZVOJKZBMZEC + NZLZGHOFHOZVMVTWGWKFHOZWOGHOZLWQWCWRWFWSWBWKDFHVNWHPZWAWJECWTVPWIAVNWHVOJ + QRSUIWEWODGHVNWLPZWDWNECXAVPWMBVNWLVOJQRSUIUJWKWOFGHHUKUNVMWPVTFGHHVMWHHT + ZWLHTZLZLZWHWLULHUOUAZHTZWPXFVOJKZVQMZECNZVTXDXGVMWHWLUMUPWPWJWNLZECNXEXJ + WJWNECUQXEXKXIECXKXIXEVOCTZLZWIWMLZVQMWIAWMBURXMXHXNVQXMXBXCVOHTXHXNUSVMX + BXCXLUTVMXBXCXLVAXECHVOVMXDUCVBWHWLVOVEVCRVDVFVGVSXJDXFHVNXFPZVRXIECXOVPX + HVQVNXFVOJQRSVHVIVJVKVL $. $} ${ @@ -118006,16 +118025,16 @@ A Cauchy sequence (as defined here, which has a rate of convergence ( cxr wcel wa cv clt wbr wn cpr wral cpnf wceq cif cle breq2 wrex wi cmnf cr csup xrmaxiflemcl eqeltrid vex elpr xrmaxifle breqtrrdi xrlenlt syldan wo wb mpbid notbid syl5ibrcom ancoms xrmaxiflemcom breqtrrd simpr syl2anc - eqtrid jaod syl5bi ralrimiv prid1g ad4antr rspcev sylancom prid2g simplll - ad4antlr simpllr simplr breq2i biimpi xrmaxiflemlub mpjaodan ex ralrimiva - adantl 3jca ) CGHZDGHZIZEGHZEAJZKLZMZACDNZOWIEKLZWIBJZKLZBWLUAZUBZAGOWGED - PQZPDUCQZCCPQZPCUCQZDWLUDKUERRRRZGFCDUFUGZWGWKAWLWIWLHWICQZWIDQZUNWGWKWIC - DAUHUIWGXDWKXEWGWKXDECKLZMZWGCESLZXGWGCXBESCDUJFUKWEWFWHXHXGUOXCCEULUMUPX - DWJXFWICEKTUQURWGWKXEEDKLZMZWGDESLZXJWGDWTPXADWRPWSCDCNUDKUERRRRZESWFWEDX - LSLDCUJUSWGEXBXLFCDUTVDVAWGWFWHXKXJUOWEWFVBXCDEULVCUPXEWJXIWIDEKTUQURVEVF - VGWGWQAGWGWIGHZIZWMWPXNWMIZWICKLZWPWIDKLZXOXPCWLHZWPWEXRWFXMWMXPCDGVHVIWO - XPBCWLWNCWIKTVJVKXOXQDWLHZWPWFXSWEXMWMXQCDGVLVNWOXQBDWLWNDWIKTVJVKXOCDWIW - EWFXMWMVMWEWFXMWMVOWGXMWMVPWMWIXBKLZXNWMXTEXBWIKFVQVRWCVSVTWAWBWD $. + eqtrid biimtrid ralrimiv prid1g ad4antr sylancom ad4antlr simplll simpllr + jaod rspcev prid2g simplr breq2i biimpi adantl xrmaxiflemlub ex ralrimiva + mpjaodan 3jca ) CGHZDGHZIZEGHZEAJZKLZMZACDNZOWIEKLZWIBJZKLZBWLUAZUBZAGOWG + EDPQZPDUCQZCCPQZPCUCQZDWLUDKUERRRRZGFCDUFUGZWGWKAWLWIWLHWICQZWIDQZUNWGWKW + ICDAUHUIWGXDWKXEWGWKXDECKLZMZWGCESLZXGWGCXBESCDUJFUKWEWFWHXHXGUOXCCEULUMU + PXDWJXFWICEKTUQURWGWKXEEDKLZMZWGDESLZXJWGDWTPXADWRPWSCDCNUDKUERRRRZESWFWE + DXLSLDCUJUSWGEXBXLFCDUTVDVAWGWFWHXKXJUOWEWFVBXCDEULVCUPXEWJXIWIDEKTUQURVM + VEVFWGWQAGWGWIGHZIZWMWPXNWMIZWICKLZWPWIDKLZXOXPCWLHZWPWEXRWFXMWMXPCDGVGVH + WOXPBCWLWNCWIKTVNVIXOXQDWLHZWPWFXSWEXMWMXQCDGVOVJWOXQBDWLWNDWIKTVNVIXOCDW + IWEWFXMWMVKWEWFXMWMVLWGXMWMVPWMWIXBKLZXNWMXTEXBWIKFVQVRVSVTWCWAWBWD $. $} ${ @@ -118721,8 +118740,8 @@ A Cauchy sequence (as defined here, which has a rate of convergence ${ $d x y z $. - $( The limit relation is function-like, and with range the complex numbers. - (Contributed by Mario Carneiro, 31-Jan-2014.) $) + $( The limit relation is function-like, and with codomian the complex + numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) $) fclim $p |- ~~> : dom ~~> --> CC $= ( vx vy vz cli cdm cc wf wfn crn wss wfun wrel cv wbr wa weq wal mpbir2an ax-gen wcel climrel climuni dffun2 funfn mpbi wex vex elrn climcl exlimiv @@ -118811,17 +118830,17 @@ A Cauchy sequence (as defined here, which has a rate of convergence ( wbr cfv wcel wa vy cli cv cmin co cabs clt cuz wral wrex crp c2 cdiv cc rphalfcl wceq breq2 rexralbidv rspccva syl2an adantr eqidd climi rexanuz2 cz adantl sylanbrc wi uztrn2 simprr ad2ant2r abssubd breq1d anbi1d climcl - an12 cr syl ad2antrr rpre ad2antlr abs3lem syl22anc sylbid anassrs syl5bi - expimpd sylan2 ralimdva reximdva mpd ralrimiva clim2c mpbird ) AGCUBQEUCZ - GRZCUDUEUFRUAUCZUGQZEDUCZUHRZUIZDJUJZUAUKUIAXBUAUKAWQUKSZTZWOFRZWPUDUEUFR - ZWQULUMUEZUGQZXEUNSZXECUDUEUFRXGUGQZTZTZEWTUIZDJUJZXBXDXHEWTUIDJUJZXKEWTU - IDJUJXNAXFBUCZUGQZEWTUIDJUJZBUKUIXGUKSZXOXCOWQUOZXRXOBXGUKXPXGUPXQXHDEJWT - XPXGXFUGUQURUSUTXDCXEXGDEFHJKAHVESXCLVAXCXSAXTVFXDWOJSZTZXEVBAFCUBQZXCPVA - VCXHXKDEHJKVDVGXDXMXADJXDWSJSZTXLWREWTXDYDWOWTSZXLWRVHZYDYETXDYAYFHWOWSJK - VIXLXIXHXJTZTYBWRXHXIXJVPYBXIYGWRXDYAXIYGWRVHXDYAXITZTZYGWPXEUDUEUFRZXGUG - QZXJTZWRYIXHYKXJYIXFYJXGUGYIXEWPXDYAXIVJZAYAWPUNSZXCXINVKZVLVMVNYIYNCUNSZ - XIWQVQSZYLWRVHYOAYPXCYHAYCYPPCFVOVRZVSYMXCYQAYHWQVTWAWPCXEWQWBWCWDWEWGWFW - HWEWIWJWKWLAUACWPDEGHIJKLMAYATWPVBYRNWMWN $. + an12 syl ad2antrr rpre ad2antlr abs3lem syl22anc anassrs expimpd biimtrid + cr sylbid sylan2 ralimdva reximdva mpd ralrimiva clim2c mpbird ) AGCUBQEU + CZGRZCUDUEUFRUAUCZUGQZEDUCZUHRZUIZDJUJZUAUKUIAXBUAUKAWQUKSZTZWOFRZWPUDUEU + FRZWQULUMUEZUGQZXEUNSZXECUDUEUFRXGUGQZTZTZEWTUIZDJUJZXBXDXHEWTUIDJUJZXKEW + TUIDJUJXNAXFBUCZUGQZEWTUIDJUJZBUKUIXGUKSZXOXCOWQUOZXRXOBXGUKXPXGUPXQXHDEJ + WTXPXGXFUGUQURUSUTXDCXEXGDEFHJKAHVESXCLVAXCXSAXTVFXDWOJSZTZXEVBAFCUBQZXCP + VAVCXHXKDEHJKVDVGXDXMXADJXDWSJSZTXLWREWTXDYDWOWTSZXLWRVHZYDYETXDYAYFHWOWS + JKVIXLXIXHXJTZTYBWRXHXIXJVPYBXIYGWRXDYAXIYGWRVHXDYAXITZTZYGWPXEUDUEUFRZXG + UGQZXJTZWRYIXHYKXJYIXFYJXGUGYIXEWPXDYAXIVJZAYAWPUNSZXCXINVKZVLVMVNYIYNCUN + SZXIWQWFSZYLWRVHYOAYPXCYHAYCYPPCFVOVQZVRYMXCYQAYHWQVSVTWPCXEWQWAWBWGWCWDW + EWHWCWIWJWKWLAUACWPDEGHIJKLMAYATWPVBYRNWMWN $. $} ${ @@ -119501,16 +119520,16 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $= clim2ser $p |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> ( A - ( seq M ( + , F ) ` N ) ) ) $= ( caddc cfv co wcel syl cc cv adantr cmin vj vx vy cseq c1 cvv cuz cz - eqid eleqtrdi peano2uz eluzelz eluzel2 serf ffvelrnd a1i wa eleqtrrdi - seqex uztrn2 sylan addcl adantl w3a wceq addass simpr eleq2i sylan2br + eqid eleqtrdi peano2uz eluzelz eluzel2 serf ffvelcdmd seqex a1i wa wf + eleqtrrdi uztrn2 sylan addcl adantl wceq addass simpr eleq2i sylan2br adantlr seq3split oveq1d syldan ffvelcdmda pncan2d eqtr2d climsubc1 - wf ) ABFLDEUDZMZUAVSLDFUELNZUDZWAUFWAUGMZWCUIZAWAEUGMZOZWAUHOAFWEOZWF - AFGWEIHUJZEFUKPZEWAULPZKAGQFVSACDEGHAWGEUHOWHEFUMPJUNZIUOZWBUFOALDWAU - SUPAUARZWCOZUQZGQWMVSAGQVSVRWNWKSAWAGOZWNWMGOAWAWEGWIHURZEWMWAGHUTVAU - OWOWMVSMZVTTNVTWMWBMZLNZVTTNWSWOWRWTVTTWOCUBUCLQDEFWMCRZQOZUBRZQOZUQX - AXCLNZQOWOXAXCVBVCXBXDUCRZQOVDXEXFLNXAXCXFLNLNVEWOXAXCXFVFVCAWNVGAWGW - NWHSAXAWEOZXADMQOZWNXGAXAGOZXHGWEXAHVHJVIVJVKVLWOVTWSAVTQOWNWLSAWCQWM - WBACDWAWCWDWJAXAWCOZXIXHAWPXJXIWQEXAWAGHUTVAJVMUNVNVOVPVQ $. + w3a ) ABFLDEUDZMZUAVSLDFUELNZUDZWAUFWAUGMZWCUIZAWAEUGMZOZWAUHOAFWEOZW + FAFGWEIHUJZEFUKPZEWAULPZKAGQFVSACDEGHAWGEUHOWHEFUMPJUNZIUOZWBUFOALDWA + UPUQAUARZWCOZURZGQWMVSAGQVSUSWNWKSAWAGOZWNWMGOAWAWEGWIHUTZEWMWAGHVAVB + UOWOWMVSMZVTTNVTWMWBMZLNZVTTNWSWOWRWTVTTWOCUBUCLQDEFWMCRZQOZUBRZQOZUR + XAXCLNZQOWOXAXCVCVDXBXDUCRZQOVRXEXFLNXAXCXFLNLNVEWOXAXCXFVFVDAWNVGAWG + WNWHSAXAWEOZXADMQOZWNXGAXAGOZXHGWEXAHVHJVIVJVKVLWOVTWSAVTQOWNWLSAWCQW + MWBACDWAWCWDWJAXAWCOZXIXHAWPXJXIWQEXAWAGHVAVBJVMUNVNVOVPVQ $. $} ${ @@ -119521,15 +119540,15 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $= clim2ser2 $p |- ( ph -> seq M ( + , F ) ~~> ( A + ( seq M ( + , F ) ` N ) ) ) $= ( vj vx caddc cfv co wcel syl cc cv vy cseq cvv cuz eleqtrdi peano2uz - c1 eqid cz eluzelz eluzel2 serf ffvelrnd seqex eleqtrrdi uztrn2 sylan - a1i syldan ffvelcdmda wa adantr addcl adantl wceq addass simpr eleq2i - w3a sylan2br adantlr seq3split comraddd climaddc1 ) ABFNDEUBZOZLNDFUG - NPZUBZVOVQUCVQUDOZVSUHZAVQEUDOZQZVQUIQAFWAQZWBAFGWAIHUEZEFUFRZEVQUJRZ - KAGSFVOACDEGHAWCEUIQWDEFUKRJULIUMZVOUCQANDEUNURAVSSLTZVRACDVQVSVTWFAC - TZVSQZWIGQZWIDOSQZAVQGQWJWKAVQWAGWEHUOEWIVQGHUPUQJUSULUTZAWHVSQZVAZWH - VOOVPWHVROAVPSQWNWGVBWMWOCMUANSDEFWHWISQZMTZSQZVAWIWQNPZSQWOWIWQVCVDW - PWRUATZSQVIWSWTNPWIWQWTNPNPVEWOWIWQWTVFVDAWNVGAWCWNWDVBAWIWAQZWLWNXAA - WKWLGWAWIHVHJVJVKVLVMVN $. + c1 eqid cz eluzelz eluzel2 ffvelcdmd seqex a1i eleqtrrdi uztrn2 sylan + serf syldan ffvelcdmda wa adantr addcl adantl w3a addass simpr eleq2i + wceq sylan2br adantlr seq3split comraddd climaddc1 ) ABFNDEUBZOZLNDFU + GNPZUBZVOVQUCVQUDOZVSUHZAVQEUDOZQZVQUIQAFWAQZWBAFGWAIHUEZEFUFRZEVQUJR + ZKAGSFVOACDEGHAWCEUIQWDEFUKRJURIULZVOUCQANDEUMUNAVSSLTZVRACDVQVSVTWFA + CTZVSQZWIGQZWIDOSQZAVQGQWJWKAVQWAGWEHUOEWIVQGHUPUQJUSURUTZAWHVSQZVAZW + HVOOVPWHVROAVPSQWNWGVBWMWOCMUANSDEFWHWISQZMTZSQZVAWIWQNPZSQWOWIWQVCVD + WPWRUATZSQVEWSWTNPWIWQWTNPNPVIWOWIWQWTVFVDAWNVGAWCWNWDVBAWIWAQZWLWNXA + AWKWLGWAWIHVHJVJVKVLVMVN $. $} $} @@ -119731,22 +119750,22 @@ Cauchy sequence (for example, fixing a rate of convergence as in (Contributed by Jim Kingdon, 23-Aug-2021.) $) climrecvg1n $p |- ( ph -> F e. dom ~~> ) $= ( vj ve vi cv cfv co clt wbr wa cn wcel cr vy caddc cuz wral wrex crp cli - cdm cdiv cmin cabs r19.21bi ad2antrr eluznn adantll ffvelrnd simplr nnrpd - wf rpdivcld rpred absdifltd mpbid ltsubaddd anbi1d ralrimiva cvg1n adantr - ad3antrrr simpr simpllr bitrd ancom bitrdi ralbidva rexbidva c1 nnuz 1zzd - cvv nnex reex fex2 syl3anc eqidd recnd ffvelcdmda clim2c climrel releldmi - a1i syl6bir sylbird impr rexlimddv ) AILZEMZUALZJLZUBNOPZWRWQWSUBNOPZQZIK - LZUCMZUDZKRUEZJUFUDZEUGUHSZUATAJUABIKCDEFGADLZEMZCLZEMZBXIUINZUBNOPZXLXJX - MUBNOPZQZCXIUCMZUDDRAXIRSZQZXPCXQXSXKXQSZQZXJXMUJNXLOPZXOQZXPYAXLXJUJNUKM - XMOPZYCXSYDCXQAYDCXQUDDRHULULYAXLXJXMYARTXKEARTEUSZXRXTFUMZXRXTXKRSAXKXIU - NUOUPZYARTXIEYFAXRXTUQZUPZYAXMYABXIABUFSXRXTGUMYAXIYHURUTVAZVBVCYAYBXNXOY - AXJXMXLYIYJYGVDVEVCVFVFVGAWRTSZXGXHAYKQZXGWQWRUJNUKMWSOPZIXDUDZKRUEZJUFUD - ZXHYLYOXFJUFYLWSUFSZQZYNXEKRYRXCRSZQZYMXBIXDYTWPXDSZQZYMXAWTQZXBUUBYMWRWS - UJNWQOPZWTQUUCUUBWQWRWSUUBRTWPEYLYEYQYSUUAAYEYKFVHZVIYSUUAWPRSZYRWPXCUNUO - UPZYLYKYQYSUUAAYKVJZVIZUUBWSYLYQYSUUAVKVAZVBUUBUUDXAWTUUBWRWSWQUUIUUJUUGV - DVEVLXAWTVMVNVOVPVOYLYPEWRUGPXHYLJWRWQKIEVQVTRVRYLVSYLYERVTSZTVTSZEVTSUUE - UUKYLWAWKUULYLWBWKRTEVTVTWCWDYLUUFQZWQWEYLWRUUHWFUUMWQYLRTWPEUUEWGWFWHEWR - UGWIWJWLWMWNWO $. + cdiv cmin cabs r19.21bi wf ad2antrr eluznn adantll ffvelcdmd simplr nnrpd + rpdivcld rpred absdifltd mpbid ltsubaddd anbi1d ralrimiva cvg1n ad3antrrr + cdm adantr simpr simpllr bitrd ancom bitrdi ralbidva rexbidva c1 cvv nnuz + 1zzd nnex a1i reex syl3anc eqidd recnd ffvelcdmda clim2c climrel releldmi + fex2 syl6bir sylbird impr rexlimddv ) AILZEMZUALZJLZUBNOPZWRWQWSUBNOPZQZI + KLZUCMZUDZKRUEZJUFUDZEUGVHSZUATAJUABIKCDEFGADLZEMZCLZEMZBXIUHNZUBNOPZXLXJ + XMUBNOPZQZCXIUCMZUDDRAXIRSZQZXPCXQXSXKXQSZQZXJXMUINXLOPZXOQZXPYAXLXJUINUJ + MXMOPZYCXSYDCXQAYDCXQUDDRHUKUKYAXLXJXMYARTXKEARTEULZXRXTFUMZXRXTXKRSAXKXI + UNUOUPZYARTXIEYFAXRXTUQZUPZYAXMYABXIABUFSXRXTGUMYAXIYHURUSUTZVAVBYAYBXNXO + YAXJXMXLYIYJYGVCVDVBVEVEVFAWRTSZXGXHAYKQZXGWQWRUINUJMWSOPZIXDUDZKRUEZJUFU + DZXHYLYOXFJUFYLWSUFSZQZYNXEKRYRXCRSZQZYMXBIXDYTWPXDSZQZYMXAWTQZXBUUBYMWRW + SUINWQOPZWTQUUCUUBWQWRWSUUBRTWPEYLYEYQYSUUAAYEYKFVIZVGYSUUAWPRSZYRWPXCUNU + OUPZYLYKYQYSUUAAYKVJZVGZUUBWSYLYQYSUUAVKUTZVAUUBUUDXAWTUUBWRWSWQUUIUUJUUG + VCVDVLXAWTVMVNVOVPVOYLYPEWRUGPXHYLJWRWQKIEVQVRRVSYLVTYLYERVRSZTVRSZEVRSUU + EUUKYLWAWBUULYLWCWBRTEVRVRWKWDYLUUFQZWQWEYLWRUUHWFUUMWQYLRTWPEUUEWGWFWHEW + RUGWIWJWLWMWNWO $. $} ${ @@ -119768,37 +119787,37 @@ Cauchy sequence (for example, fixing a rate of convergence as in climcvg1nlem $p |- ( ph -> F e. dom ~~> ) $= ( cfv wcel cn cr cc cli ci cmul co caddc wbr cdm c1 cvv nnuz 1zzd cv wa cre ffvelcdmda recld fmptd cmin cabs cdiv clt cuz wral cle wceq adantll - eluznn wf ad2antrr ffvelrnd fveq2d fvmptg syl2anc simplr oveq12d resubd - fveq2 eqtr4d subcld absrele syl eqbrtrd wi recnd eqeltrd nnrpd rpdivcld - abscld crp rpred lelttr syl3anc mpand ralimdva climrecvg1n climdm sylib - mpd nnex fex sylancl cim imcld imsubd absimle resubcld ax-icn a1i mptex - cmpt eqeltri ax-resscn fssd oveq2d adantl simpr mulcld fvmptd climmulc2 - wss replimd eqtrd climadd climrel releldmi ) AFGUAPZUBHUAPZUCUDZUEUDZUA - UFFUAUGZQAYFYHDGIFUHUIRUJAUKZAGYJQGYFUAUFACDEGABRBULZFPZUNPZSGAYLRQUMZY - MARTYLFJUOZUPMUQZKADULZFPZEULZFPZURUDZUSPZCYTUTUDZVAUFZDYTVBPZVCZERVCZY - RGPZYTGPZURUDZUSPZUUDVAUFZDUUFVCZERVCLAUUGUUNERAYTRQZUMZUUEUUMDUUFUUPYR - UUFQZUMZUULUUCVDUFZUUEUUMUURUULUUBUNPZUSPZUUCVDUURUUKUUTUSUURUUKYSUNPZU - UAUNPZURUDUUTUURUUIUVBUUJUVCURUURYRRQZUVBSQZUUIUVBVEZUUOUUQUVDAYRYTVGVF - ZUURYSUURRTYRFARTFVHZUUOUUQJVIZUVGVJZUPBYRYNUVBRSGYLYRVEZYMYSUNYLYRFVQZ - VKMVLZVMUURUUOUVCSQUUJUVCVEAUUOUUQVNZUURUUAUURRTYTFUVIUVNVJZUPBYTYNUVCR - SGYLYTVEZYMUUAUNYLYTFVQZVKMVLVMVOUURYSUUAUVJUVOVPVRZVKUURUUBTQZUVAUUCVD - UFUURYSUUAUVJUVOVSZUUBVTWAWBUURUULSQUUCSQZUUDSQZUUSUUEUMUUMWCUURUUKUURU - UKUUTTUVRUURUUTUURUUBUVTUPWDWEWHUURUUBUVTWHZUURUUDUURCYTACWIQUUOUUQKVIU - URYTUVNWFWGWJZUULUUCUUDWKWLWMWNWNWRWOGWPWQAUVHRUIQFUIQJWSRTUIFWTXAAYGUB - DHIUHUIRUJYKAHYJQHYGUAUFACDEHABRYMXBPZSHYOYMYPXCNUQZKAUUHYRHPZYTHPZURUD - ZUSPZUUDVAUFZDUUFVCZERVCLAUUGUWLERUUPUUEUWKDUUFUURUWJUUCVDUFZUUEUWKUURU - WJUUBXBPZUSPZUUCVDUURUWIUWNUSUURUWIYSXBPZUUAXBPZURUDUWNUURUWGUWPUWHUWQU - RUURUVDUWPSQZUWGUWPVEZUVGUURYSUVJXCZBYRUWEUWPRSHUVKYMYSXBUVLVKNVLZVMZUU - RUUOUWQSQUWHUWQVEUVNUURUUAUVOXCZBYTUWEUWQRSHUVPYMUUAXBUVQVKNVLVMZVOUURY - SUUAUVJUVOXDVRVKUURUVSUWOUUCVDUFUVTUUBXEWAWBUURUWJSQUWAUWBUWMUUEUMUWKWC - UURUWIUURUWIUURUWGUWHUURUWGUWPSUXBUWTWEUURUWHUWQSUXDUXCWEXFWDWHUWCUWDUW - JUUCUUDWKWLWMWNWNWRWOHWPWQUBTQZAXGXHIUIQAIBRUBYLHPZUCUDZXJZUIOBRUXGWSXI - XKXHARTYRHARSTHUWFSTXTAXLXHXMUOZAUVDUMZBYRUXGUBUWGUCUDZRITIUXHVEUXJOXHU - VKUXGUXKVEUXJUVKUXFUWGUBUCYLYRHVQXNXOAUVDXPZUXJUBUWGUXEUXJXGXHUXIXQZXRZ - XSUXJUUIARSYRGYQUOWDUXJYRIPZUXKTUXNUXMWEUXJYSUVBUBUWPUCUDZUEUDUUIUXOUEU - 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syl wf uztrn2 syl2an2r ffvelrnd abssubd zcnd ax-1cn sylancl eluzp1p1 eqid - eleq1d ralrimiva ad3antrrr eleqtrrdi rspcdva seq3m1 oveq1d adantlr syldan - npcan addcl pncan2d eqtr2d 3eqtr4d sylibd ralrimdva raleqdv rspcev syl6an - syl5 rexlimdva ralimdv mpd eqidd clim0c mpbird ) ACUEUFUGBUKZCMZNMZLUKZUH - UGZBUAUKZULMZOZUAFUIZLUMOZAUBUKZPCDUNZMZQRZYOYCYNMZSTZNMZYFUHUGZBYMULMZOZ - UOZUBUCUKZULMZOZUCFUIZLUMOZYLAYPYOUUDYNMSTNMYFUHUGUOUBUUEOUCFUILUMOZUUHAD - UPRZYNUFVDRUUIHJLUCUBYNDFGUQURLUCUBBYNDFGUSUTAUUGYKLUMAUUFYKUCFAUUDFRZUOZ - UUDVAPTZFRZUUFYGBUUMULMZOZYKUUKUUNADUUDFGVBVCZUUFYOYMVAPTZYNMZSTZNMZYFUHU - GZUBUUEOZUULUUPUUCUVBUBUUEYMUUERZUUBUVBYPUVDYMUPRYMUUARUURUUARUUBUVBVEUUD - YMVFYMVGYMYMVHYTUVBBUURUUAYCUURVIZYSUVAYFUHUVEYRUUTNUVEYQUUSYOSYCUURYNVJV - KVLVMVPVNVOVQUULUVCYGBUUOUULYCUUORZUOZUVCYCVASTZYNMZUVHVAPTZYNMZSTZNMZYFU - HUGZYGUVGUVHUUERZUVCUVNVEUULUUDUPRZUVFUVOUULUUDDULMZRZUVPUULUUDFUVQAUUKVR - ZGVSZDUUDVFWFUUDYCVTWAZUVBUVNUBUVHUUEYMUVHVIZUVAUVMYFUHUWBUUTUVLNUWBYOUVI - UUSUVKSYMUVHYNVJYMUVHVAYNPWBWCVLVMVPWFUVGUVMYEYFUHUVGUVIYQSTZNMYQUVISTZNM - 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ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) wdc cmpt cseq cli wbr w3a wrex c1 cfz co cle wex wo cio csu simpr simp-4l wf1o nfcsb1v nfeq csbeq1a eqeq12d rspc sylc simpllr simplrl simplrr sumdc ifeq1dadc mpteq2dva seqeq3d breq1d pm5.32da df-3an rexbidva f1of ad3antlr - 3bitr4g wf simplr simp-4r nnzd fznn syl mpbir2and ffvelrnd syl2anc fveq1d - wb zdcle eqeq2d exbidv orbi12d iotabidv df-sumdc 3eqtr4g ) BCJZDAUAZAEKZU - BLZUCZFKAMUDFXCUAZNGOGKZAMZDXFBPZQRZUEZXBUFZHKZUGUHZUIZEOUJZUKXBULUMZAIKZ - VAZXLXBNGSXFXBUNUHZDXFXQLZBPZQRZUEZUKUFZLZJZTZIUOZESUJZUPZHUQXDXENGOXGDXF - CPZQRZUEZXBUFZXLUGUHZUIZEOUJZXRXLXBNGSXSDXTCPZQRZUEZUKUFZLZJZTZIUOZESUJZU - PZHUQABDURACDURXAYJUUGHXAXOYQYIUUFXAXNYPEOXAXBOMZTZXDXETZXMTUUJYOTXNYPUUI - UUJXMYOUUIUUJTZXKYNXLUGUUKXJYMNXBUUKGOXIYLUUKXFOMZTZXGXHYKQUUMXGTXGXAXHYK - JZUUMXGUSXAUUHUUJUULXGUTWTUUNDXFADXHYKDXFBVBDXFCVBVCDKZXFJBXHCYKDXFBVDDXF - CVDVEVFVGUUMFAXBXFXAUUHUUJUULVHUUIXDXEUULVIUUIXDXEUULVJUUKUULUSVKVLVMVNVO - VPXDXEXMVQXDXEYOVQWAVRXAYHUUEESXAXBSMZTZYGUUDIUUQXRYFUUCUUQXRTZYEUUBXLUUR - XBYDUUAUURYCYTNUKUURGSYBYSUURXFSMZTZXSYAYRQUUTXSTZXTAMXAYAYRJZUVAXPAXFXQX - RXPAXQWBUUQUUSXSXPAXQVSVTUVAXFXPMZUUSXSUURUUSXSWCUUTXSUSUVAUUHUVCUUSXSTWL - UVAXBXAUUPXRUUSXSWDWEXFXBWFWGWHWIXAUUPXRUUSXSUTWTUVBDXTADYAYRDXTBVBDXTCVB - VCUUOXTJBYACYRDXTBVDDXTCVDVEVFVGUUTUULUUHXSUDUUTXFUURUUSUSWEUUTXBXAUUPXRU - USVHWEXFXBWMWJVLVMVNWKWNVPWOVRWPWQHABIFDEGWRHACIFDEGWRWS $. + 3bitr4g wf simplr simp-4r nnzd fznn syl mpbir2and ffvelcdmd zdcle syl2anc + wb fveq1d eqeq2d exbidv orbi12d iotabidv df-sumdc 3eqtr4g ) BCJZDAUAZAEKZ + UBLZUCZFKAMUDFXCUAZNGOGKZAMZDXFBPZQRZUEZXBUFZHKZUGUHZUIZEOUJZUKXBULUMZAIK + ZVAZXLXBNGSXFXBUNUHZDXFXQLZBPZQRZUEZUKUFZLZJZTZIUOZESUJZUPZHUQXDXENGOXGDX + FCPZQRZUEZXBUFZXLUGUHZUIZEOUJZXRXLXBNGSXSDXTCPZQRZUEZUKUFZLZJZTZIUOZESUJZ + UPZHUQABDURACDURXAYJUUGHXAXOYQYIUUFXAXNYPEOXAXBOMZTZXDXETZXMTUUJYOTXNYPUU + IUUJXMYOUUIUUJTZXKYNXLUGUUKXJYMNXBUUKGOXIYLUUKXFOMZTZXGXHYKQUUMXGTXGXAXHY + KJZUUMXGUSXAUUHUUJUULXGUTWTUUNDXFADXHYKDXFBVBDXFCVBVCDKZXFJBXHCYKDXFBVDDX + FCVDVEVFVGUUMFAXBXFXAUUHUUJUULVHUUIXDXEUULVIUUIXDXEUULVJUUKUULUSVKVLVMVNV + OVPXDXEXMVQXDXEYOVQWAVRXAYHUUEESXAXBSMZTZYGUUDIUUQXRYFUUCUUQXRTZYEUUBXLUU + RXBYDUUAUURYCYTNUKUURGSYBYSUURXFSMZTZXSYAYRQUUTXSTZXTAMXAYAYRJZUVAXPAXFXQ + XRXPAXQWBUUQUUSXSXPAXQVSVTUVAXFXPMZUUSXSUURUUSXSWCUUTXSUSUVAUUHUVCUUSXSTW + LUVAXBXAUUPXRUUSXSWDWEXFXBWFWGWHWIXAUUPXRUUSXSUTWTUVBDXTADYAYRDXTBVBDXTCV + BVCUUOXTJBYACYRDXTBVDDXTCVDVEVFVGUUTUULUUHXSUDUUTXFUURUUSUSWEUUTXBXAUUPXR + UUSVHWEXFXBWJWKVLVMVNWMWNVPWOVRWPWQHABIFDEGWRHACIFDEGWRWS $. $} ${ @@ -120239,20 +120258,20 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> ( seq M ( + , F ) ` N ) ) $= ( caddc cfv wcel cz cc wa cc0 wceq vn vm vz cseq cvv eqid eluzelz seqex cuz syl a1i eluzel2 cv cif adantl wn wi iftrue eqeltrd iffalse eqeltrdi - ex adantr 0cn wdc wo exmiddc mpjaod fvmpt2 syl2anc serf ffvelrnd addid1 - co simpr c1 cfz elfzuz cdif sseld fznuz con2d imp eldifd fveqeq2 eldifi - syl6 eldifn eqtrd vtoclga sylan2 adantlr eleq1d wral ralrimiva ad2antrr - fveq2 rspcdva addcl seq3id2 eqcomd climconst ) AGMEFUDZNZUAXCGUEGUINZXE - UFAGFUINZOZGPOKFGUGUJXCUEOAMEFUHUKAXFQGXCADEFXFXFUFAXGFPOKFGULUJADUMZXF - OZRZXHENZXHBOZCSUNZQXJXHPOZXMQOZXKXMTZXIXNAFXHUGUOXJXLXOXLUPZAXLXOUQXIA - XLXOAXLRXMCQXLXMCTAXLCSURUOIUSVBVCXQXOUQXJXQXMSQXLCSUTZVDVAUKXJXLVEXLXQ - VFJXLVGUJVHZDPXMQEHVIZVJXSUSZVKKVLZAUAUMZXEOZRZXDYCXCNYEUBUCMQEGFYCSUBU - MZQOZYFSMVNYFTYEYFVMUOAXGYDKVCAYDVOAXDQOYDYBVCAYFGVPMVNZYCVQVNOZYFENZST - ZYDYIAYFYHUINOZYKYFYHYCVRAYLRZYFPBVSZOYKYMYFPBYLYFPOAYHYFUGUOAYLYFBOZUP - AYOYLAYOYFFGVQVNZOYLUPABYPYFLVTYFFGWAWGWBWCWDXKSTYKDYFYNXHYFSEWEXHYNOZX - KXMSYQXNXOXPXHPBWFYQXMSQYQXQXMSTXHPBWHXRUJZVDVAXTVJYRWIWJUJWKWLYEYFXFOZ - RXKQOZYJQODXFYFXHYFTXKYJQXHYFEWQWMAYTDXFWNYDYSAYTDXFYAWOWPYEYSVOWRYGUCU - MZQORYFUUAMVNQOYEYFUUAWSUOWTXAXB $. + ex adantr 0cn wo exmiddc mpjaod fvmpt2 syl2anc serf ffvelcdmd co addid1 + wdc simpr c1 cfz elfzuz cdif sseld fznuz syl6 imp eldifd fveqeq2 eldifi + con2d eldifn eqtrd vtoclga sylan2 adantlr fveq2 wral ralrimiva ad2antrr + eleq1d rspcdva addcl seq3id2 eqcomd climconst ) AGMEFUDZNZUAXCGUEGUINZX + EUFAGFUINZOZGPOKFGUGUJXCUEOAMEFUHUKAXFQGXCADEFXFXFUFAXGFPOKFGULUJADUMZX + FOZRZXHENZXHBOZCSUNZQXJXHPOZXMQOZXKXMTZXIXNAFXHUGUOXJXLXOXLUPZAXLXOUQXI + AXLXOAXLRXMCQXLXMCTAXLCSURUOIUSVBVCXQXOUQXJXQXMSQXLCSUTZVDVAUKXJXLVNXLX + QVEJXLVFUJVGZDPXMQEHVHZVIXSUSZVJKVKZAUAUMZXEOZRZXDYCXCNYEUBUCMQEGFYCSUB + UMZQOZYFSMVLYFTYEYFVMUOAXGYDKVCAYDVOAXDQOYDYBVCAYFGVPMVLZYCVQVLOZYFENZS + TZYDYIAYFYHUINOZYKYFYHYCVRAYLRZYFPBVSZOYKYMYFPBYLYFPOAYHYFUGUOAYLYFBOZU + PAYOYLAYOYFFGVQVLZOYLUPABYPYFLVTYFFGWAWBWGWCWDXKSTYKDYFYNXHYFSEWEXHYNOZ + XKXMSYQXNXOXPXHPBWFYQXMSQYQXQXMSTXHPBWHXRUJZVDVAXTVIYRWIWJUJWKWLYEYFXFO + ZRXKQOZYJQODXFYFXHYFTXKYJQXHYFEWMWQAYTDXFWNYDYSAYTDXFYAWOWPYEYSVOWRYGUC + UMZQORYFUUAMVLQOYEYFUUAWSUOWTXAXB $. $} ${ @@ -120358,68 +120377,68 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> ( seq 1 ( + , G ) ` N ) ) $= ( vm vx caddc cseq cfv c1 cli cuz cfz co wf1o wf clt wiso chash wceq wb cv 1zzd nnzd fzfigd fihasheqf1od cn wcel cn0 hashfz1 3syl eqtr3d oveq2d - nnnn0 isoeq4 syl isof1o f1of nnuz ffvelrnd sseldd wa cle wbr ffvelcdmda - mpbid sylan elfzle2 cxr wss adantr cr uzssz zssre ressxr sstrdi eluzelz - cz sstri syl2anc mpbird sylanbrc cc cc0 adantl 0cnd cif eqeltrd adantlr - wn 0cn eqeltrdi wo exmiddc mpjaodan simpr sylc ad2antrr ad2antlr breq2d - wdc csb jca iftrued nfv nfcsb1v nfcv nfif nfel1 csbeq1a ifbieq1d eleq1d - eleq1 eqeltrrd weq breq1 fveq2 csbeq1d fvmptg eqtrd wi cmpt sselda ccnv - eleqtrdi eluzfz2 f1ocnvfv2 f1ocnv fzssuz a1i leisorel syl122anc elfzuzb - eqbrtrrd eluz ssrdv fsum3cvg addid2 addid1 addcl eleqtrrd iftrue simpll - ex iffalse ssneld eluzdc ffvelcdm syl2an elnnuz biimpri w3a 3jca eluzle - fmptd elfz2 wral eleq2d sselid ralrimiva rspc zdcle cdif fveqeq2 eldifi - ifcldadc elfzelz eldifn fvmpt2 vtoclga simpl nfim 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ZZ |-> if ( n e. 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ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) cn co wex neneqd pm2.21d caddc cle wbr cmpt ccom cif cseq adantr sseldd wral wss ralrimiva sumfct fveq2 simprl simprr wf fmpttd ffvelcdmda f1of syl fvco3 sylan fsum3 eqtr3d cuz nnuz eleqtrdi elnnuz biimpri ad3antrrr - adantl fco syl2anc cz w3a 1zzd nnzd simpr 3jca nnge1d sylanbrc ffvelrnd - jca elfz2 wn 0cnd zdcle ifcldadc syldan breq1 ifbieq1d eqid fvmptg fssd + adantl fco syl2anc cz w3a 1zzd nnzd simpr nnge1d jca sylanbrc ffvelcdmd + 3jca elfz2 wn 0cnd wdc zdcle ifcldadc syldan breq1 ifbieq1d eqid fvmptg eluzelz ad2antlr eqeltrd elfzle2 iftrued elfznn anim2i wb eleq1d mpbird - wdc adantlr addcl seq3clss expr exlimdv expimpd cfn wo fz1f1o mpjaod ) + fssd adantlr addcl seq3clss expr exlimdv expimpd cfn wo fz1f1o mpjaod ) ADUDUEZDEGUFZFMZDUGNZUJMZOYNUHUKZDUAPZUIZUAULZQZAYKYMADUDLUMUNAYOYSYMAY OQYRYMUAAYOYRYMAYOYRQZQZYLYNUOUBUJUBPZYNUPUQZUUCGDEURZYQUSZNZRUTZURZOVA NZFUUBDUCPZUUENZUCUFZYLUUJAUUMYLUEZUUAAESMZGDVDUUNAUUOGDAGPDMZQFSEAFSVE @@ -121056,13 +121075,13 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) UBUVOQZUVLUVMRSUVTUVLQZFSUVMAUUQUUAUVOUVLHWEUWAYPFUVHUUFUWAUVCUVEYPFUUF VKUUBUVCUVOUVLUVDTUUBUVEUVOUVLUVFTYPDFUUEYQWGWHUWAOWIMZYNWIMZUVHWIMZWJZ OUVHUPUQZUVLQZUVHYPMZUWAUWBUWCUWDUWAWKUWAYNUUBYOUVOUVLUUTTWLUWAUVHUVTUV - OUVLUUBUVOWMZVBZWLWNUWAUWFUVLUWAUVHUWJWOUVTUVLWMWRUVHOYNWSZWPWQZVCUVTUV - LWTZQXAUVTUWDUWCUVLXTZUVTUVHUWIWLUVTYNUUBYOUVOUUTVBWLUVHYNXBWHZXCZXDUBU - VHUUHUVNUJSUUIUUCUVHUEUUDUVLUUGUVMRUUCUVHYNUPXEUUCUVHUUFVHXFUUIXGXHZWHU + OUVLUUBUVOWMZVBZWLWRUWAUWFUVLUWAUVHUWJWNUVTUVLWMWOUVHOYNWSZWPWQZVCUVTUV + LWTZQXAUVTUWDUWCUVLXBZUVTUVHUWIWLUVTYNUUBYOUVOUUTVBWLUVHYNXCWHZXDZXEUBU + VHUUHUVNUJSUUIUUCUVHUEUUDUVLUUGUVMRUUCUVHYNUPXFUUCUVHUUFVHXGUUIXHXIZWHU VJUVLUVMRSUVJUVLQZYPSUVHUUFUWRDSUUEVKZUVEYPSUUFVKUUBUWSUVIUVLUUBDFSUUEU - VDAUUQUUAHVBZXITUUBUVEUVIUVLUVFTYPDSUUEYQWGWHUWRUWEUWGUWHUWRUWBUWCUWDUW - RWKUWRYNUUBYOUVIUVLUUTTWLUVIUWDUUBUVLOUVHXJXKWNUWRUWFUVLUVIUWFUUBUVLUVI - UVHUVRWOXKUVJUVLWMWRUWKWPWQUVJUWMQXAUUBUVIUVOUWNUVSUWOXDXCXLUUBUWHQZUVK + VDAUUQUUAHVBZXTTUUBUVEUVIUVLUVFTYPDSUUEYQWGWHUWRUWEUWGUWHUWRUWBUWCUWDUW + RWKUWRYNUUBYOUVIUVLUUTTWLUVIUWDUUBUVLOUVHXJXKWRUWRUWFUVLUVIUWFUUBUVLUVI + UVHUVRWNXKUVJUVLWMWOUWKWPWQUVJUWMQXAUUBUVIUVOUWNUVSUWOXEXDXLUUBUWHQZUVK FMZUVNFMZUXAUVNUVMFUXAUVLUVMRUWHUVLUUBUVHOYNXMWFZXNUXAUVTUVLUVMFMUWHUVO UUBUVHYNXOXPZUXDUWLWHXLUXAUVTUXBUXCXQUXEUVTUVKUVNFUVTUVOUVPUVQUWIUWPUWQ WHXRVOXSAUVHFMCPZFMQUVHUXFUOUKZFMUUAIYAUWTUVHSMUXFSMQUXGSMUUBUVHUXFYBWF @@ -121158,11 +121177,11 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) cfz cv wex 00id sum0 oveq12i 3eqtr4ri sumeq1 oveq12d 3eqtr4a a1i cmpt cle wa wi wbr ccom cif cseq cuz simprl nnuz eleqtrdi eqid weq ifbieq1d elnnuz breq1 biimpri adantl wf adantlr fmpttd simprr syl fco syl2anc ad2antrr cz - f1of 1zzd nnzd eluzelz ad2antlr 3jca eluzle simpr elfz2 sylanbrc ffvelrnd - w3a jca wn 0cnd wdc adantr zdcle ifcldadc fvmptd3 eqeltrd wral ffvelcdmda - simpll addcld fvmpt2 ralrimiva nffvmpt1 nfcv nfov nfeq eqeq12d rspc fvco3 - eqtr4d sylc 3eqtr4d iftrued eqcomi iffalsed wo exmiddc mpjaodan ad3antrrr - sylan ser3add fsum3 cbvmptv seqeq3 ax-mp fveq1i eqtrdi sumfct 3eqtr3d + f1of w3a 1zzd nnzd eluzelz ad2antlr 3jca eluzle simpr jca elfz2 ffvelcdmd + sylanbrc 0cnd wdc adantr zdcle ifcldadc fvmptd3 eqeltrd simpll ffvelcdmda + wn wral addcld fvmpt2 ralrimiva nffvmpt1 nfcv nfov nfeq eqeq12d rspc sylc + eqtr4d fvco3 sylan 3eqtr4d iftrued iffalsed wo exmiddc mpjaodan ad3antrrr + eqcomi ser3add fsum3 cbvmptv seqeq3 ax-mp fveq1i eqtrdi sumfct 3eqtr3d expr exlimdv expimpd cfn fz1f1o mpjaod ) ABUCKZBCDLMZEUDZBCEUDZBDEUDZLMZK ZBUENZOPZQUUMUGMZBUAUHZUFZUAUIZUTZUUFUULVAAUUFUCUUGEUDZUCCEUDZUCDEUDZLMZU UHUUKRRLMZRUVCUUTUJUVARUVBRLCEUKDEUKULUUGEUKUMBUCUUGEUNUUFUUIUVAUUJUVBLBU @@ -121174,37 +121193,37 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) UWIVJIJVKZUVSUXDUWGUXERUVRUWTUUMUSVNZUVRUWTUWFTVLUXAUWTOPZUVFUXIUXAUWTVMV OVPZUXBUXDUXERSUXBUXDUTZUUOSUWTUWFUVFUUOSUWFVQZUXAUXDUVFBSUVKVQUUOBUUPVQZ UXLUVFEBCSAEUHZBPZCSPZUVEGVRZVSZUVFUUQUXMAUUNUUQVTZUUOBUUPWFWAZUUOBSUVKUU - PWBWCWDUXKQWEPZUUMWEPZUWTWEPZWQQUWTUSVBZUXDUTUWTUUOPZUXKUYAUYBUYCUXKWGUXK - UUMUVFUUNUXAUXDUWSWDWHUXAUYCUVFUXDQUWTWIWJWKUXKUYDUXDUXAUYDUVFUXDQUWTWLWJ - UXBUXDWMZWRUWTQUUMWNWOZWPUXBUXDWSZUTZWTZUXBUYCUYBUXDXAZUXBUWTUXJWHUXBUUMU - VFUUNUXAUWSXBWHUWTUUMXCWCZXDZXEZUYMXFUXBUWTUWONZUXDUWTUWLNZRVDZSUXBIUWTUW + PWBWCWDUXKQWEPZUUMWEPZUWTWEPZWGQUWTUSVBZUXDUTUWTUUOPZUXKUYAUYBUYCUXKWHUXK + UUMUVFUUNUXAUXDUWSWDWIUXAUYCUVFUXDQUWTWJWKWLUXKUYDUXDUXAUYDUVFUXDQUWTWMWK + UXBUXDWNZWOUWTQUUMWPWRZWQUXBUXDXHZUTZWSZUXBUYCUYBUXDWTZUXBUWTUXJWIUXBUUMU + VFUUNUXAUWSXAWIUWTUUMXBWCZXCZXDZUYMXEUXBUWTUWONZUXDUWTUWLNZRVDZSUXBIUWTUW NUYQOUWOSUWOVJUXGUVSUXDUWMUYPRUXHUVRUWTUWLTVLUXJUXBUXDUYPRSUXKUUOSUWTUWLU XKBSUVNVQZUXMUUOSUWLVQUVFUYRUXAUXDUVFEBDSAUXODSPZUVEHVRZVSZWDUVFUXMUXAUXD - UXTWDZUUOBSUVNUUPWBWCUYGWPUYJUYLXDZXEZVUCXFUXBUXDUWTUVTNZRVDZUXFUYQLMZUWT + UXTWDZUUOBSUVNUUPWBWCUYGWQUYJUYLXCZXDZVUCXEUXBUXDUWTUVTNZRVDZUXFUYQLMZUWT UWCNUXCUYOLMUXBUXDVUFVUGKUYHUXKVUEUXEUYPLMZVUFVUGUXKUVFUYEVUEVUHKUVFUXAUX - DXIUYGUVFUYEUTZUWTUUPNZUVHNZVUJUVKNZVUJUVNNZLMZVUEVUHVUIVUJBPUXNUVHNZUXNU - VKNZUXNUVNNZLMZKZEBXGZVUKVUNKZUVFUUOBUWTUUPUXTXHAVUTUVEUYEAVUSEBAUXOUTZVU - OUUGVURVVBUXOUUGSPZVUOUUGKAUXOWMZVVBCDGHXJZEBUUGSUVHUVHVJXKWCVVBVUPCVUQDL + DXFUYGUVFUYEUTZUWTUUPNZUVHNZVUJUVKNZVUJUVNNZLMZVUEVUHVUIVUJBPUXNUVHNZUXNU + VKNZUXNUVNNZLMZKZEBXIZVUKVUNKZUVFUUOBUWTUUPUXTXGAVUTUVEUYEAVUSEBAUXOUTZVU + OUUGVURVVBUXOUUGSPZVUOUUGKAUXOWNZVVBCDGHXJZEBUUGSUVHUVHVJXKWCVVBVUPCVUQDL VVBUXOUXPVUPCKVVDGEBCSUVKUVKVJXKWCVVBUXOUYSVUQDKVVDHEBDSUVNUVNVJXKWCUOXTX LWDVUSVVAEVUJBEVUKVUNEBUUGVUJXMEVULVUMLEBCVUJXMELXNEBDVUJXMXOXPUXNVUJKZVU - OVUKVURVUNUXNVUJUVHTVVFVUPVULVUQVUMLUXNVUJUVKTUXNVUJUVNTUOXQXRYAUVFUXMUYE - VUEVUKKUXTUUOBUWTUVHUUPXSYJZVUIUXEVULUYPVUMLUVFUXMUYEUXEVULKUXTUUOBUWTUVK - UUPXSYJZUVFUXMUYEUYPVUMKUXTUUOBUWTUVNUUPXSYJZUOYBWCUXKUXDVUERUYFYCUXKUXFU - XEUYQUYPLUXKUXDUXERUYFYCUXKUXDUYPRUYFYCUOYBUYIRUVDVUFVUGUVDRUJYDUYIUXDVUE - RUXBUYHWMZYEUYIUXFRUYQRLUYIUXDUXERVVJYEUYIUXDUYPRVVJYEUOUPUXBUYKUXDUYHYFU + OVUKVURVUNUXNVUJUVHTVVFVUPVULVUQVUMLUXNVUJUVKTUXNVUJUVNTUOXQXRXSUVFUXMUYE + VUEVUKKUXTUUOBUWTUVHUUPYAYBZVUIUXEVULUYPVUMLUVFUXMUYEUXEVULKUXTUUOBUWTUVK + UUPYAYBZUVFUXMUYEUYPVUMKUXTUUOBUWTUVNUUPYAYBZUOYCWCUXKUXDVUERUYFYDUXKUXFU + XEUYQUYPLUXKUXDUXERUYFYDUXKUXDUYPRUYFYDUOYCUYIRUVDVUFVUGUVDRUJYJUYIUXDVUE + RUXBUYHWNZYEUYIUXFRUYQRLUYIUXDUXERVVJYEUYIUXDUYPRVVJYEUOUPUXBUYKUXDUYHYFU YLUXDYGWAYHUXBIUWTUWBVUFOUWCSUWCVJUXGUVSUXDUWAVUERUXHUVRUWTUVTTVLUXJUXBUX DVUERSUXKUUOSUWTUVTUXKBSUVHVQZUXMUUOSUVTVQAVVKUVEUXAUXDAEBUUGSVVEVSYIVUBU - UOBSUVHUUPWBWCUYGWPUYJUYLXDXEUXBUXCUXFUYOUYQLUYNVUDUOYBYKUVFUVJUUMLJOVUFU + UOBSUVHUUPWBWCUYGWQUYJUYLXCXDUXBUXCUXFUYOUYQLUYNVUDUOYCYKUVFUVJUUMLJOVUFU RZQVEZNUWEUVFBUVIVUKUBJUUPUVTUUMUVGVUJUVHTUWSUXSUVFBSUVGUVHUVFEBUUGSUVFUX - OUTCDUXQUYTXJVSXHVVGYLUUMVVMUWDVVLUWCKVVMUWDKJIOVUFUWBJIVKZUXDUVSVUEUWARU + OUTCDUXQUYTXJVSXGVVGYLUUMVVMUWDVVLUWCKVVMUWDKJIOVUFUWBJIVKZUXDUVSVUEUWARU WTUVRUUMUSVNZUWTUVRUVTTVLYMLVVLUWCQYNYOYPYQUVFUVMUWKUVPUWQLUVFUVMUUMLJOUX - FURZQVEZNUWKUVFBUVLVULUBJUUPUWFUUMUVGVUJUVKTUWSUXSUVFBSUVGUVKUXRXHVVHYLUU + FURZQVEZNUWKUVFBUVLVULUBJUUPUWFUUMUVGVUJUVKTUWSUXSUVFBSUVGUVKUXRXGVVHYLUU MVVQUWJVVPUWIKVVQUWJKJIOUXFUWHVVNUXDUVSUXEUWGRVVOUWTUVRUWFTVLYMLVVPUWIQYN YOYPYQUVFUVPUUMLJOUYQURZQVEZNUWQUVFBUVOVUMUBJUUPUWLUUMUVGVUJUVNTUWSUXSUVF - BSUVGUVNVUAXHVVIYLUUMVVSUWPVVRUWOKVVSUWPKJIOUYQUWNVVNUXDUVSUYPUWMRVVOUWTU - VRUWLTVLYMLVVRUWOQYNYOYPYQUOYBAUVJUUHKZUVEAVVCEBXGVVTAVVCEBVVEXLBUUGUBEYR - WAXBAUVQUUKKUVEAUVMUUIUVPUUJLAUXPEBXGUVMUUIKAUXPEBGXLBCUBEYRWAAUYSEBXGUVP - UUJKAUYSEBHXLBDUBEYRWAUOXBYSYTUUAUUBABUUCPUUFUUSYFFBUAUUDWAUUE $. + BSUVGUVNVUAXGVVIYLUUMVVSUWPVVRUWOKVVSUWPKJIOUYQUWNVVNUXDUVSUYPUWMRVVOUWTU + VRUWLTVLYMLVVRUWOQYNYOYPYQUOYCAUVJUUHKZUVEAVVCEBXIVVTAVVCEBVVEXLBUUGUBEYR + WAXAAUVQUUKKUVEAUVMUUIUVPUUJLAUXPEBXIUVMUUIKAUXPEBGXLBCUBEYRWAAUYSEBXIUVP + UUJKAUYSEBHXLBDUBEYRWAUOXAYSYTUUAUUBABUUCPUUFUUSYFFBUAUUDWAUUE $. $} ${ @@ -121715,53 +121734,53 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) nfralxy r19.21bi fsumcl csbeq1 sumsn syl2anc c2nd cfv cres cvv elv xpfi snfig sylancr wf1o 2ndconst syl fvres adantl mpan9 fsumf1o cop wex elxp nfcri nfan nfex opeq1 eqeq2d velsn anbi1i eqtr2 eleq2d pm5.32da bitr4id - nfv equequ1 anbi1d bitrd anbi12d nfeq2 ad2antlr exlimd syl5bi imp eqtrd - eqtrid wel cin c0 a1i syl3anc syldan fsumsplit wrex eliun xp1st eqeltrd - elsni elin sneq xpeq12d eqtri exbidv cbvex bitri nfcsb ad2antrl eqtr2di - fveq2 op2nd 3eqtrd sumeq2dv eqtr4d oveq12d disjsn eqidd unfidisj sselda - expl wn anassrs c1st simpl rexlimiva sylbi anim12i 3imtr4i noel pm2.21i - syl6bi syl5 ssrdv ss0 iunxun cbviun iunxsn uneq2i wdisj disjsnxp simprl - nfxp iunfidisj simprrl opeq1d simpll simprrr syl12anc rexlimdva 3eqtr4d - ex exlimdvv ) ABUBZCUCZGHKUDZJUDZDUCZUEZUWLJUDZUFUGZJUWKJUCZUEZGUHZUIZI - EUDZUWOJUWNGUJZUHZIEUDZUFUGZUWKUWOUKZUWLJUDZJUXGUWTUIZIEUDZUWJUWMUXBUWP - UXEUFUWJBUWMUXBULABUMSUNAUWPUXEULBAUWPUWOJUAUCZGUJZJUXKHUJZKUDZUAUDZUXE - UWOUWLUXNJUAUAUWLUOJUXLUXMKJUXKGUPZJUXKHUPUQJUAURZGUXLHUXMKJUXKGUSZUXQH - 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ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) c0 wex mul01d sumeq1 sum0 eqtrdi oveq2d eqeq12d syl5ibrcom cmpt caddc cle wbr ccom cif cseq addcl adantl ad2antrr simprl simprr adddid cuz eleqtrdi cv nnuz elnnuz biimpri wf f1of ad2antll cz w3a 1zzd nnzd eluzelz ad2antlr - 3jca eluzle simpr jca elfz2 sylanbrc fvco3 syl2anc csb ffvelrnd ralrimiva - wral ad3antrrr nfcsb1v nfel1 csbeq1a eleq1d rspc sylc eqid fvmpts eqeltrd - wn 0cnd wdc adantr zdcle ifcldadc breq1 fveq2 ifbieq1d fvmptg csbov2g syl - iftrued mulcld 3eqtrd 3eqtr4d iffalsed 3eqtr4rd wo exmiddc mpjaodan mulcl - seq3distr fmpttd ffvelcdmda sylan sumfct 3eqtr3d expr exlimdv expimpd cfn - fsum3 fz1f1o mpjaod ) ABUHKZDBCEUDZLMZBDCLMZEUDZKZBUENZOPZQUUHUFMZBUAVLZU - GZUAUIZRZAUUGUUBDSLMZSKADGUJUUBUUDUUOUUFSUUBUUCSDLUUBUUCUHCEUDSBUHCEUKCEU - LUMUNUUBUUFUHUUEEUDSBUHUUEEUKUUEEULUMUOUPAUUIUUMUUGAUUIRUULUUGUAAUUIUULUU - GAUUIUULRZRZDBIVLZEBCUQZNZIUDZLMZBUUREBUUEUQZNZIUDZUUDUUFUUQUUHURJOJVLZUU - HUSUTZUVFUVCUUKVAZNZSVBZUQZQVCNDUUHURJOUVGUVFUUSUUKVAZNZSVBZUQZQVCNZLMUVE - 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(Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) $) @@ -122593,26 +122612,26 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) 0cn oveq2d vtocl2ga mp2an eqtr2i addcani mpbi fveq2i 3eqtr4i a1i cfn wa wss csb cvv nfv nfcsb1v simplr vex simprr eldifbd simplll simprl sselda sum0 cdif syl2anc csbeq1a wral eldifad ralrimiva nfel1 eleq1d rspc sylc - ad2antrr fsumsplitsn adantr fsumcl simpr 3eqtrd nfcv wf ffvelrnd eqtr4d - nffv ex findcard2sd ) AUAUHZEFLZGMZXGEGMZFLZNUIEFLZGMZUIXJFLZNZUBUHZEFL - ZGMZXPXJFLZNZXPUGUHZUCUDZEFLZGMZYBXJFLZNZDEFLZGMZDXJFLZNUAUBUGDXGUINZXI - XMXKXNYJXHXLGXGUIEFOUEXGUIXJFOUFXGXPNZXIXRXKXSYKXHXQGXGXPEFOUEXGXPXJFOU - FXGYBNZXIYDXKYEYLXHYCGXGYBEFOUEXGYBXJFOUFXGDNZXIYHXKYIYMXHYGGXGDEFOUEXG - DXJFOUFXOAPGMZPXMXNYNYNQRZYNPQRZNYNPNYPYNYOYNPSTZYNSTUSSSPGJUJUKZULYQYQ - YNYONZUSUSBUHZCUHZQRGMZYTGMZUUAGMZQRZNZPUUAQRZGMZYNUUDQRZNYSBCPPSSYTPNZ - UUBUUHUUEUUIYTPUUAGQUMUUJUUCYNUUDQYTPGUNUOUFUUAPNZUUHYNUUIYOUUKUUGPGUUK - UUGPPQRPUUAPPQUPUQURUEUUKUUDYNYNQUUAPGUNUTUFKVAVBVCYNYNPYRYRUSVDVEXLPGE - FWCVFXJFWCVGVHAXPVITZVJZXPDVKZYADXPWDTZVJZVJZXTYFUUQXTVJZYDXSFYAEVLZGMZ - QRZYEUURYDXQUUSQRZGMZXRUUTQRZUVAUURYCUVBGUUQYCUVBNXTUUQXPYAEUUSFVMUUQFV - NZFYAEVOZAUULUUPVPZYAVMTUUQUGVQVHZUUQYADXPUUMUUNUUOVRZVSZUUQFUHZXPTZVJZ - AUVKDTESTZAUULUUPUVLVTUUQXPDUVKUUMUUNUUOWAWBIWEZFYAEWFZUUQYADTUVNFDWGZU - USSTZUUQYADXPUVIWHAUVQUULUUPAUVNFDIWIWNUVNUVRFYADFUUSSUVFWJUVKYANZEUUSS - UVPWKWLWMZWOWPUEUURXQSTZUVRUVCUVDNZUUQUWAXTUUQXPEFUVGUVOWQWPUUQUVRXTUVT - WPUUFXQUUAQRZGMZXRUUDQRZNUWBBCXQUUSSSYTXQNZUUBUWDUUEUWEYTXQUUAGQUMUWFUU - CXRUUDQYTXQGUNUOUFUUAUUSNZUWDUVCUWEUVDUWGUWCUVBGUUAUUSXQQUPUEUWGUUDUUTX - RQUUAUUSGUNUTUFKVAWEUURXRXSUUTQUUQXTWRUOWSUUQYEUVANXTUUQXPYAXJUUTFVMUVE - FUUSGFGWTUVFXDUVGUVHUVJUVMSSEGSSGXAZUVMJVHUVOXBUVSEUUSGUVPUEUUQSSUUSGUW - HUUQJVHUVTXBWOWPXCXEHXF $. + ad2antrr fsumsplitsn adantr fsumcl 3eqtrd nfcv nffv wf ffvelcdmd eqtr4d + simpr ex findcard2sd ) AUAUHZEFLZGMZXGEGMZFLZNUIEFLZGMZUIXJFLZNZUBUHZEF + LZGMZXPXJFLZNZXPUGUHZUCUDZEFLZGMZYBXJFLZNZDEFLZGMZDXJFLZNUAUBUGDXGUINZX + IXMXKXNYJXHXLGXGUIEFOUEXGUIXJFOUFXGXPNZXIXRXKXSYKXHXQGXGXPEFOUEXGXPXJFO + UFXGYBNZXIYDXKYEYLXHYCGXGYBEFOUEXGYBXJFOUFXGDNZXIYHXKYIYMXHYGGXGDEFOUEX + GDXJFOUFXOAPGMZPXMXNYNYNQRZYNPQRZNYNPNYPYNYOYNPSTZYNSTUSSSPGJUJUKZULYQY + QYNYONZUSUSBUHZCUHZQRGMZYTGMZUUAGMZQRZNZPUUAQRZGMZYNUUDQRZNYSBCPPSSYTPN + ZUUBUUHUUEUUIYTPUUAGQUMUUJUUCYNUUDQYTPGUNUOUFUUAPNZUUHYNUUIYOUUKUUGPGUU + KUUGPPQRPUUAPPQUPUQURUEUUKUUDYNYNQUUAPGUNUTUFKVAVBVCYNYNPYRYRUSVDVEXLPG + EFWCVFXJFWCVGVHAXPVITZVJZXPDVKZYADXPWDTZVJZVJZXTYFUUQXTVJZYDXSFYAEVLZGM + ZQRZYEUURYDXQUUSQRZGMZXRUUTQRZUVAUURYCUVBGUUQYCUVBNXTUUQXPYAEUUSFVMUUQF + VNZFYAEVOZAUULUUPVPZYAVMTUUQUGVQVHZUUQYADXPUUMUUNUUOVRZVSZUUQFUHZXPTZVJ + ZAUVKDTESTZAUULUUPUVLVTUUQXPDUVKUUMUUNUUOWAWBIWEZFYAEWFZUUQYADTUVNFDWGZ + UUSSTZUUQYADXPUVIWHAUVQUULUUPAUVNFDIWIWNUVNUVRFYADFUUSSUVFWJUVKYANZEUUS + SUVPWKWLWMZWOWPUEUURXQSTZUVRUVCUVDNZUUQUWAXTUUQXPEFUVGUVOWQWPUUQUVRXTUV + TWPUUFXQUUAQRZGMZXRUUDQRZNUWBBCXQUUSSSYTXQNZUUBUWDUUEUWEYTXQUUAGQUMUWFU + UCXRUUDQYTXQGUNUOUFUUAUUSNZUWDUVCUWEUVDUWGUWCUVBGUUAUUSXQQUPUEUWGUUDUUT + XRQUUAUUSGUNUTUFKVAWEUURXRXSUUTQUUQXTXDUOWRUUQYEUVANXTUUQXPYAXJUUTFVMUV + EFUUSGFGWSUVFWTUVGUVHUVJUVMSSEGSSGXAZUVMJVHUVOXBUVSEUUSGUVPUEUUQSSUUSGU + WHUUQJVHUVTXBWOWPXCXEHXF $. $} $( The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) @@ -122699,37 +122718,37 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) oveq1 csbeq1 iunxsn ineq2i ad2antrr adantl unssbd simplr disjsn disjiun eqtri wdisj sylibr syl13anc eqtr3id adantlrl iunxun uneq2i a1i wral cvv simpr simplrl unsnfi syl3anc ralrimiva ssralv sylc disjss1 iunfidisj cc - iunss1 sselda eliun rexlimdvaa syl5bi imp syldan fsumsplit eqidd sseldd - wrex anassrs fsumcl r19.21bi nfsum cbvsumi snss nfel1 eleq1d rspc sumsn - sylancr eqtrid oveq2d eqtrd syl5ibr ex a2d syl5 expcom findcard2s mpcom - mpi ) ACCLZBCDMZEFNZCDEFNZBNZOZCUHCUCPAYQUUBQZGAUAUDZCLZBUUDDMZEFNZUUDY - TBNZOZQZQARCLZREFNZRYTBNZOZQZQAKUDZCLZBUUPDMZEFNZUUPYTBNZOZQZQAUUPUBUDZ - UEZUFZCLZBUVEDMZEFNZUVEYTBNZOZQZQAUUCQUAKUBCUUDROZUUJUUOAUVLUUEUUKUUIUU - NUUDRCUGUVLUUGUULUUHUUMUVLUUFREFUVLUUFBRDMRBUUDRDUIBDUJUKSUUDRYTBULUMUN - UOUUDUUPOZUUJUVBAUVMUUEUUQUUIUVAUUDUUPCUGUVMUUGUUSUUHUUTUVMUUFUUREFBUUD - UUPDUISUUDUUPYTBULUMUNUOUUDUVEOZUUJUVKAUVNUUEUVFUUIUVJUUDUVECUGUVNUUGUV - HUUHUVIUVNUUFUVGEFBUUDUVEDUISUUDUVEYTBULUMUNUOUUDCOZUUJUUCAUVOUUEYQUUIU - UBUUDCCUGUVOUUGYSUUHUUAUVOUUFYREFBUUDCDUISUUDCYTBULUMUNUOAUUKUUNUULUTUU - MEFUPYTBUPUQVAUUPUCPZUVCUUPPURZTZAUVBUVKAUVRUVBUVKQUVBUVFUVAQAUVRTZUVKU - VFUUQUVAUVFUUPUVDCUVFUSVBZVCUVSUVFUVAUVJUVSUVFUVAUVJQUVAUVJUVSUVFTZUUSB - UVCDVDZEFNZVEVFZUUTUWCVEVFZOUUSUUTUWCVEVMUWAUVHUWDUVIUWEUWAUURUWBEUVGFA - UVQUVFUURUWBVGZROUVPAUVQTZUVFTZUWFUURBUVDDMZVGZRUWIUWBUURUWIKUVDBUUPDVD - ZMUWBBKUVDDUWKKDVHBUUPDVIZBUUPDVJZVKKUVCUWKUWBUBVLZBUUPUVCDVNZVOWCZVPUW - HBCDWDZUUQUVDCLZUUPUVDVGROZUWJROAUWQUVQUVFIVQZUVFUUQUWGUVTVRUWHUUPUVDCU - WGUVFWNZVSZUWHUVQUWSAUVQUVFVTZUUPUVCWAZWEBCDUUPUVDWBWFWGWHUVGUURUWBUFZO - UWAUVGUURUWIUFUXEBUUPUVDDWIUWIUWBUURUWPWJWCWKUWAUVEUCPZDUCPZBUVEWLZBUVE - DWDZUVGUCPUWAUVPUVCWMPZUVQUXFAUVPUVQUVFWOUXJUWAUWNWKAUVQUVFUVQUVPUXCWHZ - UUPUVCWMWPWQZAUVQUVFUXHUVPUWHUVFUXGBCWLZUXHUXAAUXMUVQUVFAUXGBCHWRVQUXGB - UVECWSWTWHAUVQUVFUXIUVPUWHUVFUWQUXIUXAUWTBUVECDXAWTWHBUVEDXBWQUWAFUDZUV - GPUXNYRPZEXCPZUWAUVGYRUXNUVFUVGYRLUVSBUVECDXDVRXEUWAUXOUXPUXOUXNDPZBCXN - ZUWAUXPBUXNCDXFAUXRUXPQUVRUVFAUXQUXPBCJXGVQXHXIXJXKUWAUVIUUTUVDYTBNZVEV - FZUWEUWAUUPUVDYTUVEBUWAUVQUWSUXKUXDWEUWAUVEXLUXLUWABUDZUVEPZUYACPZYTXCP - ZUWAUYBTUVECUYAUVSUVFUYBVTUWAUYBWNXMUWAUYDBCAUYDBCWLZUVRUVFAUYDBCAUYCTD - EFHAUYCUXQUXPJXOXPWRZVQXQXJXKAUVQUVFUXTUWEOUVPUWHUXSUWCUUTVEUWHUXSUVDUW - KEFNZKNZUWCUVDYTUYGBKKYTVHBUWKEFUWLBEVHZXRUYAUUPODUWKEFUWMSXSUWHUXJUWCX - CPZUYHUWCOUWNUWHUVCCPZUYEUYJUWHUWRUYKUXBUVCCUWNXTWEAUYEUVQUVFUYFVQUYDUY - JBUVCCBUWCXCBUWBEFBUVCDVIUYIXRYAUYAUVCOZYTUWCXCUYLDUWBEFBUVCDVJSYBYCWTU - YGUWCKUVCWMUUPUVCOUWKUWBEFUWOSYDYEYFYGWHYHUMYIYJYKYLYMYKYNYOYP $. + iunss1 sselda wrex eliun rexlimdvaa biimtrid imp syldan fsumsplit eqidd + sseldd anassrs fsumcl r19.21bi nfsum cbvsumi nfel1 eleq1d sumsn sylancr + snss rspc eqtrid oveq2d eqtrd syl5ibr ex a2d syl5 expcom findcard2s mpi + mpcom ) ACCLZBCDMZEFNZCDEFNZBNZOZCUHCUCPAYQUUBQZGAUAUDZCLZBUUDDMZEFNZUU + DYTBNZOZQZQARCLZREFNZRYTBNZOZQZQAKUDZCLZBUUPDMZEFNZUUPYTBNZOZQZQAUUPUBU + DZUEZUFZCLZBUVEDMZEFNZUVEYTBNZOZQZQAUUCQUAKUBCUUDROZUUJUUOAUVLUUEUUKUUI + UUNUUDRCUGUVLUUGUULUUHUUMUVLUUFREFUVLUUFBRDMRBUUDRDUIBDUJUKSUUDRYTBULUM + UNUOUUDUUPOZUUJUVBAUVMUUEUUQUUIUVAUUDUUPCUGUVMUUGUUSUUHUUTUVMUUFUUREFBU + UDUUPDUISUUDUUPYTBULUMUNUOUUDUVEOZUUJUVKAUVNUUEUVFUUIUVJUUDUVECUGUVNUUG + UVHUUHUVIUVNUUFUVGEFBUUDUVEDUISUUDUVEYTBULUMUNUOUUDCOZUUJUUCAUVOUUEYQUU + IUUBUUDCCUGUVOUUGYSUUHUUAUVOUUFYREFBUUDCDUISUUDCYTBULUMUNUOAUUKUUNUULUT + UUMEFUPYTBUPUQVAUUPUCPZUVCUUPPURZTZAUVBUVKAUVRUVBUVKQUVBUVFUVAQAUVRTZUV + KUVFUUQUVAUVFUUPUVDCUVFUSVBZVCUVSUVFUVAUVJUVSUVFUVAUVJQUVAUVJUVSUVFTZUU + SBUVCDVDZEFNZVEVFZUUTUWCVEVFZOUUSUUTUWCVEVMUWAUVHUWDUVIUWEUWAUURUWBEUVG + FAUVQUVFUURUWBVGZROUVPAUVQTZUVFTZUWFUURBUVDDMZVGZRUWIUWBUURUWIKUVDBUUPD + VDZMUWBBKUVDDUWKKDVHBUUPDVIZBUUPDVJZVKKUVCUWKUWBUBVLZBUUPUVCDVNZVOWCZVP + UWHBCDWDZUUQUVDCLZUUPUVDVGROZUWJROAUWQUVQUVFIVQZUVFUUQUWGUVTVRUWHUUPUVD + CUWGUVFWNZVSZUWHUVQUWSAUVQUVFVTZUUPUVCWAZWEBCDUUPUVDWBWFWGWHUVGUURUWBUF + ZOUWAUVGUURUWIUFUXEBUUPUVDDWIUWIUWBUURUWPWJWCWKUWAUVEUCPZDUCPZBUVEWLZBU + VEDWDZUVGUCPUWAUVPUVCWMPZUVQUXFAUVPUVQUVFWOUXJUWAUWNWKAUVQUVFUVQUVPUXCW + HZUUPUVCWMWPWQZAUVQUVFUXHUVPUWHUVFUXGBCWLZUXHUXAAUXMUVQUVFAUXGBCHWRVQUX + GBUVECWSWTWHAUVQUVFUXIUVPUWHUVFUWQUXIUXAUWTBUVECDXAWTWHBUVEDXBWQUWAFUDZ + UVGPUXNYRPZEXCPZUWAUVGYRUXNUVFUVGYRLUVSBUVECDXDVRXEUWAUXOUXPUXOUXNDPZBC + XFZUWAUXPBUXNCDXGAUXRUXPQUVRUVFAUXQUXPBCJXHVQXIXJXKXLUWAUVIUUTUVDYTBNZV + EVFZUWEUWAUUPUVDYTUVEBUWAUVQUWSUXKUXDWEUWAUVEXMUXLUWABUDZUVEPZUYACPZYTX + CPZUWAUYBTUVECUYAUVSUVFUYBVTUWAUYBWNXNUWAUYDBCAUYDBCWLZUVRUVFAUYDBCAUYC + TDEFHAUYCUXQUXPJXOXPWRZVQXQXKXLAUVQUVFUXTUWEOUVPUWHUXSUWCUUTVEUWHUXSUVD + UWKEFNZKNZUWCUVDYTUYGBKKYTVHBUWKEFUWLBEVHZXRUYAUUPODUWKEFUWMSXSUWHUXJUW + CXCPZUYHUWCOUWNUWHUVCCPZUYEUYJUWHUWRUYKUXBUVCCUWNYDWEAUYEUVQUVFUYFVQUYD + UYJBUVCCBUWCXCBUWBEFBUVCDVIUYIXRXTUYAUVCOZYTUWCXCUYLDUWBEFBUVCDVJSYAYEW + TUYGUWCKUVCWMUUPUVCOUWKUWBEFUWOSYBYCYFYGWHYHUMYIYJYKYLYMYKYNYPYO $. $} $( The cardinality of a disjoint indexed union. (Contributed by Mario @@ -123019,35 +123038,35 @@ Infinite sums (cont.) syl rspcdva fvmpt2d fvmptd eqtr4d nffvmpt1 nfeq1 fveq2 oveq2 fveq2d mpan9 eqeq12d simpr eluzsub syl3anc rspccva pncan3 3eqtrrd addcl seq3feq breq1d eleqtrdi sylan2br cvv iser3shft bitr4d iotabidv 3eqtr4g fmpttd ffvelcdmda - a1i df-fv isum wf addcom id eluzadd syl2anr sylan ffvelrnd 3eqtr4d sumfct - 3eqtr3d ) AHUAUEZDHBUFZPZUAUGZIUBUEZEICUFZPZUBUGZHBDUGZICEUGZAQUUCGFQUHZU - IZUJPZQUUGGUIZUJPZUUEUUIAUUMUCUEZUJUKZUCULUUOUUQUJUKZUCULUUNUUPAUURUUSUCA - UURQUUGFUMUHZUULUIZUUQUJUKUUSAUUMUVAUUQUJAUCUDQRUAUUCUUTUULAGFNMUNZAUUQUU - LUOPZSZVEZUUQUUCPZDUUQBUPZRUVEUUQHSZUVGRSZUVFUVGTUVDUVHAUVHUVDHUVCUUQKUQU - RUSZUVEUVHBRSZDHUTZUVIUVJAUVLUVDAUVKDHOVAZVBUVKUVIDUUQHDUVGRDUUQBVCVDDUCV - FBUVGRDUUQBVGVHVIVJZDUUQBHUUCRUUCVKVLVMUVNVNUUBUVCSZAUUBHSZUUDUUBUUTPZTHU - VCUUBKUQAUVPVEZUVQUUBFVOUHZUUGPZFUVSQUHZUUCPZUUDAFRSZUUBRSZUVQUVTTUVPAFMV - QZUWDUUBUVCHUULUUBVPKVRZFUUBUUGEICIVSVTIGUOPZVSJGWAWBWCWDZWEWFUVRUUHFUUFQ - UHZUUCPZTZUBIUTZUVSISUVTUWBTZAUWLUVPAUWKUBIAEUEZUUGPZFUWNQUHZUUCPZTZEIUTU - UFISZUWKAUWREIAUWNISZVEZUWOCUWQAEICUUGRAUUGWGUXAUVKCRSZDHUWPDUEUWPTZBCRLV - HAUVLUWTUVMVBUXAUWPUVCHUXAUULVSSUWPVSSUULUWPWIUKUWPUVCSUXAGFAGVSSZUWTNVBZ - AFVSSZUWTMVBZUNUXAFUWNUXGUWTUWNVSSZAUXHUWNUWGIGUWNWHJVRUSZUNUXAUULUWNFQUH - ZUWPWIUXAGUWNFUXAGUXEWJUXAUWNUXIWJUXAFUXGWJUXAUWNUWGSZGUWNWIUKUWTUXKAUWTU - XKIUWGUWNJUQWKUSGUWNWLWSWMUXAUWNFUXAUWNUXIVQUXAFUXGVQWNWOUULUWPWPWQKWRZWT - ZXAUXADUWPBCHUUCRUXAUUCWGUXCBCTUXALUSUXLUXMXBXCVAUWRUWKEUUFIEUUHUWJEICUUF - XDXEEUBVFZUWOUUHUWQUWJUWNUUFUUGXFUXNUWPUWIUUCUWNUUFFQXGZXHXJVIXIZVAVBUVRU - VSUWGIUVRUXDUXFUVOUVSUWGSAUXDUVPNVBAUXFUVPMVBUVRUUBHUVCAUVPXKKXTFGUUBXLXM - JWRUWKUWMUBUVSIUUFUVSTZUUHUVTUWJUWBUUFUVSUUGXFUXQUWIUWAUUCUUFUVSFQXGXHXJX - NVMUVRUWAUUBUUCAUWCUWDUWAUUBTUVPUWEUWFFUUBXOWFXHXPYAUUQRSUDUEZRSVEUUQUXRQ - UHRSAUUQUXRXQUSZXRXSAUCUDUUQQRUUGGFYBUUGYBSAUWHYINMAUUQUWGSZVEZUUQUUGPZEU - UQCUPZRUYAUUQISZUYCRSZUYBUYCTUXTUYDAUYDUXTIUWGUUQJUQURUSZUYAUYDUXBEIUTZUY - EUYFAUYGUXTAUXBEIUXMVAZVBUXBUYEEUUQIEUYCREUUQCVCVDEUCVFCUYCREUUQCVGVHVIVJ - ZEUUQCIUUGRUUGVKVLVMUYIVNUXSYCYDYEUCUUMUJYJUCUUOUJYJYFAUUDUAUUCUULHKUVBUV - RUUDWGAHRUUBUUCADHBROYGZYHYKAUUHUBUUGGIJNAUWSVEZUUHWGUYKUUHUWJRUXPUYKHRUW - IUUCAHRUUCYLUWSUYJVBAUWPHSZEIUTUWSUWIHSZAUYLEIUXAUWPUVCHUXAUWPUXJUVCAUWCU - WNRSZUWPUXJTUWTUWEUYNUWNUWGIGUWNVPJVRFUWNYMWFUWTUXKUXFUXJUVCSAUWTUWNIUWGU - WTYNJXTMFGUWNYOYPVNKWRVAUYLUYMEUUFIUXNUWPUWIHUXOVHXNYQYRVNYKYSAUVLUUEUUJT - UVMHBUADYTWSAUYGUUIUUKTUYHICUBEYTWSUUA $. + a1i df-fv isum wf addcom eluzadd syl2anr sylan ffvelcdmd 3eqtr4d sumfct + id 3eqtr3d ) AHUAUEZDHBUFZPZUAUGZIUBUEZEICUFZPZUBUGZHBDUGZICEUGZAQUUCGFQU + HZUIZUJPZQUUGGUIZUJPZUUEUUIAUUMUCUEZUJUKZUCULUUOUUQUJUKZUCULUUNUUPAUURUUS + UCAUURQUUGFUMUHZUULUIZUUQUJUKUUSAUUMUVAUUQUJAUCUDQRUAUUCUUTUULAGFNMUNZAUU + QUULUOPZSZVEZUUQUUCPZDUUQBUPZRUVEUUQHSZUVGRSZUVFUVGTUVDUVHAUVHUVDHUVCUUQK + UQURUSZUVEUVHBRSZDHUTZUVIUVJAUVLUVDAUVKDHOVAZVBUVKUVIDUUQHDUVGRDUUQBVCVDD + UCVFBUVGRDUUQBVGVHVIVJZDUUQBHUUCRUUCVKVLVMUVNVNUUBUVCSZAUUBHSZUUDUUBUUTPZ + THUVCUUBKUQAUVPVEZUVQUUBFVOUHZUUGPZFUVSQUHZUUCPZUUDAFRSZUUBRSZUVQUVTTUVPA + FMVQZUWDUUBUVCHUULUUBVPKVRZFUUBUUGEICIVSVTIGUOPZVSJGWAWBWCWDZWEWFUVRUUHFU + UFQUHZUUCPZTZUBIUTZUVSISUVTUWBTZAUWLUVPAUWKUBIAEUEZUUGPZFUWNQUHZUUCPZTZEI + UTUUFISZUWKAUWREIAUWNISZVEZUWOCUWQAEICUUGRAUUGWGUXAUVKCRSZDHUWPDUEUWPTZBC + RLVHAUVLUWTUVMVBUXAUWPUVCHUXAUULVSSUWPVSSUULUWPWIUKUWPUVCSUXAGFAGVSSZUWTN + VBZAFVSSZUWTMVBZUNUXAFUWNUXGUWTUWNVSSZAUXHUWNUWGIGUWNWHJVRUSZUNUXAUULUWNF + QUHZUWPWIUXAGUWNFUXAGUXEWJUXAUWNUXIWJUXAFUXGWJUXAUWNUWGSZGUWNWIUKUWTUXKAU + WTUXKIUWGUWNJUQWKUSGUWNWLWSWMUXAUWNFUXAUWNUXIVQUXAFUXGVQWNWOUULUWPWPWQKWR + ZWTZXAUXADUWPBCHUUCRUXAUUCWGUXCBCTUXALUSUXLUXMXBXCVAUWRUWKEUUFIEUUHUWJEIC + UUFXDXEEUBVFZUWOUUHUWQUWJUWNUUFUUGXFUXNUWPUWIUUCUWNUUFFQXGZXHXJVIXIZVAVBU + VRUVSUWGIUVRUXDUXFUVOUVSUWGSAUXDUVPNVBAUXFUVPMVBUVRUUBHUVCAUVPXKKXTFGUUBX + LXMJWRUWKUWMUBUVSIUUFUVSTZUUHUVTUWJUWBUUFUVSUUGXFUXQUWIUWAUUCUUFUVSFQXGXH + XJXNVMUVRUWAUUBUUCAUWCUWDUWAUUBTUVPUWEUWFFUUBXOWFXHXPYAUUQRSUDUEZRSVEUUQU + XRQUHRSAUUQUXRXQUSZXRXSAUCUDUUQQRUUGGFYBUUGYBSAUWHYINMAUUQUWGSZVEZUUQUUGP + ZEUUQCUPZRUYAUUQISZUYCRSZUYBUYCTUXTUYDAUYDUXTIUWGUUQJUQURUSZUYAUYDUXBEIUT + ZUYEUYFAUYGUXTAUXBEIUXMVAZVBUXBUYEEUUQIEUYCREUUQCVCVDEUCVFCUYCREUUQCVGVHV + IVJZEUUQCIUUGRUUGVKVLVMUYIVNUXSYCYDYEUCUUMUJYJUCUUOUJYJYFAUUDUAUUCUULHKUV + BUVRUUDWGAHRUUBUUCADHBROYGZYHYKAUUHUBUUGGIJNAUWSVEZUUHWGUYKUUHUWJRUXPUYKH + RUWIUUCAHRUUCYLUWSUYJVBAUWPHSZEIUTUWSUWIHSZAUYLEIUXAUWPUVCHUXAUWPUXJUVCAU + WCUWNRSZUWPUXJTUWTUWEUYNUWNUWGIGUWNVPJVRFUWNYMWFUWTUXKUXFUXJUVCSAUWTUWNIU + WGUWTYTJXTMFGUWNYNYOVNKWRVAUYLUYMEUUFIUXNUWPUWIHUXOVHXNYPYQVNYKYRAUVLUUEU + UJTUVMHBUADYSWSAUYGUUIUUKTUYHICUBEYSWSUUA $. $} ${ @@ -123922,26 +123941,26 @@ Infinite sums (cont.) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) $= ( caddc co cn cc adantr wcel vx vy vz clt wbr c1 cseq cfv cmin cfz cv - csu wceq wa wf nnuz 1zzd serf ffvelrnd cuz eqid peano2zd fveq2 eleq1d - nnzd wral ralrimiva peano2nnd eluznn sylan rspcdva cz cle eluzelz syl - wb zltp1le syl2anc biimpa eluz2 syl3anbrc pncan2d addcl adantl addass - w3a eleqtrdi ad2antrr simpr eleqtrrdi seq3split oveq1d eqidd fsum3ser - 3eqtr4d cc0 nnred ltp1d eqbrtrrd fzn mpbid sumeq1d sum0 eqtrdi subidd - c0 fveq2d 3eqtr2rd wo eluzle zleloe mpjaodan ) AFGUDUEZGOEUFUGZUHZFXN - UHZUIPZFUFOPZGUJPZCUKZEUHZCULZUMFGUMZAXMUNZXPGOEXRUGZUHZOPZXPUIPYFXQY - BYDXPYFYDQRFXNAQRXNUOXMADEUFQUPAUQKURZSAFQTXMMSUSYDXRUTUHZRGYEAYIRYEU - OXMAUAEXRYIYIVAAFAFMVEZVBZAUAUKZYITZUNDUKZEUHZRTZYLEUHZRTZDQYLYNYLUMY - OYQRYNYLEVCVDZAYPDQVFZYMAYPDQKVGZSAXRQTZYMYLQTAFMVHZYLXRVIVJVKURSYDXR - VLTZGVLTZXRGVMUEZGYITAUUDXMYKSAUUEXMAGFUTUHTZUUENFGVNVOZSAXMUUFAFVLTZ - UUEXMUUFVPYJUUHFGVQVRVSXRGVTWAZUSWBYDXOYGXPUIYDUAUBUCOREUFFGYLRTZUBUK - ZRTZUNYLUULOPZRTYDYLUULWCWDUUKUUMUCUKZRTWFUUNUUOOPYLUULUUOOPOPUMYDYLU - ULUUOWEWDUUJAFUFUTUHZTXMAFQUUPMUPWGSYDYLUUPTZUNZYPYRDQYLYSAYTXMUUQUUA - WHUURYLUUPQYDUUQWIUPWJVKWKWLYDYACEXRGYDXTYITZUNZYAWMUUJUUTYPYARTDQXTY - NXTUMYOYARYNXTEVCVDAYTXMUUSUUAWHUUTUUBUUSXTQTAUUBXMUUSUUCWHYDUUSWIXTX - RVIVRVKWNWOAYCUNZYBWPXPXPUIPXQUVAYBXFYACULWPUVAXSXFYACUVAGXRUDUEZXSXF - UMZUVAFGXRUDAYCWIZAFXRUDUEYCAFAFMWQWRSWSUVAUUDUUEUVBUVCVPAUUDYCYKSAUU - EYCUUHSXRGWTVRXAXBYACXCXDUVAXPAXPRTYCAQRFXNYHMUSSXEUVAXPXOXPUIUVAFGXN - UVDXGWLXHAFGVMUEZXMYCXIZAUUGUVENFGXJVOAUUIUUEUVEUVFVPYJUUHFGXKVRXAXL + csu wceq wa wf nnuz 1zzd serf ffvelcdmd cuz eqid nnzd peano2zd eleq1d + fveq2 wral ralrimiva peano2nnd eluznn sylan rspcdva cz cle eluzelz wb + syl zltp1le syl2anc biimpa eluz2 syl3anbrc pncan2d addcl w3a eleqtrdi + adantl addass ad2antrr simpr eleqtrrdi seq3split oveq1d eqidd 3eqtr4d + fsum3ser cc0 c0 nnred ltp1d eqbrtrrd mpbid sumeq1d sum0 eqtrdi subidd + fzn fveq2d 3eqtr2rd wo eluzle zleloe mpjaodan ) AFGUDUEZGOEUFUGZUHZFX + NUHZUIPZFUFOPZGUJPZCUKZEUHZCULZUMFGUMZAXMUNZXPGOEXRUGZUHZOPZXPUIPYFXQ + YBYDXPYFYDQRFXNAQRXNUOXMADEUFQUPAUQKURZSAFQTXMMSUSYDXRUTUHZRGYEAYIRYE + UOXMAUAEXRYIYIVAAFAFMVBZVCZAUAUKZYITZUNDUKZEUHZRTZYLEUHZRTZDQYLYNYLUM + YOYQRYNYLEVEVDZAYPDQVFZYMAYPDQKVGZSAXRQTZYMYLQTAFMVHZYLXRVIVJVKURSYDX + RVLTZGVLTZXRGVMUEZGYITAUUDXMYKSAUUEXMAGFUTUHTZUUENFGVNVPZSAXMUUFAFVLT + ZUUEXMUUFVOYJUUHFGVQVRVSXRGVTWAZUSWBYDXOYGXPUIYDUAUBUCOREUFFGYLRTZUBU + KZRTZUNYLUULOPZRTYDYLUULWCWFUUKUUMUCUKZRTWDUUNUUOOPYLUULUUOOPOPUMYDYL + UULUUOWGWFUUJAFUFUTUHZTXMAFQUUPMUPWESYDYLUUPTZUNZYPYRDQYLYSAYTXMUUQUU + AWHUURYLUUPQYDUUQWIUPWJVKWKWLYDYACEXRGYDXTYITZUNZYAWMUUJUUTYPYARTDQXT + YNXTUMYOYARYNXTEVEVDAYTXMUUSUUAWHUUTUUBUUSXTQTAUUBXMUUSUUCWHYDUUSWIXT + XRVIVRVKWOWNAYCUNZYBWPXPXPUIPXQUVAYBWQYACULWPUVAXSWQYACUVAGXRUDUEZXSW + QUMZUVAFGXRUDAYCWIZAFXRUDUEYCAFAFMWRWSSWTUVAUUDUUEUVBUVCVOAUUDYCYKSAU + UEYCUUHSXRGXFVRXAXBYACXCXDUVAXPAXPRTYCAQRFXNYHMUSSXEUVAXPXOXPUIUVAFGX + NUVDXGWLXHAFGVMUEZXMYCXIZAUUGUVENFGXJVPAUUIUUEUVEUVFVOYJUUHFGXKVRXAXL $. $} @@ -124037,29 +124056,29 @@ Infinite sums (cont.) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / M ) ) $= ( vi c1 cfv co cn cc wcel caddc cseq cmin cabs cdiv cmul clt cfz cv csu - cexp nnuz 1zzd serf cuz eluznn ffvelrnd subcld abscld wceq fveq2 eleq1d - syl2anc ralrimiva rspcdva nnzd peano2zd cz eluzelz syl fzfigd wa adantr - cr cle wbr cn0 nnred peano2re elfzelz adantl zred lep1d elfzle1 znn0sub - letrd wb syl2an mpbid reexpcld fsumrecl remulcld elrpd reclt1d rprecred - crp 1re difrp sylancr rpreccld rpdivcld 1nn a1i ge0p1rpd rpmulcld rpred - absge0d nndivred posdifd cvgratnnlemseq fveq2d cvgratnnlemabsle eqbrtrd - 1red resubcld cc0 cvgratnnlemfm nn0zd rpexpcld fsumge0 cvgratnnlemsumlt - rpge0d ltmul12ad lelttrd recnd nncnd nnap0d div23apd breqtrrd ) AFUADOU - BZPZEYJPZUCQZUDPZOOBUEQZOUCQZUEQZBUEQZODPZUDPZOUAQZUFQZEUEQZBOBUCQZUEQZ - UFQZUUBUUEUFQEUEQUGAYNEDPZUDPZEOUAQZFUHQZBNUIZEUCQZUKQZNUJZUFQZUUFAYMAY - KYLARSFYJACDORULAUMJUNZAERTFEUOPTZFRTLMFEUPVCUQARSEYJUUPLUQURUSAUUHUUNA - UUGACUIZDPZSTZUUGSTCREUUREUTUUSUUGSUUREDVAVBAUUTCRJVDZLVEZUSZAUUJUUMNAU - UIFAEAELVFZVGAUUQFVHTMEFVIVJVKZAUUKUUJTZVLZBUULABVNTUVFGVMUVGEUUKVOVPZU - ULVQTZUVGEUUIUUKAEVNTZUVFAELVRVMZUVGUVJUUIVNTUVKEVSVJUVGUUKUVFUUKVHTZAU - UKUUIFVTZWAWBUVGEUVKWCUVFUUIUUKVOVPAUUKUUIFWDWAWFAEVHTUVLUVHUVIWGUVFUVD - UVMEUUKWEWHWIZWJZWKZWLAUUCUUEAUUBEAUUBAYRUUAAYQBAYPAOYOUGVPZYPWPTZABOUG - VPZUVQHABABGIWMZWNWIAOVNTYOVNTUVQUVRWGWQABUVTWOOYOWRWSWIWTUVTXAAYTAYSAU - UTYSSTCROUUROUTUUSYSSUURODVAVBUVAORTAXBXCVEZUSAYSUWAXGXDXEXFZLXHZAUUEAB - UUDUVTAUUDAOBAXNZGXOAUVSXPUUDUGVPHABOGUWDXIWIWMXAXFZWLAYNUUJUUKDPNUJZUD - PUUOVOAYMUWFUDABNCDEFGHIJKLMXJXKABNCDEFGHIJKLMXLXMAUUHUUCUUNUUEUVCUWCUV - PUWEAUUGUVBXGABCDEGHIJKLXQAUUJUUMNUVEUVOUVGUUMUVGBUULABWPTUVFUVTVMUVGUU - LUVNXRXSYBXTABNCDEFGHIJKLMYAYCYDAUUBUUEEAUUBUWBYEAUUEUWEYEAELYFAELYGYHY - I $. + cexp nnuz 1zzd serf eluznn syl2anc ffvelcdmd subcld abscld fveq2 eleq1d + cuz wceq ralrimiva rspcdva nnzd peano2zd cz eluzelz fzfigd wa cr adantr + syl cle wbr cn0 nnred peano2re elfzelz zred lep1d elfzle1 letrd znn0sub + adantl wb syl2an mpbid reexpcld fsumrecl remulcld crp elrpd reclt1d 1re + rprecred difrp sylancr rpreccld rpdivcld 1nn a1i absge0d ge0p1rpd rpred + rpmulcld nndivred 1red resubcld posdifd cvgratnnlemseq cvgratnnlemabsle + cc0 fveq2d eqbrtrd cvgratnnlemfm nn0zd rpexpcld rpge0d cvgratnnlemsumlt + fsumge0 ltmul12ad lelttrd recnd nncnd nnap0d div23apd breqtrrd ) AFUADO + UBZPZEYJPZUCQZUDPZOOBUEQZOUCQZUEQZBUEQZODPZUDPZOUAQZUFQZEUEQZBOBUCQZUEQ + ZUFQZUUBUUEUFQEUEQUGAYNEDPZUDPZEOUAQZFUHQZBNUIZEUCQZUKQZNUJZUFQZUUFAYMA + YKYLARSFYJACDORULAUMJUNZAERTFEVBPTZFRTLMFEUOUPUQARSEYJUUPLUQURUSAUUHUUN + AUUGACUIZDPZSTZUUGSTCREUUREVCUUSUUGSUUREDUTVAAUUTCRJVDZLVEZUSZAUUJUUMNA + UUIFAEAELVFZVGAUUQFVHTMEFVIVNVJZAUUKUUJTZVKZBUULABVLTUVFGVMUVGEUUKVOVPZ + UULVQTZUVGEUUIUUKAEVLTZUVFAELVRVMZUVGUVJUUIVLTUVKEVSVNUVGUUKUVFUUKVHTZA + UUKUUIFVTZWFWAUVGEUVKWBUVFUUIUUKVOVPAUUKUUIFWCWFWDAEVHTUVLUVHUVIWGUVFUV + DUVMEUUKWEWHWIZWJZWKZWLAUUCUUEAUUBEAUUBAYRUUAAYQBAYPAOYOUGVPZYPWMTZABOU + GVPZUVQHABABGIWNZWOWIAOVLTYOVLTUVQUVRWGWPABUVTWQOYOWRWSWIWTUVTXAAYTAYSA + UUTYSSTCROUUROVCUUSYSSUURODUTVAUVAORTAXBXCVEZUSAYSUWAXDXEXGXFZLXHZAUUEA + BUUDUVTAUUDAOBAXIZGXJAUVSXNUUDUGVPHABOGUWDXKWIWNXAXFZWLAYNUUJUUKDPNUJZU + DPUUOVOAYMUWFUDABNCDEFGHIJKLMXLXOABNCDEFGHIJKLMXMXPAUUHUUCUUNUUEUVCUWCU + VPUWEAUUGUVBXDABCDEGHIJKLXQAUUJUUMNUVEUVOUVGUUMUVGBUULABWMTUVFUVTVMUVGU + ULUVNXRXSXTYBABNCDEFGHIJKLMYAYCYDAUUBUUEEAUUBUWBYEAUUEUWEYEAELYFAELYGYH + YI $. $} ${ @@ -124599,26 +124618,26 @@ Infinite sums (cont.) clim2prod $p |- ( ph -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` N ) x. A ) ) $= ( cmul cfv co cc wcel wceq wi fveq2 oveq2d vx vn vv cseq c1 caddc cvv cuz - eqid cz uzssz eqsstri sselid peano2zd eleqtrdi eluzel2 syl prodf ffvelrnd - seqex a1i cv wss peano2uz uzss sseqtrrdi sselda syldan ffvelcdmda eqeq12d - 3syl imbi2d eleq2i sylan2br wa adantl seq3p1 seq3-1 eqtr4d adantlr adantr - mulcl oveq1 eleqtrrdi wral ralrimiva eleq1d rspcv mpan9 simpr exp31 com12 - mulassd 3eqtrd a2d uzind4 impcom climmulc2 ) ABFLDEUDZMZCLDFUEUFNZUDZWSXA - UGXAUHMZXCUIZAFAGUJFGEUHMZUJHEUKULIUMUNZKAGOFWSACDEGHAFXEPZEUJPAFGXEIHUOZ - EFUPUQJURIUSZWSUGPALDEUTVAAXCOCVBZXBACDXAXCXDXFAXJXCPZXJGPZXJDMZOPZAXCGXJ - AXCXEGAXGXAXEPXCXEVCXHEFVDEXAVEVKZHVFVGJVHZURZVIXKAXJWSMZWTXJXBMZLNZQZAUA - VBZWSMZWTYBXBMZLNZQZRAXAWSMZWTXAXBMZLNZQZRZAUBVBZWSMZWTYLXBMZLNZQZRAYLUEU - FNZWSMZWTYQXBMZLNZQZRAYARUAUBXAXJYBXAQZYFYJAUUBYCYGYEYIYBXAWSSUUBYDYHWTLY - BXAXBSTVJVLYBYLQZYFYPAUUCYCYMYEYOYBYLWSSUUCYDYNWTLYBYLXBSTVJVLYBYQQZYFUUA - AUUDYCYRYEYTYBYQWSSUUDYDYSWTLYBYQXBSTVJVLYBXJQZYFYAAUUEYCXRYEXTYBXJWSSUUE - YDXSWTLYBXJXBSTVJVLYKXAUJPAYGWTXADMZLNYIACUCLODEFXHXJXEPZAXLXNGXEXJHVMJVN - ZXJOPUCVBZOPVOZXJUUILNOPZAXJUUIWBZVPZVQAYHUUFWTLACUCLODXAXFXPUUMVRTVSVAYL - XCPZAYPUUAAUUNYPUUARAUUNYPUUAAUUNVOZYPVOZYRYMYQDMZLNZYOUUQLNZYTUUOYRUURQY - PUUOCUCLODEYLAXCXEYLXOVGZAUUGXNUUNUUHVTUUJUUKUUOUULVPZVQWAYPUURUUSQUUOYMY - OUUQLWCVPUUPUUSWTYNUUQLNZLNZYTUUOUUSUVCQYPUUOWTYNUUQAWTOPUUNXIWAAXCOYLXBX - QVIAUUNYQGPZUUQOPZUUOYLXEPZUVDUUTUVFYQXEGEYLVDHWDUQAXNCGWEUVDUVEAXNCGJWFX - NUVECYQGXJYQQXMUUQOXJYQDSWGWHWIVHWMWAUUOYTUVCQYPUUOYSUVBWTLUUOCUCLODXAYLA - UUNWJAXKXNUUNXPVTUVAVQTWAVSWNWKWLWOWPWQWR $. + eqid uzssz eqsstri sselid peano2zd eleqtrdi eluzel2 prodf ffvelcdmd seqex + syl a1i wss peano2uz uzss 3syl sseqtrrdi sselda syldan ffvelcdmda eqeq12d + cz cv imbi2d eleq2i sylan2br wa mulcl adantl seq3p1 seq3-1 eqtr4d adantlr + adantr oveq1 eleqtrrdi wral eleq1d rspcv mpan9 mulassd simpr 3eqtrd exp31 + ralrimiva com12 a2d uzind4 impcom climmulc2 ) ABFLDEUDZMZCLDFUEUFNZUDZWSX + AUGXAUHMZXCUIZAFAGVJFGEUHMZVJHEUJUKIULUMZKAGOFWSACDEGHAFXEPZEVJPAFGXEIHUN + ZEFUOUSJUPIUQZWSUGPALDEURUTAXCOCVKZXBACDXAXCXDXFAXJXCPZXJGPZXJDMZOPZAXCGX + JAXCXEGAXGXAXEPXCXEVAXHEFVBEXAVCVDZHVEVFJVGZUPZVHXKAXJWSMZWTXJXBMZLNZQZAU + AVKZWSMZWTYBXBMZLNZQZRAXAWSMZWTXAXBMZLNZQZRZAUBVKZWSMZWTYLXBMZLNZQZRAYLUE + UFNZWSMZWTYQXBMZLNZQZRAYARUAUBXAXJYBXAQZYFYJAUUBYCYGYEYIYBXAWSSUUBYDYHWTL + YBXAXBSTVIVLYBYLQZYFYPAUUCYCYMYEYOYBYLWSSUUCYDYNWTLYBYLXBSTVIVLYBYQQZYFUU + AAUUDYCYRYEYTYBYQWSSUUDYDYSWTLYBYQXBSTVIVLYBXJQZYFYAAUUEYCXRYEXTYBXJWSSUU + EYDXSWTLYBXJXBSTVIVLYKXAVJPAYGWTXADMZLNYIACUCLODEFXHXJXEPZAXLXNGXEXJHVMJV + NZXJOPUCVKZOPVOZXJUUILNOPZAXJUUIVPZVQZVRAYHUUFWTLACUCLODXAXFXPUUMVSTVTUTY + LXCPZAYPUUAAUUNYPUUARAUUNYPUUAAUUNVOZYPVOZYRYMYQDMZLNZYOUUQLNZYTUUOYRUURQ + YPUUOCUCLODEYLAXCXEYLXOVFZAUUGXNUUNUUHWAUUJUUKUUOUULVQZVRWBYPUURUUSQUUOYM + YOUUQLWCVQUUPUUSWTYNUUQLNZLNZYTUUOUUSUVCQYPUUOWTYNUUQAWTOPUUNXIWBAXCOYLXB + XQVHAUUNYQGPZUUQOPZUUOYLXEPZUVDUUTUVFYQXEGEYLVBHWDUSAXNCGWEUVDUVEAXNCGJWM + XNUVECYQGXJYQQXMUUQOXJYQDSWFWGWHVGWIWBUUOYTUVCQYPUUOYSUVBWTLUUOCUCLODXAYL + AUUNWJAXKXNUUNXPWAUVAVRTWBVTWKWLWNWOWPWQWR $. $} ${ @@ -124636,19 +124655,19 @@ Infinite sums (cont.) clim2divap $p |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) ) $= ( cmul co cfv cdiv wcel syl cc cv vj vx vy c1 caddc cseq cli cvv cuz eqid - cz eluzelz eleq2s peano2zd eluzel2 prodf ffvelrnd recclapd seqex eleqtrdi - a1i peano2uz eleqtrrdi uztrn2 sylan ffvelcdmda syldan wa mulcl adantl w3a - wceq mulass simpr adantr eleq2i sylan2br adantlr seq3split eqcomd cc0 cap - wbr divmulapd mpbird divrecap2d eqtr3d climmulc2 climcl breqtrrd ) AMDFUD - UENZUFZUDFMDEUFZOZPNZBMNBWNPNUGABWOUAWMWLWKUHWKUIOZWPUJZAFAFGQZFUKQZIWSFE - UIOZGEFULHUMRUNZKAWNAGSFWMACDEGHAWREUKQZIXBFWTGEFUOHUMRJUPZIUQZLURWLUHQAM - DWKUSVAAUATZWPQZXEGQZXEWMOZSQAWKGQZXFXGAWKWTGAFWTQZWKWTQAFGWTIHUTZEFVBRHV - CZEXEWKGHVDVEAGSXEWMXCVFVGZAXFVHZXHWNPNZXEWLOZWOXHMNXNXOXPVLWNXPMNZXHVLXN - XHXQXNCUBUCMSDEFXECTZSQZUBTZSQZVHXRXTMNZSQXNXRXTVIVJXSYAUCTZSQVKYBYCMNXRX - TYCMNMNVLXNXRXTYCVMVJAXFVNAXJXFXKVOAXRWTQZXRDOSQZXFYDAXRGQZYEGWTXRHVPJVQV - RVSVTXNXHWNXPXMAWNSQXFXDVOZAWPSXEWLACDWKWPWQXAAXRWPQZYFYEAXIYHYFXLEXRWKGH - VDVEJVGUPVFAWNWAWBWCXFLVOZWDWEXNXHWNXMYGYIWFWGWHABWNAWMBUGWCBSQKBWMWIRXDL - WFWJ $. + cz eluzelz eleq2s peano2zd eluzel2 prodf recclapd seqex eleqtrdi peano2uz + ffvelcdmd a1i eleqtrrdi uztrn2 sylan ffvelcdmda syldan wa wceq adantl w3a + mulcl mulass simpr adantr eleq2i sylan2br adantlr seq3split cc0 divmulapd + eqcomd cap wbr mpbird divrecap2d eqtr3d climmulc2 climcl breqtrrd ) AMDFU + DUENZUFZUDFMDEUFZOZPNZBMNBWNPNUGABWOUAWMWLWKUHWKUIOZWPUJZAFAFGQZFUKQZIWSF + EUIOZGEFULHUMRUNZKAWNAGSFWMACDEGHAWREUKQZIXBFWTGEFUOHUMRJUPZIVAZLUQWLUHQA + MDWKURVBAUATZWPQZXEGQZXEWMOZSQAWKGQZXFXGAWKWTGAFWTQZWKWTQAFGWTIHUSZEFUTRH + VCZEXEWKGHVDVEAGSXEWMXCVFVGZAXFVHZXHWNPNZXEWLOZWOXHMNXNXOXPVIWNXPMNZXHVIX + NXHXQXNCUBUCMSDEFXECTZSQZUBTZSQZVHXRXTMNZSQXNXRXTVLVJXSYAUCTZSQVKYBYCMNXR + XTYCMNMNVIXNXRXTYCVMVJAXFVNAXJXFXKVOAXRWTQZXRDOSQZXFYDAXRGQZYEGWTXRHVPJVQ + VRVSWBXNXHWNXPXMAWNSQXFXDVOZAWPSXEWLACDWKWPWQXAAXRWPQZYFYEAXIYHYFXLEXRWKG + HVDVEJVGUPVFAWNVTWCWDXFLVOZWAWEXNXHWNXMYGYIWFWGWHABWNAWMBUGWDBSQKBWMWIRXD + LWFWJ $. $} ${ @@ -124740,9 +124759,9 @@ seq M ( x. , ( Z X. { 1 } ) ) = ( Z X. { 1 } ) ) $= ( co wcel cmul cfv c1 cdiv wceq wi cc vm vn vv cfz cseq cuz eluzfz2 caddc cv fveq2 oveq2d eqeq12d imbi2d eluzfz1 expcom vtoclga mpcom cz eluzel2 wa syl mulcl adantl seq3-1 3eqtr4d a1i cfzo w3a 3ad2ant3 fzofzp1 impcom 1cnd - oveq1 wf eqid prodf adantr elfzouz ffvelrnd eleq1d elfzuz adantlr cc0 cap - wbr wss elfzouz2 fzss2 sselda sylan2 anassrs prodfap0 breq1d divmuldivapd - 1t1e1 oveq1i eqtrdi eqtrd 3adant3 3ad2antl1 seq3p1 3exp com12 a2d fzind2 + oveq1 wf prodf adantr elfzouz ffvelcdmd eleq1d elfzuz adantlr cc0 cap wbr + eqid wss elfzouz2 fzss2 sselda sylan2 anassrs prodfap0 divmuldivapd 1t1e1 + breq1d oveq1i eqtrdi eqtrd 3adant3 3ad2antl1 seq3p1 3exp com12 a2d fzind2 ) FEFUDLZMZAFNDEUEZOZPFNCEUEZOZQLZRZAFEUFOZMZXGGEFUGVAAUAUIZXHOZPXPXJOZQL ZRZSAEXHOZPEXJOZQLZRZSZAUBUIZXHOZPYFXJOZQLZRZSAYFPUHLZXHOZPYKXJOZQLZRZSAX MSUAUBFEFXPERZXTYDAYPXQYAXSYCXPEXHUJYPXRYBPQXPEXJUJUKULUMXPYFRZXTYJAYQXQY @@ -124755,12 +124774,12 @@ seq M ( x. , ( Z X. { 1 } ) ) = ( Z X. { 1 } ) ) $= YKCOZNLZQLZYLYNUVAUVCYIUVBNLZUVFYJAUVCUVGRUUTYGYIUVBNVMVIAUUTUVGUVFRYJAUU TUTZUVGYIPUVDQLZNLZUVFUVHUVBUVIYINUUTAUVBUVIRZUUTYKXFMZAUVKSZEFYFVJZUUJUV MBYKXFUUEYKRZUUIUVKAUVOUUFUVBUUHUVIUUEYKDUJUVOUUGUVDPQUUEYKCUJZUKULUMUUMU - PVAVKUKUVHUVJPPNLZUVEQLUVFUVHPYHPUVDUVHVLZUVHXNTYFXJAXNTXJVNUUTABCEXNXNVO - UUNHVPVQUUTYFXNMZAYFEFVRVCZVSUVRUUTAUVDTMZUUTUVLAUWASZUVNAUUGTMZSZUWBBYKX - FUVOUWCUWAAUVOUUGUVDTUVPVTUMUULUUEXNMZUWDUUEEFWAAUWEUWCHUOVAUPVAVKUVHBCEY - FUVTAUWEUWCUUTHWBAUUTUUEEYFUDLZMZUUGWCWDWEZUUTUWGUTAUULUWHUUTUWFXFUUEUUTF - YFUFOMUWFXFWFYFEFWGYFEFWHVAWIIWJWKWLUUTAUVDWCWDWEZUUTUVLAUWISZUVNAUWHSUWJ - BYKXFUVOUWHUWIAUVOUUGUVDWCWDUVPWMUMAUULUWHIUOUPVAVKWNUVQPUVEQWOWPWQWRWSWR + PVAVKUKUVHUVJPPNLZUVEQLUVFUVHPYHPUVDUVHVLZUVHXNTYFXJAXNTXJVNUUTABCEXNXNWE + UUNHVOVPUUTYFXNMZAYFEFVQVCZVRUVRUUTAUVDTMZUUTUVLAUWASZUVNAUUGTMZSZUWBBYKX + FUVOUWCUWAAUVOUUGUVDTUVPVSUMUULUUEXNMZUWDUUEEFVTAUWEUWCHUOVAUPVAVKUVHBCEY + FUVTAUWEUWCUUTHWAAUUTUUEEYFUDLZMZUUGWBWCWDZUUTUWGUTAUULUWHUUTUWFXFUUEUUTF + YFUFOMUWFXFWFYFEFWGYFEFWHVAWIIWJWKWLUUTAUVDWBWCWDZUUTUVLAUWISZUVNAUWHSUWJ + BYKXFUVOUWHUWIAUVOUUGUVDWBWCUVPWOUMAUULUWHIUOUPVAVKWMUVQPUVEQWNWPWQWRWSWR UVABUCNTDEYFAUUTUVSYJUVTWSZAUUTUWEUUFTMYJKWTUUPUUQUVAUURVCZXAUVAYMUVEPQUV ABUCNTCEYFUWKAUUTUWEUWCYJHWTUWLXAUKVEXBXCXDXEUQ $. $} @@ -124782,15 +124801,15 @@ seq M ( x. , ( Z X. { 1 } ) ) = ( Z X. { 1 } ) ) $= ( vn cmul cseq cfv cdiv co wcel cc cuz cmpt cfz cc0 cap wbr elfzuz sylan2 c1 cv wceq wa eqid fveq2 oveq2d simpr recclapd fvmptd3 eqeltrd prodfrecap weq eleq1w anbi2d eleq1d imbi12d chvarvv breq1d fmpttd ffvelcdmda 3eqtr4d - wi divrecapd prod3fmul cz eluzel2 syl prodf ffvelrnd prodfap0 ) AGNCFOZPZ - GNMFUAPZUIMUJZDPZQRZUBZFOPZNRWAUIGNDFOZPZQRZNRGNEFOPWAWIQRAWGWJWANABDWFFG - HJBUJZFGUCRSZAWKWBSZWKDPZUDUEUFZWKFGUGZKUHZWLAWMWKWFPZUIWNQRZUKWPAWMULZMW - KWEWSWBWFTWFUMMBVAWDWNUIQWCWKDUNUOAWMUPWTWNJKUQZURZUHWTWRWSTXBXAUSUTUOABC - WFEFGHIAWBTWKWFAMWBWETAWCWBSZULZWDWTWNTSZVKXDWDTSZVKBMBMVAZWTXDXEXFXGWMXC - ABMWBVBVCZXGWNWDTWKWCDUNZVDVEJVFWTWOVKXDWDUDUEUFZVKBMXGWTXDWOXJXHXGWNWDUD - UEXIVGVEKVFUQVHVIWTWKCPZWNQRXKWSNRWKEPXKWRNRWTXKWNIJKVLLWTWRWSXKNXBUOVJVM - AWAWIAWBTGVTABCFWBWBUMZAGWBSFVNSHFGVOVPZIVQHVRAWBTGWHABDFWBXLXMJVQHVRABDF - GHJWQVSVLVJ $. + wi divrecapd prod3fmul cz eluzel2 syl prodf ffvelcdmd prodfap0 ) AGNCFOZP + ZGNMFUAPZUIMUJZDPZQRZUBZFOPZNRWAUIGNDFOZPZQRZNRGNEFOPWAWIQRAWGWJWANABDWFF + GHJBUJZFGUCRSZAWKWBSZWKDPZUDUEUFZWKFGUGZKUHZWLAWMWKWFPZUIWNQRZUKWPAWMULZM + WKWEWSWBWFTWFUMMBVAWDWNUIQWCWKDUNUOAWMUPWTWNJKUQZURZUHWTWRWSTXBXAUSUTUOAB + CWFEFGHIAWBTWKWFAMWBWETAWCWBSZULZWDWTWNTSZVKXDWDTSZVKBMBMVAZWTXDXEXFXGWMX + CABMWBVBVCZXGWNWDTWKWCDUNZVDVEJVFWTWOVKXDWDUDUEUFZVKBMXGWTXDWOXJXHXGWNWDU + DUEXIVGVEKVFUQVHVIWTWKCPZWNQRXKWSNRWKEPXKWRNRWTXKWNIJKVLLWTWRWSXKNXBUOVJV + MAWAWIAWBTGVTABCFWBWBUMZAGWBSFVNSHFGVOVPZIVQHVRAWBTGWHABDFWBXLXMJVQHVRABD + FGHJWQVSVLVJ $. $} @@ -124811,17 +124830,17 @@ seq n ( x. , F ) ~~> y ) ) $. ntrivcvgap $p |- ( ph -> seq M ( x. , F ) e. dom ~~> ) $= ( cv wbr cmul cseq cli wa wcel c1 co cc cc0 cap wex wrex cdm wceq cuz cfv cmin wo uzm1 eleq2s ad2antlr wi seqeq1 breq1d seqex breldm syl6bi adantld - vex cvv cz eluzel2 ad3antlr ad5ant15 simplr ffvelrnd climcl adantl mulcld - prodf caddc uzssz eqsstri sselid zcnd 1cnd npcand seqeq1d biimpar breldmg - clim2prod mp3an2i an32s expcom eqcomi jaoi mpcom ex exlimdv rexlimdva mpd - ) ABKZUAUBLZMEDKZNZWNOLZPZBUCZDGUDMEFNZOUEQZIAWTXBDGAWPGQZPZWSXBBXDWRXBWO - XDWRXBWPFUFZWPRUISZFUGUHZQZUJZXDWRPZXBXCXIAWRXIWPXGGFWPUKHULUMXEXJXBUNZXH - XEWRXBXDXEWRXAWNOLXBXEWQXAWNOMEWPFUOUPXAWNOMEFUQZBVAURUSUTXKXFGXGXJXFGQZX - BXDXMWRXBXAVBQXDXMPZWRPZXFXAUHZWNMSZTQXAXQOLXBXLXOXPWNXOGTXFXAXOCEFGHXCFV - CQZAXMWRXRWPXGGFWPVDHULVEACKZGQXSEUHTQXCXMWRJVFZVLXDXMWRVGZVHWRWNTQXNWNWQ - VIVJVKXOWNCEFXFGHYAXTXNMEXFRVMSZNZWNOLWRXNYCWQWNOXNYBWPMEXNWPRXNWPXNGVCWP - GXGVCHFVNVOAXCXMVGVPVQXNVRVSVTUPWAWCXAXQVBTOWBWDWEWFGXGHWGULWHWIWJUTWKWLW - M $. + vex cvv cz eluzel2 ad3antlr ad5ant15 prodf simplr ffvelcdmd climcl adantl + mulcld caddc eqsstri sselid zcnd npcand seqeq1d biimpar clim2prod breldmg + uzssz 1cnd mp3an2i an32s expcom eqcomi jaoi mpcom exlimdv rexlimdva mpd + ex ) ABKZUAUBLZMEDKZNZWNOLZPZBUCZDGUDMEFNZOUEQZIAWTXBDGAWPGQZPZWSXBBXDWRX + BWOXDWRXBWPFUFZWPRUISZFUGUHZQZUJZXDWRPZXBXCXIAWRXIWPXGGFWPUKHULUMXEXJXBUN + ZXHXEWRXBXDXEWRXAWNOLXBXEWQXAWNOMEWPFUOUPXAWNOMEFUQZBVAURUSUTXKXFGXGXJXFG + QZXBXDXMWRXBXAVBQXDXMPZWRPZXFXAUHZWNMSZTQXAXQOLXBXLXOXPWNXOGTXFXAXOCEFGHX + CFVCQZAXMWRXRWPXGGFWPVDHULVEACKZGQXSEUHTQXCXMWRJVFZVGXDXMWRVHZVIWRWNTQXNW + NWQVJVKVLXOWNCEFXFGHYAXTXNMEXFRVMSZNZWNOLWRXNYCWQWNOXNYBWPMEXNWPRXNWPXNGV + CWPGXGVCHFWBVNAXCXMVHVOVPXNWCVQVRUPVSVTXAXQVBTOWAWDWEWFGXGHWGULWHWIWMUTWJ + WKWL $. $} ${ @@ -124976,24 +124995,24 @@ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) ) \/ vj cfv wss wdc cc0 cap cif cmpt wex cfz co cle csb wo cio cprod nfra1 nfv wf1o nfan simp-4l simpr rsp sylc adantllr simpllr simplrl simplrr adantlr ifeq1dadc mpteq2da seqeq3d breq1d anbi2d exbidv rexbidva anbi12d pm5.32da - sumdc wf f1of ad3antlr simplr wb simp-4r nnzd fznn syl mpbir2and ffvelrnd - nfcsb1v nfeq csbeq1a eqeq12d rspc syl2anc mpteq2dva fveq1d eqeq2d orbi12d - zdcle iotabidv df-proddc 3eqtr4g ) BCUAZDAUBZAEJZUCUEZUFZUDJAKUGUDXKUBZLZ - FJZUHUIMZNDODJZAKZBPUJZUKZGJZQZXORMZLZFULZGXKSZNXTXJQZHJZRMZLZLZEOSZPXJUM - UNZAIJZVBZYHXJNGTYAXJUOMZDYAYNUEZBUPZPUJZUKZPQZUEZUAZLZIULZETSZUQZHURXNXP - NDOXRCPUJZUKZYAQZXORMZLZFULZGXKSZNUUIXJQZYHRMZLZLZEOSZYOYHXJNGTYPDYQCUPZP - UJZUKZPQZUEZUAZLZIULZETSZUQZHURABDUSACDUSXIUUGUVIHXIYLUUSUUFUVHXIYKUUREOX - IXJOKZLZXNYJUUQUVKXNLZYFUUNYIUUPUVLYEUUMGXKUVLYAXKKZLZYDUULFUVNYCUUKXPUVN - YBUUJXORUVNXTUUINYAUVNDOXSUUHUVLUVMDUVKXNDXIUVJDXHDAUTUVJDVAVCXNDVAVCZUVM - DVAVCUVNXQOKZLXRBCPUVLUVPXRXHUVMUVLUVPLZXRLXIXRXHXIUVJXNUVPXRVDUVQXRVEXHD - AVFVGZVHUVLUVPXRUGUVMUVQUDAXJXQXIUVJXNUVPVIUVKXLXMUVPVJUVKXLXMUVPVKUVLUVP - VEWBZVLVMVNVOVPVQVRVSUVLYGUUOYHRUVLXTUUINXJUVLDOXSUUHUVOUVQXRBCPUVRUVSVMV - NVOVPVTWAVSXIUUEUVGETXIXJTKZLZUUDUVFIUWAYOUUCUVEUWAYOLZUUBUVDYHUWBXJUUAUV - CUWBYTUVBNPUWBGTYSUVAUWBYATKZLZYPYRUUTPUWDYPLZYQAKXIYRUUTUAZUWEYMAYAYNYOY - MAYNWCUWAUWCYPYMAYNWDWEUWEYAYMKZUWCYPUWBUWCYPWFUWDYPVEUWEUVJUWGUWCYPLWGUW - EXJXIUVTYOUWCYPWHWIYAXJWJWKWLWMXIUVTYOUWCYPVDXHUWFDYQADYRUUTDYQBWNDYQCWNW - OXQYQUABYRCUUTDYQBWPDYQCWPWQWRVGUWDYAOKUVJYPUGUWDYAUWBUWCVEWIUWDXJXIUVTYO - UWCVIWIYAXJXDWSVMWTVOXAXBWAVRVSXCXEHFABIUDDEGXFHFACIUDDEGXFXG $. + sumdc wf f1of ad3antlr simplr wb simp-4r nnzd mpbir2and ffvelcdmd nfcsb1v + fznn syl nfeq csbeq1a eqeq12d rspc zdcle syl2anc mpteq2dva fveq1d orbi12d + eqeq2d iotabidv df-proddc 3eqtr4g ) BCUAZDAUBZAEJZUCUEZUFZUDJAKUGUDXKUBZL + ZFJZUHUIMZNDODJZAKZBPUJZUKZGJZQZXORMZLZFULZGXKSZNXTXJQZHJZRMZLZLZEOSZPXJU + MUNZAIJZVBZYHXJNGTYAXJUOMZDYAYNUEZBUPZPUJZUKZPQZUEZUAZLZIULZETSZUQZHURXNX + PNDOXRCPUJZUKZYAQZXORMZLZFULZGXKSZNUUIXJQZYHRMZLZLZEOSZYOYHXJNGTYPDYQCUPZ + PUJZUKZPQZUEZUAZLZIULZETSZUQZHURABDUSACDUSXIUUGUVIHXIYLUUSUUFUVHXIYKUUREO + XIXJOKZLZXNYJUUQUVKXNLZYFUUNYIUUPUVLYEUUMGXKUVLYAXKKZLZYDUULFUVNYCUUKXPUV + NYBUUJXORUVNXTUUINYAUVNDOXSUUHUVLUVMDUVKXNDXIUVJDXHDAUTUVJDVAVCXNDVAVCZUV + MDVAVCUVNXQOKZLXRBCPUVLUVPXRXHUVMUVLUVPLZXRLXIXRXHXIUVJXNUVPXRVDUVQXRVEXH + DAVFVGZVHUVLUVPXRUGUVMUVQUDAXJXQXIUVJXNUVPVIUVKXLXMUVPVJUVKXLXMUVPVKUVLUV + PVEWBZVLVMVNVOVPVQVRVSUVLYGUUOYHRUVLXTUUINXJUVLDOXSUUHUVOUVQXRBCPUVRUVSVM + VNVOVPVTWAVSXIUUEUVGETXIXJTKZLZUUDUVFIUWAYOUUCUVEUWAYOLZUUBUVDYHUWBXJUUAU + VCUWBYTUVBNPUWBGTYSUVAUWBYATKZLZYPYRUUTPUWDYPLZYQAKXIYRUUTUAZUWEYMAYAYNYO + YMAYNWCUWAUWCYPYMAYNWDWEUWEYAYMKZUWCYPUWBUWCYPWFUWDYPVEUWEUVJUWGUWCYPLWGU + WEXJXIUVTYOUWCYPWHWIYAXJWMWNWJWKXIUVTYOUWCYPVDXHUWFDYQADYRUUTDYQBWLDYQCWL + WOXQYQUABYRCUUTDYQBWPDYQCWPWQWRVGUWDYAOKUVJYPUGUWDYAUWBUWCVEWIUWDXJXIUVTY + OUWCVIWIYAXJWSWTVMXAVOXBXDWAVRVSXCXEHFABIUDDEGXFHFACIUDDEGXFXG $. $} ${ @@ -125177,21 +125196,21 @@ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) ) \/ seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) ) $= ( cfv wcel cz cc wa c1 wceq adantl vn vm vv cmul cseq cvv cuz eluzelz syl seqex a1i eluzel2 cv cif wn iftrue adantlr eqeltrd iffalse ax-1cn - eqeltrdi wdc wo exmiddc mpjaodan fvmpt2 syl2anc prodf ffvelrnd mulid1 - eqid co adantr simpr caddc cfz elfzuz cdif sseld fznuz syl6 con2d imp - eldifd fveqeq2 eldifi eldifn eqtrd sylan2 fveq2 eleq1d wral ralrimiva - vtoclga ad2antrr rspcdva mulcl seq3id2 eqcomd climconst ) AGUDEFUEZMZ - UAXAGUFGUGMZXCVKAGFUGMZNZGONKFGUHUIXAUFNAUDEFUJUKAXDPGXAADEFXDXDVKZAX - EFONZKFGULUIZADUMZXDNZQZXIEMZXIBNZCRUNZPXKXIONZXNPNZXLXNSZXJXOAFXIUHT - XKXMXPXMUOZXKXMQXNCPXMXNCSXKXMCRUPTAXMCPNXJIUQURXRXPXKXRXNRPXMCRUSZUT - VATXKXMVBXMXRVCJXMVDUIVEZDOXNPEHVFZVGXTURZVHKVIAUAUMZXCNZQZXBYCXAMYEU - BUCUDPEGFYCRUBUMZPNZYFRUDVLYFSYEYFVJTAXEYDKVMZAYDVNYEXDPGXAYEDEFXDXFA - XGYDXHVMAXJXLPNZYDYBUQVHYHVIAYFGRVOVLZYCVPVLNZYFEMZRSZYDYKAYFYJUGMNZY - MYFYJYCVQAYNQZYFOBVRZNYMYOYFOBYNYFONAYJYFUHTAYNYFBNZUOAYQYNAYQYFFGVPV - LZNYNUOABYRYFLVSYFFGVTWAWBWCWDXLRSYMDYFYPXIYFREWEXIYPNZXLXNRYSXOXPXQX - IOBWFYSXNRPYSXRXNRSXIOBWGXSUIZUTVAYAVGYTWHWNUIWIUQYEYFXDNZQYIYLPNDXDY - FXIYFSXLYLPXIYFEWJWKAYIDXDWLYDUUAAYIDXDYBWMWOYEUUAVNWPYGUCUMZPNQYFUUB - UDVLPNYEYFUUBWQTWRWSWT $. + eqid eqeltrdi wdc wo exmiddc mpjaodan fvmpt2 syl2anc ffvelcdmd mulid1 + prodf co adantr simpr caddc elfzuz cdif sseld fznuz syl6 con2d eldifd + cfz fveqeq2 eldifi eldifn eqtrd vtoclga sylan2 fveq2 eleq1d ralrimiva + imp wral ad2antrr rspcdva mulcl seq3id2 eqcomd climconst ) AGUDEFUEZM + ZUAXAGUFGUGMZXCVAAGFUGMZNZGONKFGUHUIXAUFNAUDEFUJUKAXDPGXAADEFXDXDVAZA + XEFONZKFGULUIZADUMZXDNZQZXIEMZXIBNZCRUNZPXKXIONZXNPNZXLXNSZXJXOAFXIUH + TXKXMXPXMUOZXKXMQXNCPXMXNCSXKXMCRUPTAXMCPNXJIUQURXRXPXKXRXNRPXMCRUSZU + TVBTXKXMVCXMXRVDJXMVEUIVFZDOXNPEHVGZVHXTURZVKKVIAUAUMZXCNZQZXBYCXAMYE + UBUCUDPEGFYCRUBUMZPNZYFRUDVLYFSYEYFVJTAXEYDKVMZAYDVNYEXDPGXAYEDEFXDXF + AXGYDXHVMAXJXLPNZYDYBUQVKYHVIAYFGRVOVLZYCWCVLNZYFEMZRSZYDYKAYFYJUGMNZ + YMYFYJYCVPAYNQZYFOBVQZNYMYOYFOBYNYFONAYJYFUHTAYNYFBNZUOAYQYNAYQYFFGWC + VLZNYNUOABYRYFLVRYFFGVSVTWAWMWBXLRSYMDYFYPXIYFREWDXIYPNZXLXNRYSXOXPXQ + XIOBWEYSXNRPYSXRXNRSXIOBWFXSUIZUTVBYAVHYTWGWHUIWIUQYEYFXDNZQYIYLPNDXD + YFXIYFSXLYLPXIYFEWJWKAYIDXDWNYDUUAAYIDXDYBWLWOYEUUAVNWPYGUCUMZPNQYFUU + BUDVLPNYEYFUUBWQTWRWSWT $. $} $} @@ -125246,41 +125265,41 @@ seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) ) $= syl syl2anc ancomd nnf1o oveq2d f1oeq2d chash cle wbr csb cif weq breq1 mpbird fveq2 csbeq1d ifbieq1d elnnuz biimpri wral wf f1of ad2antrr 1zzd nnzd eluzelz ad2antlr 3jca eluzle simpr cn0 nnnn0d hashfz1 fihasheqf1od - cz fzfigd eqtr3d breqtrrd jca elfz2 sylanbrc ffvelrnd ralrimiva nfcsb1v - nfel1 csbeq1a eleq1d rspc sylc wn 1cnd eqeltrrd adantr ifcldadc fvmptd3 - wdc zdcle eqeltrd eleqtrd fvco3 sylan fveq2d ffvelcdmda f1ocnvfv2 eqtrd - elfznn elfzle2 breqtrd iftrued 3eqtr4rd seq3f1o ) AKUEITUFZUQKUEHTUFUQL - 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FUVJCXMXNFUHZUVJUPCUVKUGFUVJCXOXPXQXRUVGUVIXSULXTZUVGUWIUVHXDUKZUVIYEUV - GUUEUWAWNAUXAUVFAKUVHXDUWOUWLYAYBUUEUVHYFVKZYCZYDUXCYGUVGUUEIUQUVIFUUEJ - UQZCVSZTVTZUGUVGEUUEUVNFUVMJUQZCVSZTVTZUXFVAIUGPUVRUVNUVIUXHUXETUVSUVRF - UXGUXDCUVMUUEJWDWEWFUWAUVGUVIUXETUGUWBUXDBUKUWDUXEUGUKZUWBUURBUUEJUWBUV - AUURBJWJAUVAUVFUVISWLUURBJWKVJUWBUUEUUPUURUWPAUUPUURUPUVFUVIUVDWLYHXKUW - RUWCUXJFUXDBFUXEUGFUXDCXMXNUWSUXDUPCUXEUGFUXDCXOXPXQXRUWTUXBYCZYDUXKYGA - UDUHZUUPUKZULZFUXLUUDUQZUUBUQZCVSZFUXLJUQZCVSZUXOHUQZUXLIUQZUXNFUXPUXRC - UXNUXPUXRUUCUQZUUBUQZUXRUXNUXOUYBUUBAUUPBJWJZUXMUXOUYBUPAUUPBJVGZUYDAUY - EUVASAUUPUURBJUVDVOWCUUPBJWKVJZUUPBUXLUUCJYIYJYKUXNUVBUXRBUKZUYCUXRUPAU - VBUXMRYBAUUPBUXLJUYFYLZUUPBUXRUUBYMVKYNWEZUXNUXTUXOUVHVQVRZUXQTVTZUXQUX - NEUXOUVQUYKVAHUGOUVMUXOUPZUVNUYJUVPUXQTUVMUXOUVHVQWBUYLFUVOUXPCUVMUXOUU - BWDWEWFUXNUXOUUPUKZUXOVAUKAUUPUUPUXLUUDAUUQUUPUUPUUDWJUVEUUPUUPUUDWKVJY - LZUXOKYOVJUXNUYKUXQUGUXNUYJUXQTUXNUXOKUVHVQUXNUYMUXOKVQVRUYNUXOTKYPVJAU - 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MUXLKVQVRAUXLTKYPUOUYOYQYRZUYRYGYFVUBYMYSYTAKLUUAUVCYJXE $. $} ${ @@ -125301,56 +125320,56 @@ seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) ) $= seq M ( x. , F ) ~~> ( seq 1 ( x. , G ) ` N ) ) $= ( vp vm vx cmul cseq cfv c1 cli cuz cfz co clt wiso wf1o wf chash wb cv wceq 1zzd nnzd fzfigd fihasheqf1od cn0 nnnn0d hashfz1 syl eqtr3d oveq2d - wcel isoeq4 mpbid isof1o f1of 3syl cn nnuz eleqtrdi eluzfz2 ffvelrnd wa - sseldd sselda cle wbr ccnv f1ocnvfv2 sylan elfzle2 cxr wss adantr cr cz - uzssz zssre sstri ressxr sstrdi syl2anc sylanbrc cc adantl 1cnd csb cif - eluzelz wral simpr ralrimiva ad2antrr nfcsb1v nfel1 csbeq1a eleq1d rspc - weq sylc wn wdc eleq1 ifcldadc nfcv nfv ifbieq1d fvmptf syl2an2 eqeltrd - nfif breq1 fveq2 csbeq1d ad2antlr breqtrd jca fvmptd3 eqtrd iftrued a1i + wcel isoeq4 mpbid isof1o f1of 3syl cn eleqtrdi eluzfz2 ffvelcdmd sseldd + nnuz wa sselda cle wbr ccnv f1ocnvfv2 sylan elfzle2 cxr adantr cr uzssz + wss cz zssre sstri ressxr sstrdi syl2anc sylanbrc cc adantl csb 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ffvelrnd - rpred eqid peano2nn0 syl eqidd crp cz nn0z rpexpcl syl2an cn faccl adantl - nnrpd rpdivcld cli cdm efcllem isumrpcl ltaddrpd cmin cfz recnd isumsplit - efval2 nn0cnd ax-1cn pncan sylancl oveq2d sumeq1d eleqtrdi sylan2br eqtrd - elnn0uz fsum3ser oveq1d 3eqtrd breqtrrd ) AEJDKUAZLZWSEMJNZUBLZIUCZDLZIOZ - JNZBUDLZUEAWSXDAPQEWRAIDKPRAUGAXBPSZUHZXCBXBUFNZXBUILZUJNZQABTSZXGXCXKUKA - BGULZBCDXBFUMUNZABQSXGXKQSABGUSBXBUOUNUPZUQHURAXCIDKWTXAPRXAUTZAEPSWTPSHE - VAVBZXHXCVCZXHXCXKVDXNXHXIXJABVDSXBVESXIVDSXGGXBVFBXBVGVHXHXJXGXJVISAXBVJ - VKVLVMUPAXLWRVNVOSXMBCDFVPVBZVQVRAXFPXCIOZKWTMVSNZVTNZXCIOZXDJNXEAXLXFXTU - KXMBICDFWCVBAXCIDKWTXAPRXPXQXRXHXCXOWAZXSWBAYCWSXDJAYCKEVTNZXCIOWSAYBYEXC - IAYAEKVTAETSMTSYAEUKAEHWDWEEMWFWGWHWIAXCIDKEAXBKUBLZSZUHXCVCAEPYFHRWJYGAX - GXCTSXBWMYDWKWNWLWOWPWQ $. + 0zd wa cfa cdiv wceq rpcnd eftvalcn sylan rpred reeftcl eqeltrd ffvelcdmd + serfre eqid peano2nn0 syl eqidd crp cz nn0z rpexpcl syl2an cn faccl nnrpd + adantl rpdivcld cli efcllem isumrpcl ltaddrpd cmin efval2 recnd isumsplit + cdm nn0cnd ax-1cn pncan sylancl oveq2d sumeq1d eleqtrdi sylan2br fsum3ser + cfz elnn0uz eqtrd oveq1d 3eqtrd breqtrrd ) AEJDKUAZLZWSEMJNZUBLZIUCZDLZIO + ZJNZBUDLZUEAWSXDAPQEWRAIDKPRAUGAXBPSZUHZXCBXBUFNZXBUILZUJNZQABTSZXGXCXKUK + ABGULZBCDXBFUMUNZABQSXGXKQSABGUOBXBUPUNUQZUSHURAXCIDKWTXAPRXAUTZAEPSWTPSH + EVAVBZXHXCVCZXHXCXKVDXNXHXIXJABVDSXBVESXIVDSXGGXBVFBXBVGVHXHXJXGXJVISAXBV + JVLVKVMUQAXLWRVNWBSXMBCDFVOVBZVPVQAXFPXCIOZKWTMVRNZWLNZXCIOZXDJNXEAXLXFXT + UKXMBICDFVSVBAXCIDKWTXAPRXPXQXRXHXCXOVTZXSWAAYCWSXDJAYCKEWLNZXCIOWSAYBYEX + CIAYAEKWLAETSMTSYAEUKAEHWCWDEMWEWFWGWHAXCIDKEAXBKUBLZSZUHXCVCAEPYFHRWIYGA + XGXCTSXBWMYDWJWKWNWOWPWQ $. $} $( The value of the first term of the series expansion of the exponential @@ -130137,23 +130156,23 @@ The circle constant (tau = 2 pi) 2cn vj vx vm orbi12d weq oveq2 oveq1d bitrdi oveq1 0z mul02i rspcev mp2an cbvrexv olci cn0 orcom wa cc zcn mulcom sylancl adantl wi mpan2 syl5ibcom eqid eqeq2d sylbid rexlimdva peano2z mulid2i oveq12d oveq2i eqtrdi ax-1cn - a1i df-2 adddir mp3an23 mpan addass 3eqtr4d syl2anc orim12d syl5bi nn0ind - mulcl syl ) EBFZGHZIJHZUAFZKZBLMZAFZEGHZWMKZALMZNWLOKZBLMZWQOKZALMZNEUBFZ - GHZIJHZUCFZKZUBLMZDFZEGHZXGKZDLMZNZWLXGIJHZKZBLMZWQXOKZALMZNZWLCKZBLMZWQC - KZALMZNUAUCCWMOKZWOXAWSXCYEWNWTBLWMOWLPQYEWRXBALWMOWQPQUDUAUCUEZWOXIWSXMY - FWOWLXGKZBLMXIYFWNYGBLWMXGWLPQYGXHBUBLBUBUEZWLXFXGYHWKXEIJWJXDEGUFUGRUNUH - YFWSWQXGKZALMXMYFWRYIALWMXGWQPQYIXLADLADUEWQXKXGWPXJEGUIRUNUHUDWMXOKZWOXQ - WSXSYJWNXPBLWMXOWLPQYJWRXRALWMXOWQPQUDWMCKZWOYBWSYDYKWNYABLWMCWLPQYKWRYCA - LWMCWQPQUDXCXAOLSOEGHZOKZXCUJETUKXBYMAOLWPOKWQYLOWPOEGUIRULUMUOXNXMXINXGU - PSZXTXIXMUQYNXMXQXIXSYNXLXQDLYNXJLSZURZXLEXJGHZXGKZXQYPXKYQXGYOXKYQKZYNYO - XJUSSEUSSZYSXJUTTXJEVAVBVCRYOYRXQVDYNYOWLYQIJHZKZBLMZYRXQYOUUAUUAKZUUCUUA - VGUUBUUDBXJLBDUEZWLUUAUUAUUEWKYQIJWJXJEGUFUGRULVEYRUUBXPBLYRUUAXOWLYQXGIJ - UIVHQVFVCVIVJYNXHXSUBLYNXDLSZURZWQXFIJHZKZALMZXHXSUUGXDIJHZLSZUUKEGHZUUHK - 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3ad2ant3 imp - sylbi elnnz sylanbrc ex syl6bir com13 impcom pm4.71rd bicomd rexbidva wss - nnssnn0 rexss mp1i eluzge2nn0 oddnn02np1 syl 3bitr4rd ) BCUAUCZDZAUBZEDZC - WSUDFZGUEFZBUFZHZAIJZXCAIJZXCAEJZCBUGKUPZWRXDXCAIWRWSIDZHZXCXDXJXCWTXIWRX - CWTLXCWRXIWTXCWRXBWQDZXIWTLXBBWQUHXKXIWTXKXIHWSMDZNWSOKZWTXIXLXKWSUIUJXKX - IXMXKCMDZXBMDZCXBPKZUKXIXMLZCXBULXPXNXQXOXIXPXMXIXPCGUMFZXAPKZXMXICGXACQD - ZXIRSXIUNZXIXAXICWSCIDXIUQSXIURUOUSUTXSGXAPKZXIXMXRGXAPVAVBXIYBGCVCFZWSPK - ZXMXIGQDWSQDZXTNCOKZHZYDYBTYAWSVDZYGXIXTYFRVEVHSGWSCVFVGXINYCOKZYDXMVIXIN - QDYCQDZYEYIYDHXMLXIVJYJXIVKSYHNYCWSVLVGVMVNVOVNVPVQVSVRWSVTWAWBWCWDWEWFWG - WHEIWIXGXETWRWJXCAEIWKWLWRBIDXHXFTBWMABWNWOWP $. + pm3.2i halfgt0 0red halfre ltletr mpani biimtrid com12 3ad2ant3 sylbi imp + sylbird elnnz sylanbrc ex syl6bir impcom pm4.71rd bicomd rexbidva nnssnn0 + com13 wss rexss mp1i eluzge2nn0 oddnn02np1 syl 3bitr4rd ) BCUAUCZDZAUBZED + ZCWSUDFZGUEFZBUFZHZAIJZXCAIJZXCAEJZCBUGKUPZWRXDXCAIWRWSIDZHZXCXDXJXCWTXIW + RXCWTLXCWRXIWTXCWRXBWQDZXIWTLXBBWQUHXKXIWTXKXIHWSMDZNWSOKZWTXIXLXKWSUIUJX + KXIXMXKCMDZXBMDZCXBPKZUKXIXMLZCXBULXPXNXQXOXIXPXMXIXPCGUMFZXAPKZXMXICGXAC + QDZXIRSXIUNZXIXAXICWSCIDXIUQSXIURUOUSUTXSGXAPKZXIXMXRGXAPVAVBXIYBGCVCFZWS + PKZXMXIGQDWSQDZXTNCOKZHZYDYBTYAWSVDZYGXIXTYFRVEVHSGWSCVFVGXINYCOKZYDXMVIX + INQDYCQDZYEYIYDHXMLXIVJYJXIVKSYHNYCWSVLVGVMVSVNVSVOVPVQVRWSVTWAWBWCWIWDWE + WFWGEIWJXGXETWRWHXCAEIWKWLWRBIDXHXFTBWMABWNWOWP $. $( A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) $) @@ -130352,10 +130371,10 @@ The circle constant (tau = 2 pi) sqoddm1div8z $p |- ( ( N e. ZZ /\ -. 2 || N ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) e. ZZ ) $= ( vk cz wcel c2 cdvds wbr wn wa cv cmul co caddc wceq wrex cexp cmin cdiv - c1 c8 odd2np1 biimpa eqcom sqoddm1div8 adantll mulsucdiv2z eqeltrd syl5bi - ad2antlr ex rexlimdva mpd ) ACDZEAFGHZIZEBJZKLSMLZANZBCOZAEPLSQLTRLZCDZUM - UNUSBAUAUBUOURVABCURAUQNZUOUPCDZIZVAUQAUCVDVBVAVDVBIUTUPUPSMLKLERLZCVCVBU - TVENUOAUPUDUEVCVECDUOVBUPUFUIUGUJUHUKUL $. + c1 c8 odd2np1 biimpa sqoddm1div8 adantll mulsucdiv2z ad2antlr ex biimtrid + eqcom eqeltrd rexlimdva mpd ) ACDZEAFGHZIZEBJZKLSMLZANZBCOZAEPLSQLTRLZCDZ + UMUNUSBAUAUBUOURVABCURAUQNZUOUPCDZIZVAUQAUIVDVBVAVDVBIUTUPUPSMLKLERLZCVCV + BUTVENUOAUPUCUDVCVECDUOVBUPUEUFUJUGUHUKUL $. $} $( A number which is twice an integer is even. (Contributed by AV, @@ -130772,21 +130791,21 @@ equal to the half of the predecessor of the odd number (which is an even cn cabs cmul w3a wrmo wreu divalglemnn wral nfv nfre1 oveq1 oveq1d eqeq2d wi nfim 3anbi3d cbvrexv wn simplr ad4antr simplrl ad3antrrr simpr1 simpr2 simplrr ad2antrr nnnn0d nn0ge0d absidd breqtrd simpr3 eqtr3d divalglemnqt - nnred pm2.21dd divalglemqt w3o ztri3or syl2anc mpjao3dan rexlimdva syl5bi - nnzd ex exp31 rexlimd impd ralrimivva breq2 breq1 oveq2 3anbi123d rexbidv - rmo4 sylibr reu5 sylanbrc ) BGHZAUBHZIZJCKZLMZXBAUCUAZNMZBDKZAUDOZXBPOZQZ - UEZDGRZCGRXKCGUFZXKCGUGABCDUHXAXKJEKZLMZXMXDNMZBXGXMPOZQZUEZDGRZIXBXMQZUO - ZEGUICGUIXLXAYACEGGXAXBGHZXMGHZIZIZXKXSXTYEXJXSXTUOZDGYEDUJXSXTDXRDGUKXTD - UJUPYEXFGHZXJYFXSXNXOBFKZAUDOZXMPOZQZUEZFGRYEYGIZXJIZXTXRYLDFGXFYHQZXQYKX - NXOYOXPYJBYOXGYIXMPXFYHAUDULUMUNUQURYNYLXTFGYNYHGHZIZYLXTYQYLIZXFYHNMZXTY - OYHXFNMZYRYSIYSXTYRYSSYRYSUSYSYRAXFXBXMYHYEWTYGXJYPYLWSWTYDUTVAZYMYBXJYPY - LXAYBYCYGVBVCZYMYCXJYPYLXAYBYCYGVFVCZYMYGXJYPYLYEYGSVCZYNYPYLUTZYQXNXOYKV - 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zcnd pm2.21dd syl2anc mpjao3dan rexlimdva syl5bi exp31 rexlimd ralrimivva - ex impd breq2 breq1 oveq2 3anbi123d rexbidv rmo4 sylibr reu5 ) BGHZAGHZAI - JKZLZICMZUAKZXSAUBUCZJKZBDMZANOZXSPOZQZLZDGUDZCGUDZYHCGUEZYHCGUFXOXPXQAIU - GYIXRAXOXPXQUHZUIABCDUJUKXRYHIEMZUAKZYLYAJKZBYDYLPOZQZLZDGUDZRXSYLQZULZEG - UNCGUNYJXRYTCEGGXRXSGHZYLGHZRZRZYHYRYSUUDYGYRYSULZDGUUDDUMYRYSDYQDGUOYSDU - MUPUUDYCGHZYGUUEYRYMYNBFMZANOZYLPOZQZLZFGUDUUDUUFRZYGRZYSYQUUKDFGYCUUGQZY - PUUJYMYNUUNYOUUIBUUNYDUUHYLPYCUUGANUQURUSUTVAUUMUUKYSFGUUMUUGGHZRZUUKYSUU - PUUKRZYCUUGJKZYSUUNUUGYCJKZUUQUURRUURYSUUQUURVBUUQUURVCUURUUQUURUUGVDZYCV - DZJKUUQAVDZUUTYLXSUVAXRUVBVFHZUUCUUFYGUUOUUKXRUVBGHIUVBJKZUVCXRAXOXPXQVGZ - VEXRXQUVDYKXRAXRAUVEVHVIVJUVBVKVLVMZUULUUBYGUUOUUKXRUUAUUBUUFVNSZUULUUAYG - UUOUUKXRUUAUUBUUFVQSZUUQUUGUUMUUOUUKVOZVEZUUQYCUULUUFYGUUOUUKUUDUUFVBSZVE - ZUUMXTUUOUUKUULXTYBYFVPVRUUQYLYAUVBJUUPYMYNUUJVSUUQAUUQAUULXPYGUUOUUKXOXP - XQUUCUUFVTSZVHZUUQAIUVNUUQWAUULXQYGUUOUUKXOXPXQUUCUUFWBSWCWDZWEUUQBUUTUVB - NOZYLPOZUVAUVBNOZXSPOZUUQBUUIUVQUUPYMYNUUJWFZUUQUVPUUHYLPUUQUUGAUUQUUGUVI - 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syl3an1 jca eqid biantrur necon3abii w3a dvdslegcd ex mp3an2 mpancom syl5bi - mpbid imp zred letri3d mpbir2and sylan2br wdc zdceq mpan2 exmiddc mpjaodan - wo ) ABCZADEZDAFGZAHIZEZWGJZWGWJWFWGDDFGDWHWIUAADDFUBWGWIDHIDADHUCUDUEUFUGW - KWFADUHZWJADULWFWLKZWJWHWILMZWIWHLMZWMWHANMZWNWFWPWLWFWHDNMZWPDBCZWFWQWPKSD - AUIOUJPWFWLWPWNTZWFWHBCWFWLWSWFWHWRWFWHUKCSDAUMOUNZWHAUOVCUPQWMWIDNMZWIANMZ - KZWOWFXCWLWFXAXBWFWIBCZXAAUQZWIURRWFAANMZXBAUSWFXFXBUTAAVAVBVNVDPWFWLXCWOTZ - WLDDEZWGKZJZWFXGXIADXHWGDVEVFVGXDWFXJXGTZXEXDWRWFXKSXDWRWFVHXJXGWIDAVIVJVKV - LVMVOQWFWJWNWOKUTWLWFWHWIWFWHWTVPWFWIXEVPVQPVRVSWFWGVTZWGWKWEWFWRXLSADWAWBW - GWCRWD $. + syl3an1 mpbid jca eqid biantrur necon3abii w3a dvdslegcd ex mp3an2 biimtrid + mpancom imp zred letri3d mpbir2and sylan2br wo zdceq mpan2 exmiddc mpjaodan + wdc ) ABCZADEZDAFGZAHIZEZWGJZWGWJWFWGDDFGDWHWIUAADDFUBWGWIDHIDADHUCUDUEUFUG + WKWFADUHZWJADULWFWLKZWJWHWILMZWIWHLMZWMWHANMZWNWFWPWLWFWHDNMZWPDBCZWFWQWPKS + DAUIOUJPWFWLWPWNTZWFWHBCWFWLWSWFWHWRWFWHUKCSDAUMOUNZWHAUOVCUPQWMWIDNMZWIANM + ZKZWOWFXCWLWFXAXBWFWIBCZXAAUQZWIURRWFAANMZXBAUSWFXFXBUTAAVAVBVDVEPWFWLXCWOT + ZWLDDEZWGKZJZWFXGXIADXHWGDVFVGVHXDWFXJXGTZXEXDWRWFXKSXDWRWFVIXJXGWIDAVJVKVL + VNVMVOQWFWJWNWOKUTWLWFWHWIWFWHWTVPWFWIXEVPVQPVRVSWFWGWEZWGWKVTWFWRXLSADWAWB + WGWCRWD $. $( The gcd of an integer and 0 is the integer's absolute value. Theorem 1.4(d)2 in [ApostolNT] p. 16. (Contributed by Paul Chapman, @@ -132927,23 +132946,23 @@ integers has a least element (schema form). (Contributed by NM, wb cz peano2nn0 syl2an nn0zd nn0z zsubcl zltlem1 syl2anc cc nn0cn subsub4 ax-1cn mp3an3 breq2d bitrd 3bitr2d biimpa an32s ffvelcdmda ltletr syl3anc a1d zred sylibd syland expdimp wdc 0zd zdceq sylib mpjaodan anasss expcom - wo a2d 3adant1 fnn0ind pm2.43i subidd breqtrd ffvelrnd nn0ge0d 0re letri3 - dcne mpbir2and ) ADCHZIJZXOIKLZIXOKLZAXODDMNZIKAXOXSKLZADOPZYADDKLAXTQZAD - ICHZOFAOOCUEZIOPYCOPEUAOOICUFUBUCZYEADADYEUGZUHZAUDUIZCHZDYHMNZKLZQAYCDIM - NZKLZQZABUIZCHZDYOMNZKLZQZAYOUJUKNZCHZDYTMNZKLZQZYBUDBDDYHIJZYKYMAUUEYIYC - YJYLKYHICULYHIDMUMUQUNYHYOJZYKYRAUUFYIYPYJYQKYHYOCULYHYODMUMUQUNYHYTJZYKU - UCAUUGYIUUAYJUUBKYHYTCULYHYTDMUMUQUNYHDJZYKXTAUUHYIXOYJXSKYHDCULYHDDMUMUQ - UNYNYAAYCDYLKADYCDKFYGUOADADYFUPZURUSUTYOOPZYODRLZYSUUDQYAUUJUUKSZAYRUUCA - UULYRUUCQZAUUJUUKUUMAUUJSZUUKSZUUAIJZUUMUUAIVAZUUOUUPSUUCYRUUNUUPUUKUUCUU - NUUPSZUUKUUCUURUUKIYQRLZUUAYQRLZUUCUUNUUKUUSVHZUUPUUJYOTPDTPUVAAYOVBYFYOD - VCVDVEUUPUUTUUSVHUUNUUAIYQRVFVGUUNUUTUUCVHUUPUUNUUTUUAYQUJMNZKLZUUCUUNUUA - VIPZYQVIPZUUTUVCVHUUNUUAAYDYTOPUUAOPUUJEYOVJOOYTCUFVKZVLZADVIPYOVIPUVEUUJ - ADYEVLYOVMDYOVNVKZUUAYQVOVPUUNUVBUUBUUAKADVQPZYOVQPZUVBUUBJZUUJUUIYOVRUVI - UVJUJVQPUVKVTDYOUJVSWAVKWBWCZVEWDWEWFWJUUOUUQYRUUCUUNUUQYRSUUCQUUKUUNUUQU - UAYPRLZYRUUCGUUNUVMYRSZUUTUUCUUNUUATPYPTPYQTPUVNUUTQUUNUUAUVFUGUUNYPAOOYO - CEWGUGUUNYQUVHWKUUAYPYQWHWIUVLWLWMVEWNUUOUUPWOZUUPUUQXBUUOUVDIVIPUVOUUNUV - DUUKUVGVEUUOWPUUAIWQVPUUAIXMWRWSWTXAXCXDXEWIXFADUUIXGXHAXOAOODCEYEXIZXJAX - OTPITPXPXQXRSVHAXOUVPUGXKXOIXLUBXN $. + wo a2d 3adant1 fnn0ind pm2.43i subidd breqtrd ffvelcdmd nn0ge0d mpbir2and + dcne 0re letri3 ) ADCHZIJZXOIKLZIXOKLZAXODDMNZIKAXOXSKLZADOPZYADDKLAXTQZA + DICHZOFAOOCUEZIOPYCOPEUAOOICUFUBUCZYEADADYEUGZUHZAUDUIZCHZDYHMNZKLZQAYCDI + MNZKLZQZABUIZCHZDYOMNZKLZQZAYOUJUKNZCHZDYTMNZKLZQZYBUDBDDYHIJZYKYMAUUEYIY + CYJYLKYHICULYHIDMUMUQUNYHYOJZYKYRAUUFYIYPYJYQKYHYOCULYHYODMUMUQUNYHYTJZYK + UUCAUUGYIUUAYJUUBKYHYTCULYHYTDMUMUQUNYHDJZYKXTAUUHYIXOYJXSKYHDCULYHDDMUMU + QUNYNYAAYCDYLKADYCDKFYGUOADADYFUPZURUSUTYOOPZYODRLZYSUUDQYAUUJUUKSZAYRUUC + AUULYRUUCQZAUUJUUKUUMAUUJSZUUKSZUUAIJZUUMUUAIVAZUUOUUPSUUCYRUUNUUPUUKUUCU + UNUUPSZUUKUUCUURUUKIYQRLZUUAYQRLZUUCUUNUUKUUSVHZUUPUUJYOTPDTPUVAAYOVBYFYO + DVCVDVEUUPUUTUUSVHUUNUUAIYQRVFVGUUNUUTUUCVHUUPUUNUUTUUAYQUJMNZKLZUUCUUNUU + AVIPZYQVIPZUUTUVCVHUUNUUAAYDYTOPUUAOPUUJEYOVJOOYTCUFVKZVLZADVIPYOVIPUVEUU + JADYEVLYOVMDYOVNVKZUUAYQVOVPUUNUVBUUBUUAKADVQPZYOVQPZUVBUUBJZUUJUUIYOVRUV + IUVJUJVQPUVKVTDYOUJVSWAVKWBWCZVEWDWEWFWJUUOUUQYRUUCUUNUUQYRSUUCQUUKUUNUUQ + UUAYPRLZYRUUCGUUNUVMYRSZUUTUUCUUNUUATPYPTPYQTPUVNUUTQUUNUUAUVFUGUUNYPAOOY + OCEWGUGUUNYQUVHWKUUAYPYQWHWIUVLWLWMVEWNUUOUUPWOZUUPUUQXBUUOUVDIVIPUVOUUNU + VDUUKUVGVEUUOWPUUAIWQVPUUAIXLWRWSWTXAXCXDXEWIXFADUUIXGXHAXOAOODCEYEXIZXJA + XOTPITPXPXQXRSVHAXOUVPUGXMXOIXNUBXK $. $} ${ @@ -132953,9 +132972,9 @@ integers has a least element (schema form). (Contributed by NM, 22-Jul-2021.) $) ialgrlem1st $p |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S ) $= - ( cv wcel wa c1st ccom co wceq algrflemg adantl wf adantr simprl ffvelrnd - cfv eqeltrd ) ABGZDHZCGZDHZIZIZUBUDEJKLZUBETZDUFUHUIMAUBUDEDDNOUGDDUBEADD - EPUFFQAUCUERSUA $. + ( cv wcel wa c1st ccom co cfv algrflemg adantl wf adantr simprl ffvelcdmd + wceq eqeltrd ) ABGZDHZCGZDHZIZIZUBUDEJKLZUBEMZDUFUHUITAUBUDEDDNOUGDDUBEAD + DEPUFFQAUCUERSUA $. $} ${ @@ -134090,33 +134109,33 @@ According to Wikipedia ("Least common multiple", 27-Aug-2020, simpl divides simpr anbi12d bezout oveqan12rd eqeq2d bicomd adantl nn0cnd jca oveq1 ad2antlr mul32d oveq12d cap zmulcld zaddcld gcd2n0cl nncn nnap0 cn0 cn div11ap syl3anc dividap divdirap simprd divcanap4d eqeq12d 3bitr2d - sylan9bbr anim1i ancomd bezoutr1 adantr simpll1 divmulap3 3bitr4g biimprd - eqtrd syl5bi imp32 simp2 a1dd eqeq1d sylibd sylbid exp32 com34 rexlimdvva - a1d com23 ex mpd impl rexlimdva impd ) ADEZBDEZBUDUEZUFZABUGFZAUHUIZXQBUH - UIZGZAXQHFZBXQHFZUGFZIJZXMXNXTXOABUJKXPXTCUKZXQLFZAJZCDULZUAUKZXQLFZBJZUA - DULZGYDXPXRYHXSYLXPXQDEZXMGZXRYHMXMXNYNXOXMXNGZYMXMYOXQABUMZUNZXMXNUOVEKC - 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MUWQJUUQUVHUWHQUUQUVIUWLQUWNUVHUVIXQWAVSUUQUWOUUTUWPUVAOUUQUUTXQUUQUUTUWG - QXPUVRUUOUUKXPXQUWKVDRUUQUVRUWEUWNWBZWCUUQUVAXQUUQUVAUWIQUVSUWRWCVIWOWDWE - WFUUSUVCYEYIUGFZIJZYDUVCUVBIJZUUSUWTIUVBTUUQUXAUWTNZUURUUQUUKUUOGUXBUUQUU - OUUKUUPUUOUUKXPUUOUQWGWHYEYIUUDUUFWISWJWPUUSUWSYCIUUSYEYAYIYBUGUUQYKYGYEY - AJZUUQYGUXCNYKUUQUXCYGUUQYAYEJZAYFJZUXCYGUUQAPEUVOUWFUXDUXEMUUQAXMXNXOUUO - UUKWKQUVQUWNAYEXQWLVSYEYATYFATWMWNXFWQUUQYKYGYIYBJZUUQYKUXFYGUUQUXFYKUUQY - BYIJZBYJJZUXFYKUUQBPEZUWAUWFUXGUXHMXPUXIUUOUUKXPBXMXNXOWRQRUWCUWNBYIXQWLV - SYIYBTYJBTWMWNWSWQVIWTXAXBXCXDXGXHXGXEXIXJXKXGXKXLXBXI $. + eqtrd sylan9bbr anim1i ancomd bezoutr1 biimtrid simpll1 divmulap3 3bitr4g + adantr biimprd imp32 simp2 a1dd eqeq1d sylibd sylbid exp32 com34 com23 ex + a1d rexlimdvva mpd impl rexlimdva impd ) ADEZBDEZBUDUEZUFZABUGFZAUHUIZXQB + UHUIZGZAXQHFZBXQHFZUGFZIJZXMXNXTXOABUJKXPXTCUKZXQLFZAJZCDULZUAUKZXQLFZBJZ + UADULZGYDXPXRYHXSYLXPXQDEZXMGZXRYHMXMXNYNXOXMXNGZYMXMYOXQABUMZUNZXMXNUOVE + KCXQAUPSXPYMXNGZXSYLMXMXNYRXOYOYMXNYQXMXNUQVEKUAXQBUPSURXPYHYLYDXPYGYLYDN + CDXPYEDEZGZYLYGYDYTYKYGYDNZUADXPYSYIDEZYKUUANZXPXQAUBUKZLFZBUCUKZLFZOFZJZ + UCDULUBDULZYSUUBGZUUCNZXMXNUUJXOUBUCABUSKXPUUIUULUBUCDDXPUUDDEZUUFDEZGZGZ + UUKUUIUUCUUPUUKUUIUUCNUUPUUKGZYKUUIUUAUUQYKYGUUIYDUUQYKYGUUIYDNUUQYKYGGZG + ZUUIIYEUUDLFZYIUUFLFZOFZJZYDUURUUIXQYFUUDLFZYJUUFLFZOFZJZUUQUVCUURUVGUUIU + URUVFUUHXQYGYKUVDUUEUVEUUGOYFAUUDLVFYJBUUFLVFUTVAVBUUQUVGXQUUTXQLFZUVAXQL + FZOFZJZXQXQHFZUVJXQHFZJZUVCUUQUVFUVJXQUUQUVDUVHUVEUVIOUUQYEXQUUDUUKYEPEZU + UPUUKYEYSUUBUOZQVCZXPXQPEZUUOUUKXMXNUVRXOYOXQYPVDKRZUUOUUDPEXPUUKUUOUUDUU + MUUNUOZQVGVHUUQYIXQUUFUUKYIPEZUUPUUKYIYSUUBUQZQVCZUVSUUOUUFPEXPUUKUUOUUFU + UMUUNUQZQVGVHVIVAUUQUVRUVJPEUVRXQUDVJUIZGZUVNUVKMUVSUUQUVJUUQUVHUVIUUQUUT + XQUUQYEUUDUUKYSUUPUVPVCUUOUUMXPUUKUVTVGVKZXPYMUUOUUKXMXNYMXOYQKRVKZUUQUVA + XQUUQYIUUFUUKUUBUUPUWBVCUUOUUNXPUUKUWDVGVKZUUQXQXPXQVPEZUUOUUKXMXNUWJXOYP + KZRUNVKZVLQXPUWFUUOUUKXPXQVQEZUWFABVMUWMUVRUWEXQVNXQVOVESRZXQUVJXQVRVSUUQ + UVLIUVMUVBUUQUWFUVLIJUWNXQVTSUUQUVMUVHXQHFZUVIXQHFZOFZUVBUUQUVHPEUVIPEUWF + UVMUWQJUUQUVHUWHQUUQUVIUWLQUWNUVHUVIXQWAVSUUQUWOUUTUWPUVAOUUQUUTXQUUQUUTU + WGQXPUVRUUOUUKXPXQUWKVDRUUQUVRUWEUWNWBZWCUUQUVAXQUUQUVAUWIQUVSUWRWCVIWFWD + WEWGUUSUVCYEYIUGFZIJZYDUVCUVBIJZUUSUWTIUVBTUUQUXAUWTNZUURUUQUUKUUOGUXBUUQ + UUOUUKUUPUUOUUKXPUUOUQWHWIYEYIUUDUUFWJSWOWKUUSUWSYCIUUSYEYAYIYBUGUUQYKYGY + EYAJZUUQYGUXCNYKUUQUXCYGUUQYAYEJZAYFJZUXCYGUUQAPEUVOUWFUXDUXEMUUQAXMXNXOU + UOUUKWLQUVQUWNAYEXQWMVSYEYATYFATWNWPXGWQUUQYKYGYIYBJZUUQYKUXFYGUUQUXFYKUU + QYBYIJZBYJJZUXFYKUUQBPEZUWAUWFUXGUXHMXPUXIUUOUUKXPBXMXNXOWRQRUWCUWNBYIXQW + MVSYIYBTYJBTWNWPWSWQVIWTXAXBXCXDXEXFXEXHXIXJXKXEXKXLXBXI $. $} ${ @@ -134966,10 +134985,10 @@ According to Wikipedia ("Least common multiple", 27-Aug-2020, ( P || ( Q ^ N ) -> P = Q ) ) $= ( cprime wcel cn0 cexp co cdvds wbr wceq wi cn cc0 wo wa elnn0 w3a breq2d c1 prmdvdsexpb biimpd 3expia prmnn adantl nncnd exp0d pm2.21d adantr sylbid - nprmdvds1 oveq2 imbi1d syl5ibrcom jaod syl5bi 3impia ) ADEZBDEZCFEZABCGHZIJ - ZABKZLZUTCMEZCNKZOURUSPZVDCQVGVEVDVFURUSVEVDURUSVERVBVCABCUAUBUCVGVDVFABNGH - ZIJZVCLVGVIATIJZVCVGVHTAIVGBVGBUSBMEURBUDUEUFUGSURVJVCLUSURVJVCAUKUHUIUJVFV - BVIVCVFVAVHAICNBGULSUMUNUOUPUQ $. + nprmdvds1 oveq2 imbi1d syl5ibrcom jaod biimtrid 3impia ) ADEZBDEZCFEZABCGHZ + IJZABKZLZUTCMEZCNKZOURUSPZVDCQVGVEVDVFURUSVEVDURUSVERVBVCABCUAUBUCVGVDVFABN + GHZIJZVCLVGVIATIJZVCVGVHTAIVGBVGBUSBMEURBUDUEUFUGSURVJVCLUSURVJVCAUKUHUIUJV + FVBVIVCVFVAVHAICNBGULSUMUNUOUPUQ $. $( Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) $) @@ -134990,19 +135009,19 @@ According to Wikipedia ("Least common multiple", 27-Aug-2020, (Contributed by Mario Carneiro, 6-Mar-2014.) $) prmfac1 $p |- ( ( N e. NN0 /\ P e. Prime /\ P || ( ! ` N ) ) -> P <_ N ) $= ( wcel cfa cfv cdvds wbr cle wi c1 wceq fveq2 breq2d breq2 imbi12d imbi2d - cc0 adantr cz cr vx vk cprime cv caddc co breq2i nprmdvds1 pm2.21d syl5bi - cn0 fac0 wa wo cmul facp1 wb simpr cn faccl nnzd nn0p1nn euclemma syl3anc - bitrd nn0re lep1d prmz adantl zred nnred letr mpan2d imim2d com23 syl2anc - dvdsle a1dd jaod sylbid ex a2d nn0ind 3imp ) BUKCAUCCZABDEZFGZABHGZWEAUAU - DZDEZFGZAWIHGZIZIWEAQDEZFGZAQHGZIZIWEAUBUDZDEZFGZAWRHGZIZIWEAWRJUEUFZDEZF - GZAXCHGZIZIWEWGWHIZIUAUBBWIQKZWMWQWEXIWKWOWLWPXIWJWNAFWIQDLMWIQAHNOPWIWRK - ZWMXBWEXJWKWTWLXAXJWJWSAFWIWRDLMWIWRAHNOPWIXCKZWMXGWEXKWKXEWLXFXKWJXDAFWI - XCDLMWIXCAHNOPWIBKZWMXHWEXLWKWGWLWHXLWJWFAFWIBDLMWIBAHNOPWOAJFGZWEWPWNJAF - ULUGWEXMWPAUHUIUJWRUKCZWEXBXGXNWEXBXGIXNWEUMZXEXBXFXOXEWTAXCFGZUNZXBXFIZX - OXEAWSXCUOUFZFGZXQXOXDXSAFXNXDXSKWEWRUPRMXOWEWSSCXCSCXTXQUQXNWEURXOWSXNWS - USCWEWRUTRVAXOXCXNXCUSCZWEWRVBRZVAAWSXCVCVDVEXOWTXRXPXOXBWTXFXOXAXFWTXOXA - WRXCHGZXFXOWRXNWRTCZWEWRVFRZVGXOATCYDXCTCXAYCUMXFIXOAWEASCZXNAVHVIZVJYEXO - XCYBVKAWRXCVLVDVMVNVOXOXPXFXBXOYFYAXPXFIYGYBAXCVQVPVRVSVTVOWAWBWCWD $. + cc0 adantr cz cr vx vk cn0 cprime cv caddc fac0 breq2i nprmdvds1 biimtrid + co pm2.21d wa wo cmul facp1 wb simpr cn faccl nnzd nn0p1nn euclemma bitrd + syl3anc nn0re lep1d prmz adantl zred nnred letr mpan2d imim2d dvdsle a1dd + com23 syl2anc jaod sylbid ex a2d nn0ind 3imp ) BUCCAUDCZABDEZFGZABHGZWEAU + AUEZDEZFGZAWIHGZIZIWEAQDEZFGZAQHGZIZIWEAUBUEZDEZFGZAWRHGZIZIWEAWRJUFUKZDE + ZFGZAXCHGZIZIWEWGWHIZIUAUBBWIQKZWMWQWEXIWKWOWLWPXIWJWNAFWIQDLMWIQAHNOPWIW + RKZWMXBWEXJWKWTWLXAXJWJWSAFWIWRDLMWIWRAHNOPWIXCKZWMXGWEXKWKXEWLXFXKWJXDAF + WIXCDLMWIXCAHNOPWIBKZWMXHWEXLWKWGWLWHXLWJWFAFWIBDLMWIBAHNOPWOAJFGZWEWPWNJ + AFUGUHWEXMWPAUIULUJWRUCCZWEXBXGXNWEXBXGIXNWEUMZXEXBXFXOXEWTAXCFGZUNZXBXFI + ZXOXEAWSXCUOUKZFGZXQXOXDXSAFXNXDXSKWEWRUPRMXOWEWSSCXCSCXTXQUQXNWEURXOWSXN + WSUSCWEWRUTRVAXOXCXNXCUSCZWEWRVBRZVAAWSXCVCVEVDXOWTXRXPXOXBWTXFXOXAXFWTXO + XAWRXCHGZXFXOWRXNWRTCZWEWRVFRZVGXOATCYDXCTCXAYCUMXFIXOAWEASCZXNAVHVIZVJYE + XOXCYBVKAWRXCVLVEVMVNVQXOXPXFXBXOYFYAXPXFIYGYBAXCVOVRVPVSVTVQWAWBWCWD $. $} ${ @@ -135596,9 +135615,9 @@ reduced fraction representation (no common factors, denominator ) $= ( va cq wcel cv c1st cfv c2nd cgcd co c1 wceq cdiv wa cz cn cnumer eleq1d cxp crio cdenom wreu qredeu riotacl syl cop elxp6 qnumval qdenval anbi12d - biimprd adantld syl5bi mpd ) ACDZBEZFGZUPHGZIJKLAUQURMJLNZBOPSZTZUTDZAQGZ - ODZAUAGZPDZNZUOUSBUTUBVBBAUCUSBUTUDUEVBVAVAFGZVAHGZUFLZVHODZVIPDZNZNUOVGV - AOPUGUOVMVGVJUOVGVMUOVDVKVFVLUOVCVHOBAUHRUOVEVIPBAUIRUJUKULUMUN $. + biimprd adantld biimtrid mpd ) ACDZBEZFGZUPHGZIJKLAUQURMJLNZBOPSZTZUTDZAQ + GZODZAUAGZPDZNZUOUSBUTUBVBBAUCUSBUTUDUEVBVAVAFGZVAHGZUFLZVHODZVIPDZNZNUOV + GVAOPUGUOVMVGVJUOVGVMUOVDVKVFVLUOVCVHOBAUHRUOVEVIPBAUIRUJUKULUMUN $. $( The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) $) @@ -135964,50 +135983,50 @@ reduced fraction representation (no common factors, denominator sylibr zltp1le syl2an mpbird flqlt cr cc0 zred nnred nngt0d jca ltdivmul2 nnzd zmulcld ltsubaddd zaddcld zlem1lt elfzle2 lemuldiv leaddsub dvdsmul2 flqge elfz mpbir2and pncand breqtrrd sylanbrc ex ad2antrl ltsub1dd ltdiv1 - elrab simprr nnne0d dvdsval2 lesub1dd lediv1 syl5bi adantrl adantrr eqcom - wne anbi2i zre nncnd cap nnap0d divmulap3d subadd2d bitrd 3bitr4g sylan2b - en3d entr hashen cn0 eluzle lesub1 syl3an flqword2 uznn0sub hashfz1 3syl - eqtr3d ) AUDDEKLZFUHLZUEUFZCUDKLZEKLZFUHLZUEUFZKLZUGLZUIUFZFBUJZEKLZUKMZB - CDUGLZULZUIUFZUVHAUVJUVPUMZUVIUVOUNMZAUVIUVGUDUOLZUVCUGLZUNMUVTUVOUNMUVRA - UVIUDUVGUOLZUVHUVGUOLZUGLZUVTUNAUDNOUVHNOUVGNOZUVIUWCUNMAVCZAUVCUVGAUVBAU - VANOFUPOZUVBUQOZADEADUVDURUFOZDNOZIUVDDUSUTZJVAZGUVAFVBVDZVEZAUVFAUVENOUW - FUVFUQOZAUVDEACNOZUVDNOZHCVFUTZJVAZGUVEFVBVDZVEZVAZUWTUVGUDUVHVGPAUWAUVSU - WBUVCUGAUDVHOUVGVHOUWAUVSUMVIAUVGUWTVNZUDUVGVJVKAUVCUVGAUVCUWMVNUXBVLVMVO - 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(Contributed by AV, 5-Jul-2021.) $) prm23lt5 $p |- ( ( P e. Prime /\ P < 5 ) -> ( P = 2 \/ P = 3 ) ) $= - ( cprime wcel c5 clt wbr cc0 c4 co c2 wceq c3 wo cn0 adantr c1 cz syl5bi wi - com12 wa cfz cle prmnn nnnn0d 4nn0 a1i caddc df-5 breq2i wb prmz 4z zleltp1 - sylancl biimprd imp elfz2nn0 syl3anbrc ctp cpr cun fz0to4untppr eleq2i elun - w3o eltpi cn wne nnne0 eqneqall 3syl eleq1 1nprm pm2.21i syl6bi orc a1d syl - 3jaoi elpri olc 4nprm jaoi sylbi mpd ) ABCZADEFZUAZAGHUBIZCZAJKZALKZMZWIANC - ZHNCZAHUCFZWKWGWOWHWGAAUDZUEOWPWIUFUGWGWHWQWHAHPUHIZEFZWGWQDWSAEUIUJWGWQWTW - GAQCHQCWQWTUKAULUMAHUNUOUPRUQAHURUSWKAGPJUTZLHVAZVBZCZWIWNWJXCAVCVDWGXDWNSW - HXDWGWNXDAXACZAXBCZMWGWNSZAXAXBVEXEXGXFXEAGKZAPKZWLVFXGAGPJVGXHXGXIWLWGXHWN - WGAVHCAGVIZXHWNSWRAVJXHXJWNWNAGVKTVLTXIWGPBCZWNAPBVMXKWNVNVOVPWLWNWGWLWMVQV - RVTVSXFWMAHKZMXGALHWAWMXGXLWMWNWGWMWLWBVRXLWGHBCZWNAHBVMXMWNWCVOVPWDVSWDWET - ORWF $. + ( cprime wcel c5 clt wbr cc0 c4 co c2 c3 wo cn0 adantr c1 cz biimtrid com12 + wceq wi wa cfz cle prmnn nnnn0d 4nn0 caddc df-5 breq2i prmz zleltp1 sylancl + a1i biimprd imp elfz2nn0 syl3anbrc ctp cpr cun fz0to4untppr eleq2i elun w3o + wb 4z eltpi cn wne nnne0 eqneqall 3syl eleq1 1nprm pm2.21i syl6bi orc 3jaoi + a1d syl elpri olc 4nprm jaoi sylbi mpd ) ABCZADEFZUAZAGHUBIZCZAJSZAKSZLZWIA + MCZHMCZAHUCFZWKWGWOWHWGAAUDZUENWPWIUFUMWGWHWQWHAHOUGIZEFZWGWQDWSAEUHUIWGWQW + TWGAPCHPCWQWTVEAUJVFAHUKULUNQUOAHUPUQWKAGOJURZKHUSZUTZCZWIWNWJXCAVAVBWGXDWN + TWHXDWGWNXDAXACZAXBCZLWGWNTZAXAXBVCXEXGXFXEAGSZAOSZWLVDXGAGOJVGXHXGXIWLWGXH + WNWGAVHCAGVIZXHWNTWRAVJXHXJWNWNAGVKRVLRXIWGOBCZWNAOBVMXKWNVNVOVPWLWNWGWLWMV + QVSVRVTXFWMAHSZLXGAKHWAWMXGXLWMWNWGWMWLWBVSXLWGHBCZWNAHBVMXMWNWCVOVPWDVTWDW + ERNQWF $. $( A prime is either 2 or 3 or greater than or equal to 5. (Contributed by AV, 5-Jul-2021.) $) @@ -138288,18 +138307,18 @@ given prime (or other positive integer) that divides the number. For znegcld wb negne0bd mpbid eqid pczpre syl12anc prmz zexpcl sylan dvdsnegb simpl syl2an an32s rabbidva supeq1d eqtrd simplrr pcdiv syl121anc 3eqtr4d syl eqtr4d wdc simprl 0zd zdceq syl2anc dcne sylib mpjaodan negeq eqeq12d - wo oveq2 syl5ibrcom rexlimdvva syl5bi imp ) BUAEZAUBEZBAFZGHZBAGHZIZXFACU - CZDUCZJHZIZDKUDCLUDXEXJCDAUEXEXNXJCDLKXEXKLEZXLKEZMZMZXJXNBXMFZGHZBXMGHZI - XRXTBXKFZXLJHZGHZYAXRXSYCBGXRXKXLXOXKUFEXEXPXKUGZUHXPXLUFEXEXOXLUNUIXPXLN - UJOXEXOXLUKUIULPXRXKNIZYDYAIXKNUMZXRYFMZYCXMBGYHYBXKXLJYHNFNYBXKUOYHXKNXR - YFUPZUQYIURUSPXRYGMZBYBGHZBXLGHZQHZBXKGHZYLQHZYDYAYJYKYNYLQYJYKBXLUTHZYBV - AOZDRVBZTSVCZYNYJXEYBLEZYBNUMZYKYSIXEXQYGVDZYJXKXEXOXPYGVEZVFZYJYGUUAXRYG - UPZYJXOYGUUAVGUUCXOXKYEVHWGVIZBYSDYBYSVJVKVLYJXEXOYGYNYSIUUBUUCUUEXEXOYGM - ZMZYNYPXKVAOZDRVBZTSVCZYSBUUKDXKUUKVJVKUUHTUUJYRSUUHUUIYQDRXEXLREZUUGUUIY - QVGZXEUULMYPLEZXOUUMUUGXEBLEUULUUNBVMBXLVNVOXOYGVQYPXKVPVRVSVTWAWBVLWHUSY - JXEYTUUAXPYDYMIUUBUUDUUFXEXOXPYGWCZYBXLBWDWEYJXEXOYGXPYAYOIUUBUUCUUEUUOXK - XLBWDWEWFXRYFWIZYFYGWSXRXONLEUUPXEXOXPWJXRWKXKNWLWMXKNWNWOWPWBXNXHXTXIYAX - NXGXSBGAXMWQPAXMBGWTWRXAXBXCXD $. + wo oveq2 syl5ibrcom rexlimdvva biimtrid imp ) BUAEZAUBEZBAFZGHZBAGHZIZXFA + CUCZDUCZJHZIZDKUDCLUDXEXJCDAUEXEXNXJCDLKXEXKLEZXLKEZMZMZXJXNBXMFZGHZBXMGH + ZIXRXTBXKFZXLJHZGHZYAXRXSYCBGXRXKXLXOXKUFEXEXPXKUGZUHXPXLUFEXEXOXLUNUIXPX + LNUJOXEXOXLUKUIULPXRXKNIZYDYAIXKNUMZXRYFMZYCXMBGYHYBXKXLJYHNFNYBXKUOYHXKN + XRYFUPZUQYIURUSPXRYGMZBYBGHZBXLGHZQHZBXKGHZYLQHZYDYAYJYKYNYLQYJYKBXLUTHZY + BVAOZDRVBZTSVCZYNYJXEYBLEZYBNUMZYKYSIXEXQYGVDZYJXKXEXOXPYGVEZVFZYJYGUUAXR + YGUPZYJXOYGUUAVGUUCXOXKYEVHWGVIZBYSDYBYSVJVKVLYJXEXOYGYNYSIUUBUUCUUEXEXOY + GMZMZYNYPXKVAOZDRVBZTSVCZYSBUUKDXKUUKVJVKUUHTUUJYRSUUHUUIYQDRXEXLREZUUGUU + IYQVGZXEUULMYPLEZXOUUMUUGXEBLEUULUUNBVMBXLVNVOXOYGVQYPXKVPVRVSVTWAWBVLWHU + SYJXEYTUUAXPYDYMIUUBUUDUUFXEXOXPYGWCZYBXLBWDWEYJXEXOYGXPYAYOIUUBUUCUUEUUO + XKXLBWDWEWFXRYFWIZYFYGWSXRXONLEUUPXEXOXPWJXRWKXKNWLWMXKNWNWOWPWBXNXHXTXIY + AXNXGXSBGAXMWQPAXMBGWTWRXAXBXCXD $. $} $( The prime count of an absolute value. (Contributed by Mario Carneiro, @@ -138822,20 +138841,20 @@ given prime (or other positive integer) that divides the number. For pcmpt2 $p |- ( ph -> ( P pCnt ( ( seq 1 ( x. , F ) ` M ) / ( seq 1 ( x. , F ) ` N ) ) ) = if ( ( P <_ M /\ -. P <_ N ) , B , 0 ) ) $= ( co cmin cc0 wcel cn wceq cmul c1 cseq cfv cdiv cpc cle wbr cif cprime - wn wa cz wne wf pcmptcl simprd cuz eluznn syl2anc ffvelrnd nnne0d pcdiv - nnzd syl121anc pcmpt oveq12d cn0 cv eleq1d rspcdva nn0cnd subidd adantr - prmnn syl nnred simpr eluzle letrd iftrued iftrue adantl nsyl3 iffalsed - cr 3eqtr4d iffalse oveq2d 0cnd zdcle ifcldcd subid1d sylan9eqr biantrud - cc wdc ifbid eqtrd wo exmiddc mpjaodan 3eqtrd ) ADGUAFUBUCZUDZHXDUDZUEO - UFOZDXEUFOZDXFUFOZPOZDGUGUHZCQUIZDHUGUHZCQUIZPOZXKXMUKZULZCQUIZADUJRZXE - UMRXEQUNXFSRXGXJTLAXEASSGXDASSFUOSSXDUOABEFIJUPUQZAHSRGHURUDRZGSRKNGHUS - UTZVAZVDAXEYCVBASSHXDXTKVAXEXFDVCVEAXHXLXIXNPABCDEFGIJYBLMVFABCDEFHIJKL - MVFVGAXMXOXRTXPAXMULZCCPOZQXOXRAYEQTXMACACABVHRCVHREUJDEVIDTBCVHMVJJLVK - VLZVMVNYDXLCXNCPYDXKCQYDDHGADWFRXMADAXSDSRLDVOVPZVQVNAHWFRXMAHKVQVNAGWF - RXMAGYBVQVNAXMVRZAHGUGUHZXMAYAYINHGVSVPVNVTWAXMXNCTAXMCQWBWCVGYDXQCQXQX - MYDXKXPVRYHWDWEWGAXPULZXOXLXRXPAXOXLQPOXLXPXNQXLPXMCQWHWIAXLAXKCQWPYFAW - JADUMRZGUMRXKWQADYGVDZAGYBVDDGWKUTWLWMWNYJXKXQCQYJXPXKAXPVRWOWRWSAXMWQZ - XMXPWTAYKHUMRYMYLAHKVDDHWKUTXMXAVPXBXC $. + wn wa cz wne wf pcmptcl simprd cuz eluznn syl2anc ffvelcdmd nnzd nnne0d + pcdiv syl121anc pcmpt oveq12d cv eleq1d rspcdva nn0cnd subidd adantr cr + cn0 prmnn nnred simpr eluzle letrd iftrued iftrue adantl nsyl3 iffalsed + syl 3eqtr4d iffalse oveq2d cc 0cnd wdc zdcle ifcldcd sylan9eqr biantrud + subid1d ifbid eqtrd wo exmiddc mpjaodan 3eqtrd ) ADGUAFUBUCZUDZHXDUDZUE + OUFOZDXEUFOZDXFUFOZPOZDGUGUHZCQUIZDHUGUHZCQUIZPOZXKXMUKZULZCQUIZADUJRZX + EUMRXEQUNXFSRXGXJTLAXEASSGXDASSFUOSSXDUOABEFIJUPUQZAHSRGHURUDRZGSRKNGHU + SUTZVAZVBAXEYCVCASSHXDXTKVAXEXFDVDVEAXHXLXIXNPABCDEFGIJYBLMVFABCDEFHIJK + LMVFVGAXMXOXRTXPAXMULZCCPOZQXOXRAYEQTXMACACABVORCVOREUJDEVHDTBCVOMVIJLV + JVKZVLVMYDXLCXNCPYDXKCQYDDHGADVNRXMADAXSDSRLDVPWFZVQVMAHVNRXMAHKVQVMAGV + NRXMAGYBVQVMAXMVRZAHGUGUHZXMAYAYINHGVSWFVMVTWAXMXNCTAXMCQWBWCVGYDXQCQXQ + XMYDXKXPVRYHWDWEWGAXPULZXOXLXRXPAXOXLQPOXLXPXNQXLPXMCQWHWIAXLAXKCQWJYFA + WKADUMRZGUMRXKWLADYGVBZAGYBVBDGWMUTWNWQWOYJXKXQCQYJXPXKAXPVRWPWRWSAXMWL + ZXMXPWTAYKHUMRYMYLAHKVBDHWMUTXMXAWFXBXC $. $} $d p F $. $d p ph $. $d p A $. $d p N $. @@ -138849,18 +138868,18 @@ given prime (or other positive integer) that divides the number. For sylib csbeq1 rspcv mpan9 nn0ge0d 0le0 a1i prmz cuz eluzelz adantr syl2an2 syl zdcle nnzd dcan2 sylc breq2 ifbothdc syl3anc cexp cmpt nfcv nfov nfif dcn eleq1w id ifbieq1d cbvmpt eqtri simpr pcmpt2 breqtrrd ralrimiva cq wb - oveq12d wf pcmptcl simprd eluznn syl2anc ffvelrnd znq pcz mpbird dvdsval2 - wne nnne0d ) AFUBDLUCZUDZEXMUDZUEMZXOXNUFUHZNOZAXRPUAUIZXQUGUHZQMZUARUJZA - YAUARAXSROZUKZPXSEQMZXSFQMZULZUKZCXSBUMZPUNZXTQYDPYIQMZPPQMZYHUOZPYJQMZYD - YIACKUIZBUMZSOZKRUJZYCYISOZABSOZCRUJYRHYTYQCKRYTKUPCYPSCYOBUQZURCKUSZBYPS - CYOBUTZVAVBVCZYQYSKXSRKUAUSYPYISCYOXSBVDZVAVEVFVGYLYDVHVIYDYEUOZYGUOZYMYC - XSNOZAENOZUUFXSVJZAUUIYCAEFVKUDOZUUIJFEVLVOVMXSEVPVNYDYFUOZUUGYCUUHAFNOUU - LUUJYDFAFTOZYCIVMZVQXSFVPVNYFWHVOYEYGVRVSYHYKYLYNYIPYIYJPQVTPYJPQVTWAWBYD - YPYIXSKDEFDCTCUIZROZUUOBWCUHZLUNZWDKTYOROZYOYPWCUHZLUNZWDGCKTUURUVAKUURWE - UUSCUUTLUUSCUPCYOYPWCCYOWECWCWEUUAWFCLWEWGUUBUUPUUSUUQUUTLCKRWIUUBUUOYOBY - PWCUUBWJUUCWTWKWLWMAYRYCUUDVMUUNAYCWNUUEAUUKYCJVMWOWPWQAXQWROZXRYBWSAXONO - ZXNTOUVBAXOATTEXMATTDXATTXMXAABCDGHXBXCZAUUMUUKETOIJEFXDXEXFVQZATTFXMUVDI - XFZXOXNXGXEXQUAXHVOXIAXNNOXNPXKUVCXPXRWSAXNUVFVQAXNUVFXLUVEXNXOXJWBXI $. + oveq12d pcmptcl simprd eluznn syl2anc ffvelcdmd znq pcz mpbird wne nnne0d + wf dvdsval2 ) AFUBDLUCZUDZEXMUDZUEMZXOXNUFUHZNOZAXRPUAUIZXQUGUHZQMZUARUJZ + AYAUARAXSROZUKZPXSEQMZXSFQMZULZUKZCXSBUMZPUNZXTQYDPYIQMZPPQMZYHUOZPYJQMZY + DYIACKUIZBUMZSOZKRUJZYCYISOZABSOZCRUJYRHYTYQCKRYTKUPCYPSCYOBUQZURCKUSZBYP + SCYOBUTZVAVBVCZYQYSKXSRKUAUSYPYISCYOXSBVDZVAVEVFVGYLYDVHVIYDYEUOZYGUOZYMY + CXSNOZAENOZUUFXSVJZAUUIYCAEFVKUDOZUUIJFEVLVOVMXSEVPVNYDYFUOZUUGYCUUHAFNOU + ULUUJYDFAFTOZYCIVMZVQXSFVPVNYFWHVOYEYGVRVSYHYKYLYNYIPYIYJPQVTPYJPQVTWAWBY + DYPYIXSKDEFDCTCUIZROZUUOBWCUHZLUNZWDKTYOROZYOYPWCUHZLUNZWDGCKTUURUVAKUURW + EUUSCUUTLUUSCUPCYOYPWCCYOWECWCWEUUAWFCLWEWGUUBUUPUUSUUQUUTLCKRWIUUBUUOYOB + YPWCUUBWJUUCWTWKWLWMAYRYCUUDVMUUNAYCWNUUEAUUKYCJVMWOWPWQAXQWROZXRYBWSAXON + OZXNTOUVBAXOATTEXMATTDXKTTXMXKABCDGHXAXBZAUUMUUKETOIJEFXCXDXEVQZATTFXMUVD + IXEZXOXNXFXDXQUAXGVOXHAXNNOXNPXIUVCXPXRWSAXNUVFVQAXNUVFXJUVEXNXOXLWBXH $. $} ${ @@ -139146,7 +139165,7 @@ given prime (or other positive integer) that divides the number. For eqtrd ralrimiva cbvralvw zmulcld rspcv wrex nnnn0 ad2antrr zexpcl divides simplr adantll nncnd expp1d nnexpcld mulassd eqtr4d nnzd nnne0d dvdsmulcr cn0 wne syl112anc bitrd an32s syl5ibcom rexlimdva adantlr a2d expm1t nncn - com23 nnm1nn0 ax-1cn pncan sylancl anbi2d syld anassrs ralrimdva syl5bi + com23 nnm1nn0 ax-1cn pncan sylancl anbi2d syld anassrs ralrimdva biimtrid expl nnind com12 impr rspcdva 3impia ) CFGZAFGZHZBUCGZDUDGZHZACBDIJZKJZLM ZACBDNUEJZIJZKJZLMZOZHZUUMALMZUUIUULHAEUNZUUMKJZLMZAUVCUUQKJZLMZOZHZUVBPZ UVAUVBPEFCUVCCQZUVIUVAUVBUVKUVEUUOUVHUUTUVKUVDUUNALUVCCUUMKUFRUVKUVGUUSUV @@ -139477,17 +139496,17 @@ given prime (or other positive integer) that divides the number. For $( Lemma for ~ 1arith . (Contributed by Mario Carneiro, 30-May-2014.) $) 1arithlem4 $p |- ( ph -> E. x e. NN F = ( M ` x ) ) $= ( cfv cn wcel wceq cprime cmul c1 cseq cv wrex cn0 ffvelcdmda ralrimiva - wf pcmptcl simprd ffvelrnd wral wa cpc cle wbr cc0 cif 1arithlem2 sylan - co adantr simpr fveq2 pcmpt anassrs ifeq2d wdc prmz adantl nnzd syl2anc - cz zdcle ifiddc eqtr3d iftrue wo zletric mpjaodan 3eqtrrd wb 1arithlem3 - syl wfn ffn eqfnfv syl2an mpbird rspceeqv ) AHUAFUBUCZPZQRZEWMGPZSZEBUD - ZGPZSBQUEAQQHWLAQQFUIQQWLUIACUDZEPZCFLAWTUFRZCTATUFWSEMUGUHZUJUKNULZAWP - IUDZEPZXDWOPZSZITUMZAXGITAXDTRZUNZXFXDWMUOVBZXDHUPUQZXEURUSZXEAWNXIXFXK - SXCXDDGWMJKUTVAXJWTXEXDCFHLAXACTUMXIXBVCAHQRXINVCZAXIVDWSXDEVEVFXJHXDUP - UQZXMXESZXLXJXOUNZXLXEXEUSZXMXEXQXLXEURXEAXIXOXEURSOVGVHXJXRXESZXOXJXLV - IZXSXJXDVNRZHVNRZXTXIYAAXDVJVKZXJHXNVLZXDHVOVMXLXEVPWEVCVQXLXPXJXLXEURV - RVKXJYBYAXOXLVSYDYCHXDVTVMWAWBUHATUFEUIZTUFWOUIZWPXHWCZMAWNYFXCDGWMJKWD - WEYEETWFWOTWFYGYFTUFEWGTUFWOWGITEWOWHWIVMWJBWMQWRWOEWQWMGVEWKVM $. + wf pcmptcl simprd ffvelcdmd wral wa cpc co cle wbr cc0 1arithlem2 sylan + cif adantr simpr fveq2 pcmpt anassrs ifeq2d cz prmz adantl nnzd syl2anc + wdc ifiddc syl eqtr3d iftrue wo zletric mpjaodan 3eqtrrd 1arithlem3 wfn + zdcle wb ffn eqfnfv syl2an mpbird rspceeqv ) AHUAFUBUCZPZQRZEWMGPZSZEBU + DZGPZSBQUEAQQHWLAQQFUIQQWLUIACUDZEPZCFLAWTUFRZCTATUFWSEMUGUHZUJUKNULZAW + PIUDZEPZXDWOPZSZITUMZAXGITAXDTRZUNZXFXDWMUOUPZXDHUQURZXEUSVBZXEAWNXIXFX + KSXCXDDGWMJKUTVAXJWTXEXDCFHLAXACTUMXIXBVCAHQRXINVCZAXIVDWSXDEVEVFXJHXDU + QURZXMXESZXLXJXOUNZXLXEXEVBZXMXEXQXLXEUSXEAXIXOXEUSSOVGVHXJXRXESZXOXJXL + VNZXSXJXDVIRZHVIRZXTXIYAAXDVJVKZXJHXNVLZXDHWEVMXLXEVOVPVCVQXLXPXJXLXEUS + VRVKXJYBYAXOXLVSYDYCHXDVTVMWAWBUHATUFEUIZTUFWOUIZWPXHWFZMAWNYFXCDGWMJKW + CVPYEETWDWOTWDYGYFTUFEWGTUFWOWGITEWOWHWIVMWJBWMQWRWOEWQWMGVEWKVM $. $} 1arith.2 $e |- R = { e e. ( NN0 ^m Prime ) | ( `' e " NN ) e. Fin } $. @@ -139506,10 +139525,10 @@ given prime (or other positive integer) that divides the number. For cima cfn nn0ex elmap sylibr c1 cfz wss wdc 1zzd nnz fzfigd ffn elpreima 3syl 1arithlem2 eleq1d cdvds wbr cle cz id dvdsle syl2anr pcelnn ancoms prmz cuz prmnn nnuz eleqtrdi elfz5 3imtr4d sylbid expimpd elfznn adantl - ssrdv wn prmdc syl adantr ad2antrr simpr ffvelrnd nn0zd elnndc dcan2 wo - sylc intnanrd olcd df-dc exmiddc mpjaodan dcbid mpbird ralrimiva ssfidc - syl3anc cnveq imaeq1d elrab2 sylanbrc mpbir2an adantlr adantll ralbidva - rgen ffnfv eqeq12d eqfnfv syl2an nnnn0 sylib nnred cr ad2antrl syl2anc + ssrdv prmdc syl adantr ad2antrr simpr ffvelcdmd nn0zd elnndc dcan2 sylc + wn intnanrd olcd df-dc exmiddc mpjaodan mpbird ralrimiva ssfidc syl3anc + dcbid cnveq imaeq1d elrab2 sylanbrc rgen ffnfv mpbir2an adantlr adantll + wo eqeq12d ralbidva eqfnfv syl2an nnnn0 sylib nnred cr ad2antrl syl2anc wrex pc11 3bitr4d biimpd rgen2 dff13 cexp cif caddc eqid simplbi simplr peano2nnd cc0 peano2re ltp1d simprr ltletrd nnzd zltnle mpbid biantrurd clt simprl ad3antrrr bitr4d breq1 rspccv ad2antlr mtod elnn0 1arithlem4 @@ -139523,29 +139542,29 @@ given prime (or other positive integer) that divides the number. For ZUXJUWHVTVSZUYAUXPUXLUXJWAPZUWQUYCUYDUJUWQUXJWGZUWQWBUXJUWHWCWDUXLUWQUY AUYCTUXJUWHWEWFUXLUXJVFWHMZPUWHWAPUXPUYDTUWQUXLUXJKUYGUXJWIZWJWKUXIUXJV FUWHWLWDWMWNWOWNWRUWQUXHUAUXEUWQUXFUXEPZSZUXHUXFQPZUXFUWIMZKPZSZVIZUYJU - YKUYOUYKWSZUYJUYKSZUYKVIZUYMVIZUYOUYJUYRUYKUYJUXFKPZUYRUYIUYTUWQUXFUWHW - PWQUXFWTXAZXBUYQUYLWAPUYSUYQUYLUYQQRUXFUWIUWQUXCUYIUYKUXDXCUYJUYKXDXEXF - UYLXGXAUYKUYMXHXJUYJUYPSZUYNUYNWSZXIUYOVUBVUCUYNVUBUYKUYMUYJUYPXDXKXLUY - NXMVEUYJUYRUYKUYPXIVUAUYKXNXAXOUWQUXHUYOTUYIUWQUXGUYNUWQUXCUXQUXGUYNTUX - DUXRQUXFKUWIVNVOXPXBXQXRUAUXEUXAXSXTBLZUSZKVAZVBPZUXBBUWIUWRAVUDUWINZVU - FUXAVBVUHVUEUWTKVUDUWIYAYBVQGYCYDYIHKADYJYEZUWNHIKKUWQUWJKPZSZUWLUWMVUK + YKUYOUYKXIZUYJUYKSZUYKVIZUYMVIZUYOUYJUYRUYKUYJUXFKPZUYRUYIUYTUWQUXFUWHW + PWQUXFWSWTZXAUYQUYLWAPUYSUYQUYLUYQQRUXFUWIUWQUXCUYIUYKUXDXBUYJUYKXCXDXE + UYLXFWTUYKUYMXGXHUYJUYPSZUYNUYNXIZYIUYOVUBVUCUYNVUBUYKUYMUYJUYPXCXJXKUY + NXLVEUYJUYRUYKUYPYIVUAUYKXMWTXNUWQUXHUYOTUYIUWQUXGUYNUWQUXCUXQUXGUYNTUX + DUXRQUXFKUWIVNVOXSXAXOXPUAUXEUXAXQXRBLZUSZKVAZVBPZUXBBUWIUWRAVUDUWINZVU + FUXAVBVUHVUEUWTKVUDUWIXTYAVQGYBYCYDHKADYEYFZUWNHIKKUWQUWJKPZSZUWLUWMVUK UXMUXJUWKMZNZJQOZUXTUXJUWJULUMZNZJQOZUWLUWMVUKVUMVUPJQVUKUXLSUXMUXTVULV - UOUWQUXLUXMUXTNVUJUYBYFVUJUXLVULVUONUWQUXJCDUWJEFVPYGYKYHUWQUXCQRUWKUHZ + UOUWQUXLUXMUXTNVUJUYBYGVUJUXLVULVUONUWQUXJCDUWJEFVPYHYJYKUWQUXCQRUWKUHZ UWLVUNTZVUJUXDCDUWJEFUTUXCUXQUWKQUKVUSVURUXRQRUWKVMJQUWIUWKYLYMYMUWQUWH - RPUWJRPUWMVUQTVUJUWHYNUWJYNUWHUWJJUUAYMUUBUUCUUDHIKADUUEYEUWFUWGUBLZUWI + RPUWJRPUWMVUQTVUJUWHYNUWJYNUWHUWJJUUAYMUUBUUCUUDHIKADUUEYFUWFUWGUBLZUWI NHKYTZUBAOVUIVVAUBAVUTAPZUFLZUWJVTVSZUFVUTUSZKVAZOZVVAIKVVBVUJSZVVGSZHU GCVUTUGKUGLZQPVVJVVJVUTMUUFUMVFUUGUNZDUWJVFUUHUMZJEFVVKUUIVVBQRVUTUHZVU JVVGVVBVUTUWRPZVVMVVBVVNVVFVBPZVUGVVOBVUTUWRABUBUIZVUFVVFVBVVPVUEVVEKVU - DVUTYAYBVQGYCZUUJRQVUTVCUOVDYOZXCVVIUWJVVBVUJVVGUUKZUULVVIUXLVVLUXJVTVS - ZSZSZUXJVUTMZKPZWSVWCUUMNZVWBVWDUXJUWJVTVSZVWBUWJUXJUVBVSZVWFWSZVWBUWJV - VLUXJVWBUWJVVIVUJVWAVVSXBZYPZVWBUWJYQPVVLYQPVWJUWJUUNXAVWBUXJUXLUXJKPVV + DVUTXTYAVQGYBZUUJRQVUTVCUOVDYOZXBVVIUWJVVBVUJVVGUUKZUULVVIUXLVVLUXJVTVS + ZSZSZUXJVUTMZKPZXIVWCUUMNZVWBVWDUXJUWJVTVSZVWBUWJUXJUVBVSZVWFXIZVWBUWJV + VLUXJVWBUWJVVIVUJVWAVVSXAZYPZVWBUWJYQPVVLYQPVWJUWJUUNWTVWBUXJUXLUXJKPVV IVVTUYHYRYPVWBUWJVWJUUOVVIUXLVVTUUPUUQVWBUWJWAPUYEVWGVWHTVWBUWJVWIUURUX LUYEVVIVVTUYFYRUWJUXJUUSYSUUTVWBVWDUXJVVFPZVWFVWBVWDUXLVWDSZVWKVWBUXLVW DVVIUXLVVTUVCZUVAVWBVVMVUTQUKVWKVWLTVVBVVMVUJVVGVWAVVRUVDZQRVUTVMQUXJKV UTVNVOUVEVVGVWKVWFUJVVHVWAVVDVWFUFUXJVVFVVCUXJUWJVTUVFUVGUVHWNUVIVWBVWD - VWEVWBVWCRPVWDVWEXIVWBQRUXJVUTVWNVWMXEVWCUVJYOUVLUVMUVKVVBVVFKVHVVOVVGI + VWEVWBVWCRPVWDVWEYIVWBQRUXJVUTVWNVWMXDVWCUVJYOUVLUVMUVKVVBVVFKVHVVOVVGI KYTVVBVVFVUTUVNZKVUTKUVOVVBVWOQKVVBQRVUTVVRUVPUVQUVRUVSVVBVVNVVOVVQUVTI - UFVVFUWCYSUWAYIHUBKADUWBYEKADUWDYE $. + UFVVFUWCYSUWAYDHUBKADUWBYFKADUWDYF $. $} $( Fundamental theorem of arithmetic, where a prime factorization is @@ -140096,9 +140115,9 @@ with complex numbers (gaussian integers) instead, so that we only have $( Lemma for ~ ennnfone . A direct consequence of ~ fidcenumlemrk . (Contributed by Jim Kingdon, 15-Jul-2023.) $) ennnfonelemdc $p |- ( ph -> DECID ( F ` P ) e. ( F " P ) ) $= - ( cfv cima wcel wn wo wdc com wss omelon onelssi syl wfo wf fidcenumlemrk - fof ffvelrnd df-dc sylibr ) AEFJZFEKLZUIMNUIOABCDFEPUHGHIAEPLEPQIPERSTAPD - EFAPDFUAPDFUBHPDFUDTIUEUCUIUFUG $. + ( cfv cima wcel wn wo wdc com wss omelon onelssi syl wfo wf fof ffvelcdmd + fidcenumlemrk df-dc sylibr ) AEFJZFEKLZUIMNUIOABCDFEPUHGHIAEPLEPQIPERSTAP + DEFAPDFUAPDFUBHPDFUCTIUDUEUIUFUG $. $} ${ @@ -140140,13 +140159,13 @@ with complex numbers (gaussian integers) instead, so that we only have -> ( J ` 0 ) e. { g e. ( A ^pm _om ) | dom g e. _om } ) $= ( cc0 cfv c0 cv cdm com wcel cpm co crab wceq c1 cmin ccnv cif cn0 0nn0 cvv eqid iftruei eqeltri eqeq1 fvoveq1 ifbieq2d fvmptg mp2an eqtri dmeq - 0ex eleq1d wfun cxp wss wa fun0 0ss pm3.2i wb wfo focdmex sylancr elpmg - omex sylancl mpbiri dm0 peano1 a1i elrabd eqeltrid ) AUALUBZUCEUDZUEZUF - UGZEDUFUHUIZUJWKUAUAUKZUCUAULUMUIMUNZUBZUOZUCUAUPUGWSURUGWKWSUKUQWSUCUR - WPUCWRUAUSUTZVIVABUABUDZUAUKZUCXAULUMUIWQUBZUOWSUPURLXBXBWPXCWRUCXAUAUA - VBXAUAULWQUMVCVDSVEVFWTVGAWNUCUEZUFUGZEUCWOWLUCUKWMXDUFWLUCVHVJAUCWOUGZ - UCVKZUCUFDVLZVMZVNZXGXIVOXHVPVQADURUGZUFURUGZXFXJVRAXLUFDIVSXKWCOUFDIUR - VTWAWCDUFUCURURWBWDWEXEAXDUCUFWFWGVAWHWIWJ $. + 0ex eleq1d wfun cxp wss wa fun0 0ss pm3.2i wfo omex focdmex mpsyl elpmg + wb sylancl mpbiri dm0 peano1 a1i elrabd eqeltrid ) AUALUBZUCEUDZUEZUFUG + ZEDUFUHUIZUJWKUAUAUKZUCUAULUMUIMUNZUBZUOZUCUAUPUGWSURUGWKWSUKUQWSUCURWP + UCWRUAUSUTZVIVABUABUDZUAUKZUCXAULUMUIWQUBZUOWSUPURLXBXBWPXCWRUCXAUAUAVB + XAUAULWQUMVCVDSVEVFWTVGAWNUCUEZUFUGZEUCWOWLUCUKWMXDUFWLUCVHVJAUCWOUGZUC + VKZUCUFDVLZVMZVNZXGXIVOXHVPVQADURUGZUFURUGZXFXJWCXLAUFDIVRXKVSOUFDURIVT + WAVSDUFUCURURWBWDWEXEAXDUCUFWFWGVAWHWIWJ $. $} ${ @@ -140159,12 +140178,12 @@ with complex numbers (gaussian integers) instead, so that we only have ( cv cc0 c1 caddc co cuz cfv wcel cn nnuz 0p1e1 fveq2i eqtr4i eleq2i wa com wceq c0 cmin ccnv cif cn0 eqeq1 fvoveq1 ifbieq2d nnnn0 adantl nnne0 neneqd iffalsed wf1o wf 0zd frec2uzf1od f1ocnv f1of 3syl cz 0z eluzp1m1 - biimpi sylancr ffvelrnd eqeltrd fvmptd3 sylan2br ) EUAZUBUCUDUEZUFUGZUH - ZAWGUIUHZWGLUGZUPUHUIWIWGUIUCUFUGWIUJWHUCUFUKULUMUNZAWKUOZWLWGUBUQZURWG - UCUSUEZMUTZUGZVAZUPWNBWGBUAZUBUQZURWTUCUSUEWQUGZVAWSVBLUPSWTWGUQXAWOXBW - RURWTWGUBVCWTWGUCWQUSVDVEWKWGVBUHAWGVFVGWNWSWRUPWKWSWRUQAWKWOURWRWKWGUB - WGVHVIVJVGWNUBUFUGZUPWPWQWNUPXCMVKXCUPWQVKXCUPWQVLWNBUBMWNVMRVNUPXCMVOX - CUPWQVPVQWNUBVRUHWJWPXCUHVSWKWJAWKWJWMWAVGUBWGVTWBWCWDZWEXDWDWF $. + biimpi sylancr ffvelcdmd eqeltrd fvmptd3 sylan2br ) EUAZUBUCUDUEZUFUGZU + HZAWGUIUHZWGLUGZUPUHUIWIWGUIUCUFUGWIUJWHUCUFUKULUMUNZAWKUOZWLWGUBUQZURW + GUCUSUEZMUTZUGZVAZUPWNBWGBUAZUBUQZURWTUCUSUEWQUGZVAWSVBLUPSWTWGUQXAWOXB + WRURWTWGUBVCWTWGUCWQUSVDVEWKWGVBUHAWGVFVGWNWSWRUPWKWSWRUQAWKWOURWRWKWGU + BWGVHVIVJVGWNUBUFUGZUPWPWQWNUPXCMVKXCUPWQVKXCUPWQVLWNBUBMWNVMRVNUPXCMVO + XCUPWQVPVQWNUBVRUHWJWPXCUHVSWKWJAWKWJWMWAVGUBWGVTWBWCWDZWEXDWDWF $. $} ${ @@ -140178,23 +140197,23 @@ with complex numbers (gaussian integers) instead, so that we only have ( cv cdm com wcel cpm co crab wa cfv cima cop csn cun cif cmpo wceq a1i simpr fveq2d imaeq2d eleq12d simpl dmeqd opeq12d sneqd ifbieq12d adantl weq uneq12d ssrab2 simprl sselid simprr simplrl dmeq eleq1d cvv wss wfo - wn wf omex focdmex sylancr ad2antrr elrabi elpmi syl elrab simprbi word - simpld nnord ordirr adantr ffvelrnd fsnunf syl121anc csuc df-suc peano2 - fof eqeltrrid omelon onelssi elpm2r syl22anc fdmd eqeltrd ennnfonelemdc - elrabd wdc wral ifcldadc ovmpod ) AEUBZFUBZUCZUDUEZFDUDUFUGZUHZUEZGUBZU - DUEZUIZUIZXQYDKUGYDJUJZJYDUKZUEZXQXQXQUCZYHULZUMZUNZUOZYBYGBCXQYDYAUDCU - BZJUJZJYPUKZUEZBUBZYTYTUCZYQULZUMZUNZUOZYOKYBKBCYAUDUUEUPUQYGRURBEVIZCG - VIZUIZUUEYOUQYGUUHYSYJYTUUDXQYNUUHYQYHYRYIUUHYPYDJUUFUUGUSZUTZUUHYPYDJU - UIVAVBUUFUUGVCZUUHYTXQUUCYMUUKUUHUUBYLUUHUUAYKYQYHUUHYTXQUUKVDUUJVEVFVJ - VGVHYGYBYAXQXTFYAVKAYCYEVLVMAYCYEVNZYGYJXQYNYBAYCYEYJVOYGYJWAZUIZXTYNUC - ZUDUEFYNYAXRYNUQXSUUOUDXRYNVPVQUUNDVRUEZUDVRUEZYKYKUMUNZDYNWBZUURUDVSZY - NYAUEAUUPYFUUMAUUQUDDJVTZUUPWCPUDDJVRWDWEWFUUQUUNWCURUUNYKDXQWBZYKUDUEZ - YKYKUEWAZYHDUEZUUSUUNYCUVBAYCYEUUMVOZYCUVBYKUDVSZYCXQYAUEZUVBUVGUIXTFXQ - YAWGDUDXQWHWIWMWIUUNYCUVCUVFYCUVHUVCXTUVCFXQYAFEVIXSYKUDXRXQVPVQWJWKWIZ - UUNYKWLZUVDUUNUVCUVJUVIYKWNWIYKWOWIYGUVEUUMYGUDDYDJYGUVAUDDJWBAUVAYFPWP - ZUDDJXCWIUULWQWPYKDXQUDYKYHWRWSZUUNUURUDUEZUUTUUNUVCUVMUVIUVCUURYKWTUDY - KXAYKXBXDWIZUDUURXEXFWIDUDUURYNVRVRXGXHUUNUUOUURUDUUNUURDYNUVLXIUVNXJXL - YGBCDYDJABCVIXMCDXNBDXNYFOWPUVKUULXKXOZXPUVOXJ $. + wn wf omex focdmex mpsyl ad2antrr elrabi elpmi syl simpld elrab simprbi + word ordirr adantr fof ffvelcdmd fsnunf syl121anc csuc df-suc eqeltrrid + nnord peano2 elomssom elpm2r syl22anc fdmd eqeltrd elrabd ennnfonelemdc + wdc wral ifcldadc ovmpod ) AEUBZFUBZUCZUDUEZFDUDUFUGZUHZUEZGUBZUDUEZUIZ + UIZXPYCKUGYCJUJZJYCUKZUEZXPXPXPUCZYGULZUMZUNZUOZYAYFBCXPYCXTUDCUBZJUJZJ + YOUKZUEZBUBZYSYSUCZYPULZUMZUNZUOZYNKYAKBCXTUDUUDUPUQYFRURBEVIZCGVIZUIZU + UDYNUQYFUUGYRYIYSUUCXPYMUUGYPYGYQYHUUGYOYCJUUEUUFUSZUTZUUGYOYCJUUHVAVBU + UEUUFVCZUUGYSXPUUBYLUUJUUGUUAYKUUGYTYJYPYGUUGYSXPUUJVDUUIVEVFVJVGVHYFYA + XTXPXSFXTVKAYBYDVLVMAYBYDVNZYFYIXPYMYAAYBYDYIVOYFYIWAZUIZXSYMUCZUDUEFYM + XTXQYMUQXRUUNUDXQYMVPVQUUMDVRUEZUDVRUEZYJYJUMUNZDYMWBZUUQUDVSZYMXTUEAUU + OYEUULUUPAUDDJVTZUUOWCPUDDVRJWDWEWFUUPUUMWCURUUMYJDXPWBZYJUDUEZYJYJUEWA + ZYGDUEZUURUUMYBUVAAYBYDUULVOZYBUVAYJUDVSZYBXPXTUEZUVAUVFUIXSFXPXTWGDUDX + PWHWIWJWIUUMYBUVBUVEYBUVGUVBXSUVBFXPXTFEVIXRYJUDXQXPVPVQWKWLWIZUUMYJWMZ + UVCUUMUVBUVIUVHYJXCWIYJWNWIYFUVDUULYFUDDYCJYFUUTUDDJWBAUUTYEPWOZUDDJWPW + IUUKWQWOYJDXPUDYJYGWRWSZUUMUUQUDUEZUUSUUMUVBUVLUVHUVBUUQYJWTUDYJXAYJXDX + BWIZUUQXEWIDUDUUQYMVRVRXFXGUUMUUNUUQUDUUMUUQDYMUVKXHUVMXIXJYFBCDYCJABCV + IXLCDXMBDXMYEOWOUVJUUKXKXNZXOUVNXI $. $} ${ @@ -140239,26 +140258,26 @@ with complex numbers (gaussian integers) instead, so that we only have cv cpm crab cn0 nn0uz eleqtrdi ennnfonelemj0 ennnfonelemg ennnfonelemjn cuz seqp1cd wceq fveq1i a1i c0 cmin fvoveq1 ifbieq2d peano2nn0 nn0p1gt0 eqeq1 syl gt0ne0d neneqd iffalsed nn0cn 1cnd pncand fveq2d frechashgf1o - eqtrd wf1o wf f1ocnv ax-mp f1of mp1i id ffvelrnd eqeltrd fvmptd3 eqtr2d - oveq12d 3eqtr4d cvv elexd dmexg wfo fof opexg snexg unexg ennnfonelemdc - ennnfonelemh syl2anc ifcldcd opeq1d sneqd uneq12d ifeq12d fveq2 eleq12d - dmeq imaeq2 opeq2d uneq2d ovmpog syl3anc ) AEUDUEUFZKUGZEKUGZEMUHZUGZJU - FZYKIUGZIYKUIZUJZYIYIYIUKZYMULZUMZUNZUOZAYGJLUPUQZUGZEUUAUGZYGLUGZJUFYH - YLAUBFUCUSUKURUJUCDURUTUFZVAURJLUPEAEVBUPVHUGUAVCVDABCDUCFGHIJKLMNOPQRS - TVEABCDUBUCFGHIJKLMNOPQRSTVFABCDUBFGHIJKLMNOPQRSTVGVIYHUUBVJAYGKUUATVKV - LAYIUUCYKUUDJYIUUCVJAEKUUATVKVLAUUDYGUPVJZVMYGUDVNUFZYJUGZUOZYKABYGBUSZ - UPVJZVMUUJUDVNUFYJUGZUOUUIVBLURSUUJYGVJUUKUUFUULUUHVMUUJYGUPVSUUJYGUDYJ - VNVOVPAEVBUJZYGVBUJUAEVQVTAUUMUUIURUJUAUUMUUIYKURUUMUUIUUHYKUUMUUFVMUUH - UUMYGUPUUMYGEVRWAWBWCUUMUUGEYJUUMEUDEWDUUMWEWFWGWIZUUMVBUREYJVBURYJWJZV - BURYJWKUUMURVBMWJUUOBMRWHURVBMWLWMVBURYJWNWOUUMWPWQZWRVTWSAUUMUUIYKVJUA - UUNVTWTXAXBAYIUUEUJYKURUJZYTXCUJYLYTVJAVBUUEEKABCDFGHIJKLMNOPQRSTXLUAWQ - ZAUUMUUQUAUUPVTZAYOYIYSXCAYIUUEUURXDZAYIXCUJZYRXCUJZYSXCUJUUTAYQXCUJZUV - BAYPXCUJZYMDUJUVCAUVAUVDUUTYIXCXEVTAURDYKIAURDIXFURDIWKOURDIXGVTUUSWQYP - YMXCDXHXMYQXCXIVTYIYRXCXCXJXMABCDYKINOUUSXKXNBCYIYKUUEURCUSZIUGZIUVEUIZ - UJZUUJUUJUUJUKZUVFULZUMZUNZUOYTJUVHYIYIYPUVFULZUMZUNZUOXCUUJYIVJZUVHUUJ - YIUVLUVOUVPWPZUVPUUJYIUVKUVNUVQUVPUVJUVMUVPUVIYPUVFUUJYIYAXOXPXQXRUVEYK - VJZUVHYOUVOYSYIUVRUVFYMUVGYNUVEYKIXSZUVEYKIYBXTUVRUVNYRYIUVRUVMYQUVRUVF - YMYPUVSYCXPYDVPQYEYFWI $. + eqtrd wf1o wf f1ocnv f1of mp1i ffvelcdmd eqeltrd fvmptd3 eqtr2d oveq12d + ax-mp id 3eqtr4d cvv ennnfonelemh elexd dmexg wfo fof opexg snexg unexg + syl2anc ennnfonelemdc ifcldcd opeq1d sneqd uneq12d ifeq12d fveq2 imaeq2 + dmeq eleq12d opeq2d uneq2d ovmpog syl3anc ) AEUDUEUFZKUGZEKUGZEMUHZUGZJ + UFZYKIUGZIYKUIZUJZYIYIYIUKZYMULZUMZUNZUOZAYGJLUPUQZUGZEUUAUGZYGLUGZJUFY + HYLAUBFUCUSUKURUJUCDURUTUFZVAURJLUPEAEVBUPVHUGUAVCVDABCDUCFGHIJKLMNOPQR + STVEABCDUBUCFGHIJKLMNOPQRSTVFABCDUBFGHIJKLMNOPQRSTVGVIYHUUBVJAYGKUUATVK + VLAYIUUCYKUUDJYIUUCVJAEKUUATVKVLAUUDYGUPVJZVMYGUDVNUFZYJUGZUOZYKABYGBUS + ZUPVJZVMUUJUDVNUFYJUGZUOUUIVBLURSUUJYGVJUUKUUFUULUUHVMUUJYGUPVSUUJYGUDY + JVNVOVPAEVBUJZYGVBUJUAEVQVTAUUMUUIURUJUAUUMUUIYKURUUMUUIUUHYKUUMUUFVMUU + HUUMYGUPUUMYGEVRWAWBWCUUMUUGEYJUUMEUDEWDUUMWEWFWGWIZUUMVBUREYJVBURYJWJZ + VBURYJWKUUMURVBMWJUUOBMRWHURVBMWLWTVBURYJWMWNUUMXAWOZWPVTWQAUUMUUIYKVJU + AUUNVTWRWSXBAYIUUEUJYKURUJZYTXCUJYLYTVJAVBUUEEKABCDFGHIJKLMNOPQRSTXDUAW + OZAUUMUUQUAUUPVTZAYOYIYSXCAYIUUEUURXEZAYIXCUJZYRXCUJZYSXCUJUUTAYQXCUJZU + VBAYPXCUJZYMDUJUVCAUVAUVDUUTYIXCXFVTAURDYKIAURDIXGURDIWKOURDIXHVTUUSWOY + PYMXCDXIXLYQXCXJVTYIYRXCXCXKXLABCDYKINOUUSXMXNBCYIYKUUEURCUSZIUGZIUVEUI + ZUJZUUJUUJUUJUKZUVFULZUMZUNZUOYTJUVHYIYIYPUVFULZUMZUNZUOXCUUJYIVJZUVHUU + JYIUVLUVOUVPXAZUVPUUJYIUVKUVNUVQUVPUVJUVMUVPUVIYPUVFUUJYIYAXOXPXQXRUVEY + KVJZUVHYOUVOYSYIUVRUVFYMUVGYNUVEYKIXSZUVEYKIXTYBUVRUVNYRYIUVRUVMYQUVRUV + FYMYPUVSYCXPYDVPQYEYFWI $. $} ${ @@ -140289,11 +140308,11 @@ with complex numbers (gaussian integers) instead, so that we only have Jim Kingdon, 19-Jul-2023.) $) ennnfonelemom $p |- ( ph -> dom ( H ` P ) e. _om ) $= ( vg vf cfv cdm cc0 cseq com fveq1i dmeqi co wcel cv crab ennnfonelemj0 - cpm cn0 ennnfonelemg nn0uz 0zd ennnfonelemjn seqf2 ffvelrnd wceq eleq1d - wa dmeq elrab sylib simprd eqeltrid ) AEKUDZUEEJLUFUGZUDZUEZUHVLVNEKVMT - UIUJAVNDUHUPUKZULZVOUHULZAVNUBUMZUEZUHULZUBVPUNZULVQVRVFAUQWBEVMAUCFWBU - HJLUFUQABCDUBFGHIJKLMNOPQRSTUOABCDUCUBFGHIJKLMNOPQRSTURUSAUTABCDUCFGHIJ - KLMNOPQRSTVAVBUAVCWAVRUBVNVPVSVNVDVTVOUHVSVNVGVEVHVIVJVK $. + cpm wa cn0 ennnfonelemg nn0uz ennnfonelemjn seqf2 ffvelcdmd wceq eleq1d + 0zd dmeq elrab sylib simprd eqeltrid ) AEKUDZUEEJLUFUGZUDZUEZUHVLVNEKVM + TUIUJAVNDUHUPUKZULZVOUHULZAVNUBUMZUEZUHULZUBVPUNZULVQVRUQAURWBEVMAUCFWB + UHJLUFURABCDUBFGHIJKLMNOPQRSTUOABCDUCUBFGHIJKLMNOPQRSTUSUTAVFABCDUCFGHI + JKLMNOPQRSTVAVBUAVCWAVRUBVNVPVSVNVDVTVOUHVSVNVGVEVHVIVJVK $. $} ${ @@ -140309,12 +140328,12 @@ with complex numbers (gaussian integers) instead, so that we only have -> dom ( H ` ( P + 1 ) ) = suc dom ( H ` P ) ) $= ( c1 caddc co cfv cdm csn cun csuc ccnv cop cima wcel cif ennnfonelemp1 iffalsed eqtrd dmeqd dmun eqtrdi wceq com wfo fof syl wf1o frechashgf1o - wf cn0 f1ocnv f1of mp2b a1i ffvelrnd dmsnopg uneq2d df-suc eqtr4di ) AE - UCUDUEKUFZUGZEKUFZUGZWCUHZUIZWCUJAWAWCWCEMUKZUFZIUFZULUHZUGZUIZWEAWAWBW - IUIZUGWKAVTWLAVTWHIWGUMUNZWBWLUOWLABCDEFGHIJKLMNOPQRSTUAUPAWMWBWLUBUQUR - USWBWIUTVAAWJWDWCAWHDUNWJWDVBAVCDWGIAVCDIVDVCDIVIOVCDIVEVFAVJVCEWFVJVCW - FVIZAVCVJMVGVJVCWFVGWNBMRVHVCVJMVKVJVCWFVLVMVNUAVOVOWCWHDVPVFVQURWCVRVS - $. + wf cn0 f1ocnv f1of mp2b a1i ffvelcdmd dmsnopg uneq2d df-suc eqtr4di ) A + EUCUDUEKUFZUGZEKUFZUGZWCUHZUIZWCUJAWAWCWCEMUKZUFZIUFZULUHZUGZUIZWEAWAWB + WIUIZUGWKAVTWLAVTWHIWGUMUNZWBWLUOWLABCDEFGHIJKLMNOPQRSTUAUPAWMWBWLUBUQU + RUSWBWIUTVAAWJWDWCAWHDUNWJWDVBAVCDWGIAVCDIVDVCDIVIOVCDIVEVFAVJVCEWFVJVC + WFVIZAVCVJMVGVJVCWFVGWNBMRVHVCVJMVKVJVCWFVLVMVNUAVOVOWCWHDVPVFVQURWCVRV + S $. $} ${ @@ -140326,12 +140345,12 @@ with complex numbers (gaussian integers) instead, so that we only have ennnfonelemss $p |- ( ph -> ( H ` P ) C_ ( H ` ( P + 1 ) ) ) $= ( ccnv cfv cima wcel c1 caddc co wss wceq cdm cop csn cun ennnfonelemp1 wn wa cif adantr simpr iftrued eqtrd eqimss2 syl iffalsed sseqtrrid wdc - ssun1 cn0 com wf1o frechashgf1o f1ocnv f1of mp2b ffvelrnd ennnfonelemdc - wo wf a1i exmiddc mpjaodan ) AEMUBZUCZIUCZIWDUDUEZEKUCZEUFUGUHKUCZUIZWF - UPZAWFUQZWHWGUJWIWKWHWFWGWGWGUKWEULUMZUNZURZWGAWHWNUJZWFABCDEFGHIJKLMNO - PQRSTUAUOZUSWKWFWGWMAWFUTVAVBWGWHVCVDAWJUQZWMWGWHWGWLVHWQWHWNWMAWOWJWPU - SWQWFWGWMAWJUTVEVBVFAWFVGWFWJVRABCDWDINOAVIVJEWCVIVJWCVSZAVJVIMVKVIVJWC - VKWRBMRVLVJVIMVMVIVJWCVNVOVTUAVPVQWFWAVDWB $. + ssun1 wo cn0 com wf wf1o frechashgf1o f1of mp2b ffvelcdmd ennnfonelemdc + f1ocnv a1i exmiddc mpjaodan ) AEMUBZUCZIUCZIWDUDUEZEKUCZEUFUGUHKUCZUIZW + FUPZAWFUQZWHWGUJWIWKWHWFWGWGWGUKWEULUMZUNZURZWGAWHWNUJZWFABCDEFGHIJKLMN + OPQRSTUAUOZUSWKWFWGWMAWFUTVAVBWGWHVCVDAWJUQZWMWGWHWGWLVHWQWHWNWMAWOWJWP + USWQWFWGWMAWJUTVEVBVFAWFVGWFWJVIABCDWDINOAVJVKEWCVJVKWCVLZAVKVJMVMVJVKW + CVMWRBMRVNVKVJMVSVJVKWCVOVPVTUAVQVRWFWAVDWB $. $} ${ @@ -140377,41 +140396,41 @@ with complex numbers (gaussian integers) instead, so that we only have wceq sseq12d imbi2d weq cz c0 ennnfonelem0 dm0 eqtrdi 0ss eqsstrdi cima a1i wa wn cle wbr com wf1o frechashgf1o f1of mp1i wdc wral ad2antrr wfo wne csuc wrex simplr nn0uz eleqtrrdi peano2nn0 syl ennnfonelemom adantr - wf ffvelrnd nn0red peano2re cop csn cun cif ennnfonelemp1 simpr iftrued - cr eqtrd fveq2d f1ocnv ax-mp frec2uzled mpbid f1ocnvfv2 sylancr breqtrd - 0zd id eqbrtrd lep1d letrd breqtrrd wb mpbird iffalsed dmun fof dmsnopg - uneq2d df-suc eqtr4di nnsucsssuc syl2an2r eqsstrd peano2 frec2uzsucd wo - oveq1d ennnfonelemdc exmiddc mpjaodan ex expcom a2d uzind4 eleq2s mpcom - ) EUDUEAEKUFZUGZEMUHZUFZUIZUAAUUKUJZEUKULUFZUDAUBUMZKUFZUGZUUNUUIUFZUIZ - UJAUKKUFZUGZUKUUIUFZUIZUJZAUCUMZKUFZUGZUVDUUIUFZUIZUJAUVDUNUOUPZKUFZUGZ - UVIUUIUFZUIZUJUULUBUCUKEUUNUKUSZUURUVBAUVNUUPUUTUUQUVAUVNUUOUUSUUNUKKUQ - URUUNUKUUIUQUTVAUBUCVBZUURUVHAUVOUUPUVFUUQUVGUVOUUOUVEUUNUVDKUQURUUNUVD - UUIUQUTVAUUNUVIUSZUURUVMAUVPUUPUVKUUQUVLUVPUUOUVJUUNUVIKUQURUUNUVIUUIUQ - UTVAUUNEUSZUURUUKAUVQUUPUUHUUQUUJUVQUUOUUGUUNEKUQURUUNEUUIUQUTVAUVCUKVC - UEAUUTVDUVAAUUTVDUGVDAUUSVDABCDFGHIJKLMNOPQRSTVEURVFVGUVAVHVIVKUVDUUMUE - ZAUVHUVMAUVRUVHUVMUJAUVRVLZUVHUVMUVSUVHVLZUVGIUFZIUVGVJUEZUVMUWBVMZUVTU - WBVLZUVMUVKMUFZUVLMUFZVNVOZUWDUWEUVIUWFVNUWDUWEUVDUVIUWDUWEUWDVPUDUVKMV - 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iftrued wf f1ocnv f1of 3syl frec2uzsucd f1ocnvfv2 sylancr oveq1d fveq2d - ffvelrnd peano2 syl eqtr3d df-suc imaundi wss snssd ssequn2 wb wfn fofn - sylib fnsnfv uneq2d eqeq1d mpbid mpbird cin ennnfonelemom f1osng f1oeq3 - syl2anc fof word nnord orddisj f1oun syl22anc iffalsed ennnfonelemhdmp1 - ineq2d disjsn sylibr wo ennnfonelemdc exmiddc mpjaodan ex expcom nn0ind - a2d mpcom ) EUCUDAEKUEZUFZIEMUGZUEZUHZUUQUIZUAAUBUJZKUEZUFZIUVCUUSUEZUH - ZUVDUIZUKAURKUEZUFZIURUUSUEZUHZUVIUIZUKAGUJZKUEZUFZIUVNUUSUEZUHZUVOUIZU - KAUVNULUMUNZKUEZUFZIUVTUUSUEZUHZUWAUIZUKAUVBUKUBGEUVCURUOZUVHUVMAUWFUVE - UVJUVGUVLUVDUVIUVCURKUPZUWFUVDUVIUWGUQUWFUVFUVKIUVCURUUSUPUSUTVAUBGVBZU - VHUVSAUWHUVEUVPUVGUVRUVDUVOUVCUVNKUPZUWHUVDUVOUWIUQUWHUVFUVQIUVCUVNUUSU - PUSUTVAUVCUVTUOZUVHUWEAUWJUVEUWBUVGUWDUVDUWAUVCUVTKUPZUWJUVDUWAUWKUQUWJ - UVFUWCIUVCUVTUUSUPUSUTVAUVCEUOZUVHUVBAUWLUVEUURUVGUVAUVDUUQUVCEKUPZUWLU - VDUUQUWMUQUWLUVFUUTIUVCEUUSUPUSUTVAAUVMVSVSVSUIVCAUVJVSUVLVSUVIVSABCDFG - HIJKLMNOPQRSTVDZAUVJVSUFVSAUVIVSUWNUQVEVFUVLVSUOAUVLIVSUHVSUVKVSIVSMUEZ - 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NN0 M e. dom ( H ` i ) ) $= ( vw vq va com wcel cv cfv cdm cn0 wrex wi c0 csuc eleq1 rexbidv imbi2d wceq weq c1 1nn0 csn 0ex snid cop ennnfonelem1 dmeqd wfn peano1 wfo fof - syl a1i ffvelrnd fnsng sylancr fndm eqtrd eleqtrrid fveq2 eleq2d rspcev - wf wa wdc wral ad3antrrr wne neeq1d ralbidv cbvrexv ralbii sylib simplr - ennnfonelemex word wss ad2antrr ennnfonelemom nnord simpr ordsucss sylc - wb nnsucelsuc mpbid sseldd reximdva mpd rexlimdva2 syl6ibr expcom finds - ex a2d mpcom ) MUFUGAMEUHZKUIZUJZUGZEUKULZUBAUCUHZXTUGZEUKULZUMAUNXTUGZ - EUKULZUMAGUHZXTUGZEUKULZUMAYHUOZXTUGZEUKULZUMAYBUMUCGMYCUNUSZYEYGAYNYDY - FEUKYCUNXTUPUQURUCGUTZYEYJAYOYDYIEUKYCYHXTUPUQURYCYKUSZYEYMAYPYDYLEUKYC - YKXTUPUQURYCMUSZYEYBAYQYDYAEUKYCMXTUPUQURAVAUKUGUNVAKUIZUJZUGZYGVBAUNUN - VCZYSUNVDVEAYSUNUNIUIZVFVCZUJZUUAAYRUUCABCDFGHIJKLNOPQRSTUAVGVHAUUCUUAV - IZUUDUUAUSAUNUFUGZUUBDUGUUEVJAUFDUNIAUFDIVKZUFDIWDPUFDIVLVMUUFAVJVNVOUN - UUBUFDVPVQUUAUUCVRVMVSVTYFYTEVAUKXRVAUSZXTYSUNUUHXSYRXRVAKWAVHWBWCVQYHU - FUGZAYJYMAUUIYJYMUMAUUIWEZYJYKUDUHZKUIZUJZUGZUDUKULZYMUUJYIUUOEUKUUJXRU - KUGZWEZYIWEZXTUUMUGZUDUKULUUOUURBCDXRUDFUEHIJKLNABCUTWFCDWGBDWGZUUIUUPY - IOWHZAUUGUUIUUPYIPWHZUURYHIUIZFUHIUIZWIZFHUHUOZWGZGUFULZHUFWGZUEUHZIUIZ - UVDWIZFUVFWGZUEUFULZHUFWGAUVIUUIUUPYIQWHZUVHUVNHUFUVGUVMGUEUFGUEUTZUVEU - VLFUVFUVPUVCUVKUVDYHUVJIWAWJWKWLWMWNRSTUAUUJUUPYIWOZWPUURUUSUUNUDUKUURU - UKUKUGZWEZUUSUUNUVSUUSWEZXTUOZUUMYKUVTUUMWQZUUSUWAUUMWRUVTUUMUFUGUWBUVT - BCDUUKFGHIJKLNUURUUTUVRUUSUVAWSUURUUGUVRUUSUVBWSUURUVIUVRUUSUVOWSRSTUAU - URUVRUUSWOWTUUMXAVMUVSUUSXBXTUUMXCXDUURYKUWAUGZUVRUUSUURYIUWCUUQYIXBUUR - XTUFUGYIUWCXEUURBCDXRFGHIJKLNUVAUVBUVORSTUAUVQWTYHXTXFVMXGWSXHXOXIXJXKY - LUUNEUDUKEUDUTZXTUUMYKUWDXSUULXRUUKKWAVHWBWLXLXMXPXNXQ $. + wf syl ffvelcdmd fnsng sylancr fndm eqtrd eleqtrrid fveq2 eleq2d rspcev + a1i wa wdc wral ad3antrrr wne neeq1d ralbidv ralbii sylib ennnfonelemex + cbvrexv simplr word ad2antrr ennnfonelemom nnord simpr ordsucss sylc wb + wss nnsucelsuc mpbid sseldd ex reximdva rexlimdva2 syl6ibr expcom finds + mpd a2d mpcom ) MUFUGAMEUHZKUIZUJZUGZEUKULZUBAUCUHZXTUGZEUKULZUMAUNXTUG + ZEUKULZUMAGUHZXTUGZEUKULZUMAYHUOZXTUGZEUKULZUMAYBUMUCGMYCUNUSZYEYGAYNYD + YFEUKYCUNXTUPUQURUCGUTZYEYJAYOYDYIEUKYCYHXTUPUQURYCYKUSZYEYMAYPYDYLEUKY + CYKXTUPUQURYCMUSZYEYBAYQYDYAEUKYCMXTUPUQURAVAUKUGUNVAKUIZUJZUGZYGVBAUNU + NVCZYSUNVDVEAYSUNUNIUIZVFVCZUJZUUAAYRUUCABCDFGHIJKLNOPQRSTUAVGVHAUUCUUA + VIZUUDUUAUSAUNUFUGZUUBDUGUUEVJAUFDUNIAUFDIVKZUFDIVMPUFDIVLVNUUFAVJWDVOU + NUUBUFDVPVQUUAUUCVRVNVSVTYFYTEVAUKXRVAUSZXTYSUNUUHXSYRXRVAKWAVHWBWCVQYH + UFUGZAYJYMAUUIYJYMUMAUUIWEZYJYKUDUHZKUIZUJZUGZUDUKULZYMUUJYIUUOEUKUUJXR + UKUGZWEZYIWEZXTUUMUGZUDUKULUUOUURBCDXRUDFUEHIJKLNABCUTWFCDWGBDWGZUUIUUP + YIOWHZAUUGUUIUUPYIPWHZUURYHIUIZFUHIUIZWIZFHUHUOZWGZGUFULZHUFWGZUEUHZIUI + ZUVDWIZFUVFWGZUEUFULZHUFWGAUVIUUIUUPYIQWHZUVHUVNHUFUVGUVMGUEUFGUEUTZUVE + UVLFUVFUVPUVCUVKUVDYHUVJIWAWJWKWOWLWMRSTUAUUJUUPYIWPZWNUURUUSUUNUDUKUUR + UUKUKUGZWEZUUSUUNUVSUUSWEZXTUOZUUMYKUVTUUMWQZUUSUWAUUMXEUVTUUMUFUGUWBUV + TBCDUUKFGHIJKLNUURUUTUVRUUSUVAWRUURUUGUVRUUSUVBWRUURUVIUVRUUSUVOWRRSTUA + UURUVRUUSWPWSUUMWTVNUVSUUSXAXTUUMXBXCUURYKUWAUGZUVRUUSUURYIUWCUUQYIXAUU + RXTUFUGYIUWCXDUURBCDXRFGHIJKLNUVAUVBUVORSTUAUVQWSYHXTXFVNXGWRXHXIXJXOXK + YLUUNEUDUKEUDUTZXTUUMYKUWDXSUULXRUUKKWAVHWBWOXLXMXPXNXQ $. $} ${ @@ -140722,12 +140741,12 @@ with complex numbers (gaussian integers) instead, so that we only have ( com cv cfv wcel cn0 cc0 wceq vc vr vq vp va vb vi ccom cpm co cdm cop cima csn cun cif cmpo c0 cmin ccnv cmpt cseq ciun wfo wf1o frechashgf1o c1 f1ofo ax-mp a1i foco syl2anc wne csuc wral wrex wa cfz oveq2 raleqdv - rexbidv adantr wf f1of simpr ffvelrnd rspcdva f1ocnv mp2b simprl neeq2d - fveq2 simplrr cle wbr ad2antrr peano2 syl elnn ffvelcdmi clt frec2uzltd - caddc 0zd frec2uzsucd breqtrd wb nn0leltp1 mpbird fznn0 mpbir2and fvco3 - mpd sylancr f1ocnvfv2 fveq2d 3netr4d ralrimiva neeq1d ralbidv rexlimddv - eqtrd rspcev id dmeq opeq1d sneqd uneq12d ifeq12d imaeq2 eleq12d opeq2d - uneq2d ifbieq2d cbvmpov eqeq1 fvoveq1 cbvmptv eqid cbviunv + rexbidv adantr wf f1of simpr ffvelcdmd rspcdva f1ocnv mp2b simprl fveq2 + neeq2d simplrr cle wbr ad2antrr peano2 syl elnn ffvelcdmi caddc clt 0zd + frec2uzltd mpd frec2uzsucd breqtrd wb nn0leltp1 fznn0 mpbir2and sylancr + mpbird fvco3 f1ocnvfv2 fveq2d eqtrd 3netr4d ralrimiva ralbidv rexlimddv + neeq1d rspcev id dmeq opeq1d sneqd uneq12d ifeq12d imaeq2 opeq2d uneq2d + eleq12d ifbieq2d cbvmpov eqeq1 fvoveq1 cbvmptv eqid cbviunv ennnfonelemen ) ABCDUAUBUCUDHIUHZUEUFDNUIUJZNUFOZUUBPZUUBUUDUMZQZUEOZUU HUUHUKZUUEULZUNZUOZUPZUQZUUNUERUUHSTZURUUHVGUSUJIUTZPZUPZVAZSVBZUUSUGRU GOZUUTPZVCIJARDHVDNRIVDZNDUUBVDKUVCANRIVEZUVCBIMVFZNRIVHVIVJNRDHIVKVLAU @@ -140737,20 +140756,20 @@ with complex numbers (gaussian integers) instead, so that we only have VIZVJAUVOWEZWFZWGUVPUVQRQZUWDVQZVQZUVQUUPPZNQZUWQUUBPZUVIVMZUBUVLVOZUVN UWPRNUVQUUPRNUUPWCZUWPUVDRNUUPVEUXBUVENRIWHRNUUPWDWIVJUVPUWNUWDWJZWFZUW PUWTUBUVLUWPUVHUVLQZVQZUVRUVHIPZHPZUWSUVIUXFUWAUVRUXHVMEUWCUXGUVSUXGTUV - TUXHUVRUVSUXGHWLWKUVPUWNUWDUXEWMUXFUXGUWCQZUXGRQZUXGUWBWNWOZUXFUVHNQZUX + TUXHUVRUVSUXGHWKWLUVPUWNUWDUXEWMUXFUXGUWCQZUXGRQZUXGUWBWNWOZUXFUVHNQZUX JUXFUXEUVLNQZUXLUWPUXEWEZUXFUVOUXMUVPUVOUWOUXEUWLWPZUVKWQWRZUVHUVLWSVLZ - NRUVHIUWKWTWRZUXFUXKUXGUWBVGXCUJZXAWOZUXFUXGUVLIPZUXSXAUXFUXEUXGUYAXAWO - UXNUXFBUVHUVLSIUXFXDZMUXQUXPXBXMUXFBUVKSIUYBMUXOXEXFUXFUXJUWBRQZUXKUXTX - GUXRUVPUYCUWOUXEUWMWPZUXGUWBXHVLXIUXFUYCUXIUXJUXKVQXGUYDUXGUWBXJWRXKWGU - XFUWSUWQIPZHPZUVRUXFUWJUWRUWSUYFTUWKUWPUWRUXEUXDWBNRUWQHIXLXNUXFUYEUVQH - UXFUVDUWNUYEUVQTUVEUWPUWNUXEUXCWBNRUVQIXOXNXPYBUXFUWJUXLUVIUXHTUWKUXQNR - UVHHIXLXNXQXRUVMUXAUCUWQNUVFUWQTZUVJUWTUBUVLUYGUVGUWSUVIUVFUWQUUBWLXSXT - YCVLYAXRUEUFBCUUCNUUMCOZUUBPZUUBUYHUMZQZBOZUYLUYLUKZUYIULZUNZUOZUPUUGUY + NRUVHIUWKWTWRZUXFUXKUXGUWBVGXAUJZXBWOZUXFUXGUVLIPZUXSXBUXFUXEUXGUYAXBWO + UXNUXFBUVHUVLSIUXFXCZMUXQUXPXDXEUXFBUVKSIUYBMUXOXFXGUXFUXJUWBRQZUXKUXTX + HUXRUVPUYCUWOUXEUWMWPZUXGUWBXIVLXMUXFUYCUXIUXJUXKVQXHUYDUXGUWBXJWRXKWGU + XFUWSUWQIPZHPZUVRUXFUWJUWRUWSUYFTUWKUWPUWRUXEUXDWBNRUWQHIXNXLUXFUYEUVQH + UXFUVDUWNUYEUVQTUVEUWPUWNUXEUXCWBNRUVQIXOXLXPXQUXFUWJUXLUVIUXHTUWKUXQNR + UVHHIXNXLXRXSUVMUXAUCUWQNUVFUWQTZUVJUWTUBUVLUYGUVGUWSUVIUVFUWQUUBWKYBXT + YCVLYAXSUEUFBCUUCNUUMCOZUUBPZUUBUYHUMZQZBOZUYLUYLUKZUYIULZUNZUOZUPUUGUY LUYLUYMUUEULZUNZUOZUPUUHUYLTZUUGUUHUYLUULUYSUYTYDZUYTUUHUYLUUKUYRVUAUYT UUJUYQUYTUUIUYMUUEUUHUYLYEYFYGYHYIUUDUYHTZUUGUYKUYSUYPUYLVUBUUEUYIUUFUY - JUUDUYHUUBWLZUUDUYHUUBYJYKVUBUYRUYOUYLVUBUYQUYNVUBUUEUYIUYMVUCYLYGYMYNY + JUUDUYHUUBWKZUUDUYHUUBYJYMVUBUYRUYOUYLVUBUYQUYNVUBUUEUYIUYMVUCYKYGYLYNY OMUEBRUURUYLSTZURUYLVGUSUJUUPPZUPUYTUUOVUDUUQVUEURUUHUYLSYPUUHUYLVGUUPU - SYQYNYRUUTYSUGUARUVBUAOZUUTPUVAVUFUUTWLYTUUA $. + SYQYNYRUUTYSUGUARUVBUAOZUUTPUVAVUFUUTWKYTUUA $. $} ${ @@ -140783,29 +140802,29 @@ with complex numbers (gaussian integers) instead, so that we only have ( f ` k ) =/= ( f ` j ) ) ) ) $= ( vg cn wbr cn0 cv wceq wral cfv wa wcel syl2anc wi adantl cen wdc wfo co wne cc0 cfz wrex wex nn0ennn ensymi entr mpan2 wf1o bren biimpi cz adantr - wf f1of simprl ffvelrnd nn0zd simprr zdceq wfn crn dff1o6 r19.21bi anasss - simp3bi fveq2 impbid1 dcbid mpbid ralrimivva ccnv f1ocnv syl c1 peano2nn0 - f1ofo caddc elfznn0 nn0red cle clt elfzle2 wb simplr nn0leltp1 neneqd w3a - gtned sylib simp3d ad2antrr fveqeq2 eqeq1 imbi12d eqeq2d eqeq2 rspc2v mpd - neqned ralrimiva neeq1d ralbidv rspcev cvv cnvexg elv foeq1 fveq1 neeq12d - mtod rexralbidv anbi12d spcev jca exlimddv ) CIUAJZCKUAJZALZBLZMZUBZBCNAC - 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( G ` j ) =/= ( G ` i ) ) ) $= ( cn0 cfv com wceq wcel wfo cv wne cc0 cfz co wral wrex ccnv frechashgf1o ccom wf1o f1ocnv ax-mp f1ofo foco sylancl wb foeq1 sylibr wa csuc cima wn - imaeq2 eleq2d notbid rexbidv adantr wf f1of a1i simpr ffvelrnd peano2 syl - rspcdva simprl fveq1i fvco3 sylancr eqtrid f1ocnvfv1 fveq2d eqtrd simplrr - mpan eqneltrd elfznn0 adantl wss cle wbr elfzle2 0zd frec2uzled f1ocnvfv2 - ad2antrr breq12d bitrd mpbird nnsssuc syl2anc mpbid wfun cdm wi ad3antrrr - fof ffund fdmd eleqtrrd funfvima mpd eqeltrd mtand neqned ralrimiva fveq2 - neeq1d ralbidv rspcev rexlimddv jca ) APCJUAZEUBZJQZDUBZJQZUCZDUDGUBZUEUF - ZUGZEPUHZGPUGAPCIKUIZUKZUAZYEARCIUAZPRYOUAZYQNPRYOULZYSRPKULZYTBKLUJZRPKU - MUNZPRYOUOUNPRCIYOUPUQJYPSYEYQURMPCJYPUSUNUTAYNGPAYKPTZVAZFUBZIQZIYKYOQZV - BZVCZTZVDZYNFRUUEUUGIHUBZVCZTZVDZFRUHZUULFRUHHRUUIUUMUUISZUUPUULFRUURUUOU - UKUURUUNUUJUUGUUMUUIIVEVFVGVHAUUQHRUGUUDOVIUUEUUHRTZUUIRTUUEPRYKYOPRYOVJZ - UUEYTUUTUUCPRYOVKUNZVLAUUDVMZVNZUUHVOVPVQUUEUUFRTZUULVAZVAZUUFKQZPTZUVGJQ - 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3adantl1 simpl2 equequ2 cbvral2v ssralv ralimdv sylsyld syl5bi sylc simpr - cfn cres wfun cdm fofun word ordom ordtr ax-mp trss mpsyl wceq fdmd fores - wtr sseqtrrd syl2anc vex resex foeq1 spcev foeq2 exbidv fidcenum sylanbrc - rspcev inffinp1 simprl syl2an2r eqneltrrd ex reximdva rexlimddv ralrimiva - foelrn simprr jca 3com23 3expia impbii ) CUALMZABNZOZBCPZACPZQCDRZUBZDUCZ - QCUEMZUDZYMYQYTUUAYMYQYSERZYRUFZYRFRZUGZSUHZEQUIZFQPZTZDUCZABCDEFUJZUKYMU - UKYTYMYQUUKUULULYMUUJYSDUUJYSURYMYSUUIUMUNUOUPYMQCLMUUAYMCQYMUAQLMCQLMUQC - UAQUSUTVAQCVBVJVCUUBYQUUKYMYQYTUUAVDUUBYQUUATZYTUUKYQYTUUAVEYQYTUUAVFUUMY - SUUJDYQUUAYSUUJYQYSUUAUUJYQYSUUAUDZYSUUIYQYSUUAVFUUNUUHFQUUNUUEQSZTZGRZUU - FSUHZUUHGCUUPGBCUUFUUPYQGBNZOZBCPZGCPYQYSUUAUUOVGZYPUVAAGCAGNZYOUUTBCUVCY - NUUSAGBVHVKVIVLVMYQYSUUAUUOVNYSUUAUUOUUFCVOZYQYSUVDUUAUUOYSQCYRVPZUVDQCYR - VQZUVEUUFYRVRCYRUUEVSQCYRVTWAVJZWBWCUUPHINZOZIUUFPZHUUFPZJRZUUFKRZUBZKUCZ - JQUIZUUFWMSUUPYSYQUVKYQYSUUAUUOWDZUVBYQUVIICPZHCPZYSUVKYOUVIHBNZOABHICCAH - NYNUVTAHBVHVKBINUVTUVHBIHWEVKWFYSUVDUVSUVJHCPUVKUVGYSUVRUVJHCYSUVDUVRUVJU - 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(Contributed by Jim Kingdon, 28-Oct-2023.) $) ctiunctlemudc $p |- ( ph -> A. n e. _om DECID n e. U ) $= ( vm cv wcel wdc com wral wa cfv c1st c2nd csb wn wceq eleq1 adantr cxp - dcbid wf1o f1of syl simpr ffvelrnd xp1st rspcdva wfo ad2antrr ralrimiva - wf fof nfcv nfcsb1v nfcri nfdc nfralya csbeq1a eleq2d ralbidv rspc sylc - xp2nd dcan2 wo intnanrd olcd df-dc sylibr mpjaodan 2fveq3 eleq1d fveq2d - exmiddc csbeq1d eleq12d anbi12d elrab2 wb adantl bitr4id mpbird cbvralv - ibar sylib ) AUAUBZHUCZUDZUAUEUFIUBZHUCZUDZIUEUFAXEUAUEAXCUEUCZUGZXEXCL - UHZUIUHZFUCZXKUJUHZBXLJUHZGUKZUCZUGZUDZXJXMXSXMULZXJXMUGZXMUDZXQUDZXSXJ - YBXMXJXFFUCZUDZYBIUEXLXFXLUMYDXMXFXLFUNUQAYEIUEUFXINUOXJXKUEUEUPZUCZXLU - EUCXJUEYFXCLXJUEYFLURZUEYFLVHAYHXISUOUEYFLUSUTAXIVAVBZXKUEUEVCUTVDZUOYA - XFXPUCZUDZYCIUEXNXFXNUMYKXQXFXNXPUNUQYAXODUCXFGUCZUDZIUEUFZBDUFZYLIUEUF - ZYAFDXLJAFDJVHZXIXMAFDJVEYROFDJVIUTVFXJXMVAVBAYPXIXMAYOBDQVGVFYOYQBXODY - LBIUEBUEVJYKBBIXPBXOGVKVLVMVNBUBXOUMZYNYLIUEYSYMYKYSGXPXFBXOGVOVPUQVQVR - VSYAYGXNUEUCXJYGXMYIUOXKUEUEVTUTVDXMXQWAVSXJXTUGZXRXRULZWBXSYTUUAXRYTXM - XQXJXTVAWCWDXRWEWFXJYBXMXTWBYJXMWKUTWGXJXDXRXJXDXIXRUGZXRCUBZLUHZUIUHZF - UCZUUDUJUHZBUUEJUHZGUKZUCZUGXRCXCUEHUUCXCUMZUUFXMUUJXQUUKUUEXLFUUCXCUIL - WHZWIUUKUUGXNUUIXPUUCXCUJLWHUUKBUUHXOGUUKUUEXLJUULWJWLWMWNTWOXIXRUUBWPA - XIXRXAWQWRUQWSVGXEXHUAIUEXCXFUMXDXGXCXFHUNUQWTXB $. + dcbid wf1o wf f1of syl simpr ffvelcdmd xp1st rspcdva ad2antrr ralrimiva + wfo fof nfcv nfcsb1v nfcri nfdc nfralya csbeq1a ralbidv rspc sylc xp2nd + eleq2d dcan2 wo intnanrd olcd df-dc sylibr exmiddc 2fveq3 eleq1d fveq2d + mpjaodan csbeq1d eleq12d anbi12d elrab2 wb adantl bitr4id cbvralv sylib + ibar mpbird ) AUAUBZHUCZUDZUAUEUFIUBZHUCZUDZIUEUFAXEUAUEAXCUEUCZUGZXEXC + LUHZUIUHZFUCZXKUJUHZBXLJUHZGUKZUCZUGZUDZXJXMXSXMULZXJXMUGZXMUDZXQUDZXSX + JYBXMXJXFFUCZUDZYBIUEXLXFXLUMYDXMXFXLFUNUQAYEIUEUFXINUOXJXKUEUEUPZUCZXL + UEUCXJUEYFXCLXJUEYFLURZUEYFLUSAYHXISUOUEYFLUTVAAXIVBVCZXKUEUEVDVAVEZUOY + AXFXPUCZUDZYCIUEXNXFXNUMYKXQXFXNXPUNUQYAXODUCXFGUCZUDZIUEUFZBDUFZYLIUEU + FZYAFDXLJAFDJUSZXIXMAFDJVHYROFDJVIVAVFXJXMVBVCAYPXIXMAYOBDQVGVFYOYQBXOD + YLBIUEBUEVJYKBBIXPBXOGVKVLVMVNBUBXOUMZYNYLIUEYSYMYKYSGXPXFBXOGVOVTUQVPV + QVRYAYGXNUEUCXJYGXMYIUOXKUEUEVSVAVEXMXQWAVRXJXTUGZXRXRULZWBXSYTUUAXRYTX + MXQXJXTVBWCWDXRWEWFXJYBXMXTWBYJXMWGVAWKXJXDXRXJXDXIXRUGZXRCUBZLUHZUIUHZ + FUCZUUDUJUHZBUUEJUHZGUKZUCZUGXRCXCUEHUUCXCUMZUUFXMUUJXQUUKUUEXLFUUCXCUI + LWHZWIUUKUUGXNUUIXPUUCXCUJLWHUUKBUUHXOGUUKUUEXLJUULWJWLWMWNTWOXIXRUUBWP + AXIXRXAWQWRUQXBVGXEXHUAIUEXCXFUMXDXGXCXFHUNUQWSWT $. $} ctiunct.h $e |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G @@ -141105,17 +141124,17 @@ then the Limited Principle of Omniscience (LPO) implies excluded middle. $( Lemma for ~ ctiunct . (Contributed by Jim Kingdon, 28-Oct-2023.) $) ctiunctlemf $p |- ( ph -> H : U --> U_ x e. A B ) $= ( vy cv cfv c2nd c1st csb ciun wcel wrex wfo adantr fof syl com wss wdc - wa wral adantlr cxp wf1o simpr ctiunctlemu1st ffvelrnd ralrimiva rspsbc - wf wsbc sylc wb sbcfg mpbid ctiunctlemu2nd csbeq1 eleq2d rspcev syl2anc - wceq eliun sylibr nfcv nfcsb1v csbeq1a cbviun eleqtrrdi fmptd ) AIHIUDZ - MUEZUFUEZBWJUGUEZJUEZKUHZUEZBDEUIZLAWIHUJZUSZWOUCDBUCUDZEUHZUIZWPWRWOWT - UJZUCDUKZWOXAUJWRWMDUJZWOBWMEUHZUJZXCWRFDWLJWRFDJULZFDJVIAXGWQPUMZFDJUN - UOWRBCDEFGHIJKMWIAFUPUQWQNUMZAWIFUJURIUPUTWQOUMZXHABUDDUJZGUPUQWQQVAZAX - KWIGUJURIUPUTWQRVAZAXKGEKULZWQSVAZAUPUPUPVBMVCWQTUMZUAAWQVDZVEVFZWRBWMG - UHZXEWKWNWRGEKVIZBWMVJZXSXEWNVIZWRXDXTBDUTZYAXRAYCWQAXTBDAXKUSXNXTSGEKU - NUOVGUMXTBWMDVHVKWRXDYAYBVLXRBGEKDWMVMUOVNWRBCDEFGHIJKMWIXIXJXHXLXMXOXP - UAXQVOVFXBXFUCWMDWSWMVTWTXEWOBWSWMEVPVQVRVSUCWODWTWAWBBUCDEWTUCEWCBWSEW - DBWSEWEWFWGUBWH $. + wa wf wral adantlr wf1o simpr ctiunctlemu1st ffvelcdmd ralrimiva rspsbc + cxp wsbc sylc wb sbcfg ctiunctlemu2nd wceq csbeq1 eleq2d rspcev syl2anc + mpbid eliun sylibr nfcv nfcsb1v csbeq1a cbviun eleqtrrdi fmptd ) AIHIUD + ZMUEZUFUEZBWJUGUEZJUEZKUHZUEZBDEUIZLAWIHUJZUSZWOUCDBUCUDZEUHZUIZWPWRWOW + TUJZUCDUKZWOXAUJWRWMDUJZWOBWMEUHZUJZXCWRFDWLJWRFDJULZFDJUTAXGWQPUMZFDJU + NUOWRBCDEFGHIJKMWIAFUPUQWQNUMZAWIFUJURIUPVAWQOUMZXHABUDDUJZGUPUQWQQVBZA + XKWIGUJURIUPVAWQRVBZAXKGEKULZWQSVBZAUPUPUPVIMVCWQTUMZUAAWQVDZVEVFZWRBWM + GUHZXEWKWNWRGEKUTZBWMVJZXSXEWNUTZWRXDXTBDVAZYAXRAYCWQAXTBDAXKUSXNXTSGEK + UNUOVGUMXTBWMDVHVKWRXDYAYBVLXRBGEKDWMVMUOVTWRBCDEFGHIJKMWIXIXJXHXLXMXOX + PUAXQVNVFXBXFUCWMDWSWMVOWTXEWOBWSWMEVPVQVRVSUCWODWTWAWBBUCDEWTUCEWCBWSE + WDBWSEWEWFWGUBWH $. $} ${ @@ -141130,31 +141149,31 @@ then the Limited Principle of Omniscience (LPO) implies excluded middle. ( vu vv vw vr ciun wf cv cfv wceq wrex wral wfo ctiunctlemf wcel wa nfv nfiu1 nfcri nfan nfcv nffv nfeq2 nfrexxy simplll simplr foelrn sylancom syl2anc ad4antr simpllr cop ccnv com c1st c2nd csb cxp wf1o f1ocnv f1of - syl ad5antr wss simprl sseldd simp-5l adantr simplrl ffvelrnd f1ocnvfv2 - opelxpd fveq2d op1st eqtrdi eqeltrd op2nd simprr eqtr4d csbeq1d 3eltr4d - vex csbid 2fveq3 eleq1d eleq12d anbi12d elrab2 sylanbrc fveq12d simplrr - jca eqeltrrd fvmptd3 3eqtr4rd fveq2 rspceeqv rexlimddv biimpi r19.29af2 - eliun adantl ralrimiva dffo3 ) AHBDEUIZLUJUEUKZUFUKZLULZUMZUFHUNZUEYHUO - HYHLUPABCDEFGHIJKLMNOPQRSTUAUBUQAYMUEYHAYIYHURZUSZYIEURZYMBDAYNBABUTBUE - YHBDEVAVBVCYLBUFHUDBYIYKBYJLUCBYJVDVEVFVGYOBUKZDURZUSZYPUSZYIUGUKZKULZU - MZYMUGGYSYPGEKUPZUUCUGGUNYTAYRUUDAYNYRYPVHYOYRYPVISVLUGGEYIKVJVKYTUUAGU - RZUUCUSZUSZYQUHUKZJULZUMZYMUHFUUGFDJUPZYRUUJUHFUNAUUKYNYRYPUUFPVMYOYRYP - UUFVNZUHFDYQJVJVLUUGUUHFURZUUJUSZUSZUUHUUAVOZMVPZULZHURZYIUURLULZUMYMUU - OUURVQURUURMULZVRULZFURZUVAVSULZBUVBJULZGVTZURZUSZUUSUUOVQVQWAZVQUUPUUQ - 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cfv crab cxp wfn fnmap wi fornex exlimddv fnovex mp3an2i rabexg syl cen - ax-mp wbr enref foeq1 wrex wf fof wb elmapg sylancl mpbird simpr jca ex - eximdv df-rex syl6ibr mpd cc4n ) AHCIUAZEJZKZHVOBJZDJUCZKEVOHLUBZDBHMFA - VQEVTUDMNZBHAVRHNOZVTMNZWALMMUEUFWBVOMNZHMNZWCUGWBVQWDEGVQWDWBWEVQWDUHP - HVOMVPUIUPQZUJWEWBPRVOHMMLUKULVQEVTMUMUNSHHUOUQAHPURRHVOVPVSUSAVQEVTUTZ - BHWBVQETZWGGWBWHVPVTNZVQOZETWGWBVQWJEWBVQWJWBVQOZWIVQWKWIHVOVPVAZVQWLWB - HVOVPVBQWKWDWEWIWLVCWFPVOHVPMMVDVEVFWBVQVGVHVIVJVQEVTVKVLVMSVN $. + cfv crab cxp wfn fnmap focdmex ax-mp exlimddv fnovex mp3an2i rabexg syl + wi cen wbr enref foeq1 wf fof wb elmapg sylancl mpbird simpr jca eximdv + wrex ex df-rex syl6ibr mpd cc4n ) AHCIUAZEJZKZHVOBJZDJUCZKEVOHLUBZDBHMF + AVQEVTUDMNZBHAVRHNOZVTMNZWALMMUEUFWBVOMNZHMNZWCUGWBVQWDEGVQWDWBWEVQWDUO + PHVOMVPUHUIQZUJWEWBPRVOHMMLUKULVQEVTMUMUNSHHUPUQAHPURRHVOVPVSUSAVQEVTVI + ZBHWBVQETZWGGWBWHVPVTNZVQOZETWGWBVQWJEWBVQWJWBVQOZWIVQWKWIHVOVPUTZVQWLW + BHVOVPVAQWKWDWEWIWLVBWFPVOHVPMMVCVDVEWBVQVFVGVJVHVQEVTVKVLVMSVN $. $} ${ @@ -141312,17 +141331,17 @@ then the Limited Principle of Omniscience (LPO) implies excluded middle. ( vg vz vy cn cv wcel wdc wral wa com wceq dcbid wf1o c1 ax-mp cdju eleq1 wss c1o wfo wex ccnv cbvralv crn imassrn wf1 cuz cfv cz 1z id frec2uzf1od cima nnuz f1oeq3 mpbir f1ocnv dff1o5 mpbi simpri sseqtri simplr f1of mp1i - wb wf simpr ffvelrnd rspcdva simpll f1elima mp3an2i f1ocnvfv1 mpan adantl - f1of1 eleq1d bitr3d mpbid ralrimiva sylancr sylan2b cen wbr cvv cres nnex - ssomct ssex f1ores f1oeng syl2anc enct syl adantr mpbird ) BIUCZAJZBKZLZA - IMZNOBUDUACJUECUFZODUGZBURZUDUAFJUEFUFZXFXBGJZBKZLZGIMZXJXEXMAGIXCXKPXDXL - XCXKBUBQUHXBXNNZXIOUCHJZXIKZLZHOMXJXIXHUIZOXHBUJIOXHUKZXSOPZIOXHRZXTYANOI - DRZYBYCOSULUMZDRZSUNKZYEUOYFASDYFUPEUQTIYDPYCYEVJUSIYDODUTTVAZOIDVBTZIOXH - VCVDVEVFXOXRHOXOXPOKZNZXPDUMZBKZLZXRYJXMYMGIYKXKYKPXLYLXKYKBUBQXBXNYIVGYJ - OIXPDYCOIDVKYJYGOIDVHVIXOYIVLVMZVNYJYLXQYJYKXHUMZXIKZYLXQXTYJYKIKXBYPYLVJ - YBXTYHIOXHWATZYNXBXNYIVOIOXHYKBVPVQYJYOXPXIYIYOXPPZXOYCYIYRYGOIXPDVRVSVTW - BWCQWDWEHXIFWMWFWGXBXGXJVJZXFXBBXIWHWIZYSXBBWJKBXIXHBWKZRZYTBIWLWNXTXBUUB - YQIOBXHWOVSBXIWJUUAWPWQBXICFWRWSWTXA $. + wb wf simpr ffvelcdmd rspcdva f1of1 simpll f1elima mp3an2i f1ocnvfv1 mpan + adantl eleq1d bitr3d mpbid ralrimiva ssomct sylancr sylan2b cen cres nnex + wbr cvv ssex f1ores f1oeng syl2anc enct syl adantr mpbird ) BIUCZAJZBKZLZ + AIMZNOBUDUACJUECUFZODUGZBURZUDUAFJUEFUFZXFXBGJZBKZLZGIMZXJXEXMAGIXCXKPXDX + LXCXKBUBQUHXBXNNZXIOUCHJZXIKZLZHOMXJXIXHUIZOXHBUJIOXHUKZXSOPZIOXHRZXTYANO + IDRZYBYCOSULUMZDRZSUNKZYEUOYFASDYFUPEUQTIYDPYCYEVJUSIYDODUTTVAZOIDVBTZIOX + HVCVDVEVFXOXRHOXOXPOKZNZXPDUMZBKZLZXRYJXMYMGIYKXKYKPXLYLXKYKBUBQXBXNYIVGY + JOIXPDYCOIDVKYJYGOIDVHVIXOYIVLVMZVNYJYLXQYJYKXHUMZXIKZYLXQXTYJYKIKXBYPYLV + JYBXTYHIOXHVOTZYNXBXNYIVPIOXHYKBVQVRYJYOXPXIYIYOXPPZXOYCYIYRYGOIXPDVSVTWA + WBWCQWDWEHXIFWFWGWHXBXGXJVJZXFXBBXIWIWLZYSXBBWMKBXIXHBWJZRZYTBIWKWNXTXBUU + BYQIOBXHWOVTBXIWMUUAWPWQBXICFWRWSWTXA $. $} ${ @@ -141397,40 +141416,40 @@ then the Limited Principle of Omniscience (LPO) implies excluded middle. than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.) $) nninfdclemp1 $p |- ( ph -> ( F ` U ) < ( F ` ( U + 1 ) ) ) $= - ( cn cv wcel vr va vb vp vq vs cfv c1 caddc nninfdclemf ffvelrnd sseldd - co nnred nnzd peano2zd zred peano2nnd ltp1d cuz cin cr clt cinf cle wbr - wral wa simpr elin2d eluzle syl ralrimiva inss1 sstrid wdc eleq1w dcbid - weq adantr rspcdva cz eluzdc syl2an2r dcan2 sylc elin dcbii sylibr wrex - wex breq1 rexbidv breq2 cbvrexv sylib df-rex simprl wss simprr nnltp1le - wceq wb mpbid eluz2 syl3anbrc elind ex eximdv mpd nninfdcex infregelbex - nnssre sstrdi mpbird cmpo cmpt cseq fveq1i nnuz eleqtrdi elnnuz biimpri - eqid eqidd adantl simpld fvmptd3 eqeltrd nninfdclemcl seq3p1 eqtrid a1i - eqcomi oveq12d cbvralv nnmindc syl3anc elin1d fvoveq1 infeq1d breqtrrd - ineq2d ovmpog 3eqtrd ltletrd ) AFJUGZUUGUHUIUMZFUHUIUMZJUGZAUUGAERUUGLA + ( cn cv wcel vr va vb vp vq vs c1 caddc co nninfdclemf ffvelcdmd sseldd + cfv nnred nnzd peano2zd zred peano2nnd cuz cin cr clt cinf cle wbr wral + ltp1d wa simpr elin2d eluzle syl ralrimiva sstrid wdc weq eleq1w adantr + inss1 dcbid rspcdva cz eluzdc syl2an2r dcan2 sylc elin dcbii sylibr wex + wrex wceq breq1 rexbidv breq2 cbvrexv sylib df-rex simprl wss simprr wb + nnltp1le mpbid eluz2 syl3anbrc elind eximdv mpd nninfdcex nnssre sstrdi + ex infregelbex mpbird cmpo cmpt cseq fveq1i nnuz eleqtrdi eqidd biimpri + eqid elnnuz adantl simpld fvmptd3 nninfdclemcl seq3p1 eqtrid eqcomi a1i + eqeltrd oveq12d cbvralv nnmindc syl3anc elin1d fvoveq1 infeq1d breqtrrd + ineq2d ovmpog 3eqtrd ltletrd ) AFJUMZUUGUGUHUIZFUGUHUIZJUMZAUUGAERUUGLA REFJABCDEGHIJKLMNOPUJZQUKULZUNZAUUHAUUGAUUGUULUOUPZUQZAUUJAERUUJLAREUUI - 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PAUXTUHKVCVFOYGZVTZYHUYBYIAUXOETZUESZETZVHZVHBCDEUXOUYDHIAUWSUYFLVTAUVS - UYFMVTAUWOHRVGUYFNVTAUYCUYEWRAUYCUYEWTYJYKYLAUXJUUGUXKKUXGUXJUUGXBAUUGU - XJFJUXIPXSYNYMAGUUIKKRUXHEUXRGSUUIXBKYEUUPUYAYHYOAUXAKRTUUSETUXMUUSXBUU - LAERKLUYAULAEUUQUUSAUURRWSUVPUURTZVPZBRVGZUWHUUSUURTUVDAUFSZUURTZVPZUFR - VGUYIAUYLUFRAUYJRTZVHZUYJETZUYJUUQTZVHZVPZUYLUYNUYOVPZUYPVPZUYRUYNUVRUY - SBRUYJBUFVSUVQUYOBUFEVQVRAUVSUYMMVTAUYMVIZWAAUWAUYMUYJWBTUYTUUNUYNUYJVU - AUOUUHUYJWCWDUYOUYPWEWFUYKUYQUYJEUUQWGWHWIVMUYLUYHUFBRUFBVSUYKUYGUFBUUR - VQVRYPWPUXBBUCUURYQYRYSCDUUGKRRUXFUUSUXGUUSEUXCUUGXBZVBUXEUURVCVUBUXDUU - QEUXCUUGUHUTUIYTUUCUUADSKXBUUSYEUXGYDUUDYRUUEUUBUUF $. + JUUKAFQURZUKULUNAUUGUUMVGAUUHEUUHUSUMZUTZVAVBVCZUUJVDAUUHUUSVDVEUUHUASZ + VDVEZUAUURVFAUVAUAUURAUUTUURTZVHZUUTUUQTUVAUVCEUUQUUTAUVBVIVJUUHUUTVKVL + VMAUBUCUAUURUUHAUBUCUAUURAUUREREUUQVSLVNZAUBSZUURTZVOZUBRAUVERTZVHZUVEE + TZUVEUUQTZVHZVOZUVGUVIUVJVOZUVKVOZUVMUVIBSZETZVOZUVNBRUVEBUBVPUVQUVJBUB + EVQVTAUVRBRVFZUVHMVRAUVHVIZWAAUUHWBTZUVHUVEWBTUVOUUNUVIUVEUVTUOUUHUVEWC + WDUVJUVKWEWFUVFUVLUVEEUUQWGWHWIVMAUCSZETZUUGUWBVBVEZVHZUCWJZUWBUURTZUCW + JZAUWDUCEWKZUWFAUUGISZVBVEZIEWKZUWIAHSZUWJVBVEZIEWKZUWLHRUUGUWMUUGWLUWN + UWKIEUWMUUGUWJVBWMWNNUULWAUWKUWDIUCEUWJUWBUUGVBWOWPWQUWDUCEWRWQAUWEUWGU + CAUWEUWGAUWEVHZEUUQUWBAUWCUWDWSZUWPUWAUWBWBTUUHUWBVDVEZUWBUUQTAUWAUWEUU + NVRUWPUWBUWPERUWBAERWTZUWELVRUWQULZUOUWPUWDUWRAUWCUWDXAAUUGRTZUWEUWBRTU + WDUWRXBUULUWTUUGUWBXCWDXDUUHUWBXEXFXGXMXHXIZXJAUURRVAUVDXKXLUUOXNXOAUUJ + FCDRRECSZUGUHUIUSUMZUTZVAVBVCZXPZGRKXQZUGXRZUMZUUIUXHUMZUXGUIZUUGKUXGUI + ZUUSAUUJUUIUXIUMUXLUUIJUXIPXSAUDUEUXGEUXHUGFAFRUGUSUMZQXTYAAUDSZUXNTZVH + ZUXOUXHUMKEUXQGUXOKKRUXHEUXHYDZGUDVPKYBUXPUXORTZAUXSUXPUXOYEYCYFAKETZUX + PAUXTUGKVBVEOYGZVRZYHUYBYNAUXOETZUESZETZVHZVHBCDEUXOUYDHIAUWSUYFLVRAUVS + UYFMVRAUWOHRVFUYFNVRAUYCUYEWSAUYCUYEXAYIYJYKAUXJUUGUXKKUXGUXJUUGWLAUUGU + XJFJUXIPXSYLYMAGUUIKKRUXHEUXRGSUUIWLKYBUUPUYAYHYOAUXAKRTUUSETUXMUUSWLUU + LAERKLUYAULAEUUQUUSAUURRWTUVPUURTZVOZBRVFZUWHUUSUURTUVDAUFSZUURTZVOZUFR + VFUYIAUYLUFRAUYJRTZVHZUYJETZUYJUUQTZVHZVOZUYLUYNUYOVOZUYPVOZUYRUYNUVRUY + SBRUYJBUFVPUVQUYOBUFEVQVTAUVSUYMMVRAUYMVIZWAAUWAUYMUYJWBTUYTUUNUYNUYJVU + AUOUUHUYJWCWDUYOUYPWEWFUYKUYQUYJEUUQWGWHWIVMUYLUYHUFBRUFBVPUYKUYGUFBUUR + VQVTYPWQUXBBUCUURYQYRYSCDUUGKRRUXFUUSUXGUUSEUXCUUGWLZVAUXEUURVBVUBUXDUU + QEUXCUUGUGUSUHYTUUCUUADSKWLUUSYBUXGYDUUDYRUUEUUBUUF $. $} ${ @@ -141445,21 +141464,21 @@ then the Limited Principle of Omniscience (LPO) implies excluded middle. nninfdclemlt $p |- ( ph -> ( F ` U ) < ( F ` V ) ) $= ( vw vk c1 caddc co cfz wcel cfv clt wbr cuz cz peano2nnd nnzd nnltp1le cle cn wb syl2anc mpbid eluz2 syl3anbrc eluzfz2 cv wi wceq fveq2 breq2d - syl imbi2d nninfdclemp1 a1i wa wss ad2antrr nninfdclemf ffvelrnd sseldd - cfzo wf nnred elfzoelz ad2antlr 1red cr nnge1d elfzole1 elnnz1 sylanbrc - zred letrd simpr wdc wral wrex lttrd ex expcom a2d fzind2 mpcom ) LFUCU - DUEZLUFUEUGZAFJUHZLJUHZUIUJZALXBUKUHUGZXCAXBULUGLULUGXBLUPUJZXGAXBAFRUM - ZUNALSUNAFLUIUJZXHTAFUQUGZLUQUGXJXHURRSFLUOUSUTXBLVAVBXBLVCVIAXDUAVDZJU - HZUIUJZVEAXDXBJUHZUIUJZVEZAXDUBVDZJUHZUIUJZVEAXDXRUCUDUEZJUHZUIUJZVEAXF - VEUAUBLXBLXLXBVFZXNXPAYDXMXOXDUIXLXBJVGVHVJXLXRVFZXNXTAYEXMXSXDUIXLXRJV - GVHVJXLYAVFZXNYCAYFXMYBXDUIXLYAJVGVHVJXLLVFZXNXFAYGXMXEXDUIXLLJVGVHVJXQ - XGABCDEFGHIJKMNOPQRVKVLXRXBLVSUEUGZAXTYCAYHXTYCVEAYHVMZXTYCYIXTVMZXDXSY - BYJXDYJEUQXDAEUQVNYHXTMVOZYJUQEFJAUQEJVTYHXTABCDEGHIJKMNOPQVPVOZAXKYHXT - RVOVQVRWAYJXSYJEUQXSYKYJUQEXRJYLYJXRULUGZUCXRUPUJXRUQUGYHYMAXTXRXBLWBWC - ZYJUCXBXRYJWDAXBWEUGYHXTAXBXIWAVOYJXRYNWJAUCXBUPUJYHXTAXBXIWFVOYHXBXRUP - UJAXTXRXBLWGWCWKXRWHWIZVQVRWAYJYBYJEUQYBYKYJUQEYAJYLYJXRYOUMVQVRWAYIXTW - LYJBCDEXRGHIJKYKABVDEUGWMBUQWNYHXTNVOAHVDIVDUIUJIEWOHUQWNYHXTOVOAKEUGUC - KUIUJVMYHXTPVOQYOVKWPWQWRWSWTXA $. + syl imbi2d nninfdclemp1 a1i cfzo wa wss wf nninfdclemf ffvelcdmd sseldd + ad2antrr nnred elfzoelz ad2antlr 1red zred nnge1d elfzole1 letrd elnnz1 + cr sylanbrc simpr wdc wral wrex lttrd ex expcom a2d fzind2 mpcom ) LFUC + UDUEZLUFUEUGZAFJUHZLJUHZUIUJZALXBUKUHUGZXCAXBULUGLULUGXBLUPUJZXGAXBAFRU + MZUNALSUNAFLUIUJZXHTAFUQUGZLUQUGXJXHURRSFLUOUSUTXBLVAVBXBLVCVIAXDUAVDZJ + UHZUIUJZVEAXDXBJUHZUIUJZVEZAXDUBVDZJUHZUIUJZVEAXDXRUCUDUEZJUHZUIUJZVEAX + FVEUAUBLXBLXLXBVFZXNXPAYDXMXOXDUIXLXBJVGVHVJXLXRVFZXNXTAYEXMXSXDUIXLXRJ + VGVHVJXLYAVFZXNYCAYFXMYBXDUIXLYAJVGVHVJXLLVFZXNXFAYGXMXEXDUIXLLJVGVHVJX + QXGABCDEFGHIJKMNOPQRVKVLXRXBLVMUEUGZAXTYCAYHXTYCVEAYHVNZXTYCYIXTVNZXDXS + YBYJXDYJEUQXDAEUQVOYHXTMVTZYJUQEFJAUQEJVPYHXTABCDEGHIJKMNOPQVQVTZAXKYHX + TRVTVRVSWAYJXSYJEUQXSYKYJUQEXRJYLYJXRULUGZUCXRUPUJXRUQUGYHYMAXTXRXBLWBW + CZYJUCXBXRYJWDAXBWJUGYHXTAXBXIWAVTYJXRYNWEAUCXBUPUJYHXTAXBXIWFVTYHXBXRU + PUJAXTXRXBLWGWCWHXRWIWKZVRVSWAYJYBYJEUQYBYKYJUQEYAJYLYJXRYOUMVRVSWAYIXT + WLYJBCDEXRGHIJKYKABVDEUGWMBUQWNYHXTNVTAHVDIVDUIUJIEWOHUQWNYHXTOVTAKEUGU + CKUIUJVNYHXTPVTQYOVKWPWQWRWSWTXA $. $} ${ @@ -144725,13 +144744,13 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is (Contributed by AV, 23-Jan-2020.) $) mnd1id $p |- ( I e. V -> ( 0g ` M ) = I ) $= ( va wcel cfv csn cop wceq snexg opexg mpancom syl2anc eqeq12d syl5ibrcom - cvv co syl5bi imp c0g cbs anidms grpbaseg cplusg grpplusgg snidg cv velsn - syl df-ov fvsng eqtrid oveq2 id oveq1 grpidd eqcomd ) ACFZABUAGUSEAHZAAIZ - AIZHZBAUSUTQFZVCQFZUTBUBGJACKZUSVBQFZVEVAQFZUSVGUSVHAACCLUCZVAAQCLMVBQKUJ - ZUTVCBQQDUDNUSVDVEVCBUEGJVFVJUTVCBQQDUFNACUGUSEUHZUTFZAVKVCRZVKJZVLVKAJZU - SVNEAUIZUSVNVOAAVCRZAJZUSVQVAVCGZAAAVCUKVHUSVSAJVIVAAQCULMUMZVOVMVQVKAVKA - AVCUNVOUOZOPSTUSVLVKAVCRZVKJZVLVOUSWCVPUSWCVOVRVTVOWBVQVKAVKAAVCUPWAOPSTU - QUR $. + cvv co biimtrid imp c0g cbs anidms syl grpbaseg cplusg grpplusgg snidg cv + velsn df-ov fvsng eqtrid oveq2 id oveq1 grpidd eqcomd ) ACFZABUAGUSEAHZAA + IZAIZHZBAUSUTQFZVCQFZUTBUBGJACKZUSVBQFZVEVAQFZUSVGUSVHAACCLUCZVAAQCLMVBQK + UDZUTVCBQQDUENUSVDVEVCBUFGJVFVJUTVCBQQDUGNACUHUSEUIZUTFZAVKVCRZVKJZVLVKAJ + ZUSVNEAUJZUSVNVOAAVCRZAJZUSVQVAVCGZAAAVCUKVHUSVSAJVIVAAQCULMUMZVOVMVQVKAV + KAAVCUNVOUOZOPSTUSVLVKAVCRZVKJZVLVOUSWCVPUSWCVOVRVTVOWBVQVKAVKAAVCUPWAOPS + TUQUR $. $} @@ -144928,20 +144947,20 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is mhmf1o $p |- ( F e. ( R MndHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S MndHom R ) ) ) $= ( vx vy cmhm co wcel wf1o wa cmnd cfv wceq adantr eqid syl2anc ccnv wf cv - cplusg wral c0g w3a mhmrcl2 mhmrcl1 jca f1ocnv adantl syl simpll ffvelrnd - simprl simprr mhmlin syl3anc simpr f1ocnvfv2 oveq12d eqtrd mndcl f1ocnvfv - f1of wi mpd ralrimivva mhm0 eqcomd fveq2d mndidcl f1ocnvfv1 3jca sylanbrc - ismhm wfn mhmf ffnd dff1o4 impbida ) ECDJKLZABEMZEUAZDCJKLZWCWDNZDOLZCOLZ - NZBAWEUBZHUCZIUCZDUDPZKZWEPWLWEPZWMWEPZCUDPZKZQZIBUEHBUEZDUFPZWEPZCUFPZQZ - UGWFWCWJWDWCWHWICDEUHCDEUIZUJRWGWKXAXEWGBAWEMZWKWDXGWCABEUKULBAWEVFUMZWGW - THIBBWGWLBLZWMBLZNZNZWSEPZWOQZWTXLXMWPEPZWQEPZWNKZWOXLWCWPALZWQALZXMXQQWC - WDXKUNXLBAWLWEWGWKXKXHRZWGXIXJUPZUOZXLBAWMWEXTWGXIXJUQZUOZAWRWNCDEWPWQFWR - SZWNSZURUSXLXOWLXPWMWNXLWDXIXOWLQWGWDXKWCWDUTZRZYAABWLEVATXLWDXJXPWMQYHYC - ABWMEVATVBVCXLWDWSALZXNWTVGYHXLWIXRXSYIWGWIXKWCWIWDXFRRYBYDAWRCWPWQFYEVDU - SABWSWOEVETVHVIWGXCXDEPZWEPZXDWGXBYJWEWGYJXBWCYJXBQWDCDEXBXDXDSZXBSZVJRVK - VLWGWDXDALZYKXDQYGWCYNWDWCWIYNXFACXDFYLVMUMRABXDEVNTVCVOHIBAWNWRDCWEXDXBG - FYFYEYMYLVQVPWCWFNZEAVRWEBVRWDYOABEWCABEUBWFABCDEFGVSRVTYOBAWEWFWKWCBADCW - EGFVSULVTABEWAVPWB $. + cplusg wral c0g w3a mhmrcl2 mhmrcl1 f1ocnv adantl simpll simprl ffvelcdmd + jca f1of syl simprr mhmlin syl3anc simpr f1ocnvfv2 oveq12d eqtrd wi mndcl + f1ocnvfv mpd ralrimivva mhm0 eqcomd mndidcl f1ocnvfv1 3jca ismhm sylanbrc + fveq2d wfn mhmf ffnd dff1o4 impbida ) ECDJKLZABEMZEUAZDCJKLZWCWDNZDOLZCOL + ZNZBAWEUBZHUCZIUCZDUDPZKZWEPWLWEPZWMWEPZCUDPZKZQZIBUEHBUEZDUFPZWEPZCUFPZQ + ZUGWFWCWJWDWCWHWICDEUHCDEUIZUORWGWKXAXEWGBAWEMZWKWDXGWCABEUJUKBAWEUPUQZWG + WTHIBBWGWLBLZWMBLZNZNZWSEPZWOQZWTXLXMWPEPZWQEPZWNKZWOXLWCWPALZWQALZXMXQQW + CWDXKULXLBAWLWEWGWKXKXHRZWGXIXJUMZUNZXLBAWMWEXTWGXIXJURZUNZAWRWNCDEWPWQFW + RSZWNSZUSUTXLXOWLXPWMWNXLWDXIXOWLQWGWDXKWCWDVAZRZYAABWLEVBTXLWDXJXPWMQYHY + CABWMEVBTVCVDXLWDWSALZXNWTVEYHXLWIXRXSYIWGWIXKWCWIWDXFRRYBYDAWRCWPWQFYEVF + UTABWSWOEVGTVHVIWGXCXDEPZWEPZXDWGXBYJWEWGYJXBWCYJXBQWDCDEXBXDXDSZXBSZVJRV + KVQWGWDXDALZYKXDQYGWCYNWDWCWIYNXFACXDFYLVLUQRABXDEVMTVDVNHIBAWNWRDCWEXDXB + GFYFYEYMYLVOVPWCWFNZEAVRWEBVRWDYOABEWCABEUBWFABCDEFGVSRVTYOBAWEWFWKWCBADC + WEGFVSUKVTABEWAVPWB $. $} ${ @@ -145074,11 +145093,11 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is 0subm $p |- ( G e. Mnd -> { .0. } e. ( SubMnd ` G ) ) $= ( va vb cmnd wcel csn csubmnd cfv cbs wss cv co wral eqid syl wceq velsn wa cplusg mndidcl snssd snidg anbi12i mndlid mpdan wb elsng mpbird oveq12 - eqeltrd eleq1d syl5ibrcom syl5bi ralrimivv issubm mpbir3and ) AFGZBHZAIJG - UTAKJZLBUTGZDMZEMZAUAJZNZUTGZEUTODUTOUSBVAVAABVAPZCUBZUCUSBVAGZVBVIBVAUDQ - USVGDEUTUTVCUTGZVDUTGZTVCBRZVDBRZTZUSVGVKVMVLVNDBSEBSUEUSVGVOBBVENZUTGZUS - VQVPBRZUSVJVRVIVAVEABBVHVEPZCUFUGZUSVPVAGVQVRUHUSVPBVAVTVIULVPBVAUIQUJVOV - FVPUTVCBVDBVEUKUMUNUOUPDEVAVEUTABVHCVSUQUR $. + eqeltrd eleq1d syl5ibrcom biimtrid ralrimivv issubm mpbir3and ) AFGZBHZAI + JGUTAKJZLBUTGZDMZEMZAUAJZNZUTGZEUTODUTOUSBVAVAABVAPZCUBZUCUSBVAGZVBVIBVAU + DQUSVGDEUTUTVCUTGZVDUTGZTVCBRZVDBRZTZUSVGVKVMVLVNDBSEBSUEUSVGVOBBVENZUTGZ + USVQVPBRZUSVJVRVIVAVEABBVHVEPZCUFUGZUSVPVAGVQVRUHUSVPBVAVTVIULVPBVAUIQUJV + OVFVPUTVCBVDBVEUKUMUNUOUPDEVAVEUTABVHCVSUQUR $. $} ${ @@ -145090,17 +145109,17 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is ( va vb vx vy cfv wcel wi wss cv co wral w3a wa elin imp adantl eleq1d ex csubmnd cin cmnd submrcl cbs cplusg ssinss1 3ad2ant1 ad2antrl simplbi2com c0g 3ad2ant2 com12 anbi12i oveq1 oveq2 simpl eqidd rspc2vd 3ad2ant3 simpr - adantr elind syl5bi ralrimivv 3jca eqid issubm anbi12d 3imtr4d expd mpcom - weq ) ACUBHZIZBVOIZABUCZVOIZCUDIZVPVQVSJACUEVTVPVQVSVTACUFHZKZCULHZAIZDLZ - ELZCUGHZMZAIZEANDANZOZBWAKZWCBIZWHBIZEBNDBNZOZPZVRWAKZWCVRIZFLZGLZWGMZVRI - ZGVRNFVRNZOZVPVQPVSVTWQXEVTWQPZWRWSXDWKWRVTWPWBWDWRWJABWAUHUIUJWQWSVTWKWP - WSWDWBWPWSJWJWPWDWSWMWLWDWSJWOWSWDWMWCABQUKUMUNUMRSXFXCFGVRVRWTVRIZXAVRIZ - PWTAIZWTBIZPZXAAIZXABIZPZPZXFXCXGXKXHXNWTABQXAABQUOXFXOXCXFXOPABXBXFXOXBA - IZWKXOXPJZVTWPWJWBXQWDXOWJXPXOXPWTWFWGMZAIWIDEWTXAAAADFVNZWHXRAWEWTWFWGUP - ZTEGVNZXRXBAWFXAWTWGUQZTXKXIXNXIXJURVCXOXSPZAUSXNXLXKXLXMURSUTUNVAUJRXFXO - XBBIZWQXOYDJZVTWPYEWKWOWLYEWMXOWOYDXOYDXRBIWNDEWTXABBBXSWHXRBXTTYAXRXBBYB - TXKXJXNXIXJVBVCYCBUSXNXMXKXLXMVBSUTUNVASSRVDUAVEVFVGUAVTVPWKVQWPDEWAWGACW - CWAVHZWCVHZWGVHZVIDEWAWGBCWCYFYGYHVIVJFGWAWGVRCWCYFYGYHVIVKVLVMR $. + weq adantr elind biimtrid ralrimivv 3jca eqid issubm anbi12d 3imtr4d expd + mpcom ) ACUBHZIZBVOIZABUCZVOIZCUDIZVPVQVSJACUEVTVPVQVSVTACUFHZKZCULHZAIZD + LZELZCUGHZMZAIZEANDANZOZBWAKZWCBIZWHBIZEBNDBNZOZPZVRWAKZWCVRIZFLZGLZWGMZV + RIZGVRNFVRNZOZVPVQPVSVTWQXEVTWQPZWRWSXDWKWRVTWPWBWDWRWJABWAUHUIUJWQWSVTWK + WPWSWDWBWPWSJWJWPWDWSWMWLWDWSJWOWSWDWMWCABQUKUMUNUMRSXFXCFGVRVRWTVRIZXAVR + IZPWTAIZWTBIZPZXAAIZXABIZPZPZXFXCXGXKXHXNWTABQXAABQUOXFXOXCXFXOPABXBXFXOX + BAIZWKXOXPJZVTWPWJWBXQWDXOWJXPXOXPWTWFWGMZAIWIDEWTXAAAADFVCZWHXRAWEWTWFWG + UPZTEGVCZXRXBAWFXAWTWGUQZTXKXIXNXIXJURVDXOXSPZAUSXNXLXKXLXMURSUTUNVAUJRXF + XOXBBIZWQXOYDJZVTWPYEWKWOWLYEWMXOWOYDXOYDXRBIWNDEWTXABBBXSWHXRBXTTYAXRXBB + YBTXKXJXNXIXJVBVDYCBUSXNXMXKXLXMVBSUTUNVASSRVEUAVFVGVHUAVTVPWKVQWPDEWAWGA + CWCWAVIZWCVIZWGVIZVJDEWAWGBCWCYFYGYHVJVKFGWAWGVRCWCYFYGYHVJVLVMVNR $. $} ${ @@ -145132,19 +145151,19 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is ( F o. G ) e. ( S MndHom U ) ) $= ( vx vy cmhm co wcel wa cbs cfv wf cplusg wceq c0g eqid fvco3 syl2anc w3a cmnd ccom cv wral mhmrcl2 mhmrcl1 anim12ci mhmf fco syl2an mhmlin adantll - 3expb fveq2d simpll ad2antlr simprl ffvelrnd simprr syl3anc eqtrd oveq12d - adantl mndcl sylan 3eqtr4d ralrimivva mndidcl syl mhm0 adantr 3eqtrd 3jca - ismhm sylanbrc ) DBCHIJZEABHIJZKZAUBJZCUBJZKALMZCLMZDEUCZNZFUDZGUDZAOMZIZ - WDMZWFWDMZWGWDMZCOMZIZPZGWBUEFWBUEZAQMZWDMZCQMZPZUAWDACHIJVQWAVRVTBCDUFAB - EUGZUHVSWEWPWTVQBLMZWCDNWBXBENZWEVRXBWCBCDXBRZWCRZUIWBXBABEWBRZXDUIZWBXBW - CDEUJUKVSWOFGWBWBVSWFWBJZWGWBJZKZKZWIEMZDMZWFEMZDMZWGEMZDMZWMIZWJWNXKXMXN - XPBOMZIZDMZXRXKXLXTDVRXJXLXTPZVQVRXHXIYBWBWHXSABEWFWGXFWHRZXSRZULUNUMUOXK - VQXNXBJXPXBJYAXRPVQVRXJUPXKWBXBWFEVRXCVQXJXGUQZVSXHXIURZUSXKWBXBWGEYEVSXH - XIUTZUSXBXSWMBCDXNXPXDYDWMRZULVAVBXKXCWIWBJZWJXMPYEVSVTXJYIVRVTVQXAVDZVTX - HXIYIWBWHAWFWGXFYCVEUNVFWBXBWIDESTXKWKXOWLXQWMXKXCXHWKXOPYEYFWBXBWFDESTXK - XCXIWLXQPYEYGWBXBWGDESTVCVGVHVSWRWQEMZDMZBQMZDMZWSVSXCWQWBJZWRYLPVRXCVQXG - VDVSVTYOYJWBAWQXFWQRZVIVJWBXBWQDESTVSYKYMDVRYKYMPVQABEYMWQYPYMRZVKVDUOVQY - NWSPVRBCDWSYMYQWSRZVKVLVMVNFGWBWCWHWMACWDWSWQXFXEYCYHYPYRVOVP $. + 3expb fveq2d simpll ad2antlr simprl ffvelcdmd simprr syl3anc eqtrd adantl + mndcl sylan oveq12d 3eqtr4d ralrimivva mndidcl syl mhm0 adantr 3jca ismhm + 3eqtrd sylanbrc ) DBCHIJZEABHIJZKZAUBJZCUBJZKALMZCLMZDEUCZNZFUDZGUDZAOMZI + ZWDMZWFWDMZWGWDMZCOMZIZPZGWBUEFWBUEZAQMZWDMZCQMZPZUAWDACHIJVQWAVRVTBCDUFA + BEUGZUHVSWEWPWTVQBLMZWCDNWBXBENZWEVRXBWCBCDXBRZWCRZUIWBXBABEWBRZXDUIZWBXB + WCDEUJUKVSWOFGWBWBVSWFWBJZWGWBJZKZKZWIEMZDMZWFEMZDMZWGEMZDMZWMIZWJWNXKXMX + NXPBOMZIZDMZXRXKXLXTDVRXJXLXTPZVQVRXHXIYBWBWHXSABEWFWGXFWHRZXSRZULUNUMUOX + KVQXNXBJXPXBJYAXRPVQVRXJUPXKWBXBWFEVRXCVQXJXGUQZVSXHXIURZUSXKWBXBWGEYEVSX + HXIUTZUSXBXSWMBCDXNXPXDYDWMRZULVAVBXKXCWIWBJZWJXMPYEVSVTXJYIVRVTVQXAVCZVT + XHXIYIWBWHAWFWGXFYCVDUNVEWBXBWIDESTXKWKXOWLXQWMXKXCXHWKXOPYEYFWBXBWFDESTX + KXCXIWLXQPYEYGWBXBWGDESTVFVGVHVSWRWQEMZDMZBQMZDMZWSVSXCWQWBJZWRYLPVRXCVQX + GVCVSVTYOYJWBAWQXFWQRZVIVJWBXBWQDESTVSYKYMDVRYKYMPVQABEYMWQYPYMRZVKVCUOVQ + YNWSPVRBCDWSYMYQWSRZVKVLVOVMFGWBWCWHWMACWDWSWQXFXEYCYHYPYRVNVP $. $} ${ @@ -145382,6 +145401,23 @@ an algebraic structure formed from a base set of elements (notated UEUCHSSUFUGUDUEDTOUAUB $. $} + ${ + $d B x y $. $d K x y $. $d L x y $. $d V x y $. + grppropstr.b $e |- ( Base ` K ) = B $. + grppropstr.p $e |- ( +g ` K ) = .+ $. + grppropstr.l $e |- L + = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. } $. + $( Generalize a specific 2-element group ` L ` to show that any set ` K ` + with the same (relevant) properties is also a group. (Contributed by + NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) $) + grppropstrg $p |- ( K e. V -> ( K e. Grp <-> L e. Grp ) ) $= + ( vx vy wcel cbs cfv cvv wceq eqeltrrid cplusg syl2anc eqtrid cv wfn elex + basfn funfvex funfni plusgslid slotex grpbaseg eqtr4d wa grpplusgg oveqdr + sylancr grppropd ) CEKZIJACDUOADLMZCLMZUOANKZBNKZAUPOUOAUQNFUOLNUACNKUQNK + ZUCCEUBUTNCLCLUDUEUMPZUOBCQMZNGCQEUFUGPZABDNNHUHRZUOUQAUPFVDSUIVDUOITAKJT + AKUJIJVBDQMZUOVBBVEGUOURUSBVEOVAVCABDNNHUKRSULUN $. + $} + ${ $d x y .+ $. $d y .0. $. $d x y B $. $d x y G $. $d x y ph $. $d y N $. @@ -145794,16 +145830,16 @@ an algebraic structure formed from a base set of elements (notated isgrpinv $p |- ( G e. Grp -> ( ( M : B --> B /\ A. x e. B ( ( M ` x ) .+ x ) = .0. ) <-> N = M ) ) $= ( ve cgrp wcel wf cv co wceq wral wa cfv crio grpinvval ad2antlr simpr wb - wreu simpllr simplr ffvelrnd grpinveu ad4ant13 oveq1 eqeq1d syl2anc mpbid - riota2 ex ralimdva impr wfn grpinvfng ffn ad2antrl eqfnfv syl2an2r mpbird - eqtrd grpinvf grplinv ralrimiva jca feq1 oveq1d ralbidv anbi12d syl5ibcom - fveq1 impbid ) DMNZBBEOZAPZEUAZWBCQZGRZABSZTZFERZVTWGWHVTWGTWHWBFUAZWCRZA - BSZVTWAWFWKVTWATZWEWJABWLWBBNZTZWEWJWNWETZWILPZWBCQZGRZLBUBZWCWMWIWSRWLWE - LBCDFWBGHIJKUCUDWOWEWSWCRZWNWEUEWOWCBNWRLBUGZWEWTUFWOBBWBEVTWAWMWEUHWLWMW - EUIUJVTWMXAWAWELBCDWBGHIJUKULWRWELBWCWPWCRWQWDGWPWCWBCUMUNUQUOUPVHURUSUTV - TFBVAWGEBVAZWHWKUFBDFMHKVBWAXBVTWFBBEVCVDABFEVEVFVGURVTBBFOZWIWBCQZGRZABS - ZTWHWGVTXCXFBDFHKVIVTXEABBCDFWBGHIJKVJVKVLWHXCWAXFWFBBFEVMWHXEWEABWHXDWDG - WHWIWCWBCWBFEVRVNUNVOVPVQVS $. + wreu simpllr simplr ffvelcdmd grpinveu ad4ant13 oveq1 eqeq1d riota2 mpbid + syl2anc eqtrd ex ralimdva impr wfn grpinvfng ffn ad2antrl eqfnfv syl2an2r + mpbird grpinvf grplinv ralrimiva jca feq1 fveq1 ralbidv anbi12d syl5ibcom + oveq1d impbid ) DMNZBBEOZAPZEUAZWBCQZGRZABSZTZFERZVTWGWHVTWGTWHWBFUAZWCRZ + ABSZVTWAWFWKVTWATZWEWJABWLWBBNZTZWEWJWNWETZWILPZWBCQZGRZLBUBZWCWMWIWSRWLW + ELBCDFWBGHIJKUCUDWOWEWSWCRZWNWEUEWOWCBNWRLBUGZWEWTUFWOBBWBEVTWAWMWEUHWLWM + WEUIUJVTWMXAWAWELBCDWBGHIJUKULWRWELBWCWPWCRWQWDGWPWCWBCUMUNUOUQUPURUSUTVA + VTFBVBWGEBVBZWHWKUFBDFMHKVCWAXBVTWFBBEVDVEABFEVFVGVHUSVTBBFOZWIWBCQZGRZAB + SZTWHWGVTXCXFBDFHKVIVTXEABBCDFWBGHIJKVJVKVLWHXCWAXFWFBBFEVMWHXEWEABWHXDWD + GWHWIWCWBCWBFEVNVRUNVOVPVQVS $. $} ${ @@ -146121,8 +146157,8 @@ an algebraic structure formed from a base set of elements (notated $( Closure of group subtraction. (Contributed by NM, 31-Mar-2014.) $) grpsubcl $p |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) e. B ) $= - ( cgrp wcel cxp wf co grpsubf fovrn syl3an1 ) BHIAAJACKDAIEAIDECLAIABCFGM - DEAAACNO $. + ( cgrp wcel cxp wf co grpsubf fovcdm syl3an1 ) BHIAAJACKDAIEAIDECLAIABCFG + MDEAAACNO $. $( Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.) $) @@ -146326,28 +146362,28 @@ an algebraic structure formed from a base set of elements (notated vz nfv wi oveq2 eqeq1d oveq1 anbi12d ralbidv rspcv rspcva cbvrexvw biimpi eqeq2 adantr syl ex syldc 3ad2ant3 imp rspc2va simprd expcom impl simpll1 ad2ant2r simplr simpllr simprr sgrpass syl13anc simprl anassrs id eqeq12d - oveq1d eqtr3d syl5ibcom rexlimdva syl5bi mpd simpld ancoms com12 adantllr - adantrl sylib adantrr jca expr ralrimdva reximdva exlimdd ) HUAUBZCNZEUBZ - CUCZKNZANZFOZBNZPZKEQZXFINZFOZXHPZIEQZRZBEUDZAEUDZUEZXCDNZJNZFOZXTPZGNZXT - FOZXSPZGEQZRZJEUDZDEQZCXAXDXQCXACUJXCCUFXQCUJUGYICUJXAXDXQUHXRXCRZXSXBFOZ - XBPZDEQZYIXRXCYMXQXAXCYMUKXDXCXQXEXBFOZXHPZKEQZXBXKFOZXHPZIEQZRZBEUDZYMXP - UUAAXBEACSZXOYTBEUUBXJYPXNYSUUBXIYOKEUUBXGYNXHXFXBXEFULUMTUUBXMYRIEUUBXLY - QXHXFXBXKFUNUMTUOZUPUQXCUUAYMXCUUARYNXBPZKEQZYQXBPZIEQZRZYMYTUUHBXBEBCSZY - PUUEYSUUGUUIYOUUDKEXHXBYNVATUUIYRUUFIEXHXBYQVATUOURUUEYMUUGUUEYMUUDYLKDEK - DSYNYKXBXEXSXBFUNUMUSUTVBVCVDVEVFVGYJYLYHDEYJXSEUBZRZYLYGJEUUKXTEUBZYLYGU - UKUULYLRRZYBYFUUMYQXTPZIEQZYBYJUULUUOUUJYLXRXCUULUUOXQXAXCUULRZUUOUKXDUUP - XQUUOUUPXQRYNXTPZKEQZUUOXOUURUUORYTABXBXTEEUUCBJSZYPUURYSUUOUUSYOUUQKEXHX - TYNVATUUSYRUUNIEXHXTYQVATUOVHVIVJVFVKVMUUKYLUUOYBUKUULUUOXBUINZFOZXTPZUIE - QUUKYLRZYBUUNUVBIUIEIUISYQUVAXTXKUUTXBFULUMUSUVCUVBYBUIEUVCUUTEUBZRXSUVAF - OZUVAPZUVBYBUUKYLUVDUVFUUKYLUVDRZRZYKUUTFOZUVEUVAUVHXAUUJXCUVDUVIUVEPUUKX - AUVGXAXDXQXCUUJVLVBYJUUJUVGVNXRXCUUJUVGVOUUKYLUVDVPEHXSXBFUUTLMVQVRUVHYKX - BUUTFUUKYLUVDVSWCWDVTUVBUVEYAUVAXTUVAXTXSFULUVBWAWBWEWFWGWMWHUUKUULYFYLXR - UUJUULYFXCXRUUJRUULRXEXTFOZXSPZKEQZYFXRUUJUULUVLXQXAUUJUULRZUVLUKXDUVMXQU - VLUULUUJXQUVLUKUULUUJRZXQUVLUVNXQRUVLXTXKFOZXSPZIEQZXOUVLUVQRUVJXHPZKEQZU - VOXHPZIEQZRABXTXSEEAJSZXJUVSXNUWAUWBXIUVRKEUWBXGUVJXHXFXTXEFULUMTUWBXMUVT - IEUWBXLUVOXHXFXTXKFUNUMTUOBDSZUVSUVLUWAUVQUWCUVRUVKKEXHXSUVJVATUWCUVTUVPI - EXHXSUVOVATUOVHWIVDWJWKVFVKUVKYEKGEKGSUVJYDXSXEYCXTFUNUMUSWNWLWOWPWQWRWSW - HWT $. + oveq1d eqtr3d syl5ibcom rexlimdva biimtrid adantrl mpd simpld com12 sylib + ancoms adantllr adantrr jca expr ralrimdva reximdva exlimdd ) HUAUBZCNZEU + BZCUCZKNZANZFOZBNZPZKEQZXFINZFOZXHPZIEQZRZBEUDZAEUDZUEZXCDNZJNZFOZXTPZGNZ + XTFOZXSPZGEQZRZJEUDZDEQZCXAXDXQCXACUJXCCUFXQCUJUGYICUJXAXDXQUHXRXCRZXSXBF + OZXBPZDEQZYIXRXCYMXQXAXCYMUKXDXCXQXEXBFOZXHPZKEQZXBXKFOZXHPZIEQZRZBEUDZYM + XPUUAAXBEACSZXOYTBEUUBXJYPXNYSUUBXIYOKEUUBXGYNXHXFXBXEFULUMTUUBXMYRIEUUBX + LYQXHXFXBXKFUNUMTUOZUPUQXCUUAYMXCUUARYNXBPZKEQZYQXBPZIEQZRZYMYTUUHBXBEBCS + ZYPUUEYSUUGUUIYOUUDKEXHXBYNVATUUIYRUUFIEXHXBYQVATUOURUUEYMUUGUUEYMUUDYLKD + EKDSYNYKXBXEXSXBFUNUMUSUTVBVCVDVEVFVGYJYLYHDEYJXSEUBZRZYLYGJEUUKXTEUBZYLY + GUUKUULYLRRZYBYFUUMYQXTPZIEQZYBYJUULUUOUUJYLXRXCUULUUOXQXAXCUULRZUUOUKXDU + UPXQUUOUUPXQRYNXTPZKEQZUUOXOUURUUORYTABXBXTEEUUCBJSZYPUURYSUUOUUSYOUUQKEX + HXTYNVATUUSYRUUNIEXHXTYQVATUOVHVIVJVFVKVMUUKYLUUOYBUKUULUUOXBUINZFOZXTPZU + IEQUUKYLRZYBUUNUVBIUIEIUISYQUVAXTXKUUTXBFULUMUSUVCUVBYBUIEUVCUUTEUBZRXSUV + AFOZUVAPZUVBYBUUKYLUVDUVFUUKYLUVDRZRZYKUUTFOZUVEUVAUVHXAUUJXCUVDUVIUVEPUU + KXAUVGXAXDXQXCUUJVLVBYJUUJUVGVNXRXCUUJUVGVOUUKYLUVDVPEHXSXBFUUTLMVQVRUVHY + KXBUUTFUUKYLUVDVSWCWDVTUVBUVEYAUVAXTUVAXTXSFULUVBWAWBWEWFWGWHWIUUKUULYFYL + XRUUJUULYFXCXRUUJRUULRXEXTFOZXSPZKEQZYFXRUUJUULUVLXQXAUUJUULRZUVLUKXDUVMX + QUVLUULUUJXQUVLUKUULUUJRZXQUVLUVNXQRUVLXTXKFOZXSPZIEQZXOUVLUVQRUVJXHPZKEQ + ZUVOXHPZIEQZRABXTXSEEAJSZXJUVSXNUWAUWBXIUVRKEUWBXGUVJXHXFXTXEFULUMTUWBXMU + VTIEUWBXLUVOXHXFXTXKFUNUMTUOBDSZUVSUVLUWAUVQUWCUVRUVKKEXHXSUVJVATUWCUVTUV + PIEXHXSUVOVATUOVHWJVDWMWKVFVKUVKYEKGEKGSUVJYDXSXEYCXTFUNUMUSWLWNWOWPWQWRW + SWIWT $. $( Alternate definition of a group as semigroup (with at least one element) which is also a quasigroup, i.e. a magma in which solutions ` x ` and @@ -146553,17 +146589,17 @@ an algebraic structure formed from a base set of elements (notated $( A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.) $) mhmid $p |- ( ph -> ( F ` .0. ) = ( 0g ` H ) ) $= - ( cfv va vi c0g eqid wfo wf fof cmnd wcel mndidcl ffvelrnd cv wa wceq - co simplll syl3an1 ad3antrrr simplr mhmlem mndlid fveq2d eqtr3d simpr - syl syl2anc oveq2d 3eqtr3d wrex foelrni sylan r19.29a mndrid ismgmid2 - oveq1d ) AUAJEKFTZHHUCTZNVQUDPAIJKFAIJFUEZIJFUFQIJFUGVEAGUHUIZKIUIZRI - GKMSUJZVEUKAUAULZJUIZUMZUBULZFTZWBUNZVPWBEUOZWBUNUBIWDWEIUIZUMZWGUMZV - PWFEUOZWFWHWBWKKWEDUOZFTWLWFWKBCKWEDEFIWKABULZIUICULZIUIWNWODUOFTWNFT - WOFTEUOUNAWCWIWGUPLUQZWKVSVTAVSWCWIWGRURZWAVEZWDWIWGUSZUTWKWMWEFWKVSW - IWMWEUNWQWSIDGWEKMOSVAVFVBVCWKWFWBVPEWJWGVDZVGWTVHAVRWCWGUBIVIQUBIJFW - BVJVKZVLWDWGWBVPEUOZWBUNUBIWKWFVPEUOZWFXBWBWKWEKDUOZFTXCWFWKBCWEKDEFI - WPWSWRUTWKXDWEFWKVSWIXDWEUNWQWSIDGWEKMOSVMVFVBVCWKWFWBVPEWTVOWTVHXAVL - VN $. + ( cfv va vi c0g eqid wfo wf fof syl cmnd wcel mndidcl ffvelcdmd cv wa + wceq co simplll syl3an1 ad3antrrr simplr mhmlem mndlid syl2anc fveq2d + eqtr3d simpr oveq2d 3eqtr3d wrex foelcdmi sylan r19.29a mndrid oveq1d + ismgmid2 ) AUAJEKFTZHHUCTZNVQUDPAIJKFAIJFUEZIJFUFQIJFUGUHAGUIUJZKIUJZ + RIGKMSUKZUHULAUAUMZJUJZUNZUBUMZFTZWBUOZVPWBEUPZWBUOUBIWDWEIUJZUNZWGUN + ZVPWFEUPZWFWHWBWKKWEDUPZFTWLWFWKBCKWEDEFIWKABUMZIUJCUMZIUJWNWODUPFTWN + FTWOFTEUPUOAWCWIWGUQLURZWKVSVTAVSWCWIWGRUSZWAUHZWDWIWGUTZVAWKWMWEFWKV + SWIWMWEUOWQWSIDGWEKMOSVBVCVDVEWKWFWBVPEWJWGVFZVGWTVHAVRWCWGUBIVIQUBIJ + FWBVJVKZVLWDWGWBVPEUPZWBUOUBIWKWFVPEUPZWFXBWBWKWEKDUPZFTXCWFWKBCWEKDE + FIWPWSWRVAWKXDWEFWKVSWIXDWEUOWQWSIDGWEKMOSVMVCVDVEWKWFWBVPEWTVNWTVHXA + VLVO $. $} $( The image of a monoid ` G ` under a monoid homomorphism ` F ` is a @@ -146571,40 +146607,40 @@ an algebraic structure formed from a base set of elements (notated mhmmnd $p |- ( ph -> H e. Mnd ) $= ( co wcel wa va vb vc vd vi vj vk wceq wral wrex cmnd cfv simpllr simpr cv oveq12d simp-5l syl3an1 simp-4r simplr mhmlem wf wfo fof syl ad5antr - mndcl syl3anc ffvelrnd eqeltrrd foelrni syl2an ad2antrr r19.29a simplrl - simpl simpll simplrr w3a simp-6r mndass syl13anc fveq2d simp-7l 3eqtr3d - simp1 simp2 simp3 oveq1d oveq2d simp-5r 3ad2antr3 ad4antr ralrimiva jca - sylan 3adantr3 ralrimivva eqid mndidcl simplll ad3antrrr mndlid syl2anc - c0g eqtr3d mndrid oveq1 eqeq1d ovanraleqv rspcev ismnd sylanbrc ) AUAUO - ZUBUOZERZJSZXPUCUOZERZXNXOXRERZERZUHZUCJUIZTZUBJUIUAJUIUDUOZXNERZXNUHZX - NYEERXNUHTUAJUIZUDJUJZHUKSAYDUAUBJJAXNJSZXOJSZTZTZXQYCYMUEUOZFULZXNUHZX - QUEIYMYNISZTZYPTZUFUOZFULZXOUHZXQUFIYSYTISZTZUUBTZYOUUAERZXPJUUEYOXNUUA - XOEYRYPUUCUUBUMUUDUUBUNUPUUEYNYTDRZFULZUUFJUUEBCYNYTDEFIUUEABUOZISZCUOZ - ISZUUIUUKDRFULUUIFULUUKFULERUHZAYLYQYPUUCUUBUQKURYMYQYPUUCUUBUSZYSUUCUU - BUTZVAUUEIJUUGFAIJFVBZYLYQYPUUCUUBAIJFVCZUUPPIJFVDVEZVFUUEGUKSZYQUUCUUG - ISZAUUSYLYQYPUUCUUBQVFUUNUUOIDGYNYTLNVGZVHVIVJVJYMUUBUFIUJZYQYPAUUQYKUV - 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(Contributed by Thierry Arnoux, 26-Jan-2020.) $) @@ -146625,17 +146661,17 @@ since the target of the homomorphism (operator ` O ` in our model) need 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.) $) ghmgrp $p |- ( ph -> H e. Grp ) $= ( wcel cfv wceq vf va vi cmnd cv co wrex wral cgrp grpmndd mhmmnd cminusg - c0g wa wf wfo fof syl ad3antrrr simplr grpinvcl syl2anc ffvelrnd 3adant1r - eqid sylan simpr mhmlem ad4ant13 fveq2d mhmid eqtrd oveq2d 3eqtr3rd oveq1 - grplinv eqeq1d rspcev foelrni r19.29a ralrimiva isgrp sylanbrc ) AHUDRUAU - EZUBUEZEUFZHUMSZTZUAJUGZUBJUHHUIRABCDEFGHIJKLMNOPAGQUJZUKAWIUBJAWEJRZUNZU - CUEZFSZWETZWIUCIWLWMIRZUNZWOUNZWMGULSZSZFSZJRXAWEEUFZWGTZWIWRIJWTFAIJFUOZ - WKWPWOAIJFUPZXDPIJFUQURUSWRGUIRZWPWTIRZAXFWKWPWOQUSZWLWPWOUTZIGWSWMLWSVEZ - VAZVBVCWRWTWMDUFZFSZXAWNEUFZWGXBAWPXMXNTWKWOAWPUNBCWTWMDEFIABUEZIRCUEZIRX - OXPDUFFSXOFSXPFSEUFTWPKVDAXFWPXGQXKVFAWPVGVHVIWRXMGUMSZFSZWGWRXFWPXMXRTXH - XIXFWPUNXLXQFIDGWSWMXQLNXQVEZXJVPVJVBAXRWGTWKWPWOABCDEFGHIJXQKLMNOPWJXSVK - USVLWRWNWEXAEWQWOVGVMVNWHXCUAXAJWDXATWFXBWGWDXAWEEVOVQVRVBAXEWKWOUCIUGPUC - IJFWEVSVFVTWAJEUAHWGUBMOWGVEWBWC $. + c0g wfo fof syl ad3antrrr simplr eqid grpinvcl syl2anc ffvelcdmd 3adant1r + wa sylan simpr mhmlem ad4ant13 grplinv fveq2d mhmid eqtrd oveq2d 3eqtr3rd + wf oveq1 eqeq1d rspcev foelcdmi r19.29a ralrimiva isgrp sylanbrc ) AHUDRU + AUEZUBUEZEUFZHUMSZTZUAJUGZUBJUHHUIRABCDEFGHIJKLMNOPAGQUJZUKAWIUBJAWEJRZVD + ZUCUEZFSZWETZWIUCIWLWMIRZVDZWOVDZWMGULSZSZFSZJRXAWEEUFZWGTZWIWRIJWTFAIJFV + OZWKWPWOAIJFUNZXDPIJFUOUPUQWRGUIRZWPWTIRZAXFWKWPWOQUQZWLWPWOURZIGWSWMLWSU + SZUTZVAVBWRWTWMDUFZFSZXAWNEUFZWGXBAWPXMXNTWKWOAWPVDBCWTWMDEFIABUEZIRCUEZI + RXOXPDUFFSXOFSXPFSEUFTWPKVCAXFWPXGQXKVEAWPVFVGVHWRXMGUMSZFSZWGWRXFWPXMXRT + XHXIXFWPVDXLXQFIDGWSWMXQLNXQUSZXJVIVJVAAXRWGTWKWPWOABCDEFGHIJXQKLMNOPWJXS + VKUQVLWRWNWEXAEWQWOVFVMVNWHXCUAXAJWDXATWFXBWGWDXAWEEVPVQVRVAAXEWKWOUCIUGP + UCIJFWEVSVEVTWAJEUAHWGUBMOWGUSWBWC $. $} @@ -146710,26 +146746,26 @@ since the target of the homomorphism (operator ` O ` in our model) need c1 cmpo seqeq3d eqtr4di fveq12d negeqd fveq2d ifbieq12d ifbieq2d simpll simplr c0g fn0g funfvex funfni mpan eqeltrid ad2antlr wn nnuz fvconst2g wfn wf 1zzd eqeltrd elexd adantlr simprl cplusg plusgslid slotex simprr - ovexg syl3anc feq1i sylibr ad5ant23 simp-4l sylanbrc ffvelrnd grpinvfng - seqf elnnz basfn fnex syl2anc ad3antlr znegcl ad4antr ztri3or0 ecase23d - cbs w3o cr zre lt0neg1d mpbid fvexg wdc 0zd zdclt ifcldadc zdceq ovmpod - simplll mpdan ) GUEPZHAPZQZERPZGHDUFGSUGZISGUHUIZGCUJZGUKZCUJZFUJZULZUL - ZUGYMYOYLHAEJUMUNYNYOQZUAUBGHUEAUAUOZSUGZISUUEUHUIZUUEBTUBUOZUPZUQZVFUR - ZUJZUUEUKZUUKUJZFUJZULZULZUUCDRYODUAUBUEAUUQVGUGYNUBABDUAEFRIJKLMNUSUNU - UEGUGZUUHHUGZQZUUQUUCUGUUDUUTUUFYPUUPUUBIUUTUUEGSUURUUSUTZVAUUTUUGYQUUL - UUOYRUUAUUTUUEGSUHUVAVBUUTUUEGUUKCUUTUUKBTHUPZUQZVFURZCUUTUUJUVCBVFUUTU - UIUVBTUUTUUHHUURUUSVCVDVEVHOVIZUVAVJUUTUUNYTFUUTUUMYSUUKCUVEUUTUUEGUVAV - KVJVLVMVNUNYLYMYOVOZYLYMYOVPUUDYPIUUBRYOIRPYNYPYOIEVQUJZRLVQRWGYOUVGRPZ - VRUVHREVQEVQVSVTWAWBWCUUDYPWDZQZYQYRUUARUVJYQQZTRGCYMYOTRCWHZYLUVIYQYMY - OQZTRUVDWHUVLUVMUCUDBRUVCVFTWEUVMWIYMUCUOZTPZUVNUVCUJZRPYOYMUVOQZUVPAUV - QUVPHATHUVNAWFYMUVOUTWJWKWLUVMUVNRPZUDUOZRPZQZQUVRBRPZUVTUVNUVSBUFRPUVM - UVRUVTWMYOUWBYMUWAYOBEWNUJRKEWNRWOWPWBWCUVMUVRUVTWQUVNUVSBRRRWRWSXGTRCU - VDOWTXAZXBUVKYLYQGTPYLYMYOUVIYQXCUVJYQVCGXHXDXEUVJYQWDZQZFRPZYTRPUUARPY - OUWFYNUVIUWDYOFAWGARPUWFAEFRJMXFYOAEXQUJZRJXQRWGYOUWGRPZXIUWHREXQEXQVSV - TWAWBARFXJXKXLUWETRYSCYMYOUVLYLUVIUWDUWCXBUWEYSUEPZSYSUHUIZYSTPYLUWIYMY - OUVIUWDGXMXNUWEGSUHUIZUWJUWEUWKYPYQUUDUVIUWDVPUVJUWDVCYLUWKYPYQXRYMYOUV - IUWDGXOXNXPUWEGYLGXSPYMYOUVIUWDGXTXNYAYBYSXHXDXEYTFRRYCXKUVJSUEPZYLYQYD - UVJYEYLYMYOUVIYJSGYFXKYGUUDYLUWLYPYDUVFUUDYEGSYHXKYGYIYK $. + ovexg syl3anc seqf sylibr ad5ant23 simp-4l sylanbrc ffvelcdmd grpinvfng + feq1i elnnz cbs basfn fnex syl2anc ad3antlr znegcl ad4antr w3o ztri3or0 + ecase23d zre lt0neg1d mpbid fvexg wdc 0zd simplll zdclt ifcldadc ovmpod + cr zdceq mpdan ) GUEPZHAPZQZERPZGHDUFGSUGZISGUHUIZGCUJZGUKZCUJZFUJZULZU + LZUGYMYOYLHAEJUMUNYNYOQZUAUBGHUEAUAUOZSUGZISUUEUHUIZUUEBTUBUOZUPZUQZVFU + RZUJZUUEUKZUUKUJZFUJZULZULZUUCDRYODUAUBUEAUUQVGUGYNUBABDUAEFRIJKLMNUSUN + UUEGUGZUUHHUGZQZUUQUUCUGUUDUUTUUFYPUUPUUBIUUTUUEGSUURUUSUTZVAUUTUUGYQUU + LUUOYRUUAUUTUUEGSUHUVAVBUUTUUEGUUKCUUTUUKBTHUPZUQZVFURZCUUTUUJUVCBVFUUT + UUIUVBTUUTUUHHUURUUSVCVDVEVHOVIZUVAVJUUTUUNYTFUUTUUMYSUUKCUVEUUTUUEGUVA + VKVJVLVMVNUNYLYMYOVOZYLYMYOVPUUDYPIUUBRYOIRPYNYPYOIEVQUJZRLVQRWGYOUVGRP + ZVRUVHREVQEVQVSVTWAWBWCUUDYPWDZQZYQYRUUARUVJYQQZTRGCYMYOTRCWHZYLUVIYQYM + YOQZTRUVDWHUVLUVMUCUDBRUVCVFTWEUVMWIYMUCUOZTPZUVNUVCUJZRPYOYMUVOQZUVPAU + VQUVPHATHUVNAWFYMUVOUTWJWKWLUVMUVNRPZUDUOZRPZQZQUVRBRPZUVTUVNUVSBUFRPUV + MUVRUVTWMYOUWBYMUWAYOBEWNUJRKEWNRWOWPWBWCUVMUVRUVTWQUVNUVSBRRRWRWSWTTRC + UVDOXGXAZXBUVKYLYQGTPYLYMYOUVIYQXCUVJYQVCGXHXDXEUVJYQWDZQZFRPZYTRPUUARP + YOUWFYNUVIUWDYOFAWGARPUWFAEFRJMXFYOAEXIUJZRJXIRWGYOUWGRPZXJUWHREXIEXIVS + VTWAWBARFXKXLXMUWETRYSCYMYOUVLYLUVIUWDUWCXBUWEYSUEPZSYSUHUIZYSTPYLUWIYM + YOUVIUWDGXNXOUWEGSUHUIZUWJUWEUWKYPYQUUDUVIUWDVPUVJUWDVCYLUWKYPYQXPYMYOU + VIUWDGXQXOXRUWEGYLGYIPYMYOUVIUWDGXSXOXTYAYSXHXDXEYTFRRYBXLUVJSUEPZYLYQY + CUVJYDYLYMYOUVIYESGYFXLYGUUDYLUWLYPYCUVFUUDYDGSYJXLYGYHYK $. $} $} @@ -146743,23 +146779,23 @@ since the target of the homomorphism (operator ` O ` in our model) need ( vn vx wcel cz wfn cv cc0 c0g cfv cn cvv wa syl ad2antrr vu cxp wceq clt vv wbr cplusg csn c1 cseq cneg cminusg cmpo wral elex fn0g funfvex funfni cif mpan wn wf nnuz 1zzd fvconst2g simpl eqeltrd adantll simprl plusgslid - elexd co slotex simprr ovexg syl3anc seqf adantrl simpr sylanbrc ffvelrnd - elnnz eqid grpinvfng basfn eqeltrid fnex syl2anc ad3antrrr znegcld simplr - cbs w3o ztri3or0 ecase23d zred lt0neg1d mpbid fvexg wdc 0zd simplrl zdclt - cr ifcldadc zdceq ralrimivva fnmpo mulgfvalg fneq1d mpbird ) CDIZBJAUBZKG - HJAGLZMUCZCNOZMXNUDUFZXNCUGOZPHLZUHUBZUIUJZOZXNUKZYAOZCULOZOZUSZUSZUMZXMK - ZXLYHQIZHAUNGJUNYJXLYKGHJAXLXNJIZXSAIZRZRZXOXPYGQXLXPQIZYNXOXLCQIZYPCDUOZ - NQKYQYPUPYPQCNCNUQURUTSTYOXOVAZRZXQYBYFQYTXQRZPQXNYAYOPQYAVBZYSXQXLYMUUBY - LXLYMRZUAUEXRQXTUIPVCUUCVDYMUALZPIZUUDXTOZQIXLYMUUERZUUFAUUGUUFXSAPXSUUDA - VEYMUUEVFVGVKVHUUCUUDQIZUELZQIZRZRUUHXRQIZUUJUUDUUIXRVLQIUUCUUHUUJVIXLUUL - YMUUKCUGDVJVMTUUCUUHUUJVNUUDUUIXRQQQVOVPVQVRZTUUAYLXQXNPIYOYLYSXQXLYLYMVI - ZTYTXQVSXNWBVTWAYTXQVAZRZYEQIZYDQIYFQIXLUUQYNYSUUOXLYEAKAQIZUUQACYEDEYEWC - ZWDXLYQUURYRYQACWLOZQEWLQKYQUUTQIZWEUVAQCWLCWLUQURUTWFSAQYEWGWHWIUUPPQYCY - AYOUUBYSUUOUUMTUUPYCJIZMYCUDUFZYCPIYOUVBYSUUOYOXNUUNWJTUUPXNMUDUFZUVCUUPU - VDXOXQYOYSUUOWKYTUUOVSYOUVDXOXQWMZYSUUOYOYLUVEUUNXNWNSTWOUUPXNYOXNXDIYSUU - OYOXNUUNWPTWQWRYCWBVTWAYDYEQQWSWHYTMJIZYLXQWTYTXAXLYLYMYSXBMXNXCWHXEYOYLU - VFXOWTUUNYOXAXNMXFWHXEXGGHJAYHYIQYIWCXHSXLXMBYIHAXRBGCYEDXPEXRWCXPWCUUSFX - IXJXK $. + elexd co slotex ovexg syl3anc seqf adantrl simpr elnnz sylanbrc ffvelcdmd + simprr eqid grpinvfng cbs basfn eqeltrid syl2anc ad3antrrr znegcld simplr + fnex w3o ztri3or0 ecase23d cr zred lt0neg1d mpbid fvexg wdc simplrl zdclt + 0zd ifcldadc zdceq ralrimivva fnmpo mulgfvalg fneq1d mpbird ) CDIZBJAUBZK + GHJAGLZMUCZCNOZMXNUDUFZXNCUGOZPHLZUHUBZUIUJZOZXNUKZYAOZCULOZOZUSZUSZUMZXM + KZXLYHQIZHAUNGJUNYJXLYKGHJAXLXNJIZXSAIZRZRZXOXPYGQXLXPQIZYNXOXLCQIZYPCDUO + ZNQKYQYPUPYPQCNCNUQURUTSTYOXOVAZRZXQYBYFQYTXQRZPQXNYAYOPQYAVBZYSXQXLYMUUB + YLXLYMRZUAUEXRQXTUIPVCUUCVDYMUALZPIZUUDXTOZQIXLYMUUERZUUFAUUGUUFXSAPXSUUD + AVEYMUUEVFVGVKVHUUCUUDQIZUELZQIZRZRUUHXRQIZUUJUUDUUIXRVLQIUUCUUHUUJVIXLUU + LYMUUKCUGDVJVMTUUCUUHUUJWBUUDUUIXRQQQVNVOVPVQZTUUAYLXQXNPIYOYLYSXQXLYLYMV + IZTYTXQVRXNVSVTWAYTXQVAZRZYEQIZYDQIYFQIXLUUQYNYSUUOXLYEAKAQIZUUQACYEDEYEW + CZWDXLYQUURYRYQACWEOZQEWEQKYQUUTQIZWFUVAQCWECWEUQURUTWGSAQYEWLWHWIUUPPQYC + YAYOUUBYSUUOUUMTUUPYCJIZMYCUDUFZYCPIYOUVBYSUUOYOXNUUNWJTUUPXNMUDUFZUVCUUP + UVDXOXQYOYSUUOWKYTUUOVRYOUVDXOXQWMZYSUUOYOYLUVEUUNXNWNSTWOUUPXNYOXNWPIYSU + UOYOXNUUNWQTWRWSYCVSVTWAYDYEQQWTWHYTMJIZYLXQXAYTXDXLYLYMYSXBMXNXCWHXEYOYL + UVFXOXAUUNYOXDXNMXFWHXEXGGHJAYHYIQYIWCXHSXLXMBYIHAXRBGCYEDXPEXRWCXPWCUUSF + XIXJXK $. $} ${ @@ -146879,10 +146915,10 @@ since the target of the homomorphism (operator ` O ` in our model) need mulgnnsubcl $p |- ( ( ph /\ N e. NN /\ X e. S ) -> ( N .x. X ) e. S ) $= ( cn wcel co w3a csn cxp c1 cseq cfv wceq simp2 wss 3ad2ant1 simp3 sseldd eqid mulgnn syl2anc nnuz 1zzd cv fvconst2g sylan simpl3 eqeltrd 3ad2antl1 - wa 3expb seqf ffvelrnd ) AIRSZKFSZUAZIKGTZIERKUBUCZUDUEZUFZFVJVHKDSVKVNUG - AVHVIUHZVJFDKAVHFDUIVIPUJAVHVIUKZULDEVMGHIKLNMVMUMUNUOVJRFIVMVJBCEFVLUDRU - PVJUQVJBURZRSZVDVQVLUFZKFVJVIVRVSKUGVPRKVQFUSUTAVHVIVRVAVBAVHVQFSZCURZFSZ - VDVQWAETFSZVIAVTWBWCQVEVCVFVOVGVB $. + wa 3expb seqf ffvelcdmd ) AIRSZKFSZUAZIKGTZIERKUBUCZUDUEZUFZFVJVHKDSVKVNU + GAVHVIUHZVJFDKAVHFDUIVIPUJAVHVIUKZULDEVMGHIKLNMVMUMUNUOVJRFIVMVJBCEFVLUDR + UPVJUQVJBURZRSZVDVQVLUFZKFVJVIVRVSKUGVPRKVQFUSUTAVHVIVRVAVBAVHVQFSZCURZFS + ZVDVQWAETFSZVIAVTWBWCQVEVCVFVOVGVB $. mulgnn0subcl.z $e |- .0. = ( 0g ` G ) $. mulgnn0subcl.c $e |- ( ph -> .0. e. S ) $. @@ -147398,6 +147434,42 @@ since the target of the homomorphism (operator ` O ` in our model) need OVPVQVRVSVT $. $} + ${ + $d ph a b x y $. $d B x y $. $d G a b x y $. $d H a b x y $. + $d K x y $. + mulgpropdg.m $e |- ( ph -> .x. = ( .g ` G ) ) $. + mulgpropdg.n $e |- ( ph -> .X. = ( .g ` H ) ) $. + mulgpropdg.g $e |- ( ph -> G e. V ) $. + mulgpropdg.h $e |- ( ph -> H e. W ) $. + mulgpropd.b1 $e |- ( ph -> B = ( Base ` G ) ) $. + mulgpropd.b2 $e |- ( ph -> B = ( Base ` H ) ) $. + mulgpropd.i $e |- ( ph -> B C_ K ) $. + mulgpropd.k $e |- ( ( ph /\ ( x e. K /\ y e. K ) ) -> + ( x ( +g ` G ) y ) e. K ) $. + mulgpropd.e $e |- ( ( ph /\ ( x e. K /\ y e. K ) ) -> + ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) ) $. + $( Two structures with the same group-nature have the same group multiple + function. ` K ` is expected to either be ` _V ` (when strong equality is + available) or ` B ` (when closure is available). (Contributed by Stefan + O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) $) + mulgpropdg $p |- ( ph -> .x. = .X. ) $= + ( va vb cz cbs cfv cv cc0 wceq c0g clt wbr cplusg cn csn cxp c1 cseq cneg + cminusg cif cmpo wcel w3a wa co wss wi anim12d syl imp syldan grpidpropdg + ssel 3ad2ant1 1zzd nnuz simp3 sseldd ialgrlemconst 3ad2antl1 grpinvpropdg + seqfeq3 fveq1d fveq12d ifeq12d mpoeq3dva mpoeq123dv 3eqtr3d cmg mulgfvalg + eqidd eqid eqtrd 3eqtr4d ) AUAUBUCGUDUEZUAUFZUGUHZGUIUEZUGWPUJUKZWPGULUEZ + UMUBUFZUNUOZUPUQZUEZWPURZXCUEZGUSUEZUEZUTZUTZVAZUAUBUCHUDUEZWQHUIUEZWSWPH + ULUEZXBUPUQZUEZXEXOUEZHUSUEZUEZUTZUTZVAZEFAUAUBUCDXJVAUAUBUCDYAVAXKYBAUAU + BUCDXJYAAWPUCVBZXADVBZVCZWQWRXMXIXTAYCWRXMUHYDABCDGHJKPQNOABUFZDVBZCUFZDV + BZVDZYFIVBZYHIVBZVDZYFYHWTVEZYFYHXNVEUHZAYJYMADIVFZYJYMVGRYPYGYKYIYLDIYFV + MDIYHVMVHVIVJTVKZVLVNYEWSXDXPXHXSYEWPXCXOYEBCWTXNIXBUPYEVOYEBXAIUPUMVPYED + IXAAYCYPYDRVNAYCYDVQVRVSAYCYMYNIVBYDSVTAYCYMYOYDTVTWBZWCYEXFXQXGXRAYCXGXR + UHYDABCDGHJKPQNOYQWAVNYEXEXCXOYRWCWDWEWEWFAUAUBUCDXJUCWOXJAUCWKZPAXJWKWGA + UAUBUCDYAUCXLYAYSQAYAWKWGWHAEGWIUEZXKLAGJVBYTXKUHNUBWOWTYTUAGXGJWRWOWLWTW + LWRWLXGWLYTWLWJVIWMAFHWIUEZYBMAHKVBUUAYBUHOUBXLXNUUAUAHXRKXMXLWLXNWLXMWLX + RWLUUAWLWJVIWMWN $. + $} + ${ $d x y G $. $d x y N $. $d x y S $. $d x .xb $. $d x y X $. submmulgcl.t $e |- .xb = ( .g ` G ) $. @@ -147915,6 +147987,17 @@ since the target of the homomorphism (operator ` O ` in our model) need ${ mgpbas.1 $e |- M = ( mulGrp ` R ) $. + ${ + $( Existence of the multiplication group. If ` R ` is known to be a + semiring, see ~ srgmgp . (Contributed by Jim Kingdon, + 10-Jan-2025.) $) + mgpex $p |- ( R e. V -> M e. _V ) $= + ( wcel cnx cplusg cfv cmulr cop csts co eqid mgpvalg cn cslot plusgslid + cvv wceq simpri a1i mulrslid slotex setsex mpd3an23 eqeltrd ) ACEZBAFGH + ZAIHZJKLZRAUIBCDUIMNUGUHOEZUIREUJREUKUGGUHPSUKQTUAAICUBUCUHUIACROUDUEUF + $. + $} + ${ mgpbas.2 $e |- B = ( Base ` R ) $. $( Base set of the multiplication group. (Contributed by Mario Carneiro, @@ -148537,6 +148620,1029 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Definition and basic properties of unital rings +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $c Ring $. + $c CRing $. + + $( Extend class notation with class of all (unital) rings. $) + crg $a class Ring $. + + $( Extend class notation with class of all (unital) commutative rings. $) + ccrg $a class CRing $. + + ${ + $d f p r t x y z $. + $( Define class of all (unital) rings. A unital ring is a set equipped + with two everywhere-defined internal operations, whose first one is an + additive group structure and the second one is a multiplicative monoid + structure, and where the addition is left- and right-distributive for + the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition + of a ring with identity in part Preliminaries of [Roman] p. 19. So that + the additive structure must be abelian (see ~ ringcom ), care must be + taken that in the case of a non-unital ring, the commutativity of + addition must be postulated and cannot be proved from the other + conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario + Carneiro, 27-Dec-2014.) $) + df-ring $a |- Ring = { f e. Grp | ( ( mulGrp ` f ) e. Mnd /\ + [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. + A. x e. r A. y e. r A. z e. r + ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) + /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) ) } $. + + $( Define class of all commutative rings. (Contributed by Mario Carneiro, + 7-Jan-2015.) $) + df-cring $a |- CRing = { f e. Ring | ( mulGrp ` f ) e. CMnd } $. + $} + + ${ + $d b p r t x y z B $. $d b p r t x y z .+ $. $d b p r t x y z R $. + $d r G $. $d b p r t x y z .x. $. + isring.b $e |- B = ( Base ` R ) $. + isring.g $e |- G = ( mulGrp ` R ) $. + isring.p $e |- .+ = ( +g ` R ) $. + isring.t $e |- .x. = ( .r ` R ) $. + $( The predicate "is a (unital) ring". Definition of ring with unit in + [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by + Mario Carneiro, 6-Jan-2015.) $) + isring $p |- ( R e. Ring <-> ( R e. Grp /\ G e. Mnd + /\ A. x e. B A. y e. B A. z e. B + ( ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) + /\ ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) ) ) ) $= + ( vr wcel cv co wceq wa cfv cvv vp vt vb crg cgrp cmnd wral w3a cmgp wsbc + cmulr cplusg cbs fveq2 eqtr4di eleq1d wfn basfn vex funfvex a1i plusgslid + funfni mp2an slotex elv simpl fveq2d mulrslid simpll simpllr simpr simplr + eqidd oveqd oveq123d eqeq12d anbi12d sbcied2 df-ring elrab2 3anass bitr4i + raleqbidv ) FUDNFUENZHUFNZAOZBOZCOZEPZGPZWGWHGPZWGWIGPZEPZQZWGWHEPZWIGPZW + MWHWIGPZEPZQZRZCDUGZBDUGZADUGZRZRWEWFXDUHMOZUISZUFNZWGWHWIUAOZPZUBOZPZWGW + HXKPZWGWIXKPZXIPZQZWGWHXIPZWIXKPZXNWHWIXKPZXIPZQZRZCUCOZUGZBYCUGZAYCUGZUB + XFUKSZUJZUAXFULSZUJZUCXFUMSZUJZRXEMFUEUDXFFQZXHWFYLXDYMXGHUFYMXGFUISHXFFU + IUNJUOUPYMYJXDUCYKDTYKTNZYMUMTUQXFTNYNURMUSYNTXFUMXFUMUTVCVDVAYMYKFUMSDXF + FUMUNIUOYMYCDQZRZYHXDUAYIETYITNZYPYQMXFULTVBVEVFVAYPYIFULSEYPXFFULYMYOVGV + HKUOYPXIEQZRZYFXDUBYGGTYGTNZYSYTMXFUKTVIVEVFVAYSYGFUKSGYSXFFUKYMYOYRVJVHL + UOYSXKGQZRZYEXCAYCDYMYOYRUUAVKZUUBYDXBBYCDUUCUUBYBXACYCDUUCUUBXPWOYAWTUUB + XLWKXOWNUUBWGWGXJWJXKGYSUUAVLZUUBWGVNUUBXIEWHWIYPYRUUAVMZVOVPUUBXMWLXNWMX + IEUUEUUBXKGWGWHUUDVOUUBXKGWGWIUUDVOZVPVQUUBXRWQXTWSUUBXQWPWIWIXKGUUDUUBXI + EWGWHUUEVOUUBWIVNVPUUBXNWMXSWRXIEUUEUUFUUBXKGWHWIUUDVOVPVQVRWDWDWDVSVSVSV + RABCUBMUCUAVTWAWEWFXDWBWC $. + $} + + ${ + $d r G $. $d r x y z R $. + $( A ring is a group. (Contributed by NM, 15-Sep-2011.) $) + ringgrp $p |- ( R e. Ring -> R e. Grp ) $= + ( vx vy vz crg wcel cgrp cmgp cfv cmnd cv cplusg cmulr wceq cbs wral eqid + co wa isring simp1bi ) AEFAGFAHIZJFBKZCKZDKZALIZRAMIZRUCUDUGRUCUEUGRZUFRN + UCUDUFRUEUGRUHUDUEUGRUFRNSDAOIZPCUIPBUIPBCDUIUFAUGUBUIQUBQUFQUGQTUA $. + + ringmgp.g $e |- G = ( mulGrp ` R ) $. + $( A ring is a monoid under multiplication. (Contributed by Mario + Carneiro, 6-Jan-2015.) $) + ringmgp $p |- ( R e. Ring -> G e. Mnd ) $= + ( vx vy vz crg wcel cgrp cmnd cv cplusg cfv co cmulr wceq cbs wral eqid + wa isring simp2bi ) AGHAIHBJHDKZEKZFKZALMZNAOMZNUCUDUGNUCUEUGNZUFNPUCUDUF + NUEUGNUHUDUEUGNUFNPTFAQMZREUIRDUIRDEFUIUFAUGBUISCUFSUGSUAUB $. + + $( A commutative ring is a ring whose multiplication is a commutative + monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) $) + iscrng $p |- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) ) $= + ( vr cv cmgp cfv ccmn wcel ccrg wceq fveq2 eqtr4di eleq1d df-cring elrab2 + crg ) DEZFGZHIBHIDAQJRAKZSBHTSAFGBRAFLCMNDOP $. + + $( A commutative ring's multiplication operation is commutative. + (Contributed by Mario Carneiro, 7-Jan-2015.) $) + crngmgp $p |- ( R e. CRing -> G e. CMnd ) $= + ( ccrg wcel crg ccmn iscrng simprbi ) ADEAFEBGEABCHI $. + $} + + ${ + ringgrpd.1 $e |- ( ph -> R e. Ring ) $. + $( A ring is a group. (Contributed by SN, 16-May-2024.) $) + ringgrpd $p |- ( ph -> R e. Grp ) $= + ( crg wcel cgrp ringgrp syl ) ABDEBFECBGH $. + $} + + $( A ring is a monoid under addition. (Contributed by Mario Carneiro, + 7-Jan-2015.) $) + ringmnd $p |- ( R e. Ring -> R e. Mnd ) $= + ( crg wcel ringgrp grpmndd ) ABCAADE $. + + $( A ring is a magma. (Contributed by AV, 31-Jan-2020.) $) + ringmgm $p |- ( R e. Ring -> R e. Mgm ) $= + ( crg wcel cmnd cmgm ringmnd mndmgm syl ) ABCADCAECAFAGH $. + + $( A commutative ring is a ring. (Contributed by Mario Carneiro, + 7-Jan-2015.) $) + crngring $p |- ( R e. CRing -> R e. Ring ) $= + ( ccrg wcel crg cmgp cfv ccmn eqid iscrng simplbi ) ABCADCAEFZGCAKKHIJ $. + + ${ + crngringd.1 $e |- ( ph -> R e. CRing ) $. + $( A commutative ring is a ring. (Contributed by SN, 16-May-2024.) $) + crngringd $p |- ( ph -> R e. Ring ) $= + ( ccrg wcel crg crngring syl ) ABDEBFECBGH $. + + $( A commutative ring is a group. (Contributed by SN, 16-May-2024.) $) + crnggrpd $p |- ( ph -> R e. Grp ) $= + ( crngringd ringgrpd ) ABABCDE $. + $} + + $( Restricted functionality of the multiplicative group on rings. + (Contributed by Mario Carneiro, 11-Mar-2015.) $) + mgpf $p |- ( mulGrp |` Ring ) : Ring --> Mnd $= + ( va crg cmnd cmgp cres wf wfn cv cfv wcel wral cvv wss fnmgp fnssres mp2an + ssv fvres eqid ringmgp eqeltrd rgen ffnfv mpbir2an ) BCDBEZFUEBGZAHZUEIZCJZ + ABKDLGBLMUFNBQLBDOPUIABUGBJUHUGDIZCUGBDRUGUJUJSTUAUBABCUEUCUD $. + + ${ + $d x y z B $. $d x y z R $. $d x y z .x. $. $d x y z X $. $d x y z Y $. + $d x y z .+ $. $d x y z Z $. + ringdilem.b $e |- B = ( Base ` R ) $. + ringdilem.p $e |- .+ = ( +g ` R ) $. + ringdilem.t $e |- .x. = ( .r ` R ) $. + $( Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) + (Revised by Mario Carneiro, 6-Jan-2015.) $) + ringdilem $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) + -> ( ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) + /\ ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) ) $= + ( vx vy vz wcel wa co wceq cv wral rsp crg w3a cgrp cmgp cmnd eqid isring + cfv simp3bi adantr simpr1 sylc simpr2 simpr3 simpld caovdig caovdirg jca + simprd ) CUANZEANFANGANUBOEFGBPDPEFDPEGDPZBPQEFBPGDPVAFGDPBPQUTKLMEFGABDB + AUTKRZANZLRZANZMRZANZUBZOZVBVDVFBPDPVBVDDPVBVFDPZBPQZVBVDBPVFDPVJVDVFDPBP + QZVIVKVLOZMASZVGVMVIVNLASZVEVNVIVOKASZVCVOUTVPVHUTCUCNCUDUHZUENVPKLMABCDV + QHVQUFIJUGUIUJUTVCVEVGUKVOKATULUTVCVEVGUMVNLATULUTVCVEVGUNVMMATULZUOUPUTK + LMEFGABDBAVIVKVLVRUSUQUR $. + $} + + ${ + ringcl.b $e |- B = ( Base ` R ) $. + ringcl.t $e |- .x. = ( .r ` R ) $. + $( Closure of the multiplication operation of a ring. (Contributed by NM, + 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) + ringcl $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B ) $= + ( crg wcel w3a co cmgp cfv cplusg cbs eqid 3ad2ant1 wb eleq2d mpbid simp2 + cmnd ringmgp mgpbasg simp3 mndcl syl3anc mgpplusgg oveqd eleq12d mpbird ) + BHIZDAIZEAIZJZDECKZAIZDEBLMZNMZKZUROMZIZUOURUBIZDVAIZEVAIZVBULUMVCUNBURUR + PZUCQUOUMVDULUMUNUAULUMUMVDRUNULAVADABURHVFFUDZSQTUOUNVEULUMUNUEULUMUNVER + UNULAVAEVGSQTVAUSURDEVAPUSPUFUGULUMUQVBRUNULUPUTAVAULCUSDEBCURHVFGUHUIVGU + JQUK $. + + $( A commutative ring's multiplication operation is commutative. + (Contributed by Mario Carneiro, 7-Jan-2015.) $) + crngcom $p |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> + ( X .x. Y ) = ( Y .x. X ) ) $= + ( ccrg wcel w3a cmgp cfv cplusg co ccmn wceq eqid 3ad2ant1 eleqtrd oveqd + cbs crngmgp simp2 mgpbasg simp3 cmncom syl3anc mgpplusgg 3eqtr4d ) BHIZDA + IZEAIZJZDEBKLZMLZNZEDUONZDECNEDCNUMUNOIZDUNUALZIEUSIUPUQPUJUKURULBUNUNQZU + BRUMDAUSUJUKULUCUJUKAUSPULABUNHUTFUDRZSUMEAUSUJUKULUEVASUSUOUNDEUSQUOQUFU + GUMCUODEUJUKCUOPULBCUNHUTGUHRZTUMCUOEDVBTUI $. + + $d x y B $. $d x y R $. + $( A commutative ring is a ring whose multiplication is a commutative + monoid. (Contributed by Mario Carneiro, 15-Jun-2015.) $) + iscrng2 $p |- ( R e. CRing <-> ( R e. Ring /\ + A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) ) $= + ( ccrg wcel cvv crg cv co wceq wral wa elex cfv eqid oveqd cmgp ccmn cmnd + adantr iscrng wb ringmgp cplusg iscmn mgpbasg mgpplusgg eqeq12d raleqbidv + cbs anbi2d bitr4id baibd sylan2 pm5.32da bitrid pm5.21nii ) DHIZDJIZDKIZA + LZBLZEMZVFVEEMZNZBCOZACOZPZDHQVDVCVKDKQUDVBVDDUARZUBIZPVCVLDVMVMSZUEVCVDV + NVKVDVCVMUCIZVNVKUFDVMVOUGVCVNVPVKVCVNVPVEVFVMUHRZMZVFVEVQMZNZBVMUNRZOZAW + AOZPVPVKPABWAVQVMWASVQSUIVCVKWCVPVCVJWBACWACDVMJVOFUJZVCVIVTBCWAWDVCVGVRV + HVSVCEVQVEVFDEVMJVOGUKZTVCEVQVFVEWETULUMUMUOUPUQURUSUTVA $. + + $( Associative law for multiplication in a ring. (Contributed by NM, + 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) + ringass $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> + ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) ) $= + ( crg wcel w3a cfv co wceq eqid adantr eleqtrd oveqd eqidd oveq123d cmgp + wa cplusg cmnd cbs ringmgp simpr1 simpr2 simpr3 mndass syl13anc mgpplusgg + mgpbasg 3eqtr4d ) BIJZDAJZEAJZFAJZKZUBZDEBUALZUCLZMZFVBMZDEFVBMZVBMZDECMZ + FCMDEFCMZCMUTVAUDJZDVAUELZJEVJJFVJJVDVFNUOVIUSBVAVAOZUFPUTDAVJUOUPUQURUGU + OAVJNUSABVAIVKGUMPZQUTEAVJUOUPUQURUHVLQUTFAVJUOUPUQURUIVLQVJVBVADEFVJOVBO + UJUKUTVGVCFFCVBUOCVBNUSBCVAIVKHULPZUTCVBDEVMRUTFSTUTDDVHVECVBVMUTDSUTCVBE + FVMRTUN $. + + $d u x B $. $d u x R $. $d u x .x. $. + $( The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) + (Revised by Mario Carneiro, 6-Jan-2015.) $) + ringideu $p |- ( R e. Ring -> + E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) ) $= + ( crg wcel cv co wceq wa wral wreu cfv eqid syl oveqd eqeq1d cmgp ringmgp + cplusg cmnd mndideu mgpbasg mgpplusgg anbi12d raleqbidv reubidv wb reueq1 + cbs bitrd mpbird ) DHIZBJZAJZEKZURLZURUQEKZURLZMZACNZBCOZUQURDUAPZUCPZKZU + RLZURUQVGKZURLZMZAVFUMPZNZBVMOZUPVFUDIVODVFVFQZUBABVMVGVFVMQVGQUERUPVEVNB + COZVOUPVDVNBCUPVCVLACVMCDVFHVPFUFZUPUTVIVBVKUPUSVHURUPEVGUQURDEVFHVPGUGZS + TUPVAVJURUPEVGURUQVSSTUHUIUJUPCVMLVQVOUKVRVNBCVMULRUNUO $. + $} + + ${ + ringdi.b $e |- B = ( Base ` R ) $. + ringdi.p $e |- .+ = ( +g ` R ) $. + ringdi.t $e |- .x. = ( .r ` R ) $. + $( Distributive law for the multiplication operation of a ring + (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) $) + ringdi $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> + ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) ) $= + ( crg wcel w3a wa co wceq ringdilem simpld ) CKLEALFALGALMNEFGBODOEFDOEGD + OZBOPEFBOGDOSFGDOBOPABCDEFGHIJQR $. + + $( Distributive law for the multiplication operation of a ring + (right-distributivity). (Contributed by Steve Rodriguez, + 9-Sep-2007.) $) + ringdir $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> + ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) $= + ( crg wcel w3a wa co wceq ringdilem simprd ) CKLEALFALGALMNEFGBODOEFDOEGD + OZBOPEFBOGDOSFGDOBOPABCDEFGHIJQR $. + $} + + ${ + ringidcl.b $e |- B = ( Base ` R ) $. + ringidcl.u $e |- .1. = ( 1r ` R ) $. + $( The unit element of a ring belongs to the base set of the ring. + (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, + 27-Dec-2014.) $) + ringidcl $p |- ( R e. Ring -> .1. e. B ) $= + ( crg wcel cmgp cfv c0g cbs cmnd eqid ringmgp mndidcl syl mgpbasg 3eltr4d + ringidvalg ) BFGZBHIZJIZUAKIZCATUALGUBUCGBUAUAMZNUCUAUBUCMUBMOPBCUAFUDESA + BUAFUDDQR $. + $} + + ${ + ring0cl.b $e |- B = ( Base ` R ) $. + ring0cl.z $e |- .0. = ( 0g ` R ) $. + $( The zero element of a ring belongs to its base set. (Contributed by + Mario Carneiro, 12-Jan-2014.) $) + ring0cl $p |- ( R e. Ring -> .0. e. B ) $= + ( crg wcel cgrp ringgrp grpidcl syl ) BFGBHGCAGBIABCDEJK $. + $} + + ${ + $d x y B $. $d x y I $. $d x y R $. $d x y .x. $. $d x y .1. $. + rngidm.b $e |- B = ( Base ` R ) $. + rngidm.t $e |- .x. = ( .r ` R ) $. + rngidm.u $e |- .1. = ( 1r ` R ) $. + $( Lemma for ~ ringlidm and ~ ringridm . (Contributed by NM, 15-Sep-2011.) + (Revised by Mario Carneiro, 27-Dec-2014.) $) + ringidmlem $p |- ( ( R e. Ring /\ X e. B ) + -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) ) $= + ( crg wcel wa co wceq cmgp cfv c0g eqid adantr oveq123d eqeq1d cplusg cbs + ringmgp simpr mgpbasg eleqtrd mndlrid syl2an2r mgpplusgg ringidvalg eqidd + cmnd anbi12d mpbird ) BIJZEAJZKZDECLZEMZEDCLZEMZKBNOZPOZEVBUAOZLZEMZEVCVD + LZEMZKZUOVBULJUPEVBUBOZJVIBVBVBQZUCUQEAVJUOUPUDUOAVJMUPABVBIVKFUERUFVJVDV + BEVCVJQVDQVCQUGUHUQUSVFVAVHUQURVEEUQDVCEECVDUOCVDMUPBCVBIVKGUIRZUODVCMUPB + DVBIVKHUJRZUQEUKZSTUQUTVGEUQEEDVCCVDVLVNVMSTUMUN $. + + $( The unit element of a ring is a left multiplicative identity. + (Contributed by NM, 15-Sep-2011.) $) + ringlidm $p |- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X ) $= + ( crg wcel wa co wceq ringidmlem simpld ) BIJEAJKDECLEMEDCLEMABCDEFGHNO + $. + + $( The unit element of a ring is a right multiplicative identity. + (Contributed by NM, 15-Sep-2011.) $) + ringridm $p |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X ) $= + ( crg wcel wa co wceq ringidmlem simprd ) BIJEAJKDECLEMEDCLEMABCDEFGHNO + $. + + $( Properties showing that an element ` I ` is the unity element of a ring. + (Contributed by NM, 7-Aug-2013.) $) + isringid $p |- ( R e. Ring + -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) + <-> .1. = I ) ) $= + ( vy crg wcel cfv co wceq wa wral eqid oveqd eqeq1d cmgp cbs cv wrex wreu + cplusg c0g reurex syl mgpbasg mgpplusgg anbi12d raleqbidv rexeqbidv mpbid + ringideu ismgmid eleq2d ringidvalg 3bitr4d ) CKLZFCUAMZUBMZLZFAUCZVBUFMZN + ZVEOZVEFVFNZVEOZPZAVCQZPVBUGMZFOFBLZFVEDNZVEOZVEFDNZVEOZPZABQZPEFOVAAVCVF + FJVBVMVCRVMRVFRVAJUCZVEDNZVEOZVEWADNZVEOZPZABQZJBUDZWAVEVFNZVEOZVEWAVFNZV + EOZPZAVCQZJVCUDVAWGJBUEWHAJBCDGHUPWGJBUHUIVAWGWNJBVCBCVBKVBRZGUJZVAWFWMAB + VCWPVAWCWJWEWLVAWBWIVEVADVFWAVECDVBKWOHUKZSTVAWDWKVEVADVFVEWAWQSTULUMUNUO + UQVAVNVDVTVLVABVCFWPURVAVSVKABVCWPVAVPVHVRVJVAVOVGVEVADVFFVEWQSTVAVQVIVEV + ADVFVEFWQSTULUMULVAEVMFCEVBKWOIUSTUT $. + $} + + ${ + $d B u $. $d R u $. $d X u $. $d .x. u $. + ringid.b $e |- B = ( Base ` R ) $. + ringid.t $e |- .x. = ( .r ` R ) $. + $( The multiplication operation of a unital ring has (one or more) identity + elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by + Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) $) + ringid $p |- ( ( R e. Ring /\ X e. B ) -> + E. u e. B ( ( u .x. X ) = X /\ ( X .x. u ) = X ) ) $= + ( crg wcel wa cv co wceq cur cfv eqid ringidcl adantr wb eqeq1d rspcedvd + oveq1 oveq2 anbi12d adantl ringidmlem ) CHIZEBIZJZAKZEDLZEMZEUJDLZEMZJZCN + OZEDLZEMZEUPDLZEMZJZAUPBUGUPBIUHBCUPFUPPZQRUJUPMZUOVASUIVCULURUNUTVCUKUQE + UJUPEDUBTVCUMUSEUJUPEDUCTUDUEBCDUPEFGVBUFUA $. + $} + + ${ + $d B x $. $d R x $. $d X x $. $d .x. x $. + ringadd2.b $e |- B = ( Base ` R ) $. + ringadd2.p $e |- .+ = ( +g ` R ) $. + ringadd2.t $e |- .x. = ( .r ` R ) $. + $( A ring element plus itself is two times the element. (Contributed by + Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) + (Revised by AV, 24-Aug-2021.) $) + ringadd2 $p |- ( ( R e. Ring /\ X e. B ) + -> E. x e. B ( X .+ X ) = ( ( x .+ x ) .x. X ) ) $= + ( crg wcel wa cv co wceq wrex ringid oveq12 anidms eqcomd simpll syl13anc + simpr simplr ringdir eqeq2d syl5ibr adantrd reximdva mpd ) DJKZFBKZLZAMZF + ENZFOZFUNENFOZLZABPFFCNZUNUNCNFENZOZABPABDEFGIQUMURVAABUMUNBKZLZUPVAUQUPV + AVCUSUOUOCNZOUPVDUSUPVDUSOUOFUOFCRSTVCUTVDUSVCUKVBVBULUTVDOUKULVBUAUMVBUC + ZVEUKULVBUDBCDEUNUNFGHIUEUBUFUGUHUIUJ $. + + rngo2times.u $e |- .1. = ( 1r ` R ) $. + $( A ring element plus itself is two times the element. "Two" in an + arbitrary unital ring is the sum of the unit with itself. (Contributed + by AV, 24-Aug-2021.) $) + rngo2times $p |- ( ( R e. Ring /\ A e. B ) + -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) ) $= + ( crg wcel wa co ringlidm eqcomd oveq12d wceq simpl ringidcl adantr simpr + ringdir syl13anc eqtr4d ) DKLZABLZMZAACNFAENZUICNZFFCNAENZUHAUIAUICUHUIAB + DEFAGIJOPZULQUHUFFBLZUMUGUKUJRUFUGSUFUMUGBDFGJTUAZUNUFUGUBBCDEFFAGHIUCUDU + E $. + $} + + ${ + ringacl.b $e |- B = ( Base ` R ) $. + ringacl.p $e |- .+ = ( +g ` R ) $. + $( Closure of the addition operation of a ring. (Contributed by Mario + Carneiro, 14-Jan-2014.) $) + ringacl $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B ) $= + ( crg wcel cgrp co ringgrp grpcl syl3an1 ) CHICJIDAIEAIDEBKAICLABCDEFGMN + $. + + $( Commutativity of the additive group of a ring. (Contributed by + Gérard Lang, 4-Dec-2014.) $) + ringcom $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> + ( X .+ Y ) = ( Y .+ X ) ) $= + ( wcel co wceq cfv eqid ringacl syl3anc syl13anc ringdir ringlidm syl2anc + oveq12d grpass crg w3a cur cmulr simp1 ringidcl simp2 simp3 ringdi eqtr3d + syl eqtrd 3eqtr3d cgrp ringgrp 3eqtr4d wb grprcan mpbid 3com23 grplcan ) + CUAHZDAHZEAHZUBZDDEBIZBIZDEDBIZBIZJZVFVHJZVEDDBIZEBIZVFDBIZVGVIVEVMEBIZVN + EBIZJZVMVNJZVEVLEEBIZBIZVFVFBIZVOVPVECUCKZWBBIZDCUDKZIZWCEWDIZBIZWBVFWDIZ + WHBIZVTWAVEWCVFWDIZWGWIVEVBWCAHZVCVDWJWGJVBVCVDUEZVEVBWBAHZWMWKWLVEVBWMWL + ACWBFWBLZUFUKZWOABCWBWBFGMNVBVCVDUGZVBVCVDUHZABCWDWCDEFGWDLZUIOVEVBWMWMVF + AHZWJWIJWLWOWOABCDEFGMZABCWDWBWBVFFGWRPOUJVEWEVLWFVSBVEWEWBDWDIZXABIZVLVE + VBWMWMVCWEXBJWLWOWOWPABCWDWBWBDFGWRPOVEXADXADBVEVBVCXADJWLWPACWDWBDFWRWNQ + RZXCSULVEWFWBEWDIZXDBIZVSVEVBWMWMVDWFXEJWLWOWOWQABCWDWBWBEFGWRPOVEXDEXDEB + VEVBVDXDEJWLWQACWDWBEFWRWNQRZXFSULSVEWHVFWHVFBVEVBWSWHVFJWLWTACWDWBVFFWRW + NQRZXGSUMVECUNHZVLAHZVDVDVOVTJVEVBXHWLCUOUKZVEVBVCVCXIWLWPWPABCDDFGMNZWQW + QABCVLEEFGTOVEXHWSVCVDVPWAJXJWTWPWQABCVFDEFGTOUPVEXHVMAHZVNAHZVDVQVRUQXJV + EVBXIVDXLWLXKWQABCVLEFGMNVEVBWSVCXMWLWTWPABCVFDFGMNWQABCVMVNEFGUROUSVEXHV + CVCVDVMVGJXJWPWPWQABCDDEFGTOVEXHVCVDVCVNVIJXJWPWQWPABCDEDFGTOUMVEXHWSVHAH + ZVCVJVKUQXJWTVBVDVCXNABCEDFGMUTWPABCVFVHDFGVAOUS $. + $} + + ${ + $d x y R $. + $( A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.) $) + ringabl $p |- ( R e. Ring -> R e. Abel ) $= + ( vx vy crg wcel cbs cfv cplusg eqidd ringgrp cv eqid ringcom isabld ) AD + EZBCAFGZAHGZAOPIOQIAJPQABKCKPLQLMN $. + $} + + $( A ring is a commutative monoid. (Contributed by Mario Carneiro, + 7-Jan-2015.) $) + ringcmn $p |- ( R e. Ring -> R e. CMnd ) $= + ( crg wcel cabl ccmn ringabl ablcmn syl ) ABCADCAECAFAGH $. + + ${ + $d u v w x y B $. $d u v w x y K $. $d u v w x y ph $. $d u v w x y L $. + ringpropd.1 $e |- ( ph -> B = ( Base ` K ) ) $. + ringpropd.2 $e |- ( ph -> B = ( Base ` L ) ) $. + ringpropd.3 $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> + ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) ) $. + ringpropd.4 $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> + ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) ) $. + $( If two structures have the same group components (properties), one is a + ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) + (Revised by Mario Carneiro, 6-Jan-2015.) $) + ringpropd $p |- ( ph -> ( K e. Ring <-> L e. Ring ) ) $= + ( vu vv vw wcel cfv co wceq wa wral eqid cgrp cmgp cmnd cv cplusg cbs w3a + cmulr crg simpll simprll simplrl simprlr ad2antrr eleqtrd simprr eleqtrrd + grpcld oveqrspc2v syl12anc oveq2d eqtrd simplrr adantr cvv simprl mgpbasg + wb elexd syl mndcl syl3anc mgpplusgg oveqd 3eltr4d oveq12d eqeq12d oveq1d + anbi12d anassrs ralbidva 2ralbidva raleqdv raleqbidv 3bitr3d df-3an simpr + pm5.32da 3bitr4g grppropd biimpa adantlr mpbid mndpropd anbi1d isring + bitrd ) AEUANZEUBOZUCNZKUDZLUDZMUDZEUEOZPZEUHOZPZXAXBXFPZXAXCXFPZXDPZQZXA + XBXDPZXCXFPZXIXBXCXFPZXDPZQZRZMEUFOZSZLXRSZKXRSZUGZFUANZFUBOZUCNZXAXBXCFU + EOZPZFUHOZPZXAXBYHPZXAXCYHPZYFPZQZXAXBYFPZXCYHPZYKXBXCYHPZYFPZQZRZMFUFOZS + ZLYTSZKYTSZUGZEUINFUINAYBWRWTUUCUGZUUDAWRWTRZYARUUFUUCRZYBUUEAUUFYAUUCAUU + FRZXQMDSZLDSZKDSYSMDSZLDSZKDSYAUUCUUHUUIUUKKLDDUUHXADNZXBDNZRZRXQYSMDUUHU + UOXCDNZXQYSVHUUHUUOUUPRZRZXKYMXPYRUURXGYIXJYLUURXGXAXEYHPZYIUURAUUMXEDNXG + UUSQAUUFUUQUJZUUHUUMUUNUUPUKZUURXEXRDUURXRXDEXBXCXRTZXDTZAWRWTUUQULZUURXB + DXRUUHUUMUUNUUPUMZADXRQZUUFUUQGUNZUOZUURXCDXRUUHUUOUUPUPZUVGUOURUVGUQABCD + DXFYHXAXEJUSUTUURXEYGXAYHUURAUUNUUPXEYGQUUTUVEUVIABCDDXDYFXBXCIUSUTVAVBUU + RXJXHXIYFPZYLUURAXHDNXIDNZXJUVJQUUTUURXAXBWSUEOZPZWSUFOZXHDUURWTXAUVNNZXB + UVNNZUVMUVNNAWRWTUUQVCZUURXADUVNUVAUUHDUVNQUUQUUHDXRUVNAUVFUUFGVDZUUHEVEN + ZXRUVNQZUUHEUAAWRWTVFVIZXREWSVEWSTZUVBVGZVJVBVDZUOZUURXBDUVNUVEUWDUOZUVNU + VLWSXAXBUVNTZUVLTZVKVLUURXFUVLXAXBUUHXFUVLQZUUQUUHUVSUWIUWAEXFWSVEUWBXFTZ + VMZVJVDZVNUWDVOUURXAXCUVLPZUVNXIDUURWTUVOXCUVNNZUWMUVNNUVQUWEUURXCDUVNUVI + UWDUOZUVNUVLWSXAXCUWGUWHVKVLUURXFUVLXAXCUWLVNUWDVOZABCDDXDYFXHXIIUSUTUURX + HYJXIYKYFUURAUUMUUNXHYJQUUTUVAUVEABCDDXFYHXAXBJUSUTUURAUUMUUPXIYKQUUTUVAU + VIABCDDXFYHXAXCJUSUTZVPVBVQUURXMYOXOYQUURXMXLXCYHPZYOUURAXLDNUUPXMUWRQUUT + UURXLXRDUURXRXDEXAXBUVBUVCUVDUURXADXRUVAUVGUOUVHURUVGUQUVIABCDDXFYHXLXCJU + SUTUURXLYNXCYHUURAUUMUUNXLYNQUUTUVAUVEABCDDXDYFXAXBIUSUTVRVBUURXOXIXNYFPZ + YQUURAUVKXNDNXOUWSQUUTUWPUURXBXCUVLPZUVNXNDUURWTUVPUWNUWTUVNNUVQUWFUWOUVN + UVLWSXBXCUWGUWHVKVLUURXFUVLXBXCUWLVNUWDVOABCDDXDYFXIXNIUSUTUURXIYKXNYPYFU + WQUURAUUNUUPXNYPQUUTUVEUVIABCDDXFYHXBXCJUSUTVPVBVQVSVTWAWBUUHUUJXTKDXRUVR + UUHUUIXSLDXRUVRUUHXQMDXRUVRWCWDWDUUHUULUUBKDYTADYTQZUUFHVDZUUHUUKUUALDYTU + XBUUHYSMDYTUXBWCWDWDWEWHWRWTYAWFWRWTUUCWFZWIAUUGYCYERZUUCRUUEUUDAUUFUXDUU + CAUUFWRYERUXDAWRWTYEAWRRZBCDWSYDUXEDXRUVNAUVFWRGVDUXEUVSUVTUXEEUAAWRWGVIZ + UWCVJVBUXEDYTYDUFOZAUXAWRHVDUXEFVENZYTUXGQUXEFUAAWRYCABCDEFGHIWJZWKVIZYTF + YDVEYDTZYTTZVGVJVBUXEBUDZDNCUDZDNRZRUXMUXNXFPZUXMUXNYHPZQZUXMUXNUVLPZUXMU + XNYDUEOZPZQZAUXOUXRWRJWLUXEUXRUYBVHUXOUXEUXPUXSUXQUYAUXEUVSUXPUXSQUXFUVSX + FUVLUXMUXNUWKVNVJUXEUXHUXQUYAQUXJUXHYHUXTUXMUXNFYHYDVEUXKYHTZVMVNVJVQVDWM + WNWHAWRYCYEUXIWOWQWOUXCYCYEUUCWFWIWQKLMXRXDEXFWSUVBUWBUVCUWJWPKLMYTYFFYHY + DUXLUXKYFTUYCWPWI $. + + $( If two structures have the same group components (properties), one is a + commutative ring iff the other one is. (Contributed by Mario Carneiro, + 8-Feb-2015.) $) + crngpropd $p |- ( ph -> ( K e. CRing <-> L e. CRing ) ) $= + ( crg wcel cmgp cfv ccmn wa cbs eqid wceq co ccrg mgpbasg sylan9eq adantr + ringpropd biimpa syl eqtrd cplusg adantlr mgpplusgg adantl oveqdr 3eqtr3d + cv cmulr cmnpropd pm5.32da anbi1d bitrd iscrng 3bitr4g ) AEKLZEMNZOLZPZFK + LZFMNZOLZPZEUALFUALAVFVCVIPVJAVCVEVIAVCPZBCDVDVHAVCDEQNZVDQNGVLEVDKVDRZVL + RUBUCVKDFQNZVHQNZADVNSVCHUDVKVGVNVOSAVCVGABCDEFGHIJUEZUFZVNFVHKVHRZVNRUBU + GUHVKBUOZDLCUOZDLPZPVSVTEUPNZTZVSVTFUPNZTZVSVTVDUINZTVSVTVHUINZTAWAWCWESV + CJUJVKWABCWBWFVCWBWFSAEWBVDKVMWBRUKULUMVKWABCWDWGVKVGWDWGSVQFWDVHKVRWDRUK + UGUMUNUQURAVCVGVIVPUSUTEVDVMVAFVHVRVAVB $. + $} + + ${ + $d x y K $. $d x y L $. + ringprop.b $e |- ( Base ` K ) = ( Base ` L ) $. + ringprop.p $e |- ( +g ` K ) = ( +g ` L ) $. + ringprop.m $e |- ( .r ` K ) = ( .r ` L ) $. + $( If two structures have the same ring components (properties), one is a + ring iff the other one is. (Contributed by Mario Carneiro, + 11-Oct-2013.) $) + ringprop $p |- ( K e. Ring <-> L e. Ring ) $= + ( vx vy crg wcel wtru cbs cfv wceq a1i cv cplusg co wa oveqi cmulr eqidd + wb ringpropd mptru ) AHIBHIUBJFGAKLZABJUEUAUEBKLMJCNFOZGOZAPLZQUFUGBPLZQM + JUFUEIUGUEIRRZUHUIUFUGDSNUFUGATLZQUFUGBTLZQMUJUKULUFUGESNUCUD $. + $} + + ${ + $d x .1. $. $d x y z B $. $d x y z ph $. $d x y z R $. + isringd.b $e |- ( ph -> B = ( Base ` R ) ) $. + isringd.p $e |- ( ph -> .+ = ( +g ` R ) ) $. + isringd.t $e |- ( ph -> .x. = ( .r ` R ) ) $. + isringd.g $e |- ( ph -> R e. Grp ) $. + isringd.c $e |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B ) $. + isringd.a $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> + ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) ) $. + isringd.d $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> + ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) ) $. + isringd.e $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> + ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) ) $. + isringd.u $e |- ( ph -> .1. e. B ) $. + isringd.i $e |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x ) $. + isringd.h $e |- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x ) $. + $( Properties that determine a ring. (Contributed by NM, 2-Aug-2013.) $) + isringd $p |- ( ph -> R e. Ring ) $= + ( cgrp wcel cmgp cfv cmnd cv cplusg co cmulr wceq wa cbs wral crg mgpbasg + eqid syl eqtrd mgpplusgg ismndd w3a eleq2d 3anbi123d biimpar adantr eqidd + oveqdr oveq123d 3eqtr3d jca syldan ralrimivvva isring syl3anbrc ) AGUAUBZ + GUCUDZUEUBBUFZCUFZDUFZGUGUDZUHZGUIUDZUHZVQVRWBUHZVQVSWBUHZVTUHZUJZVQVRVTU + HZVSWBUHZWEVRVSWBUHZVTUHZUJZUKZDGULUDZUMCWNUMBWNUMGUNUBMABCDEHVPIAEWNVPUL + UDZJAVOWNWOUJMWNGVPUAVPUPZWNUPZUOUQURAHWBVPUGUDZLAVOWBWRUJMGWBVPUAWPWBUPZ + USUQURNORSTUTAWMBCDWNWNWNAVQWNUBZVRWNUBZVSWNUBZVAZVQEUBZVREUBZVSEUBZVAZWM + AXGXCAXDWTXEXAXFXBAEWNVQJVBAEWNVRJVBAEWNVSJVBVCVDAXGUKZWGWLXHVQVRVSFUHZHU + HVQVRHUHZVQVSHUHZFUHWCWFPXHVQVQXIWAHWBAHWBUJXGLVEZXHVQVFAXGCDFVTKVGVHXHXJ + WDXKWEFVTAFVTUJXGKVEZAXGBCHWBLVGAXGBDHWBLVGZVHVIXHVQVRFUHZVSHUHXKVRVSHUHZ + FUHWIWKQXHXOWHVSVSHWBXLAXGBCFVTKVGXHVSVFVHXHXKWEXPWJFVTXMXNAXGCDHWBLVGVHV + IVJVKVLBCDWNVTGWBVPWQWPVTUPWSVMVN $. + + iscrngd.c $e |- ( ( ph /\ x e. B /\ y e. B ) -> + ( x .x. y ) = ( y .x. x ) ) $. + $( Properties that determine a commutative ring. (Contributed by Mario + Carneiro, 7-Jan-2015.) $) + iscrngd $p |- ( ph -> R e. CRing ) $= + ( crg wcel cmgp cfv ccmn ccrg isringd cbs wceq mgpbasg eqtrd cmulr cplusg + eqid syl mgpplusgg ismndd iscmnd iscrng sylanbrc ) AGUBUCZGUDUEZUFUCGUGUC + ABCDEFGHIJKLMNOPQRSTUHZABCEHVCAEGUIUEZVCUIUEZJAVBVEVFUJVDVEGVCUBVCUOZVEUO + UKUPULZAHGUMUEZVCUNUEZLAVBVIVJUJVDGVIVCUBVGVIUOUQUPULZABCDEHVCIVHVKNORSTU + RUAUSGVCVGUTVA $. + $} + + ${ + rngz.b $e |- B = ( Base ` R ) $. + rngz.t $e |- .x. = ( .r ` R ) $. + rngz.z $e |- .0. = ( 0g ` R ) $. + $( The zero of a unital ring is a left-absorbing element. (Contributed by + FL, 31-Aug-2009.) $) + ringlz $p |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. ) $= + ( crg wcel wa co cplusg cfv wceq cgrp ringgrp grpidcl eqid adantr w3a syl + grplid syl2anc2 oveq1d simpr ringdir syldan ringcl syl3anc grprid syl2anc + 3jca simpl eqcomd 3eqtr3d wb grplcan syl13anc mpbid ) BIJZDAJZKZEDCLZVDBM + NZLZVDEVELZOZVDEOZVCEEVELZDCLZVDVFVGVCVJEDCVAVJEOZVBVABPJZEAJZVLBQZABEFHR + ZAVEBEEFVESZHUCUDTUEVAVBVNVNVBUAVKVFOVCVNVNVBVAVNVBVAVMVNVOVPUBTZVRVAVBUF + ZUMAVEBCEEDFVQGUGUHVCVMVDAJZVDVGOVAVMVBVOTZVCVAVNVBVTVAVBUNVRVSABCEDFGUIU + JZVMVTKVGVDAVEBVDEFVQHUKUOULUPVCVMVTVNVTVHVIUQWAWBVRWBAVEBVDEVDFVQURUSUT + $. + + $( The zero of a unital ring is a right-absorbing element. (Contributed by + FL, 31-Aug-2009.) $) + ringrz $p |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. ) $= + ( crg wcel wa co cplusg cfv wceq cgrp ringgrp grpidcl grplid adantr simpr + eqid syl2anc2 oveq2d w3a 3jca ringdi syldan ringcl mpd3an3 eqcomd syl2anc + syl 3eqtr3d wb grprcan syl13anc mpbid ) BIJZDAJZKZDECLZVBBMNZLZEVBVCLZOZV + BEOZVADEEVCLZCLZVBVDVEVAVHEDCUSVHEOZUTUSBPJZEAJZVJBQZABEFHRZAVCBEEFVCUBZH + SUCTUDUSUTUTVLVLUEVIVDOVAUTVLVLUSUTUAUSVLUTUSVKVLVMVNUMTZVPUFAVCBCDEEFVOG + UGUHVAVKVBAJZVBVEOUSVKUTVMTZUSUTVLVQVPABCDEFGUIUJZVKVQKVEVBAVCBVBEFVOHSUK + ULUNVAVKVQVLVQVFVGUOVRVSVPVSAVCBVBEVBFVOUPUQUR $. + $} + + ${ + $d x y z R $. + $( Any ring is also a semiring. (Contributed by Thierry Arnoux, + 1-Apr-2018.) $) + ringsrg $p |- ( R e. Ring -> R e. SRing ) $= + ( vx vy vz crg wcel ccmn cmgp cfv cmnd cv cplusg co cmulr wceq wa cbs c0g + wral eqid csrg ringcmn ringmgp isring simp3bi ringlz ringrz jca ralrimiva + cgrp r19.26 sylanbrc issrg syl3anbrc ) AEFZAGFAHIZJFZBKZCKZDKZALIZMANIZMU + RUSVBMURUTVBMZVAMOURUSVAMUTVBMVCUSUTVBMVAMOPDAQIZSCVDSZARIZURVBMVFOZURVFV + BMVFOZPZPBVDSZAUAFAUBAUPUPTZUCUOVEBVDSZVIBVDSVJUOAUJFUQVLBCDVDVAAVBUPVDTZ + VKVATZVBTZUDUEUOVIBVDUOURVDFPVGVHVDAVBURVFVMVOVFTZUFVDAVBURVFVMVOVPUGUHUI + VEVIBVDUKULBCDVDVAAVBUPVFVMVKVNVOVPUMUN $. + $} + + ${ + ring1eq0.b $e |- B = ( Base ` R ) $. + ring1eq0.u $e |- .1. = ( 1r ` R ) $. + ring1eq0.z $e |- .0. = ( 0g ` R ) $. + $( If one and zero are equal, then any two elements of a ring are equal. + Alternately, every ring has one distinct from zero except the zero ring + containing the single element ` { 0 } ` . (Contributed by Mario + Carneiro, 10-Sep-2014.) $) + ring1eq0 $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> + ( .1. = .0. -> X = Y ) ) $= + ( crg wcel w3a wceq wa co oveq1d ringlz syl2anc eqtr4d ringlidm cmulr cfv + simpr simpl1 simpl2 eqid simpl3 3eqtr3d ex ) BJKZDAKZEAKZLZCFMZDEMUMUNNZC + DBUAUBZOZCEUPOZDEUOUQFDUPOZURUOCFDUPUMUNUCZPUOURFEUPOZUSUOCFEUPUTPUOUSFVA + UOUJUKUSFMUJUKULUNUDZUJUKULUNUEZABUPDFGUPUFZIQRUOUJULVAFMVBUJUKULUNUGZABU + PEFGVDIQRSSSUOUJUKUQDMVBVCABUPCDGVDHTRUOUJULUREMVBVEABUPCEGVDHTRUHUI $. + $} + + ${ + $d X a $. $d .0. a $. $d .1. a $. $d .x. a $. $d ph a $. + ringinvnzdiv.b $e |- B = ( Base ` R ) $. + ringinvnzdiv.t $e |- .x. = ( .r ` R ) $. + ringinvnzdiv.u $e |- .1. = ( 1r ` R ) $. + ringinvnzdiv.z $e |- .0. = ( 0g ` R ) $. + ringinvnzdiv.r $e |- ( ph -> R e. Ring ) $. + ringinvnzdiv.x $e |- ( ph -> X e. B ) $. + ringinvnzdiv.a $e |- ( ph -> E. a e. B ( a .x. X ) = .1. ) $. + $( In a unitary ring, a left invertible element is different from zero iff + ` .1. =/= .0. ` . (Contributed by FL, 18-Apr-2010.) (Revised by AV, + 24-Aug-2021.) $) + ringinvnz1ne0 $p |- ( ph -> ( X =/= .0. <-> .1. =/= .0. ) ) $= + ( co wceq wcel wa oveq2 cv wb wi ringrz sylan eqeq12 biimpd ex mpan9 syl5 + crg ringridm eqeq12d syl2anc ad2antrr impbid r19.29a necon3bid ) AFGEGAHU + AZFDPZEQZFGQZEGQZUBHBAUSBRZSZVASZVBVCVBUTUSGDPZQZVFVCFGUSDTVEVGGQZVAVHVCU + CZACUKRZVDVIMBCDUSGIJLUDUEVAVIVJVAVISVHVCUTEVGGUFUGUHUIUJVCFEDPZFGDPZQZVF + VBEGFDTAVNVBUCZVDVAAVKFBRZVOMNVKVPSZVNVBVQVLFVMGBCDEFIJKULBCDFGIJLUDUMUGU + NUOUJUPOUQUR $. + + $d Y a $. + ringinvnzdiv.y $e |- ( ph -> Y e. B ) $. + $( In a unitary ring, a left invertible element is not a zero divisor. + (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, + 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) $) + ringinvnzdiv $p |- ( ph -> ( ( X .x. Y ) = .0. <-> Y = .0. ) ) $= + ( co wceq adantr cv wrex wi wcel wa crg ringlidm syl2anc eqcomd ad3antrrr + oveq1 eqcoms adantl w3a simpr 3jca jca ringass syl eqtrd ringrz sylan9eqr + oveq2 sylan 3eqtrd exp31 rexlimdva mpd ex impbid ) AFGDRZHSZGHSZAIUAZFDRZ + ESZIBUBVLVMUCZPAVPVQIBAVNBUDZUEZVPVLVMVSVPUEZVLUEGEGDRZVNVKDRZHAGWASVRVPV + LAWAGACUFUDZGBUDZWAGSNQBCDEGJKLUGUHUIUJVTWAWBSVLVTWAVOGDRZWBVPWAWESZVSWFE + VOEVOGDUKULUMVTWCVRFBUDZWDUNZUEZWEWBSVSWIVPVSWCWHAWCVRNTVSVRWGWDAVRUOAWGV + ROTAWDVRQTUPUQTBCDVNFGJKURUSUTTVLVTWBVNHDRZHVKHVNDVCVSWJHSZVPAWCVRWKNBCDV + NHJKMVAVDTVBVEVFVGVHAVMVLVMAVKFHDRZHGHFDVCAWCWGWLHSNOBCDFHJKMVAUHVBVIVJ + $. + $} + + ${ + ringnegl.b $e |- B = ( Base ` R ) $. + ringnegl.t $e |- .x. = ( .r ` R ) $. + ringnegl.u $e |- .1. = ( 1r ` R ) $. + ringnegl.n $e |- N = ( invg ` R ) $. + ringnegl.r $e |- ( ph -> R e. Ring ) $. + ringnegl.x $e |- ( ph -> X e. B ) $. + $( Negation in a ring is the same as left multiplication by -1. + (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, + 2-Jul-2014.) $) + ringnegl $p |- ( ph -> ( ( N ` .1. ) .x. X ) = ( N ` X ) ) $= + ( cfv co wceq wcel syl syl2anc eqid cplusg c0g crg ringidcl cgrp grpinvcl + ringgrp ringdir syl13anc grprinv oveq1d ringlz eqtrd ringlidm 3eqtr3rd wb + ringcl syl3anc grpinvid1 mpbird eqcomd ) AGFNZEFNZGDOZAVBVDPZGVDCUANZOZCU + BNZPZAEVCVFOZGDOZEGDOZVDVFOZVHVGACUCQZEBQZVCBQZGBQZVKVMPLAVNVOLBCEHJUDRZA + CUEQZVOVPAVNVSLCUGRZVRBCFEHKUFSZMBVFCDEVCGHVFTZIUHUIAVKVHGDOZVHAVJVHGDAVS + VOVJVHPVTVRBVFCFEVHHWBVHTZKUJSUKAVNVQWCVHPLMBCDGVHHIWDULSUMAVLGVDVFAVNVQV + LGPLMBCDEGHIJUNSUKUOAVSVQVDBQZVEVIUPVTMAVNVPVQWELWAMBCDVCGHIUQURBVFCFGVDV + HHWBWDKUSURUTVA $. + + $( Negation in a ring is the same as right multiplication by -1. + (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, + 2-Jul-2014.) $) + rngnegr $p |- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) ) $= + ( cfv co wceq wcel syl syl2anc eqid cplusg c0g crg cgrp ringidcl grpinvcl + ringgrp ringdi syl13anc grplinv oveq2d ringrz ringridm 3eqtr3rd wb ringcl + eqtrd syl3anc grpinvid2 mpbird eqcomd ) AGFNZGEFNZDOZAVBVDPZVDGCUANZOZCUB + NZPZAGVCEVFOZDOZVDGEDOZVFOZVHVGACUCQZGBQZVCBQZEBQZVKVMPLMACUDQZVQVPAVNVRL + CUGRZAVNVQLBCEHJUERZBCFEHKUFSZVTBVFCDGVCEHVFTZIUHUIAVKGVHDOZVHAVJVHGDAVRV + QVJVHPVSVTBVFCFEVHHWBVHTZKUJSUKAVNVOWCVHPLMBCDGVHHIWDULSUQAVLGVDVFAVNVOVL + GPLMBCDEGHIJUMSUKUNAVRVOVDBQZVEVIUOVSMAVNVOVPWELMWABCDGVCHIUPURBVFCFGVDVH + HWBWDKUSURUTVA $. + $} + + ${ + ringneglmul.b $e |- B = ( Base ` R ) $. + ringneglmul.t $e |- .x. = ( .r ` R ) $. + ringneglmul.n $e |- N = ( invg ` R ) $. + ringneglmul.r $e |- ( ph -> R e. Ring ) $. + ringneglmul.x $e |- ( ph -> X e. B ) $. + ringneglmul.y $e |- ( ph -> Y e. B ) $. + $( Negation of a product in a ring. ( ~ mulneg1 analog.) (Contributed by + Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) + ringmneg1 $p |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) ) $= + ( cur cfv co crg wcel syl ringnegl wceq ringgrp ringidcl grpinvcl syl2anc + cgrp eqid ringass syl13anc oveq1d ringcl syl3anc 3eqtr3d ) ACNOZEOZFDPZGD + PZUOFGDPZDPZFEOZGDPUREOACQRZUOBRZFBRZGBRZUQUSUAKACUFRZUNBRZVBAVAVEKCUBSAV + AVFKBCUNHUNUGZUCSBCEUNHJUDUELMBCDUOFGHIUHUIAUPUTGDABCDUNEFHIVGJKLTUJABCDU + NEURHIVGJKAVAVCVDURBRKLMBCDFGHIUKULTUM $. + + $( Negation of a product in a ring. ( ~ mulneg2 analog.) (Contributed by + Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) + ringmneg2 $p |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) ) $= + ( co cur cfv crg wcel syl rngnegr wceq cgrp ringgrp eqid ringidcl syl2anc + grpinvcl ringass syl13anc ringcl syl3anc oveq2d 3eqtr3rd ) AFGDNZCOPZEPZD + NZFGUPDNZDNZUNEPFGEPZDNACQRZFBRZGBRZUPBRZUQUSUAKLMACUBRZUOBRZVDAVAVEKCUCS + AVAVFKBCUOHUOUDZUESBCEUOHJUGUFBCDFGUPHIUHUIABCDUOEUNHIVGJKAVAVBVCUNBRKLMB + CDFGHIUJUKTAURUTFDABCDUOEGHIVGJKMTULUM $. + + $( Double negation of a product in a ring. ( ~ mul2neg analog.) + (Contributed by Mario Carneiro, 4-Dec-2014.) $) + ringm2neg $p |- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) ) $= + ( cfv co cgrp wcel crg ringgrp syl2anc syl grpinvcl ringmneg1 fveq2d wceq + ringmneg2 ringcl syl3anc grpinvinv 3eqtrd ) AFENGENZDOFUKDOZENFGDOZENZENZ + UMABCDEFUKHIJKLACPQZGBQZUKBQACRQZUPKCSUAZMBCEGHJUBTUCAULUNEABCDEFGHIJKLMU + FUDAUPUMBQZUOUMUEUSAURFBQUQUTKLMBCDFGHIUGUHBCEUMHJUITUJ $. + $} + + ${ + ringsubdi.b $e |- B = ( Base ` R ) $. + ringsubdi.t $e |- .x. = ( .r ` R ) $. + ringsubdi.m $e |- .- = ( -g ` R ) $. + ringsubdi.r $e |- ( ph -> R e. Ring ) $. + ringsubdi.x $e |- ( ph -> X e. B ) $. + ringsubdi.y $e |- ( ph -> Y e. B ) $. + ringsubdi.z $e |- ( ph -> Z e. B ) $. + $( Ring multiplication distributes over subtraction. ( ~ subdi analog.) + (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, + 2-Jul-2014.) $) + ringsubdi $p |- ( ph -> + ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) ) $= + ( cfv co wcel wceq syl2anc cminusg cplusg crg ringgrp syl grpinvcl ringdi + cgrp syl13anc ringmneg2 oveq2d eqtrd grpsubval ringcl syl3anc 3eqtr4d + eqid ) AFGHCUAPZPZCUBPZQZDQZFGDQZFHDQZURPZUTQZFGHEQZDQVCVDEQZAVBVCFUSDQZU + TQZVFACUCRZFBRZGBRZUSBRZVBVJSLMNACUHRZHBRZVNAVKVOLCUDUEOBCURHIURUQZUFTBUT + CDFGUSIUTUQZJUGUIAVIVEVCUTABCDURFHIJVQLMOUJUKULAVGVAFDAVMVPVGVASNOBUTCURE + GHIVRVQKUMTUKAVCBRZVDBRZVHVFSAVKVLVMVSLMNBCDFGIJUNUOAVKVLVPVTLMOBCDFHIJUN + UOBUTCUREVCVDIVRVQKUMTUP $. + + $( Ring multiplication distributes over subtraction. ( ~ subdir analog.) + (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, + 2-Jul-2014.) $) + rngsubdir $p |- ( ph -> + ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) ) $= + ( cfv co wcel wceq syl2anc cminusg crg cgrp ringgrp eqid grpinvcl ringdir + cplusg syl ringmneg1 oveq2d eqtrd grpsubval oveq1d ringcl syl3anc 3eqtr4d + syl13anc ) AFGCUAPZPZCUHPZQZHDQZFHDQZGHDQZUSPZVAQZFGEQZHDQVDVEEQZAVCVDUTH + DQZVAQZVGACUBRZFBRZUTBRZHBRZVCVKSLMACUCRZGBRZVNAVLVPLCUDUINBCUSGIUSUEZUFT + OBVACDFUTHIVAUEZJUGURAVJVFVDVAABCDUSGHIJVRLNOUJUKULAVHVBHDAVMVQVHVBSMNBVA + CUSEFGIVSVRKUMTUNAVDBRZVEBRZVIVGSAVLVMVOVTLMOBCDFHIJUOUPAVLVQVOWALNOBCDGH + IJUOUPBVACUSEVDVEIVSVRKUMTUQ $. + $} + + ${ + $d x y B $. $d x y R $. $d x y .x. $. $d x y .X. $. $d x y X $. + $d x N $. $d x y Y $. + mulgass2.b $e |- B = ( Base ` R ) $. + mulgass2.m $e |- .x. = ( .g ` R ) $. + mulgass2.t $e |- .X. = ( .r ` R ) $. + $( An associative property between group multiple and ring multiplication. + (Contributed by Mario Carneiro, 14-Jun-2015.) $) + mulgass2 $p |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> + ( ( N .x. X ) .X. Y ) = ( N .x. ( X .X. Y ) ) ) $= + ( wcel co wceq cc0 oveq1 oveq1d eqeq12d cfv adantr syl3anc vx vy cz wi cv + crg cneg caddc w3a c0g eqid ringlz 3adant3 simp3 mulg0 syl ringcl 3eqtr4d + c1 3com23 cn0 wa cplusg cgrp simpl1 ringgrp adantl mulgp1 3ad2ant1 mulgcl + nn0z simpl2 ringdir syl13anc eqtrd syl5ibr ex cminusg fveq2 nnz ringmneg1 + cn mulgneg zindd 3exp com24 3imp2 ) BUFKZEUCKZFAKZGAKZEFCLZGDLZEFGDLZCLZM + ZWHWKWJWIWPWHWKWJWIWPUDUAUEZFCLZGDLZWQWNCLZMNFCLZGDLZNWNCLZMUBUEZFCLZGDLZ + XDWNCLZMZXDUGZFCLZGDLZXIWNCLZMZXDUSUHLZFCLZGDLZXNWNCLZMZWPWHWKWJUIZUAUBEW + QNMZWSXBWTXCXTWRXAGDWQNFCOPWQNWNCOQWQXDMZWSXFWTXGYAWRXEGDWQXDFCOPWQXDWNCO + QWQXNMZWSXPWTXQYBWRXOGDWQXNFCOPWQXNWNCOQWQXIMZWSXKWTXLYCWRXJGDWQXIFCOPWQX + IWNCOQWQEMZWSWMWTWOYDWRWLGDWQEFCOPWQEWNCOQXSBUJRZGDLZYEXBXCWHWKYFYEMWJABD + GYEHJYEUKZULUMXSXAYEGDXSWJXAYEMWHWKWJUNZACBFYEHYGIUOUPPXSWNAKZXCYEMWHWJWK + YIABDFGHJUQUTZACBWNYEHYGIUOUPURXSXDVAKZXHXRUDXHXRXSYKVBZXFWNBVCRZLZXGWNYM + LZMXFXGWNYMOYLXPYNXQYOYLXPXEFYMLZGDLZYNYLXOYPGDYLBVDKZXDUCKZWJXOYPMYLWHYR + WHWKWJYKVEZBVFZUPZYKYSXSXDVKVGZXSWJYKYHSZAYMCBXDFHIYMUKZVHTPYLWHXEAKZWJWK + YQYNMYTYLYRYSWJUUFXSYRYKWHWKYRWJUUAVIZSUUCUUDACBXDFHIVJZTUUDWHWKWJYKVLAYM + BDXEFGHUUEJVMVNVOYLYRYSYIXQYOMUUBUUCXSYIYKYJSAYMCBXDWNHIUUEVHTQVPVQXSXDWB + KZXHXMUDXHXMXSUUIVBZXFBVRRZRZXGUUKRZMXFXGUUKVSUUJXKUULXLUUMUUJXKXEUUKRZGD + LUULUUJXJUUNGDUUJYRYSWJXJUUNMXSYRUUIUUGSZUUIYSXSXDVTVGZXSWJUUIYHSZACBUUKX + DFHIUUKUKZWCTPUUJABDUUKXEGHJUURWHWKWJUUIVEUUJYRYSWJUUFUUOUUPUUQUUHTWHWKWJ + UUIVLWAVOUUJYRYSYIXLUUMMUUOUUPXSYIUUIYJSACBUUKXDWNHIUURWCTQVPVQWDWEWFWG + $. + $} + + ${ + $d Z a b c x y $. $d M a b c x y $. $d V a b c x y $. + ring1.m $e |- M = { <. ( Base ` ndx ) , { Z } >. , + <. ( +g ` ndx ) , { <. <. Z , Z >. , Z >. } >. , + <. ( .r ` ndx ) , { <. <. Z , Z >. , Z >. } >. } $. + $( The (smallest) structure representing a _zero ring_. (Contributed by + AV, 28-Apr-2019.) $) + ring1 $p |- ( Z e. V -> M e. Ring ) $= + ( va vb vc wcel cfv cv cop co wceq wa wral cplusg cvv eqid cn eqeq12d csn + vx cgrp cmgp cmnd w3a crg cnx cbs cpr snexg opexg anidms mpancom rngbaseg + vy syl syl3anc opeq2d rngplusgg preq12d grp1 eqeltrrd cmulr ctp basendxnn + sylancr cslot plusgslid simpri mulrslid tpexg eqeltrid grppropstrg mpbird + mnd1 eqidd grpbaseg syl2anc mgpbasg 3eqtr3rd rngmulrg grpplusgg mgpplusgg + wb oveqdr mndpropd df-ov fvsng eqtrid oveq2d oveq12d eqtr4d oveq1 anbi12d + oveq1d 2ralbidv ralsng oveq2 ralbidv 3bitrd mpbir2and 3jca oveqd oveq123d + raleqbidv 3anbi3d isring bitr4di mpbid ) CBHZAUCHZAUDIZUEHZEJZFJZGJZCCKZC + KZUAZLZXTLZXOXPXTLZXOXQXTLZXTLZMZYCXQXTLZYDYAXTLZMZNZGCUAZOZFYKOZEYKOZUFZ + AUGHZXKXLXNYNXKXLUHUIIZAUIIZKZUHPIZAPIZKZUJZUCHZXKYQYKKZYTXTKZUJZUUCUCXKU + UEYSUUFUUBXKYKYRYQXKYKQHZXTQHZUUIYKYRMCBUKZXKXSQHZUUIXRQHZXKUUKXKUULCCBBU + LUMZXRCQBULUNXSQUKUQZUUNYKXTAXTQQQDUOURZUSXKXTUUAYTXKUUHUUIUUIXTUUAMUUJUU + NUUNYKXTAXTQQQDUTURZUSVACUUGBUUGRZVBVCXKAQHZXLUUDWEXKAUUEUUFUHVDIZXTKZVEZ + QDXKUUEQHZUUFQHZUUTQHZUVAQHXKYQSHUUHUVBVFUUJYQYKSQULVGXKYTSHZUUIUVCPYTVHM + UVEVIVJUUNYTXTSQULVGXKUUSSHZUUIUVDVDUUSVHMUVFVKVJUUNUUSXTSQULVGUUEUUFUUTQ + QQVLURVMZYRUUAAUUCQYRRZUUARZUUCRVNUQVOXKXNUUGUEHCUUGBUUQVPXKUBUPXMUIIZXMU + UGXKUVJVQXKYKYRUUGUIIZUVJUUOXKUUHUUIYKUVKMUUJUUNYKXTUUGQQUUQVRVSXKUURYRUV + JMUVGYRAXMQXMRZUVHVTUQWAXKUBJUVJHUPJUVJHNUBUPXMPIZUUGPIZXKXTAVDIZUVNUVMXK + UUHUUIUUIXTUVOMUUJUUNUUNYKXTAXTQQQDWBURZXKUUHUUIXTUVNMUUJUUNYKXTUUGQQUUQW + CVSXKUURUVOUVMMUVGAUVOXMQUVLUVORZWDUQWAWFWGVOXKYNCCCXTLZXTLZUVRUVRXTLZMZU + VRCXTLZUVTMZXKUVSUVRUVTXKUVRCCXTXKUVRXRXTIZCCCXTWHUULXKUWDCMUUMXRCQBWIUNW + JZWKXKUVRCUVRCXTUWEUWEWLZWMXKUWBUVRUVTXKUVRCCXTUWEWPUWFWMXKYNCYAXTLZCXPXT + LZCXQXTLZXTLZMZUWHXQXTLZUWIYAXTLZMZNZGYKOZFYKOZCUWIXTLZUVRUWIXTLZMZUVRXQX + TLZUWIUWIXTLZMZNZGYKOZUWAUWCNZYMUWQECBXOCMZYJUWOFGYKYKUXGYFUWKYIUWNUXGYBU + WGYEUWJXOCYAXTWNUXGYCUWHYDUWIXTXOCXPXTWNZXOCXQXTWNZWLTUXGYGUWLYHUWMUXGYCU + WHXQXTUXHWPUXGYDUWIYAXTUXIWPTWOWQWRUWPUXEFCBXPCMZUWOUXDGYKUXJUWKUWTUWNUXC + UXJUWGUWRUWJUWSUXJYAUWICXTXPCXQXTWNZWKUXJUWHUVRUWIXTXPCCXTWSZWPTUXJUWLUXA + UWMUXBUXJUWHUVRXQXTUXLWPUXJYAUWIUWIXTUXKWKTWOWTWRUXDUXFGCBXQCMZUWTUWAUXCU + WCUXMUWRUVSUWSUVTUXMUWIUVRCXTXQCCXTWSZWKUXMUWIUVRUVRXTUXNWKTUXMUXAUWBUXBU + VTXQCUVRXTWSUXMUWIUVRUWIUVRXTUXNUXNWLTWOWRXAXBXCXKYOXLXNXOXPXQUUALZUVOLZX + OXPUVOLZXOXQUVOLZUUALZMZXOXPUUALZXQUVOLZUXRXPXQUVOLZUUALZMZNZGYROZFYROZEY + ROZUFYPXKYNUYIXLXNXKYMUYHEYKYRUUOXKYLUYGFYKYRUUOXKYJUYFGYKYRUUOXKYFUXTYIU + YEXKYBUXPYEUXSXKXOXOYAUXOXTUVOUVPXKXOVQXKXTUUAXPXQUUPXDXEXKYCUXQYDUXRXTUU + AUUPXKXTUVOXOXPUVPXDXKXTUVOXOXQUVPXDZXETXKYGUYBYHUYDXKYCUYAXQXQXTUVOUVPXK + XTUUAXOXPUUPXDXKXQVQXEXKYDUXRYAUYCXTUUAUUPUYJXKXTUVOXPXQUVPXDXETWOXFXFXFX + GEFGYRUUAAUVOXMUVHUVLUVIUVQXHXIXJ $. + $} + + $( The class of rings is not empty (it is also inhabited, as shown at + ~ ring1 ). (Contributed by AV, 29-Apr-2019.) $) + ringn0 $p |- Ring =/= (/) $= + ( vz cv cvv wcel cnx cbs cfv csn cop cplusg cmulr ctp crg c0 wne eqid ring1 + vex ne0i mp2b ) ABZCDEFGUAHIEJGUAUAIUAIHZIEKGUBILZMDMNOARUCCUAUCPQMUCST $. + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Opposite ring +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $( Introduce new constant symbols. $) + $c oppR $. $( Opposite ring $) + + $( The opposite ring operation. $) + coppr $a class oppR $. + + $( Define an opposite ring, which is the same as the original ring but with + multiplication written the other way around. (Contributed by Mario + Carneiro, 1-Dec-2014.) $) + df-oppr $a |- oppR = ( f e. _V |-> ( f sSet + <. ( .r ` ndx ) , tpos ( .r ` f ) >. ) ) $. + + ${ + $d x R $. $d x B $. $d x .x. $. $d x X $. $d x Y $. + opprval.1 $e |- B = ( Base ` R ) $. + opprval.2 $e |- .x. = ( .r ` R ) $. + opprval.3 $e |- O = ( oppR ` R ) $. + $( Value of the opposite ring. (Contributed by Mario Carneiro, + 1-Dec-2014.) $) + opprvalg $p |- ( R e. V + -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) $= + ( vx wcel coppr cfv cmulr ctpos cop csts co cvv wceq cn cnx cv df-oppr id + fveq2 eqtr4di tposeqd opeq2d oveq12d elex mulrslid simpri slotex eqeltrid + cslot a1i tposexg syl setsex syl3anc fvmptd3 eqtrid ) BEJZDBKLBUAMLZCNZOZ + PQZHVCIBIUBZVDVHMLZNZOZPQVGRKRIUCVHBSZVHBVKVFPVLUDVLVJVEVDVLVICVLVIBMLZCV + HBMUEGUFUGUHUIBEUJZVCBRJVDTJZVERJZVGRJVNVOVCMVDUOSVOUKULUPVCCRJVPVCCVMRGB + MEUKUMUNCRUQURVDVEBRRTUSUTVAVB $. + + opprmulfval.4 $e |- .xb = ( .r ` O ) $. + $( Value of the multiplication operation of an opposite ring. (Contributed + by Mario Carneiro, 1-Dec-2014.) $) + opprmulfvalg $p |- ( R e. V -> .xb = tpos .x. ) $= + ( wcel cmulr cfv ctpos cnx cop csts co cvv mulrslid opprvalg wceq tposexg + fveq2d slotex eqeltrid syl setsslid mpdan eqtr4d eqtrid ) BFKZCELMZDNZJUL + UMBOLMUNPQRZLMZUNULEUOLABDEFGHIUAUDULUNSKZUNUPUBULDSKUQULDBLMSHBLFTUEUFDS + UCUGFUNLSBTUHUIUJUK $. + + $( Value of the multiplication operation of an opposite ring. Hypotheses + eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed + by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, + 30-Aug-2015.) $) + opprmulg $p |- ( ( R e. V /\ X e. W /\ Y e. U ) + -> ( X .xb Y ) = ( Y .x. X ) ) $= + ( wcel w3a co ctpos wceq opprmulfvalg oveqd 3ad2ant1 ovtposg 3adant1 + eqtrd ) BGOZIHOZJEOZPIJCQZIJDRZQZJIDQZUFUGUIUKSUHUFCUJIJABCDFGKLMNTUAUBUG + UHUKULSUFIJDHEUCUDUE $. + + $( In a commutative ring, the opposite ring is equivalent to the original + ring. (Contributed by Mario Carneiro, 14-Jun-2015.) $) + crngoppr $p |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> + ( X .x. Y ) = ( X .xb Y ) ) $= + ( ccrg wcel w3a co crngcom opprmulg eqtr4d ) BLMFAMGAMNFGDOGFDOFGCOABDFGH + IPABCDAELAFGHIJKQR $. + $} + + ${ + opprex.o $e |- O = ( oppR ` R ) $. + $( Existence of the opposite ring. If you know that ` R ` is a ring, see + ~ opprring . (Contributed by Jim Kingdon, 10-Jan-2025.) $) + opprex $p |- ( R e. V -> O e. _V ) $= + ( wcel cnx cmulr cfv ctpos cop csts co cvv cbs opprvalg cn cslot mulrslid + eqid wceq simpri a1i slotex tposexg syl setsex mpd3an23 eqeltrd ) ACEZBAF + GHZAGHZIZJKLZMANHZAUKBCUNSUKSDOUIUJPEZULMEZUMMEUOUIGUJQTUORUAUBUIUKMEUPAG + CRUCUKMUDUEUJULACMPUFUGUH $. + $} + + ${ + $d x y z R $. $d x y z O $. + opprbas.1 $e |- O = ( oppR ` R ) $. + ${ + opprsllem.2 $e |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN ) $. + opprlem.3 $e |- ( E ` ndx ) =/= ( .r ` ndx ) $. + $( Lemma for ~ opprbasg and ~ oppraddg . (Contributed by Mario Carneiro, + 1-Dec-2014.) (Revised by AV, 6-Nov-2024.) $) + opprsllem $p |- ( R e. V -> ( E ` R ) = ( E ` O ) ) $= + ( wcel cfv cnx cmulr ctpos cop csts co cvv wceq mulrslid slotex eqid cn + tposexg syl cslot simpri setsslnid mpdan cbs opprvalg fveq2d eqtr4d ) A + DHZABIZAJKIZAKIZLZMNOZBIZCBIULUPPHZUMURQULUOPHUSAKDRSUOPUBUCDUPUNBPAFGK + UNUDQUNUAHRUEUFUGULCUQBAUHIZAUOCDUTTUOTEUIUJUK $. + $} + + ${ + opprbas.2 $e |- B = ( Base ` R ) $. + $( Base set of an opposite ring. (Contributed by Mario Carneiro, + 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.) $) + opprbasg $p |- ( R e. V -> B = ( Base ` O ) ) $= + ( wcel cbs cfv baseslid basendxnmulrndx opprsllem eqtrid ) BDGABHICHIFB + HCDEJKLM $. + $} + + ${ + oppradd.2 $e |- .+ = ( +g ` R ) $. + $( Addition operation of an opposite ring. (Contributed by Mario + Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.) $) + oppraddg $p |- ( R e. V -> .+ = ( +g ` O ) ) $= + ( wcel cplusg cfv plusgslid plusgndxnmulrndx opprsllem eqtrid ) BDGABHI + CHIFBHCDEJKLM $. + $} + + $( An opposite ring is a ring. (Contributed by Mario Carneiro, + 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) $) + opprring $p |- ( R e. Ring -> O e. Ring ) $= + ( vx vy vz crg wcel cfv cplusg eqid cv wa co opprmulg wceq syl13anc eqtrd + syl3anc 3adant3r3 cbs cmulr opprbasg oppraddg eqidd cgrp ringgrp grppropd + cur oveqdr mpbid ringcl 3com23 eqeltrd simpl simpr3 simpr2 simpr1 ringass + 3adant3r1 oveq2d oveq1d 3eqtr4rd ringdir ringacl 3adant3r2 oveq12d ringdi + w3a 3eqtr4d ringidcl syl simpr ringridm ringlidm isringd ) AGHZDEFAUAIZAJ + IZBBUBIZAUIIZVRABGCVRKZUCZVSABGCVSKZUDZVQVTUEVQAUFHBUFHAUGVQDEVRABVQVRUEW + CVQDLZVRHZELZVRHZMDEVSBJIWEUJUHUKVQWGWIVIZWFWHVTNZWHWFAUBIZNZVRVRAVTWLVRB + GVRWFWHWBWLKZCVTKZOZVQWIWGWMVRHZVRAWLWHWFWBWNULUMZUNVQWGWIFLZVRHZVIZMZWSW + HWLNZWFWLNZWSWMWLNZWFWHWSVTNZVTNZWKWSVTNZXBVQWTWIWGXDXEPVQXAUOZVQWGWIWTUP + ZVQWGWIWTUQZVQWGWIWTURZVRAWLWSWHWFWBWNUSQXBXGWFXCVTNZXDXBXFXCWFVTVQWIWTXF + XCPWGVRAVTWLVRBGVRWHWSWBWNCWOOUTZVAXBVQWGXCVRHZXMXDPXIXLXBVQWTWIXOXIXJXKV + RAWLWSWHWBWNULSVRAVTWLVRBGVRWFXCWBWNCWOOSRXBXHWMWSVTNZXEVQWGWIXHXPPWTWJWK + WMWSVTWPVBTXBVQWQWTXPXEPXIVQWGWIWQWTWRTXJVRAVTWLVRBGVRWMWSWBWNCWOOSRVCXBW + HWSVSNZWFWLNZWMWSWFWLNZVSNZWFXQVTNZWKWFWSVTNZVSNXBVQWIWTWGXRXTPXIXKXJXLVR + VSAWLWHWSWFWBWDWNVDQXBVQWGXQVRHZYAXRPXIXLVQWIWTYCWGVRVSAWHWSWBWDVEUTVRAVT + WLVRBGVRWFXQWBWNCWOOSXBWKWMYBXSVSVQWGWIWKWMPWTWPTVQWGWTYBXSPWIVRAVTWLVRBG + VRWFWSWBWNCWOOVFZVGVJXBWSWFWHVSNZWLNZXSXCVSNZYEWSVTNZYBXFVSNXBVQWTWGWIYFY + GPXIXJXLXKVRVSAWLWSWFWHWBWDWNVHQXBVQYEVRHZWTYHYFPXIVQWGWIYIWTVRVSAWFWHWBW + DVETXJVRAVTWLVRBGVRYEWSWBWNCWOOSXBYBXSXFXCVSYDXNVGVJVRAWAWBWAKZVKZVQWGMZW + AWFVTNZWFWAWLNZWFYLVQWAVRHZWGYMYNPVQWGUOZYLVQYOYPYKVLZVQWGVMZVRAVTWLVRBGV + RWAWFWBWNCWOOSVRAWLWAWFWBWNYJVNRYLWFWAVTNZWAWFWLNZWFYLVQWGYOYSYTPYPYRYQVR + AVTWLVRBGVRWFWAWBWNCWOOSVRAWLWAWFWBWNYJVORVP $. + + ${ + $d V x y $. + $( Bidirectional form of ~ opprring . (Contributed by Mario Carneiro, + 6-Dec-2014.) $) + opprringbg $p |- ( R e. V -> ( R e. Ring <-> O e. Ring ) ) $= + ( vx vy wcel crg opprring wa cfv eqid opprbasg sylan9eq cv cplusg cmulr + cbs co wceq coppr adantl eqidd oppraddg oveqdr opprmulg 3adant1l simp1l + w3a simp3 simp2 syl3anc eqtr2d 3expb ringpropd mpbird ex impbid2 ) ACGZ + AHGZBHGZABDIUSVAUTUSVAJZUTBUAKZHGZVAVDUSBVCVCLZIUBVBEFARKZAVCVBVFUCUSVA + VFBRKZVCRKVFABCDVFLZMVGBVCHVEVGLZMNVBEOZVFGZFOZVFGZJEFAPKZVCPKZUSVAVNBP + KZVOVNABCDVNLUDVPBVCHVEVPLUDNUEVBVKVMVJVLAQKZSZVJVLVCQKZSZTVBVKVMUIZVTV + LVJBQKZSZVRVAVKVMVTWCTUSVGBVSWBVFVCHVFVJVLVIWBLZVEVSLUFUGWAUSVMVKWCVRTU + SVAVKVMUHVBVKVMUJVBVKVMUKVFAWBVQVFBCVFVLVJVHVQLDWDUFULUMUNUOUPUQUR $. + $} + + ${ + $d V x y $. + oppr0.2 $e |- .0. = ( 0g ` R ) $. + $( Additive identity of an opposite ring. (Contributed by Mario + Carneiro, 1-Dec-2014.) $) + oppr0g $p |- ( R e. V -> .0. = ( 0g ` O ) ) $= + ( vy vx wcel cv cbs cfv cplusg co wceq wa wral cio eqid oveqd raleqbidv + c0g opprbasg eleq2d oppraddg eqeq1d anbi12d iotabidv cvv opprex 3eqtr4d + grpidvalg syl ) ACIZGJZAKLZIZUOHJZAMLZNZUROZURUOUSNZUROZPZHUPQZPZGRUOBK + LZIZUOURBMLZNZUROZURUOVINZUROZPZHVGQZPZGRZDBUBLZUNVFVPGUNUQVHVEVOUNUPVG + UOUPABCEUPSZUCZUDUNVDVNHUPVGVTUNVAVKVCVMUNUTVJURUNUSVIUOURUSABCEUSSZUEZ + TUFUNVBVLURUNUSVIURUOWBTUFUGUAUGUHHUPUSGACDVSWAFULUNBUIIVRVQOABCEUJHVGV + IGBUIVRVGSVISVRSULUMUK $. + $} + + ${ + $d V x y $. + oppr1.2 $e |- .1. = ( 1r ` R ) $. + $( Multiplicative identity of an opposite ring. (Contributed by Mario + Carneiro, 1-Dec-2014.) $) + oppr1g $p |- ( R e. V -> .1. = ( 1r ` O ) ) $= + ( vx vy wcel cfv cbs co wceq wa wral cio eqid eqeq1d anbi12d cvv c0g cv + cmgp cur cmulr crio opprmulg 3expa simpll simpr simplr syl3anc biancomd + ralbidva riotabidva df-riota 3eqtr3g cplusg opprex mgpex grpidvalg 3syl + opprbasg mgpbasg eqtrd eleq2d mgpplusgg oveqd raleqbidv iotabidv eqtr4d + syl 3eqtr4d ringidvalg 3eqtr4rd ) ADIZCUCJZUAJZAUCJZUAJZCUDJZBVPGUBZAKJ + ZIZWBHUBZCUEJZLZWEMZWEWBWFLZWEMZNZHWCOZNZGPZWDWBWEAUEJZLZWEMZWEWBWOLZWE + MZNZHWCOZNZGPZVRVTVPWLGWCUFXAGWCUFWNXCVPWLXAGWCVPWDNZWKWTHWCXDWEWCIZNZW + KWQWSXFWHWSWJWQXFWGWRWEVPWDXEWGWRMWCAWFWOWCCDWCWBWEWCQZWOQZEWFQZUGUHRXF + WIWPWEXFVPXEWDWIWPMVPWDXEUIXDXEUJVPWDXEUKWCAWFWOWCCDWCWEWBXGXHEXIUGULRS + UMUNUOWLGWCUPXAGWCUPUQVPVRWBVQKJZIZWBWEVQURJZLZWEMZWEWBXLLZWEMZNZHXJOZN + ZGPZWNVPCTIZVQTIVRXTMACDEUSZCVQTVQQZUTHXJXLGVQTVRXJQXLQVRQVAVBVPWMXSGVP + WDXKWLXRVPWCXJWBVPWCCKJZXJWCACDEXGVCVPYAYDXJMYBYDCVQTYCYDQVDVLVEZVFVPWK + XQHWCXJYEVPWHXNWJXPVPWGXMWEVPWFXLWBWEVPYAWFXLMYBCWFVQTYCXIVGVLZVHRVPWIX + OWEVPWFXLWEWBYFVHRSVISVJVKVPVTWBVSKJZIZWBWEVSURJZLZWEMZWEWBYILZWEMZNZHY + GOZNZGPZXCVPVSTIVTYQMAVSDVSQZUTHYGYIGVSTVTYGQYIQVTQVAVLVPXBYPGVPWDYHXAY + OVPWCYGWBWCAVSDYRXGVDZVFVPWTYNHWCYGYSVPWQYKWSYMVPWPYJWEVPWOYIWBWEAWOVSD + YRXHVGZVHRVPWRYLWEVPWOYIWEWBYTVHRSVISVJVKVMVPYAWAVRMYBCWAVQTYCWAQVNVLAB + VSDYRFVNVO $. + $} + + ${ + $d V x y $. + opprneg.2 $e |- N = ( invg ` R ) $. + $( The negative function in an opposite ring. (Contributed by Mario + Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) $) + opprnegg $p |- ( R e. V -> N = ( invg ` O ) ) $= + ( vx vy wcel cbs cfv cv cplusg co c0g wceq crio cmpt eqid grpinvfvalg + cminusg opprbasg oppraddg oveqd oppr0g riotaeqbidv mpteq12dv cvv opprex + eqeq12d syl 3eqtr4d ) ADIZGAJKZHLZGLZAMKZNZAOKZPZHUNQZRGCJKZUOUPCMKZNZC + OKZPZHVBQZRZBCUAKZUMGUNVAVBVGUNACDEUNSZUBZUMUTVFHUNVBVKUMURVDUSVEUMUQVC + UOUPUQACDEUQSZUCUDACDUSEUSSZUEUJUFUGGHUNUQABDUSVJVLVMFTUMCUHIVIVHPACDEU + IGHVBVCCVIUHVEVBSVCSVESVISTUKUL $. + $} + $} + + ${ + $d x y B $. $d x y N $. $d x y R $. $d x y X $. $d x y Y $. + mulgass3.b $e |- B = ( Base ` R ) $. + mulgass3.m $e |- .x. = ( .g ` R ) $. + mulgass3.t $e |- .X. = ( .r ` R ) $. + $( An associative property between group multiple and ring multiplication. + (Contributed by Mario Carneiro, 14-Jun-2015.) $) + mulgass3 $p |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> + ( X .X. ( N .x. Y ) ) = ( N .x. ( X .X. Y ) ) ) $= + ( vx vy crg wcel wa cfv co wceq eqid adantr cz w3a coppr cmg cbs opprring + simpr1 simpr3 opprbasg eleqtrd simpr2 mulgass2 syl13anc ringgrpd eleqtrrd + cmulr simpl mulgcld opprmulg syl3anc oveq2d 3eqtr3d eqidd ssidd cv cplusg + a1i ringacl 3expb adantlr oppraddg oveqdr mulgpropdg oveqd 3eqtr4d ) BMNZ + EUANZFANZGANZUBZOZFEGBUCPZUDPZQZDQZEFGDQZWCQZFEGCQZDQEWFCQWAWDFWBUPPZQZEG + FWIQZWCQZWEWGWAWBMNZVQGWBUEPZNFWNNWJWLRVPWMVTBWBWBSZUFTZVPVQVRVSUGZWAGAWN + VPVQVRVSUHZVPAWNRVTABWBMWOHUITZUJZWAFAWNVPVQVRVSUKZWSUJWNWBWCWIEGFWNSZWCS + ZWISZULUMWAVPWDANVRWJWERVPVTUQZWAWDWNAWAWNWCWBEGXBXCWAWBWPUNWQWTURWSUOXAA + BWIDAWBMAWDFHJWOXDUSUTWAWKWFEWCWAVPVSVRWKWFRXEWRXAABWIDAWBMAGFHJWOXDUSUTV + AVBWAWHWDFDWACWCEGWAKLACWCBWBAMMCBUDPRWAIVGWAWCVCXEWPABUEPRWAHVGWSWAAVDVP + KVEZANZLVEZANZOZXFXHBVFPZQZANZVTVPXGXIXMAXKBXFXHHXKSZVHVIVJWAXLXFXHWBVFPZ + QRXJVPVTKLXKXOXKBWBMWOXNVKVLTVMZVNVAWACWCEWFXPVNVO $. + $} + + $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# The complex numbers as an algebraic extensible structure @@ -149349,26 +150455,26 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ( vu vv vx vy vt vz vw ctb wcel cv wss cuni wi wral wel wa wrex rsp eltg2 ctg cfv ctop wal cin uniss adantl wceq unitg adantr sseqtrd eluni2 eltg2b ssel2 syl6bi imp31 an32s sylan2 an42s elssuni syl5com anim2d ad2antrl mpd - sstr2 reximdv rexlimdvaa syl5bi ralrimiv jca ex sylibrd alrimiv inss1 tg1 - sstrid simplbda syl im2anan9 reeanv 3imtr4g anandis biimpri ss2in anim12i - elin basis2 adantllr adantrrr com12 ad2antll sylanr2 rexlimdva a2d syldan - an4s imp ralrimivv cvv wb tgvalex istopg mpbir2and ) AIJZAUAUBZUCJZBKZXEL - ZXGMZXEJZNZBUDZXGCKZUEZXEJZCXEOBXEOZXDXKBXDXHXIAMZLZDEPZEKZXILZQZEARZDXIO - ZQZXJXDXHYEXDXHQZXRYDYFXIXEMZXQXHXIYGLXDXGXEUFUGXDYGXQUHXHAIUIUJUKYFYCDXI - DKZXIJDFPZFXGRYFYCFYHXGULYFYIYCFXGYFFBPZYIQQXSXTFKZLZQZEARZYCXDYIXHYJYNXH - YJQXDYIQYKXEJZYNXGXEYKUNXDYOYIYNXDYOYIYNXDYOYNDYKOYIYNNDEYKAIUMYNDYKSUOUP - UQURUSYJYNYCNYFYIYJYMYBEAYJYLYAXSYJYKXILYLYAYKXGUTXTYKXIVEVAVBVFVCVDVGVHV - IVJVKDEXIAITVLVMXDXOBCXEXEXDXGXEJZXMXEJZQZXNXQLZYIYKXNLZQZFARZDXNOZQZXOXD - YRUUDXDYRQZYSUUCYPYSXDYQYPXNXGXQXGXMVNXGAVOVPVCUUEUUBDXNXDYRYHXNJZDGPZGKZ - XGLZQZDHPZHKZXMLZQZQZHARZGARZNZUUFUUBNZXDYPYQUURXDYPQZXDYQQZQDBPZDCPZQUUJ - GARZUUNHARZQUUFUUQUUTUVBUVDUVAUVCUVEUUTUVDDXGOZUVBUVDNXDYPXGXQLUVFDGXGAIT - VQUVDDXGSVRUVAUVEDXMOZUVCUVENXDYQXMXQLUVGDHXMAITVQUVEDXMSVRVSYHXGXMWFUUJU - UNGHAAVTWAWBXDUURUUSXDUUFUUQUUBXDUUFUUQUUBNXDUUFQZUUPUUBGAUVHUUHAJZQZUUOU - UBHAUUOUVJUULAJZYHUUHUULUEZJZUVLXNLZQZUUBUUGUUKUUIUUMUVOUUGUUKQZUVMUUIUUM - QUVNUVMUVPYHUUHUULWFWCUUHXGUULXMWDWEWPUVJUVKUVOQQYIYKUVLLZQZFARZUUBUVJUVK - UVMUVSUVNXDUVIUVKUVMQUVSUUFFYHAUUHUULWGWHWIUVOUVSUUBNZUVJUVKUVNUVTUVMUVNU - VRUUAFAUVNUVQYTYIUVQUVNYTYKUVLXNVEWJVBVFUGWKVDWLVGWMVKWNWQWOVIVJVKDFXNAIT - VLWRXDXEWSJXFXLXPQWTAIXABCWSXEXBVRXC $. + sstr2 reximdv rexlimdvaa biimtrid ralrimiv jca sylibrd alrimiv tg1 sstrid + ex inss1 simplbda syl im2anan9 elin 3imtr4g anandis biimpri ss2in anim12i + reeanv an4s basis2 adantllr adantrrr com12 ad2antll sylanr2 rexlimdva a2d + imp syldan ralrimivv cvv wb tgvalex istopg mpbir2and ) AIJZAUAUBZUCJZBKZX + ELZXGMZXEJZNZBUDZXGCKZUEZXEJZCXEOBXEOZXDXKBXDXHXIAMZLZDEPZEKZXILZQZEARZDX + IOZQZXJXDXHYEXDXHQZXRYDYFXIXEMZXQXHXIYGLXDXGXEUFUGXDYGXQUHXHAIUIUJUKYFYCD + XIDKZXIJDFPZFXGRYFYCFYHXGULYFYIYCFXGYFFBPZYIQQXSXTFKZLZQZEARZYCXDYIXHYJYN + XHYJQXDYIQYKXEJZYNXGXEYKUNXDYOYIYNXDYOYIYNXDYOYNDYKOYIYNNDEYKAIUMYNDYKSUO + UPUQURUSYJYNYCNYFYIYJYMYBEAYJYLYAXSYJYKXILYLYAYKXGUTXTYKXIVEVAVBVFVCVDVGV + HVIVJVODEXIAITVKVLXDXOBCXEXEXDXGXEJZXMXEJZQZXNXQLZYIYKXNLZQZFARZDXNOZQZXO + XDYRUUDXDYRQZYSUUCYPYSXDYQYPXNXGXQXGXMVPXGAVMVNVCUUEUUBDXNXDYRYHXNJZDGPZG + KZXGLZQZDHPZHKZXMLZQZQZHARZGARZNZUUFUUBNZXDYPYQUURXDYPQZXDYQQZQDBPZDCPZQU + UJGARZUUNHARZQUUFUUQUUTUVBUVDUVAUVCUVEUUTUVDDXGOZUVBUVDNXDYPXGXQLUVFDGXGA + ITVQUVDDXGSVRUVAUVEDXMOZUVCUVENXDYQXMXQLUVGDHXMAITVQUVEDXMSVRVSYHXGXMVTUU + JUUNGHAAWFWAWBXDUURUUSXDUUFUUQUUBXDUUFUUQUUBNXDUUFQZUUPUUBGAUVHUUHAJZQZUU + OUUBHAUUOUVJUULAJZYHUUHUULUEZJZUVLXNLZQZUUBUUGUUKUUIUUMUVOUUGUUKQZUVMUUIU + UMQUVNUVMUVPYHUUHUULVTWCUUHXGUULXMWDWEWGUVJUVKUVOQQYIYKUVLLZQZFARZUUBUVJU + VKUVMUVSUVNXDUVIUVKUVMQUVSUUFFYHAUUHUULWHWIWJUVOUVSUUBNZUVJUVKUVNUVTUVMUV + NUVRUUAFAUVNUVQYTYIUVQUVNYTYKUVLXNVEWKVBVFUGWLVDWMVGWNVOWOWPWQVIVJVODFXNA + ITVKWRXDXEWSJXFXLXPQWTAIXABCWSXEXBVRXC $. $( The property ~ tgcl can be reversed: if the topology generated by ` B ` is actually a topology, then ` B ` must be a topological basis. This @@ -149622,24 +150728,24 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ( vy vz wcel wa cv wceq wss wral adantr syl eleq2 eqeq1 imbi12d elrab cvv wi cpw crab ctop cuni ctopon cfv wal cin ssrab simprl sspwuni vuniex elpw sylib sylibr wrex eluni2 r19.29 simpr elssuni eqsstrrd rexlimiva ad2antll - impr ex syl5bi jctild eqss syl6ibr sylanbrc alrimiv simprll elpwid sstrid - inss1 vex inex1 elin simprlr simprrr anim12d ineq12 inidm eqtrdi syl6 jca - anbi12i ralrimivv wb pwexg rabexg istopg mpbir2and pwidg eqidd a1d ssrab2 - 3imtr4g mpbi a1i eqssd istopon ) BDGZCBGZHZCAIZGZXFBJZTZABUAZUBZUCGZBXKUD - ZJXKBUEUFGXEXLEIZXKKZXNUDZXKGZTZEUGZXNFIZUHZXKGZFXKLEXKLZXEXREXOXNXJKZXIA - XNLZHZXEXQXIAXJXNUIXEYFXQXEYFHZXPXJGZCXPGZXPBJZTZXQYGXPBKZYHYGYDYLXEYDYEU - JXNBUKUNZXPBEULUMUOYGYIYLBXPKZHYJYGYIYNYLYIXGAXNUPZYGYNACXNUQYEYOYNTXEYDY - EYOYNYEYOHXIXGHZAXNUPYNXIXGAXNURYPYNAXNXFXNGZYPHBXFXPYQXIXGXHYQXIUSVDYQXF - XPKYPXFXNUTMVAVBNVEVCVFYMVGXPBVHVIXIYKAXPXJXFXPJXGYIXHYJXFXPCOXFXPBPQRVJV - EVFVKXEYBEFXKXKXEXNXJGZCXNGZXNBJZTZHZXTXJGZCXTGZXTBJZTZHZHZYAXJGZCYAGZYAB - JZTZHZXNXKGZXTXKGZHYBXEUUHUUMXEUUHHZUUIUULUUPYABKUUIUUPYAXNBXNXTVOUUPXNBX - EYRUUAUUGVLVMVNYABXNXTEVPVQUMUOUUJYSUUDHZUUPUUKCXNXTVRUUPUUQYTUUEHZUUKUUP - YSYTUUDUUEXEYRUUAUUGVSXEUUBUUCUUFVTWAUURYABBUHBXNBXTBWBBWCWDWEVFWFVEUUNUU - BUUOUUGXIUUAAXNXJXFXNJXGYSXHYTXFXNCOXFXNBPQRXIUUFAXTXJXFXTJXGUUDXHUUEXFXT - COXFXTBPQRWGXIUULAYAXJXFYAJXGUUJXHUUKXFYACOXFYABPQRWRWHXEXKSGZXLXSYCHWIXE - XJSGZUUSXCUUTXDBDWJMXIAXJSWKNEFSXKWLNWMXEBXMXEBXKGZBXMKXEBXJGZXDBBJZTZUVA - XCUVBXDBDWNMXEUVCXDXEBWOWPXIUVDABXJXHXGXDXHUVCXFBCOXFBBPQRVJBXKUTNXMBKZXE - XKXJKUVEXIAXJWQXKBUKWSWTXABXKXBVJ $. + impr ex biimtrid jctild eqss syl6ibr sylanbrc alrimiv inss1 elpwid sstrid + simprll vex inex1 elin simprlr simprrr anim12d ineq12 eqtrdi syl6 anbi12i + inidm jca 3imtr4g ralrimivv pwexg rabexg istopg mpbir2and pwidg eqidd a1d + wb ssrab2 mpbi a1i eqssd istopon ) BDGZCBGZHZCAIZGZXFBJZTZABUAZUBZUCGZBXK + UDZJXKBUEUFGXEXLEIZXKKZXNUDZXKGZTZEUGZXNFIZUHZXKGZFXKLEXKLZXEXREXOXNXJKZX + IAXNLZHZXEXQXIAXJXNUIXEYFXQXEYFHZXPXJGZCXPGZXPBJZTZXQYGXPBKZYHYGYDYLXEYDY + EUJXNBUKUNZXPBEULUMUOYGYIYLBXPKZHYJYGYIYNYLYIXGAXNUPZYGYNACXNUQYEYOYNTXEY + DYEYOYNYEYOHXIXGHZAXNUPYNXIXGAXNURYPYNAXNXFXNGZYPHBXFXPYQXIXGXHYQXIUSVDYQ + XFXPKYPXFXNUTMVAVBNVEVCVFYMVGXPBVHVIXIYKAXPXJXFXPJXGYIXHYJXFXPCOXFXPBPQRV + JVEVFVKXEYBEFXKXKXEXNXJGZCXNGZXNBJZTZHZXTXJGZCXTGZXTBJZTZHZHZYAXJGZCYAGZY + ABJZTZHZXNXKGZXTXKGZHYBXEUUHUUMXEUUHHZUUIUULUUPYABKUUIUUPYAXNBXNXTVLUUPXN + BXEYRUUAUUGVOVMVNYABXNXTEVPVQUMUOUUJYSUUDHZUUPUUKCXNXTVRUUPUUQYTUUEHZUUKU + UPYSYTUUDUUEXEYRUUAUUGVSXEUUBUUCUUFVTWAUURYABBUHBXNBXTBWBBWFWCWDVFWGVEUUN + UUBUUOUUGXIUUAAXNXJXFXNJXGYSXHYTXFXNCOXFXNBPQRXIUUFAXTXJXFXTJXGUUDXHUUEXF + XTCOXFXTBPQRWEXIUULAYAXJXFYAJXGUUJXHUUKXFYACOXFYABPQRWHWIXEXKSGZXLXSYCHWQ + XEXJSGZUUSXCUUTXDBDWJMXIAXJSWKNEFSXKWLNWMXEBXMXEBXKGZBXMKXEBXJGZXDBBJZTZU + VAXCUVBXDBDWNMXEUVCXDXEBWOWPXIUVDABXJXHXGXDXHUVCXFBCOXFBBPQRVJBXKUTNXMBKZ + XEXKXJKUVEXIAXJWRXKBUKWSWTXABXKXBVJ $. $} ${ @@ -151079,30 +152185,30 @@ converges to zero (in the standard topology on the reals) with this E. x e. ( ( nei ` J ) ` { P } ) ( F " x ) C_ y ) ) ) $= ( vz cfv wcel cv cima wss wrex wa syl syl2anc wb syl3anc ctopon w3a co wf ccnp csn cnei wral cnpf2 3adantl3 cnt simpll1 simpll2 simpll3 simplr ctop - 3expa cuni topontop eqid neii1 sylancom ntropn simpr adantr ffvelrnd wceq - toponuni eleqtrd snssd neiint mpbid fvexg snssg mpbird icnpimaex syl33anc - cvv simpl1 ad2antrr simprl simprrl opnneip simprrr ntrss2 sstrd reximssdv - ralrimiva ex wi simprr imass2 eleq2 imaeq2 sseq1d anbi12d rspcev syl12anc + 3expa cuni topontop neii1 sylancom ntropn simpr adantr ffvelcdmd toponuni + eqid eleqtrd snssd neiint mpbid cvv fvexg snssg mpbird icnpimaex syl33anc + simpl1 ad2antrr simprl simprrl opnneip simprrr ntrss2 reximssdv ralrimiva + sstrd jca ex wi simprr imass2 eleq2 imaeq2 sseq1d anbi12d rspcev syl12anc rexlimdvaa embantd com23 exp4a ralimdv2 imdistanda iscnp sylibrd impbid - jca ) EGUAJKZFHUAJKZCGKZUBZDCEFUEUCJZKZGHDUDZDALZMZBLZNZACUFZEUGJJZOZBCDJ - ZUFZFUGJJZUHZPZXLXNYGXLXNPZXOYFXIXJXNXOXKXIXJXNXOCDEFGHUIUQUJZYHYBBYEYHXR - YEKZPZCXPKZXQXRFUKJJZNZPZXSAYAEYKXIXJXKXNYMFKZYCYMKZYOAEOXIXJXKXNYJULXIXJ - XKXNYJUMZXIXJXKXNYJUNZXLXNYJUOZYKFUPKZXRFURZNZYPYKXJUUAYRHFUSZQZYHYJUUAUU - CUUEYDFXRUUBUUBUTZVAVBZXRFUUBUUFVCRYKYQYDYMNZYKYJUUHYHYJVDYKUUAYDUUBNUUCY - JUUHSUUEYKYCUUBYKYCHUUBYKGHCDYHXOYJYIVEYSVFYKXJHUUBVGYRHFVHQVIVJUUGYDFXRU - UBUUFVKTVLYKYCVRKZYQUUHSYKXNXKUUIYTYSCDXMGVMRYCYMVRVNQVOAYMCDEFGHVPVQYKXP - EKZYOPZPZEUPKZUUJYLXPYAKZUULXIUUMYHXIYJUUKXIXJXKXNVSVTGEUSZQYKUUJYOWAYKUU - JYLYNWBCEXPWCTUULXQYMXRYKUUJYLYNWDYKYMXRNZUUKYKUUAUUCUUPUUEUUGXRFUUBUUFWE - RVEWFWGWHXHWIXLYGXOYCXRKZCILZKZDUURMZXRNZPZIEOZWJZBFUHZPXNXLXOYFUVEXLXOPZ - YBUVDBYEFUVFYJYBWJZXRFKZUUQUVCUVFUVHUUQPZUVGUVCUVFUVIUVGUVCWJUVFUVIPZYJYB - UVCUVJUUAUVHUUQYJUVJXJUUAXIXJXKXOUVIUMUUDQUVFUVHUUQWAUVFUVHUUQWKYCFXRWCTU - VJXSUVCAYAUVJUUNXSPZPZXPEUKJJZEKZCUVMKZDUVMMZXRNZUVCUVLUUMXPEURZNZUVNUVLX - IUUMUVFXIUVIUVKXIXJXKXOVSVTZUUOQZUVLUUMUUNUVSUWAUVJUUNXSWAZXTEXPUVRUVRUTZ - VARZXPEUVRUWCVCRUVLUVOXTUVMNZUVLUUNUWEUWBUVLUUMXTUVRNUVSUUNUWESUWAUVLCUVR - UVLCGUVRUVJXKUVKXIXJXKXOUVIUNVEZUVLXIGUVRVGUVTGEVHQVIVJUWDXTEXPUVRUWCVKTV - LUVLXKUVOUWESUWFCUVMGVNQVOUVLUVPXQXRUVLUVMXPNZUVPXQNUVLUUMUVSUWGUWAUWDXPE - UVRUWCWERUVMXPDWLQUVJUUNXSWKWFUVBUVOUVQPIUVMEUURUVMVGZUUSUVOUVAUVQUURUVMC - WMUWHUUTUVPXRUURUVMDWNWOWPWQWRWSWTWIXAXBXCXDIBCDEFGHXEXFXG $. + wceq ) EGUAJKZFHUAJKZCGKZUBZDCEFUEUCJZKZGHDUDZDALZMZBLZNZACUFZEUGJJZOZBCD + JZUFZFUGJJZUHZPZXLXNYGXLXNPZXOYFXIXJXNXOXKXIXJXNXOCDEFGHUIUQUJZYHYBBYEYHX + RYEKZPZCXPKZXQXRFUKJJZNZPZXSAYAEYKXIXJXKXNYMFKZYCYMKZYOAEOXIXJXKXNYJULXIX + JXKXNYJUMZXIXJXKXNYJUNZXLXNYJUOZYKFUPKZXRFURZNZYPYKXJUUAYRHFUSZQZYHYJUUAU + UCUUEYDFXRUUBUUBVGZUTVAZXRFUUBUUFVBRYKYQYDYMNZYKYJUUHYHYJVCYKUUAYDUUBNUUC + YJUUHSUUEYKYCUUBYKYCHUUBYKGHCDYHXOYJYIVDYSVEYKXJHUUBXHYRHFVFQVHVIUUGYDFXR + UUBUUFVJTVKYKYCVLKZYQUUHSYKXNXKUUIYTYSCDXMGVMRYCYMVLVNQVOAYMCDEFGHVPVQYKX + PEKZYOPZPZEUPKZUUJYLXPYAKZUULXIUUMYHXIYJUUKXIXJXKXNVRVSGEUSZQYKUUJYOVTYKU + UJYLYNWACEXPWBTUULXQYMXRYKUUJYLYNWCYKYMXRNZUUKYKUUAUUCUUPUUEUUGXRFUUBUUFW + DRVDWGWEWFWHWIXLYGXOYCXRKZCILZKZDUURMZXRNZPZIEOZWJZBFUHZPXNXLXOYFUVEXLXOP + ZYBUVDBYEFUVFYJYBWJZXRFKZUUQUVCUVFUVHUUQPZUVGUVCUVFUVIUVGUVCWJUVFUVIPZYJY + BUVCUVJUUAUVHUUQYJUVJXJUUAXIXJXKXOUVIUMUUDQUVFUVHUUQVTUVFUVHUUQWKYCFXRWBT + UVJXSUVCAYAUVJUUNXSPZPZXPEUKJJZEKZCUVMKZDUVMMZXRNZUVCUVLUUMXPEURZNZUVNUVL + XIUUMUVFXIUVIUVKXIXJXKXOVRVSZUUOQZUVLUUMUUNUVSUWAUVJUUNXSVTZXTEXPUVRUVRVG + ZUTRZXPEUVRUWCVBRUVLUVOXTUVMNZUVLUUNUWEUWBUVLUUMXTUVRNUVSUUNUWESUWAUVLCUV + RUVLCGUVRUVJXKUVKXIXJXKXOUVIUNVDZUVLXIGUVRXHUVTGEVFQVHVIUWDXTEXPUVRUWCVJT + VKUVLXKUVOUWESUWFCUVMGVNQVOUVLUVPXQXRUVLUVMXPNZUVPXQNUVLUUMUVSUWGUWAUWDXP + EUVRUWCWDRUVMXPDWLQUVJUUNXSWKWGUVBUVOUVQPIUVMEUURUVMXHZUUSUVOUVAUVQUURUVM + CWMUWHUUTUVPXRUURUVMDWNWOWPWQWRWSWTWIXAXBXCXDIBCDEFGHXEXFXG $. $} ${ @@ -151187,20 +152293,20 @@ converges to zero (in the standard topology on the reals) with this -> ( G o. F ) e. ( ( J CnP L ) ` P ) ) $= ( vz vx vy ctop wcel ccnp co cfv wa cuni cima wss syl3anc adantr w3a ccom wf cv wrex wral ctopon simpl2 toptopon2 sylib simpl3 simprr simpl1 simprl - wi cnpf2 syl2anc cnprcl2k ffvelrnd wceq fvco3 eqeltrrd icnpimaex syl33anc - fco ad2antrr simplll adantlll simprrl imaco imass2 eqsstrid simprrr sstr2 - syl2imc anim2d reximdv mpd rexlimddv expr ralrimiva wb iscnp mpbir2and ) + wi cnpf2 fco syl2anc cnprcl2k ffvelcdmd fvco3 eqeltrrd icnpimaex syl33anc + wceq ad2antrr simplll adantlll simprrl imaco imass2 simprrr sstr2 syl2imc + eqsstrid anim2d reximdv mpd rexlimddv expr ralrimiva wb iscnp mpbir2and ) DJKZEJKZFJKZUAZBADELMNKZCABNZEFLMNKZOZOZCBUBZADFLMNKZDPZFPZWNUCZAWNNZGUDZ KZAHUDZKZWNXBQZWTRZOZHDUEZUOZGFUFZWMEPZWQCUCZWPXJBUCZWRWMEXJUGNKZFWQUGNKZ WKXKWMWFXMWEWFWGWLUHZEUIUJZWMWGXNWEWFWGWLUKFUIUJZWHWIWKULZWJCEFXJWQUPSWMD - WPUGNKZXMWIXLWMWEXSWEWFWGWLUMDUIUJZXPWHWIWKUNZABDEWPXJUPSZWPXJWQCBVEUQWMX + WPUGNKZXMWIXLWMWEXSWEWFWGWLUMDUIUJZXPWHWIWKUNZABDEWPXJUPSZWPXJWQCBUQURWMX HGFWMWTFKZXAXGWMYCXAOZOZWJIUDZKZCYFQZWTRZOZXGIEYEXMXNWJXJKZWKYCWJCNZWTKYJ - IEUEWMXMYDXPTWMXNYDXQTWMYKYDWMWPXJABYBWMXSWFWIAWPKZXTXOYAABDEWPURSZUSTWMW - KYDXRTWMYCXAUNYEWSYLWTWMWSYLUTZYDWMXLYMYOYBYNWPXJACBVAUQTWMYCXAULVBIWTWJC + IEUEWMXMYDXPTWMXNYDXQTWMYKYDWMWPXJABYBWMXSWFWIAWPKZXTXOYAABDEWPUSSZUTTWMW + KYDXRTWMYCXAUNYEWSYLWTWMWSYLVEZYDWMXLYMYOYBYNWPXJACBVAURTWMYCXAULVBIWTWJC EFXJWQVCVDYEYFEKZYJOZOZXCBXBQZYFRZOZHDUEZXGYRXSXMYMWIYPYGUUBWMXSYDYQXTVFW MXMYDYQXPVFWMYMYDYQYNVFWLYDYQWIWHWIWKYDYQVGVHYEYPYJUNYEYPYGYIVIHYFABDEWPX JVCVDYRUUAXFHDYRYTXEXCWLYDYQYTXEUOWHYTXDYHRWLYDOZYQOYIXEYTXDCYSQYHCBXBVJY - SYFCVKVLUUCYPYGYIVMXDYHWTVNVOVHVPVQVRVSVTWAWMXSXNYMWOWRXIOWBXTXQYNHGAWNDF + SYFCVKVOUUCYPYGYIVLXDYHWTVMVNVHVPVQVRVSVTWAWMXSXNYMWOWRXIOWBXTXQYNHGAWNDF WPWQWCSWD $. $} @@ -151752,30 +152858,30 @@ F C_ ( CC X. X ) ) $= ( vk vj vv cfv cc wcel wa cn wrex syl cvv vu ccom clm wbr cpm co cv cdm cuni cuz wral wi wf wss ctopon ccnp lmrcl toptopon2 sylib cnpf2 syl3anc ctop w3a nnuz 1zzd lmbr2 mpbid simp1d uniexg cnex elpm2g sylancl simpld - c1 wb fco syl2anc feq2d mpbird simprd eqsstrd mpbir2and simp2d ffvelrnd - fdmd cima simp3d adantr simprl simprr syl33anc r19.29 pm3.45 imp reximi - icnpimaex wfn ad3antrrr ffnd simplrl elssuni 3expia wceq ad2antrr fvco3 - fnfvima sylan eleq1d sylibrd simplrr sseld syld eleqtrrd jctild expimpd - simpr ralimdv expr com23 impd rexlimdva syl5 mp2and ralrimiva mpbir3and - reximdv ) ADCUBZBDMZFUCMUDYGFUIZNUEUFOZYHYIOYHUAUGZOZJUGZYGUHZOZYMYGMZY - KOZPZJKUGUJMZUKZKQRZULZUAFUKAYJYNYIYGUMZYNNUNZAUUCCUHZYIYGUMZAEUIZYIDUM - ZUUEUUGCUMZUUFAEUUGUOMOZFYIUOMOZDBEFUPUFMOZUUHAEVBOZUUJACBEUCMUDZUUMGBC - EUQSZEURUSZAFVBOZUUKHFURUSZIBDEFUUGYIUTVAZAUUIUUENUNZACUUGNUEUFOZUUIUUT - PZAUVABUUGOZBLUGZOZYMUUEOZYMCMZUVDOZPZJYSUKZKQRZULZLEUKZAUUNUVAUVCUVMVC - GALBKJCEVNUUGQUUPVDAVEZVFVGZVHAUUGTOZNTOZUVAUVBVOAUUMUVPUUOEVBVISVJUUGN - CTTVKVLVGZVMZUUEUUGYIDCVPVQZAYNUUEYIYGAUUEYIYGUVTWEZVRVSAYNUUENUWAAUUIU - UTUVRVTWAAYITOZUVQYJUUCUUDPVOAUUQUWBHFVBVISVJYINYGTTVKVLWBAUUGYIBDUUSAU - VAUVCUVMUVOWCZWDAUUBUAFAYKFOZYLUUAAUWDYLPZPZUVMUVEDUVDWFZYKUNZPZLERZUUA - AUVMUWEAUVAUVCUVMUVOWGWHUWFUUJUUKUVCUULUWDYLUWJAUUJUWEUUPWHAUUKUWEUURWH - AUVCUWEUWCWHAUULUWEIWHAUWDYLWIAUWDYLWJLYKBDEFUUGYIWPWKUVMUWJPZUVKUWHPZL - ERZUWFUUAUWKUVLUWIPZLERUWMUVLUWILEWLUWNUWLLEUVLUWIUWLUVEUVKUWHWMWNWOSUW - FUWLUUALEUWFUVDEOZPZUVKUWHUUAUWPUWHUVKUUAUWFUWOUWHUVKUUAULUWFUWOUWHPZPZ - UVJYTKQUWRUVIYRJYSUWRUVFUVHYRUWRUVFPZUVHYQYOUWSUVHYPUWGOZYQUWSUVHUVGDMZ - UWGOZUWTUWSDUUGWQZUVDUUGUNZUVHUXBULUWSUUGYIDAUUHUWEUWQUVFUUSWRWSUWSUWOU - XDUWFUWOUWHUVFWTUVDEXASUXCUXDUVHUXBUUGUVDDUVGXFXBVQUWSYPUXAUWGUWRUUIUVF - YPUXAXCAUUIUWEUWQUVSXDUUEUUGYMDCXEXGXHXIUWSUWGYKYPUWFUWOUWHUVFXJXKXLUWS - YMUUEYNUWRUVFXPAYNUUEXCUWEUWQUVFUWAWRXMXNXOXQYFXRXSXTYAYBYCXRYDAUAYHKJY - GFVNYIQUURVDUVNVFYE $. + c1 wb fco syl2anc fdmd mpbird simprd eqsstrd mpbir2and simp2d ffvelcdmd + feq2d cima simp3d adantr simprl simprr icnpimaex syl33anc r19.29 pm3.45 + imp reximi wfn ad3antrrr ffnd simplrl elssuni fnfvima 3expia wceq fvco3 + ad2antrr sylan eleq1d sylibrd simplrr sseld syld simpr eleqtrrd expimpd + jctild ralimdv reximdv expr com23 impd rexlimdva syl5 mp2and ralrimiva + mpbir3and ) ADCUBZBDMZFUCMUDYGFUIZNUEUFOZYHYIOYHUAUGZOZJUGZYGUHZOZYMYGM + ZYKOZPZJKUGUJMZUKZKQRZULZUAFUKAYJYNYIYGUMZYNNUNZAUUCCUHZYIYGUMZAEUIZYID + UMZUUEUUGCUMZUUFAEUUGUOMOZFYIUOMOZDBEFUPUFMOZUUHAEVBOZUUJACBEUCMUDZUUMG + BCEUQSZEURUSZAFVBOZUUKHFURUSZIBDEFUUGYIUTVAZAUUIUUENUNZACUUGNUEUFOZUUIU + UTPZAUVABUUGOZBLUGZOZYMUUEOZYMCMZUVDOZPZJYSUKZKQRZULZLEUKZAUUNUVAUVCUVM + VCGALBKJCEVNUUGQUUPVDAVEZVFVGZVHAUUGTOZNTOZUVAUVBVOAUUMUVPUUOEVBVISVJUU + GNCTTVKVLVGZVMZUUEUUGYIDCVPVQZAYNUUEYIYGAUUEYIYGUVTVRZWEVSAYNUUENUWAAUU + IUUTUVRVTWAAYITOZUVQYJUUCUUDPVOAUUQUWBHFVBVISVJYINYGTTVKVLWBAUUGYIBDUUS + AUVAUVCUVMUVOWCZWDAUUBUAFAYKFOZYLUUAAUWDYLPZPZUVMUVEDUVDWFZYKUNZPZLERZU + UAAUVMUWEAUVAUVCUVMUVOWGWHUWFUUJUUKUVCUULUWDYLUWJAUUJUWEUUPWHAUUKUWEUUR + WHAUVCUWEUWCWHAUULUWEIWHAUWDYLWIAUWDYLWJLYKBDEFUUGYIWKWLUVMUWJPZUVKUWHP + ZLERZUWFUUAUWKUVLUWIPZLERUWMUVLUWILEWMUWNUWLLEUVLUWIUWLUVEUVKUWHWNWOWPS + UWFUWLUUALEUWFUVDEOZPZUVKUWHUUAUWPUWHUVKUUAUWFUWOUWHUVKUUAULUWFUWOUWHPZ + PZUVJYTKQUWRUVIYRJYSUWRUVFUVHYRUWRUVFPZUVHYQYOUWSUVHYPUWGOZYQUWSUVHUVGD + MZUWGOZUWTUWSDUUGWQZUVDUUGUNZUVHUXBULUWSUUGYIDAUUHUWEUWQUVFUUSWRWSUWSUW + OUXDUWFUWOUWHUVFWTUVDEXASUXCUXDUVHUXBUUGUVDDUVGXBXCVQUWSYPUXAUWGUWRUUIU + VFYPUXAXDAUUIUWEUWQUVSXFUUEUUGYMDCXEXGXHXIUWSUWGYKYPUWFUWOUWHUVFXJXKXLU + WSYMUUEYNUWRUVFXMAYNUUEXDUWEUWQUVFUWAWRXNXPXOXQXRXSXTYAYBYCYDXSYEAUAYHK + JYGFVNYIQUURVDUVNVFYF $. $} lmcn.4 $e |- ( ph -> G e. ( J Cn K ) ) $. @@ -151875,28 +152981,28 @@ F C_ ( CC X. X ) ) $= cmpo an4s eleq1 anbi1d 2rexbidv ralxp sylibr anassrs ineq12 eqtrdi sseq2d inxp anbi2d rexbidv c1st cfv c2nd rexeqi wb 1stexg 2ndexg xpex rgenw cmpt elv vex op1std op2ndd xpeq12d mpompt eqcomi eleq2 anbi12d rexrnmpt eleq2d - sseq1 sseq1d 3bitri bitrdi raleqbidv syl5ibrcom rexlimdvva syl5bir syl5bi - ax-mp rexxp ralrimivv txbasex isbasis2g syl mpbird ) DUBOZEUBOZPZCUBOZUAQ - ZGQZOZYEHQZIQZUCZUDZPZGCRZUAYIUEZICUEHCUEZYBYMHICCYGCOZYHCOZPZYGJQZKQZSZU - GZKERZYHLQZMQZSZUGZMERZPZLDRJDRZYBYMYQUUBJDRZUUGLDRZPUUIYOUUJYPUUKUUJHCCA - BDEAQZBQZSZVIZUFZUUJHUHFJKHDEYTUUOABJKDEUUNYTYRUUMSUULYRUUMUIUUMYSYRUJUKU - LUMUNUUKICCUUPUUKIUHFLMIDEUUEUUOABLMDEUUNUUEUUCUUMSUULUUCUUMUIUUMUUDUUCUJ - UKULUMUNUOUUBUUGJLDDUPUQYBUUHYMJLDDUUHUUAUUFPZMERKERYBYRDOZUUCDOZPZPZYMUU - AUUFKMEEUPUVAUUQYMKMEEUVAYSEOZUUDEOZPZPYMUUQYDUUNOZUUNYRUUCUCZYSUUDUCZSZU - DZPZBERADRZUAUVHUEZYBUUTUVDUVLXTUUTYAUVDUVLXTUUTPZYAUVDPZPZYGYHURZUUNOZUV - IPZBERZADRZIUVGUEHUVFUEUVLUVOUVTHIUVFUVGUVMYGUVFOZUVNYHUVGOZUVTUVMUWAPYGU - ULOZUULUVFUDZPZADRZYHUUMOZUUMUVGUDZPZBERZUVTUVNUWBPXTUURUUSUWAUWFXTUURUUS - UWAUWFAYGDYRUUCUSUTVAYAUVBUVCUWBUWJYAUVBUVCUWBUWJBYHEYSUUDUSUTVAUWFUWJPUW - EUWIPZBERZADRUVTUWEUWIABDEUPUWLUVSADUWKUVRBEUWCUWGUWDUWHUVRUWCUWGPUVQUWDU - WHPUVIYGYHUULUUMVBUULUVFUUMUVGVCVDVJVEVEVFVGVJVHUVKUVTUAHIUVFUVGYDUVPUGZU - VJUVRABDEUWMUVEUVQUVIYDUVPUUNVKVLVMVNVOVJVPUUQYLUVKUAYIUVHUUQYIYTUUEUCUVH - YGYTYHUUEVQYRYSUUCUUDVTVRZUUQYLYFYEUVHUDZPZGCRZUVKUUQYKUWPGCUUQYJUWOYFUUQ - YIUVHYEUWNVSWAWBUWQUWPGUUPRZYDNQZWCWDZUWSWEWDZSZOZUXBUVHUDZPZNDESZRZUVKUW - PGCUUPFWFUXBTOZNUXFUEUWRUXGWGUXHNUXFUWTUXAUWTTONUWSTWHWMUXATONUWSTWIWMWJW - KUWPUXENGUXFUXBUUOTNUXFUXBWLUUOABNDEUXBUUNUWSUULUUMURUGZUWTUULUXAUUMUULUU - MUWSAWNZBWNZWOUULUUMUWSUXJUXKWPWQZWRWSYEUXBUGYFUXCUWOUXDYEUXBYDWTYEUXBUVH - XDXAXBXMUXEUVJNABDEUXIUXCUVEUXDUVIUXIUXBUUNYDUXLXCUXIUXBUUNUVHUXLXEXAXNXF - XGXHXIXJXKXJXLXOYBCTOYCYNWGABCDEUBUBFXPHIUAGCTXQXRXS $. + sseq1 ax-mp sseq1d 3bitri bitrdi raleqbidv syl5ibrcom rexlimdvva biimtrid + rexxp syl5bir ralrimivv txbasex isbasis2g syl mpbird ) DUBOZEUBOZPZCUBOZU + AQZGQZOZYEHQZIQZUCZUDZPZGCRZUAYIUEZICUEHCUEZYBYMHICCYGCOZYHCOZPZYGJQZKQZS + ZUGZKERZYHLQZMQZSZUGZMERZPZLDRJDRZYBYMYQUUBJDRZUUGLDRZPUUIYOUUJYPUUKUUJHC + CABDEAQZBQZSZVIZUFZUUJHUHFJKHDEYTUUOABJKDEUUNYTYRUUMSUULYRUUMUIUUMYSYRUJU + KULUMUNUUKICCUUPUUKIUHFLMIDEUUEUUOABLMDEUUNUUEUUCUUMSUULUUCUUMUIUUMUUDUUC + UJUKULUMUNUOUUBUUGJLDDUPUQYBUUHYMJLDDUUHUUAUUFPZMERKERYBYRDOZUUCDOZPZPZYM + UUAUUFKMEEUPUVAUUQYMKMEEUVAYSEOZUUDEOZPZPYMUUQYDUUNOZUUNYRUUCUCZYSUUDUCZS + ZUDZPZBERADRZUAUVHUEZYBUUTUVDUVLXTUUTYAUVDUVLXTUUTPZYAUVDPZPZYGYHURZUUNOZ + UVIPZBERZADRZIUVGUEHUVFUEUVLUVOUVTHIUVFUVGUVMYGUVFOZUVNYHUVGOZUVTUVMUWAPY + GUULOZUULUVFUDZPZADRZYHUUMOZUUMUVGUDZPZBERZUVTUVNUWBPXTUURUUSUWAUWFXTUURU + USUWAUWFAYGDYRUUCUSUTVAYAUVBUVCUWBUWJYAUVBUVCUWBUWJBYHEYSUUDUSUTVAUWFUWJP + UWEUWIPZBERZADRUVTUWEUWIABDEUPUWLUVSADUWKUVRBEUWCUWGUWDUWHUVRUWCUWGPUVQUW + DUWHPUVIYGYHUULUUMVBUULUVFUUMUVGVCVDVJVEVEVFVGVJVHUVKUVTUAHIUVFUVGYDUVPUG + ZUVJUVRABDEUWMUVEUVQUVIYDUVPUUNVKVLVMVNVOVJVPUUQYLUVKUAYIUVHUUQYIYTUUEUCU + VHYGYTYHUUEVQYRYSUUCUUDVTVRZUUQYLYFYEUVHUDZPZGCRZUVKUUQYKUWPGCUUQYJUWOYFU + UQYIUVHYEUWNVSWAWBUWQUWPGUUPRZYDNQZWCWDZUWSWEWDZSZOZUXBUVHUDZPZNDESZRZUVK + UWPGCUUPFWFUXBTOZNUXFUEUWRUXGWGUXHNUXFUWTUXAUWTTONUWSTWHWMUXATONUWSTWIWMW + JWKUWPUXENGUXFUXBUUOTNUXFUXBWLUUOABNDEUXBUUNUWSUULUUMURUGZUWTUULUXAUUMUUL + UUMUWSAWNZBWNZWOUULUUMUWSUXJUXKWPWQZWRWSYEUXBUGYFUXCUWOUXDYEUXBYDWTYEUXBU + VHXDXAXBXEUXEUVJNABDEUXIUXCUVEUXDUVIUXIUXBUUNYDUXLXCUXIUXBUUNUVHUXLXFXAXM + XGXHXIXJXKXNXKXLXOYBCTOYCYNWGABCDEUBUBFXPHIUAGCTXQXRXS $. $} ${ @@ -152009,23 +153115,23 @@ F C_ ( CC X. X ) ) $= cuni wex eltg3 bi2anan9 exdistrv an4 xpeq12i xpiundir xpiundi a1i iuneq2i uniiun 3eqtri txvalex adantr ad2antrr ssel2 anim12i an4s sylan2 ralrimiva txopn anassrs tgiun syl2anc fveq2d 3eqtr4d eleqtrd eqeltrid xpeq12 eleq1d - tgidm txval syl5ibrcom expimpd syl5bi syl5bir sylbid ralrimivv fmpo sylib - exlimdvv frnd sseqtrd 2basgeng syl3anc tgvalex 3eqtr4rd ) ACKZBDKZLZEFABE - MZFMZNZUAZUBZOPZEFAOPZBOPZXJUAZUBZOPZABUCUDZXNXOUCUDZXGXLQKZXLXQRZXQXMRXM - XRSEFXLABCDXLUEZUFZXGXKXPRYBXGXKXPABNZUGZXPXEAXNRBXORYFXKSXFACUHBDUHEFXNX - OABXJUIUJXPYEUKULXKXPUMUNXGXQXSXMXGXNXONZXSXPXGXJXSKZFXOUOEXNUOYGXSXPUPXG - YHEFXNXOXGXHXNKZXIXOKZLGMZARZXHYKUQZSZLZGURZHMZBRZXIYQUQZSZLZHURZLZYHXEYI - YPXFYJUUBGXHACUSHXIBDUSUTUUCYOUUALZHURGURXGYHYOUUAGHVAXGUUDYHGHUUDYLYRLZY - NYTLZLXGYHYLYNYRYTVBXGUUEUUFYHXGUUELZYHUUFYMYSNZXSKUUGUUHIYKJYQIMZJMZNZTZ - TZXSUUHIYKUUITZJYQUUJTZNIYKUUIUUONZTUUMYMUUNYSUUOIYKVHJYQVHVCIYKUUIUUOVDI - YKUUPUULUUPUULSUUIYKKZJYQUUJUUIVEVFVGVIUUGUUMXSOPZXSUUGXSQKZUULXSKZIYKUOU - UMUURKXGUUSUUEABCDVJZVKUUGUUTIYKUUGUUQLZUULUURXSUVBUUSUUKXSKZJYQUOUULUURK - XGUUSUUEUUQUVAVLUVBUVCJYQUUGUUQUUJYQKZUVCXGUUEUUQUVDLZUVCUUEUVELXGUUIAKZU - UJBKZLZUVCYLUUQYRUVDUVHYLUUQLUVFYRUVDLUVGYKAUUIVMYQBUUJVMVNVOUUIUUJABCDVR - VPVSVSVQJYQXSUUKQVTWAUUGUURXSSZUUQXGUVIUUEXGXMOPZXMUURXSXGYAUVJXMSYDXLQWH - UNXGXSXMOEFXLABCDYCWIZWBUVKWCVKZVKWDVQIYKXSUULQVTWAUVLWDWEUUFXJUUHXSXHYMX - IYSWFWGWJWKWLWRWMWNWOEFXNXOXJXSXPXPUEWPWQWSUVKWTXLXQQXAXBUVKXEXNQKXOQKXTX - RSXFACXCBDXCEFXQXNXOQQXQUEWIUJXD $. + tgidm txval syl5ibrcom expimpd biimtrid exlimdvv syl5bir sylbid ralrimivv + fmpo sylib frnd sseqtrd 2basgeng syl3anc tgvalex 3eqtr4rd ) ACKZBDKZLZEFA + BEMZFMZNZUAZUBZOPZEFAOPZBOPZXJUAZUBZOPZABUCUDZXNXOUCUDZXGXLQKZXLXQRZXQXMR + XMXRSEFXLABCDXLUEZUFZXGXKXPRYBXGXKXPABNZUGZXPXEAXNRBXORYFXKSXFACUHBDUHEFX + NXOABXJUIUJXPYEUKULXKXPUMUNXGXQXSXMXGXNXONZXSXPXGXJXSKZFXOUOEXNUOYGXSXPUP + XGYHEFXNXOXGXHXNKZXIXOKZLGMZARZXHYKUQZSZLZGURZHMZBRZXIYQUQZSZLZHURZLZYHXE + YIYPXFYJUUBGXHACUSHXIBDUSUTUUCYOUUALZHURGURXGYHYOUUAGHVAXGUUDYHGHUUDYLYRL + ZYNYTLZLXGYHYLYNYRYTVBXGUUEUUFYHXGUUELZYHUUFYMYSNZXSKUUGUUHIYKJYQIMZJMZNZ + TZTZXSUUHIYKUUITZJYQUUJTZNIYKUUIUUONZTUUMYMUUNYSUUOIYKVHJYQVHVCIYKUUIUUOV + DIYKUUPUULUUPUULSUUIYKKZJYQUUJUUIVEVFVGVIUUGUUMXSOPZXSUUGXSQKZUULXSKZIYKU + OUUMUURKXGUUSUUEABCDVJZVKUUGUUTIYKUUGUUQLZUULUURXSUVBUUSUUKXSKZJYQUOUULUU + RKXGUUSUUEUUQUVAVLUVBUVCJYQUUGUUQUUJYQKZUVCXGUUEUUQUVDLZUVCUUEUVELXGUUIAK + ZUUJBKZLZUVCYLUUQYRUVDUVHYLUUQLUVFYRUVDLUVGYKAUUIVMYQBUUJVMVNVOUUIUUJABCD + VRVPVSVSVQJYQXSUUKQVTWAUUGUURXSSZUUQXGUVIUUEXGXMOPZXMUURXSXGYAUVJXMSYDXLQ + WHUNXGXSXMOEFXLABCDYCWIZWBUVKWCVKZVKWDVQIYKXSUULQVTWAUVLWDWEUUFXJUUHXSXHY + MXIYSWFWGWJWKWLWMWNWOWPEFXNXOXJXSXPXPUEWQWRWSUVKWTXLXQQXAXBUVKXEXNQKXOQKX + TXRSXFACXCBDXCEFXQXNXOQQXQUEWIUJXD $. $} ${ @@ -152112,52 +153218,52 @@ F C_ ( CC X. X ) ) $= txcnp $p |- ( ph -> ( x e. X |-> <. A , B >. ) e. ( ( J CnP ( K tX L ) ) ` D ) ) $= ( cfv wcel wa vy vz vv vw vr vs vt cop cmpt ctx co ccnp cxp cima wss wrex - wf cv wi cmpo crn wral ctopon cnpf2 syl3anc fvmptelrn opelxpd fmpttd wceq - wb simpr elexd eqid fvmpt2 syl2anc opeq12d eqtr4d ralrimiva nffvmpt1 nfop - cvv nfeq fveq2 eqeq12d rspc sylc eleq1d adantr ad2antrr simplrl icnpimaex - simprl simplrr simprr jca ex opelxp reeanv 3imtr4g sylbid cin an4 biimpri - syl33anc elin a1i simpl toponss syl2an ssinss1 adantl sselda elin1d ffund - wfun cdm fdmd sseqtrrd sseldd funfvima mpd elin2d funimass4 mpbird syldan - eqeltrd adantlr xpss12 sstr2 syl2im anim12d ctop topontop syl inopn 3expb - syl5bi sylan eleq2 vex jctild expimpd imaeq2 sseq1d rspcev syl6 rexlimdvv - anbi12d expd syld ralrimivva rgen2w sseq2 anbi2d rexbidv imbi12d ralrnmpo - xpex ax-mp sylibr ctg txval txtopon tgcnp mpbir2and ) ABICDUHZUIZEFGHUJUK - ZULUKRSIJKUMZUVGUQEUVGRZUAURZSZEUBURZSZUVGUVMUNZUVKUOZTZUBFUPZUSZUAUCUDGH - UCURZUDURZUMZUTZVAZVBZABIUVFUVIABURZISZTZCDJKABICJAFIVCRSZGJVCRSZBICUIZEF - 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X ) ) $= xpss cres coeq1i fveq1i eqtrdi 3ad2ant2 simpr1 fvco3 sylan fvresd 3eqtrrd ad2antlr 3ad2ant3 eqopi syl12anc fveq2 opeq12d eqeltrrd fvmptd3 eqtr4d ex simpr eqfnfvd crn wss wfo fo1st fofn ax-mp ssv fnssres mp2an frnd mp3an2i - fnco simpl2 ffvelrnd simpl3 opelxpd fveq2d 3adantl1 syl2anc eqtrd eqtr4di - op1stg fo2nd op2ndg 3jca feq1 coeq2 eqeq2d 3anbi123d impbid eubidv mpbird - syl5ibrcom ) ADMZABHUBZACIUBZUCZABCUDZGUTZUBZHEYLUEZNZIFYLUEZNZUCZGUFYLLA - LUTZHOZYSIOZUGZUIZNZGUFZYGYHUUEYIYGUUCPMUUELAUUBDUHGUUCUJUKULYJYRUUDGYJYR - UUDYJYRUUDYJYRQZUAAYLUUCYRYLARZYJYMYOUUGYQAYKYLUMULUNYJUUCARZYRYJAYKUUCYJ - UUBYKMZLAUOZAYKUUCUBZYHYIUUJYGYHYIQUUILAYHYIYSAMZUUIYHUULQYTBMUUACMUUIYIU - ULQABYSHUPACYSIUPYTUUABCUQURUSVAVBLAYKUUBUUCUUCVCZVDUKZVEZVFUUFUAUTZAMZQZ - UUPYLOZUUPHOZUUPIOZUGZUUPUUCOZUURUUSPPUDZMZUUSSOZUUTNUUSTOZUVANUUSUVBNYRU - UQUVEYJYMYOUUQUVEYQYMUUQQYKUVDUUSBCVKAYKUUPYLUPZVGVHVIUURUUTUUPSYKVLZYLUE - ZOZUUSUVIOZUVFYRUUTUVKNZYJUUQYOYMUVMYQYOUUTUUPYNOUVKUUPHYNVJUUPYNUVJEUVIY - LJVMVNVOVPWBUUFYMUUQUVKUVLNYJYMYOYQVQZAYKUUPUVIYLVRVSUURUUSYKSYRUUQUUSYKM - ZYJYMYOUUQUVOYQUVHVHVIZVTWAUURUVAUUPTYKVLZYLUEZOZUUSUVQOZUVGYRUVAUVSNZYJU - UQYQYMUWAYOYQUVAUUPYPOUVSUUPIYPVJUUPYPUVRFUVQYLKVMVNVOWCWBUUFYMUUQUVSUVTN - UVNAYKUUPUVQYLVRVSUURUUSYKTUVPVTWAUUSUUTUVAPPWDWEZUURLUUPUUBUVBAUUCYKUUMY - SUUPNYTUUTUUAUVAYSUUPHWFYSUUPIWFWGZUUFUUQWLUURUUSUVBYKUWBUVPWHWIWJWMWKYJY - RUUDUUKHEUUCUEZNZIFUUCUEZNZUCYJUUKUWEUWGUUNYJHUVIUUCUEZUWDYJUAAHUWHYHYGHA - RYIABHUMVPUVIYKRZYJUUHUUCWNYKWOZUWHARSPRZYKPWOZUWIPPSWPUWKWQPPSWRWSYKWTZP - YKSXAXBUUOYJAYKUUCUUNXCZYKAUVIUUCXEXDYJUUQQZUUPUWHOZUVCUVIOZUVBUVIOZUUTYJ - UUKUUQUWPUWQNUUNAYKUUPUVIUUCVRVSUWOUVCUVBUVIUWOLUUPUUBUVBAUUCYKUUMUWCYJUU - QWLZUWOUUTUVABCUWOABUUPHYGYHYIUUQXFUWSXGZUWOACUUPIYGYHYIUUQXHUWSXGZXIWIZX - JUWOUWRUVBSOZUUTUWOUVBYKSYHYIUUQUVBYKMZYGYHYIUUQUXDYHUUQQUUTBMZUVACMZUXDY - IUUQQABUUPHUPACUUPIUPUUTUVABCUQURUSXKZVTUWOUXEUXFUXCUUTNUWTUXAUUTUVABCXOX - LXMWAWMEUVIUUCJVMXNYJIUVQUUCUEZUWFYJUAAIUXHYIYGIARYHACIUMWCUVQYKRZYJUUHUW - JUXHARTPRZUWLUXIPPTWPUXJXPPPTWRWSUWMPYKTXAXBUUOUWNYKAUVQUUCXEXDUWOUUPUXHO - ZUVCUVQOZUVBUVQOZUVAYJUUKUUQUXKUXLNUUNAYKUUPUVQUUCVRVSUWOUVCUVBUVQUXBXJUW - OUXMUVBTOZUVAUWOUVBYKTUXGVTUWOUXEUXFUXNUVANUWTUXAUUTUVABCXQXLXMWAWMFUVQUU - CKVMXNXRUUDYMUUKYOUWEYQUWGAYKYLUUCXSUUDYNUWDHYLUUCEXTYAUUDYPUWFIYLUUCFXTY - AYBYFYCYDYE $. + fnco simpl2 ffvelcdmd simpl3 opelxpd fveq2d 3adantl1 op1stg syl2anc eqtrd + eqtr4di fo2nd op2ndg 3jca coeq2 eqeq2d 3anbi123d syl5ibrcom impbid eubidv + feq1 mpbird ) ADMZABHUBZACIUBZUCZABCUDZGUTZUBZHEYLUEZNZIFYLUEZNZUCZGUFYLL + ALUTZHOZYSIOZUGZUIZNZGUFZYGYHUUEYIYGUUCPMUUELAUUBDUHGUUCUJUKULYJYRUUDGYJY + RUUDYJYRUUDYJYRQZUAAYLUUCYRYLARZYJYMYOUUGYQAYKYLUMULUNYJUUCARZYRYJAYKUUCY + JUUBYKMZLAUOZAYKUUCUBZYHYIUUJYGYHYIQUUILAYHYIYSAMZUUIYHUULQYTBMUUACMUUIYI + UULQABYSHUPACYSIUPYTUUABCUQURUSVAVBLAYKUUBUUCUUCVCZVDUKZVEZVFUUFUAUTZAMZQ + ZUUPYLOZUUPHOZUUPIOZUGZUUPUUCOZUURUUSPPUDZMZUUSSOZUUTNUUSTOZUVANUUSUVBNYR + UUQUVEYJYMYOUUQUVEYQYMUUQQYKUVDUUSBCVKAYKUUPYLUPZVGVHVIUURUUTUUPSYKVLZYLU + EZOZUUSUVIOZUVFYRUUTUVKNZYJUUQYOYMUVMYQYOUUTUUPYNOUVKUUPHYNVJUUPYNUVJEUVI + YLJVMVNVOVPWBUUFYMUUQUVKUVLNYJYMYOYQVQZAYKUUPUVIYLVRVSUURUUSYKSYRUUQUUSYK + MZYJYMYOUUQUVOYQUVHVHVIZVTWAUURUVAUUPTYKVLZYLUEZOZUUSUVQOZUVGYRUVAUVSNZYJ + UUQYQYMUWAYOYQUVAUUPYPOUVSUUPIYPVJUUPYPUVRFUVQYLKVMVNVOWCWBUUFYMUUQUVSUVT + NUVNAYKUUPUVQYLVRVSUURUUSYKTUVPVTWAUUSUUTUVAPPWDWEZUURLUUPUUBUVBAUUCYKUUM + YSUUPNYTUUTUUAUVAYSUUPHWFYSUUPIWFWGZUUFUUQWLUURUUSUVBYKUWBUVPWHWIWJWMWKYJ + YRUUDUUKHEUUCUEZNZIFUUCUEZNZUCYJUUKUWEUWGUUNYJHUVIUUCUEZUWDYJUAAHUWHYHYGH + ARYIABHUMVPUVIYKRZYJUUHUUCWNYKWOZUWHARSPRZYKPWOZUWIPPSWPUWKWQPPSWRWSYKWTZ + PYKSXAXBUUOYJAYKUUCUUNXCZYKAUVIUUCXEXDYJUUQQZUUPUWHOZUVCUVIOZUVBUVIOZUUTY + JUUKUUQUWPUWQNUUNAYKUUPUVIUUCVRVSUWOUVCUVBUVIUWOLUUPUUBUVBAUUCYKUUMUWCYJU + UQWLZUWOUUTUVABCUWOABUUPHYGYHYIUUQXFUWSXGZUWOACUUPIYGYHYIUUQXHUWSXGZXIWIZ + XJUWOUWRUVBSOZUUTUWOUVBYKSYHYIUUQUVBYKMZYGYHYIUUQUXDYHUUQQUUTBMZUVACMZUXD + YIUUQQABUUPHUPACUUPIUPUUTUVABCUQURUSXKZVTUWOUXEUXFUXCUUTNUWTUXAUUTUVABCXL + XMXNWAWMEUVIUUCJVMXOYJIUVQUUCUEZUWFYJUAAIUXHYIYGIARYHACIUMWCUVQYKRZYJUUHU + WJUXHARTPRZUWLUXIPPTWPUXJXPPPTWRWSUWMPYKTXAXBUUOUWNYKAUVQUUCXEXDUWOUUPUXH + OZUVCUVQOZUVBUVQOZUVAYJUUKUUQUXKUXLNUUNAYKUUPUVQUUCVRVSUWOUVCUVBUVQUXBXJU + WOUXMUVBTOZUVAUWOUVBYKTUXGVTUWOUXEUXFUXNUVANUWTUXAUUTUVABCXQXMXNWAWMFUVQU + UCKVMXOXRUUDYMUUKYOUWEYQUWGAYKYLUUCYEUUDYNUWDHYLUUCEXSXTUUDYPUWFIYLUUCFXS + XTYAYBYCYDYF $. $} ${ @@ -152256,38 +153362,38 @@ F C_ ( CC X. X ) ) $= ( wcel cvv vx vz ccn co ccom wceq wex wmo wreu cuni cfv cop cmpt ctx eqid wa cv txcnmpt oveq2i eleqtrrdi wf cnf c1st cxp wfn ffn adantr crn wss wfo cres fo1st fofn ax-mp ssv fnssres ffvelcdm opelxpi syl2an anandirs fmpttd - mp2an ffnd mp3an2i fvco3 sylan fveq2 opeq12d simpr simpll ffvelrnd simplr - frnd opelxpd fvmptd3 fveq2d fvresd op1stg syl2anc 3eqtrrd eqfnfvd reseq2i - fnco eqtrd eqtri coeq1i c2nd adantl fo2nd op2ndg jca32 eleq1 coeq2 eqeq2d - eqtr4di anbi12d spcegv sylc w3a wi ctop cntop2 unieqi txuni eqtr4id feq3d - wal syl5ib anim1d 3anass syl6ibr alrimiv weu cntop1 uniexg upxp syl2an3an - syl eumo moim df-reu eu5 bitri sylanbrc ) HFCUCUDSZIFDUCUDSZUPZGUQZFEUCUD - ZSZHAUUHUEZUFZIBUUHUEZUFZUPZUPZGUGZUUPGUHZUUOGUUIUIZUUGUAFUJZUAUQZHUKZUVA - IUKZULZUMZUUISZUVFHAUVEUEZUFZIBUVEUEZUFZUPZUPZUUQUUGUVEFCDUNUDZUCUDUUIUAC - DFHIUVEUUTUUTUOZUVEUOZUREUVMFUCMUSUTZUUGUVFUVHUVJUVPUUEUUTJHVAZUUTKIVAZUV - HUUFHFCUUTJUVNNVBZIFDUUTKUVNOVBZUVQUVRUPZHVCJKVDZVKZUVEUEZUVGUWAUBUUTHUWD - UVQHUUTVEUVRUUTJHVFVGUWCUWBVEZUWAUVEUUTVEZUVEVHUWBVIZUWDUUTVEVCTVEZUWBTVI - 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X ) ) $= cnmpt1t $p |- ( ph -> ( x e. X |-> <. A , B >. ) e. ( J Cn ( K tX L ) ) ) $= ( vy cuni cmpt cfv cop co ccn wcel wceq cv ctopon toponuni mpteq1 3syl wa - ctx simpr wf ctop cntop2 syl toptopon2 cnf2 syl3anc fvmptelrn eqid fvmpt2 - sylib syl2anc opeq12d mpteq2dva eqtr3d nfcv nffvmpt1 fveq2 cbvmpt txcnmpt - nfop eqeltrrd ) ABEMZBUAZBHCNZOZVLBHDNZOZPZNZBHCDPZNZEFGUGQRQZABHVQNZVRVT - AEHUBOSZHVKTWBVRTIHEUCBHVKVQUDUEABHVQVSAVLHSZUFZVNCVPDWEWDCFMZSVNCTAWDUHZ - ABHCWFAWCFWFUBOSZVMEFRQSZHWFVMUIIAFUJSZWHAWIWJJVMEFUKULFUMUSJVMEFHWFUNUOU - PBHCWFVMVMUQURUTWEWDDGMZSVPDTWGABHDWKAWCGWKUBOSZVOEGRQSZHWKVOUIIAGUJSZWLA - WMWNKVOEGUKULGUMUSKVOEGHWKUNUOUPBHDWKVOVOUQURUTVAVBVCAWIWMVRWASJKLFGEVMVO - VRVKVKUQBLVKVQLUAZVMOZWOVOOZPLVQVDBWPWQBHCWOVEBHDWOVEVIVLWOTVNWPVPWQVLWOV - MVFVLWOVOVFVAVGVHUTVJ $. + ctx simpr wf ctop cntop2 syl toptopon2 sylib cnf2 syl3anc fvmptelcdm eqid + fvmpt2 syl2anc opeq12d mpteq2dva eqtr3d nfcv nffvmpt1 nfop cbvmpt txcnmpt + fveq2 eqeltrrd ) ABEMZBUAZBHCNZOZVLBHDNZOZPZNZBHCDPZNZEFGUGQRQZABHVQNZVRV + TAEHUBOSZHVKTWBVRTIHEUCBHVKVQUDUEABHVQVSAVLHSZUFZVNCVPDWEWDCFMZSVNCTAWDUH + ZABHCWFAWCFWFUBOSZVMEFRQSZHWFVMUIIAFUJSZWHAWIWJJVMEFUKULFUMUNJVMEFHWFUOUP + UQBHCWFVMVMURUSUTWEWDDGMZSVPDTWGABHDWKAWCGWKUBOSZVOEGRQSZHWKVOUIIAGUJSZWL + AWMWNKVOEGUKULGUMUNKVOEGHWKUOUPUQBHDWKVOVOURUSUTVAVBVCAWIWMVRWASJKLFGEVMV + OVRVKVKURBLVKVQLUAZVMOZWOVOOZPLVQVDBWPWQBHCWOVEBHDWOVEVFVLWOTVNWPVPWQVLWO + VMVIVLWOVOVIVAVGVHUTVJ $. ${ cnmpt12f.f $e |- ( ph -> F e. ( ( K tX L ) Cn M ) ) $. @@ -152629,17 +153735,17 @@ F C_ ( CC X. X ) ) $= Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) $) cnmpt12 $p |- ( ph -> ( x e. X |-> D ) e. ( J Cn M ) ) $= - ( cmpo co cmpt ccn cv wcel wa cuni wceq ctopon cfv cnf2 syl3anc fvmptelrn - wf wi wal jca wral cxp ctx txtopon syl2anc ctop cntop2 syl toptopon2 eqid - sylib fmpo sylibr r2al adantr eleq1 bi2anan9 eleq1d imbi12d syl3c ovmpoga - spc2gv mpteq2dva cnmpt12f eqeltrrd ) ABMEFCDNOGUCZUDZUEBMHUEILUFUDABMWGHA - BUGMUHZUIZENUHZFOUHZHLUJZUHZWGHUKABMENAIMULUMUHZJNULUMUHZBMEUEZIJUFUDUHMN - WPUQPSQWPIJMNUNUOUPZABMFOAWNKOULUMUHZBMFUEZIKUFUDUHMOWSUQPTRWSIKMOUNUOUPZ - WIWJWKUIZCUGZNUHZDUGZOUHZUIZGWLUHZURZDUSCUSZXAWMWIWJWKWQWTUTZAXIWHAXGDOVA - CNVAZXIANOVBZWLWFUQZXKAJKVCUDZXLULUMUHZLWLULUMUHZWFXNLUFUDUHZXMAWOWRXOSTJ - KNOVDVEALVFUHZXPAXQXRUAWFXNLVGVHLVIVKUAWFXNLXLWLUNUOCDNOGWLWFWFVJZVLVMXGC - DNOVNVKVOXJXHXAWMURCDEFNOXBEUKZXDFUKZUIZXFXAXGWMXTXCWJYAXEWKXBENVPXDFOVPV - QYBGHWLUBVRVSWBVTCDEFNOGHWFWLUBXSWAUOWCABEFWFIJKLMPQRUAWDWE $. + ( cmpo co cmpt ccn cv wcel wa cuni wceq ctopon wf cnf2 syl3anc fvmptelcdm + cfv wal jca wral cxp ctx txtopon syl2anc ctop cntop2 toptopon2 sylib eqid + wi syl fmpo sylibr r2al adantr eleq1 bi2anan9 eleq1d imbi12d spc2gv syl3c + ovmpoga mpteq2dva cnmpt12f eqeltrrd ) ABMEFCDNOGUCZUDZUEBMHUEILUFUDABMWGH + ABUGMUHZUIZENUHZFOUHZHLUJZUHZWGHUKABMENAIMULUQUHZJNULUQUHZBMEUEZIJUFUDUHM + NWPUMPSQWPIJMNUNUOUPZABMFOAWNKOULUQUHZBMFUEZIKUFUDUHMOWSUMPTRWSIKMOUNUOUP + ZWIWJWKUIZCUGZNUHZDUGZOUHZUIZGWLUHZVJZDURCURZXAWMWIWJWKWQWTUSZAXIWHAXGDOU + TCNUTZXIANOVAZWLWFUMZXKAJKVBUDZXLULUQUHZLWLULUQUHZWFXNLUFUDUHZXMAWOWRXOST + JKNOVCVDALVEUHZXPAXQXRUAWFXNLVFVKLVGVHUAWFXNLXLWLUNUOCDNOGWLWFWFVIZVLVMXG + CDNOVNVHVOXJXHXAWMVJCDEFNOXBEUKZXDFUKZUIZXFXAXGWMXTXCWJYAXEWKXBENVPXDFOVP + VQYBGHWLUBVRVSVTWACDEFNOGHWFWLUBXSWBUOWCABEFWFIJKLMPQRUAWDWE $. $} ${ @@ -153235,8 +154341,8 @@ F C_ ( CC X. X ) ) $= by Thierry Arnoux, 7-Feb-2018.) $) psmetcl $p |- ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* ) $= - ( cpsmet cfv wcel cxp cxr wf co psmetf fovrn syl3an1 ) CDEFGDDHICJADGBDGA - BCKIGCDLABIDDCMN $. + ( cpsmet cfv wcel cxp cxr wf co psmetf fovcdm syl3an1 ) CDEFGDDHICJADGBDG + ABCKIGCDLABIDDCMN $. $d a A $. $d a b c d e $. $( The distance function of a pseudometric space is zero if its arguments @@ -153504,24 +154610,25 @@ the base set to the (extended) reals and which is nonnegative, symmetric except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.) $) isxmet2d $p |- ( ph -> D e. ( *Met ` X ) ) $= - ( wcel wa co cc0 wceq cle wbr cxr cpnf cv fovrnda 0xr xrletri3 biantrud - wb sylancl 3bitr2d w3a cr cxad cmnf caddc 3expa rexadd breqtrrd anassrs - adantl 3adantr3 pnfge syl ad2antrr oveq2 wne cicc cxp wral ffnd elxrge0 - wf wfn sylanbrc ralrimivva adantr simpr3 simpr1 fovrnd simplbi xaddpnf1 - ffnov renemnf syl2an sylan9eqr wi simpr2 simprbi syl2anc neneqd pm2.21d - ge0nemnf imp w3o elxr sylib mpjao3dan oveq1 xaddpnf2 isxmetd ) ABCDEFGH - ABUAZFLZCUAZFLZMMZWSXAENZOPZXDOQRZOXDQRZMZXFWSXAPXCXDSLZOSLXEXHUFAWSXAS - FFEHUBZUCXDOUDUGXCXGXFIUEJUHAWTXBDUAZFLZUIZMZXKWSENZUJLZXDXOXKXAENZUKNZ - QRZXOTPZXOULPZXNXPMZXQUJLZXSXQTPZXQULPZXNXPYCXSXNXPYCMZMXDXOXQUMNZXRQAX - MYFXDYGQRKUNYFXRYGPXNXOXQUOURUPUQYBYDMXDTXRQXNXDTQRZXPYDXNXIYHAWTXBXIXL - XJUSXDUTVAZVBYDYBXRXOTUKNZTXQTXOUKVCXNXOSLZXOULVDZYJTPXPXNXOOTVENZLZYKX - NXKWSYMFFEAFFVFZYMEVJZXMAEYOVKXDYMLZCFVGBFVGYPAYOSEHVHAYQBCFFXCXIXGYQXJ - IXDVIVLVMBCFFYMEVTVLVNZAWTXBXLVOZAWTXBXLVPVQZYNYKOXOQRZXOVIZVRVAZXOWAXO - VSWBWCUPYBYEXSXNYEXSWDXPXNYEXSXNXQULXNXQSLZOXQQRZXQULVDZXNXQYMLZUUDXNXK - XAYMFFEYRYSAWTXBXLWEVQZUUGUUDUUEXQVIZVRVAZXNUUGUUEUUHUUGUUDUUEUUIWFVAXQ - WJWGZWHWIVNWKYBUUDYCYDYEWLXNUUDXPUUJVNXQWMWNWOXNXTMXDTXRQXNYHXTYIVNXTXN - XRTXQUKNZTXOTXQUKWPXNUUDUUFUULTPUUJUUKXQWQWGWCUPXNYAXSXNYAXSXNXOULXNYKU - UAYLUUCXNYNUUAYTYNYKUUAUUBWFVAXOWJWGWHWIWKXNYKXPXTYAWLUUCXOWMWNWOWR $. + ( wcel wa co cc0 wceq cle wbr cxr cpnf cv fovcdmda 0xr xrletri3 sylancl + wb biantrud 3bitr2d w3a cr cxad cmnf caddc 3expa rexadd adantl breqtrrd + anassrs 3adantr3 pnfge syl ad2antrr oveq2 wne cicc cxp wf wfn wral ffnd + elxrge0 sylanbrc ralrimivva ffnov adantr simpr3 fovcdmd simplbi renemnf + simpr1 xaddpnf1 syl2an sylan9eqr simpr2 simprbi ge0nemnf syl2anc neneqd + wi pm2.21d imp w3o elxr sylib mpjao3dan oveq1 xaddpnf2 isxmetd ) ABCDEF + GHABUAZFLZCUAZFLZMMZWSXAENZOPZXDOQRZOXDQRZMZXFWSXAPXCXDSLZOSLXEXHUFAWSX + ASFFEHUBZUCXDOUDUEXCXGXFIUGJUHAWTXBDUAZFLZUIZMZXKWSENZUJLZXDXOXKXAENZUK + NZQRZXOTPZXOULPZXNXPMZXQUJLZXSXQTPZXQULPZXNXPYCXSXNXPYCMZMXDXOXQUMNZXRQ + AXMYFXDYGQRKUNYFXRYGPXNXOXQUOUPUQURYBYDMXDTXRQXNXDTQRZXPYDXNXIYHAWTXBXI + XLXJUSXDUTVAZVBYDYBXRXOTUKNZTXQTXOUKVCXNXOSLZXOULVDZYJTPXPXNXOOTVENZLZY + KXNXKWSYMFFEAFFVFZYMEVGZXMAEYOVHXDYMLZCFVIBFVIYPAYOSEHVJAYQBCFFXCXIXGYQ + XJIXDVKVLVMBCFFYMEVNVLVOZAWTXBXLVPZAWTXBXLVTVQZYNYKOXOQRZXOVKZVRVAZXOVS + XOWAWBWCUQYBYEXSXNYEXSWIXPXNYEXSXNXQULXNXQSLZOXQQRZXQULVDZXNXQYMLZUUDXN + XKXAYMFFEYRYSAWTXBXLWDVQZUUGUUDUUEXQVKZVRVAZXNUUGUUEUUHUUGUUDUUEUUIWEVA + XQWFWGZWHWJVOWKYBUUDYCYDYEWLXNUUDXPUUJVOXQWMWNWOXNXTMXDTXRQXNYHXTYIVOXT + XNXRTXQUKNZTXOTXQUKWPXNUUDUUFUULTPUUJUUKXQWQWGWCUQXNYAXSXNYAXSXNXOULXNY + KUUAYLUUCXNYNUUAYTYNYKUUAUUBWEVAXOWFWGWHWJWKXNYKXPXTYAWLUUCXOWMWNWOWR + $. $} $} @@ -153558,15 +154665,15 @@ the base set to the (extended) reals and which is nonnegative, symmetric of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) $) xmetcl $p |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* ) $= - ( cxmet cfv wcel cxp cxr wf co xmetf fovrn syl3an1 ) CDEFGDDHICJADGBDGABC - KIGCDLABIDDCMN $. + ( cxmet cfv wcel cxp cxr wf co xmetf fovcdm syl3an1 ) CDEFGDDHICJADGBDGAB + CKIGCDLABIDDCMN $. $( Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) $) metcl $p |- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR ) $= - ( cmet cfv wcel cxp cr wf co metf fovrn syl3an1 ) CDEFGDDHICJADGBDGABCKIG - CDLABIDDCMN $. + ( cmet cfv wcel cxp cr wf co metf fovcdm syl3an1 ) CDEFGDDHICJADGBDGABCKI + GCDLABIDDCMN $. $( An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, @@ -153575,15 +154682,15 @@ the base set to the (extended) reals and which is nonnegative, symmetric ( D e. ( *Met ` X ) /\ D : ( X X. X ) --> RR ) ) $= ( vx vy vz cmet cfv wcel cvv cxmet cr wf wa wrel cdm cv co cle wral cxr cxp metrel relelfvdm elexd mpan xmetrel adantr cc0 wceq wb caddc wbr cxad - simpllr simpr simplrl fovrnd simplrr breq2d ralbidva anbi2d 2ralbidva wss - rexaddd ressxr fss sylancl biantrurd bitr3d pm5.32da bitrdi isxmet anbi1d - ancom ismet 3bitr4d pm5.21nii ) ABFGHZBIHZABJGHZBBUAZKALZMZFNZVRVSUBWDVRM - BFOABFUCUDUEVTVSWBVTBJOZJNVTBWEHUFABJUCUEUDUGVSWBCPZDPZAQZUHUIWFWGUIUJZWH - EPZWFAQZWJWGAQZUKQZRULZEBSZMZDBSCBSZMZWATALZWIWHWKWLUMQZRULZEBSZMZDBSCBSZ - MZWBMZVRWCVSWRWBXEMXFVSWBWQXEVSWBMZXDWQXEXGXCWPCDBBXGWFBHZWGBHZMZMZXBWOWI - XKXAWNEBXKWJBHZMZWTWMWHRXMWKWLXMWJWFKBBAVSWBXJXLUNZXKXLUOZXGXHXIXLUPUQXMW - JWGKBBAXNXOXGXHXIXLURUQVDUSUTVAVBXGWSXDXGWBKTVCWSVSWBUOVEWAKTAVFVGVHVIVJW - BXEVNVKCDEIABVOVSVTXEWBCDEIABVLVMVPVQ $. + simpllr simpr simplrl fovcdmd simplrr rexaddd breq2d anbi2d 2ralbidva wss + ralbidva ressxr fss sylancl biantrurd bitr3d pm5.32da ancom bitrdi isxmet + ismet anbi1d 3bitr4d pm5.21nii ) ABFGHZBIHZABJGHZBBUAZKALZMZFNZVRVSUBWDVR + MBFOABFUCUDUEVTVSWBVTBJOZJNVTBWEHUFABJUCUEUDUGVSWBCPZDPZAQZUHUIWFWGUIUJZW + HEPZWFAQZWJWGAQZUKQZRULZEBSZMZDBSCBSZMZWATALZWIWHWKWLUMQZRULZEBSZMZDBSCBS + ZMZWBMZVRWCVSWRWBXEMXFVSWBWQXEVSWBMZXDWQXEXGXCWPCDBBXGWFBHZWGBHZMZMZXBWOW + IXKXAWNEBXKWJBHZMZWTWMWHRXMWKWLXMWJWFKBBAVSWBXJXLUNZXKXLUOZXGXHXIXLUPUQXM + WJWGKBBAXNXOXGXHXIXLURUQUSUTVDVAVBXGWSXDXGWBKTVCWSVSWBUOVEWAKTAVFVGVHVIVJ + WBXEVKVLCDEIABVNVSVTXEWBCDEIABVMVOVPVQ $. $( A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) $) @@ -154042,16 +155149,16 @@ the base set to the (extended) reals and which is nonnegative, symmetric ( vx cfv wcel w3a cxr cxad co wbr wa clt wb xmetcl adantr syl3anc cle cbl cxmet cin cv wn simpr3 simpr1 simpr2 xaddcld xrlenlt syl2anc mpbid simpl1 elin simpl2 elbl simpl3 anbi12d anandi bitr4di wi simpr syl22anc xmettri3 - xlt2add syl13anc xrlelttr mpand syld expimpd sylbid syl5bi mtod eq0rdv ) - AFUCHIZBFIZCFIZJZDKIZEKIZDELMZBCAMZUANZJZOZGBDAUBHZMZCEWGMZUDZWFGUEZWJIZW - CWBPNZWFWDWMUFZVSVTWAWDUGWFWBKIZWCKIZWDWNQWFDEVSVTWAWDUHZVSVTWAWDUIZUJZVS - WPWEBCAFRSZWBWCUKULUMWLWKWHIZWKWIIZOZWFWMWKWHWIUOWFXCWKFIZBWKAMZDPNZCWKAM - ZEPNZOZOZWMWFXCXDXFOZXDXHOZOXJWFXAXKXBXLWFVPVQVTXAXKQVPVQVRWEUNZVPVQVRWEU - PZWQWKABDFUQTWFVPVRWAXBXLQXMVPVQVRWEURZWRWKACEFUQTUSXDXFXHUTVAWFXDXIWMWFX - DOZXIXEXGLMZWBPNZWMXPXEKIZXGKIZVTWAXIXRVBXPVPVQXDXSWFVPXDXMSZWFVQXDXNSZWF - XDVCZBWKAFRTZXPVPVRXDXTYAWFVRXDXOSZYCCWKAFRTZWFVTXDWQSWFWAXDWRSXEXGDEVFVD - XPWCXQUANZXRWMXPVPVQVRXDYGYAYBYEYCBCWKAFVEVGXPWPXQKIWOYGXROWMVBWFWPXDWTSX - PXEXGYDYFUJWFWOXDWSSWCXQWBVHTVIVJVKVLVMVNVO $. + xlt2add syl13anc xrlelttr mpand syld expimpd sylbid biimtrid mtod eq0rdv + ) AFUCHIZBFIZCFIZJZDKIZEKIZDELMZBCAMZUANZJZOZGBDAUBHZMZCEWGMZUDZWFGUEZWJI + ZWCWBPNZWFWDWMUFZVSVTWAWDUGWFWBKIZWCKIZWDWNQWFDEVSVTWAWDUHZVSVTWAWDUIZUJZ + VSWPWEBCAFRSZWBWCUKULUMWLWKWHIZWKWIIZOZWFWMWKWHWIUOWFXCWKFIZBWKAMZDPNZCWK + AMZEPNZOZOZWMWFXCXDXFOZXDXHOZOXJWFXAXKXBXLWFVPVQVTXAXKQVPVQVRWEUNZVPVQVRW + EUPZWQWKABDFUQTWFVPVRWAXBXLQXMVPVQVRWEURZWRWKACEFUQTUSXDXFXHUTVAWFXDXIWMW + FXDOZXIXEXGLMZWBPNZWMXPXEKIZXGKIZVTWAXIXRVBXPVPVQXDXSWFVPXDXMSZWFVQXDXNSZ + WFXDVCZBWKAFRTZXPVPVRXDXTYAWFVRXDXOSZYCCWKAFRTZWFVTXDWQSWFWAXDWRSXEXGDEVF + VDXPWCXQUANZXRWMXPVPVQVRXDYGYAYBYEYCBCWKAFVEVGXPWPXQKIWOYGXROWMVBWFWPXDWT + SXPXEXGYDYFUJWFWOXDWSSWCXQWBVHTVIVJVKVLVMVNVO $. $( A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) $) @@ -154325,8 +155432,8 @@ the base set to the (extended) reals and which is nonnegative, symmetric NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) $) blssm $p |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X ) $= - ( cxmet cfv wcel cxr w3a cbl co cxp cpw wf blf fovrn syl3an1 elpwid ) ADE - FGZBDGZCHGZIBCAJFZKZDSDHLDMZUBNTUAUCUDGADOBCUDDHUBPQR $. + ( cxmet cfv wcel cxr w3a cbl co cxp cpw wf blf fovcdm syl3an1 elpwid ) AD + EFGZBDGZCHGZIBCAJFZKZDSDHLDMZUBNTUAUCUDGADOBCUDDHUBPQR $. $( The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, @@ -155236,33 +156343,33 @@ S C_ ( P ( ball ` D ) T ) ) $= vc relelfvdm sylancr elexd cpnf cicc wf cxp wfn cv wral xmetf syl ffnd wa w3a xmetge0 elxrge0 sylanbrc 3expb sylan ralrimivva ffnov fco syl2anc cop xmetcl opelxpi fvco3 syl2an df-ov fveq2i 3eqtr4g wb fveq2 eqeq1 ralrimiva - eqeq1d bibi12d adantr rspcdva 3bitrd 3adantr3 simpr3 simpr1 fovrnd simpr2 - xmeteq0 ffvelrnd ge0xaddcl xaddcld 3anrot xmettri2 sylan2br breq1 imbi12d - wi breq1d breq2 breq2d rspc2va syl21anc mpd fvoveq1 oveq1d breq12d fveq2d - oveq2 oveq2d xrletrd opelxpd oveq12d 3brtr4d isxmetd ) AUAUBUHEDUCZFAFUDU - EZAUDUFDFUDLMZFYCMUGGDFUDUIUJUKANULUMOZPEUNZFFUOZYEDUNZYGPYBUNHADYGUPUAUQ - ZUBUQZDOZYEMZUBFURUAFURYHAYGPDAYDYGPDUNZGDFUSUTZVAAYLUAUBFFAYDYIFMZYJFMZV - BZYLGYDYOYPYLYDYOYPVCYKPMNYKQRYLYIYJDFVNYIYJDFVDYKVEVFVGVHZVIUAUBFFYEDVJV - FZYGYEPEDVKVLAYQVBZYIYJYBOZNSYKELZNSZYKNSZYIYJSZYTUUAUUBNYTYIYJVMZYBLZUUF - DLZELZUUAUUBAYMUUFYGMUUGUUISYQYNYIYJFFVOYGPUUFEDVPVQYIYJYBVRYKUUHEYIYJDVR - VSVTZWEYTBUQZELZNSZUUKNSZWAZUUCUUDWABYEYKUUKYKSZUUMUUCUUNUUDUUPUULUUBNUUK - YKEWBZWEUUKYKNWCWFAUUOBYEURYQAUUOBYEIWDWGYRWHAYDYQUUDUUEWAZGYDYOYPUURYIYJ - DFWOVGVHWIAYOYPUHUQZFMZVCZVBZUUBUUSYIDOZELZUUSYJDOZELZTOZUUAUUSYIYBOZUUSY - JYBOZTOQUVBUUBUVCUVETOZELZUVGUVBYEPYKEAYFUVAHWGZAYOYPYLUUTYRWJZWPUVBYEPUV - JEUVLUVBUVCYEMZUVEYEMZUVJYEMZUVBUUSYIYEFFDAYHUVAYSWGZAYOYPUUTWKZAYOYPUUTW - LZWMZUVBUUSYJYEFFDUVQUVRAYOYPUUTWNZWMZUVCUVEWQVLZWPUVBUVDUVFUVBYEPUVCEUVL - UVTWPUVBYEPUVEEUVLUWBWPWRUVBYKUVJQRZUUBUVKQRZAYDUVAUWDGUVAYDUUTYOYPVCUWDU - UTYOYPWSYIYJUUSDFWTXAVHUVBYLUVPUUKCUQZQRZUULUWFELZQRZXDZCYEURBYEURZUWDUWE - XDZUVMUWCAUWKUVAAUWJBCYEYEJVIWGUWJUWLYKUWFQRZUUBUWHQRZXDBCYKUVJYEYEUUPUWG - UWMUWIUWNUUKYKUWFQXBUUPUULUUBUWHQUUQXEXCUWFUVJSZUWMUWDUWNUWEUWFUVJYKQXFUW - OUWHUVKUUBQUWFUVJEWBXGXCXHXIXJUVBUVNUVOUUKUWFTOELZUULUWHTOZQRZCYEURBYEURZ - UVKUVGQRZUVTUWBAUWSUVAAUWRBCYEYEKVIWGUWRUWTUVCUWFTOZELZUVDUWHTOZQRBCUVCUV - EYEYEUUKUVCSZUWPUXBUWQUXCQUUKUVCUWFETXKUXDUULUVDUWHTUUKUVCEWBXLXMUWFUVESZ - UXBUVKUXCUVGQUXEUXAUVJEUWFUVEUVCTXOXNUXEUWHUVFUVDTUWFUVEEWBXPXMXHXIXQAYOY - PUUAUUBSUUTUUJWJUVBUVHUVDUVIUVFTUVBUUSYIVMZYBLZUXFDLZELZUVHUVDUVBYMUXFYGM - UXGUXISAYMUVAYNWGZUVBUUSYIFFUVRUVSXRYGPUXFEDVPVLUUSYIYBVRUVCUXHEUUSYIDVRV - SVTUVBUUSYJVMZYBLZUXKDLZELZUVIUVFUVBYMUXKYGMUXLUXNSUXJUVBUUSYJFFUVRUWAXRY - GPUXKEDVPVLUUSYJYBVRUVEUXMEUUSYJDVRVSVTXSXTYA $. + eqeq1d bibi12d adantr rspcdva xmeteq0 3bitrd simpr3 simpr1 fovcdmd simpr2 + 3adantr3 ffvelcdmd ge0xaddcl xaddcld 3anrot xmettri2 breq1 breq1d imbi12d + sylan2br wi breq2 breq2d rspc2va syl21anc mpd fvoveq1 oveq1d oveq2 fveq2d + breq12d oveq2d xrletrd opelxpd oveq12d 3brtr4d isxmetd ) AUAUBUHEDUCZFAFU + DUEZAUDUFDFUDLMZFYCMUGGDFUDUIUJUKANULUMOZPEUNZFFUOZYEDUNZYGPYBUNHADYGUPUA + UQZUBUQZDOZYEMZUBFURUAFURYHAYGPDAYDYGPDUNZGDFUSUTZVAAYLUAUBFFAYDYIFMZYJFM + ZVBZYLGYDYOYPYLYDYOYPVCYKPMNYKQRYLYIYJDFVNYIYJDFVDYKVEVFVGVHZVIUAUBFFYEDV + JVFZYGYEPEDVKVLAYQVBZYIYJYBOZNSYKELZNSZYKNSZYIYJSZYTUUAUUBNYTYIYJVMZYBLZU + UFDLZELZUUAUUBAYMUUFYGMUUGUUISYQYNYIYJFFVOYGPUUFEDVPVQYIYJYBVRYKUUHEYIYJD + VRVSVTZWEYTBUQZELZNSZUUKNSZWAZUUCUUDWABYEYKUUKYKSZUUMUUCUUNUUDUUPUULUUBNU + UKYKEWBZWEUUKYKNWCWFAUUOBYEURYQAUUOBYEIWDWGYRWHAYDYQUUDUUEWAZGYDYOYPUURYI + YJDFWIVGVHWJAYOYPUHUQZFMZVCZVBZUUBUUSYIDOZELZUUSYJDOZELZTOZUUAUUSYIYBOZUU + SYJYBOZTOQUVBUUBUVCUVETOZELZUVGUVBYEPYKEAYFUVAHWGZAYOYPYLUUTYRWOZWPUVBYEP + UVJEUVLUVBUVCYEMZUVEYEMZUVJYEMZUVBUUSYIYEFFDAYHUVAYSWGZAYOYPUUTWKZAYOYPUU + TWLZWMZUVBUUSYJYEFFDUVQUVRAYOYPUUTWNZWMZUVCUVEWQVLZWPUVBUVDUVFUVBYEPUVCEU + VLUVTWPUVBYEPUVEEUVLUWBWPWRUVBYKUVJQRZUUBUVKQRZAYDUVAUWDGUVAYDUUTYOYPVCUW + DUUTYOYPWSYIYJUUSDFWTXDVHUVBYLUVPUUKCUQZQRZUULUWFELZQRZXEZCYEURBYEURZUWDU + WEXEZUVMUWCAUWKUVAAUWJBCYEYEJVIWGUWJUWLYKUWFQRZUUBUWHQRZXEBCYKUVJYEYEUUPU + WGUWMUWIUWNUUKYKUWFQXAUUPUULUUBUWHQUUQXBXCUWFUVJSZUWMUWDUWNUWEUWFUVJYKQXF + UWOUWHUVKUUBQUWFUVJEWBXGXCXHXIXJUVBUVNUVOUUKUWFTOELZUULUWHTOZQRZCYEURBYEU + RZUVKUVGQRZUVTUWBAUWSUVAAUWRBCYEYEKVIWGUWRUWTUVCUWFTOZELZUVDUWHTOZQRBCUVC + UVEYEYEUUKUVCSZUWPUXBUWQUXCQUUKUVCUWFETXKUXDUULUVDUWHTUUKUVCEWBXLXOUWFUVE + SZUXBUVKUXCUVGQUXEUXAUVJEUWFUVEUVCTXMXNUXEUWHUVFUVDTUWFUVEEWBXPXOXHXIXQAY + OYPUUAUUBSUUTUUJWOUVBUVHUVDUVIUVFTUVBUUSYIVMZYBLZUXFDLZELZUVHUVDUVBYMUXFY + GMUXGUXISAYMUVAYNWGZUVBUUSYIFFUVRUVSXRYGPUXFEDVPVLUUSYIYBVRUVCUXHEUUSYIDV + RVSVTUVBUUSYJVMZYBLZUXKDLZELZUVIUVFUVBYMUXKYGMUXLUXNSUXJUVBUUSYJFFUVRUWAX + RYGPUXKEDVPVLUUSYJYBVRUVEUXMEUUSYJDVRVSVTXSXTYA $. $} ${ @@ -155275,11 +156382,11 @@ S C_ ( P ( ball ` D ) T ) ) $= bdmetval $p |- ( ( ( C : ( X X. X ) --> RR* /\ R e. RR* ) /\ ( A e. X /\ B e. X ) ) -> ( A D B ) = inf ( { ( A C B ) , R } , RR* , < ) ) $= - ( cxp cxr wf wcel wa co cpr clt cinf wceq cv simprl simprr simpll xrmincl - fovrnd simplr syl2anc oveq12 preq1d infeq1d ovmpoga syl3anc ) HHJKELZGKMZ - NZCHMZDHMZNZNZUPUQCDEOZGPZKQRZKMZCDFOVBSUOUPUQUAZUOUPUQUBZUSUTKMUNVCUSCDK - HHEUMUNURUCVDVEUEUMUNURUFUTGUDUGABCDHHATZBTZEOZGPZKQRVBFKVFCSVGDSNZKVIVAQ - VJVHUTGVFCVGDEUHUIUJIUKUL $. + ( cxp cxr wf wcel wa co cpr clt cinf wceq cv simprl simprr simpll fovcdmd + simplr xrmincl syl2anc oveq12 preq1d infeq1d ovmpoga syl3anc ) HHJKELZGKM + ZNZCHMZDHMZNZNZUPUQCDEOZGPZKQRZKMZCDFOVBSUOUPUQUAZUOUPUQUBZUSUTKMUNVCUSCD + KHHEUMUNURUCVDVEUDUMUNURUEUTGUFUGABCDHHATZBTZEOZGPZKQRVBFKVFCSVGDSNZKVIVA + QVJVHUTGVFCVGDEUHUIUJIUKUL $. $( The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario @@ -155660,28 +156767,28 @@ S C_ ( P ( ball ` D ) T ) ) $= C_ ( ( F ` P ) ( ball ` D ) y ) ) ) ) $= ( vv cfv wcel wss wa wrex wi crp vu cxmet w3a ccnp co wf cv cima cbl wral crn ctopon mopntopon 3ad2ant1 ctg wceq mopnval simp3 tgcnp simpll2 simplr - 3ad2ant2 simpll3 ffvelrnd simpr blcntr cxr rpxr adantl blelrn eleq2 sseq2 - syl3anc anbi2d rexbidv imbi12d rspcv syl mpid simpl1 simplrr mopni2 sstr2 - ad2antrr imass2 syl11 reximdv syl5com expr rexlimdv syld ralrimdva simpl2 - expimpd blss 3expib r19.29r ad3antrrr ad2antrl blopn simprl sstr ad2ant2l - ancoms imaeq2 sseq1d anbi12d rspcev syl12anc rexlimdva syl5 expd ralrimdv - com23 exp4a impbid pm5.32da bitrd ) CIUBNOZDJUBNOZEIOZUCZFEGHUDUENOIJFUFZ - EFNZUAUGZOZEMUGZOZFYGUHZYEPZQZMGRZSZUADUINZUKZUJZQYCFEBUGZCUINUEZUHZYDAUG - ZYNUEZPZBTRZATUJZQYBMUAYOEFGHIJXSXTGIULNOYACGIKUMUNXTXSHYOUONUPYADHJLUQVB - XTXSHJULNOYADHJLUMVBXSXTYAURUSYBYCYPUUDYBYCQZYPUUDUUEYPUUCATUUEYTTOZQZYPY - HYIUUAPZQZMGRZUUCUUGYPYDUUAOZUUJUUGXTYDJOZUUFUUKXSXTYAYCUUFUTZUUGIJEFYBYC - UUFVAXSXTYAYCUUFVCZVDZUUEUUFVEDYDYTJVFVMUUGUUAYOOZYPUUKUUJSZSUUGXTUULYTVG - OZUUPUUMUUOUUFUURUUEYTVHVIDYDYTJVJVMYMUUQUAUUAYOYEUUAUPZYFUUKYLUUJYEUUAYD - VKUUSYKUUIMGUUSYJUUHYHYEUUAYIVLVNVOVPVQVRVSUUGUUIUUCMGUUEUUFYGGOZUUIUUCSU - UEUUFUUTQZQZYHUUHUUCUVBYHQZYRYGPZBTRZUUHUUCUVCXSUUTYHUVEUUEXSUVAYHXSXTYAY - CVTZWDUUEUUFUUTYHWAUVBYHVEBYGCEGIKWBVMUUHUVDUUBBTYSYIPUUHUUBUVDYSYIUUAWCY - RYGFWEWFWGWHWNWIWJWKWLUUEUUDYMUAYOUUEUUDYEYOOZYFYLUUEUVGYFQZUUDYLUUEUVHUU - AYEPZATRZUUDYLSUUEXTUVHUVJSXSXTYAYCWMXTUVGYFUVJAYEDYDJWOWPVRUUEUVJUUDYLUV - JUUDQUVIUUCQZATRUUEYLUVIUUCATWQUUEUVKYLATUUGUVIUUCYLUUGUVIQZUUBYLBTUVLYQT - OZUUBYLUVLUVMUUBQZQZYRGOZEYROZYSYEPZYLUVOXSYAYQVGOZUVPUUEXSUUFUVIUVNUVFWR - ZUUGYAUVIUVNUUNWDZUVMUVSUVLUUBYQVHWSCEYQGIKWTVMUVOXSYAUVMUVQUVTUWAUVLUVMU - UBXACEYQIVFVMUVNUVLUVRUUBUVIUVRUVMUUGYSUUAYEXBXCXDYKUVQUVRQMYRGYGYRUPZYHU - VQYJUVRYGYREVKUWBYIYSYEYGYRFXEXFXGXHXIWIXJWNXJXKXLWKXNXOXMXPXQXR $. + 3ad2ant2 simpll3 ffvelcdmd simpr blcntr syl3anc adantl blelrn eleq2 sseq2 + cxr rpxr anbi2d rexbidv imbi12d rspcv mpid simpl1 ad2antrr simplrr mopni2 + sstr2 imass2 syl11 reximdv syl5com expimpd expr rexlimdv ralrimdva simpl2 + syl syld blss 3expib ad3antrrr ad2antrl blopn simprl sstr ad2ant2l ancoms + r19.29r imaeq2 sseq1d anbi12d syl12anc rexlimdva syl5 expd com23 ralrimdv + rspcev exp4a impbid pm5.32da bitrd ) CIUBNOZDJUBNOZEIOZUCZFEGHUDUENOIJFUF + ZEFNZUAUGZOZEMUGZOZFYGUHZYEPZQZMGRZSZUADUINZUKZUJZQYCFEBUGZCUINUEZUHZYDAU + GZYNUEZPZBTRZATUJZQYBMUAYOEFGHIJXSXTGIULNOYACGIKUMUNXTXSHYOUONUPYADHJLUQV + BXTXSHJULNOYADHJLUMVBXSXTYAURUSYBYCYPUUDYBYCQZYPUUDUUEYPUUCATUUEYTTOZQZYP + YHYIUUAPZQZMGRZUUCUUGYPYDUUAOZUUJUUGXTYDJOZUUFUUKXSXTYAYCUUFUTZUUGIJEFYBY + CUUFVAXSXTYAYCUUFVCZVDZUUEUUFVEDYDYTJVFVGUUGUUAYOOZYPUUKUUJSZSUUGXTUULYTV + LOZUUPUUMUUOUUFUURUUEYTVMVHDYDYTJVIVGYMUUQUAUUAYOYEUUAUPZYFUUKYLUUJYEUUAY + DVJUUSYKUUIMGUUSYJUUHYHYEUUAYIVKVNVOVPVQWMVRUUGUUIUUCMGUUEUUFYGGOZUUIUUCS + UUEUUFUUTQZQZYHUUHUUCUVBYHQZYRYGPZBTRZUUHUUCUVCXSUUTYHUVEUUEXSUVAYHXSXTYA + YCVSZVTUUEUUFUUTYHWAUVBYHVEBYGCEGIKWBVGUUHUVDUUBBTYSYIPUUHUUBUVDYSYIUUAWC + YRYGFWDWEWFWGWHWIWJWNWKUUEUUDYMUAYOUUEUUDYEYOOZYFYLUUEUVGYFQZUUDYLUUEUVHU + UAYEPZATRZUUDYLSUUEXTUVHUVJSXSXTYAYCWLXTUVGYFUVJAYEDYDJWOWPWMUUEUVJUUDYLU + VJUUDQUVIUUCQZATRUUEYLUVIUUCATXDUUEUVKYLATUUGUVIUUCYLUUGUVIQZUUBYLBTUVLYQ + TOZUUBYLUVLUVMUUBQZQZYRGOZEYROZYSYEPZYLUVOXSYAYQVLOZUVPUUEXSUUFUVIUVNUVFW + QZUUGYAUVIUVNUUNVTZUVMUVSUVLUUBYQVMWRCEYQGIKWSVGUVOXSYAUVMUVQUVTUWAUVLUVM + UUBWTCEYQIVFVGUVNUVLUVRUUBUVIUVRUVMUUGYSUUAYEXAXBXCYKUVQUVRQMYRGYGYRUPZYH + UVQYJUVRYGYREVJUWBYIYSYEYGYRFXEXFXGXNXHWIXIWHXIXJXKWNXLXOXMXPXQXR $. $( Two ways to say a mapping from metric ` C ` to metric ` D ` is continuous at point ` P ` . (Contributed by NM, 11-May-2007.) (Revised @@ -155693,8 +156800,8 @@ S C_ ( P ( ball ` D ) T ) ) $= ( cfv wcel co crp wral wa wi cxmet w3a ccnp cbl cima wss wrex clt metcnp3 wf cv wbr wb wfun cdm ffun ad2antlr simpll1 simpll3 rpxr ad2antll syl3anc cxr blssm wceq fdm sseqtrrd funimass4 syl2anc elbl imbi1d impexp ad2antrr - simpl2 simplrl rpxrd simpllr adantr ffvelrnd simplr elbl2 syl22anc imbi2d - ffvelcdmda pm5.74da bitrid ralbidv2 anassrs rexbidva ralbidva pm5.32da + simpl2 simplrl simpllr adantr ffvelcdmd simplr ffvelcdmda syl22anc imbi2d + rpxrd elbl2 pm5.74da bitrid ralbidv2 anassrs rexbidva ralbidva pm5.32da bitrd ) DJUANOZEKUANOZFJOZUBZGFHIUCPNOJKGUJZGFBUKZDUDNPZUEFGNZAUKZEUDNPZU FZBQUGZAQRZSWQFCUKZDPWRUHULZWTXFGNZEPXAUHULZTZCJRZBQUGZAQRZSABDEFGHIJKLMU IWPWQXEXMWPWQSZXDXLAQXNXAQOZSXCXKBQXNXOWRQOZXCXKUMXNXOXPSZSZXCXHXBOZCWSRZ @@ -155702,8 +156809,8 @@ S C_ ( P ( ball ` D ) T ) ) $= ZWMWNWOWQXQUSZXPYCXNXOWRUTVAZDFWRJVDVBWQYBJVEWPXQJKGVFUQVGCWSXBGVHVIXRXSX JCWSJXRXFWSOZXSTXFJOZXGSZXSTZYHXJTZXRYGYIXSXRWMWOYCYGYIUMYDYEYFXFDFWRJVJV BVKYJYHXGXSTZTXRYKYHXGXSVLXRYHYLXJXRYHSZXSXIXGYMWNXAVCOWTKOXHKOXSXIUMXNWN - XQYHWMWNWOWQVNVMYMXAXNXOXPYHVOVPYMJKFGWPWQXQYHVQXRWOYHYEVRVSXRJKXFGWPWQXQ - VTWDXHEWTXAKWAWBWCWEWFWLWGWLWHWIWJWKWL $. + XQYHWMWNWOWQVNVMYMXAXNXOXPYHVOWCYMJKFGWPWQXQYHVPXRWOYHYEVQVRXRJKXFGWPWQXQ + VSVTXHEWTXAKWDWAWBWEWFWLWGWLWHWIWJWKWL $. $( Two ways to say a mapping from metric ` C ` to metric ` D ` is continuous at point ` P ` . The distance arguments are swapped compared @@ -155715,14 +156822,14 @@ S C_ ( P ( ball ` D ) T ) ) $= ( F : X --> Y /\ A. y e. RR+ E. z e. RR+ A. w e. X ( ( w C P ) < z -> ( ( F ` w ) D ( F ` P ) ) < y ) ) ) ) $= ( cfv wcel co clt wbr crp wa cxmet ccnp wf cv wi wral wrex metcnp wb wceq - w3a simpl1 ad2antrr simpl3 xmetsym syl3anc breq1d simpl2 simpllr ffvelrnd - simpr imbi12d ralbidva anassrs rexbidva pm5.32da bitrd ) DJUANOZEKUANOZFJ - OZUKZGFHIUBPNOJKGUCZFCUDZDPZBUDZQRZFGNZVMGNZEPZAUDZQRZUEZCJUFZBSUGZASUFZT - VLVMFDPZVOQRZVRVQEPZVTQRZUEZCJUFZBSUGZASUFZTABCDEFGHIJKLMUHVKVLWEWMVKVLTZ - WDWLASWNVTSOZTWCWKBSWNWOVOSOZWCWKUIWNWOWPTZTZWBWJCJWRVMJOZTZVPWGWAWIWTVNW - FVOQWTVHVJWSVNWFUJWNVHWQWSVHVIVJVLULUMWNVJWQWSVHVIVJVLUNUMZWRWSVAZFVMDJUO - UPUQWTVSWHVTQWTVIVQKOVRKOVSWHUJWNVIWQWSVHVIVJVLURUMWTJKFGVKVLWQWSUSZXAUTW - TJKVMGXCXBUTVQVREKUOUPUQVBVCVDVEVCVFVG $. + w3a simpl1 ad2antrr simpl3 simpr xmetsym syl3anc breq1d simpllr ffvelcdmd + simpl2 imbi12d ralbidva anassrs rexbidva pm5.32da bitrd ) DJUANOZEKUANOZF + JOZUKZGFHIUBPNOJKGUCZFCUDZDPZBUDZQRZFGNZVMGNZEPZAUDZQRZUEZCJUFZBSUGZASUFZ + TVLVMFDPZVOQRZVRVQEPZVTQRZUEZCJUFZBSUGZASUFZTABCDEFGHIJKLMUHVKVLWEWMVKVLT + ZWDWLASWNVTSOZTWCWKBSWNWOVOSOZWCWKUIWNWOWPTZTZWBWJCJWRVMJOZTZVPWGWAWIWTVN + WFVOQWTVHVJWSVNWFUJWNVHWQWSVHVIVJVLULUMWNVJWQWSVHVIVJVLUNUMZWRWSUOZFVMDJU + PUQURWTVSWHVTQWTVIVQKOVRKOVSWHUJWNVIWQWSVHVIVJVLVAUMWTJKFGVKVLWQWSUSZXAUT + WTJKVMGXCXBUTVQVREKUPUQURVBVCVDVEVCVFVG $. $( Two ways to say a mapping from metric ` C ` to metric ` D ` is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional @@ -155782,17 +156889,17 @@ S C_ ( P ( ball ` D ) T ) ) $= ( cfv wcel wa crp wbr wi cxr vz cxmet ccnp co cv clt wral cle metcnpi2 c2 wrex cdiv rphalfcl ad2antrl simplll simprr ctopon ctop mopntopon topontop syl simpllr simplrl cnprcl2k syl3anc rpxrd rpxr rphalflt xrlelttr expcomd - xmetcl w3a imp syl31anc wf cnpf2 ffvelrnd simplrr syl2anc imim12d anassrs - xrltle ralimdva impr breq2 rspceaimv rexlimddv ) DJUBNOZEKUBNOZPZGFHIUCUD - NOZCQOZPZPZBUEZFDUDZUAUEZUFRZWOGNZFGNZEUDZCUFRZSZBJUGZWPAUEZUHRZXACUHRZSB - JUGAQUKZUAQUABCDEFGHIJKLMUIWNWQQOZXDPPWQUJULUDZQOZWPXJUHRZXGSZBJUGZXHXIXK - WNXDWQUMZUNWNXIXDXNWNXIPXCXMBJWNXIWOJOZXCXMSWNXIXPPZPZXLWRXBXGXRWPTOZXJTO - ZWQTOZXJWQUFRZXLWRSZXRWHXPFJOZXSWHWIWMXQUOZWNXIXPUPZXRHJUQNOZIUROZWKYDXRW - HYGYEDHJLUSVAZXRIKUQNOZYHXRWIYJWHWIWMXQVBZEIKMUSVAZKIUTVAWJWKWLXQVCZFGHIJ - VDVEZWOFDJVKVEXRXJXIXKWNXPXOUNVFXIYAWNXPWQVGUNXIYBWNXPWQVHUNXSXTYAVLZYBYC - YOXLYBWRWPXJWQVIVJVMVNXRXATOZCTOXBXGSXRWIWSKOWTKOYPYKXRJKWOGXRYGYJWKJKGVO - YIYLYMFGHIJKVPVEZYFVQXRJKFGYQYNVQWSWTEKVKVEXRCWJWKWLXQVRVFXACWBVSVTWAWCWD - XFXLXGABXJQJXEXJWPUHWEWFVSWG $. + xmetcl w3a imp syl31anc wf cnpf2 ffvelcdmd simplrr xrltle syl2anc imim12d + anassrs ralimdva impr breq2 rspceaimv rexlimddv ) DJUBNOZEKUBNOZPZGFHIUCU + DNOZCQOZPZPZBUEZFDUDZUAUEZUFRZWOGNZFGNZEUDZCUFRZSZBJUGZWPAUEZUHRZXACUHRZS + BJUGAQUKZUAQUABCDEFGHIJKLMUIWNWQQOZXDPPWQUJULUDZQOZWPXJUHRZXGSZBJUGZXHXIX + KWNXDWQUMZUNWNXIXDXNWNXIPXCXMBJWNXIWOJOZXCXMSWNXIXPPZPZXLWRXBXGXRWPTOZXJT + OZWQTOZXJWQUFRZXLWRSZXRWHXPFJOZXSWHWIWMXQUOZWNXIXPUPZXRHJUQNOZIUROZWKYDXR + WHYGYEDHJLUSVAZXRIKUQNOZYHXRWIYJWHWIWMXQVBZEIKMUSVAZKIUTVAWJWKWLXQVCZFGHI + JVDVEZWOFDJVKVEXRXJXIXKWNXPXOUNVFXIYAWNXPWQVGUNXIYBWNXPWQVHUNXSXTYAVLZYBY + CYOXLYBWRWPXJWQVIVJVMVNXRXATOZCTOXBXGSXRWIWSKOWTKOYPYKXRJKWOGXRYGYJWKJKGV + OYIYLYMFGHIJKVPVEZYFVQXRJKFGYQYNVQWSWTEKVKVEXRCWJWKWLXQVRVFXACVSVTWAWBWCW + DXFXLXGABXJQJXEXJWPUHWEWFVTWG $. txmetcnp.4 $e |- L = ( MetOpen ` E ) $. $( Continuity of a binary operation on metric spaces. (Contributed by @@ -155807,9 +156914,9 @@ S C_ ( P ( ball ` D ) T ) ) $= cmpo cmopn ccnp wf wbr wi wral crp wrex wb eqid simp1 adantr simp2 xmetxp ctx simpl3 simprl simprr opelxpd metcnp syl3anc xmettx oveq1d fveq1d wceq eleq2d oveq2 breq1d fveq2 oveq2d imbi12d ralxp ad4antr simplr simpr xmetf - ad5antr syl op1stg syl2anc eqeltrd adantll fovrnd xrmaxcl preq12d supeq1d - op2ndg oveqan12d ovmpoga oveq12d eqtrd xmetcl rpxr xrmaxltsup bitrd df-ov - ad3antlr oveq12i breq1i bicomi ralbidva bitrid rexbidva anbi2d 3bitr3d + ad5antr syl op1stg syl2anc eqeltrd adantll fovcdmd op2ndg xrmaxcl preq12d + supeq1d ovmpoga oveq12d eqtrd xmetcl rpxr ad3antlr xrmaxltsup bitrd df-ov + oveqan12d oveq12i breq1i bicomi ralbidva bitrid rexbidva anbi2d 3bitr3d a1i ) GNUDUETZHOUDUETZIPUDUETZUFZENTZFOTZUGZUGZJEFUHZUAUBNOUIZYNUAUJZUKUE ZUBUJZUKUEZGULZYOUMUEZYQUMUEZHULZUNZUOUPUQZURZUSUEZMUTULZUEZTZYNPJVAZYMUC UJZUUEULZBUJZUPVBZYMJUEZUUKJUEZIULZAUJZUPVBZVCZUCYNVDZBVEVFZAVEVDZUGZJYMK @@ -155827,13 +156934,13 @@ S C_ ( P ( ball ` D ) T ) ) $= JNOUXBUXDUXFWLZUXEUXFWMZVQUXGUXMUOTUXPUOTUXSUXGUXKUXLUONNGUXGYENNUIUOGVAY HYEYKUWKUXAUXDUXFUWDWOZGNWNWPUXGUXKENUXGYIYJUXKEWCUXTUYAEFNOWQWRZUXTWSUXG UXLUVHNUXDUXFUXLUVHWCUXBUVHUVJNOWQWTZUYBWSXAUXGUXNUXOUOOOHUXGYFOOUIUOHVAY - HYFYKUWKUXAUXDUXFUWFWOZHOWNWPUXGUXNFOUXGYIYJUXNFWCUXTUYAEFNOXEWRZUYAWSUXG - UXOUVJOUXDUXFUXOUVJWCUXBUVHUVJNOXEWTZUYCWSXAUXMUXPXBWRUAUBYMUWMYNYNUUDUXR + HYFYKUWKUXAUXDUXFUWFWOZHOWNWPUXGUXNFOUXGYIYJUXNFWCUXTUYAEFNOXBWRZUYAWSUXG + UXOUVJOUXDUXFUXOUVJWCUXBUVHUVJNOXBWTZUYCWSXAUXMUXPXCWRUAUBYMUWMYNYNUUDUXR UUEUOYOYMWCZYQUWMWCZUGZUOUUCUXQUPUYLYSUXMUUBUXPUYJUYKYPUXKYRUXLGYOYMUKWGY - QUWMUKWGXFUYJUYKYTUXNUUAUXOHYOYMUMWGYQUWMUMWGXFXCXDUWCXGVSUXGUOUXQUXHUPUX - GUXMUVIUXPUVKUXGUXKEUXLUVHGUYEUYFXHUXGUXNFUXOUVJHUYHUYIXHXCXDXIWFUXGUVIUO - TZUVKUOTZUUMUOTZUXJUVLVGUXGYEYIUXDUYMUYDUXTUYBEUVHGNXJVSUXGYFYJUXFUYNUYGU - YAUYCFUVJHOXJVSUXAUYOUWLUXDUXFUUMXKXOUVIUVKUUMXLVSXMUWRUVPVGUXGUVPUWRUVOU + QUWMUKWGXOUYJUYKYTUXNUUAUXOHYOYMUMWGYQUWMUMWGXOXDXEUWCXFVSUXGUOUXQUXHUPUX + GUXMUVIUXPUVKUXGUXKEUXLUVHGUYEUYFXGUXGUXNFUXOUVJHUYHUYIXGXDXEXHWFUXGUVIUO + TZUVKUOTZUUMUOTZUXJUVLVGUXGYEYIUXDUYMUYDUXTUYBEUVHGNXIVSUXGYFYJUXFUYNUYGU + YAUYCFUVJHOXIVSUXAUYOUWLUXDUXFUUMXJXKUVIUVKUUMXLVSXMUWRUVPVGUXGUVPUWRUVOU WQUURUPUVMUUOUVNUWPIEFJXNUVHUVJJXNXPXQXRYDWIXSXSXTYAXSYBYC $. $( Continuity of a binary operation on metric spaces. (Contributed by @@ -156079,16 +157186,16 @@ S C_ ( P ( ball ` D ) T ) ) $= reopnap $p |- ( A e. RR -> { w e. RR | w =//= A } e. ( topGen ` ran (,) ) ) $= ( vx cr wcel cv cap wbr cmnf cioo co cpnf wo clt wa cxr syl simpl syl6bi - wb crab cun crn ctg cfv wi elrabi elun rexr elioomnf elioopnf jaod syl5bi - a1i reaplt ancoms breq1 elrab adantl bitr4id baibd orbi12d bitrid 3bitr4d - ibar ex pm5.21ndd eqrdv retop mnfxr iooretopg sylancr pnfxr sylancl unopn - ctop mp3an2i eqeltrd ) BDEZAFZBGHZADUAZIBJKZBLJKZUBZJUCUDUEZVSCWBWEVSCFZD - EZWGWBEZWGWEEZWIWHUFVSWAAWGDUGUNWJWGWCEZWGWDEZMZVSWHWGWCWDUHZVSWKWHWLVSWK - WHWGBNHZOZWHVSBPEZWKWPTBUIZBWGUJQZWHWORSVSWLWHBWGNHZOZWHVSWQWLXATWRBWGUKQ - ZWHWTRSULUMVSWHWIWJTVSWHOZWGBGHZWOWTMZWIWJWHVSXDXETWGBUOUPXCWIWHXDOZXDWAX - DAWGDVTWGBGUQURWHXDXFTVSWHXDVEUSUTWJWMXCXEWNXCWKWOWLWTVSWKWHWOWSVAVSWLWHW - TXBVAVBVCVDVFVGVHWFVPEVSWCWFEZWDWFEZWEWFEVIVSIPEWQXGVJWRIBVKVLVSWQLPEXHWR - VMBLVKVNWCWDWFVOVQVR $. + wb crab cun crn ctg cfv wi elrabi elun rexr elioomnf elioopnf jaod reaplt + a1i biimtrid ancoms breq1 elrab ibar adantl bitr4id baibd orbi12d 3bitr4d + ex pm5.21ndd eqrdv ctop retop mnfxr iooretopg sylancr pnfxr sylancl unopn + bitrid mp3an2i eqeltrd ) BDEZAFZBGHZADUAZIBJKZBLJKZUBZJUCUDUEZVSCWBWEVSCF + ZDEZWGWBEZWGWEEZWIWHUFVSWAAWGDUGUNWJWGWCEZWGWDEZMZVSWHWGWCWDUHZVSWKWHWLVS + WKWHWGBNHZOZWHVSBPEZWKWPTBUIZBWGUJQZWHWORSVSWLWHBWGNHZOZWHVSWQWLXATWRBWGU + KQZWHWTRSULUOVSWHWIWJTVSWHOZWGBGHZWOWTMZWIWJWHVSXDXETWGBUMUPXCWIWHXDOZXDW + AXDAWGDVTWGBGUQURWHXDXFTVSWHXDUSUTVAWJWMXCXEWNXCWKWOWLWTVSWKWHWOWSVBVSWLW + HWTXBVBVCVPVDVEVFVGWFVHEVSWCWFEZWDWFEZWEWFEVIVSIPEWQXGVJWRIBVKVLVSWQLPEXH + WRVMBLVKVNWCWDWFVOVQVR $. $} ${ @@ -156492,13 +157599,13 @@ S C_ ( P ( ball ` D ) T ) ) $= ( abs ` ( ( F ` w ) - ( F ` x ) ) ) < y ) ) ) ) $= ( cc wa co wcel cv cmin cabs cfv clt wbr wral crp sseldd ccncf wf wi wrex wss elcncf wb simplll simprl simprr abssubd breq1d simpllr simplr imbi12d - ffvelrnd anassrs ralbidva rexbidv ralbidv pm5.32da bitrd ) EHUEZFHUEZIZGE - FUAJKEFGUBZALZDLZMJNOZCLZPQZVGGOZVHGOZMJNOZBLZPQZUCZDERZCSUDZBSRZAERZIVFV - HVGMJNOZVJPQZVMVLMJNOZVOPQZUCZDERZCSUDZBSRZAERZIABCDEFGUFVEVFWAWJVEVFIZVT - WIAEWKVGEKZIZVSWHBSWMVRWGCSWMVQWFDEWKWLVHEKZVQWFUGWKWLWNIZIZVKWCVPWEWPVIW - BVJPWPVGVHWPEHVGVCVDVFWOUHZWKWLWNUIZTWPEHVHWQWKWLWNUJZTUKULWPVNWDVOPWPVLV - MWPFHVLVCVDVFWOUMZWPEFVGGVEVFWOUNZWRUPTWPFHVMWTWPEFVHGXAWSUPTUKULUOUQURUS - UTURVAVB $. + ffvelcdmd anassrs ralbidva rexbidv ralbidv pm5.32da bitrd ) EHUEZFHUEZIZG + EFUAJKEFGUBZALZDLZMJNOZCLZPQZVGGOZVHGOZMJNOZBLZPQZUCZDERZCSUDZBSRZAERZIVF + VHVGMJNOZVJPQZVMVLMJNOZVOPQZUCZDERZCSUDZBSRZAERZIABCDEFGUFVEVFWAWJVEVFIZV + TWIAEWKVGEKZIZVSWHBSWMVRWGCSWMVQWFDEWKWLVHEKZVQWFUGWKWLWNIZIZVKWCVPWEWPVI + WBVJPWPVGVHWPEHVGVCVDVFWOUHZWKWLWNUIZTWPEHVHWQWKWLWNUJZTUKULWPVNWDVOPWPVL + VMWPFHVLVCVDVFWOUMZWPEFVGGVEVFWOUNZWRUPTWPFHVMWTWPEFVHGXAWSUPTUKULUOUQURU + SUTURVAVB $. $( Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) $) @@ -156596,7 +157703,7 @@ S C_ ( P ( ball ` D ) T ) ) $= $( Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.) $) - cncffvrn $p |- ( ( C C_ CC /\ F e. ( A -cn-> B ) ) -> + cncfcdm $p |- ( ( C C_ CC /\ F e. ( A -cn-> B ) ) -> ( F e. ( A -cn-> C ) <-> F : A --> C ) ) $= ( vw vx vz vy cc wss ccncf co wcel wa cv cmin cabs cfv wral crp wf clt wi wbr wrex wb cncfrss simpl elcncf2 syl2anc cncfi 3expb ralrimivva biantrud @@ -156604,12 +157711,12 @@ S C_ ( P ( ball ` D ) T ) ) $= UBUDUCEASGTUEZHTSFASZNZVAUSAIJZUQUTVGUFURVHUQABDUGUOUQURUHFHGEACDUIUJUSVF VAURVFUQURVEFHATURVCAMVDTMVEGEABVCVDDUKULUMUOUNUP $. - $( The set of continuous functions is expanded when the range is expanded. - (Contributed by Mario Carneiro, 30-Aug-2014.) $) + $( The set of continuous functions is expanded when the codomain is + expanded. (Contributed by Mario Carneiro, 30-Aug-2014.) $) cncfss $p |- ( ( B C_ C /\ C C_ CC ) -> ( A -cn-> B ) C_ ( A -cn-> C ) ) $= - ( vf wss cc wa ccncf co cv wcel cncff adantl simpll fssd cncffvrn adantll - wf wb mpbird ex ssrdv ) BCEZCFEZGZDABHIZACHIZUEDJZUFKZUHUGKZUEUIGZUJACUHR - ZUKABCUHUIABUHRUEABUHLMUCUDUINOUDUIUJULSUCABCUHPQTUAUB $. + ( vf wss cc wa ccncf co cv wcel wf cncff adantl simpll wb cncfcdm adantll + fssd mpbird ex ssrdv ) BCEZCFEZGZDABHIZACHIZUEDJZUFKZUHUGKZUEUIGZUJACUHLZ + UKABCUHUIABUHLUEABUHMNUCUDUIOSUDUIUJULPUCABCUHQRTUAUB $. $} ${ @@ -156716,22 +157823,22 @@ S C_ ( P ( ball ` D ) T ) ) $= Mario Carneiro, 25-Aug-2014.) $) cncfco $p |- ( ph -> ( G o. F ) e. ( A -cn-> C ) ) $= ( vw vz co wcel cv cmin cabs cfv clt wi crp wa vx vy vv vu ccom ccncf wbr - wf wral cncff syl fco syl2anc adantr simprl ffvelrnd simprr cncfi syl3anc - ad2antrr simplrl simpr ad3antrrr fvoveq1 breq1d imbrov2fvoveq rspcv fvco3 - wrex wceq oveq12d fveq2d imbi2d sylibrd imp an32s imim2d anassrs ralimdva - reximdva ex mpid rexlimdva mpd ralrimivva cc wss cncfrss cncfrss2 elcncf2 - wb mpbir2and ) AFEUEZBDUFKLZBDWMUHZIMZUAMZNKOPJMZQUGZWPWMPZWQWMPZNKZOPZUB - MZQUGZRZIBUIZJSVIZUBSUIUABUIZACDFUHZBCEUHZWOAFCDUFKLZXJHCDFUJUKAEBCUFKLZX - KGBCEUJUKZBCDFEULUMAXHUAUBBSAWQBLZXDSLZTZTZUCMZWQEPZNKOPZUDMZQUGZXSFPXTFP - ZNKOPXDQUGRZUCCUIZUDSVIZXHXRXLXTCLXPYGAXLXQHUNXRBCWQEAXKXQXNUNAXOXPUOUPAX - OXPUQUDUCCDXTXDFURUSXRYFXHUDSXRYBSLZTZYFWSWPEPZXTNKOPZYBQUGZRZIBUIZJSVIZX - HYIXMXOYHYOAXMXQYHGUTAXOXPYHVAZXRYHVBJIBCWQYBEURUSYIYFYOXHRYIYFTZYNXGJSYQ - WRSLZTYMXFIBYQYRWPBLZYMXFRYQYRYSTZTYLXEWSYIYTYFYLXERZYIYTTZYFUUAUUBYFYLYJ - FPZYDNKZOPZXDQUGZRZUUAUUBYJCLYFUUGRUUBBCWPEAXKXQYHYTXNVCZYIYRYSUQZUPYEUUG - UCYJCYCYLXDQNOFYDXSYJXSYJVJYAYKYBQXSYJXTONVDVEVFVGUKUUBXEUUFYLUUBXCUUEXDQ - UUBXBUUDOUUBWTUUCXAYDNUUBXKYSWTUUCVJUUHUUIBCWPFEVHUMUUBXKXOXAYDVJUUHYIXOY - TYPUNBCWQFEVHUMVKVLVEVMVNVOVPVQVRVSVTWAWBWCWDWEABWFWGZDWFWGZWNWOXITWKAXMU - UJGBCEWHUKAXLUUKHCDFWIUKUAUBJIBDWMWJUMWL $. + wf wral wrex cncff syl fco syl2anc adantr simprl ffvelcdmd simprr syl3anc + cncfi ad2antrr simplrl simpr ad3antrrr fvoveq1 breq1d imbrov2fvoveq rspcv + wceq fvco3 oveq12d fveq2d imbi2d sylibrd imim2d anassrs ralimdva reximdva + imp an32s ex mpid rexlimdva mpd ralrimivva cc wb cncfrss cncfrss2 elcncf2 + wss mpbir2and ) AFEUEZBDUFKLZBDWMUHZIMZUAMZNKOPJMZQUGZWPWMPZWQWMPZNKZOPZU + BMZQUGZRZIBUIZJSUJZUBSUIUABUIZACDFUHZBCEUHZWOAFCDUFKLZXJHCDFUKULAEBCUFKLZ + XKGBCEUKULZBCDFEUMUNAXHUAUBBSAWQBLZXDSLZTZTZUCMZWQEPZNKOPZUDMZQUGZXSFPXTF + PZNKOPXDQUGRZUCCUIZUDSUJZXHXRXLXTCLXPYGAXLXQHUOXRBCWQEAXKXQXNUOAXOXPUPUQA + XOXPURUDUCCDXTXDFUTUSXRYFXHUDSXRYBSLZTZYFWSWPEPZXTNKOPZYBQUGZRZIBUIZJSUJZ + XHYIXMXOYHYOAXMXQYHGVAAXOXPYHVBZXRYHVCJIBCWQYBEUTUSYIYFYOXHRYIYFTZYNXGJSY + QWRSLZTYMXFIBYQYRWPBLZYMXFRYQYRYSTZTYLXEWSYIYTYFYLXERZYIYTTZYFUUAUUBYFYLY + JFPZYDNKZOPZXDQUGZRZUUAUUBYJCLYFUUGRUUBBCWPEAXKXQYHYTXNVDZYIYRYSURZUQYEUU + GUCYJCYCYLXDQNOFYDXSYJXSYJVIYAYKYBQXSYJXTONVEVFVGVHULUUBXEUUFYLUUBXCUUEXD + QUUBXBUUDOUUBWTUUCXAYDNUUBXKYSWTUUCVIUUHUUIBCWPFEVJUNUUBXKXOXAYDVIUUHYIXO + YTYPUOBCWQFEVJUNVKVLVFVMVNVSVTVOVPVQVRWAWBWCWDWEABWFWKZDWFWKZWNWOXITWGAXM + UUJGBCEWHULAXLUUKHCDFWIULUAUBJIBDWMWJUNWL $. $} ${ @@ -158148,46 +159255,46 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and crab rphalfcld readdcld ovresd cnmetdval subsub4d subidd subid1d rpge0d 0cnd absidd rphalflt rpxr elbl2 syl22anc sseldd rpap0d negsubdid negeqd cxr eqtrdi negcld addassd addid2d 3eqtr3d 3brtr4d apadd2 syl3anc elrabd - 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( F limCC B ) ) ) ) $= ( vz vd ve cc wcel wa co cfv wbr cmin cabs clt crp wss wf climc ctopon wi ccnp crest cntoptopon simpl resttopon sylancr eqeltrid 3expia sylancl imp - cnpf2 cv cap wral wrex simplr ffvelrnd ccom cxp cres simpr wb wceq simpll - cmopn cxmet cnxmet eqid metrest mpan eqtrid syl a1i xmetres2 syldan mpbid - metcnpd simprd ad3antrrr ovresd cnmetdval syl2anc abssubd biimprd adantld - sseldd 3eqtrd breq1d eqtrd biimpd imim12d ralimdva reximdva mpd ellimc3ap - mpbir2and jca ex ) AKUAZBALZMZCBDEUFNOLZAKCUBZBCOZCBUCNLZMXFXGMZXHXJXFXGX - HXFDAUDOZLZEKUDOLZXGXHUEXFDEAUGNZXLGXFXNXDXOXLLEFUHZXDXEUIAEKUJUKULXPXMXN - XGXHBCDEAKUPUMUNUOZXKXJXIKLZHUQZBURPZXSBQNROZIUQZSPZMZXSCOZXIQNROZJUQZSPZ - UEZHAUSZITUTZJTUSZXKAKBCXQXDXEXGVAZVBZXKBXSRQVCZAAVDVEZNZYBSPZXIYEYONZYGS - PZUEZHAUSZITUTZJTUSZYLXKXHUUDXKXGXHUUDMZXFXGVFXFXGXHXGUUEVGXQXFXHMZJIHYPY - OBCDEAKUUFXDDYPVJOZVHXDXEXHVIZXDDXOUUGGYOKVKOLZXDXOUUGVHVLYOYPEUUGKAYPVMF - UUGVMVNVOVPVQEYOVJOVHUUFFVRUUFUUIXDYPAVKOLVLUUHYOAKVSUKUUIUUFVLVRXDXEXHVA - ZWBVTWAWCXKUUCYKJTXKYGTLZMZUUBYJITUULYBTLZMZUUAYIHAUUNXSALZMZYDYRYTYHUUPY - CYRXTUUPYRYCUUPYQYAYBSUUPYQBXSYONZBXSQNROZYAUUPBXSYOAXKXEUUKUUMUUOYMWDUUN - UUOVFZWEUUPBKLZXSKLUUQUURVHXKUUTUUKUUMUUOXFXGXHUUTXQUUFAKBUUHUUJWKVTZWDZU - UPAKXSXKXDUUKUUMUUOXDXEXGVIZWDUUSWKZBXSYOYOVMZWFWGUUPBXSUVBUVDWHWLWMWIWJU - UPYTYHUUPYSYFYGSUUPYSXIYEQNROZYFUUPXRYEKLYSUVFVHXKXRUUKUUMUUOYNWDZUUPAKXS - CXKXHUUKUUMUUOXQWDUUSVBZXIYEYOUVEWFWGUUPXIYEUVGUVHWHWNWMWOWPWQWRWQWSXKJIH - ABXICXQUVCUVAWTXAXBXC $. + cnpf2 cv cap wral wrex simplr ffvelcdmd ccom cxp cres simpr wb cmopn wceq + simpll cxmet cnxmet eqid metrest mpan eqtrid syl a1i metcnpd syldan mpbid + xmetres2 simprd ad3antrrr ovresd sseldd cnmetdval syl2anc abssubd biimprd + 3eqtrd breq1d adantld eqtrd imim12d ralimdva reximdva ellimc3ap mpbir2and + biimpd mpd jca ex ) AKUAZBALZMZCBDEUFNOLZAKCUBZBCOZCBUCNLZMXFXGMZXHXJXFXG + XHXFDAUDOZLZEKUDOLZXGXHUEXFDEAUGNZXLGXFXNXDXOXLLEFUHZXDXEUIAEKUJUKULXPXMX + NXGXHBCDEAKUPUMUNUOZXKXJXIKLZHUQZBURPZXSBQNROZIUQZSPZMZXSCOZXIQNROZJUQZSP + ZUEZHAUSZITUTZJTUSZXKAKBCXQXDXEXGVAZVBZXKBXSRQVCZAAVDVEZNZYBSPZXIYEYONZYG + SPZUEZHAUSZITUTZJTUSZYLXKXHUUDXKXGXHUUDMZXFXGVFXFXGXHXGUUEVGXQXFXHMZJIHYP + YOBCDEAKUUFXDDYPVHOZVIXDXEXHVJZXDDXOUUGGYOKVKOLZXDXOUUGVIVLYOYPEUUGKAYPVM + FUUGVMVNVOVPVQEYOVHOVIUUFFVRUUFUUIXDYPAVKOLVLUUHYOAKWBUKUUIUUFVLVRXDXEXHV + AZVSVTWAWCXKUUCYKJTXKYGTLZMZUUBYJITUULYBTLZMZUUAYIHAUUNXSALZMZYDYRYTYHUUP + YCYRXTUUPYRYCUUPYQYAYBSUUPYQBXSYONZBXSQNROZYAUUPBXSYOAXKXEUUKUUMUUOYMWDUU + NUUOVFZWEUUPBKLZXSKLUUQUURVIXKUUTUUKUUMUUOXFXGXHUUTXQUUFAKBUUHUUJWFVTZWDZ + UUPAKXSXKXDUUKUUMUUOXDXEXGVJZWDUUSWFZBXSYOYOVMZWGWHUUPBXSUVBUVDWIWKWLWJWM + UUPYTYHUUPYSYFYGSUUPYSXIYEQNROZYFUUPXRYEKLYSUVFVIXKXRUUKUUMUUOYNWDZUUPAKX + SCXKXHUUKUUMUUOXQWDUUSVBZXIYEYOUVEWGWHUUPXIYEUVGUVHWIWNWLWTWOWPWQWPXAXKJI + HABXICXQUVCUVAWRWSXBXC $. $} ${ @@ -158293,17 +159400,17 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.) $) cnplimclemle $p |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E ) $= - ( c2 cdiv co cfv cmin cabs clt wbr wa simpr cr ffvelrnd subcld abscld - wcel cc adantr rphalfcld rpred cc0 wceq cap wn simpll syl3anc ltnsymd - syl pm2.65da wb sseldd syl2an2r mpbird fveq2d subeq0bd abs00bd rpgt0d - apti crp eqbrtrd pm2.21dd wo rphalflt wi axltwlin mpd mpjaodan ) AEUA - UBUCZIFUDZCFUDZUEUCZUFUDZUGUHZWKEUGUHZWMAWLUIZWLWMAWLUJZWNWKWGAWKUKUO - ZWLAWJAWHWIABUPIFMRULZABUPCFMNULUMUNZUQAWGUKUOZWLAWGAEPURZUSZUQWNWKUT - WGUGWNWJWNWHWIAWHUPUOWLWQUQWNICFWNICVAZICVBUHZVCZWNXCWLWNWLXCWOUQWNXC - UIZWKWGXEAWPAWLXCVDZWRVGXEAWSXFXAVGXEAXCICUEUCUFUDDUGUHZWKWGUGUHXFWNX - CUJXEAXGXFTVGSVEVFVHAIUPUOWLCUPUOZXBXDVIABUPILRVJAXHWLABUPCLNVJUQICVQ - VKVLVMVNVOWNWGAWGVRUOWLWTUQVPVSVFVTAWMUJAWGEUGUHZWLWMWAZAEVRUOXIPEWBV - GAWSEUKUOWPXIXJWCXAAEPUSWRWGEWKWDVEWEWF $. + ( c2 cdiv co cfv cmin cabs clt wbr simpr wcel ffvelcdmd subcld abscld + wa cr cc adantr rphalfcld rpred cc0 cap wn simpll syl syl3anc ltnsymd + wceq pm2.65da sseldd apti syl2an2r mpbird fveq2d subeq0bd abs00bd crp + wb rpgt0d eqbrtrd pm2.21dd wo rphalflt wi axltwlin mpd mpjaodan ) AEU + AUBUCZIFUDZCFUDZUEUCZUFUDZUGUHZWKEUGUHZWMAWLUNZWLWMAWLUIZWNWKWGAWKUOU + JZWLAWJAWHWIABUPIFMRUKZABUPCFMNUKULUMZUQAWGUOUJZWLAWGAEPURZUSZUQWNWKU + TWGUGWNWJWNWHWIAWHUPUJWLWQUQWNICFWNICVGZICVAUHZVBZWNXCWLWNWLXCWOUQWNX + CUNZWKWGXEAWPAWLXCVCZWRVDXEAWSXFXAVDXEAXCICUEUCUFUDDUGUHZWKWGUGUHXFWN + XCUIXEAXGXFTVDSVEVFVHAIUPUJWLCUPUJZXBXDVQABUPILRVIAXHWLABUPCLNVIUQICV + JVKVLVMVNVOWNWGAWGVPUJWLWTUQVRVSVFVTAWMUIAWGEUGUHZWLWMWAZAEVPUJXIPEWB + VDAWSEUOUJWPXIXJWCXAAEPUSWRWGEWKWDVEWEWF $. $} $d A d e s z $. $d B d e s z $. $d F d e s z $. $d K d e z $. @@ -158314,24 +159421,24 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and ( vz vd cfv co wcel cc clt crp ve vs cabs cmin ccom cxp cres cmopn ccnp wf cv wbr wi wral wrex wa cap cdiv breq2 imbi2d rexralbidv climc sseldd c2 wceq ellimc3ap simprd adantr rphalfcl adantl rspcdva ad5antr simpllr - mpbid ffvelrnd cnmetdval syl2anc wss simp-5r simp-4r w3a 3simpc simp1lr - eqid ovresd eqtrd simpr eqbrtrrd cnplimclemle eqbrtrd ralimdva reximdva - mpd exp31 ralrimiva cxmet cnxmet xmetres2 sylancr a1i metcnp2 mpbir2and - wb syl3anc crest metrest eqtrid oveq1d fveq1d eleqtrrd ) ADCUCUDUEZBBUF - UGZUHOZFUIPZOZCEFUIPZOADXOQZBRDUJZMUKZCXLPZNUKZSULZXSDOZCDOZXKPZUAUKZSU - LZUMZMBUNZNTUOZUATUNZJAYJUATAYFTQZUPZXSCUQULZXSCUDPUCOZYASULZUPZYCYDUDP - UCOZYFVDURPZSULZUMZMBUNZNTUOZYJYMYQYRUBUKZSULZUMZMBUNNTUOZUUCUBTYSUUDYS - VEZUUFUUANMTBUUHUUEYTYQUUDYSYRSUSUTVAAUUGUBTUNZYLAYDRQZUUIAYDDCVBPQZUUJ - UUIUPLAUBNMBCYDDJIABRCIKVCZVFVNVGVHYLYSTQAYFVIVJVKYMUUBYINTYMYATQZUPZUU - AYHMBUUNXSBQZUPZUUAYBYGUUPUUAUPZYBUPZYEYRYFSUURYCRQUUJYEYRVEUURBRXSDAXR - YLUUMUUOUUAYBJVLZUUNUUOUUAYBVMZVOUURBRCDUUSACBQZYLUUMUUOUUAYBKVLZVOYCYD - XKXKWDZVPVQUURBCYAYFDEFXSGHABRVRZYLUUMUUOUUAYBIVLZUUSUVBAUUKYLUUMUUOUUA - YBLVLAYLUUMUUOUUAYBVSYMUUMUUOUUAYBVTUUTUURYNYPWAYQYTUURYNYPWBUUPUUAYBYN - YPWCWMUURXTYOYASUURXTXSCXKPZYOUURXSCXKBUUTUVBWEUURXSRQCRQZUVFYOVEUURBRX - SUVEUUTVCAUVGYLUUMUUOUUAYBUULVLXSCXKUVCVPVQWFUUQYBWGWHWIWJWNWKWLWMWOAXL - BWPOQZXKRWPOQZUVAXQXRYKUPXCAUVIUVDUVHWQIXKBRWRWSUVIAWQWTKUANMXLXKCDXMFB - RXMWDZGXAXDXBACXPXNAEXMFUIAEFBXEPZXMHAUVIUVDUVKXMVEWQIXKXLFXMRBXLWDGUVJ - XFWSXGXHXIXJ $. + ffvelcdmd eqid cnmetdval syl2anc wss simp-5r simp-4r w3a 3simpc simp1lr + mpbid eqtrd simpr eqbrtrrd cnplimclemle eqbrtrd exp31 ralimdva reximdva + mpd ovresd ralrimiva cxmet wb cnxmet xmetres2 sylancr metcnp2 mpbir2and + a1i syl3anc crest metrest eqtrid oveq1d fveq1d eleqtrrd ) ADCUCUDUEZBBU + FUGZUHOZFUIPZOZCEFUIPZOADXOQZBRDUJZMUKZCXLPZNUKZSULZXSDOZCDOZXKPZUAUKZS + ULZUMZMBUNZNTUOZUATUNZJAYJUATAYFTQZUPZXSCUQULZXSCUDPUCOZYASULZUPZYCYDUD + PUCOZYFVDURPZSULZUMZMBUNZNTUOZYJYMYQYRUBUKZSULZUMZMBUNNTUOZUUCUBTYSUUDY + SVEZUUFUUANMTBUUHUUEYTYQUUDYSYRSUSUTVAAUUGUBTUNZYLAYDRQZUUIAYDDCVBPQZUU + JUUIUPLAUBNMBCYDDJIABRCIKVCZVFWDVGVHYLYSTQAYFVIVJVKYMUUBYINTYMYATQZUPZU + UAYHMBUUNXSBQZUPZUUAYBYGUUPUUAUPZYBUPZYEYRYFSUURYCRQUUJYEYRVEUURBRXSDAX + RYLUUMUUOUUAYBJVLZUUNUUOUUAYBVMZVNUURBRCDUUSACBQZYLUUMUUOUUAYBKVLZVNYCY + DXKXKVOZVPVQUURBCYAYFDEFXSGHABRVRZYLUUMUUOUUAYBIVLZUUSUVBAUUKYLUUMUUOUU + AYBLVLAYLUUMUUOUUAYBVSYMUUMUUOUUAYBVTUUTUURYNYPWAYQYTUURYNYPWBUUPUUAYBY + NYPWCWMUURXTYOYASUURXTXSCXKPZYOUURXSCXKBUUTUVBWNUURXSRQCRQZUVFYOVEUURBR + XSUVEUUTVCAUVGYLUUMUUOUUAYBUULVLXSCXKUVCVPVQWEUUQYBWFWGWHWIWJWKWLWMWOAX + LBWPOQZXKRWPOQZUVAXQXRYKUPWQAUVIUVDUVHWRIXKBRWSWTUVIAWRXCKUANMXLXKCDXMF + BRXMVOZGXAXDXBACXPXNAEXMFUIAEFBXEPZXMHAUVIUVDUVKXMVEWRIXKXLFXMRBXLVOGUV + JXFWTXGXHXIXJ $. $} $( A function is continuous at ` B ` iff its limit at ` B ` equals the @@ -158422,33 +159529,33 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and limccnpcntop $p |- ( ph -> ( G ` C ) e. ( ( G o. F ) limCC B ) ) $= ( cfv co wcel cc crp vz vd ve vw vp ccom climc cv cap wbr cmin cabs wa wi clt wral wrex ctopon ccnp crest wss cntoptopon resttopon sylancr eqeltrid - wf a1i cnpf2 syl3anc ctop cntoptop cnprcl2k ffvelrnd cxp cres cmopn cxmet - wceq cnxmet eqid metrest eqtrid xmetres2 metcnpd mpbid simprd simplr fssd - simplll cdm fdmd w3a limcrcl syl simp2d eqsstrrd simp3d ellimc3ap syl2anc - r19.21bi oveq2 breq1d fveq2 oveq2d imbi12d simpllr ad5antr rspcdva ovresd - simpr simpld sseldd cnmetdval abssubd 3eqtrd fvco3 fveq2d eqeltrd 3eqtr2d - 3imtr3d imim2d ralimdva reximdva mpd rexlimdva2 fco mpbir2and ) ADGPZGFUF - ZCUGQRYHSRZUAUHZCUIUJYKCUKQULPUBUHZUOUJUMZYKYIPZYHUKQULPZUCUHZUOUJZUNZUAB - UPZUBTUQZUCTUPZAESDGAHEURPZRZISURPRZGDHIUSQPRZESGVFZAHIEUTQZUUBMAUUDESVAZ - UUGUUBRILVBZKEISVCVDVEZUUDAUUIVGODGHIESVHVIZAUUCIVJRZUUEDERZUUJUULAILVKVG - ODGHIEVLVIZVMZADUDUHZULUKUFZEEVNVOZQZUEUHZUOUJZYHUUPGPZUUQQZYPUOUJZUNZUDE - UPZUETUQZUCTUPZUUAAUUFUVHAUUEUUFUVHUMOAUCUEUDUURUUQDGHIESAHUUGUURVPPZMAUU - 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cnex cnprcl2k toponuni syl eleqtrrd opelxp simpld simprd eqid cnmetdval - fovrnd simpl abssubd eqtrd cres jca ovresd sseldd 3eqtrd simprl ralrimi - rpred ltletrd r19.21bi sylc eqbrtrd oveq2 breq1d oveq2d imbi12d anbi2d + cnex cnprcl2k toponuni eleqtrrd opelxp simpld simprd fovcdmd simpl eqid + syl cnmetdval abssubd eqtrd cres jca ovresd sseldd 3eqtrd ralrimi rpred + simprl ltletrd r19.21bi sylc eqbrtrd oveq2 breq1d oveq2d imbi12d anbi2d cle cmpt cdm ex dmmptg w3a limcrcl simp2d eqsstrrd simp3d subcld abscld climc mincl simprr min1inf anbi1d oveq1 rspc2v syl3c eqbrtrrd rspceaimv min2inf exp31 breq2 ) AJKURUSUTVAZVBVCZBVDZDVEVFZUVGDVGVHZVIVNZUVEUTVFZ @@ -158506,28 +159613,28 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and OWPWQZAEPVCZUWCUVLAUXFFQVCZAUWNUWIVCUXFUXGVJAUWNMWSZUWIAMUXHWBVNVCZNWTV CZUWOUWNUXHVCAMWTVCUXIAMUWQWTUFAUWPWTVCZUWIXAVCZUWQWTVCAUXJUXJUXKNUEXBZ UXJAUXMWNZNNXCWLAPXAVCQXAVCUXLAPVRXAVRXAVCAXIWNZUCXDAQVRXAUXOUDXDPQXAXA - XEVPUWIUWPXAXFVPWMMXGXHUXNUIUWNLMNUXHXJWPAUWLUWIUXHVSUXDUWIMXKXLXMEFPQX - 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(Revised by Jim Kingdon, 27-Jun-2023.) $) dvlemap $p |- ( ( ph /\ A e. { w e. D | w =//= B } ) -> ( ( ( F ` A ) - ( F ` B ) ) / ( A - B ) ) e. CC ) $= - ( cap wbr wcel cfv cmin co cc adantr adantl ffvelrnd subcld cv crab wa wf - elrabi wss sseldd breq1 elrab simprbi subap0d divclapd ) ACBUAZDJKZBEUBLZ - UCZCFMZDFMZNOCDNOUPUQURUPEPCFAEPFUDUOGQZUOCELZAUNBCEUERZSUPEPDFUSADELUOIQ - STUPCDUPEPCAEPUFUOHQVAUGZADPLUOAEPDHIUGQZTUPCDVBVCUOCDJKZAUOUTVDUNVDBCEUM - CDJUHUIUJRUKUL $. + ( cap wbr wcel cfv cmin co cc adantr adantl ffvelcdmd subcld cv wa elrabi + crab wf wss sseldd breq1 elrab simprbi subap0d divclapd ) ACBUAZDJKZBEUDL + ZUBZCFMZDFMZNOCDNOUPUQURUPEPCFAEPFUEUOGQZUOCELZAUNBCEUCRZSUPEPDFUSADELUOI + QSTUPCDUPEPCAEPUFUOHQVAUGZADPLUOAEPDHIUGQZTUPCDVBVCUOCDJKZAUOUTVDUNVDBCEU + MCDJUHUIUJRUKUL $. $} ${ @@ -158793,19 +159900,19 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and cap cuni ctopon cntoptopon resttopon mpan topontop wb wceq toponuni mpbid sseq2d ntrss2 syl2anc sselda dvlemap fmpttd ssrab2 sstrid sseldd simpr ex ntropn simpll rabss2 limcimo moanimv sylibr eldvap mobidv mpbird sylanbrc - alrimiv dffun6 funfnd wex vex elrn dvcl exlimdv syl5bi ssrdv df-f ) AUAGU - BHZBGAUCIHZJZABUDIZXPKZUEXPUFZGLXQGXPMXOXPXOXPUGZCNZDNZXPUPZDOZCUQXPUHXMA - GLZXNXSAUIZABUJUKXOYCCXOYCXTBKZULPUMUNQZAUOIZURQQZHZYAEFNXTVJUPZFYFUSZENZ - BQXTBQPIYMXTPIUTIZVAZXTVBIHZJZDOZXOYJYPDOZVCYRXOYJYSXOYJJZDYLXTYIAYOYGFYT - EYLYNGYTFYMXTYFBXOYFGBMZYJXNUUAXMXNUUAYFALZGABVDZVEVFZRXOYFGLYJXOYFAGXNUU - BXMXNUUAUUBUUCVGVFZXMYDXNYERZVHRZXOYIYFXTXOYHVIHZYFYHVKZLZYIYFLZXOYDUUHUU - FYDYHAVLQHZUUHYGGVLQHYDUULYGYGSZVMAYGGVNVOZAYHVPTTZXOUUBUUJUUEXOYDUUBUUJV - QUUFYDAUUIYFYDUULAUUIVRUUNAYHVSTWATVTZYFYHUUIUUISZWBWCZWDZWEWFYTYLYFGYKFY - FWGUUGWHYTYFGXTUUGUUSWIXOYJWJXOYIAXTXOYIYFAUURUUEVHWDYTUUHUUJYIYHHXOUUHYJ - UUORXOUUJYJUUPRYFYHUUIUUQWLWCXMXNYJWMXOYKFYIUSYLLZYJXOUUKUUTUURYKFYIYFWNT - RUUMWOWKYJYPDWPWQXOYBYQDXOEFYFXTYAAYHBYOYGYHSUUMYOSUUFUUDUUEWRWSWTXBCDXPX - CXAXDXODXRGYAXRHYBCXEXOYAGHZCYAXPDXFXGXOYBUVACXOYBUVAXOYFXTYAABUUFUUDUUEX - HWKXIXJXKXQGXPXLXA $. + alrimiv dffun6 funfnd wex vex elrn dvcl exlimdv biimtrid ssrdv df-f ) AUA + GUBHZBGAUCIHZJZABUDIZXPKZUEXPUFZGLXQGXPMXOXPXOXPUGZCNZDNZXPUPZDOZCUQXPUHX + MAGLZXNXSAUIZABUJUKXOYCCXOYCXTBKZULPUMUNQZAUOIZURQQZHZYAEFNXTVJUPZFYFUSZE + NZBQXTBQPIYMXTPIUTIZVAZXTVBIHZJZDOZXOYJYPDOZVCYRXOYJYSXOYJJZDYLXTYIAYOYGF + YTEYLYNGYTFYMXTYFBXOYFGBMZYJXNUUAXMXNUUAYFALZGABVDZVEVFZRXOYFGLYJXOYFAGXN + UUBXMXNUUAUUBUUCVGVFZXMYDXNYERZVHRZXOYIYFXTXOYHVIHZYFYHVKZLZYIYFLZXOYDUUH + UUFYDYHAVLQHZUUHYGGVLQHYDUULYGYGSZVMAYGGVNVOZAYHVPTTZXOUUBUUJUUEXOYDUUBUU + JVQUUFYDAUUIYFYDUULAUUIVRUUNAYHVSTWATVTZYFYHUUIUUISZWBWCZWDZWEWFYTYLYFGYK + FYFWGUUGWHYTYFGXTUUGUUSWIXOYJWJXOYIAXTXOYIYFAUURUUEVHWDYTUUHUUJYIYHHXOUUH + YJUUORXOUUJYJUUPRYFYHUUIUUQWLWCXMXNYJWMXOYKFYIUSYLLZYJXOUUKUUTUURYKFYIYFW + NTRUUMWOWKYJYPDWPWQXOYBYQDXOEFYFXTYAAYHBYOYGYHSUUMYOSUUFUUDUUEWRWSWTXBCDX + PXCXAXDXODXRGYAXRHYBCXEXOYAGHZCYAXPDXFXGXOYBUVACXOYBUVAXOYFXTYAABUUFUUDUU + EXHWKXIXJXKXQGXPXLXA $. $} $( The derivative is a function. (Contributed by Mario Carneiro, @@ -158886,12 +159993,12 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and wex ccnp cdm simpl3 simpl1 sstrd simpl2 crest cnt ctop cuni cntoptop cnex wa cvv ssexg sylancl resttop sylancr ctopon cntoptopon resttopon toponuni wceq syl sseqtrd eqid ntrss2 syl2anc crab cdiv simp1 simp2 simp3 simprbda - cap eldvap sseldd ctx ffvelcdmda ffvelrnd adantr subcld a1i txtopon mp2an - ssid toponrestid cmul dvlemap ssrab2 sstrid sselda simplbda cres limcresi - cxp resmpt ax-mp oveq1i sseqtri subidd subcncntop ccncf cncfmptid syl3anc - ccn cncfmptc cncfmpt2fcntop cnmptlimc eqeltrrd sselid cop mulcncntop dvcl - oveq1 0cn opelxpi toponunii cncnpi limccnp2cntop mul01d simpr breq1 elrab - simprbi adantl subap0d divcanap1d mpteq2dva oveq1d 3eltr3d limcdifap + cap eldvap sseldd ctx ffvelcdmda ffvelcdmd adantr subcld ssid a1i txtopon + cxp mp2an toponrestid cmul dvlemap ssrab2 sstrid sselda simplbda limcresi + cres resmpt ax-mp oveq1i sseqtri subidd ccn subcncntop cncfmptid cncfmptc + ccncf syl3anc cncfmpt2fcntop oveq1 cnmptlimc eqeltrrd cop mulcncntop dvcl + sselid 0cn toponunii cncnpi limccnp2cntop mul01d simpr breq1 elrab adantl + opelxpi simprbi subap0d divcanap1d mpteq2dva oveq1d 3eltr3d limcdifap fmpttd eqtrdi eleqtrrd eqidd addcncntop addid2d npcand eqtr4d cnplimclemr feqmptd ex exlimdv eldmg ibi impel ) CKLZAKDUBZACLZUCZBUAUDZCDUEMZUFZUAUG ZDBEFUHMNOZBUUTUIZOZUURUVAUVCUAUURUVAUVCUURUVAUTZABDEFHGUVFACKUUOUUPUUQUV @@ -158901,22 +160008,22 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and UVABUVLOZUUSIJUDZBWBUFZJAVPZIUDZDNZBDNZPMZUWCBPMZVQMZQZBRMOZUURIJABUUSCUV KDUWIFUVKVMHUWIVMUUOUUPUUQVRZUUOUUPUUQVSZUUOUUPUUQVTZWCZWAWDZUVFSUWETMIAU WFUWETMZQZBRMUWEDBRMUVFIABSUWEUWFUWETFFWEMZFKKUVFUWCAOZUTZUWDUWEUVFAKUWCD - 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For the (simpler but more limited) function version, see ~ dvmulxx . 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G ) ) ( K x. L ) ) $= ( vz vw vy cmul ccom cdv wbr crest cnt cfv wcel cap crab cmin cdiv cmpt co cv climc wa eqid cc sstrd fssd eldvap mpbid simpld ctx adantr elrabi - wf adantl ffvelrnd cdm dvbss wrel wss cpm cvv a1i ssexd elpm2r syl22anc - cnex reldvg syl2anc releldm sseldd subcld breq1 elrab simprbi wi breq1d - wceq fveq2 imbi12d wral rspcdva mpd subap0d divclapd dvlemap cxp ctopon - ssidd cntoptopon txtopon mp2an toponrestid elrabd cres limcresi feqmptd - reseq1d ssrab2 resmpt ax-mp eqtrdi oveq1d sseqtrid dvcnp2cntop syl31anc - ccnp cnplimccntop simprd oveq1 oveq12d limccoap ccn cop mulcncntop dvcl - wb mpdan opelxpd toponunii fvco3 cncnpi limccnp2cntop dmdcanap2d eqtr4d - sylancr mpteq2dva eleqtrd fco mpbir2and ) ACIJUFUSZEFGUGZUHUSUICLHEUJUS - ZUKULULUMZUUJUCUDUTZCUNUIZUDLUOZUCUTZUUKULZCUUKULZUPUSZUUQCUPUSZUQUSZUR - ZCVAUSZUMAUUMJUCUUPUUQGULZCGULZUPUSZUVAUQUSZURZCVAUSUMZACJEGUHUSZUIZUUM - UVJVBUAAUCUDLCJEUULGUVIHUULVCZUBUVIVCSALKVDGOAKDVDNRVEZVFZPVGVHZVIAUUJU - CUUPUVEFULZUVFFULZUPUSZUVGUQUSZUVHUFUSZURZCVAUSUVDAUCUUPCIJUVTUVHUFHHVJ - 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X ) -> B e. CC ) $= ( cc cmpt co wf cdm wcel cvv mpbid cdv cpr cpm wss cnex a1i fmpttd elpm2r elexd syl22anc dvfgg syl2anc dmeqd wral wceq ralrimiva dmmptg eqtrd feq2d - cr syl feq1d fvmptelrn ) ABGDMAGMEBGCNZUAOZPZGMBGDNZPAVEQZMVEPZVFAEUTMUBZ - RVDMEUCORZVIHAMSRZESRGMVDPGEUDVKVLAUEUFAEVJHUIABGCMIUGLMEGVDSSUHUJEVDUKUL - AVHGMVEAVHVGQZGAVEVGKUMADFRZBGUNVMGUOAVNBGJUPBGDFUQVAURUSTAGMVEVGKVBTVC + cr syl feq1d fvmptelcdm ) ABGDMAGMEBGCNZUAOZPZGMBGDNZPAVEQZMVEPZVFAEUTMUB + ZRVDMEUCORZVIHAMSRZESRGMVDPGEUDVKVLAUEUFAEVJHUIABGCMIUGLMEGVDSSUHUJEVDUKU + LAVHGMVEAVHVGQZGAVEVGKUMADFRZBGUNVMGUOAVNBGJUPBGDFUQVAURUSTAGMVEVGKVBTVC $. ${ @@ -159623,8 +160730,8 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and snid sylibr cofmpt wi simpr simpl subap0d wb eqid oveq1 fvmptd3 id subcld cap subid breq12d mpbird ex ralrimiva dveflem eqbrtrdi 1ex 1cnd dvmptidcn eqtrd 0cnd dvmptccn dvmptsubcn 1m0e1 mpteq2i eqtr4i eqtrdi dvcoapbr 1t1e1 - breqtrdi breqdi dvmulxxbr ffvelrnd mul02d mpancom mulid2d oveq12d addid2d - fvconst2g breqtrd funbrfv mpsyl mpteq2ia wss ssid dvbsssg mp2an mpd3an23 + breqdi dvmulxxbr ffvelcdmd mul02d mpancom mulid2d oveq12d addid2d breqtrd + breqtrdi fvconst2g funbrfv mpsyl mpteq2ia wss ssid dvbsssg mp2an mpd3an23 breldmg ssriv eqssi feq2i mpbi 3eqtr4a mptru ) CDEFZDGHACAUBZUUIIZJACUUJD IZJUUIDACUUKUULUUIUCZUUJCKZUUJUULUUILZUUKUULGUUIUDZCUUIRZUUMDCCUEFKZUUQCM KZUUSCCDRZUURUFUFUGCCDMMUHUIZDUJUOZUUPCUUIUKUOUUNCCUULULZUMZBCBUBZUUJNFZD @@ -159645,9 +160752,9 @@ Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and IUUNUXASTUUIUUNUXAUXKSUUNBUUJUVFUXKCUWOCUXJUXLUXMUXNWSUXOXMXHXIUUNUUJTWBZ CUWOEFZKUUJTUXQLUUNUXPCTULZUMZUXQUUNTUXRKUXPUXSKTXJWIUUJTCUXRWDWEUUNUXQBC TSNFZJZUXSUUNBUVETUUJSCCUUNUVMWMUVNXKCBCUVEJEFBCTJZGUUNBXLPUUNUVMWNUVNXNU - UNBUUJUXMXOXPUYAUYBUXSBCUXTTXQXRBCTVGXSXTWGUUJTUXQWHWJUWHWQZYAYBYCYDUYCYE - UUNUWGSUULUPFUULUUNUWDSUWFUULUPUUNUWCUUNCCUUJUVHUWJUXMYFYGUUNUWFTUULOFUUL - UUNUWEUULTOUVSUUNUWEUULGUVTCUULUUJCYLYHVLUUNUULUVTYIXMYJUUNUULUVTYKXMYMYD + UNBUUJUXMXOXPUYAUYBUXSBCUXTTXQXRBCTVGXSXTWGUUJTUXQWHWJUWHWQZYAYBYLYCUYCYD + UUNUWGSUULUPFUULUUNUWDSUWFUULUPUUNUWCUUNCCUUJUVHUWJUXMYEYFUUNUWFTUULOFUUL + UUNUWEUULTOUVSUUNUWEUULGUVTCUULUUJCYMYGVLUUNUULUVTYHXMYIUUNUULUVTYJXMYKYC ZUUJUULUUIYNYOYPHACCUUICCUUIRZHUUQUYEUVBUUPCCUUIUUPCCCYQUURUUPCYQCYRUVACD YSYTACUUPUUNUVSUUOUUJUUPKUVTUYDUUJUULCCUUIUUBUUAUUCUUDUUEUUFPVJHACCDUUTHU GPVJUUGUUH $. @@ -162408,7 +163515,7 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is 7nn syl dcbid mpbird ad5antr 2nn simp-5l dvdsdc prm2orodd orcomd ad2antlr ecased prmnn nnnn0d nn0oddm1d2 mpbid zexpcl peano2zd zsubcld nnz ifcldadc 2z wne simp-4r neqne ad3antlr pczcl syl12anc prmdc adantl fvmptd3 eqeltrd - zmulcl seqf simplr nnabscl ffvelrnd zmulcld eqeq1d ifbid breq1d anbi12d + zmulcl seqf simplr nnabscl ffvelcdmd zmulcld eqeq1d ifbid breq1d anbi12d cmpt breq2d eleq1d ifeq12d ifeq1d mpteq2dv eqtr4di seqeq3d fveq2d fveq12d ifbieq12d df-lgs ovmpoga mpd3an3 ) AFGZDFGZDHIZAJKLZMIZMHNZDHUBUCZAHUBUCZ OZMUGZMNZDUHUIZPCMUJZUIZPLZNZFGADUKLUVIIUUNUUOOZUUPUUSUVHFUVJUUPOZUURMHFU @@ -162505,14 +163612,14 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is c2 clt wbr cneg cabs cseq lgsval lgslem2 simp3i simp2i zsqcl 1zzd zdceq cfv syl2an2r ifcldcd adantr wn simp1i simpr 0zd zdclt syldan dcan2 sylc nnuz cv wne df-ne lgsfcl2 3expa sylan2br ffvelcdmda lgslem3 adantl seqf - wf simplr neqned nnabscl ffvelrnd ifcldadc eqeltrd ) BIJZEIJZKZBEUCLEMU - DZBUFUELZNUDZNMOZEMUGUHZBMUGUHZKZNUIZNOZEUJUSZPDNUKZUSZPLZOFBCDEGULWKWL - WOXDFWKWOFJWLWKWNNMFNFJZWKWSFJZMFJZXEAFHUMZUNQZXGWKXFXGXEXHUOQWIWMIJWJN - IJWNRBUPWKUQWMNURUTVAVBWKWTFJWLVCZXCFJXDFJWKWRWSNFXFWKXFXGXEXHVDQXIWKWP - RZWQRZWRRWKWJMIJZXKWIWJVEZWKVFZEMVGSWIWJXMXLXOBMVGVHWPWQVIVJVAWKXJKZTFX - AXBXPUAUBPFDNTVKXPUQXPTFUAVLZDXJWKEMVMZTFDWBZEMVNWIWJXRXSABCDEFGHVOVPVQ - VRXQFJUBVLZFJKXQXTPLFJXPAXQXTFHVSVTWAXPWJXRXATJWIWJXJWCXPEMWKXJVEWDEWES - WFAWTXCFHVSUTWKWJXMWLRXNXOEMURSWGWH $. + wf simplr neqned nnabscl ffvelcdmd ifcldadc eqeltrd ) BIJZEIJZKZBEUCLEM + UDZBUFUELZNUDZNMOZEMUGUHZBMUGUHZKZNUIZNOZEUJUSZPDNUKZUSZPLZOFBCDEGULWKW + LWOXDFWKWOFJWLWKWNNMFNFJZWKWSFJZMFJZXEAFHUMZUNQZXGWKXFXGXEXHUOQWIWMIJWJ + NIJWNRBUPWKUQWMNURUTVAVBWKWTFJWLVCZXCFJXDFJWKWRWSNFXFWKXFXGXEXHVDQXIWKW + PRZWQRZWRRWKWJMIJZXKWIWJVEZWKVFZEMVGSWIWJXMXLXOBMVGVHWPWQVIVJVAWKXJKZTF + XAXBXPUAUBPFDNTVKXPUQXPTFUAVLZDXJWKEMVMZTFDWBZEMVNWIWJXRXSABCDEFGHVOVPV + QVRXQFJUBVLZFJKXQXTPLFJXPAXQXTFHVSVTWAXPWJXRXATJWIWJXJWCXPEMWKXJVEWDEWE + SWFAWTXCFHVSUTWKWJXMWLRXNXOEMURSWGWH $. $} $( Closure of the function ` F ` which defines the Legendre symbol at the @@ -162539,49 +163646,49 @@ mathematician Ptolemy (Claudius Ptolemaeus). 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(Contributed by Mario Carneiro, 4-Feb-2015.) $) lgsval4lem $p |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F = ( n e. NN |-> @@ -162714,12 +163821,13 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( vx vy cz wcel cn wa co cc0 clt wbr c1 cfv cmul adantl syl3anc clgs cneg cif cabs cseq wne wceq simpl nnz nnne0 lgsval4 wn nngt0 cr wi nnre ltnsym 0re sylancr mpd intnanrd iffalsed cn0 nnnn0 nn0ge0d absidd fveq2d oveq12d - nnuz 1zzd cv wf lgsfcl3 ffvelcdmda zmulcl seqf simpr ffvelrnd zcnd 3eqtrd - mulid2d ) AHIZDJIZKZADUALZDMNOZAMNOZKZPUBZPUCZDUDQZRCPUEZQZRLZPDWLQZRLWOW - DWBDHIZDMUFZWEWNUGWBWCUHZWCWPWBDUISZWCWQWBDUJSZABCDEUKTWDWJPWMWORWDWHWIPW - DWFWGWDMDNOZWFULZWCXAWBDUMSWDMUNIDUNIZXAXBUOURWCXCWBDUPSZMDUQUSUTVAVBWDWK - DWLWDDXDWDDWCDVCIWBDVDSVEVFVGVHWDWOWDWOWDJHDWLWDFGRHCPJVIWDVJWDJHFVKZCWDW - BWPWQJHCVLWRWSWTABCDEVMTVNXEHIGVKZHIKXEXFRLHIWDXEXFVOSVPWBWCVQVRVSWAVT $. + nnuz 1zzd cv lgsfcl3 ffvelcdmda zmulcl seqf ffvelcdmd zcnd mulid2d 3eqtrd + wf simpr ) AHIZDJIZKZADUALZDMNOZAMNOZKZPUBZPUCZDUDQZRCPUEZQZRLZPDWLQZRLWO + WDWBDHIZDMUFZWEWNUGWBWCUHZWCWPWBDUISZWCWQWBDUJSZABCDEUKTWDWJPWMWORWDWHWIP + WDWFWGWDMDNOZWFULZWCXAWBDUMSWDMUNIDUNIZXAXBUOURWCXCWBDUPSZMDUQUSUTVAVBWDW + KDWLWDDXDWDDWCDVCIWBDVDSVEVFVGVHWDWOWDWOWDJHDWLWDFGRHCPJVIWDVJWDJHFVKZCWD + WBWPWQJHCVTWRWSWTABCDEVLTVMXEHIGVKZHIKXEXFRLHIWDXEXFVNSVOWBWCWAVPVQVRVS + $. $} $( The value of the Legendre symbol is either -1 or 0 or 1. (Contributed by @@ -162744,29 +163852,29 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is 3eqtrd 1t1e1 iffalse intnand iffalsed oveq12d 3eqtr4a wo exmiddc mpjaodan 0re simp1 eqcomd cq pcneg syl2anc adantlr ifeq1dadc mpteq2dva seqeq3d zcn zq prmdc 3ad2ant2 absnegd fveq12d neg1cn a1i ifcldcd dcan2 sylc nnuz 1zzd - eqid lgsfcl3 ffvelcdmda zmulcl seqf nnabscl 3adant1 ffvelrnd zcnd mulassd - simp3 negne0d lgsval4 syl3anc 3eqtr4d ) ADEZBDEZBFUCZUDZBUEZFUFGZAFUFGZHZ - IUEZIJZUUAUGKZLCMCUHZUIEZAUUHUJNZUUHUUAUKNZULNZIJZUMZIUQZKZLNZUUCUUEIJZBF - UFGZUUCHZUUEIJZBUGKZLCMUUIUUJUUHBUKNZULNZIJZUMZIUQZKZLNZLNZAUUAUJNZUURABU - JNZLNYTUUQUURUVALNZUVHLNUVJYTUUFUVMUUPUVHLYTUVMUUFYTUUCUVMUUFOUUCUNZYTUUC - HZUVMUUEUVALNZUUBUUEIJZUUFUVOUURUUEUVALUUCUURUUEOYTUUCUUEIUOPUPUVOUUEUUSU - UEIJZLNZUUSIUUEJZUVPUVQUVOYRUVSUVTOZYQYRYSUUCURZYRUUSQZUWAYRFDEZUWCRBFUSU - TZUUSUUEIUVSIUUEUVRUUEOUVSUUEUUELNIUVRUUEUUELVAVBVCUVRIOUVSUUEILNUUEUVRIU - UELVAIVDVEVCVGSSUVOUVRUVAUUELUVOUUSUUTUUEIUVOUUCUUSYTUUCVHZVFVIVJUVOUVTUU - BUNZIUUEJZUVQUVOUUSUWGIUUEUVOUUSBFVKGZFUUAVKGZUWGUVOUUSUWIFBUCZUVOBFYQYRY - SUUCVLVMUVOYRUWDUUSUWIUWKHVRUWBRBFVNVOVPUVOBUVOBUWBVQZVSUVOFVTEUUAVTEUWJU - WGVRWSUVOBUWLWAFUUAWBWCWDVIUVOYRUWHUVQOZUWBYRUUBQZUWMYRUUADEZUWDUWNBWEZRU - UAFUSVOUUBIUUEWFSSWGWHUVOUUBUUDUUEIUVOUUCUUBUWFVFVIWIYTUVNHZIILNIUVMUUFWJ - UWQUURIUVAILUVNUURIOYTUUCUUEIWKPUWQUUTUUEIUWQUUCUUSYTUVNVHZWLWMWNUWQUUDUU - EIUWQUUCUUBUWRWLWMWOYTUUCQZUUCUVNWPYTYQUWDUWSYQYRYSWTZRAFUSVOZUUCWQSWRXAY - TUUGUVBUUOUVGYTUUNUVFLIYTCMUUMUVEYTUUHMEZHUUIUULUVDIYTUUIUULUVDOUXBYTUUIH - ZUUKUVCUUJULUXCUUIBXBEZUUKUVCOYTUUIVHUXCYRUXDYQYRYSUUIURBXJSBUUHXCXDVJXEU - XBUUIQYTUUHXKPXFXGXHYTBYRYQBTEYSBXIXLZXMXNWNYTUURUVAUVHYTUUCUUEITUUETEYTX - OXPZITEYTVDXPZUXAXQYTUUTUUEITUXFUXGYTUWCUWSUUTQYRYQUWCYSUWEXLUXAUUSUUCXRX - SXQYTUVHYTMDUVBUVGYTUAUBLDUVFIMXTYTYAYTMDUAUHZUVFACUVFBUVFYBZYCYDUXHDEUBU - HZDEHUXHUXJLNDEYTUXHUXJYEPYFYRYSUVBMEYQBYGYHYIYJYKWGYTYQUWOUUAFUCUVKUUQOU - WTYRYQUWOYSUWPXLYTBUXEYQYRYSYLYMACUUNUUAUUNYBYNYOYTUVLUVIUURLACUVFBUXIYNV - JYP $. + eqid lgsfcl3 ffvelcdmda seqf nnabscl 3adant1 ffvelcdmd zcnd mulassd simp3 + zmulcl negne0d lgsval4 syl3anc 3eqtr4d ) ADEZBDEZBFUCZUDZBUEZFUFGZAFUFGZH + ZIUEZIJZUUAUGKZLCMCUHZUIEZAUUHUJNZUUHUUAUKNZULNZIJZUMZIUQZKZLNZUUCUUEIJZB + FUFGZUUCHZUUEIJZBUGKZLCMUUIUUJUUHBUKNZULNZIJZUMZIUQZKZLNZLNZAUUAUJNZUURAB + UJNZLNYTUUQUURUVALNZUVHLNUVJYTUUFUVMUUPUVHLYTUVMUUFYTUUCUVMUUFOUUCUNZYTUU + CHZUVMUUEUVALNZUUBUUEIJZUUFUVOUURUUEUVALUUCUURUUEOYTUUCUUEIUOPUPUVOUUEUUS + UUEIJZLNZUUSIUUEJZUVPUVQUVOYRUVSUVTOZYQYRYSUUCURZYRUUSQZUWAYRFDEZUWCRBFUS + UTZUUSUUEIUVSIUUEUVRUUEOUVSUUEUUELNIUVRUUEUUELVAVBVCUVRIOUVSUUEILNUUEUVRI + UUELVAIVDVEVCVGSSUVOUVRUVAUUELUVOUUSUUTUUEIUVOUUCUUSYTUUCVHZVFVIVJUVOUVTU + UBUNZIUUEJZUVQUVOUUSUWGIUUEUVOUUSBFVKGZFUUAVKGZUWGUVOUUSUWIFBUCZUVOBFYQYR + YSUUCVLVMUVOYRUWDUUSUWIUWKHVRUWBRBFVNVOVPUVOBUVOBUWBVQZVSUVOFVTEUUAVTEUWJ + UWGVRWSUVOBUWLWAFUUAWBWCWDVIUVOYRUWHUVQOZUWBYRUUBQZUWMYRUUADEZUWDUWNBWEZR + UUAFUSVOUUBIUUEWFSSWGWHUVOUUBUUDUUEIUVOUUCUUBUWFVFVIWIYTUVNHZIILNIUVMUUFW + JUWQUURIUVAILUVNUURIOYTUUCUUEIWKPUWQUUTUUEIUWQUUCUUSYTUVNVHZWLWMWNUWQUUDU + UEIUWQUUCUUBUWRWLWMWOYTUUCQZUUCUVNWPYTYQUWDUWSYQYRYSWTZRAFUSVOZUUCWQSWRXA + YTUUGUVBUUOUVGYTUUNUVFLIYTCMUUMUVEYTUUHMEZHUUIUULUVDIYTUUIUULUVDOUXBYTUUI + HZUUKUVCUUJULUXCUUIBXBEZUUKUVCOYTUUIVHUXCYRUXDYQYRYSUUIURBXJSBUUHXCXDVJXE + UXBUUIQYTUUHXKPXFXGXHYTBYRYQBTEYSBXIXLZXMXNWNYTUURUVAUVHYTUUCUUEITUUETEYT + XOXPZITEYTVDXPZUXAXQYTUUTUUEITUXFUXGYTUWCUWSUUTQYRYQUWCYSUWEXLUXAUUSUUCXR + XSXQYTUVHYTMDUVBUVGYTUAUBLDUVFIMXTYTYAYTMDUAUHZUVFACUVFBUVFYBZYCYDUXHDEUB + UHZDEHUXHUXJLNDEYTUXHUXJYLPYEYRYSUVBMEYQBYFYGYHYIYJWGYTYQUWOUUAFUCUVKUUQO + UWTYRYQUWOYSUWPXLYTBUXEYQYRYSYKYMACUUNUUAUUNYBYNYOYTUVLUVIUURLACUVFBUXIYN + VJYP $. $( The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.) $) @@ -162877,20 +163985,20 @@ mathematician Ptolemy (Claudius Ptolemaeus). This particular version is ( cz wcel c2 c1 caddc co cdvds wbr cc0 cfz ax-1cn wn cmo simp1i peano2z wa c8 wi ax-mp eqeltri simp2i wb 2z dvdsadd mp2an mpbi cc addcomi eqtri zcn oveq1i df-2 add32i eqtr4i 2cn addassi breqtrri cmin wceq wo cuz cfv - elfzuz2 fzm1 syl ibi mvrraddi oveq2i eleq2s eleq2i simp3i syl5bi cn 2nn - 8nn w3a c4 cmul 4z dvdsmul2 4t2e8 breqtri dvdsmod mpan2 mp3an12 biimpar - notbid id breqtrrid nsyl pm2.21d jaod syl5 eleq1 mpbiri a1i 3pm3.2i ) E - JKLEMNOZPQAJKZLAPQZUAZUEZAUFUBOZRESOKZXLBKZUGUGEDMNOZJHDJKZXOJKDCMNOZJG - CJKZXQJKZXRLXQPQZXKXLRCSOZKZXNUGUGZFUCZCUDUHZUIZDUDUHUILLXQNOZXGPXTLYGP - QZXRXTYCFUJZLJKZXSXTYHUKULYELXQUMUNUOXGLCNOZMNOYGEYKMNEMCNOZMNOZYKEXOYM - HDYLMNDXQYLGCMXRCUPKYDCUSUHZTUQURUTURYKMMNOZCNOYMLYOCNVAUTMCMTYNTVBVCVC - UTLCMVDYNTVEURVFXMXLREMVGOZSOZKZXLEVHZVIZXKXNXMYTXMERVJVKZKXMYTUKXLREVL - XLREVMVNVOXKYRXNYSYRXLRDMVGOZSOZKZXLDVHZVIZXKXNUUFXLRDSOZYQXLUUGKZUUFUU - HDUUAKUUHUUFUKXLRDVLXLRDVMVNVOYPDRSEDMXPDUPKYFDUSUHTHVPVQVRXKUUDXNUUEUU - DYBXKXNUUCYAXLUUBCRSDCMYNTGVPVQVSXRXTYCFVTWAXKUUEXNXKLXLPQZUUEXHUUIUAXJ - XHUUIXILWBKZUFWBKZXHUUIXIUKZWCWDUUJUUKXHWELUFPQUULLWFLWGOZUFPWFJKYJLUUM - PQWHULWFLWIUNWJWKLAUFWLWMWNWPWOUUELDXLPLXQDPYIGVFUUEWQWRWSWTXAXBYSXNUGX - KYSXNEBKIXLEBXCXDXEXAXBXF $. + elfzuz2 fzm1 syl ibi mvrraddi oveq2i eleq2s eleq2i biimtrid 2nn 8nn w3a + simp3i cn c4 cmul dvdsmul2 4t2e8 breqtri dvdsmod mp3an12 notbid biimpar + 4z mpan2 id breqtrrid nsyl pm2.21d jaod syl5 eleq1 mpbiri a1i 3pm3.2i ) + EJKLEMNOZPQAJKZLAPQZUAZUEZAUFUBOZRESOKZXLBKZUGUGEDMNOZJHDJKZXOJKDCMNOZJ + GCJKZXQJKZXRLXQPQZXKXLRCSOZKZXNUGUGZFUCZCUDUHZUIZDUDUHUILLXQNOZXGPXTLYG + PQZXRXTYCFUJZLJKZXSXTYHUKULYELXQUMUNUOXGLCNOZMNOYGEYKMNEMCNOZMNOZYKEXOY + MHDYLMNDXQYLGCMXRCUPKYDCUSUHZTUQURUTURYKMMNOZCNOYMLYOCNVAUTMCMTYNTVBVCV + CUTLCMVDYNTVEURVFXMXLREMVGOZSOZKZXLEVHZVIZXKXNXMYTXMERVJVKZKXMYTUKXLREV + LXLREVMVNVOXKYRXNYSYRXLRDMVGOZSOZKZXLDVHZVIZXKXNUUFXLRDSOZYQXLUUGKZUUFU + UHDUUAKUUHUUFUKXLRDVLXLRDVMVNVOYPDRSEDMXPDUPKYFDUSUHTHVPVQVRXKUUDXNUUEU + UDYBXKXNUUCYAXLUUBCRSDCMYNTGVPVQVSXRXTYCFWDVTXKUUEXNXKLXLPQZUUEXHUUIUAX + JXHUUIXILWEKZUFWEKZXHUUIXIUKZWAWBUUJUUKXHWCLUFPQUULLWFLWGOZUFPWFJKYJLUU + MPQWOULWFLWHUNWIWJLAUFWKWPWLWMWNUUELDXLPLXQDPYIGVFUUEWQWRWSWTXAXBYSXNUG + XKYSXNEBKIXLEBXCXDXEXAXBXF $. $} $( Lemma for ~ lgsdir2 . (Contributed by Mario Carneiro, 4-Feb-2015.) $) @@ -163068,61 +164176,61 @@ mathematician Ptolemy (Claudius Ptolemaeus). 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This particular version is cn0 syl12anc expaddd eqtrd iftrue 3eqtr4rd 1t1e1 iffalse 3eqtr4a wo wdc prmdc exmiddc mpjaodan eleq1w oveq2 oveq1 ifbieq1d zexpcl 1zzd ifcldadc syl fvmptd3 prod3fmul lgsdilem2 mulcomd fveq2d eqtr4d lgsval4 neg1z a1i - zdclt dcan2 sylc ifcldcd ffvelcdmda zmulcl seqf ffvelrnd mul4d 3eqtr4d + zdclt dcan2 sylc ifcldcd ffvelcdmda zmulcl seqf ffvelcdmd mul4d 3eqtr4d ) AEFZBEFZCEFZUCZBGUDZCGUDZHZHZBCIJZGUEKZAGUEKZHZLUFZLMZUUHUGNZIDTDUHZU IFZAUUOUJJZUUOUUHOJZPJZLMZUKZLULNZIJZBGUEKZUUJHZUULLMZCGUEKZUUJHZUULLMZ IJZBUGNZIDTUUPUUQUUOBOJZPJZLMZUKZLULZNZCUGNZIDTUUPUUQUUOCOJZPJZLMZUKZLU @@ -163240,119 +164349,119 @@ mathematician Ptolemy (Claudius Ptolemaeus). 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(Contributed by Mario @@ -165772,9 +166881,20 @@ Norman Megill (2007) section 1.1.3. Megill then states, "A number of htmldef "SRing" as "SRing"; althtmldef "SRing" as "SRing"; latexdef "SRing" as "\mathrm{SRing}"; +htmldef "Ring" as + " Ring"; + althtmldef "Ring" as "Ring"; + latexdef "Ring" as "\mathrm{Ring}"; +htmldef "CRing" as + " CRing"; + althtmldef "CRing" as "CRing"; + latexdef "CRing" as "\mathrm{CRing}"; htmldef "1r" as " 1r"; althtmldef "1r" as "1r"; latexdef "1r" as "1_\mathrm{r}"; +htmldef "oppR" as "oppr"; + althtmldef "oppR" as "oppr"; + latexdef "oppR" as "\mathrm{opp}_r"; htmldef "numer" as "numer"; althtmldef "numer" as "numer"; latexdef "numer" as "\mathrm{numer}"; @@ -166382,8 +167502,8 @@ less basic results (like ~ exlimiv and ~ 19.9ht or ~ 19.23ht ). $( Weakening two hypotheses of ~ vtoclgf . (Contributed by BJ, 21-Nov-2019.) $) bj-vtoclgft $p |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) ) $= - ( wcel cvv cv wceq wi wal elex wex issetf bj-exlimmp syl5bi syl5 ) DEIDJI - ZCKDLZABMMCNZBDEOUAUBCPUCBCDFQABUBCGHRST $. + ( wcel cvv cv wceq wi wal elex wex issetf bj-exlimmp biimtrid syl5 ) DEID + JIZCKDLZABMMCNZBDEOUAUBCPUCBCDFQABUBCGHRST $. bj-vtoclgf.maj $e |- ( x = A -> ( ph -> ps ) ) $. $( Weakening two hypotheses of ~ vtoclgf . (Contributed by BJ, @@ -166757,13 +167877,13 @@ which membership in it is a bounded proposition ( ~ df-bdc ). ( c1o cpw wf cun c2o wceq wral c0 wa wcel a1i simpr adantr fvmpt2d cin cv cdif cres cfv cif fmelpw1o fmpt3d wss inss1 difssd unssd elun elin1d cmpt wo 1oex 0ex ifelpwun mpdan elin2d iftrued eqtrd 1lt2o eqeltrdi ex eldifad - eldifbd iffalsed 0lt2o jaod syl5bi imp resflem ralrimiva jca jca31 ) AEGH - ZDIECUAZECUCZJZKDWAUDIBUBZDUEZGLZBVSMZWCNLZBVTMZOABEWBCPZGNUFZVRDFWIVRPAW - BEPZOWHUGQUHZABWADEVRKWKAVSVTEVSEUIAECUJQAECUKULAWBWAPZWCKPZWLWBVSPZWBVTP - ZUPAWMWBVSVTUMAWNWMWOAWNWMAWNOZWCGKWPWCWIGWPWJWCWILZWPECWBAWNRZUNWPBEWIDG - NJHZADBEWIUOLZWNFSWIWSPZWPWJOWHGNUQURUSZQTUTWPWHGNWPECWBWRVAVBVCZVDVEVFAW - OWMAWOOZWCNKXDWCWINXDWJWQXDWBECAWORZVGXDBEWIDWSAWTWOFSXAXDWJOXBQTUTXDWHGN - XDWBECXEVHVIVCZVJVEVFVKVLVMVNAWEWGAWDBVSXCVOAWFBVTXFVOVPVQ $. + eldifbd iffalsed 0lt2o jaod biimtrid imp resflem ralrimiva jca jca31 ) AE + GHZDIECUAZECUCZJZKDWAUDIBUBZDUEZGLZBVSMZWCNLZBVTMZOABEWBCPZGNUFZVRDFWIVRP + AWBEPZOWHUGQUHZABWADEVRKWKAVSVTEVSEUIAECUJQAECUKULAWBWAPZWCKPZWLWBVSPZWBV + TPZUPAWMWBVSVTUMAWNWMWOAWNWMAWNOZWCGKWPWCWIGWPWJWCWILZWPECWBAWNRZUNWPBEWI + DGNJHZADBEWIUOLZWNFSWIWSPZWPWJOWHGNUQURUSZQTUTWPWHGNWPECWBWRVAVBVCZVDVEVF + AWOWMAWOOZWCNKXDWCWINXDWJWQXDWBECAWORZVGXDBEWIDWSAWTWOFSXAXDWJOXBQTUTXDWH + GNXDWBECXEVHVIVCZVJVEVFVKVLVMVNAWEWGAWDBVSXCVOAWFBVTXFVOVPVQ $. $} ${ @@ -166848,22 +167968,23 @@ which membership in it is a bounded proposition ( ~ df-bdc ). ( vz vy vg cv wcel wdc wral c1o wceq c0 wa c2o cfv cin cdif cmap co dcbid wrex eleq1w cbvralvw cif wf ifbid cbvmptv a1i biimpri adantl bj-charfundc cmpt ex con0 2on elmapd biimprd adantrd syld imp wb fveq1 ralbidv anbi12d - eqeq1d simprd rspcedvd syl5bi wne com wss omex 2ssom mapss mp2an cbvrexvw - wn cvv fveqeq2 1n0 eqeq1 mtbiri neqned ralimi sylbi biimpi anim12i reximi - neii ssrexv mpsyl bj-charfunr syl6ibr wstab r19.21bi stdcn sylib ralimdva - notbid wi impbid ) ABLZCMZNZBFOZXHDLZUAZPQZBFCUBZOZXMRQZBFCUCZOZSZDTFUDUE - ZUGZXKILZCMZNZIFOZAYBXJYEBIFXHYCQXIYDBICUHUFUIZAYFYBAYFSZXTXHIFYDPRUJZURZ - UAZPQZBXOOZYKRQZBXROZSZDYJYAAYFYJYAMZAYFFTYJUKZYPSZYQAYFYSYHBCYJFYJBFXIPR - UJZURQYHIBFYIYTYCXHQYDXIPRIBCUHULUMUNYFXKAXKYFYGUOUPUQZUSAYRYQYPAYQYRATFY - JUTETUTMAVAUNGVBVCVDVEVFXLYJQZXTYPVGYHUUBXPYMXSYOUUBXNYLBXOUUBXMYKPXHXLYJ - VHZVKVIUUBXQYNBXRUUBXMYKRUUCVKVIVJUPYHYRYPUUAVLVMUSVNAYBXIWCZNZBFOZXKAYBJ - LZCMZWCZNZJFOZUUFAYBUUKAYBSJCKFYBUUGKLZUAZRVOZJXOOZUUMRQZJXROZSZKVPFUDUEZ - UGZAYAUUSVQZYBUURKYAUGZUUTVPWDMTVPVQUVAVRVSTVPFWDVTWAYBXHUULUAZPQZBXOOZUV - CRQZBXROZSZKYAUGUVBXTUVHDKYAXLUULQZXPUVEXSUVGUVIXNUVDBXOUVIXMUVCPXHXLUULV - HZVKVIUVIXQUVFBXRUVIXMUVCRUVJVKVIVJWBUVHUURKYAUVEUUOUVGUUQUVEUUMPQZJXOOUU - OUVDUVKBJXOXHUUGPUULWEUIUVKUUNJXOUVKUUMRUVKUUPPRQPRWFWOUUMPRWGWHWIWJWKUVG - UUQUVFUUPBJXRXHUUGRUULWEUIWLWMWNWKUURKYAUUSWPWQUPWRUSUUEUUJBJFXHUUGQZUUDU - UIUVLXIUUHBJCUHXEUFUIWSAUUEXJBFAXHFMSXIWTZUUEXJXFAUVMBFHXAXIXBXCXDVEXG $. + eqeq1d simprd rspcedvd biimtrid wn wne com wss cvv omex 2ssom mapss mp2an + cbvrexvw fveqeq2 1n0 neii eqeq1 mtbiri neqned ralimi sylbi biimpi anim12i + reximi ssrexv mpsyl bj-charfunr notbid syl6ibr wstab r19.21bi stdcn sylib + wi ralimdva impbid ) ABLZCMZNZBFOZXHDLZUAZPQZBFCUBZOZXMRQZBFCUCZOZSZDTFUD + UEZUGZXKILZCMZNZIFOZAYBXJYEBIFXHYCQXIYDBICUHUFUIZAYFYBAYFSZXTXHIFYDPRUJZU + RZUAZPQZBXOOZYKRQZBXROZSZDYJYAAYFYJYAMZAYFFTYJUKZYPSZYQAYFYSYHBCYJFYJBFXI + PRUJZURQYHIBFYIYTYCXHQYDXIPRIBCUHULUMUNYFXKAXKYFYGUOUPUQZUSAYRYQYPAYQYRAT + FYJUTETUTMAVAUNGVBVCVDVEVFXLYJQZXTYPVGYHUUBXPYMXSYOUUBXNYLBXOUUBXMYKPXHXL + YJVHZVKVIUUBXQYNBXRUUBXMYKRUUCVKVIVJUPYHYRYPUUAVLVMUSVNAYBXIVOZNZBFOZXKAY + BJLZCMZVOZNZJFOZUUFAYBUUKAYBSJCKFYBUUGKLZUAZRVPZJXOOZUUMRQZJXROZSZKVQFUDU + EZUGZAYAUUSVRZYBUURKYAUGZUUTVQVSMTVQVRUVAVTWATVQFVSWBWCYBXHUULUAZPQZBXOOZ + UVCRQZBXROZSZKYAUGUVBXTUVHDKYAXLUULQZXPUVEXSUVGUVIXNUVDBXOUVIXMUVCPXHXLUU + LVHZVKVIUVIXQUVFBXRUVIXMUVCRUVJVKVIVJWDUVHUURKYAUVEUUOUVGUUQUVEUUMPQZJXOO + UUOUVDUVKBJXOXHUUGPUULWEUIUVKUUNJXOUVKUUMRUVKUUPPRQPRWFWGUUMPRWHWIWJWKWLU + VGUUQUVFUUPBJXRXHUUGRUULWEUIWMWNWOWLUURKYAUUSWPWQUPWRUSUUEUUJBJFXHUUGQZUU + DUUIUVLXIUUHBJCUHWSUFUIWTAUUEXJBFAXHFMSXIXAZUUEXJXEAUVMBFHXBXIXCXDXFVEXG + $. $} @@ -168110,12 +169231,12 @@ with definitions ( ` B ` is the definiendum that one wants to prove bj-indind $p |- ( ( Ind A /\ ( (/) e. B /\ A. x e. A ( x e. B -> suc x e. B ) ) ) -> Ind ( A i^i B ) ) $= ( wind c0 wcel cv csuc wi wral wa cin df-bj-ind id biimpri wal syl df-ral - elin sylibr sylanb r19.26 simpl simpr syl6an ralimi pm3.31 syl5bi anim12i - an4s alimi sylbi ) BDZECFZAGZCFZUOHZCFZIZABJZKZKZEBCLZFZUQVCFZAVCJZKZVCDV - BEBFZUNKZUQBFZABJZUTKZKZVGUMVHVKKVAVMABMVHUNVKUTVMVMNUJUAVIVDVLVFVDVIEBCS - OVLUOVCFZVEIZAPZVFVLUPVEIZABJZVPVLVJUSKZABJZVRVTVLVJUSABUBOVSVQABVSVJUPUR - VEVJUSUCVJUSUDVEVJURKUQBCSOUEUFQVRUOBFZVQIZAPVPVQABRWBVOAVNWAUPKWBVEUOBCS - WAUPVEUGUHUKULQVEAVCRTUIQAVCMT $. + elin sylibr sylanb r19.26 simpl simpr syl6an ralimi pm3.31 biimtrid alimi + an4s sylbi anim12i ) BDZECFZAGZCFZUOHZCFZIZABJZKZKZEBCLZFZUQVCFZAVCJZKZVC + DVBEBFZUNKZUQBFZABJZUTKZKZVGUMVHVKKVAVMABMVHUNVKUTVMVMNUJUAVIVDVLVFVDVIEB + CSOVLUOVCFZVEIZAPZVFVLUPVEIZABJZVPVLVJUSKZABJZVRVTVLVJUSABUBOVSVQABVSVJUP + URVEVJUSUCVJUSUDVEVJURKUQBCSOUEUFQVRUOBFZVQIZAPVPVQABRWBVOAVNWAUPKWBVEUOB + CSWAUPVEUGUHUIUKQVEAVCRTULQAVCMT $. $} ${ @@ -168414,16 +169535,16 @@ Constructive proof (from CZF). See the comment of ~ bj-bdfindis for $( A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) $) bj-nntrans $p |- ( A e. _om -> ( B e. A -> B C_ A ) ) $= - ( vx vz vy com wcel cv wss wral c0 csuc wi syl5bi wceq raleqbi1dv biimprd - nfv sseq2 biimpd ral0 csn cun wel df-suc eleq2i wo elun sssucid sstr2 mpi - imim2i elsni eqsstrdi a1i jaod ralimi2 bdcv bdss ax-bdal weq bj-bdfindisg - rgenw nfcv mp2an sseq1 rspc syl5com ) AFGZCHZAIZCAJZBAGBAIZVJKIZCKJZVJDHZ - IZCVPJZVJVPLZIZCVSJZMZDFJVIVLMVNCUAWBDFVQVTCVPVSVJVSGVJVPVPUBZUCZGZCDUDZV - QMZVTVSWDVJVPUEUFWEWFVJWCGZUGWGVTVJVPWCUHWGWFVTWHVQVTWFVQVPVSIVTVPUIZVJVP - VSUJUKULWHVTMWGWHVJVPVSVJVPUMWIUNUOUPNNUQVCVJEHZIZCWJJZVOVRWAVLEDAWKCECWJ - EURUSUTVOERVRERWAERWJKOWLVOWKVNCWJKWJKVJSPQEDVAWLVRWKVQCWJVPWJVPVJSPTWJVS - OWLWAWKVTCWJVSWJVSVJSPQEAVDVLERWJAOWLVLWKVKCWJAWJAVJSPTVBVEVKVMCBAVMCRVJB - AVFVGVH $. + ( vx vz vy com wcel cv wss wral c0 csuc wi biimtrid wceq sseq2 raleqbi1dv + nfv biimprd biimpd ral0 csn cun wel df-suc eleq2i wo sssucid sstr2 imim2i + elun mpi elsni eqsstrdi a1i jaod ralimi2 rgenw bdcv bdss ax-bdal weq nfcv + bj-bdfindisg mp2an sseq1 rspc syl5com ) AFGZCHZAIZCAJZBAGBAIZVJKIZCKJZVJD + HZIZCVPJZVJVPLZIZCVSJZMZDFJVIVLMVNCUAWBDFVQVTCVPVSVJVSGVJVPVPUBZUCZGZCDUD + ZVQMZVTVSWDVJVPUEUFWEWFVJWCGZUGWGVTVJVPWCUKWGWFVTWHVQVTWFVQVPVSIVTVPUHZVJ + VPVSUIULUJWHVTMWGWHVJVPVSVJVPUMWIUNUOUPNNUQURVJEHZIZCWJJZVOVRWAVLEDAWKCEC + WJEUSUTVAVOERVRERWAERWJKOWLVOWKVNCWJKWJKVJPQSEDVBWLVRWKVQCWJVPWJVPVJPQTWJ + VSOWLWAWKVTCWJVSWJVSVJPQSEAVCVLERWJAOWLVLWKVKCWJAWJAVJPQTVDVEVKVMCBAVMCRV + JBAVFVGVH $. $} ${ @@ -168441,16 +169562,16 @@ Constructive proof (from CZF). See the comment of ~ bj-bdfindis for natural numbers, which does not require ~ ax-setind . (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) $) bj-nnelirr $p |- ( A e. _om -> -. A e. A ) $= - ( vy vx c0 wcel wn cv csuc wi com wral syl5bi nfv wceq eleq1 eleq2 notbid - bitrd biimprd biimpd noel csn cun df-suc eleq2i wo wss bj-nntrans sucssel - elun syld vex sucid elsni eleqtrid a1i con3d ax-bdel ax-bdn elequ1 elequ2 - jaod rgen nfcv bj-bdfindisg mp2an ) DDEZFZBGZVIEZFZVIHZVLEZFZIZBJKAJEAAEZ - FZIDUAVOBJVIJEZVMVJVMVLVIVIUBZUCZEZVRVJVLVTVLVIUDUEWAVLVIEZVLVSEZUFVRVJVL - VIVSUJVRWBVJWCVRWBVLVIUGVJVIVLUHVIVIJUIUKWCVJIVRWCVIVLVIVIBULUMVLVIUNUOUP - VBLLUQVCCGZWDEZFZVHVKVNVQCBAWECCURUSVHCMVKCMVNCMWDDNZWFVHWGWEVGWGWEDWDEVG - WDDWDOWDDDPRQSWDVINZWFVKWHWEVJWHWEVIWDEVJCBCUTCBBVARQTWDVLNZWFVNWIWEVMWIW - EVLWDEVMWDVLWDOWDVLVLPRQSCAVDVQCMWDANZWFVQWJWEVPWJWEAWDEVPWDAWDOWDAAPRQTV - EVF $. + ( vy vx c0 wcel wn cv csuc com wral biimtrid nfv eleq1 eleq2 bitrd notbid + wi wceq biimprd biimpd noel csn cun df-suc eleq2i elun bj-nntrans sucssel + wo wss syld vex sucid elsni eleqtrid a1i jaod con3d ax-bdel ax-bdn elequ1 + rgen elequ2 nfcv bj-bdfindisg mp2an ) DDEZFZBGZVIEZFZVIHZVLEZFZQZBIJAIEAA + EZFZQDUAVOBIVIIEZVMVJVMVLVIVIUBZUCZEZVRVJVLVTVLVIUDUEWAVLVIEZVLVSEZUIVRVJ + VLVIVSUFVRWBVJWCVRWBVLVIUJVJVIVLUGVIVIIUHUKWCVJQVRWCVIVLVIVIBULUMVLVIUNUO + UPUQKKURVBCGZWDEZFZVHVKVNVQCBAWECCUSUTVHCLVKCLVNCLWDDRZWFVHWGWEVGWGWEDWDE + VGWDDWDMWDDDNOPSWDVIRZWFVKWHWEVJWHWEVIWDEVJCBCVACBBVCOPTWDVLRZWFVNWIWEVMW + IWEVLWDEVMWDVLWDMWDVLVLNOPSCAVDVQCLWDARZWFVQWJWEVPWJWEAWDEVPWDAWDMWDAANOP + TVEVF $. $} $( A version of ~ en2lp for natural numbers, which does not require @@ -169121,29 +170242,29 @@ a singleton (where the latter can also be thought of as representing pwle2 $p |- ( ( N e. _om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o ) $= ( com wcel c1o wa c2o wss wn c0 wceq syl ad2antrr simpr sylnib mtbid jca cpw wf1 cop cfv csn wo wf simplr f1f cxp con0 nnon 1lt2o jctil ontr1 sylc - 0lt1o opelxpi sylancl cv iunxpconst eqtri eleqtrrdi ffvelrnd elpwid df1o2 - ciun sseqtrdi pwtrufal ioran 1n0 neii 1oex 0ex opth1 wi f1veqaeq syl12anc - mto 0lt2o mtoi adantr eqeq2d 2on0 nesymi eqeq1d eqcom csuc wb 1onn peano1 - peano4 mp2an nemtbir df-2o df-1o eqeq12i mtbir neir pm2.65da mtand nntri1 - 2onn mpan2 mpbird ) DFGZBHUAZCUBZIZDJKZJDGZLZXIXKMMUCZCUDZMNZLZXNMUEZNZLZ - IZXIXKIZXPXSYAXOHMUCZCUDZMNZLZYCXQNZLZIZYAYDYFUFLZYHYAYCXQKYILYAYCHXQYAYC - HYABXGYBCYAXHBXGCUGXFXHXKUHZBXGCUIOZYAYBDHUJZBYAHDGZMHGZYBYLGYADUKGZHJGZX - KIYMXFYOXHXKDULPZYAXKYPXIXKQZUMUNHJDUOUPUQHMDHURUSBADAUTUEHUJVGYLEADHVAVB - ZVCZVDVEVFVHYCVIOYDYFVJRZYAXOIZYEYGUUBYCXNNZYDYAUUCLZXOYAUUCYBXMNZUUEHMNZ - HMVKVLHMMMVMVNVOVSYAXHYBBGZXMBGZUUCUUEVPYJYTYAXMYLBYAMDGZYNXMYLGYAYOMJGZX - KIUUIYQYAXKUUJYRVTUNMJDUOUPUQMMDHURUSYSVCZBXGYBXMCVQVRWAZWBUUBXNMYCYAXOQW - CSUUBYFJMUCZCUDZMNZLZUUNXQNZLZIZUUBYFIZUUPUURUUTMUUNNZUUOUUTXNUUNNZUVAYAU - VBLZXOYFYAUVBXMUUMNZUVDMJNJMWDWEMMJMVNVNVOVSYAXHUUHUUMBGZUVBUVDVPYJUUKYAU - UMYLBYAXKYNUUMYLGYRUQJMDHURUSYSVCZBXGXMUUMCVQVRWAZPUUTXNMUUNYAXOYFUHWFSMU - 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wf simplr fof simpr ffvelrnd eldju1st 1n0 neii eqeq1 mtbiri orim2i sylibr - df-dc ifcldadc fmpttd 2fveq3 ifbid eqcom 3imtr3i cbvmptv feq1i sylib 2onn - wo elexi elmap rspcdva eqid fvmptd3 c2nd eqcomd eqifdc orcom intnan biorf - bitr4i mpbiran2 bitrdi syl2anc biidd ex peano1 eldju2ndl simprd rexlimdva - elrab sylbid elrabd djulcl foelrn adantlr 1stinl eqtr3di iftrued sylan9eq - cinl fveq2 reximdva mpd impbid 3imtr3d df-stab exlimddv exmid1stab ) AUAA - UAIZJUDZKZUEZUVCUVDUFALUVEUBLUHZMUIZDIZUJZUVFDAEIZLUFZUVKBIZCIZUJZCNZOZEN - ZLUVMMUIZUVIUJZDNZVBZBUKUVLUVKUVGUVNUJZCNZOZENZUVJDNZFAUVGULPZUVGLUFZUVGU - VGUVNUJZCNZOZUWFUWHAUVEUBLUMUNZUOUWIUWKUVEUBLUPUVGUVGVCUVGUQZUJZUWKUVGUVG - UWNURUWOUVGUSUVGUVGUWNUTVAUWJUWOCUWNUWHUWNULPUWMUVGULVDVAUVGUVGUVNUWNVEVF - VAVGUWEUWLEUVGULUVKUVGKZUVLUWIUWDUWKUVKUVGLVHUWPUWCUWJCUVKUVGUVGUVNVIVJVK - VLVNUWBUWFUWGVBBUVGUWMUVMUVGKZUVRUWFUWAUWGUWQUVQUWEEUWQUVPUWDUVLUWQUVOUWC - CUVMUVGUVKUVNVMVJVOVJUWQUVTUVJDUWQUVSUVHKUVTUVJVPUVMUVGMVQUVSUVHLUVIVMQVJ - VRVSVTAUVJOZUVERZRZUVEVBUVFUWRHIZUCLUCIZUVISWASZJKZMJWBZWCZSZMKZHLWDZRZRZ - 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(Contributed by Jim Kingdon, 30-Jul-2022.) $) nnsf $p |- S : NN+oo --> NN+oo $= ( vj vk vf xnninf com cv wceq c1o cfv wcel c2o wss adantr adantl fveq2d - c0 cuni cif cmpt cmap co csuc wral wf wa a1i nninff nnpredcl ffvelrnd wdc - 1lt2o nndceq0 ifcldcd eqid fmptd cvv wb 2onn omex elmapg mp2an sylibr wtr - wn 1on ontrci peano2 syl df-2o eleqtrdi trsucss mpsyl sseqtrrd word simpr - iftrue nnord ordtr unisucg mpbid wne neqned nnsucpred syl2anc suceq fveq2 - 3syl sseq12d fveq1 ralbidv df-nninf elrab2 simprbi cbvralv sylib ad2antrr - rspcdva eqsstrrd eqsstrd peano3 neneqd ad2antlr iffalsed 3sstr4d mpjaodan - weq wo exmiddc eqeq1 unieq ifbieq2d fvmptg ralrimiva sylanbrc fmpti ) CHH - BIBJZTKZLXTUAZCJZMZUBZUCZADYCHNZYFOIUDUEZNZEJZUFZYFMZYJYFMZPZEIUGZYFHNYGI - OYFUHZYIYGBIYEOYFYGXTINZUIZYALYDOLONZYRUOUJYRIOYBYCYGIOYCUHZYQYCUKZQYQYBI - NYGXTULRUMYQYAUNYGXTUPRUQYFURZUSOINIUTNYIYPVAVBVCOIYFIUTVDVEVFYGYNEIYGYJI - NZUIZYKTKZLYKUAZYCMZUBZYJTKZLYJUAZYCMZUBZYLYMUUDUUIUUHUULPUUIVHZUUDUUIUIZ - UUHLUULLVGUUNUUHLUFZNUUHLPLVIVJUUNUUHOUUOUUDUUHONZUUIUUDUUELUUGOYSUUDUOUJ - 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ifbieq2d 1lt2o nninff nnpredcl ffvelrnd wdc nndceq0 ifcldcd fvmptd peano3 - wne neneqd iffalsed wtr nnord ordtr unisucg mpbid 3eqtrd ad3antlr 3eqtr3d - word eqfnfvd ex rgen2a dff13 mpbir2an ) IIAUCIIAUDEJZAKZFJZAKZLZEFMZUEZFI - NEINABCDUAXLEFIXFIOZXHIOZUFZXJXKXOXJUFZUBPXFXHXMXFPUGZXNXJXMXFQPUHUPZOZXQ - XMXSGJZUIZXFKZXTXFKZUJZGPNZYAHJZKZXTYFKZUJZGPNZYEHXFXRIHEMZYIYDGPYKYGYBYH - YCYAYFXFRXTYFXFRUKULHGUMZUNUOXFQPUQSURXNXHPUGZXMXJXNXHXROZYMXNYNYAXHKZXTX - HKZUJZGPNZYJYRHXHXRIHFMZYIYQGPYSYGYOYHYPYAYFXHRXTYFXHRUKULYLUNUOXHQPUQSUS - XPUBJZPOZUFZYTUIZXGKZUUCXIKZYTXFKZYTXHKZUUBUUCXGXIXOXJUUAUTVAUUBUUDUUCVBL - ZTUUCVCZXFKZVDZUUJUUFUUBBUUCBJZVBLZTUULVCZXFKZVDZUUKPXGQXMXGBPUUPVEZLXNXJ - UUACXFBPUUMTUUNCJZKZVDZVEZUUQIACEMZBPUUTUUPUVBUUMUUSUUOTUUNUURXFRVHVFDBPU - UPVGVIVJVKUUBUULUUCLZUFZUUMUUHUUOUUJTUVDUULUUCVBUUBUVCVLZVMZUVDUUNUUIXFUV - DUULUUCUVEVNZVOVSUUAUUCPOZXPYTVPVQZUUBUUHTUUJQTQOUUBVTVRZUUBPQUUIXFXMPQXF - UDXNXJUUAXFWAVKUUBUVHUUIPOUVIUUCWBSZWCUUBUVHUUHWDUVIUUCWESZWFWGUUBUUHTUUJ - UUBUUCVBUUAUUCVBWIXPYTWHVQWJZWKUUAUUJUUFLXPUUAUUIYTXFUUAYTWLZUUIYTLUUAYTW - 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co elmapi adantl simpll ffvelrnd elpri df2o3 eleq2s orcomd ecased exp31 - syl cpr omsinds impcom ralrimiva ) AFMZCNZOPZFQWSQRAXAAUAMZCNZOPZUDALMZ - CNZOPZUDZAXAUDUALWSXBXEPZXDXGAXIXCXFOXBXECUEUFUGXBWSPZXDXAAXJXCWTOXBWSC - UEUFUGXBQRZXHLXBUIZAXDXKXLUHZAUHZXDXCSPZXNXOSOPZOSUJUKXNXOXPXNXOUHZCDNZ - EQEMZXBRZOSULZUMZDNZSOXQCYBDXQLCEXBACUNRZXMXOJUOXKXLAXOUPZXQAXGLXBUIZXM - AXOUQXQXLAYFUDXKXLAXOURAXGLXBUSUTVJXNXOVAVBVCAXRSPXMXOKUOXQEQXSWSRZOSUL - ZUMZDNZOPZYCOPFQXBWSXBPZYJYCOYLYIYBDYLEQYHYAYLYGXTOSFUAEVDVEVFVCUFAYKFQ - UIXMXOIUOYEVGVHVKVIXNXOXDXNXCTRXOXDVLZXNQTXBCAQTCVMZXMACTQVNWCZRZYNAYPU - BMZVOZCNZYQCNZVPZUBQUIZAYDYPUUBUHJYRUCMZNZYQUUCNZVPZUBQUIUUBUCCYOUNUUCC - PZUUFUUAUBQUUGUUDYSUUEYTYRUUCCVQYQUUCCVQVRVSUCUBVTWAUTWBCTQWDWNWEXKXLAW - FWGYMXCSOWOTXCSOWHWIWJWNWKWLWMWPWQWR $. + elmapi syl adantl simpll ffvelcdmd cpr elpri df2o3 eleq2s orcomd ecased + co exp31 omsinds impcom ralrimiva ) AFMZCNZOPZFQWSQRAXAAUAMZCNZOPZUDALM + ZCNZOPZUDZAXAUDUALWSXBXEPZXDXGAXIXCXFOXBXECUEUFUGXBWSPZXDXAAXJXCWTOXBWS + CUEUFUGXBQRZXHLXBUIZAXDXKXLUHZAUHZXDXCSPZXNXOSOPZOSUJUKXNXOXPXNXOUHZCDN + ZEQEMZXBRZOSULZUMZDNZSOXQCYBDXQLCEXBACUNRZXMXOJUOXKXLAXOUPZXQAXGLXBUIZX + MAXOUQXQXLAYFUDXKXLAXOURAXGLXBUSUTVJXNXOVAVBVCAXRSPXMXOKUOXQEQXSWSRZOSU + LZUMZDNZOPZYCOPFQXBWSXBPZYJYCOYLYIYBDYLEQYHYAYLYGXTOSFUAEVDVEVFVCUFAYKF + QUIXMXOIUOYEVGVHVKVIXNXOXDXNXCTRXOXDVLZXNQTXBCAQTCVMZXMACTQVNWNZRZYNAYP + UBMZVOZCNZYQCNZVPZUBQUIZAYDYPUUBUHJYRUCMZNZYQUUCNZVPZUBQUIUUBUCCYOUNUUC + CPZUUFUUAUBQUUGUUDYSUUEYTYRUUCCVQYQUUCCVQVRVSUCUBVTWAUTWBCTQWCWDWEXKXLA + WFWGYMXCSOWHTXCSOWIWJWKWDWLWMWOWPWQWR $. $} ${ @@ -169452,21 +170573,21 @@ a singleton (where the latter can also be thought of as representing -> DECID A. k e. suc N ( Q ` ( i e. _om |-> if ( i e. k , 1o , (/) ) ) ) = 1o ) $= ( com wcel c2o xnninf cv c1o c0 wceq csuc wral wdc wi suceq raleqdv dcbid - imbi2d vw cmap cif cmpt cfv elmapi peano1 nnnninf mp1i ffvelrnd 2onn elnn - vj co sylancl 1onn nndceq csn suc0 raleqi 0ex eleq2 ifbid mpteq2dv fveq2d - eqeq1d ralsn bitri dcbii sylibr wa cun wf adantl peano2 adantr syl ralsng - wb mpbird dcan2 mpan9 ralunb df-suc exp31 a2d finds impcom ) DEFAGHUBUNFZ - BEBIZCIZFZJKUCZUDZAUEZJLZCDMZNZOZWIWPCUAIZMZNZOZPWIWPCKMZNZOZPWIWPCUMIZMZ - NZOZPWIWPCXHMZNZOZPWIWSPUAUMDWTKLZXCXFWIXNXBXEXNWPCXAXDWTKQRSTWTXGLZXCXJW - IXOXBXIXOWPCXAXHWTXGQRSTWTXHLZXCXMWIXPXBXLXPWPCXAXKWTXHQRSTWTDLZXCWSWIXQX - BWRXQWPCXAWQWTDQRSTWIBEWJKFZJKUCZUDZAUEZJLZOZXFWIYAEFZJEFZYCWIYAGFGEFZYDW - IHGXTAAGHUFZKEFXTHFWIUGBKUHUIUJUKYAGULUOUPYAJUQUOXEYBXEWPCKURZNYBWPCXDYHU - SUTWPYBCKVAWKKLZWOYAJYIWNXTAYIBEWMXSYIWLXRJKWKKWJVBVCVDVEVFVGVHVIVJXGEFZW - IXJXMYJWIXJXMYJWIVKZXJVKZWPCXHXHURZVLZNZOZXMYLXIWPCYMNZVKZOZYPYKYQOZXJYSY - KYTBEWJXHFZJKUCZUDZAUEZJLZOZYKUUDEFZYEUUFYKUUDGFYFUUGYKHGUUCAWIHGAVMYJYGV - NYKXHEFZUUCHFYJUUHWIXGVOVPZBXHUHVQUJUKUUDGULUOUPUUDJUQUOYKYQUUEYKUUHYQUUE - VSUUIWPUUECXHEWKXHLZWOUUDJUUJWNUUCAUUJBEWMUUBUUJWLUUAJKWKXHWJVBVCVDVEVFVR - VQSVTXIYQWAWBYOYRWPCXHYMWCVIVJXLYOWPCXKYNXHWDUTVIVJWEWFWGWH $. + imbi2d vw vj cmap co cif cmpt cfv elmapi nnnninf mp1i ffvelcdmd 2onn elnn + peano1 sylancl 1onn nndceq suc0 raleqi eleq2 ifbid mpteq2dv fveq2d eqeq1d + csn 0ex ralsn bitri dcbii sylibr wa cun wf adantl peano2 adantr wb ralsng + syl mpbird dcan2 mpan9 ralunb df-suc exp31 a2d finds impcom ) DEFAGHUCUDF + ZBEBIZCIZFZJKUEZUFZAUGZJLZCDMZNZOZWIWPCUAIZMZNZOZPWIWPCKMZNZOZPWIWPCUBIZM + ZNZOZPWIWPCXHMZNZOZPWIWSPUAUBDWTKLZXCXFWIXNXBXEXNWPCXAXDWTKQRSTWTXGLZXCXJ + WIXOXBXIXOWPCXAXHWTXGQRSTWTXHLZXCXMWIXPXBXLXPWPCXAXKWTXHQRSTWTDLZXCWSWIXQ + XBWRXQWPCXAWQWTDQRSTWIBEWJKFZJKUEZUFZAUGZJLZOZXFWIYAEFZJEFZYCWIYAGFGEFZYD + WIHGXTAAGHUHZKEFXTHFWIUNBKUIUJUKULYAGUMUOUPYAJUQUOXEYBXEWPCKVEZNYBWPCXDYH + URUSWPYBCKVFWKKLZWOYAJYIWNXTAYIBEWMXSYIWLXRJKWKKWJUTVAVBVCVDVGVHVIVJXGEFZ + WIXJXMYJWIXJXMYJWIVKZXJVKZWPCXHXHVEZVLZNZOZXMYLXIWPCYMNZVKZOZYPYKYQOZXJYS + YKYTBEWJXHFZJKUEZUFZAUGZJLZOZYKUUDEFZYEUUFYKUUDGFYFUUGYKHGUUCAWIHGAVMYJYG + VNYKXHEFZUUCHFYJUUHWIXGVOVPZBXHUIVSUKULUUDGUMUOUPUUDJUQUOYKYQUUEYKUUHYQUU + EVQUUIWPUUECXHEWKXHLZWOUUDJUUJWNUUCAUUJBEWMUUBUUJWLUUAJKWKXHWJUTVAVBVCVDV + RVSSVTXIYQWAWBYOYRWPCXHYMWCVIVJXLYOWPCXKYNXHWDUSVIVJWEWFWGWH $. $} ${ @@ -169543,23 +170664,23 @@ C_ if ( A. k e. suc J ( Q ` ( i e. _om 9-Aug-2022.) $) nninfsellemeq $p |- ( ph -> ( E ` Q ) = ( i e. _om |-> if ( i e. N , 1o , (/) ) ) ) $= - ( cfv c1o wceq wcel com c0 vj c2o xnninf cmap co nninfself ffvelrnd - wf a1i cv wa cif cmpt csuc wral fveq1 eqeq1d ralbidv ifbid mpteq2dv - omex mptex fvmpt syl adantr wss simpr simplr eqeltrd word nnord cvv - wb vex ordelsuc mpan 3syl ad2antrr ssralv sylc iftrued elnn syl2anc - mpbid 1onn fvmptd ralrimiva wn sucid 1n0 nesymi eleq2d fveq2d eqtrd - wrex mtbiri elequ2 notbid rspcev rexnalim iffalsed peano1 nnnninfeq - ) AUABFOZCGAUBUCUDUEZUCBFXEUCFUHACDEFHIUFUIJUGLAUAUJZXDOPQUAGAXFGRZ - UKZEXFCSCUJZDUJZRZPTULZUMZBOZPQZDEUJZUNZUOZPTULZPSXDSAXDESXSUMZQZXG - ABXERYAJHBESXMHUJZOZPQZDXQUOZPTULZUMXTXEFYBBQZESYFXSYGYEXRPTYGYDXOD - XQYGYCXNPXMYBBUPUQURUSUTIESXSVAVBVCVDZVEXHXPXFQZUKZXRPTYJXQGVFZXODG - UOZXRYJXPGRZYKYJXPXFGXHYIVGAXGYIVHVIAYMYKVMZXGYIAGSRZGVJZYNLGVKXPVL - RYPYNEVNZXPGVLVOVPVQVRWDAYLXGYIMVRXODXQGVSVTWAXHXGYOXFSRAXGVGAYOXGL - VEXFGWBWCPSRXHWEUIWFWGAEGXSTSXDSYHAXPGQZUKZXRPTYSXOWHZDXQWOZXRWHYSX - PXQRZCSXIXPRZPTULZUMZBOZPQZWHZUUAUUBYSXPYQWIUIYSUUGTPQPTWJWKYSUUFTP - YSUUFCSXIGRZPTULZUMZBOZTYSUUEUUKBYSCSUUDUUJYSUUCUUIPTYSXPGXIAYRVGWL - USUTWMAUULTQYRNVEWNUQWPYTUUHDXPXQXJXPQZXOUUGUUMXNUUFPUUMXMUUEBUUMCS - XLUUDUUMXKUUCPTDECWQUSUTWMUQWRWSWCXODXQWTVDXALTSRAXBUIWFXC $. + ( cfv c1o wceq wcel com c0 vj xnninf cmap co wf nninfself ffvelcdmd + c2o a1i cv wa cif cmpt csuc wral fveq1 eqeq1d ralbidv mpteq2dv omex + ifbid mptex fvmpt syl adantr wss simpr simplr eqeltrd wb word nnord + cvv vex ordelsuc mpan 3syl ad2antrr mpbid sylc iftrued elnn syl2anc + ssralv 1onn fvmptd ralrimiva wn wrex sucid 1n0 nesymi eleq2d fveq2d + mtbiri elequ2 notbid rspcev rexnalim iffalsed peano1 nnnninfeq + eqtrd ) AUABFOZCGAUHUBUCUDZUBBFXEUBFUEACDEFHIUFUIJUGLAUAUJZXDOPQUAG + AXFGRZUKZEXFCSCUJZDUJZRZPTULZUMZBOZPQZDEUJZUNZUOZPTULZPSXDSAXDESXSU + MZQZXGABXERYAJHBESXMHUJZOZPQZDXQUOZPTULZUMXTXEFYBBQZESYFXSYGYEXRPTY + GYDXODXQYGYCXNPXMYBBUPUQURVAUSIESXSUTVBVCVDZVEXHXPXFQZUKZXRPTYJXQGV + FZXODGUOZXRYJXPGRZYKYJXPXFGXHYIVGAXGYIVHVIAYMYKVJZXGYIAGSRZGVKZYNLG + VLXPVMRYPYNEVNZXPGVMVOVPVQVRVSAYLXGYIMVRXODXQGWDVTWAXHXGYOXFSRAXGVG + AYOXGLVEXFGWBWCPSRXHWEUIWFWGAEGXSTSXDSYHAXPGQZUKZXRPTYSXOWHZDXQWIZX + RWHYSXPXQRZCSXIXPRZPTULZUMZBOZPQZWHZUUAUUBYSXPYQWJUIYSUUGTPQPTWKWLY + SUUFTPYSUUFCSXIGRZPTULZUMZBOZTYSUUEUUKBYSCSUUDUUJYSUUCUUIPTYSXPGXIA + YRVGWMVAUSWNAUULTQYRNVEXCUQWOYTUUHDXPXQXJXPQZXOUUGUUMXNUUFPUUMXMUUE + BUUMCSXLUUDUUMXKUUCPTDECWPVAUSWNUQWQWRWCXODXQWSVDWTLTSRAXAUIWFXB $. $} ${ @@ -169575,22 +170696,22 @@ C_ if ( A. k e. suc J ( Q ` ( i e. _om cmap co csuc cbvralv elequ1 cbvmptv fveq2i eqeq1i ralbii bitri ifbi ax-mp mpteq2i eqtri ad2antlr simpll adantr simpr simplr r19.21v mpd wb sylib eqtr3id nninfsellemeq eqtr4di 3eqtr3d ex mtoi wo wf adantl - cpr elmapi syl nnnninf ffvelrnd df2o3 eleqtrdi elpri orcomd omsinds - ecased exp31 mpcom ) GOUBACOCUCZGUBZPQUDZRZBSZPTZLACOCUAUEZPQUDZRZB - SZPTZUFACOCMUEZPQUDZRZBSZPTZUFZAXKUFUAMGUAMUIZXPYAAYCXOXTPYCXNXSBYC - COXMXRYCXLXQPQUAMCUGUHUJUKULUMUAUCZGTZXPXKAYEXOXJPYEXNXIBYECOXMXHYE - XLXGPQYDGXFUNUHUJUKULUMYDOUBZYBMYDUOZAXPYFYGUPZAUPZXPXOQTZYIYJPQTZP - QUQURYIYJYKYIYJUPZBFSZBSZXOPQYLYMXNBYLYMNONUAUEZPQUDZRZXNYLBNMEFYDH - FHUSUTVAVBZEOCOCDUEZPQUDZRZHUCZSZPTZDEUCVCZUOZPQUDZRZRHYREONONMUEZP - QUDZRZUUBSZPTZMUUEUOZPQUDZRZRIHYRUUHUUPEOUUGUUOUUFUUNWBUUGUUOTUUFXS - UUBSZPTZMUUEUOUUNUUDUURDMUUEDMUIZUUCUUQPUUSUUAXSUUBUUSCOYTXRUUSYSXQ - PQDMCUGUHUJUKULVDUURUUMMUUEUUQUULPXSUUKUUBCNOXRUUJCNUIZXQUUIPQCNMVE - UHVFZVGVHVIVJUUFUUNPQVKVLVMVMVNABYRUBZYHYJJVOAYNPTYHYJKVOZYIYFYJYFY - GAVPZVQYIUUKBSZPTZMYDUOZYJYIYAMYDUOZUVGYIAUVHYHAVRYIYGAUVHUFYFYGAVS - AYAMYDVTWCWAYAUVFMYDXTUVEPXSUUKBUVAVGVHVIWCVQYLYQBSXOQXNYQBCNOXMYPU - UTXLYOPQCNUAVEUHVFZVGYIYJVRZWDWEUVIWFUKUVCUVJWGWHWIYIYJXPYIXOQPWMZU - BYJXPWJYIXOUSUVKYIUTUSXNBYIUVBUTUSBWKAUVBYHJWLBUSUTWNWOYIYFXNUTUBUV - DCYDWPWOWQWRWSXOQPWTWOXAXCXDXBXE $. + cpr elmapi syl nnnninf ffvelcdmd df2o3 eleqtrdi elpri orcomd ecased + exp31 omsinds mpcom ) GOUBACOCUCZGUBZPQUDZRZBSZPTZLACOCUAUEZPQUDZRZ + BSZPTZUFACOCMUEZPQUDZRZBSZPTZUFZAXKUFUAMGUAMUIZXPYAAYCXOXTPYCXNXSBY + CCOXMXRYCXLXQPQUAMCUGUHUJUKULUMUAUCZGTZXPXKAYEXOXJPYEXNXIBYECOXMXHY + EXLXGPQYDGXFUNUHUJUKULUMYDOUBZYBMYDUOZAXPYFYGUPZAUPZXPXOQTZYIYJPQTZ + PQUQURYIYJYKYIYJUPZBFSZBSZXOPQYLYMXNBYLYMNONUAUEZPQUDZRZXNYLBNMEFYD + HFHUSUTVAVBZEOCOCDUEZPQUDZRZHUCZSZPTZDEUCVCZUOZPQUDZRZRHYREONONMUEZ + PQUDZRZUUBSZPTZMUUEUOZPQUDZRZRIHYRUUHUUPEOUUGUUOUUFUUNWBUUGUUOTUUFX + SUUBSZPTZMUUEUOUUNUUDUURDMUUEDMUIZUUCUUQPUUSUUAXSUUBUUSCOYTXRUUSYSX + QPQDMCUGUHUJUKULVDUURUUMMUUEUUQUULPXSUUKUUBCNOXRUUJCNUIZXQUUIPQCNMV + EUHVFZVGVHVIVJUUFUUNPQVKVLVMVMVNABYRUBZYHYJJVOAYNPTYHYJKVOZYIYFYJYF + YGAVPZVQYIUUKBSZPTZMYDUOZYJYIYAMYDUOZUVGYIAUVHYHAVRYIYGAUVHUFYFYGAV + SAYAMYDVTWCWAYAUVFMYDXTUVEPXSUUKBUVAVGVHVIWCVQYLYQBSXOQXNYQBCNOXMYP + UUTXLYOPQCNUAVEUHVFZVGYIYJVRZWDWEUVIWFUKUVCUVJWGWHWIYIYJXPYIXOQPWMZ + UBYJXPWJYIXOUSUVKYIUTUSXNBYIUVBUTUSBWKAUVBYHJWLBUSUTWNWOYIYFXNUTUBU + VDCYDWPWOWQWRWSXOQPWTWOXAXBXCXDXE $. $} $} @@ -169600,21 +170721,21 @@ C_ if ( A. k e. suc J ( Q ` ( i e. _om $( Lemma for ~ nninfsel . (Contributed by Jim Kingdon, 9-Aug-2022.) $) nninfsellemeqinf $p |- ( ph -> ( E ` Q ) = ( i e. _om |-> 1o ) ) $= ( vj com cfv c1o cmpt wcel cv c0 cif wceq xnninf wf cmap co nninfself - c2o a1i ffvelrnd nninff syl ffnd wfn csn cxp fnconstg ax-mp fconstmpt - 1onn fneq1i mpbi csuc wral elequ2 ifbid mpteq2dv fveq2d eqeq1d adantr - simpr nninfsellemqall ralrimiva ad2antrr peano2 ad2antlr elnn syl2anc - wa rspcdva iftrued cvv omex mptex fveq1 ralbidv fvmptg adantl raleqdv - suceq eqeltrdi fvmptd eqidd eqid mpan2 3eqtr4d eqfnfvd ) AKLBFMZCLNOZ - ALUFWPAWPUAPLUFWPUBAUFUAUCUDZUABFWRUAFUBACDEFGHUEUGIUHWPUIUJUKWQLULZA - LNUMUNZLULZWSNLPZXAURLNLUOUPLWTWQCLNUQUSUTUGAKQZLPZVQZCLCQZDQZPZNRSZO - ZBMZNTZDXCVAZVBZNRSZNXCWPMXCWQMZXEXNNRXEXLDXMXEXGXMPZVQZCLXFXCPZNRSZO - ZBMZNTZXLKLXGXCXGTZYBXKNYDYAXJBYDCLXTXIYDXSXHNRKDCVCVDVEVFVGAYCKLVBXD - XQAYCKLXEBCDEFXCGHABWRPZXDIVHZAWPBMNTXDJVHAXDVIZVJVKVLXRXQXMLPZXGLPXE - XQVIXDYHAXQXCVMVNXGXMVOVPVRVKVSZXEEXCXLDEQZVAZVBZNRSZXOLWPLXEYEELYMOZ - VTPZWPYNTYFYOXEELYMWAWBUGGBELXJGQZMZNTZDYKVBZNRSZOYNWRVTFYPBTZELYTYMU - UAYSYLNRUUAYRXLDYKUUAYQXKNXJYPBWCVGWDVDVEHWEVPXEYJXCTZVQZYLXNNRUUCXLD - YKXMUUBYKXMTXEYJXCWHWFWGVDYGXEXONLYIURWIWJXDXPNTZAXDXBUUDURCXCNNLLWQX - FXCTNWKWQWLWEWMWFWNWO $. + c2o a1i ffvelcdmd nninff syl ffnd wfn csn cxp 1onn fnconstg fconstmpt + ax-mp fneq1i mpbi csuc wral elequ2 ifbid mpteq2dv fveq2d eqeq1d simpr + wa adantr nninfsellemqall ralrimiva ad2antrr ad2antlr syl2anc rspcdva + peano2 elnn iftrued cvv omex mptex fveq1 ralbidv fvmptg suceq raleqdv + adantl eqeltrdi fvmptd eqidd eqid mpan2 3eqtr4d eqfnfvd ) AKLBFMZCLNO + ZALUFWPAWPUAPLUFWPUBAUFUAUCUDZUABFWRUAFUBACDEFGHUEUGIUHWPUIUJUKWQLULZ + ALNUMUNZLULZWSNLPZXAUOLNLUPURLWTWQCLNUQUSUTUGAKQZLPZVIZCLCQZDQZPZNRSZ + OZBMZNTZDXCVAZVBZNRSZNXCWPMXCWQMZXEXNNRXEXLDXMXEXGXMPZVIZCLXFXCPZNRSZ + OZBMZNTZXLKLXGXCXGTZYBXKNYDYAXJBYDCLXTXIYDXSXHNRKDCVCVDVEVFVGAYCKLVBX + DXQAYCKLXEBCDEFXCGHABWRPZXDIVJZAWPBMNTXDJVJAXDVHZVKVLVMXRXQXMLPZXGLPX + EXQVHXDYHAXQXCVQVNXGXMVRVOVPVLVSZXEEXCXLDEQZVAZVBZNRSZXOLWPLXEYEELYMO + ZVTPZWPYNTYFYOXEELYMWAWBUGGBELXJGQZMZNTZDYKVBZNRSZOYNWRVTFYPBTZELYTYM + UUAYSYLNRUUAYRXLDYKUUAYQXKNXJYPBWCVGWDVDVEHWEVOXEYJXCTZVIZYLXNNRUUCXL + DYKXMUUBYKXMTXEYJXCWFWHWGVDYGXEXONLYIUOWIWJXDXPNTZAXDXBUUDUOCXCNNLLWQ + XFXCTNWKWQWLWEWMWHWNWO $. $} ${ @@ -169635,12 +170756,12 @@ C_ if ( A. k e. suc J ( Q ` ( i e. _om $( Lemma for ~ nninfomni . (Contributed by Jim Kingdon, 10-Aug-2022.) $) nninfomnilem $p |- NN+oo e. Omni $= ( vp vr xnninf wcel cv cfv c0 wceq c1o wral wo c2o cvv syl wrex cmap co - comni wb nninfex isomnimap ax-mp cpr nninfself ffvelcdmi ffvelrnd df2o3 - elmapi eleqtrdi elpri wi fveqeq2 rspcev ex simpl simpr nninfsel orim12d - wa mpd mprgbir ) IUDJZGKZHKZLZMNZGIUAZVKONGIPZQZHRIUBUCZISJVHVOHVPPUEUF - GIHSUGUHVJVPJZVJDLZVJLZMNZVSONZQZVOVQVSMOUIZJWBVQVSRWCVQIRVRVJVJRIUNVPI - VJDABCDEFUJUKZULUMUOVSMOUPTVQVTVMWAVNVQVRIJZVTVMUQWDWEVTVMVLVTGVRIVIVRM - VJURUSUTTVQWAVNVQWAVEVJABCDEGFVQWAVAVQWAVBVCUTVDVFVG $. + comni wb isomnimap ax-mp cpr elmapi nninfself ffvelcdmi ffvelcdmd df2o3 + nninfex eleqtrdi elpri wi fveqeq2 rspcev ex wa simpl simpr nninfsel mpd + orim12d mprgbir ) IUDJZGKZHKZLZMNZGIUAZVKONGIPZQZHRIUBUCZISJVHVOHVPPUEU + NGIHSUFUGVJVPJZVJDLZVJLZMNZVSONZQZVOVQVSMOUHZJWBVQVSRWCVQIRVRVJVJRIUIVP + IVJDABCDEFUJUKZULUMUOVSMOUPTVQVTVMWAVNVQVRIJZVTVMUQWDWEVTVMVLVTGVRIVIVR + MVJURUSUTTVQWAVNVQWAVAVJABCDEGFVQWAVBVQWAVCVDUTVFVEVG $. $} $} @@ -169882,11 +171003,12 @@ Limited Principle of Omniscience (LPO) implies excluded middle, so we <-> DECID A =//= B ) ) $= ( cr wcel wa clt wbr wceq w3o cap ltap 3expia cc wb syl2an wo sylibr syl6bi recn orc wdc apsym sylibrd wn df-dc syl6 olc wi w3a syl ancoms 3jaod reaplt - apti orim1i df-3or 3mix2 syl6bir jaod syl5bi impbid ) ACDZBCDZEZABFGZABHZBA - FGZIZABJGZUAZVDVEVJVFVGVDVEVIVJVDVEBAJGZVIVBVCVEVKABKLVBAMDZBMDZVIVKNVCASZB - SZABUBOUCVIVIVIUDZPZVJVIVPTVIUEZQZUFVDVFVPVJVBVLVMVFVPNVCVNVOABUNOZVPVQVJVP - VIUGVRQRVCVBVGVJUHVCVBVGVJVCVBVGUIVIVJBAKVSUJLUKULVJVQVDVHVRVDVIVHVPVDVIVEV - GPZVHABUMWAVEVFPZVGPVHVEWBVGVEVFTUOVEVFVGUPQRVDVPVFVHVTVFVEVGUQURUSUTVA $. + apti orim1i df-3or 3mix2 syl6bir jaod biimtrid impbid ) ACDZBCDZEZABFGZABHZ + BAFGZIZABJGZUAZVDVEVJVFVGVDVEVIVJVDVEBAJGZVIVBVCVEVKABKLVBAMDZBMDZVIVKNVCAS + ZBSZABUBOUCVIVIVIUDZPZVJVIVPTVIUEZQZUFVDVFVPVJVBVLVMVFVPNVCVNVOABUNOZVPVQVJ + VPVIUGVRQRVCVBVGVJUHVCVBVGVJVCVBVGUIVIVJBAKVSUJLUKULVJVQVDVHVRVDVIVHVPVDVIV + EVGPZVHABUMWAVEVFPZVGPVHVEWBVGVEVFTUOVEVFVGUPQRVDVPVFVHVTVFVEVGUQURUSUTVA + $. ${ $d A f g x $. $d G f g x $. $d G j $. $d V f g x $. @@ -169899,36 +171021,36 @@ Limited Principle of Omniscience (LPO) implies excluded middle, so we wo cmap cpr isomnimap ccnv ccom fveq1 rexbidv ralbidv orbi12d simplr cres co 012of elmapi adantl fco2 sylancr 2onn a1i simpll elmapd mpbird rspcdva wf nfv nfcv nfre1 nfra1 nfor nfralxy nfan wf1o frechashgf1o wss 0nn0 1nn0 - 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PKUXKUWRUXMPUXTRWIWJUWHPKZUYCLKZUWMUVOLSHUYDUYEYCUVPYFSTLPDYEWBYAWQWRXMXJ - XKXNXOYGYH $. + prssi mp2an ad2antlr simpr ffvelcdmd sselid f1ocnvfv2 adantr fvco3 biimpa + sylancom fveq2d wtru 0zd frec2uz0d mptru eqtrdi exp31 reximdai csuc caddc + eqtr3d df-1o fveq2i peano1 frec2uzsucd oveq1i 0p1e1 eqtri 3eqtri ralimdaa + orim12d mpd ralrimiva 2o01f cvv prexg omelon onelssi ax-mp fssd f1ocnvfv1 + ex wi f1ocnvfv 1onn impbida bitrd ) BEHZBUAHAUBZGUBZIZJKZABUCZYLLKZABMZUD + ZGNBUEUPZMZYJCUBZIZOKZABUCZUUAPKZABMZUDZCOPUFZBUEUPZMZABGEUGYIYSUUIYIYSQZ + UUFCUUHUUJYTUUHHZQZYJDUHZYTUIZIZJKZABUCZUUOLKZABMZUDZUUFUULYQUUTGYRUUNYKU + UNKZYNUUQYPUUSUVAYMUUPABUVAYLUUOJYJYKUUNUJZRUKUVAYOUURABUVAYLUUOLUVBRULUM + YIYSUUKUNUULUUNYRHBNUUNVHZUULUUGNUUMUUGUOVHBUUGYTVHZUVCADFUQUUKUVDUUJYTUU + GBURZUSBUUGNUUMYTUTVAUULNBUUNSENSHZUULVBVCYIYSUUKVDVEVFVGUULUUQUUCUUSUUEU + ULUUPUUBABUUJUUKAYIYSAYIAVIZYQAGYRAYRVJYNYPAYMABVKYOABVLVMVNVOUUKAVIVOZUU + LYJBHZUUPUUBUULUVIQZUUPQZUUAUUMIZDIZUUAOUVJUVMUUAKZUUPUVJSTDVPZUUATHUVNAD + FVQZUVJUUGTUUAOTHZPTHZUUGTVRVSVTOPTWAWBUVJBUUGYJYTUUKUVDUUJUVIUVEWCZUULUV + IWDWEWFSTUUADWGVAZWHUVKUVMJDIZOUVKUVLJDUVJUUPUVLJKUVJUUOUVLJUULUVIUVDUUOU + VLKZUVSBUUGYJUUMYTWIWKZRWJWLUWAOKZWMAODWMWNZFWOWPZWQXBWRWSUULUURUUDABUVHU + VJUURUUDUVJUURQZUVMUUAPUVJUVNUURUVTWHUWGUVMLDIZPUWGUVLLDUWGUUOUVLLUVJUWBU + URUWCWHUVJUURWDXBWLUWHJWTZDIZUWAPXAUPZPLUWIDXCXDUWJUWKKWMAJODUWEFJSHZWMXE + VCXFWPUWKOPXAUPPUWAOPXAUWFXGXHXIXJZWQXBYCXKXLXMXNYIUUIQZYQGYRUWNYKYRHZQZY + JDYKUIZIZOKZABUCZUWRPKZABMZUDZYQUWPUUFUXCCUUHUWQYTUWQKZUUCUWTUUEUXBUXDUUB + UWSABUXDUUAUWROYJYTUWQUJZRUKUXDUUDUXAABUXDUUAUWRPUXERULUMYIUUIUWOUNUWPUWQ + UUHHBUUGUWQVHZUWPNUUGDNUOVHBNYKVHZUXFADFXOUWOUXGUWNYKNBURZUSBNUUGDYKUTVAU + WPUUGBUWQXPEUUGXPHZUWPUVQUVRUXIVSVTOPTTXQWBVCYIUUIUWOVDVEVFVGUWPUWTYNUXBY + PUWPUWSYMABUWNUWOAYIUUIAUVGUUFACUUHAUUHVJUUCUUEAUUBABVKUUDABVLVMVNVOUWOAV + IVOZUWPUVIUWSYMUWPUVIQZUWSQZYLDIZUUMIZYLJUXKUXNYLKZUWSUXKUVOYLSHUXOUVPUXK + BSYJYKUXKBNSYKUWOUXGUWNUVIUXHWCZNSVRZUXKUVFUXQVBSNXRXSXTVCYAUWPUVIWDWESTY + LDYBVAZWHUXLUXNOUUMIZJUXLUXMOUUMUXKUWSUXMOKUXKUWRUXMOUWPUVIUXGUWRUXMKUXPB + NYJDYKWIWKZRWJWLUWDUXSJKZUWFUVOUWLUWDUYAYDUVPXESTJODYEWBXTWQXBWRWSUWPUXAY + OABUXJUXKUXAYOUXKUXAQZUXNYLLUXKUXOUXAUXRWHUYBUXNPUUMIZLUYBUXMPUUMUXKUXAUX + MPKUXKUWRUXMPUXTRWJWLUWHPKZUYCLKZUWMUVOLSHUYDUYEYDUVPYFSTLPDYEWBXTWQXBYCX + KXLXMXNYGYH $. $} ${ @@ -170171,52 +171293,52 @@ Principle of Omniscience (LLPO) is not yet defined in iset.mm. 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expr ex con3d mpd ralnex sylibr r19.21bi ad2antrr ffvelrnd elpri ecased - syl orcd nconstwlpolem0 fveq2d eqtrd olcd wdc cz zdceq sylancl mpjaodan - 0z exmiddc ) ADFMZNOZCUBZGMZNOZCPUCZWTQZUDWPQZAWPRZWTXAXCWSCPXCWQPSZRZW - SWRTOZXCXFQZCPXCXFCPUEZQZXGCPUCXCNDUFUGZQZXIAWPXKAXJWONANBUBZUFUGZXLFMZ - NUHZUIZXJWONUHZUIBUKDXLDOZXMXJXOXQXLDNUFUJXRXNWONXLDFULUMUNAXPBUKAXLUKS - ZXMXOXSXMRAXLUOSXOXLUPJUQVKURADEGKLUSZUTVAVBAXKXIUIWPAXHXJAXHXJAXHRZUAD - EGAPNTVCZGVDZXHKVELYAXHUAUBZGMTOZUAPUEAXHVFXFYECUAPWQYDTGVGVHVIVJVLVMVE - VNXFCPVOVPVQXEWRYBSWSXFUDXEPYBWQGAYCWPXDKVRXCXDVFVSWRNTVTWBWAURWCAXBRXA - WTAXBXAAWTWPAWTWPAWTRZWONFMZNYFDNFYFCDEGAYCWTKVELAWTVFWDWEAYGNOWTIVEWFV - LVMVBWGAWPWHZWPXBUDAWOWISNWISYHAUKWIDFHXTVSWMWONWJWKWPWNWBWL $. + ex con3d mpd ralnex sylibr r19.21bi ad2antrr ffvelcdmd elpri syl ecased + expr orcd nconstwlpolem0 fveq2d eqtrd olcd wdc cz zdceq sylancl exmiddc + 0z mpjaodan ) ADFMZNOZCUBZGMZNOZCPUCZWTQZUDWPQZAWPRZWTXAXCWSCPXCWQPSZRZ + WSWRTOZXCXFQZCPXCXFCPUEZQZXGCPUCXCNDUFUGZQZXIAWPXKAXJWONANBUBZUFUGZXLFM + ZNUHZUIZXJWONUHZUIBUKDXLDOZXMXJXOXQXLDNUFUJXRXNWONXLDFULUMUNAXPBUKAXLUK + SZXMXOXSXMRAXLUOSXOXLUPJUQWBURADEGKLUSZUTVAVBAXKXIUIWPAXHXJAXHXJAXHRZUA + DEGAPNTVCZGVDZXHKVELYAXHUAUBZGMTOZUAPUEAXHVFXFYECUAPWQYDTGVGVHVIVJVKVLV + EVMXFCPVNVOVPXEWRYBSWSXFUDXEPYBWQGAYCWPXDKVQXCXDVFVRWRNTVSVTWAURWCAXBRX + AWTAXBXAAWTWPAWTWPAWTRZWONFMZNYFDNFYFCDEGAYCWTKVELAWTVFWDWEAYGNOWTIVEWF + VKVLVBWGAWPWHZWPXBUDAWOWISNWISYHAUKWIDFHXTVRWMWONWJWKWPWLVTWN $. $} $d ph x g i j y $. $d F x y $. diff --git a/mmil.raw.html b/mmil.raw.html index fd69bfdba1..687aaedaa1 100644 --- a/mmil.raw.html +++ b/mmil.raw.html @@ -10533,9 +10533,7 @@ grppropstr - none - The forward direction seems provable; the reverse - direction might need a ` K e. _V ` condition. + ~ grppropstrg @@ -10658,9 +10656,9 @@ mulgpropd - none - presumably would be provable with ` G e. _V ` and ` H e. _V ` - hypotheses added + ~ mulgpropdg + adds ` G e. V ` and ` H e. W ` conditions and puts two + hypotheses in deduction form @@ -10758,6 +10756,128 @@ may be possible once ` gsum ` is more fully developed + + ringidss + none + the set.mm proof uses ressbas2 and ressplusg + + + + ring1ne0 + none + probably provable but the set.mm proof uses hashgt12el + + + + ringlghm , ringrghm + none + depend on GrpHom ( df-ghm ) + + + + gsummulc1 , gsummulc2 + none + depend on finSupp and gsum theorems we do not have + + + + gsummgp0 + none + depends on gsum theorems we do not have + + + + gsumdixp + none + depends on finSupp and gsum theorems we do not have + + + + prdsmgp , prdsmulrcl , prdsringd , prdscrngd , prds1 + none + depend on various structure product theorems + + + + pwsring , pws1 , pwscrng , pwsmgp + none + depend on various structure power theorems + + + + imasring + none + depends on various image structure theorems + + + + qusring2 + none + depends on various quotient structure theorems + + + + crngbinom + none + depends on various ` gsum ` theorems + + + + opprval + ~ opprvalg + + + + opprmulfval + ~ opprmulfvalg + + + + opprmul + ~ opprmulg + + + + opprlem + ~ opprsllem + + + + opprbas + ~ opprbasg + + + + oppradd + ~ oppraddg + + + + opprringb + ~ opprringbg + + + + oppr0 + ~ oppr0g + + + + oppr1 + ~ oppr1g + + + + opprneg + ~ opprnegg + + + + opprsubg + none + may be possible once SubGrp is defined + + df-drng none diff --git a/mmset.raw.html b/mmset.raw.html index 793104d601..a165b04623 100644 --- a/mmset.raw.html +++ b/mmset.raw.html @@ -5152,7 +5152,8 @@ John Wiley & Sons Inc. (1967) [QA300.A645 1967].

  • [ApostolNT] Apostol, Tom M., -Introduction to analytic number theory, Springer-Verlag, New York, Heidelberg, Berlin (1976) [QA241.A6].
    • +Introduction to analytic number theory, Springer-Verlag, New York, +Heidelberg, Berlin (1976) [QA241.A6].
  • [Baer] Baer, Reinhold, Linear Algebra and Projective Geometry, Academic Press, New York (1952) @@ -5164,6 +5165,13 @@ the American Mathematical Society, 54:481-498 (2017), DOI: 10.1090/bull/1556 .
    • +
  • + [BeauregardFraleigh] Beauregard, Raymond A., +and Fraleigh, John B., A First Course In Linear Algebra: with Optional +Introduction to Groups, Rings, and Fields, Houghton Mifflin Company, +Boston (1973). +
    • +
  • [BellMachover] Bell, J. L., and M. Machover, A Course in Mathematical Logic, North-Holland, diff --git a/set.mm b/set.mm index fa306396a1..1cc957495d 100644 --- a/set.mm +++ b/set.mm @@ -80,6 +80,7 @@ been continuously enriched since then (list of contributors below). $) SO Stefan O'Rear JO Jason Orendorff KP K P +NP Noam Pasman JPP Jon Pennant RP Richard Penner SP Stanislas Polu @@ -2378,7 +2379,7 @@ to intuitionistic (implicational) propositional calculus, it gives Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as". Instead, we will later use the biconditional connective for this purpose ( ~ df-an is its first use), as it - allows us to use logic to manipulate definitions directly. This greatly + allows to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. @@ -2413,12 +2414,12 @@ and conservative (that is, a theorem whose statement does not contain the Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as". Instead, we will later use the biconditional connective for this purpose ( ~ df-or is its - first use), as it allows us to use logic to manipulate definitions - directly. This greatly simplifies many proofs since it eliminates the - need for a separate mechanism for introducing and eliminating definitions. - Of course, we cannot use this mechanism to define the biconditional - itself, since it hasn't been introduced yet. Instead, we use a more - general form of definition, described as follows. + first use), as it allows to use logic to manipulate definitions directly. + This greatly simplifies many proofs since it eliminates the need for a + separate mechanism for introducing and eliminating definitions. Of + course, we cannot use this mechanism to define the biconditional itself, + since it hasn't been introduced yet. Instead, we use a more general form + of definition, described as follows. In its most general form, a definition is simply an assertion that introduces a new symbol (or a new combination of existing symbols, as in @@ -2659,13 +2660,6 @@ intermediate steps that are essentially incomprehensible to humans (other ( wb wi biimp syl ) ABCEBCFDBCGH $. $} - $( The antecedent of one side of a biconditional can be moved out of the - biconditional to become the antecedent of the remaining biconditional. - (Contributed by BJ, 1-Jan-2025.) $) - imbibi $p |- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) ) $= - ( wi wb biimp jarr a1d syl biimpr com23 impbidd ) ABDZCEZABCNMCDZABCDZDMCFO - PAABCGHINCABMCJKL $. - ${ mpbi.min $e |- ph $. mpbi.maj $e |- ( ph <-> ps ) $. @@ -2796,17 +2790,6 @@ intermediate steps that are essentially incomprehensible to humans (other ( biimpi syl5 ) ABCDABEGFH $. $} - ${ - syl5bi.1 $e |- ( ph <-> ps ) $. - syl5bi.2 $e |- ( ch -> ( ps -> th ) ) $. - $( A mixed syllogism inference from a nested implication and a - biconditional. Useful for substituting an embedded antecedent with a - definition. This is in the process of being renamed to ~ biimtrid so - use that name instead. (Contributed by NM, 12-Jan-1993.) $) - syl5bi $p |- ( ch -> ( ph -> th ) ) $= - ( biimpi syl5 ) ABCDABEGFH $. - $} - ${ syl5bir.1 $e |- ( ps <-> ph ) $. syl5bir.2 $e |- ( ch -> ( ps -> th ) ) $. @@ -3993,6 +3976,13 @@ the conclusion alone (see ~ pm5.1im and ~ pm5.21im ). (Contributed by ( ph -> ( ps -> ch ) ) ) $= ( wi imdi bicomi ) ABCDDABDACDDABCEF $. + $( The antecedent of one side of a biconditional can be moved out of the + biconditional to become the antecedent of the remaining biconditional. + (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, + 5-Jan-2025.) $) + imbibi $p |- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) ) $= + ( wi wb pm5.4 imbi2 bitr3id pm5.74rd ) ABDZCEZABCJAJDKACDABFJCAGHI $. + $( Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) $) pm4.8 $p |- ( ( ph -> -. ph ) <-> -. ph ) $= @@ -11570,6 +11560,22 @@ This definition (in the form of ~ dfifp2 ) appears in Section II.24 of 3orel3 $p |- ( -. ch -> ( ( ph \/ ps \/ ch ) -> ( ph \/ ps ) ) ) $= ( w3o wo wn df-3or orel2 biimtrid ) ABCDABEZCECFJABCGCJHI $. + $( Elimination of two disjuncts in a triple disjunction. (Contributed by + Scott Fenton, 9-Jun-2011.) $) + 3orel13 $p |- ( ( -. ph /\ -. ch ) -> ( ( ph \/ ps \/ ch ) -> ps ) ) $= + ( wn w3o wo 3orel3 orel1 sylan9r ) CDABCEABFADBABCGABHI $. + + ${ + 3pm3.2ni.1 $e |- -. ph $. + 3pm3.2ni.2 $e |- -. ps $. + 3pm3.2ni.3 $e |- -. ch $. + $( Triple negated disjunction introduction. (Contributed by Scott Fenton, + 20-Apr-2011.) $) + 3pm3.2ni $p |- -. ( ph \/ ps \/ ch ) $= + ( w3o wo pm3.2ni df-3or mtbir ) ABCGABHZCHLCABDEIFIABCJK $. + $} + + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Logical "nand" (Sheffer stroke) @@ -12025,7 +12031,7 @@ When the universe is infinite (as with set theory), such a (The description above applies to set theory, not predicate calculus. The purpose of introducing ` class x ` here, and not in set theory where - it belongs, is to allow us to express, i.e., "prove", the ~ weq of + it belongs, is to allow to express, i.e., "prove", the ~ weq of predicate calculus from the ~ wceq of set theory, so that we do not overload the ` = ` connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.) $) @@ -12051,7 +12057,7 @@ When the universe is infinite (as with set theory), such a ~ mmset.html#class . (The purpose of introducing ` wff A = B ` here, and not in set theory - where it belongs, is to allow us to express, i.e., "prove", the ~ weq of + where it belongs, is to allow to express, i.e., "prove", the ~ weq of predicate calculus in terms of the ~ wceq of set theory, so that we do not "overload" the ` = ` connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath @@ -14510,7 +14516,7 @@ other follows from the first two and the external(s)." Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes) #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Here we extend the language of wffs with predicate calculus, which allows us + Here we extend the language of wffs with predicate calculus, which allows to talk about individual objects in a domain of discourse (which for us will be the universe of all sets, so we call them "setvar variables") and make true/false statements about predicates, which are relationships between @@ -14856,7 +14862,7 @@ only postulated (that is, axiomatic) rule of inference of predicate $( No variable is (effectively) free in a theorem. This and later "hypothesis-building" lemmas, with labels starting - "hb...", allow us to construct proofs of formulas of the form + "hb...", allow to construct proofs of formulas of the form ` |- ( ph -> A. x ph ) ` from smaller formulas of this form. These are useful for constructing hypotheses that state " ` x ` is (effectively) not free in ` ph ` ". (Contributed by NM, 11-May-1993.) This hb* idiom @@ -16959,9 +16965,9 @@ variables and no wff metavariables (see ~ ax12wdemo for an example of $( Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of ~ cbvaldva with a disjoint variable condition, requiring fewer axioms. (Contributed by - David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) - Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf - Lammen, 10-Feb-2024.) $) + David Moews, 1-May-2017.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) Reduce axiom usage, along an idea of Gino Giotto. + (Revised by Wolf Lammen, 10-Feb-2024.) $) cbvaldvaw $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $= ( wal wi weq wb ancoms pm5.74da cbvalvw 19.21v 3bitr3i pm5.74ri ) ABDGZCE GZABHZDGACHZEGAQHARHSTDEDEIZABCAUABCJFKLMABDNACENOP $. @@ -16969,8 +16975,9 @@ variables and no wff metavariables (see ~ ax12wdemo for an example of $( Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of ~ cbvexdva with a disjoint variable condition, requiring fewer axioms. (Contributed by - David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) - Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.) $) + David Moews, 1-May-2017.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, + 10-Feb-2024.) $) cbvexdvaw $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $= ( wex wn wal weq wa notbid cbvaldvaw alnex 3bitr3g con4bid ) ABDGZCEGZABH ZDICHZEIQHRHASTDEADEJKBCFLMBDNCENOP $. @@ -16981,15 +16988,15 @@ variables and no wff metavariables (see ~ ax12wdemo for an example of cbval2vw.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $. $( Rule used to change bound variables, using implicit substitution. Version of ~ cbval2vv with more disjoint variable conditions, which - requires fewer axioms . (Contributed by NM, 4-Feb-2005.) (Revised by - Gino Giotto, 10-Jan-2024.) $) + requires fewer axioms . (Contributed by NM, 4-Feb-2005.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) $) cbval2vw $p |- ( A. x A. y ph <-> A. z A. w ps ) $= ( wal weq cbvaldvaw cbvalvw ) ADHBFHCECEIABDFGJK $. $( Rule used to change bound variables, using implicit substitution. Version of ~ cbvex2vv with more disjoint variable conditions, which - requires fewer axioms . (Contributed by NM, 26-Jul-1995.) (Revised by - Gino Giotto, 10-Jan-2024.) $) + requires fewer axioms . (Contributed by NM, 26-Jul-1995.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) $) cbvex2vw $p |- ( E. x E. y ph <-> E. z E. w ps ) $= ( wex weq cbvexdvaw cbvexvw ) ADHBFHCECEIABDFGJK $. $} @@ -17006,8 +17013,8 @@ variables and no wff metavariables (see ~ ax12wdemo for an example of cbvex4vw.2 $e |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) ) $. $( Rule used to change bound variables, using implicit substitution. Version of ~ cbvex4v with more disjoint variable conditions, which - requires fewer axioms. (Contributed by NM, 26-Jul-1995.) (Revised by - Gino Giotto, 10-Jan-2024.) $) + requires fewer axioms. (Contributed by NM, 26-Jul-1995.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) $) cbvex4vw $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $= ( wex weq wa 2exbidv cbvex2vw 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZIN HNUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $. @@ -17643,8 +17650,8 @@ disjoint variable condition ( ~ sb2 ). Theorem ~ sb6f replaces the 2sbievw.1 $e |- ( ( x = t /\ y = u ) -> ( ph <-> ps ) ) $. $( Conversion of double implicit substitution to explicit substitution. Version of ~ 2sbiev with more disjoint variable conditions, requiring - fewer axioms. (Contributed by AV, 29-Jul-2023.) (Revised by Gino - Giotto, 10-Jan-2024.) $) + fewer axioms. (Contributed by AV, 29-Jul-2023.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) $) 2sbievw $p |- ( [ t / x ] [ u / y ] ph <-> ps ) $= ( wsb weq sbiedvw sbievw ) ADEHBCFCFIABDEGJK $. $} @@ -17743,8 +17750,8 @@ disjoint variable condition ( ~ sb2 ). Theorem ~ sb6f replaces the For a general discussion of the theory of classes, see ~ mmset.html#class . - The purpose of introducing ` wff A e. B ` here is to allow us to prove - the ~ wel of predicate calculus in terms of the ~ wcel of set theory, so + The purpose of introducing ` wff A e. B ` here is to allow to prove the + ~ wel of predicate calculus in terms of the ~ wcel of set theory, so that we do not overload the ` e. ` connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables ` A ` and ` B ` are @@ -18335,9 +18342,9 @@ scheme is logically redundant (see ~ ax10w ) but is used as an auxiliary $d x y $. $( All variables are effectively bound in a distinct variable specifier. Version of ~ nfnae with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by Mario Carneiro, 11-Aug-2016.) - (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, - 25-Sep-2024.) $) + require ~ ax-13 . (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by + Wolf Lammen, 25-Sep-2024.) $) nfnaew $p |- F/ z -. A. x x = y $= ( weq wal wn hbnaev nf5i ) ABDAEFCABCGH $. @@ -19877,7 +19884,7 @@ any disjoint variable condition (but the version with a disjoint $( Conversion of implicit substitution to explicit substitution (deduction version of ~ sbiev ). Version of ~ sbied with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) - (Revised by Gino Giotto, 10-Jan-2024.) $) + Avoid ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) $) sbiedw $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $= ( wsb wi sbrim nfim1 weq wb com12 pm5.74d sbiev bitr3i pm5.74ri ) ABDEIZC ATJABJZDEIACJZABDEFKUAUBDEACDFGLDEMZABCAUCBCNHOPQRS $. @@ -20103,8 +20110,8 @@ Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, cbv2w.5 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $. $( Rule used to change bound variables, using implicit substitution. Version of ~ cbv2 with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 5-Aug-1993.) (Revised by Gino - Giotto, 10-Jan-2024.) $) + require ~ ax-13 . (Contributed by NM, 5-Aug-1993.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) $) cbv2w $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $= ( wal weq wb wi biimp syl6 cbv1v equcomi biimpr syl56 impbid ) ABDKCEKABC DEFGHIADELZBCMZBCNJBCOPQACBEDGFIHEDLUBAUCCBNEDRJBCSTQUA $. @@ -20117,15 +20124,15 @@ Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, cbvaldw.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $. $( Deduction used to change bound variables, using implicit substitution. Version of ~ cbvald with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 2-Jan-2002.) (Revised by Gino - Giotto, 10-Jan-2024.) $) + require ~ ax-13 . (Contributed by NM, 2-Jan-2002.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) $) cbvaldw $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $= ( nfv nfvd cbv2w ) ABCDEADIFGACDJHK $. $( Deduction used to change bound variables, using implicit substitution. Version of ~ cbvexd with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 2-Jan-2002.) (Revised by Gino - Giotto, 10-Jan-2024.) $) + require ~ ax-13 . (Contributed by NM, 2-Jan-2002.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) $) cbvexdw $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $= ( wex wn wal nfnd weq wb notbi syl6ib cbvaldw alnex 3bitr3g con4bid ) ABD IZCEIZABJZDKCJZEKUAJUBJAUCUDDEFABEGLADEMBCNUCUDNHBCOPQBDRCERST $. @@ -20187,14 +20194,6 @@ Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, ( wal nfal weq nfv wnf a1i wb ex cbv2w cbvalv1 ) ADLBFLCEAEDGMBCFIMCENZAB DFUBDOUBFOAFPUBHQBDPUBJQUBDFNABRKSTUA $. - $( Obsolete version of ~ cbval2v as of 14-Jan-2024. (Contributed by BJ, - 16-Jan-2019.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - cbval2vOLD $p |- ( A. x A. y ph <-> A. z A. w ps ) $= - ( wal nfal weq wi nfv nfim wb cbvalv1 19.21v pm5.74d 3bitr3i pm5.74ri - expcom ) ADLZBFLZCEAEDGMBCFIMCENZUEUFUGAOZDLUGBOZFLUGUEOUGUFOUHUIDFUGAFUG - FPHQUGBDUGDPJQDFNZUGABUGUJABRKUDUASUGADTUGBFTUBUCS $. - $( Rule used to change bound variables, using implicit substitution. Version of ~ cbvex2 with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by NM, 14-Sep-2003.) (Revised by BJ, @@ -23026,7 +23025,8 @@ of the unique existential quantifier (note that ` y ` and ` z ` need not $( Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of ~ nfeu . Version of ~ nfeud with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) $) + NM, 15-Feb-2013.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) $) nfeudw $p |- ( ph -> F/ x E! y ps ) $= ( weu wex wmo wa df-eu nfexd nfmodv nfand nfxfrd ) BDGBDHZBDIZJACBDKAPQCA BCDEFLABCDEFMNO $. @@ -23050,8 +23050,8 @@ of the unique existential quantifier (note that ` y ` and ` z ` need not nfeuw.1 $e |- F/ x ph $. $( Bound-variable hypothesis builder for the unique existential quantifier. Version of ~ nfeu with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 8-Mar-1995.) (Revised by Gino - Giotto, 10-Jan-2024.) $) + require ~ ax-13 . (Contributed by NM, 8-Mar-1995.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) $) nfeuw $p |- F/ x E! y ph $= ( weu wnf wtru nftru a1i nfeudw mptru ) ACEBFGABCCHABFGDIJK $. $( $j usage 'nfeuw' avoids 'ax-13'; $) @@ -25264,9 +25264,9 @@ the definition of class equality ( ~ df-cleq ). Its forward implication $( Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax - construction ~ cv , which allows us to substitute a setvar variable for - a class variable. See also ~ cab and ~ df-clab . Note that this is not - a rigorous justification, because ~ cv is used as part of the proof of + construction ~ cv , which allows to substitute a setvar variable for a + class variable. See also ~ cab and ~ df-clab . Note that this is not a + rigorous justification, because ~ cv is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in @@ -26028,7 +26028,7 @@ the definition of class equality ( ~ df-cleq ). Its forward implication abeq2w $p |- ( A = { x | ph } <-> A. y ( y e. A <-> ps ) ) $= ( cab wceq cv wcel wb wal dfcleq wsb df-clab sbievw bitri bibi2i albii ) EACGZHDIZEJZUATJZKZDLUBBKZDLDETMUDUEDUCBUBUCACDNBADCOABCDFPQRSQ $. - $( $j usage 'abeq2w' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12' $) + $( $j usage 'abeq2w' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $) $} @@ -26698,17 +26698,6 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim ( cab wceq wtru cv wcel wb a1i abbi2dv mptru ) CABEFGABCBHCIAJGDKLM $. $} - ${ - $d ph y $. $d ps y $. $d x y $. - $( Obsolete proof of ~ abbi as of 7-Jan-2024. (Contributed by NM, - 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof - shortened by Wolf Lammen, 16-Nov-2019.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - abbiOLD $p |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } ) $= - ( vy cab wceq cv wcel wb wal hbab1 cleqh abid bibi12i albii bitr2i ) ACEZ - BCEZFCGZQHZSRHZIZCJABIZCJCDQRACDKBCDKLUBUCCTAUABACMBCMNOP $. - $} - ${ $d x A $. $( Every class is equal to a class abstraction (the class of sets belonging @@ -27034,8 +27023,8 @@ generally appear in a single form (either definitional, but more often $( Substitution for the first argument of the membership predicate in an atomic formula (class version of ~ elsb1 ). Version of ~ clelsb1f with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by Rodolfo Medina, 28-Apr-2010.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + (Contributed by Rodolfo Medina, 28-Apr-2010.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 10-Jan-2024.) $) clelsb1fw $p |- ( [ y / x ] x e. A <-> y e. A ) $= ( vw cv wcel wsb nfcri sbco2v clelsb1 sbbii 3bitr3i ) EFCGZEAHZABHNEBHAFC GZABHBFCGNEBAAECDIJOPABEACKLEBCKM $. @@ -27222,8 +27211,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is nfabdw.2 $e |- ( ph -> F/ x ps ) $. $( Bound-variable hypothesis builder for a class abstraction. Version of ~ nfabd with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by - Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, + ~ ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.) $) nfabdw $p |- ( ph -> F/_ x { y | ps } ) $= ( vz cab nfv cv wcel weq wi wal wsb df-clab sb6 bitri nfvd nfimd nfxfrd @@ -28327,6 +28316,13 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Restricted universal and existential quantification +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + $( Extend wff notation to include restricted universal quantification. $) wral $a wff A. x e. A ph $. @@ -29739,8 +29735,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( Change the bound variable of a restricted universal quantifier using implicit substitution. Version of ~ cbvralv with a disjoint variable condition, which does not require ~ ax-10 , ~ ax-11 , ~ ax-12 , - ~ ax-13 . (Contributed by NM, 28-Jan-1997.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + ~ ax-13 . (Contributed by NM, 28-Jan-1997.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 10-Jan-2024.) $) cbvralvw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= ( cv wcel wi wal wral weq eleq1w imbi12d cbvalvw df-ral 3bitr4i ) CGEHZAI ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. @@ -29749,8 +29745,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( Change the bound variable of a restricted existential quantifier using implicit substitution. Version of ~ cbvrexv with a disjoint variable condition, which does not require ~ ax-10 , ~ ax-11 , ~ ax-12 , - ~ ax-13 . (Contributed by NM, 2-Jun-1998.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + ~ ax-13 . (Contributed by NM, 2-Jun-1998.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 10-Jan-2024.) $) cbvrexvw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= ( cv wcel wa wex wrex weq eleq1w anbi12d cbvexvw df-rex 3bitr4i ) CGEHZAI ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. @@ -29765,7 +29761,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( Change bound variables of double restricted universal quantification, using implicit substitution. Version of ~ cbvral2v with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.) $) + NM, 10-Aug-2004.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) $) cbvral2vw $p |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps ) $= ( wral weq ralbidv cbvralvw ralbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHD FMACEIJNOSTFHCBEGIKOPQ $. @@ -29780,7 +29777,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( Change bound variables of double restricted universal quantification, using implicit substitution. Version of ~ cbvrex2v with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.) $) + FL, 2-Jul-2012.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) $) cbvrex2vw $p |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps ) $= ( wrex weq rexbidv cbvrexvw rexbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHD FMACEIJNOSTFHCBEGIKOPQ $. @@ -29798,7 +29796,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( Change bound variables of triple restricted universal quantification, using implicit substitution. Version of ~ cbvral3v with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - NM, 10-May-2005.) (Revised by Gino Giotto, 10-Jan-2024.) $) + NM, 10-May-2005.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) $) cbvral3vw $p |- ( A. x e. A A. y e. B A. z e. C ph <-> A. w e. A A. v e. B A. u e. C ps ) $= ( wral weq 2ralbidv cbvralvw cbvral2vw ralbii bitri ) AGMQFLQZEKQCGMQFLQZ @@ -29884,8 +29883,9 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ralrimi.1 $e |- F/ x ph $. ralrimi.2 $e |- ( ph -> ( x e. A -> ps ) ) $. $( Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier - version). (Contributed by NM, 10-Oct-1999.) Shortened after - introduction of ~ hbralrimi . (Revised by Wolf Lammen, 4-Dec-2019.) $) + version). For a version based on fewer axioms see ~ ralrimiv . + (Contributed by NM, 10-Oct-1999.) Shortened after introduction of + ~ hbralrimi . (Revised by Wolf Lammen, 4-Dec-2019.) $) ralrimi $p |- ( ph -> A. x e. A ps ) $= ( nf5ri hbralrimi ) ABCDACEGFH $. $} @@ -29902,8 +29902,9 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ${ rexlimi.1 $e |- F/ x ps $. rexlimi.2 $e |- ( x e. A -> ( ph -> ps ) ) $. - $( Restricted quantifier version of ~ exlimi . (Contributed by NM, - 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) $) + $( Restricted quantifier version of ~ exlimi . For a version based on + fewer axioms see ~ rexlimiv . (Contributed by NM, 30-Nov-2003.) (Proof + shortened by Andrew Salmon, 30-May-2011.) $) rexlimi $p |- ( E. x e. A ph -> ps ) $= ( wi wral wrex rgen r19.23 mpbi ) ABGZCDHACDIBGMCDFJABCDEKL $. $} @@ -29954,7 +29955,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ralrimd.2 $e |- F/ x ps $. ralrimd.3 $e |- ( ph -> ( ps -> ( x e. A -> ch ) ) ) $. $( Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted - quantifier version.) (Contributed by NM, 16-Feb-2004.) $) + quantifier version.) For a version based on fewer axioms see + ~ ralrimdv . (Contributed by NM, 16-Feb-2004.) $) ralrimd $p |- ( ph -> ( ps -> A. x e. A ch ) ) $= ( cv wcel wi wal wral alrimd df-ral syl6ibr ) ABDIEJCKZDLCDEMABQDFGHNCDEO P $. @@ -29976,9 +29978,10 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is rexlimd.1 $e |- F/ x ph $. rexlimd.2 $e |- F/ x ch $. rexlimd.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $. - $( Deduction form of ~ rexlimd . (Contributed by NM, 27-May-1998.) (Proof - shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf - Lammen, 14-Jan-2020.) $) + $( Deduction form of ~ rexlimd . For a version based on fewer axioms see + ~ rexlimdv . (Contributed by NM, 27-May-1998.) (Proof shortened by + Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, + 14-Jan-2020.) $) rexlimd $p |- ( ph -> ( E. x e. A ps -> ch ) ) $= ( wnf a1i rexlimd2 ) ABCDEFCDIAGJHK $. $} @@ -30051,7 +30054,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ralbid.1 $e |- F/ x ph $. ralbid.2 $e |- ( ph -> ( ps <-> ch ) ) $. $( Formula-building rule for restricted universal quantifier (deduction - form). (Contributed by NM, 27-Jun-1998.) $) + form). For a version based on fewer axioms see ~ ralbidv . + (Contributed by NM, 27-Jun-1998.) $) ralbid $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $= ( wb cv wcel adantr ralbida ) ABCDEFABCHDIEJGKL $. $} @@ -30060,7 +30064,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is rexbid.1 $e |- F/ x ph $. rexbid.2 $e |- ( ph -> ( ps <-> ch ) ) $. $( Formula-building rule for restricted existential quantifier (deduction - form). (Contributed by NM, 27-Jun-1998.) $) + form). For a version based on fewer axioms see ~ rexbidv . + (Contributed by NM, 27-Jun-1998.) $) rexbid $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $= ( wb cv wcel adantr rexbida ) ABCDEFABCHDIEJGKL $. $} @@ -30116,9 +30121,9 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ${ $d x y $. r2alf.1 $e |- F/_ y A $. - $( Double restricted universal quantification. (Contributed by Mario - Carneiro, 14-Oct-2016.) Use ~ r2allem . (Revised by Wolf Lammen, - 9-Jan-2020.) $) + $( Double restricted universal quantification. For a version based on + fewer axioms see ~ r2al . (Contributed by Mario Carneiro, 14-Oct-2016.) + Use ~ r2allem . (Revised by Wolf Lammen, 9-Jan-2020.) $) r2alf $p |- ( A. x e. A A. y e. B ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $= ( cv wcel wi nfcri 19.21 r2allem ) ABCDEBGDHCGEHAICCBDFJKL $. @@ -30127,9 +30132,9 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ${ $d x y $. r2exf.1 $e |- F/_ y A $. - $( Double restricted existential quantification. (Contributed by Mario - Carneiro, 14-Oct-2016.) Use ~ r2exlem . (Revised by Wolf Lammen, - 10-Jan-2020.) $) + $( Double restricted existential quantification. For a version based on + fewer axioms see ~ r2ex . (Contributed by Mario Carneiro, 14-Oct-2016.) + Use ~ r2exlem . (Revised by Wolf Lammen, 10-Jan-2020.) $) r2exf $p |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $= ( wn r2alf r2exlem ) ABCDEAGBCDEFHI $. @@ -30324,53 +30329,89 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is hbra1 $p |- ( A. x e. A ph -> A. x A. x e. A ph ) $= ( wral nfra1 nf5ri ) ABCDBABCEF $. - $( Extend wff notation to include restricted existential uniqueness. $) - wreu $a wff E! x e. A ph $. + $( *** Theorems using axioms up to ax-8 and ax-11, ax-12 only: *** $) - $( Extend wff notation to include restricted "at most one". $) - wrmo $a wff E* x e. A ph $. - - $( Extend class notation to include the restricted class abstraction (class - builder). $) - crab $a class { x e. A | ph } $. - - $( Define restricted "at most one". - - Note: This notation is most often used to express that ` ph ` holds for - at most one element of a given class ` A ` . For this reading ` F/_ x A ` - is required, though, for example, asserted when ` x ` and ` A ` are - disjoint. + ${ + $d x y $. + ralcomf.1 $e |- F/_ y A $. + ralcomf.2 $e |- F/_ x B $. + $( Commutation of restricted universal quantifiers. For a version based on + fewer axioms see ~ ralcom . (Contributed by Mario Carneiro, + 14-Oct-2016.) $) + ralcomf $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $= + ( cv wcel wa wi wal wral ancomst 2albii alcom bitri r2alf 3bitr4i ) BHDIZ + CHEIZJAKZCLBLZUATJAKZBLCLZACEMBDMABDMCEMUCUDCLBLUEUBUDBCTUAANOUDBCPQABCDE + FRACBEDGRS $. - Should instead ` A ` depend on ` x ` , you rather assert at most one ` x ` - fulfilling ` ph ` happens to be contained in the corresponding - ` A ( x ) ` . This interpretation is rarely needed (see also ~ df-ral ). - (Contributed by NM, 16-Jun-2017.) $) - df-rmo $a |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) $. + $( Commutation of restricted existential quantifiers. For a version based + on fewer axioms see ~ rexcom . (Contributed by Mario Carneiro, + 14-Oct-2016.) $) + rexcomf $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $= + ( cv wcel wa wex wrex ancom anbi1i 2exbii excom bitri r2exf 3bitr4i ) BHD + IZCHEIZJZAJZCKBKZUATJZAJZBKCKZACELBDLABDLCELUDUFCKBKUGUCUFBCUBUEATUAMNOUF + BCPQABCDEFRACBEDGRS $. + $} - $( Define restricted existential uniqueness. + ${ + $d x y $. + cbvralfw.1 $e |- F/_ x A $. + cbvralfw.2 $e |- F/_ y A $. + cbvralfw.3 $e |- F/ y ph $. + cbvralfw.4 $e |- F/ x ps $. + cbvralfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Rule used to change bound variables, using implicit substitution. + Version of ~ cbvralf with a disjoint variable condition, which does not + require ~ ax-10 , ~ ax-13 . For a version not dependent on ~ ax-11 and + ax-12, see ~ cbvralvw . (Contributed by NM, 7-Mar-2004.) Avoid + ~ ax-10 , ~ ax-13 . (Revised by Gino Giotto, 23-May-2024.) $) + cbvralfw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= + ( cv wcel wi wal wral nfcri nfim weq eleq1w df-ral imbi12d cbvalv1 + 3bitr4i ) CKELZAMZCNDKELZBMZDNACEOBDEOUEUGCDUDADDCEGPHQUFBCCDEFPIQCDRUDUF + ABCDESJUAUBACETBDETUC $. + $( $j usage 'cbvralfw' avoids 'ax-10' 'ax-13'; $) + $} - Note: This notation is most often used to express that ` ph ` holds for - exactly one element of a given class ` A ` . For this reading ` F/_ x A ` - is required, though, for example, asserted when ` x ` and ` A ` are - disjoint. + ${ + $d x y $. + cbvrexfw.1 $e |- F/_ x A $. + cbvrexfw.2 $e |- F/_ y A $. + cbvrexfw.3 $e |- F/ y ph $. + cbvrexfw.4 $e |- F/ x ps $. + cbvrexfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Rule used to change bound variables, using implicit substitution. + Version of ~ cbvrexf with a disjoint variable condition, which does not + require ~ ax-13 . For a version not dependent on ~ ax-11 and ax-12, see + ~ cbvrexvw . (Contributed by FL, 27-Apr-2008.) Avoid ~ ax-10 , + ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) $) + cbvrexfw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= + ( wrex wn wral nfn weq notbid cbvralfw ralnex 3bitr3i con4bii ) ACEKZBDEK + ZALZCEMBLZDEMUALUBLUCUDCDEFGADHNBCINCDOABJPQACERBDERST $. + $( $j usage 'cbvrexfw' avoids 'ax-13'; $) + $} - Should instead ` A ` depend on ` x ` , you rather assert exactly one ` x ` - fulfilling ` ph ` happens to be contained in the corresponding - ` A ( x ) ` . This interpretation is rarely needed (see also ~ df-ral ). - (Contributed by NM, 22-Nov-1994.) $) - df-reu $a |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) $. + ${ + $d x y A $. + cbvralw.1 $e |- F/ y ph $. + cbvralw.2 $e |- F/ x ps $. + cbvralw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Rule used to change bound variables, using implicit substitution. + Version of ~ cbvralfw with more disjoint variable conditions. + (Contributed by NM, 31-Jul-2003.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 10-Jan-2024.) $) + cbvralw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= + ( nfcv cbvralfw ) ABCDECEIDEIFGHJ $. + $( $j usage 'cbvralw' avoids 'ax-13'; $) - $( Define a restricted class abstraction (class builder): ` { x e. A | ph } ` - is the class of all sets ` x ` in ` A ` such that ` ph ( x ) ` is true. - Definition of [TakeutiZaring] p. 20. + $( Rule used to change bound variables, using implicit substitution. + Version of ~ cbvrexfw with more disjoint variable conditions. + (Contributed by NM, 31-Jul-2003.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 10-Jan-2024.) $) + cbvrexw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= + ( nfcv cbvrexfw ) ABCDECEIDEIFGHJ $. + $( $j usage 'cbvrexw' avoids 'ax-13'; $) + $} - For the interpretation given in the previous paragraph to be correct, we - need to assume ` F/_ x A ` , which is the case as soon as ` x ` and ` A ` - are disjoint, which is generally the case. If ` A ` were to depend on - ` x ` , then the interpretation would be less obvious (think of the two - extreme cases ` A = { x } ` and ` A = x ` , for instance). See also - ~ df-ral . (Contributed by NM, 22-Nov-1994.) $) - df-rab $a |- { x e. A | ph } = { x | ( x e. A /\ ph ) } $. + $( *** Theorems using axioms up to ax-12 only: *** $) ${ hbral.1 $e |- ( y e. A -> A. x y e. A ) $. @@ -30390,33 +30431,21 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is nfraldw.3 $e |- ( ph -> F/ x ps ) $. $( Deduction version of ~ nfralw . Version of ~ nfrald with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - NM, 15-Feb-2013.) (Revised by Gino Giotto, 24-Sep-2024.) $) + NM, 15-Feb-2013.) Avoid ~ ax-9 , ~ ax-ext . (Revised by Gino Giotto, + 24-Sep-2024.) $) nfraldw $p |- ( ph -> F/ x A. y e. A ps ) $= ( wral cv wcel wi wal df-ral nfcrd nfimd nfald nfxfrd ) BDEIDJEKZBLZDMACB DENATCDFASBCACDEGOHPQR $. $( $j usage 'nfraldw' avoids 'ax-13' 'ax-9' 'df-cleq' 'ax-ext'; $) - $( Obsolete version of ~ nfraldw as of 24-Sep-2024. (Contributed by NM, - 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - nfraldwOLD $p |- ( ph -> F/ x A. y e. A ps ) $= - ( wral cv wcel wi wal df-ral nfcvd nfeld nfimd nfald nfxfrd ) BDEIDJZEKZB - LZDMACBDENAUBCDFAUABCACTEACTOGPHQRS $. - $( $j usage 'nfraldwOLD' avoids 'ax-13'; $) - $} - - ${ - nfrald.1 $e |- F/ y ph $. - nfrald.2 $e |- ( ph -> F/_ x A ) $. - nfrald.3 $e |- ( ph -> F/ x ps ) $. - $( Deduction version of ~ nfral . Usage of this theorem is discouraged - because it depends on ~ ax-13 . Use the weaker ~ nfraldw when possible. - (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, - 7-Oct-2016.) (New usage is discouraged.) $) - nfrald $p |- ( ph -> F/ x A. y e. A ps ) $= - ( wral cv wcel wi wal df-ral weq wn wa wnfc nfcvf adantr adantl nfeld wnf - nfimd nfald2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCMPZQZUHBCUKCUGEUJCU - GRACDSUAACERUJGTUBABCUCUJHTUDUEUF $. + $( Deduction version of ~ nfrexw . (Contributed by Mario Carneiro, + 14-Oct-2016.) Add disjoint variable condition to avoid ~ ax-13 . See + ~ nfrexd for a less restrictive version requiring more axioms. (Revised + by Gino Giotto, 20-Jan-2024.) $) + nfrexdw $p |- ( ph -> F/ x E. y e. A ps ) $= + ( wrex wn wral dfrex2 nfnd nfraldw nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGA + BCHMNMO $. + $( $j usage 'nfrexdw' avoids 'ax-13'; $) $} ${ @@ -30425,8 +30454,9 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is nfralw.2 $e |- F/ x ph $. $( Bound-variable hypothesis builder for restricted quantification. Version of ~ nfral with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 1-Sep-1999.) (Revised by Gino - Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 13-Dec-2024.) $) + require ~ ax-13 . (Contributed by NM, 1-Sep-1999.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, + 13-Dec-2024.) $) nfralw $p |- F/ x A. y e. A ph $= ( wral cv wcel nfcri nf5ri hbral nf5i ) ACDGBABCDCHDIBBCDEJKABFKLM $. $( $j usage 'nfralw' avoids 'ax-13'; $) @@ -30437,125 +30467,17 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is nfralwOLD $p |- F/ x A. y e. A ph $= ( wral wnf wtru nftru wnfc a1i nfraldw mptru ) ACDGBHIABCDCJBDKIELABHIFLM N $. - $} - - ${ - nfral.1 $e |- F/_ x A $. - nfral.2 $e |- F/ x ph $. - $( Bound-variable hypothesis builder for restricted quantification. Usage - of this theorem is discouraged because it depends on ~ ax-13 . Use the - weaker ~ nfralw when possible. (Contributed by NM, 1-Sep-1999.) - (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) $) - nfral $p |- F/ x A. y e. A ph $= - ( wral wnf wtru nftru wnfc a1i nfrald mptru ) ACDGBHIABCDCJBDKIELABHIFLMN - $. - $} - ${ - $d A y $. $d x y $. - $( Obsolete version of ~ nfra2w as of 24-Sep-2024. (Contributed by Alan - Sare, 31-Dec-2011.) (Revised by Gino Giotto, 10-Jan-2024.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - nfra2wOLDOLD $p |- F/ y A. x e. A A. y e. B ph $= - ( wral nfcv nfra1 nfralw ) ACEFCBDCDGACEHI $. - $( $j usage 'nfra2wOLDOLD' avoids 'ax-13'; $) - $} - - ${ - $d A y $. - $( Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the - antecedent is restricted. Derived automatically from ~ hbra2VD . Usage - of this theorem is discouraged because it depends on ~ ax-13 . Use the - weaker ~ nfra2w when possible. (Contributed by Alan Sare, 31-Dec-2011.) - (New usage is discouraged.) $) - nfra2 $p |- F/ y A. x e. A A. y e. B ph $= - ( wral nfcv nfra1 nfral ) ACEFCBDCDGACEHI $. - $} - - ${ - $d y z A $. $d x z $. - rgen2a.1 $e |- ( ( x e. A /\ y e. A ) -> ph ) $. - $( Generalization rule for restricted quantification. Note that ` x ` and - ` y ` are not required to be disjoint. This proof illustrates the use - of ~ dvelim . This theorem relies on the full set of axioms up to - ~ ax-ext and it should no longer be used. Usage of ~ rgen2 is highly - encouraged. (Contributed by NM, 23-Nov-1994.) (Proof shortened by - Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, - 1-Jan-2020.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - rgen2a $p |- A. x e. A A. y e. A ph $= - ( vz wral cv wcel wi wal weq wn eleq1 dvelimv alimi syl6com biimpd syli - ex pm2.61d2 df-ral sylibr rgen ) ACDGZBDBHZDIZCHZDIZAJZCKZUEUGCBLZCKZUKUM - MUGUGCKUKFHZDIUGCBFUNUFDNOUGUJCUGUIAETZPQULUJCUIULUGAULUIUGUHUFDNRUOSPUAA - CDUBUCUD $. - $} - - ${ - raleqbii.1 $e |- A = B $. - raleqbii.2 $e |- ( ps <-> ch ) $. - $( Equality deduction for restricted universal quantifier, changing both - formula and quantifier domain. Inference form. (Contributed by David - Moews, 1-May-2017.) $) - raleqbii $p |- ( A. x e. A ps <-> A. x e. B ch ) $= - ( cv wcel eleq2i imbi12i ralbii2 ) ABCDECHZDIMEIABDEMFJGKL $. - $} - - ${ - $d x y $. - nfrexd.1 $e |- F/ y ph $. - nfrexd.2 $e |- ( ph -> F/_ x A ) $. - nfrexd.3 $e |- ( ph -> F/ x ps ) $. - $( Deduction version of ~ nfrex . (Contributed by Mario Carneiro, - 14-Oct-2016.) Add disjoint variable condition to avoid ~ ax-13 . See - ~ nfrexdg for a less restrictive version requiring more axioms. - (Revised by Gino Giotto, 20-Jan-2024.) $) - nfrexd $p |- ( ph -> F/ x E. y e. A ps ) $= - ( wrex wn wral dfrex2 nfnd nfraldw nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGA - BCHMNMO $. - $( $j usage 'nfrexd' avoids 'ax-13'; $) - $} - - ${ - nfrexdg.1 $e |- F/ y ph $. - nfrexdg.2 $e |- ( ph -> F/_ x A ) $. - nfrexdg.3 $e |- ( ph -> F/ x ps ) $. - $( Deduction version of ~ nfrexg . Usage of this theorem is discouraged - because it depends on ~ ax-13 . See ~ nfrexd for a version with a - disjoint variable condition, but not requiring ~ ax-13 . (Contributed - by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.) $) - nfrexdg $p |- ( ph -> F/ x E. y e. A ps ) $= - ( wrex wn wral dfrex2 nfnd nfrald nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGAB - CHMNMO $. - $} - - ${ - $d x y $. - nfrex.1 $e |- F/_ x A $. - nfrex.2 $e |- F/ x ph $. $( Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) Add - disjoint variable condition to avoid ~ ax-13 . See ~ nfrexg for a less + disjoint variable condition to avoid ~ ax-13 . See ~ nfrex for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) $) - nfrex $p |- F/ x E. y e. A ph $= - ( wrex wnf wtru nftru wnfc a1i nfrexd mptru ) ACDGBHIABCDCJBDKIELABHIFLMN - $. - $( $j usage 'nfrex' avoids 'ax-13'; $) - $} - - ${ - nfrexg.1 $e |- F/_ x A $. - nfrexg.2 $e |- F/ x ph $. - $( Bound-variable hypothesis builder for restricted quantification. Usage - of this theorem is discouraged because it depends on ~ ax-13 . See - ~ nfrex for a version with a disjoint variable condition, but not - requiring ~ ax-13 . (Contributed by NM, 1-Sep-1999.) (Revised by Mario - Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) - (New usage is discouraged.) $) - nfrexg $p |- F/ x E. y e. A ph $= - ( wrex wnf wtru nftru wnfc a1i nfrexdg mptru ) ACDGBHIABCDCJBDKIELABHIFLM + nfrexw $p |- F/ x E. y e. A ph $= + ( wrex wnf wtru nftru wnfc a1i nfrexdw mptru ) ACDGBHIABCDCJBDKIELABHIFLM N $. + $( $j usage 'nfrexw' avoids 'ax-13'; $) $} ${ @@ -30578,427 +30500,90 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ( wral wrex cv wcel wex df-rex nfv nfra1 nfan nfex nfxfr ax-1 rsp com12 wa reximdv sylcom ralrimi ) ACEFZBDGZABDGZCEUEBHDIZUDTZBJCUDBDKUHCBUGUDCU GCLACEMNOPUECHEIZUEUFUEUIQUIUDABDUDUIAACERSUAUBUC $. - $( $j usage 'r19.12' avoids 'ax-13' 'ax-ext'; $) - $} - - ${ - rexeqbii.1 $e |- A = B $. - rexeqbii.2 $e |- ( ps <-> ch ) $. - $( Equality deduction for restricted existential quantifier, changing both - formula and quantifier domain. Inference form. (Contributed by David - Moews, 1-May-2017.) $) - rexeqbii $p |- ( E. x e. A ps <-> E. x e. B ch ) $= - ( cv wcel eleq2i anbi12i rexbii2 ) ABCDECHZDIMEIABDEMFJGKL $. - $} - - ${ - $d x y $. - ralcomf.1 $e |- F/_ y A $. - ralcomf.2 $e |- F/_ x B $. - $( Commutation of restricted universal quantifiers. (Contributed by Mario - Carneiro, 14-Oct-2016.) $) - ralcomf $p |- ( A. x e. A A. y e. B ph <-> A. y e. B A. x e. A ph ) $= - ( cv wcel wa wi wal wral ancomst 2albii alcom bitri r2alf 3bitr4i ) BHDIZ - CHEIZJAKZCLBLZUATJAKZBLCLZACEMBDMABDMCEMUCUDCLBLUEUBUDBCTUAANOUDBCPQABCDE - FRACBEDGRS $. - - $( Commutation of restricted existential quantifiers. (Contributed by - Mario Carneiro, 14-Oct-2016.) $) - rexcomf $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $= - ( cv wcel wa wex wrex ancom anbi1i 2exbii excom bitri r2exf 3bitr4i ) BHD - IZCHEIZJZAJZCKBKZUATJZAJZBKCKZACELBDLABDLCELUDUFCKBKUGUCUFBCUBUEATUAMNOUF - BCPQABCDEFRACBEDGRS $. - $} - - ${ - $d y A $. $d x A $. - $( Commutation of restricted universal quantifiers. Note that ` x ` and - ` y ` need not be disjoint (this makes the proof longer). This theorem - relies on the full set of axioms up to ~ ax-ext and it should no longer - be used. Usage of ~ ralcom is highly encouraged. (Contributed by NM, - 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) - (New usage is discouraged.) $) - ralcom2 $p |- ( A. x e. A A. y e. A ph -> A. y e. A A. x e. A ph ) $= - ( weq wal wral wi cv wcel wb eleq1w dral1 df-ral 3bitr4g wa nfnae ralrimi - nfan ex sps imbi1d bicomd imbi12d biimpd wn nfra2 nfra1 wnfc nfcvf adantr - nfcvd nfeld nfan1 rsp2 ancomsd expdimp adantll pm2.61i ) BCEZBFZACDGZBDGZ - ABDGZCDGZHVAVCVEVABIDJZVBHZBFCIZDJZVDHZCFVCVEVGVJBCVAVFVIVBVDUTVFVIKBBCDL - UAZVAVIAHZCFZVFAHZBFZVBVDVAVOVMVNVLBCVAVFVIAVKUBMUCACDNABDNOUDMVBBDNVDCDN - OUEVAUFZVCVEVPVCPZVDCDVPVCCBCCQABCDDUGSVQVIVDVQVIPABDVQVIBVPVCBBCBQVBBDUH - SVQBVHDVPBVHUIVCBCUJUKVQBDULUMUNVCVIVNVPVCVIVFAVCVFVIAABCDDUOUPUQURRTRTUS - $. + $( $j usage 'r19.12OLD' avoids 'ax-13' 'ax-ext'; $) $} ${ $d y A $. $d x B $. $d x y $. reean.1 $e |- F/ y ph $. reean.2 $e |- F/ x ps $. - $( Rearrange restricted existential quantifiers. (Contributed by NM, - 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.) $) + $( Rearrange restricted existential quantifiers. For a version based on + fewer axioms see ~ reeanv . (Contributed by NM, 27-Oct-2010.) (Proof + shortened by Andrew Salmon, 30-May-2011.) $) reean $p |- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A ph /\ E. y e. B ps ) ) $= ( cv wcel wa nfv nfan eean reeanlem ) ABCDEFCIEJZAKDIFJZBKCDPADPDLGMQBCQC LHMNO $. $} - $( Cancellation law for restricted unique existential quantification. - (Contributed by Peter Mazsa, 12-Feb-2018.) $) - reuanid $p |- ( E! x e. A ( x e. A /\ ph ) <-> E! x e. A ph ) $= - ( cv wcel wa weu wreu anabs5 eubii df-reu 3bitr4i ) BDCEZMAFZFZBGNBGNBCHABC - HONBMAIJNBCKABCKL $. - - $( Cancellation law for restricted at-most-one quantification. (Contributed - by Peter Mazsa, 24-May-2018.) $) - rmoanid $p |- ( E* x e. A ( x e. A /\ ph ) <-> E* x e. A ph ) $= - ( cv wcel wa wmo wrmo anabs5 mobii df-rmo 3bitr4i ) BDCEZMAFZFZBGNBGNBCHABC - HONBMAIJNBCKABCKL $. - - $( The setvar ` x ` is not free in ` E! x e. A ph ` . (Contributed by NM, - 19-Mar-1997.) $) - nfreu1 $p |- F/ x E! x e. A ph $= - ( wreu cv wcel wa weu df-reu nfeu1 nfxfr ) ABCDBECFAGZBHBABCILBJK $. - - $( The setvar ` x ` is not free in ` E* x e. A ph ` . (Contributed by NM, - 16-Jun-2017.) $) - nfrmo1 $p |- F/ x E* x e. A ph $= - ( wrmo cv wcel wa wmo df-rmo nfmo1 nfxfr ) ABCDBECFAGZBHBABCILBJK $. - - ${ - nfreud.1 $e |- F/ y ph $. - nfreud.2 $e |- ( ph -> F/_ x A ) $. - nfreud.3 $e |- ( ph -> F/ x ps ) $. - $( Deduction version of ~ nfreu . Usage of this theorem is discouraged - because it depends on ~ ax-13 . (Contributed by NM, 15-Feb-2013.) - (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) $) - nfreud $p |- ( ph -> F/ x E! y e. A ps ) $= - ( wreu cv wcel wa weu df-reu weq wal wn wnfc nfcvf adantr nfeld wnf nfand - adantl nfeud2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCPQZLZUHBCUKCUGEUJC - UGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $. - - $( Deduction version of ~ nfrmo . Usage of this theorem is discouraged - because it depends on ~ ax-13 . (Contributed by NM, 17-Jun-2017.) - (New usage is discouraged.) $) - nfrmod $p |- ( ph -> F/ x E* y e. A ps ) $= - ( wrmo cv wcel wa wmo df-rmo weq wal wn wnfc nfcvf adantr nfeld wnf nfand - adantl nfmod2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCPQZLZUHBCUKCUGEUJC - UGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $. - $} - ${ - $d x y $. - nfreuw.1 $e |- F/_ x A $. - nfreuw.2 $e |- F/ x ph $. - $( Bound-variable hypothesis builder for restricted uniqueness. Version of - ~ nfrmo with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, - 10-Jan-2024.) Avoid ~ ax-9 , ~ ax-ext . (Revised by Wolf Lammen, - 21-Nov-2024.) $) - nfrmow $p |- F/ x E* y e. A ph $= - ( wrmo cv wcel wa wmo df-rmo nfcri nfan nfmov nfxfr ) ACDGCHDIZAJZCKBACDL - RBCQABBCDEMFNOP $. - $( $j usage 'nfrmow' avoids 'ax-9' 'ax-13' 'ax-ext'; $) - - $( Bound-variable hypothesis builder for restricted unique existence. - Version of ~ nfreu with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 30-Oct-2010.) (Revised by Gino - Giotto, 10-Jan-2024.) Avoid ~ ax-9 , ~ ax-ext . (Revised by Wolf - Lammen, 21-Nov-2024.) $) - nfreuw $p |- F/ x E! y e. A ph $= - ( wreu cv wcel wa weu df-reu nfcri nfan nfeuw nfxfr ) ACDGCHDIZAJZCKBACDL - RBCQABBCDEMFNOP $. - $( $j usage 'nfreuw' avoids 'ax-9' 'ax-13' 'ax-ext'; $) - - $( Obsolete version of ~ nfreuw as of 21-Nov-2024. (Contributed by NM, - 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - nfreuwOLD $p |- F/ x E! y e. A ph $= - ( wreu wnf cv wcel wa weu wtru df-reu nftru nfcvd wnfc a1i nfeld nfand - nfeudw nfxfrd mptru ) ACDGZBHUDCIZDJZAKZCLMBACDNMUGBCCOMUFABMBUEDMBUEPBDQ - MERSABHMFRTUAUBUC $. - $( $j usage 'nfreuwOLD' avoids 'ax-13'; $) - - $( Obsolete version of ~ nfrmow as of 21-Nov-2024. (Contributed by NM, - 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - nfrmowOLD $p |- F/ x E* y e. A ph $= - ( wrmo cv wcel wa wmo df-rmo wnf wtru nftru nfcvd wnfc a1i nfeld nfand - nfmodv mptru nfxfr ) ACDGCHZDIZAJZCKZBACDLUGBMNUFBCCONUEABNBUDDNBUDPBDQNE - RSABMNFRTUAUBUC $. - $( $j usage 'nfrmowOLD' avoids 'ax-13'; $) - $} - - ${ - nfreu.1 $e |- F/_ x A $. - nfreu.2 $e |- F/ x ph $. - $( Bound-variable hypothesis builder for restricted unique existence. - Usage of this theorem is discouraged because it depends on ~ ax-13 . - Use the weaker ~ nfreuw when possible. (Contributed by NM, - 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) - (New usage is discouraged.) $) - nfreu $p |- F/ x E! y e. A ph $= - ( wreu wnf wtru nftru wnfc a1i nfreud mptru ) ACDGBHIABCDCJBDKIELABHIFLMN - $. - - $( Bound-variable hypothesis builder for restricted uniqueness. Usage of - this theorem is discouraged because it depends on ~ ax-13 . Use the - weaker ~ nfrmow when possible. (Contributed by NM, 16-Jun-2017.) - (New usage is discouraged.) $) - nfrmo $p |- F/ x E* y e. A ph $= - ( wrmo cv wcel wa wmo df-rmo wnf wtru nftru weq wal wn nfcvf a1i adantl - wnfc nfeld nfand nfmod2 mptru nfxfr ) ACDGCHZDIZAJZCKZBACDLUKBMNUJBCCOBCP - BQRZUJBMNULUIABULBUHDBCSBDUBULETUCABMULFTUDUAUEUFUG $. - $} - - $( An "identity" law of concretion for restricted abstraction. Special case - of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) $) - rabid $p |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) ) $= - ( cv wcel wa crab df-rab abeq2i ) BDCEAFBABCGABCHI $. - - $( Abstract builder restricted to another restricted abstract builder. - (Contributed by Thierry Arnoux, 30-Aug-2017.) $) - rabrab $p |- { x e. { x e. A | ph } | ps } = { x e. A | ( ph /\ ps ) } $= - ( cv crab wcel wa cab rabid anbi1i anass bitri abbii df-rab 3eqtr4i ) CEZAC - DFZGZBHZCIQDGZABHZHZCIBCRFUBCDFTUCCTUAAHZBHUCSUDBACDJKUAABLMNBCROUBCDOP $. - - $( Membership in a restricted abstraction, implication. (Contributed by - Glauco Siliprandi, 26-Jun-2021.) $) - rabidim1 $p |- ( x e. { x e. A | ph } -> x e. A ) $= - ( cv crab wcel rabid simplbi ) BDZABCEFICFAABCGH $. - - ${ - rabid2f.1 $e |- F/_ x A $. - $( An "identity" law for restricted class abstraction. (Contributed by NM, - 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised - by Thierry Arnoux, 13-Mar-2017.) $) - rabid2f $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $= - ( cv wcel wa cab wceq wi wal crab wral abeq2f pm4.71 bitr4i df-rab eqeq2i - wb albii df-ral 3bitr4i ) CBECFZAGZBHZIZUCAJZBKZCABCLZIABCMUFUCUDSZBKUHUD - BCDNUGUJBUCAOTPUIUECABCQRABCUAUB $. + $d z x A $. $d y A $. $d z y ph $. $d x y $. + $( Change bound variable by using a substitution. Version of ~ cbvralsv + with a disjoint variable condition, which does not require ~ ax-13 . + (Contributed by NM, 20-Nov-2005.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 10-Jan-2024.) $) + cbvralsvw $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $= + ( vz wral wsb nfv nfs1v sbequ12 cbvralw sbequ bitri ) ABDFABEGZEDFABCGZCD + FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $. + $( $j usage 'cbvralsvw' avoids 'ax-13'; $) $} ${ - $d x A $. - $( An "identity" law for restricted class abstraction. (Contributed by NM, - 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof - shortened by Wolf Lammen, 24-Nov-2024.) $) - rabid2 $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $= - ( nfcv rabid2f ) ABCBCDE $. - - $( Obsolete version of ~ rabid2 as of 24-11-2024. (Contributed by NM, - 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - rabid2OLD $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $= - ( cv wcel wa cab wceq wi wal crab abeq2 pm4.71 albii bitr4i df-rab eqeq2i - wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB - CLUFUIBUBAMNOUHUDCABCPQABCTUA $. + $d z x A $. $d y z ph $. $d y A $. $d x y $. + $( Change bound variable by using a substitution. Version of ~ cbvrexsv + with a disjoint variable condition, which does not require ~ ax-13 . + (Contributed by NM, 2-Mar-2008.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 10-Jan-2024.) $) + cbvrexsvw $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $= + ( vz wrex wsb nfv nfs1v sbequ12 cbvrexw sbequ bitri ) ABDFABEGZEDFABCGZCD + FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $. + $( $j usage 'cbvrexsvw' avoids 'ax-13'; $) $} - $( Equivalent wff's correspond to equal restricted class abstractions. - Closed theorem form of ~ rabbidva . (Contributed by NM, 25-Nov-2013.) $) - rabbi $p |- ( A. x e. A ( ps <-> ch ) - <-> { x e. A | ps } = { x e. A | ch } ) $= - ( cv wcel wa wb wal wceq wral crab abbi wi df-ral pm5.32 albii bitri df-rab - cab eqeq12i 3bitr4i ) CEDFZAGZUCBGZHZCIZUDCTZUECTZJABHZCDKZACDLZBCDLZJUDUEC - MUKUCUJNZCIUGUJCDOUNUFCUCABPQRULUHUMUIACDSBCDSUAUB $. - - $( The abstraction variable in a restricted class abstraction isn't free. - (Contributed by NM, 19-Mar-1997.) $) - nfrab1 $p |- F/_ x { x e. A | ph } $= - ( crab cv wcel wa cab df-rab nfab1 nfcxfr ) BABCDBECFAGZBHABCILBJK $. - ${ $d x y $. - nfrabw.1 $e |- F/ x ph $. - nfrabw.2 $e |- F/_ x A $. - $( A variable not free in a wff remains so in a restricted class - abstraction. Version of ~ nfrab with a disjoint variable condition, - which does not require ~ ax-13 . (Contributed by NM, 13-Oct-2003.) - (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, - 23-Nov-2024.) $) - nfrabw $p |- F/_ x { y e. A | ph } $= - ( crab cv wcel wa cab df-rab wnfc wtru nftru wnf nfcri nfan a1i nfabdw - mptru nfcxfr ) BACDGCHDIZAJZCKZACDLBUEMNUDBCCOUDBPNUCABBCDFQERSTUAUB $. - $( $j usage 'nfrabw' avoids 'ax-13'; $) - - $( Obsolete version of ~ nfrabw as of 23-Nov_2024. (Contributed by NM, - 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) - (New usage is discouraged.) (Proof modification is discouraged.) $) - nfrabwOLD $p |- F/_ x { y e. A | ph } $= - ( crab cv wcel wa cab df-rab wnfc wtru nftru wnf nfcri a1i nfand nfabdw - mptru nfcxfr ) BACDGCHDIZAJZCKZACDLBUEMNUDBCCONUCABUCBPNBCDFQRABPNERSTUAU - B $. - $( $j usage 'nfrabwOLD' avoids 'ax-13'; $) - $} - - ${ - $d x z $. $d y z $. $d z A $. - nfrab.1 $e |- F/ x ph $. - nfrab.2 $e |- F/_ x A $. - $( A variable not free in a wff remains so in a restricted class - abstraction. Usage of this theorem is discouraged because it depends on - ~ ax-13 . Use the weaker ~ nfrabw when possible. (Contributed by NM, - 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) - (New usage is discouraged.) $) - nfrab $p |- F/_ x { y e. A | ph } $= - ( vz crab cv wcel wa cab df-rab wnfc wtru nftru weq wal wn wnf eleq1w a1i - nfcri dvelimnf nfand adantl nfabd2 mptru nfcxfr ) BACDHCIDJZAKZCLZACDMBUL - NOUKBCCPBCQBRSZUKBTOUMUJABGIDJUJBCGBGDFUCGCDUAUDABTUMEUBUEUFUGUHUI $. - $} - - ${ - reubida.1 $e |- F/ x ph $. - reubida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. - $( Formula-building rule for restricted existential quantifier (deduction - form). (Contributed by Mario Carneiro, 19-Nov-2016.) $) - reubida $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $= - ( cv wcel wa weu wreu pm5.32da eubid df-reu 3bitr4g ) ADHEIZBJZDKQCJZDKBD - ELCDELARSDFAQBCGMNBDEOCDEOP $. - $} - - ${ - $d x ph $. - reubidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. - $( Formula-building rule for restricted existential quantifier (deduction - form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised - by Wolf Lammen, 14-Jan-2023.) $) - reubidva $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $= - ( cv wcel wa weu wreu pm5.32da eubidv df-reu 3bitr4g ) ADGEHZBIZDJPCIZDJB - DEKCDEKAQRDAPBCFLMBDENCDENO $. - $} - - ${ - $d x ph $. - reubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $. - $( Formula-building rule for restricted existential quantifier (deduction - form). (Contributed by NM, 17-Oct-1996.) $) - reubidv $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $= - ( wb cv wcel adantr reubidva ) ABCDEABCGDHEIFJK $. - $} - - ${ - reubiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $. - $( Formula-building rule for restricted existential quantifier (inference - form). (Contributed by NM, 14-Nov-2004.) $) - reubiia $p |- ( E! x e. A ph <-> E! x e. A ps ) $= - ( cv wcel wa weu wreu pm5.32i eubii df-reu 3bitr4i ) CFDGZAHZCIOBHZCIACDJ - BCDJPQCOABEKLACDMBCDMN $. - $} - - ${ - reubii.1 $e |- ( ph <-> ps ) $. - $( Formula-building rule for restricted existential quantifier (inference - form). (Contributed by NM, 22-Oct-1999.) $) - reubii $p |- ( E! x e. A ph <-> E! x e. A ps ) $= - ( wb cv wcel a1i reubiia ) ABCDABFCGDHEIJ $. - $} - - ${ - rmobida.1 $e |- F/ x ph $. - rmobida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. - $( Formula-building rule for restricted existential quantifier (deduction - form). (Contributed by NM, 16-Jun-2017.) $) - rmobida $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= - ( cv wcel wa wmo wrmo pm5.32da mobid df-rmo 3bitr4g ) ADHEIZBJZDKQCJZDKBD - ELCDELARSDFAQBCGMNBDEOCDEOP $. - $} - - ${ - $d x ph $. - rmobidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. - $( Formula-building rule for restricted existential quantifier (deduction - form). (Contributed by NM, 16-Jun-2017.) Avoid ~ ax-6 , ~ ax-7 , - ~ ax-12 . (Revised by Wolf Lammen, 23-Nov-2024.) $) - rmobidva $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= - ( cv wcel wa wmo wrmo pm5.32da mobidv df-rmo 3bitr4g ) ADGEHZBIZDJPCIZDJB - DEKCDEKAQRDAPBCFLMBDENCDENO $. - $( $j usage 'rmobidva' avoids 'ax-6' 'ax-7' 'ax-12'; $) - - $( Obsolete version of ~ rmobidv as of 23-Nov-2024. (Contributed by NM, - 16-Jun-2017.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - rmobidvaOLD $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= - ( nfv rmobida ) ABCDEADGFH $. - $} - - ${ - $d x ph $. - rmobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $. - $( Formula-building rule for restricted existential quantifier (deduction - form). (Contributed by NM, 16-Jun-2017.) $) - rmobidv $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= - ( wb cv wcel adantr rmobidva ) ABCDEABCGDHEIFJK $. - $} - - ${ - rmobiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $. - $( Formula-building rule for restricted existential quantifier (inference - form). (Contributed by NM, 16-Jun-2017.) $) - rmobiia $p |- ( E* x e. A ph <-> E* x e. A ps ) $= - ( cv wcel wa wmo wrmo pm5.32i mobii df-rmo 3bitr4i ) CFDGZAHZCIOBHZCIACDJ - BCDJPQCOABEKLACDMBCDMN $. + nfraldwOLD.1 $e |- F/ y ph $. + nfraldwOLD.2 $e |- ( ph -> F/_ x A ) $. + nfraldwOLD.3 $e |- ( ph -> F/ x ps ) $. + $( Obsolete version of ~ nfraldw as of 24-Sep-2024. (Contributed by NM, + 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + nfraldwOLD $p |- ( ph -> F/ x A. y e. A ps ) $= + ( wral cv wcel wi wal df-ral nfcvd nfeld nfimd nfald nfxfrd ) BDEIDJZEKZB + LZDMACBDENAUBCDFAUABCACTEACTOGPHQRS $. + $( $j usage 'nfraldwOLD' avoids 'ax-13'; $) $} ${ - rmobii.1 $e |- ( ph <-> ps ) $. - $( Formula-building rule for restricted existential quantifier (inference - form). (Contributed by NM, 16-Jun-2017.) $) - rmobii $p |- ( E* x e. A ph <-> E* x e. A ps ) $= - ( wb cv wcel a1i rmobiia ) ABCDABFCGDHEIJ $. + $d A y $. $d x y $. + $( Obsolete version of ~ nfra2w as of 24-Sep-2024. (Contributed by Alan + Sare, 31-Dec-2011.) (Revised by Gino Giotto, 10-Jan-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + nfra2wOLDOLD $p |- F/ y A. x e. A A. y e. B ph $= + ( wral nfcv nfra1 nfralw ) ACEFCBDCDGACEHI $. + $( $j usage 'nfra2wOLDOLD' avoids 'ax-13'; $) $} ${ - raleq1f.1 $e |- F/_ x A $. - raleq1f.2 $e |- F/_ x B $. - $( Equality theorem for restricted universal quantifier, with - bound-variable hypotheses instead of distinct variable restrictions. - (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, - 11-Jul-2011.) $) - raleqf $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $= - ( wceq cv wcel wi wal wral nfeq eleq2 imbi1d albid df-ral 3bitr4g ) CDGZB - HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. - - $( Equality theorem for restricted existential quantifier, with - bound-variable hypotheses instead of distinct variable restrictions. - (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, - 11-Jul-2011.) $) - rexeqf $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $= - ( wceq cv wcel wa wex wrex nfeq eleq2 anbi1d exbid df-rex 3bitr4g ) CDGZB - HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. - - $( Equality theorem for restricted unique existential quantifier, with - bound-variable hypotheses instead of distinct variable restrictions. - (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, - 11-Jul-2011.) $) - reueq1f $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $= - ( wceq cv wcel wa weu wreu nfeq eleq2 anbi1d eubid df-reu 3bitr4g ) CDGZB - HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. - - $( Equality theorem for restricted at-most-one quantifier, with - bound-variable hypotheses instead of distinct variable restrictions. - (Contributed by Alexander van der Vekens, 17-Jun-2017.) $) - rmoeq1f $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $= - ( wceq cv wcel wa wmo wrmo nfeq eleq2 anbi1d mobid df-rmo 3bitr4g ) CDGZB - HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. + $d x y z $. $d z A $. $d z ps $. $d z ph $. + cbvralfwOLD.1 $e |- F/_ x A $. + cbvralfwOLD.2 $e |- F/_ y A $. + cbvralfwOLD.3 $e |- F/ y ph $. + cbvralfwOLD.4 $e |- F/ x ps $. + cbvralfwOLD.5 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Obsolete version of ~ cbvralfw as of 23-May-2024. (Contributed by NM, + 7-Mar-2004.) (Revised by Gino Giotto, 10-Jan-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + cbvralfwOLD $p |- ( A. x e. A ph <-> A. y e. A ps ) $= + ( vz cv wcel wi wal wral wsb nfv nfcri nfim nfs1v sbequ12 imbi12d cbvalv1 + weq eleq1w nfsbv sbequ sbiev bitrdi bitri df-ral 3bitr4i ) CLEMZANZCOZDLE + MZBNZDOZACEPBDEPUPKLEMZACKQZNZKOUSUOVBCKUOKRUTVACCKEFSACKUATCKUEUNUTAVACK + EUFACKUBUCUDVBURKDUTVADDKEGSACKDHUGTURKRKDUEZUTUQVABKDEUFVCVAACDQBAKDCUHA + BCDIJUIUJUCUDUKACEULBDEULUM $. + $( $j usage 'cbvralfwOLD' avoids 'ax-13'; $) $} - ${ - $d x ph $. - raleqbidv.1 $e |- ( ph -> A = B ) $. - raleqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $. - $( Equality deduction for restricted universal quantifier. (Contributed by - NM, 6-Nov-2007.) Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 and - reduce distinct variable conditions. (Revised by Steven Nguyen, - 30-Apr-2023.) $) - raleqbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $= - ( cv wcel eleq2d imbi12d ralbidv2 ) ABCDEFADIZEJNFJBCAEFNGKHLM $. - - $( Equality deduction for restricted universal quantifier. (Contributed by - NM, 6-Nov-2007.) Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 and - reduce distinct variable conditions. (Revised by Steven Nguyen, - 30-Apr-2023.) $) - rexeqbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $= - ( cv wcel eleq2d anbi12d rexbidv2 ) ABCDEFADIZEJNFJBCAEFNGKHLM $. - $} + $( *** Theorems using axioms up to ax-9 and ax-ext only: *** $) ${ $d ph x $. $d A x $. $d B x $. @@ -31053,20 +30638,6 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is (Revised by Steven Nguyen, 30-Apr-2023.) $) rexeq $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $= ( wceq biidd rexeqbi1dv ) AABCDCDEAFG $. - - $( Equality theorem for restricted unique existential quantifier. - (Contributed by NM, 5-Apr-2004.) Remove usage of ~ ax-10 , ~ ax-11 , - and ~ ax-12 . (Revised by Steven Nguyen, 30-Apr-2023.) $) - reueq1 $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $= - ( wceq cv wcel wa weu wreu eleq2 anbi1d eubidv df-reu 3bitr4g ) CDEZBFZCG - ZAHZBIQDGZAHZBIABCJABDJPSUABPRTACDQKLMABCNABDNO $. - - $( Equality theorem for restricted at-most-one quantifier. (Contributed by - Alexander van der Vekens, 17-Jun-2017.) Remove usage of ~ ax-10 , - ~ ax-11 , and ~ ax-12 . (Revised by Steven Nguyen, 30-Apr-2023.) $) - rmoeq1 $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $= - ( wceq cv wcel wa wmo wrmo eleq2 anbi1d mobidv df-rmo 3bitr4g ) CDEZBFZCG - ZAHZBIQDGZAHZBIABCJABDJPSUABPRTACDQKLMABCNABDNO $. $} ${ @@ -31085,7 +30656,7 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ${ $d x A $. $d x B $. - raleq1d.1 $e |- ( ph -> A = B ) $. + raleqdv.1 $e |- ( ph -> A = B ) $. $( Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.) $) raleqdv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) ) $= @@ -31098,36 +30669,57 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $} ${ - $d x A $. $d x B $. - raleqd.1 $e |- ( A = B -> ( ph <-> ps ) ) $. - $( Equality deduction for restricted unique existential quantifier. - (Contributed by NM, 5-Apr-2004.) $) - reueqd $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) ) $= - ( wceq wreu reueq1 reubidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $. + raleqbii.1 $e |- A = B $. + raleqbii.2 $e |- ( ps <-> ch ) $. + $( Equality deduction for restricted universal quantifier, changing both + formula and quantifier domain. Inference form. (Contributed by David + Moews, 1-May-2017.) $) + raleqbii $p |- ( A. x e. A ps <-> A. x e. B ch ) $= + ( cv wcel eleq2i imbi12i ralbii2 ) ABCDECHZDIMEIABDEMFJGKL $. + $} - $( Equality deduction for restricted at-most-one quantifier. (Contributed - by Alexander van der Vekens, 17-Jun-2017.) $) - rmoeqd $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) ) $= - ( wceq wrmo rmoeq1 rmobidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $. + ${ + rexeqbii.1 $e |- A = B $. + rexeqbii.2 $e |- ( ps <-> ch ) $. + $( Equality deduction for restricted existential quantifier, changing both + formula and quantifier domain. Inference form. (Contributed by David + Moews, 1-May-2017.) $) + rexeqbii $p |- ( E. x e. A ps <-> E. x e. B ch ) $= + ( cv wcel eleq2i anbi12i rexbii2 ) ABCDECHZDIMEIABDEMFJGKL $. $} ${ - raleqbid.0 $e |- F/ x ph $. - raleqbid.1 $e |- F/_ x A $. - raleqbid.2 $e |- F/_ x B $. - raleqbid.3 $e |- ( ph -> A = B ) $. - raleqbid.4 $e |- ( ph -> ( ps <-> ch ) ) $. + $d A x $. $d B x $. + $( All elements of a class are elements of a class equal to this class. + (Contributed by AV, 30-Oct-2020.) $) + raleleq $p |- ( A = B -> A. x e. A x e. B ) $= + ( wceq cv wcel eleq2 biimpd ralrimiv ) BCDZAEZCFZABJKBFLBCKGHI $. + + $( Alternate proof of ~ raleleq using ~ ralel , being longer and using more + axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) + (Proof modification is discouraged.) $) + raleleqALT $p |- ( A = B -> A. x e. A x e. B ) $= + ( wceq cv wcel wral ralel id raleqdv mpbiri ) BCDZAECFZABGMACGACHLMABCLIJ + K $. + $} + + ${ + $d x ph $. + raleqbidv.1 $e |- ( ph -> A = B ) $. + raleqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $. $( Equality deduction for restricted universal quantifier. (Contributed by - Thierry Arnoux, 8-Mar-2017.) $) - raleqbid $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $= - ( wral wceq wb raleqf syl ralbid bitrd ) ABDELZBDFLZCDFLAEFMSTNJBDEFHIOPA - BCDFGKQR $. + NM, 6-Nov-2007.) Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 and + reduce distinct variable conditions. (Revised by Steven Nguyen, + 30-Apr-2023.) $) + raleqbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $= + ( cv wcel eleq2d imbi12d ralbidv2 ) ABCDEFADIZEJNFJBCAEFNGKHLM $. - $( Equality deduction for restricted existential quantifier. (Contributed - by Thierry Arnoux, 8-Mar-2017.) $) - rexeqbid $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $= - ( wrex wceq wb rexeqf syl rexbid bitrd ) ABDELZBDFLZCDFLAEFMSTNJBDEFHIOPA - BCDFGKQR $. + $( Equality deduction for restricted universal quantifier. (Contributed by + NM, 6-Nov-2007.) Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 and + reduce distinct variable conditions. (Revised by Steven Nguyen, + 30-Apr-2023.) $) + rexeqbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $= + ( cv wcel eleq2d anbi12d rexbidv2 ) ABCDEFADIZEJNFJBCAEFNGKHLM $. $} ${ @@ -31146,143 +30738,110 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $} ${ - $d A x $. $d B x $. - $( All elements of a class are elements of a class equal to this class. - (Contributed by AV, 30-Oct-2020.) $) - raleleq $p |- ( A = B -> A. x e. A x e. B ) $= - ( wceq cv wcel eleq2 biimpd ralrimiv ) BCDZAEZCFZABJKBFLBCKGHI $. - - $( Alternate proof of ~ raleleq using ~ ralel , being longer and using more - axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) - (Proof modification is discouraged.) $) - raleleqALT $p |- ( A = B -> A. x e. A x e. B ) $= - ( wceq cv wcel wral ralel id raleqdv mpbiri ) BCDZAECFZABGMACGACHLMABCLIJ - K $. - $} + $d A y $. $d ps y $. $d B x $. $d ch x $. $d x ph y $. + cbvraldva2.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $. + cbvraldva2.2 $e |- ( ( ph /\ x = y ) -> A = B ) $. + $( Rule used to change the bound variable in a restricted universal + quantifier with implicit substitution which also changes the quantifier + domain. Deduction form. (Contributed by David Moews, 1-May-2017.) $) + cbvraldva2 $p |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) ) $= + ( cv wcel wi wal wral weq wb wa simpr 19.21v df-ral eleq12d pm5.74ri + imbi12d expcom pm5.74d cbvalvw 3bitr3i 3bitr4g ) ADJZFKZBLZDMZEJZGKZCLZEM + ZBDFNCEGNAULUPAUKLZDMAUOLZEMAULLAUPLUQURDEDEOZAUKUOAUSUKUOPAUSQZUJUNBCUTU + IUMFGAUSRIUAHUCUDUEUFAUKDSAUOESUGUBBDFTCEGTUH $. - ${ - $d x y A $. - $( "At most one" element in a set. (Contributed by Thierry Arnoux, - 26-Jul-2018.) Avoid ~ ax-11 . (Revised by Wolf Lammen, - 23-Nov-2024.) $) - moel $p |- ( E* x x e. A <-> A. x e. A A. y e. A x = y ) $= - ( cv wcel wa weq wi wal wral wmo 19.21v impexp albii df-ral imbi2i eleq1w - 3bitr4i mo4 ) ADCEZBDCEZFABGZHZBIZAITUBBCJZHZAITAKUEACJUDUFATUAUBHZHZBITU - GBIZHUDUFTUGBLUCUHBTUAUBMNUEUITUBBCOPRNTUAABABCQSUEACOR $. - $( $j usage 'rmobidva' avoids 'ax-11'; $) + $( Rule used to change the bound variable in a restricted existential + quantifier with implicit substitution which also changes the quantifier + domain. Deduction form. (Contributed by David Moews, 1-May-2017.) + (Proof shortened by Wolf Lammen, 8-Jan-2025.) $) + cbvrexdva2 $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $= + ( wrex wn wral weq wa notbid cbvraldva2 ralnex 3bitr3g con4bid ) ABDFJZCE + GJZABKZDFLCKZEGLTKUAKAUBUCDEFGADEMNBCHOIPBDFQCEGQRS $. - $( Obsolete version of ~ moel as of 23-Nov-2024. (Contributed by Thierry - Arnoux, 26-Jul-2018.) (Proof modification is discouraged.) + $( Obsolete version of ~ cbvrexdva2 as of 8-Jan-2025. (Contributed by + David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, + 12-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) $) - moelOLD $p |- ( E* x x e. A <-> A. x e. A A. y e. A x = y ) $= - ( cv wcel weq wi wal wmo ralcom4 df-ral ralbii wa alcom eleq1w mo4 impexp - wral albii bitr4i 3bitr4i 3bitr4ri ) BDCEZABFZGZBHZACRUEACRZBHZUDBCRZACRA - DCEZAIZUEABCJUIUFACUDBCKLUJUCMUDGZBHAHULAHZBHUKUHULABNUJUCABABCOPUGUMBUGU - JUEGZAHUMUEACKULUNAUJUCUDQSTSUAUB $. + cbvrexdva2OLD $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $= + ( cv wcel wa wex wrex wb wi weq simpr 19.42v df-rex eleq12d ancoms pm5.32 + anbi12d pm5.32da cbvexvw 3bitr3i mpbir 3bitr4g ) ADJZFKZBLZDMZEJZGKZCLZEM + ZBDFNCEGNAUMUQOPAUMLZAUQLZOAULLZDMAUPLZEMURUSUTVADEDEQZAULUPAVBULUPOAVBLZ + UKUOBCVCUJUNFGAVBRIUAHUDUBUEUFAULDSAUPESUGAUMUQUCUHBDFTCEGTUI $. $} - $( Unrestricted "at most one" implies restricted "at most one". (Contributed - by NM, 16-Jun-2017.) $) - mormo $p |- ( E* x ph -> E* x e. A ph ) $= - ( wmo cv wcel wa wrmo moan df-rmo sylibr ) ABDBECFZAGBDABCHALBIABCJK $. - - $( Restricted uniqueness in terms of "at most one". (Contributed by NM, - 23-May-1999.) (Revised by NM, 16-Jun-2017.) $) - reu5 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) ) $= - ( cv wcel wa weu wex wmo wreu wrex wrmo df-eu df-reu df-rex anbi12i 3bitr4i - df-rmo ) BDCEAFZBGSBHZSBIZFABCJABCKZABCLZFSBMABCNUBTUCUAABCOABCRPQ $. - - $( Restricted unique existence implies restricted existence. (Contributed by - NM, 19-Aug-1999.) $) - reurex $p |- ( E! x e. A ph -> E. x e. A ph ) $= - ( wreu wrex wrmo reu5 simplbi ) ABCDABCEABCFABCGH $. - - $( Double restricted existential uniqueness, analogous to ~ 2eu2ex . - (Contributed by Alexander van der Vekens, 25-Jun-2017.) $) - 2reu2rex $p |- ( E! x e. A E! y e. B ph -> E. x e. A E. y e. B ph ) $= - ( wreu wrex reurex reximi syl ) ACEFZBDFKBDGACEGZBDGKBDHKLBDACEHIJ $. - - $( Restricted existential uniqueness implies restricted "at most one." - (Contributed by NM, 16-Jun-2017.) $) - reurmo $p |- ( E! x e. A ph -> E* x e. A ph ) $= - ( wreu wrex wrmo reu5 simprbi ) ABCDABCEABCFABCGH $. + ${ + $d ps y $. $d ch x $. $d A x y $. $d x ph y $. + cbvraldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $. + $( Rule used to change the bound variable in a restricted universal + quantifier with implicit substitution. Deduction form. (Contributed by + David Moews, 1-May-2017.) $) + cbvraldva $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $= + ( weq wa eqidd cbvraldva2 ) ABCDEFFGADEHIFJK $. - $( Restricted "at most one" in term of uniqueness. (Contributed by NM, - 16-Jun-2017.) $) - rmo5 $p |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) ) $= - ( cv wcel wa wmo wex wi wrmo wrex wreu df-rmo df-rex df-reu imbi12i 3bitr4i - weu moeu ) BDCEAFZBGTBHZTBRZIABCJABCKZABCLZITBSABCMUCUAUDUBABCNABCOPQ $. + $( Rule used to change the bound variable in a restricted existential + quantifier with implicit substitution. Deduction form. (Contributed by + David Moews, 1-May-2017.) $) + cbvrexdva $p |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) ) $= + ( weq wa eqidd cbvrexdva2 ) ABCDEFFGADEHIFJK $. + $} - $( Nonexistence implies restricted "at most one". (Contributed by NM, - 17-Jun-2017.) $) - nrexrmo $p |- ( -. E. x e. A ph -> E* x e. A ph ) $= - ( wrex wn wreu wi wrmo pm2.21 rmo5 sylibr ) ABCDZELABCFZGABCHLMIABCJK $. + $( *** Theorems using axioms up to ax-12 and ax-ext only: *** $) ${ - $d ph x $. - reueubd.1 $e |- ( ( ph /\ ps ) -> x e. V ) $. - $( Restricted existential uniqueness is equivalent to existential - uniqueness if the unique element is in the restricting class. - (Contributed by AV, 4-Jan-2021.) $) - reueubd $p |- ( ph -> ( E! x e. V ps <-> E! x ps ) ) $= - ( wreu cv wcel wa weu df-reu ex pm4.71rd eubidv bitr4id ) ABCDFCGDHZBIZCJ - BCJBCDKABQCABPABPELMNO $. - $} + raleqf.1 $e |- F/_ x A $. + raleqf.2 $e |- F/_ x B $. + $( Equality theorem for restricted universal quantifier, with + bound-variable hypotheses instead of distinct variable restrictions. + See ~ raleq for a version based on fewer axioms. (Contributed by NM, + 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.) $) + raleqf $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $= + ( wceq cv wcel wi wal wral nfeq eleq2 imbi1d albid df-ral 3bitr4g ) CDGZB + HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. - ${ - $d x y $. - cbvralfw.1 $e |- F/_ x A $. - cbvralfw.2 $e |- F/_ y A $. - cbvralfw.3 $e |- F/ y ph $. - cbvralfw.4 $e |- F/ x ps $. - cbvralfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Rule used to change bound variables, using implicit substitution. - Version of ~ cbvralf with a disjoint variable condition, which does not - require ~ ax-10 , ~ ax-13 . (Contributed by NM, 7-Mar-2004.) (Revised - by Gino Giotto, 23-May-2024.) $) - cbvralfw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= - ( cv wcel wi wal wral nfcri nfim weq eleq1w df-ral imbi12d cbvalv1 - 3bitr4i ) CKELZAMZCNDKELZBMZDNACEOBDEOUEUGCDUDADDCEGPHQUFBCCDEFPIQCDRUDUF - ABCDESJUAUBACETBDETUC $. - $( $j usage 'cbvralfw' avoids 'ax-10' 'ax-13'; $) + $( Equality theorem for restricted existential quantifier, with + bound-variable hypotheses instead of distinct variable restrictions. + See ~ rexeq for a version based on fewer axioms. (Contributed by NM, + 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) $) + rexeqf $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $= + ( wceq cv wcel wa wex wrex nfeq eleq2 anbi1d exbid df-rex 3bitr4g ) CDGZB + HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. $} ${ - $d x y z $. $d z A $. $d z ps $. $d z ph $. - cbvralfwOLD.1 $e |- F/_ x A $. - cbvralfwOLD.2 $e |- F/_ y A $. - cbvralfwOLD.3 $e |- F/ y ph $. - cbvralfwOLD.4 $e |- F/ x ps $. - cbvralfwOLD.5 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Obsolete version of ~ cbvralfw as of 23-May-2024. (Contributed by NM, - 7-Mar-2004.) (Revised by Gino Giotto, 10-Jan-2024.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - cbvralfwOLD $p |- ( A. x e. A ph <-> A. y e. A ps ) $= - ( vz cv wcel wi wal wral wsb nfv nfcri nfim nfs1v sbequ12 imbi12d cbvalv1 - weq eleq1w nfsbv sbequ sbiev bitrdi bitri df-ral 3bitr4i ) CLEMZANZCOZDLE - MZBNZDOZACEPBDEPUPKLEMZACKQZNZKOUSUOVBCKUOKRUTVACCKEFSACKUATCKUEUNUTAVACK - EUFACKUBUCUDVBURKDUTVADDKEGSACKDHUGTURKRKDUEZUTUQVABKDEUFVCVAACDQBAKDCUHA - BCDIJUIUJUCUDUKACEULBDEULUM $. - $( $j usage 'cbvralfwOLD' avoids 'ax-13'; $) + raleqbid.0 $e |- F/ x ph $. + raleqbid.1 $e |- F/_ x A $. + raleqbid.2 $e |- F/_ x B $. + raleqbid.3 $e |- ( ph -> A = B ) $. + raleqbid.4 $e |- ( ph -> ( ps <-> ch ) ) $. + $( Equality deduction for restricted universal quantifier. See ~ raleqbidv + for a version based on fewer axioms. (Contributed by Thierry Arnoux, + 8-Mar-2017.) $) + raleqbid $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $= + ( wral wceq wb raleqf syl ralbid bitrd ) ABDELZBDFLZCDFLAEFMSTNJBDEFHIOPA + BCDFGKQR $. + + $( Equality deduction for restricted existential quantifier. See + ~ rexeqbidv for a version based on fewer axioms. (Contributed by + Thierry Arnoux, 8-Mar-2017.) $) + rexeqbid $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $= + ( wrex wceq wb rexeqf syl rexbid bitrd ) ABDELZBDFLZCDFLAEFMSTNJBDEFHIOPA + BCDFGKQR $. $} ${ - $d x y $. - cbvrexfw.1 $e |- F/_ x A $. - cbvrexfw.2 $e |- F/_ y A $. - cbvrexfw.3 $e |- F/ y ph $. - cbvrexfw.4 $e |- F/ x ps $. - cbvrexfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Rule used to change bound variables, using implicit substitution. - Version of ~ cbvrexf with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by FL, 27-Apr-2008.) (Revised by Gino - Giotto, 10-Jan-2024.) $) - cbvrexfw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= - ( wrex wn wral nfn weq notbid cbvralfw ralnex 3bitr3i con4bii ) ACEKZBDEK - ZALZCEMBLZDEMUALUBLUCUDCDEFGADHNBCINCDOABJPQACERBDERST $. - $( $j usage 'cbvrexfw' avoids 'ax-13'; $) + $d x y z $. $d x z ps $. $d y z ph $. + sbralie.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Implicit to explicit substitution that swaps variables in a rectrictedly + universally quantified expression. (Contributed by NM, 5-Sep-2004.) $) + sbralie $p |- ( A. x e. y ph <-> [ y / x ] A. y e. x ps ) $= + ( vz cv wral wsb cbvralsvw sbbii raleq sbievw sbco2vv wb weq bicomd bitri + equcoms ralbii 3bitrri ) BDCGZHZCDIBDFIZFUBHZCDIUDFDGZHZACUFHZUCUECDBDFUB + JKUEUGCDUDFUBUFLMUGUDFCIZCUFHUHUDFCUFJUIACUFUIBDCIABDCFNBADCBAOCDCDPABEQS + MRTRUA $. $} + $( *** Theorems using axioms ax-13 or axc11r: *** $) + ${ $d x z $. $d y z $. $d z A $. $d z ps $. $d z ph $. cbvralf.1 $e |- F/_ x A $. @@ -31311,86 +30870,7 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $} ${ - $d x y A $. - cbvralw.1 $e |- F/ y ph $. - cbvralw.2 $e |- F/ x ps $. - cbvralw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Rule used to change bound variables, using implicit substitution. - Version of ~ cbvral with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 31-Jul-2003.) (Revised by Gino - Giotto, 10-Jan-2024.) $) - cbvralw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= - ( nfcv cbvralfw ) ABCDECEIDEIFGHJ $. - $( $j usage 'cbvralw' avoids 'ax-13'; $) - - $( Rule used to change bound variables, using implicit substitution. - Version of ~ cbvrex with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 31-Jul-2003.) (Revised by Gino - Giotto, 10-Jan-2024.) $) - cbvrexw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= - ( nfcv cbvrexfw ) ABCDECEIDEIFGHJ $. - $( $j usage 'cbvrexw' avoids 'ax-13'; $) - $} - - ${ - $d x y A $. - cbvrmow.1 $e |- F/ y ph $. - cbvrmow.2 $e |- F/ x ps $. - cbvrmow.3 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Change the bound variable of a restricted at-most-one quantifier using - implicit substitution. Version of ~ cbvrmo with a disjoint variable - condition, which does not require ~ ax-10 , ~ ax-13 . (Contributed by - NM, 16-Jun-2017.) (Revised by Gino Giotto, 23-May-2024.) $) - cbvrmow $p |- ( E* x e. A ph <-> E* y e. A ps ) $= - ( cv wcel wa wmo wrmo nfv nfan weq eleq1w anbi12d cbvmow df-rmo 3bitr4i ) - CIEJZAKZCLDIEJZBKZDLACEMBDEMUCUECDUBADUBDNFOUDBCUDCNGOCDPUBUDABCDEQHRSACE - TBDETUA $. - $( $j usage 'cbvrmow' avoids 'ax-10' 'ax-13'; $) - $} - - ${ - $d A x y z $. $d ph z $. $d ps z $. - cbvreuw.1 $e |- F/ y ph $. - cbvreuw.2 $e |- F/ x ps $. - cbvreuw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Change the bound variable of a restricted unique existential quantifier - using implicit substitution. Version of ~ cbvreu with a disjoint - variable condition, which does not require ~ ax-13 . (Contributed by - Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) - Avoid ~ ax-10 . (Revised by Wolf Lammen, 10-Dec-2024.) $) - cbvreuw $p |- ( E! x e. A ph <-> E! y e. A ps ) $= - ( wrex wrmo wa wreu cbvrexw cbvrmow anbi12i reu5 3bitr4i ) ACEIZACEJZKBDE - IZBDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $. - $( $j usage 'cbvreuw' avoids 'ax-10' 'ax-13'; $) - - $( Obsolete version of ~ cbvreuw as of 10-Dec-2024. (Contributed by Mario - Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) - (New usage is discouraged.) (Proof modification is discouraged.) $) - cbvreuwOLD $p |- ( E! x e. A ph <-> E! y e. A ps ) $= - ( vz cv wcel wa weu wreu wsb nfv sb8euv sban eubii df-reu anbi1i nfan weq - clelsb1 nfsbv eleq1w sbequ sbiev bitrdi anbi12d cbveuw 3bitri 3bitr4i - bitri ) CJEKZALZCMZDJEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZUTUPCIUPI - PQVAVDIUOACIRSVEIJEKZVCLZIMUTVDVGIVBVFVCCIEUDUASVGUSIDVFVCDVFDPACIDFUEUBU - SIPIDUCZVFURVCBIDEUFVHVCACDOBAIDCUGABCDGHUHUIUJUKUNULACETBDETUM $. - $( $j usage 'cbvreuwOLD' avoids 'ax-13'; $) - $} - - ${ - $d x y A $. - cbvrmowOLD.1 $e |- F/ y ph $. - cbvrmowOLD.2 $e |- F/ x ps $. - cbvrmowOLD.3 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Obsolete version of ~ cbvrmow as of 23-May-2024. (Contributed by NM, - 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - cbvrmowOLD $p |- ( E* x e. A ph <-> E* y e. A ps ) $= - ( wrex wreu wi wrmo cbvrexw cbvreuw imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDE - IZBDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $. - $( $j usage 'cbvrmowOLD' avoids 'ax-13'; $) - $} - - ${ - $d x z A $. $d y z A $. $d z ph $. $d z ps $. + $d x A $. $d y A $. cbvral.1 $e |- F/ y ph $. cbvral.2 $e |- F/ x ps $. cbvral.3 $e |- ( x = y -> ( ph <-> ps ) ) $. @@ -31408,56 +30888,6 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is (New usage is discouraged.) $) cbvrex $p |- ( E. x e. A ph <-> E. y e. A ps ) $= ( nfcv cbvrexf ) ABCDECEIDEIFGHJ $. - - $( Change the bound variable of a restricted unique existential quantifier - using implicit substitution. Usage of this theorem is discouraged - because it depends on ~ ax-13 . Use the weaker ~ cbvreuw when possible. - (Contributed by Mario Carneiro, 15-Oct-2016.) - (New usage is discouraged.) $) - cbvreu $p |- ( E! x e. A ph <-> E! y e. A ps ) $= - ( vz cv wcel wa weu wreu wsb nfv sb8eu sban eubii df-reu anbi1i nfsb nfan - clelsb1 weq eleq1w sbequ sbie bitrdi anbi12d cbveu bitri 3bitri 3bitr4i ) - CJEKZALZCMZDJEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZUTUPCIUPIPQVAVDIU - OACIRSVEIJEKZVCLZIMUTVDVGIVBVFVCCIEUDUASVGUSIDVFVCDVFDPACIDFUBUCUSIPIDUEZ - VFURVCBIDEUFVHVCACDOBAIDCUGABCDGHUHUIUJUKULUMACETBDETUN $. - - $( Change the bound variable of a restricted at-most-one quantifier using - implicit substitution. Usage of this theorem is discouraged because it - depends on ~ ax-13 . Use the weaker ~ cbvrmow , ~ cbvrmovw when - possible. (Contributed by NM, 16-Jun-2017.) - (New usage is discouraged.) $) - cbvrmo $p |- ( E* x e. A ph <-> E* y e. A ps ) $= - ( wrex wreu wi wrmo cbvrex cbvreu imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDEIZ - BDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $. - $} - - ${ - $d x y A $. $d y ph $. $d x ps $. - cbvrmovw.1 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Change the bound variable of a restricted at-most-one quantifier using - implicit substitution. Version of ~ cbvrmov with a disjoint variable - condition, which requires fewer axioms. (Contributed by NM, - 16-Jun-2017.) (Revised by Gino Giotto, 30-Sep-2024.) $) - cbvrmovw $p |- ( E* x e. A ph <-> E* y e. A ps ) $= - ( cv wcel wa wmo wrmo weq eleq1w anbi12d cbvmovw df-rmo 3bitr4i ) CGEHZAI - ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. - $( $j usage 'cbvrmovw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $) - - $( Change the bound variable of a restricted unique existential quantifier - using implicit substitution. Version of ~ cbvreuv with a disjoint - variable condition, which requires fewer axioms. (Contributed by NM, - 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.) $) - cbvreuvw $p |- ( E! x e. A ph <-> E! y e. A ps ) $= - ( cv wcel wa weu wreu weq eleq1w anbi12d cbveuvw df-reu 3bitr4i ) CGEHZAI - ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. - $( $j usage 'cbvreuvw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $) - - $( Obsolete version of ~ cbvreuvw as of 30-Sep-2024. (Contributed by NM, - 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - cbvreuvwOLD $p |- ( E! x e. A ph <-> E! y e. A ps ) $= - ( nfv cbvreuw ) ABCDEADGBCGFH $. - $( $j usage 'cbvreuvwOLD' avoids 'ax-13'; $) $} ${ @@ -31478,62 +30908,25 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is (Contributed by NM, 2-Jun-1998.) (New usage is discouraged.) $) cbvrexv $p |- ( E. x e. A ph <-> E. y e. A ps ) $= ( nfv cbvrex ) ABCDEADGBCGFH $. - - $( Change the bound variable of a restricted unique existential quantifier - using implicit substitution. See ~ cbvreuvw for a version without - ~ ax-13 , but extra disjoint variables. Usage of this theorem is - discouraged because it depends on ~ ax-13 . Use the weaker ~ cbvreuvw - when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario - Carneiro, 15-Oct-2016.) (New usage is discouraged.) $) - cbvreuv $p |- ( E! x e. A ph <-> E! y e. A ps ) $= - ( nfv cbvreu ) ABCDEADGBCGFH $. - - $( Change the bound variable of a restricted at-most-one quantifier using - implicit substitution. Usage of this theorem is discouraged because it - depends on ~ ax-13 . (Contributed by Alexander van der Vekens, - 17-Jun-2017.) (New usage is discouraged.) $) - cbvrmov $p |- ( E* x e. A ph <-> E* y e. A ps ) $= - ( nfv cbvrmo ) ABCDEADGBCGFH $. - $} - - ${ - $d A y $. $d ps y $. $d B x $. $d ch x $. $d x ph y $. - cbvraldva2.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $. - cbvraldva2.2 $e |- ( ( ph /\ x = y ) -> A = B ) $. - $( Rule used to change the bound variable in a restricted universal - quantifier with implicit substitution which also changes the quantifier - domain. Deduction form. (Contributed by David Moews, 1-May-2017.) $) - cbvraldva2 $p |- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) ) $= - ( cv wcel wi wal wral weq wb wa simpr 19.21v df-ral eleq12d pm5.74ri - imbi12d expcom pm5.74d cbvalvw 3bitr3i 3bitr4g ) ADJZFKZBLZDMZEJZGKZCLZEM - ZBDFNCEGNAULUPAUKLZDMAUOLZEMAULLAUPLUQURDEDEOZAUKUOAUSUKUOPAUSQZUJUNBCUTU - IUMFGAUSRIUAHUCUDUEUFAUKDSAUOESUGUBBDFTCEGTUH $. - - $( Rule used to change the bound variable in a restricted existential - quantifier with implicit substitution which also changes the quantifier - domain. Deduction form. (Contributed by David Moews, 1-May-2017.) - (Proof shortened by Wolf Lammen, 12-Aug-2023.) $) - cbvrexdva2 $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $= - ( cv wcel wa wex wrex wb wi weq simpr 19.42v df-rex eleq12d ancoms pm5.32 - anbi12d pm5.32da cbvexvw 3bitr3i mpbir 3bitr4g ) ADJZFKZBLZDMZEJZGKZCLZEM - ZBDFNCEGNAUMUQOPAUMLZAUQLZOAULLZDMAUPLZEMURUSUTVADEDEQZAULUPAVBULUPOAVBLZ - UKUOBCVCUJUNFGAVBRIUAHUDUBUEUFAULDSAUPESUGAUMUQUCUHBDFTCEGTUI $. $} ${ - $d ps y $. $d ch x $. $d A x y $. $d x ph y $. - cbvraldva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $. - $( Rule used to change the bound variable in a restricted universal - quantifier with implicit substitution. Deduction form. (Contributed by - David Moews, 1-May-2017.) $) - cbvraldva $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $= - ( weq wa eqidd cbvraldva2 ) ABCDEFFGADEHIFJK $. + $d z x A $. $d y A $. $d z y ph $. + $( Change bound variable by using a substitution. Usage of this theorem is + discouraged because it depends on ~ ax-13 . Use the weaker ~ cbvralsvw + when possible. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew + Salmon, 11-Jul-2011.) (New usage is discouraged.) $) + cbvralsv $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $= + ( vz wral wsb nfv nfs1v sbequ12 cbvral nfsb sbequ bitri ) ABDFABEGZEDFABC + GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $. - $( Rule used to change the bound variable in a restricted existential - quantifier with implicit substitution. Deduction form. (Contributed by - David Moews, 1-May-2017.) $) - cbvrexdva $p |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) ) $= - ( weq wa eqidd cbvrexdva2 ) ABCDEFFGADEHIFJK $. + $( Change bound variable by using a substitution. Usage of this theorem is + discouraged because it depends on ~ ax-13 . Use the weaker ~ cbvrexsvw + when possible. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew + Salmon, 11-Jul-2011.) (New usage is discouraged.) $) + cbvrexsv $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $= + ( vz wrex wsb nfv nfs1v sbequ12 cbvrex nfsb sbequ bitri ) ABDFABEGZEDFABC + GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $. $} ${ @@ -31549,13 +30942,7 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is cbvral2v $p |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps ) $= ( wral weq ralbidv cbvralv ralbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHDF MACEIJNOSTFHCBEGIKOPQ $. - $} - ${ - $d A x $. $d A z $. $d B w $. $d B x y $. $d B z y $. $d ch w $. - $d ch x $. $d ph z $. $d ps y $. - cbvrex2v.1 $e |- ( x = z -> ( ph <-> ch ) ) $. - cbvrex2v.2 $e |- ( y = w -> ( ch <-> ps ) ) $. $( Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ~ ax-13 . Use the weaker ~ cbvrex2vw when @@ -31585,101 +30972,649 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $} ${ - $d z x A $. $d y A $. $d z y ph $. $d x y $. - $( Change bound variable by using a substitution. Version of ~ cbvralsv - with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by NM, 20-Nov-2005.) (Revised by Gino Giotto, - 10-Jan-2024.) $) - cbvralsvw $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $= - ( vz wral wsb nfv nfs1v sbequ12 cbvralw sbequ bitri ) ABDFABEGZEDFABCGZCD - FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $. - $( $j usage 'cbvralsvw' avoids 'ax-13'; $) + $d y z A $. $d x z $. + rgen2a.1 $e |- ( ( x e. A /\ y e. A ) -> ph ) $. + $( Generalization rule for restricted quantification. Note that ` x ` and + ` y ` are not required to be disjoint. This proof illustrates the use + of ~ dvelim . This theorem relies on the full set of axioms up to + ~ ax-ext and it should no longer be used. Usage of ~ rgen2 is highly + encouraged. (Contributed by NM, 23-Nov-1994.) (Proof shortened by + Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, + 1-Jan-2020.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rgen2a $p |- A. x e. A A. y e. A ph $= + ( vz wral cv wcel wi wal weq wn eleq1 dvelimv alimi syl6com biimpd syli + ex pm2.61d2 df-ral sylibr rgen ) ACDGZBDBHZDIZCHZDIZAJZCKZUEUGCBLZCKZUKUM + MUGUGCKUKFHZDIUGCBFUNUFDNOUGUJCUGUIAETZPQULUJCUIULUGAULUIUGUHUFDNRUOSPUAA + CDUBUCUD $. $} ${ - $d z x A $. $d y z ph $. $d y A $. $d x y $. - $( Change bound variable by using a substitution. Version of ~ cbvrexsv - with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by NM, 2-Mar-2008.) (Revised by Gino Giotto, - 10-Jan-2024.) $) - cbvrexsvw $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $= - ( vz wrex wsb nfv nfs1v sbequ12 cbvrexw sbequ bitri ) ABDFABEGZEDFABCGZCD - FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $. - $( $j usage 'cbvrexsvw' avoids 'ax-13'; $) + nfrald.1 $e |- F/ y ph $. + nfrald.2 $e |- ( ph -> F/_ x A ) $. + nfrald.3 $e |- ( ph -> F/ x ps ) $. + $( Deduction version of ~ nfral . Usage of this theorem is discouraged + because it depends on ~ ax-13 . Use the weaker ~ nfraldw when possible. + (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, + 7-Oct-2016.) (New usage is discouraged.) $) + nfrald $p |- ( ph -> F/ x A. y e. A ps ) $= + ( wral cv wcel wi wal df-ral weq wn wa wnfc nfcvf adantr adantl nfeld wnf + nfimd nfald2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCMPZQZUHBCUKCUGEUJCU + GRACDSUAACERUJGTUBABCUCUJHTUDUEUF $. + + $( Deduction version of ~ nfrex . Usage of this theorem is discouraged + because it depends on ~ ax-13 . See ~ nfrexdw for a version with a + disjoint variable condition, but not requiring ~ ax-13 . (Contributed + by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.) $) + nfrexd $p |- ( ph -> F/ x E. y e. A ps ) $= + ( wrex wn wral dfrex2 nfnd nfrald nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGAB + CHMNMO $. $} ${ - $d z x A $. $d y A $. $d z y ph $. - $( Change bound variable by using a substitution. Usage of this theorem is - discouraged because it depends on ~ ax-13 . Use the weaker ~ cbvralsvw - when possible. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew - Salmon, 11-Jul-2011.) (New usage is discouraged.) $) - cbvralsv $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $= - ( vz wral wsb nfv nfs1v sbequ12 cbvral nfsb sbequ bitri ) ABDFABEGZEDFABC - GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $. + nfral.1 $e |- F/_ x A $. + nfral.2 $e |- F/ x ph $. + $( Bound-variable hypothesis builder for restricted quantification. Usage + of this theorem is discouraged because it depends on ~ ax-13 . Use the + weaker ~ nfralw when possible. (Contributed by NM, 1-Sep-1999.) + (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) $) + nfral $p |- F/ x A. y e. A ph $= + ( wral wnf wtru nftru wnfc a1i nfrald mptru ) ACDGBHIABCDCJBDKIELABHIFLMN + $. + + $( Bound-variable hypothesis builder for restricted quantification. Usage + of this theorem is discouraged because it depends on ~ ax-13 . See + ~ nfrexw for a version with a disjoint variable condition, but not + requiring ~ ax-13 . (Contributed by NM, 1-Sep-1999.) (Revised by Mario + Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) + (New usage is discouraged.) $) + nfrex $p |- F/ x E. y e. A ph $= + ( wrex wnf wtru nftru wnfc a1i nfrexd mptru ) ACDGBHIABCDCJBDKIELABHIFLMN + $. $} ${ - $d z x A $. $d y z ph $. $d y A $. - $( Change bound variable by using a substitution. Usage of this theorem is - discouraged because it depends on ~ ax-13 . Use the weaker ~ cbvrexsvw - when possible. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew - Salmon, 11-Jul-2011.) (New usage is discouraged.) $) - cbvrexsv $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $= - ( vz wrex wsb nfv nfs1v sbequ12 cbvrex nfsb sbequ bitri ) ABDFABEGZEDFABC - GZCDFAOBEDAEHABEIABEJKOPECDABECACHLPEHAECBMKN $. + $d A y $. + $( Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the + antecedent is restricted. Derived automatically from ~ hbra2VD . Usage + of this theorem is discouraged because it depends on ~ ax-13 . Use the + weaker ~ nfra2w when possible. (Contributed by Alan Sare, 31-Dec-2011.) + (New usage is discouraged.) $) + nfra2 $p |- F/ y A. x e. A A. y e. B ph $= + ( wral nfcv nfra1 nfral ) ACEFCBDCDGACEHI $. $} ${ - $d x y z $. $d x z ps $. $d y z ph $. - sbralie.1 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Implicit to explicit substitution that swaps variables in a rectrictedly - universally quantified expression. (Contributed by NM, 5-Sep-2004.) $) - sbralie $p |- ( A. x e. y ph <-> [ y / x ] A. y e. x ps ) $= - ( vz cv wral wsb cbvralsvw sbbii raleq sbievw sbco2vv wb weq bicomd bitri - equcoms ralbii 3bitrri ) BDCGZHZCDIBDFIZFUBHZCDIUDFDGZHZACUFHZUCUECDBDFUB - JKUEUGCDUDFUBUFLMUGUDFCIZCUFHUHUDFCUFJUIACUFUIBDCIABDCFNBADCBAOCDCDPABEQS - MRTRUA $. + $d y A $. $d x A $. + $( Commutation of restricted universal quantifiers. Note that ` x ` and + ` y ` need not be disjoint (this makes the proof longer). This theorem + relies on the full set of axioms up to ~ ax-ext and it should no longer + be used. Usage of ~ ralcom is highly encouraged. (Contributed by NM, + 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) + (New usage is discouraged.) $) + ralcom2 $p |- ( A. x e. A A. y e. A ph -> A. y e. A A. x e. A ph ) $= + ( weq wal wral wi cv wcel wb eleq1w dral1 df-ral 3bitr4g wa nfnae ralrimi + nfan ex sps imbi1d bicomd imbi12d biimpd wn nfra2 nfra1 wnfc nfcvf adantr + nfcvd nfeld nfan1 rsp2 ancomsd expdimp adantll pm2.61i ) BCEZBFZACDGZBDGZ + ABDGZCDGZHVAVCVEVABIDJZVBHZBFCIZDJZVDHZCFVCVEVGVJBCVAVFVIVBVDUTVFVIKBBCDL + UAZVAVIAHZCFZVFAHZBFZVBVDVAVOVMVNVLBCVAVFVIAVKUBMUCACDNABDNOUDMVBBDNVDCDN + OUEVAUFZVCVEVPVCPZVDCDVPVCCBCCQABCDDUGSVQVIVDVQVIPABDVQVIBVPVCBBCBQVBBDUH + SVQBVHDVPBVHUIVCBCUJUKVQBDULUMUNVCVIVNVPVCVIVFAVCVFVIAABCDDUOUPUQURRTRTUS + $. $} +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Restricted existential uniqueness and at-most-one quantifier +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $( Extend wff notation to include restricted existential uniqueness. $) + wreu $a wff E! x e. A ph $. + + $( Extend wff notation to include restricted "at most one". $) + wrmo $a wff E* x e. A ph $. + + $( Define restricted "at most one". Note: This notation is most often used + to express that ` ph ` holds for at most one element of a given class + ` A ` . For this reading ` F/_ x A ` is required, though, for example, + asserted when ` x ` and ` A ` are disjoint. + + Should instead ` A ` depend on ` x ` , you rather assert at most one ` x ` + fulfilling ` ph ` happens to be contained in the corresponding + ` A ( x ) ` . This interpretation is rarely needed (see also ~ df-ral ). + (Contributed by NM, 16-Jun-2017.) $) + df-rmo $a |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) $. + + $( Define restricted existential uniqueness. + + Note: This notation is most often used to express that ` ph ` holds for + exactly one element of a given class ` A ` . For this reading ` F/_ x A ` + is required, though, for example, asserted when ` x ` and ` A ` are + disjoint. + + Should instead ` A ` depend on ` x ` , you rather assert exactly one ` x ` + fulfilling ` ph ` happens to be contained in the corresponding + ` A ( x ) ` . This interpretation is rarely needed (see also ~ df-ral ). + (Contributed by NM, 22-Nov-1994.) $) + df-reu $a |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) $. + + $( *** Theorems using axioms up to ax-4 only: *** $) + + $( Restricted uniqueness in terms of "at most one". (Contributed by NM, + 23-May-1999.) (Revised by NM, 16-Jun-2017.) $) + reu5 $p |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) ) $= + ( cv wcel wa weu wex wmo wreu wrex wrmo df-eu df-reu df-rex anbi12i 3bitr4i + df-rmo ) BDCEAFZBGSBHZSBIZFABCJABCKZABCLZFSBMABCNUBTUCUAABCOABCRPQ $. + + $( Restricted existential uniqueness implies restricted "at most one." + (Contributed by NM, 16-Jun-2017.) $) + reurmo $p |- ( E! x e. A ph -> E* x e. A ph ) $= + ( wreu wrex wrmo reu5 simprbi ) ABCDABCEABCFABCGH $. + + $( Restricted unique existence implies restricted existence. (Contributed by + NM, 19-Aug-1999.) $) + reurex $p |- ( E! x e. A ph -> E. x e. A ph ) $= + ( wreu wrex wrmo reu5 simplbi ) ABCDABCEABCFABCGH $. + + $( Unrestricted "at most one" implies restricted "at most one". (Contributed + by NM, 16-Jun-2017.) $) + mormo $p |- ( E* x ph -> E* x e. A ph ) $= + ( wmo cv wcel wa wrmo moan df-rmo sylibr ) ABDBECFZAGBDABCHALBIABCJK $. + ${ - rabbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $. - $( Equivalent formulas yield equal restricted class abstractions (inference - form). (Contributed by NM, 22-May-1999.) $) - rabbiia $p |- { x e. A | ph } = { x e. A | ps } $= - ( cv wcel wa cab crab pm5.32i abbii df-rab 3eqtr4i ) CFDGZAHZCIOBHZCIACDJ + rmobiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $. + $( Formula-building rule for restricted existential quantifier (inference + form). (Contributed by NM, 16-Jun-2017.) $) + rmobiia $p |- ( E* x e. A ph <-> E* x e. A ps ) $= + ( cv wcel wa wmo wrmo pm5.32i mobii df-rmo 3bitr4i ) CFDGZAHZCIOBHZCIACDJ + BCDJPQCOABEKLACDMBCDMN $. + + $( Formula-building rule for restricted existential quantifier (inference + form). (Contributed by NM, 14-Nov-2004.) $) + reubiia $p |- ( E! x e. A ph <-> E! x e. A ps ) $= + ( cv wcel wa weu wreu pm5.32i eubii df-reu 3bitr4i ) CFDGZAHZCIOBHZCIACDJ BCDJPQCOABEKLACDMBCDMN $. $} ${ - rabbii.1 $e |- ( ph <-> ps ) $. - $( Equivalent wff's correspond to equal restricted class abstractions. - Inference form of ~ rabbidv . (Contributed by Peter Mazsa, - 1-Nov-2019.) $) - rabbii $p |- { x e. A | ph } = { x e. A | ps } $= - ( wb cv wcel a1i rabbiia ) ABCDABFCGDHEIJ $. + rmobii.1 $e |- ( ph <-> ps ) $. + $( Formula-building rule for restricted existential quantifier (inference + form). (Contributed by NM, 16-Jun-2017.) $) + rmobii $p |- ( E* x e. A ph <-> E* x e. A ps ) $= + ( wb cv wcel a1i rmobiia ) ABCDABFCGDHEIJ $. + + $( Formula-building rule for restricted existential quantifier (inference + form). (Contributed by NM, 22-Oct-1999.) $) + reubii $p |- ( E! x e. A ph <-> E! x e. A ps ) $= + ( wb cv wcel a1i reubiia ) ABCDABFCGDHEIJ $. $} + $( Cancellation law for restricted at-most-one quantification. (Contributed + by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, + 12-Jan-2025.) $) + rmoanid $p |- ( E* x e. A ( x e. A /\ ph ) <-> E* x e. A ph ) $= + ( cv wcel wa ibar bicomd rmobiia ) BDCEZAFZABCJAKJAGHI $. + + $( Cancellation law for restricted unique existential quantification. + (Contributed by Peter Mazsa, 12-Feb-2018.) (Proof shortened by Wolf + Lammen, 12-Jan-2025.) $) + reuanid $p |- ( E! x e. A ( x e. A /\ ph ) <-> E! x e. A ph ) $= + ( cv wcel wa ibar bicomd reubiia ) BDCEZAFZABCJAKJAGHI $. + + $( Obsolete version of ~ rmoanid as of 12-Jan-2025. (Contributed by Peter + Mazsa, 24-May-2018.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rmoanidOLD $p |- ( E* x e. A ( x e. A /\ ph ) <-> E* x e. A ph ) $= + ( cv wcel wa wmo wrmo anabs5 mobii df-rmo 3bitr4i ) BDCEZMAFZFZBGNBGNBCHABC + HONBMAIJNBCKABCKL $. + + $( Obsolete version of ~ reuanid as of 12-Jan-2025. (Contributed by Peter + Mazsa, 12-Feb-2018.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + reuanidOLD $p |- ( E! x e. A ( x e. A /\ ph ) <-> E! x e. A ph ) $= + ( cv wcel wa weu wreu anabs5 eubii df-reu 3bitr4i ) BDCEZMAFZFZBGNBGNBCHABC + HONBMAIJNBCKABCKL $. + + $( Double restricted existential uniqueness, analogous to ~ 2eu2ex . + (Contributed by Alexander van der Vekens, 25-Jun-2017.) $) + 2reu2rex $p |- ( E! x e. A E! y e. B ph -> E. x e. A E. y e. B ph ) $= + ( wreu wrex reurex reximi syl ) ACEFZBDFKBDGACEGZBDGKBDHKLBDACEHIJ $. + + $( *** Theorems using axioms up to ax-5 only: *** $) + ${ - rabbida.n $e |- F/ x ph $. - rabbida.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. - $( Equivalent wff's yield equal restricted class abstractions (deduction - form). Version of ~ rabbidva with disjoint variable condition replaced - by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) $) - rabbida $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $= - ( wb wral crab wceq cv wcel ex ralrimi rabbi sylib ) ABCHZDEIBDEJCDEJKARD - EFADLEMRGNOBCDEPQ $. + $d x ph $. + rmobidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. + $( Formula-building rule for restricted existential quantifier (deduction + form). (Contributed by NM, 16-Jun-2017.) Avoid ~ ax-6 , ~ ax-7 , + ~ ax-12 . (Revised by Wolf Lammen, 23-Nov-2024.) $) + rmobidva $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= + ( cv wcel wa wmo wrmo pm5.32da mobidv df-rmo 3bitr4g ) ADGEHZBIZDJPCIZDJB + DEKCDEKAQRDAPBCFLMBDENCDENO $. + $( $j usage 'rmobidva' avoids 'ax-6' 'ax-7' 'ax-12'; $) + + $( Formula-building rule for restricted existential quantifier (deduction + form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised + by Wolf Lammen, 14-Jan-2023.) $) + reubidva $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $= + ( cv wcel wa weu wreu pm5.32da eubidv df-reu 3bitr4g ) ADGEHZBIZDJPCIZDJB + DEKCDEKAQRDAPBCFLMBDENCDENO $. $} ${ - rabbid.n $e |- F/ x ph $. - rabbid.1 $e |- ( ph -> ( ps <-> ch ) ) $. - $( Version of ~ rabbidv with disjoint variable condition replaced by - nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) $) - rabbid $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $= - ( wb cv wcel adantr rabbida ) ABCDEFABCHDIEJGKL $. + $d x ph $. + rmobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $. + $( Formula-building rule for restricted existential quantifier (deduction + form). (Contributed by NM, 16-Jun-2017.) $) + rmobidv $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= + ( wb cv wcel adantr rmobidva ) ABCDEABCGDHEIFJK $. + + $( Formula-building rule for restricted existential quantifier (deduction + form). (Contributed by NM, 17-Oct-1996.) $) + reubidv $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $= + ( wb cv wcel adantr reubidva ) ABCDEABCGDHEIFJK $. + $} + + ${ + $d ph x $. + reueubd.1 $e |- ( ( ph /\ ps ) -> x e. V ) $. + $( Restricted existential uniqueness is equivalent to existential + uniqueness if the unique element is in the restricting class. + (Contributed by AV, 4-Jan-2021.) $) + reueubd $p |- ( ph -> ( E! x e. V ps <-> E! x ps ) ) $= + ( wreu cv wcel wa weu df-reu ex pm4.71rd eubidv bitr4id ) ABCDFCGDHZBIZCJ + BCJBCDKABQCABPABPELMNO $. + $} + + $( *** Theorems using axioms up to ax-8 only: *** $) + + $( Restricted "at most one" in term of uniqueness. (Contributed by NM, + 16-Jun-2017.) $) + rmo5 $p |- ( E* x e. A ph <-> ( E. x e. A ph -> E! x e. A ph ) ) $= + ( cv wcel wa wmo wex wi wrmo wrex wreu df-rmo df-rex df-reu imbi12i 3bitr4i + weu moeu ) BDCEAFZBGTBHZTBRZIABCJABCKZABCLZITBSABCMUCUAUDUBABCNABCOPQ $. + + $( Nonexistence implies restricted "at most one". (Contributed by NM, + 17-Jun-2017.) $) + nrexrmo $p |- ( -. E. x e. A ph -> E* x e. A ph ) $= + ( wrex wn wreu wi wrmo pm2.21 rmo5 sylibr ) ABCDZELABCFZGABCHLMIABCJK $. + + ${ + $d x y A $. + $( "At most one" element in a set. (Contributed by Thierry Arnoux, + 26-Jul-2018.) Avoid ~ ax-11 . (Revised by Wolf Lammen, + 23-Nov-2024.) $) + moel $p |- ( E* x x e. A <-> A. x e. A A. y e. A x = y ) $= + ( cv wcel wmo wa weq wi wal wral eleq1w mo4 r2al bitr4i ) ADCEZAFPBDCEZGA + BHZIBJAJRBCKACKPQABABCLMRABCCNO $. + $( $j usage 'rmobidva' avoids 'ax-11'; $) + $} + + ${ + $d x y A $. $d y ph $. $d x ps $. + cbvrmovw.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Change the bound variable of a restricted at-most-one quantifier using + implicit substitution. Version of ~ cbvrmov with a disjoint variable + condition, which requires fewer axioms. (Contributed by NM, + 16-Jun-2017.) (Revised by Gino Giotto, 30-Sep-2024.) $) + cbvrmovw $p |- ( E* x e. A ph <-> E* y e. A ps ) $= + ( cv wcel wa wmo wrmo weq eleq1w anbi12d cbvmovw df-rmo 3bitr4i ) CGEHZAI + ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. + $( $j usage 'cbvrmovw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $) + + $( Change the bound variable of a restricted unique existential quantifier + using implicit substitution. Version of ~ cbvreuv with a disjoint + variable condition, which requires fewer axioms. (Contributed by NM, + 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.) $) + cbvreuvw $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( cv wcel wa weu wreu weq eleq1w anbi12d cbveuvw df-reu 3bitr4i ) CGEHZAI + ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. + $( $j usage 'cbvreuvw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $) + $} + + $( *** Theorems using axioms up to ax-12 only: *** $) + + ${ + $d x y A $. + $( Obsolete version of ~ moel as of 23-Nov-2024. (Contributed by Thierry + Arnoux, 26-Jul-2018.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + moelOLD $p |- ( E* x x e. A <-> A. x e. A A. y e. A x = y ) $= + ( cv wcel weq wi wal wmo ralcom4 df-ral ralbii wa alcom eleq1w mo4 impexp + wral albii bitr4i 3bitr4i 3bitr4ri ) BDCEZABFZGZBHZACRUEACRZBHZUDBCRZACRA + DCEZAIZUEABCJUIUFACUDBCKLUJUCMUDGZBHAHULAHZBHUKUHULABNUJUCABABCOPUGUMBUGU + JUEGZAHUMUEACKULUNAUJUCUDQSTSUAUB $. + $} + + ${ + rmobida.1 $e |- F/ x ph $. + rmobida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. + $( Formula-building rule for restricted existential quantifier (deduction + form). (Contributed by NM, 16-Jun-2017.) $) + rmobida $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= + ( cv wcel wa wmo wrmo pm5.32da mobid df-rmo 3bitr4g ) ADHEIZBJZDKQCJZDKBD + ELCDELARSDFAQBCGMNBDEOCDEOP $. + + $( Formula-building rule for restricted existential quantifier (deduction + form). (Contributed by Mario Carneiro, 19-Nov-2016.) $) + reubida $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $= + ( cv wcel wa weu wreu pm5.32da eubid df-reu 3bitr4g ) ADHEIZBJZDKQCJZDKBD + ELCDELARSDFAQBCGMNBDEOCDEOP $. + $} + + ${ + $d x ph $. + rmobidvaOLD.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. + $( Obsolete version of ~ rmobidv as of 23-Nov-2024. (Contributed by NM, + 16-Jun-2017.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rmobidvaOLD $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $= + ( nfv rmobida ) ABCDEADGFH $. + $} + + ${ + $d x y A $. + cbvrmow.1 $e |- F/ y ph $. + cbvrmow.2 $e |- F/ x ps $. + cbvrmow.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Change the bound variable of a restricted at-most-one quantifier using + implicit substitution. Version of ~ cbvrmo with a disjoint variable + condition, which does not require ~ ax-10 , ~ ax-13 . (Contributed by + NM, 16-Jun-2017.) Avoid ~ ax-10 , ~ ax-13 . (Revised by Gino Giotto, + 23-May-2024.) $) + cbvrmow $p |- ( E* x e. A ph <-> E* y e. A ps ) $= + ( cv wcel wa wmo wrmo nfv nfan weq eleq1w anbi12d cbvmow df-rmo 3bitr4i ) + CIEJZAKZCLDIEJZBKZDLACEMBDEMUCUECDUBADUBDNFOUDBCUDCNGOCDPUBUDABCDEQHRSACE + TBDETUA $. + $( $j usage 'cbvrmow' avoids 'ax-10' 'ax-13'; $) + $} + + ${ + $d A x y $. + cbvreuw.1 $e |- F/ y ph $. + cbvreuw.2 $e |- F/ x ps $. + cbvreuw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Change the bound variable of a restricted unique existential quantifier + using implicit substitution. Version of ~ cbvreu with a disjoint + variable condition, which does not require ~ ax-13 . (Contributed by + Mario Carneiro, 15-Oct-2016.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) Avoid ~ ax-10 . (Revised by Wolf Lammen, + 10-Dec-2024.) $) + cbvreuw $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( wrex wrmo wa wreu cbvrexw cbvrmow anbi12i reu5 3bitr4i ) ACEIZACEJZKBDE + IZBDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $. + $( $j usage 'cbvreuw' avoids 'ax-10' 'ax-13'; $) + $} + + $( The setvar ` x ` is not free in ` E* x e. A ph ` . (Contributed by NM, + 16-Jun-2017.) $) + nfrmo1 $p |- F/ x E* x e. A ph $= + ( wrmo cv wcel wa wmo df-rmo nfmo1 nfxfr ) ABCDBECFAGZBHBABCILBJK $. + + $( The setvar ` x ` is not free in ` E! x e. A ph ` . (Contributed by NM, + 19-Mar-1997.) $) + nfreu1 $p |- F/ x E! x e. A ph $= + ( wreu cv wcel wa weu df-reu nfeu1 nfxfr ) ABCDBECFAGZBHBABCILBJK $. + + ${ + $d x y $. + nfrmow.1 $e |- F/_ x A $. + nfrmow.2 $e |- F/ x ph $. + $( Bound-variable hypothesis builder for restricted uniqueness. Version of + ~ nfrmo with a disjoint variable condition, which does not require + ~ ax-13 . (Contributed by NM, 16-Jun-2017.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 10-Jan-2024.) Avoid ~ ax-9 , ~ ax-ext . (Revised by + Wolf Lammen, 21-Nov-2024.) $) + nfrmow $p |- F/ x E* y e. A ph $= + ( wrmo cv wcel wa wmo df-rmo nfcri nfan nfmov nfxfr ) ACDGCHDIZAJZCKBACDL + RBCQABBCDEMFNOP $. + $( $j usage 'nfrmow' avoids 'ax-9' 'ax-13' 'ax-ext'; $) + + $( Bound-variable hypothesis builder for restricted unique existence. + Version of ~ nfreu with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by NM, 30-Oct-2010.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) Avoid ~ ax-9 , ~ ax-ext . + (Revised by Wolf Lammen, 21-Nov-2024.) $) + nfreuw $p |- F/ x E! y e. A ph $= + ( wreu cv wcel wa weu df-reu nfcri nfan nfeuw nfxfr ) ACDGCHDIZAJZCKBACDL + RBCQABBCDEMFNOP $. + $( $j usage 'nfreuw' avoids 'ax-9' 'ax-13' 'ax-ext'; $) + $} + + ${ + $d x y A $. + cbvrmowOLD.1 $e |- F/ y ph $. + cbvrmowOLD.2 $e |- F/ x ps $. + cbvrmowOLD.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Obsolete version of ~ cbvrmow as of 23-May-2024. (Contributed by NM, + 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + cbvrmowOLD $p |- ( E* x e. A ph <-> E* y e. A ps ) $= + ( wrex wreu wi wrmo cbvrexw cbvreuw imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDE + IZBDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $. + $( $j usage 'cbvrmowOLD' avoids 'ax-13'; $) + $} + + ${ + $d A x y z $. $d ph z $. $d ps z $. + cbvreuwOLD.1 $e |- F/ y ph $. + cbvreuwOLD.2 $e |- F/ x ps $. + cbvreuwOLD.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Obsolete version of ~ cbvreuw as of 10-Dec-2024. (Contributed by Mario + Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) + (New usage is discouraged.) (Proof modification is discouraged.) $) + cbvreuwOLD $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( vz cv wcel wa weu wreu wsb nfv sb8euv sban eubii df-reu anbi1i nfan weq + clelsb1 nfsbv eleq1w sbequ sbiev bitrdi anbi12d cbveuw 3bitri 3bitr4i + bitri ) CJEKZALZCMZDJEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZUTUPCIUPI + PQVAVDIUOACIRSVEIJEKZVCLZIMUTVDVGIVBVFVCCIEUDUASVGUSIDVFVCDVFDPACIDFUEUBU + SIPIDUCZVFURVCBIDEUFVHVCACDOBAIDCUGABCDGHUHUIUJUKUNULACETBDETUM $. + $( $j usage 'cbvreuwOLD' avoids 'ax-13'; $) + $} + + ${ + $d x y A $. $d y ph $. $d x ps $. + cbvreuvwOLD.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Obsolete version of ~ cbvreuvw as of 30-Sep-2024. (Contributed by NM, + 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + cbvreuvwOLD $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( nfv cbvreuw ) ABCDEADGBCGFH $. + $( $j usage 'cbvreuvwOLD' avoids 'ax-13'; $) + $} + + $( *** Theorems using axioms up to ax-12 and ax-ext only: *** $) + + ${ + $d x A $. $d x B $. + $( Equality theorem for restricted at-most-one quantifier. (Contributed by + Alexander van der Vekens, 17-Jun-2017.) Remove usage of ~ ax-10 , + ~ ax-11 , and ~ ax-12 . (Revised by Steven Nguyen, 30-Apr-2023.) $) + rmoeq1 $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $= + ( wceq cv wcel wa wmo wrmo eleq2 anbi1d mobidv df-rmo 3bitr4g ) CDEZBFZCG + ZAHZBIQDGZAHZBIABCJABDJPSUABPRTACDQKLMABCNABDNO $. + + $( Equality theorem for restricted unique existential quantifier. + (Contributed by NM, 5-Apr-2004.) Remove usage of ~ ax-10 , ~ ax-11 , + and ~ ax-12 . (Revised by Steven Nguyen, 30-Apr-2023.) $) + reueq1 $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $= + ( wceq cv wcel wa weu wreu eleq2 anbi1d eubidv df-reu 3bitr4g ) CDEZBFZCG + ZAHZBIQDGZAHZBIABCJABDJPSUABPRTACDQKLMABCNABDNO $. + $} + + ${ + $d x A $. $d x B $. + rmoeqd.1 $e |- ( A = B -> ( ph <-> ps ) ) $. + $( Equality deduction for restricted at-most-one quantifier. (Contributed + by Alexander van der Vekens, 17-Jun-2017.) $) + rmoeqd $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) ) $= + ( wceq wrmo rmoeq1 rmobidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $. + + $( Equality deduction for restricted unique existential quantifier. + (Contributed by NM, 5-Apr-2004.) $) + reueqd $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ps ) ) $= + ( wceq wreu reueq1 reubidv bitrd ) DEGZACDHACEHBCEHACDEILABCEFJK $. + $} + + ${ + rmoeq1f.1 $e |- F/_ x A $. + rmoeq1f.2 $e |- F/_ x B $. + $( Equality theorem for restricted at-most-one quantifier, with + bound-variable hypotheses instead of distinct variable restrictions. + (Contributed by Alexander van der Vekens, 17-Jun-2017.) $) + rmoeq1f $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $= + ( wceq cv wcel wa wmo wrmo nfeq eleq2 anbi1d mobid df-rmo 3bitr4g ) CDGZB + HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. + + $( Equality theorem for restricted unique existential quantifier, with + bound-variable hypotheses instead of distinct variable restrictions. + (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, + 11-Jul-2011.) $) + reueq1f $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $= + ( wceq cv wcel wa weu wreu nfeq eleq2 anbi1d eubid df-reu 3bitr4g ) CDGZB + HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. + $} + + ${ + $d x y $. + nfreuwOLD.1 $e |- F/_ x A $. + nfreuwOLD.2 $e |- F/ x ph $. + $( Obsolete version of ~ nfreuw as of 21-Nov-2024. (Contributed by NM, + 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + nfreuwOLD $p |- F/ x E! y e. A ph $= + ( wreu wnf cv wcel wa weu wtru df-reu nftru nfcvd wnfc a1i nfeld nfand + nfeudw nfxfrd mptru ) ACDGZBHUDCIZDJZAKZCLMBACDNMUGBCCOMUFABMBUEDMBUEPBDQ + MERSABHMFRTUAUBUC $. + $( $j usage 'nfreuwOLD' avoids 'ax-13'; $) + + $( Obsolete version of ~ nfrmow as of 21-Nov-2024. (Contributed by NM, + 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + nfrmowOLD $p |- F/ x E* y e. A ph $= + ( wrmo cv wcel wa wmo df-rmo wnf wtru nftru nfcvd wnfc a1i nfeld nfand + nfmodv mptru nfxfr ) ACDGCHZDIZAJZCKZBACDLUGBMNUFBCCONUEABNBUDDNBUDPBDQNE + RSABMNFRTUAUBUC $. + $( $j usage 'nfrmowOLD' avoids 'ax-13'; $) + $} + + $( *** Theorems using axioms ax-13 or axc11r: *** $) + + ${ + $d x z A $. $d y z A $. $d z ph $. $d z ps $. + cbvrmo.1 $e |- F/ y ph $. + cbvrmo.2 $e |- F/ x ps $. + cbvrmo.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Change the bound variable of a restricted unique existential quantifier + using implicit substitution. Usage of this theorem is discouraged + because it depends on ~ ax-13 . Use the weaker ~ cbvreuw when possible. + (Contributed by Mario Carneiro, 15-Oct-2016.) + (New usage is discouraged.) $) + cbvreu $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( vz cv wcel wa weu wreu wsb nfv sb8eu sban eubii df-reu anbi1i nfsb nfan + clelsb1 weq eleq1w sbequ sbie bitrdi anbi12d cbveu bitri 3bitri 3bitr4i ) + CJEKZALZCMZDJEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZUTUPCIUPIPQVAVDIU + OACIRSVEIJEKZVCLZIMUTVDVGIVBVFVCCIEUDUASVGUSIDVFVCDVFDPACIDFUBUCUSIPIDUEZ + VFURVCBIDEUFVHVCACDOBAIDCUGABCDGHUHUIUJUKULUMACETBDETUN $. + + $( Change the bound variable of a restricted at-most-one quantifier using + implicit substitution. Usage of this theorem is discouraged because it + depends on ~ ax-13 . Use the weaker ~ cbvrmow , ~ cbvrmovw when + possible. (Contributed by NM, 16-Jun-2017.) + (New usage is discouraged.) $) + cbvrmo $p |- ( E* x e. A ph <-> E* y e. A ps ) $= + ( wrex wreu wi wrmo cbvrex cbvreu imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDEIZ + BDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $. + $} + + ${ + $d x A $. $d y A $. $d y ph $. $d x ps $. + cbvrmov.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Change the bound variable of a restricted at-most-one quantifier using + implicit substitution. Usage of this theorem is discouraged because it + depends on ~ ax-13 . (Contributed by Alexander van der Vekens, + 17-Jun-2017.) (New usage is discouraged.) $) + cbvrmov $p |- ( E* x e. A ph <-> E* y e. A ps ) $= + ( nfv cbvrmo ) ABCDEADGBCGFH $. + + $( Change the bound variable of a restricted unique existential quantifier + using implicit substitution. See ~ cbvreuvw for a version without + ~ ax-13 , but extra disjoint variables. Usage of this theorem is + discouraged because it depends on ~ ax-13 . Use the weaker ~ cbvreuvw + when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario + Carneiro, 15-Oct-2016.) (New usage is discouraged.) $) + cbvreuv $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( nfv cbvreu ) ABCDEADGBCGFH $. + $} + + ${ + nfrmod.1 $e |- F/ y ph $. + nfrmod.2 $e |- ( ph -> F/_ x A ) $. + nfrmod.3 $e |- ( ph -> F/ x ps ) $. + $( Deduction version of ~ nfrmo . Usage of this theorem is discouraged + because it depends on ~ ax-13 . (Contributed by NM, 17-Jun-2017.) + (New usage is discouraged.) $) + nfrmod $p |- ( ph -> F/ x E* y e. A ps ) $= + ( wrmo cv wcel wa wmo df-rmo weq wal wn wnfc nfcvf adantr nfeld wnf nfand + adantl nfmod2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCPQZLZUHBCUKCUGEUJC + UGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $. + + $( Deduction version of ~ nfreu . Usage of this theorem is discouraged + because it depends on ~ ax-13 . (Contributed by NM, 15-Feb-2013.) + (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) $) + nfreud $p |- ( ph -> F/ x E! y e. A ps ) $= + ( wreu cv wcel wa weu df-reu weq wal wn wnfc nfcvf adantr nfeld wnf nfand + adantl nfeud2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCPQZLZUHBCUKCUGEUJC + UGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $. + $} + + ${ + nfrmo.1 $e |- F/_ x A $. + nfrmo.2 $e |- F/ x ph $. + $( Bound-variable hypothesis builder for restricted uniqueness. Usage of + this theorem is discouraged because it depends on ~ ax-13 . Use the + weaker ~ nfrmow when possible. (Contributed by NM, 16-Jun-2017.) + (New usage is discouraged.) $) + nfrmo $p |- F/ x E* y e. A ph $= + ( wrmo cv wcel wa wmo df-rmo wnf wtru nftru weq wal wn nfcvf a1i adantl + wnfc nfeld nfand nfmod2 mptru nfxfr ) ACDGCHZDIZAJZCKZBACDLUKBMNUJBCCOBCP + BQRZUJBMNULUIABULBUHDBCSBDUBULETUCABMULFTUDUAUEUFUG $. + + $( Bound-variable hypothesis builder for restricted unique existence. + Usage of this theorem is discouraged because it depends on ~ ax-13 . + Use the weaker ~ nfreuw when possible. (Contributed by NM, + 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) + (New usage is discouraged.) $) + nfreu $p |- F/ x E! y e. A ph $= + ( wreu wnf wtru nftru wnfc a1i nfreud mptru ) ACDGBHIABCDCJBDKIELABHIFLMN + $. $} +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Restricted class abstraction +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $( Extend class notation to include the restricted class abstraction (class + builder). $) + crab $a class { x e. A | ph } $. + + $( Define a restricted class abstraction (class builder): ` { x e. A | ph } ` + is the class of all sets ` x ` in ` A ` such that ` ph ( x ) ` is true. + Definition of [TakeutiZaring] p. 20. + + For the interpretation given in the previous paragraph to be correct, we + need to assume ` F/_ x A ` , which is the case as soon as ` x ` and ` A ` + are disjoint, which is generally the case. If ` A ` were to depend on + ` x ` , then the interpretation would be less obvious (think of the two + extreme cases ` A = { x } ` and ` A = x ` , for instance). See also + ~ df-ral . (Contributed by NM, 22-Nov-1994.) $) + df-rab $a |- { x e. A | ph } = { x | ( x e. A /\ ph ) } $. + + $( *** Theorems using axioms up to ax-9 and ax-ext only: *** $) + ${ $d x ph $. rabbidva2.1 $e |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) ) $. @@ -31699,6 +31634,31 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ALQEKBLMIFNOP $. $} + ${ + rabbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $. + $( Equivalent formulas yield equal restricted class abstractions (inference + form). (Contributed by NM, 22-May-1999.) (Proof shortened by Wolf + Lammen, 12-Jan-2025.) $) + rabbiia $p |- { x e. A | ph } = { x e. A | ps } $= + ( cv wcel pm5.32i rabbia2 ) ABCDDCFDGABEHI $. + + $( Obsolete version of ~ rabbiia as of 12-Jan-2025. (Contributed by NM, + 22-May-1999.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rabbiiaOLD $p |- { x e. A | ph } = { x e. A | ps } $= + ( cv wcel wa cab crab pm5.32i abbii df-rab 3eqtr4i ) CFDGZAHZCIOBHZCIACDJ + BCDJPQCOABEKLACDMBCDMN $. + $} + + ${ + rabbii.1 $e |- ( ph <-> ps ) $. + $( Equivalent wff's correspond to equal restricted class abstractions. + Inference form of ~ rabbidv . (Contributed by Peter Mazsa, + 1-Nov-2019.) $) + rabbii $p |- { x e. A | ph } = { x e. A | ps } $= + ( wb cv wcel a1i rabbiia ) ABCDABFCGDHEIJ $. + $} + ${ $d x ph $. rabbidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. @@ -31719,15 +31679,43 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ( wb cv wcel adantr rabbidva ) ABCDEABCGDHEIFJK $. $} + $( Swap with a membership relation in a restricted class abstraction. + (Contributed by NM, 4-Jul-2005.) $) + rabswap $p |- { x e. A | x e. B } = { x e. B | x e. A } $= + ( cv wcel ancom rabbia2 ) ADZCEZHBEZABCJIFG $. + + ${ + $d x y A $. $d y ph $. $d x ps $. + cbvrabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Rule to change the bound variable in a restricted class abstraction, + using implicit substitution. (Contributed by NM, 26-May-1999.) Require + ` x ` , ` y ` be disjoint to avoid ~ ax-11 and ~ ax-13 . (Revised by + Steven Nguyen, 4-Dec-2022.) $) + cbvrabv $p |- { x e. A | ph } = { y e. A | ps } $= + ( cv wcel wa cab crab weq eleq1w anbi12d cbvabv df-rab 3eqtr4i ) CGEHZAIZ + CJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. + $} + ${ - rabeqf.1 $e |- F/_ x A $. - rabeqf.2 $e |- F/_ x B $. - $( Equality theorem for restricted class abstractions, with bound-variable - hypotheses instead of distinct variable restrictions. (Contributed by - NM, 7-Mar-2004.) $) - rabeqf $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $= - ( wceq cv wcel wa cab crab nfeq eleq2 anbi1d abbid df-rab 3eqtr4g ) CDGZB - HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. + $d x A $. $d x ph $. + rabeqcda.1 $e |- ( ( ph /\ x e. A ) -> ps ) $. + $( When ` ps ` is always true in a context, a restricted class abstraction + is equal to the restricting class. Deduction form of ~ rabeqc . + (Contributed by Steven Nguyen, 7-Jun-2023.) $) + rabeqcda $p |- ( ph -> { x e. A | ps } = A ) $= + ( crab cv wcel wa cab df-rab ex pm4.71d abbi2dv eqtr4id ) ABCDFCGDHZBIZCJ + DBCDKAQCDAPBAPBELMNO $. + $} + + ${ + $d A x $. + rabeqc.1 $e |- ( x e. A -> ph ) $. + $( A restricted class abstraction equals the restricting class if its + condition follows from the membership of the free setvar variable in the + restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened + by SN, 15-Jan-2025.) $) + rabeqc $p |- { x e. A | ph } = A $= + ( crab wceq wtru cv wcel adantl rabeqcda mptru ) ABCECFGABCBHCIAGDJKL $. $} ${ @@ -31739,15 +31727,6 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ( cv wcel eleq2i anbi1i rabbia2 ) AABCDBFZCGKDGACDKEHIJ $. $( $j usage 'rabeqi' avoids 'ax-10' 'ax-11' 'ax-12'; $) - $( Obsolete version of ~ rabeqi as of 3-Jun-2024. (Contributed by Glauco - Siliprandi, 26-Jun-2021.) Avoid ~ ax-10 and ~ ax-11 . (Revised by Gino - Giotto, 20-Aug-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - rabeqiOLD $p |- { x e. A | ph } = { x e. B | ph } $= - ( wceq crab cv wcel wa cab nfth eleq2 anbi1d abbid df-rab 3eqtr4g ax-mp ) - CDFZABCGZABDGZFESBHZCIZAJZBKUBDIZAJZBKTUASUDUFBSBELSUCUEACDUBMNOABCPABDPQ - R $. - $( $j usage 'rabeqiOLD' avoids 'ax-10' 'ax-11'; $) $} ${ @@ -31766,6 +31745,16 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ( wceq crab rabeq syl ) ADEGBCDHBCEHGFBCDEIJ $. $} + ${ + $d A x $. $d B x $. $d ph x $. + rabeqbidva.1 $e |- ( ph -> A = B ) $. + rabeqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. + $( Equality of restricted class abstractions. (Contributed by Mario + Carneiro, 26-Jan-2017.) $) + rabeqbidva $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $= + ( crab rabbidva rabeqdv eqtrd ) ABDEICDEICDFIABCDEHJACDEFGKL $. + $} + ${ $d A x $. $d B x $. $d ph x $. rabeqbidv.1 $e |- ( ph -> A = B ) $. @@ -31773,19 +31762,43 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) $) rabeqbidv $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $= - ( crab rabeqdv rabbidv eqtrd ) ABDEIBDFICDFIABDEFGJABCDFHKL $. + ( wb cv wcel adantr rabeqbidva ) ABCDEFGABCIDJEKHLM $. $} ${ - $d A x $. $d B x $. $d ph x $. - rabeqbidva.1 $e |- ( ph -> A = B ) $. - rabeqbidva.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. - $( Equality of restricted class abstractions. (Contributed by Mario - Carneiro, 26-Jan-2017.) $) - rabeqbidva $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $= - ( crab rabbidva rabeqdv eqtrd ) ABDEICDEICDFIABCDEHJACDEFGKL $. + $d x y $. $d A y $. $d ch y $. + rabrabi.1 $e |- ( x = y -> ( ch <-> ph ) ) $. + $( Abstract builder restricted to another restricted abstract builder with + implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid + ~ ax-10 , ~ ax-11 and ~ ax-12 . (Revised by Gino Giotto, + 12-Oct-2024.) $) + rabrabi $p |- { x e. { y e. A | ph } | ps } = { x e. A | ( ch /\ ps ) } $= + ( wa crab cv wcel cab wsb df-rab eleq2i df-clab weq eleq1w wb bicomd + equcoms anbi12d sbievw 3bitri anbi1i anass bitri rabbia2 ) BCBHZDAEFIZFDJ + ZUJKZBHUKFKZCHZBHUMUIHULUNBULUKEJFKZAHZELZKUPEDMUNUJUQUKAEFNOUPDEPUPUNEDE + DQUOUMACEDFRACSDEDEQCAGTUAUBUCUDUEUMCBUFUGUH $. + $( $j usage 'rabrabi' avoids 'ax-10' 'ax-11' 'ax-12'; $) $} + $( *** Theorems using axioms up to ax-9 , ax-10, ax-ext only: *** $) + + $( The abstraction variable in a restricted class abstraction isn't free. + (Contributed by NM, 19-Mar-1997.) $) + nfrab1 $p |- F/_ x { x e. A | ph } $= + ( crab cv wcel wa cab df-rab nfab1 nfcxfr ) BABCDBECFAGZBHABCILBJK $. + + $( *** Theorems using axioms up to ax-9 , ax-12, ax-ext only: *** $) + + $( An "identity" law of concretion for restricted abstraction. Special case + of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) $) + rabid $p |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) ) $= + ( cv wcel wa crab df-rab abeq2i ) BDCEAFBABCGABCHI $. + + $( Membership in a restricted abstraction, implication. (Contributed by + Glauco Siliprandi, 26-Jun-2021.) $) + rabidim1 $p |- ( x e. { x e. A | ph } -> x e. A ) $= + ( cv crab wcel rabid simplbi ) BDZABCEFICFAABCGH $. + ${ rabeq2i.1 $e |- A = { x e. B | ph } $. $( Inference from equality of a class variable and a restricted class @@ -31794,10 +31807,94 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is ( cv wcel crab wa eleq2i rabid bitri ) BFZCGMABDHZGMDGAICNMEJABDKL $. $} - $( Swap with a membership relation in a restricted class abstraction. - (Contributed by NM, 4-Jul-2005.) $) - rabswap $p |- { x e. A | x e. B } = { x e. B | x e. A } $= - ( cv wcel ancom rabbia2 ) ADZCEZHBEZABCJIFG $. + $( Abstract builder restricted to another restricted abstract builder. + (Contributed by Thierry Arnoux, 30-Aug-2017.) $) + rabrab $p |- { x e. { x e. A | ph } | ps } = { x e. A | ( ph /\ ps ) } $= + ( cv crab wcel wa cab rabid anbi1i anass bitri abbii df-rab 3eqtr4i ) CEZAC + DFZGZBHZCIQDGZABHZHZCIBCRFUBCDFTUCCTUAAHZBHUCSUDBACDJKUAABLMNBCROUBCDOP $. + + ${ + $d A x y $. $d ph x $. $d ch y $. + rabrabiOLD.1 $e |- ( x = y -> ( ch <-> ph ) ) $. + $( Obsolete version of ~ rabrabi as of 12-Oct-2024. (Contributed by AV, + 2-Aug-2022.) Avoid ~ ax-10 and ~ ax-11 . (Revised by Gino Giotto, + 20-Aug-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rabrabiOLD $p |- + { x e. { y e. A | ph } | ps } = { x e. A | ( ch /\ ps ) } $= + ( crab wa cbvrabv rabeqi rabrab eqtr3i ) BDCDFHZHBDAEFHZHCBIDFHBDNOCADEFG + JKCBDFLM $. + $( $j usage 'rabrabiOLD' avoids 'ax-10' 'ax-11'; $) + $} + + $( *** Theorems using axioms up to ax-12, ax-ext only: *** $) + + $( Equivalent wff's correspond to equal restricted class abstractions. + Closed theorem form of ~ rabbii . (Contributed by NM, 25-Nov-2013.) $) + rabbi $p |- ( A. x e. A ( ps <-> ch ) + <-> { x e. A | ps } = { x e. A | ch } ) $= + ( cv wcel wa wb wal wceq wral crab abbi wi df-ral pm5.32 albii bitri df-rab + cab eqeq12i 3bitr4i ) CEDFZAGZUCBGZHZCIZUDCTZUECTZJABHZCDKZACDLZBCDLZJUDUEC + MUKUCUJNZCIUGUJCDOUNUFCUCABPQRULUHUMUIACDSBCDSUAUB $. + + ${ + rabbida.n $e |- F/ x ph $. + rabbida.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. + $( Equivalent wff's yield equal restricted class abstractions (deduction + form). Version of ~ rabbidva with disjoint variable condition replaced + by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) $) + rabbida $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $= + ( wb wral crab wceq cv wcel ex ralrimi rabbi sylib ) ABCHZDEIBDEJCDEJKARD + EFADLEMRGNOBCDEPQ $. + $} + + ${ + rabbid.n $e |- F/ x ph $. + rabbid.1 $e |- ( ph -> ( ps <-> ch ) ) $. + $( Version of ~ rabbidv with disjoint variable condition replaced by + nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) $) + rabbid $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $= + ( wb cv wcel adantr rabbida ) ABCDEFABCHDIEJGKL $. + $} + + ${ + rabid2f.1 $e |- F/_ x A $. + $( An "identity" law for restricted class abstraction. (Contributed by NM, + 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised + by Thierry Arnoux, 13-Mar-2017.) $) + rabid2f $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $= + ( cv wcel wa cab wceq wi wal crab wral abeq2f pm4.71 bitr4i df-rab eqeq2i + wb albii df-ral 3bitr4i ) CBECFZAGZBHZIZUCAJZBKZCABCLZIABCMUFUCUDSZBKUHUD + BCDNUGUJBUCAOTPUIUECABCQRABCUAUB $. + $} + + ${ + $d x A $. + $( An "identity" law for restricted class abstraction. (Contributed by NM, + 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof + shortened by Wolf Lammen, 24-Nov-2024.) $) + rabid2 $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $= + ( nfcv rabid2f ) ABCBCDE $. + + $( Obsolete version of ~ rabid2 as of 24-11-2024. (Contributed by NM, + 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + rabid2OLD $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $= + ( cv wcel wa cab wceq wi wal crab abeq2 pm4.71 albii bitr4i df-rab eqeq2i + wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB + CLUFUIBUBAMNOUHUDCABCPQABCTUA $. + $} + + ${ + rabeqf.1 $e |- F/_ x A $. + rabeqf.2 $e |- F/_ x B $. + $( Equality theorem for restricted class abstractions, with bound-variable + hypotheses instead of distinct variable restrictions. (Contributed by + NM, 7-Mar-2004.) $) + rabeqf $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $= + ( wceq cv wcel wa cab crab nfeq eleq2 anbi1d abbid df-rab 3eqtr4g ) CDGZB + HZCIZAJZBKTDIZAJZBKABCLABDLSUBUDBBCDEFMSUAUCACDTNOPABCQABDQR $. + $} ${ $d x y z $. $d A z $. $d ph z $. $d ps z $. @@ -31809,7 +31906,8 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of ~ cbvrab with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.) $) + Andrew Salmon, 11-Jul-2011.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) $) cbvrabw $p |- { x e. A | ph } = { y e. A | ps } $= ( vz cv wcel wa cab crab wsb nfv nfcri nfan eleq1w sbequ12 anbi12d cbvabw nfs1v weq nfsbv sbequ sbiev bitrdi eqtri df-rab 3eqtr4i ) CLEMZANZCOZDLEM @@ -31819,6 +31917,60 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is $( $j usage 'cbvrabw' avoids 'ax-13'; $) $} + ${ + $d x y $. + nfrabw.1 $e |- F/ x ph $. + nfrabw.2 $e |- F/_ x A $. + $( A variable not free in a wff remains so in a restricted class + abstraction. Version of ~ nfrab with a disjoint variable condition, + which does not require ~ ax-13 . (Contributed by NM, 13-Oct-2003.) + Avoid ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) (Proof + shortened by Wolf Lammen, 23-Nov-2024.) $) + nfrabw $p |- F/_ x { y e. A | ph } $= + ( crab cv wcel wa cab df-rab wnfc wtru nftru wnf nfcri nfan a1i nfabdw + mptru nfcxfr ) BACDGCHDIZAJZCKZACDLBUEMNUDBCCOUDBPNUCABBCDFQERSTUAUB $. + $( $j usage 'nfrabw' avoids 'ax-13'; $) + + $( Obsolete version of ~ nfrabw as of 23-Nov_2024. (Contributed by NM, + 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) + (New usage is discouraged.) (Proof modification is discouraged.) $) + nfrabwOLD $p |- F/_ x { y e. A | ph } $= + ( crab cv wcel wa cab df-rab wnfc wtru nftru wnf nfcri a1i nfand nfabdw + mptru nfcxfr ) BACDGCHDIZAJZCKZACDLBUEMNUDBCCONUCABUCBPNBCDFQRABPNERSTUAU + B $. + $( $j usage 'nfrabwOLD' avoids 'ax-13'; $) + $} + + ${ + rabeqiOLD.1 $e |- A = B $. + $( Obsolete version of ~ rabeqi as of 3-Jun-2024. (Contributed by Glauco + Siliprandi, 26-Jun-2021.) Avoid ~ ax-10 and ~ ax-11 . (Revised by Gino + Giotto, 20-Aug-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rabeqiOLD $p |- { x e. A | ph } = { x e. B | ph } $= + ( wceq crab cv wcel wa cab nfth eleq2 anbi1d abbid df-rab 3eqtr4g ax-mp ) + CDFZABCGZABDGZFESBHZCIZAJZBKUBDIZAJZBKTUASUDUFBSBELSUCUEACDUBMNOABCPABDPQ + R $. + $( $j usage 'rabeqiOLD' avoids 'ax-10' 'ax-11'; $) + $} + + $( *** Theorems using axioms ax-13 or axc11r: *** $) + + ${ + $d x z $. $d y z $. $d z A $. + nfrab.1 $e |- F/ x ph $. + nfrab.2 $e |- F/_ x A $. + $( A variable not free in a wff remains so in a restricted class + abstraction. Usage of this theorem is discouraged because it depends on + ~ ax-13 . Use the weaker ~ nfrabw when possible. (Contributed by NM, + 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) + (New usage is discouraged.) $) + nfrab $p |- F/_ x { y e. A | ph } $= + ( vz crab cv wcel wa cab df-rab wnfc wtru nftru weq wal wn wnf eleq1w a1i + nfcri dvelimnf nfand adantl nfabd2 mptru nfcxfr ) BACDHCIDJZAKZCLZACDMBUL + NOUKBCCPBCQBRSZUKBTOUMUJABGIDJUJBCGBGDFUCGCDUAUDABTUMEUBUEUFUGUHUI $. + $} + ${ $d x z $. $d y z $. $d A z $. $d ph z $. $d ps z $. cbvrab.1 $e |- F/_ x A $. @@ -31840,58 +31992,6 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is JUIUJUFUGUKACEULBDEULUM $. $} - ${ - $d x y A $. $d y ph $. $d x ps $. - cbvrabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Rule to change the bound variable in a restricted class abstraction, - using implicit substitution. (Contributed by NM, 26-May-1999.) Require - ` x ` , ` y ` be disjoint to avoid ~ ax-11 and ~ ax-13 . (Revised by - Steven Nguyen, 4-Dec-2022.) $) - cbvrabv $p |- { x e. A | ph } = { y e. A | ps } $= - ( cv wcel wa cab crab weq eleq1w anbi12d cbvabv df-rab 3eqtr4i ) CGEHZAIZ - CJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $. - $} - - ${ - $d x y $. $d A y $. $d ch y $. - rabrabi.1 $e |- ( x = y -> ( ch <-> ph ) ) $. - $( Abstract builder restricted to another restricted abstract builder with - implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid - ~ ax-10 , ~ ax-11 and ~ ax-12 . (Revised by Gino Giotto, - 12-Oct-2024.) $) - rabrabi $p |- { x e. { y e. A | ph } | ps } = { x e. A | ( ch /\ ps ) } $= - ( wa crab cv wcel cab df-rab eleq2i wsb df-clab weq eleq1w wb bitri anass - bicomd equcoms anbi12d sbievw anbi1i rabbia2 ) BCBHZDAEFIZFDJZUIKZBHUJFKZ - CHZBHULUHHUKUMBUKUJEJFKZAHZELZKZUMUIUPUJAEFMNUQUOEDOUMUODEPUOUMEDEDQUNULA - CEDFRACSDEDEQCAGUBUCUDUETTUFULCBUATUG $. - $( $j usage 'rabrabi' avoids 'ax-10' 'ax-11' 'ax-12'; $) - $} - - ${ - $d A x y $. $d ph x $. $d ch y $. - rabrabiOLD.1 $e |- ( x = y -> ( ch <-> ph ) ) $. - $( Obsolete version of ~ rabrabi as of 12-Oct-2024. (Contributed by AV, - 2-Aug-2022.) Avoid ~ ax-10 and ~ ax-11 . (Revised by Gino Giotto, - 20-Aug-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - rabrabiOLD $p |- - { x e. { y e. A | ph } | ps } = { x e. A | ( ch /\ ps ) } $= - ( crab wa cbvrabv rabeqi rabrab eqtr3i ) BDCDFHZHBDAEFHZHCBIDFHBDNOCADEFG - JKCBDFLM $. - $( $j usage 'rabrabiOLD' avoids 'ax-10' 'ax-11'; $) - $} - - ${ - $d x A $. $d x ph $. - rabeqcda.1 $e |- ( ( ph /\ x e. A ) -> ps ) $. - $( When ` ps ` is always true in a context, a restricted class abstraction - is equal to the restricting class. Deduction form of ~ rabeqc . - (Contributed by Steven Nguyen, 7-Jun-2023.) $) - rabeqcda $p |- ( ph -> { x e. A | ps } = A ) $= - ( crab cv wcel wa cab df-rab ex pm4.71d bicomd abbi1dv eqtrid ) ABCDFCGDH - ZBIZCJDBCDKARCDAQRAQBAQBELMNOP $. - $} - $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -33738,6 +33838,13 @@ general is seen either by substitution (when the variable ` V ` has no ( wcel cv wceq wa wrex r19.42v eleq1 adantr pm5.32ri bicomi baib ceqsrexv wb rexbiia pm5.32i 3bitr3i ) DEGZCHZDIZAJZJZCEKUCUFCEKZJUHUCBJUCUFCELUGUF CEUGUDEGZUFUIUFJUGUFUIUCUEUIUCSAUDDEMNOPQTUCUHBABCDEFRUAUB $. + + $( Elimination of a restricted universal quantifier, using implicit + substitution. (Contributed by Scott Fenton, 7-Dec-2020.) $) + ceqsralbv $p |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) $= + ( cv wceq wi wral wcel wn wa wrex notbid ceqsrexbv rexanali annim 3bitr3i + con4bii ) CGDHZAICEJZDEKZBIZUAALZMCENUCBLZMUBLUDLUEUFCDEUAABFOPUAACEQUCBR + ST $. $} ${ @@ -34127,11 +34234,11 @@ general is seen either by substitution (when the variable ` V ` has no ${ $d A x $. - rabeqc.1 $e |- ( x e. A -> ph ) $. - $( A restricted class abstraction equals the restricting class if its - condition follows from the membership of the free setvar variable in the - restricting class. (Contributed by AV, 20-Apr-2022.) $) - rabeqc $p |- { x e. A | ph } = A $= + rabeqcOLD.1 $e |- ( x e. A -> ph ) $. + $( Obsolete version of ~ rabeqc as of 15-Jan-2025. (Contributed by AV, + 20-Apr-2022.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rabeqcOLD $p |- { x e. A | ph } = A $= ( crab cv wcel wa cab df-rab wceq wb abeq1 pm4.71i bicomi mpgbir eqtri ) ABCEBFCGZAHZBIZCABCJTCKSRLBSBCMRSRADNOPQ $. $} @@ -34273,6 +34380,17 @@ general is seen either by substitution (when the variable ` V ` has no UHUDUILZENUJUGUKEUDACRSUIEFTUAUB $. $} + ${ + $d A y $. $d ph y $. $d x y $. + reurab.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Restricted existential uniqueness of a restricted abstraction. + (Contributed by Scott Fenton, 8-Aug-2024.) $) + reurab $p |- ( E! x e. { y e. A | ps } ch <-> E! x e. A ( ph /\ ch ) ) $= + ( cv crab wcel wa weu wreu wb weq bicomd equcoms elrab anbi1i df-reu + anass bitri eubii 3bitr4i ) DHZBEFIZJZCKZDLUEFJZACKZKZDLCDUFMUJDFMUHUKDUH + UIAKZCKUKUGULCBAEUEFBANDEDEOABGPQRSUIACUAUBUCCDUFTUJDFTUD $. + $} + ${ $d x z $. $d A z $. $( Identity used to create closed-form versions of bound-variable @@ -34485,8 +34603,8 @@ general is seen either by substitution (when the variable ` V ` has no $( Transfer existential uniqueness from a variable ` x ` to another variable ` y ` contained in expression ` A ` . Version of ~ euxfr2 with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + (Contributed by NM, 14-Nov-2004.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 10-Jan-2024.) $) euxfr2w $p |- ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) $= ( cv wceq wa wex weu wmo wi 2euswapv moani ancom mobii mpbi mpg moeq impbii biidd ceqsexv eubii bitri ) BGDHZAIZCJBKZUGBJZCKZACKUHUJUGCLZUHUJM @@ -34503,8 +34621,8 @@ general is seen either by substitution (when the variable ` V ` has no $( Transfer existential uniqueness from a variable ` x ` to another variable ` y ` contained in expression ` A ` . Version of ~ euxfr with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + (Contributed by NM, 14-Nov-2004.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 10-Jan-2024.) $) euxfrw $p |- ( E! x ph <-> E! y ps ) $= ( weu cv wceq wa wex euex ax-mp biantrur 19.41v pm5.32i exbii 3bitr2i eubii eumoi euxfr2w bitri ) ACICJEKZBLZDMZCIBDIAUGCAUEDMZALUEALZDMUGUHAUE @@ -35042,7 +35160,7 @@ Conditional equality (experimental) first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the ` CondEq ` version is parsed as - (wcdeq vx vy wph). It also allows us to give a name to the specific ternary + (wcdeq vx vy wph). It also allows to give a name to the specific ternary operation ` ( x = y -> ph ) ` . This is all used as part of a metatheorem: we want to say that @@ -35557,7 +35675,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use nfsbcdw.3 $e |- ( ph -> F/ x ps ) $. $( Deduction version of ~ nfsbcw . Version of ~ nfsbcd with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) $) + NM, 23-Nov-2005.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) $) nfsbcdw $p |- ( ph -> F/ x [. A / y ]. ps ) $= ( wsbc cab wcel df-sbc nfabdw nfeld nfxfrd ) BDEIEBDJZKACBDELACEPGABCDFHM NO $. @@ -35570,8 +35689,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use nfsbcw.2 $e |- F/ x ph $. $( Bound-variable hypothesis builder for class substitution. Version of ~ nfsbc with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by NM, 7-Sep-2014.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + ~ ax-13 . (Contributed by NM, 7-Sep-2014.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 10-Jan-2024.) $) nfsbcw $p |- F/ x [. A / y ]. ph $= ( wsbc wnf wtru nftru wnfc a1i nfsbcdw mptru ) ACDGBHIABCDCJBDKIELABHIFLM N $. @@ -35582,7 +35701,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $d x z $. $d z A $. $d y z ph $. $d x y $. $( A composition law for class substitution. Version of ~ sbcco with a disjoint variable condition, which requires fewer axioms. (Contributed - by NM, 26-Sep-2003.) (Revised by Gino Giotto, 10-Jan-2024.) $) + by NM, 26-Sep-2003.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 10-Jan-2024.) $) sbccow $p |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) $= ( vz cv wsbc cvv wcel sbcex dfsbcq wsb sbsbc sbbii sbco2vv 3bitr3ri bitri vtoclbg pm5.21nii ) ABCFGZCDGZDHIABDGZTCDJABDJTCEFZGZABUCGZUAUBEDHTCUCDKA @@ -35703,8 +35823,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use cbvsbcw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. $( Change bound variables in a wff substitution. Version of ~ cbvsbc with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by Jeff Hankins, 19-Sep-2009.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + (Contributed by Jeff Hankins, 19-Sep-2009.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 10-Jan-2024.) $) cbvsbcw $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $= ( cab wcel wsbc cbvabw eleq2i df-sbc 3bitr4i ) EACIZJEBDIZJACEKBDEKPQEABC DFGHLMACENBDENO $. @@ -35717,7 +35837,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Change the bound variable of a class substitution using implicit substitution. Version of ~ cbvsbcv with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by NM, 30-Sep-2008.) - (Revised by Gino Giotto, 10-Jan-2024.) $) + Avoid ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) $) cbvsbcvw $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $= ( cab wcel wsbc cbvabv eleq2i df-sbc 3bitr4i ) EACGZHEBDGZHACEIBDEINOEABC DFJKACELBDELM $. @@ -36675,8 +36795,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on ` A ` . Version of ~ cbvcsb with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Gino - Giotto, 10-Jan-2024.) $) + ~ ax-13 . (Contributed by Jeff Hankins, 13-Sep-2009.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 10-Jan-2024.) $) cbvcsbw $p |- [_ A / x ]_ C = [_ A / y ]_ D $= ( vz cv wcel wsbc cab csb nfcri weq eleq2d cbvsbcw abbii df-csb 3eqtr4i ) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE @@ -36729,8 +36849,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $d z A $. $d x y z $. $d y z B $. $( Composition law for chained substitutions into a class. Version of ~ csbco with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, - 10-Jan-2024.) $) + ~ ax-13 . (Contributed by NM, 10-Nov-2005.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 10-Jan-2024.) $) csbcow $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $= ( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbccow bitri abbii 3eqtr4i ) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM @@ -36858,8 +36978,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use nfcsbw.2 $e |- F/_ x B $. $( Bound-variable hypothesis builder for substitution into a class. Version of ~ nfcsb with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by Mario Carneiro, 12-Oct-2016.) - (Revised by Gino Giotto, 10-Jan-2024.) $) + require ~ ax-13 . (Contributed by Mario Carneiro, 12-Oct-2016.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 10-Jan-2024.) $) nfcsbw $p |- F/_ x [_ A / y ]_ B $= ( vz csb wnfc wtru cv wcel wsbc cab df-csb nftru a1i nfcrd nfsbcdw nfabdw nfcxfrd mptru ) ABCDHZIJAUCGKDLZBCMZGNBGCDOJUEAGGPJUDABCBPACIJEQJAGDADIJF @@ -36969,7 +37089,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $} ${ - $d x y A $. $d x y z C $. $d B y z $. + $d x y A $. $d x y C $. $d B y $. csbie.1 $e |- A e. _V $. csbie.2 $e |- ( x = A -> B = C ) $. $( Conversion of implicit substitution to explicit substitution into a @@ -38330,6 +38450,16 @@ technically classes despite morally (and provably) being sets, like ` 1 ` ( crab cv wcel wa cab df-rab ssab2 eqsstri ) ABCDBECFAGBHCABCIABCJK $. $} + ${ + $d x A $. $d x B $. $d x ph $. + rabss3d.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> x e. B ) $. + $( Subclass law for restricted abstraction. (Contributed by Thierry + Arnoux, 25-Sep-2017.) $) + rabss3d $p |- ( ph -> { x e. A | ps } C_ { x e. B | ps } ) $= + ( crab nfv nfrab1 cv wcel wa simprr jca ex rabid 3imtr4g ssrd ) ACBCDGZBC + EGZACHBCDIBCEIACJZDKZBLZUAEKZBLZUASKUATKAUCUEAUCLUDBFAUBBMNOBCDPBCEPQR $. + $} + ${ $d A x $. ssrab3.1 $e |- B = { x e. A | ph } $. @@ -38856,11 +38986,16 @@ technically classes despite morally (and provably) being sets, like ` 1 ` UCACLMDBCNOP $. $} - $( A member of a union that is not member of the first class, is member of - the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) + $( A member of a union that is not a member of the first class, is a member + of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) elunnel1 $p |- ( ( A e. ( B u. C ) /\ -. A e. B ) -> A e. C ) $= ( cun wcel wo elun biimpi orcanai ) ABCDEZABEZACEZJKLFABCGHI $. + $( A member of a union that is not a member of the second class, is a member + of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) + elunnel2 $p |- ( ( A e. ( B u. C ) /\ -. A e. C ) -> A e. B ) $= + ( cun wcel wo elun biimpi orcomd orcanai ) ABCDEZACEZABEZKMLKMLFABCGHIJ $. + ${ $d x A $. $d x B $. $d x C $. uneqri.1 $e |- ( ( x e. A \/ x e. B ) <-> x e. C ) $. @@ -40167,6 +40302,7 @@ technically classes despite morally (and provably) being sets, like ` 1 ` UOAUJUTUOVAJUJNUFTUG $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The empty set @@ -40589,7 +40725,7 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets, ( cdif c0 difid difeq2i difdif eqtr3i ) AAABZBACBAHCAADEAAFG $. ${ - $d x y $. $d y A $. $d ph y $. $d ps x $. + $d x y $. $d ph y $. $d ps x $. ab0w.1 $e |- ( x = y -> ( ph <-> ps ) ) $. $( The class of sets verifying a property is the empty class if and only if that property is a contradiction. Version of ~ ab0 using implicit @@ -40988,8 +41124,8 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets, $d x y z $. $d z A $. $d z B $. $d ph z $. $( Nest the composition of two substitutions. Version of ~ sbcnestgf with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by Mario Carneiro, 11-Nov-2016.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + (Contributed by Mario Carneiro, 11-Nov-2016.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 26-Jan-2024.) $) sbcnestgfw $p |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $= ( vz wcel wnf wal wsbc csb wb cv wi wceq dfsbcq sbceq1d cvv a1i vex nfnf1 @@ -41005,8 +41141,8 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets, $d x y z $. $d z A $. $d z B $. $d z C $. $( Nest the composition of two substitutions. Version of ~ csbnestgf with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + (Contributed by NM, 23-Nov-2005.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 26-Jan-2024.) $) csbnestgfw $p |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $= ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb abeq2i wb nfcr @@ -41021,8 +41157,8 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets, $d x y $. $d x ph $. $( Nest the composition of two substitutions. Version of ~ sbcnestg with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by NM, 27-Nov-2005.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + (Contributed by NM, 27-Nov-2005.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 26-Jan-2024.) $) sbcnestgw $p |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $= ( wcel wnf wal wsbc csb wb nfv ax-gen sbcnestgfw mpan2 ) DFGABHZCIACEJBDJ @@ -41034,8 +41170,8 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets, $d x y $. $d x C $. $( Nest the composition of two substitutions. Version of ~ csbnestg with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + (Contributed by NM, 23-Nov-2005.) Avoid ~ ax-13 . (Revised by Gino + Giotto, 26-Jan-2024.) $) csbnestgw $p |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $= ( wcel wnfc wal csb wceq nfcv ax-gen csbnestgfw mpan2 ) CFGAEHZBIACBDEJJB @@ -41048,7 +41184,8 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets, sbcco3gw.1 $e |- ( x = A -> B = C ) $. $( Composition of two substitutions. Version of ~ sbcco3g with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - NM, 27-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.) $) + NM, 27-Nov-2005.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 26-Jan-2024.) $) sbcco3gw $p |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) ) $= ( wcel wsbc csb sbcnestgw cvv wceq wb elex nfcvd csbiegf dfsbcq 3syl @@ -41584,6 +41721,11 @@ disjoint classes (see ~ disjdif ). Part of proof of Corollary 6K of ( wss cun wceq cdif ssequn1 undif2 eqeq1i bitr4i ) ABCABDZBEABAFDZBEABGLKBA BHIJ $. + $( An equality involving class union and class difference. (Contributed by + Thierry Arnoux, 26-Jun-2024.) $) + undif5 $p |- ( ( A i^i B ) = (/) -> ( ( A u. B ) \ B ) = A ) $= + ( cin c0 wceq cun cdif difun2 disjdif2 eqtrid ) ABCDEABFBGABGAABHABIJ $. + $( A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) $) @@ -42019,6 +42161,7 @@ disjoint classes (see ~ disjdif ). Part of proof of Corollary 6K of ZSUMUJVBSJVASUDUEVAUKSULSABCTABDTLABUITUFUG $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The conditional operator for classes @@ -43254,7 +43397,7 @@ will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ). $} ${ - $d x y A $. $d x y B $. $d x y C $. + $d x A $. $d x B $. $d x C $. $( A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.) $) @@ -44791,7 +44934,7 @@ will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ). $} ${ - $d A w x y $. $d A w x z $. + $d A x y $. $d A x z $. $( A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.) $) issn $p |- ( E. x e. A A. y e. A x = y -> E. z A = { z } ) $= @@ -45232,24 +45375,21 @@ will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ). ( cvv wcel cop csn cpr wceq dfopg mp2an ) AEFBEFABGAHABIIJCDABEEKL $. $} - ${ - $d x A $. $d x B $. $d x C $. - $( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) - (Revised by Mario Carneiro, 26-Apr-2015.) $) - opeq1 $p |- ( A = B -> <. A , C >. = <. B , C >. ) $= - ( wceq cvv wcel wa csn cpr c0 cif cop eleq1 anbi1d preq1 preq12d ifbieq1d - sneq dfopif 3eqtr4g ) ABDZAEFZCEFZGZAHZACIZIZJKBEFZUCGZBHZBCIZIZJKACLBCLU - AUDUIUGULJUAUBUHUCABEMNUAUEUJUFUKABRABCOPQACSBCST $. - $( $j usage 'opeq1' avoids 'ax-10' 'ax-11' 'ax-12'; $) + $( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) + (Revised by Mario Carneiro, 26-Apr-2015.) $) + opeq1 $p |- ( A = B -> <. A , C >. = <. B , C >. ) $= + ( wceq cvv wcel wa csn cpr cif cop eleq1 anbi1d sneq preq1 preq12d ifbieq1d + c0 dfopif 3eqtr4g ) ABDZAEFZCEFZGZAHZACIZIZRJBEFZUCGZBHZBCIZIZRJACKBCKUAUDU + IUGULRUAUBUHUCABELMUAUEUJUFUKABNABCOPQACSBCST $. + $( $j usage 'opeq1' avoids 'ax-10' 'ax-11' 'ax-12'; $) - $( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) - (Revised by Mario Carneiro, 26-Apr-2015.) $) - opeq2 $p |- ( A = B -> <. C , A >. = <. C , B >. ) $= - ( wceq cvv wcel wa csn cpr c0 cif cop eleq1 anbi2d preq2d ifbieq1d dfopif - preq2 3eqtr4g ) ABDZCEFZAEFZGZCHZCAIZIZJKUABEFZGZUDCBIZIZJKCALCBLTUCUHUFU - JJTUBUGUAABEMNTUEUIUDABCROPCAQCBQS $. - $( $j usage 'opeq2' avoids 'ax-10' 'ax-11' 'ax-12'; $) - $} + $( Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) + (Revised by Mario Carneiro, 26-Apr-2015.) $) + opeq2 $p |- ( A = B -> <. C , A >. = <. C , B >. ) $= + ( wceq cvv wcel wa csn cpr c0 cif eleq1 anbi2d preq2 preq2d ifbieq1d dfopif + cop 3eqtr4g ) ABDZCEFZAEFZGZCHZCAIZIZJKUABEFZGZUDCBIZIZJKCARCBRTUCUHUFUJJTU + BUGUAABELMTUEUIUDABCNOPCAQCBQS $. + $( $j usage 'opeq2' avoids 'ax-10' 'ax-11' 'ax-12'; $) $( Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) $) opeq12 $p |- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. ) $= @@ -45602,8 +45742,8 @@ there is another (different) element of the class ` V ` which is an nfunid.3 $e |- ( ph -> F/_ x A ) $. $( Deduction version of ~ nfuni . (Contributed by NM, 18-Feb-2013.) $) nfunid $p |- ( ph -> F/_ x U. A ) $= - ( vy vz cuni wel wrex cab dfuni2 nfv nfvd nfrexd nfabdw nfcxfrd ) ABCGEFH - ZFCIZEJEFCKARBEAELAQBFCAFLDAQBMNOP $. + ( vy vz cuni wel wrex cab dfuni2 nfv nfvd nfrexdw nfabdw nfcxfrd ) ABCGEF + HZFCIZEJEFCKARBEAELAQBFCAFLDAQBMNOP $. $} ${ @@ -46043,12 +46183,31 @@ there is another (different) element of the class ` V ` which is an BJPADOAEBCOAKLMN $. $} + + ${ + $d y ph $. $d x y A $. + $( Two ways of saying a set is an element of the intersection of a class. + (Contributed by NM, 30-Aug-1993.) Put in closed form. (Revised by RP, + 13-Aug-2020.) $) + elintabg $p |- + ( A e. V -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) ) $= + ( vy wcel cab cint cv wral wi wal elintg eleq2w ralab2 bitrdi ) CDFCABGZH + FCEIFZEQJACBIFZKBLECQDMARSEBEBCNOP $. + $} + ${ $d A x y $. $d ph y $. - inteqab.1 $e |- A e. _V $. + elintab.ex $e |- A e. _V $. $( Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.) $) elintab $p |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) $= + ( cvv wcel cab cint cv wi wal wb elintabg ax-mp ) CEFCABGHFACBIFJBKLDABCE + MN $. + + $( Obsolete version of ~ elintab as of 17-Jan-2025. (Contributed by NM, + 30-Aug-1993.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + elintabOLD $p |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) $= ( vy cab cint wcel cv wi wal elint nfsab1 nfv nfim weq eleq1w abid bitrdi eleq2w imbi12d cbvalv1 bitri ) CABFZGHEIZUDHZCUEHZJZEKACBIZHZJZBKECUDDLUH UKEBUFUGBABEMUGBNOUKENEBPZUFAUGUJULUFUIUDHAEBUDQABRSEBCTUAUBUC $. @@ -46659,8 +46818,8 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and ~ ax-13 . See ~ nfiung for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) $) nfiun $p |- F/_ y U_ x e. A B $= - ( vz ciun cv wcel wrex cab df-iun nfcri nfrex nfab nfcxfr ) BACDHGIDJZACK - ZGLAGCDMSBGRBACEBGDFNOPQ $. + ( vz ciun cv wcel wrex cab df-iun nfcri nfrexw nfab nfcxfr ) BACDHGIDJZAC + KZGLAGCDMSBGRBACEBGDFNOPQ $. $( $j usage 'nfiun' avoids 'ax-13'; $) $( Bound-variable hypothesis builder for indexed intersection. @@ -46684,8 +46843,8 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and ~ ax-13 . (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.) $) nfiung $p |- F/_ y U_ x e. A B $= - ( vz ciun cv wcel wrex cab df-iun nfcri nfrexg nfabg nfcxfr ) BACDHGIDJZA - CKZGLAGCDMSBGRBACEBGDFNOPQ $. + ( vz ciun cv wcel wrex cab df-iun nfcri nfrex nfabg nfcxfr ) BACDHGIDJZAC + KZGLAGCDMSBGRBACEBGDFNOPQ $. $( Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ~ ax-13 . See ~ nfiin @@ -47043,8 +47202,16 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and JACBMKL $. $( An indexed union of singletons recovers the index set. (Contributed by - NM, 6-Sep-2005.) $) + NM, 6-Sep-2005.) (Proof shortened by SN, 15-Jan-2025.) $) iunid $p |- U_ x e. A { x } = A $= + ( vy cv csn ciun wcel wrex df-iun clel5 velsn rexbii bitr4i abbi2i eqtr4i + cab weq ) ABADZEZFCDZSGZABHZCPBACBSIUBCBTBGCAQZABHUBABTJUAUCABCRKLMNO $. + $( $j usage 'iunid' avoids 'ax-10' 'ax-11' 'ax-12'; $) + + $( Obsolete version of ~ iunid as of 15-Jan-2025. (Contributed by NM, + 6-Sep-2005.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + iunidOLD $p |- U_ x e. A { x } = A $= ( vy cv csn ciun wceq cab wcel df-sn equcom abbii eqtri a1i iuneq2i iunab wrex risset abid2 3eqtr2i ) ABADZEZFABUACDZGZCHZFZBABUBUEUBUEGUABIUBUCUAG ZCHUECUAJUGUDCCAKLMNOUFUDABQZCHUCBIZCHBUDACBPUIUHCAUCBRLCBSTM $. @@ -47622,8 +47789,8 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and nfdisjw.2 $e |- F/_ y B $. $( Bound-variable hypothesis builder for disjoint collection. Version of ~ nfdisj with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by - Gino Giotto, 26-Jan-2024.) $) + ~ ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 26-Jan-2024.) $) nfdisjw $p |- F/ y Disj_ x e. A B $= ( vz wdisj cv wcel wa wmo wal dfdisj2 wnf wtru nftru nfcvd wnfc a1i nfeld nfcri nfand nfmodv mptru nfal nfxfr ) ACDHAIZCJZGIDJZKZALZGMBAGCDNULBGULB @@ -50293,10 +50460,9 @@ holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] ${ $d w x y z $. $( Alternate proof of ~ dtru using ~ ax-pow instead of ~ ax-pr . - (Contributed by NM, 7-Nov-2006.) Avoid ~ ax-13 . (Revised by Gino - Giotto, 5-Sep-2023.) Avoid ~ ax-12 . (Revised by Rohan Ridenour, - 9-Oct-2024.) (Proof modification is discouraged.) - (New usage is discouraged.) $) + (Contributed by NM, 7-Nov-2006.) Avoid ~ ax-13 . (Revised by BJ, + 31-May-2019.) Avoid ~ ax-12 . (Revised by Rohan Ridenour, 9-Oct-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) dtruALT2 $p |- -. A. x x = y $= ( vw vz cv wceq wn wex wal wel wcel elALT2 ax-nul elequ1 notbid spw ax7v1 wa con3d spimevw eximii exdistrv mpbir2an ax9v2 com12 con3dimp 2eximi weq @@ -50305,7 +50471,7 @@ holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] VIFZVEVLACDNOPUAVCVFCDUBUCVGVKCDVCVJVEVJVCVECDAUDUEUFUGVKVBCDDBUHZVKVBUIV NVKVIUSFZGZVBVNVJVODBCUJOVPVAACVMUTVOACBQSTUKVNGZVBVKVQVAADURVDFUTVNADBQS TULUMUNUOUTAUPUQ $. - $( $j usage 'dtruALT2' avoids 'ax-10 'ax-11' 'ax-12' 'ax-13' 'ax-ext' + $( $j usage 'dtruALT2' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-ext' 'ax-sep' 'ax-pr'; $) $} @@ -50382,8 +50548,8 @@ That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with Note that ~ dvdemo2 is partially bundled, in that the pairs of setvar variables ` x , y ` and ` y , z ` need not be disjoint, and in spite of that, its proof does not require any of the auxiliary axioms ~ ax-10 , - ~ ax-11 , ~ ax-12 , ~ ax-13 . (Contributed by NM, 1-Dec-2006.) - (Revised by BJ, 13-Jan-2024.) $) + ~ ax-11 , ~ ax-12 , ~ ax-13 . (Contributed by NM, 1-Dec-2006.) Avoid + ~ ax-13 . (Revised by BJ, 13-Jan-2024.) $) dvdemo2 $p |- E. x ( x = y -> z e. x ) $= ( wel weq wi elALT2 ax-1 eximii ) CADZABEZJFACAGJKHI $. $} @@ -51073,18 +51239,30 @@ That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with $. $} + $( A singleton built on a setvar is a set. (Contributed by BJ, + 15-Jan-2025.) $) + vsnex $p |- { x } e. _V $= + ( cv csn cpr cvv dfsn2 zfpair2 eqeltri ) ABZCIIDEIFAAGH $. + ${ $d x A $. - $( A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using - Extensionality, Separation, Null Set, and Pairing. See also ~ snexALT . - (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, - 19-May-2013.) (Proof modification is discouraged.) $) - snex $p |- { A } e. _V $= - ( vx cvv wcel csn cpr dfsn2 cv wceq preq12 anidms eleq1d zfpair2 eqeltrid - vtoclg wn c0 snprc biimpi 0ex eqeltrdi pm2.61i ) ACDZAEZCDUCUDAAFZCAGBHZU - FFZCDUECDBACUFAIZUGUECUHUGUEIUFUFAAJKLBBMONUCPZUDQCUIUDQIARSTUAUB $. + $( A singleton built on a set is a set. Special case of ~ snex which does + not require ~ ax-nul and is intuitionistically valid. (Contributed by + NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract + from ~ snex . (Revised by BJ, 15-Jan-2025.) $) + snexg $p |- ( A e. V -> { A } e. _V ) $= + ( vx cv csn cvv wcel wceq sneq eleq1d vsnex vtoclg ) CDZEZFGAEZFGCABMAHNO + FMAIJCKL $. $} + $( A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using + Extensionality, Separation, Null Set, and Pairing. See also ~ snexALT . + (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, + 19-May-2013.) $) + snex $p |- { A } e. _V $= + ( cvv wcel csn snexg wn c0 wceq snprc biimpi 0ex eqeltrdi pm2.61i ) ABCZADZ + BCABENFZOGBPOGHAIJKLM $. + ${ $d x A $. $d x y B $. $( The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] @@ -51100,21 +51278,83 @@ That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with $} ${ - $d x A $. - $( If a class is a set, then it is a member of a set. (Contributed by BJ, - 3-Apr-2019.) $) - sels $p |- ( A e. V -> E. x A e. x ) $= - ( wcel csn cv wex snidg snex eleq2 spcev syl ) BCDBBEZDZBAFZDZAGBCHPNAMBI - OMBJKL $. + $d x y z $. + $( There exist two sets, one a member of the other. + + This theorem looks similar to ~ el , but its meaning is different. It + only depends on the axioms ~ ax-mp to ~ ax-4 , ~ ax-6 , and ~ ax-pr . + This theorem does not exclude that these two sets could actually be one + single set containing itself. That two different sets exist is proved + by ~ exexneq . (Contributed by SN, 23-Dec-2024.) $) + exel $p |- E. y E. x x e. y $= + ( vz weq wo wel wi wal wex ax-pr ax6ev pm2.07 eximii exim mpi ) ACDZPEZAB + FZGAHZRAIZBCCBAJSQAITPQAACKPLMQRANOM $. + $( $j usage 'exel' avoids 'ax-5' 'ax-7' 'ax-8' 'ax-9' 'ax-10' 'ax-11' + 'ax-12' 'ax-13' 'ax-ext' 'ax-sep' 'ax-nul' 'ax-pow'; $) + $} + + ${ + $d x y z $. + $( There exist two different sets. (Contributed by NM, 7-Nov-2006.) Avoid + ~ ax-13 . (Revised by BJ, 31-May-2019.) Avoid ~ ax-8 . (Revised by + SN, 21-Sep-2023.) Avoid ~ ax-12 . (Revised by Rohan Ridenour, + 9-Oct-2024.) Use ~ ax-pr instead of ~ ax-pow . (Revised by + BTernaryTau, 3-Dec-2024.) Extract this result from the proof of + ~ dtru . (Revised by BJ, 2-Jan-2025.) $) + exexneq $p |- E. x E. y -. x = y $= + ( vz wel wex wn wal wa weq ax-nul exdistrv mpbir2an wi ax9v1 eximdv df-ex + exel syl6ib imnan sylib con2i 2eximi ax-mp ) CADZCEZCBDZFCGZHZBEAEZABIZFZ + BEAEUIUEAEUGBECAQBCJUEUGABKLUHUKABUJUHUJUEUGFZMUHFUJUEUFCEULUJUDUFCABCNOU + FCPRUEUGSTUAUBUC $. + $( $j usage 'exexneq' avoids 'ax-7' 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13' + 'ax-ext' 'ax-sep' 'ax-pow'; $) + $} + + ${ + $d w x y z $. + $( Given any set (the " ` y ` " in the statement), there exists a set not + equal to it. + + The same statement without disjoint variable condition is false, since + we do not have ` E. x -. x = x ` . This theorem is proved directly from + set theory axioms (no class definitions) and does not depend on + ~ ax-ext , ~ ax-sep , or ~ ax-pow nor auxiliary logical axiom schemes + ~ ax-10 to ~ ax-13 . See ~ dtruALT for a shorter proof using more + axioms, and ~ dtruALT2 for a proof using ~ ax-pow instead of ~ ax-pr . + (Contributed by NM, 7-Nov-2006.) Avoid ~ ax-13 . (Revised by BJ, + 31-May-2019.) Avoid ~ ax-8 . (Revised by SN, 21-Sep-2023.) Avoid + ~ ax-12 . (Revised by Rohan Ridenour, 9-Oct-2024.) Use ~ ax-pr instead + of ~ ax-pow . (Revised by BTernaryTau, 3-Dec-2024.) Extract this + result from the proof of ~ dtru . (Revised by BJ, 2-Jan-2025.) $) + exneq $p |- E. x -. x = y $= + ( vz vw weq wn wex exexneq equeuclr con3d ax7v1 spimevw syl6 a1d exlimivv + wi pm2.61i ax-mp ) CDEZFZDGCGABEZFZAGZCDHTUCCDDBEZTUCPUDTCBEZFZUCUDUESDCB + IJUFUBACACEUAUEACBKJLMUDFZUCTUGUBADADEUAUDADBKJLNQOR $. + $( $j usage 'exneq' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-ext' + 'ax-sep' 'ax-pow' 'df-clab' 'df-cleq' 'df-clel'; $) + $} + + ${ + $d x y $. + $( Given any set (the " ` y ` " in the statement), not all sets are equal + to it. The same statement without disjoint variable condition is false + since it contradicts ~ stdpc6 . The same comments and revision history + concerning axiom usage as in ~ exneq apply. (Contributed by NM, + 7-Nov-2006.) Extract ~ exneq as an intermediate result. (Revised by + BJ, 2-Jan-2025.) $) + dtru $p |- -. A. x x = y $= + ( weq wn wex wal exneq exnal mpbi ) ABCZDAEJAFDABGJAHI $. + $( $j usage 'dtru' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-ext' + 'ax-sep' 'ax-pow' 'df-clab' 'df-cleq' 'df-clel'; $) $} ${ $d x y z $. - $( Every set is an element of some other set. See ~ elALT for a shorter + $( Any set is an element of some other set. See ~ elALT for a shorter proof using more axioms, and see ~ elALT2 for a proof that uses ~ ax-9 and ~ ax-pow instead of ~ ax-pr . (Contributed by NM, 4-Jan-2002.) - (Proof shortened by Andrew Salmon, 25-Jul-2011.) Avoid ~ ax-9 , - ~ ax-pow . (Revised by BTernaryTau, 2-Dec-2024.) $) + (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ~ ax-pr instead of + ~ ax-9 and ~ ax-pow . (Revised by BTernaryTau, 2-Dec-2024.) $) el $p |- E. y x e. y $= ( vz weq wo wel wi ax-pr pm4.25 imbi1i albii elequ1 equsalvw bitr3i exbii wal wex mpbi ) CADZSEZCBFZGZCPZBQABFZBQAABCHUCUDBUCSUAGZCPUDUEUBCSTUASIJK @@ -51122,69 +51362,96 @@ That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with $( $j usage 'el' avoids 'ax-9' 'ax-ext' 'ax-sep' 'ax-nul' 'ax-pow'; $) $} + ${ + $d x y A $. + $( If a class is a set, then it is a member of a set. (Contributed by NM, + 4-Jan-2002.) Generalize from the proof of ~ elALT . (Revised by BJ, + 3-Apr-2019.) Avoid ~ ax-sep , ~ ax-nul , ~ ax-pow . (Revised by + BTernaryTau, 15-Jan-2025.) $) + sels $p |- ( A e. V -> E. x A e. x ) $= + ( vy wel wex cv wcel wceq eleq1 exbidv el vtoclg ) DAEZAFBAGZHZAFDBCDGZBI + NPAQBOJKDALM $. + $( $j usage 'sels' avoids 'ax-sep' 'ax-nul' 'ax-pow'; $) + $} + + ${ + $d x A $. + $( Alternate proof of ~ sels , requiring ~ ax-sep but not using ~ el (which + is proved from it as ~ elALT ). (especially when the proof of ~ el is + inlined in ~ sels ). (Contributed by NM, 4-Jan-2002.) Generalize from + the proof of ~ elALT . (Revised by BJ, 3-Apr-2019.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + selsALT $p |- ( A e. V -> E. x A e. x ) $= + ( wcel csn cv wex snidg cvv snexg eleq2 spcedv syl ) BCDBBEZDZBAFZDZAGBCH + OQOAINBNJBNHPNBKLM $. + $( $j usage 'sels' avoids 'ax-nul' 'ax-pow'; $) + $} + ${ $d x y $. - $( Alternate proof of ~ el , shorter but requiring more axioms. - (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) + $( Alternate proof of ~ el , shorter but requiring ~ ax-sep . (Contributed + by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) $) elALT $p |- E. y x e. y $= - ( cv csn wcel wex vex snid snex eleq2 spcev ax-mp ) ACZMDZEZMBCZEZBFMAGHQ - OBNMIPNMJKL $. + ( cv cvv wcel wel wex vex selsALT ax-mp ) ACZDEABFBGAHBKDIJ $. + $( $j usage 'sels' avoids 'ax-nul' 'ax-pow'; $) $} ${ $d w x y z $. - $( At least two sets exist (or in terms of first-order logic, the universe - of discourse has two or more objects). Note that we may not substitute - the same variable for both ` x ` and ` y ` (as indicated by the distinct - variable requirement), for otherwise we would contradict ~ stdpc6 . - - This theorem is proved directly from set theory axioms (no set theory - definitions) and does not use ~ ax-ext , ~ ax-sep , or ~ ax-pow . See - ~ dtruALT for a shorter proof using these axioms, and see ~ dtruALT2 for - a proof that uses ~ ax-pow instead of ~ ax-pr . - - The proof makes use of dummy variables ` z ` and ` w ` which do not - appear in the final theorem. They must be distinct from each other and - from ` x ` and ` y ` . In other words, if we were to substitute ` x ` - for ` z ` throughout the proof, the proof would fail. (Contributed by - NM, 7-Nov-2006.) Avoid ~ ax-13 . (Revised by Gino Giotto, 5-Sep-2023.) - Avoid ~ ax-12 . (Revised by Rohan Ridenour, 9-Oct-2024.) Use ~ ax-pr - instead of ~ ax-pow . (Revised by BTernaryTau, 3-Dec-2024.) $) - dtru $p |- -. A. x x = y $= + $( Obsolete proof of ~ dtru as of 01-Jan-2025. (Contributed by NM, + 7-Nov-2006.) Avoid ~ ax-13 . (Revised by BJ, 31-May-2019.) Avoid + ~ ax-12 . (Revised by Rohan Ridenour, 9-Oct-2024.) Use ~ ax-pr instead + of ~ ax-pow . (Revised by BTernaryTau, 3-Dec-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + dtruOLD $p |- -. A. x x = y $= ( vw vz weq wn wex wal wel wa el ax-nul elequ1 notbid spw eximii exdistrv ax7v1 con3d spimevw mpbir2an ax9v2 con3dimp 2eximi equequ2 syl6bi pm2.61i com12 wi a1d exlimivv mp2b exnal mpbi ) ABEZFZAGZUOAHFACIZADIZFZJZDGCGZCD EZFZDGCGUQVBURCGUTDGACKUTAHUTDDALUTCDIZFACACEZUSVEACDMNOPURUTCDQUAVAVDCDU RVCUSVCURUSCDAUBUHUCUDVDUQCDDBEZVDUQUIVGVDCBEZFZUQVGVCVHDBCUENVIUPACVFUOV HACBRSTUFVGFZUQVDVJUPADADEUOVGADBRSTUJUGUKULUOAUMUN $. - $( $j usage 'dtru' avoids 'ax-10 'ax-11' 'ax-12' 'ax-13' 'ax-ext' 'ax-sep' - 'ax-pow'; $) $} - $( A singleton of a set belongs to the power class of a class containing the - set. (Contributed by Alan Sare, 25-Aug-2011.) $) + $( A singleton of a set is a member of the powerclass of a class if and only + if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) + Put in closed form and avoid ~ ax-nul . (Revised by BJ, 17-Jan-2025.) $) + snelpwg $p |- ( A e. V -> ( A e. B <-> { A } e. ~P B ) ) $= + ( wcel csn wss cpw snssg cvv wb snexg elpwg syl bitr4d ) ACDZABDAEZBFZPBGDZ + ABCHOPIDRQJACKPBILMN $. + + $( If a set is a member of a class, then the singleton of that set is a + member of the powerclass of that class. (Contributed by Alan Sare, + 25-Aug-2011.) $) snelpwi $p |- ( A e. B -> { A } e. ~P B ) $= + ( wcel csn cpw snelpwg ibi ) ABCADBECABBFG $. + + $( Obsolete version of ~ snelpwi as of 17-Jan-2025. (Contributed by NM, + 28-May-1995.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + snelpwiOLD $p |- ( A e. B -> { A } e. ~P B ) $= ( wcel csn wss cpw snssi snex elpw sylibr ) ABCADZBEKBFCABGKBAHIJ $. ${ - snelpw.1 $e |- A e. _V $. - $( A singleton of a set belongs to the power class of a class containing - the set. (Contributed by NM, 1-Apr-1998.) $) + snelpw.ex $e |- A e. _V $. + $( A singleton of a set is a member of the powerclass of a class if and + only if that set is a member of that class. (Contributed by NM, + 1-Apr-1998.) $) snelpw $p |- ( A e. B <-> { A } e. ~P B ) $= - ( wcel csn wss cpw snss snex elpw bitr4i ) ABDAEZBFLBGDABCHLBAIJK $. + ( cvv wcel csn cpw wb snelpwg ax-mp ) ADEABEAFBGEHCABDIJ $. $} - $( A pair of two sets belongs to the power class of a class containing those - two sets and vice versa. (Contributed by AV, 8-Jan-2020.) $) + $( An unordered pair of two sets is a member of the powerclass of a class if + and only if the two sets are members of that class. (Contributed by AV, + 8-Jan-2020.) $) prelpw $p |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } e. ~P C ) ) $= ( wcel wa cpr wss cpw prssg prex elpw bitr4di ) ADFBEFGACFBCFGABHZCIOCJFABC DEKOCABLMN $. - $( A pair of two sets belongs to the power class of a class containing those - two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) (Proof shortened - by AV, 23-Oct-2021.) $) + $( If two sets are members of a class, then the unordered pair of those two + sets is a member of the powerclass of that class. (Contributed by Thierry + Arnoux, 10-Mar-2017.) (Proof shortened by AV, 23-Oct-2021.) $) prelpwi $p |- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C ) $= ( wcel wa cpr cpw prelpw ibi ) ACDBCDEABFCGDABCCCHI $. @@ -51194,9 +51461,9 @@ This theorem is proved directly from set theory axioms (no set theory singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.) $) rext $p |- ( A. z ( x e. z -> y e. z ) -> x = y ) $= - ( cv wcel wi wal csn wceq vsnid snex eleq2 imbi12d spcv mpi velsn equcomi - sylbi syl ) ADZCDZEZBDZUAEZFZCGZUCTHZEZTUCIZUFTUGEZUHAJUEUJUHFCUGTKUAUGIU - BUJUDUHUAUGTLUAUGUCLMNOUHUCTIUIBTPBAQRS $. + ( cv wcel wal csn wceq vsnid vsnex eleq2 imbi12d spcv velsn equcomi sylbi + wi mpi syl ) ADZCDZEZBDZUAEZQZCFZUCTGZEZTUCHZUFTUGEZUHAIUEUJUHQCUGAJUAUGH + UBUJUDUHUAUGTKUAUGUCKLMRUHUCTHUIBTNBAOPS $. $} ${ @@ -51206,9 +51473,9 @@ This theorem is proved directly from set theory axioms (no set theory Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) $) sspwb $p |- ( A C_ B <-> ~P A C_ ~P B ) $= - ( vx wss cpw sspw cv csn wcel ssel snex elpw vex snss bitr4i ssrdv impbii - 3imtr3g ) ABDAEZBEZDZABFUACABUACGZHZSIZUCTIZUBAIZUBBIZSTUCJUDUCADUFUCAUBK - ZLUBACMZNOUEUCBDUGUCBUHLUBBUINORPQ $. + ( vx wss cpw sspw csn wcel ssel vsnex elpw vex snss bitr4i 3imtr3g impbii + cv ssrdv ) ABDAEZBEZDZABFUACABUACQZGZSHZUCTHZUBAHZUBBHZSTUCIUDUCADUFUCACJ + ZKUBACLZMNUEUCBDUGUCBUHKUBBUIMNORP $. $} ${ @@ -51263,10 +51530,22 @@ This theorem is proved directly from set theory axioms (no set theory ${ $d x A $. - intid.1 $e |- A e. _V $. $( The intersection of all sets to which a set belongs is the singleton of - that set. (Contributed by NM, 5-Jun-2009.) $) - intid $p |- |^| { x | A e. x } = { A } $= + that set. (Contributed by NM, 5-Jun-2009.) Put in closed form and + avoid ~ ax-nul . (Revised by BJ, 17-Jan-2025.) $) + intidg $p |- ( A e. V -> |^| { x | A e. x } = { A } ) $= + ( wcel cab cint csn wss cvv snexg snidg eleq2 elabd intss1 syl wal ax-gen + cv wi id elintabg mpbiri snssd eqssd ) BCDZBARZDZAEZFZBGZUEUJUHDUIUJHUEUG + BUJDAUJIBCJBCKUFUJBLMUJUHNOUEBUIUEBUIDUGUGSZAPUKAUGTQUGABCUAUBUCUD $. + $} + + ${ + $d x A $. + intid.1 $e |- A e. _V $. + $( Obsolete version of ~ intidg as of 18-Jan-2025. (Contributed by NM, + 5-Jun-2009.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + intidOLD $p |- |^| { x | A e. x } = { A } $= ( cv wcel cab cint csn cvv wss snex eleq2 intmin3 ax-mp wi elintab mpgbir snid id snssi eqssi ) BADZEZAFGZBHZUEIEUDUEJBKUCBUEEAUEIUBUEBLBCRMNBUDEZU EUDJUFUCUCOAUCABCPUCSQBUDTNUA $. @@ -51278,8 +51557,8 @@ This theorem is proved directly from set theory axioms (no set theory (Contributed by NM, 30-Dec-1996.) $) moabex $p |- ( E* x ph -> { x | ph } e. _V ) $= ( vy wmo weq wi wal wex cab cvv wcel df-mo cv csn abss velsn imbi2i albii - wss bitri snex ssex sylbir exlimiv sylbi ) ABDABCEZFZBGZCHABIZJKZABCLUHUJ - CUHUICMZNZSZUJUMABMULKZFZBGUHABULOUOUGBUNUFABUKPQRTUIULUKUAUBUCUDUE $. + wss bitri vsnex ssex sylbir exlimiv sylbi ) ABDABCEZFZBGZCHABIZJKZABCLUHU + JCUHUICMZNZSZUJUMABMULKZFZBGUHABULOUOUGBUNUFABUKPQRTUIULCUAUBUCUDUE $. $} $( Restricted "at most one" existence implies a restricted class abstraction @@ -51298,10 +51577,10 @@ This theorem is proved directly from set theory axioms (no set theory $( A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.) $) nnullss $p |- ( A =/= (/) -> E. x ( x C_ A /\ x =/= (/) ) ) $= - ( vy c0 wne cv wcel wex wss wa n0 csn vex snss snnz snex wceq sseq1 neeq1 - sylbi anbi12d spcev mpan2 exlimiv ) BDECFZBGZCHAFZBIZUGDEZJZAHZCBKUFUKCUF - UELZBIZUKUEBCMZNUMULDEZUKUEUNOUJUMUOJAULUEPUGULQUHUMUIUOUGULBRUGULDSUAUBU - CTUDT $. + ( vy c0 wne cv wcel wex wss wa csn snss snnz vsnex wceq sseq1 neeq1 sylbi + n0 vex anbi12d spcev mpan2 exlimiv ) BDECFZBGZCHAFZBIZUGDEZJZAHZCBSUFUKCU + FUEKZBIZUKUEBCTZLUMULDEZUKUEUNMUJUMUOJAULCNUGULOUHUMUIUOUGULBPUGULDQUAUBU + CRUDR $. $} ${ @@ -51311,13 +51590,13 @@ This theorem is proved directly from set theory axioms (no set theory exss $p |- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) ) $= ( vz wrex cv wcel wa cab wex wss crab wne df-rab rabn0 sylbi wsb df-clab c0 neeq1i n0 3bitr3i csn vex snss ssab2 sstr2 mpi simpr weq equsb1v velsn - sbbii mpbir jctil sban bitri eleq2i 3bitri 3imtr4i ne0d sylib sseq1 rexeq - snex wceq anbi12d spcev syl2anc exlimiv ) ABDFZEGZBGZDHZAIZBJZHZEKZCGZDLZ - ABVTFZIZCKZABDMZTNVQTNVLVSWEVQTABDOUAABDPEVQUBUCVRWDEVRVMUDZDLZABWFFZWDVR - WFVQLZWGVMVQEUEUFWIVQDLWGABDUGWFVQDUHUIQVRABWFMZTNWHVRWJVMVOBERZABERZIZVN - WFHZBERZWLIZVRVMWJHZWMWLWOWKWLUJWOBEUKZBERBEULWNWRBEBVMUMUNUOUPVRVPBERWMV - PEBSVOABEUQURWQVMWNAIZBJZHWSBERWPWJWTVMABWFOUSWSEBSWNABEUQUTVAVBABWFPVCWC - WGWHICWFVMVFVTWFVGWAWGWBWHVTWFDVDABVTWFVEVHVIVJVKQ $. + sbbii mpbir jctil sban bitri eleq2i 3bitri 3imtr4i ne0d sylib vsnex sseq1 + wceq rexeq anbi12d spcev syl2anc exlimiv ) ABDFZEGZBGZDHZAIZBJZHZEKZCGZDL + ZABVTFZIZCKZABDMZTNVQTNVLVSWEVQTABDOUAABDPEVQUBUCVRWDEVRVMUDZDLZABWFFZWDV + RWFVQLZWGVMVQEUEUFWIVQDLWGABDUGWFVQDUHUIQVRABWFMZTNWHVRWJVMVOBERZABERZIZV + NWFHZBERZWLIZVRVMWJHZWMWLWOWKWLUJWOBEUKZBERBEULWNWRBEBVMUMUNUOUPVRVPBERWM + VPEBSVOABEUQURWQVMWNAIZBJZHWSBERWPWJWTVMABWFOUSWSEBSWNABEUQUTVAVBABWFPVCW + CWGWHICWFEVDVTWFVFWAWGWBWHVTWFDVEABVTWFVGVHVIVJVKQ $. $} $( An ordered pair of classes is a set. Exercise 7 of [TakeutiZaring] p. 16. @@ -52097,6 +52376,21 @@ necessary if all involved classes exist as sets (i.e. are not proper ( cop csn wbr wcel wa wceq df-br opex elsn opthg2 bitrid ) EFABGZHZIEFGZSJZ ACJBDJKZEALFBLKZEFSMUATRLUBUCTREFNOEFABCDPQQ $. + ${ + brtp.1 $e |- X e. _V $. + brtp.2 $e |- Y e. _V $. + $( A necessary and sufficient condition for two sets to be related under a + binary relation which is an unordered triple. (Contributed by Scott + Fenton, 8-Jun-2011.) $) + brtp $p |- ( X { <. A , B >. , <. C , D >. , <. E , F >. } Y <-> + ( ( X = A /\ Y = B ) \/ + ( X = C /\ Y = D ) \/ + ( X = E /\ Y = F ) ) ) $= + ( cop ctp wbr wcel wceq w3o wa df-br opex opth eltp 3orbi123i 3bitri ) GH + ABKZCDKZEFKZLZMGHKZUGNUHUDOZUHUEOZUHUFOZPGAOHBOQZGCOHDOQZGEOHFOQZPGHUGRUH + UDUEUFGHSUAUIULUJUMUKUNGHABIJTGHCDIJTGHEFIJTUBUC $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -52108,8 +52402,8 @@ necessary if all involved classes exist as sets (i.e. are not proper $d x y z $. $d ph z $. $( The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of ~ opabid with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 14-Apr-1995.) (Revised by Gino - Giotto, 26-Jan-2024.) $) + require ~ ax-13 . (Contributed by NM, 14-Apr-1995.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 26-Jan-2024.) $) opabidw $p |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) $= ( vz cv cop wceq wa wex copab opex copsexgw bicomd df-opab elab2 ) DEZBEZ CEZFZGZAHCIBIZADSABCJQRKTAUAABCPLMABCDNO $. @@ -52156,12 +52450,12 @@ necessary if all involved classes exist as sets (i.e. are not proper $( Restricted existential quantification over an ordered-pair class abstraction. (Contributed by AV, 8-Nov-2023.) $) rexopabb $p |- ( E. o e. O ps <-> E. x E. y ( ph /\ ch ) ) $= - ( wrex copab wa wex rexeqi cv wcel impcom nfv nfrex exlimi wceq elopab wi - cop simprr biimpd adantr ex 2eximdv sylanb rexlimiva nfopab1 nfopab2 wsbc - jca opabidw opex sbcie rspesbca syl2anbr impbii bitri ) BFGJBFADEKZJZACLZ - EMZDMZBFGVCHNVDVGBVGFVCFOZVCPVHDOZEOZUDZUAZALZEMDMZBVGADEVHUBBVNVGBVMVEDE - BVMVEBVMLACBVLAUEVMBCVLBCUCAVLBCIUFUGQUOUHUIQUJUKVFVDDBDFVCADEULBDRSVEVDE - BEFVCADEUMBERSAVKVCPBFVKUNVDCADEUPBCFVKVIVJUQIURBFVKVCUSUTTTVAVB $. + ( wrex copab wa wex rexeqi cv wcel impcom nfv nfrexw exlimi elopab simprr + cop wceq wi biimpd adantr jca ex 2eximdv sylanb rexlimiva nfopab1 nfopab2 + wsbc opabidw opex sbcie rspesbca syl2anbr impbii bitri ) BFGJBFADEKZJZACL + ZEMZDMZBFGVCHNVDVGBVGFVCFOZVCPVHDOZEOZUCZUDZALZEMDMZBVGADEVHUABVNVGBVMVED + EBVMVEBVMLACBVLAUBVMBCVLBCUEAVLBCIUFUGQUHUIUJQUKULVFVDDBDFVCADEUMBDRSVEVD + EBEFVCADEUNBERSAVKVCPBFVKUOVDCADEUPBCFVKVIVJUQIURBFVKVCUSUTTTVAVB $. $} ${ @@ -52382,8 +52676,8 @@ necessary if all involved classes exist as sets (i.e. are not proper $d x y $. $( Equivalence of ordered pair abstraction subclass and implication. Version of ~ ssopab2b with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by NM, 27-Dec-1996.) (Revised by Gino - Giotto, 26-Jan-2024.) $) + require ~ ax-13 . (Contributed by NM, 27-Dec-1996.) Avoid ~ ax-13 . + (Revised by Gino Giotto, 26-Jan-2024.) $) ssopab2bw $p |- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <-> A. x A. y ( ph -> ps ) ) $= ( copab wss wal nfopab1 nfss nfopab2 cop wcel ssel opabidw 3imtr3g alrimi @@ -52393,8 +52687,8 @@ necessary if all involved classes exist as sets (i.e. are not proper $( Equivalence of ordered pair abstraction equality and biconditional. Version of ~ eqopab2b with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by Mario Carneiro, 4-Jan-2017.) - (Revised by Gino Giotto, 26-Jan-2024.) $) + require ~ ax-13 . (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 26-Jan-2024.) $) eqopab2bw $p |- ( { <. x , y >. | ph } = { <. x , y >. | ps } <-> A. x A. y ( ph <-> ps ) ) $= ( copab wss wa wi wal wceq wb ssopab2bw anbi12i eqss 2albiim 3bitr4i ) AC @@ -53241,6 +53535,22 @@ We have not yet defined relations ( ~ df-rel ), but here we introduce a few OQUPUQUR $. $} + $( Trichotomy law for strict orderings. (Contributed by Scott Fenton, + 8-Dec-2021.) $) + sotrine $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> + ( B =/= C <-> ( B R C \/ C R B ) ) ) $= + ( wor wcel wa wbr wo wceq wn sotrieq bicomd necon1abid ) ADEBAFCAFGGZBCDHCB + DHIZBCOBCJPKABCDLMN $. + + $( Transitivity law for strict orderings. (Contributed by Scott Fenton, + 24-Nov-2021.) $) + sotr3 $p |- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> + ( ( X R Y /\ -. Z R Y ) -> X R Z ) ) $= + ( wor wcel w3a wa wbr wn wceq wo wb simp3 simp2 jca sotric sylan2 con2bid + adantr wi breq2 biimprcd adantl sotr expdimp jaod sylbird expimpd ) ABFZCAG + ZDAGZEAGZHZIZCDBJZEDBJZKZCEBJZUPUQIZUSEDLZDEBJZMZUTUPVDUSNUQUPURVDUOUKUNUMI + URVDKNUOUNUMULUMUNOULUMUNPQAEDBRSTUAVAVBUTVCUQVBUTUBUPVBUTUQEDCBUCUDUEUPUQV + CUTACDEBUFUGUHUIUJ $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -53463,9 +53773,9 @@ dummy variables (but note that their equivalence uses ~ ax-sep ). Carneiro, 14-Oct-2016.) $) nffr $p |- F/ x R Fr A $= ( va vc vb wfr cv wss c0 wne wa wbr wn wral wrex wi nfcv wal nfss nfv nfn - df-fr nfan nfbr nfralw nfrex nfim nfal nfxfr ) BCIFJZBKZUMLMZNZGJZHJZCOZP - ZGUMQZHUMRZSZFUAAFHGBCUEVCAFUPVBAUNUOAAUMBAUMTZEUBUOAUCUFVAAHUMVDUTAGUMVD - USAAUQURCAUQTDAURTUGUDUHUIUJUKUL $. + df-fr nfan nfbr nfralw nfrexw nfim nfal nfxfr ) BCIFJZBKZUMLMZNZGJZHJZCOZ + PZGUMQZHUMRZSZFUAAFHGBCUEVCAFUPVBAUNUOAAUMBAUMTZEUBUOAUCUFVAAHUMVDUTAGUMV + DUSAAUQURCAUQTDAURTUGUDUHUIUJUKUL $. $( Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, @@ -53844,7 +54154,7 @@ of the restricted converse (see ~ cnvrescnv ). (Contributed by NM, $} ${ - $d x y z w A $. $d x y z w B $. $d x y z w C $. + $d x y z A $. $d x y z B $. $d x y z C $. $( Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.) $) xpeq1 $p |- ( A = B -> ( A X. C ) = ( B X. C ) ) $= @@ -54943,11 +55253,11 @@ of the restricted converse (see ~ cnvrescnv ). (Contributed by NM, of its arguments. (Contributed by NM, 13-Sep-2006.) $) xpsspw $p |- ( A X. B ) C_ ~P ~P ( A u. B ) $= ( vx vy cxp cun cpw relxp cv wcel csn cpr wss snssi syl elpw sylibr sylib - wa vex opelxp ssun3 snex adantr df-pr ssun4 anim12i unss eqsstrid zfpair2 - cop jca prex dfop eleq1i prss 3bitr4ri sylbi relssi ) CDABEZABFZGZGZABHCI - ZDIZUKZUTJVDAJZVEBJZSZVFVCJZVDVEABUAVIVDKZVBJZVDVELZVBJZSZVJVIVLVNVGVLVHV - GVKVAMZVLVGVKAMVPVDANVKABUBOZVKVAVDUCZPQUDVIVMVAMVNVIVMVKVEKZFZVAVDVEUEVI - VPVSVAMZSVTVAMVGVPVHWAVQVHVSBMWAVEBNVSBAUFOUGVKVSVAUHRUIVMVACDUJZPQULVKVM + wa vex cop opelxp ssun3 vsnex adantr df-pr ssun4 anim12i eqsstrid zfpair2 + unss jca prex dfop eleq1i prss 3bitr4ri sylbi relssi ) CDABEZABFZGZGZABHC + IZDIZUAZUTJVDAJZVEBJZSZVFVCJZVDVEABUBVIVDKZVBJZVDVELZVBJZSZVJVIVLVNVGVLVH + VGVKVAMZVLVGVKAMVPVDANVKABUCOZVKVACUDZPQUEVIVMVAMVNVIVMVKVEKZFZVAVDVEUFVI + VPVSVAMZSVTVAMVGVPVHWAVQVHVSBMWAVEBNVSBAUGOUHVKVSVAUKRUIVMVACDUJZPQULVKVM LZVCJWCVBMVJVOWCVBVKVMUMPVFWCVCVDVECTDTUNUOVKVMVBVRWBUPUQRURUS $. $} @@ -59292,6 +59602,17 @@ singleton of the first member (with no sethood assumptions on ` B ` ). AUADRLST $. $} + ${ + snres0.1 $e |- B e. _V $. + $( Condition for restriction of a singleton to be empty. (Contributed by + Scott Fenton, 9-Aug-2024.) $) + snres0 $p |- ( ( { <. A , B >. } |` C ) = (/) <-> -. A e. C ) $= + ( cop csn cres c0 wceq cdm cin wcel wn wrel wb relres reldm0 ax-mp dmsnop + dmres ineq2i eqtri eqeq1i disjsn 3bitri ) ABEFZCGZHIZUGJZHIZCAFZKZHIACLMU + GNUHUJOUFCPUGQRUIULHUICUFJZKULUFCTUMUKCABDSUAUBUCCAUDUE $. + $} + + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The Predecessor Class @@ -59504,18 +59825,15 @@ singleton of the first member (with no sethood assumptions on ` B ` ). TLVQVNWGWEQVPAABCVTXETVHVIVJ $. $} - ${ - $d R z $. $d A z $. $d X z $. $d Y z $. - $( Property of the predecessor class for partial orders. (Contributed by - Scott Fenton, 28-Apr-2012.) (Proof shortened by Scott Fenton, - 28-Oct-2024.) $) - predpo $p |- ( ( R Po A /\ X e. A ) -> - ( Y e. Pred ( R , A , X ) -> - Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) ) $= - ( wpo wcel wa cpred wss cxp cin ccom cid cres wceq dfpo2 simprbi ad2antrr - c0 simpr simplr predtrss syl3anc ex ) ABEZCAFZGZDABCHZFZABDHUHIZUGUIGBAAJ - KZUKLBIZUIUFUJUEULUFUIUEBMANKSOULABPQRUGUITUEUFUIUAABCDUBUCUD $. - $} + $( Property of the predecessor class for partial orders. (Contributed by + Scott Fenton, 28-Apr-2012.) (Proof shortened by Scott Fenton, + 28-Oct-2024.) $) + predpo $p |- ( ( R Po A /\ X e. A ) -> + ( Y e. Pred ( R , A , X ) -> + Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) ) $= + ( wpo wcel wa cpred wss cxp cin ccom cres wceq dfpo2 simprbi ad2antrr simpr + cid c0 simplr predtrss syl3anc ex ) ABEZCAFZGZDABCHZFZABDHUHIZUGUIGBAAJKZUK + LBIZUIUFUJUEULUFUIUEBSAMKTNULABOPQUGUIRUEUFUIUAABCDUBUCUD $. $( Property of the predecessor class for strict total orders. (Contributed by Scott Fenton, 11-Feb-2011.) $) @@ -59652,6 +59970,7 @@ singleton of the first member (with no sethood assumptions on ` B ` ). RAVDCRUQVBURVCUSVEABCTAVDCTUTVA $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Well-founded induction (variant) @@ -59779,7 +60098,7 @@ singleton of the first member (with no sethood assumptions on ` B ` ). $) ${ - $d x y A $. $d x y B $. $d x y R $. + $d y A $. $d y B $. $d y R $. $( All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) @@ -59878,8 +60197,7 @@ singleton of the first member (with no sethood assumptions on ` B ` ). $} ${ - $d A w y $. $d A z $. $d ph w $. $d ph z $. $d R w y $. $d R z $. - $d w y z $. + $d A y z $. $d ph z $. $d R y z $. wfisg.1 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) ) $. $( Well-Ordered Induction Schema. If a property passes from all elements @@ -60818,7 +61136,7 @@ with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] $. $( An ordinal number is a transitive class. (Contributed by NM, - 11-Jun-1994.) $) + 11-Jun-1994.) Put in closed form. (Resised by BJ, 28-Dec-2024.) $) ontr $p |- ( A e. On -> Tr A ) $= ( con0 wcel word wtr eloni ordtr syl ) ABCADAEAFAGH $. @@ -60835,9 +61153,10 @@ with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] onordi $p |- Ord A $= ( con0 wcel word eloni ax-mp ) ACDAEBAFG $. - $( An ordinal number is a transitive class. (Contributed by NM, - 11-Jun-1994.) $) - ontrci $p |- Tr A $= + $( Obsolete version of ~ ontr as of 28-Dec-2024. (Contributed by NM, + 11-Jun-1994.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + ontrciOLD $p |- Tr A $= ( con0 wcel wtr ontr ax-mp ) ACDAEBAFG $. $( An ordinal number is not a member of itself. Theorem 7M(c) of @@ -61011,7 +61330,8 @@ Definite description binder (inverted iota) nfiotadw.2 $e |- ( ph -> F/ x ps ) $. $( Deduction version of ~ nfiotaw . Version of ~ nfiotad with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by - NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.) $) + NM, 18-Feb-2013.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 26-Jan-2024.) $) nfiotadw $p |- ( ph -> F/_ x ( iota y ps ) ) $= ( vz cio weq wb wal cab cuni dfiota2 nfv nfvd nfbid nfald nfabdw nfunid nfcxfrd ) ACBDHBDGIZJZDKZGLZMBDGNACUEAUDCGAGOAUCCDEABUBCFAUBCPQRSTUA $. @@ -61023,8 +61343,8 @@ Definite description binder (inverted iota) nfiotaw.1 $e |- F/ x ph $. $( Bound-variable hypothesis builder for the ` iota ` class. Version of ~ nfiota with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by NM, 23-Aug-2011.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + ~ ax-13 . (Contributed by NM, 23-Aug-2011.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 26-Jan-2024.) $) nfiotaw $p |- F/_ x ( iota y ph ) $= ( cio wnfc wtru nftru wnf a1i nfiotadw mptru ) BACEFGABCCHABIGDJKL $. $( $j usage 'nfiotaw' avoids 'ax-13'; $) @@ -61061,8 +61381,8 @@ Definite description binder (inverted iota) cbviotaw.3 $e |- F/ x ps $. $( Change bound variables in a description binder. Version of ~ cbviota with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + (Contributed by Andrew Salmon, 1-Aug-2011.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 26-Jan-2024.) $) cbviotaw $p |- ( iota x ph ) = ( iota y ps ) $= ( vw vz weq wb wal cab cuni cio nfv nfbi equequ1 bibi12d cbvalv1 nfs1v cv wsb sbequ12 nfsbv sbhypf bitri abbii unieqi dfiota2 3eqtr4i ) ACHJZKZCLZH @@ -61205,7 +61525,7 @@ Definite description binder (inverted iota) iotaval $p |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y ) $= ( weq wb wal cab cv csn wceq cio abbi1 df-sn eqtr4di iotaval2 syl ) ABCDZ EBFZABGZCHZIZJABKTJRSQBGUAAQBLBTMNABCOP $. - $( $j usage 'iotaval' avoids 'ax-10', 'ax-11', 'ax-12'; $) + $( $j usage 'iotaval' avoids 'ax-10' 'ax-11' 'ax-12'; $) $} ${ @@ -61232,7 +61552,7 @@ Definite description binder (inverted iota) $} ${ - $d ph z $. $d ps z $. $d x y z $. + $d ph z $. $d x y z $. $( Obsolete version of ~ iotaval as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) $) @@ -61350,7 +61670,7 @@ Definite description binder (inverted iota) iota2df.4 $e |- F/ x ph $. iota2df.5 $e |- ( ph -> F/ x ch ) $. iota2df.6 $e |- ( ph -> F/_ x B ) $. - $( A condition that allows us to represent "the unique element such that + $( A condition that allows to represent "the unique element such that ` ph ` " with a class expression ` A ` . (Contributed by NM, 30-Dec-2014.) $) iota2df $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $= @@ -61360,7 +61680,7 @@ Definite description binder (inverted iota) $} $d x B $. $d x ph $. $d x ch $. - $( A condition that allows us to represent "the unique element such that + $( A condition that allows to represent "the unique element such that ` ph ` " with a class expression ` A ` . (Contributed by NM, 30-Dec-2014.) $) iota2d $p |- ( ph -> ( ch <-> ( iota x ps ) = B ) ) $= @@ -61858,21 +62178,21 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). funopg $p |- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B ) $= ( vu vt vx vy vz vw vv wcel cop wceq cv weq wi cpr vex cvv funeqd imbi12d wfun opeq1 eqeq1 opeq2 eqeq2 csn wa wex wrel funrel relop sylib opth opid - preq1i dfop preq2i snex zfpair2 3eqtr4ri eqeq2i bitr3i wal dffun4 simprbi - prid1 eleq2 mpbiri prid2 jca w3a opeq12 3adant3 eleq1d 3adant2 anbi12d wb - opex eqeq12 3adant1 spc3gv mp3an syl2im biimtrid dfsn2 preq2 eqeq2d eqtr3 - eqtr2id expcom syl6bi com13 imp sylcom exlimdvv mpd vtocl2g 3impia ) ACLB - DLABMZUCZABNZEOZFOZMZUCZEFPZQAXEMZUCZAXENZQXBXCQEFABCDXDANZXGXJXHXKXLXFXI - XDAXEUDUAXDAXEUEUBXEBNZXJXBXKXCXMXIXAXEBAUFUAXEBAUGUBXGXDGOZUHZNZXEXNHOZR - ZNZUIZHUJGUJZXHXGXFUKZYAXFULGHXDXEESZFSZUMUNXGXTXHGHXGXTGHPZXHXTXFXNXNMZX - NXQMZRZNZXGYEXTXFXOXRMZNYIXDXEXOXRYCYDUOYJYHXFYFXOXRRZRXOUHZYKRYHYJYFYLYK - XNGSZUPUQYGYKYFXNXQYMHSZURUSXOXRXNUTGHVAURVBVCVDXGIOZJOZMZXFLZYOKOZMZXFLZ - UIZJKPZQZKVEJVEIVEZYIYFXFLZYGXFLZUIZYEXGYBUUEIJKXFVFVGYIUUFUUGYIUUFYFYHLY - FYGXNXNVTVHXFYHYFVIVJYIUUGYGYHLYFYGXNXQVTVKXFYHYGVIVJVLXNTLZUUIXQTLUUEUUH + preq1i dfop preq2i zfpair2 3eqtr4ri eqeq2i bitr3i wal dffun4 simprbi opex + vsnex prid1 eleq2 mpbiri jca w3a opeq12 3adant3 eleq1d 3adant2 anbi12d wb + prid2 eqeq12 3adant1 spc3gv mp3an syl2im dfsn2 preq2 eqtr2id eqeq2d eqtr3 + biimtrid expcom syl6bi com13 imp sylcom exlimdvv mpd vtocl2g 3impia ) ACL + BDLABMZUCZABNZEOZFOZMZUCZEFPZQAXEMZUCZAXENZQXBXCQEFABCDXDANZXGXJXHXKXLXFX + IXDAXEUDUAXDAXEUEUBXEBNZXJXBXKXCXMXIXAXEBAUFUAXEBAUGUBXGXDGOZUHZNZXEXNHOZ + RZNZUIZHUJGUJZXHXGXFUKZYAXFULGHXDXEESZFSZUMUNXGXTXHGHXGXTGHPZXHXTXFXNXNMZ + XNXQMZRZNZXGYEXTXFXOXRMZNYIXDXEXOXRYCYDUOYJYHXFYFXOXRRZRXOUHZYKRYHYJYFYLY + KXNGSZUPUQYGYKYFXNXQYMHSZURUSXOXRGVHGHUTURVAVBVCXGIOZJOZMZXFLZYOKOZMZXFLZ + UIZJKPZQZKVDJVDIVDZYIYFXFLZYGXFLZUIZYEXGYBUUEIJKXFVEVFYIUUFUUGYIUUFYFYHLY + FYGXNXNVGVIXFYHYFVJVKYIUUGYGYHLYFYGXNXQVGVTXFYHYGVJVKVLXNTLZUUIXQTLUUEUUH YEQZQYMYMYNUUDUUJIJKXNXNXQTTTIGPZJGPZKHPZVMZUUBUUHUUCYEUUNYRUUFUUAUUGUUNY QYFXFUUKUULYQYFNUUMYOYPXNXNVNVOVPUUNYTYGXFUUKUUMYTYGNUULYOYSXNXQVNVQVPVRU - ULUUMUUCYEVSUUKYPXNYSXQWAWBUBWCWDWEWFXPXSYEXHQYEXSXPXHYEXSXEXONZXPXHQYEXR - XOXEYEXOXNXNRXRXNWGXNXQXNWHWKWIXPUUOXHXDXEXOWJWLWMWNWOWPWQWRWSWT $. + ULUUMUUCYEVSUUKYPXNYSXQWAWBUBWCWDWEWKXPXSYEXHQYEXSXPXHYEXSXEXONZXPXHQYEXR + XOXEYEXOXNXNRXRXNWFXNXQXNWGWHWIXPUUOXHXDXEXOWJWLWMWNWOWPWQWRWSWT $. $} ${ @@ -62027,7 +62347,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). ${ $d x y A $. $d x y B $. $( The converse singleton of an ordered pair is a function. This is - equivalent to ~ funsn via ~ cnvsn , but stating it this way allows us to + equivalent to ~ funsn via ~ cnvsn , but stating it this way allows to skip the sethood assumptions on ` A ` and ` B ` . (Contributed by NM, 30-Apr-2015.) $) funcnvsn $p |- Fun `' { <. A , B >. } $= @@ -64070,8 +64390,8 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). CKUFUDTAUCCLMNOPABCRBAUARS $. $( A relation is a one-to-one onto function iff its converse is a one-to-one - onto function with domain and range interchanged. (Contributed by NM, - 8-Dec-2003.) $) + onto function with domain and codomain/range interchanged. (Contributed + by NM, 8-Dec-2003.) $) f1ocnvb $p |- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) ) $= ( wrel wf1o ccnv f1ocnv wceq wb dfrel2 f1oeq1 sylbi syl5ib impbid2 ) CDZABC @@ -64419,7 +64739,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). $} ${ - $d A x y $. $d F x y $. + $d A x $. $d F x $. $( A function's value at a proper class is the empty set. See ~ fvprcALT for a proof that uses ~ ax-pow instead of ~ ax-pr . (Contributed by NM, 20-May-1998.) Avoid ~ ax-pow . (Revised by BTernaryTau, 3-Aug-2024.) @@ -64765,10 +65085,35 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). IAFZDGZHZIUJUIDUMUJJULUMDULUKKDLMNOUJPZUKUMUIUKULSUNBQIZUIQIBUIARZUIUKIUOUJ BAUATUNBAUBUPUJUPUJABUCUDTBUIAQQUEUFUGUH $. - $( The result of a function value is always a subset of the union of the - range, even if it is invalid and thus empty. (Contributed by Stefan - O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) $) - fvssunirn $p |- ( F ` X ) C_ U. ran F $= + $( If the value of a function is not null, the value is an element of the + range of the function. (Contributed by Alexander van der Vekens, + 22-Jul-2018.) (Proof shortened by SN, 13-Jan-2025.) $) + fvn0fvelrn $p |- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F ) $= + ( cfv c0 wne crn csn cun wcel wn fvrn0 nelsn elunnel2 sylancr ) BACZDEOAFZD + GZHIOQIJOPIABKODLOPQMN $. + + $( A function value is a subset of the union of the range. (An artifact of + our function value definition, compare ~ elfvdm ). (Contributed by + Thierry Arnoux, 13-Nov-2016.) Remove functionhood antecedent. (Revised + by SN, 10-Jan-2025.) $) + elfvunirn $p |- ( B e. ( F ` A ) -> B e. U. ran F ) $= + ( cfv wcel crn cuni c0 wne wss ne0i fvn0fvelrn elssuni 3syl sseld pm2.43i ) + BACDZEZBCFZGZERQTBRQHIQSEQTJQBKCALQSMNOP $. + + ${ + $d F x $. $d X x $. + $( The result of a function value is always a subset of the union of the + range, even if it is invalid and thus empty. (Contributed by Stefan + O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof + shortened by SN, 13-Jan-2025.) $) + fvssunirn $p |- ( F ` X ) C_ U. ran F $= + ( vx cfv crn cuni cv elfvunirn ssriv ) CBADAEFBCGAHI $. + $} + + $( Obsolete version of ~ fvssunirn as of 13-Jan-2025. (Contributed by Stefan + O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + fvssunirnOLD $p |- ( F ` X ) C_ U. ran F $= ( cfv crn c0 csn cun cuni wcel wss fvrn0 elssuni ax-mp uniun 0ex uneq2i un0 unisn 3eqtri sseqtri ) BACZADZEFZGZHZUBHZUAUDIUAUEJABKUAUDLMUEUFUCHZGUFEGUF UBUCNUGEUFEORPUFQST $. @@ -65853,7 +66198,8 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). set of the class abstraction depends on the argument of the function. Version of ~ elfvmptrab1 with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by Alexander van der Vekens, - 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.) $) + 15-Jul-2018.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 26-Jan-2024.) $) elfvmptrab1w $p |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) $= ( cdm wcel cfv csb wa crab cvv wceq nfcv elfvdm wsbc dmmptss sseli rabexg @@ -66627,8 +66973,8 @@ in the range of the function (the implication "to the right" is always ralrnmptw.2 $e |- ( y = B -> ( ps <-> ch ) ) $. $( A restricted quantifier over an image set. Version of ~ ralrnmpt with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 26-Jan-2024.) $) ralrnmptw $p |- ( A. x e. A B e. V -> ( A. y e. ran F ps <-> A. x e. A ch ) ) $= ( vw vz wcel wral cv cfv wsbc wb syl nfv crn fnmpt dfsbcq nfsbc1v sbceq2a @@ -66647,8 +66993,8 @@ in the range of the function (the implication "to the right" is always rexrnmptw.2 $e |- ( y = B -> ( ps <-> ch ) ) $. $( A restricted quantifier over an image set. Version of ~ rexrnmpt with a disjoint variable condition, which does not require ~ ax-13 . - (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 26-Jan-2024.) $) rexrnmptw $p |- ( A. x e. A B e. V -> ( E. y e. ran F ps <-> E. x e. A ch ) ) $= ( wcel wral wn crn wrex cv wceq notbid ralrnmptw dfrex2 3bitr4g ) FHKCELZ @@ -66687,7 +67033,7 @@ in the range of the function (the implication "to the right" is always ${ f0cl.1 $e |- F : A --> B $. f0cl.2 $e |- (/) e. B $. - $( Unconditional closure of a function when the range includes the empty + $( Unconditional closure of a function when the codomain includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.) $) f0cli $p |- ( F ` C ) e. B $= ( wcel cfv ffvelcdmi cdm fdmi eleq2i wn c0 ndmfv eqeltrdi sylnbir pm2.61i @@ -67489,10 +67835,10 @@ pairs as classes (in set.mm, the Kuratowski encoding). A more fvressn $p |- ( X e. V -> ( ( F |` { X } ) ` X ) = ( F ` X ) ) $= ( wcel csn snidg fvresd ) CBDCCEACBFG $. - $( If the value of a function is not null, the value is an element of the - range of the function. (Contributed by Alexander van der Vekens, - 22-Jul-2018.) $) - fvn0fvelrn $p |- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F ) $= + $( Obsolete version of ~ fvn0fvelrn as of 13-Jan-2025. (Contributed by + Alexander van der Vekens, 22-Jul-2018.) (New usage is discouraged.) + (Proof modification is discouraged.) $) + fvn0fvelrnOLD $p |- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F ) $= ( cfv c0 wne cdm wcel csn cres wfun wa crn fvfundmfvn0 wi eldmressnsn pm3.2 fvelrn syl ex com13 mpd imp fvressn eleq1d fvrnressn sylbid impcom 3syl ) B ACZDEBAFZGZABHIZJZKBULCZULLZGZUKKZUIALGZBAMUKUMUQUKBULFGZUMUQNBAOUMUSUKUQUM @@ -68530,7 +68876,8 @@ pairs as classes (in set.mm, the Kuratowski encoding). A more ${ $d x y A $. $d x y F $. $( Membership in the union of the range of a function. See ~ elunirnALT - for a shorter proof which uses ~ ax-pow . (Contributed by NM, + for a shorter proof which uses ~ ax-pow . See ~ elfvunirn for a more + general version of the reverse direction. (Contributed by NM, 24-Sep-2006.) $) elunirn $p |- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) ) $= @@ -68557,9 +68904,10 @@ pairs as classes (in set.mm, the Kuratowski encoding). A more ${ $d x A $. $d x B $. $d x F $. - $( Condition for the membership in the union of the range of a function. - (Contributed by Thierry Arnoux, 13-Nov-2016.) $) - elunirn2 $p |- ( ( Fun F /\ B e. ( F ` A ) ) -> B e. U. ran F ) $= + $( Obsolete version of ~ elfvunirn as of 12-Jan-2025. (Contributed by + Thierry Arnoux, 13-Nov-2016.) (New usage is discouraged.) + (Proof modification is discouraged.) $) + elunirn2OLD $p |- ( ( Fun F /\ B e. ( F ` A ) ) -> B e. U. ran F ) $= ( vx wfun cfv wcel wa crn cuni cdm wrex elfvdm wceq eleq2d rspcev mpancom cv fveq2 adantl wb elunirn adantr mpbird ) CEZBACFZGZHBCIJGZBDRZCFZGZDCKZ LZUGUMUEAULGUGUMBACMUKUGDAULUIANUJUFBUIACSOPQTUEUHUMUAUGDBCUBUCUD $. @@ -68783,8 +69131,8 @@ more bytes (305 versus 282). (Contributed by AV, 3-Feb-2021.) VBVCKULAUPURCDAVEVGWAUPURPVRQLVTWBVDECDFTUJUKSUN $. $d A x y z $. $d B x y z $. $d C x y z $. $d F x y z $. $d R x y z $. - $( The range of a 1-1 function from a set with three different elements has - (at least) three different elements. (Contributed by AV, + $( The codomain of a 1-1 function from a set with three different elements + has (at least) three different elements. (Contributed by AV, 20-Mar-2019.) $) f1dom3el3dif $p |- ( ph -> E. x e. R E. y e. R E. z e. R ( x =/= y /\ x =/= z /\ y =/= z ) ) $= @@ -70086,6 +70434,67 @@ complete lattice has a (least) fixpoint. Here we specialize this $. $} + $( The value of a restricted function at a class is either the empty set or + the value of the unrestricted function at that class. (Contributed by + Scott Fenton, 4-Sep-2011.) $) + fvresval $p |- ( ( ( F |` B ) ` A ) = ( F ` A ) \/ + ( ( F |` B ) ` A ) = (/) ) $= + ( wcel wn wo cres cfv wceq c0 exmid fvres nfvres orim12i ax-mp ) ABDZPEZFAC + BGHZACHIZRJIZFPKPSQTABCLABCMNO $. + + $( If ` (/) ` is not part of the range of a function ` F ` , then ` A ` is in + the domain of ` F ` iff ` ( F `` A ) =/= (/) ` . (Contributed by Scott + Fenton, 7-Dec-2021.) $) + funeldmb $p |- ( ( Fun F /\ -. (/) e. ran F ) -> + ( A e. dom F <-> ( F ` A ) =/= (/) ) ) $= + ( wfun c0 crn wcel wn wa cdm cfv wi fvelrn ex adantr wb eleq1 adantl sylibd + wceq con3d impancom ndmfv impbid1 necon2abid ) BCZDBEZFZGZHZABIFZABJZDUIUKD + SZUJGZUEULUHUMUEULHZUJUGUNUJUKUFFZUGUEUJUOKULUEUJUOABLMNULUOUGOUEUKDUFPQRTU + AABUBUCUD $. + + ${ + $d F x y $. $d G x y $. $d X x y $. $d Y x y $. + $( Law for adjoining an element to restrictions of functions. (Contributed + by Scott Fenton, 6-Dec-2021.) $) + eqfunresadj $p |- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\ + ( Y e. dom F /\ Y e. dom G /\ ( F ` Y ) = ( G ` Y ) ) ) -> + ( F |` ( X u. { Y } ) ) = ( G |` ( X u. { Y } ) ) ) $= + ( vx vy wfun wa cres wceq cdm wcel cfv w3a cun relres cv wbr wo wb simp33 + breq 3ad2ant2 eqeq1d simp1l simp31 funbrfvb syl2anc simp1r simp32 3bitr3d + csn velsn bibi12d syl5ibrcom biimtrid pm5.32d vex 3bitr4g orbi12d resundi + breq1 brresi breqi brun bitri eqbrrdiv ) AGZBGZHZACIZBCIZJZDAKLZDBKLZDAMZ + DBMZJZNZNZEFACDULZOZIZBWBIZAWBPBWBPVTEQZFQZVKRZWEWFAWAIZRZSZWEWFVLRZWEWFB + WAIZRZSZWEWFWCRZWEWFWDRZVTWGWKWIWMVMVJWGWKTVSWEWFVKVLUBUCVTWEWALZWEWFARZH + WQWEWFBRZHWIWMVTWQWRWSWQWEDJZVTWRWSTZEDUMVTXAWTDWFARZDWFBRZTVTVPWFJZVQWFJ + ZXBXCVTVPVQWFVJVMVNVOVRUAUDVTVHVNXDXBTVHVIVMVSUEVJVMVNVOVRUFDWFAUGUHVTVIV + OXEXCTVHVIVMVSUIVJVMVNVOVRUJDWFBUGUHUKWTWRXBWSXCWEDWFAVBWEDWFBVBUNUOUPUQW + AWEWFAFURZVCWAWEWFBXFVCUSUTWOWEWFVKWHOZRWJWEWFWCXGACWAVAVDWEWFVKWHVEVFWPW + EWFVLWLOZRWNWEWFWDXHBCWAVAVDWEWFVLWLVEVFUSVG $. + $} + + $( Law for equality of restriction to successors. This is primarily useful + when ` X ` is an ordinal, but it does not require that. (Contributed by + Scott Fenton, 6-Dec-2021.) $) + eqfunressuc $p |- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\ + ( X e. dom F /\ X e. dom G /\ ( F ` X ) = ( G ` X ) ) ) -> + ( F |` suc X ) = ( G |` suc X ) ) $= + ( wfun wa cres wceq cdm wcel cfv w3a csn eqfunresadj df-suc reseq2i 3eqtr4g + cun csuc ) ADBDEACFBCFGCAHICBHICAJCBJGKKACCLQZFBSFACRZFBTFABCCMTSACNZOTSBUA + OP $. + + ${ + $d A y $. $d B x y $. $d C x y $. $d F x y $. + $( Condition for subset of an intersection of an image. (Contributed by + Scott Fenton, 16-Aug-2024.) $) + fnssintima $p |- ( ( F Fn A /\ B C_ A ) -> ( C C_ |^| ( F " B ) <-> + A. x e. B C C_ ( F ` x ) ) ) $= + ( vy cima cint wss cv wcel wi wal wfn wa cfv wral bitri wceq albii df-ral + ssint wrex fvelimab imbi1d albidv ralcom4 eqcom imbi1i fvex sseq2 ceqsalv + ralbii r19.23v 3bitr3ri bitrdi bitrid ) DECGZHIZFJZURKZDUTIZLZFMZEBNCBIOZ + DAJZEPZIZACQZUSVBFURQVDFDURUBVBFURUARVEVDVGUTSZACUCZVBLZFMZVIVEVCVLFVEVAV + KVBABCUTEUDUEUFVJVBLZFMZACQVNACQZFMVIVMVNAFCUGVOVHACVOUTVGSZVBLZFMVHVNVRF + VJVQVBVGUTUHUITVBVHFVGVFEUJUTVGDUKULRUMVPVLFVJVBACUNTUOUPUQ $. + $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -70212,7 +70621,8 @@ Restricted iota (description binder) nfriotadw.3 $e |- ( ph -> F/_ x A ) $. $( Deduction version of ~ nfriota with a disjoint variable condition, which contrary to ~ nfriotad does not require ~ ax-13 . (Contributed by NM, - 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.) $) + 18-Feb-2013.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 26-Jan-2024.) $) nfriotadw $p |- ( ph -> F/_ x ( iota_ y e. A ps ) ) $= ( vw crio cv wcel wa cio df-riota wal wnfc adantr wnf df-nfc weq wn nfcvd nfnaew adantl nfeld nfand nfiotadw ex nfiota1 biidd drnf1v albidv 3bitr4g @@ -70230,8 +70640,8 @@ Restricted iota (description binder) cbvriotaw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. $( Change bound variable in a restricted description binder. Version of ~ cbvriota with a disjoint variable condition, which does not require - ~ ax-13 . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, - 26-Jan-2024.) $) + ~ ax-13 . (Contributed by NM, 18-Mar-2013.) Avoid ~ ax-13 . (Revised + by Gino Giotto, 26-Jan-2024.) $) cbvriotaw $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $= ( vz cv wcel wa cio crio weq eleq1w anbi12d nfv nfan cbviotaw wsb sbequ12 nfs1v sbhypf nfsbv eqtri df-riota 3eqtr4i ) CJEKZALZCMZDJZEKZBLZDMZACENBD @@ -70412,7 +70822,7 @@ Restricted iota (description binder) riota2f.1 $e |- F/_ x B $. riota2f.2 $e |- F/ x ps $. riota2f.3 $e |- ( x = B -> ( ph <-> ps ) ) $. - $( This theorem shows a condition that allows us to represent a descriptor + $( This theorem shows a condition that allows to represent a descriptor with a class expression ` B ` . (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) $) riota2f $p |- ( ( B e. A /\ E! x e. A ph ) -> @@ -70424,7 +70834,7 @@ Restricted iota (description binder) ${ $d x ps $. $d x A $. $d x B $. riota2.1 $e |- ( x = B -> ( ph <-> ps ) ) $. - $( This theorem shows a condition that allows us to represent a descriptor + $( This theorem shows a condition that allows to represent a descriptor with a class expression ` B ` . (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.) $) riota2 $p |- ( ( B e. A /\ E! x e. A ph ) -> @@ -70635,6 +71045,17 @@ Restricted iota (description binder) wn ) DCEZOZABCFZABCGZCEZABCHQRTROZTOQUATPUASDCABCIJKLMN $. $} + ${ + $d A y $. $d ph y $. $d x y $. + riotarab.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Restricted iota of a restricted abstraction. (Contributed by Scott + Fenton, 8-Aug-2024.) $) + riotarab $p |- ( iota_ x e. { y e. A | ps } ch ) = + ( iota_ x e. A ( ph /\ ch ) ) $= + ( cv crab wcel wa cio crio wb weq bicomd equcoms elrab anbi1i df-riota + anass bitri iotabii 3eqtr4i ) DHZBEFIZJZCKZDLUEFJZACKZKZDLCDUFMUJDFMUHUKD + UHUIAKZCKUKUGULCBAEUEFBANDEDEOABGPQRSUIACUAUBUCCDUFTUJDFTUD $. + $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -70903,8 +71324,8 @@ ordered pairs (for use in defining operations). This is a special case $d x y z $. $( The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of ~ oprabid with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by Mario Carneiro, 20-Mar-2013.) - (Revised by Gino Giotto, 26-Jan-2024.) $) + require ~ ax-13 . (Contributed by Mario Carneiro, 20-Mar-2013.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 26-Jan-2024.) $) oprabidw $p |- ( <. <. x , y >. , z >. e. { <. <. x , y >. , z >. | ph } <-> ph ) $= ( vw va vt vr vs cv cop wceq wa wex wi vex weq 19.8a anim2i eximi eqvinop @@ -71370,7 +71791,7 @@ ordered pairs (for use in defining operations). This is a special case $( Equivalence of ordered pair abstraction subclass and biconditional. Version of ~ eqoprab2b with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by Mario Carneiro, 4-Jan-2017.) - (Revised by Gino Giotto, 26-Jan-2024.) $) + Avoid ~ ax-13 . (Revised by Gino Giotto, 26-Jan-2024.) $) eqoprab2bw $p |- ( { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph <-> ps ) ) $= ( coprab wss wa wi wal nfoprab1 nfss nfoprab2 nfoprab3 wcel ssel oprabidw @@ -73446,7 +73867,8 @@ result of an operator (deduction version). (Contributed by Paul the base set of the class abstraction depends on the first operand. Version of ~ elovmporab1 with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by Alexander van der Vekens, - 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.) $) + 15-Jul-2018.) Avoid ~ ax-13 . (Revised by Gino Giotto, + 26-Jan-2024.) $) elovmporab1w $p |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) ) $= ( cvv wcel wa csb cv wceq nfcv nfel1 co w3a crab elmpocl wsbc cmpo csbeq1 @@ -74333,9 +74755,9 @@ result of an operator (deduction version). (Contributed by Paul ${ $d x y A $. $d x y B $. $( Define a relation which corresponds to proper subsethood ~ df-pss on - sets. This allows us to use proper subsethood with general concepts - that require relations, such as strict ordering, see ~ sorpss . - (Contributed by Stefan O'Rear, 2-Nov-2014.) $) + sets. This allows to use proper subsethood with general concepts that + require relations, such as strict ordering, see ~ sorpss . (Contributed + by Stefan O'Rear, 2-Nov-2014.) $) df-rpss $a |- [C.] = { <. x , y >. | x C. y } $. $( The proper subset relation is a relation. (Contributed by Stefan @@ -74657,9 +75079,9 @@ result of an operator (deduction version). (Contributed by Paul (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof shortened by BJ, 5-Dec-2021.) $) snnex $p |- { x | E. y x = { y } } e/ _V $= - ( cv csn cvv wcel wa wceq wex cab wnel wal abnex df-nel sylibr snex vsnid - wn pm3.2i mpg ) BCZDZEFZUAUBFZGZACUBHBIAJZEKZBUEBLUFEFRUGBAUBEMUFENOUCUDU - APBQST $. + ( cv csn cvv wcel wa wceq wex cab wnel wn abnex df-nel sylibr vsnex vsnid + wal pm3.2i mpg ) BCZDZEFZUAUBFZGZACUBHBIAJZEKZBUEBRUFEFLUGBAUBEMUFENOUCUD + BPBQST $. $} ${ @@ -74841,7 +75263,8 @@ result of an operator (deduction version). (Contributed by Paul $d x y z $. $( The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of - [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.) Avoid ~ ax-un . + [Mendelson] p. 244. For a shorter proof requiring ~ ax-un , see + ~ epweonALT . (Contributed by NM, 1-Nov-2003.) Avoid ~ ax-un . (Revised by BTernaryTau, 30-Nov-2024.) $) epweon $p |- _E We On $= ( vx vy vz con0 cep wwe wfr wor cv wbr w3o wral wn wa wcel wel word eloni @@ -74857,14 +75280,15 @@ result of an operator (deduction version). (Contributed by Paul ${ $d x y $. - $( Obsolete version of ~ epweon as of 30-Nov-2024. (Contributed by NM, - 1-Nov-2003.) (Proof modification is discouraged.) + $( Alternate proof of ~ epweon , shorter but requiring ~ ax-un . + (Contributed by NM, 1-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.) $) - epweonOLD $p |- _E We On $= + epweonALT $p |- _E We On $= ( vx vy con0 cep wwe wfr cv wbr weq w3o wral onfr wcel eloni wa ordtri3or word wel epel biid 3orbi123i sylibr syl2an rgen2 dfwe2 mpbir2an ) CDECDFA GZBGZDHZABIZUHUGDHZJZBCKACKLULABCCUGCMUGQZUHQZULUHCMUGNUHNUMUNOABRZUJBARZ JULUGUHPUIUOUJUJUKUPBUGSUJTAUHSUAUBUCUDABCDUEUF $. + $( $j usage 'epweonALT' avoids 'ax-reg'; $) $} $( The class of all ordinal numbers is ordinal. Proposition 7.12 of @@ -74932,6 +75356,13 @@ be a set (so instead it is a proper class). Here we prove the denial of ( word con0 wss ordon wa wcel wceq ordeleqon biimpi adantr ordsseleq mpbird wo mpan2 ) ABZCBZACDZEPQFRACGACHNZPSQPSAIJKACLMO $. + $( A class is ordinal iff it is a subclass of ` On ` and transitive. + (Contributed by Scott Fenton, 21-Nov-2021.) $) + dford5 $p |- ( Ord A <-> ( A C_ On /\ Tr A ) ) $= + ( word con0 wss wtr wa ordsson ordtr jca cep wwe epweon wess df-ord biimpri + mpi ancoms sylan impbii ) ABZACDZAEZFTUAUBAGAHIUAAJKZUBTUACJKUCLACJMPUBUCTT + UBUCFANOQRS $. + $( An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) $) onss $p |- ( A e. On -> A C_ On ) $= @@ -75144,12 +75575,32 @@ be a set (so instead it is a proper class). Here we prove the denial of VDVGUJUKULUMUNUOVBVFVISZVAVBVDTJVRAUPBVDVETUQURUSUT $. $} + $( The successor of an ordinal class is an ordinal class. (Contributed by + NM, 6-Jun-1994.) Extract and adapt from a subproof of ~ suceloni . + (Revised by BTernaryTau, 6-Jan-2025.) (Proof shortened by BJ, + 11-Jan-2025.) $) + ordsuci $p |- ( Ord A -> Ord suc A ) $= + ( word csuc wtr con0 wss ordtr suctr syl csn cun df-suc ordsson cvv wcel wa + elon2 snssi sylbir c0 wceq snprc biimpi 0ss eqsstrdi adantl pm2.61dan unssd + wn eqsstrid ordon a1i trssord syl3anc ) ABZACZDZUPEFEBZUPBUOADUQAGAHIUOUPAA + JZKEALUOAUSEAMUOANOZUSEFZUOUTPAEOVAAQAERSUTUIZVAUOVBUSTEVBUSTUAAUBUCEUDUEUF + UGUHUJURUOUKULUPEUMUN $. + $( $j usage 'ordsuci' avoids 'ax-un'; $) + + $( If the successor of an ordinal number exists, it is an ordinal number. + This variation of ~ suceloni does not require ~ ax-un . (Contributed by + BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.) $) + sucexeloni $p |- ( ( A e. On /\ suc A e. V ) -> suc A e. On ) $= + ( con0 wcel csuc word cvv eloni ordsuci syl elex elong biimparc syl2an ) AC + DZAEZFZPGDZPCDZPBDOAFQAHAIJPBKRSQPGLMN $. + $( $j usage 'sucexeloni' avoids 'ax-un'; $) + ${ $d A x $. - $( If the successor of an ordinal number exists, it is an ordinal number. - This variation of ~ suceloni does not require ~ ax-un . (Contributed by - BTernaryTau, 30-Nov-2024.) $) - sucexeloni $p |- ( ( A e. On /\ suc A e. V ) -> suc A e. On ) $= + $( Obsolete version of ~ sucexeloni as of 6-Jan-2025. (Contributed by + BTernaryTau, 30-Nov-2024.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + sucexeloniOLD $p |- ( ( A e. On /\ suc A e. V ) -> suc A e. On ) $= ( vx con0 wcel csuc wa word wtr wss cv wral csn wo onelss wi velsn eqimss wceq sylbi a1i orim12d cun df-suc eleq2i elun oridm 3imtr3g sssucid sstr2 bitr2i syl6mpi ralrimiv dftr3 sylibr onss snssi unssd eqsstrid ordon 3exp @@ -75158,7 +75609,6 @@ be a set (so instead it is a proper class). Here we prove the denial of VTVTNVSVTVJWAVTWCVTAVQOWCVTPVJWCVQASVTCAQVQARTUAUBVSVQAWBUCZEWDVKWEVQAUDZ UEVQAWBUFUKVTUGUHAUIVQAVKUJULUMCVKUNUOVJVKWEDWFVJAWBDAUPADUQURUSVOVPDHZVN UTVOVPWGVNVKDVBVAVCVDVEVLVMVNVFVJVKBVGVHVI $. - $( $j usage 'sucexeloni' avoids 'ax-un'; $) $} $( The successor of an ordinal number is an ordinal number. Proposition 7.24 @@ -75183,9 +75633,19 @@ be a set (so instead it is a proper class). Here we prove the denial of UTVBVCVIVJTDVKVLVDACVEVJTVFVGVH $. $} - $( The successor of an ordinal class is ordinal. (Contributed by NM, - 3-Apr-1995.) $) + $( A class is ordinal if and only if its successor is ordinal. (Contributed + by NM, 3-Apr-1995.) Avoid ~ ax-un . (Revised by BTernaryTau, + 6-Jan-2025.) $) ordsuc $p |- ( Ord A <-> Ord suc A ) $= + ( word csuc ordsuci cvv wcel wi sucidg ordelord ex syl5com wn sucprc eqcomd + wceq wb ordeq syl biimprd pm2.61i impbii ) ABZACZBZADAEFZUDUBGUEAUCFZUDUBAE + HUDUFUBUCAIJKUELZUBUDUGAUCOUBUDPUGUCAAMNAUCQRSTUA $. + $( $j usage 'ordsuc' avoids 'ax-un'; $) + + $( Obsolete version of ~ ordsuc as of 6-Jan-2025. (Contributed by NM, + 3-Apr-1995.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + ordsucOLD $p |- ( Ord A <-> Ord suc A ) $= ( cvv wcel word csuc wb elong suceloni eloni syl syl6bir sucidg ordelord ex con0 syl5com impbid wn wceq sucprc eqcomd ordeq pm2.61i ) ABCZADZAEZDZFZUDU EUGUDUEAOCZUGABGUIUFOCUGAHUFIJKUDAUFCZUGUEABLUGUJUEUFAMNPQUDRZAUFSUHUKUFAAT @@ -75378,6 +75838,13 @@ be a set (so instead it is a proper class). Here we prove the denial of ( con0 wlim word c0 wne cuni wceq ordon onn0 unon eqcomi df-lim mpbir3an ) ABACADEAAFZGHINAJKALM $. + $( An ordinal number is either its own union (if zero or a limit ordinal) or + the successor of its union. (Contributed by NM, 13-Jun-1994.) Put in + closed form. (Revised by BJ, 11-Jan-2025.) $) + onuniorsuc $p |- ( A e. On -> ( A = U. A \/ A = suc U. A ) ) $= + ( con0 wcel word cuni wceq csuc wo eloni orduniorsuc syl ) ABCADAAEZFALGFHA + IAJK $. + ${ onssi.1 $e |- A e. On $. $( An ordinal number is a subset of ` On ` . (Contributed by NM, @@ -75390,26 +75857,27 @@ be a set (so instead it is a proper class). Here we prove the denial of onsuci $p |- suc A e. On $= ( con0 wcel csuc suceloni ax-mp ) ACDAECDBAFG $. - $( An ordinal number is either its own union (if zero or a limit ordinal) - or the successor of its union. (Contributed by NM, 13-Jun-1994.) $) - onuniorsuci $p |- ( A = U. A \/ A = suc U. A ) $= - ( word cuni wceq csuc wo onordi orduniorsuc ax-mp ) ACAADZEAKFEGABHAIJ $. + $( Obsolete version of ~ onuniorsuc as of 11-Jan-2025. (Contributed by NM, + 13-Jun-1994.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + onuniorsuciOLD $p |- ( A = U. A \/ A = suc U. A ) $= + ( con0 wcel cuni wceq csuc wo onuniorsuc ax-mp ) ACDAAEZFAKGFHBAIJ $. ${ $d x A $. $( A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.) $) onuninsuci $p |- ( A = U. A <-> -. E. x e. On A = suc x ) $= - ( cv csuc wceq con0 wrex cuni wn wa wcel onirri cun csn wtr sylancr syl - id sylib df-suc eqeq2i unieq sylbi uniun unisnv uneq2i eqtri eqtrdi wss - tron eleq1 mpbii trsuc word eloni ordtr df-tr eqtrd sylan9eqr vex sucid - ssequn1 eleq2 mpbiri adantr eqeltrd mto rexlimivw onuni onuniorsuci ori - imnani ax-mp suceq rspceeqv impbii con2bii ) BADZEZFZAGHZBBIZFZWBWDJZWA - WEAGWAWDWAWDKZBBLBCMWFBVSBWDWABWCVSWDSWAWCVSIZVSNZVSWAWCVSVSOZNZIZWHWAB - WJFWCWKFVTWJBVSUAUBBWJUCUDWKWGWIIZNWHVSWIUEWLVSWGAUFUGUHUIWAWGVSUJZWHVS - FWAVSGLZWMWAGPVTGLZWNUKWABGLZWOCBVTGULUMGVSUNQWNVSPZWMWNVSUOWQVSUPVSUQR - VSURTRWGVSVCTUSUTWAVSBLZWDWAWRVSVTLVSAVAVBBVTVSVDVEVFVGVHVMVIWEWCGLZBWC - EZFZWBWPWSCBVJVNWDXABCVKVLAWCGVTWTBVSWCVOVPQVQVR $. + ( cv csuc wceq con0 wrex cuni wn wa onirri id cun csn wtr sylancr sylib + wcel ax-mp df-suc eqeq2i unieq sylbi uniun unisnv uneq2i eqtri wss tron + eqtrdi eleq1 mpbii trsuc ontr df-tr ssequn1 eqtrd sylan9eqr sucid eleq2 + syl vex mpbiri adantr eqeltrd mto imnani rexlimivw onuni onuniorsuc ori + wo suceq rspceeqv impbii con2bii ) BADZEZFZAGHZBBIZFZWAWCJZVTWDAGVTWCVT + WCKZBBSBCLWEBVRBWCVTBWBVRWCMVTWBVRIZVRNZVRVTWBVRVROZNZIZWGVTBWIFWBWJFVS + WIBVRUAUBBWIUCUDWJWFWHIZNWGVRWHUEWKVRWFAUFUGUHUKVTWFVRUIZWGVRFVTVRGSZWL + VTGPVSGSZWMUJVTBGSZWNCBVSGULUMGVRUNQWMVRPWLVRUOVRUPRVBWFVRUQRURUSVTVRBS + ZWCVTWPVRVSSVRAVCUTBVSVRVAVDVEVFVGVHVIWDWBGSZBWBEZFZWAWOWQCBVJTWCWSWOWC + WSVMCBVKTVLAWBGVSWRBVRWBVNVOQVPVQ $. $} ${ @@ -75578,6 +76046,20 @@ be a set (so instead it is a proper class). Here we prove the denial of CUMVJCDVRVLUNCBUOACVJUPUQURVBUSBCUTVA $. $} + ${ + $d ph y z $. $d x y z $. + $( A closed form of ~ tfis . (Contributed by Scott Fenton, 8-Jun-2011.) $) + tfisg $p |- ( A. x e. On ( A. y e. x [. y / x ]. ph -> ph ) -> + A. x e. On ph ) $= + ( vz cv wsbc wral wi con0 crab wceq wss wcel ssrab2 wa dfss3 nfcv elrabsf + simprbi nfsbc1v ralimi sylbi nfralw nfim sbceq1a imbi12d rspc impcom syl5 + raleq simpr jctild syl6ibr ralrimiva tfi sylancr eqcomd rabid2 sylib ) AB + CEZFZCBEZGZAHZBIGZIABIJZKABIGVEVFIVEVFILDEZVFLZVGVFMZHZDIGVFIKABINVEVJDIV + EVGIMZOZVHVKABVGFZOVIVLVHVMVKVHVACVGGZVLVMVHUTVFMZCVGGVNCVGVFPVOVACVGVOUT + IMVAABUTIBIQZRSUAUBVKVEVNVMHZVDVQBVGIVNVMBVABCVGBVGQABUTTUCABVGTUDVBVGKVC + VNAVMVACVBVGUJABVGUEUFUGUHUIVEVKUKULABVGIVPRUMUNDVFUOUPUQABIURUS $. + $} + ${ $d w y z ph $. $d w x y z $. tfis.1 $e |- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) ) $. @@ -75847,7 +76329,6 @@ last three are the basis and the induction hypotheses (for successor and UCFCDOUDULHUMUEFCHUMUEZQUMUFZUKFCHUMUGUPFUOAPUDUHUI $. $} - $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The natural numbers (i.e., finite ordinals) @@ -76074,6 +76555,7 @@ also define a subset of the complex numbers ( ~ df-nn ) with analogous $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Peano's postulates @@ -76473,11 +76955,11 @@ Axiom of Replacement (unlike ~ resfunexg ). (Contributed by NM, a set. (Contributed by Mario Carneiro, 27-Apr-2015.) $) soex $p |- ( ( R Or A /\ R e. V ) -> A e. _V ) $= ( vx wor wcel wa cvv c0 wceq simpr 0ex eqeltrdi wne cv wex cun unexg wss - n0 csn cdm snex dmexg rnexg syl2anc sylancr ad2antlr cdif sossfld adantlr - crn ssundif sylibr ssexd ex exlimdv imp sylan2b pm2.61dane ) ABEZBCFZGZAH - FZAIVCAIJZGAIHVCVEKLMAINVCDOZAFZDPZVDDATVCVHVDVCVGVDDVCVGVDVCVGGZAVFUAZBU - BZBULZQZQZHVBVNHFZVAVGVBVJHFVMHFZVOVFUCVBVKHFVLHFVPBCUDBCUEVKVLHHRUFVJVMH - HRUGUHVIAVJUIVMSZAVNSVAVGVQVBAVFBUJUKAVJVMUMUNUOUPUQURUSUT $. + n0 csn cdm crn vsnex dmexg rnexg syl2anc sylancr ad2antlr sossfld adantlr + cdif ssundif sylibr ssexd ex exlimdv imp sylan2b pm2.61dane ) ABEZBCFZGZA + HFZAIVCAIJZGAIHVCVEKLMAINVCDOZAFZDPZVDDATVCVHVDVCVGVDDVCVGVDVCVGGZAVFUAZB + UBZBUCZQZQZHVBVNHFZVAVGVBVJHFVMHFZVODUDVBVKHFVLHFVPBCUEBCUFVKVLHHRUGVJVMH + HRUHUIVIAVJULVMSZAVNSVAVGVQVBAVFBUJUKAVJVMUMUNUOUPUQURUSUT $. $} ${ @@ -76606,8 +77088,8 @@ Axiom of Replacement (unlike ~ resfunexg ). (Contributed by NM, AUCUAUEUAUBKLMABCNRUGUBUEGZBAIUJUFULBAUCUBUEUAUBOLMABCPRS $. $} - $( A function with bounded domain and range is a set. This version of ~ fex - is proven without the Axiom of Replacement ~ ax-rep , but depends on + $( A function with bounded domain and codomain is a set. This version of + ~ fex is proven without the Axiom of Replacement ~ ax-rep , but depends on ~ ax-un , which is not required for the proof of ~ fex . (Contributed by Mario Carneiro, 24-Jun-2015.) $) fex2 $p |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V ) $= @@ -76846,8 +77328,8 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this ex focdmex syl6ci imp ) ABDEZBCFZAGFZUDUEDHZGFZUGADIZJZUFUDUEUHUDUGBKUEUHUD ABDABDLMUGBCNOTUDUGAUIPUJABDQUGAUIRSUGAGUIUAUBUC $. - $( The range of a 1-1 onto function is a set iff its domain is a set. - (Contributed by AV, 21-Mar-2019.) $) + $( The codomain/range of a 1-1 onto function is a set iff its domain is a + set. (Contributed by AV, 21-Mar-2019.) $) f1ovv $p |- ( F : A -1-1-onto-> B -> ( A e. _V <-> B e. _V ) ) $= ( wf1o cvv wcel wfo f1ofo focdmex syl5com wf1 wi f1of1 f1dmex ex syl impbid ) ABCDZAEFZBEFZRABCGSTABCHABECIJRABCKZTSLABCMUATSABECNOPQ $. @@ -76941,11 +77423,11 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this (Contributed by Jeff Madsen, 2-Sep-2009.) $) abrexex2g $p |- ( ( A e. V /\ A. x e. A { y | ph } e. W ) -> { y | E. x e. A ph } e. _V ) $= - ( vz wcel cab wral wa wrex cv cvv wsb nfv nfcv nfs1v nfrex weq abbii ciun - sbequ12 rexbidv cbvabw df-clab rexbii eqtr4i df-iun eqeltrrid eqeltrid - iunexg ) DEHACIZFHBDJKZABDLZCIZGMUMHZBDLZGIZNUPACGOZBDLZGIUSUOVACGUOGPUTC - BDCDQACGRSCGTAUTBDACGUCUDUEURVAGUQUTBDAGCUFUGUAUHUNUSBDUMUBNBGDUMUIBDUMEF - ULUJUK $. + ( vz wcel cab wral wa wrex cv cvv wsb nfv nfcv nfs1v nfrexw weq eqeltrrid + sbequ12 rexbidv cbvabw df-clab rexbii abbii eqtr4i df-iun iunexg eqeltrid + ciun ) DEHACIZFHBDJKZABDLZCIZGMUMHZBDLZGIZNUPACGOZBDLZGIUSUOVACGUOGPUTCBD + CDQACGRSCGTAUTBDACGUBUCUDURVAGUQUTBDAGCUEUFUGUHUNUSBDUMULNBGDUMUIBDUMEFUJ + UAUK $. $} ${ @@ -76959,15 +77441,15 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this ( vz vv vw cv wcel wa cvv wex cop wceq exbii bitri copab csn cab cxp ciun 19.42v an12 elxp excom velsn anbi1i opeq1 eqeq2d anbi1d equsexvw nfv nfan nfsab1 opeq2 wsb df-clab wb sbequ12 equcoms bitr4id anbi12d 3bitri anbi2i - cbvexv1 3bitr4ri wrex eliun df-rex elopab 3bitr4i eqriv wral snex sylancr - xpexg ralrimiva iunexg syl2anc eqeltrrid ) ACLZEMZBNZCDUAZCEWEUBZBDUCZUDZ - UEZOIWLWHWFILZWKMZNZCPZWMWEDLZQZRZWGNZDPZCPWMWLMZWMWHMWOXACWFWSBNZNZDPWFX - CDPZNXAWOWFXCDUFWTXDDWSWFBUGSWNXEWFWNWMJLZKLZQZRZXFWIMZXGWJMZNNZKPJPZWMWE - XGQZRZXKNZKPZXEJKWMWIWJUHXMXLJPZKPXQXLJKUIXRXPKXRXFWERZXIXKNZNZJPXPXLYAJX - LXJXTNYAXIXJXKUGXJXSXTJWEUJUKTSXTXPJCXSXIXOXKXSXHXNWMXFWEXGULUMUNUOTSTXPX - CKDXOXKDXODUPBDKURUQXCKUPXGWQRZXOWSXKBYBXNWRWMXGWQWEUSUMYBXKBDKUTZBBKDVAB - YCVBDKBDKVCVDVEVFVIVGVHVJSXBWNCEVKWPCWMEWKVLWNCEVMTWGCDWMVNVOVPAEFMWKOMZC - EVQWLOMGAYDCEAWFNWIOMWJOMYDWEVRHWIWJOOVTVSWACEWKFOWBWCWD $. + cbvexv1 3bitr4ri wrex eliun df-rex 3bitr4i eqriv wral vsnex xpexg sylancr + elopab ralrimiva iunexg syl2anc eqeltrrid ) ACLZEMZBNZCDUAZCEWEUBZBDUCZUD + ZUEZOIWLWHWFILZWKMZNZCPZWMWEDLZQZRZWGNZDPZCPWMWLMZWMWHMWOXACWFWSBNZNZDPWF + XCDPZNXAWOWFXCDUFWTXDDWSWFBUGSWNXEWFWNWMJLZKLZQZRZXFWIMZXGWJMZNNZKPJPZWMW + EXGQZRZXKNZKPZXEJKWMWIWJUHXMXLJPZKPXQXLJKUIXRXPKXRXFWERZXIXKNZNZJPXPXLYAJ + XLXJXTNYAXIXJXKUGXJXSXTJWEUJUKTSXTXPJCXSXIXOXKXSXHXNWMXFWEXGULUMUNUOTSTXP + XCKDXOXKDXODUPBDKURUQXCKUPXGWQRZXOWSXKBYBXNWRWMXGWQWEUSUMYBXKBDKUTZBBKDVA + BYCVBDKBDKVCVDVEVFVIVGVHVJSXBWNCEVKWPCWMEWKVLWNCEVMTWGCDWMVTVNVOAEFMWKOMZ + CEVPWLOMGAYDCEAWFNWIOMWJOMYDCVQHWIWJOOVRVSWACEWKFOWBWCWD $. $} ${ @@ -76982,15 +77464,15 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this 19.42v an12 ancom anbi2i 2exbii velsn anbi1i opeq2 eqeq2d anbi1d equsexvw elxp nfv nfsab1 nfan opeq1 wsb df-clab wb sbequ12 equcoms bitr4id anbi12d cbvexv1 3bitri 3bitr4ri excom wrex eliun df-rex elopab 3bitr4i eqriv wral - snex xpexg sylancl ralrimiva iunexg syl2anc eqeltrrid ) ADLZEMZBNZCDUAZDE - BCUBZWGUCZUDZUEZOIWNWJWHILZWMMZNZDPZWOCLZWGQZRZWINZDPCPZWOWNMZWOWJMWRXBCP - ZDPXCWQXEDWHXABNZNZCPWHXFCPZNXEWQWHXFCUFXBXGCXAWHBUGSWPXHWHWPWOJLZKLZQZRZ - XJWLMZXIWKMZNZNZKPZJPZWOXIWGQZRZXNNZJPXHWPXLXNXMNZNZKPJPXRJKWOWKWLUQYCXPJ - KYBXOXLXNXMUHUIUJTXQYAJXQXJWGRZXLXNNZNZKPYAXPYFKXPXMYENYFXLXMXNUGXMYDYEKW - GUKULTSYEYAKDYDXLXTXNYDXKXSWOXJWGXIUMUNUOUPTSYAXFJCXTXNCXTCURBCJUSUTXFJUR - XIWSRZXTXAXNBYGXSWTWOXIWSWGVAUNYGXNBCJVBZBBJCVCBYHVDCJBCJVEVFVGVHVIVJUIVK - SXBDCVLTXDWPDEVMWRDWOEWMVNWPDEVOTWICDWOVPVQVRAEFMWMOMZDEVSWNOMGAYIDEAWHNW - KOMWLOMYIHWGVTWKWLOOWAWBWCDEWMFOWDWEWF $. + vsnex xpexg sylancl ralrimiva iunexg syl2anc eqeltrrid ) ADLZEMZBNZCDUAZD + EBCUBZWGUCZUDZUEZOIWNWJWHILZWMMZNZDPZWOCLZWGQZRZWINZDPCPZWOWNMZWOWJMWRXBC + PZDPXCWQXEDWHXABNZNZCPWHXFCPZNXEWQWHXFCUFXBXGCXAWHBUGSWPXHWHWPWOJLZKLZQZR + ZXJWLMZXIWKMZNZNZKPZJPZWOXIWGQZRZXNNZJPXHWPXLXNXMNZNZKPJPXRJKWOWKWLUQYCXP + JKYBXOXLXNXMUHUIUJTXQYAJXQXJWGRZXLXNNZNZKPYAXPYFKXPXMYENYFXLXMXNUGXMYDYEK + WGUKULTSYEYAKDYDXLXTXNYDXKXSWOXJWGXIUMUNUOUPTSYAXFJCXTXNCXTCURBCJUSUTXFJU + RXIWSRZXTXAXNBYGXSWTWOXIWSWGVAUNYGXNBCJVBZBBJCVCBYHVDCJBCJVEVFVGVHVIVJUIV + KSXBDCVLTXDWPDEVMWRDWOEWMVNWPDEVOTWICDWOVPVQVRAEFMWMOMZDEVSWNOMGAYIDEAWHN + WKOMWLOMYIHDVTWKWLOOWAWBWCDEWMFOWDWEWF $. $} ${ @@ -77003,15 +77485,15 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this ( vz vv vw cv wcel wa cvv wex cop wceq an12 exbii bitri eqeq2d ciun copab csn cab cxp 19.42v elxp excom velsn anbi1i opeq1 anbi1d equsexvw nfv nfan nfsab1 opeq2 wsb df-clab wb sbequ12 equcoms bitr4id anbi12d 3bitri anbi2i - cbvexv1 3bitr4ri wrex eliun df-rex elopab 3bitr4i eqriv wral snex sylancr - xpexg rgen iunexg mp2an eqeltrri ) BDBJZUCZACUDZUEZUAZWCDKZALZBCUBZMGWGWJ - WHGJZWFKZLZBNZWKWCCJZOZPZWILZCNZBNWKWGKZWKWJKWMWSBWHWQALZLZCNWHXACNZLWSWM - WHXACUFWRXBCWQWHAQRWLXCWHWLWKHJZIJZOZPZXDWDKZXEWEKZLLZINHNZWKWCXEOZPZXILZ - INZXCHIWKWDWEUGXKXJHNZINXOXJHIUHXPXNIXPXDWCPZXGXILZLZHNXNXJXSHXJXHXRLXSXG - XHXIQXHXQXRHWCUIUJSRXRXNHBXQXGXMXIXQXFXLWKXDWCXEUKTULUMSRSXNXAICXMXICXMCU - NACIUPUOXAIUNXEWOPZXMWQXIAXTXLWPWKXEWOWCUQTXTXIACIURZAAICUSAYAUTCIACIVAVB - VCVDVGVEVFVHRWTWLBDVIWNBWKDWFVJWLBDVKSWIBCWKVLVMVNDMKWFMKZBDVOWGMKEYBBDWH - WDMKWEMKYBWCVPFWDWEMMVRVQVSBDWFMMVTWAWB $. + cbvexv1 3bitr4ri wrex eliun df-rex 3bitr4i eqriv wral vsnex xpexg sylancr + elopab rgen iunexg mp2an eqeltrri ) BDBJZUCZACUDZUEZUAZWCDKZALZBCUBZMGWGW + JWHGJZWFKZLZBNZWKWCCJZOZPZWILZCNZBNWKWGKZWKWJKWMWSBWHWQALZLZCNWHXACNZLWSW + MWHXACUFWRXBCWQWHAQRWLXCWHWLWKHJZIJZOZPZXDWDKZXEWEKZLLZINHNZWKWCXEOZPZXIL + ZINZXCHIWKWDWEUGXKXJHNZINXOXJHIUHXPXNIXPXDWCPZXGXILZLZHNXNXJXSHXJXHXRLXSX + GXHXIQXHXQXRHWCUIUJSRXRXNHBXQXGXMXIXQXFXLWKXDWCXEUKTULUMSRSXNXAICXMXICXMC + UNACIUPUOXAIUNXEWOPZXMWQXIAXTXLWPWKXEWOWCUQTXTXIACIURZAAICUSAYAUTCIACIVAV + BVCVDVGVEVFVHRWTWLBDVIWNBWKDWFVJWLBDVKSWIBCWKVRVLVMDMKWFMKZBDVNWGMKEYBBDW + HWDMKWEMKYBBVOFWDWEMMVPVQVSBDWFMMVTWAWB $. $} ${ @@ -77458,21 +77940,21 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) $( The ` 1st ` function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $) fo1st $p |- 1st : _V -onto-> _V $= - ( vx vy cvv c1st wfo wfn crn wceq cv csn cdm cuni snex dmex df-1st fnmpti - uniex wrex cab wcel rnmpt vex cop opex op1sta eqcomi sneq unieqd rspceeqv - dmeqd mp2an 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZKZLZDUT - USURMNQAOZPUQBIZVAHACRZBSCABCVADVBUAVDBCVCCTVDBUBZVCVCUCZCTVCVFJZKZLZHVDV - CVCUDVIVCVCVCVEVEUEUFAVFCVAVIVCURVFHZUTVHVJUSVGURVFUGUJUHUIUKULUMUNCCDUOU + ( vx vy cvv c1st wfo wfn crn wceq cv csn cdm cuni vsnex dmex uniex df-1st + fnmpti wrex cab wcel rnmpt vex cop opex op1sta eqcomi sneq dmeqd rspceeqv + unieqd mp2an 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZKZLZDU + TUSAMNOAPZQUQBIZVAHACRZBSCABCVADVBUAVDBCVCCTVDBUBZVCVCUCZCTVCVFJZKZLZHVDV + CVCUDVIVCVCVCVEVEUEUFAVFCVAVIVCURVFHZUTVHVJUSVGURVFUGUHUJUIUKULUMUNCCDUOU P $. $( The ` 2nd ` function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) $) fo2nd $p |- 2nd : _V -onto-> _V $= - ( vx vy cvv c2nd wfo wfn crn wceq cv csn cuni snex rnex uniex df-2nd wrex - fnmpti cab rnmpt wcel opex op2nda eqcomi sneq rneqd unieqd rspceeqv mp2an - vex cop 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZGZKZDUSURUQ - LMNAOZQUPBIZUTHACPZBRCABCUTDVASVCBCVBCTVCBUIZVBVBUJZCTVBVEJZGZKZHVCVBVBUA - VHVBVBVBVDVDUBUCAVECUTVHVBUQVEHZUSVGVIURVFUQVEUDUEUFUGUHUKULUMCCDUNUO $. + ( vx vy cvv c2nd wfo wfn crn wceq csn cuni vsnex rnex uniex df-2nd fnmpti + cv wrex cab rnmpt wcel vex cop opex op2nda eqcomi sneq rneqd unieqd mp2an + rspceeqv 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAPZIZGZJZDUSURA + KLMANZOUPBPZUTHACQZBRCABCUTDVASVCBCVBCTVCBUAZVBVBUBZCTVBVEIZGZJZHVCVBVBUC + VHVBVBVBVDVDUDUEAVECUTVHVBUQVEHZUSVGVIURVFUQVEUFUGUHUJUIUKULUMCCDUNUO $. $} $( Uniqueness condition for the binary relation ` 1st ` . (Contributed by @@ -78221,10 +78703,10 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) $( Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) $) mpoexxg $p |- ( ( A e. R /\ A. x e. A B e. S ) -> F e. _V ) $= - ( wcel wral wa wfun cdm cvv mpofun cv csn cxp sylancr ciun dmmpossx xpexg - wss snex mpan ralimi iunexg sylan2 ssexg funex ) CFJZDGJZACKZLZHMHNZOJZHO - JABCDEHIPUOUPACAQZRZDSZUAZUDVAOJZUQABCDEHIUBUNULUTOJZACKVBUMVCACUSOJUMVCU - RUEUSDOGUCUFUGACUTFOUHUIUPVAOUJTOHUKT $. + ( wcel wral wa wfun cdm cvv mpofun cv csn cxp sylancr ciun dmmpossx vsnex + wss xpexg mpan ralimi iunexg sylan2 ssexg funex ) CFJZDGJZACKZLZHMHNZOJZH + OJABCDEHIPUOUPACAQRZDSZUAZUDUTOJZUQABCDEHIUBUNULUSOJZACKVAUMVBACUROJUMVBA + UCURDOGUEUFUGACUSFOUHUIUPUTOUJTOHUKT $. $d x B $. $( Existence of an operation class abstraction (special case). @@ -78867,12 +79349,12 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) $) cnvf1o $p |- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A ) $= - ( vy wrel ccnv cv csn cuni cmpt cvv eqid wcel wa snex cnvex a1i cnvf1olem - uniex wceq wb relcnv simpr sylancr dfrel2 eleq2 sylbi anbi1d adantr mpbid - impbida f1od ) BDZACBBEZAFZGZEZHZCFZGZEZHZABUQIZJJVBKUQJLULUNBLZMUPUOUNNO - RPVAJLULURUMLZMUTUSURNORPULVCURUQSZMZVDUNVASMZBUNURQULVGMZUNUMEZLZVEMZVFV - HUMDVGVKBUAULVGUBUMURUNQUCULVKVFTVGULVJVCVEULVIBSVJVCTBUDVIBUNUEUFUGUHUIU - JUK $. + ( vy wrel ccnv cv csn cuni cmpt cvv eqid wcel vsnex cnvex uniex cnvf1olem + wa a1i wceq wb relcnv simpr sylancr dfrel2 eleq2 sylbi anbi1d adantr f1od + mpbid impbida ) BDZACBBEZAFZGZEZHZCFZGZEZHZABUQIZJJVBKUQJLULUNBLZQUPUOAMN + ORVAJLULURUMLZQUTUSCMNORULVCURUQSZQZVDUNVASQZBUNURPULVGQZUNUMEZLZVEQZVFVH + UMDVGVKBUAULVGUBUMURUNPUCULVKVFTVGULVJVCVEULVIBSVJVCTBUDVIBUNUEUFUGUHUJUK + UI $. $} ${ @@ -79484,6 +79966,175 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Ordering Ordinal Sequences +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + ${ + $d A f x $. $d G f x $. $d X x $. + orderseqlem.1 $e |- F = { f | E. x e. On f : x --> A } $. + $( Lemma for ~ poseq and ~ soseq . The function value of a sequence is + either in ` A ` or null. (Contributed by Scott Fenton, 8-Jun-2011.) $) + orderseqlem $p |- ( G e. F -> ( G ` X ) e. ( A u. { (/) } ) ) $= + ( wcel cv wf con0 wrex cfv c0 csn cun wceq feq1 wss syl rexbidv ibi unss1 + elab2g crn frn fvrn0 ssel mpisyl rexlimivw ) EDHZAIZBEJZAKLZFEMZBNOZPZHZU + KUNULBCIZJZAKLUNCEDDUSEQUTUMAKULBUSERUAGUDUBUMURAKUMEUEZUPPZUQSZUOVBHURUM + VABSVCULBEUFVABUPUCTEFUGVBUQUOUHUIUJT $. + $} + + ${ + $d A b f x $. $d a b c f g t w x y z $. $d F a b c f g t w x z $. + $d R f g t w x z $. $d S a b c $. + poseq.1 $e |- R Po ( A u. { (/) } ) $. + poseq.2 $e |- F = { f | E. x e. On f : x --> A } $. + poseq.3 $e |- S = { <. f , g >. | ( ( f e. F /\ g e. F ) /\ + E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ + ( f ` x ) R ( g ` x ) ) ) } $. + $( A partial ordering of ordinal sequences. (Contributed by Scott Fenton, + 8-Jun-2011.) $) + poseq $p |- S Po F $= + ( vz vw wbr wa wral cfv con0 wrex anbi12d va vb vc vt wpo cv wn wcel wceq + wi w3a c0 csn cun cab feq2 cbvrexvw abbii eqtri orderseqlem poirr sylancr + wf intnand adantr nrexdv imnan vex weq eleq1w anbi1d fveq1 eqeq1d ralbidv + mpbi breq1d rexbidv anbi2d eqeq2d breq2d mtbir raleq fveq2 breq12d bitrid + brab simplll simplrr an4 2rexbii reeanv bitri word eloni ordtri3or syl2an + wel w3o simp1l wss onelss imp adantll ssralv anim2d r19.26 syl6ibr ralimi + eqtr syl adantrd 3impia eqeq12d rspcv breq2 biimpd com3l ad2ant2lr impcom + syl6 3adant1 rspcev syl12anc 3exp ad2antrr ad2antlr ad2antll 3jca anim12i + potr anassrs sylan2 exp32 imbi1d syl5ibcom simp1r adantlr anim1d sylbir + a1d breq1 biimprd ad2ant2rl 3jaod mpd rexlimivv jca31 an4s syl2anb sylibr + pm3.2i a1i rgen3 df-po mpbir ) HEUEUAUFZUUPENZUGZUUPUBUFZENZUUSUCUFZENZOZ + UUPUVAENZUJZOZUCHPUBHPUAHPUVFUAUBUCHHHUVFUUPHUHZUUSHUHZUVAHUHZUKUURUVEUUQ + 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On f : x --> A } $. + soseq.3 $e |- S = { <. f , g >. | ( ( f e. F /\ g e. F ) /\ + E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ + ( f ` x ) R ( g ` x ) ) ) } $. + soseq.4 $e |- -. (/) e. A $. + $( A linear ordering of ordinal sequences. (Contributed by Scott Fenton, + 8-Jun-2011.) $) + soseq $p |- S Or F $= + ( wral wcel wa cfv wceq con0 wrex wi va vb vp wor wpo wbr weq w3o csn cun + vq cv c0 sopo ax-mp poseq wo wn eleq1w anbi1d fveq1 eqeq1d ralbidv breq1d + anbi12d rexbidv anbi2d eqeq2d breq2d brabg bianabs ancoms notbid ralinexa + wb orbi12d andi eqcom ralbii anbi1i orbi2i bitri rexbii r19.43 xchbinx wf + cab feq2 cbvrexvw abbii eqtri orderseqlem sotrieq mpan syl2an imbi2d wsbc + vex fveq2 eqeq12d sbcie imbi1i tfisg sylbir feq1 elab2 anbi12i reeanv wss + bitr4i onss ssralv syl ad2antlr rspcv a1i ffvelcdm cdm eloni ordirr eleq2 + fdm word biimparc sylan ndmfv eleq1 ex com23 sylan2 exp4b imp32 syldd imp + syl5 mtoi syld jcad wfn ffn eqtr2 biimprd 3syl adantlr eqtr biimpd expcom + ad2antrr adantll ordtri3or adantr 3orel13 syl6ci eqfnfv2 adantl rexlimivv + sylibrd sylbi sylbird syl5bir sylbid orrd df-3or bitr2i sylib rgen2 df-so + 3orcomb mpbir2an ) HEUDHEUEUAULZUBULZEUFZUAUBUGZUVKUVJEUFZUHZUBHMUAHMABCD + EFGHCUMUIUJZDUDZUVPDUEIUVPDUNUOJKUPUVOUAUBHHUVJHNZUVKHNZOZUVLUVNUQZUVMUQZ + UVOUVTUWAUVMUVTUWAURBULZUVJPZUWCUVKPZQZBAULZMZUWGUVJPZUWGUVKPZDUFZOZARSZU + WEUWDQZBUWGMZUWJUWIDUFZOZARSZUQZURZUVMUVTUWAUWSUVTUVLUWMUVNUWRUVTUVLUWMFU + LZHNZGULZHNZOZUWCUXAPZUWCUXCPZQZBUWGMZUWGUXAPZUWGUXCPZDUFZOZARSZOZUVRUXDO + ZUWDUXGQZBUWGMZUWIUXKDUFZOZARSZOUVTUWMOFGUVJUVKHHEFUAUGZUXEUXPUXNUYAUYBUX + BUVRUXDFUAHUSUTUYBUXMUXTARUYBUXIUXRUXLUXSUYBUXHUXQBUWGUYBUXFUWDUXGUWCUXAU + VJVAVBVCUYBUXJUWIUXKDUWGUXAUVJVAVDVEVFVEGUBUGZUXPUVTUYAUWMUYCUXDUVSUVRGUB + HUSVGUYCUXTUWLARUYCUXRUWHUXSUWKUYCUXQUWFBUWGUYCUXGUWEUWDUWCUXCUVKVAVHVCUY + CUXKUWJUWIDUWGUXCUVKVAVIVEVFVEKVJVKUVSUVRUVNUWRVOUVSUVROZUVNUWRUXOUVSUXDO + ZUWEUXGQZBUWGMZUWJUXKDUFZOZARSZOUYDUWROFGUVKUVJHHEFUBUGZUXEUYEUXNUYJUYKUX + BUVSUXDFUBHUSUTUYKUXMUYIARUYKUXIUYGUXLUYHUYKUXHUYFBUWGUYKUXFUWEUXGUWCUXAU + VKVAVBVCUYKUXJUWJUXKDUWGUXAUVKVAVDVEVFVEGUAUGZUYEUYDUYJUWRUYLUXDUVRUVSGUA + HUSVGUYLUYIUWQARUYLUYGUWOUYHUWPUYLUYFUWNBUWGUYLUXGUWDUWEUWCUXCUVJVAVHVCUY + LUXKUWIUWJDUWGUXCUVJVAVIVEVFVEKVJVKVLVPVMUWTUWHUWKUWPUQZURZTZARMZUVTUVMUY + PUWHUYMOZARSZUWSUWHUYMARVNUYRUWLUWQUQZARSUWSUYQUYSARUYQUWLUWHUWPOZUQUYSUW + HUWKUWPVQUYTUWQUWLUWHUWOUWPUWFUWNBUWGUWDUWEVRVSVTWAWBWCUWLUWQARWDWBWEUVTU + YPUWHUWIUWJQZTZARMZUVMUVTVUBUYOARUVTVUAUYNUWHUVRUWIUVPNZUWJUVPNZVUAUYNVOZ + UVSBCFHUVJUWGHUWGCUXAWFZARSZFWGUWCCUXAWFZBRSZFWGJVUHVUJFVUGVUIABRUWGUWCCU + XAWHWIWJWKZWLBCFHUVKUWGVUKWLUVQVUDVUEOVUFIUVPUWIUWJDWMWNWOWPVCVUCVUAARMZU + VTUVMVUCVUAAUWCWQZBUWGMZVUATZARMVULVUOVUBARVUNUWHVUAVUMUWFBUWGVUAUWFAUWCB + WRABUGUWIUWDUWJUWEUWGUWCUVJWSUWGUWCUVKWSWTXAVSXBVSVUAABXCXDUVTUCULZCUVJWF + ZUKULZCUVKWFZOZUKRSUCRSZVULUVMTZUVTVUQUCRSZVUSUKRSZOVVAUVRVVCUVSVVDUVRUWG + CUVJWFZARSZVVCVUHVVFFUVJHUAWRUYBVUGVVEARUWGCUXAUVJXEVFJXFVVEVUQAUCRUWGVUP + CUVJWHWIWBUVSUWGCUVKWFZARSZVVDVUHVVHFUVKHUBWRUYKVUGVVGARUWGCUXAUVKXEVFJXF + VVGVUSAUKRUWGVURCUVKWHWIWBXGVUQVUSUCUKRRXHXJVUTVVBUCUKRRVUPRNZVURRNZOZVUT + VVBVVKVUTOZVULUCUKUGZVUAAVUPMZOZUVMVVLVULVVMVVNVVLVULVUPVURNZURZVURVUPNZU + RZOVVPVVMVVRUHZVVMVVLVULVVQVVSVVLVULVUAAVURMZVVQVVJVULVWATZVVIVUTVVJVURRX + IVWBVURXKVUAAVURRXLXMXNVVLVWAVVQVVLVWAOVVPUMCNZLVVLVWAVVPVWCTVVLVVPVWAVWC + VVLVVPVWAVUPUVJPZVUPUVKPZQZVWCVVPVWAVWFTTVVLVUAVWFAVUPVURAUCUGUWIVWDUWJVW + EUWGVUPUVJWSUWGVUPUVKWSWTXOXPVVKVUQVUSVVPVWFVWCTZTVVKVUQVUSVVPVWGVUSVVPOV + WECNZVVKVUQOVWGVURCVUPUVKXQVVIVUQVWHVWGTZVVJVUQVVIUVJXRZVUPQZVWIVUPCUVJYB + VVIVWKOZVWFVWHVWCVWLVUPVWJNZURZVWDUMQZVWFVWHVWCTZTVVIVUPVUPNZURZVWKVWNVVI + VUPYCZVWRVUPXSZVUPXTXMVWKVWNVWRVWKVWMVWQVWJVUPVUPYAVMYDYEVUPUVJYFVWOVWFVW + PVWOVWFOUMVWEQZVWPVWDUMVWEUUAVXAVWCVWHUMVWECYGUUBXMYHUUCYIYJUUDYOYKYLYMYI + YNYPYHYQVVLVULVVNVVSVVIVULVVNTZVVJVUTVVIVUPRXIVXBVUPXKVUAAVUPRXLXMUUHZVVL + VVNVVSVVLVVNOVVRVWCLVVLVVNVVRVWCTVVLVVRVVNVWCVVLVVRVVNVURUVJPZVURUVKPZQZV + WCVVRVVNVXFTTVVLVUAVXFAVURVUPAUKUGUWIVXDUWJVXEUWGVURUVJWSUWGVURUVKWSWTXOX + PVVKVUQVUSVVRVXFVWCTZTZVVKVUSVUQVXHVVKVUSVUQVVRVXGVUQVVROVXDCNZVVKVUSOVXG + VUPCVURUVJXQVUSVVKUVKXRZVURQZVXIVXGTZVURCUVKYBVVJVXKVXLVVIVVJVXKOVXEUMQZV + XLVVJVURVURNZURZVXKVXMVVJVURYCZVXOVURXSZVURXTXMVXOVXKOVURVXJNZURZVXMVXKVX + SVXOVXKVXRVXNVXJVURVURYAVMYDVURUVKYFXMYEVXMVXFVXIVWCVXFVXMVXIVWCTZVXFVXMO + VXDUMQZVXTVXDVXEUMUUEVYAVXIVWCVXDUMCYGUUFXMUUGYIXMUUIYJYOYKYIYLYMYIYNYPYH + YQYRVVKVVTVUTVVIVWSVXPVVTVVJVWTVXQVUPVURUUJWOUUKVVPVVMVVRUULUUMVXCYRVUTUV + MVVOVOZVVKVUQUVJVUPYSUVKVURYSVYBVUSVUPCUVJYTVURCUVKYTAVUPVURUVJUVKUUNWOUU + OUUQYHUUPUURYOUUSUUTUVAUVBUVOUVLUVNUVMUHUWBUVLUVMUVNUVHUVLUVNUVMUVCUVDUVE + UVFUAUBHEUVGUVI $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The support of functions @@ -80512,8 +81163,8 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( ctpos cdm ccnv c0 csn cun crn cxp wss wrel tposssxp relxp relss mp2 ) A BZACDEFGZAHZIZJSKPKALQRMPSNO $. - $( Value of the transposition at a pair ` <. A , B >. ` . (Contributed by - Mario Carneiro, 10-Sep-2015.) $) + $( Value of the transposition at an ordered pair ` <. A , B >. ` . + (Contributed by Mario Carneiro, 10-Sep-2015.) $) brtpos2 $p |- ( B e. V -> ( A tpos F B <-> ( A e. ( `' dom F u. { (/) } ) /\ U. `' { A } F B ) ) ) $= ( vy vx wcel cvv wbr cdm ccnv csn cuni wa wi a1i wb cv wceq bitri reltpos @@ -80530,7 +81181,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and $( The behavior of ` tpos ` when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair - when the arguments are proper classes). This allows us to eliminate + when the arguments are proper classes). This allows to eliminate sethood hypotheses on ` A , B ` in ~ brtpos . (Contributed by Mario Carneiro, 10-Sep-2015.) $) brtpos0 $p |- ( A e. V -> ( (/) tpos F A <-> (/) F A ) ) $= @@ -80721,7 +81372,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and HZCIZAJZEZUKFZBGZHABCKULBUKKUFUJUMUOUFUGUMUIACLMUFUGUIUOUFUGHZUOUIUPUNUHB UPCNZDZUNUHGUGURUFUGUQAACQOPCRSUAUBUCUDABCTULBUKTUE $. - $( The domain and range of a transposition. (Contributed by NM, + $( The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) $) tposf2 $p |- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) ) $= ( wrel wf ccnv ctpos wa crn wfo tposfo2 wfn ffn dffn4 sylib impel fof syl @@ -80746,14 +81397,14 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( wrel wf1 wfo wa ccnv ctpos wf1o tposf12 tposfo2 anim12d df-f1o 3imtr4g ) ADZABCEZABCFZGAHZBCIZEZSBTFZGABCJSBTJPQUARUBABCKABCLMABCNSBTNO $. - $( The domain and range of a transposition. (Contributed by NM, + $( The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.) $) tposfo $p |- ( F : ( A X. B ) -onto-> C -> tpos F : ( B X. A ) -onto-> C ) $= ( cxp wfo ccnv ctpos wrel wi relxp tposfo2 ax-mp wceq cnvxp foeq2 sylib wb ) ABEZCDFZSGZCDHZFZBAEZCUBFZSITUCJABKSCDLMUAUDNUCUERABOUAUDCUBPMQ $. - $( The domain and range of a transposition. (Contributed by NM, + $( The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.) $) tposf $p |- ( F : ( A X. B ) --> C -> tpos F : ( B X. A ) --> C ) $= ( cxp wf ccnv ctpos wrel wi relxp tposf2 ax-mp cnvxp feq2i sylib ) ABEZCD @@ -81519,7 +82170,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= WQVCPLABVDWFWDWTWQVEVBZWEWTWFXBABWPVFVGVHWQASXAABWPVIVJWGAWHVKZVMVNZXCASZ XDTXCBWPUHVMQPXCVOWGXDTZXEXDXFAWHVPWGXDVQVRPAXCBVSVTWACAWBWC $. - $d X x y z u v a b c f g h $. + $d X z $. $( Law of well-founded recursion over a partial order, part two. Now we establish the value of ` F ` within ` A ` . (Contributed by Scott Fenton, 11-Sep-2023.) (Proof shortened by Scott Fenton, @@ -81620,6 +82271,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= $( $j usage 'fprresex' avoids 'ax-rep'; $) $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Well-ordered recursion @@ -81664,7 +82316,6 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= LPUHUOUGEICDBILEUIUJUPUKULUMUNUQURUSUTVA $. $} - $( General equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.) $) @@ -82341,6 +82992,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= DFEUDUH $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Functions on ordinals; strictly monotone ordinal functions @@ -82767,9 +83419,9 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= ${ $d A x $. $d B x $. $d F x $. - $( The range of a strictly monotone ordinal function dominates the domain. - (Contributed by Mario Carneiro, 13-Mar-2013.) $) - smorndom $p |- ( ( F : A --> B /\ Smo F /\ Ord B ) -> A C_ B ) $= + $( The codomain of a strictly monotone ordinal function dominates the + domain. (Contributed by Mario Carneiro, 13-Mar-2013.) $) + smocdmdom $p |- ( ( F : A --> B /\ Smo F /\ Ord B ) -> A C_ B ) $= ( vx wf wsmo word w3a cv wcel wa cfv wss wfn simpl1 simpl2 smodm2 syl2anc ffnd ordelord sylancom simpl3 simpr syl3anc ffvelcdm 3ad2antl1 ordtr2 imp smogt syl22anc ex ssrdv ) ABCEZCFZBGZHZDABUPDIZAJZUQBJZUPURKZUQGZUOUQUQCL @@ -83390,7 +84042,7 @@ other applications (for instance in intuitionistic set theory) need it. $( Define a recursive definition generator on ` On ` (the class of ordinal numbers) with characteristic function ` F ` and initial value ` I ` . This combines functions ` F ` in ~ tfr1 and ` G ` in ~ tz7.44-1 into one - definition. This rather amazing operation allows us to define, with + definition. This rather amazing operation allows to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in @@ -83683,6 +84335,7 @@ other applications (for instance in intuitionistic set theory) need it. AUAUBUCUTGURUKUDUEUFCDUQAGURUNAUPUGURUHUIUJ $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Finite recursion @@ -83691,7 +84344,7 @@ other applications (for instance in intuitionistic set theory) need it. $( We use the next 3 theorems for finite recursion. Note that we take advantage of what we have done for transfinite recursion, which allows - us to explicitly specify the function and moreover not require the + to explicitly specify the function and moreover not require the existence of omega. $) $( The function generated by finite recursive definition generation is a @@ -84185,6 +84838,13 @@ other applications (for instance in intuitionistic set theory) need it. 2on0 $p |- 2o =/= (/) $= ( c2o c1o csuc c0 df-2o nsuceq0 eqnetri ) ABCDEBFG $. + $( Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, + 6-Jan-2025.) $) + ord3 $p |- Ord 3o $= + ( c3o word c2o csuc con0 wcel 2on eloni ordsuci mp2b wceq df-3o ordeq ax-mp + wb mpbir ) ABZCDZBZCEFCBSGCHCIJARKQSOLARMNP $. + $( $j usage 'ord3' avoids 'ax-un'; $) + $( Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) $) 3on $p |- 3o e. On $= @@ -86873,6 +87533,7 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so LWIEBVMVQXQYHWLVRWAVSWDWMYIAEXRWJYHWLWDYBYCWJYHPXQYDYEBWIVNSVTWBVPWC $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Equivalence relations and classes @@ -87856,7 +88517,7 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so UABVGVABVGUABUTEUBZUCVCUTUHZVDVGEVAVCUTFUDBUTEUEZUFUGUIUJAEJTZBIUKUTITVAJ TZAVMBILULVMVNBUTIBVAJVJUMVKEVAJVLUNUOUPMNBUTVBEUQUR $. - $( The domain and range of the function ` F ` . (Contributed by Mario + $( The domain and codomain of the function ` F ` . (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.) $) qliftf $p |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) ) $= ( vy wfun cv cec cmpt crn wf wceq cqs qliftlem fliftf wrex cab df-qs eqid @@ -89284,8 +89945,8 @@ the first case of his notation (simple exponentiation) and subscript it nfixpw.2 $e |- F/_ y B $. $( Bound-variable hypothesis builder for indexed Cartesian product. Version of ~ nfixp with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by Mario Carneiro, 15-Oct-2016.) - (Revised by Gino Giotto, 26-Jan-2024.) $) + require ~ ax-13 . (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid + ~ ax-13 . (Revised by Gino Giotto, 26-Jan-2024.) $) nfixpw $p |- F/_ y X_ x e. A B $= ( vz cixp cv wcel cab wfn cfv nfcv wnfc wtru nfab a1i mptru wnf wa df-ixp wral nfel nffn wal df-ral nftru nffvd nfeld nfimd nfald nfxfr nfan nfcxfr @@ -89774,15 +90435,17 @@ the first case of his notation (simple exponentiation) and subscript it ${ $d f x y A $. $d f x y B $. $d y C $. $( Equinumerosity relation. This variation of ~ bren does not require the - Axiom of Union. (Contributed by BTernaryTau, 23-Sep-2024.) $) + Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a + subproof of ~ bren . (Revised by BTernaryTau, 23-Sep-2024.) $) breng $p |- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) ) $= ( vx vy cv wf1o wex cen wceq f1oeq2 exbidv f1oeq3 df-en brabg ) FHZGHZCHZ IZCJASTIZCJABTIZCJFGABDEKRALUAUBCRASTMNSBLUBUCCSBATONFGCPQ $. $( $j usage 'breng' avoids 'ax-un'; $) - $( Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) (Proof - shortened by BTernaryTau, 23-Sep-2024.) $) + $( Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) Extract + ~ breng as an intermediate result. (Revised by BTernaryTau, + 23-Sep-2024.) $) bren $p |- ( A ~~ B <-> E. f f : A -1-1-onto-> B ) $= ( cen wbr cvv wcel wa wf1o wex encv wfn f1ofn cdm fndm vex dmex eqeltrrdi cv syl crn wfo wceq f1ofo forn rnex jca exlimiv breng pm5.21nii ) ABDEAFG @@ -89801,15 +90464,17 @@ the first case of his notation (simple exponentiation) and subscript it UP $. $( Dominance relation. This variation of ~ brdomg does not require the - Axiom of Union. (Contributed by BTernaryTau, 29-Nov-2024.) $) + Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a + subproof of ~ brdomg . (Revised by BTernaryTau, 29-Nov-2024.) $) brdom2g $p |- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) $= ( vx vy cv wf1 wex cdom wceq f1eq2 exbidv f1eq3 df-dom brabg ) FHZGHZCHZI ZCJASTIZCJABTIZCJFGABDEKRALUAUBCRASTMNSBLUBUCCSBATONFGCPQ $. $( $j usage 'brdom2g' avoids 'ax-un'; $) - $( Dominance relation. (Contributed by NM, 15-Jun-1998.) (Proof shortened - by BTernaryTau, 29-Nov-2024.) $) + $( Dominance relation. (Contributed by NM, 15-Jun-1998.) Extract + ~ brdom2g as an intermediate result. (Revised by BTernaryTau, + 29-Nov-2024.) $) brdomg $p |- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) $= ( cvv wcel cdom wbr cv wf1 wex wb wi brdom2g ex wn reldom brrelex1i f1f wf cdm fdm vex dmex eqeltrrdi syl exlimiv pm5.21ni a1d pm2.61i ) AEFZBCFZ @@ -91739,11 +92404,11 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily domunsn $p |- ( A ~< B -> ( A u. { C } ) ~<_ B ) $= ( vz csdm wbr cv wcel csn cun cdom c0 wceq wn wex sdom0 breq2 sylib cvv wa mtbiri con2i neq0 cdif domdifsn adantr cen en2sn elvd endom syl biimpi - snprc snex 0dom eqbrtrdi pm2.61i disjdifr undom mpan2 sylancl uncom simpr - cin wss snssd undif eqtrid breqtrd exlimddv ) ABEFZDGZBHZACIZJZBKFDVKBLMZ - NVMDOVPVKVPVKALEFAPBLAEQUAUBDBUCRVKVMTZVOBVLIZUDZVRJZBKVQAVSKFZVNVRKFZVOV - TKFZVKWAVMABVLUEUFCSHZWBWDVNVRUGFZWBWDWEDCVLSSUHUIVNVRUJUKWDNZVNLVRKWFVNL - MCUMULVRVLUNUOUPUQWAWBTVSVRVDLMWCVRBURAVSVNVRUSUTVAVQVTVRVSJZBVSVRVBVQVRB + snprc vsnex eqbrtrdi pm2.61i cin disjdifr undom mpan2 sylancl uncom simpr + 0dom wss snssd undif eqtrid breqtrd exlimddv ) ABEFZDGZBHZACIZJZBKFDVKBLM + ZNVMDOVPVKVPVKALEFAPBLAEQUAUBDBUCRVKVMTZVOBVLIZUDZVRJZBKVQAVSKFZVNVRKFZVO + VTKFZVKWAVMABVLUEUFCSHZWBWDVNVRUGFZWBWDWEDCVLSSUHUIVNVRUJUKWDNZVNLVRKWFVN + LMCUMULVRDUNVDUOUPWAWBTVSVRUQLMWCVRBURAVSVNVRUSUTVAVQVTVRVSJZBVSVRVBVQVRB VEWGBMVQVLBVKVMVCVFVRBVGRVHVIVJ $. $( There exists a mapping from a set onto any (nonempty) set that it @@ -91751,25 +92416,25 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily fodomr $p |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B ) $= ( vz vg c0 cdom wa cvv wcel cv wex adantl wne crn cun sylib wceq wfun cdm wbr wf1 wfo reldom brrelex2i wb brrelex1i 0sdomg n0 bitrdi biimpac brdomi - csdm syl wi ccnv cdif csn cxp difexg snex xpexg sylancl cnvex jctil unexb - vex wfn cin wf df-f1 simprbi fconst ffun ax-mp jctir df-rn eqcomi ineq12i - snnz dmxp disjdif eqtri funun dmun uneq1i uneq2i 3eqtr2i wss undif eqtrid - f1f frnd df-fn sylanbrc rnun dfdm4 f1dm eqtr3id uneq1d xpeq1 eqtrdi rneqd - 0xp rn0 0ss eqsstrdi a1d rnxp adantr snssi eqsstrd ex pm2.61ine sylan9eqr - ssequn2 df-fo foeq1 spcegv syl2im expdimp exlimdv syl3c ) FBUMUAZBAGUAZHA - IJZDKZBJZDLZBAEKZUBZELZABCKZUCZCLZYEYFYDBAGUDUEMYEYDYIYEBIJZYDYIUFBAGUDUG - YPYDBFNYIBIUHDBUIUJUNUKYEYLYDBAEULMYFYHYLYOUOZDYFYHYQYFYHHYKYOEYFYHYKYOYF - YJUPZAYJOZUQZYGURZUSZPZIJZYHYKHZABUUCUCZYOYFYRIJZUUBIJZHUUDYFUUHUUGYFYTIJ - UUAIJUUHAYSIUTYGVAYTUUAIIVBVCYJEVGVDVEYRUUBVFQUUEUUCAVHZUUCOZBRUUFUUEUUCS - ZUUCTZARZUUIYKUUKYHYKYRSZUUBSZHYRTZUUBTZVIZFRUUKYKUUNUUOYKBAYJVJUUNBAYJVK - VLYTUUAUUBVJUUOYTYGDVGZVMYTUUAUUBVNVOVPUURYSYTVIFUUPYSUUQYTYSUUPYJVQZVRUU - AFNUUQYTRYGUUSVTYTUUAWAVOZVSYSAWBWCYRUUBWDVCMYKUUMYHYKUULYSYTPZAUULUUPUUQ - PYSUUQPUVBYRUUBWEYSUUPUUQUUTWFUUQYTYSUVAWGWHYKYSAWIUVBARYKBAYJBAYJWLWMYSA - WJQWKMUUCAWNWOUUEUUJYROZUUBOZPZBYRUUBWPYKYHUVEBUVDPZBYKUVCBUVDYKUVCYJTBYJ - WQBAYJWRWSWTYHUVDBWIZUVFBRYHUVGUOYTFYTFRZUVGYHUVHUVDFBUVHUVDFOFUVHUUBFUVH - UUBFUUAUSFYTFUUAXAUUAXDXBXCXEXBBXFXGXHYTFNZYHUVGUVIYHHUVDUUABUVIUVDUUARYH - YTUUAXIXJYHUUABWIUVIYGBXKMXLXMXNUVDBXPQXOWKABUUCXQWOYNUUFCUUCIABYMUUCXRXS - XTYAYBXMYBYC $. + csdm syl wi ccnv csn cxp difexg vsnex xpexg sylancl vex cnvex jctil unexb + cdif wfn cin df-f1 simprbi fconst ffun ax-mp jctir df-rn eqcomi snnz dmxp + ineq12i disjdif eqtri funun dmun uneq1i uneq2i 3eqtr2i wss f1f frnd undif + wf eqtrid df-fn sylanbrc rnun dfdm4 eqtr3id uneq1d xpeq1 0xp eqtrdi rneqd + f1dm rn0 0ss eqsstrdi a1d rnxp adantr eqsstrd pm2.61ine ssequn2 sylan9eqr + snssi ex df-fo foeq1 spcegv syl2im expdimp exlimdv syl3c ) FBUMUAZBAGUAZH + AIJZDKZBJZDLZBAEKZUBZELZABCKZUCZCLZYEYFYDBAGUDUEMYEYDYIYEBIJZYDYIUFBAGUDU + GYPYDBFNYIBIUHDBUIUJUNUKYEYLYDBAEULMYFYHYLYOUOZDYFYHYQYFYHHYKYOEYFYHYKYOY + FYJUPZAYJOZVGZYGUQZURZPZIJZYHYKHZABUUCUCZYOYFYRIJZUUBIJZHUUDYFUUHUUGYFYTI + JUUAIJUUHAYSIUSDUTYTUUAIIVAVBYJEVCVDVEYRUUBVFQUUEUUCAVHZUUCOZBRUUFUUEUUCS + ZUUCTZARZUUIYKUUKYHYKYRSZUUBSZHYRTZUUBTZVIZFRUUKYKUUNUUOYKBAYJWLUUNBAYJVJ + VKYTUUAUUBWLUUOYTYGDVCZVLYTUUAUUBVMVNVOUURYSYTVIFUUPYSUUQYTYSUUPYJVPZVQUU + AFNUUQYTRYGUUSVRYTUUAVSVNZVTYSAWAWBYRUUBWCVBMYKUUMYHYKUULYSYTPZAUULUUPUUQ + PYSUUQPUVBYRUUBWDYSUUPUUQUUTWEUUQYTYSUVAWFWGYKYSAWHUVBARYKBAYJBAYJWIWJYSA + WKQWMMUUCAWNWOUUEUUJYROZUUBOZPZBYRUUBWPYKYHUVEBUVDPZBYKUVCBUVDYKUVCYJTBYJ + WQBAYJXDWRWSYHUVDBWHZUVFBRYHUVGUOYTFYTFRZUVGYHUVHUVDFBUVHUVDFOFUVHUUBFUVH + UUBFUUAURFYTFUUAWTUUAXAXBXCXEXBBXFXGXHYTFNZYHUVGUVIYHHUVDUUABUVIUVDUUARYH + YTUUAXIXJYHUUABWHUVIYGBXOMXKXPXLUVDBXMQXNWMABUUCXQWOYNUUFCUUCIABYMUUCXRXS + XTYAYBXPYBYC $. $} ${ @@ -92310,8 +92975,29 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily $) $( Lemma for ~ rexdif1en and ~ dif1en . (Contributed by BTernaryTau, - 18-Aug-2024.) $) - dif1enlem $p |- ( ( F e. V /\ M e. _om /\ F : A -1-1-onto-> suc M ) -> + 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to + avoid ~ ax-un . (Revised by BTernaryTau, 5-Jan-2025.) $) + dif1enlem $p |- ( ( ( F e. V /\ A e. W /\ M e. On ) /\ + F : A -1-1-onto-> suc M ) -> ( A \ { ( `' F ` M ) } ) ~~ M ) $= + ( wcel con0 wf1o cfv csn cdif cres wa wfo adantr wceq wb mpbird c0 cvv csuc + w3a ccnv cen wbr sucidg wfun dff1o3 simprbi f1ofo wfn f1ofn fnresdm 3syl wf + foeq1 wne f1ocnvdm f1ocnvfv2 snidg adantl eqeltrd fressnfv biimp3ar syl3anc + cin disjsn con2bii sylib fnresdisj syl neqned foconst syl2anc resdif sylan2 + wn mtbid word eloni orddif f1oeq3d ancoms 3ad2antl3 difexg f1oen4g syl3anl1 + resexg syl3anl2 syldan ) BDFZAEFZCGFZUBACUAZBHZACBUCZIZJZKZCBWSLZHZWSCUDUEZ + WMWKWOXAWLWOWMXAWOWMMXAWSWNCJZKZWTHZWMWOCWNFZXECGUFWOXFMZWPUGZAWNBALZNZWRXC + BWRLZNZXEWOXHXFWOAWNBNZXHAWNBUHUIOWOXJXFWOXJXMAWNBUJWOBAUKZXIBPXJXMQAWNBULZ + ABUMAWNXIBUPUNROXGWRXCXKUOZXKSUQXLXGXNWQAFZWQBIZXCFZXPWOXNXFXOOAWNCBURZXGXR + CXCAWNCBUSXFCXCFWOCWNUTVAVBXNXQXPXSAWQXCBVCVDVEXGXKSXGAWRVFSPZXKSPZXGXQYAVQ + XTYAXQAWQVGVHVIWOYAYBQZXFWOXNYCXOAWRBVJVKOVRVLWRCXKVMVNAWRWNXCBVOVEVPWMXAXE + QWOWMCXDWSWTWMCVSCXDPCVTCWAVKWBVARWCWDWLWKWSTFZWMXAXBAWREWEWKWTTFYDWMXAXBBW + SDWHWSCWTTTGWFWGWIWJ $. + $( $j usage 'dif1enlem' avoids 'ax-pow' 'ax-un'; $) + + $( Obsolete version of ~ dif1enlem as of 5-Jan-2025. (Contributed by + BTernaryTau, 18-Aug-2024.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + dif1enlemOLD $p |- ( ( F e. V /\ M e. _om /\ F : A -1-1-onto-> suc M ) -> ( A \ { ( `' F ` M ) } ) ~~ M ) $= ( wcel com wf1o cfv csn cdif cres wa wfo adantr wb mpbird c0 adantl syl3anc wceq csuc w3a ccnv cen wbr simp1 sucidg wfun dff1o3 simprbi f1ofo wfn f1ofn @@ -92325,50 +93011,100 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily WJUTRVAXJXMXLXOAWMWSBVBVCSXCXGQXCAWNVDQTZXGQTZXCXMXQVPXPXQXMAWMVFVGVHWKXQXR OZXBWKXJXSXKAWNBVIVJNVKVLWNCXGVMVNAWNWJWSBVOSVQWIWQXAOWKWICWTWOWPWICVRCWTTC VSCVTVJWARPWBWCWHWPWDEWQWRBWODWEWOCWPWDWFWGVN $. - $( $j usage 'dif1enlem' avoids 'ax-pow'; $) ${ $d A f x $. $d M f x $. - $( If a set is equinumerous to a nonzero finite ordinal, then there exists - an element in that set such that removing it leaves the set equinumerous - to the predecessor of that ordinal. (Contributed by BTernaryTau, - 26-Aug-2024.) $) - rexdif1en $p |- ( ( M e. _om /\ A ~~ suc M ) -> + $( If a set is equinumerous to a nonzero ordinal, then there exists an + element in that set such that removing it leaves the set equinumerous to + the predecessor of that ordinal. (Contributed by BTernaryTau, + 26-Aug-2024.) Generalize to all ordinals and avoid ~ ax-un . (Revised + by BTernaryTau, 5-Jan-2025.) $) + rexdif1en $p |- ( ( M e. On /\ A ~~ suc M ) -> + E. x e. A ( A \ { x } ) ~~ M ) $= + ( vf con0 wcel csuc cen wbr cv csn cdif wrex wi cvv encv simpld wa ancoms + sylan wf1o wex wb breng syl ibi cfv sucidg f1ocnvdm adantll vex dif1enlem + ccnv mp3anl1 wceq sneq difeq2d breq1d rspcev syl2anc exlimdv syl5 syldbl2 + ex ) CEFZBCGZHIZBAJZKZLZCHIZABMZVGVEVGVLNZVGBOFZVEVMVGVNVFOFZBVFPZQVGBVFD + JZUAZDUBZVNVERZVLVGVSVGVNVORVGVSUCVPBVFDOOUDUEUFVTVRVLDVTVRVLVTVRRCVQUMUG + ZBFZBWAKZLZCHIZVLVEVRWBVNVECVFFZVRWBCEUHVRWFWBBVFCVQUISTUJVQOFVNVEVRWEDUK + BVQCOOULUNVKWEAWABVHWAUOZVJWDCHWGVIWCBVHWAUPUQURUSUTVDVAVBTSVC $. + $( $j usage 'rexdif1en' avoids 'ax-pow' 'ax-un'; $) + $} + + ${ + $d A f x $. $d M f x $. + $( Obsolete version of ~ rexdif1en as of 5-Jan-2025. (Contributed by + BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rexdif1enOLD $p |- ( ( M e. _om /\ A ~~ suc M ) -> E. x e. A ( A \ { x } ) ~~ M ) $= ( vf csuc cen wbr com wcel cv wf1o wex csn cdif wrex bren 19.42v ccnv cvv - wa cfv sucidg f1ocnvdm ancoms sylan vex dif1enlem mp3an1 wceq sneq breq1d - difeq2d rspcev syl2anc exlimiv sylbir sylan2b ) BCEZFGCHIZBURDJZKZDLZBAJZ - MZNZCFGZABOZBURDPUSVBTUSVATZDLVGUSVADQVHVGDVHCUTRUAZBIZBVIMZNZCFGZVGUSCUR - IZVAVJCHUBVAVNVJBURCUTUCUDUEUTSIUSVAVMDUFBUTCSUGUHVFVMAVIBVCVIUIZVEVLCFVO - VDVKBVCVIUJULUKUMUNUOUPUQ $. - $( $j usage 'rexdif1en' avoids 'ax-pow'; $) + wa cfv sucidg f1ocnvdm ancoms sylan dif1enlemOLD mp3an1 wceq sneq difeq2d + vex breq1d rspcev syl2anc exlimiv sylbir sylan2b ) BCEZFGCHIZBURDJZKZDLZB + AJZMZNZCFGZABOZBURDPUSVBTUSVATZDLVGUSVADQVHVGDVHCUTRUAZBIZBVIMZNZCFGZVGUS + CURIZVAVJCHUBVAVNVJBURCUTUCUDUEUTSIUSVAVMDUKBUTCSUFUGVFVMAVIBVCVIUHZVEVLC + FVOVDVKBVCVIUIUJULUMUNUOUPUQ $. $} ${ $d A f $. $d M f $. $d X f $. - $( If a set ` A ` is equinumerous to the successor of a natural number - ` M ` , then ` A ` with an element removed is equinumerous to ` M ` . - For a proof with fewer symbols using ~ ax-pow , see ~ dif1enALT . + $( If a set ` A ` is equinumerous to the successor of an ordinal ` M ` , + then ` A ` with an element removed is equinumerous to ` M ` . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, - 16-Aug-2015.) Avoid ~ ax-pow . (Revised by BTernaryTau, - 26-Aug-2024.) $) - dif1en $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) -> + 16-Aug-2015.) Avoid ~ ax-pow . (Revised by BTernaryTau, 26-Aug-2024.) + Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.) $) + dif1en $p |- ( ( M e. On /\ A ~~ suc M /\ X e. A ) -> + ( A \ { X } ) ~~ M ) $= + ( vf cen wbr wcel con0 csn cdif w3a cvv wf1o wi wa ccnv cfv cpr cop jca + csuc simp1 encv simpld 3anim1i cv wex bren cres wceq wfun sucidg f1ocnvdm + cun 3adant2 f1ofvswap syld3an3 wb f1ocnvfv2 opeq2d preq1d f1oeq1d syl3an3 + uneq2d mpbid 3adant3r1 f1ofun opex prid1 elun2 ax-mp funopfv mpi f1ocnvfv + simpr2 syl2anc mpd sneqd difeq2d simpr1 3simpc anim2i 3anass sylibr simpl + 3syl simpr3 simpr vex prex unex dif1enlem mp3anl1 sylan2 eqbrtrrd exlimiv + resex ex sylbi sylc 3comr ) ABUAZEFZCAGZBHGZACIZJZBEFZXCXDXEKXCALGZXDXEKZ + XHXCXDXEUBXCXIXDXEXCXIXBLGAXBUCUDUEXCAXBDUFZMZDUGXJXHNZAXBDUHXLXMDXLXJXHX + LXJOZABXKACBXKPQZRJZUIZCBSZXOCXKQSZRZUNZPQZIZJZXGBEXNYCXFAXNYBCXNCYAQBUJZ + YBCUJZXNAXBYAMZYAUKZYEXLXDXEYGXIXEXLXDBXBGZYGBHULXLXDYIKAXBXQCXOXKQZSZXSR + ZUNZMZYGXLXDYIXOAGZYNXLYIYOXDAXBBXKUMUOAXBXKCXOUPUQXLYIYNYGURXDXLYIOZAXBY + MYAYPYLXTXQYPYKXRXSYPYJBCAXBBXKUSUTVAVDVBUOVEVCZVFZAXBYAVGYHXRYAGZYEXRXTG + YSXRXSCBVHVIXRXTXQVJVKCBYAVLVMWFXNYGXDYEYFNYRXLXIXDXEVOAXBCBYAVNVPVQVRVSX + NXIXLXDXEKZOZXIXEOZYTOYDBEFZXNXIYTXLXIXDXEVTXNXLXDXEOZOYTXJUUDXLXIXDXEWAW + BXLXDXEWCWDTUUAUUBYTUUAXIXEXIYTWEXIXLXDXEWGTXIYTWHTYTUUBYGUUCYQYALGXIXEYG + UUCXQXTXKXPDWIWQXRXSWJWKAYABLLWLWMWNWFWOWRWPWSWTXA $. + $( $j usage 'dif1en' avoids 'ax-pow'; $) + $} + + $( If a set ` A ` is equinumerous to the successor of a natural number + ` M ` , then ` A ` with an element removed is equinumerous to ` M ` . See + also ~ dif1ennnALT . (Contributed by BTernaryTau, 6-Jan-2025.) $) + dif1ennn $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) -> + ( A \ { X } ) ~~ M ) $= + ( com wcel con0 csuc cen wbr csn cdif nnon dif1en syl3an1 ) BDEBFEABGHICAEA + CJKBHIBLABCMN $. + $( $j usage 'dif1ennn' avoids 'ax-pow'; $) + + ${ + $d A f $. $d M f $. $d X f $. + $( Obsolete version of ~ dif1en as of 6-Jan-2025. (Contributed by Jeff + Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid + ~ ax-pow . (Revised by BTernaryTau, 26-Aug-2024.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + dif1enOLD $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) -> ( A \ { X } ) ~~ M ) $= ( vf cen wbr wcel com csn cdif wf1o wex w3a 3anass ccnv cfv cpr cop cun wa csuc bren 19.41v exbii 3bitr4i cres wceq sucidg wfun 3adant2 f1ofvswap cv f1ocnvdm syld3an3 wb f1ocnvfv2 opeq2d preq1d uneq2d f1oeq1d mpbid opex f1ofun prid1 elun2 ax-mp funopfv mpi 3syl wi f1ocnvfv syl2anc mpd syl3an3 - simp2 sneqd difeq2d cvv vex resex prex dif1enlem mp3an2i eqbrtrrd exlimiv - unex simp3 sylbir syl3an1b 3comr ) ABUAZEFZCAGZBHGZACIZJZBEFZWLAWKDULZKZD - LZWMWNWQAWKDUBWTWMWNMZWSWMWNMZDLZWQWSWMWNTZTZDLWTXDTXCXAWSXDDUCXBXEDWSWMW - NNUDWTWMWNNUEXBWQDXBABWRACBWROPZQJZUFZCBRZXFCWRPRZQZSZOPZIZJZWPBEXBXNWOAX - BXMCWNWSWMBWKGZXMCUGZBHUHZWSWMXPMZCXLPBUGZXQXSAWKXLKZXLUIZXTXSAWKXHCXFWRP - ZRZXJQZSZKZYAWSWMXPXFAGZYGWSXPYHWMAWKBWRUMUJAWKWRCXFUKUNWSXPYGYAUOWMWSXPT - ZAWKYFXLYIYEXKXHYIYDXIXJYIYCBCAWKBWRUPUQURUSUTUJVAZAWKXLVCYBXIXLGZXTXIXKG - YKXIXJCBVBVDXIXKXHVEVFCBXLVGVHVIXSYAWMXTXQVJYJWSWMXPVOAWKCBXLVKVLVMVNVPVQ - XLVRGXBWNYAXOBEFXHXKWRXGDVSVTXIXJWAWFWSWMWNWGWNWSWMXPYAXRYJVNAXLBVRWBWCWD - WEWHWIWJ $. - $( $j usage 'dif1en' avoids 'ax-pow'; $) + simp2 sneqd difeq2d cvv vex resex prex unex dif1enlemOLD mp3an2i eqbrtrrd + simp3 exlimiv sylbir syl3an1b 3comr ) ABUAZEFZCAGZBHGZACIZJZBEFZWLAWKDULZ + KZDLZWMWNWQAWKDUBWTWMWNMZWSWMWNMZDLZWQWSWMWNTZTZDLWTXDTXCXAWSXDDUCXBXEDWS + WMWNNUDWTWMWNNUEXBWQDXBABWRACBWROPZQJZUFZCBRZXFCWRPRZQZSZOPZIZJZWPBEXBXNW + OAXBXMCWNWSWMBWKGZXMCUGZBHUHZWSWMXPMZCXLPBUGZXQXSAWKXLKZXLUIZXTXSAWKXHCXF + WRPZRZXJQZSZKZYAWSWMXPXFAGZYGWSXPYHWMAWKBWRUMUJAWKWRCXFUKUNWSXPYGYAUOWMWS + XPTZAWKYFXLYIYEXKXHYIYDXIXJYIYCBCAWKBWRUPUQURUSUTUJVAZAWKXLVCYBXIXLGZXTXI + XKGYKXIXJCBVBVDXIXKXHVEVFCBXLVGVHVIXSYAWMXTXQVJYJWSWMXPVOAWKCBXLVKVLVMVNV + PVQXLVRGXBWNYAXOBEFXHXKWRXGDVSVTXIXJWAWBWSWMWNWFWNWSWMXPYAXRYJVNAXLBVRWCW + DWEWGWHWIWJ $. $} ${ @@ -92387,16 +93123,16 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily findcard $p |- ( A e. Fin -> ta ) $= ( vw wcel cen wbr com vv cfn cv wrex isfi wi wal csuc breq2 imbi1d albidv c0 wceq en0 mpbiri sylbi ax-gen w3a wa peano2 rspcev sylan sylibr 3adant2 - wral csn cdif dif1en 3expa vex difexi breq1 imbi12d syl5com ralrimdva imp - spcv an32s 3impa sylc 3exp alrimdv cbvalvw syl6ibr finds1 19.21bi vtoclga - rexlimiv ) AEFIUBMFUCZUBQWIPUCZRSZPTUDAPWIUEWKAPTWJTQWKAUFZFWLFUGWIULRSZA - UFZFUGWIUAUCZRSZAUFZFUGZWIWOUHZRSZAUFZFUGZPUAWJULUMZWLWNFXCWKWMAWJULWIRUI - UJUKWJWOUMZWLWQFXDWKWPAWJWOWIRUIUJUKWJWSUMZWLXAFXEWKWTAWJWSWIRUIUJUKWNFWM - WIULUMZAWIUNXFABNJUOUPUQWOTQZWRGUCZWSRSZDUFZGUGXBXGWRXJGXGWRXIDXGWRXIURXH - UBQZCHXHVEZDXGXIXKWRXGXIUSZXHWJRSZPTUDZXKXGWSTQXIXOWOUTXNXIPWSTWJWSXHRUIV - AVBPXHUEVCVDXGWRXIXLXGXIWRXLXMWRXLXMWRCHXHXMHUCZXHQZUSXHXPVFZVGZWORSZWRCX - GXIXQXTXHWOXPVHVIWQXTCUFFXSXHXRGVJVKWIXSUMWPXTACWIXSWORVLKVMVQVNVOVPVRVSO - VTWAWBXAXJFGWIXHUMWTXIADWIXHWSRVLLVMWCWDWEWFWHUPWG $. + wral csn cdif dif1ennn 3expa vex breq1 imbi12d spcv syl5com ralrimdva imp + difexi an32s sylc alrimdv cbvalvw syl6ibr finds1 19.21bi rexlimiv vtoclga + 3impa 3exp ) AEFIUBMFUCZUBQWIPUCZRSZPTUDAPWIUEWKAPTWJTQWKAUFZFWLFUGWIULRS + ZAUFZFUGWIUAUCZRSZAUFZFUGZWIWOUHZRSZAUFZFUGZPUAWJULUMZWLWNFXCWKWMAWJULWIR + UIUJUKWJWOUMZWLWQFXDWKWPAWJWOWIRUIUJUKWJWSUMZWLXAFXEWKWTAWJWSWIRUIUJUKWNF + WMWIULUMZAWIUNXFABNJUOUPUQWOTQZWRGUCZWSRSZDUFZGUGXBXGWRXJGXGWRXIDXGWRXIUR + XHUBQZCHXHVEZDXGXIXKWRXGXIUSZXHWJRSZPTUDZXKXGWSTQXIXOWOUTXNXIPWSTWJWSXHRU + IVAVBPXHUEVCVDXGWRXIXLXGXIWRXLXMWRXLXMWRCHXHXMHUCZXHQZUSXHXPVFZVGZWORSZWR + CXGXIXQXTXHWOXPVHVIWQXTCUFFXSXHXRGVJVQWIXSUMWPXTACWIXSWORVKKVLVMVNVOVPVRW + GOVSWHVTXAXJFGWIXHUMWTXIADWIXHWSRVKLVLWAWBWCWDWEUPWF $. $( $j usage 'findcard' avoids 'ax-pow'; $) $} @@ -92415,24 +93151,25 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily (Contributed by Jeff Madsen, 8-Jul-2010.) Avoid ~ ax-pow . (Revised by BTernaryTau, 26-Aug-2024.) $) findcard2 $p |- ( A e. Fin -> ta ) $= - ( vw vv cen wbr wi cfn cv wcel com wrex isfi wal csuc breq2 imbi1d albidv - c0 wceq weq en0 mpbiri sylbi ax-gen wsbc csn cdif rexdif1en wss cun snssi - wa uncom undif biimpi eqtrid vex difexi breq1 anbi2d uneq1 sbceq1d imbi2d - imbi12d spvv rspe sylibr pm2.27 adantl sylsyld syl5 snex unex sbcie vtocl - syl6ibr dfsbcq syl5ib 3syl com12 rexlimdv adantr mpd ex com23 alrimdv nfv - expd nfsbc1v nfim sbceq1a cbvalv1 finds1 19.21bi rexlimiv vtoclga ) AEFIU - AMFUBZUAUCXKPUBZRSZPUDUEAPXKUFXMAPUDXLUDUCXMATZFXNFUGXKULRSZATZFUGXKQUBZR - SZATZFUGZXKXQUHZRSZATZFUGZPQXLULUMZXNXPFYEXMXOAXLULXKRUIUJUKPQUNZXNXSFYFX - MXRAXLXQXKRUIUJUKXLYAUMZXNYCFYGXMYBAXLYAXKRUIUJUKXPFXOXKULUMZAXKUOYHABNJU - PUQURXQUDUCZXTXLYARSZAFXLUSZTZPUGYDYIXTYLPYIYJXTYKYIYJXTYKTZYIYJVFXLHUBZU - TZVAZXQRSZHXLUEZYMHXLXQVBYIYRYMTYJYIYQYMHXLYNXLUCZYIYQYMTYSYIYQYMYSYOXLVC - ZYPYOVDZXLUMZYIYQVFZYMTYNXLVEYTUUAYOYPVDZXLYPYOVGYTUUDXLUMYOXLVHVIVJUUCXT - AFUUAUSZTZUUBYMYIGUBZXQRSZVFZXTAFUUGYOVDZUSZTZTUUCUUFTGYPXLYOPVKVLUUGYPUM - ZUUIUUCUULUUFUUMUUHYQYIUUGYPXQRVMVNUUMUUKUUEXTUUMAFUUJUUAUUGYPYOVOVPVQVRU - UIXTDUUKXTUUHCTZUUIDXSUUNFGFGUNXRUUHACXKUUGXQRVMKVRVSUUIUUGUAUCZUUNCDUUIU - UHQUDUEUUOUUHQUDVTQUUGUFWAUUHUUNCTYIUUHCWBWCOWDWEADFUUJUUGYOGVKYNWFWGLWHW - JWIUUBUUEYKXTAFUUAXLWKVQWLWMXBWNWOWPWQWRWSWTYCYLFPYCPXAYJYKFYJFXAAFXLXCXD - FPUNYBYJAYKXKXLYARVMAFXLXEVRXFWJXGXHXIUQXJ $. + ( vw cen wbr wi wceq vv cfn cv wcel com wrex isfi wal breq2 imbi1d albidv + c0 csuc en0 mpbiri sylbi ax-gen wsbc wa csn cdif con0 rexdif1en sylan wss + cun snssi uncom undif biimpi eqtrid vex difexi breq1 anbi2d uneq1 sbceq1d + nnon imbi2d imbi12d spvv rspe sylibr pm2.27 sylsyld syl5 vsnex unex sbcie + adantl syl6ibr vtocl dfsbcq syl5ib 3syl expd com12 rexlimdv adantr mpd ex + com23 alrimdv nfv nfsbc1v sbceq1a cbvalv1 finds1 19.21bi rexlimiv vtoclga + nfim ) AEFIUBMFUCZUBUDXMPUCZQRZPUEUFAPXMUGXOAPUEXNUEUDXOASZFXPFUHXMULQRZA + SZFUHXMUAUCZQRZASZFUHZXMXSUMZQRZASZFUHZPUAXNULTZXPXRFYGXOXQAXNULXMQUIUJUK + XNXSTZXPYAFYHXOXTAXNXSXMQUIUJUKXNYCTZXPYEFYIXOYDAXNYCXMQUIUJUKXRFXQXMULTZ + AXMUNYJABNJUOUPUQXSUEUDZYBXNYCQRZAFXNURZSZPUHYFYKYBYNPYKYLYBYMYKYLYBYMSZY + KYLUSXNHUCZUTZVAZXSQRZHXNUFZYOYKXSVBUDYLYTXSVRHXNXSVCVDYKYTYOSYLYKYSYOHXN + YPXNUDZYKYSYOSUUAYKYSYOUUAYQXNVEZYRYQVFZXNTZYKYSUSZYOSYPXNVGUUBUUCYQYRVFZ + XNYRYQVHUUBUUFXNTYQXNVIVJVKUUEYBAFUUCURZSZUUDYOYKGUCZXSQRZUSZYBAFUUIYQVFZ + URZSZSUUEUUHSGYRXNYQPVLVMUUIYRTZUUKUUEUUNUUHUUOUUJYSYKUUIYRXSQVNVOUUOUUMU + UGYBUUOAFUULUUCUUIYRYQVPVQVSVTUUKYBDUUMYBUUJCSZUUKDYAUUPFGXMUUITXTUUJACXM + UUIXSQVNKVTWAUUKUUIUBUDZUUPCDUUKUUJUAUEUFUUQUUJUAUEWBUAUUIUGWCUUJUUPCSYKU + UJCWDWJOWEWFADFUULUUIYQGVLHWGWHLWIWKWLUUDUUGYMYBAFUUCXNWMVSWNWOWPWQWRWSWT + XAXBXCYEYNFPYEPXDYLYMFYLFXDAFXNXEXLXMXNTYDYLAYMXMXNYCQVNAFXNXFVTXGWKXHXIX + JUPXK $. $( $j usage 'findcard2' avoids 'ax-pow'; $) $} @@ -92690,14 +93427,14 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily (Revised by BTernaryTau, 7-Sep-2024.) $) pwfilem $p |- ( ~P b e. Fin -> ~P ( b u. { x } ) e. Fin ) $= ( vd cv cpw cfn wcel csn cun cdif pwundif crn wss wfun cima wceq sylib wa - funmpt2 cdm vex snex unex dmmpti imaeq2i eqtr3i imafi eqeltrrid mpan wrex - imadmrn eldifi elpwun undif1 elpwunsn snssd ssequn2 eqtr2id uneq1 syl2anc - rspceeqv elrnmptd ssriv ssfi sylancl unfi mpancom eqeltrid ) CGZHZIJZVLAG - ZKZLHZVQVMMZVMLZIVLVPNVRIJZVNVSIJVNBOZIJZVRWAPVTBQZVNWBDVMDGZVPLZBEUBWCVN - UAWABVMRZIBBUCZRWFWAWGVMBDVMWEBWDVPDUDVOUEZUFEUGUHBUNUIBVMUJUKULFVRWAFGZV - RJZDVMWEWIBVQEWJWIVPMZVMJZWIWKVPLZSWIWESDVMUMWJWIVQJWLWIVQVMUOZWIVLVPWHUP - TWJWMWIVPLZWIWIVPUQWJVPWIPWOWISWJVOWIWIVLVOURUSVPWIUTTVADWKVMWEWMWIWDWKVP - VBVDVCWNVEVFWAVRVGVHVRVMVIVJVK $. + funmpt2 cdm vsnex unex dmmpti imaeq2i imadmrn eqtr3i imafi eqeltrrid mpan + wrex eldifi elpwun undif1 elpwunsn snssd ssequn2 eqtr2id rspceeqv syl2anc + vex uneq1 elrnmptd ssriv ssfi sylancl unfi mpancom eqeltrid ) CGZHZIJZVLA + GZKZLHZVQVMMZVMLZIVLVPNVRIJZVNVSIJVNBOZIJZVRWAPVTBQZVNWBDVMDGZVPLZBEUBWCV + NUAWABVMRZIBBUCZRWFWAWGVMBDVMWEBWDVPDVCAUDZUEEUFUGBUHUIBVMUJUKULFVRWAFGZV + RJZDVMWEWIBVQEWJWIVPMZVMJZWIWKVPLZSWIWESDVMUMWJWIVQJWLWIVQVMUNZWIVLVPWHUO + TWJWMWIVPLZWIWIVPUPWJVPWIPWOWISWJVOWIWIVLVOUQURVPWIUSTUTDWKVMWEWMWIWDWKVP + VDVAVBWNVEVFWAVRVGVHVRVMVIVJVK $. $( $j usage 'pwfilem' avoids 'ax-pow'; $) $} @@ -93064,10 +93801,10 @@ proved without using the Axiom of Power Sets (unlike ~ domsdomtr ). successor minus any element of the successor. (Contributed by NM, 26-May-1998.) Avoid ~ ax-pow . (Revised by BTernaryTau, 23-Sep-2024.) $) phplem1 $p |- ( ( A e. _om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) $= - ( com wcel csuc wa csn cen wbr simpl peano2 enrefnn syl adantr simpr dif1en - cdif syl3anc cfn wb nnfi ensymfib 3syl mpbird ) ACDZBAEZDZFZAUFBGQZHIZUIAHI - ZUHUEUFUFHIZUGUKUEUGJZUEULUGUEUFCDULAKUFLMNUEUGOUFABPRUHUEASDUJUKTUMAUAAUIU - BUCUD $. + ( com wcel csuc wa csn cdif cen wbr simpl peano2 enrefnn syl simpr dif1ennn + adantr syl3anc cfn wb nnfi ensymfib 3syl mpbird ) ACDZBAEZDZFZAUFBGHZIJZUIA + IJZUHUEUFUFIJZUGUKUEUGKZUEULUGUEUFCDULALUFMNQUEUGOUFABPRUHUEASDUJUKTUMAUAAU + IUBUCUD $. $( $j usage 'phplem1' avoids 'ax-pow'; $) ${ @@ -93354,7 +94091,7 @@ proved without using the Axiom of Power Sets (unlike ~ domsdomtr ). $} ${ - $d x y A $. $d x y B $. + $d x A $. $d x B $. $( Obsolete version of ~ php2 as of 20-Nov-2024. (Contributed by NM, 31-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) $) @@ -93507,11 +94244,11 @@ Finite sets (cont.) $( Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid - ~ ax-pow . (Revised by BTernaryTau, 4-Dec-2024.) $) + ~ ax-pow . (Revised by BTernaryTau, 4-Dec-2024.) (Proof shortened by BJ, + 11-Jan-2025.) $) sucdom $p |- ( A e. _om -> ( A ~< B <-> suc A ~<_ B ) ) $= - ( com wcel csdm wbr csuc cdom sucdom2 wi php4 cfn sdomdomtrfi syl3an1 mpdan - nnfi 3expia impbid2 ) ACDZABEFZAGZBHFZABISAUAEFZUBTJAKSUCUBTSALDUCUBTAPAUAB - MNQOR $. + ( com wcel csdm wbr csuc cdom sucdom2 nnfi php4 sdomdomtrfi syl2anc impbid2 + cfn wi 3expia ) ACDZABEFZAGZBHFZABIRAODZATEFZUASPAJAKUBUCUASATBLQMN $. $( $j usage 'sucdom' avoids 'ax-pow'; $) $( Obsolete version of ~ sucdom as of 4-Dec-2024. (Contributed by Mario @@ -93540,9 +94277,9 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, ${ $d A x $. - $( Strict dominance over 0 is the same as dominance over 1. (Contributed - by NM, 28-Sep-2004.) Avoid ~ ax-un . (Revised by BTernaryTau, - 7-Dec-2024.) $) + $( Strict dominance over 0 is the same as dominance over 1. For a shorter + proof requiring ~ ax-un , see 0sdom1domALT . (Contributed by NM, + 28-Sep-2004.) Avoid ~ ax-un . (Revised by BTernaryTau, 7-Dec-2024.) $) 0sdom1dom $p |- ( (/) ~< A <-> 1o ~<_ A ) $= ( vx c0 csdm wbr cvv wcel c1o cdom relsdom brrelex2i reldom wne 0sdomg cv wex n0 csn wss cen snssi df1o2 0ex en2sn mp2an eqbrtri endom ax-mp domssr @@ -93554,27 +94291,30 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, $( $j usage '0sdom1dom' avoids 'ax-pow' 'ax-un'; $) $} - $( Obsolete version of ~ 0sdom1dom as of 7-Dec-2024. (Contributed by NM, - 28-Sep-2004.) (Proof modification is discouraged.) + $( Alternate proof of ~ 0sdom1dom , shorter but requiring ~ ax-un . + (Contributed by NM, 28-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.) $) - 0sdom1domOLD $p |- ( (/) ~< A <-> 1o ~<_ A ) $= + 0sdom1domALT $p |- ( (/) ~< A <-> 1o ~<_ A ) $= ( c0 csdm wbr csuc cdom c1o wcel wb peano1 sucdom ax-mp df-1o breq1i bitr4i com ) BACDZBEZAFDZGAFDBPHQSIJBAKLGRAFMNO $. + $( $j usage '0sdom1domALT' avoids 'ax-pow'; $) - $( Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, - 4-Apr-2007.) Avoid ~ ax-un . (Revised by BTernaryTau, 8-Dec-2024.) $) + $( Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof + requiring ~ ax-un , see ~ 1sdom2ALT . (Contributed by NM, 4-Apr-2007.) + Avoid ~ ax-un . (Revised by BTernaryTau, 8-Dec-2024.) $) 1sdom2 $p |- 1o ~< 2o $= ( c1o c2o csdm wbr cdom cen wn wne 2on0 2oex 0sdom mpbir 0sdom1dom mpbi csn c0 snnen2o df1o2 breq1i mtbir brsdom mpbir2an ) ABCDABEDZABFDZGPBCDZUCUEBPH IBJKLBMNUDPOZBFDPQAUFBFRSTABUAUB $. $( $j usage '1sdom2' avoids 'ax-pow' 'ax-un'; $) - $( Obsolete version of ~ 1sdom2 as of 8-Dec-2024. (Contributed by NM, - 4-Apr-2007.) (Proof modification is discouraged.) + $( Alternate proof of ~ 1sdom2 , shorter but requiring ~ ax-un . + (Contributed by NM, 4-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.) $) - 1sdom2OLD $p |- 1o ~< 2o $= + 1sdom2ALT $p |- 1o ~< 2o $= ( c1o csuc c2o csdm com wcel wbr 1onn php4 ax-mp df-2o breqtrri ) AABZCDAEF AMDGHAIJKL $. + $( $j usage '1sdom2ALT' avoids 'ax-pow'; $) ${ $d A x $. $d A f $. @@ -93817,21 +94557,68 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, UBUEIABKZUAUFABLUGUAUFABMNOPRST $. $( The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] - p. 136. (Contributed by NM, 2-Jun-1998.) $) + p. 136. (Contributed by NM, 2-Jun-1998.) Avoid ~ ax-pow . (Revised by + BTernaryTau, 2-Jan-2025.) $) ominf $p |- -. _om e. Fin $= + ( vx com cfn wcel wn wi cv cen wbr wrex isfi wpss wss wne wa wb nnord ordom + word ordelssne sylancl ibi df-pss sylibr nnfi ensymfib syl biimpar syl2an2r + pssinf rexlimiva sylbi pm2.01 ax-mp ) BCDZUOEZFUPUOBAGZHIZABJUPABKURUPABUQB + DZUQBLZURUQBHIZUPUSUQBMUQBNOZUTUSVBUSUQSBSUSVBPUQQRUQBTUAUBUQBUCUDUSVAURUSU + QCDVAURPUQUEUQBUFUGUHUQBUJUIUKULUOUMUN $. + $( $j usage 'ominf' avoids 'ax-pow'; $) + + $( Obsolete version of ~ ominf as of 2-Jan-2025. (Contributed by NM, + 2-Jun-1998.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + ominfOLD $p |- -. _om e. Fin $= ( vx com cfn wcel wn wi cv cen wbr wrex isfi wpss wss wne wa wb nnord ordom word ordelssne ibi df-pss sylibr ensym pssinf syl2an rexlimiva sylbi pm2.01 sylancl ax-mp ) BCDZULEZFUMULBAGZHIZABJUMABKUOUMABUNBDZUNBLZUNBHIUMUOUPUNBM UNBNOZUQUPURUPUNSBSUPURPUNQRUNBTUJUAUNBUBUCBUNUDUNBUEUFUGUHULUIUK $. ${ - $d A f m x y z $. $d A m n x y $. $d f g m x y z $. + $d A m n x y $. $d f g m x z $. $d A f m x y z $. $( Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has _countably infinite_ subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. - (Contributed by Mario Carneiro, 15-Jan-2013.) $) + (Contributed by Mario Carneiro, 15-Jan-2013.) Avoid ~ ax-pow . + (Revised by BTernaryTau, 2-Jan-2025.) $) isinf $p |- ( -. A e. Fin -> A. n e. _om E. x ( x C_ A /\ x ~~ n ) ) $= + ( vm vy vf vz vg wcel cv wss cen wbr wa wex com c0 wceq wi wf1o cfn breq2 + wn anbi2d exbidv wb sseq1 adantl breq1 sylan9bbr anbi12d cbvexdvaw peano1 + csuc 0ss enrefnn ax-mp 0ex spcev mp2an a1i w3a cdif ssdif0 eqss wrex rspe + biimpa sylan2 isfi sylibr sylanbr ex sylan2br 3impd com12 con3d bren neq0 + expcom csn cun eldifi snssd unss biimpi ad2ant2r cop cin vex f1osn eldifn + jctr disjsn word nnord orddisj syl anim12i syl2an df-suc f1oeq3 snex unex + f1oun f1oeq1 sylbir adantll syl2anc exlimiv sylbi com13 3imp21 syld com3l + 3expia finds2 ralrimiv ) BUAIZUCZAJZBKZYACJZLMZNZAOZCPYCPIXTYFYFYBYAQLMZN + ZAOZYBYADJZLMZNZAOZEJZBKZYNYJUNZLMZNZEOZXTCDYCQRZYEYHAYTYDYGYBYCQYALUBUDU + EYCYJRZYEYLAUUAYDYKYBYCYJYALUBUDUEYCYPRZYEYRAEUUBYAYNRZNYBYOYDYQUUCYBYOUF + UUBYAYNBUGUHUUCYDYNYCLMUUBYQYAYNYCLUIYCYPYNLUBUJUKULYIXTQBKZQQLMZYIBUOQPI + UUEUMQUPUQYHUUDUUENAQURYAQRYBUUDYGUUEYAQBUGYAQQLUIUKUSUTVAYMYJPIZXTYSYLUU + FXTYSSZSAYBYKUUFUUGYBYKUUFVBZXTBYAVCZQRZUCZYSUUHUUJXSUUJUUHXSUUJYBYKUUFXS + YBUUJYKUUFXSSZSZUUJYBBYAKZUUMBYAVDYBUUNNZYKUULUUOYABRZYKUULYABVEUUFUUPYKN + ZXSUUFUUQNBYJLMZDPVFZXSUUQUUFUURUUSUUPYKUURYABYJLUIVHUURDPVGVIDBVJVKVTVLV + MVNVTVOVPVQYKYBUUFUUKYSSZYKYAYJFJZTZFOYBUUFUUTSZSZYAYJFVRUVBUVDFYBUVBUVCU + UKUUFYBUVBNZYSUUKGJZUUIIZGOUUFUVEYSSZSZGUUIVSUVGUVIGUVGUUFUVHUVEUVGUUFNZY + SUVEUVJNYAUVFWAZWBZBKZUVLYPLMZYSYBUVGUVMUVBUUFUVGYBUVKBKZUVMUVGUVFBUVFBYA + WCWDYBUVONUVMYAUVKBWEWFVIWGUVBUVJUVNYBUVBUVJNUVLYJYJWAZWBZUVAUVFYJWHZWAZW + BZTZUVNUVBUVBUVKUVPUVSTZNYAUVKWIQRZYJUVPWIQRZNUWAUVJUVBUWBUVFYJGWJDWJWKWM + UVGUWCUUFUWDUVGUVFYAIUCUWCUVFBYAWLYAUVFWNVKUUFYJWOUWDYJWPYJWQWRWSYAYJUVKU + VPUVAUVSXEWTUWAUVLYPUVTTZUVNYPUVQRUWEUWAUFYJXAYPUVQUVLUVTXBUQUWEUVLYPHJZT + ZHOUVNUWGUWEHUVTUVAUVSFWJUVRXCXDUVLYPUWFUVTXFUSUVLYPHVRVKXGWRXHYRUVMUVNNE + UVLYAUVKAWJUVFXCXDYNUVLRYOUVMYQUVNYNUVLBUGYNUVLYPLUIUKUSXIVTVMXJXKXLVTXJX + KXMXNXPXJXOXQVPXR $. + $( $j usage 'isinf' avoids 'ax-pow'; $) + $} + + ${ + $d A f m x y z $. $d A m n x y $. $d f g m x y z $. + $( Obsolete version of ~ isinf as of 2-Jan-2025. (Contributed by Mario + Carneiro, 15-Jan-2013.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + isinfOLD $p |- ( -. A e. Fin -> A. n e. _om E. x ( x C_ A /\ x ~~ n ) ) $= ( vm vy vf vz vg wcel cv wss cen wbr wa wex com c0 wceq wi wf1o cfn breq2 wn csuc anbi2d exbidv wb sseq1 adantl sylan9bbr anbi12d cbvexdvaw 0ss 0ex breq1 enref spcev mp2an a1i cdif ssdif0 eqss wrex biimpa rspe sylan2 isfi @@ -93986,8 +94773,25 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, $( Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario - Carneiro, 27-Feb-2014.) $) + Carneiro, 27-Feb-2014.) Avoid ~ ax-pow . (Revised by BTernaryTau, + 4-Jan-2025.) $) f1finf1o $p |- ( ( A ~~ B /\ B e. Fin ) -> + ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) ) $= + ( cen wbr cfn wcel wa wf1 wf1o wfo simpr wfn crn wceq wf csdm wi sylanbrc + ex f1f adantl ffnd simpll wne wpss wn wss wb frnd df-pss baib php3 ad2antlr + enfii ancoms f1f1orn f1oenfi syl2an cdom endom domsdomtrfi syl3an2 syl2an2r + syl 3expia syld sdomnen syl6 sylbird necon4ad df-fo df-f1o f1of1 impbid1 + mpd ) ABDEZBFGZHZABCIZABCJZVSVTWAVSVTHZVTABCKZWAVSVTLWBCAMCNZBOZWCWBABCVTAB + CPVSABCUAUBZUCWBVQWEVQVRVTUDWBVQWDBWBWDBUEZWDBUFZVQUGZWBWDBUHZWHWGUIWBABCWF + UJWHWJWGWDBUKULVEWBWHABQEZWIWBWHWDBQEZWKVRWHWLRVQVTVRWHWLBWDUMTUNVSAFGZVTAW + DDEZWLWKRVRVQWMABUOUPZVSWMAWDCJWNVTWOABCUQAWDCURUSWMWNWLWKWNWMAWDUTEWLWKAWD + VAAWDBVBVCVFVDVGABVHVIVJVKVPABCVLSABCVMSTABCVNVO $. + $( $j usage 'f1finf1o' avoids 'ax-pow'; $) + + $( Obsolete version of ~ f1finf1o as of 4-Jan-2025. (Contributed by Jeff + Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + f1finf1oOLD $p |- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) ) $= ( cen wbr cfn wcel wa wf1 wf1o wfo simpr wfn crn adantl syl cvv ex sylanbrc csdm wceq wf f1f ffnd simpll wne wpss wn wb frnd df-pss baib cmap co simplr @@ -94008,9 +94812,23 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, GZEAHBDAIQPQPFCDJBFCKLMABNO $. $} - $( A set with one element is a singleton. (Contributed by FL, - 18-Aug-2008.) $) - en1eqsn $p |- ( ( A e. B /\ B ~~ 1o ) -> B = { A } ) $= + ${ + $d A x $. $d B x $. + $( A set with one element is a singleton. (Contributed by FL, + 18-Aug-2008.) Avoid ~ ax-pow , ~ ax-un . (Revised by BTernaryTau, + 4-Jan-2025.) $) + en1eqsn $p |- ( ( A e. B /\ B ~~ 1o ) -> B = { A } ) $= + ( vx c1o cen wbr wcel csn wceq cv wex wi en1 eleq2 elsni sneqd syl6bi imp + eqtr3 syldan ex exlimiv sylbi impcom ) BDEFZABGZBAHZIZUEBCJZHZIZCKUFUHLZC + BMUKULCUKUFUHUKUFUGUJIZUHUKUFUMUKUFAUJGZUMBUJANUNAUIAUIOPQRBUGUJSTUAUBUCU + D $. + $( $j usage 'en1eqsn' avoids 'ax-pow' 'ax-un'; $) + $} + + $( Obsolete version of ~ en1eqsn as of 4-Jan-2025. (Contributed by FL, + 18-Aug-2008.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + en1eqsnOLD $p |- ( ( A e. B /\ B ~~ 1o ) -> B = { A } ) $= ( wcel c1o cen wbr csn cfn wss wceq com 1onn ssid ssnnfi mp2an enfii adantl wa mpan snssi adantr ensn1g ensym entr syl2an fisseneq syl3anc eqcomd ) ABC ZBDEFZRZAGZBUKBHCZULBIZULBEFZULBJUJUMUIDHCZUJUMDKCDDIUPLDMDDNOBDPSQUIUNUJAB @@ -94025,11 +94843,10 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, ${ $d A x $. $d X x $. $d M x $. - $( Alternate proof of ~ dif1en with fewer symbols using ~ ax-pow . - (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, - 16-Aug-2015.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - dif1enALT $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) -> + $( Alternate proof of ~ dif1ennn using ~ ax-pow . (Contributed by Jeff + Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + dif1ennnALT $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) -> ( A \ { X } ) ~~ M ) $= ( vx com wcel csuc cen wbr csn wrex wa cfn peano2 breq2 isfi wceq wi cin c0 w3a cdif cv rspcev sylibr sylan diffi sylib syl 3adant3 cun en2sn elvd @@ -94058,20 +94875,39 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, $} ${ - $d x A $. $d x N $. - enp1i.1 $e |- M e. _om $. + $d A x $. $d M x $. + enp1i.1 $e |- Ord M $. enp1i.2 $e |- N = suc M $. enp1i.3 $e |- ( ( A \ { x } ) ~~ M -> ph ) $. enp1i.4 $e |- ( x e. A -> ( ph -> ps ) ) $. - $( Proof induction for ~ en2i and related theorems. (Contributed by Mario - Carneiro, 5-Jan-2016.) $) + $( Proof induction for ~ en2 and related theorems. (Contributed by Mario + Carneiro, 5-Jan-2016.) Generalize to all ordinals and avoid ~ ax-pow , + ~ ax-un . (Revised by BTernaryTau, 6-Jan-2025.) $) enp1i $p |- ( A ~~ N -> E. x ps ) $= + ( cen wbr csuc wex breq2i wrex wcel cvv 3syl sylbi cv cdif con0 word encv + csn wb simprd sssucid ssexg mpan elong mpbiri rexdif1en mpancom reximi wa + wss df-rex imp eximi ) DFKLDEMZKLZBCNZFVBDKHOVCDCUAZUFUBEKLZCDPZACDPZVDEU + CQZVCVGVCVIEUDZGVCVBRQZERQZVIVJUGVCDRQVKDVBUEUHEVBURVKVLEUIEVBRUJUKERULSU + MCDEUNUOVFACDIUPVHVEDQZAUQZCNVDACDUSVNBCVMABJUTVATST $. + $( $j usage 'enp1i' avoids 'ax-pow' 'ax-un'; $) + $} + + ${ + $d x A $. $d x N $. + enp1iOLD.1 $e |- M e. _om $. + enp1iOLD.2 $e |- N = suc M $. + enp1iOLD.3 $e |- ( ( A \ { x } ) ~~ M -> ph ) $. + enp1iOLD.4 $e |- ( x e. A -> ( ph -> ps ) ) $. + $( Obsolete version of ~ enp1i as of 6-Jan-2025. (Contributed by Mario + Carneiro, 5-Jan-2016.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + enp1iOLD $p |- ( A ~~ N -> E. x ps ) $= ( cen wbr cv wcel wex c0 wceq wn csuc sylib wne nsuceq0 breq1 en0 eqtr3id - ensym syl6bi necon3ad mpi con2i neq0 wi breq2i csn cdif com dif1en mp3an1 - wa syl ex sylbi sylcom eximdv mpd ) DFKLZCMZDNZCOZBCOVFDPQZRVIVJVFVJESZPU - AVFREUBVJVFVKPVJVFPFKLZVKPQDPFKUCVLVKFPHVLFPKLFPQPFUFFUDTUEUGUHUIUJCDUKTV - FVHBCVFVHABVFDVKKLZVHAULFVKDKHUMVMVHAVMVHUSDVGUNUOEKLZAEUPNVMVHVNGDEVGUQU - RIUTVAVBJVCVDVE $. + ensym syl6bi necon3ad mpi con2i neq0 wi breq2i wa csn com dif1ennn mp3an1 + cdif syl ex sylbi sylcom eximdv mpd ) DFKLZCMZDNZCOZBCOVFDPQZRVIVJVFVJESZ + PUAVFREUBVJVFVKPVJVFPFKLZVKPQDPFKUCVLVKFPHVLFPKLFPQPFUFFUDTUEUGUHUIUJCDUK + TVFVHBCVFVHABVFDVKKLZVHAULFVKDKHUMVMVHAVMVHUNDVGUOUSEKLZAEUPNVMVHVNGDEVGU + QURIUTVAVBJVCVDVE $. $} ${ @@ -94079,9 +94915,10 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, $( A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.) $) en2 $p |- ( A ~~ 2o -> E. x E. y A = { x , y } ) $= - ( cv csn cdif wceq wex cpr c1o c2o 1onn df-2o cen wbr en1 biimpi enp1ilem - wcel df-pr eximdv enp1i ) CADZEFZBDZEZGZBHZCUCUEIZGZBHACJKLMUDJNOUHBUDPQU - CCSUGUJBACUFUIUCUETRUAUB $. + ( csn cdif wceq wex cpr c1o c2o 1on onordi df-2o cen wbr en1 biimpi df-pr + cv wcel enp1ilem eximdv enp1i ) CASZDEZBSZDZFZBGZCUDUFHZFZBGACIJIKLMUEINO + UIBUEPQUDCTUHUKBACUGUJUDUFRUAUBUC $. + $( $j usage 'en2' avoids 'ax-pow' 'ax-un'; $) $} ${ @@ -94089,9 +94926,10 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, $( A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.) $) en3 $p |- ( A ~~ 3o -> E. x E. y E. z A = { x , y , z } ) $= - ( cv csn cdif cpr wceq wex ctp c2o c3o 2onn df-3o en2 wcel tpass enp1ilem - 2eximdv enp1i ) DAEZFGZBEZCEZHZIZCJBJDUBUDUEKZIZCJBJADLMNOBCUCPUBDQUGUIBC - ADUFUHUBUDUERSTUA $. + ( cv csn cdif cpr wceq wex ctp c2o c3o 2on onordi df-3o en2 wcel enp1ilem + tpass 2eximdv enp1i ) DAEZFGZBEZCEZHZIZCJBJDUCUEUFKZIZCJBJADLMLNOPBCUDQUC + DRUHUJBCADUGUIUCUEUFTSUAUB $. + $( $j usage 'en3' avoids 'ax-pow' 'ax-un'; $) $} ${ @@ -94100,9 +94938,10 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, Carneiro, 5-Jan-2016.) $) en4 $p |- ( A ~~ 4o -> E. x E. y E. z E. w A = ( { x , y } u. { z , w } ) ) $= - ( cv csn cdif ctp wceq wex cpr cun c3o c4o 3onn df-4o en3 wcel qdassr + ( cv csn cdif ctp wceq wex cpr cun c3o c4o ord3 df-4o en3 wcel qdassr enp1ilem eximdv 2eximdv enp1i ) EAFZGHZBFZCFZDFZIZJZDKZCKBKEUEUGLUHUILMZJ ZDKZCKBKAENOPQBCDUFRUEESZULUOBCUPUKUNDAEUJUMUEUGUHUITUAUBUCUD $. + $( $j usage 'en4' avoids 'ax-pow' 'ax-un'; $) $} ${ @@ -94121,38 +94960,79 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, ( vw cen wbr wi c0 vv cfn wcel com wrex isfi wal csuc breq2 imbi1d albidv cv wceq weq en0 mpbiri sylbi ax-gen wne wa nsuceq0 breq1 anbi2d wb peano1 wsbc peano2 nneneq sylancr biimpa eqcomd syl6bi com12 necon3d mpi wel wex - ex n0 csn cdif dif1en 3expia wss cun snssi uncom biimpi eqtrid vex difexi - undif uneq1 sbceq1d imbi2d imbi12d spvv rspe sylibr pm2.27 adantl sylsyld - syl5 snex unex sbcie syl6ibr vtocl dfsbcq syl5ib 3syl expd adantr exlimdv - mpdd biimtrid com23 alrimdv nfsbc1v nfim sbceq1a cbvalv1 19.21bi rexlimiv - nfv finds1 vtoclga ) AEFIUBMFULZUBUCYHPULZQRZPUDUEAPYHUFYJAPUDYIUDUCYJASZ - FYKFUGYHTQRZASZFUGYHUAULZQRZASZFUGZYHYNUHZQRZASZFUGZPUAYITUMZYKYMFUUBYJYL - AYITYHQUIUJUKPUAUNZYKYPFUUCYJYOAYIYNYHQUIUJUKYIYRUMZYKYTFUUDYJYSAYIYRYHQU - IUJUKYMFYLYHTUMZAYHUOUUEABNJUPUQURYNUDUCZYQYIYRQRZAFYIVFZSZPUGUUAUUFYQUUI - PUUFUUGYQUUHUUFUUGYITUSZYQUUHSZUUFUUGUUJUUFUUGUTZYRTUSUUJYNVAUULYITYRTUUB - UULYRTUMZUUBUULUUFTYRQRZUTZUUMUUBUUGUUNUUFYITYRQVBVCUUOTYRUUFUUNTYRUMZUUF - TUDUCYRUDUCUUNUUPVDVEYNVGTYRVHVIVJVKVLVMVNVOVRUUFUUGUUJUUKSUUJHPVPZHVQUUL - UUKHYIVSUULUUQUUKHUULUUQYIHULZVTZWAZYNQRZUUKUUFUUGUUQUVAYIYNUURWBWCUUFUUQ - UVAUUKSZSUUGUUQUUFUVBUUQUUFUVAUUKUUQUUSYIWDZUUTUUSWEZYIUMZUUFUVAUTZUUKSUU - RYIWFUVCUVDUUSUUTWEZYIUUTUUSWGUVCUVGYIUMUUSYIWLWHWIUVFYQAFUVDVFZSZUVEUUKU - UFGULZYNQRZUTZYQAFUVJUUSWEZVFZSZSUVFUVISGUUTYIUUSPWJWKUVJUUTUMZUVLUVFUVOU - VIUVPUVKUVAUUFUVJUUTYNQVBVCUVPUVNUVHYQUVPAFUVMUVDUVJUUTUUSWMWNWOWPUVLYQDU - VNYQUVKCSZUVLDYPUVQFGFGUNYOUVKACYHUVJYNQVBKWPWQUVLUVJUBUCZUVQCDUVLUVKUAUD - UEUVRUVKUAUDWRUAUVJUFWSUVKUVQCSUUFUVKCWTXAOXBXCADFUVMUVJUUSGWJUURXDXELXFX - GXHUVEUVHUUHYQAFUVDYIXIWOXJXKXLVMXMXOXNXPVRXOXQXRYTUUIFPYTPYEUUGUUHFUUGFY - EAFYIXSXTFPUNYSUUGAUUHYHYIYRQVBAFYIYAWPYBXGYFYCYDUQYG $. - $} - - ${ - $d w x y z $. $d w y z ph $. $d w x A $. $d w x ta $. $d x ch $. + ex n0 csn dif1enOLD 3expia wss cun snssi uncom undif biimpi eqtrid difexi + cdif vex uneq1 sbceq1d imbi2d imbi12d spvv rspe sylibr pm2.27 adantl syl5 + sylsyld snex unex sbcie syl6ibr vtocl dfsbcq syl5ib 3syl expd adantr mpdd + exlimdv biimtrid com23 alrimdv nfv nfsbc1v sbceq1a cbvalv1 finds1 19.21bi + nfim rexlimiv vtoclga ) AEFIUBMFULZUBUCYHPULZQRZPUDUEAPYHUFYJAPUDYIUDUCYJ + ASZFYKFUGYHTQRZASZFUGYHUAULZQRZASZFUGZYHYNUHZQRZASZFUGZPUAYITUMZYKYMFUUBY + JYLAYITYHQUIUJUKPUAUNZYKYPFUUCYJYOAYIYNYHQUIUJUKYIYRUMZYKYTFUUDYJYSAYIYRY + HQUIUJUKYMFYLYHTUMZAYHUOUUEABNJUPUQURYNUDUCZYQYIYRQRZAFYIVFZSZPUGUUAUUFYQ + UUIPUUFUUGYQUUHUUFUUGYITUSZYQUUHSZUUFUUGUUJUUFUUGUTZYRTUSUUJYNVAUULYITYRT + UUBUULYRTUMZUUBUULUUFTYRQRZUTZUUMUUBUUGUUNUUFYITYRQVBVCUUOTYRUUFUUNTYRUMZ + UUFTUDUCYRUDUCUUNUUPVDVEYNVGTYRVHVIVJVKVLVMVNVOVRUUFUUGUUJUUKSUUJHPVPZHVQ + UULUUKHYIVSUULUUQUUKHUULUUQYIHULZVTZWKZYNQRZUUKUUFUUGUUQUVAYIYNUURWAWBUUF + UUQUVAUUKSZSUUGUUQUUFUVBUUQUUFUVAUUKUUQUUSYIWCZUUTUUSWDZYIUMZUUFUVAUTZUUK + SUURYIWEUVCUVDUUSUUTWDZYIUUTUUSWFUVCUVGYIUMUUSYIWGWHWIUVFYQAFUVDVFZSZUVEU + UKUUFGULZYNQRZUTZYQAFUVJUUSWDZVFZSZSUVFUVISGUUTYIUUSPWLWJUVJUUTUMZUVLUVFU + VOUVIUVPUVKUVAUUFUVJUUTYNQVBVCUVPUVNUVHYQUVPAFUVMUVDUVJUUTUUSWMWNWOWPUVLY + QDUVNYQUVKCSZUVLDYPUVQFGFGUNYOUVKACYHUVJYNQVBKWPWQUVLUVJUBUCZUVQCDUVLUVKU + AUDUEUVRUVKUAUDWRUAUVJUFWSUVKUVQCSUUFUVKCWTXAOXCXBADFUVMUVJUUSGWLUURXDXEL + XFXGXHUVEUVHUUHYQAFUVDYIXIWOXJXKXLVMXMXNXOXPVRXNXQXRYTUUIFPYTPXSUUGUUHFUU + GFXSAFYIXTYEFPUNYSUUGAUUHYHYIYRQVBAFYIYAWPYBXGYCYDYFUQYG $. + $} + + ${ + $d ch x $. $d ta w x $. $d x y z $. $d A w x $. $d ph w y z $. findcard3.1 $e |- ( x = y -> ( ph <-> ch ) ) $. findcard3.2 $e |- ( x = A -> ( ph <-> ta ) ) $. findcard3.3 $e |- ( y e. Fin -> ( A. x ( x C. y -> ph ) -> ch ) ) $. $( Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed - by Mario Carneiro, 13-Dec-2013.) $) + by Mario Carneiro, 13-Dec-2013.) Avoid ~ ax-pow . (Revised by + BTernaryTau, 7-Jan-2025.) $) findcard3 $p |- ( A e. Fin -> ta ) $= + ( vw vz cfn wcel cen wbr wi wal wa com wrex cv isfi wral con0 nnon eleq1w + breq2 imbi1d albidv imbi12d wpss rspe sylibr 19.21v ralbii ralcom4 bitr3i + weq wss pssss ssfi syl2an csdm simprl wb nnfi ensymfib syl biimpar adantl + sylib simplll php3 sylan adantr simpllr cdom endom 3adant2 sdomdom sylan2 + 3adant3 syld3an1 sdomdomtrfi syl3an3 syl3anc syl3an1 syl3an2 nnsdomo word + domfi domsdomtrfi nnord ordelpss bitr4d syl2anc mpbid simpr jca2 reximdv2 + ex mpd r19.29 expcom ordom ordelss mpan ad2antrr sseld impd syl6 rexlimdv + pm2.27 syld com23 biimtrid sylsyld impancom alrimiv breq1 cbvalvw syl6ibr + alimdv a1i tfis2 mpcom rgen sylbi wceq spcgv rexlimdvw ) FLMZDUAZJUAZNOZA + PZDQZFYNNOZRZJSTZCYLYRJSTZYTJFUBYQJSUCUUAYTYQJSYNUDMZYNSMZYQYNUEUUCYQPZKU + AZSMZYMUUENOZAPZDQZPZJKJKURZUUCUUFYQUUIJKSUFUUKYPUUHDUUKYOUUGAYNUUEYMNUGU + HUIUJUUJKYNUCZUUDPUUBUULUUCEUAZYNNOZBPZEQZYQUUCUULUUPUUCUULRUUOEUUCUUNUUL + BUUCUUNRZUUMLMZUULYMUUMUKZAPZDQZBUUQUUNJSTUURUUNJSULJUUMUBUMZUULUUFUUHPZK + YNUCZDQZUUQUVAUULUVCDQZKYNUCUVEUVFUUJKYNUUFUUHDUNUOUVCKDYNUPUQUUQUVDUUTDU + UQUUSUVDAUUQUUSUVDAPUUQUUSRZUVDUVCUUGRZKYNTZAUVGUUGKYNTZUVDUVIPUVGUUGKSTZ + UVJUUQUURYMUUMUSZUVKUUSUVBYMUUMUTUURUVLRYMLMZUVKUUMYMVAKYMUBVKVBUVGUUGUUG + KSYNUVGUUFUUGRZUUEYNMZUUGUVGUVNUVOUVGUVNRZUUEYNVCOZUVOUVPUUFUUEYMNOZYMYNV + COZUVQUVGUUFUUGVDZUVNUVRUVGUUFUVRUUGUUFUUELMZUVRUUGVEUUEVFZUUEYMVGVHVIVJU + VPUUCYMUUMVCOZUUNUVSUUCUUNUUSUVNVLZUVGUWCUVNUUQUURUUSUWCUVBUUMYMVMVNVOUUC + UUNUUSUVNVPUUNUUCUWCUUMYNVQOZUVSUUMYNVRUVMUWCUUCUWEUVSUURUWCUUCUWEUVMUUCU + WEUURUWCUUCYNLMUWEUURYNVFYNUUMWKVNVSUURUWCUVMUWEUWCUURYMUUMVQOUVMYMUUMVTU + UMYMWKWAWBWCYMUUMYNWDWCWEWFUVRUUFUUEYMVQOZUVSUVQUUEYMVRUUFUWAUWFUVSUVQUWB + UUEYMYNWLWGWHWFUVPUUFUUCUVQUVOVEUVTUWDUUFUUCRUVQUUEYNUKZUVOUUEYNWIUUFUUEW + JYNWJUVOUWGVEUUCUUEWMYNWMUUEYNWNVBWOWPWQXAUUFUUGWRWSWTXBUVDUVJUVIUVCUUGKY + NXCXDVHUVGUVHAKYNUVGUVOUUFUVHAPUVGYNSUUEUUCYNSUSZUUNUUSSWJUUCUWHXESYNXFXG + XHXIUUFUVCUUGAUUFUUHXMXJXKXLXNXAXOYCXPIXQXRXSXDYPUUODEDEURYOUUNABYMUUMYNN + XTGUJYAYBYDYEYFYGYQYRJSXCXGYHYLYSCJSYLYQYRCYPYRCPDFLYMFYIYOYRACYMFYNNXTHU + JYJXJYKXB $. + $( $j usage 'findcard3' avoids 'ax-pow'; $) + $} + + ${ + $d w x y z $. $d w y z ph $. $d w x A $. $d w x ta $. $d x ch $. + findcard3OLD.1 $e |- ( x = y -> ( ph <-> ch ) ) $. + findcard3OLD.2 $e |- ( x = A -> ( ph <-> ta ) ) $. + findcard3OLD.3 $e |- ( y e. Fin -> ( A. x ( x C. y -> ph ) -> ch ) ) $. + $( Obsolete version of ~ findcard3 as of 7-Jan-2025. (Contributed by Mario + Carneiro, 13-Dec-2013.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + findcard3OLD $p |- ( A e. Fin -> ta ) $= ( vw vz cfn wcel cen wbr wi wal wa com wrex cv isfi wral con0 nnon eleq1w wceq breq2 imbi1d albidv imbi12d wpss sylibr 19.21v ralbii ralcom4 bitr3i rspe wss pssss ssfi sylib csdm ensym ad2antll php3 sylan simpllr sdomentr @@ -94446,11 +95326,25 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, $} ${ - $d x A $. + $d A x $. $( Omega strictly dominates a natural number. Example 3 of [Enderton] - p. 146. In order to avoid the Axiom of infinity, we include it as a - hypothesis. (Contributed by NM, 15-Jun-1998.) $) + p. 146. In order to avoid the Axiom of Infinity, we include it as part + of the antecedent. See ~ nnsdom for the version without this sethood + requirement. (Contributed by NM, 15-Jun-1998.) Avoid ~ ax-pow . + (Revised by BTernaryTau, 7-Jan-2025.) $) nnsdomg $p |- ( ( _om e. _V /\ A e. _om ) -> A ~< _om ) $= + ( vx com cvv wcel wa cdom wbr cen csdm wss word ordom ordelss mpan adantr + wn cfn nnfi ssdomfi2 syl3an1 mpd3an3 ancoms ominf wb ensymfib syl cv wrex + breq2 rspcev isfi sylibr ex sylbid mtoi adantl brsdom sylanbrc ) CDEZACEZ + FACGHZACIHZQZACJHVAUTVBVAUTACKZVBVAVEUTCLVAVEMCANOPVAAREZUTVEVBASZACDTUAU + BUCVAVDUTVAVCCREZUDVAVCCAIHZVHVAVFVCVIUEVGACUFUGVAVIVHVAVIFCBUHZIHZBCUIVH + VKVIBACVJACIUJUKBCULUMUNUOUPUQACURUS $. + $( $j usage 'nnsdomg' avoids 'ax-pow'; $) + + $( Obsolete version of ~ nnsdomg as of 7-Jan-2025. (Contributed by NM, + 15-Jun-1998.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + nnsdomgOLD $p |- ( ( _om e. _V /\ A e. _om ) -> A ~< _om ) $= ( vx com cvv wcel wa cdom wbr cen csdm wss ssdomg word ordom ordelss mpan wn impel cfn ominf ensym wrex breq2 rspcev isfi sylibr syl5 adantl brsdom cv ex mtoi sylanbrc ) CDEZACEZFACGHZACIHZQZACJHUNACKZUPUOACDLCMUOUSNCAOPR @@ -94468,13 +95362,40 @@ singletons being the empty set ( ` A e/ _V ` ). (Contributed by AV, $} $( An infinite set strictly dominates a natural number. (Contributed by NM, - 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) $) + 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) Avoid ~ ax-pow . + (Revised by BTernaryTau, 7-Jan-2025.) $) infsdomnn $p |- ( ( _om ~<_ A /\ B e. _om ) -> B ~< A ) $= + ( com cdom wbr wcel cfn csdm nnfi adantl cvv reldom brrelex1i nnsdomg sylan + wa simpl sdomdomtrfi syl3anc ) CADEZBCFZPBGFZBCHEZTBAHEUAUBTBIJTCKFUAUCCADL + MBNOTUAQBCARS $. + $( $j usage 'infsdomnn' avoids 'ax-pow'; $) + + $( Obsolete version of ~ infsdomnn as of 7-Jan-2025. (Contributed by NM, + 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + infsdomnnOLD $p |- ( ( _om ~<_ A /\ B e. _om ) -> B ~< A ) $= ( com cdom wbr wcel csdm cvv reldom brrelex1i nnsdomg sylan simpl sdomdomtr wa syl2anc ) CADEZBCFZOBCGEZQBAGEQCHFRSCADIJBKLQRMBCANP $. - $( An infinite set is not empty. (Contributed by NM, 23-Oct-2004.) $) - infn0 $p |- ( _om ~<_ A -> A =/= (/) ) $= + ${ + $d A f $. + $( An infinite set is not empty. For a shorter proof using ~ ax-un , see + ~ infn0ALT . (Contributed by NM, 23-Oct-2004.) Avoid ~ ax-un . + (Revised by BTernaryTau, 8-Jan-2025.) $) + infn0 $p |- ( _om ~<_ A -> A =/= (/) ) $= + ( vf com cdom wbr cv wf1 wex c0 wne brdomi wceq wcel peano1 wf1o ccnv crn + wa wn f1f1orn adantr wss frnd sseq0 sylan f1oeq3d mpbid f1ocnv noel f1o00 + f1f simprbi eleq2d mtbiri 3syl mt2 imnani neqned exlimiv syl ) CADECABFZG + ZBHAIJZCABKVBVCBVBAIVBAILZVBVDRZICMZNVECIVAOZICVAPZOZVFSVECVAQZVAOZVGVBVK + VDCAVATUAVEVJICVAVBVJAUBVDVJILVBCAVACAVAUKUCVJAUDUEUFUGCIVAUHVIVFIIMIUIVI + CIIVIVHILCILCVHUJULUMUNUOUPUQURUSUT $. + $( $j usage 'infn0' avoids 'ax-pow' 'ax-un'; $) + $} + + $( Shorter proof of ~ infn0 using ~ ax-un . (Contributed by NM, + 23-Oct-2004.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + infn0ALT $p |- ( _om ~<_ A -> A =/= (/) ) $= ( com cdom wbr c0 csdm wne wcel peano1 infsdomnn mpan2 cvv reldom brrelex2i wb 0sdomg syl mpbid ) BACDZEAFDZAEGZSEBHTIAEJKSALHTUAOBACMNALPQR $. @@ -94575,11 +95496,20 @@ Axiom of Infinity (see comments in ~ fin2inf ). (Contributed by NM, unfi impcom ) BCDZACDZEABFZCDZESUBTUBSTUBSGUABHZCDZTUABQUDABHZCDZTUCUECABIJ UFTSABKLMNOPR $. + $( The Cartesian product of two finite sets is finite. (Contributed by Jeff + Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) Avoid + ~ ax-pow . (Revised by BTernaryTau, 10-Jan-2025.) $) + xpfi $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin ) $= + ( cfn wcel wa cun cpw cxp wss unfi pwfi bitri sylib xpsspw ssfi sylancl ) A + CDBCDEZABFZGZGZCDZABHZTIUBCDQRCDZUAABJUCSCDUARKSKLMABNTUBOP $. + $( $j usage 'xpfi' avoids 'ax-pow'; $) + ${ $d A w x y z $. $d B w x y z $. - $( The Cartesian product of two finite sets is finite. (Contributed by - Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) $) - xpfi $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin ) $= + $( Obsolete version of ~ xpfi as of 10-Jan-2025. (Contributed by Jeff + Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + xpfiOLD $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin ) $= ( vx vy vz vw cfn wcel cxp cv wi c0 csn cdif wceq xpeq1 eleq1d imbi2d cvv wa 0xp 0fin eqeltri a1i wral wn wex neq0 sneq difeq2d xpeq1d rspcv adantl pm2.27 ad2antlr cen wbr c2nd cres wf1o snex xpexg id vex 2ndconst f1oen2g @@ -94886,8 +95816,8 @@ Axiom of Infinity (see comments in ~ fin2inf ). (Contributed by NM, ${ $d w x y z A $. $d w y z B $. - $( If a function has a finite domain, its range is finite. Theorem 37 of - [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.) $) + $( If an onto function has a finite domain, its codomain/range is finite. + Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.) $) fofi $p |- ( ( A e. Fin /\ F : A -onto-> B ) -> B e. Fin ) $= ( cfn wcel wfo cdom wbr fodomfi domfi syldan ) ADEABCFBAGHBDEABCIABJK $. @@ -95204,18 +96134,18 @@ of Infinity (see comments in ~ fin2inf ). (Contributed by NM, wex nfsbc1v sbceq1a cbvrexw ac6sfi sylan2b crn wfo wfn ffn ad2antrl dffn4 simpll sylib fofi syl2anc frn wi fnfvelrn sylan rspesbca syl ralimdva imp ex adantl simpr simprr weq fveq2 sbceq1d bitrd cbvralw r19.21bi ralrimiva - wceq rexbidv ralrn mpbird nfcv nfrex sylibr sseq1 rexeq ralbidv 3anbi123d - wb raleq rspcev syl13anc exlimddv 3adant2 ) DKLZACEMZBDNZGOZEPZACWTMZBDNZ - ABDMZCWTNZQZGKMZEFLWQWSRZDEHOZUAZACBOZXIUBZSZBDNZRZXGHWSWQACIOZSZIEMZBDNX - OHUEWRXRBDAXQCIEAITACXPUFZACXPUGZUHUCXQXMBIDEHACXPXLUDUIUJXHXORZXIUKZKLZY - BEPZACYBMZBDNZXDCYBNZXGYAWQDYBXIULZYCWQWSXOUQYAXIDUMZYHXJYIXHXNDEXIUNZUOZ - DXIUPURDYBXIUSUTXJYDXHXNDEXIVAUOXOYFXHXJXNYFXJXMYEBDXJXKDLZRXLYBLZXMYEVBX - JYIYLYMYJDXKXIVCVDYMXMYEACXLYBVEVIVFVGVHVJYAXQBDMZIYBNZYGYAYOACJOZXIUBZSZ - BDMZJDNZYAYSJDYAYPDLZRUUAYRBYPSZYSYAUUAVKYAUUBJDYAXNUUBJDNXHXJXNVLXMUUBBJ - DXMJTYRBYPUFBJVMZXMYRUUBUUCACXLYQXKYPXIVNVOYRBYPUGVPVQURVRYRBYPDVEUTVSYAY - IYOYTWKYKYNYSIJDXIXPYQVTXQYRBDACXPYQUDWAWBVFWCXDYNCIYBXDITXQCBDCDWDXSWECI - VMAXQBDXTWAVQWFXFYDYFYGQGYBKWTYBVTZXAYDXCYFXEYGWTYBEWGUUDXBYEBDACWTYBWHWI - XDCWTYBWLWJWMWNWOWP $. + wb wceq rexbidv ralrn mpbird nfcv nfrexw sylibr sseq1 rexeq ralbidv raleq + 3anbi123d rspcev syl13anc exlimddv 3adant2 ) DKLZACEMZBDNZGOZEPZACWTMZBDN + ZABDMZCWTNZQZGKMZEFLWQWSRZDEHOZUAZACBOZXIUBZSZBDNZRZXGHWSWQACIOZSZIEMZBDN + XOHUEWRXRBDAXQCIEAITACXPUFZACXPUGZUHUCXQXMBIDEHACXPXLUDUIUJXHXORZXIUKZKLZ + YBEPZACYBMZBDNZXDCYBNZXGYAWQDYBXIULZYCWQWSXOUQYAXIDUMZYHXJYIXHXNDEXIUNZUO + ZDXIUPURDYBXIUSUTXJYDXHXNDEXIVAUOXOYFXHXJXNYFXJXMYEBDXJXKDLZRXLYBLZXMYEVB + XJYIYLYMYJDXKXIVCVDYMXMYEACXLYBVEVIVFVGVHVJYAXQBDMZIYBNZYGYAYOACJOZXIUBZS + ZBDMZJDNZYAYSJDYAYPDLZRUUAYRBYPSZYSYAUUAVKYAUUBJDYAXNUUBJDNXHXJXNVLXMUUBB + JDXMJTYRBYPUFBJVMZXMYRUUBUUCACXLYQXKYPXIVNVOYRBYPUGVPVQURVRYRBYPDVEUTVSYA + YIYOYTVTYKYNYSIJDXIXPYQWAXQYRBDACXPYQUDWBWCVFWDXDYNCIYBXDITXQCBDCDWEXSWFC + IVMAXQBDXTWBVQWGXFYDYFYGQGYBKWTYBWAZXAYDXCYFXEYGWTYBEWHUUDXBYEBDACWTYBWIW + JXDCWTYBWKWLWMWNWOWP $. $( $j usage 'indexfi' avoids 'ax-ac' 'ax-ac2'; $) $} @@ -96165,54 +97095,54 @@ intersections of elements of the argument (see ~ elfi2 ). 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(Contributed TVUSTXVTVUTFXVQVWSVUDVUSWYBXUCWUHXVPVWTXUCVXAWYAXUEXFWCZXVPVUDVUSTXVKXVPV UDWYIVUSVUDWYIWHZVUSVWSXMXCWBYEXVTVUSWWJYHYFYTXWDVXCWUPYCXWAVWIVWSVUDUXJV XCWUPUXKUXLUXMXVQVWSUJFVXCVWSUXNHVWSVUDSZQOZVXCXVSSQOZXWBXWCUWDXVQVUSVWSV - WEVUBVWQWYBXVPUXPXWGUXQXVQVWSVXCXVQVWSVXCGHZWYAVWSVXCUXNHXVQVWKXWLEVUTVWS + WEVUBVWQWYBXVPUYEXWGUXPXVQVWSVXCXVQVWSVXCGHZWYAVWSVXCUXNHXVQVWKXWLEVUTVWS EUEULZVUCVWSVWJVXCGXWMVNVUCVWSVWIVOVPXWFXVQXUCVWSVUTFXWGVWSVUSUEVRXNYFYTW - UHVXBWYAXVPUWRVWSVXCUXRUXSUXTXVPXWJXVKXVPVUDWYITZXWJXWHXWNXWIVUDVWSSQVWSV - UDYKVUDVUSVWSUYEXEWAWBXWKXVQVXCWUPYJVJVWSVXCVUDXVSUYAYNYOWUHXVRXVSOZWYBXV + UHVXBWYAXVPUWRVWSVXCUXQUXRUXSXVPXWJXVKXVPVUDWYITZXWJXWHXWNXWIVUDVWSSQVWSV + UDYKVUDVUSVWSUXTXEWAWBXWKXVQVXCWUPYJVJVWSVXCVUDXVSUYAYNYOWUHXVRXVSOZWYBXV PVWPXWOVWGVWLVWPWVMWUPVXCWVRXDWIXFXHXVPXVNXVROXVKXVPXVNWUPWYKSXVRXVPXVMWU PWYKXVPXVLVUDVWIXVPXWNXVLVUDOXWHVUDWYIXGRWQWRWUPVUAVXCWSUYBWBXHXIXVOXVHDE - XVIWVSVUDVUCXVNXVGGWVTWVSXVNWYMVUDUBXVGWYIWYKVWIVUDXQVUDVUCWYMVOUVKVPXJRX + XVIWVSVUDVUCXVNXVGGWVTWVSXVNWYMVUDUBXVGWYIWYKVWIVUDXQVUDVUCWYMVOUVGVPXJRX KWYDWYMWYIVUAUKZIZFZVUFEXVIJZXVEBVULMZNZDXWQJZXVJXVFNZXWRWYMXWPTWYMWYLXWP VWIWYLXLZWYKVUATWYLXWPTVUAVXCXMWYKVUAWYIUVHUVIUVJWYMXWPVWIWYLVYIUVLXNVTWY DVWCWYIVUSVBZXXBXUOVXBXXEVWRWYAVXBXVBQVDZXXEVWTXXFVXAVWTXVBVWSQVWTXVBVWSV - USSZVWSVUSVWSYKVWTXUCXXGVWSOXUEVWSVUSXGRXEUYCUYDXXEXVBQXVBQOXXEYSVUSVWSUY - FUYGUYHRWIVWBXXEXXBNUAWYIVUSVWSWWJWGVUGWYIOZVVSXXEVUQXXBVUGWYIVUSWLXXHVUN + USSZVWSVUSVWSYKVWTXUCXXGVWSOXUEVWSVUSXGRXEUYCUYFXXEXVBQXVBQOXXEYSVUSVWSUY + DUYGUYHRWIVWBXXEXXBNUAWYIVUSVWSWWJWGVUGWYIOZVVSXXEVUQXXBVUGWYIVUSWLXXHVUN XXADVUPXWQXXHVUOXWPVUGWYIVUAUMUNXXHVUIXWSVUMXWTXXHVUFEVUHXVIVUGWYIUOUPXXH - VUKXVEBVULVUGWYIKVUJUSUQURUTURUVOWNXXAXXCDWYMXWQVUDWYMOZXWSXVJXWTXVFXXIVU - FXVHEXVIXXIVUEXVGVUCGVUDWYMVUCXPXRXSXXIXVEBVULWYNVUDWYMUOXTURVQUVPYAXVEWY + VUKXVEBVULVUGWYIKVUJUSUQURUTURUWAWNXXAXXCDWYMXWQVUDWYMOZXWSXVJXWTXVFXXIVU + FXVHEXVIXXIVUEXVGVUCGVUDWYMVUCXPXRXSXXIXVEBVULWYNVUDWYMUOXTURVQUVOYAXVEWY JBUGWYNWYIKVUJVXPVFYBRWYDVWPXUCWYHWYOVWNNZNXURXUFVWPXUCPZWYEXXJUFWYGXXKVX LWYGFZWYEPZPZWYJVWNUGWYNXXNVXPWYNFZWYJPZPZVXLVXPYCZVWMFZVUSKXXRLZVWNXXQXX RVWITXXSXXQVXLVXPVWIXXMVXLVWITZXXKXXPXXLXYAWYEXXLVXLWYFVWIVXLWYFWHZVWIVWS @@ -96267,13 +97197,13 @@ intersections of elements of the argument (see ~ elfi2 ). (Contributed XYHXXQVWSXYDVXLYGZWYIXYEVXPYGZVWSWYISQOZXYDXYESQOZXYJXXMXYKXXKXXPWYEXYKXX LVWSKVXLYIWBWCWYJXYLXXNXXOWYIKVXPYIUVNXYMXXQVWSVUSYJVJXXQXYDVXCTZVXCXYESZ QOXYNXXMXYOXXKXXPXXLXYOWYEXXLXYDWYFWTZVXCXXLVXLWYFTXYDXYQTXYBVXLWYFXAWAVW - IVWSUYJUYKXOWCXXQXYPXYEVXCSZQVXCXYEYKXXQXYEWYKTZWYKVXCSQOXYRQOXXOXYSXXNWY + IVWSUYNUYJXOWCXXQXYPXYEVXCSZQVXCXYEYKXXQXYEWYKTZWYKVXCSQOXYRQOXXOXYSXXNWY JXXOXYEWYLWTZWYKXXOVXPWYLTXYEXYTTXXOVXPWYMWYLXYCXXDXCVXPWYLXAWAWYIWYKXBXC WIVXCVUAUWBXYEWYKVXCYLYMXEXYDVXCXYEYLYPVWSXYDWYIXYEVXLVXPUWCYNXXQXYIVUSXY FXXRXXKXYIVUSOZXXMXXPXUCYUAVWPXUCYUAVWSVUSUWFWJWBXFUWGYOVUSXYFXXRUWHWAXYF - UWIVUSXYFKXXRUWJYMVVBXXTBXXRVWMVUSKVUJXXRVFVGYPYRYRYPUYLUWKYQUYMUYNUYOUYP + UWIVUSXYFKXXRUWJYMVVBXXTBXXRVWMVUSKVUJXXRVFVGYPYRYRYPUYKUWKYQUYOUYLUYMUYP UYQVWOVVDUDDVVFUDDULZVWLVVAVWNVVCYUBVWKVUFEVUTYUBVWJVUEVUCGVWIVUDVUCXPXRX - SYUBVVBBVWMVULVWIVUDUOXTURXJRUYRUYSUWMUYT $. + SYUBVVBBVWMVULVWIVUDUOXTURXJRUYRUYSUWNUYT $. $} ${ @@ -97288,12 +98218,12 @@ all reals (greatest real number) for which all positive reals are greater. nfoi $p |- F/_ x OrdIso ( R , A ) $= ( vh vu vv vj vw vz vt va cvv cv wbr wral con0 c0 nfcv coi wwe wse wa crn wn crab crio cmpt crecs cima wrex cres df-oi nfwe nfse nfan nfralw nfrabw - 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(Contributed by Stefan O'Rear, 11-Feb-2015.) $) @@ -98483,31 +99413,31 @@ of choice (over ZF), although it is know to imply dependent choice. ccnv csn ssun2 wb f1ofn elpreima wn wi elun df-or bitri ad2antlr ad2antrr eldifn simprr sselid fvco3 eldifi adantl snssd xpss1 simprl sseldd fvresd wfn xp1st eqeltrd elsni eqtrd ffnd a1i eqeltrrd expr mtod imim1d biimtrid - impd sylbid ssrdv wf1 f1ocnv f1of1 snex xpexg f1imaen2g syl22anc xpsnen2g - fnfvima entr domen1 mpbid olcd exlimdv pm2.61dne exlimiv sylbi ) AAUBZBCU - CZFGUUJUUIFGZABHGZACIGZUOZUUIUUJUDUUKUUJUUIDJZUEZDUFUUNUUJUUIDUGUUPUUNDUU - PUUNAKUUIUHZUUOUIZBUJZUKZUPUUTUPLAUUSMZUUPUUNAUUSULUUPUVAUUNUUPUVANZUULUU - MUVBAUUSHGZUUSBHGZUULUVBAUUSIGZUVCUVBUUSOTZUVAUVEUUPUVFUVAUUPUUSAOUUPAUUI - UMOAUNUUPUUIOUUPUUIUUOUQZOUUPUUJUUIUUOURUVGUUILUUJUUIUUOUSUUJUUIUUOUTPUUO - DVAZVHVBVCVDZUUPUUSUURUQAUURBVEUUPUUJAUURUUPUUIAUUQVFUUJUUIUUOVFZUUJAUURV - FAAVGUUJUUIUUOVIZUUJUUIAUUQUUOVJQZVKVLVMZRUUPUVAVNAUUSOWAVOAUUSVPPUUPUVDU - VAUUPUURVQBOTZUVFUVDUUPUUJAUURUVLVRUUPBUUJMZUUJOTZUVNBCVSZUUPUUJUUOUMOUUJ - 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(Contributed by NM, 19-Aug-1993.) $) elirrv $p |- -. x e. x $= - ( vz vy cv wcel csn wn wral wrex cvv wex snex eleq1w vsnid speivw zfregcl - mp2 weq velsn wi equcoms com12 biimtrid notbid rspccv mt2i nsyli ralrimiv - ax9 con2d ralnex sylib mt2 ) ADZUNEZBDUNFZEZGZBCDZHZCUPIZUPJEUSUPEZCKVAUN - LVBUNUPEZCACAUPMANZOCBUPJPQUOUTGZCUPHVAGUOVECUPUOUTVBUOVBUNUSEZUTVBCARZUO - VFCUNSVGUOVFUOVFTACACAUIUAUBUCUTVFVCVDURVCGBUNUSBARUQVCBAUPMUDUEUFUGUJUHU + ( vz vy cv wcel csn wn wral cvv wex vsnex eleq1w vsnid speivw zfregcl mp2 + wrex weq velsn wi ax9 equcoms com12 biimtrid notbid rspccv nsyli ralrimiv + mt2i con2d ralnex sylib mt2 ) ADZUNEZBDUNFZEZGZBCDZHZCUPQZUPIEUSUPEZCJVAA + KVBUNUPEZCACAUPLAMZNCBUPIOPUOUTGZCUPHVAGUOVECUPUOUTVBUOVBUNUSEZUTVBCARZUO + VFCUNSVGUOVFUOVFTACACAUAUBUCUDUTVFVCVDURVCGBUNUSBARUQVCBAUPLUEUFUIUGUJUHU TCUPUKULUM $. $( $j usage 'elirrv' avoids 'ax-13'; $) $} @@ -98714,6 +99644,12 @@ is that it denies the existence of a set containing itself ( ~ elirrv ). ( cv wnel cab cvv vprc nelir wceq wb ruv neleq1 ax-mp mpbir ) ABZNCADZECZEE CZEEFGOEHPQIAJOEEKLM $. + $( A class is disjoint from its singleton. A consequence of regularity. + (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, + 4-Apr-2019.) $) + disjcsn $p |- ( A i^i { A } ) = (/) $= + ( csn cin c0 wceq wcel wn elirr disjsn mpbir ) AABCDEAAFGAHAAIJ $. + ${ $d x y A $. $( The membership relation is well-founded on any class. (Contributed by @@ -101026,6 +101962,7 @@ a Cantor normal form (and injectivity, together with coherence UM $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Transitive closure of a relation @@ -101069,10 +102006,10 @@ a Cantor normal form (and injectivity, together with coherence form. (Contributed by Scott Fenton, 26-Oct-2024.) $) nfttrcld $p |- ( ph -> F/_ x t++ R ) $= ( vf vn vy vz va cttrcl cv csuc wfn c0 cfv wceq wa nfv nfcvd nfvd wbr w3a - wral wex com cdif wrex copab df-ttrcl nfbrd nfraldw nf3and nfrexd nfopabd - c1o nfexd nfcxfrd ) ABCJEKZFKZLMZNUROGKPUSUROHKPQZIKZUROZVBLUROZCUAZIUSUC - ZUBZEUDZFUEUOUFZUGZGHUHGHCEIFUIAVJGHBAGRAHRAVHBFVIAFRABVISAVGBEAERAUTVAVF - BAUTBTAVABTAVEBIUSAIRABUSSABVCVDCABVCSDABVDSUJUKULUPUMUNUQ $. + wral wex com c1o cdif copab df-ttrcl nfbrd nfraldw nf3and nfrexdw nfopabd + wrex nfexd nfcxfrd ) ABCJEKZFKZLMZNUROGKPUSUROHKPQZIKZUROZVBLUROZCUAZIUSU + CZUBZEUDZFUEUFUGZUOZGHUHGHCEIFUIAVJGHBAGRAHRAVHBFVIAFRABVISAVGBEAERAUTVAV + FBAUTBTAVABTAVEBIUSAIRABUSSABVCVDCABVCSDABVDSUJUKULUPUMUNUQ $. $} ${ @@ -101130,7 +102067,7 @@ a Cantor normal form (and injectivity, together with coherence $} ${ - $d R x y z f n m $. + $d R x y f n m $. $( If ` R ` is a relation, then it is a subclass of its transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.) $) ssttrcl $p |- ( Rel R -> R C_ t++ R ) $= @@ -101700,17 +102637,17 @@ a Cantor normal form (and injectivity, together with coherence wfr csn wal w3a snssi anim2i ssin vex snss bitr4i sylib inss2 inex1 epfrc ne0d mp3an2 elin anbi1i anass bitri elinel1 ancri trel inass incom ineq2i eqtri eleq2i bitr2i ne0i sylbi ex syl6 expd com34 impd biimtrid com23 imp - exlimdv necon4d anim2d expimpd reximdv2 expcomd impcom 3adant3 snex tz9.1 - exlimiiv exlimiv ) BFGZBUAUBZAHZBIZFJZABKZWMCHZBLZCMWNWRNZCBOWTXACWSUCZDH - ZPZXCQZXBEHZPXFQRXCXFPNEUDZUEWTXANZDXDXEXHXGXEXDXHXEXDWTXAXDWTRZXCBIZFGZX - EXAXIXJWSXIXDXBBPZRZWSXJLZWTXLXDWSBUFUGXMXBXJPXNXBXCBUHWSXJCUIUJUKULUPXEW - NXKWRWNXKRXJWOIZFJZAXJKZXEWRWNXJBPXKXQXCBUMABXJXCBDUIUNUOUQXEXPWQAXJBWOXJ - LZXPRZADSZWOBLZXPRZRZXEYAWQRZXSXTYARZXPRYCXRYEXPWOXCBURUSXTYAXPUTVAXEXTYB - YDXEXTRZXPWQYAYFWPFXOFXEXTWPFGZXOFGZNXEYGXTYHYGXFWPLZEMXEXTYHNZEWPOXEYIYJ - EYIEASZYIRXEYJYIYKXFWOBVBVCXEYKYIYJXEYKXTYIYHXEYKXTYIYHNZXEYKXTREDSZYLXCX - FWOVDYMYIYHYMYIRZXFXOLZYHYOXFXCWPIZLYNXOYPXFXOXCBWOIZIYPXCBWOVEYQWPXCBWOV - FVGVHVIXFXCWPURVJXOXFVKVLVMVNVOVPVQTWAVRVSVTWBWCWDVRWETWFTVOWGWHDEXBWSWIW - JWKWLVLWG $. + exlimdv necon4d anim2d expimpd expcomd impcom 3adant3 vsnex tz9.1 exlimiv + reximdv2 exlimiiv ) BFGZBUAUBZAHZBIZFJZABKZWMCHZBLZCMWNWRNZCBOWTXACWSUCZD + HZPZXCQZXBEHZPXFQRXCXFPNEUDZUEWTXANZDXDXEXHXGXEXDXHXEXDWTXAXDWTRZXCBIZFGZ + XEXAXIXJWSXIXDXBBPZRZWSXJLZWTXLXDWSBUFUGXMXBXJPXNXBXCBUHWSXJCUIUJUKULUPXE + WNXKWRWNXKRXJWOIZFJZAXJKZXEWRWNXJBPXKXQXCBUMABXJXCBDUIUNUOUQXEXPWQAXJBWOX + JLZXPRZADSZWOBLZXPRZRZXEYAWQRZXSXTYARZXPRYCXRYEXPWOXCBURUSXTYAXPUTVAXEXTY + BYDXEXTRZXPWQYAYFWPFXOFXEXTWPFGZXOFGZNXEYGXTYHYGXFWPLZEMXEXTYHNZEWPOXEYIY + JEYIEASZYIRXEYJYIYKXFWOBVBVCXEYKYIYJXEYKXTYIYHXEYKXTYIYHNZXEYKXTREDSZYLXC + XFWOVDYMYIYHYMYIRZXFXOLZYHYOXFXCWPIZLYNXOYPXFXOXCBWOIZIYPXCBWOVEYQWPXCBWO + VFVGVHVIXFXCWPURVJXOXFVKVLVMVNVOVPVQTWAVRVSVTWBWCWDVRWKTWETVOWFWGDEXBCWHW + IWLWJVLWF $. $} ${ @@ -102558,7 +103495,7 @@ of the previous layer (and the union of previous layers when the ${ $d x y z $. - $( The domain and range of the ` rank ` function. (Contributed by Mario + $( The domain and codomain of the ` rank ` function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.) $) rankf $p |- rank : U. ( R1 " On ) --> On $= ( vx vy vz cr1 con0 cima cuni crnk wf wfn cv cfv wcel wceq eqtri mpbir2an @@ -105079,9 +106016,9 @@ several of their earlier lemmas available (which would otherwise be ULAKZBKZLMNZUJUKUNABOUJUKUQUNPZUJUOQIJZUPQIJZURUKACRBDRUSUTSZUQUNVAUQUOUPTZ HIJUNUOUPUBUMVBHIABUCUDUEUFUGUAUHUI $. - $( If an unordered pair has two elements they are different. (Contributed by - FL, 14-Feb-2010.) Avoid ~ ax-pow , ~ ax-un . (Revised by BTernaryTau, - 30-Dec-2024.) $) + $( If an unordered pair has two elements, then they are different. + (Contributed by FL, 14-Feb-2010.) Avoid ~ ax-pow , ~ ax-un . (Revised by + BTernaryTau, 30-Dec-2024.) $) pr2ne $p |- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o <-> A =/= B ) ) $= ( wcel wa cpr c2o cen wbr wne csn snnen2o dfsn2 preq2 eqtr2id breq1d mtbiri wceq necon2ai enpr2 3expia impbid2 ) ACEZBDEZFABGZHIJZABKZUGABABSZUGALZHIJA @@ -105124,12 +106061,12 @@ several of their earlier lemmas available (which would otherwise be and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) $) en2eleq $p |- ( ( X e. P /\ P ~~ 2o ) -> P = { X , U. ( P \ { X } ) } ) $= ( wcel c2o cen wbr wa csn cdif cuni cpr cfn wss wceq com adantl wne syl3anc - 2onn c1o nnfi ax-mp enfi mpbiri simpl csuc 1onn simpr df-2o breqtrdi dif1en - mp3an2i en1uniel syl eldifsn sylib simpld prssd simprd necomd enpr2 syl2anc - ensym entr fisseneq eqcomd ) BACZADEFZGZBABHIZJZKZAVIALCZVLAMVLAEFZVLANVHVM - VGVHVMDLCZDOCVOSDUAUBADUCUDPVIBVKAVGVHUEZVIVKACZVKBQZVIVKVJCZVQVRGVIVJTEFZV - STOCVIATUFZEFVGVTUGVIADWAEVGVHUHUIUJVPATBUKULVJUMUNVKABUOUPZUQZURVIVLDEFZDA - EFZVNVIVGVQBVKQWDVPWCVIVKBVIVQVRWBUSUTBVKAAVARVHWEVGADVCPVLDAVDVBVLAVERVF + 2onn c1o nnfi ax-mp enfi mpbiri simpl simpr df-2o breqtrdi dif1ennn mp3an2i + csuc 1onn en1uniel syl eldifsn sylib simpld prssd simprd necomd enpr2 ensym + entr syl2anc fisseneq eqcomd ) BACZADEFZGZBABHIZJZKZAVIALCZVLAMVLAEFZVLANVH + VMVGVHVMDLCZDOCVOSDUAUBADUCUDPVIBVKAVGVHUEZVIVKACZVKBQZVIVKVJCZVQVRGVIVJTEF + ZVSTOCVIATUKZEFVGVTULVIADWAEVGVHUFUGUHVPATBUIUJVJUMUNVKABUOUPZUQZURVIVLDEFZ + DAEFZVNVIVGVQBVKQWDVPWCVIVKBVIVQVRWBUSUTBVKAAVARVHWEVGADVBPVLDAVCVDVLAVERVF $. $( Taking the other element twice in a pair gets back to the original @@ -105137,12 +106074,12 @@ several of their earlier lemmas available (which would otherwise be en2other2 $p |- ( ( X e. P /\ P ~~ 2o ) -> U. ( P \ { U. ( P \ { X } ) } ) = X ) $= ( wcel c2o cen wbr csn cdif cuni cpr en2eleq prcom eqtrdi difeq1d difprsnss - wa eqsstrdi wne simpl c1o com csuc 1onn simpr df-2o breqtrdi dif1en mp3an2i - en1uniel eldifsni 3syl necomd eldifsn snssd eqssd unieqd wceq unisng adantr - sylanbrc eqtrd ) BACZADEFZPZAABGZHZIZGZHZIVEIZBVDVIVEVDVIVEVDVIVGBJZVHHVEVD - AVKVHVDABVGJVKABKBVGLMNVGBOQVDBVIVDVBBVGRBVICVBVCSZVDVGBVDVFTEFZVGVFCVGBRTU - ACVDATUBZEFVBVMUCVDADVNEVBVCUDUEUFVLATBUGUHVFUIVGABUJUKULBAVGUMUTUNUOUPVBVJ - BUQVCBAURUSVA $. + wa eqsstrdi wne simpl c1o com csuc simpr breqtrdi dif1ennn mp3an2i en1uniel + 1onn df-2o eldifsni 3syl necomd eldifsn sylanbrc snssd unieqd unisng adantr + eqssd wceq eqtrd ) BACZADEFZPZAABGZHZIZGZHZIVEIZBVDVIVEVDVIVEVDVIVGBJZVHHVE + VDAVKVHVDABVGJVKABKBVGLMNVGBOQVDBVIVDVBBVGRBVICVBVCSZVDVGBVDVFTEFZVGVFCVGBR + TUACVDATUBZEFVBVMUHVDADVNEVBVCUCUIUDVLATBUEUFVFUGVGABUJUKULBAVGUMUNUOUSUPVB + VJBUTVCBAUQURVA $. ${ $d A m $. $d X m $. @@ -107069,27 +108006,27 @@ is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The dfac5lem2 $p |- ( <. w , g >. e. U. A <-> ( w e. h /\ g e. w ) ) $= ( cv cuni wcel wceq wrex wa wel wex exbii 3bitri anbi2i vex bitri cop wne c0 csn cxp cab unieqi eleq2i eluniab r19.42v bitr2i rexcom4 df-rex bitr3i - anass ancom ne0i pm4.71i bitr4i snex xpex eleq2 ceqsexv opelxp velsn an12 - equcom anbi1i elequ1 anbi12d ) AHZEHZUAZDIZJVMBHZUCUBZVOCHZUDZVQUEZKZCFHZ - LZMZBUFZIZJZCFNZVMVOJZVPMZVTMZBOZMZCOZAFNZEANZMZVNWEVMDWDGUGUHWFWHWCMZBOW - JCWALZBOZWMWCBVMUIWQWRBWRWIWBMWQWIVTCWAUJWHVPWBUOUKPWSWKCWALWMWJCBWAULWKC - WAUMUNQWMVQVKKZWGECNZMZMZCOWPWLXCCWLWGVMVSJZMWGWTXAMZMXCWKXDWGWKVTWHMZBOX - DWJXFBWJVTWIMXFWIVTUPWHWIVTWHVPVOVMUQURRUSPWHXDBVSVRVQVQUTCSVAVOVSVMVBVCT - RXDXEWGXDVKVRJZXAMXEVKVLVRVQVDXGWTXAXGVKVQKWTAVQVEACVGTVHTRWGWTXAVFQPXBWP - CVKASWTWGWNXAWOCAFVIVQVKVLVBVJVCTQ $. + anass ancom pm4.71i bitr4i vsnex eleq2 ceqsexv opelxp velsn equcom anbi1i + ne0i xpex an12 elequ1 anbi12d ) AHZEHZUAZDIZJVMBHZUCUBZVOCHZUDZVQUEZKZCFH + ZLZMZBUFZIZJZCFNZVMVOJZVPMZVTMZBOZMZCOZAFNZEANZMZVNWEVMDWDGUGUHWFWHWCMZBO + WJCWALZBOZWMWCBVMUIWQWRBWRWIWBMWQWIVTCWAUJWHVPWBUOUKPWSWKCWALWMWJCBWAULWK + CWAUMUNQWMVQVKKZWGECNZMZMZCOWPWLXCCWLWGVMVSJZMWGWTXAMZMXCWKXDWGWKVTWHMZBO + XDWJXFBWJVTWIMXFWIVTUPWHWIVTWHVPVOVMVFUQRURPWHXDBVSVRVQCUSCSVGVOVSVMUTVAT + RXDXEWGXDVKVRJZXAMXEVKVLVRVQVBXGWTXAXGVKVQKWTAVQVCACVDTVETRWGWTXAVHQPXBWP + CVKASWTWGWNXAWOCAFVIVQVKVLUTVJVATQ $. $( Lemma for ~ dfac5 . (Contributed by NM, 12-Apr-2004.) $) dfac5lem3 $p |- ( ( { w } X. w ) e. A <-> ( w =/= (/) /\ w e. h ) ) $= - ( cv csn cxp c0 wne wceq wrex wa wcel vex xpeq2 crn rneq snnz neeq1 eqeq1 - cab snex xpex rexbidv anbi12d elab eleq2i xp0 eqtrdi ax-mp 3eqtr3g impbii - rnxp rn0 necon3bii wex df-rex sneq xpeq1d eqtrd equcom bitri exbii elequ1 - anbi1ci equsexvw 3bitrri anbi12i 3bitr4i ) AGZHZVLIZBGZJKZVOCGZHZVQIZLZCE - GZMZNZBUCZOVNJKZVNVSLZCWAMZNZVNDOVLJKZVLWAOZNWCWHBVNVMVLVLUDAPZUEVOVNLZVP - WEWBWGVOVNJUAWLVTWFCWAVOVNVSUBUFUGUHDWDVNFUIWIWEWJWGVLJVNJVLJLZVNJLZWMVNV - MJIJVLJVMQVMUJUKWNVNRZJRVLJVNJSVMJKWOVLLVLWKTVMVLUOULZUPUMUNUQWGVQWAOZWFN - ZCURVQVLLZWQNZCURWJWFCWAUSWRWTCWFWSWQWFVLVQLZWSWFXAWFWOVSRZVLVQVNVSSWPVRJ - KXBVQLVQCPTVRVQUOULUMXAVNVRVLIVSXAVMVRVLVLVQUTVAVLVQVRQVBUNACVCVDVGVEWQWJ - CACAEVFVHVIVJVK $. + ( cv csn cxp c0 wne wceq wrex wa wcel vex xpeq2 crn rneq snnz vsnex neeq1 + cab xpex eqeq1 rexbidv anbi12d elab eleq2i xp0 eqtrdi rnxp 3eqtr3g impbii + ax-mp rn0 necon3bii df-rex xpeq1d eqtrd equcom bitri anbi1ci exbii elequ1 + wex sneq equsexvw 3bitrri anbi12i 3bitr4i ) AGZHZVLIZBGZJKZVOCGZHZVQIZLZC + EGZMZNZBUCZOVNJKZVNVSLZCWAMZNZVNDOVLJKZVLWAOZNWCWHBVNVMVLAUAAPZUDVOVNLZVP + WEWBWGVOVNJUBWLVTWFCWAVOVNVSUEUFUGUHDWDVNFUIWIWEWJWGVLJVNJVLJLZVNJLZWMVNV + MJIJVLJVMQVMUJUKWNVNRZJRVLJVNJSVMJKWOVLLVLWKTVMVLULUOZUPUMUNUQWGVQWAOZWFN + ZCVFVQVLLZWQNZCVFWJWFCWAURWRWTCWFWSWQWFVLVQLZWSWFXAWFWOVSRZVLVQVNVSSWPVRJ + KXBVQLVQCPTVRVQULUOUMXAVNVRVLIVSXAVMVRVLVLVQVGUSVLVQVRQUTUNACVAVBVCVDWQWJ + CACAEVEVHVIVJVK $. ${ dfac5lem.2 $e |- B = ( U. A i^i y ) $. @@ -107104,32 +108041,32 @@ is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The simprbi bitri bi2anan9 exdistrv bitr4di biimpac sylan2b adantrr exlimiv bitrdi velsn opeq1 sylan9req syl exlimivv syl6bi syl6 eqeq12 sylibrd ex opth1 rexlimivw rexlimdvw syl2an syl5bir necon1ad alrimdv disj1 syl6ibr - imp rgen2 cvv cuni cpw vuniex xpex pwex wss elssuni xpss12 syl2anc snex - snssi sylibr eleq1 syl5ibrcom rexlimiv adantl abssi ssexi eqeltri raleq - elpw raleqbi1dv exbidv imbi12d spcv sylbi mp2ani ) ADOZUBUCZDIPZYPEOZUC - ZYPYSUDUBQZULZEIPZDIPZFOZYPCOZUDRFUEZDIPZCSZYQDIYQYPGOZUBUCZUUJHOZUFZUU - LUGZQZHKOZUHZTZGUIZIYPUUSRYQYPUUNQZHUUPUHZUURYQUVATZGYPDUJZGDUMZUUKYQUU - QUVAUUJYPUBUKUVDUUOUUTHUUPUUJYPUUNUNUOUPZUQURLUSVEUUBDEIIYPIRZYSIRZTZYT - BDUTZBEUTZVAULZBVBUUAUVHYTUVKBUVHUVKYPYSUVKVAUVIUVJTZUVHDEUMZUVIUVJVCUV - FUVAYSUAOZUFZUVNUGZQZUAUUPUHZUVLUVMULZUVGUVFYQUVAUURUVBGYPIUVCUVELVDVQU - VGYSUUNQZHUUPUHZUVRUVGYSUBUCZUWAUURUWBUWATGYSIEUJGEUMZUUKUWBUUQUWAUUJYS - UBUKUWCUUOUVTHUUPUUJYSUUNUNUOUPLVDVQUVTUVQHUAUUPHUAUMZUUNUVPYSUWDUUNUVO - UULUGUVPUWDUUMUVOUULUULUVNVFVGUULUVNUVOVHVIZVJVKVLUVAUVRUVSUVAUVQUVSUAU - UPUUTUVQUVSULHUUPUUTUVQUVSUUTUVQTZUVLUUNUVPQZUVMUWFUVLUWDUWGUWFUVLBOZUU - JUUEVMZQZUUJUUMRZFHUTZTTZGSZUWHUUJUUFVMZQZUUJUVORZCUAUTZTTZGSZTZCSFSZUW - DUWFUVLUWNFSZUWTCSZTUXBUUTUVIUXCUVQUVJUXDUUTUVIUWHUUNRZUXCYPUUNUWHVNUXE - UWMFSGSUXCGFUWHUUMUULVOUWMGFVPVRWFUVQUVJUWHUVPRZUXDYSUVPUWHVNUXFUWSCSGS - UXDGCUWHUVOUVNVOUWSGCVPVRWFVSUWNUWTFCVTWAUXAUWDFCUXAUULUUEVMZUVNUUFVMZQ - UWDUWNUWTUXGUWHUXHUWMUWHUXGQZGUWJUWKUXIUWLUWKUWJGHUMZUXIGUULWGUXJUWJUXI - UXJUWIUXGUWHUUJUULUUEWHVJWBWCWDWEUWSUWHUXHQZGUWPUWQUXKUWRUWQUWPGUAUMZUX - KGUVNWGUXLUWPUXKUXLUWOUXHUWHUUJUVNUUFWHVJWBWCWDWEWIUULUUEUVNUUFHUJZFUJW - QWJWKWLUWEWMYPUUNYSUVPWNWOWPWRWSXFWTXAXBXCBYPYSXDXEXGAYQDUWHPZUUBEUWHPZ - DUWHPZTZUUGDUWHPZCSZULZBVBYRUUDTZUUIULZNUXTUYBBIIUUSXHLUUSUUPUUPXIZUGZX - JZUYDUUPUYCKUJKXKXLXMUURGUYEUUQUUJUYERZUUKUUOUYFHUUPHKUTZUYFUUOUUNUYERZ - UYGUUNUYDXNZUYHUYGUUMUUPXNUULUYCXNUYIUULUUPXSUULUUPXOUUMUUPUULUYCXPXQUU - NUYDUUMUULUULXRUXMXLYIXTUUJUUNUYEYAYBYCYDYEYFYGUWHIQZUXQUYAUXSUUIUYJUXN - YRUXPUUDYQDUWHIYHUXOUUCDUWHIUUBEUWHIYHYJUPUYJUXRUUHCUUGDUWHIYHYKYLYMYNY - O $. + imp rgen2 cvv cuni cpw vuniex xpex pwex wss snssi elssuni syl2anc vsnex + xpss12 elpw sylibr eleq1 syl5ibrcom rexlimiv adantl abssi ssexi eqeltri + raleq raleqbi1dv exbidv imbi12d spcv sylbi mp2ani ) ADOZUBUCZDIPZYPEOZU + CZYPYSUDUBQZULZEIPZDIPZFOZYPCOZUDRFUEZDIPZCSZYQDIYQYPGOZUBUCZUUJHOZUFZU + ULUGZQZHKOZUHZTZGUIZIYPUUSRYQYPUUNQZHUUPUHZUURYQUVATZGYPDUJZGDUMZUUKYQU + UQUVAUUJYPUBUKUVDUUOUUTHUUPUUJYPUUNUNUOUPZUQURLUSVEUUBDEIIYPIRZYSIRZTZY + TBDUTZBEUTZVAULZBVBUUAUVHYTUVKBUVHUVKYPYSUVKVAUVIUVJTZUVHDEUMZUVIUVJVCU + VFUVAYSUAOZUFZUVNUGZQZUAUUPUHZUVLUVMULZUVGUVFYQUVAUURUVBGYPIUVCUVELVDVQ + UVGYSUUNQZHUUPUHZUVRUVGYSUBUCZUWAUURUWBUWATGYSIEUJGEUMZUUKUWBUUQUWAUUJY + SUBUKUWCUUOUVTHUUPUUJYSUUNUNUOUPLVDVQUVTUVQHUAUUPHUAUMZUUNUVPYSUWDUUNUV + OUULUGUVPUWDUUMUVOUULUULUVNVFVGUULUVNUVOVHVIZVJVKVLUVAUVRUVSUVAUVQUVSUA + UUPUUTUVQUVSULHUUPUUTUVQUVSUUTUVQTZUVLUUNUVPQZUVMUWFUVLUWDUWGUWFUVLBOZU + UJUUEVMZQZUUJUUMRZFHUTZTTZGSZUWHUUJUUFVMZQZUUJUVORZCUAUTZTTZGSZTZCSFSZU + WDUWFUVLUWNFSZUWTCSZTUXBUUTUVIUXCUVQUVJUXDUUTUVIUWHUUNRZUXCYPUUNUWHVNUX + EUWMFSGSUXCGFUWHUUMUULVOUWMGFVPVRWFUVQUVJUWHUVPRZUXDYSUVPUWHVNUXFUWSCSG + SUXDGCUWHUVOUVNVOUWSGCVPVRWFVSUWNUWTFCVTWAUXAUWDFCUXAUULUUEVMZUVNUUFVMZ + QUWDUWNUWTUXGUWHUXHUWMUWHUXGQZGUWJUWKUXIUWLUWKUWJGHUMZUXIGUULWGUXJUWJUX + IUXJUWIUXGUWHUUJUULUUEWHVJWBWCWDWEUWSUWHUXHQZGUWPUWQUXKUWRUWQUWPGUAUMZU + XKGUVNWGUXLUWPUXKUXLUWOUXHUWHUUJUVNUUFWHVJWBWCWDWEWIUULUUEUVNUUFHUJZFUJ + WQWJWKWLUWEWMYPUUNYSUVPWNWOWPWRWSXFWTXAXBXCBYPYSXDXEXGAYQDUWHPZUUBEUWHP + ZDUWHPZTZUUGDUWHPZCSZULZBVBYRUUDTZUUIULZNUXTUYBBIIUUSXHLUUSUUPUUPXIZUGZ + XJZUYDUUPUYCKUJKXKXLXMUURGUYEUUQUUJUYERZUUKUUOUYFHUUPHKUTZUYFUUOUUNUYER + ZUYGUUNUYDXNZUYHUYGUUMUUPXNUULUYCXNUYIUULUUPXOUULUUPXPUUMUUPUULUYCXSXQU + UNUYDUUMUULHXRUXMXLXTYAUUJUUNUYEYBYCYDYEYFYGYHUWHIQZUXQUYAUXSUUIUYJUXNY + RUXPUUDYQDUWHIYIUXOUUCDUWHIUUBEUWHIYIYJUPUYJUXRUUHCUUGDUWHIYIYKYLYMYNYO + $. $( Lemma for ~ dfac5 . (Contributed by NM, 12-Apr-2004.) $) dfac5lem5 $p |- ( ph -> @@ -107665,17 +108602,17 @@ our Axiom of Choice (in the form of ~ ac2 ). The proof does not depend ( vv vu cv cin wcel weu wi wral wex wel wn wa wceq eleq2d eubidv cuni csn ineq2 imbi2d ralbidv cbvexvw indi c0 elssuni ssneld disjsn syl6ibr impcom cun uneq2d un0 eqtrdi eqtr2id ralbidva wo vsnid olci elun mpbir sseld mpi - con3i biantrurd bitrd snex eleq1 notbid anbi12d spcev syl6bi vuniex eleq2 - vex unex exbidv wal nalset alexn spi vtocl exlimiiv exlimiv sylbi exsimpr - impbii ) AEHZDHZCHZIZJZEKZLZDBHZMZCNZCBOZPZWSQZCNZWTAWKWLFHZIZJZEKZLZDWRM - ZFNXDWSXJCFWMXERZWQXIDWRXKWPXHAXKWOXGEXKWNXFWKWMXEWLUCSTUDUEUFXJXDFGHZWRU - AZJZPZXJXDLGXOXJXEXLUBZUNZWRJZPZAWKWLXQIZJZEKZLZDWRMZQZXDXOXJYDYEXOXIYCDW - RXODBOZQZXHYBAYGXGYAEYGXFXTWKYGXTXFWLXPIZUNZXFWLXEXPUGYGYIXFUHUNXFYGYHUHX - FYFXOYHUHRZYFXOGDOPYJYFWLXMXLWLWRUIUJWLXLUKULUMUOXFUPUQURSTUDUSXOXSYDXRXN - XRXLXQJZXNYKGFOZXLXPJZUTYMYLGVAVBXLXEXPVCVDXRXQXMXLXQWRUIVEVFVGVHVIXCYECX - QXEXPFVRXLVJVSWMXQRZXBXSWSYDYNXAXRWMXQWRVKVLYNWQYCDWRYNWPYBAYNWOYAEYNWNXT - WKWMXQWLUCSTUDUEVMVNVOGCOZPZGNZXOGNCXMBVPWMXMRZYPXOGYRYOXNWMXMXLVQVLVTYQC - YQCWAYOGWACNPCGWBYOCGWCVDWDWEWFWGWHXBWSCWIWJ $. + con3i biantrurd bitrd vsnex unex eleq1 notbid anbi12d spcev syl6bi vuniex + vex eleq2 exbidv wal nalset alexn spi vtocl exlimiiv exlimiv sylbi impbii + exsimpr ) AEHZDHZCHZIZJZEKZLZDBHZMZCNZCBOZPZWSQZCNZWTAWKWLFHZIZJZEKZLZDWR + MZFNXDWSXJCFWMXERZWQXIDWRXKWPXHAXKWOXGEXKWNXFWKWMXEWLUCSTUDUEUFXJXDFGHZWR + UAZJZPZXJXDLGXOXJXEXLUBZUNZWRJZPZAWKWLXQIZJZEKZLZDWRMZQZXDXOXJYDYEXOXIYCD + WRXODBOZQZXHYBAYGXGYAEYGXFXTWKYGXTXFWLXPIZUNZXFWLXEXPUGYGYIXFUHUNXFYGYHUH + XFYFXOYHUHRZYFXOGDOPYJYFWLXMXLWLWRUIUJWLXLUKULUMUOXFUPUQURSTUDUSXOXSYDXRX + NXRXLXQJZXNYKGFOZXLXPJZUTYMYLGVAVBXLXEXPVCVDXRXQXMXLXQWRUIVEVFVGVHVIXCYEC + XQXEXPFVRGVJVKWMXQRZXBXSWSYDYNXAXRWMXQWRVLVMYNWQYCDWRYNWPYBAYNWOYAEYNWNXT + WKWMXQWLUCSTUDUEVNVOVPGCOZPZGNZXOGNCXMBVQWMXMRZYPXOGYRYOXNWMXMXLVSVMVTYQC + YQCWAYOGWACNPCGWBYOCGWCVDWDWEWFWGWHXBWSCWJWI $. $( Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, @@ -109533,63 +110470,63 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality ( con0 wcel wa cfv wss vt vs word wrex wral wsmo w3a wex csuc cdm wfo cep cv wf wiso wf1o cvv wwe ssrab3 ssexg mpan onss sstrid epweon mpisyl oiiso wess syl2anc ad2antlr isof1o f1ofo 3syl fof fss sylancl oion simplr eloni - syl smoiso2 syl2an biimpar simprd syl21anc smorndom syl3anc onsssuc mpbid - wb adantrr ccom vex oiexg coexg sylancr fcod simpr ordsson ad2antrr simpl - simprl ffvelcdm wfn ffn smoel2 sylan wi wceq fveq2 eleq2d raleqbi1dv crab - jca weq eleq1d cbvralvw bitrid cbvrabv eqtri elrab2 simprbi rspccv ordtr1 - sylc ancomsd imp fvco3 syl2an2r 3eltr4d ralrimivva issmo2 syl13anc sseq2d - c0 cint ex wn ffvelcdmd mpd ralbidv rabn0 ssrab2 inteqi eqeltrid sylan2br - wne onint elrab sylib adantl simpr2 simp3 eleq2i simp21 simp1l fssd ontr1 - simp22 simp1r 3impib ontri1 simp23 simpl1 sstr anim12i sylibr onnmin expr - rabid syl31anc sylbird con4d biimtrid 3expia ralrimiv expcom 3expib com13 - sylanbrc syld com23 imp31 foelrn eleq1 biimpcd ad5ant24 sylancom reximdva - syl6 sylibrd ralimdv impr 3jca smoeq fveq1 rexbidv 3anbi123d spcedv rexeq - feq1 feq2 3anbi13d exbidv rspcev exlimdv ) FUCZGPQZRZGFIUMZUNZCUMZDUMZUXI - SZTZDGUDZCFUEZRZAUMZFJUMZUNZUXSUFZUXKEUMZUXSSZTZEUXRUDZCFUEZUGZJUHZAGUIZU - DZIUXHUXQUYJUXHUXQRZLUJZUYIQZUYLFUXSUNZUYAUYDEUYLUDZCFUEZUGZJUHZUYJUXHUXJ - UYMUXPUXHUXJRZUYLGTZUYMUYSUYLGLUNZLUFZGUCZUYTUYSUYLHLUKZVUAUYSUYLHULULLUO - ZUYLHLUPVUDUXGVUEUXFUXJUXGHUQQZHULURZVUEHGTZUXGVUFUXMBUMZUXISZQZDVUIUEZBG - HMUSZHGPUTVAZUXGHPTZPULURVUGUXGHGPVUMGVBZVCZVDHPULVGVEHULLUQOVFVHVIZUYLHU - LULLVJUYLHLVKVLZVUDUYLHLUNZVUHVUAUYLHLVMZVUMUYLHGLVNVOVSZUYSUYLPQZUXGVUEV - UBUXGVVCUXFUXJUXGVUFVVCVUNHULLUQOVPVSVIZUXFUXGUXJVQZVURVVCUXGRZVUERVUDVUB - VVFVUDVUBRZVUEVVCUYLUCZVUOVVGVUEWIUXGUYLVRZVUQUYLHLVTWAWBWCWDZUXGVUCUXFUX - JGVRVIUYLGLWEWFUYSVVCUXGUYTUYMWIVVDVVEUYLGWGVHWHWJUYKUYQUYLFUXILWKZUNZVVK - UFZUXKUYBVVKSZTZEUYLUDZCFUEZUGJUQVVKUYKUXIUQQLUQQZVVKUQQIWLUXGVVRUXFUXQUX - GVUFVVRVUNHULLUQOWMVSVIUXILUQUQWNWOUYKVVLVVMVVQUYKUYLGFUXILUXHUXJUXPXAUXH - UXJVUAUXPVVBWJWPUXHUXJVVMUXPUYSVVLFPTZVVHUAUMZVVKSZUBUMZVVKSZQZUAVWBUEUBU - YLUEZVVMUYSUYLGFUXILUXHUXJWQVVBWPUXFVVSUXGUXJFWRZWSUYSVVCVVHVVDVVIVSUYSVW - 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The cofinality ( vc vb vd wcel cfn crpss wpo sylancr cv wbr wn cin wrex cvv wss wa wne c0 cpw ccnv wfr porpss cnvpo mpbi pwfi biimpi frfi wral inss2 pwexg ssexg 0fin 0elpw elini ne0ii fri mpanr12 sylan ex elinel1 ralnex csn cun adantr - snfi unfi sylancl elinel2 elpwid snssi ad2antrl vex snex unex elpw sylibr - unssd elind wpss disjsn biimpri disjpss ad2antll brcnv brrpss bitri breq1 - wceq snnz rspcev syl2anc expr con1d biimtrid impancom ssrdv ssfi syl2an2r - rexlimiva syl6 impbid2 ) ABFZAGFZAUAZHUBZUCZXEXFXGIZXFGFZXHXFHIXIXFUDXFHU - EUFXEXJAUGUHXFXGUIJXDXHCKZDKZXGLZMCGXFNZUJZDXNOZXEXDXHXPXDXNPFZXHXPXDXNXF - QZXFPFXQGXFUKZABULXNXFPUMJXQXHRXRXNTSXPXSTXNTGXFUNAUOUPUQDCXFXNPXGURUSUTV - AXOXEDXNXLXNFZXLGFZXOAXLQXEXLGXFVBZXTXOREAXLXTEKZAFZXOYCXLFZXOXMCXNOZMXTY - DRZYEXMCXNVCYGYEYFXTYDYEMZYFXTYDYHRZRZXLYCVDZVEZXNFYLXLXGLZYFYJGXFYLYJYAY - KGFYLGFXTYAYIYBVFYCVGXLYKVHVIYJYLAQYLXFFYJXLYKAXTXLAQYIXTXLAXLGXFVJVKVFYD - YKAQXTYHYCAVLVMVSYLAXLYKDVNZYCVOVPZVQVRVTYJXLYLWAZYMYHYPXTYDYHXLYKNTWJZYK - TSYPYQYHXLYCWBWCYCEVNWKXLYKWDVIWEYMXLYLHLYPYLXLHYOYNWFXLYLYOWGWHVRXMYMCYL - XNXKYLXLXGWIWLWMWNWOWPWQWRXLAWSWTXAXBXC $. + snfi unfi sylancl elinel2 elpwid snssi ad2antrl unssd vex vsnex unex elpw + sylibr elind wpss wceq disjsn biimpri disjpss ad2antll brcnv brrpss bitri + snnz breq1 rspcev syl2anc expr con1d biimtrid impancom syl2an2r rexlimiva + ssrdv ssfi syl6 impbid2 ) ABFZAGFZAUAZHUBZUCZXEXFXGIZXFGFZXHXFHIXIXFUDXFH + UEUFXEXJAUGUHXFXGUIJXDXHCKZDKZXGLZMCGXFNZUJZDXNOZXEXDXHXPXDXNPFZXHXPXDXNX + FQZXFPFXQGXFUKZABULXNXFPUMJXQXHRXRXNTSXPXSTXNTGXFUNAUOUPUQDCXFXNPXGURUSUT + VAXOXEDXNXLXNFZXLGFZXOAXLQXEXLGXFVBZXTXOREAXLXTEKZAFZXOYCXLFZXOXMCXNOZMXT + YDRZYEXMCXNVCYGYEYFXTYDYEMZYFXTYDYHRZRZXLYCVDZVEZXNFYLXLXGLZYFYJGXFYLYJYA + YKGFYLGFXTYAYIYBVFYCVGXLYKVHVIYJYLAQYLXFFYJXLYKAXTXLAQYIXTXLAXLGXFVJVKVFY + DYKAQXTYHYCAVLVMVNYLAXLYKDVOZEVPVQZVRVSVTYJXLYLWAZYMYHYPXTYDYHXLYKNTWBZYK + TSYPYQYHXLYCWCWDYCEVOWJXLYKWEVIWFYMXLYLHLYPYLXLHYOYNWGXLYLYOWHWIVSXMYMCYL + XNXKYLXLXGWKWLWMWNWOWPWQWTXLAXAWRWSXBXC $. $( A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, @@ -112019,17 +112956,17 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes) ( con0 wss cwdom wbr wa wcel cpw cdom word cep cvv 3syl wf adantr syl3anc syl cdm char cfv oicl wb cuni csuc relwdom brrelex1i adantl uniexg sucexg wsmo oif onsucuni fss sylancr crn wceq oismo simpld ssorduni ordsuc sylib - smorndom ssexd elong mpbiri csdm canth2g sdomdom cen wf1o wiso wwe epweon - simpl wess mpisyl epse oiiso2 sylancl isof1o simprd f1oeq3d mpbid f1oen2g + smocdmdom ssexd elong mpbiri csdm canth2g sdomdom cen wf1o wiso wwe simpl + wse epweon wess mpisyl oiiso2 sylancl isof1o simprd f1oeq3d mpbid f1oen2g endom domwdom wdomtr sylancom wdompwdom domtr syl2anc elharval sylanbrc - wse ) AEFZABGHZIZCUAZEJZXABKZLHZXAXCUBUCJWTXBXAMZANCDUDWTXAOJZXBXEUEWTXAA - UFZUGZOWTAOJZXGOJXHOJWSXIWRABGUHUIUJZAOUKXGOULPWTXAXHCQZCUMZXHMZXAXHFWTXA - ACQAXHFZXKANCDUNWRXNWSAUORXAAXHCUPUQWTXLCURZAUSZWRXLXPIWSACDUTRZVAWTXGMZX - MWRXRWSAVBRXGVCVDXAXHCVESVFZXAOVGTVHZWTXAXAKZLHZYAXCLHZXDWTXFXAYAVIHYBXSX - AOVJXAYAVKPWTXABGHZYCWRWSXAAGHZYDWTXAAVLHZXAALHYEWTXBXIXAACVMZYFXTXJWTXAX - OCVMZYGWTXAXONNCVNZYHWTANVOZANWQYIWTWRENVOYJWRWSVQVPAENVRVSAVTANCDWAWBXAX - ONNCWCTWTXOAXACWTXLXPXQWDWEWFXAACEOWGSXAAWHXAAWIPXAABWJWKXABWLTXAYAXCWMWN - XCXAWOWP $. + epse ) AEFZABGHZIZCUAZEJZXABKZLHZXAXCUBUCJWTXBXAMZANCDUDWTXAOJZXBXEUEWTXA + AUFZUGZOWTAOJZXGOJXHOJWSXIWRABGUHUIUJZAOUKXGOULPWTXAXHCQZCUMZXHMZXAXHFWTX + AACQAXHFZXKANCDUNWRXNWSAUORXAAXHCUPUQWTXLCURZAUSZWRXLXPIWSACDUTRZVAWTXGMZ + XMWRXRWSAVBRXGVCVDXAXHCVESVFZXAOVGTVHZWTXAXAKZLHZYAXCLHZXDWTXFXAYAVIHYBXS + XAOVJXAYAVKPWTXABGHZYCWRWSXAAGHZYDWTXAAVLHZXAALHYEWTXBXIXAACVMZYFXTXJWTXA + XOCVMZYGWTXAXONNCVNZYHWTANVOZANVQYIWTWRENVOYJWRWSVPVRAENVSVTAWQANCDWAWBXA + XONNCWCTWTXOAXACWTXLXPXQWDWEWFXAACEOWGSXAAWHXAAWIPXAABWJWKXABWLTXAYAXCWMW + NXCXAWOWP $. $} ${ @@ -112050,21 +112987,21 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes) c1st coi xpexg 3adant3 nfv nfra1 nfan rsp onelss imp adantrl wfo crn wsmo wi wf oismo syl ad2antrl simprd oif jctil dffo2 dffo3 simprbi 3syl 3impia ssrexv sylc 3exp sylan9r reximdai 3adant1 nfcv nfcsb1v nfoi nfeq2 csbeq1a - nffv nfrex oieq2 eqtrid fveq1d eqeq2d rexbidv cbvrexw syl6ib eliun op1std - cop vex csbeq1d op2ndd fveq12d rexxp 3imtr4g wdomd hsmexlem1 syl2anc ) AF - MZCNMZBNUBMZDUCZCMZOZGAUDZUEZGABUFZNUGZYHACUHZUIUJEUCYJUBUKPMYFXTYIYAYFBN - UGZGAUDYIYEYKGAYBYKYDBNULZUMUNGABNUOUPUQYGJUAYHYJURUAQZUSPZGYMVAPZBUTZRVB - ZPZXTYAYJURMYFACFNVCVDYGJQZYHMZYSYRSZUAYJTZYGYSBMZGATZYSKQZGLQZBUTZRVBZPZ - SZKCTZLATZYTUUBYGUUDYSUUEDPZSZKCTZGATZUULYAYFUUDUUPVOXTYAYFOUUCUUOGAYAYFG - YAGVEYEGAVFVGYFGQZAMYEYAUUCUUOVOYEGAVHYAYEUUCUUOYAYEUUCUEYCCUGZUUNKYCTZUU - OYAYEUURUUCYAYDUURYBYAYDUURCYCVIVJVKVDYAYEUUCUUSYAYEOZYCBDVLZUUSJBUDZUUCU - USVOUUTYCBDVPZDVMBSZOUVAUUTUVDUVCUUTDVNZUVDYBUVEUVDOZYAYDYBYKUVFYLBDHVQVR - VSVTBRDHWAWBYCBDWCUPUVAUVCUVBKJYCBDWDWEUUSJBVHWFWGUUNKYCCWHWIWJWKWLWMUUOU - UKGLAUUOLVEUUJGKCGCWNGYSUUIGUUEUUHGUUGRGRWNGUUFBWOWPGUUEWNWSWQWTUUQUUFSZU - UNUUJKCUVGUUMUUIYSUVGUUEDUUHUVGDBRVBZUUHHUVGBUUGSUVHUUHSGUUFBWRBUUGRXAVRX - BXCXDXEXFXGGYSABXHUUAUUJUALKACYMUUFUUEXJSZYRUUIYSUVIYNUUEYQUUHUVIYPUUGSYQ - UUHSUVIGYOUUFBUUFUUEYMLXKZKXKZXIXLYPUUGRXAVRUUFUUEYMUVJUVKXMXNXDXOXPVJXQY - HYJEIXRXS $. + nfrexw oieq2 eqtrid fveq1d eqeq2d rexbidv cbvrexw syl6ib eliun cop op1std + nffv vex csbeq1d op2ndd fveq12d rexxp 3imtr4g wdomd hsmexlem1 syl2anc ) A + FMZCNMZBNUBMZDUCZCMZOZGAUDZUEZGABUFZNUGZYHACUHZUIUJEUCYJUBUKPMYFXTYIYAYFB + NUGZGAUDYIYEYKGAYBYKYDBNULZUMUNGABNUOUPUQYGJUAYHYJURUAQZUSPZGYMVAPZBUTZRV + BZPZXTYAYJURMYFACFNVCVDYGJQZYHMZYSYRSZUAYJTZYGYSBMZGATZYSKQZGLQZBUTZRVBZP + ZSZKCTZLATZYTUUBYGUUDYSUUEDPZSZKCTZGATZUULYAYFUUDUUPVOXTYAYFOUUCUUOGAYAYF + GYAGVEYEGAVFVGYFGQZAMYEYAUUCUUOVOYEGAVHYAYEUUCUUOYAYEUUCUEYCCUGZUUNKYCTZU + UOYAYEUURUUCYAYDUURYBYAYDUURCYCVIVJVKVDYAYEUUCUUSYAYEOZYCBDVLZUUSJBUDZUUC + UUSVOUUTYCBDVPZDVMBSZOUVAUUTUVDUVCUUTDVNZUVDYBUVEUVDOZYAYDYBYKUVFYLBDHVQV + RVSVTBRDHWAWBYCBDWCUPUVAUVCUVBKJYCBDWDWEUUSJBVHWFWGUUNKYCCWHWIWJWKWLWMUUO + UUKGLAUUOLVEUUJGKCGCWNGYSUUIGUUEUUHGUUGRGRWNGUUFBWOWPGUUEWNXJWQWSUUQUUFSZ + UUNUUJKCUVGUUMUUIYSUVGUUEDUUHUVGDBRVBZUUHHUVGBUUGSUVHUUHSGUUFBWRBUUGRWTVR + XAXBXCXDXEXFGYSABXGUUAUUJUALKACYMUUFUUEXHSZYRUUIYSUVIYNUUEYQUUHUVIYPUUGSY + QUUHSUVIGYOUUFBUUFUUEYMLXKZKXKZXIXLYPUUGRWTVRUUFUUEYMUVJUVKXMXNXCXOXPVJXQ + YHYJEIXRXS $. $( Lemma for ~ hsmex . Clear ` I ` hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g., @@ -112098,40 +113035,40 @@ version of the pigeonhole principle (for aleph-null pigeons and 2 holes) ( wcel cfv crnk ve vf cv com cdm wral c0 cima cep csuc wceq fveq2 imaeq2d coi oieq2 syl eqtrid dmeqd eleq12d ralbidv weq cpw char wss cwdom wbr crn con0 imassrn cr1 cuni wf rankf frn ax-mp cvv ituni0 elv imaeq2i wfun ffun - sstri vex wdomimag mp2an cdom csn sneq fveq2d raleqdv elrab2 simprbi snex - ctc wi tcid vsnid sselii breq1 rspcv domwdom 3syl wdomtr sylancr eqbrtrid - eqid hsmexlem1 hsmexlem7 eleqtrrdi rgen wa nfra1 nfv nfan cxp ciun imaiun - ituniiun eqtri dmeqi ad2antll hsmexlem9 fveq1d eleq1d simpll ssrab3 sseli - ad2antrl r1elssi sselda snssi tcss tcel mp1i sstrd mpan9 sylanbrc adantll - ssralv rspcdva fvex funimaex elpw mpbir jctil ralrimiva syl21anc eqeltrid - hsmexlem3 hsmexlem8 eleqtrrd expr ralrimi expcom finds1 r19.21bi ) KUCZUD - RGUEZUUQFSZRZLDUUTLDUFTUGLUCZESZSZUHZUIUNZUEZUGFSZRZLDUFTUAUCZUVBSZUHZUIU - NZUEZUVIFSZRZLDUFZTUVIUJZUVBSZUHZUIUNZUEZUVQFSZRZLDUFZKUAUUQUGUKZUUTUVHLD - UWEUURUVFUUSUVGUWEGUVEUWEGTUUQUVBSZUHZUIUNZUVEQUWEUWGUVDUKUWHUVEUKUWEUWFU - VCTUUQUGUVBULUMUWGUVDUIUOUPUQURUUQUGFULUSUTKUAVAZUUTUVOLDUWIUURUVMUUSUVNU - WIGUVLUWIGUWHUVLQUWIUWGUVKUKUWHUVLUKUWIUWFUVJTUUQUVIUVBULUMUWGUVKUIUOUPUQ - URUUQUVIFULUSUTUUQUVQUKZUUTUWCLDUWJUURUWAUUSUWBUWJGUVTUWJGUWHUVTQUWJUWGUV - SUKUWHUVTUKUWJUWFUVRTUUQUVQUVBULUMUWGUVSUIUOUPUQURUUQUVQFULUSUTUVHLDUVADR - ZUVFHVBVCSZUVGUWKUVDVHVDUVDHVEVFUVFUWLRUVDTVGZVHTUVCVIVJVHUHVKZVHTVLZUWMV - HVDVMUWNVHTVNVOZWBUWKUVDTUVAUHZHVEUVCUVATUVCUVAUKLABUVAEVPOVQVRVSUWKUWQUV - AVEVFZUVAHVEVFZUWQHVEVFTVTZUVAVPRUWRUWOUWTVMUWNVHTWAVOZLWCZUVATVPWDWEUWKJ - UCZHWFVFZJUVAWGZWNSZUFZUVAHWFVFZUWSUWKUVAUWNRZUXGUXDJIUCZWGZWNSZUFZUXGIUV - AUWNDILVAZUXDJUXLUXFUXNUXKUXEWNUXJUVAWHWIWJPWKWLZUVAUXFRUXGUXHWOUXEUXFUVA - UXEVPRUXEUXFVDUVAWMZUXEVPWPVOLWQZWRUXDUXHJUVAUXFUXCUVAHWFWSWTVOUVAHXAXBZU - WQUVAHXCXDXEUVDHUVEUVEXFXGXDCFHNXHXIXJUVPUVIUDRZUWDUVPUXSXKUWCLDUVPUXSLUV - OLDXLUXSLXMXNUVPUXSUWKUWCUVPUXSUWKXKZXKZUWAHUVNXOVBVCSZUWBUYAUWAUBUVATUVI - UBUCZESZSZUHZXPZUIUNZUEZUYBUVTUYHUVSUYGUKUVTUYHUKUVSTUBUVAUYEXPZUHUYGUVRU - YJTUVRUYJUKLABUVAUVIEVPUBOXRVRVSUBTUVAUYEXQXSUVSUYGUIUOVOXTUYAUWSUVNVHRZU - YFVHVBRZUYFUIUNZUEZUVNRZXKZUBUVAUFUYIUYBRUWKUWSUVPUXSUXRYAUXSUYKUVPUWKCFH - UANYBYHUYAUYPUBUVAUYAUYCUVARZXKZUYOUYLUYRUVOUYOLDUYCLUBVAZUVMUYNUVNUYSUVL - UYMUYSUVKUYFUKUVLUYMUKUYSUVJUYETUYSUVIUVBUYDUVAUYCEULYCUMUVKUYFUIUOUPURYD - UVPUXTUYQYEUXTUYQUYCDRZUVPUWKUYQUYTUXSUWKUYQXKUYCUWNRUXDJUYCWGZWNSZUFZUYT - UWKUVAUWNUYCUWKUXIUVAUWNVDDUWNUVAUXMIUWNDPYFYGUVAYIUPYJUWKUXGUYQVUCUXOUYQ - VUBUXFVDUXGVUCWOUYQVUBUVAWNSZUXFUYQVUAUVAVDVUBVUDVDUYCUVAYKUVAVUAUXBYLUPU - VAUXERVUDUXFVDUYQUXQUXEUVAUXPYMYNYOUXDJVUBUXFYSUPYPUXMVUCIUYCUWNDIUBVAZUX - DJUXLVUBVUEUXKVUAWNUXJUYCWHWIWJPWKYQYRYRYTUYLUYFVHVDUYFUWMVHTUYEVIUWPWBUY - FVHUWTUYFVPRUXATUYEUVIUYDUUAUUBVOUUCUUDUUEUUFUVAUYFUVNHUYMUYHUBUYMXFUYHXF - UUIUUGUUHUXSUWBUYBUKUVPUWKCFHUANUUJYHUUKUULUUMUUNUUOUUP $. + sstri vex wdomimag mp2an cdom csn ctc fveq2d raleqdv elrab2 simprbi vsnex + sneq wi tcid vsnid sselii breq1 domwdom 3syl wdomtr sylancr eqbrtrid eqid + rspcv hsmexlem1 hsmexlem7 eleqtrrdi rgen nfra1 nfv nfan cxp ciun ituniiun + imaiun eqtri dmeqi ad2antll hsmexlem9 ad2antrl fveq1d eleq1d simpll sseli + wa ssrab3 r1elssi sselda snssi tcss tcel mp1i sstrd ssralv mpan9 sylanbrc + adantll rspcdva fvex funimaex mpbir ralrimiva hsmexlem3 syl21anc eqeltrid + elpw jctil hsmexlem8 eleqtrrd expr ralrimi expcom finds1 r19.21bi ) KUCZU + DRGUEZUUQFSZRZLDUUTLDUFTUGLUCZESZSZUHZUIUNZUEZUGFSZRZLDUFTUAUCZUVBSZUHZUI + UNZUEZUVIFSZRZLDUFZTUVIUJZUVBSZUHZUIUNZUEZUVQFSZRZLDUFZKUAUUQUGUKZUUTUVHL + DUWEUURUVFUUSUVGUWEGUVEUWEGTUUQUVBSZUHZUIUNZUVEQUWEUWGUVDUKUWHUVEUKUWEUWF + UVCTUUQUGUVBULUMUWGUVDUIUOUPUQURUUQUGFULUSUTKUAVAZUUTUVOLDUWIUURUVMUUSUVN + UWIGUVLUWIGUWHUVLQUWIUWGUVKUKUWHUVLUKUWIUWFUVJTUUQUVIUVBULUMUWGUVKUIUOUPU + QURUUQUVIFULUSUTUUQUVQUKZUUTUWCLDUWJUURUWAUUSUWBUWJGUVTUWJGUWHUVTQUWJUWGU + VSUKUWHUVTUKUWJUWFUVRTUUQUVQUVBULUMUWGUVSUIUOUPUQURUUQUVQFULUSUTUVHLDUVAD + RZUVFHVBVCSZUVGUWKUVDVHVDUVDHVEVFUVFUWLRUVDTVGZVHTUVCVIVJVHUHVKZVHTVLZUWM + VHVDVMUWNVHTVNVOZWBUWKUVDTUVAUHZHVEUVCUVATUVCUVAUKLABUVAEVPOVQVRVSUWKUWQU + VAVEVFZUVAHVEVFZUWQHVEVFTVTZUVAVPRUWRUWOUWTVMUWNVHTWAVOZLWCZUVATVPWDWEUWK + JUCZHWFVFZJUVAWGZWHSZUFZUVAHWFVFZUWSUWKUVAUWNRZUXGUXDJIUCZWGZWHSZUFZUXGIU + VAUWNDILVAZUXDJUXLUXFUXNUXKUXEWHUXJUVAWNWIWJPWKWLZUVAUXFRUXGUXHWOUXEUXFUV + AUXEVPRUXEUXFVDLWMZUXEVPWPVOLWQZWRUXDUXHJUVAUXFUXCUVAHWFWSXFVOUVAHWTXAZUW + QUVAHXBXCXDUVDHUVEUVEXEXGXCCFHNXHXIXJUVPUVIUDRZUWDUVPUXSYGUWCLDUVPUXSLUVO + LDXKUXSLXLXMUVPUXSUWKUWCUVPUXSUWKYGZYGZUWAHUVNXNVBVCSZUWBUYAUWAUBUVATUVIU + BUCZESZSZUHZXOZUIUNZUEZUYBUVTUYHUVSUYGUKUVTUYHUKUVSTUBUVAUYEXOZUHUYGUVRUY + JTUVRUYJUKLABUVAUVIEVPUBOXPVRVSUBTUVAUYEXQXRUVSUYGUIUOVOXSUYAUWSUVNVHRZUY + FVHVBRZUYFUIUNZUEZUVNRZYGZUBUVAUFUYIUYBRUWKUWSUVPUXSUXRXTUXSUYKUVPUWKCFHU + ANYAYBUYAUYPUBUVAUYAUYCUVARZYGZUYOUYLUYRUVOUYOLDUYCLUBVAZUVMUYNUVNUYSUVLU + YMUYSUVKUYFUKUVLUYMUKUYSUVJUYETUYSUVIUVBUYDUVAUYCEULYCUMUVKUYFUIUOUPURYDU + VPUXTUYQYEUXTUYQUYCDRZUVPUWKUYQUYTUXSUWKUYQYGUYCUWNRUXDJUYCWGZWHSZUFZUYTU + WKUVAUWNUYCUWKUXIUVAUWNVDDUWNUVAUXMIUWNDPYHYFUVAYIUPYJUWKUXGUYQVUCUXOUYQV + UBUXFVDUXGVUCWOUYQVUBUVAWHSZUXFUYQVUAUVAVDVUBVUDVDUYCUVAYKUVAVUAUXBYLUPUV + AUXERVUDUXFVDUYQUXQUXEUVAUXPYMYNYOUXDJVUBUXFYPUPYQUXMVUCIUYCUWNDIUBVAZUXD + JUXLVUBVUEUXKVUAWHUXJUYCWNWIWJPWKYRYSYSYTUYLUYFVHVDUYFUWMVHTUYEVIUWPWBUYF + VHUWTUYFVPRUXATUYEUVIUYDUUAUUBVOUUHUUCUUIUUDUVAUYFUVNHUYMUYHUBUYMXEUYHXEU + UEUUFUUGUXSUWBUYBUKUVPUWKCFHUANUUJYBUUKUULUUMUUNUUOUUP $. $( Lemma for ~ hsmex . Combining the above constraints, along with ~ itunitc and ~ tcrank , gives an effective constraint on the rank of @@ -112266,38 +113203,38 @@ enough that it can be proven using DC (see ~ axcc ). It is, however, axcc2lem $p |- E. g ( g Fn _om /\ A. n e. _om ( ( F ` n ) =/= (/) -> ( g ` n ) e. ( F ` n ) ) ) $= ( vz c0 wne cfv wcel wi com wa cvv wceq vk va cv crn wral wfn c2nd fnmpti - wex fvex w3a csn cxp snex xpex fvmpt2 mpan2 vex snnz iftrue neeq1d mpbiri - cif 0ex wn iffalse eqnetrd pm2.61i p0ex ifex xpnz biimpi sylancr fnfvelrn - neqne mpan neeq1 id eleq12d imbi12d rspccv syl5 mpdi impcom eleq2d adantr - fveq2 wb mpbid xp2nd syl 3adant3 3ad2ant1 eqcomd ifnefalse 3ad2ant3 eqtrd - 3eltr3d 3expia expcom ralrimiv omex fnex mp2an fneq1 fveq1 eleq1d ralbidv - imbi2d anbi12d spcev wf1 cen wbr wf a1i fmptd sneq xpeq12d fvmpt3i adantl - ax-mp eqeq2d eqeq1d xp11 sneqr syl6bi sylbid rgen2 dff13 mpbir2an f1f1orn - wf1o f1oen ensym 3syl cmpt rneqi cdm eqeltri wfun dmmptg mprg funrnex mp2 - funmpt breq1 raleq exbidv ax-cc vtocl mp2b exlimiiv ) KUCZLMZUUNBUCZNZUUN - OZPZKAUDZUEZCUCZQUFZDUCZENZLMZUVDUVBNZUVEOZPZDQUEZRZCUIZBUVAFQUFZUVFUVDFN - ZUVEOZPZDQUEZUVLDQUVDANZUUPNZUGNZFUVSUGUJZJUHZUVAUVPDQUVDQOZUVAUVPUWCUVAU - VFUVOUWCUVAUVFUKZUVTUVDGNZUVNUVEUWCUVAUVTUWEOZUVFUWCUVARZUVSUVDULZUWEUMZO - ZUWFUWGUVSUVROZUWJUVAUWCUWKUVAUWCUVRLMZUWKUWCUVRUWILUWCUWISOZUVRUWITZUWHU - WEUVDUNUVDGUJUOZDQUWISAIUPUQZUWCUWHLMZUWELMZUWILMZUVDDURZUSZUWCUWRUVELTZL - ULZUVEVCZLMZUXBUXEUXBUXEUXCLMLVDUSUXBUXDUXCLUXBUXCUVEUTVAVBUXBVEUXDUVELUX - BUXCUVEVFUVELVOVGVHUWCUWEUXDLUWCUXDSOUWEUXDTZUXBUXCUVEVIUVDEUJVJDQUXDSGHU - PUQZVAVBZUWQUWRRUWSUWHUWEVKVLVMVGUWCUVRUUTOZUVAUWLUWKPZAQUFUWCUXIDQUWIAUW - OIUHQUVDAVNVPUUSUXJKUVRUUTUUNUVRTZUUOUWLUURUWKUUNUVRLVQUXKUUQUVSUUNUVRUUN - UVRUUPWGUXKVRVSVTWAWBWCWDUWCUWKUWJWHUVAUWCUVRUWIUVSUWPWEWFWIUVSUWHUWEWJWK - WLUWDUVNUVTUWCUVAUVNUVTTZUVFUWCUVTSOUXLUWADQUVTSFJUPUQWMWNUWDUWEUXDUVEUWC - UVAUXFUVFUXGWMUVFUWCUXDUVETUVAUVELUXCUVEWOWPWQWRWSWTXAUVKUVMUVQRCFUVMQSOZ - FSOUWBXBQSFXCXDUVBFTZUVCUVMUVJUVQQUVBFXEUXNUVIUVPDQUXNUVHUVOUVFUXNUVGUVNU - VEUVDUVBFXFXGXIXHXJXKVMQSAXLZUUTQXMXNZUVABUIZUXOQSAXOZUVRUAUCZANZTZUVDUXS - TZPZUAQUEDQUEUXMUXRXBUXMDQUWISAUWMUXMUWCRUWOXPIXQYBUYCDUAQQUWCUXSQOZRZUYA - UVRUXSULZUXSGNZUMZTZUYBUYEUXTUYHUVRUYDUXTUYHTUWCDUXSUWIUYHQAUYBUWHUYFUWEU - YGUVDUXSXRUVDUXSGWGXSIUWOXTYAYCUYEUYIUWIUYHTZUYBUYEUVRUWIUYHUWCUWNUYDUWPW - FYDUWCUYJUYBPUYDUWCUYJUWHUYFTZUWEUYGTZRZUYBUWCUWQUWRUYJUYMWHUXAUXHUWHUWEU - YFUYGYEVMUYKUYBUYLUVDUXSUWTYFWFYGWFYHYHYIDUAQSAYJYKUXOQUUTAYMQUUTXMXNUXPQ - SAYLQUUTAXBYNQUUTYOYPUBUCZQXMXNZUUSKUYNUEZBUIZPUXPUXQPUBUUTUUTDQUWIYQZUDZ - SAUYRIYRUYRYSZSOUYRUUAUYSSOUYTQSUWMUYTQTDQDQUWISUUBUWMUWCUWOXPUUCXBYTDQUW - IUUFSUYRUUDUUEYTUYNUUTTZUYOUXPUYQUXQUYNUUTQXMUUGVUAUYPUVABUUSKUYNUUTUUHUU - IVTUBKBUUJUUKUULUUM $. + wex fvex w3a csn cxp vsnex xpex fvmpt2 mpan2 vex cif iftrue neeq1d mpbiri + snnz 0ex wn iffalse neqne eqnetrd pm2.61i p0ex ifex xpnz sylancr fnfvelrn + biimpi mpan neeq1 fveq2 id eleq12d imbi12d rspccv syl5 mpdi impcom eleq2d + adantr mpbid xp2nd syl 3adant3 3ad2ant1 eqcomd ifnefalse 3ad2ant3 3eltr3d + wb eqtrd 3expia expcom ralrimiv omex fnex mp2an fneq1 fveq1 eleq1d imbi2d + ralbidv anbi12d spcev wf1 cen wbr wf a1i fmptd ax-mp sneq xpeq12d fvmpt3i + adantl eqeq2d eqeq1d xp11 sneqr syl6bi sylbid rgen2 mpbir2an wf1o f1f1orn + dff13 f1oen ensym 3syl cmpt rneqi cdm eqeltri wfun dmmptg mprg funmpt mp2 + funrnex breq1 raleq exbidv ax-cc vtocl mp2b exlimiiv ) KUCZLMZUUNBUCZNZUU + NOZPZKAUDZUEZCUCZQUFZDUCZENZLMZUVDUVBNZUVEOZPZDQUEZRZCUIZBUVAFQUFZUVFUVDF + NZUVEOZPZDQUEZUVLDQUVDANZUUPNZUGNZFUVSUGUJZJUHZUVAUVPDQUVDQOZUVAUVPUWCUVA + UVFUVOUWCUVAUVFUKZUVTUVDGNZUVNUVEUWCUVAUVTUWEOZUVFUWCUVARZUVSUVDULZUWEUMZ + OZUWFUWGUVSUVROZUWJUVAUWCUWKUVAUWCUVRLMZUWKUWCUVRUWILUWCUWISOZUVRUWITZUWH + UWEDUNUVDGUJUOZDQUWISAIUPUQZUWCUWHLMZUWELMZUWILMZUVDDURZVCZUWCUWRUVELTZLU + LZUVEUSZLMZUXBUXEUXBUXEUXCLMLVDVCUXBUXDUXCLUXBUXCUVEUTVAVBUXBVEUXDUVELUXB + UXCUVEVFUVELVGVHVIUWCUWEUXDLUWCUXDSOUWEUXDTZUXBUXCUVEVJUVDEUJVKDQUXDSGHUP + UQZVAVBZUWQUWRRUWSUWHUWEVLVOVMVHUWCUVRUUTOZUVAUWLUWKPZAQUFUWCUXIDQUWIAUWO + IUHQUVDAVNVPUUSUXJKUVRUUTUUNUVRTZUUOUWLUURUWKUUNUVRLVQUXKUUQUVSUUNUVRUUNU + VRUUPVRUXKVSVTWAWBWCWDWEUWCUWKUWJWQUVAUWCUVRUWIUVSUWPWFWGWHUVSUWHUWEWIWJW + KUWDUVNUVTUWCUVAUVNUVTTZUVFUWCUVTSOUXLUWADQUVTSFJUPUQWLWMUWDUWEUXDUVEUWCU + VAUXFUVFUXGWLUVFUWCUXDUVETUVAUVELUXCUVEWNWOWRWPWSWTXAUVKUVMUVQRCFUVMQSOZF + SOUWBXBQSFXCXDUVBFTZUVCUVMUVJUVQQUVBFXEUXNUVIUVPDQUXNUVHUVOUVFUXNUVGUVNUV + EUVDUVBFXFXGXHXIXJXKVMQSAXLZUUTQXMXNZUVABUIZUXOQSAXOZUVRUAUCZANZTZUVDUXST + ZPZUAQUEDQUEUXMUXRXBUXMDQUWISAUWMUXMUWCRUWOXPIXQXRUYCDUAQQUWCUXSQOZRZUYAU + VRUXSULZUXSGNZUMZTZUYBUYEUXTUYHUVRUYDUXTUYHTUWCDUXSUWIUYHQAUYBUWHUYFUWEUY + GUVDUXSXSUVDUXSGVRXTIUWOYAYBYCUYEUYIUWIUYHTZUYBUYEUVRUWIUYHUWCUWNUYDUWPWG + YDUWCUYJUYBPUYDUWCUYJUWHUYFTZUWEUYGTZRZUYBUWCUWQUWRUYJUYMWQUXAUXHUWHUWEUY + FUYGYEVMUYKUYBUYLUVDUXSUWTYFWGYGWGYHYHYIDUAQSAYMYJUXOQUUTAYKQUUTXMXNUXPQS + AYLQUUTAXBYNQUUTYOYPUBUCZQXMXNZUUSKUYNUEZBUIZPUXPUXQPUBUUTUUTDQUWIYQZUDZS + AUYRIYRUYRYSZSOUYRUUAUYSSOUYTQSUWMUYTQTDQDQUWISUUBUWMUWCUWOXPUUCXBYTDQUWI + UUDSUYRUUFUUEYTUYNUUTTZUYOUXPUYQUXQUYNUUTQXMUUGVUAUYPUVABUUSKUYNUUTUUHUUI + WAUBKBUUJUUKUULUUM $. $} ${ @@ -114463,16 +115400,16 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario iunfo $p |- ( 2nd |` T ) : T -onto-> U_ x e. A B $= ( vy vz ciun c2nd wfo wfn wceq cvv wss mp2an cv cfv wrex wcel eliun fo2nd cres crn wf fof ffn mp2b ssv fnssres cima df-ima eleq2i bitri xp2nd eleq1 - csn syl5ib reximdv biimtrid impcom rexlimiva nfiu1 nfcxfr nfrex wa ssiun2 - cxp nfv cop adantr simpr vsnid opelxp mpbiran sylibr sseldd eleqtrrdi vex - op2nd fveqeq2 rspcev sylancl ex rexlimi impbii wb fvelimab 3bitr4i eqtr3i - eqriv df-fo mpbir2an ) DABCHZIDUBZJWNDKZWNUCZWMLIMKZDMNZWOMMIJMMIUDWQUAMM - IUEMMIUFUGZDUHZMDIUIOIDUJZWPWMIDUKFXAWMGPZIQZFPZLZGDRZXDCSZABRZXDXASZXDWM - SXFXHXEXHGDXEXBDSZXHXJXBAPZUPZCVGZSZABRZXEXHXJXBABXMHZSXODXPXBEULAXBBXMTU - MXEXNXGABXNXCCSXEXGXBXLCUNXCXDCUOUQURUSUTVAXGXFABXEAGDADXPEABXMVBVCXEAVHV - DXKBSZXGXFXQXGVEZXKXDVIZDSXSIQXDLZXFXRXSXPDXRXMXPXSXQXMXPNXGABXMVFVJXRXGX - SXMSZXQXGVKYAXKXLSXGAVLXKXDXLCVMVNVOVPEVQXKXDAVRFVRVSXEXTGXSDXBXSXDIVTWAW - BWCWDWEWQWRXIXFWFWSWTGMDXDIWGOAXDBCTWHWJWIDWMWNWKWL $. + csn cxp syl5ib reximdv biimtrid impcom rexlimiva nfcxfr nfv nfrexw wa cop + nfiu1 ssiun2 adantr simpr vsnid opelxp mpbiran sylibr eleqtrrdi vex op2nd + sseldd fveqeq2 rspcev sylancl ex rexlimi impbii wb fvelimab 3bitr4i eqriv + eqtr3i df-fo mpbir2an ) DABCHZIDUBZJWNDKZWNUCZWMLIMKZDMNZWOMMIJMMIUDWQUAM + MIUEMMIUFUGZDUHZMDIUIOIDUJZWPWMIDUKFXAWMGPZIQZFPZLZGDRZXDCSZABRZXDXASZXDW + MSXFXHXEXHGDXEXBDSZXHXJXBAPZUPZCUQZSZABRZXEXHXJXBABXMHZSXODXPXBEULAXBBXMT + UMXEXNXGABXNXCCSXEXGXBXLCUNXCXDCUOURUSUTVAVBXGXFABXEAGDADXPEABXMVHVCXEAVD + VEXKBSZXGXFXQXGVFZXKXDVGZDSXSIQXDLZXFXRXSXPDXRXMXPXSXQXMXPNXGABXMVIVJXRXG + XSXMSZXQXGVKYAXKXLSXGAVLXKXDXLCVMVNVOVSEVPXKXDAVQFVQVRXEXTGXSDXBXSXDIVTWA + WBWCWDWEWQWRXIXFWFWSWTGMDXDIWGOAXDBCTWHWIWJDWMWNWKWL $. iundomg.2 $e |- ( ph -> U_ x e. A ( C ^m B ) e. AC_ A ) $. iundomg.3 $e |- ( ph -> A. x e. A B ~<_ C ) $. @@ -114483,12 +115420,12 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario ( vy vg cfv wf1 wral wa wcel wb cvv syl wceq vf vz cmap co ciun cv wf csb wex cxp cdom wbr wacn brdomi adantl f1f reldom brrelex2i brrelex1i elmapd wrex syl5ibr ssiun2 adantr sseld syld ancrd eximdv df-rex sylibr ralimiaa - wss mpd nfv nfiu1 nfcv nfcsb1v nff1 nfrex weq csbeq1a f1eq2 rexbidv sylib - cbvralw f1eq1 acni3 syl2anc fveq2 bitrd c0 wn wne acnrcl r19.2z rexlimivw - df-ne expcom xpexg syl6an syl5bir xpeq1 0xp 0ex eqeltri eqeltrdi pm2.61d2 - wi c1st c2nd cop csn eleq2i eliun bitri r19.29 xp1st ad2antll elsni simpl - eqeltrd fveq2d fveq1d ad2antrl xp2nd ffvelcdmd opelxpd rexlimiva biimtrid - ex fvex opth simpr eqeq2d djussxp eqsstri simprl sselid simpll csbeq1d + wss mpd nfv nfiu1 nfcv nfcsb1v nfrexw csbeq1a f1eq2 rexbidv cbvralw sylib + nff1 weq f1eq1 acni3 syl2anc fveq2 bitrd c0 wn wne df-ne acnrcl wi r19.2z + rexlimivw expcom xpexg syl6an syl5bir xpeq1 0xp eqeltri eqeltrdi pm2.61d2 + 0ex c1st c2nd cop csn eleq2i eliun bitri r19.29 xp1st elsni simpl eqeltrd + ad2antll fveq2d fveq1d ad2antrl xp2nd ffvelcdmd opelxpd rexlimiva ex fvex + biimtrid opth simpr eqeq2d djussxp eqsstri simprl sselid simpll csbeq1d rspc sylc nfel2 eqcomd eleqtrd rexlimi imp adantrr ralrimiv eleq12d sylan rspccva adantrl eleqtrrd f1fveq bitr3d simprr xpopth bitrid dom2d syl5com syl12anc pm5.32da adantld exlimdv ) ACBCEDUCUDZUEZUAUFZUGZBJUFZDUHZEUVJUV @@ -114497,31 +115434,31 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario UIZUWNUWGUWOUWJDEKUNUOUWKUWDUWMKUWKUWDUWLUWKUWDUVTUVFPZUWLUWDUWPUWKDEUVTU GZDEUVTUPUWGUWPUWQQUWJUWGEDUVTRRDEUKUQURZDEUKUQUSUTUOVBUWKUVFUVGUVTUWJUVF UVGVLUWGBCUVFVCVDVEVFVGVHVMUWDKUVGVIVJVKSUWEUWBBJCUWEJVNUWABKUVGBCUVFVOBU - VKEUVTBUVTVPBUVJDVQZBEVPZVRVSBJVTZUWDUWAKUVGUXADUVKTZUWDUWAQBUVJDWAZDUVKE - UVTWBSWCWEWDUWAUVMJKCUAUVGUVKEUVTUVLWFWGWHAUVOUVRUAAUVNUVRUVIUVNDEUWIUVHL - ZMZBCNZAUVRUXEUVMBJCUXEJVNBUVKEUVLBUVLVPUWSUWTVRUXAUXEDEUVLMZUVMUXAUXDUVL - TUXEUXGQUWIUVJUVHWIDEUXDUVLWFSUXAUXBUXGUVMQUXCDUVKEUVLWBSWJWEAUVQRPZUXFUV - RACWKTZUXHUXIWLCWKWMZAUXHCWKWQACRPZUXJERPZUXHAUVSUXKHCUVGWNSAUWHUXJUXLXHI - UXJUWHUXLUXJUWHOUWGBCVAUXLUWGBCWOUWGUXLBCUWRWPSWRSCERRWSWTXAUXIUVQWKEUJZR - CWKEXBUXMWKREXCXDXEXFXGUXFJUBFUVQUVJXILZUVJXJLZUXNUVHLZLZXKZUBUFZXILZUXSX + VKEUVTBUVTVPBUVJDVQZBEVPZWDVRBJWEZUWDUWAKUVGUXADUVKTZUWDUWAQBUVJDVSZDUVKE + UVTVTSWAWBWCUWAUVMJKCUAUVGUVKEUVTUVLWFWGWHAUVOUVRUAAUVNUVRUVIUVNDEUWIUVHL + ZMZBCNZAUVRUXEUVMBJCUXEJVNBUVKEUVLBUVLVPUWSUWTWDUXAUXEDEUVLMZUVMUXAUXDUVL + TUXEUXGQUWIUVJUVHWIDEUXDUVLWFSUXAUXBUXGUVMQUXCDUVKEUVLVTSWJWBAUVQRPZUXFUV + RACWKTZUXHUXIWLCWKWMZAUXHCWKWNACRPZUXJERPZUXHAUVSUXKHCUVGWOSAUWHUXJUXLWPI + UXJUWHUXLUXJUWHOUWGBCVAUXLUWGBCWQUWGUXLBCUWRWRSWSSCERRWTXAXBUXIUVQWKEUJZR + CWKEXCUXMWKREXDXHXEXFXGUXFJUBFUVQUVJXILZUVJXJLZUXNUVHLZLZXKZUBUFZXILZUXSX JLZUXTUVHLZLZXKZRUVJFPZUVJUWIXLZDUJZPZBCVAZUXFUXRUVQPZUYEUVJBCUYGUEZPUYIF UYKUVJGXMBUVJCUYGXNXOZUXFUYIUYJUXFUYIOZUXEUYHOZBCVAZUYJUXEUYHBCXPZUYNUYJB - CUWJUYNOZUXNUXQCEUYQUXNUWICUYQUXNUYFPZUXNUWITUYHUYRUWJUXEUVJUYFDXQXRUXNUW - IXSSZUWJUYNXTYAUYQUXQUXOUXDLEUYQUXOUXPUXDUYQUXNUWIUVHUYSYBYCUYQDEUXOUXDUX - EDEUXDUGUWJUYHDEUXDUPYDUYHUXODPUWJUXEUVJUYFDYEXRZYFYAYGYHSYJYIUXFUYEUXSFP - ZOZUXRUYDTZJUBVTZQVUCUXNUXTTZUXQUYCTZOZUXFVUBOZVUDUXNUXQUXTUYCUVJXIYKUXOU - XPYKYLVUHVUGVUEUXOUYATZOZVUDVUHVUEVUFVUIVUHVUEOZUXQUYAUXPLZTZVUFVUIVUKVUL + CUWJUYNOZUXNUXQCEUYQUXNUWICUYQUXNUYFPZUXNUWITUYHUYRUWJUXEUVJUYFDXQYAUXNUW + IXRSZUWJUYNXSXTUYQUXQUXOUXDLEUYQUXOUXPUXDUYQUXNUWIUVHUYSYBYCUYQDEUXOUXDUX + EDEUXDUGUWJUYHDEUXDUPYDUYHUXODPUWJUXEUVJUYFDYEYAZYFXTYGYHSYIYKUXFUYEUXSFP + ZOZUXRUYDTZJUBWEZQVUCUXNUXTTZUXQUYCTZOZUXFVUBOZVUDUXNUXQUXTUYCUVJXIYJUXOU + XPYJYLVUHVUGVUEUXOUYATZOZVUDVUHVUEVUFVUIVUHVUEOZUXQUYAUXPLZTZVUFVUIVUKVUL UYCUXQVUKUYAUXPUYBVUKUXNUXTUVHVUHVUEYMZYBYCYNVUKBUXNDUHZEUXPMZUXOVUOPZUYA VUOPVUMVUIQVUKUXNCPZUXFVUPVUKUVJCRUJZPZVURVUHVUTVUEVUHFVUSUVJFUYKVUSGBCDY OYPZUXFUYEVUAYQYRZVDUVJCRXQSUXFVUBVUEYSUXEVUPBUXNCBVUOEUXPBUXPVPBUXNDVQZU - WTVRUWIUXNTZUXEDEUXPMZVUPVVDUXDUXPTUXEVVEQUWIUXNUVHWIDEUXDUXPWFSVVDDVUOTZ - VVEVUPQBUXNDWAZDVUOEUXPWBSWJUUAUUBVUHVUQVUEUXFUYEVUQVUAUXFUYEVUQUYEUYIUXF + WTWDUWIUXNTZUXEDEUXPMZVUPVVDUXDUXPTUXEVVEQUWIUXNUVHWIDEUXDUXPWFSVVDDVUOTZ + VVEVUPQBUXNDVSZDVUOEUXPVTSWJUUAUUBVUHVUQVUEUXFUYEVUQVUAUXFUYEVUQUYEUYIUXF VUQUYLUXFUYIVUQUYMUYOVUQUYPUYNVUQBCBUXOVUOVVCUUCUWJUYNVUQUYQUXODVUOUYTUYQ - VVDVVFUYQUXNUWIUYSUUDVVGSUUEYJUUFSYJYIZUUGUUHVDVUKUYABUXTDUHZVUOVUHUYAVVI + VVDVVFUYQUXNUWIUYSUUDVVGSUUEYIUUFSYIYKZUUGUUHVDVUKUYABUXTDUHZVUOVUHUYAVVI PZVUEUXFVUAVVJUYEUXFVUQJFNVUAVVJUXFVUQJFVVHUUIVUQVVJJUXSFVUDUXOUYAVUOVVIU VJUXSXJWIVUDBUXNUXTDUVJUXSXIWIYTUUJUULUUKUUMVDVUKBUXNUXTDVUNYTUUNVUOEUXOU YAUXPUUOUVBUUPUVCVUHVUTUXSVUSPVUJVUDQVVBVUHFVUSUXSVVAUXFUYEVUAUUQYRUVJUXS - CRCRUURWHWJUUSYJUUTUVAXAUVDUVEVM $. + CRCRUURWHWJUUSYIUUTUVAXBUVDUVEVM $. iundomg.4 $e |- ( ph -> ( A X. C ) e. AC_ U_ x e. A B ) $. $( An upper bound for the cardinality of an indexed union, with explicit @@ -116717,27 +117654,27 @@ set maps to an element of the set (so that it cannot be extended without stronger form of ~ canth2 . Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.) $) canthwe $p |- ( A e. V -> A ~< O ) $= - ( vv vf wcel wbr cvv cxp wss cv wwe wa csn c0 wceq eqid vu vy vw va vs vz - cdom cen csdm cpw w3a copab simp1 velpw sylibr simp2 xpss12 syl2anc sstrd - wn jca ssopab2i df-xp 3sstr4i pwexg sqxpexg pwexd xpexd ssexg sylancr cop - simpr snssd 0ss a1i wrel rel0 br0 wesn mpbiri mp1i snex 0ex simpl sqxpeqd + ( vu vv vf wcel wbr cvv cxp wss cv wwe wa csn c0 wceq vy vw va vs vz cdom + cen wn csdm cpw w3a copab simp1 velpw sylibr simp2 syl2anc sstrd ssopab2i + jca df-xp 3sstr4i pwexg sqxpexg pwexd xpexd ssexg sylancr cop simpr snssd + xpss12 0ss a1i wrel rel0 br0 wesn mpbiri mp1i vsnex simpl sqxpeqd sseq12d sseq1d weeq2 weeq1 sylan9bb 3anbi123d opelopaba syl3anbrc eleqtrrdi ex wb - sseq12d opth2 mpbiran2 sneqbg elv bitri 2a1i mpd wf1o wex wf1 cin co ccnv - dom2d cima wsbc wral cdm cuni fpwwe2cbv canthwelem f1of1 nsyl nexdv ensym - cfv bren sylib brsdom sylanbrc ) BDIZBCUGJZBCUHJZUTBCUIJYGCKIZYHYGCBUJZBB - LZUJZLZMYNKIYJANZBMZENZYOYOLZMZYOYQOZUKZAEULZYOYKIZYQYMIZPZAEULCYNUUAUUEA - 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cdif mp2an ax-mp eqbrtri xpen sylancl snex pwex cun uncom wss simpr snssd - undif eqtrid difexd canth2g domunsn 3syl eqbrtrrd xpdom1g sylancr endomtr - pwdjuen domentr cin a1i disjdifr endjudisj syl3anc breqtrd exlimddv domtr - ensymd ) CADEZAFGZAFHZIEZWTAJZIEZWSXBIEWRCFDEXAUAAFUBUCWRBUDZAKZXCBWRALMZ - UEXEBUFXFWRXFWRCLDECUGALCDUHUIUJBAULSWRXEUKZWTAXDNZVDZXHGZJZIEZXKXBOEZXCX - GWTXIJZXHJZHZIEZXPXKOEXLXGWTAXOHZOEZXRXPIEZXQXGAAOEZFXOOEXSXGAPKZYAWRYBXE - CADUMUNUOZAPUPQFLNZJZXOOFLYDUQYEURUSUTYDXHOEZYEXOOELPKXDPKYFVABVBLXDPPVCV - EYDXHRVFVGAAFXOVHVIXGXOPKAXNIEXTXHXDVJZVKXGXIXHVLZAXNIXGYHXHXIVLZAXIXHVMX - GXHAVNYIAMXGXDAWRXEVOVPXHAVQSVRZXGXIPKZXIXNDEYHXNIEXGAXHPYCVSZXIPVTXIXNXD - WAWBWCAXNXOPWDWEWTXRXPWFTXGXKXPXGYKXHPKZXKXPOEYLYGXIXHPPWGVIWQWTXPXKWHTXG - XJAOEXMXGXJYHAOXGYKYMXIXHWILMZXJYHOEYLYMXGYGWJYNXGXHAWKWJXIXHPPWLWMYJWNXJ - ARQWTXKXBWHTWOWSWTXBWPT $. + cdif mp2an ax-mp eqbrtri xpen sylancl vsnex pwex uncom simpr snssd eqtrid + cun wss undif difexd canth2g domunsn 3syl xpdom1g sylancr endomtr pwdjuen + eqbrtrrd ensymd domentr cin a1i disjdifr endjudisj syl3anc exlimddv domtr + breqtrd ) CADEZAFGZAFHZIEZWTAJZIEZWSXBIEWRCFDEXAUAAFUBUCWRBUDZAKZXCBWRALM + ZUEXEBUFXFWRXFWRCLDECUGALCDUHUIUJBAULSWRXEUKZWTAXDNZVDZXHGZJZIEZXKXBOEZXC + XGWTXIJZXHJZHZIEZXPXKOEXLXGWTAXOHZOEZXRXPIEZXQXGAAOEZFXOOEXSXGAPKZYAWRYBX + ECADUMUNUOZAPUPQFLNZJZXOOFLYDUQYEURUSUTYDXHOEZYEXOOELPKXDPKYFVABVBLXDPPVC + VEYDXHRVFVGAAFXOVHVIXGXOPKAXNIEXTXHBVJZVKXGXIXHVPZAXNIXGYHXHXIVPZAXIXHVLX + GXHAVQYIAMXGXDAWRXEVMVNXHAVRSVOZXGXIPKZXIXNDEYHXNIEXGAXHPYCVSZXIPVTXIXNXD + WAWBWGAXNXOPWCWDWTXRXPWETXGXKXPXGYKXHPKZXKXPOEYLYGXIXHPPWFVIWHWTXPXKWITXG + XJAOEXMXGXJYHAOXGYKYMXIXHWJLMZXJYHOEYLYMXGYGWKYNXGXHAWLWKXIXHPPWMWNYJWQXJ + ARQWTXKXBWITWOWSWTXBWPT $. $} ${ @@ -117577,7 +118514,7 @@ Tarski classes ( ~ df-tsk ), and Grothendieck universes ( ~ df-gru ). We $( ` _om ` is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow ` _om ` as an inaccessible cardinal, - but this choice allows us to reuse our results for inaccessibles for + but this choice allows to reuse our results for inaccessibles for ` _om ` .) (Contributed by Mario Carneiro, 29-May-2014.) $) omina $p |- _om e. Inacc $= ( vx com cina wcel c0 wne ccf cfv wceq cv cpw csdm wbr wral peano1 cfom cfn @@ -118116,9 +119053,9 @@ prove that every set is contained in a weak universe in ZF (see Tarski-Grothendieck axiom, contrary to ~ grothtsk . (Contributed by Mario Carneiro, 2-Jan-2017.) $) uniwun $p |- U. WUni = _V $= - ( vx vu cwun cuni cvv wceq cv wcel eqv csn wss wrex snex wunex eluni2 vex - ax-mp snss rexbii bitri mpbir mpgbir ) CDZEFAGZUCHZAAUCIUEUDJZBGZKZBCLZUF - EHUIUDMBUFENQUEUDUGHZBCLUIBUDCOUJUHBCUDUGAPRSTUAUB $. + ( vx vu cwun cuni cvv wceq wcel eqv csn wss wrex vsnex wunex ax-mp eluni2 + cv vex snss rexbii bitri mpbir mpgbir ) CDZEFAPZUCGZAAUCHUEUDIZBPZJZBCKZU + FEGUIALBUFEMNUEUDUGGZBCKUIBUDCOUJUHBCUDUGAQRSTUAUB $. wunex3.u $e |- U = ( R1 ` ( ( rank ` A ) +o _om ) ) $. $( Construct a weak universe from a given set. This version of ~ wunex has @@ -118224,12 +119161,12 @@ prove that every set is contained in a weak universe in ZF (see $( The class of all Tarski classes. Tarski classes is a phrase coined by Grzegorz Bancerek in his article _Tarski's Classes and Ranks_, Journal of Formalized Mathematics, Vol 1, No 3, May-August 1990. A Tarski class - is a set whose existence is ensured by Tarski's axiom A (see ~ ax-groth + is a set whose existence is ensured by Tarski's Axiom A (see ~ ax-groth and the equivalent axioms). Axiom A was first presented in Tarski's - article _Ueber unerreichbare Kardinalzahlen_. Tarski introduced the - axiom A to enable ZFC to manage inaccessible cardinals. Later - Grothendieck introduced the concept of Grothendieck universes and showed - they were equal to transitive Tarski classes. (Contributed by FL, + article _Ueber unerreichbare Kardinalzahlen_. Tarski introduced Axiom A + to allow reasoning with inaccessible cardinals in ZFC. Later, + Grothendieck introduced the concept of (Grothendieck) universes and + showed they were exactly transitive Tarski classes. (Contributed by FL, 30-Dec-2010.) $) df-tsk $a |- Tarski = { y | ( A. z e. y ( ~P z C_ y /\ E. w e. y ~P z C_ w ) /\ A. z e. ~P y ( z ~~ y \/ z e. y ) ) } $. @@ -118571,33 +119508,33 @@ prove that every set is contained in a weak universe in ZF (see 0elon fndmi rankonid necon3i cima cuni wb rankvaln necon1ai 0sdom1dom vex breq2 0sdom bitr3i vtoclbg mpbird eqbrtrd rexlimiva domnsym csn cab nlim0 adantl wss limeq r1elssi sselda ranksnb rankelb limsuc sylibd imp eqeltrd - con4i eleq1a rexlimdva abssdv ciun snex dfiun2 iunid eqtr3i snwf abrexexg - fveq2i 3syl eleq1 sseq1 r1elss rankuni2b eqtr3id fvex abrexco eqtri cfslb - unieqi eqtr2di mpd3an23 2fveq3 breq12 mpdan rexeq abbidv mpancom cmpt crn - imbi12d eqid rnmpt ccrd wfo cfon sdomdom ondomen sylancr wfn fnmpti dffn4 - fodomnum mpisyl eqbrtrrid vtoclg domtr syl6an pm2.01d 3jaoi ax-mp ) AFGZH - IZUUMBUAZUBZIZBJKZUUMUCLZUUMUDZMZUEZAUUMNGZOPZUFZUUMJLUVBAUGBUUMUHUNUUNUV - EUURUVAUUNUVDAHOPAUIUUNUVCHAOUUNUVCHNGHUUMHNUJUOUKULUMUURUVCAQPZUVEUUQUVF - BJUUOJLZUUQMUVCRAQUUQUVGUVCUUPNGRUUMUUPNUJUUOUPUQUUQRAQPZUVGUUQUUMHURZUVH - UUQUVIUUPHURUUOUSUUMUUPHUTVAUVIUVHAHURZAHUUMHAHIUUMHFGZHAHFUJHVBVCZLUVKHI - HJUVLVFJVBVDVGVEHVHUNUKVIUVIAVBJVJVKZLZUVHUVJVLUVNUUMHAVMZVNRCUAZQPZUVPHU - RZUVHUVJCAUVMUVPARQVQUVPAHUTUVQHUVPOPUVRUVPVOUVPCVPVRVSVTTWATWHWBWCUVCAWD - ZTUUTUVEUUSUUTUVDUUTUVCDUAZUUOWEZFGZIZBAKZDWFZQPZUVDUWEAQPZUVEUUTUWEUUMWI - UWEVKZUUMIZUWFUUTUWDDUUMUUTUWCUVTUUMLZBAUUTUUOALZMZUWBUUMLUWCUWJSUWLUWBUU - OFGZUBZUUMUWLUUOUVMLZUWBUWNIUUTAUVMUUOUUTUVNAUVMWIUVNUUTUVNUFUUNUUTUFUVOU - UNUUTHUDWGUUMHWJUMTWSZAWKZTWLUUOWMTUUTUWKUWNUUMLZUUTUWKUWMUUMLZUWRUUTUVNU - WKUWSSUWPUUOAWNTUUMUWMWOWPWQWRUWBUUMUVTWTTXAXBUUTUVNUWIUWPUVNUUMEUVPUWAIZ - BAKZCWFZEUAZFGZXCZUWHUVNUUMUXBVKZFGZUXEUXFAFBAUWAXCUXFABCAUWAUUOXDZXEBAXF - XGXJUVNUXBUVMLZUXGUXEIUVNUXIUXBUVMWIZUVNUXACUVMUVNUWTUVPUVMLZBAUVNUWKMUWO - UWAUVMLUWTUXKSUVNAUVMUUOUWQWLUUOXHUWAUVMUVPWTXKXAXBUVNUXBUCLUXIUXJVLBCAUW - AUVMXIUXCUVMLUXCUVMWIUXIUXJEUXBUCUXCUXBUVMXLUXCUXBUVMXMUXCEVPXNVTTWAEUXBX - OTXPUXEUVTUXDIEUXBKDWFZVKUWHEDUXBUXDUXCFXQXEUXLUWEDECBAUWAUXDUWBUXHUXCUWA - FUJXRYAXSYBTUUMUWEAFXQXTYCUUTUVNUVDUWGSZUWPUVPUVPFGZNGZOPZUWCBUVPKZDWFZUV - PQPZSUXMCAUVMUVPAIZUXPUVDUXSUWGUXTUXOUVCIUXPUVDVLUVPANFYDUVPAUXOUVCOYEYFU - XRUWEIUXTUXSUWGVLUXTUXQUWDDUWCBUVPAYGYHUXRUWEUVPAQYEYIYLUXPUXRBUVPUWBYJZY - KZUVPQBDUVPUWBUYAUYAYMZYNUXPUVPYOVCLZUVPUYBUYAYPZUYBUVPQPUXPUXOJLUVPUXOQP - UYDUXNYQUVPUXOYRUXOUVPYSYTUYAUVPUUAUYEBUVPUWBUYAUWAFXQUYCUUBUVPUYAUUCUNUV - PUYBUYAUUDUUEUUFUUGTUWFUWGMUVFUVEUVCUWEAUUHUVSTUUIUUJWHUUKUUL $. + con4i eleq1a rexlimdva abssdv ciun vsnex dfiun2 eqtr3i snwf 3syl abrexexg + iunid fveq2i eleq1 sseq1 r1elss eqtr3id fvex abrexco unieqi eqtri eqtr2di + rankuni2b cfslb mpd3an23 2fveq3 breq12 mpdan rexeq abbidv mpancom imbi12d + cmpt crn eqid rnmpt ccrd wfo cfon sdomdom ondomen sylancr fnmpti fodomnum + wfn dffn4 mpisyl eqbrtrrid vtoclg domtr syl6an pm2.01d 3jaoi ax-mp ) AFGZ + HIZUUMBUAZUBZIZBJKZUUMUCLZUUMUDZMZUEZAUUMNGZOPZUFZUUMJLUVBAUGBUUMUHUNUUNU + VEUURUVAUUNUVDAHOPAUIUUNUVCHAOUUNUVCHNGHUUMHNUJUOUKULUMUURUVCAQPZUVEUUQUV + FBJUUOJLZUUQMUVCRAQUUQUVGUVCUUPNGRUUMUUPNUJUUOUPUQUUQRAQPZUVGUUQUUMHURZUV + HUUQUVIUUPHURUUOUSUUMUUPHUTVAUVIUVHAHURZAHUUMHAHIUUMHFGZHAHFUJHVBVCZLUVKH + IHJUVLVFJVBVDVGVEHVHUNUKVIUVIAVBJVJVKZLZUVHUVJVLUVNUUMHAVMZVNRCUAZQPZUVPH + URZUVHUVJCAUVMUVPARQVQUVPAHUTUVQHUVPOPUVRUVPVOUVPCVPVRVSVTTWATWHWBWCUVCAW + DZTUUTUVEUUSUUTUVDUUTUVCDUAZUUOWEZFGZIZBAKZDWFZQPZUVDUWEAQPZUVEUUTUWEUUMW + IUWEVKZUUMIZUWFUUTUWDDUUMUUTUWCUVTUUMLZBAUUTUUOALZMZUWBUUMLUWCUWJSUWLUWBU + UOFGZUBZUUMUWLUUOUVMLZUWBUWNIUUTAUVMUUOUUTUVNAUVMWIUVNUUTUVNUFUUNUUTUFUVO + UUNUUTHUDWGUUMHWJUMTWSZAWKZTWLUUOWMTUUTUWKUWNUUMLZUUTUWKUWMUUMLZUWRUUTUVN + UWKUWSSUWPUUOAWNTUUMUWMWOWPWQWRUWBUUMUVTWTTXAXBUUTUVNUWIUWPUVNUUMEUVPUWAI + ZBAKZCWFZEUAZFGZXCZUWHUVNUUMUXBVKZFGZUXEUXFAFBAUWAXCUXFABCAUWABXDZXEBAXJX + FXKUVNUXBUVMLZUXGUXEIUVNUXIUXBUVMWIZUVNUXACUVMUVNUWTUVPUVMLZBAUVNUWKMUWOU + WAUVMLUWTUXKSUVNAUVMUUOUWQWLUUOXGUWAUVMUVPWTXHXAXBUVNUXBUCLUXIUXJVLBCAUWA + UVMXIUXCUVMLUXCUVMWIUXIUXJEUXBUCUXCUXBUVMXLUXCUXBUVMXMUXCEVPXNVTTWAEUXBYA + TXOUXEUVTUXDIEUXBKDWFZVKUWHEDUXBUXDUXCFXPXEUXLUWEDECBAUWAUXDUWBUXHUXCUWAF + UJXQXRXSXTTUUMUWEAFXPYBYCUUTUVNUVDUWGSZUWPUVPUVPFGZNGZOPZUWCBUVPKZDWFZUVP + QPZSUXMCAUVMUVPAIZUXPUVDUXSUWGUXTUXOUVCIUXPUVDVLUVPANFYDUVPAUXOUVCOYEYFUX + RUWEIUXTUXSUWGVLUXTUXQUWDDUWCBUVPAYGYHUXRUWEUVPAQYEYIYJUXPUXRBUVPUWBYKZYL + ZUVPQBDUVPUWBUYAUYAYMZYNUXPUVPYOVCLZUVPUYBUYAYPZUYBUVPQPUXPUXOJLUVPUXOQPU + YDUXNYQUVPUXOYRUXOUVPYSYTUYAUVPUUCUYEBUVPUWBUYAUWAFXPUYCUUAUVPUYAUUDUNUVP + UYBUYAUUBUUEUUFUUGTUWFUWGMUVFUVEUVCUWEAUUHUVSTUUIUUJWHUUKUUL $. $} ${ @@ -122149,25 +123086,25 @@ B C_ ( A +P. C ) ) $= elprnq pm2.43i dmrecnq 0nnq ndmfvrcl ltrnq prcdnq biimtrid alrimiv abeq2i exanali bitri con2bii sylib eximi exlimdv nss 3imtr4g ltrelnq brel simpld wb ex sylbi ssriv jctil dfpss3 ltsonq sotri anim1d com12 nfe1 nfab nfcxfr - cab nfv nfrex 19.8a simpll expcom ltbtwnnq df-rex impcom exlimi rgen elnp - adantll sylanblrc ) CUCGZHDUAZDIUAZJFUBZAUBZKLZYKDGZUDZFUEZYLYKKLZFDUFZJZ - ADUGDUCGYHYIYJYHDHUOZYIYHCIUAYLIGZYLCGZMZJZANYTCUHACIUIUUDYTAUUDYNFNZYTUU - DYKYLOPZKLZFNZUUEUUAUUHUUCUUAUUFIGZUUHYLUJFUUFUKQULUUDUUGYNFUUAUUCUUGYNUD - UUAUUGUUCYNUUAUUGUUCJZYKBUBZKLZUUKOPZCGZMZJZBNZYNUUAUUJUUGUUFOPZCGZMZJZUU - QUUAUUTUUCUUGUUAUUSUUBUUAUURYLCYLUPRUMUNUUPUVABUUFYLOUQUUKUUFURZUULUUGUUO - UUTUUKUUFYKKUSUVBUUNUUSUVBUUMUURCUUKUUFOUTRUMVAVBVCYLUUKKLZUUOJZBNZUUQAYK - DFVDYLYKURZUVDUUPBUVFUVCUULUUOYLYKUUKKVEVFVGEVHZVIVJVKVLVMFDVNSVOVPDVQSYH - DIVRZIDVRMZJYJYHUVIUVHYHCHUOZUVICVSYHYKCGZFNUUAYLDGZMZJZANZUVJUVIYHUVKUVO - FYHUVKUVOYHUVKJZYHUUFCGZJZANZUVOUVPUVSUVPUVPYHYKOPZOPZCGZJZUVSUVPYKIGZUWC - UVPXACYKVTUWDUWBUVKYHUWDUWAYKCYKUPRUNQUVRUWCAUVTYKOUQYLUVTURZUVQUWBYHUWEU - UFUWACYLUVTOUTRUNVBVCWAUVRUVNAUVRUUAUVMUVRUUIUUACUUFVTYLIOWBWCWDQUVRUVCUU - NUDZBUEZUVMUVRUWFBUVCUUMUUFKLUVRUUNYLUUKWECUUFUUMWFWGWHUVLUWGUVLUVEUWGMUV - EADEWIZUVCUUNBWJWKWLWMTWNQXBWOFCVNAIDWPWQVMADIUVLUVEUUAUWHUVDUUABUVCUUAUU - OUVCUUAUUKIGYLUUKIIKWRWSWTULVOXCXDXEDIXFSTYSADUVLYPYRUVLYOFYMUVLYNYMUVEUU - QUVLYNYMUVDUUPBYMUVCUULUUOYMUVCUULYKYLUUKKIXGWRXHXBXIVLUWHUVGWQXJWHUVLUVE - YRUWHUVDYRBYQBFDBDUVEAXNEUVEBAUVDBXKXLXMYQBXOXPUUOUVCYRUUOYQUULJZFNYNYQJZ - FNUVCYRUUOUWIUWJFUWIUUOUWJUWIUUOJYNYQUULUUOYNYQUUPUUQYNUUPBXQUVGSYFYQUULU - UOXRTXSVLFYLUUKXTYQFDYAWQYBYCXCTYDAFDYEYG $. + cab nfrexw 19.8a adantll simpll expcom ltbtwnnq df-rex impcom exlimi rgen + nfv elnp sylanblrc ) CUCGZHDUAZDIUAZJFUBZAUBZKLZYKDGZUDZFUEZYLYKKLZFDUFZJ + ZADUGDUCGYHYIYJYHDHUOZYIYHCIUAYLIGZYLCGZMZJZANYTCUHACIUIUUDYTAUUDYNFNZYTU + UDYKYLOPZKLZFNZUUEUUAUUHUUCUUAUUFIGZUUHYLUJFUUFUKQULUUDUUGYNFUUAUUCUUGYNU + DUUAUUGUUCYNUUAUUGUUCJZYKBUBZKLZUUKOPZCGZMZJZBNZYNUUAUUJUUGUUFOPZCGZMZJZU + UQUUAUUTUUCUUGUUAUUSUUBUUAUURYLCYLUPRUMUNUUPUVABUUFYLOUQUUKUUFURZUULUUGUU + OUUTUUKUUFYKKUSUVBUUNUUSUVBUUMUURCUUKUUFOUTRUMVAVBVCYLUUKKLZUUOJZBNZUUQAY + KDFVDYLYKURZUVDUUPBUVFUVCUULUUOYLYKUUKKVEVFVGEVHZVIVJVKVLVMFDVNSVOVPDVQSY + HDIVRZIDVRMZJYJYHUVIUVHYHCHUOZUVICVSYHYKCGZFNUUAYLDGZMZJZANZUVJUVIYHUVKUV + OFYHUVKUVOYHUVKJZYHUUFCGZJZANZUVOUVPUVSUVPUVPYHYKOPZOPZCGZJZUVSUVPYKIGZUW + CUVPXACYKVTUWDUWBUVKYHUWDUWAYKCYKUPRUNQUVRUWCAUVTYKOUQYLUVTURZUVQUWBYHUWE + UUFUWACYLUVTOUTRUNVBVCWAUVRUVNAUVRUUAUVMUVRUUIUUACUUFVTYLIOWBWCWDQUVRUVCU + UNUDZBUEZUVMUVRUWFBUVCUUMUUFKLUVRUUNYLUUKWECUUFUUMWFWGWHUVLUWGUVLUVEUWGMU + VEADEWIZUVCUUNBWJWKWLWMTWNQXBWOFCVNAIDWPWQVMADIUVLUVEUUAUWHUVDUUABUVCUUAU + UOUVCUUAUUKIGYLUUKIIKWRWSWTULVOXCXDXEDIXFSTYSADUVLYPYRUVLYOFYMUVLYNYMUVEU + UQUVLYNYMUVDUUPBYMUVCUULUUOYMUVCUULYKYLUUKKIXGWRXHXBXIVLUWHUVGWQXJWHUVLUV + EYRUWHUVDYRBYQBFDBDUVEAXNEUVEBAUVDBXKXLXMYQBYEXOUUOUVCYRUUOYQUULJZFNYNYQJ + ZFNUVCYRUUOUWIUWJFUWIUUOUWJUWIUUOJYNYQUULUUOYNYQUUPUUQYNUUPBXPUVGSXQYQUUL + UUOXRTXSVLFYLUUKXTYQFDYAWQYBYCXCTYDAFDYFYG $. $( Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) @@ -123399,9 +124336,9 @@ B C_ ( A +P. C ) ) $= ~ qsid , which shows that the coset of the converse membership relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that ` CC ` is a quotient - set, even though it is not (compare ~ df-c ), and allows us to reuse some - of the equivalence class lemmas we developed for the transition from - positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) + set, even though it is not (compare ~ df-c ), and allows to reuse some of + the equivalence class lemmas we developed for the transition from positive + reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) $) dfcnqs $p |- CC = ( ( R. X. R. ) /. `' _E ) $= ( cc cnr cxp cep ccnv cqs df-c qsid eqtr4i ) ABBCZJDEFGJHI $. @@ -134887,40 +135824,40 @@ Nonnegative integers (as a subset of complex numbers) ZEFBPEFACJBAPBDKACKZGALQMNO $. $} - $( Two ways to write the support of a function on ` NN0 ` . (Contributed by - Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.) $) - frnnn0supp $p |- ( ( I e. V /\ F : I --> NN0 ) - -> ( F supp 0 ) = ( `' F " NN ) ) $= + $( Two ways to write the support of a function into ` NN0 ` . (Contributed + by Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.) $) + fcdmnn0supp $p |- ( ( I e. V /\ F : I --> NN0 ) + -> ( F supp 0 ) = ( `' F " NN ) ) $= ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq cvv c0ex fsuppeq wi mpan2 imp dfn2 imaeq2i eqtr4di ) BCDZBEAFZGAHIJZAKZEHLMZNZUIONUFUGUHUKPZ UFHQDUGULTREABCQHSUAUBOUJUIUCUDUE $. - $( A function on ` NN0 ` is finitely supported iff its support is finite. + $( A function into ` NN0 ` is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019.) $) - frnnn0fsupp $p |- ( ( I e. V /\ F : I --> NN0 ) - -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) $= + fcdmnn0fsupp $p |- ( ( I e. V /\ F : I --> NN0 ) + -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) $= ( wcel cn0 wf wa cc0 cfsupp wbr ccnv csn cdif cima cfn cn wb cvv wi c0ex ffsuppbi mpan2 imp dfn2 imaeq2i eleq1i bitr4di ) BCDZBEAFZGAHIJZAKZEHLMZNZO DZUKPNZODUHUIUJUNQZUHHRDUIUPSTEABCRHUAUBUCUOUMOPULUKUDUEUFUG $. - $( Version of ~ frnnn0supp avoiding ~ ax-rep by assuming ` F ` is a set + $( Version of ~ fcdmnn0supp avoiding ~ ax-rep by assuming ` F ` is a set rather than its domain ` I ` . (Contributed by SN, 5-Aug-2024.) $) - frnnn0suppg $p |- ( ( F e. V /\ F : I --> NN0 ) - -> ( F supp 0 ) = ( `' F " NN ) ) $= + fcdmnn0suppg $p |- ( ( F e. V /\ F : I --> NN0 ) + -> ( F supp 0 ) = ( `' F " NN ) ) $= ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq cvv c0ex fsuppeqg wi mpan2 imp dfn2 imaeq2i eqtr4di ) ACDZBEAFZGAHIJZAKZEHLMZNZUIONUFUGUHUKPZ UFHQDUGULTREABCQHSUAUBOUJUIUCUDUE $. - $( $j usage 'frnnn0suppg' avoids 'ax-rep'; $) + $( $j usage 'fcdmnn0suppg' avoids 'ax-rep'; $) - $( Version of ~ frnnn0fsupp avoiding ~ ax-rep by assuming ` F ` is a set + $( Version of ~ fcdmnn0fsupp avoiding ~ ax-rep by assuming ` F ` is a set rather than its domain ` I ` . (Contributed by SN, 5-Aug-2024.) $) - frnnn0fsuppg $p |- ( ( F e. V /\ F : I --> NN0 ) - -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) $= + fcdmnn0fsuppg $p |- ( ( F e. V /\ F : I --> NN0 ) + -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) $= ( wcel cn0 wf wa cc0 cfsupp wbr csupp co cfn ccnv cn cima wfun wb ffun cvv - simpl c0ex funisfsupp mp3an3 syl2an2 frnnn0suppg eleq1d bitrd ) ACDZBEAFZGZ - AHIJZAHKLZMDZANOPZMDUJAQZUIUIULUNRZBEASUIUJUAUPUIHTDUQUBACTHUCUDUEUKUMUOMAB - CUFUGUH $. - $( $j usage 'frnnn0fsuppg' avoids 'ax-rep'; $) + simpl c0ex funisfsupp mp3an3 syl2an2 fcdmnn0suppg eleq1d bitrd ) ACDZBEAFZG + ZAHIJZAHKLZMDZANOPZMDUJAQZUIUIULUNRZBEASUIUJUAUPUIHTDUQUBACTHUCUDUEUKUMUOMA + BCUFUGUH $. + $( $j usage 'fcdmnn0fsuppg' avoids 'ax-rep'; $) ${ nnnn0d.1 $e |- ( ph -> A e. NN ) $. @@ -137506,7 +138443,7 @@ nonnegative integers (cont.)". $) ( vj cv cle wbr cz crab cuz wceq breq1 rabbidv df-uz zex rabex fvmpt ) CB CDZADZEFZAGHBREFZAGHGIQBJSTAGQBREKLCAMTAGNOP $. - $( The domain and range of the upper integers function. (Contributed by + $( The domain and codomain of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) 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(Contributed by Mario Carneiro, 26-Feb-2016.) $) nfwrd $p |- F/_ x Word S $= - ( vl vw cword cc0 cv cfzo co wf cn0 wrex cab df-word nfcv nff nfab nfcxfr - nfrex ) ABFGDHIJZBEHZKZDLMZENEBDOUDAEUCADLALPAUABUBAUBPAUAPCQTRS $. + ( vl vw cword cc0 cv cfzo co wf cn0 wrex cab df-word nfcv nff nfrexw nfab + nfcxfr ) ABFGDHIJZBEHZKZDLMZENEBDOUDAEUCADLALPAUABUBAUBPAUAPCQRST $. $} ${ @@ -157793,17 +158730,6 @@ computer programs (as last() or lastChar()), the terminology used for VMVHVIVKVGCRVJDRUGVMUHUIVFVMUJUKVAVETGZVOVEQURVCTGVDTGVPUTACSBDSVCVDUMULV EUNUOUP $. - $( Obsolete version of ~ ccatlen as of 1-Jan-2024. The length of a - concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - ccatlenOLD $p |- ( ( S e. Word B /\ T e. Word B ) -> - ( # ` ( S ++ T ) ) = ( ( # ` S ) + ( # ` T ) ) ) $= - ( vx cword wcel wa cconcat co chash cfv cc0 caddc cfzo cmin wceq fvex cn0 - cv lencl cif cmpt ccatfval fveq2d wfn ifex eqid fnmpti hashfn mp1i syl2an - nn0addcl hashfzo0 syl 3eqtrd ) BAEZFZCUPFZGZBCHIZJKDLBJKZCJKZMIZNIZDSZLVA - NIFZVEBKZVEVAOIZCKZUAZUBZJKZVDJKZVCUSUTVKJDBCUPUPUCUDVKVDUEVLVMPUSDVDVJVK - VFVGVIVEBQVHCQUFVKUGUHVDVKUIUJUSVCRFZVMVCPUQVARFVBRFVNURABTACTVAVBULUKVCU - MUNUO $. $( The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) (Revised by JJ, 18-Jan-2024.) $) @@ -158385,16 +159311,6 @@ computer programs (as last() or lastChar()), the terminology used for s1len oveq12i 1p1e2 eqtri eqtrdi mp2an ) ACZDEZFZBCZUEFZUDUGGHIJZKLAMBMUFUH NUIUDIJZUGIJZOHZKDDUDUGPULQQOHKUJQUKQOARBRSTUAUBUC $. - $( Obsolete version of ~ ccat2s1len as of 14-Jan-2024. The length of the - concatenation of two singleton words. (Contributed by Alexander van der - Vekens, 22-Sep-2018.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - ccat2s1lenOLD $p |- ( ( X e. V /\ Y e. V ) - -> ( # ` ( <" X "> ++ <" Y "> ) ) = 2 ) $= - ( wcel cs1 cword cconcat co chash c2 wceq s1cl wa caddc ccatlenOLD c1 s1len - cfv oveq12i 1p1e2 eqtri eqtrdi syl2an ) BADBEZAFZDZCEZUEDZUDUGGHIRZJKCADBAL - CALUFUHMUIUDIRZUGIRZNHZJAUDUGOULPPNHJUJPUKPNBQCQSTUAUBUC $. - $( The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) $) ccatw2s1cl $p |- ( ( W e. Word V /\ X e. V /\ Y e. V ) @@ -161928,12 +162844,23 @@ the symbol at any position is repeated at multiples of L (modulo the $} ${ - $d V n y $. $d W n w y $. + $d W n w $. $( The class of (different!) words resulting by cyclically shifting something (not necessarily a word) is a set. (Contributed by AV, - 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) $) + 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) (Proof + shortened by SN, 15-Jan-2025.) $) cshwsexa $p |- { w e. Word V | E. n e. ( 0 ..^ ( # ` W ) ) ( W cyclShift n ) = w } e. _V $= + ( cv ccsh co wceq cc0 chash cfv cfzo wrex cword crab cab cvv eqcom rexbii + abbii ovex abrexex eqeltri rabssab ssexi ) DBEFGZAEZHZBIDJKZLGZMZACNZOUKA + PZUMUGUFHZBUJMZAPQUKUOAUHUNBUJUFUGRSTBAUJUFIUILUAUBUCUKAULUDUE $. + + $d V n y $. $d W n w y $. + $( Obsolete version of ~ cshwsexa as of 15-Jan-2025. (Contributed by AV, + 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + cshwsexaOLD $p |- { w e. Word V | E. n e. ( 0 ..^ ( # ` W ) ) + ( W cyclShift n ) = w } e. _V $= ( vy cv ccsh co wceq cc0 cfzo wrex wcel wa wex cab cvv abbii wtru wi crab chash cfv cword df-rab r19.42v bicomi df-rex 3eqtri abid2 ovexi weq ovexd wal tru pm3.2i eqtr3 ex eqcoms adantl com12 ad2antlr alrimiv spcimedv imp @@ -170030,8 +170957,8 @@ Superior limit (lim sup) ${ $d x y z $. - $( The limit relation is function-like, and with range the complex numbers. - (Contributed by Mario Carneiro, 31-Jan-2014.) $) + $( The limit relation is function-like, and with codomain the complex + numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) $) fclim $p |- ~~> : dom ~~> --> CC $= ( vx vy vz cli cdm cc wf wfn crn wss wfun wrel cv wbr wa weq wal mpbir2an ax-gen wcel climrel climuni dffun2 funfn mpbi wex vex elrn climcl exlimiv @@ -172963,7 +173890,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ nfsum1 $p |- F/_ k sum_ k e. A B $= ( vm vn vx vf cv cfv caddc cz csb cc0 cmpt cseq cli c1 cn nfcv csu cuz wa wss wcel cif wbr wrex cfz co wf1o wceq wex wo cio nfss nfcri nfcsb1v nfif - df-sum nfmpt nfseq nfbr nfan nfrex nff1o nffv nfeq2 nfex nfiotaw nfcxfr + df-sum nfmpt nfseq nfbr nfan nfrexw nff1o nffv nfeq2 nfex nfiotaw nfcxfr nfor ) CABCUAAEIZUBJZUDZKFLFIZAUEZCVPBMZNUFZOZVMPZGIZQUGZUCZELUHZRVMUIUJZ AHIZUKZWBVMKFSCVPWGJZBMZOZRPZJZULZUCZHUMZESUHZUNZGUOGABHCEFUTWRCGWEWQCWDC ELCLTZVOWCCCAVNDCVNTUPCWAWBQCKVTVMCVMTZCKTZCFLVSWSVQCVRNCFADUQCVPBURCNTUS @@ -172983,7 +173910,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ nfsum $p |- F/_ x sum_ k e. A B $= ( vm vn vz vf cv cfv caddc cz csb cc0 cli c1 cn nfcv csu cuz wss wcel cif cmpt cseq wbr wa wrex cfz co wf1o wceq wex wo cio df-sum nfss nfcsbw nfif - nfcri nfmpt nfseq nfbr nfan nfrex nff1o nffv nfeq2 nfex nfiotaw nfcxfr + nfcri nfmpt nfseq nfbr nfan nfrexw nff1o nffv nfeq2 nfex nfiotaw nfcxfr nfor ) ABCDUABGKZUBLZUCZMHNHKZBUDZDVRCOZPUEZUFZVOUGZIKZQUHZUIZGNUJZRVOUKU LZBJKZUMZWDVOMHSDVRWILZCOZUFZRUGZLZUNZUIZJUOZGSUJZUPZIUQIBCJDGHURWTAIWGWS AWFAGNANTZVQWEAABVPEAVPTUSAWCWDQAMWBVOAVOTZAMTZAHNWAXAVSAVTPAHBEVBADVRCAV @@ -173004,12 +173931,12 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ nfsumOLD $p |- F/_ x sum_ k e. A B $= ( vm vn vz vf cv cfv caddc cz csb cc0 cli c1 cn nfcv csu cuz wss wcel cif cmpt cseq wbr wa wrex cfz co wf1o wceq wex wo cio df-sum nfss nfcri nfcsb - nfif nfmpt nfseq nfbr nfan nfrex nff1o nffv nfeq2 nfex nfor nfiota nfcxfr - ) ABCDUABGKZUBLZUCZMHNHKZBUDZDVRCOZPUEZUFZVOUGZIKZQUHZUIZGNUJZRVOUKULZBJK - ZUMZWDVOMHSDVRWILZCOZUFZRUGZLZUNZUIZJUOZGSUJZUPZIUQIBCJDGHURWTAIWGWSAWFAG - NANTZVQWEAABVPEAVPTUSAWCWDQAMWBVOAVOTZAMTZAHNWAXAVSAVTPAHBEUTADVRCAVRTFVA - APTVBVCVDAQTAWDTVEVFVGWRAGSASTZWQAJWJWPAAWHBWIAWITAWHTEVHAWDWOAVOWNAMWMRA - RTXCAHSWLXDADWKCAWKTFVAVCVDXBVIVJVFVKVGVLVMVN $. + nfif nfmpt nfseq nfbr nfan nfrexw nff1o nffv nfeq2 nfex nfiota nfcxfr + nfor ) ABCDUABGKZUBLZUCZMHNHKZBUDZDVRCOZPUEZUFZVOUGZIKZQUHZUIZGNUJZRVOUKU + LZBJKZUMZWDVOMHSDVRWILZCOZUFZRUGZLZUNZUIZJUOZGSUJZUPZIUQIBCJDGHURWTAIWGWS + AWFAGNANTZVQWEAABVPEAVPTUSAWCWDQAMWBVOAVOTZAMTZAHNWAXAVSAVTPAHBEUTADVRCAV + RTFVAAPTVBVCVDAQTAWDTVEVFVGWRAGSASTZWQAJWJWPAAWHBWIAWITAWHTEVHAWDWOAVOWNA + MWMRARTXCAHSWLXDADWKCAWKTFVAVCVDXBVIVJVFVKVGVNVLVM $. $} ${ @@ -176082,73 +177009,73 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ sylibr simpllr eqcomd uneq1 eqeq2d syl5ibrcom ssneldd sylan2 eqtrd 3eqtrd 3eqtr4d syldan fsumcl subcld eqtr4d disjdif ssun1 sspwi undif mpbi eqcomi subsub4d simpll snfi unfi expcld simplr fsumsplit cmpt intunsn eqid unss1 - snex elpw ssun2 snss mpbir ssel syl2imc mtod eldifd eldifi uncom sseqtrdi - elpwid ssundif elpw2 elpwunsn ad2antll snssd ssequn2 undif1 difsnb difeq1 - ad2antrl difun2 impbid f1o2d fvmpt fsumf1o cbvsumv expp1d hashunsng sseli - wi mulcomd mulm1d oveq1d mulneg1d eqtrid fsumneg sselda negsubd ralrimdva - elv ex cbvralvw syl6ibr findcard2s rspccva ) AEFDUFZGHZUXHAUGZIZGHZJKZAUH - ZUIUJZCUFZGHZUKKZUXHUXPULZIZGHZLKZCMZNZDEUMZBEFBGHZBUXJIZGHZJKZUXNUXRBUXS - IZGHZLKZCMZNZUXIUXHUAUFZUGZIZGHZJKZUYOUHZUYBCMZNZDEUMUXIUNJKZOUOZUYBCMZNZ - DEUMUXIUXHUBUFZUGZIZGHZJKZVUGUHZUYBCMZNZDEUMZUXIUXHVUHUCUFZUPZIZGHZJKZVUG - VUPUOZUPZUHZUYBCMZNZDEUMZUYEUAUBUCAUYOONZVUBVUFDEVVGUYSVUCVUAVUEVVGUYRUNU - XIJVVGUYROGHZUNVVGUYQOGVVGUYQUXHOIOVVGUYPOUXHVVGUYPOUGOUYOOURUQPUSUXHUTPQ - 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ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ nfcprod1 $p |- F/_ k prod_ k e. A B $= ( vm vy vn vx vf cv cfv cmul cz c1 cseq cli wrex cn nfcv nfseq cprod wcel cuz wss cc0 wne cif cmpt wbr wa wex w3a cfz co wf1o csb wceq df-prod nfss - wo cio nfv nfbr nfan nfex nfrex nf3an nff1o nfcsb1v nfmpt nffv nfeq2 nfor - nfmpt1 nfiotaw nfcxfr ) CABCUAAEJZUCKZUDZFJZUEUFZLCMCJAUBBNUGZUHZGJZOZVTP - UIZUJZFUKZGVRQZLWCVQOZHJZPUIZULZEMQZNVQUMUNZAIJZUOZWKVQLGRCWDWPKZBUPZUHZN - OZKZUQZUJZIUKZERQZUTZHVAHFABICEGURXGCHWNXFCWMCEMCMSVSWIWLCCAVRDCVRSZUSWHC - GVRXHWGCFWAWFCWACVBCWEVTPCLWCWDCWDSCLSZCMWBVNZTCPSZCVTSVCVDVEVFCWJWKPCLWC - VQCVQSZXIXJTXKCWKSVCVGVFXECERCRSZXDCIWQXCCCWOAWPCWPSCWOSDVHCWKXBCVQXACLWT - NCNSXICGRWSXMCWRBVIVJTXLVKVLVDVEVFVMVOVP $. + cio nfv nfmpt1 nfbr nfan nfex nfrexw nf3an nff1o nfcsb1v nfmpt nffv nfeq2 + wo nfor nfiotaw nfcxfr ) CABCUAAEJZUCKZUDZFJZUEUFZLCMCJAUBBNUGZUHZGJZOZVT + PUIZUJZFUKZGVRQZLWCVQOZHJZPUIZULZEMQZNVQUMUNZAIJZUOZWKVQLGRCWDWPKZBUPZUHZ + NOZKZUQZUJZIUKZERQZVMZHUTHFABICEGURXGCHWNXFCWMCEMCMSVSWIWLCCAVRDCVRSZUSWH + CGVRXHWGCFWAWFCWACVACWEVTPCLWCWDCWDSCLSZCMWBVBZTCPSZCVTSVCVDVEVFCWJWKPCLW + CVQCVQSZXIXJTXKCWKSVCVGVFXECERCRSZXDCIWQXCCCWOAWPCWPSCWOSDVHCWKXBCVQXACLW + TNCNSXICGRWSXMCWRBVIVJTXLVKVLVDVEVFVNVOVP $. $} ${ @@ -178346,14 +179273,14 @@ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ nfcprod $p |- F/_ x prod_ k e. A B $= ( vm vz vn vy vf cv cfv cmul cz c1 cseq cli cn nfcv cprod cuz wss cc0 wne wcel cif cmpt wbr wa wex wrex w3a cfz co wf1o csb wceq wo cio df-prod nfv - nfss nfcri nfif nfmpt nfseq nfbr nfan nfex nfrex nf3an nff1o nfcsbw nfeq2 - nffv nfor nfiotaw nfcxfr ) ABCDUABGLZUBMZUCZHLZUDUEZNDODLBUFZCPUGZUHZILZQ - ZWCRUIZUJZHUKZIWAULZNWGVTQZJLZRUIZUMZGOULZPVTUNUOZBKLZUPZWOVTNISDWHWTMZCU - QZUHZPQZMZURZUJZKUKZGSULZUSZJUTJHBCKDGIVAXKAJWRXJAWQAGOAOTZWBWMWPAABWAEAW - ATZVCWLAIWAXMWKAHWDWJAWDAVBAWIWCRANWGWHAWHTANTZADOWFXLWEACPADBEVDFAPTZVEV - FZVGARTZAWCTVHVIVJVKAWNWORANWGVTAVTTZXNXPVGXQAWOTVHVLVKXIAGSASTZXHAKXAXGA - AWSBWTAWTTAWSTEVMAWOXFAVTXEANXDPXOXNAISXCXSADXBCAXBTFVNVFVGXRVPVOVIVJVKVQ - VRVS $. + nfss nfcri nfif nfmpt nfseq nfbr nfan nfex nfrexw nf3an nff1o nfcsbw nffv + nfeq2 nfor nfiotaw nfcxfr ) ABCDUABGLZUBMZUCZHLZUDUEZNDODLBUFZCPUGZUHZILZ + QZWCRUIZUJZHUKZIWAULZNWGVTQZJLZRUIZUMZGOULZPVTUNUOZBKLZUPZWOVTNISDWHWTMZC + UQZUHZPQZMZURZUJZKUKZGSULZUSZJUTJHBCKDGIVAXKAJWRXJAWQAGOAOTZWBWMWPAABWAEA + WATZVCWLAIWAXMWKAHWDWJAWDAVBAWIWCRANWGWHAWHTANTZADOWFXLWEACPADBEVDFAPTZVE + VFZVGARTZAWCTVHVIVJVKAWNWORANWGVTAVTTZXNXPVGXQAWOTVHVLVKXIAGSASTZXHAKXAXG + AAWSBWTAWTTAWSTEVMAWOXFAVTXEANXDPXOXNAISXCXSADXBCAXBTFVNVFVGXRVOVPVIVJVKV + QVRVS $. $} ${ @@ -184040,7 +184967,7 @@ and cancel them ( ~ rpnnen2lem10 ), since these are the same for both VQGUIIRUHGULRGUMUNRVMGUOUPUQURUSVCVPVKUISMVKUDGVOTTMTUTSVAVDVKTUTSMOVBVDV OVEVFVGVHVIUS $. - $( Lemma for ~ ruc . Domain and range of the interval sequence. + $( Lemma for ~ ruc . Domain and codomain of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.) $) ruclem6 $p |- ( ph -> G : NN0 --> ( RR X. RR ) ) $= ( cr cc0 cfv c1 cop wcel wceq co vz vw cn0 cxp cseq wf fveq1i cz 0z ax-mp @@ -189080,44 +190007,6 @@ obviously not a bijection (by Cantor's theorem ~ canth2 ), and in fact PVQQVRVS $. $} - ${ - $d M k n $. $d N k n $. - $( Obsolete proof of ~ gcdmultiple as of 12-Jan-2024. The GCD of a - multiple of a number is the number itself. (Contributed by Scott - Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - gcdmultipleOLD $p |- ( ( M e. NN /\ N e. NN ) - -> ( M gcd ( M x. N ) ) = M ) $= - ( vk vn cn wcel cmul co cgcd wi c1 caddc oveq2 oveq2d eqeq1d imbi2d eqtrd - wceq cz cc cv weq nncn mulid1d cabs cfv nnz gcdid syl nnre nn0ge0d absidd - nnnn0 wa zmulcl syl2an gcdaddm mp3an2ani ax-1cn adddi mp3an3 mulcom mpan2 - 1z adantr eqtr4d biimpd expcom a2d nnind impcom ) BEFAEFZAABGHZIHZARZVLAA - CUAZGHZIHZARZJVLAAKGHZIHZARZJVLAADUAZGHZIHZARZJVLAAWCKLHZGHZIHZARZJVLVOJC - DBVPKRZVSWBVLWKVRWAAWKVQVTAIVPKAGMNOPCDUBZVSWFVLWLVRWEAWLVQWDAIVPWCAGMNOP - VPWGRZVSWJVLWMVRWIAWMVQWHAIVPWGAGMNOPVPBRZVSVOVLWNVRVNAWNVQVMAIVPBAGMNOPV - LWAAAIHZAVLVTAAIVLAAUCZUDNVLWOAUEUFZAVLASFZWOWQRAUGZAUHUIVLAAUJVLAAUMUKUL - QQWCEFZVLWFWJVLWTWFWJJVLWTUNZWFWJXAWEWIAXAWEAWDKAGHZLHZIHZWIKSFVLWRWTWDSF - ZWEXDRVDWSVLWRWCSFXEWTWSWCUGAWCUOUPKAWDUQURXAWHXCAIVLATFZWCTFZWHXCRWTWPWC - UCXFXGUNZWHWDVTLHZXCXFXGKTFZWHXIRUSAWCKUTVAXHVTXBWDLXFVTXBRZXGXFXJXKUSAKV - BVCVENQUPNVFOVGVHVIVJVK $. - $} - - $( Obsolete proof of ~ gcdmultiplez as of 12-Jan-2024. Extend ~ gcdmultiple - so ` N ` can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) - (Revised by Mario Carneiro, 19-Apr-2014.) (New usage is discouraged.) - (Proof modification is discouraged.) $) - gcdmultiplezOLD $p |- ( ( M e. NN /\ N e. ZZ ) - -> ( M gcd ( M x. N ) ) = M ) $= - ( cn wcel cz wa cmul co cgcd wceq cc0 oveq2 oveq2d eqeq1d cabs cfv cc eqtrd - adantr syl wne nncn zcn absmul syl2an nnre nnnn0 nn0ge0d absidd oveq1d nnzd - simpll nnz zmulcl sylan gcdabs2 syl2anc nnabscl gcdmultiple anassrs 3eqtr3d - sylan2 mul01 cn0 nn0gcdid0 pm2.61ne ) ACDZBEDZFZAABGHZIHZAJAAKGHZIHZAJBKBKJ - ZVKVMAVNVJVLAIBKAGLMNVIBKUAZFZAVJOPZIHZAABOPZGHZIHZVKAVIVRWAJVOVIVQVTAIVIVQ - AOPZVSGHZVTVGAQDZBQDVQWCJVHAUBZBUCABUDUEVGWCVTJVHVGWBAVSGVGAAUFVGAAUGZUHUIU - JSRMSVPAEDZVJEDZVRVKJVPAVGVHVOULUKVIWHVOVGWGVHWHAUMABUNUOSVJAUPUQVGVHVOWAAJ - ZVHVOFVGVSCDWIBURAVSUSVBUTVAVIVMAKIHZAVGVMWJJZVHVGWDWKWEWDVLKAIAVCMTSVGWJAJ - ZVHVGAVDDWLWFAVETSRVF $. - $( A positive integer ` A ` is equal to its gcd with an integer ` B ` if and only if ` A ` divides ` B ` . Generalization of ~ gcdeq . (Contributed by AV, 1-Jul-2020.) $) @@ -200658,7 +201547,7 @@ consisting of identical symbols by at least one position (and not by as prmlem2.23 $e |- -. ; 2 3 || N $. $( Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that - 2 and 3 are prime and 4 is not allows us to cover the numbers less than + 2 and 3 are prime and 4 is not allows to cover the numbers less than ` 5 ^ 2 = 2 5 ` . Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another @@ -201985,7 +202874,7 @@ implemented as a function (a set of ordered pairs)". The current csts $a class sSet $. ${ - $d e s w x $. + $d e s $. $( Set a component of an extensible structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, ~ df-ress adjusts the base set to match its second @@ -202396,6 +203285,7 @@ class of all (base sets of) groups is proper. (Contributed by Mario LLAMZCAMEAMUBLFNCAOUAELAUAECDBPLGCDBHQRST $. $} + $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Structure component indices @@ -207381,7 +208271,7 @@ the closure of the set with the element removed (Section 0.6 in ${ $d F c x s $. $d C c x s $. $d X c x s $. $d U c x s $. $d V c x s $. - $( The domain and range of the function expression for Moore closures. + $( The domain and codomain of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.) $) mrcflem $p |- ( C e. ( Moore ` X ) -> ( x e. ~P X |-> |^| { s e. C | x C_ s } ) : ~P X --> C ) $= @@ -207812,9 +208702,8 @@ the closure of the set with the element removed (Section 0.6 in subsets ` S ` of the base set such that ` z ` is in the closure of ` ( S u. { y } ) ` but not in the closure of ` S ` , ` y ` is in the closure of ` ( S u. { z } ) ` (Definition 3.1.9 in [FaureFrolicher] - p. 57 to 58.) This theorem allows us to construct substitution - instances of this definition. (Contributed by David Moews, - 1-May-2017.) $) + p. 57 to 58.) This theorem allows to construct substitution instances + of this definition. (Contributed by David Moews, 1-May-2017.) $) mreexd $p |- ( ph -> Y e. ( N ` ( S u. { Z } ) ) ) $= ( csn cun cfv wcel cv cdif wral cpw sselpwd wceq wa adantr ad2antrr simpr simplr uneq12d fveq2d eleqtrrd wn neleqtrrd eldifd simpllr eleq12d rspcdv @@ -207983,38 +208872,38 @@ f C_ ( N ` ( g u. h ) ) /\ ( f u. h ) e. I ) -> csn wral wss animorrl mreexexlem3d wne n0 biimpi adantl wn mreexexlem2d wex simpr w3a 3anass cvv ad2antrr elfvexd simpr2 difsnb ssdifssd ssdifd sylib difun1 sseqtrrdi simpr1 uncom uneq2i unass difsnid uneq1d eqtr3id - eqsstrrd eqtrid syl fveq2d sseqtrrd simpr3 csuc wo com wi simplr dif1en - 3anan12 sylbir expcom syl2anc orim12d mpd mreexexlemd ad3antrrr difss2d - wal ssexd simprl simplr1 snssd unssd sselpwd ad3antlr cin simprrl en2sn - elpwid el2v a1i disjdifr ssdifin0 unen syl22anc eqbrtrrd eqtr2i simprrr - eqeltrrid rexlimddv breq2 uneq1 eleq1d anbi12d rspcev syl12anc sylan2br - adantlr exlimddv pm2.61dane ) AIHUKZULUMZUUKKUNZLUOZUPZHJUQZURZIUSAIUSU - TZUPBCDHIJKLNOPADOVAVBUOZUURQVCRSABUKZPUKZCUKVEUNNVBUOCUVAUUTVEUNNVBUVA - NVBVDVFBOVFPOUQVFZUURTVCAIOKVDZVGZUURUAVCAJUVCVGZUURUBVCAIJKUNZNVBZVGZU - URUCVCAIKUNLUOZUURUDVCAUURJUSUTVHVIAIUSVJZUPUHUKZIUOZUUQUHUVJUVLUHVPZAU - VJUVMUHIVKVLVMAUVLUUQUVJAUVLUPZUIUKZIUVKVEZVDZUOVNZUVQKUVOVEZUNZUNLUOZU - PZUUQUIJUVNBCDUIIJKLNOUVKPAUUSUVLQVCRSAUVBUVLTVCAUVDUVLUAVCAUVEUVLUBVCA - 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B ) --> ~P ( ( B X. B ) X. _V ) ) $= ( vx cxp cv csn chom cfv cvv cpw eqid homafval wcel wa wss adantl sylancl - snssi ssv xpss12 snex fvex xpex elpw sylibr fmpt3d ) AHBBIZHJZKZUMCLMZMZI - ZULNIZOZDAHBCDUOEFGUOPQAUMULRZSZUQURTZUQUSRVAUNULTZUPNTVBUTVCAUMULUCUAUPU - DUNULUPNUEUBUQURUNUPUMUFUMUOUGUHUIUJUK $. + snssi ssv xpss12 vsnex fvex xpex elpw sylibr fmpt3d ) AHBBIZHJZKZUMCLMZMZ + IZULNIZOZDAHBCDUOEFGUOPQAUMULRZSZUQURTZUQUSRVAUNULTZUPNTVBUTVCAUMULUCUAUP + UDUNULUPNUEUBUQURUNUPHUFUMUOUGUHUIUJUK $. homaval.j $e |- J = ( Hom ` C ) $. homaval.x $e |- ( ph -> X e. B ) $. @@ -220604,11 +221493,11 @@ Partially ordered sets (posets) In our formalism of extensible structures, the base set of a poset ` f ` is denoted by ` ( Base `` f ) ` and its partial order by ` ( le `` f ) ` (for "less than or equal to"). The quantifiers ` E. b E. r ` provide a - notational shorthand to allow us to refer to the base and ordering - relation as ` b ` and ` r ` in the definition rather than having to - repeat ` ( Base `` f ) ` and ` ( le `` f ) ` throughout. These - quantifiers can be eliminated with ~ ceqsex2v and related theorems. - (Contributed by NM, 18-Oct-2012.) $) + notational shorthand to allow to refer to the base and ordering relation + as ` b ` and ` r ` in the definition rather than having to repeat + ` ( Base `` f ) ` and ` ( le `` f ) ` throughout. These quantifiers can + be eliminated with ~ ceqsex2v and related theorems. (Contributed by NM, + 18-Oct-2012.) $) df-poset $a |- Poset = { f | E. b E. r ( b = ( Base ` f ) /\ r = ( le ` f ) @@ -227447,52 +228336,6 @@ proposition to be be proved (the first four hypotheses tell its values IABDCOYQUCFGUUNUVJYIUWAADTZUVSADTYIYCUWQYAYCYDYHXRADWGSYIUWAUVSADUWCXAXBX QYIABECOYSUCFGUUNUWHYIUWNAETZUWIAETYIYDUWRYAYCYDYHXTAEWGSYIUWNUWIAEUWPXAX BXQXOXS $. - - $( Obsolete version of ~ gsumccat as of 13-Jan-2024. Homomorphic property - of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised - by Mario Carneiro, 1-Oct-2015.) (New usage is discouraged.) - (Proof modification is discouraged.) $) - gsumccatOLD $p |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> - ( G gsum ( W ++ X ) ) = ( ( G gsum W ) .+ ( G gsum X ) ) ) $= - ( wcel co cgsu wceq c0 cfv oveq2d wa caddc c1 cc0 cn0 syl vx vy cword w3a - vz cmnd cconcat c0g oveq1 oveq2 eqid gsum0 eqtrdi oveq1d eqeq12d wne cmin - chash cseq simpl1 cn lennncl 3ad2antl2 adantrr 3ad2antl3 adantrl nnaddcld - cuz nnm1nn0 nn0uz eleqtrdi cfzo wf cfz simpl2 simpl3 syl2anc wrdf ccatlen - ccatcl cz nnzd zaddcld fzoval eqtrd feq2d mpbid gsumval2 oveq12d cv mndcl - 3expb sylan mndass uzid uzaddcl nncnd 1cnd addsubassd ax-1cn npcan fveq2d - sylancl 3eltr4d ffvelcdmda simpll2 simpll3 eleq2d biimpar syl3anc seqfveq - seqsplit ccatval1 addid2d eqtr4d seqeq1d addcomd addsubd fveq12d ccatval3 - cc eqcomd seqshft2 anassrs ccatrid gsumwcl 3adant3 adantr mndrid pm2.61ne - ccatlid 3ad2ant3 simp1 mndlid 3imp3i2an ) CUFHZDAUCZHZEYQHZUDZCDEUGIZJIZC - DJIZCEJIZBIZKZCLEUGIZJIZCUHMZUUDBIZKDLDLKZUUBUUHUUEUUJUUKUUAUUGCJDLEUGUIN - UUKUUCUUIUUDBUUKUUCCLJIZUUIDLCJUJCUUIUUIUKZULZUMUNUOYTDLUPZOZUUFCDLUGIZJI - ZUUCUUIBIZKELELKZUUBUURUUEUUSUUTUUAUUQCJELDUGUJNUUTUUDUUIUUCBUUTUUDUULUUI - ELCJUJUUNUMNUOYTUUOELUPZUUFYTUUOUVAOZOZUUBDURMZEURMZPIZQUQIZBUUARUSZMZUUE - UVCABUUACRUVGUFFGYPYRYSUVBUTZUVCUVGSRVHMZUVCUVFVAHUVGSHUVCUVDUVEYTUUOUVDV - AHZUVAYRYPUUOUVLYSADVBVCVDZYTUVAUVEVAHZUUOYSYPUVAUVNYRAEVBVEVFZVGUVFVITVJ - VKUVCRUUAURMZVLIZAUUAVMZRUVGVNIZAUUAVMUVCUUAYQHZUVRUVCYRYSUVTYPYRYSUVBVOZ - YPYRYSUVBVPZADEVTVQAUUAVRTUVCUVQUVSAUUAUVCUVQRUVFVLIZUVSUVCUVPUVFRVLUVCYR - YSUVPUVFKUWAUWBAADEVSVQNUVCUVFWAHUWCUVSKUVCUVDUVEUVCUVDUVMWBZUVCUVEUVOWBZ - WCRUVFWDTWEWFWGZWHUVCUUEUVDQUQIZBDRUSMZUVEQUQIZBERUSMZBIZUVIUVCUUCUWHUUDU - WJBUVCABDCRUWGUFFGUVJUVCUWGSUVKUVCUVLUWGSHUVMUVDVITVJVKZUVCRUVDVLIZADVMZR - UWGVNIZADVMUVCYRUWNUWAADVRTUVCUWMUWOADUVCUVDWAHZUWMUWOKUWDRUVDWDTZWFWGWHU - VCABECRUWIUFFGUVJUVCUWISUVKUVCUVNUWISHZUVOUVEVITZVJVKZUVCRUVEVLIZAEVMZRUW - IVNIZAEVMUVCYSUXBUWBAEVRTUVCUXAUXCAEUVCUVEWAHUXAUXCKUWERUVEWDTZWFWGWHWIUV - CUVIUWGUVHMZUVGBUUAUWGQPIZUSZMZBIUWKUVCUAUBUEBAUUARUWGUVGUVCYPUAWJZAHZUBW - JZAHZOUXIUXKBIZAHZUVJYPUXJUXLUXNABCUXIUXKFGWKWLWMUVCYPUXJUXLUEWJZAHUDUXMU - XOBIUXIUXKUXOBIBIKUVJABCUXIUXKUXOFGWNWMUVCUVDUWIPIZUVDVHMZUVGUXFVHMUVCUVD - UXQHZUWRUXPUXQHUVCUWPUXRUWDUVDWOTUWSUWIUVDUVDWPVQUVCUVDUVEQUVCUVDUVMWQZUV - CUVEUVOWQZUVCWRZWSUVCUXFUVDVHUVCUVDYAHQYAHUXFUVDKUXSWTUVDQXAXCZXBXDUWLUVC - UVSAUXIUUAUWFXEXLUVCUXEUWHUXHUWJBUVCBUAUUADRUWGUWLUVCUXIUWOHZOYRYSUXIUWMH - ZUXIUUAMUXIDMKYPYRYSUVBUYCXFYPYRYSUVBUYCXGUVCUYDUYCUVCUWMUWOUXIUWQXHXIAAD - EUXIXMXJXKUVCUXHUWIUVDPIZBUUARUVDPIZUSZMUWJUVCUVGUYEUXGUYGUVCUXFUYFBUUAUV - CUXFUVDUYFUYBUVCUVDUXSXNXOXPUVCUVGUVEUVDPIZQUQIUYEUVCUVFUYHQUQUVCUVDUVEUX - SUXTXQUNUVCUVEUVDQUXTUXSUYAXRWEXSUVCBUAEUUAUVDRUWIUWTUWDUVCUXIUXCHZOZUXIU - VDPIUUAMZUXIEMZUYJYRYSUXIUXAHZUYKUYLKYPYRYSUVBUYIXFYPYRYSUVBUYIXGUVCUYMUY - IUVCUXAUXCUXIUXDXHXIADEUXIXTXJYBYCXOWIWEXOXOYDUUPUURUUCUUSUUPUUQDCJUUPYRU - UQDKYPYRYSUUOVOADYETNUUPYPUUCAHZUUSUUCKYPYRYSUUOUTYTUYNUUOYPYRUYNYSACDFYF - YGYHABCUUCUUIFGUUMYIVQXOYJYTUUHUUDUUJYTUUGECJYSYPUUGEKYRAEYKYLNYPYRYSYPUU - DAHUUJUUDKYPYRYSYMACEFYFABCUUDUUIFGUUMYNYOXOYJ $. $} ${ @@ -229097,7 +229940,7 @@ with the same (relevant) properties is also a group. (Contributed by grpss.s $e |- G C_ R $. grpss.f $e |- Fun R $. $( Show that a structure extending a constructed group (e.g., a ring) is - also a group. This allows us to prove that a constructed potential ring + also a group. This allows to prove that a constructed potential ring ` R ` is a group before we know that it is also a ring. (Theorem ~ ringgrp , on the other hand, requires that we know in advance that ` R ` is a ring.) (Contributed by NM, 11-Oct-2013.) $) @@ -235240,11 +236083,11 @@ operation is a permutation group (group consisting of permutations), see cop cts ctp hasheq0 cefmnd csymg 0symgefmndeq eqcomi fveq2 eqtrid 3eqtr4a wa adantl eqid efmnd adantr cmap co csn 0map0sn0 id oveq12d cv cab fveq2d wf1o symgbas symgbas0 3eqtr3g opeq2d tpeq1d 3eqtrd ex sylbid wex hash1snb - snex eleq1 mpbiri syl snsymgefmndeq eqtr4di eqtr4i 3eqtr3d exlimiv syl6bi - efmndbas cin csts cdif cres cun wss wn ssnpss symgpssefmnd psseq12i nsyl3 - wpss sylibr fvexd f1osetex eqeltri symgval ressval2 syl3anc inex2 setsval - a1i ovex sylancl cpr reseq1d uneq1d eqidd ccom cmpo mpoex cpw cxp cpt wne - fvexi basendxnplusgndx necomi tsetndxnbasendx tpres uncom tpass wi ssrdv + vsnex mpbiri snsymgefmndeq eqtr4di efmndbas eqtr4i 3eqtr3d exlimiv syl6bi + eleq1 syl cin csts cdif cres cun wss wn wpss ssnpss symgpssefmnd psseq12i + sylibr nsyl3 fvexd f1osetex eqeltri symgval ressval2 syl3anc ovex setsval + a1i inex2 sylancl cpr reseq1d eqidd ccom cmpo mpoex cpw cxp cpt fvexi wne + uneq1d basendxnplusgndx necomi tsetndxnbasendx tpres uncom tpass wi ssrdv symgbasmap df-ss sylib 3jaod mpd ) BJPZBUAQZUBRZUUHUCRZUCUUHUDUEZUFGUGUHQ ZCUKZUGUIQZDUKZUGULQZHUKZUMZRZBJUJUUGUUIUUSUUJUUKUUGUUIBSRZUUSBJUNUUGUUTU USUUGUUTVBZGBUOQZUULIUKZUUOUUQUMZUURUUTGUVBRUUGUUTSUPQZSUOQZGUVBUVFUVEUQU @@ -235252,19 +236095,19 @@ operation is a permutation group (group consisting of permutations), see AUVCUUMUUOUUQUVAICUULUVAIBBVGVHZCMUUTUVLCRUUGUUTSSVGVHSVIZUVLCVJUUTBSBSVG UUTVKZUVNVLUUTCBBAVMZVPAVNZUVMLUUTGUHQZUVEUHQUVPUVMUUTGUVEUHUVHVOABUVQGKU VQVDVQZVRVSUTVAVCUTVTWAWBWCWDUUGUUJBUVOVIZRZAWEUUSBJAWFUVTUUSAUVTUVBUVDGU - 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com eqbrtrd simpr dif1en mp3an2i weu euen1b eumo sylbi wi f1omvdmvd eleq2 - syl ex ad2antll difeq1 eleq2d 3imtr4d imp ad4ant24 fvex pm3.2i moi mp3an1 - eleq1 syl12anc adantlr wn fnelnfp sylan bitrd necon2bbid eqtr4d pm2.61dan - wne biimpar eqfnfvd ) AABFZAACFZGZBHIJZKLMZCHIJZWMNZGZGZDABCWJBAOZWKWQAAB - PUAZWKCAOZWJWQAACPUBZWRDUCZAQZGZXCWOQZXCBUDZXCCUDZNZWRXFXIXDWRXFGZEUCZWOX - CUEZIZQZEUFZXGXMQZXHXMQZXIXJXMRLMZXORUMQXJWORUGZLMXFXRUHXJWOWMXSLWLWNWPXF - UIXJWMKXSLWLWNWPXFUJUKULUNWRXFUOWORXCUPUQXRXNEURXOEXMUSXNEUTVAVEWRXFXPWRX - CWMQZXGWMXLIZQZXFXPWJXTYBVBWKWQWJXTYBABXCVCVFUAWPXFXTSWLWNWOWMXCVDVGWPXPY - BSWLWNWPXMYAXGWOWMXLVHVIVGVJVKWKXFXQWJWQACXCVCVLXGTQZXHTQZGXOXPXQGXIYCYDX - CBVMXCCVMVNXNXPXQEXGXHTTXKXGXMVQXKXHXMVQVOVPVRVSXEXFVTZGXGXCXHXEXGXCNYEXE - XFXGXCXEXFXTXGXCWGZXEWOWMXCWLWNWPXDUIVIWRWSXDXTYFSWTABXCWAWBWCWDWHXEXHXCN - YEXEXFXHXCWRXAXDXFXHXCWGSXBACXCWAWBWDWHWEWFWI $. + com eqbrtrd simpr dif1ennn mp3an2i weu euen1b eumo sylbi syl wi f1omvdmvd + ex eleq2 ad2antll difeq1 eleq2d 3imtr4d ad4ant24 fvex pm3.2i eleq1 mp3an1 + imp moi syl12anc adantlr wn fnelnfp sylan bitrd necon2bbid biimpar eqtr4d + wne pm2.61dan eqfnfvd ) AABFZAACFZGZBHIJZKLMZCHIJZWMNZGZGZDABCWJBAOZWKWQA + ABPUAZWKCAOZWJWQAACPUBZWRDUCZAQZGZXCWOQZXCBUDZXCCUDZNZWRXFXIXDWRXFGZEUCZW + OXCUEZIZQZEUFZXGXMQZXHXMQZXIXJXMRLMZXORUMQXJWORUGZLMXFXRUHXJWOWMXSLWLWNWP + XFUIXJWMKXSLWLWNWPXFUJUKULUNWRXFUOWORXCUPUQXRXNEURXOEXMUSXNEUTVAVBWRXFXPW + RXCWMQZXGWMXLIZQZXFXPWJXTYBVCWKWQWJXTYBABXCVDVEUAWPXFXTSWLWNWOWMXCVFVGWPX + PYBSWLWNWPXMYAXGWOWMXLVHVIVGVJVPWKXFXQWJWQACXCVDVKXGTQZXHTQZGXOXPXQGXIYCY + DXCBVLXCCVLVMXNXPXQEXGXHTTXKXGXMVNXKXHXMVNVQVOVRVSXEXFVTZGXGXCXHXEXGXCNYE + XEXFXGXCXEXFXTXGXCWGZXEWOWMXCWLWNWPXDUIVIWRWSXDXTYFSWTABXCWAWBWCWDWEXEXHX + CNYEXEXFXHXCWRXAXDXFXHXCWGSXBACXCWAWBWDWEWFWHWI $. $( If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, @@ -236282,11 +237125,11 @@ operation is a permutation group (group consisting of permutations), see 16-Aug-2015.) $) pmtrf $p |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( T ` P ) : D --> D ) $= ( vz wcel wss c2o cen wbr w3a cv csn cdif cuni cif cfv wa c1o pmtrval com - simpll2 csuc 1onn simpll3 df-2o breqtrdi simpr dif1en mp3an2i eldifi 3syl - en1uniel sseldd wn simplr ifclda fmpt3d ) ADGZBAHZBIJKZLZFAFMZBGZBVDNZOZP - ZVDQABCRFABCDEUAVCVDAGZSZVEVHVDAVJVESZBAVHUTVAVBVIVEUCVKVGTJKZVHVGGVHBGTU - BGVKBTUDZJKVEVLUEVKBIVMJUTVAVBVIVEUFUGUHVJVEUIBTVDUJUKVGUNVHBVFULUMUOVCVI - VEUPUQURUS $. + simpll2 csuc 1onn simpll3 df-2o breqtrdi dif1ennn mp3an2i en1uniel eldifi + simpr 3syl sseldd wn simplr ifclda fmpt3d ) ADGZBAHZBIJKZLZFAFMZBGZBVDNZO + ZPZVDQABCRFABCDEUAVCVDAGZSZVEVHVDAVJVESZBAVHUTVAVBVIVEUCVKVGTJKZVHVGGVHBG + TUBGVKBTUDZJKVEVLUEVKBIVMJUTVAVBVIVEUFUGUHVJVEUMBTVDUIUJVGUKVHBVFULUNUOVC + VIVEUPUQURUS $. $( A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.) $) @@ -236294,14 +237137,14 @@ operation is a permutation group (group consisting of permutations), see dom ( ( T ` P ) \ _I ) = P ) $= ( vz wcel wss c2o cen wbr cfv cdif wne crab cin wceq 3syl wa c1o cv pmtrf w3a cid cdm wf wfn ffn fndifnfp csn cif pmtrfv neeq1d wb iffalse necon1ai - cuni iftrue adantl csuc 1onn simpl3 df-2o breqtrdi simpr mp3an2i en1uniel - com dif1en eldifsni eqnetrd ex impbid2 bitrd rabbidva incom dfin5 eqtr4di - adantr eqtri simp2 df-ss sylib 3eqtrd ) ADGZBAHZBIJKZUCZBCLZUDMUEZFUAZWIL - ZWKNZFAOZBAPZBWHAAWIUFWIAUGWJWNQABCDEUBAAWIUHFAWIUIRWHWNWKBGZFAOZWOWHWMWP - FAWHWKAGZSZWMWPBWKUJMZUQZWKUKZWKNZWPWSWLXBWKABCDWKEULUMWHXCWPUNWRWHXCWPWP - XBWKWPXAWKUOUPWHWPXCWHWPSZXBXAWKWPXBXAQWHWPXAWKURUSXDWTTJKZXAWTGXAWKNTVHG - XDBTUTZJKWPXEVAXDBIXFJWEWFWGWPVBVCVDWHWPVEBTWKVIVFWTVGXABWKVJRVKVLVMVSVNV - OWOABPWQBAVPFABVQVTVRWHWFWOBQWEWFWGWABAWBWCWD $. + cuni iftrue adantl csuc 1onn simpl3 df-2o breqtrdi simpr dif1ennn mp3an2i + com en1uniel eldifsni eqnetrd ex impbid2 bitrd rabbidva incom dfin5 eqtri + adantr eqtr4di simp2 df-ss sylib 3eqtrd ) ADGZBAHZBIJKZUCZBCLZUDMUEZFUAZW + ILZWKNZFAOZBAPZBWHAAWIUFWIAUGWJWNQABCDEUBAAWIUHFAWIUIRWHWNWKBGZFAOZWOWHWM + WPFAWHWKAGZSZWMWPBWKUJMZUQZWKUKZWKNZWPWSWLXBWKABCDWKEULUMWHXCWPUNWRWHXCWP + WPXBWKWPXAWKUOUPWHWPXCWHWPSZXBXAWKWPXBXAQWHWPXAWKURUSXDWTTJKZXAWTGXAWKNTV + HGXDBTUTZJKWPXEVAXDBIXFJWEWFWGWPVBVCVDWHWPVEBTWKVFVGWTVIXABWKVJRVKVLVMVSV + NVOWOABPWQBAVPFABVQVRVTWHWFWOBQWEWFWGWABAWBWCWD $. $} ${ @@ -236375,19 +237218,19 @@ operation is a permutation group (group consisting of permutations), see ( vx wcel wf wfn cdif cfv cen wbr wceq syl wa c1o sylan eqtrd cid cdm cvv ccom cres wss c2o w3a pmtrfrn simpld pmtrf simprd feq1d mpbird fco anidms eqid ffn 3syl fnresi a1i cv csn cif pmtrffv iftrue sylan9eq fveq2d simpll - cuni simp2d ad2antrr csuc 1onn simp3d df-2o breqtrdi simpr dif1en mp3an2i - com en1uniel eldifad sseldd syl2anc adantr en2other2 ancoms wn wb fnelnfp - wne ffnd necon2bbid biimpar fveq2 pm2.61dan fvresi adantl 3eqtr4d eqfnfvd - id fvco2 ) DBHZGADDUDZUAAUEZXDAADIZAAXEIZXEAJXDXGAADUAKUBZCLZIZXDAUCHZXIA - UFZXIUGMNZUHZXKXDXODXJOZAXIBCDEFXIUQZUIZUJZAXICUCEUKPXDAADXJXDXOXPXRULUMU - NZXGXHAAADDUOUPAAXEURUSXFAJXDAUTVAXDGVBZAHZQZYADLZDLZYAYAXELZYAXFLZYCYAXI - HZYEYAOZYCYHQZYEXIYAVCZKZVJZDLZYAYJYDYMDYCYHYDYHYMYAVDYMAXIBCDYAEFXQVEYHY - MYAVFVGVHYJYNYMXIHZXIYMVCKVJZYMVDZYAYJXDYMAHYNYQOXDYBYHVIYJXIAYMXDXMYBYHX - DXLXMXNXSVKVLYJYMXIYKYJYLRMNZYMYLHRWAHYJXIRVMZMNZYHYRVNXDYTYBYHXDXIUGYSMX - DXLXMXNXSVOZVPVQVLYCYHVRXIRYAVSVTYLWBPWCZWDAXIBCDYMEFXQVEWEYJYQYPYAYJYOYQ - YPOUUBYOYPYMVFPYCXNYHYPYAOZXDXNYBUUAWFYHXNUUCXIYAWGWHSTTTYCYHWIZQYDYAOZYI - YCUUEUUDYCYHYDYAXDDAJZYBYHYDYAWLWJXDAADXTWMZADYAWKSWNWOUUEYEYDYAYDYADWPUU - EXBTPWQXDUUFYBYFYEOUUGADDYAXCSYBYGYAOXDAYAWRWSWTXA $. + cuni simp2d ad2antrr com csuc 1onn simp3d df-2o breqtrdi dif1ennn mp3an2i + simpr en1uniel eldifad sseldd syl2anc adantr en2other2 ancoms wn wne ffnd + wb fnelnfp necon2bbid biimpar fveq2 pm2.61dan fvco2 fvresi adantl 3eqtr4d + id eqfnfvd ) DBHZGADDUDZUAAUEZXDAADIZAAXEIZXEAJXDXGAADUAKUBZCLZIZXDAUCHZX + IAUFZXIUGMNZUHZXKXDXODXJOZAXIBCDEFXIUQZUIZUJZAXICUCEUKPXDAADXJXDXOXPXRULU + MUNZXGXHAAADDUOUPAAXEURUSXFAJXDAUTVAXDGVBZAHZQZYADLZDLZYAYAXELZYAXFLZYCYA + XIHZYEYAOZYCYHQZYEXIYAVCZKZVJZDLZYAYJYDYMDYCYHYDYHYMYAVDYMAXIBCDYAEFXQVEY + HYMYAVFVGVHYJYNYMXIHZXIYMVCKVJZYMVDZYAYJXDYMAHYNYQOXDYBYHVIYJXIAYMXDXMYBY + HXDXLXMXNXSVKVLYJYMXIYKYJYLRMNZYMYLHRVMHYJXIRVNZMNZYHYRVOXDYTYBYHXDXIUGYS + MXDXLXMXNXSVPZVQVRVLYCYHWAXIRYAVSVTYLWBPWCZWDAXIBCDYMEFXQVEWEYJYQYPYAYJYO + YQYPOUUBYOYPYMVFPYCXNYHYPYAOZXDXNYBUUAWFYHXNUUCXIYAWGWHSTTTYCYHWIZQYDYAOZ + YIYCUUEUUDYCYHYDYAXDDAJZYBYHYDYAWJWLXDAADXTWKZADYAWMSWNWOUUEYEYDYAYDYADWP + UUEXBTPWQXDUUFYBYFYEOUUGADDYAWRSYBYGYAOXDAYAWSWTXAXC $. $( A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.) $) @@ -240159,17 +241002,6 @@ U C_ ( T .(+) U ) ) $= ( csubg cfv wcel csubmnd co wceq subgsubm smndlsmidm syl ) BCEFGBCHFGBBAI BJBCKABCDLM $. - $( Obsolete proof of ~ lsmidm as of 13-Jan-2024. Subgroup sum is - idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario - Carneiro, 21-Jun-2014.) (New usage is discouraged.) - (Proof modification is discouraged.) $) - lsmidmOLD $p |- ( U e. ( SubGrp ` G ) -> ( U .(+) U ) = U ) $= - ( vx vy csubg cfv wcel co cv cplusg cmpo crn wceq eqid lsmval anidms wral - cbs cxp subgcl 3expb ralrimivva fmpo sylib frnd eqsstrd wss lsmub1 eqssd - wf ) BCGHIZBBAJZBUMUNEFBBEKZFKZCLHZJZMZNZBUMUNUTOEFCTHZUQABBCVAPUQPZDQRUM - BBUAZBUSUMURBIZFBSEBSVCBUSULUMVDEFBBUMUOBIUPBIVDUQBCUOUPVBUBUCUDEFBBURBUS - USPUEUFUGUHUMBUNUIABBCDUJRUK $. - $( The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) $) lsmlub $p |- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ @@ -246294,27 +247126,27 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements Carneiro, 28-Dec-2014.) $) gsum2d2 $p |- ( ph -> ( G gsum ( j e. A , k e. C |-> X ) ) = ( G gsum ( j e. A |-> ( G gsum ( k e. C |-> X ) ) ) ) ) $= - ( vm vn vx vy cmpo cgsu co cv csn cxp ciun cima cmpt wcel wral snex xpexg - cvv wa sylancr ralrimiva iunexg syl2anc wrel relxp rgenw reliun mpbir a1i - cdm cop wex eldm2 wceq eliunxp opth1 ad2antrl simprrl eqeltrd ex exlimdvv - vex biimtrid exlimdv ssrdv wf ralrimivva eqid fmpox sylib gsum2d2lem nfcv - gsum2d nfiu1 nfima nfmpo1 nfov nfmpt sneq imaeq2d mpteq12dv oveq2d cbvmpt - oveq1 elimasn bitri baib eqrdv mpteq1d nfmpo2 eqtrdi adantl simprl simprr - opeliunxp oveq2 ovmpt4g syl3anc anassrs mpteq2dva eqtrd eqtrid ) AHFGBDKU - EZUFUGHUABHUBFBFUHZUIZDUJZUKZUAUHZUIZULZYHUBUHZYCUGZUMZUFUGZUMZUFUGHFBHGD - KUMZUFUGZUMZUFUGAYGCBUAUBYCHURILMNOABIUNYFURUNZFBUOYGURUNPAYSFBAYDBUNZUSZ - YEURUNDJUNYSYDUPQYEDURJUQUTVAFBYFIURVBVCYGVDZAUUBYFVDZFBUOUUCFBYEDVEVFFBY - FVGVHVIPAUCYGVJZBUCUHZUUDUNUUEUDUHZVKZYGUNZUDVLAUUEBUNZUDUUEYGUCWBZVMAUUH - UUIUDUUHUUGYDGUHZVKZVNZYTUUKDUNZUSZUSZGVLFVLAUUIFGBDUUGVOAUUPUUIFGAUUPUUI - AUUPUSUUEYDBUUMUUEYDVNAUUOUUEUUFYDUUKUUJUDWBVPVQAUUMYTUUNVRVSVTWAWCWDWCWE - AKCUNZGDUOFBUOYGCYCWFAUUQFGBDRWGFGBDKCYCYCWHZWIWJABCDEFGHIJKLMNOPQRSTWKWM - AYOYRHUFAYOFBHUBYGYEULZYDYKYCUGZUMZUFUGZUMYRUAFBYNUVBFHYMUFFHWLFUFWLFUBYJ - YLFYGYIFBYFWNFYIWLWOFYHYKYCFYHWLFGBDKWPFYKWLWQWRWQUAUVBWLYHYDVNZYMUVAHUFU - VCUBYJYLUUSUUTUVCYIYEYGYHYDWSWTYHYDYKYCXDXAXBXCAFBUVBYQUUAUVAYPHUFUUAUVAG - DYDUUKYCUGZUMZYPYTUVAUVEVNAYTUVAUBDUUTUMUVEYTUBUUSDUUTYTGUUSDUUKUUSUNZYTU - UNUVFUULYGUNUUOYGYDUUKFWBGWBXEFBDUUKXOXFXGXHXIUBGDUUTUVDGYDYKYCGYDWLFGBDK - XJGYKWLWQUBUVDWLYKUUKYDYCXPXCXKXLUUAGDUVDKAYTUUNUVDKVNZAUUOUSYTUUNUUQUVGA - YTUUNXMAYTUUNXNRFGBDKYCCUURXQXRXSXTYAXBXTYBXBYA $. + ( vm vn vx vy cmpo cgsu co cv csn cxp ciun cima cmpt cvv wcel vsnex xpexg + wral wa sylancr ralrimiva iunexg syl2anc relxp rgenw reliun mpbir a1i cdm + wrel cop wex vex eldm2 eliunxp opth1 ad2antrl simprrl eqeltrd ex exlimdvv + wceq biimtrid exlimdv ssrdv ralrimivva eqid fmpox sylib gsum2d2lem gsum2d + wf nfcv nfiu1 nfima nfmpo1 nfov nfmpt sneq imaeq2d oveq1 mpteq12dv oveq2d + cbvmpt elimasn bitri baib eqrdv mpteq1d nfmpo2 oveq2 eqtrdi adantl simprl + opeliunxp simprr ovmpt4g syl3anc anassrs mpteq2dva eqtrd eqtrid ) AHFGBDK + UEZUFUGHUABHUBFBFUHZUIZDUJZUKZUAUHZUIZULZYHUBUHZYCUGZUMZUFUGZUMZUFUGHFBHG + DKUMZUFUGZUMZUFUGAYGCBUAUBYCHUNILMNOABIUOYFUNUOZFBURYGUNUOPAYSFBAYDBUOZUS + ZYEUNUODJUOYSFUPQYEDUNJUQUTVAFBYFIUNVBVCYGVJZAUUBYFVJZFBURUUCFBYEDVDVEFBY + FVFVGVHPAUCYGVIZBUCUHZUUDUOUUEUDUHZVKZYGUOZUDVLAUUEBUOZUDUUEYGUCVMZVNAUUH + UUIUDUUHUUGYDGUHZVKZWBZYTUUKDUOZUSZUSZGVLFVLAUUIFGBDUUGVOAUUPUUIFGAUUPUUI + AUUPUSUUEYDBUUMUUEYDWBAUUOUUEUUFYDUUKUUJUDVMVPVQAUUMYTUUNVRVSVTWAWCWDWCWE + AKCUOZGDURFBURYGCYCWLAUUQFGBDRWFFGBDKCYCYCWGZWHWIABCDEFGHIJKLMNOPQRSTWJWK + AYOYRHUFAYOFBHUBYGYEULZYDYKYCUGZUMZUFUGZUMYRUAFBYNUVBFHYMUFFHWMFUFWMFUBYJ + YLFYGYIFBYFWNFYIWMWOFYHYKYCFYHWMFGBDKWPFYKWMWQWRWQUAUVBWMYHYDWBZYMUVAHUFU + VCUBYJYLUUSUUTUVCYIYEYGYHYDWSWTYHYDYKYCXAXBXCXDAFBUVBYQUUAUVAYPHUFUUAUVAG + DYDUUKYCUGZUMZYPYTUVAUVEWBAYTUVAUBDUUTUMUVEYTUBUUSDUUTYTGUUSDUUKUUSUOZYTU + UNUVFUULYGUOUUOYGYDUUKFVMGVMXEFBDUUKXOXFXGXHXIUBGDUUTUVDGYDYKYCGYDWMFGBDK + XJGYKWMWQUBUVDWMYKUUKYDYCXKXDXLXMUUAGDUVDKAYTUUNUVDKWBZAUUOUSYTUUNUUQUVGA + YTUUNXNAYTUUNXPRFGBDKYCCUURXQXRXSXTYAXCXTYBXCYA $. gsumcom2.d $e |- ( ph -> D e. Y ) $. gsumcom2.c $e |- ( ph -> @@ -246325,37 +247157,37 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements gsumcom2 $p |- ( ph -> ( G gsum ( j e. A , k e. C |-> X ) ) = ( G gsum ( k e. D , j e. E |-> X ) ) ) $= ( vz vx vy cmpo cgsu co cv csn cxp ciun ccnv cuni cmpt ccom cvv wcel wral - wa snex xpexg sylancr ralrimiva iunexg syl2anc ralrimivva eqid gsum2d2lem - wf fmpox sylib wf1o wrel relxp rgenw reliun mpbir cnvf1o ax-mp relcnv cop - wb wi nfv nfiu1 nfcnv nfel2 nfbi nfim weq eleq1d bibi12d imbi2d opeliunxp - opeq2 3bitr4g vex opelcnv bitr4di chvarfv eqrelrdv f1oeq3d mpbiri gsumf1o - opeq1 cfv csb wceq sneq cnveqd unieqd opswap eqtrdi df-ov eqtr4di mpomptx - fveq2d nfcv nfcsb1v nfxp csbeq1a xpeq12d cbviun mpteq1i nfmpo2 nfov oveq2 - nfmpo1 oveq1 sylan9eq cbvmpox 3eqtr4i f1of syl fmpt sylibr eqidd fveq2 ex - feqmptd fmptcof w3a ovmpt4g 3expia sylcom sylbird 3impib eqcomd mpoeq3dva - 3eqtr4a oveq2d eqtrd ) AJGHBDMUIZUJUKJUUQUFHEHULZUMZIUNZUOZUFULZUMZUPZUQZ - URZUSZUJUKJHGEIMUIZUJUKAGBGULZUMZDUNZUOZCUVAUUQJUVFUTOPQRABKVAUVKUTVAZGBV - BUVLUTVASAUVMGBAUVIBVAZVCUVJUTVADLVAUVMUVIVDTUVJDUTLVEVFVGGBUVKKUTVHVIAMC - VAZHDVBGBVBUVLCUUQVMAUVOGHBDUAVJGHBDMCUUQUUQVKZVNVOZABCDFGHJKLMOPQRSTUAUB - UCVLAUVAUVLUVFVPZUVAUVAUPZUVFVPZUVAVQZUVTUWAUUTVQZHEVBUWBHEUUSIVRVSHEUUTV - TWAUFUVAWBWCAUVLUVSUVAUVFAUGUHUVLUVSUVLVQUVKVQZGBVBUWCGBUVJDVRVSGBUVKVTWA - UVAWDAUGULZUURWEZUVLVAZUWEUVSVAZWFZWGZAUWDUHULZWEZUVLVAZUWKUVSVAZWFZWGHUH - AUWNHAHWHUWLUWMHUWLHWHHUWKUVSHUVAHEUUTWIWJWKWLWMHUHWNZUWHUWNAUWOUWFUWLUWG - UWMUWOUWEUWKUVLUURUWJUWDWSZWOUWOUWEUWKUVSUWPWOWPWQAUVIUURWEZUVLVAZUWQUVSV - AZWFZWGUWIGUGAUWHGAGWHUWFUWGGGUWEUVLGBUVKWIWKUWGGWHWLWMGUGWNZUWTUWHAUXAUW - RUWFUWSUWGUXAUWQUWEUVLUVIUWDUURXIZWOUXAUWQUWEUVSUXBWOWPWQAUWRUURUVIWEUVAV - AZUWSAUVNUURDVAZVCZUUREVAZUVIIVAZVCZUWRUXCUEGBDUURWRHEIUVIWRWTUVIUURUVAGX - AHXAXBXCXDXDXEXFXGZXHAUVGUVHJUJAUFUVAUVEUUQXJZURZHGEIUVIUURUUQUKZUIZUVGUV - HUFUGEUWDUMZHUWDIXKZUNZUOZUXJURUGUHEUXOUWJUWDUUQUKZUIUXKUXMUGUHUFEUXOUXJU - XRUVBUWKXLZUXJUWJUWDWEZUUQXJUXRUXSUVEUXTUUQUXSUVEUWKUMZUPZUQUXTUXSUVDUYBU - XSUVCUYAUVBUWKXMXNXOUWDUWJXPXQYAUWJUWDUUQXRXSXTUFUVAUXQUXJHUGEUUTUXPUGUUT - YBHUXNUXOHUXNYBHUWDIYCZYDHUGWNZUUSUXNIUXOUURUWDXMHUWDIYEZYFYGYHHGUGUHEIUX - LUXOUXRUGIYBUYCUGUXLYBUHUXLYBHUWJUWDUUQHUWJYBGHBDMYIHUWDYBYJGUWJUWDUUQGUW - JYBGHBDMYLGUWDYBYJUYEUYDGUHWNUXLUVIUWDUUQUKUXRUURUWDUVIUUQYKUVIUWJUWDUUQY - MYNYOYPAUFUGUVAUVLUVEUWDUUQXJUXJUVFUUQAUVAUVLUVFVMZUVEUVLVAUFUVAVBAUVRUYF - UXIUVAUVLUVFYQYRUFUVAUVLUVEUVFUVFVKYSYTAUVFUUAAUGUVLCUUQUVQUUDUWDUVEUUQUU - BUUEAHGEIMUXLAUXFUXGUUFUXLMAUXFUXGUXLMXLZAUXHUXEUYGUEAUXEUVOUYGAUXEUVOUAU - UCUVNUXDUVOUYGGHBDMUUQCUVPUUGUUHUUIUUJUUKUULUUMUUNUUOUUP $. + wa vsnex xpexg sylancr ralrimiva iunexg syl2anc wf ralrimivva fmpox sylib + eqid gsum2d2lem wf1o wrel relxp rgenw reliun mpbir cnvf1o ax-mp relcnv wb + cop nfv nfiu1 nfcnv nfel2 nfbi nfim weq opeq2 eleq1d bibi12d imbi2d opeq1 + wi opeliunxp 3bitr4g vex opelcnv bitr4di chvarfv eqrelrdv f1oeq3d gsumf1o + mpbiri cfv csb wceq sneq cnveqd unieqd opswap eqtrdi fveq2d df-ov eqtr4di + mpomptx nfcv nfcsb1v nfxp csbeq1a xpeq12d cbviun nfmpo2 nfov nfmpo1 oveq2 + mpteq1i oveq1 sylan9eq cbvmpox 3eqtr4i f1of syl fmpt sylibr eqidd feqmptd + fmptcof w3a ovmpt4g 3expia sylcom sylbird 3impib eqcomd mpoeq3dva 3eqtr4a + fveq2 ex oveq2d eqtrd ) AJGHBDMUIZUJUKJUUQUFHEHULZUMZIUNZUOZUFULZUMZUPZUQ + ZURZUSZUJUKJHGEIMUIZUJUKAGBGULZUMZDUNZUOZCUVAUUQJUVFUTOPQRABKVAUVKUTVAZGB + VBUVLUTVASAUVMGBAUVIBVAZVCUVJUTVADLVAUVMGVDTUVJDUTLVEVFVGGBUVKKUTVHVIAMCV + AZHDVBGBVBUVLCUUQVJAUVOGHBDUAVKGHBDMCUUQUUQVNZVLVMZABCDFGHJKLMOPQRSTUAUBU + CVOAUVAUVLUVFVPZUVAUVAUPZUVFVPZUVAVQZUVTUWAUUTVQZHEVBUWBHEUUSIVRVSHEUUTVT + WAUFUVAWBWCAUVLUVSUVAUVFAUGUHUVLUVSUVLVQUVKVQZGBVBUWCGBUVJDVRVSGBUVKVTWAU + VAWDAUGULZUURWFZUVLVAZUWEUVSVAZWEZWSZAUWDUHULZWFZUVLVAZUWKUVSVAZWEZWSHUHA + UWNHAHWGUWLUWMHUWLHWGHUWKUVSHUVAHEUUTWHWIWJWKWLHUHWMZUWHUWNAUWOUWFUWLUWGU + WMUWOUWEUWKUVLUURUWJUWDWNZWOUWOUWEUWKUVSUWPWOWPWQAUVIUURWFZUVLVAZUWQUVSVA + ZWEZWSUWIGUGAUWHGAGWGUWFUWGGGUWEUVLGBUVKWHWJUWGGWGWKWLGUGWMZUWTUWHAUXAUWR + UWFUWSUWGUXAUWQUWEUVLUVIUWDUURWRZWOUXAUWQUWEUVSUXBWOWPWQAUWRUURUVIWFUVAVA + ZUWSAUVNUURDVAZVCZUUREVAZUVIIVAZVCZUWRUXCUEGBDUURWTHEIUVIWTXAUVIUURUVAGXB + HXBXCXDXEXEXFXGXIZXHAUVGUVHJUJAUFUVAUVEUUQXJZURZHGEIUVIUURUUQUKZUIZUVGUVH + UFUGEUWDUMZHUWDIXKZUNZUOZUXJURUGUHEUXOUWJUWDUUQUKZUIUXKUXMUGUHUFEUXOUXJUX + RUVBUWKXLZUXJUWJUWDWFZUUQXJUXRUXSUVEUXTUUQUXSUVEUWKUMZUPZUQUXTUXSUVDUYBUX + SUVCUYAUVBUWKXMXNXOUWDUWJXPXQXRUWJUWDUUQXSXTYAUFUVAUXQUXJHUGEUUTUXPUGUUTY + BHUXNUXOHUXNYBHUWDIYCZYDHUGWMZUUSUXNIUXOUURUWDXMHUWDIYEZYFYGYLHGUGUHEIUXL + UXOUXRUGIYBUYCUGUXLYBUHUXLYBHUWJUWDUUQHUWJYBGHBDMYHHUWDYBYIGUWJUWDUUQGUWJ + YBGHBDMYJGUWDYBYIUYEUYDGUHWMUXLUVIUWDUUQUKUXRUURUWDUVIUUQYKUVIUWJUWDUUQYM + YNYOYPAUFUGUVAUVLUVEUWDUUQXJUXJUVFUUQAUVAUVLUVFVJZUVEUVLVAUFUVAVBAUVRUYFU + XIUVAUVLUVFYQYRUFUVAUVLUVEUVFUVFVNYSYTAUVFUUAAUGUVLCUUQUVQUUBUWDUVEUUQUUM + UUCAHGEIMUXLAUXFUXGUUDUXLMAUXFUXGUXLMXLZAUXHUXEUYGUEAUXEUVOUYGAUXEUVOUAUU + NUVNUXDUVOUYGGHBDMUUQCUVPUUEUUFUUGUUHUUIUUJUUKUULUUOUUP $. $} ${ @@ -247909,110 +248741,110 @@ G dom DProd ( j e. 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(Contributed by Mario Carneiro, 25-Apr-2016.) $) @@ -250092,7 +250924,7 @@ elements and the subgroup containing only the identity ( ~ simpgnsgbid ). addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and ~ unitgrp shows that we get a group if we restrict to the elements that have - inverses. This allows us to formalize such notions as "the multiplication + inverses. This allows to formalize such notions as "the multiplication operation of a ring is a monoid" ( ~ ringmgp ) or "the multiplicative identity" in terms of the identity of a monoid ( ~ df-ur ). (Contributed by Mario Carneiro, 21-Dec-2014.) $) @@ -250258,21 +251090,58 @@ monoid has the same (if any) scalars as the original ring. Mostly to $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Ring unit + Ring unity (multiplicative identity) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + In Wikipedia "Identity element", see + ~ https://en.wikipedia.org/wiki/Identity_element (18-Jan-2025): + "... an identity with respect to multiplication is called a multiplicative + identity (often denoted as 1). ... The distinction between additive and + multiplicative identity is used most often for sets that support both binary + operations, such as rings, integral domains, and fields. The multiplicative + identity is often called unity in the latter context (a ring with unity). + This should not be confused with a unit in ring theory, which is any element + having a multiplicative inverse. By its own definition, unity itself is + necessarily a unit." + + Calling the multiplicative identity of a ring a unity is taken from the + definition of a ring with unity in section 17.3 of [BeauregardFraleigh] + p. 135, "A ring ( R , + , . ) is a ring with unity if R is not the zero ring + and ( R , . ) is a monoid. In this case, the identity element of ( R , . ) + is denoted by 1 and is called the unity of R." This definition of a "ring + with unity" corresponds to our definition of a unital ring (see ~ df-ring ). + + Some authors call the multiplicative identity "unit" or "unit element" (for + example in section I, 2.2 of [BourbakiAlg1] p. 14, definition in section 1.3 + of [Hall] p. 4, or in section I, 1 of [Lang] p. 3), whereas other authors use + the term "unit" for an element having a multiplicative inverse (for example + in section 17.3 of [BeauregardFraleigh] p. 135, in definition in [Roman] + p. 26, or even in section II, 1 of [Lang] p. 84). Sometimes, the + multiplicative identity is simply called "one" (see, for example, chapter 8 + in [Schechter] p. 180). + + To avoid this ambiguity of the term "unit", also mentioned in Wikipedia, we + call the multiplicative identity of a structure with a multiplication + (usually a ring) a "ring unity", or straightly "multiplicative identity". + + The term "unit" will be used for an element having a multiplicative inverse + (see ~ df-unit ), and we have "the ring unity is a unit", see ~ 1unit . + $) $c 1r $. - $( Extend class notation with ring unit. $) + $( Extend class notation with ring unity. $) cur $a class 1r $. $( Define the multiplicative identity, i.e., the monoid identity ( ~ df-0g ) of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This - definition works by transferring the multiplicative operation from the - ` .r ` slot to the ` +g ` slot and then looking at the element which is - then the ` 0g ` element, that is an identity with respect to the operation - which started out in the ` .r ` slot. + multiplicative identity is also called "ring unity" or "unity element". + + This definition works by transferring the multiplicative operation from + the ` .r ` slot to the ` +g ` slot and then looking at the element which + is then the ` 0g ` element, that is an identity with respect to the + operation which started out in the ` .r ` slot. See also ~ dfur2 , which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under @@ -250308,9 +251177,9 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This $( --.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Semirings --.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $c SRing $. @@ -250442,7 +251311,7 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This UAUB $. $d u x B $. $d u x R $. $d u x .x. $. - $( The unit element of a semiring is unique. (Contributed by NM, + $( The unity element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) $) srgideu $p |- ( R e. SRing -> @@ -250492,7 +251361,7 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This ${ srgidcl.b $e |- B = ( Base ` R ) $. srgidcl.u $e |- .1. = ( 1r ` R ) $. - $( The unit element of a semiring belongs to the base set of the semiring. + $( The unity element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) $) srgidcl $p |- ( R e. SRing -> .1. e. B ) $= @@ -250524,14 +251393,14 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This ringidval sylan ) BIJBKLZMJEAJDECNEOEDCNEOPBUDUDQZRACUDEDABUDUEFSBCUDUEGT BDUDUEHUBUAUC $. - $( The unit element of a semiring is a left multiplicative identity. + $( The unity element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.) $) srglidm $p |- ( ( R e. SRing /\ X e. B ) -> ( .1. .x. X ) = X ) $= ( csrg wcel wa co wceq srgidmlem simpld ) BIJEAJKDECLEMEDCLEMABCDEFGHNO $. - $( The unit element of a semiring is a right multiplicative identity. + $( The unity element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.) $) srgridm $p |- ( ( R e. SRing /\ X e. B ) -> ( X .x. .1. ) = X ) $= @@ -250824,10 +251693,10 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This ` ( N _C k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ) `. Note that the binomial theorem also holds in the non-unital case (that is, in - a "rg") and actually, the additive unit is not needed in its proof either. - Therefore, it can be proven in even more general cases. An example is the - "rg" (resp. "rg without a zero") of integrable nonnegative (resp. positive) - functions on ` RR `. + a "rg") and actually, the additive identity is not needed in its proof + either. Therefore, it can be proven in even more general cases. An example + is the "rg" (resp. "rg without a zero") of integrable nonnegative (resp. + positive) functions on ` RR `. Special cases of the binomial theorem are ~ csrgbinom (binomial theorem for commutative semirings) and ~ crngbinom (binomial theorem for commutative @@ -251113,7 +251982,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be isring.g $e |- G = ( mulGrp ` R ) $. isring.p $e |- .+ = ( +g ` R ) $. isring.t $e |- .x. = ( .r ` R ) $. - $( The predicate "is a (unital) ring". Definition of ring with unit in + $( The predicate "is a (unital) ring". Definition of "ring with unit" in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) $) isring $p |- ( R e. Ring <-> ( R e. Grp /\ G e. Mnd @@ -251206,12 +252075,12 @@ the additive structure must be abelian (see ~ ringcom ), care must be ${ $d x y z B $. $d x y z R $. $d x y z .x. $. $d x y z X $. $d x y z Y $. $d x y z .+ $. $d x y z Z $. - ringi.b $e |- B = ( Base ` R ) $. - ringi.p $e |- .+ = ( +g ` R ) $. - ringi.t $e |- .x. = ( .r ` R ) $. + ringdilem.b $e |- B = ( Base ` R ) $. + ringdilem.p $e |- .+ = ( +g ` R ) $. + ringdilem.t $e |- .x. = ( .r ` R ) $. $( Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) - ringi $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) + ringdilem $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) /\ ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) ) $= ( vx vy vz wcel wa co wceq cv wral rsp crg w3a cgrp cmgp cmnd eqid isring @@ -251258,8 +252127,8 @@ the additive structure must be abelian (see ~ ringcom ), care must be UAUB $. $d u x B $. $d u x R $. $d u x .x. $. - $( The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) - (Revised by Mario Carneiro, 6-Jan-2015.) $) + $( The unity element of a ring is unique. (Contributed by NM, + 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) ringideu $p |- ( R e. Ring -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) ) $= ( crg wcel cmgp cfv cmnd cv co wceq wa wral wreu eqid ringmgp mndideu syl @@ -251275,22 +252144,22 @@ the additive structure must be abelian (see ~ ringcom ), care must be (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) $) ringdi $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) ) $= - ( crg wcel w3a wa co wceq ringi simpld ) CKLEALFALGALMNEFGBODOEFDOEGDOZBO - PEFBOGDOSFGDOBOPABCDEFGHIJQR $. + ( crg wcel w3a wa co wceq ringdilem simpld ) CKLEALFALGALMNEFGBODOEFDOEGD + OZBOPEFBOGDOSFGDOBOPABCDEFGHIJQR $. $( Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) $) ringdir $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) $= - ( crg wcel w3a wa co wceq ringi simprd ) CKLEALFALGALMNEFGBODOEFDOEGDOZBO - PEFBOGDOSFGDOBOPABCDEFGHIJQR $. + ( crg wcel w3a wa co wceq ringdilem simprd ) CKLEALFALGALMNEFGBODOEFDOEGD + OZBOPEFBOGDOSFGDOBOPABCDEFGHIJQR $. $} ${ ringidcl.b $e |- B = ( Base ` R ) $. ringidcl.u $e |- .1. = ( 1r ` R ) $. - $( The unit element of a ring belongs to the base set of the ring. + $( The unity element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) $) ringidcl $p |- ( R e. Ring -> .1. e. B ) $= @@ -251320,13 +252189,13 @@ the additive structure must be abelian (see ~ ringcom ), care must be ringidval sylan ) BIJBKLZMJEAJDECNEOEDCNEOPBUDUDQZRACUDEDABUDUEFSBCUDUEGT BDUDUEHUBUAUC $. - $( The unit element of a ring is a left multiplicative identity. + $( The unity element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) $) ringlidm $p |- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X ) $= ( crg wcel wa co wceq ringidmlem simpld ) BIJEAJKDECLEMEDCLEMABCDEFGHNO $. - $( The unit element of a ring is a right multiplicative identity. + $( The unity element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) $) ringridm $p |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X ) $= ( crg wcel wa co wceq ringidmlem simprd ) BIJEAJKDECLEMEDCLEMABCDEFGHNO @@ -251375,8 +252244,8 @@ the additive structure must be abelian (see ~ ringcom ), care must be rngo2times.u $e |- .1. = ( 1r ` R ) $. $( A ring element plus itself is two times the element. "Two" in an - arbitrary unital ring is the sum of the unit with itself. (Contributed - by AV, 24-Aug-2021.) $) + arbitrary unital ring is the sum of the unity element with itself. + (Contributed by AV, 24-Aug-2021.) $) rngo2times $p |- ( ( R e. Ring /\ A e. B ) -> ( A .+ A ) = ( ( .1. .+ .1. ) .x. A ) ) $= ( crg wcel wa co ringlidm eqcomd oveq12d wceq simpl ringidcl adantr simpr @@ -251641,7 +252510,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be ringinvnzdiv.r $e |- ( ph -> R e. Ring ) $. ringinvnzdiv.x $e |- ( ph -> X e. B ) $. ringinvnzdiv.a $e |- ( ph -> E. a e. B ( a .x. X ) = .1. ) $. - $( In a unitary ring, a left invertible element is different from zero iff + $( In a unital ring, a left invertible element is different from zero iff ` .1. =/= .0. ` . (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) $) ringinvnz1ne0 $p |- ( ph -> ( X =/= .0. <-> .1. =/= .0. ) ) $= @@ -251654,7 +252523,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be $d Y a $. ringinvnzdiv.y $e |- ( ph -> Y e. B ) $. - $( In a unitary ring, a left invertible element is not a zero divisor. + $( In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) $) ringinvnzdiv $p |- ( ph -> ( ( X .x. Y ) = .0. <-> Y = .0. ) ) $= @@ -252109,7 +252978,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be prds1.i $e |- ( ph -> I e. W ) $. prds1.s $e |- ( ph -> S e. V ) $. prds1.r $e |- ( ph -> R : I --> Ring ) $. - $( Value of the ring unit in a structure family product. (Contributed by + $( Value of the ring unity in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.) $) prds1 $p |- ( ph -> ( 1r o. R ) = ( 1r ` Y ) ) $= ( vx vy c0g cmgp ccom cfv cur eqid crg cprds co cmnd cres wf mgpf sylancr @@ -252134,7 +253003,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be ${ pws1.y $e |- Y = ( R ^s I ) $. pws1.o $e |- .1. = ( 1r ` R ) $. - $( Value of the ring unit in a structure power. (Contributed by Mario + $( Value of the ring unity in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.) $) pws1 $p |- ( ( R e. Ring /\ I e. V ) -> ( I X. { .1. } ) = ( 1r ` Y ) ) $= ( crg wcel cur cfv csca csn cxp ccom eqid cvv wfn c0g cmgp wa cprds simpr @@ -252546,13 +253415,13 @@ the additive structure must be abelian (see ~ ringcom ), care must be $( Introduce new constant symbols. $) $c ||r $. $( Ring divisibility relation $) - $c Unit $. $( Ring unit $) + $c Unit $. $( Units in a ring $) $c Irred $. $( Ring irreducibles $) $( Ring divisibility relation. $) cdsr $a class ||r $. - $( Ring unit. $) + $( Units in a ring. $) cui $a class Unit $. $( Ring irreducibles. $) @@ -252966,7 +253835,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be unitinvcl.3 $e |- .x. = ( .r ` R ) $. unitinvcl.4 $e |- .1. = ( 1r ` R ) $. - $( A unit times its inverse is the identity. (Contributed by Mario + $( A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.) $) unitlinv $p |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) .x. X ) = .1. ) $= ( crg wcel wa cfv co cmgp cress wceq eqid cvv c0g cgrp unitgrp unitgrpbas @@ -252975,7 +253844,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be VAEFVBACVAGVCUDCTLBVAUENRCAUFGUGCBUTVATVCABUTUTSIUHUIUJVBSACVAEGVCHUKUOUL UQDVBRURACDVAGVCJUMUNUP $. - $( A unit times its inverse is the identity. (Contributed by Mario + $( A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.) $) unitrinv $p |- ( ( R e. Ring /\ X e. U ) -> ( X .x. ( I ` X ) ) = .1. ) $= ( crg wcel wa cfv co cmgp cress wceq eqid cvv c0g cgrp unitgrp unitgrpbas @@ -252988,7 +253857,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be ${ 1rinv.1 $e |- I = ( invr ` R ) $. 1rinv.2 $e |- .1. = ( 1r ` R ) $. - $( The inverse of the identity is the identity. (Contributed by Mario + $( The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015.) $) 1rinv $p |- ( R e. Ring -> ( I ` .1. ) = .1. ) $= ( crg wcel cfv cmulr cbs wceq cui 1unit ringinvcl mpdan ringlidm unitrinv @@ -253564,9 +254433,10 @@ irreducible elements (see ~ df-irred ). (Contributed by Mario Carneiro, ${ $d F f $. $d R f r s $. $d S f r s $. $d V f r s $. $d W f r s $. - $( An isomorphism of rings is a homomorphism whose converse is also a - homomorphism . (Contributed by AV, 22-Oct-2019.) $) - isrim0 $p |- ( ( R e. V /\ S e. W ) -> ( F e. ( R RingIso S ) + $( Obsolete version of ~ isrim0 as of 12-Jan-2025. (Contributed by AV, + 22-Oct-2019.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + isrim0OLD $p |- ( ( R e. V /\ S e. W ) -> ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) ) ) $= ( vf vr vs wcel wa crs co cv ccnv crh crab cvv wceq a1i adantl cmpo rabex df-rngiso oveq12 ancoms eleq2d rabeqbidv adantr ovmpod cnveq eleq1d elrab @@ -253581,6 +254451,20 @@ irreducible elements (see ~ df-irred ). (Contributed by Mario Carneiro, rimrcl $p |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) ) $= ( vr vs vf cvv cv ccnv crh co wcel crab crs df-rngiso elmpocl ) DEGGFHIEH ZDHZJKLFRQJKMABNCFEDOP $. + + $( A ring isomorphism is a homomorphism whose converse is also a + homomorphism. Compare ~ isgim2 . (Contributed by AV, 22-Oct-2019.) + Remove sethood antecedent. (Revised by SN, 10-Jan-2025.) $) + isrim0 $p |- ( F e. ( R RingIso S ) <-> + ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) ) $= + ( vf vr vs crs co wcel cvv wa crh ccnv crg elexd cv crab wceq a1i oveq12 + rimrcl rhmrcl1 rhmrcl2 jca adantr df-rngiso adantl ancoms rabeqbidv simpl + cmpo eleq2d simpr ovex rabex ovmpod cnveq eleq1d elrab bitrdi pm5.21nii ) + CABGHZIZAJIZBJIZKZCABLHZIZCMZBALHZIZKZABCUAVHVFVKVHVDVEVHANABCUBOVHBNABCU + COUDUEVFVCCDPZMZVJIZDVGQZIVLVFVBVPCVFEFABJJVNFPZEPZLHZIZDVRVQLHZQZVPGJGEF + JJWBUKRVFDFEUFSVFVRARZVQBRZKZKZVTVODWAVGWEWAVGRVFVRAVQBLTUGWFVSVJVNWEVSVJ + RZVFWDWCWGVQBVRALTUHUGULUIVDVEUJVDVEUMVPJIVFVODVGABLUNUOSUPULVOVKDCVGVMCR + VNVIVJVMCUQURUSUTVA $. $} $( A ring homomorphism is an additive group homomorphism. (Contributed by @@ -253692,32 +254576,45 @@ irreducible elements (see ~ df-irred ). (Contributed by Mario Carneiro, ABCDEFGVCSVDWHBAVKVLBAVKVBVIBADCVKGFVCVEVDABEVFVAVH $. $( An isomorphism of rings is a bijective homomorphism. (Contributed by - AV, 22-Oct-2019.) $) - isrim $p |- ( ( R e. V /\ S e. W ) -> ( F e. ( R RingIso S ) + AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, + 12-Jan-2025.) $) + isrim $p |- ( F e. ( R RingIso S ) + <-> ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) ) $= + ( crs co wcel crh ccnv wa wf1o isrim0 rhmf1o bicomd pm5.32i bitri ) ECDHI + JECDKIJZELDCKIJZMTABENZMCDEOTUAUBTUBUAABCDEFGPQRS $. + + $( Obsolete version of ~ isrim as of 12-Jan-2025. (Contributed by AV, + 22-Oct-2019.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + isrimOLD $p |- ( ( R e. V /\ S e. W ) -> ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) ) ) $= - ( wcel wa crs co crh ccnv wf1o isrim0 wb wi rhmf1o bicomd pm5.32d bitrd - a1i ) CFJDGJKZECDLMJECDNMJZEODCNMJZKUFABEPZKCDEFGQUEUFUGUHUFUGUHRSUEUFUHU - GABCDEHITUAUDUBUC $. + ( wcel wa crs co crh ccnv wf1o isrim0OLD wb wi rhmf1o bicomd a1i pm5.32d + bitrd ) CFJDGJKZECDLMJECDNMJZEODCNMJZKUFABEPZKCDEFGQUEUFUGUHUFUGUHRSUEUFU + HUGABCDEHITUAUBUCUD $. $( An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.) $) rimf1o $p |- ( F e. ( R RingIso S ) -> F : B -1-1-onto-> C ) $= - ( cvv wcel wa crs co wf1o rimrcl crh isrim simpr syl6bi mpcom ) CHIDHIJZE - CDKLIZABEMZCDENTUAECDOLIZUBJUBABCDEHHFGPUCUBQRS $. + ( crs co wcel crh wf1o isrim simprbi ) ECDHIJECDKIJABELABCDEFGMN $. - $( An isomorphism of rings is a homomorphism. (Contributed by AV, - 22-Oct-2019.) $) - rimrhm $p |- ( F e. ( R RingIso S ) -> F e. ( R RingHom S ) ) $= - ( cvv wcel wa crs co crh rimrcl wf1o isrim simpl syl6bi mpcom ) CHIDHIJZE - CDKLIZECDMLIZCDENTUAUBABEOZJUBABCDEHHFGPUBUCQRS $. + $( Obsolete version of ~ rimrhm as of 12-Jan-2025. (Contributed by AV, + 22-Oct-2019.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + rimrhmOLD $p |- ( F e. ( R RingIso S ) -> F e. ( R RingHom S ) ) $= + ( crs co wcel crh wf1o isrim simplbi ) ECDHIJECDKIJABELABCDEFGMN $. $} + $( A ring isomorphism is a homomorphism. Compare ~ gimghm . (Contributed by + AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.) $) + rimrhm $p |- ( F e. ( R RingIso S ) -> F e. ( R RingHom S ) ) $= + ( crs co wcel crh ccnv isrim0 simplbi ) CABDEFCABGEFCHBAGEFABCIJ $. + $( An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019.) $) rimgim $p |- ( F e. ( R RingIso S ) -> F e. ( R GrpIso S ) ) $= - ( crs co wcel cghm cbs cfv wf1o cgim crh eqid rimrhm rhmghm rimf1o sylanbrc - syl isgim ) CABDEFZCABGEFZAHIZBHIZCJCABKEFTCABLEFUAUBUCABCUBMZUCMZNABCORUBU - CABCUDUEPUBUCABCUDUESQ $. + ( crs co wcel cghm cbs cfv wf1o cgim crh rimrhm rhmghm eqid rimf1o sylanbrc + syl isgim ) CABDEFZCABGEFZAHIZBHIZCJCABKEFTCABLEFUAABCMABCNRUBUCABCUBOZUCOZ + PUBUCABCUDUESQ $. $( The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) $) @@ -253887,11 +254784,10 @@ irreducible elements (see ~ df-irred ). (Contributed by Mario Carneiro, JM's mathbox. (Contributed by AV, 24-Dec-2019.) $) brric2 $p |- ( R ~=r S <-> ( ( R e. Ring /\ S e. Ring ) /\ E. f f e. ( R RingIso S ) ) ) $= - ( cric wbr crs co c0 wne cv wcel wex crg wa brric cvv crh cmgp cfv eqid - n0 rimrcl ccnv isrim0 cghm cmhm isrhm simplbi adantr syl6bi mpcom exlimiv - pm4.71ri 3bitri ) ABDEABFGZHICJZUOKZCLZAMKBMKNZURNABOCUOUAURUSUQUSCAPKBPK - NZUQUSABUPUBUTUQUPABQGKZUPUCBAQGKZNUSABUPPPUDVAUSVBVAUSUPABUEGKUPARSZBRSZ - UFGKNABUPVCVDVCTVDTUGUHUIUJUKULUMUN $. + ( cric wbr crs co c0 wne cv wcel wex crg wa brric n0 crh cmgp cfv eqid + rimrhm cghm cmhm isrhm simplbi syl exlimiv pm4.71ri 3bitri ) ABDEABFGZHIC + JZUJKZCLZAMKBMKNZUMNABOCUJPUMUNULUNCULUKABQGKZUNABUKUAUOUNUKABUBGKUKARSZB + RSZUCGKNABUKUPUQUPTUQTUDUEUFUGUHUI $. $( If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.) $) @@ -253901,6 +254797,82 @@ irreducible elements (see ~ df-irred ). (Contributed by Mario Carneiro, ABUAPSQT $. $} + ${ + $d c y A $. $d c y B $. $d c y F $. $d c R $. $d c y S $. $d c X $. + $d c .|| $. + rhmdvdsr.x $e |- X = ( Base ` R ) $. + rhmdvdsr.m $e |- .|| = ( ||r ` R ) $. + rhmdvdsr.n $e |- ./ = ( ||r ` S ) $. + $( A ring homomorphism preserves the divisibility relation. (Contributed + by Thierry Arnoux, 22-Oct-2017.) $) + rhmdvdsr $p |- ( ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) + /\ A .|| B ) -> ( F ` A ) ./ ( F ` B ) ) $= + ( vy vc co wcel wa cfv wceq wrex syl2anc crh w3a wbr cbs cv simpl1 simpl2 + cmulr eqid rhmf ffvelcdmda simpll1 ralrimiva adantr rhmmul syl3anc dvdsr2 + wral simpr biimpac r19.29 simpl fveq2d eqtr3d reximi syl eqeq1d rexlimivw + oveq1 rspcev dvdsr sylanbrc ) GEFUANOZAHOZBHOZUBZABCUCZPZAGQZFUDQZOZLUEZV + SFUHQZNZBGQZRZLVTSZVSWEDUCVRVMVNWAVMVNVOVQUFVMVNVOVQUGZVMHVTAGHVTEFGIVTUI + ZUJZUKTVRMUEZGQZVTOZWLVSWCNZWERZPZMHSZWGVRWMMHURWOMHSZWQVRWMMHVRWKHOZPZVM + WSWMVMVNVOVQWSULZVRWSUSZVMHVTWKGWJUKTUMVRWKAEUHQZNZGQZWNRZMHURZXDBRZMHSZW + RVRXFMHWTVMWSVNXFXAXBVRVNWSWHUNWKAEFXCWCGHIXCUIZWCUIZUOUPUMVRVQVNXIVPVQUS + WHVNVQXIMHCEXCABIJXJUQUTTXGXIPXFXHPZMHSWRXFXHMHVAXLWOMHXLXEWNWEXFXHVBXLXD + BGXFXHUSVCVDVEVFTWMWOMHVATWPWGMHWFWOLWLVTWBWLRWDWNWEWBWLVSWCVIVGVJVHVFLVT + DFWCVSWEWIKXKVKVL $. + $} + + ${ + $d x y F $. $d x y R $. $d x y S $. + $( A ring homomorphism is also a ring homomorphism for the opposite rings. + (Contributed by Thierry Arnoux, 27-Oct-2017.) $) + rhmopp $p |- ( F e. ( R RingHom S ) + -> F e. ( ( oppR ` R ) RingHom ( oppR ` S ) ) ) $= + ( vx vy co wcel coppr cfv cbs cmulr cur eqid opprringb sylib oppr1 eqcomi + crg cv wa rhmrcl1 rhmrcl2 rhm1 wceq simpl simprr opprbas eleqtrrdi simprl + crh rhmmul syl3anc opprmul fveq2i 3eqtr4g cgrp wf cplusg wral ringgrp syl + cghm rhmf rhmghm ad2antrr simplr simpr ghmlin ralrimiva jca jca31 oppradd + isghm sylibr isrhm2d ) CABUJFGZDEAHIZJIZVQBHIZVQKIZVSKIZVQLIZCVSLIZVRMWBM + WCMVTMZWAMZVPARGVQRGZABCUAAVQVQMZNOZVPBRGVSRGZABCUBBVSVSMZNOZABWBCWCALIZW + BAWLVQWGWLMPQBLIZWCBWMVSWJWMMPQUCVPDSZVRGZESZVRGZTZTZWPWNAKIZFZCIZWPCIZWN + CIZBKIZFZWNWPVTFZCIXDXCWAFWSVPWPAJIZGZWNXHGZXBXFUDVPWRUEWSWPVRXHVPWOWQUFX + HAVQWGXHMZUGZUHWSWNVRXHVPWOWQUIXLUHWPWNABWTXECXHXKWTMZXEMZUKULXGXACXHAVTW + TVQWNWPXKXMWGWDUMUNBJIZBWAXEVSXDXCXOMZXNWJWEUMUOVPVQUPGZVSUPGZTXHXOCUQZWN + WPAURIZFCIXDXCBURIZFUDZEXHUSZDXHUSZTZTCVQVSVBFGVPXQXRYEVPWFXQWHVQUTVAVPWI + XRWKVSUTVAVPXSYDXHXOABCXKXPVCVPYCDXHVPXJTZYBEXHYFXITCABVBFGZXJXIYBVPYGXJX + IABCVDVEVPXJXIVFYFXIVGXTYAABWNCWPXHXKXTMZYAMZVHULVIVIVJVKEDXTYAVQVSCXHXOX + LXOBVSWJXPUGXTAVQWGYHVLYABVSWJYIVLVMVNVO $. + $} + + $( Ring homomorphisms preserve unit elements. (Contributed by Thierry + Arnoux, 23-Oct-2017.) $) + elrhmunit $p |- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) + -> ( F ` A ) e. ( Unit ` S ) ) $= + ( crh co wcel cui cfv wa cur cdsr wbr coppr isunit rhmdvdsr syl31anc adantr + eqid wb cbs simpl unitss simpr sselid crg rhmrcl1 ringidcl 3syl simpld rhm1 + sylib breq2d mpbid rhmopp simprd opprbas sylanbrc ) DBCEFGZABHIZGZJZADIZCKI + ZCLIZMZVCVDCNIZLIZMZVCCHIZGVBVCBKIZDIZVEMZVFVBUSABUAIZGZVKVNGZAVKBLIZMZVMUS + VAUBZVBUTVNAVNBUTVNSZUTSZUCUSVAUDZUEZVBUSBUFGVPVSBCDUGVNBVKVTVKSZUHUIZVBVRA + VKBNIZLIZMZVBVAVRWHJWBVQBWFUTVKWGAWAWDVQSZWFSZWGSZOULZUJAVKVQVEBCDVNVTWIVES + ZPQUSVMVFTVAUSVLVDVCVEBCVKDVDWDVDSZUKZUMRUNVBVCVLVHMZVIVBDWFVGEFGZVOVPWHWPU + SWQVABCDUORWCWEVBVRWHWLUPAVKWGVHWFVGDVNVNBWFWJVTUQWKVHSZPQUSWPVITVAUSVLVDVC + VHWOUMRUNVECVGVJVDVHVCVJSWNWMVGSWROUR $. + + $( Ring homomorphisms preserve the inverse of unit elements. (Contributed by + Thierry Arnoux, 23-Oct-2017.) $) + rhmunitinv $p |- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) + -> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) $= + ( co wcel cui cfv cinvr cmulr wceq cur eqid unitlinv sylan unitinvcl sselid + crg adantr elrhmunit crh rhmrcl1 fveq2d cbs simpl unitss simpr syl3anc rhm1 + wa rhmmul 3eqtr3d rhmrcl2 syl2anc eqtr4d cmgp cress cgrp unitgrp syl syldan + unitgrpbas cvv cplusg fvex mgpplusg ressplusg ax-mp grprcan syl13anc mpbid + wb ) DBCUAEFZABGHZFZUJZABIHZHZDHZADHZCJHZEZVTCIHZHZVTWAEZKZVSWDKZVPWBCLHZWE + VPVRABJHZEZDHZBLHZDHZWBWHVPWJWLDVMBRFZVOWJWLKBCDUBZBWIVNWLVQAVNMZVQMZWIMZWL + MZNOUCVPVMVRBUDHZFAWTFWKWBKVMVOUEVPVNWTVRWTBVNWTMZWPUFZVMWNVOVRVNFZWOBVNVQA + WPWQPOZQVPVNWTAXBVMVOUGQVRABCWIWADWTXAWRWAMZUKUHVMWMWHKVOBCWLDWHWSWHMZUISUL + VPCRFZVTCGHZFZWEWHKVMXGVOBCDUMZSZABCDTZCWAXHWHWCVTXHMZWCMZXEXFNUNUOVPCUPHZX + HUQEZURFZVSXHFZWDXHFZXIWFWGVLVMXQVOVMXGXQXJCXHXPXMXPMZUSUTSVMVOXCXRXDVRBCDT + VAVPXGXIXSXKXLCXHWCVTXMXNPUNXLXHWAXPVSWDVTCXHXPXMXTVBXHVCFWAXPVDHKCGVEXHWAX + OXPVCXTCWAXOXOMXEVFVGVHVIVJVK $. + $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# @@ -253970,15 +254942,35 @@ irreducible elements (see ~ df-irred ). (Contributed by Mario Carneiro, ( cdr wcel crg cui cfv cbs c0g csn cdif wceq eqid isdrng simplbi ) ABCADCAE FZAGFZAHFZIJKPAOQPLOLQLMN $. - $( A division ring is a group. (Contributed by NM, 8-Sep-2011.) $) + ${ + drngringd.1 $e |- ( ph -> R e. DivRing ) $. + $( A division ring is a ring. (Contributed by SN, 16-May-2024.) $) + drngringd $p |- ( ph -> R e. Ring ) $= + ( cdr wcel crg drngring syl ) ABDEBFECBGH $. + + $( A division ring is a group (deduction form). (Contributed by SN, + 16-May-2024.) $) + drnggrpd $p |- ( ph -> R e. Grp ) $= + ( drngringd ringgrpd ) ABABCDE $. + $} + + $( A division ring is a group (closed form). (Contributed by NM, + 8-Sep-2011.) $) drnggrp $p |- ( R e. DivRing -> R e. Grp ) $= - ( cdr wcel crg cgrp drngring ringgrp syl ) ABCADCAECAFAGH $. + ( cdr wcel id drnggrpd ) ABCZAFDE $. $( A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) $) isfld $p |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) ) $= ( cdr ccrg cfield df-field elin2 ) ABCDEF $. + ${ + fldcrngd.1 $e |- ( ph -> R e. Field ) $. + $( A field is a commutative ring. (Contributed by SN, 23-Nov-2024.) $) + fldcrngd $p |- ( ph -> R e. CRing ) $= + ( cfield wcel ccrg cdr isfld simprbi syl ) ABDEZBFEZCKBGELBHIJ $. + $} + ${ $d x B $. $d x G $. $d x R $. $d x .0. $. isdrng2.b $e |- B = ( Base ` R ) $. @@ -254064,7 +255056,7 @@ nonzero elements form a group under multiplication (from which it drngid.z $e |- .0. = ( 0g ` R ) $. drngid.u $e |- .1. = ( 1r ` R ) $. drngid.g $e |- G = ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) $. - $( A division ring's unit is the identity element of its multiplicative + $( A division ring's unity is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.) $) drngid $p |- ( R e. DivRing -> .1. = ( 0g ` G ) ) $= ( cdr wcel cmgp cfv cui cress co c0g crg wceq eqid drngring unitgrpid syl @@ -254076,7 +255068,7 @@ nonzero elements form a group under multiplication (from which it ${ drngunz.z $e |- .0. = ( 0g ` R ) $. drngunz.u $e |- .1. = ( 1r ` R ) $. - $( A division ring's unit is different from its zero. (Contributed by NM, + $( A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.) $) drngunz $p |- ( R e. DivRing -> .1. =/= .0. ) $= ( cdr wcel cbs cfv wne cui wa crg drngring eqid 1unit syl drngunit simprd @@ -254940,11 +255932,15 @@ nonzero elements form a group under multiplication (from which it ${ $d s w $. - $( A sub-division-ring is a subset of a division ring's set which is a - division ring under the induced operation. If the overring is - commutative this is a field; no special consideration is made of the - fields in the center of a skew field. (Contributed by Stefan O'Rear, - 3-Oct-2015.) $) + $( Define the function associating with a ring the set of its + sub-division-rings. A sub-division-ring of a ring is a subset of its + base set which is a division ring when equipped with the induced + structure (sum, multiplication, zero, and unity). If a ring is + commutative (resp., a field), then its sub-division-rings are + commutative (resp., are fields) ( ~ fldsdrgfld ), so we do not make a + specific definition for subfields. (Contributed by Stefan O'Rear, + 3-Oct-2015.) TODO: extend this definition to a function with domain + ` _V ` or at least ` Ring ` and not only ` DivRing ` . $) df-sdrg $a |- SubDRing = ( w e. DivRing |-> { s e. ( SubRing ` w ) | ( w |`s s ) e. DivRing } ) $. $} @@ -254993,6 +255989,16 @@ nonzero elements form a group under multiplication (from which it B $. $} + $( A sub-division-ring of a field is itself a field, so it is a subfield. We + can therefore use ` SubDRing ` to express subfields. (Contributed by + Thierry Arnoux, 11-Jan-2025.) $) + fldsdrgfld $p |- ( ( F e. Field /\ A e. ( SubDRing ` F ) ) + -> ( F |`s A ) e. Field ) $= + ( cfield wcel csdrg cfv wa cress co ccrg csubrg issdrg simp3bi adantl isfld + cdr simprbi simp2bi eqid subrgcrng syl2an sylanbrc ) BCDZABEFDZGBAHIZPDZUEJ + DZUECDUDUFUCUDBPDZABKFDZUFBALZMNUCBJDZUIUGUDUCUHUKBOQUDUHUIUFUJRABUEUESTUAU + EOUB $. + ${ $d a b V $. $d a E $. $d a b X $. $d a b Y $. $( Construction of a closure rule from a one-parameter _partial_ operation. @@ -255168,7 +256174,7 @@ nonzero elements form a group under multiplication (from which it ${ $d R s $. primefld0cl.1 $e |- .0. = ( 0g ` R ) $. - $( The prime field contains the neutral element of the division ring. + $( The prime field contains the zero element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.) $) primefld0cl $p |- ( R e. DivRing -> .0. e. |^| ( SubDRing ` R ) ) $= ( vs cdr wcel csdrg cfv csubg wss c0 wne cv wi csubrg cress co issdrg syl @@ -255180,8 +256186,8 @@ nonzero elements form a group under multiplication (from which it ${ $d R s $. primefld1cl.1 $e |- .1. = ( 1r ` R ) $. - $( The prime field contains the multiplicative neutral element of the - division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.) $) + $( The prime field contains the unity element of the division ring. + (Contributed by Thierry Arnoux, 22-Aug-2023.) $) primefld1cl $p |- ( R e. DivRing -> .1. e. |^| ( SubDRing ` R ) ) $= ( vs cdr wcel csdrg cfv cint csubrg wss c0 wne cv wi cress issdrg simp2bi co a1i ssrdv cbs eqid sdrgid ne0d subrgint syl2anc subrg1cl syl ) AEFZAGH @@ -256346,7 +257352,7 @@ Absolute value (abstract algebra) lmod1cl.f $e |- F = ( Scalar ` W ) $. lmod1cl.k $e |- K = ( Base ` F ) $. lmod1cl.u $e |- .1. = ( 1r ` F ) $. - $( The ring unit in a left module belongs to the ring base set. + $( The ring unity in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmod1cl $p |- ( W e. LMod -> .1. e. K ) $= @@ -256358,8 +257364,9 @@ Absolute value (abstract algebra) lmodvs1.f $e |- F = ( Scalar ` W ) $. lmodvs1.s $e |- .x. = ( .s ` W ) $. lmodvs1.u $e |- .1. = ( 1r ` F ) $. - $( Scalar product with ring unit. ( ~ ax-hvmulid analog.) (Contributed by - NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) + $( Scalar product with the ring unity. ( ~ ax-hvmulid analog.) + (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, + 19-Jun-2014.) $) lmodvs1 $p |- ( ( W e. LMod /\ X e. V ) -> ( .1. .x. X ) = X ) $= ( clmod wcel wa cbs cfv co wceq eqid w3a cplusg simpl lmod1cl simpr cmulr adantr lmodlema simprrd syl122anc ) EKLZFDLZMUIBCNOZLZULUJUJBFAPZFQZUIUJU @@ -263732,7 +264739,7 @@ such that every prime ideal contains a prime element (this is a cnring 0cn cnfldbas cnfldadd eqid grpid mp2an mpbi eqcomi ) ABCZDDDEFDGZU EDGZHAIJZDKJUFUGLAMJUHPANOQKEADUERSUETUAUBUCUD $. - $( One is the unit element of the field of complex numbers. (Contributed + $( One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) cnfld1 $p |- 1 = ( 1r ` CCfld ) $= ( vx ccnfld cur cfv c1 cc wcel cv cmul wceq wral ax-1cn mulid2 mulid1 jca @@ -264437,15 +265444,15 @@ According to Wikipedia ("Integer", 25-May-2019, ( cvv wcel cmul czring cmulr cfv wceq zex ccnfld df-zring cnfldmul ressmulr cz ax-mp ) MABCDEFGHMIDCAJKLN $. - $( The neutral element of the ring of integers. (Contributed by Thierry - Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) $) + $( The zero element of the ring of integers. (Contributed by Thierry Arnoux, + 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) $) zring0 $p |- 0 = ( 0g ` ZZring ) $= ( ccnfld cmnd wcel cc0 cz wss czring c0g cfv wceq ccrg crg crngring ringmnd cc cncrng mp2b 0z zsscn df-zring cnfldbas cnfld0 ress0g mp3an ) ABCZDECEOFD GHIJAKCALCUEPAMANQRSEOAGDTUAUBUCUD $. - $( The multiplicative neutral element of the ring of integers. (Contributed - by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) $) + $( The unity element of the ring of integers. (Contributed by Thierry + Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) $) zring1 $p |- 1 = ( 1r ` ZZring ) $= ( cz ccnfld csubrg cfv wcel c1 czring cur wceq zsubrg df-zring cnfld1 ax-mp subrg1 ) ABCDEFGHDIJABGFKLNM $. @@ -266194,9 +267201,8 @@ According to Wikipedia ("Integer", 25-May-2019, zrhpsgnevpm.y $e |- Y = ( ZRHom ` R ) $. zrhpsgnevpm.s $e |- S = ( pmSgn ` N ) $. zrhpsgnevpm.o $e |- .1. = ( 1r ` R ) $. - $( The sign of an even permutation embedded into a ring is the - multiplicative neutral element of the ring. (Contributed by SO, - 9-Jul-2018.) $) + $( The sign of an even permutation embedded into a ring is the unity + element of the ring. (Contributed by SO, 9-Jul-2018.) $) zrhpsgnevpm $p |- ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> ( ( Y o. S ) ` F ) = .1. ) $= ( crg wcel cfn cevpm cfv w3a c1 cbs co wceq eqid ccom csymg cmgp cneg cpr @@ -266209,8 +267215,8 @@ According to Wikipedia ("Integer", 25-May-2019, zrhpsgnodpm.p $e |- P = ( Base ` ( SymGrp ` N ) ) $. zrhpsgnodpm.i $e |- I = ( invg ` R ) $. $( The sign of an odd permutation embedded into a ring is the additive - inverse of the multiplicative neutral element of the ring. (Contributed - by SO, 9-Jul-2018.) $) + inverse of the unity element of the ring. (Contributed by SO, + 9-Jul-2018.) $) zrhpsgnodpm $p |- ( ( R e. Ring /\ N e. Fin /\ F e. ( P \ ( pmEven ` N ) ) ) -> ( ( Y o. S ) ` F ) = ( I ` .1. ) ) $= ( wcel cfv c1 co wceq eqid czring crg cfn cevpm cdif w3a ccom cneg ccnfld @@ -266528,15 +267534,15 @@ According to Wikipedia ("Integer", 25-May-2019, ( cr cvv wcel cmul crefld cmulr wceq reex ccnfld df-refld cnfldmul ressmulr cfv ax-mp ) ABCDEFMGHAIEDBJKLN $. - $( The neutral element of the field of reals. (Contributed by Thierry - Arnoux, 1-Nov-2017.) $) + $( The zero element of the field of reals. (Contributed by Thierry Arnoux, + 1-Nov-2017.) $) re0g $p |- 0 = ( 0g ` RRfld ) $= ( ccnfld cmnd wcel cc0 cr wss crefld c0g cfv wceq ccrg crg crngring ringmnd cc cncrng mp2b 0re ax-resscn df-refld cnfldbas cnfld0 ress0g mp3an ) ABCZDE CEOFDGHIJAKCALCUEPAMANQRSEOAGDTUAUBUCUD $. - $( The multiplicative neutral element of the field of reals. (Contributed by - Thierry Arnoux, 1-Nov-2017.) $) + $( The unity element of the field of reals. (Contributed by Thierry Arnoux, + 1-Nov-2017.) $) re1r $p |- 1 = ( 1r ` RRfld ) $= ( cr ccnfld csubrg cfv wcel c1 crefld cur wceq cdr resubdrg simpli df-refld cnfld1 subrg1 ax-mp ) ABCDEZFGHDIQGJEKLABGFMNOP $. @@ -266592,11 +267598,11 @@ According to Wikipedia ("Integer", 25-May-2019, cnfldcj ) ABCDEFGHIAJEDBKNLM $. $( The real numbers form a star ring. (Contributed by Thierry Arnoux, - 19-Apr-2019.) $) - recrng $p |- RRfld e. *Ring $= - ( vx crefld csr wcel wtru cr ccj rebase refldcj ccrg cdr cfield refld isfld - wa mpbi simpri a1i cv cfv wceq cjre adantl idsrngd mptru ) BCDEAFBGHIBJDZEB - KDZUFBLDUGUFOMBNPQRASZFDUHGTUHUAEUHUBUCUDUE $. + 19-Apr-2019.) (Proof shortened by Thierry Arnoux, 11-Jan-2025.) $) + resrng $p |- RRfld e. *Ring $= + ( vx crefld csr wcel cr ccj rebase refldcj cfield refld a1i fldcrngd cv cfv + wtru wceq cjre adantl idsrngd mptru ) BCDOAEBFGHOBBIDOJKLAMZEDUAFNUAPOUAQRS + T $. ${ $d F x $. $d I x $. $d V x $. @@ -268118,7 +269124,7 @@ S C_ ( ._|_ ` ( ._|_ ` S ) ) ) $= obsipid.h $e |- ., = ( .i ` W ) $. obsipid.f $e |- F = ( Scalar ` W ) $. obsipid.u $e |- .1. = ( 1r ` F ) $. - $( A basis element has unit length. (Contributed by Mario Carneiro, + $( A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.) $) obsipid $p |- ( ( B e. ( OBasis ` W ) /\ A e. B ) -> ( A ., A ) = .1. ) $= ( cobs cfv wcel wa co wceq c0g cif cbs eqid obsip 3anidm23 iftruei eqtrdi @@ -268272,8 +269278,8 @@ with modules (over rings) instead of vector spaces (over fields) allows for a are usually regarded as "over a ring" in this part. Unless otherwise stated, the rings of scalars need not be commutative - (see ~ df-cring ), but the existence of a multiplicative neutral element is - always assumed (our rings are unital, see ~ df-ring ). + (see ~ df-cring ), but the existence of a unity element is always assumed + (our rings are unital, see ~ df-ring ). For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under @@ -269470,9 +270476,9 @@ is used for the (special) unit vectors forming the standard basis of free uvcff.u $e |- U = ( R unitVec I ) $. uvcff.y $e |- Y = ( R freeLMod I ) $. uvcff.b $e |- B = ( Base ` Y ) $. - $( Domain and range of the unit vector generator; ring condition required - to be sure 1 and 0 are actually in the ring. (Contributed by Stefan - O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) $) + $( Domain and codomain of the unit vector generator; ring condition + required to be sure 1 and 0 are actually in the ring. (Contributed by + Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) $) uvcff $p |- ( ( R e. Ring /\ I e. W ) -> U : I --> B ) $= ( vi vj crg wcel wa cfv c0g eqid cv cvv a1i weq cur cif cmpt uvcfval cmap cbs co cfsupp wbr wf ringidcl ring0cl ifcld ad3antrrr fmpttd wb fvex mpan @@ -270833,7 +271839,7 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, |^| { t e. ( ( SubRing ` w ) i^i ( LSubSp ` w ) ) | s C_ t } ) ) $. $( Every unital algebra contains a canonical homomorphic image of its ring - of scalars as scalar multiples of the unit. This names the + of scalars as scalar multiples of the unity element. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.) $) df-ascl $a |- algSc = ( w e. _V |-> ( x e. ( Base ` ( Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) ) $. @@ -271757,17 +272763,17 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, antecedent. (Revised by SN, 7-Aug-2024.) $) psrbagfsupp $p |- ( F e. D -> F finSupp 0 ) $= ( wcel cc0 cfsupp wbr ccnv cn cima cfn cn0 wf cvv wa id psrbagf ffnd wb - fndmexd psrbag biimpa mpancom simprd frnnn0fsuppg mpdan mpbird ) CAFZCGHI - ZCJKLMFZUJDNCOZULDPFZUJUMULQZUJDCAUJRUJDNCABCDESZTUBUNUJUOABCDPEUCUDUEUFU - JUMUKULUAUPCDAUGUHUI $. + fndmexd psrbag biimpa mpancom simprd fcdmnn0fsuppg mpdan mpbird ) CAFZCGH + IZCJKLMFZUJDNCOZULDPFZUJUMULQZUJDCAUJRUJDNCABCDESZTUBUNUJUOABCDPEUCUDUEUF + UJUMUKULUAUPCDAUGUHUI $. $( Obsolete version of ~ psrbagfsupp as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) $) psrbagfsuppOLD $p |- ( ( X e. D /\ I e. V ) -> X finSupp 0 ) $= ( wcel wa cc0 cfsupp wbr ccnv cn cima cfn cn0 wf psrbag biimpac simprd wb - simpr psrbagfOLD ancoms frnnn0fsupp syl2anc mpbird ) EAGZCDGZHZEIJKZELMNO - GZUJCPEQZULUIUHUMULHABECDFRSTUJUIUMUKULUAUHUIUBUIUHUMABECDFUCUDECDUEUFUG + simpr psrbagfOLD ancoms fcdmnn0fsupp syl2anc mpbird ) EAGZCDGZHZEIJKZELMN + OGZUJCPEQZULUIUHUMULHABECDFRSTUJUIUMUKULUAUHUIUBUIUHUMABECDFUCUDECDUEUFUG $. ${ @@ -271777,11 +272783,11 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, snifpsrbag $p |- ( ( I e. V /\ N e. NN0 ) -> ( y e. I |-> if ( y = X , N , 0 ) ) e. D ) $= ( wcel cn0 wa cv wceq cc0 cif cmpt a1i adantr cvv wb wf ccnv cima simpr - cn cfn 0nn0 ifcld fmpttd cfsupp c0ex eqid sniffsupp frnnn0fsupp adantlr - wbr id bicomd mpdan mpbird psrbag mpbir2and ) DFIZEJIZKZADALZGMZENOZPZB - IZDJVIUAZVIUBUEUCUFIZVEADVHJVEVHJIVFDIVEVGENJVCVDUDNJIVEUGQUHRUIZVEVLVI - NUJUPZVCVNVDVCAEVIDFSGNVCUQNSIVCUKQVIULUMRVEVKVLVNTVMVEVKKVNVLVCVKVNVLT - VDVIDFUNUOURUSUTVCVJVKVLKTVDBCVIDFHVARVB $. + cn cfn 0nn0 ifcld fmpttd cfsupp wbr id c0ex eqid sniffsupp fcdmnn0fsupp + adantlr bicomd mpdan mpbird psrbag mpbir2and ) DFIZEJIZKZADALZGMZENOZPZ + BIZDJVIUAZVIUBUEUCUFIZVEADVHJVEVHJIVFDIVEVGENJVCVDUDNJIVEUGQUHRUIZVEVLV + INUJUKZVCVNVDVCAEVIDFSGNVCULNSIVCUMQVIUNUORVEVKVLVNTVMVEVKKVNVLVCVKVNVL + TVDVIDFUPUQURUSUTVCVJVKVLKTVDBCVIDFHVARVB $. $} $( The constant function equal to zero is a finite bag. (Contributed by @@ -271798,16 +272804,16 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, psrbaglesupp $p |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' G " NN ) C_ ( `' F " NN ) ) $= ( vc va vb vx wcel cn0 cle wbr cc0 cvv wceq cv cfv wa wf cofr w3a ccnv cn - cima csupp co cdm cin wral relopabiv brrelex1i 3ad2ant3 simp2 frnnn0suppg - df-ofr syl2anc cdif eldifi simp3 ffnd 3ad2ant1 simp1 inidm eqidd ofrfvalg - psrbagf r19.21bi sylan2 wss eqimss syl c0ex a1i suppssrg breqtrd ffvelcdm - mpbid syl2an nn0ge0d nn0red 0re letri3 sylancl mpbir2and suppss eqsstrrd - cr wb ) CAKZELDUAZDCMUBZNZUCZDUDUEUFZDOUGUHZCUDUEUFZWODPKZWLWQWPQWNWKWSWL - DCWMGRZHRZSWTIRZSMNGXAUIXBUIUJUKHIWMGMHIUQULUMUNZWKWLWNUOZDEPUPURWOELJDWR + cima csupp cdm cin df-ofr relopabiv brrelex1i 3ad2ant3 simp2 fcdmnn0suppg + co wral syl2anc cdif eldifi simp3 ffnd psrbagf 3ad2ant1 simp1 inidm eqidd + ofrfvalg mpbid r19.21bi sylan2 wss syl c0ex a1i suppssrg breqtrd ffvelcdm + eqimss syl2an nn0ge0d nn0red 0re letri3 sylancl mpbir2and suppss eqsstrrd + cr wb ) CAKZELDUAZDCMUBZNZUCZDUDUEUFZDOUGUPZCUDUEUFZWODPKZWLWQWPQWNWKWSWL + DCWMGRZHRZSWTIRZSMNGXAUHXBUHUIUQHIWMGMHIUJUKULUMZWKWLWNUNZDEPUOURWOELJDWR OXDWOJRZEWRUSKZTZXEDSZOQZXHOMNZOXHMNZXGXHXECSZOMXFWOXEEKZXHXLMNZXEEWRUTZW - OXNJEWOWNXNJEUKWKWLWNVAWOJEEXHXLMEDCPAWOELDXDVBWOELCWKWLELCUAZWNABCEFVHVC - ZVBXCWKWLWNVDZEVEWOXMTZXHVFXSXLVFVGVSVIVJWOELPCAWRXEOXQWOCOUGUHZWRQZXTWRV - KWOWKXPYAXRXQCEAUPURXTWRVLVMXROPKWOVNVOVPVQXGXHWOWLXMXHLKXFXDXOELXEDVRVTZ + OXNJEWOWNXNJEUQWKWLWNVAWOJEEXHXLMEDCPAWOELDXDVBWOELCWKWLELCUAZWNABCEFVCVD + ZVBXCWKWLWNVEZEVFWOXMTZXHVGXSXLVGVHVIVJVKWOELPCAWRXEOXQWOCOUGUPZWRQZXTWRV + LWOWKXPYAXRXQCEAUOURXTWRVSVMXROPKWOVNVOVPVQXGXHWOWLXMXHLKXFXDXOELXEDVRVTZ WAXGXHWIKOWIKXIXJXKTWJXGXHYBWBWCXHOWDWEWFWGWH $. $( $j usage 'psrbaglesupp' avoids 'ax-rep'; $) @@ -271816,17 +272822,17 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, (Proof modification is discouraged.) $) psrbaglesuppOLD $p |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' G " NN ) C_ ( `' F " NN ) ) $= - ( vx wcel cn0 wf cle wbr wa ccnv cn cima cc0 csupp wceq w3a co frnnn0supp - cofr 3ad2antr2 simpr2 cv cdif cfv eldifi wral simpr3 psrbagfOLD 3ad2antr1 - ffnd simpl inidm eqidd ofrfval mpbid r19.21bi sylan2 cvv eqimss 3syl c0ex - wss jca a1i suppssr breqtrd ffvelcdm syl2an nn0ge0d cr nn0red 0re sylancl - wb letri3 mpbir2and suppss eqsstrrd ) EFIZCAIZEJDKZDCLUDMZUAZNZDOPQZDRSUB - ZCOPQZWDWEWFWKWJTWGDEFUCUEWIEJHDWLRWDWEWFWGUFZWIHUGZEWLUHIZNZWNDUIZRTZWQR - LMZRWQLMZWPWQWNCUIZRLWOWIWNEIZWQXALMZWNEWLUJZWIXCHEWIWGXCHEUKWDWEWFWGULWI - HEEWQXALEDCFFWIEJDWMUOWIEJCWDWFWEEJCKZWGABCEFGUMUNZUOWDWHUPZXGEUQWIXBNZWQ - URXHXAURUSUTVAVBWIEJVCCFWLWNRXFWIWDXENCRSUBZWLTXIWLVGWIWDXEXGXFVHCEFUCXIW - LVDVEXGRVCIWIVFVIVJVKWPWQWIWFXBWQJIWOWMXDEJWNDVLVMZVNWPWQVOIRVOIWRWSWTNVS - WPWQXJVPVQWQRVTVRWAWBWC $. + ( vx wcel cn0 wf cle wbr wa ccnv cn cima cc0 csupp wceq cofr co 3ad2antr2 + w3a fcdmnn0supp simpr2 cv cdif cfv eldifi wral psrbagfOLD 3ad2antr1 simpl + simpr3 ffnd inidm eqidd ofrfval mpbid r19.21bi sylan2 cvv wss eqimss 3syl + jca c0ex a1i suppssr breqtrd ffvelcdm syl2an nn0ge0d cr nn0red 0re letri3 + wb sylancl mpbir2and suppss eqsstrrd ) EFIZCAIZEJDKZDCLUAMZUDZNZDOPQZDRSU + BZCOPQZWDWEWFWKWJTWGDEFUEUCWIEJHDWLRWDWEWFWGUFZWIHUGZEWLUHIZNZWNDUIZRTZWQ + RLMZRWQLMZWPWQWNCUIZRLWOWIWNEIZWQXALMZWNEWLUJZWIXCHEWIWGXCHEUKWDWEWFWGUOW + IHEEWQXALEDCFFWIEJDWMUPWIEJCWDWFWEEJCKZWGABCEFGULUMZUPWDWHUNZXGEUQWIXBNZW + QURXHXAURUSUTVAVBWIEJVCCFWLWNRXFWIWDXENCRSUBZWLTXIWLVDWIWDXEXGXFVGCEFUEXI + WLVEVFXGRVCIWIVHVIVJVKWPWQWIWFXBWQJIWOWMXDEJWNDVLVMZVNWPWQVOIRVOIWRWSWTNV + SWPWQXJVPVQWQRVRVTWAWBWC $. $( The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, @@ -271853,13 +272859,13 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, (Revised by SN, 7-Aug-2024.) $) psrbagaddcl $p |- ( ( F e. D /\ G e. D ) -> ( F oF + G ) e. D ) $= ( vx vy wcel wa caddc co cn0 wf cfn cvv cv adantl cc0 csupp cof ccnv cima - cn nn0addcl psrbagf adantr simpl ffnd fndmexd inidm wceq ovex frnnn0suppg - off sylancr cun cfsupp wbr psrbagfsupp fsuppunfi 0nn0 a1i suppofssd ssfid - 00id eqeltrrd wb psrbag syl mpbir2and ) CAIZDAIZJZCDKUAZLZAIZEMVPNZVPUBUD - UCZOIZVNGHEEEKMMMCDPPGQZMIHQZMIJWAWBKLMIVNWAWBUERVLEMCNVMABCEFUFUGZVMEMDN - VLABDEFUFRZVNECAVLVMUHVNEMCWCUIUJZWEEUKUOZVNVPSTLZVSOVNVPPIVRWGVSULCDVOUM - WFVPEPUNUPVNCSTLDSTLUQWGVNCDSVLCSURUSVMABCEFUTUGVMDSURUSVLABDEFUTRVAVNEMC - DPKSWESMIVNVBVCWCWDSSKLSULVNVFVCVDVEVGVNEPIVQVRVTJVHWEABVPEPFVIVJVK $. + cn nn0addcl psrbagf adantr simpl ffnd fndmexd inidm off wceq fcdmnn0suppg + ovex sylancr cun cfsupp wbr psrbagfsupp fsuppunfi 0nn0 a1i 00id suppofssd + ssfid eqeltrrd wb psrbag syl mpbir2and ) CAIZDAIZJZCDKUAZLZAIZEMVPNZVPUBU + DUCZOIZVNGHEEEKMMMCDPPGQZMIHQZMIJWAWBKLMIVNWAWBUERVLEMCNVMABCEFUFUGZVMEMD + NVLABDEFUFRZVNECAVLVMUHVNEMCWCUIUJZWEEUKULZVNVPSTLZVSOVNVPPIVRWGVSUMCDVOU + OWFVPEPUNUPVNCSTLDSTLUQWGVNCDSVLCSURUSVMABCEFUTUGVMDSURUSVLABDEFUTRVAVNEM + CDPKSWESMIVNVBVCWCWDSSKLSUMVNVDVCVEVFVGVNEPIVQVRVTJVHWEABVPEPFVIVJVK $. $( Obsolete version of ~ psrbagaddcl as of 7-Aug-2024. (Contributed by Mario Carneiro, 9-Jan-2015.) (New usage is discouraged.) @@ -271868,18 +272874,18 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, ( F oF + G ) e. D ) $= ( vx wcel caddc co cn0 wf ccnv cfn wa cc0 csupp syl2anc cvv vy w3a cof cn cima cv nn0addcl adantl simp2 wb psrbag 3ad2ant1 mpbid simpld simp3 simp1 - inidm off cmpt wceq frnnn0supp fvexd feqmptd offval2 oveq1d eqtr3d simprd - cfv cun eqeltrd unfi cdif wss ssun1 a1i c0ex suppssr ssun2 oveq12d eqtrdi - 00id suppss2 ssfid mpbir2and ) EFIZCAIZDAIZUBZCDJUCKZAIZELWIMZWINUDUEZOIZ - WHHUAEEEJLLLCDFFHUFZLIUAUFZLIPWNWOJKLIWHWNWOUGUHWHELCMZCNUDUEZOIZWHWFWPWR - PZWEWFWGUIWEWFWFWSUJWGABCEFGUKULUMZUNZWHELDMZDNUDUEZOIZWHWGXBXDPZWEWFWGUO - WEWFWGXEUJWGABDEFGUKULUMZUNZWEWFWGUPZXHEUQURZWHWLHEWNCVHZWNDVHZJKZUSZQRKZ - OWHWIQRKZWLXNWHWEWKXOWLUTXHXIWIEFVASWHWIXMQRWHHEXJXKJCDFTTXHWHWNEIPZWNCVB - XPWNDVBWHHELCXAVCWHHELDXGVCVDVEVFWHCQRKZDQRKZVIZXNWHXQOIXROIXSOIWHXQWQOWH - WEWPXQWQUTXHXACEFVASWHWPWRWTVGVJWHXRXCOWHWEXBXRXCUTXHXGDEFVASWHXBXDXFVGVJ - XQXRVKSWHEXLHFXSQWHWNEXSVLIPZXLQQJKQXTXJQXKQJWHELTCFXSWNQXAXQXSVMWHXQXRVN - VOXHQTIWHVPVOZVQWHELTDFXSWNQXGXRXSVMWHXRXQVRVOXHYAVQVSWAVTXHWBWCVJWEWFWJW - KWMPUJWGABWIEFGUKULWD $. + inidm off cfv cmpt fcdmnn0supp fvexd feqmptd offval2 oveq1d eqtr3d simprd + wceq cun eqeltrd unfi cdif wss ssun1 a1i c0ex suppssr oveq12d 00id eqtrdi + ssun2 suppss2 ssfid mpbir2and ) EFIZCAIZDAIZUBZCDJUCKZAIZELWIMZWINUDUEZOI + ZWHHUAEEEJLLLCDFFHUFZLIUAUFZLIPWNWOJKLIWHWNWOUGUHWHELCMZCNUDUEZOIZWHWFWPW + RPZWEWFWGUIWEWFWFWSUJWGABCEFGUKULUMZUNZWHELDMZDNUDUEZOIZWHWGXBXDPZWEWFWGU + OWEWFWGXEUJWGABDEFGUKULUMZUNZWEWFWGUPZXHEUQURZWHWLHEWNCUSZWNDUSZJKZUTZQRK + ZOWHWIQRKZWLXNWHWEWKXOWLVHXHXIWIEFVASWHWIXMQRWHHEXJXKJCDFTTXHWHWNEIPZWNCV + BXPWNDVBWHHELCXAVCWHHELDXGVCVDVEVFWHCQRKZDQRKZVIZXNWHXQOIXROIXSOIWHXQWQOW + HWEWPXQWQVHXHXACEFVASWHWPWRWTVGVJWHXRXCOWHWEXBXRXCVHXHXGDEFVASWHXBXDXFVGV + JXQXRVKSWHEXLHFXSQWHWNEXSVLIPZXLQQJKQXTXJQXKQJWHELTCFXSWNQXAXQXSVMWHXQXRV + NVOXHQTIWHVPVOZVQWHELTDFXSWNQXGXRXSVMWHXRXQWAVOXHYAVQVRVSVTXHWBWCVJWEWFWJ + WKWMPUJWGABWIEFGUKULWD $. $( The analogue of the statement " ` 0 <_ G <_ F ` implies ` 0 <_ F - G <_ F ` " for finite bags. (Contributed by Mario Carneiro, @@ -271932,18 +272938,18 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, cuz nn0uz adantr ffvelcdmda nn0zd elfz5 syl2anc ralbidva ffnd simpl inidm eqidd ofrfvalg bitr4d psrbaglecl 3expia pm4.71rd 3bitrrd pm5.21ndd eqtrid ex abbi1dv ccnv cn cima cmap wceq cnveq imaeq1d eleq1d simprbi fzfid cdif - elrab2 csn csupp frnnn0suppg mpdan eqimss id suppssrg oveq2d fz0sn eqtrdi - c0ex ixpfi2 eqeltrd ) DBHZAIZDJUAUBZABUCZGEKGIZDLZMNZUDZOXQXTXRBHZXSPZAUE - YDXSABUFXQYFAYDXQEQXRUGZYFXRYDHZXQYEYGXSYEYGUPXQBCXREFUHRUIYHYGUPXQYHXRGE - QUDZHYGYDYIXRYCQSZYDYISGEGEYCQUJYJYAEHZYBUKRULUMGEQXRAUQZUNUORXQYGYFYHURX - QYGPZYHYAXRLZYCHZGEUSZXSYFYMXREUTZYHYPURYGYQXQEQXRVAVBZYHYQYPGEYCXRYLVCVD - VEYMYPYNYBJUBZGEUSXSYMYOYSGEYMYKPZYNKVJLZHYBVFHYOYSURYTYNQUUAYGYKYNQHXQEQ - YAXRVGVHVKVIYTYBYMEQYADXQEQDUGZYGBCDEFUHZVLVMVNYNKYBVOVPVQYMGEEYNYBJEXRDT - BYRXQDEUTYGXQEQDUUCVRVLXRTHYMYLRXQYGVSEVTYTYNWAYTYBWAWBWCYMXSYEXQYGXSYEBC - DXREFWDWEWFWGWJWHWKWIXQGEYCDWLZWMWNZKXQDQEWONZHUUEOHZCIZWLZWMWNZOHUUGCDUU - FBUUHDWPZUUJUUEOUUKUUIUUDWMUUHDWQWRWSFXCWTXQYKPKYBXAXQYAEUUEXBHPZYCKXDZWP - YCUUMSUULYCKKMNUUMUULYBKKMXQEQTDBUUEYAKUUCXQDKXENZUUEWPZUUNUUESXQUUBUUOUU - CDEBXFXGUUNUUEXHVEXQXIKTHXQXNRXJXKXLXMYCUUMXHVEXOXP $. + elrab2 csn csupp fcdmnn0suppg mpdan eqimss id c0ex suppssrg oveq2d eqtrdi + fz0sn ixpfi2 eqeltrd ) DBHZAIZDJUAUBZABUCZGEKGIZDLZMNZUDZOXQXTXRBHZXSPZAU + EYDXSABUFXQYFAYDXQEQXRUGZYFXRYDHZXQYEYGXSYEYGUPXQBCXREFUHRUIYHYGUPXQYHXRG + EQUDZHYGYDYIXRYCQSZYDYISGEGEYCQUJYJYAEHZYBUKRULUMGEQXRAUQZUNUORXQYGYFYHUR + XQYGPZYHYAXRLZYCHZGEUSZXSYFYMXREUTZYHYPURYGYQXQEQXRVAVBZYHYQYPGEYCXRYLVCV + DVEYMYPYNYBJUBZGEUSXSYMYOYSGEYMYKPZYNKVJLZHYBVFHYOYSURYTYNQUUAYGYKYNQHXQE + QYAXRVGVHVKVIYTYBYMEQYADXQEQDUGZYGBCDEFUHZVLVMVNYNKYBVOVPVQYMGEEYNYBJEXRD + TBYRXQDEUTYGXQEQDUUCVRVLXRTHYMYLRXQYGVSEVTYTYNWAYTYBWAWBWCYMXSYEXQYGXSYEB + CDXREFWDWEWFWGWJWHWKWIXQGEYCDWLZWMWNZKXQDQEWONZHUUEOHZCIZWLZWMWNZOHUUGCDU + UFBUUHDWPZUUJUUEOUUKUUIUUDWMUUHDWQWRWSFXCWTXQYKPKYBXAXQYAEUUEXBHPZYCKXDZW + PYCUUMSUULYCKKMNUUMUULYBKKMXQEQTDBUUEYAKUUCXQDKXENZUUEWPZUUNUUESXQUUBUUOU + UCDEBXFXGUUNUUEXHVEXQXIKTHXQXJRXKXLXNXMYCUUMXHVEXOXP $. $( Obsolete version of ~ psrbaglefi as of 6-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) @@ -271956,20 +272962,20 @@ the same dimension over the same (nonzero) ring. (Contributed by AV, wi cz ffvelcdm adantll nn0uz eleqtrdi psrbagfOLD ffvelcdmda nn0zd syl2anc elfz5 ralbidva ffnd simpll inidm eqidd ofrfval bitr4d psrbagleclOLD 3exp2 imp31 pm4.71rd 3bitrrd ex pm5.21ndd abbi1dv cmap simpr wceq cnveq imaeq1d - eqtrid eleq1d elrab2 simprd fzfid cdif csn cvv jca frnnn0supp eqimss 3syl - csupp c0ex suppssr oveq2d fz0sn eqtrdi ixpfi2 eqeltrd ) EFIZDBIZJZAKZDLUA - UBZABUCZHEMHKZDNZUDOZUFZPYEYHYFBIZYGJZAUGYLYGABUEYEYNAYLYEEQYFURZYNYFYLIZ - YEYMYOYGYEYMYOYFUHRUIPIZJZYOYCYMYRSYDBCYFEFGUJUKYOYQULUMUNYPYOVLYEYPYFHEQ - UFZIYOYLYSYFYKQTZYLYSTHEHEYKQUOYTYIEIZYJUPUQUSUTHEQYFAVAZVBVCUQYEYOYNYPSY - EYOJZYPYIYFNZYKIZHEVDZYGYNUUCYFEVEZYPUUFSYOUUGYEEQYFVFVGZYPUUGUUFHEYKYFUU - BVHVIVJUUCUUFUUDYJLUBZHEVDYGUUCUUEUUIHEUUCUUAJZUUDMVKNZIYJVMIUUEUUISUUJUU - DQUUKYOUUAUUDQIYEEQYIYFVNVOVPVQUUJYJUUCEQYIDYEEQDURZYOBCDEFGVRZUKZVSVTUUD - MYJWBWAWCUUCHEEUUDYJLEYFDFFUUHUUCEQDUUNWDYCYDYOWEZUUOEWFUUJUUDWGUUJYJWGWH - WIUUCYGYMYCYDYOYGYMVLYCYDYOYGYMBCDYFEFGWJWKWLWMWNWOWPWQXCYEHEYKDUHZRUIZMY - EDQEWROZIZUUQPIZYEYDUUSUUTJYCYDWSCKZUHZRUIZPIUUTCDUURBUVADWTZUVCUUQPUVDUV - BUUPRUVADXAXBXDGXEVCXFYEUUAJMYJXGYEYIEUUQXHIJZYKMXIZWTYKUVFTUVEYKMMUDOUVF - UVEYJMMUDYEEQXJDFUUQYIMUUMYEYCUULJDMXOOZUUQWTUVGUUQTYEYCUULYCYDULZUUMXKDE - FXLUVGUUQXMXNUVHMXJIYEXPUQXQXRXSXTYKUVFXMVJYAYB $. + eqtrid eleq1d elrab2 simprd fzfid cdif csn cvv csupp jca fcdmnn0supp 3syl + eqimss c0ex suppssr oveq2d fz0sn eqtrdi ixpfi2 eqeltrd ) EFIZDBIZJZAKZDLU + AUBZABUCZHEMHKZDNZUDOZUFZPYEYHYFBIZYGJZAUGYLYGABUEYEYNAYLYEEQYFURZYNYFYLI + ZYEYMYOYGYEYMYOYFUHRUIPIZJZYOYCYMYRSYDBCYFEFGUJUKYOYQULUMUNYPYOVLYEYPYFHE + QUFZIYOYLYSYFYKQTZYLYSTHEHEYKQUOYTYIEIZYJUPUQUSUTHEQYFAVAZVBVCUQYEYOYNYPS + YEYOJZYPYIYFNZYKIZHEVDZYGYNUUCYFEVEZYPUUFSYOUUGYEEQYFVFVGZYPUUGUUFHEYKYFU + UBVHVIVJUUCUUFUUDYJLUBZHEVDYGUUCUUEUUIHEUUCUUAJZUUDMVKNZIYJVMIUUEUUISUUJU + UDQUUKYOUUAUUDQIYEEQYIYFVNVOVPVQUUJYJUUCEQYIDYEEQDURZYOBCDEFGVRZUKZVSVTUU + DMYJWBWAWCUUCHEEUUDYJLEYFDFFUUHUUCEQDUUNWDYCYDYOWEZUUOEWFUUJUUDWGUUJYJWGW + HWIUUCYGYMYCYDYOYGYMVLYCYDYOYGYMBCDYFEFGWJWKWLWMWNWOWPWQXCYEHEYKDUHZRUIZM + YEDQEWROZIZUUQPIZYEYDUUSUUTJYCYDWSCKZUHZRUIZPIUUTCDUURBUVADWTZUVCUUQPUVDU + VBUUPRUVADXAXBXDGXEVCXFYEUUAJMYJXGYEYIEUUQXHIJZYKMXIZWTYKUVFTUVEYKMMUDOUV + FUVEYJMMUDYEEQXJDFUUQYIMUUMYEYCUULJDMXKOZUUQWTUVGUUQTYEYCUULYCYDULZUUMXLD + EFXMUVGUUQXOXNUVHMXJIYEXPUQXQXRXSXTYKUVFXOVJYAYB $. psrbagconf1o.s $e |- S = { y e. D | y oR <_ F } $. $( The complement of a bag is a bag. (Contributed by Mario Carneiro, @@ -274347,84 +275353,84 @@ series in the subring which are also polynomials (in the parent ring). = ( G gsum ( k e. I |-> ( ( Y ` k ) .^ ( V ` k ) ) ) ) ) $= ( vi vw vz va vb cv wceq cif cmpt ccnv cn cima wcel cfv cc0 co cgsu cn0 wf cfn wa wb psrbag syl mpbid simpld feqmptd iftrue adantl wn eldif cvv - cdif wss frnnn0supp syl2anc eqimss c0ex a1i suppssr ifeq2d ifid eqtr3di - csupp sylan2br pm2.61dan mpteq2dva eqtr4d eqeq2d ifbid mpteq2dv wel csn - wi c0 cur sseq1 eleq2 iffalsed mpteq1 eqtrdi oveq2d eqid eqeq12d imbi2d - imbi12d weq caddc adantr crg ffvelcdmda 0nn0 ifcl sylancl fmpttd sstrid - suppss2 eqidd nn0cnd iftrued oveq12d simpr 3eqtr4d wo eqtrd wral oveq1d - fveq2 mplcoe5lem mvrcl mulgnn0cl syl3anc a2d cres cnvimass fssdm simprd - anassrs cxp cun noel mtbiri fconstmpt eqtr4di mpt0 ringidval gsum0 mpl1 - a1d ssun1 sstr2 ax-mp imim1i cmulr oveq1 cof cbs simprll ssfid eqeltrrd - eldifn mpbir2and ssun2 simprr vex sylibr ffvelcdmd snifpsrbag mplmonmul + cdif csupp wss fcdmnn0supp syl2anc eqimss c0ex a1i suppssr ifid eqtr3di + ifeq2d sylan2br pm2.61dan mpteq2dva eqtr4d eqeq2d ifbid mpteq2dv wi wel + c0 csn sseq1 eleq2 iffalsed mpteq1 eqtrdi oveq2d eqeq12d imbi12d imbi2d + cur eqid weq caddc adantr ffvelcdmda 0nn0 sylancl fmpttd suppss2 sstrid + crg ifcl eqidd nn0cnd iftrued oveq12d simpr 3eqtr4d wo eqtrd wral fveq2 + oveq1d mplcoe5lem mvrcl mulgnn0cl syl3anc a2d anassrs cres fssdm simprd + cnvimass cxp cun noel mtbiri fconstmpt eqtr4di mpt0 ringidval gsum0 a1d + mpl1 ssun1 sstr2 ax-mp imim1i cmulr oveq1 cof cbs eldifn ssfid eqeltrrd + simprll mpbir2and simprr vex snss sylibr ffvelcdmd snifpsrbag mplmonmul mplcoe3 offval2 addid2d simprlr ad2antrr eqneltrd fveq2d sselid addid1d - snss elsni velsn sylnib orcom bitri biorf bitr4id 3eqtr3rd ccntz mgpbas - elun mgpplusg cmnd mplring ringmgp cplusg cbvral2vw sylib sselda syldan - adantlr gsumzunsnd syl5ibr expr syl5 findcard2s mpcom mpd resmptd ssidd - expcom eldifi sylan2 mulg0 cfsupp wbr mptexd funmpt fvexd suppssfifsupp - wfun syl32anc gsumzres 3eqtr2d ) ACDCUMZOUNZGPUOZUPCDUYKUHLUHUMZOUQURUS - 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UYFUXFWBUYBUWTUYBVDUXEUWLUWMUWNYRXGUWOYEAUXSUXHVUCUTUNUOLNOVUCKUBSVVAUWPY + KUWQYTAUXSUXCIUXAUNBIJKNUKUWRYKUWS $. $} ${ @@ -280244,7 +281250,7 @@ represented as an element of (the base set of) ` ( ( 1 ... n ) Mat R ) ` . set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia ~ https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). - Its determinant is the ring unit, see ~ mdet0fv0 . + Its determinant is the ring unity, see ~ mdet0fv0 . $) @@ -282157,10 +283163,8 @@ represented as an element of (the base set of) ` ( ( 1 ... n ) Mat R ) ` . $( There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) $) mat1rngiso $p |- ( ( R e. Ring /\ E e. V ) -> F e. ( R RingIso A ) ) $= - ( crg wcel wa crs co crh wf1o mat1rhm mat1f1o wb csn snfi matring sylancr - cfn simpl isrim syldan mpbir2and ) DOPZEIPZQZFDBRSPZFDBTSPZGCFUAZABCDEFGH - IJKLMNUBABCDEFGHIJKLMNUCUNUOBOPZUQURUSQUDUPEUEZUIPUNUTEUFUNUOUJBDVAKUGUHG - CDBFOOJLUKULUM $. + ( crg wcel wa crh co wf1o crs mat1rhm mat1f1o isrim sylanbrc ) DOPEIPQFDB + RSPGCFTFDBUASPABCDEFGHIJKLMNUBABCDEFGHIJKLMNUCGCDBFJLUDUE $. $} ${ @@ -283084,12 +284088,10 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) 29-Dec-2019.) $) scmatrngiso $p |- ( ( N e. Fin /\ N =/= (/) /\ R e. Ring ) -> F e. ( R RingIso S ) ) $= - ( wcel crg cfv cfn c0 wne w3a crs crh wf1o scmatrhm 3adant2 scmatf1o wceq - co cbs scmatstrbas f1oeq3d mpbird wa simp3 csubrg c0g scmatsrng subrgring - wb eqid syl isrim 3imp3i2an mpbir2and ) JUARZJUBUCZDSRZUDZGDEUEULRZGDEUFU - LRZIEUMTZGUGZVIVKVNVJABCDEFGHIJKLMNOPQUHUIVLVPICGUGABCDFGHIJKLMNOPUJVLVOC - IGVIVKVOCUKVJBCDEJLPQUNUIUOUPVIVJVKVKESRZVMVNVPUQVCVIVJVKURVIVKUQCBUSTRVQ - BBUMTZDCIJDUTTZLVRVDKVSVDPVACBEQVBVEIVODEGSSKVOVDVFVGVH $. + ( wcel co wf1o cfn wne crg w3a crh cbs cfv scmatrhm 3adant2 scmatf1o wceq + c0 crs scmatstrbas f1oeq3d mpbird eqid isrim sylanbrc ) JUARZJULUBZDUCRZU + DZGDEUESRZIEUFUGZGTZGDEUMSRUTVBVDVAABCDEFGHIJKLMNOPQUHUIVCVFICGTABCDFGHIJ + KLMNOPUJVCVECIGUTVBVECUKVABCDEJLPQUNUIUOUPIVEDEGKVEUQURUS $. $} ${ @@ -284157,7 +285159,7 @@ is alternating (see ~ mdetralt ) and normalized (see ~ mdet1 ). ${ $d R m p x $. $( The determinant function for 0-dimensional matrices on a given ring is - the function mapping the empty set to the unit of that ring. + the function mapping the empty set to the unity element of that ring. (Contributed by AV, 28-Feb-2019.) $) mdet0pr $p |- ( R e. Ring -> ( (/) maDet R ) = { <. (/) , ( 1r ` R ) >. } ) $= @@ -284184,16 +285186,16 @@ is alternating (see ~ mdetralt ) and normalized (see ~ mdet1 ). $( The determinant function for 0-dimensional matrices on a given ring is a bijection from the singleton containing the empty set (empty matrix) - onto the singleton containing the unit of that ring. (Contributed by - AV, 28-Feb-2019.) $) + onto the singleton containing the unity element of that ring. + (Contributed by AV, 28-Feb-2019.) $) mdet0f1o $p |- ( R e. Ring -> ( (/) maDet R ) : { (/) } -1-1-onto-> { ( 1r ` R ) } ) $= ( crg wcel c0 cmdat co cur cfv cop csn wceq wf1o mdet0pr 0ex f1osn f1oeq1 fvex mpbiri syl ) ABCDAEFZDAGHZIJZKZDJZUAJZTLZAMUCUFUDUEUBLDUANAGQOUDUETU BPRS $. - $( The determinant of the empty matrix on a given ring is the unit of that - ring . (Contributed by AV, 28-Feb-2019.) $) + $( The determinant of the empty matrix on a given ring is the unity element + of that ring. (Contributed by AV, 28-Feb-2019.) $) mdet0fv0 $p |- ( R e. Ring -> ( ( (/) maDet R ) ` (/) ) = ( 1r ` R ) ) $= ( crg wcel c0 cmdat cfv cur cop csn mdet0pr fveq1d 0ex fvex fvsn eqtrdi co ) ABCZDDAEPZFDDAGFZHIZFSQDRTAJKDSLAGMNO $. @@ -288515,14 +289517,12 @@ whose entries are constant polynomials (or "scalar polynomials", see ring isomorphism. (Contributed by AV, 19-Nov-2019.) $) m2cpmrngiso $p |- ( ( N e. Fin /\ R e. CRing ) -> T e. ( A RingIso U ) ) $= - ( wcel wa cfv crg vm cfn ccrg crs crh cbs wf1o m2cpmrhm crngring m2cpmf1o - co anim2i wss wceq cv eqid cpmatpmat 3expia ssrdv ressbas2 eqcomd f1oeq3d - syl mpbird cgrp wb matring csubg cpmatsubgpmat subggrp 3syl isrim syl2anc - mpbir2and ) IUBQZDUCQZRZFAGUDUKQZFAGUEUKQZHGUFSZFUGZAHBCDEFGIJKLMNOPUHVQV - ODTQZRZWAVPWBVODUIULZWCWAHEFUGADEFHIJKLMUJWCVTEHFWCEVTWCEBUFSZUMEVTUNWCUA - EWEVOWBUAUOZEQWFWEQWEBCDEWFITJNOWEUPZUQURUSEWEGBPWGUTVCVAVBVDVCVQATQZGVEQ - ZVRVSWARVFVQWCWHWDADILVGVCVQWCEBVHSQWIWDBCDEIJNOVIEBGPVJVKHVTAGFTVEMVTUPV - LVMVN $. + ( vm wcel wa co cfn ccrg crh cbs cfv wf1o m2cpmrhm crngring m2cpmf1o wceq + crs crg wss cv eqid cpmatpmat 3expia ssrdv ressbas2 eqcomd f1oeq3d mpbird + syl sylan2 isrim sylanbrc ) IUARZDUBRZSFAGUCTRHGUDUEZFUFZFAGUKTRAHBCDEFGI + JKLMNOPUGVHVGDULRZVJDUHVGVKSZVJHEFUFADEFHIJKLMUIVLVIEHFVLEVIVLEBUDUEZUMEV + IUJVLQEVMVGVKQUNZERVNVMRVMBCDEVNIULJNOVMUOZUPUQUREVMGBPVOUSVCUTVAVBVDHVIA + GFMVIUOVEVF $. $} ${ @@ -290478,11 +291478,9 @@ over matrices (over the same ring). (Contributed by AV, 5-Dec-2019.) $) $( The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.) $) pm2mprngiso $p |- ( ( N e. Fin /\ R e. Ring ) -> T e. ( C RingIso Q ) ) $= - ( cfn wcel crg wa co cbs cfv eqid crs crh wf1o pm2mprhm cmgp cmg pm2mpf1o - cvsca cv1 wb pmatring matring ply1ring syl isrim syl2anc mpbir2and ) GMNE - ONPZFBDUAQNZFBDUBQNZBRSZDRSZFUCZABCDEFGHIJKLUDAVABCDEFDUESUFSZDUHSZVBGAUI - SZHIVATZVETVDTVFTJKVBTZLUGURBONDONZUSUTVCPUJBCEGHIUKURAONVIAEGJULDAKUMUNV - AVBBDFOOVGVHUOUPUQ $. + ( cfn wcel crg wa co cbs cfv eqid crh wf1o crs pm2mprhm cmgp cmg pm2mpf1o + cvsca cv1 isrim sylanbrc ) GMNEONPFBDUAQNBRSZDRSZFUBFBDUCQNABCDEFGHIJKLUD + AULBCDEFDUESUFSZDUHSZUMGAUISZHIULTZUOTUNTUPTJKUMTZLUGULUMBDFUQURUJUK $. $} ${ @@ -297976,7 +298974,7 @@ converges to zero (in the standard topology on the reals) with this lmbr.2 $e |- ( ph -> J e. ( TopOn ` X ) ) $. $( Express the binary relation "sequence ` F ` converges to point ` P ` " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. - The condition ` F C_ ( CC X. X ) ` allows us to use objects more general + The condition ` F C_ ( CC X. X ) ` allows to use objects more general than sequences when convenient; see the comment in ~ df-lm . (Contributed by Mario Carneiro, 14-Nov-2013.) $) lmbr $p |- ( ph -> ( F ( ~~>t ` J ) P <-> @@ -299782,15 +300780,15 @@ require the space to be Hausdorff (which would make it the same as T_3), ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.) $) dishaus $p |- ( A e. V -> ~P A e. Haus ) $= - ( vx vy vu vv wcel cv cin c0 wceq w3a wrex wral csn wss snssd snex elpw + ( vx vy vu vv wcel cv cin c0 wceq w3a wrex wral csn wss snssd vsnex elpw wa cpw ctop wne wi cha distop simplrl sylibr simplrr vsnid disjsn2 adantl a1i eleq2 ineq1 3anbi13d ineq2 3anbi23d rspc2ev syl113anc ralrimivva cuni eqeq1d ex unipw eqcomi ishaus sylanbrc ) ABGZAUAZUBGCHZDHZUCZVKEHZGZVLFHZ GZVNVPIZJKZLZFVJMEVJMZUDZDANCANVJUEGABUFVIWBCDAAVIVKAGZVLAGZTTZVMWAWEVMTZ - VKOZVJGZVLOZVJGZVKWGGZVLWIGZWGWIIZJKZWAWFWGAPWHWFVKAVIWCWDVMUGQWGAVKRSUHW - FWIAPWJWFVLAVIWCWDVMUIQWIAVLRSUHWKWFCUJUMWLWFDUJUMVMWNWEVKVLUKULVTWKWLWNL - WKVQWGVPIZJKZLEFWGWIVJVJVNWGKZVOWKVSWPVQVNWGVKUNWQVRWOJVNWGVPUOVCUPVPWIKZ - VQWLWPWNWKVPWIVLUNWRWOWMJVPWIWGUQVCURUSUTVDVACDFEVJAVJVBAAVEVFVGVH $. + VKOZVJGZVLOZVJGZVKWGGZVLWIGZWGWIIZJKZWAWFWGAPWHWFVKAVIWCWDVMUGQWGACRSUHWF + WIAPWJWFVLAVIWCWDVMUIQWIADRSUHWKWFCUJUMWLWFDUJUMVMWNWEVKVLUKULVTWKWLWNLWK + VQWGVPIZJKZLEFWGWIVJVJVNWGKZVOWKVSWPVQVNWGVKUNWQVRWOJVNWGVPUOVCUPVPWIKZVQ + WLWPWNWKVPWIVLUNWRWOWMJVPWIWGUQVCURUSUTVDVACDFEVJAVJVBAAVEVFVGVH $. $} ${ @@ -300021,15 +301019,15 @@ require the space to be Hausdorff (which would make it the same as T_3), by Mario Carneiro, 19-Mar-2015.) $) discmp $p |- ( A e. Fin <-> ~P A e. Comp ) $= ( vy vx cfn wcel cpw ccmp ctop cin distop pwfi biimpi elind fincmp syl cv - cuni wceq wrex wss csn cmpt crn wa simpr snex elpw sylibr fmpttd frnd cab - snssd ciun eqid rnmpt unieqi dfiun2 iunid 3eqtr2ri eqcomi cmpcov mpd3an23 - a1i unipw elinel2 elinel1 elpwid wral snfi rgenw fmpt mpbi frn mp1i sstrd - wf unifi syl2anc eleq1 syl5ibrcom rexlimiv impbii ) ADEZAFZGEZWCWDHDIEWEW - CHDWDADJWCWDDEAKLMWDNOWEABPZQZRZBCACPZUAZUBZUCZFZDIZSZWCWEWLWDTAWLQZRZWOW - EAWDWKWECAWJWDWEWIAEZUDZWJATWJWDEWSWIAWEWRUEULWJAWIUFZUGUHUIUJWQWEWPWFWJR - CASBUKZQCAWJUMAWLXACBAWJWKWKUNZUOUPCBAWJWTUQCAURUSVCWLWDABWDQAAVDUTVAVBWH - WCBWNWFWNEZWCWHWGDEZXCWFDEWFDTXDWFWMDVEXCWFWLDXCWFWLWFWMDVFVGADWKVPZWLDTX - CWJDEZCAVHXEXFCAWIVIVJCADWJWKXBVKVLADWKVMVNVOWFVQVRAWGDVSVTWAOWB $. + cuni wceq wrex wss csn cmpt crn simpr snssd vsnex elpw sylibr fmpttd frnd + wa cab ciun eqid rnmpt unieqi dfiun2 iunid 3eqtr2ri unipw eqcomi mpd3an23 + a1i cmpcov elinel2 elinel1 elpwid wf wral snfi rgenw fmpt mpbi mp1i sstrd + frn unifi syl2anc eleq1 syl5ibrcom rexlimiv impbii ) ADEZAFZGEZWCWDHDIEWE + WCHDWDADJWCWDDEAKLMWDNOWEABPZQZRZBCACPZUAZUBZUCZFZDIZSZWCWEWLWDTAWLQZRZWO + WEAWDWKWECAWJWDWEWIAEZUKZWJATWJWDEWSWIAWEWRUDUEWJACUFZUGUHUIUJWQWEWPWFWJR + CASBULZQCAWJUMAWLXACBAWJWKWKUNZUOUPCBAWJWTUQCAURUSVCWLWDABWDQAAUTVAVDVBWH + WCBWNWFWNEZWCWHWGDEZXCWFDEWFDTXDWFWMDVEXCWFWLDXCWFWLWFWMDVFVGADWKVHZWLDTX + CWJDEZCAVIXEXFCAWIVJVKCADWJWKXBVLVMADWKVPVNVOWFVQVRAWGDVSVTWAOWB $. cmpsub.1 $e |- X = U. J $. $( Lemma for ~ cmpsub . 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(Contributed by Jeff @@ -300814,15 +301812,15 @@ require the space to be Hausdorff (which would make it the same as T_3), ( wcel vx cv ciun cin wex wn c0 wne n0 sylib elin wrex eliun nfan crest wa nfv co adantr ctopon cfv ad2antrr wss simprr inelcm syl2anc ad2antrl cconn wceq ssiun2 ad2antlr sscond sstrd wb inss1 toponss sstrid reldisj - cdif syl mpbird nconnsubb expr mt2d an4s rexlimd syl5bi expimpd exlimdv - cun exp32 mpd ) AUAUBZHFBCUCZUDZTZUAUEZDETZUFZAWOUGUHWQPUAWOUIUJAWPWSUA - WPWMHTZWMWNTZUPAWSWMHWNUKAWTXAWSXAWMCTZFBULAWTUPZWSFWMBCUMXCXBWSFBAWTFS - WTFUQUNWSFUQXCFUBBTZXBWSAXDWTXBWSAXDUPZWTXBUPZUPWRGCUOURVHTZXEXGXFMUSXE - XFWRXGUFXEXFWRUPZUPZCEGHIAGIUTVATZXDXHJVBZXECIVCXHKUSAEGTZXDXHNVBZAHGTX - DXHOVBXIWRDCTZECUDUGUHXEXFWRVDXEXNXHLUSDECVEVFXFHCUDUGUHXEWRWMHCVEVGXIE - HUDZCUDUGVIZXOICVSZVCZXIXOIWNVSZXQAXOXSVCXDXHQVBXICWNIXDCWNVCAXHFBCVJVK - ZVLVMXIXOIVCXPXRVNXIXOEIEHVOXIXJXLEIVCXKXMEGIVPVFVQXOCIVRVTWAXICWNEHWJZ - XTAWNYAVCXDXHRVBVMWBWCWDWEWKWFWGWHWGWIWL $. + cdif syl mpbird nconnsubb expr mt2d an4s exp32 rexlimd biimtrid expimpd + cun exlimdv mpd ) AUAUBZHFBCUCZUDZTZUAUEZDETZUFZAWOUGUHWQPUAWOUIUJAWPWS + UAWPWMHTZWMWNTZUPAWSWMHWNUKAWTXAWSXAWMCTZFBULAWTUPZWSFWMBCUMXCXBWSFBAWT + FSWTFUQUNWSFUQXCFUBBTZXBWSAXDWTXBWSAXDUPZWTXBUPZUPWRGCUOURVHTZXEXGXFMUS + XEXFWRXGUFXEXFWRUPZUPZCEGHIAGIUTVATZXDXHJVBZXECIVCXHKUSAEGTZXDXHNVBZAHG + TXDXHOVBXIWRDCTZECUDUGUHXEXFWRVDXEXNXHLUSDECVEVFXFHCUDUGUHXEWRWMHCVEVGX + IEHUDZCUDUGVIZXOICVSZVCZXIXOIWNVSZXQAXOXSVCXDXHQVBXICWNIXDCWNVCAXHFBCVJ + VKZVLVMXIXOIVCXPXRVNXIXOEIEHVOXIXJXLEIVCXKXMEGIVPVFVQXOCIVRVTWAXICWNEHW + JZXTAWNYAVCXDXHRVBVMWBWCWDWEWFWGWHWIWHWKWL $. $} $d k ph $. @@ -300861,15 +301859,15 @@ require the space to be Hausdorff (which would make it the same as T_3), mpan9 ctopon cfv cin c0 wne cun wi wex n0 w3a cpr ciun cuni uniiun simpl1 toponmax syl simpl2l simpl2r uniprg syl2anc eqtr3id oveq2d wo elpr simpl2 vex sseq1 simpl3 elin sylib eleq2 simpr eleq1d iunconn eqeltrrd ex 3expia - oveq2 exlimdv syl5bi 3impia ) CDUAUBGZADHZBDHZIZABUCZUDUEZCAJKZLGZCBJKZLG - ZIZCABUFZJKZLGZUGZWHEMZWGGZEUHWCWFIZWQEWGUIWTWSWQEWCWFWSWQWCWFWSUJZWMWPXA - WMIZCFABUKZFMZULZJKWOLXBXEWNCJXBXEXCUMZWNFXCUNXBANGBNGXFWNOXBADCXBWCDCGWC - WFWSWMUOZDCUPUQZWDWEWCWSWMURPXBBDCXHWDWEWCWSWMUSPABNNUTVAVBVCXBXCXDWRFCDX - GXDXCGZXBXDAOZXDBOZVDZXDDHZXDABFVGVEZXBWFXLXMWCWFWSWMVFXJWDXMXKWEXJXMWDXD - ADVHQXKXMWEXDBDVHQRTSXIXBXLWRXDGZXNXBWRAGZWRBGZIZXLXOXBWSXRWCWFWSWMVIWRAB - VJVKXJXPXOXKXQXJXOXPXDAWRVLQXKXOXQXDBWRVLQRTSXIXBXLCXDJKZLGZXNXBWMXLXTXAW - MVMXJWJXTXKWLXJXTWJXJXSWILXDACJVSVNQXKXTWLXKXSWKLXDBCJVSVNQRTSVOVPVQVRVTW - AWB $. + oveq2 exlimdv biimtrid 3impia ) CDUAUBGZADHZBDHZIZABUCZUDUEZCAJKZLGZCBJKZ + LGZIZCABUFZJKZLGZUGZWHEMZWGGZEUHWCWFIZWQEWGUIWTWSWQEWCWFWSWQWCWFWSUJZWMWP + XAWMIZCFABUKZFMZULZJKWOLXBXEWNCJXBXEXCUMZWNFXCUNXBANGBNGXFWNOXBADCXBWCDCG + WCWFWSWMUOZDCUPUQZWDWEWCWSWMURPXBBDCXHWDWEWCWSWMUSPABNNUTVAVBVCXBXCXDWRFC + DXGXDXCGZXBXDAOZXDBOZVDZXDDHZXDABFVGVEZXBWFXLXMWCWFWSWMVFXJWDXMXKWEXJXMWD + XDADVHQXKXMWEXDBDVHQRTSXIXBXLWRXDGZXNXBWRAGZWRBGZIZXLXOXBWSXRWCWFWSWMVIWR + ABVJVKXJXPXOXKXQXJXOXPXDAWRVLQXKXOXQXDBWRVLQRTSXIXBXLCXDJKZLGZXNXBWMXLXTX + AWMVMXJWJXTXKWLXJXTWJXJXSWILXDACJVSVNQXKXTWLXKXSWKLXDBCJVSVNQRTSVOVPVQVRV + TWAWB $. $( The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.) $) @@ -301246,11 +302244,11 @@ require the space to be Hausdorff (which would make it the same as T_3), 2ndcrest $p |- ( ( J e. 2ndc /\ A e. V ) -> ( J |`t A ) e. 2ndc ) $= ( vx vy wcel c2ndc crest co cv com cdom wbr ctg cfv wceq ctb wrex syl2anc wa is2ndc simplr simpll restbas ad2antlr cin cmpt crn restval 1stcrestlem - tgrest adantl eqbrtrd 2ndci eqeltrrd oveq1 eleq1d syl5ibcom syl5bi impcom - expimpd rexlimdva ) ACFZBGFZBAHIZGFZVDDJZKLMZVGNOZBPZTZDQRVCVFDBUAVCVKVFD - QVCVGQFZTZVHVJVFVMVHTZVIAHIZGFVJVFVNVGAHIZNOZVOGVNVLVCVQVOPVCVLVHUBZVCVLV - HUCZAVGQCUKSVNVPQFZVPKLMVQGFVLVTVCVHAVGUDUEVNVPEVGEJAUFZUGUHZKLVNVLVCVPWB - PVRVSEAVGQCUISVHWBKLMVMEVGWAUJULUMVPUNSUOVJVOVEGVIBAHUPUQURVAVBUSUT $. + tgrest adantl eqbrtrd 2ndci eqeltrrd syl5ibcom expimpd rexlimdva biimtrid + oveq1 eleq1d impcom ) ACFZBGFZBAHIZGFZVDDJZKLMZVGNOZBPZTZDQRVCVFDBUAVCVKV + FDQVCVGQFZTZVHVJVFVMVHTZVIAHIZGFVJVFVNVGAHIZNOZVOGVNVLVCVQVOPVCVLVHUBZVCV + LVHUCZAVGQCUKSVNVPQFZVPKLMVQGFVLVTVCVHAVGUDUEVNVPEVGEJAUFZUGUHZKLVNVLVCVP + WBPVRVSEAVGQCUISVHWBKLMVMEVGWAUJULUMVPUNSUOVJVOVEGVIBAHUTVAUPUQURUSVB $. $} ${ @@ -301275,55 +302273,55 @@ require the space to be Hausdorff (which would make it the same as T_3), bastg sseqtrrdi sstrd simprrl simprr ad2antlr eleqtrrd simprrr tg2 simprl sseqtrd sseldd simplrl weq rspcev syl12anc syl3anbrc wi sseq12d rexlimddv fveq2 ex eqeq2i biimpi eqtr2di wfn ffn fnfvelrn simplll rspcv op1st op2nd - 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(Contributed by Mario Carneiro, 21-Mar-2015.) (Proof @@ -301484,27 +302482,27 @@ require the space to be Hausdorff (which would make it the same as T_3), by Mario Carneiro, 13-Apr-2015.) $) dis2ndc $p |- ( X ~<_ _om <-> ~P X e. 2ndc ) $= ( vx vy vz vw vb com cdom wbr cvv wcel c2ndc cv csn wceq wss wel wrex ctb - wa cpw ctex pwexr cmpt crn cen wb wf1o wf1 snex 2a1i sneqr impbii dom2lem - vex sneq f1f1orn syl f1oeng mpdan domen1 ctg cfv ctop distop simpr snelpw - wral sylib fmpttd frnd elpwi ad2antrl simprr sseldd rspceeqv syl2anc eqid - eqidd elrnmpt ax-mp sylibr vsnid a1i snssd eleq2 sseq1 anbi12d ralrimivva - rspcev syl12anc basgen2 syl3anc adantr eqeltrd tgclb 2ndci sylan eqeltrrd - is2ndc simplrr eleqtrrd tg2 sylancl simprrl eqssd simprl rexlimddv ssdomg - simprrr mpsyl domtr rexlimdva2 syl5bi imp impbida bitrd pm5.21nii ) AGHIZ - AJKZAUAZLKZAUBALUCXTXSBABMZNZUDZUEZGHIZYBXTAYFUFIZXSYGUGXTAYFYEUHZYHXTAJY - EUIYIXTBCAJYDCMZNZYDJKXTYCAKZYCUJUKYDYKOZYCYJOZUGXTYLYJAKTYMYNYCYJBUOZULY - CYJUPUMUKUNAJYEUQURAYFJYEUSUTAYFGVAURXTYGYBXTYGTYFVBVCZYALXTYPYAOZYGXTYAV - 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eqtrd cin distop adantr wi elpwi adantl ssralv syl simprl sstrd snex elpw - sylibr elind snidg ad2antrl simpll simprr eqeltrd eleq1d anbi12d syl12anc - eleq2 oveq2 rspcev expr ralimdva syld imp an32s islly sylanbrc impbida ) + eqtrd cin distop adantr elpwi adantl ssralv syl simprl sstrd vsnex sylibr + elpw elind snidg ad2antrl simpll simprr eqeltrd eleq2 oveq2 eleq1d rspcev + wi anbi12d syl12anc expr ralimdva syld imp an32s islly sylanbrc impbida ) DCGZDHZBUAGZAIZUBZHZBGZADJZWRWTKZXDADXFXADGZKZEIZXBLZXAXIGZWSXIMNZBGZUCZE WSOZXDXHWTXBWSGZXAXBGZXOWRWTXGPXHXGXPXFXGUDXADAUEUFUGXQXHAUHUIEBXAXBWSUJU KXHXNXDEWSXHXNXDXHXNKZXLXCBXRXLWSXBMNZXCXRXIXBWSMXRXIXBXHXJXKXMULXRXAXIXH XJXKXMUMQUNUOXRWRXBDLZXSXCRZWRWTXGXNUPXRXADXFXGXNPQDXBCUQZTVDXHXJXKXMURUS UTVAVBSWRXEKZWSVCGZXAFIZGZWSYEMNZBGZKZFWSXIHZVEZOZAXIJZEWSJWTWRYDXEDCVFVG - YCYMEWSWRXIWSGZXEYMWRYNKZXEYMYOXEXDAXIJZYMYOXIDLZXEYPVHYNYQWRXIDVIVJZXDAX - IDVKVLYOXDYLAXIYOXKXDYLYOXKXDKZKZXBYKGXQXSBGZYLYTWSYJXBYTXTXPYTXBXIDYTXAX - IYOXKXDVMQZYOYQYSYRVGVNZXBDXAVOZVPVQYTXBXILXBYJGUUBXBXIUUDVPVQVRXKXQYOXDX - 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(Contributed by Mario Carneiro, 20-Mar-2015.) $) @@ -302142,11 +303140,11 @@ arbitrary neighborhoods (such as "locally compact", which is actually 21-Mar-2015.) $) dis1stc $p |- ( X e. V -> ~P X e. 1stc ) $= ( vx wcel cpw c1stc clly cv csn wral c2ndc ctg cfv ctop cvv ax-mp com wbr - cfn mpbi wceq snex distop ctb cdom topbas csdm snfi pwfi isfinite sdomdom - tgtop 2ndci mp2an eqeltrri 2ndc1stc rgenw dislly mpbiri lly1stc eleqtrdi - ) BADZBEZFGZFVBVCVDDCHZIZEZFDZCBJVHCBVGKDVHVGLMZVGKVGNDZVIVGUAVFODVJVEUBV - FOUCPZVGULPVGUDDZVGQUERZVIKDVJVLVKVGUFPVGQUGRZVMVGSDZVNVFSDVOVEUHVFUITVGU - JTVGQUKPVGUMUNUOVGUPPUQCFABURUSUTVA $. + cfn mpbi wceq vsnex distop tgtop ctb cdom csdm snfi pwfi isfinite sdomdom + topbas 2ndci mp2an eqeltrri 2ndc1stc rgenw dislly mpbiri lly1stc eleqtrdi + ) BADZBEZFGZFVBVCVDDCHZIZEZFDZCBJVHCBVGKDVHVGLMZVGKVGNDZVIVGUAVFODVJCUBVF + OUCPZVGUDPVGUEDZVGQUFRZVIKDVJVLVKVGULPVGQUGRZVMVGSDZVNVFSDVOVEUHVFUITVGUJ + TVGQUKPVGUMUNUOVGUPPUQCFABURUSUTVA $. $} ${ @@ -302489,21 +303487,21 @@ arbitrary neighborhoods (such as "locally compact", which is actually cdif cun cuni cin wne crab wral cpw locfinnei ralrimiva cmpcov2 sylan2 wi elfpw ciun simplrr eldifsn wex ineq1 neeq1d simplrl simprr inelcm syl2anc elrabd elunii eleqtrrdi ancoms adantl locfinbas ad3antrrr eleqtrrd simplr - eleqtrd eluni2 sylib reximddv expr exlimdv n0 eliun 3imtr4g expimpd ssrdv - syl5bi iunfi ssfi expcom sylan9 sylan2b rexlimdva snfi unfi sylancl ssun1 - ex mpd undif1 sseqtrri jca ctop cmptop finlocfin 3expib syl impbid ) BUAL - ZABUBUCLZAMLZCDNZOZXKXLXOXKXLOZXMXNXPAPUDZUEZXQUFZMLZAXSQXMXPXRMLZXQMLXTX - PCGRZUGZNZHRZIRZUHZPUIZHAUJZMLZIYBUKZOZGBULMUHZSZYAXLXKJITYJOIBSZJCUKYNXL - YOJCAJRZIBCHEUMUNYJJIBCGEUOUPXPYLYAGYMYBYMLXPYBBQZYBMLZOZYLYAUQYBBURXPYSO - ZYDYKYAYTYDOZYRXRIYBYIUSZQZYKYAUQXPYQYRYDUTUUAJXRUUBYPXRLYPALZYPPUIZOUUAY - PUUBLZYPAPVAUUAUUDUUEUUFUUAUUDOZKJTZKVBYPYILZIYBSZUUEUUFUUGUUHUUJKUUAUUDU - UHUUJUUAUUDUUHOZOZKITZUUIIYBUULIGTZUUMOZOZYHYPYFUHZPUIZHYPAYEYPNYGUUQPYEY - PYFVCVDUUAUUDUUHUUOVEUUPUUHUUMUURUUAUUDUUHUUOUTUULUUNUUMVFKRZYPYFVGVHVIUU - LUUSYCLUUMIYBSUULUUSCYCUULUUSDCUUKUUSDLZUUAUUHUUDUUTUUHUUDOUUSAUGDUUSYPAV - JFVKVLVMXPXNYSYDUUKXLXNXKABCDEFVNVMZVOVPYTYDUUKVQVRIUUSYBVSVTWAWBWCKYPWDI - YPYBYIWEWFWGWIWHYRYKUUBMLZUUCYAYRYKUVBIYBYIWJWTUVBUUCYAUUBXRWKWLWMVHWGWNW - OXAPWPXRXQWQWRAAXQUFXSAXQWSAXQXBXCXSAWKWRUVAXDWTXKBXELZXOXLUQBXFUVCXMXNXL - ABCDEFXGXHXIXJ $. + eleqtrd eluni2 sylib reximddv expr exlimdv eliun 3imtr4g expimpd biimtrid + n0 ssrdv iunfi ssfi expcom sylan9 sylan2b rexlimdva mpd snfi unfi sylancl + ex ssun1 undif1 sseqtrri jca ctop cmptop finlocfin 3expib syl impbid ) BU + ALZABUBUCLZAMLZCDNZOZXKXLXOXKXLOZXMXNXPAPUDZUEZXQUFZMLZAXSQXMXPXRMLZXQMLX + TXPCGRZUGZNZHRZIRZUHZPUIZHAUJZMLZIYBUKZOZGBULMUHZSZYAXLXKJITYJOIBSZJCUKYN + XLYOJCAJRZIBCHEUMUNYJJIBCGEUOUPXPYLYAGYMYBYMLXPYBBQZYBMLZOZYLYAUQYBBURXPY + SOZYDYKYAYTYDOZYRXRIYBYIUSZQZYKYAUQXPYQYRYDUTUUAJXRUUBYPXRLYPALZYPPUIZOUU + AYPUUBLZYPAPVAUUAUUDUUEUUFUUAUUDOZKJTZKVBYPYILZIYBSZUUEUUFUUGUUHUUJKUUAUU + DUUHUUJUUAUUDUUHOZOZKITZUUIIYBUULIGTZUUMOZOZYHYPYFUHZPUIZHYPAYEYPNYGUUQPY + EYPYFVCVDUUAUUDUUHUUOVEUUPUUHUUMUURUUAUUDUUHUUOUTUULUUNUUMVFKRZYPYFVGVHVI + UULUUSYCLUUMIYBSUULUUSCYCUULUUSDCUUKUUSDLZUUAUUHUUDUUTUUHUUDOUUSAUGDUUSYP + AVJFVKVLVMXPXNYSYDUUKXLXNXKABCDEFVNVMZVOVPYTYDUUKVQVRIUUSYBVSVTWAWBWCKYPW + HIYPYBYIWDWEWFWGWIYRYKUUBMLZUUCYAYRYKUVBIYBYIWJWTUVBUUCYAUUBXRWKWLWMVHWFW + NWOWPPWQXRXQWRWSAAXQUFXSAXQXAAXQXBXCXSAWKWSUVAXDWTXKBXELZXOXLUQBXFUVCXMXN + XLABCDEFXGXHXIXJ $. $} ${ @@ -302513,12 +303511,12 @@ arbitrary neighborhoods (such as "locally compact", which is actually (Contributed by Thierry Arnoux, 9-Jan-2020.) $) unisngl $p |- X = U. C $= ( vy cuni cv csn wceq wrex cab unieqi wcel wa simpl simpr eleqtrd exlimiv - wex eqid snex eleq2 eqeq1 anbi12d spcev mpan2 impbii equcom 3bitri rexbii - velsn r19.42v exbii rexcom4 eluniab 3bitr4ri risset 3bitr4i eqriv eqtr2i + wex eqid vsnex eleq2 eqeq1 anbi12d spcev mpan2 impbii velsn equcom 3bitri + rexbii r19.42v exbii rexcom4 eluniab 3bitr4ri risset 3bitr4i eqriv eqtr2i ) CGBHZAHZIZJZADKZBLZGZDCVGEMFVHDFHZVBNZVEOZBTZADKZVCVIJZADKVIVHNZVIDNVLV - NADVLVIVDNZVIVCJVNVLVPVKVPBVKVIVBVDVJVEPVJVEQRSVPVDVDJZVLVDUAVKVPVQOBVDVC - UBVEVJVPVEVQVBVDVIUCVBVDVDUDUEUFUGUHFVCULFAUIUJUKVKADKZBTVJVFOZBTVMVOVRVS - BVJVEADUMUNVKABDUOVFBVIUPUQAVIDURUSUTVA $. + NADVLVIVDNZVIVCJVNVLVPVKVPBVKVIVBVDVJVEPVJVEQRSVPVDVDJZVLVDUAVKVPVQOBVDAU + BVEVJVPVEVQVBVDVIUCVBVDVDUDUEUFUGUHFVCUIFAUJUKULVKADKZBTVJVFOZBTVMVOVRVSB + VJVEADUMUNVKABDUOVFBVIUPUQAVIDURUSUTVA $. $( The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.) $) @@ -302616,45 +303614,45 @@ arbitrary neighborhoods (such as "locally compact", which is actually uneq2 csn frn wfo wfn simprrl ffnd fofi snfi unfi sylancl simplrr simprrr dffn4 iuneq2 eqtrid eqtrd ssun2 vsnid sselii fvssunirn ssun1 unissi sstri sylanbrc unssi mpbir eqsstrdi sstrd uniss eqssd expr 3syld com23 rexlimdv - mpdd rexlimdvaa syl5bi pm2.61dne syl3an3 com24 iscmp baibr sylibd impbid2 - ralrimdv ) AUCJZAUDJZBCKZLZMZDKZUXHNZBUXKLZMZOZDUGUHZPZCAUIZQZUXGUXQCUXRU - XHUXRJUXHANZUXGUXQUXHAUJUXGUXTUXJUXPUXGUXTUXJUKUXNDUXHUIRULZUHUXPUXHABDEU - MUXNUXODUYAUGUXKUYAJUXLUXKRJZOUXNUXKUGJZUXOOZUXKUXHUNUYBUXLUXNUYDUYBUXLUX - NUYDUYBUYCUXOUXKUOURUPUQUSUTSVAVBVCUXFUXSBUAKZLZMZBUBKZLZMUBUYEUIZRULZUHZ - PZUAUXRQZUXGUXFUXSUYMUAUXRUXFUYGUYEUXRJZUXSUYLUXFUYGUYOUXSUYLPZUYOUXFUYGU - YEANZUYPUYEAUJUXFUYGUYQUKZUYPBTUYRBTMZUXSUYLTUYKJUYSUYLTUYJRUYEVDVEVFUBTU - YKUYITBUYHTMUYITLTUYHTVIVJVGVHVKVLBTVMUEKZBJZUEVNUYRUYPUEBVOUYRVUAUYPUEUY - RVUAUEFVPZFUYEUHZUYPUYRVUAUYTUYFJZVUCUYRVUAVUDUYRBUYFUYTUXFUYGUYQWCVQVRFU - 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( R Cn T ) | ( h " K ) C_ V } ) ) ) $= wn 1st2nd2 crab xp2nd cmpt ccnv cima mptpreima ccn ctopon toptopon2 sylib xp1st cnmptc cnmptid cnmpt1t cnima wrex wne simprl elunii eqeltrrd rspcev opeq1 rabn0 sylibr cnclima connclo elrab simprbi opeq2 eqeltrd expr ssrdv - elin2d eqssd ex exlimdv syl5bi orrd vex elpr syl6ibr isconn2 sylanbrc ) A - FGZBFGZHZABUBIZUCGZXMXMUDJZUEZKXMUFZUGZUHXMFGXJAUCGZBUCGZXNXKAUIZBUIZABUJ - UKXLCXPXRXLCLZXPGZYCKMZYCXQMZULZYCXRGXLYDYGXLYDHZYEYFYEVEDLZYCGZDUMYHYFDY - CUNYHYJYFDYHYJYFYHYJHZYCXQYKYCXMGZYCXQUHYKXMXOYCXLYDYJUOUPYCXMUQNYKUAXQYC - YHYJUALZXQGZYMYCGYHYJYNHZHZYMYMURJZYMUSJZOZYCYPYMAUFZBUFZUTZGZYMYSMYPYMXQ - UUBYHYJYNVAYPXSXTUUBXQMYPXJXSXJXKYDYOVBZYANZYPXKXTXJXKYDYOVCZYBNZABYTUUAY - TPZUUAPZVDQZSZYMYTUUAVFNYPYRYQELZOZYCGZEUUAVGZGZYSYCGZYPYRUUAUUOYPUUCYRUU - AGZUUKYMYTUUAVHNYPUUOBUUAUUIUUFYPUUOEUUAUUMVIZVJYCVKZBEUUAUUMYCUUSUUSPVLZ - YPUUSBXMVMIGZYLUUTBGYPEYQUULBABUUAYPXTBUUAVNJGUUGBVOVPZYPEYQBAUUAYTUVCYPX - SAYTVNJGUUEAVOVPZYPUUCYQYTGZUUKYMYTUUAVQNZVRYPEBUUAUVCVSVTZYPXMXOYCXLYDYO - UOZUPZYCUUSBXMWAQRYPUUNEUUAWBZUUOKWCYPYIUSJZUUAGZYQUVKOZYCGZUVJYPYIUUBGZU - VLYPYIXQUUBYPYJYLYIXQGYHYJYNWDZUVIYIYCXMWEQUUJSZYIYTUUAVHNZYPYQUULUVKOZYC - GZEYTVGZGZUVNYPYQYTUWAUVFYPUWAAYTUUHUUDYPUWAEYTUVSVIZVJYCVKZAEYTUVSYCUWCU - WCPVLZYPUWCAXMVMIGZYLUWDAGYPEUULUVKAABYTUVDYPEAYTUVDVSYPEUVKABYTUUAUVDUVC - UVRVRVTZUVIYCUWCAXMWAQRYPUVTEYTWBZUWAKWCYPYIURJZYTGZUWIUVKOZYCGZUWHYPUVOU - WJUVQYIYTUUAVQNYPYIUWKYCYPUVOYIUWKMUVQYIYTUUAVFNUVPWFUVTUWLEUWIYTUULUWIMU - VSUWKYCUULUWIUVKWHTWGQUVTEYTWIWJYPUWAUWDAUDJZUWEYPUWFYCXOGZUWDUWMGUWGYPXM - XOYCUVHWSZYCUWCAXMWKQRWLSUWBUVEUVNUVTUVNEYQYTUULYQMUVSUVMYCUULYQUVKWHTWMW - NNUUNUVNEUVKUUAUULUVKMUUMUVMYCUULUVKYQWOTWGQUUNEUUAWIWJYPUUOUUTBUDJZUVAYP - UVBUWNUUTUWPGUVGUWOYCUUSBXMWKQRWLSUUPUURUUQUUNUUQEYRUUAUULYRMUUMYSYCUULYR - YQWOTWMWNNWPWQWRWTXAXBXCXDXAYCKXQCXEXFXGWRXMXQXQPXHXI $. + elin2d eqssd ex exlimdv biimtrid orrd vex elpr syl6ibr isconn2 sylanbrc ) + AFGZBFGZHZABUBIZUCGZXMXMUDJZUEZKXMUFZUGZUHXMFGXJAUCGZBUCGZXNXKAUIZBUIZABU + JUKXLCXPXRXLCLZXPGZYCKMZYCXQMZULZYCXRGXLYDYGXLYDHZYEYFYEVEDLZYCGZDUMYHYFD + YCUNYHYJYFDYHYJYFYHYJHZYCXQYKYCXMGZYCXQUHYKXMXOYCXLYDYJUOUPYCXMUQNYKUAXQY + CYHYJUALZXQGZYMYCGYHYJYNHZHZYMYMURJZYMUSJZOZYCYPYMAUFZBUFZUTZGZYMYSMYPYMX + QUUBYHYJYNVAYPXSXTUUBXQMYPXJXSXJXKYDYOVBZYANZYPXKXTXJXKYDYOVCZYBNZABYTUUA + YTPZUUAPZVDQZSZYMYTUUAVFNYPYRYQELZOZYCGZEUUAVGZGZYSYCGZYPYRUUAUUOYPUUCYRU + UAGZUUKYMYTUUAVHNYPUUOBUUAUUIUUFYPUUOEUUAUUMVIZVJYCVKZBEUUAUUMYCUUSUUSPVL + ZYPUUSBXMVMIGZYLUUTBGYPEYQUULBABUUAYPXTBUUAVNJGUUGBVOVPZYPEYQBAUUAYTUVCYP + XSAYTVNJGUUEAVOVPZYPUUCYQYTGZUUKYMYTUUAVQNZVRYPEBUUAUVCVSVTZYPXMXOYCXLYDY + OUOZUPZYCUUSBXMWAQRYPUUNEUUAWBZUUOKWCYPYIUSJZUUAGZYQUVKOZYCGZUVJYPYIUUBGZ + UVLYPYIXQUUBYPYJYLYIXQGYHYJYNWDZUVIYIYCXMWEQUUJSZYIYTUUAVHNZYPYQUULUVKOZY + CGZEYTVGZGZUVNYPYQYTUWAUVFYPUWAAYTUUHUUDYPUWAEYTUVSVIZVJYCVKZAEYTUVSYCUWC + UWCPVLZYPUWCAXMVMIGZYLUWDAGYPEUULUVKAABYTUVDYPEAYTUVDVSYPEUVKABYTUUAUVDUV + CUVRVRVTZUVIYCUWCAXMWAQRYPUVTEYTWBZUWAKWCYPYIURJZYTGZUWIUVKOZYCGZUWHYPUVO + UWJUVQYIYTUUAVQNYPYIUWKYCYPUVOYIUWKMUVQYIYTUUAVFNUVPWFUVTUWLEUWIYTUULUWIM + UVSUWKYCUULUWIUVKWHTWGQUVTEYTWIWJYPUWAUWDAUDJZUWEYPUWFYCXOGZUWDUWMGUWGYPX + MXOYCUVHWSZYCUWCAXMWKQRWLSUWBUVEUVNUVTUVNEYQYTUULYQMUVSUVMYCUULYQUVKWHTWM + WNNUUNUVNEUVKUUAUULUVKMUUMUVMYCUULUVKYQWOTWGQUUNEUUAWIWJYPUUOUUTBUDJZUVAY + PUVBUWNUUTUWPGUVGUWOYCUUSBXMWKQRWLSUUPUURUUQUUNUUQEYRUUAUULYRMUUMYSYCUULY + RYQWOTWMWNNWPWQWRWTXAXBXCXDXAYCKXQCXEXFXGWRXMXQXQPXHXI $. $} ${ @@ -308543,37 +309541,37 @@ coincides with the usual topological product (generated by a base of weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.) $) ptcmpfi $p |- ( ( A e. Fin /\ F : A --> Comp ) -> ( Xt_ ` F ) e. Comp ) $= - ( vx cfn wcel ccmp cres cpt cfv wceq syl fveq2d wss wi ctop cvv cuni eqid - cv c0 vy vz vu vv wf wa wfn ffn fnresdm adantl ssid csn sseq1 reseq2 res0 + ( vx vz cfn wcel ccmp cres cpt cfv wceq fveq2d wss wi cv c0 ctop cvv cuni + eqid vy vu vv wf wa wfn ffn fnresdm syl adantl ssid csn sseq1 reseq2 res0 cun eqtrdi eleq1d imbi2d imbi12d cin 0ex pttop mp2an cpw cixp ptuni ixp0x f0 snfi eqeltri eqeltrri pwfi mpbi pwuni ssfi elini fincmp ax-mp wn ssun1 2a1i id sstrid imim1i ctx co chmph cmpo chmeo resabs1 eqcomi fveq2i ssun2 - wbr vex snex unex a1i simplr cmptop ssriv fss sylancl simprr eqidd simprl - fssresd disjsn sylibr ptunhmeo hmphi ad2antlr snss fnressn syl2anc ctopon - cmpt ffvelcdmd sselid toptopon2 sylib pt1hmeo cmphmph sylc eqeltrd expcom - cop txcmp sylsyld a2d ex syl5 findcard2s mpi anabsi5 eqeltrrd ) ADEZAFBUE - ZUFZBAGZHIZBHIFYTUUABHYSUUABJZYRYSBAUGZUUCAFBUHZABUIKUJLYRYSUUBFEZYRAAMZY - TUUFNZAUKCSZAMZYTBUUIGZHIZFEZNZNTAMZYTTHIZFEZNZNUASZAMZYTBUUSGZHIZFEZNZNZ - UUSUBSZULZUPZAMZYTBUVHGZHIZFEZNZNZUUGUUHNCUAUBAUUITJZUUJUUOUUNUURUUITAUMU - VOUUMUUQYTUVOUULUUPFUVOUUKTHUVOUUKBTGTUUITBUNBUOUQLURUSUTUUIUUSJZUUJUUTUU - NUVDUUIUUSAUMUVPUUMUVCYTUVPUULUVBFUVPUUKUVAHUUIUUSBUNLURUSUTUUIUVHJZUUJUV - IUUNUVMUUIUVHAUMUVQUUMUVLYTUVQUULUVKFUVQUUKUVJHUUIUVHBUNLURUSUTUUIAJZUUJU - UGUUNUUHUUIAAUMUVRUUMUUFYTUVRUULUUBFUVRUUKUUAHUUIABUNLURUSUTUUQUUOYTUUPOD - VAEUUQUUPODTPEZTOTUEZUUPOEVBOVIZTTPVCVDUUPQZVEZDEZUUPUWCMUUPDEUWBDEUWDCTU - UITIQZVFZUWBDUVSUVTUWFUWBJVBUWACTTUUPPUUPRVGVDUWFTULDCUWEVHTVJVKVLUWBVMVN - UUPVOUWCUUPVPVDVQUUPVRVSWBUVFUUSEVTZUVEUVNNUUSDEUVEUVIUVDNUWGUVNUVIUUTUVD - UVIUUSUVHAUUSUVGWAZUVIWCWDWEUWGUVIUVDUVMUWGUVIUVDUVMNUWGUVIUFZYTUVCUVLYTU - WIUVCUVLNYTUWIUFZUVBBUVGGZHIZWFWGZUVKWHWOZUVCUWMFEZUVLUWJUCUDUVBQZUWLQZUC - SUDSUPWIZUWMUVKWJWGEUWNUWJUCUDUUSUVGUVHUVJUWRUVKUVBUWLPUWPUWQUWPRUWQRUVKR - UVAUVJUUSGZHUWSUVAUUSUVHMUWSUVAJUWHBUUSUVHWKVSWLWMUWKUVJUVGGZHUWTUWKUVGUV - HMUWTUWKJUVGUUSWNZBUVGUVHWKVSWLWMUWRRUVHPEUWJUUSUVGUAWPUVFWQWRWSUWJAOUVHB - UWJYSFOMAOBUEYRYSUWIWTZCFOUUIXAXBZAFOBXCXDYTUWGUVIXEZXHUWJUVHXFUWJUWGUUSU - VGVATJYTUWGUVIXGUUSUVFXIXJXKUWRUWMUVKXLKUWJUWLFEZUVCUWONUWJUWLUVFUVFBIZYH - ULZHIZFUWJUWKUXGHUWJUUDUVFAEZUWKUXGJYSUUDYRUWIUUEXMUWJUVGAMUXIUWJUVGUVHAU - XAUXDWDUVFAUBWPZXNXJZAUVFBXOXPLUWJUXFUXHWHWOZUXFFEUXHFEUWJCUXFQZUVFUUIYHU - LXRZUXFUXHWJWGEUXLUWJCUVFUXFUXHPUXMUXHRUVFPEUWJUXJWSUWJUXFOEUXFUXMXQIEUWJ - FOUXFUXCUWJAFUVFBUXBUXKXSZXTUXFYAYBYCUXNUXFUXHXLKUXOUXFUXHYDYEYFUVCUXEUWO - UVBUWLYIYGKUWMUVKYDYJYGYKYLYKYMUJYNYOYPYQ $. + wbr vex vsnex a1i simplr cmptop ssriv sylancl simprr fssresd eqidd simprl + unex fss disjsn sylibr ptunhmeo hmphi cop ad2antlr fnressn syl2anc ctopon + snss cmpt ffvelcdmd sselid toptopon2 sylib pt1hmeo cmphmph eqeltrd expcom + sylc txcmp sylsyld a2d ex syl5 findcard2s mpi anabsi5 eqeltrrd ) AEFZAGBU + DZUEZBAHZIJZBIJGYTUUABIYSUUABKZYRYSBAUFZUUCAGBUGZABUHUIUJLYRYSUUBGFZYRAAM + ZYTUUFNZAUKCOZAMZYTBUUIHZIJZGFZNZNPAMZYTPIJZGFZNZNUAOZAMZYTBUUSHZIJZGFZNZ + NZUUSDOZULZUPZAMZYTBUVHHZIJZGFZNZNZUUGUUHNCUADAUUIPKZUUJUUOUUNUURUUIPAUMU + VOUUMUUQYTUVOUULUUPGUVOUUKPIUVOUUKBPHPUUIPBUNBUOUQLURUSUTUUIUUSKZUUJUUTUU + NUVDUUIUUSAUMUVPUUMUVCYTUVPUULUVBGUVPUUKUVAIUUIUUSBUNLURUSUTUUIUVHKZUUJUV + IUUNUVMUUIUVHAUMUVQUUMUVLYTUVQUULUVKGUVQUUKUVJIUUIUVHBUNLURUSUTUUIAKZUUJU + UGUUNUUHUUIAAUMUVRUUMUUFYTUVRUULUUBGUVRUUKUUAIUUIABUNLURUSUTUUQUUOYTUUPQE + VAFUUQUUPQEPRFZPQPUDZUUPQFVBQVIZPPRVCVDUUPSZVEZEFZUUPUWCMUUPEFUWBEFUWDCPU + UIPJSZVFZUWBEUVSUVTUWFUWBKVBUWACPPUUPRUUPTVGVDUWFPULECUWEVHPVJVKVLUWBVMVN + UUPVOUWCUUPVPVDVQUUPVRVSWBUVFUUSFVTZUVEUVNNUUSEFUVEUVIUVDNUWGUVNUVIUUTUVD + UVIUUSUVHAUUSUVGWAZUVIWCWDWEUWGUVIUVDUVMUWGUVIUVDUVMNUWGUVIUEZYTUVCUVLYTU + WIUVCUVLNYTUWIUEZUVBBUVGHZIJZWFWGZUVKWHWOZUVCUWMGFZUVLUWJUBUCUVBSZUWLSZUB + OUCOUPWIZUWMUVKWJWGFUWNUWJUBUCUUSUVGUVHUVJUWRUVKUVBUWLRUWPUWQUWPTUWQTUVKT + UVAUVJUUSHZIUWSUVAUUSUVHMUWSUVAKUWHBUUSUVHWKVSWLWMUWKUVJUVGHZIUWTUWKUVGUV + HMUWTUWKKUVGUUSWNZBUVGUVHWKVSWLWMUWRTUVHRFUWJUUSUVGUAWPDWQXGWRUWJAQUVHBUW + JYSGQMAQBUDYRYSUWIWSZCGQUUIWTXAZAGQBXHXBYTUWGUVIXCZXDUWJUVHXEUWJUWGUUSUVG + VAPKYTUWGUVIXFUUSUVFXIXJXKUWRUWMUVKXLUIUWJUWLGFZUVCUWONUWJUWLUVFUVFBJZXMU + LZIJZGUWJUWKUXGIUWJUUDUVFAFZUWKUXGKYSUUDYRUWIUUEXNUWJUVGAMUXIUWJUVGUVHAUX + AUXDWDUVFADWPZXRXJZAUVFBXOXPLUWJUXFUXHWHWOZUXFGFUXHGFUWJCUXFSZUVFUUIXMULX + SZUXFUXHWJWGFUXLUWJCUVFUXFUXHRUXMUXHTUVFRFUWJUXJWRUWJUXFQFUXFUXMXQJFUWJGQ + UXFUXCUWJAGUVFBUXBUXKXTZYAUXFYBYCYDUXNUXFUXHXLUIUXOUXFUXHYEYHYFUVCUXEUWOU + VBUWLYIYGUIUWMUVKYEYJYGYKYLYKYMUJYNYOYPYQ $. $} ${ @@ -309074,16 +310072,16 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, cfbas cfv cint wpss wal wral bi2.04 wb ibar adantr bitr4id albidv bitr4di df-ral 0nelfb notbid syl5ibrcom necon2ad imp ssn0 ex syl5com ssint notbii a1dd rexnal bitr4i w3a cin fbasssin sspsstr sylan2b expcom reximdv 3expia - nssinpss rexlimdv syl5bi r19.29 id rexlimivw syl pm2.61d sylbid com12 a1i - syl6 findcard3 3impia ) CAUAUBGZBCGZBHGZCUCZIJZWLWJWKKZWNWJDLZCGZKZWNMZWJ - ELZCGZKZWNMZWOWNMDEBWPWTNZWRXBWNXDWQXAWJWPWTCOPQWPBNZWRWOWNXEWQWKWJWPBCOP - QWPWTUDZWSMZDUEZXCMWTHGXBXHWNXBXHXFWNMZDCUFZWNXBXHWQXIMZDUEXJXBXGXKDXBXGW - RXIMXKXFWRWNUGXBWQWRXIWJWQWRUHXAWJWQUIUJQUKULXIDCUNUMXBWTWMRZXJWNMZXBXLWN - XJXBWTIJZXLWNWJXAXNWJXAWTIWJXASWTINZICGZSACUOXOXAXPWTICOUPUQURUSXLXNWNWTW - MUTVAVBVEXBXLSZXFDCTZXMXQWTFLZRZSZFCTZXBXRXQXTFCUFZSYBXLYCFWTCVCVDXTFCVFV - GXBYAXRFCWJXAXSCGZYAXRMWJXAYDVHWPWTXSVIZRZDCTYAXRDWTXSCAVJYAYFXFDCYFYAXFY - AYFYEWTUDXFWTXSVPWPYEWTVKVLVMVNVBVOVQVRXJXRWNXJXRKXIXFKZDCTWNXIXFDCVSYGWN - DCXIXFWNXIVTUSWAWBVMWGWCWDWEWFWHWEWI $. + nssinpss rexlimdv biimtrid r19.29 rexlimivw syl syl6 pm2.61d sylbid com12 + id a1i findcard3 3impia ) CAUAUBGZBCGZBHGZCUCZIJZWLWJWKKZWNWJDLZCGZKZWNMZ + WJELZCGZKZWNMZWOWNMDEBWPWTNZWRXBWNXDWQXAWJWPWTCOPQWPBNZWRWOWNXEWQWKWJWPBC + OPQWPWTUDZWSMZDUEZXCMWTHGXBXHWNXBXHXFWNMZDCUFZWNXBXHWQXIMZDUEXJXBXGXKDXBX + GWRXIMXKXFWRWNUGXBWQWRXIWJWQWRUHXAWJWQUIUJQUKULXIDCUNUMXBWTWMRZXJWNMZXBXL + WNXJXBWTIJZXLWNWJXAXNWJXAWTIWJXASWTINZICGZSACUOXOXAXPWTICOUPUQURUSXLXNWNW + TWMUTVAVBVEXBXLSZXFDCTZXMXQWTFLZRZSZFCTZXBXRXQXTFCUFZSYBXLYCFWTCVCVDXTFCV + FVGXBYAXRFCWJXAXSCGZYAXRMWJXAYDVHWPWTXSVIZRZDCTYAXRDWTXSCAVJYAYFXFDCYFYAX + FYAYFYEWTUDXFWTXSVPWPYEWTVKVLVMVNVBVOVQVRXJXRWNXJXRKXIXFKZDCTWNXIXFDCVSYG + WNDCXIXFWNXIWFUSVTWAVMWBWCWDWEWGWHWEWI $. $} ${ @@ -309099,16 +310097,16 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, cfbas cfv cpw wnel cin wrex ssrab2 cuni eqimss2i sspwuni mpbir a1i topopn sstri anim1i 3adant3 ne0d ss0 necon3ai intnand df-nel notbii bitr2i sylib 3ad2ant3 anbi12i simpl simprll simprrl inopn syl3anc ssin biimpi ad2ant2l - adantl jca 3ad2antl1 ssid rspcev sylancl ex syl5bi ralrimivv 3jca isfbas2 - sseq1 wb syl 3ad2ant1 mpbir2and ) CUAIZBDJZBKLZMZBANZJZACUBZDUCUDIZWSDUEZ - JZWSKLZKWSUFZFNZGNZHNZUGZJZFWSUHZHWSOGWSOZMZXBWPWSCXAWRACUICXAJCUJZDJDXME - UKCDULUMUPUNWPXCXDXKWPWSDWPDCIZWNPZDWSIWMWNXOWOWMXNWNCDEUOZUQURWRWNADCWQD - BQRSUSWPKCIZBKJZPZTZXDWPXRXQWOWMXRTWNXRBKBUTVAVGVBXDKWSIZTXTKWSVCYAXSWRXR - AKCWQKBQRVDVEVFWPXJGHWSWSXFWSIZXGWSIZPXFCIZBXFJZPZXGCIZBXGJZPZPZWPXJYBYFY - CYIWRYEAXFCWQXFBQRWRYHAXGCWQXGBQRVHWPYJXJWPYJPZXHWSIZXHXHJZXJYKXHCIZBXHJZ - PZYLWMWNYJYPWOWMYJPZYNYOYQWMYDYGYNWMYJVIWMYDYEYIVJWMYFYGYHVKXFXGCVLVMYJYO - WMYEYHYOYDYGYEYHPYOBXFXGVNVOVPVQVRVSWRYOAXHCWQXHBQRSXHVTXIYMFXHWSXEXHXHWH - WAWBWCWDWEWFWMWNWTXBXLPWIZWOWMXNYRXPGHFCDWSWGWJWKWL $. + adantl jca 3ad2antl1 ssid sseq1 rspcev sylancl ex biimtrid ralrimivv 3jca + wb isfbas2 syl 3ad2ant1 mpbir2and ) CUAIZBDJZBKLZMZBANZJZACUBZDUCUDIZWSDU + EZJZWSKLZKWSUFZFNZGNZHNZUGZJZFWSUHZHWSOGWSOZMZXBWPWSCXAWRACUICXAJCUJZDJDX + MEUKCDULUMUPUNWPXCXDXKWPWSDWPDCIZWNPZDWSIWMWNXOWOWMXNWNCDEUOZUQURWRWNADCW + QDBQRSUSWPKCIZBKJZPZTZXDWPXRXQWOWMXRTWNXRBKBUTVAVGVBXDKWSIZTXTKWSVCYAXSWR + XRAKCWQKBQRVDVEVFWPXJGHWSWSXFWSIZXGWSIZPXFCIZBXFJZPZXGCIZBXGJZPZPZWPXJYBY + FYCYIWRYEAXFCWQXFBQRWRYHAXGCWQXGBQRVHWPYJXJWPYJPZXHWSIZXHXHJZXJYKXHCIZBXH + JZPZYLWMWNYJYPWOWMYJPZYNYOYQWMYDYGYNWMYJVIWMYDYEYIVJWMYFYGYHVKXFXGCVLVMYJ + YOWMYEYHYOYDYGYEYHPYOBXFXGVNVOVPVQVRVSWRYOAXHCWQXHBQRSXHVTXIYMFXHWSXEXHXH + WAWBWCWDWEWFWGWMWNWTXBXLPWHZWOWMXNYRXPGHFCDWSWIWJWKWL $. $} ${ @@ -309255,9 +310253,9 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) $) filtop $p |- ( F e. ( Fil ` X ) -> X e. F ) $= ( vx cfil cfv wcel c0 wne cfbas filfbas fbasne0 syl cv wex n0 wss filelss - wa wi mpd ssid filss 3exp2 imp mpi ex exlimdv syl5bi ) ABDEFZAGHZBAFZUIAB - IEFUJABJBAKLUJCMZAFZCNUIUKCAOUIUMUKCUIUMUKUIUMRZULBPZUKULABQUNBBPZUOUKSZB - UAUIUMUPUQSUIUMUPUOUKULBABUBUCUDUETUFUGUHT $. + wa wi mpd ssid filss 3exp2 imp mpi ex exlimdv biimtrid ) ABDEFZAGHZBAFZUI + ABIEFUJABJBAKLUJCMZAFZCNUIUKCAOUIUMUKCUIUMUKUIUMRZULBPZUKULABQUNBBPZUOUKS + ZBUAUIUMUPUQSUIUMUPUOUKULBABUBUCUDUETUFUGUHT $. $} ${ @@ -309312,15 +310310,15 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, ( wss c0 wcel wa wsbc isfildlem cpw wn w3a cv wrex wi wral cin cfil velpw biimpri adantr syl6bi ssrdv simpr mtod ssid jctil mpbird 3jca elpwi simp2 jctild adantld wb 3ad2ant1 3imtr4d 3expa impancom rexlimdva syl5 ralrimiv - cfv ex ssinss1 ad2antrr a1i 3expb expimpd syl5bi anbi12d ralrimivv isfil2 - an4 jcad syl3anbrc ) AGFUAZOZPGQZUBZFGQZUCEUDZDUDZOZEGUEWMGQZUFZDWGUGWMWL - UHZGQZEGUGDGUGGFUIVMQAWHWJWKACGWGACUDZGQWSFOZBRWSWGQZIWTXABXAWTCFUJUKULUM - UNAWIBCPSZLAWIPFOZXBRXBABCFPGHIJTXCXBUOUMUPAWKFFOZBCFSZRAXEXDKFUQURABCFFG - HIJTUSUTAWPDWGWMWGQWMFOZAWPWMFVAAXFWPAXFRZWNWOEGXGWNWLGQZWOAXFWNXHWOUFAXF - WNUCZWLFOZBCWLSZRZXFBCWMSZRZXHWOXIXKXNXJXIXKXMXFMAXFWNVBVCVDAXFXHXLVEWNAB - CFWLGHIJTZVFAXFWOXNVEWNABCFWMGHIJTZVFVGVHVIVJVNVKVLAWRDEGGAXNXLRZWQFOZBCW - QSZRWOXHRWRAXQXRXSXQXRUFAXFXRXMXLWMWLFVOVPVQXQXFXJRZXMXKRZRAXSXFXMXJXKWDA - XTYAXSAXFXJYAXSUFNVRVSVTWEAWOXNXHXLXPXOWAABCFWQGHIJTVGWBDEGFWCWF $. + cfv ex ssinss1 ad2antrr a1i 3expb expimpd biimtrid jcad anbi12d ralrimivv + an4 isfil2 syl3anbrc ) AGFUAZOZPGQZUBZFGQZUCEUDZDUDZOZEGUEWMGQZUFZDWGUGWM + WLUHZGQZEGUGDGUGGFUIVMQAWHWJWKACGWGACUDZGQWSFOZBRWSWGQZIWTXABXAWTCFUJUKUL + UMUNAWIBCPSZLAWIPFOZXBRXBABCFPGHIJTXCXBUOUMUPAWKFFOZBCFSZRAXEXDKFUQURABCF + FGHIJTUSUTAWPDWGWMWGQWMFOZAWPWMFVAAXFWPAXFRZWNWOEGXGWNWLGQZWOAXFWNXHWOUFA + XFWNUCZWLFOZBCWLSZRZXFBCWMSZRZXHWOXIXKXNXJXIXKXMXFMAXFWNVBVCVDAXFXHXLVEWN + ABCFWLGHIJTZVFAXFWOXNVEWNABCFWMGHIJTZVFVGVHVIVJVNVKVLAWRDEGGAXNXLRZWQFOZB + CWQSZRWOXHRWRAXQXRXSXQXRUFAXFXRXMXLWMWLFVOVPVQXQXFXJRZXMXKRZRAXSXFXMXJXKW + DAXTYAXSAXFXJYAXSUFNVRVSVTWAAWOXNXHXLXPXOWBABCFWQGHIJTVGWCDEGFWEWF $. $} ${ @@ -309462,18 +310460,18 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, cfbas wne w3o notbid w3a 3ioran df-3an bitr2i bitr4di nesym ralbii ralnex bitri fbasfip anim12i biantrurd bitr2id ssfii ssralv ralimdv sylan9 ineq1 neeq1d ineq2 cbvral2vw fbssfi r19.29 ss2in sseq2 ss0 syl6bi syl5com com13 - necon3d ex rexlimivw impancom expimpd com12 syl2an an4s ralrimdvva syl5bi - imp impbid 3bitrd ) CEUEIZJZDFUEIZJZKZLCDUAUBIJZMZLCUBIZJZMZLDUBIZJZMZKZL - ANZBNZUCZOZBXAPZAWRPZMZKZXGLUFZBXAQZAWRQZXMBDQZACQZWOWQWSXBXJUGZMZXLWOWPX - RABLCDWKWMUDUHXSWTXCXKUIXLWSXBXJUJWTXCXKUKULUMXOXKWOXLXOXIMZAWRQXKXNXTAWR - XNXHMZBXAQXTXMYABXAXGLUNUOXHBXAUPUQUOXIAWRUPUQWOXDXKWLWTWNXCCEURDFURUSUTV - AWOXOXQWLXOXNACQZWNXQWLCWRRXOYBSCWKVBXNACWRVCTWNXNXPACWNDXARXNXPSDWMVBXMB - DXAVCTVDVEXQGNZHNZUCZLUFZHDQZGCQZWOXOXMYFYCXFUCZLUFABGHCDXEYCOXGYILXEYCXF - VFVGXFYDOYIYELXFYDYCVHVGVIWOYHXMABWRXAWLXEWRJZWNXFXAJZYHXMSZWLYJKYCXERZGC - PZYDXFRZHDPZYLWNYKKGXECEVJHXFDFVJYHYNYPKXMYHYNYPXMYHYNKYGYMKZGCPYPXMSZYGY - MGCVKYQYRGCYGYPYMXMYGYPKYFYOKZHDPYMXMSZYFYOHDVKYSYTHDYFYOYTYMYOYFXMYMYOYF - XMSYMYOKZXGLYELUUAYEXGRZXGLOZYELOZYCXEYDXFVLUUCUUBYELRUUDXGLYEVMYEVNVOVPV - RVSVQWHVTTWAVTTWBWCWDWEWFWGWIWJ $. + necon3d ex imp rexlimivw impancom expimpd syl2an an4s ralrimdvva biimtrid + com12 impbid 3bitrd ) CEUEIZJZDFUEIZJZKZLCDUAUBIJZMZLCUBIZJZMZLDUBIZJZMZK + ZLANZBNZUCZOZBXAPZAWRPZMZKZXGLUFZBXAQZAWRQZXMBDQZACQZWOWQWSXBXJUGZMZXLWOW + PXRABLCDWKWMUDUHXSWTXCXKUIXLWSXBXJUJWTXCXKUKULUMXOXKWOXLXOXIMZAWRQXKXNXTA + WRXNXHMZBXAQXTXMYABXAXGLUNUOXHBXAUPUQUOXIAWRUPUQWOXDXKWLWTWNXCCEURDFURUSU + TVAWOXOXQWLXOXNACQZWNXQWLCWRRXOYBSCWKVBXNACWRVCTWNXNXPACWNDXARXNXPSDWMVBX + MBDXAVCTVDVEXQGNZHNZUCZLUFZHDQZGCQZWOXOXMYFYCXFUCZLUFABGHCDXEYCOXGYILXEYC + XFVFVGXFYDOYIYELXFYDYCVHVGVIWOYHXMABWRXAWLXEWRJZWNXFXAJZYHXMSZWLYJKYCXERZ + GCPZYDXFRZHDPZYLWNYKKGXECEVJHXFDFVJYHYNYPKXMYHYNYPXMYHYNKYGYMKZGCPYPXMSZY + GYMGCVKYQYRGCYGYPYMXMYGYPKYFYOKZHDPYMXMSZYFYOHDVKYSYTHDYFYOYTYMYOYFXMYMYO + YFXMSYMYOKZXGLYELUUAYEXGRZXGLOZYELOZYCXEYDXFVLUUCUUBYELRUUDXGLYEVMYEVNVOV + PVRVSVQVTWATWBWATWCWHWDWEWFWGWIWJ $. $} ${ @@ -309668,31 +310666,31 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, ( vx vy cfil cfv wcel cuni c0 cun cin wss wral wa wn wo elun syl wceq cvv csn cconn id filunibas fveq2d eleqtrrd ctop ccld cpr cv wi wal wex simpll nss ssel2 adantll sylib orcomd ord impr uniss ad2antlr uniun unisn uneq2i - 0ex un0 3eqtrri sseqtrrdi elssuni ad2antrl filss syl13anc elun1 ex syl5bi - exlimdv uni0b ssun2 sselii eleq1 mpbiri sylbir pm2.61d2 alrimiv w3a filin - snid 3expa ralrimiva elsni ineq2 eqtrdi eqeltrdi rgen ralun sylancl ineq1 - in0 0in ralrimivw wb p0ex unexg mpan2 istopg mpbir2and cdif cfbas filfbas - cldopn fbncp sylan pm2.21d a1i13 jaod adantl ssdif0 biimpri eqss simplbi2 - imp syl2im syl5 orim12d expimpd elin elpr 3imtr4g ssrdv isconn2 sylanbrc - vex ) ABEFZGZAAHZEFZGZAIUAZJZUBGZYPAYOYRYPUCYPYQBEABUDUEUFYSUUAUGGZUUAUUA - UHFZKZIYQUIZLUUBYSUUCCUJZUUALZUUGHZUUAGZUKZCULZUUGDUJZKZUUAGZDUUAMZCUUAMZ - YSUUKCYSUUHUUJYSUUHNZUUGYTLZUUJUUSOUUMUUGGZUUMYTGZOZNZDUMUURUUJDUUGYTUOUU - RUVCUUJDUURUVCUUJUURUVCNZUUIAGZUUJUVDYSUUMAGZUUIYQLUUMUUILZUVEYSUUHUVCUNU - URUUTUVBUVFUURUUTNZUVAUVFUVHUVFUVAUVHUUMUUAGZUVFUVAPUUHUUTUVIYSUUGUUAUUMU - PUQUUMAYTQURUSUTVAUVDUUIUUAHZYQUUHUUIUVJLYSUVCUUGUUAVBVCUVJYQYTHZJYQIJYQA - YTVDUVKIYQIVGVEVFYQVHVIZVJUUTUVGUURUVBUUMUUGVKVLUUMUUIAYQVMVNUUIAYTVORVPV - RVQUUSUUIISZUUJUUGVSUVMUUJIUUAGYTUUAIYTAVTIVGWIWAZUUIIUUAWBWCWDWEVPWFYSUU - PCAMUUPCYTMUUQYSUUPCAYSUUGAGZNZUUODAMUUODYTMUUPUVPUUODAYSUVOUVFUUOYSUVOUV - FWGUUNAGUUOUUGUUMAYQWHUUNAYTVORWJWKUUODYTUVAUUMISZUUOUUMIWLUVQUUNIUUAUVQU - UNUUGIKIUUMIUUGWMUUGWTWNUVNWORWPUUODAYTWQWRWKUUPCYTUUGYTGZUUGISZUUPUUGIWL - ZUVSUUODUUAUVSUUNIUUAUVSUUNIUUMKIUUGIUUMWSUUMXAWNUVNWOXBRWPUUPCAYTWQWRYSU - UATGZUUCUULUUQNXCYSYTTGUWAXDAYTYRTXEXFCDTUUAXGRXHYSCUUEUUFYSUUGUUAGZUUGUU - DGZNUVSUUGYQSZPZUUGUUEGUUGUUFGYSUWBUWCUWEUWCYQUUGXIZAGZUWFYTGZPZYSUWBNZUW - EUWCUWFUUAGUWIUUGUUAYQUVLXLUWFAYTQURUWJUWGUVSUWHUWDYSUWBUWGUVSUKZUWBUVOUV - RPYSUWKUUGAYTQYSUVOUWKUVRYSUVOUWKUVPUWGUVSYSAYQXJFGUVOUWGOAYQXKUUGYQAYQXM - XNXOVPYSUVRUWGUVSUVTXPXQVQYCUWHUWFISZUWJUWDUWFIWLUWJUUGYQLZUWLYQUUGLZUWDU - WBUWMYSUWBUUGUVJYQUUGUUAVKUVLVJXRUWNUWLYQUUGXSXTUWDUWMUWNUUGYQYAYBYDYEYFY - EYGUUGUUAUUDYHUUGIYQCYNYIYJYKUUAYQUVLYLYMR $. + 0ex 3eqtrri sseqtrrdi elssuni ad2antrl filss syl13anc ex exlimdv biimtrid + un0 elun1 uni0b ssun2 snid sselii eleq1 mpbiri pm2.61d2 alrimiv w3a filin + sylbir 3expa ralrimiva elsni ineq2 in0 eqtrdi eqeltrdi rgen ralun sylancl + ineq1 0in ralrimivw p0ex unexg mpan2 istopg mpbir2and cdif cldopn filfbas + wb cfbas fbncp sylan pm2.21d jaod imp adantl ssdif0 biimpri eqss simplbi2 + a1i13 syl2im syl5 orim12d expimpd elin vex 3imtr4g ssrdv isconn2 sylanbrc + elpr ) ABEFZGZAAHZEFZGZAIUAZJZUBGZYPAYOYRYPUCYPYQBEABUDUEUFYSUUAUGGZUUAUU + AUHFZKZIYQUIZLUUBYSUUCCUJZUUALZUUGHZUUAGZUKZCULZUUGDUJZKZUUAGZDUUAMZCUUAM + ZYSUUKCYSUUHUUJYSUUHNZUUGYTLZUUJUUSOUUMUUGGZUUMYTGZOZNZDUMUURUUJDUUGYTUOU + URUVCUUJDUURUVCUUJUURUVCNZUUIAGZUUJUVDYSUUMAGZUUIYQLUUMUUILZUVEYSUUHUVCUN + UURUUTUVBUVFUURUUTNZUVAUVFUVHUVFUVAUVHUUMUUAGZUVFUVAPUUHUUTUVIYSUUGUUAUUM + UPUQUUMAYTQURUSUTVAUVDUUIUUAHZYQUUHUUIUVJLYSUVCUUGUUAVBVCUVJYQYTHZJYQIJYQ + AYTVDUVKIYQIVGVEVFYQVQVHZVIUUTUVGUURUVBUUMUUGVJVKUUMUUIAYQVLVMUUIAYTVRRVN + VOVPUUSUUIISZUUJUUGVSUVMUUJIUUAGYTUUAIYTAVTIVGWAWBZUUIIUUAWCWDWIWEVNWFYSU + UPCAMUUPCYTMUUQYSUUPCAYSUUGAGZNZUUODAMUUODYTMUUPUVPUUODAYSUVOUVFUUOYSUVOU + VFWGUUNAGUUOUUGUUMAYQWHUUNAYTVRRWJWKUUODYTUVAUUMISZUUOUUMIWLUVQUUNIUUAUVQ + UUNUUGIKIUUMIUUGWMUUGWNWOUVNWPRWQUUODAYTWRWSWKUUPCYTUUGYTGZUUGISZUUPUUGIW + LZUVSUUODUUAUVSUUNIUUAUVSUUNIUUMKIUUGIUUMWTUUMXAWOUVNWPXBRWQUUPCAYTWRWSYS + UUATGZUUCUULUUQNXKYSYTTGUWAXCAYTYRTXDXECDTUUAXFRXGYSCUUEUUFYSUUGUUAGZUUGU + UDGZNUVSUUGYQSZPZUUGUUEGUUGUUFGYSUWBUWCUWEUWCYQUUGXHZAGZUWFYTGZPZYSUWBNZU + WEUWCUWFUUAGUWIUUGUUAYQUVLXIUWFAYTQURUWJUWGUVSUWHUWDYSUWBUWGUVSUKZUWBUVOU + VRPYSUWKUUGAYTQYSUVOUWKUVRYSUVOUWKUVPUWGUVSYSAYQXLFGUVOUWGOAYQXJUUGYQAYQX + MXNXOVNYSUVRUWGUVSUVTYCXPVPXQUWHUWFISZUWJUWDUWFIWLUWJUUGYQLZUWLYQUUGLZUWD + UWBUWMYSUWBUUGUVJYQUUGUUAVJUVLVIXRUWNUWLYQUUGXSXTUWDUWMUWNUUGYQYAYBYDYEYF + YEYGUUGUUAUUDYHUUGIYQCYIYNYJYKUUAYQUVLYLYMR $. $} ${ @@ -309714,31 +310712,31 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, ax-mp bitr4i fbasssin 3expb 3ad2antl1 adantrr rspceeqv mpan2 ad2antrl vex funimaex bitrid ad2antrr mpbird imass2 inss1 inss2 ssini ineq12 sseqtrrid ad2antll ad2antlr sstrd sseq1 rspcev syl2anc adantlrl rexlimddv rexlimdvv - exp32 syl5bi ralrimivv 3jca isfbas2 3ad2ant3 mpbir2and ) BFUIUEKZFGDVBZGE - KZUFZCGUIUEKZCGUGZLZCMUHZMCUJZUANZUBNZUCNZUKZLZUACOZUCCULUBCULZUFZUUPCABD - ANZUMZUNZUOZUURHUUPBUURUVLUUPABUVKUURUUPUVJBKZPZUVKGEUUMUUNUUOUVNUPUVOUUN - UVKGLUUMUUNUUOUVNUQFGDUVJURUSUTVAVCVDUUPUUTUVAUVHUUPCUVMMCUVMQUUPHVEUUPUV - LVFZMUHUVMMUHUUPUVPBMUUPDVGZUVKRKZABULUVPBQUUNUUMUVQUUOFGDVMVHZUVQUVRABDU - VJBVIVJABUVKRVKVNUUMUUNBMUHUUOFBVLVOVPUVPMUVMMUVLVQVRVSVPUUPMUVKQZSZABULZ - UVAUUPUWAABUUPUVNUVJFLZUVJMQZSZPZUWAUUPUVNUWCUWEUUMUUNUVNUWCVTUUOUUMUVNUW - CFBUVJWAWDVOUUMUUNUVNUWEVTUUOUUMUWDUVNUUMUVNSUWDMBKZSFBWBUWDUVNUWGUVJMBWE - WCWFWGVOWHUUPUWFDVFZUVJUKZMQZSZUWAUUPUWCUWEUWKUUPUWCPZUWJUWDUWLUWJUWDUWLU - WIUVJMUWLUVJUWHLZUWIUVJQUUPUWMUWCUUPUWHFUVJUUNUUMUWHFQUUOFGDWOVHWIWJUVJUW - HWKVSWLWMWNWPUVTUWJUVTUVKMQUWJMUVKWQDUVJWRXFWSWTXAXBMCKZSUVTABOZSUVAUWBUW - NUWOUWNMUVMKZUWOCUVMMHXCMRKUWPUWOTXDABUVKMUVLRUVLXGZXEXQXFWSMCXHUVTABXIXJ - XKUUPUVGUBUCCCUVCCKZUVDCKZPZUVCDINZUMZQZUVDDJNZUMZQZPZJBOIBOZUUPUVGUWTUXC - IBOZUXFJBOZPUXHUWRUXIUWSUXJUWRUVCUVMKZUXICUVMUVCHXCUXKUXITUBIBUXBUVCUVLRA - IBUVKUXBUVJUXADXLXMXEXNXFUWSUVDUVMKZUXJCUVMUVDHXCUXLUXJTUCJBUXEUVDUVLRAJB - UVKUXEUVJUXDDXLXMXEXNXFXOUXCUXFIJBBXPXRUUPUXGUVGIJBBUUPUXABKZUXDBKZPZUXGU - VGUUPUXOUXGPPUDNZUXAUXDUKZLZUVGUDBUUPUXOUXRUDBOZUXGUUMUUNUXOUXSUUOUUMUXMU - XNUXSUDUXAUXDBFXSXTYAYBUUPUXGUXPBKZUXRPZUVGUXOUUPUXGPZUYAPZDUXPUMZCKZUYDU - VELZUVGUYCUYEUYDUVKQABOZUXTUYGUYBUXRUXTUYDUYDQUYGUYDXGAUXPBUVKUYDUYDUVJUX - PDXLYCYDYEUUPUYEUYGTUXGUYAUYEUYDUVMKZUUPUYGCUVMUYDHXCUUPUVQUYDRKUYHUYGTUV - SDUXPUDYFYGABUVKUYDUVLRUWQXEVNYHYIYJUYCUYDDUXQUMZUVEUXRUYDUYILUYBUXTUXPUX - QDYKYQUYCUXBUXEUKZUYIUVEUYIUXBUXEUXQUXALUYIUXBLUXAUXDYLUXQUXADYKXQUXQUXDL - UYIUXELUXAUXDYMUXQUXDDYKXQYNUXGUVEUYJQUUPUYAUVCUXBUVDUXEYOYRYPYSUVFUYFUAU - YDCUVBUYDUVEYTUUAUUBUUCUUDUUFUUEUUGUUHUUIUUOUUMUUQUUSUVIPTUUNUBUCUAEGCUUJ - UUKUUL $. + exp32 biimtrid ralrimivv 3jca isfbas2 3ad2ant3 mpbir2and ) BFUIUEKZFGDVBZ + GEKZUFZCGUIUEKZCGUGZLZCMUHZMCUJZUANZUBNZUCNZUKZLZUACOZUCCULUBCULZUFZUUPCA + BDANZUMZUNZUOZUURHUUPBUURUVLUUPABUVKUURUUPUVJBKZPZUVKGEUUMUUNUUOUVNUPUVOU + UNUVKGLUUMUUNUUOUVNUQFGDUVJURUSUTVAVCVDUUPUUTUVAUVHUUPCUVMMCUVMQUUPHVEUUP + UVLVFZMUHUVMMUHUUPUVPBMUUPDVGZUVKRKZABULUVPBQUUNUUMUVQUUOFGDVMVHZUVQUVRAB + DUVJBVIVJABUVKRVKVNUUMUUNBMUHUUOFBVLVOVPUVPMUVMMUVLVQVRVSVPUUPMUVKQZSZABU + LZUVAUUPUWAABUUPUVNUVJFLZUVJMQZSZPZUWAUUPUVNUWCUWEUUMUUNUVNUWCVTUUOUUMUVN + UWCFBUVJWAWDVOUUMUUNUVNUWEVTUUOUUMUWDUVNUUMUVNSUWDMBKZSFBWBUWDUVNUWGUVJMB + WEWCWFWGVOWHUUPUWFDVFZUVJUKZMQZSZUWAUUPUWCUWEUWKUUPUWCPZUWJUWDUWLUWJUWDUW + LUWIUVJMUWLUVJUWHLZUWIUVJQUUPUWMUWCUUPUWHFUVJUUNUUMUWHFQUUOFGDWOVHWIWJUVJ + UWHWKVSWLWMWNWPUVTUWJUVTUVKMQUWJMUVKWQDUVJWRXFWSWTXAXBMCKZSUVTABOZSUVAUWB + UWNUWOUWNMUVMKZUWOCUVMMHXCMRKUWPUWOTXDABUVKMUVLRUVLXGZXEXQXFWSMCXHUVTABXI + XJXKUUPUVGUBUCCCUVCCKZUVDCKZPZUVCDINZUMZQZUVDDJNZUMZQZPZJBOIBOZUUPUVGUWTU + XCIBOZUXFJBOZPUXHUWRUXIUWSUXJUWRUVCUVMKZUXICUVMUVCHXCUXKUXITUBIBUXBUVCUVL + RAIBUVKUXBUVJUXADXLXMXEXNXFUWSUVDUVMKZUXJCUVMUVDHXCUXLUXJTUCJBUXEUVDUVLRA + JBUVKUXEUVJUXDDXLXMXEXNXFXOUXCUXFIJBBXPXRUUPUXGUVGIJBBUUPUXABKZUXDBKZPZUX + GUVGUUPUXOUXGPPUDNZUXAUXDUKZLZUVGUDBUUPUXOUXRUDBOZUXGUUMUUNUXOUXSUUOUUMUX + MUXNUXSUDUXAUXDBFXSXTYAYBUUPUXGUXPBKZUXRPZUVGUXOUUPUXGPZUYAPZDUXPUMZCKZUY + DUVELZUVGUYCUYEUYDUVKQABOZUXTUYGUYBUXRUXTUYDUYDQUYGUYDXGAUXPBUVKUYDUYDUVJ + UXPDXLYCYDYEUUPUYEUYGTUXGUYAUYEUYDUVMKZUUPUYGCUVMUYDHXCUUPUVQUYDRKUYHUYGT + UVSDUXPUDYFYGABUVKUYDUVLRUWQXEVNYHYIYJUYCUYDDUXQUMZUVEUXRUYDUYILUYBUXTUXP + UXQDYKYQUYCUXBUXEUKZUYIUVEUYIUXBUXEUXQUXALUYIUXBLUXAUXDYLUXQUXADYKXQUXQUX + DLUYIUXELUXAUXDYMUXQUXDDYKXQYNUXGUVEUYJQUUPUYAUVCUXBUVDUXEYOYRYPYSUVFUYFU + AUYDCUVBUYDUVEYTUUAUUBUUCUUDUUFUUEUUGUUHUUIUUOUUMUUQUUSUVIPTUUNUBUCUAEGCU + UJUUKUUL $. $} ${ @@ -309749,31 +310747,31 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, filuni $p |- ( ( F C_ ( Fil ` X ) /\ F =/= (/) /\ A. f e. F A. g e. F ( f u. g ) e. F ) -> U. F e. ( Fil ` X ) ) $= ( vx vy vh c0 cv wcel wel wrex eluni2 wi wa ex syl wsbc sbcel1v bitri cfv - cfil wss wne cun wral w3a cuni cvv ssel2 filelss 3ad2ant1 syl5bi pm4.71rd - rexlimdva fvprc necon1ai 3adant3 filtop a1d ralimdva r19.2z sylan9 3impia - ssn0 sylibr wn 0nelfil ralrimiva notbii ralnex bitr4i simp13 r19.29 simp1 - simpl syl2an simprrr simpl2 simpl3 syl13anc expr reximdva syl3an1 3imtr4g - filss syld cin simp11 elun1 elun2 anim12i wceq eleq2 anbi12d rspcev an12s - sylan2 ad2antlr syl9r impr ancom2s rexlimiva 3expib syl2im syland anbi12i - imp filin isfild ) CDUBUAZUCZCHUDZAIZBIZUEZCJZBCUFZACUFZUGZEIZCUHZJZEFEDY - BUIXTYCYADUCZYCEAKZACLZXTYDAYACMZXLXMYFYDNXSXLYEYDACXLXNCJZOZXNXKJZYEYDNC - XKXNUJZYJYEYDYAXNDUKPQUOULUMUNXLXMDUIJZXSXLXMOXKHUDYLCXKVEYLXKHDUBUPUQQUR - XTDYBJZYCEDRXTDXNJZACLZYMXLXMXSYOXLXSYNACUFZXMYOXLXRYNACYIYNXRYIYJYNYKXND - USQUTVAXMYPYOYNACVBPVCVDADCMVFEDYBSVFXTHXNJZVGZACUFZYCEHRZVGZXLXMYSXSXLYR - ACYIYJYRYKXNDVHQVIULUUAYQACLZVGYSYTUUBYTHYBJUUBEHYBSAHCMTVJYQACVKVLVFXTFI - ZDUCZYAUUCUCZUGZYFFAKZACLZYCEYARZYCEUUCRZUUFYFXRYEOZACLZUUHUUFXSYFUULNXLX - MXSUUDUUEVMXSYFUULXRYEACVNPQXTXLUUDUUEUULUUHNXLXMXSVOXLUUDUUEUGZUUKUUGACU - UMYHUUKUUGUUMYHUUKOZOYJYEUUDUUEUUGUUMXLYHYJUUNXLUUDUUEVOYHUUKVPYKVQUUMYHX - RYEVRXLUUDUUEUUNVSXLUUDUUEUUNVTYAUUCXNDWFWAWBWCWDWGUUIYCYFEYAYBSZYGTUUJUU - CYBJUUHEUUCYBSAUUCCMTZWEXTUUDYDUGZUUHEBKZBCLZOUUCYAWHZGIZJZGCLZUUJUUIOYCE - UUTRZUUQUUHXRUUGOZACLZUUSUVCUUQXSUUHUVFNXLXMXSUUDYDVMXSUUHUVFXRUUGACVNPQU - UQXLUVFUUSOFGKZEGKZOZGCLZUVCXLXMXSUUDYDWIUVFUUSUVJUVEUUSUVJNZACYHUUGXRUVK - YHUUGXRUVKXRUUSXQUUROZBCLZYHUUGOZUVJXRUUSUVMXQUURBCVNPUVNUVLUVJBCUUGUVLUV - JNYHXOCJUUGUVLUVJXQUUGUURUVJUUGUUROXQUUCXPJZYAXPJZOZUVJUUGUVOUURUVPUUCXNX - OWJYAXOXNWKWLUVIUVQGXPCUVAXPWMUVGUVOUVHUVPUVAXPUUCWNUVAXPYAWNWOWPWRWQPWSU - OWTXAXBXCXHXLUVIUVBGCXLUVACJOUVAXKJZUVIUVBNCXKUVAUJUVRUVGUVHUVBUUCYAUVADX - IXDQWCXEXFUUJUUHUUIUUSUUPUUIYCUUSUUOBYACMTXGUVDUUTYBJUVCEUUTYBSGUUTCMTWEX - J $. + cfil wss wne cun wral w3a cuni ssel2 rexlimdva 3ad2ant1 biimtrid pm4.71rd + cvv filelss ssn0 fvprc necon1ai 3adant3 filtop a1d ralimdva r19.2z sylan9 + 3impia sylibr wn ralrimiva notbii ralnex bitr4i simp13 r19.29 simp1 simpl + 0nelfil syl2an simprrr simpl2 simpl3 filss syl13anc expr reximdva syl3an1 + syld 3imtr4g simp11 elun1 elun2 anim12i eleq2 anbi12d rspcev sylan2 an12s + cin wceq ad2antlr syl9r impr ancom2s rexlimiva filin 3expib syl2im syland + imp anbi12i isfild ) CDUBUAZUCZCHUDZAIZBIZUEZCJZBCUFZACUFZUGZEIZCUHZJZEFE + DYBUNXTYCYADUCZYCEAKZACLZXTYDAYACMZXLXMYFYDNXSXLYEYDACXLXNCJZOZXNXKJZYEYD + NCXKXNUIZYJYEYDYAXNDUOPQUJUKULUMXLXMDUNJZXSXLXMOXKHUDYLCXKUPYLXKHDUBUQURQ + USXTDYBJZYCEDRXTDXNJZACLZYMXLXMXSYOXLXSYNACUFZXMYOXLXRYNACYIYNXRYIYJYNYKX + NDUTQVAVBXMYPYOYNACVCPVDVEADCMVFEDYBSVFXTHXNJZVGZACUFZYCEHRZVGZXLXMYSXSXL + YRACYIYJYRYKXNDVPQVHUKUUAYQACLZVGYSYTUUBYTHYBJUUBEHYBSAHCMTVIYQACVJVKVFXT + FIZDUCZYAUUCUCZUGZYFFAKZACLZYCEYARZYCEUUCRZUUFYFXRYEOZACLZUUHUUFXSYFUULNX + LXMXSUUDUUEVLXSYFUULXRYEACVMPQXTXLUUDUUEUULUUHNXLXMXSVNXLUUDUUEUGZUUKUUGA + CUUMYHUUKUUGUUMYHUUKOZOYJYEUUDUUEUUGUUMXLYHYJUUNXLUUDUUEVNYHUUKVOYKVQUUMY + HXRYEVRXLUUDUUEUUNVSXLUUDUUEUUNVTYAUUCXNDWAWBWCWDWEWFUUIYCYFEYAYBSZYGTUUJ + UUCYBJUUHEUUCYBSAUUCCMTZWGXTUUDYDUGZUUHEBKZBCLZOUUCYAWQZGIZJZGCLZUUJUUIOY + CEUUTRZUUQUUHXRUUGOZACLZUUSUVCUUQXSUUHUVFNXLXMXSUUDYDVLXSUUHUVFXRUUGACVMP + QUUQXLUVFUUSOFGKZEGKZOZGCLZUVCXLXMXSUUDYDWHUVFUUSUVJUVEUUSUVJNZACYHUUGXRU + VKYHUUGXRUVKXRUUSXQUUROZBCLZYHUUGOZUVJXRUUSUVMXQUURBCVMPUVNUVLUVJBCUUGUVL + UVJNYHXOCJUUGUVLUVJXQUUGUURUVJUUGUUROXQUUCXPJZYAXPJZOZUVJUUGUVOUURUVPUUCX + NXOWIYAXOXNWJWKUVIUVQGXPCUVAXPWRUVGUVOUVHUVPUVAXPUUCWLUVAXPYAWLWMWNWOWPPW + SUJWTXAXBXCXHXLUVIUVBGCXLUVACJOUVAXKJZUVIUVBNCXKUVAUIUVRUVGUVHUVBUUCYAUVA + DXDXEQWDXFXGUUJUUHUUIUUSUUPUUIYCUUSUUOBYACMTXIUVDUUTYBJUVCEUUTYBSGUUTCMTW + GXJ $. $} ${ @@ -310101,31 +311099,31 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, A. f e. ( Fil ` X ) ( F C_ f -> F = f ) ) ) $= ( vx vy cfv wcel cv wss wceq wi wral wa wn cin c0 wne cvv syl ad2antrr wo cufil cfil ufilfil ufilmax ralrimiva jca cdif cpw simpl velpw csn cun cfi - 3expia cfg simpll snex unexg sylancl ssfii cfbas filsspw biimpri ad2antlr - snssd unssd ssun2 vex snnz ssn0 mp2an a1i simpr ineq2 neeq1d ralsn ralbii - co sylibr wb filfbas simplr inss2 filtop adantr ineq1 rspcva sylan snfbas - sylancr syl3anc fbunfip syl2anc mpbird w3a fsubbas mpbir3and sstrd unssad - ssfg fgcl sseq2 eqeq2 rspcv 3syl mpid vsnid sselii sseldd syl5ibrcom syld - imbi12d eleq2 impancom an32s con3d rexnal nne filelss reldisj difss filss - wrex 3exp2 mpii imp sylbid syl5bi rexlimdva syl5bir orrd sylan2b sylanbrc - isufil impbii ) BCUBFGZBCUCFZGZBAHZIZBYTJZKZAYRLZMZYQYSUUDBCUDYQUUCAYRYQY - TYRGUUAUUBBYTCUEUOUFUGUUEYSDHZBGZCUUFUHZBGZUAZDCUIZLYQYSUUDUJUUEUUJDUUKUU - FUUKGZUUEUUFCIZUUJDCUKZUUEUUMMZUUGUUIUUOUUGNEHZUUFOZPQZEBLZNZUUIUUOUUSUUG - YSUUMUUDUUSUUGKYSUUMMZUUSUUDUUGUVAUUSMZUUDBCBUUFULZUMZUNFZUPVSZJZUUGUVBUU - DBUVFIZUVGUVBBUVCUVFUVBUVDUVEUVFUVBUVDRGZUVDUVEIUVBYSUVCRGUVIYSUUMUUSUQUU - FURBUVCYRRUSUTUVDRVASUVBUVECVBFZGZUVEUVFIUVBUVKUVDUUKIZUVDPQZPUVEGNZUVBBU - VCUUKYSBUUKIUUMUUSBCVCTUVBUUFUUKUUMUULYSUUSUULUUMUUNVDVEVFVGUVMUVBUVCUVDI - UVCPQUVMUVCBVHZUUFDVIZVJUVCUVDVKVLVMUVBUVNUUPYTOZPQZAUVCLZEBLZUVBUUSUVTUV - AUUSVNUVSUUREBUVRUURAUUFUVPYTUUFJUVQUUQPYTUUFUUPVOVPVQVRVTUVBBUVJGZUVCUVJ - GZUVNUVTWAYSUWAUUMUUSBCWBTUVBUUMUUFPQZCBGZUWBYSUUMUUSWCUVBCUUFOZUUFIUWEPQ - ZUWCCUUFWDUVAUWDUUSUWFYSUWDUUMBCWEZWFUURUWFECBUUPCJUUQUWEPUUPCUUFWGVPWHWI - UWEUUFVKWKYSUWDUUMUUSUWGTZUUFCBWJWLEABUVCCCWMWNWOUVBUWDUVKUVLUVMUVNWPWAUW - HUVDBCWQSWRZUVECXASWSZWTUVBUVKUVFYRGUUDUVHUVGKZKUWIUVECXBUUCUWKAUVFYRYTUV - FJUUAUVHUUBUVGYTUVFBXCYTUVFBXDXMXEXFXGUVBUUGUVGUUFUVFGUVBUVDUVFUUFUWJUUFU - VDGUVBUVCUVDUUFUVODXHXIVMXJBUVFUUFXNXKXLXOXPXQYSUUTUUIKUUDUUMUUTUURNZEBYD - YSUUIUUREBXRYSUWLUUIEBUWLUUQPJZYSUUPBGZMZUUIUUQPXSUWOUWMUUPUUHIZUUIUWOUUP - CIUWMUWPWAUUPBCXTUUPUUFCYASYSUWNUWPUUIKZYSUWNUUHCIZUWQCUUFYBYSUWNUWRUWPUU - IUUPUUHBCYCYEYFYGYHYIYJYKTXLYLYMUFDBCYOYNYP $. + 3expia co simpll vsnex unexg sylancl ssfii cfbas filsspw biimpri ad2antlr + cfg snssd unssd ssun2 vex snnz ssn0 mp2an simpr ineq2 neeq1d ralsn ralbii + a1i sylibr filfbas simplr inss2 filtop adantr ineq1 rspcva sylancr snfbas + wb sylan syl3anc fbunfip syl2anc mpbird w3a fsubbas mpbir3and ssfg unssad + sstrd fgcl sseq2 eqeq2 imbi12d rspcv vsnid sselii sseldd eleq2 syl5ibrcom + 3syl mpid syld impancom an32s con3d wrex rexnal nne filelss reldisj difss + filss 3exp2 imp sylbid biimtrid rexlimdva syl5bir sylan2b isufil sylanbrc + mpii orrd impbii ) BCUBFGZBCUCFZGZBAHZIZBYTJZKZAYRLZMZYQYSUUDBCUDYQUUCAYR + YQYTYRGUUAUUBBYTCUEUOUFUGUUEYSDHZBGZCUUFUHZBGZUAZDCUIZLYQYSUUDUJUUEUUJDUU + KUUFUUKGZUUEUUFCIZUUJDCUKZUUEUUMMZUUGUUIUUOUUGNEHZUUFOZPQZEBLZNZUUIUUOUUS + UUGYSUUMUUDUUSUUGKYSUUMMZUUSUUDUUGUVAUUSMZUUDBCBUUFULZUMZUNFZVFUPZJZUUGUV + BUUDBUVFIZUVGUVBBUVCUVFUVBUVDUVEUVFUVBUVDRGZUVDUVEIUVBYSUVCRGUVIYSUUMUUSU + QDURBUVCYRRUSUTUVDRVASUVBUVECVBFZGZUVEUVFIUVBUVKUVDUUKIZUVDPQZPUVEGNZUVBB + UVCUUKYSBUUKIUUMUUSBCVCTUVBUUFUUKUUMUULYSUUSUULUUMUUNVDVEVGVHUVMUVBUVCUVD + IUVCPQUVMUVCBVIZUUFDVJZVKUVCUVDVLVMVSUVBUVNUUPYTOZPQZAUVCLZEBLZUVBUUSUVTU + VAUUSVNUVSUUREBUVRUURAUUFUVPYTUUFJUVQUUQPYTUUFUUPVOVPVQVRVTUVBBUVJGZUVCUV + JGZUVNUVTWJYSUWAUUMUUSBCWATUVBUUMUUFPQZCBGZUWBYSUUMUUSWBUVBCUUFOZUUFIUWEP + QZUWCCUUFWCUVAUWDUUSUWFYSUWDUUMBCWDZWEUURUWFECBUUPCJUUQUWEPUUPCUUFWFVPWGW + KUWEUUFVLWHYSUWDUUMUUSUWGTZUUFCBWIWLEABUVCCCWMWNWOUVBUWDUVKUVLUVMUVNWPWJU + WHUVDBCWQSWRZUVECWSSXAZWTUVBUVKUVFYRGUUDUVHUVGKZKUWIUVECXBUUCUWKAUVFYRYTU + VFJUUAUVHUUBUVGYTUVFBXCYTUVFBXDXEXFXLXMUVBUUGUVGUUFUVFGUVBUVDUVFUUFUWJUUF + UVDGUVBUVCUVDUUFUVODXGXHVSXIBUVFUUFXJXKXNXOXPXQYSUUTUUIKUUDUUMUUTUURNZEBX + RYSUUIUUREBXSYSUWLUUIEBUWLUUQPJZYSUUPBGZMZUUIUUQPXTUWOUWMUUPUUHIZUUIUWOUU + PCIUWMUWPWJUUPBCYAUUPUUFCYBSYSUWNUWPUUIKZYSUWNUUHCIZUWQCUUFYCYSUWNUWRUWPU + UIUUPUUHBCYDYEYNYFYGYHYIYJTXNYOYKUFDBCYLYMYP $. $( An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.) $) @@ -310177,21 +311175,21 @@ contains the choice as a hypothesis (in the assumption that ` ~P ~P X ` cfil ccrd cdm wpss wn crab cufil crpss wor w3a cuni wal simpr rabss velpw filsspw a1d mprgbir ssnum sylancl ssid jctr adantr cun simpr1 ssrab sylib ne0d simpld simpr2 simpr3 sorpssun ralrimivva syl filuni syl3anc n0 ssel2 - wex simprd ssuni sylancom ex exlimdv syl5bi elrabd alrimiv zornn0g ralrab - sylc weq simpll sstr2 imim1d df-pss simplbi2 necon1bd a2i ralimdv adantll - syl6 imp isufil2 sylanbrc simplr jca syl2anb reximi2 ) BCUAGZHZCIZIZUBUCZ - HZJZAKZDKZUDZUEZDBEKZLZEXIUFZMZAYBNZBXPLZACUGGZNXOYBXMHZYBOPZFKZYBLZYIOPZ - YIUHUIZUJZYIUKZYBHZQZFULZYDXOXNYBXLLZYGXJXNUMYRYAXTXLHZQEXIYAEXIXLUNXTXIH - ZYSYAYTXTXKLYSXTCUPEXKUORUQURXLYBUSUTXJYHXNXJYBBXJXJBBLZJBYBHXJUUABVAVBYA - UUAEBXIXTBBSTRVHVCXJYQXNXJYPFXJYMYOXJYMJZYABYNLZEYNXIXTYNBSUUBYIXILZYKXTX - QVDYIHZDYIMEYIMZYNXIHUUBUUDYAEYIMZUUBYJUUDUUGJXJYJYKYLVEZYAEXIYIVFVGVIXJY - JYKYLVJZUUBYLUUFXJYJYKYLVKYLUUEEDYIYIYIXTXQVLVMVNEDYICVOVPUUBYJYKUUCUUHUU - IYKXQYIHZDVSYJUUCDYIVQYJUUJUUCDYJUUJUUCYJUUJBXQLZUUCYJUUJJZXQXIHZUUKUULXQ - YBHUUMUUKJYIYBXQVRYAUUKEXQXIXTXQBSZTVGVTBXQYIWAWBWCWDWEWJWFWCWGVCADFYBWHV - PYCYEAYBYFXPYBHXPXIHZYEJZUUKXSQZDXIMZXPYFHZYEJYCYAYEEXPXIXTXPBSTYAUUKXSDE - XIUUNWIUUPUURJZUUSYEUUTUUOXPXQLZADWKZQZDXIMZUUSUUOYEUURWLYEUURUVDUUOYEUUR - UVDYEUUQUVCDXIYEUUQUVAXSQUVCYEUVAUUKXSBXPXQWMWNUVAXSUVBUVAXRXPXQXRUVAXPXQ - PXPXQWOWPWQWRXAWSXBWTDXPCXCXDUUOYEUURXEXFXGXHVN $. + wex simprd ssuni sylancom ex exlimdv biimtrid sylc elrabd alrimiv zornn0g + ralrab weq simpll sstr2 imim1d simplbi2 necon1bd a2i syl6 ralimdv adantll + df-pss imp isufil2 sylanbrc simplr jca syl2anb reximi2 ) BCUAGZHZCIZIZUBU + CZHZJZAKZDKZUDZUEZDBEKZLZEXIUFZMZAYBNZBXPLZACUGGZNXOYBXMHZYBOPZFKZYBLZYIO + PZYIUHUIZUJZYIUKZYBHZQZFULZYDXOXNYBXLLZYGXJXNUMYRYAXTXLHZQEXIYAEXIXLUNXTX + IHZYSYAYTXTXKLYSXTCUPEXKUORUQURXLYBUSUTXJYHXNXJYBBXJXJBBLZJBYBHXJUUABVAVB + YAUUAEBXIXTBBSTRVHVCXJYQXNXJYPFXJYMYOXJYMJZYABYNLZEYNXIXTYNBSUUBYIXILZYKX + TXQVDYIHZDYIMEYIMZYNXIHUUBUUDYAEYIMZUUBYJUUDUUGJXJYJYKYLVEZYAEXIYIVFVGVIX + JYJYKYLVJZUUBYLUUFXJYJYKYLVKYLUUEEDYIYIYIXTXQVLVMVNEDYICVOVPUUBYJYKUUCUUH + UUIYKXQYIHZDVSYJUUCDYIVQYJUUJUUCDYJUUJUUCYJUUJBXQLZUUCYJUUJJZXQXIHZUUKUUL + XQYBHUUMUUKJYIYBXQVRYAUUKEXQXIXTXQBSZTVGVTBXQYIWAWBWCWDWEWFWGWCWHVCADFYBW + IVPYCYEAYBYFXPYBHXPXIHZYEJZUUKXSQZDXIMZXPYFHZYEJYCYAYEEXPXIXTXPBSTYAUUKXS + DEXIUUNWJUUPUURJZUUSYEUUTUUOXPXQLZADWKZQZDXIMZUUSUUOYEUURWLYEUURUVDUUOYEU + URUVDYEUUQUVCDXIYEUUQUVAXSQUVCYEUVAUUKXSBXPXQWMWNUVAXSUVBUVAXRXPXQXRUVAXP + XQPXPXQXAWOWPWQWRWSXBWTDXPCXCXDUUOYEUURXEXFXGXHVN $. $( A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via ~ numth3 .) (Contributed by Jeff Hankins, 2-Dec-2009.) @@ -310280,35 +311278,35 @@ contains the choice as a hypothesis (in the assumption that ` ~P ~P X ` mpd fbunfip syl2anc w3a mpbir3and fgcl filssufil r19.29 biimp simpll snex fsubbas unexg sylancl ssfii ssfg sstrd unssad sstr2 sseldd a2i syl56 impd imim1d rexlimdvw syl5 mpan2d an32s snidg elun2 adantlr simprl ufilb con4d - simpllr mpdd ssrdv eqssd simplr eqeltrd rexlimdva2 syl5bi impbid2 ) BCUBE - ZFZBCUCEZFZBAGZHZAUVFUDZUVGUVIUVHBIZJZAUVFKZUVJUVGUVLAUVFUVHUVFFUVGUVHUVD - FZUVLUVHCUEUVGUVNLZUVIUVKUVOUVILBUVHUVGUVNUVIBUVHIBUVHCUFUGUHMUIUJUVGBBHZ - UVMUVJBUMUVIUVPAUVFBUVHBBUKULUNUOUVJUVIUVHDGZIZNZAUVFKZDUVFUPUVEUVGUVIADU - VFUQUVEUVTUVGDUVFUVEUVQUVFFZLZUVTLZBUVQUVFUWCBUVQUWAUVTBUVQHZUVEUVSUWDAUV - QUVFUVRUVIUVSUWDUVRUVIUVSUVRUVIURUSUVHUVQBUKUTVAVBUWCVDUVQBUWCVDGZUVQFZUW - ECHZUWEBFZUWAUWFUWGJZUVEUVTUWAUVQUVDFZUWIUVQCUEUWJUWFUWGUWEUVQCVEMOVCUWCU - WGUWFUWHUWCUWGUWFUWHJUWCUWGLUWHUWFUWCUWGUWHPZUWFPZUWCUWGUWKLZLZUWLCUWEVFZ - UVQFZUWNCBUWOUAZVGZVHEZVIVJZUVQUWOUWBUWMUVTUWTUVQHZUWBUWMLZUVTUXAUXBUVTUW - 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(Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, @@ -310864,40 +311862,41 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the cvv cdm cnvimass fdm sseqtrid 3ad2ant3 ssexd elrnmpt mpbird ssid ad2antrr wfun ffun funimass3 sylancl mpbiri sseq1d jcad elv ad3antrrr imassrn ssin rspcev sylanblc elin jctir eleq2d biimpar fvimacnv biimpa funfvima2 eleq1 - cin sylc imbi12d rexlimdva impcomd syl5bi ssrdv eqssd filin com23 eqeltrd - 3exp imp31 exp32 eleq1d imbi2d impbid imp44 simprr simprlr filss syl13anc - syl5ibrcom exp44 rnelfmlem simpl3 elfm bitr4d eqrdv fnfvelrn ) EAHZCDUEIZ - HZEDBUFZUGZCDBUHUIZUJZHZBUJZCHZUURUVAUAJZUUSIZCKZUAEUKIZLZUVCUURUUSUVGULZ - UVAUVHMUURUVGUUOUUSUURDCHZUUNUUQUVGUUOUUSUFUUPUUNUVJUUQCDUMZUNZUUNUUPUUQU - OUUNUUPUUQUPCABDEUSUQURZUAUVGCUUSUTNUURUVFUVCUAUVGUUPUUQUVDUVGHZUVFUVCOZO - ZUUNUUPUVJUUQUVPUVKUVJUUQPZUVNUVOUVQUVNPZUVBUVEHUVFUVCUVRBEQZUVBUVEUUQUVS - UVBKZUVJUVNUUQEUVBBVAZUVTUUQBEULZUWAEDBVBZEBVCVDEUVBBVENVFUVRUVJUVNUUQEEU - VDVKUIZHZUVSUVEHUVJUUQUVNVGUVQUVNVHUVJUUQUVNVIUVNUWEUVQUVNUWDEUEIHUWEUVDE - 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adantrr syl5bi ssrdv filss syl13anc exp32 imaeq2 sseq1d imbi1d syl5ibrcom - rexlimdva ) AINZEUCBNZUDZUEZEXHUDZCNZOZXMGOZXMFPZQZQZBFAXIFPZRZXRXKEXJUDZ - XMOZXQQXTYBXOXPXTYBXORZRZFGUFSPZXIEUGZUHZFPZXOYGXMOXPAYEXSYCKTZYDYEXSYFFP - ZYHYIAXSYCUIAYJXSYCADGEUJUKSZFYFMAEHUDZYFYKAHGEULZHYFEUMZYLYFUELYMEHUNZYN - HGEUOZHEUPUQHYFEURUSAGFPZDHUTSPZYMHHDVAUKZPZYLYKPAYEYQKFGVBVCJLAYRYSHUFSP - YTJDHVEYSHVBUSFDHEYSGHYSVFVDVGVHVITXIYFFGVJVKXTYBXOVLYDUAYGXMUANZYGPUUAXI - PZUUAYFPZRZYDUUAXMPZUUAXIYFVMXTYBUUDUUEQXOXTYBRZUUCUUBUUEUUFUUCUBNZESZUUA - UEZUBHVNZUUBUUEQZAUUCUUJWFZXSYBAYMYOUULLYPUBHUUAEVOUSTUUFUUIUUKUBHXTYBUUG - HPZUUIUUKQXTYBUUMRZRZUUHXIPZUUHXMPZQUUIUUKUUOUUPUUGXJPZUUQUUOEVPZUUGEWGZP - UUPUURWFAUUSXSUUNAHGELVQTZUUOUUGHUUTXTYBUUMVLAUUTHUEXSUUNAHGELVRTVSUUGXIE - VTWAUUOUURUUHYAPZUUQUUOUUSXJUUTOUURUVBQUVAEXIWBXJUUGEWCWDYBUVBUUQQXTUUMYA - XMUUHWEWHWIWJUUIUUPUUBUUQUUEUUHUUAXIWKUUHUUAXMWKWLWMWNWOWJWPWQWRWSYGXMFGW - TXAXBXKXNYBXQXKXLYAXMXHXJEXCXDXEXFXG $. + adantrr biimtrid ssrdv syl13anc imaeq2 sseq1d imbi1d syl5ibrcom rexlimdva + filss exp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emma for ~ fmfnfm . (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) $) @@ -310968,45 +311967,45 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the vy cv cfil filelss ex cfbas mptexg rnexg unexg syl2anc unssbd wceq imaeq2 ssfii eqid rspceeqv mpan2 adantl cdm elfvdm cnvimass fssdm elrnmpt mpbird wb ssexd sseldd wf wfun ffun ssid funimass2 sylancl sseq1d rspcev cin w3o - elfiun fmfnfmlem1 fmfnfmlem3 eleq2d fmfnfmlem2 syl5bi sylbid bitrdi sylan - jcad elv fbssfi ad3antrrr cfm co w3a cfg filtop 3jca ssfg imaelfm adantrr - sselda jca filin 3expa simprr elin fvelima simplrr simprl syl2an ad2antrr - wel ssel2 fbelss fdmd sseqtrrd fvimacnv biimpd impr ad2ant2rl elind inss2 - sstri funfvima2 sylc anassrs expr eleq1 rexlimdva imbi1d syl5ibrcom ssrdv - imbi12d syl5ibcom syld impd adantrl sstrd filss syl13anc exp32 rexlimdvaa - ineq2 imaeq2d imp syldan rexlimdvva 3jaod rexlimdv impcomd impbid ) ACULZ - FNZUVAGOZEIULZUCZUVAOZIDBFEUDZBULZUCZUEZUFZUGZUHUIZUJZPAUVBUVCUVNAFGUMUIZ - NZUVBUVCQKUVPUVBUVCUVAFGUNUORAUVBUVNAUVBPZUVGUVAUCZUVMNEUVRUCZUVAOZUVNUVQ - 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(Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by @@ -311019,35 +312018,35 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ffn unssd ssun1 fbasne0 ssn0 sylancr cin wral wb cvv elrnmpt elv ad2antrr 0nelfil adantr 3jca ssfg sselda syl2anc filin 3expa sylan eleq1 syl5ibcom w3a jca mtod wex neq0 elin wi wfun ffun fvelima ad3antrrr fbelss sseqtrrd - fdmd fvimacnv inelcm adantl sylbid imbi1d rexlimdva syld impd exlimdv mpd - syl5bi ineq2 neeq1d syl5ibrcom expimpd ralrimivv fbunfip mpbird mpbir3and - ex fsubbas unexg ssfii unssad sstrd syl3anc fmfnfmlem4 bitr4d eqrdv eqtrd - elfm fmfg sseq2 fveq2 eqeq2d anbi12d rspcev syl12anc ) AGBLEDUCLUDZUEZUKZ - UFZUGZUHNZUIUJZGULNZOZBUUSPZEUUSFDUMUJZNZQZBCUDZPZEUVFUVCNZQZRZCUUTUNAUUR - GUONZOZUVAAUVLUUQGUPZPZUUQSUQZSUUROURZABUUPUVMABUVKOZBUVMPHGBUSTAUUPUVKOZ - UUPUVMPAGUOUTZOZEFULNOZGFDVFZDUFZEOUVRAUVQUVTHBGUOVAZTIJABUVCNZEUWCKADGUE - ZUWCUWEAUWBGUWCDVBZUWFUWCQJUWBDGVCUWGGFDVQGDVDVEGUWCDVGVHAFEOZUVQUWBGGBUI - UJZOZUWFUWEOAUWAUWHIEFVITZHJAUVQUWIUUTOUWJHBGVJUWIGVIVHEBGDUWIFGUWIVKZVLV - MVNVOLUVSDEFGVPVMZGUUPUSTVRABUUQPBSUQZUVOBUUPVSAUVQUWNHGBVTTBUUQWAWBAUVPU - AUDZMUDZWCZSUQZMUUPWDUABWDZAUWRUAMBUUPAUWOBOZUWPUUPOZUWRUXAUWPUUNQZLEUNZA - UWTRZUWRUXAUXCWEMLEUUNUWPUUOWFUUOVKWGWHUXDUXBUWRLEUXDUUMEOZRZUWRUXBUWOUUN - WCZSUQZUXFDUWOUEZUUMWCZSQZURZUXHUXFUXKSEOZAUXMURZUWTUXEAUWAUXNIEFWJTWIUXF - UXJEOZUXKUXMUXDUWAUXIEOZRUXEUXOUXDUWAUXPAUWAUWTIWKUXDUWEEUXIAUWEEPUWTKWKU - XDUWHUVQUWBXAZUWOUWIOUXIUWEOAUXQUWTAUWHUVQUWBUWKHJWLWKABUWIUWOAUVQBUWIPHB - GWMTWNEBUWODUWIFGUWLVLWOVOXBUWAUXPUXEUXOUXIUUMEFWPWQWRUXJSEWSWTXCUXLUWPUX - JOZMXDUXFUXHMUXJXEUXFUXRUXHMUXRUWPUXIOZUWPUUMOZRUXFUXHUWPUXIUUMXFUXFUXSUX - TUXHUXFUXSUBUDZDNZUWPQZUBUWOUNZUXTUXHXGZAUXSUYDXGZUWTUXEAUWBDXHZUYFJGFDXI - ZUYGUXSUYDUBUWPUWODXJYNVHWIUXFUYCUYEUBUWOUXFUYAUWOOZRZUYBUUMOZUXHXGUYCUYE - UYJUYKUYAUUNOZUXHUYJUYGUYADUTZOUYKUYLWEAUYGUWTUXEUYIAUWBUYGJUYHTXKUXFUWOU - YMUYAUXDUWOUYMPUXEUXDUWOGUYMAUVQUWTUWOGPHGBUWOXLWRAUYMGQUWTAGFDJXNWKXMWKW - NUYAUUMDXOWOUYIUYLUXHXGUXFUYIUYLUXHUYAUWOUUNXPYNXQXRUYCUYKUXTUXHUYBUWPUUM - WSXSWTXTYAYBYEYCYEYDUXBUWQUXGSUWPUUNUWOYFYGYHXTYEYIYJAUVQUVRUVPUWSWEHUWMU - AMBUUPGGYKWOYLAUVQUVTUVLUVNUVOUVPXAWEHUWDUUQUVSGYOVHYMZUURGVJTABUURUUSABU - UPUURAUUQWFOZUUQUURPAUVQUVRUYOHUWMBUUPUVKUVKYPWOUUQWFYQTYRAUVLUURUUSPUYNU - URGWMTYSAEUURUVCNZUVDAMEUYPAUWPEOUWPFPUXIUWPPUAUURUNRZUWPUYPOZALMBDEFGUAH - IJKUUAAUWHUVLUWBUYRUYQWEUWKUYNJUAUWPUUREDFGUUEYTUUBUUCAUWHUVLUWBUYPUVDQUW - KUYNJUUREDUUSFGUUSVKUUFYTUUDUVJUVBUVERCUUSUUTUVFUUSQZUVGUVBUVIUVEUVFUUSBU - UGUYSUVHUVDEUVFUUSUVCUUHUUIUUJUUKUUL $. + ex fdmd fvimacnv inelcm adantl sylbid imbi1d rexlimdva syld impd biimtrid + exlimdv mpd neeq1d syl5ibrcom expimpd ralrimivv fbunfip fsubbas mpbir3and + ineq2 mpbird unexg ssfii unssad sstrd syl3anc fmfnfmlem4 elfm bitr4d fmfg + eqrdv eqtrd sseq2 fveq2 eqeq2d anbi12d rspcev syl12anc ) AGBLEDUCLUDZUEZU + KZUFZUGZUHNZUIUJZGULNZOZBUUSPZEUUSFDUMUJZNZQZBCUDZPZEUVFUVCNZQZRZCUUTUNAU + URGUONZOZUVAAUVLUUQGUPZPZUUQSUQZSUUROURZABUUPUVMABUVKOZBUVMPHGBUSTAUUPUVK + OZUUPUVMPAGUOUTZOZEFULNOZGFDVFZDUFZEOUVRAUVQUVTHBGUOVAZTIJABUVCNZEUWCKADG + UEZUWCUWEAUWBGUWCDVBZUWFUWCQJUWBDGVCUWGGFDVQGDVDVEGUWCDVGVHAFEOZUVQUWBGGB + UIUJZOZUWFUWEOAUWAUWHIEFVITZHJAUVQUWIUUTOUWJHBGVJUWIGVIVHEBGDUWIFGUWIVKZV + LVMVNVOLUVSDEFGVPVMZGUUPUSTVRABUUQPBSUQZUVOBUUPVSAUVQUWNHGBVTTBUUQWAWBAUV + PUAUDZMUDZWCZSUQZMUUPWDUABWDZAUWRUAMBUUPAUWOBOZUWPUUPOZUWRUXAUWPUUNQZLEUN + ZAUWTRZUWRUXAUXCWEMLEUUNUWPUUOWFUUOVKWGWHUXDUXBUWRLEUXDUUMEOZRZUWRUXBUWOU + UNWCZSUQZUXFDUWOUEZUUMWCZSQZURZUXHUXFUXKSEOZAUXMURZUWTUXEAUWAUXNIEFWJTWIU + XFUXJEOZUXKUXMUXDUWAUXIEOZRUXEUXOUXDUWAUXPAUWAUWTIWKUXDUWEEUXIAUWEEPUWTKW + KUXDUWHUVQUWBXAZUWOUWIOUXIUWEOAUXQUWTAUWHUVQUWBUWKHJWLWKABUWIUWOAUVQBUWIP + HBGWMTWNEBUWODUWIFGUWLVLWOVOXBUWAUXPUXEUXOUXIUUMEFWPWQWRUXJSEWSWTXCUXLUWP + UXJOZMXDUXFUXHMUXJXEUXFUXRUXHMUXRUWPUXIOZUWPUUMOZRUXFUXHUWPUXIUUMXFUXFUXS + UXTUXHUXFUXSUBUDZDNZUWPQZUBUWOUNZUXTUXHXGZAUXSUYDXGZUWTUXEAUWBDXHZUYFJGFD + XIZUYGUXSUYDUBUWPUWODXJXNVHWIUXFUYCUYEUBUWOUXFUYAUWOOZRZUYBUUMOZUXHXGUYCU + YEUYJUYKUYAUUNOZUXHUYJUYGUYADUTZOUYKUYLWEAUYGUWTUXEUYIAUWBUYGJUYHTXKUXFUW + OUYMUYAUXDUWOUYMPUXEUXDUWOGUYMAUVQUWTUWOGPHGBUWOXLWRAUYMGQUWTAGFDJXOWKXMW + KWNUYAUUMDXPWOUYIUYLUXHXGUXFUYIUYLUXHUYAUWOUUNXQXNXRXSUYCUYKUXTUXHUYBUWPU + UMWSXTWTYAYBYCYDYEYDYFUXBUWQUXGSUWPUUNUWOYNYGYHYAYDYIYJAUVQUVRUVPUWSWEHUW + MUAMBUUPGGYKWOYOAUVQUVTUVLUVNUVOUVPXAWEHUWDUUQUVSGYLVHYMZUURGVJTABUURUUSA + BUUPUURAUUQWFOZUUQUURPAUVQUVRUYOHUWMBUUPUVKUVKYPWOUUQWFYQTYRAUVLUURUUSPUY + NUURGWMTYSAEUURUVCNZUVDAMEUYPAUWPEOUWPFPUXIUWPPUAUURUNRZUWPUYPOZALMBDEFGU + AHIJKUUAAUWHUVLUWBUYRUYQWEUWKUYNJUAUWPUUREDFGUUBYTUUCUUEAUWHUVLUWBUYPUVDQ + UWKUYNJUUREDUUSFGUUSVKUUDYTUUFUVJUVBUVERCUUSUUTUVFUUSQZUVGUVBUVIUVEUVFUUS + BUUGUYSUVHUVDEUVFUUSUVCUUHUUIUUJUUKUUL $. $} ${ @@ -311344,14 +312343,14 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the hausflimi $p |- ( J e. Haus -> E* x x e. ( J fLim F ) ) $= ( vy vu vv cha wcel cv cflim co wa wceq wi wal wne c0 wrex flimelbas syl wmo cin w3a cuni simpl simprll eqid simprlr simprr syl13anc df-3an simprl - hausnei hausflimlem 3expa sylanl1 a1d necon4d expimpd rexlimdvva mpd expr - syl5bi necon1bd pm2.18d ex alrimivv eleq1w mo4 sylibr ) CGHZAIZCBJKZHZDIZ - VMHZLZVLVOMZNZDOAOVNAUAVKVSADVKVQVRVKVQLZVRVTVRVLVOVKVQVLVOPZVRVKVQWALZLZ - VLEIZHZVOFIZHZWDWFUBZQMZUCZFCRECRZVRWCVKVLCUDZHZVOWLHZWAWKVKWBUEWCVNWMVKV - NVPWAUFVLBCWLWLUGZSTWCVPWNVKVNVPWAUHVOBCWLWOSTVKVQWAUIVLVOFECWLWOUMUJWCWJ - VREFCCWJWEWGLZWILWCWDCHWFCHLZLZVRWEWGWIUKWRWPWIVRWRWPLZVLVOWHQWSWHQPZWAWC - VQWQWPWTVKVQWAULVQWQWPWTVLVOWDBCWFUNUOUPUQURUSVCUTVAVBVDVEVFVGVNVPADADVMV - HVIVJ $. + hausnei hausflimlem 3expa sylanl1 a1d necon4d expimpd biimtrid rexlimdvva + mpd expr necon1bd pm2.18d ex alrimivv eleq1w mo4 sylibr ) CGHZAIZCBJKZHZD + IZVMHZLZVLVOMZNZDOAOVNAUAVKVSADVKVQVRVKVQLZVRVTVRVLVOVKVQVLVOPZVRVKVQWALZ + LZVLEIZHZVOFIZHZWDWFUBZQMZUCZFCRECRZVRWCVKVLCUDZHZVOWLHZWAWKVKWBUEWCVNWMV + KVNVPWAUFVLBCWLWLUGZSTWCVPWNVKVNVPWAUHVOBCWLWOSTVKVQWAUIVLVOFECWLWOUMUJWC + WJVREFCCWJWEWGLZWILWCWDCHWFCHLZLZVRWEWGWIUKWRWPWIVRWRWPLZVLVOWHQWSWHQPZWA + WCVQWQWPWTVKVQWAULVQWQWPWTVLVOWDBCWFUNUOUPUQURUSUTVAVBVCVDVEVFVGVNVPADADV + MVHVIVJ $. flimcf.1 $e |- X = U. J $. $( A condition for a topology to be Hausdorff in terms of filters. A @@ -311528,34 +312527,34 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the wn syl12anc eqeq1 anbi2d rexbidv elab sylibr wi ad3antrrr simplr simprr ad2antlr simprl inopn syl3anc simpr2 elind clsndisj syl32anc sylib elin wex n0 anbi1i bitri biimpri adantll ad2antll simpll simpr3 elinel1 nsyl - minel syl2anc nelneq2 sylnib expr syl5bi exlimdv mpd anassrs nan nrexdv - eqcom sylnibr nelne1 ex rexlimdvva necon4d anbi1d impbid1 dom2lem f1eq1 - abbidv ax-mp ) EUANZBFUBZOZBEUCUDUDZBUEZUEZAUULACPZGQZCQZBRZSZOZCETZGUF - ZUGZUHZUULUUNDUHZUUKAJUULUUNUVBJCPZUUSOZCETZGUFZUUKUVBUUNNZAQZUULNZUUKU - VJUVBUUMUBZUVAGUUMUUTUUPUUMNZCEUUSUVNUUOUUSUVNUURUUMNZUVOUURBUBUUQBUIUU - RBUUQBCUJUKULUMUUPUURUUMUNUOUPUQURUUKBUTNZUUMUTNUVJUVMVAUUJUUIUVPUUIUUJ - FENZUVPUUIEUSNZUVQEVBZEFHVCVDBFEVEVFVGBUTVHUVBUUMUTVIVJUOVKUUKUVLJQZUUL - NZOZUVBUVISZAJVLZVAUUKUWBOZUWCUWDUWEUVKUVTUVBUVIUWEUVKUVTVMZUVBUVIVMZUW - EUWFOZAKPZJLPZKQZLQZRVNSZVOZLETKETZUWGUWHUUIUVKFNUVTFNUWFUWOUUIUUJUWBUW - FVPUWHUULFUVKUUKUULFUBZUWBUWFUUIUVRUUJUWPUVSBEFHVQVRVSZUUKUVLUWAUWFVTWA - UWHUULFUVTUWQUUKUVLUWAUWFWBWAUWEUWFWDUVKUVTLKEFHWCWEUWHUWNUWGKLEEUWHUWK - ENZUWLENZOZUWNUWGUWHUWTUWNOZOZUWKBRZUVBNZUXCUVINZWNUWGUXBUUOUXCUURSZOZC - ETZUXDUXBUWRUWIUXCUXCSZUXHUWHUWRUWSUWNWFUWIUWJUWMUWTUWHWGUXBUXCWHUXGUWI - UXIOCUWKECKVLZUUOUWIUXFUXICKAWIUXJUURUXCUXCUUQUWKBWJWKWLWMWOUVAUXHGUXCU - WKBKUJUKZUUPUXCSZUUTUXGCEUXLUUSUXFUUOUUPUXCUURWPZWQWRWSWTUXBUVFUXFOZCET - ZUXEUXBUXNCEUXBUUQENZOZUXNWNXAUXQUVFOUXFWNZXAUXBUXPUVFUXRUXBUXPUVFOZOZM - QZUWLUUQRZBRZNZMXOZUXRUXTUYCVNVMZUYEUXTUVRUUJUWAUYBENZUVTUYBNUYFUWEUVRU - WFUXAUXSUUIUVRUUJUWBUVSVSXBZUWEUUJUWFUXAUXSUUIUUJUWBXCXBUWEUWAUWFUXAUXS - UUKUVLUWAXDXBUXTUVRUWSUXPUYGUYHUXAUWSUWHUXSUWRUWSUWNXCXEUXBUXPUVFXFUWLU - UQEXGXHUXTUWLUUQUVTUXAUWJUWHUXSUWTUWIUWJUWMXIXEUXBUXPUVFXDXJUVTBUYBEFHX - KXLMUYCXPXMUXTUYDUXRMUYDMLPZMCPZOZUYABNZOZUXTUXRUYDUYAUYBNZUYLOUYMUYAUY - BBXNUYNUYKUYLUYAUWLUUQXNXQXRUXBUXSUYMUXRUXBUXSUYMOZOZUURUXCSZUXFUYPUYAU - URNZUYAUXCNZWNZUYQWNUYMUYRUXBUXSUYJUYLUYRUYIUYRUYJUYLOUYAUUQBXNXSXTYAUY - PUYIUWMUYTUYMUYIUXBUXSUYIUYJUYLYBYAUXAUWMUWHUYOUWTUWIUWJUWMYCXEUYIUWMOM - KPUYSUYAUWLUWKYFUYAUWKBYDYEYGUYAUURUXCYHYGUURUXCYQYIYJYKYLYMYNUXQUVFUXF - YOUMYPUVHUXOGUXCUXKUXLUVGUXNCEUXLUUSUXFUVFUXMWQWRWSYRUXCUVBUVIYSYGYJUUA - YMYTUUBUWDUVAUVHGUWDUUTUVGCEUWDUUOUVFUUSUVKUVTUUQUNUUCWRUUGUUDYTUUEDUVC - SUVEUVDVAIUULUUNDUVCUUFUUHWT $. + minel syl2anc nelneq2 eqcom sylnib expr biimtrid exlimdv anassrs nrexdv + mpd nan sylnibr nelne1 ex rexlimdvva necon4d anbi1d impbid1 f1eq1 ax-mp + abbidv dom2lem ) EUANZBFUBZOZBEUCUDUDZBUEZUEZAUULACPZGQZCQZBRZSZOZCETZG + UFZUGZUHZUULUUNDUHZUUKAJUULUUNUVBJCPZUUSOZCETZGUFZUUKUVBUUNNZAQZUULNZUU + KUVJUVBUUMUBZUVAGUUMUUTUUPUUMNZCEUUSUVNUUOUUSUVNUURUUMNZUVOUURBUBUUQBUI + UURBUUQBCUJUKULUMUUPUURUUMUNUOUPUQURUUKBUTNZUUMUTNUVJUVMVAUUJUUIUVPUUIU + UJFENZUVPUUIEUSNZUVQEVBZEFHVCVDBFEVEVFVGBUTVHUVBUUMUTVIVJUOVKUUKUVLJQZU + ULNZOZUVBUVISZAJVLZVAUUKUWBOZUWCUWDUWEUVKUVTUVBUVIUWEUVKUVTVMZUVBUVIVMZ + UWEUWFOZAKPZJLPZKQZLQZRVNSZVOZLETKETZUWGUWHUUIUVKFNUVTFNUWFUWOUUIUUJUWB + UWFVPUWHUULFUVKUUKUULFUBZUWBUWFUUIUVRUUJUWPUVSBEFHVQVRVSZUUKUVLUWAUWFVT + WAUWHUULFUVTUWQUUKUVLUWAUWFWBWAUWEUWFWDUVKUVTLKEFHWCWEUWHUWNUWGKLEEUWHU + WKENZUWLENZOZUWNUWGUWHUWTUWNOZOZUWKBRZUVBNZUXCUVINZWNUWGUXBUUOUXCUURSZO + ZCETZUXDUXBUWRUWIUXCUXCSZUXHUWHUWRUWSUWNWFUWIUWJUWMUWTUWHWGUXBUXCWHUXGU + WIUXIOCUWKECKVLZUUOUWIUXFUXICKAWIUXJUURUXCUXCUUQUWKBWJWKWLWMWOUVAUXHGUX + CUWKBKUJUKZUUPUXCSZUUTUXGCEUXLUUSUXFUUOUUPUXCUURWPZWQWRWSWTUXBUVFUXFOZC + ETZUXEUXBUXNCEUXBUUQENZOZUXNWNXAUXQUVFOUXFWNZXAUXBUXPUVFUXRUXBUXPUVFOZO + ZMQZUWLUUQRZBRZNZMXOZUXRUXTUYCVNVMZUYEUXTUVRUUJUWAUYBENZUVTUYBNUYFUWEUV + RUWFUXAUXSUUIUVRUUJUWBUVSVSXBZUWEUUJUWFUXAUXSUUIUUJUWBXCXBUWEUWAUWFUXAU + XSUUKUVLUWAXDXBUXTUVRUWSUXPUYGUYHUXAUWSUWHUXSUWRUWSUWNXCXEUXBUXPUVFXFUW + LUUQEXGXHUXTUWLUUQUVTUXAUWJUWHUXSUWTUWIUWJUWMXIXEUXBUXPUVFXDXJUVTBUYBEF + HXKXLMUYCXPXMUXTUYDUXRMUYDMLPZMCPZOZUYABNZOZUXTUXRUYDUYAUYBNZUYLOUYMUYA + UYBBXNUYNUYKUYLUYAUWLUUQXNXQXRUXBUXSUYMUXRUXBUXSUYMOZOZUURUXCSZUXFUYPUY + AUURNZUYAUXCNZWNZUYQWNUYMUYRUXBUXSUYJUYLUYRUYIUYRUYJUYLOUYAUUQBXNXSXTYA + UYPUYIUWMUYTUYMUYIUXBUXSUYIUYJUYLYBYAUXAUWMUWHUYOUWTUWIUWJUWMYCXEUYIUWM + OMKPUYSUYAUWLUWKYFUYAUWKBYDYEYGUYAUURUXCYHYGUURUXCYIYJYKYLYMYPYNUXQUVFU + XFYQUMYOUVHUXOGUXCUXKUXLUVGUXNCEUXLUUSUXFUVFUXMWQWRWSYRUXCUVBUVIYSYGYKU + UAYPYTUUBUWDUVAUVHGUWDUUTUVGCEUWDUUOUVFUUSUVKUVTUUQUNUUCWRUUGUUDYTUUHDU + VCSUVEUVDVAIUULUUNDUVCUUEUUFWT $. $} $( If ` X ` is a Hausdorff space, then the cardinality of the closure of a @@ -311679,16 +312678,16 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the A. o e. J ( A e. o -> E. s e. B ( F " s ) C_ o ) ) ) ) $= ( vt cfv wcel co cv cima wss wrex wi wa ctopon cfbas wf cflf wral cfil wb w3a cfg fgcl eqeltrid isflf syl3an2 eleq2i elfg sstr2 imass2 syl11 adantl - 3ad2ant2 reximdv ex com23 adantld sylbid adantr syl5bi rexlimdv sseqtrrdi - ssfg sselda 3ad2antl2 ad2ant2r simprr weq imaeq2 sseq1d rspcev rexlimdvaa - syl2anc impbid imbi2d ralbidv pm5.32da bitrd ) EGUALMZBHUBLMZHGDUCZUHZADE - FUDNLMZAGMZACOZMZDKOZPZWLQZKFRZSZCEUEZTZWKWMDIOZPZWLQZIBRZSZCEUEZTWGWFFHU - FLZMWHWJWTUGWGFHBUINZXGJBHUJUKACDEFGHKULUMWIWKWSXFWIWKTZWRXECEXIWQXDWMXIW - QXDXIWPXDKFWNFMWNXHMZXIWPXDSZFXHWNJUNWIXJXKSWKWIXJWNHQZXAWNQZIBRZTZXKWGWF - XJXOUGWHIWNBHUOUTWIXNXKXLWIWPXNXDWIWPXNXDSWIWPTXMXCIBWPXMXCSWIXBWOQWPXCXM - XBWOWLUPXAWNDUQURUSVAVBVCVDVEVFVGVHXIXCWQIBXIXABMZXCTTXAFMZXCWQWIXPXQWKXC - WGWFXPXQWHWGBFXAWGBXHFBHVJJVIVKVLVMXIXPXCVNWPXCKXAFKIVOWOXBWLWNXADVPVQVRV - TVSWAWBWCWDWE $. + 3ad2ant2 reximdv ex com23 adantld sylbid biimtrid rexlimdv ssfg sseqtrrdi + adantr sselda ad2ant2r simprr weq imaeq2 sseq1d rspcev syl2anc rexlimdvaa + 3ad2antl2 impbid imbi2d ralbidv pm5.32da bitrd ) EGUALMZBHUBLMZHGDUCZUHZA + DEFUDNLMZAGMZACOZMZDKOZPZWLQZKFRZSZCEUEZTZWKWMDIOZPZWLQZIBRZSZCEUEZTWGWFF + HUFLZMWHWJWTUGWGFHBUINZXGJBHUJUKACDEFGHKULUMWIWKWSXFWIWKTZWRXECEXIWQXDWMX + IWQXDXIWPXDKFWNFMWNXHMZXIWPXDSZFXHWNJUNWIXJXKSWKWIXJWNHQZXAWNQZIBRZTZXKWG + WFXJXOUGWHIWNBHUOUTWIXNXKXLWIWPXNXDWIWPXNXDSWIWPTXMXCIBWPXMXCSWIXBWOQWPXC + XMXBWOWLUPXAWNDUQURUSVAVBVCVDVEVJVFVGXIXCWQIBXIXABMZXCTTXAFMZXCWQWIXPXQWK + XCWGWFXPXQWHWGBFXAWGBXHFBHVHJVIVKVTVLXIXPXCVMWPXCKXAFKIVNWOXBWLWNXADVOVPV + QVRVSWAWBWCWDWE $. $} ${ @@ -311702,15 +312701,15 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the A. o e. B ( A e. o -> E. s e. L ( F " s ) C_ o ) ) ) ) $= ( vu cfv wcel cv wss wrex wi wral wa ctop ctopon cfil wf w3a cflf co cima isflf ctg raleqi ctb simpl1 topontop syl eqeltrrid tgclb sylibr bastg weq - eleq2w sseq2 rexbidv imbi12d cbvralvw ssralv syl5bi 3syl tg2 r19.29 simpl - simpr sstr2 syl5com reximdv embantd impcom rexlimivw ex expdimp ralrimiva - syl5 impbid1 bitrid pm5.32da bitrd ) EGUALMZFHUBLMZHGDUCZUDZADEFUEUFLMAGM - ZAKNZMZDINUGZWKOZIFPZQZKERZSWJACNZMZWMWROZIFPZQZCBRZSAKDEFGHIUHWIWJWQXCWQ - WPKBUILZRZWIWJSZXCWPKEXDJUJXFXEXCXFBUKMZBXDOZXEXCQXFXDTMXGXFXDETJXFWFETMW - FWGWHWJULGEUMUNUOBUPUQBUKURXEXBCXDRXHXCWPXBKCXDKCUSZWLWSWOXAKCAUTXIWNWTIF - WKWRWMVAVBVCVDXBCBXDVEVFVGXCWPKXDXCWKXDMZWLWOXJWLSWSWRWKOZSZCBPZXCWOCWKBA - VHXCXMWOXCXMSXBXLSZCBPWOXBXLCBVIXNWOCBXLXBWOXLWSXAWOWSXKVJXLWTWNIFXLXKWTW - NWSXKVKWMWRWKVLVMVNVOVPVQUNVRWAVSVTWBWCWDWE $. + eleq2w sseq2 rexbidv cbvralvw ssralv biimtrid 3syl tg2 r19.29 simpl simpr + imbi12d sstr2 syl5com reximdv embantd rexlimivw ex syl5 expdimp ralrimiva + impcom impbid1 bitrid pm5.32da bitrd ) EGUALMZFHUBLMZHGDUCZUDZADEFUEUFLMA + GMZAKNZMZDINUGZWKOZIFPZQZKERZSWJACNZMZWMWROZIFPZQZCBRZSAKDEFGHIUHWIWJWQXC + WQWPKBUILZRZWIWJSZXCWPKEXDJUJXFXEXCXFBUKMZBXDOZXEXCQXFXDTMXGXFXDETJXFWFET + MWFWGWHWJULGEUMUNUOBUPUQBUKURXEXBCXDRXHXCWPXBKCXDKCUSZWLWSWOXAKCAUTXIWNWT + IFWKWRWMVAVBVKVCXBCBXDVDVEVFXCWPKXDXCWKXDMZWLWOXJWLSWSWRWKOZSZCBPZXCWOCWK + BAVGXCXMWOXCXMSXBXLSZCBPWOXBXLCBVHXNWOCBXLXBWOXLWSXAWOWSXKVIXLWTWNIFXLXKW + TWNWSXKVJWMWRWKVLVMVNVOWAVPUNVQVRVSVTWBWCWDWE $. $} ${ @@ -312651,51 +313650,51 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the elunii eldifsni elinel2 elfir syl13anc filfi eleqtrd imbi12d syl5ibrcom eleq2 eleq1 rexlimdva sylbid imp32 adantrrr elssuni fibas tgtopon ax-mp ctopon ctb eqeltrdi fiuni eqtrd fveq2d eleqtrrd toponuni sseqtrrd filss - 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Comp -> E. x ( J = ( topGen ` ( fi ` x ) ) /\ A. c e. ~P x ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) ) $= ( ccmp wcel cfi cfv ctg wceq cv cuni cpw cfn cin wrex wral wa wi wex ctop - cmptop fitop fveq2d tgtop eqtr2d velpw cmpcov 3exp syl5bi ralrimiv 2fveq3 - syl wss eqeq2d pweq raleqdv anbi12d spcegv mp2and ) BGHZBBIJZKJZLZCDMZNLZ - CEMNLEVGOPQRZUAZDBOZSZBAMZIJKJZLZVJDVMOZSZTZAUBVCBUCHZVFBUDVSVEBKJBVSVDBK - BUEUFBUGUHUOVCVJDVKVGVKHVGBUPZVCVJDBUIVCVTVHVIVGBCEFUJUKULUMVRVFVLTABGVMB - LZVOVFVQVLWAVNVEBVMBKIUNUQWAVJDVPVKVMBURUSUTVAVB $. + cmptop fitop fveq2d tgtop eqtr2d syl cmpcov 3exp biimtrid ralrimiv 2fveq3 + wss velpw eqeq2d pweq raleqdv anbi12d spcegv mp2and ) BGHZBBIJZKJZLZCDMZN + LZCEMNLEVGOPQRZUAZDBOZSZBAMZIJKJZLZVJDVMOZSZTZAUBVCBUCHZVFBUDVSVEBKJBVSVD + BKBUEUFBUGUHUIVCVJDVKVGVKHVGBUOZVCVJDBUPVCVTVHVIVGBCEFUJUKULUMVRVFVLTABGV + MBLZVOVFVQVLWAVNVEBVMBKIUNUQWAVJDVPVKVMBURUSUTVAVB $. $( Lemma for ~ alexsubALT . Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff @@ -312771,40 +313770,40 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the wo ineq1d anbi12d elrab velsn orbi12i bitri elpwi adantr 0ss sseq1 mpbiri jaoi sylbi syl6 ralrimiv unissb sylibr ad2antlr vuniex uni0b wne disjssun elpw notbii biimpcd necon3bd n0 elin anbi2i simprrl simpl syl2anc exlimiv - wex ssuni ad2antrl syl5bi imp elfpw unieq eqtrdi necon3bi simplrr simprlr - uni0 simprr finsschain syl22anc expr 0elpw elini eqeq2d notbid rspccv mpi - 0fin velpw syl5bir expd com23 ad2antll a1i ss0 syl6bi biimprcd a1dd syl9r - com34 jaod sylan9r sylan ad2ant2lr sylib ex adantrl rexlimdv syld impcomd - cbvralvw jca32 orcom elsn df-or 3bitr4i alrimiv fvex pwex rabex p0ex unex - bitr2i zorn syl ) EAMZUAUBZUCUBNFIMZUDNFJMUDNJUVBUEOUFUGPIUUTUEUHGMZUVAUE - ZQUIZFHMZUDZNZULZHUVCUEZOUFZUHZRZUJMZUVCBMZSZUVIHUVOUEZOUFZUHZRZBUVDUMZTU - NZUOZSZUVNUPUQZRZUVNUDZUWCQZPZUJURDMCMUSULCUWCUHDUWCUGUVMUWIUJUVMUWFUWHUV - MUWFRZUWGTNZULZUWGUVDQZUVCUWGSZUVIHUWGUEZOUFZUHZRZRZPZUWHUWJUWLUWSUWJUWLR - 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If a point is covered by a collection taken from the base with no finite subcover, a set from the subbase can be @@ -312826,71 +313825,71 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the bitr4i uniun unisnv uneq2i eqtri eqsstrd difss unissi sseq2 mpbiri sseldd ad3antrrr elssuni syl ctb unieq 3ad2ant1 cvv eqtr4di sseqtrd unssd uneq1d eqssd rspceeqv syl2anc eqeq2d ad2antrl ralrimiva sylibr wal alinexa rspcv - ex weq eleq2 eluni notbid syl9r syl5bi imp31 ad2antrr eximi eliun orbi12i - anim2i velsn bitri syl6 exp32 elpwid ad2antlr simprl fibas unitg 3eqtr4rd - elinel1 fiuni expr expd rexlimdv ralimdva elinel2 ac6sfi ffvelcdm simplrr - ciun iunss snssd elin2d iunfi syl2an ad4ant14 ad2ant2lr snfi unfi sylancl - imbi2i albii unieqd id uneq12d biimpd orbi1i imbi1i 3bitri eleq1w ralbidv - fveq2 imbi12d spvv eleq2d alrimdv imim1d a1dd com14 com23 ralrimdv elint2 - syld syl6ibr wb sylibrd orrd orc equid equequ1 anbi12d spcev mpan2 anbi2i - olc jaoi exbii ad5ant25 ad2ant2l ssrdv nfra1 nfv rsp eqimss2 ssun3 expcom - 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If any cover taken from a subbase has a finite subcover, any cover taken from the corresponding base has a finite @@ -312908,63 +313907,63 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the dfrex2 con2d a1d 3ad2ant2 adantr impd wb unissi fiuni elv ctb fibas unitg impr ax-mp eqtr4i eqtr4id eqtr4di 3ad2ant1 sseqtrid syl ad2antrl ad4antlr wex nss ad2antrr ad2antlr simprl sseldd sselid sylibr ssun1 notbid adantl - wel wne ex syl5bir expd con3d eqimss2 psseq2 rexlimdv syl5bi eqcom bitrid - eqss baib mtbid sstr2 df-rex bitr4i eluni2 cint elpwi elfi alexsubALTlem3 - con3rr3 sseld el2v ssfii elinel1 elpwid snssd unssd elpw2 sstrdi cbvralvw - biimpi ad2antll jca32 elun1 vsnid intss1 impcom ad4ant24 adantrrr adantll - fvex elun2 simprlr elin elunii sylc com23 exp32 imp55 eleq1w elequ1 spcev - expr sylancr necon3bi jctil df-pss syl2anc rexlimddv exp45 mpdd biantrurd - rspcev 3syld bitr3d simpll3 simplr elrabd rspcv 0elpw 0fin elini eqeltrdi - id syl5 necon2ad psseq1 0pss 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elpwid snssd unssd fvex elpw2 cbvralvw + sstrdi biimpi ad2antll jca32 elun1 vsnid intss1 ad4ant24 adantrrr adantll + elun2 impcom simprlr elin elunii sylc expr com23 exp32 imp55 eleq1w spcev + elequ1 sylancr necon3bi jctil df-pss rspcev syl2anc rexlimddv exp45 3syld + mpdd biantrurd bitr3d simpll3 simplr elrabd rspcv id 0elpw elini eqeltrdi + 0fin syl5 necon2ad psseq1 0pss sylibrd nsyld jaod mpd con4d 3exp ralrimdv + bitrdi ) BALZUEUFZUGUFZMZCFLZNZMZCGLZNZMZGUYEUHZOUJZUKZPZFUYAUHZULZCDLZNZ + MZCELZNZMZEUYQUHZOUJZUKZPZDUYBUHZUYDUYPUYQVUGQZVUFUYDUYPVUHUMZVUEUYSVUERV + UBRZEVUDULZVUIUYSRZVUBEVUDUNVUIVUKVULVUIVUKSZILZJLZUOZRZJUYQUALZTZVUJEVUR + UHZOUJZULZSZUAVUGUPZUQURZVCZULZIVVFUKVULAUAJIBCDEFGHUSVUMVVGVULIVVFVUNVVF + QZVUNVUGQZUYQVUNTZVUJEVUNUHZOUJZULZSZSZVUNUQMZUTZVUMVVGVULPZVVHVUNVVDQZVU + NVVEQZUTVVQVUNVVDVVEVAVVSVVOVVTVVPVVCVVNUAVUNVUGUAIVNZVUSVVJVVBVVMVURVUNU + YQVBVWAVUJEVVAVVLVWAVUTVVKOVURVUNVOVDVEVFVGIUQVHVIVJVUMVVOVVRVVPVUIVVOVVR + 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MZUYQVUDQVUKVULWWIUYQUQVUDWWIUXDUQVUCOUYQUXEUXHUXFUXGVUJVULEUYQVUDEDVNZVU + BUYSWWJVUAUYRCUYTUYQWAWBYIWFUXIUXJYCWWDWWEUQUYQUOZWWHVVPWWEWWKXCVUMVUNUQU + YQUXKYJUYQUXLUXTUXMUXNYMUXOYTYSUXPYMYNUXQUXRUXS $. $( The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. @@ -312977,44 +313976,44 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( vb vy vz vt vw wcel cv wceq cuni cfn wrex wi wa wss va ccmp cfi cfv ctg vf cpw cin wral wex alexsubALTlem1 alexsubALTlem4 ctop velpw w3a crab wel wb eleq2 3ad2ant3 eluni ssel ex syl6bi sylan9r 3impia sseq2 rspcev anim2d - tg2 reximdv syld 3expia com23 exlimdv syl5bi 3adant3 sylbid elunii expcom - impd biimprd syl9r rexlimdva rexlimdvw impbid elunirab bitr4di eqrdv fvex - ssrab2 elpw2 mpbir unieq eqeq2d pweq ineq1d rexeqdv imbi12d rspcv syl5com - ax-mp elfpw wf sseq1 rexbidv simprbi syl6 ralrimiv ac6sfi syl5 adantl crn - elrab simprll frn syl cdom wbr simplr wfo wfn dffn4 sylib adantr ad2antrl - ffn fodomfi syl2anc jca elin vex anbi1i bitr2i simprr ciun uniiun simprlr - domfi ss2iun eqsstrid fniunfv sseqtrd eqsstrd sstrd uniss sseqtrrdi eqssd - 3syl simpll2 exp32 rexlimdv 3exp com34 syl7bi ralrimdv fibas eleq1 mpbiri - ctb tgcl jctild iscmp syl6ibr imp exlimiv impbii ) BUBLZBAMZUCUDZUEUDZNZC - DMZOZNZCEMZOZNZEUVMUGZPUHZQZRZDUVIUGUIZSZAUJABCDEFUKUWDUVHAUVLUWCUVHUVLUW - CCUAMZOZNZCGMZOZNZGUWEUGZPUHZQZRZUAUVJUGZUIZUVHABCUAGDEFULUVLUWPBUMLZUWBD - BUGZUIZSUVHUVLUWPUWSUWQUVLUWPUWBDUWRUVMUWRLUVMBTZUVLUWPUWBDBUNUVLUWTUWPUW - BUVLUWTUVOUWPUWAUVLUWTUVOUWPUWARUVLUWTUVOUOZUWPUWJGHMZIMZTZIUVMQZHUVJUPZU - GZPUHZQZUWAUXACUXFOZNZUWPUXIUXAJCUXJUXAJMZCLZJHUQZUXESZHUVJQZUXLUXJLUXAUX - MUXPUXAUXMUXLUVNLZUXPUVOUVLUXMUXQURUWTCUVNUXLUSUTZUVLUWTUXQUXPRUVOUXQJKUQ - ZKDUQZSZKUJUVLUWTSZUXPKUXLUVMVAUYBUYAUXPKUYBUXSUXTUXPUYBUXTUXSUXPUVLUWTUX - TUXSUXPRUVLUWTUXTUOZUXSUXNUXBKMZTZSZHUVJQZUXPUVLUWTUXTUXSUYGRZUWTUXTUYDBL - 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syl2anc domfi jca elin vex anbi1i bitr2i simprr ciun uniiun + simprlr ss2iun fniunfv 3syl sseqtrd eqsstrd simpll2 sstrd uniss sseqtrrdi + eqsstrid eqssd exp32 rexlimdv 3exp com34 syl7bi ralrimdv fibas tgcl eleq1 + ctb mpbiri jctild iscmp syl6ibr imp exlimiv impbii ) BUBLZBAMZUCUDZUEUDZN + ZCDMZOZNZCEMZOZNZEUVMUGZPUHZQZRZDUVIUGUIZSZAUJABCDEFUKUWDUVHAUVLUWCUVHUVL + UWCCUAMZOZNZCGMZOZNZGUWEUGZPUHZQZRZUAUVJUGZUIZUVHABCUAGDEFULUVLUWPBUMLZUW + BDBUGZUIZSUVHUVLUWPUWSUWQUVLUWPUWBDUWRUVMUWRLUVMBTZUVLUWPUWBDBUNUVLUWTUWP + UWBUVLUWTUVOUWPUWAUVLUWTUVOUWPUWARUVLUWTUVOUOZUWPUWJGHMZIMZTZIUVMQZHUVJUP + ZUGZPUHZQZUWAUXACUXFOZNZUWPUXIUXAJCUXJUXAJMZCLZJHUQZUXESZHUVJQZUXLUXJLUXA + UXMUXPUXAUXMUXLUVNLZUXPUVOUVLUXMUXQURUWTCUVNUXLUSUTZUVLUWTUXQUXPRUVOUXQJK + UQZKDUQZSZKUJUVLUWTSZUXPKUXLUVMVAUYBUYAUXPKUYBUXSUXTUXPUYBUXTUXSUXPUVLUWT + UXTUXSUXPRUVLUWTUXTUOZUXSUXNUXBKMZTZSZHUVJQZUXPUVLUWTUXTUXSUYGRZUWTUXTUYD + BLZUVLUYHUVMBUYDVBUVLUYIUYDUVKLZUYHBUVKUYDUSUYJUXSUYGHUYDUVJUXLVJVCVDVEVF + 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AJUIZKUJZUKZULUMUN + ZUOZJEUPZUQUDUIZVNZIUIZUXFUJZUXHKUJZUKZLUKZURZUSZIUXEUTZWFZHUIZEVAZUXHU + XQUJZUXMUSZIEUTZWFZHVBZUDAUXEUQUSZIUXEUEUIZUXMUSZUEUQUPZVCZVDVEZUSZUYFU + EUQVFZIUXEUTUXPUDVBAEMUSZUYDQUXDJEMVGVHAIUXEUXKVCZUYIUSUYHUYMVIZUYJABCD + EFGIJKMNOPQRSTUAUBVJAUYGUXKVIZIUXEUTUYNAUYOIUXEAUYFUEUQUXKUYFAUYEUXKUSZ + UYEUQUSUYEUXKUXLVKZVLVMVOIUXEUYGUXKVPVHUYMUYHVQVRAUYKIUXEUXHUXEUSAUXHEU + SZUYKUXDJUXHEVSAUYRWFZUYFUEVBZUYKUYSUXMWGVTZUOUYTUYSVUANBUIZUKZVTBGWAWB + WCZVFZAVUEUOUYRUBWDVUAUXKUXLVIZUYSVUEUXKUXLWEUYSVUFVUEUYSVUFWFZUXKUFUIZ + UKZVTZVUEUFLWAWBWCZVUGUXJWHUSZLUXJVIZUXKUXLVTVUJUFVUKVFUYSVULVUFAEWHUXH + KRWIWDVUMVUGCNUXHCUIZUJZWJZWKZDUIZWLZGUSZDUXJLUCWMZWSVUGUXKUXLUYSVUFWNV + UMUXLUXKVIVUGVVALUXJWOWPWQLUXJUXKUFUXKWRWTXAVUGVUHVUKUSZVUJWFZWFZUGVUHV + UQUGUIZWLZWJZXBZVUDUSZNVVHUKZVTVUEVVDVVHGVIVVHWBUSZVVIVVDVUHGVVGVVDUGVU + HVVFGVVDVVEVUHUSZWFVVELUSZVVFGUSZVVDVUHLVVEVVBVUHLVIZVUGVUJVVBVVOVUHWBU + SZVUHLXJZXCXDXEVVMVVEUXJUSVVNVUTVVNDVVEUXJLDUGXFVUSVVFGVURVVEVUQXGXHUCX + 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(Contributed by Mario Carneiro, 26-Aug-2015.) $) ptcmplem4 $p |- -. ph $= @@ -313289,13 +314288,13 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( vf ctop wcel wa wf wss cfv crest co cflf cvv cdm ccl csn cnei ciun cuni cxp cpm ccnext cmpt wceq cnextval adantr simpr dmeqd simplrl eqtrd fveq2d cv fdmd oveq2d fveq12d xpeq2d iuneq12d uniexg ad2antlr eqid feq23i biimpi - ad2antrr ad2antrl sseq2i ad2antll elpm2r syl22anc fvex snex iunex fvmptd - xpex a1i ) EKLZFKLZMZBCDNZBGOZMZMZJDAJUSZUAZEUBPZPZAUSZUCZWIFWNEUDPPZWJQR - ZSRZPZUGZUEZABWKPZWNDFWOBQRZSRZPZUGZUEZFUFZEUFZUHRZEFUIRZTWDXJJXIWTUJUKWG - AJEFULUMWHWIDUKZMZAWLXAWSXEXLWJBWKXLWJDUABXLWIDWHXKUNZUOXLBCDWDWEWFXKUPUT - UQZURXLWRXDWNXLWIDWQXCXLWPXBFSXLWJBWOQXNVAVAXMVBVCVDWHXGTLZXHTLZBXGDNZBXH - OZDXILWCXOWBWGFKVEVFWBXPWCWGEKVEVJWEXQWDWFWEXQBCBXGDBVGIVHVIVKWFXRWDWEWFX - RGXHBHVLVIVMXGXHBDTTVNVOXFTLWHAXAXEBWKVPWNXDWMVQDXCVPVTVRWAVS $. + ad2antrr ad2antrl sseq2i ad2antll elpm2r syl22anc fvex vsnex iunex fvmptd + xpex a1i ) EKLZFKLZMZBCDNZBGOZMZMZJDAJUSZUAZEUBPZPZAUSUCZWIFWMEUDPPZWJQRZ + SRZPZUGZUEZABWKPZWMDFWNBQRZSRZPZUGZUEZFUFZEUFZUHRZEFUIRZTWDXIJXHWSUJUKWGA + JEFULUMWHWIDUKZMZAWLWTWRXDXKWJBWKXKWJDUABXKWIDWHXJUNZUOXKBCDWDWEWFXJUPUTU + QZURXKWQXCWMXKWIDWPXBXKWOXAFSXKWJBWNQXMVAVAXLVBVCVDWHXFTLZXGTLZBXFDNZBXGO + ZDXHLWCXNWBWGFKVEVFWBXOWCWGEKVEVJWEXPWDWFWEXPBCBXFDBVGIVHVIVKWFXQWDWEWFXQ + GXGBHVLVIVMXFXGBDTTVNVOXETLWHAWTXDBWKVPWMXCAVQDXBVPVTVRWAVS $. $} ${ @@ -314257,31 +315256,31 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the difeq1d c0g cec eqid eqgid cvv ovexi cgrp tgpgrp grpidcl ecelqsg eqeltrrd cqg sylancr snssd ssequn2 sylib eqtrid unieqd wer eqger a1i eqtrd eqtr3id uniqs2 eqtr3d cin c0 wb difss unissi sseqtrid cpw cv wne wn adantr qsdisj - df-ne simpr ord disj2 syl6ib syl5bi expimpd eldifsn velpw 3imtr4g sspwuni - ssrdv sylibr uneqdifeq syl2anc mpbid ctop topontop simpl3 diffi wrex elqs - vex cplusg co cmpt cima simpll2 subgrcl subgss eqglact simplr tgplacthmeo - syl3anc chmeo sylan sseqtrd hmeocld eqeltrd syl5ibrcom rexlimdva ssdifssd - eleq1 unicld cldopn ex wi opnsubg 3expia 3adant3 impbid ) CUALZBCUBMZLZEA - UCZUDLZUFZBDUEMZLZBDLZUUKUUMUUNUUKUUMNZDOZUUIBUGZPZOZPZBDUUOEUUSPZUUTBUUO - EUUPUUSUUODEUHMLZEUUPQUUOUUFUVBUUFUUHUUJUUMUIZCDEFHUJRZEDUKRZUQUUOUUSBSZE - QZUVABQZUUOUUSUUQOZSZUVFEUUOUVIBUUSUUOUUHUVIBQUUFUUHUUJUUMULZBUUGUMRUNUUO - UVJUURUUQSZOZEUURUUQUOUUOUVMUUIOZEUUOUVLUUIUUOUVLUUIUUQSZUUIUUIUUQUPUUOUU - QUUITUVOUUIQUUOBUUIUUOCURMZAUSZBUUIUUOUUHUVQBQUVKACEBUVPHGUVPUTZVARUUOAVB - LZUVPELZUVQUUILACBVIGVCZUUOCVDLZUVTUUOUUFUWBUVCCVERECUVPHUVRVFREUVPAVBVGV - JVHZVKUUQUUIVLVMVNVOUUOEAVBUUOUUHEAVPZUVKACEBHGVQRZUVSUUOUWAVRWAZVSVTWBUU - OUUSETUUSBWCWDQZUVGUVHWEUUOUVNUUSEUURUUIUUIUUQWFWGUWFWHUUOUUSVBBPZTZUWGUU - OUURUWHWIZTUWIUUOIUURUWJUUOIWJZUUILZUWKBWKZNUWKUWHTZUWKUURLUWKUWJLUUOUWLU - 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X , w e. X |-> [ ( z ( -g ` G ) w ) ] ( G ~QG Y ) ) $. @@ -315488,11 +316487,11 @@ discrete topology (which is Hausdorff), this theorem can be used to turn ( vx cfv wcel c0 wceq syl adantr ctsu co ccld ctop ctgp tgptopon topontop ctopon 0cld eleq1 syl5ibrcom wne cv wex n0 wa csn ccl wf simpr tgptsmscls ccmn cuni wss tgptps tsmscl toponuni sseqtrd sselda snssd clscld syl2an2r - ctps eqid eqeltrd ex exlimdv syl5bi pm2.61dne ) AEDUAUBZFUCOZPZVTQAWBVTQR - QWAPZAFUDPZWCAFCUHOPZWDAEUEPZWEKEFCIHUFSZCFUGSZFUISVTQWAUJUKVTQULNUMZVTPZ - NUNAWBNVTUOAWJWBNAWJWBAWJUPZVTWIUQZFUROOZWAWKBCDEFGWIHIAEVBPWJJTAWFWJKTAB - GPWJLTABCDUSWJMTAWJUTVAAWDWJWLFVCZVDWMWAPWHWKWIWNAVTWNWIAVTCWNABCDEGHJAWF - EVMPKEVESLMVFAWECWNRWGCFVGSVHVIVJWLFWNWNVNVKVLVOVPVQVRVS $. + ctps eqid eqeltrd ex exlimdv biimtrid pm2.61dne ) AEDUAUBZFUCOZPZVTQAWBVT + QRQWAPZAFUDPZWCAFCUHOPZWDAEUEPZWEKEFCIHUFSZCFUGSZFUISVTQWAUJUKVTQULNUMZVT + PZNUNAWBNVTUOAWJWBNAWJWBAWJUPZVTWIUQZFUROOZWAWKBCDEFGWIHIAEVBPWJJTAWFWJKT + ABGPWJLTABCDUSWJMTAWJUTVAAWDWJWLFVCZVDWMWAPWHWKWIWNAVTWNWIAVTCWNABCDEGHJA + WFEVMPKEVESLMVFAWECWNRWGCFVGSVHVIVJWLFWNWNVNVKVLVOVPVQVRVS $. $} ${ @@ -315582,59 +316581,59 @@ discrete topology (which is Hausdorff), this theorem can be used to turn reseq2 cmnd cmnmnd simplr jca ovexd c0g fvexi fsuppmptdm gsumcl velsn ovres sylanbr oveq1 eqtrd gsumsn syl3anc snssd xpss12 fssresd sylancr snfi xpfi fdmfifsupp gsumxp 3eqtr4rd elrnmpti cabl ad3antrrr ablnncan - 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JXBZVYCEWCYLZUVKVYEVXTVUPVYBUVLYMVXTVUFLKUYQVRZVYCVSZVKZVLVIZQVIZPVNZ - HOVPZVYKAUYPVUNWUOVYLVUMWUNHOAVVRUYPVUMWUNVOAVVRVQZUYPVQZVUMLVUCVYCVK - ZVLVIZVUJVNZWUNWUQVYDVUMWUTVOAUYPVYDVVRWUHYNVULWUTUSVYCUYNUYSVYCXHZVU - LVUKWUTWVAUYTVULVUKUVMWVAVYCUYRUYSUYRVYBVYCUYOUYQUVNVYBVUPUVOUVPWVAUV - QUVRUYTVUKUVSWIWVAVUEWUSVUJWVAVUDWURLVLUYSVYCVUCUWBYOYPUVTUWAWIWUQWUT - WULVUJVNZWUNWUQWUSWULVUJWUQLVAWUILIVYCVVKVUAWUKVIZVJZVLVIZVJVLVIZLIVY - CVUBVJZVLVIZWULWUSWUQLUWCVNZVVRWVHDVNWVFWVHXHWUQVWDWVIAVWDVVRUYPUBYQZ - LUWDWIAVVRUYPUWEWUQVYCDWVGLVCTUAUJWVJAUYPVYNVVRWUGYNZWUQIVYCVUBDWUQVU - AVYCVNZVWGVWHWUQVYCEVUAAUYPVYMVVRWUAYNZWEWUPVWGVWHUYPWUPVWJVVSVQVWGVW - HWUPVWJVVSAVWJVVRUFWHVWBUWFVWJVVSVWGVWHVWKWOXLYNYFWPWUQIVYCWVGYDYDVUB - TWVGWFWVKWUQWVLVQUYQVUAKUWGTYDVNWUQTLUWHUJUWIYMZUWJUWKWVEDWVHVALUYQOU - AVAHYTZWVDWVGLVLWVOIVYCWVCVUBWVOWVLVQWVCVVKVUAKVIZVUBWVOVVKWUIVNWVLWV - CWVPXHVAUYQUWLVVKVUAWUIVYCKUWMUWNWVOWVPVUBXHWVLVVKUYQVUAKUWOWHUWPXMYO - UWQUWRWUQWUIDVYCVAIWUKLVCVCTUAUJWVJWUIVCVNZWUQUYQUXCZYMWVKWUQVWIDWUJK - AVWJVVRUYPUFYQWUQWUICVHVYMWUJVWIVHWUQUYQCWUPVVSUYPVWBWHUWSWVMWUICVYCE - UWTXBUXAZWUQWUJDWUKYDTWVSWUQWVQVYNWUJVCVNWVRWVKWUIVYCUXDUXBWVNUXEUXFW - UQWURWVGLVLWUQIEVYCVUBWVMXCYOUXGYPWVBWULVUHXHZUTPVTWUQWUNUTPVUHWULVUI - VXRVUFVUGQYEUXHWUQWVTWUNUTPWUQVUGPVNZVQZWUNWVTVUFVUHQVIZPVNWWBWWCVUGP - WWBDLQVUFVUGUAULALUXIVNZVVRUYPWWAAVXJVWDWWDVXKUBLUXLYLUXJWUPVXFUYPWWA - AVVRVVSVXFVWBVXHYFYQWUQPDVUGAVWOVVRUYPVWRYQWEUXKWUQWWAUXMYBWVTWUMWWCP - WULVUHVUFQYIYPUXNUXOUXPUYBUXQUXRUXSUXTWUNVYJHBOHBYTZWUMVYIPWWEVUFVUTW - ULVYHQUYQVUSMUYCWWEWUKVYGLVLWWEWUJVYFKWWEWUIVVAVYCUYQVUSUYAUYDYRYOUYE - YPUYFYSVVHVYEVYKVQJVYCUYNVUQVYCXHZVURVYEVVGVYKVUQVYCVUPUYKWWFVVFVYJBO - WWFVVEVYIPWWFVVDVYHVUTQWWFVVCVYGLVLWWFVVBVYFKVUQVYCVVAUYLYRYOYOYPYGUY - GUYHUYIUYJ $. + isabl simpr syl5ibrcom rexlimdva biimtrid sylbid an32s ralimdva fveq2 + syld impr xpeq1d oveq12d cbvralvw sseq2 xpeq2 anbi12d rspcev syl12anc + sneq exlimddv ) AOEVBZVCVDZURVEZVFZHVEZUYOVGZUSVEZVHZLIEUYQIVEZKVIZVJ + ZUYSVKZVLVIZUTPUYQMVGZUTVEZQVIZVJZVMZVNZVOZUSUYNVPZHOVPZVQZFVMZJVEZVH + ZBVEZMVGZLKVUSVRZVUQVSZVKZVLVIZQVIZPVNZBOVPZVQZJUYNVTZURAOVCVNZVAVEZU + YSVHZVUKVOZUSUYNVPZVAUYNVTZHOVPVUOURWAACVBZVCOUOWBZAVVOHOAUYQOVNZUYQC + VNZVVOAOCUYQAOVVPVCVDVNZOCVHZUOVVTVWAVVJOCWCWDWIWEZAVVSVQZUSVAEDVUFUY + NVUJVUCLNSUAUIUYNWFALWGVNZVVSUBWHALWJVNZVVSALWKVNZVWEUCLWLWIWHAESVNVV + SUEWHVWCIEVUBDAVVSVUAEVNZVUBDVNZACEVSZDKVFZVVSVWGVWHUFUYQVUADCEKWMZWN + WOWPUHVWCUTDVUHVJZPWQZVUJNVWCVWMVWLPVKZVMVUJVWLPWRVWCVWNVUIVWCUTDPVUH + APDVHZVVSANDWTVGVNZPNVNZVWOAVWFVWPUCLNDUIUAWSWIUMPNDXAXBZWHXCXDXEVWCV + WLNNXFVIZVNVWQVWMNVNVWCVWLVADVUFVVKGVIZVJZLXJVGZXGZVWSVWCVWLUTDVUFVUG + VXBVGZGVIZVJVXCVWCUTDVUHVXEVWCVUFDVNZVUGDVNZVUHVXEXHACDUYQMUGXKZDGLVX + BQVUFVUGUAUKVXBWFZULXIXLXMVWCUTVADDVXDVWTVXEVXBVXAVWCLXNVNZVXGVXDDVNA + VXJVVSAVWFVXJUCLXOWIZWHZDLVXBVUGUAVXIXPXLVWCUTDDVXBVWCVXJDDVXBVFVXLDL + VXBUAVXIXQWIXSVWCVXAXRVVKVXDVUFGYIXTYAVWCVXBVWSVNZVXAVWSVNZVXCVWSVNVW + CVWFVXMAVWFVVSUCWHZLVXBNUIVXIUUAWIVWCVWFVXFVXNVXOVXHVAVUFGVXALNDVXAWF + UAUKUIUUHXBVXBVXANNNUUBXBYBAVWQVVSUMWHPVWLNNUUCXBYCVWCVUFTQVIZVUFVUJV + WCVXJVXFVXPVUFXHVXLVXHDLQVUFTUAUJULUUDXBVWCTPVNZVXPYDVNVXPVUJVNAVXQVV + SUNWHVUFTQYEUTPVUHVXPTVUIYDVUIWFZVUGTVUFQYIUUEUUFYCUUGYFUUIVVNVUMHVAO + UYNURVVKUYRXHZVVMVULUSUYNVXSVVLUYTVUKVVKUYRUYSUUJUUKYGUULXBAVUOVQZUYO + VMZUUMZVUPUUNZUYNVNZVUPVYCVHZVUTLKVVAVYCVSZVKZVLVIZQVIZPVNZBOVPZVVIAU + YPVYDVUNAUYPVQZVYCEVHZVYCVCVNZVYDVYLVYBVUPEVYLVYAUYMVHVYBEVHVYLVYAUYN + UYMUYPVYAUYNVHAOUYNUYOUURYHZUYMVCUUOYJVYAEUUPYSAVUPEVHUYPAVUPVWIVMZEA + FVWIVBZVCVDVNZFVWIVHZVUPVYPVHUQVYRVYSFVCVNZFVWIWCWDFVWIUUQYKCEUUSYJWH + UUTZVYLVYBVCVNZVUPVCVNZVYNVYLVYAVCVNZVYAVCVHWUBVYLVVJOVYAUYOUVAZWUDAV + VJUYPVVQWHVYLUYOOUVBZWUEUYPWUFAOUYNUYOUVCYHOUYOUVDYSOVYAUYOUVEXBVYLVY + AUYNVCVYOUYMVCUVFYJVYAUVGXBAWUCUYPAVYRVYTWUCUQFVYQVCUVHFUVIYKWHVYBVUP + UVJXBZVYCEWCYLZUVKVYEVXTVUPVYBUVLYMVXTVUFLKUYQVRZVYCVSZVKZVLVIZQVIZPV + NZHOVPZVYKAUYPVUNWUOVYLVUMWUNHOAVVRUYPVUMWUNVOAVVRVQZUYPVQZVUMLVUCVYC + VKZVLVIZVUJVNZWUNWUQVYDVUMWUTVOAUYPVYDVVRWUHYNVULWUTUSVYCUYNUYSVYCXHZ + VULVUKWUTWVAUYTVULVUKUVMWVAVYCUYRUYSUYRVYBVYCUYOUYQUVNVYBVUPUVOUVPWVA + UVQUVRUYTVUKUVSWIWVAVUEWUSVUJWVAVUDWURLVLUYSVYCVUCUWBYOYPUVTUWAWIWUQW + UTWULVUJVNZWUNWUQWUSWULVUJWUQLVAWUILIVYCVVKVUAWUKVIZVJZVLVIZVJVLVIZLI + VYCVUBVJZVLVIZWULWUSWUQLUWCVNZVVRWVHDVNWVFWVHXHWUQVWDWVIAVWDVVRUYPUBY + QZLUWDWIAVVRUYPUWEWUQVYCDWVGLVCTUAUJWVJAUYPVYNVVRWUGYNZWUQIVYCVUBDWUQ + VUAVYCVNZVWGVWHWUQVYCEVUAAUYPVYMVVRWUAYNZWEWUPVWGVWHUYPWUPVWJVVSVQVWG + VWHWUPVWJVVSAVWJVVRUFWHVWBUWFVWJVVSVWGVWHVWKWOXLYNYFWPWUQIVYCWVGYDYDV + UBTWVGWFWVKWUQWVLVQUYQVUAKUWGTYDVNWUQTLUWHUJUWIYMZUWJUWKWVEDWVHVALUYQ + OUAVAHYTZWVDWVGLVLWVOIVYCWVCVUBWVOWVLVQWVCVVKVUAKVIZVUBWVOVVKWUIVNWVL + WVCWVPXHVAUYQUWLVVKVUAWUIVYCKUWMUWNWVOWVPVUBXHWVLVVKUYQVUAKUWOWHUWPXM + YOUWQUWRWUQWUIDVYCVAIWUKLVCVCTUAUJWVJWUIVCVNZWUQUYQUXCZYMWVKWUQVWIDWU + JKAVWJVVRUYPUFYQWUQWUICVHVYMWUJVWIVHWUQUYQCWUPVVSUYPVWBWHUWSWVMWUICVY + CEUWTXBUXAZWUQWUJDWUKYDTWVSWUQWVQVYNWUJVCVNWVRWVKWUIVYCUXDUXBWVNUXEUX + FWUQWURWVGLVLWUQIEVYCVUBWVMXCYOUXGYPWVBWULVUHXHZUTPVTWUQWUNUTPVUHWULV + UIVXRVUFVUGQYEUXHWUQWVTWUNUTPWUQVUGPVNZVQZWUNWVTVUFVUHQVIZPVNWWBWWCVU + GPWWBDLQVUFVUGUAULALUXIVNZVVRUYPWWAAVXJVWDWWDVXKUBLUXLYLUXJWUPVXFUYPW + WAAVVRVVSVXFVWBVXHYFYQWUQPDVUGAVWOVVRUYPVWRYQWEUXKWUQWWAUXMYBWVTWUMWW + CPWULVUHVUFQYIYPUXNUXOUXPUXQUYAUXRUXSUYBWUNVYJHBOHBYTZWUMVYIPWWEVUFVU + TWULVYHQUYQVUSMUXTWWEWUKVYGLVLWWEWUJVYFKWWEWUIVVAVYCUYQVUSUYKUYCYRYOU + YDYPUYEYSVVHVYEVYKVQJVYCUYNVUQVYCXHZVURVYEVVGVYKVUQVYCVUPUYFWWFVVFVYJ + BOWWFVVEVYIPWWFVVDVYHVUTQWWFVVCVYGLVLWWFVVBVYFKVUQVYCVVAUYGYRYOYOYPYG + UYHUYIUYJUYL $. $} tsmsxp.4 $e |- ( ph -> A. c e. S A. d e. T ( c .+ d ) e. U ) $. @@ -316329,17 +317328,17 @@ unit group (that is, the nonzero numbers) to the field. (Contributed wrex ibi adantl simp2d ne0d simp3d cop opelidres elv biimpri r19.2z mpan2 rgen ad2antrr ne0i rexlimivw ssn0 syl2anc nelrdva df-nel sylibr vex inex2 wn pwid a1i elind 3jca xpexd isfbas mpbir2and wex elin velpw anbi2i bitri - n0 exbii an32s expimpd exlimdv syl5bi isfil sylanbrc ) BEFZABUAGHZIZABBUB - ZUCGHZACUDZJZKZEFZXCAHZUEZCXAJZLAXAUFGHWTXBAXIMZAEFZEAUGZADUDZXCKZJZKZEFZ - CALZDALZNZWTXJXAAHZXMXCMZXGUEZCXILZXNAHZCALZUHBUIZXMMZXMUJAHZXCXCUKXMMCAU - NZNZNZDALZWSXJYAYMNZWRWSYNWSBOHWSYNPABUAULZCDAOBUMSUOUPZTWTXKXLXSWTAXAWTX - JYAYMYPUQURWTEAHVQXLWTDAEWTXMAHZIZYHYGEFZXMEFYRYHYIYJYRYDYFYKWTYLDAWTXJYA - YMYPUSQZUSTYRXCXCUTZYGHZCBUNZYSWRUUCWSYQWRUUBCBLUUCUUBCBUUBXCBHZUUBUUDPCX - CBOVAVBVCVFUUBCBVDVEVGUUBYSCBYGUUAVHVISYGXMVJVKVLEAVMVNWTXRDAYRXQCAYRXGIZ - XPXNUUEAXOXNYRYECAYRYDYFYKYTUQQXNXOHUUEXNXCXMCVOVPVRVSVTURRRWAWSXBXJXTIPZ - WRWSXAOHUUFWSBBOOYOYOWBDCOXAAWCSUPWDWTXHCXIXFYQYBIZDWEZWTXCXIHZIZXGXFXMXE - HZDWEUUHDXEWJUUKUUGDUUKYQXMXDHZIUUGXMAXDWFUULYBYQDXCWGWHWIWKWIUUJUUGXGDUU - JYQYBXGWTYQUUIYCYRYCCXIYRYDYFYKYTTQWLWMWNWORCAXAWPWQ $. + n0 exbii an32s expimpd exlimdv biimtrid isfil sylanbrc ) BEFZABUAGHZIZABB + UBZUCGHZACUDZJZKZEFZXCAHZUEZCXAJZLAXAUFGHWTXBAXIMZAEFZEAUGZADUDZXCKZJZKZE + FZCALZDALZNZWTXJXAAHZXMXCMZXGUEZCXILZXNAHZCALZUHBUIZXMMZXMUJAHZXCXCUKXMMC + AUNZNZNZDALZWSXJYAYMNZWRWSYNWSBOHWSYNPABUAULZCDAOBUMSUOUPZTWTXKXLXSWTAXAW + TXJYAYMYPUQURWTEAHVQXLWTDAEWTXMAHZIZYHYGEFZXMEFYRYHYIYJYRYDYFYKWTYLDAWTXJ + YAYMYPUSQZUSTYRXCXCUTZYGHZCBUNZYSWRUUCWSYQWRUUBCBLUUCUUBCBUUBXCBHZUUBUUDP + CXCBOVAVBVCVFUUBCBVDVEVGUUBYSCBYGUUAVHVISYGXMVJVKVLEAVMVNWTXRDAYRXQCAYRXG + IZXPXNUUEAXOXNYRYECAYRYDYFYKYTUQQXNXOHUUEXNXCXMCVOVPVRVSVTURRRWAWSXBXJXTI + PZWRWSXAOHUUFWSBBOOYOYOWBDCOXAAWCSUPWDWTXHCXIXFYQYBIZDWEZWTXCXIHZIZXGXFXM + XEHZDWEUUHDXEWJUUKUUGDUUKYQXMXDHZIUUGXMAXDWFUULYBYQDXCWGWHWIWKWIUUJUUGXGD + UUJYQYBXGWTYQUUIYCYRYCCXIYRYDYFYKYTTQWLWMWNWORCAXAWPWQ $. $} $( A uniform structure cannot be empty. (Contributed by Thierry Arnoux, @@ -316493,16 +317492,11 @@ unit group (that is, the nonzero numbers) to the field. (Contributed UPQSUNUTURVAJZVBUSVAEUTVDEUNUSURKZVAUOUPUAVEBUOUKZKZVAVFURBAUBZUCVFBEVGVAEB UOUDZVFBUEUFRRUSVAURUGTURBEVDVBEUNURVFBVHVIRURBVAUHTUIUJ $. - ${ - $d x U $. $d x X $. - $( First direction for ~ ustbas . (Contributed by Thierry Arnoux, - 16-Nov-2017.) $) - elrnust $p |- ( U e. ( UnifOn ` X ) -> U e. U. ran UnifOn ) $= - ( vx cust cfv wcel cv cdm wrex crn cuni elfvdm wceq eleq2d rspcev mpancom - fveq2 cvv wfn wfun wb ustfn fnfun elunirn mp2b sylibr ) ABDEZFZACGZDEZFZC - DHZIZADJKFZBULFUHUMABDLUKUHCBULUIBMUJUGAUIBDQNOPDRSDTUNUMUAUBRDUCCADUDUEU - F $. - $} + $( Obsolete version of ~ elfvunirn as of 12-Jan-2025. (Contributed by + Thierry Arnoux, 16-Nov-2017.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + elrnustOLD $p |- ( U e. ( UnifOn ` X ) -> U e. U. ran UnifOn ) $= + ( cust elfvunirn ) BACD $. ${ $d v w U $. $d v w X $. @@ -316535,9 +317529,9 @@ unit group (that is, the nonzero numbers) to the field. (Contributed Arnoux, 16-Nov-2017.) $) ustbas $p |- ( U e. U. ran UnifOn <-> U e. ( UnifOn ` X ) ) $= ( vx cust crn cuni wcel cfv cv cdm wrex cvv wfun ustfn fnfun elunirn mp2b - wfn wb ustbas2 eqtr4di fveq2d eleq2d ibi rexlimivw sylbi elrnust impbii ) - AEFGHZABEIZHZUJADJZEIZHZDEKZLZULEMSENUJUQTOMEPDAEQRUOULDUPUOULUOUNUKAUOUM - BEUOUMAGKBAUMUACUBUCUDUEUFUGABUHUI $. + wfn wb ustbas2 eqtr4di fveq2d eleq2d ibi rexlimivw sylbi elfvunirn impbii + ) AEFGHZABEIZHZUJADJZEIZHZDEKZLZULEMSENUJUQTOMEPDAEQRUOULDUPUOULUOUNUKAUO + UMBEUOUMAGKBAUMUACUBUCUDUEUFUGBAEUHUI $. $} $( Lemma for ~ ustuqtop . (Contributed by Thierry Arnoux, 5-Dec-2017.) $) @@ -316637,10 +317631,10 @@ unit group (that is, the nonzero numbers) to the field. (Contributed A. x e. a E. v e. U ( v " { x } ) C_ a } ) $= ( vu cust cfv wcel cv csn cima wrex wral cuni cdm cpw crab cvv wceq cutop wss crn df-utop wa simpr unieqd dmeqd ustbas2 adantr eqtr4d pweqd rexeqdv - ralbidv rabeqbidv elrnust elfvex pwexg rabexg 3syl fvmptd2 ) CDGHIZFCBJAJ - KLEJZUBZBFJZMZAVCNZEVEOZPZQZRVDBCMZAVCNZEDQZRZGUCOUASABFEUDVBVECTZUEZVGVL - EVJVMVPVIDVPVICOZPZDVPVHVQVPVECVBVOUFZUGUHVBDVRTVOCDUIUJUKULVPVFVKAVCVPVD - BVECVSUMUNUOCDUPVBDSIVMSIVNSICDGUQDSURVLEVMSUSUTVA $. + ralbidv rabeqbidv elfvunirn elfvex pwexg rabexg 3syl fvmptd2 ) CDGHIZFCBJ + AJKLEJZUBZBFJZMZAVCNZEVEOZPZQZRVDBCMZAVCNZEDQZRZGUCOUASABFEUDVBVECTZUEZVG + VLEVJVMVPVIDVPVICOZPZDVPVHVQVPVECVBVOUFZUGUHVBDVRTVOCDUIUJUKULVPVFVKAVCVP + VDBVECVSUMUNUODCGUPVBDSIVMSIVNSICDGUQDSURVLEVMSUSUTVA $. $} ${ @@ -317257,9 +318251,9 @@ unit group (that is, the nonzero numbers) to the field. (Contributed sSet <. ( TopSet ` ndx ) , ( unifTop ` U ) >. ) ) $= ( vu cust cfv wcel cnx cbs cv cuni cdm cop cunif cpr cutop csts co opeq2d cts crn ctus cvv df-tus wceq wa simpr unieqd dmeqd preq12d fveq2d oveq12d - elrnust ovexd fvmptd2 ) ABDEFZCAGHEZCIZJZKZLZGMEZUQLZNZGSEZUQOEZLZPQUPAJZ - KZLZVAALZNZVDAOEZLZPQDTJUAUBCUCUOUQAUDZUEZVCVKVFVMPVOUTVIVBVJVOUSVHUPVOUR - VGVOUQAUOVNUFZUGUHRVOUQAVAVPRUIVOVEVLVDVOUQAOVPUJRUKABULUOVKVMPUMUN $. + elfvunirn ovexd fvmptd2 ) ABDEFZCAGHEZCIZJZKZLZGMEZUQLZNZGSEZUQOEZLZPQUPA + JZKZLZVAALZNZVDAOEZLZPQDTJUAUBCUCUOUQAUDZUEZVCVKVFVMPVOUTVIVBVJVOUSVHUPVO + URVGVOUQAUOVNUFZUGUHRVOUQAVAVPRUIVOVEVLVDVOUQAOVPUJRUKBADULUOVKVMPUMUN $. tuslem.k $e |- K = ( toUnifSp ` U ) $. $( Lemma for ~ tusbas , ~ tusunif , and ~ tustopn . (Contributed by @@ -317393,17 +318387,17 @@ unit group (that is, the nonzero numbers) to the field. (Contributed ucnval $p |- ( ( U e. ( UnifOn ` X ) /\ V e. ( UnifOn ` Y ) ) -> ( U uCn V ) = { f e. ( Y ^m X ) | A. s e. V E. r e. U A. x e. X A. y e. X ( x r y -> ( f ` x ) s ( f ` y ) ) } ) $= - ( vu vv cfv wcel co cv cuni cdm wral cmap wceq cust wa cucn wbr wrex crab - wi crn cvv elrnust adantr adantl ovex rabex a1i simpr dmeqd simpl oveq12d - unieqd raleqdv raleqbidv rexeqbidv df-ucn ovmpoga syl3anc ustbas2 rexbidv - rabeqbidv oveqan12rd ralbidv eqtr4d ) CFUALMZEGUALMZUBZCEUCNZAOZBOZIOUDVQ - DOZLVRVSLHOUDUGZBCPZQZRZAWBRZICUEZHERZDEPZQZWBSNZUFZVTBFRZAFRZICUEZHERZDG - FSNZUFVOCUAUHPZMZEWPMZWJUIMZVPWJTVMWQVNCFUJUKVNWRVMEGUJULWSVOWFDWIWHWBSUM - UNUOJKCEWPWPVTBJOZPZQZRZAXBRZIWTUEZHKOZRZDXFPZQZXBSNZUFWJUCUIWTCTZXFETZUB - ZXGWFDXJWIXMXIWHXBWBSXMXHWGXMXFEXKXLUPZUTUQXMXAWAXMWTCXKXLURZUTUQZUSXMXEW - EHXFEXNXMXDWDIWTCXOXMXCWCAXBWBXPXMVTBXBWBXPVAVBVCVBVIABKJDHIVDVEVFVOWNWFD - WOWIVNVMGWHFWBSEGVGCFVGZVJVOWMWEHEVOWLWDICVOWKWCAFWBVMFWBTVNXQUKZVOVTBFWB - XRVAVBVHVKVIVL $. + ( vu cust cfv wcel co cv cuni cdm wral cmap wceq vv wa cucn wbr wrex crab + crn cvv elfvunirn adantr adantl ovex rabex a1i simpr unieqd dmeqd oveq12d + wi simpl raleqdv raleqbidv rexeqbidv rabeqbidv ovmpoga syl3anc oveqan12rd + df-ucn ustbas2 rexbidv ralbidv eqtr4d ) CFKLMZEGKLMZUBZCEUCNZAOZBOZIOUDVQ + DOZLVRVSLHOUDUSZBCPZQZRZAWBRZICUEZHERZDEPZQZWBSNZUFZVTBFRZAFRZICUEZHERZDG + FSNZUFVOCKUGPZMZEWPMZWJUHMZVPWJTVMWQVNFCKUIUJVNWRVMGEKUIUKWSVOWFDWIWHWBSU + LUMUNJUACEWPWPVTBJOZPZQZRZAXBRZIWTUEZHUAOZRZDXFPZQZXBSNZUFWJUCUHWTCTZXFET + ZUBZXGWFDXJWIXMXIWHXBWBSXMXHWGXMXFEXKXLUOZUPUQXMXAWAXMWTCXKXLUTZUPUQZURXM + XEWEHXFEXNXMXDWDIWTCXOXMXCWCAXBWBXPXMVTBXBWBXPVAVBVCVBVDABUAJDHIVHVEVFVOW + NWFDWOWIVNVMGWHFWBSEGVICFVIZVGVOWMWEHEVOWLWDICVOWKWCAFWBVMFWBTVNXQUJZVOVT + BFWBXRVAVBVJVKVDVL $. $} ${ @@ -317646,12 +318640,12 @@ unit group (that is, the nonzero numbers) to the field. (Contributed iscfilu $p |- ( U e. ( UnifOn ` X ) -> ( F e. ( CauFilU ` U ) <-> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) ) $= ( vf vu cust cfv wcel ccfilu cuni cdm cfbas cv wrex wral wa crab wceq cxp - wss crn elrnust unieq dmeqd raleq rabeqbidv df-cfilu fvex rabex fvmpt syl - fveq2d eleq2d rexeq ralbidv elrab bitrdi ustbas2 anbi1d bitr4d ) BDHIJZCB - KIZJZCBLZMZNIZJZEOZVJUAAOUBZECPZABQZRZCDNIZJZVMRVCVECVKEFOZPZABQZFVHSZJVN - VCVDVTCVCBHUCLZJVDVTTBDUDGBVRAGOZQZFWBLZMZNIZSVTWAKWBBTZWCVSFWFVHWGWEVGNW - GWDVFWBBUEUFUNVRAWBBUGUHAGFEUIVSFVHVGNUJUKULUMUOVSVMFCVHVQCTVRVLABVKEVQCU - PUQURUSVCVPVIVMVCVOVHCVCDVGNBDUTUNUOVAVB $. + wss crn elfvunirn unieq dmeqd fveq2d raleq rabeqbidv df-cfilu rabex fvmpt + fvex syl eleq2d rexeq ralbidv elrab bitrdi ustbas2 anbi1d bitr4d ) BDHIJZ + CBKIZJZCBLZMZNIZJZEOZVJUAAOUBZECPZABQZRZCDNIZJZVMRVCVECVKEFOZPZABQZFVHSZJ + VNVCVDVTCVCBHUCLZJVDVTTDBHUDGBVRAGOZQZFWBLZMZNIZSVTWAKWBBTZWCVSFWFVHWGWEV + GNWGWDVFWBBUEUFUGVRAWBBUHUIAGFEUJVSFVHVGNUMUKULUNUOVSVMFCVHVQCTVRVLABVKEV + QCUPUQURUSVCVPVIVMVCVOVHCVCDVGNBDUTUGUOVAVB $. $} ${ @@ -318966,114 +319960,114 @@ the base 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and which is nonnegative, symmetric ( vx cfv wcel w3a cxr cxad co cle wbr wa clt xmetcl adantr syl3anc cbl cv cxmet cin wn simpr3 simpr1 simpr2 xaddcld xrlenltd mpbid wb simpl1 simpl2 elin elbl simpl3 anbi12d bitr4di simpr xlt2add syl22anc xmettri3 syl13anc - anandi wi xrlelttr mpand syld expimpd sylbid syl5bi mtod eq0rdv ) AFUCHIZ - BFIZCFIZJZDKIZEKIZDELMZBCAMZNOZJZPZGBDAUAHZMZCEWFMZUDZWEGUBZWIIZWBWAQOZWE - WCWLUEVRVSVTWCUFWEWAWBWEDEVRVSVTWCUGZVRVSVTWCUHZUIZVRWBKIZWDBCAFRSZUJUKWK - WJWGIZWJWHIZPZWEWLWJWGWHUOWEWTWJFIZBWJAMZDQOZCWJAMZEQOZPZPZWLWEWTXAXCPZXA - XEPZPXGWEWRXHWSXIWEVOVPVSWRXHULVOVPVQWDUMZVOVPVQWDUNZWMWJABDFUPTWEVOVQVTW - SXIULXJVOVPVQWDUQZWNWJACEFUPTURXAXCXEVEUSWEXAXFWLWEXAPZXFXBXDLMZWAQOZWLXM - XBKIZXDKIZVSVTXFXOVFXMVOVPXAXPWEVOXAXJSZWEVPXAXKSZWEXAUTZBWJAFRTZXMVOVQXA - XQXRWEVQXAXLSZXTCWJAFRTZWEVSXAWMSWEVTXAWNSXBXDDEVAVBXMWBXNNOZXOWLXMVOVPVQ - XAYDXRXSYBXTBCWJAFVCVDXMWPXNKIWAKIZYDXOPWLVFWEWPXAWQSXMXBXDYAYCUIWEYEXAWO - SWBXNWAVGTVHVIVJVKVLVMVN $. + anandi wi xrlelttr mpand syld expimpd sylbid biimtrid mtod eq0rdv ) AFUCH + IZBFIZCFIZJZDKIZEKIZDELMZBCAMZNOZJZPZGBDAUAHZMZCEWFMZUDZWEGUBZWIIZWBWAQOZ + WEWCWLUEVRVSVTWCUFWEWAWBWEDEVRVSVTWCUGZVRVSVTWCUHZUIZVRWBKIZWDBCAFRSZUJUK + WKWJWGIZWJWHIZPZWEWLWJWGWHUOWEWTWJFIZBWJAMZDQOZCWJAMZEQOZPZPZWLWEWTXAXCPZ + XAXEPZPXGWEWRXHWSXIWEVOVPVSWRXHULVOVPVQWDUMZVOVPVQWDUNZWMWJABDFUPTWEVOVQV + TWSXIULXJVOVPVQWDUQZWNWJACEFUPTURXAXCXEVEUSWEXAXFWLWEXAPZXFXBXDLMZWAQOZWL + XMXBKIZXDKIZVSVTXFXOVFXMVOVPXAXPWEVOXAXJSZWEVPXAXKSZWEXAUTZBWJAFRTZXMVOVQ + XAXQXRWEVQXAXLSZXTCWJAFRTZWEVSXAWMSWEVTXAWNSXBXDDEVAVBXMWBXNNOZXOWLXMVOVP + VQXAYDXRXSYBXTBCWJAFVCVDXMWPXNKIWAKIZYDXOPWLVFWEWPXAWQSXMXBXDYAYCUIWEYEXA + WOSWBXNWAVGTVHVIVJVKVLVMVN $. $( A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) $) @@ -319709,11 +320703,11 @@ the base set to the (extended) reals and which is nonnegative, symmetric ( ( P ( ball ` D ) R ) =/= (/) <-> 0 < R ) ) $= ( vx cxmet cfv wcel cxr w3a cbl co c0 cc0 clt wbr wa 3expa 3adantl3 wi cv wne wex elbl cle xmetge0 0xr xmetcl simpl3 xrlelttr mp3an2i mpand expimpd - n0 sylbid exlimdv syl5bi xblcntr ne0d expr 3impa impbid ) ADFGHZBDHZCIHZJ - ZBCAKGLZMUBZNCOPZVHEUAZVGHZEUCVFVIEVGUNVFVKVIEVFVKVJDHZBVJALZCOPZQVIVJABC - DUDVFVLVNVIVFVLQZNVMUEPZVNVIVCVDVLVPVEVCVDVLVPBVJADUFRSNIHVOVMIHZVEVPVNQV - ITUGVCVDVLVQVEVCVDVLVQBVJADUHRSVCVDVEVLUINVMCUJUKULUMUOUPUQVCVDVEVIVHTVCV - DQVEVIVHVCVDVEVIQZVHVCVDVRJVGBABCDURUSRUTVAVB $. + n0 sylbid exlimdv biimtrid xblcntr ne0d expr 3impa impbid ) ADFGHZBDHZCIH + ZJZBCAKGLZMUBZNCOPZVHEUAZVGHZEUCVFVIEVGUNVFVKVIEVFVKVJDHZBVJALZCOPZQVIVJA + BCDUDVFVLVNVIVFVLQZNVMUEPZVNVIVCVDVLVPVEVCVDVLVPBVJADUFRSNIHVOVMIHZVEVPVN + QVITUGVCVDVLVQVEVCVDVLVQBVJADUHRSVCVDVEVLUINVMCUJUKULUMUOUPUQVCVDVEVIVHTV + CVDQVEVIVHVCVDVEVIQZVHVCVDVRJVGBABCDURUSRUTVAVB $. $( A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) $) @@ -321420,86 +322414,86 @@ S C_ ( P ( ball ` D ) T ) ) $= w3a wa wex wi wne co cxr cxmet wb syl adantr simprl syl3anc cds cbs cxp cixp cres cmpt 3ad2ant1 ovex ffvelcdmda ralrimiva feqmptd oveq2d eqtrid eqid cprds fveq2d eleqtrd blopn 2fveq3 eqtr4di fveq2 sylan eqtrd eleq1d - fveq1 anbi12d ixpeq2dva syl121anc sylbid eqtrdi rspcev syl5bi sseqtrrdi - weq raleqdv ccnv cima ctop ctps xmstps eqeltrd ctopon eqeltrrd ad2antrr - cfi cnveqd crp ccom cpt ffnd dffn2 fnfco ptval cab wal prdsxmslem1 blrn - eldifsn cc0 clt wbr simprr xbln0 mptexd rgenw fnmpt mp1i xmsxmet simp2l - prdsbascl r19.21bi simp2r sqxpeqd reseq12d eqidd oveq123d fvmpt xmstopn - adantl fvco3 3eltr4d oveqd cbvmptv oveq2i simp3 prdsbl ralbidv sylan9eq - fneq1 eqeq2d spcegv 3impib 3expia simpr neeq1d df-3an ral0 difeq2 difid - sylancl biantrud bitr4id eqeq1 bi2anan9 3imtr4d rexlimdvva impd alrimiv - exbidv ex ssab sylibr cmpo cun fnssres mp2an fvres tpstop rgen mpbir2an - ssv ffnfv 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(Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) $) metuval $p |- ( D e. ( PsMet ` X ) -> ( metUnif ` D ) = ( ( X X. X ) filGen ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) ) $= - ( vd vx vy vz vu vw vv cpsmet wcel cv cdm cxp crp co crn cfg wceq cfv cc0 - ccnv cico cima cmpt cmetu cvv df-metu simpr dmeqd psmetdmdm adantr eqtr4d - cuni wa sqxpeqd simplr cnveqd imaeq1d mpteq2dva rneqd oveq12d wrex elfvdm - fveq2 eleq2d rspcev mpancom wfun cxad cle wbr wral cxr cmap crab df-psmet - wb funmpt2 elunirn ax-mp sylibr ovexd fvmptd2 ) ABKUAZLZDADMZNZNZWJOZCPWH - UCZUBCMZUDQZUEZUFZRZSQBBOZCPAUCZWNUEZUFZRZSQKRUOZUGUHCDUIWGWHATZUPZWKWRWQ - XBSXEWJBXEWJANZNZBXEWIXFXEWHAWGXDUJUKUKWGBXGTXDABULUMUNUQXEWPXAXECPWOWTXE - WMPLZUPZWLWSWNXIWHAWGXDXHURUSUTVAVBVCWGAEMZKUAZLZEKNZVDZAXCLZBXMLWGXNABKV - EXLWGEBXMXJBTXKWFAXJBKVFVGVHVIKVJXOXNVSFUHGMZXPHMZQUBTXPIMZXQQJMZXPXQQXSX - RXQQVKQVLVMJFMZVNIXTVNUPGXTVNHVOXTXTOVPQVQKFGIJHVRVTEAKWAWBWCWGWRXBSWDWE - $. + ( vd cpsmet cfv wcel cv cdm cxp crp ccnv co cima cmpt crn cfg wceq dmeqd + wa cc0 cico cuni cmetu cvv df-metu psmetdmdm adantr eqtr4d sqxpeqd simplr + simpr cnveqd imaeq1d mpteq2dva rneqd oveq12d elfvunirn ovexd fvmptd2 ) AB + EFGZDADHZIZIZVDJZCKVBLZUACHZUBMZNZOZPZQMBBJZCKALZVHNZOZPZQMEPUCUDUECDUFVA + VBARZTZVEVLVKVPQVRVDBVRVDAIZIZBVRVCVSVRVBAVAVQULSSVABVTRVQABUGUHUIUJVRVJV + OVRCKVIVNVRVGKGZTZVFVMVHWBVBAVAVQWAUKUMUNUOUPUQBAEURVAVLVPQUSUT $. $} ${ @@ -325930,12 +326919,12 @@ Normed space homomorphisms (bounded linear operators) ( vx cr wcel cxr wa cbl cfv co ccnv cima cec wss rexr cvv wb simpr blssec cxmet xrsxmet eqid mp3an1 sylan wbr vex simpl elecg sylancr w3a xmeterval cv ax-mp wceq simplll eqeltrrd simplr3 simplr1 simplr2 xrsdsreclb syl3anc - wne mpbid simprd pm2.61dane ex syl5bi sylbid ssrdv sstrd ) BFGZCHGZIZBCAJ - KLZBAMFNZOZFVMBHGZVNVPVRPZBQAHUBKGZVSVNVTADUCZABVQCHVQUDZUAUEUFVOEVRFVOEU - NZVRGZBWDVQUGZWDFGZVOWDRGVMWEWFSEUHVMVNUIWDBVQRFUJUKWFVSWDHGZBWDALFGZULZV - OWGWAWFWJSWBBWDAVQHWCUMUOVOWJWGVOWJIZWGBWDWKBWDUPZIBWDFWKWLTVMVNWJWLUQURW - KBWDVDZIZVMWGWNWIVMWGIZVSWHWIVOWMUSWNVSWHWMWIWOSVSWHWIVOWMUTVSWHWIVOWMVAW - KWMTBWDADVBVCVEVFVGVHVIVJVKVL $. + wne mpbid simprd pm2.61dane ex biimtrid sylbid ssrdv sstrd ) BFGZCHGZIZBC + AJKLZBAMFNZOZFVMBHGZVNVPVRPZBQAHUBKGZVSVNVTADUCZABVQCHVQUDZUAUEUFVOEVRFVO + EUNZVRGZBWDVQUGZWDFGZVOWDRGVMWEWFSEUHVMVNUIWDBVQRFUJUKWFVSWDHGZBWDALFGZUL + ZVOWGWAWFWJSWBBWDAVQHWCUMUOVOWJWGVOWJIZWGBWDWKBWDUPZIBWDFWKWLTVMVNWJWLUQU + RWKBWDVDZIZVMWGWNWIVMWGIZVSWHWIVOWMUSWNVSWHWMWIWOSVSWHWIVOWMUTVSWHWIVOWMV + AWKWMTBWDADVBVCVEVFVGVHVIVJVKVL $. xrsmopn.1 $e |- J = ( MetOpen ` D ) $. $( The metric on the extended reals generates a topology, but this does not @@ -326193,54 +327182,54 @@ Normed space homomorphisms (bounded linear operators) suprubd ifboth min2 eqbrtrid w3a wb elicc2 mpbir3and cmin ltsubrpd c0 breqtrdi wne resubcld suprlub syl31anc wa wi weq oveq2 sseq1d rexbidv elrab2 unieq sseq2d cbvrexvw csn cun simpr1 syl adantrr simprr sseldd - 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(Contributed by Mario Carneiro, 13-Jun-2014.) $) @@ -326400,18 +327389,18 @@ Normed space homomorphisms (bounded linear operators) wa cioo crn ctg cfv crest cconn reconnlem1 ralrimivva ex wne cdif w3a cun c0 wn anbi12i exdistrv simplll simprll elin2d sseldd simprlr cle wbr csup wi clt simplrl simpllr simpr eqid reconnlem2 incom eqsstrid sseq2i sylnib - n0 uncom lecasei exp32 exlimdvv syl5bir syl5bi 3impd ralrimdvva ctopon wb - expd retopon connsub mpan sylibrd impbid ) CHIZUAUBUCUDZCUEJUFKZALZBLZMJC - IZBCNACNZWNWPWTWNWPTWSABCCCWQWRUGUHUIWNWTDLZCOZUNUJZELZCOZUNUJZXAXDOZHCUK - ZIZULCXAXDUMZIZUOZVFZEWONDWONZWPWNWTXMDEWOWOWNXAWOKZXDWOKZTZTZWTXMXRWTTZX - CXFXIXLXSXCXFXIXLVFZXCXFTFLZXBKZFPZGLZXEKZGPZTZXSXTXCYCXFYFFXBVQGXEVQUPYG - YBYETZGPFPXSXTYBYEFGUQXSYHXTFGXSYHXIXLXSYHXITZTZXLYAYDYJCHYAWNXQWTYIURZYJ - XACYAXSYBYEXIUSZUTVAYJCHYDYKYJXDCYDXSYBYEXIVBZUTVAYJYAYDVCVDZTABCYAYDXAYA - YDMJOHVGVEZXAXDYJWNYNYKQXSXOYIYNWNXOXPWTVHZRXSXPYIYNWNXOXPWTSZRXRWTYIYNVI - YJYBYNYLQYJYEYNYMQXSYHXIYNSYJYNVJYOVKVLYJYDYAVCVDZTZCXDXAUMZIXKYSABCYDYAX - DYDYAMJOHVGVEZXDXAYJWNYRYKQXSXPYIYRYQRXSXOYIYRYPRXRWTYIYRVIYJYEYRYMQYJYBY - RYLQYSXDXAOXGXHXDXAVMXSYHXIYRSVNYJYRVJUUAVKVLYTXJCXDXAVRVOVPVSVTWAWBWCWHW - DUIWEWOHWFUDKWNWPXNWGWIDECWOHWJWKWLWM $. + n0 uncom lecasei exp32 exlimdvv syl5bir biimtrid expd 3impd ralrimdvva wb + ctopon retopon connsub mpan sylibrd impbid ) CHIZUAUBUCUDZCUEJUFKZALZBLZM + JCIZBCNACNZWNWPWTWNWPTWSABCCCWQWRUGUHUIWNWTDLZCOZUNUJZELZCOZUNUJZXAXDOZHC + UKZIZULCXAXDUMZIZUOZVFZEWONDWONZWPWNWTXMDEWOWOWNXAWOKZXDWOKZTZTZWTXMXRWTT + ZXCXFXIXLXSXCXFXIXLVFZXCXFTFLZXBKZFPZGLZXEKZGPZTZXSXTXCYCXFYFFXBVQGXEVQUP + YGYBYETZGPFPXSXTYBYEFGUQXSYHXTFGXSYHXIXLXSYHXITZTZXLYAYDYJCHYAWNXQWTYIURZ + YJXACYAXSYBYEXIUSZUTVAYJCHYDYKYJXDCYDXSYBYEXIVBZUTVAYJYAYDVCVDZTABCYAYDXA + YAYDMJOHVGVEZXAXDYJWNYNYKQXSXOYIYNWNXOXPWTVHZRXSXPYIYNWNXOXPWTSZRXRWTYIYN + VIYJYBYNYLQYJYEYNYMQXSYHXIYNSYJYNVJYOVKVLYJYDYAVCVDZTZCXDXAUMZIXKYSABCYDY + AXDYDYAMJOHVGVEZXDXAYJWNYRYKQXSXPYIYRYQRXSXOYIYRYPRXRWTYIYRVIYJYEYRYMQYJY + BYRYLQYSXDXAOXGXHXDXAVMXSYHXIYRSVNYJYRVJUUAVKVLYTXJCXDXAVRVOVPVSVTWAWBWCW + DWEUIWFWOHWHUDKWNWPXNWGWIDECWOHWJWKWLWM $. $} ${ @@ -326447,22 +327436,22 @@ Normed space homomorphisms (bounded linear operators) cxr recnd eqtrd ltsubrpd eqbrtrrd ltaddrpd addassd breqtrd iccssioo domtr syl22anc syl2anc wceq eqid bl2ioo breqtrrd sylan simplll simpr sylc cmopn syl2an2r wrex tgioo eleq2i rexmet mopni2 mp3an1 sylanb r19.29a ex exlimdv - cxmet syl5bi imp sbth ) ADUAUBEZFZAUCUDZGHZAULUEZYGAGHZAYGIHYFAJGHZJYGIHY - HJKFYFAJLYKUFYFAYEUGJAYEUHUIUJZAJKMUMUKAJYGUNUOYFYIYJYIBUPZAFZBUQYFYJBAUR - YFYNYJBYFYNYJYFYNNZYMCUPZUSOUTJJVAVBZVGEPZALZYJCQYOYPQFZNZYGYRGHZYSYRAGHZ - YJYOYMJFZYTUUBYFAJYMYLVCUUDYTNZYGYMYPOPZYMYPRPZDPZYRGUUEYGYMYPVDVEPZOPZYM - UUIRPZVFPZGHZUULUUHGHZYGUUHGHUUEYGVRVHVFPZGHUUOUULIHZUUMVIUUEUUJJFZUUKJFZ - UUJUUKSHUUPYTUUDUUIJFZUUQYTUUIYPVJZVKZYMUUIVLTZYTUUDUUSUURUVAYMUUIVMTZUUE - UUJYMUUKUVBUUDYTVNZUVCYTUUDUUIQFZUUJYMSHUUTYMUUIVOTYTUUDUVEYMUUKSHUUTYMUU - IVPTVQUUJUUKVSVTYGUUOUULUNWAUUHKFUUEUULUUHLZUUNUUFUUGDWBUUEUUFWIFUUGWIFUU - FUUJSHUUKUUGSHUVFUUEUUFYTUUDYPJFZUUFJFYPWCZYMYPVLTWDUUEUUGYTUUDUVGUUGJFUV - HYMYPVMTWDUUEUUJUUIOPZUUFUUJSUUEUVIYMUUIUUIRPZOPUUFUUEYMUUIUUIUUEYMUVDWJZ - UUEUUIYTUUSUUDUVAWEWJZUVLWFUUEUVJYPYMOUUEYPUUEYPYTUVGUUDUVHWEWJWGZWHWKUUE - UUJUUIUVBYTUVEUUDUUTWEZWLWMUUEUUKUUKUUIRPZUUGSUUEUUKUUIUVCUVNWNUUEUVOYMUV - JRPUUGUUEYMUUIUUIUVKUVLUVLWOUUEUVJYPYMRUVMWHWKWPUUFUUGUUJUUKWQWSUULUUHKMU - MYGUULUUHWRWTYTUUDUVGYRUUHXAUVHYMYPYQYQXBZXCTXDXEUUAYSNYFYSUUCYFYNYTYSXFU - UAYSXGYRAYEMXHYGYRAWRXJYFAYQXIEZFZYNYSCQXKZYEUVQAYQUVQUVPUVQXBZXLXMYQJYAE - FUVRYNUVSYQUVPXNCAYQYMUVQJUVTXOXPXQXRXSXTYBYCAYGYDXJ $. + cxmet biimtrid imp sbth ) ADUAUBEZFZAUCUDZGHZAULUEZYGAGHZAYGIHYFAJGHZJYGI + HYHJKFYFAJLYKUFYFAYEUGJAYEUHUIUJZAJKMUMUKAJYGUNUOYFYIYJYIBUPZAFZBUQYFYJBA + URYFYNYJBYFYNYJYFYNNZYMCUPZUSOUTJJVAVBZVGEPZALZYJCQYOYPQFZNZYGYRGHZYSYRAG + HZYJYOYMJFZYTUUBYFAJYMYLVCUUDYTNZYGYMYPOPZYMYPRPZDPZYRGUUEYGYMYPVDVEPZOPZ + YMUUIRPZVFPZGHZUULUUHGHZYGUUHGHUUEYGVRVHVFPZGHUUOUULIHZUUMVIUUEUUJJFZUUKJ + FZUUJUUKSHUUPYTUUDUUIJFZUUQYTUUIYPVJZVKZYMUUIVLTZYTUUDUUSUURUVAYMUUIVMTZU + UEUUJYMUUKUVBUUDYTVNZUVCYTUUDUUIQFZUUJYMSHUUTYMUUIVOTYTUUDUVEYMUUKSHUUTYM + UUIVPTVQUUJUUKVSVTYGUUOUULUNWAUUHKFUUEUULUUHLZUUNUUFUUGDWBUUEUUFWIFUUGWIF + UUFUUJSHUUKUUGSHUVFUUEUUFYTUUDYPJFZUUFJFYPWCZYMYPVLTWDUUEUUGYTUUDUVGUUGJF + UVHYMYPVMTWDUUEUUJUUIOPZUUFUUJSUUEUVIYMUUIUUIRPZOPUUFUUEYMUUIUUIUUEYMUVDW + JZUUEUUIYTUUSUUDUVAWEWJZUVLWFUUEUVJYPYMOUUEYPUUEYPYTUVGUUDUVHWEWJWGZWHWKU + UEUUJUUIUVBYTUVEUUDUUTWEZWLWMUUEUUKUUKUUIRPZUUGSUUEUUKUUIUVCUVNWNUUEUVOYM + UVJRPUUGUUEYMUUIUUIUVKUVLUVLWOUUEUVJYPYMRUVMWHWKWPUUFUUGUUJUUKWQWSUULUUHK + MUMYGUULUUHWRWTYTUUDUVGYRUUHXAUVHYMYPYQYQXBZXCTXDXEUUAYSNYFYSUUCYFYNYTYSX + FUUAYSXGYRAYEMXHYGYRAWRXJYFAYQXIEZFZYNYSCQXKZYEUVQAYQUVQUVPUVQXBZXLXMYQJY + AEFUVRYNUVSYQUVPXNCAYQYMUVQJUVTXOXPXQXRXSXTYBYCAYGYDXJ $. $} $( A countable subset of the reals has empty interior. 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A subset of ` CC ` is compact @@ -329756,9 +330745,9 @@ topological space to the reals is bounded (above). (Boundedness below phtpc01 $p |- ( F ( ~=ph ` J ) G -> ( ( F ` 0 ) = ( G ` 0 ) /\ ( F ` 1 ) = ( G ` 1 ) ) ) $= ( vh cphtpc cfv wbr cii ccn co wcel cphtpy c0 wne w3a cc0 wceq c1 isphtpc - wa cv wex n0 simpll simplr simpr phtpy01 ex exlimdv syl5bi 3impia sylbi ) - ABCEFGAHCIJZKZBUMKZABCLFJZMNZOPAFPBFQRAFRBFQTZABCSUNUOUQURUQDUAZUPKZDUBUN - UOTZURDUPUCVAUTURDVAUTURVAUTTABUSCUNUOUTUDUNUOUTUEVAUTUFUGUHUIUJUKUL $. + wa cv wex n0 simpll simplr simpr phtpy01 ex exlimdv biimtrid 3impia sylbi + ) ABCEFGAHCIJZKZBUMKZABCLFJZMNZOPAFPBFQRAFRBFQTZABCSUNUOUQURUQDUAZUPKZDUB + UNUOTZURDUPUCVAUTURDVAUTURVAUTTABUSCUNUOUTUDUNUOUTUEVAUTUFUGUHUIUJUKUL $. $} ${ @@ -330653,13 +331642,13 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) ( vx vy cfv wcel wa wceq cphtpc cima wss cii ccn co cv wbr wrex vex cc0 elima c1 cphtpy c0 wne w3a simpr isphtpc sylib adantrl phtpc01 ad2antll simp2d simpld om1elbas biimpa adantrr eqtr3d simprd simp3d wb mpbir3and - adantr rexlimdvaa syl5bi ssrdv simp1 syl6bi jca ) ADUAQZEUBZEUCEUDDUEUF - ZUCAOWBEOUGZWBRPUGZWDWAUHZPEUIAWDERZPWDWAEOUJULAWFWGPEAWEERZWFSZSZWGWDW - CRZUKWDQZHTZUMWDQZHTZWJWEWCRZWKWEWDDUNQUFUOUPZAWFWPWKWQUQZWHAWFSWFWRAWF - URWEWDDUSUTVAVDWJUKWEQZWLHWJWSWLTZUMWEQZWNTZWFWTXBSAWHWEWDDVBVCZVEWJWPW - SHTZXAHTZAWHWPXDXEUQZWFAWHXFAEWEDFGHLJKNVFVGVHZVDVIWJXAWNHWJWTXBXCVJWJW - PXDXEXGVKVIAWGWKWMWOUQZVLWIAEWDDFGHLJKNVFZVNVMVOVPVQAOEWCAWGXHWKXIWKWMW - OVRVSVQVT $. + adantr rexlimdvaa biimtrid ssrdv simp1 syl6bi jca ) ADUAQZEUBZEUCEUDDUE + UFZUCAOWBEOUGZWBRPUGZWDWAUHZPEUIAWDERZPWDWAEOUJULAWFWGPEAWEERZWFSZSZWGW + DWCRZUKWDQZHTZUMWDQZHTZWJWEWCRZWKWEWDDUNQUFUOUPZAWFWPWKWQUQZWHAWFSWFWRA + WFURWEWDDUSUTVAVDWJUKWEQZWLHWJWSWLTZUMWEQZWNTZWFWTXBSAWHWEWDDVBVCZVEWJW + PWSHTZXAHTZAWHWPXDXEUQZWFAWHXFAEWEDFGHLJKNVFVGVHZVDVIWJXAWNHWJWTXBXCVJW + JWPXDXEXGVKVIAWGWKWMWOUQZVLWIAEWDDFGHLJKNVFZVNVMVOVPVQAOEWCAWGXHWKXIWKW + MWOVRVSVQVT $. $( Another way to write the loop space base in terms of the base of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.) $) @@ -331205,13 +332194,13 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) ${ $d f k w F $. $d f k w K $. $d f k w W $. $( Define the class of subcomplex modules, which are left modules over a - subring of the field of complex numbers ` CCfld ` , which allows us to - use the complex addition, multiplication, etc. in theorems about - subcomplex modules. Since the field of complex numbers is commutative - and so are its subrings (see ~ subrgcrng ), left modules over such - subrings are the same as right modules, see ~ rmodislmod . Therefore, - we drop the word "left" from "subcomplex left module". (Contributed by - Mario Carneiro, 16-Oct-2015.) $) + subring of the field of complex numbers ` CCfld ` , which allows to use + the complex addition, multiplication, etc. in theorems about subcomplex + modules. Since the field of complex numbers is commutative and so are + its subrings (see ~ subrgcrng ), left modules over such subrings are the + same as right modules, see ~ rmodislmod . Therefore, we drop the word + "left" from "subcomplex left module". (Contributed by Mario Carneiro, + 16-Oct-2015.) $) df-clm $a |- CMod = { w e. LMod | [. ( Scalar ` w ) / f ]. [. ( Base ` f ) / k ]. ( f = ( CCfld |`s k ) /\ k e. ( SubRing ` CCfld ) ) } $. @@ -331448,7 +332437,7 @@ and so are its subrings (see ~ subrgcrng ), left modules over such ${ clmvs1.v $e |- V = ( Base ` W ) $. clmvs1.s $e |- .x. = ( .s ` W ) $. - $( Scalar product with ring unit. ( ~ lmodvs1 analog.) (Contributed by + $( Scalar product with ring unity. ( ~ lmodvs1 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.) $) clmvs1 $p |- ( ( W e. CMod /\ X e. V ) -> ( 1 .x. X ) = X ) $= ( cclm wcel wa c1 co csca cfv cur wceq eqid clm1 adantr oveq1d clmod @@ -333704,49 +334693,49 @@ of the norm of the sum of two orthogonal vectors (i.e., whose inner c2 wb clvec cdr phllvec lvecdrng cphsubrglem simp2d inss2 eqsstrdi adantr ipcl 3anidm23 sylan sseldd sqrtcld sqeq0 sqsqrtd cclm clm0 eqeq12d bitr3d phclm ipeq0 3bitrd caddc cle simp1d wbr 3anass simpr2 recnd jca ex eleq2d - recn elin rbaib syl5bi adantlr simprl simprr 3expb sylancr 3jca tcphnmval - cabs syl2an2r wf sqrtf sylan9bb adantrr pm5.32rd bitr4di 3imtr4d grpsubcl - bitrdi imp tcphcphlem1 oveqan12d 3brtr4d tngngpd cnnrg simp3d tcphcphlem2 - subrgnrg eqeltrd lmodvscl csubg subrgsubg cnfldnm subgnm2 oveq12d 3eqtr4d - fveq1d eqtrd tcphbas tcphvsca tcphsca isnlm sylanbrc elrege0 anbi2i bitri - ralrimivva ralrimiv wfun ffun ax-mp inss1 sstrid fdmi sseqtrrdi funimass4 - cdm mpbird cnex tngnm syl2anc eqcomd tcphip iscph syl3anbrc ) ADUCQZDUDQZ - CUECUFRZUGUHZSZUIUJUWPUKULUMUHZUNZUOUWPUPZDVGRZUAGUAVHZUXCEUHZUJRZUQZSDUR - 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( *Met ` X ) ) $. $( Express the binary relation "sequence ` F ` converges to point ` P ` " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The - condition ` F C_ ( CC X. X ) ` allows us to use objects more general - than sequences when convenient; see the comment in ~ df-lm . - (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, - 1-May-2014.) $) + condition ` F C_ ( CC X. X ) ` allows to use objects more general than + sequences when convenient; see the comment in ~ df-lm . (Contributed by + NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.) $) lmmbr $p |- ( ph -> ( F ( ~~>t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. x e. RR+ E. y e. ran ZZ>= ( F |` y ) : y --> ( P ( ball ` D ) x ) ) ) ) $= @@ -334506,10 +335494,9 @@ is an accumulation point (limit point) of subset ` y ` ". @) $( Express the binary relation "sequence ` F ` converges to point ` P ` " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The - condition ` F C_ ( CC X. X ) ` allows us to use objects more general - than sequences when convenient; see the comment in ~ df-lm . - (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, - 1-May-2014.) $) + condition ` F C_ ( CC X. X ) ` allows to use objects more general than + sequences when convenient; see the comment in ~ df-lm . (Contributed by + NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.) $) lmmbr2 $p |- ( ph -> ( F ( ~~>t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < x ) ) ) ) $= @@ -334772,28 +335759,28 @@ is an accumulation point (limit point) of subset ` y ` ". @) mpancom cdiv simpll rphalfcl cfil3i simprr cxr rpxr ad2antlr blssm cin c0 wex simpllr rpxrd blopn blcntr fclsopni syl13anc n0 elin cr simplrl rpred wne blhalf syl22anc sselda adantrr simprl wb blcom mpbid sstrd ex exlimdv - syl5bi mpd rexlimddv ad2ant2r toponss ralrimiva flimopn syl2anc mpbir2and - filss expr ssrdv flimfcls a1i eqssd ) BAUEKLZCBUFMZCBUGMZYKGYLYMYKGNZYLLZ - YNYMLZYKYOOZYPYNDLZYNHNZLZYSBLZUHZHCUBZYQYNCUCZDYOYNUUDLYKYNBCUUDUUDUIUJU - KYQCDULKLZDUUDUMYQADPKZLZUUEYKUUGYOYKAAUNUNZPKZUUFYKAPUOUCZLAUUILUAUUJUAN - ZYSYSUPURUQYNUSMQHUDNUTGRUBUDUUKUNUNVAKVBUEBAGHUDUAVCVDAVEVIDUUHPFVJVFZSZ - ACDEVGVHZDCVKVHVLZYQUUBHCYQYSCLZYTUUAYQUUPYTOZOZYNINZAVMKZMZYSQZUUAIRYQUU - GUUQUVBIRUTZUUMUUGUUPYTUVCIYSAYNCDEVNVOVPUURUUSRLZUVBOZOBDVAKLZUVABLZYSDQ - ZUVBUUAYQUVFUUQUVEYKUVFYOUUGYKUVFUULABDVQVTSZVRYQUVDUVGUUQUVBYQUVDOZYSUUS - VSWAMZVSWAMZUUTMZBLZUVGHDUVJUUGYKUVLRLZUVNHDUTYQUUGUVDUUMSZYKYOUVDWBUVJUV - KRLZUVOUVDUVQYQUUSWCUKZUVKWCVHHAUVLBDWDTUVJYSDLZUVNOZOZUVFUVNUVADQZUVMUVA - QZUVGYQUVFUVDUVTUVIVRUVJUVSUVNWEZUWAUUGYRUUSWFLZUWBUVJUUGUVTUVPSZYQYRUVDU - VTUUOVRZUVDUWEYQUVTUUSWGWHAYNUUSDWITUWAJNZYNUVKUUTMZUVMWJZLZJWLZUWCUWAUWJ - WKXDZUWLUWAYOUWICLZYNUWILZUVNUWMYKYOUVDUVTWMUWAUUGYRUVKWFLZUWNUWFUWGUWAUV - KUVJUVQUVTUVRSZWNZAYNUVKCDEWOTUWAUUGYRUVQUWOUWFUWGUWQAYNUVKDWPTUWDYNUVMUW - IBCWQWRJUWJWSVIUWAUWKUWCJUWKUWHUWILZUWHUVMLZOZUWAUWCUWHUWIUVMWTUWAUXAUWCU - WAUXAOZUVMUWHUVKUUTMZUVAUXBUUGUVSUVKXALUWTUVMUXCQUWAUUGUXAUWFSZUVJUVSUVNU - XAXBUXBUVKUWAUVQUXAUWQSZXCUWAUWSUWTWEUVKADYSUWHXEXFUXBUUGUWHDLZUUSXALYNUX - CLZUXCUVAQUXDUWAUWSUXFUWTUWAUWIDUWHUWAUUGYRUWPUWIDQUWFUWGUWRAYNUVKDWITXGX - HZUXBUUSYQUVDUVTUXAWMXCUXBUWSUXGUWAUWSUWTXIUXBUUGUWPYRUXFUWSUXGXJUXDUXBUV - KUXEWNUWAYRUXAUWGSUXHUWHAYNUVKDXKXFXLUUSADUWHYNXEXFXMXNXPXOXQUVMUVABDYEWR - XRXSUURUVHUVEYQUUEUUQUVHUUNUUEUUPUVHYTYSCDXTXHVPSUURUVDUVBWEUVAYSBDYEWRXR - YFYAYQUUEUVFYPYRUUCOXJUUNUVIHYNBCDYBYCYDXNYGYMYLQYKBCYHYIYJ $. + biimtrid mpd filss rexlimddv ad2ant2r toponss ralrimiva flimopn mpbir2and + expr syl2anc ssrdv flimfcls a1i eqssd ) BAUEKLZCBUFMZCBUGMZYKGYLYMYKGNZYL + LZYNYMLZYKYOOZYPYNDLZYNHNZLZYSBLZUHZHCUBZYQYNCUCZDYOYNUUDLYKYNBCUUDUUDUIU + JUKYQCDULKLZDUUDUMYQADPKZLZUUEYKUUGYOYKAAUNUNZPKZUUFYKAPUOUCZLAUUILUAUUJU + ANZYSYSUPURUQYNUSMQHUDNUTGRUBUDUUKUNUNVAKVBUEBAGHUDUAVCVDAVEVIDUUHPFVJVFZ + SZACDEVGVHZDCVKVHVLZYQUUBHCYQYSCLZYTUUAYQUUPYTOZOZYNINZAVMKZMZYSQZUUAIRYQ + UUGUUQUVBIRUTZUUMUUGUUPYTUVCIYSAYNCDEVNVOVPUURUUSRLZUVBOZOBDVAKLZUVABLZYS + DQZUVBUUAYQUVFUUQUVEYKUVFYOUUGYKUVFUULABDVQVTSZVRYQUVDUVGUUQUVBYQUVDOZYSU + USVSWAMZVSWAMZUUTMZBLZUVGHDUVJUUGYKUVLRLZUVNHDUTYQUUGUVDUUMSZYKYOUVDWBUVJ + UVKRLZUVOUVDUVQYQUUSWCUKZUVKWCVHHAUVLBDWDTUVJYSDLZUVNOZOZUVFUVNUVADQZUVMU + VAQZUVGYQUVFUVDUVTUVIVRUVJUVSUVNWEZUWAUUGYRUUSWFLZUWBUVJUUGUVTUVPSZYQYRUV + DUVTUUOVRZUVDUWEYQUVTUUSWGWHAYNUUSDWITUWAJNZYNUVKUUTMZUVMWJZLZJWLZUWCUWAU + WJWKXDZUWLUWAYOUWICLZYNUWILZUVNUWMYKYOUVDUVTWMUWAUUGYRUVKWFLZUWNUWFUWGUWA + UVKUVJUVQUVTUVRSZWNZAYNUVKCDEWOTUWAUUGYRUVQUWOUWFUWGUWQAYNUVKDWPTUWDYNUVM + UWIBCWQWRJUWJWSVIUWAUWKUWCJUWKUWHUWILZUWHUVMLZOZUWAUWCUWHUWIUVMWTUWAUXAUW + CUWAUXAOZUVMUWHUVKUUTMZUVAUXBUUGUVSUVKXALUWTUVMUXCQUWAUUGUXAUWFSZUVJUVSUV + NUXAXBUXBUVKUWAUVQUXAUWQSZXCUWAUWSUWTWEUVKADYSUWHXEXFUXBUUGUWHDLZUUSXALYN + UXCLZUXCUVAQUXDUWAUWSUXFUWTUWAUWIDUWHUWAUUGYRUWPUWIDQUWFUWGUWRAYNUVKDWITX + GXHZUXBUUSYQUVDUVTUXAWMXCUXBUWSUXGUWAUWSUWTXIUXBUUGUWPYRUXFUWSUXGXJUXDUXB + UVKUXEWNUWAYRUXAUWGSUXHUWHAYNUVKDXKXFXLUUSADUWHYNXEXFXMXNXPXOXQUVMUVABDXR + WRXSXTUURUVHUVEYQUUEUUQUVHUUNUUEUUPUVHYTYSCDYAXHVPSUURUVDUVBWEUVAYSBDXRWR + XSYEYBYQUUEUVFYPYRUUCOXJUUNUVIHYNBCDYCYFYDXNYGYMYLQYKBCYHYIYJ $. $} ${ @@ -334815,9 +335802,9 @@ is an accumulation point (limit point) of subset ` y ` ". @) $( Express the property " ` F ` is a Cauchy sequence of metric ` D ` ". Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition - ` F C_ ( CC X. X ) ` allows us to use objects more general than - sequences when convenient; see the comment in ~ df-lm . (Contributed by - NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.) $) + ` F C_ ( CC X. X ) ` allows to use objects more general than sequences + when convenient; see the comment in ~ df-lm . (Contributed by NM, + 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.) $) iscau $p |- ( D e. ( *Met ` X ) -> ( F e. ( Cau ` D ) <-> ( F e. ( X ^pm CC ) /\ A. x e. RR+ E. k e. ZZ ( F |` ( ZZ>= ` k ) ) : ( ZZ>= ` k ) --> ( ( F ` k ) ( ball ` D ) x ) ) ) ) $= @@ -334903,27 +335890,27 @@ is an accumulation point (limit point) of subset ` y ` ". @) wrex simpr eleqtrdi eluzelz uzid 3syl fveq2 oveq1d breq1d raleqbidv rspcv wceq syl adantr oveq2d cbvralvw ralimi eleq1d syl2im imp r19.26 ad3antrrr cxmet simplr simprr xmetsym syl3anc expimpd ralimdv syl5bir expd impancom - cz biimpd mpd syl5bi syld imdistanda df-3an ralbii reximdva anim2d sylbid - 3imtr4g wss uzssz eqsstri ssrexv ax-mp anim2i iscau2 syl5ibr impbid simpl - uztrn2 adantrl adantrr oveq12d 3anbi23d anassrs ralbidva rexbidva ralbidv - wb jca sylan2 anbi2d bitrd ) AHEUBUCQZHJUDUEUFQZGUGZHUHQZYEHUCZJQZYGFUGZH - UCZEUFZBUGZRUIZUJZGYIUKUCZSZFKUOZBULSZTZYDYFCJQZCDEUFZYLRUIZUJZGYOSZFKUOZ - BULSZTAYCYSAYCYDYFYHYGUAUGZHUCZEUFZYLRUIZUAYEUKUCZSZUJZGYOSZFKUOZBULSZTYS - ABEFGUAHIJKLMNUMAUUPYRYDAUUOYQBULAUUNYPFKAYIKQZTZYFYHTZUULTZGYOSZUUSYMTZG - YOSZUUNYPUURUUSGYOSZUULGYOSZTUVDYMGYOSZTUVAUVCUURUVDUVEUVFUURUVDTZUVEYJUU - 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ressbas2 eleq12d biimpd expimpd syl5bi imp 3adant1 eqeltrd syl2anc mstopn - metsscmetcld eleqtrrd ) DIJZAEKZCUAJZUDZADLMZEENZOZUBMZPMZBPMWBWEEUEMJZWE - AANZOZAQMZJAWGJVSVTWHWAWEDEGWERZUCUFWBWJCLMZWIOZWKWBWJWCWIOWNWBWCWIWDVTVS - WIWDKZWAVTWOAEAEUGUHUIUJWBWCWMWIWBAUOJZWCWMSVTVSWPWAVTADTMZKWPEWQAGUKAWQD - TULUMUPUIAWCDCUOFWCRUNUQURUSVTWAWNWKJZVSVTWAWRWACIJZWMCTMZWTNZOZWTQMZJZUT - VTWRXBCWTWTRXBRVAVTWSXDWRVTWSUTZXDWRXEXBWNXCWKXEXAWIWMXEWTAXEAWTVTAWTSWSA - ECDFGVGVBVCZVDVEXEWTAQXFVFVHVIVJVKVLVMVNWEWFEAWFRVQVOWBBWFPVSVTBWFSWAWEBD - EHGWLVPUFVFVR $. + ressbas2 eleq12d biimpd expimpd biimtrid imp 3adant1 eqeltrd metsscmetcld + syl2anc mstopn eleqtrrd ) DIJZAEKZCUAJZUDZADLMZEENZOZUBMZPMZBPMWBWEEUEMJZ + WEAANZOZAQMZJAWGJVSVTWHWAWEDEGWERZUCUFWBWJCLMZWIOZWKWBWJWCWIOWNWBWCWIWDVT + VSWIWDKZWAVTWOAEAEUGUHUIUJWBWCWMWIWBAUOJZWCWMSVTVSWPWAVTADTMZKWPEWQAGUKAW + QDTULUMUPUIAWCDCUOFWCRUNUQURUSVTWAWNWKJZVSVTWAWRWACIJZWMCTMZWTNZOZWTQMZJZ + UTVTWRXBCWTWTRXBRVAVTWSXDWRVTWSUTZXDWRXEXBWNXCWKXEXAWIWMXEWTAXEAWTVTAWTSW + SAECDFGVGVBVCZVDVEXEWTAQXFVFVHVIVJVKVLVMVNWEWFEAWFRVOVPWBBWFPVSVTBWFSWAWE + BDEHGWLVQUFVFVR $. $( The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) $) @@ -336908,7 +337895,7 @@ need not be complete (for instance if the given set is infinite rrxnm $p |- ( I e. V -> ( f e. B |-> ( sqrt ` ( RRfld gsum ( x e. I |-> ( ( f ` x ) ^ 2 ) ) ) ) ) = ( norm ` H ) ) $= ( vh vg wcel crefld co cfv cbs cv cip wceq eqid wa cfrlm ctcph csqrt cmpt - cnm c2 cexp cgsu clmod cgrp crg csr recrng srngring frlmlmod mpan lmodgrp + cnm c2 cexp cgsu clmod cgrp crg csr resrng srngring frlmlmod mpan lmodgrp ax-mp tchnmfval 3syl rrxval fveq2d tcphbas 3eqtr4g cr cmap cc0 cfsupp wbr crab rrxbase ssrab2 eqsstrdi sselda cmul cvv cmpo rrxip tcphip a1i adantr 3eqtr4rd simprl fveq1d simprr oveq12d cc elmapi adantl ffvelcdmda adantlr @@ -336975,7 +337962,7 @@ need not be complete (for instance if the given set is infinite -> ( f e. B , g e. B |-> ( sqrt ` ( RRfld gsum ( x e. I |-> ( ( ( f ` x ) - ( g ` x ) ) ^ 2 ) ) ) ) ) = ( dist ` H ) ) $= ( vh wcel cfv crefld co c2 cexp cmpt wceq eqid cr cds cfrlm ctcph cnm csg - ccom cv cmin cgsu csqrt cmpo rrxval fveq2d clmod cgrp crg recrng srngring + ccom cv cmin cgsu csqrt cmpo rrxval fveq2d clmod cgrp crg resrng srngring csr ax-mp frlmlmod mpan lmodgrp tcphds 3syl cxp wf cbs grpsubf cc0 cfsupp wbr cmap crab rrxbase rebase frlmbas eqtrd sqxpeqd feq23d mpbird fovcdmda re0g wfn ffnd fnov sylib rrxnm eqtr2d fveq1 oveq1d mpteq2dv oveq2d fmpoco @@ -337037,7 +338024,7 @@ need not be complete (for instance if the given set is infinite + ( C x. ( Y ` i ) ) ) ) ) $= ( co wceq crefld cfrlm ctcph cfv cvsca cplusg cv cmul caddc wral rrxval wcel syl fveq2d eqtrid oveqd oveq123d eqeq2d cr tcphbas 3eqtr4g eleqtrd - cbs eqid csr recrng srngring mp1i rebase tcphvsca eqcomi remulr replusg + cbs eqid csr resrng srngring mp1i rebase tcphvsca eqcomi remulr replusg crg tchplusg frlmvplusgscavalb bitrd ) AMBKFUDZDLFUDZEUDZUEMBKUFIUGUDZU HUIZUJUIZUDZDLWHUDZWGUKUIZUDZUEGULZMUIBWMKUIUMUDDWMLUIUMUDUNUDUEGIUOAWE WLMAWCWIWDWJEWKAEHUKUIWKUBAHWGUKAIJUQHWGUEQHIJNUPURZUSUTAFWHBKAFHUJUIWH @@ -338206,11 +339193,11 @@ Hilbert space (in the algebraic sense, meaning that all algebraically Mario Carneiro, 17-Oct-2015.) $) hlhil $p |- ( W e. CHil -> W e. Hil ) $= ( chl wcel cphl cpj cfv cdm ccss wceq chil hlphl wss eqid pjcss syl ctopn - vx clss ccld cin cbs cldcss2 cv wa elin pjth2 3expib syl5bi ssrdv eqsstrd - eqssd ishil sylanbrc ) ABCZADCZAEFZGZAHFZIAJCAKZUNUQURUNUOUQURLUSURUPAUPM - ZURMZNOUNURARFZAPFZSFZTZUQURVCVBAUAFZAVFMVCMZVBMZVAUBUNQVEUQQUCZVECVIVBCZ - VIVDCZUDUNVIUQCZVIVBVDUEUNVJVKVLVIVCUPVBAVGVHUTUFUGUHUIUJUKURAUPUTVAULUM - $. + vx clss ccld cin cbs cldcss2 cv wa elin pjth2 3expib biimtrid ssrdv eqssd + eqsstrd ishil sylanbrc ) ABCZADCZAEFZGZAHFZIAJCAKZUNUQURUNUOUQURLUSURUPAU + PMZURMZNOUNURARFZAPFZSFZTZUQURVCVBAUAFZAVFMVCMZVBMZVAUBUNQVEUQQUCZVECVIVB + CZVIVDCZUDUNVIUQCZVIVBVDUEUNVJVKVLVIVCUPVBAVGVHUTUFUGUHUIUKUJURAUPUTVAULU + M $. $} $( End $[ set-top.mm $] $) @@ -338998,7 +339985,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically SZGDIZRSZDVIVIPFGPUNVKVMDVIUOUQVHCPURUSDVIGUTVACAVLBVIVIVBZVCVDCABVIVNVEV F $. - $( The domain and range of the outer volume function. (Contributed by + $( The domain and codomain of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) $) ovolf $p |- vol* : ~P RR --> ( 0 [,] +oo ) $= ( vx vf vy cr cpw cc0 cpnf cicc co covol cv wcel ccom crn wss cxr clt cle @@ -339183,29 +340170,29 @@ Hilbert space (in the algebraic sense, meaning that all algebraically mp2an fveq2 eqtr4di fmptco cmet cnmet met0 sylancr mpteq2dva eqtrd adantr 0xr fconstmpt seqeq3d cz 1z nnuz ser0f rneqd c0 wne 1nn ne0i rnxp supeq1d mp2b wor xrltso breqtrd ovolge0 ovolcl xrletri3 sylancl mpbir2and exlimdv - supsn ex syl5bi ensym impel ) ADUBZEAUCFZAUDGZHIZAEUCFUVJEAUAUEZUFZUAUGUV - IUVLEAUAUHUVIUVNUVLUAUVIUVNUVLUVIUVNJZUVLUVKHKFZHUVKKFZUVOUVKUIUJUKULZUVM - UVMLUMUNZULZMUOZUPZNOUQZHKUVOEKDDURZUSZUVSPZAUTUVSULUPVAUBZUVKUWCKFUVOUWF - EUWEBEBUEZUVMGZUWILUNZVBZPUVOBEUWIUWIVCZUWEUWKUVOUWHEQZJZKUWDUWLUWNUWIUWI - KFUWLKQUWNUWIUWNADUWIUVIUVNUWMVDUVOEAUWHUVMUVNEAUVMPUVIEAUVMVEVFZVGVJZVHZ - UWIUWIKVIVKUWNUWIUWIDDUWPUWPVLVMBEUWJUWLUWJUWLLGZUWLUWIUWILVNUWLRQZUWRUWL - IUWIUWIVOZUWLRVPVQVRVSZVTUVOEUWEUVSUWKUVOBEUWIUWILUVMUVMRTTERQUVOWBWCUWNU - WIUWPWAZUXBUVOBEAUVMUWOWDZUXCWEZWFWGZUVOUWGUWHUVSGZWHGZCUEZKFZUXHUXFWIGZK - FZJZBEWJZCAWKZUVOUXMCAUVOUXHAQZUWIUXHIZBEWJZUXMUVOUXHUVMUPZQZUXOUXQUVOUXR - AUXHUVOEAUVMWLZUXRAIUVNUXTUVIEAUVMWOVFEAUVMWMWPWNUVOUVMEWQZUXSUXQWRUVNUYA - UVIEAUVMWSVFBEUXHUVMWTWPXAUVOUXPUXLBEUWNUXGUWIKFZUWIUXJKFZJUXPUXLUWNUYBUY - CUWNUXGUWIUWIKUWNUXGUWLWHGUWIUWNUXFUWLWHUVOUWMUXFUWHBEUWLVBZGZUWLUVOUWHUV - SUYDUVOUVSUWKUYDUXDUXASZXBUWMUWSUYEUWLIUWTBEUWLRUYDUYDXGXCXDXEZXFUWIUWIUW - HUVMXHZUYHXISUWQXJUWNUWIUWIUXJKUWQUWNUXJUWLWIGUWIUWNUXFUWLWIUYGXFUWIUWIUY - HUYHXKSXLXMUXPUYBUXIUYCUXKUWIUXHUXGKXNUWIUXHUXJKXOXRXPXQXSXTUVIUVNUWFUWGU - XNWRUXECABUVSYAYBWGAUWAUVSUWAXGYCYDUVOUWCHYEZNOUQZHUVONUWBUYIOUVOUWBEUYIU - RZUPZUYIUVOUWAUYKUVOUWAUIUYKMUOZUYKUVOUVTUYKUIMUVOUVTBEHVBZUYKUVOUVTBEUWI - UWIUVRUNZVBUYNUVOBCETTURZUWLUXHUVRGZUYOUVSUVRUWNUWIUWITTUXBUXBVLUYFUVOCUY - PDUVRUYPDUVRPZUVOTDUJPUYPTUKPUYRYFYGUYPTDUJUKYHYIWCWDUXHUWLIUYQUWLUVRGUYO - UXHUWLUVRYJUWIUWIUVRVNYKYLUVOBEUYOHUWNUVRTYMGQUWITQUYOHIYNUXBUWIUVRTYOYPY - QYRBEHUUAYKUUBMUUCQUYMUYKIUUDMEUUEUUFVQSUUGMEQEUUHUUIUYLUYIIUUJEMUUKEUYIU - ULUUNSUUMNOUUOHNQZUYJHIUUPYTNHOUVDYISUUQUVIUVQUVNAUURYSUVOUVKNQZUYSUVLUVP - UVQJWRUVIUYTUVNAUUSYSYTUVKHUUTUVAUVBUVEUVCUVFAEUVGUVH $. + supsn ex biimtrid ensym impel ) ADUBZEAUCFZAUDGZHIZAEUCFUVJEAUAUEZUFZUAUG + UVIUVLEAUAUHUVIUVNUVLUAUVIUVNUVLUVIUVNJZUVLUVKHKFZHUVKKFZUVOUVKUIUJUKULZU + VMUVMLUMUNZULZMUOZUPZNOUQZHKUVOEKDDURZUSZUVSPZAUTUVSULUPVAUBZUVKUWCKFUVOU + WFEUWEBEBUEZUVMGZUWILUNZVBZPUVOBEUWIUWIVCZUWEUWKUVOUWHEQZJZKUWDUWLUWNUWIU + WIKFUWLKQUWNUWIUWNADUWIUVIUVNUWMVDUVOEAUWHUVMUVNEAUVMPUVIEAUVMVEVFZVGVJZV + HZUWIUWIKVIVKUWNUWIUWIDDUWPUWPVLVMBEUWJUWLUWJUWLLGZUWLUWIUWILVNUWLRQZUWRU + WLIUWIUWIVOZUWLRVPVQVRVSZVTUVOEUWEUVSUWKUVOBEUWIUWILUVMUVMRTTERQUVOWBWCUW + NUWIUWPWAZUXBUVOBEAUVMUWOWDZUXCWEZWFWGZUVOUWGUWHUVSGZWHGZCUEZKFZUXHUXFWIG + ZKFZJZBEWJZCAWKZUVOUXMCAUVOUXHAQZUWIUXHIZBEWJZUXMUVOUXHUVMUPZQZUXOUXQUVOU + XRAUXHUVOEAUVMWLZUXRAIUVNUXTUVIEAUVMWOVFEAUVMWMWPWNUVOUVMEWQZUXSUXQWRUVNU + YAUVIEAUVMWSVFBEUXHUVMWTWPXAUVOUXPUXLBEUWNUXGUWIKFZUWIUXJKFZJUXPUXLUWNUYB + UYCUWNUXGUWIUWIKUWNUXGUWLWHGUWIUWNUXFUWLWHUVOUWMUXFUWHBEUWLVBZGZUWLUVOUWH + UVSUYDUVOUVSUWKUYDUXDUXASZXBUWMUWSUYEUWLIUWTBEUWLRUYDUYDXGXCXDXEZXFUWIUWI + UWHUVMXHZUYHXISUWQXJUWNUWIUWIUXJKUWQUWNUXJUWLWIGUWIUWNUXFUWLWIUYGXFUWIUWI + UYHUYHXKSXLXMUXPUYBUXIUYCUXKUWIUXHUXGKXNUWIUXHUXJKXOXRXPXQXSXTUVIUVNUWFUW + GUXNWRUXECABUVSYAYBWGAUWAUVSUWAXGYCYDUVOUWCHYEZNOUQZHUVONUWBUYIOUVOUWBEUY + IURZUPZUYIUVOUWAUYKUVOUWAUIUYKMUOZUYKUVOUVTUYKUIMUVOUVTBEHVBZUYKUVOUVTBEU + WIUWIUVRUNZVBUYNUVOBCETTURZUWLUXHUVRGZUYOUVSUVRUWNUWIUWITTUXBUXBVLUYFUVOC + UYPDUVRUYPDUVRPZUVOTDUJPUYPTUKPUYRYFYGUYPTDUJUKYHYIWCWDUXHUWLIUYQUWLUVRGU + YOUXHUWLUVRYJUWIUWIUVRVNYKYLUVOBEUYOHUWNUVRTYMGQUWITQUYOHIYNUXBUWIUVRTYOY + PYQYRBEHUUAYKUUBMUUCQUYMUYKIUUDMEUUEUUFVQSUUGMEQEUUHUUIUYLUYIIUUJEMUUKEUY + IUULUUNSUUMNOUUOHNQZUYJHIUUPYTNHOUVDYISUUQUVIUVQUVNAUURYSUVOUVKNQZUYSUVLU + VPUVQJWRUVIUYTUVNAUUSYSYTUVKHUUTUVAUVBUVEUVCUVFAEUVGUVH $. $} $( The rational numbers have 0 outer Lebesgue measure. 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$( The domain and range of the Lebesgue measure function. (Contributed by - Mario Carneiro, 19-Mar-2014.) $) + $( The domain and codomain of the Lebesgue measure function. 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(Contributed by Mario Carneiro, 4-Jul-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) $) @@ -341208,19 +342195,19 @@ Hilbert space (in the algebraic sense, meaning that all algebraically U_ n e. 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A | @@ -342480,27 +343467,27 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Mario Carneiro, 26-Mar-2015.) $) @@ -342859,46 +343846,46 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Mario Carneiro, 16-Jun-2014.) $) vitalilem3 $p |- ( ph -> Disj_ m e. NN ( T ` m ) ) $= @@ -343381,18 +344368,18 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Mario Carneiro, 19-Jun-2014.) $) @@ -345519,86 +346506,86 @@ or are almost disjoint (the interiors are disjoint). 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Verify that ` G ` describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, @@ -348977,45 +349964,45 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Mario Carneiro, 12-Aug-2014.) $) @@ -350087,45 +351074,45 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by eqtrdi sylancr evthicc breqan12rd ralbidva adantl ioossicc ssralv ax-mp syl6bi syldan fveq1d sstrid breq1d cbvralvw dvferm subeq0d vex elpr wne ne0i ndmioo necon1ai 3syl iooss2 dvferm1 eqbrtrrd suble0d elicc2 simp3d - fvmpt breqtrrd simp2d eqbrtrd iooss1 dvferm2 breqtrd subge0d syl5bi cun - jaod elun prunioo syl3anc eleq2d bitr3id biimpar mpjaod syld reximdva + fvmpt breqtrrd simp2d eqbrtrd dvferm2 breqtrd subge0d jaod biimtrid cun + iooss1 elun prunioo syl3anc eleq2d bitr3id biimpar mpjaod syld reximdva mpd ) AUAUCZHIJUDRZUEZUFZBUCZUXGUFZUGUHZUAUXFUIZBUXFUJZUXISGUNRZUFZFUKZ BUXFUJAUXMUXJUXHUGUHUAUXFUIBUXFUJABUABUAIJUXGADEULRZSIDEUMZKUOZAUXQSJUX RLUOZAIJUXSUXTOUPZAUXGUXFSUQRTZUXFSUXGURZAUXQSUXFHACUXQCUCZGUFZFUYDUSRZ @@ -353451,10 +354438,10 @@ or are almost disjoint (the interiors are disjoint). (Contributed by PUWGWUEXQVXAUGUHVXOWUEXQVWTVXAUGWUEUBIEUXIHUXQAVVBVWAWUDVUIXPUYTWUEUXRV OWUEUXIJIEULRZWUFAJWUGTZVWAWUDAWUHJSTZVYAVYNUXTOVYOAIYHTZVYFWUHWUIVYAVY NYIWCAIUXSYOZVYLIEJYJVLYKXPYLAWUGUXQVEZVWAWUDAVYDDIUGUHWULVYIADIVYIWUKV - YCYPDIEUWHVLXPWUEUXIUXQVUSVWBVWPWUDVWRVNAVUTVWAWUDVVTXPVIWUEVXGUBIUXIUL - RZUIVXHWUEVWGVXHVWBVXKVWGYAVXIYCWUEVXGUBWUMVWFWUEUXIJIULWUFWNYQXLUWIVWB - VXBWUDVXDVNUWJWUEUXOFVWBVWQWUDVWSVNZAUYMVWAWUDVUEXPZUWKWLWUEUXOFWUNWUOY - SYTYDUWNUWLAVWJVWMYEZVWAWUPUXIVWFVWLUWMZTAVWAUXIVWFVWLUWOAWUQUXFUXIAWUJ + YCYPDIEUWNVLXPWUEUXIUXQVUSVWBVWPWUDVWRVNAVUTVWAWUDVVTXPVIWUEVXGUBIUXIUL + RZUIVXHWUEVWGVXHVWBVXKVWGYAVXIYCWUEVXGUBWUMVWFWUEUXIJIULWUFWNYQXLUWHVWB + VXBWUDVXDVNUWIWUEUXOFVWBVWQWUDVWSVNZAUYMVWAWUDVUEXPZUWJWLWUEUXOFWUNWUOY + SYTYDUWKUWLAVWJVWMYEZVWAWUPUXIVWFVWLUWMZTAVWAUXIVWFVWLUWOAWUQUXFUXIAWUJ VYEIJUGUHWUQUXFUKWUKVYJUYAIJUWPUWQUWRUWSUWTUXAUXBUXCUXD $. $( Lemma for ~ dvivth . (Contributed by Mario Carneiro, 20-Feb-2015.) $) @@ -353528,37 +354515,37 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Mario Carneiro, 19-Feb-2015.) $) @@ -353775,52 +354762,52 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by UNUUQUURUVAWQYTUVBALYTYGUVBALUJZUUHYTYPUUTYGUVCVFUUFUUGYFAXSWTXATXBXCWHXD YTXTAUULYCYTUUKYAYBYTYFBUUIXEXFXHTWEXGXIXJWHYDDLUKXKXLXM $. - $( The domain and range of the coefficient function. (Contributed by Mario - Carneiro, 22-Jul-2014.) $) + $( The domain and codomain of the coefficient function. (Contributed by + Mario Carneiro, 22-Jul-2014.) $) coef $p |- ( F e. ( Poly ` S ) -> A : NN0 --> ( S u. { 0 } ) ) $= ( vx vn cply cfv wcel cn0 cc0 csn cun wf cv cle wbr ccnv cc cdif cima cz wral wrex dgrlem simpld ) CBGHIJBKLZMANEOFOPQEARSUGTUAUCFUBUDEABFCDUEUF $. - $( The domain and range of the coefficient function. (Contributed by Mario - Carneiro, 22-Jul-2014.) $) + $( The domain and codomain of the coefficient function. (Contributed by + Mario Carneiro, 22-Jul-2014.) $) coef2 $p |- ( ( F e. ( Poly ` S ) /\ 0 e. S ) -> A : NN0 --> S ) $= ( cply cfv wcel cc0 wa cn0 csn cun wf coef adantr wss simpr snssd ssequn2 wceq sylib feq3d mpbid ) CBEFGZHBGZIZJBHKZLZAMZJBAMUDUIUEABCDNOUFUHBAJUFU GBPUHBTUFHBUDUEQRUGBSUAUBUC $. - $( The domain and range of the coefficient function. (Contributed by Mario - Carneiro, 22-Jul-2014.) $) + $( The domain and codomain of the coefficient function. (Contributed by + Mario Carneiro, 22-Jul-2014.) $) coef3 $p |- ( F e. ( Poly ` S ) -> A : NN0 --> CC ) $= ( cply cfv wcel cc cc0 cn0 wf plyssc sseli 0cn coef2 sylancl ) CBEFZGCHEF ZGIHGJHAKQRCBLMNAHCDOP $. @@ -360420,33 +361408,33 @@ or are almost disjoint (the interiors are disjoint). 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necon3bd mpd cxmet cxr cnxmet 1xr elbl3 mpanl12 expr 3impb syl5bi rabssdv - orrd elun sylibr eqsstrid ssundif sylib jca syl2anc ) AEUHKZLEMNZODKZDOUA - ZUBLOPQUCZUDUERZUFZUIHIUUOUUPUIZUUQUVAUVBOSKZLELTRZMNZUUQUVBUGUVBLLUVDMUJ - UVBEUVBEUUOUUPUKULZUMZUNOBUOZQRZPUEZEOUVHPUEZQRZTRZMNZUVEBOSDUVHOUPZUVJLU - VMUVDMUVOUVIUVOUVIOOQRZLUVHOOQUQURVAUSUVOUVLLETUVOUVLUVPLUVOUVKOOQUVOUVKO - PUEZOUVHOPUTVBVAVCURVAVCVDJVEVFUVBDUURUUTVGZUFUVAUVBDUVNBSVHUVRJUVBUVNBSU - VRUVBUVHSKZUVNVIZUVHUURKZUVHUUTKZVJUVHUVRKUVTUWAUWBUWAVMUVHOVKZUVTUWBUWAU - VHOBOVNVLUVBUVSUVNUWCUWBVOUVBUVSUVNUIZUWCUWBUVBUWDUWCUIZUIZUWBUVHLUUSRZOV - PNZUWFUWGUVKOVPUWFUWGUVHLQRZPUEZUVKUWFUVSLSKZUWGUWJUPUVBUVSUVNUWCVQZVRUVH - LUUSUUSWEVSVTUWFUWIUVHPUWFUVHUWLWAWBWCUWFUVKOUWFUVHUWLWDZUWFWFZUWFUVKOMNZ - EOWGRZUVKTRZUWPOTRZMNZUWFEUVKTRZUVKWGRZUWPUWQUWRMUWFUXAUWPMNUVKUWPUWTQRZM - NUWFUVKUVMOWGRZUXBMUWFUVKOQRZUVMMNUVKUXCMNUWFUXDUVJUVMUWFUVKUHKZOUHKZUXDU - HKUWMWHUVKOWIVTUWFUVIUWFUVCUVSUVISKZWJUWLOUVHWPWKZWDZUWFEUVLUUOUUPUWEWLZU - WFUXFUXEUVLUHKWHUWMOUVKWIWKWMZUWFUXDUVHOQRPUEZUVJMUWFUXDUVKUVQQRZUXLMUVQO - 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(Contributed by Mario Carneiro, 31-Mar-2015.) $) abelthlem3 $p |- ( ( ph /\ X e. S ) -> @@ -365826,31 +366815,31 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the wceq c4 cioo elioore syl resubcld 4re eliooord simprd 2t2e4 wne 0red 2pos wa c0 simpld lttrd elrpd pilem1 sylanbrc ne0d infrecl mp3an13 rpre adantl wn wo letric ord cioc ad2antlr rpgt0 simpr cxr w3a wb 0xr elioc2 sin02gt0 - syl3anbrc gt0ne0d syld necon4bd syl5bi ralrimiv infregelb syl31anc mpbird - ex expimpd infrelb syl3anc letrd pm3.2i lemul2 mpbid ltletrd posdifd ccos - eqbrtrrid cc recnd sinsub oveq1d coscld mul02d eqtrd oveq2d eqtrdi mul01d - syl2anc sin2t 2t0e0 oveq12d eqbrtrid eqeltrid leaddsub readdcld ledivmul - 0m0e0 ) AUABUFJZKUBJCLMZUULKCNJZLMZAUUOUAUUNBUCJZLMZAUAOUDUGPUEUHZUIZQRUJ - ZUUPLUKAUUSQULZHUMZIUMZLMZIUUSUNZHQUTZUUPUUSSZUUTUUPLMUVAAUUSOQOUURUOUPUQ - ZURZUVFAPQSPUVCLMZIUUSUNZUVFVAUVJIUUSUVCUUSSUVCUVCOUURUSVBVCUVEUVKHPQUVBP - VLUVDUVJIUUSUVBPUVCLVDVEVFVGZURZAUUPOSUUPUDTZPVLUVGAUUPAUUNBAKQSZCQSZUUNQ - SZVHACEVIZKCVJVKZABKVMVNJSZBQSZDBKVMVOVPZVQABUUNRMPUUPRMABVMUUNUWBVMQSAVR - URUVSAKBRMZBVMRMZAUVTUWCUWDWEDBKVMVSVPZVTAVMKKNJZUUNLWAAKCLMZUWFUUNLMZAKU - UTCUVOAVHURZAUUSWFWBZUUTQSZAUUSBABOSBUDTZPVLBUUSSABUWBAPKBAWCUWIUWBPKRMZA - WDURAUWCUWDUWEWGWHWIFBWJWKWLZUVAUWJUVFUWKUVHUVLHIUUSWMWNVPZUVRAKUUTLMZKUV - BLMZHUUSUNZAUWQHUUSUVBUUSSUVBOSZUVBUDTZPVLZWEAUWQUVBWJAUWSUXAUWQAUWSWEZUW - QUWTPUXBUWQWQUVBKLMZUWTPWBZUXBUWQUXCUXBUVOUVBQSZUWQUXCWRVHUWSUXEAUVBWOZWP - KUVBWSVKWTUXBUXCUXDUXBUXCWEZUWTUXGUVBPKXAJSZPUWTRMUXGUXEPUVBRMZUXCUXHUWSU - XEAUXCUXFXBUWSUXIAUXCUVBXCXBUXBUXCXDPXESUVOUXHUXEUXIUXCXFXGXHVHPKUVBXIVGX - KUVBXJVPXLXTXMXNYAXOXPAUVAUWJUVFUVOUWPUWRXGUVIUWNUVMUWIHIHUUSKXQXRXSAUVAU - VFCUUSSZUUTCLMUVIUVMACOSCUDTZPVLUXJEGCWJWKHICUUSYBYCYDAUVOUVPUVOUWMWEZUWG - UWHXGUWIUVRUXLAUVOUWMVHWDYEURZKCKYFYCYGYKYHABUUNUWBUVSYIYGWIAUVNUUNUDTZBY - JTZNJZUUNYJTZUWLNJZUCJZPAUUNYLSBYLSUVNUXSVLAUUNUVSYMZABUWBYMZUUNBYNUUBAUX - SPPUCJPAUXPPUXRPUCAUXPPUXONJPAUXNPUXONAUXNKUXKCYJTZNJZNJZPACYLSUXNUYDVLAC - UVRYMZCUUCVPAUYDKPNJPAUYCPKNAUYCPUYBNJPAUXKPUYBNGYOAUYBACUYEYPYQYRYSUUDYT - YRYOAUXOABUYAYPYQYRAUXRUXQPNJPAUWLPUXQNFYSAUXQAUUNUXTYPUUAYRUUEUUKYTYRUUP - WJWKHIUUPUUSYBYCUUFAUAQSUWAUVQUUOUUQXGAUAUUTQUKUWOUUGZUWBUVSUABUUNUUHYCXS - AUULQSUVPUXLUUMUUOXGAUABUYFUWBUUIUVRUXMUULCKUUJYCXS $. + syl3anbrc gt0ne0d syld necon4bd expimpd biimtrid ralrimiv syl31anc mpbird + ex infregelb infrelb syl3anc letrd pm3.2i mpbid eqbrtrrid ltletrd posdifd + lemul2 ccos cc recnd sinsub oveq1d coscld mul02d eqtrd oveq2d eqtrdi + syl2anc sin2t mul01d oveq12d eqbrtrid eqeltrid leaddsub readdcld ledivmul + 2t0e0 0m0e0 ) AUABUFJZKUBJCLMZUULKCNJZLMZAUUOUAUUNBUCJZLMZAUAOUDUGPUEUHZU + IZQRUJZUUPLUKAUUSQULZHUMZIUMZLMZIUUSUNZHQUTZUUPUUSSZUUTUUPLMUVAAUUSOQOUUR + UOUPUQZURZUVFAPQSPUVCLMZIUUSUNZUVFVAUVJIUUSUVCUUSSUVCUVCOUURUSVBVCUVEUVKH + PQUVBPVLUVDUVJIUUSUVBPUVCLVDVEVFVGZURZAUUPOSUUPUDTZPVLUVGAUUPAUUNBAKQSZCQ + SZUUNQSZVHACEVIZKCVJVKZABKVMVNJSZBQSZDBKVMVOVPZVQABUUNRMPUUPRMABVMUUNUWBV + MQSAVRURUVSAKBRMZBVMRMZAUVTUWCUWDWEDBKVMVSVPZVTAVMKKNJZUUNLWAAKCLMZUWFUUN + LMZAKUUTCUVOAVHURZAUUSWFWBZUUTQSZAUUSBABOSBUDTZPVLBUUSSABUWBAPKBAWCUWIUWB + PKRMZAWDURAUWCUWDUWEWGWHWIFBWJWKWLZUVAUWJUVFUWKUVHUVLHIUUSWMWNVPZUVRAKUUT + LMZKUVBLMZHUUSUNZAUWQHUUSUVBUUSSUVBOSZUVBUDTZPVLZWEAUWQUVBWJAUWSUXAUWQAUW + SWEZUWQUWTPUXBUWQWQUVBKLMZUWTPWBZUXBUWQUXCUXBUVOUVBQSZUWQUXCWRVHUWSUXEAUV + BWOZWPKUVBWSVKWTUXBUXCUXDUXBUXCWEZUWTUXGUVBPKXAJSZPUWTRMUXGUXEPUVBRMZUXCU + XHUWSUXEAUXCUXFXBUWSUXIAUXCUVBXCXBUXBUXCXDPXESUVOUXHUXEUXIUXCXFXGXHVHPKUV + BXIVGXKUVBXJVPXLXTXMXNXOXPXQAUVAUWJUVFUVOUWPUWRXGUVIUWNUVMUWIHIHUUSKYAXRX + SAUVAUVFCUUSSZUUTCLMUVIUVMACOSCUDTZPVLUXJEGCWJWKHICUUSYBYCYDAUVOUVPUVOUWM + WEZUWGUWHXGUWIUVRUXLAUVOUWMVHWDYEURZKCKYJYCYFYGYHABUUNUWBUVSYIYFWIAUVNUUN + UDTZBYKTZNJZUUNYKTZUWLNJZUCJZPAUUNYLSBYLSUVNUXSVLAUUNUVSYMZABUWBYMZUUNBYN + UUAAUXSPPUCJPAUXPPUXRPUCAUXPPUXONJPAUXNPUXONAUXNKUXKCYKTZNJZNJZPACYLSUXNU + YDVLACUVRYMZCUUBVPAUYDKPNJPAUYCPKNAUYCPUYBNJPAUXKPUYBNGYOAUYBACUYEYPYQYRY + SUUJYTYRYOAUXOABUYAYPYQYRAUXRUXQPNJPAUWLPUXQNFYSAUXQAUUNUXTYPUUCYRUUDUUKY + TYRUUPWJWKHIUUPUUSYBYCUUEAUAQSUWAUVQUUOUUQXGAUAUUTQUKUWOUUFZUWBUVSUABUUNU + UGYCXSAUULQSUVPUXLUUMUUOXGAUABUYFUWBUUHUVRUXMUULCKUUIYCXS $. $} ${ @@ -367591,7 +368580,7 @@ imaginary part lies in the interval (-pi, pi]. See $} $( The range of the natural logarithm function, also the principal domain of - the exponential function. This allows us to write the longer class + the exponential function. This allows to write the longer class expression as simply ` ran log ` . (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.) $) logrn $p |- ran log = ( `' Im " ( -u _pi (,] _pi ) ) $= @@ -369636,7 +370625,7 @@ imaginary part lies in the interval (-pi, pi]. See XKBWEWBTVRWTWQDEYFXRYIJYEWQXJWCWDQWFWGWHWIWJWK $. $} - $( The complex power function allows us to write n-th roots via the idiom + $( The complex power function allows to write n-th roots via the idiom ` A ^c ( 1 / N ) ` . (Contributed by Mario Carneiro, 6-May-2015.) $) cxproot $p |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^c ( 1 / N ) ) ^ N ) = A ) $= @@ -369653,13 +370642,13 @@ imaginary part lies in the interval (-pi, pi]. See ( cc wcel wa cmul co ccxp cexp wceq cneg cxpmul2 ad4ant13 c1 syl3anc oveq2d cn0 cdiv mulcld cc0 wne cz cr wo elznn0 3expia simplll simplr simprr simprl wi recnd mulneg2d negeqd negnegd eqtrd simpllr negcld cxpneg eqtr3d expneg2 - cxpcl 3eqtr4d expr jaod expimpd syl5bi impr ) ADEZAUAUBZFZBDEZCUCEZABCGHZIH - ZABIHZCJHZKZVNCUDEZCREZCLZREZUEZFVLVMFZVSCUFWEVTWDVSWEVTFWAVSWCVJVMWAVSULVK - VTVJVMWAVSABCMUGNWEVTWCVSWEVTWCFZFZOABWBGHZIHZSHZOVQWBJHZSHZVPVRWGWIWKOSWGV - JVMWCWIWKKVJVKVMWFUHZVLVMWFUIZWEVTWCUJZABWBMPQWGAWHLZIHZVPWJWGWPVOAIWGWPVOL - ZLVOWGWHWRWGBCWNWGCWEVTWCUKUMZUNUOWGVOWGBCWNWSTUPUQQWGVJVKWHDEWQWJKWMVJVKVM - WFURWGBWBWNWGCWSUSTAWHUTPVAWGVQDEZCDEWCVRWLKVJVMWTVKWFABVCNWSWOVQCVBPVDVEVF - VGVHVI $. + cxpcl 3eqtr4d expr jaod expimpd biimtrid impr ) ADEZAUAUBZFZBDEZCUCEZABCGHZ + IHZABIHZCJHZKZVNCUDEZCREZCLZREZUEZFVLVMFZVSCUFWEVTWDVSWEVTFWAVSWCVJVMWAVSUL + VKVTVJVMWAVSABCMUGNWEVTWCVSWEVTWCFZFZOABWBGHZIHZSHZOVQWBJHZSHZVPVRWGWIWKOSW + GVJVMWCWIWKKVJVKVMWFUHZVLVMWFUIZWEVTWCUJZABWBMPQWGAWHLZIHZVPWJWGWPVOAIWGWPV + OLZLVOWGWHWRWGBCWNWGCWEVTWCUKUMZUNUOWGVOWGBCWNWSTUPUQQWGVJVKWHDEWQWJKWMVJVK + VMWFURWGBWBWNWGCWSUSTAWHUTPVAWGVQDEZCDEWCVRWLKVJVMWTVKWFABVCNWSWOVQCVBPVDVE + VFVGVHVI $. $( Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.) $) @@ -370329,46 +371318,46 @@ the complex square root function (the branch cut is in the same place for absid simp3l mp1i min2 halflt1 cxplt3d cmul rpcnne0d recid oveq2d cxp1d cxpmuld 3eqtr3d simprd cxplt2 3expia anassrs sylan2br expr eleq2s fveq2 ralrimiva rpne0d re0 eqtrdi necon3i 0cxpd adantl abs00bd simpllr rpgt0d - fvoveq1 breq1d syl5ibcom a1dd ralrimdva sylbid expimpd ralrimiv anbi12d - jaod syl5bi imbi1d 2ralbidv rspcev ) ABRZEUARZUBZCUARIUCZUDUEZCSTZAJUCZ - UFUGZUDUEZCSTZUBZUWMUWPUHUGUDUEZESTZUIZJBUJZIUKULUMUGZUJZUWNKUCZSTZUWRU - XGSTZUBZUXBUIZJBUJIUXEUJZKUAUOUWLCDEUNDUPUGZUHUGZUQTZDUXNURZUAQUWLUXODU - XNUAUWLDAUSUEZUNUQTZUXQUNURZUTUPUGZUAPUWLUXSUWLUXQUARZUNUARUXSUARZUWJUY - AUWKUWJAVARZUYAUWJAUSVBUAVCZRZUYCUYAUBZBUYDALVDVAVEUSVFZUSVAVGZUYEUYFVK - VHVAVEUSVIZVAAUAUSVJVLVMZVNVOVPUXRUXQUNUAVQWBZVRVSZUWLEUXMUWJUWKVTUWLUX - MUWLDUYLWAWCWDZWEVSUWLUXDIUXEUWMUXERUWMVERZUKUWMUQTZUBZUWLUXDUWMWFUWLUY - NUYOUXDUWLUYNUBZUYOUKUWMSTZUKUWMWGZWHZUXDUWLUKVERUYNUYOUYTVKUWLWIUKUWMW - 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There are branch points at WSBCUMWPYPXEAWFSHWRUPRUQVEYCYEWEWGXFXPXNYHXFXNYFWHXFXNYFWIWJWKWLXFXOXPYGYHW MWNVRAVPWO $. - $( Domain and range of the arcsin function. (Contributed by Mario Carneiro, - 31-Mar-2015.) $) + $( Domain and codomain of the arcsin function. (Contributed by Mario + Carneiro, 31-Mar-2015.) $) asinf $p |- arcsin : CC --> CC $= ( vx cc ci cneg cv cmul co c1 cexp cmin csqrt caddc clog casin df-asin wcel c2 cfv mulcl sylancr negicn ax-icn mpan ax-1cn subcl sqrtcld addcld asinlem @@ -373703,8 +374692,8 @@ already know is total (except at ` 0 ` ). There are branch points at asincl $p |- ( A e. CC -> ( arcsin ` A ) e. CC ) $= ( cc casin asinf ffvelcdmi ) BBACDE $. - $( Domain and range of the arccos function. (Contributed by Mario Carneiro, - 31-Mar-2015.) $) + $( Domain and codoamin of the arccos function. (Contributed by Mario + Carneiro, 31-Mar-2015.) $) acosf $p |- arccos : CC --> CC $= ( vx cc cpi c2 cdiv co casin cfv cmin cacos df-acos wcel picn halfcl asincl cv ax-mp subcl sylancr fmpti ) ABBCDEFZAPZGHZIFZJAKUBBLUABLZUCBLUDBLCBLUEMC @@ -373764,8 +374753,8 @@ already know is total (except at ` 0 ` ). There are branch points at ZMPZUMUNPZUPMPULUMEDZUNEDUSUTSAUIZUAUMUNUBQULURUPMULURUMIRHZUPULVAIEDZURVCP VBTUMIUCQULVAVDVCUPPVBTUMIUDQUJUEUFUGUHUK $. - $( Domain and range of the arctan function. (Contributed by Mario Carneiro, - 31-Mar-2015.) $) + $( Domain and codoamin of the arctan function. (Contributed by Mario + Carneiro, 31-Mar-2015.) $) atanf $p |- arctan : ( CC \ { -u _i , _i } ) --> CC $= ( vx cc ci cneg cpr co c1 cmul cmin clog cfv catan wcel ax-icn ax-1cn mulcl cc0 wne sylancr logcld cdif c2 cdiv cv caddc df-atan cdm ovex dmmpti eleq2i @@ -373918,13 +374907,13 @@ already know is total (except at ` 0 ` ). There are branch points at KWLWM $. $( The arcsine function composed with ` sin ` is equal to the identity. This - plus ~ sinasin allow us to view ` sin ` and ` arcsin ` as inverse - operations to each other. For ease of use, we have not defined precisely - the correct domain of correctness of this identity; in addition to the - main region described here it is also true for _some_ points on the branch - cuts, namely when ` A = ( _pi / 2 ) - _i y ` for nonnegative real ` y ` - and also symmetrically at ` A = _i y - ( _pi / 2 ) ` . In particular, - when restricted to reals this identity extends to the closed interval + plus ~ sinasin allow to view ` sin ` and ` arcsin ` as inverse operations + to each other. For ease of use, we have not defined precisely the correct + domain of correctness of this identity; in addition to the main region + described here it is also true for _some_ points on the branch cuts, + namely when ` A = ( _pi / 2 ) - _i y ` for nonnegative real ` y ` and also + symmetrically at ` A = _i y - ( _pi / 2 ) ` . In particular, when + restricted to reals this identity extends to the closed interval ` [ -u ( _pi / 2 ) , ( _pi / 2 ) ] ` , not just the open interval (see ~ reasinsin ). (Contributed by Mario Carneiro, 2-Apr-2015.) $) asinsin $p |- ( ( A e. CC /\ ( Re ` A ) e. @@ -375991,60 +376980,60 @@ already know is total (except at ` 0 ` ). 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Here we round out the argument of ~ wilthlem2 with @@ -378821,40 +379810,40 @@ sum notation (which we used for its unordered summing capabilities) into nnuz eleqtrdi eluzfz2 wa cmul wn simpl elfzelz prmnn fzm1ndvds sylan eqid adantl prmdiv syl3anc simpld ralrimiva ovex pwid eleq2 raleqbi1dv anbi12d cpw elrab2 mpbiran sylanbrc cfn eleq1 reseq2 oveq2d oveq1d eqeq1d imbi12d - fzfi imbi2d wpss wal bi2.04 pm2.27 com34 syl5bi alimdv df-ral syl6ibr w3a - simp1 simp3 simp2 wilthlem2 3exp syldc a1i findcard3 mpd cnfld1 ringidval - ax-mp wb ccrg ccmn cncrng crngmgp mp1i csubrg csubmnd zsubrg subrgsubm wf - wss wf1o f1oi f1of fzssz fss mp2an cvv 1ex fdmfifsupp cseq csupp 1z mpbid - gsumsubmcl znegcl moddvds ccom fcoi1 fveq1i fvres eqtrid seqfveq cnfldbas - ccntz mgpbas cnfldmul mgpplusg crg cmnd cnring ringmgp zsscn cntzcmnf wf1 - f1of1 crn cdm suppssdm dmresi sseqtri sseqtrri gsumval3 facnn 3eqtr4d cn0 - rnresi nnm1nn0 faccld nncnd ax-1cn subneg sylancl eqtrd breqtrd ) DUBIZDE - JKDKLMZUFMZUCZUDMZKUGZLMZUWEUHNZKUIMZUJUWDUWHDOMZUWIDOMZPZDUWJUJUKZUWDUWF - 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This is Metamath 100 fzfi dvdsssfz1 sstrid ssfi sylancr ) ACDZEAFGZHDBIAJKZBLMZUDNUFHDEARUCUFU EBCMZUDLCNUFUGNOUEBLCPQABSTUDUFUAUB $. - $( Domain and range of the Chebyshev function. (Contributed by Mario + $( Domain and codoamin of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) $) chtf $p |- theta : RR --> RR $= ( vx vp cr cc0 cv cicc co cprime cin clog cfv csu ccht df-cht ppifi wa cn @@ -380241,7 +381230,7 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 DZEBFGDZHZAIGZJAKLMNZOGZBAPLZMNZOGUMASDZUNUPQUKUSULAUAZRAUBUCUMUOUROUMUOB AUDGZPLZMNZURUKUSULUOVCQUTABUEUFUKVCURQULUKVBUQMUKVAABPAUGUHUIRTUJT $. - $( Domain and range of the prime-counting function pi. (Contributed by + $( Domain and codomain of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.) $) ppif $p |- ppi : RR --> NN0 $= ( vx cr cn0 cc0 cv cicc co cprime cin chash cppi df-ppi wcel ppifi hashcl @@ -380281,10 +381270,10 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 muval2 $p |- ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) $= ( cn wcel cmu cfv cc0 wne c1 cneg cv cdvds wbr cprime crab cexp co eqeq1d - wceq wo chash wn df-ne c2 wrex cif ifeqor muval orbi12d mpbiri ord syl5bi - imp ) ACDZAEFZGHZUOIJBKZALMBNOUAFPQZSZUPUOGSZUBUNUSUOGUCUNUTUSUNUTUSTUQUD - PQALMBNUEZGURUFZGSZVBURSZTVAGURUGUNUTVCUSVDUNUOVBGABUHZRUNUOVBURVERUIUJUK - ULUM $. + wceq wo chash wn df-ne c2 wrex cif ifeqor orbi12d mpbiri ord biimtrid imp + muval ) ACDZAEFZGHZUOIJBKZALMBNOUAFPQZSZUPUOGSZUBUNUSUOGUCUNUTUSUNUTUSTUQ + UDPQALMBNUEZGURUFZGSZVBURSZTVAGURUGUNUTVCUSVDUNUOVBGABUMZRUNUOVBURVERUHUI + UJUKUL $. $( Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.) $) @@ -380338,15 +381327,15 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( cn wcel cmu cfv cc0 wne wa wceq cpc co cprime wbr wb cn0 wo c1 syl2anr cv wral cdvds nnnn0 syl2an ad2ant2r eleq1 wn dfbi3 cle ad4ant124 nnle1eq1 pc11 sqfpc syl5ibcom simprl adantr simplrr simpr syl3anc anim12d eqtr3 id - syl6 simpll pccl elnn0 sylib ord jaod syl5bi impbid2 pcelnn bibi12d bitrd - ralbidva ) ADEZAFGHIZJZBDEZBFGHIZJZJZABKZCUAZALMZWEBLMZKZCNUBZWEAUCOZWEBU - COZPZCNUBVQVTWDWIPZVRWAVQAQEBQEWMVTAUDBUDABCUMUEUFWCWHWLCNWCWENEZJZWHWFDE - ZWGDEZPZWLWOWHWRWFWGDUGWRWPWQJZWPUHZWQUHZJZRWOWHWPWQUIWOWSWHXBWOWSWFSKZWG - SKZJWHWOWPXCWQXDWOWFSUJOZWPXCVQVRWNXEWBAWEUNUKWFULUOWOWGSUJOZWQXDWOVTWAWN - XFWCVTWNVSVTWAUPZUQVSVTWAWNURWCWNUSBWEUNUTWGULUOVAWFWGSVBVDWOXBWFHKZWGHKZ - JWHWOWTXHXAXIWOWPXHWOWFQEZWPXHRWNWNVQXJWCWNVCZVQVRWBVEZWEAVFTWFVGVHVIWOWQ - XIWOWGQEZWQXIRWNWNVTXMWCXKXGWEBVFTWGVGVHVIVAWFWGHVBVDVJVKVLWOWPWJWQWKWNWN - VQWPWJPWCXKXLWEAVMTWNWNVTWQWKPWCXKXGWEBVMTVNVOVPVO $. + syl6 simpll pccl elnn0 sylib ord biimtrid impbid2 pcelnn bibi12d ralbidva + jaod bitrd ) ADEZAFGHIZJZBDEZBFGHIZJZJZABKZCUAZALMZWEBLMZKZCNUBZWEAUCOZWE + BUCOZPZCNUBVQVTWDWIPZVRWAVQAQEBQEWMVTAUDBUDABCUMUEUFWCWHWLCNWCWENEZJZWHWF + DEZWGDEZPZWLWOWHWRWFWGDUGWRWPWQJZWPUHZWQUHZJZRWOWHWPWQUIWOWSWHXBWOWSWFSKZ + WGSKZJWHWOWPXCWQXDWOWFSUJOZWPXCVQVRWNXEWBAWEUNUKWFULUOWOWGSUJOZWQXDWOVTWA + WNXFWCVTWNVSVTWAUPZUQVSVTWAWNURWCWNUSBWEUNUTWGULUOVAWFWGSVBVDWOXBWFHKZWGH + KZJWHWOWTXHXAXIWOWPXHWOWFQEZWPXHRWNWNVQXJWCWNVCZVQVRWBVEZWEAVFTWFVGVHVIWO + WQXIWOWGQEZWQXIRWNWNVTXMWCXKXGWEBVFTWGVGVHVIVAWFWGHVBVDVOVJVKWOWPWJWQWKWN + WNVQWPWJPWCXKXLWEAVLTWNWNVTWQWKPWCXKXGWEBVLTVMVPVNVP $. $( The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.) $) @@ -381540,20 +382529,20 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 dvdsmod uzid simpl dvdsprm sylancr bitrd simpr syl5ibrcom clt wn 2lt4 2re breq2 4re ltnlei mpbi pm2.21i syl6 sylbid imbi1d syl5ibcom com3r dvdsmul1 3nn df-3 peano2uz eqeltri 3lt4 3re eleq1a a1d 3jaoi oveq1i eleq2i syl5bir - simpri jaod syl5bi pm3.2i ) BHIJZAUAKZLAIJZUFZAHUBMZBNUCMKZYJONUDZKZPPBNI - JZNHIJYFYNBOUGMZNOUGMZIJCHYOYPICHIJZYIYJCNUCMZKZYMPPZDUEFUHUIBNOBEUJZUKUL - UMUNZNHUKUOUSUPBNHUUAUKUOUQURYKYJBUTZYJYONUCMZKZVAZYIYMNBVBVCKZYKUUFQUUGB - VDKNVDKYNBEVENVFVNUUBBNVGVHYJBNVIVJYIUUCYMUUERBSJZTBSJZBYLKZVKYIUUCYMPZPZ - GUUHUULUUIUUJYIUUCUUHYMYIRYJSJZYMPUUCUUHYMPYIUUMRAUTZYMYIUUMRASJZUUNRVLKZ - HVLKZYIAVDKZUUMUUOQZVMVOYGUURYHAVPVQZUUPUUQUURVRRHSJUUSRTRVSMZHSTVDKZRVDK - ZRUVASJVTWATRWBURWCWDRAHWGWEWFYIRRVBVCZKZYGUUOUUNQUVCUVEWARWHVJZYGYHWIZAR - WJWKWLYIUUNLRIJZYMYIUVHUUNYHYGYHWMZRALIWSWNUVHYMRLWOJUVHWPWQRLWRWTXAXBXCX - DXEUUCUUMUUHYMYJBRSWSXFXGXHYIUUCUUIYMYITYJSJZYMPUUCUUIYMPYIUVJTAUTZYMYIUV - JTASJZUVKTVLKZUUQYIUURUVJUVLQZXJVOUUTUVMUUQUURVRTHSJUVNTUVAHSUVBUVCTUVASJ - VTWATRXIURWCWDTAHWGWEWFYITUVDKYGUVLUVKQTROUGMZUVDXKUVEUVOUVDKUVFRRXLVJXMU - VGATWJWKWLYIUVKLTIJZYMYIUVPUVKYHUVITALIWSWNUVPYMTLWOJUVPWPXNTLXOWTXAXBXCX - DXEUUCUVJUUIYMYJBTSWSXFXGXHUUJUUKYIBYLYJXPXQXRVJUUEYSYIYMYRUUDYJCYONUCFXS - XTYQYTDYBYAYCYDYE $. + simpri jaod biimtrid pm3.2i ) BHIJZAUAKZLAIJZUFZAHUBMZBNUCMKZYJONUDZKZPPB + NIJZNHIJYFYNBOUGMZNOUGMZIJCHYOYPICHIJZYIYJCNUCMZKZYMPPZDUEFUHUIBNOBEUJZUK + ULUMUNZNHUKUOUSUPBNHUUAUKUOUQURYKYJBUTZYJYONUCMZKZVAZYIYMNBVBVCKZYKUUFQUU + GBVDKNVDKYNBEVENVFVNUUBBNVGVHYJBNVIVJYIUUCYMUUERBSJZTBSJZBYLKZVKYIUUCYMPZ + PZGUUHUULUUIUUJYIUUCUUHYMYIRYJSJZYMPUUCUUHYMPYIUUMRAUTZYMYIUUMRASJZUUNRVL + KZHVLKZYIAVDKZUUMUUOQZVMVOYGUURYHAVPVQZUUPUUQUURVRRHSJUUSRTRVSMZHSTVDKZRV + DKZRUVASJVTWATRWBURWCWDRAHWGWEWFYIRRVBVCZKZYGUUOUUNQUVCUVEWARWHVJZYGYHWIZ + ARWJWKWLYIUUNLRIJZYMYIUVHUUNYHYGYHWMZRALIWSWNUVHYMRLWOJUVHWPWQRLWRWTXAXBX + CXDXEUUCUUMUUHYMYJBRSWSXFXGXHYIUUCUUIYMYITYJSJZYMPUUCUUIYMPYIUVJTAUTZYMYI + UVJTASJZUVKTVLKZUUQYIUURUVJUVLQZXJVOUUTUVMUUQUURVRTHSJUVNTUVAHSUVBUVCTUVA + SJVTWATRXIURWCWDTAHWGWEWFYITUVDKYGUVLUVKQTROUGMZUVDXKUVEUVOUVDKUVFRRXLVJX + MUVGATWJWKWLYIUVKLTIJZYMYIUVPUVKYHUVITALIWSWNUVPYMTLWOJUVPWPXNTLXOWTXAXBX + CXDXEUUCUVJUUIYMYJBTSWSXFXGXHUUJUUKYIBYLYJXPXQXRVJUUEYSYIYMYRUUDYJCYONUCF + XSXTYQYTDYBYAYCYDYE $. $} $( A prime greater than ` 3 ` does not divide ` 2 ` or ` 3 ` , so its residue @@ -381942,44 +382931,44 @@ particular proof approach is due to Cauchy (1821). 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This is Metamath 100 ( cz wcel c2 c1 caddc co cdvds wbr cc0 cfz ax-1cn wn cmo simp1i peano2z wa c8 wi ax-mp eqeltri simp2i wb 2z dvdsadd mp2an mpbi cc addcomi eqtri zcn oveq1i df-2 add32i eqtr4i 2cn addassi breqtrri cmin wceq wo cuz cfv - elfzuz2 fzm1 syl ibi mvrraddi oveq2i eleq2s eleq2i simp3i syl5bi cn 2nn - 8nn w3a c4 cmul 4z dvdsmul2 4t2e8 breqtri dvdsmod mpan2 mp3an12 biimpar - notbid id breqtrrid nsyl pm2.21d jaod syl5 eleq1 mpbiri a1i 3pm3.2i ) E - JKLEMNOZPQAJKZLAPQZUAZUEZAUFUBOZRESOKZXLBKZUGUGEDMNOZJHDJKZXOJKDCMNOZJG - CJKZXQJKZXRLXQPQZXKXLRCSOZKZXNUGUGZFUCZCUDUHZUIZDUDUHUILLXQNOZXGPXTLYGP - QZXRXTYCFUJZLJKZXSXTYHUKULYELXQUMUNUOXGLCNOZMNOYGEYKMNEMCNOZMNOZYKEXOYM - HDYLMNDXQYLGCMXRCUPKYDCUSUHZTUQURUTURYKMMNOZCNOYMLYOCNVAUTMCMTYNTVBVCVC - UTLCMVDYNTVEURVFXMXLREMVGOZSOZKZXLEVHZVIZXKXNXMYTXMERVJVKZKXMYTUKXLREVL - XLREVMVNVOXKYRXNYSYRXLRDMVGOZSOZKZXLDVHZVIZXKXNUUFXLRDSOZYQXLUUGKZUUFUU - HDUUAKUUHUUFUKXLRDVLXLRDVMVNVOYPDRSEDMXPDUPKYFDUSUHTHVPVQVRXKUUDXNUUEUU - DYBXKXNUUCYAXLUUBCRSDCMYNTGVPVQVSXRXTYCFVTWAXKUUEXNXKLXLPQZUUEXHUUIUAXJ - XHUUIXILWBKZUFWBKZXHUUIXIUKZWCWDUUJUUKXHWELUFPQUULLWFLWGOZUFPWFJKYJLUUM - PQWHULWFLWIUNWJWKLAUFWLWMWNWPWOUUELDXLPLXQDPYIGVFUUEWQWRWSWTXAXBYSXNUGX - KYSXNEBKIXLEBXCXDXEXAXBXF $. + elfzuz2 fzm1 syl ibi mvrraddi oveq2i eleq2s eleq2i biimtrid 2nn 8nn w3a + simp3i cn c4 cmul dvdsmul2 4t2e8 breqtri dvdsmod mp3an12 notbid biimpar + 4z mpan2 id breqtrrid nsyl pm2.21d jaod syl5 eleq1 mpbiri a1i 3pm3.2i ) + EJKLEMNOZPQAJKZLAPQZUAZUEZAUFUBOZRESOKZXLBKZUGUGEDMNOZJHDJKZXOJKDCMNOZJ + GCJKZXQJKZXRLXQPQZXKXLRCSOZKZXNUGUGZFUCZCUDUHZUIZDUDUHUILLXQNOZXGPXTLYG + PQZXRXTYCFUJZLJKZXSXTYHUKULYELXQUMUNUOXGLCNOZMNOYGEYKMNEMCNOZMNOZYKEXOY + MHDYLMNDXQYLGCMXRCUPKYDCUSUHZTUQURUTURYKMMNOZCNOYMLYOCNVAUTMCMTYNTVBVCV + CUTLCMVDYNTVEURVFXMXLREMVGOZSOZKZXLEVHZVIZXKXNXMYTXMERVJVKZKXMYTUKXLREV + LXLREVMVNVOXKYRXNYSYRXLRDMVGOZSOZKZXLDVHZVIZXKXNUUFXLRDSOZYQXLUUGKZUUFU + UHDUUAKUUHUUFUKXLRDVLXLRDVMVNVOYPDRSEDMXPDUPKYFDUSUHTHVPVQVRXKUUDXNUUEU + UDYBXKXNUUCYAXLUUBCRSDCMYNTGVPVQVSXRXTYCFWDVTXKUUEXNXKLXLPQZUUEXHUUIUAX + JXHUUIXILWEKZUFWEKZXHUUIXIUKZWAWBUUJUUKXHWCLUFPQUULLWFLWGOZUFPWFJKYJLUU + MPQWOULWFLWHUNWIWJLAUFWKWPWLWMWNUUELDXLPLXQDPYIGVFUUEWQWRWSWTXAXBYSXNUG + XKYSXNEBKIXLEBXCXDXEXAXBXF $. $} $( Lemma for ~ lgsdir2 . (Contributed by Mario Carneiro, 4-Feb-2015.) $) @@ -385384,25 +386373,25 @@ particular proof approach is due to Cauchy (1821). This is Metamath 100 ( cz wcel wa c8 cmo co c3 c5 cmul c1 wceq 3z a1i eqtr4di modmul12d oveq1i 3cn wtru cpr cneg c7 wo ovex elpr anbi12i simpll simplr crp elrpii simprl 8pos lgsdir2lem1 simpri simpli simprr orcd ex znegcl mp1i mulneg1i eqtrdi - 8re olcd mulneg2i mul2negi ccased syl5bi imp sylibr caddc df-9 8cn ax-1cn - c9 addcomi eqtri 3t3e9 mulid2i oveq2i 3eqtr4i cr 1z modcyc mp3an nnmulcli - 1re 3nn nnzi eqidd mptru mulcli mulm1i 3eqtr3i preq12i eleqtrdi ) ACDZBCD - ZEZAFGHZIJUAZDZBFGHZXBDZEZEZABKHZFGHZIIKHZFGHZXJUBZFGHZUAZLUCUAXGXIXKMZXI - XMMZUDZXIXNDWTXFXQXFXAIMZXAJMZUDZXDIMZXDJMZUDZEWTXQXCXTXEYCXAIJAFGUEUFXDI - JBFGUEUFUGWTXRYAXSYBXQWTXRYAEZXQWTYDEZXOXPYEAIBIFWRWSYDUHICDZYENOZWRWSYDU - IYGFUJDZYEFVDUMUKZOYEXAIIFGHZWTXRYAULYJIMZIUBZFGHZJMZLFGHZLMZLUBZFGHZUCMZ - EZYKYNEZUNUOZUPZPYEXDIYJWTXRYAUQUUCPQURUSWTXSYAEZXQWTUUDEZXPXOUUEXIYLIKHZ - FGHXMUUEAYLBIFWRWSUUDUHYFYLCDZUUENIUTZVAWRWSUUDUIYFUUENOYHUUEYIOUUEXAJYMW - TXSYAULYKYNUUBUOZPUUEXDIYJWTXSYAUQUUCPQUUFXLFGIISSVBRVCVEUSWTXRYBEZXQWTUU - JEZXPXOUUKXIIYLKHZFGHXMUUKAIBYLFWRWSUUJUHYFUUKNOWRWSUUJUIYFUUGUUKNUUHVAYH - UUKYIOUUKXAIYJWTXRYBULUUCPUUKXDJYMWTXRYBUQUUIPQUULXLFGIISSVFRVCVEUSWTXSYB - EZXQWTUUMEZXOXPUUNXIYLYLKHZFGHXKUUNAYLBYLFWRWSUUMUHYFUUGUUNNUUHVAZWRWSUUM - UIUUPYHUUNYIOUUNXAJYMWTXSYBULUUIPUUNXDJYMWTXSYBUQUUIPQUUOXJFGIISSVGRVCURU - SVHVIVJXIXKXMXHFGUEUFVKXKLXMUCXKYOLXKLLFKHZVLHZFGHZYOXJUURFGVPLFVLHZXJUUR - VPFLVLHUUTVMFLVNVOVQVRVSUUQFLVLFVNVTWAWBRLWCDYHLCDZUUSYOMWHYIWDLFLWEWFVRZ - YPYSYTUUAUNUPZUPVRXMYRUCYQXJKHZFGHZYQLKHZFGHZXMYRUVEUVGMTYQYQXJLFUVAYQCDT - WDLUTVAZUVHXJCDTXJIIWIWIWGWJOUVATWDOYHTYIOTYRWKXKYOMTUVBOQWLUVDXLFGXJIISS - WMWNRUVFYQFGLVOWNRWOYPYSUVCUOVRWPWQ $. + 8re olcd mulneg2i mul2negi ccased biimtrid imp sylibr caddc c9 8cn ax-1cn + df-9 addcomi eqtri 3t3e9 mulid2i oveq2i 3eqtr4i cr 1re 1z modcyc nnmulcli + mp3an 3nn nnzi eqidd mptru mulcli mulm1i 3eqtr3i preq12i eleqtrdi ) ACDZB + CDZEZAFGHZIJUAZDZBFGHZXBDZEZEZABKHZFGHZIIKHZFGHZXJUBZFGHZUAZLUCUAXGXIXKMZ + XIXMMZUDZXIXNDWTXFXQXFXAIMZXAJMZUDZXDIMZXDJMZUDZEWTXQXCXTXEYCXAIJAFGUEUFX + DIJBFGUEUFUGWTXRYAXSYBXQWTXRYAEZXQWTYDEZXOXPYEAIBIFWRWSYDUHICDZYENOZWRWSY + DUIYGFUJDZYEFVDUMUKZOYEXAIIFGHZWTXRYAULYJIMZIUBZFGHZJMZLFGHZLMZLUBZFGHZUC + MZEZYKYNEZUNUOZUPZPYEXDIYJWTXRYAUQUUCPQURUSWTXSYAEZXQWTUUDEZXPXOUUEXIYLIK + HZFGHXMUUEAYLBIFWRWSUUDUHYFYLCDZUUENIUTZVAWRWSUUDUIYFUUENOYHUUEYIOUUEXAJY + MWTXSYAULYKYNUUBUOZPUUEXDIYJWTXSYAUQUUCPQUUFXLFGIISSVBRVCVEUSWTXRYBEZXQWT + UUJEZXPXOUUKXIIYLKHZFGHXMUUKAIBYLFWRWSUUJUHYFUUKNOWRWSUUJUIYFUUGUUKNUUHVA + YHUUKYIOUUKXAIYJWTXRYBULUUCPUUKXDJYMWTXRYBUQUUIPQUULXLFGIISSVFRVCVEUSWTXS + YBEZXQWTUUMEZXOXPUUNXIYLYLKHZFGHXKUUNAYLBYLFWRWSUUMUHYFUUGUUNNUUHVAZWRWSU + UMUIUUPYHUUNYIOUUNXAJYMWTXSYBULUUIPUUNXDJYMWTXSYBUQUUIPQUUOXJFGIISSVGRVCU + RUSVHVIVJXIXKXMXHFGUEUFVKXKLXMUCXKYOLXKLLFKHZVLHZFGHZYOXJUURFGVMLFVLHZXJU + URVMFLVLHUUTVPFLVNVOVQVRVSUUQFLVLFVNVTWAWBRLWCDYHLCDZUUSYOMWDYIWELFLWFWHV + RZYPYSYTUUAUNUPZUPVRXMYRUCYQXJKHZFGHZYQLKHZFGHZXMYRUVEUVGMTYQYQXJLFUVAYQC + DTWELUTVAZUVHXJCDTXJIIWIWIWGWJOUVATWEOYHTYIOTYRWKXKYOMTUVBOQWLUVDXLFGXJII + SSWMWNRUVFYQFGLVOWNRWOYPYSUVCUOVRWPWQ $. $( The Legendre symbol is completely multiplicative at ` 2 ` . (Contributed by Mario Carneiro, 4-Feb-2015.) $) @@ -386113,17 +387102,17 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and ex csn cdif clgs cv cexp wn lgsqrmod imp cn eldifi prmnn syl zsqcl adantl cmin simplll moddvds syl3anc w3a nnzd 3jca dvdssub2 sylan bicom simpr 2nn a1i prmdvdsexp biimparc bianir ad2antlr dvdsmod0 lgsprme0 sylan2 wne 0ne1 - eqeq1 eqneqall mpi syl6bi syl6bir com23 ad2antrl syl5com mpcom syl5bi 2a1 - syld pm2.61i sylbid ancld reximdva mpd ) BDEZCFGUAZUBEZHZBCUCIZJKZAUDZGUE - IZCLIBCLIZKZCWTMNZUFZHZADOZWQWSHZXCADOZXGWQWSXIABCUGUHXHXCXFADXHWTDEZHZXC - XEXKXCCXABUOIMNZXEXKCUIEZXADEZWNXCXLPWPXMWNWSXJWPCFEZXMCFWOUJZCUKULZQXJXN - XHWTUMUNZWNWPWSXJUPZXABCUQURXDXKXLXERZRXDXKXTXDXKHZXLCXAMNZCBMNZPZXEYAXLY - DYACDEZXNWNUSZXLYDXKYFXDXKYEXNWNWPYEWNWSXJWPCXQUTQXRXSVAUNCXABVBVCTYDYCYB - PZYAXEYBYCVDYBYAYGXERXKYBXDXKXOXJGUIEZYBXDPWPXOWNWSXJXPQXHXJVEYHXKVFVGWTC - GVHURVIYBYGYAXEYBYGYAXERYBYGHYCYAXEYBYCVJXHYCXERXDXJXHYCXBSKZXEXHXMYCYIRW - 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(Contributed by AV, 5-Jul-2021.) $) @@ -386467,26 +387456,26 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and cz wi elfz2 cfl oveq1i breq1i nnre 4re cc0 wne 4ne0 redivcld fllelt flcld c4 syl zred peano2re zre adantr ltleletr syl3anc adantld mpd wb rehalfcld expd imp 2re remulcld 2pos pm3.2i lediv1 nncn 2cnne0 divdiv1 2t2e4 oveq2i - cc eqtrdi 2cnd 2ne0 divcan4d breqan12rd bitrd mpbird exp31 com23 3ad2ant3 - syl5bi com12 impcom sylbi elfzelz lenlt syl2an mpbid sylan iffalsed eqtrd - zcn cuz wss cn0 gausslemma2dlem0d nn0p1nn nnuz fzss1 sselda ovexd fvmptd2 - eleqtrdi ralrimiva ) AEUAZDUBCYNLUCMZUDMZUEEGNUFMZFUGMZAYNYROZPZBYNBUAZLU - CMZCLUJMZUHQZUUBCUUBUDMZUIZYPNFUGMZDUKJYTUUAYNUEZPZUUFYOUUCUHQZYOYPUIZYPU - UHUUFUUKUEYTUUHUUDUUJUUBUUEYOYPUUHUUBYOUUCUHUUAYNLUCULZUMUULUUHUUBYOCUDUU - LUNUOUPUUIUUJYOYPYTUUJUQZUUHACUROZYSUUMACHUSUUNYSPUUCYORQZUUMYSUUNUUOYSYQ - VAOZFVAOZYNVAOZUTZYQYNRQZYNFRQZPZPUUNUUOVBZYNYQFVCUVBUUSUVCUUTUUSUVCVBUVA - UUSUUTUVCUURUUPUUTUVCVBUUQUUTCVOUJMZVDUBZNUFMZYNRQZUURUVCYQUVFYNRGUVENUFK - VEVFUURUUNUVGUUOUURUUNUVGUUOUURUUNPZUVGPUUOUVDYNRQZUVHUVGUVIUVHUVEUVDRQZU - VDUVFUHQZPZUVGUVIVBZUVHUVDSOZUVLUUNUVNUURUUNCVOCVGZVOSOUUNVHTVOVIVJUUNVKT - VLZUPZUVDVMVPUVHUVKUVMUVJUVHUVKUVGUVIUVHUVNUVFSOZYNSOZUVKUVGPUVIVBUVQUUNU - VRUURUUNUVESOUVRUUNUVEUUNUVDUVPVNVQUVEVRVPUPUURUVSUUNYNVSZVTUVDUVFYNWAWBW - GWCWDWHUVHUUOUVIWEUVGUVHUUOUUCLUJMZYOLUJMZRQZUVIUVHUUCSOZYOSOZLSOZVILUHQZ - PZUUOUWCWEUUNUWDUURUUNCUVOWFZUPUURUWEUUNUURYNLUVTUWFUURWITWJVTUWHUVHUWFUW - GWIWKWLTUUCYOLWMWBUUNUURUWAUVDUWBYNRUUNUWACLLUCMZUJMZUVDUUNCWSOLWSOLVIVJZ - PZUWMUWAUWKUECWNUWMUUNWOTZUWNCLLWPWBUWJVOCUJWQWRWTUURYNLYNYAUURXAUWLUURXB - TXCXDXEVTXFXGXHXJXIXKVTXLXMXLUUNUWDUWEUUOUUMWEYSUWIYSYNLYSYNYNYQFXNVQUWFY - SWITWJUUCYOXOXPXQXRVTXSXTAYRUUGYNAYQNYBUBZOZYRUUGYCAGYDOZUWPACGHKYEUWQYQU - RUWOGYFYGYLVPYQNFYHVPYIYTCYOUDYJYKYM $. + cc eqtrdi 2cnd 2ne0 divcan4d breqan12rd bitrd mpbird exp31 com23 biimtrid + zcn 3ad2ant3 com12 impcom sylbi elfzelz lenlt syl2an mpbid sylan iffalsed + eqtrd cuz wss cn0 gausslemma2dlem0d nn0p1nn eleqtrdi fzss1 sselda fvmptd2 + nnuz ovexd ralrimiva ) AEUAZDUBCYNLUCMZUDMZUEEGNUFMZFUGMZAYNYROZPZBYNBUAZ + LUCMZCLUJMZUHQZUUBCUUBUDMZUIZYPNFUGMZDUKJYTUUAYNUEZPZUUFYOUUCUHQZYOYPUIZY + PUUHUUFUUKUEYTUUHUUDUUJUUBUUEYOYPUUHUUBYOUUCUHUUAYNLUCULZUMUULUUHUUBYOCUD + UULUNUOUPUUIUUJYOYPYTUUJUQZUUHACUROZYSUUMACHUSUUNYSPUUCYORQZUUMYSUUNUUOYS + YQVAOZFVAOZYNVAOZUTZYQYNRQZYNFRQZPZPUUNUUOVBZYNYQFVCUVBUUSUVCUUTUUSUVCVBU + VAUUSUUTUVCUURUUPUUTUVCVBUUQUUTCVOUJMZVDUBZNUFMZYNRQZUURUVCYQUVFYNRGUVENU + FKVEVFUURUUNUVGUUOUURUUNUVGUUOUURUUNPZUVGPUUOUVDYNRQZUVHUVGUVIUVHUVEUVDRQ + ZUVDUVFUHQZPZUVGUVIVBZUVHUVDSOZUVLUUNUVNUURUUNCVOCVGZVOSOUUNVHTVOVIVJUUNV + KTVLZUPZUVDVMVPUVHUVKUVMUVJUVHUVKUVGUVIUVHUVNUVFSOZYNSOZUVKUVGPUVIVBUVQUU + NUVRUURUUNUVESOUVRUUNUVEUUNUVDUVPVNVQUVEVRVPUPUURUVSUUNYNVSZVTUVDUVFYNWAW + BWGWCWDWHUVHUUOUVIWEUVGUVHUUOUUCLUJMZYOLUJMZRQZUVIUVHUUCSOZYOSOZLSOZVILUH + QZPZUUOUWCWEUUNUWDUURUUNCUVOWFZUPUURUWEUUNUURYNLUVTUWFUURWITWJVTUWHUVHUWF + UWGWIWKWLTUUCYOLWMWBUUNUURUWAUVDUWBYNRUUNUWACLLUCMZUJMZUVDUUNCWSOLWSOLVIV + JZPZUWMUWAUWKUECWNUWMUUNWOTZUWNCLLWPWBUWJVOCUJWQWRWTUURYNLYNXJUURXAUWLUUR + XBTXCXDXEVTXFXGXHXIXKXLVTXMXNXMUUNUWDUWEUUOUUMWEYSUWIYSYNLYSYNYNYQFXOVQUW + FYSWITWJUUCYOXPXQXRXSVTXTYAAYRUUGYNAYQNYBUBZOZYRUUGYCAGYDOZUWPACGHKYEUWQY + QURUWOGYFYKYGVPYQNFYHVPYIYTCYOUDYLYJYM $. $d H k $. $d M k $. $d P k $. $( Lemma 4 for ~ gausslemma2d . (Contributed by AV, 16-Jun-2021.) $) @@ -386631,17 +387620,17 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and cn simpr modmul1 syl3anc ex zcnd nncnd mul32d caddc nn0cnd 2timesd oveq2d neg1cn expaddd nn0zd m1expeven 3eqtr3d oveq1d mulid2d 3eqtrd eqeq12d cmin cc cdiv oveq2i oveq1i eqeq1i 2z lgsvalmod eqeq1d gausslemma2dlem0i sylbid - syl5bi syld mpd ) AMUAZGNOZPENOZQOZCROZMSZPCUGOZXPSZABCDEFGHIJKLUBAXTXSMC - ROZSZYBAMYCXSAYCMACUCTZMCUDUHZUEZYCMSACUFPUIZUJTZCUFTZYGHCUFYHUKYJYEYFYJC - CULUMCUNUOUPCVAUQURUSAYDXRXPQOZCROZMXPQOZCROZSZYBAYDYOAYDUEXRUCTZMUCTZUEZ - XPUTTZCVBTZUEZYDYOAYRYDAYPYQAXRAXPXQAXOUTTGVCTYSVDACEFGHKILVEZXOGVFVGZAXQ - APEPVTTAVHVJAEACEHIVIVKVLZVMVNVOAVPUOVQAUUAYDAYSYTUUCACACHVRVSUOVQAYDWAXR - MXPCWBWCWDAYOXQCROZXPCROZSZYBAYLUUEYNUUFAYKXQCRAYKXPXPQOZXQQOMXQQOXQAXPXQ - XPAXPUUCWEZAXQUUDWFZUUIWGAUUHMXQQAXOGGWHOZNOXOPGQOZNOZUUHMAUUKUULXONAUULU - UKAGAGUUBWIWJURWKAXOGGXOXBTAWLVJUUBUUBWMAGUTTUUMMSAGUUBWNGWOUQWPWQAXQUUJW - RWSWQAYMXPCRAXPUUIWRWQWTUUGPCMXAOPXCOZNOZCROZUUFSZAYBUUEUUPUUFXQUUOCREUUN - PNIXDXEXFAUUQYACROZUUFSYBAUUPUURUUFAUURUUPAPUTTYIUURUUPSXGHPCXHVGURXIACEF - GHKILXJXKXLXKXMXKXN $. + biimtrid syld mpd ) AMUAZGNOZPENOZQOZCROZMSZPCUGOZXPSZABCDEFGHIJKLUBAXTXS + MCROZSZYBAMYCXSAYCMACUCTZMCUDUHZUEZYCMSACUFPUIZUJTZCUFTZYGHCUFYHUKYJYEYFY + JCCULUMCUNUOUPCVAUQURUSAYDXRXPQOZCROZMXPQOZCROZSZYBAYDYOAYDUEXRUCTZMUCTZU + EZXPUTTZCVBTZUEZYDYOAYRYDAYPYQAXRAXPXQAXOUTTGVCTYSVDACEFGHKILVEZXOGVFVGZA + XQAPEPVTTAVHVJAEACEHIVIVKVLZVMVNVOAVPUOVQAUUAYDAYSYTUUCACACHVRVSUOVQAYDWA + XRMXPCWBWCWDAYOXQCROZXPCROZSZYBAYLUUEYNUUFAYKXQCRAYKXPXPQOZXQQOMXQQOXQAXP + XQXPAXPUUCWEZAXQUUDWFZUUIWGAUUHMXQQAXOGGWHOZNOXOPGQOZNOZUUHMAUUKUULXONAUU + LUUKAGAGUUBWIWJURWKAXOGGXOXBTAWLVJUUBUUBWMAGUTTUUMMSAGUUBWNGWOUQWPWQAXQUU + JWRWSWQAYMXPCRAXPUUIWRWQWTUUGPCMXAOPXCOZNOZCROZUUFSZAYBUUEUUPUUFXQUUOCREU + UNPNIXDXEXFAUUQYACROZUUFSYBAUUPUURUUFAUURUUPAPUTTYIUURUUPSXGHPCXHVGURXIAC + EFGHKILXJXKXLXKXMXKXN $. $} $( @@ -387944,28 +388933,28 @@ multiple of the prime (in which case it is ` 0 ` , see ~ lgsne0 ) and <-> 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) ) ) $= ( cz wcel c2 cdvds wbr c8 co wceq cmul caddc wb wa cc a1i adantr cr syl c4 vk wn cmo w3a cv wrex cexp c1 cmin cdiv crp wi cn 8nn nnrp ax-mp eqcom - modmuladdim syl5bi mpan2 imp 3adant2 zcn 8cn mulcomd adantl oveq1d eqeq2d - simpr zmodcld nn0zd 3ad2ant1 eleq1 3ad2ant3 mpbird 2lgsoddprmlem1 syl3anc - id breq2d 2z simp1 nn0red resqcl peano2rem 8re cc0 8pos gt0ne0ii redivcld - wne cn0 nn0z nnzi zsqcl zmulcld zmulcl ancoms zaddcld 4z jca dvdsmul1 4cn - 4t2e8 2cn mulcomi eqtr3i mulassd eqtrd adddid eqtr4d breqtrrd dvdsaddre2b - zcnd ex mp3an2ani bitr4d sylbid rexlimdva mpd ) BCDZEBFGUBZABHUCIZJZUDZBU - AUEZHKIZALIZJZUACUFZEBEUGIUHUIIHUJIZFGZEAEUGIZUHUIIZHUJIZFGZMZXTYCYIYAXTY - CYIXTHUKDZYCYIULHUMDZYQUNHUOUPYCYBAJXTYQNYIAYBUQBAUAHURUSUTVAVBYDYHYPUACY - DYECDZNZYHBHYEKIZALIZJZYPYTYGUUBBYTYFUUAALYSYFUUAJYDYSYEHYEVCHODYSVDPVEVF - VGVHYTUUCYPYTUUCNZYKEHYEEUGIZKIZEYEAKIZKIZLIZYNLIZFGZYOUUDYJUUJEFUUDYSACD - ZUUCYJUUJJYTYSUUCYDYSVIQYTUULUUCYDUULYSYDUULYBCDZXTYAUUMYCXTYBXTBHXTVRYRX - TUNPVJVKVLYCXTUULUUMMYAAYBCVMVNVOQQYTUUCVIYEABVPVQVSECDZYTYNRDZUUCUUICDZE - UUIFGZNZYOUUKMVTYDUUOYSYDARDZUUOYDUUSYBRDZYDYBYDBHXTYAYCWAYRYDUNPVJZWBYCX - TUUSUUTMYAAYBRVMVNVOUUSYMHUUSYLRDYMRDAWCYLWDSHRDUUSWEPHWFWJUUSHWEWGWHPWIS - QYTUURUUCYDYSUURYDAWKDZYSUURULZYDUVBYBWKDZUVAYCXTUVBUVDMYAAYBWKVMVNVOUVBU - ULUVCAWLUULYSUURUULYSNZUUPUUQUVEUUFUUHUVEHUUEHCDUVEHUNWMPYSUUECDUULYEWNZV - FZWOUVEEUUGUUNUVEVTPZYSUULUUGCDYEAWPZWQZWOWRUVEEETUUEKIZUUGLIZKIZUUIFUVEU - UNUVLCDZNEUVMFGUVEUUNUVNUVHUVEUVKUUGUVETUUETCDUVEWSPUVGWOZUVJWRWTEUVLXASU - VEUUIEUVKKIZUUHLIUVMUVEUUFUVPUUHLUVEUUFETKIZUUEKIUVPUVEHUVQUUEKHUVQJUVETE - KIHUVQXCTEXBXDXEXFPVGUVEETUUEEODUVEXDPZTODUVEXBPYSUUEODUULYSUUEUVFXMVFXGX - HVGUVEEUVKUUGUVRUVEUVKUVOXMYSUULUUGODYSUULNUUGUVIXMWQXIXJXKWTXNSSVAQEYNUU - IXLXOXPXNXQXRXS $. + modmuladdim biimtrid mpan2 imp 3adant2 zcn 8cn adantl oveq1d eqeq2d simpr + mulcomd zmodcld nn0zd 3ad2ant1 eleq1 mpbird 2lgsoddprmlem1 syl3anc breq2d + id 3ad2ant3 2z nn0red resqcl peano2rem 8re cc0 wne 8pos gt0ne0ii redivcld + simp1 cn0 nn0z nnzi zsqcl zmulcld zmulcl ancoms zaddcld 4z dvdsmul1 4t2e8 + jca 4cn 2cn mulcomi eqtr3i zcnd mulassd eqtrd adddid breqtrrd dvdsaddre2b + eqtr4d ex mp3an2ani bitr4d sylbid rexlimdva mpd ) BCDZEBFGUBZABHUCIZJZUDZ + BUAUEZHKIZALIZJZUACUFZEBEUGIUHUIIHUJIZFGZEAEUGIZUHUIIZHUJIZFGZMZXTYCYIYAX + TYCYIXTHUKDZYCYIULHUMDZYQUNHUOUPYCYBAJXTYQNYIAYBUQBAUAHURUSUTVAVBYDYHYPUA + CYDYECDZNZYHBHYEKIZALIZJZYPYTYGUUBBYTYFUUAALYSYFUUAJYDYSYEHYEVCHODYSVDPVI + VEVFVGYTUUCYPYTUUCNZYKEHYEEUGIZKIZEYEAKIZKIZLIZYNLIZFGZYOUUDYJUUJEFUUDYSA + CDZUUCYJUUJJYTYSUUCYDYSVHQYTUULUUCYDUULYSYDUULYBCDZXTYAUUMYCXTYBXTBHXTVRY + RXTUNPVJVKVLYCXTUULUUMMYAAYBCVMVSVNQQYTUUCVHYEABVOVPVQECDZYTYNRDZUUCUUICD + ZEUUIFGZNZYOUUKMVTYDUUOYSYDARDZUUOYDUUSYBRDZYDYBYDBHXTYAYCWJYRYDUNPVJZWAY + CXTUUSUUTMYAAYBRVMVSVNUUSYMHUUSYLRDYMRDAWBYLWCSHRDUUSWDPHWEWFUUSHWDWGWHPW + ISQYTUURUUCYDYSUURYDAWKDZYSUURULZYDUVBYBWKDZUVAYCXTUVBUVDMYAAYBWKVMVSVNUV + BUULUVCAWLUULYSUURUULYSNZUUPUUQUVEUUFUUHUVEHUUEHCDUVEHUNWMPYSUUECDUULYEWN + ZVEZWOUVEEUUGUUNUVEVTPZYSUULUUGCDYEAWPZWQZWOWRUVEEETUUEKIZUUGLIZKIZUUIFUV + EUUNUVLCDZNEUVMFGUVEUUNUVNUVHUVEUVKUUGUVETUUETCDUVEWSPUVGWOZUVJWRXBEUVLWT + SUVEUUIEUVKKIZUUHLIUVMUVEUUFUVPUUHLUVEUUFETKIZUUEKIUVPUVEHUVQUUEKHUVQJUVE + TEKIHUVQXATEXCXDXEXFPVFUVEETUUEEODUVEXDPZTODUVEXCPYSUUEODUULYSUUEUVFXGVEX + HXIVFUVEEUVKUUGUVRUVEUVKUVOXGYSUULUUGODYSUULNUUGUVIXGWQXJXMXKXBXNSSVAQEYN + UUIXLXOXPXNXQXRXS $. $} $( Lemma 1 for ~ 2lgsoddprmlem3 . (Contributed by AV, 20-Jul-2021.) $) @@ -388998,20 +389987,20 @@ to the second component (see, for example, ~ 2sqreunnltb and cv cle clt 2sqreulem1 wne weq oveq1 oveq2d adantr nn0cn 2times eqcomd syl sqcld adantl ad2antrl eqtrd eqeq1d eleq1 anbi12d nn0z 2nn0 zexpcl sylancl cc cz 2mulprm oveq2 2t1e2 eqtrdi oveq1d cr crp cc0 2re cn nnrp ax-mp 0le2 - 4nn 2lt4 modid mp4an 1ne2 eqcom eqneqall com12 syl5bi syl6bi impcomd expd - com34 eqcoms com14 imp31 sylbid expimpd 2a1 pm2.61ine pm4.71d nn0re ltlen - syl2an bibi2d mpbird ex pm5.32rd reubidva mpbid ) ADEZAFGHZIJZKZBUAZCUAZU - BLZXNMNHZXOMNHZOHZAJZKZCPQZBPQXNXOUCLZXTKZCPQZBPQABCUDXMYBYEBPXMXNPEZKZYA - YDCPYGXOPEZKZXTXPYCYIXTXPYCRZYIXTKZYJXPXPXOXNUEZKZRZYKXPYLYKXPYLSZSXOXNCB - UFZYIXTYOYPYIKZXTMXQTHZAJZYOYQXSYRAYQXSXQXQOHZYRYPXSYTJYIYPXRXQXQOXOXNMNU - GUHUIYGYTYRJZYPYHYFUUAXMYFXQVEEZUUAYFXNXNUJUNUUBYRYTXQUKULUMUOUPUQURYGYSY - OSZYPYHXJXLYFUUCYSXLYFXJYOXLYFXJYOSSSAYRAYRJZXLXJYFYOUUDXLXJYFYOSZUUDXLXJ - KYRFGHZIJZYRDEZKZUUEUUDXLUUGXJUUHUUDXKUUFIAYRFGUGURAYRDUSUTYFUUIYOYFUUHUU - GYOYFUUHXQIJZUUGYOSYFXQVFEZUUHUUJRYFXNVFEMPEUUKXNVAVBXNMVCVDXQVGUMUUJUUGM - IJZYOUUJUUFMIUUJUUFMFGHZMUUJYRMFGUUJYRMITHMXQIMTVHVIVJVKMVLEFVMEZVNMUBLMF - UCLUUMMJVOFVPEUUNVTFVQVRVSWAMFWBWCVJURIMUEZUULYOSWDUULIMJZUUOYOMIWEUUPUUO - YOYOIMWFWGWHVRWIWIWJWGWIWKWLWMWNWOUPWPWQYLYKXPWRWSWTYIYJYNRXTYIYCYMXPYGXN - VLEZXOVLEYCYMRYHYFUUQXMXNXAUOXOXAXNXOXBXCXDUIXEXFXGXHXHXI $. + 4nn 2lt4 modid mp4an 1ne2 eqcom eqneqall com12 biimtrid syl6bi expd com34 + impcomd eqcoms com14 imp31 sylbid expimpd pm2.61ine pm4.71d syl2an bibi2d + 2a1 nn0re ltlen mpbird ex pm5.32rd reubidva mpbid ) ADEZAFGHZIJZKZBUAZCUA + ZUBLZXNMNHZXOMNHZOHZAJZKZCPQZBPQXNXOUCLZXTKZCPQZBPQABCUDXMYBYEBPXMXNPEZKZ + YAYDCPYGXOPEZKZXTXPYCYIXTXPYCRZYIXTKZYJXPXPXOXNUEZKZRZYKXPYLYKXPYLSZSXOXN + CBUFZYIXTYOYPYIKZXTMXQTHZAJZYOYQXSYRAYQXSXQXQOHZYRYPXSYTJYIYPXRXQXQOXOXNM + NUGUHUIYGYTYRJZYPYHYFUUAXMYFXQVEEZUUAYFXNXNUJUNUUBYRYTXQUKULUMUOUPUQURYGY + SYOSZYPYHXJXLYFUUCYSXLYFXJYOXLYFXJYOSSSAYRAYRJZXLXJYFYOUUDXLXJYFYOSZUUDXL + XJKYRFGHZIJZYRDEZKZUUEUUDXLUUGXJUUHUUDXKUUFIAYRFGUGURAYRDUSUTYFUUIYOYFUUH + UUGYOYFUUHXQIJZUUGYOSYFXQVFEZUUHUUJRYFXNVFEMPEUUKXNVAVBXNMVCVDXQVGUMUUJUU + GMIJZYOUUJUUFMIUUJUUFMFGHZMUUJYRMFGUUJYRMITHMXQIMTVHVIVJVKMVLEFVMEZVNMUBL + MFUCLUUMMJVOFVPEUUNVTFVQVRVSWAMFWBWCVJURIMUEZUULYOSWDUULIMJZUUOYOMIWEUUPU + UOYOYOIMWFWGWHVRWIWIWLWGWIWJWKWMWNWOUPWPWQYLYKXPXBWRWSYIYJYNRXTYIYCYMXPYG + XNVLEZXOVLEYCYMRYHYFUUQXMXNXCUOXOXCXNXOXDWTXAUIXEXFXGXHXHXI $. $( Lemma for ~ 2sqreultb . (Contributed by AV, 10-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 18-Jun-2023.) $) @@ -389092,20 +390081,20 @@ to the second component (see, for example, ~ 2sqreunnltb and cv cle clt 2sqreunnlem1 wne oveq1 oveq2d adantr cc nncn 2times eqcomd syl sqcld adantl ad2antrl eqtrd eqeq1d anbi12d cz cn0 nnz 2nn0 zexpcl sylancl eleq1 2mulprm oveq2 2t1e2 eqtrdi oveq1d cr crp cc0 2re 4nn nnrp 0le2 2lt4 - ax-mp modid mp4an eqcom eqneqall com12 syl5bi syl6bi impcomd com34 eqcoms - 1ne2 expd com14 imp31 sylbid expimpd pm2.61ine pm4.71d nnre syl2an bibi2d - 2a1 ltlen mpbird ex pm5.32rd reubidva mpbid ) ADEZAFGHZIJZKZBUAZCUAZUBLZX - MMNHZXNMNHZOHZAJZKZCPQZBPQXMXNUCLZXSKZCPQZBPQABCUDXLYAYDBPXLXMPEZKZXTYCCP - YFXNPEZKZXSXOYBYHXSXOYBRZYHXSKZYIXOXOXNXMUEZKZRZYJXOYKYJXOYKSZSXNXMXNXMJZ - YHXSYNYOYHKZXSMXPTHZAJZYNYPXRYQAYPXRXPXPOHZYQYOXRYSJYHYOXQXPXPOXNXMMNUFUG - UHYFYSYQJZYOYGYEYTXLYEXPUIEZYTYEXMXMUJUNUUAYQYSXPUKULUMUOUPUQURYFYRYNSZYO - YGXIXKYEUUBYRXKYEXIYNXKYEXIYNSSSAYQAYQJZXKXIYEYNUUCXKXIYEYNSZUUCXKXIKYQFG - HZIJZYQDEZKZUUDUUCXKUUFXIUUGUUCXJUUEIAYQFGUFURAYQDVFUSYEUUHYNYEUUGUUFYNYE - UUGXPIJZUUFYNSYEXPUTEZUUGUUIRYEXMUTEMVAEUUJXMVBVCXMMVDVEXPVGUMUUIUUFMIJZY - NUUIUUEMIUUIUUEMFGHZMUUIYQMFGUUIYQMITHMXPIMTVHVIVJVKMVLEFVMEZVNMUBLMFUCLU - ULMJVOFPEUUMVPFVQVTVRVSMFWAWBVJURIMUEZUUKYNSWKUUKIMJZUUNYNMIWCUUOUUNYNYNI - MWDWEWFVTWGWGWHWEWGWLWIWJWMWNUPWOWPYKYJXOXBWQWRYHYIYMRXSYHYBYLXOYFXMVLEZX - NVLEYBYLRYGYEUUPXLXMWSUOXNWSXMXNXCWTXAUHXDXEXFXGXGXH $. + ax-mp modid mp4an 1ne2 eqcom eqneqall com12 biimtrid syl6bi impcomd com34 + expd eqcoms com14 imp31 sylbid expimpd 2a1 pm2.61ine pm4.71d ltlen syl2an + nnre bibi2d mpbird ex pm5.32rd reubidva mpbid ) ADEZAFGHZIJZKZBUAZCUAZUBL + ZXMMNHZXNMNHZOHZAJZKZCPQZBPQXMXNUCLZXSKZCPQZBPQABCUDXLYAYDBPXLXMPEZKZXTYC + CPYFXNPEZKZXSXOYBYHXSXOYBRZYHXSKZYIXOXOXNXMUEZKZRZYJXOYKYJXOYKSZSXNXMXNXM + JZYHXSYNYOYHKZXSMXPTHZAJZYNYPXRYQAYPXRXPXPOHZYQYOXRYSJYHYOXQXPXPOXNXMMNUF + UGUHYFYSYQJZYOYGYEYTXLYEXPUIEZYTYEXMXMUJUNUUAYQYSXPUKULUMUOUPUQURYFYRYNSZ + YOYGXIXKYEUUBYRXKYEXIYNXKYEXIYNSSSAYQAYQJZXKXIYEYNUUCXKXIYEYNSZUUCXKXIKYQ + FGHZIJZYQDEZKZUUDUUCXKUUFXIUUGUUCXJUUEIAYQFGUFURAYQDVFUSYEUUHYNYEUUGUUFYN + YEUUGXPIJZUUFYNSYEXPUTEZUUGUUIRYEXMUTEMVAEUUJXMVBVCXMMVDVEXPVGUMUUIUUFMIJ + ZYNUUIUUEMIUUIUUEMFGHZMUUIYQMFGUUIYQMITHMXPIMTVHVIVJVKMVLEFVMEZVNMUBLMFUC + LUULMJVOFPEUUMVPFVQVTVRVSMFWAWBVJURIMUEZUUKYNSWCUUKIMJZUUNYNMIWDUUOUUNYNY + NIMWEWFWGVTWHWHWIWFWHWKWJWLWMWNUPWOWPYKYJXOWQWRWSYHYIYMRXSYHYBYLXOYFXMVLE + ZXNVLEYBYLRYGYEUUPXLXMXBUOXNXBXMXNWTXAXCUHXDXEXFXGXGXH $. $( Lemma for ~ 2sqreunnltb . (Contributed by AV, 11-Jun-2023.) The prime needs not be odd, as observed by WL. (Revised by AV, 18-Jun-2023.) $) @@ -389138,11 +390127,11 @@ to the second component (see, for example, ~ 2sqreunnltb and 2sqreulem3 $p |- ( ( A e. NN0 /\ ( B e. NN0 /\ C e. NN0 ) ) -> ( ( ( ph /\ ( ( A ^ 2 ) + ( B ^ 2 ) ) = P ) /\ ( ps /\ ( ( A ^ 2 ) + ( C ^ 2 ) ) = P ) ) -> B = C ) ) $= - ( cn0 wcel c2 cexp co caddc wceq wa wi w3a wb eqeq2 eqcoms adantld eqcom ex - adantl 2sqreulem2 syl5bi adantr sylbid impd 3expb ) CGHZDGHZEGHZACIJKZDIJKL - KZFMZNZBUMEIJKLKZFMZNZNDEMZOUJUKULPZUPUSUTVAUOUSUTOZAVAUOVBVAUONZURUTBVCURU - QUNMZUTUOURVDQZVAVEFUNFUNUQRSUCVAVDUTOUOVDUNUQMVAUTUQUNUACDEUDUEUFUGTUBTUHU - I $. + ( cn0 wcel c2 cexp co caddc wceq wa wi w3a wb eqeq2 eqcoms adantld biimtrid + adantl eqcom 2sqreulem2 adantr sylbid ex impd 3expb ) CGHZDGHZEGHZACIJKZDIJ + KLKZFMZNZBUMEIJKLKZFMZNZNDEMZOUJUKULPZUPUSUTVAUOUSUTOZAVAUOVBVAUONZURUTBVCU + RUQUNMZUTUOURVDQZVAVEFUNFUNUQRSUBVAVDUTOUOVDUNUQMVAUTUQUNUCCDEUDUAUEUFTUGTU + HUI $. ${ $d P b c $. $d a b c $. $d ps c $. @@ -391480,56 +392469,56 @@ a multiplicative function (but not completely multiplicative). wceq syl eqbrtrrd cexp rprege0d sqrtsq eqeltrd oveq1d sumeq2dv oveq2d fveq2 rpred fznnfl simplbda syl2anc mpbir2and oveq1 eqeltrrd eqtrd wn sqsqrtd ad2antrr letrd nnrecred logsqrt rpsqrtcl harmoniclbnd cif crn - abscld wrex eqid ovex elrnmpti eleq1d syl5ibrcom rexlimdva syl5bi imp - iftrued wf1o nnsqcld le2sq breqtrd ex wi sq11 dom2lem f1f1orn fvmpt3i - wf1 f1f frn 3syl sselda 1re ifcli rerpdivcl syldan fsumf1o cdif eldif - nncnd sqrtle elrnmpt1s expr con3d impr sylan2b iffalsed eldifi sylan2 - 0re rpcnne0d div0 fsumss 3eqtr3d dchrisum0flb lediv1dd fsumle eqbrtrd - cbs leabsd o1le o1mul2 mto ) ABUGBUHZUIUJZUKZULUMBUNABUGUOUXEUOUPUQZV - AUQZUKUXFULABUGUXHUXEAUXDUGUMZURZUXEUOUXJUXEUXIUXEUSUMZAUXDUTZVBZVMUX - JVCZUOVDVEUXJVFVGVHVIABUGUOUXGVJUXNUXJUXGUXJUXEUXMVKZVMZBUGUOUKULUMZA - UGUSVLUOVJUMUXQVNVOBUGUOVPVQVGABUGVRUXDVSUJZVTUQZFUHZGUJZUXTWAUJZUPUQ - ZFWBZUXGVRWCAWGUDUYDWCUMUXJUXSUYCFWDVGUXPAUXIVRUXDWEWFZURZURZUXGWHUJU - XGUYDWHUJZWEUYGUXGAUXIUXGUSUMUYEUXOWIZUYGUXKVDUXEWEWFUOUSUMZVDUOWJWFZ - VDUXGWEWFUXIUXKAUYEUXLWKUYGVDVRUIUJZUXEWEWLUYGUYEUYLUXEWEWFZAUXIUYEWM - UYGVRUGUMUXIUYEUYMWNWOAUXIUYEWPZVRUXDWQWSWRWTUYJUYGXIVGUYKUYGXAVGUXEU - OXBXCXDUYGUXGUYDUYHUYIUYGUXSUYCFUYGVRUXRXEZUYGUXTUXSUMZURZUYAUYBUYGXF - 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NN , X ( n ) / n ` is nonzero for all non-principal Dirichlet characters (i.e. the assumption ` X e. W ` is - contradictory). This is the key result that allows us to eliminate - the conditionals from ~ dchrmusum2 and ~ dchrvmasumif . Lemma 9.4.4 - of [Shapiro], p. 382. (Contributed by Mario Carneiro, - 12-May-2016.) $) + contradictory). This is the key result that allows to eliminate the + conditionals from ~ dchrmusum2 and ~ dchrvmasumif . Lemma 9.4.4 of + [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.) $) dchrisum0 $p |- -. ph $= ( co vx vk vb vi vz vd vc vt va cn cdvds wbr crab cfv csu cmpt eqid csn cv cdif cdiv cc0 wceq ssrab3 difss sstri sselid dchrisum0re crp cfl cfz @@ -392227,9 +393215,9 @@ a multiplicative function (but not completely multiplicative). ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - T ) ) <_ ( C / y ) ) $. $( The sum ` sum_ n e. NN , X ( n ) / n ` is nonzero for all non-principal Dirichlet characters (i.e. the assumption ` X e. W ` - is contradictory). This is the key result that allows us to - eliminate the conditionals from ~ dchrmusum2 and ~ dchrvmasumif . - Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, + is contradictory). This is the key result that allows to eliminate + the conditionals from ~ dchrmusum2 and ~ dchrvmasumif . Lemma 9.4.4 + of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.) $) dchrisumn0 $p |- ( ph -> T =/= 0 ) $= ( vm cc0 wceq wa cn cv cfv cdiv co csu csn cdif crab wcel adantr eqid @@ -396727,73 +397715,73 @@ a multiplicative function (but not completely multiplicative). breq2d jad nrexdv pm2.65da bitrdi rpgt0 ltletr rexanali cbvralvw simprl simplr pntrval oveq1d chpcl rpcn rpne0 divsubdird dividd 3eqtrrd simprr mpbir2and simplrl sseldd simp-4r rerpdivcl mpancom resubcl lelttr exp32 - embantd ralimdv2 syl5bi reximdva anassrs impancom expimpd rexlimdva mpd - com24 ssrexv mpsyl rgen 1cnd rlim2 ) ABUDBUEZUFUGZVUNUHRZUIUOUJSCUEZVUN - UKSZVUPUOULRZUMUGZUAUEZUNSZUPZBUDUQZCURUSZUAUDUQAVVEUAUDUDURUTZAVVAUDTZ - VAZVVDCUDUSZVVEVBVVHDUEZIUGZVVJUHRZUMUGZFUEZUKSZDVUQVCVDRZUQZCUDUSZVVAV - VNUKSZVEZVAZFVLGVFRZUSZVVIVVHVVRVVSUPFVWBUQZVEVWCVVHVWDVLJURUNVGZUNSZAV - WFVEVVGAVWFEUEZVWEULRZUMUGZVVAUNSZVWGUBVHUBUEZHVWKVIVJRZVKRZULRZUIZUGZV - WEVWOUGZULRZUMUGZHVWEVIVJRZVKRZUNSZUPZEVHUQZUAUDUSZAVWFVAZVWOVHVHVMRZTV - 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(Contributed by Mario Carneiro, @@ -397810,1203 +398798,5073 @@ a multiplicative function (but not completely multiplicative). $) $( End $[ set-numfunc.mm $] $) - -$( Begin $[ set-tarskigeom.mm $] $) +$( Begin $[ set-surreals.mm $] $) $( ############################################################################### - ELEMENTARY GEOMETRY + SURREAL NUMBERS ############################################################################### - - This part develops elementary geometry based on Tarski's axioms, following - [Schwabhauser]. Tarski's geometry is a first-order theory with one sort, the - "points". It has two primitive notions, the ternary predicate of - "betweenness" and the quaternary predicate of "congruence". To adapt this - theory to the framework of set.mm, and to be able to talk of *a* Tarski - structure as a space satisfying the given axioms, we use the following - definition, stated informally: - - A Tarski structure ` f ` is a set (of points) ` ( Base `` f ) ` together with - functions ` ( Itv `` f ) ` and ` ( dist `` f ) ` on - ` ( ( Base `` f ) X. ( Base `` f ) ) ` satisfying certain axioms (given in - Definitions ~ df-trkg _et sequentes_). This allows us to treat a Tarski - structure as a special kind of extensible structure (see ~ df-struct ). - - The translation to and from Tarski's treatment is as follows (given, again, - informally). - - Suppose that one is given an extensible structure ` f ` . One defines a - betweenness ternary predicate Btw by positing that, for any - ` x , y , z e. ( Base `` f ) ` , one has "Btw ` x y z ` " if and only if - ` y e. x ( Itv `` f ) z ` , and a congruence quaternary predicate Congr by - positing that, for any ` x , y , z , t e. ( Base `` f ) ` , one has - "Congr ` x y z t ` " if and only if - ` x ( dist `` f ) y = z ( dist `` f ) t ` . - It is easy to check that if ` f ` satisfies our Tarski axioms, then Btw and - Congr satisfy Tarski's Tarski axioms when ` ( Base `` f ) ` is interpreted as - the universe of discourse. - - Conversely, suppose that one is given a set ` a ` , a ternary predicate Btw, - and a quaternary predicate Congr. One defines the extensible structure ` f ` - such that ` ( Base `` f ) ` is ` a ` , and ` ( Itv `` f ) ` is the function - which associates with each ` <. x , y >. e. ( a X. a ) ` the set of points - ` z e. a ` such that "Btw ` x z y ` ", and ` ( dist `` f ) ` is the function - which associates with each ` <. x , y >. e. ( a X. a ) ` the set of ordered - pairs ` <. z , t >. e. ( a X. a ) ` such that "Congr ` x y z t ` ". It is - easy to check that if Btw and Congr satisfy Tarski's Tarski axioms when ` a ` - is interpreted as the universe of discourse, then ` f ` satisfies our Tarski - axioms. - - We intentionally choose to represent congruence (without loss of generality) - as ` x ( dist `` f ) y = z ( dist `` f ) t ` instead of "Congr ` x y z t ` ", - as it is more convenient. It is always possible to define ` dist ` for any - particular geometry to produce equal results when conguence is desired, and - in many cases there is an obvious interpretation of "distance" between - two points that can be useful in other situations. Encoding congruence as - an equality of distances makes it easier to use these theorems in cases where - there is a preferred distance function. We prove that representing a - congruence relationship using a distance in the form - ` x ( dist `` f ) y = z ( dist `` f ) t ` causes no loss of generality in - ~ tgjustc1 and ~ tgjustc2 , which in turn are supported by - ~ tgjustf and ~ tgjustr . - - A similar representation of congruence (using a "distance" function) - is used in Axiom A1 of [Beeson2016] p. 5, which discusses how a large number - of formalized proofs were found in Tarskian Geometry using OTTER. Their - detailed proofs in Tarski Geometry, along with other information, are - available at ~ https://www.michaelbeeson.com/research/FormalTarski/ . - - Most theorems are in deduction form, as this is a very general, simple, and - convenient format to use in Metamath. An assertion in deduction form can be - easily converted into an assertion in inference form (removing the - antecedents ` ph -> ` ) by insert a ` T. -> ` in each hypothesis, using - ~ a1i , then using ~ mptru to remove the final ` T. -> ` prefix. In some - cases we represent, without loss of generality, an implication antecedent in - [Schwabhauser] as a hypothesis. The implication can be retrieved from the - by using ~ simpr , the theorem as stated, and ~ ex . - - For descriptions of individual axioms, we refer to the specific definitions - below. A particular feature of Tarski's axioms is modularity, so by using - various subsets of the set of axioms, we can define the classes of "absolute - dimensionless Tarski structures" ( ~ df-trkg ), of "Euclidean dimensionless - Tarski structures" ( ~ df-trkge ) and of "Tarski structures of dimension no - less than N" ( ~ df-trkgld ). - - In this system, angles are not a primitive notion, but instead a derived - notion (see ~ df-cgra and ~ iscgra ). To maintain its simplicity, in this - system congruence between shapes (a finite sequence of points) is the case - where corresponding segments between all corresponding points are congruent. - This includes triangles (a shape of 3 distinct points). Note that this - definition has no direct regard for angles. For more details and rationale, - see ~ df-cgrg . - - The first section is devoted to the definitions of these various structures. - The second section ("Tarskian geometry") develops the synthetic treatment of - geometry. The remaining sections prove that real Euclidean spaces and - complex Hilbert spaces, with intended interpretations, are Euclidean Tarski - structures. - - Most of the work in this part is due to Thierry Arnoux, with earlier work by - Mario Carneiro and Scott Fenton. See also the credits in the comment of each - statement. - $) $( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Definition and Tarski's Axioms of Geometry -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Definitions and initial properties +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $c TarskiG Itv LineG $. - $c TarskiGC TarskiGB TarskiGCB TarskiGE TarskiGDim>= $. - - $( Extends class notation with the class of Tarski geometries. $) - cstrkg $a class TarskiG $. - $( Extends class notation with the class of geometries fulfilling the - congruence axioms. $) - cstrkgc $a class TarskiGC $. - $( Extends class notation with the class of geometries fulfilling the - betweenness axioms. $) - cstrkgb $a class TarskiGB $. - $( Extends class notation with the class of geometries fulfilling the - congruence and betweenness axioms. $) - cstrkgcb $a class TarskiGCB $. - $( Extends class notation with the relation for geometries fulfilling the - lower dimension axioms. $) - cstrkgld $a class TarskiGDim>= $. - $( Extends class notation with the class of geometries fulfilling Euclid's - axiom. $) - cstrkge $a class TarskiGE $. - - $( Declare the syntax for the Interval (segment) index extractor. $) - citv $a class Itv $. - - $( Declare the syntax for the Line function. $) - clng $a class LineG $. - - $( Define the Interval (segment) index extractor for Tarski geometries. - (Contributed by Thierry Arnoux, 24-Aug-2017.) Use its index-independent - form ~ itvid instead. (New usage is discouraged.) $) - df-itv $a |- Itv = Slot ; 1 6 $. - - $( Define the line index extractor for geometries. (Contributed by Thierry - Arnoux, 27-Mar-2019.) Use its index-independent form ~ lngid instead. - (New usage is discouraged.) $) - df-lng $a |- LineG = Slot ; 1 7 $. - - $( Index value of the Interval (segment) slot. Use ~ ndxarg . (Contributed - by Thierry Arnoux, 24-Aug-2017.) (New usage is discouraged.) $) - itvndx $p |- ( Itv ` ndx ) = ; 1 6 $= - ( citv c1 c6 cdc df-itv 1nn0 6nn decnncl ndxarg ) ABCDEBCFGHI $. - - $( Index value of the "line" slot. Use ~ ndxarg . (Contributed by Thierry - Arnoux, 27-Mar-2019.) (New usage is discouraged.) $) - lngndx $p |- ( LineG ` ndx ) = ; 1 7 $= - ( clng c1 c7 cdc df-lng 1nn0 7nn decnncl ndxarg ) ABCDEBCFGHI $. + $( Set up the new syntax for surreal numbers. $) + $c No { 1o , 2o } } $. + $} ${ - trkgstr.w $e |- W = { <. ( Base ` ndx ) , U >. , - <. ( dist ` ndx ) , D >. , <. ( Itv ` ndx ) , I >. } $. - $( Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, - 24-Aug-2017.) $) - trkgstr $p |- W Struct <. 1 , ; 1 6 >. $= - ( cnx cbs cfv cop cds citv c1 c6 cdc c2 1nn 2nn0 1nn0 decnncl 6nn basendx - ctp cstr 1lt10 declti 2nn dsndx 2lt6 declt itvndx strle3 eqbrtri ) DFGHZB - IFJHZAIFKHZCIUBLLMNZIUCEUMUNUOLLONUPBACPUALOLPQRUDUELORUFSUGLOMRQTUHUILMR - TSUJUKUL $. + $d f g x y $. + $( Next, we introduce surreal less-than, a comparison relation over the + surreals by lexicographically ordering them. (Contributed by Scott + Fenton, 9-Jun-2011.) $) + df-slt $a |- . | ( ( f e. No /\ g e. No ) /\ + E. x e. On ( + A. y e. x ( f ` y ) = ( g ` y ) /\ + ( f ` x ) { <. 1o , (/) >. , + <. 1o , 2o >. , + <. (/) , 2o >. } ( g ` x ) ) ) } $. + $} - $( The base set of a Tarski geometry. (Contributed by Thierry Arnoux, - 24-Aug-2017.) $) - trkgbas $p |- ( U e. V -> U = ( Base ` W ) ) $= - ( cbs c1 c6 cdc cop trkgstr baseid cnx cfv csn cds citv ctp snsstp1 strfv - sseqtrri ) BEGDHHIJKABCEFLMNGOBKZPUCNQOAKZNROCKZSEUCUDUETFUBUA $. + $( Finally, we introduce the birthday function. This function maps each + surreal to an ordinal. In our implementation, this is the domain of the + sign function. The important properties of this function are established + later. (Contributed by Scott Fenton, 11-Jun-2011.) $) + df-bday $a |- bday = ( x e. No |-> dom x ) $. - $( The measure of a distance in a Tarski geometry. (Contributed by Thierry - Arnoux, 24-Aug-2017.) $) - trkgdist $p |- ( D e. V -> D = ( dist ` W ) ) $= - ( cds c1 c6 cdc cop trkgstr dsid cnx cfv csn cbs citv ctp snsstp2 strfv - sseqtrri ) AEGDHHIJKABCEFLMNGOAKZPNQOBKZUCNROCKZSEUDUCUETFUBUA $. + ${ + $d A f x $. + $( Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). + (Contributed by Scott Fenton, 11-Jun-2011.) $) + elno $p |- ( A e. No <-> E. x e. On A : x --> { 1o , 2o } ) $= + ( vf csur wcel cvv cv c1o c2o cpr wf con0 wrex elex ancoms rexlimiva wceq + fex feq1 rexbidv df-no elab2g pm5.21nii ) BDEBFEZAGZHIJZBKZALMZBDNUGUDALU + GUELEUDUEUFLBROPUEUFCGZKZALMUHCBDFUIBQUJUGALUEUFUIBSTCAUAUBUC $. + $} - $( The congruence relation in a Tarski geometry. (Contributed by Thierry - Arnoux, 24-Aug-2017.) $) - trkgitv $p |- ( I e. V -> I = ( Itv ` W ) ) $= - ( citv c1 c6 cdc cop trkgstr itvid cnx cfv csn cbs cds ctp snsstp3 strfv - sseqtrri ) CEGDHHIJKABCEFLMNGOCKZPNQOBKZNROAKZUCSEUDUEUCTFUBUA $. + ${ + $d A f g x y $. $d B f g x y $. + $( The value of the surreal less-than relation. (Contributed by Scott + Fenton, 14-Jun-2011.) $) + sltval $p |- ( ( A e. No /\ B e. No ) -> + ( A E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ + ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , + <. (/) , 2o >. } ( B ` x ) ) ) ) $= + ( vf vg csur wcel wa cslt wbr cfv wceq wral cop con0 wrex fveq1 anbi12d + cv c1o c0 c2o ctp eleq1 anbi1d eqeq1d ralbidv breq1d anbi2d eqeq2d breq2d + rexbidv df-slt brabg bianabs ) CGHZDGHZIZCDJKBTZCLZUTDLZMZBATZNZVDCLZVDDL + ZUAUBOUAUCOUBUCOUDZKZIZAPQZETZGHZFTZGHZIZUTVLLZUTVNLZMZBVDNZVDVLLZVDVNLZV + HKZIZAPQZIUQVOIZVAVRMZBVDNZVFWBVHKZIZAPQZIUSVKIEFCDGGJVLCMZVPWFWEWKWLVMUQ + VOVLCGUEUFWLWDWJAPWLVTWHWCWIWLVSWGBVDWLVQVAVRUTVLCRUGUHWLWAVFWBVHVDVLCRUI + SUMSVNDMZWFUSWKVKWMVOURUQVNDGUEUJWMWJVJAPWMWHVEWIVIWMWGVCBVDWMVRVBVAUTVND + RUKUHWMWBVGVFVHVDVNDRULSUMSABEFUNUOUP $. $} ${ - $d a b c d f g n p i j s t u v x y z $. - $( Define the class of geometries fulfilling the congruence axioms of - reflexivity, identity and transitivity. These are axioms A1 to A3 of - [Schwabhauser] p. 10. With our distance based notation for congruence, - transitivity of congruence boils down to transitivity of equality and is - already given by ~ eqtr , so it is not listed in this definition. - (Contributed by Thierry Arnoux, 24-Aug-2017.) $) - df-trkgc $a |- TarskiGC = { f | [. ( Base ` f ) / p ]. - [. ( dist ` f ) / d ]. - ( A. x e. p A. y e. p ( x d y ) = ( y d x ) - /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) ) } $. + $d A x $. + $( The value of the birthday function within the surreals. (Contributed by + Scott Fenton, 14-Jun-2011.) $) + bdayval $p |- ( A e. No -> ( bday ` A ) = dom A ) $= + ( vx csur wcel cdm cvv cbday cfv wceq dmexg cv dmeq df-bday fvmptg mpdan + ) ACDAEZFDAGHPIACJBABKZEPCFGQALBMNO $. + $} - $( Define the class of geometries fulfilling the 3 betweenness axioms in - Tarski's Axiomatization of Geometry: identity, Axiom A6 of - [Schwabhauser] p. 11, axiom of Pasch, Axiom A7 of [Schwabhauser] p. 12, - and continuity, Axiom A11 of [Schwabhauser] p. 13. (Contributed by - Thierry Arnoux, 24-Aug-2017.) $) - df-trkgb $a |- TarskiGB = { f | [. ( Base ` f ) / p ]. - [. ( Itv ` f ) / i ]. - ( A. x e. p A. y e. p ( y e. ( x i x ) -> x = y ) - /\ A. x e. p A. y e. p A. z e. p A. u e. p A. v e. p - ( ( u e. ( x i z ) /\ v e. ( y i z ) ) - -> E. a e. p ( a e. ( u i y ) /\ a e. ( v i x ) ) ) - /\ A. s e. ~P p A. t e. ~P p - ( E. a e. p A. x e. s A. y e. t x e. ( a i y ) - -> E. b e. p A. x e. s A. y e. t b e. ( x i y ) ) ) - } $. + ${ + $d A x $. + $( A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.) $) + nofun $p |- ( A e. No -> Fun A ) $= + ( vx csur wcel cv c1o c2o cpr wf con0 wrex wfun elno ffun rexlimivw sylbi + ) ACDBEZFGHZAIZBJKALZBAMSTBJQRANOP $. - $( Define the class of geometries fulfilling the five segment axiom, Axiom - A5 of [Schwabhauser] p. 11, and segment construction axiom, Axiom A4 of - [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 14-Mar-2019.) $) - df-trkgcb $a |- TarskiGCB = { f | [. ( Base ` f ) / p ]. - [. ( dist ` f ) / d ]. [. ( Itv ` f ) / i ]. - ( A. x e. p A. y e. p A. z e. p A. u e. p - A. a e. p A. b e. p A. c e. p A. v e. p - ( ( ( x =/= y /\ y e. ( x i z ) /\ b e. ( a i c ) ) - /\ ( ( ( x d y ) = ( a d b ) /\ ( y d z ) = ( b d c ) ) - /\ ( ( x d u ) = ( a d v ) /\ ( y d u ) = ( b d v ) ) ) ) - -> ( z d u ) = ( c d v ) ) - /\ A. x e. p A. y e. p A. a e. p A. b e. p E. z e. p - ( y e. ( x i z ) /\ ( y d z ) = ( a d b ) ) ) - } $. + $( The domain of a surreal is an ordinal. (Contributed by Scott Fenton, + 16-Jun-2011.) $) + nodmon $p |- ( A e. No -> dom A e. On ) $= + ( vx csur wcel cv c1o c2o cpr wf con0 wrex cdm elno fdm biimprcd rexlimiv + eleq1d sylbi ) ACDBEZFGHZAIZBJKALZJDZBAMUAUCBJUAUCSJDUAUBSJSTANQOPR $. - $( Define the class of geometries fulfilling Euclid's axiom, Axiom A10 of - [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 14-Mar-2019.) $) - df-trkge $a |- TarskiGE = { f | [. ( Base ` f ) / p ]. - [. ( Itv ` f ) / i ]. - A. x e. p A. y e. p A. z e. p A. u e. p A. v e. p - ( ( u e. ( x i v ) /\ u e. ( y i z ) /\ x =/= u ) -> E. a e. p - E. b e. p ( y e. ( x i a ) /\ z e. ( x i b ) /\ v e. ( a i b ) ) ) - } $. + $( The range of a surreal is a subset of the surreal signs. (Contributed + by Scott Fenton, 16-Jun-2011.) $) + norn $p |- ( A e. No -> ran A C_ { 1o , 2o } ) $= + ( vx csur wcel cv c1o c2o cpr con0 wrex crn wss elno frn rexlimivw sylbi + wf ) ACDBEZFGHZAQZBIJAKSLZBAMTUABIRSANOP $. + $} - $( Define the class of geometries fulfilling the lower dimension axiom for - dimension ` n ` . For such geometries, there are three non-colinear - points that are equidistant from ` n - 1 ` distinct points. Derived - from remarks in _Tarski's System of Geometry_, Alfred Tarski and Steven - Givant, Bulletin of Symbolic Logic, Volume 5, Number 2 (1999), 175-214. - (Contributed by Scott Fenton, 22-Apr-2013.) (Revised by Thierry Arnoux, - 23-Nov-2019.) $) - df-trkgld $a |- TarskiGDim>= = { <. g , n >. | - [. ( Base ` g ) / p ]. [. ( dist ` g ) / d ]. [. ( Itv ` g ) / i ]. - E. f ( f : ( 1 ..^ n ) -1-1-> p /\ E. x e. p E. y e. p E. z e. p - ( A. j e. ( 2 ..^ n ) ( ( ( f ` 1 ) d x ) = ( ( f ` j ) d x ) /\ - ( ( f ` 1 ) d y ) = ( ( f ` j ) d y ) /\ - ( ( f ` 1 ) d z ) = ( ( f ` j ) d z ) ) - /\ -. ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) ) ) - } $. + $( A surreal is a function over its birthday. (Contributed by Scott Fenton, + 16-Jun-2011.) $) + nofnbday $p |- ( A e. No -> A Fn ( bday ` A ) ) $= + ( csur wcel wfun cdm cbday cfv wceq wfn nofun bdayval eqcomd df-fn sylanbrc + ) ABCZADAEZAFGZHAQIAJOQPAKLAQMN $. - $( Define the class of Tarski geometries. A Tarski geometry is a set of - points, equipped with a betweenness relation (denoting that a point lies - on a line segment between two other points) and a congruence relation - (denoting equality of line segment lengths). + $( The domain of a surreal has the ordinal property. (Contributed by Scott + Fenton, 16-Jun-2011.) $) + nodmord $p |- ( A e. No -> Ord dom A ) $= + ( csur wcel cdm con0 word nodmon eloni syl ) ABCADZECJFAGJHI $. - Here, we are using the following:
      -
    • for congruence, ` ( x .- y ) = ( z .- w ) ` where - ` .- = ( dist `` W ) ` -
    • for betweenness, ` y e. ( x I z ) ` , where ` I = ( Itv `` W ) ` -
    - With this definition, the axiom A2 is actually equivalent to the - transitivity of equality, ~ eqtrd .

    + ${ + $d x A $. + $( An alternative condition for membership in ` No ` . (Contributed by + Scott Fenton, 21-Mar-2012.) $) + elno2 $p |- ( A e. No <-> + ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) ) $= + ( vx csur wcel wfun cdm con0 crn c1o c2o cpr wss w3a nofun nodmon norn wf + 3jca wa sylibr wrex simp2 wfn simpl funfnd anim1i 3impa df-f feq2 syl2anc + cv rspcev elno impbii ) ACDZAEZAFZGDZAHIJKZLZMZUOUPURUTANAOAPRVABUKZUSAQZ + BGUAZUOVAURUQUSAQZVDUPURUTUBVAAUQUCZUTSZVEUPURUTVGUPURSZVFUTVHAUPURUDUEUF + UGUQUSAUHTVCVEBUQGVBUQUSAUIULUJBAUMTUN $. + $} - Tarski originally had more axioms, but later reduced his list to 11:

      -
    • A1 A kind of reflexivity for the congruence relation - ( ` TarskiGC ` ) -
    • A2 Transitivity for the congruence relation ( ` TarskiGC ` ) -
    • A3 Identity for the congruence relation ( ` TarskiGC ` ) -
    • A4 Axiom of segment construction ( ` TarskiGCB ` ) -
    • A5 5-segment axiom ( ` TarskiGCB ` ) -
    • A6 Identity for the betweenness relation ( ` TarskiGB ` ) -
    • A7 Axiom of Pasch ( ` TarskiGB ` ) -
    • A8 Lower dimension axiom ` ( ``' TarskiGDim>= " { 2 } ) ` -
    • A9 Upper dimension axiom ` ( _V \ ( ``' TarskiGDim>= " { 3 } ) ) ` -
    • A10 Euclid's axiom ( ` TarskiGE ` ) -
    • A11 Axiom of continuity ( ` TarskiGB ` ) -
    + $( Another condition for membership in ` No ` . (Contributed by Scott + Fenton, 14-Apr-2012.) $) + elno3 $p |- ( A e. No <-> ( A : dom A --> { 1o , 2o } /\ dom A e. On ) ) $= + ( wfun cdm con0 wcel crn c1o c2o cpr wss w3a wa csur 3anan32 elno2 wfn df-f + wf funfn anbi1i bitr4i 3bitr4i ) ABZACZDEZAFGHIZJZKUCUGLZUELAMEUDUFARZUELUC + UEUGNAOUIUHUEUIAUDPZUGLUHUDUFAQUCUJUGASTUATUB $. - Our definition is split into 5 parts:
      -
    • congruence axioms ` TarskiGC ` (which metric spaces fulfill) -
    • betweenness axioms ` TarskiGB ` -
    • congruence and betweenness axioms ` TarskiGCB ` -
    • upper and lower dimension axioms ` TarskiGDim>= ` -
    • axiom of Euclid / parallel postulate ` TarskiGE ` -
    + ${ + $d A a x y $. $d B a x y $. + $( Alternate expression for surreal less-than. Two surreals obey surreal + less-than iff they obey the sign ordering at the first place they + differ. (Contributed by Scott Fenton, 17-Jun-2011.) $) + sltval2 $p |- ( ( A e. No /\ B e. No ) -> + ( A + ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) + { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } + ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) + ) ) $= + ( vy vx wcel wa wbr cfv wceq c1o c0 c2o con0 wne fvex mtbiri adantl fveq2 + wn csur cslt cv wral cop ctp wrex crab cint sltval cvv w3o brtp 1n0 eqeq1 + neii fvprc nsyl2 adantr 2on0 3jaoi sylbi sylib onelon expcom syl5 neeq12d + onintrab onnminsb com12 syldc df-ne con2bii syl6ibr ralrimiv jca ex impac + anass raleq breq12d anbi12d rspcev syl wi eqeq12 csuc 1on 0elon necon3bid + suc11 mp2an mpbir df-2o df-1o eqeq12i nemtbir eqcom bitrdi nesymi 3imtr4i + wb elrab biimpri adantlr ssrab2 ne0i onint sylancr nfrab1 nfint nfcv nffv + nfne elrabf simprbi eqeq12d rspccv ad2antlr simpll oninton ontri1 syl2anc + wss mtod mpbird intss1 eqssd syldan sylan2 fveq2d biimpd pm2.43d rexlimiv + expimpd impbid1 bitr4d ) AUAFBUAFGZABUBHDUCZAIZYSBIZJZDEUCZUDZUUCAIZUUCBI + ZKLUEKMUELMUEUFZHZGZENUGZCUCZAIZUUKBIZOZCNUHZUIZAIZUUPBIZUUGHZEDABUJYRUUS + UUJYRUUSUUJYRUUSGZUUPNFZUUBDUUPUDZUUSGZGZUUJUUTUVAUVBGZUUSGUVDYRUUSUVEYRU + USUVEUUTUVAUVBUUSUVAYRUUSUUPUKFZUVAUUSUUQKJZUURLJZGZUVGUURMJZGZUUQLJZUVJG + ZULUVFKLKMLMUUQUURUUPAPUUPBPUMUVIUVFUVKUVMUVGUVFUVHUVGUVLUVFUVGUVLKLJZKLU + NUPZUUQKLUOQUUPAUQURZUSUVGUVFUVJUVPUSUVJUVFUVLUVJUVHUVFUVJUVHMLJMLUTUPUUR + MLUOQUUPBUQURRVAVBUUNCVHVCRZUUTUUBDUUPUUTYSUUPFZYTUUAOZTZUUBUVRUUTYSNFZUV + TUUTUVAUVRUWAUVQUVAUVRUWAUUPYSVDVEVFUWAUVRUVTUUNUVSCYSUUKYSJUULYTUUMUUAUU + KYSASUUKYSBSVGVIVJVKUVSUUBYTUUAVLVMVNVOVPVQVRUVAUVBUUSVSVCUUIUVCEUUPNUUCU + UPJZUUDUVBUUHUUSUUBDUUCUUPVTUWBUUEUUQUUFUURUUGUUCUUPASUUCUUPBSWAWBWCWDVQU + UIUUSENUUCNFZUUDUUHUUSUWCUUDGZUUHUUSUWDUUHUUHUUSWEUWDUUHGZUUHUUSUWEUUEUUQ + UUFUURUUGUWEUUCUUPAUUHUWDUUEUUFOZUWBUUEKJZUUFLJGZUWGUUFMJZGZUUELJUWIGZULU + UEUUFJZTZUUHUWFUWHUWMUWJUWKUWHUWLUVNUVOUUEKUUFLWFQUWJUWLMKJZUWNKWGZLWGZUW + OUWPOZKLOZUNKNFZLNFZUWQUWRXBWHWIUWSUWTGUWOUWPKLKLWKWJWLWMMUWOKUWPWNWOWPWQ + UWJUWLKMJUWNUUEKUUFMWFKMWRWSQUWKUWLLMJMLUTWTUUELUUFMWFQVAKLKMLMUUEUUFUUCA + PUUCBPUMUUEUUFVLXAUWDUWFUUCUUOFZUWBUWCUWFUXAUUDUXAUWCUWFGUUNUWFCUUCNUUKUU + CJUULUUEUUMUUFUUKUUCASUUKUUCBSVGXCXDXEUWDUXAGZUUCUUPUXBUUCUUPYDZUUPUUCFZT + ZUXBUXDUUQUURJZUXBUUQUUROZUXFTUXBUUPUUOFZUXGUXBUUONYDZUUOLOZUXHUUNCNXFZUX + AUXJUWDUUOUUCXGZRUUOXHXIUXHUVAUXGUUNUXGCUUPNCUUOUUNCNXJXKZCNXLCUUQUURCUUP + ACAXLUXMXMCUUPBCBXLUXMXMXNUUKUUPJUULUUQUUMUURUUKUUPASUUKUUPBSVGXOXPWDUUQU + URVLVCUUDUXDUXFWEUWCUXAUUBUXFDUUPUUCYSUUPJYTUUQUUAUURYSUUPASYSUUPBSXQXRXS + YEUXBUWCUVAUXCUXEXBUWCUUDUXAXTUXAUVAUWDUXAUXIUXJUVAUXKUXLUUOYAXIRUUCUUPYB + YCYFUXAUUPUUCYDUWDUUCUUOYGRYHYIYJZYKUWEUUCUUPBUXNYKWAYLVQYMYOYNYPYQ $. + $} - So our definition of a Tarskian Geometry includes the 3 axioms for the - quaternary congruence relation (A1, A2, A3), the 3 axioms for the - ternary betweenness relation (A6, A7, A11), and the 2 axioms of - compatibility of the congruence and the betweenness relations (A4,A5). + $( The function value of a surreal is either a sign or the empty set. + (Contributed by Scott Fenton, 22-Jun-2011.) $) + nofv $p |- ( A e. No -> ( ( A ` X ) = (/) \/ ( A ` X ) = 1o \/ + ( A ` X ) = 2o ) ) $= + ( csur wcel cfv c0 wceq c1o c2o wo w3o cdm wn pm2.1 wi ndmfv a1i wfun crn + wa cpr nofun norn fvelrn ssel syl5com impancom 1oex con0 elexi elpr2 syl6ib + wss 2on syl2anc orim12d mpi 3orass sylibr ) ACDZBAEZFGZVAHGZVAIGZJZJZVBVCVD + KUTBALDZMZVGJVFVGNUTVHVBVGVEVHVBOUTBAPQUTARZASZHIUAZUMZVGVEOAUBAUCVIVLTVGVA + VKDZVEVIVGVLVMVIVGTVAVJDVLVMBAUDVJVKVAUEUFUGVAHIUHIUIUNUJUKULUOUPUQVBVCVDUR + US $. - It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 - (Lower dimension axiom) and A9 (Upper dimension axiom) so the number - of dimensions of the geometry it formalizes is not constrained. + $( ` (/) ` is not a surreal sign. (Contributed by Scott Fenton, + 16-Jun-2011.) $) + nosgnn0 $p |- -. (/) e. { 1o , 2o } $= + ( c0 c1o c2o cpr wcel wceq wo 1n0 nesymi csuc wne necom df-2o neeq2i bitr4i + nsuceq0 mpbi neii pm3.2ni 0ex elpr mtbir ) ABCDEABFZACFZGUCUDBAHIACBJZAKZAC + KZBPUFAUEKUGUEALCUEAMNOQRSABCTUAUB $. - Considering A2 as one of the 3 axioms for the quaternary congruence - relation is somewhat conventional, because the transitivity of the - congruence relation is automatically given by our choice to take the - distance as this congruence relation in our definition of Tarski - geometries. - (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry - Arnoux, 27-Apr-2019.) $) - df-trkg $a |- TarskiG = ( ( TarskiGC i^i TarskiGB ) - i^i ( TarskiGCB i^i { f | [. ( Base ` f ) / p ]. [. ( Itv ` f ) / i ]. - ( LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | - ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } ) ) $. + ${ + nosgnn0i.1 $e |- X e. { 1o , 2o } $. + $( If ` X ` is a surreal sign, then it is not null. (Contributed by Scott + Fenton, 3-Aug-2011.) $) + nosgnn0i $p |- (/) =/= X $= + ( c0 wceq c1o c2o cpr wcel nosgnn0 eleq1 mpbiri mto neir ) CACADZCEFGZHZI + NPAOHBCAOJKLM $. $} ${ - $d d f g i n p G $. $d a b c d f g i j n p s t u v x y z I $. - $d a b c d f g i j n p s t u v x y z P $. - $d a b c d f g i j n p u v x y z .- $. - istrkg.p $e |- P = ( Base ` G ) $. - istrkg.d $e |- .- = ( dist ` G ) $. - istrkg.i $e |- I = ( Itv ` G ) $. - $( Property of being a Tarski geometry - congruence part. (Contributed by - Thierry Arnoux, 14-Mar-2019.) $) - istrkgc $p |- ( G e. TarskiGC <-> ( G e. _V /\ - ( A. x e. P A. y e. P ( x .- y ) = ( y .- x ) /\ A. x e. P - A. y e. P A. z e. P ( ( x .- y ) = ( z .- z ) -> x = y ) ) ) ) $= - ( vd vp vf cv co wceq wral wa wcel raleqbidva wi cds cfv wsbc cbs cstrkgc - simpl eqcomd adantr simpllr oveqd eqeq12d oveqdr anbi12d sbcie2s df-trkgc - imbi1d elab4g ) ANZBNZKNZOZUTUSVAOZPZBLNZQZAVEQZVBCNZVHVAOZPZUSUTPZUAZCVE - QZBVEQZAVEQZRZKMNZUBUCUDLVQUEUCUDUSUTGOZUTUSGOZPZBDQZADQZVRVHVHGOZPZVKUAZ - CDQZBDQZADQZRZMEUFWIVPMDGUEUBELKHIVEDPZVAGPZRZWBVGWHVOWLWAVFADVEWLVEDWJWK - UGUHZWLUSDSZRZVTVDBDVEWLDVEPZWNWMUIZWOUTDSZRZVRVBVSVCWSGVAUSUTWSVAGWJWKWN - WRUJUHZUKWSGVAUTUSWTUKULTTWLWGVNADVEWMWOWFVMBDVEWQWSWEVLCDVEWOWPWRWQUIWSV - HDSZRZWDVJVKXBVRVBWCVIWSXAABGVAWTUMWSXACCGVAWTUMULUQTTTUNUOABCMLKUPUR $. - - $( Property of being a Tarski geometry - betweenness part. (Contributed by - Thierry Arnoux, 14-Mar-2019.) $) - istrkgb $p |- ( G e. TarskiGB <-> ( G e. _V /\ - ( A. x e. P A. y e. P ( y e. ( x I x ) -> x = y ) - /\ A. x e. P A. y e. P A. z e. P A. u e. P A. v e. P - ( ( u e. ( x I z ) /\ v e. ( y I z ) ) - -> E. a e. P ( a e. ( u I y ) /\ a e. ( v I x ) ) ) - /\ A. s e. ~P P A. t e. ~P P - ( E. a e. P A. x e. s A. y e. t x e. ( a I y ) - -> E. b e. P A. x e. s A. y e. t b e. ( x I y ) ) ) ) ) $= - ( co wcel wral wa vi vp vf cv wceq wrex cpw w3a citv cfv wsbc cbs cstrkgb - simpl eqcomd adantr simpllr oveqd eleq2d imbi1d raleqbidva simp-6r oveqdr - wi anbi12d rexeqbidva imbi12d ad2antrr simp-4r 2ralbidv 3anbi123d sbcie2s - pweqd df-trkgb elab4g ) BUDZAUDZVQUAUDZQZRZVQVPUEZVDZBUBUDZSZAWCSZEUDZVQC - UDZVRQZRZDUDZVPWGVRQZRZTZLUDZWFVPVRQZRZWNWJVQVRQZRZTZLWCUFZVDZDWCSZEWCSZC - WCSZBWCSZAWCSZVQWNVPVRQZRZBFUDZSAKUDZSZLWCUFZMUDZVQVPVRQZRZBXISAXJSZMWCUF - ZVDZFWCUGZSZKXSSZUHZUAUCUDZUIUJUKUBYCULUJUKVPVQVQIQZRZWAVDZBGSZAGSZWFVQWG - IQZRZWJVPWGIQZRZTZWNWFVPIQZRZWNWJVQIQZRZTZLGUFZVDZDGSZEGSZCGSZBGSZAGSZVQW - NVPIQZRZBXISAXJSZLGUFZXMVQVPIQZRZBXISAXJSZMGUFZVDZFGUGZSZKUUOSZUHZUCHUMUU - RYBUCGIULUIHUBUANPWCGUEZVRIUEZTZYHWEUUEXFUUQYAUVAYGWDAGWCUVAWCGUUSUUTUNUO - ZUVAVQGRZTZYFWBBGWCUVAGWCUEZUVCUVBUPZUVDVPGRZTZYEVTWAUVHYDVSVPUVHIVRVQVQU - VHVRIUUSUUTUVCUVGUQUOURUSUTVAVAUVAUUDXEAGWCUVBUVDUUCXDBGWCUVFUVHUUBXCCGWC - UVDUVEUVGUVFUPZUVHWGGRZTZUUAXBEGWCUVHUVEUVJUVIUPZUVKWFGRZTZYTXADGWCUVKUVE - UVMUVLUPZUVNWJGRZTZYMWMYSWTUVQYJWIYLWLUVQYIWHWFUVQIVRVQWGUVQVRIUUSUUTUVCU - VGUVJUVMUVPVBUOZURUSUVQYKWKWJUVQIVRVPWGUVRURUSVEUVQYRWSLGWCUVNUVEUVPUVOUP - UVQWNGRZTZYOWPYQWRUVTYNWOWNUVQUVSEBIVRUVRVCUSUVTYPWQWNUVQUVSDAIVRUVRVCUSV - EVFVGVAVAVAVAVAUVAUUPXTKUUOXSUVAGWCUVBVMZUVAXJUUORZTZUUNXRFUUOXSUVAUUOXSU - EUWBUWAUPUWCXIUUORZTZUUIXLUUMXQUWEUUHXKLGWCUVAUVEUWBUWDUVBVHZUWEUVSTZUUGX - HABXJXIUWGUUFXGVQUWGIVRWNVPUWGVRIUUSUUTUWBUWDUVSVIUOURUSVJVFUWEUULXPMGWCU - WFUWEXMGRZTZUUKXOABXJXIUWIUUJXNXMUWIIVRVQVPUWIVRIUUSUUTUWBUWDUWHVIUOURUSV - JVFVGVAVAVKVLABCDEFUCUAKUBLMVNVO $. - - $( Property of being a Tarski geometry - congruence and betweenness part. - (Contributed by Thierry Arnoux, 14-Mar-2019.) $) - istrkgcb $p |- ( G e. TarskiGCB <-> ( G e. _V /\ - ( A. x e. P A. y e. P A. z e. P A. u e. P - A. a e. P A. b e. P A. c e. P A. v e. P - ( ( ( x =/= y /\ y e. ( x I z ) /\ b e. ( a I c ) ) - /\ ( ( ( x .- y ) = ( a .- b ) /\ ( y .- z ) = ( b .- c ) ) - /\ ( ( x .- u ) = ( a .- v ) /\ ( y .- u ) = ( b .- v ) ) ) ) - -> ( z .- u ) = ( c .- v ) ) - /\ A. x e. P A. y e. P A. a e. P A. b e. P E. z e. P - ( y e. ( x I z ) /\ ( y .- z ) = ( a .- b ) ) ) ) ) $= - ( co wceq wa wral oveqd vi vd vp vf cv wne wcel w3a wi wrex citv cfv wsbc - cds cstrkgcb simp1 eqcomd adantr ad6antr simpll3 3anbi23d simpll2 eqeq12d - cbs eleq2d anbi12d imbi12d raleqbidva ad3antrrr rexeqbidva sbcie3s elab4g - df-trkgcb ) AUEZBUEZUFZVOVNCUEZUAUEZPZUGZKUEZJUEZLUEZVRPZUGZUHZVNVOUBUEZP - ZWBWAWGPZQZVOVQWGPZWAWCWGPZQZRZVNEUEZWGPZWBDUEZWGPZQZVOWOWGPZWAWQWGPZQZRZ - RZRZVQWOWGPZWCWQWGPZQZUIZDUCUEZSZLXJSZKXJSZJXJSZEXJSZCXJSZBXJSZAXJSZVTWKW - IQZRZCXJUJZKXJSZJXJSZBXJSZAXJSZRZUAUDUEZUKULUMUBYGUNULUMUCYGVDULUMVPVOVNV - QHPZUGZWAWBWCHPZUGZUHZVNVOIPZWBWAIPZQZVOVQIPZWAWCIPZQZRZVNWOIPZWBWQIPZQZV - OWOIPZWAWQIPZQZRZRZRZVQWOIPZWCWQIPZQZUIZDFSZLFSZKFSZJFSZEFSZCFSZBFSZAFSZY - IYPYNQZRZCFUJZKFSZJFSZBFSZAFSZRZUDGUOUVHYFUDFIHVDUNUKGUCUBUAMNOXJFQZWGIQZ - VRHQZUHZUUTXRUVGYEUVLUUSXQAFXJUVLXJFUVIUVJUVKUPUQZUVLVNFUGZRZUURXPBFXJUVL - FXJQZUVNUVMURZUVOVOFUGZRZUUQXOCFXJUVOUVPUVRUVQURZUVSVQFUGZRZUUPXNEFXJUVSU - VPUWAUVTURZUWBWOFUGZRZUUOXMJFXJUWBUVPUWDUWCURZUWEWBFUGZRZUUNXLKFXJUWEUVPU - WGUWFURUWHWAFUGZRZUUMXKLFXJUVLUVPUVNUVRUWAUWDUWGUWIUVMUSUWJWCFUGZRZUULXID - FXJUVOUVPUVRUWAUWDUWGUWIUWKUVQUSUWLWQFUGZRZUUHXEUUKXHUWNYLWFUUGXDUWNYIVTY - KWEVPUWNYHVSVOUWNHVRVNVQUWNVRHUVSUVKUWAUWDUWGUWIUWKUWMUVIUVJUVKUVNUVRUTZU - SUQZTVEUWNYJWDWAUWNHVRWBWCUWPTVEVAUWNYSWNUUFXCUWNYOWJYRWMUWNYMWHYNWIUWNIW - GVNVOUWNWGIUVSUVJUWAUWDUWGUWIUWKUWMUVIUVJUVKUVNUVRVBZUSUQZTUWNIWGWBWAUWRT - VCUWNYPWKYQWLUWNIWGVOVQUWRTUWNIWGWAWCUWRTVCVFUWNUUBWSUUEXBUWNYTWPUUAWRUWN - IWGVNWOUWRTUWNIWGWBWQUWRTVCUWNUUCWTUUDXAUWNIWGVOWOUWRTUWNIWGWAWQUWRTVCVFV - FVFUWNUUIXFUUJXGUWNIWGVQWOUWRTUWNIWGWCWQUWRTVCVGVHVHVHVHVHVHVHVHUVLUVFYDA - FXJUVMUVOUVEYCBFXJUVQUVSUVDYBJFXJUVTUVSUWGRZUVCYAKFXJUVSUVPUWGUVTURZUWSUW - IRZUVBXTCFXJUWSUVPUWIUWTURUXAUWARZYIVTUVAXSUXBYHVSVOUXBHVRVNVQUXBVRHUVSUV - KUWGUWIUWAUWOVIUQTVEUXBYPWKYNWIUXBIWGVOVQUXBWGIUVSUVJUWGUWIUWAUWQVIUQZTUX - BIWGWBWAUXCTVCVFVJVHVHVHVHVFVKABCDEUDUAUCJKLUBVMVL $. + $d A x y $. $d B x y $. + $( The restriction of a surreal to an ordinal is still a surreal. + (Contributed by Scott Fenton, 4-Sep-2011.) $) + noreson $p |- ( ( A e. No /\ B e. On ) -> ( A |` B ) e. No ) $= + ( vy vx csur wcel con0 wa cv c1o c2o cpr cres wf wrex elno wi onin fresin + cin feq2 rspcev syl2an an32s ex rexlimiva imp sylanb sylibr ) AEFZBGFZHCI + ZJKLZABMZNZCGOZUNEFUJDIZUMANZDGOZUKUPDAPUSUKUPURUKUPQDGUQGFZURHUKUPUTUKUR + UPUTUKHUQBTZGFVAUMUNNZUPURUQBRUQUMABSUOVBCVAGULVAUMUNUAUBUCUDUEUFUGUHCUNP + UI $. + $} - $( Property of fulfilling Euclid's axiom. (Contributed by Thierry Arnoux, - 14-Mar-2019.) $) - istrkge $p |- ( G e. TarskiGE <-> ( G e. _V /\ - A. x e. P A. y e. P A. z e. P A. u e. P A. v e. P - ( ( u e. ( x I v ) /\ u e. ( y I z ) /\ x =/= u ) -> E. a e. P - E. b e. P ( y e. ( x I a ) /\ z e. ( x I b ) /\ v e. ( a I b ) ) ) ) ) $= - ( cv co wcel wral wa adantr vi vp vf wne w3a wrex wi citv cfv cbs cstrkge - wsbc wceq simpl eqcomd simp-6r oveqd eleq2d 3anbi12d 3anbi123d rexeqbidva - ad2antrr imbi12d raleqbidva sbcie2s df-trkge elab4g ) EOZAOZDOZUAOZPZQZVH - BOZCOZVKPZQZVIVHUDZUEZVNVIJOZVKPZQZVOVIKOZVKPZQZVJVTWCVKPZQZUEZKUBOZUFZJW - IUFZUGZDWIRZEWIRZCWIRZBWIRZAWIRZUAUCOZUHUIULUBWRUJUIULVHVIVJHPZQZVHVNVOHP - ZQZVRUEZVNVIVTHPZQZVOVIWCHPZQZVJVTWCHPZQZUEZKFUFZJFUFZUGZDFRZEFRZCFRZBFRZ - AFRZUCGUKXRWQUCFHUJUHGUBUALNWIFUMZVKHUMZSZXQWPAFWIYAWIFXSXTUNUOZYAVIFQZSZ - XPWOBFWIYAFWIUMZYCYBTZYDVNFQZSZXOWNCFWIYDYEYGYFTZYHVOFQZSZXNWMEFWIYHYEYJY - ITZYKVHFQZSZXMWLDFWIYKYEYMYLTZYNVJFQZSZXCVSXLWKYQWTVMXBVQVRYQWSVLVHYQHVKV - IVJYQVKHXSXTYCYGYJYMYPUPZUOZUQURYQXAVPVHYQHVKVNVOYSUQURUSYQXKWJJFWIYNYEYP - YOTZYQVTFQZSZXJWHKFWIYQYEUUAYTTUUBWCFQZSZXEWBXGWEXIWGUUDXDWAVNUUDHVKVIVTU - UDVKHYQXTUUAUUCYRVBUOZUQURUUDXFWDVOUUDHVKVIWCUUEUQURUUDXHWFVJUUDHVKVTWCUU - EUQURUTVAVAVCVDVDVDVDVDVEABCDEUCUAUBJKVFVG $. + ${ + $d A a $. $d B a $. + $( If ` A + ( A |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V ) ) $= + ( csur wcel wa wbr cfv c1o c0 cop c2o wceq fvex wn fvprc neii eqeq1 eqcom + bitrdi cslt cv wne con0 crab cint ctp cvv sltval2 w3o brtp 1n0 mtbiri syl + con4i adantr 2on0 adantl 3jaoi sylbi syl6bi ) ADEBDEFABUAGCUBZAHVBBHUCCUD + UEUFZAHZVCBHZIJKILKJLKUGGZVCUHEZABCUIVFVDIMZVEJMZFZVHVELMZFZVDJMZVKFZUJVG + IJILJLVDVEVCANVCBNUKVJVGVLVNVHVGVIVGVHVGOZVMVHOVCAPVMVHIJMZIJULQVMVHJIMVP + VDJIRJISTUMUNUOZUPVHVGVKVQUPVKVGVMVGVKVOVIVKOVCBPVIVKLJMZLJUQQVIVKJLMVRVE + JLRJLSTUMUNUOURUSUTVA $. + $} - $( Building lines from the segment property. (Contributed by Thierry - Arnoux, 14-Mar-2019.) $) - istrkgl $p |- ( G e. { f | [. ( Base ` f ) / p ]. [. ( Itv ` f ) / i ]. - ( LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | - ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } <-> - ( G e. _V /\ ( LineG ` G ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | - ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) ) ) $= - ( cv clng cfv co wcel wceq wa csn cdif w3o crab cmpo citv wsbc cbs eqcomd - cab simpl adantr difeq1d simpr oveqd eleq2d 3orbi123d mpoeq123dva sbcie2s - rabeqbidv eqeq2d fveqeq2 bitrd eqid elab4g ) ENZOPZABJNZVHANZUAZUBZCNZVIB - NZFNZQZRZVIVLVMVNQZRZVMVIVLVNQZRZUCZCVHUDZUEZSZFVFUFPUGJVFUHPUGZGOPABDDVJ - UBZVLVIVMHQZRZVIVLVMHQZRZVMVIVLHQZRZUCZCDUDZUEZSZEGWEEUJZVFGSWEVGWOSZWPWR - WDEDHUHUFGJFKMVHDSZVNHSZTZWOWCVGXAABDWFWNVHVKWBXAVHDWSWTUKUIZXAVIDRZTDVHV - JXADVHSXCXBULUMXAWNWBSXCVMWFRTXAWMWACDVHXBXAWHVPWJVRWLVTXAWGVOVLXAHVNVIVM - XAVNHWSWTUNUIZUOUPXAWIVQVIXAHVNVLVMXDUOUPXAWKVSVMXAHVNVIVLXDUOUPUQUTULURV - AUSVFGWOOVBVCWQVDVE $. + ${ + $d A a x y $. $d B a x y $. $d X a x y $. + $( If the restrictions of two surreals to a given ordinal obey surreal + less-than, then so do the two surreals themselves. (Contributed by + Scott Fenton, 4-Sep-2011.) $) + sltres $p |- ( ( A e. No /\ B e. No /\ X e. On ) -> + ( ( A |` X ) A = ` 2 ) ) -> ( G TarskiGDim>= N <-> - E. f ( f : ( 1 ..^ N ) -1-1-> P /\ E. x e. P E. y e. P E. z e. P - ( A. j e. ( 2 ..^ N ) ( ( ( f ` 1 ) .- x ) = ( ( f ` j ) .- x ) /\ - ( ( f ` 1 ) .- y ) = ( ( f ` j ) .- y ) /\ - ( ( f ` 1 ) .- z ) = ( ( f ` j ) .- z ) ) - /\ -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) ) - ) ) $= - ( cv cfzo co wceq wrex oveqd vn vp vd vi vg c1 wf1 cfv w3a c2 wral w3o wn - wcel wex citv wsbc cds cbs cuz cstrkgld eqidd simp1 eqcomd f1eq123d simp2 - eqeq12d 3anbi123d ralbidv simp3 eleq2d 3orbi123d notbid anbi12d rexeqbidv - wa exbidv sbcie3s oveq2 raleqdv anbi1d rexbidv 2rexbidv df-trkgld brabg ) - UFUAOZPQZUBOZEOZUGZUFWIUHZAOZUCOZQZFOWIUHZWLWMQZRZWKBOZWMQZWOWRWMQZRZWKCO - ZWMQZWOXBWMQZRZUIZFUJWFPQZUKZXBWLWRUDOZQZUNZWLXBWRXIQZUNZWRWLXBXIQZUNZULZ - UMZVPZCWHSZBWHSZAWHSZVPZEUOZUDUEOZUPUHUQUCYDURUHUQUBYDUSUHUQWGDWIUGZWKWLI - QZWOWLIQZRZWKWRIQZWOWRIQZRZWKXBIQZWOXBIQZRZUIZFXGUKZXBWLWRHQZUNZWLXBWRHQZ - UNZWRWLXBHQZUNZULZUMZVPZCDSZBDSZADSZVPZEUOZUFJPQZDWIUGZYOFUJJPQZUKZUUDVPZ - CDSZBDSADSZVPZEUOUEUAGJKUJUTUHVAUUJYCUEDIHUSURUPGUBUCUDLMNWHDRZWMIRZXIHRZ - UIZUUIYBEUVBYEWJUUHYAUVBWGWGDWHWIWIUVBWIVBUVBWGVBUVBWHDUUSUUTUVAVCVDZVEUV - BUUGXTADWHUVCUVBUUFXSBDWHUVCUVBUUEXRCDWHUVCUVBYPXHUUDXQUVBYOXFFXGUVBYHWQY - KXAYNXEUVBYFWNYGWPUVBIWMWKWLUVBWMIUUSUUTUVAVFVDZTUVBIWMWOWLUVDTVGUVBYIWSY - JWTUVBIWMWKWRUVDTUVBIWMWOWRUVDTVGUVBYLXCYMXDUVBIWMWKXBUVDTUVBIWMWOXBUVDTV - GVHVIUVBUUCXPUVBYRXKYTXMUUBXOUVBYQXJXBUVBHXIWLWRUVBXIHUUSUUTUVAVJVDZTVKUV - BYSXLWLUVBHXIXBWRUVETVKUVBUUAXNWRUVBHXIWLXBUVETVKVLVMVNVOVOVOVNVQVRWFJRZU - UIUUREUVFYEUULUUHUUQUVFWGUUKDDWIWIUVFWIVBWFJUFPVSUVFDVBVEUVFUUFUUPABDDUVF - UUEUUOCDUVFYPUUNUUDUVFYOFXGUUMWFJUJPVSVTWAWBWCVNVQABCEUEUDFUAUBUCWDWE $. + $( The Cartesian product of an ordinal and ` { 1o } ` is a surreal. + (Contributed by Scott Fenton, 12-Jun-2011.) $) + noxp1o $p |- ( A e. On -> ( A X. { 1o } ) e. No ) $= + ( con0 wcel c1o csn cxp cdm c2o cpr wf csur 1oex prid1 fconst6 c0 wceq snnz + wne dmxp ax-mp feq2i mpbir a1i eleq1i biimpri elno3 sylanbrc ) ABCZADEZFZGZ + DHIZUJJZUKBCZUJKCUMUHUMAULUJJADULDHLMNUKAULUJUIORUKAPDLQAUISTZUAUBUCUNUHUKA + BUOUDUEUJUFUG $. - $( Property of fulfilling the lower dimension 2 axiom. (Contributed by - Thierry Arnoux, 20-Nov-2019.) $) - istrkg2ld $p |- ( G e. V - -> ( G TarskiGDim>= 2 <-> E. x e. P E. y e. P E. z e. P - -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) ) $= - ( vf vj wcel c2 c1 co wceq wa wrex cstrkgld wbr cfzo wf1 cfv w3a wral w3o - cv wn wex cuz wb cz 2z ax-mp istrkgld mpan2 r19.41v ancom rexbii 3bitr3ri - uzid exbii rexcom4 simpr reximi adantl exlimiv cmpt csn cop wss 1ex f1osn - wf1o vex f1of1 mp1i snssi f1ss syl2anc wtru fzo12sn mpteq1i fmptsn eqtr4i - cvv mp2an eqidd f1eq123d mptru sylibr c0 ral0 fzo0 raleqi mpbir jctl ovex - a1i mptex f1eq1 nfmpt1 nfeq2 nfan simpll fveq1d eqeq12d 3anbi123d ralbida - nfv oveq1d anbi1d 2rexbidva anbi12d syl2an impbida rexbiia 3bitr2i bitrdi - spcev ) EHNZEOUAUBZPOUCQZDLUIZUDZPYFUEZAUIZGQZMUIZYFUEZYIGQZRZYHBUIZGQZYL - YOGQZRZYHCUIZGQZYLYSGQZRZUFZMOOUCQZUGZYSYIYOFQNYIYSYOFQNYOYIYSFQNUHUJZSZC - DTZBDTZADTZSZLUKZUUFCDTZBDTZADTZYCOOULUENZYDUULUMOUNNUUPUOOVCUPABCDLMEFGO - HIJKUQURUULYGUUISZADTZLUKUUQLUKZADTUUOUUKUURLUUIYGSZADTUUJYGSUURUUKUUIYGA - DUSUUTUUQADUUIYGUTVAUUJYGUTVBVDUUQALDVEUUSUUNADYIDNZUUSUUNUUSUUNUVAUUQUUN - LUUIUUNYGUUHUUMBDUUGUUFCDUUEUUFVFVGVGVHVIVHUVAYEDMYEYIVJZUDZPUVBUEZYIGQZY - KUVBUEZYIGQZRZUVDYOGQZUVFYOGQZRZUVDYSGQZUVFYSGQZRZUFZMUUDUGZUUFSZCDTZBDTZ - UUSUUNUVAPVKZDPYIVLVKZUDZUVCUVAUVTYIVKZUWAUDZUWCDVMUWBUVTUWCUWAVPUWDUVAPY - IVNAVQZVOUVTUWCUWAVRVSYIDVTUVTUWCDUWAWAWBUVCUWBUMWCYEUVTDDUVBUWAUVBUWARWC - UVBMUVTYIVJZUWAMYEUVTYIWDWEPWHNYIWHNUWAUWFRVNUWEMPYIWHWHWFWIWGXAYEUVTRWCW - DXAWCDWJWKWLWMUUMUVRBDUUFUVQCDUUFUVPUVPUVOMWNUGUVOMWOUVOMUUDWNOWPWQWRWSVG - VGUUQUVCUVSSLUVBMYEYIPOUCWTXBYFUVBRZYGUVCUUIUVSYEDYFUVBXCUWGUUGUVQBCDDUWG - YODNYSDNSZSZUUEUVPUUFUWIUUCUVOMUUDUWGUWHMMYFUVBMYEYIXDXEUWHMXLXFUWIYKUUDN - ZSZYNUVHYRUVKUUBUVNUWKYJUVEYMUVGUWKYHUVDYIGUWKPYFUVBUWGUWHUWJXGZXHZXMUWKY - LUVFYIGUWKYKYFUVBUWLXHZXMXIUWKYPUVIYQUVJUWKYHUVDYOGUWMXMUWKYLUVFYOGUWNXMX - IUWKYTUVLUUAUVMUWKYHUVDYSGUWMXMUWKYLUVFYSGUWNXMXIXJXKXNXOXPYBXQXRXSXTYA - $. + ${ + $d A x $. $d B x $. + $( Lemma for ~ nosepon . Consider a case of proper subset domain. + (Contributed by Scott Fenton, 21-Sep-2020.) $) + noseponlem $p |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> + -. A. x e. On ( A ` x ) = ( B ` x ) ) $= + ( csur wcel cdm w3a cv cfv wne con0 wrex wceq wral wn nodmon 3ad2ant1 syl + c0 fveq2 word nodmord ordirr ndmfv c1o c2o nosgnn0 elno3 simplbi 3ad2ant2 + cpr wf simp3 ffvelcdmd eleq1 syl5ibcom mtoi neqned necomd eqnetrd neeq12d + rspcev syl2anc df-ne rexbii rexnal bitri sylib ) BDEZCDEZBFZCFZEZGZAHZBIZ + VOCIZJZAKLZVPVQMZAKNOZVNVKKEZVKBIZVKCIZJZVSVIVJWBVMBPQVNWCSWDVNVKVKEOZWCS + MVIVJWFVMVIVKUAWFBUBVKUCRQVKBUDRVNWDSVNWDSVNWDSMZSUEUFUKZEZUGVNWDWHEWGWIV + NVLWHVKCVJVIVLWHCULZVMVJWJVLKECUHUIUJVIVJVMUMUNWDSWHUOUPUQURUSUTVRWEAVKKV + OVKMVPWCVQWDVOVKBTVOVKCTVAVBVCVSVTOZAKLWAVRWKAKVPVQVDVEVTAKVFVGVH $. - $( Property of fulfilling the lower dimension 3 axiom. (Contributed by - Thierry Arnoux, 12-Jul-2020.) $) - istrkg3ld $p |- ( G e. V - -> ( G TarskiGDim>= 3 <-> E. u e. P E. v e. P ( u =/= v - /\ E. x e. P E. y e. P E. z e. P ( ( ( u .- x ) = ( v .- x ) - /\ ( u .- y ) = ( v .- y ) /\ ( u .- z ) = ( v .- z ) ) - /\ -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) ) ) ) $= - ( wcel c3 c1 co wceq c2 wrex vf vj cstrkgld wbr cfzo wf1 cfv w3a wral w3o - cv wn wa wex cpr wne cuz wb cz cle 3z 2re 3re 2lt3 ltleii eluz1i mpbir2an - 2z istrkgld mpan2 fzo13pr f1eq2 ax-mp anbi1i exbii a1i 1z oveq1 3anbi123d - eqeq1d anbi1d rexbidv 2rexbidv eqeq2d csn caddc 2p1e3 oveq2i fzosn eqtr3i - 1ne2 raleqi fveq2 oveq1d ralsng bitri bitr4di f1prex mp3an 3bitrd ) GJNZG - OUCUDZPOUEQZFUAUKZUFZPXDUGZAUKZIQZUBUKZXDUGZXGIQZRZXFBUKZIQZXJXMIQZRZXFCU - KZIQZXJXQIQZRZUHZUBSOUEQZUIZXQXGXMHQNXGXQXMHQNXMXGXQHQNUJULZUMZCFTZBFTAFT - ZUMZUAUNZPSUOZFXDUFZYGUMZUAUNZEUKZDUKZUPYNXGIQZYOXGIQZRZYNXMIQZYOXMIQZRZY - NXQIQZYOXQIQZRZUHZYDUMZCFTZBFTAFTZUMDFTEFTZXAOSUQUGNZXBYIURUUJOUSNSOUTUDV - ASOVBVCVDVESOVHVFVGABCFUAUBGHIOJKLMVIVJYIYMURXAYHYLUAXEYKYGXCYJRXEYKURVKX - CYJFXDVLVMVNVOVPYMUUIURZXAPUSNSUSNZPSUPUUKVQVHWKYGUUHXHYQRZXNYTRZXRUUCRZU - HZYDUMZCFTZBFTAFTEDPSFUAUSUSYNXFRZUUGUURABFFUUSUUFUUQCFUUSUUEUUPYDUUSYRUU - MUUAUUNUUDUUOUUSYPXHYQYNXFXGIVRVTUUSYSXNYTYNXFXMIVRVTUUSUUBXRUUCYNXFXQIVR - VTVSWAWBWCYOSXDUGZRZUURYFABFFUVAUUQYECFUVAUUPYCYDUVAUUPXHUUTXGIQZRZXNUUTX - MIQZRZXRUUTXQIQZRZUHZYCUVAUUMUVCUUNUVEUUOUVGUVAYQUVBXHYOUUTXGIVRWDUVAYTUV - DXNYOUUTXMIVRWDUVAUUCUVFXRYOUUTXQIVRWDVSYCYAUBSWEZUIZUVHYAUBYBUVISSPWFQZU - EQZYBUVIUVKOSUEWGWHUULUVLUVIRVHSWIVMWJWLUULUVJUVHURVHYAUVHUBSUSXISRZXLUVC - XPUVEXTUVGUVMXKUVBXHUVMXJUUTXGIXISXDWMZWNWDUVMXOUVDXNUVMXJUUTXMIUVNWNWDUV - MXSUVFXRUVMXJUUTXQIUVNWNWDVSWOVMWPWQWAWBWCWRWSVPWT $. + $( Given two unequal surreals, the minimal ordinal at which they differ is + an ordinal. (Contributed by Scott Fenton, 21-Sep-2020.) $) + nosepon $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> + |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. On ) $= + ( csur wcel wne w3a cfv con0 wrex wa wn wceq wral cdm word nodmord 3expia + wo 3impia cv crab cint df-ne rexbii notbii dfral2 bitr4i ordtri3or syl2an + w3o 3orass or12 bitri sylib ord noseponlem wi eqcom ralbii sylnibr ancoms + jaod syld con4d wss ordsson ssralv 3syl adantr wb nofun 3ad2ant1 3ad2ant2 + wfun eqfunfv syl2anc mpbir2and biimtrid necon1ad onintrab2 ) BDEZCDEZBCFZ + GAUAZBHZWECHZFZAIJZWHAIUBUCIEWBWCWDWIWBWCKZWIBCWILZWFWGMZAINZWJBCMZWKWLLZ + AIJZLWMWIWPWHWOAIWFWGUDUEUFWLAIUGUHWBWCWMWNWBWCWMGZWNBOZCOZMZWLAWRNZWBWCW + MWTWJWTWMWJWTLWRWSEZWSWREZSZWMLZWJWTXDWJXBWTXCUKZWTXDSZWBWRPZWSPXFWCBQZCQ + WRWSUIUJXFXBWTXCSSXGXBWTXCULXBWTXCUMUNUOUPWJXBXEXCWBWCXBXEABCUQRWCWBXCXEU + RWCWBXCXEWCWBXCGWGWFMZAINWMACBUQWLXJAIWFWGUSUTVARVBVCVDVETWBWCWMXAWBWMXAU + RZWCWBXHWRIVFXKXIWRVGWLAWRIVHVIVJTWQBVOZCVOZWNWTXAKVKWBWCXLWMBVLVMWCWBXMW + MCVLVNABCVPVQVRRVSVTTWHAWAUO $. $} ${ - $d f i p x y z $. $d a b c v z A $. $d b c v z B $. $d c v C $. - $d a b c s t u v x y z I $. $d a b c s t u v x y z P $. $d a b s t x S $. - $d a b t x y T $. $d a b c u v U $. $d a b c u v x y z X $. - $d a b c u v y z Y $. $d a b c u v z Z $. $d a b v V $. - $d a b c u v x y z .- $. - axtrkg.p $e |- P = ( Base ` G ) $. - axtrkg.d $e |- .- = ( dist ` G ) $. - axtrkg.i $e |- I = ( Itv ` G ) $. - axtrkg.g $e |- ( ph -> G e. TarskiG ) $. - ${ - axtgcgrrflx.1 $e |- ( ph -> X e. P ) $. - axtgcgrrflx.2 $e |- ( ph -> Y e. P ) $. - $( Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. - (Contributed by Thierry Arnoux, 14-Mar-2019.) $) - axtgcgrrflx $p |- ( ph -> ( X .- Y ) = ( Y .- X ) ) $= - ( vx vy cv co wceq cstrkgc wcel vf vp vz vi cstrkg cstrkgb cin cstrkgcb - wral clng cfv csn cdif w3o crab cmpo citv cbs cab df-trkg inss1 eqsstri - wsbc sstri sselid wi cvv istrkgc simprbi simpld syl oveq1 oveq2 eqeq12d - wa rspc2v syl2anc mpd ) ANPZOPZEQZVTVSEQZRZOBUINBUIZFGEQZGFEQZRZACSTZWD - AUESCUESUFUGZUHUAPZUJUKNOUBPZWKVSULUMUCPZVSVTUDPZQTVSWLVTWMQTVTVSWLWMQT - UNUCWKUOUPRUDWJUQUKVCUBWJURUKVCUAUSUGZUGZSNOUCUAUDUBUTWOWISWIWNVASUFVAV - DVBKVEWHWDWAWLWLEQRVSVTRVFUCBUIOBUINBUIZWHCVGTWDWPVONOUCBCDEHIJVHVIVJVK - AFBTGBTWDWGVFLMWCWGFVTEQZVTFEQZRNOFGBBVSFRWAWQWBWRVSFVTEVLVSFVTEVMVNVTG - RWQWEWRWFVTGFEVMVTGFEVLVNVPVQVR $. - $} - - ${ - axtgcgrid.1 $e |- ( ph -> X e. P ) $. - axtgcgrid.2 $e |- ( ph -> Y e. P ) $. - axtgcgrid.3 $e |- ( ph -> Z e. P ) $. - axtgcgrid.4 $e |- ( ph -> ( X .- Y ) = ( Z .- Z ) ) $. - $( Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. - (Contributed by Thierry Arnoux, 14-Mar-2019.) $) - axtgcgrid $p |- ( ph -> X = Y ) $= - ( vx co wceq wcel vy vz vf vp vi cv wi wral cstrkgc cstrkg cin cstrkgcb - cstrkgb clng cfv csn cdif w3o crab cmpo citv wsbc cbs cab df-trkg inss1 - sstri eqsstri sselid cvv wa istrkgc simprbi simprd oveq1 eqeq1d imbi12d - syl eqeq1 oveq2 eqeq2 id oveq12d eqeq2d imbi1d rspc3v syl3anc mp2d ) AQ - UFZUAUFZERZUBUFZWLERZSZWIWJSZUGZUBBUHUABUHQBUHZFGERZHHERZSZFGSZACUITZWQ - AUJUICUJUIUMUKZULUCUFZUNUOQUAUDUFZXEWIUPUQWLWIWJUEUFZRTWIWLWJXFRTWJWIWL - XFRTURUBXEUSUTSUEXDVAUOVBUDXDVCUOVBUCVDUKZUKZUIQUAUBUCUEUDVEXHXCUIXCXGV - FUIUMVFVGVHLVIXBWKWJWIERSUABUHQBUHZWQXBCVJTXIWQVKQUAUBBCDEIJKVLVMVNVRPA - FBTGBTHBTWQWTXAUGZUGMNOWPXJFWJERZWMSZFWJSZUGWRWMSZXAUGQUAUBFGHBBBWIFSZW - NXLWOXMXOWKXKWMWIFWJEVOVPWIFWJVSVQWJGSZXLXNXMXAXPXKWRWMWJGFEVTVPWJGFWAV - QWLHSZXNWTXAXQWMWSWRXQWLHWLHEXQWBZXRWCWDWEWFWGWH $. - $} - - ${ - axtgsegcon.1 $e |- ( ph -> X e. P ) $. - axtgsegcon.2 $e |- ( ph -> Y e. P ) $. - axtgsegcon.3 $e |- ( ph -> A e. P ) $. - axtgsegcon.4 $e |- ( ph -> B e. P ) $. - $( Axiom of segment construction, Axiom A4 of [Schwabhauser] p. 11. As - discussed in Axiom 4 of [Tarski1999] p. 178, "The intuitive content - [is that] given any line segment ` A B ` , one can construct a line - segment congruent to it, starting at any point ` Y ` and going in the - direction of any ray containing ` Y ` . The ray is determined by the - point ` Y ` and a second point ` X ` , the endpoint of the ray. The - other endpoint of the line segment to be constructed is just the point - ` z ` whose existence is asserted." (Contributed by Thierry Arnoux, - 15-Mar-2019.) $) - axtgsegcon $p |- ( ph - -> E. z e. P ( Y e. ( X I z ) /\ ( Y .- z ) = ( A .- B ) ) ) $= - ( co wral va vb vy vx vf vp vi vc vu vv cv wcel wceq wa cstrkgcb cstrkg - wrex cstrkgc cstrkgb cin clng cfv csn cdif w3o crab cmpo citv cbs inss2 - wsbc cab df-trkg inss1 sstri eqsstri sselid wne w3a wi istrkgcb simprbi - cvv simprd syl oveq1 eleq2d anbi1d rexbidv 2ralbidv eleq1 eqeq1d rspc2v - anbi12d syl2anc mpd eqeq2d anbi2d oveq2 ) AJIBUKZGSZULZJWTHSZUAUKZUBUKZ - HSZUMZUNZBEUQZUBETUAETZXBXCCDHSZUMZUNZBEUQZAUCUKZUDUKZWTGSZULZXOWTHSZXF - UMZUNZBEUQZUBETUAETZUCETUDETZXJAFUOULZYDAUPUOFUPURUSUTZUOUEUKZVAVBUDUCU - FUKZYHXPVCVDWTXPXOUGUKZSULXPWTXOYISULXOXPWTYISULVEBYHVFVGUMUGYGVHVBVKUF - YGVIVBVKUEVLZUTZUTZUOUDUCBUEUGUFVMYLYKUOYFYKVJUOYJVNVOVPNVQYEXPXOVRXRXE - XDUHUKZGSULVSXPXOHSXFUMXSXEYMHSUMUNXPUIUKZHSXDUJUKZHSUMXOYNHSXEYOHSUMUN - UNUNWTYNHSYMYOHSUMVTUJETUHETUBETUAETUIETBETUCETUDETZYDYEFWCULYPYDUNUDUC - BUJUIEFGHUAUBUHKLMWAWBWDWEAIEULJEULYDXJVTOPYCXJXOXAULZXTUNZBEUQZUBETUAE - TUDUCIJEEXPIUMZYBYSUAUBEEYTYAYRBEYTXRYQXTYTXQXAXOXPIWTGWFWGWHWIWJXOJUMZ - YSXIUAUBEEUUAYRXHBEUUAYQXBXTXGXOJXAWKUUAXSXCXFXOJWTHWFWLWNWIWJWMWOWPACE - ULDEULXJXNVTQRXIXNXBXCCXEHSZUMZUNZBEUQUAUBCDEEXDCUMZXHUUDBEUUEXGUUCXBUU - EXFUUBXCXDCXEHWFWQWRWIXEDUMZUUDXMBEUUFUUCXLXBUUFUUBXKXCXEDCHWSWQWRWIWMW - OWP $. - $} - - ${ - axtg5seg.1 $e |- ( ph -> X e. P ) $. - axtg5seg.2 $e |- ( ph -> Y e. P ) $. - axtg5seg.3 $e |- ( ph -> Z e. P ) $. - axtg5seg.4 $e |- ( ph -> A e. P ) $. - axtg5seg.5 $e |- ( ph -> B e. P ) $. - axtg5seg.6 $e |- ( ph -> C e. P ) $. - axtg5seg.7 $e |- ( ph -> U e. P ) $. - axtg5seg.8 $e |- ( ph -> V e. P ) $. - axtg5seg.9 $e |- ( ph -> X =/= Y ) $. - axtg5seg.10 $e |- ( ph -> Y e. ( X I Z ) ) $. - axtg5seg.11 $e |- ( ph -> B e. ( A I C ) ) $. - axtg5seg.12 $e |- ( ph -> ( X .- Y ) = ( A .- B ) ) $. - axtg5seg.13 $e |- ( ph -> ( Y .- Z ) = ( B .- C ) ) $. - axtg5seg.14 $e |- ( ph -> ( X .- U ) = ( A .- V ) ) $. - axtg5seg.15 $e |- ( ph -> ( Y .- U ) = ( B .- V ) ) $. - $( Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two - triangles ` X Z U ` and ` A C V ` , a point ` Y ` on ` X Z ` , and a - point ` B ` on ` A C ` . If all corresponding line segments except - for ` Z U ` and ` C V ` are congruent ( i.e., ` X Y .~ A B ` , - ` Y Z .~ B C ` , ` X U .~ A V ` , and ` Y U .~ B V ` ), then ` Z U ` - and ` C V ` are also congruent. As noted in Axiom 5 of [Tarski1999] - p. 178, "this axiom is similar in character to the well-known theorems - of Euclidean geometry that allow one to conclude, from hypotheses - about the congruence of certain corresponding sides and angles in two - triangles, the congruence of other corresponding sides and angles." - (Contributed by Thierry Arnoux, 14-Mar-2019.) $) - axtg5seg $p |- ( ph -> ( Z .- U ) = ( C .- V ) ) $= - ( vc vv vb va vu vx vy vz vf vp vi wne co wcel cv wceq wa wral cstrkgcb - w3a wi cstrkg cstrkgc cstrkgb cin clng cfv csn cdif crab cmpo citv wsbc - w3o cbs cab df-trkg inss2 inss1 sstri eqsstri wrex cvv istrkgcb simprbi - sselid simpld neeq1 oveq1 eleq2d 3anbi12d eqeq1d anbi1d anbi12d ralbidv - imbi1d 2ralbidv neeq2 eleq1 oveq2 anbi2d 3anbi2d imbi12d rspc3v syl3anc - syl mpd 3anbi3d eqeq2d 3jca jca jca32 rspc2v syl2anc mp2d ) AKLVDZLKMHV - EZVFZCBUMVGZHVEZVFZVLZKLIVEZBCIVEZVHZLMIVEZCYKIVEZVHZVIZKFIVEZBUNVGZIVE - ZVHZLFIVEZCUUCIVEZVHZVIZVIZVIZMFIVEZYKUUCIVEZVHZVMZUNEVJUMEVJZYHYJCBDHV - EZVFZVLZYQYRCDIVEZVHZVIZUUBBJIVEZVHZUUFCJIVEZVHZVIZVIZVIZUULDJIVEZVHZAY - HYJUOVGZUPVGZYKHVEZVFZVLZYOUVMUVLIVEZVHZYRUVLYKIVEZVHZVIZKUQVGZIVEZUVMU - UCIVEZVHZLUWBIVEZUVLUUCIVEZVHZVIZVIZVIZMUWBIVEZUUMVHZVMZUNEVJZUMEVJZUOE - VJZUPEVJUQEVJZUUPAURVGZUSVGZVDZUWTUWSUTVGZHVEZVFZUVOVLZUWSUWTIVEZUVQVHZ - UWTUXBIVEZUVSVHZVIZUWSUWBIVEZUWDVHZUWTUWBIVEZUWGVHZVIZVIZVIZUXBUWBIVEZU - UMVHZVMZUNEVJZUMEVJUOEVJZUPEVJUQEVJZUTEVJUSEVJUREVJZUWRAGVKVFZUYDAVNVKG - VNVOVPVQZVKVAVGZVRVSURUSVBVGZUYHUWSVTWAUXBUWSUWTVCVGZVEVFUWSUXBUWTUYIVE - VFUWTUWSUXBUYIVEVFWFUTUYHWBWCVHVCUYGWDVSWEVBUYGWGVSWEVAWHZVQZVQZVKURUSU - TVAVCVBWIUYLUYKVKUYFUYKWJVKUYJWKWLWMQWRUYEUYDUXDUXHUVQVHVIUTEWNUOEVJUPE - VJUSEVJUREVJZUYEGWOVFUYDUYMVIURUSUTUNUQEGHIUPUOUMNOPWPWQWSXRAKEVFLEVFME - VFUYDUWRVMRSTUYCUWRKUWTVDZUWTKUXBHVEZVFZUVOVLZKUWTIVEZUVQVHZUXIVIZUWEUX - NVIZVIZVIZUXSVMZUNEVJZUMEVJUOEVJZUPEVJUQEVJYHLUYOVFZUVOVLZUVRLUXBIVEZUV - SVHZVIZUWIVIZVIZUXSVMZUNEVJZUMEVJUOEVJZUPEVJUQEVJURUSUTKLMEEEUWSKVHZUYB - VUFUQUPEEVUQUYAVUEUOUMEEVUQUXTVUDUNEVUQUXQVUCUXSVUQUXEUYQUXPVUBVUQUXAUY - NUXDUYPUVOUWSKUWTWTVUQUXCUYOUWTUWSKUXBHXAXBXCVUQUXJUYTUXOVUAVUQUXGUYSUX - IVUQUXFUYRUVQUWSKUWTIXAXDXEVUQUXLUWEUXNVUQUXKUWCUWDUWSKUWBIXAXDXEXFXFXH - XGXIXIUWTLVHZVUFVUPUQUPEEVURVUEVUOUOUMEEVURVUDVUNUNEVURVUCVUMUXSVURUYQV - UHVUBVULVURUYNYHUYPVUGUVOUWTLKXJUWTLUYOXKXCVURUYTVUKVUAUWIVURUYSUVRUXIV - UJVURUYRYOUVQUWTLKIXLXDVURUXHVUIUVSUWTLUXBIXAXDXFVURUXNUWHUWEVURUXMUWFU - WGUWTLUWBIXAXDXMXFXFXHXGXIXIUXBMVHZVUPUWQUQUPEEVUSVUOUWOUOUMEEVUSVUNUWN - UNEVUSVUMUWKUXSUWMVUSVUHUVPVULUWJVUSVUGYJYHUVOVUSUYOYILUXBMKHXLXBXNVUSV - UKUWAUWIVUSVUJUVTUVRVUSVUIYRUVSUXBMLIXLXDXMXEXFVUSUXRUWLUUMUXBMUWBIXAXD - XOXGXIXIXPXQXSAFEVFBEVFCEVFUWRUUPVMUDUAUBUWPUUPUVPUWAUUBUWDVHZUUFUWGVHZ - VIZVIZVIZUUNVMZUNEVJUMEVJYHYJUVLYLVFZVLZYOBUVLIVEZVHZUVTVIZUUEVVAVIZVIZ - VIZUUNVMZUNEVJUMEVJUQUPUOFBCEEEUWBFVHZUWNVVEUMUNEEVVOUWKVVDUWMUUNVVOUWJ - VVCUVPVVOUWIVVBUWAVVOUWEVUTUWHVVAVVOUWCUUBUWDUWBFKIXLXDVVOUWFUUFUWGUWBF - LIXLXDXFXMXMVVOUWLUULUUMUWBFMIXLXDXOXIUVMBVHZVVEVVNUMUNEEVVPVVDVVMUUNVV - PUVPVVGVVCVVLVVPUVOVVFYHYJVVPUVNYLUVLUVMBYKHXAXBXTVVPUWAVVJVVBVVKVVPUVR - VVIUVTVVPUVQVVHYOUVMBUVLIXAYAXEVVPVUTUUEVVAVVPUWDUUDUUBUVMBUUCIXAYAXEXF - XFXHXIUVLCVHZVVNUUOUMUNEEVVQVVMUUKUUNVVQVVGYNVVLUUJVVQVVFYMYHYJUVLCYLXK - XTVVQVVJUUAVVKUUIVVQVVIYQUVTYTVVQVVHYPYOUVLCBIXLYAVVQUVSYSYRUVLCYKIXAYA - XFVVQVVAUUHUUEVVQUWGUUGUUFUVLCUUCIXAYAXMXFXFXHXIXPXQXSAUUSUVBUVGAYHYJUU - RUFUGUHYBAYQUVAUIUJYCAUVDUVFUKULYCYDADEVFJEVFUUPUVIUVKVMZVMUCUEUUOVVRUU - SUVBUUIVIZVIZUULDUUCIVEZVHZVMUMUNDJEEYKDVHZUUKVVTUUNVWBVWCYNUUSUUJVVSVW - CYMUURYHYJVWCYLUUQCYKDBHXLXBXTVWCUUAUVBUUIVWCYTUVAYQVWCYSUUTYRYKDCIXLYA - XMXEXFVWCUUMVWAUULYKDUUCIXAYAXOUUCJVHZVVTUVIVWBUVKVWDVVSUVHUUSVWDUUIUVG - UVBVWDUUEUVDUUHUVFVWDUUDUVCUUBUUCJBIXLYAVWDUUGUVEUUFUUCJCIXLYAXFXMXMVWD - VWAUVJUULUUCJDIXLYAXOYEYFYG $. - $} + noextend.1 $e |- X e. { 1o , 2o } $. + $( Extending a surreal by one sign value results in a new surreal. + (Contributed by Scott Fenton, 22-Nov-2021.) $) + noextend $p |- ( A e. No -> ( A u. { <. dom A , X >. } ) e. No ) $= + ( csur wcel cdm cop csn cun wfun con0 crn c1o wss cin wceq cvv syl eqtrid + c0 c2o cpr nofun dmexg funsng sylancl elexi dmsnop ineq2i wn word nodmord + ordirr disjsn sylibr funun syl21anc uneq2i df-suc 3eqtr4i nodmon suceloni + csuc dmun eqeltrid rnun rnsnopg uneq2d norn snssi unssd eqsstrd syl3anbrc + mp1i elno2 ) ADEZAAFZBGHZIZJZVSFZKEVSLZMUAUBZNVSDEVPAJVRJZVQVRFZOZTPVTAUC + VPVQQEZBWCEZWDADUDZCVQBQWCUEUFVPWFVQVQHZOZTWEWJVQVQBBWCCUGUHZUIVPVQVQEUJZ + WKTPVPVQUKWMAULVQUMRVQVQUNUOSAVRUPUQVPWAVQVCZKVQWEIVQWJIWAWNWEWJVQWLURAVR + VDVQUSUTVPVQKEWNKEAVAVQVBRVEVPWBALZBHZIZWCVPWBWOVRLZIWQAVRVFVPWRWPWOVPWGW + RWPPWIVQBQVGRVHSVPWOWPWCAVIWHWPWCNVPCBWCVJVNVKVLVSVOVM $. - ${ - axtgbtwnid.1 $e |- ( ph -> X e. P ) $. - axtgbtwnid.2 $e |- ( ph -> Y e. P ) $. - axtgbtwnid.3 $e |- ( ph -> Y e. ( X I X ) ) $. - $( Identity of Betweenness. Axiom A6 of [Schwabhauser] p. 11. - (Contributed by Thierry Arnoux, 15-Mar-2019.) $) - axtgbtwnid $p |- ( ph -> X = Y ) $= - ( vy vx cv co wcel wral vf vp vz vi vu vv va vt vs vb wi cstrkgb cstrkg - wceq cstrkgc cin cstrkgcb clng cfv csn cdif w3o crab cmpo citv wsbc cbs - cab df-trkg inss1 inss2 sstri eqsstri sselid wa cpw cvv istrkgb simprbi - wrex w3a simp1d syl id oveq12d eleq2d eqeq1 imbi12d eleq1 eqeq2 syl2anc - rspc2v mp2d ) AOQZPQZWODRZSZWOWNUNZUKZOBTPBTZGFFDRZSZFGUNZACULSZWTAUMUL - CUMUOULUPZUQUAQZURUSPOUBQZXGWOUTVAUCQZWOWNUDQZRSWOXHWNXIRSWNWOXHXIRSVBU - CXGVCVDUNUDXFVEUSVFUBXFVGUSVFUAVHUPZUPZULPOUCUAUDUBVIXKXEULXEXJVJUOULVK - VLVMKVNXDWTUEQZWOXHDRSUFQZWNXHDRSVOUGQZXLWNDRSXNXMWODRSVOUGBVTUKUFBTUEB - TUCBTOBTPBTZWOXNWNDRSOUHQZTPUIQZTUGBVTUJQWOWNDRSOXPTPXQTUJBVTUKUHBVPZTU - IXRTZXDCVQSWTXOXSWAPOUCUFUEUHBCDEUIUGUJHIJVRVSWBWCNAFBSGBSWTXBXCUKZUKLM - WSXTWNXASZFWNUNZUKPOFGBBWOFUNZWQYAWRYBYCWPXAWNYCWOFWOFDYCWDZYDWEWFWOFWN - WGWHWNGUNYAXBYBXCWNGXAWIWNGFWJWHWLWKWM $. - $} + $( Extend a surreal by a sequence of ordinals. (Contributed by Scott + Fenton, 30-Nov-2021.) $) + noextendseq $p |- ( ( A e. No /\ B e. On ) -> + ( A u. ( ( B \ dom A ) X. { X } ) ) e. No ) $= + ( csur wcel con0 wa cdm cun wfun crn wss mp2b cin c0 wceq eqtri adantr wi + cdif csn cxp c1o c2o cpr nofun wfn fnconstg fnfun wne dmxp ineq2i disjdif + snnzg funun mpan2 sylancl dmun uneq2i nodmon undif eleq1a adantl biimtrid + ssdif0 uneq2 un0 eqtrdi eleq1d biimprcd wo eloni ordtri2or2 syl2an mpjaod + word sylan eqeltrid rnun rnxpss snssi ax-mp sstri sylanblc eqsstrid elno2 + norn unss syl3anbrc ) AEFZBGFZHZABAIZUAZCUBZUCZJZKZWRIZGFWRLZUDUEUFZMWREF + WKWSWLWKAKZWQKZWSAUGCXBFZWQWOUHXDDWOCXBUIWOWQUJNXCXDHWNWQIZOZPQWSXGWNWOOP + XFWOWNXEWPPUKXFWOQDCXBUOWOWPULNZUMWNBUNRAWQUPUQURSWMWTWNWOJZGWTWNXFJXIAWQ + USXFWOWNXHUTRWKWNGFZWLXIGFZAVAXJWLHZWNBMZXKBWNMZXMXIBQZXLXKWNBVBWLXOXKTXJ + BGXIVCVDVEXNWOPQZXLXKBWNVFXJXPXKTWLXPXKXJXPXIWNGXPXIWNPJWNWOPWNVGWNVHVIVJ + VKSVEXJWNVQBVQXMXNVLWLWNVMBVMWNBVNVOVPVRVSWMXAALZWQLZJZXBAWQVTWMXQXBMZXRX + BMXSXBMWKXTWLAWHSXRWPXBWOWPWAXEWPXBMDCXBWBWCWDXQXRXBWIWEWFWRWGWJ $. - ${ - axtgpasch.1 $e |- ( ph -> X e. P ) $. - axtgpasch.2 $e |- ( ph -> Y e. P ) $. - axtgpasch.3 $e |- ( ph -> Z e. P ) $. - axtgpasch.4 $e |- ( ph -> U e. P ) $. - axtgpasch.5 $e |- ( ph -> V e. P ) $. - axtgpasch.6 $e |- ( ph -> U e. ( X I Z ) ) $. - axtgpasch.7 $e |- ( ph -> V e. ( Y I Z ) ) $. - $( Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given - triangle ` X Y Z ` , point ` U ` in segment ` X Z ` , and point ` V ` - in segment ` Y Z ` , there exists a point ` a ` on both the segment - ` U Y ` and the segment ` V X ` . This axiom is essentially a subset - of the general Pasch axiom. The general Pasch axiom asserts that on a - plane "a line intersecting a triangle in one of its sides, and not - intersecting any of the vertices, must intersect one of the other two - sides" (per the discussion about Axiom 7 of [Tarski1999] p. 179). The - (general) Pasch axiom was used implicitly by Euclid, but never stated; - Moritz Pasch discovered its omission in 1882. As noted in the - Metamath book, this means that the omission of Pasch's axiom from - Euclid went unnoticed for 2000 years. Only the inner Pasch algorithm - is included as an axiom; the "outer" form of the Pasch axiom can be - proved using the inner form (see theorem 9.6 of [Schwabhauser] p. 69 - and the brief discussion in axiom 7.1 of [Tarski1999] p. 180). - (Contributed by Thierry Arnoux, 15-Mar-2019.) $) - axtgpasch $p |- ( ph - -> E. a e. P ( a e. ( U I Y ) /\ a e. ( V I X ) ) ) $= - ( vu vv vx vz vy vf vp vi vt vs vb co wcel cv wa wrex wi cstrkgb cstrkg - wral cstrkgc cin cstrkgcb clng cfv csn cdif w3o crab cmpo wceq citv cbs - wsbc cab df-trkg inss1 inss2 eqsstri sselid cpw cvv w3a istrkgb simprbi - sstri simp2d oveq1 eleq2d anbi1d oveq2 rexbidv imbi12d 2ralbidv anbi12d - syl anbi2d imbi1d rspc3v syl3anc mpd eleq1 rspc2v syl2anc mp2and ) ACHJ - EUNZUOZGIJEUNZUOZKUPZCIEUNZUOZXLGHEUNZUOZUQZKBURZUAUBAUCUPZXHUOZUDUPZXJ - UOZUQZXLXSIEUNZUOZXLYAHEUNZUOZUQZKBURZUSZUDBVBUCBVBZXIXKUQZXRUSZAXSUEUP - ZUFUPZEUNZUOZYAUGUPZYOEUNZUOZUQZXLXSYREUNZUOZXLYAYNEUNZUOZUQZKBURZUSZUD - BVBUCBVBZUFBVBUGBVBUEBVBZYKADUTUOZUUJAVAUTDVAVCUTVDZVEUHUPZVFVGUEUGUIUP - ZUUNYNVHVIYOYNYRUJUPZUNUOYNYOYRUUOUNUOYRYNYOUUOUNUOVJUFUUNVKVLVMUJUUMVN - VGVPUIUUMVOVGVPUHVQVDZVDZUTUEUGUFUHUJUIVRUUQUULUTUULUUPVSVCUTVTWHWAOWBU - UKYRYNYNEUNUOYNYRVMUSUGBVBUEBVBZUUJYNXLYREUNUOUGUKUPZVBUEULUPZVBKBURUMU - PYNYREUNUOUGUUSVBUEUUTVBUMBURUSUKBWCZVBULUVAVBZUUKDWDUOUURUUJUVBWEUEUGU - FUDUCUKBDEFULKUMLMNWFWGWIWRAHBUOIBUOJBUOUUJYKUSPQRUUIYKXSHYOEUNZUOZYTUQ - ZUUCYGUQZKBURZUSZUDBVBUCBVBUVDYAIYOEUNZUOZUQZYIUSZUDBVBUCBVBUEUGUFHIJBB - BYNHVMZUUHUVHUCUDBBUVMUUAUVEUUGUVGUVMYQUVDYTUVMYPUVCXSYNHYOEWJWKWLUVMUU - FUVFKBUVMUUEYGUUCUVMUUDYFXLYNHYAEWMWKWSWNWOWPYRIVMZUVHUVLUCUDBBUVNUVEUV - KUVGYIUVNYTUVJUVDUVNYSUVIYAYRIYOEWJWKWSUVNUVFYHKBUVNUUCYEYGUVNUUBYDXLYR - IXSEWMWKWLWNWOWPYOJVMZUVLYJUCUDBBUVOUVKYCYIUVOUVDXTUVJYBUVOUVCXHXSYOJHE - WMWKUVOUVIXJYAYOJIEWMWKWQWTWPXAXBXCACBUOGBUOYKYMUSSTYJYMXIYBUQZXNYGUQZK - BURZUSUCUDCGBBXSCVMZYCUVPYIUVRUVSXTXIYBXSCXHXDWLUVSYHUVQKBUVSYEXNYGUVSY - DXMXLXSCIEWJWKWLWNWOYAGVMZUVPYLUVRXRUVTYBXKXIYAGXJXDWSUVTUVQXQKBUVTYGXP - XNUVTYFXOXLYAGHEWJWKWSWNWOXEXFXCXG $. - $} + $d A x y $. $d X x y $. + $( Calculate the place where a surreal and its extension differ. + (Contributed by Scott Fenton, 22-Nov-2021.) $) + noextenddif $p |- ( A e. No -> + |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } = + dom A ) $= + ( vy wcel cv cfv wne con0 wss c0 wn wceq syl sylibr fveq2 wa wo 3ad2ant1 + csur cdm cop csn cun crab cint nodmon nosgnn0i a1i word nodmord ndmfv wfn + ordirr cin nofun funfn sylib c1o c2o cpr fnsng sylancl disjsn snidg fvun2 + wfun syl112anc fvsng eqtrd 3netr4d neeq12d onintss sylc wi wb eloni eqcom + ordtri2 orbi1i orcom notbii bitrdi ordsseleq notbid ancoms bitr4d syl2anr + bitri w3a simp3 fvun1 eqcomd 3expia sylbird necon1ad ralrimiva weq ralrab + wral ssint eqssd ) BUAFZAGZBHZXEBBUBZCUCUDZUEZHZIZAJUFZUGZXGXDXGJFZXGBHZX + GXIHZIZXMXGKBUHZXDLCXOXPLCIXDCDUIUJXDXGXGFMZXOLNXDXGUKZXSBULZXGUOOZXGBUMO + XDXPXGXHHZCXDBXGUNZXHXGUDZUNZXGYEUPLNZXGYEFZXPYCNXDBVHYDBUQBURUSZXDXNCUTV + AVBZFZYFXRDXGCJYJVCVDZXDXSYGYBXGXGVEPZXDXNYHXRXGJVFOXGYEBXHXGVGVIXDXNYKYC + CNXRDXGCJYJVJVDVKVLXKXQAXGXEXGNXFXOXJXPXEXGBQXEXGXIQVMVNVOXDXGEGZKZEXLXAZ + XGXMKXDYNBHZYNXIHZIZYOVPZEJXAYPXDYTEJXDYNJFZRZYOYQYRUUBYOMZYNXGFZYQYRNZUU + AYNUKZXTUUDUUCVQXDYNVRYAUUFXTRZUUDXGYNFZXGYNNZSZMZUUCUUGUUDYNXGNZUUHSZMUU + KYNXGVTUUMUUJUUMUUIUUHSUUJUULUUIUUHYNXGVSWAUUIUUHWBWJWCWDXTUUFUUCUUKVQXTU + UFRYOUUJXGYNWEWFWGWHWIXDUUAUUDUUEXDUUAUUDWKZYRYQUUNYDYFYGUUDYRYQNXDUUAYDU + UDYITXDUUAYFUUDYLTXDUUAYGUUDYMTXDUUAUUDWLXGYEBXHYNWMVIWNWOWPWQWRXKYSYOEAJ + AEWSXFYQXJYRXEYNBQXEYNXIQVMWTPEXGXLXBPXC $. + $} - ${ - axtgcont.1 $e |- ( ph -> S C_ P ) $. - axtgcont.2 $e |- ( ph -> T C_ P ) $. - $( Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. This axiom - (scheme) asserts that any two sets ` S ` and ` T ` (of points) such - that the elements of ` S ` precede the elements of ` T ` with respect - to some point ` a ` (that is, ` x ` is between ` a ` and ` y ` - whenever ` x ` is in ` X ` and ` y ` is in ` Y ` ) are separated by - some point ` b ` ; this is explained in Axiom 11 of [Tarski1999] - p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.) $) - axtgcont1 $p |- ( ph -> - ( E. a e. P A. x e. S A. y e. T x e. ( a I y ) - -> E. b e. P A. x e. S A. y e. T b e. ( x I y ) ) ) $= - ( cv wcel wral vt vs vf vp vz vi vu vv co wi cpw cstrkgb cstrkg cstrkgc - wrex cin cstrkgcb clng cfv csn cdif w3o crab cmpo wceq citv cbs df-trkg - wsbc cab inss1 inss2 sstri eqsstri sselid wa cvv istrkgb simprbi simp3d - w3a syl wss wb fvexi ssex elpwg mpbird raleq rexbidv imbi12d rexralbidv - 3syl rspc2v syl2anc mpd ) ABRZJRZCRZHUISZCUARZTZBUBRZTZJDUOZKRWQWSHUISZ - CXATZBXCTZKDUOZUJZUADUKZTUBXKTZWTCFTZBETJDUOZXFCFTZBETKDUOZUJZAGULSZXLA - UMULGUMUNULUPZUQUCRZURUSBCUDRZYAWQUTVAUERZWQWSUFRZUISWQYBWSYCUISWSWQYBY - CUISVBUEYAVCVDVEUFXTVFUSVIUDXTVGUSVIUCVJUPZUPZULBCUEUCUFUDVHYEXSULXSYDV - KUNULVLVMVNOVOXRWSWQWQHUISWQWSVEUJCDTBDTZUGRZWQYBHUISUHRZWSYBHUISVPWRYG - WSHUISWRYHWQHUISVPJDUOUJUHDTUGDTUEDTCDTBDTZXLXRGVQSYFYIXLWABCUEUHUGUADG - HIUBJKLMNVRVSVTWBAEXKSZFXKSZXLXQUJAYJEDWCZPAYLEVQSYJYLWDPEDDGVGLWEZWFED - VQWGWMWHAYKFDWCZQAYNFVQSYKYNWDQFDYMWFFDVQWGWMWHXJXQXBBETZJDUOZXGBETZKDU - OZUJUBUAEFXKXKXCEVEZXEYPXIYRYSXDYOJDXBBXCEWIWJYSXHYQKDXGBXCEWIWJWKXAFVE - ZYPXNYRXPYTXBXMJBDEWTCXAFWIWLYTXGXOKBDEXFCXAFWIWLWKWNWOWP $. + ${ + $d A x $. + $( Extending a surreal with a negative sign results in a smaller surreal. + (Contributed by Scott Fenton, 22-Nov-2021.) $) + noextendlt $p |- ( A e. No -> ( A u. { <. dom A , 1o >. } ) A e. P ) $. - axtgcont.4 $e |- ( ( ph /\ u e. S /\ v e. T ) -> u e. ( A I v ) ) $. - $( Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more - information see ~ axtgcont1 . (Contributed by Thierry Arnoux, - 16-Mar-2019.) $) - axtgcont $p |- ( ph -> E. b e. P A. x e. S A. y e. T b e. ( x I y ) ) $= - ( va cv co wcel wral wrex 3expb ralrimivva wceq weq simplr simpll simpr - wa oveq12d eleq12d cbvraldva rspcev syl2anc axtgcont1 mpd ) ABUCZUBUCZC - UCZKUDZUEZCIUFZBHUFZUBGUGZMUCVCVEKUDUECIUFBHUFMGUGAFGUEEUCZFDUCZKUDZUEZ - DIUFZEHUFZVJTAVNEDHIAVKHUEVLIUEVNUAUHUIVIVPUBFGVDFUJZVHVOBEHVQBEUKZUOZV - GVNCDIVSCDUKZUOZVCVKVFVMVQVRVTULWAVDFVEVLKVQVRVTUMVSVTUNUPUQURURUSUTABC - GHIJKLUBMNOPQRSVAVB $. - $} + $( Extending a surreal with a positive sign results in a bigger surreal. + (Contributed by Scott Fenton, 22-Nov-2021.) $) + noextendgt $p |- ( A e. No -> A . } ) ) $= + ( vx csur wcel c2o cop csn wbr cfv con0 c1o c0 wceq wa syl wfn 2on sylibr + sylancl fvex cdm cun cslt cv wne crab cint ctp w3o wn word nodmord ordirr + ndmfv cin wfun nofun funfn sylib nodmon fnsng snidg fvun2 syl112anc fvsng + disjsn eqtrd jca 3mix3d brtp elexi noextenddif fveq2d 3brtr4d wb noextend + prid2 sltval2 mpdan mpbird ) ACDZAAAUAZEFGZUBZUCHZBUDZAIWFWDIUEBJUFUGZAIZ + WGWDIZKLFKEFLEFUHZHZWAWBAIZWBWDIZWHWIWJWAWLKMZWMLMNZWNWMEMZNZWLLMZWPNZUIW + LWMWJHWAWSWOWQWAWRWPWAWBWBDUJZWRWAWBUKWTAULWBUMOZWBAUNOWAWMWBWCIZEWAAWBPZ + WCWBGZPZWBXDUOLMZWBXDDZWMXBMWAAUPXCAUQAURUSWAWBJDZEJDZXEAUTZQWBEJJVASWAWT + XFXAWBWBVFRWAXHXGXJWBJVBOWBXDAWCWBVCVDWAXHXIXBEMXJQWBEJJVESVGVHVIKLKELEWL + WMWBATWBWDTVJRWAWGWBABAEKEEJQVKVQZVLZVMWAWGWBWDXLVMVNWAWDCDWEWKVOAEXKVPAW + DBVRVSVT $. $} ${ - $d u v x y z .- $. $d u v x y z I $. $d u v x y z P $. - axtrkge.p $e |- P = ( Base ` G ) $. - axtrkge.d $e |- .- = ( dist ` G ) $. - axtrkge.i $e |- I = ( Itv ` G ) $. - ${ - axtglowdim2.v $e |- ( ph -> G e. V ) $. - axtglowdim2.g $e |- ( ph -> G TarskiGDim>= 2 ) $. - $( Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] - p. 13. There exist 3 non-colinear points. (Contributed by Thierry - Arnoux, 20-Nov-2019.) $) - axtglowdim2 $p |- ( ph -> E. x e. P E. y e. P E. z e. P - -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) $= - ( c2 cstrkgld cv co wcel wrex wbr w3o wn wb istrkg2ld syl mpbid ) AFOPU - AZDQZBQZCQZGRSUJUIUKGRSUKUJUIGRSUBUCDETCETBETZNAFISUHULUDMBCDEFGHIJKLUE - UFUG $. - $} + $d A x y $. $d B x y $. $d X x y $. + $( Given ` A ` less-than or equal to ` B ` , equal to ` B ` up to ` X ` , + and ` A ( X ) = 2o ` , then ` B ( X ) = 2o ` . (Contributed by Scott + Fenton, 6-Dec-2021.) $) + nolesgn2o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ + ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B + ( B ` X ) = 2o ) $= + ( vy vx csur wcel con0 w3a cres wceq cfv c2o wa wbr wn c0 c1o wo cop cslt + w3o simpl2 nofv syl 3orel3 syl5com cv wral ctp simp13 fveq1 eqcomd adantr + simpr fvresd 3eqtr3d ralrimiva 3ad2ant2 simprr ancld orim12d 3impia 3mix3 + wrex a1d 3mix2 jaoi fvex brtp sylibr raleq fveq2 breq12d anbi12d syl12anc + rspcev wb simp12 simp11 sltval syl2anc mpbird 3expia syld con1d ) AFGZBFG + ZCHGZIZACJZBCJZKZCALZMKZNZBAUAOZPCBLZMKZWJWPNZWSWQWTWSPZWRQKZWRRKZSZWQWTX + BXCWSUBZXAXDWTWHXEWGWHWIWPUCBCUDUEXBXCWSUFUGWJWPXDWQWJWPXDIZWQDUHZBLZXGAL + ZKZDEUHZUIZXKBLZXKALZRQTRMTQMTUJZOZNZEHVEZXFWIXJDCUIZWRWNXOOZXRWGWHWIWPXD + UKWPWJXSXDWMXSWOWMXJDCWMXGCGZNZXGWLLZXGWKLZXHXIWMYCYDKYAWMYDYCXGWKWLULUMU + NYBXGCBWMYAUOZUPYBXGCAYEUPUQURUNUSXFXCWNQKNZXCWONZXBWONZUBZXTXFYHYGSZYIWJ + WPXDYJWTXBYHXCYGWTXBWOWTWOXBWJWMWOUTZVFVAWTXCWOWTWOXCYKVFVAVBVCYHYIYGYHYF + YGVDYGYFYHVGVHUERQRMQMWRWNCBVICAVIVJVKXQXSXTNECHXKCKZXLXSXPXTXJDXKCVLYLXM + WRXNWNXOXKCBVMXKCAVMVNVOVQVPXFWHWGWQXRVRWGWHWIWPXDVSWGWHWIWPXDVTEDBAWAWBW + CWDWEWFVC $. - ${ - $d u v x y z .- $. $d u v x y z I $. $d u v x y z P $. $d u v z Z $. - $d u v x y z X $. $d u v y z Y $. $d u v x y z U $. $d v x y z V $. - axtgupdim2.x $e |- ( ph -> X e. P ) $. - axtgupdim2.y $e |- ( ph -> Y e. P ) $. - axtgupdim2.z $e |- ( ph -> Z e. P ) $. - axtgupdim2.u $e |- ( ph -> U e. P ) $. - axtgupdim2.v $e |- ( ph -> V e. P ) $. - axtgupdim2.0 $e |- ( ph -> U =/= V ) $. - axtgupdim2.1 $e |- ( ph -> ( U .- X ) = ( V .- X ) ) $. - axtgupdim2.2 $e |- ( ph -> ( U .- Y ) = ( V .- Y ) ) $. - axtgupdim2.3 $e |- ( ph -> ( U .- Z ) = ( V .- Z ) ) $. - axtgupdim2.w $e |- ( ph -> G e. V ) $. - axtgupdim2.g $e |- ( ph -> -. G TarskiGDim>= 3 ) $. - $( Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] - p. 13. Three points ` X ` , ` Y ` and ` Z ` equidistant to two given - two points ` U ` and ` V ` must be colinear. (Contributed by Thierry - Arnoux, 29-May-2019.) (Revised by Thierry Arnoux, 11-Jul-2020.) $) - axtgupdim2 $p |- ( ph -> - ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) $= - ( vx vy vz vu vv co wcel w3o wceq wn w3a wa wi cv wral wrex c3 cstrkgld - wne wbr istrkg3ld syl mtbid ralnex2 sylibr neeq1 oveq1 eqeq1d 3anbi123d - wb anbi1d rexbidv 2rexbidv anbi12d notbid neeq2 eqeq2d rspc2v mpd imnan - syl2anc ralnex3 eqeq12d 3anbi1d eleq2d 3orbi123d 3anbi2d 3anbi3d rspc3v - oveq2 eleq1 syl3anc mp3and notnotrd ) AJHIEUJZUKZHJIEUJZUKZIHJEUJZUKZUL - ZACHFUJZGHFUJZUMZCIFUJZGIFUJZUMZCJFUJZGJFUJZUMZXEUNZUNZTUAUBAXHXKXNUOZX - OUPZUNZXQXPUQACUEURZFUJZGXTFUJZUMZCUFURZFUJZGYDFUJZUMZCUGURZFUJZGYHFUJZ - UMZUOZYHXTYDEUJZUKZXTYHYDEUJZUKZYDXTYHEUJZUKZULZUNZUPZUNZUGBUSUFBUSUEBU - SZXSAUUAUGBUTZUFBUTUEBUTZUNZUUCACGVCZUUFSAUUGUUEUPZUNZUUGUUFUQAUHURZUIU - RZVCZUUJXTFUJZUUKXTFUJZUMZUUJYDFUJZUUKYDFUJZUMZUUJYHFUJZUUKYHFUJZUMZUOZ - YTUPZUGBUTZUFBUTUEBUTZUPZUNZUIBUSUHBUSZUUIAUVFUIBUTUHBUTZUNUVHADVAVBVDZ - UVIUDADGUKUVJUVIVNUCUEUFUGUIUHBDEFGKLMVEVFVGUVFUHUIBBVHVIACBUKGBUKUVHUU - IUQQRUVGUUICUUKVCZYAUUNUMZYEUUQUMZYIUUTUMZUOZYTUPZUGBUTZUFBUTUEBUTZUPZU - NUHUICGBBUUJCUMZUVFUVSUVTUULUVKUVEUVRUUJCUUKVJUVTUVDUVQUEUFBBUVTUVCUVPU - GBUVTUVBUVOYTUVTUUOUVLUURUVMUVAUVNUVTUUMYAUUNUUJCXTFVKVLUVTUUPYEUUQUUJC - YDFVKVLUVTUUSYIUUTUUJCYHFVKVLVMVOVPVQVRVSUUKGUMZUVSUUHUWAUVKUUGUVRUUEUU - KGCVTUWAUVQUUDUEUFBBUWAUVPUUAUGBUWAUVOYLYTUWAUVLYCUVMYGUVNYKUWAUUNYBYAU - UKGXTFVKWAUWAUUQYFYEUUKGYDFVKWAUWAUUTYJYIUUKGYHFVKWAVMVOVPVQVRVSWBWEWCU - UGUUEWDVIWCUUAUEUFUGBBBWFVIAHBUKIBUKJBUKUUCXSUQNOPUUBXSXHYGYKUOZYHHYDEU - JZUKZHYOUKZYDHYHEUJZUKZULZUNZUPZUNXHXKYKUOZYHWSUKZHYHIEUJZUKZIUWFUKZULZ - UNZUPZUNUEUFUGHIJBBBXTHUMZUUAUWJUWSYLUWBYTUWIUWSYCXHYGYKUWSYAXFYBXGXTHC - FWNXTHGFWNWGWHUWSYSUWHUWSYNUWDYPUWEYRUWGUWSYMUWCYHXTHYDEVKWIXTHYOWOUWSY - QUWFYDXTHYHEVKWIWJVSVRVSYDIUMZUWJUWRUWTUWBUWKUWIUWQUWTYGXKXHYKUWTYEXIYF - XJYDICFWNYDIGFWNWGWKUWTUWHUWPUWTUWDUWLUWEUWNUWGUWOUWTUWCWSYHYDIHEWNWIUW - TYOUWMHYDIYHEWNWIYDIUWFWOWJVSVRVSYHJUMZUWRXRUXAUWKXQUWQXOUXAYKXNXHXKUXA - YIXLYJXMYHJCFWNYHJGFWNWGWLUXAUWPXEUXAUWLWTUWNXBUWOXDYHJWSWOUXAUWMXAHYHJ - IEVKWIUXAUWFXCIYHJHEWNWIWJVSVRVSWMWPWCXQXOWDVIWQWR $. - $} + $( Given ` A ` less-than or equal to ` B ` , equal to ` B ` up to ` X ` , + and ` A ( X ) = 2o ` , then ` ( A |`` suc X ) = ( B |`` suc X ) ` . + (Contributed by Scott Fenton, 6-Dec-2021.) $) + nolesgn2ores $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ + ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B + ( A |` suc X ) = ( B |` suc X ) ) $= + ( vx csur wcel con0 w3a cres wceq cfv c2o wa wn cdm syl wne fvresd wfun + c0 cslt wbr csuc cv wral cin dmres wss simp11 nodmord ndmfv c1o 2on elexi + word prid2 nosgnn0i neeq1 mpbiri neneqd con4i adantl 3ad2ant2 df-ss sylib + ordsucss eqtrid simp12 nolesgn2o eqtr4d eleq2d wo vex elsuc simp2l fveq1d + sylc adantr simpr 3eqtr3d ex simp2r fveq2 eqeq12d syl5ibrcom biimtrid imp + jaod 3eqtr4d sylbid ralrimiv nofun funres 3syl eqfunfv syl2anc mpbir2and + wb ) AEFZBEFZCGFZHZACIZBCIZJZCAKZLJZMZBAUAUBNZHZACUCZIZBXKIZJZXLOZXMOZJZD + UDZXLKZXRXMKZJZDXOUEZXJXOXKXPXJXOXKAOZUFZXKAXKUGXJXKYCUHZYDXKJXJYCUOZCYCF + ZYEXJWSYFWSWTXAXHXIUIZAUJPXHXBYGXIXGYGXEYGXGYGNXFTJZXGNCAUKYIXFLYIXFLQTLQ + ZLULLLGUMUNUPUQZXFTLURUSUTPVAVBVCCYCVFVQXKYCVDVEVGZXJXPXKBOZUFZXKBXKUGXJX + KYMUHZYNXKJXJYMUOZCYMFZYOXJWTYPWSWTXAXHXIVHZBUJPXJCBKZLJZYQABCVIZYQYTYQNY + STJZYTNCBUKUUBYSLUUBYSLQYJYKYSTLURUSUTPVAPCYMVFVQXKYMVDVEVGVJXJYADXOXJXRX + OFXRXKFZYAXJXOXKXRYLVKXJUUCYAXJUUCMZXRAKZXRBKZXSXTXJUUCUUEUUFJZUUCXRCFZXR + CJZVLXJUUGXRCDVMVNXJUUHUUGUUIXJUUHUUGXJUUHMZXRXCKZXRXDKZUUEUUFXJUUKUULJUU + HXJXRXCXDXBXEXGXIVOVPVRUUJXRCAXJUUHVSZRUUJXRCBUUMRVTWAXJUUGUUIXFYSJXJXFLY + SXBXEXGXIWBUUAVJUUIUUEXFUUFYSXRCAWCXRCBWCWDWEWHWFWGUUDXRXKAXJUUCVSZRUUDXR + XKBUUNRWIWAWJWKXJXLSZXMSZXNXQYBMWRXJWSASUUOYHAWLXKAWMWNXJWTBSUUPYRBWLXKBW + MWNDXLXMWOWPWQ $. - ${ - $d a b u v x y z I $. $d a b u v x y z P $. $d a b v V $. - $d a b u v U $. $d a b u v x y z X $. $d a b u v y z Y $. - $d a b u v z Z $. $d a b u v x y z .- $. - axtgeucl.g $e |- ( ph -> G e. TarskiGE ) $. - axtgeucl.1 $e |- ( ph -> X e. P ) $. - axtgeucl.2 $e |- ( ph -> Y e. P ) $. - axtgeucl.3 $e |- ( ph -> Z e. P ) $. - axtgeucl.4 $e |- ( ph -> U e. P ) $. - axtgeucl.5 $e |- ( ph -> V e. P ) $. - axtgeucl.6 $e |- ( ph -> U e. ( X I V ) ) $. - axtgeucl.7 $e |- ( ph -> U e. ( Y I Z ) ) $. - axtgeucl.8 $e |- ( ph -> X =/= U ) $. - $( Euclid's Axiom. Axiom A10 of [Schwabhauser] p. 13. This is - equivalent to Euclid's parallel postulate when combined with other - axioms. (Contributed by Thierry Arnoux, 16-Mar-2019.) $) - axtgeucl $p |- ( ph -> E. a e. P E. b e. P - ( Y e. ( X I a ) /\ Z e. ( X I b ) /\ V e. ( a I b ) ) ) $= - ( vu vv vx vy vz co wcel wne cv w3a wrex wi wral cvv cstrkge wa istrkge - sylib simprd wceq oveq1 eleq2d neeq1 3anbi13d 3anbi12d 2rexbidv imbi12d - 2ralbidv 3anbi2d eleq1 3anbi1d oveq2 rspc3v syl3anc mpd neeq2 3anbi123d - imbi1d 3anbi3d rspc2v syl2anc mp3and ) ACHGEUJZUKZCIJEUJZUKZHCULZIHKUMZ - EUJZUKZJHLUMZEUJZUKZGWLWOEUJZUKZUNZLBUOKBUOZUBUCUDAUEUMZHUFUMZEUJZUKZXB - WIUKZHXBULZUNZWNWQXCWRUKZUNZLBUOKBUOZUPZUFBUQUEBUQZWHWJWKUNZXAUPZAXBUGU - MZXCEUJZUKZXBUHUMZUIUMZEUJZUKZXPXBULZUNZXSXPWLEUJZUKZXTXPWOEUJZUKZXIUNZ - LBUOKBUOZUPZUFBUQUEBUQZUIBUQUHBUQUGBUQZXMADURUKZYMADUSUKYNYMUTPUGUHUIUF - UEBDEFKLMNOVAVBVCAHBUKIBUKJBUKYMXMUPQRSYLXMXEYBXGUNZXSWMUKZXTWPUKZXIUNZ - LBUOKBUOZUPZUFBUQUEBUQXEXBIXTEUJZUKZXGUNZWNYQXIUNZLBUOKBUOZUPZUFBUQUEBU - QUGUHUIHIJBBBXPHVDZYKYTUEUFBBUUGYDYOYJYSUUGXRXEYCXGYBUUGXQXDXBXPHXCEVEV - FXPHXBVGVHUUGYIYRKLBBUUGYFYPYHYQXIUUGYEWMXSXPHWLEVEVFUUGYGWPXTXPHWOEVEV - FVIVJVKVLXSIVDZYTUUFUEUFBBUUHYOUUCYSUUEUUHYBUUBXEXGUUHYAUUAXBXSIXTEVEVF - VMUUHYRUUDKLBBUUHYPWNYQXIXSIWMVNVOVJVKVLXTJVDZUUFXLUEUFBBUUIUUCXHUUEXKU - UIUUBXFXEXGUUIUUAWIXBXTJIEVPVFVMUUIUUDXJKLBBUUIYQWQWNXIXTJWPVNVMVJVKVLV - QVRVSACBUKGBUKXMXOUPTUAXLXOCXDUKZWJWKUNZXKUPUEUFCGBBXBCVDZXHUUKXKUULXEU - UJXFWJXGWKXBCXDVNXBCWIVNXBCHVTWAWBXCGVDZUUKXNXKXAUUMUUJWHWJWKUUMXDWGCXC - GHEVPVFVOUUMXJWTKLBBUUMXIWSWNWQXCGWRVNWCVJVKWDWEVSWF $. - $} - $} + $( Given ` A ` greater than or equal to ` B ` , equal to ` B ` up to + ` X ` , and ` A ( X ) = 1o ` , then ` B ( X ) = 1o ` . (Contributed by + Scott Fenton, 9-Aug-2024.) $) + nogesgn1o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ + ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A + ( B ` X ) = 1o ) $= + ( vy vx csur wcel con0 w3a cres wceq cfv c1o wa wbr wn c0 c2o wo cop cslt + w3o simpl2 nofv syl 3orel2 syl5com cv wral wrex simp13 fveq1 adantr simpr + ctp fvresd 3eqtr3d ralrimiva 3ad2ant2 simp2r simp3 andi sylib 3mix1 3mix2 + jca jaoi fvex brtp sylibr raleq breq12d anbi12d rspcev syl12anc wb simp11 + fveq2 simp12 sltval syl2anc mpbird 3expia syld con1d 3impia ) AFGZBFGZCHG + ZIZACJZBCJZKZCALZMKZNZABUAOZPCBLZMKZWJWPNZWSWQWTWSPZWRQKZWRRKZSZWQWTXBWSX + CUBZXAXDWTWHXEWGWHWIWPUCBCUDUEXBWSXCUFUGWJWPXDWQWJWPXDIZWQDUHZALZXGBLZKZD + EUHZUIZXKALZXKBLZMQTMRTQRTUOZOZNZEHUJZXFWIXJDCUIZWNWRXOOZXRWGWHWIWPXDUKWP + WJXSXDWMXSWOWMXJDCWMXGCGZNZXGWKLZXGWLLZXHXIWMYCYDKYAXGWKWLULUMYBXGCAWMYAU + NZUPYBXGCBYEUPUQURUMUSXFWOXBNZWOXCNZWNQKXCNZUBZXTXFYFYGSZYIXFWOXDNYJXFWOX + DWJWMWOXDUTWJWPXDVAVFWOXBXCVBVCYFYIYGYFYGYHVDYGYFYHVEVGUEMQMRQRWNWRCAVHCB + VHVIVJXQXSXTNECHXKCKZXLXSXPXTXJDXKCVKYKXMWNXNWRXOXKCAVRXKCBVRVLVMVNVOXFWG + WHWQXRVPWGWHWIWPXDVQWGWHWIWPXDVSEDABVTWAWBWCWDWEWF $. + $( Given ` A ` greater than or equal to ` B ` , equal to ` B ` up to + ` X ` , and ` A ( X ) = 1o ` , then + ` ( A |`` suc X ) = ( B |`` suc X ) ` . (Contributed by Scott Fenton, + 6-Dec-2021.) $) + nogesgn1ores $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ + ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A + ( A |` suc X ) = ( B |` suc X ) ) $= + ( vx csur wcel w3a cres wceq cfv c1o wa wn cdm cin syl c0 wne fvresd wfun + con0 cslt wbr csuc cv wral dmres wss word simp11 nodmord ndmfv 1n0 necomi + neeq1 mpbiri neneqd con4i adantl 3ad2ant2 ordsucss df-ss eqtrid nogesgn1o + sylc sylib simp12 eqtr4d eleq2d wo vex elsuc simpl2l fveq1d simpr 3eqtr3d + simp2r fveq2 eqeq12d syl5ibrcom jaod biimtrid imp 3eqtr4d sylbid ralrimiv + ex wb nofun funresd eqfunfv syl2anc mpbir2and ) AEFZBEFZCUAFZGZACHZBCHZIZ + CAJZKIZLZABUBUCMZGZACUDZHZBXFHZIZXGNZXHNZIZDUEZXGJZXMXHJZIZDXJUFZXEXJXFXK + XEXJXFANZOZXFAXFUGXEXFXRUHZXSXFIXEXRUIZCXRFZXTXEWNYAWNWOWPXCXDUJZAUKPXCWQ + YBXDXBYBWTYBXBYBMXAQIZXBMCAULYDXAKYDXAKRQKRZKQUMUNZXAQKUOUPUQPURUSUTCXRVA + VEXFXRVBVFVCZXEXKXFBNZOZXFBXFUGXEXFYHUHZYIXFIXEYHUIZCYHFZYJXEWOYKWNWOWPXC + XDVGZBUKPXECBJZKIZYLABCVDZYLYOYLMYNQIZYOMCBULYQYNKYQYNKRYEYFYNQKUOUPUQPUR + PCYHVAVEXFYHVBVFVCVHXEXPDXJXEXMXJFXMXFFZXPXEXJXFXMYGVIXEYRXPXEYRLZXMAJZXM + BJZXNXOXEYRYTUUAIZYRXMCFZXMCIZVJXEUUBXMCDVKVLXEUUCUUBUUDXEUUCUUBXEUUCLZXM + WRJXMWSJYTUUAUUEXMWRWSWTXBWQXDUUCVMVNUUEXMCAXEUUCVOZSUUEXMCBUUFSVPWGXEUUB + UUDXAYNIXEXAKYNWQWTXBXDVQYPVHUUDYTXAUUAYNXMCAVRXMCBVRVSVTWAWBWCYSXMXFAXEY + RVOZSYSXMXFBUUGSWDWGWEWFXEXGTXHTXIXLXQLWHXEXFAXEWNATYCAWIPWJXEXFBXEWOBTYM + BWIPWJDXGXHWKWLWM $. + $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Justification for the congruence notation + Ordering =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ - $d A r u v x y z $. $d F r u v x y z $. $d V u v $. - $( Given any function ` F ` , equality of the image by ` F ` is an - equivalence relation. (Contributed by Thierry Arnoux, 25-Jan-2023.) $) - tgjustf $p |- ( A e. V -> E. r ( r Er A - /\ A. x e. A A. y e. A ( x r y <-> ( F ` x ) = ( F ` y ) ) ) ) $= - ( vu vv wcel cv wa cfv wceq wbr weq simpl fveq2d simpr eqeq12d brab2a wer - vz copab wb wral relopabv ancom eqcom anbi12i eqid 3bitr4i biimpi simplll - simprlr simplr simprr eqtrd jca31 3imtr4i biantru pm4.24 iseri baib rgen2 - wex cvv wi id simprll opabex2 ereq1 breqd bibi1d 2ralbidva anbi12d spcegv - syl mp2ani ) CEIZCGJZCIZHJZCIZKVTDLZWBDLZMZKZGHUCZUAZAJZBJZWHNZWJDLZWKDLZ - MZUDZBCUEACUEZCFJZUAZWJWKWRNZWOUDZBCUEACUEZKZFVEZABUBCWHWGGHUFWLWKWJWHNZW - JCIZWKCIZKZWOKZXGXFKZWNWMMZKWLXEXHXJWOXKXFXGUGWMWNUHUIWFWOGHWJWKCCWHGAOZH - BOZKZWDWMWEWNXNVTWJDXLXMPQXNWBWKDXLXMRQSWHUJZTZWFXKGHWKWJCCWHGBOZHAOZKZWD - WNWEWMXSVTWKDXQXRPQXSWBWJDXQXRRQSXOTUKULXIXGUBJZCIZKZWNXTDLZMZKZKZXFYAKWM - YCMZKWLWKXTWHNZKWJXTWHNYFXFYAYGXFXGWOYEUMXIXGYAYDUNYFWMWNYCXHWOYEUOXIYBYD - UPUQURWLXIYHYEXPWFYDGHWKXTCCWHXQHUBOZKZWDWNWEYCYJVTWKDXQYIPQYJWBXTDXQYIRQ - SXOTUIWFYGGHWJXTCCWHXLYIKZWDWMWEYCYKVTWJDXLYIPQYKWBXTDXLYIRQSXOTUSXFXFKZY - LWMWMMZKXFWJWJWHNYMYLWMUJUTXFVAWFYMGHWJWJCCWHXLXRKZWDWMWEWMYNVTWJDXLXRPQY - NWBWJDXLXRRQSXOTUKVBWPABCCWLXHWOXPVCVDVSWHVFIWIWQKZXDVGVSWGGHCCEEVSVHZYPV - SWAWCWFVIVSWAWCWFUNVJXCYOFWHVFWRWHMZWSWIXBWQCWRWHVKYQXAWPABCCYQXHKZWTWLWO - YRWRWHWJWKYQXHPVLVMVNVOVPVQVR $. - $} - - ${ - $d A f u x y $. $d R f u x y $. $d V u x y $. - $( Given any equivalence relation ` R ` , one can define a function ` f ` - such that all elements of an equivalence classe of ` R ` have the same - image by ` f ` . (Contributed by Thierry Arnoux, 25-Jan-2023.) $) - tgjustr $p |- ( ( A e. V /\ R Er A ) -> E. f ( f Fn A - /\ A. x e. A A. y e. A ( x R y <-> ( f ` x ) = ( f ` y ) ) ) ) $= - ( vu wcel wa cv cec wfn cfv wceq wb wral cvv syl adantr ralrimiva wer wbr - cmpt wex erex ecexg eqid fnmpt simpllr simpr erth cqs eceq1 ecelqsg sylan - impcom fvmptd3 adantlr eqeq12d bitr4d wi mptexg fneq1 simpl fveq1d bibi2d - 2ralbidva anbi12d spcegv mp2and ) CFHZCDUAZIZGCGJZDKZUCZCLZAJZBJZDUBZVRVP - MZVSVPMZNZOZBCPZACPZEJZCLZVTVRWGMZVSWGMZNZOZBCPACPZIZEUDZVMVOQHZGCPVQVMWP - GCVMWPVNCHVMDQHZWPVLVKWQCDFUEUPZVNQDUFRSTGCVOVPQVPUGZUHRVMWEACVMVRCHZIZWD - BCXAVSCHZIZVTVRDKZVSDKZNWCXCVRVSDCVKVLWTXBUIXAWTXBVMWTUJZSUKXCWAXDWBXEXAW - AXDNXBXAGVRVOXDCVPCDULZWSVNVRDUMXFVMWQWTXDXGHWRCVRDQUNUOUQSVMXBWBXENWTVMX - BIGVSVOXECVPXGWSVNVSDUMVMXBUJVMWQXBXEXGHWRCVSDQUNUOUQURUSUTTTVMVPQHZVQWFI - ZWOVAVKXHVLGCVOFVBSWNXIEVPQWGVPNZWHVQWMWFCWGVPVCXJWLWDABCCXJWTXBIZIZWKWCV - TXLWIWAWJWBXLVRWGVPXJXKVDZVEXLVSWGVPXMVEUSVFVGVHVIRVJ $. - $} - - ${ - $d .- r u v w x y z $. $d P r u v w x y z $. - tgjustc1.p $e |- P = ( Base ` G ) $. - tgjustc1.d $e |- .- = ( dist ` G ) $. - $( A justification for using distance equality instead of the textbook - relation on pairs of points for congruence. (Contributed by Thierry - Arnoux, 29-Jan-2023.) $) - tgjustc1 $p |- E. r ( r Er ( P X. P ) - /\ A. w e. P A. x e. P A. y e. P A. z e. P - ( <. w , x >. r <. y , z >. <-> ( w .- x ) = ( y .- z ) ) ) $= - ( vu vv cv wbr cfv wceq wb wral wa wcel cxp wer cop co cvv wex fvexi xpex - cbs tgjustf ax-mp simplrl simplrr opelxpd simprl simprr breq1 fveq2 df-ov - simpll eqtr4di eqeq1d bibi12d breq2 eqeq2d rspc2va syl21anc anim2i eximii - ralrimivva ) EEUAZHMZUBZKMZLMZVLNZVNGOZVOGOZPZQZLVKRKVKRZSZVMDMZAMZUCZBMZ - CMZUCZVLNZWCWDGUDZWFWGGUDZPZQZCERBERZAERDERZSHVKUETWBHUFEEEFUIIUGZWPUHKLV - KGUEHUJUKWAWOVMWAWNDAEEWAWCETZWDETZSZSZWMBCEEWTWFETZWGETZSZSZWEVKTWHVKTWA - WMXDWCWDEEWAWQWRXCULWAWQWRXCUMUNXDWFWGEEWTXAXBUOWTXAXBUPUNWAWSXCUTVTWMWEV - OVLNZWJVRPZQKLWEWHVKVKVNWEPZVPXEVSXFVNWEVOVLUQXGVQWJVRXGVQWEGOWJVNWEGURWC - WDGUSVAVBVCVOWHPZXEWIXFWLVOWHWEVLVDXHVRWKWJXHVRWHGOWKVOWHGURWFWGGUSVAVEVC - VFVGVJVJVHVI $. + $d x y z $. + $( Lemma for ~ sltso . The "sign expansion" binary relation totally orders + the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.) $) + sltsolem1 $p |- { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } + Or ( { 1o , 2o } u. { (/) } ) $= + ( vx vy vz c1o c2o c0 cop cv wbr wcel wceq wa w3o eqtr2 brtp expcom 3jaod + mto ex ad2ant2lr ctp wor cpr csn cun wn 1n0 neii con0 wb 0elon csuc df-2o + 1on df-1o eqeq12i suc11 bitrid nemtbir ancoms nsuceq0 eqeq1i 3pm3.2ni vex + mp2an mtbir a1i w3a pm2.21i ad2ant2rl 3mix2 3jaoi imp syl2anb sylibr eltp + wi eqtr3 3mix2d 3mix1d 3mix1 3mix3 biid 3orbi123i issoi df-tp soeq2 ax-mp + 3mix3d mpbi ) DEFUAZDFGDEGFEGUAZUBZDEUCFUDUEZWLUBZABCWKWLAHZWPWLIZUFWPWKJ + ZWQWPDKZWPFKZLZWSWPEKZLZWTXBLZMXAXCXDXADFKZDFUGUHZWPDFNRXCEDKZXGDFUGDUIJZ + FUIJZXGXEUJUNUKXGDULZFULZKXHXILXEEXJDXKUMUOUPDFUQURVEUSZXBWSXGWPEDNUTRXDE + FKZXMXJFDVAEXJFUMVBUSZXBWTXMWPEFNUTRVCDFDEFEWPWPAVDZXOOVFVGWPBHZWLIZXPCHZ + WLIZLZWPXRWLIZVQWRXPWKJZXRWKJVHXTWSXRFKZLZWSXREKZLZWTYELZMZYAXQWSXPFKZLZW + SXPEKZLZWTYKLZMZXPDKZYCLZYOYELZYIYELZMZYHXSDFDEFEWPXPXOBVDZOZDFDEFEXPXRYT + CVDZOYNYSYHYJYSYHVQYLYMYJYPYHYQYRYPYJYHYOYIYHYCWSYOYILZYHUUCXEXFXPDFNRVIZ + VJPYQYJYHYOYIYHYEWSUUDVJPYJYRYHWSYEYHYIYIYFYDYGVKVJSQYLYPYHYQYRYLYPYHYKYO + YHWSYCYKYOLZYHUUEXGXLXPEDNRVIZTSYLYQYHYKYOYHWSYEUUFTSYLYRYHYKYIYHWSYEYKYI + LZYHUUGXMXNXPEFNRVIZTSQYMYPYHYQYRYMYPYHYKYOYHWTYCUUFTSYMYQYHYKYOYHWTYEUUF + TSYMYRYHYKYIYHWTYEUUHTSQVLVMVNDFDEFEWPXRXOUUBOVOVGWRYBLYNWPXPKZYOWTLZYOXB + LZYIXBLZMZMZXQUUIXPWPWLIZMWRWSXBWTMZYOYKYIMZUUNYBWPDEFXOVPXPDEFYTVPUUPUUQ + UUNWSUUQUUNVQXBWTWSYOUUNYKYIWSYOUUNWSYOLUUIYNUUMWPXPDVRVSSWSYKUUNYLYNUUIU + UMYLYJYMVKVTSWSYIUUNYJYNUUIUUMYJYLYMWAVTSQXBYOUUNYKYIYOXBUUNUUKUUMYNUUIUU + KUUJUULVKWIPXBYKUUNXBYKLUUIYNUUMWPXPEVRVSSYIXBUUNUULUUMYNUUIUULUUJUUKWBWI + PQWTYOUUNYKYIYOWTUUNUUJUUMYNUUIUUJUUKUULWAWIPWTYKUUNYMYNUUIUUMYMYJYLWBVTS + WTYIUUNWTYILUUIYNUUMWPXPFVRVSSQVLVMVNXQYNUUIUUIUUOUUMUUAUUIWCDFDEFEXPWPYT + XOOWDVOWEWKWNKWMWOUJDEFWFWKWNWLWGWHWJ $. $} ${ - $d P d u v w x y z $. $d R d u v w x y z $. - tgjustc2.p $e |- P = ( Base ` G ) $. - tgjustc2.d $e |- R Er ( P X. P ) $. - $( A justification for using distance equality instead of the textbook - relation on pairs of points for congruence. (Contributed by Thierry - Arnoux, 29-Jan-2023.) $) - tgjustc2 $p |- E. d ( d Fn ( P X. P ) - /\ A. w e. P A. x e. P A. y e. P A. z e. P - ( <. w , x >. R <. y , z >. <-> ( w d x ) = ( y d z ) ) ) $= - ( vu vv cv wbr cfv wceq wb wral wa wcel cxp wfn cop cvv wer wex cbs fvexi - co tgjustr mp2an simplrl simplrr opelxpd simprl simprr simpll breq1 fveq2 - xpex df-ov eqtr4di eqeq1d bibi12d breq2 eqeq2d syl21anc ralrimivva anim2i - rspc2va eximii ) HMZEEUAZUBZKMZLMZFNZVOVLOZVPVLOZPZQZLVMRKVMRZSZVNDMZAMZU - CZBMZCMZUCZFNZWDWEVLUIZWGWHVLUIZPZQZCERBERZAERDERZSHVMUDTVMFUEWCHUFEEEGUG - IUHZWQUTJKLVMFHUDUJUKWBWPVNWBWODAEEWBWDETZWEETZSZSZWNBCEEXAWGETZWHETZSZSZ - WFVMTWIVMTWBWNXEWDWEEEWBWRWSXDULWBWRWSXDUMUNXEWGWHEEXAXBXCUOXAXBXCUPUNWBW - TXDUQWAWNWFVPFNZWKVSPZQKLWFWIVMVMVOWFPZVQXFVTXGVOWFVPFURXHVRWKVSXHVRWFVLO - WKVOWFVLUSWDWEVLVAVBVCVDVPWIPZXFWJXGWMVPWIWFFVEXIVSWLWKXIVSWIVLOWLVPWIVLU - SWGWHVLVAVBVFVDVJVGVHVHVIVK $. + $d f g x y $. + $( Less-than totally orders the surreals. Axiom O of [Alling] p. 184. + (Contributed by Scott Fenton, 9-Jun-2011.) $) + sltso $p |- On $= + ( vx vy csur con0 cbday wfo wfn crn wceq cdm wcel wral dmexg rgen df-bday + cv cvv mptfng mpbi c1o wrex cab rnmpt csn cxp noxp1o 1oex snnz dmxp ax-mp + wne eqcomi dmeq rspceeqv sylancl nodmon eleq1a syl rexlimiv impbii abbi2i + c0 wi eqtr4i df-fo mpbir2an ) CDEFECGZEHZDIAPZJZQKZACLVGVKACVICMNACVJEAOZ + RSVHBPZVJIZACUAZBUBDABCVJEVLUCVOBDVMDKZVOVPVMTUDZUEZCKVMVRJZIVOVMUFVSVMVQ + VBUKVSVMITUGUHVMVQUIUJULAVRCVJVSVMVIVRUMUNUOVNVPACVICKVJDKVNVPVCVIUPVJDVM + UQURUSUTVAVDCDEVEVF $. + $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Congruence + Density =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) + $( The value of a surreal at its birthday is ` (/) ` . (Contributed by Scott + Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) $) + fvnobday $p |- ( A e. No -> ( A ` ( bday ` A ) ) = (/) ) $= + ( csur wcel cbday cfv wn c0 wceq bdayval word nodmord ordirr eqneltrd ndmfv + cdm syl ) ABCZADEZAOZCFRAEGHQRSSAIQSJSSCFAKSLPMRANP $. + ${ - tkgeom.p $e |- P = ( Base ` G ) $. - tkgeom.d $e |- .- = ( dist ` G ) $. - tkgeom.i $e |- I = ( Itv ` G ) $. - tkgeom.g $e |- ( ph -> G e. TarskiG ) $. + $d A x $. $d B x $. + $( Lemma for ~ nosepne . (Contributed by Scott Fenton, 24-Nov-2021.) $) + nosepnelem $p |- ( ( A e. No /\ B e. No /\ A + ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= + ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) $= + ( csur wcel wbr cfv wne con0 wa c1o cop c2o wceq fvex simpl simpr neeq12d + c0 mpbiri cslt cv crab cint ctp sltval2 w3o brtp csuc df-2o df-1o eqeq12i + 1n0 wb 1on 0elon suc11 mp2an bitri necon3bii mpbir necomi 2on elexi prid2 + nosgnn0i 3jaoi sylbi syl6bi 3impia ) BDEZCDEZBCUAFZAUBZBGVNCGHAIUCUDZBGZV + OCGZHZVKVLJVMVPVQKSLKMLSMLUEFZVRBCAUFVSVPKNZVQSNZJZVTVQMNZJZVPSNZWCJZUGVR + KSKMSMVPVQVOBOVOCOUHWBVRWDWFWBVRKSHZUMWBVPKVQSVTWAPVTWAQRTWDVRKMHMKMKHWGU + MMKKSMKNKUIZSUIZNZKSNZMWHKWIUJUKULKIESIEWJWKUNUOUPKSUQURUSUTVAVBWDVPKVQMV + TWCPVTWCQRTWFVRSMHMKMMIVCVDVEVFWFVPSVQMWEWCPWEWCQRTVGVHVIVJ $. - ${ - tgcgrcomimp.a $e |- ( ph -> A e. P ) $. - tgcgrcomimp.b $e |- ( ph -> B e. P ) $. - tgcgrcomimp.c $e |- ( ph -> C e. P ) $. - tgcgrcomimp.d $e |- ( ph -> D e. P ) $. - $( Congruence commutes on the RHS. Theorem 2.5 of [Schwabhauser] p. 27. - (Contributed by David A. Wheeler, 29-Jun-2020.) $) - tgcgrcomimp $p |- ( ph -> - ( ( A .- B ) = ( C .- D ) -> ( A .- B ) = ( D .- C ) ) ) $= - ( co wceq axtgcgrrflx eqeq2d biimpd ) ABCIRZDEIRZSUCEDIRZSAUDUEUCAFGHID - EJKLMPQTUAUB $. - $} + $( The value of two non-equal surreals at the first place they differ is + different. (Contributed by Scott Fenton, 24-Nov-2021.) $) + nosepne $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> + ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= + ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) $= + ( csur wcel wne cv cfv con0 crab cint wa cslt wbr wo wor nosepnelem necom + 3expia fveq2i wb sltso sotrine mpan wi rabbii inteqi neeq12i bitri sylibr + w3a ancoms jaod sylbid 3impia ) BDEZCDEZBCFZAGZBHZUSCHZFZAIJZKZBHZVDCHZFZ + UPUQLZURBCMNZCBMNZOZVGDMPVHURVKUAUBDBCMUCUDVHVIVGVJUPUQVIVGABCQSUQUPVJVGU + EUQUPVJVGUQUPVJUKVAUTFZAIJZKZCHZVNBHZFZVGACBQVGVPVOFVQVEVPVFVOVDVNBVCVMVB + VLAIUTVARUFUGZTVDVNCVRTUHVPVORUIUJSULUMUNUO $. + $} - ${ - tgcgrcomr.a $e |- ( ph -> A e. P ) $. - tgcgrcomr.b $e |- ( ph -> B e. P ) $. - tgcgrcomr.c $e |- ( ph -> C e. P ) $. - tgcgrcomr.d $e |- ( ph -> D e. P ) $. - tgcgrcomr.6 $e |- ( ph -> ( A .- B ) = ( C .- D ) ) $. - $( Congruence commutes on the RHS. Variant of Theorem 2.5 of - [Schwabhauser] p. 27, but in a convenient form for a common case. - (Contributed by David A. Wheeler, 29-Jun-2020.) $) - tgcgrcomr $p |- ( ph -> ( A .- B ) = ( D .- C ) ) $= - ( co axtgcgrrflx eqtrd ) ABCISDEISEDISRAFGHIDEJKLMPQTUA $. + ${ + $d A x $. $d B x $. + $( If the value of a surreal at a separator is ` 1o ` then the surreal is + lesser. (Contributed by Scott Fenton, 7-Dec-2021.) $) + nosep1o $p |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ + ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> + A ( B .- A ) = ( C .- D ) ) $= - ( co axtgcgrrflx eqtr3d ) ABCISCBISDEISAFGHIBCJKLMNOTRUA $. - $} + ${ + $d A x $. $d B x $. + $( If the value of a surreal at a separator is ` 2o ` then the surreal is + greater. (Contributed by Scott Fenton, 7-Dec-2021.) $) + nosep2o $p |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ + ( A ` |^| { x e. On | ( B ` x ) =/= ( A ` x ) } ) = 2o ) -> + B A e. P ) $. - tgcgrcomlr.b $e |- ( ph -> B e. P ) $. - tgcgrcomlr.c $e |- ( ph -> C e. P ) $. - tgcgrcomlr.d $e |- ( ph -> D e. P ) $. - tgcgrcomlr.6 $e |- ( ph -> ( A .- B ) = ( C .- D ) ) $. - $( Congruence commutes on both sides. (Contributed by Thierry Arnoux, - 23-Mar-2019.) $) - tgcgrcomlr $p |- ( ph -> ( B .- A ) = ( D .- C ) ) $= - ( co axtgcgrrflx 3eqtr3d ) ABCISDEISCBISEDISRAFGHIBCJKLMNOTAFGHIDEJKLMP - QTUA $. + ${ + $d A x $. $d B x $. + $( Lemma for ~ nosepdm . (Contributed by Scott Fenton, 24-Nov-2021.) $) + nosepdmlem $p |- ( ( A e. No /\ B e. No /\ A + |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) ) $= + ( csur wcel wbr cfv wne con0 cdm wo wn wi wa c1o c0 cop c2o wceq syl cslt + w3a cv crab cint cun ctp sltval2 w3o fvex brtp df-3or ndmfv 1oex nosgnn0i + prid1 neeq1 mpbiri intnanrd ioran sylanbrc orel1 biimtrid 2on elexi prid2 + neneqd adantl con4i syl6 ex com23 sylbid 3impia orrd elun sylibr ) BDEZCD + EZBCUAFZUBZAUCZBGWBCGHAIUDUEZBJZEZWCCJZEZKWCWDWFUFEWAWEWGVRVSVTWELZWGMZVR + VSNZVTWCBGZWCCGZOPQORQPRQUGFZWIBCAUHWMWKOSZWLPSZNZWNWLRSZNZWKPSZWQNZUIZWJ + WIOPORPRWKWLWCBUJWCCUJUKWJWHXAWGWJWHXAWGMWJWHNZXAWTWGXAWPWRKZWTKZXBWTWPWR + WTULXBXCLZXDWTMWHXEWJWHWSXEWCBUMWSWPLWRLXEWSWNWOWSWKOWSWKOHPOHOORUNUPUOWK + POUQURVGZUSWSWNWQXFUSWPWRUTVATVHXCWTVBTVCWQWGWSWGWQWGLZWLRXGWOWLRHZWCCUMW + OXHPRHRORRIVDVEVFUOWLPRUQURTVGVIVHVJVKVLVCVMVNVOWCWDWFVPVQ $. - $( Congruence and equality. (Contributed by Thierry Arnoux, - 27-Aug-2019.) $) - tgcgreqb $p |- ( ph -> ( A = B <-> C = D ) ) $= - ( wcel adantr wa cstrkg co simpr oveq1d eqtr3d axtgcgrid eqtrd impbida - wceq ) ABCUJZDEUJZAUKUAZFGHIDECJKLAGUBSZUKMTADFSUKPTAEFSZUKQTACFSZUKOTU - MBCIUCZDEIUCZCCIUCAUQURUJZUKRTUMBCCIAUKUDUEUFUGAULUAZFGHIBCEJKLAUNULMTA - BFSULNTAUPULOTAUOULQTUTUQUREEIUCAUSULRTUTDEEIAULUDUEUHUGUI $. + $( The first place two surreals differ is an element of the larger of their + domains. (Contributed by Scott Fenton, 24-Nov-2021.) $) + nosepdm $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> + |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) ) $= + ( csur wcel wne cv cfv con0 crab cint cdm cun wa wbr wo wor wb nosepdmlem + cslt sltso sotrine 3expa simplr simpll simpr syl3anc necom rabbii 3eltr4g + mpan inteqi uncom jaodan ex sylbid 3impia ) BDEZCDEZBCFZAGZBHZVACHZFZAIJZ + KZBLZCLZMZEZURUSNZUTBCTOZCBTOZPZVJDTQVKUTVNRUADBCTUBUKVKVNVJVKVLVJVMURUSV + LVJABCSUCVKVMNZVCVBFZAIJZKZVHVGMZVFVIVOUSURVMVRVSEURUSVMUDURUSVMUEVKVMUFA + CBSUGVEVQVDVPAIVBVCUHUIULVGVHUMUJUNUOUPUQ $. + $} - ${ - tgcgreq.1 $e |- ( ph -> A = B ) $. - $( Congruence and equality. (Contributed by Thierry Arnoux, - 27-Aug-2019.) $) - tgcgreq $p |- ( ph -> C = D ) $= - ( wceq tgcgreqb mpbid ) ABCTDETSABCDEFGHIJKLMNOPQRUAUB $. - $} + ${ + $d A x $. $d B x $. $d X x $. + $( The values of two surreals at a point less than their separators are + equal. (Contributed by Scott Fenton, 6-Dec-2021.) $) + nosepeq $p |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ + X e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> + ( A ` X ) = ( B ` X ) ) $= + ( csur wcel wne w3a cv cfv con0 crab cint wa wn wceq nosepon onelon sylan + fveq2 simpr neeq12d onnminsb sylc df-ne con2bii sylibr ) BEFCEFBCGHZDAIZB + JZUICJZGZAKLMZFZNZDBJZDCJZGZOZUPUQPZUODKFZUNUSUHUMKFUNVAABCQUMDRSUHUNUAUL + URADUIDPUJUPUKUQUIDBTUIDCTUBUCUDURUTUPUQUEUFUG $. + $} - ${ - tgcgrneq.1 $e |- ( ph -> A =/= B ) $. - $( Congruence and equality. (Contributed by Thierry Arnoux, - 27-Aug-2019.) $) - tgcgrneq $p |- ( ph -> C =/= D ) $= - ( wne tgcgreqb necon3bid mpbid ) ABCTDETSABCDEABCDEFGHIJKLMNOPQRUAUBU - C $. - $} - $} + ${ + $d A x $. $d B x $. + $( Given two non-equal surreals, their separator is less-than or equal to + the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. + (Contributed by Scott Fenton, 6-Dec-2021.) $) + nosepssdm $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> + |^| { x e. On | ( A ` x ) =/= ( B ` x ) } C_ dom A ) $= + ( csur wcel wne cfv con0 cdm wn wceq wa c0 wi word nodmord 3ad2ant1 ndmfv + wss syl w3a cv crab cint nosepne neneqd ordn2lp sylibr imp nosepeq simpl1 + imnan ordirr 3syl eqeq1d eqcom bitrdi crn wb simpl2 nofun c1o c2o nosgnn0 + wfun cpr norn sseld mtoi funeldmb syl2anc necon2bbid ordtr1 expdimp con3d + 3ad2ant2 sylbid mpd eqtr4d mtand nosepon nodmon ontri1 mpbird ) BDEZCDEZB + CFZUAZAUBZBGWICGFAHUCUDZBIZSZWKWJEZJZWHWMWJBGZWJCGZKWHWOWPABCUEUFWHWMLZWO + MWPWQWJWKEZJZWOMKWHWMWSWHWMWRLJZWMWSNWHWKOZWTWEWFXAWGBPZQWKWJUGTWMWRULUHU + IWJBRTWQWJCIZEZJZWPMKWQWKBGZWKCGZKZXEABCWKUJWQXHXGMKZXEWQXHMXGKXIWQXFMXGW + QXAWKWKEJXFMKWQWEXAWEWFWGWMUKXBTWKUMWKBRUNUOMXGUPUQWQXIWKXCEZJXEWQXJXGMWQ + CVEZMCURZEZJXJXGMFUSWQWFXKWEWFWGWMUTZCVATWQXMMVBVCVFZEVDWQXLXOMWQWFXLXOSX + NCVGTVHVIWKCVJVKVLWQXDXJWHWMXDXJWHXCOZWMXDLXJNWFWEXPWGCPVPWKWJXCVMTVNVOVQ + VQVRWJCRTVSVTWHWJHEWKHEZWLWNUSABCWAWEWFXQWGBWBQWJWKWCVKWD $. + $} - ${ - $d x .- $. $d x A $. $d x B $. $d x I $. $d x P $. $d x ph $. - tgcgrtriv.1 $e |- ( ph -> A e. P ) $. - tgcgrtriv.2 $e |- ( ph -> B e. P ) $. - $( Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] - p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) $) - tgcgrtriv $p |- ( ph -> ( A .- A ) = ( B .- B ) ) $= - ( vx cv co wcel wceq wa ad2antrr cstrkg simplr simprr oveq2d axtgsegcon - axtgcgrid eqtrd r19.29a ) ABCNOZFPQZBUIGPZCCGPZRZSZBBGPZULRNDAUIDQZSZUN - SZUOUKULURBUIBGURDEFGBUICHIJAEUAQUPUNKTABDQUPUNLTAUPUNUBACDQUPUNMTUQUJU - MUCZUFUDUSUGANCCDEFGCBHIJKMLMMUEUH $. - $} + ${ + $d A a $. $d B a $. + $( Lemma for ~ nodense . Show that a particular abstraction is an ordinal. + (Contributed by Scott Fenton, 16-Jun-2011.) $) + nodenselem4 $p |- ( ( ( A e. No /\ B e. No ) /\ A + |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On ) $= + ( csur wcel wa cslt wbr wne cv con0 crab cint simpll simplr wn wceq sltso + cfv wor sonr mpan adantr breq2 notbid syl5ibcom necon2ad nosepon syl3anc + imp ) ADEZBDEZFZABGHZFUKULABIZCJZASUPBSICKLMKEUKULUNNUKULUNOUMUNUOUMUNABU + MAAGHZPZABQZUNPUKURULDGTUKURRDAGUAUBUCUSUQUNABAGUDUEUFUGUJCABUHUI $. + $} - ${ - tgcgrextend.a $e |- ( ph -> A e. P ) $. - tgcgrextend.b $e |- ( ph -> B e. P ) $. - tgcgrextend.c $e |- ( ph -> C e. P ) $. - tgcgrextend.d $e |- ( ph -> D e. P ) $. - tgcgrextend.e $e |- ( ph -> E e. P ) $. - tgcgrextend.f $e |- ( ph -> F e. P ) $. - ${ - tgcgrextend.1 $e |- ( ph -> B e. ( A I C ) ) $. - tgcgrextend.2 $e |- ( ph -> E e. ( D I F ) ) $. - tgcgrextend.3 $e |- ( ph -> ( A .- B ) = ( D .- E ) ) $. - tgcgrextend.4 $e |- ( ph -> ( B .- C ) = ( E .- F ) ) $. - $( Link congruence over a pair of line segments. Theorem 2.11 of - [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) - (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.) $) - tgcgrextend $p |- ( ph -> ( A .- C ) = ( D .- F ) ) $= - ( co wceq wa adantr simpr oveq1d cstrkg tgcgreq 3eqtr4d wne tgcgrtriv - wcel tgcgrcomlr axtg5seg pm2.61dane ) ABDKUFZEHKUFZUGBCABCUGZUHZCDKUF - ZGHKUFZVAVBAVEVFUGZVCUEUIVDBCDKAVCUJZUKVDEGHKVDBCEGFIJKLMNAIULUQZVCOU - IABFUQZVCPUIACFUQZVCQUIAEFUQZVCSUIAGFUQZVCTUIABCKUFEGKUFUGZVCUDUIVHUM - UKUNABCUOZUHZDBHEFIJKLMNAVIVOOUIZADFUQVORUIZAVJVOPUIZAHFUQVOUAUIZAVLV - OSUIZVPEGHFBIJKEBCDLMNVQVSAVKVOQUIZVRWAAVMVOTUIZVTVSWAAVOUJACBDJUFUQV - OUBUIAGEHJUFUQVOUCUIAVNVOUDUIZAVGVOUEUIVPBEFIJKLMNVQVSWAUPVPBCEGFIJKL - MNVQVSWBWAWCWDURUSURUT $. - $} + ${ + $d A a $. $d B a $. + $( Lemma for ~ nodense . If the birthdays of two distinct surreals are + equal, then the ordinal from ~ nodenselem4 is an element of that + birthday. (Contributed by Scott Fenton, 16-Jun-2011.) $) + nodenselem5 $p |- ( ( ( A e. No /\ B e. No ) /\ + ( ( bday ` A ) = ( bday ` B ) /\ A + |^| { a e. On | ( + A ` a ) =/= ( B ` a ) } e. + ( bday ` A ) ) $= + ( csur wcel wa cbday cfv wceq cslt wbr cv wne con0 cdm cun wn bdayval syl + crab cint simpll simplr wi wor sltso sonr breq2 notbid syl5ibcom necon2ad + mpan adantr adantrl nosepdm syl3anc simprl uneq2d eqtr3di uneq12d 3eqtr3d + imp unidm eleqtrd eleqtrrd ) ADEZBDEZFZAGHZBGHZIZABJKZFZFZCLZAHVOBHMCNTUA + ZAOZVIVNVPVQBOZPZVQVNVFVGABMZVPVSEVFVGVMUBZVFVGVMUCZVHVLVTVKVHVLVTVFVLVTU + DVGVFVLABVFAAJKZQZABIZVLQDJUEVFWDUFDAJUGULWEWCVLABAJUHUIUJUKUMVBUNCABUOUP + VNVIVJPZVIVSVQVNVIVIPWFVIVNVIVJVIVHVKVLUQURVIVCUSVNVIVQVJVRVNVFVIVQIWAARS + ZVNVGVJVRIWBBRSUTWGVAVDWGVE $. + $} - ${ - tgsegconeq.1 $e |- ( ph -> D =/= A ) $. - tgsegconeq.2 $e |- ( ph -> A e. ( D I E ) ) $. - tgsegconeq.3 $e |- ( ph -> A e. ( D I F ) ) $. - tgsegconeq.4 $e |- ( ph -> ( A .- E ) = ( B .- C ) ) $. - tgsegconeq.5 $e |- ( ph -> ( A .- F ) = ( B .- C ) ) $. - $( Two points that satisfy the conclusion of ~ axtgsegcon are - identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] - p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) $) - tgsegconeq $p |- ( ph -> E = F ) $= - ( co eqidd eqtr4d tgcgrextend axtg5seg eqcomd axtgcgrid ) AFIJKGHGLMN - OTUATAGGKUGGHKUGAEBGFGIJKHEBGLMNOSPTSPTTUAUBUCUCAEBKUGUHZABGKUGZUHAEB - GEFBHIJKLMNOSPTSPUAUCUDUNAUOCDKUGBHKUGUEUFUIZUJUPUKULUM $. - $} - $} + ${ + $d A a $. $d B a $. + $( The restriction of a surreal to the abstraction from ~ nodenselem4 is + still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.) $) + nodenselem6 $p |- ( ( ( A e. No /\ B e. No ) /\ + ( ( bday ` A ) = ( bday ` B ) /\ A + ( A |` |^| { a e. On | ( + A ` a ) =/= ( B ` a ) } ) e. No ) $= + ( csur wcel wa cbday cfv wceq cslt wbr cv con0 crab cint cres nodenselem4 + wne simpll adantrl noreson syl2anc ) ADEZBDEZFZAGHBGHIZABJKZFZFUCCLZAHUIB + HRCMNOZMEZAUJPDEUCUDUHSUEUGUKUFABCQTAUJUAUB $. + $} + + ${ + $d A a $. $d B a $. $d C a $. + $( Lemma for ~ nodense . ` A ` and ` B ` are equal at all elements of the + abstraction. (Contributed by Scott Fenton, 17-Jun-2011.) $) + nodenselem7 $p |- ( ( ( A e. No /\ B e. No ) /\ + ( ( bday ` A ) = ( bday ` B ) /\ A + ( C e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } + -> ( A ` C ) = ( B ` C ) ) ) $= + ( csur wcel wa cbday cfv wceq cslt wbr cv wne con0 crab cint w3a simpll + wn simplr wor sltso sonr mpan breq2 syl5ibcom necon2ad imp ad2ant2rl 3jca + notbid nosepeq sylan ex ) AEFZBEFZGAHIBHIJZABKLZGZGZCDMZAIVBBINDOPQFZCAIC + BIJZVAUPUQABNZRVCVDVAUPUQVEUPUQUTSUPUQUTUAUPUSVEUQURUPUSVEUPUSABUPAAKLZTZ + ABJZUSTEKUBUPVGUCEAKUDUEVHVFUSABAKUFULUGUHUIUJUKDABCUMUNUO $. + $} + + ${ + $d A a $. $d B a $. + $( Lemma for ~ nodense . Give a condition for surreal less-than when two + surreals have the same birthday. (Contributed by Scott Fenton, + 19-Jun-2011.) $) + nodenselem8 $p |- ( ( A e. No /\ B e. No /\ + ( bday ` A ) = ( bday ` B ) ) -> + ( A + ( ( A ` + |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = + 1o /\ + ( B ` + |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = + 2o ) ) ) $= + ( csur wcel cbday cfv wceq wbr c1o c2o wa wi 3impia c0 cop wn nosgnn0 crn + fvex w3a cslt cv wne con0 crab cint nodenselem5 exp32 ctp sltval2 3adant3 + wb w3o brtp eleq2 biimpd cpr nofnbday fnfvelrn eleq1 syl5ibcom sylan norn + wfn sseld adantr syld mtoi ex adantl syl9r imp intnand 3ad2antl1 intnanrd + 3orel13 syl2anc com23 biimtrid sylbid mpdd 3mix2 sylibr syl5ibr impbid ) + ADEZBDEZAFGZBFGZHZUAZABUBIZCUCZAGWNBGUDCUEUFUGZAGZJHZWOBGZKHZLZWLWMWOWIEZ + WTWGWHWKWMXAMWGWHLZWKWMXAABCUHUINWLWMWPWRJOPJKPOKPUJIZXAWTMZWGWHWMXCUMWKA + BCUKULZXCWQWROHZLZWTWPOHZWSLZUNZWLXDJOJKOKWPWRWOATWOBTUOZWLXAXJWTWLXAXJWT + MZWLXALZXGQXIQXLXMXFWQWLXAXFQZWGWHWKXAXNMWKXAWOWJEZXBXNWKXAXOWIWJWOUPUQWH + XOXNMWGWHXOXNWHXOLZXFOJKURZEZRXPXFOBSZEZXRWHBWJVEZXOXFXTMBUSYAXOLWRXSEXFX + TWJWOBUTWROXSVAVBVCWHXTXRMXOWHXSXQOBVDVFVGVHVIVJVKVLNVMVNXMXHWSWGWHXAXHQW + KWGXALZXHXRRYBXHOASZEZXRWGAWIVEZXAXHYDMAUSYEXALWPYCEXHYDWIWOAUTWPOYCVAVBV + CWGYDXRMXAWGYCXQOAVDVFVGVHVIVOVPXGWTXIVQVRVJVSVTWAWBWTWMWLXCWTXJXCWTXGXIW + CXKWDXEWEWF $. + $} + ${ + $d A a x y $. $d B a x y $. + $( Given two distinct surreals with the same birthday, there is an older + surreal lying between the two of them. Axiom SD of [Alling] p. 184. + (Contributed by Scott Fenton, 16-Jun-2011.) $) + nodense $p |- ( ( ( A e. No /\ B e. No ) /\ + ( ( bday ` A ) = ( bday ` B ) /\ A + E. x e. No ( ( bday ` x ) e. ( bday ` A ) /\ + A suc U. ( bday " A ) e. On ) $= + ( wcel cbday cima cuni word cvv wa csuc con0 wfun csur wfo bdayfo funimaexg + fofun ax-mp mpan uniexd wss crn imassrn wceq sseqtri ssorduni jctil sucelon + forn elon2 bitr3i sylib ) ABCZDAEZFZGZUOHCZIZUOJKCZUMUQUPUMUNHDLZUMUNHCMKDN + ZUTOMKDQRDABPSTUNKUAUPUNDUBZKDAUCVAVBKUDOMKDUIRUEUNUFRUGURUOKCUSUOUJUOUHUKU + L $. + + $( Lemma for ~ nolt02o . If ` A ( X ) ` is undefined with ` A ` surreal and + ` X ` ordinal, then ` dom A C_ X ` . (Contributed by Scott Fenton, + 6-Dec-2021.) $) + nolt02olem $p |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> + dom A C_ X ) $= + ( csur wcel con0 cfv c0 wceq w3a cdm wss wn c1o c2o cpr nosgnn0 wa 3ad2ant1 + a1i simpl3 crn simpl1 norn wfun nofun fvelrn sylan sseldd eqeltrrd mtand wb + syl nodmon simp2 ontri1 syl2anc mpbird ) ACDZBEDZBAFZGHZIZAJZBKZBVCDZLZVBVE + GMNOZDZVHLVBPSVBVEQZUTGVGURUSVAVETVIAUAZVGUTVIURVJVGKURUSVAVEUBAUCULVBAUDZV + EUTVJDURUSVKVAAUERBAUFUGUHUIUJVBVCEDZUSVDVFUKURUSVLVAAUMRURUSVAUNVCBUOUPUQ + $. + + ${ + $d A x y $. $d B x y $. $d X x y $. + $( Given ` A ` less-than ` B ` , equal to ` B ` up to ` X ` , and undefined + at ` X ` , then ` B ( X ) = 2o ` . (Contributed by Scott Fenton, + 6-Dec-2021.) $) + nolt02o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ + ( ( A |` X ) = ( B |` X ) /\ A + ( B ` X ) = 2o ) $= + ( vy vx csur wcel con0 wceq cslt wbr wa cfv c0 c2o c1o syl2anc bitri w3o + wn w3a cres simp11 wor sltso sonr mpan syl simp2r syl5ibrcom mtod simpl2l + breq2 wrel cdm wss simpl11 nofun funrel simpl13 simpl3 nolt02olem syl3anc + wfun 3syl relssres simpl12 simpr 3eqtr3d mtand cv wral cop ctp wi wrex wb + simp12 sltval mpbid df-an rexbii rexnal sylib 1oex prid1 nosgnn0i simpll3 + neii simplr eqeq1 anbi1d eqtr2 syl6bi com12 mtoi simplrr fveq2 rspcv sylc + eqeq12d simprl adantr ontri1 mpbird wo onsseleq ancoms csuc df-1o eqeq12i + mto df-2o 0elon 1on suc11 mp2an nemtbir 2on elexi prid2 3pm3.2i fvex brtp + 3oran con2bii fveq1d breq2d mtbii breq12d notbid syl5ibcom intnanr intnan + mpbi fvres 0ex mtbiri jaod 3orrot sylbid mpd expr ralrimiva nofv ecase23d + ) AFGZBFGZCHGZUAZACUBZBCUBZIZABJKZLZCAMZNIZUAZCBMZOIZUUSNIZUUSPIZUURUVAAB + IZUURUVCAAJKZUURUUGUVDTZUUGUUHUUIUUOUUQUCZFJUDUUGUVEUEFAJUFUGUHUURUVDUVCU + UNUUJUUMUUNUUQUIZABAJUMUJUKUURUVALZUUKUULABUUMUUNUUJUUQUVAULUVHAUNZAUOCUP + ZUUKAIUVHUUGAVDUVIUUGUUHUUIUUOUUQUVAUQZAURAUSVEUVHUUGUUIUUQUVJUVKUUGUUHUU + IUUOUUQUVAUTZUUJUUOUUQUVAVAACVBVCACVFQUVHBUNZBUOCUPZUULBIUVHUUHBVDUVMUUGU + UHUUIUUOUUQUVAVGZBURBUSVEUVHUUHUUIUVAUVNUVOUVLUURUVAVHBCVBVCBCVFQVIVJUURU + VBDVKZAMZUVPBMZIZDEVKZVLZUVTAMZUVTBMZPNVMPOVMNOVMVNZKZTZVOZEHVLZUURUWAUWE + LZEHVPZUWHTZUURUUNUWJUVGUURUUGUUHUUNUWJVQUVFUUGUUHUUIUUOUUQVRZEDABVSQVTUW + JUWGTZEHVPUWKUWIUWMEHUWAUWEWAWBUWGEHWCRWDUURUVBLZUWGEHUWNUVTHGZUWAUWFUWNU + WOUWALZLZUVTCUPZUWFUWQUWRCUVTGZTZUWQUWSUUPUUSIZUWQUXANPIZNPPPOWEWFWGZWIZU + WQUUQUVBUXAUXBVOUUJUUOUUQUVBUWPWHZUURUVBUWPWJZUXAUUQUVBLZUXBUXAUXGUVAUVBL + UXBUXAUUQUVAUVBUUPUUSNWKWLUUSNPWMWNWOQWPUWQUWSLUWSUWAUXAUWQUWSVHUWNUWOUWA + UWSWQUVSUXADCUVTUVPCIUVQUUPUVRUUSUVPCAWRUVPCBWRXAWSWTVJUWQUWOUUIUWRUWTVQU + WNUWOUWAXBZUWNUUIUWPUUGUUHUUIUUOUUQUVBUTXCZUVTCXDQXEUWQUWRUVTCGZUVTCIZXFZ + UWFUWQUWOUUIUWRUXLVQUXHUXIUVTCXGQUWQUXJUWFUXKUWQUVTUUKMZUVTUULMZUWDKZTUXJ + UWFUWQUXMUXMUWDKZUXOUXMPIZUXMNIZLZTZUXQUXMOIZLZTZUXRUYALZTZUAZUXPTUXTUYCU + YEUXSUXBUXDUXRUXQUXBUXMNPWMXHXLUYBPOIZUYGNPUXCUYGNXIZPXIZIZUXBPUYHOUYIXJX + MXKNHGPHGUYJUXBVQXNXONPXPXQRXRZUXMPOWMXLUYDNOINOOPOOHXSXTYAWGWIUXMNOWMXLY + BUXPUYFUXPUXSUYBUYDSUYFTPNPONOUXMUXMUVTUUKYCZUYLYDUXSUYBUYDYERYFYOUWQUXMU + XNUXMUWDUWQUVTUUKUULUWNUUMUWPUUMUUNUUJUUQUVBULXCYGYHYIUXJUXOUWEUXJUXMUWBU + XNUWCUWDUVTCAYPUVTCBYPYJYKYLUWQUWFUXKUUPUUSUWDKZTUWQUYMNPUWDKZUXBPNIZLZTZ + UXBUYGLZTZNNIZUYGLZTZUAZUYNTUYQUYSVUBUXBUYOUXDYMUXBUYGUXDYMUYGUYTUYKYNYBU + YNVUCUYNUYPUYRVUASVUCTPNPONONPYQWEYDUYPUYRVUAYERYFYOUWQUUPNUUSPUWDUXEUXFY + JYRUXKUWEUYMUXKUWBUUPUWCUUSUWDUVTCAWRUVTCBWRYJYKUJYSUUAUUBUUCUUDVJUURUVAU + VBUUTSZUUTUVAUVBSZUURUUHVUDUWLBCUUEUHVUDUVBUUTUVASVUEUVAUVBUUTYTUVBUUTUVA + YTRWDUUF $. + + $( Given ` A ` greater than ` B ` , equal to ` B ` up to ` X ` , and + ` B ( X ) ` undefined, then ` A ( X ) = 1o ` . (Contributed by Scott + Fenton, 9-Aug-2024.) $) + nogt01o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ + ( ( A |` X ) = ( B |` X ) /\ A + ( A ` X ) = 1o ) $= + ( vy vx csur wcel con0 w3a wceq cslt wbr wa cfv c0 c1o c2o wn syl syl2anc + cres wor sltso simp11 sonr sylancr simpl2r simpl2l wrel cdm simpl11 nofun + wss wfun funrel simpl13 simpr nolt02olem syl3anc relssres simpl12 3eqtr3d + simpl3 breqtrrd mtand cv wral cop ctp wi simp2r wb simp12 sltval ralinexa + wrex mpbid con2bii sylib 1n0 neii eqtr2 mto csuc word wne eqeltrri onordi + df-2o 1oex sucid nordeq mp2an eqnetri nesymi 2on0 3pm3.2i fvex brtp 3oran + 2on bitri mpbi adantr fveq1d breq2d mtbii fvres breq12d syl5ibcom intnanr + w3o notbid intnan 2oex 0ex simplr simpll3 mtbiri fveq2 syl5ibrcom eqeq12d + wo rspccv ad2antll eqcom syl6ib sylibd mtoi simprl ontri1 onsseleq bitr3d + mpjaod expr ralrimiva nofv 3orcoma ecase23d ) AFGZBFGZCHGZIZACUAZBCUAZJZA + BKLZMZCBNZOJZIZCANZPJZUULOJZUULQJZUUKUUNAAKLZUUKFKUBYTUUPRUCYTUUAUUBUUHUU + JUDZFAKUEUFUUKUUNMZABAKUUFUUGUUCUUJUUNUGUURUUDUUEABUUFUUGUUCUUJUUNUHUURAU + IZAUJCUMZUUDAJUURYTUUSYTUUAUUBUUHUUJUUNUKZYTAUNUUSAULAUOSSUURYTUUBUUNUUTU + VAYTUUAUUBUUHUUJUUNUPZUUKUUNUQACURUSACUTTUURBUIZBUJCUMZUUEBJUURUUAUVCYTUU + AUUBUUHUUJUUNVAZUUABUNUVCBULBUOSSUURUUAUUBUUJUVDUVEUVBUUCUUHUUJUUNVCBCURU + SBCUTTVBVDVEUUKUUODVFZANZUVFBNZJZDEVFZVGZUVJANZUVJBNZPOVHPQVHOQVHVIZLZRZV + JZEHVGZUUKUVKUVOMEHVPZUVRRUUKUUGUVSUUCUUFUUGUUJVKUUKYTUUAUUGUVSVLUUQYTUUA + UUBUUHUUJVMEDABVNTVQUVRUVSUVKUVOEHVOVRVSUUKUUOMZUVQEHUVTUVJHGZUVKUVPUVTUW + AUVKMZMZUVJCGZUVPUVJCJZUWCUVJUUDNZUVJUUENZUVNLZRUWDUVPUWCUWFUWFUVNLZUWHUW + FPJZUWFOJZMZRZUWJUWFQJZMZRZUWKUWNMZRZIZUWIRUWMUWPUWRUWLPOJPOVTWAUWFPOWBWC + UWOPQJQPQPWDZPWIUWTWEPUWTGUWTPWFUWTQUWTHWIXAWGWHPWJWKUWTPWLWMWNZWOUWFPQWB + WCUWQOQJZQOWPWOZUWFOQWBWCWQUWIUWSUWIUWLUWOUWQXLUWSRPOPQOQUWFUWFUVJUUDWRZU + XDWSUWLUWOUWQWTXBVRXCUWCUWFUWGUWFUVNUWCUVJUUDUUEUVTUUFUWBUUFUUGUUCUUJUUOU + HXDXEXFXGUWDUWHUVOUWDUWFUVLUWGUVMUVNUVJCAXHUVJCBXHXIXMXJUWCUVPUWEUULUUIUV + NLZRUWCUXEQOUVNLZQPJZOOJZMZRZUXGUXBMZRZQOJZUXBMZRZIZUXFRUXJUXLUXOUXGUXHQP + UXAWAXKUXBUXGUXCXNUXBUXMUXCXNWQUXFUXPUXFUXIUXKUXNXLUXPRPOPQOQQOXOXPWSUXIU + XKUXNWTXBVRXCUWCUULQUUIOUVNUUKUUOUWBXQZUUCUUHUUJUUOUWBXRZXIXSUWEUVOUXEUWE + UVLUULUVMUUIUVNUVJCAXTUVJCBXTXIXMYAUWCCUVJGZRZUWDUWEYCZUWCUXSUXBUXCUWCUXS + UUIUULJZUXBUWCUXSUULUUIJZUYBUVKUXSUYCVJUVTUWAUVIUYCDCUVJUVFCJUVGUULUVHUUI + UVFCAXTUVFCBXTYBYDYEUULUUIYFYGUWCUUIOUULQUXRUXQYBYHYIUWCUWAUUBUXTUYAVLUVT + UWAUVKYJUVTUUBUWBYTUUAUUBUUHUUJUUOUPXDUWAUUBMUVJCUMUXTUYAUVJCYKUVJCYLYMTV + QYNYOYPVEUUKUUNUUMUUOXLZUUMUUNUUOXLUUKYTUYDUUQACYQSUUNUUMUUOYRVSYS $. + $} + + ${ + $d S g x $. $d U g x $. $d A g x $. + $( Restriction law for surreals. Lemma 2.1.4 of [Lipparini] p. 3. + (Contributed by Scott Fenton, 5-Dec-2021.) $) + noresle $p |- ( ( ( U e. No /\ S e. No ) /\ + ( dom U C_ A /\ dom S C_ A /\ + A. g e. A -. ( S |` suc g ) + -. S E* x e. S A. y e. S -. x E* x e. S A. y e. S -. y + E* x E. u e. A ( G e. dom u /\ + A. v e. A ( -. v ( u |` suc G ) = ( v |` suc G ) ) /\ + ( u ` G ) = x ) ) $= + ( vp vy csur cv wcel cslt wbr wn wceq wi cfv wrex wa weq adantl csuc cres + wss cdm w3a wal reeanv breq1 notbid reseq1 eqeq2d imbi12d simprr2 simprll + wmo rspcdva eqcom syl6ib simprl2 simprlr wo simpl sseldd wor sltso soasym + wral syl2anc pm4.62 sylib mpjaod fveq1d simprl1 sucidg syl fvresd 3eqtr3d + mpan simprl3 simprr3 expr rexlimdvva syl5bir eqeq2 3anbi3d rexbidv eleq2d + alrimivv dmeq breq2 eqeq1d ralbidv fveq1 3anbi123d cbvrexvw bitrdi sylibr + mo4 ) DHUCZECIZUDZJZBIZWTKLZMZWTEUAZUBZXCXFUBZNZOZBDVGZEWTPZAIZNZUEZCDQZE + FIZUDZJZXCXQKLZMZXQXFUBZXHNZOZBDVGZEXQPZGIZNZUEZFDQZRZAGSZOZGUFAUFXPAUOWS + YMAGYKXOYIRZFDQCDQWSYLXOYICFDDUGWSYNYLCFDDWSWTDJZXQDJZRZYNYLWSYQYNRZRZXLY + FXMYGYSEXGPEYBPXLYFYSEXGYBYSWTXQKLZMZXGYBNZXQWTKLZMZYSUUAYBXGNZUUBYSYDUUA + UUEOBDWTBCSZYAUUAYCUUEUUFXTYTXCWTXQKUHUIUUFXHXGYBXCWTXFUJUKULYRYEWSXSYEYH + XOYQUMTWSYOYPYNUNZUPYBXGUQURYSXJUUDUUBOBDXQBFSZXEUUDXIUUBUUHXDUUCXCXQWTKU + HUIUUHXHYBXGXCXQXFUJUKULYRXKWSXBXKXNYIYQUSTWSYOYPYNUTZUPYSYTUUDOZUUAUUDVA + YSWTHJZXQHJZUUJYSDHWTWSYRVBZUUGVCYSDHXQUUMUUIVCHKVDUUKUULRUUJVEHKWTXQVFVR + VHYTUUCVIVJVKVLYSEXFWTYSXBEXFJYRXBWSXBXKXNYIYQVMTEXAVNVOZVPYSEXFXQUUNVPVQ + YRXNWSXBXKXNYIYQVSTYRYHWSXSYEYHXOYQVTTVQWAWBWCWHXPYJAGYLXPXBXKXLYGNZUEZCD + QYJYLXOUUPCDYLXNUUOXBXKXMYGXLWDWEWFUUPYICFDCFSZXBXSXKYEUUOYHUUQXAXREWTXQW + IWGUUQXJYDBDUUQXEYAXIYCUUQXDXTWTXQXCKWJUIUUQXGYBXHWTXQXFUJWKULWLUUQXLYFYG + EWTXQWMWKWNWOWPWRWQ $. + + $( In any class of surreals, there is at most one value of the prefix + property. (Contributed by Scott Fenton, 8-Aug-2024.) $) + noinfprefixmo $p |- ( A C_ No -> + E* x E. u e. A ( G e. dom u /\ + A. v e. A ( -. u ( u |` suc G ) = ( v |` suc G ) ) /\ + ( u ` G ) = x ) ) $= + ( vp vy csur cv wcel cslt wbr wn wceq wi cfv wrex wa weq adantl csuc cres + wss cdm w3a wal reeanv breq2 notbid reseq1 eqeq2d imbi12d simprl2 simprlr + wmo rspcdva simprr2 simprll eqcom syl6ib wo simpl sseldd wor sltso soasym + wral mpan syl2anc imor sylib mpjaod fveq1d simprl1 sucidg 3eqtr3d simprl3 + syl fvresd simprr3 expr rexlimdvva syl5bir alrimivv eqeq2 3anbi3d rexbidv + dmeq eleq2d breq1 eqeq1d ralbidv fveq1 3anbi123d cbvrexvw bitrdi sylibr + mo4 ) DHUCZECIZUDZJZWTBIZKLZMZWTEUAZUBZXCXFUBZNZOZBDVGZEWTPZAIZNZUEZCDQZE + FIZUDZJZXQXCKLZMZXQXFUBZXHNZOZBDVGZEXQPZGIZNZUEZFDQZRZAGSZOZGUFAUFXPAUOWS + YMAGYKXOYIRZFDQCDQWSYLXOYICFDDUGWSYNYLCFDDWSWTDJZXQDJZRZYNYLWSYQYNRZRZXLY + FXMYGYSEXGPEYBPXLYFYSEXGYBYSWTXQKLZMZXGYBNZXQWTKLZMZYSXJUUAUUBOBDXQBFSZXE + UUAXIUUBUUEXDYTXCXQWTKUHUIUUEXHYBXGXCXQXFUJUKULYRXKWSXBXKXNYIYQUMTWSYOYPY + NUNZUPYSUUDYBXGNZUUBYSYDUUDUUGOBDWTBCSZYAUUDYCUUGUUHXTUUCXCWTXQKUHUIUUHXH + XGYBXCWTXFUJUKULYRYEWSXSYEYHXOYQUQTWSYOYPYNURZUPYBXGUSUTYSYTUUDOZUUAUUDVA + YSWTHJZXQHJZUUJYSDHWTWSYRVBZUUIVCYSDHXQUUMUUFVCHKVDUUKUULRUUJVEHKWTXQVFVH + VIYTUUDVJVKVLVMYSEXFWTYSXBEXFJYRXBWSXBXKXNYIYQVNTEXAVOVRZVSYSEXFXQUUNVSVP + YRXNWSXBXKXNYIYQVQTYRYHWSXSYEYHXOYQVTTVPWAWBWCWDXPYJAGYLXPXBXKXLYGNZUEZCD + QYJYLXOUUPCDYLXNUUOXBXKXMYGXLWEWFWGUUPYICFDCFSZXBXSXKYEUUOYHUUQXAXREWTXQW + HWIUUQXJYDBDUUQXEYAXIYCUUQXDXTWTXQXCKWJUIUUQXGYBXHWTXQXFUJWKULWLUUQXLYFYG + EWTXQWMWKWNWOWPWRWQ $. + + $} + + ${ + $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d A d $. $d a e $. + $d A e $. $d A f $. $d a g $. $d A g $. $d a u $. $d A u $. $d a v $. + $d A v $. $d a x $. $d A x $. $d a y $. $d A y $. $d b x $. $d b y $. + $d c d $. $d c e $. $d c f $. $d c g $. $d c u $. $d c v $. $d c x $. + $d c y $. $d d e $. $d d f $. $d d u $. $d d v $. $d d y $. $d e f $. + $d e g $. $d e u $. $d e v $. $d e x $. $d e y $. $d f g $. $d f u $. + $d f v $. $d f x $. $d f y $. $d g u $. $d g v $. $d g x $. $d g y $. + $d u v $. $d u x $. $d u y $. $d v y $. $d x y $. + nosupcbv.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Lemma to change bound variables in a surreal supremum. (Contributed by + Scott Fenton, 9-Aug-2024.) $) + nosupcbv $p |- S = if ( E. a e. A A. b e. A -. a . } ) , + ( c e. { d | E. e e. A ( d e. dom e /\ + A. f e. A ( -. f ( e |` suc d ) = ( f |` suc d ) ) ) } + |-> ( iota a E. e e. A ( c e. dom e /\ + A. f e. A ( -. f ( e |` suc c ) = ( f |` suc c ) ) /\ + ( e ` c ) = a ) ) ) ) $= + ( cv cslt wral wrex cres wceq wbr wn crio cdm c2o cop csn wcel csuc wi wa + cun cab cfv w3a cio cmpt cif breq1 notbid ralbidv breq2 cbvralvw cbvrexvw + bitrdi cbvriotavw dmeqi opeq1i sneqi uneq12i eleq1w suceq reseq2d eqeq12d + weq imbi2d fveqeq2 3anbi123d rexbidv iotabidv eqeq2 3anbi3d eleq2d reseq1 + dmeq eqeq1d imbi12d eqeq2d fveq1 cbviotavw eqtrdi cbvmptv anbi12d mpteq1i + cbvabv eqtri ifbieq12i ) FAOZBOZPUAZUBZBEQZAERZXBAEUCZXDUDZUEUFZUGZULZIWS + DOZUDZUHZCOZXIPUAZUBZXIWSUIZSZXLXOSZTZUJZCEQZUKZDERZBUMZIOZXJUHZXNXIYDUIZ + SZXLYFSZTZUJZCEQZYDXIUNWRTZUOZDERZAUPZUQZURJOZKOZPUAZUBZKEQZJERZUUAJEUCZU + UCUDZUEUFZUGZULZLMOZGOZUDZUHZHOZUUIPUAZUBZUUIUUHUIZSZUULUUOSZTZUJZHEQZUKZ + GERZMUMZLOZUUJUHZUUNUUIUVDUIZSZUULUVFSZTZUJZHEQZUVDUUIUNZYQTZUOZGERZJUPZU + QZURNXCUUBXHYPUUGUVQXBUUAAJEAJVOZXBYQWSPUAZUBZBEQUUAUVRXAUVTBEUVRWTUVSWRY + QWSPUSUTVAUVTYTBKEBKVOUVSYSWSYRYQPVBUTVCVEZVDXDUUCXGUUFXBUUAAJEUWAVFZXFUU + EXEUUDUEXDUUCUWBVGVHVIVJYPLYCUVPUQUVQILYCYOUVPILVOZYOUVDXJUHZXNXIUVFSZXLU + VFSZTZUJZCEQZUVDXIUNZWRTZUOZDERZAUPUVPUWCYNUWMAUWCYMUWLDEUWCYEUWDYKUWIYLU + WKILXJVKUWCYJUWHCEUWCYIUWGXNUWCYGUWEYHUWFUWCYFUVFXIYDUVDVLZVMUWCYFUVFXLUW + NVMVNVPVAYDUVDWRXIVQVRVSVTUWMUVOAJUVRUWMUWDUWIUWJYQTZUOZDERUVOUVRUWLUWPDE + UVRUWKUWOUWDUWIWRYQUWJWAWBVSUWPUVNDGEDGVOZUWDUVEUWIUVKUWOUVMUWQXJUUJUVDXI + UUIWEZWCUWQUWIXLUUIPUAZUBZUVGUWFTZUJZCEQUVKUWQUWHUXBCEUWQXNUWTUWGUXAUWQXM + UWSXIUUIXLPVBUTZUWQUWEUVGUWFXIUUIUVFWDWFWGVAUXBUVJCHECHVOZUWTUUNUXAUVIUXD + UWSUUMXLUULUUIPUSUTZUXDUWFUVHUVGXLUULUVFWDWHWGVCVEUWQUWJUVLYQUVDXIUUIWIWF + VRVDVEWJWKWLLYCUVCUVPYBUVBBMBMVOZYBUUHXJUHZXNXIUUOSZXLUUOSZTZUJZCEQZUKZDE + RUVBUXFYAUXMDEUXFXKUXGXTUXLBMXJVKUXFXSUXKCEUXFXRUXJXNUXFXPUXHXQUXIUXFXOUU + OXIWSUUHVLZVMUXFXOUUOXLUXNVMVNVPVAWMVSUXMUVADGEUWQUXGUUKUXLUUTUWQXJUUJUUH + UWRWCUWQUXLUWTUUPUXITZUJZCEQUUTUWQUXKUXPCEUWQXNUWTUXJUXOUXCUWQUXHUUPUXIXI + UUIUUOWDWFWGVAUXPUUSCHEUXDUWTUUNUXOUURUXEUXDUXIUUQUUPXLUULUUOWDWHWGVCVEWM + VDVEWOWNWPWQWP $. + $} + + ${ + $d A a b x y z g v u $. + nosupno.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( The next several theorems deal with a surreal "supremum". This surreal + will ultimately be shown to bound ` A ` below and bound the restriction + of any surreal above. We begin by showing that the given expression + actually defines a surreal number. (Contributed by Scott Fenton, + 5-Dec-2021.) $) + nosupno $p |- ( ( A C_ No /\ A e. V ) -> S e. No ) $= + ( va wcel csur cvv wa wrex cres wceq syl con0 cbday vb vz wss elex cv wbr + cslt wn wral crio cdm c2o cop csn cun csuc wi cab cfv w3a cio cmpt iftrue + cif adantr simprl wreu wrmo nomaxmo ad2antrl reu5 sylanbrc riotacl sseldd + simpl c1o 2on elexi prid2 noextend eqeltrd iffalse wfun crn funmpt iotaex + cpr a1i eqid dmmpti word ssel2 nodmon onss sseld adantrd rexlimdva abssdv + wtr wel wal simplr adantlr ontr1 mpand reseq1 onelon sylan suceloni eloni + simpllr ordsucelsuc mpbid onelss sylc resabs1d eqeq12d syl5ib expimpd vex + weq eleq1w suceq reseq2d imbi2d ralbidv anbi12d rexbidv elab bdayfo ax-mp + wb mpan adantl reximi eqeltrrd eqeltrid wex eqeq2 3anbi3d imim2d reximdva + ralimdv jcad anbi2i 3imtr4g alrimivv dftr2 sylibr cima cuni wfo funimaexg + dford5 fofun uniexd ss2abi bdayval wfn fofn fnfvima mp3an1 elssuni sstrid + ssexd elong mpbird rnmpt weu wmo spcev mp3an3 rexcom4 sylib nosupprefixmo + fvex df-eu iota2 eqcom bitrdi simprr3 adantrr norn simprr1 fvelrn syl2anc + nofun rexlimdvaa sylbird sylan2b eqsstrid syl3anbrc pm2.61ian sylan2 + elno2 ) EHKELUCZEMKZFLKEHUDUWPUWQNZFAUEZBUEZUGUFUHBEUIZAEOZUXAAEUJZUXCUKU + LUMUNUOZGUWTDUEZUKZKZCUEZUXEUGUFUHZUXEUWTUPZPZUXHUXJPZQZUQZCEUIZNZDEOZBUR + ZGUEZUXFKZUXIUXEUXSUPZPZUXHUYAPZQZUQZCEUIZUXSUXEUSZUWSQZUTZDEOZAVAZVBZVDZ + LIUXBUWRUYMLKUXBUWRNZUYMUXDLUXBUYMUXDQUWRUXBUXDUYLVCVEUYNUXCLKUXDLKUYNELU + XCUXBUWPUWQVFUYNUXAAEVGZUXCEKUYNUXBUXAAEVHZUYOUXBUWRVOUWPUYPUXBUWQABEVIVJ + UXAAEVKVLUXAAEVMRVNUXCULVPULULSVQVRVSVTRWAUXBUHZUWRNZUYMUYLLUYQUYMUYLQUWR + UXBUXDUYLWBVEUYRUYLWCZUYLUKZSKZUYLWDZVPULWGZUCZUYLLKUYSUYRGUXRUYKWEWHUWRV + UAUYQUWRUYTUXRSGUXRUYKUYLUYJAWFUYLWIZWJUWRUXRSKZUXRWKZUWPVUGUWQUWPUXRSUCU + XRWSZVUGUWPUXQBSUWPUXPUWTSKZDEUWPUXEEKZNZUXGVUIUXOVUKUXFSUWTVUKUXFSKZUXFS + UCVUKUXELKZVULELUXEWLZUXEWMRZUXFWNRWOWPWQWRUWPJUAWTZUAUEZUXRKZNZJUEZUXRKZ + UQZUAXAJXAVUHUWPVVBJUAUWPVUPVUQUXFKZUXIUXEVUQUPZPZUXHVVDPZQZUQZCEUIZNZDEO + ZNVUTUXFKZUXIUXEVUTUPZPZUXHVVMPZQZUQZCEUIZNZDEOZVUSVVAUWPVUPVVKVVTUWPVUPN + ZVVJVVSDEVWAVUJNZVVJVVLVVRVWBVVCVVLVVIVWBVUPVVCVVLUWPVUPVUJXBVWBVULVUPVVC + NVVLUQUWPVUJVULVUPVUOXCZVUTVUQUXFXDRXEWPVWBVVCVVIVVRVWBVVCNZVVHVVQCEVWDVV + GVVPUXIVVGVVEVVMPZVVFVVMPZQVWDVVPVVEVVFVVMXFVWDVWEVVNVWFVVOVWDUXEVVMVVDVW + DVVDSKZVVMVVDKZVVMVVDUCVWDVUQSKZVWGVWBVULVVCVWIVWCUXFVUQXGXHZVUQXIRVWDVUP + VWHUWPVUPVUJVVCXKVWDVUQWKZVUPVWHYLVWDVWIVWKVWJVUQXJRVUTVUQXLRXMVVDVVMXNXO + ZXPVWDUXHVVMVVDVWLXPXQXRUUAUUCXSUUDUUBXSVURVVKVUPUXQVVKBVUQUAXTBUAYAZUXPV + VJDEVWMUXGVVCUXOVVIBUAUXFYBVWMUXNVVHCEVWMUXMVVGUXIVWMUXKVVEUXLVVFVWMUXJVV + DUXEUWTVUQYCZYDVWMUXJVVDUXHVWNYDXQYEYFYGYHYIUUEUXQVVTBVUTJXTBJYAZUXPVVSDE + VWOUXGVVLUXOVVRBJUXFYBVWOUXNVVQCEVWOUXMVVPUXIVWOUXKVVNUXLVVOVWOUXJVVMUXEU + WTVUTYCZYDVWOUXJVVMUXHVWPYDXQYEYFYGYHYIUUFUUGJUAUXRUUHUUIUXRUUNVLVEUWRUXR + MKVUFVUGYLUWRUXRTEUUJZUUKZMUWQVWRMKUWPUWQVWQMTWCZUWQVWQMKLSTUULZVWSYJLSTU + UOYKTEMUUMYMUUPYNUWRUXRUXGDEOZBURZVWRUXQVXABUXPUXGDEUXGUXOVOYOUUQUWPVXBVW + RUCUWQUWPVXABVWRUWPUXGUWTVWRKDEVUKUXFVWRUWTVUKUXFVWQKUXFVWRUCVUKUXETUSZUX + FVWQVUKVUMVXCUXFQVUNUXEUURRTLUUSZUWPVUJVXCVWQKVWTVXDYJLSTUUTYKLETUXEUVAUV + BYPUXFVWQUVCRWOWQWRVEUVDUVEUXRMUVFRUVGYQYNUWPVUDUYQUWQUWPVUBUBUEZUYKQZGUX + ROZUBURVUCGUBUXRUYKUYLVUEUVHUWPVXGUBVUCUWPVXFVXEVUCKZGUXRUXSUXRKUWPUXTUYF + NZDEOZVXFVXHUQUXQVXJBUXSGXTBGYAZUXPVXIDEVXKUXGUXTUXOUYFBGUXFYBVXKUXNUYECE + VXKUXMUYDUXIVXKUXKUYBUXLUYCVXKUXJUYAUXEUWTUXSYCZYDVXKUXJUYAUXHVXLYDXQYEYF + YGYHYIUWPVXJNZVXFUXTUYFUYGVXEQZUTZDEOZVXHVXMVXPUYKVXEQZVXFVXMUYJAUVIZVXPV + XQYLZVXMUYJAYRZUYJAUVJZVXRVXJVXTUWPVXJUYIAYRZDEOVXTVXIVYBDEUXTUYFUYGUYGQZ + VYBUYGWIUYIUXTUYFVYCUTAUYGUXSUXEUVPUWSUYGQUYHVYCUXTUYFUWSUYGUYGYSYTUVKUVL + YOUYIDAEUVMUVNYNUWPVYAVXJACDEUXSUVOVEUYJAUVQVLVXEMKVXRVXSUBXTUYJVXPAVXEMA + UBYAZUYIVXODEVYDUYHVXNUXTUYFUWSVXEUYGYSYTYHUVRYMRUYKVXEUVSUVTUWPVXPVXHUQV + XJUWPVXOVXHDEUWPVUJVXONNZUYGVXEVUCUXTUYFVXNVUJUWPUWAVYEUXEWDZVUCUYGVYEVUM + VYFVUCUCUWPVUJVUMVXOVUNUWBZUXEUWCRVYEUXEWCZUXTUYGVYFKVYEVUMVYHVYGUXEUWGRU + XTUYFVXNVUJUWPUWDUXSUXEUWEUWFVNYPUWHVEUWIUWJWQWRUWKVJUYLUWOUWLWAUWMYQUWN + $. + $} + + ${ + $d A g $. $d A p $. $d A q $. $d A u $. $d A v $. $d A y $. $d A z $. + $d g u $. $d g v $. $d g y $. $d p q $. $d p u $. $d p v $. $d p y $. + $d p z $. $d q u $. $d q v $. $d q y $. $d q z $. $d u v $. $d u y $. + $d u z $. $d v y $. $d v z $. $d y z $. + nosupdm.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( The domain of the surreal supremum when there is no maximum. The + primary point of this theorem is to change bound variable. (Contributed + by Scott Fenton, 6-Dec-2021.) $) + nosupdm $p |- ( -. E. x e. A A. y e. A -. x + dom S = { z | E. p e. A ( z e. dom p /\ + A. q e. A ( -. q ( p |` suc z ) = ( q |` suc z ) ) ) } ) $= + ( cv cslt wn wral wrex cdm cres wceq wi wbr wcel csuc wa cab cfv w3a cmpt + cio crio c2o cop csn cun cif iffalse eqtrid iotaex eqid dmmpti eqtrdi weq + dmeqd dmeq eleq2d breq1 notbid reseq1 eqeq2d imbi12d breq2 eqeq1d ralbidv + cbvralvw bitrid anbi12d cbvrexvw eleq1w reseq2d eqeq12d imbi2d rexbidv + suceq cbvabv ) ALZBLZMUANBFOZAFPZNZGQZWFELZQZUBZDLZWKMUAZNZWKWFUCZRZWNWQR + ZSZTZDFOZUDZEFPZBUEZCLZJLZQZUBZILZXGMUAZNZXGXFUCZRZXJXMRZSZTZIFOZUDZJFPZC + UEWIWJHXEHLZWLUBWPWKYAUCZRWNYBRSTDFOYAWKUFWESUGEFPZAUIZUHZQXEWIGYEWIGWHWG + AFUJZYFQUKULUMUNZYEUOYEKWHYGYEUPUQVCHXEYDYEYCAURYEUSUTVAXDXTBCXDWFXHUBZXL + XGWQRZXJWQRZSZTZIFOZUDZJFPBCVBZXTXCYNEJFEJVBZWMYHXBYMYPWLXHWFWKXGVDVEXBXJ + WKMUAZNZWRYJSZTZIFOYPYMXAYTDIFDIVBZWPYRWTYSUUAWOYQWNXJWKMVFVGUUAWSYJWRWNX + JWQVHVIVJVNYPYTYLIFYPYRXLYSYKYPYQXKWKXGXJMVKVGYPWRYIYJWKXGWQVHVLVJVMVOVPV + QYOYNXSJFYOYHXIYMXRBCXHVRYOYLXQIFYOYKXPXLYOYIXNYJXOYOWQXMXGWFXFWCZVSYOWQX + MXJUUBVSVTWAVMVPWBVOWDVA $. + $} + + ${ + $d A g u v x y $. $d O u y $. + nosupbday.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Birthday bounding law for surreal supremum. (Contributed by Scott + Fenton, 5-Dec-2021.) $) + nosupbday $p |- ( ( ( A C_ No /\ A e. _V ) /\ + ( O e. On /\ ( bday " A ) C_ O ) ) -> + ( bday ` S ) C_ O ) $= + ( csur wss wcel wa cbday cfv cdm wceq adantr syl cv cvv con0 cima nosupno + bdayval cslt wbr wn wral wrex crio csuc c2o cop csn cun cres cab w3a cmpt + cio cif iftrue eqtrid dmeqd 2oex dmsnop uneq2i dmun df-suc 3eqtr4i eqtrdi + wi word simprrl eloni simprll wreu wrmo simpl nomaxmo adantl reu5 adantrr + sylanbrc riotacl sseldd simprrr wfn wfo bdayfo fofn ax-mp fnfvima mp3an2i + eqeltrrd ordsucss sylc eqsstrd iffalse iotaex eqid simplrl ssel2 ad4ant14 + dmmpti simplrr mp3an1 onelss sseld adantrd rexlimdva abssdv pm2.61ian ) E + JKZEUALZMZHUBLZNEUCZHKZMZMZFNOZFPZHYBFJLZYCYDQXQYEYAABCDEFGUAIUDRFUESATZB + TZUFUGUHBEUIZAEUJZYBYDHKYIYBMZYDYHAEUKZPZULZHYIYDYMQYBYIYDYKYLUMUNUOZUPZP + ZYMYIFYOYIFYIYOGYGDTZPZLZCTZYQUFUGUHZYQYGULZUQYTUUBUQQVMCEUIZMZDEUJZBURZG + TZYRLUUAYQUUGULZUQYTUUHUQQVMCEUIUUGYQOYFQUSDEUJZAVAZUTZVBZYOIYIYOUUKVCVDV + EYLYNPZUPYLYLUOZUPYPYMUUMUUNYLYLUMVFVGVHYKYNVIYLVJVKVLRYJHVNZYLHLYMHKYJXR + UUOYIXQXRXTVOHVPSYJYKNOZYLHYJYKJLUUPYLQYJEJYKYIXOXPYAVQZYJYHAEVRZYKELZYIX + QUURYAYIXQMYIYHAEVSZUURYIXQVTXQUUTYIXOUUTXPABEWARWBYHAEWCWEWDYHAEWFSZWGYK + UESYJXSHUUPYIXQXRXTWHNJWIZYJXOUUSUUPXSLJUBNWJUVBWKJUBNWLWMZUUQUVAJENYKWNW + OWGWPYLHWQWRWSYIUHZYBMYDUUFHUVDYDUUFQYBUVDYDUUKPUUFUVDFUUKUVDFUULUUKIYIYO + UUKWTVDVEGUUFUUJUUKUUIAXAUUKXBXFVLRYBUUFHKUVDYBUUEBHYBUUDYGHLZDEYBYQELZMZ + YSUVEUUCUVGYRHYGUVGXRYRHLYRHKXQXRXTUVFXCUVGYQNOZYRHUVGYQJLZUVHYRQXOUVFUVI + XPYAEJYQXDXEYQUESUVGXSHUVHXQXRXTUVFXGXOUVFUVHXSLZXPYAUVBXOUVFUVJUVCJENYQW + NXHXEWGWPHYRXIWRXJXKXLXMWBWSXNWS $. + $} + + ${ + $d A g $. $d A p $. $d A u $. $d A v $. $d A x $. $d A y $. $d G g $. + $d G p $. $d g u $. $d G u $. $d g v $. $d G v $. $d g x $. $d G x $. + $d g y $. $d G y $. $d p u $. $d p v $. $d U p $. $d U u $. $d u v $. + $d U v $. $d u x $. $d U x $. $d u y $. $d v x $. $d v y $. + nosupfv.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( The value of surreal supremum when there is no maximum. (Contributed by + Scott Fenton, 5-Dec-2021.) $) + nosupfv $p |- ( ( -. E. x e. A A. y e. A -. x ( U |` suc G ) = ( v |` suc G ) ) ) ) -> + ( S ` G ) = ( U ` G ) ) $= + ( vp cv cslt wn wral wrex wcel cres wceq w3a wbr csur wss cvv wa cdm csuc + wi cfv cab cio cmpt crio c2o cop csn cun cif iffalse eqtrid fveq1d simp32 + 3ad2ant1 eleq2d breq2 notbid reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev + dmeq 3impb weq cbvrexvw sylibr eleq1 suceq reseq2d eqeq12d imbi2d rexbidv + 3ad2ant3 elabd 3anbi123d iotabidv eqid iotaex fvmpt syl simp1 simp2 simp3 + fveqeq2 eqidd fveq1 syl13anc weu fvex wex wmo eqeq2 3anbi3d mp3an3 reximi + wb spcev rexcom4 sylib nosupprefixmo adantr 3ad2ant2 df-eu sylanbrc iota2 + sylancr mpbid 3eqtrd ) ALZBLZMUANBEOZAEPZNZEUBUCZEUDQZUEZGEQZIGUFZQZCLZGM + UAZNZGIUGZRZYKYNRZSZUHZCEOZTZTZIFUIZIHYADLZUFZQZYKUUCMUAZNZUUCYAUGZRZYKUU + HRZSZUHZCEOZUEZDEPZBUJZHLZUUDQZUUGUUCUUQUGZRZYKUUSRZSZUHZCEOZUUQUUCUIXTSZ + TZDEPZAUKZULZUIZIUUDQZUUGUUCYNRZYPSZUHZCEOZIUUCUIZXTSZTZDEPZAUKZIGUIZYDYG + UUBUVJSYTYDIFUVIYDFYCYBAEUMZUWBUFUNUOUPUQZUVIURUVIJYCUWCUVIUSUTVAVCUUAIUU + PQUVJUVTSUUAUUOUVKUVOUEZDEPZBIYIYDYGYHYJYSVBYTYDUWEYGYTIKLZUFZQZYKUWFMUAZ + NZUWFYNRZYPSZUHZCEOZUEZKEPZUWEYHYJYSUWPUWOYJYSUEKGEUWFGSZUWHYJUWNYSUWQUWG + YIIUWFGVMVDUWQUWMYRCEUWQUWJYMUWLYQUWQUWIYLUWFGYKMVEVFUWQUWKYOYPUWFGYNVGVH + VIVJVKVLVNUWDUWODKEDKVOZUVKUWHUVOUWNUWRUUDUWGIUUCUWFVMVDUWRUVNUWMCEUWRUUG + UWJUVMUWLUWRUUFUWIUUCUWFYKMVEVFUWRUVLUWKYPUUCUWFYNVGVHVIVJVKVPVQZWDYAISZU + UNUWDDEUWTUUEUVKUUMUVOYAIUUDVRUWTUULUVNCEUWTUUKUVMUUGUWTUUIUVLUUJYPUWTUUH + YNUUCYAIVSZVTUWTUUHYNYKUXAVTWAWBVJVKWCWEHIUVHUVTUUPUVIUUQISZUVGUVSAUXBUVF + UVRDEUXBUURUVKUVDUVOUVEUVQUUQIUUDVRUXBUVCUVNCEUXBUVBUVMUUGUXBUUTUVLUVAYPU + XBUUSYNUUCUUQIVSZVTUXBUUSYNYKUXCVTWAWBVJUUQIXTUUCWOWFWCWGUVIWHUVSAWIWJWKU + UAUVKUVOUVPUWASZTZDEPZUVTUWASZYTYDUXFYGYTYHYJYSUWAUWASZUXFYHYJYSWLYHYJYSW + MYHYJYSWNYTUWAWPUXEYJYSUXHTDGEUUCGSZUVKYJUVOYSUXDUXHUXIUUDYIIUUCGVMVDUXIU + VNYRCEUXIUUGYMUVMYQUXIUUFYLUUCGYKMVEVFUXIUVLYOYPUUCGYNVGVHVIVJUXIUVPUWAUW + AIUUCGWQVHWFVLWRWDUUAUWAUDQUVSAWSZUXFUXGXGIGWTUUAUVSAXAZUVSAXBZUXJYTYDUXK + YGYTUWEUXKUWSUWEUVRAXAZDEPUXKUWDUXMDEUVKUVOUVPUVPSZUXMUVPWHUVRUVKUVOUXNTA + UVPIUUCWTXTUVPSUVQUXNUVKUVOXTUVPUVPXCXDXHXEXFUVRDAEXIXJWKWDYGYDUXLYTYEUXL + YFACDEIXKXLXMUVSAXNXOUVSUXFAUWAUDXTUWASZUVRUXEDEUXOUVQUXDUVKUVOXTUWAUVPXC + XDWCXPXQXRXS $. + $} + + ${ + $d A a $. $d a g $. $d A g $. $d A p $. $d a u $. $d A u $. $d a v $. + $d A v $. $d a x $. $d A x $. $d a y $. $d A y $. $d G a $. $d G p $. + $d g u $. $d G u $. $d g v $. $d G v $. $d g x $. $d g y $. $d G y $. + $d p u $. $d p v $. $d S a $. $d U a $. $d U p $. $d U u $. $d u v $. + $d U v $. $d u x $. $d U x $. $d u y $. $d v x $. $d v y $. $d x y $. + nosupres.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( A restriction law for surreal supremum when there is no maximum. + (Contributed by Scott Fenton, 5-Dec-2021.) $) + nosupres $p |- ( ( -. E. x e. A A. y e. A -. x ( U |` suc G ) = ( v |` suc G ) ) ) ) -> + ( S |` suc G ) = ( U |` suc G ) ) $= + ( vp cv cslt wral wcel wa cdm cres wceq wi wbr wrex csur wss cvv csuc w3a + va cfv cin dmres word nosupno 3ad2ant2 nodmord syl cab dmeq eleq2d notbid + wn breq2 reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev weq cbvrexvw sylibr + 3impb wb eleq1 suceq reseq2d eqeq12d imbi2d rexbidv elabg mpbird 3ad2ant3 + cio cmpt crio c2o cop csn cun cif iffalse eqtrid dmeqd iotaex eqid dmmpti + eqtrdi 3ad2ant1 eleqtrrd ordsucss df-ss sylib simp2l simp31 sseldd simp32 + sylc eqtr4d simpl1 simpl2 simpl31 sselda con0 nodmon onelon syl2anc eloni + ordsuc imp simpl33 resabs1 syl5ib imim2d ralimdv nosupfv syl113anc fvresd + simpr 3eqtr4d ex sylbid ralrimiv wfun nofun funres 3syl eqfunfv mpbir2and + ) ALZBLZMUAVABENZAEUBZVAZEUCUDZEUEOZPZGEOZIGQZOZCLZGMUAZVAZGIUFZRZUUJUUMR + ZSZTZCENZUGZUGZFUUMRZUUNSZUVAQZUUNQZSZUHLZUVAUIZUVFUUNUIZSZUHUVCNZUUTUVCU + UMUVDUUTUVCUUMFQZUJZUUMFUUMUKUUTUUMUVKUDZUVLUUMSUUTUVKULZIUVKOUVMUUTFUCOZ + UVNUUFUUCUVOUUSABCDEFHUEJUMUNZFUOUPUUTIYTDLZQZOZUUJUVQMUAZVAZUVQYTUFZRZUU + JUWBRZSZTZCENZPZDEUBZBUQZUVKUUSUUCIUWJOZUUFUUSUWKIUVROZUWAUVQUUMRZUUOSZTZ + CENZPZDEUBZUUSIKLZQZOZUUJUWSMUAZVAZUWSUUMRZUUOSZTZCENZPZKEUBZUWRUUGUUIUUR + UXIUXHUUIUURPKGEUWSGSZUXAUUIUXGUURUXJUWTUUHIUWSGURUSUXJUXFUUQCEUXJUXCUULU + XEUUPUXJUXBUUKUWSGUUJMVBUTUXJUXDUUNUUOUWSGUUMVCVDVEVFVGVHVLUWQUXHDKEDKVIZ + UWLUXAUWPUXGUXKUVRUWTIUVQUWSURUSUXKUWOUXFCEUXKUWAUXCUWNUXEUXKUVTUXBUVQUWS + UUJMVBUTUXKUWMUXDUUOUVQUWSUUMVCVDVEVFVGVJVKUUIUUGUWKUWRVMUURUWIUWRBIUUHYT + ISZUWHUWQDEUXLUVSUWLUWGUWPYTIUVRVNUXLUWFUWOCEUXLUWEUWNUWAUXLUWCUWMUWDUUOU + XLUWBUUMUVQYTIVOZVPUXLUWBUUMUUJUXMVPVQVRVFVGVSVTUNWAWBUUCUUFUVKUWJSUUSUUC + UVKHUWJHLZUVROUWAUVQUXNUFZRUUJUXORSTCENUXNUVQUIYSSUGDEUBZAWCZWDZQUWJUUCFU + XRUUCFUUBUUAAEWEZUXSQWFWGWHWIZUXRWJUXRJUUBUXTUXRWKWLWMHUWJUXQUXRUXPAWNUXR + WOWPWQWRWSIUVKWTXGUUMUVKXAXBWLZUUTUVDUUMUUHUJZUUMGUUMUKUUTUUMUUHUDZUYBUUM + SUUTUUHULZUUIUYCUUTGUCOZUYDUUTEUCGUUCUUDUUEUUSXCUUCUUFUUGUUIUURXDXEZGUOUP + UUCUUFUUGUUIUURXFZIUUHWTXGZUUMUUHXAXBWLXHUUTUVIUHUVCUUTUVFUVCOUVFUUMOZUVI + UUTUVCUUMUVFUYAUSUUTUYIUVIUUTUYIPZUVFFUIZUVFGUIZUVGUVHUYJUUCUUFUUGUVFUUHO + UULGUVFUFZRZUUJUYMRZSZTZCENZUYKUYLSUUCUUFUUSUYIXIUUCUUFUUSUYIXJUUGUUIUURU + UCUUFUYIXKUUTUUMUUHUVFUYHXLUYJUYMUUMUDZUURUYRUUTUYIUYSUUTUUMULZUYIUYSTUUT + IULZUYTUUTIXMOZVUAUUTUUHXMOZUUIVUBUUTUYEVUCUYFGXNUPUYGUUHIXOXPIXQUPIXRXBU + VFUUMWTUPXSUUGUUIUURUUCUUFUYIXTUYSUUQUYQCEUYSUUPUYPUULUUPUUNUYMRZUUOUYMRZ + SUYSUYPUUNUUOUYMVCUYSVUDUYNVUEUYOGUYMUUMYAUUJUYMUUMYAVQYBYCYDXGABCDEFGHUV + FJYEYFUYJUVFUUMFUUTUYIYHZYGUYJUVFUUMGVUFYGYIYJYKYLUUTUVAYMZUUNYMZUVBUVEUV + JPVMUUTUVOFYMVUGUVPFYNUUMFYOYPUUTUYEGYMVUHUYFGYNUUMGYOYPUHUVAUUNYQXPYR $. + $} + + ${ + nosupbnd1.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + + ${ + $d A g $. $d A h $. $d A p $. $d A u $. $d A v $. $d A x $. + $d A y $. $d g u $. $d g v $. $d g x $. $d g y $. $d h p $. + $d h u $. $d h v $. $d h x $. $d h y $. $d p u $. $d p v $. + $d p x $. $d p y $. $d S h $. $d S p $. $d U h $. $d U p $. + $d u v $. $d U v $. $d u x $. $d u y $. $d v x $. $d v y $. + $d x y $. + $( Lemma for ~ nosupbnd1 . Establish a soft upper bound. (Contributed + by Scott Fenton, 5-Dec-2021.) $) + nosupbnd1lem1 $p |- ( ( -. E. x e. A A. y e. A -. x -. S + ( W |` dom S ) = S ) $= + ( cv cslt wbr wn csur cvv wcel wa cres wceq wral wss cdm w3a simp3rr wi + wrex con0 simp2l simp3rl sseldd simp3ll nosupno 3ad2ant2 nodmon syl3anc + sltres mtod simp3lr breq2d mtbid nosupbnd1lem1 syld3an3 noreson syl2anc + syl wb wor sltso sotrieq2 mpan mpbir2and ) AKBKLMNBEUAAEUGNZEOUBZEPQZRZ + GEQZGFUCZSZFTZRZIEQZIGLMZNZRZRZUDZIVRSZFTZWHFLMZNZFWHLMNZWGWHVSLMZWJWGW + MWCWBWDWAVMVPUEWGIOQZGOQVRUHQZWMWCUFWGEOIVMVNVOWFUIZWBWDWAVMVPUJZUKZWGE + OGWPVQVTWEVMVPULUKWGFOQZWOVPVMWSWFABCDEFHPJUMUNZFUOVFZIGVRUQUPURWGVSFWH + LVQVTWEVMVPUSUTVAVMVPWFWBWLWQABCDEFIHJVBVCWGWHOQZWSWIWKWLRVGZWGWNWOXBWR + XAIVRVDVEWTOLVHXBWSRXCVIOWHFLVJVKVEVL $. + $} + + ${ + $d A g $. $d A p $. $d A q $. $d A u $. $d A v $. $d A x $. + $d A y $. $d A z $. $d g u $. $d g v $. $d g x $. $d g y $. + $d p q $. $d p u $. $d p v $. $d p y $. $d p z $. $d q u $. + $d q v $. $d q x $. $d q y $. $d q z $. $d S p $. $d S q $. + $d S z $. $d U p $. $d U q $. $d u v $. $d u x $. $d u y $. + $d u z $. $d v x $. $d v y $. $d v z $. $d x y $. $d y z $. + $( Lemma for ~ nosupbnd1 . If ` U ` is a prolongment of ` S ` and in + ` A ` , then ` ( U `` dom S ) ` is not ` 2o ` . (Contributed by Scott + Fenton, 6-Dec-2021.) $) + nosupbnd1lem3 $p |- ( ( -. E. x e. A A. y e. A -. x + ( U ` dom S ) =/= 2o ) $= + ( vp vq cv wn csur wcel wa cres wceq c2o adantr vz cslt wbr wss cvv cdm + wral wrex w3a word nosupno 3ad2ant2 nodmord ordirr 3syl csuc wi simpl3l + cfv ndmfv wne c1o con0 2on elexi prid2 nosgnn0i neeq1 mpbiri neneqd syl + con4i adantl simpl2l sseldd simprl nodmon simpl3r simpll1 simpll2 simpr + c0 simpll3 nosupbnd1lem2 syl112anc eqtr4d simplr nolesgn2ores syl321anc + simprr expr ralrimiva eleq2d breq2 notbid reseq1 eqeq1d imbi12d ralbidv + dmeq anbi12d rspcev syl12anc cab nosupdm 3ad2ant1 eleq1 reseq2d eqeq12d + wb suceq imbi2d rexbidv elabg bitrd mpbird mtand neqned ) ALBLUBUCMBEUG + AEUHMZENUDZEUEOZPZGEOZGFUFZQZFRZPZUIZYDGUSZSYHYISRZYDYDOZYHFNOZYDUJYKMY + BXSYLYGABCDEFHUEIUKULZFUMYDUNUOYHYJPZYKYDJLZUFZOZKLZYOUBUCZMZYOYDUPZQZY + RUUAQZRZUQZKEUGZPZJEUHZYNYCYDGUFZOZYRGUBUCZMZGUUAQZUUCRZUQZKEUGZUUHYCYF + XSYBYJURZYJUUJYHUUJYJUUJMYIWBRZYJMYDGUTUURYISUURYISVAWBSVASVBSSVCVDVEVF + VGYIWBSVHVIVJVKVLVMYNUUOKEYNYREOZUULUUNYNUUSUULPZPZGNOYRNOYDVCOZYEYRYDQ + ZRYJUULUUNUVAENGYNXTUUTXTYAXSYGYJVNTZYNYCUUTUUQTVOUVAENYRUVDYNUUSUULVPV + OUVAYLUVBYNYLUUTYHYLYJYMTTFVQZVKUVAYEFUVCYNYFUUTYCYFXSYBYJVRTUVAXSYBYGU + UTUVCFRXSYBYGYJUUTVSXSYBYGYJUUTVTXSYBYGYJUUTWCYNUUTWAABCDEFGHYRIWDWEWFY + HYJUUTWGYNUUSUULWJGYRYDWHWIWKWLUUGUUJUUPPJGEYOGRZYQUUJUUFUUPUVFYPUUIYDY + OGWTWMUVFUUEUUOKEUVFYTUULUUDUUNUVFYSUUKYOGYRUBWNWOUVFUUBUUMUUCYOGUUAWPW + QWRWSXAXBXCYHYKUUHXJYJYHYKYDUALZYPOZYTYOUVGUPZQZYRUVIQZRZUQZKEUGZPZJEUH + ZUAXDZOZUUHXSYBYKUVRXJYGXSYDUVQYDABUACDEFHKJIXEWMXFYHYLUVBUVRUUHXJYMUVE + UVPUUHUAYDVCUVGYDRZUVOUUGJEUVSUVHYQUVNUUFUVGYDYPXGUVSUVMUUEKEUVSUVLUUDY + TUVSUVJUUBUVKUUCUVSUVIUUAYOUVGYDXKZXHUVSUVIUUAYRUVTXHXIXLWSXAXMXNUOXOTX + PXQXR $. + $} + + ${ + $d A g $. $d A u $. $d A v $. $d A w $. $d A x $. $d A y $. + $d g u $. $d g v $. $d g x $. $d g y $. $d S w $. $d U u $. + $d u v $. $d u w $. $d U w $. $d u x $. $d u y $. $d v w $. + $d v x $. $d v y $. $d w x $. $d w y $. $d x y $. + $( Lemma for ~ nosupbnd1 . If ` U ` is a prolongment of ` S ` and in + ` A ` , then ` ( U `` dom S ) ` is not undefined. (Contributed by + Scott Fenton, 6-Dec-2021.) $) + nosupbnd1lem4 $p |- ( ( -. E. x e. A A. y e. A -. x + ( U ` dom S ) =/= (/) ) $= + ( vw cv cslt wbr wn wrex csur wcel wa wceq adantr wral wss cvv cdm cres + w3a cfv c2o wne simpl1 simpl2 simprl simpl3 simprr simp2l simp3l sseldd + c0 wi simpl2l wor sltso soasym syl2an2r mpd jca nosupbnd1lem2 syl112anc + mpan nosupbnd1lem3 neneqd imnan sylib nrexdv simpl3l weq breq2 cbvrexvw + expr breq1 rexbidv bitrid dfrex2 ralbii ralnex 3bitri sylibr rspcv sylc + cbvralvw nosupno 3ad2ant2 nodmon simpl3r simpll1 simpll2 simpll3 eqtr4d + con0 syl simplr nolt02o syl321anc ancld reximdva mtand neqned ) AKZBKZL + MZNBEUAZAEONZEPUBZEUCQZRZGEQZGFUDZUEZFSZRZUFZXQGUGZURYAYBURSZGJKZLMZXQY + DUGZUHSZRZJEOZYAYHJEYAYDEQZRYEYGNZUSYHNYAYJYEYKYAYJYERZRZYFUHYMXLXOYJYD + XQUEZFSZYFUHUIXLXOXTYLUJZXLXOXTYLUKZYAYJYEULZYMXLXOXTYJYDGLMNZRZYOYPYQX + LXOXTYLUMYMYJYSYRYMYEYSYAYJYEUNYAGPQZYLYDPQZYEYSUSZYAEPGXLXMXNXTUOXLXOX + PXSUPUQYMEPYDXMXNXLXTYLUTYRUQPLVAUUAUUBRUUCVBPLGYDVCVIZVDVEVFABCDEFGHYD + IVGZVHABCDEFYDHIVJVHVKVSYEYGVLVMVNYAYCRZYEJEOZYIUUFXPDKZYDLMZJEOZDEUAZU + UGXPXSXLXOYCVOZUUFXLUUKXLXOXTYCUJUUKXJBEOZAEUAXKNZAEUAXLUUJUUMDAEUUJUUH + XILMZBEODAVPZUUMUUIUUOJBEYDXIUUHLVQVRUUPUUOXJBEUUHXHXILVTWAWBWJUUMUUNAE + XJBEWCWDXKAEWEWFWGUUJUUGDGEUUHGSUUIYEJEUUHGYDLVTWAWHWIUUFYEYHJEUUFYJRYE + YGUUFYJYEYGUUFYLRZUUAUUBXQWSQZXRYNSYEYCYGUUFUUAYLUUFEPGXMXNXLXTYCUTZUUL + UQZTUUQEPYDUUFXMYLUUSTUUFYJYEULZUQZUUQFPQZUURUUFUVCYLYAUVCYCXOXLUVCXTAB + CDEFHUCIWKWLTTFWMWTUUQXRFYNUUFXSYLXPXSXLXOYCWNTUUQXLXOXTYTYOXLXOXTYCYLW + OXLXOXTYCYLWPXLXOXTYCYLWQUUQYJYSUVAUUQYEYSUUFYJYEUNZUUFUUAYLUUBUUCUUTUV + BUUDVDVEVFUUEVHWRUVDYAYCYLXAGYDXQXBXCVSXDXEVEXFXG $. + $} + + ${ + $d A a $. $d A g $. $d a p $. $d A p $. $d a u $. $d A u $. + $d a v $. $d A v $. $d A x $. $d a y $. $d A y $. $d a z $. + $d A z $. $d g u $. $d g v $. $d g x $. $d g y $. $d p u $. + $d p v $. $d p y $. $d p z $. $d S a $. $d S p $. $d S z $. + $d U p $. $d u v $. $d u x $. $d u y $. $d u z $. $d U z $. + $d v x $. $d v y $. $d v z $. $d x y $. $d x z $. $d y z $. + $( Lemma for ~ nosupbnd1 . If ` U ` is a prolongment of ` S ` and in + ` A ` , then ` ( U `` dom S ) ` is not ` 1o ` . (Contributed by Scott + Fenton, 6-Dec-2021.) $) + nosupbnd1lem5 $p |- ( ( -. E. x e. A A. y e. A -. x + ( U ` dom S ) =/= 1o ) $= + ( vz vp wn c1o wceq wi csur wcel wa cres c0 va cv cslt wbr cdm cfv wral + wrex wss cvv w3a word nosupno 3ad2ant2 adantl nodmord 3syl csuc simpr3l + wne ordirr adantr ndmfv c2o 1oex prid1 nosgnn0i neeq1 mpbiri neneqd syl + con4i simp2l simp3l sseldd nofun simpl2l simpll simpl3r simpll1 simpll2 + wfun syl2an simpll3 simprl nosupbnd1lem2 syl112anc eqtr4d simplr simprr + ssel2 eqfunressuc syl213anc expr a2d ralimdva impcom anassrs dmeq breq2 + eleq2d notbid reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev syl12anc cab + wb simplr1 nosupdm con0 nodmon eleq1 suceq reseq2d eqeq12d imbi2d elabg + rexbidv bitrd mpbird mtand neqned rexanali wo simpl nofv 3orel2 syl5com + w3o imdistanda simpl1 simpl2 simpl3 simpr nosupbnd1lem4 pm2.21d expimpd + nosupbnd1lem3 jaod syldc anasss rexlimiva imp sylanbr pm2.61ian ) JUBZG + UCUDZLZFUEZUUJUFZMNZOZJEUGZAUBBUBUCUDLBEUGAEUHLZEPUIZEUJQZRZGEQZGUUMSZF + NZRZUKZUUMGUFZMUTZUUQUVFRZUVGMUVIUVGMNZUUMUUMQZUVIFPQZUUMULUVKLUVFUVLUU + QUVAUURUVLUVEABCDEFHUJIUMUNUOZFUPUUMVAUQUVIUVJRZUVKUUMKUBZUEZQZUUJUVOUC + UDZLZUVOUUMURZSZUUJUVTSZNZOZJEUGZRZKEUHZUVNUVBUUMGUEZQZUULGUVTSZUWBNZOZ + JEUGZUWGUVIUVBUVJUVBUVDUURUVAUUQUSVBUVJUWIUVIUWIUVJUWILUVGTNZUVJLUUMGVC + UWNUVGMUWNUVHTMUTZMMVDVEVFVGZUVGTMVHVIVJVKVLZUOUUQUVFUVJUWMUVFUVJRZUUQU + WMUWRUUPUWLJEUWRUUJEQZRUULUUOUWKUWRUWSUULUUOUWKOUWRUWSUULRZUUOUWKUWRUWT + UUORZRZGWBZUUJWBZUVCUUJUUMSZNUWIUUMUUJUEQZUVGUUNNUWKUXBGPQZUXCUWRUXGUXA + UVFUXGUVJUVFEPGUURUUSUUTUVEVMZUURUVAUVBUVDVNVOVBVBGVPVKUXBUUJPQZUXDUWRU + USUWSUXIUXAUUSUUTUURUVEUVJVQUWSUULUUOVREPUUJWKZWCUUJVPVKUXBUVCFUXEUWRUV + DUXAUVBUVDUURUVAUVJVSVBUXBUURUVAUVEUWTUXEFNZUURUVAUVEUVJUXAVTUURUVAUVEU + VJUXAWAUURUVAUVEUVJUXAWDUWRUWTUUOWEABCDEFGHUUJIWFZWGWHUWRUWIUXAUVJUWIUV + FUWQUOVBUXAUXFUWRUUOUXFUWTUXFUUOUXFLUUNTNZUUOLZUUMUUJVCUXMUUNMUXMUUNMUT + UWOUWPUUNTMVHVIVJVKVLUOUOUXBUVGMUUNUVFUVJUXAWIUWRUWTUUOWJWHGUUJUUMWLWMW + NWNWOWPWQWRUWFUWIUWMRKGEUVOGNZUVQUWIUWEUWMUXOUVPUWHUUMUVOGWSXAUXOUWDUWL + JEUXOUVSUULUWCUWKUXOUVRUUKUVOGUUJUCWTXBUXOUWAUWJUWBUVOGUVTXCXDXEXFXGXHX + IUVNUVKUUMUAUBZUVPQZUVSUVOUXPURZSZUUJUXRSZNZOZJEUGZRZKEUHZUAXJZQZUWGUVN + UURUVKUYGXKUURUVAUVEUUQUVJXLUURUUMUYFUUMABUACDEFHJKIXMXAVKUVNUVLUUMXNQU + YGUWGXKUVIUVLUVJUVMVBFXOUYEUWGUAUUMXNUXPUUMNZUYDUWFKEUYHUXQUVQUYCUWEUXP + UUMUVPXPUYHUYBUWDJEUYHUYAUWCUVSUYHUXSUWAUXTUWBUYHUXRUVTUVOUXPUUMXQZXRUY + HUXRUVTUUJUYIXRXSXTXFXGYBYAUQYCYDYEYFUUQLUULUXNRZJEUHZUVFUVHUULUUOJEYGU + YKUVFUVHUYJUVFUVHOZJEUWSUULUXNUYLUVFUWTUXNRUWTUXMUUNVDNZYHZRUVHUVFUWTUX + NUYNUVFUWTRZUXMUUOUYMYMZUXNUYNUYOUXIUYPUVFUUSUWSUXIUWTUXHUWSUULYIUXJWCU + UJUUMYJVKUXMUUOUYMYKYLYNUVFUWTUYNUVHUYOUXMUVHUYMUYOUXMUVHUYOUUNTUYOUURU + VAUWSUXKUUNTUTUURUVAUVEUWTYOZUURUVAUVEUWTYPZUVFUWSUULWEZUYOUURUVAUVEUWT + UXKUYQUYRUURUVAUVEUWTYQUVFUWTYRUXLWGZABCDEFUUJHIYSWGVJYTUYOUYMUVHUYOUUN + VDUYOUURUVAUWSUXKUUNVDUTUYQUYRUYSUYTABCDEFUUJHIUUBWGVJYTUUCUUAUUDUUEUUF + UUGUUHUUI $. + $} + + ${ + $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. + $d g v $. $d g x $. $d g y $. $d U u $. $d u v $. $d U v $. + $d u x $. $d u y $. $d v x $. $d v y $. $d x y $. + $( Lemma for ~ nosupbnd1 . Establish a hard upper bound when there is no + maximum. (Contributed by Scott Fenton, 6-Dec-2021.) $) + nosupbnd1lem6 $p |- ( ( -. E. x e. A A. y e. A -. x ( U |` dom S ) + ( U |` dom S ) + -. ( Z |` suc dom U ) . } ) ) $= + ( vx vq wcel cslt wbr wn wa csur c2o cfv con0 wceq syl c1o c0 vp wral wss + cvv w3a cdm csuc cres cop csn cun simp1l simp3 breq1 rspcv sylc crab cint + cv wne simpl21 simpl1l sseldd simpl23 simp21 sltso sonr mpan breq2 notbid + wor syl5ibcom con2d neqned nosepssdm syl3anc wo wb nosepon nodmon syl2anc + imp onsseleq ctp wrex adantr sylan simpr weq fveq2 neeq12d onnminsb df-ne + onelon con2bii sylibr simplr ontr1 mp2and fvresd eqtr4d ralrimiva sltval2 + wi mpbid breqtrrd raleq breq12d anbi12d rspcev syl12anc sltval mpbird cxp + noreson cin df-res 2on xpsng sylancl ineq1d 3syl eqtrid nofun eqtrd sylib + wfun mpd eleq2d ex sylbid nosgnn0i mpbiri neneqd intnanrd w3o adantl fvex + ndmfv 3orrot incom word nodmord ordirr disjsn xpdisj1 eqtr3d resundir un0 + uneq2d eqcomi 3eqtr4g wrel resdm sssucid resabs1 mp1i 3brtr4d elexi prid2 + funrel noextend sucelon sltres soasym df-suc reseq2i resundi eqtri rabbii + dmres necom inteqi necomd eqsstrid eqsstrrd df-ss biimpar ralrimiv funres + nesym eqfunfv mpbir2and wfn funfn 1oex prid1 neeq1d 3brtr3d brtp ecase23d + 3bitri simprd neeq1 con4i fnressn opeq2d sneqd uneq12d eqnbrtrd jaodan + mpdan ) CBHZCAUSIJKABUBZLZBMUCZBUDHZDMHZUEZEUSZDIJZEBUBZUEZCDIJZDCUFZUGZU + HZCUXONUIZUJZUKZIJKZUXMUXCUXLUXNUXCUXDUXIUXLULZUXEUXIUXLUMUXKUXNECBUXJCDI + UNUOUPUXMUXNLZFUSZCOZUYDDOZUTZFPUQZURZUXOUCZUYAUYCCMHZUXHCDUTZUYJUYCBMCUX + FUXGUXHUXEUXLUXNVAUXCUXDUXIUXLUXNVBVCZUXFUXGUXHUXEUXLUXNVDZUYCCDUXMUXNCDQ + ZKUXMUYOUXNUXMCCIJZKZUYOUXNKUXMUYKUYQUXMBMCUXEUXFUXGUXHUXLVEUYBVCMIVKZUYK + UYQVFMCIVGVHRUYOUYPUXNCDCIVIVJVLVMWBVNZFCDVOVPUYCUYJUYIUXOHZUYIUXOQZVQZUY + AUYCUYIPHZUXOPHZUYJVUBVRUYCUYKUXHUYLVUCUYMUYNUYSFCDVSZVPUYCUYKVUDUYMCVTZR + ZUYIUXOWCWAUYCVUBUYAUYCUYTUYAVUAUYCUYTLZUXTUXQIJZUYAVUHUXTUXOUHZUXQUXOUHZ + IJZVUIVUHCDUXOUHZVUJVUKIVUHCVUMIJZGUSZCOZVUOVUMOZQZGUAUSZUBZVUSCOZVUSVUMO + ZSTUISNUITNUIWDZJZLZUAPWEZVUHVUCVURGUYIUBZUYICOZUYIVUMOZVVCJZVVFVUHUYKUXH + UYLVUCUYCUYKUYTUYMWFZUYCUXHUYTUYNWFZUYCUYLUYTUYSWFVUEVPZVUHVURGUYIVUHVUOU + YIHZLZVUPVUODOZVUQVVOVUPVVPUTZKZVUPVVPQZVVOVUOPHZVVNVVRVUHVUCVVNVVTVVMUYI + VUOWNWGVUHVVNWHZUYGVVQFVUOFGWIUYEVUPUYFVVPUYDVUOCWJUYDVUODWJWKWLZUPVVQVVS + VUPVVPWMWOWPVVOVUOUXODVVOVVNUYTVUOUXOHZVWAUYCUYTVVNWQVVOVUDVVNUYTLVWCXDVU + HVUDVVNUYCVUDUYTVUGWFZWFVUOUYIUXOWRRWSWTXAXBVUHVVHUYIDOZVVIVVCVUHUXNVVHVW + EVVCJZUXMUXNUYTWQVUHUYKUXHUXNVWFVRZVVKVVLCDFXCZWAXEVUHUYIUXODUYCUYTWHWTXF + VVEVVGVVJLUAUYIPVUSUYIQZVUTVVGVVDVVJVURGVUSUYIXGVWIVVAVVHVVBVVIVVCVUSUYIC + WJVUSUYIVUMWJXHXIXJXKVUHUYKVUMMHZVUNVVFVRVVKVUHUXHVUDVWJVVLVWDDUXOXOWAUAG + CVUMXLWAXMVUHVUJCUXOUHZCVUHVWKUXSUXOUHZUKVWKTUKZVUJVWKVUHVWLTVWKVUHVWLUXS + UXOUDXNZXPZTUXSUXOXQVUHUXOUJZNUJZXNZVWNXPZVWOTVUHVWRUXSVWNVUHVUDNPHVWRUXS + QVWDXRUXONPPXSXTYAVUHVWPUXOXPZTQVWSTQVUHVWTUXOVWPXPZTVWPUXOUUAVUHUXOUXOHK + ZVXATQVUHUYKUXOUUBZVXBVVKCUUCZUXOUUDZYBUXOUXOUUEWPYCVWPUXOVWQUDUUFRUUGYCU + UJCUXSUXOUUHVWMVWKVWKUUIUUKUULVUHCYGZCUUMVWKCQVUHUYKVXFVVKCYDZRCUVACUUNYB + YEUXOUXPUCVUKVUMQVUHUXOUUODUXOUXPUUPUUQUURVUHUXTMHZUXQMHZVUDVULVUIXDUYCVX + HUYTUYCUYKVXHUYMCNSNNPXRUUSUUTZUVBZRWFZUYCVXIUYTUYCUXHUXPPHZVXIUYNUYCVUDV + XMVUGUXOUVCYFDUXPXOWAWFZVWDUXTUXQUXOUVDVPYHVUHVXHVXIVUIUYAXDZVXLVXNUYRVXH + VXILVXOVFMIUXTUXQUVEVHWAYHUYCVUALZUXQUXTUXTIVXPUXQVUMDVWPUHZUKZUXTUXQDUXO + VWPUKZUHVXRUXPVXSDUXOUVFUVGDUXOVWPUVHUVIVXPVUMCVXQUXSVXPVUMCQZVUMUFZUXOQZ + VUQVUPQZGVYAUBZVXPVYAUXODUFZXPZUXODUXOUVKVXPUXOVYEUCVYFUXOQVXPUXOUYIVYEUY + CVUAWHZVXPUYIUYFUYEUTZFPUQZURZVYEUYHVYIUYGVYHFPUYEUYFUVLUVJUVMVXPUXHUYKDC + UTVYJVYEUCUYCUXHVUAUYNWFZUYCUYKVUAUYMWFZVXPCDUYCUYLVUAUYSWFUVNFDCVOVPUVOU + VPUXOVYEUVQYFYCZVXPVYCGVYAVXPVUOVYAHVWCVYCVXPVYAUXOVUOVYMYIVXPVWCVYCVXPVW + CLZVUQVVPVUPVYNVUOUXODVXPVWCWHWTVYNVVRVVPVUPQZVYNVVTVVNVVRVXPVUDVWCVVTVXP + UYKVUDVYLVUFRUXOVUOWNWGVXPVVNVWCVXPUYIUXOVUOVYGYIUVRVWBUPVVQVYOVUPVVPUWAW + OWPYEYJYKUVSVXPVUMYGZVXFVXTVYBVYDLVRVXPUXHDYGZVYPVYKDYDZUXODUVTYBVXPUYKVX + FVYLVXGRGVUMCUWBWAUWCVXPVXQUXOUXODOZUIZUJZUXSVXPDVYEUWDZUXOVYEHZVXQWUAQVX + PVYQWUBVXPUXHVYQVYKVYRRDUWEYFVXPVYSNQZWUCVXPUXOCOZTQZWUDVXPWUFWUDLZWUESQZ + VYSTQZLZWUHWUDLZVXPWUHWUIVXPWUESVXPWUESUTTSUTSSNUWFUWGYLVXPWUETSVXPVXCVXB + WUFVXPUYKVXCVYLVXDRVXEUXOCYSYBUWHYMYNZYOVXPWUHWUDWULYOVXPWUEVYSVVCJZWUGWU + JWUKYPZVXPVVHVWEWUEVYSVVCVXPUXNVWFUXMUXNVUAWQVXPUYKUXHVWGVYLVYKVWHWAXEVUA + VVHWUEQUYCUYIUXOCWJYQVUAVWEVYSQUYCUYIUXODWJYQUWIWUMWUJWUKWUGYPWUKWUGWUJYP + WUNSTSNTNWUEVYSUXOCYRUXODYRUWJWUJWUKWUGYTWUKWUGWUJYTUWLYFUWKUWMZWUCWUDWUC + KWUIWUDKUXODYSWUIVYSNWUIVYSNUTTNUTNVXJYLVYSTNUWNYMYNRUWORVYEUXODUWPWAVXPV + YTUXRVXPVYSNUXOWUOUWQUWRYEUWSYCVXPUYKVXHUXTUXTIJKZVYLVXKUYRVXHWUPVFMUXTIV + GVHYBUWTUXAYJYKYHUXB $. + $} + + ${ + nosupbnd2.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + + ${ + $d A a $. $d a g $. $d A g $. $d a p $. $d A p $. $d A q $. + $d a u $. $d A u $. $d a v $. $d A v $. $d a x $. $d A x $. + $d a y $. $d A y $. $d g p $. $d g q $. $d g u $. $d g v $. + $d g x $. $d g y $. $d p q $. $d p u $. $d p v $. $d p x $. + $d p y $. $d q u $. $d q v $. $d q y $. $d S a $. $d S g $. + $d S p $. $d u v $. $d u x $. $d u y $. $d v x $. $d v y $. + $d x y $. $d Z a $. $d Z g $. $d Z p $. $d Z x $. + + $( Bounding law from above for the surreal supremum. Proposition 4.3 of + [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.) $) + nosupbnd2 $p |- ( ( A C_ No /\ A e. _V /\ Z e. No ) -> + ( A. a e. A a -. ( Z |` dom S ) . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Change bound variables for surreal infimum. (Contributed by Scott + Fenton, 9-Aug-2024.) $) + noinfcbv $p |- T = if ( E. a e. B A. b e. B -. b . } ) , + ( c e. { b | E. d e. B ( b e. dom d /\ + A. e e. B ( -. d ( d |` suc b ) = ( e |` suc b ) ) ) } + |-> ( iota a E. d e. B ( c e. dom d /\ + A. e e. B ( -. d ( d |` suc c ) = ( e |` suc c ) ) /\ + ( d ` c ) = a ) ) ) ) $= + ( cv cslt wral wrex cres wceq wi wbr wn crio cdm c1o cop csn wcel csuc wa + cun cab cfv w3a cio cmpt cif breq2 notbid ralbidv breq1 cbvralvw cbvrexvw + bitrdi cbvriotavw dmeqi opeq1i sneqi uneq12i eleq1w suceq reseq2d eqeq12d + weq imbi2d fveqeq2 3anbi123d rexbidv iotabidv eqeq2 3anbi3d eleq2d reseq1 + dmeq eqeq1d imbi12d eqeq2d fveq1 cbviotavw eqtrdi cbvmptv anbi12d mpteq1i + cbvabv eqtri ifbieq12i ) FBNZANZOUAZUBZBEPZAEQZXAAEUCZXCUDZUEUFZUGZUKZHWQ + DNZUDZUHZXHCNZOUAZUBZXHWQUIZRZXKXNRZSZTZCEPZUJZDEQZBULZHNZXIUHZXMXHYCUIZR + ZXKYERZSZTZCEPZYCXHUMWRSZUNZDEQZAUOZUPZUQJNZINZOUAZUBZJEPZIEQZYTIEUCZUUBU + DZUEUFZUGZUKZKYPLNZUDZUHZUUGGNZOUAZUBZUUGYPUIZRZUUJUUMRZSZTZGEPZUJZLEQZJU + LZKNZUUHUHZUULUUGUVBUIZRZUUJUVDRZSZTZGEPZUVBUUGUMZYQSZUNZLEQZIUOZUPZUQMXB + UUAXGYOUUFUVOXAYTAIEAIVNZXAWQYQOUAZUBZBEPYTUVPWTUVRBEUVPWSUVQWRYQWQOURUSU + TUVRYSBJEBJVNZUVQYRWQYPYQOVAUSVBVDZVCXCUUBXFUUEXAYTAIEUVTVEZXEUUDXDUUCUEX + CUUBUWAVFVGVHVIYOKYBUVNUPUVOHKYBYNUVNHKVNZYNUVBXIUHZXMXHUVDRZXKUVDRZSZTZC + EPZUVBXHUMZWRSZUNZDEQZAUOUVNUWBYMUWLAUWBYLUWKDEUWBYDUWCYJUWHYKUWJHKXIVJUW + BYIUWGCEUWBYHUWFXMUWBYFUWDYGUWEUWBYEUVDXHYCUVBVKZVLUWBYEUVDXKUWMVLVMVOUTY + CUVBWRXHVPVQVRVSUWLUVMAIUVPUWLUWCUWHUWIYQSZUNZDEQUVMUVPUWKUWODEUVPUWJUWNU + WCUWHWRYQUWIVTWAVRUWOUVLDLEDLVNZUWCUVCUWHUVIUWNUVKUWPXIUUHUVBXHUUGWDZWBUW + PUWHUUGXKOUAZUBZUVEUWESZTZCEPUVIUWPUWGUXACEUWPXMUWSUWFUWTUWPXLUWRXHUUGXKO + VAUSZUWPUWDUVEUWEXHUUGUVDWCWEWFUTUXAUVHCGECGVNZUWSUULUWTUVGUXCUWRUUKXKUUJ + UUGOURUSZUXCUWEUVFUVEXKUUJUVDWCWGWFVBVDUWPUWIUVJYQUVBXHUUGWHWEVQVCVDWIWJW + KKYBUVAUVNYAUUTBJUVSYAYPXIUHZXMXHUUMRZXKUUMRZSZTZCEPZUJZDEQUUTUVSXTUXKDEU + VSXJUXEXSUXJBJXIVJUVSXRUXICEUVSXQUXHXMUVSXOUXFXPUXGUVSXNUUMXHWQYPVKZVLUVS + XNUUMXKUXLVLVMVOUTWLVRUXKUUSDLEUWPUXEUUIUXJUURUWPXIUUHYPUWQWBUWPUXJUWSUUN + UXGSZTZCEPUURUWPUXIUXNCEUWPXMUWSUXHUXMUXBUWPUXFUUNUXGXHUUGUUMWCWEWFUTUXNU + UQCGEUXCUWSUULUXMUUPUXDUXCUXGUUOUUNXKUUJUUMWCWGWFVBVDWLVCVDWNWMWOWPWO $. + $} + + ${ + $d B a b x y z g v u $. $d V g z $. + noinfno.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( The next several theorems deal with a surreal "infimum". This surreal + will ultimately be shown to bound ` B ` above and bound the restriction + of any surreal below. We begin by showing that the given expression + actually defines a surreal number. (Contributed by Scott Fenton, + 8-Aug-2024.) $) + noinfno $p |- ( ( B C_ No /\ B e. V ) -> T e. No ) $= + ( va vb csur wcel wa wrex cres wceq syl con0 cbday vz wss cv cslt wn wral + wbr crio cdm c1o cop csn cun csuc wi cab cfv w3a cio iftrue adantr simprl + cmpt cif wreu wrmo nominmo ad2antrl reu5 sylanbrc riotacl sseldd c2o 1oex + simpl noextend eqeltrd iffalse wfun crn cpr funmpt a1i iotaex eqid dmmpti + prid1 word wtr nodmon simprrl onelon syl2anc rexlimdvaa abssdv wel simplr + wal ssel2 adantlr ontr1 mpand adantrd reseq1 sylan sucelon sylib wb eloni + simpllr ordsucelsuc onelss resabs1d eqeq12d syl5ib imim2d ralimdv expimpd + mpbid sylc vex weq eleq1w reseq2d imbi2d ralbidv anbi12d rexbidv elab cvv + suceq bdayfo ax-mp mpan adantl eqeltrrd eqeltrid wex eqeq2 3anbi3d anbi2i + jcad reximdva 3imtr4g alrimivv dftr2 sylibr cima cuni wfo fofun funimaexg + dford5 uniexd bdayval wfn fofn fnfvima mp3an1 adantrr elssuni ssexd elong + mpbird rnmpt rexab weu wmo fvex spcev mp3an3 reximi rexcom4 noinfprefixmo + df-eu iota2 eqcom bitrdi simprr3 simprr1 fvelrn sylbird biimtrid eqsstrid + norn nofun exlimdv elno2 syl3anbrc pm2.61ian ) ELUBZEHMZNZFBUCZAUCZUDUGUE + BEUFZAEOZUWPAEUHZUWRUIUJUKULUMZGUWNDUCZUIZMZUWTCUCZUDUGUEZUWTUWNUNZPZUXCU + XEPZQZUOZCEUFZNZDEOZBUPZGUCZUXAMZUXDUWTUXNUNZPZUXCUXPPZQZUOZCEUFZUXNUWTUQ + ZUWOQZURZDEOZAUSZVCZVDZLIUWQUWMUYHLMUWQUWMNZUYHUWSLUWQUYHUWSQUWMUWQUWSUYG + UTVAUYIUWRLMUWSLMUYIELUWRUWQUWKUWLVBUYIUWPAEVEZUWREMUYIUWQUWPAEVFZUYJUWQU + WMVOUWKUYKUWQUWLABEVGVHUWPAEVIVJUWPAEVKRVLUWRUJUJVMVNWGVPRVQUWQUEZUWMNZUY + HUYGLUYLUYHUYGQUWMUWQUWSUYGVRVAUYMUYGVSZUYGUIZSMUYGVTZUJVMWAZUBUYGLMUYNUY + MGUXMUYFWBWCUYMUYOUXMSGUXMUYFUYGUYEAWDUYGWEZWFUYMUXMSMZUXMWHZUWKUYTUYLUWL + UWKUXMSUBUXMWIZUYTUWKUXLBSUWKUXKUWNSMZDEUWKUWTEMZUXKNZNZUXASMZUXBVUBVUEUW + TLMZVUFVUEELUWTUWKVUDVOUWKVUCUXKVBVLZUWTWJZRUWKVUCUXBUXJWKZUXAUWNWLWMWNWO + UWKJKWPZKUCZUXMMZNZJUCZUXMMZUOZKWRJWRVUAUWKVUQJKUWKVUKVULUXAMZUXDUWTVULUN + ZPZUXCVUSPZQZUOZCEUFZNZDEOZNVUOUXAMZUXDUWTVUOUNZPZUXCVVHPZQZUOZCEUFZNZDEO + ZVUNVUPUWKVUKVVFVVOUWKVUKNZVVEVVNDEVVPVUCNZVVEVVGVVMVVQVURVVGVVDVVQVUKVUR + VVGUWKVUKVUCWQVVQVUFVUKVURNVVGUOVVQVUGVUFUWKVUCVUGVUKELUWTWSWTVUIRZVUOVUL + UXAXARXBXCVVQVURVVDVVMVVQVURNZVVCVVLCEVVSVVBVVKUXDVVBVUTVVHPZVVAVVHPZQVVS + VVKVUTVVAVVHXDVVSVVTVVIVWAVVJVVSUWTVVHVUSVVSVUSSMZVVHVUSMZVVHVUSUBVVSVULS + MZVWBVVQVUFVURVWDVVRUXAVULWLXEZVULXFXGVVSVUKVWCUWKVUKVUCVURXJVVSVULWHZVUK + VWCXHVVSVWDVWFVWEVULXIRVUOVULXKRXSVUSVVHXLXTZXMVVSUXCVVHVUSVWGXMXNXOXPXQX + RUUBUUCXRVUMVVFVUKUXLVVFBVULKYABKYBZUXKVVEDEVWHUXBVURUXJVVDBKUXAYCVWHUXIV + VCCEVWHUXHVVBUXDVWHUXFVUTUXGVVAVWHUXEVUSUWTUWNVULYKZYDVWHUXEVUSUXCVWIYDXN + YEYFYGYHYIUUAUXLVVOBVUOJYABJYBZUXKVVNDEVWJUXBVVGUXJVVMBJUXAYCVWJUXIVVLCEV + WJUXHVVKUXDVWJUXFVVIUXGVVJVWJUXEVVHUWTUWNVUOYKZYDVWJUXEVVHUXCVWKYDXNYEYFY + GYHYIUUDUUEJKUXMUUFUUGUXMUUMVJVHUYMUXMYJMUYSUYTXHUYMUXMTEUUHZUUIZYJUWMVWM + YJMZUYLUWLVWNUWKUWLVWLYJTVSZUWLVWLYJMLSTUUJZVWOYLLSTUUKYMTEHUULYNUUNYOYOU + WKUXMVWMUBUYLUWLUWKUXLBVWMUWKUXKUWNVWMMDEVUEUXAVWMUWNVUEUXAVWLMUXAVWMUBVU + EUWTTUQZUXAVWLVUEVUGVWQUXAQVUHUWTUUORUWKVUCVWQVWLMZUXKTLUUPZUWKVUCVWRVWPV + WSYLLSTUUQYMLETUWTUURUUSUUTYPUXAVWLUVARVUJVLWNWOVHUVBUXMYJUVCRUVDYQUYMUYP + UAUCZUYFQZGUXMOZUAUPUYQGUAUXMUYFUYGUYRUVEUYMVXBUAUYQVXBUXOUYANZDEOZVXANZG + YRUYMVWTUYQMZUXLVXDVXAGBBGYBZUXKVXCDEVXGUXBUXOUXJUYABGUXAYCVXGUXIUXTCEVXG + UXHUXSUXDVXGUXFUXQUXGUXRVXGUXEUXPUWTUWNUXNYKZYDVXGUXEUXPUXCVXHYDXNYEYFYGY + HUVFUYMVXEVXFGUYMVXDVXAVXFUYMVXDNZVXAUXOUYAUYBVWTQZURZDEOZVXFVXIVXLUYFVWT + QZVXAVXIUYEAUVGZVXLVXMXHZVXIUYEAYRZUYEAUVHZVXNVXDVXPUYMVXDUYDAYRZDEOVXPVX + CVXRDEUXOUYAUYBUYBQZVXRUYBWEUYDUXOUYAVXSURAUYBUXNUWTUVIUWOUYBQUYCVXSUXOUY + AUWOUYBUYBYSYTUVJUVKUVLUYDDAEUVMXGYOUYMVXQVXDUWKVXQUYLUWLACDEUXNUVNVHVAUY + EAUVOVJVWTYJMVXNVXOUAYAUYEVXLAVWTYJAUAYBZUYDVXKDEVXTUYCVXJUXOUYAUWOVWTUYB + YSYTYHUVPYNRUYFVWTUVQUVRUYMVXLVXFUOZVXDUWKVYAUYLUWLUWKVXKVXFDEUWKVUCVXKNZ + NZUYBVWTUYQUXOUYAVXJVUCUWKUVSVYCUWTVTZUYQUYBVYCVUGVYDUYQUBVYCELUWTUWKVYBV + OUWKVUCVXKVBVLZUWTUWERVYCUWTVSZUXOUYBVYDMVYCVUGVYFVYEUWTUWFRUXOUYAVXJVUCU + WKUVTUXNUWTUWAWMVLYPWNVHVAUWBXRUWGUWCWOUWDUYGUWHUWIVQUWJYQ $. + $} + + ${ + $d B g $. $d B p q u v y z $. $d g u $. $d g v $. $d g y $. + noinfdm.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Next, we calculate the domain of ` T ` . This is mostly to change bound + variables. (Contributed by Scott Fenton, 8-Aug-2024.) $) + noinfdm $p |- ( -. E. x e. B A. y e. B -. y + dom T = { z | E. p e. B ( z e. dom p /\ A. q e. B ( -. p + ( p |` suc z ) = ( q |` suc z ) ) ) } ) $= + ( cv cslt wn wral wrex cdm cres wceq wi wbr wcel csuc wa cab cfv w3a cmpt + cio crio c1o cop csn cun cif iffalse eqtrid iotaex eqid dmmpti weq eleq1w + dmeqd reseq2d eqeq12d imbi2d ralbidv anbi12d rexbidv eleq2d notbid reseq1 + suceq dmeq breq1 eqeq1d imbi12d breq2 eqeq2d cbvralvw bitrdi cbvabv eqtri + cbvrexvw eqtrdi ) BLZALZMUANBFOZAFPZNZGQHWFELZQZUBZWKDLZMUAZNZWKWFUCZRZWN + WQRZSZTZDFOZUDZEFPZBUEZHLZWLUBWPWKXFUCZRWNXGRSTDFOXFWKUFWGSUGEFPZAUIZUHZQ + ZCLZJLZQZUBZXMILZMUAZNZXMXLUCZRZXPXSRZSZTZIFOZUDZJFPZCUEZWJGXJWJGWIWHAFUJ + ZYHQUKULUMUNZXJUOXJKWIYIXJUPUQVCXKXEYGHXEXIXJXHAURXJUSUTXDYFBCBCVAZXDXLWL + UBZWPWKXSRZWNXSRZSZTZDFOZUDZEFPYFYJXCYQEFYJWMYKXBYPBCWLVBYJXAYODFYJWTYNWP + YJWRYLWSYMYJWQXSWKWFXLVMZVDYJWQXSWNYRVDVEVFVGVHVIYQYEEJFEJVAZYKXOYPYDYSWL + XNXLWKXMVNVJYSYPXMWNMUAZNZXTYMSZTZDFOYDYSYOUUCDFYSWPUUAYNUUBYSWOYTWKXMWNM + VOVKYSYLXTYMWKXMXSVLVPVQVGUUCYCDIFDIVAZUUAXRUUBYBUUDYTXQWNXPXMMVRVKUUDYMY + AXTWNXPXSVLVSVQVTWAVHWDWAWBWCWE $. + $} + + ${ + $d B g $. $d B p $. $d B q $. $d B u $. $d B v $. $d B x $. $d B y $. + $d B z $. $d g u $. $d g v $. $d g x $. $d g y $. $d O p $. $d O z $. + $d p q $. $d p u $. $d p v $. $d p y $. $d p z $. $d q u $. $d q v $. + $d q y $. $d q z $. $d u v $. $d u x $. $d u y $. $d u z $. $d V g $. + $d V p $. $d v x $. $d v y $. $d v z $. $d V z $. $d x y $. $d y z $. + noinfbday.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Birthday bounding law for surreal infimum. (Contributed by Scott + Fenton, 8-Aug-2024.) $) + noinfbday $p |- ( ( ( B C_ No /\ B e. V ) /\ + ( O e. On /\ ( bday " B ) C_ O ) ) -> ( bday ` T ) C_ O ) $= + ( csur wss wcel wa cbday cdm wceq syl cv cres vz vp con0 cima cfv noinfno + vq bdayval adantr cslt wbr wn wral wrex crio csuc c1o cop csn cun cab w3a + wi cio cmpt cif iftrue eqtrid dmeqd 1oex dmsnop uneq2i dmun df-suc eqtrdi + 3eqtr4i word simprrl eloni simprll wreu wrmo nominmo reu5 sylanbrc sseldd + simpl riotacl simprrr wfn wfo bdayfo fofn ax-mp fnfvima eqeltrrd ordsucss + mp3an2i sylc eqsstrd noinfdm simplrl ssel2 simplrr mp3an1 ordelss syl2anc + ad4ant14 sseld adantrd rexlimdva abssdv adantl pm2.61ian ) EKLZEIMZNZHUCM + ZOEUDZHLZNZNZFOUEZFPZHXQYCYDQZYAXQFKMYEABCDEFGIJUFFUHRUIBSZASZUJUKULBEUMZ + AEUNZYBYDHLYIYBNZYDYHAEUOZPZUPZHYIYDYMQYBYIYDYKYLUQURUSZUTZPZYMYIFYOYIFYI + YOGYFDSZPZMYQCSZUJUKULZYQYFUPZTYSUUATQVCCEUMNDEUNBVAGSZYRMYTYQUUBUPZTYSUU + CTQVCCEUMUUBYQUEYGQVBDEUNAVDVEZVFYOJYIYOUUDVGVHVIYLYNPZUTYLYLUSZUTYPYMUUE + UUFYLYLUQVJVKVLYKYNVMYLVNVPVOUIYJHVQZYLHMYMHLYJXRUUGYIXQXRXTVRHVSZRYJYKOU + EZYLHYJYKKMUUIYLQYJEKYKYIXOXPYAVTZYJYHAEWAZYKEMZYJYIYHAEWBZUUKYIYBWGYJXOU + UMUUJABEWCRYHAEWDWEYHAEWHRZWFYKUHRYJXSHUUIYIXQXRXTWIOKWJZYJXOUULUUIXSMKUC + OWKUUOWLKUCOWMWNZUUJUUNKEOYKWOWRWFWPYLHWQWSWTYIULZYBNYDUASZUBSZPZMZUUSUGS + ZUJUKULUUSUURUPZTUVBUVCTQVCUGEUMZNZUBEUNZUAVAZHUUQYDUVGQYBABUACDEFGUGUBJX + AUIYBUVGHLUUQYBUVFUAHYBUVEUURHMZUBEYBUUSEMZNZUVAUVHUVDUVJUUTHUURUVJUUGUUT + HMUUTHLUVJXRUUGXQXRXTUVIXBUUHRUVJUUSOUEZUUTHUVJUUSKMZUVKUUTQXOUVIUVLXPYAE + KUUSXCXHUUSUHRUVJXSHUVKXQXRXTUVIXDXOUVIUVKXSMZXPYAUUOXOUVIUVMUUPKEOUUSWOX + EXHWFWPHUUTXFXGXIXJXKXLXMWTXNWT $. + $} + + ${ + $d B g $. $d B u $. $d B v $. $d B x $. $d B y $. $d G g $. $d g u $. + $d G u $. $d g v $. $d G v $. $d g x $. $d G x $. $d g y $. $d G y $. + $d U u $. $d u v $. $d U v $. $d u x $. $d U x $. $d u y $. $d v x $. + $d v y $. + noinffv.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( The value of surreal infimum when there is no minimum. (Contributed by + Scott Fenton, 8-Aug-2024.) $) + noinffv $p |- ( ( -. E. x e. B A. y e. B -. y + ( U |` suc G ) = ( v |` suc G ) ) ) ) -> ( T ` G ) = ( U ` G ) ) $= + ( cv cslt wral wrex wcel cres wceq w3a cfv wbr wn csur wss wa cdm csuc wi + cab cio cmpt crio c1o cop csn cun cif iffalse eqtrid fveq1d 3ad2ant1 dmeq + eleq2d breq1 notbid reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev 3ad2ant3 + 3impb simp32 eleq1 suceq reseq2d eqeq12d imbi2d rexbidv elabg syl fveqeq2 + mpbird 3anbi123d iotabidv eqid iotaex fvmpt simp1 simp2 simp3 eqidd fveq1 + wb syl13anc cvv weu fvex wex wmo eqeq2 3anbi3d spcev mp3an3 rexcom4 sylib + reximi simp2l noinfprefixmo df-eu sylanbrc iota2 sylancr mpbid 3eqtrd ) B + LZALZMUAUBBENZAEOZUBZEUCUDZEJPZUEZGEPZIGUFZPZGCLZMUAZUBZGIUGZQZYHYKQZRZUH + ZCENZSZSZIFTZIHXQDLZUFZPZYTYHMUAZUBZYTXQUGZQZYHUUEQZRZUHZCENZUEZDEOZBUIZH + LZUUAPZUUDYTUUNUGZQZYHUUPQZRZUHZCENZUUNYTTXRRZSZDEOZAUJZUKZTZIUUAPZUUDYTY + KQZYMRZUHZCENZIYTTZXRRZSZDEOZAUJZIGTZYAYDYSUVGRYQYAIFUVFYAFXTXSAEULZUVSUF + UMUNUOUPZUVFUQUVFKXTUVTUVFURUSUTVAYRIUUMPZUVGUVQRYRUWAUVHUVLUEZDEOZYQYAUW + CYDYEYGYPUWCUWBYGYPUEDGEYTGRZUVHYGUVLYPUWDUUAYFIYTGVBVCZUWDUVKYOCEUWDUUDY + JUVJYNUWDUUCYIYTGYHMVDVEUWDUVIYLYMYTGYKVFVGVHVIZVJVKVMZVLYRYGUWAUWCWOYAYD + YEYGYPVNUULUWCBIYFXQIRZUUKUWBDEUWHUUBUVHUUJUVLXQIUUAVOUWHUUIUVKCEUWHUUHUV + JUUDUWHUUFUVIUUGYMUWHUUEYKYTXQIVPZVQUWHUUEYKYHUWIVQVRVSVIVJVTWAWBWDHIUVEU + VQUUMUVFUUNIRZUVDUVPAUWJUVCUVODEUWJUUOUVHUVAUVLUVBUVNUUNIUUAVOUWJUUTUVKCE + UWJUUSUVJUUDUWJUUQUVIUURYMUWJUUPYKYTUUNIVPZVQUWJUUPYKYHUWKVQVRVSVIUUNIXRY + TWCWEVTWFUVFWGUVPAWHWIWBYRUVHUVLUVMUVRRZSZDEOZUVQUVRRZYQYAUWNYDYQYEYGYPUV + RUVRRZUWNYEYGYPWJYEYGYPWKYEYGYPWLYQUVRWMUWMYGYPUWPSDGEUWDUVHYGUVLYPUWLUWP + UWEUWFUWDUVMUVRUVRIYTGWNVGWEVKWPVLYRUVRWQPUVPAWRZUWNUWOWOIGWSYRUVPAWTZUVP + AXAZUWQYQYAUWRYDYQUWCUWRUWGUWCUVOAWTZDEOUWRUWBUWTDEUVHUVLUVMUVMRZUWTUVMWG + UVOUVHUVLUXASAUVMIYTWSXRUVMRUVNUXAUVHUVLXRUVMUVMXBXCXDXEXHUVODAEXFXGWBVLY + RYBUWSYAYBYCYQXIACDEIXJWBUVPAXKXLUVPUWNAUVRWQXRUVRRZUVOUWMDEUXBUVNUWLUVHU + VLXRUVRUVMXBXCVTXMXNXOXP $. + $} + + ${ + $d a b $. $d a c $. $d a g $. $d a u $. $d a v $. $d a x $. $d a y $. + $d B a $. $d B b $. $d b c $. $d B c $. $d B g $. $d b u $. $d B u $. + $d b v $. $d B v $. $d B x $. $d b y $. $d B y $. $d c u $. $d c v $. + $d c y $. $d G a $. $d G b $. $d G c $. $d g u $. $d g v $. $d G v $. + $d g x $. $d g y $. $d T a $. $d U a $. $d U b $. $d U c $. $d U u $. + $d u v $. $d U v $. $d u x $. $d U x $. $d u y $. $d V a $. $d V g $. + $d v x $. $d v y $. $d x y $. + noinfres.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( The restriction of surreal infimum when there is no minimum. + (Contributed by Scott Fenton, 8-Aug-2024.) $) + noinfres $p |- ( ( -. E. x e. B A. y e. B -. y + ( U |` suc G ) = ( v |` suc G ) ) ) ) -> + ( T |` suc G ) = ( U |` suc G ) ) $= + ( va vb vc cv wral wcel cres wceq syl cslt wbr wn wrex csur wss wa cdm wi + csuc w3a cfv cin dmres word noinfno 3ad2ant2 nodmord simp31 simp32 simp33 + cab dmeq eleq2d breq1 notbid reseq1 eqeq1d imbi12d ralbidv breq2 cbvralvw + weq eqeq2d bitrdi anbi12d rspcev syl12anc wb eleq1 reseq2d eqeq12d imbi2d + suceq rexbidv elabg mpbird noinfdm 3ad2ant1 eleqtrrd ordsucss df-ss sylib + sylc eqtrid con0 simp2l sseldd nodmon eqtr4d simpl1 simpl2 simpl31 sselda + eloni adantr simpl32 onelon syl2anc sucelon simpl33 resabs1 syl5ib imim2d + simpr ralimdv noinffv syl113anc fvresd 3eqtr4d sylbid ralrimiv wfun nofun + ex funresd eqfunfv mpbir2and ) BOAOUAUBUCBEPAEUDUCZEUEUFZEJQZUGZGEQZIGUHZ + QZGCOZUAUBZUCZGIUJZRZYPYSRZSZUIZCEPZUKZUKZFYSRZYTSZUUGUHZYTUHZSZLOZUUGULZ + UULYTULZSZLUUIPZUUFUUIYSUUJUUFUUIYSFUHZUMZYSFYSUNUUFYSUUQUFZUURYSSUUFUUQU + OZIUUQQUUSUUFFUEQZUUTYLYIUVAUUEABCDEFHJKUPUQZFURTUUFIUULMOZUHZQZUVCNOZUAU + BZUCZUVCUULUJZRZUVFUVIRZSZUIZNEPZUGZMEUDZLVBZUUQUUFIUVQQZIUVDQZUVHUVCYSRZ + UVFYSRZSZUIZNEPZUGZMEUDZUUFYMYOUUDUWFYIYLYMYOUUDUSZYIYLYMYOUUDUTZYIYLYMYO + UUDVAUWEYOUUDUGMGEUVCGSZUVSYOUWDUUDUWIUVDYNIUVCGVCVDUWIUWDGUVFUAUBZUCZYTU + WASZUIZNEPUUDUWIUWCUWMNEUWIUVHUWKUWBUWLUWIUVGUWJUVCGUVFUAVEVFUWIUVTYTUWAU + VCGYSVGVHVIVJUWMUUCNCENCVMZUWKYRUWLUUBUWNUWJYQUVFYPGUAVKVFUWNUWAUUAYTUVFY + PYSVGVNVIVLVOVPVQVRUUFYOUVRUWFVSUWHUVPUWFLIYNUULISZUVOUWEMEUWOUVEUVSUVNUW + DUULIUVDVTUWOUVMUWCNEUWOUVLUWBUVHUWOUVJUVTUVKUWAUWOUVIYSUVCUULIWDZWAUWOUV + IYSUVFUWPWAWBWCVJVPWEWFTWGYIYLUUQUVQSUUEABLCDEFHNMKWHWIWJIUUQWKWNYSUUQWLW + MWOZUUFUUJYSYNUMZYSGYSUNUUFYSYNUFZUWRYSSUUFYNUOZYOUWSUUFYNWPQZUWTUUFGUEQZ + UXAUUFEUEGYIYJYKUUEWQUWGWRZGWSZTYNXETUWHIYNWKWNZYSYNWLWMWOWTUUFUUOLUUIUUF + UULUUIQUULYSQZUUOUUFUUIYSUULUWQVDUUFUXFUUOUUFUXFUGZUULFULZUULGULZUUMUUNUX + GYIYLYMUULYNQYRGUVIRZYPUVIRZSZUIZCEPZUXHUXISYIYLUUEUXFXAYIYLUUEUXFXBYMYOU + UDYIYLUXFXCUUFYSYNUULUXEXDUXGUVIYSUFZUUDUXNUXGYSUOZUXFUXOUXGYSWPQZUXPUXGI + WPQZUXQUXGUXAYOUXRUXGUXBUXAUUFUXBUXFUXCXFUXDTYMYOUUDYIYLUXFXGYNIXHXIIXJWM + YSXETUUFUXFXOZUULYSWKWNYMYOUUDYIYLUXFXKUXOUUCUXMCEUXOUUBUXLYRUUBYTUVIRZUU + AUVIRZSUXOUXLYTUUAUVIVGUXOUXTUXJUYAUXKGUVIYSXLYPUVIYSXLWBXMXNXPWNABCDEFGH + UULJKXQXRUXGUULYSFUXSXSUXGUULYSGUXSXSXTYEYAYBUUFUUGYCZYTYCZUUHUUKUUPUGVSU + UFUVAUYBUVBUVAYSFFYDYFTUUFUXBUYCUXCUXBYSGGYDYFTLUUGYTYGXIYH $. + $} + + ${ + $d B g $. $d B h $. $d B p $. $d B q $. $d B u $. $d B v $. $d B x $. + $d B y $. $d g u $. $d g v $. $d g x $. $d g y $. $d h p $. $d h q $. + $d h u $. $d h v $. $d h x $. $d h y $. $d p q $. $d p u $. $d p v $. + $d p x $. $d p y $. $d q u $. $d q v $. $d q y $. $d T h $. $d T p $. + $d U h $. $d U p $. $d u v $. $d U v $. $d u x $. $d u y $. $d V g $. + $d V h $. $d V p $. $d v x $. $d v y $. $d x y $. + noinfbnd1.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Lemma for ~ noinfbnd1 . Establish a soft lower bound. (Contributed by + Scott Fenton, 9-Aug-2024.) $) + noinfbnd1lem1 $p |- ( ( -. E. x e. B A. y e. B -. y -. ( U |` dom T ) ( W |` dom T ) = T ) $= + ( cv cslt wbr wn csur wcel wa cres wceq wss cdm w3a simp3rl noinfbnd1lem1 + wral wrex syld3an3 simp3rr con0 wi simp2l simp3ll sseldd noinfno 3ad2ant2 + nodmon syl sltres syl3anc mtod simp3lr breq1d mtbid noreson syl2anc sltso + wb wor sotrieq2 mpan mpbir2and ) BLALMNOBEUFAEUGOZEPUAZEIQZRZGEQZGFUBZSZF + TZRZJEQZGJMNZOZRZRZUCZJVRSZFTZWHFMNOZFWHMNZOZVMVPWFWBWJWBWDWAVMVPUDZABCDE + FJHIKUEUHWGVSWHMNZWKWGWNWCWBWDWAVMVPUIWGGPQJPQZVRUJQZWNWCUKWGEPGVMVNVOWFU + LZVQVTWEVMVPUMUNWGEPJWQWMUNZWGFPQZWPVPVMWSWFABCDEFHIKUOUPZFUQURZGJVRUSUTV + AWGVSFWHMVQVTWEVMVPVBVCVDWGWHPQZWSWIWJWLRVHZWGWOWPXBWRXAJVRVEVFWTPMVIXBWS + RXCVGPWHFMVJVKVFVL $. + + $d B z $. $d p z $. $d q z $. $d T q $. $d T z $. $d U q $. $d u z $. + $d v z $. $d y z $. $d q x $. $d V q $. + $( Lemma for ~ noinfbnd1 . If ` U ` is a prolongment of ` T ` and in + ` B ` , then ` ( U `` dom T ) ` is not ` 1o ` . (Contributed by Scott + Fenton, 9-Aug-2024.) $) + noinfbnd1lem3 $p |- ( ( -. E. x e. B A. y e. B -. y + ( U ` dom T ) =/= 1o ) $= + ( vp vq wn csur wcel wa cres wceq c1o adantr vz cv cslt wbr wral wrex wss + cdm w3a cfv word noinfno 3ad2ant2 nodmord ordirr 3syl wi simpl3l c0 ndmfv + wne 1n0 necomi neeq1 mpbiri neneqd syl con4i adantl simpl2l sseldd simprl + csuc nodmon simpl3r simpll1 simpll2 simpll3 simpr noinfbnd1lem2 syl112anc + con0 eqtr4d simplr simprr nogesgn1ores syl321anc expr eleq2d breq1 notbid + ralrimiva dmeq reseq1 eqeq1d imbi12d ralbidv anbi12d syl12anc cab noinfdm + rspcev wb 3ad2ant1 eleq1 suceq reseq2d eqeq12d imbi2d rexbidv elabg bitrd + mpbird mtand neqned ) BUBAUBUCUDMBEUEAEUFMZENUGZEIOZPZGEOZGFUHZQZFRZPZUIZ + YAGUJZSYEYFSRZYAYAOZYEFNOZYAUKYHMXSXPYIYDABCDEFHIJULUMZFUNYAUOUPYEYGPZYHY + AKUBZUHZOZYLLUBZUCUDZMZYLYAVMZQZYOYRQZRZUQZLEUEZPZKEUFZYKXTYAGUHZOZGYOUCU + DZMZGYRQZYTRZUQZLEUEZUUEXTYCXPXSYGURZYGUUGYEUUGYGUUGMYFUSRZYGMYAGUTUUOYFS + UUOYFSVAUSSVASUSVBVCYFUSSVDVEVFVGVHVIYKUULLEYKYOEOZUUIUUKYKUUPUUIPZPZGNOZ + YONOYAWBOZYBYOYAQZRYGUUIUUKYKUUSUUQYKENGXQXRXPYDYGVJZUUNVKTUURENYOYKXQUUQ + UVBTYKUUPUUIVLVKYKUUTUUQYKYIUUTYEYIYGYJTFVNZVGTUURYBFUVAYKYCUUQXTYCXPXSYG + VOTUURXPXSYDUUQUVAFRXPXSYDYGUUQVPXPXSYDYGUUQVQXPXSYDYGUUQVRYKUUQVSABCDEFG + HIYOJVTWAWCYEYGUUQWDYKUUPUUIWEGYOYAWFWGWHWLUUDUUGUUMPKGEYLGRZYNUUGUUCUUMU + VDYMUUFYAYLGWMWIUVDUUBUULLEUVDYQUUIUUAUUKUVDYPUUHYLGYOUCWJWKUVDYSUUJYTYLG + YRWNWOWPWQWRXBWSYEYHUUEXCYGYEYHYAUAUBZYMOZYQYLUVEVMZQZYOUVGQZRZUQZLEUEZPZ + KEUFZUAWTZOZUUEXPXSYHUVPXCYDXPYAUVOYAABUACDEFHLKJXAWIXDYEYIUUTUVPUUEXCYJU + VCUVNUUEUAYAWBUVEYARZUVMUUDKEUVQUVFYNUVLUUCUVEYAYMXEUVQUVKUUBLEUVQUVJUUAY + QUVQUVHYSUVIYTUVQUVGYRYLUVEYAXFZXGUVQUVGYRYOUVRXGXHXIWQWRXJXKUPXLTXMXNXO + $. + + $d B w $. $d T w $. $d U w $. $d U x $. $d U y $. $d v w $. $d V w $. + $d w x $. $d w y $. + + $( Lemma for ~ noinfbnd1 . If ` U ` is a prolongment of ` T ` and in + ` B ` , then ` ( U `` dom T ) ` is not undefined. (Contributed by Scott + Fenton, 9-Aug-2024.) $) + noinfbnd1lem4 $p |- ( ( -. E. x e. B A. y e. B -. y + ( U ` dom T ) =/= (/) ) $= + ( vw cslt wbr wn wrex csur wcel wa wceq adantr cv wral wss cdm w3a cfv c0 + cres c1o wi simpl1 simpl2 simprl simpl3 simp2l sselda simp3l sseldd sltso + wne soasym mpan syl2anc impr noinfbnd1lem2 syl112anc noinfbnd1lem3 neneqd + wor jca expr imnan sylib nrexdv breq2 rexbidv dfral2 ralnex rexbii sylibr + xchbinxr simpl3l rspcdva cbvrexvw con0 simpl2l noinfno syl nodmon simpll1 + breq1 simpll2 simpll3 simprr mpd simpl3r eqtr4d simplr syl321anc reximdva + nogt01o ancld mtand neqned ) BUAZAUAZLMZNBEUBZAEOZNZEPUCZEIQZRZGEQZGFUDZU + HZFSZRZUEZXOGUFZUGXSXTUGSZKUAZGLMZXOYBUFZUISZRZKEOZXSYFKEXSYBEQZRZYCYENZU + JYFNXSYHYCYJXSYHYCRZRZYDUIYLXJXMYHYBXOUHZFSZYDUIUTXJXMXRYKUKZXJXMXRYKULZX + SYHYCUMZYLXJXMXRYHGYBLMNZRZYNYOYPXJXMXRYKUNYLYHYRYQXSYHYCYRYIYBPQZGPQZYCY + RUJZXSEPYBXJXKXLXRUOZUPXSUUAYHXSEPGUUCXJXMXNXQUQURTPLVIYTUUARUUBUSPLYBGVA + VBZVCVDVJABCDEFGHIYBJVEZVFABCDEFYBHIJVGVFVHVKYCYEVLVMVNXSYARZYCKEOZYGUUFX + EGLMZBEOZUUGUUFXGBEOZUUIAEGXFGSXGUUHBEXFGXELVOVPUUFXJUUJAEUBZXJXMXRYAUKUU + KUUJNZAEOXIUUJAEVQXHUULAEXGBEVRVSWAVTXNXQXJXMYAWBZWCUUHYCBKEXEYBGLWKWDVMU + UFYCYFKEUUFYHRYCYEUUFYHYCYEUUFYKRZYTUUAXOWEQZYMXPSYCYAYEUUNEPYBUUFXKYKXKX + LXJXRYAWFTZUUFYHYCUMZURZUUNEPGUUPUUFXNYKUUMTURZUUNFPQZUUOUUFUUTYKUUFXMUUT + XJXMXRYAULABCDEFHIJWGWHTFWIWHUUNYMFXPUUNXJXMXRYSYNXJXMXRYAYKWJXJXMXRYAYKW + LXJXMXRYAYKWMUUNYHYRUUQUUNYCYRUUFYHYCWNZUUNYTUUAUUBUURUUSUUDVCWOVJUUEVFUU + FXQYKXNXQXJXMYAWPTWQUVAXSYAYKWRYBGXOXAWSVKXBWTWOXCXD $. + + $d a p $. $d a u $. $d a v $. $d a y $. $d a z $. $d B a $. $d T a $. + $d U z $. $d V z $. $d x z $. + + $( Lemma for ~ noinfbnd1 . If ` U ` is a prolongment of ` T ` and in + ` B ` , then ` ( U `` dom T ) ` is not ` 2o ` . (Contributed by Scott + Fenton, 9-Aug-2024.) $) + noinfbnd1lem5 $p |- ( ( -. E. x e. B A. y e. B -. y + ( U ` dom T ) =/= 2o ) $= + ( vz wn c2o wceq csur wcel wa cres syl c0 va vp cv cslt wbr cdm wral wrex + cfv wi wss w3a wne word noinfno 3ad2ant2 adantl nodmord ordirr cab adantr + csuc simpr3l ndmfv 2on0 necomi neeq1 mpbiri neneqd simpl2l simpl3l sseldd + con4i wfun nofun simprll simpl3r simpll1 simpll2 simpll3 simprl syl112anc + noinfbnd1lem2 eqtr4d simplr simprr eqfunressuc syl213anc expr a2d anassrs + ralimdva impcom eleq2d breq1 notbid reseq1 eqeq1d imbi12d ralbidv anbi12d + dmeq rspcev syl12anc wb nodmon eleq1 suceq reseq2d eqeq12d imbi2d rexbidv + elabg mpbird noinfdm 3ad2ant1 mtand neqned rexanali simpr1 simpr2 simplll + con0 c1o simpr3 simpll noinfbnd1lem4 pm2.21d noinfbnd1lem3 w3o wo simpr2l + nofv 3orel3 sylc mpjaod ex anasss rexlimiva imp sylanbr pm2.61ian ) GKUCZ + UDUEZLZFUFZUUCUIZMNZUJZKEUGZBUCAUCUDUELBEUGAEUHLZEOUKZEIPZQZGEPZGUUFRZFNZ + QZULZUUFGUIZMUMZUUJUUSQZUUTMUVBUUTMNZUUFUUFPZUVBUUFUNZUVDLUVBFOPZUVEUUSUV + FUUJUUNUUKUVFUURABCDEFHIJUOUPUQZFURSUUFUSSUVBUVCQZUVDUUFUAUCZUBUCZUFZPZUV + JUUCUDUEZLZUVJUVIVBZRZUUCUVORZNZUJZKEUGZQZUBEUHZUAUTZPZUVHUWDUUFUVKPZUVNU + VJUUFVBZRZUUCUWFRZNZUJZKEUGZQZUBEUHZUVHUUOUUFGUFZPZUUEGUWFRZUWHNZUJZKEUGZ + UWMUVBUUOUVCUUOUUQUUKUUNUUJVCVAUVCUWOUVBUWOUVCUWOLZUUTMUWTUUTTNZUVAUUFGVD + UXAUVATMUMZMTVEVFZUUTTMVGVHSVIVMZUQUUJUUSUVCUWSUUSUVCQZUUJUWSUXEUUIUWRKEU + XEUUCEPZQUUEUUHUWQUXEUXFUUEUUHUWQUJUXEUXFUUEQZUUHUWQUXEUXGUUHQZQZGVNZUUCV + NZUUPUUCUUFRZNUWOUUFUUCUFPZUUTUUGNUWQUXIGOPUXJUXIEOGUXEUULUXHUULUUMUUKUUR + UVCVJVAZUXEUUOUXHUUOUUQUUKUUNUVCVKVAVLGVOSUXIUUCOPZUXKUXIEOUUCUXNUXEUXFUU + EUUHVPVLUUCVOSUXIUUPFUXLUXEUUQUXHUUOUUQUUKUUNUVCVQVAUXIUUKUUNUURUXGUXLFNZ + UUKUUNUURUVCUXHVRUUKUUNUURUVCUXHVSUUKUUNUURUVCUXHVTUXEUXGUUHWAABCDEFGHIUU + CJWCZWBWDUXIUVCUWOUUSUVCUXHWEZUXDSUXIUUHUXMUXEUXGUUHWFZUXMUUHUXMLZUUGMUXT + UUGTNZUUGMUMZUUFUUCVDUYAUYBUXBUXCUUGTMVGVHSVIVMSUXIUUTMUUGUXRUXSWDGUUCUUF + WGWHWIWIWJWLWMWKUWLUWOUWSQUBGEUVJGNZUWEUWOUWKUWSUYCUVKUWNUUFUVJGXBWNUYCUW + JUWRKEUYCUVNUUEUWIUWQUYCUVMUUDUVJGUUCUDWOWPUYCUWGUWPUWHUVJGUWFWQWRWSWTXAX + CXDUVHUUFYCPZUWDUWMXEUVBUYDUVCUVBUVFUYDUVGFXFSVAUWBUWMUAUUFYCUVIUUFNZUWAU + WLUBEUYEUVLUWEUVTUWKUVIUUFUVKXGUYEUVSUWJKEUYEUVRUWIUVNUYEUVPUWGUVQUWHUYEU + VOUWFUVJUVIUUFXHZXIUYEUVOUWFUUCUYFXIXJXKWTXAXLXMSXNUVHUUFUWCUUFUVBUUFUWCN + ZUVCUUSUYGUUJUUKUUNUYGUURABUACDEFHKUBJXOXPUQVAWNXNXQXRUUJLUUEUUHLZQZKEUHZ + UUSUVAUUEUUHKEXSUYJUUSUVAUYIUUSUVAUJZKEUXFUUEUYHUYKUXGUYHQZUUSUVAUYLUUSQZ + UYAUVAUUGYDNZUYMUYAUVAUYMUUGTUYMUUKUUNUXFUXPUUGTUMUYLUUKUUNUURXTZUYLUUKUU + NUURYAZUXFUUEUYHUUSYBZUYMUUKUUNUURUXGUXPUYOUYPUYLUUKUUNUURYEUXGUYHUUSYFUX + QWBZABCDEFUUCHIJYGWBVIYHUYMUYNUVAUYMUUGYDUYMUUKUUNUXFUXPUUGYDUMUYOUYPUYQU + YRABCDEFUUCHIJYIWBVIYHUYMUYHUYAUYNUUHYJZUYAUYNYKUXGUYHUUSWEUYMUXOUYSUYMEO + UUCUULUUMUUKUURUYLYLUYQVLUUCUUFYMSUYAUYNUUHYNYOYPYQYRYSYTUUAUUB $. + + $( Lemma for ~ noinfbnd1 . Establish a hard lower bound when there is no + minimum. (Contributed by Scott Fenton, 9-Aug-2024.) $) + noinfbnd1lem6 $p |- ( ( -. E. x e. B A. y e. B -. y T + T + -. ( U u. { <. dom U , 1o >. } ) . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Bounding law from below for the surreal infimum. Analagous to + proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, + 9-Aug-2024.) $) + noinfbnd2 $p |- ( ( B C_ No /\ B e. V /\ Z e. No ) -> + ( A. b e. B Z -. T . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + nosupinfsep.2 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Given two sets of surreals, a surreal ` W ` separates them iff its + restriction to the maximum of ` dom S ` and ` dom T ` separates them. + Corollary 4.4 of [Lipparini] p. 7. (Contributed by Scott Fenton, + 9-Aug-2024.) $) + nosupinfsep $p |- ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) + /\ W e. No ) -> ( ( A. a e. A a + ( A. a e. A a . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + noetasuplem.2 $e |- Z = + ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) ) $. + + ${ + $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. + $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. + $d v x $. $d v y $. $d x y $. + $( Lemma for ~ noeta . Establish that our final surreal really is a + surreal. (Contributed by Scott Fenton, 6-Dec-2021.) $) + noetasuplem1 $p |- ( ( A C_ No /\ A e. _V /\ B e. _V ) -> Z e. No ) $= + ( csur wss cvv wcel w3a cbday cima cuni c1o cdm csn cxp nosupno 3adant3 + csuc cdif cun con0 bdayimaon 3ad2ant3 1oex noextendseq syl2anc eqeltrid + c2o prid1 ) ELMZENOZFNOZPZIGQFRSUFZGUAUGTUBUCUHZLKVAGLOZVBUIOZVCLOURUSV + DUTABCDEGHNJUDUEUTURVEUSFNUJUKGVBTTUPULUQUMUNUO $. + $} + + ${ + $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. + $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. + $d v x $. $d v y $. $d x y $. + $( Lemma for ~ noeta . The restriction of ` Z ` to ` dom S ` is ` S ` . + (Contributed by Scott Fenton, 9-Aug-2024.) $) + noetasuplem2 $p |- ( ( A C_ No /\ A e. _V /\ B e. _V ) -> + ( Z |` dom S ) = S ) $= + ( csur cvv wcel cdm cres c0 cun c1o wceq wss w3a cima cuni csuc csn cxp + cbday cdif reseq1i resundir cin dmres wne 1oex snnz dmxp ineq2i disjdif + ax-mp 3eqtri wrel relres reldm0 mpbir uneq2i wfun nosupno 3adant3 nofun + wb syl funrel resdm 3syl uneq1d un0 eqtrdi eqtrid ) ELUAZEMNZFMNZUBZIGO + ZPZGWDPZQRZGWEGUHFUCUDUEZWDUIZSUFZUGZRZWDPWFWKWDPZRWGIWLWDKUJGWKWDUKWMQ + WFWMQTZWMOZQTZWOWDWKOZULWDWIULQWKWDUMWQWIWDWJQUNWQWITSUOUPWIWJUQUTURWDW + HUSVAWMVBWNWPVKWKWDVCWMVDUTVEVFVAWCWGGQRGWCWFGQWCGVGZGVBWFGTWCGLNZWRVTW + AWSWBABCDEGHMJVHVIGVJVLGVMGVNVOVPGVQVRVS $. + $} + + ${ + $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. + $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. + $d v x $. $d v y $. $d X u $. $d X v $. $d X x $. $d x y $. + $d X y $. + $( Lemma for ~ noeta . ` Z ` is an upper bound for ` A ` . Part of + Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, + 4-Dec-2021.) $) + noetasuplem3 $p |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> + X A. b e. B Z . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + noetainflem.2 $e |- W = ( T u. + ( ( suc U. ( bday " A ) \ dom T ) X. { 2o } ) ) $. + $( Lemma for ~ noeta . Establish that this particular construction gives a + surreal. (Contributed by Scott Fenton, 9-Aug-2024.) $) + noetainflem1 $p |- ( ( A e. _V /\ B C_ No /\ B e. _V ) -> W e. No ) $= + ( cvv wcel csur wss w3a cbday cima cuni c2o csuc cdm cdif csn cxp noinfno + cun con0 3adant1 bdayimaon 3ad2ant1 c1o 2oex noextendseq syl2anc eqeltrid + prid2 ) ELMZFNOZFLMZPZIGQERSUAZGUBUCTUDUEUGZNKVAGNMZVBUHMZVCNMUSUTVDURABC + DFGHLJUFUIURUSVEUTELUJUKGVBTULTUMUQUNUOUP $. + + $( Lemma for ~ noeta . The restriction of ` W ` to the domain of ` T ` is + ` T ` . (Contributed by Scott Fenton, 9-Aug-2024.) $) + noetainflem2 $p |- ( ( B C_ No /\ B e. _V ) -> + ( W |` dom T ) = T ) $= + ( csur cvv wcel cdm cres cun eqtri c0 wceq wss wa cima cuni csuc cdif c2o + cbday csn cxp reseq1i resundir wfun wrel noinfno nofun syl resdm 3syl cin + funrel dmres wne 2oex snnz dmxp ax-mp ineq2i disjdif wb relres reldm0 a1i + mpbir uneq12d un0 eqtrdi eqtrid ) FLUAFMNUBZIGOZPZGVTPZUHEUCUDUEZVTUFZUGU + IZUJZVTPZQZGWAGWFQZVTPWHIWIVTKUKGWFVTULRVSWHGSQGVSWBGWGSVSGUMZGUNWBGTVSGL + NWJABCDFGHMJUOGUPUQGVAGURUSWGSTZVSWKWGOZSTZWLVTWFOZUTZSWFVTVBWOVTWDUTSWNW + DVTWESVCWNWDTUGVDVEWDWEVFVGVHVTWCVIRRWGUNWKWMVJWFVTVKWGVLVGVNVMVOGVPVQVR + $. + + $d Y v $. $d Y x $. $d Y y $. + $( Lemma for ~ noeta . ` W ` bounds ` B ` below . (Contributed by Scott + Fenton, 9-Aug-2024.) $) + noetainflem3 $p |- ( ( ( A e. _V /\ B C_ No /\ B e. _V ) /\ Y e. B ) -> + W A. a e. A a . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + noetalem1.2 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + noetalem1.3 $e |- Z = + ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) ) $. + noetalem1.4 $e |- W = ( T u. + ( ( suc U. ( bday " A ) \ dom T ) X. { 2o } ) ) $. + $( Lemma for ~ noeta . Either ` S ` or ` T ` satisfies the final + condition. (Contributed by Scott Fenton, 9-Aug-2024.) $) + noetalem1 $p |- ( ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) /\ + A. a e. A A. b e. B a + ( ( S e. No /\ ( A. a e. A a . } ) , + ( g e. { y | E. u e. A ( y e. dom u /\ + A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. A ( g e. dom u /\ + A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + noetalem2.2 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , + ( g e. { y | E. u e. B ( y e. dom u /\ + A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } + |-> ( iota x E. u e. B ( g e. dom u /\ + A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ + ( u ` g ) = x ) ) ) ) $. + $( Lemma for ~ noeta . The full statement of the theorem with hypotheses + in place. (Contributed by Scott Fenton, 10-Aug-2024.) $) + noetalem2 $p |- ( ( ( ( A C_ No /\ A e. V ) /\ ( B C_ No /\ B e. W ) /\ + A. a e. A A. b e. B a + E. c e. No ( A. a e. A a + E. z e. No ( A. x e. A x -. A + ( ( A A ( A -. B ( A + ( A = B <-> ( -. A + ( A =/= B <-> ( A ( A <_s B <-> -. B ( A -. B <_s A ) ) $= + ( csur wcel wa csle wbr cslt wn wb slenlt ancoms con2bid ) ACDZBCDZEBAFGZAB + HGZONPQIJBAKLM $. + + $( Surreal less-than or equal in terms of less-than. (Contributed by Scott + Fenton, 8-Dec-2021.) $) + sleloe $p |- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> + ( A + ( A = B <-> ( A <_s B /\ B <_s A ) ) ) $= + ( csur wcel wa wceq cslt wbr wn csle slttrieq2 slenlt ancoms anbi12d bitrdi + wb ancom bitr4d ) ACDZBCDZEZABFABGHIZBAGHIZEZABJHZBAJHZEZABKUAUGUCUBEUDUAUE + UCUFUBABLTSUFUBPBALMNUCUBQOR $. + + $( Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) $) + sltletr $p |- ( ( A e. No /\ B e. No /\ C e. No ) -> + ( ( A A + ( ( A <_s B /\ B A + ( ( A <_s B /\ B <_s C ) -> A <_s C ) ) $= + ( csur wcel w3a csle wbr cslt wn wa sltletr 3coml expcomd imp con3d expimpd + wi wb slenlt 3adant1 anbi2d 3adant2 3imtr4d ) ADEZBDEZCDEZFZABGHZCBIHZJZKCA + IHZJZUIBCGHZKACGHZUHUIUKUMUHUIKULUJUHUIULUJRUHULUIUJUGUEUFULUIKUJRCABLMNOPQ + UHUNUKUIUFUGUNUKSUEBCTUAUBUEUGUOUMSUFACTUCUD $. + + ${ + slttrd.1 $e |- ( ph -> A e. No ) $. + slttrd.2 $e |- ( ph -> B e. No ) $. + slttrd.3 $e |- ( ph -> C e. No ) $. + ${ + slttrd.4 $e |- ( ph -> A B A A B <_s C ) $. + $( Surreal less-than is transitive. (Contributed by Scott Fenton, + 8-Dec-2021.) $) + sltletrd $p |- ( ph -> A A <_s B ) $. + slelttrd.5 $e |- ( ph -> B A A <_s B ) $. + sletrd.5 $e |- ( ph -> B <_s C ) $. + $( Surreal less-than or equal is transitive. (Contributed by Scott + Fenton, 8-Dec-2021.) $) + sletrd $p |- ( ph -> A <_s C ) $= + ( csle wbr csur wcel wa wi sletr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABL + MCLMDLMSTNUAOEFGBCDPQR $. + $} + $} + + $( Surreal less-than or equal is reflexive. Theorem 0(iii) of [Conway] + p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) $) + slerflex $p |- ( A e. No -> A <_s A ) $= + ( csur wcel csle wbr cslt wn sltirr wb slenlt anidms mpbird ) ABCZAADEZAAFE + GZAHMNOIAAJKL $. + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Birthday Theorems +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $( The birthday function is a function. (Contributed by Scott Fenton, + 14-Jun-2011.) $) + bdayfun $p |- Fun bday $= + ( csur con0 cbday wfo wfun bdayfo fofun ax-mp ) ABCDCEFABCGH $. + + $( The birthday function is a function over ` No ` . (Contributed by Scott + Fenton, 30-Jun-2011.) $) + bdayfn $p |- bday Fn No $= + ( csur con0 cbday wfo wfn bdayfo fofn ax-mp ) ABCDCAEFABCGH $. + + $( The birthday function's domain is ` No ` . (Contributed by Scott Fenton, + 14-Jun-2011.) $) + bdaydm $p |- dom bday = No $= + ( csur con0 cbday wfo wf bdayfo fof ax-mp fdmi ) ABCABCDABCEFABCGHI $. + + $( The birthday function's range is ` On ` . (Contributed by Scott Fenton, + 14-Jun-2011.) $) + bdayrn $p |- ran bday = On $= + ( csur con0 cbday wfo crn wceq bdayfo forn ax-mp ) ABCDCEBFGABCHI $. + + $( The value of the birthday function is always an ordinal. (Contributed by + Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, + 8-Dec-2021.) $) + bdayelon $p |- ( bday ` A ) e. On $= + ( csur con0 cbday wfo wf bdayfo fof ax-mp 0elon f0cli ) BCADBCDEBCDFGBCDHIJ + K $. + + ${ + $d A w x y z $. $d X w x y z $. $d Y w y z $. + $( Lemma for ~ nocvxmin . Given two birthday-minimal elements of a convex + class of surreals, they are not comparable. (Contributed by Scott + Fenton, 30-Jun-2011.) $) + nocvxminlem $p |- ( ( A C_ No /\ A. x e. A A. y e. A + A. z e. No ( ( x z e. A ) ) -> + ( ( ( X e. A /\ Y e. A ) /\ + ( ( bday ` X ) = |^| ( bday " A ) /\ + ( bday ` Y ) = |^| ( bday " A ) ) ) -> + -. X z e. A ) ) -> + E! w e. A ( bday ` w ) = |^| ( bday " A ) ) $= + ( vt c0 wne csur wss cv cslt wbr wa wcel wi wral cbday cfv imp w3a bdayrn + cima cint wceq wrex weq wreu con0 crn imassrn sseqtri wex cdm bdaydm wfun + sseq2i bdayfun funfvima2 mpan sylbir elex2 syl6 exlimdv n0 3imtr4g impcom + onint sylancr wb wfn bdayfn fvelimab adantl mpbid 3adant3 wn ssel anim12d + ad2ant2r nocvxminlem anbi12i biimtrid slttrieq2 biimpar ralrimivv 3adant1 + ancom syl12anc exp32 fveqeq2 reu4 sylanbrc ) EGHZEIJZAKZCKZLMWQBKLMNWQEOP + CIQBEQAEQZUADKZRSREUCZUDZUEZDEUFZXBFKZRSXAUEZNZDFUGZPZFEQDEQZXBDEUHWNWOXC + WRWNWONZXAWTOZXCXJWTUIJWTGHZXKWTRUJUIREUKUBULWOWNXLWOWPEOZAUMWSWTODUMZWNX + LWOXMXNAWOXMWPRSZWTOZXNWOERUNZJZXMXPPZXQIEUOUQRUPXRXSUREWPRUSUTVADXOWTVBV + CVDAEVEDWTVEVFVGWTVHVIWOXKXCVJZWNRIVKWOXTVLDIEXARVMUTVNVOVPWOWRXIWNWOWRNZ + XHDFEEYAWSEOZXDEOZNZXFXGYAYDXFNZNWSIOZXDIOZNZWSXDLMVQZXDWSLMVQZXGWOYDYHWR + XFWOYDYHWOYBYFYCYGEIWSVREIXDVRVSTVTYAYEYIABCEWSXDWATYAYEYJYEYCYBNZXEXBNZN + YAYJYDYKXFYLYBYCWHXBXEWHWBABCEXDWSWAWCTYHXGYIYJNWSXDWDWEWIWJWFWGXBXEDFEWS + XDXARWKWLWM $. + $} + + $( The surreal numbers are a proper class. (Contributed by Scott Fenton, + 16-Jun-2011.) $) + noprc $p |- -. No e. _V $= + ( csur cvv wcel con0 onprc cbday wfo bdayfo focdmex mpi mto ) ABCZDBCZELADF + GMHADBFIJK $. + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Conway cuts +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $( Declare new constant symbols. $) + $c <. | ( a C_ No /\ b C_ No /\ + A. x e. a A. y e. b x + ( iota_ x e. { y e. No | ( a < + E. z e. No ( A. x e. A x ( ( A e. _V /\ B e. _V ) /\ + ( A C_ No /\ B C_ No /\ A. x e. A A. y e. B x A e. _V ) $= + ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simpll sylbi + cv ) ABEFAGHZBGHZIAJKBJKCRDRLFDBMCAMNZISCDABOSTUAPQ $. + + $( The second argument of surreal set less-than exists. (Contributed by + Scott Fenton, 8-Dec-2021.) $) + ssltex2 $p |- ( A < B e. _V ) $= + ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simplr sylbi + cv ) ABEFAGHZBGHZIAJKBJKCRDRLFDBMCAMNZITCDABOSTUAPQ $. + + $( The first argument of surreal set is a set of surreals. (Contributed by + Scott Fenton, 8-Dec-2021.) $) + ssltss1 $p |- ( A < A C_ No ) $= + ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simpr1 sylbi + cv ) ABEFAGHBGHIZAJKZBJKZCRDRLFDBMCAMZNITCDABOSTUAUBPQ $. + + $( The second argument of surreal set is a set of surreals. (Contributed + by Scott Fenton, 8-Dec-2021.) $) + ssltss2 $p |- ( A < B C_ No ) $= + ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simpr2 sylbi + cv ) ABEFAGHBGHIZAJKZBJKZCRDRLFDBMCAMZNIUACDABOSTUAUBPQ $. + + $( The separation property of surreal set less-than. (Contributed by Scott + Fenton, 8-Dec-2021.) $) + ssltsep $p |- ( A < A. x e. A A. y e. B x A e. V ) $. + ssltd.2 $e |- ( ph -> B e. W ) $. + ssltd.3 $e |- ( ph -> A C_ No ) $. + ssltd.4 $e |- ( ph -> B C_ No ) $. + ssltd.5 $e |- ( ( ph /\ x e. A /\ y e. B ) -> x A < X A < X e. A ) $. + ssltsepcd.3 $e |- ( ph -> Y e. B ) $. + $( Two elements of separated sets obey less-than. Deduction form of + ~ ssltsepc . (Contributed by Scott Fenton, 25-Sep-2024.) $) + ssltsepcd $p |- ( ph -> X C < A < (/) < A < + E! x e. { y e. No | ( A < ( A |s B ) = + ( iota_ x e. { y e. No | ( A < + ( ( A |s B ) e. No /\ A < ( A |s B ) e. No ) $= + ( csslt wbr cscut co csur wcel csn scutcut simp1d ) ABCDABEFZGHALIZCDMBCD + ABJK $. + + ${ + scutcld.1 $e |- ( ph -> A < ( A |s B ) e. No ) $= + ( csslt wbr cscut co csur wcel scutcl syl ) ABCEFBCGHIJDBCKL $. + $} + + $( The birthday of the surreal cut is equal to the minimum birthday in the + gap. (Contributed by Scott Fenton, 8-Dec-2021.) $) + scutbday $p |- ( A < + ( bday ` ( A |s B ) ) = + |^| ( bday " { x e. No | ( A < ( ( L |s R ) = X <-> + ( L < ( ( L |s R ) = X <-> + ( L < + ( bday ` X ) C_ ( bday ` y ) ) ) ) ) $= + ( vx vz csslt wbr csur wcel wa wceq csn cbday cv wss wi wral wal bitri co + cscut cfv crab cima cint eqscut eqss wb sneq breq2d breq1d anbi12d elrab3 + w3a adantl biimpar wfn bdayfn ssrab2 fnfvima mp3an12 intss1 3syl biantrud + ssint wrex fvelimab mp2an weq rexrab imbi1i r19.23v anbi1ci impexp ralbii + eqcom 3bitr2i albii df-ral ralcom4 3bitr4i sseq2 ceqsalv bitr3di pm5.32da + fvex imbi2d bitrid df-3an 3bitr4g bitrd ) CBGHZDIJZKZCBUBUADLCDMZGHZWPBGH + ZDNUCZNCEOZMZGHZXABGHZKZEIUDZUEZUFZLZUOZWQWRCAOZMZGHZXKBGHZKZWSXJNUCZPZQZ + AIRZUOZEBCDUGWOWQWRKZXHKXTXRKXIXSWOXTXHXRXHWSXGPZXGWSPZKZWOXTKZXRWSXGUHYD + YAYCXRYDYBYAYDDXEJZWSXFJZYBWOYEXTWNYEXTUIWMXDXTEDIWTDLZXBWQXCWRYGXAWPCGWT + DUJZUKYGXAWPBGYHULUMUNUPUQNIURZXEIPZYEYFUSXDEIUTZIXENDVAVBWSXFVCVDVEYAWSF + OZPZFXFRZXRFWSXFVFYNYLXOLZXNYMQZQZFSZAIRZXRYLXFJZYMQZFSYQAIRZFSYNYSUUAUUB + FUUAXNXOYLLZKZAIVGZYMQUUDYMQZAIRUUBYTUUEYMYTUUCAXEVGZUUEYIYJYTUUGUIUSYKAI + XEYLNVHVIXDXNUUCAEIEAVJZXBXLXCXMUUHXAXKCGWTXJUJZUKUUHXAXKBGUUIULUMVKTVLUU + DYMAIVMUUFYQAIUUFYOXNKZYMQYQUUDUUJYMUUCYOXNXOYLVQVNVLYOXNYMVOTVPVRVSYMFXF + VTYQAFIWAWBYRXQAIYPXQFXOXJNWGYOYMXPXNYLXOWSWCWHWDVPTTWEWIWFWQWRXHWJWQWRXR + WJWKWL $. + $} + + ${ + $d A x y z $. $d B x y z $. $d C x y z $. + $( Transitive law for surreal set less-than. (Contributed by Scott Fenton, + 9-Dec-2021.) $) + sslttr $p |- ( ( A < A < ( A u. B ) < A < ( ( A u. C ) |s ( B u. D ) ) = ( A |s B ) ) $= + ( vx vy vz csslt wbr cscut csn cv cbday wa csur wceq wcel syl syl2anc wss + w3a cun cfv crab cima cint crio simp1 scutcut simp2d simp2 ssltun1 simp3d + co simp3 ssltun2 c0 wne ovex snnz sslttr mp3an3 scutval wral wrex vex weq + elima sneq breq2d breq1d anbi12d rexrab bitri wi wb simplr bdayfn fnbrfvb + wfn mpan simpll1 scutbday simprl ssun1 sssslt1 mpan2 simprr sssslt2 elrab + sylanbrc ssrab2 fnfvima mp3an12 intss1 eqsstrd sseq2 biimpd com12 sylbird + ex impd rexlimdva biimtrid ralrimiv ssint sylibr simp1d eqssd wreu conway + jca fveqeq2 riota2 mpbid eqcomd eqtr4d ) ABHIZCABJUNZKZHIZXTDHIZUAZACUBZB + DUBZJUNZELZMUCMYDFLZKZHIZYIYEHIZNZFOUDZUEZUFZPZEYMUGZXSYCYDYEHIZYFYQPYCYD + XTHIZXTYEHIZYRYCAXTHIZYAYSYCXSOQZUUAXTBHIZYCXRUUBUUAUUCUAXRYAYBUHABUIRZUJ + XRYAYBUKACXTULSZYCUUCYBYTYCUUBUUAUUCUUDUMXRYAYBUOXTBDUPSZYSYTXTUQURYRXSAB + JUSUTYDXTYEVAVBSZEFYDYEVCRYCYQXSYCXSMUCZYOPZYQXSPZYCUUHYOYCUUHYGTZEYNVDUU + HYOTYCUUKEYNYGYNQZYDGLZKZHIZUUNYEHIZNZUUMYGMIZNZGOVEZYCUUKUULUURGYMVEUUTG + YGMYMEVFVHYLUUQUURGFOFGVGZYJUUOYKUUPUVAYIUUNYDHYHUUMVIZVJUVAYIUUNYEHUVBVK + VLVMVNYCUUSUUKGOYCUUMOQZNZUUQUURUUKUVDUUQUURUUKVOUVDUUQNZUURUUMMUCZYGPZUU + KUVEUVCUVGUURVPZYCUVCUUQVQZMOVTZUVCUVHVROUUMYGMVSWARUVEUUHUVFTZUVGUUKVOUV + EUUHMAYIHIZYIBHIZNZFOUDZUEZUFZUVFUVEXRUUHUVQPXRYAYBUVCUUQWBFABWCRUVEUVFUV + PQZUVQUVFTUVEUUMUVOQZUVRUVEUVCAUUNHIZUUNBHIZNZUVSUVIUVEUVTUWAUVEUUOUVTUVD + UUOUUPWDUUOAYDTUVTACWEYDUUNAWFWGRUVEUUPUWAUVDUUOUUPWHUUPBYETUWABDWEUUNYEB + WIWGRXLUVNUWBFUUMOUVAUVLUVTUVMUWAUVAYIUUNAHUVBVJUVAYIUUNBHUVBVKVLWJWKUVJU + VOOTUVSUVRVRUVNFOWLOUVOMUUMWMWNRUVFUVPWORWPUVGUVKUUKUVGUVKUUKUVFYGUUHWQWR + WSRWTXAXBXCXDXEEUUHYNXFXGYCUUHYNQZYOUUHTYCXSYMQZUWCYCUUBYSYTNZUWDYCUUBUUA + UUCUUDXHYCYSYTUUEUUFXLYLUWEFXSOYHXSPZYJYSYKYTUWFYIXTYDHYHXSVIZVJUWFYIXTYE + HUWGVKVLWJWKZUVJYMOTUWDUWCVRYLFOWLOYMMXSWMWNRUUHYNWORXIYCUWDYPEYMXJZUUIUU + JVPUWHYCYRUWIUUGEFYDYEXKRYPUUIEYMXSYGXSYOMXMXNSXOXPXQ $. + $} + + ${ + $d a b c x y $. + $( The domain of the surreal cut operation is all separated surreal sets. + (Contributed by Scott Fenton, 8-Dec-2021.) $) + dmscut $p |- dom |s = < No $= + ( va vb vx vy vz csslt csur cscut wf wfn wceq csn cima cbday wbr mpbir2an + cv wrex wcel vex crn wss wfun cdm cpw cfv wa crab cint crio mpofun dmscut + df-scut df-fn cab rnmpo wi elimasn df-br bitr4i co scutval eqeltrrd sylbi + cop scutcl eleq1a syl adantl rexlimivv abssi eqsstri df-f ) FGHIHFJZHUAZG + UBVNHUCHUDFKABGUEZFAQZLMZCQNUFNVQDQLZFOVSBQZFOUGDGUHZMUIKCWAUJZHCDABUMZUK + ULHFUNPVOEQZWBKZBVRRAVPRZEUOGABEVPVRWBHWCUPWFEGWEWDGSZABVPVRVTVRSZWEWGUQZ + VQVPSWHWBGSZWIWHVQVTFOZWJWHVQVTVEFSWKFVQVTATBTURVQVTFUSUTWKVQVTHVAWBGCDVQ + VTVBVQVTVFVCVDWBGWDVGVHVIVJVKVLFGHVMP $. + $} + + ${ + $d A x y z $. $d B x y z $. $d O x y z $. + $( A restatement of ~ noeta using set less-than. (Contributed by Scott + Fenton, 10-Aug-2024.) $) + etasslt $p |- ( ( A < + E. x e. No ( A < + E. x e. No ( A < ( bday ` ( A |s B ) ) C_ O ) $= + ( vx vy csslt wbr con0 wcel cbday cun cima wss w3a cv csn cfv csur wa syl + cscut etasslt crab cint wceq simpl1 scutbday wfn bdayfn ssrab2 weq breq2d + co sneq breq1d anbi12d simprl simprr1 simprr2 jca elrabd fnfvima mp3an12i + intss1 eqsstrd simprr3 sstrd rexlimddv ) ABFGZCHIZJABKLCMZNZADOZPZFGZVNBF + GZVMJQZCMZNZABUAUMJQZCMDRDABCUBVLVMRIZVSSZSZVTVQCWCVTJAEOZPZFGZWEBFGZSZER + UCZLZUDZVQWCVIVTWKUEVIVJVKWBUFEABUGTWCVQWJIZWKVQMJRUHWIRMWCVMWIIWLUIWHERU + JWCWHVOVPSEVMREDUKZWFVOWGVPWMWEVNAFWDVMUNZULWMWEVNBFWNUOUPVLWAVSUQWCVOVPV + OVPVRWAVLURVOVPVRWAVLUSUTVARWIJVMVBVCVQWJVDTVEVOVPVRWAVLVFVGVH $. + $} + + ${ + $d A x y $. $d B x y $. + $( An upper bound on the birthday of a surreal cut. (Contributed by Scott + Fenton, 10-Dec-2021.) $) + scutbdaybnd2 $p |- ( A < + ( bday ` ( A |s B ) ) C_ suc U. ( bday " ( A u. B ) ) ) $= + ( vx vy csslt wbr csn cbday cfv cun cima cuni csuc wss w3a csur wrex wcel + cv wa co etasslt2 crab cint wceq scutbday adantr wfn bdayfn ssrab2 simprl + cscut simprr1 simprr2 jca weq sneq breq2d breq1d anbi12d sylanbrc fnfvima + elrab mp3an12i intss1 syl eqsstrd simprr3 sstrd rexlimdvaa mpd ) ABEFZACS + ZGZEFZVNBEFZVMHIZHABJKLMZNZOZCPQABULUAHIZVRNZCABUBVLVTWBCPVLVMPRZVTTZTZWA + VQVRWEWAHADSZGZEFZWGBEFZTZDPUCZKZUDZVQVLWAWMUEWDDABUFUGWEVQWLRZWMVQNHPUHW + KPNWEVMWKRZWNUIWJDPUJWEWCVOVPTZWOVLWCVTUKWEVOVPVOVPVSWCVLUMVOVPVSWCVLUNUO + WJWPDVMPDCUPZWHVOWIVPWQWGVNAEWFVMUQZURWQWGVNBEWRUSUTVCVAPWKHVMVBVDVQWLVEV + FVGVOVPVSWCVLVHVIVJVK $. + $} + + $( An upper bound on the birthday of a surreal cut when it is a limit + birthday. (Contributed by Scott Fenton, 7-Aug-2024.) $) + scutbdaybnd2lim $p |- ( ( A < + ( bday ` ( A |s B ) ) C_ U. ( bday " ( A u. B ) ) ) $= + ( csslt wbr cscut co cbday cfv wlim wa cun cima cuni wss wcel adantr cvv wb + word con0 csuc scutbdaybnd2 wceq wfun bdayfun ssltex1 ssltex2 unexg syl2anc + wne wn funimaexg sylancr uniexd nlimsucg syl wi limeq biimpcd adantl neqned + mtod bdayelon onordi imassrn bdayrn sseqtri ssorduni ax-mp ordsuc ordelssne + crn mpbi mp2an sylanbrc a1i ordsssuc sylancl mpbird ) ABCDZABEFZGHZIZJZWBGA + BKZLZMZNZWBWGUAZOZWDWBWINZWBWIUJZWJVTWKWCABUBPWDWBWIWDWBWIUCZWIIZWDWGQOZWNU + KVTWOWCVTWFQVTGUDWEQOZWFQOUEVTAQOBQOWPABUFABUGABQQUHUIGWEQULUMUNPWGQUOUPWCW + MWNUQVTWMWCWNWBWIURUSUTVBVAWBSWISZWJWKWLJRWBWAVCZVDWGSZWQWFTNWSWFGVLTGWEVEV + FVGWFVHVIZWGVJVMWBWIVKVNVOWDWBTOZWSWHWJRXAWDWRVPWTWBWGVQVRVS $. + + ${ + $d A x y $. $d B x y $. $d X x y $. + $( If a surreal lies in a gap and is not equal to the cut, its birthday is + greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.) $) + scutbdaylt $p |- ( ( X e. No /\ ( A < ( bday ` ( A |s B ) ) e. ( bday ` X ) ) $= + ( vy vx csur wcel csn csslt wbr wa wne cbday cfv cv wceq syl3anc sylanbrc + wss syl cscut co w3a crab cima cint simp2l simp2r snnzg 3ad2ant1 scutbday + c0 sslttr wfn bdayfn ssrab2 simp1 simp2 sneq breq2d anbi12d elrab fnfvima + breq1d mp3an12i intss1 crio simprl simprr adantr eqeq1d eqcom bitrdi wreu + eqsstrd biimpa wb biimpri conway fveqeq2 riota2 syl2an2r mpbid scutval ex + eqtr4d necon3d 3impia con0 bdayelon onelpss mp2an ) CFGZACHZIJZWNBIJZKZCA + BUAUBZLZUCZWRMNZCMNZSZXAXBLZXAXBGZWTXAMADOZHZIJZXGBIJZKZDFUDZUEZUFZXBWTAB + IJZXAXMPZWTWOWPWNULLZXNWMWOWPWSUGWMWOWPWSUHWMWQXPWSCFUIZUJAWNBUMZQDABUKZT + WTXBXLGZXMXBSMFUNXKFSWTCXKGZXTUOXJDFUPWTWMWQYAWMWQWSUQWMWQWSURXJWQDCFXFCP + ZXHWOXIWPYBXGWNAIXFCUSZUTYBXGWNBIYCVDVAVBZRFXKMCVCVEXBXLVFTVOWMWQWSXDWMWQ + KZXAXBCWRYEXAXBPZCWRPYEYFKZCEOZMNXMPZEXKVGZWRYGXBXMPZCYJPZYEYFYKYEYFXMXBP + YKYEXAXMXBYEXNXOYEWOWPXPXNWMWOWPVHWMWOWPVIWMXPWQXQVJXRQZXSTVKXMXBVLVMVPYE + YAYFYIEXKVNZYKYLVQYAYEYDVRYGXNYNYEXNYFYMVJZEDABVSTYAYNKYKYJCPYLYIYKEXKCYH + CXMMVTWAYJCVLVMWBWCYGXNWRYJPYOEDABWDTWFWEWGWHXAWIGXBWIGXEXCXDKVQWRWJCWJXA + XBWKWLR $. + $} + + ${ + $d A a b c d $. $d B a b c d $. $d C a b c d $. $d D a b c d $. + $d X a d $. $d Y a d $. + $( A comparison law for surreals considered as cuts of sets of surreals. + Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem + 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 11-Dec-2021.) $) + slerec $p |- ( ( ( A < + ( X <_s Y <-> ( A. d e. D X + ( X ( E. c e. C X <_s c \/ E. b e. B b <_s Y ) ) ) $= + ( csslt wbr wa cscut cslt csle wral wn wb csur wcel syl2anc co wceq cv wo + wrex simplr simpll simprr simprl slerec syl22anc ancom bitrdi csn scutcut + simp1d ad2antlr eqeltrd ad2antrr slenlt wss sselda adantr sltnle ralbidva + ssltss1 ssltss2 anbi12d ralnex anbi12i ioran bitr4i 3bitr3d con4bid ) ABI + JZCDIJZKZEABLUAZUBZFCDLUAZUBZKZKZEFMJZEHUCZNJZHCUEZGUCZFNJZGBUEZUDZWCFENJ + ZWEEMJZHCOZFWHMJZGBOZKZWDPZWKPZWCWLWPWNKZWQWCVPVOWAVSWLWTQVOVPWBUFVOVPWBU + GVQVSWAUHZVQVSWAUIZCDABFEHGUJUKWPWNULUMWCFRSZERSZWLWRQWCFVTRXAVPVTRSZVOWB + VPXECVTUNZIJXFDIJCDUOUPUQURZWCEVRRXBVOVRRSZVPWBVOXHAVRUNZIJXIBIJABUOUPUSU + RZFEUTTWCWQWFPZHCOZWIPZGBOZKZWSWCWNXLWPXNWCWMXKHCWCWECSZKWERSXDWMXKQWCCRW + EVPCRVAVOWBCDVFUQVBWCXDXPXJVCWEEVDTVEWCWOXMGBWCWHBSZKXCWHRSWOXMQWCXCXQXGV + CWCBRWHVOBRVAVPWBABVGUSVBFWHVDTVEVHXOWGPZWJPZKWSXLXRXNXSWFHCVIWIGBVIVJWGW + JVKVLUMVMVN $. + $} + + ${ + $d A x $. $d B x $. + $( If ` A ` preceeds ` B ` , then ` A ` and ` B ` are disjoint. + (Contributed by Scott Fenton, 18-Sep-2024.) $) + ssltdisj $p |- ( A < ( A i^i B ) = (/) ) $= + ( vx csslt wbr cv wcel wn wral c0 wceq wa cslt csur ssltss1 sselda sltirr + cin syl ssltsepc 3expa mtand ralrimiva disj sylibr ) ABDEZCFZBGZHZCAIABRJ + KUFUICAUFUGAGZLZUHUGUGMEZUKUGNGULHUFANUGABOPUGQSUFUJUHULABUGUGTUAUBUCCABU + DUE $. + $} + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Zero and One +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $( Define new constants. $) + $c 0s 1s $. + + $( Declare the class syntax for surreal zero. $) + c0s $a class 0s $. + + $( Declare the class syntax for surreal one. $) + c1s $a class 1s $. + + $( Define surreal zero. This is the simplest cut of surreal number sets. + Definition from [Conway] p. 17. (Contributed by Scott Fenton, + 7-Aug-2024.) $) + df-0s $a |- 0s = ( (/) |s (/) ) $. + + $( Define surreal one. This is the simplest number greater than surreal + zero. Definition from [Conway] p. 18. (Contributed by Scott Fenton, + 7-Aug-2024.) $) + df-1s $a |- 1s = ( { 0s } |s (/) ) $. + + $( Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) $) + 0sno $p |- 0s e. No $= + ( c0s c0 cscut co csur df-0s csslt wbr wcel cpw 0elpw nulssgt ax-mp eqeltri + scutcl ) ABBCDZEFBBGHZPEIBEJIQEKBLMBBOMN $. + + $( Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) $) + 1sno $p |- 1s e. No $= + ( c1s c0s csn c0 cscut csur df-1s csslt wbr wcel 0sno snelpwi ax-mp nulssgt + co cpw scutcl eqeltri ) ABCZDEOZFGSDHIZTFJSFPJZUABFJUBKBFLMSNMSDQMR $. + + ${ + $d x y $. + $( Calculate the birthday of surreal zero. (Contributed by Scott Fenton, + 7-Aug-2024.) $) + bday0s $p |- ( bday ` 0s ) = (/) $= + ( vx c0s cbday cfv c0 cv csn csslt wa csur crab cima cint con0 cscut wcel + wbr co nulssgt 3eqtri df-0s fveq2i cpw wceq 0elpw scutbday mp2b eqtri cdm + crn snelpwi nulsslt jca rabeqc bdaydm eqtr4i imaeq2i imadmrn bdayrn inton + syl inteqi ) BCDZCEAFZGZHQZVEEHQZIZAJKZLZMZNMEVCEEORZCDZVKBVLCUAUBEJUCZPE + EHQVMVKUDJUEESAEEUFUGUHVJNVJCCUIZLCUJNVIVOCVIJVOVHAJVDJPVEVNPZVHVDJUKVPVF + VGVEULVESUMVAUNUOUPUQCURUSTVBUTT $. + $} + + ${ + $d x y $. + $( Surreal zero is less than surreal one. Theorem from [Conway] p. 7. + (Contributed by Scott Fenton, 7-Aug-2024.) $) + 0slt1s $p |- 0s ( ( bday ` X ) = (/) <-> X = 0s ) ) $= + ( vx csur wcel cbday cfv c0 c0s wa cscut co df-0s csn csslt wbr cv adantr + wceq syl nulssgt wss wi cpw snelpwi nulsslt id 0ss eqsstrdi a1d ralrimivw + wral adantl wb 0elpw ax-mp eqscut2 mpan mpbir3and eqtr2id ex fveq2 bday0s + w3a eqtrdi impbid1 ) ACDZAEFZGRZAHRZVFVHVIVFVHIZHGGJKZALVJVKARZGAMZNOZVMG + NOZGBPZMZNOVQGNOIZVGVPEFZUAZUBZBCUKZVFVNVHVFVMCUCZDZVNACUDZVMUESQVFVOVHVF + WDVOWEVMTSQVJWABCVHWAVFVHVTVRVHVGGVSVHUFVSUGUHUIULUJVFVLVNVOWBVCUMZVHGGNO + ZVFWFGWCDWGCUNGTUOBGGAUPUQQURUSUTVIVGHEFGAHEVAVBVDVE $. + $} + + ${ + $d x y z $. + $( The birthday of surreal one is ordinal one. (Contributed by Scott + Fenton, 8-Aug-2024.) $) + bday1s $p |- ( bday ` 1s ) = 1o $= + ( vx vy vz cbday cfv c0s csn c1o cima csslt wbr csur wcel 0sno ax-mp wceq + c0 wss wral cslt c1s cscut df-1s fveq2i cun cuni csuc cpw snelpwi nulssgt + co scutbdaybnd2 un0 imaeq2i wfn bdayfn fnsnfv mp2an bday0s 3eqtr2i unieqi + sneqi 0ex unisn eqtri suceq df-1o eqtr4i sseqtri cv wa crab wb fnssintima + cint ssrab2 weq sneq breq2d breq1d anbi12d elrab wn sltirr breq2 necon2ai + wne mtbiri bday0b necon3bid syl5ibr word bdayelon onordi ordge1n0 syl6ibr + ssltsep ralsn ralbii elexi breq1 bitri sylib impel adantrr sylbi scutbday + vex mprgbir sseqtrri eqssi ) UADEFGZQUBUKZDEZHUAXMDUCUDXNHXNDXLQUEZIZUFZU + GZHXLQJKZXNXRRXLLUHMZXSFLMZXTNFLUIOXLUJOZXLQULOXRQUGZHXQQPXRYCPXQQGZUFQXP + YDXPDXLIZFDEZGZYDXOXLDXLUMUNDLUOZYAYGYEPUPNLFDUQURYFQUSVBUTVAQVCVDVEXQQVF + OVGVHVIHDXLAVJZGZJKZYJQJKZVKZALVLZIVOZXNHYORZHBVJZDEZRZBYNYHYNLRYPYSBYNSV + MUPYMALVPBLYNHDVNURYQYNMYQLMZXLYQGZJKZUUAQJKZVKZVKYSYMUUDAYQLABVQZYKUUBYL + UUCUUEYJUUAXLJYIYQVRZVSUUEYJUUAQJUUFVTWAWBYTUUBYSUUCYTFYQTKZYSUUBYTUUGYRQ + WGZYSUUGUUHYTYQFWGUUGYQFYQFPUUGFFTKZYAUUIWCNFWDOYQFFTWEWHWFYTYRQYQFYQWIWJ + WKYRWLYSUUHVMYRYQWMWNYRWOOWPUUBYICVJZTKZCUUASZAXLSZUUGACXLUUAWQUUMYIYQTKZ + AXLSUUGUULUUNAXLUUKUUNCYQBXHUUJYQYITWEWRWSUUNUUGAFFLNWTYIFYQTXAWRXBXCXDXE + XFXIXSXNYOPYBAXLQXGOXJXKVE $. + $} + + +$( End $[ set-surreals.mm $] $) + +$( Begin $[ set-tarskigeom.mm $] $) +$( +############################################################################### + ELEMENTARY GEOMETRY +############################################################################### + + This part develops elementary geometry based on Tarski's axioms, following + [Schwabhauser]. Tarski's geometry is a first-order theory with one sort, the + "points". It has two primitive notions, the ternary predicate of + "betweenness" and the quaternary predicate of "congruence". To adapt this + theory to the framework of set.mm, and to be able to talk of *a* Tarski + structure as a space satisfying the given axioms, we use the following + definition, stated informally: + + A Tarski structure ` f ` is a set (of points) ` ( Base `` f ) ` together with + functions ` ( Itv `` f ) ` and ` ( dist `` f ) ` on + ` ( ( Base `` f ) X. ( Base `` f ) ) ` satisfying certain axioms (given in + Definitions ~ df-trkg _et sequentes_). This allows to treat a Tarski + structure as a special kind of extensible structure (see ~ df-struct ). + + The translation to and from Tarski's treatment is as follows (given, again, + informally). + + Suppose that one is given an extensible structure ` f ` . One defines a + betweenness ternary predicate Btw by positing that, for any + ` x , y , z e. ( Base `` f ) ` , one has "Btw ` x y z ` " if and only if + ` y e. x ( Itv `` f ) z ` , and a congruence quaternary predicate Congr by + positing that, for any ` x , y , z , t e. ( Base `` f ) ` , one has + "Congr ` x y z t ` " if and only if + ` x ( dist `` f ) y = z ( dist `` f ) t ` . + It is easy to check that if ` f ` satisfies our Tarski axioms, then Btw and + Congr satisfy Tarski's Tarski axioms when ` ( Base `` f ) ` is interpreted as + the universe of discourse. + + Conversely, suppose that one is given a set ` a ` , a ternary predicate Btw, + and a quaternary predicate Congr. One defines the extensible structure ` f ` + such that ` ( Base `` f ) ` is ` a ` , and ` ( Itv `` f ) ` is the function + which associates with each ` <. x , y >. e. ( a X. a ) ` the set of points + ` z e. a ` such that "Btw ` x z y ` ", and ` ( dist `` f ) ` is the function + which associates with each ` <. x , y >. e. ( a X. a ) ` the set of ordered + pairs ` <. z , t >. e. ( a X. a ) ` such that "Congr ` x y z t ` ". It is + easy to check that if Btw and Congr satisfy Tarski's Tarski axioms when ` a ` + is interpreted as the universe of discourse, then ` f ` satisfies our Tarski + axioms. + + We intentionally choose to represent congruence (without loss of generality) + as ` x ( dist `` f ) y = z ( dist `` f ) t ` instead of "Congr ` x y z t ` ", + as it is more convenient. It is always possible to define ` dist ` for any + particular geometry to produce equal results when conguence is desired, and + in many cases there is an obvious interpretation of "distance" between + two points that can be useful in other situations. Encoding congruence as + an equality of distances makes it easier to use these theorems in cases where + there is a preferred distance function. We prove that representing a + congruence relationship using a distance in the form + ` x ( dist `` f ) y = z ( dist `` f ) t ` causes no loss of generality in + ~ tgjustc1 and ~ tgjustc2 , which in turn are supported by + ~ tgjustf and ~ tgjustr . + + A similar representation of congruence (using a "distance" function) + is used in Axiom A1 of [Beeson2016] p. 5, which discusses how a large number + of formalized proofs were found in Tarskian Geometry using OTTER. Their + detailed proofs in Tarski Geometry, along with other information, are + available at ~ https://www.michaelbeeson.com/research/FormalTarski/ . + + Most theorems are in deduction form, as this is a very general, simple, and + convenient format to use in Metamath. An assertion in deduction form can be + easily converted into an assertion in inference form (removing the + antecedents ` ph -> ` ) by insert a ` T. -> ` in each hypothesis, using + ~ a1i , then using ~ mptru to remove the final ` T. -> ` prefix. In some + cases we represent, without loss of generality, an implication antecedent in + [Schwabhauser] as a hypothesis. The implication can be retrieved from the + by using ~ simpr , the theorem as stated, and ~ ex . + + For descriptions of individual axioms, we refer to the specific definitions + below. A particular feature of Tarski's axioms is modularity, so by using + various subsets of the set of axioms, we can define the classes of "absolute + dimensionless Tarski structures" ( ~ df-trkg ), of "Euclidean dimensionless + Tarski structures" ( ~ df-trkge ) and of "Tarski structures of dimension no + less than N" ( ~ df-trkgld ). + + In this system, angles are not a primitive notion, but instead a derived + notion (see ~ df-cgra and ~ iscgra ). To maintain its simplicity, in this + system congruence between shapes (a finite sequence of points) is the case + where corresponding segments between all corresponding points are congruent. + This includes triangles (a shape of 3 distinct points). Note that this + definition has no direct regard for angles. For more details and rationale, + see ~ df-cgrg . + + The first section is devoted to the definitions of these various structures. + The second section ("Tarskian geometry") develops the synthetic treatment of + geometry. The remaining sections prove that real Euclidean spaces and + complex Hilbert spaces, with intended interpretations, are Euclidean Tarski + structures. + + Most of the work in this part is due to Thierry Arnoux, with earlier work by + Mario Carneiro and Scott Fenton. See also the credits in the comment of each + statement. + +$) + + +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Definition and Tarski's Axioms of Geometry +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + + $c TarskiG Itv LineG $. + $c TarskiGC TarskiGB TarskiGCB TarskiGE TarskiGDim>= $. + + $( Extends class notation with the class of Tarski geometries. $) + cstrkg $a class TarskiG $. + $( Extends class notation with the class of geometries fulfilling the + congruence axioms. $) + cstrkgc $a class TarskiGC $. + $( Extends class notation with the class of geometries fulfilling the + betweenness axioms. $) + cstrkgb $a class TarskiGB $. + $( Extends class notation with the class of geometries fulfilling the + congruence and betweenness axioms. $) + cstrkgcb $a class TarskiGCB $. + $( Extends class notation with the relation for geometries fulfilling the + lower dimension axioms. $) + cstrkgld $a class TarskiGDim>= $. + $( Extends class notation with the class of geometries fulfilling Euclid's + axiom. $) + cstrkge $a class TarskiGE $. + + $( Declare the syntax for the Interval (segment) index extractor. $) + citv $a class Itv $. + + $( Declare the syntax for the Line function. $) + clng $a class LineG $. + + $( Define the Interval (segment) index extractor for Tarski geometries. + (Contributed by Thierry Arnoux, 24-Aug-2017.) Use its index-independent + form ~ itvid instead. (New usage is discouraged.) $) + df-itv $a |- Itv = Slot ; 1 6 $. + + $( Define the line index extractor for geometries. (Contributed by Thierry + Arnoux, 27-Mar-2019.) Use its index-independent form ~ lngid instead. + (New usage is discouraged.) $) + df-lng $a |- LineG = Slot ; 1 7 $. + + $( Index value of the Interval (segment) slot. Use ~ ndxarg . (Contributed + by Thierry Arnoux, 24-Aug-2017.) (New usage is discouraged.) $) + itvndx $p |- ( Itv ` ndx ) = ; 1 6 $= + ( citv c1 c6 cdc df-itv 1nn0 6nn decnncl ndxarg ) ABCDEBCFGHI $. + + $( Index value of the "line" slot. Use ~ ndxarg . (Contributed by Thierry + Arnoux, 27-Mar-2019.) (New usage is discouraged.) $) + lngndx $p |- ( LineG ` ndx ) = ; 1 7 $= + ( clng c1 c7 cdc df-lng 1nn0 7nn decnncl ndxarg ) ABCDEBCFGHI $. + + $( Utility theorem: index-independent form of ~ df-itv . (Contributed by + Thierry Arnoux, 24-Aug-2017.) $) + itvid $p |- Itv = Slot ( Itv ` ndx ) $= + ( citv c1 c6 cdc df-itv 1nn0 6nn decnncl ndxid ) ABCDEBCFGHI $. + + $( Utility theorem: index-independent form of ~ df-lng . (Contributed by + Thierry Arnoux, 27-Mar-2019.) $) + lngid $p |- LineG = Slot ( LineG ` ndx ) $= + ( clng c1 c7 cdc df-lng 1nn0 7nn decnncl ndxid ) ABCDEBCFGHI $. + + $( The slots ` Base ` , ` +g ` , ` .s ` and ` dist ` are different from the + slot ` Itv ` . Formerly part of ~ ttglem and proofs using it. + (Contributed by AV, 29-Oct-2024.) $) + slotsinbpsd $p |- ( ( ( Itv ` ndx ) =/= ( Base ` ndx ) + /\ ( Itv ` ndx ) =/= ( +g ` ndx ) ) + /\ ( ( Itv ` ndx ) =/= ( .s ` ndx ) + /\ ( Itv ` ndx ) =/= ( dist ` ndx ) ) ) $= + ( cnx citv cfv cbs wne wa c1 c6 cdc itvndx 6nn0 1nn0 declti gtneii neeqtrri + 1nn eqnetri c2 2nn0 pm3.2i cplusg cvsca cds 1re 1lt10 basendx 2re 2lt10 6re + plusgndx 6lt10 vscandx 2nn decnncl nnrei 6nn 2lt6 declt dsndx ) ABCZADCZEZU + TAUACZEZFUTAUBCZEZUTAUCCZEZFVBVDUTGHIZVAJVIGVAGVIUDGHGPKLUEMNUFOQUTVIVCJVIR + VCRVIUGGHRPKSUHMNUJOQTVFVHUTVIVEJVIHVEHVIUIGHHPKKUKMNULOQUTVIVGJVIGRIZVGVJV + IVJGRLUMUNUOGRHLSUPUQURNUSOQTT $. + + $( The slots ` Base ` , ` +g ` , ` .s ` and ` dist ` are different from the + slot ` LineG ` . Formerly part of ~ ttglem and proofs using it. + (Contributed by AV, 29-Oct-2024.) $) + slotslnbpsd $p |- ( ( ( LineG ` ndx ) =/= ( Base ` ndx ) + /\ ( LineG ` ndx ) =/= ( +g ` ndx ) ) + /\ ( ( LineG ` ndx ) =/= ( .s ` ndx ) + /\ ( LineG ` ndx ) =/= ( dist ` ndx ) ) ) $= + ( cnx clng cfv wne wa c1 c7 cdc lngndx 1nn 7nn0 1nn0 declti gtneii neeqtrri + eqnetri c2 2nn0 pm3.2i c6 cbs cplusg cvsca cds 1lt10 basendx 2lt10 plusgndx + 1re 2re 6re 6nn0 6lt10 vscandx 2nn decnncl nnrei 7nn 2lt7 declt dsndx ) ABC + ZAUACZDZVBAUBCZDZEVBAUCCZDZVBAUDCZDZEVDVFVBFGHZVCIVKFVCFVKUIFGFJKLUEMNUFOPV + BVKVEIVKQVEQVKUJFGQJKRUGMNUHOPSVHVJVBVKVGIVKTVGTVKUKFGTJKULUMMNUNOPVBVKVIIV + KFQHZVIVLVKVLFQLUOUPUQFQGLRURUSUTNVAOPSS $. + + $( The slot for the line is not the slot for the Interval (segment) in an + extensible structure. Formerly part of proof for ~ ttgval . (Contributed + by AV, 9-Nov-2024.) $) + lngndxnitvndx $p |- ( LineG ` ndx ) =/= ( Itv ` ndx ) $= + ( cnx clng cfv citv wne c1 c7 cdc c6 1nn0 6nn decnncl nnrei 6nn0 6lt7 declt + 7nn gtneii lngndx itvndx neeq12i mpbir ) ABCZADCZEFGHZFIHZEUFUEUFFIJKLMFIGJ + NQOPRUCUEUDUFSTUAUB $. + + ${ + trkgstr.w $e |- W = { <. ( Base ` ndx ) , U >. , + <. ( dist ` ndx ) , D >. , <. ( Itv ` ndx ) , I >. } $. + $( Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, + 24-Aug-2017.) $) + trkgstr $p |- W Struct <. 1 , ; 1 6 >. $= + ( cnx cbs cfv cop cds citv c1 c6 cdc c2 1nn 2nn0 1nn0 decnncl 6nn basendx + ctp cstr 1lt10 declti 2nn dsndx 2lt6 declt itvndx strle3 eqbrtri ) DFGHZB + IFJHZAIFKHZCIUBLLMNZIUCEUMUNUOLLONUPBACPUALOLPQRUDUELORUFSUGLOMRQTUHUILMR + TSUJUKUL $. + + $( The base set of a Tarski geometry. (Contributed by Thierry Arnoux, + 24-Aug-2017.) $) + trkgbas $p |- ( U e. V -> U = ( Base ` W ) ) $= + ( cbs c1 c6 cdc cop trkgstr baseid cnx cfv csn cds citv ctp snsstp1 strfv + sseqtrri ) BEGDHHIJKABCEFLMNGOBKZPUCNQOAKZNROCKZSEUCUDUETFUBUA $. + + $( The measure of a distance in a Tarski geometry. (Contributed by Thierry + Arnoux, 24-Aug-2017.) $) + trkgdist $p |- ( D e. V -> D = ( dist ` W ) ) $= + ( cds c1 c6 cdc cop trkgstr dsid cnx cfv csn cbs citv ctp snsstp2 strfv + sseqtrri ) AEGDHHIJKABCEFLMNGOAKZPNQOBKZUCNROCKZSEUDUCUETFUBUA $. + + $( The congruence relation in a Tarski geometry. (Contributed by Thierry + Arnoux, 24-Aug-2017.) $) + trkgitv $p |- ( I e. V -> I = ( Itv ` W ) ) $= + ( citv c1 c6 cdc cop trkgstr itvid cnx cfv csn cbs cds ctp snsstp3 strfv + sseqtrri ) CEGDHHIJKABCEFLMNGOCKZPNQOBKZNROAKZUCSEUDUEUCTFUBUA $. + $} + + ${ + $d a b c d f g n p i j s t u v x y z $. + $( Define the class of geometries fulfilling the congruence axioms of + reflexivity, identity and transitivity. These are axioms A1 to A3 of + [Schwabhauser] p. 10. With our distance based notation for congruence, + transitivity of congruence boils down to transitivity of equality and is + already given by ~ eqtr , so it is not listed in this definition. + (Contributed by Thierry Arnoux, 24-Aug-2017.) $) + df-trkgc $a |- TarskiGC = { f | [. ( Base ` f ) / p ]. + [. ( dist ` f ) / d ]. + ( A. x e. p A. y e. p ( x d y ) = ( y d x ) + /\ A. x e. p A. y e. p A. z e. p ( ( x d y ) = ( z d z ) -> x = y ) ) } $. + + $( Define the class of geometries fulfilling the 3 betweenness axioms in + Tarski's Axiomatization of Geometry: identity, Axiom A6 of + [Schwabhauser] p. 11, axiom of Pasch, Axiom A7 of [Schwabhauser] p. 12, + and continuity, Axiom A11 of [Schwabhauser] p. 13. (Contributed by + Thierry Arnoux, 24-Aug-2017.) $) + df-trkgb $a |- TarskiGB = { f | [. ( Base ` f ) / p ]. + [. ( Itv ` f ) / i ]. + ( A. x e. p A. y e. p ( y e. ( x i x ) -> x = y ) + /\ A. x e. p A. y e. p A. z e. p A. u e. p A. v e. p + ( ( u e. ( x i z ) /\ v e. ( y i z ) ) + -> E. a e. p ( a e. ( u i y ) /\ a e. ( v i x ) ) ) + /\ A. s e. ~P p A. t e. ~P p + ( E. a e. p A. x e. s A. y e. t x e. ( a i y ) + -> E. b e. p A. x e. s A. y e. t b e. ( x i y ) ) ) + } $. + + $( Define the class of geometries fulfilling the five segment axiom, Axiom + A5 of [Schwabhauser] p. 11, and segment construction axiom, Axiom A4 of + [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 14-Mar-2019.) $) + df-trkgcb $a |- TarskiGCB = { f | [. ( Base ` f ) / p ]. + [. ( dist ` f ) / d ]. [. ( Itv ` f ) / i ]. + ( A. x e. p A. y e. p A. z e. p A. u e. p + A. a e. p A. b e. p A. c e. p A. v e. p + ( ( ( x =/= y /\ y e. ( x i z ) /\ b e. ( a i c ) ) + /\ ( ( ( x d y ) = ( a d b ) /\ ( y d z ) = ( b d c ) ) + /\ ( ( x d u ) = ( a d v ) /\ ( y d u ) = ( b d v ) ) ) ) + -> ( z d u ) = ( c d v ) ) + /\ A. x e. p A. y e. p A. a e. p A. b e. p E. z e. p + ( y e. ( x i z ) /\ ( y d z ) = ( a d b ) ) ) + } $. + + $( Define the class of geometries fulfilling Euclid's axiom, Axiom A10 of + [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 14-Mar-2019.) $) + df-trkge $a |- TarskiGE = { f | [. ( Base ` f ) / p ]. + [. ( Itv ` f ) / i ]. + A. x e. p A. y e. p A. z e. p A. u e. p A. v e. p + ( ( u e. ( x i v ) /\ u e. ( y i z ) /\ x =/= u ) -> E. a e. p + E. b e. p ( y e. ( x i a ) /\ z e. ( x i b ) /\ v e. ( a i b ) ) ) + } $. + + $( Define the class of geometries fulfilling the lower dimension axiom for + dimension ` n ` . For such geometries, there are three non-colinear + points that are equidistant from ` n - 1 ` distinct points. Derived + from remarks in _Tarski's System of Geometry_, Alfred Tarski and Steven + Givant, Bulletin of Symbolic Logic, Volume 5, Number 2 (1999), 175-214. + (Contributed by Scott Fenton, 22-Apr-2013.) (Revised by Thierry Arnoux, + 23-Nov-2019.) $) + df-trkgld $a |- TarskiGDim>= = { <. g , n >. | + [. ( Base ` g ) / p ]. [. ( dist ` g ) / d ]. [. ( Itv ` g ) / i ]. + E. f ( f : ( 1 ..^ n ) -1-1-> p /\ E. x e. p E. y e. p E. z e. p + ( A. j e. ( 2 ..^ n ) ( ( ( f ` 1 ) d x ) = ( ( f ` j ) d x ) /\ + ( ( f ` 1 ) d y ) = ( ( f ` j ) d y ) /\ + ( ( f ` 1 ) d z ) = ( ( f ` j ) d z ) ) + /\ -. ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) ) ) + } $. + + $( Define the class of Tarski geometries. A Tarski geometry is a set of + points, equipped with a betweenness relation (denoting that a point lies + on a line segment between two other points) and a congruence relation + (denoting equality of line segment lengths). + + Here, we are using the following:
      +
    • for congruence, ` ( x .- y ) = ( z .- w ) ` where + ` .- = ( dist `` W ) ` +
    • for betweenness, ` y e. ( x I z ) ` , where ` I = ( Itv `` W ) ` +
    + With this definition, the axiom A2 is actually equivalent to the + transitivity of equality, ~ eqtrd .

    + + Tarski originally had more axioms, but later reduced his list to 11:

      +
    • A1 A kind of reflexivity for the congruence relation + ( ` TarskiGC ` ) +
    • A2 Transitivity for the congruence relation ( ` TarskiGC ` ) +
    • A3 Identity for the congruence relation ( ` TarskiGC ` ) +
    • A4 Axiom of segment construction ( ` TarskiGCB ` ) +
    • A5 5-segment axiom ( ` TarskiGCB ` ) +
    • A6 Identity for the betweenness relation ( ` TarskiGB ` ) +
    • A7 Axiom of Pasch ( ` TarskiGB ` ) +
    • A8 Lower dimension axiom ` ( ``' TarskiGDim>= " { 2 } ) ` +
    • A9 Upper dimension axiom ` ( _V \ ( ``' TarskiGDim>= " { 3 } ) ) ` +
    • A10 Euclid's axiom ( ` TarskiGE ` ) +
    • A11 Axiom of continuity ( ` TarskiGB ` ) +
    + + Our definition is split into 5 parts:
      +
    • congruence axioms ` TarskiGC ` (which metric spaces fulfill) +
    • betweenness axioms ` TarskiGB ` +
    • congruence and betweenness axioms ` TarskiGCB ` +
    • upper and lower dimension axioms ` TarskiGDim>= ` +
    • axiom of Euclid / parallel postulate ` TarskiGE ` +
    + + So our definition of a Tarskian Geometry includes the 3 axioms for the + quaternary congruence relation (A1, A2, A3), the 3 axioms for the + ternary betweenness relation (A6, A7, A11), and the 2 axioms of + compatibility of the congruence and the betweenness relations (A4,A5). + + It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 + (Lower dimension axiom) and A9 (Upper dimension axiom) so the number + of dimensions of the geometry it formalizes is not constrained. + + Considering A2 as one of the 3 axioms for the quaternary congruence + relation is somewhat conventional, because the transitivity of the + congruence relation is automatically given by our choice to take the + distance as this congruence relation in our definition of Tarski + geometries. + (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry + Arnoux, 27-Apr-2019.) $) + df-trkg $a |- TarskiG = ( ( TarskiGC i^i TarskiGB ) + i^i ( TarskiGCB i^i { f | [. ( Base ` f ) / p ]. [. ( Itv ` f ) / i ]. + ( LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | + ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } ) ) $. + $} + + ${ + $d d f g i n p G $. $d a b c d f g i j n p s t u v x y z I $. + $d a b c d f g i j n p s t u v x y z P $. + $d a b c d f g i j n p u v x y z .- $. + istrkg.p $e |- P = ( Base ` G ) $. + istrkg.d $e |- .- = ( dist ` G ) $. + istrkg.i $e |- I = ( Itv ` G ) $. + $( Property of being a Tarski geometry - congruence part. (Contributed by + Thierry Arnoux, 14-Mar-2019.) $) + istrkgc $p |- ( G e. TarskiGC <-> ( G e. _V /\ + ( A. x e. P A. y e. P ( x .- y ) = ( y .- x ) /\ A. x e. P + A. y e. P A. z e. P ( ( x .- y ) = ( z .- z ) -> x = y ) ) ) ) $= + ( vd vp vf cv co wceq wral wa wcel raleqbidva wi cds cfv wsbc cbs cstrkgc + simpl eqcomd adantr simpllr oveqd eqeq12d oveqdr anbi12d sbcie2s df-trkgc + imbi1d elab4g ) ANZBNZKNZOZUTUSVAOZPZBLNZQZAVEQZVBCNZVHVAOZPZUSUTPZUAZCVE + QZBVEQZAVEQZRZKMNZUBUCUDLVQUEUCUDUSUTGOZUTUSGOZPZBDQZADQZVRVHVHGOZPZVKUAZ + CDQZBDQZADQZRZMEUFWIVPMDGUEUBELKHIVEDPZVAGPZRZWBVGWHVOWLWAVFADVEWLVEDWJWK + UGUHZWLUSDSZRZVTVDBDVEWLDVEPZWNWMUIZWOUTDSZRZVRVBVSVCWSGVAUSUTWSVAGWJWKWN + WRUJUHZUKWSGVAUTUSWTUKULTTWLWGVNADVEWMWOWFVMBDVEWQWSWEVLCDVEWOWPWRWQUIWSV + HDSZRZWDVJVKXBVRVBWCVIWSXAABGVAWTUMWSXACCGVAWTUMULUQTTTUNUOABCMLKUPUR $. + + $( Property of being a Tarski geometry - betweenness part. (Contributed by + Thierry Arnoux, 14-Mar-2019.) $) + istrkgb $p |- ( G e. TarskiGB <-> ( G e. _V /\ + ( A. x e. P A. y e. P ( y e. ( x I x ) -> x = y ) + /\ A. x e. P A. y e. P A. z e. P A. u e. P A. v e. P + ( ( u e. ( x I z ) /\ v e. ( y I z ) ) + -> E. a e. P ( a e. ( u I y ) /\ a e. ( v I x ) ) ) + /\ A. s e. ~P P A. t e. ~P P + ( E. a e. P A. x e. s A. y e. t x e. ( a I y ) + -> E. b e. P A. x e. s A. y e. t b e. ( x I y ) ) ) ) ) $= + ( co wcel wral wa vi vp vf cv wceq wrex cpw w3a citv cfv wsbc cbs cstrkgb + simpl eqcomd adantr simpllr oveqd eleq2d imbi1d raleqbidva simp-6r oveqdr + wi anbi12d rexeqbidva imbi12d ad2antrr simp-4r 2ralbidv 3anbi123d sbcie2s + pweqd df-trkgb elab4g ) BUDZAUDZVQUAUDZQZRZVQVPUEZVDZBUBUDZSZAWCSZEUDZVQC + UDZVRQZRZDUDZVPWGVRQZRZTZLUDZWFVPVRQZRZWNWJVQVRQZRZTZLWCUFZVDZDWCSZEWCSZC + WCSZBWCSZAWCSZVQWNVPVRQZRZBFUDZSAKUDZSZLWCUFZMUDZVQVPVRQZRZBXISAXJSZMWCUF + ZVDZFWCUGZSZKXSSZUHZUAUCUDZUIUJUKUBYCULUJUKVPVQVQIQZRZWAVDZBGSZAGSZWFVQWG + IQZRZWJVPWGIQZRZTZWNWFVPIQZRZWNWJVQIQZRZTZLGUFZVDZDGSZEGSZCGSZBGSZAGSZVQW + NVPIQZRZBXISAXJSZLGUFZXMVQVPIQZRZBXISAXJSZMGUFZVDZFGUGZSZKUUOSZUHZUCHUMUU + RYBUCGIULUIHUBUANPWCGUEZVRIUEZTZYHWEUUEXFUUQYAUVAYGWDAGWCUVAWCGUUSUUTUNUO + ZUVAVQGRZTZYFWBBGWCUVAGWCUEZUVCUVBUPZUVDVPGRZTZYEVTWAUVHYDVSVPUVHIVRVQVQU + VHVRIUUSUUTUVCUVGUQUOURUSUTVAVAUVAUUDXEAGWCUVBUVDUUCXDBGWCUVFUVHUUBXCCGWC + UVDUVEUVGUVFUPZUVHWGGRZTZUUAXBEGWCUVHUVEUVJUVIUPZUVKWFGRZTZYTXADGWCUVKUVE + UVMUVLUPZUVNWJGRZTZYMWMYSWTUVQYJWIYLWLUVQYIWHWFUVQIVRVQWGUVQVRIUUSUUTUVCU + VGUVJUVMUVPVBUOZURUSUVQYKWKWJUVQIVRVPWGUVRURUSVEUVQYRWSLGWCUVNUVEUVPUVOUP + UVQWNGRZTZYOWPYQWRUVTYNWOWNUVQUVSEBIVRUVRVCUSUVTYPWQWNUVQUVSDAIVRUVRVCUSV + EVFVGVAVAVAVAVAUVAUUPXTKUUOXSUVAGWCUVBVMZUVAXJUUORZTZUUNXRFUUOXSUVAUUOXSU + EUWBUWAUPUWCXIUUORZTZUUIXLUUMXQUWEUUHXKLGWCUVAUVEUWBUWDUVBVHZUWEUVSTZUUGX + HABXJXIUWGUUFXGVQUWGIVRWNVPUWGVRIUUSUUTUWBUWDUVSVIUOURUSVJVFUWEUULXPMGWCU + WFUWEXMGRZTZUUKXOABXJXIUWIUUJXNXMUWIIVRVQVPUWIVRIUUSUUTUWBUWDUWHVIUOURUSV + JVFVGVAVAVKVLABCDEFUCUAKUBLMVNVO $. + + $( Property of being a Tarski geometry - congruence and betweenness part. + (Contributed by Thierry Arnoux, 14-Mar-2019.) $) + istrkgcb $p |- ( G e. TarskiGCB <-> ( G e. _V /\ + ( A. x e. P A. y e. P A. z e. P A. u e. P + A. a e. P A. b e. P A. c e. P A. v e. P + ( ( ( x =/= y /\ y e. ( x I z ) /\ b e. ( a I c ) ) + /\ ( ( ( x .- y ) = ( a .- b ) /\ ( y .- z ) = ( b .- c ) ) + /\ ( ( x .- u ) = ( a .- v ) /\ ( y .- u ) = ( b .- v ) ) ) ) + -> ( z .- u ) = ( c .- v ) ) + /\ A. x e. P A. y e. P A. a e. P A. b e. P E. z e. P + ( y e. ( x I z ) /\ ( y .- z ) = ( a .- b ) ) ) ) ) $= + ( co wceq wa wral oveqd vi vd vp vf cv wne wcel w3a wi wrex citv cfv wsbc + cds cstrkgcb simp1 eqcomd adantr ad6antr simpll3 3anbi23d simpll2 eqeq12d + cbs eleq2d anbi12d imbi12d raleqbidva ad3antrrr rexeqbidva sbcie3s elab4g + df-trkgcb ) AUEZBUEZUFZVOVNCUEZUAUEZPZUGZKUEZJUEZLUEZVRPZUGZUHZVNVOUBUEZP + ZWBWAWGPZQZVOVQWGPZWAWCWGPZQZRZVNEUEZWGPZWBDUEZWGPZQZVOWOWGPZWAWQWGPZQZRZ + RZRZVQWOWGPZWCWQWGPZQZUIZDUCUEZSZLXJSZKXJSZJXJSZEXJSZCXJSZBXJSZAXJSZVTWKW + IQZRZCXJUJZKXJSZJXJSZBXJSZAXJSZRZUAUDUEZUKULUMUBYGUNULUMUCYGVDULUMVPVOVNV + QHPZUGZWAWBWCHPZUGZUHZVNVOIPZWBWAIPZQZVOVQIPZWAWCIPZQZRZVNWOIPZWBWQIPZQZV + OWOIPZWAWQIPZQZRZRZRZVQWOIPZWCWQIPZQZUIZDFSZLFSZKFSZJFSZEFSZCFSZBFSZAFSZY + IYPYNQZRZCFUJZKFSZJFSZBFSZAFSZRZUDGUOUVHYFUDFIHVDUNUKGUCUBUAMNOXJFQZWGIQZ + VRHQZUHZUUTXRUVGYEUVLUUSXQAFXJUVLXJFUVIUVJUVKUPUQZUVLVNFUGZRZUURXPBFXJUVL + FXJQZUVNUVMURZUVOVOFUGZRZUUQXOCFXJUVOUVPUVRUVQURZUVSVQFUGZRZUUPXNEFXJUVSU + VPUWAUVTURZUWBWOFUGZRZUUOXMJFXJUWBUVPUWDUWCURZUWEWBFUGZRZUUNXLKFXJUWEUVPU + WGUWFURUWHWAFUGZRZUUMXKLFXJUVLUVPUVNUVRUWAUWDUWGUWIUVMUSUWJWCFUGZRZUULXID + FXJUVOUVPUVRUWAUWDUWGUWIUWKUVQUSUWLWQFUGZRZUUHXEUUKXHUWNYLWFUUGXDUWNYIVTY + KWEVPUWNYHVSVOUWNHVRVNVQUWNVRHUVSUVKUWAUWDUWGUWIUWKUWMUVIUVJUVKUVNUVRUTZU + SUQZTVEUWNYJWDWAUWNHVRWBWCUWPTVEVAUWNYSWNUUFXCUWNYOWJYRWMUWNYMWHYNWIUWNIW + GVNVOUWNWGIUVSUVJUWAUWDUWGUWIUWKUWMUVIUVJUVKUVNUVRVBZUSUQZTUWNIWGWBWAUWRT + VCUWNYPWKYQWLUWNIWGVOVQUWRTUWNIWGWAWCUWRTVCVFUWNUUBWSUUEXBUWNYTWPUUAWRUWN + IWGVNWOUWRTUWNIWGWBWQUWRTVCUWNUUCWTUUDXAUWNIWGVOWOUWRTUWNIWGWAWQUWRTVCVFV + FVFUWNUUIXFUUJXGUWNIWGVQWOUWRTUWNIWGWCWQUWRTVCVGVHVHVHVHVHVHVHVHUVLUVFYDA + FXJUVMUVOUVEYCBFXJUVQUVSUVDYBJFXJUVTUVSUWGRZUVCYAKFXJUVSUVPUWGUVTURZUWSUW + IRZUVBXTCFXJUWSUVPUWIUWTURUXAUWARZYIVTUVAXSUXBYHVSVOUXBHVRVNVQUXBVRHUVSUV + KUWGUWIUWAUWOVIUQTVEUXBYPWKYNWIUXBIWGVOVQUXBWGIUVSUVJUWGUWIUWAUWQVIUQZTUX + BIWGWBWAUXCTVCVFVJVHVHVHVHVFVKABCDEUDUAUCJKLUBVMVL $. + + $( Property of fulfilling Euclid's axiom. (Contributed by Thierry Arnoux, + 14-Mar-2019.) $) + istrkge $p |- ( G e. TarskiGE <-> ( G e. _V /\ + A. x e. P A. y e. P A. z e. P A. u e. P A. v e. P + ( ( u e. ( x I v ) /\ u e. ( y I z ) /\ x =/= u ) -> E. a e. P + E. b e. P ( y e. ( x I a ) /\ z e. ( x I b ) /\ v e. ( a I b ) ) ) ) ) $= + ( cv co wcel wral wa adantr vi vp vf wne w3a wrex wi citv cfv cbs cstrkge + wsbc wceq simpl eqcomd simp-6r oveqd eleq2d 3anbi12d 3anbi123d rexeqbidva + ad2antrr imbi12d raleqbidva sbcie2s df-trkge elab4g ) EOZAOZDOZUAOZPZQZVH + BOZCOZVKPZQZVIVHUDZUEZVNVIJOZVKPZQZVOVIKOZVKPZQZVJVTWCVKPZQZUEZKUBOZUFZJW + IUFZUGZDWIRZEWIRZCWIRZBWIRZAWIRZUAUCOZUHUIULUBWRUJUIULVHVIVJHPZQZVHVNVOHP + ZQZVRUEZVNVIVTHPZQZVOVIWCHPZQZVJVTWCHPZQZUEZKFUFZJFUFZUGZDFRZEFRZCFRZBFRZ + AFRZUCGUKXRWQUCFHUJUHGUBUALNWIFUMZVKHUMZSZXQWPAFWIYAWIFXSXTUNUOZYAVIFQZSZ + XPWOBFWIYAFWIUMZYCYBTZYDVNFQZSZXOWNCFWIYDYEYGYFTZYHVOFQZSZXNWMEFWIYHYEYJY + ITZYKVHFQZSZXMWLDFWIYKYEYMYLTZYNVJFQZSZXCVSXLWKYQWTVMXBVQVRYQWSVLVHYQHVKV + IVJYQVKHXSXTYCYGYJYMYPUPZUOZUQURYQXAVPVHYQHVKVNVOYSUQURUSYQXKWJJFWIYNYEYP + YOTZYQVTFQZSZXJWHKFWIYQYEUUAYTTUUBWCFQZSZXEWBXGWEXIWGUUDXDWAVNUUDHVKVIVTU + UDVKHYQXTUUAUUCYRVBUOZUQURUUDXFWDVOUUDHVKVIWCUUEUQURUUDXHWFVJUUDHVKVTWCUU + EUQURUTVAVAVCVDVDVDVDVDVEABCDEUCUAUBJKVFVG $. + + $( Building lines from the segment property. (Contributed by Thierry + Arnoux, 14-Mar-2019.) $) + istrkgl $p |- ( G e. { f | [. ( Base ` f ) / p ]. [. ( Itv ` f ) / i ]. + ( LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | + ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } <-> + ( G e. _V /\ ( LineG ` G ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | + ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) ) ) $= + ( cv clng cfv co wcel wceq wa csn cdif w3o crab cmpo citv wsbc cbs eqcomd + cab simpl adantr difeq1d simpr oveqd eleq2d 3orbi123d mpoeq123dva sbcie2s + rabeqbidv eqeq2d fveqeq2 bitrd eqid elab4g ) ENZOPZABJNZVHANZUAZUBZCNZVIB + NZFNZQZRZVIVLVMVNQZRZVMVIVLVNQZRZUCZCVHUDZUEZSZFVFUFPUGJVFUHPUGZGOPABDDVJ + UBZVLVIVMHQZRZVIVLVMHQZRZVMVIVLHQZRZUCZCDUDZUEZSZEGWEEUJZVFGSWEVGWOSZWPWR + WDEDHUHUFGJFKMVHDSZVNHSZTZWOWCVGXAABDWFWNVHVKWBXAVHDWSWTUKUIZXAVIDRZTDVHV + JXADVHSXCXBULUMXAWNWBSXCVMWFRTXAWMWACDVHXBXAWHVPWJVRWLVTXAWGVOVLXAHVNVIVM + XAVNHWSWTUNUIZUOUPXAWIVQVIXAHVNVLVMXDUOUPXAWKVSVMXAHVNVIVLXDUOUPUQUTULURV + AUSVFGWOOVBVCWQVDVE $. + + $d f g j n p x y z N $. + $( Property of fulfilling the lower dimension ` N ` axiom. (Contributed by + Thierry Arnoux, 20-Nov-2019.) $) + istrkgld $p |- ( ( G e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( G TarskiGDim>= N <-> + E. f ( f : ( 1 ..^ N ) -1-1-> P /\ E. x e. P E. y e. P E. z e. P + ( A. j e. ( 2 ..^ N ) ( ( ( f ` 1 ) .- x ) = ( ( f ` j ) .- x ) /\ + ( ( f ` 1 ) .- y ) = ( ( f ` j ) .- y ) /\ + ( ( f ` 1 ) .- z ) = ( ( f ` j ) .- z ) ) + /\ -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) ) + ) ) $= + ( cv cfzo co wceq wrex oveqd vn vp vd vi vg c1 wf1 cfv w3a c2 wral w3o wn + wcel wex citv wsbc cds cbs cuz cstrkgld eqidd simp1 eqcomd f1eq123d simp2 + eqeq12d 3anbi123d ralbidv simp3 eleq2d 3orbi123d notbid anbi12d rexeqbidv + wa exbidv sbcie3s oveq2 raleqdv anbi1d rexbidv 2rexbidv df-trkgld brabg ) + UFUAOZPQZUBOZEOZUGZUFWIUHZAOZUCOZQZFOWIUHZWLWMQZRZWKBOZWMQZWOWRWMQZRZWKCO + ZWMQZWOXBWMQZRZUIZFUJWFPQZUKZXBWLWRUDOZQZUNZWLXBWRXIQZUNZWRWLXBXIQZUNZULZ + UMZVPZCWHSZBWHSZAWHSZVPZEUOZUDUEOZUPUHUQUCYDURUHUQUBYDUSUHUQWGDWIUGZWKWLI + QZWOWLIQZRZWKWRIQZWOWRIQZRZWKXBIQZWOXBIQZRZUIZFXGUKZXBWLWRHQZUNZWLXBWRHQZ + UNZWRWLXBHQZUNZULZUMZVPZCDSZBDSZADSZVPZEUOZUFJPQZDWIUGZYOFUJJPQZUKZUUDVPZ + CDSZBDSADSZVPZEUOUEUAGJKUJUTUHVAUUJYCUEDIHUSURUPGUBUCUDLMNWHDRZWMIRZXIHRZ + UIZUUIYBEUVBYEWJUUHYAUVBWGWGDWHWIWIUVBWIVBUVBWGVBUVBWHDUUSUUTUVAVCVDZVEUV + BUUGXTADWHUVCUVBUUFXSBDWHUVCUVBUUEXRCDWHUVCUVBYPXHUUDXQUVBYOXFFXGUVBYHWQY + KXAYNXEUVBYFWNYGWPUVBIWMWKWLUVBWMIUUSUUTUVAVFVDZTUVBIWMWOWLUVDTVGUVBYIWSY + JWTUVBIWMWKWRUVDTUVBIWMWOWRUVDTVGUVBYLXCYMXDUVBIWMWKXBUVDTUVBIWMWOXBUVDTV + GVHVIUVBUUCXPUVBYRXKYTXMUUBXOUVBYQXJXBUVBHXIWLWRUVBXIHUUSUUTUVAVJVDZTVKUV + BYSXLWLUVBHXIXBWRUVETVKUVBUUAXNWRUVBHXIWLXBUVETVKVLVMVNVOVOVOVNVQVRWFJRZU + UIUUREUVFYEUULUUHUUQUVFWGUUKDDWIWIUVFWIVBWFJUFPVSUVFDVBVEUVFUUFUUPABDDUVF + UUEUUOCDUVFYPUUNUUDUVFYOFXGUUMWFJUJPVSVTWAWBWCVNVQABCEUEUDFUAUBUCWDWE $. + + $( Property of fulfilling the lower dimension 2 axiom. (Contributed by + Thierry Arnoux, 20-Nov-2019.) $) + istrkg2ld $p |- ( G e. V + -> ( G TarskiGDim>= 2 <-> E. x e. P E. y e. P E. z e. P + -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) ) $= + ( vf vj wcel c2 c1 co wceq wa wrex cstrkgld wbr cfzo wf1 cfv w3a wral w3o + cv wn wex cuz wb cz 2z ax-mp istrkgld mpan2 r19.41v ancom rexbii 3bitr3ri + uzid exbii rexcom4 simpr reximi adantl exlimiv cmpt csn cop wss 1ex f1osn + wf1o vex f1of1 mp1i snssi f1ss syl2anc wtru fzo12sn mpteq1i fmptsn eqtr4i + cvv mp2an eqidd f1eq123d mptru sylibr c0 ral0 fzo0 raleqi mpbir jctl ovex + a1i mptex f1eq1 nfmpt1 nfeq2 nfan simpll fveq1d eqeq12d 3anbi123d ralbida + nfv oveq1d anbi1d 2rexbidva anbi12d syl2an impbida rexbiia 3bitr2i bitrdi + spcev ) EHNZEOUAUBZPOUCQZDLUIZUDZPYFUEZAUIZGQZMUIZYFUEZYIGQZRZYHBUIZGQZYL + YOGQZRZYHCUIZGQZYLYSGQZRZUFZMOOUCQZUGZYSYIYOFQNYIYSYOFQNYOYIYSFQNUHUJZSZC + DTZBDTZADTZSZLUKZUUFCDTZBDTZADTZYCOOULUENZYDUULUMOUNNUUPUOOVCUPABCDLMEFGO + HIJKUQURUULYGUUISZADTZLUKUUQLUKZADTUUOUUKUURLUUIYGSZADTUUJYGSUURUUKUUIYGA + DUSUUTUUQADUUIYGUTVAUUJYGUTVBVDUUQALDVEUUSUUNADYIDNZUUSUUNUUSUUNUVAUUQUUN + LUUIUUNYGUUHUUMBDUUGUUFCDUUEUUFVFVGVGVHVIVHUVAYEDMYEYIVJZUDZPUVBUEZYIGQZY + KUVBUEZYIGQZRZUVDYOGQZUVFYOGQZRZUVDYSGQZUVFYSGQZRZUFZMUUDUGZUUFSZCDTZBDTZ + UUSUUNUVAPVKZDPYIVLVKZUDZUVCUVAUVTYIVKZUWAUDZUWCDVMUWBUVTUWCUWAVPUWDUVAPY + IVNAVQZVOUVTUWCUWAVRVSYIDVTUVTUWCDUWAWAWBUVCUWBUMWCYEUVTDDUVBUWAUVBUWARWC + UVBMUVTYIVJZUWAMYEUVTYIWDWEPWHNYIWHNUWAUWFRVNUWEMPYIWHWHWFWIWGXAYEUVTRWCW + DXAWCDWJWKWLWMUUMUVRBDUUFUVQCDUUFUVPUVPUVOMWNUGUVOMWOUVOMUUDWNOWPWQWRWSVG + VGUUQUVCUVSSLUVBMYEYIPOUCWTXBYFUVBRZYGUVCUUIUVSYEDYFUVBXCUWGUUGUVQBCDDUWG + YODNYSDNSZSZUUEUVPUUFUWIUUCUVOMUUDUWGUWHMMYFUVBMYEYIXDXEUWHMXLXFUWIYKUUDN + ZSZYNUVHYRUVKUUBUVNUWKYJUVEYMUVGUWKYHUVDYIGUWKPYFUVBUWGUWHUWJXGZXHZXMUWKY + LUVFYIGUWKYKYFUVBUWLXHZXMXIUWKYPUVIYQUVJUWKYHUVDYOGUWMXMUWKYLUVFYOGUWNXMX + IUWKYTUVLUUAUVMUWKYHUVDYSGUWMXMUWKYLUVFYSGUWNXMXIXJXKXNXOXPYBXQXRXSXTYA + $. + + $( Property of fulfilling the lower dimension 3 axiom. (Contributed by + Thierry Arnoux, 12-Jul-2020.) $) + istrkg3ld $p |- ( G e. V + -> ( G TarskiGDim>= 3 <-> E. u e. P E. v e. P ( u =/= v + /\ E. x e. P E. y e. P E. z e. P ( ( ( u .- x ) = ( v .- x ) + /\ ( u .- y ) = ( v .- y ) /\ ( u .- z ) = ( v .- z ) ) + /\ -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) ) ) ) $= + ( wcel c3 c1 co wceq c2 wrex vf vj cstrkgld wbr cfzo wf1 cfv w3a wral w3o + cv wn wa wex cpr wne cuz wb cz cle 3z 2re 3re 2lt3 ltleii eluz1i mpbir2an + 2z istrkgld mpan2 fzo13pr f1eq2 ax-mp anbi1i exbii a1i 1z oveq1 3anbi123d + eqeq1d anbi1d rexbidv 2rexbidv eqeq2d csn caddc 2p1e3 oveq2i fzosn eqtr3i + 1ne2 raleqi fveq2 oveq1d ralsng bitri bitr4di f1prex mp3an 3bitrd ) GJNZG + OUCUDZPOUEQZFUAUKZUFZPXDUGZAUKZIQZUBUKZXDUGZXGIQZRZXFBUKZIQZXJXMIQZRZXFCU + KZIQZXJXQIQZRZUHZUBSOUEQZUIZXQXGXMHQNXGXQXMHQNXMXGXQHQNUJULZUMZCFTZBFTAFT + ZUMZUAUNZPSUOZFXDUFZYGUMZUAUNZEUKZDUKZUPYNXGIQZYOXGIQZRZYNXMIQZYOXMIQZRZY + NXQIQZYOXQIQZRZUHZYDUMZCFTZBFTAFTZUMDFTEFTZXAOSUQUGNZXBYIURUUJOUSNSOUTUDV + ASOVBVCVDVESOVHVFVGABCFUAUBGHIOJKLMVIVJYIYMURXAYHYLUAXEYKYGXCYJRXEYKURVKX + CYJFXDVLVMVNVOVPYMUUIURZXAPUSNSUSNZPSUPUUKVQVHWKYGUUHXHYQRZXNYTRZXRUUCRZU + HZYDUMZCFTZBFTAFTEDPSFUAUSUSYNXFRZUUGUURABFFUUSUUFUUQCFUUSUUEUUPYDUUSYRUU + MUUAUUNUUDUUOUUSYPXHYQYNXFXGIVRVTUUSYSXNYTYNXFXMIVRVTUUSUUBXRUUCYNXFXQIVR + VTVSWAWBWCYOSXDUGZRZUURYFABFFUVAUUQYECFUVAUUPYCYDUVAUUPXHUUTXGIQZRZXNUUTX + MIQZRZXRUUTXQIQZRZUHZYCUVAUUMUVCUUNUVEUUOUVGUVAYQUVBXHYOUUTXGIVRWDUVAYTUV + DXNYOUUTXMIVRWDUVAUUCUVFXRYOUUTXQIVRWDVSYCYAUBSWEZUIZUVHYAUBYBUVISSPWFQZU + EQZYBUVIUVKOSUEWGWHUULUVLUVIRVHSWIVMWJWLUULUVJUVHURVHYAUVHUBSUSXISRZXLUVC + XPUVEXTUVGUVMXKUVBXHUVMXJUUTXGIXISXDWMZWNWDUVMXOUVDXNUVMXJUUTXMIUVNWNWDUV + MXSUVFXRUVMXJUUTXQIUVNWNWDVSWOVMWPWQWAWBWCWRWSVPWT $. + $} + + ${ + $d f i p x y z $. $d a b c v z A $. $d b c v z B $. $d c v C $. + $d a b c s t u v x y z I $. $d a b c s t u v x y z P $. $d a b s t x S $. + $d a b t x y T $. $d a b c u v U $. $d a b c u v x y z X $. + $d a b c u v y z Y $. $d a b c u v z Z $. $d a b v V $. + $d a b c u v x y z .- $. + axtrkg.p $e |- P = ( Base ` G ) $. + axtrkg.d $e |- .- = ( dist ` G ) $. + axtrkg.i $e |- I = ( Itv ` G ) $. + axtrkg.g $e |- ( ph -> G e. TarskiG ) $. + ${ + axtgcgrrflx.1 $e |- ( ph -> X e. P ) $. + axtgcgrrflx.2 $e |- ( ph -> Y e. P ) $. + $( Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. + (Contributed by Thierry Arnoux, 14-Mar-2019.) $) + axtgcgrrflx $p |- ( ph -> ( X .- Y ) = ( Y .- X ) ) $= + ( vx vy cv co wceq cstrkgc wcel vf vp vz vi cstrkg cstrkgb cin cstrkgcb + wral clng cfv csn cdif w3o crab cmpo citv cbs cab df-trkg inss1 eqsstri + wsbc sstri sselid wi cvv istrkgc simprbi simpld syl oveq1 oveq2 eqeq12d + wa rspc2v syl2anc mpd ) ANPZOPZEQZVTVSEQZRZOBUINBUIZFGEQZGFEQZRZACSTZWD + AUESCUESUFUGZUHUAPZUJUKNOUBPZWKVSULUMUCPZVSVTUDPZQTVSWLVTWMQTVTVSWLWMQT + UNUCWKUOUPRUDWJUQUKVCUBWJURUKVCUAUSUGZUGZSNOUCUAUDUBUTWOWISWIWNVASUFVAV + DVBKVEWHWDWAWLWLEQRVSVTRVFUCBUIOBUINBUIZWHCVGTWDWPVONOUCBCDEHIJVHVIVJVK + AFBTGBTWDWGVFLMWCWGFVTEQZVTFEQZRNOFGBBVSFRWAWQWBWRVSFVTEVLVSFVTEVMVNVTG + RWQWEWRWFVTGFEVMVTGFEVLVNVPVQVR $. + $} + + ${ + axtgcgrid.1 $e |- ( ph -> X e. P ) $. + axtgcgrid.2 $e |- ( ph -> Y e. P ) $. + axtgcgrid.3 $e |- ( ph -> Z e. P ) $. + axtgcgrid.4 $e |- ( ph -> ( X .- Y ) = ( Z .- Z ) ) $. + $( Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. + (Contributed by Thierry Arnoux, 14-Mar-2019.) $) + axtgcgrid $p |- ( ph -> X = Y ) $= + ( vx co wceq wcel vy vz vf vp vi cv wi wral cstrkgc cstrkg cin cstrkgcb + cstrkgb clng cfv csn cdif w3o crab cmpo citv wsbc cbs cab df-trkg inss1 + sstri eqsstri sselid cvv wa istrkgc simprbi simprd oveq1 eqeq1d imbi12d + syl eqeq1 oveq2 eqeq2 id oveq12d eqeq2d imbi1d rspc3v syl3anc mp2d ) AQ + UFZUAUFZERZUBUFZWLERZSZWIWJSZUGZUBBUHUABUHQBUHZFGERZHHERZSZFGSZACUITZWQ + AUJUICUJUIUMUKZULUCUFZUNUOQUAUDUFZXEWIUPUQWLWIWJUEUFZRTWIWLWJXFRTWJWIWL + XFRTURUBXEUSUTSUEXDVAUOVBUDXDVCUOVBUCVDUKZUKZUIQUAUBUCUEUDVEXHXCUIXCXGV + FUIUMVFVGVHLVIXBWKWJWIERSUABUHQBUHZWQXBCVJTXIWQVKQUAUBBCDEIJKVLVMVNVRPA + FBTGBTHBTWQWTXAUGZUGMNOWPXJFWJERZWMSZFWJSZUGWRWMSZXAUGQUAUBFGHBBBWIFSZW + NXLWOXMXOWKXKWMWIFWJEVOVPWIFWJVSVQWJGSZXLXNXMXAXPXKWRWMWJGFEVTVPWJGFWAV + QWLHSZXNWTXAXQWMWSWRXQWLHWLHEXQWBZXRWCWDWEWFWGWH $. + $} + + ${ + axtgsegcon.1 $e |- ( ph -> X e. P ) $. + axtgsegcon.2 $e |- ( ph -> Y e. P ) $. + axtgsegcon.3 $e |- ( ph -> A e. P ) $. + axtgsegcon.4 $e |- ( ph -> B e. P ) $. + $( Axiom of segment construction, Axiom A4 of [Schwabhauser] p. 11. As + discussed in Axiom 4 of [Tarski1999] p. 178, "The intuitive content + [is that] given any line segment ` A B ` , one can construct a line + segment congruent to it, starting at any point ` Y ` and going in the + direction of any ray containing ` Y ` . The ray is determined by the + point ` Y ` and a second point ` X ` , the endpoint of the ray. The + other endpoint of the line segment to be constructed is just the point + ` z ` whose existence is asserted." (Contributed by Thierry Arnoux, + 15-Mar-2019.) $) + axtgsegcon $p |- ( ph + -> E. z e. P ( Y e. ( X I z ) /\ ( Y .- z ) = ( A .- B ) ) ) $= + ( co wral va vb vy vx vf vp vi vc vu vv cv wcel wceq wa cstrkgcb cstrkg + wrex cstrkgc cstrkgb cin clng cfv csn cdif w3o crab cmpo citv cbs inss2 + wsbc cab df-trkg inss1 sstri eqsstri sselid wne w3a wi istrkgcb simprbi + cvv simprd syl oveq1 eleq2d anbi1d rexbidv 2ralbidv eleq1 eqeq1d rspc2v + anbi12d syl2anc mpd eqeq2d anbi2d oveq2 ) AJIBUKZGSZULZJWTHSZUAUKZUBUKZ + HSZUMZUNZBEUQZUBETUAETZXBXCCDHSZUMZUNZBEUQZAUCUKZUDUKZWTGSZULZXOWTHSZXF + UMZUNZBEUQZUBETUAETZUCETUDETZXJAFUOULZYDAUPUOFUPURUSUTZUOUEUKZVAVBUDUCU + FUKZYHXPVCVDWTXPXOUGUKZSULXPWTXOYISULXOXPWTYISULVEBYHVFVGUMUGYGVHVBVKUF + YGVIVBVKUEVLZUTZUTZUOUDUCBUEUGUFVMYLYKUOYFYKVJUOYJVNVOVPNVQYEXPXOVRXRXE + XDUHUKZGSULVSXPXOHSXFUMXSXEYMHSUMUNXPUIUKZHSXDUJUKZHSUMXOYNHSXEYOHSUMUN + UNUNWTYNHSYMYOHSUMVTUJETUHETUBETUAETUIETBETUCETUDETZYDYEFWCULYPYDUNUDUC + BUJUIEFGHUAUBUHKLMWAWBWDWEAIEULJEULYDXJVTOPYCXJXOXAULZXTUNZBEUQZUBETUAE + TUDUCIJEEXPIUMZYBYSUAUBEEYTYAYRBEYTXRYQXTYTXQXAXOXPIWTGWFWGWHWIWJXOJUMZ + YSXIUAUBEEUUAYRXHBEUUAYQXBXTXGXOJXAWKUUAXSXCXFXOJWTHWFWLWNWIWJWMWOWPACE + ULDEULXJXNVTQRXIXNXBXCCXEHSZUMZUNZBEUQUAUBCDEEXDCUMZXHUUDBEUUEXGUUCXBUU + EXFUUBXCXDCXEHWFWQWRWIXEDUMZUUDXMBEUUFUUCXLXBUUFUUBXKXCXEDCHWSWQWRWIWMW + OWP $. + $} + + ${ + axtg5seg.1 $e |- ( ph -> X e. P ) $. + axtg5seg.2 $e |- ( ph -> Y e. P ) $. + axtg5seg.3 $e |- ( ph -> Z e. P ) $. + axtg5seg.4 $e |- ( ph -> A e. P ) $. + axtg5seg.5 $e |- ( ph -> B e. P ) $. + axtg5seg.6 $e |- ( ph -> C e. P ) $. + axtg5seg.7 $e |- ( ph -> U e. P ) $. + axtg5seg.8 $e |- ( ph -> V e. P ) $. + axtg5seg.9 $e |- ( ph -> X =/= Y ) $. + axtg5seg.10 $e |- ( ph -> Y e. ( X I Z ) ) $. + axtg5seg.11 $e |- ( ph -> B e. ( A I C ) ) $. + axtg5seg.12 $e |- ( ph -> ( X .- Y ) = ( A .- B ) ) $. + axtg5seg.13 $e |- ( ph -> ( Y .- Z ) = ( B .- C ) ) $. + axtg5seg.14 $e |- ( ph -> ( X .- U ) = ( A .- V ) ) $. + axtg5seg.15 $e |- ( ph -> ( Y .- U ) = ( B .- V ) ) $. + $( Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two + triangles ` X Z U ` and ` A C V ` , a point ` Y ` on ` X Z ` , and a + point ` B ` on ` A C ` . If all corresponding line segments except + for ` Z U ` and ` C V ` are congruent ( i.e., ` X Y .~ A B ` , + ` Y Z .~ B C ` , ` X U .~ A V ` , and ` Y U .~ B V ` ), then ` Z U ` + and ` C V ` are also congruent. As noted in Axiom 5 of [Tarski1999] + p. 178, "this axiom is similar in character to the well-known theorems + of Euclidean geometry that allow one to conclude, from hypotheses + about the congruence of certain corresponding sides and angles in two + triangles, the congruence of other corresponding sides and angles." + (Contributed by Thierry Arnoux, 14-Mar-2019.) $) + axtg5seg $p |- ( ph -> ( Z .- U ) = ( C .- V ) ) $= + ( vc vv vb va vu vx vy vz vf vp vi wne co wcel cv wceq wa wral cstrkgcb + w3a wi cstrkg cstrkgc cstrkgb cin clng cfv csn cdif crab cmpo citv wsbc + w3o cbs cab df-trkg inss2 inss1 sstri eqsstri wrex cvv istrkgcb simprbi + sselid simpld neeq1 oveq1 eleq2d 3anbi12d eqeq1d anbi1d anbi12d ralbidv + imbi1d 2ralbidv neeq2 eleq1 oveq2 anbi2d 3anbi2d imbi12d rspc3v syl3anc + syl mpd 3anbi3d eqeq2d 3jca jca jca32 rspc2v syl2anc mp2d ) AKLVDZLKMHV + EZVFZCBUMVGZHVEZVFZVLZKLIVEZBCIVEZVHZLMIVEZCYKIVEZVHZVIZKFIVEZBUNVGZIVE + ZVHZLFIVEZCUUCIVEZVHZVIZVIZVIZMFIVEZYKUUCIVEZVHZVMZUNEVJUMEVJZYHYJCBDHV + EZVFZVLZYQYRCDIVEZVHZVIZUUBBJIVEZVHZUUFCJIVEZVHZVIZVIZVIZUULDJIVEZVHZAY + HYJUOVGZUPVGZYKHVEZVFZVLZYOUVMUVLIVEZVHZYRUVLYKIVEZVHZVIZKUQVGZIVEZUVMU + UCIVEZVHZLUWBIVEZUVLUUCIVEZVHZVIZVIZVIZMUWBIVEZUUMVHZVMZUNEVJZUMEVJZUOE + VJZUPEVJUQEVJZUUPAURVGZUSVGZVDZUWTUWSUTVGZHVEZVFZUVOVLZUWSUWTIVEZUVQVHZ + UWTUXBIVEZUVSVHZVIZUWSUWBIVEZUWDVHZUWTUWBIVEZUWGVHZVIZVIZVIZUXBUWBIVEZU + UMVHZVMZUNEVJZUMEVJUOEVJZUPEVJUQEVJZUTEVJUSEVJUREVJZUWRAGVKVFZUYDAVNVKG + VNVOVPVQZVKVAVGZVRVSURUSVBVGZUYHUWSVTWAUXBUWSUWTVCVGZVEVFUWSUXBUWTUYIVE + VFUWTUWSUXBUYIVEVFWFUTUYHWBWCVHVCUYGWDVSWEVBUYGWGVSWEVAWHZVQZVQZVKURUSU + TVAVCVBWIUYLUYKVKUYFUYKWJVKUYJWKWLWMQWRUYEUYDUXDUXHUVQVHVIUTEWNUOEVJUPE + VJUSEVJUREVJZUYEGWOVFUYDUYMVIURUSUTUNUQEGHIUPUOUMNOPWPWQWSXRAKEVFLEVFME + VFUYDUWRVMRSTUYCUWRKUWTVDZUWTKUXBHVEZVFZUVOVLZKUWTIVEZUVQVHZUXIVIZUWEUX + NVIZVIZVIZUXSVMZUNEVJZUMEVJUOEVJZUPEVJUQEVJYHLUYOVFZUVOVLZUVRLUXBIVEZUV + SVHZVIZUWIVIZVIZUXSVMZUNEVJZUMEVJUOEVJZUPEVJUQEVJURUSUTKLMEEEUWSKVHZUYB + VUFUQUPEEVUQUYAVUEUOUMEEVUQUXTVUDUNEVUQUXQVUCUXSVUQUXEUYQUXPVUBVUQUXAUY + NUXDUYPUVOUWSKUWTWTVUQUXCUYOUWTUWSKUXBHXAXBXCVUQUXJUYTUXOVUAVUQUXGUYSUX + IVUQUXFUYRUVQUWSKUWTIXAXDXEVUQUXLUWEUXNVUQUXKUWCUWDUWSKUWBIXAXDXEXFXFXH + XGXIXIUWTLVHZVUFVUPUQUPEEVURVUEVUOUOUMEEVURVUDVUNUNEVURVUCVUMUXSVURUYQV + UHVUBVULVURUYNYHUYPVUGUVOUWTLKXJUWTLUYOXKXCVURUYTVUKVUAUWIVURUYSUVRUXIV + UJVURUYRYOUVQUWTLKIXLXDVURUXHVUIUVSUWTLUXBIXAXDXFVURUXNUWHUWEVURUXMUWFU + WGUWTLUWBIXAXDXMXFXFXHXGXIXIUXBMVHZVUPUWQUQUPEEVUSVUOUWOUOUMEEVUSVUNUWN + UNEVUSVUMUWKUXSUWMVUSVUHUVPVULUWJVUSVUGYJYHUVOVUSUYOYILUXBMKHXLXBXNVUSV + UKUWAUWIVUSVUJUVTUVRVUSVUIYRUVSUXBMLIXLXDXMXEXFVUSUXRUWLUUMUXBMUWBIXAXD + XOXGXIXIXPXQXSAFEVFBEVFCEVFUWRUUPVMUDUAUBUWPUUPUVPUWAUUBUWDVHZUUFUWGVHZ + VIZVIZVIZUUNVMZUNEVJUMEVJYHYJUVLYLVFZVLZYOBUVLIVEZVHZUVTVIZUUEVVAVIZVIZ + VIZUUNVMZUNEVJUMEVJUQUPUOFBCEEEUWBFVHZUWNVVEUMUNEEVVOUWKVVDUWMUUNVVOUWJ + VVCUVPVVOUWIVVBUWAVVOUWEVUTUWHVVAVVOUWCUUBUWDUWBFKIXLXDVVOUWFUUFUWGUWBF + LIXLXDXFXMXMVVOUWLUULUUMUWBFMIXLXDXOXIUVMBVHZVVEVVNUMUNEEVVPVVDVVMUUNVV + PUVPVVGVVCVVLVVPUVOVVFYHYJVVPUVNYLUVLUVMBYKHXAXBXTVVPUWAVVJVVBVVKVVPUVR + VVIUVTVVPUVQVVHYOUVMBUVLIXAYAXEVVPVUTUUEVVAVVPUWDUUDUUBUVMBUUCIXAYAXEXF + XFXHXIUVLCVHZVVNUUOUMUNEEVVQVVMUUKUUNVVQVVGYNVVLUUJVVQVVFYMYHYJUVLCYLXK + XTVVQVVJUUAVVKUUIVVQVVIYQUVTYTVVQVVHYPYOUVLCBIXLYAVVQUVSYSYRUVLCYKIXAYA + XFVVQVVAUUHUUEVVQUWGUUGUUFUVLCUUCIXAYAXMXFXFXHXIXPXQXSAUUSUVBUVGAYHYJUU + RUFUGUHYBAYQUVAUIUJYCAUVDUVFUKULYCYDADEVFJEVFUUPUVIUVKVMZVMUCUEUUOVVRUU + SUVBUUIVIZVIZUULDUUCIVEZVHZVMUMUNDJEEYKDVHZUUKVVTUUNVWBVWCYNUUSUUJVVSVW + CYMUURYHYJVWCYLUUQCYKDBHXLXBXTVWCUUAUVBUUIVWCYTUVAYQVWCYSUUTYRYKDCIXLYA + XMXEXFVWCUUMVWAUULYKDUUCIXAYAXOUUCJVHZVVTUVIVWBUVKVWDVVSUVHUUSVWDUUIUVG + UVBVWDUUEUVDUUHUVFVWDUUDUVCUUBUUCJBIXLYAVWDUUGUVEUUFUUCJCIXLYAXFXMXMVWD + VWAUVJUULUUCJDIXLYAXOYEYFYG $. + $} + + ${ + axtgbtwnid.1 $e |- ( ph -> X e. P ) $. + axtgbtwnid.2 $e |- ( ph -> Y e. P ) $. + axtgbtwnid.3 $e |- ( ph -> Y e. ( X I X ) ) $. + $( Identity of Betweenness. Axiom A6 of [Schwabhauser] p. 11. + (Contributed by Thierry Arnoux, 15-Mar-2019.) $) + axtgbtwnid $p |- ( ph -> X = Y ) $= + ( vy vx cv co wcel wral vf vp vz vi vu vv va vt vs vb wi cstrkgb cstrkg + wceq cstrkgc cin cstrkgcb clng cfv csn cdif w3o crab cmpo citv wsbc cbs + cab df-trkg inss1 inss2 sstri eqsstri sselid wa cpw cvv istrkgb simprbi + wrex w3a simp1d syl id oveq12d eleq2d eqeq1 imbi12d eleq1 eqeq2 syl2anc + rspc2v mp2d ) AOQZPQZWODRZSZWOWNUNZUKZOBTPBTZGFFDRZSZFGUNZACULSZWTAUMUL + CUMUOULUPZUQUAQZURUSPOUBQZXGWOUTVAUCQZWOWNUDQZRSWOXHWNXIRSWNWOXHXIRSVBU + CXGVCVDUNUDXFVEUSVFUBXFVGUSVFUAVHUPZUPZULPOUCUAUDUBVIXKXEULXEXJVJUOULVK + VLVMKVNXDWTUEQZWOXHDRSUFQZWNXHDRSVOUGQZXLWNDRSXNXMWODRSVOUGBVTUKUFBTUEB + TUCBTOBTPBTZWOXNWNDRSOUHQZTPUIQZTUGBVTUJQWOWNDRSOXPTPXQTUJBVTUKUHBVPZTU + IXRTZXDCVQSWTXOXSWAPOUCUFUEUHBCDEUIUGUJHIJVRVSWBWCNAFBSGBSWTXBXCUKZUKLM + WSXTWNXASZFWNUNZUKPOFGBBWOFUNZWQYAWRYBYCWPXAWNYCWOFWOFDYCWDZYDWEWFWOFWN + WGWHWNGUNYAXBYBXCWNGXAWIWNGFWJWHWLWKWM $. + $} + + ${ + axtgpasch.1 $e |- ( ph -> X e. P ) $. + axtgpasch.2 $e |- ( ph -> Y e. P ) $. + axtgpasch.3 $e |- ( ph -> Z e. P ) $. + axtgpasch.4 $e |- ( ph -> U e. P ) $. + axtgpasch.5 $e |- ( ph -> V e. P ) $. + axtgpasch.6 $e |- ( ph -> U e. ( X I Z ) ) $. + axtgpasch.7 $e |- ( ph -> V e. ( Y I Z ) ) $. + $( Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given + triangle ` X Y Z ` , point ` U ` in segment ` X Z ` , and point ` V ` + in segment ` Y Z ` , there exists a point ` a ` on both the segment + ` U Y ` and the segment ` V X ` . This axiom is essentially a subset + of the general Pasch axiom. The general Pasch axiom asserts that on a + plane "a line intersecting a triangle in one of its sides, and not + intersecting any of the vertices, must intersect one of the other two + sides" (per the discussion about Axiom 7 of [Tarski1999] p. 179). The + (general) Pasch axiom was used implicitly by Euclid, but never stated; + Moritz Pasch discovered its omission in 1882. As noted in the + Metamath book, this means that the omission of Pasch's axiom from + Euclid went unnoticed for 2000 years. Only the inner Pasch algorithm + is included as an axiom; the "outer" form of the Pasch axiom can be + proved using the inner form (see theorem 9.6 of [Schwabhauser] p. 69 + and the brief discussion in axiom 7.1 of [Tarski1999] p. 180). + (Contributed by Thierry Arnoux, 15-Mar-2019.) $) + axtgpasch $p |- ( ph + -> E. a e. P ( a e. ( U I Y ) /\ a e. ( V I X ) ) ) $= + ( vu vv vx vz vy vf vp vi vt vs vb co wcel cv wa wrex wi cstrkgb cstrkg + wral cstrkgc cin cstrkgcb clng cfv csn cdif w3o crab cmpo wceq citv cbs + wsbc cab df-trkg inss1 inss2 eqsstri sselid cpw cvv w3a istrkgb simprbi + sstri simp2d oveq1 eleq2d anbi1d oveq2 rexbidv imbi12d 2ralbidv anbi12d + syl anbi2d imbi1d rspc3v syl3anc mpd eleq1 rspc2v syl2anc mp2and ) ACHJ + EUNZUOZGIJEUNZUOZKUPZCIEUNZUOZXLGHEUNZUOZUQZKBURZUAUBAUCUPZXHUOZUDUPZXJ + UOZUQZXLXSIEUNZUOZXLYAHEUNZUOZUQZKBURZUSZUDBVBUCBVBZXIXKUQZXRUSZAXSUEUP + ZUFUPZEUNZUOZYAUGUPZYOEUNZUOZUQZXLXSYREUNZUOZXLYAYNEUNZUOZUQZKBURZUSZUD + BVBUCBVBZUFBVBUGBVBUEBVBZYKADUTUOZUUJAVAUTDVAVCUTVDZVEUHUPZVFVGUEUGUIUP + ZUUNYNVHVIYOYNYRUJUPZUNUOYNYOYRUUOUNUOYRYNYOUUOUNUOVJUFUUNVKVLVMUJUUMVN + VGVPUIUUMVOVGVPUHVQVDZVDZUTUEUGUFUHUJUIVRUUQUULUTUULUUPVSVCUTVTWHWAOWBU + UKYRYNYNEUNUOYNYRVMUSUGBVBUEBVBZUUJYNXLYREUNUOUGUKUPZVBUEULUPZVBKBURUMU + PYNYREUNUOUGUUSVBUEUUTVBUMBURUSUKBWCZVBULUVAVBZUUKDWDUOUURUUJUVBWEUEUGU + FUDUCUKBDEFULKUMLMNWFWGWIWRAHBUOIBUOJBUOUUJYKUSPQRUUIYKXSHYOEUNZUOZYTUQ + ZUUCYGUQZKBURZUSZUDBVBUCBVBUVDYAIYOEUNZUOZUQZYIUSZUDBVBUCBVBUEUGUFHIJBB + BYNHVMZUUHUVHUCUDBBUVMUUAUVEUUGUVGUVMYQUVDYTUVMYPUVCXSYNHYOEWJWKWLUVMUU + FUVFKBUVMUUEYGUUCUVMUUDYFXLYNHYAEWMWKWSWNWOWPYRIVMZUVHUVLUCUDBBUVNUVEUV + KUVGYIUVNYTUVJUVDUVNYSUVIYAYRIYOEWJWKWSUVNUVFYHKBUVNUUCYEYGUVNUUBYDXLYR + IXSEWMWKWLWNWOWPYOJVMZUVLYJUCUDBBUVOUVKYCYIUVOUVDXTUVJYBUVOUVCXHXSYOJHE + WMWKUVOUVIXJYAYOJIEWMWKWQWTWPXAXBXCACBUOGBUOYKYMUSSTYJYMXIYBUQZXNYGUQZK + BURZUSUCUDCGBBXSCVMZYCUVPYIUVRUVSXTXIYBXSCXHXDWLUVSYHUVQKBUVSYEXNYGUVSY + DXMXLXSCIEWJWKWLWNWOYAGVMZUVPYLUVRXRUVTYBXKXIYAGXJXDWSUVTUVQXQKBUVTYGXP + XNUVTYFXOXLYAGHEWJWKWSWNWOXEXFXCXG $. + $} + + ${ + axtgcont.1 $e |- ( ph -> S C_ P ) $. + axtgcont.2 $e |- ( ph -> T C_ P ) $. + $( Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. This axiom + (scheme) asserts that any two sets ` S ` and ` T ` (of points) such + that the elements of ` S ` precede the elements of ` T ` with respect + to some point ` a ` (that is, ` x ` is between ` a ` and ` y ` + whenever ` x ` is in ` X ` and ` y ` is in ` Y ` ) are separated by + some point ` b ` ; this is explained in Axiom 11 of [Tarski1999] + p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.) $) + axtgcont1 $p |- ( ph -> + ( E. a e. P A. x e. S A. y e. T x e. ( a I y ) + -> E. b e. P A. x e. S A. y e. T b e. ( x I y ) ) ) $= + ( cv wcel wral vt vs vf vp vz vi vu vv co wi cpw cstrkgb cstrkg cstrkgc + wrex cin cstrkgcb clng cfv csn cdif w3o crab cmpo wceq citv cbs df-trkg + wsbc cab inss1 inss2 sstri eqsstri sselid wa cvv istrkgb simprbi simp3d + w3a syl wss wb fvexi ssex elpwg mpbird raleq rexbidv imbi12d rexralbidv + 3syl rspc2v syl2anc mpd ) ABRZJRZCRZHUISZCUARZTZBUBRZTZJDUOZKRWQWSHUISZ + CXATZBXCTZKDUOZUJZUADUKZTUBXKTZWTCFTZBETJDUOZXFCFTZBETKDUOZUJZAGULSZXLA + UMULGUMUNULUPZUQUCRZURUSBCUDRZYAWQUTVAUERZWQWSUFRZUISWQYBWSYCUISWSWQYBY + CUISVBUEYAVCVDVEUFXTVFUSVIUDXTVGUSVIUCVJUPZUPZULBCUEUCUFUDVHYEXSULXSYDV + KUNULVLVMVNOVOXRWSWQWQHUISWQWSVEUJCDTBDTZUGRZWQYBHUISUHRZWSYBHUISVPWRYG + WSHUISWRYHWQHUISVPJDUOUJUHDTUGDTUEDTCDTBDTZXLXRGVQSYFYIXLWABCUEUHUGUADG + HIUBJKLMNVRVSVTWBAEXKSZFXKSZXLXQUJAYJEDWCZPAYLEVQSYJYLWDPEDDGVGLWEZWFED + VQWGWMWHAYKFDWCZQAYNFVQSYKYNWDQFDYMWFFDVQWGWMWHXJXQXBBETZJDUOZXGBETZKDU + OZUJUBUAEFXKXKXCEVEZXEYPXIYRYSXDYOJDXBBXCEWIWJYSXHYQKDXGBXCEWIWJWKXAFVE + ZYPXNYRXPYTXBXMJBDEWTCXAFWIWLYTXGXOKBDEXFCXAFWIWLWKWNWOWP $. + + $d u v ph $. $d u v S $. $d u v T $. $d u v x y A $. + axtgcont.3 $e |- ( ph -> A e. P ) $. + axtgcont.4 $e |- ( ( ph /\ u e. S /\ v e. T ) -> u e. ( A I v ) ) $. + $( Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more + information see ~ axtgcont1 . (Contributed by Thierry Arnoux, + 16-Mar-2019.) $) + axtgcont $p |- ( ph -> E. b e. P A. x e. S A. y e. T b e. ( x I y ) ) $= + ( va cv co wcel wral wrex 3expb ralrimivva wceq weq simplr simpll simpr + wa oveq12d eleq12d cbvraldva rspcev syl2anc axtgcont1 mpd ) ABUCZUBUCZC + UCZKUDZUEZCIUFZBHUFZUBGUGZMUCVCVEKUDUECIUFBHUFMGUGAFGUEEUCZFDUCZKUDZUEZ + DIUFZEHUFZVJTAVNEDHIAVKHUEVLIUEVNUAUHUIVIVPUBFGVDFUJZVHVOBEHVQBEUKZUOZV + GVNCDIVSCDUKZUOZVCVKVFVMVQVRVTULWAVDFVEVLKVQVRVTUMVSVTUNUPUQURURUSUTABC + GHIJKLUBMNOPQRSVAVB $. + $} + $} + + ${ + $d u v x y z .- $. $d u v x y z I $. $d u v x y z P $. + axtrkge.p $e |- P = ( Base ` G ) $. + axtrkge.d $e |- .- = ( dist ` G ) $. + axtrkge.i $e |- I = ( Itv ` G ) $. + ${ + axtglowdim2.v $e |- ( ph -> G e. V ) $. + axtglowdim2.g $e |- ( ph -> G TarskiGDim>= 2 ) $. + $( Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] + p. 13. There exist 3 non-colinear points. (Contributed by Thierry + Arnoux, 20-Nov-2019.) $) + axtglowdim2 $p |- ( ph -> E. x e. P E. y e. P E. z e. P + -. ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) ) $= + ( c2 cstrkgld cv co wcel wrex wbr w3o wn wb istrkg2ld syl mpbid ) AFOPU + AZDQZBQZCQZGRSUJUIUKGRSUKUJUIGRSUBUCDETCETBETZNAFISUHULUDMBCDEFGHIJKLUE + UFUG $. + $} + + ${ + $d u v x y z .- $. $d u v x y z I $. $d u v x y z P $. $d u v z Z $. + $d u v x y z X $. $d u v y z Y $. $d u v x y z U $. $d v x y z V $. + axtgupdim2.x $e |- ( ph -> X e. P ) $. + axtgupdim2.y $e |- ( ph -> Y e. P ) $. + axtgupdim2.z $e |- ( ph -> Z e. P ) $. + axtgupdim2.u $e |- ( ph -> U e. P ) $. + axtgupdim2.v $e |- ( ph -> V e. P ) $. + axtgupdim2.0 $e |- ( ph -> U =/= V ) $. + axtgupdim2.1 $e |- ( ph -> ( U .- X ) = ( V .- X ) ) $. + axtgupdim2.2 $e |- ( ph -> ( U .- Y ) = ( V .- Y ) ) $. + axtgupdim2.3 $e |- ( ph -> ( U .- Z ) = ( V .- Z ) ) $. + axtgupdim2.w $e |- ( ph -> G e. V ) $. + axtgupdim2.g $e |- ( ph -> -. G TarskiGDim>= 3 ) $. + $( Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] + p. 13. Three points ` X ` , ` Y ` and ` Z ` equidistant to two given + two points ` U ` and ` V ` must be colinear. (Contributed by Thierry + Arnoux, 29-May-2019.) (Revised by Thierry Arnoux, 11-Jul-2020.) $) + axtgupdim2 $p |- ( ph -> + ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) $= + ( vx vy vz vu vv co wcel w3o wceq wn w3a wa wi cv wral wrex c3 cstrkgld + wne wbr istrkg3ld syl mtbid ralnex2 sylibr neeq1 oveq1 eqeq1d 3anbi123d + wb anbi1d rexbidv 2rexbidv anbi12d notbid neeq2 eqeq2d rspc2v mpd imnan + syl2anc ralnex3 eqeq12d 3anbi1d eleq2d 3orbi123d 3anbi2d 3anbi3d rspc3v + oveq2 eleq1 syl3anc mp3and notnotrd ) AJHIEUJZUKZHJIEUJZUKZIHJEUJZUKZUL + ZACHFUJZGHFUJZUMZCIFUJZGIFUJZUMZCJFUJZGJFUJZUMZXEUNZUNZTUAUBAXHXKXNUOZX + OUPZUNZXQXPUQACUEURZFUJZGXTFUJZUMZCUFURZFUJZGYDFUJZUMZCUGURZFUJZGYHFUJZ + UMZUOZYHXTYDEUJZUKZXTYHYDEUJZUKZYDXTYHEUJZUKZULZUNZUPZUNZUGBUSUFBUSUEBU + SZXSAUUAUGBUTZUFBUTUEBUTZUNZUUCACGVCZUUFSAUUGUUEUPZUNZUUGUUFUQAUHURZUIU + RZVCZUUJXTFUJZUUKXTFUJZUMZUUJYDFUJZUUKYDFUJZUMZUUJYHFUJZUUKYHFUJZUMZUOZ + YTUPZUGBUTZUFBUTUEBUTZUPZUNZUIBUSUHBUSZUUIAUVFUIBUTUHBUTZUNUVHADVAVBVDZ + UVIUDADGUKUVJUVIVNUCUEUFUGUIUHBDEFGKLMVEVFVGUVFUHUIBBVHVIACBUKGBUKUVHUU + IUQQRUVGUUICUUKVCZYAUUNUMZYEUUQUMZYIUUTUMZUOZYTUPZUGBUTZUFBUTUEBUTZUPZU + NUHUICGBBUUJCUMZUVFUVSUVTUULUVKUVEUVRUUJCUUKVJUVTUVDUVQUEUFBBUVTUVCUVPU + GBUVTUVBUVOYTUVTUUOUVLUURUVMUVAUVNUVTUUMYAUUNUUJCXTFVKVLUVTUUPYEUUQUUJC + YDFVKVLUVTUUSYIUUTUUJCYHFVKVLVMVOVPVQVRVSUUKGUMZUVSUUHUWAUVKUUGUVRUUEUU + KGCVTUWAUVQUUDUEUFBBUWAUVPUUAUGBUWAUVOYLYTUWAUVLYCUVMYGUVNYKUWAUUNYBYAU + UKGXTFVKWAUWAUUQYFYEUUKGYDFVKWAUWAUUTYJYIUUKGYHFVKWAVMVOVPVQVRVSWBWEWCU + UGUUEWDVIWCUUAUEUFUGBBBWFVIAHBUKIBUKJBUKUUCXSUQNOPUUBXSXHYGYKUOZYHHYDEU + JZUKZHYOUKZYDHYHEUJZUKZULZUNZUPZUNXHXKYKUOZYHWSUKZHYHIEUJZUKZIUWFUKZULZ + UNZUPZUNUEUFUGHIJBBBXTHUMZUUAUWJUWSYLUWBYTUWIUWSYCXHYGYKUWSYAXFYBXGXTHC + FWNXTHGFWNWGWHUWSYSUWHUWSYNUWDYPUWEYRUWGUWSYMUWCYHXTHYDEVKWIXTHYOWOUWSY + QUWFYDXTHYHEVKWIWJVSVRVSYDIUMZUWJUWRUWTUWBUWKUWIUWQUWTYGXKXHYKUWTYEXIYF + XJYDICFWNYDIGFWNWGWKUWTUWHUWPUWTUWDUWLUWEUWNUWGUWOUWTUWCWSYHYDIHEWNWIUW + TYOUWMHYDIYHEWNWIYDIUWFWOWJVSVRVSYHJUMZUWRXRUXAUWKXQUWQXOUXAYKXNXHXKUXA + YIXLYJXMYHJCFWNYHJGFWNWGWLUXAUWPXEUXAUWLWTUWNXBUWOXDYHJWSWOUXAUWMXAHYHJ + IEVKWIUXAUWFXCIYHJHEWNWIWJVSVRVSWMWPWCXQXOWDVIWQWR $. + $} + + ${ + $d a b u v x y z I $. $d a b u v x y z P $. $d a b v V $. + $d a b u v U $. $d a b u v x y z X $. $d a b u v y z Y $. + $d a b u v z Z $. $d a b u v x y z .- $. + axtgeucl.g $e |- ( ph -> G e. TarskiGE ) $. + axtgeucl.1 $e |- ( ph -> X e. P ) $. + axtgeucl.2 $e |- ( ph -> Y e. P ) $. + axtgeucl.3 $e |- ( ph -> Z e. P ) $. + axtgeucl.4 $e |- ( ph -> U e. P ) $. + axtgeucl.5 $e |- ( ph -> V e. P ) $. + axtgeucl.6 $e |- ( ph -> U e. ( X I V ) ) $. + axtgeucl.7 $e |- ( ph -> U e. ( Y I Z ) ) $. + axtgeucl.8 $e |- ( ph -> X =/= U ) $. + $( Euclid's Axiom. Axiom A10 of [Schwabhauser] p. 13. This is + equivalent to Euclid's parallel postulate when combined with other + axioms. (Contributed by Thierry Arnoux, 16-Mar-2019.) $) + axtgeucl $p |- ( ph -> E. a e. P E. b e. P + ( Y e. ( X I a ) /\ Z e. ( X I b ) /\ V e. ( a I b ) ) ) $= + ( vu vv vx vy vz co wcel wne cv w3a wrex wi wral cvv cstrkge wa istrkge + sylib simprd wceq oveq1 eleq2d neeq1 3anbi13d 3anbi12d 2rexbidv imbi12d + 2ralbidv 3anbi2d eleq1 3anbi1d oveq2 rspc3v syl3anc mpd neeq2 3anbi123d + imbi1d 3anbi3d rspc2v syl2anc mp3and ) ACHGEUJZUKZCIJEUJZUKZHCULZIHKUMZ + EUJZUKZJHLUMZEUJZUKZGWLWOEUJZUKZUNZLBUOKBUOZUBUCUDAUEUMZHUFUMZEUJZUKZXB + WIUKZHXBULZUNZWNWQXCWRUKZUNZLBUOKBUOZUPZUFBUQUEBUQZWHWJWKUNZXAUPZAXBUGU + MZXCEUJZUKZXBUHUMZUIUMZEUJZUKZXPXBULZUNZXSXPWLEUJZUKZXTXPWOEUJZUKZXIUNZ + LBUOKBUOZUPZUFBUQUEBUQZUIBUQUHBUQUGBUQZXMADURUKZYMADUSUKYNYMUTPUGUHUIUF + UEBDEFKLMNOVAVBVCAHBUKIBUKJBUKYMXMUPQRSYLXMXEYBXGUNZXSWMUKZXTWPUKZXIUNZ + LBUOKBUOZUPZUFBUQUEBUQXEXBIXTEUJZUKZXGUNZWNYQXIUNZLBUOKBUOZUPZUFBUQUEBU + QUGUHUIHIJBBBXPHVDZYKYTUEUFBBUUGYDYOYJYSUUGXRXEYCXGYBUUGXQXDXBXPHXCEVEV + FXPHXBVGVHUUGYIYRKLBBUUGYFYPYHYQXIUUGYEWMXSXPHWLEVEVFUUGYGWPXTXPHWOEVEV + FVIVJVKVLXSIVDZYTUUFUEUFBBUUHYOUUCYSUUEUUHYBUUBXEXGUUHYAUUAXBXSIXTEVEVF + VMUUHYRUUDKLBBUUHYPWNYQXIXSIWMVNVOVJVKVLXTJVDZUUFXLUEUFBBUUIUUCXHUUEXKU + UIUUBXFXEXGUUIUUAWIXBXTJIEVPVFVMUUIUUDXJKLBBUUIYQWQWNXIXTJWPVNVMVJVKVLV + QVRVSACBUKGBUKXMXOUPTUAXLXOCXDUKZWJWKUNZXKUPUEUFCGBBXBCVDZXHUUKXKUULXEU + UJXFWJXGWKXBCXDVNXBCWIVNXBCHVTWAWBXCGVDZUUKXNXKXAUUMUUJWHWJWKUUMXDWGCXC + GHEVPVFVOUUMXJWTKLBBUUMXIWSWNWQXCGWRVNWCVJVKWDWEVSWF $. + $} + $} + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Justification for the congruence notation +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + ${ + $d A r u v x y z $. $d F r u v x y z $. $d V u v $. + $( Given any function ` F ` , equality of the image by ` F ` is an + equivalence relation. (Contributed by Thierry Arnoux, 25-Jan-2023.) $) + tgjustf $p |- ( A e. V -> E. r ( r Er A + /\ A. x e. A A. y e. A ( x r y <-> ( F ` x ) = ( F ` y ) ) ) ) $= + ( vu vv wcel cv wa cfv wceq wbr weq simpl fveq2d simpr eqeq12d brab2a wer + vz copab wb wral relopabv ancom eqcom anbi12i eqid 3bitr4i biimpi simplll + simprlr simplr simprr eqtrd jca31 3imtr4i biantru pm4.24 iseri baib rgen2 + wex cvv wi id simprll opabex2 ereq1 breqd bibi1d 2ralbidva anbi12d spcegv + syl mp2ani ) CEIZCGJZCIZHJZCIZKVTDLZWBDLZMZKZGHUCZUAZAJZBJZWHNZWJDLZWKDLZ + MZUDZBCUEACUEZCFJZUAZWJWKWRNZWOUDZBCUEACUEZKZFVEZABUBCWHWGGHUFWLWKWJWHNZW + JCIZWKCIZKZWOKZXGXFKZWNWMMZKWLXEXHXJWOXKXFXGUGWMWNUHUIWFWOGHWJWKCCWHGAOZH + BOZKZWDWMWEWNXNVTWJDXLXMPQXNWBWKDXLXMRQSWHUJZTZWFXKGHWKWJCCWHGBOZHAOZKZWD + WNWEWMXSVTWKDXQXRPQXSWBWJDXQXRRQSXOTUKULXIXGUBJZCIZKZWNXTDLZMZKZKZXFYAKWM + YCMZKWLWKXTWHNZKWJXTWHNYFXFYAYGXFXGWOYEUMXIXGYAYDUNYFWMWNYCXHWOYEUOXIYBYD + UPUQURWLXIYHYEXPWFYDGHWKXTCCWHXQHUBOZKZWDWNWEYCYJVTWKDXQYIPQYJWBXTDXQYIRQ + SXOTUIWFYGGHWJXTCCWHXLYIKZWDWMWEYCYKVTWJDXLYIPQYKWBXTDXLYIRQSXOTUSXFXFKZY + LWMWMMZKXFWJWJWHNYMYLWMUJUTXFVAWFYMGHWJWJCCWHXLXRKZWDWMWEWMYNVTWJDXLXRPQY + NWBWJDXLXRRQSXOTUKVBWPABCCWLXHWOXPVCVDVSWHVFIWIWQKZXDVGVSWGGHCCEEVSVHZYPV + SWAWCWFVIVSWAWCWFUNVJXCYOFWHVFWRWHMZWSWIXBWQCWRWHVKYQXAWPABCCYQXHKZWTWLWO + YRWRWHWJWKYQXHPVLVMVNVOVPVQVR $. + $} + + ${ + $d A f u x y $. $d R f u x y $. $d V u x y $. + $( Given any equivalence relation ` R ` , one can define a function ` f ` + such that all elements of an equivalence classe of ` R ` have the same + image by ` f ` . (Contributed by Thierry Arnoux, 25-Jan-2023.) $) + tgjustr $p |- ( ( A e. V /\ R Er A ) -> E. f ( f Fn A + /\ A. x e. A A. y e. A ( x R y <-> ( f ` x ) = ( f ` y ) ) ) ) $= + ( vu wcel wa cv cec wfn cfv wceq wb wral cvv syl adantr ralrimiva wer wbr + cmpt wex erex ecexg eqid fnmpt simpllr simpr erth cqs eceq1 ecelqsg sylan + impcom fvmptd3 adantlr eqeq12d bitr4d wi mptexg fneq1 simpl fveq1d bibi2d + 2ralbidva anbi12d spcegv mp2and ) CFHZCDUAZIZGCGJZDKZUCZCLZAJZBJZDUBZVRVP + MZVSVPMZNZOZBCPZACPZEJZCLZVTVRWGMZVSWGMZNZOZBCPACPZIZEUDZVMVOQHZGCPVQVMWP + GCVMWPVNCHVMDQHZWPVLVKWQCDFUEUPZVNQDUFRSTGCVOVPQVPUGZUHRVMWEACVMVRCHZIZWD + BCXAVSCHZIZVTVRDKZVSDKZNWCXCVRVSDCVKVLWTXBUIXAWTXBVMWTUJZSUKXCWAXDWBXEXAW + AXDNXBXAGVRVOXDCVPCDULZWSVNVRDUMXFVMWQWTXDXGHWRCVRDQUNUOUQSVMXBWBXENWTVMX + BIGVSVOXECVPXGWSVNVSDUMVMXBUJVMWQXBXEXGHWRCVSDQUNUOUQURUSUTTTVMVPQHZVQWFI + ZWOVAVKXHVLGCVOFVBSWNXIEVPQWGVPNZWHVQWMWFCWGVPVCXJWLWDABCCXJWTXBIZIZWKWCV + TXLWIWAWJWBXLVRWGVPXJXKVDZVEXLVSWGVPXMVEUSVFVGVHVIRVJ $. + $} + + ${ + $d .- r u v w x y z $. $d P r u v w x y z $. + tgjustc1.p $e |- P = ( Base ` G ) $. + tgjustc1.d $e |- .- = ( dist ` G ) $. + $( A justification for using distance equality instead of the textbook + relation on pairs of points for congruence. (Contributed by Thierry + Arnoux, 29-Jan-2023.) $) + tgjustc1 $p |- E. r ( r Er ( P X. P ) + /\ A. w e. P A. x e. P A. y e. P A. z e. P + ( <. w , x >. r <. y , z >. <-> ( w .- x ) = ( y .- z ) ) ) $= + ( vu vv cv wbr cfv wceq wb wral wa wcel cxp wer cop co cvv wex fvexi xpex + cbs tgjustf ax-mp simplrl simplrr opelxpd simprl simprr breq1 fveq2 df-ov + simpll eqtr4di eqeq1d bibi12d breq2 eqeq2d rspc2va syl21anc anim2i eximii + ralrimivva ) EEUAZHMZUBZKMZLMZVLNZVNGOZVOGOZPZQZLVKRKVKRZSZVMDMZAMZUCZBMZ + CMZUCZVLNZWCWDGUDZWFWGGUDZPZQZCERBERZAERDERZSHVKUETWBHUFEEEFUIIUGZWPUHKLV + KGUEHUJUKWAWOVMWAWNDAEEWAWCETZWDETZSZSZWMBCEEWTWFETZWGETZSZSZWEVKTWHVKTWA + WMXDWCWDEEWAWQWRXCULWAWQWRXCUMUNXDWFWGEEWTXAXBUOWTXAXBUPUNWAWSXCUTVTWMWEV + OVLNZWJVRPZQKLWEWHVKVKVNWEPZVPXEVSXFVNWEVOVLUQXGVQWJVRXGVQWEGOWJVNWEGURWC + WDGUSVAVBVCVOWHPZXEWIXFWLVOWHWEVLVDXHVRWKWJXHVRWHGOWKVOWHGURWFWGGUSVAVEVC + VFVGVJVJVHVI $. + $} + + ${ + $d P d u v w x y z $. $d R d u v w x y z $. + tgjustc2.p $e |- P = ( Base ` G ) $. + tgjustc2.d $e |- R Er ( P X. P ) $. + $( A justification for using distance equality instead of the textbook + relation on pairs of points for congruence. (Contributed by Thierry + Arnoux, 29-Jan-2023.) $) + tgjustc2 $p |- E. d ( d Fn ( P X. P ) + /\ A. w e. P A. x e. P A. y e. P A. z e. P + ( <. w , x >. R <. y , z >. <-> ( w d x ) = ( y d z ) ) ) $= + ( vu vv cv wbr cfv wceq wb wral wa wcel cxp wfn cop cvv wer wex cbs fvexi + co tgjustr mp2an simplrl simplrr opelxpd simprl simprr simpll breq1 fveq2 + xpex df-ov eqtr4di eqeq1d bibi12d breq2 eqeq2d syl21anc ralrimivva anim2i + rspc2va eximii ) HMZEEUAZUBZKMZLMZFNZVOVLOZVPVLOZPZQZLVMRKVMRZSZVNDMZAMZU + CZBMZCMZUCZFNZWDWEVLUIZWGWHVLUIZPZQZCERBERZAERDERZSHVMUDTVMFUEWCHUFEEEGUG + IUHZWQUTJKLVMFHUDUJUKWBWPVNWBWODAEEWBWDETZWEETZSZSZWNBCEEXAWGETZWHETZSZSZ + WFVMTWIVMTWBWNXEWDWEEEWBWRWSXDULWBWRWSXDUMUNXEWGWHEEXAXBXCUOXAXBXCUPUNWBW + TXDUQWAWNWFVPFNZWKVSPZQKLWFWIVMVMVOWFPZVQXFVTXGVOWFVPFURXHVRWKVSXHVRWFVLO + WKVOWFVLUSWDWEVLVAVBVCVDVPWIPZXFWJXGWMVPWIWFFVEXIVSWLWKXIVSWIVLOWLVPWIVLU + SWGWHVLVAVBVFVDVJVGVHVHVIVK $. + $} + + +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Tarskian Geometry +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Congruence +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + ${ + tkgeom.p $e |- P = ( Base ` G ) $. + tkgeom.d $e |- .- = ( dist ` G ) $. + tkgeom.i $e |- I = ( Itv ` G ) $. + tkgeom.g $e |- ( ph -> G e. TarskiG ) $. + + ${ + tgcgrcomimp.a $e |- ( ph -> A e. P ) $. + tgcgrcomimp.b $e |- ( ph -> B e. P ) $. + tgcgrcomimp.c $e |- ( ph -> C e. P ) $. + tgcgrcomimp.d $e |- ( ph -> D e. P ) $. + $( Congruence commutes on the RHS. Theorem 2.5 of [Schwabhauser] p. 27. + (Contributed by David A. Wheeler, 29-Jun-2020.) $) + tgcgrcomimp $p |- ( ph -> + ( ( A .- B ) = ( C .- D ) -> ( A .- B ) = ( D .- C ) ) ) $= + ( co wceq axtgcgrrflx eqeq2d biimpd ) ABCIRZDEIRZSUCEDIRZSAUDUEUCAFGHID + EJKLMPQTUAUB $. + $} + + ${ + tgcgrcomr.a $e |- ( ph -> A e. P ) $. + tgcgrcomr.b $e |- ( ph -> B e. P ) $. + tgcgrcomr.c $e |- ( ph -> C e. P ) $. + tgcgrcomr.d $e |- ( ph -> D e. P ) $. + tgcgrcomr.6 $e |- ( ph -> ( A .- B ) = ( C .- D ) ) $. + $( Congruence commutes on the RHS. Variant of Theorem 2.5 of + [Schwabhauser] p. 27, but in a convenient form for a common case. + (Contributed by David A. Wheeler, 29-Jun-2020.) $) + tgcgrcomr $p |- ( ph -> ( A .- B ) = ( D .- C ) ) $= + ( co axtgcgrrflx eqtrd ) ABCISDEISEDISRAFGHIDEJKLMPQTUA $. + + $( Congruence commutes on the LHS. Variant of Theorem 2.5 of + [Schwabhauser] p. 27, but in a convenient form for a common case. + (Contributed by David A. Wheeler, 29-Jun-2020.) $) + tgcgrcoml $p |- ( ph -> ( B .- A ) = ( C .- D ) ) $= + ( co axtgcgrrflx eqtr3d ) ABCISCBISDEISAFGHIBCJKLMNOTRUA $. + $} + + ${ + tgcgrcomlr.a $e |- ( ph -> A e. P ) $. + tgcgrcomlr.b $e |- ( ph -> B e. P ) $. + tgcgrcomlr.c $e |- ( ph -> C e. P ) $. + tgcgrcomlr.d $e |- ( ph -> D e. P ) $. + tgcgrcomlr.6 $e |- ( ph -> ( A .- B ) = ( C .- D ) ) $. + $( Congruence commutes on both sides. (Contributed by Thierry Arnoux, + 23-Mar-2019.) $) + tgcgrcomlr $p |- ( ph -> ( B .- A ) = ( D .- C ) ) $= + ( co axtgcgrrflx 3eqtr3d ) ABCISDEISCBISEDISRAFGHIBCJKLMNOTAFGHIDEJKLMP + QTUA $. + + $( Congruence and equality. (Contributed by Thierry Arnoux, + 27-Aug-2019.) $) + tgcgreqb $p |- ( ph -> ( A = B <-> C = D ) ) $= + ( wcel adantr wa cstrkg co simpr oveq1d eqtr3d axtgcgrid eqtrd impbida + wceq ) ABCUJZDEUJZAUKUAZFGHIDECJKLAGUBSZUKMTADFSUKPTAEFSZUKQTACFSZUKOTU + MBCIUCZDEIUCZCCIUCAUQURUJZUKRTUMBCCIAUKUDUEUFUGAULUAZFGHIBCEJKLAUNULMTA + BFSULNTAUPULOTAUOULQTUTUQUREEIUCAUSULRTUTDEEIAULUDUEUHUGUI $. + + ${ + tgcgreq.1 $e |- ( ph -> A = B ) $. + $( Congruence and equality. (Contributed by Thierry Arnoux, + 27-Aug-2019.) $) + tgcgreq $p |- ( ph -> C = D ) $= + ( wceq tgcgreqb mpbid ) ABCTDETSABCDEFGHIJKLMNOPQRUAUB $. + $} + + ${ + tgcgrneq.1 $e |- ( ph -> A =/= B ) $. + $( Congruence and equality. (Contributed by Thierry Arnoux, + 27-Aug-2019.) $) + tgcgrneq $p |- ( ph -> C =/= D ) $= + ( wne tgcgreqb necon3bid mpbid ) ABCTDETSABCDEABCDEFGHIJKLMNOPQRUAUBU + C $. + $} + $} + + ${ + $d x .- $. $d x A $. $d x B $. $d x I $. $d x P $. $d x ph $. + tgcgrtriv.1 $e |- ( ph -> A e. P ) $. + tgcgrtriv.2 $e |- ( ph -> B e. P ) $. + $( Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] + p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) $) + tgcgrtriv $p |- ( ph -> ( A .- A ) = ( B .- B ) ) $= + ( vx cv co wcel wceq wa ad2antrr cstrkg simplr simprr oveq2d axtgsegcon + axtgcgrid eqtrd r19.29a ) ABCNOZFPQZBUIGPZCCGPZRZSZBBGPZULRNDAUIDQZSZUN + SZUOUKULURBUIBGURDEFGBUICHIJAEUAQUPUNKTABDQUPUNLTAUPUNUBACDQUPUNMTUQUJU + MUCZUFUDUSUGANCCDEFGCBHIJKMLMMUEUH $. + $} + + ${ + tgcgrextend.a $e |- ( ph -> A e. P ) $. + tgcgrextend.b $e |- ( ph -> B e. P ) $. + tgcgrextend.c $e |- ( ph -> C e. P ) $. + tgcgrextend.d $e |- ( ph -> D e. P ) $. + tgcgrextend.e $e |- ( ph -> E e. P ) $. + tgcgrextend.f $e |- ( ph -> F e. P ) $. + ${ + tgcgrextend.1 $e |- ( ph -> B e. ( A I C ) ) $. + tgcgrextend.2 $e |- ( ph -> E e. ( D I F ) ) $. + tgcgrextend.3 $e |- ( ph -> ( A .- B ) = ( D .- E ) ) $. + tgcgrextend.4 $e |- ( ph -> ( B .- C ) = ( E .- F ) ) $. + $( Link congruence over a pair of line segments. Theorem 2.11 of + [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) + (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.) $) + tgcgrextend $p |- ( ph -> ( A .- C ) = ( D .- F ) ) $= + ( co wceq wa adantr simpr oveq1d cstrkg tgcgreq 3eqtr4d wne tgcgrtriv + wcel tgcgrcomlr axtg5seg pm2.61dane ) ABDKUFZEHKUFZUGBCABCUGZUHZCDKUF + ZGHKUFZVAVBAVEVFUGZVCUEUIVDBCDKAVCUJZUKVDEGHKVDBCEGFIJKLMNAIULUQZVCOU + IABFUQZVCPUIACFUQZVCQUIAEFUQZVCSUIAGFUQZVCTUIABCKUFEGKUFUGZVCUDUIVHUM + UKUNABCUOZUHZDBHEFIJKLMNAVIVOOUIZADFUQVORUIZAVJVOPUIZAHFUQVOUAUIZAVLV + OSUIZVPEGHFBIJKEBCDLMNVQVSAVKVOQUIZVRWAAVMVOTUIZVTVSWAAVOUJACBDJUFUQV + OUBUIAGEHJUFUQVOUCUIAVNVOUDUIZAVGVOUEUIVPBEFIJKLMNVQVSWAUPVPBCEGFIJKL + MNVQVSWBWAWCWDURUSURUT $. + $} + + ${ + tgsegconeq.1 $e |- ( ph -> D =/= A ) $. + tgsegconeq.2 $e |- ( ph -> A e. ( D I E ) ) $. + tgsegconeq.3 $e |- ( ph -> A e. ( D I F ) ) $. + tgsegconeq.4 $e |- ( ph -> ( A .- E ) = ( B .- C ) ) $. + tgsegconeq.5 $e |- ( ph -> ( A .- F ) = ( B .- C ) ) $. + $( Two points that satisfy the conclusion of ~ axtgsegcon are + identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] + p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) $) + tgsegconeq $p |- ( ph -> E = F ) $= + ( co eqidd eqtr4d tgcgrextend axtg5seg eqcomd axtgcgrid ) AFIJKGHGLMN + OTUATAGGKUGGHKUGAEBGFGIJKHEBGLMNOSPTSPTTUAUBUCUCAEBKUGUHZABGKUGZUHAEB + GEFBHIJKLMNOSPTSPUAUCUDUNAUOCDKUGBHKUGUEUFUIZUJUPUKULUM $. + $} + $} + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Betweenness =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) @@ -407637,7 +412495,6 @@ and angles (6 parts). This is often the actual textbook definition of ZUDUELZMUDINGZLUDIRGZLMZMUHOUGUHUKPQSITGZUEULUFLZULUELZMULUILULUJLMZMUN UAUMUNUOPQSUBUC $. - $( Obsolete proof of ~ ttgplusg as of 29-Oct-2024. The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) @@ -408613,51 +413470,51 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to necon3bid syl3anc bitrid resubcld oveq1 ralimi ralbi syl oveq2 wo df-3or andir subadd2 bicomd div23d eqeq2d divmuld mulcomd eqeq1d ralbidva 3simpb 3bitr2d simpl2 simpl1 simpl3 3ad2ant1 redivcld colinearalglem4 syl5ibrcom - recnd biimpar sylbird syldan syld syl5bi sylbid pm2.61dne pm4.71rd orbi1i - rexlimdvaa anbi1i bitri 3bitr4i bitr2di ) AFUAHZIZBUVMIZCUVMIZJZABCUBUEKZ - BCAUBUEKZCABUBUEKZUCDUDZBHZUWAAHZLMZUWACHZUWCLMZNMZOPKZDUFFVOMZQZUWDEUDZC - HZUWKAHZLMZNMZUWKBHZUWMLMZUWFNMZRZEUWIQDUWIQZSZUWEUWBLMZUWCUWBLMZNMZOPKZD - UWIQZUWTSZUWCUWELMZUWBUWELMZNMZOPKZDUWIQZUWTSZUCZUWTUVQUVRUXAUVSUXGUVTUXM - ABCDEFUGUVQUVSUXFUXBUWMUWPLMNMUWLUWPLMUXCNMREUWIQDUWIQZSZUXGUVOUVPUVNUVSU - XPTBCADEFUGUHUVQUXOUWTUXFUVOUVPUVNUXOUWTTBCADEFUIUHUJUKUVPUVNUVOUVTUXMTUV - PUVNUVOJZUVTUXLUXHUWPUWLLMNMUWMUWLLMUXINMREUWIQDUWIQZSUXMCABDEFUGUXQUXRUW - TUXLCABDEFULUJUKUMUNUVQUWTUWJUXFUXLUCZUWTSZUXNUVQUWTUXSUVQUWTUXSUOZCAUVQC - ARZUXSUWTUVQUYBUWJUXSUVQUXIUWEUWELMZNMZOPKZDUWIQZUYBUWJUVOUVPUYFUVNUVOUVP - 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This proof indirectly refers to ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) } $. $( Lemma for ~ axcont . The idea here is to set up a mapping ` F ` that - will allow us to transfer ~ dedekind to two sets of points. Here, we - set up ` F ` and show its domain and range. (Contributed by Scott + will allow to transfer ~ dedekind to two sets of points. Here, we set + up ` F ` and show its domain and codomain. (Contributed by Scott Fenton, 17-Jun-2013.) $) axcontlem2 $p |- ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) -> @@ -410664,117 +415521,117 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to ralbi rexbidv syldan wss 3simpa 3imtr4i ssriv ssrexv mpsyl jaodan sylan2b anasss r19.26 sylbir anim12i simpl2 ad2ant2rl ad2ant2l addsubeq4d nnncan1 eqtr2 simpl3 mp3an1 ad2antlr subdird eqeq12d mulcan1g 3bitr2d syl12anc wn - ancoms r19.32v neneqd mtbid orel2 subeq0 sylibd syl5bi sylbid sylan2 syl5 - ralrimivva df-reu fnopabg fveere resubcl remulcl 3adant3 3adant2 readdcld - reu4 cmpt simpll1 mptelee letric 0red 1red 0lt1 ltletrd divelunit gt0ne0d - a1i 3eqtr3d eqeltrd npncan2 3eqtr2rd fveq2 eqid ovex fvmpt ralbiia bitrdi - ex 3jca anbi1i rgen sylancl orim12d fveq1 eqeq1d ssrab3 sseli bitr3i an12 - eqtr3 opabbii eqtri cnveqi cnvopab dff1o4 ) GUUALZHGUUBUDZLZDVVKLZUEZHDUF - ZMZFCUGZFUHZNUUCUUDOZUGZCVVSFUUEVVPBUIZVVSLZEUIZAUIZUDZPVWAUJOZVWCHUDZQOZ - VWAVWCDUDZQOZROZSZEPGUUFOZTZMZBUKZACTVVQVVPVWPACVVPVWDCLZMZVWNBVVSULZVWPV - WRVWNBVVSUMZVWNVWEPUAUIZUJOZVWGQOZVXAVWIQOZROZSZEVWMTZMZBUAUNZUOZUAVVSTBV - 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(Contributed by AV, 25-Nov-2020.) $) @@ -413940,10 +418797,10 @@ could still hold ( ` { M , N } ` would be either ` { M } ` or umgrpredgv $p |- ( ( G e. UMGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) ) $= ( cumgr wcel cpr wa cvtx cfv cpw chash c2 wceq cedg eleq2i syl edgumgr wi - sylan2b wne hashprdifel eqcomi pweqi prelpw biimprd syl5bi 3adant3 impcom - w3a eqid ) BHIZCDJZAIZKUPBLMZNZIZUPOMPQZKZCEIDEIKZUQUOUPBRMZIVBAVDUPGSUPB - UAUCVAUTVCVACUPIZDUPIZCDUDZUMUTVCUBZCDUPUPUNUEVEVFVHVGUTUPENZIZVEVFKZVCUS - VIUPUREEURFUFUGSVKVCVJCDEUPUPUHUIUJUKTULT $. + sylan2b wne eqid hashprdifel eqcomi pweqi prelpw biimprd biimtrid 3adant3 + w3a impcom ) BHIZCDJZAIZKUPBLMZNZIZUPOMPQZKZCEIDEIKZUQUOUPBRMZIVBAVDUPGSU + PBUAUCVAUTVCVACUPIZDUPIZCDUDZUMUTVCUBZCDUPUPUEUFVEVFVHVGUTUPENZIZVEVFKZVC + USVIUPUREEURFUGUHSVKVCVJCDEUPUPUIUJUKULTUNT $. $d A a b c $. $d E a c $. $d G c $. $( For a vertex incident to an edge there is another vertex incident to the @@ -414022,35 +418879,35 @@ could still hold ( ` { M , N } ` would be either ` { M } ` or 3exp syl2an mpd syl5bir orrd anandi bicomi notbii orbi2i 3bitri ralrimiva ianor sylibr inrab eqeq1i rabeq0 ralrimivva eleq1w anbi2d rabbidv hashiun bitri disjor eqcomd oveq1d iunfi syl2anc fveqeq2 elrab eldifn eliun fveq2 - 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DUVBYQYSUUIUVGUVIOUUDUUHUVGUVIPUUGABCDEFGHYTUUAUUBUUC $. + syl5ibr intnand ralnex eleq2d anbi12d ralbii 3bitr2i disjr hashun syl3anc + biimtrid ralrimiv edglnl 3adant2 fveq2d 3eqtr2d ) DUCJZFKJZCKJZLZEFJZUDZF + EUGZUEZEBMZCNZJZAMZUUMJZLZBCUHZUIZONAUJZUUMUUJPZBUURUIZONZUKULAUUKUUSUMZO + NZUVCUKULZUVDUVBUNZONZUUNBUURUIZONUUIUUTUVEUVCUKUUIUVEUUTUUIAUUKUUSUUGUUD + UUKKJZUUHUUEUVJUUFFUUJUOQUPZUUIUUSKJZUUOUUKJZUUGUUDUVLUUHUUFUVLUUEUUFUURK + JZUVLCVAZUUQBUURUQURUSUPQZUUIAIUTZUUSUUNIMZUUMJZLZBUURUIZVBZVCPZVDZIUUKRA + UUKRAUUKUUSVEUUIUWDAIUUKUUKUUIUVMUVRUUKJZLZLZUVQUWCUWGUVQSZUWCUWGUWHLZUUQ + UVTLZSZBUURRZUWCUWIUWKBUURUWIUULUURJZLZUUNSZUUPSZUVSSZVDZVDZUWKUWNUWOUWRU + WOSUUNUWNUWRUUNVFUWNUUNUWRUWNUUNLZUWPUWQUWPSUUPUWTUWQUUPVFUWNUUNUUPUWQTZU + WNUUMUAMZUBMZVGZPZUBFVHUAFVHZUUNUXATZUWIUWMUXFUWGUWMUXFTZUWHUUIUXHUWFUUDU + UGUXHUUHUUDUWMUXFUUDUWMUUMDVINZJZUXFUUDCVNZUWMUXJUUDDVJJUXKDVKCDHVLURCDUU + LHVMVOUUMUXIDFUAUBGUXIVPVQVRVSVTQQWEUWIUXFUXGTZUWMUWGUWHUXLUWFUWHUXLTZUUI + UVMUUOEWFZUVREWFZUXMUWEUUOFEWAUVRFEWAUXNUXOLZUWHUXLUXPUWHLZUXEUXGUAUBFFUX + QUXEUXGTUXBFJUXCFJLUXQUXGUXEEUXDJZUUOUXDJZUVRUXDJZSZTZTZUXPUWHUYCUXPUXSUX + RUWHUYAUXPUXSUXRUWHUYATUXPUXSUXRUDUXTUVQUXPUXSUXRUXTUVQTZUXPUXRUXSUYDUXPU + XRUXSUXTUVQUXRUXSUXTUDUXPUVQUXBUXCEUUOUVRWBWCWDWGWHWIWPWJWEUXEUUNUXRUXAUY + BUUMUXDEWKUXEUUPUXSUWQUYAUUMUXDUUOWKUXEUVSUXTUUMUXDUVRWKWLWMWMWNQWOVSWQUS + WEQWRWEWSWTVSWSWTUWKUUNUUPUVSLZLZSUWOUYESZVDUWSUWJUYFUYFUWJUUNUUPUVSXAXBX + CUUNUYEXGUYGUWRUWOUUPUVSXGXDXEXHXFUWCUWJBUURUIZVCPUWLUWBUYHVCUUQUVTBUURXI + XJUWJBUURXKXQXHVSWTXLUUKUUSUWAAIUVQUUQUVTBUURUVQUUPUVSUUNAIUUMXMXNXOXRXHX + PXSXTUUIUVDKJZUVBKJZUVDUVBVBVCPZUVHUVFPUUIUVJUVLAUUKRUYIUVKUUIUVLAUUKUVPX + FAUUKUUSYAYBUUGUUDUYJUUHUUFUYJUUEUUFUVNUYJUVOUVABUURUQURUSUPUUIUFMZUVDJZS + ZUFUVBRUYKUUIUYNUFUVBUYLUVBJUYLUURJZUYLCNZUUJPZLZUUIUYNUVAUYQBUYLUURUULUY + LUUJCYCYDUUIUYRUYNUUIUYRLZUYOEUYPJZUUOUYPJZLZLZSZAUUKRZUYNUYSVUDAUUKUYSUV + MLZVUBUYOVUFVUAUYTUYSUVMVUASZUYRUVMVUGTZUUIUYQVUHUYOUVMVUGUYQUUOUUJJZSUUO + FUUJYEUYQVUAVUIUYPUUJUUOWKWLYHUSUSWEYIYIXFUYNUYLUUSJZAUUKVHZSVUJSZAUUKRVU + EUYMVUKAUYLUUKUUSYFXCVUJAUUKYJVULVUDAUUKVUJVUCUUQVUBBUYLUURBUFUTZUUNUYTUU + PVUAVUMUUMUYPEUULUYLCYGZYKVUMUUMUYPUUOVUNYKYLYDXCYMYNXHVSYRYSUFUVDUVBYOXH + UVDUVBYPYQUUIUVGUVIOUUDUUHUVGUVIPUUGABCDEFGHYTUUAUUBUUC $. $} ${ @@ -414319,11 +419176,11 @@ unordered pairs of elements of V (edge set). (Contributed by Alexander /\ Fun ( iEdg ` H ) ) -> H e. UMGraph ) $= ( wcel cvtx cfv cedg wbr ciedg w3a wceq wss cvv wb fvex eqid cumgr cdm cv wfun chash c2 cpw crab wf wi isausgr mp2an crn edgval sseq1d funfn biimpi - a1i 3ad2ant3 simp2 df-f sylanbrc 3exp sylbid syl5bi 3imp isumgrs 3ad2ant1 - wfn mpbird ) EFHZEIJZEKJZDLZEMJZUDZNEUAHZVOUBZAUCUEJUFOAVLUGUHZVOUIZVKVNV - PVTVNVMVSPZVKVPVTUJZVLQHVMQHVNWAREISEKSABCVMDVLQQGUKULVKWAVOUMZVSPZWBVKVM - WCVSVMWCOVKEUNURUOVKWDVPVTVKWDVPNVOVRVIZWDVTVPVKWEWDVPWEVOUPUQUSVKWDVPUTV - RVSVOVAVBVCVDVEVFVKVNVQVTRVPAFVOEVLVLTVOTVGVHVJ $. + a1i wfn 3ad2ant3 df-f sylanbrc 3exp sylbid biimtrid 3imp isumgrs 3ad2ant1 + simp2 mpbird ) EFHZEIJZEKJZDLZEMJZUDZNEUAHZVOUBZAUCUEJUFOAVLUGUHZVOUIZVKV + NVPVTVNVMVSPZVKVPVTUJZVLQHVMQHVNWAREISEKSABCVMDVLQQGUKULVKWAVOUMZVSPZWBVK + VMWCVSVMWCOVKEUNURUOVKWDVPVTVKWDVPNVOVRUSZWDVTVPVKWEWDVPWEVOUPUQUTVKWDVPV + IVRVSVOVAVBVCVDVEVFVKVNVQVTRVPAFVOEVLVLTVOTVGVHVJ $. $d H f $. $d x W $. ausgrusgri.1 $e |- O = { f | f : dom f -1-1-> ran f } $. @@ -414334,13 +419191,13 @@ unordered pairs of elements of V (edge set). (Contributed by Alexander /\ ( iEdg ` H ) e. O ) -> H e. USGraph ) $= ( wcel cvtx cfv cedg ciedg w3a wceq wf1 cvv fvex wbr cusgr cdm chash crab cv c2 cpw wss wi isausgr mp2an crn edgval a1i sseq1d cab eleq2i dmeq rneq - wb id f1eq123d elab sylbb 3ad2ant3 simp2 f1ssr syl2anc 3exp sylbid syl5bi - 3imp eqid isusgrs 3ad2ant1 mpbird ) FHKZFLMZFNMZEUAZFOMZGKZPFUBKZWBUCZAUF - UDMUGQAVSUHUEZWBRZVRWAWCWGWAVTWFUIZVRWCWGUJZVSSKVTSKWAWHVAFLTFNTABCVTEVSS - SIUKULVRWHWBUMZWFUIZWIVRVTWJWFVTWJQVRFUNUOUPVRWKWCWGVRWKWCPWEWJWBRZWKWGWC - VRWLWKWCWBDUFZUCZWMUMZWMRZDUQZKWLGWQWBJURWPWLDWBFOTWMWBQZWNWEWOWJWMWBWRVB - WMWBUSWMWBUTVCVDVEVFVRWKWCVGWEWJWFWBVHVIVJVKVLVMVRWAWDWGVAWCAHWBFVSVSVNWB - VNVOVPVQ $. + wb id f1eq123d elab sylbb 3ad2ant3 simp2 f1ssr syl2anc 3exp biimtrid 3imp + sylbid eqid isusgrs 3ad2ant1 mpbird ) FHKZFLMZFNMZEUAZFOMZGKZPFUBKZWBUCZA + UFUDMUGQAVSUHUEZWBRZVRWAWCWGWAVTWFUIZVRWCWGUJZVSSKVTSKWAWHVAFLTFNTABCVTEV + SSSIUKULVRWHWBUMZWFUIZWIVRVTWJWFVTWJQVRFUNUOUPVRWKWCWGVRWKWCPWEWJWBRZWKWG + WCVRWLWKWCWBDUFZUCZWMUMZWMRZDUQZKWLGWQWBJURWPWLDWBFOTWMWBQZWNWEWOWJWMWBWR + VBWMWBUSWMWBUTVCVDVEVFVRWKWCVGWEWJWFWBVHVIVJVMVKVLVRWAWDWGVAWCAHWBFVSVSVN + WBVNVOVPVQ $. $( The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 2-Jan-2020.) @@ -415109,30 +419966,30 @@ two elements (the endvertices of the corresponding edge). (Contributed c0 a1i eqtrid wf f1f syl feqresmpt eqcomd eqtrd wrex cab wfun ciedg cuhgr ushgruhgr uhgrfun funeqi sylibr dfimafn fveq2 eleq2d elrab w3a crn fvelrn wi simpl eqcomi rneqi eleq2i syl2an 3adant3 wb eleq1 eqcoms 3ad2ant3 cedg - mpbird edgval 3ad2ant1 eleq2 biimpcd adantl 3imp jca 3exp syl5bi rexlimdv - wfn funfnd fvelrnb dmeqi biimpi fveq1i eqeq2i adantld imp eqeq1i reximdv2 - ex com23 sylbid impd impbid vex eqeq2 rexbidv elab elrab2 eqrdv f1oeq123d - 3bitr4g ) HUBSZJKSZUCZBCGUDJEUNZIUEZSZEIUFZUGZIUUEUHZIUUEUIZUDZYTUUDHUJUE - ZUKVFULUMZIUOZUUEUUDUPZUUHYRUUKYSIHUUIUUIUQMURZUSUUCEUUDUTZUUDUUJUUEIVAVB - YTBUUECUUFGUUGYTGAUUEAUNZIUEZVCZUUGYTGABUUPVCUUQQYTABUUPUUEUUPBUUETYTOVGZ - YTUUOBSUCUUPVDVEVHYTUUGUUQYRUUGUUQTYSYRAUUDUUJUUEIYRUUKUUDUUJIVIUUMUUDUUJ - IVJVKUULYRUUNVGVLUSVMVNUURYTUUFCYTUUFRUNZIUEZDUNZTZRUUEVOZDVPZCYTIVQZUULU - UFUVDTYRUVEYSYRHVRUEZVQZUVEYRHVSSUVGHVTUVFHUVFUQWAVKZIUVFMWBWCUSZUUNRDUUE - IWDVBYTUAUVDCYTUUTUAUNZTZRUUEVOZUVJFSZJUVJSZUCZUVJUVDSUVJCSYTUVLUVOYTUVKU - VORUUEUUSUUESZUUSUUDSZJUUTSZUCZYTUVKUVOWKUUCUVREUUSUUDUUAUUSTUUBUUTJUUAUU - SIWEWFWGZYTUVSUVKUVOYTUVSUVKWHZUVMUVNUWAUVMUVJUVFWIZSZUWAUWCUUTUWBSZYTUVS - UWDUVKYTUVEUVQUWDUVSUVIUVQUVRWLUVEUVQUCUUTIWIZSUWDUUSIWJUWBUWEUUTUVFIIUVF - MWMZWNWOWCWPWQUVKYTUWCUWDWRZUVSUWGUVJUUTUVJUUTUWBWSWTXAXCYTUVSUVMUWCWRZUV - KYRUWHYSYRFUWBUVJYRFHXBUEZUWBLUWIUWBTYRHXDVGVHWFZUSXEXCYTUVSUVKUVNUVSUVKU - VNWKZWKYTUVRUWKUVQUVKUVRUVNUUTUVJJXFXGXHVGXIXJXKXLXMYRUVOUVLWKYSYRUVMUVNU - VLYRUVMUWCUVNUVLWKZUWJYRUWCUUSUVFUEZUVJTZRUVFUFZVOZUWLYRUVFUWOXNUWCUWPWRY - RUVFUVHXORUWOUVJUVFXPVKYRUVNUWPUVLYRUVNUWPUVLWKYRUVNUCZUWNUVKRUWOUUEUWQUU - SUWOSZUWNUCZUVPUVKUCUWQUWSUCZUVPUVKUWTUVSUVPUWTUVQUVRUWSUVQUWQUWRUVQUWNUW - RUVQUWOUUDUUSUVFIUWFXQWOXRUSXHUWQUWSUVRUWQUWNUVRUWRUVNUWNUVRWKYRUWNUVNUVR - UWNUVJUUTJUVJUUTTZUVJUWMUVJUWMTUXAUWMUUTUVJUUSUVFIUWFXSZXTXRWTWFXGXHYAYBX - JUVTWCUWSUVKUWQUWNUVKUWRUWNUVKUWMUUTUVJUXBYCXRXHXHXJYEYDYEYFYGYGYHUSYIUVC - UVLDUVJUAYJUVAUVJTUVBUVKRUUEUVAUVJUUTYKYLYMJUVASUVNDUVJFCUVAUVJJXFPYNYQYO - VNVMYPXC $. + mpbird edgval 3ad2ant1 biimpcd adantl 3imp jca 3exp biimtrid rexlimdv wfn + eleq2 funfnd fvelrnb dmeqi biimpi fveq1i eqeq2i imp eqeq1i reximdv2 com23 + adantld ex sylbid impd impbid vex eqeq2 rexbidv elab elrab2 3bitr4g eqrdv + f1oeq123d ) HUBSZJKSZUCZBCGUDJEUNZIUEZSZEIUFZUGZIUUEUHZIUUEUIZUDZYTUUDHUJ + UEZUKVFULUMZIUOZUUEUUDUPZUUHYRUUKYSIHUUIUUIUQMURZUSUUCEUUDUTZUUDUUJUUEIVA + VBYTBUUECUUFGUUGYTGAUUEAUNZIUEZVCZUUGYTGABUUPVCUUQQYTABUUPUUEUUPBUUETYTOV + GZYTUUOBSUCUUPVDVEVHYTUUGUUQYRUUGUUQTYSYRAUUDUUJUUEIYRUUKUUDUUJIVIUUMUUDU + UJIVJVKUULYRUUNVGVLUSVMVNUURYTUUFCYTUUFRUNZIUEZDUNZTZRUUEVOZDVPZCYTIVQZUU + LUUFUVDTYRUVEYSYRHVRUEZVQZUVEYRHVSSUVGHVTUVFHUVFUQWAVKZIUVFMWBWCUSZUUNRDU + UEIWDVBYTUAUVDCYTUUTUAUNZTZRUUEVOZUVJFSZJUVJSZUCZUVJUVDSUVJCSYTUVLUVOYTUV + KUVORUUEUUSUUESZUUSUUDSZJUUTSZUCZYTUVKUVOWKUUCUVREUUSUUDUUAUUSTUUBUUTJUUA + UUSIWEWFWGZYTUVSUVKUVOYTUVSUVKWHZUVMUVNUWAUVMUVJUVFWIZSZUWAUWCUUTUWBSZYTU + VSUWDUVKYTUVEUVQUWDUVSUVIUVQUVRWLUVEUVQUCUUTIWIZSUWDUUSIWJUWBUWEUUTUVFIIU + VFMWMZWNWOWCWPWQUVKYTUWCUWDWRZUVSUWGUVJUUTUVJUUTUWBWSWTXAXCYTUVSUVMUWCWRZ + UVKYRUWHYSYRFUWBUVJYRFHXBUEZUWBLUWIUWBTYRHXDVGVHWFZUSXEXCYTUVSUVKUVNUVSUV + KUVNWKZWKYTUVRUWKUVQUVKUVRUVNUUTUVJJXNXFXGVGXHXIXJXKXLYRUVOUVLWKYSYRUVMUV + NUVLYRUVMUWCUVNUVLWKZUWJYRUWCUUSUVFUEZUVJTZRUVFUFZVOZUWLYRUVFUWOXMUWCUWPW + RYRUVFUVHXORUWOUVJUVFXPVKYRUVNUWPUVLYRUVNUWPUVLWKYRUVNUCZUWNUVKRUWOUUEUWQ + UUSUWOSZUWNUCZUVPUVKUCUWQUWSUCZUVPUVKUWTUVSUVPUWTUVQUVRUWSUVQUWQUWRUVQUWN + UWRUVQUWOUUDUUSUVFIUWFXQWOXRUSXGUWQUWSUVRUWQUWNUVRUWRUVNUWNUVRWKYRUWNUVNU + VRUWNUVJUUTJUVJUUTTZUVJUWMUVJUWMTUXAUWMUUTUVJUUSUVFIUWFXSZXTXRWTWFXFXGYEY + AXIUVTWCUWSUVKUWQUWNUVKUWRUWNUVKUWMUUTUVJUXBYBXRXGXGXIYFYCYFYDYGYGYHUSYIU + VCUVLDUVJUAYJUVAUVJTUVBUVKRUUEUVAUVJUUTYKYLYMJUVASUVNDUVJFCUVAUVJJXNPYNYO + YPVNVMYQXC $. $( In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. @@ -415163,30 +420020,30 @@ two elements (the endvertices of the corresponding edge). (Contributed c0 a1i eqtrid wf f1f syl feqresmpt eqtr4d wrex wfun ciedg cuhgr ushgruhgr cab uhgrfun funeqi sylibr dfimafn wi fveqeq2 elrab w3a simpl fvelrn rneqi crn eqcomi eleqtrrdi syl2an 3adant3 wb eqcoms 3ad2ant3 mpbird cedg edgval - eleq1 eleq2d 3ad2ant1 biimpcd adantl 3imp jca 3exp syl5bi rexlimdv funfnd - eqeq1 wfn fvelrnb dmeqi eleq2i biimpi fveq1i eqeq2i eqeq1d adantld eqeq1i - imp ex reximdv2 com23 sylbid impd impbid vex eqeq2 rexbidv elrab2 3bitr4g - elab eqrdv eqtr2d f1oeq123d ) HUBRZJKRZSZBCGUCEUMZIUDJUEZTZEIUFZUGZIUUFUH - ZIUUFUIZUCZUUAUUEHUJUDZUKVEUEULZIUNZUUFUUEUOZUUIYSUULYTIHUUJUUJUPMUQZURUU - DEUUEUSZUUEUUKUUFIUTVAUUABUUFCUUGGUUHUUAGAUUFAUMZIUDZVBZUUHUUAGABUUQVBUUR - PUUAABUUQUUFUUQBUUFTUUANVFZUUAUUPBRSUUQVCVDVGYSUUHUURTYTYSAUUEUUKUUFIYSUU - LUUEUUKIVHUUNUUEUUKIVIVJUUMYSUUOVFVKURVLUUSUUAUUGQUMZIUDZDUMZTZQUUFVMZDVR - ZCUUAIVNZUUMUUGUVETYSUVFYTYSHVOUDZVNZUVFYSHVPRUVHHVQUVGHUVGUPVSVJZIUVGMVT - WAURZUUOQDUUFIWBVAUUAUAUVECUUAUVAUAUMZTZQUUFVMZUVKFRZUVKUUCTZSZUVKUVERUVK - CRUUAUVMUVPUUAUVLUVPQUUFUUTUUFRZUUTUUERZUVAUUCTZSZUUAUVLUVPWCUUDUVSEUUTUU - EUUBUUTUUCIWDWEZUUAUVTUVLUVPUUAUVTUVLWFZUVNUVOUWBUVNUVKUVGWJZRZUWBUWDUVAU - WCRZUUAUVTUWEUVLUUAUVFUVRUWEUVTUVJUVRUVSWGUVFUVRSUVAIWJUWCUUTIWHUVGIIUVGM - WKZWIWLWMWNUVLUUAUWDUWEWOZUVTUWGUVKUVAUVKUVAUWCXAWPWQWRUUAUVTUVNUWDWOZUVL - YSUWHYTYSFUWCUVKYSFHWSUDZUWCLUWIUWCTYSHWTVFVGXBZURXCWRUUAUVTUVLUVOUVTUVLU - VOWCZWCUUAUVSUWKUVRUVLUVSUVOUVAUVKUUCXLXDXEVFXFXGXHXIXJYSUVPUVMWCYTYSUVNU - VOUVMYSUVNUWDUVOUVMWCZUWJYSUWDUUTUVGUDZUVKTZQUVGUFZVMZUWLYSUVGUWOXMUWDUWP - WOYSUVGUVIXKQUWOUVKUVGXNVJYSUVOUWPUVMYSUVOUWPUVMWCYSUVOSZUWNUVLQUWOUUFUWQ - UUTUWORZUWNSZUVQUVLSUWQUWSSZUVQUVLUWTUVTUVQUWTUVRUVSUWSUVRUWQUWRUVRUWNUWR - UVRUWOUUEUUTUVGIUWFXOXPXQURXEUWQUWSUVSUWQUWNUVSUWRUVOUWNUVSWCYSUWNUVOUVSU - WNUVKUVAUUCUVKUVATZUVKUWMUVKUWMTUXAUWMUVAUVKUUTUVGIUWFXRZXSXQWPXTXDXEYAYC - XGUWAWAUWSUVLUWQUWNUVLUWRUWNUVLUWMUVAUVKUXBYBXQXEXEXGYDYEYDYFYGYGYHURYIUV - DUVMDUVKUAYJUVBUVKTUVCUVLQUUFUVBUVKUVAYKYLYOUVBUUCTUVODUVKFCUVBUVKUUCXLOY - MYNYPYQYRWR $. + eleq1 eleq2d 3ad2ant1 eqeq1 biimpcd adantl 3imp jca biimtrid rexlimdv wfn + 3exp funfnd fvelrnb eleq2i biimpi fveq1i eqeq2i eqeq1d adantld imp eqeq1i + dmeqi ex reximdv2 com23 sylbid impd impbid vex eqeq2 rexbidv elab 3bitr4g + elrab2 eqrdv eqtr2d f1oeq123d ) HUBRZJKRZSZBCGUCEUMZIUDJUEZTZEIUFZUGZIUUF + UHZIUUFUIZUCZUUAUUEHUJUDZUKVEUEULZIUNZUUFUUEUOZUUIYSUULYTIHUUJUUJUPMUQZUR + UUDEUUEUSZUUEUUKUUFIUTVAUUABUUFCUUGGUUHUUAGAUUFAUMZIUDZVBZUUHUUAGABUUQVBU + URPUUAABUUQUUFUUQBUUFTUUANVFZUUAUUPBRSUUQVCVDVGYSUUHUURTYTYSAUUEUUKUUFIYS + UULUUEUUKIVHUUNUUEUUKIVIVJUUMYSUUOVFVKURVLUUSUUAUUGQUMZIUDZDUMZTZQUUFVMZD + VRZCUUAIVNZUUMUUGUVETYSUVFYTYSHVOUDZVNZUVFYSHVPRUVHHVQUVGHUVGUPVSVJZIUVGM + VTWAURZUUOQDUUFIWBVAUUAUAUVECUUAUVAUAUMZTZQUUFVMZUVKFRZUVKUUCTZSZUVKUVERU + VKCRUUAUVMUVPUUAUVLUVPQUUFUUTUUFRZUUTUUERZUVAUUCTZSZUUAUVLUVPWCUUDUVSEUUT + UUEUUBUUTUUCIWDWEZUUAUVTUVLUVPUUAUVTUVLWFZUVNUVOUWBUVNUVKUVGWJZRZUWBUWDUV + AUWCRZUUAUVTUWEUVLUUAUVFUVRUWEUVTUVJUVRUVSWGUVFUVRSUVAIWJUWCUUTIWHUVGIIUV + GMWKZWIWLWMWNUVLUUAUWDUWEWOZUVTUWGUVKUVAUVKUVAUWCXAWPWQWRUUAUVTUVNUWDWOZU + VLYSUWHYTYSFUWCUVKYSFHWSUDZUWCLUWIUWCTYSHWTVFVGXBZURXCWRUUAUVTUVLUVOUVTUV + LUVOWCZWCUUAUVSUWKUVRUVLUVSUVOUVAUVKUUCXDXEXFVFXGXHXLXIXJYSUVPUVMWCYTYSUV + NUVOUVMYSUVNUWDUVOUVMWCZUWJYSUWDUUTUVGUDZUVKTZQUVGUFZVMZUWLYSUVGUWOXKUWDU + WPWOYSUVGUVIXMQUWOUVKUVGXNVJYSUVOUWPUVMYSUVOUWPUVMWCYSUVOSZUWNUVLQUWOUUFU + WQUUTUWORZUWNSZUVQUVLSUWQUWSSZUVQUVLUWTUVTUVQUWTUVRUVSUWSUVRUWQUWRUVRUWNU + WRUVRUWOUUEUUTUVGIUWFYCXOXPURXFUWQUWSUVSUWQUWNUVSUWRUVOUWNUVSWCYSUWNUVOUV + SUWNUVKUVAUUCUVKUVATZUVKUWMUVKUWMTUXAUWMUVAUVKUUTUVGIUWFXQZXRXPWPXSXEXFXT + YAXHUWAWAUWSUVLUWQUWNUVLUWRUWNUVLUWMUVAUVKUXBYBXPXFXFXHYDYEYDYFYGYGYHURYI + UVDUVMDUVKUAYJUVBUVKTUVCUVLQUUFUVBUVKUVAYKYLYMUVBUUCTUVODUVKFCUVBUVKUUCXD + OYOYNYPYQYRWR $. $} ${ @@ -415286,10 +420143,10 @@ two elements (the endvertices of the corresponding edge). (Contributed $( The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.) $) uhgr0v0e $p |- ( ( G e. UHGraph /\ V = (/) ) -> E = (/) ) $= - ( cuhgr wcel c0 wceq wa ciedg cfv cvtx wi eqeq1i uhgr0vb biimpd ex syl5bi - pm2.43a imp wb cedg uhgriedg0edg0 bitrid adantr mpbird ) BFGZCHIZJAHIZBKL - HIZUHUIUKUIUHUKUIBMLZHIZUHUHUKNZCULHDOUHUMUNUHUMJUHUKBFPQRSTUAUHUJUKUBUIU - JBUCLZHIUHUKAUOHEOBUDUEUFUG $. + ( cuhgr wcel c0 wceq wa ciedg cfv cvtx wi eqeq1i uhgr0vb biimtrid pm2.43a + biimpd ex imp wb cedg uhgriedg0edg0 bitrid adantr mpbird ) BFGZCHIZJAHIZB + KLHIZUHUIUKUIUHUKUIBMLZHIZUHUHUKNZCULHDOUHUMUNUHUMJUHUKBFPSTQRUAUHUJUKUBU + IUJBUCLZHIUHUKAUOHEOBUDUEUFUG $. $( The size of a hypergraph with no vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, @@ -415452,16 +420309,16 @@ two elements (the endvertices of the corresponding edge). (Contributed -> ( Edg ` G ) = (/) ) $= ( vv chash cfv c1 wceq c0 c2 cle wbr cvv wn cc0 cuhgr wcel cdm cedg ciedg wf w3a wa a1i dmeqi cv cpw crab csn wex wi cvtx fvexi hash1snb ax-mp pweq - wb rabeqdv wral cpr clt 2pos 0re 2re ltnlei mpbi 1lt2 1re 0ex fveq2 hash0 - snex eqtrdi breq2d notbid hashsng ralpr mpbir2an pwsn raleqi mpbir rabeq0 - elv a1d exlimiv sylbi impcom eqtrid feq123d biimp3a simplbi uhgriedg0edg0 - f00 syl 3ad2ant1 mpbird ) CUAUBZEJKLMZDUCZBDUFZUGZCUDKNMZCUEKZNMZXFXHUCZN - XHUFZXIXBXCXEXKXBXCUHZXDXJBNDXHDXHMXLGUIXDXJMXLDXHGUJUIXLBOAUKZJKZPQZAEUL - ZUMZNHXCXBXQNMZXCEIUKZUNZMZIUOZXBXRUPZERUBXCYBVBECUQFURERIUSUTYAYCIYAXRXB - YAXQXOAXTULZUMZNYAXOAXPYDEXTVAVCYENMXOSZAYDVDZYGYFANXTVEZVDZYIOTPQZSZOLPQ - ZSZTOVFQYKVGTOVHVIVJVKLOVFQYMVLLOVMVIVJVKYFYKYMANXTVNXSVQXMNMZXOYJYNXNTOP - YNXNNJKTXMNJVOVPVRVSVTXMXTMZXOYLYOXNLOPYOXNXTJKZLXMXTJVOYPLMIXSRWAWHVRVSV - TWBWCYFAYDYHXSWDWEWFXOAYDWGWFVRWIWJWKWLWMWNWOXKXIXJNMXJXHWRWPWSXBXCXGXIVB + wb rabeqdv wral cpr clt 2pos 0re 2re ltnlei mpbi 1lt2 1re 0ex vsnex fveq2 + hash0 eqtrdi breq2d notbid hashsng elv ralpr mpbir2an raleqi mpbir rabeq0 + a1d exlimiv sylbi impcom eqtrid feq123d biimp3a f00 simplbi uhgriedg0edg0 + pwsn syl 3ad2ant1 mpbird ) CUAUBZEJKLMZDUCZBDUFZUGZCUDKNMZCUEKZNMZXFXHUCZ + NXHUFZXIXBXCXEXKXBXCUHZXDXJBNDXHDXHMXLGUIXDXJMXLDXHGUJUIXLBOAUKZJKZPQZAEU + LZUMZNHXCXBXQNMZXCEIUKZUNZMZIUOZXBXRUPZERUBXCYBVBECUQFURERIUSUTYAYCIYAXRX + BYAXQXOAXTULZUMZNYAXOAXPYDEXTVAVCYENMXOSZAYDVDZYGYFANXTVEZVDZYIOTPQZSZOLP + QZSZTOVFQYKVGTOVHVIVJVKLOVFQYMVLLOVMVIVJVKYFYKYMANXTVNIVOXMNMZXOYJYNXNTOP + YNXNNJKTXMNJVPVQVRVSVTXMXTMZXOYLYOXNLOPYOXNXTJKZLXMXTJVPYPLMIXSRWAWBVRVSV + TWCWDYFAYDYHXSWRWEWFXOAYDWGWFVRWHWIWJWKWLWMWNXKXIXJNMXJXHWOWPWSXBXCXGXIVB XECWQWTXA $. $} @@ -415849,13 +420706,13 @@ two elements (the endvertices of the corresponding edge). (Contributed ( wcel cdm w3a cfv chash c2 wceq cpw crab 3ad2ant2 wss eqid csubgr wbr cv cumgr c0 cdif fveqeq2 cuhgr umgruhgr simp1 simp3 subgruhgredgd ciedg wfun csn uhgrfun syl cvtx cedg subgrprop2 simp2d funssfv eqcomd syl3anc fveq2d - 3ad2ant1 simp2 wi dmeqi eleq2i subgreldmiedg syl5bi 3imp umgredg2 syl2anc - ex a1d eqtrd elrabd prprrab eleqtrdi ) ACUAUBZCUDIZFDJZIZKZFDLZBUCZMLNOZB - EPZUEUOUFZQWIBWJQWFWIWGMLZNOBWGWKWHWGNMUGWFACDEFGHWCWBCUHIZWECUIZRWBWCWEU - JWBWCWEUKZULWFWLFCUMLZLZMLZNWFWGWQMWFWPUNZDWPSZWEWGWQOWCWBWSWEWCWMWSWNWPC - WPTZUPUQRWBWCWTWEWBAURLZCURLZSWTAUSLZXBPSXCWPAXDCDXBXBTXCTZHXAXDTUTVAVFWO - WSWTWEKWQWGFWPDVBVCVDVEWFWCFWPJIZWRNOWBWCWEVGWBWCWEXFWBWEXFVHWCWEFAUMLZJZ - IZWBXFWDXHFDXGHVIVJWBXIXFACFVKVPVLVQVMWPCXCFXEXAVNVOVRVSBEVTWA $. + 3ad2ant1 simp2 wi dmeqi eleq2i subgreldmiedg ex biimtrid a1d 3imp syl2anc + umgredg2 eqtrd elrabd prprrab eleqtrdi ) ACUAUBZCUDIZFDJZIZKZFDLZBUCZMLNO + ZBEPZUEUOUFZQWIBWJQWFWIWGMLZNOBWGWKWHWGNMUGWFACDEFGHWCWBCUHIZWECUIZRWBWCW + EUJWBWCWEUKZULWFWLFCUMLZLZMLZNWFWGWQMWFWPUNZDWPSZWEWGWQOWCWBWSWEWCWMWSWNW + PCWPTZUPUQRWBWCWTWEWBAURLZCURLZSWTAUSLZXBPSXCWPAXDCDXBXBTXCTZHXAXDTUTVAVF + WOWSWTWEKWQWGFWPDVBVCVDVEWFWCFWPJIZWRNOWBWCWEVGWBWCWEXFWBWEXFVHWCWEFAUMLZ + JZIZWBXFWDXHFDXGHVIVJWBXIXFACFVKVLVMVNVOWPCXCFXEXAVQVPVRVSBEVTWA $. $} ${ @@ -415947,13 +420804,13 @@ two elements (the endvertices of the corresponding edge). (Contributed wrex cres cuhgr uhgrfun funres 3syl funeqd mpbird elrnrexdmb syl adantr wb fveq1d cin dmeqd dmres eqtrdi eleq2d elinel1 syl6bi imp fvresd eqtrd wa wss elinel2 uhgrss syl2an2r pweqd fvex elpw eqeltrd eleq1 syl5ibrcom - bitrdi rexlimdva sylbid syl5bi ssrdv ) ANCUAPZCUBPZQZNUCZWHRWKCUDPZUEZR - ZAWKWJRZWHWMWKCUFUGAWNWKOUCZWLPZUHZOWLSZUIZWOAWLTZWNWTUTAXADBUJZTZAEUKR - ZDTXCMDEIULBDUMUNAWLXBLUOUPOWLWKUQURAWRWOOWSAWPWSRZVLZWOWRWQWJRXFWQWPDP - ZWJXFWQWPXBPXGXFWPWLXBAWLXBUHXELUSVAXFWPBDAXEWPBRZAXEWPBDSZVBZRZXHAWSXJ - WPAWSXBSXJAWLXBLVCDBVDVEVFZWPBXIVGVHVIVJVKXFXGWJRZXGFVMZAXDXEWPXIRZXNMA - XEXOAXEXKXOXLWPBXIVNVHVIDWPEFHIVOVPXFXMXGFQZRZXNAXMXQUTXEAWJXPXGAWIFKVQ - VFUSXGFWPDVRVSWCUPVTWKWQWJWAWBWDWEWFWG $. + bitrdi rexlimdva sylbid biimtrid ssrdv ) ANCUAPZCUBPZQZNUCZWHRWKCUDPZUE + ZRZAWKWJRZWHWMWKCUFUGAWNWKOUCZWLPZUHZOWLSZUIZWOAWLTZWNWTUTAXADBUJZTZAEU + KRZDTXCMDEIULBDUMUNAWLXBLUOUPOWLWKUQURAWRWOOWSAWPWSRZVLZWOWRWQWJRXFWQWP + DPZWJXFWQWPXBPXGXFWPWLXBAWLXBUHXELUSVAXFWPBDAXEWPBRZAXEWPBDSZVBZRZXHAWS + XJWPAWSXBSXJAWLXBLVCDBVDVEVFZWPBXIVGVHVIVJVKXFXGWJRZXGFVMZAXDXEWPXIRZXN + MAXEXOAXEXKXOXLWPBXIVNVHVIDWPEFHIVOVPXFXMXGFQZRZXNAXMXQUTXEAWJXPXGAWIFK + VQVFUSXGFWPDVRVSWCUPVTWKWQWJWAWBWDWEWFWG $. $( A spanning subgraph ` S ` of a hypergraph ` G ` is actually a subgraph of ` G ` . A subgraph ` S ` of a graph ` G ` which has the same @@ -416079,16 +420936,16 @@ called a _spanning subgraph_ (see section I.1 in [Bollobas] p. 2 and ( vc vj wcel wa wss cfv wceq cv cvv cuhgr csubgr wbr cdif ciedg cres cima csn cdm cpw difssd uhgrspan1lem3 resresdm mp1i wrex wi uhgrfun fvelima ex wfun adantr wnel eqidd fveq2 neleq12d elrab2 fvexd uhgrss ad2ant2r simprr - syl weq elpwdifsn syl3anc wb eleq1 eqcoms syl5ibrcom syl5bi rexlimdv syld - ssrdv w3a cop opex eqeltri a1i cvtx uhgrspan1lem2 eqcomi eqid cedg edgval - crn rneqi df-ima 3eqtr4ri issubgr sylan2 mpbir3and ) DUANZFGNZOZADUBUCZGF - UHZUDZGPZAUEQZEXHUIUFRZECUGZXFUJZPZXCGXEUKXHECUFZRXIXCABCDEFGHIJKULZCEXHU - MUNXCLXJXKXCLSZXJNZMSZEQZXORZMCUOZXOXKNZXAXPXTUPZXBXAEUTZYBEDIUQYCXPXTMXO - CEURUSVKVAXCXSYAMCXQCNXQEUIZNZFXRVBZOZXCXSYAUPZFBSZEQZVBYFBXQYDCBMVLZFFYJ - XRYKFVCYIXQEVDVEJVFXCYGYHXCYGOZYAXSXRXKNZYLXRTNXRGPZYFYMYLXQEVGXAYEYNXBYF - EXQDGHIVHVIXCYEYFVJFXRGTVMVNYAYMVOXOXRXOXRXKVPVQVRUSVSVTWAWBXBXAATNZXDXGX - IXLWCVOYOXBAXFXMWDTKXFXMWEWFWGGEATXJDXHXFUAAWHQXFABCDEFGHIJKWIWJHXHWKIXHW - NXMWNAWLQXJXHXMXNWOAWMECWPWQWRWSWT $. + syl weq elpwdifsn syl3anc eleq1 eqcoms syl5ibrcom biimtrid rexlimdv ssrdv + wb syld w3a cop opex eqeltri a1i cvtx uhgrspan1lem2 eqcomi eqid crn rneqi + cedg edgval df-ima 3eqtr4ri issubgr sylan2 mpbir3and ) DUANZFGNZOZADUBUCZ + GFUHZUDZGPZAUEQZEXHUIUFRZECUGZXFUJZPZXCGXEUKXHECUFZRXIXCABCDEFGHIJKULZCEX + HUMUNXCLXJXKXCLSZXJNZMSZEQZXORZMCUOZXOXKNZXAXPXTUPZXBXAEUTZYBEDIUQYCXPXTM + XOCEURUSVKVAXCXSYAMCXQCNXQEUIZNZFXRVBZOZXCXSYAUPZFBSZEQZVBYFBXQYDCBMVLZFF + YJXRYKFVCYIXQEVDVEJVFXCYGYHXCYGOZYAXSXRXKNZYLXRTNXRGPZYFYMYLXQEVGXAYEYNXB + YFEXQDGHIVHVIXCYEYFVJFXRGTVMVNYAYMWAXOXRXOXRXKVOVPVQUSVRVSWBVTXBXAATNZXDX + GXIXLWCWAYOXBAXFXMWDTKXFXMWEWFWGGEATXJDXHXFUAAWHQXFABCDEFGHIJKWIWJHXHWKIX + HWLXMWLAWNQXJXHXMXNWMAWOECWPWQWRWSWT $. $} ${ @@ -416105,17 +420962,17 @@ called a _spanning subgraph_ (see section I.1 in [Bollobas] p. 2 and ( vj wcel wa cv chash cfv c0 syl wi adantr cupgr cres crn cima c2 cle wbr csn cdif cpw crab df-ima wss wral cdm wnel wceq wb fveq2 neleq2 elrab2 wf upgrf ffvelcdm breq1d elrab wne eldifsn simpl elpwi simpr elpwdifsn sylbi - syl3anc imp eldifsni sylanbrc elrabd a1d com23 imp4b syl5bi ralrimiv wfun - ex cuhgr upgruhgr uhgrfun ssrab3 funimass4 sylancl mpbird eqsstrrid ) DUA - LZEFLZMZBCUBUCBCUDZGNZOPZUEUFUGZGFEUHUIUJZQUHZUIZUKZBCULWPWQXDUMZKNZBPZXD - LZKCUNZWPXHKCXFCLXFBUOZLZEXGUPZMWPXHEANZBPZUPZXLAXFXJCXMXFUQXNXGUQXOXLURX - MXFBUSXNXGEUTRJVAWNWOXKXLXHWNXJWTGFUJZXBUIZUKZBVBZWOXKXLXHSZSSGBDFHIVCXSX - KWOXTXSXKWOXTSZXSXKMXGXRLZYAXJXRXFBVDYBXGXQLZXGOPZUEUFUGZMZYAWTYEGXGXQWRX - GUQWSYDUEUFWRXGOUSVEZVFYFXTWOYFXLXHYFXLMZWTYEGXGXCYGYHXGXALZXGQVGZXGXCLYF - XLYIYCXLYISZYEYCXGXPLZYJMYKXGXPQVHYLYKYJYLXLYIYLXLMYLXGFUMZXLYIYLXLVIYLYM - XLXGFVJTYLXLVKEXGFXPVLVNWETVMTVOYFYJXLYCYJYEXGXPQVPTTXGXAQVHVQYFYEXLYCYEV - KTVRWEVSVMRWEVTRWAWBWCWPBWDZCXJUMXEXIURWNYNWOWNDWFLYNDWGBDIWHRTXOAXJCJWIK - CXDBWJWKWLWM $. + syl3anc ex imp eldifsni sylanbrc elrabd a1d com23 imp4b biimtrid ralrimiv + wfun cuhgr upgruhgr uhgrfun ssrab3 funimass4 sylancl mpbird eqsstrrid ) D + UALZEFLZMZBCUBUCBCUDZGNZOPZUEUFUGZGFEUHUIUJZQUHZUIZUKZBCULWPWQXDUMZKNZBPZ + XDLZKCUNZWPXHKCXFCLXFBUOZLZEXGUPZMWPXHEANZBPZUPZXLAXFXJCXMXFUQXNXGUQXOXLU + RXMXFBUSXNXGEUTRJVAWNWOXKXLXHWNXJWTGFUJZXBUIZUKZBVBZWOXKXLXHSZSSGBDFHIVCX + SXKWOXTXSXKWOXTSZXSXKMXGXRLZYAXJXRXFBVDYBXGXQLZXGOPZUEUFUGZMZYAWTYEGXGXQW + RXGUQWSYDUEUFWRXGOUSVEZVFYFXTWOYFXLXHYFXLMZWTYEGXGXCYGYHXGXALZXGQVGZXGXCL + YFXLYIYCXLYISZYEYCXGXPLZYJMYKXGXPQVHYLYKYJYLXLYIYLXLMYLXGFUMZXLYIYLXLVIYL + YMXLXGFVJTYLXLVKEXGFXPVLVNVOTVMTVPYFYJXLYCYJYEXGXPQVQTTXGXAQVHVRYFYEXLYCY + EVKTVSVOVTVMRVOWARWBWCWDWPBWEZCXJUMXEXIURWNYNWOWNDWFLYNDWGBDIWHRTXOAXJCJW + IKCXDBWJWKWLWM $. $( Lemma for ~ umgrres and ~ usgrres . (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) $) @@ -416124,14 +420981,14 @@ called a _spanning subgraph_ (see section I.1 in [Bollobas] p. 2 and ( vj wcel wa cv chash cfv c2 syl wi adantr cumgr cres cima wceq cdif crab crn csn cpw df-ima wss wral cdm weq wb fveq2 neleq2 elrab2 umgrf ffvelcdm wf fveqeq2 elrab simpll elpwi simpr elpwdifsn syl3anc elrabd ex a1d sylbi - wnel com23 imp4b syl5bi ralrimiv cuhgr umgruhgr uhgrfun funimass4 sylancl - wfun ssrab3 mpbird eqsstrrid ) DUALZEFLZMZBCUBUGBCUCZGNZOPQUDZGFEUHUEUIZU - FZBCUJWIWJWNUKZKNZBPZWNLZKCULZWIWRKCWPCLWPBUMZLZEWQVMZMWIWREANZBPZVMZXBAW - PWTCAKUNXDWQUDXEXBUOXCWPBUPXDWQEUQRJURWGWHXAXBWRWGWTWLGFUIZUFZBVAZWHXAXBW - RSZSSGBDFHIUSXHXAWHXIXHXAWHXISZXHXAMWQXGLZXJWTXGWPBUTXKWQXFLZWQOPQUDZMZXJ - WLXMGWQXFWKWQQOVBZVCXNXIWHXNXBWRXNXBMZWLXMGWQWMXOXPXLWQFUKZXBWQWMLXLXMXBV - DXNXQXBXLXQXMWQFVETTXNXBVFEWQFXFVGVHXNXMXBXLXMVFTVIVJVKVLRVJVNRVOVPVQWIBW - CZCWTUKWOWSUOWGXRWHWGDVRLXRDVSBDIVTRTXEAWTCJWDKCWNBWAWBWEWF $. + wnel com23 imp4b biimtrid ralrimiv wfun umgruhgr uhgrfun ssrab3 funimass4 + cuhgr sylancl mpbird eqsstrrid ) DUALZEFLZMZBCUBUGBCUCZGNZOPQUDZGFEUHUEUI + ZUFZBCUJWIWJWNUKZKNZBPZWNLZKCULZWIWRKCWPCLWPBUMZLZEWQVMZMWIWREANZBPZVMZXB + AWPWTCAKUNXDWQUDXEXBUOXCWPBUPXDWQEUQRJURWGWHXAXBWRWGWTWLGFUIZUFZBVAZWHXAX + BWRSZSSGBDFHIUSXHXAWHXIXHXAWHXISZXHXAMWQXGLZXJWTXGWPBUTXKWQXFLZWQOPQUDZMZ + XJWLXMGWQXFWKWQQOVBZVCXNXIWHXNXBWRXNXBMZWLXMGWQWMXOXPXLWQFUKZXBWQWMLXLXMX + BVDXNXQXBXLXQXMWQFVETTXNXBVFEWQFXFVGVHXNXMXBXLXMVFTVIVJVKVLRVJVNRVOVPVQWI + BVRZCWTUKWOWSUOWGXRWHWGDWCLXRDVSBDIVTRTXEAWTCJWAKCWNBWBWDWEWF $. $d N p $. $d S p $. $d V p $. upgrres.s $e |- S = <. ( V \ { N } ) , ( E |` F ) >. $. @@ -416225,20 +421082,20 @@ this vertex from a simple graph (see ~ uhgrspan1 ) is a simple graph. cupgr wf wf1o f1oi f1of mp1i ffdmd wnel wral wss simpr adantr cedg eleq2i wi cvtx wne w3a edgupgr sseqtrrdi 3ad2ant1 syl sylan2b ad4ant13 elpwdifsn elpwi syl3anc wn simpl biimpi simp2d syl2an nelsn eldifd ralrimiva sylibr - rabss eqsstrid elrabi ciedg crn edgval eqtri eqid upgrf frnd sseld syl5bi - ex weq fveq2 breq1d elrab simprbi syl6 syl5com eleq2s impcom ssrabdv fssd - cvv wb cop opex eqeltri upgrres1lem2 eqcomi upgrres1lem3 isupgr mpbird ) - EUJNZFGNZOZAUJNZUADUBZUCZLPZQRZSTUDZLGFUEUFZUGZUHUEZUFZUIZYDUKZYBYEDYMYDY - BDDYDDDYDULDDYDUKYBDUMDDYDUNUOUPYBYHLYLDYBDFBPZUQZBCUIZYLJYBYPYOYLNZVDZBC - URYQYLUSYBYSBCYBYOCNZOZYPYRUUAYPOZYOYJYKUUBYTYOGUSZYPYOYJNUUAYTYPYBYTUTVA - XTYTUUCYAYPYTXTYOEVBRZNZUUCCUUDYOIVCZXTUUEOZYOEVERZUGNZYOUHVFZYOQRSTUDZVG - UUCYOEVHZUUIUUJUUCUUKUUIYOUUHGYOUUHVOHVIVJVKVLVMUUAYPUTFYOGCVNVPUUBUUJYOY - KNVQUUAUUJYPYBXTUUEUUJYTXTYAVRYTUUEUUFVSUUGUUIUUJUUKUULVTWAVAYOUHWBVKWCWR - WDYPBCYLWFWEWGYFDNYBYHYBYHVDYFYQDYFYQNYFCNZYBYHYPBYFCWHXTUUMYHVDYAXTUUMYF - MPZQRZSTUDZMGUGYKUFZUIZNZYHUUMYFEWIRZWJZNXTUUSCUVAYFCUUDUVAIEWKWLVCXTUVAU - URYFXTUUTUCUURUUTMUUTEGHUUTWMWNWOWPWQUUSYFUUQNYHUUPYHMYFUUQMLWSUUOYGSTUUN - YFQWTXAXBXCXDVAXEJXFXGXHXIAXJNYCYNXKYBAYIYDXLXJKYIYDXMXNLXJYDAYIAVERYIABC - DEFGHIJKXOXPAWIRYDABCDEFGHIJKXQXPXRUOXS $. + ex rabss eqsstrid elrabi ciedg crn edgval eqtri eqid upgrf sseld biimtrid + frnd weq fveq2 breq1d simprbi syl6 syl5com eleq2s impcom ssrabdv fssd cvv + elrab wb cop opex eqeltri upgrres1lem2 eqcomi upgrres1lem3 isupgr mpbird + ) EUJNZFGNZOZAUJNZUADUBZUCZLPZQRZSTUDZLGFUEUFZUGZUHUEZUFZUIZYDUKZYBYEDYMY + DYBDDYDDDYDULDDYDUKYBDUMDDYDUNUOUPYBYHLYLDYBDFBPZUQZBCUIZYLJYBYPYOYLNZVDZ + BCURYQYLUSYBYSBCYBYOCNZOZYPYRUUAYPOZYOYJYKUUBYTYOGUSZYPYOYJNUUAYTYPYBYTUT + VAXTYTUUCYAYPYTXTYOEVBRZNZUUCCUUDYOIVCZXTUUEOZYOEVERZUGNZYOUHVFZYOQRSTUDZ + VGUUCYOEVHZUUIUUJUUCUUKUUIYOUUHGYOUUHVOHVIVJVKVLVMUUAYPUTFYOGCVNVPUUBUUJY + OYKNVQUUAUUJYPYBXTUUEUUJYTXTYAVRYTUUEUUFVSUUGUUIUUJUUKUULVTWAVAYOUHWBVKWC + WFWDYPBCYLWGWEWHYFDNYBYHYBYHVDYFYQDYFYQNYFCNZYBYHYPBYFCWIXTUUMYHVDYAXTUUM + YFMPZQRZSTUDZMGUGYKUFZUIZNZYHUUMYFEWJRZWKZNXTUUSCUVAYFCUUDUVAIEWLWMVCXTUV + AUURYFXTUUTUCUURUUTMUUTEGHUUTWNWOWRWPWQUUSYFUUQNYHUUPYHMYFUUQMLWSUUOYGSTU + UNYFQWTXAXJXBXCVAXDJXEXFXGXHAXINYCYNXKYBAYIYDXLXIKYIYDXMXNLXIYDAYIAVERYIA + BCDEFGHIJKXOXPAWJRYDABCDEFGHIJKXQXPXRUOXS $. $( A multigraph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of @@ -416416,10 +421273,10 @@ this vertex from a simple graph (see ~ uhgrspan1 ) is a simple graph. ( cid cv wnel cop cedg cfv crab cres cfn wcel wa cfusgr wb eqid w3a ciedg residfi cusgr fusgrusgr usgredgffibi 3ad2ant2 simp2 opvtxfv eqcomd eleq2d cvtx biimpa usgrfilem 3imp3i2an opiedgfv eleq1d 3ad2ant1 3bitr3rd biimprd - syl syl5bi ) GBFHIFCAJZKLZMZNOPVEOPZCDPAEPQZVCRPZBCPZUAZAOPZVEUCVJVKVFVJV - DOPZVCUBLZOPZVFVKVHVGVLVNSZVIVHVCUDPVOVCUEVDVCVMVMTVDTZUFVAUGVGVHVIVHBVCU - LLZPZVLVFSVGVHVIUHVGVIVRVGCVQBVGVQCACDEUIUJUKUMFVDVEVCBVQVQTVPVETUNUOVGVH - VNVKSVIVGVMAOACDEUPUQURUSUTVB $. + syl biimtrid ) GBFHIFCAJZKLZMZNOPVEOPZCDPAEPQZVCRPZBCPZUAZAOPZVEUCVJVKVFV + JVDOPZVCUBLZOPZVFVKVHVGVLVNSZVIVHVCUDPVOVCUEVDVCVMVMTVDTZUFVAUGVGVHVIVHBV + CULLZPZVLVFSVGVHVIUHVGVIVRVGCVQBVGVQCACDEUIUJUKUMFVDVEVCBVQVQTVPVETUNUOVG + VHVNVKSVIVGVMAOACDEUPUQURUSUTVB $. $} ${ @@ -416625,14 +421482,14 @@ function instead of the edges themselves (see also ~ nbgrval ). ( wcel wa cv wss wceq wi wn c0 adantr cfv simpl cuhgr cnbgr cpr wrex cdif co csn crab nbgrval a1d wnel df-nel nbgrnvtx0 sylbir wral cvtx cpw eleq2i cedg biimpi edguhgr syl2an velpw eqcomi sseq2i bitri sstr wb prssg bicomd - cvv syl6bi syl5com ex com13 ad3antlr syl5bi mpd rexlimdva con3rr3 expdimp - elvd ralrimiv rabeq0 sylibr eqtr4d pm2.61i ) EFJZDUAJZEGJZKZDEUBUFZEBLZUC - ZALZMZACUDZBFEUGUEZUHZNZOWHWTWKABCDEFHIUIUJWHPZWKWTXAWKKZWLQWSXAWLQNZWKXA - EFUKXCEFULDFEHUMUNRXBWQPZBWRUOWSQNXBXDBWRXAWKWMWRJZXDWKXEKZWQWHXFWPWHACXF - WOCJZKZWODUPSZUQJZWPWHOZXFWIWODUSSZJZXJXGWKWIXEWIWJTRXGXMCXLWOIURUTWODVAV - BXJWOFMZXHXKXJWOXIMXNAXIVCXIFWOFXIHVDVEVFWJXNXKOWIXEXGWPXNWJWHWPXNWJWHOWP - XNKWNFMZWJWHWNWOFVGWJXOWHWMFJZKZWHWJXOXQVHBWJWMVKJKXQXOEWMFGVKVIVJWBWHXPT - VLVMVNVOVPVQVRVSVTWAWCWQBWRWDWEWFVNWG $. + cvv elvd syl6bi syl5com com13 ad3antlr biimtrid rexlimdva con3rr3 expdimp + ex mpd ralrimiv rabeq0 sylibr eqtr4d pm2.61i ) EFJZDUAJZEGJZKZDEUBUFZEBLZ + UCZALZMZACUDZBFEUGUEZUHZNZOWHWTWKABCDEFHIUIUJWHPZWKWTXAWKKZWLQWSXAWLQNZWK + XAEFUKXCEFULDFEHUMUNRXBWQPZBWRUOWSQNXBXDBWRXAWKWMWRJZXDWKXEKZWQWHXFWPWHAC + XFWOCJZKZWODUPSZUQJZWPWHOZXFWIWODUSSZJZXJXGWKWIXEWIWJTRXGXMCXLWOIURUTWODV + AVBXJWOFMZXHXKXJWOXIMXNAXIVCXIFWOFXIHVDVEVFWJXNXKOWIXEXGWPXNWJWHWPXNWJWHO + WPXNKWNFMZWJWHWNWOFVGWJXOWHWMFJZKZWHWJXOXQVHBWJWMVKJKXQXOEWMFGVKVIVJVLWHX + PTVMVNWAVOVPVQWBVRVSVTWCWQBWRWDWEWFWAWG $. $d E n v x $. $d G e x $. $d G v $. $d N n v x $. $d V v x $. $( The set of neighbors of a vertex in a pseudograph. (Contributed by AV, @@ -416745,13 +421602,13 @@ function instead of the edges themselves (see also ~ nbgrval ). ( ve wcel cv cpr wceq wa wss wi cfv cpw sylbi com13 ex cuhgr cnbgr co wne wrex w3a nbgrel cvtx cedg eleq2i edguhgr sylan2b eqeq1i pweq eleq2d velpw wb bitrdi adantl prcom sseq1i eqss eleq1a a1i sylbir ad2antlr sylbid mpid - impancom com14 rexlimdv 3impia com12 syl5bi ) BUAIZCDJZEJZKZLZVPBVQUBUCIZ - VRAIZVTVPCIVQCIMZVPVQUDZVQVPKZHJZNZHAUEZUFZVOVSMZWAHABVPCVQFGUGWHWIWAWBWC - WGWIWAOZWBWCMZWFWJHAWIWEAIZWFWKWAVOWLVSWFWKWAOZOZVOWLMZVSWEBUHPZQZIZWNWLV - OWEBUIPZIWRAWSWEGUJWEBUKULWOVSWRWNOWOVSMWRWEVRNZWNVSWRWTUQZWOVSWPVRLZXACW - PVRFUMXBWRWEVRQZIWTXBWQXCWEWPVRUNUOHVRUPURRUSWLWTWNOVOVSWFWTWLWMWFVRWENZW - TWLWMOZOWDVRWEVQVPUTVAXDWTXEXDWTMVRWELZXEVRWEVBWKWLXFWAWLXFWAOOWKWEAVRVCV - DSVETRSVFVGTVHVIVJVKVLVMVNVL $. + impancom com14 rexlimdv 3impia com12 biimtrid ) BUAIZCDJZEJZKZLZVPBVQUBUC + IZVRAIZVTVPCIVQCIMZVPVQUDZVQVPKZHJZNZHAUEZUFZVOVSMZWAHABVPCVQFGUGWHWIWAWB + WCWGWIWAOZWBWCMZWFWJHAWIWEAIZWFWKWAVOWLVSWFWKWAOZOZVOWLMZVSWEBUHPZQZIZWNW + LVOWEBUIPZIWRAWSWEGUJWEBUKULWOVSWRWNOWOVSMWRWEVRNZWNVSWRWTUQZWOVSWPVRLZXA + CWPVRFUMXBWRWEVRQZIWTXBWQXCWEWPVRUNUOHVRUPURRUSWLWTWNOVOVSWFWTWLWMWFVRWEN + ZWTWLWMOZOWDVRWEVQVPUTVAXDWTXEXDWTMVRWELZXEVRWEVBWKWLXFWAWLXFWAOOWKWEAVRV + CVDSVETRSVFVGTVHVIVJVKVLVMVNVL $. $( If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. @@ -416765,21 +421622,22 @@ function instead of the edges themselves (see also ~ nbgrval ). id necomd adantr ad2antlr prcom eleq1i biimpi eqimssi a1i rspcedvd nbgrel sseq2 syl3anbrc simplrl jca ex nbuhgr2vtx1edgblem 3exp adantld imp impbid wi eleq1 difeq1 raleqdv raleqbidv vex difeq2d oveq2 eleq2d ralpr difprsn1 - sneq ralsn bitrdi difprsn2 anbi12d bitrid sylan9bbr bibi12d mpbird syl5bi - rexlimdvva ) DUAKZEUBUCUDLZECKZBMZDAMZNUEZKZBEXEUFZOZPZAEPZQZXBHMZIMZUGZE - XMXNUHZLZRZIESHESZXAXLEUIKXBXSQEDUJFUKEUIHIULUMXAXRXLHIEEXAXMEKZXNEKZRZRZ - XRXLYCXRRZXLXPCKZXNDXMNUEZKZXMDXNNUEZKZRZQZYDYEYJYDYEYJYDYERZYGYIYLYAXTRZ - XNXMUGZXPJMZUNZJCSZYGYCYMXRYEYCXTYAXAYBUOZUPUQXRYNYCYEXOYNXQXOXMXNXOURUSU - TVAYEYQYDYEYPXPXNXMUHZUNZJYSCYEYSCKXPYSCXMXNVBZVCVDYOYSLYPYTQYEYOYSXPVITY - TYEXPYSUUAVEVFVGTJCDXNEXMFGVHVJYLYBXOYSYOUNZJCSZYIYCYBXRYEYRUQYCXOXQYEVKY - EUUCYDYEUUBYSXPUNZJXPCYEURYOXPLUUBUUDQYEYOXPYSVITUUDYEYSXPXNXMVBVEVFVGTJC - DXMEXNFGVHVJVLVMYDYIYEYGYCXRYIYEVSZYCXQUUEXOXAXQUUEVSYBXAXQYIYECDEHIFGVNV - OUTVPVQVPVRXRXLYKQYCXRXCYEXKYJXQXCYEQXOEXPCVTTXQXKXGBXPXHOZPZAXPPZXOYJXQX - JUUGAEXPXQURXQXGBXIUUFEXPXHWAWBWCUUHXDYFKZBXPXMUFZOZPZXDYHKZBXPXNUFZOZPZR - XOYJUUGUULUUPAXMXNHWDZIWDZXEXMLZXGUUIBUUFUUKUUSXHUUJXPXEXMWJWEUUSXFYFXDXE - XMDNWFWGWCXEXNLZXGUUMBUUFUUOUUTXHUUNXPXEXNWJWEUUTXFYHXDXEXNDNWFWGWCWHXOUU - LYGUUPYIXOUULUUIBUUNPYGXOUUIBUUKUUNXMXNWIWBUUIYGBXNUURXDXNYFVTWKWLXOUUPUU - MBUUJPYIXOUUMBUUOUUJXMXNWMWBUUMYIBXMUUQXDXMYHVTWKWLWNWOWPWQTWRVMWTWSVQ $. + sneq bitrdi difprsn2 anbi12d bitrid sylan9bbr bibi12d rexlimdvva biimtrid + ralsn mpbird ) DUAKZEUBUCUDLZECKZBMZDAMZNUEZKZBEXEUFZOZPZAEPZQZXBHMZIMZUG + ZEXMXNUHZLZRZIESHESZXAXLEUIKXBXSQEDUJFUKEUIHIULUMXAXRXLHIEEXAXMEKZXNEKZRZ + RZXRXLYCXRRZXLXPCKZXNDXMNUEZKZXMDXNNUEZKZRZQZYDYEYJYDYEYJYDYERZYGYIYLYAXT + RZXNXMUGZXPJMZUNZJCSZYGYCYMXRYEYCXTYAXAYBUOZUPUQXRYNYCYEXOYNXQXOXMXNXOURU + SUTVAYEYQYDYEYPXPXNXMUHZUNZJYSCYEYSCKXPYSCXMXNVBZVCVDYOYSLYPYTQYEYOYSXPVI + TYTYEXPYSUUAVEVFVGTJCDXNEXMFGVHVJYLYBXOYSYOUNZJCSZYIYCYBXRYEYRUQYCXOXQYEV + KYEUUCYDYEUUBYSXPUNZJXPCYEURYOXPLUUBUUDQYEYOXPYSVITUUDYEYSXPXNXMVBVEVFVGT + JCDXMEXNFGVHVJVLVMYDYIYEYGYCXRYIYEVSZYCXQUUEXOXAXQUUEVSYBXAXQYIYECDEHIFGV + NVOUTVPVQVPVRXRXLYKQYCXRXCYEXKYJXQXCYEQXOEXPCVTTXQXKXGBXPXHOZPZAXPPZXOYJX + QXJUUGAEXPXQURXQXGBXIUUFEXPXHWAWBWCUUHXDYFKZBXPXMUFZOZPZXDYHKZBXPXNUFZOZP + ZRXOYJUUGUULUUPAXMXNHWDZIWDZXEXMLZXGUUIBUUFUUKUUSXHUUJXPXEXMWJWEUUSXFYFXD + XEXMDNWFWGWCXEXNLZXGUUMBUUFUUOUUTXHUUNXPXEXNWJWEUUTXFYHXDXEXNDNWFWGWCWHXO + UULYGUUPYIXOUULUUIBUUNPYGXOUUIBUUKUUNXMXNWIWBUUIYGBXNUURXDXNYFVTWSWKXOUUP + UUMBUUJPYIXOUUMBUUOUUJXMXNWLWBUUMYIBXMUUQXDXMYHVTWSWKWMWNWOWPTWTVMWQWRVQ + $. $} ${ @@ -416806,13 +421664,13 @@ function instead of the edges themselves (see also ~ nbgrval ). -> ( G NeighbVtx N ) = (/) ) $= ( vn cvv wcel cuhgr cv wnel cedg cfv wral wa cnbgr c0 wceq wi eqid expcom wn co cpr wss wrex cvtx csn cdif crab nbuhgr adantlr df-nel prssg syl6bir - wel simpl ad2antlr con3d syl5bi ralimdva ralnex sylib expd impcom expdimp - imp ralrimiv rabeq0 sylibr eqtrd id intnand nbgrprc0 syl a1d pm2.61i ) CE - FZBGFZCAHZIZABJKZLZMZBCNUAZOPZQWBVPWDWBVPMZWCCDHZUBVRUCZAVTUDZDBUEKZCUFUG - ZUHZOVQVPWCWKPWAADVTBCWIEWIRVTRUIUJWEWHTZDWJLWKOPWEWLDWJWBVPWFWJFZWLWAVQV - PWMMZWLQWAVQWNWLVQWNMZWAWLWOWAMWGTZAVTLZWLWOWAWQWOVSWPAVTVSCVRFZTWOVRVTFZ - MZWPCVRUKWTWGWRWNWGWRQVQWSWNWGWRDAUNZMWRCWFVREWJULWRXAUOUMUPUQURUSVEWGAVT - UTVASVBVCVDVFWHDWJVGVHVISVPTZWDWBXBBEFZVPMTWDXBVPXCXBVJVKBCVLVMVNVO $. + wel simpl ad2antlr con3d biimtrid ralimdva imp ralnex expd impcom expdimp + sylib ralrimiv rabeq0 sylibr eqtrd id intnand nbgrprc0 syl a1d pm2.61i ) + CEFZBGFZCAHZIZABJKZLZMZBCNUAZOPZQWBVPWDWBVPMZWCCDHZUBVRUCZAVTUDZDBUEKZCUF + UGZUHZOVQVPWCWKPWAADVTBCWIEWIRVTRUIUJWEWHTZDWJLWKOPWEWLDWJWBVPWFWJFZWLWAV + QVPWMMZWLQWAVQWNWLVQWNMZWAWLWOWAMWGTZAVTLZWLWOWAWQWOVSWPAVTVSCVRFZTWOVRVT + FZMZWPCVRUKWTWGWRWNWGWRQVQWSWNWGWRDAUNZMWRCWFVREWJULWRXAUOUMUPUQURUSUTWGA + VTVAVESVBVCVDVFWHDWJVGVHVISVPTZWDWBXBBEFZVPMTWDXBVPXCXBVJVKBCVLVMVNVO $. $} ${ @@ -416998,13 +421856,13 @@ function instead of the edges themselves (see also ~ nbgrval ). ( wcel wa cpr wceq weu wreu cfv eleq2i biimpi ad2antrl simprr cvtx simpll cusgr cv cedg usgredg2vtxeu syl3anc df-reu wi prcom eqeq2i eleq1d biimpcd adantld imp jca simpl eqid usgrpredgv simpld syl2an impbida eubidv biimpd - syl5bi mpd wb cnbgr co nbusgreledg bitrid anbi1d ad2antrr mpbird sylibr ) - DUCJZEGJZKZACJZEAJZKZKZBUDZFJZAEWCLZMZKZBNZWFBFOWBWHWCELZCJZWFKZBNZWBWFBD - UAPZOZWLWBVPADUEPZJZVTWNVPVQWAUBZVSWPVRVTVSWPCWOAHQRSVRVSVTTBADEUFUGWNWCW - MJZWFKZBNZWBWLWFBWMUHWBWTWLWBWSWKBWBWSWKWBWSKWJWFWBWSWJWBWFWJWRVSWFWJUIVR - VTWFVSWJWFAWICWFAWIMWEWIAEWCUJUKRULUMSUNUOWBWRWFTUPWBWKKWRWFWBVPWJWRWKWQW - JWFUQVPWJKWREWMJCDWCEWMHWMURUSUTVAWBWJWFTUPVBVCVDVEVFWBWGWKBVPWGWKVGVQWAV - PWDWJWFWDWCDEVHVIZJVPWJFXAWCIQCDEWCHVJVKVLVMVCVNWFBFUHVO $. + biimtrid mpd wb cnbgr co nbusgreledg bitrid anbi1d ad2antrr mpbird sylibr + ) DUCJZEGJZKZACJZEAJZKZKZBUDZFJZAEWCLZMZKZBNZWFBFOWBWHWCELZCJZWFKZBNZWBWF + BDUAPZOZWLWBVPADUEPZJZVTWNVPVQWAUBZVSWPVRVTVSWPCWOAHQRSVRVSVTTBADEUFUGWNW + CWMJZWFKZBNZWBWLWFBWMUHWBWTWLWBWSWKBWBWSWKWBWSKWJWFWBWSWJWBWFWJWRVSWFWJUI + VRVTWFVSWJWFAWICWFAWIMWEWIAEWCUJUKRULUMSUNUOWBWRWFTUPWBWKKWRWFWBVPWJWRWKW + QWJWFUQVPWJKWREWMJCDWCEWMHWMURUSUTVAWBWJWFTUPVBVCVDVEVFWBWGWKBVPWGWKVGVQW + AVPWDWJWFWDWCDEVHVIZJVPWJFXAWCIQCDEWCHVJVKVLVMVCVNWFBFUHVO $. $} ${ @@ -417020,13 +421878,13 @@ function instead of the edges themselves (see also ~ nbgrval ). nbusgredgeu0 $p |- ( ( ( G e. USGraph /\ U e. V ) /\ M e. N ) -> E! i e. I i = { U , M } ) $= ( wcel wa cv weu wreu cnbgr co cusgr wceq simpll eleq2i wb nbgrsym biimpd - cpr a1i syl5bi imp nbusgredgeu syl2anc df-reu sylib anass prid1g ad2antlr - eleq2 syl5ibrcom pm4.71rd bicomd anbi2d bitrid eubidv mpbird elrab2 eubii - anbi1i bitri sylibr ) EUANZAINZOZGHNZOZCPZDNZAVQNZOZVQAGUHZUBZOZCQZWBCFRZ - VPWDVRWBOZCQZVPWBCDRZWGVPVLAEGSTNZWHVLVMVOUCVNVOWIVOGEASTZNZVNWIHWJGLUDVN - WKWIWKWIUEVNEAGUFUIUGUJUKCDEAGKULUMWBCDUNUOVPWCWFCWCVRVSWBOZOVPWFVRVSWBUP - VPWLWBVRVPWBWLVPWBVSVPVSWBAWANZVMWMVLVOAGIUQURVQWAAUSUTVAVBVCVDVEVFWEVQFN - ZWBOZCQWDWBCFUNWOWCCWNVTWBABPZNVSBVQDFWPVQAUSMVGVIVHVJVK $. + cpr a1i biimtrid imp nbusgredgeu df-reu sylib anass prid1g ad2antlr eleq2 + syl2anc syl5ibrcom bicomd anbi2d bitrid eubidv mpbird elrab2 anbi1i eubii + pm4.71rd bitri sylibr ) EUANZAINZOZGHNZOZCPZDNZAVQNZOZVQAGUHZUBZOZCQZWBCF + RZVPWDVRWBOZCQZVPWBCDRZWGVPVLAEGSTNZWHVLVMVOUCVNVOWIVOGEASTZNZVNWIHWJGLUD + VNWKWIWKWIUEVNEAGUFUIUGUJUKCDEAGKULUSWBCDUMUNVPWCWFCWCVRVSWBOZOVPWFVRVSWB + UOVPWLWBVRVPWBWLVPWBVSVPVSWBAWANZVMWMVLVOAGIUPUQVQWAAURUTVIVAVBVCVDVEWEVQ + FNZWBOZCQWDWBCFUMWOWCCWNVTWBABPZNVSBVQDFWPVQAURMVFVGVHVJVK $. $d E n $. $d G e n $. $d I e n $. $d N e n $. $d U n $. $d V e n $. ${ @@ -417040,11 +421898,11 @@ is a bijection ( 1-1 onto function) in a simple graph. (Contributed -> F : N -1-1-onto-> I ) $= ( wcel wa cv cpr wral adantr cusgr wceq wreu wf1o co eleq2i nbusgreledg cnbgr wb prcom eleq1i biimpi adantl prid1g eleq2 elrab2 sylanbrc sylbid - ex syl5bi ralrimiv rabeq2i edgnbusgreu sylan2b ralrimiva f1ompt ) FUAOZ - AIOZPZACQZRZGOZCHSBQZVKUBCHUCZBGSHGEUDVIVLCHVJHOVJFAUHUEZOZVIVLHVOVJLUF - VIVPVJARZDOZVLVGVPVRUIVHDFAVJKUGTVIVRVLVIVRPVKDOZAVKOZVLVRVSVIVRVSVQVKD - VJAUJUKULUMVIVTVRVHVTVGAVJIUNUMTAVMOZVTBVKDGVMVKAUOMUPUQUSURUTVAVIVNBGV - MGOVIVMDOWAPVNWABGDMVBVMCDFAHIKLVCVDVECBHGVKENVFUQ $. + ex biimtrid ralrimiv rabeq2i edgnbusgreu sylan2b ralrimiva f1ompt ) FUA + OZAIOZPZACQZRZGOZCHSBQZVKUBCHUCZBGSHGEUDVIVLCHVJHOVJFAUHUEZOZVIVLHVOVJL + UFVIVPVJARZDOZVLVGVPVRUIVHDFAVJKUGTVIVRVLVIVRPVKDOZAVKOZVLVRVSVIVRVSVQV + KDVJAUJUKULUMVIVTVRVHVTVGAVJIUNUMTAVMOZVTBVKDGVMVKAUOMUPUQUSURUTVAVIVNB + GVMGOVIVMDOWAPVNWABGDMVBVMCDFAHIKLVCVDVECBHGVKENVFUQ $. $} $d I f $. $d N f n $. $d U f $. @@ -417540,10 +422398,10 @@ other vertices (of the graph), or equivalently, if all other vertices -> ( ( # ` ( G NeighbVtx U ) ) = ( ( # ` V ) - 1 ) -> U e. ( UnivVtx ` G ) ) ) $= ( vv cfusgr wcel wa cnbgr co chash cfv c1 cmin wceq cuvtx cdif nbgrssovtx - csn cv sseli wne eldifsn wi nbusgrvtxm1 syl5bi impbid2 eqrdv wb uvtxnbgrb - imp ad2antlr mpbird ex ) BFGZACGZHZBAIJZKLCKLMNJOZABPLGZUQUSHZUTURCASQZOZ - VAEURVBVAETZURGZVDVBGZURVBVDBCADRUAVFVDCGVDAUBHZVAVEVDCAUCUQUSVGVEUDABVDC - DUEUKUFUGUHUPUTVCUIUOUSBACDUJULUMUN $. + csn cv sseli wne eldifsn nbusgrvtxm1 imp biimtrid impbid2 eqrdv uvtxnbgrb + wi wb ad2antlr mpbird ex ) BFGZACGZHZBAIJZKLCKLMNJOZABPLGZUQUSHZUTURCASQZ + OZVAEURVBVAETZURGZVDVBGZURVBVDBCADRUAVFVDCGVDAUBHZVAVEVDCAUCUQUSVGVEUJABV + DCDUDUEUFUGUHUPUTVCUKUOUSBACDUIULUMUN $. $( A universal vertex has ` n - 1 ` neighbors in a finite simple graph with ` n ` vertices. A biconditional version of ~ nbusgrvtxm1uvtx resp. @@ -417733,15 +422591,15 @@ property of a complete graph (see also ~ cplgr0v ), but cannot be an cusgr cdif chash cpw crab iscusgredg wss cedg cvtx usgredgss pweqi rabeqi c2 3sstr4g wne wrex elss2prb weq sneq difeq2d preq2 raleqbidv rspcv simpr necom biimpi anim12i eldifsn sylibr preq1 syl id prcom eqtr2di biimpd a1d - ad2antll com23 3syld ex rexlimivv com13 imp syl5bi ssrdv eqssd sylbi ) CU - BKCUDKZGLZHLZMZBKZGDWMUCZUEZNZHDNZOZBALUFPUPQZADUGZUHZQHGBCDEFUIWTBXCWKBX - CUJWSWKCUKPXAACULPZUGZUHBXCACUMFXAAXBXEDXDEUNUOUQRWTUAXCBUALZXCKILZJLZURZ - XFXGXHMZQZOZJDUSIDUSZWTXFBKZIJAXFDUTWKWSXMXNSXMWSWKXNXLWSWKXNSZSZIJDDXGDK - ZXHDKZOZXLXPXSXLOZWSWLXGMZBKZGDXGUCZUEZNZXHXGMZBKZXOXSWSYESZXLXQYHXRWRYEH - XGDHIVAZWOYBGWQYDYIWPYCDWMXGVBVCYIWNYABWMXGWLVDTVEVFRRXTXHYDKZYEYGSXTXRXH - XGURZOYJXSXRXLYKXQXRVGXIYKXKXIYKXGXHVHVIRVJXHDXGVKVLYBYGGXHYDGJVAYAYFBWLX - HXGVMTVFVNXTWKYGXNXKWKYGXNSZSXSXIXKYLWKXKYGXNXKYFXFBXKXFXJYFXKVOXGXHVPVQT - VRVSVTWAWBWCWDWEWFWGWHWIWJ $. + ad2antll com23 3syld ex rexlimivv com13 imp biimtrid ssrdv eqssd sylbi ) + CUBKCUDKZGLZHLZMZBKZGDWMUCZUEZNZHDNZOZBALUFPUPQZADUGZUHZQHGBCDEFUIWTBXCWK + BXCUJWSWKCUKPXAACULPZUGZUHBXCACUMFXAAXBXEDXDEUNUOUQRWTUAXCBUALZXCKILZJLZU + RZXFXGXHMZQZOZJDUSIDUSZWTXFBKZIJAXFDUTWKWSXMXNSXMWSWKXNXLWSWKXNSZSZIJDDXG + DKZXHDKZOZXLXPXSXLOZWSWLXGMZBKZGDXGUCZUEZNZXHXGMZBKZXOXSWSYESZXLXQYHXRWRY + EHXGDHIVAZWOYBGWQYDYIWPYCDWMXGVBVCYIWNYABWMXGWLVDTVEVFRRXTXHYDKZYEYGSXTXR + XHXGURZOYJXSXRXLYKXQXRVGXIYKXKXIYKXGXHVHVIRVJXHDXGVKVLYBYGGXHYDGJVAYAYFBW + LXHXGVMTVFVNXTWKYGXNXKWKYGXNSZSXSXIXKYLWKXKYGXNXKYFXFBXKXFXJYFXKVOXGXHVPV + QTVRVSVTWAWBWCWDWEWFWGWHWIWJ $. $} $( The null graph (with no vertices and no edges) represented by the empty @@ -418259,12 +423117,12 @@ property of a complete graph (see also ~ cplgr0v ), but cannot be an cusgrfi $p |- ( ( G e. ComplUSGraph /\ E e. Fin ) -> V e. Fin ) $= ( vn vv ve vp ccusgr wcel cfn wn cv wi wa cpr wceq wrex expcom crab eqeq1 nfielex wne cpw csn cdif cmpt weq anbi2d rexbidv cbvrabv eqid cusgrfilem3 - notbid biimpac cedg cfv cusgrfilem1 eleq1i ssfi syl5bi con3d com23 adantl - wss syl mpd exlimddv com12 con4d imp ) BJKZALKZCLKZVMVOVNVOMZVMVNMZVPFNZC - KZVMVQOZFFCUCVPVSPGNZVRUDZHNZWAVRQZRZPZGCSZHCUEZUAZLKZMZVTVSVPWKVSVOWJIWI - ICVRUFUGINZVRQUHZBVRCGDWGWBWLWDRZPZGCSHIWHHIUIZWFWOGCWPWEWNWBWCWLWDUBUJUK - ULZWMUMUNUOUPVSWKVTOVPVSVMWKVQVMVSWKVQOZVMVSPWIBUQURZVFZWRIWIBVRCGDWQUSWT - VNWJVNWSLKZWTWJAWSLEUTXAWTWJWSWIVATVBVCVGTVDVEVHVIVJVKVL $. + notbid biimpac cedg cfv wss cusgrfilem1 eleq1i ssfi biimtrid con3d adantl + syl com23 mpd exlimddv com12 con4d imp ) BJKZALKZCLKZVMVOVNVOMZVMVNMZVPFN + ZCKZVMVQOZFFCUCVPVSPGNZVRUDZHNZWAVRQZRZPZGCSZHCUEZUAZLKZMZVTVSVPWKVSVOWJI + WIICVRUFUGINZVRQUHZBVRCGDWGWBWLWDRZPZGCSHIWHHIUIZWFWOGCWPWEWNWBWCWLWDUBUJ + UKULZWMUMUNUOUPVSWKVTOVPVSVMWKVQVMVSWKVQOZVMVSPWIBUQURZUSZWRIWIBVRCGDWQUT + WTVNWJVNWSLKZWTWJAWSLEVAXAWTWJWSWIVBTVCVDVFTVGVEVHVIVJVKVL $. $} ${ @@ -418283,14 +423141,14 @@ property of a complete graph (see also ~ cplgr0v ), but cannot be an ( va vb vn vk wcel wa cv cpr wceq wi wral cusgr ccusgr wne wrex usgredg csn cdif iscusgredg weq sneq difeq2d preq2 eleq1d raleqbidv rspcv simpl ve necomd anim2i eldifsn sylibr preq1 prcom eqeq2i eqcom sylbb ad2antll - syl biimpd syld syl9 impl adantld syl5bi ex rexlimivv impancom ssrdv ) - CUANZDUBNZOUQABVSUQPZANZVTWABNZVSWBOJPZKPZUCZWAWDWEQZRZOZKEUDJEUDVTWCSZ - WAACEJKFGUEWIWJJKEEWDENZWEENZOZWIWJVTDUANZLPZMPZQZBNZLEWPUFZUGZTZMETZOW - MWIOZWCMLBDEHIUHXCXBWCWNWKWLWIXBWCSWKXBWOWDQZBNZLEWDUFZUGZTZWLWIOZWCXAX - HMWDEMJUIZWRXELWTXGXJWSXFEWPWDUJUKXJWQXDBWPWDWOULUMUNUOXIXHWEWDQZBNZWCX - IWEXGNZXHXLSXIWLWEWDUCZOXMWIXNWLWIWDWEWFWHUPURUSWEEWDUTVAXEXLLWEXGLKUIX - DXKBWOWEWDVBUMUOVHWHXLWCSWLWFWHXLWCWHXKWABWHWAXKRXKWARWGXKWAWDWEVCVDWAX - KVEVFUMVIVGVJVKVLVMVNVOVPVHVQVR $. + syl biimpd syld syl9 impl adantld biimtrid ex rexlimivv impancom ssrdv + ) CUANZDUBNZOUQABVSUQPZANZVTWABNZVSWBOJPZKPZUCZWAWDWEQZRZOZKEUDJEUDVTWC + SZWAACEJKFGUEWIWJJKEEWDENZWEENZOZWIWJVTDUANZLPZMPZQZBNZLEWPUFZUGZTZMETZ + OWMWIOZWCMLBDEHIUHXCXBWCWNWKWLWIXBWCSWKXBWOWDQZBNZLEWDUFZUGZTZWLWIOZWCX + AXHMWDEMJUIZWRXELWTXGXJWSXFEWPWDUJUKXJWQXDBWPWDWOULUMUNUOXIXHWEWDQZBNZW + CXIWEXGNZXHXLSXIWLWEWDUCZOXMWIXNWLWIWDWEWFWHUPURUSWEEWDUTVAXEXLLWEXGLKU + IXDXKBWOWEWDVBUMUOVHWHXLWCSWLWFWHXLWCWHXKWABWHWAXKRXKWARWGXKWAWDWEVCVDW + AXKVEVFUMVIVGVJVKVLVMVNVOVPVHVQVR $. $( A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, @@ -418887,14 +423745,14 @@ property of a complete graph (see also ~ cplgr0v ), but cannot be an 1loopgrnb0 $p |- ( ph -> ( G NeighbVtx N ) = (/) ) $= ( vv cfv wcel csn cdif c0 wceq eleq2d mpbird wa cnbgr co cv cpr cedg cvtx crab cupgr cuspgr 1loopgruspgr uspgrupgr eqid nbupgr syl2anc wral difeq1d - syl wn eldifsn adantr simpr preqsnd syl6bi necon3ad expimpd syl5bi sylbid - wne imp 1loopgredg prex elsn bitrdi notbid ralrimiva rabeq0 sylibr eqtrd - wb ) ACDUAUBZDKUCZUDZCUELZMZKCUFLZDNZOZUGZPACUHMZDWEMZVTWHQACUIMWIABCDEFG - HIJUJCUKUQAWJDEMZIAWEEDGRSKWCCDWEWEULWCULUMUNAWDURZKWGUOWHPQAWLKWGAWAWGMZ - TWLWBWFQZURZAWMWOAWMWAEWFOZMZWOAWGWPWAAWEEWFGUPRWQWAEMZWADVHZTAWOWAEDUSAW - RWSWOAWRTZWNWADWTWNDDQZWADQZTXBWTDWADEEAWKWRIUTAWRVAVBXAXBVAVCVDVEVFVGVIA - WLWOVSWMAWDWNAWDWBWFNZMWNAWCXCWBABCDEFGHIJVJRWBWFDWAVKVLVMVNUTSVOWDKWGVPV - QVR $. + syl wn wne eldifsn adantr preqsnd syl6bi necon3ad expimpd biimtrid sylbid + simpr wb 1loopgredg prex elsn bitrdi notbid ralrimiva rabeq0 sylibr eqtrd + imp ) ACDUAUBZDKUCZUDZCUELZMZKCUFLZDNZOZUGZPACUHMZDWEMZVTWHQACUIMWIABCDEF + GHIJUJCUKUQAWJDEMZIAWEEDGRSKWCCDWEWEULWCULUMUNAWDURZKWGUOWHPQAWLKWGAWAWGM + ZTWLWBWFQZURZAWMWOAWMWAEWFOZMZWOAWGWPWAAWEEWFGUPRWQWAEMZWADUSZTAWOWAEDUTA + WRWSWOAWRTZWNWADWTWNDDQZWADQZTXBWTDWADEEAWKWRIVAAWRVHVBXAXBVHVCVDVEVFVGVS + AWLWOVIWMAWDWNAWDWBWFNZMWNAWCXCWBABCDEFGHIJVJRWBWFDWAVKVLVMVNVASVOWDKWGVP + VQVR $. $d A a e $. $d G a e $. $d N a e $. $d V a e $. $d X a e $. $d ph a e $. @@ -419335,11 +424193,11 @@ a loop (see ~ uspgr1v1eop ) is a singleton of a singleton. (Contributed by AV, 23-Dec-2020.) $) usgrvd0nedg $p |- ( ( G e. USGraph /\ U e. V ) -> ( ( ( VtxDeg ` G ) ` U ) = 0 -> -. E. v e. V { U , v } e. E ) ) $= - ( cusgr wcel wa cvtxdg cfv cc0 wceq cv cpr crab wn cvv syl5bi wrex eqeq1d - chash vtxdusgradjvtx c0 cvtx fvexi rabex hasheq0 ax-mp wral rabeq0 ralnex - wb wi biimpi a1i sylbid ) DHIBEIJZBDKLLZMNBAOPCIZAEQZUCLZMNZVAAEUARZUSUTV - CMABCDEFGUDUBVDVBUENZUSVEVBSIVDVFUNVAAEEDUFFUGUHVBSUIUJVFVARAEUKZUSVEVAAE - ULVGVEUOUSVGVEVAAEUMUPUQTTUR $. + ( cusgr wcel wa cvtxdg cfv cc0 wceq cv cpr crab wn cvv biimtrid eqeq1d c0 + chash wrex vtxdusgradjvtx wb cvtx fvexi rabex hasheq0 ax-mp rabeq0 ralnex + wral wi biimpi a1i sylbid ) DHIBEIJZBDKLLZMNBAOPCIZAEQZUCLZMNZVAAEUDRZUSU + TVCMABCDEFGUEUAVDVBUBNZUSVEVBSIVDVFUFVAAEEDUGFUHUIVBSUJUKVFVARAEUNZUSVEVA + AEULVGVEUOUSVGVEVAAEUMUPUQTTUR $. $d E e v $. $d G e $. $d V e $. $( If every vertex in a hypergraph has degree 0, there is no edge in the @@ -419350,13 +424208,13 @@ a loop (see ~ uspgr1v1eop ) is a singleton of a singleton. (Contributed ( ve wcel cv cfv wceq wral wn c0 wa wrex wne wex wss ex cuhgr cvtxdg eqid cc0 vtxduhgr0edgnel ralnex bitr4di ralbidva ralcom ralnex2 bitri cvtx cpw simpr csn cdif cedg eleq2i uhgredgn0 sylan2b eldifsn elpwi sseq2i ssn0rex - wi sylbir syl imp sylbi jca eximdv df-rex 3imtr4g con3d syl5bi nne syl6ib - n0 sylbid ) CUAHZAIZCUBJZJUDKZADLWAGIZHZMZGBLZADLZBNKZVTWCWGADVTWADHOWCWE - GBPMWGWBWAGBCDEFWBUCUEWEGBUFUGUHVTWHBNQZMZWIWHWEADPZGBPZMZVTWKWHWFADLGBLW - NWFAGDBUIWEGABDUJUKVTWJWMVTWDBHZGRWOWLOZGRWJWMVTWOWPGVTWOWPVTWOOZWOWLVTWO - UNWQWDCULJZUMZNUOUPHZWLWOVTWDCUQJZHWTBXAWDFURWDCUSUTWTWDWSHZWDNQZOWLWDWSN - VAXBXCWLXBWDWRSZXCWLVEZWDWRVBXDWDDSZXEDWRWDEVCXFXCWLAWDDVDTVFVGVHVIVGVJTV - KGBVRWLGBVLVMVNVOBNVPVQVS $. + wi sylbir syl imp sylbi jca eximdv n0 df-rex con3d biimtrid syl6ib sylbid + 3imtr4g nne ) CUAHZAIZCUBJZJUDKZADLWAGIZHZMZGBLZADLZBNKZVTWCWGADVTWADHOWC + WEGBPMWGWBWAGBCDEFWBUCUEWEGBUFUGUHVTWHBNQZMZWIWHWEADPZGBPZMZVTWKWHWFADLGB + LWNWFAGDBUIWEGABDUJUKVTWJWMVTWDBHZGRWOWLOZGRWJWMVTWOWPGVTWOWPVTWOOZWOWLVT + WOUNWQWDCULJZUMZNUOUPHZWLWOVTWDCUQJZHWTBXAWDFURWDCUSUTWTWDWSHZWDNQZOWLWDW + SNVAXBXCWLXBWDWRSZXCWLVEZWDWRVBXDWDDSZXEDWRWDEVCXFXCWLAWDDVDTVFVGVHVIVGVJ + TVKGBVLWLGBVMVRVNVOBNVSVPVQ $. $( If every vertex in a simple graph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) @@ -421111,15 +425969,15 @@ in common (for j=1, ... , k). In contrast to the definition in Aksoy et cdm cfz wf cv c1 caddc csn wss wif iswlkg wn wo df-ifp dfsn2 preq2 eqtrid cfzo eqeq2d biimpa a1d cedg upgredginwlk adantrr imp simp-4l simpr adantr eqid wne adantl fvexd neqne 3jca upgredgpr syl31anc eqcomd exp31 mpd jaoi - com12 syl5bi ifpprsnss impbid1 ralbidva pm5.32da df-3an 3bitr4g bitrd ) D - IJZCADUAKUBCEUEUCJZLCUDKZUFMFAUGZBUHZAKZWQUIUJMZAKZNZWQCKEKZWRUKZNZWRWTOZ - XBULZUMZBLWOVAMZPZQZWNWPXBXENZBXHPZQZABCDEFIGHUNWMWNWPRZXIRXNXLRXJXMWMXNX - IXLWMXNRZXGXKBXHXOWQXHJZRZXGXKXGXAXDRZXAUOZXFRZUPZXQXKXAXDXFUQYAXQXKXRXQX - KSXTXRXKXQXAXDXKXAXCXEXBXAXCWRWROXEWRURWRWTWRUSUTVBVCVDXQXTXKXQXBDVEKZJZX - TXKSXOXPYCWMWNXPYCSWPYBCDEWQHYBVLZVFVGVHXQYCXTXKXQYCRZXTRZXEXBYFWMYCXFWRT - JZWTTJZWRWTVMZQZXEXBNWMXNXPYCXTVIYEYCXTXQYCVJVKXTXFYEXSXFVJVNXTYJYEXSYJXF - XSYGYHYIXSWQAVOXSWSAVOWRWTVPVQVKVNWRWTXBTYBDFTGYDVRVSVTWAWBWDWCWDWEWRWTXB - WFWGWHWIWNWPXIWJWNWPXLWJWKWL $. + com12 biimtrid ifpprsnss impbid1 ralbidva pm5.32da df-3an 3bitr4g bitrd ) + DIJZCADUAKUBCEUEUCJZLCUDKZUFMFAUGZBUHZAKZWQUIUJMZAKZNZWQCKEKZWRUKZNZWRWTO + ZXBULZUMZBLWOVAMZPZQZWNWPXBXENZBXHPZQZABCDEFIGHUNWMWNWPRZXIRXNXLRXJXMWMXN + XIXLWMXNRZXGXKBXHXOWQXHJZRZXGXKXGXAXDRZXAUOZXFRZUPZXQXKXAXDXFUQYAXQXKXRXQ + XKSXTXRXKXQXAXDXKXAXCXEXBXAXCWRWROXEWRURWRWTWRUSUTVBVCVDXQXTXKXQXBDVEKZJZ + XTXKSXOXPYCWMWNXPYCSWPYBCDEWQHYBVLZVFVGVHXQYCXTXKXQYCRZXTRZXEXBYFWMYCXFWR + TJZWTTJZWRWTVMZQZXEXBNWMXNXPYCXTVIYEYCXTXQYCVJVKXTXFYEXSXFVJVNXTYJYEXSYJX + FXSYGYHYIXSWQAVOXSWSAVOWRWTVPVQVKVNWRWTXBTYBDFTGYDVRVSVTWAWBWDWCWDWEWRWTX + BWFWGWHWIWNWPXIWJWNWPXLWJWKWL $. $} ${ @@ -421150,9 +426008,10 @@ in common (for j=1, ... , k). In contrast to the definition in Aksoy et /\ A. k e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) $= ( cupgr wcel cwlks cfv cdm cword cc0 co wbr chash cfz wf cv c1 caddc wceq - cpr cfzo wral w3a c1st c2nd wlkcpr breq12i bitr4i upgriswlk biimpd syl5bi - imp ) DLMZGDNOZMZCEPQMRCUAOZUBSFAUCBUDZCOEOVEAOVEUEUFSAOUHUGBRVDUISUJUKZV - CCAVBTZVAVFVCGULOZGUMOZVBTVGDGUNCVHAVIVBJKUOUPVAVGVFABCDEFHIUQURUSUT $. + cpr cfzo wral w3a c1st c2nd wlkcpr breq12i bitr4i upgriswlk biimtrid imp + biimpd ) DLMZGDNOZMZCEPQMRCUAOZUBSFAUCBUDZCOEOVEAOVEUEUFSAOUHUGBRVDUISUJU + KZVCCAVBTZVAVFVCGULOZGUMOZVBTVGDGUNCVHAVIVBJKUOUPVAVGVFABCDEFHIUQUTURUS + $. $} ${ @@ -421201,35 +426060,35 @@ in common (for j=1, ... , k). In contrast to the definition in Aksoy et eqcoms bi2anan9r eqeq2 biimpd syl6bi com13 ral2imi 3ad2ant3 syl2and 3imp1 wf sylbir eqcom crn wf1 cedg wf1o uspgrf1oedg f1of1 eqidd edgval f1eq123d eqcomi mpbird 3ad2ant1 wlkelwrd eleq2d wrdsymbcl expcom f1veqaeq syl2an2r - jcad syl2an syl5bi ralimdva 3syld expimpd pm4.71d 3bitr4d ) DUAGZBDUBHZGZ - CYIGZIZEBUCHZUDHZJZUEZECUCHZUDHZJZAUFZYMHZYTYQHZJZAKELMZNZYTBUGHZHZYTCUGH - ZHZJZAKEUHMZNZUEZYSUULIZUUEIZBCJZUUNUUMUUOUIYPYSUUEUULUJUKYLYOUUPUUMUIZYH - YJYKYOUUQABCDEULUMUNYPUUNUUEYPYSUULUUEYPYSIZUULUUGYTVFUOMZUUFHZUPZUUIUUSU - UHHZUPZJZAUUDNZUUBDUQHZHZUUAUVFHZJZAUUDNZUUEUURUULUVEUURUULIZUUTUVBJZAUUD - NZUVEUVKFUFZVFUOMZUUFHZUVOUUHHZJZFUUDNUVMUVKUVRFUUDUURUVNUUDGZUULUVRUURUV - SIZUUJUVRAUVOUUKUVSUVOUUKGUURKEUVNURUSYTUVOJZUUJUVRUIUVTUWAUUGUVPUUIUVQYT - UVOUUFUTYTUVOUUHUTVAUSVGVBVCUVLUVRAFUUDYTUVNJUUTUVPUVBUVQYTUVNVFUUFUOVDYT - UVNVFUUHUOVDVAVEVHUURUULUVMUVEOZUURUULUUJAUUDNZUWBUUDUUKVIUULUWCOUURKEVJU - UJAUUDUUKVKVLUURUWCUVMUVEUWCUVMIUUJUVLIZAUUDNUURUVEUUJUVLAUUDVMUURUWDUVDA - UUDUWDUVDOUURUUGUUTUUIUVBVRUKVNVOVPVQPVSQYHYLYOYSUVEUVJOZYHYJYMUVFVTZWAZG - ZKYNUHMDWBHZUUFWSZUVHUVAJZAKYNLMZNZUEZYKYQUWGGZKYRUHMUWIUUHWSZUVGUVCJZAKY - RLMZNZUEZYOYSUWEOOZYHDWCGZYJUWNODWDZUXBYJUWNUUFAYMDUVFUWIBUWIRZUVFRZYMRZU - UFRZWEQWFYHUXBYKUWTOUXCUXBYKUWTUUHAYQDUVFUWICUXDUXEYQRZUUHRZWEQWFUWNUWTIZ - UXAOYHUXJYOYSUWEUWNUWTYOYSIZUWEOZUWMUWHUWTUXLOUWJUWTUWMUXLUWSUWOUWMUXLOUW - PUWSUWMUXLUXKUWSUWMIZUWEUXKUXMUWQAUUDNZUWKAUUDNZIZUWEYSUWSUXNYOUWMUXOYSUW - QAUWRUUDUWRUUDJYREYREKLWGWIWHYOUWKAUWLUUDUWLUUDJYNEYNEKLWGWIWHWJUXPUWQUWK - IZAUUDNUWEUWQUWKAUUDVMUXQUVDUVIAUUDUWQUWKUVDUVIOUVDUWKUWQUVIUVDUWKUVHUVCJ - ZUWQUVIOUVAUVCUVHWKUXRUWQUVIUWQUVIUIUVCUVHUVCUVHUVGWKWIWLWMWNPWOWTWMSQWPS - WPPVPUKWQWRUURUVIUUCAUUDUVIUVHUVGJZUURYTUUDGZIUUCUVGUVHXAUURUWFUVFXBZUVFX - CZUXTUUAUWFGZUUBUWFGZIZUXSUUCOYPUYBYSYHYLUYBYOYHUYBUWFDXDHZUVFXCZYHUWFUYF - UVFXEUYGUVFDUXEXFUWFUYFUVFXGWFYHUWFUWFUYAUYFUVFUVFYHUVFXHYHUWFXHUYAUYFJYH - UYFUYADXIXKUKXJXLXMTUURUXTUYEYPYSUXTUYEOZYLYOYSUYHOZYHYLYOUYIYLYOYSUYHYLU - XKUXTUYEYJUWHUWJIZUWOUWPIZUXKUXTIZUYEOZYKUUFYMDUVFUWIBUXDUXEUXFUXGXNUUHYQ - DUVFUWICUXDUXEUXHUXIXNUYJUYKUYMUWHUYKUYMOUWJUYKUWHUYMUWOUWHUYMOUWPUWOUWHU - YMUWOUWHIUYLUYCUYDUWHUYLUYCOUWOUYLUWHUYCUXKUXTUWHUYCOZYOUXTUYNOYSYOUXTYTU - WLGZUYNYOUUDUWLYTEYNKLWGXOUWHUYOUYCYTUWFYMXPXQWMTPSUSUWOUYLUYDOUWHUYLUWOU - YDUXKUXTUWOUYDOZYSUXTUYPOYOYSUXTYTUWRGZUYPYSUUDUWRYTEYRKLWGXOUWOUYQUYDYTU - WFYQXPXQWMUSPSTXTQTSTPYAVPVPPUNPPUWFUYAUUAUUBUVFXRXSYBYCYDYEYFYG $. + jcad syl2an biimtrid ralimdva 3syld expimpd pm4.71d 3bitr4d ) DUAGZBDUBHZ + GZCYIGZIZEBUCHZUDHZJZUEZECUCHZUDHZJZAUFZYMHZYTYQHZJZAKELMZNZYTBUGHZHZYTCU + GHZHZJZAKEUHMZNZUEZYSUULIZUUEIZBCJZUUNUUMUUOUIYPYSUUEUULUJUKYLYOUUPUUMUIZ + YHYJYKYOUUQABCDEULUMUNYPUUNUUEYPYSUULUUEYPYSIZUULUUGYTVFUOMZUUFHZUPZUUIUU + SUUHHZUPZJZAUUDNZUUBDUQHZHZUUAUVFHZJZAUUDNZUUEUURUULUVEUURUULIZUUTUVBJZAU + UDNZUVEUVKFUFZVFUOMZUUFHZUVOUUHHZJZFUUDNUVMUVKUVRFUUDUURUVNUUDGZUULUVRUUR + UVSIZUUJUVRAUVOUUKUVSUVOUUKGUURKEUVNURUSYTUVOJZUUJUVRUIUVTUWAUUGUVPUUIUVQ + YTUVOUUFUTYTUVOUUHUTVAUSVGVBVCUVLUVRAFUUDYTUVNJUUTUVPUVBUVQYTUVNVFUUFUOVD + YTUVNVFUUHUOVDVAVEVHUURUULUVMUVEOZUURUULUUJAUUDNZUWBUUDUUKVIUULUWCOUURKEV + JUUJAUUDUUKVKVLUURUWCUVMUVEUWCUVMIUUJUVLIZAUUDNUURUVEUUJUVLAUUDVMUURUWDUV + DAUUDUWDUVDOUURUUGUUTUUIUVBVRUKVNVOVPVQPVSQYHYLYOYSUVEUVJOZYHYJYMUVFVTZWA + ZGZKYNUHMDWBHZUUFWSZUVHUVAJZAKYNLMZNZUEZYKYQUWGGZKYRUHMUWIUUHWSZUVGUVCJZA + KYRLMZNZUEZYOYSUWEOOZYHDWCGZYJUWNODWDZUXBYJUWNUUFAYMDUVFUWIBUWIRZUVFRZYMR + ZUUFRZWEQWFYHUXBYKUWTOUXCUXBYKUWTUUHAYQDUVFUWICUXDUXEYQRZUUHRZWEQWFUWNUWT + IZUXAOYHUXJYOYSUWEUWNUWTYOYSIZUWEOZUWMUWHUWTUXLOUWJUWTUWMUXLUWSUWOUWMUXLO + UWPUWSUWMUXLUXKUWSUWMIZUWEUXKUXMUWQAUUDNZUWKAUUDNZIZUWEYSUWSUXNYOUWMUXOYS + UWQAUWRUUDUWRUUDJYREYREKLWGWIWHYOUWKAUWLUUDUWLUUDJYNEYNEKLWGWIWHWJUXPUWQU + WKIZAUUDNUWEUWQUWKAUUDVMUXQUVDUVIAUUDUWQUWKUVDUVIOUVDUWKUWQUVIUVDUWKUVHUV + CJZUWQUVIOUVAUVCUVHWKUXRUWQUVIUWQUVIUIUVCUVHUVCUVHUVGWKWIWLWMWNPWOWTWMSQW + PSWPPVPUKWQWRUURUVIUUCAUUDUVIUVHUVGJZUURYTUUDGZIUUCUVGUVHXAUURUWFUVFXBZUV + FXCZUXTUUAUWFGZUUBUWFGZIZUXSUUCOYPUYBYSYHYLUYBYOYHUYBUWFDXDHZUVFXCZYHUWFU + YFUVFXEUYGUVFDUXEXFUWFUYFUVFXGWFYHUWFUWFUYAUYFUVFUVFYHUVFXHYHUWFXHUYAUYFJ + YHUYFUYADXIXKUKXJXLXMTUURUXTUYEYPYSUXTUYEOZYLYOYSUYHOZYHYLYOUYIYLYOYSUYHY + LUXKUXTUYEYJUWHUWJIZUWOUWPIZUXKUXTIZUYEOZYKUUFYMDUVFUWIBUXDUXEUXFUXGXNUUH + YQDUVFUWICUXDUXEUXHUXIXNUYJUYKUYMUWHUYKUYMOUWJUYKUWHUYMUWOUWHUYMOUWPUWOUW + HUYMUWOUWHIUYLUYCUYDUWHUYLUYCOUWOUYLUWHUYCUXKUXTUWHUYCOZYOUXTUYNOYSYOUXTY + TUWLGZUYNYOUUDUWLYTEYNKLWGXOUWHUYOUYCYTUWFYMXPXQWMTPSUSUWOUYLUYDOUWHUYLUW + OUYDUXKUXTUWOUYDOZYSUXTUYPOYOYSUXTYTUWRGZUYPYSUUDUWRYTEYRKLWGXOUWOUYQUYDY + TUWFYQXPXQWMUSPSTXTQTSTPYAVPVPPUNPPUWFUYAUUAUUBUVFXRXSYBYCYDYEYFYG $. $} ${ @@ -421258,14 +426117,14 @@ in common (for j=1, ... , k). In contrast to the definition in Aksoy et -> A = B ) $= ( wcel cfv wa c2nd wceq c1st chash wbr wi wlkcpr c1 cmin adantl wlklenvm1 co simpl anim12i cuspgr cwlks w3a wlkcl fveq2 oveq1d eqcomd wb eqeqan12rd - cn0 adantr mpbird anim2i exp44 mpcom sylbi imp31 eqidd simpr uspgr2wlkeq2 - syl5bi 3adant1 syl3anc ex com23 3impia mpd ) CUADZACUBEZDZBVIDZFZAGEZBGEZ - HZUCAIEZJEZUJDZBIEZJEZVQHZFZABHZVLVOWBVHVJVKVOWBVJVPVMVIKZVKVOWBLZLCAMVKV - SVNVIKZWDWECBMVRWDWFWELVMVPCUDVRWDWFVOWBWDWFFZVOFZWAVRWHWAVNJEZNORZVMJEZN - ORZHZVOWMWGVOWLWJVOWKWINOVMVNJUEUFUGPWGWAWMUHVOWFWDVTWJVQWLVNVSCQVMVPCQUI - UKULUMUNUOVAUPUQVBVHVLVOWBWCLVHVLFZWBVOWCWNWBVOWCLZWNWBFVHVRFVJVQVQHZFVKW - AFWOWNVHWBVRVHVLSVRWASTWNVJWBWPVLVJVHVJVKSPWBVQURTWNVKWBWAVLVKVHVJVKUSPVR - WAUSTABCVQUTVCVDVEVFVG $. + adantr mpbird anim2i exp44 mpcom biimtrid sylbi imp31 3adant1 eqidd simpr + cn0 uspgr2wlkeq2 syl3anc ex com23 3impia mpd ) CUADZACUBEZDZBVIDZFZAGEZBG + EZHZUCAIEZJEZVADZBIEZJEZVQHZFZABHZVLVOWBVHVJVKVOWBVJVPVMVIKZVKVOWBLZLCAMV + KVSVNVIKZWDWECBMVRWDWFWELVMVPCUDVRWDWFVOWBWDWFFZVOFZWAVRWHWAVNJEZNORZVMJE + ZNORZHZVOWMWGVOWLWJVOWKWINOVMVNJUEUFUGPWGWAWMUHVOWFWDVTWJVQWLVNVSCQVMVPCQ + UIUJUKULUMUNUOUPUQURVHVLVOWBWCLVHVLFZWBVOWCWNWBVOWCLZWNWBFVHVRFVJVQVQHZFV + KWAFWOWNVHWBVRVHVLSVRWASTWNVJWBWPVLVJVHVJVKSPWBVQUSTWNVKWBWAVLVKVHVJVKUTP + VRWAUTTABCVQVBVCVDVEVFVG $. $} ${ @@ -421300,11 +426159,11 @@ in common (for j=1, ... , k). In contrast to the definition in Aksoy et ( cvtx cfv c0 wceq cwlks wcel c1st c2nd wa wbr wlkcpr ciedg cc0 wf eqid jca wi cz cdm cword chash cfz co wlkf wlkp feq3 f00 bitrdi cn0 clt nn0z sylancr wb 0z fzn nn0nlt0 pm2.21d sylbird com12 adantl lencl impel simpll ex syl6bi - impcomd syl5 syl5bi imp ) ACDZEFZBAGDZHZBIDZEFZBJDZEFZKZVOVPVRVNLZVMVTABMWA - VPANDZUAZUBHZOVPUCDZUDUEZVLVRPZKVMVTWAWDWGVRVPAWBWBQUFVRVPAVLVLQUGRVMWGWDVT - VMWGVSWFEFZKZWDVTSVMWGWFEVRPWIVLEWFVRUHWFVRUIUJWIWDVTWIWDKVQVSWIWEUKHZVQWDW - HWJVQSVSWJWHVQWJWHWEOULLZVQWJOTHWETHWKWHUOUPWEUMOWEUQUNWJWKVQWEURUSUTVAVBWC - VPVCVDVSWHWDVERVFVGVHVIVJVK $. + impcomd syl5 biimtrid imp ) ACDZEFZBAGDZHZBIDZEFZBJDZEFZKZVOVPVRVNLZVMVTABM + WAVPANDZUAZUBHZOVPUCDZUDUEZVLVRPZKVMVTWAWDWGVRVPAWBWBQUFVRVPAVLVLQUGRVMWGWD + VTVMWGVSWFEFZKZWDVTSVMWGWFEVRPWIVLEWFVRUHWFVRUIUJWIWDVTWIWDKVQVSWIWEUKHZVQW + DWHWJVQSVSWJWHVQWJWHWEOULLZVQWJOTHWETHWKWHUOUPWEUMOWEUQUNWJWKVQWEURUSUTVAVB + WCVPVCVDVSWHWDVERVFVGVHVIVJVK $. ${ $d G w $. @@ -421346,25 +426205,6 @@ in common (for j=1, ... , k). In contrast to the definition in Aksoy et VSVJVMCVCUIUJVSVJUKULRVRVTVPVRVGVONVHVGTKVNVGUMUNVNVOTKVHVNVOVMCUOUPRVRUQUR USUTVAVB $. - $( Obsolete version of ~ wlklenvclwlk as of 14-Jan-2024. The number of - vertices in a walk equals the length of the walk after it is "closed" - (i.e. enhanced by an edge from its last vertex to its first vertex). - (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, - 2-May-2021.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - wlklenvclwlkOLD $p |- ( ( W e. Word ( Vtx ` G ) /\ 1 <_ ( # ` W ) ) - -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. ( Walks ` G ) - -> ( # ` F ) = ( # ` W ) ) ) $= - ( cc0 cfv cs1 cconcat co wcel c1 chash wbr wa wceq wi caddc cc nn0cn adantr - cn0 cwlks cvtx cword df-br wlklenvp1 wlkcl wrdsymb1 s1cld ccatlenOLD syldan - cop cle s1len a1i oveq2d eqtrd eqeq1d lencl eqcom adantl 1cnd biimpd syl5bi - addcan2d ex com23 syl sylbid com3l sylc sylbir com12 ) ACDCEZFZGHZUKBUAEZIZ - CBUBEZUCZIZJCKEZULLZMZAKEZWANZVQAVOVPLZWCWEOZAVOVPUDWFVOKEZWDJPHZNZWDTIZWGV - OABUEVOABUFWCWJWKWEWCWJWAJPHZWINZWKWEOZWCWHWLWIWCWHWAVNKEZPHZWLVTWBVNVSIWHW - PNWCVMVRVRCUGUHVRCVNUIUJWCWOJWAPWOJNWCVMUMUNUOUPUQVTWMWNOZWBVTWATIZWQVRCURW - RWKWMWEWRWKWMWEOWMWIWLNZWRWKMZWEWLWIUSWTWSWEWTWDWAJWKWDQIWRWDRUTWRWAQIWKWAR - SWTVAVDVBVCVEVFVGSVHVIVJVKVL $. - ${ $d A a b f g p $. $d B a b f g p $. $d G a b f g p $. $d V f g p $. wlkson.v $e |- V = ( Vtx ` G ) $. @@ -421734,18 +426574,18 @@ segment of the walk (of length ` N ` ) forms a walk on the subgraph ( vx cv cfv wceq c1 caddc co ciedg w3a cc0 cfzo wcel wral wlkp1lem5 wa wi cfz elfzofz adantl fveq2 eqeq12d rspcv syl imp fzofzp1 cop csn cun adantr fveq1i wne wn fzonel eleq1 mtbii con2d neqned fvunsn eqtrid fveq12d chash - a1i cdm oveq2i eleq2i cword cwlks wbr wlkf wrdsymbcl ex syl5bi syl5ibrcom - con3d mpid eqtrd 3jca mpidan ralrimiva ) AGULZEUMZXJDUMZUNZXJUOUPUQZEUMZX - NDUMZUNZXJKUMZFURUMZUMZXJIUMZLUMZUNZUSZGUTMVAUQZAXJYEVBZUKULZEUMZYGDUMZUN - ZUKUTMVGUQZVCZYDABCDEFUKHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJVDAYFVEZYLVEXMXQ - YCYMYLXMYMXJYKVBZYLXMVFYFYNAXJUTMVHVIYJXMUKXJYKYGXJUNYHXKYIXLYGXJEVJYGXJD - VJVKVLVMVNYMYLXQYMXNYKVBZYLXQVFYFYOAUTMXJVOVIYJXQUKXNYKYGXNUNYHXOYIXPYGXN - EVJYGXNDVJVKVLVMVNYMYCYLYMXTYALBHVPVQVRZUMZYBYMXRYAXSYPAXSYPUNYFUGVSYMXRX - JIMBVPVQVRZUMZYAXJKYRUHVTYMMXJWAYSYAUNYMMXJAYFMXJUNZWBAYTYFYTYFWBVFAYTMYE - VBYFUTMWCMXJYEWDWEWLWFVNWGIMBXJWHVMWIWJYMBYAWAYQYBUNYMBYAAYFBYAUNZWBZAYFB - LWMZVBZWBZUUBUBAYFUUEUUBVFYMUUAUUDYMUUDUUAYAUUCVBZAYFUUFYFXJUTIWKUMZVAUQZ - VBZAUUFYEUUHXJMUUGUTVAUDWNWOAIUUCWPVBZUUIUUFVFAIDJWQUMWRUUJUCDIJLQWSVMUUJ - UUIUUFXJUUCIWTXAVMXBVNBYAUUCWDXCXDXAXEVNWGLBHYAWHVMXFVSXGXHXI $. + a1i cdm oveq2i eleq2i cword cwlks wbr wrdsymbcl biimtrid syl5ibrcom con3d + wlkf ex mpid eqtrd 3jca mpidan ralrimiva ) AGULZEUMZXJDUMZUNZXJUOUPUQZEUM + ZXNDUMZUNZXJKUMZFURUMZUMZXJIUMZLUMZUNZUSZGUTMVAUQZAXJYEVBZUKULZEUMZYGDUMZ + UNZUKUTMVGUQZVCZYDABCDEFUKHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJVDAYFVEZYLVEXM + XQYCYMYLXMYMXJYKVBZYLXMVFYFYNAXJUTMVHVIYJXMUKXJYKYGXJUNYHXKYIXLYGXJEVJYGX + JDVJVKVLVMVNYMYLXQYMXNYKVBZYLXQVFYFYOAUTMXJVOVIYJXQUKXNYKYGXNUNYHXOYIXPYG + XNEVJYGXNDVJVKVLVMVNYMYCYLYMXTYALBHVPVQVRZUMZYBYMXRYAXSYPAXSYPUNYFUGVSYMX + RXJIMBVPVQVRZUMZYAXJKYRUHVTYMMXJWAYSYAUNYMMXJAYFMXJUNZWBAYTYFYTYFWBVFAYTM + YEVBYFUTMWCMXJYEWDWEWLWFVNWGIMBXJWHVMWIWJYMBYAWAYQYBUNYMBYAAYFBYAUNZWBZAY + FBLWMZVBZWBZUUBUBAYFUUEUUBVFYMUUAUUDYMUUDUUAYAUUCVBZAYFUUFYFXJUTIWKUMZVAU + QZVBZAUUFYEUUHXJMUUGUTVAUDWNWOAIUUCWPVBZUUIUUFVFAIDJWQUMWRUUJUCDIJLQXCVMU + UJUUIUUFXJUUCIWSXDVMWTVNBYAUUCWDXAXBXDXEVNWGLBHYAWHVMXFVSXGXHXI $. $d N k $. $d P k $. $d Q k $. $( Lemma for ~ wlkp1 . (Contributed by AV, 6-Mar-2021.) $) @@ -421925,16 +426765,16 @@ segment of the walk (of length ` N ` ) forms a walk on the subgraph caddc wne cc0 cfzo wral cword cfz cvtx csn cpr wss wif wb wlkv eqid iswlk ifptru adantr simplr wrdsymbcl ad4ant14 ffvelcdmd fveq2 breq2d elrab fvex w3a syl hashsng ax-mp breq2i clt 1lt2 wn ltnlei pm2.21 sylbi syl6bi com12 - 1re 2re adantl a1i syl5bi mpd sylbid ex neqne 2a1d pm2.61i ralimdva com23 - 3impia pm2.43i imp ) DBEUAIJZFUBZKAUCZLIZMJZAGUDZUEZFUFZCUCZBIZXJNUGOBIZU - HZCUIDLIZUJOZUKZXBXIXPPZXBXBDXCULQZUIXNUMOEUNIZBUFZXKXLRZXJDIZFIZXKUOZRZX - KXLUPYCUQZURZCXOUKZVMZXQXBESQDSQBSQVMXBYIUSBDEUTBSCDEFXSSSXSVAHVBVNXRXTYH - XQXRXTTZXIYHXPYJXIYHXPPYJXITZYGXMCXOYAYKXJXOQZTZYGXMPZPYAYMYNYAYMTYGYEXMY - AYGYEUSYMYAYEYFVCVDYMYEXMPZYAYMYCXHQZYOYMXCXHYBFYJXIYLVEXRYLYBXCQXTXIXJXC - DVFVGVHYPYCXGQZKYCLIZMJZTZYMYOXFYSAYCXGXDYCRXEYRKMXDYCLVIVJVKYTYOPYMYSYOY - QYEYSXMYEYSKYDLIZMJZXMYEYRUUAKMYCYDLVIVJUUBKNMJZXMUUANKMXKSQUUANRXJBVLXKS - VOVPVQNKVRJZUUCXMPZVSUUDUUCVTUUENKWFWGWAUUCXMWBWCVPWCWDWEWHWIWJWKWHWLWMYA - VTXMYMYGXKXLWNWOWPWQWMWRWSWDWTXA $. + 1re 2re adantl a1i biimtrid mpd sylbid neqne 2a1d pm2.61i ralimdva 3impia + ex com23 pm2.43i imp ) DBEUAIJZFUBZKAUCZLIZMJZAGUDZUEZFUFZCUCZBIZXJNUGOBI + ZUHZCUIDLIZUJOZUKZXBXIXPPZXBXBDXCULQZUIXNUMOEUNIZBUFZXKXLRZXJDIZFIZXKUOZR + ZXKXLUPYCUQZURZCXOUKZVMZXQXBESQDSQBSQVMXBYIUSBDEUTBSCDEFXSSSXSVAHVBVNXRXT + YHXQXRXTTZXIYHXPYJXIYHXPPYJXITZYGXMCXOYAYKXJXOQZTZYGXMPZPYAYMYNYAYMTYGYEX + MYAYGYEUSYMYAYEYFVCVDYMYEXMPZYAYMYCXHQZYOYMXCXHYBFYJXIYLVEXRYLYBXCQXTXIXJ + XCDVFVGVHYPYCXGQZKYCLIZMJZTZYMYOXFYSAYCXGXDYCRXEYRKMXDYCLVIVJVKYTYOPYMYSY + OYQYEYSXMYEYSKYDLIZMJZXMYEYRUUAKMYCYDLVIVJUUBKNMJZXMUUANKMXKSQUUANRXJBVLX + KSVOVPVQNKVRJZUUCXMPZVSUUDUUCVTUUENKWFWGWAUUCXMWBWCVPWCWDWEWHWIWJWKWHWLWR + YAVTXMYMYGXKXLWMWNWOWPWRWSWQWDWTXA $. lfgriswlk.v $e |- V = ( Vtx ` G ) $. $( Conditions for a pair of functions to be a walk in a loop-free graph. @@ -422524,16 +427364,16 @@ where f enumerates the (indices of the) different edges, and p cima cdm cword cfz cvtx caddc csn wss wif istrl eqid iswlkg anbi1d bitrid wf wn wne pthdadjvtx ad5ant245 neneqd wb ifpfal adantl prsshashgt1 sylbid neqne syl31anc mpdan ralimdva ex com23 exp31 com24 3impia exp4c imp com14 - syl6bi 3imp com12 syl5bi 3ad2ant1 mpcom pm2.43i ) CADUAGHZICJGZUBHZUCBUDZ - CGZEGZJGUEHZBKXDLMZUFZXCXEXKNZDOPZCOPZAOPZQZXCXCXLNZXCCADUGGHZXPACDUHACDU - IUJXMXNXCXQNXOXCCADUKGHZAIXDLMZULUMUNZAKXDUOUSAXTUSUPUQRZQZXMXQACDURYCXMX - QXSYAYBXMXQNXMYAYBXSXQXMXSYBYAXQXMXSCEUTVAPZKXDVBMDVCGZAVMZXFAGZXFIVDMZAG - ZRZXHYGVERZYGYIUOXHVFZVGZBXJUFZQZCUMUNZSZYBYAXQNNZXSXRYPSXMYQACDVHXMXRYOY - PABCDEYEOYEVIFVJVKVLYOYPYRYOYPYBYAXQYDYFYNYPYBSYASZXQNYDYFSZXCYSYNXLYTXCY - SYNXLNYTXCSYSSZXEYNXKUUAXEYNXKNUUAXESZYMXIBXJUUBXFXJPZSZYJVNZYMXINUUDYGYI - XCXEUUCYGYIVOZYTYSACDXFVPVQVRUUDUUESYMYLXIUUEYMYLVSUUDYJYKYLVTWAUUEYLXINZ - UUDUUEYGOPYIOPUUFXHOPUUGUUEXFATUUEYHATYGYIWDUUEXGETYGYIXHOOOWBWEWAWCWFWGW - HWIWJWKWLWMWNWPWKWOWQWRWSWTXAXBWN $. + syl6bi 3imp com12 biimtrid 3ad2ant1 mpcom pm2.43i ) CADUAGHZICJGZUBHZUCBU + DZCGZEGZJGUEHZBKXDLMZUFZXCXEXKNZDOPZCOPZAOPZQZXCXCXLNZXCCADUGGHZXPACDUHAC + DUIUJXMXNXCXQNXOXCCADUKGHZAIXDLMZULUMUNZAKXDUOUSAXTUSUPUQRZQZXMXQACDURYCX + MXQXSYAYBXMXQNXMYAYBXSXQXMXSYBYAXQXMXSCEUTVAPZKXDVBMDVCGZAVMZXFAGZXFIVDMZ + AGZRZXHYGVERZYGYIUOXHVFZVGZBXJUFZQZCUMUNZSZYBYAXQNNZXSXRYPSXMYQACDVHXMXRY + OYPABCDEYEOYEVIFVJVKVLYOYPYRYOYPYBYAXQYDYFYNYPYBSYASZXQNYDYFSZXCYSYNXLYTX + CYSYNXLNYTXCSYSSZXEYNXKUUAXEYNXKNUUAXESZYMXIBXJUUBXFXJPZSZYJVNZYMXINUUDYG + YIXCXEUUCYGYIVOZYTYSACDXFVPVQVRUUDUUESYMYLXIUUEYMYLVSUUDYJYKYLVTWAUUEYLXI + NZUUDUUEYGOPYIOPUUFXHOPUUGUUEXFATUUEYHATYGYIWDUUEXGETYGYIXHOOOWBWEWAWCWFW + GWHWIWJWKWLWMWNWPWKWOWQWRWSWTXAXBWN $. $( A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.) $) @@ -422568,11 +427408,11 @@ where f enumerates the (indices of the) different edges, and p -> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) $= ( cspths cfv wbr chash cc0 wne ctrls ccnv wfun isspth cfz wcel wceq syl jca wa wi co cvtx wf1 wf cwlks trliswlk eqid wlkp anim1i df-f1 sylibr cn0 wlkcl - nn0fz0 biimpi 0elfz 3syl adantr eqcom f1veqaeq syl5bi necon3d sylbi imp ) B - ACDEFZBGEZHIZHAEZVFAEZIZVEBACJEFZAKLZSZVGVJTABCMVMVHVIVFHVMHVFNUAZCUBEZAUCZ - VFVNOZHVNOZSZSZVHVIPZVFHPZTVMVPVSVMVNVOAUDZVLSVPVKWCVLVKBACUEEFZWCABCUFZABC - VOVOUGUHQUIVNVOAUJUKVKVSVLVKWDVFULOZVSWEABCUMWFVQVRWFVQVFUNUOVFUPRUQURRWAVI - VHPVTWBVHVIUSVNVOVFHAUTVAQVBVCVD $. + nn0fz0 biimpi 0elfz 3syl adantr eqcom f1veqaeq biimtrid necon3d sylbi imp ) + BACDEFZBGEZHIZHAEZVFAEZIZVEBACJEFZAKLZSZVGVJTABCMVMVHVIVFHVMHVFNUAZCUBEZAUC + ZVFVNOZHVNOZSZSZVHVIPZVFHPZTVMVPVSVMVNVOAUDZVLSVPVKWCVLVKBACUEEFZWCABCUFZAB + CVOVOUGUHQUIVNVOAUJUKVKVSVLVKWDVFULOZVSWEABCUMWFVQVRWFVQVFUNUOVFUPRUQURRWAV + IVHPVTWBVHVIUSVNVOVFHAUTVAQVBVCVD $. $( A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, @@ -422605,36 +427445,36 @@ where f enumerates the (indices of the) different edges, and p simpl preq12b dff13 elfzofz fveqeq2 eqeq1 eqeq2d eqeq2 rspc2v syl2an a1dd imbi12d com14 cn0 hashcl a1i fzofzp1 anim12d1 imp anim12d wb oveq1 eqeq1d expimpd elfzonn0 c2 cc nn0cn add1p1 wne addn0nid syl3anc eqneqall syl5com - 2cnd 2ne0 sylbid syld ex com3l expd com34 jaoi adantld syl5bi com23 sylbi - com15 impcom ralrimivv adantlr sylanbrc df-f1 sylib syl2anc ) CDUAZUBJZKC - UCLZUDMZEAUEZBUFZCLDLZYJALZYJNOMALZUGZPZBKYGUHMZUIZCUJUKZQZYFCULJZYPYECUM - ZYIYSQYECUNYECUOYTUUARZYIYQYRUUBYIYQRZYRUUBUUCRZUUAYRRZYRUUDYPYECUEZUUEUU - DUUAFUFZCLZGUFZCLZPZFGSZQZGYPUIFYPUIZUUFUUBUUAUUCYTUUAUPUQYTUUCUUNUUAYTUU - CRUUMFGYPYPUUCYTUUGYPJZUUIYPJZRZUUMQZYIYQYTUURQUUQYQYTYIUUMUUQYQUUHDLZUUG - ALZUUGNOMZALZUGZPZUUJDLZUUIALZUUINOMZALZUGZPZRZYTYIUUMQQUUOYQUVDUUPUVJYOU - VDBUUGYPBFSZYKUUSYNUVCYJUUGDCURUVLYLUUTYMUVBYJUUGAUSYJUUGNAOUTVAVBVCYOUVJ - BUUIYPBGSZYKUVEYNUVIYJUUIDCURUVMYLUVFYMUVHYJUUIAUSYJUUINAOUTVAVBVCVDUUKUV - 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necon3d 3jca jctild com23 mpdd syl5bi fveq2 neeq2d oveq2 fzo0to2pr eqtrdi - raleqdv feq2d imbi1d imbi12d syl5ibrcom com24 3impd sylbid imp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necon3d 3jca jctild com23 mpdd biimtrid fveq2 neeq2d oveq2 eqtrdi raleqdv + fzo0to2pr feq2d imbi1d imbi12d syl5ibrcom com24 3impd sylbid imp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} $( Lemma 1 for ~ usgr2wlkspth . (Contributed by Alexander van der Vekens, @@ -422973,20 +427813,20 @@ where f enumerates the (indices of the) different edges, and p cfz w3a syl eqidd oveq2 fzo0to2pr eqtrdi f1eq123d raleqdv 2wlklem anbi12d bitrdi adantl c0ex pm3.2i 0ne1 f12dfv mp2an cedg wf1o usgrf1oedg f1of1 id 1ex prid1 a1i ffvelcdmd prid2 jca anim1ci necon3d simpl simpr preq1 prcom - f1veqaeq neeq12d necon3i syl6bi com12 syl6 expcom impd com23 mpcom syl5bi - a1d adantr sylbid 3adant2 expdcom imp ) CUCEZBUDFZGHZBACUEFUAZIAFZGAFZJZK - XGXJXIXMXGXJIXHLMZCUFFZUGZBNZIXHUOMCUHFZAUBZDUIZBFXOFXTAFXTOUJMAFPHZDXNUK - ZUPZXIXMKXGCULEXJYCQCUMADBCXOXRXRRXORZUNUQXGXIYCXMYCXGXIXMXQYBXGXISZXMKXS - YEXQYBSZXMYEYFIOPZXPBNZIBFZXOFZXKOAFZPZHZOBFZXOFZYKXLPZHZSZSZXMXIYFYSQXGX - IXQYHYBYRXIXNYGXPXPBBXIBURXIXNIGLMYGXHGILUSUTVAZXIXPURVBXIYBYADYGUKYRXIYA - DXNYGYTVCADXOBVDVFVEVGXGYSXMKXIXGYHYRXMYHYGXPBUBZYIYNJZSZXGYRXMKZITEZOTEZ - SIOJYHUUCQUUEUUFVHVRVIVJYGXPTBTIOYGRVKVLXPCVMFZXOVNZXGUUCUUDKZUUGCXOYDUUG - RVOUUHXPUUGXONZXGUUIKXPUUGXOVPUUJUUCXGUUDUUJUUAUUBXGUUDKZUUAUUJUUBUUKKUUA - UUJSZUUBYJYOJZUUKUULYJYOYIYNUULUUJYIXPEZYNXPEZSZSYJYOHYIYNHKUUAUUPUUJUUAU - UNUUOUUAYGXPIBUUAVQZIYGEUUAIOVHVSVTWAUUAYGXPOBUUQOYGEUUAIOVRWBVTWAWCWDXPU - UGYIYNXOWJUQWEUUMUUDXGYRUUMXMYRUUMYLYPJXMYRYJYLYOYPYMYQWFYMYQWGWKXKXLYLYP - XKXLHYLXLYKPYPXKXLYKWHXLYKWIVAWLWMWNXAWOWPWQWRUQWSWTWQXBXCWNXDXEWRXCWRXF - $. + f1veqaeq neeq12d necon3i syl6bi com12 a1d syl6 expcom impd com23 biimtrid + mpcom adantr sylbid 3adant2 expdcom imp ) CUCEZBUDFZGHZBACUEFUAZIAFZGAFZJ + ZKXGXJXIXMXGXJIXHLMZCUFFZUGZBNZIXHUOMCUHFZAUBZDUIZBFXOFXTAFXTOUJMAFPHZDXN + UKZUPZXIXMKXGCULEXJYCQCUMADBCXOXRXRRXORZUNUQXGXIYCXMYCXGXIXMXQYBXGXISZXMK + XSYEXQYBSZXMYEYFIOPZXPBNZIBFZXOFZXKOAFZPZHZOBFZXOFZYKXLPZHZSZSZXMXIYFYSQX + GXIXQYHYBYRXIXNYGXPXPBBXIBURXIXNIGLMYGXHGILUSUTVAZXIXPURVBXIYBYADYGUKYRXI + YADXNYGYTVCADXOBVDVFVEVGXGYSXMKXIXGYHYRXMYHYGXPBUBZYIYNJZSZXGYRXMKZITEZOT + EZSIOJYHUUCQUUEUUFVHVRVIVJYGXPTBTIOYGRVKVLXPCVMFZXOVNZXGUUCUUDKZUUGCXOYDU + UGRVOUUHXPUUGXONZXGUUIKXPUUGXOVPUUJUUCXGUUDUUJUUAUUBXGUUDKZUUAUUJUUBUUKKU + UAUUJSZUUBYJYOJZUUKUULYJYOYIYNUULUUJYIXPEZYNXPEZSZSYJYOHYIYNHKUUAUUPUUJUU + AUUNUUOUUAYGXPIBUUAVQZIYGEUUAIOVHVSVTWAUUAYGXPOBUUQOYGEUUAIOVRWBVTWAWCWDX + PUUGYIYNXOWJUQWEUUMUUDXGYRUUMXMYRUUMYLYPJXMYRYJYLYOYPYMYQWFYMYQWGWKXKXLYL + YPXKXLHYLXLYKPYPXKXLYKWHXLYKWIVAWLWMWNWOWPWQWRWSUQXAWTWRXBXCWNXDXEWSXCWSX + F $. $} $( In a simple graph, any trail of length 2 is a simple path. (Contributed @@ -423151,8 +427991,8 @@ where f enumerates the (indices of the) different edges, and p syld sylibr sylan fzo0sn0fzo1 caddc cuz 1zzd c2 1p1e2 2z eqeltri ltaddsub wb w3a bicomd 2re sylancr sylbid imp eluz2 syl3anbrc fzosplitsnm1 raleqdv uneq2d eqtrd ralunb anbi2i bitri bitrdi eqcomi oveq2i raleqi fvres eqcomd - adantl neeq12d biimpd imim2d ralimdva syl5bi adantrd adantld mpd sylanbrc - dff14a df-f1 sylib simprd wn funcnv0 nn0zd peano2zm zred lenltd biimpar + adantl neeq12d biimpd imim2d ralimdva biimtrid adantrd adantld mpd dff14a + sylanbrc df-f1 sylib simprd wn funcnv0 nn0zd peano2zm zred lenltd biimpar c0 eqbrtrid eqeltrid fzon mpbird reseq2d res0 eqtrdi cnveqd funeqd mpbiri jca pm2.61dan ) AIBUFJZIUAKZUBLZBICMKZUCZUDZUEZAUUONZUUPOUUQUGZUUSUUTUUPO UUQUHZUVAUUSNUUTUVADUIZEUIZUJZUVCUUQJZUVDUUQJZUJZUKZEUUPPZDUUPPZUVBAUVAUU @@ -423171,7 +428011,7 @@ where f enumerates the (indices of the) different edges, and p IXJXKXLXMUUTUWKUVKUWFUUTUWHUVKUWJUWHUWCDUUPPUUTUVKUWCDUWGUUPUUNCIMCUUNGXN XOXPUUTUWCUVJDUUPUUTUVCUUPQZNZUWBUVIEUUPUXRUVDUUPQZNZUWAUVHUVEUXTUWAUVHUX TUVSUVFUVTUVGUXRUVSUVFSZUXSUXQUYAUUTUXQUVFUVSUVCUUPBXQXRXSVGUXSUVTUVGSUXR - UXSUVGUVTUVDUUPBXQXRXSXTYAYBYCYCYDYEYFXBYGDEUUPOUUQYIYHUUPOUUQYJYKYLAUUOY + UXSUVGUVTUVDUUPBXQXRXSXTYAYBYCYCYDYEYFXBYGDEUUPOUUQYHYIUUPOUUQYJYKYLAUUOY MZNZUUSYTUDZUEYNUYCUURUYDUYCUUQYTUYCUUQBYTUCYTUYCUUPYTBUYCUUPYTSZCIVKLZUY CCUUNIVKGAUUNIVKLUYBAUUNIAUUNAUVOUUNTQAUUMUWSYOUUMYPRZYQAVSYRYSUUAUYCUXJC TQZNZUYEUYFWQAUYIUYBAUXJUYHAWKACUUNTGUYGUUBUUKVGUYIUYFUYEICUUCWSRUUDUUEBU @@ -423185,18 +428025,18 @@ where f enumerates the (indices of the) different edges, and p cima wn wnel cv wrex 3ad2ant1 ralcom clt elfzo1 nnne0 necomd sylbi adantl wbr neeq1 expd cle cr nnre adantr ltlend simpr syl6bi 3impia jaoi 3adant1 impcom imp lbfzo0 biimpri eleq1 cmin fzo0end eqeltrid fveq2 imbi12d rspcv - neeq1d syl mpid nesym syl6ib ralimdva syl5bi ralnex sylib wfun cword wrdf - mpd wf ffun 3syl fvelima ex mtod df-nel sylibr ) ABUAJZKLZFMNZFCNZUBZUCZF - BJZBUDCUEUFZUGZLZUHXKXMUIXJXNEUJZBJZXKNZEXLUKZXJXQUHZEXLOZXRUHXJDUJZXOPZY - ABJZXPPZQZEXLODMXEUEUFZOZXTAXFYGXIIULYGYEDYFOZEXLOXJXTYEDEYFXLUMXJYHXSEXL - XJXOXLLZRZYHXKXPPZXSYJYHFXOPZYKXJYIYLXFXIYIYLQZAXIXFYMXGXFYMQXHXGXFYIYLXF - YIRZYLXGMXOPZYIYOXFYIXOKLZCKLZXOCUNUTZUCZYOCXOUOZYPYQYOYRYPXOMXOUPUQULURU - SFMXOVASVBXHXFYIYLYNYLXHCXOPZYIUUAXFYIYSUUAYTYPYQYRUUAYPYQRZYRXOCVCUTZUUA - RUUAUUBXOCYPXOVDLYQXOVEVFYQCVDLYPCVEUSVGUUCUUAVHVIVJURUSFCXOVASVBVKVMVLVN - YJFYFLZYHYLYKQZQXJUUDYIXFXIUUDAXIXFUUDXGXFUUDQXHXFUUDXGMYFLZUUFXFXEVOVPFM - YFVQSXFUUDXHCYFLXFCXEUDVRUFYFHXEVSVTFCYFVQSVKVMVLVFYEUUEDFYFYAFNZYBYLYDYK - YAFXOVAUUGYCXKXPYAFBWAWDWBWCWEWFXKXPWGWHWIWJWPXQEXLWKWLXJBWMZXNXRQAXFUUHX - IABTWNLYFTBWQUUHGTBWOYFTBWRWSULUUHXNXREXKXLBWTXAWEXBXKXMXCXD $. + neeq1d syl mpid nesym syl6ib ralimdva biimtrid ralnex sylib wfun cword wf + mpd wrdf ffun 3syl fvelima ex mtod df-nel sylibr ) ABUAJZKLZFMNZFCNZUBZUC + ZFBJZBUDCUEUFZUGZLZUHXKXMUIXJXNEUJZBJZXKNZEXLUKZXJXQUHZEXLOZXRUHXJDUJZXOP + ZYABJZXPPZQZEXLODMXEUEUFZOZXTAXFYGXIIULYGYEDYFOZEXLOXJXTYEDEYFXLUMXJYHXSE + XLXJXOXLLZRZYHXKXPPZXSYJYHFXOPZYKXJYIYLXFXIYIYLQZAXIXFYMXGXFYMQXHXGXFYIYL + XFYIRZYLXGMXOPZYIYOXFYIXOKLZCKLZXOCUNUTZUCZYOCXOUOZYPYQYOYRYPXOMXOUPUQULU + RUSFMXOVASVBXHXFYIYLYNYLXHCXOPZYIUUAXFYIYSUUAYTYPYQYRUUAYPYQRZYRXOCVCUTZU + UARUUAUUBXOCYPXOVDLYQXOVEVFYQCVDLYPCVEUSVGUUCUUAVHVIVJURUSFCXOVASVBVKVMVL + VNYJFYFLZYHYLYKQZQXJUUDYIXFXIUUDAXIXFUUDXGXFUUDQXHXFUUDXGMYFLZUUFXFXEVOVP + FMYFVQSXFUUDXHCYFLXFCXEUDVRUFYFHXEVSVTFCYFVQSVKVMVLVFYEUUEDFYFYAFNZYBYLYD + YKYAFXOVAUUGYCXKXPYAFBWAWDWBWCWEWFXKXPWGWHWIWJWPXQEXLWKWLXJBWMZXNXRQAXFUU + HXIABTWNLYFTBWOUUHGTBWQYFTBWRWSULUUHXNXREXKXLBWTXAWEXBXKXMXCXD $. $( Lemma 2 for ~ pthd . (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 10-Feb-2021.) $) @@ -423425,10 +428265,10 @@ the walk is closed if v(n) = v0". -> ( ( P ` 0 ) = ( P ` 1 ) /\ { ( P ` 0 ) } e. ( Edg ` G ) ) ) $= ( ciedg cfv wfun cclwlks wbr chash c1 wceq cc0 csn cedg wa cwlks wi isclwlk wcel fveq2 eqeq2d anbi2d simp2r simp3 simp2l simpr anim2i 3adant3 wlkl1loop - w3a syl21anc jca 3exp sylbid com13 syl5bi 3imp ) CDEFZBACGEHZBIEZJKZLAEZJAE - ZKZVBMCNESZOZUSBACPEHZVBUTAEZKZOZURVAVFQABCRVAVJURVFVAVJVGVDOZURVFQVAVIVDVG - VAVHVCVBUTJATUAUBVAVKURVFVAVKURUJZVDVEVAVGVDURUCVLURVGVAVDOZVEVAVKURUDVAVGV - DURUEVAVKVMURVKVDVAVGVDUFUGUHABCUIUKULUMUNUOUPUQ $. + w3a syl21anc jca 3exp sylbid com13 biimtrid 3imp ) CDEFZBACGEHZBIEZJKZLAEZJ + AEZKZVBMCNESZOZUSBACPEHZVBUTAEZKZOZURVAVFQABCRVAVJURVFVAVJVGVDOZURVFQVAVIVD + VGVAVHVCVBUTJATUAUBVAVKURVFVAVKURUJZVDVEVAVGVDURUCVLURVGVAVDOZVEVAVKURUDVAV + GVDURUEVAVKVMURVKVDVAVGVDUFUGUHABCUIUKULUMUNUOUPUQ $. $( @@ -423723,20 +428563,20 @@ According to Wikipedia ("Cycle (graph theory)", w3a lep1d 1red readdcld letr mpan2d syld sylbid syl2an 3impia sylbi eluz2 syl3anc syl3anbrc syl fzss1 sselda fvex ifex a1i fvmptd3 wn elfz2 anim12i zre simprr simpl resubcld ltp1d simprl ltletr mpand ltnled sylibd 3adant1 - expcom syl5com com13 adantr impcom com12 syl5bi imp iffalsed eqtrd ) AFGE - UAJZKLJZGUBJZMZNZFDOFXPPQZFELJZCOZYBGUAJZCOZUCZYEXTBFBURZXPPQZYGELJZCOZYI - GUAJCOZUCYFRGUBJZDUDIYGFUEZYHYAYJYKYCYEYGFXPPUGYGFECLUFYMYIYBGCUAYGFELUHU - IUJAXRYLFAXQRUKOMZXRYLULAEKGUMJMZYNHYORUSMXQUSMZRXQPQZYNYOUNYOXPYOGEEKGUO - EKGUPZUQUTYOEVAMZGVAMZEGVBQZVKYQGEVCYSYTUUAYQYSESMZGSMZUUAYQTYTEVDGVDUUBU - UCNZUUARXPVBQZYQEGVEUUDUUERXPPQZYQUUDRSMZXPSMZUUEUUFTUUDVFZUUCUUBUUHGEVGV - HZRXPVIVJUUDUUFXPXQPQZYQUUDXPUUJVLUUDUUGUUHXQSMZUUFUUKNYQTUUIUUJUUDXPKUUJ - UUDVMVNRXPXQVOWCVPVQVRVSVTWARXQWBWDWEXQRGWFWEWGYFUDMXTYAYCYEYBCWHYDCWHWIW - JWKXTYAYCYEAXSYAWLZAYOXSUUMTHXSYPGUSMZFUSMZVKZXQFPQZFGPQZNZNZYOUUMFXQGWMU - UTYOUUMUUSUUPYOUUMTZUUQUUPUVATUURYOUUPUUQUUMYOEUSMZUUPUUQUUMTZYRUUNUUOUVB - UVCTZYPUUOUUNUVDUVBUUOUUNNZUVCUVBUUBFSMZUUCNZUVCUVEEWOUUOUVFUUNUUCFWOGWOW - NUUBUVGNZUUQXPFVBQZUUMUVHXPXQVBQZUUQUVIUVHXPUVHGEUUBUVFUUCWPUUBUVGWQWRZWS - UVHUUHUULUVFUVJUUQNUVITUVKUVHXPKUVKUVHVMVNUUBUVFUUCWTZXPXQFXAWCXBUVHXPFUV - KUVLXCXDVSXFVHXEXGXHXIXJXKXLWEXMXNXO $. + expcom syl5com com13 adantr impcom com12 biimtrid imp iffalsed eqtrd ) AF + GEUAJZKLJZGUBJZMZNZFDOFXPPQZFELJZCOZYBGUAJZCOZUCZYEXTBFBURZXPPQZYGELJZCOZ + YIGUAJCOZUCYFRGUBJZDUDIYGFUEZYHYAYJYKYCYEYGFXPPUGYGFECLUFYMYIYBGCUAYGFELU + HUIUJAXRYLFAXQRUKOMZXRYLULAEKGUMJMZYNHYORUSMXQUSMZRXQPQZYNYOUNYOXPYOGEEKG + UOEKGUPZUQUTYOEVAMZGVAMZEGVBQZVKYQGEVCYSYTUUAYQYSESMZGSMZUUAYQTYTEVDGVDUU + BUUCNZUUARXPVBQZYQEGVEUUDUUERXPPQZYQUUDRSMZXPSMZUUEUUFTUUDVFZUUCUUBUUHGEV + GVHZRXPVIVJUUDUUFXPXQPQZYQUUDXPUUJVLUUDUUGUUHXQSMZUUFUUKNYQTUUIUUJUUDXPKU + UJUUDVMVNRXPXQVOWCVPVQVRVSVTWARXQWBWDWEXQRGWFWEWGYFUDMXTYAYCYEYBCWHYDCWHW + IWJWKXTYAYCYEAXSYAWLZAYOXSUUMTHXSYPGUSMZFUSMZVKZXQFPQZFGPQZNZNZYOUUMFXQGW + MUUTYOUUMUUSUUPYOUUMTZUUQUUPUVATUURYOUUPUUQUUMYOEUSMZUUPUUQUUMTZYRUUNUUOU + VBUVCTZYPUUOUUNUVDUVBUUOUUNNZUVCUVBUUBFSMZUUCNZUVCUVEEWOUUOUVFUUNUUCFWOGW + OWNUUBUVGNZUUQXPFVBQZUUMUVHXPXQVBQZUUQUVIUVHXPUVHGEUUBUVFUUCWPUUBUVGWQWRZ + WSUVHUUHUULUVFUVJUUQNUVITUVKUVHXPKUVKUVHVMVNUUBUVFUUCWTZXPXQFXAWCXBUVHXPF + UVKUVLXCXDVSXFVHXEXGXHXIXJXKXLWEXMXNXO $. $d F i $. $d I i $. $d N i $. $d P i $. $d S i $. $d ph i j $. $d ph j x $. @@ -423806,55 +428646,55 @@ According to Wikipedia ("Cycle (graph theory)", cc0 wi cc zcnd adantl 1cnd adantr elfzoel2 2addsubd eqcomd cn0 cn clt wbr w3a elfzo1 cuz cz cle nnz 3ad2ant2 3ad2ant1 zaddcld elfzo2 eluz2 zre nnre cr anim12i simplr simpll resubcld lep1d 1red readdcld simpr syl3anc mpand - 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(Contributed by AV, 12-Mar-2021.) $) @@ -423991,45 +428831,45 @@ According to Wikipedia ("Cycle (graph theory)", nn0re mpcom ad2antrl cvv elnn0z cn elfzo0 anim12ci 3ad2ant2 simpr3 syl2an nnre jca32 syl2anb le2add lesubadd ifclda exp32 fmptd crctcshlem2 wlkprop feq2d eqcomi oveq2i fzo1fzo0n0 simplbi2 simplll wkslem1 cbvralvw crctprop - raleqi ctrls fveq2i eqeq2i eqcomd raleqdv syl5bi com23 crctcshlem3 iswlk - crctcshwlkn0lem7 ) AEUDUEZSZHDGUFUGZUHZHIUIZUJZTZUDHUKUGZUQULZKDUMZUAUNZD - UGZUWHUOURULDUGZUPUWHHUGIUGZUWIUSUPUWIUWJUTUWKVAVBZUAUDUWEVCULZVDZVEZUVSU - WDUWGUWNAUWDUVRAHFEVFULZUWCQAFUWCTZUWPUWCTAFCGVGUGUHZFCUVTUHZUWQNCFGVHZCF - GIMVIVKEUWBFVJVLVMVNAUWGUVRAUWGUDJUQULZKDUMABUXABUNZJEVOULZVPUHZUXBEURULZ - CUGZUXEJVOULZCUGZVQZKDAUXBUXATZUXIKTZUWSAUXJUXKVRZAUWRUWSNUWTVLUWSUDFUKUG - ZUQULZKCUMZAUXLVRCFGKLVSUXOAUXJUXKUXOAUXJSZSZUXDUXFUXHKUXQUXDSUXNKUXECUXO - UXPUXDVTUXPUXDUXEUXNTZUXOUXPUXDSUXEWATZUXMWATZUXEUXMVPUHZVEZUXRAEUDJVCULZ - TZUXJUXDUYBPUYDUXJSZUXDSZUXSUXTUYAUYEUXSUXDUYEUXBEUXJUXBWATZUYDUXBJWBXCUY - 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F , P >. ` results in a walk ` <. H , Q >. ` . (Contributed by AV, 10-Mar-2021.) $) @@ -424639,14 +429479,14 @@ sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) cedg cwwlksn chash cwwlks crab cn0 nnnn0 wwlksn wn wne cvtx cword cpr w3a cfzo eqid iswwlks cc nncn pncan1 id eqeltrd wb oveq1 eleq1d adantr mpbird lbfzo0 sylibr fveq2 fv0p1e1 preq12d rspcdv eleq2 noel pm2.21i syldc com12 - syl6bi 3ad2ant3 syl5bi expimpd ax-1 pm2.61i ralrimiva rabeq0 eqtrd ) AUAE - ZFGZBHIZJZBAUBKZCLZUCEZBMNKZGZCAUDEZUEZFWJWLWRGZWIWJBUFIWSBUGCABUHOPWKWPU - IZCWQQWRFGWKWTCWQWPWKWMWQIZJZWTRWPWKXAWTXAWMFUJZWMAUKEZULIZDLZWMEZXFMNKWM - EZUMZWHIZDSWNMTKZUOKZQZUNZWPWKJZWTDWHAXDWMXDUPWHUPUQXNXOWTXMXCXOWTRXEXOXM - SWMEZMWMEZUMZWHIZWTXOXJXSDSXLXOXKHIZSXLIXOXTWOMTKZHIZWKYBWPWJYBWIWJYABHWJ - BURIYABGBUSBUTOWJVAVBPPWPXTYBVCWKWPXKYAHWNWOMTVDVEVFVGXKVHVIXFSGZXJXSVCXO - YCXIXRWHYCXGXPXHXQXFSWMVJWMXFVKVLVEPVMWKXSWTRZWPWIYDWJWIXSXRFIZWTWHFXRVNY - EWTXRVOVPVSVFPVQVTVRWAWBWTXBWCWDWEWPCWQWFVIWG $. + syl6bi 3ad2ant3 biimtrid expimpd ax-1 pm2.61i ralrimiva rabeq0 eqtrd ) AU + AEZFGZBHIZJZBAUBKZCLZUCEZBMNKZGZCAUDEZUEZFWJWLWRGZWIWJBUFIWSBUGCABUHOPWKW + PUIZCWQQWRFGWKWTCWQWPWKWMWQIZJZWTRWPWKXAWTXAWMFUJZWMAUKEZULIZDLZWMEZXFMNK + WMEZUMZWHIZDSWNMTKZUOKZQZUNZWPWKJZWTDWHAXDWMXDUPWHUPUQXNXOWTXMXCXOWTRXEXO + XMSWMEZMWMEZUMZWHIZWTXOXJXSDSXLXOXKHIZSXLIXOXTWOMTKZHIZWKYBWPWJYBWIWJYABH + WJBURIYABGBUSBUTOWJVAVBPPWPXTYBVCWKWPXKYAHWNWOMTVDVEVFVGXKVHVIXFSGZXJXSVC + XOYCXIXRWHYCXGXPXHXQXFSWMVJWMXFVKVLVEPVMWKXSWTRZWPWIYDWJWIXSXRFIZWTWHFXRV + NYEWTXRVOVPVSVFPVQVTVRWAWBWTXBWCWDWEWPCWQWFVIWG $. $} $( A walk of a fixed length as word is a walk (in an undirected graph) as @@ -424804,18 +429644,18 @@ sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) wf ancoms adantr edgval a1i eleq2d ralbidv biimpd wlkiswwlks2lem6 sylsyld eleq1 fveq2 oveq2d feq2d fveq1 fveqeq2d raleqbidv 3anbi123d imbi2d adantl 3jca wb mpbird spcimedv com23 3impia impcom imp cupgr uspgrupgr upgriswlk - ex expd syl exbidv syl5bi mpcom com12 ) ACUBFGZCUCGZBHZACUDFUAZBUEZCIGACU - FFZUGZGZJZXGXHXKKZCXLAXLLZUHXGAUOUIZXNDHZAFXSMUJNAFUKZCULFZGZDOAPFZMUMNZQ - NZRZSZXOXPDYACXLAXQYALUNXOYGXPXOYGJZXHXKYHXHJZXKXICUPFZUQUGZGZOXIPFZURNZX - LAVIZXSXIFZYJFXTTZDOYMQNZRZSZBUEZYHXHUUAYGXOXHUUAKYGXOXHUUAXRXNYFXOXHJZUU - AKXRXNJZUUBYFUUAUUCUUBYFUUAKUUCUUBJZYTYFBEYEEHZAFUUEMUJNAFUKYJUSFZUTZIYEI - GUUGIGUUDOYDQVAEYEUUFIVBVCUUDXIUUGTZJZYFYTKZYFUUGYKGZOUUGPFZURNZXLAVIZXSU - UGFZYJFXTTZDOUULQNZRZSZKZUUIXHXNMYCVDUAZSZYFXTYJVEZGZDYERZUUSUUDUVBUUHUUD - XHXNUVAUUCXOXHVFXRXNUUBVGUUCUVAUUBXNXRUVAAXMVHVJVKWIVKUUIYFUVEUUIYBUVDDYE - UUIYAUVCXTYAUVCTUUICVLVMVNVOVPEADYJUUGCXLUUGLYJLZVQVRUUHUUJUUTWJUUDUUHYTU - USYFUUHYLUUKYOUUNYSUURXIUUGYKVSUUHYNUUMXLAUUHYMUULOURXIUUGPVTZWAWBUUHYQUU - PDYRUUQUUHYMUULOQUVGWAUUHYPUUOXTYJXSXIUUGWCWDWEWFWGWHWKWLWTWMWNXAWOWPYIXJ - YTBXHXJYTWJZYHXHCWQGUVHCWRADXICYJXLXQUVFWSXBWHXCWKWTWTXDXEXF $. + ex expd syl exbidv biimtrid mpcom com12 ) ACUBFGZCUCGZBHZACUDFUAZBUEZCIGA + CUFFZUGZGZJZXGXHXKKZCXLAXLLZUHXGAUOUIZXNDHZAFXSMUJNAFUKZCULFZGZDOAPFZMUMN + ZQNZRZSZXOXPDYACXLAXQYALUNXOYGXPXOYGJZXHXKYHXHJZXKXICUPFZUQUGZGZOXIPFZURN + ZXLAVIZXSXIFZYJFXTTZDOYMQNZRZSZBUEZYHXHUUAYGXOXHUUAKYGXOXHUUAXRXNYFXOXHJZ + UUAKXRXNJZUUBYFUUAUUCUUBYFUUAKUUCUUBJZYTYFBEYEEHZAFUUEMUJNAFUKYJUSFZUTZIY + EIGUUGIGUUDOYDQVAEYEUUFIVBVCUUDXIUUGTZJZYFYTKZYFUUGYKGZOUUGPFZURNZXLAVIZX + SUUGFZYJFXTTZDOUULQNZRZSZKZUUIXHXNMYCVDUAZSZYFXTYJVEZGZDYERZUUSUUDUVBUUHU + UDXHXNUVAUUCXOXHVFXRXNUUBVGUUCUVAUUBXNXRUVAAXMVHVJVKWIVKUUIYFUVEUUIYBUVDD + YEUUIYAUVCXTYAUVCTUUICVLVMVNVOVPEADYJUUGCXLUUGLYJLZVQVRUUHUUJUUTWJUUDUUHY + TUUSYFUUHYLUUKYOUUNYSUURXIUUGYKVSUUHYNUUMXLAUUHYMUULOURXIUUGPVTZWAWBUUHYQ + UUPDYRUUQUUHYMUULOQUVGWAUUHYPUUOXTYJXSXIUUGWCWDWEWFWGWHWKWLWTWMWNXAWOWPYI + XJYTBXHXJYTWJZYHXHCWQGUVHCWRADXICYJXLXQUVFWSXBWHXCWKWTWTXDXEXF $. $} ${ @@ -424844,24 +429684,24 @@ sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) ( vi vx cfv wcel cv co cc0 cfzo wral wex wa wceq wi syl adantl ex adantr cwwlks c0 wne cvtx cword c1 caddc cpr cedg chash cmin w3a cupgr cwlks wbr eqid iswwlks ciedg cdm cfz wf wrex crn edgval wfun cuhgr upgruhgr uhgrfun - 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(Contributed by Alexander van der Vekens, n0 wwlknon cwwlks chash caddc iswwlksn cpths spthonisspth spthispth cwlks w3a cspths pthiswlk wlklenvm1 3syl oveq1 eqeq2d simpr cc nncn eqtrd nnne0 pncan1 eqnetrd spthonepeq necon3bid syl5ibrcom expcom com23 com13 exlimiv - syl6bi mpd adantl 3ad2ant1 syl5bi impd 3impia sylbi impcom ) CDBAUCGGZUAH - ZBUBIZCDHZWOEUDZWNIZEUEWPWQJZEWNUNWSWTEWSBUFIZAUGIZKZCAUHLZIDXDIKZWRCDBAU - IGGIZFUDZWRCDAUJLGMZFUEZKZVDWTCDFABXDWRXDUKULXCXEXJWTXCXEKZXFXIWTXFWRBAUM - GIZNWRLCOZBWRLDOZVDZXKXIWTJZCDABWRUOXOXKXPXLXMXKXPJXNXKXLXPXCXLXPJZXEXAXQ - XBXAXLWRAUPLIZWRUQLZBPURGZOZKXPABWRUSYAXPXRXIYAWTXHYAWTJZFXHXGUQLZXSPQGZO - ZYBXHXGWRAVELMXGWRAUTLMZYECDWRXGAVAWRXGAVBYFXGWRAVCLMYEWRXGAVFWRXGAVGRVHY - AYEXHWTYAYEYCXTPQGZOZXHWTJYAYDYGYCXSXTPQVIVJYHWPXHWQWPYHXHWQJWPYHKZWQXHYC - NHYIYCBNYIYCYGBWPYHVKWPYGBOZYHWPBVLIYJBVMBVPRSVNWPBNHYHBVOSVQXHCDYCNCDWRX - GAVRVSVTWAWBWEWCWFWDTWGWESSTWHTWIWJWKRWDWLWM $. + syl6bi mpd adantl 3ad2ant1 biimtrid impd 3impia sylbi impcom ) CDBAUCGGZU + AHZBUBIZCDHZWOEUDZWNIZEUEWPWQJZEWNUNWSWTEWSBUFIZAUGIZKZCAUHLZIDXDIKZWRCDB + AUIGGIZFUDZWRCDAUJLGMZFUEZKZVDWTCDFABXDWRXDUKULXCXEXJWTXCXEKZXFXIWTXFWRBA + UMGIZNWRLCOZBWRLDOZVDZXKXIWTJZCDABWRUOXOXKXPXLXMXKXPJXNXKXLXPXCXLXPJZXEXA + XQXBXAXLWRAUPLIZWRUQLZBPURGZOZKXPABWRUSYAXPXRXIYAWTXHYAWTJZFXHXGUQLZXSPQG + ZOZYBXHXGWRAVELMXGWRAUTLMZYECDWRXGAVAWRXGAVBYFXGWRAVCLMYEWRXGAVFWRXGAVGRV + HYAYEXHWTYAYEYCXTPQGZOZXHWTJYAYDYGYCXSXTPQVIVJYHWPXHWQWPYHXHWQJWPYHKZWQXH + YCNHYIYCBNYIYCYGBWPYHVKWPYGBOZYHWPBVLIYJBVMBVPRSVNWPBNHYHBVOSVQXHCDYCNCDW + RXGAVRVSVTWAWBWEWCWFWDTWGWESSTWHTWIWJWKRWDWLWM $. $} ${ @@ -426147,14 +430987,14 @@ between two (different) vertices. (Contributed by Alexander van der ( cvtx cfv wne wa wcel w3a cumgr cpr 3anass sylanbrc umgr2adedgwlklem syl simprd simpld wss ssid sseqtrrid jca eqid wi fveq2 eqcoms eqeq1d eqtr2 ex syl6bi com13 eqcom prcom eqeq2i bitri umgrpredgv anim12d preqr1g eqneqall - wceq sylc syl6ci syl5bi syld neqne pm2.61d1 2spthd ) ABCDEGHIJKHUAUBZONAB - CUCCDUCUDZBWDUEZCWDUEZDWDUEZUFZAHUGUEZBCUHZFUEZCDUHZFUEZUFZWEWIUDAWJWLWNU - DZWOPQWJWLWNUIUJBCDFHLUKULZUMAWEWIWQUNAWKJIUBZUOWMKIUBZUOAWKWKWRWKUPRUQAW - MWMWSWMUPSUQURWDUSZMAJKVPZJKUCZAXAWMWKVPZXBAWRWKVPZWSWMVPZXAXCUTRSXAXEXDX - CXAXEWRWMVPZXDXCUTXAWSWRWMWSWRVPKJKJIVAVBVCXFXDXCWRWMWKVDVEVFVGVQXCWKDCUH - ZVPZAXBXCWKWMVPXHWMWKVHWMXGWKCDVIVJVKAXHBDVPZBDUCXBAWFWHUDZXHXIUTAWJWPXJP - QWJWLWFWNWHWJWLWFWJWLUDWFWGFHBCWDWTLVLUNVEWJWNWHWJWNUDWGWHFHCDWDWTLVLUMVE - VMVQBDCWDWDVNULTXBBDVOVRVSVTJKWAWBTWC $. + wceq sylc syl6ci biimtrid syld neqne pm2.61d1 2spthd ) ABCDEGHIJKHUAUBZON + ABCUCCDUCUDZBWDUEZCWDUEZDWDUEZUFZAHUGUEZBCUHZFUEZCDUHZFUEZUFZWEWIUDAWJWLW + NUDZWOPQWJWLWNUIUJBCDFHLUKULZUMAWEWIWQUNAWKJIUBZUOWMKIUBZUOAWKWKWRWKUPRUQ + AWMWMWSWMUPSUQURWDUSZMAJKVPZJKUCZAXAWMWKVPZXBAWRWKVPZWSWMVPZXAXCUTRSXAXEX + DXCXAXEWRWMVPZXDXCUTXAWSWRWMWSWRVPKJKJIVAVBVCXFXDXCWRWMWKVDVEVFVGVQXCWKDC + UHZVPZAXBXCWKWMVPXHWMWKVHWMXGWKCDVIVJVKAXHBDVPZBDUCXBAWFWHUDZXHXIUTAWJWPX + JPQWJWLWFWNWHWJWLWFWJWLUDWFWGFHBCWDWTLVLUNVEWJWNWHWJWNUDWGWHFHCDWDWTLVLUM + VEVMVQBDCWDWDVNULTXBBDVOVRVSVTJKWAWBTWC $. $} ${ @@ -427228,46 +432068,46 @@ that the word does not contain the terminating vertex p(n) of the walk, oveq1 csn elsni pm2.24d fzo01 eleq2s syl6bi adantld wne df-ne nn0re simpr cr 2re leltned elfzo0 simp-4l cz nn0z zsubcld posdifd biimpa elnnz com23 2z sylanbrc ad5ant24 peano2zm zltlem1 syl2an subsub4d 1p1e2 breq2d bitr2i - 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(Contributed by Alexander van der Vekens, @@ -427380,42 +432220,42 @@ that the word does not contain the terminating vertex p(n) of the walk, ( wcel wa cc0 cfv co c2 cle c1 wceq wi imp syl adantr adantl cdm cword wf wf1 chash cfz wbr cv caddc cpr cfzo wral clsw crn cmin w3a wfn f1fn dffn3 sylib cn0 lencl ffn fnfz0hash syl2an ffz0iswrd lsw ad6antr ad4antlr eqcom - fvoveq1 nn0cn pncand eqcomd fveqeq2d biimpd syl5bi adantld 3eqtrd cuz wss - 1cnd cz nn0z peano2zm nn0re lem1d eluz2 syl3anbrc fzoss2 ssralv 3syl wrdf - simpr simpll fzossrbm1 sselda ffvelcdmd exp31 ad3antrrr biimpi syl5ibrcom - eleq1d ralimdva syldc impcom cn wb breq2 cr 2re a1i 1red lesubaddd breq1i - 2m1e1 elnnnn0c simplbi2 sylbird sylbid lbfzo0 sylibr fzoend fveq2 preq12d - 2fveq3 eqeq12d rspcdv npcand fveq2d preq2d eqeq2d com12 com3r imp31 preq2 - syl6bi mpbird 3jca exp41 com13 mpcom mpd expcom com14 impcomd sylan 3imp - ) DUAZBDUDZEUUIUBGZHIEUEJZUFKZFAUCZLAUEJZMUGZHZCUHZEJZDJZUURAJZUURNUIKAJZ - UJZOZCIUULUKKZULZIAJZUULAJZOZHZAUMJZUVGOZUVCDUNZGZCIUULNUOKZUKKZULZUVOAJZ - 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(Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) $) @@ -427607,15 +432447,15 @@ that the word does not contain the terminating vertex p(n) of the walk, chash cpfx cclwwlk clsw cc0 c1st cwlks cn w3a clwlkclwwlkflem cclwlks wex isclwlk fvex breq1 spcev sylbir 3adant3 cvtx cword simpl wlkpwrd 3ad2ant1 wb wi cn0 elnnnn0c caddc nn0re 1e2m1 breq1i biimpi 2re 1re mp3an12 syl5ib - lesubadd syl wlklenvp1 adantr breq2d sylibrd expimpd a1d 3imp clwlkclwwlk - syl5bi ciedg syl3anc mpbid sylan2 simprd fmptd ) DUAIZEBEUBZUCJZWTUEJZKUD - LUFLZDUGJZCWRWSBIZMWTUHJUIWTJZNZXBXCIZXDWRWSUJJZWTDUKJOZXEXHUEJZWTJNZXJUL - IZUMZXFXGMZAXHWTBWSDFXHPWTPUNWRXMMZHUBZWTDUOJZOZHUPZXNXMXSWRXIXKXSXLXIXKM - XHWTXQOZXSWTXHDUQXRXTHXHWSUJURXPXHWTXQUSUTVAVBQXOWRWTDVCJZVDIZRXASOZXSXNV - HWRXMVEXMYBWRXIXKYBXLWTXHDYAYAPZVFVGQXMYCWRXIXKXLYCXIXLYCVIXKXLXJVJIZKXJS - OZMXIYCXJVKXIYEYFYCXIYEMZYFRXJKVLLZSOZYCYEYFYIVIZXIYEXJTIZYJXJVMYFRKUDLZX - JSOZYKYIYFYMKYLXJSVNVOVPRTIKTIYKYMYIVHVQVRRKXJWAVSVTWBQYGXAYHRSXIXAYHNYEW - TXHDWCWDWEWFWGWKWHWIQWTHDWLJZDYAYDYNPWJWMWNWOWPGWQ $. + lesubadd syl wlklenvp1 adantr breq2d sylibrd expimpd biimtrid clwlkclwwlk + a1d 3imp ciedg syl3anc mpbid sylan2 simprd fmptd ) DUAIZEBEUBZUCJZWTUEJZK + UDLUFLZDUGJZCWRWSBIZMWTUHJUIWTJZNZXBXCIZXDWRWSUJJZWTDUKJOZXEXHUEJZWTJNZXJ + ULIZUMZXFXGMZAXHWTBWSDFXHPWTPUNWRXMMZHUBZWTDUOJZOZHUPZXNXMXSWRXIXKXSXLXIX + KMXHWTXQOZXSWTXHDUQXRXTHXHWSUJURXPXHWTXQUSUTVAVBQXOWRWTDVCJZVDIZRXASOZXSX + NVHWRXMVEXMYBWRXIXKYBXLWTXHDYAYAPZVFVGQXMYCWRXIXKXLYCXIXLYCVIXKXLXJVJIZKX + JSOZMXIYCXJVKXIYEYFYCXIYEMZYFRXJKVLLZSOZYCYEYFYIVIZXIYEXJTIZYJXJVMYFRKUDL + ZXJSOZYKYIYFYMKYLXJSVNVOVPRTIKTIYKYMYIVHVQVRRKXJWAVSVTWBQYGXAYHRSXIXAYHNY + EWTXHDWCWDWEWFWGWHWJWKQWTHDWLJZDYAYDYNPWIWMWNWOWPGWQ $. $d C f w $. $d F c f w $. $( ` F ` is a function from the nonempty closed walks onto the closed walks @@ -427630,21 +432470,21 @@ that the word does not contain the terminating vertex p(n) of the walk, simpr3 clwlkclwwlkfolem syl3anc 3expa ovex fveq2 2fveq3 oveq12d vex op2nd oveq1d fveq2i oveq1i oveq12i eqtrdi fvmptg sylancl ad2antrr oveq2d simpll wrdlenccats1lenm1 wrdsymb1 s1cld eqidd pfxccatid 3eqtrrd 3adant1 ad2antlr - ex eqeq2d imbi2d mpbird rspcimedv pm2.43b syl5bi exlimdv 3expib com23 imp - sylbird ralrimiva dffo3 sylanbrc ) DUAIZBDUBJZCUDAKZEKZCJZLZEBUEZAYFUFBYF - CUCABCDEFGUGYEYKAYFYEYGYFIZMYGDUHJZUIZIZNYGOJZUJUKZMZYKYLYRYEYLULYPUMUKZY - RDYGUNYLDUOIZYOYGUPUQZURYSYRPZDYMYGYMUSZUTYOYTUUBUUAYOYSYQYOYSYQYOYPVAIYS - YQVBYOYPYMYGVCVDYPVEVJVFVGVHVJVKVIYEYLYRYKPYEYRYLYKYEYOYQYLYKPYEYOYQURZYL - HKZYGULYGJZVLZVMQZDVNJZUKZHVOYKYGHDVPJZDYMUUCUUKUSVQUUDUUJYKHUUJUUEUUHVRZ - UUIIZUUDYKUUEUUHUUIVSUUDUUMYKUUMUUDUUMYKPUUMUUDMZYJUUMEUULBUUNYOYQUUMUULB - IZUUMYEYOYQVTUUMYEYOYQWBUUMUUDWAABHDYGFWCZWDUUNYHUULLZMUUMYJPZUUMYGUULCJZ - LZPZUUDUVAUUMUUQYOYQUVAYEYRUUMUUTYRUUMMZUUSUUHUUHOJZNRQZSQZUUHYPSQZYGUVBU - UOUVEUOIUUSUVELYOYQUUMUUOUUPWEUUHUVDSWFEUULYHTJZUVGOJZNRQZSQZUVEBUOCUUQUV - JUULTJZUVKOJZNRQZSQUVEUUQUVGUVKUVIUVMSYHUULTWGUUQUVHUVLNRYHUULOTWHWLWIUVK - UUHUVMUVDSUUEUUHHWJYGUUGVMWFWKZUVLUVCNRUVKUUHOUVNWMWNWOWPGWQWRUVBUVDYPUUH - SYOUVDYPLYQUUMUUFYMYGXBWSWTUVBYOUUGYNIYPYPLUVFYGLYOYQUUMXAUVBUUFYMUVBYRUU - FYMIYRUUMWAYMYGXCVJXDUVBYPXEYGUUGYPYMXFWDXGXJXHXIUUQUURUVAVBUUNUUQYJUUTUU - MUUQYIUUSYGYHUULCWGXKXLVIXMXNXJXOXPXQYAXRXSXTVKYBEABYFCYCYD $. + ex eqeq2d imbi2d mpbird rspcimedv pm2.43b biimtrid exlimdv sylbird 3expib + com23 imp ralrimiva dffo3 sylanbrc ) DUAIZBDUBJZCUDAKZEKZCJZLZEBUEZAYFUFB + YFCUCABCDEFGUGYEYKAYFYEYGYFIZMYGDUHJZUIZIZNYGOJZUJUKZMZYKYLYRYEYLULYPUMUK + ZYRDYGUNYLDUOIZYOYGUPUQZURYSYRPZDYMYGYMUSZUTYOYTUUBUUAYOYSYQYOYSYQYOYPVAI + YSYQVBYOYPYMYGVCVDYPVEVJVFVGVHVJVKVIYEYLYRYKPYEYRYLYKYEYOYQYLYKPYEYOYQURZ + YLHKZYGULYGJZVLZVMQZDVNJZUKZHVOYKYGHDVPJZDYMUUCUUKUSVQUUDUUJYKHUUJUUEUUHV + RZUUIIZUUDYKUUEUUHUUIVSUUDUUMYKUUMUUDUUMYKPUUMUUDMZYJUUMEUULBUUNYOYQUUMUU + LBIZUUMYEYOYQVTUUMYEYOYQWBUUMUUDWAABHDYGFWCZWDUUNYHUULLZMUUMYJPZUUMYGUULC + JZLZPZUUDUVAUUMUUQYOYQUVAYEYRUUMUUTYRUUMMZUUSUUHUUHOJZNRQZSQZUUHYPSQZYGUV + BUUOUVEUOIUUSUVELYOYQUUMUUOUUPWEUUHUVDSWFEUULYHTJZUVGOJZNRQZSQZUVEBUOCUUQ + UVJUULTJZUVKOJZNRQZSQUVEUUQUVGUVKUVIUVMSYHUULTWGUUQUVHUVLNRYHUULOTWHWLWIU + VKUUHUVMUVDSUUEUUHHWJYGUUGVMWFWKZUVLUVCNRUVKUUHOUVNWMWNWOWPGWQWRUVBUVDYPU + UHSYOUVDYPLYQUUMUUFYMYGXBWSWTUVBYOUUGYNIYPYPLUVFYGLYOYQUUMXAUVBUUFYMUVBYR + UUFYMIYRUUMWAYMYGXCVJXDUVBYPXEYGUUGYPYMXFWDXGXJXHXIUUQUURUVAVBUUNUUQYJUUT + UUMUUQYIUUSYGYHUULCWGXKXLVIXMXNXJXOXPXQXRXSXTYAVKYBEABYFCYCYD $. $d C x y $. $d F x y $. $d G c x y $. $d i x y $. $d w x y $. $( ` F ` is a one-to-one function from the nonempty closed walks into the @@ -427890,21 +432730,21 @@ that the word does not contain the terminating vertex p(n) of the walk, ( vm vk cv cvv wcel wa wi cfv wceq ccsh co wbr vex w3a erclwwlkeqlen wrex chash 3adant3 3adant1 cclwwlk cc0 cfz wb erclwwlkeq simpr1 simplr2 eqeq2d oveq2 cbvrexvw cvtx cword c0 wne eqid clwwlkbp simp2d ad2antlr cshwcsh2id - simpr exp5l imp41 rexlimdva syl7bi syl5bi exp31 com15 impcom com13 3impia - rexlimdva2 3jca 3adant2 syl5ibrcom com24 ex com4t sylbid com25 mpdd mp3an - impd ) ALZMNZBLZMNZCLZMNZWKWMFUAZWMWOFUAZOWKWOFUAZPAUBBUBCUBWLWNWPUCZWQWR - WSWTWQWKUFQWMUFQZRZWRWSPWLWNWQXBPWPDEFWKGHWMMMIUDUGWTWRXBWQWSWTWRXAWOUFQZ - RZXBWQWSPPZWNWPWRXDPWLDEFWMGHWOMMIUDUHWTWRWMHUIQZNZWOXFNZWMWOGLZSTZRZGUJX - CUKTZUEZUCZXDXEPWNWPWRXNULWLDEFWMGHWOMMIUMUHWTWQXDXBXNWSWTWQWKXFNZXGWKWMX - ISTZRZGUJXAUKTZUEZUCZXDXBXNWSPZPPWLWNWQXTULWPDEFWKGHWMMMIUMUGXDXBWTXTYAXD - XBWTXTYAPPXDXBOZXNXTWTWSYBXNXTWTWSPYBXNOZXTOZWSWTXOXHWKXJRGXLUEZUCZYDXOXH - YEYCXOXGXSUNXGXHXMYBXTUOXTYCYEXOXGXSYCYEPYCXSXOXGOZYEXNYBXSYGYEPPZXHXMYBY - HPZXGXMXHYIYGXHYBXSXMYEYGXHYBXSXMYEPZPXSWKWMJLZSTZRZJXRUEZYGXHOZYBOZYJXQY - MGJXRXIYKRXPYLWKXIYKWMSUQUPURXMWMWOKLZSTZRZKXLUEZYPYNYEXKYSGKXLXIYQRXJYRW - MXIYQWOSUQUPURYPYMYTYEPJXRYPYKXRNZOYMOYSYEKXLYPUUAYMYQXLNZYSYEPYPUUAYMUUB - YSYEYPABCKJGHUSQZXHWOUUCUTNZYGYBXHHMNUUDWOVAVBHUUCWOUUCVCVDVEVFYOYBVHVGVI - VJVKVSVLVMVNVOVPUHVPVQVRVPVTWLWPWSYFULWNDEFWKGHWOMMIUMWAWBVNWCWDWEWFWGWFW - HWCWHWJWI $. + simpr exp5l imp41 rexlimdva rexlimdva2 syl7bi biimtrid exp31 com15 impcom + com13 3impia 3jca 3adant2 syl5ibrcom com24 ex com4t com25 mpdd impd mp3an + sylbid ) ALZMNZBLZMNZCLZMNZWKWMFUAZWMWOFUAZOWKWOFUAZPAUBBUBCUBWLWNWPUCZWQ + WRWSWTWQWKUFQWMUFQZRZWRWSPWLWNWQXBPWPDEFWKGHWMMMIUDUGWTWRXBWQWSWTWRXAWOUF + QZRZXBWQWSPPZWNWPWRXDPWLDEFWMGHWOMMIUDUHWTWRWMHUIQZNZWOXFNZWMWOGLZSTZRZGU + JXCUKTZUEZUCZXDXEPWNWPWRXNULWLDEFWMGHWOMMIUMUHWTWQXDXBXNWSWTWQWKXFNZXGWKW + MXISTZRZGUJXAUKTZUEZUCZXDXBXNWSPZPPWLWNWQXTULWPDEFWKGHWMMMIUMUGXDXBWTXTYA + XDXBWTXTYAPPXDXBOZXNXTWTWSYBXNXTWTWSPYBXNOZXTOZWSWTXOXHWKXJRGXLUEZUCZYDXO + XHYEYCXOXGXSUNXGXHXMYBXTUOXTYCYEXOXGXSYCYEPYCXSXOXGOZYEXNYBXSYGYEPPZXHXMY + BYHPZXGXMXHYIYGXHYBXSXMYEYGXHYBXSXMYEPZPXSWKWMJLZSTZRZJXRUEZYGXHOZYBOZYJX + QYMGJXRXIYKRXPYLWKXIYKWMSUQUPURXMWMWOKLZSTZRZKXLUEZYPYNYEXKYSGKXLXIYQRXJY + RWMXIYQWOSUQUPURYPYMYTYEPJXRYPYKXRNZOYMOYSYEKXLYPUUAYMYQXLNZYSYEPYPUUAYMU + UBYSYEYPABCKJGHUSQZXHWOUUCUTNZYGYBXHHMNUUDWOVAVBHUUCWOUUCVCVDVEVFYOYBVHVG + VIVJVKVLVMVNVOVPVQUHVQVRVSVQVTWLWPWSYFULWNDEFWKGHWOMMIUMWAWBVOWCWDWEWJWFW + JWGWCWGWHWI $. $d G x $. $d .~ x y z $. $( ` .~ ` is an equivalence relation over the set of closed walks (defined @@ -428387,33 +433227,33 @@ of fixed length (as words) is also finite. (Contributed by Alexander jctil nnre lep1d syl5ibr cop csubstr pfxval ad2ant2rl ad2ant2l swrdspsleq breq2 3adant3 bitrd syl112anc lbfzo0 biimpri syl sylbid eqeqan12rd mpbird rspcv imp simpr jca32 clt 1red nngt0 0lt1 addgt0d pfxsuff1eqwrdeq expdcom - a1i 3jca exp31 imp31 com12 syl5bi ralrimivva dff13 sylanbrc ) FUCJZCFEUDK - ZDUEUAUFZDLZUBUFZDLZMZYPYRMZNZUBCUGUACUGCYODUQABCDEFGHUHYNUUBUAUBCCYNYPCJ - ZYRCJZOZOYTYPFPKZYRFPKZMZUUAUUEYTUUHQYNUUCUUDYQUUFYSUUGABCDEFYPGHUIABCDEF - YRGHUIUJRYNUUEUUHUUANZUUEYPFEUKKZJZYPULLZSYPLZMZOZYRUUJJZYRULLZSYRLZMZOZO - ZYNUUIUUCUUOUUDUUTAUFZULLZSUVBLZMZUUNAYPUUJCUVBYPMUVCUULUVDUUMUVBYPULUMSU - VBYPURUNGUOUVEUUSAYRUUJCUVBYRMUVCUUQUVDUURUVBYRULUMSUVBYRURUNGUOUPUVAYNUU - IUUKUUNUUTYNUUINZUUKYPEUSLZUTZJZYPVALZFTVBKZMZIUFZYPLZUVMTVBKZYPLVCEVDLZJ - ISFVEKZUGZVFUUNUUTUVFNNZIUVPEFUVGYPUVGVJZUVPVJZVGUVIUVLUVSUVRUUTUVIUVLOZU - UNUVFUUPUUSUWBUUNOZUVFNZUUPYRUVHJZYRVALZUVKMZUVMYRLZUVOYRLVCUVPJIUVQUGZVF - UUSUWDNZIUVPEFUVGYRUVTUWAVGUWEUWGUWJUWIUWEUWGOZUUSUWDYNUWKUUSOZUWCUUIYNUW - LUWCOZUUHUUAYNUWMOZUUHOZUUAUVJUWFMZYPUVJTVKKZPKZYRUWQPKZMZUULUUQMZOOZUWOU - WPUWTUXAUWMUWPYNUUHUWMUVJUVKUWFUWLUVIUVLUUNVHUWEUWGUUSUWCVIVLVMUWNUUHUWTU - WMYNUUHUWTQZUWCYNUXCNZUWLUVLUXDUVIUUNUVLYNUXCUVLYNOZUUFUWRUUGUWSUXEFUWQYP - PYNUVLFUVKTVKKZUWQYNFVNJZTVNJZFUXFMFWAVOUXGUXHOUXFFFTVPVQVRUVLUWQUXFUVJUV - KTVKWBVQVSZVTUXEFUWQYRPUXIVTUNWCVMRWDWEUWOUXAUUMUURMZUWNUUHUXJUWNUUHUVNUW - HMZIUVQUGZUXJUWNUVIUWEOZSWFJZFWFJZOZFUVJWGWHZFUWFWGWHZUUHUXLQUWMUXMYNUWLU - WEUWCUVIUWEUWGUUSWIZUVIUVLUUNWIZWJRYNUXPUWMYNUXOUXNFWKWLWNWMUWMYNUXQUWCYN - UXQNZUWLUVLUYAUVIUUNYNUXQUVLFUVKWGWHZYNFFWOZWPZUVJUVKFWGXDWQVMRWDUWMYNUXR - UWLYNUXRNZUWCUWGUYEUWEUUSYNUXRUWGUYBUYDUWFUVKFWGXDWQVMWMWDUXMUXPUXQUXROZV - FUUHYPSFWRZWSKZYRUYGWSKZMZUXLUXMUXPUUHUYJQUYFUXMUXPOUUFUYHUUGUYIUVIUXOUUF - UYHMUWEUXNYPFUVHWTXAUWEUXOUUGUYIMUVIUXNYRFUVHWTXBUNXEYRISFUVGYPXCXFXGUWNS - UVQJZUXLUXJNYNUYKUWMUYKYNFXHXIWMUXKUXJISUVQUVMSMUVNUUMUWHUURUVMSYPUMUVMSY - RUMUNXNXJXKXOUWMUXAUXJQYNUUHUWCUWLUULUUMUUQUURUWBUUNXPUWKUUSXPXLVMXMXQUWO - UVIUWESUVJXRWHZVFZUUAUXBQUWNUYMUUHUWNUVIUWEUYLUWMUVIYNUWCUVIUWLUXTRRUWMUW - EYNUWLUWEUWCUXSWMRUWMYNUYLUWCYNUYLNZUWLUVLUYNUVIUUNYNUYLUVLSUVKXRWHYNFTUY - CYNXSFXTSTXRWHYNYAYEYBUVJUVKSXRXDWQVMRWDYFWMYRUVGYPYCXJXMYGYDWCXEXJXOYDXE - XJYHYIYJXOXKYKUAUBCYODYLYM $. + a1i 3jca exp31 imp31 com12 biimtrid ralrimivva dff13 sylanbrc ) FUCJZCFEU + DKZDUEUAUFZDLZUBUFZDLZMZYPYRMZNZUBCUGUACUGCYODUQABCDEFGHUHYNUUBUAUBCCYNYP + CJZYRCJZOZOYTYPFPKZYRFPKZMZUUAUUEYTUUHQYNUUCUUDYQUUFYSUUGABCDEFYPGHUIABCD + EFYRGHUIUJRYNUUEUUHUUANZUUEYPFEUKKZJZYPULLZSYPLZMZOZYRUUJJZYRULLZSYRLZMZO + ZOZYNUUIUUCUUOUUDUUTAUFZULLZSUVBLZMZUUNAYPUUJCUVBYPMUVCUULUVDUUMUVBYPULUM + SUVBYPURUNGUOUVEUUSAYRUUJCUVBYRMUVCUUQUVDUURUVBYRULUMSUVBYRURUNGUOUPUVAYN + UUIUUKUUNUUTYNUUINZUUKYPEUSLZUTZJZYPVALZFTVBKZMZIUFZYPLZUVMTVBKZYPLVCEVDL + ZJISFVEKZUGZVFUUNUUTUVFNNZIUVPEFUVGYPUVGVJZUVPVJZVGUVIUVLUVSUVRUUTUVIUVLO + ZUUNUVFUUPUUSUWBUUNOZUVFNZUUPYRUVHJZYRVALZUVKMZUVMYRLZUVOYRLVCUVPJIUVQUGZ + VFUUSUWDNZIUVPEFUVGYRUVTUWAVGUWEUWGUWJUWIUWEUWGOZUUSUWDYNUWKUUSOZUWCUUIYN + UWLUWCOZUUHUUAYNUWMOZUUHOZUUAUVJUWFMZYPUVJTVKKZPKZYRUWQPKZMZUULUUQMZOOZUW + OUWPUWTUXAUWMUWPYNUUHUWMUVJUVKUWFUWLUVIUVLUUNVHUWEUWGUUSUWCVIVLVMUWNUUHUW + TUWMYNUUHUWTQZUWCYNUXCNZUWLUVLUXDUVIUUNUVLYNUXCUVLYNOZUUFUWRUUGUWSUXEFUWQ + YPPYNUVLFUVKTVKKZUWQYNFVNJZTVNJZFUXFMFWAVOUXGUXHOUXFFFTVPVQVRUVLUWQUXFUVJ + UVKTVKWBVQVSZVTUXEFUWQYRPUXIVTUNWCVMRWDWEUWOUXAUUMUURMZUWNUUHUXJUWNUUHUVN + UWHMZIUVQUGZUXJUWNUVIUWEOZSWFJZFWFJZOZFUVJWGWHZFUWFWGWHZUUHUXLQUWMUXMYNUW + LUWEUWCUVIUWEUWGUUSWIZUVIUVLUUNWIZWJRYNUXPUWMYNUXOUXNFWKWLWNWMUWMYNUXQUWC + YNUXQNZUWLUVLUYAUVIUUNYNUXQUVLFUVKWGWHZYNFFWOZWPZUVJUVKFWGXDWQVMRWDUWMYNU + XRUWLYNUXRNZUWCUWGUYEUWEUUSYNUXRUWGUYBUYDUWFUVKFWGXDWQVMWMWDUXMUXPUXQUXRO + ZVFUUHYPSFWRZWSKZYRUYGWSKZMZUXLUXMUXPUUHUYJQUYFUXMUXPOUUFUYHUUGUYIUVIUXOU + UFUYHMUWEUXNYPFUVHWTXAUWEUXOUUGUYIMUVIUXNYRFUVHWTXBUNXEYRISFUVGYPXCXFXGUW + NSUVQJZUXLUXJNYNUYKUWMUYKYNFXHXIWMUXKUXJISUVQUVMSMUVNUUMUWHUURUVMSYPUMUVM + SYRUMUNXNXJXKXOUWMUXAUXJQYNUUHUWCUWLUULUUMUUQUURUWBUUNXPUWKUUSXPXLVMXMXQU + WOUVIUWESUVJXRWHZVFZUUAUXBQUWNUYMUUHUWNUVIUWEUYLUWMUVIYNUWCUVIUWLUXTRRUWM + UWEYNUWLUWEUWCUXSWMRUWMYNUYLUWCYNUYLNZUWLUVLUYNUVIUUNYNUYLUVLSUVKXRWHYNFT + UYCYNXSFXTSTXRWHYNYAYEYBUVJUVKSXRXDWQVMRWDYFWMYRUVGYPYCXJXMYGYDWCXEXJXOYD + XEXJYHYIYJXOXKYKUAUBCYODYLYM $. $d D p $. $d F p $. $d G i p w x $. $d N p $. $( Lemma 4 for ~ clwwlkf1o : F is an onto function. (Contributed by @@ -428894,27 +433734,27 @@ closed walk as word of the same length (in an undirected graph). chash cfv 3adant1 cc0 cfz wrex wb erclwwlkneq simpr1 simplr2 oveq2 eqeq2d cbvrexvw cclwwlkn cvtx cword eqid clwwlknbp biimpi simpl2im eleq2s simpld eqcom ad2antlr adantl simprr eqcoms adantr sylan9eq eleq2d anbi1d anbi12d - cshwcsh2id rexeqdv 3imtr4d mpancom exp5l imp41 rexlimdva ex syl7bi syl5bi - exp31 com15 impcom com13 3impia 3jca 3adant2 syl5ibrcom com24 com4t com25 - sylbid mpdd impd mp3an ) AMZNOZBMZNOZCMZNOZXEXGFUCZXGXIFUCZPXEXIFUCZQAUDB - UDCUDXFXHXJUEZXKXLXMXNXKXEUHUIXGUHUIZRZXLXMQXFXHXKXPQXJDEFXEXGGHIJNNKLUFU - GXNXLXPXKXMXNXLXOXIUHUIZRZXPXKXMQQZXHXJXLXRQXFDEFXGXIGHIJNNKLUFUJXNXLXGJO - ZXIJOZXGXIGMZSTZRZGUKIULTZUMZUEZXRXSQXHXJXLYGUNXFDEFXGXIGHIJNNKLUOUJXNXKX - RXPYGXMXNXKXEJOZXTXEXGYBSTZRZGYEUMZUEZXRXPYGXMQZQQXFXHXKYLUNXJDEFXEXGGHIJ - NNKLUOUGXRXPXNYLYMXRXPXNYLYMQQXRXPPZYGYLXNXMYNYGYLXNXMQYNYGPZYLPZXMXNYHYA - XEYCRZGYEUMZUEZYPYHYAYRYOYHXTYKUPXTYAYFYNYLUQYLYOYRYHXTYKYOYRQYOYKYHXTPZY - 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simplbiim 3jca ex eqid clwwlknbp syl11 syl5bi scshwfzeqfzo oveq2 cbvrexvw - syl imp eqcom bitrdi rexbidv bitrid cbvrabv adantr orcomd fveqeq2 orbi12d - cshwshash adantl mpbird eleq1 rexeqdv rabbidv eqeq2 orbi2d imbi12d eleq2s - sylbid rexlimdva com12 syl6bi pm2.43i ) DHCUBZMZGUCMZDNUDZUHOZYBGOZUIZXTY - AYEPZXTXTDKQZUAQZEQZUERZOZESGUFRTZKHUGZOZUAHTZYFUAKABDCEFGHXSIJUJYAYOYEYA - YNYEUAHYAYHHMZUTZYNDYLKFUKUDZULZUGZOZYEYQYMYTDYQYMYLKGFUMRZUGZYTHUUBOYMUU - COYQIYLKHUUBUNUOYAGUPMZYHUUBMZUUCYTOYPYAGGVAZUQYPUUEHUUBYHIURZUSKEFGYHVBV - CVDVEYQUUADYKESGVFRZTZKYSUGZOZYEYQYTUUJDYQYHYSMZYHVGVHZGYHNUDZOZVMZYTUUJO - YAYPUUPYAGVIMZYPUUPPUUFYPUUEUUQUUPUUGUULUUNGOZUTZUUQUUPUUEUUSUUQUUPUUSUUQ - UTZUULUUMUUOUULUURUUQVJUUQUUSUUMUUQUUDGSVHZUUSUUMPGVKUUSUVAUUMUUSGSYHVGUU - RGSOZUUNSOZUULYHVGOUVBUVCVLGUUNGUUNSVNVOYHYSVPVQVRVSWCVTUUTUUNGUULUURUUQW - AWBWDWEFGYRYHYRWFWGZWHWIWMWNKEGYRYHWJWMVEYPYAUUKYEPZYAUVEPZYHUUBHUUEUUSUV - FUVDUUSUVFUUNUCMZDYKESUUNVFRZTZKYSUGZOZYCYBUUNOZUIZPZPZUULUVOUURUULUVGUVN - UULUVGUTZUVKUVMUVPUVKUTZUVMUVJNUDZUHOZUVRUUNOZUIZUVQUVTUVSUVPUVTUVSUIUVKA - 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FriendGraph /\ ( U e. D /\ W e. D ) /\ U =/= W ) -> ( ( { U , A } e. E /\ { W , A } e. 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D ) -> E! y e. N { x , y } e. E ) $= ( cv wcel wa cpr wss wreu cfrgr wne w3a adantr cnbgr eleq2i nbgrisvtx a1i - co wi syl5bi imp wnel elnelne2 expcom syl 3jca frcond1 cusgr frgrusgr weu - sylc prex ancom bitr3i anbi2i cumgr usgrumgr umgrpredgv simprd ex adantld - prss pm4.71rd bitr4id nbusgreledg bitr2id anbi1d bitrd eubidv biimpd 3syl - df-reu 3imtr4g mpd ) ABUBZEUCZUDZWMCUBZUEZWPKUEZUEFUFZCIUGZWQFUCZCHUGZWOG - UHUCZWMIUCZKIUCZWMKUIZUJWTAXCWNTUKWOXDXEXFAWNXDWNWMGJULUPZUCZAXDEXGWMNUMX - HXDUQAGJWMILUNUOURUSAXEWNQUKAWNXFAKEUTZWNXFUQSWNXIXFWMKEVAVBVCUSVDWMKFGIC - LMVEVIAWTXBUQZWNAXCGVFUCZXJTGVGXKWPIUCZWSUDZCVHZWPHUCZXAUDZCVHZWTXBXKXNXQ - XKXMXPCXKXMWRFUCZXAUDZXPXKXMXLXSUDXSWSXSXLWSXAXRUDXSWQWRFWMWPVJWPKVJVTXAX - RVKVLVMXKXSXLXKXAXLXRXKGVNUCZXAXLUQGVOXTXAXLXTXAUDXDXLFGWMWPILMVPVQVRVCVS - WAWBXKXRXOXAXOWPGKULUPZUCXKXRHYAWPOUMFGKWPMWCWDWEWFWGWHWSCIWJXACHWJWKWIUK - WL $. + co biimtrid imp wnel elnelne2 expcom syl 3jca frcond1 sylc cusgr frgrusgr + wi prex prss ancom bitr3i anbi2i cumgr usgrumgr umgrpredgv simprd adantld + ex pm4.71rd bitr4id nbusgreledg bitr2id anbi1d bitrd eubidv biimpd df-reu + weu 3imtr4g 3syl mpd ) ABUBZEUCZUDZWMCUBZUEZWPKUEZUEFUFZCIUGZWQFUCZCHUGZW + OGUHUCZWMIUCZKIUCZWMKUIZUJWTAXCWNTUKWOXDXEXFAWNXDWNWMGJULUPZUCZAXDEXGWMNU + MXHXDVHAGJWMILUNUOUQURAXEWNQUKAWNXFAKEUSZWNXFVHSWNXIXFWMKEUTVAVBURVCWMKFG + ICLMVDVEAWTXBVHZWNAXCGVFUCZXJTGVGXKWPIUCZWSUDZCWIZWPHUCZXAUDZCWIZWTXBXKXN + XQXKXMXPCXKXMWRFUCZXAUDZXPXKXMXLXSUDXSWSXSXLWSXAXRUDXSWQWRFWMWPVIWPKVIVJX + AXRVKVLVMXKXSXLXKXAXLXRXKGVNUCZXAXLVHGVOXTXAXLXTXAUDXDXLFGWMWPILMVPVQVSVB + VRVTWAXKXRXOXAXOWPGKULUPZUCXKXRHYAWPOUMFGKWPMWBWCWDWEWFWGWSCIWHXACHWHWJWK + UKWL $. $d D n $. $d E n $. $d G n $. $d N n y $. $d V n $. $d Y n $. $d ph n $. $d n x $. @@ -433141,19 +437981,19 @@ follows that deg(v) >= 2 for every node v of a friendship graph". -> { ( iota_ y e. N { x , y } e. E ) } = ( ( G NeighbVtx x ) i^i N ) ) $= ( vn cv wcel wa cnbgr co cin cpr crio csn ineq2i wceq wrex wne w3a adantr - cfrgr eleq2i wi nbgrisvtx a1i syl5bi imp wnel elnelne2 sylan2 ancoms 3jca - frcond3 sylc cvv vex elinsn mpan cusgr frgrusgr nbusgreledg eleq1i bitrdi - prcom biimpd 3syl com12 eqcomi biimpi adantl frgrncvvdeqlem2 preq2 eleq1d - wb riota2 syl2an mpbid sylan eqcomd sneqd eqeq1 mpbird ex rexlimivw mpcom - wreu eqtr2id ) ABUCZEUDZUEZGXEUFUGZHUHXHGKUFUGZUHZXECUCZUIZFUDZCHUJZUKZHX - IXHOULXJUBUCZUKZUMZUBIUNZXGXJXOUMZXGGURUDZXEIUDZKIUDZXEKUOZUPXSAYAXFTUQXG - YBYCYDAXFYBXFXEGJUFUGZUDZAYBEYEXENUSYFYBUTAGJXEILVAVBVCVDAYCXFQUQXFAYDAXF - KEVEYDSXEKEVFVGVHVIUBXEKFGILMVJVKXRXGXTUTUBIXRXGXTXRXGUEZXTXQXOUMZYGXPXNY - GXNXPXRXPXHUDZXPXIUDZUEZXGXNXPUMZXPVLUDXRYKUBVMXPXHXIVLVNVOYKXGUEXEXPUIZF - UDZYLYKXGYNYIXGYNUTYJXGYIYNAYIYNUTZXFAYAGVPUDZYOTGVQYPYIYNYPYIXPXEUIZFUDY - NFGXEXPMVRYQYMFXPXEWAVSVTWBWCUQWDUQVDYKXPHUDZXMCHXCYNYLWKXGYJYRYIYJYRXIHX - PHXIOWEUSWFWGABCDEFGHIJKLMNOPQRSTUAWHXMYNCHXPXKXPUMXLYMFXKXPXEWIWJWLWMWNW - OWPWQXRXTYHWKXGXJXQXOWRUQWSWTXAXBXD $. + cfrgr eleq2i wi nbgrisvtx a1i biimtrid imp elnelne2 sylan2 ancoms frcond3 + wnel 3jca sylc elinsn mpan cusgr frgrusgr nbusgreledg prcom eleq1i bitrdi + cvv vex biimpd 3syl com12 wreu eqcomi biimpi adantl frgrncvvdeqlem2 preq2 + wb eleq1d riota2 syl2an mpbid sylan eqcomd sneqd eqeq1 ex rexlimivw mpcom + mpbird eqtr2id ) ABUCZEUDZUEZGXEUFUGZHUHXHGKUFUGZUHZXECUCZUIZFUDZCHUJZUKZ + HXIXHOULXJUBUCZUKZUMZUBIUNZXGXJXOUMZXGGURUDZXEIUDZKIUDZXEKUOZUPXSAYAXFTUQ + XGYBYCYDAXFYBXFXEGJUFUGZUDZAYBEYEXENUSYFYBUTAGJXEILVAVBVCVDAYCXFQUQXFAYDA + XFKEVIYDSXEKEVEVFVGVJUBXEKFGILMVHVKXRXGXTUTUBIXRXGXTXRXGUEZXTXQXOUMZYGXPX + NYGXNXPXRXPXHUDZXPXIUDZUEZXGXNXPUMZXPVTUDXRYKUBWAXPXHXIVTVLVMYKXGUEXEXPUI + ZFUDZYLYKXGYNYIXGYNUTYJXGYIYNAYIYNUTZXFAYAGVNUDZYOTGVOYPYIYNYPYIXPXEUIZFU + DYNFGXEXPMVPYQYMFXPXEVQVRVSWBWCUQWDUQVDYKXPHUDZXMCHWEYNYLWKXGYJYRYIYJYRXI + HXPHXIOWFUSWGWHABCDEFGHIJKLMNOPQRSTUAWIXMYNCHXPXKXPUMXLYMFXKXPXEWJWLWMWNW + OWPWQWRXRXTYHWKXGXJXQXOWSUQXCWTXAXBXD $. $d D x $. $d N x y $. $d ph x $. $( Lemma 4 for ~ frgrncvvdeq . The mapping of neighbors to neighbors is a @@ -433233,26 +438073,26 @@ the common neighbor of x and a(x)". (Contributed by Alexander van der by AV, 12-Feb-2022.) $) frgrncvvdeqlem9 $p |- ( ph -> A : D -onto-> N ) $= ( vn vm wf cv cfv wceq wrex wral wfo frgrncvvdeqlem4 wa wex cpr cfrgr wne - wcel w3a wreu adantr cnbgr co eleq2i nbgrisvtx a1i syl5bi frgrncvvdeqlem1 - wi imp wnel df-nel nelelne sylbi syl 3jca jca frcond2 reurex df-rex sylib - wn 3syl cusgr frgrusgr nbusgreledg bicomd biimpa sylibr ad2ant2rl biimpar - wb a1d expimpd cin elin csn simpl crio cmpt preq1 riotabidv cbvmptv eqtri - eleq1d frgrncvvdeqlem5 eleq2 eqcoms elsni syl6bi expcom sylbir ex adantld - com3r com14 mpd eximdv ralrimiva dffo3 sylanbrc ) AEHDUDUBUEZUCUEZDUFZUGZ - UCEUHZUBHUIEHDUJABCDEFGHIJKLMNOPQRSTUAUKAYEUBHAYAHUQZULZYBEUQZYDULZUCUMZY - EYGYBIUQZYAYBUNFUQZYBJUNFUQZULZULZUCUMZYJYGGUOUQZYAIUQZJIUQZYAJUPZURZULYN - UCIUSZYPYGYQUUAAYQYFTUTYGYRYSYTAYFYRYFYAGKVAVBZUQZAYRHUUCYAOVCUUDYRVHAGKY - AILVDVEVFVIAYSYFPUTAYFYTAJHVJZYFYTVHZABCDEFGHIJKLMNOPQRSTUAVGUUEJHUQWAUUF - JHVKJHYAVLVMVNVIVOVPYQUUAUUBYAJFGIUCLMVQVIUUBYNUCIUHYPYNUCIVRYNUCIVSVTWBY - GYOYIUCYGYNYIYKYGYNYIYGYNULZYHYDAYMYHYFYLAYMULZYBGJVAVBZUQZYHAYMUUJAYQGWC - UQZYMUUJWKTGWDZUUKUUJYMFGJYBMWEWFWBWGEUUIYBNVCWHZWIUUGYAGYBVAVBZUQZYDYGYN - UUOAYNUUOVHZYFAYQUUKUUPTUULUUKYLYMUUOUUKYLULUUOYMUUKUUOYLFGYBYAMWEWJWLWMW - BUTVIYGYNUUOYDVHZYGYMUUQYLAYFYMUUQVHUUOYFYMAYDUUOYFYMAYDVHVHZUUOYFULYAUUN - HWNZUQZUURYAUUNHWOYMAUUTYDAYMUUTYDVHZUUHAYHULYCWPZUUSUGZUVAUUHAYHAYMWQUUM - VPAUCCDEFGHIJKLMNOPQRSTDBEBUEZCUEZUNZFUQZCHWRZWSUCEYBUVEUNZFUQZCHWRZWSUAB - UCEUVHUVKUVDYBUGZUVGUVJCHUVLUVFUVIFUVDYBUVEWTXDXAXBXCXEUVCUUTYAUVBUQZYDUU - TUVMWKUUSUVBUUSUVBYAXFXGYAYCXHXIWBXJXNXKXLXOVIXMVIXPVPXLXMXQXPYDUCEVSWHXR - UCUBEHDXSXT $. + wcel w3a adantr cnbgr co eleq2i wi nbgrisvtx a1i biimtrid frgrncvvdeqlem1 + wreu imp wnel wn df-nel nelelne sylbi syl jca frcond2 reurex df-rex sylib + 3jca cusgr wb frgrusgr nbusgreledg bicomd biimpa sylibr ad2ant2rl biimpar + 3syl a1d expimpd cin elin simpl crio preq1 eleq1d riotabidv cbvmptv eqtri + csn cmpt frgrncvvdeqlem5 eleq2 eqcoms elsni syl6bi expcom com3r sylbir ex + com14 adantld mpd eximdv ralrimiva dffo3 sylanbrc ) AEHDUDUBUEZUCUEZDUFZU + GZUCEUHZUBHUIEHDUJABCDEFGHIJKLMNOPQRSTUAUKAYEUBHAYAHUQZULZYBEUQZYDULZUCUM + ZYEYGYBIUQZYAYBUNFUQZYBJUNFUQZULZULZUCUMZYJYGGUOUQZYAIUQZJIUQZYAJUPZURZUL + YNUCIVHZYPYGYQUUAAYQYFTUSYGYRYSYTAYFYRYFYAGKUTVAZUQZAYRHUUCYAOVBUUDYRVCAG + KYAILVDVEVFVIAYSYFPUSAYFYTAJHVJZYFYTVCZABCDEFGHIJKLMNOPQRSTUAVGUUEJHUQVKU + UFJHVLJHYAVMVNVOVIWAVPYQUUAUUBYAJFGIUCLMVQVIUUBYNUCIUHYPYNUCIVRYNUCIVSVTW + KYGYOYIUCYGYNYIYKYGYNYIYGYNULZYHYDAYMYHYFYLAYMULZYBGJUTVAZUQZYHAYMUUJAYQG + WBUQZYMUUJWCTGWDZUUKUUJYMFGJYBMWEWFWKWGEUUIYBNVBWHZWIUUGYAGYBUTVAZUQZYDYG + YNUUOAYNUUOVCZYFAYQUUKUUPTUULUUKYLYMUUOUUKYLULUUOYMUUKUUOYLFGYBYAMWEWJWLW + MWKUSVIYGYNUUOYDVCZYGYMUUQYLAYFYMUUQVCUUOYFYMAYDUUOYFYMAYDVCVCZUUOYFULYAU + UNHWNZUQZUURYAUUNHWOYMAUUTYDAYMUUTYDVCZUUHAYHULYCXCZUUSUGZUVAUUHAYHAYMWPU + UMVPAUCCDEFGHIJKLMNOPQRSTDBEBUEZCUEZUNZFUQZCHWQZXDUCEYBUVEUNZFUQZCHWQZXDU + ABUCEUVHUVKUVDYBUGZUVGUVJCHUVLUVFUVIFUVDYBUVEWRWSWTXAXBXEUVCUUTYAUVBUQZYD + UUTUVMWCUUSUVBUUSUVBYAXFXGYAYCXHXIWKXJXKXLXMXNVIXOVIXPVPXMXOXQXPYDUCEVSWH + XRUCUBEHDXSXT $. $( Lemma 10 for ~ frgrncvvdeq . (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, @@ -433299,16 +438139,16 @@ then they have the same degree (i.e., the same number of adjacent a1i imp 3adant1 cv csn cdif wral frgrncvvdeq oveq2 neleq2 fveqeq2 imbi12d syl neleq1 eqeq2d simpll sneq difeq2d adantl simpr biimpi anim12i eldifsn wb necom sylibr rspc2vd wn nnel nbgrsym cusgr frgrusgr nbusgreledg biimpd - syl5bi a1d expcom sylbi eqneqall 2a1d ja com12 syld com3l mpcom expd 3imp - com34 mpd ) CUALZEDLZFDLZMZEANZFANZOZUBEFOZEFUCBLZWRXAXBWOWRXAXBWREFWSWTE - FPWSWTPZQWREFAUDUFUEUGUHWOWRXAXBXCQWOWRXBXAXCWOWRXBXAXCQZJUIZCKUIZRSZTZXG - ANXFANZPZQZJDXGUJZUKZULKDULZWOWRXBMZXEQKJACDGHUMXPXOWOXEXPXOFCERSZTZXDQZW - OXEQZXPXSXFXQTZWSXJPZQXLKJEFDXNDEUJZUKZXGEPZXIYAXKYBYEXHXQPXIYAVIXGECRUNX - HXQXFUOURXGEXJAUPUQXFFPZYAXRYBXDXFFXQUSYFXJWTWSXFFAUDUTUQWPWQXBVAYEXNYDPX - PYEXMYCDXGEVBVCVDXPWQFEOZMFYDLWRWQXBYGWPWQVEXBYGEFVJVFVGFDEVHVKVLXSXPXTXR - XDXPXTQZXRVMFXQLZYHFXQVNYIXTXPWOYIXEWOYIMXCXAWOYIXCYIECFRSLZWOXCCEFVOWOYJ - XCWOCVPLYJXCVICVQBCFEIVRURVSVTUGWAWBWAWCXDXEXPWOXCWSWTWDWEWFWGWHWIWJWKWMW - LWN $. + biimtrid a1d expcom sylbi eqneqall 2a1d com12 syld com3l mpcom expd com34 + ja 3imp mpd ) CUALZEDLZFDLZMZEANZFANZOZUBEFOZEFUCBLZWRXAXBWOWRXAXBWREFWSW + TEFPWSWTPZQWREFAUDUFUEUGUHWOWRXAXBXCQWOWRXBXAXCWOWRXBXAXCQZJUIZCKUIZRSZTZ + XGANXFANZPZQZJDXGUJZUKZULKDULZWOWRXBMZXEQKJACDGHUMXPXOWOXEXPXOFCERSZTZXDQ + ZWOXEQZXPXSXFXQTZWSXJPZQXLKJEFDXNDEUJZUKZXGEPZXIYAXKYBYEXHXQPXIYAVIXGECRU + NXHXQXFUOURXGEXJAUPUQXFFPZYAXRYBXDXFFXQUSYFXJWTWSXFFAUDUTUQWPWQXBVAYEXNYD + PXPYEXMYCDXGEVBVCVDXPWQFEOZMFYDLWRWQXBYGWPWQVEXBYGEFVJVFVGFDEVHVKVLXSXPXT + XRXDXPXTQZXRVMFXQLZYHFXQVNYIXTXPWOYIXEWOYIMXCXAWOYIXCYIECFRSLZWOXCCEFVOWO + YJXCWOCVPLYJXCVICVQBCFEIVRURVSVTUGWAWBWAWCXDXEXPWOXCWSWTWDWEWLWFWGWHWIWJW + KWMWN $. $( If a friendship graph has two vertices with the same degree and two other vertices with different degrees, then there is a 4-cycle in the @@ -433424,11 +438264,11 @@ then they have the same degree (i.e., the same number of adjacent ( vv wcel wceq cv wral cfrgr csn cdif wa frgrwopreglem4 ralcom wi snidg cpr adantr wb eleq2 adantl mpbird preq2 prcom eqtrdi eleq1d ralbidv syl rspcv wss cfv ssrab3 ssdifim mp2an difeq2 eqtrid raleqdv sylibd syl5com - syl5bi 3impib ) GUAQZJIQZDJUBZRZJBSZUIZFQZBIVPUCZTZVNVRPSZUIZFQZPDTBCTZ - VOVQUDZWBACDEFGHIBPKLMNOUEWFWEBCTZPDTZWGWBWEBPCDUFWGWIVTBCTZWBWGJDQZWIW - JUGWGWKJVPQZVOWLVQJIUHUJVQWKWLUKVODVPJULUMUNWHWJPJDWCJRZWEVTBCWMWDVSFWM - WDVRJUIVSWCJVRUOVRJUPUQURUSVAUTWGVTBCWAWGCIDUCZWACIVBDICUCRCWNRASEVCHRA - ICMVDNCDIVEVFVQWNWARVODVPIVGUMVHVIVJVLVKVM $. + biimtrid 3impib ) GUAQZJIQZDJUBZRZJBSZUIZFQZBIVPUCZTZVNVRPSZUIZFQZPDTBC + TZVOVQUDZWBACDEFGHIBPKLMNOUEWFWEBCTZPDTZWGWBWEBPCDUFWGWIVTBCTZWBWGJDQZW + IWJUGWGWKJVPQZVOWLVQJIUHUJVQWKWLUKVODVPJULUMUNWHWJPJDWCJRZWEVTBCWMWDVSF + WMWDVRJUIVSWCJVRUOVRJUPUQURUSVAUTWGVTBCWAWGCIDUCZWACIVBDICUCRCWNRASEVCH + RAICMVDNCDIVEVFVQWNWARVODVPIVGUMVHVIVJVLVKVM $. $d V v $. $( According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, @@ -433438,10 +438278,10 @@ then they have the same degree (i.e., the same number of adjacent -> E. v e. V A. w e. ( V \ { v } ) { v , w } e. E ) $= ( wcel cfv wceq cv wrex cfrgr chash c1 cpr csn cdif wral wex cvtx fvexi cvv wb rabex2 hash1snb ax-mp wi exsnrex wss ssrab3 ssrexv frgrwopregasn - 3expia reximdva syl5com sylbi com12 syl5bi imp ) HUAPZDUBQUCRZCSZBSUDGP - BJVKUEZUFUGZCJTZVJDVLRZCUHZVIVNDUKPVJVPULASFQIRZAJDMJHUIKUJUMDUKCUNUOVP - VIVNVPVOCDTZVIVNUPCDUQVRVOCJTZVIVNDJURVRVSUPVQAJDMUSVOCDJUTUOVIVOVMCJVI - VKJPVOVMABDEFGHIJVKKLMNOVAVBVCVDVEVFVGVH $. + 3expia reximdva syl5com sylbi com12 biimtrid imp ) HUAPZDUBQUCRZCSZBSUD + GPBJVKUEZUFUGZCJTZVJDVLRZCUHZVIVNDUKPVJVPULASFQIRZAJDMJHUIKUJUMDUKCUNUO + VPVIVNVPVOCDTZVIVNUPCDUQVRVOCJTZVIVNDJURVRVSUPVQAJDMUSVOCDJUTUOVIVOVMCJ + VIVKJPVOVMABDEFGHIJVKKLMNOVAVBVCVDVEVFVGVH $. $( According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander @@ -433450,10 +438290,10 @@ then they have the same degree (i.e., the same number of adjacent -> E. v e. V A. w e. ( V \ { v } ) { v , w } e. E ) $= ( wcel wceq cv wrex cvv cfrgr chash cfv c1 cpr cdif wral frgrwopreglem1 csn wex wb simpri hash1snb ax-mp wi exsnrex difss eqsstri frgrwopregbsn - wss ssrexv 3expia reximdva syl5com sylbi com12 syl5bi imp ) HUAPZEUBUCU - DQZCRZBRUEGPBJVKUIZUFUGZCJSZVJEVLQZCUJZVIVNETPZVJVPUKDTPVQADEFHIJKLMNUH - ULETCUMUNVPVIVNVPVOCESZVIVNUOCEUPVRVOCJSZVIVNEJUTVRVSUOEJDUFJNJDUQURVOC - EJVAUNVIVOVMCJVIVKJPVOVMABDEFGHIJVKKLMNOUSVBVCVDVEVFVGVH $. + wss ssrexv 3expia reximdva syl5com sylbi com12 biimtrid imp ) HUAPZEUBU + CUDQZCRZBRUEGPBJVKUIZUFUGZCJSZVJEVLQZCUJZVIVNETPZVJVPUKDTPVQADEFHIJKLMN + UHULETCUMUNVPVIVNVPVOCESZVIVNUOCEUPVRVOCJSZVIVNEJUTVRVSUOEJDUFJNJDUQURV + OCEJVAUNVIVOVMCJVIVKJPVOVMABDEFGHIJVKKLMNOUSVBVCVDVEVFVGVH $. $d A x z $. $d D z $. $d G z $. $d K z $. $d V z $. $d y z $. $( Lemma for ~ frgrwopreglem5 . (Contributed by AV, 5-Feb-2022.) $) @@ -433463,12 +438303,12 @@ then they have the same degree (i.e., the same number of adjacent /\ ( D ` a ) =/= ( D ` b ) /\ ( D ` x ) =/= ( D ` y ) ) ) $= ( vz cv wcel cfv wa wceq wne rabeq2i fveqeq2 elrab2 eqtr3 expcom adantl - wi com12 simplbiim syl5bi imp adantr frgrwopreglem3 ad2ant2r crab eqtri - cbvrabv ad2ant2l 3jca ) JRZCSZARZCSZUAZKRZDSZBRZDSZUAZUAVCETZVEETZUBZVM - VHETUCZVNVJETUCZVGVOVLVDVFVOVFVEISZVNHUBZUAZVDVOVSACINUDVDVCISVMHUBZVTV - OUJVSWAAVCICVEVCHEUENUFVTWAVOVSWAVOUJVRWAVSVOVMVNHUGUHUIUKULUMUNUOVDVIV - PVFVKACDEGHIVCVHLMNOUPUQVFVKVQVDVIQCDEGHIVEVJLMCVSAIURQRZETHUBZQIURNVSW - CAQIVEWBHEUEUTUSOUPVAVB $. + wi com12 simplbiim biimtrid adantr frgrwopreglem3 ad2ant2r crab cbvrabv + imp eqtri ad2ant2l 3jca ) JRZCSZARZCSZUAZKRZDSZBRZDSZUAZUAVCETZVEETZUBZ + VMVHETUCZVNVJETUCZVGVOVLVDVFVOVFVEISZVNHUBZUAZVDVOVSACINUDVDVCISVMHUBZV + TVOUJVSWAAVCICVEVCHEUENUFVTWAVOVSWAVOUJVRWAVSVOVMVNHUGUHUIUKULUMUSUNVDV + IVPVFVKACDEGHIVCVHLMNOUOUPVFVKVQVDVIQCDEGHIVEVJLMCVSAIUQQRZETHUBZQIUQNV + SWCAQIVEWBHEUEURUTOUOVAVB $. $d A a y $. $d B a b y z $. $d E x z $. $( Lemma 5 for ~ frgrwopreg . If ` A ` as well as ` B ` contain at least @@ -433514,23 +438354,23 @@ then they have the same degree (i.e., the same number of adjacent /\ ( { x , y } e. E /\ { y , a } e. E ) ) ) $= ( wcel wa wrex wi vz cfrgr chash cfv clt wbr wne cpr w3a simpllr anim1i c1 cv wral frgrwopreglem4 wceq preq1 eleq1d ralbidv cbvralvw rsp2 com12 - ad2ant2r syl5bi imp prcom ad2ant2lr eqeltrid jca expcom syl adantr impl - sylbi preq2 rspc2v ad2ant2l impcom ad2ant2rl 3jca reximdvva exp31 com24 - imp31 com13 cvv frgrwopreglem1 hashgt12el im2anan9 ax-mp syl11 3impib - ex ) GUBQZULCUCUDUEUFZULDUCUDUEUFZJUMZAUMZUGZKUMZBUMZUGZRZWQWTUHZFQZWTW - RUHZFQZRZWRXAUHZFQZXAWQUHZFQZRZUIZBDSKDSZACSJCSZWSACSJCSZXBBDSKDSZRZWNX - PWOWPRZXQXRWNXPTWNXRXQXPWNXRXQXPTWNXRRWSXOJACCWNXRWQCQZWRCQZRZWSXOTWNWS - YCXRXOWNWSYCXRXOTWNWSRZYCRZXBXNKBDDYEWTDQZXADQZRZRZXBXNYIXBRXCXHXMYIWSX - BWNWSYCYHUJUKYIXHXBYDYCYHXHWNYCYHRZXHTZWSWNUAUMZWTUHZFQZKDUNZUACUNZYKAC - DEFGHIUAKLMNOPUOZYJYPXHYJYPRZXEXGYJYPXEYPXEKDUNZJCUNZYJXEYOYSUAJCYLWQUP - ZYNXEKDUUAYMXDFYLWQWTUQURZUSUTYAYFYTXETYBYGYTYAYFRXEXEJKCDVAVBVCVDVEYRX - FWRWTUHZFWTWRVFYJYPUUCFQZYBYFYPUUDTYAYGYPYBYFRZUUDYPUUDKDUNZACUNUUEUUDT - YOUUFUAACYLWRUPZYNUUDKDUUGYMUUCFYLWRWTUQURZUSUTUUDAKCDVAVNVBVGVEVHVIVJV - KVLVMVLYIXMXBYDYCYHXMWNYJXMTZWSWNYPUUIYQYPYJXMYPYJRZXJXLYJYPXJYBYGYPXJT - YAYFYNXJUUDUAKWRXACDUUHWTXAUPZUUCXIFWTXAWRVOURVPVQVRUUJXKWQXAUHZFXAWQVF - YJYPUULFQZYAYGYPUUMTYBYFYNUUMXEUAKWQXACDUUBUUKXDUULFWTXAWQVOURVPVSVRVHV - IWMVKVLVMVLVTWMWAWBWCWDWAWMWEVECWFQZDWFQZRXTXSTACDEGHILMNOWGUUNWOXQUUOW - PXRUUNWOXQCWFJAWHWMUUOWPXRDWFKBWHWMWIWJWKWL $. + ad2ant2r biimtrid imp prcom sylbi ad2ant2lr eqeltrid expcom adantr impl + jca syl preq2 rspc2v ad2ant2l impcom ad2ant2rl ex reximdvva exp31 com24 + 3jca imp31 com13 frgrwopreglem1 hashgt12el im2anan9 ax-mp syl11 3impib + cvv ) GUBQZULCUCUDUEUFZULDUCUDUEUFZJUMZAUMZUGZKUMZBUMZUGZRZWQWTUHZFQZWT + WRUHZFQZRZWRXAUHZFQZXAWQUHZFQZRZUIZBDSKDSZACSJCSZWSACSJCSZXBBDSKDSZRZWN + XPWOWPRZXQXRWNXPTWNXRXQXPWNXRXQXPTWNXRRWSXOJACCWNXRWQCQZWRCQZRZWSXOTWNW + SYCXRXOWNWSYCXRXOTWNWSRZYCRZXBXNKBDDYEWTDQZXADQZRZRZXBXNYIXBRXCXHXMYIWS + XBWNWSYCYHUJUKYIXHXBYDYCYHXHWNYCYHRZXHTZWSWNUAUMZWTUHZFQZKDUNZUACUNZYKA + CDEFGHIUAKLMNOPUOZYJYPXHYJYPRZXEXGYJYPXEYPXEKDUNZJCUNZYJXEYOYSUAJCYLWQU + PZYNXEKDUUAYMXDFYLWQWTUQURZUSUTYAYFYTXETYBYGYTYAYFRXEXEJKCDVAVBVCVDVEYR + XFWRWTUHZFWTWRVFYJYPUUCFQZYBYFYPUUDTYAYGYPYBYFRZUUDYPUUDKDUNZACUNUUEUUD + TYOUUFUAACYLWRUPZYNUUDKDUUGYMUUCFYLWRWTUQURZUSUTUUDAKCDVAVGVBVHVEVIVMVJ + VNVKVLVKYIXMXBYDYCYHXMWNYJXMTZWSWNYPUUIYQYPYJXMYPYJRZXJXLYJYPXJYBYGYPXJ + TYAYFYNXJUUDUAKWRXACDUUHWTXAUPZUUCXIFWTXAWRVOURVPVQVRUUJXKWQXAUHZFXAWQV + FYJYPUULFQZYAYGYPUUMTYBYFYNUUMXEUAKWQXACDUUBUUKXDUULFWTXAWQVOURVPVSVRVI + VMVTVNVKVLVKWDVTWAWBWCWEWAVTWFVECWMQZDWMQZRXTXSTACDEGHILMNOWGUUNWOXQUUO + WPXRUUNWOXQCWMJAWHVTUUOWPXRDWMKBWHVTWIWJWKWL $. $} $d A a b y $. $d B a b x y $. $d D y $. $d G a b y $. $d V y $. @@ -433670,18 +438510,18 @@ third vertex being the middle vertex of a (simple) path/walk of length 2 wne w3a c2 chash cfv weu wal frgr2wwlkn0 elwwlks2ons3 anbi12i frgr2wwlkeu co c1 wreu wral eleq1d reu4 anbi1d equequ1 imbi12d anbi2d equequ2 rspc2va c0 pm3.35 equcoms adantr wb eqeq12 adantl mpbird ex syl com23 exp4b com13 - equcomd imp expcom simplbiim rexlimdva syl5bi alrimivv eqeuel syl2anc cvv - impl impd ovex euhash1 mp1i ) CUALADLBDLMABUCUDZABUECUBUNZUNZUFUGUONZFOZW - PLZFUHZWNWPVFUCWSGOZWPLZMZFGPZQZGUIFUIWTABCDEUJWNXEFGXCWRAHOZBRZNZXGWPLZM - ZHDSZXAAIOZBRZNZXMWPLZMZIDSZMZWNXDWSXKXBXQABCDWRHEUKABCDXAIEUKULWNAJOZBRZ - WPLZJDUPZXRXDQABCDJEUMYBXKXQXDYBXJXQXDQHDYBXFDLZMZXQXJXDYDXPXJXDQZIDYBYCX - LDLZXPYEQZYBYAJDSYAAKOZBRZWPLZMZJKPZQZKDUQJDUQZYCYFMZYGQYAYJJKDYLXTYIWPAX - SBYHTURUSYOYNYGYOYNMXIXOMZHIPZQZYGYMYRXIYJMZHKPZQJKXFXLDDJHPZYKYSYLYTUUAY - AXIYJUUAXTXGWPAXSBXFTURUTJHKVAVBKIPZYSYPYTYQUUBYJXOXIUUBYIXMWPAYHBXLTURVC - KIHVDVBVEXJXPYRXDXHXIXPYRXDQZQXPXIXHUUCXNXOXIXHUUCQZQXIXOXNUUDXIXOXNXHUUC - YPYRXNXHMZXDYPYRUUEXDQZYPYRMYQUUFYPYQVGYQUUEXDYQUUEMZGFUUGGFPZXMXGNZYQUUI - UUEUUIIHAXLBXFTVHVIUUEUUHUUIVJYQXAXMWRXGVKVLVMVSVNVOVNVPVQVRVTVRVTVRVOWAW - BWIWCVPWCWJVOWDWEFGWPWFWGWPWHLWQWTVJWNABWOWKWPWHFWLWMVM $. + equcomd imp expcom simplbiim impl rexlimdva impd biimtrid alrimivv eqeuel + syl2anc cvv ovex euhash1 mp1i ) CUALADLBDLMABUCUDZABUECUBUNZUNZUFUGUONZFO + ZWPLZFUHZWNWPVFUCWSGOZWPLZMZFGPZQZGUIFUIWTABCDEUJWNXEFGXCWRAHOZBRZNZXGWPL + ZMZHDSZXAAIOZBRZNZXMWPLZMZIDSZMZWNXDWSXKXBXQABCDWRHEUKABCDXAIEUKULWNAJOZB + RZWPLZJDUPZXRXDQABCDJEUMYBXKXQXDYBXJXQXDQHDYBXFDLZMZXQXJXDYDXPXJXDQZIDYBY + CXLDLZXPYEQZYBYAJDSYAAKOZBRZWPLZMZJKPZQZKDUQJDUQZYCYFMZYGQYAYJJKDYLXTYIWP + AXSBYHTURUSYOYNYGYOYNMXIXOMZHIPZQZYGYMYRXIYJMZHKPZQJKXFXLDDJHPZYKYSYLYTUU + AYAXIYJUUAXTXGWPAXSBXFTURUTJHKVAVBKIPZYSYPYTYQUUBYJXOXIUUBYIXMWPAYHBXLTUR + VCKIHVDVBVEXJXPYRXDXHXIXPYRXDQZQXPXIXHUUCXNXOXIXHUUCQZQXIXOXNUUDXIXOXNXHU + UCYPYRXNXHMZXDYPYRUUEXDQZYPYRMYQUUFYPYQVGYQUUEXDYQUUEMZGFUUGGFPZXMXGNZYQU + UIUUEUUIIHAXLBXFTVHVIUUEUUHUUIVJYQXAXMWRXGVKVLVMVSVNVOVNVPVQVRVTVRVTVRVOW + AWBWCWDVPWDWEVOWFWGFGWPWHWIWPWJLWQWTVJWNABWOWKWPWJFWLWMVM $. $( In a friendship graph, there is exactly one simple path of length 2 between two different vertices. (Contributed by Alexander van der @@ -434215,22 +439055,22 @@ last vertex (which is the first vertex) itself ("walking forth and back" -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X C N ) ) ) $= ( wcel cfv w3a co wa wi vi cusgr cuz cnbgr cs1 cconcat cclwwlkn cmin cpfx c3 wceq cpr cedg cclwwlknon simpl2 nbgrisvtx ad2antll simpl3 nbgrsym eqid - c2 c1 nbusgreledg biimpd syl5bi 3ad2ant1 imp simprl eleqtrdi clwwlknonex2 - adantld syl311anc cword chash caddc cc0 cfzo wral clsw eleq2i wne uz3m2nn - cv wb nnne0d clwwlknonel syl 3ad2ant3 bitrid 3simpa adantr simp32 anim12i - simpl33 3jca 3exp1 3adant3 com12 sylbid imp32 numclwwlk1lem2foalem eleq1a - idd 3anim123d mpd extwwlkfabel mpbir2and ex ) FUBOZJHOZGUJUCPOZQZIEOZKFJU - DRZOZSZIJUEUFRKUEUFRZJGCROZXLXPSZXRXQGFUGROZXQGVAUHRZUIRZEOZGVBUHRXQPZXNO - ZYAXQPJUKZQZXSXJKHOZXKJKULFUMPZOZIJYAFUNPRZOZXTXIXJXKXPUOXOYHXLXMFJKHLUPZ - UQXIXJXKXPURXLXPYJXIXJXPYJTXKXIXOYJXMXOJFKUDROZXIYJFJKUSXIYNYJYIFKJYIUTZV - CVDVEVKVFVGXSIEYKXLXMXOVHZNVIYIFGHIJKLYOVJVLXSYBIUKZYDKUKZYFQZYGXSIHVMOZI - VNPZYAUKZSZXJYHSZXKQZYSXLXMXOUUEXLXMYTUAWCZIPUUFVBVORIPULYIOUAVPUUAVBUHRV - QRVRZIVSPVPIPZULYIOZQZUUBUUHJUKZQZXOUUETZXMYLXLUULEYKINVTXKXIYLUULWDZXJXK - YAVPWAUUNXKYAGWBWEUAYIFYAHIJLYOWFWGWHWIUULXLUUMUUJUUBXLUUMTZUUKUUJUUBUUOY - TUUGUUBUUOTUUIYTUUBXLXOUUEYTUUBXLQZXOSUUCUUDXKUUPUUCXOYTUUBXLWJWKUUPXJXOY - HYTUUBXIXJXKWLYMWMXIXJXKYTUUBXOWNWOWPVFVGWQWRWSWTGHIJKXAWGXSYQYCYRYEYFYFX - SXMYQYCTYPIEYBXBWGXOYRYETXLXMKXNYDXBUQXSYFXCXDXEXLXRXTYGSWDXPABCDEFGHXQJL - MNXFWKXGXH $. + c2 nbusgreledg biimpd biimtrid adantld 3ad2ant1 imp eleqtrdi clwwlknonex2 + c1 simprl syl311anc cword chash cv caddc cc0 cfzo wral clsw eleq2i wb wne + uz3m2nn nnne0d clwwlknonel syl 3ad2ant3 bitrid 3simpa adantr anim12i 3jca + simp32 simpl33 3exp1 3adant3 com12 sylbid numclwwlk1lem2foalem eleq1a idd + imp32 3anim123d mpd extwwlkfabel mpbir2and ex ) FUBOZJHOZGUJUCPOZQZIEOZKF + JUDRZOZSZIJUEUFRKUEUFRZJGCROZXLXPSZXRXQGFUGROZXQGVAUHRZUIRZEOZGVJUHRXQPZX + NOZYAXQPJUKZQZXSXJKHOZXKJKULFUMPZOZIJYAFUNPRZOZXTXIXJXKXPUOXOYHXLXMFJKHLU + PZUQXIXJXKXPURXLXPYJXIXJXPYJTXKXIXOYJXMXOJFKUDROZXIYJFJKUSXIYNYJYIFKJYIUT + ZVBVCVDVEVFVGXSIEYKXLXMXOVKZNVHYIFGHIJKLYOVIVLXSYBIUKZYDKUKZYFQZYGXSIHVMO + ZIVNPZYAUKZSZXJYHSZXKQZYSXLXMXOUUEXLXMYTUAVOZIPUUFVJVPRIPULYIOUAVQUUAVJUH + RVRRVSZIVTPVQIPZULYIOZQZUUBUUHJUKZQZXOUUETZXMYLXLUULEYKINWAXKXIYLUULWBZXJ + XKYAVQWCUUNXKYAGWDWEUAYIFYAHIJLYOWFWGWHWIUULXLUUMUUJUUBXLUUMTZUUKUUJUUBUU + OYTUUGUUBUUOTUUIYTUUBXLXOUUEYTUUBXLQZXOSUUCUUDXKUUPUUCXOYTUUBXLWJWKUUPXJX + OYHYTUUBXIXJXKWNYMWLXIXJXKYTUUBXOWOWMWPVFVGWQWRWSXCGHIJKWTWGXSYQYCYRYEYFY + FXSXMYQYCTYPIEYBXAWGXOYRYETXLXMKXNYDXAUQXSYFXBXDXEXLXRXTYGSWBXPABCDEFGHXQ + JLMNXFWKXGXH $. $d C u $. $d F u $. $d G u w $. $d N u $. $d V u $. $d X u $. ${ @@ -434275,36 +439115,36 @@ last vertex (which is the first vertex) itself ("walking forth and back" 3ad2ant2 simprd oveq1d oveq2d biimpcd impcom fveq2d adantl lsw sylan9eq fvoveq1 syl ad3antrrr oveq1 eqcoms eqeq2d mpbird biimpd adantld clt wbr 3jca clwwlknwrd clwwlknlen cz eluz2b1 simplbiim syl2imc 3adant3 syl3anc - 2swrd2eqwrdeq mpbir2and 3exp 3ad2ant3 sylbid syl5bi ralrimivva sylanbrc - breq2 dff13 ) HUCPZKJPZIUDUEQPZUFZKIDUGZGHKUHUGUMZEUIUAUJZEQZUBUJZEQZRZ - YTUUBRZUKZUBYRULUAYRULYRYSEUNABCDEFGHIJKLMNOUOYQUUFUAUBYRYRYQYTYRPZUUBY - RPZSZSZUUDYTIUPUQUGZURUGZIVBUQUGZYTQZVCZUUBUUKURUGZUUMUUBQZVCZRZUUEUUJU - UAUUOUUCUURUUGUUAUUORYQUUHABCDEFGHIJYTKLMNOUSUTUUHUUCUURRYQUUGABCDEFGHI - JUUBKLMNOUSVAVDUUSUULUUPRZUUNUUQRZSZUUJUUEUULUUNUUPUUQYTUUKURVEUUMYTVFV - GYQUUIUVBUUEUKZYQUUIYTIHVHUGZPZVMYTQZKRZSZUUKYTQZKRZSZUUBUVDPZVMUUBQZKR - ZSZUUKUUBQZKRZSZSZUVCYOYPUUIUVSVIZYNYPYOIUPUEQPZUVTIVJZYOUWASZUUGUVKUUH - UVRUWCUUGYTKIHVKQUGZPZUVJSUVKABDFHIJYTKMVLUWEUVHUVJHIYTKVNVOVPUWCUUHUUB - UWDPZUVQSUVRABDFHIJUUBKMVLUWFUVOUVQHIUUBKVNVOVPVQVRWCYPYNUVSUVCUKYOYPUV - SUVBUUEYPUVSUVBUFZUUEYTVSQZUUBVSQZRZYTUWHUPUQUGZURUGZUUBUWKURUGZRZUWKYT - QZUWKUUBQZRZYTVTQZUUBVTQZRZUFZUVSYPUWJUVBUVKYTJWAZPZUWHIRZSZUVGUVIUVFRZ - SZSZUUBUXBPZUWIIRZSZUVNUVPUVMRZSZSZUWJUVRUVKUXEUVGUXFUVHUXEUVJUVEUXEUVG - HIJYTLWBZTZTUVHUVGUVJUVEUVGWDZTUVKUVIKUVFUVHUVJWDZUVHKUVFRUVJUVHUVFKUXQ - 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This theorem holds even for k=0, but not for n=2, since ( wcel cfv chash cc0 wceq wa c0 cop csn wb wi wbr cvv adantr wf c1st c2nd cclwlks wal crab cwlks clwlkwlk wlkop syl w3a fvex hasheq0 ax-mp 3ad2ant3 cv biimpi cfz co adantl breq1d 1vgrex 0clwlk bitrd fz0sn c0ex fsn2 simprr - feq2i opeq2d sneqd eqtrd ex syl5bi sylbid com23 3imp opeq12d 3exp bitr4di - eleq1 df-br eqeq1 imbi2d imbi12d syl5ibr mpcom com12 impd eqidd a1i snidg - fsnd 0clwlkv mpd3an23 hash0 fvsng jca32 0ex op1std fveqeq2d op2ndd fveq1d - mpan snex eqeq1d anbi12d syl5ibrcom impbid alrimiv rabeqsn sylibr ) DCFZA - UOZBUCGZFZXMUAGZHGIJZIXMUBGZGZDJZKZKZXMLIDMZNZMZJZOZAUDYAAXNUEYENJXLYGAXL - YBYFXLXOYAYFXOXLYAYFPZXMXPXRMZJZXOXLYHPXOXMBUFGFYJBXMUGBXMUHUIYJXLXOYHXLX - OYHPYJXPXRXNQZYAYIYEJZPZPXLYKYAYLXLYKYAUJXPLXRYDYAXLXPLJZYKXQYNXTXQYNXPRF - XQYNOXMUAUKXPRULUMUPSZUNXLYKYAXRYDJZXLYAYKYPXLYAYKYPPXLYAKZYKIIUQURZCXRTZ - YPYQYKLXRXNQZYSYQXPLXRXNYAYNXLYOUSUTXLYTYSOZYAXLBRFUUABDCEVAXRBCREVBUISVC - YSINZCXRTZYQYPYRUUBCXRVDVHUUCXSCFZXRIXSMZNZJZKZYQYPICXRVEVFYQUUHYPYQUUHKZ - XRUUFYDYQUUDUUGVGUUIUUEYCUUIXSDIYQXTUUHXLXQXTVGSVIVJVKVLVMVMVNVLVOVPVQVRY - JXOYKYHYMYJXOYIXNFYKXMYIXNVTXPXRXNWAVSYJYFYLYAXMYIYEWBWCWDWEVOWFWGWHXLYBY - FLYDXNQZLHGIJZIYDGZDJZKZKXLUUJUUKUUMXLLLJUUBDNZYDTUUJXLLWIXLIDRUUOIRFZXLV - EWJDCWKWLYDLBCDEWMWNUUKXLWOWJUUPXLUUMVEIDRCWPXCWQYFXOUUJYAUUNYFXOYEXNFUUJ - XMYEXNVTLYDXNWAVSYFXQUUKXTUUMYFXPLIHLYDXMWRYCXDZWSWTYFXSUULDYFIXRYDLYDXMW - RUUQXAXBXEXFXFXGXHXIYAAXNYEXJXK $. + feq2i opeq2d sneqd eqtrd ex biimtrid sylbid com23 3imp opeq12d 3exp eleq1 + df-br bitr4di eqeq1 imbi12d syl5ibr mpcom com12 impd eqidd a1i snidg fsnd + imbi2d 0clwlkv mpd3an23 hash0 fvsng mpan jca32 0ex op1std fveqeq2d op2ndd + snex fveq1d eqeq1d anbi12d syl5ibrcom impbid alrimiv rabeqsn sylibr ) DCF + ZAUOZBUCGZFZXMUAGZHGIJZIXMUBGZGZDJZKZKZXMLIDMZNZMZJZOZAUDYAAXNUEYENJXLYGA + XLYBYFXLXOYAYFXOXLYAYFPZXMXPXRMZJZXOXLYHPXOXMBUFGFYJBXMUGBXMUHUIYJXLXOYHX + LXOYHPYJXPXRXNQZYAYIYEJZPZPXLYKYAYLXLYKYAUJXPLXRYDYAXLXPLJZYKXQYNXTXQYNXP + RFXQYNOXMUAUKXPRULUMUPSZUNXLYKYAXRYDJZXLYAYKYPXLYAYKYPPXLYAKZYKIIUQURZCXR + TZYPYQYKLXRXNQZYSYQXPLXRXNYAYNXLYOUSUTXLYTYSOZYAXLBRFUUABDCEVAXRBCREVBUIS + VCYSINZCXRTZYQYPYRUUBCXRVDVHUUCXSCFZXRIXSMZNZJZKZYQYPICXRVEVFYQUUHYPYQUUH + KZXRUUFYDYQUUDUUGVGUUIUUEYCUUIXSDIYQXTUUHXLXQXTVGSVIVJVKVLVMVMVNVLVOVPVQV + RYJXOYKYHYMYJXOYIXNFYKXMYIXNVSXPXRXNVTWAYJYFYLYAXMYIYEWBWLWCWDVOWEWFWGXLY + BYFLYDXNQZLHGIJZIYDGZDJZKZKXLUUJUUKUUMXLLLJUUBDNZYDTUUJXLLWHXLIDRUUOIRFZX + LVEWIDCWJWKYDLBCDEWMWNUUKXLWOWIUUPXLUUMVEIDRCWPWQWRYFXOUUJYAUUNYFXOYEXNFU + UJXMYEXNVSLYDXNVTWAYFXQUUKXTUUMYFXPLIHLYDXMWSYCXCZWTXAYFXSUULDYFIXRYDLYDX + MWSUUQXBXDXEXFXFXGXHXIYAAXNYEXJXK $. $d G w $. $d K w $. $( There are k walks of length 2 on each vertex ` X ` in a k-regular simple @@ -435957,7 +440797,7 @@ to the label of the original theorem (for instance, ~ con3i is the associated with ~ imim1 .

    "Deduction form" is the preferred form for theorems because this form - allows us to easily use the theorem in places where (in traditional + allows to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem (see below) would be used. We call this approach "deduction style". In contrast, we usually avoid theorems in "inference form" when that @@ -436010,7 +440850,7 @@ corresponding deduction (its associated inference). See for instance

  • Recursion. We define recursive functions using various "recursion constructors". - These allow us to define, with compact direct definitions, functions + These allow to define, with compact direct definitions, functions that are usually defined in textbooks with indirect self-referencing recursive definitions. This produces compact definition and much simpler proofs, and greatly reduces the risk of creating unsound @@ -439040,9 +443880,9 @@ This law is thought to have originated with Aristotle (_Metaphysics_, $( Define the class of planar incidence geometries. We use Hilbert's axioms and adapt them to planar geometry. We use ` e. ` for the incidence relation. We could have used a generic binary relation, but - using ` e. ` allows us to reuse previous results. Much of what follows - is directly borrowed from Aitken, _Incidence-Betweenness Geometry_, - 2008, ~ http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf . + using ` e. ` allows to reuse previous results. Much of what follows is + directly borrowed from Aitken, _Incidence-Betweenness Geometry_, 2008, + ~ http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf . The class ` Plig ` is the class of planar incidence geometries, where a planar incidence geometry is defined as a set of lines satisfying three @@ -443601,12 +448441,12 @@ a normed complex vector space (normally a Hilbert space). @) -> E. x e. X x =/= Z ) $= ( cnv wcel wne wn wceq wral bitr3i cfv w3a cv wrex ralnex nne ralbii cn0v csn cxp wfn wa fveq2 cba eqid lno0 sylan9eqr ralimdv lnof jctild fconstfv - ex ffnd fvex fconst2 syl6ib 0ofval 3adant3 eqeq2d sylibrd syl5bi necon1ad - wf imp ) CMNZFMNZBDNZUAZBEOAUBZHOZAGUCZVQVTBEVTPZVRHQZAGRZVQBEQZWAVSPZAGR - WCVSAGUDWEWBAGVRHUEUFSVQWCBGFUGTZUHZUIZQZWDVQWCBGUJZVRBTZWFQZAGRZUKZWIVQW - CWMWJVQWBWLAGVQWBWLWBVQWKHBTWFVRHBULHBCDFGFUMTZWFIWOUNZJWFUNZLUOUPVAUQVQG - WOBBCDFGWOIWPLURVBUSWNGWGBVLWIAGWFBUTGWFBFUGVCVDSVEVQEWHBVNVOEWHQVPCEFGWF - IWQKVFVGVHVIVJVKVM $. + ex ffnd wf fconst2 syl6ib 0ofval 3adant3 eqeq2d sylibrd biimtrid necon1ad + fvex imp ) CMNZFMNZBDNZUAZBEOAUBZHOZAGUCZVQVTBEVTPZVRHQZAGRZVQBEQZWAVSPZA + GRWCVSAGUDWEWBAGVRHUEUFSVQWCBGFUGTZUHZUIZQZWDVQWCBGUJZVRBTZWFQZAGRZUKZWIV + QWCWMWJVQWBWLAGVQWBWLWBVQWKHBTWFVRHBULHBCDFGFUMTZWFIWOUNZJWFUNZLUOUPVAUQV + QGWOBBCDFGWOIWPLURVBUSWNGWGBVCWIAGWFBUTGWFBFUGVLVDSVEVQEWHBVNVOEWHQVPCEFG + WFIWQKVFVGVHVIVJVKVM $. $} ${ @@ -449892,6 +454732,9 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the htmldef "[:]" as "[:]"; althtmldef "[:]" as "[:]"; latexdef "[:]" as "[:]"; +htmldef "fldGen" as "fldGen"; + althtmldef "fldGen" as "fldGen"; + latexdef "fldGen" as "\mathrm{fldGen}"; htmldef "repr" as "repr"; althtmldef "repr" as "repr"; @@ -450803,12 +455646,12 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the htmldef "ErALTV" as ' ErALTV '; althtmldef "ErALTV" as ' ErALTV '; latexdef "ErALTV" as "\mathrm{ErALTV}"; -htmldef "MembErs" as ' MembErs '; - althtmldef "MembErs" as ' MembErs '; - latexdef "MembErs" as "\mathrm{MembErs}"; -htmldef "MembEr" as ' MembEr '; - althtmldef "MembEr" as ' MembEr '; - latexdef "MembEr" as "\mathrm{MembEr}"; +htmldef "CoMembErs" as ' CoMembErs '; + althtmldef "CoMembErs" as ' CoMembErs '; + latexdef "CoMembErs" as "\mathrm{CoMembErs}"; +htmldef "CoMembEr" as ' CoMembEr '; + althtmldef "CoMembEr" as ' CoMembEr '; + latexdef "CoMembEr" as "\mathrm{CoMembEr}"; htmldef "Funss" as ' Funss '; althtmldef "Funss" as ' Funss '; latexdef "Funss" as "\mathrm{Funss}"; @@ -450833,6 +455676,21 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the htmldef "ElDisj" as ' ElDisj '; althtmldef "ElDisj" as ' ElDisj '; latexdef "ElDisj" as "\mathrm{ElDisj}"; +htmldef "AntisymRel" as ' AntisymRel '; + althtmldef "AntisymRel" as ' AntisymRel '; + latexdef "AntisymRel" as "\mathrm{AntisymRel}"; +htmldef "Parts" as ' Parts '; + althtmldef "Parts" as ' Parts '; + latexdef "Parts" as "{\rm Parts}"; +htmldef "Part" as ' Part '; + althtmldef "Part" as ' Part '; + latexdef "Part" as "{\rm Part}"; +htmldef "MembParts" as ' MembParts '; + althtmldef "MembParts" as ' MembParts '; + latexdef "MembParts" as "{\rm MembParts}"; +htmldef "MembPart" as ' MembPart '; + althtmldef "MembPart" as ' MembPart '; + latexdef "MembPart" as "{\rm MembPart}"; /* End of Peter Mazsa's mathbox */ /* Mathbox of Rodolfo Medina */ @@ -453372,9 +458230,9 @@ have been translated into extensible structure versions (or are not useful). $( Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of ` A ` and ` B ` is usually written - ` <. A , B >. ` but our operation notation allows us to use existing - theorems about operations and also eliminates ambiguity with the - definition of an ordered pair ~ df-op . $) + ` <. A , B >. ` but our operation notation allows to use existing theorems + about operations and also eliminates ambiguity with the definition of an + ordered pair ~ df-op . $) csp $a class .ih $. $( Register inner product in Hilbert space as a primitive expression (lacking @@ -454516,12 +459374,11 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). $( Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ` * `` x ` is the complex conjugate ~ cjval of ` x ` . In the literature, the inner product of ` A ` and ` B ` is usually written - ` <. A , B >. ` , but our operation notation ~ co allows us to use - existing theorems about operations and also avoids a clash with the - definition of an ordered pair ~ df-op . Physicists use ` <. B | A >. ` , - called Dirac bra-ket notation, to represent this operation; see comments - in ~ df-bra . (Contributed by NM, 29-Jul-1999.) - (New usage is discouraged.) $) + ` <. A , B >. ` , but our operation notation ~ co allows to use existing + theorems about operations and also avoids a clash with the definition of + an ordered pair ~ df-op . Physicists use ` <. B | A >. ` , called Dirac + bra-ket notation, to represent this operation; see comments in ~ df-bra . + (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) $) ax-his1 $a |- ( ( A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) ) $. @@ -456401,8 +461258,8 @@ to a member of the subspace (Definition of complete subspace in [Beran] ${ $d x y f $. - $( The unit Hilbert lattice element (which is all of Hilbert space) belongs - to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. + $( The Hilbert lattice one (which is all of Hilbert space) belongs to the + Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.) $) helch $p |- ~H e. CH $= ( vf vx vy chba cch wcel csh cn cv wf chli wbr wa co wral cc pm3.2i rgen2 @@ -457866,15 +462723,15 @@ to a member of the subspace (Definition of complete subspace in [Beran] shseli wb w3a hvsubadd syl3an eqcom bitrdi 3expb shsvsi eleqtrdi shlesb1i shscomi biimpi eleq2d syl5ib eleq1 biimpd sylan9 anim2d syl6ibr ex ancoms com13 anasss sylbird imp shincli shsvai syl5ibr expd com12 ad2antrl exp31 - syld rexlimdvv syl5bi impd ssrdv ) ACUAZGABJKZCLZABCLZJKZGMZWENWHWDNZWHCN - ZOWCWHWGNZWHWDCPWCWIWJWKWJWIWCWKWIWHHMZIMZUBKZQZIBUCHAUCWJWCWKRZHIABWHDEU - EWJWOWPHIABWJWLANZWMBNZOZWOWPWJWSOZWOOWCWMWFNZWKWTWOWCXARZWTWOWHWLUDKZWMQ - ZXBWJWQWRXDWOUFWJWQWRUGXDWNWHQZWOWJWHSNWQWLSNWRWMSNXDXEUFWHCFTWLADTWMBETW - HWLWMUHUIWNWHUJUKULWJWQWRXDXBRZWRWJWQOZXFWCXDWRXGOZXAWCXDXHXARWCXDOZXHWRW - MCNZOXAXIXGXJWRWCXGXCCNZXDXJXGXCACJKZNWCXKXGXCCAJKXLCAWHWLFDUMCAFDUPUNWCX - LCXCWCXLCQACDFUOUQURUSXDXKXJXCWMCUTVAVBVCWMBCPVDVEVGVFVHVIVJWTWOXAWKRZWQW - OXMRWJWRWOWQXMWOWQXAWKWQXAOWKWOWNWGNAWFWLWMDBCEFVKVLWHWNWGUTVMVNVOVPVJVRV - QVSVTVGWAVTWB $. + syld rexlimdvv biimtrid impd ssrdv ) ACUAZGABJKZCLZABCLZJKZGMZWENWHWDNZWH + CNZOWCWHWGNZWHWDCPWCWIWJWKWJWIWCWKWIWHHMZIMZUBKZQZIBUCHAUCWJWCWKRZHIABWHD + EUEWJWOWPHIABWJWLANZWMBNZOZWOWPWJWSOZWOOWCWMWFNZWKWTWOWCXARZWTWOWHWLUDKZW + MQZXBWJWQWRXDWOUFWJWQWRUGXDWNWHQZWOWJWHSNWQWLSNWRWMSNXDXEUFWHCFTWLADTWMBE + TWHWLWMUHUIWNWHUJUKULWJWQWRXDXBRZWRWJWQOZXFWCXDWRXGOZXAWCXDXHXARWCXDOZXHW + RWMCNZOXAXIXGXJWRWCXGXCCNZXDXJXGXCACJKZNWCXKXGXCCAJKXLCAWHWLFDUMCAFDUPUNW + CXLCXCWCXLCQACDFUOUQURUSXDXKXJXCWMCUTVAVBVCWMBCPVDVEVGVFVHVIVJWTWOXAWKRZW + QWOXMRWJWRWOWQXMWOWQXAWKWQXAOWKWOWNWGNAWFWLWMDBCEFVKVLWHWNWGUTVMVNVOVPVJV + RVQVSVTVGWAVTWB $. $( The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, @@ -458274,12 +463131,12 @@ to a member of the subspace (Definition of complete subspace in [Beran] shlub $p |- ( ( A e. SH /\ B e. SH /\ C e. CH ) -> ( ( A C_ C /\ B C_ C ) <-> ( A vH B ) C_ C ) ) $= ( csh wcel cch w3a wss wa chj cort cfv chba shss 3ad2ant3 syl2anc wceq sstr - syl sylan co cun unss wi simp1 simp2 unssd chss occon2 syl5bi shjval eqcomd - ococ sseq12d sylibrd shub1 shub2 jca ex impbid ) ADEZBDEZCFEZGZACHZBCHZIZAB - JUAZCHZVDVGABUBZKLKLZCKLKLZHZVIVGVJCHZVDVMABCUCVDVJMHCMHZVNVMUDVDABMVDVAAMH - VAVBVCUEZANSVDVBBMHVAVBVCUFZBNSUGVCVAVOVBCUHOVJCUIPUJVDVHVKCVLVDVAVBVHVKQVP - VQABUKPVDVLCVCVAVLCQVBCUMOULUNUOVDVIVGVDVIIVEVFVDAVHHZVIVEVDVAVBVRVPVQABUPP - AVHCRTVDBVHHZVIVFVDVBVAVSVQVPBAUQPBVHCRTURUSUT $. + syl sylan co cun unss wi simp1 simp2 unssd chss occon2 biimtrid shjval ococ + eqcomd sseq12d sylibrd shub1 shub2 jca ex impbid ) ADEZBDEZCFEZGZACHZBCHZIZ + ABJUAZCHZVDVGABUBZKLKLZCKLKLZHZVIVGVJCHZVDVMABCUCVDVJMHCMHZVNVMUDVDABMVDVAA + MHVAVBVCUEZANSVDVBBMHVAVBVCUFZBNSUGVCVAVOVBCUHOVJCUIPUJVDVHVKCVLVDVAVBVHVKQ + VPVQABUKPVDVLCVCVAVLCQVBCULOUMUNUOVDVIVGVDVIIVEVFVDAVHHZVIVEVDVAVBVRVPVQABU + PPAVHCRTVDBVHHZVIVFVDVBVAVSVQVPBAUQPBVHCRTURUSUT $. ${ shlub.1 $e |- A e. SH $. @@ -458886,7 +463743,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] chjoi $p |- ( A vH ( _|_ ` A ) ) = ~H $= ( chba wss cort cfv chj co wceq chssii ssjo ax-mp ) ACDAAEFGHCIABJAKL $. - $( Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) + $( Join with Hilbert lattice one. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) $) chj1i $p |- ( A vH ~H ) = ~H $= ( chba chj co helch chjcli chssii chub2i eqssi ) ACDEZCKACBFGHCAFBIJ $. @@ -459765,34 +464622,34 @@ equals the join of their closures (double orthocomplements). wa shmulcl mp3an13 shaddcl syl3an3 mp3an1 choccli pjhclii spansnmul pjpji mpan adantl oveq2i ax-hvdistr1 mp3an23 eqtrid oveq2d cheli mpan2 ax-hvass hvmulcl jca 3expb syl2an eqtr4d rspceov syl3anc sylibr oveq2 eqeq2d eleq1 - biimpa biimparc exp43 rexlimdv syl5bi rexlimiv sylbi negcl syl3an2 ancoms - cneg syl c0v c1 hvm1neg hvnegid sylancl ax-hvcom syl2anc 3eqtr3d hvaddid2 - oveq1d hvaddcl anim12i hvadd4 impbii chssii spanuni spanid oveq1i 3eqtr4i - eqriv ax-mp eqtri ) ABUBZUCJZKLZABAUDJZUEJJZUBZUCJZKLZAYBUFUCJZAYGUFUCJZU - AYDYIUAMZYDNZYLYINZYMYLEMZFMZOLZPZFYCQZEAQYNEFAYCYLACUGZBRNZYBRUHZYCUINDB - RUJZYBUKULZUMYSYNEAYOANZYRYNFYCYPYCNYPGMZBSLZPZGTQUUEYRYNUNZGBYPDUOUUEUUH - UUIGTUUEUUFTNZUUHYRYNUUEUUJUQZYOUUGOLZYINZYLUULPZYNUUHYRUQUUKUULHMIMOLZPI - YHQHAQZUUMUUKYOUUFBAUEJJZSLZOLZANZUUFYFSLZYHNZUULUUSUVAOLZPUUPAUINZUUEUUJ - 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syl syl2anc sylibd exp31 com23 imp syld com3r expd rexlimdv syl5bi sylcom - snssi chssii occon2i ococi sseqtrdi syl6 anc2li eqss syl6ibr necon1d neor - sylibr ) ABUDZGHGHZIZAXBJAKLZMAXBLZXDUAXCAKAXBXCAKJZXCXBAIZNXEXCXFXGXCXFB - AOZXGXFEUBZPJZEAUCXCXHEACUEXCXJXHEAXCXIAOZXIXBOZXJXHMZAXBXIUFXLXIFUBZBQRZ - LZFSUCXKXMFXIBDUGXKXPXMFSXKXNSOZXPXMXQXPNZXJXKXHXRXJXNTJZXKXHMZXPXJXSMXQX - PXNTXIPXNTLZXIPLXPXOPLYAXOTBQRZPXNTBQUHBUIOZYBPLDBUJUKULXIXOPUOUMUNUPXQXP - XSXTMXQXSXPXTXQXSXPXTXQXSNZXPNZXKUQXNURRZXIQRZAOZXHYDXKYHMZXPYDYFSOZYIXNU - SZYJXKYHAUTOYJXKYHACVAYFXIAVBVCVDVQVEYEYGBAXPYDYGYFXOQRZBXIXOYFQVFYDYLUQB - QRZBYDYFXNVGRZBQRZYLYMYDYJXQYOYLLZYKXQXSVHYJXQYCYPDYFXNBVIVJVRYDYNUQBQXNV - KVLVMYCYMBLDBVNUKULVOVPVSVTWAWBWCWDWEWFWGWHWFWGXHXBAGHGHZAXHXAAIXBYQIBAWI - XAAYCXAUIIDBUIWIUKACWJWKVQACWLWMWNWOAXBWPWQWRXDAXBWSWT $. + syl2anc sylibd exp31 com23 syld com3r expd rexlimdv biimtrid sylcom snssi + syl chssii occon2i ococi sseqtrdi syl6 anc2li eqss syl6ibr necon1d sylibr + imp neor ) ABUDZGHGHZIZAXBJAKLZMAXBLZXDUAXCAKAXBXCAKJZXCXBAIZNXEXCXFXGXCX + FBAOZXGXFEUBZPJZEAUCXCXHEACUEXCXJXHEAXCXIAOZXIXBOZXJXHMZAXBXIUFXLXIFUBZBQ + RZLZFSUCXKXMFXIBDUGXKXPXMFSXKXNSOZXPXMXQXPNZXJXKXHXRXJXNTJZXKXHMZXPXJXSMX + QXPXNTXIPXNTLZXIPLXPXOPLYAXOTBQRZPXNTBQUHBUIOZYBPLDBUJUKULXIXOPUOUMUNUPXQ + XPXSXTMXQXSXPXTXQXSXPXTXQXSNZXPNZXKUQXNURRZXIQRZAOZXHYDXKYHMZXPYDYFSOZYIX + NUSZYJXKYHAUTOYJXKYHACVAYFXIAVBVCVDWHVEYEYGBAXPYDYGYFXOQRZBXIXOYFQVFYDYLU + QBQRZBYDYFXNVGRZBQRZYLYMYDYJXQYOYLLZYKXQXSVHYJXQYCYPDYFXNBVIVJVQYDYNUQBQX + NVKVLVMYCYMBLDBVNUKULVOVPVRVSVTWSWAWBWCWDWEWFWDWEXHXBAGHGHZAXHXAAIXBYQIBA + WGXAAYCXAUIIDBUIWGUKACWIWJWHACWKWLWMWNAXBWOWPWQXDAXBWTWR $. $} $( A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) @@ -466513,7 +471370,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $( Value of the converse of the bra function. Based on the Riesz Lemma ~ riesz4 , this very important theorem not only justifies the Dirac - bra-ket notation, but allows us to extract a unique vector from any + bra-ket notation, but allows to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" _all_ of the information contained in any entire continuous linear functional (mapping from ` ~H ` to @@ -468116,7 +472973,7 @@ Positive operators (cont.) VEHUQURUSUT $. $( The norm of a Hilbert-space-valued state equals one iff the state value - equals the state value of the lattice unit. (Contributed by NM, + equals the state value of the lattice one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.) $) hst1h $p |- ( ( S e. CHStates /\ A e. CH ) -> ( ( normh ` ( S ` A ) ) = 1 <-> ( S ` A ) = ( S ` ~H ) ) ) $= @@ -468184,7 +473041,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = SVCUQUSGURRVALUQUSNUQURVALMUSABCOSPUAUMUOVBUNUPBCUBQUCUMUOVEVDUDZUNUPUMUO GUTUEEVATEZVFBCUFUTUGVGRTEVFUHVARULUIUJQUK $. - $( A Hilbert-space-valued state orthogonal to the state of the lattice unit + $( A Hilbert-space-valued state orthogonal to the state of the lattice one is zero. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.) $) hstoh $p |- ( ( S e. CHStates /\ A e. CH /\ ( ( S ` A ) .ih ( S ` ~H ) ) = @@ -468311,8 +473168,8 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = (New usage is discouraged.) $) stm1ri $p |- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( S ` B ) = 1 ) ) $= - ( cin cfv c1 wceq cst wcel incom fveq2i eqeq1i stm1i syl5bi ) ABFZCGZHIBA - FZCGZHICJKBCGHIRTHQSCABLMNBACEDOP $. + ( cin cfv c1 wceq cst wcel incom fveq2i eqeq1i stm1i biimtrid ) ABFZCGZHI + BAFZCGZHICJKBCGHIRTHQSCABLMNBACEDOP $. $( Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.) $) @@ -468353,19 +473210,19 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = stadd3i $p |- ( S e. States -> ( ( ( ( S ` A ) + ( S ` B ) ) + ( S ` C ) ) = 3 -> ( S ` A ) = 1 ) ) $= ( wcel cfv caddc co c3 wceq c1 cr mpi clt wbr cle 1re cst recnd addassd - cch stcl eqeq1d wn eqcom wne wi readdcld ltne ex syl necon2bd syl5bi wa - readdcli a1i 1red le2addd leadd2dd adantr w3a ltadd1 biimpd syl3anc imp - stle1 readdcl sylancl lelttr mp2and c2 df-3 df-2 oveq1i addassi 3eqtrri - ax-1cn breqtrdi con3d wo wb leloe mpbid ord 3syld sylbid ) DUAHZADIZBDI - ZJKCDIZJKZLMWKWLWMJKZJKZLMZWKNMZWJWNWPLWJWKWLWMWJWKWJAUDHZWKOHZEADUEPZU - BWJWLWJBUDHZWLOHFBDUEPZUBWJWMWJCUDHZWMOHGCDUEPZUBUCUFWJWQWPLQRZUGZWKNQR - ZUGWRWQLWPMWJXGWPLUHWJXFLWPWJWPOHZXFLWPUIZUJWJWKWOXAWJWLWMXCXEUKZUKZXIX - FXJWPLULUMUNUOUPWJXHXFWJXHXFWJXHUQZWPNNNJKZJKZLQXMWPWKXNJKZSRZXPXOQRZWP - XOQRZWJXQXHWJWOXNWKXKXNOHZWJNNTTURZUSZXAWJWLWMNNXCXEWJUTZYCWJXBWLNSRFBD - VIPWJXDWMNSRGCDVIPVAVBVCWJXHXRWJWTNOHZXTXHXRUJXAYCYBWTYDXTVDXHXRWKNXNVE - VFVGVHWJXQXRUQXSUJZXHWJXIXPOHZXOOHZYEXLWJWTXTYFXAYAWKXNVJVKYGWJNXNTYAUR - USWPXPXOVLVGVCVMLVNNJKXNNJKXOVOVNXNNJVPVQNNNVTVTVTVRVSWAUMWBWJXHWRWJWKN - SRZXHWRWCZWJWSYHEADVIPWJWTYDYHYIWDXATWKNWEVKWFWGWHWI $. + cch stcl eqeq1d wn eqcom wi readdcld ltne ex necon2bd biimtrid readdcli + wne syl wa a1i 1red stle1 le2addd leadd2dd adantr ltadd1 biimpd syl3anc + w3a readdcl sylancl lelttr mp2and c2 df-3 oveq1i ax-1cn addassi 3eqtrri + imp df-2 breqtrdi con3d wo wb leloe mpbid ord 3syld sylbid ) DUAHZADIZB + DIZJKCDIZJKZLMWKWLWMJKZJKZLMZWKNMZWJWNWPLWJWKWLWMWJWKWJAUDHZWKOHZEADUEP + ZUBWJWLWJBUDHZWLOHFBDUEPZUBWJWMWJCUDHZWMOHGCDUEPZUBUCUFWJWQWPLQRZUGZWKN + QRZUGWRWQLWPMWJXGWPLUHWJXFLWPWJWPOHZXFLWPUPZUIWJWKWOXAWJWLWMXCXEUJZUJZX + IXFXJWPLUKULUQUMUNWJXHXFWJXHXFWJXHURZWPNNNJKZJKZLQXMWPWKXNJKZSRZXPXOQRZ + WPXOQRZWJXQXHWJWOXNWKXKXNOHZWJNNTTUOZUSZXAWJWLWMNNXCXEWJUTZYCWJXBWLNSRF + BDVAPWJXDWMNSRGCDVAPVBVCVDWJXHXRWJWTNOHZXTXHXRUIXAYCYBWTYDXTVHXHXRWKNXN + VEVFVGVSWJXQXRURXSUIZXHWJXIXPOHZXOOHZYEXLWJWTXTYFXAYAWKXNVIVJYGWJNXNTYA + UOUSWPXPXOVKVGVDVLLVMNJKXNNJKXOVNVMXNNJVTVONNNVPVPVPVQVRWAULWBWJXHWRWJW + KNSRZXHWRWCZWJWSYHEADVAPWJWTYDYHYIWDXATWKNWEVJWFWGWHWI $. $} $} @@ -469365,18 +474222,18 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = ( vy cin cv wss chj co wa wceq wi cch wral wcel sseq1 wb mp3an23 cmd sstr wbr sseq2 anbi12d oveq1 ineq1d eqeq12d imbi12d rspccv impexp bitr2i inss2 chlub biimpd mpan2i chub2i mpan2 syl6 chub2 mpan jctild chjcl jcad chjass - chincli chjcomi chabs1i oveq2i eqtrdi chjidmi imim12d syl5bi syl5com mdbr - eqtri ralrimiv mp2an sylibr ax-1 ralimi sylbi impbii ) BCGZAHZIZWEBCJKZIZ - LZWECIZWEBJKZCGZWEWDJKZMZNZNZAOPZBCUAUCZWQFHZCIZWSBJKZCGZWSWDJKZMZNZFOPZW - RWQXEFOWQXCOQZWDXCIZXCWGIZLZXCCIZXCBJKZCGZXCWDJKZMZNZNZNZWSOQZXEWPXQAXCOW - EXCMZWIXJWOXPXTWFXHWHXIWEXCWDUDWEXCWGRUEXTWJXKWNXOWEXCCRXTWLXMWMXNXTWKXLC - WEXCBJUFUGWEXCWDJUFUHUIUIUJXRXGXJLZXKLZXONZXSXEYCYAXPNXRYAXKXOUKXGXJXPUKU - LXSWTYBXOXDXSWTYAXKXSWTXJXGXSWTXIXHXSWTXKXIXSWTWDCIZXKBCUMXSWTYDLZXKXSWDO - QZCOQZYEXKSBCDEVFZEWSWDCUNTUOUPZXKCWGIXICBEDUQXCCWGUBURUSYFXSXHYHWDWSUTVA - VBXSYFXGYHWSWDVCURVBYIVDXSXOXDXSXMXBXNXCXSXLXACXSXLWSWDBJKZJKZXAXSYFBOQZX - LYKMYHDWSWDBVETYJBWSJYJBWDJKBWDBYHDVGBCDEVHVPVIVJUGXSXNWSWDWDJKZJKZXCXSYF - YFXNYNMYHYHWSWDWDVETYMWDWSJWDYHVKVIVJUHUOVLVMVNVQYLYGWRXFSDEFBCVOVRVSWRWO - AOPZWQYLYGWRYOSDEABCVOVRWOWPAOWOWIVTWAWBWC $. + chincli chjcomi chabs1i eqtri oveq2i eqtrdi chjidmi imim12d biimtrid mdbr + syl5com ralrimiv mp2an sylibr ax-1 ralimi sylbi impbii ) BCGZAHZIZWEBCJKZ + IZLZWECIZWEBJKZCGZWEWDJKZMZNZNZAOPZBCUAUCZWQFHZCIZWSBJKZCGZWSWDJKZMZNZFOP + ZWRWQXEFOWQXCOQZWDXCIZXCWGIZLZXCCIZXCBJKZCGZXCWDJKZMZNZNZNZWSOQZXEWPXQAXC + OWEXCMZWIXJWOXPXTWFXHWHXIWEXCWDUDWEXCWGRUEXTWJXKWNXOWEXCCRXTWLXMWMXNXTWKX + LCWEXCBJUFUGWEXCWDJUFUHUIUIUJXRXGXJLZXKLZXONZXSXEYCYAXPNXRYAXKXOUKXGXJXPU + KULXSWTYBXOXDXSWTYAXKXSWTXJXGXSWTXIXHXSWTXKXIXSWTWDCIZXKBCUMXSWTYDLZXKXSW + DOQZCOQZYEXKSBCDEVFZEWSWDCUNTUOUPZXKCWGIXICBEDUQXCCWGUBURUSYFXSXHYHWDWSUT + VAVBXSYFXGYHWSWDVCURVBYIVDXSXOXDXSXMXBXNXCXSXLXACXSXLWSWDBJKZJKZXAXSYFBOQ + ZXLYKMYHDWSWDBVETYJBWSJYJBWDJKBWDBYHDVGBCDEVHVIVJVKUGXSXNWSWDWDJKZJKZXCXS + YFYFXNYNMYHYHWSWDWDVETYMWDWSJWDYHVLVJVKUHUOVMVNVPVQYLYGWRXFSDEFBCVOVRVSWR + WOAOPZWQYLYGWRYOSDEABCVOVRWOWPAOWOWIVTWAWBWC $. $( If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, @@ -469414,13 +474271,13 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = ( vx cin wbr wss chj co wa wceq wi cch chabs1i eqtri oveq1 ineq1d 3eqtr4a wcel ccv cv wral cmd anass chub2i mpan2 pm4.71ri anbi2i bitr4i wo chincli sstr cvnbtwn4 mp3an12 impcom chjcomi ineq1i chjidmi eqtr4i chabs2i oveq2i - incom 3eqtr2i jaoi syl6 syl5bi exp4b ralrimiv mdsl1i sylib ) ABFZBUAGZVLE - UBZHZVNABIJZHZKZVNBHZVNAIJZBFZVNVLIJZLZMMZENUCABUDGVMWDENVMVNNTZVRVSWCVRV - SKZVOVSKZVMWEKZWCWFVOVQVSKZKWGVOVQVSUEVSWIVOVSVQVSBVPHVQBADCUFVNBVPUMUGUH - UIUJWHWGVNVLLZVNBLZUKZWCWEVMWGWLMZVLNTBNTWEVMWMMABCDULZDVLBVNUNUOUPWJWCWK - WJVLAIJZBFZVLVLIJZWAWBWPVLWQWOABWOAVLIJAVLAWNCUQABCDOPURVLWNUSUTWJVTWOBVN - VLAIQRVNVLVLIQSWKBAIJZBFZBVLIJZWAWBWSBWRFZWTWRBVCXABBBAFZIJWTBADCVABADCOX - BVLBIBAVCVBVDPWKVTWRBVNBAIQRVNBVLIQSVEVFVGVHVIEABCDVJVK $. + incom 3eqtr2i jaoi syl6 biimtrid exp4b ralrimiv mdsl1i sylib ) ABFZBUAGZV + LEUBZHZVNABIJZHZKZVNBHZVNAIJZBFZVNVLIJZLZMMZENUCABUDGVMWDENVMVNNTZVRVSWCV + RVSKZVOVSKZVMWEKZWCWFVOVQVSKZKWGVOVQVSUEVSWIVOVSVQVSBVPHVQBADCUFVNBVPUMUG + UHUIUJWHWGVNVLLZVNBLZUKZWCWEVMWGWLMZVLNTBNTWEVMWMMABCDULZDVLBVNUNUOUPWJWC + WKWJVLAIJZBFZVLVLIJZWAWBWPVLWQWOABWOAVLIJAVLAWNCUQABCDOPURVLWNUSUTWJVTWOB + VNVLAIQRVNVLVLIQSWKBAIJZBFZBVLIJZWAWBWSBWRFZWTWRBVCXABBBAFZIJWTBADCVABADC + OXBVLBIBAVCVBVDPWKVTWRBVNBAIQRVNBVLIQSVEVFVGVHVIEABCDVJVK $. $} ${ @@ -469875,41 +474732,41 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = sylancom eqtr3d rspceeqv rexlimdva2 syld sylibrd spansneleq eqcom sylan9r syl6ib adantlrl adantrr adantll ax-hvcom eqeq1d bitr4d adantrl cpr spanpr adantlrr oveq12 cph cun df-pr fveq2i snssi spanun syl2an spansnch spansnj - 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(New usage is discouraged.) $) chjatom $p |- ( ( A e. CH /\ B e. HAtoms ) -> ( A +H B ) = ( A vH B ) ) $= ( vx cch wcel cat cph co chj wceq cv c0v wne csn cspn cfv wa chba oveq2 - wi wrex atom1d spansnj eqeq12d syl5ibr expd adantl com3l rexlimdv syl5bi - imp ) ADEZBFEZABGHZABIHZJZUMCKZLMZBUQNOPZJZQZCRUAULUPCBUBULVAUPCRVAULUQRE - ZUPUTULVBUPTTURUTULVBUPULVBQUPUTAUSGHZAUSIHZJAUQUCUTUNVCUOVDBUSAGSBUSAISU - DUEUFUGUHUIUJUK $. + wi wrex atom1d spansnj eqeq12d syl5ibr adantl com3l rexlimdv biimtrid imp + expd ) ADEZBFEZABGHZABIHZJZUMCKZLMZBUQNOPZJZQZCRUAULUPCBUBULVAUPCRVAULUQR + EZUPUTULVBUPTTURUTULVBUPULVBQUPUTAUSGHZAUSIHZJAUQUCUTUNVCUOVDBUSAGSBUSAIS + UDUEUKUFUGUHUIUJ $. $} ${ @@ -470493,19 +475350,19 @@ r C_ ( p vH q ) ) ) -> ( r = p \/ r = q ) ) $= df-ne simpll simprl cch wb atelch chsscon3 sylancl biimpa sylan2 ancoms atne0 adantr sseq1 bicomd chssoc syl sylan9bbr an32s ex necon3d adantlr sstr mpd syldan adantrl superpos syl3anc df-3an neanior anbi1i bitri wi - chirredlem4 anassrs pm2.24d com23 impd syl5bi rexlimdva rexlimivv sylbi - an4s mt4 fveq2 ococi choc0 3eqtr3g orim2i ax-mp ) BHIZBJKZHIZLZXDBUBIZL - 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syl5bi adantl eqsstrrd chincl mpan2 chslej ad2antrl sstrd ralrimiv dmdbr2 - exp32 mp2an sylibr ) ABUAIZABUDIZJZBEKZLZXRXPMZXRAMZBUDIZLZNZEOUBZABUCUEZ - XQYDEOXQXROPZXSYCXQYGXSQZQZXTYABUAIZYBYIXTXRXOMZYJXQYKXTJYHXOXPXRUFUGYHYK - YJLXQYHFYKYJFKZYKPFEUHZYLXOPZQYHYLYJPZYLXRXOUIYHYMYNYOYNYLGKZHKZUJIZJZHBR - ZGARZYHYMQZYOGHABYLCDUKUUBUUAYTGYARZYOUUBUUAGEUHZYTQZGARUUCUUBYTUUEGAUUBY - PAPZQZYTUUDYSQZHBRUUEUUGYSUUHHBUUGYQBPZYSUUHUUGUUIQZYSQZUUDYSUUKYLYQULIZX - RPZUUDUUJUUMYSUUBUUIUUMUUFYGXSYMUUIUUMYGYMXSUUIUUMNYGYMXSUUIUUMXSUUIQHEUH - ZYGYMUUMBXRYQUMYGXRSPZYMUUNUUMNNXRUNZUUOYMUUNUUMYLYQXRUOUPURUQUSUTVAVBUGU - UJYSUUMUUDTZUUJYSUULYPJZUUQUUBUUFUUIUURYSTZUUBYLVCPZUUFYPVCPZUUIYQVCPZUUS - YGYMUUTXSYLXRVDVBYPACVEYQBDVEUUTUVBUVAUUSUUTUVBUVAVFUURYQYPUJIZYLJZYSYLYQ - YPVGUVBUVAUVDYSTUUTUVBUVAQZUVDYRYLJYSUVEUVCYRYLYQYPVHVJYRYLVIVKVLVMVNVOVP - UULYPXRVQVRWAVSUUJYSVTWBWCWDUUDYSHBWEWFWGYTUUEGYAAYPYAPZYTQUUFUUDQZYTQUUF - UUEQUVFUVGYTUVFUUDUUFQUVGYPXRAUIUUDUUFWHWIWLUUFUUDYTWJWIWKWMUUBYASPZBSPYO - UUCTYGUVHXSYMYGUUOASPUVHUUPACWNXRAWOWPWQBDWNGHYABYLWRWPWSXBWTXBXAXCXDYGYJ - YBLZXQXSYGYAOPZBOPZUVIYGAOPZUVJCXRAXEXFDYABXGWPXHXIXLXJUVLUVKYFYETCDEABXK - XMXN $. + biimtrid adantl eqsstrrd chincl mpan2 ad2antrl sstrd exp32 ralrimiv mp2an + chslej dmdbr2 sylibr ) ABUAIZABUDIZJZBEKZLZXRXPMZXRAMZBUDIZLZNZEOUBZABUCU + EZXQYDEOXQXROPZXSYCXQYGXSQZQZXTYABUAIZYBYIXTXRXOMZYJXQYKXTJYHXOXPXRUFUGYH + YKYJLXQYHFYKYJFKZYKPFEUHZYLXOPZQYHYLYJPZYLXRXOUIYHYMYNYOYNYLGKZHKZUJIZJZH + BRZGARZYHYMQZYOGHABYLCDUKUUBUUAYTGYARZYOUUBUUAGEUHZYTQZGARUUCUUBYTUUEGAUU + BYPAPZQZYTUUDYSQZHBRUUEUUGYSUUHHBUUGYQBPZYSUUHUUGUUIQZYSQZUUDYSUUKYLYQULI + ZXRPZUUDUUJUUMYSUUBUUIUUMUUFYGXSYMUUIUUMYGYMXSUUIUUMNYGYMXSUUIUUMXSUUIQHE + UHZYGYMUUMBXRYQUMYGXRSPZYMUUNUUMNNXRUNZUUOYMUUNUUMYLYQXRUOUPURUQUSUTVAVBU + GUUJYSUUMUUDTZUUJYSUULYPJZUUQUUBUUFUUIUURYSTZUUBYLVCPZUUFYPVCPZUUIYQVCPZU + USYGYMUUTXSYLXRVDVBYPACVEYQBDVEUUTUVBUVAUUSUUTUVBUVAVFUURYQYPUJIZYLJZYSYL + YQYPVGUVBUVAUVDYSTUUTUVBUVAQZUVDYRYLJYSUVEUVCYRYLYQYPVHVJYRYLVIVKVLVMVNVO + VPUULYPXRVQVRWAVSUUJYSVTWBWCWDUUDYSHBWEWFWGYTUUEGYAAYPYAPZYTQUUFUUDQZYTQU + UFUUEQUVFUVGYTUVFUUDUUFQUVGYPXRAUIUUDUUFWHWIWLUUFUUDYTWJWIWKWMUUBYASPZBSP + YOUUCTYGUVHXSYMYGUUOASPUVHUUPACWNXRAWOWPWQBDWNGHYABYLWRWPWSXBWTXBXAXCXDYG + YJYBLZXQXSYGYAOPZBOPZUVIYGAOPZUVJCXRAXEXFDYABXLWPXGXHXIXJUVLUVKYFYETCDEAB + XMXKXN $. $( Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.) $) @@ -470906,20 +475763,20 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, w3a hvsubadd eqcom bitrdi syl3an 3expa shsvsi eleq1 syl5ibcom com4r imp31 wn sylbird adantrr shscli elspansn5 ax-mp mpdd oveq2 ax-hvaddid sylan9eqr adantl sylan eqeq2d adantll biimpac biimparc biimpri ancoms expr ad2antrl - sylan2 mpd a1d com23 com4l imp4c exp4a expd rexlimdvv sylbid imp4b syl5bi - ex ssrdv wss shsub1 ssrind eqssd ) CHIZCABUBJZIVGZKZBCUCUDUEZUBJZAUFZBAUF - ZXHUAXKXLUALZXKIXMXJIZXMAIZKXHXMXLIZXMXJAUGXEXGXNXOXPXEXNXGXOXPMZXEXNXMFL - ZGLZNJZOZGXIUHFBUHZXGXQMZXEBPIZXIPIZXNYBQBEUIZCUJZFGBXIXMUKULXEYAYCFGBXIX - EXRBIZXSXIIZKZYAYCXOXEYJYAKZXGXPXOYKXEXGXPMXOYKXEXGXPXOYHYIYAXHXPMZYAXOYH - YIYLYAXOYHYIYLMYAXOYHKZKZXHYIXPYNXHYIXPMYNXHKZYIXSROZXPYOYIXSXFIZYPYNXEYI - YQMZXGYAYMXEYRYMXEYIYAYQYMXEYIYAYQMYMXEYIKZKYAXMXRUMJZXSOZYQXOYHYSUUAYAQZ - XOXMHIZYHXRHIZYSXSHIZUUBXMADUNXRBEUNZCXSUOUUCUUDUUEUPUUAXTXMOYAXMXRXSUQXT - XMURUSUTVAYMUUAYQMYSYMYTXFIUUAYQABXMXRADUIZYFVBYTXSXFVCVDSVHTVEVFVIXHYIYQ - YPMMYNXHYIYQYPXFPIXHYIYQKKYPMABUUGYFVJXFCXSVKVLTVQVMYNYIYPXPMZMXHYNUUHYIY - AYMYPXPYAYMYPKZKXMXROZXPUUIYAUUJYHYPYAUUJQXOYHYPKXTXRXMYHUUDYPXTXROUUFYPU - UDXTXRRNJXRXSRXRNVNXRVOVPVRVSVTWAYMUUJXPMYAYPXOYHUUJXPYHUUJKXOXMBIZXPUUJU - UKYHXMXRBVCWBUUKXOXPXPUUKXOKXMBAUGWCWDWGWEWFWHWEWISVMWSWJTWKWLWMWJWKWNWOW - PWJWQWRWTXEXLXKXAXGXEBXJAXEYDYEBXJXAYFYGBXIXBULXCSXD $. + sylan2 mpd a1d ex com23 com4l imp4c exp4a rexlimdvv sylbid imp4b biimtrid + expd ssrdv wss shsub1 ssrind eqssd ) CHIZCABUBJZIVGZKZBCUCUDUEZUBJZAUFZBA + UFZXHUAXKXLUALZXKIXMXJIZXMAIZKXHXMXLIZXMXJAUGXEXGXNXOXPXEXNXGXOXPMZXEXNXM + FLZGLZNJZOZGXIUHFBUHZXGXQMZXEBPIZXIPIZXNYBQBEUIZCUJZFGBXIXMUKULXEYAYCFGBX + IXEXRBIZXSXIIZKZYAYCXOXEYJYAKZXGXPXOYKXEXGXPMXOYKXEXGXPXOYHYIYAXHXPMZYAXO + YHYIYLYAXOYHYIYLMYAXOYHKZKZXHYIXPYNXHYIXPMYNXHKZYIXSROZXPYOYIXSXFIZYPYNXE + YIYQMZXGYAYMXEYRYMXEYIYAYQYMXEYIYAYQMYMXEYIKZKYAXMXRUMJZXSOZYQXOYHYSUUAYA + QZXOXMHIZYHXRHIZYSXSHIZUUBXMADUNXRBEUNZCXSUOUUCUUDUUEUPUUAXTXMOYAXMXRXSUQ + XTXMURUSUTVAYMUUAYQMYSYMYTXFIUUAYQABXMXRADUIZYFVBYTXSXFVCVDSVHTVEVFVIXHYI + YQYPMMYNXHYIYQYPXFPIXHYIYQKKYPMABUUGYFVJXFCXSVKVLTVQVMYNYIYPXPMZMXHYNUUHY + IYAYMYPXPYAYMYPKZKXMXROZXPUUIYAUUJYHYPYAUUJQXOYHYPKXTXRXMYHUUDYPXTXROUUFY + PUUDXTXRRNJXRXSRXRNVNXRVOVPVRVSVTWAYMUUJXPMYAYPXOYHUUJXPYHUUJKXOXMBIZXPUU + JUUKYHXMXRBVCWBUUKXOXPXPUUKXOKXMBAUGWCWDWGWEWFWHWEWISVMWJWKTWLWMWNWKWLWSW + OWPWKWQWRWTXEXLXKXAXGXEBXJAXEYDYEBXJXAYFYGBXIXBULXCSXD $. $( Lemma for ~ sumdmdi . (Contributed by NM, 23-Dec-2004.) (New usage is discouraged.) $) @@ -471203,10 +476060,10 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, cdj3lem2a $p |- ( ( C e. ( A +H B ) /\ ( A i^i B ) = 0H ) -> ( S ` C ) e. A ) $= ( vv vu cin wceq co wcel cfv cv wrex wa c0h cph cva shseli w3a cdj3lem2 - simp1 eqeltrd 3expa fveq2 eleq1d syl5ibr com13 rexlimdvv syl5bi impcom - expd ) DEMUANZFDEUBOPZFGQZDPZUSFKRZLRZUCOZNZLESKDSURVAKLDEFHIUDURVEVAKL - DEVEVBDPZVCEPZTZURVAVEVHURVAVHURTVAVEVDGQZDPZVFVGURVJVFVGURUEVIVBDABCDE - VBVCGHIJUFVFVGURUGUHUIVEUTVIDFVDGUJUKULUQUMUNUOUP $. + simp1 eqeltrd 3expa fveq2 eleq1d syl5ibr expd rexlimdvv biimtrid impcom + com13 ) DEMUANZFDEUBOPZFGQZDPZUSFKRZLRZUCOZNZLESKDSURVAKLDEFHIUDURVEVAK + LDEVEVBDPZVCEPZTZURVAVEVHURVAVHURTVAVEVDGQZDPZVFVGURVJVFVGURUEVIVBDABCD + EVBVCGHIJUFVFVGURUGUHUIVEUTVIDFVDGUJUKULUMUQUNUOUP $. $( Lemma for ~ cdj3i . The first-component function ` S ` is bounded if the subspaces are completely disjoint. (Contributed by NM, @@ -471260,10 +476117,10 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, cdj3lem3a $p |- ( ( C e. ( A +H B ) /\ ( A i^i B ) = 0H ) -> ( T ` C ) e. B ) $= ( vv vu cin wceq co wcel cfv cv wrex wa c0h cph cva shseli w3a cdj3lem3 - simp2 eqeltrd 3expa fveq2 eleq1d syl5ibr com13 rexlimdvv syl5bi impcom - expd ) DEMUANZFDEUBOPZFGQZEPZUSFKRZLRZUCOZNZLESKDSURVAKLDEFHIUDURVEVAKL - DEVEVBDPZVCEPZTZURVAVEVHURVAVHURTVAVEVDGQZEPZVFVGURVJVFVGURUEVIVCEABCDE - VBVCGHIJUFVFVGURUGUHUIVEUTVIEFVDGUJUKULUQUMUNUOUP $. + simp2 eqeltrd 3expa fveq2 eleq1d syl5ibr expd rexlimdvv biimtrid impcom + com13 ) DEMUANZFDEUBOPZFGQZEPZUSFKRZLRZUCOZNZLESKDSURVAKLDEFHIUDURVEVAK + LDEVEVBDPZVCEPZTZURVAVEVHURVAVHURTVAVEVDGQZEPZVFVGURVJVFVGURUEVIVCEABCD + EVBVCGHIJUFVFVGURUGUHUIVEUTVIEFVDGUJUKULUMUQUNUOUP $. $( Lemma for ~ cdj3i . The second-component function ` T ` is bounded if the subspaces are completely disjoint. (Contributed by NM, @@ -471321,37 +476178,37 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, sheli normcl anim12i hvaddcl syl2an remulcl sylan2 adantlr adantll le2add syl syl12anc wb cdj3lem2 fveq2d breq1d cdj3lem3 3expa ancoms recnd adddir recn syl3an 3imtr4d syld ralrimdvva readdcl oveq1d fvoveq1 oveq2 2ralbidv - cc cbvral2vw bitrid rspcev syl2and syl5bi rexlimdvva 3impib impbii ) UEGU - FZUGUHZCUFZUIUJZDUFZUIUJZUKULZYMYOYQUMULUIUJZUNULZSUHZDJUOCIUOZUPZGTURZIJ - UQUSUTZABVAUUEUUFABGCDIJMNVBUUEYNHUFZKUJUIUJZYMUUGUIUJZUNULZSUHZHIJVCULZU - OZUPZGTURZACDEFGHIJKMNOVDQVEUUEYNUUGLUJUIUJZUUJSUHZHUULUOZUPZGTURZBCDEFGH - IJLMNPVFRVEVJUUFABUUEABUPZUEUAUFZUGUHZUUHUVBUUIUNULZSUHZHUULUOZUPZUEUBUFZ - UGUHZUUPUVHUUIUNULZSUHZHUULUOZUPZUPZUBTURUATURZUUFUUEUVAUVGUATURZUVMUBTUR - ZUPUVOAUVPBUVQAUUOUVPQUUNUVGGUATGUAVGZYNUVCUUMUVFYMUVBUEUGVHUVRUUKUVEHUUL - UVRUUJUVDUUHSYMUVBUUIUNVIVKVLVMVNVOBUUTUVQRUUSUVMGUBTGUBVGZYNUVIUURUVLYMU - VHUEUGVHUVSUUQUVKHUULUVSUUJUVJUUPSYMUVHUUIUNVIVKVLVMVNVOVPUVGUVMUAUBTTVQV - 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A /\ x e. B ) ` . opreu2reuALT $p |- ( ( E! a e. A E. b e. B ch /\ E! b e. B E. a e. A ch ) -> E! p e. ( A X. B ) ph ) $= ( vx vy wrex wa wceq wcel nfre1 nfv nfan nfra1 nfcv simpr wreu cv wi wral - cxp 2reu4 cop wsbc w3a simpllr simplr opelxpi syl2anc nfsbc1v nfsbc nfrex - nfral rspa ad5ant23 imp syl21anc simprd simpld sbceq1a adantllr r19.29af2 - biimpa simplll c1st cfv c2nd 1st2nd2 ad2antlr xp1st xp2nd eqcom eqopi syl - wb bicomd ancoms ex syl2anbr impcom ad4ant24 simpl eqeq1d anbi12d imbi12d - adantl rspc2daf com12 anabsi7 opeq12d eqtrd ralrimiva 3jca opsbc2ie eqreu - r19.29ffa sylbi ) BGDKZFCUABFCKGDUALXBFCKZBFUBZIUBZMZGUBZJUBZMZLZUCZGDUDZ - FCUDZJDKICKLAECDUEZUAZBFGIJCDUFXCXMXOIJCDXCXECNZLZXHDNZLZXMLZXEXHUGZXNNZB - FXEUHZGXHUHZAEUBZYAMZUCZEXNUDZUIXOXTYBYDYHXTXPXRYBXCXPXRXMUJXQXRXMUKXEXHC - DULUMXTXBYDFCXSXMFXQXRFXCXPFXBFCOXPFPQXRFPQXLFCRQYCFGXHFXHSBFXEUNUOXTXDCN - ZLZXBLBYDGDYJXBGXTYIGXSXMGXQXRGXCXPGXBGFCGCSBGDOUPXPGPQXRGPQXLGFCGCSXKGDR - UQQYIGPQBGDOQYCGXHUNYJXGDNZBYDXBYJYKLZBLZXIYCYDYMXFXIYMXLYKBXJXMYIXLXSYKB - XLFCURUSYJYKBUKYLBTXLYKLZBXJXKGDURUTVAVBYMXFBYCYMXFXIYMXLYKBXJXMYIXLXSYKB - XLFCURUSYJYKBUKYLBTYNBXJXKGDURUTVAVCYLBTXFBYCBFXEVDVGUMXIYCYDYCGXHVDVGUMV - EYJXBTVFXCXPXRXMVHVFXTYGEXNXTYEXNNZLZAYFYPALZYEYEVIVJZYEVKVJZUGZYAYOYEYTM - XTAYECDVLVMYQYRXEYSXHYQYRXEMZYSXHMZYPAUUAUUBLZYQAUUCYQXKAUUCUCZFGYRYSCDYP - AFXTYOFXSXMFXQXRFXCXPFXBFCOXPFPQXRFPQXLFCRQYOFPQAFPQYPAGXTYOGXSXMGXQXRGXC - XPGXBGFCGCSBGDOUPXPGPQXRGPQXLGFCGCSXKGDRUQQYOGPQAGPQUUDFPUUDGPYOYRCNZXTAY - ECDVNVMYOYSDNZXTAYECDVOVMYQXDYRMZXGYSMZLZLZBAXJUUCYOUUIBAVSZXTAUUIYOUUKUU - GYRXDMZYSXGMZYOUUKUCZUUHYRXDVPYSXGVPUULUUMLZYOUUKYOUUOUUKYOUUOLZABUUPYEXD - XGUGMZABVSZYEXDXGCDVQHVRVTWAWBWCWDWEUUIXJUUCVSZYQUUIXFUUAXIUUBUUIXDYRXEUU - GUUHWFWGUUIXGYSXHUUGUUHTWGWHWJWIXSXMYOAUJWKWLWMVCYQUUAUUBYPAUUCYQAUUCYQXK - UUDFGYRYSCDYPAFXTYOFXSXMFXQXRFXCXPFXBFCOXPFPQXRFPQXLFCRQYOFPQAFPQYPAGXTYO - GXSXMGXQXRGXCXPGXBGFCGCSBGDOUPXPGPQXRGPQXLGFCGCSXKGDRUQQYOGPQAGPQUUDFPUUD - GPYOUUEXTAYECDVNVMYOUUFXTAYECDVOVMUUJBAXJUUCYOUUIUUKXTAUUIYOUUKUUGUULUUMU - UNUUHYRXDVPYSXGVPUUOYOUUKYOUUOUUKUUPABUUPUUQUURYEXDXGCDVQHVRVTWAWBWCWDWEU - UIUUSYQUUIXFUUAXIUUBUUIXDYRXEUUGUUHWFWGUUIXGYSXHUUGUUHTWGWHWJWIXSXMYOAUJW - KWLWMVBWNWOWBWPWQAYDEXNYAABIJEFGHWRWSVRWTXA $. + cxp 2reu4 cop wsbc w3a simpllr simplr opelxpi syl2anc nfsbc1v nfsbc nfral + nfrexw rspa ad5ant23 imp syl21anc simprd simpld biimpa adantllr r19.29af2 + sbceq1a simplll c1st cfv c2nd 1st2nd2 ad2antlr xp1st xp2nd wb eqcom eqopi + syl bicomd ancoms ex syl2anbr impcom ad4ant24 simpl eqeq1d anbi12d adantl + imbi12d rspc2daf com12 anabsi7 opeq12d eqtrd ralrimiva opsbc2ie r19.29ffa + 3jca eqreu sylbi ) BGDKZFCUABFCKGDUALXBFCKZBFUBZIUBZMZGUBZJUBZMZLZUCZGDUD + ZFCUDZJDKICKLAECDUEZUAZBFGIJCDUFXCXMXOIJCDXCXECNZLZXHDNZLZXMLZXEXHUGZXNNZ + BFXEUHZGXHUHZAEUBZYAMZUCZEXNUDZUIXOXTYBYDYHXTXPXRYBXCXPXRXMUJXQXRXMUKXEXH + CDULUMXTXBYDFCXSXMFXQXRFXCXPFXBFCOXPFPQXRFPQXLFCRQYCFGXHFXHSBFXEUNUOXTXDC + NZLZXBLBYDGDYJXBGXTYIGXSXMGXQXRGXCXPGXBGFCGCSBGDOUQXPGPQXRGPQXLGFCGCSXKGD + RUPQYIGPQBGDOQYCGXHUNYJXGDNZBYDXBYJYKLZBLZXIYCYDYMXFXIYMXLYKBXJXMYIXLXSYK + BXLFCURUSYJYKBUKYLBTXLYKLZBXJXKGDURUTVAVBYMXFBYCYMXFXIYMXLYKBXJXMYIXLXSYK + BXLFCURUSYJYKBUKYLBTYNBXJXKGDURUTVAVCYLBTXFBYCBFXEVGVDUMXIYCYDYCGXHVGVDUM + VEYJXBTVFXCXPXRXMVHVFXTYGEXNXTYEXNNZLZAYFYPALZYEYEVIVJZYEVKVJZUGZYAYOYEYT + MXTAYECDVLVMYQYRXEYSXHYQYRXEMZYSXHMZYPAUUAUUBLZYQAUUCYQXKAUUCUCZFGYRYSCDY + PAFXTYOFXSXMFXQXRFXCXPFXBFCOXPFPQXRFPQXLFCRQYOFPQAFPQYPAGXTYOGXSXMGXQXRGX + CXPGXBGFCGCSBGDOUQXPGPQXRGPQXLGFCGCSXKGDRUPQYOGPQAGPQUUDFPUUDGPYOYRCNZXTA + YECDVNVMYOYSDNZXTAYECDVOVMYQXDYRMZXGYSMZLZLZBAXJUUCYOUUIBAVPZXTAUUIYOUUKU + UGYRXDMZYSXGMZYOUUKUCZUUHYRXDVQYSXGVQUULUUMLZYOUUKYOUUOUUKYOUUOLZABUUPYEX + DXGUGMZABVPZYEXDXGCDVRHVSVTWAWBWCWDWEUUIXJUUCVPZYQUUIXFUUAXIUUBUUIXDYRXEU + UGUUHWFWGUUIXGYSXHUUGUUHTWGWHWIWJXSXMYOAUJWKWLWMVCYQUUAUUBYPAUUCYQAUUCYQX + KUUDFGYRYSCDYPAFXTYOFXSXMFXQXRFXCXPFXBFCOXPFPQXRFPQXLFCRQYOFPQAFPQYPAGXTY + OGXSXMGXQXRGXCXPGXBGFCGCSBGDOUQXPGPQXRGPQXLGFCGCSXKGDRUPQYOGPQAGPQUUDFPUU + DGPYOUUEXTAYECDVNVMYOUUFXTAYECDVOVMUUJBAXJUUCYOUUIUUKXTAUUIYOUUKUUGUULUUM + UUNUUHYRXDVQYSXGVQUUOYOUUKYOUUOUUKUUPABUUPUUQUURYEXDXGCDVRHVSVTWAWBWCWDWE + UUIUUSYQUUIXFUUAXIUUBUUIXDYRXEUUGUUHWFWGUUIXGYSXHUUGUUHTWGWHWIWJXSXMYOAUJ + WKWLWMVBWNWOWBWPWSAYDEXNYAABIJEFGHWQWTVSWRXA $. $} @@ -472279,16 +477136,6 @@ Class abstractions (a.k.a. class builders) ( csn wss wcel c0 wceq wo sssn snnzg neneqd pm2.21d sneqrg jaod imp sylan2b ) ADZBDZEACFZRGHZRSHZIZABHZRBJTUCUDTUAUDUBTUAUDTRGACKLMABCNOPQ $. - ${ - $d x A $. $d x B $. $d x ph $. - rabss3d.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> x e. B ) $. - $( Subclass law for restricted abstraction. (Contributed by Thierry - Arnoux, 25-Sep-2017.) $) - rabss3d $p |- ( ph -> { x e. A | ps } C_ { x e. B | ps } ) $= - ( crab nfv nfrab1 cv wcel wa simprr jca ex rabid 3imtr4g ssrd ) ACBCDGZBC - EGZACHBCDIBCEIACJZDKZBLZUAEKZBLZUASKUATKAUCUEAUCLUDBFAUBBMNOBCDPBCEPQR $. - $} - $( Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.) $) inin $p |- ( A i^i ( A i^i B ) ) = ( A i^i B ) $= @@ -472324,11 +477171,6 @@ Class abstractions (a.k.a. class builders) ( wceq wss wa cdif c0 eqss ssdif0 anbi12i sylbbr ) ABCABDZBADZEABFGCZBAFGCZ EABHLNMOABIBAIJK $. - $( An equality involving class union and class difference. (Contributed by - Thierry Arnoux, 26-Jun-2024.) $) - undif5 $p |- ( ( A i^i B ) = (/) -> ( ( A u. B ) \ B ) = A ) $= - ( cin c0 wceq cun cdif difun2 disjdif2 eqtrid ) ABCDEABFBGABGAABHABIJ $. - $( Two ways to express equality relative to a class ` A ` . (Contributed by Thierry Arnoux, 23-Jun-2024.) $) indifbi $p |- ( ( A i^i B ) = ( A i^i C ) <-> ( A \ B ) = ( A \ C ) ) $= @@ -472796,10 +477638,10 @@ Class abstractions (a.k.a. class builders) iunrnmptss $p |- ( ph -> U_ y e. ran ( x e. A |-> B ) C C_ U_ x e. A D ) $= ( vz cv wcel cmpt wrex cab ciun wa wex df-iun wceq wral wb ralrimiva eqid crn df-rex elrnmptg syl anbi1d exbidv r19.41v eleq2d biimpa reximi sylbir - exlimiv syl6bi syl5bi ss2abdv 3sstr4g ) AKLZFMZCBDENZUFZOZKPVBGMZBDOZKPCV - EFQBDGQAVFVHKVFCLZVEMZVCRZCSZAVHVCCVEUGAVLVIEUAZBDOZVCRZCSVHAVKVOCAVJVNVC - AEHMZBDUBVJVNUCAVPBDJUDBDEVIVDHVDUEUHUIUJUKVOVHCVOVMVCRZBDOVHVMVCBDULVQVG - BDVMVCVGVMFGVBIUMUNUOUPUQURUSUTCKVEFTBKDGTVA $. + exlimiv syl6bi biimtrid ss2abdv 3sstr4g ) AKLZFMZCBDENZUFZOZKPVBGMZBDOZKP + CVEFQBDGQAVFVHKVFCLZVEMZVCRZCSZAVHVCCVEUGAVLVIEUAZBDOZVCRZCSVHAVKVOCAVJVN + VCAEHMZBDUBVJVNUCAVPBDJUDBDEVIVDHVDUEUHUIUJUKVOVHCVOVMVCRZBDOVHVMVCBDULVQ + VGBDVMVCVGVMFGVBIUMUNUOUPUQURUSUTCKVEFTBKDGTVA $. $} ${ @@ -472953,11 +477795,11 @@ Class abstractions (a.k.a. class builders) x =/= Y ) -> ( B i^i C ) = (/) ) $= ( vy vz cv wcel wa cin c0 wceq wo weq csb wral eqeq1d wdisj df-ne disjors wn equequ1 csbeq1 csbid eqtrdi ineq1d orbi12d eqeq2 csbhypf ineq2d rspc2v - wne nfcv syl5bi impcom ord 3impia ) ABCUAZAJZBKEBKLZVBEUOZCDMZNOZVDVBEOZU - DVAVCLZVFVBEUBVHVGVFVCVAVGVFPZVAHIQZAHJZCRZAIJZCRZMZNOZPZIBSHBSVCVIABCHIU - CVQVIAIQZCVNMZNOZPHIVBEBBHAQZVJVRVPVTHAIUEWAVOVSNWAVLCVNWAVLAVBCRCAVKVBCU - FACUGUHUITUJVMEOZVRVGVTVFVMEVBUKWBVSVENWBVNDCAIECDAEUPFGULUMTUJUNUQURUSUQ - UT $. + wne nfcv biimtrid impcom ord 3impia ) ABCUAZAJZBKEBKLZVBEUOZCDMZNOZVDVBEO + ZUDVAVCLZVFVBEUBVHVGVFVCVAVGVFPZVAHIQZAHJZCRZAIJZCRZMZNOZPZIBSHBSVCVIABCH + IUCVQVIAIQZCVNMZNOZPHIVBEBBHAQZVJVRVPVTHAIUEWAVOVSNWAVLCVNWAVLAVBCRCAVKVB + CUFACUGUHUITUJVMEOZVRVGVTVFVMEVBUKWBVSVENWBVNDCAIECDAEUPFGULUMTUJUNUQURUS + UQUT $. $( Property of a disjoint collection: if ` B ( x ) ` and ` B ( Y ) = D ` have a common element ` Z ` , then ` x = Y ` . (Contributed by Thierry @@ -473010,11 +477852,11 @@ Class abstractions (a.k.a. class builders) ( Z e. B /\ Z e. C ) ) -> x = Y ) $= ( vy vz wcel wa cv cin c0 wceq wo weq csb wdisj wne wral disjorsf equequ1 inelcm csbeq1 csbid eqtrdi ineq1d eqeq1d orbi12d eqeq2 nfcv ineq2d rspc2v - csbhypf syl5bi impcom ord necon1ad 3impia syl3an3 ) FCLFDLMABCUAZANZBLEBL - MZCDOZPUBZVEEQZFCDUFVDVFVHVIVDVFMZVIVGPVJVIVGPQZVFVDVIVKRZVDJKSZAJNZCTZAK - NZCTZOZPQZRZKBUCJBUCVFVLABCJKGUDVTVLAKSZCVQOZPQZRJKVEEBBJASZVMWAVSWCJAKUE - WDVRWBPWDVOCVQWDVOAVECTCAVNVECUGACUHUIUJUKULVPEQZWAVIWCVKVPEVEUMWEWBVGPWE - VQDCAKECDAEUNHIUQUOUKULUPURUSUTVAVBVC $. + csbhypf biimtrid impcom ord necon1ad 3impia syl3an3 ) FCLFDLMABCUAZANZBLE + BLMZCDOZPUBZVEEQZFCDUFVDVFVHVIVDVFMZVIVGPVJVIVGPQZVFVDVIVKRZVDJKSZAJNZCTZ + AKNZCTZOZPQZRZKBUCJBUCVFVLABCJKGUDVTVLAKSZCVQOZPQZRJKVEEBBJASZVMWAVSWCJAK + UEWDVRWBPWDVOCVQWDVOAVECTCAVNVECUGACUHUIUJUKULVPEQZWAVIWCVKVPEVEUMWEWBVGP + WEVQDCAKECDAEUNHIUQUOUKULUPURUSUTVAVBVC $. $} ${ @@ -473152,23 +477994,23 @@ Class abstractions (a.k.a. class builders) ( vx vm cn ciun c1 cv cfzo wcel wrex cr clt nfcv nfcri cdif csb crab cinf co wa cuz cfv wss wne ssrab2 nnuz sseqtri rabn0 biimpri infssuzcl sylancr c0 nfrab1 nfinf nfcsb1 wceq csbeq1a eleq2d elrabf sylib simpld simprd wbr - nnred ltnrd eliun nfrex nfrabw nfbr ad2antrr eleqtrrdi ad2antlr cle simpr - elfzouz sylanbrc elfzolt2 lelttrd exp31 rexlimd syl5bi mtod eldifd csbeq1 - infssuzle nfeq2 oveq2 eqidd iuneq12df difeq12d rspcev syl2anc nfv nfcsb1v - nfiun nfdif iuneq1d cbvrexw sylibr eldifi reximi impbii 3bitr4i eqriv - nfov ) HDJAKZDJACLDMZNUEZBKZUAZKZHMZAOZDJPZXRXPOZDJPZXRXLOXRXQOXTYBXTXRDI - MZAUBZCLYCNUEZBKZUAZOZIJPZYBXTXSDJUCZQRUDZJOZXRDYKAUBZCLYKNUEZBKZUAZOZYIX - TYLXRYMOZXTYKYJOZYLYRUFXTYJLUGUHZUIZYJURUJZYSYJJYTXSDJUKULUMZUUBXTXSDJUNU - OYJLUPUQXSYRDYKJDYJQRXSDJUSDQSDRSUTZDJSZDHYMDYKAUUDVATXMYKVBAYMXRDYKAVCVD - VEVFZVGZXTXRYMYOXTYLYRUUFVHXTXRYOOZYKYKRVIZXTYKXTYKUUGVJZVKUUHXRBOZCYNPXT - UUICXRYNBVLXTUUKUUICYNXSCDJCJSZCHAETZVMCYKYKRCYJQRXSCDJUUMUULVNCQSCRSZUTZ - UUNUUOVOXTCMZYNOZUUKUUIXTUUQUFZUUKUFZYKUUPYKXTYKQOUUQUUKUUJVPZUUSUUPUUQUU - PJOZXTUUKUUQUUPYTJUUPLYKWAULVQVRZVJUUTUUSUUAUUPYJOZYKUUPVSVIUUCUUSUVAUUKU - VCUVBUURUUKVTXSUUKDUUPJDUUPSUUEDHBFTXMUUPVBABXRGVDVEWBUUPYJLWKUQUUQUUPYKR - VIXTUUKUUPLYKWCVRWDWEWFWGWHWIYHYQIYKJYCYKVBZYGYPXRUVDYDYMYFYODYCYKAWJUVDC - YEYNBBCYCYKUUOWLCYESCLYKNCLSCNSUUOXKYCYKLNWMUVDBWNWOWPVDWQWRYAYHDIJYAIWSD - HYGDYDYFDYCAWTCDYEBDYESFXAXBTXMYCVBZXPYGXRUVEAYDXOYFDYCAVCUVECXNYEBXMYCLN - WMXCWPVDXDXEYAXSDJXRAXOXFXGXHDXRJAVLDXRJXPVLXIXJ $. + nnred ltnrd eliun nfrexw nfrabw ad2antrr elfzouz eleqtrrdi ad2antlr simpr + nfbr cle sylanbrc infssuzle elfzolt2 lelttrd rexlimd biimtrid mtod eldifd + exp31 csbeq1 nfeq2 nfov oveq2 eqidd iuneq12df difeq12d rspcev syl2anc nfv + nfcsb1v nfiun nfdif iuneq1d cbvrexw sylibr eldifi reximi impbii 3bitr4i + eqriv ) HDJAKZDJACLDMZNUEZBKZUAZKZHMZAOZDJPZXRXPOZDJPZXRXLOXRXQOXTYBXTXRD + IMZAUBZCLYCNUEZBKZUAZOZIJPZYBXTXSDJUCZQRUDZJOZXRDYKAUBZCLYKNUEZBKZUAZOZYI + XTYLXRYMOZXTYKYJOZYLYRUFXTYJLUGUHZUIZYJURUJZYSYJJYTXSDJUKULUMZUUBXTXSDJUN + UOYJLUPUQXSYRDYKJDYJQRXSDJUSDQSDRSUTZDJSZDHYMDYKAUUDVATXMYKVBAYMXRDYKAVCV + DVEVFZVGZXTXRYMYOXTYLYRUUFVHXTXRYOOZYKYKRVIZXTYKXTYKUUGVJZVKUUHXRBOZCYNPX + TUUICXRYNBVLXTUUKUUICYNXSCDJCJSZCHAETZVMCYKYKRCYJQRXSCDJUUMUULVNCQSCRSZUT + ZUUNUUOVTXTCMZYNOZUUKUUIXTUUQUFZUUKUFZYKUUPYKXTYKQOUUQUUKUUJVOZUUSUUPUUQU + UPJOZXTUUKUUQUUPYTJUUPLYKVPULVQVRZVJUUTUUSUUAUUPYJOZYKUUPWAVIUUCUUSUVAUUK + UVCUVBUURUUKVSXSUUKDUUPJDUUPSUUEDHBFTXMUUPVBABXRGVDVEWBUUPYJLWCUQUUQUUPYK + RVIXTUUKUUPLYKWDVRWEWJWFWGWHWIYHYQIYKJYCYKVBZYGYPXRUVDYDYMYFYODYCYKAWKUVD + CYEYNBBCYCYKUUOWLCYESCLYKNCLSCNSUUOWMYCYKLNWNUVDBWOWPWQVDWRWSYAYHDIJYAIWT + DHYGDYDYFDYCAXACDYEBDYESFXBXCTXMYCVBZXPYGXRUVEAYDXOYFDYCAVCUVECXNYEBXMYCL + NWNXDWQVDXEXFYAXSDJXRAXOXGXHXIDXRJAVLDXRJXPVLXJXK $. $( A disjoint union is disjoint. Cf. ~ iundisj2 . (Contributed by Thierry Arnoux, 30-Dec-2016.) $) @@ -474681,9 +479523,9 @@ its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 23-May-2017.) $) rnmposs $p |- ( A. x e. A A. y e. B C e. D -> ran F C_ D ) $= ( vz wcel wral crn cv wceq wrex rnmpo abeq2i wa 2r19.29 wi eleq1 biimparc - a1i rexlimivv syl ex syl5bi ssrdv ) EFJZBDKACKZIGLZFIMZUKJULENZBDOACOZUJU - LFJZUNIUKABICDEGHPQUJUNUOUJUNRUIUMRZBDOACOUOUIUMABCDSUPUOABCDUPUOTAMCJBMD - JRUMUOUIULEFUAUBUCUDUEUFUGUH $. + a1i rexlimivv syl ex biimtrid ssrdv ) EFJZBDKACKZIGLZFIMZUKJULENZBDOACOZU + JULFJZUNIUKABICDEGHPQUJUNUOUJUNRUIUMRZBDOACOUOUIUMABCDSUPUOABCDUPUOTAMCJB + MDJRUMUOUIULEFUAUBUCUDUEUFUGUH $. $} ${ @@ -475029,7 +479871,7 @@ its graph has a given second element (that is, function value). $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- - Isomorphisms - misc. add. + Isomorphisms - misc. additions -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) @@ -476234,43 +481076,43 @@ its graph has a given second element (that is, function value). fdmi xaddcld pm5.32i nfvd ad2antrr simprl supxrub simprr xle2add syl22anc sseldd mp2and fvelima mpan adantl eqeq1d rexxp sylib ancom 2rexbii biimpa r19.29d2r breq1 reximi 19.9d2r sylbi ralrimiva w3a simplr simpr xlt2addrd - 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X |-> ( G gsum ( F |` C ) ) ) ) ) $= ( vy vz cgsu co c2nd cv csn ciun cres ccom cima cmpt 2ndresdjuf1o gsumf1o - cxp eqid cvv wcel wral snex a1i adantr ssidd eqsstrd iunss sylib r19.21bi - wa wss ssexd xpexd ralrimiva iunexg syl2anc relxp reliun sylibr cdm dmiun - wrel dmxpss rgenw ss2iun ax-mp eqsstri iunid sseqtri wfn wf wfo fo2nd fof - ssv fssres mp2an ffn mp1i djussxp2 imass2 wceq c0 ima0 0xp eqtrdi imaeq2d - xpeq1 iuneq1 0iun 3eqtr4a adantl wne 2ndimaxp pm2.61dane sseqtrid resssxp - eqtrd dff2 sylanbrc fcod 2ndresdju c0g fvexi fexd fsuppco csb cfv nfcsb1v - gsum2d csbeq1a iunsnima2 cop df-ov ad2antrr wrex simplr eleq2d vex oveq2d - nfcv crn vsnid biimpa opelxpd nfxp nfel2 sneq xpeq12d rspce fvco3d fvresd - eliun op2nd fveq2d eqtrid mpteq12dva xpeq2d rnss syl rnxpss sstrdi sstrid - imassrn eqsstrrd feqresmpt eqtr4d mpteq2dva reseq2d cbvmpt eqtr4di 3eqtrd - nfres nfov ) AGFUCUDGFUEBJBUFZUGZEUOZUHZUIZUJZUCUDGUAJGUBUVPUAUFZUGZUKZUV - 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imass2 wceq c0 ima0 xpeq1 0xp imaeq2d + eqtrdi iuneq1 0iun 3eqtr4a wne 2ndimaxp pm2.61dane eqtrd sseqtrid resssxp + adantl dff2 sylanbrc fcod 2ndresdju c0g fvexi fexd fsuppco gsum2d csb cfv + nfcsb1v csbeq1a iunsnima2 cop ad2antrr wrex simplr eleq2d nfcv vex oveq2d + df-ov crn vsnid biimpa opelxpd nfxp nfel2 sneq xpeq12d rspce eliun fvco3d + fvresd op2nd fveq2d eqtrid mpteq12dva imassrn xpeq2d rnxpss sstrdi sstrid + rnss syl eqsstrrd feqresmpt eqtr4d mpteq2dva nfres reseq2d cbvmpt eqtr4di + nfov 3eqtrd ) AGFUCUDGFUEBJBUFZUGZEUOZUHZUIZUJZUCUDGUAJGUBUVPUAUFZUGZUKZU + VSUBUFZUVRUDZULZUCUDZULZUCUDGBJGFEUIZUCUDZULZUCUDACDUVPFGUVQHKLMNOQRABCEU + VPHIJUVPUPZOPSTUMUNAUVPDJUAUBUVRGUQIKLMNAJIURUVOUQURZBJUSUVPUQURPAUWKBJAU + VMJURZUTZUVNEUQUQUVNUQURUWMBVAVBUWMECHACHURUWLOVCAECVDZBJABJEUHZCVDUWNBJU + SAUWOCCTACVEVIBJECVFVGVHVJZVKVLBJUVOIUQVMVNAUVOVOZBJUSUVPVOAUWQBJUWQUWMUV + NEVPVBVLBJUVOVQVRPUVPVSZJVDAUWRBJUVNUHZJUWRBJUVOVSZUHZUWSBJUVOVTUWTUVNVDZ + 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eleqtrrdi tg2 cxp wfn ioof ovelrn mp2b inss1 sstrdi simp-4l ffund + ad3antrrr biimtrid mpd simprr ad2antrl ctps cha xrsbas fssres syl2an frnd + cbs ressbas2 xrge0cmn adantl fmpttd supxrcl eqeltrd 0ss 0fin mpbir2an 0cn + res0 cdif csubmnd xrge0subm xrex difexg simpl eldifsn 3imtr4i ssriv xrs10 + ressabs eqtr4i subm0 sylanbrc elrest elinel1 reex elrestr mp3an12 xrtgioo + gsum0 ctg eleqtrrdi tg2 cxp wfn ioof ffn ovelrn mp2b inss1 sstrdi simp-4l wfun c0ex resfifsupp ssfi eliooord simprd xrlelttrd elioo1 anassrs simpld - supxrlub rgenw cbvmptv breq2 rexrnmptw anbi12d imbi1d rexlimdvva rexlimdv - sseq1 cioc eqeltrrd pnfnei simprl simp-5l pnfge pnfxr elioc1 ltpnf simplr - ffn rexr rexlimddv wo xrnemnf mpjaodan imbi2d rexralbidv imbi12d ralrimiv - syl5 rexlimdva cts xrstset resstset topnval xrstps resstps mpbir2and ctsr - eltsms letsr ordthaus mp1i resthaus haustsms2 ) ACEDUGLZMZVUFCUHUIAVUGCNU - JULLZMZCUAUMZMZUBUMZUCUMZUNZEDVUMUOZUPLZVUJMZUQZUCBURZUSUTZVAUBVUTVBZUQZU - 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(Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) $) slmd1cl $p |- ( W e. SLMod -> .1. e. K ) $= @@ -484124,9 +488966,9 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a slmdvs1.f $e |- F = ( Scalar ` W ) $. slmdvs1.s $e |- .x. = ( .s ` W ) $. slmdvs1.u $e |- .1. = ( 1r ` F ) $. - $( Scalar product with ring unit. ( ~ ax-hvmulid analog.) (Contributed by - NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised - by Thierry Arnoux, 1-Apr-2018.) $) + $( Scalar product with ring unity. ( ~ ax-hvmulid analog.) (Contributed + by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) + (Revised by Thierry Arnoux, 1-Apr-2018.) $) slmdvs1 $p |- ( ( W e. SLMod /\ X e. V ) -> ( .1. .x. X ) = X ) $= ( cslmd wcel wa cfv co wceq eqid w3a c0g cplusg simpl slmd1cl simpr cmulr cbs adantr slmdlema simprd simp2d syl122anc ) EKLZFDLZMUKBCUENZLZUNULULBF @@ -484337,8 +489179,8 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a rngurd.z $e |- ( ph -> .1. e. B ) $. rngurd.i $e |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x ) $. rngurd.j $e |- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x ) $. - $( Deduce the unit of a ring from its properties. (Contributed by Thierry - Arnoux, 6-Sep-2016.) $) + $( Deduce the unity element of a ring from its properties. (Contributed by + Thierry Arnoux, 6-Sep-2016.) $) rngurd $p |- ( ph -> .1. = ( 1r ` R ) ) $= ( ve wcel co wceq wa wral oveqd eqeq1d anbi12d cur cfv cbs cmulr cio eqid cv dfur2 eleqtrd jca ralrimiva adantr raleqbidva mpbid wb wi eleq2d simpr @@ -484632,6 +489474,43 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) + ${ + sdrgdvcl.i $e |- ./ = ( /r ` R ) $. + sdrgdvcl.0 $e |- .0. = ( 0g ` R ) $. + sdrgdvcl.a $e |- ( ph -> A e. ( SubDRing ` R ) ) $. + sdrgdvcl.x $e |- ( ph -> X e. A ) $. + sdrgdvcl.y $e |- ( ph -> Y e. A ) $. + sdrgdvcl.1 $e |- ( ph -> Y =/= .0. ) $. + $( A sub-division-ring is closed under the ring division operation. + (Contributed by Thierry Arnoux, 15-Jan-2025.) $) + sdrgdvcl $p |- ( ph -> ( X ./ Y ) e. A ) $= + ( co cfv wcel cdr wceq eqid syl cress cbs crg cui csubrg csdrg w3a issdrg + cdvr sylib simp3d drngringd simp2d subrgbas eleqtrd c0g subrg0 neeqtrd wa + wne drngunit biimpar syl12anc dvrcl syl3anc subrgdv 3eltr4d ) AEFDBUANZUI + OZNZVHUBOZEFCNZBAVHUCPEVKPFVHUDOZPZVJVKPAVHADQPZBDUEOPZVHQPZABDUFOPVOVPVQ + UGJDBUHUJZUKZULAEBVKKAVPBVKRAVOVPVQVRUMZBDVHVHSZUNTZUOAVQFVKPZFVHUPOZUTZV + NVSAFBVKLWBUOAFGWDMAVPGWDRVTBDVHGWAIUQTURVQVNWCWEUSVKVHVMFWDVKSZVMSZWDSVA + VBVCZVKVIVHVMEFWFWGVISZVDVEAVPEBPVNVLVJRVTKWHBCDVHVMVIEFWAHWGWIVFVEWBVG + $. + $} + + ${ + sdrginvcl.i $e |- I = ( invr ` R ) $. + sdrginvcl.0 $e |- .0. = ( 0g ` R ) $. + $( A sub-division-ring is closed under the ring inverse operation. + (Contributed by Thierry Arnoux, 15-Jan-2025.) $) + sdrginvcl $p |- ( ( A e. ( SubDRing ` R ) /\ X e. A /\ X =/= .0. ) + -> ( I ` X ) e. A ) $= + ( csdrg cfv wcel wne w3a cress co cinvr cbs cdr wceq eqid syl c0g eleqtrd + csubrg issdrg biimpi 3ad2ant1 simp3d simp2 simp2d subrgbas subrg0 neeqtrd + simp3 drnginvrcl syl3anc cui wa drngunit biimpar syl12anc syl2anc 3eltr4d + subrginv ) ABHIJZDAJZDEKZLZDBAMNZOIZIZVHPIZDCIZAVGVHQJZDVKJZDVHUAIZKZVJVK + JVGBQJZABUCIJZVMVDVEVQVRVMLZVFVDVSBAUDUEUFZUGZVGDAVKVDVEVFUHVGVRAVKRVGVQV + RVMVTUIZABVHVHSZUJTZUBZVGDEVOVDVEVFUMVGVREVORWBABVHEWCGUKTULZVKVHVIDVOVKS + ZVOSZVISZUNUOVGVRDVHUPIZJZVLVJRWBVGVMVNVPWKWAWEWFVMWKVNVPUQVKVHWJDVOWGWJS + ZWHURUSUTABVHWJCVIDWCFWLWIVCVAWDVB $. + $} + ${ $d R s $. primefldchr.1 $e |- P = ( R |`s |^| ( SubDRing ` R ) ) $. @@ -484645,6 +489524,153 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a $} +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Field extensions generated by a set +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $c fldGen $. + + $( Syntax for a function generating sub-fields. $) + cfldgen $a class fldGen $. + + ${ + $d f s a $. + $( Define a function generating the smallest sub-division-ring of a given + ring containing a given set. If the base structure is a division ring, + then this is also a division ring (see ~ fldgensdrg ). If the base + structure is a field, this is a subfield (see ~ fldgenfld and + ~ fldsdrgfld ). In general this will be used in the context of fields, + hence the name ` fldGen ` . (Contributed by Saveliy Skresanov and + Thierry Arnoux, 9-Jan-2025.) $) + df-fldgen $a |- fldGen = ( f e. _V , s e. _V |-> + |^| { a e. ( SubDRing ` f ) | s C_ a } ) $. + $} + + ${ + $d B a $. $d F a f s x $. $d S a f s x $. $d T a $. $d ph a x $. + fldgenval.1 $e |- B = ( Base ` F ) $. + fldgenval.2 $e |- ( ph -> F e. DivRing ) $. + ${ + fldgenval.3 $e |- ( ph -> S C_ B ) $. + $( Value of the field generating function: ` ( F fldGen S ) ` is the + smallest sub-division-ring of ` F ` containing ` S ` . (Contributed + by Thierry Arnoux, 11-Jan-2025.) $) + fldgenval $p |- ( ph + -> ( F fldGen S ) = |^| { a e. ( SubDRing ` F ) | S C_ a } ) $= + ( vf vs cvv wcel cv wss csdrg cfv crab cint cfldgen wceq co elexd fvexi + cdr cbs a1i ssexd sdrgid syl wb sseq2 adantl rspcedvd intexrab sylib wa + simpl fveq2d simpr sseq1d rabeqbidv inteqd df-fldgen ovmpoga syl3anc + wrex ) ADKLCKLCEMZNZEDOPZQZRZKLZDCSUAVKTADUDGUBACBKBKLABDUEFUCUFHUGAVHE + VIVFVLAVHCBNZEBVIADUDLBVILGBDFUHUIVGBTVHVMUJAVGBCUKULHUMVHEVIUNUOIJDCKK + JMZVGNZEIMZOPZQZRVKSKVPDTZVNCTZUPZVRVJWAVOVHEVQVIWAVPDOVSVTUQURWAVNCVGV + SVTUSUTVAVBIJEVCVDVE $. + + $( The field generated by a set of elements contains those elements. See + ~ lspssid . (Contributed by Thierry Arnoux, 15-Jan-2025.) $) + fldgenssid $p |- ( ph -> S C_ ( F fldGen S ) ) $= + ( va cv wss csdrg cfv crab cint cfldgen co ssintub fldgenval sseqtrrid + ) ACHIJHDKLZMNCDCOPHCTQABCDHEFGRS $. + + $( A generated subfield is a sub-division-ring. (Contributed by Thierry + Arnoux, 11-Jan-2025.) $) + fldgensdrg $p |- ( ph -> ( F fldGen S ) e. ( SubDRing ` F ) ) $= + ( va vx co cv wss cfv cdr wcel cress crg wa issdrg syl crab cint csubrg + cfldgen csdrg fldgenval drngringd eqid sseq2 elrab biimpi adantl simpld + cur simp2bi ex ssrdv sdrgid elrabd simp3bi subdrgint intss1 wi subrg1cl + ne0d wral ad2antlr ralrimiva elintrab sylibr issubrg syl22anc syl3anbrc + fvex biimpri eqeltrd ) ADCUDJCHKZLZHDUEMZUAZUBZVSABCDHEFGUFADNOZWADUCMZ + OZDWAPJZNOWAVSOFADQOZWEQOZWABLZDUNMZWAOZWDADFUGAWEADVTWEIWEUHFAIVTWCAIK + ZVTOZWKWCOZAWLRZWKVSOZWMWNWOCWKLZWLWOWPRZAWLWQVRWPHWKVSVQWKCUIUJUKULUMZ + WOWBWMDWKPJNOZDWKSZUOTUPUQAVTBAVRCBLHBVSVQBCUIAWBBVSOFBDEURTGUSZVEWNWOW + SWRWOWBWMWSWTUTTVAZUGABVTOWHXABVTVBTAVRWIVQOZVCZHVSVFWJAXDHVSAVQVSOZRVR + XCXEXCAVRXEVQWCOZXCXEWBXFDVQPJNODVQSUOVQDWIWIUHZVDTVGUPVHVRHWIVSDUNVNVI + VJWDWFWGRWHWJRRWABDWIEXGVKVOVLXBDWASVMVP $. + + fldgenss.t $e |- ( ph -> T C_ S ) $. + $( Generated subfields preserve subset ordering. ( see ~ lspss and + ~ spanss ) (Contributed by Thierry Arnoux, 15-Jan-2025.) $) + fldgenss $p |- ( ph -> ( F fldGen T ) C_ ( F fldGen S ) ) $= + ( va cv wss csdrg crab cint cfldgen co adantr sstrd fldgenval cfv wi wa + wcel simpr ex ss2rabdv intss syl 3sstr4d ) ADJKZLZJEMUAZNZOZCUKLZJUMNZO + ZEDPQECPQAUQUNLUOURLAUPULJUMAUPULUBUKUMUDAUPULAUPUCDCUKADCLUPIRAUPUESUF + RUGUQUNUHUIABDEJFGADCBIHSTABCEJFGHTUJ $. + $} + + ${ + fldgenidfld.s $e |- ( ph -> S e. ( SubDRing ` F ) ) $. + $( The subfield generated by a subfield is the subfield itself. + (Contributed by Thierry Arnoux, 15-Jan-2025.) $) + fldgenidfld $p |- ( ph -> ( F fldGen S ) = S ) $= + ( va cfldgen co cv wss csdrg cfv crab cint wcel sdrgss syl fldgenval + wceq intmin eqtrd ) ADCIJCHKLHDMNZOPZCABCDHEFACUDQZCBLGBDCERSTAUFUECUAG + HCUDUBSUC $. + $} + + $( The subfield of a field ` F ` generated by the whole base set of ` F ` + is ` F ` itself. (Contributed by Thierry Arnoux, 11-Jan-2025.) $) + fldgenid $p |- ( ph -> ( F fldGen B ) = B ) $= + ( va cfldgen co cv wss csdrg cfv crab cint ssidd fldgenval wcel sseq2 cdr + syl sdrgid elrabd intss1 wi wral wa ralrimiva ssintrab sylibr eqssd eqtrd + idd ) ACBGHBFIZJZFCKLZMZNZBABBCFDEABOZPAUQBABUPQUQBJAUNBBJFBUOUMBBRACSQBU + OQEBCDUATURUBBUPUCTAUNUNUDZFUOUEBUQJAUSFUOAUMUOQUFUNULUGUNFBUOUHUIUJUK $. + $} + + ${ + fldgenfld.1 $e |- B = ( Base ` F ) $. + fldgenfld.2 $e |- ( ph -> F e. Field ) $. + fldgenfld.3 $e |- ( ph -> S C_ B ) $. + $( A generated subfield is a field. (Contributed by Thierry Arnoux, + 11-Jan-2025.) $) + fldgenfld $p |- ( ph -> ( F |`s ( F fldGen S ) ) e. Field ) $= + ( cfield wcel cfldgen co csdrg cfv cress cdr ccrg wa isfld sylib simpld + fldgensdrg fldsdrgfld syl2anc ) ADHIZDCJKZDLMIDUENKHIFABCDEADOIZDPIZAUDUF + UGQFDRSTGUAUEDUBUC $. + $} + + ${ + $d .1. a $. $d B a $. $d R a $. $d a ph $. + primefldgen1.b $e |- B = ( Base ` R ) $. + primefldgen1.1 $e |- .1. = ( 1r ` R ) $. + primefldgen1.r $e |- ( ph -> R e. DivRing ) $. + $( The prime field of a division ring is the subfield generated by the + multiplicative identity element. In general, we should write "prime + division ring", but since most later usages are in the case where the + ambient ring is commutative, we keep the term "prime field". + (Contributed by Thierry Arnoux, 11-Jan-2025.) $) + primefldgen1 $p |- ( ph -> |^| ( SubDRing ` R ) = ( R fldGen { .1. } ) ) $= + ( va csdrg cfv cint csn cv wss crab co wcel cdr syl snssd cfldgen wral wa + wceq csubrg issdrg simp2bi subrg1cl adantl ralrimiva rabid2 sylibr inteqd + cress crg drngringd ringidcl fldgenval eqtr4d ) ACIJZKDLZHMZNZHUTOZKCVAUA + PAUTVDAVCHUTUBUTVDUDAVCHUTAVBUTQZUCDVBVEDVBQZAVEVBCUEJQZVFVECRQVGCVBUNPRQ + CVBUFUGVBCDFUHSUITUJVCHUTUKULUMABVACHEGADBACUOQDBQACGUPBCDEFUQSTURUS $. + $} + + ${ + $d p q z $. + $( The field of rational numbers ` QQ ` is generated by ` 1 ` in + ` CCfld ` , that is, ` QQ ` is the prime field of ` CCfld ` . + (Contributed by Thierry Arnoux, 15-Jan-2025.) $) + 1fldgenq $p |- ( CCfld fldGen { 1 } ) = QQ $= + ( vz vp vq ccnfld c1 co cq wceq wtru cc cnfldbas cdr a1i wss cz cfv cv cn + wcel mptru csn cfldgen cndrng qsscn snssi ax-mp zssq sstri fldgenss csdrg + 1z csubrg cress qsubdrg simpli simpri issdrg mpbir3an fldgenidfld sseqtrd + cdiv wrex elq wa cc0 cnflddiv cnfld0 sstrdi fldgensdrg cmg cmul cnfldmulg + ax-1cn mpan2 cr zre ax-1rid syl eqtrd csubg w3a mpbi subrgsubg fldgenssid + simp2i 1ex snss sylibr eqid subgmulgcl mp3an13 eqeltrrd adantr nnz adantl + ssriv sselid wne nnne0 sdrgdvcl eleq1 syl5ibrcom rexlimivv sylbi eqssd ) + DEUAZUBFZGHIXGGIXGDGUBFGIJGXFDKDLSZIUCMZGJNIUDMXFGNIXFOGEOSXFONUKEOUEUFUG + UHMZUIIJGDKXIGDUJPZSZIXLXHGDULPZSZDGUMFLSZUCXNXOUNUOXNXOUNUPDGUQURMUSUTGX + GNIAGXGAQZGSXPBQZCQZVAFZHZCRVBBOVBXPXGSZBCXPVCXTYABCORXQOSZXRRSZVDZYAXTXS + XGSYDXGVADXQXRVEVFVGXGXKSZYDYEIJXFDKXIIXFGJXJUDVHZVITZMYBXQXGSYCYBXQEDVJP + ZFZXQXGYBYIXQEVKFZXQYBEJSYIYJHVMXQEVLVNYBXQVOSYJXQHXQVPXQVQVRVSXGDVTPSZYB + EXGSZYIXGSXGXMSZYKXHYMDXGUMFLSZYEXHYMYNWAYGDXGUQWBWEXGDWCUFYLIXFXGNYLIJXF + DKXIYFWDEXGWFWGWHTXGYHDXQEYHWIWJWKWLZWMYDOXGXRBOXGYOWPYCXROSYBXRWNWOWQYCX + RVEWRYBXRWSWOWTXPXSXGXAXBXCXDWPMXET $. + $} + + $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Totally ordered rings and fields @@ -484904,7 +489930,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a orng0le1.2 $e |- .1. = ( 1r ` F ) $. ${ orng0le1.3 $e |- .<_ = ( le ` F ) $. - $( In an ordered ring, the ring unit is positive. (Contributed by + $( In an ordered ring, the ring unity is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.) $) orng0le1 $p |- ( F e. oRing -> .0. .<_ .1. ) $= ( corng wcel cmulr cfv cbs wbr crg orngring eqid ringidcl syl orngsqr @@ -484916,8 +489942,8 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a ${ ofld0lt1.3 $e |- .< = ( lt ` F ) $. $( TODO - this can be generalized to ordered division rings $) - $( In an ordered field, the ring unit is strictly positive. (Contributed - by Thierry Arnoux, 21-Jan-2018.) $) + $( In an ordered field, the ring unity is strictly positive. + (Contributed by Thierry Arnoux, 21-Jan-2018.) $) ofldlt1 $p |- ( F e. oField -> .0. .< .1. ) $= ( cofld wcel wbr cple cfv wne corng cfield isofld simprbi eqid fvexi cvv orng0le1 syl cdr ofldfld ccrg isfld simplbi drngunz 3syl necomd c0g @@ -485057,65 +490083,6 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) - ${ - $d c y A $. $d c y B $. $d c y F $. $d c R $. $d c y S $. $d c X $. - $d c .|| $. - rhmdvdsr.x $e |- X = ( Base ` R ) $. - rhmdvdsr.m $e |- .|| = ( ||r ` R ) $. - rhmdvdsr.n $e |- ./ = ( ||r ` S ) $. - $( A ring homomorphism preserves the divisibility relation. (Contributed - by Thierry Arnoux, 22-Oct-2017.) $) - rhmdvdsr $p |- ( ( ( F e. ( R RingHom S ) /\ A e. X /\ B e. X ) - /\ A .|| B ) -> ( F ` A ) ./ ( F ` B ) ) $= - ( vy vc co wcel wa cfv wceq wrex syl2anc crh w3a wbr cbs cv simpl1 simpl2 - cmulr eqid rhmf ffvelcdmda simpll1 ralrimiva adantr rhmmul syl3anc dvdsr2 - wral simpr biimpac r19.29 simpl fveq2d eqtr3d reximi syl eqeq1d rexlimivw - oveq1 rspcev dvdsr sylanbrc ) GEFUANOZAHOZBHOZUBZABCUCZPZAGQZFUDQZOZLUEZV - SFUHQZNZBGQZRZLVTSZVSWEDUCVRVMVNWAVMVNVOVQUFVMVNVOVQUGZVMHVTAGHVTEFGIVTUI - ZUJZUKTVRMUEZGQZVTOZWLVSWCNZWERZPZMHSZWGVRWMMHURWOMHSZWQVRWMMHVRWKHOZPZVM - WSWMVMVNVOVQWSULZVRWSUSZVMHVTWKGWJUKTUMVRWKAEUHQZNZGQZWNRZMHURZXDBRZMHSZW - RVRXFMHWTVMWSVNXFXAXBVRVNWSWHUNWKAEFXCWCGHIXCUIZWCUIZUOUPUMVRVQVNXIVPVQUS - WHVNVQXIMHCEXCABIJXJUQUTTXGXIPXFXHPZMHSWRXFXHMHVAXLWOMHXLXEWNWEXFXHVBXLXD - BGXFXHUSVCVDVEVFTWMWOMHVATWPWGMHWFWOLWLVTWBWLRWDWNWEWBWLVSWCVIVGVJVHVFLVT - DFWCVSWEWIKXKVKVL $. - $} - - ${ - $d x y F $. $d x y R $. $d x y S $. - $( A ring homomorphism is also a ring homomorphism for the opposite rings. - (Contributed by Thierry Arnoux, 27-Oct-2017.) $) - rhmopp $p |- ( F e. ( R RingHom S ) - -> F e. ( ( oppR ` R ) RingHom ( oppR ` S ) ) ) $= - ( vx vy co wcel coppr cfv cbs cmulr cur eqid opprringb sylib oppr1 eqcomi - crg cv wa rhmrcl1 rhmrcl2 rhm1 wceq simpl simprr opprbas eleqtrrdi simprl - crh rhmmul syl3anc opprmul fveq2i 3eqtr4g cgrp wf cplusg wral ringgrp syl - cghm rhmf rhmghm ad2antrr simplr simpr ghmlin ralrimiva jca jca31 oppradd - isghm sylibr isrhm2d ) CABUJFGZDEAHIZJIZVQBHIZVQKIZVSKIZVQLIZCVSLIZVRMWBM - WCMVTMZWAMZVPARGVQRGZABCUAAVQVQMZNOZVPBRGVSRGZABCUBBVSVSMZNOZABWBCWCALIZW - BAWLVQWGWLMPQBLIZWCBWMVSWJWMMPQUCVPDSZVRGZESZVRGZTZTZWPWNAKIZFZCIZWPCIZWN - CIZBKIZFZWNWPVTFZCIXDXCWAFWSVPWPAJIZGZWNXHGZXBXFUDVPWRUEWSWPVRXHVPWOWQUFX - HAVQWGXHMZUGZUHWSWNVRXHVPWOWQUIXLUHWPWNABWTXECXHXKWTMZXEMZUKULXGXACXHAVTW - TVQWNWPXKXMWGWDUMUNBJIZBWAXEVSXDXCXOMZXNWJWEUMUOVPVQUPGZVSUPGZTXHXOCUQZWN - WPAURIZFCIXDXCBURIZFUDZEXHUSZDXHUSZTZTCVQVSVBFGVPXQXRYEVPWFXQWHVQUTVAVPWI - XRWKVSUTVAVPXSYDXHXOABCXKXPVCVPYCDXHVPXJTZYBEXHYFXITCABVBFGZXJXIYBVPYGXJX - IABCVDVEVPXJXIVFYFXIVGXTYAABWNCWPXHXKXTMZYAMZVHULVIVIVJVKEDXTYAVQVSCXHXOX - LXOBVSWJXPUGXTAVQWGYHVLYABVSWJYIVLVMVNVO $. - $} - - $( Ring homomorphisms preserve unit elements. (Contributed by Thierry - Arnoux, 23-Oct-2017.) $) - elrhmunit $p |- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) - -> ( F ` A ) e. ( Unit ` S ) ) $= - ( crh co wcel cui cfv wa cur cdsr wbr coppr isunit rhmdvdsr syl31anc adantr - eqid wb cbs simpl unitss simpr sselid crg rhmrcl1 ringidcl 3syl simpld rhm1 - sylib breq2d mpbid rhmopp simprd opprbas sylanbrc ) DBCEFGZABHIZGZJZADIZCKI - ZCLIZMZVCVDCNIZLIZMZVCCHIZGVBVCBKIZDIZVEMZVFVBUSABUAIZGZVKVNGZAVKBLIZMZVMUS - VAUBZVBUTVNAVNBUTVNSZUTSZUCUSVAUDZUEZVBUSBUFGVPVSBCDUGVNBVKVTVKSZUHUIZVBVRA - VKBNIZLIZMZVBVAVRWHJWBVQBWFUTVKWGAWAWDVQSZWFSZWGSZOULZUJAVKVQVEBCDVNVTWIVES - ZPQUSVMVFTVAUSVLVDVCVEBCVKDVDWDVDSZUKZUMRUNVBVCVLVHMZVIVBDWFVGEFGZVOVPWHWPU - SWQVABCDUORWCWEVBVRWHWLUPAVKWGVHWFVGDVNVNBWFWJVTUQWKVHSZPQUSWPVITVAUSVLVDVC - VHWOUMRUNVECVGVJVDVHVCVJSWNWMVGSWROUR $. - $( @{ rhmdvr.u @e |- U = ( Unit ` R ) @. @@ -485150,23 +490117,6 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a GIUOVLJWGEFILWGTZUPUNWCUQVCVGVIVKURVCVGVIVKUSWGDFVQHVPVHVJWHKMWEUTVAVB $. $} - $( Ring homomorphisms preserve the inverse of unit elements. (Contributed by - Thierry Arnoux, 23-Oct-2017.) $) - rhmunitinv $p |- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) - -> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) $= - ( co wcel cui cfv cinvr cmulr wceq cur eqid unitlinv sylan unitinvcl sselid - crg adantr elrhmunit crh rhmrcl1 fveq2d cbs simpl unitss simpr syl3anc rhm1 - wa rhmmul 3eqtr3d rhmrcl2 syl2anc eqtr4d cmgp cress cgrp unitgrp syl syldan - unitgrpbas cvv cplusg fvex mgpplusg ressplusg ax-mp grprcan syl13anc mpbid - wb ) DBCUAEFZABGHZFZUJZABIHZHZDHZADHZCJHZEZVTCIHZHZVTWAEZKZVSWDKZVPWBCLHZWE - VPVRABJHZEZDHZBLHZDHZWBWHVPWJWLDVMBRFZVOWJWLKBCDUBZBWIVNWLVQAVNMZVQMZWIMZWL - MZNOUCVPVMVRBUDHZFAWTFWKWBKVMVOUEVPVNWTVRWTBVNWTMZWPUFZVMWNVOVRVNFZWOBVNVQA - WPWQPOZQVPVNWTAXBVMVOUGQVRABCWIWADWTXAWRWAMZUKUHVMWMWHKVOBCWLDWHWSWHMZUISUL - VPCRFZVTCGHZFZWEWHKVMXGVOBCDUMZSZABCDTZCWAXHWHWCVTXHMZWCMZXEXFNUNUOVPCUPHZX - HUQEZURFZVSXHFZWDXHFZXIWFWGVLVMXQVOVMXGXQXJCXHXPXMXPMZUSUTSVMVOXCXRXDVRBCDT - VAVPXGXIXSXKXLCXHWCVTXMXNPUNXLXHWAXPVSWDVTCXHXPXMXTVBXHVCFWAXPVDHKCGVEXHWAX - OXPVCXTCWAXOXOMXEVFVGVHVIVJVK $. - $( @{ rhmuntghm.1 @e |- G = ( ( mulGrp ` R ) |`s ( Unit ` R ) ) @. @@ -485876,6 +490826,45 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) + ${ + fermltlchr.z $e |- P = ( chr ` F ) $. + fermltlchr.b $e |- B = ( Base ` F ) $. + fermltlchr.p $e |- .^ = ( .g ` ( mulGrp ` F ) ) $. + fermltlchr.1 $e |- A = ( ( ZRHom ` F ) ` E ) $. + fermltlchr.2 $e |- ( ph -> P e. Prime ) $. + fermltlchr.3 $e |- ( ph -> E e. ZZ ) $. + fermltlchr.4 $e |- ( ph -> F e. CRing ) $. + $( A generalization of Fermat's little theorem in a commutative ring ` F ` + of prime characteristic. See ~ fermltl . (Contributed by Thierry + Arnoux, 9-Jan-2024.) $) + fermltlchr $p |- ( ph -> ( P .^ A ) = A ) $= + ( cfv wceq co cz wcel eqid czrh wa ccnfld cmgp cress cmg cn0 cprime prmnn + cexp nnnn0d syl adantr cminusg cbs zsscn cnfldbas mgpbas sseqtri cmnd c0g + cc wss crg cnring ringmgp ax-mp c1 cnfld1 ringidval eqeltrri ress0g mp3an + 1z ressmulgnn0 syl2anc zcnd cnfldexp eqtrd fveq2d czring crngringd zrhrhm + cmhm crh zringmpg rhmmhm ressbas2 mhmmulg syl3anc cmin zringsubgval cdvds + csg zexpcld wbr cmo cc0 zred cn nnrpd fermltl modsub12d zcn subidd oveq1d + eqidd crp 0mod 3eqtrd zsubcld dvdsval3 mpbird chrdvds mpbid cghm zringbas + wb rhmghm ghmsub 3eqtr3rd cgrp crnggrpd wf rhmf ffvelcdmd grpsubeq0 oveq2 + 3eqtr3d adantl simpr 3eqtr4d mpan2 ) ABEGUAOZOZPZDBFQZBPKAYPUBZDYOFQZYOYQ + BYRDEUCUDOZRUEQZUFOZQZYNOZEDUJQZYNOZYSYOYRUUCUUEYNYRUUCDEYTUFOZQZUUEYRDUG + SZERSZUUCUUHPAUUIYPADUHSZUUILUUKDDUIZUKULZUMZAUUJYPMUMZRYTUUAYTUNOZUUGDEU + UATZRVBYTUOOUPVBUCYTYTTZUQURZUSUUGTUUPTYTUTSZYTVAOZRSRVBVCZUVAUUAVAOPUCVD + SUUTVEUCYTUURVFVGVHUVARUCVHYTUURVIVJVNVKUPRVBYTUUAUVAUUQUUSUVATVLVMVOVPYR + EVBSUUIUUHUUEPYREUUOVQUUNEDVRVPVSVTYRYNUUAGUDOZWDQSZUUIUUJUUDYSPAUVDYPAYN + WAGWEQSZUVDAGVDSZUVEAGNWBZGYNYNTZWCULZWAGYNUUAUVCWFUVCTWGULUMUUNUUORUUBFY + NUUAUVCDEUVBRUUAUOOPUPRVBUUAYTUUQUUSWHVGUUBTJWIWJAUUFYOPZYPAUUFYOGWNOZQZG + VAOZPZUVJAUUEEWKQZYNOZUUEEWAWNOZQZYNOZUVMUVLAUVOUVRYNAUUERSZUUJUVOUVRPAED + MUUMWOZMUVQUUEEUVQTZWLVPVTADUVOWMWPZUVPUVMPZAUWCUVODWQQZWRPZAUWEEEWKQZDWQ + QWRDWQQZWRAUUEEEEDAUUEUWAWSAEMWSZUWIUWIADAUUKDWTSZLUULULZXAZAUUKUUJUUEDWQ + QEDWQQZPLMEDXBVPAUWMXGXCAUWGWRDWQAUUJUWGWRPMUUJEEXDXEULXFADXHSUWHWRPUWLDX + IULXJAUWJUVORSZUWCUWFXRUWKAUUEEUWAMXKZDUVOXLVPXMAUVFUWNUWCUWDXRUVGUWODGYN + UVOUVMHUVHUVMTZXNVPXOAYNWAGXPQSZUVTUUJUVSUVLPAUVEUWQUVIWAGYNXSULUWAMRWAGU + UEYNUVQUVKEXQUWBUVKTZXTWJYAAGYBSUUFGUOOZSYOUWSSUVNUVJXRAGNYCARUWSUUEYNAUV + ERUWSYNYDUVIRUWSWAGYNXQUWSTZYEULZUWAYFARUWSEYNUXAMYFUWSGUVKUUFYOUVMUWTUWP + UWRYGWJXOUMYIYPYQYSPABYODFYHYJAYPYKYLYM $. + $} + ${ $d .^ a $. $d A a $. $d B a $. $d P a $. $d Z a $. znfermltl.z $e |- Z = ( Z/nZ ` P ) $. @@ -486601,8 +491590,7 @@ The sumset operation can be used for both group (additive) operations and $} ${ - $d B i j x $. $d E g $. $d F g $. $d F h $. $d G i j k $. $d H g $. - $d H h x $. $d N i j k $. $d S h $. $d g ph $. $d h ph x $. $d k x $. + $d B x $. $d F h $. $d H h x $. $d S h $. $d h ph x $. qusima.b $e |- B = ( Base ` G ) $. qusima.q $e |- Q = ( G /s ( G ~QG N ) ) $. qusima.p $e |- .(+) = ( LSSum ` G ) $. @@ -486675,10 +491663,9 @@ The sumset operation can be used for both group (additive) operations and ${ $d .(+) a h x $. $d .(+) k $. $d B a h x $. $d B k $. $d E a f h x $. - $d E q $. $d F f h x $. $d G a f h x $. $d G h k x $. $d N a h x $. - $d N k $. $d N q $. $d Q a f h x $. $d Q q $. $d S a f h x $. - $d S k $. $d T a f h x $. $d V f h $. $d W f h $. $d a f h ph x $. - $d f h q $. $d k ph $. + $d F f h x $. $d G a f h x $. $d G h k x $. $d N a h x $. $d N k $. + $d Q a f h x $. $d S a f h x $. $d S k $. $d T a f h x $. $d V f h $. + $d W f h $. $d a f h ph x $. $d k ph $. nsgmgc.b $e |- B = ( Base ` G ) $. nsgmgc.s $e |- S = { h e. ( SubGrp ` G ) | N C_ h } $. nsgmgc.t $e |- T = ( SubGrp ` Q ) $. @@ -488298,6 +493285,40 @@ such that every prime ideal contains a prime element (this is a UULWNUUHUUIUQTWCWHYIWSWLKYJYIWRWJWKWQABWPWJCYPYCWOWESQWFWG $. $} + ${ + ply1fermltlchr.w $e |- W = ( Poly1 ` F ) $. + ply1fermltlchr.x $e |- X = ( var1 ` F ) $. + ply1fermltlchr.l $e |- .+ = ( +g ` W ) $. + ply1fermltlchr.n $e |- N = ( mulGrp ` W ) $. + ply1fermltlchr.t $e |- .^ = ( .g ` N ) $. + ply1fermltlchr.c $e |- C = ( algSc ` W ) $. + ply1fermltlchr.a $e |- A = ( C ` ( ( ZRHom ` F ) ` E ) ) $. + ply1fermltlchr.p $e |- P = ( chr ` F ) $. + ply1fermltlchr.f $e |- ( ph -> F e. CRing ) $. + ply1fermltlchr.1 $e |- ( ph -> P e. Prime ) $. + ply1fermltlchr.2 $e |- ( ph -> E e. ZZ ) $. + $( Fermat's little theorem for polynomials in a commutative ring ` F ` of + characteristic ` P ` prime: we have the polynomial equation + ` ( X + A ) ^ P = ( ( X ^ P ) + A ) ` . (Contributed by Thierry Arnoux, + 9-Jan-2025.) $) + ply1fermltlchr $p |- ( ph -> ( P .^ ( X .+ A ) ) = ( ( P .^ X ) .+ A ) ) $= + ( cchr cfv co cbs eqid cmg cmgp fveq2i ccrg wcel ply1crng syl cprime wceq + eqtri ply1chr eqtr4di eqeltrd crg crngringd vr1cl cz czring crh wf zrhrhm + czrh zringbas rhmf 3syl ffvelcdmd ply1sclcl syl2anc freshmansdream oveq1d + eqeltrid cmhm cn0 csca casa ply1assa asclrhm cgrp crnggrpd ply1sca rhmmhm + eleqtrrd prmnn mgpbas mhmmulg syl3anc a1i oveq2d eqtr4d fermltlchr fveq2d + cn nnnn0 3eqtr2d oveq12d 3eqtr3d ) AJUCUDZKBEUEZGUEXDKGUEZXDBGUEZEUEDXEGU + EDKGUEZBEUEAJUFUDZXDEJGKBXIUGZNGIUHUDJUIUDZUHUDPIXKUHOUJUQXDUGAHUKULZJUKU + LTJHLUMUNAXDDUOAXDHUCUDZDAXLXDXMUPTJHLURUNSUSZUAUTAHVAULZKXIULAHTVBZXIJHK + MLXJVCUNABFHVIUDZUDZCUDZXIRAXOXRHUFUDZULZXSXIULXPAVDXTFXQAXOXQVEHVFUEULVD + XTXQVGXPHXQXQUGVHVDXTVEHXQVJXTUGZVKVLUBVMZCXIJHXRXTLQYBXJVNVOVRVPAXDDXEGX + NVQAXFXHXGBEAXDDKGXNVQAXGDBGUEZDXRHUIUDZUHUDZUEZCUDZBAXDDBGXNVQAYHDXSGUEZ + YDACYEIVSUEULZDVTULZYAYHYIUPACHJVFUEZULYJACJWAUDZJVFUEZYLAXLJWBULCYNULTJH + LWCCYMJQYMUGWDVLAHYMJVFAHWEULHYMUPAHTWFJHWELWGUNVQWIHJCYEIYEUGZOWHUNADUOU + LDWSULYKUADWJDWTVLYCXTYFGCYEIDXRXTHYEYOYBWKYFUGZPWLWMABXSDGBXSUPARWNWOWPA + YHXSBAYGXRCAXRXTDFYFHSYBYPXRUGUAUBTWQWRRUSXAXBXC $. + $} + ${ ply1fermltl.z $e |- Z = ( Z/nZ ` P ) $. ply1fermltl.w $e |- W = ( Poly1 ` Z ) $. @@ -488313,22 +493334,11 @@ such that every prime ideal contains a prime element (this is a ` ( X + A ) ^ P = ( ( X ^ P ) + A ) ` modulo ` P ` . (Contributed by Thierry Arnoux, 24-Jul-2024.) $) ply1fermltl $p |- ( ph -> ( P .^ ( X .+ A ) ) = ( ( P .^ X ) .+ A ) ) $= - ( cchr cfv co cbs eqid cmg cmgp fveq2i eqtri ccrg wcel cprime prmnn nnnn0 - cn cn0 zncrng 4syl ply1crng syl ply1chr znchr eqtrd eqeltrd crg crngringd - wceq vr1cl czrh cz czring crh wf zrhrhm zringbas rhmf ffvelcdmd ply1sclcl - 3syl syl2anc eqeltrid freshmansdream cmhm csca casa ply1assa asclrhm cgrp - oveq1d crnggrpd ply1sca eleqtrrd rhmmhm mgpbas mhmmulg syl3anc a1i oveq2d - eqtr4d znfermltl fveq2d eqtr4di 3eqtr2d oveq12d 3eqtr3d ) AIUBUCZJBEUDZGU - DXGJGUDZXGBGUDZEUDDXHGUDDJGUDZBEUDAIUEUCZXGEIGJBXLUFZOGHUGUCIUHUCZUGUCQHX - NUGPUIUJXGUFAKUKULZIUKULADUMULZDUPULZDUQULZXOTDUNZDUOZDKLURUSZIKMUTVAAXGD - UMAXGKUBUCZDAXOXGYBVHYAIKMVBVAAXPXQXRYBDVHTXSXTDKLVCUSVDZTVEAKVFULZJXLULA - KYAVGZXLIKJNMXMVIVAABFKVJUCZUCZCUCZXLSAYDYGKUEUCZULZYHXLULYEAVKYIFYFAYDYF - VLKVMUDULVKYIYFVNYEKYFYFUFVOVKYIVLKYFVPYIUFZVQVTUAVRZCXLIKYGYIMRYKXMVSWAW - BWCAXGDXHGYCWJAXIXKXJBEAXGDJGYCWJAXJDBGUDZDYGKUHUCZUGUCZUDZCUCZBAXGDBGYCW - JAYQDYHGUDZYMACYNHWDUDULZXRYJYQYRVHACKIVMUDZULYSACIWEUCZIVMUDZYTAXOIWFULC - UUBULYAIKMWGCUUAIRUUAUFWHVTAKUUAIVMAKWIULKUUAVHAKYAWKIKWIMWLVAWJWMKICYNHY - NUFZPWNVAAXPXQXRTXSXTVTYLYIYOGCYNHDYGYIKYNUUCYKWOYOUFZQWPWQABYHDGBYHVHASW - RWSWTAYQYHBAYPYGCAXPYJYPYGVHTYLYGYIDYOKLYKUUDXAWAXBSXCXDXEXF $. + ( cchr cfv co eqid cprime wcel cn ccrg prmnn nnnn0 zncrng 4syl wceq znchr + cn0 eqeltrd ply1fermltlchr oveq1d 3eqtr3d ) AKUBUCZJBEUDZGUDVAJGUDZBEUDDV + BGUDDJGUDZBEUDABCVAEFGKHIJMNOPQRSVAUEADUFUGZDUHUGZDUPUGZKUIUGTDUJZDUKZDKL + ULUMAVADUFAVEVFVGVADUNTVHVIDKLUOUMZTUQUAURAVADVBGVJUSAVCVDBEAVADJGVJUSUSU + T $. $} @@ -488343,8 +493353,8 @@ such that every prime ideal contains a prime element (this is a sral1r.a $e |- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) $. sral1r.1 $e |- ( ph -> .1. = ( 1r ` W ) ) $. sral1r.s $e |- ( ph -> S C_ ( Base ` W ) ) $. - $( The multiplicative neutral element of a subring algebra. (Contributed - by Thierry Arnoux, 24-Jul-2023.) $) + $( The unity element of a subring algebra. (Contributed by Thierry Arnoux, + 24-Jul-2023.) $) sra1r $p |- ( ph -> .1. = ( 1r ` A ) ) $= ( vx vy cur cfv cbs eqidd srabase cv wcel wa cmulr sramulr oveqdr eqtrd rngidpropd ) ADEKLBKLGAIJEMLZEBAUDNABCEFHOAIPUDQJPUDQRIJESLBSLABCEFHTUAUC @@ -491315,62 +496325,62 @@ such that every prime ideal contains a prime element (this is a impbida 3bitr4g dfiun3g 3eqtrd elrnmpt ssrexv sylc nfmpt1 nfrn nfcri rspa simp-5r sseqtrd ralrimi unissb sylibr exp31 reximdai rnex mptex mpbir2and eqsstrd rnexg isref ffun ad6antlr imafi simp3 cnvex fvmpt 3ad2ant2 fnmpti - wfo dffn4 mpbi rabfodom cbvrabv breqtrdi rabex nfrab1 nfel1 nfrex ad5antr - rabid sylanbrc fveqeq2 uniinn0 adantl r19.29af2 ssdomg mpsyl domtr adantr - ss2rabdv dffn3 ssrab2 fimarab mpan2 breqtrrd domfi imdistanda imp simplll - 3syl simp3d r19.21bi syl3anbrc dmeq rneq 3anbi123d breq1d eleq1d syl32anc - funeq spcev expl exlimdv ) AEBUAULZUMZMULZUYDUYBUEZPZMEUFZQZUAUGZCULZUHZU - YJUIZBPZUYJUJZDPZUKZUYNBUNUOZUYNDUPUEZRZQZQZCUGZABUQZEUQZPZUYIAEBUNUOZVUE - 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$d d w x y z $. $d d x y D $. $d d x y X $. + $d d x y D $. $d d x y X $. $( Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.) $) metidval $p |- ( D e. ( PsMet ` X ) -> ( ~Met ` D ) = { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } ) $= - ( vd vz vw cpsmet cfv wcel cv cdm wa co cc0 wceq copab cvv eleq2d wral wb - crn cuni cmetid cmpt df-metid simpr dmeqd psmetdmdm adantr eqtr4d anbi12d - a1i oveqd eqeq1d opabbidv wrex elfvdm fveq2 mpancom wfun cxad cle wbr cxr - rspcev cxp cmap df-psmet funmpt2 elunirn ax-mp sylibr wss opabssxp elfvex - crab xpexd ssexg sylancr fvmptd ) CDHIZJZECAKZEKZLZLZJZBKZWGJZMZWDWIWENZO - PZMZABQZWDDJZWIDJZMZWDWICNZOPZMZABQZHUBUCZUDRUDEXCWOUEPWCABEUFUMWCWECPZMZ - WNXAABXEWKWRWMWTXEWHWPWJWQXEWGDWDXEWGCLZLZDXEWFXFXEWECWCXDUGZUHUHWCDXGPXD - CDUIUJUKZSXEWGDWIXISULXEWLWSOXEWECWDWIXHUNUOULUPWCCWDHIZJZAHLZUQZCXCJZDXL - JWCXMCDHURXKWCADXLWDDPXJWBCWDDHUSSVFUTHVAXNXMUAARWIWIWENOPWIFKZWENGKZWIWE - NXPXOWENVBNVCVDGWDTFWDTMBWDTEVEWDWDVGVHNVQHABFGEVIVJACHVKVLVMWCXBDDVGZVNX - QRJXBRJWTABDDVOWCDDRRCDHVPZXRVRXBXQRVSVTWA $. + ( vd cpsmet cfv wcel cv cdm wa co cc0 wceq copab crn dmeqd eleq2d anbi12d + cvv cuni cmetid df-metid simpr psmetdmdm adantr eqtr4d opabbidv elfvunirn + oveqd eqeq1d cxp wss opabssxp elfvex xpexd ssexg sylancr fvmptd2 ) CDFGHZ + ECAIZEIZJZJZHZBIZVDHZKZVAVFVBLZMNZKZABOVADHZVFDHZKZVAVFCLZMNZKZABOZFPUAUB + TABEUCUTVBCNZKZVKVQABVTVHVNVJVPVTVEVLVGVMVTVDDVAVTVDCJZJZDVTVCWAVTVBCUTVS + UDZQQUTDWBNVSCDUEUFUGZRVTVDDVFWDRSVTVIVOMVTVBCVAVFWCUJUKSUHDCFUIUTVRDDULZ + UMWETHVRTHVPABDDUNUTDDTTCDFUOZWFUPVRWETUQURUS $. $( As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.) $) @@ -492425,21 +497431,17 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). pstmval $p |- ( D e. ( PsMet ` X ) -> ( pstoMet ` D ) = ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x D y ) } ) ) $= - ( vd vc cpsmet cfv wcel cv cdm co wceq wrex cvv cmetid cqs cab cuni cpstm - cmpo crn cmpt df-pstm a1i psmetdmdm adantr dmeq dmeqd adantl eqtr4d qseq1 - wa syl fveq2 eqtr4id qseq2 eqtr2d mpoeq12 syl2anc w3a simp1r oveqd eqeq2d - 2rexbidv abbidv unieqd mpoeq3dva elfvdm eleq2d rspcev mpancom wfun wb cc0 - eqtrd cxad cle wbr wral cxr cxp cmap crab df-psmet funmpt2 elunirn sylibr - ax-mp elfvex qsexg mpoexga fvmptd ) DFLMZNZJDGHJOZPZPZXAUAMZUBZXECOZAOZBO - ZXAQZRZBHOZSAGOZSZCUCZUDZUFZGHFEUBZXQXFXGXHDQZRZBXKSAXLSZCUCZUDZUFZLUGUDZ - UETUEJYDXPUHRWTABCGHJUIUJWTXADRZURZXPGHXQXQXOUFZYCYFXEXQRZYHXPYGRYFXQXCEU - BZXEYFFXCRXQYIRYFFDPZPZXCWTFYKRYEDFUKULYEXCYKRWTYEXBYJXADUMUNUOUPFXCEUQUS - YEYIXERZWTYEEXDRYLYEEDUAMXDIXADUAUTVAEXDXCVBUSUOVCZYMGHXEXEXQXQXOVDVEYFGH - XQXQXOYBYFXLXQNZXKXQNZVFZXNYAYPXMXTCYPXJXSABXLXKYPXIXRXFYPXADXGXHWTYEYNYO - VGVHVIVJVKVLVMWAWTDXGLMZNZALPZSZDYDNZFYSNWTYTDFLVNYRWTAFYSXGFRYQWSDXGFLUT - VOVPVQLVRUUAYTVSATXLXLXAQVTRXLXKXAQKOZXLXAQUUBXKXAQWBQWCWDKXGWEHXGWEURGXG - WEJWFXGXGWGWHQWILAGHKJWJWKADLWLWNWMWTXQTNZUUCYCTNWTFTNUUCDFLWOFETWPUSZUUD - GHXQXQYBTTWQVEWR $. + ( vd cpsmet cfv wcel cv cdm cmetid cqs wceq wrex cvv co cab cuni cmpo crn + cpstm df-pstm psmetdmdm adantr dmeq dmeqd adantl eqtr4d qseq1 syl eqtr4id + fveq2 qseq2d eqtr2d mpoeq12 syl2anc w3a simp1r oveqd eqeq2d abbidv unieqd + wa 2rexbidv mpoeq3dva eqtrd elfvunirn elfvex qsexg mpoexga fvmptd2 ) DFKL + MZJDGHJNZOZOZVRPLZQZWBCNZANZBNZVRUAZRZBHNZSAGNZSZCUBZUCZUDZGHFEQZWNWCWDWE + DUAZRZBWHSAWISZCUBZUCZUDZKUEUCUFTABCGHJUGVQVRDRZVHZWMGHWNWNWLUDZWTXBWBWNR + ZXDWMXCRXBWNVTEQZWBXBFVTRWNXERXBFDOZOZVTVQFXGRXADFUHUIXAVTXGRVQXAVSXFVRDU + JUKULUMFVTEUNUOXAXEWBRVQXAEWAVTXAEDPLWAIVRDPUQUPURULUSZXHGHWBWBWNWNWLUTVA + XBGHWNWNWLWSXBWIWNMZWHWNMZVBZWKWRXKWJWQCXKWGWPABWIWHXKWFWOWCXKVRDWDWEVQXA + XIXJVCVDVEVIVFVGVJVKFDKVLVQWNTMZXLWTTMVQFTMXLDFKVMFETVNUOZXMGHWNWNWSTTVOV + AVP $. $d a b e f z .~ $. $d a b e f x y z A $. $d a b e f x y z B $. $d e f z D $. $d e f z X $. @@ -492756,22 +497758,22 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). syl oveq12d oveq1d mptru df1stres 3eqtr4ri wral wfn rgen2 fnmpo ax-mp wfo fmpoco fo1st fofn xp1st crn wrex cab rnmpo adantr eqeltrd rexlimivv abssi eqsstri sselid resubd cnre2csqlem imval crim df-im fvoveq1 df2ndres fo2nd - 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WCSWAHXQVOUVGUUOVCPVFXRWSWJABOOXSWLUVNTTMWRMTWNXTTTMXAWQUVGOOYBUVSUVGUVQU - WCUWDYCXNYDYEYE $. + ex xp2nd imsubd anim12d biimtrid ) FEIJZCUBKZYFCLKZUCKZEMJZCUBKZYJCLKZUCK + ZUEZNZFIOOUEZUDZUFYIUGZMYPUDZUFYMUGZUHZNZEYPNFYPNCUINUJZFDJEDJUBKZPJUKJCU + LUMZUUDQJUKJCULUMZUNZYIOUOZYMOUOZYOUUBUPYGYHUTYKYLUTUUHUUIUNYNUUAFYIYMOOU + QURUSUUBFYRNZFYTNZUNUUCUUGFYRYTVAUUCUUJUUEUUKUUFHUACDIPEFABOOAVBZVCBVBZVD + KZLKZUUOVNJZLKZVEVFKZVGZABOOUULVGPDVHZYQABOOUURUULUULONZUUMONZUNZUUOPJZUU + RUULUVCUUORNZUVDUURSUVCUULUUNUVCUULUVAUVBVIVJUVCVCUUMVCRNUVCVKVOUVCUUMUVA + UVBVLVJVMVPZUUOVQWGUULUUMVRVSVTUUTUUSSWAABHOORUUOHVBZUVGVNJZLKZVEVFKZUURD + PUVCUVEWAUVFWBZDABOOUUOVGSWAGVOZPHRUVJWCSWAHWDVOUVGUUOSZUVIUUQVEVFUVMUVGU + UOUVHUUPLUVMWEUVGUUOVNWFWHWIWSWJABOOWKWLUVEBOWMAOWMDYPWNUVEABOOUVFWOABOOU + UODRGWPWQZTTIWRITWNWTTTIXAWQUVGOOXBUVGDXCZNZUAVBZUVONZUNZUVGUVQUVSUVORUVG + UVOUVMBOXDAOXDZHXERABHOOUUODGXFUVTHRUVMUVGRNZABOOUVCUVMUWAUVCUVMUNUVGUUOR + UVCUVMVLUVCUVEUVMUVFXGXHYAXIXJXKZUVPUVRVIXLZUVSUVORUVQUWBUVPUVRVLXLZXMXNH + UACDMQEFABOOUUOVCVFKPJZVGZABOOUUMVGQDVHZYSABOOUWEUUMUVCUUOQJZUWEUUMUVCUVE + UWHUWESUVFUUOXOWGUULUUMXPVSVTUWGUWFSWAABHOORUUOUVGVCVFKPJZUWEDQUVKUVLQHRU + WIWCSWAHXQVOUVGUUOVCPVFXRWSWJABOOXSWLUVNTTMWRMTWNXTTTMXAWQUVGOOYBUVSUVGUV + QUWCUWDYCXNYDYEYE $. $} ${ @@ -493020,27 +498022,27 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). cbvrexvw bitr3i c0 ctop ordttop 0opn syl5ibrcom wne rabn0 notbid bitri wo ad3antrrr ad2antrr sselda simpllr trleile syl3anc wi rabss r19.21bi an32s ord impr sylan2b brinxp ancoms rabbidva eleqtrrd ordtopn1 eqeltrd anassrs - an4 eqtr4d expr syld rexlimdva pm2.61dne rexlimdvaa pm2.61d ralrimiva cbs - syl5bi rabexg ralrimivw ineq1 eleq1d ralrnmptw mpbird ) AFUAZGUBZJGGUCZUB - ZUDUEZPZFDHEUAZDUAZJUFZULZEHUGZUHZUIUJZUUFGUBZYTPZDHUJZAUUJDHAUUCHPZQZUUI - UUEEGUGZYTUUMUUIUUEEHGUBZUGZUUNUUEEHGUKUUMUUOGRZUUPUUNRAUUQUULAGHUMZUUQNG - HUNUOUPUUEEUUOGUQSURUUMUUEEGUJZUUNYTPZUUMGYTPZUUSUUTAUVAUULAYTGUSUEZPUVAA - YTYSUTZUSUEZUVBAYSTPZYTUVDPUVEAJYRJIVAUEZHHUCZUBTLUVFUVGIVAVBVPVCVPZVDZYS - TUVCUVCVEZVFSAUVCGUSAIVGPZUURUVCGRZAIVHPZIVIPUVKMIVJIVKVLNGHIJKLVMVNZVOVQ - GYTVRSUPUUSGUUNRUVAUUTVSUUEEGVTGUUNYTWAWBWCUUSULZBUAZUUCJUFZBGWDZUUMUUTUV - OUUDEGWDUVRUUDEGWEUUDUVQEBGUUBUVPUUCJWFWGWHUUMUVQUUTBGUUMUVPGPZUVQQZQZUUT - UUNWIUWAUUTUUNWIRWIYTPZUUMUWBUVTUUMYTWJPZUWBAUWCUULAUVEUWCUVIYSTWKSUPYTWL - SUPUUNWIYTWAWMUUNWIWNZCUAZUUCJUFZULZCGWDZUWAUUTUWDUUEEGWDUWHUUEEGWOUUEUWG - ECGUUBUWERUUDUWFUUBUWEUUCJWFWPWGWQUWAUWGUUTCGUWAUWEGPZQZUWGUUCUWEJUFZUUTU - WJUWFUWKUWJUVMUWEHPUULUWFUWKWRAUVMUULUVTUWIMWSUWAGHUWEAUURUULUVTNWTXAAUUL - UVTUWIXBHIJUWEUUCKLXCXDXIUWAUWIUWKUUTUUMUVTUWIUWKQZUUTUUMUVTUWLQZQZUUNUUB - 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(Contributed by Thierry @@ -493791,31 +498793,31 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). elioc1 biimpa syl2anc ex ralimdva reximdva fveq2 raleqdv cbvrexvw syl6ibr simp2d imim12d rspcimdv imp mpd ralrimdva simplll pnfneige0 simplr sseldd simpllr r19.29r simp-4l uznnssnn jca ffvelcdmda eqeltrrd ad2antrr biimpar - sselid pnfge syl13anc adantlr syl21anc syl5bi expimpd syl5 syl12anc exp31 - rexlimdva impbid bitrd ) AFLGUBMNZLUAUCZOZCUVCOZEKUCZUDMZUEZKPUFZUGZUAGUE - ZBUCZCUHNZEDUCZUDMZUEZDPUFZBQUEZAUVBLRLUJUIZOZUVKSUVKAUACLKEFGVNUVSPGUVSU - KMZOAGULUMMZUVSUNUIZUWAGUOUVSUPUIZUQMUWCHUVSUWDUWBUOUWDURUSUTVAZUWBTUKMOU - VSTVBUWCUWAOVCRLVDZUVSUWBTVEVFVGVHVIAVJIJVKUVTUVKRTOZLTOZRLULNUVTVLVMVORL - VPVQVRVSAUVKUVRAUVKUVQBQAUVLQOZSZUVKUVQUWJUVKSLUVLLVTUIZRLVTUIZWAZOZUVQUW - IUWNAUVKUWILUWKOZLUWLOZSUWNUWIUWOUWPUWIUVLTOZUWHUVLLUHNUWOUVLWBUWHUWIVMVH - UVLWCUVLLWDWFUWGUWHRLUHNUWPVLVMWERLWDVQWGLUWKUWLWHWIWJUWJUVKUWNUVQUGZUWJU - VJUWRUAUWMGUWMGOUWJUWMUVSWAZUWCUWMGUWBWKOZUVSWLOUWMUWBOZUWSUWCOWMRLUJWNUW - TUWKUWBOUWLUWBOUXAWMUVLWORWOUWKUWLUWBWQVQUWMUVSUWBWKWLWPVQUVSUWMWAZUWMUWS - UWMUVSVBUXBUWMWRUWMUWLUVSUWKUWLWSRLWTXAUWMUVSXBXHUVSUWMXCXDUWEXEVHUWJUVCU - WMWRZSZUWNUVDUVIUVQUXDUVDUWNUXCUVDUWNXIUWJUVCUWMLXFXGXJUXDUVIUVMEUVGUEZKP - UFZUVQUXDUVHUXEKPUXDUVFPOZSZUVEUVMEUVGUXHEUCZUVGOZSZUVEUVMUXKUVESZUWQCUWK - OZUVMUXLUVLAUWIUXCUXGUXJUVEXKXLUXLCUWMOZUXMUXLCUVCUWMUXKUVEXMUWJUXCUXGUXJ - UVEXNXOUXNUXMCUWLOCUWKUWLWHXPXQUWQUXMSCTOZUVMCLULNZUWQUXMUXOUVMUXPXRZUWQU - WHUXMUXQXIVMUVLLCXTXSZYAYJYBYCYDYEUVPUXEDKPUVNUVFWRUVMEUVOUVGUVNUVFUDYFYG - YHZYIYKYLYMYNYCYOAUVRUVJUAGAUVCGOZSZUVRUVDUVIUYAUVRSZUVDSZAUWKUVCVBZBQUFZ - UVRUVIAUXTUVRUVDYPUYCUXTUVDUYEAUXTUVRUVDYTUYBUVDXMBUVCGHYQYBUYAUVRUVDYRAU - YEUVRSZUVIUYFUYDUVQSZBQUFAUVIUYDUVQBQUUAAUYGUVIBQUWJUYDUVQUVIUVQUXFUWJUYD - SZUVIUXSUYHUXEUVHKPUYHUXGSZUVMUVEEUVGUYIUXJSZAUXIPOZSZUWIUYDUVMUVEUGUYJAU - YKAUWIUYDUXGUXJUUBUYJUVGPUXIUXGUVGPVBUYHUXJUVFUUCWJUYIUXJXMYSUUDAUWIUYDUX - GUXJXNUWJUYDUXGUXJYTUYLUWISZUYDSZUVMUVEUYNUVMSUWKUVCCUYMUYDUVMYRUYMUVMUXM - UYDUYMUVMSZUWQUXOUVMUXPUXMUYOUVLUYLUWIUVMYRXLUYLUXOUWIUVMUYLUVSTCUWFUYLUX - IFMCUVSJAPUVSUXIFIUUEUUFUUIUUGZUYMUVMXMUYOUXOUXPUYPCUUJXQUWQUXMUXQUXRUUHU - UKUULYSYCUUMYDYEUUNUUOUUSUUPYMUUQUURYOUUTUVA $. + sselid syl13anc adantlr syl21anc biimtrid expimpd rexlimdva syl5 syl12anc + pnfge exp31 impbid bitrd ) AFLGUBMNZLUAUCZOZCUVCOZEKUCZUDMZUEZKPUFZUGZUAG + UEZBUCZCUHNZEDUCZUDMZUEZDPUFZBQUEZAUVBLRLUJUIZOZUVKSUVKAUACLKEFGVNUVSPGUV + SUKMZOAGULUMMZUVSUNUIZUWAGUOUVSUPUIZUQMUWCHUVSUWDUWBUOUWDURUSUTVAZUWBTUKM + OUVSTVBUWCUWAOVCRLVDZUVSUWBTVEVFVGVHVIAVJIJVKUVTUVKRTOZLTOZRLULNUVTVLVMVO + RLVPVQVRVSAUVKUVRAUVKUVQBQAUVLQOZSZUVKUVQUWJUVKSLUVLLVTUIZRLVTUIZWAZOZUVQ + UWIUWNAUVKUWILUWKOZLUWLOZSUWNUWIUWOUWPUWIUVLTOZUWHUVLLUHNUWOUVLWBUWHUWIVM + VHUVLWCUVLLWDWFUWGUWHRLUHNUWPVLVMWERLWDVQWGLUWKUWLWHWIWJUWJUVKUWNUVQUGZUW + JUVJUWRUAUWMGUWMGOUWJUWMUVSWAZUWCUWMGUWBWKOZUVSWLOUWMUWBOZUWSUWCOWMRLUJWN + UWTUWKUWBOUWLUWBOUXAWMUVLWORWOUWKUWLUWBWQVQUWMUVSUWBWKWLWPVQUVSUWMWAZUWMU + WSUWMUVSVBUXBUWMWRUWMUWLUVSUWKUWLWSRLWTXAUWMUVSXBXHUVSUWMXCXDUWEXEVHUWJUV + CUWMWRZSZUWNUVDUVIUVQUXDUVDUWNUXCUVDUWNXIUWJUVCUWMLXFXGXJUXDUVIUVMEUVGUEZ + KPUFZUVQUXDUVHUXEKPUXDUVFPOZSZUVEUVMEUVGUXHEUCZUVGOZSZUVEUVMUXKUVESZUWQCU + WKOZUVMUXLUVLAUWIUXCUXGUXJUVEXKXLUXLCUWMOZUXMUXLCUVCUWMUXKUVEXMUWJUXCUXGU + XJUVEXNXOUXNUXMCUWLOCUWKUWLWHXPXQUWQUXMSCTOZUVMCLULNZUWQUXMUXOUVMUXPXRZUW + QUWHUXMUXQXIVMUVLLCXTXSZYAYJYBYCYDYEUVPUXEDKPUVNUVFWRUVMEUVOUVGUVNUVFUDYF + YGYHZYIYKYLYMYNYCYOAUVRUVJUAGAUVCGOZSZUVRUVDUVIUYAUVRSZUVDSZAUWKUVCVBZBQU + FZUVRUVIAUXTUVRUVDYPUYCUXTUVDUYEAUXTUVRUVDYTUYBUVDXMBUVCGHYQYBUYAUVRUVDYR + AUYEUVRSZUVIUYFUYDUVQSZBQUFAUVIUYDUVQBQUUAAUYGUVIBQUWJUYDUVQUVIUVQUXFUWJU + YDSZUVIUXSUYHUXEUVHKPUYHUXGSZUVMUVEEUVGUYIUXJSZAUXIPOZSZUWIUYDUVMUVEUGUYJ + AUYKAUWIUYDUXGUXJUUBUYJUVGPUXIUXGUVGPVBUYHUXJUVFUUCWJUYIUXJXMYSUUDAUWIUYD + UXGUXJXNUWJUYDUXGUXJYTUYLUWISZUYDSZUVMUVEUYNUVMSUWKUVCCUYMUYDUVMYRUYMUVMU + XMUYDUYMUVMSZUWQUXOUVMUXPUXMUYOUVLUYLUWIUVMYRXLUYLUXOUWIUVMUYLUVSTCUWFUYL + UXIFMCUVSJAPUVSUXIFIUUEUUFUUIUUGZUYMUVMXMUYOUXOUXPUYPCUURXQUWQUXMUXQUXRUU + HUUJUUKYSYCUULYDYEUUMUUNUUOUUPYMUUQUUSYOUUTUVA $. $} ${ @@ -494006,8 +499008,8 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). ${ $d x y G $. $d x y W $. zlm1.1 $e |- .1. = ( 1r ` G ) $. - $( Unit of a ` ZZ ` -module (if present). (Contributed by Thierry - Arnoux, 8-Nov-2017.) $) + $( Unity element of a ` ZZ ` -module (if present). (Contributed by + Thierry Arnoux, 8-Nov-2017.) $) zlm1 $p |- .1. = ( 1r ` W ) $= ( vx vy cur cfv wceq wtru cbs eqid a1i zlmbas cv wcel wa cmulr zlmmulr oveqd rngidpropd mptru eqtri ) ABHIZCHIZEUEUFJKFGBLIZBCUGUGJKUGMZNUGCLI @@ -494235,7 +499237,7 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). ( crg wcel cchr cfv cc0 wceq cz wf1 ccnv csn cima zrhchr zrhf1ker bitrd ) BHIBJKLMNACOCPDQRLQMABCDEFGSABCDEFGTUA $. - $( The preimage by ` ZRHom ` of the unit of a division ring is + $( The preimage by ` ZRHom ` of the units of a division ring is ` ( ZZ \ { 0 } ) ` . (Contributed by Thierry Arnoux, 22-Oct-2017.) $) zrhunitpreima $p |- ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( `' L " ( Unit ` R ) ) = ( ZZ \ { 0 } ) ) $= @@ -494383,7 +499385,7 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). WTXLKCVMZXMWRWOWSXKBCDWOGWOSZVNCXKDGXKSZVOVPTWFXMWOAHZXLWOKXNWFCVQHXQCVRA CWOEXOVSTABCXKWOEFXPVTWAWCWDWEWC $. - $( The image of ` 1 ` by the ` QQHom ` homomorphism is the ring's unit. + $( The image of ` 1 ` by the ` QQHom ` homomorphism is the ring unity. (Contributed by Thierry Arnoux, 22-Oct-2017.) $) qqh1 $p |- ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( ( QQHom ` R ) ` 1 ) = ( 1r ` R ) ) $= @@ -496928,63 +501930,63 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). 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It would be eqid syl21anc ovex elrnmpti breq2d rspcedvd 3anassrs ex jca breq1d notbid nfbr ralbidv elpwi sselda adantrr eqidd esumval biimpi sstrd iunss sylibr cdm mpbir oveq2d ralrimi adantllr cima iccssxr ccmn vex nfiu1 cop simp-4r - xrltso elsnxp adantll eliun rnmptss xpexg sylancr iunexg elrnmpt1 nfesum1 - snex nfrex ad2antrr 3ad2antr3 esumlub imbi1d anbi12d rspcedv supcl simprr - mpd esum2dlem anassrs eqtr3d iunss1 wb mpbid mtbid mpbird nfel1 nfsup w3a - simpr1l simpr1r breqtrd dmss dmiun sseqtrdi dmxpss snssi rgen sstrdi dmex - 3syl dmfi elind mpteq1 elrnmpt1s nfov simp-4l nfpw nfin wrel rgenw df-rel - xpss gsummpt2d imass1 iunsnima sseqtrd imaexg ax-mp mpan esumlef mpteq2da - imafi2 eqtrd 3brtr3d xrltletrd ad2antlr suplub imp breq2 cbvrexvw r19.29a - id eqsupd 3eqtr4d ) AUACUFZUGUHZUIUJUKUOULZUMULZFUAUNZDEGUPZUQZURULZUQZUS - ZQRUTPFCFUNZVAZDVBZVCZUFZUGUHZUYHBPUNZHUQZURULZUQZUSZQRUTZCUYJFUPUYRHBUPZ - AUBUCQUYNVUFRQRVDAUUGVEZAUDUBUCQVUERVUHAVUGUBUNZRVFZVKZUBVUEVGZVUIVUGRVFZ - VUIUCUNZRVFZUCVUEVHZVLZUBQVGZSZUDUNZVUIRVFZVKZUBVUEVGZVUIVUTRVFZVUPVLZUBQ - VGZSZUDQVHAVULVURAVUKUBVUEAVUIVUETZSZVUIVUCVIZVUKPUYTAVVHPAPVJZPVUIVUEPVU - IVMZPVUDPUYTVUCVNVOZVPVQVVIVUAUYTTZSZVVJSZVUIQTZVUGQTZVUIVUGVRVFZVUKVVPVU - EQVUIAVUEQVSZVVHVVNVVJAVUCQTZPUYTVGVVTAVWAPUYTAVVNSZUYGQVUCUJUKUUAZVWBUYG - BUYHVUAHVTUYHUUBTZVWBWAVEZVWBUYSUGVUAAVVNWDZWBZVWBHUYGTZBVUAVWBBUNZVUATZS - ZAVWIUYRTZVWHAVVNVWJWCVWKVUAUYRVWIVWKVUAUYSTZVUAUYRVSZVWBVWMVWJVWBUYSUGVU - AVWFWEZWFVUAUYRPUUCZWNZWGVWBVWJWDWHAVWLSZVWIUYQTZVWHFCAVWLFAFVJZFVWIUYRFV - WIVMFCUYQUUDZVPVQVWRUYOCTZSVWSSZVWIUYOGUNZUUEVIZVWHGDVXCGVJGHUYGKGUYGVMVP - VXCVXDDTZSZVXESZHEUYGVXEHEVIVXGLWIVXHAVXBVXFEUYGTZAVWLVXBVWSVXFVXEWJVWRVX - BVWSVXFVXEUUFVXCVXFVXEWKOWLWOVXBVWSVXEGDVHZVWRVXBVWSVXJGDCUYOVWIUUHWPUUIW - MVWRVWLVWSFCVHAVWLWDFVWICUYQUUJWGWQZWRZWSWTXAWSPUYTVUCQVUDVUDXMZUUKXCXBAV - VHVVNVVJXDWHAVVRVVHVVNVVJAUYGQVUGVWCAUYRXETZVWHBUYRVGVUGUYGTACITUYQXETZFC - VGVXNMAVXOFCAVXBSZUYPXETDJTZVXOUYOUUQNUYPDXEJUULUUMWSFCUYQIXEUUNWRZAVWHBU + xrltso elsnxp adantll eliun rnmptss vsnex sylancr iunexg elrnmpt1 nfesum1 + xpexg nfrexw ad2antrr 3ad2antr3 esumlub imbi1d anbi12d rspcedv mpd simprr + supcl esum2dlem anassrs eqtr3d iunss1 wb mpbid mtbid mpbird nfel1 simpr1l + nfsup w3a simpr1r breqtrd dmss dmiun sseqtrdi dmxpss snssi rgen dmex 3syl + sstrdi dmfi elind mpteq1 elrnmpt1s nfov simp-4l nfpw nfin wrel xpss rgenw + df-rel gsummpt2d imass1 iunsnima sseqtrd imaexg ax-mp mpan esumlef imafi2 + id mpteq2da eqtrd 3brtr3d xrltletrd ad2antlr suplub breq2 cbvrexvw eqsupd + imp r19.29a 3eqtr4d ) AUACUFZUGUHZUIUJUKUOULZUMULZFUAUNZDEGUPZUQZURULZUQZ + USZQRUTPFCFUNZVAZDVBZVCZUFZUGUHZUYHBPUNZHUQZURULZUQZUSZQRUTZCUYJFUPUYRHBU + PZAUBUCQUYNVUFRQRVDAUUGVEZAUDUBUCQVUERVUHAVUGUBUNZRVFZVKZUBVUEVGZVUIVUGRV + FZVUIUCUNZRVFZUCVUEVHZVLZUBQVGZSZUDUNZVUIRVFZVKZUBVUEVGZVUIVUTRVFZVUPVLZU + BQVGZSZUDQVHAVULVURAVUKUBVUEAVUIVUETZSZVUIVUCVIZVUKPUYTAVVHPAPVJZPVUIVUEP + VUIVMZPVUDPUYTVUCVNVOZVPVQVVIVUAUYTTZSZVVJSZVUIQTZVUGQTZVUIVUGVRVFZVUKVVP + VUEQVUIAVUEQVSZVVHVVNVVJAVUCQTZPUYTVGVVTAVWAPUYTAVVNSZUYGQVUCUJUKUUAZVWBU + YGBUYHVUAHVTUYHUUBTZVWBWAVEZVWBUYSUGVUAAVVNWDZWBZVWBHUYGTZBVUAVWBBUNZVUAT + ZSZAVWIUYRTZVWHAVVNVWJWCVWKVUAUYRVWIVWKVUAUYSTZVUAUYRVSZVWBVWMVWJVWBUYSUG + VUAVWFWEZWFVUAUYRPUUCZWNZWGVWBVWJWDWHAVWLSZVWIUYQTZVWHFCAVWLFAFVJZFVWIUYR + FVWIVMFCUYQUUDZVPVQVWRUYOCTZSVWSSZVWIUYOGUNZUUEVIZVWHGDVXCGVJGHUYGKGUYGVM + VPVXCVXDDTZSZVXESZHEUYGVXEHEVIVXGLWIVXHAVXBVXFEUYGTZAVWLVXBVWSVXFVXEWJVWR + VXBVWSVXFVXEUUFVXCVXFVXEWKOWLWOVXBVWSVXEGDVHZVWRVXBVWSVXJGDCUYOVWIUUHWPUU + IWMVWRVWLVWSFCVHAVWLWDFVWICUYQUUJWGWQZWRZWSWTXAWSPUYTVUCQVUDVUDXMZUUKXCXB + AVVHVVNVVJXDWHAVVRVVHVVNVVJAUYGQVUGVWCAUYRXETZVWHBUYRVGVUGUYGTACITUYQXETZ + FCVGVXNMAVXOFCAVXBSZUYPXETDJTZVXOFUULNUYPDXEJUUQUUMWSFCUYQIXEUUNWRZAVWHBU YRVXKWSUYRHBXEBUYRVMZXFWRXAZXBVVPVUIVUCVUGVRVVOVVJWDVVOVUCVUGVRVFZVVJAVVN VYAVVHVWBVUAHBUPZVUCVUGVRVWBVUAHBVWBBVJZBVUAVMVWGVXLXGZVWBVUAHUYRBXEVYCAV XNVVNVXRWFAVWLVWHVVNVXKXHVWBVWMVWNVWOVWQWGZXIXJXHWFXKVVQVVRSVVSVUKVUIVUGX @@ -497036,7 +502038,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be MZBRVMZUYRHBVXSUUPYDVQAVXNVVQVUMVXRUUSAVVQVUMVWLVWHAVVQVWLVWHVUMVXKUUTXSA VVQVUMWKVYGVUMWDUVAWMXTWSYAAVVGVUSUDVUGQVXTAVUTVUGVIZSZVVCVULVVFVURVYSVVB VUKUBVUEVYSVVAVUJVYSVUTVUGVUIRAVYRWDZYBYCYEVYSVVEVUQUBQVYSVVDVUMVUPVYSVUT - VUGVUIRVYTXQUVBYEUVCUVDUVGZUVEAVUIUYNTZSZVUIUYLVIZVUFVUIRVFZVKZUAUYFAWUBU + VUGVUIRVYTXQUVBYEUVCUVDUVEZUVGAVUIUYNTZSZVUIUYLVIZVUFVUIRVFZVKZUAUYFAWUBU AAUAVJUAVUIUYNUAVUIVMUAUYMUAUYFUYLVNVOVPVQWUCUYIUYFTZSZWUDSZWUFVUFUYLRVFZ VKZWUHWUKWUDAWUGWUKWUBAWUGSZVUGUYLRVFZWUJWULUYLVUGVRVFZWUMVKZWULFUYIUYQVC ZHBUPZUYLVUGVRWULUYIUYJFUPWUQUYLWULBUYIDEFGHUYFJKLAWUGWDZWULUYOUYITZSZAVX @@ -497050,15 +502052,15 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be CYIYJZWFYBUVNXHWFWUIWUEWUJWUIVUIUYLVUFRWUHWUDWDXQYCUVOWUBWUDUAUYFVHZAWUBW VJUAUYFUYLVUIUYMUYMXMZUYHUYKURXOXPYKWIWQAVVQVUIVUFRVFZSZSZVUIUEUNZRVFZVUO UCUYNVHZUEVUEWVNWVOVUETZSZWVPSZWVOVUCVIZWVQPUYTWVSWVPPWVNWVRPAWVMPVVKVVQW - VLPPVUIQVVLUVPPVUIVUFRVVLPRVMZPVUEQRVVMPQVMWWBUVQYDVQVQPWVOVUEPWVOVMVVMVP + VLPPVUIQVVLUVPPVUIVUFRVVLPRVMZPVUEQRVVMPQVMWWBUVRYDVQVQPWVOVUEPWVOVMVVMVP VQWVPPVJVQWVTVVNSZWWASZVYGWVLSVUIVUCRVFZVVNWVQWWDVYGWVLWWDAVVQAWVMWVRWVPV - VNWWAWJWVSWVPVVNWWAVVQAWVMWVRWVPVVNWWAUVRZVVQVVQWVLWVRWWFAUVSXSXSYAWVSWVP + VNWWAWJWVSWVPVVNWWAVVQAWVMWVRWVPVVNWWAUVSZVVQVVQWVLWVRWWFAUVQXSXSYAWVSWVP VVNWWAWVLAWVMWVRWWFWVLVVQWVLWVRWWFAUVTXSXSYAWWDVUIWVOVUCRWVSWVPVVNWWAXDWW CWWAWDUWAWVTVVNWWAWKVYGWWEVVNWVQWVLVYGWWESZVVNSZVUOVUIUYHFVUAYOZUYJUQZURU LZRVFUCWWKUYNWWHWWIUYFTWWKXETZWWKUYNTWWHUYEUGWWIWWHVWMVWNWWIUYETZWWHUYSUG VUAWWGVVNWDZWEZVUAUYRYFZVWNWWICVSZWWMVWNWWIFCUYQYOZVCZCVWNWWIUYRYOWWSVUAU YRUWBFCUYQUWCUWDWWSCVSWWRCVSZFCVGWWTFCVXBWWRUYPCWWRUYPVSVXBUYPDUWEVEUYOCU - WFYLUWGFCWWRCYMYPUWHZWWICVUAVWPUWIZWNYNUWJWWHVUAUGTZWWIUGTZWWHUYSUGVUAWWN + WFYLUWGFCWWRCYMYPUWJZWWICVUAVWPUWHZWNYNUWIWWHVUAUGTZWWIUGTZWWHUYSUGVUAWWN WBZVUAUWKZXCZUWLWWLWWHUYHWWJURXOVEUAUYFUYLWWKWWIUYMXEWVKUYIWWIVIUYKWWJUYH URFUYIWWIUYJUWMYQUWNWRWWHVUNWWKVIZSVUNWWKVUIRWWHWXHWDXQWWHVUIVUCWWKAVVQWW EVVNXDWWHUYGQVUCVWCWWHUYGBUYHVUAHVTVWDWWHWAVEZWXEWWHVWHBVUAWWGVVNBVYGWWEB @@ -497069,8 +502071,8 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be SZAVXBWVFAVVNWXKWCZVWBWWICUYOVWBVWNWWQVYEWXAXCYGZWVGWRZYSYSXTYRWTXAVYGWWE VVNWKVYGVVNVUCWWKVRVFZWWEAVVNWXPVVQVWBVUCUYHFWWIUYHGVUAUYPYTZEUQURULZUQZU RULZWWKVRVWBBFGVUAUYGHEUYHKAVVNFVWTWXJVQZVTLVWBVUAXEXEVBZVSZVUAUWSVWBVWNW - YCVYEVWNVUAUYRWYBVWNUYBUYRWYBVSZVWNWYDUYQWYBVSZFCVGWYEFCUYPDUXBUWTFCUYQWY - BYMYPVEYLXCVUAUXAYNVWGVWEVXLUXCVWBWWIWXQEGUPZFUPZWWIUYJFUPWXTWWKVRVWBWWIW + YCVYEVWNVUAUYRWYBVWNUXLUYRWYBVSZVWNWYDUYQWYBVSZFCVGWYEFCUYPDUWTUXAFCUYQWY + BYMYPVEYLXCVUAUXBYNVWGVWEVXLUXCVWBWWIWXQEGUPZFUPZWWIUYJFUPWXTWWKVRVWBWWIW YFUYJFXEWYAFWWIVMZWWIXETVWBWXBVEWXLVXIGWXQVGZWYFUYGTZWXLVXIGWXQWXLVXDWXQT ZSAVXBVXFVXIWXLAWYKWXMWFWXLVXBWYKWXNWFWXLWXQDVXDWXLWXQUYRUYPYTZDWXLVWNWXQ WYLVSVWBVWNWXKVYEWFVUAUYRUYPUXDXCWXLAVXBWYLDVIWXMWXNAFCDIJMNUXEWRUXFZYGOW @@ -497078,9 +502080,9 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be QEDGJWXLGVJZWXLAVXBVXQWXMWXNNWRWXLVXFSAVXBVXFVXIWXLAVXFWXMWFWXLVXBVXFWXNW FWXLVXFWDOWLWYMXIUXJVWBWYGUYHFWWIWYFUQZURULWXTVWBWWIWYFFWYAWYHVWBWXCWXDVW GWXFXCZWYQXGVWBWYSWXSUYHURVWBFWWIWYFWXRWYAWXLWXQEGWYRWYPWXLWXCWXQUGTVWBWX - CWXKVWGWFVUAUYPUXLXCWYNXGUXKYQUXMVWBWWIUYJFWYAWYHWYTWXOXGUXNXKXHXHUXOXRYS - XNWVRWWAPUYTVHZWVNWVPWVRXUAPUYTVUCWVOVUDVXMVYFXPYKUXPWQWVNVUPWVPUEVUEVHAW - VMVUPAUDUBUCQVUEVUIRVUHWUAUXQUXRVUOWVPUCUEVUEVUNWVOVUIRUXSUXTWGUYAUYCAUAC + CWXKVWGWFVUAUYPUXKXCWYNXGUXMYQUXNVWBWWIUYJFWYAWYHWYTWXOXGUXOXKXHXHUXPXRYS + XNWVRWWAPUYTVHZWVNWVPWVRXUAPUYTVUCWVOVUDVXMVYFXPYKUXQWQWVNVUPWVPUEVUEVHAW + VMVUPAUDUBUCQVUEVUIRVUHWUAUXRUYBVUOWVPUCUEVUEVUNWVOVUIRUXSUXTWGUYCUYAAUAC UYJUYLFIVWTFCVMMWVGWULUYLYIYJWVIUYD $. $} @@ -497101,33 +502103,33 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cv adantrlr nfrn nff1o nfralw nfan nfss simpr fveq2d simplr simpld simprd simp-4r ad2antrr 2fveq3 eqeq12d rspcva eqtr3d f1ocnvfv1 3eqtr2rd wfn wrex id f1ofn simpllr fvelrnb biimpa r19.29a simprr sselda eliun r19.29af cpnf - sylib cc0 cicc co nfel adantllr biimpi adantl adantlr esumf1o eqcomd snex - a1i xpexd c1st nfcsbw simplll rspcsbela xp1st elsni xp2nd sylbi r19.29af2 - reximi esummono eqbrtrrd cop vex op2ndd anasss esum2d breqtrrd exlimddv ) - AEBCUAZLVNZUBZUUAUCZUHVNZUUANUDNZUUDOZUHYTUEZPZUUBEBEVNZUFZCUGZUAZUIZPZYT - DFUJZBCDFUJEUJZUKULLAYTUULUUAUMZUUGPZLUNUUNLUNABCLEUHGHIJUOUURUUNLUURUUHU - UMUUQUUCUUGYTUULUUAUPUQUUQUUMUUGUUQYTUULUUAYTUULUUAURUSUTVBVAVCAUUNPZUUOU - ULFMVNZUDNZDVDZMUJZUUPUKUUSUUBUVBMUJZUUOUVCUKUUSUUOUVDUUSYTDUUBUVBFMUUAVE - ZUVAQUUSMVFZMDRFUVADVGZMYTRMUUBRMUVERFUVADVHZAYTQSZUUNABGSZCHSZEBUEUVIIAU - VKEBJVIEBCGHVJTUTAUUCUUMUUBYTUVEUCZUUGAUUCUUMPPUUCUVLAUUCUUMVKYTUUBUUAVLV - CVOUUSUUTUUBSZPZUUTUUKSZUUTUVENZUVAOZEBUUSUVMEAUUNEAEVFZUUHUUMEUUCUUGEEYT - UUBUUAEUUARZEBCVMZEUUAUVSVPVQUUFEUHYTUVTUUFEVFVRVSEUUBUULEUUBREBUUKVMZVTV - SVSUVMEVFVSUVNUUIBSZPUVOPZFVNZUUANZUUTOZUVQFYTUWCUWDYTSZPZUWFPZUVAUWDUWEU - VENZUVPUWIUWEUDNZUVAUWDUWIUWEUUTUDUWHUWFWAZWBUWIUWGUUGUWKUWDOZUWCUWGUWFWC - ZUWCUUGUWGUWFUWCUUCUUGUWCUUHUUMAUUNUVMUWBUVOWFWDZWEWGUUFUWMUHUWDYTUUDUWDO - ZUUEUWKUUDUWDUUDUWDUDUUAWHUWPWPWIWJTWKUWIUUCUWGUWJUWDOUWCUUCUWGUWFUWCUUCU - UGUWOWDZWGUWNYTUUBUWDUUAWLTUWIUWEUUTUVEUWLWBWMUWCUUAYTWNZUVMUWFFYTWOZUWCU - UCUWRUWQYTUUBUUAWQVCUUSUVMUWBUVOWRUWRUVMUWSFYTUUTUUAWSWTTXAUVNUUTUULSZUVO - EBWOZUUSUUBUULUUTAUUHUUMXBXCEUUTBUUKXDZXGXEAUWGDXHXFXIXJZSZUUNAUWGPUWDCSZ - UXDEBAUWGEUVREUWDYTEUWDRUVTXKVSAUWBUXEUXDUWGKXLUWGUXEEBWOZAUWGUXFEUWDBCXD - XMXNXEXOXPXQUUSUUBUVBUULMQUVFAUULQSZUUNAUVJUUKQSZEBUEUXGIAUXHEBAUWBPZUUJC - QHUUJQSUXIUUIXRXSJXTVIEBUUKGQVJTUTAUWTUVBUXCSZUUNAUWTPZUUTYANZUUIOZUVACSZ - PZUXJEBAUWTEUVREUUTUULEUUTRUWAXKVSEUVBUXCEFUVADEUVAREDRYBEUXCRXKUXKUWBPZU - XOPZUXNUXDFCUEZUXJUXPUXMUXNXBUXQAUWBUXRAUWTUWBUXOYCUXKUWBUXOWCUXIUXDFCKVI - TFUVACDUXCYDTUWTUXOEBWOZAUWTUXAUXSUXBUVOUXOEBUVOUXMUXNUVOUXLUUJSUXMUUTUUJ - CYEUXLUUIYFVCUUTUUJCYGVBYJYHXNYIXOAUUCUUMUUMUUGAUUCUUMXBVOYKYLAUUPUVCOUUN + sylib cc0 cicc co adantllr biimpi adantl adantlr esumf1o eqcomd vsnex a1i + nfel xpexd c1st nfcsbw rspcsbela xp1st elsni xp2nd reximi sylbi r19.29af2 + simplll esummono eqbrtrrd cop vex op2ndd anasss esum2d breqtrrd exlimddv + ) AEBCUAZLVNZUBZUUAUCZUHVNZUUANUDNZUUDOZUHYTUEZPZUUBEBEVNZUFZCUGZUAZUIZPZ + YTDFUJZBCDFUJEUJZUKULLAYTUULUUAUMZUUGPZLUNUUNLUNABCLEUHGHIJUOUURUUNLUURUU + HUUMUUQUUCUUGYTUULUUAUPUQUUQUUMUUGUUQYTUULUUAYTUULUUAURUSUTVBVAVCAUUNPZUU + OUULFMVNZUDNZDVDZMUJZUUPUKUUSUUBUVBMUJZUUOUVCUKUUSUUOUVDUUSYTDUUBUVBFMUUA + VEZUVAQUUSMVFZMDRFUVADVGZMYTRMUUBRMUVERFUVADVHZAYTQSZUUNABGSZCHSZEBUEUVII + AUVKEBJVIEBCGHVJTUTAUUCUUMUUBYTUVEUCZUUGAUUCUUMPPUUCUVLAUUCUUMVKYTUUBUUAV + LVCVOUUSUUTUUBSZPZUUTUUKSZUUTUVENZUVAOZEBUUSUVMEAUUNEAEVFZUUHUUMEUUCUUGEE + YTUUBUUAEUUARZEBCVMZEUUAUVSVPVQUUFEUHYTUVTUUFEVFVRVSEUUBUULEUUBREBUUKVMZV + TVSVSUVMEVFVSUVNUUIBSZPUVOPZFVNZUUANZUUTOZUVQFYTUWCUWDYTSZPZUWFPZUVAUWDUW + EUVENZUVPUWIUWEUDNZUVAUWDUWIUWEUUTUDUWHUWFWAZWBUWIUWGUUGUWKUWDOZUWCUWGUWF + WCZUWCUUGUWGUWFUWCUUCUUGUWCUUHUUMAUUNUVMUWBUVOWFWDZWEWGUUFUWMUHUWDYTUUDUW + DOZUUEUWKUUDUWDUUDUWDUDUUAWHUWPWPWIWJTWKUWIUUCUWGUWJUWDOUWCUUCUWGUWFUWCUU + CUUGUWOWDZWGUWNYTUUBUWDUUAWLTUWIUWEUUTUVEUWLWBWMUWCUUAYTWNZUVMUWFFYTWOZUW + CUUCUWRUWQYTUUBUUAWQVCUUSUVMUWBUVOWRUWRUVMUWSFYTUUTUUAWSWTTXAUVNUUTUULSZU + VOEBWOZUUSUUBUULUUTAUUHUUMXBXCEUUTBUUKXDZXGXEAUWGDXHXFXIXJZSZUUNAUWGPUWDC + SZUXDEBAUWGEUVREUWDYTEUWDRUVTXSVSAUWBUXEUXDUWGKXKUWGUXEEBWOZAUWGUXFEUWDBC + XDXLXMXEXNXOXPUUSUUBUVBUULMQUVFAUULQSZUUNAUVJUUKQSZEBUEUXGIAUXHEBAUWBPZUU + JCQHUUJQSUXIEXQXRJXTVIEBUUKGQVJTUTAUWTUVBUXCSZUUNAUWTPZUUTYANZUUIOZUVACSZ + PZUXJEBAUWTEUVREUUTUULEUUTRUWAXSVSEUVBUXCEFUVADEUVAREDRYBEUXCRXSUXKUWBPZU + XOPZUXNUXDFCUEZUXJUXPUXMUXNXBUXQAUWBUXRAUWTUWBUXOYJUXKUWBUXOWCUXIUXDFCKVI + TFUVACDUXCYCTUWTUXOEBWOZAUWTUXAUXSUXBUVOUXOEBUVOUXMUXNUVOUXLUUJSUXMUUTUUJ + CYDUXLUUIYEVCUUTUUJCYFVBYGYHXMYIXNAUUCUUMUUMUUGAUUCUUMXBVOYKYLAUUPUVCOUUN AMBCDEFUVBGHUVGUUTUUIUWDYMOZDUVBUXTUWDUVAODUVBOUXTUVAUWDUUIUWDUUTEYNFYNYO - XQUVHVCXQIJAUWBUXEUXDKYPYQUTYRYS $. + XPUVHVCXPIJAUWBUXEUXDKYPYQUTYRYS $. $} $( End $[ set-mbox-ta-esum.mm $] $) @@ -498630,7 +503632,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry $} ${ - $d x y M $. $d m s x y $. + $d x y M $. $( The domain of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 19-Feb-2018.) $) dmmeas $p |- ( M e. U. ran measures -> dom M e. U. ran sigAlgebra ) $= @@ -498640,14 +503642,12 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry $( The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.) $) - measbasedom $p |- ( M e. U. ran measures - <-> M e. ( measures ` dom M ) ) $= - ( vx vy vs vm cmeas crn cuni wcel cfv cc0 wf c0 wceq cv cesum wi cpw wral - w3a cdm cpnf cicc co com wbr wdisj wa csiga isrnmeas simprd dmmeas ismeas - cdom wb syl mpbird wfun cab df-meas funmpt2 elunirn2 mpan impbii ) AFGHIZ - AAUAZFJIZVEVGVFKUBUCUDZALMAJKNBOZUEUNUFCVICOZUGUHZVIHZAJVIVJAJCPNQBVFRSTZ - VEVFUIGHZIZVMBCAUJUKVEVOVGVMUOAULBCVFAUMUPUQFURVGVEDVNDOZVHEOZLMVQJKNVKVL - VQJVIVJVQJCPNQBVPRSTEUSFBCEDUTVAVFAFVBVCVD $. + measbasedom $p |- ( M e. U. ran measures <-> M e. ( measures ` dom M ) ) $= + ( vx vy cmeas crn cuni wcel cdm cfv cc0 cpnf cicc co wf wceq com cdom wbr + c0 cv wdisj wa cesum cpw wral w3a csiga isrnmeas simprd dmmeas ismeas syl + wi wb mpbird elfvunirn impbii ) ADEFGZAAHZDIGZURUTUSJKLMANSAIJOBTZPQRCVAC + TZUAUBVAFAIVAVBAICUCOUMBUSUDUEUFZURUSUGEFGZVCBCAUHUIURVDUTVCUNAUJBCUSAUKU + LUOUSADUPUQ $. $} ${ @@ -499082,30 +504082,30 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry ( vy vz cfv wcel wa cc0 co cv cxdiv c0 wceq wi cvv simplr syl oveq1d cpnf cmeas crp cicc cmpt wf com cdom wbr wdisj cuni cesum cpw wral w3a wfn crn wss wfun cdm funmpt ovex rgenw dmmptg ax-mp mpbir2an a1i wrex wb vex eqid - df-fn elrnmpt measfrge0 ffvelcdm sylan xrpxdivcld eleq1a rexlimdva syl5bi - adantlr ssrdv sylanbrc csiga measbase 0elsiga adantr jctir fveq2 measvnul - df-f fvmptg xdiv0rp sylan9eq eqtrd simpll simprl 3jca simplll simpr elpwg - simprr ssel2 sylanb syl2anc measvxrge0 esumdivc 3ad2antr1 ad2antrr simpr1 - simpr2 sigaclcu syl3anc fvmpt3i jca simpr3 3expia ralrimiva r19.21bi sylc - measvun esumeq2dv 3eqtr4d ex ismeas biimprd mpd ) DCUBGZHZBUCHZIZCJUAUDKZ - ACALZDGZBMKZUEZUFZNYPGZJOZELZUGUHUIZFYTFLZUJZIZYTUKZYPGZYTUUBYPGZFULZOZPZ - ECUMZUNZUOZYPYHHZYKYQYSUULYKYPCUPZYPUQZYLURYQUUOYKUUOYPUSYPUTCOZACYOVAYOQ - HZACUNUUQUURACYNBMVBZVCACYOQVDVEYPCVLVFVGYKEUUPYLYTUUPHZYTYOOZACVHZYKYTYL - HZYTQHZUUTUVBVIEVJZACYOYTYPQYPVKZVMVEYKUVAUVCACYKYMCHZIZYOYLHUVAUVCPUVHYN - BYIUVGYNYLHZYJYICYLDUFUVGUVICDVNCYLYMDVOVPWAYIYJUVGRVQYOYLYTVRSVSVTWBCYLY - PWKWCYKYRNDGZBMKZJYKNCHZUVKQHZIYRUVKOYKUVLUVMYIUVLYJYICWDUQUKHZUVLCDWEZCW - FSWGUVJBMVBWHANYOUVKCQYPYMNOYNUVJBMYMNDWITUVFWLSYIYJUVKJBMKJYIUVJJBMCDWJT - BWMWNWOYKUUJEUUKYKYTUUKHZIZUUDUUIUVQUUDIZYKUVPUUAUUCUOZUUIYKUVPUUDWPUVRUV - PUUAUUCYKUVPUUDRUVQUUAUUCWQUVQUUAUUCXBWRYKUVSIZYTUUBDGZFULZBMKZYTUWABMKZF - ULZUUFUUHYKUUAUVPUWCUWEOUUCUVQYTUWABFQUVDUVQUVEVGUVQUUBYTHZIZYIUUBCHZUWAY - LHYIYJUVPUWFWSUWGUVPUWFUWHYKUVPUWFRUVQUWFWTUVPYTCURZUWFUWHUVDUVPUWIVIUVEY - TCQXAVEYTCUUBXCXDZXEUUBCDXFXEYIYJUVPRXGXHUVTUUFUUEDGZBMKZUWCUVTUUECHZUUFU - WLOUVTUVNUVPUUAUWMYIUVNYJUVSUVOXIYKUVPUUAUUCXJZYKUVPUUAUUCXKZYTCXLXMAUUEY - OUWLCYPYMUUEOYNUWKBMYMUUEDWITUVFUUSXNSUVTUWKUWBBMUVTYIUVPIUUDUWKUWBOZUVTY - IUVPYIYJUVSWPUWNXOUVTUUAUUCUWOYKUVPUUAUUCXPXOYIUUDUWPPZEUUKYIUWQEUUKYIUVP - UUDUWPFYTCDYAXQXRXSXTTWOUVTUVPUUHUWEOUWNUVPYTUUGUWDFUVPUWFIUWHUUGUWDOUWJA - UUBYOUWDCYPYMUUBOYNUWABMYMUUBDWITUVFUUSXNSYBSYCXEYDXRWRYIUUMUUNPYJYIUUNUU - MYIUVNUUNUUMVIUVOEFCYPYESYFWGYG $. + df-fn elrnmpt measfrge0 ffvelcdm sylan adantlr xrpxdivcld eleq1a biimtrid + rexlimdva ssrdv sylanbrc csiga measbase 0elsiga adantr jctir fveq2 fvmptg + df-f measvnul xdiv0rp eqtrd simpll simprl simprr 3jca simplll simpr elpwg + sylan9eq ssel2 sylanb syl2anc measvxrge0 esumdivc 3ad2antr1 simpr1 simpr2 + ad2antrr sigaclcu syl3anc fvmpt3i jca simpr3 measvun 3expia r19.21bi sylc + ralrimiva esumeq2dv 3eqtr4d ex ismeas biimprd mpd ) DCUBGZHZBUCHZIZCJUAUD + KZACALZDGZBMKZUEZUFZNYPGZJOZELZUGUHUIZFYTFLZUJZIZYTUKZYPGZYTUUBYPGZFULZOZ + PZECUMZUNZUOZYPYHHZYKYQYSUULYKYPCUPZYPUQZYLURYQUUOYKUUOYPUSYPUTCOZACYOVAY + OQHZACUNUUQUURACYNBMVBZVCACYOQVDVEYPCVLVFVGYKEUUPYLYTUUPHZYTYOOZACVHZYKYT + YLHZYTQHZUUTUVBVIEVJZACYOYTYPQYPVKZVMVEYKUVAUVCACYKYMCHZIZYOYLHUVAUVCPUVH + YNBYIUVGYNYLHZYJYICYLDUFUVGUVICDVNCYLYMDVOVPVQYIYJUVGRVRYOYLYTVSSWAVTWBCY + LYPWKWCYKYRNDGZBMKZJYKNCHZUVKQHZIYRUVKOYKUVLUVMYIUVLYJYICWDUQUKHZUVLCDWEZ + CWFSWGUVJBMVBWHANYOUVKCQYPYMNOYNUVJBMYMNDWITUVFWJSYIYJUVKJBMKJYIUVJJBMCDW + LTBWMXBWNYKUUJEUUKYKYTUUKHZIZUUDUUIUVQUUDIZYKUVPUUAUUCUOZUUIYKUVPUUDWOUVR + UVPUUAUUCYKUVPUUDRUVQUUAUUCWPUVQUUAUUCWQWRYKUVSIZYTUUBDGZFULZBMKZYTUWABMK + ZFULZUUFUUHYKUUAUVPUWCUWEOUUCUVQYTUWABFQUVDUVQUVEVGUVQUUBYTHZIZYIUUBCHZUW + AYLHYIYJUVPUWFWSUWGUVPUWFUWHYKUVPUWFRUVQUWFWTUVPYTCURZUWFUWHUVDUVPUWIVIUV + EYTCQXAVEYTCUUBXCXDZXEUUBCDXFXEYIYJUVPRXGXHUVTUUFUUEDGZBMKZUWCUVTUUECHZUU + FUWLOUVTUVNUVPUUAUWMYIUVNYJUVSUVOXKYKUVPUUAUUCXIZYKUVPUUAUUCXJZYTCXLXMAUU + EYOUWLCYPYMUUEOYNUWKBMYMUUEDWITUVFUUSXNSUVTUWKUWBBMUVTYIUVPIUUDUWKUWBOZUV + TYIUVPYIYJUVSWOUWNXOUVTUUAUUCUWOYKUVPUUAUUCXPXOYIUUDUWPPZEUUKYIUWQEUUKYIU + VPUUDUWPFYTCDXQXRYAXSXTTWNUVTUVPUUHUWEOUWNUVPYTUUGUWDFUVPUWFIUWHUUGUWDOUW + JAUUBYOUWDCYPYMUUBOYNUWABMYMUUBDWITUVFUUSXNSYBSYCXEYDYAWRYIUUMUUNPYJYIUUN + UUMYIUVNUUNUUMVIUVOEFCYPYESYFWGYG $. $} @@ -499627,16 +504627,11 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry mpbid syl ) ADCIZBIZJKLZUFUEDMAUGDNHOPBLHCQZADBCRKLUGUHSGAHBCDEFTUCUADUEU FUBUD $. - $( The predicate to be a measurable function. (Contributed by Thierry - Arnoux, 30-Jan-2017.) $) - isanmbfm $p |- ( ph -> F e. U. ran MblFnM ) $= - ( vt vs vx cv cuni cmap co wcel wral wa crn wrex cmbfm csiga ismbfm mpbid - ccnv cima wceq unieq oveq2d eleq2d ralbidv anbi12d oveq1d rspc2ev syl3anc - eleq2 raleq elunirnmbfm sylibr ) ADHKZLZIKZLZMNZOZDUDJKUEZVAOZJUSPZQZHUAR - LZSIVISZDTRLOABVIOCVIODCLZBLZMNZOZVEBOZJCPZQZVJEFADBCTNOVQGAJBCDEFUBUCVHV - QDUTVLMNZOZVOJUSPZQIHBCVIVIVABUFZVDVSVGVTWAVCVRDWAVBVLUTMVABUGUHUIWAVFVOJ - USVABVEUOUJUKUSCUFZVSVNVTVPWBVRVMDWBUTVKVLMUSCUGULUIVOJUSCUPUKUMUNJHDIUQU - R $. + $( Obsolete version of ~ isanmbfm as of 13-Jan-2025. (Contributed by + Thierry Arnoux, 30-Jan-2017.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + isanmbfmOLD $p |- ( ph -> F e. U. ran MblFnM ) $= + ( cmbfm co crn cuni ovssunirn sselid ) ABCHIHJKDHBCLGM $. $d x A $. mbfmcnvima.4 $e |- ( ph -> A e. T ) $. @@ -499649,15 +504644,32 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry $} ${ - mbfmbfm.1 $e |- ( ph -> M e. U. ran measures ) $. - mbfmbfm.2 $e |- ( ph -> J e. Top ) $. - mbfmbfm.3 $e |- ( ph -> F e. ( dom M MblFnM ( sigaGen ` J ) ) ) $. + isanmbfm.1 $e |- ( ph -> F e. ( S MblFnM T ) ) $. + $( The predicate to be a measurable function. (Contributed by Thierry + Arnoux, 30-Jan-2017.) Remove hypotheses. (Revised by SN, + 13-Jan-2025.) $) + isanmbfm $p |- ( ph -> F e. U. ran MblFnM ) $= + ( cmbfm co crn cuni ovssunirn sselid ) ABCFGFHIDFBCJEK $. + $} + + ${ + mbfmbfmOLD.1 $e |- ( ph -> M e. U. ran measures ) $. + mbfmbfmOLD.2 $e |- ( ph -> J e. Top ) $. + mbfmbfmOLD.3 $e |- ( ph -> F e. ( dom M MblFnM ( sigaGen ` J ) ) ) $. $( A measurable function to a Borel Set is measurable. (Contributed by - Thierry Arnoux, 24-Jan-2017.) $) + Thierry Arnoux, 24-Jan-2017.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + mbfmbfmOLD $p |- ( ph -> F e. U. ran MblFnM ) $= + ( cdm csigagen cfv isanmbfm ) ADHCIJBGK $. + $} + + ${ + mbfmbfm.1 $e |- ( ph -> F e. ( dom M MblFnM ( sigaGen ` J ) ) ) $. + $( A measurable function to a Borel Set is measurable. (Contributed by + Thierry Arnoux, 24-Jan-2017.) Remove hypotheses. (Revised by SN, + 13-Jan-2025.) $) mbfmbfm $p |- ( ph -> F e. U. ran MblFnM ) $= - ( cdm csigagen cfv cmeas cuni wcel csiga measbasedom biimpi measbase 3syl - crn ctop sgsiga isanmbfm ) ADHZCIJBADKSLMZDUCKJMZUCNSLMEUDUEDOPUCDQRACTFU - AGUB $. + ( cdm csigagen cfv isanmbfm ) ADFCGHBEI $. $} ${ @@ -499962,18 +504974,18 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = ( RR X. RR ) $= ( vz cr cv cpnf cico co cxp cmpt cuni wss wcel pnfxr syl ovex reex cfv wa xpex crn cun unissb wo elun cpw eqid rnmptss cxr icossre mpan2 xpss1 elpw - sylibr mprg sseli elpwid xpss2 jaoi mprgbir wfun c1st funmpt cvv c2nd clt - sylbi wbr rexr a1i ltpnf lbico1 syl3anc anim1i anim2i elxp7 3imtr4i xp1st - wceq oveq1 xpeq1d fvmpt eleqtrrd elunirn2 sylancr ssriv ssun3 ax-mp uniun - sseqtrri eqssi ) ADAEZFGHZDIZJZUAZBDDBEZFGHZIZJZUAZUBZKZDDIZXCXDLCEZXDLZC - XBCXBXDUCXEXBMXEWPMZXEXAMZUDXFXEWPXAUEXGXFXHXGXEXDWPXDUFZXEWNXIMZWPXILADA - DWNXIWOWOUGZUHWLDMZWNXDLZXJXLWMDLZXMXLFUIMZXNNWLFUJUKWMDDULOWNXDWMDWLFGPQ - TUMUNUOUPUQXHXEXDXAXIXEWSXIMZXAXILBDBDWSXIWTWTUGUHWQDMZWSXDLZXPXQWRDLZXRX - QXOXSNWQFUJUKWRDDUROWSXDDWRQWQFGPTUMUNUOUPUQUSVGUTXDWPKZXAKZUBZXCXDXTLXDY - BLCXDXTXEXDMZWOVAXEXEVBRZWORZMXEXTMADWNVCYCXEYDFGHZDIZYEXEVDVDIMZYDDMZXEV - ERDMZSZSYHYDYFMZYJSZSYCXEYGMYKYMYHYIYLYJYIYDUIMXOYDFVFVHYLYDVIXOYINVJYDVK - YDFVLVMVNVOXEDDVPXEYFDVPVQYCYIYEYGVSXEDDVRAYDWNYGDWOWLYDVSWMYFDWLYDFGVTWA - XKYFDYDFGPQTWBOWCYDXEWOWDWEWFXDXTYAWGWHWPXAWIWJWK $. + sylibr mprg sseli xpss2 jaoi sylbi mprgbir c1st cvv c2nd clt wbr rexr a1i + elpwid ltpnf lbico1 syl3anc anim1i anim2i elxp7 3imtr4i wceq xp1st xpeq1d + oveq1 fvmpt eleqtrrd elfvunirn ssriv ssun3 ax-mp uniun sseqtrri eqssi ) A + DAEZFGHZDIZJZUAZBDDBEZFGHZIZJZUAZUBZKZDDIZWTXALCEZXALZCWSCWSXAUCXBWSMXBWM + MZXBWRMZUDXCXBWMWRUEXDXCXEXDXBXAWMXAUFZXBWKXFMZWMXFLADADWKXFWLWLUGZUHWIDM + ZWKXALZXGXIWJDLZXJXIFUIMZXKNWIFUJUKWJDDULOWKXAWJDWIFGPQTUMUNUOUPVHXEXBXAW + RXFXBWPXFMZWRXFLBDBDWPXFWQWQUGUHWNDMZWPXALZXMXNWODLZXOXNXLXPNWNFUJUKWODDU + QOWPXADWOQWNFGPTUMUNUOUPVHURUSUTXAWMKZWRKZUBZWTXAXQLXAXSLCXAXQXBXAMZXBXBV + ARZWLRZMXBXQMXTXBYAFGHZDIZYBXBVBVBIMZYADMZXBVCRDMZSZSYEYAYCMZYGSZSXTXBYDM + YHYJYEYFYIYGYFYAUIMXLYAFVDVEYIYAVFXLYFNVGYAVIYAFVJVKVLVMXBDDVNXBYCDVNVOXT + YFYBYDVPXBDDVQAYAWKYDDWLWIYAVPWJYCDWIYAFGVSVRXHYCDYAFGPQTVTOWAYAXBWLWBOWC + XAXQXRWDWEWMWRWFWGWH $. $} ${ @@ -500248,20 +505260,20 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry mpoex eqeltri rnex xpex fnex mp2an elpw2 mpbir a1i wn wral rex0 rexeq mtbiri ralrimivw rabeq0 sylibr unieqd uni0 eqtrdi 0ss eqsstrdi elequ2 wne sseq1 anbi12d rexbidv elrab simpr reximi r19.9rzv syl5ibr adantld - syl5bi ralrimiv unissb pm2.61ine dya2iocnei simpl ssel2 ancoms adantl - elequ1 anbi1d rspcev syl2anc jca simprl reximi2 eluni2 eqelssd eqeq1d - syl unieq ) DHHUBUCPZMNUDZNQZDUEZUJZMDRZNEUFZUGZYEUHZPZYFUKZDSZIQZUKZ - DSZIYGRYHXSYHYFYEUEYDNYEUIYFYEEEGUFZYNULZUMYOUNPEUNPABCEFGHJKLUOYNYNG - GAFUPUPAQZUQFQURUCZUSUCYPUTVAUCYQUSUCVBUCZVCUNKAFUPUPYRVDVDVEVFVGZYSV - HYOUNEVIVJVGVKVLVMXSUAYIDYIDUEZXSYTDTDTSZYITDUUAYITUKTUUAYFTUUAYDVNZN - YEVOYFTSUUAUUBNYEUUAYDYCMTRYCMVPYCMDTVQVRVSYDNYEVTWAWBWCWDDWEWFDTWHZO - QZDUEZOYFVOYTUUCUUEOYFUUDYFPZUUDYEPZMOUDZUUEUJZMDRZUJZUUCUUEYDUUJNUUD - YEYAUUDSZYCUUIMDUULXTUUHYBUUENOMWGYAUUDDWIWJWKWLZUUCUUJUUEUUGUUJUUEUU - CUUEMDRUUIUUEMDUUHUUEWMWNUUEMDWOWPWQWRWSOYFDWTWAXAVMXSUAQZDPZUJZUAOUD - ZOYFRZUUNYIPUUPUUQUUEUJZOYERUURABCDEFGHUUNOJKLXBUUSUUQOYEYFUUGUUSUJZU - UFUUQUUTUUKUUFUUTUUGUUJUUGUUSXCUUTUUOUUSUUJUUSUUOUUGUUEUUQUUOUUDDUUNX - DXEXFUUGUUSWMUUIUUSMUUNDMQUUNSUUHUUQUUEMUAOXGXHXIXJXKUUMWAUUGUUQUUEXL - XKXMXQOUUNYFXNWAXOYMYJIYFYGYKYFSYLYIDYKYFXRXPXIXJ $. + biimtrid unissb pm2.61ine dya2iocnei simpl ssel2 ancoms adantl elequ1 + ralrimiv anbi1d rspcev syl2anc jca simprl reximi2 eluni2 unieq eqeq1d + syl eqelssd ) DHHUBUCPZMNUDZNQZDUEZUJZMDRZNEUFZUGZYEUHZPZYFUKZDSZIQZU + KZDSZIYGRYHXSYHYFYEUEYDNYEUIYFYEEEGUFZYNULZUMYOUNPEUNPABCEFGHJKLUOYNY + NGGAFUPUPAQZUQFQURUCZUSUCYPUTVAUCYQUSUCVBUCZVCUNKAFUPUPYRVDVDVEVFVGZY + SVHYOUNEVIVJVGVKVLVMXSUAYIDYIDUEZXSYTDTDTSZYITDUUAYITUKTUUAYFTUUAYDVN + ZNYEVOYFTSUUAUUBNYEUUAYDYCMTRYCMVPYCMDTVQVRVSYDNYEVTWAWBWCWDDWEWFDTWH + ZOQZDUEZOYFVOYTUUCUUEOYFUUDYFPZUUDYEPZMOUDZUUEUJZMDRZUJZUUCUUEYDUUJNU + UDYEYAUUDSZYCUUIMDUULXTUUHYBUUENOMWGYAUUDDWIWJWKWLZUUCUUJUUEUUGUUJUUE + UUCUUEMDRUUIUUEMDUUHUUEWMWNUUEMDWOWPWQWRXGOYFDWSWAWTVMXSUAQZDPZUJZUAO + UDZOYFRZUUNYIPUUPUUQUUEUJZOYERUURABCDEFGHUUNOJKLXAUUSUUQOYEYFUUGUUSUJ + ZUUFUUQUUTUUKUUFUUTUUGUUJUUGUUSXBUUTUUOUUSUUJUUSUUOUUGUUEUUQUUOUUDDUU + NXCXDXEUUGUUSWMUUIUUSMUUNDMQUUNSUUHUUQUUEMUAOXFXHXIXJXKUUMWAUUGUUQUUE + XLXKXMXQOUUNYFXNWAXRYMYJIYFYGYKYFSYLYIDYKYFXOXPXIXJ $. $} ${ @@ -500655,131 +505667,131 @@ strict in the case where the sets B(x) overlap. 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Value of the sum of a finite partition ` A ` @@ -502771,47 +507783,47 @@ strict in the case where the sets B(x) overlap. 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(Contributed by Thierry (Contributed by Thierry Arnoux, 21-Jan-2017.) $) orvcval4 $p |- ( ph -> ( X oRVC R A ) = ( `' X " { y e. U. J | y R A } ) ) $= - ( cima cuni co wceq ctop wf wcel cvv ccnv wbr cab cin corvc crab wfun crn - cv wss csigagen sgsiga isanmbfm mbfmfun mbfmf elex unisg 3syl feq3d mpbid + ( cima cuni co wceq wf ctop wcel cvv ccnv wbr cab cin corvc crab wfun crn + cv wss csigagen isanmbfm mbfmfun sgsiga mbfmf elex unisg 3syl feq3d mpbid cfv frnd fimacnvinrn2 syl2anc cmbfm orvcval dfrab2 a1i imaeq2d 3eqtr4d ) AHUAZBUICDUBZBUCZMZVKVMFNZUDZMZHCDUEOVKVLBVOUFZMAHUGHUHVOUJVNVQPAHAEFUKVA - ZHIAFQJULZKUMUNZAENZVOHAWBVSNZHRWBVOHRAEVSHIVTKUOAWCVOHWBAFQSFTSWCVOPJFQU - PFTUQURUSUTVBVMVOHVCVDABCDEVSVEOGHWAKLVFAVRVPVKVRVPPAVLBVOVGVHVIVJ $. + ZHKULUMZAENZVOHAWAVSNZHQWAVOHQAEVSHIAFRJUNKUOAWBVOHWAAFRSFTSWBVOPJFRUPFTU + QURUSUTVBVMVOHVCVDABCDEVSVEOGHVTKLVFAVRVPVKVRVPPAVLBVOVGVHVIVJ $. ${ orvcoel.5 $e |- ( ph -> { y e. U. J | y R A } e. J ) $. @@ -508068,7 +513080,8 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry signshn0 3eqtr4d eqtr4d ffvelcdm sylancr s1eqd subidi c0ex prid1 wn df-ne mul0ord - wi df-or syl5bi necon1d resubcld signstf fzossfz cz nnzd fzval3 eleqtrd + wi df-or biimtrid necon1d resubcld signstf fzossfz cz nnzd fzval3 + eleqtrd 0cn signstlen rpne0 necomd neeq1d subne0d signstfvneq0 pm2.21d mulneg2d neneqd w3a rpgt0 mulgt0d breqtrd lt0neg2d eqbrtrd 3jca swrd0lenOLD signhv0 @@ -512384,17 +517397,6 @@ conditions of the Five Segment Axiom ( ~ axtg5seg ). See ~ df-ofs . ( cvv wcel cop csn wfun funsng mpan ) ADEBDEABFGHCABDDIJ $. $} - ${ - $d A x $. - $( First-order logic and set theory. (Contributed by Jonathan Ben-Naim, - 3-Jun-2011.) (New usage is discouraged.) $) - bnj521 $p |- ( A i^i { A } ) = (/) $= - ( vx csn cin c0 wne wn wceq cv wcel wex elirr wa elin velsn eleq1 biimpac - sylan2b sylbi exlimiv mto n0 mtbir nne mpbi ) AACZDZEFZGUGEHUHBIZUGJZBKZU - KAAJZALUJULBUJUIAJZUIUFJZMULUIAUFNUNUMUIAHZULBAOUOUMULUIAAPQRSTUABUGUBUCU - GEUDUE $. - $} - ${ bnj524.1 $e |- ( ph <-> ps ) $. bnj524.2 $e |- A e. _V $. @@ -512685,10 +517687,10 @@ conditions of the Five Segment Axiom ( ~ axtg5seg ). See ~ df-ofs . $( First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) $) bnj927 $p |- ( ( p = suc n /\ f Fn n ) -> G Fn p ) $= - ( cv csuc wceq wfn wa csn cun cop simpr vex fnsn a1i cin c0 bnj521 fneq1i - fnund sylibr df-suc eqeq2i biimpi adantr fneq2d mpbird ) EHZCHZIZJZBHZUMK - ZLZDULKDUMUMMZNZKZURUPUMAOMZNZUTKVAURUMUSUPVBUOUQPVBUSKURUMACQGRSUMUSTUAJ - URUMUBSUDUTDVCFUCUEURULUTDUOULUTJZUQUOVDUNUTULUMUFUGUHUIUJUK $. + ( cv csuc wceq wfn wa csn cun cop simpr vex fnsn a1i cin c0 disjcsn fnund + fneq1i sylibr df-suc eqeq2i biimpi adantr fneq2d mpbird ) EHZCHZIZJZBHZUM + KZLZDULKDUMUMMZNZKZURUPUMAOMZNZUTKVAURUMUSUPVBUOUQPVBUSKURUMACQGRSUMUSTUA + JURUMUBSUCUTDVCFUDUEURULUTDUOULUTJZUQUOVDUNUTULUMUFUGUHUIUJUK $. $} ${ @@ -514246,12 +519248,12 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a n ) $= ( cv wfn cvv w-bnj15 csn cun wceq w-bnj17 bnj422 bnj251 bitri cfv c-bnj14 ciun cop wcel wfun wral fvex bnj518 iunexg sylancr vex bnj519 syl dmsnopg - wa cdm bnj1422 cin c0 bnj521 mpan2 sylan2 fneq1i sylibr fneq2 syl5ibr imp - fnun sylbi ) DEUAZAIRZHRZWAUBZUCZUDZFRZWASZUEZWDWFVSAVDZVDZVDZJVTSZWGWDWF - VSAUEWJVSAWDWFUFWDWFVSAUGUHWDWIWKWIWKWDJWCSZWIWEWACKRZWEUIZDECRUJZUKZULUB - ZUCZWCSZWLWHWFWQWBSZWSWHWQWBWHWPTUMZWQUNWHWNTUMWOTUMCWNUOXAWMWEUPLMABCDEF - GHKNOQUQCWNWOTTURUSZWAWPHUTVAVBWHXAWQVEWBUDXBWAWPTVCVBVFWFWTVDWAWBVGVHUDW - SWAVIWAWBWEWQVQVJVKWCJWRPVLVMVTWCJVNVOVPVR $. + wa cdm bnj1422 cin c0 disjcsn fnun mpan2 sylan2 fneq1i sylibr syl5ibr imp + fneq2 sylbi ) DEUAZAIRZHRZWAUBZUCZUDZFRZWASZUEZWDWFVSAVDZVDZVDZJVTSZWGWDW + FVSAUEWJVSAWDWFUFWDWFVSAUGUHWDWIWKWIWKWDJWCSZWIWEWACKRZWEUIZDECRUJZUKZULU + BZUCZWCSZWLWHWFWQWBSZWSWHWQWBWHWPTUMZWQUNWHWNTUMWOTUMCWNUOXAWMWEUPLMABCDE + FGHKNOQUQCWNWOTTURUSZWAWPHUTVAVBWHXAWQVEWBUDXBWAWPTVCVBVFWFWTVDWAWBVGVHUD + WSWAVIWAWBWEWQVJVKVLWCJWRPVMVNVTWCJVQVOVPVR $. $} ${ @@ -515130,10 +520132,10 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) $) bnj873 $p |- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) } $= - ( cv wfn w3a wrex cab nfv wsbc nfsbc1v nfcv nfxfr nf3an nfrex weq sbceq1a - fneq1 bitr4di 3anbi123d rexbidv cbvabw eqtri ) CEMZGMZNZABOZGDPZEQFMZUNNZ - HIOZGDPZFQJUQVAEFUQFRUTEGDEDUAUSHIEUSERHAEURSZEKAEURTUBIBEURSZELBEURTUBUC - UDEFUEZUPUTGDVDUOUSAHBIUNUMURUGVDAVBHAEURUFKUHVDBVCIBEURUFLUHUIUJUKUL $. + ( cv wfn w3a wrex cab nfv wsbc nfsbc1v nfcv nfxfr nf3an weq fneq1 sbceq1a + nfrexw bitr4di 3anbi123d rexbidv cbvabw eqtri ) CEMZGMZNZABOZGDPZEQFMZUNN + ZHIOZGDPZFQJUQVAEFUQFRUTEGDEDUAUSHIEUSERHAEURSZEKAEURTUBIBEURSZELBEURTUBU + CUGEFUDZUPUTGDVDUOUSAHBIUNUMURUEVDAVBHAEURUFKUHVDBVCIBEURUFLUHUIUJUKUL $. $} ${ @@ -515160,17 +520162,17 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a ( vw w-bnj15 wcel wa cvv cv wrex wi wal wex bnj865 wss wfn w3a cab bnj873 wel df-rex 19.29 an12 df-3an anbi1i 3bitr4i wsbc bnj581 bitr3i weu bnj864 exancom sylbb nfeu1 nfe1 nfan nfsbc1v nfv nfim weq sbceq1a elequ1 imbi12d - id imbi2d eupick chvarfv syl2an syl5bir ex embantd sylbir expimpd exlimdv - impd syl5bi syl5 expdimp abssdv eqsstrid vex ssex syl mpi ) GJUJZOGUKZULZ - EHUMUKZUHECDKUIUNZUOZUPZNUQZUIURXMABCDFUIGIJKMNOSTUAUCUDUSEXQXMUIEXQXMEXQ - ULZHXNUTXMXRHLUNZNUNZVAPQVBZNIUOZLVCXNABHIKLNPQUBUEUFVDXRYBLXNYBXTIUKZYAU - LZNURZXRLUIVEZYANIVFEXQYEYFXQYEULXPYDULZNUREYFXPYDNVGEYGYFNYGYCXPYAULZULE - YFXPYCYAVHEYCYHYFEYCULZCYHYFUPXJXKYCVBXLYCULCYIXJXKYCVIUCEXLYCUHVJVKCXPYA - YFCCXOYAYFUPZCWICXOYJYADKXSVLZCXOULYFYKRYAUGABDKLNPQRUDUEUFUGVMVNCDKVOZDK - UIVEZULZKURZYKYFUPZXOABCDFGIJKMNOSTUAUCUDVPXOYMDULKURYODKXNVFYMDKVQVRYLYO - ULZDYMUPZUPYQYPUPKLYQYPKYLYOKDKVSYNKVTWAYKYFKDKXSWBYFKWCWDWDKLWEZYRYPYQYS - DYKYMYFDKXSWFKLUIWGWHWJDYMKWKWLWMWNWOWPWTWQWRXAWSXBXCXAXDXEHXNUIXFXGXHWOW - SXIWQ $. + imbi2d eupick chvarfv syl2an syl5bir embantd impd sylbir expimpd biimtrid + id ex exlimdv syl5 expdimp abssdv eqsstrid vex ssex syl mpi ) GJUJZOGUKZU + LZEHUMUKZUHECDKUIUNZUOZUPZNUQZUIURXMABCDFUIGIJKMNOSTUAUCUDUSEXQXMUIEXQXME + XQULZHXNUTXMXRHLUNZNUNZVAPQVBZNIUOZLVCXNABHIKLNPQUBUEUFVDXRYBLXNYBXTIUKZY + AULZNURZXRLUIVEZYANIVFEXQYEYFXQYEULXPYDULZNUREYFXPYDNVGEYGYFNYGYCXPYAULZU + LEYFXPYCYAVHEYCYHYFEYCULZCYHYFUPXJXKYCVBXLYCULCYIXJXKYCVIUCEXLYCUHVJVKCXP + YAYFCCXOYAYFUPZCWSCXOYJYADKXSVLZCXOULYFYKRYAUGABDKLNPQRUDUEUFUGVMVNCDKVOZ + DKUIVEZULZKURZYKYFUPZXOABCDFGIJKMNOSTUAUCUDVPXOYMDULKURYODKXNVFYMDKVQVRYL + YOULZDYMUPZUPYQYPUPKLYQYPKYLYOKDKVSYNKVTWAYKYFKDKXSWBYFKWCWDWDKLWEZYRYPYQ + YSDYKYMYFDKXSWFKLUIWGWHWIDYMKWJWKWLWMWTWNWOWPWQWRXAXBXCWRXDXEHXNUIXFXGXHW + TXAXIWP $. $} ${ @@ -516051,15 +521053,15 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a bnj1014 $p |- ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ _trCl ( X , A , R ) ) $= ( cv wcel nfv cdm wa cfv c-bnj18 wss wi wfn w3a wrex cab nfcv bnj911 nf5i - nfrex nfab nfcxfr nfcri nfan nfim nf5ri wb weq eleq1w anbi2d fveq2 sseq1d - imbi12d equcoms bnj1317 dmeq eleq2d anbi12d fveq1 ssiun2 bnj882 sseqtrrdi - ciun sylan9ssr chvarfv speivw bnj1131 ) IRZESZKRZWBUAZSZUBZWDWBUCZDGMUDZU - EZUFZJWKJWGWJJWCWFJJIEJEHRZLRUGABUHZLFUIZHUJQWNJHWMJLFJFUKWMJABCDGHJLMNOU - LUMUNUOUPUQWFJTURWJJTUSUTWKWCJRZWESZUBZWOWBUCZWIUEZUFZJKWKWTVAKJKJVBZWGWQ - WJWSXAWFWPWCKJWEVCVDXAWHWRWIWDWOWBVEVFVGVHWLESZWOWLUAZSZUBZWOWLUCZWIUEZUF - WTHIWQWSHWCWPHWCHWNHIEQVIUMWPHTURWSHTUSHIVBZXEWQXGWSXHXBWCXDWPHIEVCXHXCWE - WOWLWBVJVKVLXHXFWRWIWOWLWBVMVFVGXDXBXFJXCXFVQZWIJXCXFVNXBXIHEXIVQWIHEXIVN - ABCDEFGHJLMNOPQVOVPVRVSVTWA $. + nfrexw nfab nfcxfr nfcri nfan nfim nf5ri weq eleq1w anbi2d sseq1d imbi12d + wb fveq2 equcoms bnj1317 dmeq eleq2d anbi12d ciun ssiun2 bnj882 sseqtrrdi + fveq1 sylan9ssr chvarfv speivw bnj1131 ) IRZESZKRZWBUAZSZUBZWDWBUCZDGMUDZ + UEZUFZJWKJWGWJJWCWFJJIEJEHRZLRUGABUHZLFUIZHUJQWNJHWMJLFJFUKWMJABCDGHJLMNO + ULUMUNUOUPUQWFJTURWJJTUSUTWKWCJRZWESZUBZWOWBUCZWIUEZUFZJKWKWTVFKJKJVAZWGW + QWJWSXAWFWPWCKJWEVBVCXAWHWRWIWDWOWBVGVDVEVHWLESZWOWLUAZSZUBZWOWLUCZWIUEZU + FWTHIWQWSHWCWPHWCHWNHIEQVIUMWPHTURWSHTUSHIVAZXEWQXGWSXHXBWCXDWPHIEVBXHXCW + EWOWLWBVJVKVLXHXFWRWIWOWLWBVQVDVEXDXBXFJXCXFVMZWIJXCXFVNXBXIHEXIVMWIHEXIV + NABCDEFGHJLMNOPQVOVPVRVSVTWA $. $} ${ @@ -516591,18 +521593,18 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a w-bnj19 simpr2 wfn bnj1254 3ad2ant3 ad2antrl adantr bnj1232 simpr1 bnj923 anim1i ancomd syl2anc elnn syl cdm adantl ad2antlr bnj951 c-bnj14 simp2bi wi cvv 3ad2ant2 simp3 anim12i ciun bnj256 wal bnj1112 biimpi eleq1 anbi2d - 19.21bi fveqeq2 imbi12d syl5ibr imp31 sylbi df-bnj19 ssralv syl5bi impcom - wral iunss sylibr sseq1 biimpar syl2an bnj1023 ) MUGZUHUIZDECUJZSUKZUKZXP - NUGZOUGZULZXLXQUMZUNZXQLUGZUOZIUPZUJZUKZXLYBUOZIUPZNXPYEWCXPYFWCNABCDEFIJ - LMNOPRSUAUCUDUEUFUQXPYEURUSYFYABXQUTULZXLXRULZVAZHIKVBZYDUKZYHYABYIYJYFXP - XSYAYDVCXPBYEXNBXMSCDBECXRJULZYBXRVDZABUAVEVFVGVHYFXSXRUTULZUKZYIYFYNXSYQ - XPYNYEXNYNXMSCDYNECYNYOABUAVIVFVGVHXPXSYAYDVJYNXSUKYPXSYNYPXSJOUCVKVLVMVN - XQXRVOVPXOYJXMYESYJXNSYJFYBPULXLYBVQULUEVIVRVSVTXPYLYEYDXNYLXMSEDYLCEIWDU - LYLHKQWAIUPUBWBWEVGXSYAYDWFWGYKYGGYCHKGUGWAZWHZUNZYSIUPZYHYMYKYABUKYIYJUK - ZUKYTYABYIYJWIYABUUBYTBUUBYTWCYAYIXTXRULZUKZXTYBUOYSUNZWCZBUUFNBUUFNWJBGH - KLMNOTWKWLWOYAUUBUUDYTUUEYAYJUUCYIXLXTXRWMWNXLXTYSYBWPWQWRWSWTYMYRIUPZGYC - XEZUUAYDYLUUHYLUUGGIXEYDUUHGHIKXAUUGGYCIXBXCXDGYCYRIXFXGYTYHUUAYGYSIXHXIX - JVNXK $. + 19.21bi fveqeq2 imbi12d syl5ibr imp31 sylbi wral df-bnj19 ssralv biimtrid + impcom iunss sylibr sseq1 biimpar syl2an bnj1023 ) MUGZUHUIZDECUJZSUKZUKZ + XPNUGZOUGZULZXLXQUMZUNZXQLUGZUOZIUPZUJZUKZXLYBUOZIUPZNXPYEWCXPYFWCNABCDEF + IJLMNOPRSUAUCUDUEUFUQXPYEURUSYFYABXQUTULZXLXRULZVAZHIKVBZYDUKZYHYABYIYJYF + XPXSYAYDVCXPBYEXNBXMSCDBECXRJULZYBXRVDZABUAVEVFVGVHYFXSXRUTULZUKZYIYFYNXS + YQXPYNYEXNYNXMSCDYNECYNYOABUAVIVFVGVHXPXSYAYDVJYNXSUKYPXSYNYPXSJOUCVKVLVM + VNXQXRVOVPXOYJXMYESYJXNSYJFYBPULXLYBVQULUEVIVRVSVTXPYLYEYDXNYLXMSEDYLCEIW + DULYLHKQWAIUPUBWBWEVGXSYAYDWFWGYKYGGYCHKGUGWAZWHZUNZYSIUPZYHYMYKYABUKYIYJ + UKZUKYTYABYIYJWIYABUUBYTBUUBYTWCYAYIXTXRULZUKZXTYBUOYSUNZWCZBUUFNBUUFNWJB + GHKLMNOTWKWLWOYAUUBUUDYTUUEYAYJUUCYIXLXTXRWMWNXLXTYSYBWPWQWRWSWTYMYRIUPZG + YCXAZUUAYDYLUUHYLUUGGIXAYDUUHGHIKXBUUGGYCIXCXDXEGYCYRIXFXGYTYHUUAYGYSIXHX + IXJVNXK $. $} ${ @@ -516642,12 +521644,12 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a bnj1123 $p |- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) ) $= ( cv wcel wsbc cdm wa cfv wss wi sbcbii wb cvv wfn w3a wrex cab nfcv csuc - nfv c-bnj14 ciun wceq com bnj1095 nf5i nf3an nfrex nfab nfcxfr nfcri nfan - nfim weq eleq1w anbi2d fveq2 sseq1d imbi12d sbciegf elv 3bitri ) NCJKSZUA - ISZMTZJSZVTUBZTZUCZWBVTUDZFUEZUFZJVSUAZWAVSWCTZUCZVSVTUDZFUEZUFZRCWHJVSQU - GWIWNUHKWHWNJVSUIWKWMJWAWJJJIMJMVTLSZUJZABUKZLGULZIUMPWRJIWQJLGJGUNWPABJW - PJUPAJUPBJBWBUOZWOTWSVTUDDWFEHDSUQURUSUFJUTOVAVBVCVDVEVFVGWJJUPVHWMJUPVIJ - KVJZWEWKWGWMWTWDWJWAJKWCVKVLWTWFWLFWBVSVTVMVNVOVPVQVR $. + nfv c-bnj14 ciun wceq com bnj1095 nf5i nf3an nfrexw nfab nfcxfr nfan nfim + nfcri weq eleq1w anbi2d fveq2 sseq1d imbi12d sbciegf elv 3bitri ) NCJKSZU + AISZMTZJSZVTUBZTZUCZWBVTUDZFUEZUFZJVSUAZWAVSWCTZUCZVSVTUDZFUEZUFZRCWHJVSQ + UGWIWNUHKWHWNJVSUIWKWMJWAWJJJIMJMVTLSZUJZABUKZLGULZIUMPWRJIWQJLGJGUNWPABJ + WPJUPAJUPBJBWBUOZWOTWSVTUDDWFEHDSUQURUSUFJUTOVAVBVCVDVEVFVIWJJUPVGWMJUPVH + JKVJZWEWKWGWMWTWDWJWAJKWCVKVLWTWFWLFWBVSVTVMVNVOVPVQVR $. $} ${ @@ -517498,9 +522500,9 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) $) bnj1307 $p |- ( w e. C -> A. x w e. C ) $= - ( cv wfn cfv wceq wral wa wrex cab nfcii nfv nfra1 nfan nfrex nfab nfcxfr - nfcrii ) ABDADEKZHKZLZAKUGMGFMNZAUHOZPZHCQZERIUMAEULAHCABCJSUIUKAUIATUJAU - HUAUBUCUDUEUF $. + ( cv wfn cfv wceq wral wa wrex cab nfcii nfv nfra1 nfrexw nfcxfr nfcrii + nfan nfab ) ABDADEKZHKZLZAKUGMGFMNZAUHOZPZHCQZERIUMAEULAHCABCJSUIUKAUIATU + JAUHUAUEUBUFUCUD $. $} ${ @@ -518081,15 +523083,15 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a bnj1445 $p |- ( si -> A. d si ) $= ( cv c-bnj14 wcel cdm csn c-bnj18 cun wceq w-bnj17 w3a wa wn wral w-bnj15 wbr c0 wne nfv wex crab wfn cfv wrex cab nfre1 nfab nfcri nfan nfxfr nfex - nfcxfr nfcv nfrabw nfne nfralw nf3an nf5ri bnj1351 nf5i wsbc nfsbcw nfrex - nfn ax-5 bnj982 hbxfrbi ) HIKVMZMSJVMZVNZVOZTVMZOVOZYCVPZXSVQMSXSVRVSVTZW - AUHVJIYBYDYFUHIUHIGYCUCVOZLVMZYEVOZWBUHVIGYGYIUHGDYHMSXTVRZVOZWCUHVHDYKUH - DCYHUAVOZWCZUHVFYMUHCYLUHCUHCBXTPVOZXSXTSWGWDZKPWEZWBUHUPBYNYPUHBMSWFZPWH - WIZWCUHUOYQYRUHYQUHWJUHPWHUHPETWKZWDZJMWLUNYTUHJMYSUHEUHTEYDYEXTVQYJVSVTZ - WCUHUMYDUUAUHUHTOUHOYCUHVMZWMXTYCWNUFUBWNVTJUUBWEWCZUHNWOZTWPULUUDUHTUUCU - HNWQWRXCWSZUUAUHWJWTXAZXBXOUHMXDXEXCZUHWHXDXFWTXAUHJPUUGWSYOUHKPUUGYOUHWJ - XGXHXAXIXJXKXAYKUHWJWTXAUHTUCUHUCUIKYAWOZTWPURUUHUHTUIUHKYAUHYAXDUIEJXSXL - UHUQEUHJXSUHXSXDUUFXMXAXNWRXCWSYIUHWJXHXAXIYBUHXPYDUHUUEXIYFUHXPXQXR $. + nfcxfr nfn nfcv nfrabw nfne nfralw nf3an nf5ri bnj1351 nf5i nfsbcw nfrexw + wsbc ax-5 bnj982 hbxfrbi ) HIKVMZMSJVMZVNZVOZTVMZOVOZYCVPZXSVQMSXSVRVSVTZ + WAUHVJIYBYDYFUHIUHIGYCUCVOZLVMZYEVOZWBUHVIGYGYIUHGDYHMSXTVRZVOZWCUHVHDYKU + HDCYHUAVOZWCZUHVFYMUHCYLUHCUHCBXTPVOZXSXTSWGWDZKPWEZWBUHUPBYNYPUHBMSWFZPW + HWIZWCUHUOYQYRUHYQUHWJUHPWHUHPETWKZWDZJMWLUNYTUHJMYSUHEUHTEYDYEXTVQYJVSVT + ZWCUHUMYDUUAUHUHTOUHOYCUHVMZWMXTYCWNUFUBWNVTJUUBWEWCZUHNWOZTWPULUUDUHTUUC + UHNWQWRXCWSZUUAUHWJWTXAZXBXDUHMXEXFXCZUHWHXEXGWTXAUHJPUUGWSYOUHKPUUGYOUHW + JXHXIXAXJXKXLXAYKUHWJWTXAUHTUCUHUCUIKYAWOZTWPURUUHUHTUIUHKYAUHYAXEUIEJXSX + OUHUQEUHJXSUHXSXEUUFXMXAXNWRXCWSYIUHWJXIXAXJYBUHXPYDUHUUEXJYFUHXPXQXR $. $} ${ @@ -518117,13 +523119,13 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a 3-Jun-2011.) (New usage is discouraged.) $) bnj1446 $p |- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) ) $= ( cv cfv wceq cop csn cun cuni c-bnj14 wrex cab nfcv wsbc wcel c-bnj18 wa - cdm wral nfre1 nfab nfcxfr nfcri nfan nfxfr nfsbcw nfrex nfuni cres nfres - wfn nfv nfop nffv nfsn nfun nfeq nf5ri ) FUOZLUPZQOUPZUQTTWLWMTWKLTLKDUOZ - SOUPZURZUSZUTUMTKWQTKPVAUKTPTPUAEGMWNVBZVCZNVDUJWSTNUATEWRTWRVEZUACDEUOZV - FTUICTDXATXAVECNUOZIVGZXBVJWNUSGMWNVHUTUQZVITUEXCXDTTNITIXBTUOZWCWNXBUPRO - UPUQDXEVKVIZTHVCZNVDUDXGTNXFTHVLVMVNVOXDTWDVPVQVRVQVSVMVNVTVNZTWPTWNWOTWN - VEZTSOTOVEZTSWNKWRWAZURULTWNXKXITKWRXHWTWBWEVNWFWEWGWHVNZTWKVEZWFTQOXJTQW - KLGMWKVBZWAZURUNTWKXOXMTLXNXLTXNVEWBWEVNWFWIWJ $. + cdm wfn wral nfre1 nfab nfcxfr nfcri nfan nfxfr nfsbcw nfrexw nfuni nfres + nfv cres nfop nffv nfsn nfun nfeq nf5ri ) FUOZLUPZQOUPZUQTTWLWMTWKLTLKDUO + ZSOUPZURZUSZUTUMTKWQTKPVAUKTPTPUAEGMWNVBZVCZNVDUJWSTNUATEWRTWRVEZUACDEUOZ + VFTUICTDXATXAVECNUOZIVGZXBVJWNUSGMWNVHUTUQZVITUEXCXDTTNITIXBTUOZVKWNXBUPR + OUPUQDXEVLVIZTHVCZNVDUDXGTNXFTHVMVNVOVPXDTWCVQVRVSVRVTVNVOWAVOZTWPTWNWOTW + NVEZTSOTOVEZTSWNKWRWDZURULTWNXKXITKWRXHWTWBWEVOWFWEWGWHVOZTWKVEZWFTQOXJTQ + WKLGMWKVBZWDZURUNTWKXOXMTLXNXLTXNVEWBWEVOWFWIWJ $. $} ${ @@ -518400,20 +523402,20 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a ( wcel wss cv c-bnj14 wral csn c-bnj18 cun wex wn wbr bnj1212 bnj1147 a1i snssd unssd eqsstrid wa elsni adantl bnj602 syl w-bnj15 c0 simplbi bnj835 wne bnj906 syl2anc ad2antrr eqsstrd ssun4 sseqtrrdi simpr bnj1213 bnj1125 - wceq syl3anc sstrd wo bnj1424 mpjaodan ralrimiva wsbc cvv wb snex bnj1149 - bnj893 eqeltrid weq sseq1d cbvralvw anbi2i sbcbii bitrdi sseq1 raleqbi1dv - bnj1454 sseq2 anbi12d sbcieg bitrd mpbir2and ) BOHUQZOGURZGMFUSZUTZOURZFO - VAZBODUSZVBZGMYGVCZVDZGUPBYHYIGBYGGCNVEVFABEUSYGMVGVFEJVAZDGJUGUIVHZVKYIG - URBGMYGVIZVJVLVMBYEFOBYCOUQZVNZYCYHUQZYEYCYIUQZYOYPVNZYDYIURZYEYRYDGMYGUT - ZYIYRYCYGWMZYDYTWMYPUUAYOYCYGVOVPGMYCYGVQVRBYTYIURZYNYPBGMVSZYGGUQZUUBAYG - JUQYKUUCBUIAUUCJVTWCUHWAWBZYLGMYGWDWEWFWGYSYDYJOYDYIYHWHUPWIZVRYOYQVNZYSY - EUUGYDGMYCVCZYIUUGUUCYCGUQYDUUHURBUUCYNYQUUEWFZUUGFYIGYMYOYQWJZWKGMYCWDWE - UUGUUCUUDYQUUHYIURUUIBUUDYNYQYLWFUUJGMYGYCWLWNWOUUFVRYNYPYQWPBOYHYIYCUPWQ - VPWRWSBYAUAUSZGURZYDUUKURZFUUKVAZVNZUAOWTZYBYFVNZBYAUULYTUUKURZDUUKVAZVNZ - UAOWTZUUPBOXAUQZYAUVAXBBOYJXAUPBYHYIYHXAUQBYGXCVJBUUCUUDYIXAUQUUEYLGMYGXE - WEXDXFZUUTUAHOUCXOVRUUTUUOUAOUUSUUNUULUURUUMDFUUKDFXGYTYDUUKGMYGYCVQXHXIX - JXKXLBUVBUUPUUQXBUVCUUOUUQUAOXAUUKOWMUULYBUUNYFUUKOGXMUUMYEFUUKOUUKOYDXPX - NXQXRVRXSXT $. + wceq syl3anc sstrd wo bnj1424 mpjaodan ralrimiva wsbc cvv wb vsnex bnj893 + bnj1149 eqeltrid bnj1454 sseq1d cbvralvw anbi2i sbcbii bitrdi sseq1 sseq2 + weq raleqbi1dv anbi12d sbcieg bitrd mpbir2and ) BOHUQZOGURZGMFUSZUTZOURZF + OVAZBODUSZVBZGMYGVCZVDZGUPBYHYIGBYGGCNVEVFABEUSYGMVGVFEJVAZDGJUGUIVHZVKYI + GURBGMYGVIZVJVLVMBYEFOBYCOUQZVNZYCYHUQZYEYCYIUQZYOYPVNZYDYIURZYEYRYDGMYGU + TZYIYRYCYGWMZYDYTWMYPUUAYOYCYGVOVPGMYCYGVQVRBYTYIURZYNYPBGMVSZYGGUQZUUBAY + GJUQYKUUCBUIAUUCJVTWCUHWAWBZYLGMYGWDWEWFWGYSYDYJOYDYIYHWHUPWIZVRYOYQVNZYS + YEUUGYDGMYCVCZYIUUGUUCYCGUQYDUUHURBUUCYNYQUUEWFZUUGFYIGYMYOYQWJZWKGMYCWDW + EUUGUUCUUDYQUUHYIURUUIBUUDYNYQYLWFUUJGMYGYCWLWNWOUUFVRYNYPYQWPBOYHYIYCUPW + QVPWRWSBYAUAUSZGURZYDUUKURZFUUKVAZVNZUAOWTZYBYFVNZBYAUULYTUUKURZDUUKVAZVN + ZUAOWTZUUPBOXAUQZYAUVAXBBOYJXAUPBYHYIYHXAUQBDXCVJBUUCUUDYIXAUQUUEYLGMYGXD + WEXEXFZUUTUAHOUCXGVRUUTUUOUAOUUSUUNUULUURUUMDFUUKDFXOYTYDUUKGMYGYCVQXHXIX + JXKXLBUVBUUPUUQXBUVCUUOUUQUAOXAUUKOWMUULYBUUNYFUUKOGXMUUMYEFUUKOUUKOYDXNX + PXQXRVRXSXT $. $} ${ @@ -518469,12 +523471,12 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a 3-Jun-2011.) (New usage is discouraged.) $) bnj1467 $p |- ( w e. Q -> A. d w e. Q ) $= ( cv cfv cop csn cun cuni c-bnj14 wrex cab nfcv wsbc wcel c-bnj18 wceq wa - cdm wral nfre1 nfab nfcxfr nfcri nfan nfxfr nfsbcw nfrex nfuni cres nfres - wfn nfv nfop nffv nfsn nfun nfcrii ) SFLSLKDUMZROUNZUOZUPZUQULSKWKSKPURUJ - SPSPTEGMWHUSZUTZNVAUIWMSNTSEWLSWLVBZTCDEUMZVCSUHCSDWOSWOVBCNUMZIVDZWPVHWH - UPGMWHVEUQVFZVGSUDWQWRSSNISIWPSUMZWAWHWPUNQOUNVFDWSVIVGZSHUTZNVAUCXASNWTS - HVJVKVLVMWRSWBVNVOVPVOVQVKVLVRVLZSWJSWHWISWHVBZSROSOVBSRWHKWLVSZUOUKSWHXD - XCSKWLXBWNVTWCVLWDWCWEWFVLWG $. + cdm wfn wral nfre1 nfab nfcxfr nfcri nfan nfxfr nfsbcw nfrexw nfuni nfres + nfv cres nfop nffv nfsn nfun nfcrii ) SFLSLKDUMZROUNZUOZUPZUQULSKWKSKPURU + JSPSPTEGMWHUSZUTZNVAUIWMSNTSEWLSWLVBZTCDEUMZVCSUHCSDWOSWOVBCNUMZIVDZWPVHW + HUPGMWHVEUQVFZVGSUDWQWRSSNISIWPSUMZVIWHWPUNQOUNVFDWSVJVGZSHUTZNVAUCXASNWT + SHVKVLVMVNWRSWAVOVPVQVPVRVLVMVSVMZSWJSWHWISWHVBZSROSOVBSRWHKWLWBZUOUKSWHX + DXCSKWLXBWNVTWCVMWDWCWEWFVMWG $. $} ${ @@ -518510,23 +523512,23 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a ( vw wcel cv wfn cfv wceq wral wrex wsbc c-bnj14 cres cop wex elexd eleq1 cvv fneq2 raleq anbi12d wss bnj1317 nfcii nfel2 bnj1467 nfcv nffn bnj1446 wa nf5i nfralw nf5ri jca32 bnj1465 mpdan df-rex sylibr wb bnj1466 bnj1448 - nfan nfrex nfeq2 fneq1 fveq1 reseq1 opeq2d eqtr4di fveq2d eqeq12d ralbidv - rexbid bnj1468 syl mpbird fveq2 id bnj602 reseq2d opeq12d eqtrid cbvralvw - anbi2i rexbii sbcbii bnj1454 ) BLIVBZNVCZUAVCZVDZDVCZYGVEZSPVEZVFZDYHVGZW - HZUAHVHZNLVIZBYIFVCZYGVEZYRYGGMYRVJZVKZVLZPVEZVFZFYHVGZWHZUAHVHZNLVIZYQBU - UHLYHVDZYRLVEZRPVEZVFZFYHVGZWHZUAHVHZBYHHVBZUUNWHZUAVMZUUOBOVPVBUURBOHUTV - NUUQOHVBZLOVDZUULFOVGZWHZWHZBUAOVPYHOVFZUUPUUSUUNUVBYHOHVOUVDUUIUUTUUMUVA - YHOLVQUULFYHOVRVSVSUVCUAUUSUVBUAUAOHUAVAHYHGVTGMYJVJZYHVTDYHVGWHUAVAHUCWA - WBWCUUTUVAUAUAOLUAVALABCDEVAGHIJKLMNPQSTUAUBUCUDUEUFUGUHUIUJUKULUMUNWDWBZ - UAOWEZWFUULUAFOUVGUULUAABCDEFGHIJKLMNPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOW - GWIWJWTWTWKBUUSUUTUVAUTUSURWLWMWNUUNUAHWOWPBLVPVBZUUHUUOWQUQUUGUUONVALVPU - UONUUNNUAHNHWEUUIUUMNNYHLNVALABCDEVAGHIJKLMNPQSTUAUBUCUDUEUFUGUHUIUJUKULU - MUNWRZWBNYHWEZWFUULNFYHUVJUULNABCDEFGHIJKLMNPQRSTUAUBUCUDUEUFUGUHUIUJUKUL - UMUNUOWSWIWJWTXAWKYGLVFZUUFUUNUAHUAYGLUVFXBUVKYIUUIUUEUUMYHYGLXCUVKUUDUUL - FYHUVKYSUUJUUCUUKYRYGLXDUVKUUBRPUVKUUBYRLYTVKZVLRUVKUUAUVLYRYGLYTXEXFUOXG - 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cs1 wal cdm crn edgval eleq2i elrnrexdm eqcom rexbii syl6ib syl5bi df-rex - wb lp1cycl 3expib eximdv syld s1len ax-gen 19.29r mpan2 syl6 imp cvv cvtx - wne cdif uhgredgn0 eldifsni snnzb s2fv0 alrimiv syl2anc exbii cword s1cli - c0 cpw breq1 fveqeq2 3anbi12d rspcev mpan rexex exlimiv s2cli breq2 fveq1 - eqeq1d 3anbi13d eximi ex impbid ) CUBFZBGZDGZCUCHZIZYIJHZKLZMYJHZALZNZDOZ - BOZAPZCUDHZFZYSYHUUBYQYHUUBQBDYQYHYOPZUUAFZUUBYLYNYHUUDQZYPYLYNRZYIYJCUEH - IZYNYOKYJHZLZNZUUEUUFUUGUUIYNNZUUJUUFUUGUUIRZYNRZUUKYLYNUUMUUFUULYNUUFYIY - JCUFHIZUUIRZUULYLUUNYOYMYJHZLZRZYNUUOYJYICUGYNUURUUOYNUUQUUIUUNYNUUPUUHYO - YMKYJUHUIUJUMUKUUNUUGUUIYJYICULUNSUNUAUUGUUIYNUOVDUUGUUIYNUPUQUUGYNUUIUUE - CURHZUSZUUGYNUUIRZRUUDYHUUTUUGUVAUUDYJYICUTVAUUSCUUSVBZVCZVEVFSVGYPYLUUDU - UBWGYNYPUUCYTUUAYOAVHVIVJVKVLVMYHUUBYSYHUUBRZYIAAVNZYKIZYNMUVEHZALZNZBOZY - SUVDEGZVOZUVEYKIZUVLJHKLZUVHNZEOZUVJUVDUVMUVNRZUVHRZEOZUVPUVDUVQEOZUVHEVP - UVSYHUUBUVTYHUUBUVMEOZUVTYHUUBUVKUUSVQZFZUVKUUSHZYTLZRZEOZUWAYHUUBUWEEUWB - TZUWGYHUUTUUBUWHQUVCUUBYTUUSVRZFZUUTUWHUUAUWIYTCVSVTUUTUWJYTUWDLZEUWBTUWH - 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(Contributed by cacycgr ccycls wn isacycgr c2 cle c1 cmin co cn0 cxnn0 3nn0 cvtx hashxnn0 fvexi ax-mp xnn0lem1lt mp2an cxr rexri xnn0xr xrlenlt breq1i cusgr3cyclex 3re 3m1e2 3bitr3i 3ne0 neeq1 mpbiri hasheq0 necon3bii sylib anim2i 2eximi - elv ex syl5bi con1d sylbid cusgr wi cusgrusgr usgrcyclgt2v 3expib syl6ibr - syl exlimdvv con2d sylibrd impbid ) AFGZAUBGZBHIZJKLZWMWNDUAZEUAZAUCILZWQ - MNZOZEPDPZUDZWPDAFEUEZWMWPXBWPUDZUFWOKLZWMXBJWOUGLZJUHUIUJZWOKLZXEXFJUKGW - OULGZXGXIQUMBRGXJBAUNCUPBRUOUQZJWOURUSJUTGWOUTGZXGXEQJVFVAXJXLXKWOVBUQJWO - VCUSXHUFWOKVGVDVHZWMXFXBWMXFOWSWQHIZJSZOZEPDPXBDABECVEXPXADEXOWTWSXOXNTNZ - WTXOXQJTNVIXNJTVJVKXNTWQMXNTSWQMSQDWQRVLVQVMVNVOVPWHVRVSVTWAWMWPXCWNWMXBW - PWMXAXEDEWMXAXFXEWMAWBGZXAXFWCAWDXRWSWTXFWRWQABCWEWFWHXMWGWIWJXDWKWL $. + elv syl biimtrid con1d sylbid cusgr cusgrusgr usgrcyclgt2v 3expib syl6ibr + ex wi exlimdvv con2d sylibrd impbid ) AFGZAUBGZBHIZJKLZWMWNDUAZEUAZAUCILZ + WQMNZOZEPDPZUDZWPDAFEUEZWMWPXBWPUDZUFWOKLZWMXBJWOUGLZJUHUIUJZWOKLZXEXFJUK + GWOULGZXGXIQUMBRGXJBAUNCUPBRUOUQZJWOURUSJUTGWOUTGZXGXEQJVFVAXJXLXKWOVBUQJ + WOVCUSXHUFWOKVGVDVHZWMXFXBWMXFOWSWQHIZJSZOZEPDPXBDABECVEXPXADEXOWTWSXOXNT + NZWTXOXQJTNVIXNJTVJVKXNTWQMXNTSWQMSQDWQRVLVQVMVNVOVPVRWGVSVTWAWMWPXCWNWMX + BWPWMXAXEDEWMXAXFXEWMAWBGZXAXFWHAWCXRWSWTXFWRWQABCWDWEVRXMWFWIWJXDWKWL $. $} $( A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, @@ -520691,32 +525693,32 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by wf eqtrid simplrr fveq2 neeq12d rspcv sylc eqnetrd necomd simpllr necon3d id f1ocnvfv mpd ralrimiva cbvralvw jca ex vex f1oeq1 fveq1 neeq1d ralbidv anbi12d elab coex cnvex 3imtr4g wb anbi12i wfo f1ofo adantrr f1ofn simplr - wfn simprrl cocan2 syl3anc f1of1 simprll cocan1 bitrd syl5bi exlimdv mp2d - wf1 dom2d enfii ancoms hashdom mpbird derangval 3brtr4d ) CDUCUDZDIJZKZCC - FLZMZBLZYFNZYHOZBCPZKZFUFZUGNZDDYFMZYJBDPZKZFUFZUGNZCENZDENZUHYEYNYSUHUDZ - YMYRUIUDZYECDHLZMZHUJZYRIJZUUCYEYCUUFYCYDUKCDHULUMYDUUGYCYPDFUNUOZYEUUEUU - GUUCUPZHYEUUEUUIYEUUEKZUAUBYMYRUUDUALZQZUUDUQZQZUUDUBLZQZUUMQZIUUJCCUUKMZ - YHUUKNZYHOZBCPZKZDDUUNMZYHUUNNZYHOZBDPZKZUUKYMJZUUNYRJUUJUVBUVGUUJUVBKZUV - CUVFUVICDUULMZDCUUMMZUVCUUEUURUVJYEUVACCDUUDUUKURUSZUUEUVKYEUVBCDUUDUTZVA - ZDCDUULUUMURRUVIUELZUUNNZUVOOZUEDPUVFUVIUVQUEDUVIUVODJZKZUVPUVOUUKUUMQZNZ - UUDNZUVOUVSUVPUVOUUDUVTQZNZUWBUVOUUNUWCUUDUUKUUMVBVCUVIDCUVTVIZUVRUWDUWBS - UVIDCUVTMZUWEUVIUURUVKUWFUUJUURUVAVDUVNDCCUUKUUMURRDCUVTVETZDCUVOUUDUVTVF - VGVJUVSUVOUUMNZUWAOUWBUVOOUVSUWAUWHUVSUWAUWHUUKNZUWHUVIDCUUMVIZUVRUWAUWIS - UVIUVKUWJUVNDCUUMVETZDCUVOUUKUUMVFVGUVSUWHCJUVAUWIUWHOZUVIDCUVOUUMUWKVHUU - JUURUVAUVRVKUUTUWLBUWHCYHUWHSZUUSUWIYHUWHYHUWHUUKVLUWMVTVMVNVOVPVQUVSUWBU - VOUWHUWAUVSUUEUWACJUWBUVOSUWHUWASUPYEUUEUVBUVRVRUVIDCUVOUVTUWGVHCDUWAUVOU - UDWARVSWBVPWCUVQUVEUEBDUVOYHSZUVPUVDUVOYHUVOYHUUNVLUWNVTVMWDUMWEWFYLUVBFU - UKUAWGZYFUUKSZYGUURYKUVACCYFUUKWHUWPYJUUTBCUWPYIUUSYHYHYFUUKWIWJWKWLWMZYQ - UVGFUUNUULUUMUUDUUKHWGZUWOWNUUDUWRWOWNYFUUNSZYOUVCYPUVFDDYFUUNWHUWSYJUVEB - DUWSYIUVDYHYHYFUUNWIWJWKWLWMWPUVHUUOYMJZKUVBCCUUOMZYHUUONZYHOZBCPZKZKZUUJ - UUNUUQSZUUKUUOSZWQZUVHUVBUWTUXEUWQYLUXEFUUOUBWGYFUUOSZYGUXAYKUXDCCYFUUOWH - UXJYJUXCBCUXJYIUXBYHYHYFUUOWIWJWKWLWMWRUUJUXFUXIUUJUXFKZUXGUULUUPSZUXHUXK - DCUUMWSZUULCXDZUUPCXDZUXGUXLWQUXKUVKUXMUUEUVKYEUXFUVMVADCUUMWTTUXKUVJUXNU - UJUVBUVJUXEUVLXACDUULXBTUXKCDUUPMZUXOUXKUUEUXAUXPYEUUEUXFXCUUJUVBUXAUXDXE - ZCCDUUDUUOURRCDUUPXBTDCUUMUULUUPXFXGUXKCDUUDXOZCCUUKVIZCCUUOVIZUXLUXHWQUU - EUXRYEUXFCDUUDXHVAUXKUURUXSUUJUURUVAUXEXICCUUKVETUXKUXAUXTUXQCCUUOVETCCDU - UDUUKUUOXJXGXKWFXLXPWFXMXNYEYMIJZUUGUUBUUCWQYECIJZUYAYDYCUYBCDXQXRZYKCFUN - TUUHYMYRIXSRXTYEUYBYTYNSUYCABCEFGYATYDUUAYSSYCABDEFGYAUOYB $. + wfn simprrl cocan2 syl3anc wf1 f1of1 simprll bitrd biimtrid dom2d exlimdv + cocan1 mp2d enfii ancoms hashdom mpbird derangval 3brtr4d ) CDUCUDZDIJZKZ + CCFLZMZBLZYFNZYHOZBCPZKZFUFZUGNZDDYFMZYJBDPZKZFUFZUGNZCENZDENZUHYEYNYSUHU + DZYMYRUIUDZYECDHLZMZHUJZYRIJZUUCYEYCUUFYCYDUKCDHULUMYDUUGYCYPDFUNUOZYEUUE + UUGUUCUPZHYEUUEUUIYEUUEKZUAUBYMYRUUDUALZQZUUDUQZQZUUDUBLZQZUUMQZIUUJCCUUK + MZYHUUKNZYHOZBCPZKZDDUUNMZYHUUNNZYHOZBDPZKZUUKYMJZUUNYRJUUJUVBUVGUUJUVBKZ + UVCUVFUVICDUULMZDCUUMMZUVCUUEUURUVJYEUVACCDUUDUUKURUSZUUEUVKYEUVBCDUUDUTZ + VAZDCDUULUUMURRUVIUELZUUNNZUVOOZUEDPUVFUVIUVQUEDUVIUVODJZKZUVPUVOUUKUUMQZ + NZUUDNZUVOUVSUVPUVOUUDUVTQZNZUWBUVOUUNUWCUUDUUKUUMVBVCUVIDCUVTVIZUVRUWDUW + BSUVIDCUVTMZUWEUVIUURUVKUWFUUJUURUVAVDUVNDCCUUKUUMURRDCUVTVETZDCUVOUUDUVT + VFVGVJUVSUVOUUMNZUWAOUWBUVOOUVSUWAUWHUVSUWAUWHUUKNZUWHUVIDCUUMVIZUVRUWAUW + ISUVIUVKUWJUVNDCUUMVETZDCUVOUUKUUMVFVGUVSUWHCJUVAUWIUWHOZUVIDCUVOUUMUWKVH + UUJUURUVAUVRVKUUTUWLBUWHCYHUWHSZUUSUWIYHUWHYHUWHUUKVLUWMVTVMVNVOVPVQUVSUW + BUVOUWHUWAUVSUUEUWACJUWBUVOSUWHUWASUPYEUUEUVBUVRVRUVIDCUVOUVTUWGVHCDUWAUV + OUUDWARVSWBVPWCUVQUVEUEBDUVOYHSZUVPUVDUVOYHUVOYHUUNVLUWNVTVMWDUMWEWFYLUVB + FUUKUAWGZYFUUKSZYGUURYKUVACCYFUUKWHUWPYJUUTBCUWPYIUUSYHYHYFUUKWIWJWKWLWMZ + YQUVGFUUNUULUUMUUDUUKHWGZUWOWNUUDUWRWOWNYFUUNSZYOUVCYPUVFDDYFUUNWHUWSYJUV + EBDUWSYIUVDYHYHYFUUNWIWJWKWLWMWPUVHUUOYMJZKUVBCCUUOMZYHUUONZYHOZBCPZKZKZU + UJUUNUUQSZUUKUUOSZWQZUVHUVBUWTUXEUWQYLUXEFUUOUBWGYFUUOSZYGUXAYKUXDCCYFUUO + WHUXJYJUXCBCUXJYIUXBYHYHYFUUOWIWJWKWLWMWRUUJUXFUXIUUJUXFKZUXGUULUUPSZUXHU + XKDCUUMWSZUULCXDZUUPCXDZUXGUXLWQUXKUVKUXMUUEUVKYEUXFUVMVADCUUMWTTUXKUVJUX + NUUJUVBUVJUXEUVLXACDUULXBTUXKCDUUPMZUXOUXKUUEUXAUXPYEUUEUXFXCUUJUVBUXAUXD + XEZCCDUUDUUOURRCDUUPXBTDCUUMUULUUPXFXGUXKCDUUDXHZCCUUKVIZCCUUOVIZUXLUXHWQ + UUEUXRYEUXFCDUUDXIVAUXKUURUXSUUJUURUVAUXEXJCCUUKVETUXKUXAUXTUXQCCUUOVETCC + DUUDUUKUUOXOXGXKWFXLXMWFXNXPYEYMIJZUUGUUBUUCWQYECIJZUYAYDYCUYBCDXQXRZYKCF + UNTUUHYMYRIXSRXTYEUYBYTYNSUYCABCEFGYATYDUUAYSSYCABDEFGYAUOYB $. $( The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario @@ -521207,7 +526209,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by LHSHYDYFHNUYBYGWMWC $. $} - $( A two-term recurrence for the subfactorial. This theorem allows us to + $( A two-term recurrence for the subfactorial. This theorem allows to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and ~ subfac0 , ~ subfac1 . (Contributed by Mario Carneiro, 23-Jan-2015.) $) @@ -521590,28 +526592,28 @@ property of an acyclic graph (see also ~ acycgr0v ). 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NN ) $. erdszelem.s $e |- ( ph -> S e. NN ) $. @@ -522335,21 +527337,21 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by co wn adantl cicc wf iiuni iiconn a1i cdif ccld simprr eqtrid wss elssuni uncom wb syl incom uneqdifeq mpbid ctop pconntop ad3antrrr opncld 0elunit eqeltrrd eqeltrd conncn 1elunit ffvelcdm sylancl inelcm pm2.21ddne neqned - expr pm2.01d rexlimddv exp32 exlimdvv syl5bi impd ralrimivva ctopon sylib - toptopon dfconn2 mpbird ) AUAGZAUBGZBHZIJZCHZIJZXMXOUCZIKZUDZXMXOUEZAUFZJ - ZULZCAUGBAUGZXKYCBCAAXSXNXPLZXRLXKXMAGZXOAGZLZLZYBXNXPXRUHYIYEXRYBYEDHZXM - GZEHZXOGZLZEMDMZYIXRYBULZYEYKDMZYMEMZLYOXNYQXPYRDXMUMEXOUMUIYKYMDEUJUKYIY - NYPDEYIYNXRYBYIYNXRLZLZNFHZOZYJKZPUUAOZYLKZLZYBFQAUNVDZYTXKYJYAGZYLYAGZUU - FFUUGUOXKYHYSUPYTYKYFUUHYIYKYMXRUQZXKYFYGYSURZYJXMAUSRYTYMYGUUIYIYKYMXRUT - ZXKYFYGYSTZYLXOAUSRYJYLFAYAYAVAZVBVCYTUUAUUGGZUUFLZLZXTYAUUQXTYAKZYTUUPUU - RUURVEZYTUUPUURLZLZUUSXQIYIYNXRUUTTZUVAYLXMGYMXQIJUVAUUDYLXMUUTUUEYTUUOUU - CUUEUURTVFUVANPVGVDZXMUUAVHPUVCGUUDXMGUVANXMUUAQAUVCVIQUBGUVAVJVKYTUUOUUF - UURUQYTYFUUTUUKSUVAYAXOVLZXMAVMOZUVAXOXMUEZYAKZUVDXMKZUVAUVFXTYAXOXMVRYTU - UPUURVNVOUVAXOYAVPZXOXMUCZIKUVGUVHVSUVAYGUVIYTYGUUTUUMSZXOAVQVTUVAUVJXQIX - OXMWAUVBVOXOXMYAWBRWCUVAAWDGZYGUVDUVEGXKUVLYHYSUUTAWEZWFUVKXOAYAUUNWGRWIN - UVCGUVAWHVKUVAUUBYJXMUUTUUCYTUUOUUCUUEUURURVFYTYKUUTUUJSWJWKWLUVCXMPUUAWM - WNWIYTYMUUTUULSYLXMXOWORWPWRWSWQWTXAXBXCXDXCXEXKAYAXFOGZXLYDVSXKUVLUVNUVM - AYAUUNXHXGBCAYAXIVTXJ $. + expr pm2.01d rexlimddv exp32 exlimdvv biimtrid ralrimivva ctopon toptopon + impd sylib dfconn2 mpbird ) AUAGZAUBGZBHZIJZCHZIJZXMXOUCZIKZUDZXMXOUEZAUF + ZJZULZCAUGBAUGZXKYCBCAAXSXNXPLZXRLXKXMAGZXOAGZLZLZYBXNXPXRUHYIYEXRYBYEDHZ + XMGZEHZXOGZLZEMDMZYIXRYBULZYEYKDMZYMEMZLYOXNYQXPYRDXMUMEXOUMUIYKYMDEUJUKY + IYNYPDEYIYNXRYBYIYNXRLZLZNFHZOZYJKZPUUAOZYLKZLZYBFQAUNVDZYTXKYJYAGZYLYAGZ + UUFFUUGUOXKYHYSUPYTYKYFUUHYIYKYMXRUQZXKYFYGYSURZYJXMAUSRYTYMYGUUIYIYKYMXR + UTZXKYFYGYSTZYLXOAUSRYJYLFAYAYAVAZVBVCYTUUAUUGGZUUFLZLZXTYAUUQXTYAKZYTUUP + UURUURVEZYTUUPUURLZLZUUSXQIYIYNXRUUTTZUVAYLXMGYMXQIJUVAUUDYLXMUUTUUEYTUUO + UUCUUEUURTVFUVANPVGVDZXMUUAVHPUVCGUUDXMGUVANXMUUAQAUVCVIQUBGUVAVJVKYTUUOU + UFUURUQYTYFUUTUUKSUVAYAXOVLZXMAVMOZUVAXOXMUEZYAKZUVDXMKZUVAUVFXTYAXOXMVRY + TUUPUURVNVOUVAXOYAVPZXOXMUCZIKUVGUVHVSUVAYGUVIYTYGUUTUUMSZXOAVQVTUVAUVJXQ + IXOXMWAUVBVOXOXMYAWBRWCUVAAWDGZYGUVDUVEGXKUVLYHYSUUTAWEZWFUVKXOAYAUUNWGRW + INUVCGUVAWHVKUVAUUBYJXMUUTUUCYTUUOUUCUUEUURURVFYTYKUUTUUJSWJWKWLUVCXMPUUA + WMWNWIYTYMUUTUULSYLXMXOWORWPWRWSWQWTXAXBXCXGXCXDXKAYAXEOGZXLYDVSXKUVLUVNU + VMAYAUUNXFXHBCAYAXIVTXJ $. $( The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) $) @@ -522884,24 +527886,24 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by inss1 wi cxp cpw wf wfn ioof ffn ovelrn mp2b simprrr sstrid ineq1d oveq2d simprrl ioossre cconn eqid resconn reconn bitrd ax-mp mpbi ssralv ralimdv ioosconn syld mp1i inss2 iccconn iccssre syl mpbid ad2antrr mpsyl 2ralbii - ssin r19.26-2 bitr3i sylanbrc sstri sylibr eqeltrd exp32 rexlimdvw syl5bi - 3jca reximdvai ctb retopbas bastg syl6 mpd ralrimivva ctop cvv retop ovex - ssrexv subislly mp2an ) AHIBHIJZCKZABUCLZUDZDKZMZEKZXRIZNUEZUFUGZXTOLZPIZ - UHZCYFQZEYAXSUDZRDYFRZYFXSOLZPUIIZXQYJDEYFYKXQYAYFIZYCYKIZJZJZYDXRYAMZJZC - YEQZYJYRYOYCYAIUUAXQYOYPUJYRYKYAYCYAXSUPXQYOYPUKULCYAYEYCUMUNYRUUAYICYEQZ - YJYRYTYICYEXRYEIZXRUAKZUBKZNLZUOZUBSQZUASQZYRYTYIUQZSSURZHUSZNUTNUUKVAUUC - UUITVBUUKUULNVCUAUBSSXRNVDVEYRUUHUUJUASYRUUGUUJUBSYRUUGYTYIYRUUGYTJZJZYBY - DYHUUNXTXRYAXRXSUPYRUUGYDYSVFVGYRUUGYDYSVJUUNYGYFUUFXSUDZOLZPUUNXTUUOYFOU - UNXRUUFXSYRUUGYTUJVHVIUUNFKGKUCLZUUOMZGUUORFUUORZUUPPIZUUNUUQUUFMZGUUORZF - UUORZUUQXSMZGUUORZFUUORZUUSUVAGUUFRZFUUFRZUVCUUNYFUUFOLZPIZUVHUUDUUEWAUUF - HMZUVJUVHTUUDUUEVKZUVKUVJUVIVLIUVHUUFUVIUVIVMVNFGUUFVOVPVQVRUUOUUFMZUVHUV - CUQUUFXSUPZUVMUVHUVBFUUFRUVCUVMUVGUVBFUUFUVAGUUOUUFVSVTUVBFUUOUUFVSWBVQWC - UUOXSMZUUNUVDGXSRZFXSRZUVFUUFXSWDXQUVQYQUUMXQYMVLIZUVQABWEXQXSHMUVRUVQTAB - WFFGXSVOWGWHWIUVOUVQUVEFXSRUVFUVOUVPUVEFXSUVDGUUOXSVSVTUVEFUUOXSVSWBWJUUS - UVAUVDJZGUUORFUUORUVCUVFJUVSUURFGUUOUUOUUQUUFXSWLWKUVAUVDFGUUOUUOWMWNWOUU - OHMZUUTUUSTUUOUUFHUVNUVLWPUVTUUTUUPVLIUUSUUOUUPUUPVMVNFGUUOVOVPVQWQWRXBWS - WTWTXAXCYEXDIYEYFMUUBYJUQXEYEXDXFYICYEYFXNVEXGXHXIYFXJIXSXKIYNYLTXLABUCXM - DECPXSYFXKXOXPWQ $. + ssin r19.26-2 bitr3i sylanbrc sstri eqeltrd 3jca exp32 rexlimdvw biimtrid + sylibr reximdvai ctb retopbas bastg ssrexv syl6 mpd ralrimivva ctop retop + cvv ovex subislly mp2an ) AHIBHIJZCKZABUCLZUDZDKZMZEKZXRIZNUEZUFUGZXTOLZP + IZUHZCYFQZEYAXSUDZRDYFRZYFXSOLZPUIIZXQYJDEYFYKXQYAYFIZYCYKIZJZJZYDXRYAMZJ + ZCYEQZYJYRYOYCYAIUUAXQYOYPUJYRYKYAYCYAXSUPXQYOYPUKULCYAYEYCUMUNYRUUAYICYE + QZYJYRYTYICYEXRYEIZXRUAKZUBKZNLZUOZUBSQZUASQZYRYTYIUQZSSURZHUSZNUTNUUKVAU + UCUUITVBUUKUULNVCUAUBSSXRNVDVEYRUUHUUJUASYRUUGUUJUBSYRUUGYTYIYRUUGYTJZJZY + BYDYHUUNXTXRYAXRXSUPYRUUGYDYSVFVGYRUUGYDYSVJUUNYGYFUUFXSUDZOLZPUUNXTUUOYF + OUUNXRUUFXSYRUUGYTUJVHVIUUNFKGKUCLZUUOMZGUUORFUUORZUUPPIZUUNUUQUUFMZGUUOR + ZFUUORZUUQXSMZGUUORZFUUORZUUSUVAGUUFRZFUUFRZUVCUUNYFUUFOLZPIZUVHUUDUUEWAU + UFHMZUVJUVHTUUDUUEVKZUVKUVJUVIVLIUVHUUFUVIUVIVMVNFGUUFVOVPVQVRUUOUUFMZUVH + UVCUQUUFXSUPZUVMUVHUVBFUUFRUVCUVMUVGUVBFUUFUVAGUUOUUFVSVTUVBFUUOUUFVSWBVQ + WCUUOXSMZUUNUVDGXSRZFXSRZUVFUUFXSWDXQUVQYQUUMXQYMVLIZUVQABWEXQXSHMUVRUVQT + ABWFFGXSVOWGWHWIUVOUVQUVEFXSRUVFUVOUVPUVEFXSUVDGUUOXSVSVTUVEFUUOXSVSWBWJU + USUVAUVDJZGUUORFUUORUVCUVFJUVSUURFGUUOUUOUUQUUFXSWLWKUVAUVDFGUUOUUOWMWNWO + UUOHMZUUTUUSTUUOUUFHUVNUVLWPUVTUUTUUPVLIUUSUUOUUPUUPVMVNFGUUOVOVPVQXBWQWR + WSWTWTXAXCYEXDIYEYFMUUBYJUQXEYEXDXFYICYEYFXGVEXHXIXJYFXKIXSXMIYNYLTXLABUC + XNDECPXSYFXMXOXPXB $. $( The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) $) @@ -523119,15 +528121,15 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by csn crest chmeo w3a cvmsi 3anass cpw crab pwexg difexg rabexg 3syl imaeq2 id eqeq2d oveq2 oveq2d eleq2d anbi2d ralbidv anbi12d rabbidv fvmptg unieq syl2anr eqeq1d difeq1 raleqdv anbi1d raleqbi1dv elrab elpw2g adantr bitrd - eldifsn wb bitrid biimprd expimpd syl5bi impbid2 ) CJMZEFDUAZMZFIMZECUBZE - NUCZOZEUDZHUEZFUFZPZBQZAQUGNPZAEXBUJZUHZRZHXBUIZCXBUKSZIFUKSZULSZMZOZBERZ - OZUMZABCDEFGHIKLUNXOWNWQXNOZOWKWMWNWQXNUOWKWNXPWMWKWNOZWMXPXQWMEKQZUDZWTP - ZXCAXRXDUHZRZXKOZBXRRZOZKCUPZNUJZUHZUQZMZXPXQWLYIEWNWNYITMZWLYIPWKWNVCWKY - FTMYHTMYKCJURYFYGTUSYEKYHTUTVAGFXSWSGQZUFZPZYBXGXHIYLUKSZULSZMZOZBXRRZOZK - YHUQYIITDYLFPZYTYEKYHUUAYNXTYSYDUUAYMWTXSYLFWSVBVDUUAYRYCBXRUUAYQXKYBUUAY - PXJXGUUAYOXIXHULYLFIUKVEVFVGVHVIVJVKLVLVNVGYJEYHMZXNOXQXPYEXNKEYHXREPZXTX - AYDXMUUCXSWRWTXREVMVOYCXLBXREUUCYBXFXKUUCXCAYAXEXREXDVPVQVRVSVJVTXQUUBWQX - NUUBEYFMZWPOXQWQEYFNWDXQUUDWOWPWKUUDWOWEWNECJWAWBVRWFVRWFWCWGWHWIWJ $. + eldifsn wb bitrid biimprd expimpd biimtrid impbid2 ) CJMZEFDUAZMZFIMZECUB + ZENUCZOZEUDZHUEZFUFZPZBQZAQUGNPZAEXBUJZUHZRZHXBUIZCXBUKSZIFUKSZULSZMZOZBE + RZOZUMZABCDEFGHIKLUNXOWNWQXNOZOWKWMWNWQXNUOWKWNXPWMWKWNOZWMXPXQWMEKQZUDZW + TPZXCAXRXDUHZRZXKOZBXRRZOZKCUPZNUJZUHZUQZMZXPXQWLYIEWNWNYITMZWLYIPWKWNVCW + KYFTMYHTMYKCJURYFYGTUSYEKYHTUTVAGFXSWSGQZUFZPZYBXGXHIYLUKSZULSZMZOZBXRRZO + ZKYHUQYIITDYLFPZYTYEKYHUUAYNXTYSYDUUAYMWTXSYLFWSVBVDUUAYRYCBXRUUAYQXKYBUU + AYPXJXGUUAYOXIXHULYLFIUKVEVFVGVHVIVJVKLVLVNVGYJEYHMZXNOXQXPYEXNKEYHXREPZX + TXAYDXMUUCXSWRWTXREVMVOYCXLBXREUUCYBXFXKUUCXCAYAXEXREXDVPVQVRVSVJVTXQUUBW + QXNUUBEYFMZWPOXQWQEYFNWDXQUUDWOWPWKUUDWOWEWNECJWAWBVRWFVRWFWCWGWHWIWJ $. $( An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, @@ -523198,22 +528200,23 @@ property of an acyclic graph (see also ~ acycgr0v ). 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(Contributed by Mario Carneiro, 7-Jul-2015.) $) @@ -523227,50 +528230,50 @@ property of an acyclic graph (see also ~ acycgr0v ). 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(Contributed by Mario Carneiro, 7-Jul-2015.) $) @@ -523324,25 +528327,25 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by cin simprr simprl cvmsiota syl13anc cvmshmeo ctop cvmtop1 simpllr elrestr simpld syl3anc hmeoima eqeltrrid cvmtop2 ad2antrr ad2antll restopn2 mpbid wb cvmsrcl wfn ffnd inss1 sstrid simplr simprd elind fnfvima imass2 eleq2 - mp1i sseq1 anbi12d rspcev expr exlimdv syl5bi expimpd rexlimdvw ralrimiva - syl12anc mpd eleq1 anbi1d rexbidv ralima mpbird eltop2 ) HEIUDOPZCEPZQZHC - UEZIPZMRZNRZPZYKYHSZQZNIUFZMYHUGZYGYPUARZHUHZYKPZYMQZNIUFZUACUGZYGUUAUACY - GYQCPZQZYRUBRZPZUUEFUHZUIUKZQZUBIUFZUUAUUDYEYRIUJZPUUJYEYFUUCULZUUDDUUKYQ - HYGDUUKHUMZUUCYGHEIUNOPZUUMYEUUNYFEHIUOTHEIDUUKLUUKVCZUPUQZTYGCDYQYFCDSZY - EYFCEUJDCEURLUSZUTZVAZVBUBABEYRFGHIUUKJKUUOVDVEUUDUUIUUAUBIUUDUUFUUHUUAUU - HUCRZUUGPZUCVFUUDUUFQZUUAUCUUGVGUVCUVBUUAUCUUDUUFUVBUUAUUDUUFUVBQZQZHCYQY - JPMUVAVHZVPZUEZIPZYRUVHPZUVHYHSZUUAUVEUVIUVHUUESZUVEUVHIUUEVIOZPZUVIUVLQZ - UVEUVHHUVFVJZUVGUEZUVMUVGUVFSUVQUVHVKCUVFVLHUVGUVFVMVNUVEUVPEUVFVIOZUVMVO - OPZUVGUVRPZUVQUVMPUVEUVBUVFUVAPZUVSUUDUUFUVBVQZUVEUWAYQUVFPZUVEYEUVBYQDPZ - UUFUWAUWCQUUDYEUVDUULTZUWBUUDUWDUVDUUTTUUDUUFUVBVRMABYQDEFUVAUUEGHIUVFJKL - UVFVCVSVTZWFZABUVFEFUVAUUEGHIJKWAVEUVEEWBPZUWAYFUVTUVEYEUWHUWEEHIWCUQUWGY - EYFUUCUVDWDZCUVFEWBUVAWEWGUVGUVPUVRUVMWHVEWIUVEIWBPZUUEIPZUVNUVOWOYGUWJUU - CUVDYEUWJYFEHIWJTZWKUVBUWKUUDUUFABEFUVAUUEGHIJKWPWLUUEUVHIWMVEWNWFUVEHDWQ - ZUVGDSYQUVGPUVJYGUWMUUCUVDYGDUUKHUUPWRZWKUVEUVGCDCUVFWSZUVEYFUUQUWIUURUQW - TUVECUVFYQYGUUCUVDXAUVEUWAUWCUWFXBXCDUVGHYQXDWGUVGCSUVKUVEUWOUVGCHXEXGYTU - VJUVKQNUVHIYKUVHVKYSUVJYMUVKYKUVHYRXFYKUVHYHXHXIXJXQXKXLXMXNXOXRXPYGUWMUU - QYPUUBWOUWNUUSYOUUAMUADCHYJYRVKZYNYTNIUWPYLYSYMYJYRYKXSXTYAYBVEYCYGUWJYIY - PWOUWLMNYHIYDUQYC $. + mp1i sseq1 anbi12d rspcev syl12anc exlimdv biimtrid expimpd rexlimdvw mpd + expr ralrimiva eleq1 anbi1d rexbidv ralima mpbird eltop2 ) HEIUDOPZCEPZQZ + HCUEZIPZMRZNRZPZYKYHSZQZNIUFZMYHUGZYGYPUARZHUHZYKPZYMQZNIUFZUACUGZYGUUAUA + CYGYQCPZQZYRUBRZPZUUEFUHZUIUKZQZUBIUFZUUAUUDYEYRIUJZPUUJYEYFUUCULZUUDDUUK + YQHYGDUUKHUMZUUCYGHEIUNOPZUUMYEUUNYFEHIUOTHEIDUUKLUUKVCZUPUQZTYGCDYQYFCDS + ZYEYFCEUJDCEURLUSZUTZVAZVBUBABEYRFGHIUUKJKUUOVDVEUUDUUIUUAUBIUUDUUFUUHUUA + UUHUCRZUUGPZUCVFUUDUUFQZUUAUCUUGVGUVCUVBUUAUCUUDUUFUVBUUAUUDUUFUVBQZQZHCY + QYJPMUVAVHZVPZUEZIPZYRUVHPZUVHYHSZUUAUVEUVIUVHUUESZUVEUVHIUUEVIOZPZUVIUVL + QZUVEUVHHUVFVJZUVGUEZUVMUVGUVFSUVQUVHVKCUVFVLHUVGUVFVMVNUVEUVPEUVFVIOZUVM + VOOPZUVGUVRPZUVQUVMPUVEUVBUVFUVAPZUVSUUDUUFUVBVQZUVEUWAYQUVFPZUVEYEUVBYQD + PZUUFUWAUWCQUUDYEUVDUULTZUWBUUDUWDUVDUUTTUUDUUFUVBVRMABYQDEFUVAUUEGHIUVFJ + KLUVFVCVSVTZWFZABUVFEFUVAUUEGHIJKWAVEUVEEWBPZUWAYFUVTUVEYEUWHUWEEHIWCUQUW + GYEYFUUCUVDWDZCUVFEWBUVAWEWGUVGUVPUVRUVMWHVEWIUVEIWBPZUUEIPZUVNUVOWOYGUWJ + UUCUVDYEUWJYFEHIWJTZWKUVBUWKUUDUUFABEFUVAUUEGHIJKWPWLUUEUVHIWMVEWNWFUVEHD + WQZUVGDSYQUVGPUVJYGUWMUUCUVDYGDUUKHUUPWRZWKUVEUVGCDCUVFWSZUVEYFUUQUWIUURU + QWTUVECUVFYQYGUUCUVDXAUVEUWAUWCUWFXBXCDUVGHYQXDWGUVGCSUVKUVEUWOUVGCHXEXGY + TUVJUVKQNUVHIYKUVHVKYSUVJYMUVKYKUVHYRXFYKUVHYHXHXIXJXKXQXLXMXNXOXPXRYGUWM + UUQYPUUBWOUWNUUSYOUUAMUADCHYJYRVKZYNYTNIUWPYLYSYMYJYRYKXSXTYAYBVEYCYGUWJY + IYPWOUWLMNYHIYDUQYC $. cvmfolem.2 $e |- X = U. J $. $( Lemma for ~ cvmfo . (Contributed by Mario Carneiro, 13-Feb-2015.) $) @@ -523351,7 +528354,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by cvmcn c0 wne cvmcov ex wex n0 cvmsn0 ad2antll sylib cres ccnv wss simprlr cuni cvmsss simprr sseldd elssuni sseqtrrdi simpll cvmsf1o syl3anc f1ocnv wf1o f1of simprll ffvelcdmd f1ocnvfv2 syl2anc fvres eqtr3d fveq2 rspceeqv - 3syl expr exlimdv syl5bi expimpd rexlimdva syld ralrimiv dffo3 sylanbrc + 3syl expr exlimdv biimtrid expimpd rexlimdva syld ralrimiv dffo3 sylanbrc mpd ) GDHUCUDQZCIGUEZNRZORZGSZUFOCUGZNIUHCIGUIWRGDHUJUDQWSDGHUMGDHCILMUKU LWRXCNIWRWTIQZWTPRZQZXEESZUNUOZTZPHUGZXCWRXDXJPABDWTEFGHIJKMUPUQWRXIXCPHW RXEHQZTZXFXHXCXHUARZXGQZUAURXLXFTZXCUAXGUSXOXNXCUAXLXFXNXCXLXFXNTZTZUBRZX @@ -523464,37 +528467,37 @@ property of an acyclic graph (see also ~ acycgr0v ). 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(Contributed by ex simprl1 simprr1 xpss12 sstrd 3jcad anim12i jca2 syl5bir rexlimdvva mp2and simp3l1 simp3l2 simpl1l simpl1r simpld eqeltrid simprd simpl2l crio df-ov simpl2r simp3rl simp3rr simp3l3 simprr cvmlift2lem9 3expia - rexlimdvaa 3jca reximdvva expr exlimdv syl5bi expimpd rexlimdvw ) AST - URZOUSZUNUTZVAZUVRKUSZVBVCZVDZUNQVEZSGUTZVAZTFUTZVAZRUWDEUTZVFVGZVLVH - VHVIVJZUWIVKVJIVMVJVAZEUWFVERUWDUWFVGZVLUWJUWLVKVJIVMVJVAZVNZVOZFVHVE - GVHVEZANIQVPVJVAZUVQQVQZVAUWCUEAVRVSVTVJZUWSVGZUWRUVPOAOUWJQVMVJVAZUW - TUWROWGZUFOUWJQUWTUWRVHVHUWSUWSWAWAWBWBWCUWRWDZWEWFZASTUWSUWSULUMWHZW - IUNUCUBIUVQKMNQUWRUAUKUXCWJWKAUWBUWPUNQAUVSUWAUWPUWAUOUTZUVTVAZUOWLAU - VSVDZUWPUOUVTWMUXHUXGUWPUOAUVSUXGUWPAUVSUXGVDZVDZUWEUWGUWLOWNUVRWOZWP - ZVOZVHUWDVKVJZWQVAZVHUWFVKVJZWQVAZVDZVDZFVHVEZGVHVEZUWPUXJSUPUTZVAZTU - QUTZVAZVDZUYBUYDVGZUXKWPZVDZUQVHVEUPVHVEZUYAUXJDUTZUYGVAZUYHVDZUQVHVE - UPVHVEZUYJDUXKUVPUYKUVPWSZUYMUYIUPUQVHVHUYOUYLUYFUYHUYOUYLUVPUYGVAUYF - UYKUVPUYGWRSTUYBUYDWTXAXBXCUXJUXKUWJVAZUYNDUXKXDZUXJUXAUVRQVAZUYPAUXA - UXIUFXEUXGUYRAUVSUCUBIKUXFUVRMNQUAUKXFXGUVROUWJQXHWKVHXIVAZUYSUYPUYQX - TWAWAUPUQUXKVHVHXIXIDXJXKXLUXJUVPUXKVAZUVPUWTVAZUVSAVUAUXIUXEXEAUVSUX - GXMUXJUXBOUWTXNUYTVUAUVSVDXTAUXBUXIUXDXEUWTUWROYAUWTUVPUVROXOXPXQXRUX - JUYIUYAUPUQVHVHUXJUYBVHVAZUYDVHVAZVDVDZUYIUYAVUDUYIVDZUWDUYBWPZUWEUXO - VOZGVHVEZUWFUYDWPZUWGUXQVOZFVHVEZUYAVHWQXSVAZVUEVUBUYCVUHYBUXJVUBVUCU - YIYCVUDUYCUYEUYHYDGWQSUYBVHYEYFVULVUEVUCUYEVUKYBUXJVUBVUCUYIYGVUDUYCU - YEUYHYHFWQTUYDVHYEYFVUHVUKVDVUGVUJVDZFVHVEZGVHVEVUEUYAVUGVUJGFVHVHYIV - UEVUNUXTGVHVUEVUMUXSFVHVUEVUMUXMUXRVUEVUMUWEUWGUXLVUMUWEVNVUEVUFUWEUX - OVUJYJYKVUMUWGVNVUEVUGVUIUWGUXQYLYKVUEVUMUXLVUEVUMVDZUWLUYGUXKVUOVUFV - UIUWLUYGWPVUFUWEUXOVUJVUEUUAVUIUWGUXQVUGVUEUUBUWDUYBUWFUYDUUCWKVUDUYF - UYHVUMYGUUDYTUUEVUGUXOVUJUXQVUFUWEUXOYMVUIUWGUXQYMUUFUUGYNYNUUHUUJYTU - UIYOUXJUXSUWOGFVHVHUXJUWDVHVAZUWFVHVAZVDZUXSUWOUXJVURUXSVOZUWEUWGUWNU - WEUWGUXLUXRUXJVURUUKZUWEUWGUXLUXRUXJVURUULZVUSUWKUWMEUWFVUSUWHUWFVAZU - WKVDZVDZBCDHIJKUXFUWDLMNOPQRUVRUWFSTRVJUYDVAUQUXFUUSZSTUWHUAUQUBUCUDV - VDAUWQAUXIVURUXSVVCUUMZUEWFVVDAUXAVVFUFWFVVDAJHVAVVFUGWFVVDAJNUSVRVRO - VJWSVVFUHWFUIUJUKVVDSTOVJUVQUVRSTOUUTVVDUVSUXGAUXIVURUXSVVCUUNZUUOUUP - VVDUVSUXGVVGUUQVUPVUQUXJUXSVVCUURVUPVUQUXJUXSVVCUVAVVDUXOUXNYPVAUXNYQ - VAVUSUXOVVCUXOUXQUXMUXJVURUVBXEUXNYRUXNYSXPVVDUXQUXPYPVAUXPYQVAVUSUXQ - VVCUXOUXQUXMUXJVURUVCXEUXPYRUXPYSXPVUSUWEVVCVUTXEVUSUWGVVCVVAXEVUSUXL - VVCUWEUWGUXLUXRUXJVURUVDXEVUSVVBUWKXMVUSVVBUWKUVEVVEWDUVFUVHUVIUVGUVJ - YOUVKUVLUVMUVNUVOYO $. + rexlimdvaa 3jca reximdvva expr exlimdv biimtrid expimpd rexlimdvw ) A + STURZOUSZUNUTZVAZUVRKUSZVBVCZVDZUNQVEZSGUTZVAZTFUTZVAZRUWDEUTZVFVGZVL + VHVHVIVJZUWIVKVJIVMVJVAZEUWFVERUWDUWFVGZVLUWJUWLVKVJIVMVJVAZVNZVOZFVH + VEGVHVEZANIQVPVJVAZUVQQVQZVAUWCUEAVRVSVTVJZUWSVGZUWRUVPOAOUWJQVMVJVAZ + UWTUWROWGZUFOUWJQUWTUWRVHVHUWSUWSWAWAWBWBWCUWRWDZWEWFZASTUWSUWSULUMWH + ZWIUNUCUBIUVQKMNQUWRUAUKUXCWJWKAUWBUWPUNQAUVSUWAUWPUWAUOUTZUVTVAZUOWL + AUVSVDZUWPUOUVTWMUXHUXGUWPUOAUVSUXGUWPAUVSUXGVDZVDZUWEUWGUWLOWNUVRWOZ + WPZVOZVHUWDVKVJZWQVAZVHUWFVKVJZWQVAZVDZVDZFVHVEZGVHVEZUWPUXJSUPUTZVAZ + TUQUTZVAZVDZUYBUYDVGZUXKWPZVDZUQVHVEUPVHVEZUYAUXJDUTZUYGVAZUYHVDZUQVH + VEUPVHVEZUYJDUXKUVPUYKUVPWSZUYMUYIUPUQVHVHUYOUYLUYFUYHUYOUYLUVPUYGVAU + YFUYKUVPUYGWRSTUYBUYDWTXAXBXCUXJUXKUWJVAZUYNDUXKXDZUXJUXAUVRQVAZUYPAU + XAUXIUFXEUXGUYRAUVSUCUBIKUXFUVRMNQUAUKXFXGUVROUWJQXHWKVHXIVAZUYSUYPUY + QXTWAWAUPUQUXKVHVHXIXIDXJXKXLUXJUVPUXKVAZUVPUWTVAZUVSAVUAUXIUXEXEAUVS + UXGXMUXJUXBOUWTXNUYTVUAUVSVDXTAUXBUXIUXDXEUWTUWROYAUWTUVPUVROXOXPXQXR + UXJUYIUYAUPUQVHVHUXJUYBVHVAZUYDVHVAZVDVDZUYIUYAVUDUYIVDZUWDUYBWPZUWEU + XOVOZGVHVEZUWFUYDWPZUWGUXQVOZFVHVEZUYAVHWQXSVAZVUEVUBUYCVUHYBUXJVUBVU + CUYIYCVUDUYCUYEUYHYDGWQSUYBVHYEYFVULVUEVUCUYEVUKYBUXJVUBVUCUYIYGVUDUY + CUYEUYHYHFWQTUYDVHYEYFVUHVUKVDVUGVUJVDZFVHVEZGVHVEVUEUYAVUGVUJGFVHVHY + IVUEVUNUXTGVHVUEVUMUXSFVHVUEVUMUXMUXRVUEVUMUWEUWGUXLVUMUWEVNVUEVUFUWE + UXOVUJYJYKVUMUWGVNVUEVUGVUIUWGUXQYLYKVUEVUMUXLVUEVUMVDZUWLUYGUXKVUOVU + FVUIUWLUYGWPVUFUWEUXOVUJVUEUUAVUIUWGUXQVUGVUEUUBUWDUYBUWFUYDUUCWKVUDU + YFUYHVUMYGUUDYTUUEVUGUXOVUJUXQVUFUWEUXOYMVUIUWGUXQYMUUFUUGYNYNUUHUUJY + TUUIYOUXJUXSUWOGFVHVHUXJUWDVHVAZUWFVHVAZVDZUXSUWOUXJVURUXSVOZUWEUWGUW + NUWEUWGUXLUXRUXJVURUUKZUWEUWGUXLUXRUXJVURUULZVUSUWKUWMEUWFVUSUWHUWFVA + ZUWKVDZVDZBCDHIJKUXFUWDLMNOPQRUVRUWFSTRVJUYDVAUQUXFUUSZSTUWHUAUQUBUCU + DVVDAUWQAUXIVURUXSVVCUUMZUEWFVVDAUXAVVFUFWFVVDAJHVAVVFUGWFVVDAJNUSVRV + ROVJWSVVFUHWFUIUJUKVVDSTOVJUVQUVRSTOUUTVVDUVSUXGAUXIVURUXSVVCUUNZUUOU + UPVVDUVSUXGVVGUUQVUPVUQUXJUXSVVCUURVUPVUQUXJUXSVVCUVAVVDUXOUXNYPVAUXN + YQVAVUSUXOVVCUXOUXQUXMUXJVURUVBXEUXNYRUXNYSXPVVDUXQUXPYPVAUXPYQVAVUSU + XQVVCUXOUXQUXMUXJVURUVCXEUXPYRUXPYSXPVUSUWEVVCVUTXEVUSUWGVVCVVAXEVUSU + XLVVCUWEUWGUXLUXRUXJVURUVDXEVUSVVBUWKXMVUSVVBUWKUVEVVEWDUVFUVHUVIUVGU + VJYOUVKUVLUVMUVNUVOYO $. $} ${ @@ -524588,24 +529591,24 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by wel vex adantl jca df-xp mp2an cop cres crest wi w3a cuni cdif adantr c0 cmpt eqid simprr simprl cvmlift2lem10 simplrl simplrr syl22anc wal eleqtrrdi syl2anr ad3antrrr sneq xpeq2d reseq2d oveq2d oveq1d eleq12d - simpr weq syl5bi cvmlift2lem11 impbid syl2anc sseq12i ssopab2bw bitri - opelxpi rexlimdvva mpd eleq1d ralrimiva sylib cvmlift2lem1 syl elrab2 - ex cmpo ctopon iitopon cvv sylancr c1 cicc cvmlift2lem5 iunid xpiundi - wf xpeq2i eqtr3i cconn iiconn ccld inss1 cnt ccmp iicmp txtopi neiss2 - cnei mpan snss a1d rexlimiv simpl ssopab2i 3sstr4i txunii ntropn ccnv - cima chmeo ccvm txopn elunii opnneip syl3anc simplr2 cbvrexvw simplr3 - cpw rspe alrimivv ssntr simpr1 simpr2 opeq2 ralsn anassrs dfss3 eleq1 - sseldd ralxp txtube ntrss2 sstr mpan2 r2al ralcom ralimi bicom rexbii - 3bitr2i ralbii sylbi rabeq2i ad3antlr elssuni sseqtrrdi sselda sseq1d - baib bibi12d syl5ibr ralimdva syl5 anim2d ssrab2 eqsstri isclo sselid - reximdva 0elunit wrel relxp opelxp df-3an wfn ffnd fnov reseq1d snssd - id resmpo cvmlift2lem8 sylan syldan elsni eqeq1d syl5ibrcom mpoeq3dva - simplll 3impia 3eqtrd cnmpt1st cvmlift2lem2 simp1d cnmpt21f cnmpt2res - ccom txrest mp4an oveq1i eqeltrd rspcev xpss12 restuni eleqtrd cncnpi - snex expcom isopn3i eleqtrrd cnprest sylibrd embantd expimpd sylanbrc - fveq2 eleq2d opeq2d relssdv ne0d inss2 connclo eqtr3di iunss eqsstrid - rabid2 sseqtrdi ssrab simprbi txtopon cvmtop1 toptopon mpbir2and - cncnp ) AQURURUSUTZIVAUTVBZVCUUAUUBUTZVWHVDZHQUUFZQDVIZVWFIVEUTZVFZVB + simpr biimtrid cvmlift2lem11 impbid syl2anc sseq12i ssopab2bw opelxpi + weq bitri rexlimdvva mpd eleq1d ralrimiva sylib cvmlift2lem1 syl cmpo + ex elrab2 ctopon iitopon cvv sylancr c1 wf cvmlift2lem5 iunid xpiundi + cicc xpeq2i eqtr3i cconn iiconn ccld inss1 cnt ccmp iicmp txtopi cnei + neiss2 mpan snss a1d rexlimiv simpl ssopab2i 3sstr4i txunii ccnv cima + ntropn chmeo cpw ccvm elunii opnneip syl3anc simplr2 cbvrexvw simplr3 + txopn rspe ssntr simpr1 simpr2 sseldd opeq2 ralsn anassrs dfss3 eleq1 + alrimivv ralxp txtube ntrss2 mpan2 ralcom 3bitr2i ralimi bicom rexbii + sstr r2al ralbii sylbi rabeq2i baib ad3antlr elssuni sseqtrrdi sselda + sseq1d bibi12d syl5ibr ralimdva anim2d reximdva ssrab2 eqsstri sselid + syl5 isclo 0elunit wrel relxp opelxp id df-3an wfn ffnd reseq1d snssd + fnov resmpo simplll cvmlift2lem8 sylan syldan elsni eqeq1d syl5ibrcom + 3impia mpoeq3dva 3eqtrd cnmpt1st ccom cvmlift2lem2 cnmpt21f cnmpt2res + simp1d snex txrest mp4an oveq1i eqeltrd rspcev xpss12 restuni eleqtrd + cncnpi expcom isopn3i eleqtrrd cnprest sylibrd embantd expimpd eleq2d + fveq2 sylanbrc opeq2d relssdv ne0d inss2 connclo eqtr3di rabid2 iunss + eqsstrid sseqtrdi ssrab simprbi txtopon cvmtop1 toptopon mpbir2and + cncnp ) AQURURUSUTZIVAUTVBZVCUUAUUFUTZVWHVDZHQUUBZQDVIZVWFIVEUTZVFZVB ZDVWIVGZABCDHIJLMNOPQUAUBUCUDUEUFUGUUCZAVWIVWNDVWIVHZVJZVWOAVWIRVWQAV WITVWHVWHTVIZVKZVDZVLZRVWHTVWHVWTVLZVDVWIVXBVXCVWHVWHTVWHUUDUUGTVWHVW TVWHUUEUUHAVXARVJZTVWHVGZVXBRVJAVWHVXDTVWHVHZVMVXEAGVWHVXFAGURVWHVNUR @@ -524613,74 +529616,74 @@ property of an acyclic graph (see also ~ acycgr0v ). 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(Contributed by ( vy vw cv cfv cmpt ccom wcel wa cc0 wceq c1 cii ccn crio w3a wrex eqid co cvmlift3lem4 df-3an ccvm ad3antrrr simplr cnco syl2anc simprl fveq2d mpbii cicc wf iiuni cnf syl 0elunit sylancl 3eqtr4rd cvmliftiota simp2d - fvco3 fveq1d simp1d 1elunit simprr eqtrd 3eqtr3d fveqeq2 expimpd syl5bi - syl5ibcom rexlimdva mpd mpteq2dva cvmlift3lem3 ffvelcdmda feqmptd cvmcn - cuni 3syl fveq2 fmptco 3eqtr4d ) AUFOUFUHZKUIZIUIZUJUFOXGJUIZUJIKUKJAUF - OXIXJAXGOULZUMZUNGUHZUIZNUOZUPXMUIZXGUOZUPIHUHZUKJXMUKZUOUNXRUIFUOUMHUQ - EURVCZUSZUIZXHUOZUTZGUQMURVCZVAZXIXJUOZXLXHXHUOYFXHVBABCXHDEFGHIJKLMNXG - OPQRSTUAUBUCUDUEVDVMXLYDYGGYEYDXOXQUMZYCUMXLXMYEULZUMZYGXOXQYCVEYJYHYCY - GYJYHUMZYBIUIZXJUOYCYGYKUPIYAUKZUIZUPXSUIZYLXJYKUPYMXSYKYAXTULZYMXSUOZU - NYAUIFUOZYKDEFHIXSYALPYAVBAIELVFVCULZXKYIYHRVGYKYIJMLURVCULZXSUQLURVCUL - XLYIYHVHZAYTXKYIYHUBVGXMJUQMLVIVJAFDULXKYIYHUCVGYKXNJUIZNJUIZUNXSUIZFIU - IZYKXNNJYJXOXQVKVLYKUNUPVNVCZOXMVOZUNUUFULUUDUUBUOYKYIUUGUUAXMUQMUUFOVP - QVQVRZVSUUFOUNJXMWDVTAUUEUUCUOXKYIYHUDVGWAWBZWCWEYKUUFDYAVOZUPUUFULZYNY - LUOYKYPUUJYKYPYQYRUUIWFYAUQEUUFDVPPVQVRWGUUFDUPIYAWDVTYKYOXPJUIZXJYKUUG - UUKYOUULUOUUHWGUUFOUPJXMWDVTYKXPXGJYJXOXQWHVLWIWJYBXHXJIWKWNWLWMWOWPWQA - UFUGODXHUGUHZIUIXIKIAODXGKABCDEFGHIJKLMNOPQRSTUAUBUCUDUEWRZWSAUFODKUUNW - TAUGDLXBZIAYSIELURVCULDUUOIVOREILXAIELDUUOPUUOVBZVQXCWTUUMXHIXDXEAUFOUU - OJAYTOUUOJVOUBJMLOUUOQUUPVQVRWTXF $. + fveq1d simp1d 1elunit simprr 3eqtr3d fveqeq2 syl5ibcom expimpd biimtrid + fvco3 rexlimdva mpd mpteq2dva cvmlift3lem3 ffvelcdmda feqmptd cuni 3syl + eqtrd cvmcn fveq2 fmptco 3eqtr4d ) AUFOUFUHZKUIZIUIZUJUFOXGJUIZUJIKUKJA + UFOXIXJAXGOULZUMZUNGUHZUIZNUOZUPXMUIZXGUOZUPIHUHZUKJXMUKZUOUNXRUIFUOUMH + UQEURVCZUSZUIZXHUOZUTZGUQMURVCZVAZXIXJUOZXLXHXHUOYFXHVBABCXHDEFGHIJKLMN + XGOPQRSTUAUBUCUDUEVDVMXLYDYGGYEYDXOXQUMZYCUMXLXMYEULZUMZYGXOXQYCVEYJYHY + CYGYJYHUMZYBIUIZXJUOYCYGYKUPIYAUKZUIZUPXSUIZYLXJYKUPYMXSYKYAXTULZYMXSUO + ZUNYAUIFUOZYKDEFHIXSYALPYAVBAIELVFVCULZXKYIYHRVGYKYIJMLURVCULZXSUQLURVC + ULXLYIYHVHZAYTXKYIYHUBVGXMJUQMLVIVJAFDULXKYIYHUCVGYKXNJUIZNJUIZUNXSUIZF + IUIZYKXNNJYJXOXQVKVLYKUNUPVNVCZOXMVOZUNUUFULUUDUUBUOYKYIUUGUUAXMUQMUUFO + VPQVQVRZVSUUFOUNJXMWMVTAUUEUUCUOXKYIYHUDVGWAWBZWCWDYKUUFDYAVOZUPUUFULZY + NYLUOYKYPUUJYKYPYQYRUUIWEYAUQEUUFDVPPVQVRWFUUFDUPIYAWMVTYKYOXPJUIZXJYKU + UGUUKYOUULUOUUHWFUUFOUPJXMWMVTYKXPXGJYJXOXQWGVLXBWHYBXHXJIWIWJWKWLWNWOW + PAUFUGODXHUGUHZIUIXIKIAODXGKABCDEFGHIJKLMNOPQRSTUAUBUCUDUEWQZWRAUFODKUU + NWSAUGDLWTZIAYSIELURVCULDUUOIVOREILXCIELDUUOPUUOVBZVQXAWSUUMXHIXDXEAUFO + UUOJAYTOUUOJVOUBJMLOUUOQUUPVQVRWSXF $. cvmlift3lem7.s $e |- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ @@ -525130,23 +530133,23 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by n0 crest cpconn w3a ccnv cpw cnlly ad2antrr simprr cvmsrcl cnima simplr cima simprl wfn wb ffn elpreima 4syl mpbir2and nlly2i syl3anc ad3antrrr crio csconn simprll elpwid simprr3 simprlr simprr2 simprr1 cvmlift3lem7 - wceq expr rexlimdvva mpd exlimdv syl5bi rexlimdvw ralrimiva ctopon ctop - expimpd sconntop toptopon sylib cvmtop1 cncnp ) AMOEURUSUTZQDMVAZMULVBZ - OEVCUSVDUTZULQVEZABCDEFHIKLMNOPQUAUBUCUDUEUFUGUHUIUJVFAYMULQAYLQUTZVGZY - LLVDZUMVBZUTZYRGVDZVIVHZVGZUMNVJZYMYPKENVKUSUTZYQNVLZUTUUCAUUDYOUCVMAQU - UEYLLALONURUSUTZQUUELVAZUGLONQUUEUBUUEVNZVOZVPVQUMTSEYQGJKNUUERUKUUHVRV - SYPUUBYMUMNYPYSUUAYMUUAUNVBZYTUTZUNVTYPYSVGZYMUNYTWBUULUUKYMUNYPYSUUKYM - YPYSUUKVGZVGZYLUOVBZUTZUUOUPVBZWAZOUUQWCUSWDUTZWEZUOOVJUPLWFYRWNZWGZVJZ - YMUUNOWDWHUTZUVAOUTZYLUVAUTZUVCAUVDYOUUMUEWIUUNUUFYRNUTZUVEAUUFYOUUMUGW - IZUUNUUKUVGYPYSUUKWJZTSEGUUJYRJKNRUKWKVPYRLONWLVSUUNUVFYOYSAYOUUMWMYPYS - UUKWOZUUNUUFUUGLQWPUVFYOYSVGWQUVHUUIQUUELWRQYLYRLWSWTXAUOWDYLUVAOUPXBXC - UUNUUTYMUPUOUVBOUUNUUQUVBUTZUUOOUTZVGZUUTYMUUNUVMUUTVGZVGZBCYRDEFGUUJHI - JKLMNOUUQPUUOYLMVDUQVBUTUQUUJXEZYLQRUQSTUAUBAUUDYOUUMUVNUCXDAOXFUTZYOUU - MUVNUDXDAUVDYOUUMUVNUEXDAPQUTYOUUMUVNUFXDAUUFYOUUMUVNUGXDAFDUTYOUUMUVNU - HXDAFKVDPLVDXNYOUUMUVNUIXDUJUKUUNYSUVNUVJVMUUNUUKUVNUVIVMUVOUUQUVAUUNUV - KUVLUUTXGXHUVPVNUUPUURUUSUVMUUNXIUUNUVKUVLUUTXJUUPUURUUSUVMUUNXKUUPUURU - USUVMUUNXLXMXOXPXQXOXRXSYDXTXQYAAOQYBVDUTZEDYBVDUTZYJYKYNVGWQAOYCUTZUVR - AUVQUVTUDOYEVPOQUBYFYGAEYCUTZUVSAUUDUWAUCEKNYHVPEDUAYFYGULMOEQDYIVSXA + wceq rexlimdvva mpd exlimdv biimtrid expimpd rexlimdvw ralrimiva ctopon + expr ctop sconntop toptopon sylib cvmtop1 cncnp ) AMOEURUSUTZQDMVAZMULV + BZOEVCUSVDUTZULQVEZABCDEFHIKLMNOPQUAUBUCUDUEUFUGUHUIUJVFAYMULQAYLQUTZVG + ZYLLVDZUMVBZUTZYRGVDZVIVHZVGZUMNVJZYMYPKENVKUSUTZYQNVLZUTUUCAUUDYOUCVMA + QUUEYLLALONURUSUTZQUUELVAZUGLONQUUEUBUUEVNZVOZVPVQUMTSEYQGJKNUUERUKUUHV + RVSYPUUBYMUMNYPYSUUAYMUUAUNVBZYTUTZUNVTYPYSVGZYMUNYTWBUULUUKYMUNYPYSUUK + YMYPYSUUKVGZVGZYLUOVBZUTZUUOUPVBZWAZOUUQWCUSWDUTZWEZUOOVJUPLWFYRWNZWGZV + JZYMUUNOWDWHUTZUVAOUTZYLUVAUTZUVCAUVDYOUUMUEWIUUNUUFYRNUTZUVEAUUFYOUUMU + GWIZUUNUUKUVGYPYSUUKWJZTSEGUUJYRJKNRUKWKVPYRLONWLVSUUNUVFYOYSAYOUUMWMYP + YSUUKWOZUUNUUFUUGLQWPUVFYOYSVGWQUVHUUIQUUELWRQYLYRLWSWTXAUOWDYLUVAOUPXB + XCUUNUUTYMUPUOUVBOUUNUUQUVBUTZUUOOUTZVGZUUTYMUUNUVMUUTVGZVGZBCYRDEFGUUJ + HIJKLMNOUUQPUUOYLMVDUQVBUTUQUUJXEZYLQRUQSTUAUBAUUDYOUUMUVNUCXDAOXFUTZYO + UUMUVNUDXDAUVDYOUUMUVNUEXDAPQUTYOUUMUVNUFXDAUUFYOUUMUVNUGXDAFDUTYOUUMUV + NUHXDAFKVDPLVDXNYOUUMUVNUIXDUJUKUUNYSUVNUVJVMUUNUUKUVNUVIVMUVOUUQUVAUUN + UVKUVLUUTXGXHUVPVNUUPUURUUSUVMUUNXIUUNUVKUVLUUTXJUUPUURUUSUVMUUNXKUUPUU + RUUSUVMUUNXLXMYCXOXPYCXQXRXSXTXPYAAOQYBVDUTZEDYBVDUTZYJYKYNVGWQAOYDUTZU + VRAUVQUVTUDOYEVPOQUBYFYGAEYDUTZUVSAUUDUWAUCEKNYHVPEDUAYFYGULMOEQDYIVSXA $. $( Lemma for ~ cvmlift2 . (Contributed by Mario Carneiro, @@ -526069,18 +531072,18 @@ codes is an increasing chain (with respect to inclusion). 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(Contributed simpr wb adantl eqidd rspcedvd funeldmdif mpbird ex eleq2i eqcomd 3imtr4d difeq12d eleq2d imp oveq1 eqeq2d rexbidv id goaleq12d orbi12d wrel releqi satfrel sylibr 1stdm sylan eleqtrrd rexlimdva2 orim1d orim12d releldmdifi - sylbid eqcomi rexeqi r19.41v releldm2 bitrd reximdv syl5bir expd rexlimdv - wi syl5bi sylbird expimpd syld eqtrd cres cun rabex2 simplr ancri sssucid - dmopab3rexdif funeqi biimpi 3sstr3g difeq12i simpl ad4ant13 r19.29an eqid - oveq2 eqcoms biimpa reximdva biimpcd impbid abbidv ineq12d fmlasucdisj + sylbid eqcomi rexeqi r19.41v releldm2 bitrd reximdv biimtrid syl5bir expd + wi rexlimdv sylbird expimpd syld eqtrd rabex2 ancri sssucid dmopab3rexdif + cres cun simplr funeqi biimpi difeq12i simpl ad4ant13 oveq2 r19.29an eqid + 3sstr3g eqcoms biimpa reximdva biimpcd impbid abbidv ineq12d fmlasucdisj com23 ) LUESZKMSZJNSZTZTZLUFZHUGZUHZTZUVLUQZAUIZEUIZUJUGZDUIZUJUGZUKULZUM ZBUIZFUMTZDUVLUNUVPUVRIUIZUOZUMZUWCGUMTIUEUNUPEUVLLHUGZURZUNUWDDUWIUNEUWH UNUPABUSUQZUTUVKVAUGZUVPUAUIZUBUIZUKULZUMZUBUWKUNZUVPUWLUWEUOZUMZIUEUNZUP @@ -527260,24 +532263,24 @@ codes is an increasing chain (with respect to inclusion). 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U -> Y e. U ) ) $= ( vx wcel cmpst cfv cmpps cima wa wceq ccnv eqid mthmval eleq2i ax-mp wfn wb wf msrf ffn elpreima bitri eleq1 wrex wfun ffun fvelima mpan rexlimiva - cv mthmi syl syl6bi adantld syl5bi ) DCIZDBJKZIZDAKZABLKZMZIZNZVDEAKZOZEC - IZVADAPVFMZIZVHCVLDABCVEFVEQZGRSAVBUAZVMVHUBVBVBAUCZVOVBABVBQFUDZVBVBAUET - VBDVFAUFTUGVJVGVKVCVJVGVIVFIZVKVDVIVFUHVRHUOZAKVIOZHVEUIZVKAUJZVRWAVPWBVQ - VBVBAUKTHVIVEAULUMVTVKHVEABCVEVSEFVNGUPUNUQURUSUT $. + cv mthmi syl syl6bi adantld biimtrid ) DCIZDBJKZIZDAKZABLKZMZIZNZVDEAKZOZ + ECIZVADAPVFMZIZVHCVLDABCVEFVEQZGRSAVBUAZVMVHUBVBVBAUCZVOVBABVBQFUDZVBVBAU + ETVBDVFAUFTUGVJVGVKVCVJVGVIVFIZVKVDVIVFUHVRHUOZAKVIOZHVEUIZVKAUJZVRWAVPWB + VQVBVBAUKTHVIVEAULUMVTVKHVEABCVEVSEFVNGUPUNUQURUSUT $. $( If two statements have the same reduct then one is a theorem iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.) $) @@ -529827,9083 +534830,4853 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership TZTWBZYLZEYMDYMZVUOVXFYLZVGVXSVXTYLVHVIVXRVXTDEUYFVUNYPYPVXGUYFWCZVXHVU NWCZWDZVXIVUOVXQVXFVXGUYFVXHVUNUXAYNVYCVXPVXETVYCVXLVVNVXOVWBVYCVXKVVMU CVYCVXJVUQUWHVYCVXGUYFPVYAVYBYQYEYEYEVYCVXNVWAUCVYCVXMVVBUWHVYCVXHVUNPV - YAVYBYRYEYEYEUVIUVJUVRUVKUVLWAYDYBYSYDAVXDVVLVVTVVFAVXDVVLVVTVVFABVLZCV - LZTXRZVUPVYDPVMZKVMZUCVMZVQZVVAVYEPVMZKVMZUCVMZVQZWHZWDZVVFYLAVXDVVLVVT - WHZWDZVVFYLBCVVHVVPVKYTFYTVYDVVHWCZVYEVVPWCZWDZVYPVYRVVFWUAVYOVYQAWUAVY - FVXDVYJVVLVYNVVTVYDVVHVYEVVPTYNWUAVYIVVKVUPWUAVYHVVJUCWUAVYGVVIKWUAVYDV - VHPVYSVYTYQYEYEYEYIWUAVYMVVSVVAWUAVYLVVRUCWUAVYKVVQKWUAVYEVVPPVYSVYTYRY - EYEYEYIUVMUVNUVOVCUVPUVQUVSXBUVTYSUWAUWBUWCUWDUWEAKOXDUWMUWNUWPWDXLAOOK - UYNXFOUWIUWKKXPWAUWF $. - $} - - $d m o p s w z ph $. - $( The closure is closed under application of provable pre-statements. - (Compare ~ mclsax .) This theorem is what justifies the treatment of - theorems as "equivalent" to axioms once they have been proven: the - composition of one theorem in the proof of another yields a theorem. - (Contributed by Mario Carneiro, 18-Jul-2016.) $) - mclspps $p |- ( ph -> ( S ` P ) e. ( K C B ) ) $= - ( vz vw vm vo vs vp wfn ccnv co cima wcel cfv crn msubf syl ffnd cmax wss - wf wceq cfn cotp cmpst w3a eqid mppspst sselid elmpst sylib simp1d simpld - wa simp2d cv wral ralrimiva wfun wb ffund fdmd sseqtrrd funimass5 syl2anc - cdm mpbird cmfs mvhf ffvelcdmda elpreima adantr mpbir2and cun wbr cxp wal - 3ad2ant1 3ad2antl1 simp21 simp22 simp23 mclsppslem mclsind elmpps simprbi - wi simp3 sseldd simplbda ) AIKVDZHIVENEFVFZVGZVHZHIVIYGVHZAKKIAIOVJZVHZKK - IVPUNOJKIUIUCVKVLZVMZAPQFVFZYHHAURUSDJVNVIZQFGYHJUTVAKLPORSVBVCUBUCUDUEAP - GVOZPVEPVQZAYQYRWIZQKVOZQVRVHZWIZHKVHZAPQHVSZJVTVIZVHZYSUUBUUCWAAMUUEUUDU - UEJMUUEWBZUHWCUMWDHPUUEJKQGUBUCUUGWEWFZWGWHAYTUUAAYSUUBUUCUUHWJWHZYPWBUIU - JUKULAQYHVOZBWKZIVIYGVHZBQWLZAUULBQUOWMAIWNQIXAZVOUUJUUMWOAKKIYMWPAQKUUNU - UIAKKIYMWQWRBQYGIWSWTXBADWKZRVHZWIUUOLVIZYHVHZUUQKVHZUUQIVIYGVHZARKUUOLAJ - XCVHZRKLVPUEJKLRUJUCUKXDVLXEUPAUURUUSUUTWIWOZUUPAYFUVBYNKUUQYGIXFVLXGXHAU - TWKZVAWKZVCWKVSYPVHZVBWKZYKVHZUVFUVDLVJXIVGYHVOZWAZURWKZUSWKZUVCXJUVJLVIU - VFVISVIUVKLVIUVFVISVIXKPVOYBUSXLURXLZWABCURUSDEFGHIJUTVAKLMNOPQRSVBVCTUAU - BUCUDAUVIUVAUVLUEXMAUVINGVOUVLUFXMAUVIEKVOUVLUGXMUHUIUJUKULAUVIUUDMVHZUVL - UMXMAUVIYLUVLUNXMAUVIUUKQVHUULUVLUOXNAUVIUUPUUTUVLUPXNAUVIUUKCWKZPXJTWKZU - UKLVIIVISVIVHUAWKZUVNLVIIVISVIVHWAUVOUVPNXJUVLUQXNAUVEUVGUVHUVLXOAUVEUVGU - VHUVLXPAUVEUVGUVHUVLXQAUVIUVLYCXRXSAUVMHYOVHZUMUVMUUFUVQHFPUUEJQMUUGUHUDX - TYAVLYDYFYIUUCYJKHYGIXFYEWT $. - $} - - $( - @{ - mthmco.d @e |- D = ( mDV ` T ) @. - mthmco.e @e |- E = ( mEx ` T ) @. - mthmco.1 @e |- ( ph -> T e. mFS ) @. - mthmco.2 @e |- ( ph -> K C_ D ) @. - mthmco.3 @e |- ( ph -> B C_ E ) @. - mthmco.j @e |- U = ( mThm ` T ) @. - mthmco.l @e |- L = ( mSubst ` T ) @. - mthmco.v @e |- V = ( mVR ` T ) @. - mthmco.h @e |- H = ( mVH ` T ) @. - mthmco.w @e |- W = ( mVars ` T ) @. - mthmco.4 @e |- ( ph -> <. M , O , P >. e. U ) @. - mthmco.5 @e |- ( ph -> S e. ran L ) @. - mthmco.6 @e |- ( ( ph /\ x e. O ) -> - <. K , B , ( S ` x ) >. e. U ) @. - mthmco.7 @e |- ( ( ph /\ v e. V ) -> - <. K , B , ( S ` ( H ` v ) ) >. e. U ) @. - mthmco.8 @e |- ( ( ph /\ ( x M y /\ a e. ( W ` ( S ` ( H ` x ) ) ) /\ - b e. ( W ` ( S ` ( H ` y ) ) ) ) ) -> a K b ) @. - @( The application of a theorem to a collection of theorems is a theorem. - (Compare ~ mclsax .) This is what justifies the treatment of - theorems as "equivalent" to axioms once they have been proven: the - composition of one theorem in the proof of another yields a theorem. - - This theorem has a similar statement to ~ mclspps , but deals with the - elimination of dummy variables instead of raw composition. The - assumption ~ mfsinf that there are enough variables of each type is - essential for this proof. @) - mthmco @p |- ( ph -> <. K , B , ( S ` P ) >. e. U ) @= - ? @. - @} - $) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Grammatical formal systems -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $c m0St $. - $c mSA $. - $c mWGFS $. - $c mSyn $. - $c mESyn $. - $c mGFS $. - $c mTree $. - $c mST $. - $c mSAX $. - $c mUFS $. - - $( Mapping expressions to statements. $) - cm0s $a class m0St $. - - $( The set of syntax axioms. $) - cmsa $a class mSA $. - - $( The set of weakly grammatical formal systems. $) - cmwgfs $a class mWGFS $. - - $( The syntax typecode function. $) - cmsy $a class mSyn $. - - $( The syntax typecode function for expressions. $) - cmesy $a class mESyn $. - - $( The set of grammatical formal systems. $) - cmgfs $a class mGFS $. - - $( The set of proof trees. $) - cmtree $a class mTree $. - - $( The set of syntax trees. $) - cmst $a class mST $. - - $( The indexing set for a syntax axiom. $) - cmsax $a class mSAX $. - - $( The set of unambiguous formal sytems. $) - cmufs $a class mUFS $. - - $( Define a function mapping expressions to statements. (Contributed by - Mario Carneiro, 14-Jul-2016.) $) - df-m0s $a |- m0St = ( a e. _V |-> <. (/) , (/) , a >. ) $. - - ${ - $d a c d e h m o p r s t x y $. - $( Define the set of syntax axioms. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-msa $a |- mSA = ( t e. _V |-> { a e. ( mEx ` t ) | - ( ( m0St ` a ) e. ( mAx ` t ) /\ ( 1st ` a ) e. ( mVT ` t ) /\ - Fun ( `' ( 2nd ` a ) |` ( mVR ` t ) ) ) } ) $. - - $( Define the set of weakly grammatical formal systems. (Contributed by - Mario Carneiro, 14-Jul-2016.) $) - df-mwgfs $a |- mWGFS = { t e. mFS | A. d A. h A. a - ( ( <. d , h , a >. e. ( mAx ` t ) /\ ( 1st ` a ) e. ( mVT ` t ) ) -> - E. s e. ran ( mSubst ` t ) a e. ( s " ( mSA ` t ) ) ) } $. - - $( Define the syntax typecode function. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-msyn $a |- mSyn = Slot 6 $. - - $( Define the syntax typecode function for expressions. (Contributed by - Mario Carneiro, 12-Jun-2023.) $) - df-mesyn $a |- mESyn = ( t e. _V |-> - ( c e. ( mTC ` t ) , e e. ( mREx ` t ) |-> - ( ( ( mSyn ` t ) ` c ) m0St e ) ) ) $. - - $( Define the set of grammatical formal systems. (Contributed by Mario - Carneiro, 12-Jun-2023.) $) - df-mgfs $a |- mGFS = { t e. mWGFS | - ( ( mSyn ` t ) : ( mTC ` t ) --> ( mVT ` t ) /\ - A. c e. ( mVT ` t ) ( ( mSyn ` t ) ` c ) = c /\ - A. d A. h A. a ( <. d , h , a >. e. ( mAx ` t ) -> - A. e e. ( h u. { a } ) - ( ( mESyn ` t ) ` e ) e. ( mPPSt ` t ) ) ) } $. - - $( Define the set of proof trees. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-mtree $a |- mTree = ( t e. _V |-> - ( d e. ~P ( mDV ` t ) , h e. ~P ( mEx ` t ) |-> - |^| { r | ( A. e e. ran ( mVH ` t ) e r <. ( m0St ` e ) , (/) >. /\ - A. e e. h e r <. ( ( mStRed ` t ) ` <. d , h , e >. ) , (/) >. /\ - A. m A. o A. p ( <. m , o , p >. e. ( mAx ` t ) -> - A. s e. ran ( mSubst ` t ) ( A. x A. y ( x m y -> - ( ( ( mVars ` t ) ` ( s ` ( ( mVH ` t ) ` x ) ) ) X. - ( ( mVars ` t ) ` ( s ` ( ( mVH ` t ) ` y ) ) ) ) C_ d ) -> - ( { ( s ` p ) } X. X_ e e. ( o u. ( ( mVH ` t ) " - U. ( ( mVars ` t ) " ( o u. { p } ) ) ) ) ( r " { ( s ` e ) } ) ) C_ r - ) ) ) } ) ) $. - - $( Define the function mapping syntax expressions to syntax trees. - (Contributed by Mario Carneiro, 14-Jul-2016.) $) - df-mst $a |- mST = ( t e. _V |-> - ( ( (/) ( mTree ` t ) (/) ) |` ( ( mEx ` t ) |` ( mVT ` t ) ) ) ) $. - - $( Define the indexing set for a syntax axiom's representation in a tree. - (Contributed by Mario Carneiro, 14-Jul-2016.) $) - df-msax $a |- mSAX = ( t e. _V |-> - ( p e. ( mSA ` t ) |-> ( ( mVH ` t ) " ( ( mVars ` t ) ` p ) ) ) ) $. - $} - - $( Define the set of unambiguous formal systems. (Contributed by Mario - Carneiro, 14-Jul-2016.) $) - df-mufs $a |- mUFS = { t e. mGFS | Fun ( mST ` t ) } $. - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Models of formal systems -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $c mUV $. - $c mVL $. - $c mVSubst $. - $c mFresh $. - $c mFRel $. - $c mEval $. - $c mMdl $. - $c mUSyn $. - $c mGMdl $. - $c mItp $. - $c mFromItp $. - - $( The universe of a model. $) - cmuv $a class mUV $. - - $( The set of valuations. $) - cmvl $a class mVL $. - - $( Substitution for a valuation. $) - cmvsb $a class mVSubst $. - - $( The freshness relation of a model. $) - cmfsh $a class mFresh $. - - $( The set of freshness relations. $) - cmfr $a class mFRel $. - - $( The evaluation function of a model. $) - cmevl $a class mEval $. - - $( The set of models. $) - cmdl $a class mMdl $. - - $( The syntax function applied to elements of the model. $) - cusyn $a class mUSyn $. - - $( The set of models in a grammatical formal system. $) - cgmdl $a class mGMdl $. - - $( The interpretation function of the model. $) - cmitp $a class mItp $. - - $( The evaluation function derived from the interpretation. $) - cmfitp $a class mFromItp $. - - $( Define the universe of a model. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-muv $a |- mUV = Slot 7 $. - - $( Define the freshness relation of a model. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-mfsh $a |- mFresh = Slot ; 1 9 $. - - $( Define the evaluation function of a model. (Contributed by Mario - Carneiro, 14-Jul-2016.) $) - df-mevl $a |- mEval = Slot ; 2 0 $. - - ${ - $d a c d e f g h i m n p r s t u v w x y z $. - $( Define the set of valuations. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-mvl $a |- mVL = ( t e. _V |-> X_ v e. ( mVR ` t ) - ( ( mUV ` t ) " { ( ( mType ` t ) ` v ) } ) ) $. - - $( Define substitution applied to a valuation. (Contributed by Mario - Carneiro, 14-Jul-2016.) $) - df-mvsb $a |- mVSubst = ( t e. _V |-> { <. <. s , m >. , x >. | - ( ( s e. ran ( mSubst ` t ) /\ m e. ( mVL ` t ) ) /\ - A. v e. ( mVR ` t ) m dom ( mEval ` t ) ( s ` ( ( mVH ` t ) ` v ) ) /\ - x = ( v e. ( mVR ` t ) |-> - ( m ( mEval ` t ) ( s ` ( ( mVH ` t ) ` v ) ) ) ) ) } ) $. - - $( Define the set of freshness relations. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-mfrel $a |- mFRel = ( t e. _V |-> - { r e. ~P ( ( mUV ` t ) X. ( mUV ` t ) ) | ( `' r = r /\ - A. c e. ( mVT ` t ) A. w e. ( ~P ( mUV ` t ) i^i Fin ) - E. v e. ( ( mUV ` t ) " { c } ) w C_ ( r " { v } ) ) } ) $. - - $( Define the set of models of a formal system. (Contributed by Mario - Carneiro, 14-Jul-2016.) $) - df-mdl $a |- mMdl = { t e. mFS | - [. ( mUV ` t ) / u ]. [. ( mEx ` t ) / x ]. - [. ( mVL ` t ) / v ]. [. ( mEval ` t ) / n ]. [. ( mFresh ` t ) / f ]. - ( ( u C_ ( ( mTC ` t ) X. _V ) /\ f e. ( mFRel ` t ) /\ - n e. ( u ^pm ( v X. ( mEx ` t ) ) ) ) /\ A. m e. v - ( ( A. e e. x ( n " { <. m , e >. } ) C_ ( u " { ( 1st ` e ) } ) /\ - A. y e. ( mVR ` t ) <. m , ( ( mVH ` t ) ` y ) >. n ( m ` y ) /\ - A. d A. h A. a ( <. d , h , a >. e. ( mAx ` t ) -> - ( ( A. y A. z ( y d z -> ( m ` y ) f ( m ` z ) ) /\ - h C_ ( dom n " { m } ) ) -> m dom n a ) ) ) /\ - ( A. s e. ran ( mSubst ` t ) A. e e. ( mEx ` t ) - A. y ( <. s , m >. ( mVSubst ` t ) y -> - ( n " { <. m , ( s ` e ) >. } ) = ( n " { <. y , e >. } ) ) /\ - A. p e. v A. e e. x - ( ( m |` ( ( mVars ` t ) ` e ) ) = ( p |` ( ( mVars ` t ) ` e ) ) -> - ( n " { <. m , e >. } ) = ( n " { <. p , e >. } ) ) /\ - A. y e. u A. e e. x ( ( m " ( ( mVars ` t ) ` e ) ) C_ ( f " { y } ) -> - ( n " { <. m , e >. } ) C_ ( f " { y } ) ) ) ) ) } $. - - $( Define the syntax typecode function for the model universe. - (Contributed by Mario Carneiro, 14-Jul-2016.) $) - df-musyn $a |- mUSyn = ( t e. _V |-> ( v e. ( mUV ` t ) |-> - <. ( ( mSyn ` t ) ` ( 1st ` v ) ) , ( 2nd ` v ) >. ) ) $. - - $( Define the set of models of a grammatical formal system. (Contributed - by Mario Carneiro, 14-Jul-2016.) $) - df-gmdl $a |- mGMdl = { t e. ( mGFS i^i mMdl ) | - ( A. c e. ( mTC ` t ) ( ( mUV ` t ) " { c } ) C_ - ( ( mUV ` t ) " { ( ( mSyn ` t ) ` c ) } ) /\ - A. v e. ( mUV ` c ) A. w e. ( mUV ` c ) ( v ( mFresh ` t ) w <-> - v ( mFresh ` t ) ( ( mUSyn ` t ) ` w ) ) /\ - A. m e. ( mVL ` t ) A. e e. ( mEx ` t ) - ( ( mEval ` t ) " { <. m , e >. } ) = - ( ( ( mEval ` t ) " { <. m , ( ( mESyn ` t ) ` e ) >. } ) i^i - ( ( mUV ` t ) " { ( 1st ` e ) } ) ) ) } $. - - $( Define the interpretation function for a model. (Contributed by Mario - Carneiro, 14-Jul-2016.) $) - df-mitp $a |- mItp = ( t e. _V |-> ( a e. ( mSA ` t ) |-> - ( g e. X_ i e. ( ( mVars ` t ) ` a ) - ( ( mUV ` t ) " { ( ( mType ` t ) ` i ) } ) |-> - ( iota x E. m e. ( mVL ` t ) ( g = ( m |` ( ( mVars ` t ) ` a ) ) /\ - x = ( m ( mEval ` t ) a ) ) ) ) ) ) $. - - $( Define a function that produces the evaluation function, given the - interpretation function for a model. (Contributed by Mario Carneiro, - 14-Jul-2016.) $) - df-mfitp $a |- mFromItp = ( t e. _V |-> ( f e. X_ a e. ( mSA ` t ) - ( ( ( mUV ` t ) " { ( ( 1st ` t ) ` a ) } ) ^m - X_ i e. ( ( mVars ` t ) ` a ) - ( ( mUV ` t ) " { ( ( mType ` t ) ` i ) } ) ) |-> - ( iota_ n e. ( ( mUV ` t ) ^pm ( ( mVL ` t ) X. ( mEx ` t ) ) ) - A. m e. ( mVL ` t ) - ( A. v e. ( mVR ` t ) <. m , ( ( mVH ` t ) ` v ) >. n ( m ` v ) /\ - A. e A. a A. g ( e ( mST ` t ) <. a , g >. -> - <. m , e >. n ( f ` ( i e. ( ( mVars ` t ) ` a ) |-> - ( m n ( g ` ( ( mVH ` t ) ` i ) ) ) ) ) ) /\ - A. e e. ( mEx ` t ) ( n " { <. m , e >. } ) = - ( ( n " { <. m , ( ( mESyn ` t ) ` e ) >. } ) i^i - ( ( mUV ` t ) " { ( 1st ` e ) } ) ) ) ) ) ) $. - $} - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Splitting fields -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $( Introduce new constant symbols. $) - $c IntgRing $. $( Integral subring of a ring $) - $c cplMetSp $. $( Completion of a metric space $) - $c HomLimB $. $( Direct limit auxiliaries $) - $c HomLim $. $( Direct limit structure $) - $c polyFld $. $( Polynomial extension field $) - $c splitFld1 $. $( Splitting field for a polynomial auxiliary $) - $c splitFld $. $( Splitting field for a polynomial $) - $c polySplitLim $. $( Splitting field for a sequence of polynomials $) - - $( Integral subring of a ring. $) - citr $a class IntgRing $. - - $( Completion of a metric space. $) - ccpms $a class cplMetSp $. - - $( Embeddings for a direct limit. $) - chlb $a class HomLimB $. - - $( Direct limit structure. $) - chlim $a class HomLim $. - - $( Polynomial extension field. $) - cpfl $a class polyFld $. - - $( Splitting field for a single polynomial (auxiliary). $) - csf1 $a class splitFld1 $. - - $( Splitting field for a finite set of polynomials. $) - csf $a class splitFld $. - - $( Splitting field for a sequence of polynomials. $) - cpsl $a class polySplitLim $. - - ${ - $d b e f g h i j m n p q r s t v w x y z $. - $( Define the subring of elements of ` r ` integral over ` s ` in a ring. - (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-irng $a |- IntgRing = ( r e. _V , s e. _V |-> - U_ f e. ( Monic1p ` ( r |`s s ) ) ( `' f " { ( 0g ` r ) } ) ) $. - - $( A function which completes the given metric space. (Contributed by - Mario Carneiro, 2-Dec-2014.) $) - df-cplmet $a |- cplMetSp = ( w e. _V |-> - [_ ( ( w ^s NN ) |`s ( Cau ` ( dist ` w ) ) ) / r ]_ - [_ ( Base ` r ) / v ]_ [_ { <. f , g >. | ( { f , g } C_ v /\ - A. x e. RR+ E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : - ( ZZ>= ` j ) --> ( ( g ` j ) ( ball ` ( dist ` w ) ) x ) ) } / e ]_ - ( ( r /s e ) sSet { <. ( dist ` ndx ) , { <. <. x , y >. , z >. | - E. p e. v E. q e. v ( ( x = [ p ] e /\ y = [ q ] e ) /\ - ( p oF ( dist ` r ) q ) ~~> z ) } >. } ) ) $. - - $( The input to this function is a sequence (on ` NN ` ) of homomorphisms - ` F ( n ) : R ( n ) --> R ( n + 1 ) ` . The resulting structure is the - direct limit of the direct system so defined. This function returns the - pair ` <. S , G >. ` where ` S ` is the terminal object and ` G ` is a - sequence of functions such that ` G ( n ) : R ( n ) --> S ` and - ` G ( n ) = F ( n ) o. G ( n + 1 ) ` . (Contributed by Mario Carneiro, - 2-Dec-2014.) $) - df-homlimb $a |- HomLimB = ( f e. _V |-> - [_ U_ n e. NN ( { n } X. dom ( f ` n ) ) / v ]_ - [_ |^| { s | ( s Er v /\ ( x e. v |-> <. ( ( 1st ` x ) + 1 ) , - ( ( f ` ( 1st ` x ) ) ` ( 2nd ` x ) ) >. ) C_ s ) } / e ]_ - <. ( v /. e ) , ( n e. NN |-> - ( x e. dom ( f ` n ) |-> [ <. n , x >. ] e ) ) >. ) $. - - $( The input to this function is a sequence (on ` NN ` ) of structures - ` R ( n ) ` and homomorphisms ` F ( n ) : R ( n ) --> R ( n + 1 ) ` . - The resulting structure is the direct limit of the direct system so - defined, and maintains any structures that were present in the original - objects. TODO: generalize to directed sets? (Contributed by Mario - Carneiro, 2-Dec-2014.) $) - df-homlim $a |- HomLim = ( r e. _V , f e. _V |-> - [_ ( HomLimB ` f ) / e ]_ [_ ( 1st ` e ) / v ]_ [_ ( 2nd ` e ) / g ]_ - ( { <. ( Base ` ndx ) , v >. , - <. ( +g ` ndx ) , U_ n e. NN ran ( x e. dom ( g ` n ) , - y e. dom ( g ` n ) |-> <. <. ( ( g ` n ) ` x ) , ( ( g ` n ) ` y ) >. , - ( ( g ` n ) ` ( x ( +g ` ( r ` n ) ) y ) ) >. ) >. , - <. ( .r ` ndx ) , U_ n e. NN ran ( x e. dom ( g ` n ) , - y e. dom ( g ` n ) |-> <. <. ( ( g ` n ) ` x ) , ( ( g ` n ) ` y ) >. , - ( ( g ` n ) ` ( x ( .r ` ( r ` n ) ) y ) ) >. ) >. } u. - { <. ( TopOpen ` ndx ) , { s e. ~P v | - A. n e. NN ( `' ( g ` n ) " s ) e. ( TopOpen ` ( r ` n ) ) } >. , - <. ( dist ` ndx ) , U_ n e. NN ran ( x e. dom ( ( g ` n ) ` n ) , - y e. dom ( ( g ` n ) ` n ) |-> <. <. ( ( g ` n ) ` x ) , - ( ( g ` n ) ` y ) >. , ( x ( dist ` ( r ` n ) ) y ) >. ) >. , - <. ( le ` ndx ) , U_ n e. NN ( `' ( g ` n ) o. - ( ( le ` ( r ` n ) ) o. ( g ` n ) ) ) >. } ) ) $. - - $( Define the field extension that augments a field with the root of the - given irreducible polynomial, and extends the norm if one exists and the - extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-plfl $a |- polyFld = ( r e. _V , p e. _V |-> [_ ( Poly1 ` r ) / s ]_ - [_ ( ( RSpan ` s ) ` { p } ) / i ]_ - [_ ( z e. ( Base ` r ) |-> - [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ - <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. - ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , - [_ ( z e. ( Base ` t ) |-> - ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ - ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. ) $. - - $( Temporary construction for the splitting field of a polynomial. The - inputs are a field ` r ` and a polynomial ` p ` that we want to split, - along with a tuple ` j ` in the same format as the output. The output - is a tuple ` <. S , F >. ` where ` S ` is the splitting field and ` F ` - is an injective homomorphism from the original field ` r ` . - - The function works by repeatedly finding the smallest monic irreducible - factor, and extending the field by that factor using the ` polyFld ` - construction. We keep track of a total order in each of the splitting - fields so that we can pick an element definably without needing global - choice. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-sfl1 $a |- splitFld1 = ( r e. _V , j e. _V |-> ( p e. ( Poly1 ` r ) |-> - ( rec ( ( s e. _V , f e. _V |-> [_ ( mPoly ` s ) / m ]_ - [_ { g e. ( ( Monic1p ` s ) i^i ( Irred ` m ) ) | - ( g ( ||r ` m ) ( p o. f ) /\ 1 < ( s deg1 g ) ) } / b ]_ - if ( ( ( p o. f ) = ( 0g ` m ) \/ b = (/) ) , <. s , f >. , - [_ ( glb ` b ) / h ]_ [_ ( s polyFld h ) / t ]_ - <. ( 1st ` t ) , ( f o. ( 2nd ` t ) ) >. ) ) , j ) ` - ( card ` ( 1 ... ( r deg1 p ) ) ) ) ) ) $. - - $( Define the splitting field of a finite collection of polynomials, given - a total ordered base field. The output is a tuple ` <. S , F >. ` where - ` S ` is the totally ordered splitting field and ` F ` is an injective - homomorphism from the original field ` r ` . (Contributed by Mario - Carneiro, 2-Dec-2014.) $) - df-sfl $a |- splitFld = ( r e. _V , p e. _V |-> - ( iota x E. f ( f Isom < , ( lt ` r ) ( ( 1 ... ( # ` p ) ) , p ) /\ - x = ( seq 0 ( ( e e. _V , g e. _V |-> - ( ( r splitFld1 e ) ` g ) ) , ( f u. - { <. 0 , <. r , ( _I |` ( Base ` r ) ) >. >. } ) ) ` - ( # ` p ) ) ) ) ) $. - - $( Define the direct limit of an increasing sequence of fields produced by - pasting together the splitting fields for each sequence of polynomials. - That is, given a ring ` r ` , a strict order on ` r ` , and a sequence - ` p : NN --> ( ~P r i^i Fin ) ` of finite sets of polynomials to split, - we construct the direct limit system of field extensions by splitting - one set at a time and passing the resulting construction to ` HomLim ` . - (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-psl $a |- polySplitLim = ( r e. _V , - p e. ( ( ~P ( Base ` r ) i^i Fin ) ^m NN ) |-> - [_ ( 1st o. seq 0 ( ( g e. _V , q e. _V |-> - [_ ( 1st ` g ) / e ]_ [_ ( 1st ` e ) / s ]_ - [_ ( s splitFld ran ( x e. q |-> ( x o. ( 2nd ` g ) ) ) ) / f ]_ - <. f , ( ( 2nd ` g ) o. ( 2nd ` f ) ) >. ) , - ( p u. { <. 0 , <. <. r , (/) >. , - ( _I |` ( Base ` r ) ) >. >. } ) ) ) / f ]_ - ( ( 1st o. ( f shift 1 ) ) HomLim ( 2nd o. f ) ) ) $. - $} - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - p-adic number fields -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $( Introduce new constant symbols. $) - $c ZRing $. $( Integral elements of a ring $) - $c GF $. $( Galois finite field $) - $c GF_oo $. $( Galois limit field $) - $c ~Qp $. $( Equivalence relation for Qp $) - $c /Qp $. $( Representative of equivalence relation $) - $c Qp $. $( p-adic rational numbers $) - $c Zp $. $( p-adic integers $) - $c _Qp $. $( Algebraic completion of Qp $) - $c Cp $. $( Metric completion of _Qp $) - - $( Integral elements of a ring. $) - czr $a class ZRing $. - - $( Galois finite field. $) - cgf $a class GF $. - - $( Galois limit field. $) - cgfo $a class GF_oo $. - - $( Equivalence relation for ~ df-qp . $) - ceqp $a class ~Qp $. - - $( Equivalence relation representatives for ~ df-qp . $) - crqp $a class /Qp $. - - $( The set of ` p ` -adic rational numbers. $) - cqp $a class Qp $. - - $( The set of ` p ` -adic integers. (Not to be confused with ~ czn .) $) - czp $a class Zp $. - - $( Algebraic completion of the ` p ` -adic rational numbers. $) - cqpa $a class _Qp $. - - $( Metric completion of ` _Qp ` . $) - ccp $a class Cp $. - - ${ - $d b d f g h k n p r s x y $. - $( Define the subring of integral elements in a ring. (Contributed by - Mario Carneiro, 2-Dec-2014.) $) - df-zrng $a |- ZRing = ( r e. _V |-> - ( r IntgRing ran ( ZRHom ` r ) ) ) $. - - $( Define the Galois finite field of order ` p ^ n ` . (Contributed by - Mario Carneiro, 2-Dec-2014.) $) - df-gf $a |- GF = ( p e. Prime , n e. NN |-> [_ ( Z/nZ ` p ) / r ]_ - ( 1st ` ( r splitFld { [_ ( Poly1 ` r ) / s ]_ [_ ( var1 ` r ) / x ]_ - ( ( ( p ^ n ) ( .g ` ( mulGrp ` s ) ) x ) ( -g ` s ) x ) } ) ) ) $. - - $( Define the Galois field of order ` p ^ +oo ` , as a direct limit of the - Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-gfoo $a |- GF_oo = ( p e. Prime |-> [_ ( Z/nZ ` p ) / r ]_ - ( r polySplitLim ( n e. NN |-> - { [_ ( Poly1 ` r ) / s ]_ [_ ( var1 ` r ) / x ]_ - ( ( ( p ^ n ) ( .g ` ( mulGrp ` s ) ) x ) ( -g ` s ) x ) } ) ) ) $. - - $( Define an equivalence relation on ` ZZ ` -indexed sequences of integers - such that two sequences are equivalent iff the difference is equivalent - to zero, and a sequence is equivalent to zero iff the sum - ` sum_ k <_ n f ( k ) ( p ^ k ) ` is a multiple of ` p ^ ( n + 1 ) ` for - every ` n ` . (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-eqp $a |- ~Qp = ( p e. Prime |-> - { <. f , g >. | ( { f , g } C_ ( ZZ ^m ZZ ) /\ - A. n e. ZZ sum_ k e. ( ZZ>= ` -u n ) ( ( ( f ` -u k ) - ( g ` -u k ) ) / - ( p ^ ( k + ( n + 1 ) ) ) ) e. ZZ ) } ) $. - - $( There is a unique element of ` ( ZZ ^m ( 0 ... ( p - 1 ) ) ) ` ` ~Qp ` - -equivalent to any element of ` ( ZZ ^m ZZ ) ` , if the sequences are - zero for sufficiently large negative values; this function selects that - element. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-rqp $a |- /Qp = ( p e. Prime |-> ( ~Qp i^i [_ { f e. ( ZZ ^m ZZ ) | - E. x e. ran ZZ>= ( `' f " ( ZZ \ { 0 } ) ) C_ x } / y ]_ - ( y X. ( y i^i ( ZZ ^m ( 0 ... ( p - 1 ) ) ) ) ) ) ) $. - - $( Define the ` p ` -adic completion of the rational numbers, as a normed - field structure with a total order (that is not compatible with the - operations). (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by - AV, 10-Oct-2021.) $) - df-qp $a |- Qp = ( p e. Prime |-> - [_ { h e. ( ZZ ^m ( 0 ... ( p - 1 ) ) ) | - E. x e. ran ZZ>= ( `' h " ( ZZ \ { 0 } ) ) C_ x } / b ]_ - ( ( { <. ( Base ` ndx ) , b >. , - <. ( +g ` ndx ) , - ( f e. b , g e. b |-> ( ( /Qp ` p ) ` ( f oF + g ) ) ) >. , - <. ( .r ` ndx ) , - ( f e. b , g e. b |-> ( ( /Qp ` p ) ` ( n e. ZZ |-> - sum_ k e. ZZ ( ( f ` k ) x. ( g ` ( n - k ) ) ) ) ) ) >. } u. - { <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ b /\ - sum_ k e. ZZ ( ( f ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) < - sum_ k e. ZZ ( ( g ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) ) } >. } ) - toNrmGrp ( f e. b |-> if ( f = ( ZZ X. { 0 } ) , 0 , - ( p ^ -u inf ( ( `' f " ( ZZ \ { 0 } ) ) , RR , < ) ) ) ) ) ) $. - - $( Define the ` p ` -adic integers, as a subset of the ` p ` -adic - rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-zp $a |- Zp = ( ZRing o. Qp ) $. - - $( Define the completion of the ` p ` -adic rationals. Here we simply - define it as the splitting field of a dense sequence of polynomials - (using as the ` n ` -th set the collection of polynomials with degree - less than ` n ` and with coefficients ` < ( p ^ n ) ` ). Krasner's - lemma will then show that all monic polynomials have splitting fields - isomorphic to a sufficiently close Eisenstein polynomial from the list, - and unramified extensions are generated by the polynomial - ` x ^ ( p ^ n ) - x ` , which is in the list. Thus, every finite - extension of ` Qp ` is a subfield of this field extension, so it is - algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-qpa $a |- _Qp = ( p e. Prime |-> [_ ( Qp ` p ) / r ]_ - ( r polySplitLim ( n e. NN |-> { f e. ( Poly1 ` r ) | - ( ( r deg1 f ) <_ n /\ A. d e. ran ( coe1 ` f ) - ( `' d " ( ZZ \ { 0 } ) ) C_ ( 0 ... n ) ) } ) ) ) $. - - $( Define the metric completion of the algebraic completion of the ` p ` - -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - df-cp $a |- Cp = ( cplMetSp o. _Qp ) $. - $} - - $( TODO list $) - - $( change *mpt2* -> *mpo* $) - $( use more symbol variables like .+ $) - $( Uniform spaces $) - $( Characterization of perfectly normal spaces $) - $( Hausdorff quotient $) - $( add mpt2mptf, fmpt2i $) - -$( (End of Mario Carneiro's mathbox.) $) -$( End $[ set-mbox-mc.mm $] $) - - -$( Begin $[ set-mbox-fc.mm $] $) -$( Skip $[ set-main.mm $] $) -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Filip Cernatescu -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - - I hope someone will enjoy solving (proving) the simple equations, - inequalities, and calculations from this mathbox. I have proved these - problems (theorems) using the Milpgame proof assistant. (It can be - downloaded from ~ https://us.metamath.org/other/milpgame/milpgame.html .) - -$) - - $( Practice problem 1. Clues: ~ 5p4e9 ~ 3p2e5 ~ eqtri ~ oveq1i . - (Contributed by Filip Cernatescu, 16-Mar-2019.) - (Proof modification is discouraged.) $) - problem1 $p |- ( ( 3 + 2 ) + 4 ) = 9 $= - ( c3 c2 caddc co c4 c5 c9 3p2e5 oveq1i 5p4e9 eqtri ) ABCDZECDFECDGLFECHIJK - $. - - $( Practice problem 2. Clues: ~ oveq12i ~ adddiri ~ add4i ~ mulcli ~ recni - ~ 2re ~ 3eqtri ~ 10re ~ 5re ~ 1re ~ 4re ~ eqcomi ~ 5p4e9 ~ oveq1i ~ df-3 . - (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, - 9-Sep-2021.) (Proof modification is discouraged.) $) - problem2 $p |- ( ( ( 2 x. ; 1 0 ) + 5 ) + ( ( 1 x. ; 1 0 ) + 4 ) ) - = ( ( 3 x. ; 1 0 ) + 9 ) $= - ( c2 c1 cc0 cdc cmul co c5 caddc c4 c9 c3 2re recni 10re mulcli 5re 1re 4re - eqcomi oveq1i add4i adddiri 5p4e9 oveq12i df-3 3eqtri ) ABCDZEFZGHFBUGEFZIH - FHFUHUIHFZGIHFZHFABHFZUGEFZJHFKUGEFZJHFUHGUIIAUGALMZUGNMZOGPMBUGBQMZUPOIRMU - AUJUMUKJHUMUJABUGUOUQUPUBSUCUDUMUNJHULKUGEKULUESTTUF $. - - ${ - problem3.1 $e |- A e. CC $. - problem3.2 $e |- ( A + 3 ) = 4 $. - $( Practice problem 3. Clues: ~ eqcomi ~ eqtri ~ subaddrii ~ recni ~ 4re - ~ 3re ~ 1re ~ df-4 ~ addcomi . (Contributed by Filip Cernatescu, - 16-Mar-2019.) (Proof modification is discouraged.) $) - problem3 $p |- A = 1 $= - ( c1 c4 c3 cmin co 4re recni 3re 1re caddc eqcomi subaddrii addcomi eqtri - df-4 ) DADEFGHZASDEFDEIJZFKJZDLJEFDMHRNONEFATUABFAMHAFMHEFAUABPCQOQN $. - $} - - ${ - problem4.1 $e |- A e. CC $. - problem4.2 $e |- B e. CC $. - problem4.3 $e |- ( A + B ) = 3 $. - problem4.4 $e |- ( ( 3 x. A ) + ( 2 x. B ) ) = 7 $. - $( Practice problem 4. Clues: ~ pm3.2i ~ eqcomi ~ eqtri ~ subaddrii - ~ recni ~ 7re ~ 6re ~ ax-1cn ~ df-7 ~ ax-mp ~ oveq1i ~ 3cn ~ 2cn ~ df-3 - ~ mulid2i ~ subdiri ~ mp3an ~ mulcli ~ subadd23 ~ oveq2i ~ oveq12i - ~ 3t2e6 ~ mulcomi ~ subcli ~ biimpri ~ subadd2i . (Contributed by Filip - Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) $) - problem4 $p |- ( A = 1 /\ B = 2 ) $= - ( c1 wceq c2 c7 c6 cmin co caddc eqcomi c3 cmul 3cn 2cn eqtri ax-1cn df-7 - 7re recni 6re subaddrii df-3 oveq1i mulid2i subdiri mulcli subadd23 mp3an - cc wcel 3t2e6 mulcomi oveq12i subcli oveq2i subadd2i biimpri ax-mp pm3.2i - ) AGHBIHGAGJKLMZAVEGJKGJUCUDZKUEUDZUAJKGNMUBOUFOAKNMZJHZVEAHZVHPAQMZIBQMZ - NMZJVHVKIAQMZLMZKNMZVMAVOKNAPILMZAQMZVOVRAVRGAQMAVQGAQPIGRSUAPIGNMZUGOZUF - UHACUITOPIARSCUJTUHVPVKKVNLMZNMZVMVKUNUOVNUNUOKUNUOVPWBHPARCUKIASCUKVGVKV - NKULUMVMWBVLWAVKNWAVLWAPIQMZAIQMZLMZVLWEWAWCKWDVNLUPAICSUQUROWEPALMZIQMZV - LWGWEPAIRCSUJOWGIWFQMZVLWHWGIWFSPARCUSUQOVLWHBWFIQWFBPABRCDEUFOZUTOTTTOUT - OTTFTVJVIJKAVFVGCVAVBVCTOZBWFIWIWFPGLMZIAGPLWJUTVSPHZWKIHZVTWMWLPGIRUASVA - VBVCTTVD $. - $} - - ${ - problem5.1 $e |- A e. RR $. - problem5.2 $e |- ( ( 2 x. A ) + 3 ) < 9 $. - $( Practice problem 5. Clues: ~ 3brtr3i ~ mpbi ~ breqtri ~ ltaddsubi - ~ remulcli ~ 2re ~ 3re ~ 9re ~ eqcomi ~ mvlladdi 3cn ~ 6cn ~ eqtr3i - ~ 6p3e9 ~ addcomi ~ ltdiv1ii ~ 6re ~ nngt0i ~ 2nn ~ divcan3i ~ recni - ~ 2cn ~ 2ne0 ~ mpbir ~ eqtri ~ mulcomi ~ 3t2e6 ~ divmuli . (Contributed - by Filip Cernatescu, 16-Mar-2019.) - (Proof modification is discouraged.) $) - problem5 $p |- A < 3 $= - ( c2 cmul co cdiv c6 c3 clt wbr c9 caddc 2re mpbi 3cn 6cn eqcomi 2cn 2ne0 - cmin remulcli 3re 9re ltaddsubi 6p3e9 addcomi eqtr3i mvlladdi breqtri 6re - 2nn nngt0i ltdiv1ii recni divcan3i wceq mulcomi 3t2e6 eqtri divmuli mpbir - 3brtr3i ) DAEFZDGFZHDGFZAIJVDHJKVEVFJKVDLIUAFZHJVDIMFLJKVDVGJKCVDILDANBUB - ZUCUDUEOHVGIHLPQLIHMFZHIMFLVIUFHIQPUGUHRUIRUJVDHDVHUKNDULUMUNOADABUOSTUPV - FIUQDIEFZHUQVJIDEFHDISPURUSUTHDIQSPTVAVBVC $. - $} - - ${ - quad3.1 $e |- X e. CC $. - quad3.2 $e |- A e. CC $. - quad3.3 $e |- A =/= 0 $. - quad3.4 $e |- B e. CC $. - quad3.5 $e |- C e. CC $. - quad3.6 $e |- ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 $. - $( Variant of quadratic equation with discriminant expanded. (Contributed - by Filip Cernatescu, 19-Oct-2019.) $) - quad3 $p |- ( X = ( ( -u B + ( sqrt ` ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) - ) ) / ( 2 x. A ) ) \/ X = ( ( -u B - ( sqrt ` ( ( B ^ 2 ) - - ( 4 x. ( A x. C ) ) ) ) ) / ( 2 x. A ) ) ) $= - ( c2 cmul co cdiv caddc cmin wceq 2cn oveq2i oveq1i cexp c4 csqrt cneg wo - cfv mulcli 2ne0 mulne0i divcli addcli sqmuli binom2i sqcli divdiri div23i - divcan3i oveq12i eqtr2i mulcomi divcan2i divassi cc cc0 wa pm3.2i divdiv1 - wcel wne mp3an eqtri 3eqtr2i addassi eqcomi pncan3oi df-neg eqtr4i negcli - 3eqtr3i addcomi sqdivi 4cn 4ne0 divmuldivi c1 dividi eqtr3i mulm1i neg1cn - mulid2i mulassi 3eqtri 2t2e4 sqvali eqnetri negsubi subcli eqsqrtor ax-mp - wb mpbi sqrtcl divmuli eqcom bitr3i subadd2i divneg eqeq2i 3bitri orbi12i - ) KALMZDBXKNMZOMZLMZBKUAMZUBACLMZLMZPMZUCUFZQZXNXSUDZQZUEZDBUDZXSOMXKNMZQ - ZDYDXSPMZXKNMZQZUEXNKUAMZXRQZYCYJXKKUAMZXMKUAMZLMYLXRYLNMZLMXRXKXMKARFUGZ - DXLEBXKHYOKARFUHGUIZUJZUKZULYMYNYLLYMCUDZANMZXLKUAMZOMZUUAYTOMZYNYMDKUAMZ - KDXLLMZLMZOMZUUAOMUUBDXLEYQUMUUGYTUUAOUUDBANMZDLMZOMZAUUDLMZBDLMZOMZANMZU - UGYTUUNUUKANMZUULANMZOMUUJUUKUULAAUUDFDEUNZUGZBDHEUGZFGUOUUOUUDUUPUUIOUUD - AUUQFGUQBDAHEFGUPURUSUUIUUFUUDOUUIDUUHLMZKUUTKNMZLMUUFUUHDBAHFGUJZEUTUUTK - DUUHEUVBUGRUHVAUVAUUEKLUVADUUHKNMZLMUUEDUUHKEUVBRUHVBUVCXLDLUVCBAKLMZNMZX - LBVCVHZAVCVHZAVDVIZVEKVCVHZKVDVIZVEUVCUVEQHUVGUVHFGVFUVIUVJRUHVFBAKVGVJUV - DXKBNAKFRUTSVKSVKSVLSUUMYSANUUMUUKUULCOMOMZCPMZYSUVLUUMCOMZCPMUUMUVKUVMCP - UVMUVKUUKUULCUURUUSIVMVNTUUMCUUKUULUURUUSUKIVOUSUVLVDCPMYSUVKVDCPJTCVPVQV - KTVSTVKYTUUAYSACIVRZFGUJZXLYQUNVTUUCXOYLNMZXQUDZYLNMZOMXOUVQOMZYLNMYNUUAU - VPYTUVROBXKHYOYPWAUBALMZUVTNMZYTLMZUVTYSLMZUVTALMZNMYTUVRUVTUVTYSAUBAWBFU - GZUWEUVNFUBAWBFWCGUIZGWDWEYTLMUWBYTWEUWAYTLUWAWEUVTUWEUWFWFVNTYTUVOWJWGUW - CUVQUWDYLNUWCWEUDZUVTCLMZLMZUWGXQLMUVQUWCUVTUWGLMZCLMZUWGUVTLMZCLMUWIUWCU - VTUWGCLMZLMUWKYSUWMUVTLUWMYSCIWHVNSUVTUWGCUWEWIIWKVQUWJUWLCLUVTUWGUWEWIUT - TUWGUVTCWIUWEIWKWLUWHXQUWGLUBACWBFIWKSXQUBXPWBACFIUGUGZWHWLUWDXKXKLMZYLUW - DKXKLMZALMZXKKLMZALMUWOUWDKKLMZALMZALMUWQUVTUWTALUBUWSALUWSUBWMVNTTUWTUWP - ALKKARRFWKTVKUWPUWRALKXKRYOUTTXKKAYORFWKWLXKYOWNZVQURVSURXOUVQYLBHUNZXQUW - NVRXKYOUNZYLUWOVDUXAXKXKYOYOYPYPUIWOZUOUVSXRYLNXOXQUXBUWNWPTVLWLSXRYLXOXQ - UXBUWNWQZUXCUXDVAWLXNVCVHZXRVCVHZVEYKYCWTUXFUXGXKXMYOYRUGUXEVFXNXRWRWSXAX - TYFYBYIXTXMXSXKNMZQZDUXHXLPMZQZYFXTUXHXMQUXIXSXKXMUXGXSVCVHUXEXRXBWSZYOYR - YPXCUXHXMXDXEUXIUXJDQUXKUXHXLDXSXKUXLYOYPUJZYQEXFUXJDXDXEUXJYEDUXJYDXKNMZ - UXHOMZYEUXHXLUDZOMUXHUXNOMUXJUXOUXPUXNUXHOUVFXKVCVHXKVDVIUXPUXNQHYOYPBXKX - GVJZSUXHXLUXMYQWPUXHUXNUXMYDXKBHVRZYOYPUJZVTVSYDXSXKUXRUXLYOYPUOVQXHXIYBX - MYAXKNMZQZDUXTXLPMZQZYIYBUXTXMQUYAYAXKXMXSUXLVRZYOYRYPXCUXTXMXDXEUYAUYBDQ - UYCUXTXLDYAXKUYDYOYPUJZYQEXFUYBDXDXEUYBYHDUYBUXNUXTOMZYDYAOMZXKNMYHUXTUXP - OMUXTUXNOMUYBUYFUXPUXNUXTOUXQSUXTXLUYEYQWPUXTUXNUYEUXSVTVSYDYAXKUXRUYDYOY - PUOUYGYGXKNYDXSUXRUXLWPTVLXHXIXJXA $. - $} - -$( (End of Filip Cernatescu's mathbox.) $) -$( End $[ set-mbox-fc.mm $] $) - - -$( Begin $[ set-mbox-pc.mm $] $) -$( Skip $[ set-main.mm $] $) -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Paul Chapman -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# -$) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Propositional calculus -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Prime numbers -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Finite set induction -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - The ` # ` (set size) function -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Real and complex numbers (cont.) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - ${ - $d k m $. - $( Utility lemma to convert between ` m <_ k ` and ` k e. ( ZZ>= `` m ) ` - in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.) $) - climuzcnv $p |- ( m e. NN -> ( ( k e. ( ZZ>= ` m ) -> ph ) <-> - ( k e. NN -> ( m <_ k -> ph ) ) ) ) $= - ( cv cn wcel cuz cfv wi cle wbr wa elnnuz uztrn sylan2b sylibr expcom nnz - c1 cz eluzle a1i jcad biimpri syl3an1 syl3an2 3expib impbid imbi1d impexp - w3a eluz2 bitrdi ) CDZEFZBDZUNGHFZAIUPEFZUNUPJKZLZAIURUSAIIUOUQUTAUOUQUTU - OUQURUSUQUOURUQUOLUPSGHZFZURUOUQUNVAFVBUNMUNUPSNOUPMPQUQUSIUOUNUPUAUBUCUO - URUSUQURUOUPTFZUSUQUPRUOUNTFZVCUSUQUNRUQVDVCUSUKUNUPULUDUEUFUGUHUIURUSAUJ - UM $. - $} - - ${ - $d k w x y z F $. $d k w y z H $. $d k z M $. $d k w y z ph $. - $d k G $. - sinccvg.1 $e |- ( ph -> F : NN --> ( RR \ { 0 } ) ) $. - sinccvg.2 $e |- ( ph -> F ~~> 0 ) $. - sinccvg.3 $e |- G = ( x e. ( RR \ { 0 } ) |-> ( ( sin ` x ) / x ) ) $. - sinccvg.4 $e |- H = ( x e. CC |-> ( 1 - ( ( x ^ 2 ) / 3 ) ) ) $. - sinccvg.5 $e |- ( ph -> M e. NN ) $. - sinccvg.6 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> - ( abs ` ( F ` k ) ) < 1 ) $. - $( ` ( ( sin `` x ) / x ) ~~> 1 ` as (real) ` x ~~> 0 ` . (Contributed by - Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, - 21-May-2014.) $) - sinccvglem $p |- ( ph -> ( G o. F ) ~~> 1 ) $= - ( c1 cc0 wcel cc co c3 cdiv vy vz vw ccom cvv cuz cfv eqid nnzd cli cv c2 - wfun cexp cmin funmpt2 cn cr csn cdif wf nnex sylancl cofunexg sylancr wa - fex wne adantr eluznn sylan ffvelcdmd eldifsn sylib simpld recnd sqcl 3cn - ax-1cn 3ne0 divcl mp3an23 syl subcl fmpti ccncf crp cabs clt wi wral wrex - wbr cmpt ccnfld ctopn ccn wtru ctopon cnfldtopon 1cnd cnmptc divccn mp2an - a1i sqcn oveq1 cnmpt11 ctx subcn cnmpt12f mptru cncfcn1 cncfi mp3an1 wceq - 3eltr4i fvco3 syldan climcn1lem 0cn oveq1d eqtrdi oveq2d fvmpt csin eqtrd - ovex 1re eqeltrd fveq2 id oveq12d resincld simprd cmul eqbrtrd cneg ltled - cle sq0i div0i 1m0e1 1ex breqtrdi resqcld nndivre resubcl redivcld abscld - ax-mp 3nn subdird mulid2d caddc df-3 oveq2i cn0 2nn0 expp1 absresq eqtrid - div23d eqtr2d cioc absrpcld rpgt0d ltle mpd cxr w3a wb syl3anbrc sin01bnd - 0xr elioc2 ltmuldivd mpbid sinneg eqeq1d syl5ibrcom absord mpjaod breqtrd - div2negd 3brtr4d mulid1d breqtrrd ltdivmuld mpbird eqbrtrrd climsqz ) ANC - FDUDZEDUDZGUEGUFUGZUWOUHZAGLUIZAUWMOFUGZNUJAUAUBUCOCDUWMFGUEUWOUWPIAFUMDU - EPZUWMUEPBQNBUKZULUNRZSTRZUORZFKUPAUQUROUSUTZDVAZUQUEPUWSHVBUQUXDUEDVGVCZ - FDUEVDVEUWQACUKZUWOPZVFZUXGDUGZUXIUXJURPZUXJOVHZUXIUXJUXDPZUXKUXLVFUXIUQU - XDUXGDAUXEUXHHVIAGUQPUXHUXGUQPZLUXGGVJVKZVLZUXJUROVMVNZVOZVPZBQQUXCFKUWTQ - PZNQPUXBQPZUXCQPVSUXTUXAQPZUYAUWTVQUYBSQPZSOVHZUYAVRVTUXASWAWBWCNUXBWDVEW - EFQQWFRZPOQPZUAUKZWGPUCUKZOUORWHUGUBUKWIWMUYHFUGUWRUORWHUGUYGWIWMWJUCQWKU - BWGWLBQUXCWNZWOWPUGZUYJWQRZFUYEUYIUYKPWRBNUXBUOUYJUYJUYJUYJQUYJQWSUGPWRUY - JUYJUHZWTXEZWRBNUYJUYJQQUYMUYMWRXAXBWRBUAUXAUYGSTRZUXBUYJUYJUYJQQUYMBQUXA - WNUYKPWRBUYJUYLXFXEUYMUAQUYNWNUYKPZWRUYCUYDUYOVRVTUASUYJUYLXCXDXEUYGUXAST - XGXHUOUYJUYJXIRUYJWQRPWRUYJUYLXJXEXKXLKUYJUYLXMXQUBUCQQOUYGFXNXOAUXHUXNUX - GUWMUGZUXJFUGZXPZUXOAUXEUXNUYRHUQUXDUXGFDXRVKXSZXTUYFUWRNXPYABOUXCNQFUWTO - XPZUXCNOUORNUYTUXBONUOUYTUXBOSTROUYTUXAOSTUWTUUAYBSVRVTUUBYCYDUUCYCKUUDYE - UUKUUEAEUMUWSUWNUEPBUXDUWTYFUGZUWTTRZEJUPUXFEDUEVDVEUXIUYPNUXJULUNRZSTRZU - ORZURUXIUYPUYQVUEUYSUXIUXJQPZUYQVUEXPUXSBUXJUXCVUEQFUWTUXJXPZUXBVUDNUOVUG - UXAVUCSTUWTUXJULUNXGYBYDKNVUDUOYHYEWCYGZUXINURPZVUDURPZVUEURPYIUXIVUCURPS - UQPVUJUXIUXJUXRUUFZUULVUCSUUGVCZNVUDUUHVEZYJUXIUXGUWNUGZUXJYFUGZUXJTRZURU - XIVUNUXJEUGZVUPAUXHUXNVUNVUQXPZUXOAUXEUXNVURHUQUXDUXGEDXRVKXSUXIUXMVUQVUP - XPUXPBUXJVUBVUPUXDEVUGVUAVUOUWTUXJTUWTUXJYFYKVUGYLYMJVUOUXJTYHYEWCYGZUXIV - UOUXJUXIUXJUXRYNZUXRUXIUXKUXLUXQYOZUUIZYJUXIVUEVUPUYPVUNYTUXIVUEVUPVUMVVB - UXIVUEUXJWHUGZYFUGZVVCTRZVUPWIUXIVUEVVCYPRZVVDWIWMVUEVVEWIWMUXIVVFVVCVVCS - UNRZSTRZUORZVVDWIUXIVVFNVVCYPRZVUDVVCYPRZUORVVIUXINVUDVVCUXIXAUXIVUDVULVP - UXIVVCUXIUXJUXSUUJZVPZUUMUXIVVJVVCVVKVVHUOUXIVVCVVMUUNUXIVVHVUCVVCYPRZSTR - VVKUXIVVGVVNSTUXIVVGVVCULNUUORZUNRZVVNSVVOVVCUNUUPUUQUXIVVPVVCULUNRZVVCYP - RZVVNUXIVVCQPULUURPVVPVVRXPVVMUUSVVCULUUTVCUXIVVQVUCVVCYPUXIUXKVVQVUCXPUX - RUXJUVAWCYBYGUVBYBUXIVUCVVCSUXIVUCVUKVPVVMUYCUXIVRXEUYDUXIVTXEUVCUVDYMYGU - XIVVIVVDWIWMZVVDVVCWIWMZUXIVVCONUVERPZVVSVVTVFUXIVVCURPZOVVCWIWMZVVCNYTWM - ZVWAVVLUXIVVCUXIUXJUXSVVAUVFZUVGUXIVVCNWIWMZVWDMUXIVWBVUIVWFVWDWJVVLYIVVC - NUVHVCUVIOUVJPVUIVWAVWBVWCVWDUVKUVLUVOYIONVVCUVPXDUVMVVCUVNWCZVOYQUXIVUEV - VDVVCVUMUXIVVCVVLYNZVWEUVQUVRUXIVVCUXJXPZVVEVUPXPZVVCUXJYRZXPZVWIVWJWJUXI - VWIVVDVUOVVCUXJTVVCUXJYFYKVWIYLYMXEUXIVWJVWLVWKYFUGZVWKTRZVUPXPUXIVWNVUOY - RZVWKTRVUPUXIVWMVWOVWKTUXIVUFVWMVWOXPUXSUXJUVSWCYBUXIVUOUXJUXIVUOVUTVPUXS - VVAUWEYGVWLVVEVWNVUPVWLVVDVWMVVCVWKTVVCVWKYFYKVWLYLYMUVTUWAUXIUXJUXRUWBUW - CZUWDYSVUHVUSUWFUXIVUNVUPNYTVUSUXIVUPNVVBVUIUXIYIXEZUXIVVEVUPNWIVWPUXIVVE - NWIWMVVDVVCNYPRZWIWMUXIVVDVVCVWRWIUXIVVSVVTVWGYOUXIVVCVVMUWGUWHUXIVVDNVVC - VWHVWQVWEUWIUWJUWKYSYQUWL $. - $} - - ${ - $d j k n x F $. - $( ` ( ( sin `` x ) / x ) ~~> 1 ` as (real) ` x ~~> 0 ` . (Contributed by - Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, - 21-May-2014.) $) - sinccvg $p |- ( ( F : NN --> ( RR \ { 0 } ) /\ F ~~> 0 ) -> - ( ( x e. ( RR \ { 0 } ) |-> ( ( sin ` x ) / x ) ) o. F ) ~~> 1 ) $= - ( vk vj vn cn cc0 cli wbr wa cv cfv cabs c1 clt cdiv co cmpt wcel eqid cr - csn cdif wf cuz wral csin ccom nnuz 1zzd crp 1rp eqidd simpr climi0 cc c2 - a1i cexp cmin simpll simplr simprl simprr weq 2fveq3 breq1d rspccva sylan - c3 sinccvglem rexlimddv ) FUAGUBUCZBUDZBGHIZJZCKZBLZMLZNOIZCDKZUELZUFZAVM - AKZUGLWDPQRZBUHNHIDFVPVRNDCBNFUIVPUJNUKSVPULURVPVQFSJVRUMVNVOUNUOVPWAFSZW - CJZJZAEBWEAUPNWDUQUSQVJPQUTQRZWAVNVOWGVAVNVOWGVBWETWITVPWFWCVCWHWCEKZWBSW - JBLMLZNOIZVPWFWCVDVTWLCWJWBCEVEVSWKNOVQWJMBVFVGVHVIVKVL $. - $} - - ${ - $d n y $. $d k P $. $d k n R $. - circum.1 $e |- A = ( ( 2 x. _pi ) / n ) $. - circum.2 $e |- - P = ( n e. NN |-> ( ( 2 x. n ) x. ( R x. ( sin ` ( A / 2 ) ) ) ) ) $. - circum.3 $e |- R e. RR $. - $( The circumference of a circle of radius ` R ` , defined as the limit as - ` n ~~> +oo ` of the perimeter of an inscribed n-sided isogons, is - ` ( ( 2 x. _pi ) x. R ) ` . (Contributed by Paul Chapman, 10-Nov-2012.) - (Proof shortened by Mario Carneiro, 21-May-2014.) $) - circum $p |- P ~~> ( ( 2 x. _pi ) x. R ) $= - ( c2 cpi cmul co wtru cn cdiv csin cfv cr cc0 wcel cc vk vy c1 cli wbr cv - cmpt cvv nnuz 1zzd csn cdif ccom wne wa crp pirp rpdivcl sylancr rprene0d - nnrp eldifsn sylibr adantl eqidd wceq fveq2 id oveq12d wf eqid fmpti pire - fmptco recni divcnv mp1i sinccvg eqbrtrrd 2re remulcli nnex mptex eqeltri - a1i resincld eldifsni redivcld fco mp2an mptru feq1i mpbi ffvelcdmi recnd - eldifi nncn nnne0 divassd oveq1d simpr nndivre 2ne0 divcan3d eqtrd fveq2d - rpne0d divcan2d eqtr4d oveq2d oveq2 ovex fvmpt mulassd mulcl mul4d mul32d - eqeltrrd 3eqtr2d eqtrid 3eqtr4d climmulc2 mulid1i breqtri ) BHIJKZCJKZUCJ - KZYFUDBYGUDUELUCYFUADMIDUFZNKZOPZYINKZUGZBUCUHMUILUJLUBQRUKZULZUBUFZOPZYO - NKZUGZDMYIUGZUMZYLUCUDLDUBMYNYIYQYKYSYRYHMSZYIYNSZLUUAYIQSYIRUNUOUUBUUAYI - UUAIUPSZYHUPSYIUPSUQYHVAIYHURUSUTYIQRVBVCZVDLYSVELYRVEYOYIVFZYPYJYOYINYOY - IOVGUUEVHVIVNZLMYNYSVJZYSRUDUEZYTUCUDUEDMYNYIYSYSVKUUDVLZITSZUUHLIVMVOZID - VPVQUBYSVRUSVSYFTSLYFYECHIVTVMWAGWAVOZWEBUHSLBDMHYHJKZCAHNKZOPZJKZJKZUGUH - FDMUUQWBWCWDWELUAUFZMSZUOZUURYLPZUUSUVAQSLMQUURYLMQYTVJZMQYLVJYNQYRVJUUGU - VBUBYNQYQYRYRVKYOYNSZYPYOUVCYOYOQYMWPZWFUVDYOQRWGWHVLUUIMYNQYRYSWIWJMQYTY - LYTYLVFUUFWKWLWMWNVDWOZUUTHUURJKZCYEUURNKZHNKZOPZJKZJKZYFIUURNKZOPZUVLNKZ - JKZUURBPZYFUVAJKUUTUVKUVFCUVLJKZUVNJKZJKUVFUVQJKZUVNJKUVOUUTUVJUVRUVFJUUT - UVJCUVLUVNJKZJKUVRUUTUVIUVTCJUUTUVIUVMUVTUUTUVHUVLOUUTUVHHUVLJKZHNKUVLUUT - UVGUWAHNUUTHIUURHTSZUUTHVTVOZWEZUUJUUTUUKWEZUUSUURTSZLUURWQVDZUUSUURRUNLU - URWRVDZWSWTUUTUVLHUUTUVLUUTIQSUUSUVLQSVMLUUSXAIUURXBUSZWOZUWDHRUNUUTXCWEX - DXEXFUUTUVMUVLUUTUVMUUTUVLUWIWFWOUWJUUTUVLUUTUUCUURUPSZUVLUPSUQUUSUWKLUUR - VAVDIUURURUSXGXHXIXJUUTCUVLUVNCTSZUUTCGVOZWEZUWJUUTUVAUVNTUUSUVAUVNVFLDUU - RYKUVNMYLYHUURVFZYJUVMYIUVLNUWOYIUVLOYHUURINXKZXFUWPVIYLVKUVMUVLNXLXMVDZU - VEXRZXNXIXJUUTUVFUVQUVNUUTUWBUWFUVFTSUWCUWGHUURXOUSUUTUWLUVLTSUVQTSUWMUWJ - CUVLXOUSUWRXNUUTUVSYFUVNJUUTUVSHCJKZUURUVLJKZJKZYFUUTHUURCUVLUWDUWGUWNUWJ - XPUUTUXAUWSIJKYFUUTUWTIUWSJUUTIUURUWEUWGUWHXHXJUUTHCIUWDUWNUWEXQXEXEWTXSU - USUVPUVKVFLDUURUUQUVKMBUWOUUMUVFUUPUVJJYHUURHJXKUWOUUOUVICJUWOUUNUVHOUWOA - UVGHNUWOAYEYHNKUVGEYHUURYENXKXTWTXFXJVIFUVFUVJJXLXMVDUUTUVAUVNYFJUWQXJYAY - BWKYFUULYCYD $. - $} - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Miscellaneous theorems -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $( Membership in a curtailed finite sequence of integers. (Contributed by - Paul Chapman, 17-Nov-2012.) $) - elfzm12 $p |- ( N e. NN -> - ( M e. ( 1 ... ( N - 1 ) ) -> M e. ( 1 ... N ) ) ) $= - ( cn wcel c1 cmin co cfz cz cuz cfv wss nnz cle wbr zre lem1d peano2zm eluz - wb mpancom mpbird fzss2 3syl sseld ) BCDZEBEFGZHGZEBHGZAUFBIDZBUGJKDZUHUILB - MUJUKUGBNOZUJBBPQUGIDUJUKULTBRUGBSUAUBUGEBUCUDUE $. - - ${ - $d F k $. $d N k $. - nn0seqcvg.1 $e |- F : NN0 --> NN0 $. - nn0seqcvg.2 $e |- N = ( F ` 0 ) $. - nn0seqcvg.3 $e |- ( k e. NN0 -> ( ( F ` ( k + 1 ) ) =/= 0 -> - ( F ` ( k + 1 ) ) < ( F ` k ) ) ) $. - $( A strictly-decreasing nonnegative integer sequence with initial term - ` N ` reaches zero by the ` N ` th term. Inference version. - (Contributed by Paul Chapman, 31-Mar-2011.) $) - nn0seqcvg $p |- ( F ` N ) = 0 $= - ( c1 wceq cfv cc0 eqid cn0 wf a1i cv wcel caddc co wne clt wbr nn0seqcvgd - wi adantl ax-mp ) GGHZCBIJHGKUFABCLLBMUFDNCJBIHUFENAOZLPUGGQRBIZJSUHUGBIT - UAUCUFFUDUBUE $. - $} - - $( Division of both sides of 'less than or equal to' by a nonnegative number. - (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) - (Proof modification is discouraged.) $) - lediv2aALT $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ - ( C e. RR /\ 0 <_ C ) ) -> - ( A <_ B -> ( C / B ) <_ ( C / A ) ) ) $= - ( cr wcel cc0 clt wbr wa cle w3a c1 cdiv co anim12i ancoms syl recn adantr - cc wi wne gt0ne0 rereccl syldan 3adant3 simp3 df-3an sylanbrc lemul2a ex wb - cmul lerec wceq jca 3anass sylibr divrec 3adant1 3adant2 breq12d 3imtr4d ) - ADEZFAGHZIZBDEZFBGHZIZCDEZFCJHZIZKZLBMNZLAMNZJHZCVNUMNZCVOUMNZJHZABJHZCBMNZ - CAMNZJHVMVNDEZVODEZVLKZVPVSUAVMWCWDIZVLWEVFVIWFVLVIVFWFVIWCVFWDVGVHBFUBZWCB - UCZBUDUEVDVEAFUBZWDAUCZAUDUEOPUFVFVIVLUGWCWDVLUHUIWEVPVSVNVOCUJUKQVFVIVTVPU - LVLABUNUFVMWAVQWBVRJVIVLWAVQUOZVFVLVIWKVLVIIZCTEZBTEZWGKZWKWLWMWNWGIZIWOVLW - MVIWPVJWMVKCRSZVIWNWGVGWNVHBRSWHUPOWMWNWGUQURCBUSQPUTVFVLWBVRUOZVIVLVFWRVLV - FIZWMATEZWIKZWRWSWMWTWIIZIXAVLWMVFXBWQVFWTWIVDWTVEARSWJUPOWMWTWIUQURCAUSQPV - AVBVC $. - - ${ - abs2sqlei.1 $e |- A e. CC $. - abs2sqlei.2 $e |- B e. CC $. - $( The absolute values of two numbers compare as their squares. - (Contributed by Paul Chapman, 7-Sep-2007.) $) - abs2sqlei $p |- ( ( abs ` A ) <_ ( abs ` B ) <-> - ( ( abs ` A ) ^ 2 ) <_ ( ( abs ` B ) ^ 2 ) ) $= - ( cc0 cabs cfv cle wbr c2 cexp co wb absge0i abscli le2sqi mp2an ) EAFGZH - IEBFGZHIRSHIRJKLSJKLHIMACNBDNRSACOBDOPQ $. - $} - - ${ - abs2sqlti.1 $e |- A e. CC $. - abs2sqlti.2 $e |- B e. CC $. - $( The absolute values of two numbers compare as their squares. - (Contributed by Paul Chapman, 7-Sep-2007.) $) - abs2sqlti $p |- ( ( abs ` A ) < ( abs ` B ) <-> - ( ( abs ` A ) ^ 2 ) < ( ( abs ` B ) ^ 2 ) ) $= - ( cc0 cabs cfv cle wbr clt c2 cexp co wb absge0i abscli lt2sqi mp2an ) EA - FGZHIEBFGZHISTJISKLMTKLMJINACOBDOSTACPBDPQR $. - $} - - $( The absolute values of two numbers compare as their squares. (Contributed - by Paul Chapman, 7-Sep-2007.) $) - abs2sqle $p |- ( ( A e. CC /\ B e. CC ) -> - ( ( abs ` A ) <_ ( abs ` B ) <-> - ( ( abs ` A ) ^ 2 ) <_ ( ( abs ` B ) ^ 2 ) ) ) $= - ( cc wcel cabs cfv cle wbr c2 cexp co wb cc0 cif wceq breq1d bibi12d breq2d - fveq2 0cn oveq1d oveq1 syl elimel abs2sqlei dedth2h ) ACDZBCDZAEFZBEFZGHZUI - IJKZUJIJKZGHZLUGAMNZEFZUJGHZUPIJKZUMGHZLUPUHBMNZEFZGHZURVAIJKZGHZLABMMAUOOZ - UKUQUNUSVEUIUPUJGAUOESZPVEULURUMGVEUIUPIJVFUAPQBUTOZUQVBUSVDVGUJVAUPGBUTESZ - RVGUJVAOZUSVDLVHVIUMVCURGUJVAIJUBRUCQUOUTAMCTUDBMCTUDUEUF $. - - $( The absolute values of two numbers compare as their squares. (Contributed - by Paul Chapman, 7-Sep-2007.) $) - abs2sqlt $p |- ( ( A e. CC /\ B e. CC ) -> - ( ( abs ` A ) < ( abs ` B ) <-> - ( ( abs ` A ) ^ 2 ) < ( ( abs ` B ) ^ 2 ) ) ) $= - ( cc wcel cabs cfv clt wbr c2 cexp co wb cc0 cif wceq breq1d bibi12d breq2d - fveq2 0cn oveq1d oveq1 syl elimel abs2sqlti dedth2h ) ACDZBCDZAEFZBEFZGHZUI - IJKZUJIJKZGHZLUGAMNZEFZUJGHZUPIJKZUMGHZLUPUHBMNZEFZGHZURVAIJKZGHZLABMMAUOOZ - UKUQUNUSVEUIUPUJGAUOESZPVEULURUMGVEUIUPIJVFUAPQBUTOZUQVBUSVDVGUJVAUPGBUTESZ - RVGUJVAOZUSVDLVHVIUMVCURGUJVAIJUBRUCQUOUTAMCTUDBMCTUDUEUF $. - - ${ - abs2difi.1 $e |- A e. CC $. - abs2difi.2 $e |- B e. CC $. - $( Difference of absolute values. (Contributed by Paul Chapman, - 7-Sep-2007.) $) - abs2difi $p |- ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) $= - ( cc wcel cabs cfv cmin co cle wbr abs2dif mp2an ) AEFBEFAGHBGHIJABIJGHKL - CDABMN $. - $} - - ${ - abs2difabsi.1 $e |- A e. CC $. - abs2difabsi.2 $e |- B e. CC $. - $( Absolute value of difference of absolute values. (Contributed by Paul - Chapman, 7-Sep-2007.) $) - abs2difabsi $p |- ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ - ( abs ` ( A - B ) ) $= - ( cc wcel cabs cfv cmin co cle wbr abs2difabs mp2an ) AEFBEFAGHBGHIJGHABI - JGHKLCDABMN $. - $} - -$( (End of Paul Chapman's mathbox.) $) -$( End $[ set-mbox-pc.mm $] $) - - -$( Begin $[ set-mbox-df.mm $] $) -$( Skip $[ set-main.mm $] $) -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Drahflow -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - - This is the mathbox of Jens-Wolfhard Schicke-Uffmann, reachable at - drahflow@gmx.de / drahflow.name - -$) - -$( (End of Drahflow's mathbox.) $) -$( End $[ set-mbox-df.mm $] $) - - -$( Begin $[ set-mbox-sf.mm $] $) -$( Skip $[ set-main.mm $] $) -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Scott Fenton -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# -$) - - $( Set up a new wff var $) - $v al $. $( Greek alpha $) - - $( Let variable ` al ` be a wff. $) - walpha $f wff al $. - - $( Declare some new variables. $) - $v xO $. - $v xL $. - $v xR $. - $v yO $. - $v yL $. - $v yR $. - $( Let ` xO ` be an individual variable. $) - vxo $f setvar xO $. - $( Let ` xL ` be an individual variable. $) - vxl $f setvar xL $. - $( Let ` xR ` be an individual variable. $) - vxr $f setvar xR $. - $( Let ` yO ` be an individual variable. $) - vyo $f setvar yO $. - $( Let ` yL ` be an individual variable. $) - vyl $f setvar yL $. - $( Let ` yR ` be an individual variable. $) - vyr $f setvar yR $. - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - ZFC Axioms in primitive form -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $( ~ ax-ext without distinct variable conditions or defined symbols. - (Contributed by Scott Fenton, 13-Oct-2010.) $) - axextprim $p |- -. A. x -. ( ( x e. y -> x e. z ) -> - ( ( x e. z -> x e. y ) -> y = z ) ) $= - ( wel wb weq wi wex wn wal axextnd wa dfbi2 imbi1i impexp bitri exbii df-ex - mpbi ) ABDZACDZEZBCFZGZAHZTUAGZUATGZUCGGZIAJIZABCKUEUHAHUIUDUHAUDUFUGLZUCGU - HUBUJUCTUAMNUFUGUCOPQUHARPS $. - - $( ~ ax-rep without distinct variable conditions or defined symbols. - (Contributed by Scott Fenton, 13-Oct-2010.) $) - axrepprim $p |- -. A. x -. ( -. A. y -. A. z ( ph -> z = y ) -> - A. z -. ( ( A. y z e. x -> -. A. x ( A. z x e. y -> - -. A. y ph ) ) -> -. ( -. A. x ( A. z x e. y -> -. A. y ph ) - -> A. y z e. x ) ) ) $= - ( weq wi wal wex wel wa wb wn axrepnd df-ex df-an exbii exnal bitri bibi2i - dfbi1 albii imbi12i mpbi ) ADCEFDGZCHZDBICGZBCIDGZACGZJZBHZKZDGZFZBHZUDLCGL - ZUFUGUHLFZBGLZFUQUFFLFLZDGZFZLBGLZABCDMUNUTBHVAUMUTBUEUOULUSUDCNUKURDUKUFUQ - KURUJUQUFUJUPLZBHUQUIVBBUGUHOPUPBQRSUFUQTRUAUBPUTBNRUC $. - - $( ~ ax-un without distinct variable conditions or defined symbols. - (Contributed by Scott Fenton, 13-Oct-2010.) $) - axunprim $p |- -. A. x -. A. y ( -. A. x ( y e. x -> -. x e. z ) - -> y e. x ) $= - ( wel wa wex wi wal axunnd df-an exbii exnal bitri imbi1i albii df-ex mpbi - wn ) BADZACDZEZAFZSGZBHZAFZSTRGZAHRZSGZBHZRAHRZABCIUEUIAFUJUDUIAUCUHBUBUGSU - BUFRZAFUGUAUKASTJKUFALMNOKUIAPMQ $. - - $( ~ ax-pow without distinct variable conditions or defined symbols. - (Contributed by Scott Fenton, 13-Oct-2010.) $) - axpowprim $p |- ( A. x -. A. y ( A. x ( -. A. z -. x e. y -> A. y x e. z ) - -> y e. x ) -> x = y ) $= - ( weq wel wn wal wi wex axpownd df-ex imbi1i albii exbii bitri sylib con4i - ) ABDZABEZFCGFZACEBGZHZAGZBAEZHZBGZFAGZRFSCIZUAHZAGZUDHZBGZAIZUGFZABCJUMUFA - IUNULUFAUKUEBUJUCUDUIUBAUHTUASCKLMLMNUFAKOPQ $. - - $( ~ ax-reg without distinct variable conditions or defined symbols. - (Contributed by Scott Fenton, 13-Oct-2010.) $) - axregprim $p |- ( x e. y -> - -. A. x ( x e. y -> -. A. z ( z e. x -> -. z e. y ) ) ) $= - ( wel wn wi wal wa wex axregnd df-an exbii exnal bitri sylib ) ABDZPCADCBDE - FCGZHZAIZPQEFZAGEZABCJSTEZAIUARUBAPQKLTAMNO $. - - $( ~ ax-inf without distinct variable conditions or defined symbols. - (New usage is discouraged.) (Contributed by Scott Fenton, - 13-Oct-2010.) $) - axinfprim $p |- -. A. x -. ( y e. z -> -. ( y e. x -> - -. A. y ( y e. x -> -. A. z ( y e. z -> -. z e. x ) ) ) ) $= - ( wel wa wex wi wn axinfnd df-an exbii exnal bitri imbi2i albii anbi2i mpbi - wal df-ex ) BCDZBADZUATCADZEZCFZGZBRZEZGZAFZTUAUATUBHGZCRHZGZBRZHGHZGZHARHZ - ABCIUIUOAFUPUHUOAUGUNTUGUAUMEUNUFUMUAUEULBUDUKUAUDUJHZCFUKUCUQCTUBJKUJCLMNO - PUAUMJMNKUOASMQ $. - - $( ~ ax-ac without distinct variable conditions or defined symbols. - (New usage is discouraged.) (Contributed by Scott Fenton, - 26-Oct-2010.) $) - axacprim $p |- -. A. x -. A. y A. z ( A. x -. ( y e. z -> -. z e. w ) -> - -. A. w -. A. y -. ( ( -. A. w ( y e. z -> ( z e. w -> - ( y e. w -> -. w e. x ) ) ) -> y = w ) -> -. ( y = w -> - -. A. w ( y e. z -> ( z e. w -> ( y e. w -> -. w e. x ) - ) ) ) ) ) $= - ( wel wa wal wex wb wi wn axacnd df-an albii annim anbi2i exbii bitri df-ex - weq anass pm4.63 bitr3i 3bitr2i exnal bibi1i dfbi1 imbi12i 2albii mpbi ) BC - EZCDEZFZAGZUMBDEZDAEZFZFZDHZBDTZIZBGZDHZJZCGBGZAHZUKULKJKZAGZUKULUOUPKJZJZJ - ZDGKZUTJUTVLJKJKZBGZKDGKZJZCGBGZKAGKZABCDLVFVQAHVRVEVQAVDVPBCUNVHVCVOUMVGAU - KULMNVCVNDHVOVBVNDVAVMBVAVLUTIVMUSVLUTUSVKKZDHVLURVSDURUKULUQFZFUKVJKZFVSUK - ULUQUAWAVTUKWAULVIKZFVTULVIOWBUQULUOUPUBPUCPUKVJOUDQVKDUERUFVLUTUGRNQVNDSRU - HUIQVQASRUJ $. - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Untangled classes -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - ${ - $d x A $. - $( We call a class "untanged" if all its members are not members of - themselves. The term originates from Isbell (see citation in ~ dfon2 ). - Using this concept, we can avoid a lot of the uses of the Axiom of - Regularity. Here, we prove a series of properties of untanged classes. - First, we prove that an untangled class is not a member of itself. - (Contributed by Scott Fenton, 28-Feb-2011.) $) - untelirr $p |- ( A. x e. A -. x e. x -> -. A e. A ) $= - ( wel wn wral wcel cv wceq eleq1 eleq2 bitrd notbid rspccv pm2.01d ) AACZ - DZABEBBFZPQDABBAGZBHZOQSOBRFQRBRIRBBJKLMN $. - $} - - ${ - $d x y A $. - $( The union of a class is untangled iff all its members are untangled. - (Contributed by Scott Fenton, 28-Feb-2011.) $) - untuni $p |- ( A. x e. U. A -. x e. x - <-> A. y e. A A. x e. y -. x e. x ) $= - ( cv cuni wcel wel wn wal wral wrex r19.23v albii ralcom4 eluni2 3bitr4ri - wi imbi1i df-ral ralbii 3bitr4i ) ADZCEZFZAAGHZQZAIZABGZUEQZAIZBCJZUEAUCJ - UEABDZJZBCJUIBCJZAIUHBCKZUEQZAIUKUGUNUPAUHUEBCLMUIBACNUFUPAUDUOUEBUBCORMP - UEAUCSUMUJBCUEAULSTUA $. - $} - - ${ - $d x A $. $d x y $. - untsucf.1 $e |- F/_ y A $. - $( If a class is untangled, then so is its successor. (Contributed by - Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, - 11-Dec-2016.) $) - untsucf $p |- ( A. x e. A -. x e. x -> A. y e. suc A -. y e. y ) $= - ( wel wn wral csuc nfv nfralw cv wcel wceq vex elsuc elequ1 elequ2 notbid - wo bitrd rspccv untelirr eleq1 eleq2 syl5ibrcom jaod syl5bi ralrimi ) AAE - ZFZACGZBBEZFZBCHZUJBACDUJBIJBKZUNLUOCLZUOCMZSUKUMUOCBNOUKUPUMUQUJUMAUOCAK - UOMZUIULURUIBAEULABAPABBQTRUAUKUMUQCCLZFACUBUQULUSUQULCUOLUSUOCUOUCUOCCUD - TRUEUFUGUH $. - $} - - $( The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) - (Proof shortened by Andrew Salmon, 27-Aug-2011.) $) - unt0 $p |- A. x e. (/) -. x e. x $= - ( wel wn ral0 ) AABCAD $. - - ${ - $d x y A $. - $( If there is an untangled element of a class, then the intersection of - the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) $) - untint $p |- ( E. x e. A A. y e. x -. y e. y -> - A. y e. |^| A -. y e. y ) $= - ( wel wn cv wral cint wcel wss wi intss1 ssralv syl rexlimiv ) BBDEZBAFZG - ZPBCHZGZACQCISQJRTKQCLPBSQMNO $. - $} - - ${ - $d x A $. - $( If ` A ` is well-founded by ` _E ` , then it is untangled. (Contributed - by Scott Fenton, 1-Mar-2011.) $) - efrunt $p |- ( _E Fr A -> A. x e. A -. x e. x ) $= - ( cep wfr cv wcel wn wa wbr frirr epel sylnib ralrimiva ) BCDZAEZOFZGABNO - BFHOOCIPBOCJAOKLM $. - $} - - ${ - $d x y A $. - $( A transitive class is untangled iff its elements are. (Contributed by - Scott Fenton, 7-Mar-2011.) $) - untangtr $p |- ( Tr A -> - ( A. x e. A -. x e. x <-> - A. x e. A A. y e. x -. y e. y ) ) $= - ( wtr wel wn wral cv cuni wss df-tr ssralv sylbi weq elequ1 elequ2 notbid - wi bitrd cbvralvw untuni bitri syl6ib untelirr ralimi impbid1 ) CDZAAEZFZ - ACGZBBEZFZBAHZGZACGZUGUJUIACIZGZUOUGUPCJUJUQRCKUIAUPCLMUQULBUPGUOUIULABUP - ABNZUHUKURUHBAEUKABAOABBPSQTBACUAUBUCUNUIACBUMUDUEUF $. - $} - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Extra propositional calculus theorems -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - ${ - 3pm3.2ni.1 $e |- -. ph $. - 3pm3.2ni.2 $e |- -. ps $. - 3pm3.2ni.3 $e |- -. ch $. - $( Triple negated disjunction introduction. (Contributed by Scott Fenton, - 20-Apr-2011.) $) - 3pm3.2ni $p |- -. ( ph \/ ps \/ ch ) $= - ( w3o wo pm3.2ni df-3or mtbir ) ABCGABHZCHLCABDEIFIABCJK $. - $} - - ${ - 3jaodd.1 $e |- ( ph -> ( ps -> ( ch -> et ) ) ) $. - 3jaodd.2 $e |- ( ph -> ( ps -> ( th -> et ) ) ) $. - 3jaodd.3 $e |- ( ph -> ( ps -> ( ta -> et ) ) ) $. - $( Double deduction form of ~ 3jaoi . (Contributed by Scott Fenton, - 20-Apr-2011.) $) - 3jaodd $p |- ( ph -> ( ps -> ( ( ch \/ th \/ ta ) -> et ) ) ) $= - ( w3o wi com3r 3jaoi com3l ) CDEJABFCABFKKDEABCFGLABDFHLABEFILMN $. - $} - - $( Closed form of ~ 3ori . (Contributed by Scott Fenton, 20-Apr-2011.) $) - 3orit $p |- ( ( ph \/ ps \/ ch ) <-> ( ( -. ph /\ -. ps ) -> ch ) ) $= - ( w3o wo wn wi wa df-3or df-or ioran imbi1i 3bitri ) ABCDABEZCENFZCGAFBFHZC - GABCINCJOPCABKLM $. - - $( A biconditional in the antecedent is the same as two implications. - (Contributed by Scott Fenton, 12-Dec-2010.) $) - biimpexp $p |- ( ( ( ph <-> ps ) -> ch ) - <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> ch ) ) ) $= - ( wb wi wa dfbi2 imbi1i impexp bitri ) ABDZCEABEZBAEZFZCELMCEEKNCABGHLMCIJ - $. - - $( Elimination of two disjuncts in a triple disjunction. (Contributed by - Scott Fenton, 9-Jun-2011.) $) - 3orel13 $p |- ( ( -. ph /\ -. ch ) -> ( ( ph \/ ps \/ ch ) -> ps ) ) $= - ( wn w3o wo 3orel3 orel1 sylan9r ) CDABCEABFADBABCGABHI $. - - ${ - $d A b $. $d C b $. - $( Ordinal less than is equivalent to having an ordinal between them. - (Contributed by Scott Fenton, 8-Aug-2024.) $) - onelssex $p |- ( ( A e. On /\ C e. On ) -> - ( A e. C <-> E. b e. C A C_ b ) ) $= - ( con0 wcel wa cv wss wrex ssid sseq2 rspcev mpan2 ontr2 expcomd rexlimdv - impbid2 ) ADEBDEFZABEZACGZHZCBIZSAAHZUBAJUAUCCABTAAKLMRUASCBRUATBESATBNOP - Q $. - $} - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Misc. Useful Theorems -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $( Two classes are unequal iff their intersection is a proper subset of one - of them. (Contributed by Scott Fenton, 23-Feb-2011.) $) - nepss $p |- ( A =/= B <-> ( ( A i^i B ) C. A \/ ( A i^i B ) C. B ) ) $= - ( wne cin wss wa wo wpss wn wceq nne neeq1 biimprcd orrd jctl inidm necon3i - syl5bi adantl df-pss inss1 inss2 orim12i syl ineq2 ineq1 eqtrdi jaoi impbii - eqtr3di orbi12i bitr4i ) ABCZABDZAEZUNACZFZUNBEZUNBCZFZGZUNAHZUNBHZGUMVAUMU - PUSGVAUMUPUSUPIUNAJZUMUSUNAKVDUSUMUNABLMRNUPUQUSUTUPUOABUAOUSURABUBOUCUDUQU - MUTUPUMUOABUNAABJZAADUNAABAUEAPUJQSUSUMURABUNBVEUNBBDBABBUFBPUGQSUHUIVBUQVC - UTUNATUNBTUKUL $. - - ${ - 3ccased.1 $e |- ( ph -> ( ( ch /\ et ) -> ps ) ) $. - 3ccased.2 $e |- ( ph -> ( ( ch /\ ze ) -> ps ) ) $. - 3ccased.3 $e |- ( ph -> ( ( ch /\ si ) -> ps ) ) $. - 3ccased.4 $e |- ( ph -> ( ( th /\ et ) -> ps ) ) $. - 3ccased.5 $e |- ( ph -> ( ( th /\ ze ) -> ps ) ) $. - 3ccased.6 $e |- ( ph -> ( ( th /\ si ) -> ps ) ) $. - 3ccased.7 $e |- ( ph -> ( ( ta /\ et ) -> ps ) ) $. - 3ccased.8 $e |- ( ph -> ( ( ta /\ ze ) -> ps ) ) $. - 3ccased.9 $e |- ( ph -> ( ( ta /\ si ) -> ps ) ) $. - $( Triple disjunction form of ~ ccased . (Contributed by Scott Fenton, - 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) $) - 3ccased $p |- ( ph -> ( ( ( ch \/ th \/ ta ) /\ ( et \/ ze \/ si ) ) -> - ps ) ) $= - ( wa com12 3jaodan w3o wi 3jaoian ) CDEUAFGHUAZRABCUDABUBZDECFUEGHACFRBIS - ACGRBJSACHRBKSTDFUEGHADFRBLSADGRBMSADHRBNSTEFUEGHAEFRBOSAEGRBPSAEHRBQSTUC - S $. - $} - - ${ - $d R x y z $. $d A x y z $. - $( Expansion of the definition of a strict order. (Contributed by Scott - Fenton, 6-Jun-2016.) $) - dfso3 $p |- ( R Or A <-> - A. x e. A A. y e. A A. z e. A - ( -. x R x /\ - ( ( x R y /\ y R z ) -> x R z ) /\ - ( x R y \/ x = y \/ y R x ) ) ) $= - ( cv wbr wn wa wi weq w3o wral w3a wor wcel c0 wne wb ralbii r19.27zv syl - ne0i ralbiia df-3an 2ralbii df-po anbi1i df-so r19.26-2 3bitr4i 3bitr4ri - wpo ) AFZUNEGHZUNBFZEGZUPCFZEGIUNUREGJZIZUQABKUPUNEGLZIZCDMZBDMZADMUTCDMZ - VAIZBDMZADMZUOUSVANZCDMZBDMADMDEOZVDVGADVCVFBDUPDPDQRVCVFSDUPUCUTVACDUAUB - UDTVJVCABDDVIVBCDUOUSVAUETUFDEUMZVABDMADMZIVEBDMADMZVMIVKVHVLVNVMABCDEUGU - HABDEUIVEVAABDDUJUKUL $. - $} - - $( A binary relation involving unordered triples. (Contributed by Scott - Fenton, 7-Jun-2016.) $) - brtpid1 $p |- A { <. A , B >. , C , D } B $= - ( cop ctp wbr wcel opex tpid1 df-br mpbir ) ABABEZCDFZGMNHMCDABIJABNKL $. - - $( A binary relation involving unordered triples. (Contributed by Scott - Fenton, 7-Jun-2016.) $) - brtpid2 $p |- A { C , <. A , B >. , D } B $= - ( cop ctp wbr wcel opex tpid2 df-br mpbir ) ABCABEZDFZGMNHCMDABIJABNKL $. - - $( A binary relation involving unordered triples. (Contributed by Scott - Fenton, 7-Jun-2016.) $) - brtpid3 $p |- A { C , D , <. A , B >. } B $= - ( cop ctp wbr wcel opex tpid3 df-br mpbir ) ABCDABEZFZGMNHCDMABIJABNKL $. - - ${ - $d A x $. $d B x $. $d ps x $. - ceqsrexv2.1 $e |- ( x = A -> ( ph <-> ps ) ) $. - $( Alternate elimitation of a restricted existential quantifier, using - implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) $) - ceqsrexv2 $p |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) ) $= - ( ceqsrexbv ) ABCDEFG $. - $} - - ${ - $d x V $. $d A y $. $d ps y $. $d x y $. - iota5f.1 $e |- F/ x ph $. - iota5f.2 $e |- F/_ x A $. - iota5f.3 $e |- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) ) $. - $( A method for computing iota. (Contributed by Scott Fenton, - 13-Dec-2017.) $) - iota5f $p |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A ) $= - ( vy wcel wa cv wceq wb wal cio nfel1 nfan wi eqeq2 alrimi bibi2d imbi12d - weq nfeq2 albid iotaval vtoclg adantl mpd ) ADEJZKZBCLZDMZNZCOZBCPZDMZULU - OCAUKCFCDEGQRHUAUKUPURSZABCIUDZNZCOZUQILZMZSUSIDEVCDMZVBUPVDURVEVAUOCCVCD - GUEVEUTUNBVCDUMTUBUFVCDUQTUCBCIUGUHUIUJ $. - $} - - ${ - $d x A $. $d x B $. $d ps x $. - ceqsralv2.1 $e |- ( x = A -> ( ph <-> ps ) ) $. - $( Alternate elimination of a restricted universal quantifier, using - implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.) $) - ceqsralv2 $p |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) $= - ( cv wceq wi wral wcel wn wa wrex notbid ceqsrexv2 rexanali annim 3bitr3i - con4bii ) CGDHZAICEJZDEKZBIZUAALZMCENUCBLZMUBLUDLUEUFCDEUAABFOPUAACEQUCBR - ST $. - $} - - $( A class is ordinal iff it is a subclass of ` On ` and transitive. - (Contributed by Scott Fenton, 21-Nov-2021.) $) - dford5 $p |- ( Ord A <-> ( A C_ On /\ Tr A ) ) $= - ( word con0 wss wtr wa ordsson ordtr jca cep wwe epweon wess df-ord biimpri - mpi ancoms sylan impbii ) ABZACDZAEZFTUAUBAGAHIUAAJKZUBTUACJKUCLACJMPUBUCTT - UBUCFANOQRS $. - - $( Closed form of ~ ja . Proved using the completeness script. - (Proof modification is discouraged.) (Contributed by Scott Fenton, - 13-Dec-2021.) $) - jath $p |- ( ( -. ph -> ch ) -> ( ( ps -> ch ) -> - ( ( ph -> ps ) -> ch ) ) ) $= - ( wn wi jcn pm2.21 imim2 ax-mp ax-1 pm2.61 ) ADZLCEZBCEZABEZCEZEZEZEZRLCDZR - EZEZSLTMDZEZEZUBLCFUDUAEZUEUBEUCREUFMQGUCRTHIUDUALHIIUAREZUBSECREZUGCQEZUHC - PEZUICOJPQEZUJUIEPNJZPQCHIIQREZUIUHEQMJZQRCHIICRKIZUARLHIIAREZSREABDZREZEZU - PAUQQEZEZUSAUQPEZEZVAAUQODZEZEZVCABFVEVBEZVFVCEVDPEVGOCGVDPUQHIVEVBAHIIVBUT - EZVCVAEUKVHULPQUQHIVBUTAHIIUTUREZVAUSEUMVIUNQRUQHIUTURAHIIURREZUSUPEBREZVJB - UAEZVKBTQEZEZVLBTNDZEZEZVNBCFVPVMEZVQVNEVOQEVRNPGVOQTHIVPVMBHIIVMUAEZVNVLEU - MVSUNQRTHIVMUABHIIUGVLVKEUOUARBHIIBRKIURRAHIIARKII $. - - ${ - $d A y $. $d ph y $. $d x y $. - riotarab.1 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Restricted iota of a restricted abstraction. (Contributed by Scott - Fenton, 8-Aug-2024.) $) - riotarab $p |- ( iota_ x e. { y e. A | ps } ch ) = - ( iota_ x e. A ( ph /\ ch ) ) $= - ( cv crab wcel wa cio crio wb weq bicomd equcoms elrab anbi1i df-riota - anass bitri iotabii 3eqtr4i ) DHZBEFIZJZCKZDLUEFJZACKZKZDLCDUFMUJDFMUHUKD - UHUIAKZCKUKUGULCBAEUEFBANDEDEOABGPQRSUIACUAUBUCCDUFTUJDFTUD $. - $} - - ${ - $d A y $. $d ph y $. $d x y $. - reurab.1 $e |- ( x = y -> ( ph <-> ps ) ) $. - $( Restricted existential uniqueness of a restricted abstraction. - (Contributed by Scott Fenton, 8-Aug-2024.) $) - reurab $p |- ( E! x e. { y e. A | ps } ch <-> E! x e. A ( ph /\ ch ) ) $= - ( cv crab wcel wa weu wreu wb weq bicomd equcoms elrab anbi1i df-reu - anass bitri eubii 3bitr4i ) DHZBEFIZJZCKZDLUEFJZACKZKZDLCDUFMUJDFMUHUKDUH - UIAKZCKUKUGULCBAEUEFBANDEDEOABGPQRSUIACUAUBUCCDUFTUJDFTUD $. - $} - - ${ - snres0.1 $e |- B e. _V $. - $( Condition for restriction of a singleton to be empty. (Contributed by - Scott Fenton, 9-Aug-2024.) $) - snres0 $p |- ( ( { <. A , B >. } |` C ) = (/) <-> -. A e. C ) $= - ( cop csn cres c0 wceq cdm cin wcel wn wrel wb relres reldm0 ax-mp dmsnop - dmres ineq2i eqtri eqeq1i disjsn 3bitri ) ABEFZCGZHIZUGJZHIZCAFZKZHIACLMU - GNUHUJOUFCPUGQRUIULHUICUFJZKULUFCTUMUKCABDSUAUBUCCAUDUE $. - $} - - ${ - $d A y $. $d B x y $. $d C x y $. $d F x y $. - $( Condition for subset of an intersection of an image. (Contributed by - Scott Fenton, 16-Aug-2024.) $) - fnssintima $p |- ( ( F Fn A /\ B C_ A ) -> ( C C_ |^| ( F " B ) <-> - A. x e. B C C_ ( F ` x ) ) ) $= - ( vy cima cint wss cv wcel wi wal wfn wa cfv wral bitri wceq albii df-ral - ssint wrex fvelimab imbi1d albidv ralcom4 eqcom imbi1i fvex sseq2 ceqsalv - ralbii r19.23v 3bitr3ri bitrdi bitrid ) DECGZHIZFJZURKZDUTIZLZFMZEBNCBIOZ - DAJZEPZIZACQZUSVBFURQVDFDURUBVBFURUARVEVDVGUTSZACUCZVBLZFMZVIVEVCVLFVEVAV - KVBABCUTEUDUEUFVJVBLZFMZACQVNACQZFMVIVMVNAFCUGVOVHACVOUTVGSZVBLZFMVHVNVRF - VJVQVBVGUTUHUITVBVHFVGVFEUJUTVGDUKULRUMVPVLFVJVBACUNTUOUPUQ $. - $} - - ${ - $d a b $. $d a ph $. $d a ps $. $d a x $. $d a y $. $d b ph $. - $d b ps $. $d b x $. $d b y $. $d ph y $. $d ps x $. $d x y $. - $( Cross product of two class abstractions. (Contributed by Scott Fenton, - 19-Aug-2024.) $) - xpab $p |- ( { x | ph } X. { y | ps } ) = { <. x , y >. | ( ph /\ ps ) } $= - ( va vb cab cxp wa copab wcel wsbc wbr wsb df-clab sban sbsbc sbv 3bitr3i - cv relxp relopabv anbi12i anbi1i sbbii anbi2i bitri bitr4i brabsb 3bitr4i - brxp eqid eqbrriv ) EFACGZBDGZHZABIZCDJZUNUOUAUQCDUBETZUNKZFTZUOKZIZUQDVA - LZCUSLZUSVAUPMUSVAURMVCACENZBDFNZIZVEUTVFVBVGAECOBFDOUCVDCENAVGIZCENZVEVH - VDVICEUQDFNADFNZVGIVDVIABDFPUQDFQVKAVGADFRUDSUEVDCEQVJVFVGCENZIVHAVGCEPVL - VGVFVGCERUFUGSUHUSVAUNUOUKUQCDUSVAURURULUIUJUM $. - $} - - ${ - $d A x $. $d A y $. $d B x $. $d B y $. $d B z $. $d ph x $. - $d x y $. $d x z $. $d y z $. - $( A version of ~ ralxp with explicit substitution. (Contributed by Scott - Fenton, 21-Aug-2024.) $) - ralxpes $p |- ( A. x e. ( A X. B ) [. ( 1st ` x ) / y ]. - [. ( 2nd ` x ) / z ]. ph <-> A. y e. A A. z e. B ph ) $= - ( cv c2nd cfv wsbc c1st nfsbc1v nfcv nfsbcw nfv sbcopeq1a ralxpf ) ADBGZH - IZJZCRKIZJABCDEFTCUALTDCUADUAMADSLNABOACDRPQ $. - $} - - $( Ordered triple membership in a triple cross product. (Contributed by - Scott Fenton, 21-Aug-2024.) $) - ot2elxp $p |- ( <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) <-> - ( A e. D /\ B e. E /\ C e. F ) ) $= - ( cop cxp wcel wa w3a opelxp anbi1i df-3an 3bitr4i ) ABGZDEHZIZCFIZJADIZBEI - ZJZSJPCGQFHITUASKRUBSABDELMPCQFLTUASNO $. - - ${ - ot21st.1 $e |- A e. _V $. - ot21st.2 $e |- B e. _V $. - ot21st.3 $e |- C e. _V $. - $( Extract the first member of an ordered triple. Deduction version. - (Contributed by Scott Fenton, 21-Aug-2024.) $) - ot21std $p |- ( X = <. <. A , B >. , C >. -> ( 1st ` ( 1st ` X ) ) = A ) $= - ( cop wceq c1st cfv opex op1std fveq2d op1st eqtrdi ) DABHZCHIZDJKZJKQJKA - RSQJQCDABLGMNABEFOP $. - - $( Extract the second member of an ordered triple. Deduction version. - (Contributed by Scott Fenton, 21-Aug-2024.) $) - ot22ndd $p |- ( X = <. <. A , B >. , C >. -> ( 2nd ` ( 1st ` X ) ) = B ) $= - ( cop wceq c1st cfv c2nd opex op1std fveq2d op2nd eqtrdi ) DABHZCHIZDJKZL - KRLKBSTRLRCDABMGNOABEFPQ $. - $} - - ${ - otthne.1 $e |- A e. _V $. - otthne.2 $e |- B e. _V $. - otthne.3 $e |- C e. _V $. - $( Contrapositive of the ordered triple theorem. (Contributed by Scott - Fenton, 21-Aug-2024.) $) - otthne $p |- ( <. <. A , B >. , C >. =/= <. <. D , E >. , F >. <-> - ( A =/= D \/ B =/= E \/ C =/= F ) ) $= - ( cop wne wo w3o opthne orbi1i opex df-3or 3bitr4i ) ABJZDEJZKZCFKZLADKZB - EKZLZUBLSCJTFJKUCUDUBMUAUEUBABDEGHNOSCTFABPINUCUDUBQR $. - $} - - ${ - $d A x y z p $. $d B x y z p $. $d C x y z p $. $d D x y z p $. - $( Membership in a triple cross product. (Contributed by Scott Fenton, - 21-Aug-2024.) $) - elxpxp $p |- ( A e. ( ( B X. C ) X. D ) <-> - E. x e. B E. y e. C E. z e. D A = <. <. x , y >. , z >. ) $= - ( vp cxp wcel cv cop wceq wrex elxp2 opeq1 eqeq2d rexbidv rexxp bitri ) D - EFIZGIJDHKZCKZLZMZCGNZHUANDAKBKLZUCLZMZCGNZBFNAENHCDUAGOUFUJHABEFUBUGMZUE - UICGUKUDUHDUBUGUCPQRST $. - $} - - ${ - $d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $. - $( Version of ~ elrel for triple cross products. (Contributed by Scott - Fenton, 21-Aug-2024.) $) - elxpxpss $p |- ( ( R C_ ( ( B X. C ) X. D ) /\ A e. R ) -> - E. x E. y E. z A = <. <. x , y >. , z >. ) $= - ( cxp wss wcel wa cv cop wceq wex wrex rexex reximi syl ssel2 elxpxp - sylbi ) HEFIGIZJDHKLDUDKZDAMBMNCMNOZCPZBPZAPZHUDDUAUEUFCGQZBFQZAEQZUIABCD - EFGUBULUHAEQUIUKUHAEUKUGBFQUHUJUGBFUFCGRSUGBFRTSUHAERTUCT $. - $} - - ${ - $d A w $. $d A x $. $d A y $. $d A z $. $d B w $. $d B x $. $d B y $. - $d B z $. $d C w $. $d C x $. $d C y $. $d C z $. $d w x $. $d w y $. - $d w z $. $d x y $. $d x z $. $d y z $. - ral3xpf.1 $e |- F/ y ph $. - ral3xpf.2 $e |- F/ z ph $. - ral3xpf.3 $e |- F/ w ph $. - ral3xpf.4 $e |- F/ x ps $. - ral3xpf.5 $e |- ( x = <. <. y , z >. , w >. -> ( ph <-> ps ) ) $. - $( Restricted for all over a triple cross product. (Contributed by Scott - Fenton, 22-Aug-2024.) $) - ralxp3f $p |- ( A. x e. ( ( A X. B ) X. C ) ph <-> - A. y e. A A. z e. B A. w e. C ps ) $= - ( wral cv wi wal wrex ralbii cxp wcel cop wceq df-ral r19.23 bitri elxpxp - imbi1i 3bitr4ri albii ralcom4 opex ceqsal bitr3i 3bitri ) ACGHUAIUAZOCPZU - QUBZAQZCRURDPEPUCZFPZUCZUDZAQZFIOZEHOZDGOZCRZBFIOZEHOZDGOZACUQUEUTVHCVDFI - SZEHSZAQZDGOVNDGSZAQVHUTVNADGJUFVGVODGVGVMAQZEHOVOVFVQEHVDAFILUFTVMAEHKUF - UGTUSVPADEFURGHIUHUIUJUKVIVGCRZDGOVLVGDCGULVRVKDGVRVFCRZEHOVKVFECHULVSVJE - HVSVECRZFIOVJVEFCIULVTBFIABCVCMVAVBUMNUNTUOTUOTUOUP $. - $} - - ${ - $d A p $. $d A x $. $d A y $. $d A z $. $d B p $. $d B x $. $d B y $. - $d B z $. $d C p $. $d C x $. $d C y $. $d C z $. $d p ps $. - $d p x $. $d p y $. $d p z $. $d ph x $. $d ph y $. $d ph z $. - $d x y $. $d x z $. $d y z $. - ralxp3.1 $e |- ( p = <. <. x , y >. , z >. -> ( ph <-> ps ) ) $. - $( Restricted for-all over a triple cross product. (Contributed by Scott - Fenton, 21-Aug-2024.) $) - ralxp3 $p |- ( A. p e. ( ( A X. B ) X. C ) ph <-> - A. x e. A A. y e. B A. z e. C ps ) $= - ( nfv ralxp3f ) ABICDEFGHACKADKAEKBIKJL $. - $} - - $( Equality theorem for substitution of a class for an ordered triple. - (Contributed by Scott Fenton, 22-Aug-2024.) $) - sbcoteq1a $p |- ( A = <. <. x , y >. , z >. -> - ( [. ( 1st ` ( 1st ` A ) ) / x ]. [. ( 2nd ` ( 1st ` A ) ) / y ]. - [. ( 2nd ` A ) / z ]. ph <-> ph ) ) $= - ( cv cop wceq c2nd cfv wsbc c1st opex vex op2ndd eqcomd sbceq1a syl ot22ndd - wb ot21std 3bitrrd ) EBFZCFZGZDFZGHZAADEIJZKZUICELJZIJZKZULBUJLJZKZUGUFUHHA - UITUGUHUFUEUFEUCUDMDNZOPADUHQRUGUDUKHUIULTUGUKUDUCUDUFEBNZCNZUOSPUICUKQRUGU - CUMHULUNTUGUMUCUCUDUFEUPUQUOUAPULBUMQRUB $. - - ${ - $d A w $. $d A x $. $d A y $. $d A z $. $d B w $. $d B x $. $d B y $. - $d B z $. $d C w $. $d C x $. $d C y $. $d C z $. $d ph x $. - $d w x $. $d w y $. $d w z $. $d x y $. $d x z $. $d y z $. - $( Restricted for-all over a triple cross product with explicit - substitution. (Contributed by Scott Fenton, 22-Aug-2024.) $) - ralxp3es $p |- ( A. x e. ( ( A X. B ) X. C ) - [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ]. - [. ( 2nd ` x ) / w ]. ph <-> - A. y e. A A. z e. B A. w e. C ph ) $= - ( cv c2nd cfv wsbc c1st nfsbc1v nfcv nfsbcw nfv sbcoteq1a ralxp3f ) AEBIZ - JKZLZDTMKZJKZLZCUCMKZLABCDEFGHUECUFNUEDCUFDUFOUBDUDNPUEECUFEUFOUBEDUDEUDO - AEUANPPABQACDETRS $. - $} - - $( The union of two ordinals is in a third iff both of the first two are. - (Contributed by Scott Fenton, 10-Sep-2024.) $) - onunel $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A u. B ) e. C <-> - ( A e. C /\ B e. C ) ) ) $= - ( con0 wcel w3a wss cun wa wb wceq biimpi eleq1d adantl ontr2 expdimp bitrd - wi word eloni ssequn1 3adant2 pm4.71rd ssequn2 3adant1 wo ordtri2or2 syl2an - pm4.71d 3adant3 mpjaodan ) ADEZBDEZCDEZFZABGZABHZCEZACEZBCEZIZJBAGZUOUPIZUR - UTVAUPURUTJUOUPUQBCUPUQBKABUALMNVCUTUSUOUPUTUSULUNUPUTIUSRUMABCOUBPUCQUOVBI - ZURUSVAVBURUSJUOVBUQACVBUQAKBAUDLMNVDUSUTUOVBUSUTUMUNVBUSIUTRULBACOUEPUIQUL - UMUPVBUFZUNULASBSVEUMATBTABUGUHUJUK $. - - ${ - $d A x $. $d B x y $. $d F x y $. $d ph y $. $d ps x $. - imaeqsex.1 $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $. - $( Substitute a function value into an existential quantifier over an - image. (Contributed by Scott Fenton, 27-Sep-2024.) $) - imaeqsexv $p |- ( ( F Fn A /\ B C_ A ) -> - ( E. x e. ( F " B ) ph <-> E. y e. B ps ) ) $= - ( wfn wss wa cima wrex cv cfv wceq wex wcel df-rex exbii fvelimab rexcom4 - anbi1d exbidv bitrid eqcom anbi1i ceqsexv rexbii r19.41v 3bitr3ri bitrdi - fvex bitri ) GEIFEJKZACGFLZMZDNZGOZCNZPZDFMZAKZCQZBDFMZUQUTUPRZAKZCQUOVDA - CUPSUOVGVCCUOVFVBADEFUTGUAUCUDUEVAAKZCQZDFMVHDFMZCQVEVDVHDCFUBVIBDFVIUTUS - PZAKZCQBVHVLCVAVKAUSUTUFUGTABCUSURGUMHUHUNUIVJVCCVAADFUJTUKUL $. - - $( Substitute a function value into a universal quantifier over an image. - (Contributed by Scott Fenton, 27-Sep-2024.) $) - imaeqsalv $p |- ( ( F Fn A /\ B C_ A ) -> - ( A. x e. ( F " B ) ph <-> A. y e. B ps ) ) $= - ( wfn wss wa wn cima wrex wral cv cfv wceq notbid dfral2 imaeqsexv - 3bitr4g ) GEIFEJKZALZCGFMZNZLBLZDFNZLACUEOBDFOUCUFUHUDUGCDEFGCPDPGQRABHSU - ASACUETBDFTUB $. - $} - - ${ - $d A x $. - $( The union of a finite ordinal is a finite ordinal. (Contributed by - Scott Fenton, 17-Oct-2024.) $) - nnuni $p |- ( A e. _om -> U. A e. _om ) $= - ( vx com wcel c0 wceq cv csuc wrex cuni nn0suc unieq uni0 eqtrdi eqeltrdi - wo peano1 word nnord syl ordunisuc id eqeltrd eleq1d syl5ibrcom rexlimiv - jaoi ) ACDAEFZABGZHZFZBCIZPAJZCDZBAKUHUNULUHUMECUHUMEJEAELMNQOUKUNBCUICDZ - UNUKUJJZCDUOUPUICUOUIRUPUIFUISUIUATUOUBUCUKUMUPCAUJLUDUEUFUGT $. - $} - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Properties of real and complex numbers -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - ${ - sqdivzi.1 $e |- A e. CC $. - sqdivzi.2 $e |- B e. CC $. - $( Distribution of square over division. (Contributed by Scott Fenton, - 7-Jun-2013.) $) - sqdivzi $p |- ( B =/= 0 -> - ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) ) $= - ( cc0 wne cdiv co c2 cexp c1 cif oveq2 oveq1d oveq1 oveq2d eqeq12d ax-1cn - wceq cc ifcli elimne0 sqdivi dedth ) BEFZABGHZIJHZAIJHZBIJHZGHZSAUEBKLZGH - ZIJHZUHUKIJHZGHZSBKBUKSZUGUMUJUOUPUFULIJBUKAGMNUPUIUNUHGBUKIJOPQAUKCUEBKT - DRUABUBUCUD $. - $} - - ${ - $d N x $. $d M x $. - $( The supremum of a finite sequence of integers. (Contributed by Scott - Fenton, 8-Aug-2013.) $) - supfz $p |- ( N e. ( ZZ>= ` M ) -> sup ( ( M ... N ) , ZZ , < ) = N ) $= - ( vx cuz cfv wcel cz cfz co clt wor cr wss zssre ltso mp2 a1i eluzelz wbr - soss eluzfz2 cv wa cle wn elfzle2 adantl wb elfzelz zred eluzelre syl2anr - lenlt mpbid supmax ) BADEFZCGABHIZBJGJKZUPGLMLJKURNOGLJTPQABRABUAUPCUBZUQ - FZUCUSBUDSZBUSJSUEZUTVAUPUSABUFUGUTUSLFBLFVAVBUHUPUTUSUSABUIUJABUKUSBUMUL - UNUO $. - - $( The infimum of a finite sequence of integers. (Contributed by Scott - Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.) $) - inffz $p |- ( N e. ( ZZ>= ` M ) -> inf ( ( M ... N ) , ZZ , < ) = M ) $= - ( vx cuz cfv wcel cz cfz co clt wor wss zssre ltso soss mp2 a1i wbr zred - cr eluzel2 eluzfz1 cv wa cle wn elfzle1 adantl elfzelz lenlt syl2an mpbid - wb infmin ) BADEFZCGABHIZAJGJKZUOGTLTJKUQMNGTJOPQABUAZABUBUOCUCZUPFZUDAUS - UERZUSAJRUFZUTVAUOUSABUGUHUOATFUSTFVAVBUMUTUOAURSUTUSUSABUISAUSUJUKULUN - $. - $} - - $( The sequence ` ( 0 ... ( N - 1 ) ) ` is empty iff ` N ` is zero. - (Contributed by Scott Fenton, 16-May-2014.) $) - fz0n $p |- ( N e. NN0 -> ( ( 0 ... ( N - 1 ) ) = (/) <-> N = 0 ) ) $= - ( cn0 wcel c1 cmin co cc0 clt wbr cfz c0 wceq cz wb nn0z sylancr cle bitr3d - 0z cr peano2zm syl fzn cn wo elnn0 wn nnge1 1re wa subge0 0re resubcl lenlt - nnre sylancl mpbid nnne0 neneqd 2falsed cneg oveq1 eqtr4di neg1lt0 eqbrtrdi - df-neg id 2thd jaoi sylbi ) ABCZADEFZGHIZGVLJFKLZAGLZVKGMCVLMCZVMVNNSVKAMCV - PAOAUAUBGVLUCPVKAUDCZVOUEVMVONZAUFVQVRVOVQVMVOVQDAQIZVMUGZAUHVQATCZDTCZVSVT - NAUOUIWAWBUJZGVLQIZVSVTADUKWCGTCVLTCWDVTNULADUMGVLUNPRUPUQVQAGAURUSUTVOVMVO - VOVLDVAZGHVOVLGDEFWEAGDEVBDVFVCVDVEVOVGVHVIVJR $. - - ${ - $d F f $. $d A f $. $d B f $. - $( Value of a sequence shifted by ` A ` . (Contributed by Scott Fenton, - 16-Dec-2017.) $) - shftvalg $p |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> - ( ( F shift A ) ` B ) = ( F ` ( B - A ) ) ) $= - ( vf wcel cc cshi co cmin wceq wa cv wi oveq1 fveq1d fveq1 eqeq12d imbi2d - cfv vex shftval vtoclg 3impib ) CDFAGFZBGFZBCAHIZTZBAJIZCTZKZUEUFLZBEMZAH - IZTZUIUMTZKZNULUKNECDUMCKZUQUKULURUOUHUPUJURBUNUGUMCAHOPUIUMCQRSABUMEUAUB - UCUD $. - $} - - ${ - $d A k $. $d A m $. $d B k $. $d B m $. $d F k $. $d k m $. - $d k ph $. $d M k $. $d m ph $. $d Z k $. - divcnvlin.1 $e |- Z = ( ZZ>= ` M ) $. - divcnvlin.2 $e |- ( ph -> M e. ZZ ) $. - divcnvlin.3 $e |- ( ph -> A e. CC ) $. - divcnvlin.4 $e |- ( ph -> B e. ZZ ) $. - divcnvlin.5 $e |- ( ph -> F e. V ) $. - divcnvlin.6 $e |- ( ( ph /\ k e. Z ) -> - ( F ` k ) = ( ( k + A ) / ( k + B ) ) ) $. - $( Limit of the ratio of two linear functions. (Contributed by Scott - Fenton, 17-Dec-2017.) $) - divcnvlin $p |- ( ph -> F ~~> 1 ) $= - ( vm c1 co caddc cn wcel cli wbr cz cv cmin cdiv cmpt cc0 wa cc nncn zcnd - adantl subcld adantr wne nnne0 divdird dividd oveq1d eqtrd mpteq2dva nnuz - cvv 1zzd divcnv syl 1cnd nnex mptex a1i divcld fmpttd ffvelcdmda cfv wceq - weq oveq2 oveq2d eqid ovex fvmpt eqtr4d climaddc2 eqbrtrd cuz cres resmpt - wss nnssz ax-mp reseq2i eqtr3i 1p0e1 breq12i wb climres mp2an bitri sylib - zex eluzelz eleq2s ppncand zaddcld oveq1 oveq12d 3eqtr4d climshft2 mpbird - 1z id ) AEPUAUBOUCOUDZBCUEQZRQZXMUFQZUGZPUAUBZAOSXPUGZPUHRQZUAUBZXRAXSOSP - XNXMUFQZRQZUGZXTUAAOSXPYCAXMSTZUIZXPXMXMUFQZYBRQYCYFXMXNXMYEXMUJTAXMUKUMZ - AXNUJTZYEABCKACLULUNZUOZYHYEXMUHUPAXMUQUMZURYFYGPYBRYFXMYHYLUSUTVAVBAUHPD - OSYBUGZYDPVDSVCAVEAYIYMUHUAUBYJXNOVFVGAVHYDVDTAOSYCVIVJVKASUJDUDZYMAOSYBU - JYFXNXMYKYHYLVLVMVNYNSTZYNYDVOZPYNYMVOZRQZVPAYOYPPXNYNUFQZRQZYROYNYCYTSYD - ODVQYBYSPRXMYNXNUFVRZVSYDVTPYSRWAWBYOYQYSPROYNYBYSSYMUUAYMVTXNYNUFWAWBVSW - CUMWDWEYAXQPWFVOZWGZPUAUBZXRXSUUCXTPUAXQSWGZXSUUCSUCWIUUEXSVPWJOUCSXPWHWK - SUUBXQVCWLWMWNWOPUCTXQVDTZUUDXRWPXKOUCXPXAVJZPXQPVDWQWRWSWTAPDEXQCFGVDHIJ - LMUUFAUUGVKAYNHTZUIZYNCRQZXNRQZUUJUFQZYNBRQZUUJUFQUUJXQVOZYNEVOUUIUUKUUMU - UJUFUUIYNCBUUHYNUJTAUUHYNYNUCTZYNFWFVOHFYNXBIXCZULUMUUICACUCTUUHLUOZULABU - JTUUHKUOXDUTUUIUUJUCTUUNUULVPUUIYNCUUHUUOAUUPUMUUQXEOUUJXPUULUCXQXMUUJVPZ - XOUUKXMUUJUFXMUUJXNRXFUURXLXGXQVTUUKUUJUFWAWBVGNXHXIXJ $. - $} + YAVYBYRYEYEYEUVIUVJUVRUVKUVLWAYDYBYSYDAVXDVVLVVTVVFAVXDVVLVVTVVFABVLZCV + LZTXRZVUPVYDPVMZKVMZUCVMZVQZVVAVYEPVMZKVMZUCVMZVQZWHZWDZVVFYLAVXDVVLVVT + WHZWDZVVFYLBCVVHVVPVKYTFYTVYDVVHWCZVYEVVPWCZWDZVYPVYRVVFWUAVYOVYQAWUAVY + FVXDVYJVVLVYNVVTVYDVVHVYEVVPTYNWUAVYIVVKVUPWUAVYHVVJUCWUAVYGVVIKWUAVYDV + VHPVYSVYTYQYEYEYEYIWUAVYMVVSVVAWUAVYLVVRUCWUAVYKVVQKWUAVYEVVPPVYSVYTYRY + EYEYEYIUVMUVNUVOVCUVPUVQUVSXBUVTYSUWAUWBUWCUWDUWEAKOXDUWMUWNUWPWDXLAOOK + UYNXFOUWIUWKKXPWAUWF $. + $} - ${ - $d A k $. $d B k $. $d F k $. $d F m $. $d k m $. $d k ph $. - $d M k $. $d Z k $. $d Z m $. - climlec3.1 $e |- Z = ( ZZ>= ` M ) $. - climlec3.2 $e |- ( ph -> M e. ZZ ) $. - climlec3.3 $e |- ( ph -> B e. RR ) $. - climlec3.4 $e |- ( ph -> F ~~> A ) $. - climlec3.5 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $. - climlec3.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ B ) $. - $( Comparison of a constant to the limit of a sequence. (Contributed by - Scott Fenton, 5-Jan-2018.) $) - climlec3 $p |- ( ph -> A <_ B ) $= - ( vm cle wbr cneg cfv cc0 wcel cv cmpt renegcld cmin co cli cvv cuz fvexi - 0cnd mptex a1i wa recnd eqid weq fveq2 negeqd simpr fvmptd3 df-neg eqtrdi - climsubc2 breqtrrdi adantr lenegd mpbid breqtrrd climlec2 climrecl mpbird - cr eqeltrd ) ABCOPCQZBQZOPAVNVODNGNUAZERZQZUBZFGHIACJUCAVSSBUDUEVOUFABSDE - VSFUGGHIKAUJVSUGTANGVRGFUHHUIUKULADUAZGTZUMZVTERZLUNWBVTVSRZWCQZSWCUDUEWB - NVTVRWEGVSVLVSUONDUPVQWCVPVTEUQURAWAUSWBWCLUCZUTZWCVAVBVCBVAVDWBWDWEVLWGW - FVMWBVNWEWDOWBWCCOPVNWEOPMWBWCCLACVLTWAJVEVFVGWGVHVIABCABDEFGHIKLVJJVFVK - $. + $d m o p s w z ph $. + $( The closure is closed under application of provable pre-statements. + (Compare ~ mclsax .) This theorem is what justifies the treatment of + theorems as "equivalent" to axioms once they have been proven: the + composition of one theorem in the proof of another yields a theorem. + (Contributed by Mario Carneiro, 18-Jul-2016.) $) + mclspps $p |- ( ph -> ( S ` P ) e. ( K C B ) ) $= + ( vz vw vm vo vs vp wfn ccnv co cima wcel cfv crn msubf syl ffnd cmax wss + wf wceq cfn cotp cmpst w3a eqid mppspst sselid elmpst sylib simp1d simpld + wa simp2d cv wral ralrimiva wfun wb ffund fdmd sseqtrrd funimass5 syl2anc + cdm mpbird cmfs mvhf ffvelcdmda elpreima adantr mpbir2and cun wbr cxp wal + 3ad2ant1 3ad2antl1 simp21 simp22 simp23 mclsppslem mclsind elmpps simprbi + wi simp3 sseldd simplbda ) AIKVDZHIVENEFVFZVGZVHZHIVIYGVHZAKKIAIOVJZVHZKK + IVPUNOJKIUIUCVKVLZVMZAPQFVFZYHHAURUSDJVNVIZQFGYHJUTVAKLPORSVBVCUBUCUDUEAP + GVOZPVEPVQZAYQYRWIZQKVOZQVRVHZWIZHKVHZAPQHVSZJVTVIZVHZYSUUBUUCWAAMUUEUUDU + UEJMUUEWBZUHWCUMWDHPUUEJKQGUBUCUUGWEWFZWGWHAYTUUAAYSUUBUUCUUHWJWHZYPWBUIU + JUKULAQYHVOZBWKZIVIYGVHZBQWLZAUULBQUOWMAIWNQIXAZVOUUJUUMWOAKKIYMWPAQKUUNU + UIAKKIYMWQWRBQYGIWSWTXBADWKZRVHZWIUUOLVIZYHVHZUUQKVHZUUQIVIYGVHZARKUUOLAJ + XCVHZRKLVPUEJKLRUJUCUKXDVLXEUPAUURUUSUUTWIWOZUUPAYFUVBYNKUUQYGIXFVLXGXHAU + TWKZVAWKZVCWKVSYPVHZVBWKZYKVHZUVFUVDLVJXIVGYHVOZWAZURWKZUSWKZUVCXJUVJLVIU + VFVISVIUVKLVIUVFVISVIXKPVOYBUSXLURXLZWABCURUSDEFGHIJUTVAKLMNOPQRSVBVCTUAU + BUCUDAUVIUVAUVLUEXMAUVINGVOUVLUFXMAUVIEKVOUVLUGXMUHUIUJUKULAUVIUUDMVHZUVL + UMXMAUVIYLUVLUNXMAUVIUUKQVHUULUVLUOXNAUVIUUPUUTUVLUPXNAUVIUUKCWKZPXJTWKZU + UKLVIIVISVIVHUAWKZUVNLVIIVISVIVHWAUVOUVPNXJUVLUQXNAUVEUVGUVHUVLXOAUVEUVGU + VHUVLXPAUVEUVGUVHUVLXQAUVIUVLYCXRXSAUVMHYOVHZUMUVMUUFUVQHFPUUEJQMUUGUHUDX + TYAVLYDYFYIUUCYJKHYGIXFYEWT $. $} - $( Calculate the logarithm of ` _i ` . (Contributed by Scott Fenton, - 13-Apr-2020.) $) - logi $p |- ( log ` _i ) = ( _i x. ( _pi / 2 ) ) $= - ( ci clog cfv cpi c2 co cmul wceq cc wcel cc0 ax-icn clt wbr halfpire pipos - wb pire ax-mp wtru cdiv ce efhalfpi wne crn ine0 cneg cim cle recni lt0neg2 - mulcli mpbi halfpos2 renegcli 0re lttri mp2an cre reim rere eqtr3i breqtrri - cr caddc a1i ltaddposd mpbii picn times2i breqtrrdi ltdivmul2d mpbird mptru - crp 2rp ltleii eqbrtri ellogrn mpbir3an logeftb mp3an mpbir ) ABCADEUAFZGFZ - HZWEUBCAHZUCAIJAKUDWEBUEJZWFWGQLUFWHWEIJDUGZWEUHCZMNWJDUINAWDLWDOUJZULWIWDW - JMWIKMNZKWDMNZWIWDMNKDMNZWLPDVDJZWNWLQRDUKSUMWNWMPWOWNWMQRDUNSUMWIKWDDRUOUP - OUQURWDUSCZWJWDWDIJWPWJHWKWDUTSWDVDJWPWDHOWDVASVBZVCWJWDDUIWQWDDORWDDMNZTWR - DDEGFZMNTDDDVEFZWSMTWNDWTMNPTDDWOTRVFZXAVGVHDVIVJVKTDDEXAXAEVOJTVPVFVLVMVNV - QVRWEVSVTAWEWAWBWC $. - - $( ` _i ` raised to itself is real. (Contributed by Scott Fenton, - 13-Apr-2020.) $) - iexpire $p |- ( _i ^c _i ) e. RR $= - ( ci ccxp co c1 cneg cpi c2 cdiv cmul ce cfv cr clog cc wcel cc0 wne ax-icn - wceq halfpire ine0 cxpef mp3an logi oveq2i recni mulassi ixi oveq1i 3eqtr2i - fveq2i eqtri neg1rr remulcli reefcl ax-mp eqeltri ) AABCZDEZFGHCZICZJKZLURA - AMKZICZJKZVBANOZAPQVFURVESRUARAAUBUCVDVAJVDAAUTICZICAAICZUTICVAVCVGAIUDUEAA - UTRRUTTUFUGVHUSUTIUHUIUJUKULVALOVBLOUSUTUMTUNVAUOUPUQ $. - - $( The binomial coefficent over negative one is zero. (Contributed by Scott - Fenton, 29-May-2020.) $) - bcneg1 $p |- ( N e. NN0 -> ( N _C -u 1 ) = 0 ) $= - ( cn0 wcel c1 cneg cbc co cc0 cfz cfa cfv cmin cmul cdiv cif wceq neg1z cle - cz wbr bcval mpan2 wa clt wn neg1lt0 neg1rr 0re ltnlei mpbi intnanr wb nn0z - 0z elfz mp3an12 syl mtbiri iffalsed eqtrd ) ABCZADEZFGZVBHAIGCZAJKAVBLGJKVB - JKMGNGZHOZHVAVBSCZVCVFPQVBAUAUBVAVDVEHVAVDHVBRTZVBARTZUCZVHVIVBHUDTVHUEUFVB - HUGUHUIUJUKVAASCZVDVJULZAUMVGHSCVKVLQUNVBHAUOUPUQURUSUT $. - - $( The proportion of one bionmial coefficient to another with ` N ` decreased - by 1. (Contributed by Scott Fenton, 23-Jun-2020.) $) - bcm1nt $p |- ( ( N e. NN /\ K e. ( 0 ... ( N - 1 ) ) ) -> - ( N _C K ) = ( ( ( N - 1 ) _C K ) x. ( N / ( N - K ) ) ) ) $= - ( cn wcel cc0 c1 cmin co cfz wa caddc cbc cdiv cmul wceq bcp1n adantl simpl - nncnd oveq1d 1cnd npcand oveq12d oveq2d 3eqtr3d ) BCDZAEBFGHZIHDZJZUGFKHZAL - HZUGALHZUJUJAGHZMHZNHZBALHULBBAGHZMHZNHUHUKUOOUFAUGPQUIUJBALUIBFUIBUFUHRSUI - UAUBZTUIUNUQULNUIUJBUMUPMURUIUJBAGURTUCUDUE $. - - ${ - $d N n m j k $. - $( A product identity for binomial coefficents. (Contributed by Scott - Fenton, 23-Jun-2020.) $) - bcprod $p |- ( N e. NN -> - prod_ k e. ( 1 ... ( N - 1 ) ) ( ( N - 1 ) _C k ) = - prod_ k e. ( 1 ... ( N - 1 ) ) ( k ^ ( ( 2 x. k ) - N ) ) ) $= - ( c1 cmin co cfz cbc cprod cmul cexp wceq oveq2d oveq1d adantr prodeq12dv - c2 wcel cn cdiv nncnd vm vn vj cv cc0 caddc oveq1 1m1e0 eqtrdi fz10 oveq2 - c0 eqeq12d weq prod0 eqtr4i wa cfa cfv nncn 1cnd pncand cuz elnnuz biimpi - simpr cn0 nnnn0 elfzelz bccl syl2an nn0cnd fprodm1 bcnn syl fzfid fprodcl - cz mulid1d fz1ssfz0 bcm1nt sylan2 prodeq2dv nnm1nn0 clt wbr elfznn adantl - sseli cc nnred nn0red cr cle elfzle2 ltm1d lelttrd wb simpl nnsub syl2anc - nnre mpbid nnne0d divcld fprodmul fproddiv chash cfn fzfi sylancr hashfz1 - fprodconst eqtr2d fprodfac nnz 1zzd nn0zd fprodrev nncand prodeq1d 3eqtrd - id oveq12d eqtr4d 2nn a1i nnmulcld nnzd peano2nn zsubcld expclzd subsub4d - expm1d eqtr3d 3eqtr4d 2timesd mvrladdd faccl expcld div32d eqtrd ex nnind - ) CUAUDZCDEZFEZUUFAUDZGEZAHZUUGUUHPUUHIEZUUEDEZJEZAHZKULUEUUHGEZAHZULUUHU - UKCDEZJEZAHZKCUBUDZCDEZFEZUVAUUHGEZAHZUVBUUHUUKUUTDEZJEZAHZKZCUUTCUFEZCDE - ZFEZUVJUUHGEZAHZUVKUUHUUKUVIDEZJEZAHZKZCBCDEZFEZUVRUUHGEZAHZUVSUUHUUKBDEZ - JEZAHZKUAUBBUUECKZUUJUUPUUNUUSUWEUUGULUUIUUOAUWEUUGCUEFEULUWEUUFUECFUWEUU - FCCDEUEUUECCDUGUHUIZLUJUIZUWEUUIUUOKUUHUUGQZUWEUUFUEUUHGUWFMNOUWEUUGULUUM - UURAUWGUWEUUMUURKUWHUWEUULUUQUUHJUUECUUKDUKLNOUMUAUBUNZUUJUVDUUNUVGUWIUUG - UVBUUIUVCAUWIUUFUVACFUUEUUTCDUGZLZUWIUUIUVCKUWHUWIUUFUVAUUHGUWJMNOUWIUUGU - VBUUMUVFAUWKUWIUUMUVFKUWHUWIUULUVEUUHJUUEUUTUUKDUKLNOUMUUEUVIKZUUJUVMUUNU - VPUWLUUGUVKUUIUVLAUWLUUFUVJCFUUEUVICDUGZLZUWLUUIUVLKUWHUWLUUFUVJUUHGUWMMN - OUWLUUGUVKUUMUVOAUWNUWLUUMUVOKUWHUWLUULUVNUUHJUUEUVIUUKDUKLNOUMUUEBKZUUJU - WAUUNUWDUWOUUGUVSUUIUVTAUWOUUFUVRCFUUEBCDUGZLZUWOUUIUVTKUWHUWOUUFUVRUUHGU - WPMNOUWOUUGUVSUUMUWCAUWQUWOUUMUWCKUWHUWOUULUWBUUHJUUEBUUKDUKLNOUMUUPCUUSU - UOAUOUURAUOUPUUTRQZUVHUVQUWRUVHUQZUVDUUTUVAJEZUVAURUSZSEZIEZUVGUXBIEZUVMU - VPUWSUVDUVGUXBIUWRUVHVFMUWRUVMUXCKUVHUWRUVMCUUTFEZUUTUUHGEZAHUVBUXFAHZUUT - UUTGEZIEZUXCUWRUVKUXEUVLUXFAUWRUVJUUTCFUWRUUTCUUTUTZUWRVAZVBZLZUWRUVLUXFK - UUHUVKQUWRUVJUUTUUHGUXLMNOUWRUXFUXHACUUTUWRUUTCVCUSQUUTVDVEZUWRUUHUXEQZUQ - ZUXFUWRUUTVGQZUUHVRQZUXFVGQZUXOUUTVHZUUHCUUTVIUUHUUTVJZVKVLUUHUUTUUTGUKVM - UWRUXIUXGCIEUXGUXCUWRUXHCUXGIUWRUXQUXHCKUXTUUTVNVOLUWRUXGUWRUVBUXFAUWRCUV - AVPZUWRUUHUVBQZUQZUXFUWRUXQUXRUXSUYCUXTUUHCUVAVIZUYAVKVLVQVSUWRUXGUVBUVCU - UTUUTUUHDEZSEZIEZAHUVDUVBUYGAHZIEUXCUWRUVBUXFUYHAUYCUWRUUHUEUVAFEZQUXFUYH - KUVBUYJUUHUVAVTWIUUHUUTWAWBWCUWRUVBUVCUYGAUYBUYDUVCUWRUVAVGQZUXRUVCVGQUYC - UUTWDZUYEUUHUVAVJVKVLUYDUUTUYFUWRUUTWJQZUYCUXJNZUYDUYFUYDUUHUUTWEWFZUYFRQ - ZUYDUUHUVAUUTUYDUUHUYCUUHRQZUWRUUHUVAWGWHZWKUYDUVAUWRUYKUYCUYLNWLUWRUUTWM - QUYCUUTXBNZUYCUUHUVAWNWFUWRUUHCUVAWOWHUYDUUTUYSWPWQUYDUYQUWRUYOUYPWRUYRUW - RUYCWSUUHUUTWTXAXCZTZUYDUYFUYTXDZXEXFUWRUYIUXBUVDIUWRUYIUVBUUTAHZUVBUYFAH - ZSEUXBUWRUVBUUTUYFAUYBUYNVUAVUBXGUWRUWTVUCUXAVUDSUWRVUCUUTUVBXHUSZJEZUWTU - WRUVBXIQUYMVUCVUFKCUVAXJUXJUVBUUTAXMXKUWRVUEUVAUUTJUWRUYKVUEUVAKUYLUVAXLV - OLXNUWRUXAUVBUCUDZUCHZUUTUVADEZUVAFEZUYFAHVUDUWRUYKUXAVUHKUYLUVAUCXOVOUWR - VUGUYFUCAUUTCUVAUUTXPZUWRXQUWRUVAUYLXRUWRVUGUVBQZUQVUGVULVUGRQUWRVUGUVAWG - WHTVUGUYFKYCXSUWRVUJUVBUYFAUWRVUICUVAFUWRUUTCUXJUXKXTMYAYBYDYELYBYBYBNUWR - UVPUXDKUVHUWRUVPUXEUVOAHUVBUVOAHZUUTPUUTIEZUVIDEZJEZIEZUXDUWRUVKUXEUVOAUX - MYAUWRUVOVUPACUUTUXNUXPUUHUVNUXPUUHUXOUYQUWRUUHUUTWGWHZTUXPUUHVURXDUXPUUK - UVIUXPUUKUXPPUUHPRQZUXPYFYGVURYHYIUXPUVIUWRUVIRQUXOUUTYJNYIYKYLAUBUNZUUHU - UTUVNVUOJVUTYCVUTUUKVUNUVIDUUHUUTPIUKMYDVMUWRVUQUVGUXASEZUWTIEUXDUWRVUMVV - AVUPUWTIUWRUVBUVFUUHSEZAHUVGUVBUUHAHZSEVUMVVAUWRUVBUVFUUHAUYBUYDUUHUVEUYD - UUHUYRTZUYDUUHUYRXDZUYDUUKUUTUYDUUKUYDPUUHVUSUYDYFYGUYRYHZYIUWRUUTVRQUYCV - UKNYKZYLZVVDVVEXGUWRUVBUVOVVBAUYDUUHUVECDEZJEUVOVVBUYDVVIUVNUUHJUYDUUKUUT - CUYDUUKVVFTUYNUYDVAYMLUYDUUHUVEVVDVVEVVGYNYOWCUWRUXAVVCUVGSUWRUYKUXAVVCKU - YLUVAAXOVOLYPUWRVUOUVAUUTJUWRVUNUUTDEZCDEVUOUVAUWRVUNUUTCUWRVUNUWRPUUTVUS - UWRYFYGUWRYCYHTUXJUXKYMUWRVVJUUTCDUWRVUNUUTUUTUXJUXJUWRUUTUXJYQYRMYOLYDUW - RUVGUXAUWTUWRUVBUVFAUYBVVHVQUWRUXAUWRUYKUXARQUYLUVAYSVOZTUWRUUTUVAUXJUYLY - TUWRUXAVVKXDUUAUUBYBNYPUUCUUD $. - $} + $( + @{ + mthmco.d @e |- D = ( mDV ` T ) @. + mthmco.e @e |- E = ( mEx ` T ) @. + mthmco.1 @e |- ( ph -> T e. mFS ) @. + mthmco.2 @e |- ( ph -> K C_ D ) @. + mthmco.3 @e |- ( ph -> B C_ E ) @. + mthmco.j @e |- U = ( mThm ` T ) @. + mthmco.l @e |- L = ( mSubst ` T ) @. + mthmco.v @e |- V = ( mVR ` T ) @. + mthmco.h @e |- H = ( mVH ` T ) @. + mthmco.w @e |- W = ( mVars ` T ) @. + mthmco.4 @e |- ( ph -> <. M , O , P >. e. U ) @. + mthmco.5 @e |- ( ph -> S e. ran L ) @. + mthmco.6 @e |- ( ( ph /\ x e. O ) -> + <. K , B , ( S ` x ) >. e. U ) @. + mthmco.7 @e |- ( ( ph /\ v e. V ) -> + <. K , B , ( S ` ( H ` v ) ) >. e. U ) @. + mthmco.8 @e |- ( ( ph /\ ( x M y /\ a e. ( W ` ( S ` ( H ` x ) ) ) /\ + b e. ( W ` ( S ` ( H ` y ) ) ) ) ) -> a K b ) @. + @( The application of a theorem to a collection of theorems is a theorem. + (Compare ~ mclsax .) This is what justifies the treatment of + theorems as "equivalent" to axioms once they have been proven: the + composition of one theorem in the proof of another yields a theorem. - ${ - $d N n m k $. $d C n m k $. - $( A column-sum rule for binomial coefficents. (Contributed by Scott - Fenton, 24-Jun-2020.) $) - bccolsum $p |- ( ( N e. NN0 /\ C e. NN0 ) -> - sum_ k e. ( 0 ... N ) ( k _C C ) = ( ( N + 1 ) _C ( C + 1 ) ) ) $= - ( cn0 wcel cc0 cfz co cbc csu c1 caddc wceq wi oveq2 sumeq1d oveq1 oveq1d - eqeq12d imbi2d vm vn cv 0p1e1 eqtrdi weq cz 0nn0 nn0z bccl sylancr nn0cnd - cc 0z fsum1 cn wo elnn0 cfa cfv cmin cmul cdiv cif cle wbr 1red ltaddrp2d - wn nnrp peano2nn nnred ltnled mpbid elfzle2 nsyl iffalsed 1nn0 nnzd bcval - clt bc0k 3eqtr4rd bcnn ax-mp eqtr4i oveq2d 3eqtr4a sylbi eqtrd wa elnn0uz - jaoi cuz biimpi adantr elfznn0 adantl simplr nn0zd syl2anc fsump1 id 1cnd - nn0cn pncand eqcomd oveqan12rd peano2nn0 bcpasc syl2an 3eqtrd a2d nn0ind - exp31 imp ) CDEADEZFCGHZBUCZAIHZBJZCKLHZAKLHZIHZMZXQFUAUCZGHZXTBJZYFKLHZY - CIHZMZNXQFFGHZXTBJZKYCIHZMZNXQFUBUCZGHZXTBJZYPKLHZYCIHZMZNXQFYSGHZXTBJZYS - KLHZYCIHZMZNXQYENUAUBCYFFMZYKYOXQUUGYHYMYJYNUUGYGYLXTBYFFFGOPUUGYIKYCIUUG - YIFKLHZKYFFKLQUDUERSTUAUBUFZYKUUAXQUUIYHYRYJYTUUIYGYQXTBYFYPFGOPUUIYIYSYC - IYFYPKLQRSTYFYSMZYKUUFXQUUJYHUUCYJUUEUUJYGUUBXTBYFYSFGOPUUJYIUUDYCIYFYSKL - QRSTYFCMZYKYEXQUUKYHYAYJYDUUKYGXRXTBYFCFGOPUUKYIYBYCIYFCKLQRSTXQYMFAIHZYN - XQFUGEUULUMEYMUULMUNXQUULXQFDEZAUGEZUULDEUHAUIAFUJUKULXTUULBFXSFAIQUOUKXQ - AUPEZAFMZUQUULYNMZAURUUOUUQUUPUUOYCFKGHEZKUSUTKYCVAHUSUTYCUSUTVBHVCHZFVDZ - FYNUULUUOUURUUSFUUOYCKVEVFZUURUUOKYCWAVFUVAVIUUOKAUUOVGZAVJVHUUOKYCUVBUUO - YCAVKZVLVMVNYCFKVOVPVQUUOKDEZYCUGEZYNUUTMVRUUOYCUVCVSYCKVTUKAWBWCUUPFFIHZ - KKIHZUULYNUVFKUVGUUMUVFKMUHFWDWEUVDUVGKMVRKWDWEWFAFFIOUUPYCKKIUUPYCUUHKAF - KLQUDUEWGWHWMWIWJYPDEZXQUUAUUFUVHXQUUAUUFUVHXQWKZUUAWKUUCYRYSAIHZLHZYTYSY - CKVAHZIHZLHZUUEUVIUUCUVKMUUAUVIXTUVJBFYPUVHYPFWNUTEZXQUVHUVOYPWLWOWPUVIXS - UUBEZWKZXTUVQXSDEZUUNXTDEUVPUVRUVIXSYSWQWRUVQAUVHXQUVPWSWTAXSUJXAULXSYSAI - QXBWPUUAUVIYRYTUVJUVMLUUAXCUVIUVMUVJUVIUVLAYSIUVIAKXQAUMEUVHAXEWRUVIXDXFW - GXGXHUVIUVNUUEMZUUAUVHYSDEUVEUVSXQYPXIXQYCAXIWTYCYSXJXKWPXLXOXMXNXP $. - $} + This theorem has a similar statement to ~ mclspps , but deals with the + elimination of dummy variables instead of raw composition. The + assumption ~ mfsinf that there are enough variables of each type is + essential for this proof. @) + mthmco @p |- ( ph -> <. K , B , ( S ` P ) >. e. U ) @= + ? @. + @} + $) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Infinite products + Grammatical formal systems =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $( - @{ - @d A w x @. @d B w x @. @d C w x @. @d ph x @. @d R w x @. - @d w z @. @d X w y @. @d x y @. @d x z @. @d X z @. @d Y w x @. - ntrivcvgvaulem1.1 @e |- ( ( ph /\ x e. A ) -> - E. y e. B A. z e. C Y R x ) @. - @( Lemma for ~ ntrivcvgcau . Specialization of a particular form - while avoiding a distinctness requirement between ` x ` and ` X ` . - (Contributed by Scott Fenton, 30-Dec-2017.) @) - ntrivcvgcaulem1 @p |- ( ( ph /\ X e. A ) -> - E. y e. B A. z e. C Y R X ) @= - ( vw cv wbr wral wrex wcel breq2 ralbidv rexbidv ralrimiva weq sylib wceq - cbvralv rspcv mpan9 ) AJLMZHNZDGOZCFPZLEOZIEQJIHNZDGOZCFPZAJBMZHNZDGOZCFP - ZBEOULAUSBEKUAUSUKBLEBLUBZURUJCFUTUQUIDGUPUHJHRSTUEUCUKUOLIEUHIUDZUJUNCFV - AUIUMDGUHIJHRSTUFUG @. - @} - - @{ - ntrivcvgcaulem2.1 @e |- Z = ( ZZ>= ` M ) @. - ntrivcvgcaulem2.2 @e |- ( ph -> K e. Z ) @. - ntrivcvgcaulem2.3 @e |- ( ( ph /\ k e. Z ) -> A e. CC ) @. - ntrivcvgcaulem2.4 @e |- ( ( ph /\ m e. ( ZZ>= ` K ) ) -> - ( abs ` ( prod_ k e. ( K ... m ) A - 1 ) ) < ( 1 / 2 ) ) @. - - @{ - @d K k @. @d k m @. @d k ph @. - @( Lemma for ~ ntrivcvgcau . A product whose absolute difference - from one is less than one half has an absolute value less than two. - (Contributed by Scott Fenton, 30-Dec-2017.) @) - ntrivcvgcaulem2 @p |- ( ( ph /\ m e. ( ZZ>= ` K ) ) -> - ( abs ` prod_ k e. ( K ... m ) A ) < 2 ) @= - ( cfv wcel co c1 caddc cabs c2 a1i cr cv cuz wa cfz cprod cmin clt elfzuz - fzfid cc uztrn2 syl2an syldan adantlr fprodcl ax-1cn npcand fveq2d abscld - eqeltrd subcld abs1 1re eqeltri readdcld 2re abstrid rereccli wbr halflt1 - cdiv 2ne0 lttrd ltadd1dd oveq2i df-2 3brtr4g lelttrd eqbrtrrd ) ADUAZEUBL - ZMZUCZEVTUDNZBCUEZOUFNZOPNZQLZWEQLRUGWCWGWEQWCWEOWCWDBCWCEVTUIACUAZWDMZBU - JMZWBAWJWIGMZWKAEGMWIWAMWLWJIWIEVTUHFWIEGHUKULJUMUNUOZOUJMWCUPSZUQZURWCWH - WFQLZOQLZPNZRWCWGWCWGWEUJWOWMUTUSWCWPWQWCWFWCWEOWMWNVAZUSZWQTMWCWQOTVBVCV - DSVERTMWCVFSWCWFOWSWNVGWCWPOPNOOPNWRRUGWCWPOOWTOTMWCVCSZXAWCWPORVKNZOWTXB - TMWCRVFVLVHSXAKXBOUGVIWCVJSVMVNWQOWPPVBVOVPVQVRVS @. - @} + $c m0St $. + $c mSA $. + $c mWGFS $. + $c mSyn $. + $c mESyn $. + $c mGFS $. + $c mTree $. + $c mST $. + $c mSAX $. + $c mUFS $. - @{ - @d A m @. @d K k @. @d k m @. @d K m @. @d k ph @. @d m ph @. - @( Lemma for ~ ntrivcvgcau . A product whose absolute difference from - one is less than one half has no zero values. (Contributed - by Scott Fenton, 30-Dec-2017.) @) - ntrivcvgcaulem3 @p |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A =/= 0 ) @= - ( cfv wcel cc0 wceq co c1 cmin cabs clt cv cuz wne csb wi wa cfz cprod c2 - wral cdiv wbr fzfid elfzuz uztrn2 syl2an syldan fprodcl adantr ax-1cn a1i - cc subcld abscld cr 2re 2ne0 rereccli ltnsymd halflt1 cneg absnegi df-neg - fveq2i abs1 3eqtr3i breqtrri oveq1 fveq2d breqtrrid nsyl adantlr fprodm1s - cmul simpr oveq2 sylan9eqr eqtrd mtand neqned ralrimiva nfv nfcsb1v nfcv - mul01d nfne weq csbeq1a neeq1d cbvral rsp sylbir syl imp ) ACUAZEUBLZMZBN - UCZACDUAZBUDZNUCZDXFUJZXGXHUEZAXKDXFAXIXFMZUFZXJNXOXJNOZEXIUGPZBCUHZNOZXO - QUIUKPZXRQRPZSLZTULXSXOYBXTXOYAXOXRQAXRVBMXNAXQBCAEXIUMAXEXQMZXEGMZBVBMZA - EGMZXGYDYCIXEEXIUNFXEEGHUOZUPJUQZURUSQVBMXOUTVAVCVDXTVEMXOUIVFVGVHVAKVIXS - XTNQRPZSLZYBTXTQYJTVJQVKZSLQSLYJQQUTVLYKYISQVMVNVOVPVQXSYAYISXRNQRVRVSVTW - AXOXPUFXREXIQRPZUGPZBCUHZXJWDPZNXOXRYOOXPXOBCEXIAXNWEAYCYEXNYHWBWCUSXPXOY - OYNNWDPNXJNYNWDWFXOYNXOYMBCXOEYLUMAXEYMMZYEXNAYPYDYEAYFXGYDYPIXEEYLUNYGUP - JUQWBURWOWGWHWIWJWKXLXHCXFUJXMXHXKCDXFXHDWLCXJNCXIBWMCNWNWPCDWQBXJNCXIBWR - WSWTXHCXFXAXBXCXD @. - @} + $( Mapping expressions to statements. $) + cm0s $a class m0St $. - ntrivcvgcaulem4.5 @e |- ( ( ph /\ x e. RR+ ) -> - E. n e. Z A. q e. ( ZZ>= ` n ) - ( abs ` ( prod_ k e. ( n ... q ) A - 1 ) ) < x ) @. + $( The set of syntax axioms. $) + cmsa $a class mSA $. - ntrivcvgcaulem4 @p |- ( ( ph /\ x e. RR+ ) -> - E. j e. ( ZZ>= ` K ) A. p e. ( ZZ>= ` j ) - ( abs ` ( prod_ k e. ( ( j + 1 ) ... p ) A - 1 ) ) < ( x / 2 ) ) @= ? @. + $( The set of weakly grammatical formal systems. $) + cmwgfs $a class mWGFS $. - @{ - @d j p @. @d j ph @. @d j x @. @d K p @. @d p ph @. @d p x @. @d A m @. - @d j k @. @d j m @. @d K k @. @d k m @. @d K m @. @d k p @. @d k ph @. - @d k x @. @d m ph @. + $( The syntax typecode function. $) + cmsy $a class mSyn $. - @( Lemma for ~ ntrivcvgcau . Translate final hypothesis into - a Cauchy-like criterion. (Contributed by Scott Fenton, 30-Dec-2017.) @) - ntrivcvgcaulem5 @p |- ( ( ph /\ x e. RR+ ) -> - E. j e. Z A. p e. ( ZZ>= ` j ) - ( abs ` ( prod_ k e. ( K ... p ) A - prod_ k e. ( K ... j ) A ) ) - < x ) @= - ( wcel wa co cv crp c1 caddc cfz cprod cmin cabs cfv c2 cdiv clt wbr wral - cuz wrex ntrivcvgcaulem4 wss eleqtrdi uzss sseqtrrdi adantr sseld adantrd - syl wi w3a cmul fzfid simplll simpll elfzuz uztrn2 syl2an syl2anc fprodcl - cr cc abscld 3adant3 simpl peano2uzs ax-1cn subcld simplr rphalfcld rpred - 2re a1i cc0 cle absge0d simprl eleq1 anbi2d prodeq1d oveq1d fveq2d breq1d - weq oveq2 imbi12d chvarv ntrivcvgcaulem2 simp3 ltmul12ad wceq c0 eluzelre - cin ad2antrl fzdisj cun simprr elfzuzb sylanbrc fzsplit fprodsplit muls1d - ltp1d eqtr4d absmuld eqtr2d simp1r rpcn 2cn 2ne0 divcan2d 3brtr3d anassrs - wne 3expia ralimdva expimpd jcad reximdv2 mpd ) ABUAZUBRZSZDUAZUCUDTZLUAZ - UETZCEUFZUCUGTZUHUIZYRUJUKTZULUMZLUUAUOUIZUNZDHUOUIZUPHUUCUETZCEUFZHUUAUE - TZCEUFZUGTZUHUIZYRULUMZLUUJUNZDJUPABCDEFGHIJKLMNOPQUQYTUUKUUTDUULJYTUUAUU - LRZUUKSUUAJRZUUTYTUVAUVBUUKYTUULJUUAAUULJURYSAUULIUOUIZJAHUVCRUULUVCURAHJ - UVCNMUSIHUTVEMVAVBVCVDYTUVAUUKUUTYTUVASUUIUUSLUUJYTUVAUUCUUJRZUUIUUSVFYTU - VAUVDSZUUIUUSYTUVEUUIVGZUUPUHUIZUUGVHTZUJUUHVHTZUURYRULUVFUVGUJUUGUUHYTUV - EUVGVQRUUIYTUVESZUUPUVJUUOCEUVJHUUAVIUVJEUAZUUORZSAUVKJRZCVRRZAYSUVEUVLVJ - UVJHJRZUVKUULRZUVMUVLUVJAUVOAYSUVEVKZNVEZUVKHUUAVLIUVKHJMVMZVNOVOVPZVSVTU - JVQRUVFWHWIYTUVEUUGVQRUUIUVJUUFUVJUUEUCUVJUUDCEUVJUUBUUCVIUVJUVKUUDRZSAUV - MUVNAYSUVEUWAVJUVJUUBJRZUVKUUBUOUIRUVMUWAUVJUVBUWBYTUVOUVAUVBUVEAUVOYSNVB - UVAUVDWAIUUAHJMVMVNIUUAJMWBVEUVKUUBUUCVLIUVKUUBJMVMVNOVOVPZUCVRRUVJWCWIWD - ZVSVTYTUVEUUHVQRUUIUVJUUHUVJYRAYSUVEWEWFWGVTYTUVEWJUVGWKUMUUIUVJUUPUVTWLV - TYTUVEUVGUJULUMZUUIUVJAUVAUWEUVQYTUVAUVDWMZACEDHIJMNOAFUAZUULRZSZHUWGUETZ - CEUFZUCUGTZUHUIZUCUJUKTZULUMZVFAUVASZUUPUCUGTZUHUIZUWNULUMZVFFDFDWTZUWIUW - PUWOUWSUWTUWHUVAAUWGUUAUULWNWOUWTUWMUWRUWNULUWTUWLUWQUHUWTUWKUUPUCUGUWTUW - JUUOCEUWGUUAHUEXAWPWQWRWSXBPXCXDVOVTYTUVEWJUUGWKUMUUIUVJUUFUWDWLVTYTUVEUU - IXEXFYTUVEUVHUURXGUUIUVJUURUUPUUFVHTZUHUIUVHUVJUUQUXAUHUVJUUQUUPUUEVHTZUU - PUGTUXAUVJUUNUXBUUPUGUVJUUOUUDCUUMEUVJUUAUUBULUMUUOUUDXJXHXGUVJUUAUVAUUAV - QRYTUVDHUUAXIXKXTHUUAUUBUUCXLVEUVJUUAUUMRZUUMUUOUUDXMXGUVJUVAUVDUXCUWFYTU - VAUVDXNUUAHUUCXOXPUUAHUUCXQVEUVJHUUCVIUVJUVKUUMRZSAUVMUVNAYSUVEUXDVJUVJUV - OUVPUVMUXDUVRUVKHUUCVLUVSVNOVOXRWQUVJUUPUUEUVTUWCXSYAWRUVJUUPUUFUVTUWDYBY - CVTUVFYSUVIYRXGAYSUVEUUIYDYSYRUJYRYEUJVRRYSYFWIUJWJYKYSYGWIYHVEYIYLYJYMYN - YOYPYQ @. - @} - @} - $) + $( The syntax typecode function for expressions. $) + cmesy $a class mESyn $. - ${ - $d F j $. $d F k $. $d F n $. $d j k $. $d j n $. $d j ph $. - $d k ph $. $d M j $. $d M k $. $d M n $. $d n ph $. $d Z k $. - iprodefisumlem.1 $e |- Z = ( ZZ>= ` M ) $. - iprodefisumlem.2 $e |- ( ph -> M e. ZZ ) $. - iprodefisumlem.3 $e |- ( ph -> F : Z --> CC ) $. - $( Lemma for ~ iprodefisum . (Contributed by Scott Fenton, - 11-Feb-2018.) $) - iprodefisumlem $p |- ( ph -> seq M ( x. , ( exp o. F ) ) = - ( exp o. seq M ( + , F ) ) ) $= - ( vk vj cmul ce caddc cc wcel cfv wceq fvco3 syl wi co vn ccom cseq cv wa - sylan ffvelcdmda efcl eqeltrd prodf ffnd wfn eff ax-mp serf fnfco sylancr - wf ffn cuz c1 fveq2 2fveq3 eqeq12d imbi2d weq uzid eleqtrrdi syl2anc seq1 - cz fveq2d 3eqtr4d a1i oveq1 3ad2ant3 adantl peano2uz adantr oveq2d expcom - w3a eqcomi eleq2s imp ffvelcdmd efadd eqtr4d 3adant3 eqtrd seqp1 3exp a2d - uzind4 impcom eqfnfvd ) AHDJKBUBZCUCZKLBCUCZUBZADMWRAHWQCDEFAHUDZDNZUEZXA - WQOZXABOZKOZMADMBURZXBXDXFPGDMXAKBQUFXCXEMNXFMNADMXABGUGZXEUHRUIUJUKAKMUL - ZDMWSURZWTDULMMKURXIUMMMKUSUNAHBCDEFXHUOZMDKWSUPUQXCXAWROZXAWSOKOZXAWTOZX - BAXLXMPZAXOSZXACUTOZDAIUDZWROZXRWSOKOZPZSACWROZCWSOZKOZPZSZAUAUDZWROZYGWS - OZKOZPZSAYGVALTZWROZYLWSOZKOZPZSXPIUACXAXRCPZYAYEAYQXSYBXTYDXRCWRVBXRCKWS - VCVDVEIUAVFZYAYKAYRXSYHXTYJXRYGWRVBXRYGKWSVCVDVEXRYLPZYAYPAYSXSYMXTYOXRYL - WRVBXRYLKWSVCVDVEIHVFZYAXOAYTXSXLXTXMXRXAWRVBXRXAKWSVCVDVEYFCVKNZACWQOZCB - OZKOZYBYDAXGCDNUUBUUDPGACXQDAUUACXQNFCVGREVHDMCKBQVIAUUAYBUUBPFJWQCVJRAYC - UUCKAUUAYCUUCPFLBCVJRVLVMVNYGXQNZAYKYPUUEAYKYPUUEAYKWBZYHYLWQOZJTZYIYLBOZ - LTZKOZYMYOUUFUUHYJUUGJTZUUKYKUUEUUHUULPAYHYJUUGJVOVPUUEAUULUUKPYKUUEAUEZU - ULYJUUIKOZJTZUUKUUMUUGUUNYJJUUMXGYLDNZUUGUUNPAXGUUEGVQZUUEUUPAUUEYLXQDCYG - VREVHVSZDMYLKBQVIVTUUMYIMNZUUIMNUUKUUOPUUEAUUSAUUSSYGDXQAYGDNUUSADMYGWSXK - UGWADXQEWCWDWEUUMDMYLBUUQUURWFYIUUIWGVIWHWIWJUUEAYMUUHPZYKUUEUUTAJWQCYGWK - VSWIUUEAYOUUKPYKUUMYNUUJKUUEYNUUJPALBCYGWKVSVLWIVMWLWMWNEWDWOAXJXBXNXMPXK - DMXAKWSQUFWHWP $. - $} + $( The set of grammatical formal systems. $) + cmgfs $a class mGFS $. - ${ - $d F k $. $d k ph $. $d M k $. $d Z k $. $d F j $. $d j k $. - $d Z j $. $d j ph $. $d M j $. - iprodefisum.1 $e |- Z = ( ZZ>= ` M ) $. - iprodefisum.2 $e |- ( ph -> M e. ZZ ) $. - iprodefisum.3 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B ) $. - iprodefisum.4 $e |- ( ( ph /\ k e. Z ) -> B e. CC ) $. - iprodefisum.5 $e |- ( ph -> seq M ( + , F ) e. dom ~~> ) $. - $( Applying the exponential function to an infinite sum yields an infinite - product. (Contributed by Scott Fenton, 11-Feb-2018.) $) - iprodefisum $p |- ( ph -> prod_ k e. Z ( exp ` B ) = - ( exp ` sum_ k e. Z B ) ) $= - ( vj ce cfv cc wcel syl caddc cseq cli cmpt ccom csu cc0 wne isumcl efne0 - cv cmul ccncf co efcn a1i wa wceq fveq2 eqid fvex adantl eqeltrd serf cuz - fvmpt eqcomi eleq2s seqfeq cdm wbr climdm eqbrtrd climcl climcncf cbvmptv - sylib fmptd iprodefisumlem isum fveq2d 3brtr4d wf fvco3 sylan 3eqtrd efcl - iprodn0 ) ABMNZCMLFLUHZDNZUAZUBZEFBCUCZMNZFGHAWKOPWLUDUEABCDEFGHIJKUFWKUG - QAMRWIESZUBRDESZTNZMNUIWJESWLTAOOWOMWMEFGHMOOUJUKPAULUMACWIEFGHACUHZFPZUN - ZWPWINZWPDNZOWQWSWTUOZALWPWHWTFWIWGWPDUPZWIUQWPDURVCZUSZWRWTBOIJUTZUTVAAW - MWNWOTARCWIDEHWPEVBNZPXAAXAWPFXFXCFXFGVDVEUSVFAWNTVGPWNWOTVHZKWNVIVNZVJAX - GWOOPXHWOWNVKQVLAWIEFGHACFWTOWIXELCFWHWTXBVMVOZVPAWKWOMABCDEFGHIJVQVRVSWR - WPWJNZWSMNZWTMNWFAFOWIVTWQXJXKUOXIFOWPMWIWAWBWRWSWTMXDVRWRWTBMIVRWCWRBOPW - FOPJBWDQWE $. - $} + $( The set of proof trees. $) + cmtree $a class mTree $. - ${ - $d A j $. $d A k $. $d A z $. $d j k $. $d j ph $. $d k ph $. - $d k z $. - iprodgam.1 $e |- ( ph -> A e. ( CC \ ( ZZ \ NN ) ) ) $. - $( An infinite product version of Euler's gamma function. (Contributed by - Scott Fenton, 12-Feb-2018.) $) - iprodgam $p |- ( ph -> ( _G ` A ) = - ( prod_ k e. NN ( ( ( 1 + ( 1 / k ) ) ^c A ) / ( 1 + ( A / k ) ) ) / - A ) ) $= - ( vj cfv ce cn c1 cdiv co caddc cc wcel wceq clog cmul cmin oveq12d eqtrd - vz clgam cgam cv ccxp cprod cz cdif eflgam oveq1 fvoveq1d sumeq2sdv fveq2 - syl csu df-lgam ovex fvmpt fveq2d cmpt nnuz 1zzd weq id oveq2d oveq2 eqid - adantl wa eldifad adantr peano2nn nncnd nncn cc0 wne nnne0 divcld divne0d - nnne0d logcld mulcld 1cnd addcld simpr dmgmdivn0 cseq cli wbr cdm lgamcvg - subcld seqex breldm isumcl dmgmn0 efsub syl2anc iprodefisum dividd oveq1d - divdird crp 1rp nnrpd rpreccld rpaddcld rpcnd rpne0d cxpefd eflog addcomd - a1i eqtr4d prodeq2dv eqtr3d ) ABUBFZGFZBUCFZHIICUDZJKZLKZBUEKZIBXTJKZLKZJ - KZCUFZBJKZABMUGHUHZUHZNZXRXSODBUIUNAXRHBXTILKZXTJKZPFZQKZYDILKZPFZRKZCUOZ - BPFZRKZGFZYHAXQUUAGAYKXQUUAODUABHUAUDZYNQKZUUCXTJKZILKPFZRKZCUOZUUCPFZRKU - UAYJUBUUCBOZUUHYSUUIYTRUUJHUUGYRCUUJUUDYOUUFYQRUUCBYNQUJUUJUUEYDIPLUUCBXT - JUJUKSULUUCBPUMSUACUPYSYTRUQURUNUSAUUBYSGFZYTGFZJKZYHAYSMNYTMNUUBUUMOAYRC - EHBEUDZILKZUUNJKZPFZQKZBUUNJKZILKPFZRKZUTZIHVAAVBZXTHNZXTUVBFYROAEXTUVAYR - HUVBECVCZUURYOUUTYQRUVEUUQYNBQUVEUUPYMPUVEUUOYLUUNXTJUUNXTILUJUVEVDSUSVEU - VEUUSYDIPLUUNXTBJVFUKSUVBVGZYOYQRUQURVHZAUVDVIZYOYQUVHBYNABMNZUVDABMYIDVJ - ZVKZUVHYMUVHYLXTUVHYLUVDYLHNAXTVLVHZVMZUVDXTMNAXTVNVHZUVDXTVOVPAXTVQVHZVR - UVHYLXTUVMUVNUVHYLUVLVTUVOVSWAWBZUVHYPUVHYDIUVHBXTUVKUVNUVOVRZUVHWCZWDZUV - HBXTAYKUVDDVKAUVDWEZWFZWAZWLZALUVBIWGZXQYTLKZWHWIUWDWHWJNABEUVBUVFDWKUWDU - WEWHLUVBIWMXQYTLUQWNUNZWOABUVJABDWPZWAYSYTWQWRAUUKYGUULBJAHYRGFZCUFUUKYGA - YRCUVBIHVAUVCUVGUWCUWFWSAHUWHYFCUVHUWHYOGFZYQGFZJKZYFUVHYOMNYQMNUWHUWKOUV - PUWBYOYQWQWRUVHUWIYCUWJYEJUVHUWIBYBPFZQKZGFYCUVHYOUWMGUVHYNUWLBQUVHYMYBPU - VHYMXTXTJKZYALKYBUVHXTIXTUVNUVRUVNUVOXBUVHUWNIYALUVHXTUVNUVOWTXATUSVEUSUV - HYBBUVHYBUVHIYAIXCNUVHXDXMUVHXTUVHXTUVTXEXFXGZXHUVHYBUWOXIUVKXJXNUVHUWJYP - YEUVHYPMNYPVOVPUWJYPOUVSUWAYPXKWRUVHYDIUVQUVRXLTSTXOXPAUVIBVOVPUULBOUVJUW - GBXKWRSTTXP $. - $} + $( The set of syntax trees. $) + cmst $a class mST $. + $( The indexing set for a syntax axiom. $) + cmsax $a class mSAX $. -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Factorial limits -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) + $( The set of unambiguous formal sytems. $) + cmufs $a class mUFS $. - ${ - $d a b $. $d a k $. $d a n $. $d b n $. $d b x $. $d k n $. $d M a $. - $d M b $. $d M k $. $d M n $. $d M x $. - $( Lemma for ~ faclim . Closed form for a particular sequence. - (Contributed by Scott Fenton, 15-Dec-2017.) $) - faclimlem1 $p |- ( M e. NN0 -> - seq 1 ( x. , ( n e. NN |-> ( ( ( 1 + ( M / n ) ) x. ( 1 + ( 1 / n ) ) ) / - ( 1 + ( ( M + 1 ) / n ) ) ) ) ) = - ( x e. NN |-> ( ( M + 1 ) x. ( ( x + 1 ) / ( x + ( M + 1 ) ) ) ) ) ) $= - ( vb wcel cv cmul cn c1 cdiv co caddc cfv wceq oveq1 oveq12d oveq2d oveq2 - 3eqtrd rpcnd va vk cn0 cmpt cseq wral wa wi fveq2 cz 1z seq1 ax-mp eqtrdi - eqeq12d imbi2d weq 1nn eqid ovex fvmpt nn0cn div1d 1div1e1 oveq2i nn0p1nn - nncnd 1cnd addcomd oveq1d cc ax-1cn addcli nnaddcld nnne0d divassd eqtrid - a1i cuz seqp1 nnuz eleq2s adantr adantl peano2nn syl nnrpd simpl rpaddcld - rpdivcld crp cr nn0re nndivred recnd cc0 cle wbr divge0d ge0p1rpd eqeltrd - nn0ge0 1rp rpreccld rpmulcld mulassd rpne0d divcan5d mul12d mulid1d simpr - adddid nn0cnd divcan2d addassd eqtrd eqtr3d divcan1d 3eqtr3rd exp31 nnind - a2d impcom eqtr4d ralrimiva wfn wb seqfn fneq2i mpbir fnmpti eqfnfv mp2an - sylibr ) CUCEZDFZGBHICBFZJKZLKZIIYQJKZLKZGKZICILKZYQJKZLKZJKZUDZIUEZMZYPA - HUUCAFZILKZUUJUUCLKZJKZGKZUDZMZNZDHUFZUUHUUONZYOUUQDHYOYPHEZUGUUIUUCYPILK - ZYPUUCLKZJKZGKZUUPUUTYOUUIUVDNZYOUAFZUUHMZUUCUVFILKZUVFUUCLKZJKZGKZNZUHYO - IUUGMZUUCIILKZIUUCLKZJKZGKZNZUHYOUBFZUUHMZUUCUVSILKZUVSUUCLKZJKZGKZNZUHYO - UWAUUHMZUUCUWAILKZUWAUUCLKZJKZGKZNZUHYOUVEUHUAUBYPUVFINZUVLUVRYOUWLUVGUVM - UVKUVQUWLUVGIUUHMZUVMUVFIUUHUIIUJEZUWMUVMNUKGUUGIULUMUNUWLUVJUVPUUCGUWLUV - HUVNUVIUVOJUVFIILOUVFIUUCLOPQUOUPUAUBUQZUVLUWEYOUWOUVGUVTUVKUWDUVFUVSUUHU - IUWOUVJUWCUUCGUWOUVHUWAUVIUWBJUVFUVSILOUVFUVSUUCLOPQUOUPUVFUWANZUVLUWKYOU - WPUVGUWFUVKUWJUVFUWAUUHUIUWPUVJUWIUUCGUWPUVHUWGUVIUWHJUVFUWAILOUVFUWAUUCL - OPQUOUPUADUQZUVLUVEYOUWQUVGUUIUVKUVDUVFYPUUHUIUWQUVJUVCUUCGUWQUVHUVAUVIUV - BJUVFYPILOUVFYPUUCLOPQUOUPYOUVMICIJKZLKZIIIJKZLKZGKZIUUCIJKZLKZJKZUVQIHEZ - UVMUXENURBIUUFUXEHUUGYQINZUUBUXBUUEUXDJUXGYSUWSUUAUXAGUXGYRUWRILYQICJRQUX - GYTUWTILYQIIJRQPUXGUUDUXCILYQIUUCJRQPUUGUSZUXBUXDJUTVAUMYOUXEICLKZUVNGKZU - VOJKUUCUVNGKZUVOJKUVQYOUXBUXJUXDUVOJYOUWSUXIUXAUVNGYOUWRCILYOCCVBZVCQUXAU - VNNYOUWTIILVDVEVRPYOUXCUUCILYOUUCYOUUCCVFZVGZVCQPYOUXJUXKUVOJYOUXIUUCUVNG - YOICYOVHUXLVIVJVJYOUUCUVNUVOUXNUVNVKEYOIIVLVLVMVRYOUVOYOIUUCUXFYOURVRUXMV - NZVGYOUVOUXOVOVPSVQUVSHEZYOUWEUWKUXPYOUWEUWKUXPYOUGZUWEUGUWFUVTUWAUUGMZGK - ZUWDUXRGKZUWJUXQUWFUXSNZUWEUXPUYAYOUYAUVSIVSMZHGUUGIUVSVTWAWBWCWCUWEUXSUX - TNUXQUVTUWDUXRGOWDUXQUXTUWJNUWEUXQUXTUWDICUWAJKZLKZIIUWAJKZLKZGKZIUUCUWAJ - KZLKZJKZGKZUUCUWCUYJGKZGKUWJUXPUXTUYKNYOUXPUXRUYJUWDGUXPUWAHEZUXRUYJNUVSW - EZBUWAUUFUYJHUUGYQUWANZUUBUYGUUEUYIJUYOYSUYDUUAUYFGUYOYRUYCILYQUWACJRQUYO - YTUYEILYQUWAIJRQPUYOUUDUYHILYQUWAUUCJRQPUXHUYGUYIJUTVAWFQWCUXQUUCUWCUYJUX - QUUCYOUUCHEUXPUXMWDZVGZUXQUWCUXQUWAUWBUXQUWAUXPUYMYOUYNWCZWGZUXQUVSUUCUXQ - UVSUXPYOWHZWGUXQUUCUYPWGZWIZWJZTZUXQUYJUXQUYGUYIUXQUYDUYFUXQUYDUYCILKWKUX - QIUYCUXQVHZUXQUYCUXQCUWAYOCWLEUXPCWMWDZUYRWNZWOZVIUXQUYCVUGUXQCUWAVUFUYSY - OWPCWQWRUXPCXBWDWSWTXAZUXQIUYEIWKEUXQXCVRZUXQUWAUYSXDZWIZXEZUXQIUYHVUJUXQ - UUCUWAVUAUYSWJZWIZWJTXFUXQUYLUWIUUCGUXQUWAUWCUYGGKZGKZUWAUYIGKZJKVUPUYIJK - UWIUYLUXQVUPUYIUWAUXQVUPUXQUWCUYGVUCVUMXETUXQUYIVUOTZUXQUWAUYRVGZUXQUYIVU - OXGZUXQUWAUYRVOZXHUXQVUQUWGVURUWHJUXQVUQUWCUWAUYGGKZGKUWCUWBUYFGKZGKZUWGU - XQUWAUWCUYGVUTVUDUXQUYGVUMTZXIUXQVVCVVDUWCGUXQUWAUYDGKZUYFGKVVCVVDUXQUWAU - YDUYFVUTUXQUYDVUITUXQUYFVULTZXFUXQVVGUWBUYFGUXQVVGUWAIGKZUWAUYCGKZLKUWACL - KZUWBUXQUWAIUYCVUTVUEVUHXLUXQVVIUWAVVJCLUXQUWAVUTXJZUXQCUWAUXQCUXPYOXKXMZ - VUTVVBXNPUXQVVKUVSUXILKUWBUXQUVSICUXQUVSUYTVGVUEVVMXOUXQUXIUUCUVSLUXQICVU - EVVMVIQXPSVJXQQUXQUWCUWBGKZUYFGKZVVEUWGUXQUWCUWBUYFVUDUXQUWBVUBTZVVHXFUXQ - VVOUWAUYFGKVVIUWAUYEGKZLKUWGUXQVVNUWAUYFGUXQUWAUWBVUTVVPUXQUWBVUBXGXRVJUX - QUWAIUYEVUTVUEUXQUYEVUKTXLUXQVVIUWAVVQILVVLUXQIUWAVUEVUTVVBXNPSXQSUXQVURV - VIUWAUYHGKZLKUWHUXQUWAIUYHVUTVUEUXQUYHVUNTXLUXQVVIUWAVVRUUCLVVLUXQUUCUWAU - YQVUTVVBXNPXPPUXQUWCUYGUYIVUDVVFVUSVVAVPXSQSWCSXTYBYAYCUUTUUPUVDNYOAYPUUN - UVDHUUOADUQZUUMUVCUUCGVVSUUKUVAUULUVBJUUJYPILOUUJYPUUCLOPQUUOUSZUUCUVCGUT - VAWDYDYEUUHHYFZUUOHYFUUSUURYGVWAUUHUYBYFZUWNVWBUKGUUGIYHUMHUYBUUHWAYIYJAH - UUNUUOUUCUUMGUTVVTYKDHUUHUUOYLYMYN $. - $} + $( Define a function mapping expressions to statements. (Contributed by + Mario Carneiro, 14-Jul-2016.) $) + df-m0s $a |- m0St = ( a e. _V |-> <. (/) , (/) , a >. ) $. ${ - $d k m $. $d M k $. $d M m $. $d M n $. - $( Lemma for ~ faclim . Show a limit for the inductive step. (Contributed - by Scott Fenton, 15-Dec-2017.) $) - faclimlem2 $p |- ( M e. NN0 -> - seq 1 ( x. , - ( n e. NN |-> ( ( ( 1 + ( M / n ) ) x. ( 1 + ( 1 / n ) ) ) / - ( 1 + ( ( M + 1 ) / n ) ) ) ) ) ~~> ( M + 1 ) ) $= - ( vm vk wcel cmul cn c1 cv cdiv co caddc cmpt cli cvv nnuz nnex mptex a1i - adantl cn0 cseq faclimlem1 1zzd 1cnd nn0p1nn nnzd cfv wceq weq oveq1 eqid - oveq12d ovex fvmpt divcnvlin nncnd cc wa peano2nn nnred nnaddcld nndivred - simpr adantr recnd fmpttd oveq2d eqtr4d climmulc2 mulid1d breqtrd eqbrtrd - ffvelcdmda ) BUAEZFAGHBAIZJKLKHHVPJKLKFKHBHLKZVPJKLKJKMHUBCGVQCIZHLKZVRVQ - LKZJKZFKZMZVQNCABUCVOWCVQHFKVQNVOHVQDCGWAMZWCHOGPVOUDZVOHVQDWDHOGPWEVOUEV - OVQBUFZUGWDOEVOCGWAQRSDIZGEZWGWDUHZWGHLKZWGVQLKZJKZUIVOCWGWAWLGWDCDUJZVSW - JVTWKJVRWGHLUKVRWGVQLUKUMZWDULWJWKJUNUOZTUPVOVQWFUQZWCOEVOCGWBQRSVOGURWGW - DVOCGWAURVOVRGEZUSZWAWRVSVTWRVSWQVSGEVOVRUTTVAWRVRVQVOWQVDVOVQGEWQWFVEVBV - CVFVGVNWHWGWCUHZVQWIFKZUIVOWHWSVQWLFKZWTCWGWBXAGWCWMWAWLVQFWNVHWCULVQWLFU - NUOWHWIWLVQFWOVHVITVJVOVQWPVKVLVM $. - $} + $d a c d e h m o p r s t x y $. + $( Define the set of syntax axioms. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-msa $a |- mSA = ( t e. _V |-> { a e. ( mEx ` t ) | + ( ( m0St ` a ) e. ( mAx ` t ) /\ ( 1st ` a ) e. ( mVT ` t ) /\ + Fun ( `' ( 2nd ` a ) |` ( mVR ` t ) ) ) } ) $. - $( Lemma for ~ faclim . Algebraic manipulation for the final induction. - (Contributed by Scott Fenton, 15-Dec-2017.) $) - faclimlem3 $p |- ( ( M e. NN0 /\ B e. NN ) -> - ( ( ( 1 + ( 1 / B ) ) ^ ( M + 1 ) ) / ( 1 + ( ( M + 1 ) / B ) ) ) = - ( ( ( ( 1 + ( 1 / B ) ) ^ M ) / ( 1 + ( M / B ) ) ) x. - ( ( ( 1 + ( M / B ) ) x. ( 1 + ( 1 / B ) ) ) / ( 1 + ( ( M + 1 ) / B ) ) ) - ) ) $= - ( cn0 wcel cn c1 cdiv co caddc cexp cmul crp 1rp a1i adantl rpaddcld rpne0d - rpcnd oveq1d rpdivcld wa nnrp rpreccld simpl expp1d cz nn0z rpexpcl syl2anr - 1cnd nn0nndivcl addcomd nn0ge0div ge0p1rpd eqeltrd divcan1d mulassd 3eqtr2d - recnd rpmulcld nn0p1nn nnrpd adantr divassd eqtrd ) BCDZAEDZUAZFFAGHZIHZBFI - HZJHZFVKAGHZIHZGHVJBJHZFBAGHZIHZGHZVQVJKHZKHZVNGHVRVSVNGHKHVHVLVTVNGVHVLVOV - JKHVRVQKHZVJKHVTVHVJBVHVJVHFVIFLDZVHMNZVGVILDVFVGAAUBZUCZOPZRZVFVGUDUEVHWAV - OVJKVHVOVQVHVOVGVJLDBUFDVOLDVFVGFVIWBVGMNWEPBUGVJBUHUIZRVHVQVHVQVPFIHLVHFVP - VHUJVHVPBAUKZUSULVHVPWIBAUMUNUOZRZVHVQWJQUPSVHVRVQVJVHVRVHVOVQWHWJTRZWKWGUQ - URSVHVRVSVNWLVHVSVHVQVJWJWFUTRVHVNVHFVMWCVHVKAVFVKLDVGVFVKBVAVBVCVGALDVFWDO - TPZRVHVNWMQVDVE $. + $( Define the set of weakly grammatical formal systems. (Contributed by + Mario Carneiro, 14-Jul-2016.) $) + df-mwgfs $a |- mWGFS = { t e. mFS | A. d A. h A. a + ( ( <. d , h , a >. e. ( mAx ` t ) /\ ( 1st ` a ) e. ( mVT ` t ) ) -> + E. s e. ran ( mSubst ` t ) a e. ( s " ( mSA ` t ) ) ) } $. - ${ - $d A a b m n x $. - faclim.1 $e |- F = - ( n e. NN |-> ( ( ( 1 + ( 1 / n ) ) ^ A ) / ( 1 + ( A / n ) ) ) ) $. - $( An infinite product expression relating to factorials. Originally due - to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) $) - faclim $p |- ( A e. NN0 -> seq 1 ( x. , F ) ~~> ( ! ` A ) ) $= - ( wcel cmul c1 cseq cn cdiv co caddc cexp cfa cfv cli wceq cc0 oveq12d cc - va vm vb vx cn0 cmpt seqeq3 ax-mp wbr oveq2 oveq1 oveq2d mpteq2dv seqeq3d - cv fveq2 fac0 eqtrdi breq12d weq csn cxp 1red nnrecre readdcld recnd nncn - exp0d nnne0 div0d 1p0e1 1div1e1 fconstmpt eqtr4i wtru nnuz 1zzd climprod1 - mpteq2ia mptru eqbrtri wa cvv simpr seqex faclimlem2 adantr elnnuz biimpi - a1i cuz adantl cfz wf crp nnrp rpreccld rpaddcld nn0z rpexpcld nn0nndivcl - cz 1cnd addcomd nn0ge0div ge0p1rpd eqeltrd rpdivcld rpcnd fmpttd ffvelcdm - 1rp elfznn syl2an adantlr mulcl seqcl rpmulcld nn0p1nn rpdivcl faclimlem3 - nnrpd oveq1d eqid fvmpt 3eqtr4d sylan2 prodfmul climmul facp1 breqtrrd ex - ovex nn0ind eqbrtrid ) AUEEFCGHZFBIGGBUOZJKZLKZAMKZGAYQJKZLKZJKZUFZGHZANO - ZPCUUDQYPUUEQDFCUUDGUGUHFBIYSUAUOZMKZGUUGYQJKZLKZJKZUFZGHZUUGNOZPUIFBIYSR - MKZGRYQJKZLKZJKZUFZGHZGPUIFBIYSUBUOZMKZGUVAYQJKZLKZJKZUFZGHZUVANOZPUIZFBI - YSUVAGLKZMKZGUVJYQJKZLKZJKZUFZGHZUVJNOZPUIZUUEUUFPUIUAUBAUUGRQZUUMUUTUUNG - PUVSUULUUSFGUVSBIUUKUURUVSUUHUUOUUJUUQJUUGRYSMUJUVSUUIUUPGLUUGRYQJUKULSUM - UNUVSUUNRNOGUUGRNUPUQURUSUAUBUTZUUMUVGUUNUVHPUVTUULUVFFGUVTBIUUKUVEUVTUUH - UVBUUJUVDJUUGUVAYSMUJUVTUUIUVCGLUUGUVAYQJUKULSUMUNUUGUVANUPUSUUGUVJQZUUMU - VPUUNUVQPUWAUULUVOFGUWABIUUKUVNUWAUUHUVKUUJUVMJUUGUVJYSMUJUWAUUIUVLGLUUGU - VJYQJUKULSUMUNUUGUVJNUPUSUUGAQZUUMUUEUUNUUFPUWBUULUUDFGUWBBIUUKUUCUWBUUHY - TUUJUUBJUUGAYSMUJUWBUUIUUAGLUUGAYQJUKULSUMUNUUGANUPUSUUTFIGVAVBZGHZGPUUSU - WCQUUTUWDQUUSBIGUFUWCBIUURGYQIEZUURGGJKGUWEUUOGUUQGJUWEYSUWEYSUWEGYRUWEVC - YQVDVEVFVHUWEUUQGRLKGUWEUUPRGLUWEYQYQVGYQVIVJULVKURSVLURVSBIGVMVNFUUSUWCG - UGUHUWDGPUIVOGIVPVOVQVRVTWAUVAUEEZUVIUVRUWFUVIWBZUVPUVHUVJFKZUVQPUWGUVHUV - JUAUVGFBIUVDYSFKZUVMJKZUFZGHZUVPGWCIVPUWGVQUWFUVIWDUVPWCEUWGFUVOGWEWJUWFU - WLUVJPUIUVIBUVAWFWGUWFUUGIEZUUGUVGOZTEUVIUWFUWMWBZUCUDFTUVFGUUGUWMUUGGWKO - EZUWFUWMUWPUUGWHWIWLZUWFUCUOZGUUGWMKEZUWRUVFOZTEZUWMUWFITUVFWNUWRIEZUXAUW - SUWFBIUVETUWFUWEWBZUVEUXCUVBUVDUXCYSUVAUXCGYRGWOEUXCXLWJZUWEYRWOEUWFUWEYQ - YQWPZWQWLWRZUWFUVAXBEUWEUVAWSWGWTUXCUVDUVCGLKWOUXCGUVCUXCXCUXCUVCUVAYQXAZ - VFXDUXCUVCUXGUVAYQXEXFXGZXHXIXJUWRUUGXMZITUWRUVFXKXNXOZUWRTEUDUOZTEWBUWRU - XKFKTEUWOUWRUXKXPWLZXQXOUWFUWMUUGUWLOZTEUVIUWOUCUDFTUWKGUUGUWQUWFUWSUWRUW - KOZTEZUWMUWFITUWKWNUXBUXOUWSUWFBIUWJTUXCUWJUXCUWIUVMUXCUVDYSUXHUXFXRUXCGU - VLUXDUWFUVJWOEYQWOEUVLWOEUWEUWFUVJUVAXSYBUXEUVJYQXTXNWRXHXIXJUXIITUWRUWKX - KXNXOZUXLXQXOUWFUWMUUGUVPOUWNUXMFKQUVIUWOUCUVFUWKUVOGUUGUWQUXJUXPUWFUWSUW - RUVOOZUWTUXNFKZQZUWMUWSUWFUXBUXSUXIUWFUXBWBGGUWRJKZLKZUVJMKZGUVJUWRJKZLKZ - JKZUYAUVAMKZGUVAUWRJKZLKZJKZUYHUYAFKZUYDJKZFKZUXQUXRUWRUVAYAUXBUXQUYEQUWF - BUWRUVNUYEIUVOBUCUTZUVKUYBUVMUYDJUYMYSUYAUVJMUYMYRUXTGLYQUWRGJUJULZYCUYMU - VLUYCGLYQUWRUVJJUJULZSUVOYDUYBUYDJYMYEWLUXBUXRUYLQUWFUXBUWTUYIUXNUYKFBUWR - UVEUYIIUVFUYMUVBUYFUVDUYHJUYMYSUYAUVAMUYNYCUYMUVCUYGGLYQUWRUVAJUJULZSUVFY - DUYFUYHJYMYEBUWRUWJUYKIUWKUYMUWIUYJUVMUYDJUYMUVDUYHYSUYAFUYPUYNSUYOSUWKYD - UYJUYDJYMYESWLYFYGXOYHXOYIUWFUVQUWHQUVIUVAYJWGYKYLYNYO $. - $} + $( Define the syntax typecode function. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-msyn $a |- mSyn = Slot 6 $. - ${ - $d A k x $. - $( An infinite product expression for factorial. (Contributed by Scott - Fenton, 15-Dec-2017.) $) - iprodfac $p |- ( A e. NN0 -> ( ! ` A ) = - prod_ k e. NN ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) ) $= - ( vx cn0 wcel cn c1 cv cdiv co caddc cexp cprod cfa cfv cmpt oveq2 oveq2d - nnuz crp 1zzd facne0 eqid faclim wceq oveq1d oveq12d ovex fvmpt adantl wa - weq 1rp a1i simpr nnrpd rpreccld rpaddcld nn0z adantr rpexpcld nn0nndivcl - cz recnd addcomd nn0ge0div ge0p1rpd eqeltrd rpdivcld rpcnd iprodn0 eqcomd - 1cnd ) ADEZFGGBHZIJZKJZALJZGAVOIJZKJZIJZBMANOZVNWABCFGGCHZIJZKJZALJZGAWCI - JZKJZIJZPZGWBFSVNUAAUBACWJWJUCZUDVOFEZVOWJOWAUEVNCVOWIWAFWJCBULZWFVRWHVTI - WMWEVQALWMWDVPGKWCVOGIQRUFWMWGVSGKWCVOAIQRUGWKVRVTIUHUIUJVNWLUKZWAWNVRVTW - NVQAWNGVPGTEWNUMUNWNVOWNVOVNWLUOUPUQURVNAVCEWLAUSUTVAWNVTVSGKJTWNGVSWNVMW - NVSAVOVBZVDVEWNVSWOAVOVFVGVHVIVJVKVL $. - $} + $( Define the syntax typecode function for expressions. (Contributed by + Mario Carneiro, 12-Jun-2023.) $) + df-mesyn $a |- mESyn = ( t e. _V |-> + ( c e. ( mTC ` t ) , e e. ( mREx ` t ) |-> + ( ( ( mSyn ` t ) ` c ) m0St e ) ) ) $. - ${ - $d a m $. $d a n $. $d k m $. $d k n $. $d M a $. $d m n $. $d M n $. - faclim2.1 $e |- F = - ( n e. NN |-> - ( ( ( ! ` n ) x. ( ( n + 1 ) ^ M ) ) / ( ! ` ( n + M ) ) ) ) $. - $( Another factorial limit due to Euler. (Contributed by Scott Fenton, - 17-Dec-2017.) $) - faclim2 $p |- ( M e. NN0 -> F ~~> 1 ) $= - ( wcel cn cfa cfv c1 caddc co cexp cmul cdiv cli wceq oveq2 oveq12d nncnd - cc0 va vm vk cn0 cmpt wbr oveq2d fveq2d mpteq2dv breq1d weq wtru cvv nnuz - cv 1zzd nnex mptex a1i 1cnd fveq2 oveq1 oveq1d fvoveq1 eqid ovex peano2nn - fvmpt exp0d nnnn0 faccl mulid1d eqtrd addid1d nnne0d dividd 3eqtrd adantl - syl nncn climconst mptru wa simpr nn0p1nn nnzd divcnvlin adantr cc nnnn0d - nnexpcl sylan ancoms nnmulcld nnred nnnn0addcl nndivred fmpttd ffvelcdmda - recnd adantlr nnaddcld simpl expp1d mulassd eqtr4d nn0addcld facp1 nn0cnd - addassd 3eqtr3d divmuldivd 3eqtr4d climmul 1t1e1 breqtrdi nn0ind eqbrtrid - ex ) CUDEBAFAUOZGHZXTIJKZCLKZMKZXTCJKZGHZNKZUEZIODAFYAYBUAUOZLKZMKZXTYIJK - ZGHZNKZUEZIOUFAFYAYBTLKZMKZXTTJKZGHZNKZUEZIOUFZAFYAYBUBUOZLKZMKZXTUUCJKZG - HZNKZUEZIOUFZAFYAYBUUCIJKZLKZMKZXTUUKJKZGHZNKZUEZIOUFZYHIOUFUAUBCYITPZYOU - UAIOUUSAFYNYTUUSYKYQYMYSNUUSYJYPYAMYITYBLQUGUUSYLYRGYITXTJQUHRUIUJUAUBUKZ - YOUUIIOUUTAFYNUUHUUTYKUUEYMUUGNUUTYJUUDYAMYIUUCYBLQUGUUTYLUUFGYIUUCXTJQUH - RUIUJYIUUKPZYOUUQIOUVAAFYNUUPUVAYKUUMYMUUONUVAYJUULYAMYIUUKYBLQUGUVAYLUUN - GYIUUKXTJQUHRUIUJYICPZYOYHIOUVBAFYNYGUVBYKYDYMYFNUVBYJYCYAMYICYBLQUGUVBYL - YEGYICXTJQUHRUIUJUUBULIUBUUAIUMFUNULUPUUAUMEULAFYTUQURUSULUTUUCFEZUUCUUAH - ZIPULUVCUVDUUCGHZUUKTLKZMKZUUCTJKZGHZNKZUVEUVENKIAUUCYTUVJFUUAAUBUKZYQUVG - YSUVINUVKYAUVEYPUVFMXTUUCGVAUVKYBUUKTLXTUUCIJVBVCRXTUUCTGJVDRUUAVEUVGUVIN - VFVHUVCUVGUVEUVIUVENUVCUVGUVEIMKUVEUVCUVFIUVEMUVCUUKUVCUUKUUCVGSVIUGUVCUV - EUVCUVEUVCUUCUDEZUVEFEUUCVJUUCVKVSZSZVLVMUVCUVHUUCGUVCUUCUUCVTVNUHRUVCUVE - UVNUVCUVEUVMVOVPVQVRWAWBUVLUUJUURUVLUUJWCZUUQIIMKIOUVOIIUCUUIAFYBUUNNKZUE - ZUUQIUMFUNUVOUPUVLUUJWDUUQUMEUVOAFUUPUQURUSUVLUVQIOUFUUJUVLIUUKUCUVQIUMFU - NUVLUPUVLUTUVLUUKUUCWEZWFUVQUMEUVLAFUVPUQURUSUCUOZFEZUVSUVQHZUVSIJKZUVSUU - KJKZNKZPUVLAUVSUVPUWDFUVQAUCUKZYBUWBUUNUWCNXTUVSIJVBZXTUVSUUKJVBRUVQVEUWB - UWCNVFVHZVRWGWHUVLUVTUVSUUIHZWIEUUJUVLFWIUVSUUIUVLAFUUHWIUVLXTFEZWCZUUHUW - JUUEUUGUWJUUEUWJYAUUDUWJXTUDEYAFEUWJXTUVLUWIWDZWJXTVKVSUWIUVLUUDFEZUWIYBF - EZUVLUWLXTVGZYBUUCWKWLWMWNWOUWJUUFUDEUUGFEUWJUUFUWIUVLUUFFEXTUUCWPWMWJUUF - VKVSWQWTWRWSXAUVLUVTUWAWIEUUJUVLFWIUVSUVQUVLAFUVPWIUWJUVPUWJYBUUNUWJYBUWI - UWMUVLUWNVRWOUWJXTUUKUWKUVLUUKFEZUWIUVRWHXBWQWTWRWSXAUVLUVTUVSUUQHZUWHUWA - MKZPUUJUVLUVTWCZUVSGHZUWBUUKLKZMKZUWCGHZNKZUWSUWBUUCLKZMKZUVSUUCJKZGHZNKZ - UWDMKZUWPUWQUWRUXCUXEUWBMKZUXGUWCMKZNKUXIUWRUXAUXJUXBUXKNUWRUXAUWSUXDUWBM - KZMKUXJUWRUWTUXLUWSMUWRUWBUUCUWRUWBUVTUWBFEZUVLUVSVGZVRSZUVLUVTXCZXDUGUWR - UWSUXDUWBUWRUWSUWRUVSUDEUWSFEUWRUVSUVLUVTWDZWJZUVSVKVSZSUWRUXDUVTUVLUXDFE - ZUVTUXMUVLUXTUXNUWBUUCWKWLWMZSUXOXEXFUWRUXFIJKZGHZUXGUYBMKZUXBUXKUWRUXFUD - EZUYCUYDPUWRUVSUUCUXRUXPXGZUXFXHVSUWRUYBUWCGUWRUVSUUCIUWRUVSUXQSUWRUUCUXP - XIUWRUTXJZUHUWRUYBUWCUXGMUYGUGXKRUWRUXEUXGUWBUWCUWRUXEUWRUWSUXDUXSUYAWNSU - WRUXGUWRUYEUXGFEUYFUXFVKVSZSUXOUWRUWCUWRUVSUUKUXQUVLUWOUVTUVRWHXBZSUWRUXG - UYHVOUWRUWCUYIVOXLXFUVTUWPUXCPUVLAUVSUUPUXCFUUQUWEUUMUXAUUOUXBNUWEYAUWSUU - LUWTMXTUVSGVAZUWEYBUWBUUKLUWFVCRXTUVSUUKGJVDRUUQVEUXAUXBNVFVHVRUVTUWQUXIP - UVLUVTUWHUXHUWAUWDMAUVSUUHUXHFUUIUWEUUEUXEUUGUXGNUWEYAUWSUUDUXDMUYJUWEYBU - WBUUCLUWFVCRXTUVSUUCGJVDRUUIVEUXEUXGNVFVHUWGRVRXMXAXNXOXPXSXQXR $. - $} + $( Define the set of grammatical formal systems. (Contributed by Mario + Carneiro, 12-Jun-2023.) $) + df-mgfs $a |- mGFS = { t e. mWGFS | + ( ( mSyn ` t ) : ( mTC ` t ) --> ( mVT ` t ) /\ + A. c e. ( mVT ` t ) ( ( mSyn ` t ) ` c ) = c /\ + A. d A. h A. a ( <. d , h , a >. e. ( mAx ` t ) -> + A. e e. ( h u. { a } ) + ( ( mESyn ` t ) ` e ) e. ( mPPSt ` t ) ) ) } $. + $( Define the set of proof trees. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-mtree $a |- mTree = ( t e. _V |-> + ( d e. ~P ( mDV ` t ) , h e. ~P ( mEx ` t ) |-> + |^| { r | ( A. e e. ran ( mVH ` t ) e r <. ( m0St ` e ) , (/) >. /\ + A. e e. h e r <. ( ( mStRed ` t ) ` <. d , h , e >. ) , (/) >. /\ + A. m A. o A. p ( <. m , o , p >. e. ( mAx ` t ) -> + A. s e. ran ( mSubst ` t ) ( A. x A. y ( x m y -> + ( ( ( mVars ` t ) ` ( s ` ( ( mVH ` t ) ` x ) ) ) X. + ( ( mVars ` t ) ` ( s ` ( ( mVH ` t ) ` y ) ) ) ) C_ d ) -> + ( { ( s ` p ) } X. X_ e e. ( o u. ( ( mVH ` t ) " + U. ( ( mVars ` t ) " ( o u. { p } ) ) ) ) ( r " { ( s ` e ) } ) ) C_ r + ) ) ) } ) ) $. -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Greatest common divisor and divisibility -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) + $( Define the function mapping syntax expressions to syntax trees. + (Contributed by Mario Carneiro, 14-Jul-2016.) $) + df-mst $a |- mST = ( t e. _V |-> + ( ( (/) ( mTree ` t ) (/) ) |` ( ( mEx ` t ) |` ( mVT ` t ) ) ) ) $. - $( Swap the second and third arguments of a gcd. (Contributed by Scott - Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $) - gcd32 $p |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> - ( ( A gcd B ) gcd C ) = ( ( A gcd C ) gcd B ) ) $= - ( cz wcel w3a cgcd co wceq gcdcom 3adant1 oveq2d gcdass 3com23 3eqtr4d ) AD - EZBDEZCDEZFZABCGHZGHACBGHZGHZABGHCGHACGHBGHZSTUAAGQRTUAIPBCJKLCBAMPRQUCUBIB - CAMNO $. + $( Define the indexing set for a syntax axiom's representation in a tree. + (Contributed by Mario Carneiro, 14-Jul-2016.) $) + df-msax $a |- mSAX = ( t e. _V |-> + ( p e. ( mSA ` t ) |-> ( ( mVH ` t ) " ( ( mVars ` t ) ` p ) ) ) ) $. + $} - $( Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) - (Revised by Mario Carneiro, 19-Apr-2014.) $) - gcdabsorb $p |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) gcd B ) = ( A gcd B - ) ) $= - ( cz wcel wa cgcd cabs cfv wceq gcdass 3anidm23 gcdid oveq2d adantl gcdabs2 - co 3eqtrd ) ACDZBCDZEABFPZBFPZABBFPZFPZABGHZFPZTRSUAUCIBBAJKSUCUEIRSUBUDAFB - LMNBAOQ $. + $( Define the set of unambiguous formal systems. (Contributed by Mario + Carneiro, 14-Jul-2016.) $) + df-mufs $a |- mUFS = { t e. mGFS | Fun ( mST ` t ) } $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Properties of relationships + Models of formal systems =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - ${ - brtp.1 $e |- X e. _V $. - brtp.2 $e |- Y e. _V $. - $( A condition for a binary relation over an unordered triple. - (Contributed by Scott Fenton, 8-Jun-2011.) $) - brtp $p |- ( X { <. A , B >. , <. C , D >. , <. E , F >. } Y <-> - ( ( X = A /\ Y = B ) \/ - ( X = C /\ Y = D ) \/ - ( X = E /\ Y = F ) ) ) $= - ( cop ctp wbr wcel wceq w3o wa df-br opex opth eltp 3orbi123i 3bitri ) GH - ABKZCDKZEFKZLZMGHKZUGNUHUDOZUHUEOZUHUFOZPGAOHBOQZGCOHDOQZGEOHFOQZPGHUGRUH - UDUEUFGHSUAUIULUJUMUKUNGHABIJTGHCDIJTGHEFIJTUBUC $. - $} - - ${ - $d A x y $. - dftr6.1 $e |- A e. _V $. - $( A potential definition of transitivity for sets. (Contributed by Scott - Fenton, 18-Mar-2012.) $) - dftr6 $p |- ( Tr A <-> A e. ( _V \ ran ( ( _E o. _E ) \ _E ) ) ) $= - ( vx vy wel cv wcel wa wi wal cep cdif wn cvv wbr wex epeli anbi12i exbii - 3bitri ccom crn wtr elrn brdif vex brco epel notbii 19.41v exanali bitr3i - bitri exnal con2bii dftr2 eldif mpbiran 3bitr4i ) CDEZDFZAGZHZCFZAGZIDJZC - JZAKKUAZKLZUBZGZMZAUCANVJLGZVKVGVKVDAVIOZCPVFMZCPVGMCAVIBUDVNVOCVNVDAVHOZ - VDAKOZMZHVCDPZVEMZHZVOVDAVHKUEVPVSVRVTVPVDVAKOZVAAKOZHZDPVSDVDAKKCUFBUGWD - VCDWBUTWCVBDVDUHVAABQRSUMVQVEVDABQUIRWAVCVTHDPVOVCVTDUJVCVEDUKULTSVFCUNTU - OCDAUPVMANGVLBANVJUQURUS $. - $} + $c mUV $. + $c mVL $. + $c mVSubst $. + $c mFresh $. + $c mFRel $. + $c mEval $. + $c mMdl $. + $c mUSyn $. + $c mGMdl $. + $c mItp $. + $c mFromItp $. - ${ - $d A x $. $d B x $. $d R x $. - coep.1 $e |- A e. _V $. - coep.2 $e |- B e. _V $. - $( Composition with the membership relation. (Contributed by Scott Fenton, - 18-Feb-2013.) $) - coep $p |- ( A ( _E o. R ) B <-> E. x e. B A R x ) $= - ( cv wbr cep wex wcel ccom wrex epeli anbi1ci exbii brco df-rex 3bitr4i - wa ) BAGZDHZUACIHZTZAJUACKZUBTZAJBCIDLHUBACMUDUFAUCUEUBUACFNOPABCIDEFQUBA - CRS $. + $( The universe of a model. $) + cmuv $a class mUV $. - $( Composition with the converse membership relation. (Contributed by - Scott Fenton, 18-Feb-2013.) $) - coepr $p |- ( A ( R o. `' _E ) B <-> E. x e. A x R B ) $= - ( cv cep ccnv wbr wa wex wcel ccom wrex vex brcnv epeli bitri anbi1i brco - exbii df-rex 3bitr4i ) BAGZHIZJZUECDJZKZALUEBMZUHKZALBCDUFNJUHABOUIUKAUGU - JUHUGUEBHJUJBUEHEAPQUEBERSTUBABCDUFEFUAUHABUCUD $. - $} + $( The set of valuations. $) + cmvl $a class mVL $. - ${ - $d R x y z $. $d A x y z $. - $( A quantifier-free definition of a well-founded relationship. - (Contributed by Scott Fenton, 11-Apr-2011.) $) - dffr5 $p |- ( R Fr A <-> - ( ~P A \ { (/) } ) C_ ran ( _E \ ( _E o. `' R ) ) ) $= - ( vx vz vy cv c0 cdif wcel cep wi wal wss wa wbr wrex anbi12i bitri vex - wn cpw csn ccnv ccom crn wne wral wfr eldif velpw necon3bbii wex wel epel - velsn brdif coep brcnv rexbii dfrex2 3bitrri con1bii exbii df-rex 3bitr4i - elrn imbi12i albii dfss2 df-fr 3bitr4ri ) CFZAUAZGUBZHZIZVLJJBUCZUDZHZUEZ - IZKZCLVLAMZVLGUFZNZDFZEFZBOZTDVLUGZEVLPZKZCLVOVTMABUHWBWKCVPWEWAWJVPVLVMI - ZVLVNIZTZNWEVLVMVNUIWLWCWNWDCAUJWMVLGCGUOUKQRWGVLVSOZEULECUMZWINZEULWAWJW - OWQEWOWGVLJOZWGVLVROZTZNWQWGVLJVRUPWRWPWTWICWGUNWIWSWSWGWFVQOZDVLPWHDVLPW - ITDWGVLVQESZCSZUQXAWHDVLWGWFBXBDSURUSWHDVLUTVAVBQRVCEVLVSXCVFWIEVLVDVEVGV - HCVOVTVICEDABVJVK $. - $} + $( Substitution for a valuation. $) + cmvsb $a class mVSubst $. - ${ - $d A x y $. $d R x y $. - $( Quantifier-free definition of a strict order. (Contributed by Scott - Fenton, 22-Feb-2013.) $) - dfso2 $p |- ( R Or A <-> - ( R Po A /\ ( A X. A ) C_ ( R u. ( _I u. `' R ) ) ) ) $= - ( vx vy wor wpo cv wbr wral wa cid cun wcel wi wal wo brun bitr2i 3bitr4i - vex weq w3o cxp ccnv wss df-so cop opelxp ideq brcnv orbi12i orbi2i df-br - 3orass imbi12i 2albii wrel wb relxp ssrel ax-mp r2al anbi2i bitr4i ) ABEA - BFZCGZDGZBHZCDUAZVGVFBHZUBZDAICAIZJVEAAUCZBKBUDZLZLZUEZJCDABUFVQVLVEVFVGU - GZVMMZVRVPMZNZDOCOZVFAMVGAMJZVKNZDOCOVQVLWAWDCDVSWCVTVKVFVGAAUHVKVFVGVPHZ - VTVHVIVJPZPVHVFVGVOHZPVKWEWFWGVHWGVFVGKHZVFVGVNHZPWFVFVGKVNQWHVIWIVJVFVGD - TZUIVFVGBCTWJUJUKRULVHVIVJUNVFVGBVOQSVFVGVPUMRUOUPVMUQVQWBURAAUSCDVMVPUTV - AVKCDAAVBSVCVD $. - $} + $( The freshness relation of a model. $) + cmfsh $a class mFresh $. - ${ - $d b ch $. $d e et $. $d a b c d e f g h p q P $. $d a ps $. $d g si $. - $d a b c d e f g h p q x A $. $d p q x ph $. $d a b c d e f g h x rh $. - $d a b c d e f g h p q x B $. $d a b c d e f g h p q x C $. $d d ta $. - $d a b c d e f g h p q x D $. $d a b c d e f g h p q x E $. $d c th $. - $d a b c d e f g h p q x F $. $d a b c d e f g h p q x G $. $d f ze $. - $d a b c d e f g h p q x H $. $d a b c d e f g h p q x S $. - $d a b c d e f g h x Q $. $d a b c d e f g h x X $. - br8.1 $e |- ( a = A -> ( ph <-> ps ) ) $. - br8.2 $e |- ( b = B -> ( ps <-> ch ) ) $. - br8.3 $e |- ( c = C -> ( ch <-> th ) ) $. - br8.4 $e |- ( d = D -> ( th <-> ta ) ) $. - br8.5 $e |- ( e = E -> ( ta <-> et ) ) $. - br8.6 $e |- ( f = F -> ( et <-> ze ) ) $. - br8.7 $e |- ( g = G -> ( ze <-> si ) ) $. - br8.8 $e |- ( h = H -> ( si <-> rh ) ) $. - br8.9 $e |- ( x = X -> P = Q ) $. - br8.10 $e |- R = { <. p , q >. | - E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P E. e e. P - E. f e. P E. g e. P E. h e. P ( p = <. <. a , b >. , <. c , d >. >. /\ - q = <. <. e , f >. , <. g , h >. >. /\ ph ) } $. - $( Substitution for an eight-place predicate. (Contributed by Scott - Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) $) - br8 $p |- ( ( ( X e. S /\ A e. Q /\ B e. Q ) /\ - ( C e. Q /\ D e. Q /\ E e. Q ) /\ - ( F e. Q /\ G e. Q /\ H e. Q ) ) -> - ( <. <. A , B >. , <. C , D >. >. R <. <. E , F >. , <. G , H >. >. <-> - rh ) ) $= - ( cop wbr wceq wrex wcel opex eqeq1 3anbi1d rexbidv 2rexbidv 3anbi2d brab - cv w3a wa wi wb opth vex sylan9bb eqcoms biimp3a a1i rexlimdva rexlimdvva - sylbi simpl11 simpl12 simpl13 simpl21 simpl22 simpl23 simpl31 eqidd simpr - simpl32 simpl33 opeq2d eqeq2d 3anbi23d rspc2ev syl113anc 3anbi13d rspc3ev - opeq1 opeq1d syl31anc rexeqdv rexeqbidv rspcev syl2anc ex impbid bitrid - opeq2 ) KLVDZMNVDZVDZUCUDVDZUEUFVDZVDZQVEYAUJVPZUKVPZVDZULVPZUMVPZVDZVDZV - FZYDSVPZTVPZVDZUAVPZUBVPZVDZVDZVFZAVQZUBOVGZUAOVGZTOVGZSOVGZUMOVGZULOVGZU - KOVGZUJOVGZJRVGZUGRVHZKPVHZLPVHZVQZMPVHZNPVHZUCPVHZVQZUDPVHZUEPVHZUFPVHZV - QZVQZIUIVPZYKVFZUHVPZYSVFZAVQZUBOVGZUAOVGTOVGZSOVGUMOVGZULOVGUKOVGZUJOVGJ - RVGYLUVGAVQZUBOVGZUAOVGTOVGZSOVGUMOVGZULOVGUKOVGZUJOVGJRVGUUJUIUHYAYDQXSX - TVIYBYCVIUVDYAVFZUVLUVQJUJROUVRUVKUVPUKULOOUVRUVJUVOUMSOOUVRUVIUVNTUAOOUV - RUVHUVMUBOUVRUVEYLUVGAUVDYAYKVJVKVLVMVMVMVMUVFYDVFZUVQUUHJUJROUVSUVPUUFUK - ULOOUVSUVOUUDUMSOOUVSUVNUUBTUAOOUVSUVMUUAUBOUVSUVGYTYLAUVFYDYSVJVNVLVMVMV - MVMVCVOUVCUUJIUVCUUHIJUJROUVCJVPZRVHYEOVHVRVRZUUFIUKULOOUWAYFOVHYHOVHVRVR - ZUUDIUMSOOUWBYIOVHYMOVHVRVRZUUBITUAOOUWCYNOVHYPOVHVRVRZUUAIUBOUUAIVSUWDYQ - OVHVRYLYTAIYLAEYTIAEVTZYKYAYKYAVFYGXSVFZYJXTVFZVRUWEYGYJXSXTYEYFVIYHYIVIW - AUWFACUWGEUWFYEKVFZYFLVFZVRACVTYEYFKLUJWBUKWBWAUWHABUWICUNUOWCWIUWGYHMVFZ - YINVFZVRCEVTYHYIMNULWBUMWBWAUWJCDUWKEUPUQWCWIWCWIWDEIVTZYSYDYSYDVFYOYBVFZ - YRYCVFZVRUWLYOYRYBYCYMYNVIYPYQVIWAUWMEGUWNIUWMYMUCVFZYNUDVFZVREGVTYMYNUCU - DSWBTWBWAUWOEFUWPGURUSWCWIUWNYPUEVFZYQUFVFZVRGIVTYPYQUEUFUAWBUBWBWAUWQGHU - WRIUTVAWCWIWCWIWDWCWEWFWGWHWHWHWHUVCIUUJUVCIVRZUUKUUAUBPVGZUAPVGZTPVGZSPV - GZUMPVGZULPVGZUKPVGZUJPVGZUUJUUKUULUUMUURUVBIWJUWSUULUUMUUOYAXSMYIVDZVDZV - FZYTDVQZUBPVGZUAPVGZTPVGZSPVGUMPVGZUXGUUKUULUUMUURUVBIWKUUKUULUUMUURUVBIW - LUUOUUPUUQUUNUVBIWMUWSUUPUUQUUSYAYAVFZYDYBYRVDZVFZGVQZUBPVGUAPVGZUXOUUOUU - PUUQUUNUVBIWNUUOUUPUUQUUNUVBIWOUUSUUTUVAUUNUURIWPUWSUUTUVAUXPYDYDVFZIUXTU - USUUTUVAUUNUURIWSUUSUUTUVAUUNUURIWTUWSYAWQUWSYDWQUVCIWRUXSUXPUYAIVQUXPYDY - BUEYQVDZVDZVFZHVQUAUBUEUFPPUWQUXRUYDGHUXPUWQUXQUYCYDUWQYRUYBYBYPUEYQXHXAX - BUTXCUWRUYDUYAHIUXPUWRUYCYDYDUWRUYBYCYBYQUFUEXRXAXBVAXCXDXEUXMUXTUXPYTEVQ - ZUBPVGUAPVGUXPYDUCYNVDZYRVDZVFZFVQZUBPVGUAPVGUMSTNUCUDPPPUWKUXKUYEUAUBPPU - WKUXJUXPDEYTUWKUXIYAYAUWKUXHXTXSYINMXRXAXBUQXFVMUWOUYEUYIUAUBPPUWOYTUYHEF - UXPUWOYSUYGYDUWOYOUYFYRYMUCYNXHXIXBURXCVMUWPUYIUXSUAUBPPUWPUYHUXRFGUXPUWP - UYGUXQYDUWPUYFYBYRYNUDUCXRXIXBUSXCVMXGXJUXDUXOYAKYFVDZYJVDZVFZYTBVQZUBPVG - ZUAPVGTPVGZSPVGUMPVGYAXSYJVDZVFZYTCVQZUBPVGZUAPVGTPVGZSPVGUMPVGUJUKULKLMP - PPUWHUXBUYOUMSPPUWHUWTUYNTUAPPUWHUUAUYMUBPUWHYLUYLABYTUWHYKUYKYAUWHYGUYJY - JYEKYFXHXIXBUNXFVLVMVMUWIUYOUYTUMSPPUWIUYNUYSTUAPPUWIUYMUYRUBPUWIUYLUYQBC - YTUWIUYKUYPYAUWIUYJXSYJYFLKXRXIXBUOXFVLVMVMUWJUYTUXNUMSPPUWJUYSUXLTUAPPUW - JUYRUXKUBPUWJUYQUXJCDYTUWJUYPUXIYAUWJYJUXHXSYHMYIXHXAXBUPXFVLVMVMXGXJUUIU - XGJUGRUVTUGVFZUUHUXFUJOPVBVUAUUGUXEUKOPVBVUAUUFUXDULOPVBVUAUUEUXCUMOPVBVU - AUUDUXBSOPVBVUAUUCUXATOPVBVUAUUBUWTUAOPVBVUAUUAUBOPVBXKXLXLXLXLXLXLXLXMXN - XOXPXQ $. - $} + $( The set of freshness relations. $) + cmfr $a class mFRel $. - ${ - $d b ch $. $d e et $. $d a b c d e f p q P $. $d p q x ph $. $d a ps $. - $d a b c d e f p q x A $. $d a b c d e f p q x B $. $d a b c d e f x Q $. - $d a b c d e f p q x C $. $d a b c d e f p q x D $. $d a b c d e f x X $. - $d a b c d e f p q x E $. $d d ta $. $d c th $. $d a b c d e f x ze $. - $d a b c d e f p q x F $. $d a b c d e f p q x S $. - br6.1 $e |- ( a = A -> ( ph <-> ps ) ) $. - br6.2 $e |- ( b = B -> ( ps <-> ch ) ) $. - br6.3 $e |- ( c = C -> ( ch <-> th ) ) $. - br6.4 $e |- ( d = D -> ( th <-> ta ) ) $. - br6.5 $e |- ( e = E -> ( ta <-> et ) ) $. - br6.6 $e |- ( f = F -> ( et <-> ze ) ) $. - br6.7 $e |- ( x = X -> P = Q ) $. - br6.8 $e |- R = { <. p , q >. | - E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P E. e e. P - E. f e. P ( p = <. a , <. b , c >. >. /\ - q = <. d , <. e , f >. >. /\ ph ) } $. - $( Substitution for a six-place predicate. (Contributed by Scott Fenton, - 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) $) - br6 $p |- ( ( X e. S /\ - ( A e. Q /\ B e. Q /\ C e. Q ) /\ - ( D e. Q /\ E e. Q /\ F e. Q ) ) -> - ( <. A , <. B , C >. >. R <. D , <. E , F >. >. <-> ze ) ) $= - ( cop wbr cv wceq w3a wrex wcel opex eqeq1 eqcom 3anbi1d rexbidv 2rexbidv - bitrdi 3anbi2d brab wa wi wb vex opth sylan9bb sylbi rexlimdva rexlimdvva - biimp3a a1i simpl1 simpl2 opeq1 eqeq1d 3anbi23d opeq2d eqid pm3.2i df-3an - opeq2 mpbiran rspc3ev 3ad2antl3 3anbi13d syl2anc rexeqdv rexeqbidv rspcev - ex impbid bitrid ) IJKUPZUPZLSTUPZUPZOUQUDURZUEURZUFURZUPZUPZXEUSZUGURZQU - RZRURZUPZUPZXGUSZAUTZRMVAZQMVAZUGMVAZUFMVAZUEMVAZUDMVAZHPVAZUAPVBZINVBJNV - BKNVBUTZLNVBSNVBTNVBUTZUTZGUCURZXLUSZUBURZXRUSZAUTZRMVAZQMVAUGMVAZUFMVAUE - MVAZUDMVAHPVAXMYOAUTZRMVAZQMVAUGMVAZUFMVAUEMVAZUDMVAHPVAYGUCUBXEXGOIXDVCL - XFVCYLXEUSZYSUUCHUDPMUUDYRUUBUEUFMMUUDYQUUAUGQMMUUDYPYTRMUUDYMXMYOAUUDYMX - EXLUSXMYLXEXLVDXEXLVEVIVFVGVHVHVHYNXGUSZUUCYEHUDPMUUEUUBYCUEUFMMUUEUUAYAU - GQMMUUEYTXTRMUUEYOXSXMAUUEYOXGXRUSXSYNXGXRVDXGXRVEVIVJVGVHVHVHUOVKYKYGGYK - YEGHUDPMYKHURZPVBXHMVBVLVLZYCGUEUFMMUUGXIMVBXJMVBVLVLZYAGUGQMMUUHXNMVBXOM - VBVLVLZXTGRMXTGVMUUIXPMVBVLXMXSAGXMADXSGXMXHIUSZXKXDUSZVLADVNXHXKIXDUDVOX - IXJVCVPUUJABUUKDUHUUKXIJUSZXJKUSZVLBDVNXIXJJKUEVOUFVOVPUULBCUUMDUIUJVQVRV - QVRXSXNLUSZXQXFUSZVLDGVNXNXQLXFUGVOXOXPVCVPUUNDEUUOGUKUUOXOSUSZXPTUSZVLEG - VNXOXPSTQVORVOVPUUPEFUUQGULUMVQVRVQVRVQWAWBVSVTVTVTYKGYGYKGVLZYHXTRNVAZQN - VAZUGNVAZUFNVAZUENVAZUDNVAZYGYHYIYJGWCUURYIXEXEUSZXSDUTZRNVAZQNVAUGNVAZUV - DYHYIYJGWDYJYHGUVHYIUVFGUVELXQUPZXGUSZEUTUVELSXPUPZUPZXGUSZFUTZUGQRLSTNNN - UUNXSUVJDEUVEUUNXRUVIXGXNLXQWEWFUKWGUUPUVJUVMEFUVEUUPUVIUVLXGUUPXQUVKLXOS - XPWEWHWFULWGUUQUVNUVEXGXGUSZGUTZGUUQUVMUVOFGUVEUUQUVLXGXGUUQUVKXFLXPTSWLW - HWFUMWGUVPUVEUVOVLGUVEUVOXEWIXGWIWJUVEUVOGWKWMVIWNWOUVAUVHIXKUPZXEUSZXSBU - TZRNVAZQNVAUGNVAIJXJUPZUPZXEUSZXSCUTZRNVAZQNVAUGNVAUDUEUFIJKNNNUUJUUSUVTU - GQNNUUJXTUVSRNUUJXMUVRABXSUUJXLUVQXEXHIXKWEWFUHWPVGVHUULUVTUWEUGQNNUULUVS - UWDRNUULUVRUWCBCXSUULUVQUWBXEUULXKUWAIXIJXJWEWHWFUIWPVGVHUUMUWEUVGUGQNNUU - MUWDUVFRNUUMUWCUVECDXSUUMUWBXEXEUUMUWAXDIXJKJWLWHWFUJWPVGVHWNWQYFUVDHUAPU - UFUAUSZYEUVCUDMNUNUWFYDUVBUEMNUNUWFYCUVAUFMNUNUWFYBUUTUGMNUNUWFYAUUSQMNUN - UWFXTRMNUNWRWSWSWSWSWSWTWQXAXBXC $. - $} + $( The evaluation function of a model. $) + cmevl $a class mEval $. - ${ - $d a b c d p q x A $. $d a b c d p q x B $. $d b ch $. $d a b c d x Q $. - $d a b c d p q x C $. $d a b c d p q x D $. $d a ps $. $d a b c d x X $. - $d a b c d p q P $. $d a b c d p q x S $. $d a b c d x ta $. $d c th $. - $d p q x ph $. - br4.1 $e |- ( a = A -> ( ph <-> ps ) ) $. - br4.2 $e |- ( b = B -> ( ps <-> ch ) ) $. - br4.3 $e |- ( c = C -> ( ch <-> th ) ) $. - br4.4 $e |- ( d = D -> ( th <-> ta ) ) $. - br4.5 $e |- ( x = X -> P = Q ) $. - br4.6 $e |- R = { <. p , q >. | - E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P - ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) } $. - $( Substitution for a four-place predicate. (Contributed by Scott Fenton, - 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.) $) - br4 $p |- ( ( X e. S /\ - ( A e. Q /\ B e. Q ) /\ - ( C e. Q /\ D e. Q ) ) -> - ( <. A , B >. R <. C , D >. <-> ta ) ) $= - ( cop wbr cv wceq w3a wrex wcel wa eqeq1 3anbi1d rexbidv 2rexbidv 3anbi2d - opex brab wi wb vex opth sylan9bb eqcoms biimp3a a1i rexlimdva rexlimdvva - sylbi simpl1 simpl2l simpl2r simpl3l simpl3r eqidd simpr 3anbi23d rspc2ev - opeq1 eqeq2d opeq2 syl113anc 3anbi13d syl3anc rexeqdv rexeqbidv rspcev ex - syl2anc impbid bitrid ) GHUHZIJUHZMUIWPRUJZSUJZUHZUKZWQTUJZUAUJZUHZUKZAUL - ZUAKUMZTKUMZSKUMZRKUMZFNUMZONUNZGLUNZHLUNZUOZILUNZJLUNZUOZULZEQUJZWTUKZPU - JZXDUKZAULZUAKUMZTKUMSKUMZRKUMFNUMXAYCAULZUAKUMZTKUMSKUMZRKUMFNUMXKQPWPWQ - MGHVAIJVAXTWPUKZYFYIFRNKYJYEYHSTKKYJYDYGUAKYJYAXAYCAXTWPWTUPUQURUSUSYBWQU - KZYIXIFRNKYKYHXGSTKKYKYGXFUAKYKYCXEXAAYBWQXDUPUTURUSUSUGVBXSXKEXSXIEFRNKX - SFUJZNUNWRKUNUOUOZXGESTKKYMWSKUNXBKUNUOUOZXFEUAKXFEVCYNXCKUNUOXAXEAEXAACX - EEACVDZWTWPWTWPUKWRGUKZWSHUKZUOYOWRWSGHRVESVEVFYPABYQCUBUCVGVMVHCEVDZXDWQ - XDWQUKXBIUKZXCJUKZUOYRXBXCIJTVEUAVEVFYSCDYTEUDUEVGVMVHVGVIVJVKVLVLXSEXKXS - EUOZXLXFUALUMZTLUMZSLUMZRLUMZXKXLXOXREVNUUAXMXNWPWPUKZXECULZUALUMTLUMZUUE - XMXNXLXREVOXMXNXLXREVPUUAXPXQUUFWQWQUKZEUUHXPXQXLXOEVQXPXQXLXOEVRUUAWPVSU - UAWQVSXSEVTUUGUUFUUIEULUUFWQIXCUHZUKZDULTUAIJLLYSXEUUKCDUUFYSXDUUJWQXBIXC - WCWDUDWAYTUUKUUIDEUUFYTUUJWQWQXCJIWEWDUEWAWBWFUUCUUHWPGWSUHZUKZXEBULZUALU - MTLUMRSGHLLYPXFUUNTUALLYPXAUUMABXEYPWTUULWPWRGWSWCWDUBWGUSYQUUNUUGTUALLYQ - UUMUUFBCXEYQUULWPWPWSHGWEWDUCWGUSWBWHXJUUEFONYLOUKZXIUUDRKLUFUUOXHUUCSKLU - FUUOXGUUBTKLUFUUOXFUAKLUFWIWJWJWJWKWMWLWNWO $. - $} + $( The set of models. $) + cmdl $a class mMdl $. - ${ - $d A x y z $. $d B x y z $. - $( Another distributive law of converse over class composition. - (Contributed by Scott Fenton, 3-May-2014.) $) - cnvco1 $p |- `' ( `' A o. B ) = ( `' B o. A ) $= - ( vx vy vz ccnv ccom relcnv relco cv wbr wa wex cop wcel vex brcnv bicomi - anbi12ci opelco exbii opelcnv bitri 3bitr4i eqrelriiv ) CDAFZBGZFZBFZAGZU - GHUIAIDJZEJZBKZULCJZUFKZLZEMZUNULAKZULUKUIKZLZEMUNUKNZUHOZVAUJOUPUTEUMUSU - OURUSUMULUKBEPZDPZQRULUNAVCCPZQSUAVBUKUNNUGOUQUNUKUGVEVDUBEUKUNUFBVDVETUC - EUNUKUIAVEVDTUDUE $. + $( The syntax function applied to elements of the model. $) + cusyn $a class mUSyn $. - $( Another distributive law of converse over class composition. - (Contributed by Scott Fenton, 3-May-2014.) $) - cnvco2 $p |- `' ( A o. `' B ) = ( B o. `' A ) $= - ( vx vy vz ccnv ccom relcnv relco cv wbr wa wex cop wcel vex brcnv bicomi - anbi12ci opelco exbii opelcnv bitri 3bitr4i eqrelriiv ) CDABFZGZFZBAFZGZU - GHBUIIDJZEJZUFKZULCJZAKZLZEMZUNULUIKZULUKBKZLZEMUNUKNZUHOZVAUJOUPUTEUMUSU - OURUKULBDPZEPZQURUOUNULACPZVDQRSUAVBUKUNNUGOUQUNUKUGVEVCUBEUKUNAUFVCVETUC - EUNUKBUIVEVCTUDUE $. - $} + $( The set of models in a grammatical formal system. $) + cgmdl $a class mGMdl $. - ${ - $d A x y z p $. $d B x y z p $. - $( Quantifier-free definition of membership in a domain. (Contributed by - Scott Fenton, 21-Jan-2017.) $) - eldm3 $p |- ( A e. dom B <-> ( B |` { A } ) =/= (/) ) $= - ( vx vy vp vz cdm wcel cvv csn cres c0 wne wceq cv eleq1 cop wex wa exbii - elex wn snprc reseq2 res0 eqtrdi sylbi necon1ai reseq2d neeq1d dfclel vex - sneq eldm2 n0 wrex elres pm5.32i opeq1 eqeq2d anbi1d bitr3id exbidv rexsn - weq bitri excom 3bitri 3bitr4i vtoclbg pm5.21nii ) ABGZHZAIHZBAJZKZLMZAVL - UAVNVPLVNUBVOLNZVPLNAUCVRVPBLKLVOLBUDBUEUFUGUHCOZVLHZBVSJZKZLMZVMVQCAIVSA - VLPVSANZWBVPLWDWAVOBVSAUMUIUJVSDOZQZBHZDREOZWFNZWHBHZSZERZDRZVTWCWGWLDEWF - BUKTDVSBCULZUNWCWHWBHZERWKDRZERWMEWBUOWOWPEWOWHFOZWEQZNZWRBHZSZDRZFWAUPWP - FDWHBWAUQXBWPFVSWNFCVEZXAWKDXAWSWJSXCWKWSWJWTWHWRBPURXCWSWIWJXCWRWFWHWQVS - WEUSUTVAVBVCVDVFTWKEDVGVHVIVJVK $. - $} + $( The interpretation function of the model. $) + cmitp $a class mItp $. - $( Quantifier-free definition of membership in a range. (Contributed by - Scott Fenton, 21-Jan-2017.) $) - elrn3 $p |- ( A e. ran B <-> ( B i^i ( _V X. { A } ) ) =/= (/) ) $= - ( crn wcel ccnv cdm csn cres wne cvv cxp cin df-rn eleq2i eldm3 wceq ineq2i - c0 cnvxp cnvin df-res 3eqtr4ri eqeq1i wrel wb cnveq0 ax-mp bitr4i necon3bii - relinxp 3bitri ) ABCZDABEZFZDUMAGZHZRIBJUOKZLZRIULUNABMNAUMOUPRURRUPRPUREZR - PZURRPZUPUSRUMUQEZLUMUOJKZLUSUPVBVCUMJUOSQBUQTUMUOUAUBUCURUDVAUTUEJUOBUJURU - FUGUHUIUK $. + $( The evaluation function derived from the interpretation. $) + cmfitp $a class mFromItp $. - ${ - $d R x y z $. $d A x y z $. + $( Define the universe of a model. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-muv $a |- mUV = Slot 7 $. - $( The converse of a partial ordering is still a partial ordering. - (Contributed by Scott Fenton, 13-Jun-2018.) $) - pocnv $p |- ( R Po A -> `' R Po A ) $= - ( vx vy vz wpo ccnv cv wcel wa wbr poirr vex brcnv sylnibr wi 3anrev potr - w3a sylan2b anbi12ci 3imtr4g ispod ) ABFZCDEABGZUDCHZAIZJUFUFBKUFUFUEKAUF - BLUFUFBCMZUHNOUDUGDHZAIZEHZAIZSZJUKUIBKZUIUFBKZJZUKUFBKZUFUIUEKZUIUKUEKZJ - UFUKUEKUMUDULUJUGSUPUQPUGUJULQAUKUIUFBRTURUOUSUNUFUIBUHDMZNUIUKBUTEMZNUAU - FUKBUHVANUBUC $. + $( Define the freshness relation of a model. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-mfsh $a |- mFresh = Slot ; 1 9 $. - $( The converse of a strict ordering is still a strict ordering. - (Contributed by Scott Fenton, 13-Jun-2018.) $) - socnv $p |- ( R Or A -> `' R Or A ) $= - ( wor ccnv cnvso biimpi ) ABCABDCABEF $. - $} + $( Define the evaluation function of a model. (Contributed by Mario + Carneiro, 14-Jul-2016.) $) + df-mevl $a |- mEval = Slot ; 2 0 $. ${ - sotrd.1 $e |- ( ph -> R Or A ) $. - sotrd.2 $e |- ( ph -> X e. A ) $. - sotrd.3 $e |- ( ph -> Y e. A ) $. - sotrd.4 $e |- ( ph -> Z e. A ) $. - sotrd.5 $e |- ( ph -> X R Y ) $. - sotrd.6 $e |- ( ph -> Y R Z ) $. - $( Transitivity law for strict orderings, deduction form. (Contributed by - Scott Fenton, 24-Nov-2021.) $) - sotrd $p |- ( ph -> X R Z ) $= - ( wbr wor wcel wa wi sotr syl13anc mp2and ) ADECMZEFCMZDFCMZKLABCNDBOEBOF - BOUAUBPUCQGHIJBDEFCRST $. - $} + $d a c d e f g h i m n p r s t u v w x y z $. + $( Define the set of valuations. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-mvl $a |- mVL = ( t e. _V |-> X_ v e. ( mVR ` t ) + ( ( mUV ` t ) " { ( ( mType ` t ) ` v ) } ) ) $. - $( Transitivity law for strict orderings. (Contributed by Scott Fenton, - 24-Nov-2021.) $) - sotr3 $p |- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> - ( ( X R Y /\ -. Z R Y ) -> X R Z ) ) $= - ( wor wcel w3a wa wbr wn wceq wo wb simp3 simp2 jca sotric sylan2 con2bid - adantr wi breq2 biimprcd adantl sotr expdimp jaod sylbird expimpd ) ABFZCAG - ZDAGZEAGZHZIZCDBJZEDBJZKZCEBJZUPUQIZUSEDLZDEBJZMZUTUPVDUSNUQUPURVDUOUKUNUMI - URVDKNUOUNUMULUMUNOULUMUNPQAEDBRSTUAVAVBUTVCUQVBUTUBUPVBUTUQEDCBUCUDUEUPUQV - CUTACDEBUFUGUHUIUJ $. + $( Define substitution applied to a valuation. (Contributed by Mario + Carneiro, 14-Jul-2016.) $) + df-mvsb $a |- mVSubst = ( t e. _V |-> { <. <. s , m >. , x >. | + ( ( s e. ran ( mSubst ` t ) /\ m e. ( mVL ` t ) ) /\ + A. v e. ( mVR ` t ) m dom ( mEval ` t ) ( s ` ( ( mVH ` t ) ` v ) ) /\ + x = ( v e. ( mVR ` t ) |-> + ( m ( mEval ` t ) ( s ` ( ( mVH ` t ) ` v ) ) ) ) ) } ) $. - $( Trichotomy law for strict orderings. (Contributed by Scott Fenton, - 8-Dec-2021.) $) - sotrine $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> - ( B =/= C <-> ( B R C \/ C R B ) ) ) $= - ( wor wcel wa wbr wo wceq wn sotrieq bicomd necon1abid ) ADEBAFCAFGGZBCDHCB - DHIZBCOBCJPKABCDLMN $. + $( Define the set of freshness relations. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-mfrel $a |- mFRel = ( t e. _V |-> + { r e. ~P ( ( mUV ` t ) X. ( mUV ` t ) ) | ( `' r = r /\ + A. c e. ( mVT ` t ) A. w e. ( ~P ( mUV ` t ) i^i Fin ) + E. v e. ( ( mUV ` t ) " { c } ) w C_ ( r " { v } ) ) } ) $. - ${ - $d F x y $. $d G x y $. $d X x y $. $d Y x y $. - $( Law for adjoining an element to restrictions of functions. (Contributed - by Scott Fenton, 6-Dec-2021.) $) - eqfunresadj $p |- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\ - ( Y e. dom F /\ Y e. dom G /\ ( F ` Y ) = ( G ` Y ) ) ) -> - ( F |` ( X u. { Y } ) ) = ( G |` ( X u. { Y } ) ) ) $= - ( vx vy wfun wa cres wceq cdm wcel cfv w3a cun relres cv wbr wo wb simp33 - breq 3ad2ant2 eqeq1d simp1l simp31 funbrfvb syl2anc simp1r simp32 3bitr3d - csn velsn breq1 bibi12d syl5ibrcom syl5bi pm5.32d 3bitr4g orbi12d resundi - vex brresi breqi brun bitri eqbrrdiv ) AGZBGZHZACIZBCIZJZDAKLZDBKLZDAMZDB - MZJZNZNZEFACDULZOZIZBWBIZAWBPBWBPVTEQZFQZVKRZWEWFAWAIZRZSZWEWFVLRZWEWFBWA - IZRZSZWEWFWCRZWEWFWDRZVTWGWKWIWMVMVJWGWKTVSWEWFVKVLUBUCVTWEWALZWEWFARZHWQ - WEWFBRZHWIWMVTWQWRWSWQWEDJZVTWRWSTZEDUMVTXAWTDWFARZDWFBRZTVTVPWFJZVQWFJZX - BXCVTVPVQWFVJVMVNVOVRUAUDVTVHVNXDXBTVHVIVMVSUEVJVMVNVOVRUFDWFAUGUHVTVIVOX - EXCTVHVIVMVSUIVJVMVNVOVRUJDWFBUGUHUKWTWRXBWSXCWEDWFAUNWEDWFBUNUOUPUQURWAW - EWFAFVBZVCWAWEWFBXFVCUSUTWOWEWFVKWHOZRWJWEWFWCXGACWAVAVDWEWFVKWHVEVFWPWEW - FVLWLOZRWNWEWFWDXHBCWAVAVDWEWFVLWLVEVFUSVG $. - $} + $( Define the set of models of a formal system. (Contributed by Mario + Carneiro, 14-Jul-2016.) $) + df-mdl $a |- mMdl = { t e. mFS | + [. ( mUV ` t ) / u ]. [. ( mEx ` t ) / x ]. + [. ( mVL ` t ) / v ]. [. ( mEval ` t ) / n ]. [. ( mFresh ` t ) / f ]. + ( ( u C_ ( ( mTC ` t ) X. _V ) /\ f e. ( mFRel ` t ) /\ + n e. ( u ^pm ( v X. ( mEx ` t ) ) ) ) /\ A. m e. v + ( ( A. e e. x ( n " { <. m , e >. } ) C_ ( u " { ( 1st ` e ) } ) /\ + A. y e. ( mVR ` t ) <. m , ( ( mVH ` t ) ` y ) >. n ( m ` y ) /\ + A. d A. h A. a ( <. d , h , a >. e. ( mAx ` t ) -> + ( ( A. y A. z ( y d z -> ( m ` y ) f ( m ` z ) ) /\ + h C_ ( dom n " { m } ) ) -> m dom n a ) ) ) /\ + ( A. s e. ran ( mSubst ` t ) A. e e. ( mEx ` t ) + A. y ( <. s , m >. ( mVSubst ` t ) y -> + ( n " { <. m , ( s ` e ) >. } ) = ( n " { <. y , e >. } ) ) /\ + A. p e. v A. e e. x + ( ( m |` ( ( mVars ` t ) ` e ) ) = ( p |` ( ( mVars ` t ) ` e ) ) -> + ( n " { <. m , e >. } ) = ( n " { <. p , e >. } ) ) /\ + A. y e. u A. e e. x ( ( m " ( ( mVars ` t ) ` e ) ) C_ ( f " { y } ) -> + ( n " { <. m , e >. } ) C_ ( f " { y } ) ) ) ) ) } $. - $( Law for equality of restriction to successors. This is primarily useful - when ` X ` is an ordinal, but it does not require that. (Contributed by - Scott Fenton, 6-Dec-2021.) $) - eqfunressuc $p |- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\ - ( X e. dom F /\ X e. dom G /\ ( F ` X ) = ( G ` X ) ) ) -> - ( F |` suc X ) = ( G |` suc X ) ) $= - ( wfun wa cres wceq cdm wcel cfv w3a csn eqfunresadj df-suc reseq2i 3eqtr4g - cun csuc ) ADBDEACFBCFGCAHICBHICAJCBJGKKACCLQZFBSFACRZFBTFABCCMTSACNZOTSBUA - OP $. + $( Define the syntax typecode function for the model universe. + (Contributed by Mario Carneiro, 14-Jul-2016.) $) + df-musyn $a |- mUSyn = ( t e. _V |-> ( v e. ( mUV ` t ) |-> + <. ( ( mSyn ` t ) ` ( 1st ` v ) ) , ( 2nd ` v ) >. ) ) $. - $( If ` (/) ` is not part of the range of a function ` F ` , then ` A ` is in - the domain of ` F ` iff ` ( F `` A ) =/= (/) ` . (Contributed by Scott - Fenton, 7-Dec-2021.) $) - funeldmb $p |- ( ( Fun F /\ -. (/) e. ran F ) -> - ( A e. dom F <-> ( F ` A ) =/= (/) ) ) $= - ( wfun c0 crn wcel wn wa cdm cfv wi fvelrn ex adantr wb eleq1 adantl sylibd - wceq con3d impancom ndmfv impbid1 necon2abid ) BCZDBEZFZGZHZABIFZABJZDUIUKD - SZUJGZUEULUHUMUEULHZUJUGUNUJUKUFFZUGUEUJUOKULUEUJUOABLMNULUOUGOUEUKDUFPQRTU - AABUBUCUD $. + $( Define the set of models of a grammatical formal system. (Contributed + by Mario Carneiro, 14-Jul-2016.) $) + df-gmdl $a |- mGMdl = { t e. ( mGFS i^i mMdl ) | + ( A. c e. ( mTC ` t ) ( ( mUV ` t ) " { c } ) C_ + ( ( mUV ` t ) " { ( ( mSyn ` t ) ` c ) } ) /\ + A. v e. ( mUV ` c ) A. w e. ( mUV ` c ) ( v ( mFresh ` t ) w <-> + v ( mFresh ` t ) ( ( mUSyn ` t ) ` w ) ) /\ + A. m e. ( mVL ` t ) A. e e. ( mEx ` t ) + ( ( mEval ` t ) " { <. m , e >. } ) = + ( ( ( mEval ` t ) " { <. m , ( ( mESyn ` t ) ` e ) >. } ) i^i + ( ( mUV ` t ) " { ( 1st ` e ) } ) ) ) } $. - ${ - $d A y z $. $d B y z $. $d F y z $. $d X y z $. - elintfv.1 $e |- X e. _V $. - $( Membership in an intersection of function values. (Contributed by Scott - Fenton, 9-Dec-2021.) $) - elintfv $p |- ( ( F Fn A /\ B C_ A ) -> - ( X e. |^| ( F " B ) <-> A. y e. B X e. ( F ` y ) ) ) $= - ( vz cima cint wcel cv wi wal wfn wss wa cfv wral elint wceq wrex r19.23v - imbi1d bitr4di albidv ralcom4 eqcom imbi1i albii fvex eleq2 ceqsalv bitri - fvelimab ralbii bitr3i bitrdi bitrid ) EDCHZIJGKZUSJZEUTJZLZGMZDBNCBOPZEA - KZDQZJZACRZGEUSFSVEVDVGUTTZVBLZACRZGMZVIVEVCVLGVEVCVJACUAZVBLVLVEVAVNVBAB - CUTDUNUCVJVBACUBUDUEVMVKGMZACRVIVKAGCUFVOVHACVOUTVGTZVBLZGMVHVKVQGVJVPVBV - GUTUGUHUIVBVHGVGVFDUJUTVGEUKULUMUOUPUQUR $. + $( Define the interpretation function for a model. (Contributed by Mario + Carneiro, 14-Jul-2016.) $) + df-mitp $a |- mItp = ( t e. _V |-> ( a e. ( mSA ` t ) |-> + ( g e. X_ i e. ( ( mVars ` t ) ` a ) + ( ( mUV ` t ) " { ( ( mType ` t ) ` i ) } ) |-> + ( iota x E. m e. ( mVL ` t ) ( g = ( m |` ( ( mVars ` t ) ` a ) ) /\ + x = ( m ( mEval ` t ) a ) ) ) ) ) ) $. + + $( Define a function that produces the evaluation function, given the + interpretation function for a model. (Contributed by Mario Carneiro, + 14-Jul-2016.) $) + df-mfitp $a |- mFromItp = ( t e. _V |-> ( f e. X_ a e. ( mSA ` t ) + ( ( ( mUV ` t ) " { ( ( 1st ` t ) ` a ) } ) ^m + X_ i e. ( ( mVars ` t ) ` a ) + ( ( mUV ` t ) " { ( ( mType ` t ) ` i ) } ) ) |-> + ( iota_ n e. ( ( mUV ` t ) ^pm ( ( mVL ` t ) X. ( mEx ` t ) ) ) + A. m e. ( mVL ` t ) + ( A. v e. ( mVR ` t ) <. m , ( ( mVH ` t ) ` v ) >. n ( m ` v ) /\ + A. e A. a A. g ( e ( mST ` t ) <. a , g >. -> + <. m , e >. n ( f ` ( i e. ( ( mVars ` t ) ` a ) |-> + ( m n ( g ` ( ( mVH ` t ) ` i ) ) ) ) ) ) /\ + A. e e. ( mEx ` t ) ( n " { <. m , e >. } ) = + ( ( n " { <. m , ( ( mESyn ` t ) ` e ) >. } ) i^i + ( ( mUV ` t ) " { ( 1st ` e ) } ) ) ) ) ) ) $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Properties of functions and mappings + Splitting fields =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $( A condition for subset trichotomy for functions. (Contributed by Scott - Fenton, 19-Apr-2011.) $) - funpsstri $p |- ( ( Fun H /\ ( F C_ H /\ G C_ H ) /\ - ( dom F C_ dom G \/ dom G C_ dom F ) ) -> - ( F C. G \/ F = G \/ G C. F ) ) $= - ( wfun wss wa cdm wo w3a wpss wceq w3o cres funssres anim12d ssres2 orim12i - wi ex sseq12 wb ancoms orbi12d syl5ib syl6 3imp sspsstri sylib ) CDZACEZBCE - ZFZAGZBGZEZUNUMEZHZIABEZBAEZHZABJABKBAJLUIULUQUTUIULCUMMZAKZCUNMZBKZFZUQUTR - UIUJVBUKVDUIUJVBCANSUIUKVDCBNSOUQVAVCEZVCVAEZHVEUTUOVFUPVGUMUNCPUNUMCPQVEVF - URVGUSVAAVCBTVDVBVGUSUAVCBVAATUBUCUDUEUFABUGUH $. + $( Introduce new constant symbols. $) + $c IntgRing $. $( Integral subring of a ring $) + $c cplMetSp $. $( Completion of a metric space $) + $c HomLimB $. $( Direct limit auxiliaries $) + $c HomLim $. $( Direct limit structure $) + $c polyFld $. $( Polynomial extension field $) + $c splitFld1 $. $( Splitting field for a polynomial auxiliary $) + $c splitFld $. $( Splitting field for a polynomial $) + $c polySplitLim $. $( Splitting field for a sequence of polynomials $) - ${ - $d F p x y z $. $d G p x y z $. - $( If a class ` F ` is a proper subset of a function ` G ` , then - ` dom F C. dom G ` . (Contributed by Scott Fenton, 20-Apr-2011.) $) - fundmpss $p |- ( Fun G -> ( F C. G -> dom F C. dom G ) ) $= - ( vx vy vz vp wpss wss wn wa wi syl cv wbr wex wcel adantl ex wal exlimdv - wfun cdm pssss dmss a1i cdif c0 wne pssdif n0 wrel funrel reldif cop wceq - sylib elrel eleq1 df-br bitr4di biimpcd 2eximdv difss ssbri eximi simprbi - mpd adantr brdif ssbrd ad2antlr weq dffun2 2sp breq2 biimprd syl6 simplbi - sps expd impel adantlr com23 mpdd mtod jcad eximdv nss vex notbii anbi12i - syld eldm exbii bitri sylibr dfpss3 syl6ibr ) BUAZABGZAUBZBUBZHZXBXAHIZJX - AXBGWSWTXCXDWTXCKWSWTABHXCABUCZABUDLUEWSWTXDWSWTJZCMZDMZBNZDOZXGEMZANZEOZ - IZJZCOZXDXFFMZBAUFZPZFOZXPWTXTWSWTXRUGUHXTABUIFXRUJUPQXFXSXPFXFXSXGXHXRNZ - DOZCOZXPWSXSYCKZWTWSXRUKZYDWSBUKZYEBULBAUMLYEXSYCYEXSJZXQXGXHUNZUOZDOCOYC - CDXQXRUQYGYIYACDXSYIYAKYEYIXSYAYIXSYHXRPYAXQYHXRURXGXHXRUSUTVAQVBVGRLVHXF - YBXOCXFYBXJXNYBXJKXFYAXIDXRBXGXHBAVCVDVEUEXFYAXNDXFYAXNXFYAJZXMXGXHANZYAY - KIZXFYAXIYLXGXHBAVIZVFQYJXLYKEYJXLXGXKBNZYKWTXLYNKWSYAWTABXGXKXEVJVKYJYNX - LYKWSYAYNXLYKKZKZWTWSXIYPYAWSXIYNYOWSXIYNJZDEVLZYOWSYQYRKZESDSZCSZYSWSYFU - UACDEBVMVFYTYSCYSDEVNVSLYRYKXLXHXKXGAVOVPVQVTYAXIYLYMVRWAWBWCWDTWERTWFWGW - LTVGXDXGXBPZXGXAPZIZJZCOXPCXBXAWHUUEXOCUUBXJUUDXNDXGBCWIZWMUUCXMEXGAUUFWM - WJWKWNWOWPRWFXAXBWQWR $. - $} + $( Integral subring of a ring. $) + citr $a class IntgRing $. - $( The value of a function at a restriction is either null or the same as the - function itself. (Contributed by Scott Fenton, 4-Sep-2011.) $) - fvresval $p |- ( ( ( F |` B ) ` A ) = ( F ` A ) \/ - ( ( F |` B ) ` A ) = (/) ) $= - ( wcel wn wo cres cfv wceq c0 exmid fvres nfvres orim12i ax-mp ) ABDZPEZFAC - BGHZACHIZRJIZFPKPSQTABCLABCMNO $. + $( Completion of a metric space. $) + ccpms $a class cplMetSp $. - $( Given two functions with equal domains, equality only requires one - direction of the subset relationship. (Contributed by Scott Fenton, - 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.) $) - funsseq $p |- ( ( Fun F /\ Fun G /\ dom F = dom G ) -> - ( F = G <-> F C_ G ) ) $= - ( wfun cdm wceq w3a eqimss wa cres simpl3 reseq2d funssres 3ad2antl2 simpl2 - wss wrel funrel resdm 3syl 3eqtr3d ex impbid2 ) ACZBCZADZBDZEZFZABEZABOZABG - UHUJUIUHUJHZBUEIZBUFIZABUKUEUFBUCUDUGUJJKUDUCUJULAEUGBALMUKUDBPUMBEUCUDUGUJ - NBQBRSTUAUB $. + $( Embeddings for a direct limit. $) + chlb $a class HomLimB $. - ${ - $d A x y z $. $d B x y z $. $d C x y z $. $d F x y z $. - fununiq.1 $e |- A e. _V $. - fununiq.2 $e |- B e. _V $. - fununiq.3 $e |- C e. _V $. - $( The uniqueness condition of functions. (Contributed by Scott Fenton, - 18-Feb-2013.) $) - fununiq $p |- ( Fun F -> ( ( A F B /\ A F C ) -> B = C ) ) $= - ( vx vy vz cv wbr wa wi wal wceq cvv wcel wb breq12 wfun wrel weq 3adant3 - dffun2 w3a 3adant2 anbi12d eqeq12 3adant1 imbi12d spc3gv mp3an simplbiim - ) DUADUBHKZIKZDLZUOJKZDLZMZIJUCZNZJOIOHOZABDLZACDLZMZBCPZNZHIJDUEAQRBQRCQ - RVCVHNEFGVBVHHIJABCQQQUOAPZUPBPZURCPZUFZUTVFVAVGVLUQVDUSVEVIVJUQVDSVKUOAU - PBDTUDVIVKUSVESVJUOAURCDTUGUHVJVKVAVGSVIUPBURCUIUJUKULUMUN $. - $} + $( Direct limit structure. $) + chlim $a class HomLim $. - ${ - funbreq.1 $e |- A e. _V $. - funbreq.2 $e |- B e. _V $. - funbreq.3 $e |- C e. _V $. - $( An equality condition for functions. (Contributed by Scott Fenton, - 18-Feb-2013.) $) - funbreq $p |- ( ( Fun F /\ A F B ) -> ( A F C <-> B = C ) ) $= - ( wfun wbr wa wceq fununiq expdimp wi breq2 biimpcd adantl impbid ) DHZAB - DIZJACDIZBCKZSTUAUBABCDEFGLMTUBUANSUBTUABCADOPQR $. - $} + $( Polynomial extension field. $) + cpfl $a class polyFld $. - ${ - br1steq.1 $e |- A e. _V $. - br1steq.2 $e |- B e. _V $. - $( Uniqueness condition for the binary relation ` 1st ` . (Contributed by - Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, - 3-May-2015.) $) - br1steq $p |- ( <. A , B >. 1st C <-> C = A ) $= - ( cvv wcel cop c1st wbr wceq wb br1steqg mp2an ) AFGBFGABHCIJCAKLDEABCFFM - N $. + $( Splitting field for a single polynomial (auxiliary). $) + csf1 $a class splitFld1 $. - $( Uniqueness condition for the binary relation ` 2nd ` . (Contributed by - Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, - 3-May-2015.) $) - br2ndeq $p |- ( <. A , B >. 2nd C <-> C = B ) $= - ( cvv wcel cop c2nd wbr wceq wb br2ndeqg mp2an ) AFGBFGABHCIJCBKLDEABCFFM - N $. - $} + $( Splitting field for a finite set of polynomials. $) + csf $a class splitFld $. + + $( Splitting field for a sequence of polynomials. $) + cpsl $a class polySplitLim $. ${ - $d A p x y z $. - $( Definition of domain in terms of ` 1st ` and image. (Contributed by - Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) - (Proof shortened by Peter Mazsa, 2-Oct-2022.) $) - dfdm5 $p |- dom A = ( ( 1st |` ( _V X. _V ) ) " A ) $= - ( vx vy vp vz c1st cvv cv cop wcel wex wbr wa wceq excom vex bitri anbi1i - exbii 3bitr4i cdm cxp cres cima breq1 eleq1 anbi12d br1steq equcom bitrdi - opex ceqsexv opeq1 eleq1d 3bitr3ri ancom anass brresi elvv 19.41vv elima2 - eldm2 eqriv ) BAUAZFGGUBZUCZAUDZBHZCHZIZAJZCKZDHZAJZVMVHVFLZMZDKZVHVDJVHV - GJVMEHZVIIZNZVMVHFLZVNMZMZEKZDKZCKWDCKZDKVLVQWDCDOVKWECWCDKZEKVRVHNZVSAJZ - MZEKWEVKWGWJEWBWJDVSVRVIUKVTWBVSVHFLZWIMWJVTWAWKVNWIVMVSVHFUEVMVSAUFUGWKW - HWIWKVHVRNWHVRVIVHEPCPUHBEUIQRUJULSWCEDOWIVKEVHBPZWHVSVJAVRVHVIUMUNULUOSV - PWFDVPVOVNMZWFVNVOUPVTEKCKZWAMZVNMWNWBMWMWFWNWAVNUQVOWOVNVOVMVEJZWAMWOVEV - MVHFWLURWPWNWAWPVTCKEKWNECVMUSVTECOQRQRVTWBCEUTTQSTCVHAWLVBDVHVFAWLVATVC - $. + $d b e f g h i j m n p q r s t v w x y z $. + $( Define the subring of elements of ` r ` integral over ` s ` in a ring. + (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-irng $a |- IntgRing = ( r e. _V , s e. _V |-> + U_ f e. ( Monic1p ` ( r |`s s ) ) ( `' f " { ( 0g ` r ) } ) ) $. - $( Definition of range in terms of ` 2nd ` and image. (Contributed by - Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) - (Proof shortened by Peter Mazsa, 2-Oct-2022.) $) - dfrn5 $p |- ran A = ( ( 2nd |` ( _V X. _V ) ) " A ) $= - ( vx vy vp vz c2nd cvv cv cop wcel wex wbr wa wceq excom vex bitri anbi1i - exbii 3bitr4i crn cxp cres cima breq1 eleq1 anbi12d br2ndeq equcom bitrdi - opex ceqsexv opeq2 eleq1d 3bitr3ri ancom anass brresi elvv 19.41vv elima2 - elrn2 eqriv ) BAUAZFGGUBZUCZAUDZCHZBHZIZAJZCKZDHZAJZVMVIVFLZMZDKZVIVDJVIV - GJVMVHEHZIZNZVMVIFLZVNMZMZEKZDKZCKWDCKZDKVLVQWDCDOVKWECWCDKZEKVRVINZVSAJZ - MZEKWEVKWGWJEWBWJDVSVHVRUKVTWBVSVIFLZWIMWJVTWAWKVNWIVMVSVIFUEVMVSAUFUGWKW - HWIWKVIVRNWHVHVRVICPEPUHBEUIQRUJULSWCEDOWIVKEVIBPZWHVSVJAVRVIVHUMUNULUOSV - PWFDVPVOVNMZWFVNVOUPVTEKCKZWAMZVNMWNWBMWMWFWNWAVNUQVOWOVNVOVMVEJZWAMWOVEV - MVIFWLURWPWNWACEVMUSRQRVTWBCEUTTQSTCVIAWLVBDVIVFAWLVATVC $. - $} + $( A function which completes the given metric space. (Contributed by + Mario Carneiro, 2-Dec-2014.) $) + df-cplmet $a |- cplMetSp = ( w e. _V |-> + [_ ( ( w ^s NN ) |`s ( Cau ` ( dist ` w ) ) ) / r ]_ + [_ ( Base ` r ) / v ]_ [_ { <. f , g >. | ( { f , g } C_ v /\ + A. x e. RR+ E. j e. ZZ ( f |` ( ZZ>= ` j ) ) : + ( ZZ>= ` j ) --> ( ( g ` j ) ( ball ` ( dist ` w ) ) x ) ) } / e ]_ + ( ( r /s e ) sSet { <. ( dist ` ndx ) , { <. <. x , y >. , z >. | + E. p e. v E. q e. v ( ( x = [ p ] e /\ y = [ q ] e ) /\ + ( p oF ( dist ` r ) q ) ~~> z ) } >. } ) ) $. - ${ - $d A z $. $d B z $. $d C z $. $d D z $. - $( Alternate way of saying that an ordered pair is in a composition. - (Contributed by Scott Fenton, 6-May-2018.) $) - opelco3 $p |- ( <. A , B >. e. ( C o. D ) <-> - B e. ( C " ( D " { A } ) ) ) $= - ( vz cop ccom wcel wbr csn cima df-br cvv wa relco wn c0 ima0 wex wb wceq - brrelex12i snprc noel imaeq2 imaeq2d eqtri eqtrdi eleq2d sylbi con4i elex - imaeq2i mtbiri jca wrex df-rex elimasng elvd bitr4di adantr anbi1d exbidv - cv bitr2id brcog elimag adantl 3bitr4d pm5.21nii bitr3i ) ABFCDGZHABVLIZB - CDAJZKZKZHZABVLLVMAMHZBMHZNZVQABVLCDOUBVQVRVSVRVQVRPVNQUAZVQPAUCWAVQBQHBU - DWAVPQBWAVPCDQKZKZQWAVOWBCVNQDUEUFWCCQKQWBQCDRUMCRUGUHUIUNUJUKBVPULUOVTAE - VDZDIZWDBCIZNZESZWFEVOUPZVMVQWIWDVOHZWFNZESVTWHWFEVOUQVTWKWGEVTWJWEWFVRWJ - WETVSVRWJAWDFDHZWEVRWJWLTEDAWDMMURUSAWDDLUTVAVBVCVEEABCDMMVFVSVQWITVREBCV - OMVGVHVIVJVK $. - $} + $( The input to this function is a sequence (on ` NN ` ) of homomorphisms + ` F ( n ) : R ( n ) --> R ( n + 1 ) ` . The resulting structure is the + direct limit of the direct system so defined. This function returns the + pair ` <. S , G >. ` where ` S ` is the terminal object and ` G ` is a + sequence of functions such that ` G ( n ) : R ( n ) --> S ` and + ` G ( n ) = F ( n ) o. G ( n + 1 ) ` . (Contributed by Mario Carneiro, + 2-Dec-2014.) $) + df-homlimb $a |- HomLimB = ( f e. _V |-> + [_ U_ n e. NN ( { n } X. dom ( f ` n ) ) / v ]_ + [_ |^| { s | ( s Er v /\ ( x e. v |-> <. ( ( 1st ` x ) + 1 ) , + ( ( f ` ( 1st ` x ) ) ` ( 2nd ` x ) ) >. ) C_ s ) } / e ]_ + <. ( v /. e ) , ( n e. NN |-> + ( x e. dom ( f ` n ) |-> [ <. n , x >. ] e ) ) >. ) $. - ${ - $d A x $. $d B p x y z $. $d R p x y z $. - $( Quantifier-free expression saying that a class is a member of an image. - (Contributed by Scott Fenton, 8-May-2018.) $) - elima4 $p |- ( A e. ( R " B ) <-> ( R i^i ( B X. { A } ) ) =/= (/) ) $= - ( vx vp vy vz wcel cvv csn cxp cin c0 wne wceq cv wex wa 3bitri exbii xp0 - cima elex xpeq2 eqtrdi ineq2d in0 necon3i snnzb sylibr sneq xpeq2d neeq1d - eleq1 cop elin ancom anbi1i anass 2exbii 19.41vv bitr3i exrot3 weq anbi2d - elxp opex ceqsexv an12 velsn vex opeq2 eleq1d n0 elima3 vtoclbg pm5.21nii - 3bitr4ri ) ACBUBZHZAIHZCBAJZKZLZMNZAVSUCWEWBMNWAWBMWDMWBMOZWDCMLMWFWCMCWF - WCBMKMWBMBUDBUAUEUFCUGUEUHAUIUJDPZVSHZCBWGJZKZLZMNZVTWEDAIWGAVSUNWGAOZWKW - DMWMWJWCCWMWIWBBWGAUKULUFUMEPZWKHZEQZFPZBHZWQWGUOZCHZRZFQZWLWHWPWNWQGPZUO - ZOZWRXCWIHZRZRZGQFQZWNCHZRZEQZXEXGXJRZRZEQZGQZFQZXBWOXKEWOXJWNWJHZRXRXJRX - KWNCWJUPXJXRUQXRXIXJFGWNBWIVFURSTXLXNGQFQZEQXQXSXKEXSXHXJRZGQFQXKXTXNFGXE - XGXJUSUTXHXJFGVAVBTXNEFGVCVBXPXAFXPXGXDCHZRZGQGDVDZWRYARZRZGQXAXOYBGXMYBE - XDWQXCVGXEXJYAXGWNXDCUNVEVHTYBYEGYBWRXFYARRXFYDRYEWRXFYAUSWRXFYAVIXFYCYDG - WGVJURSTYDXAGWGDVKZYCYAWTWRYCXDWSCXCWGWQVLVMVEVHSTSEWKVNFWGCBYFVOVRVPVQ - $. - $} + $( The input to this function is a sequence (on ` NN ` ) of structures + ` R ( n ) ` and homomorphisms ` F ( n ) : R ( n ) --> R ( n + 1 ) ` . + The resulting structure is the direct limit of the direct system so + defined, and maintains any structures that were present in the original + objects. TODO: generalize to directed sets? (Contributed by Mario + Carneiro, 2-Dec-2014.) $) + df-homlim $a |- HomLim = ( r e. _V , f e. _V |-> + [_ ( HomLimB ` f ) / e ]_ [_ ( 1st ` e ) / v ]_ [_ ( 2nd ` e ) / g ]_ + ( { <. ( Base ` ndx ) , v >. , + <. ( +g ` ndx ) , U_ n e. NN ran ( x e. dom ( g ` n ) , + y e. dom ( g ` n ) |-> <. <. ( ( g ` n ) ` x ) , ( ( g ` n ) ` y ) >. , + ( ( g ` n ) ` ( x ( +g ` ( r ` n ) ) y ) ) >. ) >. , + <. ( .r ` ndx ) , U_ n e. NN ran ( x e. dom ( g ` n ) , + y e. dom ( g ` n ) |-> <. <. ( ( g ` n ) ` x ) , ( ( g ` n ) ` y ) >. , + ( ( g ` n ) ` ( x ( .r ` ( r ` n ) ) y ) ) >. ) >. } u. + { <. ( TopOpen ` ndx ) , { s e. ~P v | + A. n e. NN ( `' ( g ` n ) " s ) e. ( TopOpen ` ( r ` n ) ) } >. , + <. ( dist ` ndx ) , U_ n e. NN ran ( x e. dom ( ( g ` n ) ` n ) , + y e. dom ( ( g ` n ) ` n ) |-> <. <. ( ( g ` n ) ` x ) , + ( ( g ` n ) ` y ) >. , ( x ( dist ` ( r ` n ) ) y ) >. ) >. , + <. ( le ` ndx ) , U_ n e. NN ( `' ( g ` n ) o. + ( ( le ` ( r ` n ) ) o. ( g ` n ) ) ) >. } ) ) $. - $( The value of the converse of ` 1st ` restricted to a singleton. - (Contributed by Scott Fenton, 2-Jul-2020.) $) - fv1stcnv $p |- ( ( X e. A /\ Y e. V ) -> - ( `' ( 1st |` ( A X. { Y } ) ) ` X ) = <. X , Y >. ) $= - ( wcel wa c1st csn cxp cres ccnv cfv cop wceq wbr wb cvv bitrd mpbird wf1o - snidg anim2i eqid jctir opex brcnvg mpan2 brres adantr opelxp anbi1i anbi2d - br1steqg bitrid wfn 1stconst f1ocnv f1ofn 3syl simpl fnbrfvb syl2an2 ) CAEZ - DBEZFZCGADHZIZJZKZLCDMZNZCVJVIOZVEVLVCDVFEZFZCCNZFZVEVNVOVDVMVCDBUAUBCUCUDV - EVLVJVGEZVJCGOZFZVPVCVLVSPVDVCVLVJCVHOZVSVCVJQEVLVTPCDUECVJAQVHUFUGVGVJCGAU - HRUIVSVNVRFVEVPVQVNVRCDAVFUJUKVEVRVOVNCDCABUMULUNRSVDVIAUOZVCVCVKVLPVDVGAVH - TAVGVITWAADBUPVGAVHUQAVGVIURUSVCVDUTACVJVIVAVBS $. + $( Define the field extension that augments a field with the root of the + given irreducible polynomial, and extends the norm if one exists and the + extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-plfl $a |- polyFld = ( r e. _V , p e. _V |-> [_ ( Poly1 ` r ) / s ]_ + [_ ( ( RSpan ` s ) ` { p } ) / i ]_ + [_ ( z e. ( Base ` r ) |-> + [ ( z ( .s ` s ) ( 1r ` s ) ) ] ( s ~QG i ) ) / f ]_ + <. [_ ( s /s ( s ~QG i ) ) / t ]_ ( ( t toNrmGrp ( iota_ n e. + ( AbsVal ` t ) ( n o. f ) = ( norm ` r ) ) ) sSet <. ( le ` ndx ) , + [_ ( z e. ( Base ` t ) |-> + ( iota_ q e. z ( r deg1 q ) < ( r deg1 p ) ) ) / g ]_ + ( `' g o. ( ( le ` s ) o. g ) ) >. ) , f >. ) $. - $( The value of the converse of ` 2nd ` restricted to a singleton. - (Contributed by Scott Fenton, 2-Jul-2020.) $) - fv2ndcnv $p |- ( ( X e. V /\ Y e. A ) -> - ( `' ( 2nd |` ( { X } X. A ) ) ` Y ) = <. X , Y >. ) $= - ( wcel wa c2nd csn cxp cres ccnv cfv cop wceq wbr wb wf1o cvv adantl bitrd - snidg anim1i eqid jctir wfn 2ndconst adantr f1ocnv f1ofn 3syl sylancom opex - fnbrfvb brcnvg mpan2 brres opelxp anbi1i br2ndeqg anbi2d bitrid mpbird ) CB - EZDAEZFZDGCHZAIZJZKZLCDMZNZCVFEZVDFZDDNZFZVEVMVNVCVLVDCBUAUBDUCUDVEVKDVJVIO - ZVOVCVDVIAUEZVKVPPVEVGAVHQZAVGVIQVQVCVRVDCABUFUGVGAVHUHAVGVIUIUJADVJVIUMUKV - EVPVJDVHOZVOVDVPVSPZVCVDVJREVTCDULDVJARVHUNUOSVEVSVJVGEZVJDGOZFZVOVDVSWCPVC - VGVJDGAUPSWCVMWBFVEVOWAVMWBCDVFAUQURVEWBVNVMCDDBAUSUTVATTTVB $. + $( Temporary construction for the splitting field of a polynomial. The + inputs are a field ` r ` and a polynomial ` p ` that we want to split, + along with a tuple ` j ` in the same format as the output. The output + is a tuple ` <. S , F >. ` where ` S ` is the splitting field and ` F ` + is an injective homomorphism from the original field ` r ` . - ${ - $d R x y $. $d A x y $. - $( The image is unaffected by intersection with the domain. (Contributed - by Scott Fenton, 17-Dec-2021.) $) - imaindm $p |- ( R " A ) = ( R " ( A i^i dom R ) ) $= - ( vx vy cima cdm cin cv wbr wrex wcel wa vex breldm pm4.71ri rexbii rexin - bitr4i elima 3bitr4i eqriv ) CBAEZBABFZGZEZDHZCHZBIZDAJZUHDUDJZUGUBKUGUEK - UIUFUCKZUHLZDAJUJUHULDAUHUKUFUGBDMCMZNOPUHDAUCQRDUGBAUMSDUGBUDUMSTUA $. + The function works by repeatedly finding the smallest monic irreducible + factor, and extending the field by that factor using the ` polyFld ` + construction. We keep track of a total order in each of the splitting + fields so that we can pick an element definably without needing global + choice. (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-sfl1 $a |- splitFld1 = ( r e. _V , j e. _V |-> ( p e. ( Poly1 ` r ) |-> + ( rec ( ( s e. _V , f e. _V |-> [_ ( mPoly ` s ) / m ]_ + [_ { g e. ( ( Monic1p ` s ) i^i ( Irred ` m ) ) | + ( g ( ||r ` m ) ( p o. f ) /\ 1 < ( s deg1 g ) ) } / b ]_ + if ( ( ( p o. f ) = ( 0g ` m ) \/ b = (/) ) , <. s , f >. , + [_ ( glb ` b ) / h ]_ [_ ( s polyFld h ) / t ]_ + <. ( 1st ` t ) , ( f o. ( 2nd ` t ) ) >. ) ) , j ) ` + ( card ` ( 1 ... ( r deg1 p ) ) ) ) ) ) $. + + $( Define the splitting field of a finite collection of polynomials, given + a total ordered base field. The output is a tuple ` <. S , F >. ` where + ` S ` is the totally ordered splitting field and ` F ` is an injective + homomorphism from the original field ` r ` . (Contributed by Mario + Carneiro, 2-Dec-2014.) $) + df-sfl $a |- splitFld = ( r e. _V , p e. _V |-> + ( iota x E. f ( f Isom < , ( lt ` r ) ( ( 1 ... ( # ` p ) ) , p ) /\ + x = ( seq 0 ( ( e e. _V , g e. _V |-> + ( ( r splitFld1 e ) ` g ) ) , ( f u. + { <. 0 , <. r , ( _I |` ( Base ` r ) ) >. >. } ) ) ` + ( # ` p ) ) ) ) ) $. + + $( Define the direct limit of an increasing sequence of fields produced by + pasting together the splitting fields for each sequence of polynomials. + That is, given a ring ` r ` , a strict order on ` r ` , and a sequence + ` p : NN --> ( ~P r i^i Fin ) ` of finite sets of polynomials to split, + we construct the direct limit system of field extensions by splitting + one set at a time and passing the resulting construction to ` HomLim ` . + (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-psl $a |- polySplitLim = ( r e. _V , + p e. ( ( ~P ( Base ` r ) i^i Fin ) ^m NN ) |-> + [_ ( 1st o. seq 0 ( ( g e. _V , q e. _V |-> + [_ ( 1st ` g ) / e ]_ [_ ( 1st ` e ) / s ]_ + [_ ( s splitFld ran ( x e. q |-> ( x o. ( 2nd ` g ) ) ) ) / f ]_ + <. f , ( ( 2nd ` g ) o. ( 2nd ` f ) ) >. ) , + ( p u. { <. 0 , <. <. r , (/) >. , + ( _I |` ( Base ` r ) ) >. >. } ) ) ) / f ]_ + ( ( 1st o. ( f shift 1 ) ) HomLim ( 2nd o. f ) ) ) $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Set induction (or epsilon induction) + p-adic number fields =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - ${ - $d ph y z $. $d x y z $. - setinds.1 $e |- ( A. y e. x [. y / x ]. ph -> ph ) $. - $( Principle of set induction (or ` _E ` -induction). If a property passes - from all elements of ` x ` to ` x ` itself, then it holds for all - ` x ` . (Contributed by Scott Fenton, 10-Mar-2011.) $) - setinds $p |- ph $= - ( vz cv cvv wcel vex cab wss wi wceq setind wsbc wral dfss3 df-sbc ralbii - nfsbc1v nfcv nfralw nfim raleq sbceq1a imbi12d chvarfv sylbir sylbi sylib - mpg eqcomi abeq2i mpbi ) BFZGHABIABGABJZGEFZUPKZUQUPHZLUPGMEEUPNURABUQOZU - SURCFZUPHZCUQPZUTCUQUPQVCABVAOZCUQPZUTVDVBCUQABVARSVDCUOPZALVEUTLBEVEUTBV - DBCUQBUQUAABVATUBABUQTUCUOUQMVFVEAUTVDCUOUQUDABUQUEUFDUGUHUIABUQRUJUKULUM - UN $. - $} - - ${ - $d x y $. $d ph y $. - setinds2f.1 $e |- F/ x ps $. - setinds2f.2 $e |- ( x = y -> ( ph <-> ps ) ) $. - setinds2f.3 $e |- ( A. y e. x ps -> ph ) $. - $( ` _E ` induction schema, using implicit substitution. (Contributed by - Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, - 11-Dec-2016.) $) - setinds2f $p |- ph $= - ( cv wsbc wral wsb sbsbc sbiev bitr3i ralbii sylbi setinds ) ACDACDHIZDCH - ZJBDSJARBDSRACDKBACDLABCDEFMNOGPQ $. - $} + $( Introduce new constant symbols. $) + $c ZRing $. $( Integral elements of a ring $) + $c GF $. $( Galois finite field $) + $c GF_oo $. $( Galois limit field $) + $c ~Qp $. $( Equivalence relation for Qp $) + $c /Qp $. $( Representative of equivalence relation $) + $c Qp $. $( p-adic rational numbers $) + $c Zp $. $( p-adic integers $) + $c _Qp $. $( Algebraic completion of Qp $) + $c Cp $. $( Metric completion of _Qp $) - ${ - $d x y $. $d ph y $. $d ps x $. - setinds2.1 $e |- ( x = y -> ( ph <-> ps ) ) $. - setinds2.2 $e |- ( A. y e. x ps -> ph ) $. - $( ` _E ` induction schema, using implicit substitution. (Contributed by - Scott Fenton, 10-Mar-2011.) $) - setinds2 $p |- ph $= - ( nfv setinds2f ) ABCDBCGEFH $. - $} + $( Integral elements of a ring. $) + czr $a class ZRing $. + $( Galois finite field. $) + cgf $a class GF $. -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Ordinal numbers -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) + $( Galois limit field. $) + cgfo $a class GF_oo $. - ${ - $d x y z A $. - $( A class of transitive sets is partially ordered by ` _E ` . - (Contributed by Scott Fenton, 15-Oct-2010.) $) - elpotr $p |- ( A. z e. A Tr z -> _E Po A ) $= - ( vx vy wel wa wi wal wral wtr cep wpo alral alimi syl ralcom ralbii epel - cv wbr ralimi bitri sylib dftr2 df-po anbi12i imbi12i elirrv mtbir bitr3i - wn biantrur 2ralbii bitr4i 3imtr4i ) CDEZDAEZFZCAEZGZDHZCHZABIZUTABIZDBIZ - CBIZASZJZABIBKLZVCUTDBIZCBIZABIZVFVBVKABVBVJCHVKVAVJCUTDBMNVJCBMOUAVLVJAB - IZCBIVFVJACBBPVMVECBUTADBBPQUBUCVHVBABCDVGUDQVICSZVNKTZUKZVNDSZKTZVQVGKTZ - FZVNVGKTZGZFZABIZDBICBIVFCDABKUEVDWDCDBBUTWCABUTWBWCVTURWAUSVRUPVSUQDVNRA - VQRUFAVNRUGVPWBVOCCECUHCVNRUIULUJQUMUNUO $. - $} + $( Equivalence relation for ~ df-qp . $) + ceqp $a class ~Qp $. - $( Given ~ ax-reg , an ordinal is a transitive class totally ordered by the - membership relation. (Contributed by Scott Fenton, 28-Jan-2011.) $) - dford5reg $p |- ( Ord A <-> ( Tr A /\ _E Or A ) ) $= - ( word wtr cep wwe wa wor df-ord wfr zfregfr df-we mpbiran anbi2i bitri ) A - BACZADEZFOADGZFAHPQOPADIQAJADKLMN $. + $( Equivalence relation representatives for ~ df-qp . $) + crqp $a class /Qp $. - ${ - $d x y $. $d y ph $. $d y ps $. - $( Lemma for ~ dfon2 . (Contributed by Scott Fenton, 28-Feb-2011.) $) - dfon2lem1 $p |- Tr U. { x | ( ph /\ Tr x /\ ps ) } $= - ( vy cv wtr w3a cab cuni truni wcel wsbc nfsbc1v nfv vex weq sbceq1a treq - nf3an 3anbi123d elabf simp2bi mprg ) DEZFZACEZFZBGZCHZIFDUIDUIJUDUIKACUDL - ZUEBCUDLZUHUJUEUKGCUDUJUEUKCACUDMUECNBCUDMSDOCDPAUJUGUEBUKACUDQUFUDRBCUDQ - TUAUBUC $. - $} + $( The set of ` p ` -adic rational numbers. $) + cqp $a class Qp $. - ${ - $d x A $. - $( Lemma for ~ dfon2 . (Contributed by Scott Fenton, 28-Feb-2011.) $) - dfon2lem2 $p |- U. { x | ( x C_ A /\ ph /\ ps ) } C_ A $= - ( cv wss w3a cab cpw cuni simp1 ss2abi df-pw sseqtrri sspwuni mpbi ) CEDF - ZABGZCHZDIZFSJDFSQCHTRQCQABKLCDMNSDOP $. - $} + $( The set of ` p ` -adic integers. (Not to be confused with ~ czn .) $) + czp $a class Zp $. - ${ - $d A x z w t $. + $( Algebraic completion of the ` p ` -adic rational numbers. $) + cqpa $a class _Qp $. - $( Lemma for ~ dfon2 . All sets satisfying the new definition are - transitive and untangled. (Contributed by Scott Fenton, - 25-Feb-2011.) $) - dfon2lem3 $p |- ( A e. V -> ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) -> - ( Tr A /\ A. z e. A -. z e. z ) ) ) $= - ( vw vt wcel cv wtr wa wi wel wn wral wss w3a wceq sseq1 treq cvv wal cab - wpss cuni untelirr wrex eluni2 raleq 3anbi123d elequ1 elequ2 bitrd notbid - vex elab cbvralvw biimpi 3ad2ant3 sylbi rsp syl rexlimiv dfon2lem2 dfpss2 - mprg dfon2lem1 ssexg mpan psseq1 anbi12d eleq1 imbi12d spcgv imp csuc csn - sylan snssi cun unss df-suc sseq1i sylbb2 sylancr suctr ax-mp mprgbir nfv - untuni nfra1 nf3an nfab nfuni untsucf raleqf cbvabv elab2g biimprd sucexg - nfcv nfsuc syl11 mp3an23 com12 elssuni sucssel syl5 syld mpd syl6 syl5bir - mpan2i mpani mt3i pm3.2i mpbii ex ) CDGZAHZCUCZXSIZJZXSCGZKZAUAZCIZBBLZMZ - BCNZJZXRYEJZEHZCOZYLIZFFLZMZFYLNZPZEUBZUDZCQZYJYKUUAYTYTGZYHUUBMBYTBYTUEB - HZYTGBALZAYSUFYHAUUCYSUGUUDYHAYSXSYSGZYHBXSNZUUDYHKUUEXSCOZYAYPFXSNZPZUUF - YRUUIEXSAUNYLXSQYMUUGYNYAYQUUHYLXSCRYLXSSYPFYLXSUHUIUOUUHUUGUUFYAUUHUUFYP - YHFBXSFHUUCQZYOYGUUJYOBFLYGFBFUJFBBUKULUMUPUQURUSZYHBXSUTVAVBUSVEYKYTCOZU - UAMZUUBYNYQECVCZUULUUMJYTCUCZYKUUBYTCVDYKUUOYTIZUUBYMYQEVFZYKUUOUUPJZYTCG - ZUUBXRYTTGZYEUURUUSKZUULXRUUTUUNYTCDVGVHUUTYEUVAYDUVAAYTTXSYTQZYBUURYCUUS - UVBXTUUOYAUUPXSYTCVIXSYTSVJXSYTCVKVLVMVNVQUUSYTVOZCOZUUBUUSUULYTVPZCOZUVD - UUNYTCVRUULUVFJYTUVEVSZCOUVDYTUVECVTUVCUVGCYTWAWBWCWDUUSUVDUVCYSGZUUBUVDU - USUVHUVDUVCIZYPFUVCNZUUSUVHKUUPUVIUUQYTWEWFYHBYTNZUVJUVKUUFAYSBAYSWIUUKWG - ZBFYTFYSYRFEYMYNYQFYMFWHYNFWHYPFYLWJWKWLWMZWNWFUVCTGZUVDUVIUVJPZUVHUUSUVN - UVHUVOUUCCOZUUCIZYPFUUCNZPZUVOBUVCYSTUUCUVCQUVPUVDUVQUVIUVRUVJUUCUVCCRUUC - UVCSYPFUUCUVCFUUCWTFYTUVMXAWOUIYRUVSEBYLUUCQYMUVPYNUVQYQUVRYLUUCCRYLUUCSY - PFYLUUCUHUIWPWQWRYTCWSXBXCXDUVHUVCYTOUUSUUBUVCYSXEYTYTCXFXGXHXIXJXLXKXMXN - UUAUUPUVKJYJUUPUVKUUQUVLXOUUAUUPYFUVKYIYTCSYHBYTCUHVJXPVAXQ $. - $} + $( Metric completion of ` _Qp ` . $) + ccp $a class Cp $. ${ - $d A x y z $. $d B x y z $. - dfon2lem4.1 $e |- A e. _V $. - dfon2lem4.2 $e |- B e. _V $. - $( Lemma for ~ dfon2 . If two sets satisfy the new definition, then one is - a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.) $) - dfon2lem4 $p |- ( ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) /\ - A. y ( ( y C. B /\ Tr y ) -> y e. B ) ) -> - ( A C_ B \/ B C_ A ) ) $= - ( vz cv wpss wtr wa wcel wi wal wceq wo wss wn cvv eleq1 inss1 sseli wral - cin wel dfon2lem3 ax-mp simprd eleq2 bitrd notbid rspccv syl syl5 pm2.01d - adantr elin sylnib simpld trin syl2an inex1 psseq1 anbi12d imbi12d mpan2d - treq spcv adantl anim12d mtod ianor sylib sspss mpbi inss2 orel1 orc syl6 - olc jaoa mp2ani df-ss sseqin2 orbi12i sylibr ) AHZCIZWGJZKZWGCLZMZANZBHZD - IZWNJZKZWNDLZMZBNZKZCDUDZCOZXBDOZPZCDQZDCQZPXAXBCIZRZXBDIZRZPZXEXAXHXJKZR - XLXAXMXBCLZXBDLZKZXAXBXBLZXPXAXQXQXNXAXQRZXBCXBCDUAZUBWMXNXRMZWTWMGGUEZRZ - GCUCZXTWMCJZYCCSLWMYDYCKMEAGCSUFUGZUHYBXRGXBCGHZXBOZYAXQYGYAXBYFLXQYFXBYF - TYFXBXBUIUJUKULUMUPUNUOXBCDUQURXAXHXNXJXOXAXHXBJZXNWMYDDJZYHWTWMYDYCYEUSW - TYIYBGDUCZDSLWTYIYJKMFBGDSUFUGUSCDUTVAZWMXHYHKZXNMZWTWLYMAXBCDEVBZWGXBOZW - JYLWKXNYOWHXHWIYHWGXBCVCWGXBVGVDWGXBCTVEVHUPVFXAXJYHXOYKWTXJYHKZXOMZWMWSY - QBXBYNWNXBOZWQYPWRXOYRWOXJWPYHWNXBDVCWNXBVGVDWNXBDTVEVHVIVFVJVKXHXJVLVMXL - XHXCPZXJXDPZXEXBCQYSXSXBCVNVOXBDQYTCDVPXBDVNVOXIYSXEXKYTXIYSXCXEXHXCVQXCX - DVRVSXKYTXDXEXJXDVQXDXCVTVSWAWBUMXFXCXGXDCDWCDCWDWEWF $. - $} + $d b d f g h k n p r s x y $. + $( Define the subring of integral elements in a ring. (Contributed by + Mario Carneiro, 2-Dec-2014.) $) + df-zrng $a |- ZRing = ( r e. _V |-> + ( r IntgRing ran ( ZRHom ` r ) ) ) $. - ${ - $d A x y z $. $d B x y z $. - dfon2lem5.1 $e |- A e. _V $. - dfon2lem5.2 $e |- B e. _V $. - $( Lemma for ~ dfon2 . Two sets satisfying the new definition also satisfy - trichotomy with respect to ` e. ` . (Contributed by Scott Fenton, - 25-Feb-2011.) $) - dfon2lem5 $p |- ( ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) /\ - A. y ( ( y C. B /\ Tr y ) -> y e. B ) ) -> - ( A e. B \/ A = B \/ B e. A ) ) $= - ( vz cv wpss wtr wa wcel wi wal wceq wn wo w3o bitri cvv dfon2lem4 dfpss2 - wss eqcom notbii anbi2i orbi12i andir bitr4i orcom dfon2lem3 ax-mp simpld - wel wral psseq1 treq anbi12d eleq1 imbi12d spcv expcomd imp mpan9 orim12d - sylan2 syl5bi syl5bir mpand 3orrot 3orass df-or sylibr ) AHZCIZVNJZKZVNCL - ZMZANZBHZDIZWAJZKZWADLZMZBNZKZCDOZPZDCLZCDLZQZMZWLWIWKRZWHCDUCZDCUCZQZWJW - MABCDEFUAWRWJKZCDIZDCIZQZWHWMXBWPWJKZWQWJKZQWSWTXCXAXDCDUBXAWQDCOZPZKXDDC - UBXFWJWQXEWIDCUDUEUFSUGWPWQWJUHUIXBXAWTQWHWMWTXAUJWHXAWKWTWLWGVTDJZXAWKMZ - WGXGGGUNPZGDUOZDTLWGXGXJKMFBGDTUKULUMVTXGXHVTXAXGWKVSXAXGKZWKMADFVNDOZVQX - KVRWKXLVOXAVPXGVNDCUPVNDUQURVNDCUSUTVAVBVCVFVTCJZWGWTWLMVTXMXIGCUOZCTLVTX - MXNKMEAGCTUKULUMWGWTXMWLWFWTXMKZWLMBCEWACOZWDXOWEWLXPWBWTWCXMWACDUPWACUQU - RWACDUSUTVAVBVDVEVGVHVIWOWIWKWLRZWNWLWIWKVJXQWIWMQWNWIWKWLVKWIWMVLSSVM $. - $} + $( Define the Galois finite field of order ` p ^ n ` . (Contributed by + Mario Carneiro, 2-Dec-2014.) $) + df-gf $a |- GF = ( p e. Prime , n e. NN |-> [_ ( Z/nZ ` p ) / r ]_ + ( 1st ` ( r splitFld { [_ ( Poly1 ` r ) / s ]_ [_ ( var1 ` r ) / x ]_ + ( ( ( p ^ n ) ( .g ` ( mulGrp ` s ) ) x ) ( -g ` s ) x ) } ) ) ) $. - ${ - $d S x y z w s t $. - $( Lemma for ~ dfon2 . A transitive class of sets satisfying the new - definition satisfies the new definition. (Contributed by Scott Fenton, - 25-Feb-2011.) $) - dfon2lem6 $p |- ( ( Tr S /\ - A. x e. S A. z ( ( z C. x /\ Tr z ) -> z e. x ) ) -> - A. y ( ( y C. S /\ Tr y ) -> y e. S ) ) $= - ( vs vw vt wtr cv wpss wa wel wi wal wral wcel weq wn syl imbi12d wo cdif - wss pssss ssralv impcom adantrr ad2ant2lr psseq2 anbi1d elequ2 albidv imp - rspccv eldifi rspcv psseq1 treq anbi12d elequ1 syl6ib ad2ant2l adantr w3o - cbvalvw vex dfon2lem5 3orrot 3orass bitri eleq1a nsyli adantll orel1 trss - elndif eldifn ssel con3d syl5com syl9 adantl imp31 mpd syl5bi syl5 mp2and - syl9r ssrdv dfpss2 spvv expd com23 syl6 com3l adantld imp32 syl5bir mpand - orrd anassrs ralrimiva wrex wne pssdif r19.2z ad2antrl eleq1w syl5ibr a1i - ex c0 trel syl7 ad2antrr jaod rexlimdv syld alrimiv ) DHZCIZAIZJZYAHZKZCA - LZMZCNZADOZKZBIZDJZYKHZKZYKDPZMBYJYNYOYJYNKZBEQZBELZUAZEDYKUBZOZYOYPYSEYT - YJYNEIZYTPZYSYJYNUUCKZKZYQYRUUEYKUUBUCZYQRZYRUUEFYKUUBUUEFBLZFELZUUEUUHKZ - YAFIZJZYDKZCFLZMZCNZGIZUUBJZUUQHZKZGELZMZGNZUUIUUEUUHUUPUUEYHAYKOZUUHUUPM - YIYNUVDXTUUCYIYLUVDYMYLYIUVDYLYKDUCYIUVDMYKDUDYHAYKDUESUFUGUHYHUUPAUUKYKA - FQZYGUUOCUVEYEUUMYFUUNUVEYCUULYDYBUUKYAUIUJAFCUKTULUNSUMUUEUVCUUHYIUUCUVC - XTYNUUCYIUVCUUCYIYAUUBJZYDKZCELZMZCNZUVCUUCUUBDPZYIUVJMUUBDYKUOZYHUVJAUUB - DAEQZYGUVICUVMYEUVGYFUVHUVMYCUVFYDYBUUBYAUIUJAECUKTULUPSZUVIUVBCGCGQZUVGU - UTUVHUVAUVOUVFUURYDUUSYAUUQUUBUQYAUUQURUSCGEUTTVEVAUFVBVCUUPUVCKUUIFEQZEF - LZVDZUUJUUICGUUKUUBFVFEVFVGUVRUVPUVQUUIUAZUAZUUJUUIUVRUVPUVQUUIVDUVTUUIUV - PUVQVHUVPUVQUUIVIVJUUJUVPRZUVTUUIMUUDUUHUWAYJUUCUUHUWAYNUUCUUHUWAUUCUVPUU - KYTPUUHUUBYTUUKVKUUKYKDVPVLUMVMVMUWAUVTUVSUUJUUIUVPUVSVNUUJUVQRZUVSUUIMUU - DUUHUWBYJYNUUCUUHUWBYMUUCUUHUWBMMYLYMUUHUUKYKUCZUUCUWBYKUUKVOUUCEBLZRUWCU - WBUUBDYKVQUWCUVQUWDUUKYKUUBVRVSVTWAWBWCVMUVQUUIVNSWHWDWEWFWGXKWIUUFUUGKYK - UUBJZUUEYRYKUUBWJYJYNUUCUWEYRMZYIYNUUCUWFMZMXTYIYMUWGYLUUCYIYMUWFUUCYIUVJ - YMUWFMUVNUVJUWEYMYRUVJUWEYMYRUVIUWEYMKZYRMCBCBQZUVGUWHUVHYRUWIUVFUWEYDYMY - AYKUUBUQYAYKURUSCBEUTTWKWLWMWNWOWPWBWQWRWSWTXAXBYPUUAYSEYTXCZYOYLUUAUWJMZ - YJYMYLYTXLXDZUWKYKDXEUWLUUAUWJYSEYTXFXKSXGYPYSYOEYTYPYSUUCYOYPYQUUCYOMZYR - YQUWMMYPUUCYOYQUVKUVLBEDXHXIXJXTYRUWMMYIYNUUCUVKXTYRYOUVLXTYRUVKYODYKUUBX - MWLXNXOXPWMXQXRWDXKXS $. - $} + $( Define the Galois field of order ` p ^ +oo ` , as a direct limit of the + Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-gfoo $a |- GF_oo = ( p e. Prime |-> [_ ( Z/nZ ` p ) / r ]_ + ( r polySplitLim ( n e. NN |-> + { [_ ( Poly1 ` r ) / s ]_ [_ ( var1 ` r ) / x ]_ + ( ( ( p ^ n ) ( .g ` ( mulGrp ` s ) ) x ) ( -g ` s ) x ) } ) ) ) $. - ${ - $d A x y z w s t u $. $d B x y z w t $. - dfon2lem7.1 $e |- A e. _V $. - $( Lemma for ~ dfon2 . All elements of a new ordinal are new ordinals. - (Contributed by Scott Fenton, 25-Feb-2011.) $) - dfon2lem7 $p |- ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) -> - ( B e. A -> A. y ( ( y C. B /\ Tr y ) -> y e. B ) ) ) $= - ( vw vt vz vu wpss wtr wa wcel wi wal wss wel wral sylbi imbi12d w3a cuni - vs cv cab wceq csuc csn wn weq elequ1 elequ2 bitrd notbid cbvralvw biimpi - ralimi untuni sylibr vex sseq1 treq raleq 3anbi123d elab dfon2lem3 simprd - cvv ax-mp untelirr syl 3ad2ant3 psseq2 anbi1d albidv 3anbi3i abbii unieqi - mprg eleq2i sylnib dfon2lem2 ssexi snss mtbi intnan cun df-suc unss mtbir - sseq1i bitr4i dfon2lem1 suctr wo elsuc wrex eluni2 rspccv anbi12d cbvalvw - nfa1 psseq1 syl6ib rexlimi rexlimiv rgen dfon2lem6 mp2an eleq2 jaoi sucex - mpbiri elssuni sylbir mp3an23 ralbii bitrdi cbvabv sseqtrdi syl5bir mpani - a1i syl5bi mtoi eleq1 spcv mpan2i mtod dfpss2 biimpri mpan nsyl2 rsp 3syl - mpbii ) AUDZCJZYQKZLZYQCMZNZAOZFUDZCPZUUDKZBUDZGUDZJZUUGKZLZBGQZNZBOZGUUD - RZUAZFUEZUBZCUFZUUGHUDZJZUUJLZBHQZNZBOZHCRZDCMUUGDJZUUJLZUUGDMZNZBOZNUUCU - URCJZUUSUUCUVLUURCMZUUCUVMUURUGZUUEUUFUUGIUDZJZUUJLZBIQZNZBOZIUUDRZUAZFUE - ZUBZPZUWEUURUWDPZUURUHZUWDPZLZUWHUWFUURUWDMZUWHHHQZUIZHUURRZUWJUIGGQZUIZG - YQRZUWMAUUQUWPAUUQRUWLHYQRZAUUQRUWMUWPUWQAUUQUWPUWQUWOUWLGHYQGHUJZUWNUWKU - WRUWNHGQUWKGHGUKGHHULUMUNUOUPUQHAUUQURUSYQUUQMZYQCPZYSUUNGYQRZUAZUWPUUPUX - BFYQAUTZFAUJZUUEUWTUUFYSUUOUXAUUDYQCVAZUUDYQVBZUUNGUUDYQVCZVDVEZUXAUWTUWP - YSUUNUWOGYQUUNIIQUIIUUHRZUWOUUNUUHKZUXIUUHVHMUUNUXJUXILNGUTBIUUHVHVFVIVGI - UUHVJVKUQVLSVSUWMUURUURMUWJHUURVJUURUWDUURUUQUWCUUPUWBFUUOUWAUUEUUFUUNUVT - GIUUDGIUJZUUMUVSBUXKUUKUVQUULUVRUXKUUIUVPUUJUUHUVOUUGVMVNGIBULTVOZUOVPVQV - RVTWAVIUURUWDUURCEUUFUUOFCWBZWCZWDWEWFUWEUURUWGWGZUWDPUWIUVNUXOUWDUURWHZW - KUURUWGUWDWIWLWJUVMUWGCPZUUCUWEUURCUXNWDUUCUURCPZUXQUWEUXMUXRUXQLZUVNCPZU - UCUWEUXTUXOCPUXSUVNUXOCUXPWKUURUWGCWIWLUXTUWENUUCUXTUVNUCUDZCPZUYAKZYQUVO - JZYSLZAIQZNZAOZIUYARZUAZUCUEZUBZUWDUXTUVNKZUYHIUVNRZUVNUYLPZUURKZUYMUUEUU - OFWMZUURWNVIUYHIUVNUVOUVNMUVOUURMZUVOUURUFZWOUYHUVOUURIUTWPUYRUYHUYSUYRIA - QZAUUQWQZUYHAUVOUUQWRZUYTUYHAUUQUYGAXBUWSUXBUYTUYHNZUXHUXAUWTVUCYSUXAUYTU - VTUYHUUNUVTGUVOYQUXLWSZUVSUYGBABAUJZUVQUYEUVRUYFVUEUVPUYDUUJYSUUGYQUVOXCU - UGYQVBWTBAIUKTXAXDVLSXESUYSUYHYQUURJZYSLZYQUURMZNZAOZUYPUUTUVOJZUUTKZLZHI - QZNZHOZIUURRVUJUYQVUPIUURUYRVUAVUPVUBUYTVUPAUUQUWSUXBUYTVUPNZUXHUXAUWTVUQ - YSUXAUYTUVTVUPVUDUVSVUOBHBHUJZUVQVUMUVRVUNVURUVPVUKUUJVULUUGUUTUVOXCUUGUU - TVBWTBHIUKTXAXDVLSXFSXGIAHUURXHXIUYSUYGVUIAUYSUYEVUGUYFVUHUYSUYDVUFYSUVOU - URYQVMVNUVOUURYQXJTVOXMXKSXGUXTUYMUYNUAZUVNUYKMUYOUYJVUSUCUVNUURUXNXLUYAU - VNUFUYBUXTUYCUYMUYIUYNUYAUVNCVAUYAUVNVBUYHIUYAUVNVCVDVEUVNUYKXNXOXPUYKUWC - UYJUWBUCFUCFUJZUYBUUEUYCUUFUYIUWAUYAUUDCVAUYAUUDVBVUTUYIUYHIUUDRUWAUYHIUY - AUUDVCUYHUVTIUUDUYGUVSABABUJZUYEUVQUYFUVRVVAUYDUVPYSUUJYQUUGUVOXCYQUUGVBW - TABIUKTXAXQXRVDXSVRXTYCYAYBYDYEUUCUVLUYPUVMUYQUUBUVLUYPLZUVMNAUURUXNYQUUR - UFZYTVVBUUAUVMVVCYRUVLYSUYPYQUURCXCYQUURVBWTYQUURCYFTYGYHYIUXRUUSUIZUVLUX - MUVLUXRVVDLUURCYJYKYLYMUUSUVEHUURRUVFUVEHUURUUTUURMHAQZAUUQWQUVEAUUTUUQWR - VVEUVEAUUQUWSUWTYSUVEHYQRZUAZVVEUVENZUUPVVGFYQUXCUXDUUEUWTUUFYSUUOVVFUXEU - XFUXDUUOUXAVVFUXGUUNUVEGHYQUWRUUMUVDBUWRUUKUVBUULUVCUWRUUIUVAUUJUUHUUTUUG - VMVNGHBULTVOUOXRVDVEVVFUWTVVHYSUVEHYQYNVLSXFSXGUVEHUURCVCYPUVEUVKHDCUUTDU - FZUVDUVJBVVIUVBUVHUVCUVIVVIUVAUVGUUJUUTDUUGVMVNUUTDUUGXJTVOWSYO $. - $} + $( Define an equivalence relation on ` ZZ ` -indexed sequences of integers + such that two sequences are equivalent iff the difference is equivalent + to zero, and a sequence is equivalent to zero iff the sum + ` sum_ k <_ n f ( k ) ( p ^ k ) ` is a multiple of ` p ^ ( n + 1 ) ` for + every ` n ` . (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-eqp $a |- ~Qp = ( p e. Prime |-> + { <. f , g >. | ( { f , g } C_ ( ZZ ^m ZZ ) /\ + A. n e. ZZ sum_ k e. ( ZZ>= ` -u n ) ( ( ( f ` -u k ) - ( g ` -u k ) ) / + ( p ^ ( k + ( n + 1 ) ) ) ) e. ZZ ) } ) $. - ${ - $d A x y z w t $. - $( Lemma for ~ dfon2 . The intersection of a nonempty class ` A ` of new - ordinals is itself a new ordinal and is contained within ` A ` - (Contributed by Scott Fenton, 26-Feb-2011.) $) - dfon2lem8 $p |- ( ( A =/= (/) /\ - A. x e. A A. y ( ( y C. x /\ Tr y ) -> y e. x ) ) -> - ( A. z ( ( z C. |^| A /\ Tr z ) -> z e. |^| A ) /\ - |^| A e. A ) ) $= - ( vt vw cv wpss wtr wa wel wi wal wral wcel wn cvv syl sylbi wceq c0 cint - wne vex dfon2lem3 ax-mp simpld ralimi trint adantl cuni dfon2lem7 alrimiv - df-ral 19.21v albii bitr4i impexp 2albii eluni2 biimpi imim1i alimi alcom - wrex wex 19.23v df-rex imbi1i bitri 3imtr4i sylbir intssuni ssralv adantr - wss mpd dfon2lem6 imp simprd untelirr adantlr risset notbii ralnex psseq2 - intex eqcom anbi1d elequ2 imbi12d albidv rspccv intss1 dfpss2 psseq1 treq - anbi12d eleq1 spcgv expd exp4b com45 com23 syl5 mpdd mpid syl7bi ralrimiv - syl5bir ralim syl5bi wb elintg ad2antrr sylibrd mt3d ex ancld mp2and ) DU - AUCZBGZAGZHZYBIZJZBAKZLZBMZADNZJZDUBZIZEGZFGZHYNIJEFKLEMZFYLNZCGZYLHYRIJY - RYLOLCMZYLDOZJZYJYMYAYJYCIZADNYMYIUUBADYIUUBCCKPCYCNZYCQOYIUUBUUCJLAUDZBC - YCQUEUFUGUHADUIRUJYKYPFDUKZNZYQYJUUFYAYJFAKZYPLZFMZADNZUUFYIUUIADYIUUHFBE - YCYOUUDULUMUHUUJYCDOZUUHLZFMZAMZUUFUUJUUKUUILZAMUUNUUIADUNUUMUUOAUUKUUHFU - OUPUQUUNUUKUUGJZYPLZFMAMZUUFUUQUULAFUUKUUGYPURUSUUGADVEZYPLZFMZYOUUEOZYPL - ZFMUURUUFUUTUVCFUVBUUSYPUVBUUSAYODUTVAVBVCUURUUQAMZFMUVAUUQAFVDUVDUUTFUVD - UUPAVFZYPLUUTUUPYPAVGUUSUVEYPUUGADVHVIUQUPVJYPFUUEUNVKVLSRUJYAUUFYQLZYJYA - YLUUEVPUVFDVMYPFYLUUEVNRVOVQYMYQJYSYKUUAFCEYLVRYKYSYTYKYSYTYKYSJZYTYLYLOZ - YAYSUVHPZYJYAYSJZEEKPEYLNZUVIUVJYMUVKYAYSYMUVKJZYAYLQOZYSUVLLDWGZCEYLQUES - VSZVTEYLWARWBUVGYTPZYLYNOZEDNZUVHUVPYNYLTZPZEDNZUVGUVRUVPUVSEDVEZPUWAYTUW - BEYLDWCWDUVSEDWEUQUVGUVTUVQLZEDNUWAUVRLUVGUWCEDUVTYLYNTZPZUVGYNDOZUVQUVSU - WDYNYLWHWDUVGUWFYMUWEUVQLZYAYSYMYJUVJYMUVKUVOUGWBYKUWFYMUWGLZLYSYKUWFYBYN - HZYEJZBEKZLZBMZUWHYJUWFUWMLYAYIUWMAYNDYCYNTZYHUWLBUWNYFUWJYGUWKUWNYDUWIYE - YCYNYBWFWIAEBWJWKWLWMUJYAUWFUWMUWHLZLYJUWFYLYNVPZYAUWOYNDWNYAUWMUWPUWHYAU - WMUWPUWEYMUVQYAUWMUWPUWEYMUVQLZUWPUWEJYLYNHZYAUWMJZUWQYLYNWOUWSUWRYMUVQYA - UWMUWRYMJZUVQLZYAUVMUWMUXALUVNUWLUXABYLQYBYLTZUWJUWTUWKUVQUXBUWIUWRYEYMYB - YLYNWPYBYLWQWRYBYLYNWSWKWTSVSXAXJXBXCXDXEVOXFVOXGXHXIUVTUVQEDXKRXLYAUVHUV - RXMZYJYSYAUVMUXCUVNEYLDQXNSXOXPXQXRXSXEXT $. - $} + $( There is a unique element of ` ( ZZ ^m ( 0 ... ( p - 1 ) ) ) ` ` ~Qp ` + -equivalent to any element of ` ( ZZ ^m ZZ ) ` , if the sequences are + zero for sufficiently large negative values; this function selects that + element. (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-rqp $a |- /Qp = ( p e. Prime |-> ( ~Qp i^i [_ { f e. ( ZZ ^m ZZ ) | + E. x e. ran ZZ>= ( `' f " ( ZZ \ { 0 } ) ) C_ x } / y ]_ + ( y X. ( y i^i ( ZZ ^m ( 0 ... ( p - 1 ) ) ) ) ) ) ) $. - ${ - $d A x y z w t u $. - $( Lemma for ~ dfon2 . A class of new ordinals is well-founded by ` _E ` . - (Contributed by Scott Fenton, 3-Mar-2011.) $) - dfon2lem9 $p |- ( A. x e. A A. y ( ( y C. x /\ Tr y ) -> y e. x ) - -> _E Fr A ) $= - ( vz vt vw vu cv wpss wtr wa wel wi wal wral wss wn wrex cep wcel wne wfr - c0 ssralv cint dfon2lem8 simprd intss1 simpld cvv wceq dfon2lem3 untelirr - intex imp syl eleq1 notbid syl5ibcom a1dd trss eqss simplbi2com syl6 con3 - com23 pm2.61d sylanb syldan ralrimiv eleq2 ralbidv rspcev syl2anc syl6com - syl5 expcom impd alrimiv wbr df-fr epel notbii ralbii rexbii imbi2i albii - bitri sylibr ) BHZAHIWJJKBALMBNZACOZDHZCPZWMUCUAZKZEFLZQZEWMOZFWMRZMZDNZC - SUBZWLXADWLWNWOWTWNWLWKAWMOZWOWTMWKAWMCUDWOXDWTWOXDKZWMUEZWMTZEHZXFTZQZEW - MOZWTXEGHZXFIXLJKXLXFTMGNZXGABGWMUFZUGXEXJEWMEDLXFXHPZXEXJXHWMUHWOXDXMXOX - JMZXEXMXGXNUIWOXFUJTZXMXPWMUNXQXMKZXFXHUKZXPXRXSXJXOXRXFXFTZQZXSXJXRAALQA - XFOZYAXRXFJZYBXQXMYCYBKGAXFUJULUOZUGAXFUMUPXSXTXIXFXHXFUQURUSUTXRXOXSQZXJ - XRXOXIXSMYEXJMXRXIXOXSXRXIXHXFPZXOXSMXRYCXIYFMXRYCYBYDUIXFXHVAUPXSXOYFXFX - HVBVCVDVFXIXSVEVDVFVGVHVIVPVJWSXKFXFWMFHZXFUKZWRXJEWMYHWQXIYGXFXHVKURVLVM - VNVQVOVRVSXCWPXHYGSVTZQZEWMOZFWMRZMZDNXBDFECSWAYMXADYLWTWPYKWSFWMYJWREWMY - IWQFXHWBWCWDWEWFWGWHWI $. - $} + $( Define the ` p ` -adic completion of the rational numbers, as a normed + field structure with a total order (that is not compatible with the + operations). (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by + AV, 10-Oct-2021.) $) + df-qp $a |- Qp = ( p e. Prime |-> + [_ { h e. ( ZZ ^m ( 0 ... ( p - 1 ) ) ) | + E. x e. ran ZZ>= ( `' h " ( ZZ \ { 0 } ) ) C_ x } / b ]_ + ( ( { <. ( Base ` ndx ) , b >. , + <. ( +g ` ndx ) , + ( f e. b , g e. b |-> ( ( /Qp ` p ) ` ( f oF + g ) ) ) >. , + <. ( .r ` ndx ) , + ( f e. b , g e. b |-> ( ( /Qp ` p ) ` ( n e. ZZ |-> + sum_ k e. ZZ ( ( f ` k ) x. ( g ` ( n - k ) ) ) ) ) ) >. } u. + { <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ b /\ + sum_ k e. ZZ ( ( f ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) < + sum_ k e. ZZ ( ( g ` -u k ) x. ( ( p + 1 ) ^ -u k ) ) ) } >. } ) + toNrmGrp ( f e. b |-> if ( f = ( ZZ X. { 0 } ) , 0 , + ( p ^ -u inf ( ( `' f " ( ZZ \ { 0 } ) ) , RR , < ) ) ) ) ) ) $. - ${ - $d x y z w t u v $. - $( ` On ` consists of all sets that contain all its transitive proper - subsets. This definition comes from J. R. Isbell, "A Definition of - Ordinal Numbers", American Mathematical Monthly, vol 67 (1960), pp. - 51-52. (Contributed by Scott Fenton, 20-Feb-2011.) $) - dfon2 $p |- On = { x | A. y ( ( y C. x /\ Tr y ) -> y e. x ) } $= - ( vz vw vu vt vv cv cab wpss wtr wa wel wi wal cep wral vex weq imbi12d - con0 word df-on wss wne tz7.7 df-pss bitr4di exbiri com23 impd alrimiv wn - wwe cvv wcel dfon2lem3 simpld wfr w3o dfon2lem7 ralrimiv dfon2lem9 psseq2 - ax-mp anbi1d elequ2 albidv psseq1 anbi12d elequ1 cbvalvw bitrdi dfon2lem5 - treq anim12d syl6 ralrimivv jca syl wbr dfwe2 epel biid 3orbi123i 2ralbii - rspccv anbi2i bitri sylibr df-ord sylanbrc impbii abbii eqtri ) UAAHZUBZA - IBHZWPJZWRKZLBAMZNZBOZAIAUCWQXCAWQXCWQXBBWQWSWTXAWQWTWSXAWQWTXAWSWQWTLXAW - RWPUDWRWPUELWSWPWRUFWRWPUGUHUIUJUKULXCWPKZWPPUNZWQXCXDCCMUMCWPQZWPUOUPXCX - DXFLNARZBCWPUOUQVEURXCWPPUSZCDMZCDSZDCMZUTZDWPQCWPQZLZXEXCEHZFHZJZXOKZLZE - FMZNZEOZFWPQZXNXCYBFWPBEWPXPXGVAVBYCXHXMFEWPVCYCXLCDWPWPYCCAMZDAMZLGHZCHZ - JZYFKZLZGCMZNZGOZWRDHZJZWTLZBDMZNZBOZLXLYCYDYMYEYSYBYMFYGWPFCSZYBXOYGJZXR - LZECMZNZEOYMYTYAUUDEYTXSUUBXTUUCYTXQUUAXRXPYGXOVDVFFCEVGTVHUUDYLEGEGSZUUB - YJUUCYKUUEUUAYHXRYIXOYFYGVIXOYFVOVJEGCVKTVLVMWGYBYSFYNWPFDSZYBXOYNJZXRLZE - DMZNZEOYSUUFYAUUJEUUFXSUUHXTUUIUUFXQUUGXRXPYNXOVDVFFDEVGTVHUUJYREBEBSZUUH - YPUUIYQUUKUUGYOXRWTXOWRYNVIXOWRVOVJEBDVKTVLVMWGVPGBYGYNCRDRVNVQVRVSVTXEXH - YGYNPWAZXJYNYGPWAZUTZDWPQCWPQZLXNCDWPPWBUUOXMXHUUNXLCDWPWPUULXIXJXJUUMXKD - YGWCXJWDCYNWCWEWFWHWIWJWPWKWLWMWNWO $. - $} + $( Define the ` p ` -adic integers, as a subset of the ` p ` -adic + rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-zp $a |- Zp = ( ZRing o. Qp ) $. - ${ - $d F g $. $d I g $. - $( The value of the recursive definition generator at ` (/) ` when the base - value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) - (Revised by Mario Carneiro, 19-Apr-2014.) $) - rdgprc0 $p |- ( -. I e. _V -> ( rec ( F , I ) ` (/) ) = (/) ) $= - ( vg cvv wcel wn c0 crdg cfv cv wceq cdm wlim crn cuni cif cmpt cres eqid - unieqd con0 0elon rdgval ax-mp res0 fveq2i eqtri eqeq1 wb dmeq limeq rneq - syl fveq12d fveq2d ifbieq12d ifbieq2d eleq1d elrab2 iftruei eleq1i biimpi - id dmmpt simplbiim ndmfv nsyl5 eqtrid ) BDEZFGABHZIZGCDCJZGKZBVLLZMZVLNZO - ZVNOZVLIZAIZPZPZQZIZGVKVJGRZWCIZWDGUAEVKWFKUBBGCAUCUDWEGWCVJUEUFUGGWCLZEZ - VIWDGKWHGDEGGKZBGLZMZGNZOZWJOZGIZAIZPZPZDEZVIWBDEWSCGDWGVMWBWRDVMVMWIWAWQ - BVLGGUHVMVOWKVQVTWMWPVMVNWJKVOWKUIVLGUJZVNWJUKUMVMVPWLVLGULTVMVSWOAVMVRWN - VLGVMVCVMVNWJWTTUNUOUPUQURCDWBWCWCSVDUSWSVIWRBDWIBWQGSUTVAVBVEGWCVFVGVH - $. - $} + $( Define the completion of the ` p ` -adic rationals. Here we simply + define it as the splitting field of a dense sequence of polynomials + (using as the ` n ` -th set the collection of polynomials with degree + less than ` n ` and with coefficients ` < ( p ^ n ) ` ). Krasner's + lemma will then show that all monic polynomials have splitting fields + isomorphic to a sufficiently close Eisenstein polynomial from the list, + and unramified extensions are generated by the polynomial + ` x ^ ( p ^ n ) - x ` , which is in the list. Thus, every finite + extension of ` Qp ` is a subfield of this field extension, so it is + algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-qpa $a |- _Qp = ( p e. Prime |-> [_ ( Qp ` p ) / r ]_ + ( r polySplitLim ( n e. NN |-> { f e. ( Poly1 ` r ) | + ( ( r deg1 f ) <_ n /\ A. d e. ran ( coe1 ` f ) + ( `' d " ( ZZ \ { 0 } ) ) C_ ( 0 ... n ) ) } ) ) ) $. - ${ - $d F x y z $. $d I x y z $. - $( The value of the recursive definition generator when ` I ` is a proper - class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario - Carneiro, 19-Apr-2014.) $) - rdgprc $p |- ( -. I e. _V -> rec ( F , I ) = rec ( F , (/) ) ) $= - ( vx vz vy cvv wcel cv crdg cfv c0 wceq con0 wral wi fveq2 eqeq12d imbi2d - weq wb csuc rdgprc0 0ex rdg0 eqtr4di rdgsuc syl5ibr imim2d wlim cres word - r19.21v wss limord ordsson wfn rdgfnon fvreseq mpanl12 3syl cima cuni crn - wn rneq df-ima 3eqtr4g unieqd vex rdglim mpan sylbird syl5bi tfinds com12 - wa ralrimiv eqfnfv mp2an sylibr ) BFGVDZCHZABIZJZWBAKIZJZLZCMNZWCWELZWAWG - CMWBMGWAWGWADHZWCJZWJWEJZLZOZWAKWCJZKWEJZLZOWAEHZWCJZWRWEJZLZOZWAWRUAZWCJ - ZXCWEJZLZOWAWGODEWBWJKLZWMWQWAXGWKWOWLWPWJKWCPWJKWEPQRDESZWMXAWAXHWKWSWLW - TWJWRWCPWJWRWEPQRWJXCLZWMXFWAXIWKXDWLXEWJXCWCPWJXCWEPQRDCSZWMWGWAXJWKWDWL - WFWJWBWCPWJWBWEPQRWAWOKWPABUBKAUCUDUEWRMGZXAXFWAXAXFXKWSAJZWTAJZLWSWTAPXK - XDXLXEXMBWRAUFKWRAUFQUGUHXBEWJNWAXAEWJNZOWJUIZWNWAXAEWJULXOXNWMWAXOXNWCWJ - UJZWEWJUJZLZWMXOWJUKWJMUMZXRXNTZWJUNWJUOWCMUPZWEMUPZXSXTBAUQZKAUQZEMWJWCW - EURUSUTXRWMXOWCWJVAZVBZWEWJVAZVBZLZXRYEYGXRXPVCXQVCYEYGXPXQVEWCWJVFWEWJVF - VGVHWJFGZXOWMYITDVIYJXOVPWKYFWLYHBWJFAVJKWJFAVJQVKUGVLUHVMVNVOVQYAYBWIWHT - YCYDCMWCWEVRVSVT $. + $( Define the metric completion of the algebraic completion of the ` p ` + -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) $) + df-cp $a |- Cp = ( cplMetSp o. _Qp ) $. $} - ${ - $d F f $. $d f g $. $d F g $. $d f i $. $d F i $. $d f x $. $d F x $. - $d f y $. $d F y $. $d g i $. $d g x $. $d g y $. $d I f $. $d I i $. - $d i x $. $d I x $. $d i y $. $d I y $. $d x y $. $d f z $. $d F z $. - $d i z $. $d y z $. - $( Alternate definition of the recursive function generator when ` I ` is a - set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario - Carneiro, 19-Apr-2014.) $) - dfrdg2 $p |- ( I e. V -> rec ( F , I ) = - U. { f | E. x e. On ( f Fn x /\ - A. y e. x ( f ` y ) = - if ( y = (/) , I , - if ( Lim y , U. ( f " y ) , - ( F ` ( f ` U. y ) ) ) ) ) } ) $= - ( vg vz cv cfv c0 wceq cuni cif wa con0 cvv wcel fveq2d ifbieq2d crdg wfn - wlim cima wral wrex cab rdgeq2 ifeq1 eqeq2d ralbidv anbi2d rexbidv abbidv - vi unieqd eqeq12d cdm crn cmpt crecs cres df-rdg dfrecs3 wb w3a vex resex - eqeq1 wrel relres reldm0 ax-mp bitrdi dmeq limeq syl rneq eqtr4di fveq12d - df-ima id ifbieq12d eqid imaexg uniex fvex fvmpt cin dmres wss onelss imp - ifex 3adant2 fndm 3ad2ant2 sseqtrrd df-ss sylib eqtrid unieq onelon eloni - word csuc ordzsl iftrue eqtr4d sucid fvres ordunisuc 3eqtr4a nsuceq0 neii - w3o iffalsei wn nlimsucg iffalse eqtri 3eqtr4g reseq2 syl5ibrcom rexlimiv - mp2b wne df-lim simp2bi neneqd iffalsed 3jaoi sylbi sylan9eqr mpdan 3expa - 3eqtr4d ralbidva pm5.32da rexbiia abbii unieqi 3eqtri vtoclg ) DUOIZUAZCI - ZAIZUBZBIZUUGJZUUJKLZUUEUUJUCZUUGUUJUDZMZUUJMZUUGJZDJZNZNZLZBUUHUEZOZAPUF - ZCUGZMZLDEUAZUUIUUKUULEUUSNZLZBUUHUEZOZAPUFZCUGZMZLUOEFUUEELZUUFUVGUVFUVN - UUEEDUHUVOUVEUVMUVOUVDUVLCUVOUVCUVKAPUVOUVBUVJUUIUVOUVAUVIBUUHUVOUUTUVHUU - KUULUUEEUUSUIUJUKULUMUNUPUQUUFGQGIZKLZUUEUVPURZUCZUVPUSZMZUVRMZUVPJZDJZNZ - NZUTZVAUUIUUKUUGUUJVBZUWGJZLZBUUHUEZOZAPUFZCUGZMUVFGDUUEVCABCUWGVDUWNUVEU - WMUVDCUWLUVCAPUUHPRZUUIUWKUVBUWOUUIOUWJUVABUUHUWOUUIUUJUUHRZUWJUVAVEUWOUU - IUWPVFZUWIUUTUUKUWQUWIUWHURZKLZUUEUWRUCZUUOUWRMZUWHJZDJZNZNZUUTUWHQRUWIUX - ELUUGUUJCVGZVHGUWHUWFUXEQUWGUVPUWHLZUVQUWSUWEUXDUUEUXGUVQUWHKLZUWSUVPUWHK - VIUWHVJUXHUWSVEUUGUUJVKUWHVLVMVNUXGUVSUWTUWAUWDUUOUXCUXGUVRUWRLUVSUWTVEUV - PUWHVOZUVRUWRVPVQUXGUVTUUNUXGUVTUWHUSUUNUVPUWHVRUUGUUJWAVSUPUXGUWCUXBDUXG - UWBUXAUVPUWHUXGWBUXGUVRUWRUXIUPVTSWCTUWGWDUWSUUEUXDUOVGUWTUUOUXCUUNUUGQRU - UNQRUXFUUGUUJQWEVMWFUXBDWGWNWNWHVMUWQUWRUUJLZUXEUUTLUWQUWRUUJUUGURZWIZUUJ - UUGUUJWJUWQUUJUXKWKUXLUUJLUWQUUJUUHUXKUWOUWPUUJUUHWKZUUIUWOUWPUXMUUHUUJWL - WMWOUUIUWOUXKUUHLUWPUUHUUGWPWQWRUUJUXKWSWTXAUXJUWQUXEUULUUEUUMUUOUUPUWHJZ - DJZNZNZUUTUXJUWSUULUXDUXPUUEUWRUUJKVIUXJUWTUUMUXCUXOUUOUWRUUJVPUXJUXBUXND - UXJUXAUUPUWHUWRUUJXBSSTTUWQUUJXEZUXQUUTLZUWOUWPUXRUUIUWOUWPOUUJPRUXRUUHUU - JXCUUJXDVQWOUXRUULUUJHIZXFZLZHPUFZUUMXPUXSHUUJXGUULUXSUYCUUMUULUXQUUEUUTU - ULUUEUXPXHUULUUEUUSXHXIUYBUXSHPUXTPRZUXSUYBUYAKLZUUEUYAUCZUUOUYAMZUUGUYAV - BZJZDJZNZNZUYEUUEUYFUUOUYGUUGJZDJZNZNZLUYDUYJUYNUYLUYPUYDUYIUYMDUYDUXTUYH - JZUXTUUGJZUYIUYMUXTUYARUYQUYRLUXTHVGZXJUXTUYAUUGXKVMUYDUYGUXTUYHUYDUXTXEU - YGUXTLUXTXDUXTXLVQZSUYDUYGUXTUUGUYTSXMSUYLUYKUYJUYEUUEUYKUYAKUXTXNXOZXQUX - TQRZUYFXRZUYKUYJLUYSUXTQXSZUYFUUOUYJXTYFYAUYPUYOUYNUYEUUEUYOVUAXQVUBVUCUY - OUYNLUYSVUDUYFUUOUYNXTYFYAYBUYBUXQUYLUUTUYPUYBUULUYEUXPUYKUUEUUJUYAKVIZUY - BUUMUYFUXOUYJUUOUUJUYAVPZUYBUXNUYIDUYBUUPUYGUWHUYHUUJUYAUUGYCUUJUYAXBZVTS - TTUYBUULUYEUUSUYOUUEVUEUYBUUMUYFUURUYNUUOVUFUYBUUQUYMDUYBUUPUYGUUGVUGSSTT - UQYDYEUUMUXQUUSUUTUUMUXPUUOUXQUUSUUMUUOUXOXHUUMUULUUEUXPUUMUUJKUUMUXRUUJK - YGUUJUUPLUUJYHYIYJZYKUUMUUOUURXHYQUUMUULUUEUUSVUHYKXIYLYMVQYNYOXAUJYPYRYS - YTUUAUUBUUCUUD $. - $} + $( TODO list $) - ${ - $d F f $. $d f x $. $d F x $. $d f y $. $d F y $. $d I f $. $d I x $. - $d I y $. $d x y $. - $( Generalization of ~ dfrdg2 to remove sethood requirement. (Contributed - by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, - 19-Apr-2014.) $) - dfrdg3 $p |- rec ( F , I ) = - U. { f | E. x e. On ( f Fn x /\ - A. y e. x ( f ` y ) = - if ( y = (/) , if ( I e. _V , I , (/) ) , - if ( Lim y , U. ( f " y ) , - ( F ` ( f ` U. y ) ) ) ) ) } $= - ( cvv wcel crdg cv cfv c0 wceq cif cuni wral wa con0 wrex cab dfrdg2 wlim - wfn cima iftrue ifeq1d eqeq2d ralbidv anbi2d rexbidv abbidv unieqd eqtr4d - wn 0ex ax-mp rdgprc iffalse 3eqtr4a pm2.61i ) EFGZDEHZCIZAIZUBZBIZVBJZVEK - LZUTEKMZVEUAVBVEUCNVENVBJDJMZMZLZBVCOZPZAQRZCSZNZLUTVAVDVFVGEVIMZLZBVCOZP - ZAQRZCSZNVPABCDEFTUTVOWBUTVNWACUTVMVTAQUTVLVSVDUTVKVRBVCUTVJVQVFUTVGVHEVI - UTEKUDUEUFUGUHUIUJUKULUTUMZDKHZVDVFVGKVIMZLZBVCOZPZAQRZCSZNZVAVPKFGWDWKLU - NABCDKFTUODEUPWCVOWJWCVNWICWCVMWHAQWCVLWGVDWCVKWFBVCWCVJWEVFWCVGVHKVIUTEK - UQUEUFUGUHUIUJUKURUS $. - $} + $( change *mpt2* -> *mpo* $) + $( use more symbol variables like .+ $) + $( Uniform spaces $) + $( Characterization of perfectly normal spaces $) + $( Hausdorff quotient $) + $( add mpt2mptf, fmpt2i $) + +$( (End of Mario Carneiro's mathbox.) $) +$( End $[ set-mbox-mc.mm $] $) +$( Begin $[ set-mbox-fc.mm $] $) +$( Skip $[ set-main.mm $] $) $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Defined equality axioms -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Filip Cernatescu +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + + I hope someone will enjoy solving (proving) the simple equations, + inequalities, and calculations from this mathbox. I have proved these + problems (theorems) using the Milpgame proof assistant. (It can be + downloaded from ~ https://us.metamath.org/other/milpgame/milpgame.html .) + $) - $( A version of ~ ax-ext for use with defined equality. (Contributed by - Scott Fenton, 12-Dec-2010.) $) - axextdfeq $p |- E. z ( ( z e. x -> z e. y ) - -> ( ( z e. y -> z e. x ) -> ( x e. w -> y e. w ) ) ) $= - ( wel wb wi wex weq axextnd ax8 imim2i eximii biimpexp exbii mpbi ) CAEZCBE - ZFZADEBDEGZGZCHQRGRQGTGGZCHSABIZGUACCABJUCTSABDKLMUAUBCQRTNOP $. + $( Practice problem 1. Clues: ~ 5p4e9 ~ 3p2e5 ~ eqtri ~ oveq1i . + (Contributed by Filip Cernatescu, 16-Mar-2019.) + (Proof modification is discouraged.) $) + problem1 $p |- ( ( 3 + 2 ) + 4 ) = 9 $= + ( c3 c2 caddc co c4 c5 c9 3p2e5 oveq1i 5p4e9 eqtri ) ABCDZECDFECDGLFECHIJK + $. - $( A version of ~ ax-8 for use with defined equality. (Contributed by Scott - Fenton, 12-Dec-2010.) $) - ax8dfeq $p |- E. z ( ( z e. x -> z e. y ) -> ( w e. x -> w e. y ) ) $= - ( weq wel wi ax6e ax8 equcoms imim12d eximii ) CDEZCAFZCBFZGDAFZDBFZGGCCDHM - PNOQPNGDCDCAIJCDBIKL $. + $( Practice problem 2. Clues: ~ oveq12i ~ adddiri ~ add4i ~ mulcli ~ recni + ~ 2re ~ 3eqtri ~ 10re ~ 5re ~ 1re ~ 4re ~ eqcomi ~ 5p4e9 ~ oveq1i ~ df-3 . + (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, + 9-Sep-2021.) (Proof modification is discouraged.) $) + problem2 $p |- ( ( ( 2 x. ; 1 0 ) + 5 ) + ( ( 1 x. ; 1 0 ) + 4 ) ) + = ( ( 3 x. ; 1 0 ) + 9 ) $= + ( c2 c1 cc0 cdc cmul co c5 caddc c4 c9 c3 2re recni 10re mulcli 5re 1re 4re + eqcomi oveq1i add4i adddiri 5p4e9 oveq12i df-3 3eqtri ) ABCDZEFZGHFBUGEFZIH + FHFUHUIHFZGIHFZHFABHFZUGEFZJHFKUGEFZJHFUHGUIIAUGALMZUGNMZOGPMBUGBQMZUPOIRMU + AUJUMUKJHUMUJABUGUOUQUPUBSUCUDUMUNJHULKUGEKULUESTTUF $. + + ${ + problem3.1 $e |- A e. CC $. + problem3.2 $e |- ( A + 3 ) = 4 $. + $( Practice problem 3. Clues: ~ eqcomi ~ eqtri ~ subaddrii ~ recni ~ 4re + ~ 3re ~ 1re ~ df-4 ~ addcomi . (Contributed by Filip Cernatescu, + 16-Mar-2019.) (Proof modification is discouraged.) $) + problem3 $p |- A = 1 $= + ( c1 c4 c3 cmin co 4re recni 3re 1re caddc eqcomi subaddrii addcomi eqtri + df-4 ) DADEFGHZASDEFDEIJZFKJZDLJEFDMHRNONEFATUABFAMHAFMHEFAUABPCQOQN $. + $} + + ${ + problem4.1 $e |- A e. CC $. + problem4.2 $e |- B e. CC $. + problem4.3 $e |- ( A + B ) = 3 $. + problem4.4 $e |- ( ( 3 x. A ) + ( 2 x. B ) ) = 7 $. + $( Practice problem 4. Clues: ~ pm3.2i ~ eqcomi ~ eqtri ~ subaddrii + ~ recni ~ 7re ~ 6re ~ ax-1cn ~ df-7 ~ ax-mp ~ oveq1i ~ 3cn ~ 2cn ~ df-3 + ~ mulid2i ~ subdiri ~ mp3an ~ mulcli ~ subadd23 ~ oveq2i ~ oveq12i + ~ 3t2e6 ~ mulcomi ~ subcli ~ biimpri ~ subadd2i . (Contributed by Filip + Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) $) + problem4 $p |- ( A = 1 /\ B = 2 ) $= + ( c1 wceq c2 c7 c6 cmin co caddc eqcomi c3 cmul 3cn 2cn eqtri ax-1cn df-7 + 7re recni 6re subaddrii df-3 oveq1i mulid2i subdiri mulcli subadd23 mp3an + cc wcel 3t2e6 mulcomi oveq12i subcli oveq2i subadd2i biimpri ax-mp pm3.2i + ) AGHBIHGAGJKLMZAVEGJKGJUCUDZKUEUDZUAJKGNMUBOUFOAKNMZJHZVEAHZVHPAQMZIBQMZ + NMZJVHVKIAQMZLMZKNMZVMAVOKNAPILMZAQMZVOVRAVRGAQMAVQGAQPIGRSUAPIGNMZUGOZUF + UHACUITOPIARSCUJTUHVPVKKVNLMZNMZVMVKUNUOVNUNUOKUNUOVPWBHPARCUKIASCUKVGVKV + NKULUMVMWBVLWAVKNWAVLWAPIQMZAIQMZLMZVLWEWAWCKWDVNLUPAICSUQUROWEPALMZIQMZV + LWGWEPAIRCSUJOWGIWFQMZVLWHWGIWFSPARCUSUQOVLWHBWFIQWFBPABRCDEUFOZUTOTTTOUT + OTTFTVJVIJKAVFVGCVAVBVCTOZBWFIWIWFPGLMZIAGPLWJUTVSPHZWKIHZVTWMWLPGIRUASVA + VBVCTTVD $. + $} ${ - $d w x $. $d w y $. $d z w $. - $( ~ ax-ext with distinctors instead of distinct variable conditions. - (Contributed by Scott Fenton, 13-Dec-2010.) $) - axextdist $p |- ( ( -. A. z z = x /\ -. A. z z = y ) - -> ( A. z ( z e. x <-> z e. y ) -> x = y ) ) $= - ( vw weq wal wn wa wel wb nfnae nfan wnfc nfcvf adantr nfcrd adantl nfbid - cv elequ1 wi bibi12d a1i cbvald axextg syl6bir ) CAECFGZCBECFGZHZCAIZCBIZ - JZCFDAIZDBIZJZDFABEUIUOULDCUGUHCCACKCBCKLUIUMUNCUICDASZUGCUPMUHCANOPUICDB - SZUHCUQMUGCBNQPRDCEZUOULJUAUIURUMUJUNUKDCATDCBTUBUCUDABDUEUF $. - - $( ~ axextb with distinctors instead of distinct variable conditions. - (Contributed by Scott Fenton, 13-Dec-2010.) $) - axextbdist $p |- ( ( -. A. z z = x /\ -. A. z z = y ) - -> ( x = y <-> A. z ( z e. x <-> z e. y ) ) ) $= - ( weq wal wn wa wel wb wi axc9 imp nfnae nfan elequ2 alimd syld axextdist - a1i impbid ) CADCEFZCBDCEFZGZABDZCAHCBHIZCEZUCUDUDCEZUFUAUBUDUGJABCKLUCUD - UECUAUBCCACMCBCMNUDUEJUCABCOSPQABCRT $. + problem5.1 $e |- A e. RR $. + problem5.2 $e |- ( ( 2 x. A ) + 3 ) < 9 $. + $( Practice problem 5. Clues: ~ 3brtr3i ~ mpbi ~ breqtri ~ ltaddsubi + ~ remulcli ~ 2re ~ 3re ~ 9re ~ eqcomi ~ mvlladdi 3cn ~ 6cn ~ eqtr3i + ~ 6p3e9 ~ addcomi ~ ltdiv1ii ~ 6re ~ nngt0i ~ 2nn ~ divcan3i ~ recni + ~ 2cn ~ 2ne0 ~ mpbir ~ eqtri ~ mulcomi ~ 3t2e6 ~ divmuli . (Contributed + by Filip Cernatescu, 16-Mar-2019.) + (Proof modification is discouraged.) $) + problem5 $p |- A < 3 $= + ( c2 cmul co cdiv c6 c3 clt wbr c9 caddc 2re mpbi 3cn 6cn eqcomi 2cn 2ne0 + cmin remulcli 3re 9re ltaddsubi 6p3e9 addcomi eqtr3i mvlladdi breqtri 6re + 2nn nngt0i ltdiv1ii recni divcan3i wceq mulcomi 3t2e6 eqtri divmuli mpbir + 3brtr3i ) DAEFZDGFZHDGFZAIJVDHJKVEVFJKVDLIUAFZHJVDIMFLJKVDVGJKCVDILDANBUB + ZUCUDUEOHVGIHLPQLIHMFZHIMFLVIUFHIQPUGUHRUIRUJVDHDVHUKNDULUMUNOADABUOSTUPV + FIUQDIEFZHUQVJIDEFHDISPURUSUTHDIQSPTVAVBVC $. $} ${ - $d x y $. - 19.12b.1 $e |- F/ y ph $. - 19.12b.2 $e |- F/ x ps $. - $( Version of ~ 19.12vv with not-free hypotheses, instead of distinct - variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) - (Revised by Mario Carneiro, 11-Dec-2016.) $) - 19.12b $p |- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) ) $= - ( wi wal wex 19.21 exbii nfal 19.36 albii bitr2i 3bitri ) ABGZDHZCIABDHZG - ZCIACHZSGZQCIZDHZRTCABDEJKASCBCDFLMUDUABGZDHUBUCUEDABCFMNUABDADCELJOP $. + quad3.1 $e |- X e. CC $. + quad3.2 $e |- A e. CC $. + quad3.3 $e |- A =/= 0 $. + quad3.4 $e |- B e. CC $. + quad3.5 $e |- C e. CC $. + quad3.6 $e |- ( ( A x. ( X ^ 2 ) ) + ( ( B x. X ) + C ) ) = 0 $. + $( Variant of quadratic equation with discriminant expanded. (Contributed + by Filip Cernatescu, 19-Oct-2019.) $) + quad3 $p |- ( X = ( ( -u B + ( sqrt ` ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) + ) ) / ( 2 x. A ) ) \/ X = ( ( -u B - ( sqrt ` ( ( B ^ 2 ) - + ( 4 x. ( A x. C ) ) ) ) ) / ( 2 x. A ) ) ) $= + ( c2 cmul co cdiv caddc cmin wceq 2cn oveq2i oveq1i cexp c4 csqrt cneg wo + cfv mulcli 2ne0 mulne0i divcli addcli sqmuli binom2i sqcli divdiri div23i + divcan3i oveq12i eqtr2i mulcomi divcan2i divassi cc cc0 wa pm3.2i divdiv1 + wcel wne mp3an eqtri 3eqtr2i addassi eqcomi pncan3oi df-neg eqtr4i negcli + 3eqtr3i addcomi sqdivi 4cn 4ne0 divmuldivi c1 dividi eqtr3i mulm1i neg1cn + mulid2i mulassi 3eqtri 2t2e4 sqvali eqnetri negsubi subcli eqsqrtor ax-mp + wb mpbi sqrtcl divmuli eqcom bitr3i subadd2i divneg eqeq2i 3bitri orbi12i + ) KALMZDBXKNMZOMZLMZBKUAMZUBACLMZLMZPMZUCUFZQZXNXSUDZQZUEZDBUDZXSOMXKNMZQ + ZDYDXSPMZXKNMZQZUEXNKUAMZXRQZYCYJXKKUAMZXMKUAMZLMYLXRYLNMZLMXRXKXMKARFUGZ + DXLEBXKHYOKARFUHGUIZUJZUKZULYMYNYLLYMCUDZANMZXLKUAMZOMZUUAYTOMZYNYMDKUAMZ + KDXLLMZLMZOMZUUAOMUUBDXLEYQUMUUGYTUUAOUUDBANMZDLMZOMZAUUDLMZBDLMZOMZANMZU + UGYTUUNUUKANMZUULANMZOMUUJUUKUULAAUUDFDEUNZUGZBDHEUGZFGUOUUOUUDUUPUUIOUUD + AUUQFGUQBDAHEFGUPURUSUUIUUFUUDOUUIDUUHLMZKUUTKNMZLMUUFUUHDBAHFGUJZEUTUUTK + DUUHEUVBUGRUHVAUVAUUEKLUVADUUHKNMZLMUUEDUUHKEUVBRUHVBUVCXLDLUVCBAKLMZNMZX + LBVCVHZAVCVHZAVDVIZVEKVCVHZKVDVIZVEUVCUVEQHUVGUVHFGVFUVIUVJRUHVFBAKVGVJUV + DXKBNAKFRUTSVKSVKSVLSUUMYSANUUMUUKUULCOMOMZCPMZYSUVLUUMCOMZCPMUUMUVKUVMCP + UVMUVKUUKUULCUURUUSIVMVNTUUMCUUKUULUURUUSUKIVOUSUVLVDCPMYSUVKVDCPJTCVPVQV + KTVSTVKYTUUAYSACIVRZFGUJZXLYQUNVTUUCXOYLNMZXQUDZYLNMZOMXOUVQOMZYLNMYNUUAU + VPYTUVROBXKHYOYPWAUBALMZUVTNMZYTLMZUVTYSLMZUVTALMZNMYTUVRUVTUVTYSAUBAWBFU + GZUWEUVNFUBAWBFWCGUIZGWDWEYTLMUWBYTWEUWAYTLUWAWEUVTUWEUWFWFVNTYTUVOWJWGUW + CUVQUWDYLNUWCWEUDZUVTCLMZLMZUWGXQLMUVQUWCUVTUWGLMZCLMZUWGUVTLMZCLMUWIUWCU + VTUWGCLMZLMUWKYSUWMUVTLUWMYSCIWHVNSUVTUWGCUWEWIIWKVQUWJUWLCLUVTUWGUWEWIUT + TUWGUVTCWIUWEIWKWLUWHXQUWGLUBACWBFIWKSXQUBXPWBACFIUGUGZWHWLUWDXKXKLMZYLUW + DKXKLMZALMZXKKLMZALMUWOUWDKKLMZALMZALMUWQUVTUWTALUBUWSALUWSUBWMVNTTUWTUWP + ALKKARRFWKTVKUWPUWRALKXKRYOUTTXKKAYORFWKWLXKYOWNZVQURVSURXOUVQYLBHUNZXQUW + NVRXKYOUNZYLUWOVDUXAXKXKYOYOYPYPUIWOZUOUVSXRYLNXOXQUXBUWNWPTVLWLSXRYLXOXQ + UXBUWNWQZUXCUXDVAWLXNVCVHZXRVCVHZVEYKYCWTUXFUXGXKXMYOYRUGUXEVFXNXRWRWSXAX + TYFYBYIXTXMXSXKNMZQZDUXHXLPMZQZYFXTUXHXMQUXIXSXKXMUXGXSVCVHUXEXRXBWSZYOYR + YPXCUXHXMXDXEUXIUXJDQUXKUXHXLDXSXKUXLYOYPUJZYQEXFUXJDXDXEUXJYEDUXJYDXKNMZ + UXHOMZYEUXHXLUDZOMUXHUXNOMUXJUXOUXPUXNUXHOUVFXKVCVHXKVDVIUXPUXNQHYOYPBXKX + GVJZSUXHXLUXMYQWPUXHUXNUXMYDXKBHVRZYOYPUJZVTVSYDXSXKUXRUXLYOYPUOVQXHXIYBX + MYAXKNMZQZDUXTXLPMZQZYIYBUXTXMQUYAYAXKXMXSUXLVRZYOYRYPXCUXTXMXDXEUYAUYBDQ + UYCUXTXLDYAXKUYDYOYPUJZYQEXFUYBDXDXEUYBYHDUYBUXNUXTOMZYDYAOMZXKNMYHUXTUXP + OMUXTUXNOMUYBUYFUXPUXNUXTOUXQSUXTXLUYEYQWPUXTUXNUYEUXSVTVSYDYAXKUXRUYDYOY + PUOUYGYGXKNYDXSUXRUXLWPTVLXHXIXJXA $. $} - $( There is always a set not in ` y ` . (Contributed by Scott Fenton, - 13-Dec-2010.) $) - exnel $p |- E. x -. x e. y $= - ( wel wn wex elirrv nfth weq ax8 con3d spime ax-mp ) BBCZDZABCZDZAEBFZNPABN - AQGABHOMABBIJKL $. +$( (End of Filip Cernatescu's mathbox.) $) +$( End $[ set-mbox-fc.mm $] $) - ${ - $d x z $. $d y z $. - $( Distinctors in terms of membership. (NOTE: this only works with - relations where we can prove ~ el and ~ elirrv .) (Contributed by Scott - Fenton, 15-Dec-2010.) $) - distel $p |- ( -. A. y y = x <-> -. A. y -. x e. y ) $= - ( vz weq wal wn wel wex el df-ex nfnae dveel1 nf5d nfnd elequ2 notbid a1i - wb wi cbvald bitrid mpbii elirrv elequ1 mtbii alimi con3i impbii ) BADZBE - ZFZABGZFZBEZFZUKACGZCHZUOACIUQUPFZCEZFUKUOUPCJUKUSUNUKURUMCBBABKZUKUPBUKU - PBUTBACLMNCBDZURUMRSUKVAUPULCBAOPQTPUAUBUJUNUIUMBUIBBGULBUCBABUDUEUFUGUH - $. - $} - $( ~ axextnd as a biconditional. (Contributed by Scott Fenton, - 14-Dec-2010.) $) - axextndbi $p |- E. z ( x = y <-> ( z e. x <-> z e. y ) ) $= - ( weq wel wb wex wi wa axextnd elequ2 jctl eximii dfbi2 exbii mpbir ) ABDZC - AECBEFZFZCGQRHZRQHZIZCGUAUBCCABJUATABCKLMSUBCQRNOP $. +$( Begin $[ set-mbox-pc.mm $] $) +$( Skip $[ set-main.mm $] $) +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Paul Chapman +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Hypothesis builders + Propositional calculus =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $( A more general form of ~ hbnt . (Contributed by Scott Fenton, - 13-Dec-2010.) $) - hbntg $p |- ( A. x ( ph -> A. x ps ) -> ( -. ps -> A. x -. ph ) ) $= - ( wn wal wi axc7 con1i con3 al2imi syl5 ) BDBCEZDZCEZALFZCEADZCENBBCGHOMPCA - LIJK $. - - $( A more general and closed form of ~ hbim . (Contributed by Scott Fenton, - 13-Dec-2010.) $) - hbimtg $p |- ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> - ( ( ch -> ps ) -> A. x ( ph -> th ) ) ) $= - ( wal wi wa wn hbntg pm2.21 alimi syl6 adantr ala1 imim2i adantl jad ) ACEF - GEFZBDEFZGZHCBADGZEFZSCIZUCGUASUDAIZEFUCACEJUEUBEADKLMNUABUCGSTUCBDAEOPQR - $. - $( A more general and closed form of ~ hbal . (Contributed by Scott Fenton, - 13-Dec-2010.) $) - hbaltg $p |- ( A. x ( ph -> A. y ps ) -> ( A. x ph -> A. y A. x ps ) ) $= - ( wal wi alim ax-11 syl6 ) ABDEZFCEACEJCEBCEDEAJCGBCDHI $. +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Prime numbers +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) - ${ - hbg.1 $e |- ( ph -> A. x ps ) $. - $( A more general form of ~ hbn . (Contributed by Scott Fenton, - 13-Dec-2010.) $) - hbng $p |- ( -. ps -> A. x -. ph ) $= - ( wal wi wn hbntg mpg ) ABCEFBGAGCEFCABCHDI $. - ${ - hbg.2 $e |- ( ch -> A. x th ) $. - $( A more general form of ~ hbim . (Contributed by Scott Fenton, - 13-Dec-2010.) $) - hbimg $p |- ( ( ps -> ch ) -> A. x ( ph -> th ) ) $= - ( wal wi ax-gen hbimtg mp2an ) ABEHIZEHCDEHIBCIADIEHIMEFJGACBDEKL $. - $} - $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Finite set induction +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - (Trans)finite Recursion Theorems + The ` # ` (set size) function =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - ${ - $d ph y z $. $d x y z $. - $( A closed form of ~ tfis . (Contributed by Scott Fenton, 8-Jun-2011.) $) - tfisg $p |- ( A. x e. On ( A. y e. x [. y / x ]. ph -> ph ) -> - A. x e. On ph ) $= - ( vz cv wsbc wral wi con0 crab wceq wss wcel ssrab2 wa dfss3 nfcv elrabsf - simprbi nfsbc1v ralimi sylbi nfralw nfim sbceq1a imbi12d rspc impcom syl5 - raleq simpr jctild syl6ibr ralrimiva tfi sylancr eqcomd rabid2 sylib ) AB - CEZFZCBEZGZAHZBIGZIABIJZKABIGVEVFIVEVFILDEZVFLZVGVFMZHZDIGVFIKABINVEVJDIV - EVGIMZOZVHVKABVGFZOVIVLVHVMVKVHVACVGGZVLVMVHUTVFMZCVGGVNCVGVFPVOVACVGVOUT - IMVAABUTIBIQZRSUAUBVKVEVNVMHZVDVQBVGIVNVMBVABCVGBVGQABUTTUCABVGTUDVBVGKVC - VNAVMVACVBVGUJABVGUEUFUGUHUIVEVKUKULABVGIVPRUMUNDVFUOUPUQABIURUS $. - $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Well-Founded Induction + Real and complex numbers (cont.) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ - $d A p $. $d A q $. $d A w $. $d A x $. $d A y $. $d A z $. $d B p $. - $d B q $. $d B w $. $d B x $. $d B y $. $d B z $. $d ch q $. - $d ch y $. $d p ph $. $d p q $. $d p x $. $d p y $. $d ph q $. - $d ph z $. $d ps w $. $d ps x $. $d q w $. $d q x $. $d q y $. - $d q z $. $d R p $. $d R q $. $d R w $. $d R x $. $d R y $. $d R z $. - $d ta y $. $d th x $. $d w x $. $d w y $. $d w z $. $d X x $. - $d x y $. $d X y $. $d x z $. $d Y y $. $d y z $. - frpoins3xpg.1 $e |- ( ( x e. A /\ y e. B ) -> - ( A. z A. w ( <. z , w >. e. Pred ( R , ( A X. B ) , <. x , y >. ) -> ch ) - -> ph ) ) $. - frpoins3xpg.2 $e |- ( x = z -> ( ph <-> ps ) ) $. - frpoins3xpg.3 $e |- ( y = w -> ( ps <-> ch ) ) $. - frpoins3xpg.4 $e |- ( x = X -> ( ph <-> th ) ) $. - frpoins3xpg.5 $e |- ( y = Y -> ( th <-> ta ) ) $. - $( Special case of founded partial induction over a cross product. - (Contributed by Scott Fenton, 22-Aug-2024.) $) - frpoins3xpg $p |- ( ( ( R Fr ( A X. B ) /\ R Po ( A X. B ) /\ - R Se ( A X. B ) ) /\ ( X e. A /\ Y e. B ) ) -> ta ) $= - ( vq vp wcel wa cxp wfr wpo wse w3a wral c2nd cfv wsbc c1st cop wceq wrex - cv cpred wi elxp2 nfcv nfsbc1v nfralw nfv nfsbcw wal weq sbcbidv cbvsbcvw - nfim sbcbii bitri ralbii impexp cin elin wss predss sseqin2 eleq2i bitr3i - imbi1i albii df-ral 3bitr4ri eleq1 sbcopeq1a imbi12d ralxpf opelxp 2albii - mpbi r2al 3bitri syl5bi predeq3 raleqdv syl5ibrcom ex rexlimd sylbi fveq2 - rexlimi sbceq1d sbceqbid frpoins2g ralxpes sylib rspc2va sylan2 ancoms ) - MJUBNKUBUCZJKUDZLUEXMLUFXMLUGUHZEXNXLAGKUIFJUIZEXNAGUAUQZUJUKZULZFXPUMUKZ - ULZUAXMUIXOXTAGTUQZUJUKZULZFYAUMUKZULZUATXMLXPXMUBXPFUQZGUQZUNZUOZGKUPZFJ - UPYETXMLXPURZUIZXTUSZFGXPJKUTYJYMFJYLXTFYEFTYKFYKVAYCFYDVBVCXRFXSVBVJYFJU - BZYIYMGKYNGVDYLXTGYEGTYKGYKVAYCGFYDGYDVAAGYBVBVEVCXRGFXSGXSVAAGXQVBVEVJYN - YGKUBZYIYMUSYNYOUCZYMYIYETXMLYHURZUIZAUSYRHUQZIUQZUNZYQUBZCUSZIVFHVFZYPAY - RCIYBULZHYDULZTYQUIZYAYQUBZUUFUSZTXMUIZUUDYEUUFTYQYEBGYBULZHYDULUUFYCUUKF - HYDFHVGABGYBPVHVIUUKUUEHYDBCGIYBQVIVKVLVMYAXMUBZUUIUSZTVFUUITVFUUJUUGUUMU - UITUUMUULUUHUCZUUFUSUUIUULUUHUUFVNUUNUUHUUFUUNYAXMYQVOZUBUUHYAXMYQVPUUOYQ - YAYQXMVQUUOYQUOXMLYHVRYQXMVSWLZVTWAWBWAWCUUITXMWDUUFTYQWDWEUUJUUCIKUIHJUI - YSJUBYTKUBUCZUUCUSZIVFHVFUUDUUIUUCTHIJKUUHUUFHUUHHVDUUEHYDVBVJUUHUUFIUUHI - VDUUEIHYDIYDVACIYBVBVEVJUUCTVDYAUUAUOUUHUUBUUFCYAUUAYQWFCHIYAWGWHWIUUCHIJ - KWMUURUUCHIUURUUAXMUBZUUBUCZCUSZUUCUVAUUSUUCUSUURUUSUUBCVNUUSUUQUUCYSYTJK - WJWBVLUUTUUBCUUTUUAUUOUBUUBUUAXMYQVPUUOYQUUAUUPVTWAWBWAWKWNWNOWOYIYLYRXTA - YIYETYKYQXMLXPYHWPWQAFGXPWGWHWRWSWTXCXAUATVGZXRYCFXSYDXPYAUMXBUVBAGXQYBXP - YAUJXBXDXEXFAUAFGJKXGXHAEDFGMNJKRSXIXJXK $. + $d k m $. + $( Utility lemma to convert between ` m <_ k ` and ` k e. ( ZZ>= `` m ) ` + in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.) $) + climuzcnv $p |- ( m e. NN -> ( ( k e. ( ZZ>= ` m ) -> ph ) <-> + ( k e. NN -> ( m <_ k -> ph ) ) ) ) $= + ( cv cn wcel cuz cfv wi cle wbr wa elnnuz uztrn sylan2b sylibr expcom nnz + c1 cz eluzle a1i jcad biimpri syl3an1 syl3an2 3expib impbid imbi1d impexp + w3a eluz2 bitrdi ) CDZEFZBDZUNGHFZAIUPEFZUNUPJKZLZAIURUSAIIUOUQUTAUOUQUTU + OUQURUSUQUOURUQUOLUPSGHZFZURUOUQUNVAFVBUNMUNUPSNOUPMPQUQUSIUOUNUPUAUBUCUO + URUSUQURUOUPTFZUSUQUPRUOUNTFZVCUSUQUNRUQVDVCUSUKUNUPULUDUEUFUGUHUIURUSAUJ + UM $. $} ${ - $d A p $. $d A q $. $d A x $. $d A y $. $d A z $. $d B p $. $d B q $. - $d B x $. $d B y $. $d B z $. $d C p $. $d C q $. $d C x $. $d C y $. - $d C z $. $d et y $. $d p ph $. $d p q $. $d p x $. $d p y $. - $d p z $. $d ph q $. $d q x $. $d q y $. $d q z $. $d R p $. - $d R q $. $d ta x $. $d X x $. $d x y $. $d X y $. $d x z $. - $d X z $. $d Y y $. $d y z $. $d Y z $. $d Z z $. $d z ze $. - $d A t $. $d A u $. $d A w $. $d B t $. $d B u $. $d B w $. $d C t $. - $d C u $. $d C w $. $d ch u $. $d ch y $. $d ph w $. $d ps t $. - $d ps x $. $d q t $. $d q th $. $d q u $. $d q w $. $d R t $. - $d R u $. $d R w $. $d R x $. $d R y $. $d R z $. $d t u $. $d t w $. - $d t x $. $d t y $. $d t z $. $d th x $. $d th y $. $d th z $. - $d u w $. $d u x $. $d u y $. $d u z $. $d w x $. $d w y $. $d w z $. - frpoins3xp3g.1 $e |- ( ( x e. A /\ y e. B /\ z e. C ) -> - ( A. w A. t A. u - ( <. <. w , t >. , u >. e. Pred ( R , ( ( A X. B ) X. C ) , - <. <. x , y >. , z >. ) -> th ) -> ph ) ) $. - frpoins3xp3g.2 $e |- ( x = w -> ( ph <-> ps ) ) $. - frpoins3xp3g.3 $e |- ( y = t -> ( ps <-> ch ) ) $. - frpoins3xp3g.4 $e |- ( z = u -> ( ch <-> th ) ) $. - frpoins3xp3g.5 $e |- ( x = X -> ( ph <-> ta ) ) $. - frpoins3xp3g.6 $e |- ( y = Y -> ( ta <-> et ) ) $. - frpoins3xp3g.7 $e |- ( z = Z -> ( et <-> ze ) ) $. - $( Special case of founded partial recursion over a triple cross product. - (Contributed by Scott Fenton, 22-Aug-2024.) $) - frpoins3xp3g $p |- ( ( ( R Fr ( ( A X. B ) X. C ) - /\ R Po ( ( A X. B ) X. C ) - /\ R Se ( ( A X. B ) X. C ) ) /\ ( X e. A /\ Y e. B /\ Z e. C ) ) - -> ze ) $= - ( vp vq cxp wfr wpo wse w3a wral wcel cv c2nd cfv wsbc c1st cpred sbcbidv - weq cbvsbcvw sbcbii bitri ralbii cop wceq wrex wi elxpxp nfv nfsbc1v nfim - nfcv nfsbcw wa wal impexp cin elin wss predss sseqin2 eleq2i bitr3i albii - mpbi imbi1i r3al sbcoteq1a imbi12d ralxp3f ot2elxp 2albii 3bitr4ri df-ral - eleq1 syl5bi predeq3 raleqdv syl5ibrcom 3expia rexlimd sylbi 2fveq3 fveq2 - ex rexlimi sbceq1d sbceqbid frpoins2g ralxp3es sylib rspc3v mpan9 ) NOUJP - UJZQUKXSQULXSQUMUNZAJPUOIOUOHNUOZRNUPSOUPTPUPUNGXTAJUHUQZURUSZUTZIYBVAUSZ - URUSZUTZHYEVAUSZUTZUHXSUOYAYIAJUIUQZURUSZUTZIYJVAUSZURUSZUTZHYMVAUSZUTZUH - UIXSQYQUIXSQYBVBZUODLYKUTZMYNUTZKYPUTZUIYRUOZYBXSUPZYIYQUUAUIYRYQBJYKUTZI - YNUTZKYPUTUUAYOUUEHKYPHKVDZYLUUDIYNUUFABJYKUBVCVCVEUUEYTKYPUUECJYKUTZMYNU - TYTUUDUUGIMYNIMVDBCJYKUCVCVEUUGYSMYNCDJLYKUDVEVFVGVFVGVHUUCYBHUQZIUQZVIJU - QZVIZVJZJPVKZIOVKZHNVKUUBYIVLZHIJYBNOPVMUUNUUOHNUUBYIHUUBHVNYGHYHVOVPUUHN - UPZUUMUUOIOUUPIVNUUBYIIUUBIVNYGIHYHIYHVQYDIYFVOVRVPUUPUUIOUPZUUMUUOVLUUPU - UQVSZUULUUOJPUURJVNUUBYIJUUBJVNYGJHYHJYHVQYDJIYFJYFVQAJYCVOVRVRVPUUPUUQUU - JPUPZUULUUOVLUUPUUQUUSUNZUUOUULUUAUIXSQUUKVBZUOZAVLUVBKUQZMUQZVILUQZVIZUV - AUPZDVLZLVTZMVTKVTZUUTAYJXSUPZYJUVAUPZUUAVLZVLZUIVTZUVMUIVTUVJUVBUVNUVMUI - UVNUVKUVLVSZUUAVLUVMUVKUVLUUAWAUVPUVLUUAUVPYJXSUVAWBZUPUVLYJXSUVAWCUVQUVA - YJUVAXSWDUVQUVAVJXSQUUKWEUVAXSWFWJZWGWHWKWHWIUVJUVMUIXSUOZUVOUVHLPUOMOUOK - NUOUVCNUPUVDOUPUVEPUPUNZUVHVLZLVTZMVTKVTUVSUVJUVHKMLNOPWLUVMUVHUIKMLNOPUV - LUUAKUVLKVNYTKYPVOVPUVLUUAMUVLMVNYTMKYPMYPVQYSMYNVOVRVPUVLUUALUVLLVNYTLKY - PLYPVQYSLMYNLYNVQDLYKVOVRVRVPUVHUIVNYJUVFVJUVLUVGUUADYJUVFUVAWTDKMLYJWMWN - WOUVIUWBKMUVHUWALUVHUVFXSUPZUVHVLZUWAUVHUWCUVGVSZDVLUWDUWEUVGDUWEUVFUVQUP - UVGUVFXSUVAWCUVQUVAUVFUVRWGWHWKUWCUVGDWAWHUWCUVTUVHUVCUVDUVENOPWPWKVGWIWQ - WRUVMUIXSWSVGUUAUIUVAWSWRUAXAUULUUBUVBYIAUULUUAUIYRUVAXSQYBUUKXBXCAHIJYBW - MWNXDXEXFXJXFXKXGXAUHUIVDZYGYOHYHYPYBYJVAVAXHUWFYDYLIYFYNYBYJURVAXHUWFAJY - CYKYBYJURXIXLXMXMXNAUHHIJNOPXOXPAGEFHIJRSTNOPUEUFUGXQXR $. + $d k w x y z F $. $d k w y z H $. $d k z M $. $d k w y z ph $. + $d k G $. + sinccvg.1 $e |- ( ph -> F : NN --> ( RR \ { 0 } ) ) $. + sinccvg.2 $e |- ( ph -> F ~~> 0 ) $. + sinccvg.3 $e |- G = ( x e. ( RR \ { 0 } ) |-> ( ( sin ` x ) / x ) ) $. + sinccvg.4 $e |- H = ( x e. CC |-> ( 1 - ( ( x ^ 2 ) / 3 ) ) ) $. + sinccvg.5 $e |- ( ph -> M e. NN ) $. + sinccvg.6 $e |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> + ( abs ` ( F ` k ) ) < 1 ) $. + $( ` ( ( sin `` x ) / x ) ~~> 1 ` as (real) ` x ~~> 0 ` . (Contributed by + Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, + 21-May-2014.) $) + sinccvglem $p |- ( ph -> ( G o. F ) ~~> 1 ) $= + ( c1 cc0 wcel cc co c3 cdiv vy vz vw ccom cvv cuz cfv eqid nnzd cli cv c2 + wfun cexp cmin funmpt2 cn cr csn cdif wf nnex sylancl cofunexg sylancr wa + fex wne adantr eluznn sylan ffvelcdmd eldifsn sylib simpld recnd sqcl 3cn + ax-1cn 3ne0 divcl mp3an23 syl subcl fmpti ccncf crp cabs clt wi wral wrex + wbr cmpt ccnfld ctopn ccn wtru ctopon cnfldtopon 1cnd cnmptc divccn mp2an + a1i sqcn oveq1 cnmpt11 ctx subcn cnmpt12f mptru cncfcn1 cncfi mp3an1 wceq + 3eltr4i fvco3 syldan climcn1lem 0cn oveq1d eqtrdi oveq2d fvmpt csin eqtrd + ovex 1re eqeltrd fveq2 id oveq12d resincld simprd cmul eqbrtrd cneg ltled + cle sq0i div0i 1m0e1 1ex breqtrdi resqcld nndivre resubcl redivcld abscld + ax-mp 3nn subdird mulid2d caddc df-3 oveq2i cn0 2nn0 expp1 absresq eqtrid + div23d eqtr2d cioc absrpcld rpgt0d ltle mpd cxr w3a wb syl3anbrc sin01bnd + 0xr elioc2 ltmuldivd mpbid sinneg eqeq1d syl5ibrcom absord mpjaod breqtrd + div2negd 3brtr4d mulid1d breqtrrd ltdivmuld mpbird eqbrtrrd climsqz ) ANC + FDUDZEDUDZGUEGUFUGZUWOUHZAGLUIZAUWMOFUGZNUJAUAUBUCOCDUWMFGUEUWOUWPIAFUMDU + EPZUWMUEPBQNBUKZULUNRZSTRZUORZFKUPAUQUROUSUTZDVAZUQUEPUWSHVBUQUXDUEDVGVCZ + FDUEVDVEUWQACUKZUWOPZVFZUXGDUGZUXIUXJURPZUXJOVHZUXIUXJUXDPZUXKUXLVFUXIUQU + XDUXGDAUXEUXHHVIAGUQPUXHUXGUQPZLUXGGVJVKZVLZUXJUROVMVNZVOZVPZBQQUXCFKUWTQ + PZNQPUXBQPZUXCQPVSUXTUXAQPZUYAUWTVQUYBSQPZSOVHZUYAVRVTUXASWAWBWCNUXBWDVEW + EFQQWFRZPOQPZUAUKZWGPUCUKZOUORWHUGUBUKWIWMUYHFUGUWRUORWHUGUYGWIWMWJUCQWKU + BWGWLBQUXCWNZWOWPUGZUYJWQRZFUYEUYIUYKPWRBNUXBUOUYJUYJUYJUYJQUYJQWSUGPWRUY + JUYJUHZWTXEZWRBNUYJUYJQQUYMUYMWRXAXBWRBUAUXAUYGSTRZUXBUYJUYJUYJQQUYMBQUXA + WNUYKPWRBUYJUYLXFXEUYMUAQUYNWNUYKPZWRUYCUYDUYOVRVTUASUYJUYLXCXDXEUYGUXAST + XGXHUOUYJUYJXIRUYJWQRPWRUYJUYLXJXEXKXLKUYJUYLXMXQUBUCQQOUYGFXNXOAUXHUXNUX + GUWMUGZUXJFUGZXPZUXOAUXEUXNUYRHUQUXDUXGFDXRVKXSZXTUYFUWRNXPYABOUXCNQFUWTO + XPZUXCNOUORNUYTUXBONUOUYTUXBOSTROUYTUXAOSTUWTUUAYBSVRVTUUBYCYDUUCYCKUUDYE + UUKUUEAEUMUWSUWNUEPBUXDUWTYFUGZUWTTRZEJUPUXFEDUEVDVEUXIUYPNUXJULUNRZSTRZU + ORZURUXIUYPUYQVUEUYSUXIUXJQPZUYQVUEXPUXSBUXJUXCVUEQFUWTUXJXPZUXBVUDNUOVUG + UXAVUCSTUWTUXJULUNXGYBYDKNVUDUOYHYEWCYGZUXINURPZVUDURPZVUEURPYIUXIVUCURPS + UQPVUJUXIUXJUXRUUFZUULVUCSUUGVCZNVUDUUHVEZYJUXIUXGUWNUGZUXJYFUGZUXJTRZURU + XIVUNUXJEUGZVUPAUXHUXNVUNVUQXPZUXOAUXEUXNVURHUQUXDUXGEDXRVKXSUXIUXMVUQVUP + XPUXPBUXJVUBVUPUXDEVUGVUAVUOUWTUXJTUWTUXJYFYKVUGYLYMJVUOUXJTYHYEWCYGZUXIV + UOUXJUXIUXJUXRYNZUXRUXIUXKUXLUXQYOZUUIZYJUXIVUEVUPUYPVUNYTUXIVUEVUPVUMVVB + UXIVUEUXJWHUGZYFUGZVVCTRZVUPWIUXIVUEVVCYPRZVVDWIWMVUEVVEWIWMUXIVVFVVCVVCS + UNRZSTRZUORZVVDWIUXIVVFNVVCYPRZVUDVVCYPRZUORVVIUXINVUDVVCUXIXAUXIVUDVULVP + UXIVVCUXIUXJUXSUUJZVPZUUMUXIVVJVVCVVKVVHUOUXIVVCVVMUUNUXIVVHVUCVVCYPRZSTR + VVKUXIVVGVVNSTUXIVVGVVCULNUUORZUNRZVVNSVVOVVCUNUUPUUQUXIVVPVVCULUNRZVVCYP + RZVVNUXIVVCQPULUURPVVPVVRXPVVMUUSVVCULUUTVCUXIVVQVUCVVCYPUXIUXKVVQVUCXPUX + RUXJUVAWCYBYGUVBYBUXIVUCVVCSUXIVUCVUKVPVVMUYCUXIVRXEUYDUXIVTXEUVCUVDYMYGU + XIVVIVVDWIWMZVVDVVCWIWMZUXIVVCONUVERPZVVSVVTVFUXIVVCURPZOVVCWIWMZVVCNYTWM + ZVWAVVLUXIVVCUXIUXJUXSVVAUVFZUVGUXIVVCNWIWMZVWDMUXIVWBVUIVWFVWDWJVVLYIVVC + NUVHVCUVIOUVJPVUIVWAVWBVWCVWDUVKUVLUVOYIONVVCUVPXDUVMVVCUVNWCZVOYQUXIVUEV + VDVVCVUMUXIVVCVVLYNZVWEUVQUVRUXIVVCUXJXPZVVEVUPXPZVVCUXJYRZXPZVWIVWJWJUXI + VWIVVDVUOVVCUXJTVVCUXJYFYKVWIYLYMXEUXIVWJVWLVWKYFUGZVWKTRZVUPXPUXIVWNVUOY + RZVWKTRVUPUXIVWMVWOVWKTUXIVUFVWMVWOXPUXSUXJUVSWCYBUXIVUOUXJUXIVUOVUTVPUXS + VVAUWEYGVWLVVEVWNVUPVWLVVDVWMVVCVWKTVVCVWKYFYKVWLYLYMUVTUWAUXIUXJUXRUWBUW + CZUWDYSVUHVUSUWFUXIVUNVUPNYTVUSUXIVUPNVVBVUIUXIYIXEZUXIVVEVUPNWIVWPUXIVVE + NWIWMVVDVVCNYPRZWIWMUXIVVDVVCVWRWIUXIVVSVVTVWGYOUXIVVCVVMUWGUWHUXIVVDNVVC + VWHVWQVWEUWIUWJUWKYSYQUWL $. + $} + + ${ + $d j k n x F $. + $( ` ( ( sin `` x ) / x ) ~~> 1 ` as (real) ` x ~~> 0 ` . (Contributed by + Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, + 21-May-2014.) $) + sinccvg $p |- ( ( F : NN --> ( RR \ { 0 } ) /\ F ~~> 0 ) -> + ( ( x e. ( RR \ { 0 } ) |-> ( ( sin ` x ) / x ) ) o. F ) ~~> 1 ) $= + ( vk vj vn cn cc0 cli wbr wa cv cfv cabs c1 clt cdiv co cmpt wcel eqid cr + csn cdif wf cuz wral csin ccom nnuz 1zzd crp 1rp eqidd simpr climi0 cc c2 + a1i cexp cmin simpll simplr simprl simprr weq 2fveq3 breq1d rspccva sylan + c3 sinccvglem rexlimddv ) FUAGUBUCZBUDZBGHIZJZCKZBLZMLZNOIZCDKZUELZUFZAVM + AKZUGLWDPQRZBUHNHIDFVPVRNDCBNFUIVPUJNUKSVPULURVPVQFSJVRUMVNVOUNUOVPWAFSZW + CJZJZAEBWEAUPNWDUQUSQVJPQUTQRZWAVNVOWGVAVNVOWGVBWETWITVPWFWCVCWHWCEKZWBSW + JBLMLZNOIZVPWFWCVDVTWLCWJWBCEVEVSWKNOVQWJMBVFVGVHVIVKVL $. + $} + + ${ + $d n y $. $d k P $. $d k n R $. + circum.1 $e |- A = ( ( 2 x. _pi ) / n ) $. + circum.2 $e |- + P = ( n e. NN |-> ( ( 2 x. n ) x. ( R x. ( sin ` ( A / 2 ) ) ) ) ) $. + circum.3 $e |- R e. RR $. + $( The circumference of a circle of radius ` R ` , defined as the limit as + ` n ~~> +oo ` of the perimeter of an inscribed n-sided isogons, is + ` ( ( 2 x. _pi ) x. R ) ` . (Contributed by Paul Chapman, 10-Nov-2012.) + (Proof shortened by Mario Carneiro, 21-May-2014.) $) + circum $p |- P ~~> ( ( 2 x. _pi ) x. R ) $= + ( c2 cpi cmul co wtru cn cdiv csin cfv cr cc0 wcel cc vk vy c1 cli wbr cv + cmpt cvv nnuz 1zzd csn cdif ccom wne wa crp pirp rpdivcl sylancr rprene0d + nnrp eldifsn sylibr adantl eqidd wceq fveq2 id oveq12d wf eqid fmpti pire + fmptco recni divcnv mp1i sinccvg eqbrtrrd 2re remulcli nnex mptex eqeltri + a1i resincld eldifsni redivcld fco mp2an mptru feq1i mpbi ffvelcdmi recnd + eldifi nncn nnne0 divassd oveq1d simpr nndivre 2ne0 divcan3d eqtrd fveq2d + rpne0d divcan2d eqtr4d oveq2d oveq2 ovex fvmpt mulassd mulcl mul4d mul32d + eqeltrrd 3eqtr2d eqtrid 3eqtr4d climmulc2 mulid1i breqtri ) BHIJKZCJKZUCJ + KZYFUDBYGUDUELUCYFUADMIDUFZNKZOPZYINKZUGZBUCUHMUILUJLUBQRUKZULZUBUFZOPZYO + NKZUGZDMYIUGZUMZYLUCUDLDUBMYNYIYQYKYSYRYHMSZYIYNSZLUUAYIQSYIRUNUOUUBUUAYI + UUAIUPSZYHUPSYIUPSUQYHVAIYHURUSUTYIQRVBVCZVDLYSVELYRVEYOYIVFZYPYJYOYINYOY + IOVGUUEVHVIVNZLMYNYSVJZYSRUDUEZYTUCUDUEDMYNYIYSYSVKUUDVLZITSZUUHLIVMVOZID + VPVQUBYSVRUSVSYFTSLYFYECHIVTVMWAGWAVOZWEBUHSLBDMHYHJKZCAHNKZOPZJKZJKZUGUH + FDMUUQWBWCWDWELUAUFZMSZUOZUURYLPZUUSUVAQSLMQUURYLMQYTVJZMQYLVJYNQYRVJUUGU + VBUBYNQYQYRYRVKYOYNSZYPYOUVCYOYOQYMWPZWFUVDYOQRWGWHVLUUIMYNQYRYSWIWJMQYTY + LYTYLVFUUFWKWLWMWNVDWOZUUTHUURJKZCYEUURNKZHNKZOPZJKZJKZYFIUURNKZOPZUVLNKZ + JKZUURBPZYFUVAJKUUTUVKUVFCUVLJKZUVNJKZJKUVFUVQJKZUVNJKUVOUUTUVJUVRUVFJUUT + UVJCUVLUVNJKZJKUVRUUTUVIUVTCJUUTUVIUVMUVTUUTUVHUVLOUUTUVHHUVLJKZHNKUVLUUT + UVGUWAHNUUTHIUURHTSZUUTHVTVOZWEZUUJUUTUUKWEZUUSUURTSZLUURWQVDZUUSUURRUNLU + URWRVDZWSWTUUTUVLHUUTUVLUUTIQSUUSUVLQSVMLUUSXAIUURXBUSZWOZUWDHRUNUUTXCWEX + DXEXFUUTUVMUVLUUTUVMUUTUVLUWIWFWOUWJUUTUVLUUTUUCUURUPSZUVLUPSUQUUSUWKLUUR + VAVDIUURURUSXGXHXIXJUUTCUVLUVNCTSZUUTCGVOZWEZUWJUUTUVAUVNTUUSUVAUVNVFLDUU + RYKUVNMYLYHUURVFZYJUVMYIUVLNUWOYIUVLOYHUURINXKZXFUWPVIYLVKUVMUVLNXLXMVDZU + VEXRZXNXIXJUUTUVFUVQUVNUUTUWBUWFUVFTSUWCUWGHUURXOUSUUTUWLUVLTSUVQTSUWMUWJ + CUVLXOUSUWRXNUUTUVSYFUVNJUUTUVSHCJKZUURUVLJKZJKZYFUUTHUURCUVLUWDUWGUWNUWJ + XPUUTUXAUWSIJKYFUUTUWTIUWSJUUTIUURUWEUWGUWHXHXJUUTHCIUWDUWNUWEXQXEXEWTXSU + USUVPUVKVFLDUURUUQUVKMBUWOUUMUVFUUPUVJJYHUURHJXKUWOUUOUVICJUWOUUNUVHOUWOA + UVGHNUWOAYEYHNKUVGEYHUURYENXKXTWTXFXJVIFUVFUVJJXLXMVDUUTUVAUVNYFJUWQXJYAY + BWKYFUULYCYD $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Ordering Cross Products, Part 2 + Miscellaneous theorems =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - ${ - xpord2.1 $e |- T = { <. x , y >. | ( x e. ( A X. B ) /\ y e. ( A X. B ) - /\ ( ( ( 1st ` x ) R ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ - ( ( 2nd ` x ) S ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ - x =/= y ) ) } $. + $( Membership in a curtailed finite sequence of integers. (Contributed by + Paul Chapman, 17-Nov-2012.) $) + elfzm12 $p |- ( N e. NN -> + ( M e. ( 1 ... ( N - 1 ) ) -> M e. ( 1 ... N ) ) ) $= + ( cn wcel c1 cmin co cfz cz cuz cfv wss nnz cle wbr zre lem1d peano2zm eluz + wb mpancom mpbird fzss2 3syl sseld ) BCDZEBEFGZHGZEBHGZAUFBIDZBUGJKDZUHUILB + MUJUKUGBNOZUJBBPQUGIDUJUKULTBRUGBSUAUBUGEBUCUDUE $. - ${ - $d a x $. $d A x $. $d a y $. $d A y $. $d b x $. $d B x $. - $d b y $. $d B y $. $d c x $. $d c y $. $d d x $. $d d y $. - $d R x $. $d R y $. $d S x $. $d S y $. $d x y $. - $( Lemma for cross product ordering. Calculate the value of the cross - product relationship. (Contributed by Scott Fenton, 19-Aug-2024.) $) - xpord2lem $p |- ( <. a , b >. T <. c , d >. <-> - ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) /\ - ( ( a R c \/ a = c ) /\ ( b S d \/ b = d ) /\ - ( a =/= c \/ b =/= d ) ) ) ) $= - ( cv wcel cfv wbr wceq wo wne w3a cxp c1st c2nd wa cop weq eleq1 opelxp - bitrdi vex op1std breq1d eqeq1d orbi12d op2ndd neeq1 3anbi123d 3anbi13d - opex breq2d eqeq2d neeq2 opthne 3anbi23d brab ) AMZCDUAZNZBMZVGNZVFUBOZ - VIUBOZEPZVKVLQZRZVFUCOZVIUCOZFPZVPVQQZRZVFVISZTZTHMZCNIMZDNUDZVJWCVLEPZ - WCVLQZRZWDVQFPZWDVQQZRZWCWDUEZVISZTZTWEJMZCNKMZDNUDZWCWOEPZHJUFZRZWDWPF - PZIKUFZRZWCWOSWDWPSRZTZTABWLWOWPUEZGWCWDUSWOWPUSVFWLQZVHWEWBWNVJXGVHWLV - GNWEVFWLVGUGWCWDCDUHUIXGVOWHVTWKWAWMXGVMWFVNWGXGVKWCVLEWCWDVFHUJZIUJZUK - ZULXGVKWCVLXJUMUNXGVRWIVSWJXGVPWDVQFWCWDVFXHXIUOZULXGVPWDVQXKUMUNVFWLVI - UPUQURVIXFQZVJWQWNXEWEXLVJXFVGNWQVIXFVGUGWOWPCDUHUIXLWHWTWKXCWMXDXLWFWR - WGWSXLVLWOWCEWOWPVIJUJZKUJZUKZUTXLVLWOWCXOVAUNXLWIXAWJXBXLVQWPWDFWOWPVI - XMXNUOZUTXLVQWPWDXPVAUNXLWMWLXFSXDVIXFWLVBWCWDWOWPXHXIVCUIUQVDLVE $. - $} + ${ + $d F k $. $d N k $. + nn0seqcvg.1 $e |- F : NN0 --> NN0 $. + nn0seqcvg.2 $e |- N = ( F ` 0 ) $. + nn0seqcvg.3 $e |- ( k e. NN0 -> ( ( F ` ( k + 1 ) ) =/= 0 -> + ( F ` ( k + 1 ) ) < ( F ` k ) ) ) $. + $( A strictly-decreasing nonnegative integer sequence with initial term + ` N ` reaches zero by the ` N ` th term. Inference version. + (Contributed by Paul Chapman, 31-Mar-2011.) $) + nn0seqcvg $p |- ( F ` N ) = 0 $= + ( c1 wceq cfv cc0 eqid cn0 wf a1i cv wcel caddc co wne clt wbr nn0seqcvgd + wi adantl ax-mp ) GGHZCBIJHGKUFABCLLBMUFDNCJBIHUFENAOZLPUGGQRBIZJSUHUGBIT + UAUCUFFUDUBUE $. + $} - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a p $. - $d A p $. $d a ph $. $d a q $. $d A q $. $d a r $. $d A r $. - $d a s $. $d A s $. $d a t $. $d A t $. $d a u $. $d A u $. - $d A x $. $d A y $. $d B a $. $d B b $. $d b c $. $d B c $. - $d b p $. $d B p $. $d b ph $. $d b q $. $d B q $. $d b r $. - $d B r $. $d b s $. $d B s $. $d b t $. $d B t $. $d b u $. - $d B u $. $d B x $. $d B y $. $d c p $. $d c ph $. $d c q $. - $d c r $. $d c s $. $d c t $. $d c u $. $d p ph $. $d p q $. - $d p r $. $d p s $. $d p t $. $d p u $. $d p x $. $d p y $. - $d ph q $. $d ph r $. $d ph s $. $d ph t $. $d ph u $. $d q r $. - $d q s $. $d q t $. $d q u $. $d q x $. $d q y $. $d r s $. - $d r t $. $d r u $. $d r x $. $d R x $. $d r y $. $d R y $. - $d s t $. $d s u $. $d s x $. $d S x $. $d s y $. $d S y $. - $d T a $. $d T b $. $d T c $. $d T p $. $d T q $. $d T r $. - $d T s $. $d T t $. $d t u $. $d T u $. $d t x $. $d t y $. - $d u x $. $d u y $. $d x y $. - poxp2.1 $e |- ( ph -> R Po A ) $. - poxp2.2 $e |- ( ph -> S Po B ) $. - $( Another way of partially ordering a cross product of two classes. - (Contributed by Scott Fenton, 19-Aug-2024.) $) - poxp2 $p |- ( ph -> T Po ( A X. B ) ) $= - ( vp vq vr vs vt wbr wrex wa wo va vb vc vu cxp cv wcel wn cop wceq weq - elxp2 wne w3a equid pm3.2i neorian con2bii simp3 mto xpord2lem mtbir wb - mpbi breq12 anidms mtbiri rexlimivw sylbi adantl 3reeanv rexbii 2rexbii - wi 3anbi123i 3bitr4ri df-3an simpr1l adantr simpr2r jca simpr2l simpr3r - wpo ad2antrr simpr1r potr syl13anc orc syl6 expd syl6bir simprl1 mpjaod - breq1 a1i breq2 equequ2 orbi12d simprr1 simpr3l simprl2 simprr2 simprr3 - syl5ibcom sylib equequ1 neeq1 3anbi123d anbi1d anbi2d orcom ordi equcom - orbi2i bitri anbi12i 3bitri sylanbrc po2nr syl12anc orcnd com12 syl5bir - syl6bi mt3d 3jca syl2ani 3adant3 bitrdi 3adant1 anbi12d 3adant2 imbi12d - ex syl5ibrcom sylan2br anassrs rexlimdvva imp sylan2b ispod ) AUAUBUCDE - UEZHUAUFZUUCUGZUUDUUDHQZUHZAUUEUUDLUFZMUFZUIZUJZMERZLDRZUUGLMUUDDEULZUU - LUUGLDUUKUUGMEUUKUUFUUJUUJHQZUUOUUHDUGZUUIEUGZSZUURUUHUUHFQLLUKZTZUUIUU - IGQMMUKZTZUUHUUHUMUUIUUIUMTZUNZUNZUVEUVDUVDUVCUUSUVASZUVCUHUUSUVALUOMUO - UPUVCUVFUUHUUHUUIUUIUQURVDUUTUVBUVCUSUTUURUURUVDUSUTBCDEFGHLMLMIVAVBUUK - UUFUUOVCUUDUUJUUDUUJHVEVFVGVHVHVIVJUUEUBUFZUUCUGZUCUFZUUCUGZUNZAUUKUVGN - UFZOUFZUIZUJZUVIPUFZUDUFZUIZUJZUNZUDEROERZMERZPDRZNDRLDRZUUDUVGHQZUVGUV - IHQZSZUUDUVIHQZVNZUULUVOOERZUVSUDERZUNZPDRZNDRLDRUUMUWJNDRZUWKPDRZUNUWD - UVKUULUWJUWKLNPDDDVKUWCUWMLNDDUWBUWLPDUUKUVOUVSMOUDEEEVKVLVMUUEUUMUVHUW - NUVJUWOUUNNOUVGDEULPUDUVIDEULVOVPAUWDUWIAUWCUWILNDDAUUPUVLDUGZSZSUWAUWI - PMDEAUWQUVPDUGZUUQSZUWAUWIVNAUWQUWSSZSUVTUWIOUDEEAUWTUVMEUGZUVQEUGZSZUV - TUWIVNZUWTUXCSAUWQUWSUXCUNZUXDUWQUWSUXCVQAUXESZUWIUVTUURUWPUXASZUUHUVLF - QZLNUKZTZUUIUVMGQZMOUKZTZUUHUVLUMZUUIUVMUMZTZUNZUNZUXGUWRUXBSZUVLUVPFQZ - NPUKZTZUVMUVQGQZOUDUKZTZUVLUVPUMUVMUVQUMTZUNZUNZSZUURUXSUUHUVPFQZLPUKZT - ZUUIUVQGQZMUDUKZTZUUHUVPUMUUIUVQUMTZUNZUNZVNUXRUXFUXQUYGUYRUYHUURUXGUXQ - USUXGUXSUYGUSUXFUXQUYGSZUYRUXFUYSSZUURUXSUYQUYTUUPUUQUXFUUPUYSUUPUWPUWS - UXCAVRVSZUXFUUQUYSUWRUUQUWQUXCAVTVSZWAUYTUWRUXBUXFUWRUYSUWRUUQUWQUXCAWB - ZVSZUXFUXBUYSUXAUXBUWQUWSAWCZVSZWAUYTUYLUYOUYPUYTUXTUYLUYAUYTUXHUXTUYLV - NZUXIUYTUXHUXTUYLUYTUXHUXTSZUYJUYLUYTDFWDZUUPUWPUWRVUHUYJVNAVUIUXEUYSJW - EVUAUXFUWPUYSUUPUWPUWSUXCAWFZVSVUDDUUHUVLUVPFWGWHUYJUYKWIZWJWKUXIVUGVNU - YTUXIUXTUYJUYLUUHUVLUVPFWOVUKWLWPUXJUXMUXPUYGUXFWMZWNUYTUXJUYAUYLVULUYA - UXHUYJUXIUYKUVLUVPUUHFWQNPLWRWSXEUYBUYEUYFUXQUXFWTWNUYTUYCUYOUYDUYTUXKU - YCUYOVNZUXLUYTUXKUYCUYOUYTUXKUYCSZUYMUYOUYTEGWDZUUQUXAUXBVUNUYMVNAVUOUX - EUYSKWEVUBUXFUXAUYSUXAUXBUWQUWSAXAZVSVUFEUUIUVMUVQGWGWHUYMUYNWIZWJWKUXL - VUMVNUYTUXLUYCUYMUYOUUIUVMUVQGWOVUQWLWPUXJUXMUXPUYGUXFXBZWNUYTUXMUYDUYO - VURUYDUXKUYMUXLUYNUVMUVQUUIGWQOUDMWRWSXEUYBUYEUYFUXQUXFXCWNUYTUYPUYAUYD - SZUYTUYFVUSUHUYBUYEUYFUXQUXFXDUVLUVPUVMUVQUQXFUYPUHUYKUYNSZUYTVUSUYPVUT - UUHUVPUUIUVQUQURVUTUYTVUSVUTUYTUXFUVPUVLFQZPNUKZTZUVQUVMGQZUDOUKZTZUVPU - VLUMZUVQUVMUMZTZUNZUYGSZSZVUSVUTUYSVVKUXFVUTUXQVVJUYGVUTUXJVVCUXMVVFUXP - VVIUYKUXJVVCVCUYNUYKUXHVVAUXIVVBUUHUVPUVLFWOLPNXGWSVSUYNUXMVVFVCUYKUYNU - XKVVDUXLVVEUUIUVQUVMGWOMUDOXGWSVJVUTUXNVVGUXOVVHUYKUXNVVGVCUYNUUHUVPUVL - XHVSUYNUXOVVHVCUYKUUIUVQUVMXHVJWSXIXJXKVVLUYAUYDVVLVVAUXTSZUYAVVLVVCUYB - VVMUYATZVVCVVFVVIUYGUXFWMUYBUYEUYFVVJUXFWTVVNUYAVVMTUYAVVATZUYAUXTTZSVV - CUYBSVVMUYAXLUYAVVAUXTXMVVOVVCVVPUYBVVOVVAUYATVVCUYAVVAXLUYAVVBVVANPXNX - OXPUYAUXTXLXQXRXSVVLVUIUWRUWPVVMUHAVUIUXEVVKJWEUXFUWRVVKVUCVSUXFUWPVVKV - UJVSDUVPUVLFXTYAYBVVLVVDUYCSZUYDVVLVVFUYEVVQUYDTZVVCVVFVVIUYGUXFXBUYBUY - EUYFVVJUXFXCVVRUYDVVQTUYDVVDTZUYDUYCTZSVVFUYESVVQUYDXLUYDVVDUYCXMVVSVVF - VVTUYEVVSVVDUYDTVVFUYDVVDXLUYDVVEVVDOUDXNXOXPUYDUYCXLXQXRXSVVLVUOUXBUXA - VVQUHAVUOUXEVVKKWEUXFUXBVVKVUEVSUXFUXAVVKVUPVSEUVQUVMGXTYAYBWAYEYCYDYFY - GYGYOYHUVTUWGUYIUWHUYRUVTUWEUXRUWFUYHUVTUWEUUJUVNHQZUXRUUKUVOUWEVWAVCUV - SUUDUUJUVGUVNHVEYIBCDEFGHLMNOIVAYJUVTUWFUVNUVRHQZUYHUVOUVSUWFVWBVCUUKUV - GUVNUVIUVRHVEYKBCDEFGHNOPUDIVAYJYLUVTUWHUUJUVRHQZUYRUUKUVSUWHVWCVCUVOUU - DUUJUVIUVRHVEYMBCDEFGHLMPUDIVAYJYNYPYQYRYSYRYSYSYTUUAUUB $. - $} + $( Division of both sides of 'less than or equal to' by a nonnegative number. + (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) + (Proof modification is discouraged.) $) + lediv2aALT $p |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ + ( C e. RR /\ 0 <_ C ) ) -> + ( A <_ B -> ( C / B ) <_ ( C / A ) ) ) $= + ( cr wcel cc0 clt wbr wa cle w3a c1 cdiv co anim12i ancoms syl recn adantr + cc wi wne gt0ne0 rereccl syldan 3adant3 simp3 df-3an sylanbrc lemul2a ex wb + cmul lerec wceq jca 3anass sylibr divrec 3adant1 3adant2 breq12d 3imtr4d ) + ADEZFAGHZIZBDEZFBGHZIZCDEZFCJHZIZKZLBMNZLAMNZJHZCVNUMNZCVOUMNZJHZABJHZCBMNZ + CAMNZJHVMVNDEZVODEZVLKZVPVSUAVMWCWDIZVLWEVFVIWFVLVIVFWFVIWCVFWDVGVHBFUBZWCB + UCZBUDUEVDVEAFUBZWDAUCZAUDUEOPUFVFVIVLUGWCWDVLUHUIWEVPVSVNVOCUJUKQVFVIVTVPU + LVLABUNUFVMWAVQWBVRJVIVLWAVQUOZVFVLVIWKVLVIIZCTEZBTEZWGKZWKWLWMWNWGIZIWOVLW + MVIWPVJWMVKCRSZVIWNWGVGWNVHBRSWHUPOWMWNWGUQURCBUSQPUTVFVLWBVRUOZVIVLVFWRVLV + FIZWMATEZWIKZWRWSWMWTWIIZIXAVLWMVFXBWQVFWTWIVDWTVEARSWJUPOWMWTWIUQURCAUSQPV + AVBVC $. - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. - $d a e $. $d a p $. $d A p $. $d a ph $. $d a q $. $d A q $. - $d a s $. $d A s $. $d B a $. $d b c $. $d B c $. $d B d $. - $d B e $. $d B p $. $d B q $. $d b s $. $d B s $. $d c d $. - $d c e $. $d c s $. $d d e $. $d d s $. $d e s $. $d p q $. - $d p s $. $d ph s $. $d q s $. $d R a $. $d R b $. $d R c $. - $d S c $. $d S d $. $d S e $. $d T a $. $d T p $. $d T q $. - $d T s $. $d A e $. $d a f $. $d A f $. $d a x $. $d A x $. - $d a y $. $d A y $. $d b e $. $d b f $. $d B f $. $d b q $. - $d B x $. $d B y $. $d c f $. $d c p $. $d c ph $. $d c q $. - $d c x $. $d c y $. $d d f $. $d d q $. $d e f $. $d e ph $. - $d e q $. $d e x $. $d e y $. $d f ph $. $d f q $. $d f s $. - $d f x $. $d f y $. $d ph q $. $d R e $. $d R f $. $d R q $. - $d R x $. $d R y $. $d S f $. $d S q $. $d S x $. $d S y $. - $d T c $. $d T e $. $d T f $. $d x y $. - frxp2.1 $e |- ( ph -> R Fr A ) $. - frxp2.2 $e |- ( ph -> S Fr B ) $. - $( Another way of giving a founded order to a cross product of two - classes. (Contributed by Scott Fenton, 19-Aug-2024.) $) - frxp2 $p |- ( ph -> T Fr ( A X. B ) ) $= - ( va vc ve cv c0 wa wn wi wcel vs vq vp vb vd cxp wss wne wbr wral wrex - vf wal wfr cdm dmss ad2antrl dmxpss sstrdi simprr wrel wceq relxp relss - wb mpi reldm0 syl necon3bid mpbid adantr df-fr sylib dmex sseq1 anbi12d - vex neeq1 raleq rexeqbi1dv imbi12d spcv mp2and csn cima crn rnss rnxpss - imassrn sstrid simprl cin imadisj disjsn bitri necon2abii imaex elimasn - ad2antrr cop wel wex elrel sylan weq wo w3a breq1 notbid simplrr adantl - biimpri rspcdva intnanrd opeq1 eleq1d anbi2d 3anass olc biantrurd neirr - orbi1d biorf bitr4di nonconne biorfi bitr4i bitrdi bitr3d bitrid mpbiri - ax-mp andir impcom opeldm simpr neneqd ioran sylanbrc rexlimddv ralbidv - intn3an1d pm2.61dane intn3an3d eleq1 xpord2lem com12 exlimdvv ralrimiva - mpd breq2 rspcev syl2anc ex alrimiv sylibr ) AUAOZDEUFZUGZUUQPUHZQZUBOZ - UCOZHUIZRZUBUUQUJZUCUUQUKZSZUAUMUURHUNAUVHUAAUVAUVGAUVAQZUDOZLOZFUIZRZU - DUUQUOZUJZUVGLUVNUVIUVNDUGZUVNPUHZUVOLUVNUKZUVIUVNUURUOZDUUSUVNUVSUGAUU - TUUQUURUPUQDEURUSUVIUUTUVQAUUSUUTUTUVIUUQPUVNPUVIUUQVAZUUQPVBUVNPVBVEUU - SUVTAUUTUUSUURVAUVTDEVCUUQUURVDVFUQZUUQVGVHVIVJUVIMOZDUGZUWBPUHZQZUVMUD - UWBUJZLUWBUKZSZMUMZUVPUVQQZUVRSZUVIDFUNZUWIAUWLUVAJVKMLUDDFVLVMUWHUWKMU - VNUUQUAVQZVNUWBUVNVBZUWEUWJUWGUVRUWNUWCUVPUWDUVQUWBUVNDVOUWBUVNPVRVPUWF - UVOLUWBUVNUVMUDUWBUVNVSVTWAWBVHWCUVIUVKUVNTZUVOQZQZUEOZUWBGUIZRZUEUUQUV - KWDZWEZUJZUVGMUXBUWQUXBEUGZUXBPUHZUXCMUXBUKZUWQUXBUUQWFZEUUQUXAWIUWQUXG - UURWFZEUVIUXGUXHUGZUWPUUSUXIAUUTUUQUURWGUQVKDEWHUSWJUWQUWOUXEUVIUWOUVOW - KUWOUXBPUXBPVBUVNUXAWLPVBUWORUUQUXAWMUVNUVKWNWOWPVMUWQNOZEUGZUXJPUHZQZU - WTUEUXJUJZMUXJUKZSZNUMZUXDUXEQZUXFSZAUXQUVAUWPAEGUNUXQKNMUEEGVLVMWSUXPU - XSNUXBUUQUXAUWMWQUXJUXBVBZUXMUXRUXOUXFUXTUXKUXDUXLUXEUXJUXBEVOUXJUXBPVR - VPUXNUXCMUXJUXBUWTUEUXJUXBVSVTWAWBVHWCUWQUWBUXBTZUXCQZQZUVKUWBWTZUUQTZU - VBUYDHUIZRZUBUUQUJZUVGUYCUYAUYEUWQUYAUXCWKUUQUVKUWBLVQZMVQWRVMUYCUYGUBU - UQUYCUBUAXAZQZUVBUXJULOZWTZVBZULXBNXBZUYGUYCUVTUYJUYOUVIUVTUWPUYBUWAWSN - ULUVBUUQXCXDUYKUYNUYGNULUYNUYKUYGUYNUYKUYGSUYCUYMUUQTZQZUXJDTUYLETQZUVK - DTUWBETQZUXJUVKFUIZNLXEZXFZUYLUWBGUIZULMXEZXFZUXJUVKUHZUYLUWBUHZXFZXGZX - GZRZSUYQVUIUYRUYSUYQVUIRZUXJUVKVUAUYQVULVUAUYQVULSUYCUVKUYLWTZUUQTZQZVU - CVUGQZRZSVUOVUCVUGVUOUWTVUCRUEUXBUYLUEULXEUWSVUCUWRUYLUWBGXHXIUWQUYAUXC - VUNXJVUNUYLUXBTZUYCVURVUNUUQUVKUYLUYIULVQZWRXLXKXMXNVUAUYQVUOVULVUQVUAU - YPVUNUYCVUAUYMVUMUUQUXJUVKUYLXOXPXQVUAVUIVUPVUIVUBVUEVUHQZQZVUAVUPVUBVU - EVUHXRVUAVUTVVAVUPVUAVUBVUTVUAUYTXSXTVUAVUTVUEVUGQZVUPVUAVUHVUGVUEVUAVU - HUVKUVKUHZVUGXFZVUGVUAVUFVVCVUGUXJUVKUVKVRYBVVCRVUGVVDVEUVKYAVVCVUGYCYL - YDXQVVBVUPVUDVUGQZXFVUPVUCVUDVUGYMVVEVUPUYLUWBYEYFYGYHYIYJXIWAYKYNUYQVU - FQZVUBVUEVUHVVFUYTRZVUARVUBRVVFUVMVVGUDUVNUXJUDNXEUVLUYTUVJUXJUVKFXHXIU - YCUVOUYPVUFUVIUWOUVOUYBXJWSUYQUXJUVNTZVUFUYPVVHUYCUXJUYLUUQNVQVUSYOXKVK - XMVVFUXJUVKUYQVUFYPYQUYTVUAYRYSUUBUUCUUDUYNUYKUYQUYGVUKUYNUYJUYPUYCUVBU - YMUUQUUEXQUYNUYFVUJUYNUYFUYMUYDHUIVUJUVBUYMUYDHXHBCDEFGHNULLMIUUFYHXIWA - YKUUGUUHUUJUUIUVFUYHUCUYDUUQUVCUYDVBZUVEUYGUBUUQVVIUVDUYFUVCUYDUVBHUUKX - IUUAUULUUMYTYTUUNUUOUAUCUBUURHVLUUP $. - $} + ${ + abs2sqlei.1 $e |- A e. CC $. + abs2sqlei.2 $e |- B e. CC $. + $( The absolute values of two numbers compare as their squares. + (Contributed by Paul Chapman, 7-Sep-2007.) $) + abs2sqlei $p |- ( ( abs ` A ) <_ ( abs ` B ) <-> + ( ( abs ` A ) ^ 2 ) <_ ( ( abs ` B ) ^ 2 ) ) $= + ( cc0 cabs cfv cle wbr c2 cexp co wb absge0i abscli le2sqi mp2an ) EAFGZH + IEBFGZHIRSHIRJKLSJKLHIMACNBDNRSACOBDOPQ $. + $} - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. - $d A d $. $d a e $. $d A e $. $d a x $. $d A x $. $d a y $. - $d A y $. $d B a $. $d B b $. $d b c $. $d B c $. $d b d $. - $d B d $. $d b e $. $d B e $. $d b x $. $d B x $. $d b y $. - $d B y $. $d c d $. $d c e $. $d c x $. $d c y $. $d d e $. - $d d x $. $d d y $. $d R a $. $d R b $. $d R c $. $d R d $. - $d R e $. $d R x $. $d R y $. $d S a $. $d S b $. $d S c $. - $d S d $. $d S e $. $d S x $. $d S y $. $d T a $. $d T b $. - $d T c $. $d T d $. $d T e $. $d X a $. $d X b $. $d x y $. - $d Y b $. - $( Calculate the predecessor class in ~ frxp2 . (Contributed by Scott - Fenton, 22-Aug-2024.) $) - xpord2pred $p |- ( ( X e. A /\ Y e. B ) -> - Pred ( T , ( A X. B ) , <. X , Y >. ) = - ( ( ( Pred ( R , A , X ) u. { X } ) X. ( Pred ( S , B , Y ) u. { Y } ) ) - \ { <. X , Y >. } ) ) $= - ( va vb vc vd cpred csn wceq wcel wa wo ve cxp cv cop cun opeq1 predeq3 - cdif syl sneq uneq12d xpeq1d sneqd difeq12d eqeq12d opeq2 xpeq2d predel - wi a1i eldifi predss snssi unssd xpss12 syl2an sseld syl5 wrex wb elxp2 - wss wbr cvv elpred ax-mp opelxpi adantl biantrurd weq wne w3a xpord2lem - opex eldif opelxp elun vex elv velsn orbi12i bitri anbi12i notbii df-ne - elsn opthne 3bitr2i simprl orbi1d simprr 3anbi12d df-3an bitr2di bitrid - pm3.22 bitr4di bitr2d bitr3d eleq1 bibi12d syl5ibrcom rexlimdvva syl5bi - wn pm5.21ndd eqrdv vtocl2ga ) CDUBZGKUCZLUCZUDZOZCEXTOZXTPZUEZDFYAOZYAP - ZUEZUBZYBPZUHZQXSGHYAUDZOZCEHOZHPZUEZYIUBZYMPZUHZQXSGHIUDZOZYQDFIOZIPZU - EZUBZUUAPZUHZQKLHICDXTHQZYCYNYLYTUUIYBYMQYCYNQXTHYAUFZXSGYBYMUGUIUUIYJY - RYKYSUUIYFYQYIUUIYDYOYEYPCEXTHUGXTHUJUKULUUIYBYMUUJUMUNUOYAIQZYNUUBYTUU - HUUKYMUUAQYNUUBQYAIHUPZXSGYMUUAUGUIUUKYRUUFYSUUGUUKYIUUEYQUUKYGUUCYHUUD - DFYAIUGYAIUJUKUQUUKYMUUAUULUMUNUOXTCRZYADRZSZUAYCYLUUOUAUCZXSRZUUPYCRZU - UPYLRZUURUUQUSUUOXSGYBUUPURUTUUSUUPYJRUUOUUQUUPYJYKVAUUOYJXSUUPUUMYFCVL - YIDVLYJXSVLUUNUUMYDYECYDCVLUUMCEXTVBUTXTCVCVDUUNYGYHDYGDVLUUNDFYAVBUTYA - DVCVDYFCYIDVEVFVGVHUUQUUPMUCZNUCZUDZQZNDVIMCVIUUOUURUUSVJZMNUUPCDVKUUOU - VCUVDMNCDUUOUUTCRZUVADRZSZSZUVDUVCUVBYCRZUVBYLRZVJUVIUVBXSRZUVBYBGVMZSZ - UVHUVJYBVNRUVIUVMVJXTYAWDXSVNGYBUVBUUTUVAWDZVOVPUVHUVLUVMUVJUVHUVKUVLUV - GUVKUUOUUTUVACDVQVRVSUVLUVGUUOUUTXTEVMZMKVTZTZUVAYAFVMZNLVTZTZUUTXTWAUV - AYAWATZWBZWBZUVHUVJABCDEFGMNKLJWCUVHUVJUWBUWCUVJUVEUVOSZUVPTZUVFUVRSZUV - STZSZUWASZUVHUWBUVJUVBYJRZUVBYKRZXOZSUWIUVBYJYKWEUWJUWHUWLUWAUWJUUTYFRZ - UVAYIRZSUWHUUTUVAYFYIWFUWMUWEUWNUWGUWMUUTYDRZUUTYERZTUWEUUTYDYEWGUWOUWD - UWPUVPUWOUWDVJKCVNEXTUUTMWHZVOWIMXTWJWKWLUWNUVAYGRZUVAYHRZTUWGUVAYGYHWG - UWRUWFUWSUVSUWRUWFVJLDVNFYAUVANWHZVOWINYAWJWKWLWMWLUWLUVBYBQZXOUVBYBWAU - WAUWKUXAUVBYBUVNWPWNUVBYBWOUUTUVAXTYAUWQUWTWQWRWMWLUVHUWBUWEUWGUWAWBUWI - UVHUVQUWEUVTUWGUWAUVHUVOUWDUVPUVHUVEUVOUUOUVEUVFWSVSWTUVHUVRUWFUVSUVHUV - FUVRUUOUVEUVFXAVSWTXBUWEUWGUWAXCXDXEUVHUWBUVGUUOSZUWBSUWCUVHUXBUWBUUOUV - GXFVSUVGUUOUWBXCXGXHXEXIXEUVCUURUVIUUSUVJUUPUVBYCXJUUPUVBYLXJXKXLXMXNXP - XQXR $. - $} + ${ + abs2sqlti.1 $e |- A e. CC $. + abs2sqlti.2 $e |- B e. CC $. + $( The absolute values of two numbers compare as their squares. + (Contributed by Paul Chapman, 7-Sep-2007.) $) + abs2sqlti $p |- ( ( abs ` A ) < ( abs ` B ) <-> + ( ( abs ` A ) ^ 2 ) < ( ( abs ` B ) ^ 2 ) ) $= + ( cc0 cabs cfv cle wbr clt c2 cexp co wb absge0i abscli lt2sqi mp2an ) EA + FGZHIEBFGZHISTJISKLMTKLMJINACOBDOSTACPBDPQR $. + $} - ${ - $d A a $. $d a b $. $d A b $. $d a p $. $d A p $. $d a ph $. - $d A x $. $d A y $. $d B a $. $d B b $. $d b p $. $d B p $. - $d b ph $. $d B x $. $d B y $. $d p ph $. $d R x $. $d R y $. - $d S x $. $d S y $. $d T a $. $d T b $. $d T p $. $d x y $. - sexp2.1 $e |- ( ph -> R Se A ) $. - sexp2.2 $e |- ( ph -> S Se B ) $. - $( Condition for the relationship in ~ frxp2 to be set-like. - (Contributed by Scott Fenton, 19-Aug-2024.) $) - sexp2 $p |- ( ph -> T Se ( A X. B ) ) $= - ( vp va vb cv cpred cvv wcel wse csn cxp wral cop wceq wrex wa cun cdif - elxp2 xpord2pred adantl setlikespec ancoms sylan adantrr snex a1i unexd - adantrl xpexd difexd eqeltrd predeq3 eleq1d syl5ibrcom rexlimdvva dfse3 - syl5bi ralrimiv sylibr ) ADEUAZHLOZPZQRZLVKUBVKHSAVNLVKVLVKRVLMOZNOZUCZ - UDZNEUEMDUEAVNMNVLDEUIAVRVNMNDEAVODRZVPERZUFZUFZVNVRVKHVQPZQRWBWCDFVOPZ - VOTZUGZEGVPPZVPTZUGZUAZVQTZUHZQWAWCWLUDABCDEFGHVOVPIUJUKWBWJWKQWBWFWIQQ - WBWDWEQQAVSWDQRZVTADFSZVSWMJVSWNWMDFVOULUMUNUOWEQRWBVOUPUQURWBWGWHQQAVT - WGQRZVSAEGSZVTWOKVTWPWOEGVPULUMUNUSWHQRWBVPUPUQURUTVAVBVRVMWCQVKHVLVQVC - VDVEVFVHVILVKHVGVJ $. - $} + $( The absolute values of two numbers compare as their squares. (Contributed + by Paul Chapman, 7-Sep-2007.) $) + abs2sqle $p |- ( ( A e. CC /\ B e. CC ) -> + ( ( abs ` A ) <_ ( abs ` B ) <-> + ( ( abs ` A ) ^ 2 ) <_ ( ( abs ` B ) ^ 2 ) ) ) $= + ( cc wcel cabs cfv cle wbr c2 cexp co wb cc0 cif wceq breq1d bibi12d breq2d + fveq2 0cn oveq1d oveq1 syl elimel abs2sqlei dedth2h ) ACDZBCDZAEFZBEFZGHZUI + IJKZUJIJKZGHZLUGAMNZEFZUJGHZUPIJKZUMGHZLUPUHBMNZEFZGHZURVAIJKZGHZLABMMAUOOZ + UKUQUNUSVEUIUPUJGAUOESZPVEULURUMGVEUIUPIJVFUAPQBUTOZUQVBUSVDVGUJVAUPGBUTESZ + RVGUJVAOZUSVDLVHVIUMVCURGUJVAIJUBRUCQUOUTAMCTUDBMCTUDUEUF $. - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. - $d A d $. $d a ps $. $d a ta $. $d A x $. $d A y $. $d B a $. - $d B b $. $d b c $. $d B c $. $d b ch $. $d b d $. $d B d $. - $d b et $. $d B x $. $d B y $. $d c d $. $d c ph $. $d c th $. - $d d ps $. $d R c $. $d R d $. $d R x $. $d R y $. $d S c $. - $d S d $. $d S x $. $d S y $. $d T a $. $d T b $. $d T c $. - $d T d $. $d X a $. $d X b $. $d x y $. $d Y b $. - xpord2ind.1 $e |- R Fr A $. - xpord2ind.2 $e |- R Po A $. - xpord2ind.3 $e |- R Se A $. - xpord2ind.4 $e |- S Fr B $. - xpord2ind.5 $e |- S Po B $. - xpord2ind.6 $e |- S Se B $. - xpord2ind.7 $e |- ( a = c -> ( ph <-> ps ) ) $. - xpord2ind.8 $e |- ( b = d -> ( ps <-> ch ) ) $. - xpord2ind.9 $e |- ( a = c -> ( th <-> ch ) ) $. - xpord2ind.11 $e |- ( a = X -> ( ph <-> ta ) ) $. - xpord2ind.12 $e |- ( b = Y -> ( ta <-> et ) ) $. - xpord2ind.i $e |- ( ( a e. A /\ b e. B ) -> - ( ( A. c e. Pred ( R , A , a ) A. d e. Pred ( S , B , b ) ch /\ - A. c e. Pred ( R , A , a ) ps /\ - A. d e. Pred ( S , B , b ) th ) -> ph ) ) $. - $( Induction over the cross product ordering. Note that the - substitutions cover all possible cases of membership in the - predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.) $) - xpord2ind $p |- ( ( X e. A /\ Y e. B ) -> et ) $= - ( cxp wfr wpo wse w3a wcel wa wtru a1i frxp2 poxp2 sexp2 3jca mptru cop - cv cpred wi wal wne wo csn cun wral cdif xpord2pred eleq2d imbi1d eldif - wn opelxp wceq opex elsn notbii df-ne vex opthne 3bitr2i anbi12i imbi1i - bitri impexp bitrdi 2albidv r2al bitr4di wss ssun1 ssralv ax-mp syl weq - ralimi predpoirr eleq1w mtbiri necon2ai olcd pm2.27 ralimia ssun2 neeq1 - orbi2d wb equcoms bicomd imbi12d ralsn ralbii sylib orcd orbi1d ralbidv - syl5 sylbid frpoins3xpg mpan ) IJUMZMUNZYKMUOZYKMUPZUQZNIUROJURUSFYOUTY - LYMYNUTGHIJKLMTIKUNUTUAVAJLUNUTUDVAVBUTGHIJKLMTIKUOZUTUBVAJLUOZUTUEVAVC - UTGHIJKLMTIKUPUTUCVAJLUPUTUFVAVDVEVFABCEFPQRSIJMNOPVHZIURQVHZJURUSZRVHZ - SVHZVGZYKMYRYSVGZVIZURZCVJZSVKRVKZUUAYRVLZUUBYSVLZVMZCVJZSJLYSVIZYSVNZV - OZVPZRIKYRVIZYRVNZVOZVPZAYTUUHUUAUUSURUUBUUOURUSZUULVJZSVKRVKUUTYTUUGUV - BRSYTUUGUUCUUSUUOUMZUUDVNZVQZURZCVJZUVBYTUUFUVFCYTUUEUVEUUCGHIJKLMYRYST - VRVSVTUVGUVAUUKUSZCVJUVBUVFUVHCUVFUUCUVCURZUUCUVDURZWBZUSUVHUUCUVCUVDWA - UVIUVAUVKUUKUUAUUBUUSUUOWCUVKUUCUUDWDZWBUUCUUDVLUUKUVJUVLUUCUUDUUAUUBWE - WFWGUUCUUDWHUUAUUBYRYSRWISWIWJWKWLWNWMUVAUUKCWOWNWPWQUULRSUUSUUOWRWSUUT - CSUUMVPZRUUQVPZBRUUQVPZDSUUMVPZUQYTAUUTUVNUVOUVPUUTUULSUUMVPZRUUQVPZUVN - UUTUUPRUUQVPZUVRUUQUUSWTUUTUVSVJUUQUURXAUUPRUUQUUSXBXCZUUPUVQRUUQUUMUUO - WTUUPUVQVJUUMUUNXAUULSUUMUUOXBXCZXFXDUVQUVMRUUQUULCSUUMUUBUUMURZUUKUULC - VJUWBUUJUUIUWBUUBYSSQXEZUWBYSUUMURZYQUWDWBUEJLYSXGXCSQUUMXHXIXJZXKUUKCX - LXDXMXFXDUUTUUIYSYSVLZVMZBVJZRUUQVPZUVOUUTUULSUUNVPZRUUQVPZUWIUUTUVSUWK - UVTUUPUWJRUUQUUNUUOWTUUPUWJVJUUNUUMXNUULSUUNUUOXBXCXFXDUWJUWHRUUQUULUWH - SYSQWIUWCUUKUWGCBUWCUUJUWFUUIUUBYSYSXOXPUWCBCBCXQQSUHXRXSXTYAYBYCUWHBRU - UQUUAUUQURZUWGUWHBVJUWLUUIUWFUWLUUAYRRPXEZUWLYRUUQURZYPUWNWBUBIKYRXGXCR - PUUQXHXIXJYDUWGBXLXDXMXDUUTYRYRVLZUUJVMZDVJZSUUMVPZUVPUUTUVQRUURVPZUWRU - UTUUPRUURVPZUWSUURUUSWTUUTUWTVJUURUUQXNUUPRUURUUSXBXCUUPUVQRUURUWAXFXDU - VQUWRRYRPWIUWMUULUWQSUUMUWMUUKUWPCDUWMUUIUWOUUJUUAYRYRXOYEUWMDCDCXQPRUI - XRXSXTYFYAYCUWQDSUUMUWBUWPUWQDVJUWBUUJUWOUWEXKUWPDXLXDXMXDVEULYGYHUGUHU - JUKYIYJ $. - $} + $( The absolute values of two numbers compare as their squares. (Contributed + by Paul Chapman, 7-Sep-2007.) $) + abs2sqlt $p |- ( ( A e. CC /\ B e. CC ) -> + ( ( abs ` A ) < ( abs ` B ) <-> + ( ( abs ` A ) ^ 2 ) < ( ( abs ` B ) ^ 2 ) ) ) $= + ( cc wcel cabs cfv clt wbr c2 cexp co wb cc0 cif wceq breq1d bibi12d breq2d + fveq2 0cn oveq1d oveq1 syl elimel abs2sqlti dedth2h ) ACDZBCDZAEFZBEFZGHZUI + IJKZUJIJKZGHZLUGAMNZEFZUJGHZUPIJKZUMGHZLUPUHBMNZEFZGHZURVAIJKZGHZLABMMAUOOZ + UKUQUNUSVEUIUPUJGAUOESZPVEULURUMGVEUIUPIJVFUAPQBUTOZUQVBUSVDVGUJVAUPGBUTESZ + RVGUJVAOZUSVDLVHVIUMVCURGUJVAIJUBRUCQUOUTAMCTUDBMCTUDUEUF $. + ${ + abs2difi.1 $e |- A e. CC $. + abs2difi.2 $e |- B e. CC $. + $( Difference of absolute values. (Contributed by Paul Chapman, + 7-Sep-2007.) $) + abs2difi $p |- ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) $= + ( cc wcel cabs cfv cmin co cle wbr abs2dif mp2an ) AEFBEFAGHBGHIJABIJGHKL + CDABMN $. $} ${ - xpord3.1 $e |- U = { <. x , y >. | ( x e. ( ( A X. B ) X. C ) - /\ y e. ( ( A X. B ) X. C ) - /\ ( ( ( ( 1st ` ( 1st ` x ) ) R ( 1st ` ( 1st ` y ) ) \/ - ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` y ) ) ) /\ - ( ( 2nd ` ( 1st ` x ) ) S ( 2nd ` ( 1st ` y ) ) \/ - ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` y ) ) ) /\ - ( ( 2nd ` x ) T ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) ) /\ - x =/= y ) ) } $. + abs2difabsi.1 $e |- A e. CC $. + abs2difabsi.2 $e |- B e. CC $. + $( Absolute value of difference of absolute values. (Contributed by Paul + Chapman, 7-Sep-2007.) $) + abs2difabsi $p |- ( abs ` ( ( abs ` A ) - ( abs ` B ) ) ) <_ + ( abs ` ( A - B ) ) $= + ( cc wcel cabs cfv cmin co cle wbr abs2difabs mp2an ) AEFBEFAGHBGHIJGHABI + JGHKLCDABMN $. + $} - ${ - $d a x $. $d A x $. $d a y $. $d A y $. $d b x $. $d B x $. - $d b y $. $d B y $. $d c x $. $d C x $. $d c y $. $d C y $. - $d d x $. $d d y $. $d e x $. $d e y $. $d f x $. $d f y $. - $d R x $. $d R y $. $d S x $. $d S y $. $d T x $. $d T y $. - $d x y $. - $( Lemma for triple ordering. Calculate the value of the relationship. - (Contributed by Scott Fenton, 21-Aug-2024.) $) - xpord3lem $p |- ( <. <. a , b >. , c >. U <. <. d , e >. , f >. <-> - ( ( a e. A /\ b e. B /\ c e. C ) /\ ( d e. A /\ e e. B /\ f e. C ) /\ - ( ( ( a R d \/ a = d ) /\ ( b S e \/ b = e ) /\ ( c T f \/ c = f ) ) /\ - ( a =/= d \/ b =/= e \/ c =/= f ) ) ) ) $= - ( wbr wcel wo w3a cv cop cxp weq wne wa w3o c1st cfv wceq c2nd opex vex - ot21std breq1d eqeq1d orbi12d ot22ndd op2ndd 3anbi123d anbi12d 3anbi13d - eleq1 neeq1 breq2d eqeq2d 3anbi23d brab ot2elxp otthne anbi2i 3anbi123i - neeq2 bitri ) LUAZMUAZUBZNUAZUBZOUAZJUAZUBZKUAZUBZIQVSCDUCEUCZRZWDWERZV - OVTFQZLOUDZSZVPWAGQZMJUDZSZVRWCHQZNKUDZSZTZVSWDUEZUFZTZVOCRVPDRVRERTZVT - CRWADRWCERTZWQVOVTUEVPWAUEVRWCUEUGZUFZTAUAZWERZBUAZWERZXEUHUIZUHUIZXGUH - UIZUHUIZFQZXJXLUJZSZXIUKUIZXKUKUIZGQZXPXQUJZSZXEUKUIZXGUKUIZHQZYAYBUJZS - ZTZXEXGUEZUFZTWFXHVOXLFQZVOXLUJZSZVPXQGQZVPXQUJZSZVRYBHQZVRYBUJZSZTZVSX - GUEZUFZTWTABVSWDIVQVRULWBWCULXEVSUJZXFWFYHYTXHXEVSWEVCUUAYFYRYGYSUUAXOY - KXTYNYEYQUUAXMYIXNYJUUAXJVOXLFVOVPVRXELUMZMUMZNUMZUNZUOUUAXJVOXLUUEUPUQ - UUAXRYLXSYMUUAXPVPXQGVOVPVRXEUUBUUCUUDURZUOUUAXPVPXQUUFUPUQUUAYCYOYDYPU - UAYAVRYBHVQVRXEVOVPULUUDUSZUOUUAYAVRYBUUGUPUQUTXEVSXGVDVAVBXGWDUJZXHWGY - TWSWFXGWDWEVCUUHYRWQYSWRUUHYKWJYNWMYQWPUUHYIWHYJWIUUHXLVTVOFVTWAWCXGOUM - ZJUMZKUMZUNZVEUUHXLVTVOUULVFUQUUHYLWKYMWLUUHXQWAVPGVTWAWCXGUUIUUJUUKURZ - VEUUHXQWAVPUUMVFUQUUHYOWNYPWOUUHYBWCVRHWBWCXGVTWAULUUKUSZVEUUHYBWCVRUUN - VFUQUTXGWDVSVMVAVGPVHWFXAWGXBWSXDVOVPVRCDEVIVTWAWCCDEVIWRXCWQVOVPVRVTWA - WCUUBUUCUUDVJVKVLVN $. - $} +$( (End of Paul Chapman's mathbox.) $) +$( End $[ set-mbox-pc.mm $] $) - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. - $d A d $. $d a e $. $d A e $. $d a f $. $d A f $. $d a g $. - $d A g $. $d a h $. $d A h $. $d a i $. $d A i $. $d a j $. - $d A j $. $d a k $. $d A k $. $d a l $. $d A l $. $d a ph $. - $d A x $. $d A y $. $d B a $. $d B b $. $d b c $. $d B c $. - $d b d $. $d B d $. $d b e $. $d B e $. $d b f $. $d B f $. - $d b g $. $d B g $. $d b h $. $d B h $. $d b i $. $d B i $. - $d b j $. $d B j $. $d b k $. $d B k $. $d b l $. $d B l $. - $d b ph $. $d B x $. $d B y $. $d C a $. $d C b $. $d C c $. - $d c d $. $d C d $. $d c e $. $d C e $. $d c f $. $d C f $. - $d c g $. $d C g $. $d c h $. $d C h $. $d c i $. $d C i $. - $d c j $. $d C j $. $d c k $. $d C k $. $d c l $. $d C l $. - $d c ph $. $d C x $. $d C y $. $d d e $. $d d f $. $d d g $. - $d d h $. $d d i $. $d d j $. $d d k $. $d d l $. $d d ph $. - $d d x $. $d d y $. $d e f $. $d e g $. $d e h $. $d e i $. - $d e j $. $d e k $. $d e l $. $d e ph $. $d e x $. $d e y $. - $d f g $. $d f h $. $d f i $. $d f j $. $d f k $. $d f l $. - $d f ph $. $d f x $. $d f y $. $d g h $. $d g i $. $d g j $. - $d g k $. $d g l $. $d g ph $. $d g x $. $d g y $. $d h i $. - $d h j $. $d h k $. $d h l $. $d h ph $. $d h x $. $d h y $. - $d i j $. $d i k $. $d i l $. $d i ph $. $d i x $. $d i y $. - $d j k $. $d j l $. $d j ph $. $d j x $. $d j y $. $d k l $. - $d k ph $. $d k x $. $d k y $. $d l ph $. $d l x $. $d l y $. - $d R x $. $d R y $. $d S x $. $d S y $. $d T x $. $d T y $. - $d U a $. $d U b $. $d U c $. $d U d $. $d U e $. $d U f $. - $d U g $. $d U h $. $d U i $. $d U j $. $d U k $. $d U l $. - $d x y $. - poxp3.1 $e |- ( ph -> R Po A ) $. - poxp3.2 $e |- ( ph -> S Po B ) $. - poxp3.3 $e |- ( ph -> T Po C ) $. - $( Triple cross product partial ordering. (Contributed by Scott Fenton, - 21-Aug-2024.) $) - poxp3 $p |- ( ph -> U Po ( ( A X. B ) X. C ) ) $= - ( vd ve wbr wrex wa wo va vb vc vf vg vh vi vj vk vl cxp cv wcel wn cop - elxpxp wi w3a weq wne w3o neirr 3pm3.2ni intnan simp3 mto breq12 anidms - wceq wb xpord3lem bitrdi mtbiri rexlimivw a1i rexlimivv sylbi 3anbi123i - adantl 3reeanv rexbii 2rexbii bitr4i simprl1 simprr2 wpo adantr simpl11 - bitri simpr11 simpr21 potr syl13anc orc syl6 expd breq1 syl6bir simp3l1 - ad2antrl mpjaod breq2 equequ2 orbi12d syl5ibcom simpl12 simpr12 simpr22 - ad2antll simp3l2 simpl13 simpr13 simpr23 simp3l3 3jca simpr3r 3orbi123i - df-ne 3ianor sylib con2bii equequ1 equcom ordir sylanbrc po2nr syl12anc - imp orcnd ex orbi2i 3anim123d syl5bir jca 3adant3 3adant1 rexlimdvw a1d - mt3d rexlimdvv anbi12d 3adant2 imbi12d syl5ibrcom syl5bi ispod ) AUAUBU - CDEUKFUKZJUAULZUUGUMZUUHUUHJQZUNZAUUIUUHOULZPULZUOUDULZUOZVIZUDFRZPERZO - DRZUUKOPUDUUHDEFUPZUUQUUKOPDEUUQUUKUQUULDUMZUUMEUMZSUUPUUKUDFUUPUUJUVAU - VBUUNFUMZURZUVDUULUULGQOOUSTUUMUUMHQPPUSTUUNUUNIQUDUDUSTURZUULUULUTZUUM - UUMUTZUUNUUNUTZVAZSZURZUVKUVJUVIUVEUVFUVGUVHUULVBUUMVBUUNVBVCVDUVDUVDUV - JVEVFUUPUUJUUOUUOJQZUVKUUPUUJUVLVJUUHUUOUUHUUOJVGVHBCDEFGHIJPUDOPUDOKVK - VLVMVNVOVPVQVSAUUIUBULZUUGUMZUCULZUUGUMZURZUUHUVMJQZUVMUVOJQZSZUUHUVOJQ - ZUQZUVQUUPUVMUEULZUFULZUOUGULZUOZVIZUVOUHULZUIULZUOUJULZUOZVIZURZUJFRZU - GFRUDFRZUIERZUFERPERZUHDRZUEDRODRZAUWBUVQUUSUWGUGFRZUFERZUEDRZUWLUJFRZU - IERZUHDRZURZUWSUUIUUSUVNUXBUVPUXEUUTUEUFUGUVMDEFUPUHUIUJUVODEFUPVRUWSUU - RUXAUXDURZUHDRZUEDRODRUXFUWRUXHOUEDDUWQUXGUHDUWQUUQUWTUXCURZUIERZUFERPE - RUXGUWPUXJPUFEEUWOUXIUIEUUPUWGUWLUDUGUJFFFVTWAWBUUQUWTUXCPUFUIEEEVTWIWA - WBUURUXAUXDOUEUHDDDVTWIWCAUWRUWBOUEDDAUWRUWBUQUVAUWCDUMZSAUWQUWBUHDAUWP - UWBPUFEEAUWPUWBUQUVBUWDEUMZSAUWOUWBUIEAUWNUWBUDUGFFAUWNUWBUQUVCUWEFUMZS - AUWMUWBUJFAUWBUWMUVDUXKUXLUXMURZUULUWCGQZOUEUSZTZUUMUWDHQZPUFUSZTZUUNUW - EIQZUDUGUSZTZURUULUWCUTUUMUWDUTUUNUWEUTVAZSZURZUXNUWHDUMZUWIEUMZUWJFUMZ - URZUWCUWHGQZUEUHUSZTZUWDUWIHQZUFUIUSZTZUWEUWJIQZUGUJUSZTZURZUWCUWHUTZUW - DUWIUTZUWEUWJUTZVAZSZURZSZUVDUYJUULUWHGQZOUHUSZTZUUMUWIHQZPUIUSZTZUUNUW - JIQZUDUJUSZTZURZUULUWHUTZUUMUWIUTZUUNUWJUTZVAZSZURZUQAVUGVVCAVUGSZUVDUY - JVVBUVDUXNUYEVUFAWDUXNUYJVUEUYFAWEVVDVUQVVAVVDVUJVUMVUPVVDUYKVUJUYLVVDU - XOUYKVUJUQZUXPVVDUXOUYKVUJVVDUXOUYKSZVUHVUJVVDDGWFZUVAUXKUYGVVFVUHUQAVV - GVUGLWGZVUGUVAAUVAUVBUVCUXNUYEVUFWHVSVUGUXKAUXKUXLUXMUYJVUEUYFWJVSZVUGU - YGAUYGUYHUYIUXNVUEUYFWKVSZDUULUWCUWHGWLWMVUHVUIWNZWOWPUXPVVEUQVVDUXPUYK - VUHVUJUULUWCUWHGWQVVKWRVOUYFUXQAVUFUXQUXTUYCUYDUVDUXNWSWTZXAVVDUXQUYLVU - JVVLUYLUXOVUHUXPVUIUWCUWHUULGXBUEUHOXCXDXEVUFUYMAUYFUYMUYPUYSVUDUXNUYJW - SXIZXAVVDUYNVUMUYOVVDUXRUYNVUMUQZUXSVVDUXRUYNVUMVVDUXRUYNSZVUKVUMVVDEHW - FZUVBUXLUYHVVOVUKUQAVVPVUGMWGZVUGUVBAUVAUVBUVCUXNUYEVUFXFVSVUGUXLAUXKUX - LUXMUYJVUEUYFXGVSZVUGUYHAUYGUYHUYIUXNVUEUYFXHVSZEUUMUWDUWIHWLWMVUKVULWN - ZWOWPUXSVVNUQVVDUXSUYNVUKVUMUUMUWDUWIHWQVVTWRVOUYFUXTAVUFUXQUXTUYCUYDUV - DUXNXJWTZXAVVDUXTUYOVUMVWAUYOUXRVUKUXSVULUWDUWIUUMHXBUFUIPXCXDXEVUFUYPA - UYFUYMUYPUYSVUDUXNUYJXJXIZXAVVDUYQVUPUYRVVDUYAUYQVUPUQZUYBVVDUYAUYQVUPV - VDUYAUYQSZVUNVUPVVDFIWFZUVCUXMUYIVWDVUNUQAVWEVUGNWGZVUGUVCAUVAUVBUVCUXN - UYEVUFXKVSVUGUXMAUXKUXLUXMUYJVUEUYFXLVSZVUGUYIAUYGUYHUYIUXNVUEUYFXMVSZF - UUNUWEUWJIWLWMVUNVUOWNZWOWPUYBVWCUQVVDUYBUYQVUNVUPUUNUWEUWJIWQVWIWRVOUY - FUYCAVUFUXQUXTUYCUYDUVDUXNXNWTZXAVVDUYCUYRVUPVWJUYRUYAVUNUYBVUOUWEUWJUU - NIXBUGUJUDXCXDXEVUFUYSAUYFUYMUYPUYSVUDUXNUYJXNXIZXAXOVVDVVAUYLUYOUYRURZ - VVDVUDVWLUNZVUGVUDAUYTVUDUXNUYJUYFXPVSVUDUYLUNZUYOUNZUYRUNZVAVWMVUAVWNV - UBVWOVUCVWPUWCUWHXRUWDUWIXRUWEUWJXRXQUYLUYOUYRXSWCXTVVAUNVUIVULVUOURZVV - DVWLVVAVWQVVAVUIUNZVULUNZVUOUNZVAVWQUNVURVWRVUSVWSVUTVWTUULUWHXRUUMUWIX - RUUNUWJXRXQVUIVULVUOXSWCYAVVDVUIUYLVULUYOVUOUYRVVDVUIUYLVVDVUISZUWHUWCG - QZUYKSZUYLVXAVXBUYLTZUYMVXCUYLTVVDVUIVXDVVDUXQVUIVXDVVLVUIUXOVXBUXPUYLU - ULUWHUWCGWQVUIUXPUHUEUSUYLOUHUEYBUHUEYCVLXDXEYHVVDUYMVUIVVMWGVXBUYKUYLY - DYEVVDVXCUNZVUIVVDVVGUYGUXKVXEVVHVVJVVIDUWHUWCGYFYGWGYIYJVVDVULUYOVVDVU - LSZUWIUWDHQZUYNSZUYOVXFVXGUYOTZUYPVXHUYOTVVDVULVXIVVDUXTVULVXIVWAVULUXR - VXGUXSUYOUUMUWIUWDHWQVULUXSUIUFUSUYOPUIUFYBUIUFYCVLXDXEYHVVDUYPVULVWBWG - VXGUYNUYOYDYEVVDVXHUNZVULVVDVVPUYHUXLVXJVVQVVSVVREUWIUWDHYFYGWGYIYJVVDV - UOUYRVVDVUOSZUWJUWEIQZUYQSZUYRVXKVXLUYRTZUYSVXMUYRTVXKVXLUJUGUSZTZVXNVV - DVUOVXPVVDUYCVUOVXPVWJVUOUYAVXLUYBVXOUUNUWJUWEIWQUDUJUGYBXDXEYHVXOUYRVX - LUJUGYCYKXTVVDUYSVUOVWKWGVXLUYQUYRYDYEVVDVXMUNZVUOVVDVWEUYIUXMVXQVWFVWH - VWGFUWJUWEIYFYGWGYIYJYLYMYSYNXOYJUWMUVTVUGUWAVVCUWMUVRUYFUVSVUFUWMUVRUU - OUWFJQZUYFUUPUWGUVRVXRVJUWLUUHUUOUVMUWFJVGYOBCDEFGHIJUFUGOPUDUEKVKVLUWM - UVSUWFUWKJQZVUFUWGUWLUVSVXSVJUUPUVMUWFUVOUWKJVGYPBCDEFGHIJUIUJUEUFUGUHK - VKVLUUAUWMUWAUUOUWKJQZVVCUUPUWLUWAVXTVJUWGUUHUUOUVOUWKJVGUUBBCDEFGHIJUI - UJOPUDUHKVKVLUUCUUDYQYRYTYQYRYTYQYRYTUUEYHUUF $. - $} - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. - $d A d $. $d a e $. $d a f $. $d a g $. $d A g $. $d a p $. - $d A p $. $d a ph $. $d a q $. $d A q $. $d a s $. $d A s $. - $d B a $. $d B b $. $d b c $. $d B c $. $d b d $. $d b e $. - $d B e $. $d b f $. $d b g $. $d B g $. $d b p $. $d B p $. - $d b ph $. $d b q $. $d B q $. $d b s $. $d B s $. $d C a $. - $d C b $. $d C c $. $d c d $. $d c e $. $d c f $. $d C f $. - $d c g $. $d C g $. $d c p $. $d C p $. $d c ph $. $d c q $. - $d C q $. $d c s $. $d C s $. $d d g $. $d d s $. $d e g $. - $d e s $. $d f g $. $d f s $. $d g s $. $d p q $. $d p s $. - $d ph s $. $d q s $. $d R a $. $d R b $. $d R c $. $d R d $. - $d R g $. $d S b $. $d S c $. $d S e $. $d S g $. $d T c $. - $d T f $. $d T g $. $d U a $. $d U b $. $d U c $. $d U p $. - $d U q $. $d U s $. $d A h $. $d A i $. $d B h $. $d B i $. - $d C h $. $d C i $. $d d q $. $d e q $. $d f q $. $d g h $. - $d g i $. $d g q $. $d h i $. $d h q $. $d i q $. $d ph q $. - $d R q $. $d S q $. $d T q $. $d a h $. $d a i $. $d a x $. - $d A x $. $d a y $. $d A y $. $d b h $. $d b i $. $d b x $. - $d B x $. $d b y $. $d B y $. $d c h $. $d c i $. $d c x $. - $d C x $. $d c y $. $d C y $. $d d h $. $d d i $. $d e h $. - $d e i $. $d f h $. $d f i $. $d g ph $. $d g x $. $d g y $. - $d h ph $. $d h s $. $d h x $. $d h y $. $d i ph $. $d i s $. - $d i x $. $d i y $. $d R h $. $d R i $. $d R x $. $d R y $. - $d S h $. $d S i $. $d S x $. $d S y $. $d T h $. $d T i $. - $d T x $. $d T y $. $d U g $. $d U h $. $d U i $. $d x y $. - frxp3.1 $e |- ( ph -> R Fr A ) $. - frxp3.2 $e |- ( ph -> S Fr B ) $. - frxp3.3 $e |- ( ph -> T Fr C ) $. - $( Give foundedness over a triple cross product. (Contributed by Scott - Fenton, 21-Aug-2024.) $) - frxp3 $p |- ( ph -> U Fr ( ( A X. B ) X. C ) ) $= - ( vg c0 wa wn wi wcel vs vq vp vd va ve vb vf vc vh vi cxp wss wne wral - cv wbr wrex wal wfr cdm dmss ad2antrl dmxpss sstrdi simprr wrel wceq wb - syl simprl relxp relss mpisyl reldm0 necon3bid mpbid df-fr sylib adantr - vex dmex sseq1 neeq1 anbi12d raleq rexeqbi1dv imbi12d spcv csn cima crn - mp2and imassrn rnss rnxpss sstrid imadisj disjsn bitri necon2abii imaex - cin ad2antrr cop elimasn ad3antrrr opex wel wex w3a weq wo adantl breq1 - notbid simplrr sylibr rspcdva simpr ioran sylanbrc intn3an3d pm2.61dane - neneqd opeq1d eleq1d anbi2d orbi1d neirr orel1 olc impbii bitrdi mpbiri - intnanrd ax-mp impcom opeldm rexlimddv simplrl elxpxpss sylan w3o df-ne - con2bii biimpi intnand simplr opeq2 intn3an2d opeq1 3orass bitrid eleq1 - intn3an1d xpord3lem com12 exlimdv exlimdvv mpd ralrimiva ralbidv rspcev - breq2 syl2anc ex alrimiv ) AUAUPZDEULZFULZUMZUVIPUNZQZUBUPZUCUPZJUQZRZU - BUVIUOZUCUVIURZSZUAUSUVKJUTAUWAUAAUVNUVTAUVNQZUDUPZUEUPZGUQZRZUDUVIVAZV - AZUOZUVTUEUWHUWBUWHDUMZUWHPUNZUWIUEUWHURZUWBUWHUVJVAZDUWBUWGUVJUMZUWHUW - MUMUWBUWGUVKVAZUVJUVLUWGUWOUMAUVMUVIUVKVBVCUVJFVDVEZUWGUVJVBVJDEVDVEUWB - UWGPUNZUWKUWBUVMUWQAUVLUVMVFUWBUVIPUWGPUWBUVIVGZUVIPVHUWGPVHZVIUWBUVLUV - KVGUWRAUVLUVMVKUVJFVLUVIUVKVMVNUVIVOVJVPVQUWBUWGPUWHPUWBUWGVGZUWSUWHPVH - VIUWBUWNUVJVGUWTUWPDEVLUWGUVJVMVNUWGVOVJVPVQUWBOUPZDUMZUXAPUNZQZUWFUDUX - AUOZUEUXAURZSZOUSZUWJUWKQZUWLSZAUXHUVNADGUTUXHLOUEUDDGVRVSVTUXGUXJOUWHU - WGUVIUAWAZWBZWBUXAUWHVHZUXDUXIUXFUWLUXMUXBUWJUXCUWKUXAUWHDWCUXAUWHPWDWE - UXEUWIUEUXAUWHUWFUDUXAUWHWFWGWHWIVJWMUWBUWDUWHTZUWIQZQZUFUPZUGUPZHUQZRZ - UFUWGUWDWJZWKZUOZUVTUGUYBUXPUYBEUMZUYBPUNZUYCUGUYBURZUXPUYBUWGWLZEUWGUY - AWNUWBUYGEUMUXOUWBUYGUVJWLZEUWBUWNUYGUYHUMUWPUWGUVJWOVJDEWPVEVTWQUXPUXN - UYEUWBUXNUWIVKUXNUYBPUYBPVHUWHUYAXCPVHUXNRUWGUYAWRUWHUWDWSWTXAVSAUYDUYE - QZUYFSZUVNUXOAUXAEUMZUXCQZUXTUFUXAUOZUGUXAURZSZOUSZUYJAEHUTUYPMOUGUFEHV - RVSUYOUYJOUYBUWGUYAUXLXBUXAUYBVHZUYLUYIUYNUYFUYQUYKUYDUXCUYEUXAUYBEWCUX - AUYBPWDWEUYMUYCUGUXAUYBUXTUFUXAUYBWFWGWHWIVJXDWMUXPUXRUYBTZUYCQZQZUHUPZ - UIUPZIUQZRZUHUVIUWDUXRXEZWJZWKZUOZUVTUIVUGUYTVUGFUMZVUGPUNZVUHUIVUGURZU - WBVUIUXOUYSUWBVUGUVIWLZFUVIVUFWNUWBVULUVKWLZFUVLVULVUMUMAUVMUVIUVKWOVCU - VJFWPVEWQXDUYTVUEUWGTZVUJUYTUYRVUNUXPUYRUYCVKUWGUWDUXRUEWAZUGWAXFVSVUNV - UGPVUGPVHUWGVUFXCPVHVUNRUVIVUFWRUWGVUEWSWTXAVSAVUIVUJQZVUKSZUVNUXOUYSAU - XAFUMZUXCQZVUDUHUXAUOZUIUXAURZSZOUSZVUQAFIUTVVCNOUIUHFIVRVSVVBVUQOVUGUV - IVUFUXKXBUXAVUGVHZVUSVUPVVAVUKVVDVURVUIUXCVUJUXAVUGFWCUXAVUGPWDWEVUTVUH - UIUXAVUGVUDUHUXAVUGWFWGWHWIVJXGWMUYTVUBVUGTZVUHQZQZVUEVUBXEZUVITZUVOVVH - JUQZRZUBUVIUOZUVTVVGVVEVVIUYTVVEVUHVKUVIVUEVUBUWDUXRXHZUIWAXFVSVVGVVKUB - UVIVVGUBUAXIZQZUVOUXAUJUPZXEZUKUPZXEZVHZUKXJZUJXJOXJZVVKVVGUVLVVNVWBUXP - UVLUYSVVFAUVLUVMUXOUUAXDOUJUKUVODEFUVIUUBUUCVVOVWAVVKOUJVVOVVTVVKUKVVTV - VOVVKVVTVVOVVKSVVGVVSUVITZQZUXADTVVPETVVRFTXKZUWDDTUXRETVUBFTXKZUXAUWDG - UQZOUEXLZXMZVVPUXRHUQZUJUGXLZXMZVVRVUBIUQZUKUIXLZXMZXKZUXAUWDUNZVVPUXRU - NZVVRVUBUNZUUDZQZXKZRZSVWDVXAVWEVWFVWDVXARZUXAUWDVWHVWDVXDVWHVWDVXDSVVG - UWDVVPXEZVVRXEZUVITZQZVWPVWRVWSXMZQZRZSZVXHVXKVVPUXRVWKVXHVXKVWKVXLVVGV - UEVVRXEZUVITZQZVWPVWSQZRZSVXOVXQVVRVUBVXOVWNQVWSVWPVWNVWSRZVXOVWNVXRVWS - VWNVVRVUBUUEUUFUUGXNUUHVXOVWSQZVWPVWSVXSVWOVWIVWLVXSVWMRZVWNRVWORVXSVUD - VXTUHVUGVVRUHUKXLVUCVWMVUAVVRVUBIXOXPVXOVUHVWSUYTVVEVUHVXNXQVTVXSVXNVVR - VUGTVVGVXNVWSUUIUVIVUEVVRVVMUKWAZXFXRXSVXSVVRVUBVXOVWSXTYEVWMVWNYAYBYCY - PYDVWKVXHVXOVXKVXQVWKVXGVXNVVGVWKVXFVXMUVIVWKVXEVUEVVRVVPUXRUWDUUJYFYGY - HVWKVXJVXPVWKVXIVWSVWPVWKVXIUXRUXRUNZVWSXMZVWSVWKVWRVYBVWSVVPUXRUXRWDYI - VYCVWSVYBRVYCVWSSUXRYJVYBVWSYKYQVWSVYBYLYMYNYHXPWHYOYRVXHVWRQZVWPVXIVYD - VWLVWIVWOVYDVWJRZVWKRVWLRVYDUXTVYEUFUYBVVPUFUJXLUXSVWJUXQVVPUXRHXOXPVVG - UYCVXGVWRUXPUYRUYCVVFXQXDVYDVXEUWGTZVVPUYBTVXHVYFVWRVXGVYFVVGVXEVVRUVIU - WDVVPXHVYAYSXNVTUWGUWDVVPVUOUJWAZXFXRXSVYDVVPUXRVXHVWRXTYEVWJVWKYAYBUUK - YPYDVWHVWDVXHVXDVXKVWHVWCVXGVVGVWHVVSVXFUVIVWHVVQVXEVVRUXAUWDVVPUULYFYG - YHVWHVXAVXJVWHVWTVXIVWPVWHVWTUWDUWDUNZVXIXMZVXIVWTVWQVXIXMVWHVYIVWQVWRV - WSUUMVWHVWQVYHVXIUXAUWDUWDWDYIUUNVYIVXIVYHRVYIVXISUWDYJVYHVXIYKYQVXIVYH - YLYMYNYHXPWHYOYRVWDVWQQZVWPVWTVYJVWIVWLVWOVYJVWGRZVWHRVWIRVYJUWFVYKUDUW - HUXAUDOXLUWEVWGUWCUXAUWDGXOXPUYTUWIVVFVWCVWQUWBUXNUWIUYSXQXGVWDUXAUWHTZ - VWQVWDVVQUWGTZVYLVWCVYMVVGVVQVVRUVIUXAVVPXHVYAYSXNUXAVVPUWGOWAVYGYSVJVT - XSVYJUXAUWDVWDVWQXTYEVWGVWHYAYBUUPYPYDYCVVTVVOVWDVVKVXCVVTVVNVWCVVGUVOV - VSUVIUUOYHVVTVVJVXBVVTVVJVVSVVHJUQVXBUVOVVSVVHJXOBCDEFGHIJUGUIOUJUKUEKU - UQYNXPWHYOUURUUSUUTUVAUVBUVSVVLUCVVHUVIUVPVVHVHZUVRVVKUBUVIVYNUVQVVJUVP - VVHUVOJUVEXPUVCUVDUVFYTYTYTUVGUVHUAUCUBUVKJVRXR $. - $} +$( Begin $[ set-mbox-df.mm $] $) +$( Skip $[ set-main.mm $] $) +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Drahflow +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. - $d A d $. $d a e $. $d A e $. $d a f $. $d A f $. $d a q $. - $d A q $. $d a x $. $d A x $. $d a y $. $d A y $. $d B a $. - $d B b $. $d b c $. $d B c $. $d b d $. $d B d $. $d b e $. - $d B e $. $d b f $. $d B f $. $d b q $. $d B q $. $d b x $. - $d B x $. $d b y $. $d B y $. $d C a $. $d C b $. $d C c $. - $d c d $. $d C d $. $d c e $. $d C e $. $d c f $. $d C f $. - $d c q $. $d C q $. $d c x $. $d C x $. $d c y $. $d C y $. - $d d e $. $d d f $. $d d q $. $d d x $. $d d y $. $d e f $. - $d e q $. $d e x $. $d e y $. $d f q $. $d f x $. $d f y $. - $d R a $. $d R b $. $d R c $. $d R d $. $d R e $. $d R f $. - $d R q $. $d R x $. $d R y $. $d S a $. $d S b $. $d S c $. - $d S d $. $d S e $. $d S f $. $d S q $. $d S x $. $d S y $. - $d T a $. $d T b $. $d T c $. $d T d $. $d T e $. $d T f $. - $d T q $. $d T x $. $d T y $. $d U a $. $d U b $. $d U c $. - $d U d $. $d U e $. $d U f $. $d U q $. $d X a $. $d X b $. - $d X c $. $d x y $. $d Y b $. $d Y c $. $d Z c $. - $( Calculate the predecsessor class for the triple order. (Contributed - by Scott Fenton, 21-Aug-2024.) $) - xpord3pred $p |- ( ( X e. A /\ Y e. B /\ Z e. C ) -> - Pred ( U , ( ( A X. B ) X. C ) , <. <. X , Y >. , Z >. ) = - ( ( ( ( Pred ( R , A , X ) u. { X } ) X. - ( Pred ( S , B , Y ) u. { Y } ) ) X. - ( Pred ( T , C , Z ) u. { Z } ) ) \ - { <. <. X , Y >. , Z >. } ) ) $= - ( cxp cop cpred csn wceq wcel wa va vb vc vq vd ve vf cv cun cdif opeq1 - opeq1d predeq3 syl sneq uneq12d xpeq1d sneqd difeq12d eqeq12d opeq2 w3a - xpeq2d cin wbr wrex wb elxpxp wi df-3an weq wo wne wn eldif opelxp elun - w3o cvv vex elpred velsn orbi12i bitri anbi12i df-ne otthne bitr3i opex - elv elsn simpr1 biantrurd orbi1d simpr2 simpr3 3anbi123d bitrdi bitr4id - xchnxbir anbi1d pm3.22 bitr2d bitrid breq1 xpord3lem bibi12d syl5ibrcom - eleq1 sylan2br anassrs rexlimdva rexlimdvva syl5bi pm5.32d elin 3bitr4g - ax-mp eqrdv wss predss a1i snssi unssd 3ad2ant1 3ad2ant2 xpss12 syl2anc - 3ad2ant3 ssdifssd sseqin2 sylib eqtrd vtocl3ga ) CDNZENZIUAUHZUBUHZOZUC - UHZOZPZCFYQPZYQQZUIZDGYRPZYRQZUIZNZEHYTPZYTQZUIZNZUUAQZUJZRYPIJYROZYTOZ - PZCFJPZJQZUIZUUHNZUULNZUUQQZUJZRYPIJKOZYTOZPZUVADGKPZKQZUIZNZUULNZUVGQZ - UJZRYPIUVFLOZPZUVLEHLPZLQZUIZNZUVPQZUJZRUAUBUCJKLCDEYQJRZUUBUURUUOUVEUW - DUUAUUQRUUBUURRUWDYSUUPYTYQJYRUKULZYPIUUAUUQUMUNUWDUUMUVCUUNUVDUWDUUIUV - BUULUWDUUEUVAUUHUWDUUCUUSUUDUUTCFYQJUMYQJUOUPUQUQUWDUUAUUQUWEURUSUTYRKR - ZUURUVHUVEUVOUWFUUQUVGRUURUVHRUWFUUPUVFYTYRKJVAULZYPIUUQUVGUMUNUWFUVCUV - MUVDUVNUWFUVBUVLUULUWFUUHUVKUVAUWFUUFUVIUUGUVJDGYRKUMYRKUOUPVCUQUWFUUQU - VGUWGURUSUTYTLRZUVHUVQUVOUWCUWHUVGUVPRUVHUVQRYTLUVFVAZYPIUVGUVPUMUNUWHU - VMUWAUVNUWBUWHUULUVTUVLUWHUUJUVRUUKUVSEHYTLUMYTLUOUPVCUWHUVGUVPUWIURUSU - TYQCSZYRDSZYTESZVBZUUBYPUUOVDZUUOUWMUDUUBUWNUWMUDUHZYPSZUWOUUAIVEZTZUWP - UWOUUOSZTUWOUUBSZUWOUWNSUWMUWPUWQUWSUWPUWOUEUHZUFUHZOZUGUHZOZRZUGEVFZUF - DVFUECVFUWMUWQUWSVGZUEUFUGUWOCDEVHUWMUXGUXHUEUFCDUWMUXACSZUXBDSZTZTUXFU - XHUGEUWMUXKUXDESZUXFUXHVIZUXKUXLTUWMUXIUXJUXLVBZUXMUXIUXJUXLVJUWMUXNTZU - XHUXFUXNUWMUXAYQFVEZUEUAVKZVLZUXBYRGVEZUFUBVKZVLZUXDYTHVEZUGUCVKZVLZVBZ - UXAYQVMUXBYRVMUXDYTVMVRZTZVBZUXEUUOSZVGUYHUXNUWMTZUYGTZUXOUYIUXNUWMUYGV - JUXOUYIUYGUYKUXOUYIUXIUXPTZUXQVLZUXJUXSTZUXTVLZTZUXLUYBTZUYCVLZTZUYFTZU - YGUYIUXEUUMSZUXEUUNSZVNZTUYTUXEUUMUUNVOVUAUYSVUCUYFVUAUXCUUISZUXDUULSZT - UYSUXCUXDUUIUULVPVUDUYPVUEUYRVUDUXAUUESZUXBUUHSZTUYPUXAUXBUUEUUHVPVUFUY - MVUGUYOVUFUXAUUCSZUXAUUDSZVLUYMUXAUUCUUDVQVUHUYLVUIUXQVUHUYLVGUACVSFYQU - XAUEVTZWAWJUEYQWBWCWDVUGUXBUUFSZUXBUUGSZVLUYOUXBUUFUUGVQVUKUYNVULUXTVUK - UYNVGUBDVSGYRUXBUFVTZWAWJUFYRWBWCWDWEWDVUEUXDUUJSZUXDUUKSZVLUYRUXDUUJUU - KVQVUNUYQVUOUYCVUNUYQVGUCEVSHYTUXDUGVTZWAWJUGYTWBWCWDWEWDUXEUUARZUYFVUB - VUQVNUXEUUAVMUYFUXEUUAWFUXAUXBUXDYQYRYTVUJVUMVUPWGWHUXEUUAUXCUXDWIWKWTW - EWDUXOUYEUYSUYFUXOUYEUYMUYOUYRVBUYSUXOUXRUYMUYAUYOUYDUYRUXOUXPUYLUXQUXO - UXIUXPUWMUXIUXJUXLWLWMWNUXOUXSUYNUXTUXOUXJUXSUWMUXIUXJUXLWOWMWNUXOUYBUY - QUYCUXOUXLUYBUWMUXIUXJUXLWPWMWNWQUYMUYOUYRVJWRXAWSUXOUYJUYGUWMUXNXBWMXC - XDUXFUWQUYHUWSUYIUXFUWQUXEUUAIVEUYHUWOUXEUUAIXEABCDEFGHIUBUCUEUFUGUAMXF - WRUWOUXEUUOXIXGXHXJXKXLXMXNXOUUAVSSUWTUWRVGYSYTWIYPVSIUUAUWOUDVTWAXRUWO - YPUUOXPXQXSUWMUUOYPXTUWNUUORUWMUUMYPUUNUWMUUIYOXTZUULEXTZUUMYPXTUWMUUEC - XTZUUHDXTZVURUWJUWKVUTUWLUWJUUCUUDCUUCCXTUWJCFYQYAYBYQCYCYDYEUWKUWJVVAU - WLUWKUUFUUGDUUFDXTUWKDGYRYAYBYRDYCYDYFUUECUUHDYGYHUWLUWJVUSUWKUWLUUJUUK - EUUJEXTUWLEHYTYAYBYTEYCYDYIUUIYOUULEYGYHYJUUOYPYKYLYMYN $. - $} + This is the mathbox of Jens-Wolfhard Schicke-Uffmann, reachable at + drahflow@gmx.de / drahflow.name - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a p $. - $d A p $. $d a ph $. $d A x $. $d A y $. $d B a $. $d B b $. - $d b c $. $d B c $. $d b p $. $d B p $. $d b ph $. $d B x $. - $d B y $. $d C a $. $d C b $. $d C c $. $d c p $. $d C p $. - $d c ph $. $d C x $. $d C y $. $d p ph $. $d R x $. $d R y $. - $d S x $. $d S y $. $d T x $. $d T y $. $d U a $. $d U b $. - $d U c $. $d U p $. $d x y $. - sexp3.1 $e |- ( ph -> R Se A ) $. - sexp3.2 $e |- ( ph -> S Se B ) $. - sexp3.3 $e |- ( ph -> T Se C ) $. - $( Show that the triple order is set-like. (Contributed by Scott Fenton, - 21-Aug-2024.) $) - sexp3 $p |- ( ph -> U Se ( ( A X. B ) X. C ) ) $= - ( vp va vb cpred cvv wcel vc cxp cv wral wse cop wceq wrex elxpxp wa wi - w3a df-3an csn cdif xpord3pred adantl simpr1 adantr setlikespec syl2anc - cun a1i unexd simpr2 xpexd simpr3 difexd eqeltrd predeq3 eleq1d biimprd - snex com12 sylan2br anassrs rexlimdva rexlimdvva syl5bi ralrimiv sylibr - syl dfse3 ) ADEUBFUBZJOUCZRZSTZOWDUDWDJUEAWGOWDWEWDTWEPUCZQUCZUFUAUCZUF - ZUGZUAFUHZQEUHPDUHAWGPQUAWEDEFUIAWMWGPQDEAWHDTZWIETZUJZUJWLWGUAFAWPWJFT - ZWLWGUKZWPWQUJAWNWOWQULZWRWNWOWQUMAWSUJZWDJWKRZSTZWRWTXADGWHRZWHUNZVBZE - HWIRZWIUNZVBZUBZFIWJRZWJUNZVBZUBZWKUNZUOZSWSXAXOUGABCDEFGHIJWHWIWJKUPUQ - WTXMXNSWTXIXLSSWTXEXHSSWTXCXDSSWTWNDGUEZXCSTAWNWOWQURAXPWSLUSDGWHUTVAXD - STWTWHVMVCVDWTXFXGSSWTWOEHUEZXFSTAWNWOWQVEAXQWSMUSEHWIUTVAXGSTWTWIVMVCV - DVFWTXJXKSSWTWQFIUEZXJSTAWNWOWQVGAXRWSNUSFIWJUTVAXKSTWTWJVMVCVDVFVHVIWL - XBWGWLWGXBWLWFXASWDJWEWKVJVKVLVNWBVOVPVQVRVSVTOWDJWCWA $. - $} +$) - ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. - $d A d $. $d a e $. $d A e $. $d a f $. $d A f $. $d a ps $. - $d a rh $. $d a th $. $d A x $. $d A y $. $d B a $. $d B b $. - $d b c $. $d B c $. $d b ch $. $d b d $. $d B d $. $d b e $. - $d B e $. $d b f $. $d B f $. $d b mu $. $d b th $. $d B x $. - $d B y $. $d C a $. $d C b $. $d C c $. $d c d $. $d C d $. - $d c e $. $d C e $. $d c f $. $d C f $. $d c la $. $d c th $. - $d C x $. $d C y $. $d ch f $. $d d e $. $d d f $. $d d ph $. - $d d ta $. $d e et $. $d e f $. $d e ps $. $d e ze $. $d f si $. - $d R d $. $d R e $. $d R f $. $d R x $. $d R y $. $d S d $. - $d S e $. $d S f $. $d S x $. $d S y $. $d T d $. $d T e $. - $d T f $. $d T x $. $d T y $. $d U a $. $d U b $. $d U c $. - $d U d $. $d U e $. $d U f $. $d X a $. $d X b $. $d X c $. - $d x y $. $d Y b $. $d Y c $. $d Z c $. - xpord3ind.1 $e |- R Fr A $. - xpord3ind.2 $e |- R Po A $. - xpord3ind.3 $e |- R Se A $. - xpord3ind.4 $e |- S Fr B $. - xpord3ind.5 $e |- S Po B $. - xpord3ind.6 $e |- S Se B $. - xpord3ind.7 $e |- T Fr C $. - xpord3ind.8 $e |- T Po C $. - xpord3ind.9 $e |- T Se C $. - xpord3ind.10 $e |- ( a = d -> ( ph <-> ps ) ) $. - xpord3ind.11 $e |- ( b = e -> ( ps <-> ch ) ) $. - xpord3ind.12 $e |- ( c = f -> ( ch <-> th ) ) $. - xpord3ind.13 $e |- ( a = d -> ( ta <-> th ) ) $. - xpord3ind.14 $e |- ( b = e -> ( et <-> ta ) ) $. - xpord3ind.15 $e |- ( b = e -> ( ze <-> th ) ) $. - xpord3ind.16 $e |- ( c = f -> ( si <-> ta ) ) $. - xpord3ind.17 $e |- ( a = X -> ( ph <-> rh ) ) $. - xpord3ind.18 $e |- ( b = Y -> ( rh <-> mu ) ) $. - xpord3ind.19 $e |- ( c = Z -> ( mu <-> la ) ) $. - xpord3ind.i $e |- ( ( a e. A /\ b e. B /\ c e. C ) -> - ( ( ( A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b ) - A. f e. Pred ( T , C , c ) th /\ - A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b ) ch /\ - A. d e. Pred ( R , A , a ) A. f e. Pred ( T , C , c ) ze ) - /\ - ( A. d e. Pred ( R , A , a ) ps /\ - A. e e. Pred ( S , B , b ) A. f e. Pred ( T , C , c ) ta /\ - A. e e. Pred ( S , B , b ) si ) /\ - A. f e. Pred ( T , C , c ) et ) - -> ph ) ) $. - $( Induction over the triple cross product ordering. Note that the - substitutions cover all possible cases of membership in the - predecessor class. (Contributed by Scott Fenton, 4-Sep-2024.) $) - xpord3ind $p |- ( ( X e. A /\ Y e. B /\ Z e. C ) -> la ) $= - ( cxp wfr wpo wse w3a wcel wtru a1i frxp3 mptru poxp3 sexp3 3pm3.2i cop - cv cpred wal wne w3o csn cun wral cdif xpord3pred eleq2d imbi1d eldifsn - wi ot2elxp vex otthne anbi12i bitri imbi1i impexp bitr2i bitr4di albidv - wa 2albidv wss ssun1 ssralv ax-mp ralimi syl wn predpoirr eleq1w mtbiri - weq necon2ai 3mix3d pm2.27 ralimia ssun2 biidd 3orbi123d bicomd equcoms - neeq1 wb imbi12d ralsn sylib 3mix2d ralbidv ralbii 3jca 2ralbidv equcom - r3al 2ralimi 3syl 2ralbii bicom 3imtr3i 3mix1d syl5 sylbid frpoins3xp3g - mpan ) NOVKPVKZTVLZUUMTVMZUUMTVNZVOUCNVPUDOVPUEPVPVOKUUNUUOUUPUUNVQLMNO - PQRSTUJNQVLVQUKVRORVLVQUNVRPSVLVQUQVRVSVTUUOVQLMNOPQRSTUJNQVMZVQULVRORV - MZVQUOVRPSVMZVQURVRWAVTUUPVQLMNOPQRSTUJNQVNVQUMVRORVNVQUPVRPSVNVQUSVRWB - VTWCABCDIJKUFUGUHUIUBUANOPTUCUDUEUFWEZNVPUGWEZOVPUHWEZPVPVOZUIWEZUAWEZW - DUBWEZWDZUUMTUUTUVAWDUVBWDZWFZVPZDWRZUBWGZUAWGUIWGZUVDUUTWHZUVEUVAWHZUV - FUVBWHZWIZDWRZUBPSUVBWFZUVBWJZWKZWLZUAORUVAWFZUVAWJZWKZWLZUINQUUTWFZUUT - WJZWKZWLZAUVCUVMUVDUWIVPUVEUWEVPUVFUWAVPVOZUVRWRZUBWGZUAWGUIWGUWJUVCUVL - UWMUIUAUVCUVKUWLUBUVCUVKUVGUWIUWEVKUWAVKZUVHWJWMZVPZDWRZUWLUVCUVJUWPDUV - CUVIUWOUVGLMNOPQRSTUUTUVAUVBUJWNWOWPUWQUWKUVQXIZDWRUWLUWPUWRDUWPUVGUWNV - PZUVGUVHWHZXIUWRUVGUWNUVHWQUWSUWKUWTUVQUVDUVEUVFUWIUWEUWAWSUVDUVEUVFUUT - UVAUVBUIWTUAWTUBWTXAXBXCXDUWKUVQDXEXFXGXHXJUVRUIUAUBUWIUWEUWAUUBXGUWJDU - BUVSWLZUAUWCWLUIUWGWLZCUAUWCWLZUIUWGWLZGUBUVSWLZUIUWGWLZVOZBUIUWGWLZEUB - UVSWLZUAUWCWLZHUAUWCWLZVOZFUBUVSWLZVOUVCAUWJUXGUXLUXMUWJUXBUXDUXFUWJUWF - UIUWGWLZUVRUBUVSWLZUAUWCWLZUIUWGWLUXBUWGUWIXKUWJUXNWRUWGUWHXLUWFUIUWGUW - IXMXNZUWFUXPUIUWGUWFUWBUAUWCWLZUXPUWCUWEXKUWFUXRWRUWCUWDXLUWBUAUWCUWEXM - XNZUWBUXOUAUWCUVSUWAXKUWBUXOWRUVSUVTXLUVRUBUVSUWAXMXNZXOXPZXOUXOUXAUIUA - UWGUWCUVRDUBUVSUVFUVSVPZUVQUVRDWRUYBUVPUVNUVOUYBUVFUVBUBUHYAZUYBUVBUVSV - PZUUSUYDXQURPSUVBXRXNUBUHUVSXSXTYBZYCUVQDYDXPYEUUCUUDUWJUVNUVOUVBUVBWHZ - WIZCWRZUAUWCWLZUIUWGWLZUXDUWJUVRUBUVTWLZUAUWCWLZUIUWGWLZUYJUWJUXNUYMUXQ - UWFUYLUIUWGUWFUXRUYLUXSUWBUYKUAUWCUVTUWAXKUWBUYKWRUVTUVSYFUVRUBUVTUWAXM - XNZXOXPZXOXPUYKUYHUIUAUWGUWCUVRUYHUBUVBUHWTZUYCUVQUYGDCUYCUVNUVNUVOUVOU - VPUYFUYCUVNYGUYCUVOYGZUVFUVBUVBYKZYHDCYLUHUBUHUBYAZCDVBYIYJYMYNZUUEYOUY - IUXCUIUWGUYHCUAUWCUVEUWCVPZUYGUYHCWRVUAUVOUVNUYFVUAUVEUVAUAUGYAZVUAUVAU - WCVPZUURVUCXQUOORUVAXRXNUAUGUWCXSXTYBZYPUYGCYDXPYEXOXPUWJUVNUVAUVAWHZUV - PWIZGWRZUBUVSWLZUIUWGWLZUXFUWJUXOUAUWDWLZUIUWGWLZVUIUWJUXNVUKUXQUWFVUJU - IUWGUWFUWBUAUWDWLZVUJUWDUWEXKUWFVULWRUWDUWCYFUWBUAUWDUWEXMXNZUWBUXOUAUW - DUXTXOXPZXOXPVUJVUHUIUWGUXOVUHUAUVAUGWTZVUBUVRVUGUBUVSVUBUVQVUFDGVUBUVN - UVNUVOVUEUVPUVPVUBUVNYGZUVEUVAUVAYKZVUBUVPYGZYHUGUAYAZGDYLVUBDGYLVEUGUA - UUAGDUUFUUGYMYQYNYRYOVUHUXEUIUWGVUGGUBUVSUYBVUFVUGGWRUYBUVPUVNVUEUYEYCV - UFGYDXPYEXOXPYSUWJUXHUXJUXKUWJUVNVUEUYFWIZBWRZUIUWGWLZUXHUWJUYKUAUWDWLZ - UIUWGWLZVVBUWJUXNVVDUXQUWFVVCUIUWGUWFVULVVCVUMUWBUYKUAUWDUYNXOXPXOXPVVC - VVAUIUWGVVCUYHUAUWDWLVVAUYKUYHUAUWDUYTYRUYHVVAUAUVAVUOVUBUYGVUTCBVUBUVN - UVNUVOVUEUYFUYFVUPVUQVUBUYFYGYHCBYLUGUAVUSBCVAYIYJYMYNXCYRYOVVABUIUWGUV - DUWGVPZVUTVVABWRVVEUVNVUEUYFVVEUVDUUTUIUFYAZVVEUUTUWGVPZUUQVVGXQULNQUUT - XRXNUIUFUWGXSXTYBUUHVUTBYDXPYEXPUWJUUTUUTWHZUVOUVPWIZEWRZUBUVSWLZUAUWCW - LZUXJUWJUXPUIUWHWLZVVLUWJUWFUIUWHWLZVVMUWHUWIXKUWJVVNWRUWHUWGYFUWFUIUWH - UWIXMXNZUWFUXPUIUWHUYAXOXPUXPVVLUIUUTUFWTZVVFUVRVVJUAUBUWCUVSVVFUVQVVID - EVVFUVNVVHUVOUVOUVPUVPUVDUUTUUTYKVVFUVOYGVVFUVPYGYHVVFEDEDYLUFUIVCYJYIY - MZYTYNYOVVKUXIUAUWCVVJEUBUVSUYBVVIVVJEWRUYBUVPVVHUVOUYEYCVVIEYDXPYEXOXP - UWJVVHUVOUYFWIZHWRZUAUWCWLZUXKUWJUYLUIUWHWLZVVTUWJVVNVWAVVOUWFUYLUIUWHU - YOXOXPVWAVVJUBUVTWLZUAUWCWLZVVTUYLVWCUIUUTVVPVVFUVRVVJUAUBUWCUVTVVQYTYN - VWBVVSUAUWCVVJVVSUBUVBUYPUYCVVIVVREHUYCVVHVVHUVOUVOUVPUYFUYCVVHYGUYQUYR - YHEHYLUHUBUYSHEVFYIYJYMYNYRXCYOVVSHUAUWCVUAVVRVVSHWRVUAUVOVVHUYFVUDYPVV - RHYDXPYEXPYSUWJVVHVUEUVPWIZFWRZUBUVSWLZUXMUWJVUJUIUWHWLZVWFUWJVVNVWGVVO - UWFVUJUIUWHVUNXOXPVWGVVKUAUWDWLZVWFVUJVWHUIUUTVVPVVFUVRVVJUAUBUWDUVSVVQ - YTYNVVKVWFUAUVAVUOVUBVVJVWEUBUVSVUBVVIVWDEFVUBVVHVVHUVOVUEUVPUVPVUBVVHY - GVUQVURYHVUBFEFEYLUGUAVDYJYIYMYQYNXCYOVWEFUBUVSUYBVWDVWEFWRUYBUVPVVHVUE - UYEYCVWDFYDXPYEXPYSVJUUIUUJUTVAVBVGVHVIUUKUUL $. - $} - $} +$( (End of Drahflow's mathbox.) $) +$( End $[ set-mbox-df.mm $] $) +$( Begin $[ set-mbox-sf.mm $] $) +$( Skip $[ set-main.mm $] $) $( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Ordering Ordinal Sequences -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Scott Fenton +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $) - ${ - $d A f x $. $d G f x $. $d X x $. - orderseqlem.1 $e |- F = { f | E. x e. On f : x --> A } $. - $( Lemma for ~ poseq and ~ soseq . The function value of a sequene is - either in ` A ` or null. (Contributed by Scott Fenton, 8-Jun-2011.) $) - orderseqlem $p |- ( G e. F -> ( G ` X ) e. ( A u. { (/) } ) ) $= - ( wcel cv wf con0 wrex cfv c0 csn cun wceq feq1 wss syl rexbidv ibi unss1 - elab2g crn frn fvrn0 ssel mpisyl rexlimivw ) EDHZAIZBEJZAKLZFEMZBNOZPZHZU - KUNULBCIZJZAKLUNCEDDUSEQUTUMAKULBUSERUAGUDUBUMURAKUMEUEZUPPZUQSZUOVBHURUM - VABSVCULBEUFVABUPUCTEFUGVBUQUOUHUIUJT $. - $} + $( Set up a new wff var $) + $v al $. $( Greek alpha $) - ${ - $d A b f x $. $d a b c f g t w x y z $. $d F a b c f g t w x z $. - $d R f g t w x z $. $d S a b c $. - poseq.1 $e |- R Po ( A u. { (/) } ) $. - poseq.2 $e |- F = { f | E. x e. On f : x --> A } $. - poseq.3 $e |- S = { <. f , g >. | ( ( f e. F /\ g e. F ) /\ - E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ - ( f ` x ) R ( g ` x ) ) ) } $. - $( A partial ordering of sequences of ordinals. (Contributed by Scott - Fenton, 8-Jun-2011.) $) - poseq $p |- S Po F $= - ( vz vw wbr wa wral cfv con0 wrex anbi12d va vb vc vt wpo cv wn wcel wceq - wi w3a c0 csn cun cab feq2 cbvrexvw abbii eqtri orderseqlem poirr sylancr - wf intnand adantr nrexdv imnan vex weq eleq1w anbi1d fveq1 eqeq1d ralbidv - mpbi breq1d rexbidv anbi2d eqeq2d breq2d mtbir raleq fveq2 breq12d bitrid - brab simplll simplrr an4 2rexbii reeanv bitri word eloni ordtri3or syl2an - wel w3o simp1l wss onelss imp adantll ssralv anim2d r19.26 syl6ibr ralimi - eqtr syl adantrd 3impia eqeq12d rspcv breq2 biimpd com3l ad2ant2lr impcom - syl6 3adant1 rspcev syl12anc 3exp ad2antrr ad2antlr ad2antll 3jca anim12i - potr anassrs sylan2 exp32 imbi1d syl5ibcom simp1r adantlr anim1d sylbir - a1d breq1 biimprd ad2ant2rl 3jaod mpd rexlimivv jca31 an4s syl2anb sylibr - pm3.2i a1i rgen3 df-po mpbir ) HEUEUAUFZUUPENZUGZUUPUBUFZENZUUSUCUFZENZOZ - UUPUVAENZUJZOZUCHPUBHPUAHPUVFUAUBUCHHHUVFUUPHUHZUUSHUHZUVAHUHZUKUURUVEUUQ - UVGUVGOZBUFZUUPQZUVLUIZBAUFZPZUVNUUPQZUVPDNZOZARSZOZUVJUVSUGZUJUVTUGUVGUW - AUVGUVGUVRARUVGUVRUGUVNRUHUVGUVQUVOUVGCULUMUNZDUEZUVPUWBUHUVQUGIUBCFHUUPU - VNHUVNCFUFZVCZARSZFUOUUSCUWDVCZUBRSZFUOJUWFUWHFUWEUWGAUBRUVNUUSCUWDUPUQUR - USUTUWBUVPDVAVBVDVEVFVEUVJUVSVGVOUWDHUHZGUFZHUHZOZUVKUWDQZUVKUWJQZUIZBUVN - PZUVNUWDQZUVNUWJQZDNZOZARSZOZUVGUWKOZUVLUWNUIZBUVNPZUVPUWRDNZOZARSZOUVTFG - UUPUUPEUAVHZUXIFUAVIZUWLUXCUXAUXHUXJUWIUVGUWKFUAHVJVKZUXJUWTUXGARUXJUWPUX - EUWSUXFUXJUWOUXDBUVNUXJUWMUVLUWNUVKUWDUUPVLVMZVNUXJUWQUVPUWRDUVNUWDUUPVLV - PTVQTGUAVIZUXCUVJUXHUVSUXMUWKUVGUVGGUAHVJVRUXMUXGUVRARUXMUXEUVOUXFUVQUXMU - XDUVMBUVNUXMUWNUVLUVLUVKUWJUUPVLVSVNUXMUWRUVPUVPDUVNUWJUUPVLVTTVQTKWFWAUV - CUVGUVIOZUVLUVKUVAQZUIZBUDUFZPZUXQUUPQZUXQUVAQZDNZOZUDRSZOZUVDUUTUVGUVHOZ - UVLUVKUUSQZUIZBLUFZPZUYHUUPQZUYHUUSQZDNZOZLRSZOZUVHUVIOZUYFUXOUIZBMUFZPZU - YRUUSQZUYRUVAQZDNZOZMRSZOZUYDUVBUXBUXCUXDBUYHPZUYJUYHUWJQZDNZOZLRSZOUYOFG - UUPUUSEUXIUBVHZUXJUWLUXCUXAVUJUXKUXAUWOBUYHPZUYHUWDQZVUGDNZOZLRSUXJVUJUWT - VUOALRALVIZUWPVULUWSVUNUWOBUVNUYHWBVUPUWQVUMUWRVUGDUVNUYHUWDWCUVNUYHUWJWC - WDTUQUXJVUOVUILRUXJVULVUFVUNVUHUXJUWOUXDBUYHUXLVNUXJVUMUYJVUGDUYHUWDUUPVL - 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WCVWDVWFVWPUYRUYHXAXBXCVWPVVQUYGUYQOZBUYHPZVWLVWPVVQUYIUYQBUYHPZOZVWRVWPU - YSVWSUYIUYQBUYHUYRXDXEUYGUYQBUYHXFZXGVWQUXPBUYHUVLUYFUXOXIZXHZXTXJXKXLVWF - VVSVWNVWEVVSVWFVWNUYSUYLVWFVWNUJZUYIVUBUYSUYLVXDVWFUYSUYLVWNVWFUYSUYKVWMU - IZUYLVWNUJUYQVXEBUYHUYRBLVIUYFUYKUXOVWMUVKUYHUUSWCUVKUYHUVAWCXMXNVXEUYLVW - NUYKVWMUYJDXOXPXTXQXBXRXSYAUYBVWLVWNOZUDUYHRUDLVIZUXRVWLUYAVWNUXPBUXQUYHW - BVXGUXSUYJUXTVWMDUXQUYHUUPWCUXQUYHUVAWCWDTYBZYCYTYDVWCVWGVWJUJVWDVWCVWRUY - LUYKVWMDNZOZOZVWAUJVWGVWJVWCVXKVVOUYCVXKVVOOVWCVXFUYCVWRVXJVVOVXFVWRVWLVX - JVVOOVWNVXCVVOVXJVWNVVOUWCUYJUWBUHZUYKUWBUHZVWMUWBUHZUKVXJVWNUJIVVOVXLVXM - VXNUVGVXLUVHUYPACFHUUPUYHJUTYEUVHVXMUVGUYPACFHUUSUYHJUTYFUVIVXNUYEUVHACFH - UVAUYHJUTYGYHUWBUYJUYKVWMDYJVBXSYIYKVXHYLYMVWGVXKVVSVWAVWGVWRVVQVXJVVRVWR - VWTVWGVVQVXAVWGVWSUYSUYIUYQBUYHUYRWBVRWEVWGVXIVUBUYLVWGUYKUYTVWMVUADUYHUY - RUUSWCUYHUYRUVAWCWDVRTYNYOVEVWEVWHVVSVWAVWEVWHVVSUKZUYCVVOVXOVWDUXPBUYRPZ - UYRUUPQZVUADNZUYCVWCVWDVWHVVSYPVWEVWHVVSVXPVWEVWHOUYRUYHWTZVVSVXPUJVWCVWH - VXSVWDVWCVWHVXSUYHUYRXAXBYQVXSVVQVXPVVRVXSVVQUYGBUYRPZUYSOZVXPVXSUYIVXTUY - SUYGBUYRUYHXDYRVYAVWQBUYRPVXPUYGUYQBUYRXFVWQUXPBUYRVXBXHYSXTXKXJXLVWHVVSV - XRVWEVVSVWHVXRUYIVUBVWHVXRUJZUYSUYLUYIVUBVYBVWHUYIVUBVXRVWHUYIVXQUYTUIZVU - BVXRUJUYGVYCBUYRUYHBMVIUVLVXQUYFUYTUVKUYRUUPWCUVKUYRUUSWCXMXNVYCVXRVUBVXQ - UYTVUADUUAUUBXTXQXBUUCXSYAUYBVXPVXROUDUYRRUDMVIZUXRVXPUYAVXRUXPBUXQUYRWBV - YDUXSVXQUXTVUADUXQUYRUUPWCUXQUYRUVAWCWDTYBYCYTYDUUDUUEUUFYSXSUUGUUHUUIUXB - UXCUXDBUXQPZUXSUXQUWJQZDNZOZUDRSZOUYDFGUUPUVAEUXIVVEUXJUWLUXCUXAVYIUXKUXA - UWOBUXQPZUXQUWDQZVYFDNZOZUDRSUXJVYIUWTVYMAUDRAUDVIZUWPVYJUWSVYLUWOBUVNUXQ - WBVYNUWQVYKUWRVYFDUVNUXQUWDWCUVNUXQUWJWCWDTUQUXJVYMVYHUDRUXJVYJVYEVYLVYGU - XJUWOUXDBUXQUXLVNUXJVYKUXSVYFDUXQUWDUUPVLVPTVQWETVVLUXCUXNVYIUYCVVLUWKUVI - UVGVVMVRVVLVYHUYBUDRVVLVYEUXRVYGUYAVVLUXDUXPBUXQVVLUWNUXOUVLVVNVSVNVVLVYF - UXTUXSDUXQUWJUVAVLVTTVQTKWFUUJUUKUULUUMUAUBUCHEUUNUUO $. - $} + $( Let variable ` al ` be a wff. $) + walpha $f wff al $. - ${ - $d a b p q y $. $d A f p q x $. $d A y $. $d F a b f g x $. - $d f g x y $. $d R f g x $. $d S a b $. - soseq.1 $e |- R Or ( A u. { (/) } ) $. - soseq.2 $e |- F = { f | E. x e. On f : x --> A } $. - soseq.3 $e |- S = { <. f , g >. | ( ( f e. F /\ g e. F ) /\ - E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ - ( f ` x ) R ( g ` x ) ) ) } $. - soseq.4 $e |- -. (/) e. A $. - $( A linear ordering of sequences of ordinals. (Contributed by Scott - Fenton, 8-Jun-2011.) $) - soseq $p |- S Or F $= - ( wral wcel wa cfv wceq con0 wrex wi va vb vp wor wpo wbr weq w3o csn cun - vq cv c0 sopo ax-mp poseq wo wn eleq1w anbi1d fveq1 eqeq1d ralbidv breq1d - anbi12d rexbidv anbi2d eqeq2d breq2d brabg bianabs ancoms notbid ralinexa - wb orbi12d andi eqcom ralbii anbi1i orbi2i bitri rexbii r19.43 xchbinx wf - cab feq2 cbvrexvw abbii eqtri orderseqlem sotrieq mpan syl2an imbi2d wsbc - vex fveq2 eqeq12d sbcie imbi1i tfisg sylbir feq1 elab2 anbi12i reeanv wss - bitr4i onss ssralv syl ad2antlr rspcv a1i ffvelcdm cdm eloni ordirr eleq2 - fdm word biimparc sylan ndmfv eleq1 ex com23 sylan2 exp4b imp32 syldd imp - syl5 mtoi syld jcad wfn ffn eqtr2 biimprd 3syl adantlr eqtr biimpd expcom - ad2antrr adantll ordtri3or adantr 3orel13 syl6ci eqfnfv2 adantl rexlimivv - sylibrd sylbi sylbird syl5bir sylbid orrd df-3or bitr2i sylib rgen2 df-so - 3orcomb mpbir2an ) HEUDHEUEUAULZUBULZEUFZUAUBUGZUVKUVJEUFZUHZUBHMUAHMABCD - EFGHCUMUIUJZDUDZUVPDUEIUVPDUNUOJKUPUVOUAUBHHUVJHNZUVKHNZOZUVLUVNUQZUVMUQZ - UVOUVTUWAUVMUVTUWAURBULZUVJPZUWCUVKPZQZBAULZMZUWGUVJPZUWGUVKPZDUFZOZARSZU - WEUWDQZBUWGMZUWJUWIDUFZOZARSZUQZURZUVMUVTUWAUWSUVTUVLUWMUVNUWRUVTUVLUWMFU - LZHNZGULZHNZOZUWCUXAPZUWCUXCPZQZBUWGMZUWGUXAPZUWGUXCPZDUFZOZARSZOZUVRUXDO - ZUWDUXGQZBUWGMZUWIUXKDUFZOZARSZOUVTUWMOFGUVJUVKHHEFUAUGZUXEUXPUXNUYAUYBUX - BUVRUXDFUAHUSUTUYBUXMUXTARUYBUXIUXRUXLUXSUYBUXHUXQBUWGUYBUXFUWDUXGUWCUXAU - VJVAVBVCUYBUXJUWIUXKDUWGUXAUVJVAVDVEVFVEGUBUGZUXPUVTUYAUWMUYCUXDUVSUVRGUB - HUSVGUYCUXTUWLARUYCUXRUWHUXSUWKUYCUXQUWFBUWGUYCUXGUWEUWDUWCUXCUVKVAVHVCUY - CUXKUWJUWIDUWGUXCUVKVAVIVEVFVEKVJVKUVSUVRUVNUWRVOUVSUVROZUVNUWRUXOUVSUXDO - ZUWEUXGQZBUWGMZUWJUXKDUFZOZARSZOUYDUWROFGUVKUVJHHEFUBUGZUXEUYEUXNUYJUYKUX - BUVSUXDFUBHUSUTUYKUXMUYIARUYKUXIUYGUXLUYHUYKUXHUYFBUWGUYKUXFUWEUXGUWCUXAU - VKVAVBVCUYKUXJUWJUXKDUWGUXAUVKVAVDVEVFVEGUAUGZUYEUYDUYJUWRUYLUXDUVRUVSGUA - HUSVGUYLUYIUWQARUYLUYGUWOUYHUWPUYLUYFUWNBUWGUYLUXGUWDUWEUWCUXCUVJVAVHVCUY - LUXKUWIUWJDUWGUXCUVJVAVIVEVFVEKVJVKVLVPVMUWTUWHUWKUWPUQZURZTZARMZUVTUVMUY - PUWHUYMOZARSZUWSUWHUYMARVNUYRUWLUWQUQZARSUWSUYQUYSARUYQUWLUWHUWPOZUQUYSUW - HUWKUWPVQUYTUWQUWLUWHUWOUWPUWFUWNBUWGUWDUWEVRVSVTWAWBWCUWLUWQARWDWBWEUVTU - YPUWHUWIUWJQZTZARMZUVMUVTVUBUYOARUVTVUAUYNUWHUVRUWIUVPNZUWJUVPNZVUAUYNVOZ - UVSBCFHUVJUWGHUWGCUXAWFZARSZFWGUWCCUXAWFZBRSZFWGJVUHVUJFVUGVUIABRUWGUWCCU - XAWHWIWJWKZWLBCFHUVKUWGVUKWLUVQVUDVUEOVUFIUVPUWIUWJDWMWNWOWPVCVUCVUAARMZU - VTUVMVUCVUAAUWCWQZBUWGMZVUATZARMVULVUOVUBARVUNUWHVUAVUMUWFBUWGVUAUWFAUWCB - WRABUGUWIUWDUWJUWEUWGUWCUVJWSUWGUWCUVKWSWTXAVSXBVSVUAABXCXDUVTUCULZCUVJWF - ZUKULZCUVKWFZOZUKRSUCRSZVULUVMTZUVTVUQUCRSZVUSUKRSZOVVAUVRVVCUVSVVDUVRUWG - CUVJWFZARSZVVCVUHVVFFUVJHUAWRUYBVUGVVEARUWGCUXAUVJXEVFJXFVVEVUQAUCRUWGVUP - CUVJWHWIWBUVSUWGCUVKWFZARSZVVDVUHVVHFUVKHUBWRUYKVUGVVGARUWGCUXAUVKXEVFJXF - VVGVUSAUKRUWGVURCUVKWHWIWBXGVUQVUSUCUKRRXHXJVUTVVBUCUKRRVUPRNZVURRNZOZVUT - VVBVVKVUTOZVULUCUKUGZVUAAVUPMZOZUVMVVLVULVVMVVNVVLVULVUPVURNZURZVURVUPNZU - RZOVVPVVMVVRUHZVVMVVLVULVVQVVSVVLVULVUAAVURMZVVQVVJVULVWATZVVIVUTVVJVURRX - IVWBVURXKVUAAVURRXLXMXNVVLVWAVVQVVLVWAOVVPUMCNZLVVLVWAVVPVWCTVVLVVPVWAVWC - VVLVVPVWAVUPUVJPZVUPUVKPZQZVWCVVPVWAVWFTTVVLVUAVWFAVUPVURAUCUGUWIVWDUWJVW - EUWGVUPUVJWSUWGVUPUVKWSWTXOXPVVKVUQVUSVVPVWFVWCTZTVVKVUQVUSVVPVWGVUSVVPOV - WECNZVVKVUQOVWGVURCVUPUVKXQVVIVUQVWHVWGTZVVJVUQVVIUVJXRZVUPQZVWIVUPCUVJYB - VVIVWKOZVWFVWHVWCVWLVUPVWJNZURZVWDUMQZVWFVWHVWCTZTVVIVUPVUPNZURZVWKVWNVVI - VUPYCZVWRVUPXSZVUPXTXMVWKVWNVWRVWKVWMVWQVWJVUPVUPYAVMYDYEVUPUVJYFVWOVWFVW - PVWOVWFOUMVWEQZVWPVWDUMVWEUUAVXAVWCVWHUMVWECYGUUBXMYHUUCYIYJUUDYOYKYLYMYI - YNYPYHYQVVLVULVVNVVSVVIVULVVNTZVVJVUTVVIVUPRXIVXBVUPXKVUAAVUPRXLXMUUHZVVL - VVNVVSVVLVVNOVVRVWCLVVLVVNVVRVWCTVVLVVRVVNVWCVVLVVRVVNVURUVJPZVURUVKPZQZV - WCVVRVVNVXFTTVVLVUAVXFAVURVUPAUKUGUWIVXDUWJVXEUWGVURUVJWSUWGVURUVKWSWTXOX - PVVKVUQVUSVVRVXFVWCTZTZVVKVUSVUQVXHVVKVUSVUQVVRVXGVUQVVROVXDCNZVVKVUSOVXG - VUPCVURUVJXQVUSVVKUVKXRZVURQZVXIVXGTZVURCUVKYBVVJVXKVXLVVIVVJVXKOVXEUMQZV - XLVVJVURVURNZURZVXKVXMVVJVURYCZVXOVURXSZVURXTXMVXOVXKOVURVXJNZURZVXMVXKVX - SVXOVXKVXRVXNVXJVURVURYAVMYDVURUVKYFXMYEVXMVXFVXIVWCVXFVXMVXIVWCTZVXFVXMO - VXDUMQZVXTVXDVXEUMUUEVYAVXIVWCVXDUMCYGUUFXMUUGYIXMUUIYJYOYKYIYLYMYIYNYPYH - YQYRVVKVVTVUTVVIVWSVXPVVTVVJVWTVXQVUPVURUUJWOUUKVVPVVMVVRUULUUMVXCYRVUTUV - MVVOVOZVVKVUQUVJVUPYSUVKVURYSVYBVUSVUPCUVJYTVURCUVKYTAVUPVURUVJUVKUUNWOUU - OUUQYHUUPUURYOUUSUUTUVAUVBUVOUVLUVNUVMUHUWBUVLUVMUVNUVHUVLUVNUVMUVCUVDUVE - UVFUAUBHEUVGUVI $. - $} + $( Declare some new variables. $) + $v xO $. + $v xL $. + $v xR $. + $v yO $. + $v yL $. + $v yR $. + $( Let ` xO ` be an individual variable. $) + vxo $f setvar xO $. + $( Let ` xL ` be an individual variable. $) + vxl $f setvar xL $. + $( Let ` xR ` be an individual variable. $) + vxr $f setvar xR $. + $( Let ` yO ` be an individual variable. $) + vyo $f setvar yO $. + $( Let ` yL ` be an individual variable. $) + vyl $f setvar yL $. + $( Let ` yR ` be an individual variable. $) + vyr $f setvar yR $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Well-founded zero, successor, and limits + ZFC Axioms in primitive form =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $c wsuc WLim $. + $( ~ ax-ext without distinct variable conditions or defined symbols. + (Contributed by Scott Fenton, 13-Oct-2010.) $) + axextprim $p |- -. A. x -. ( ( x e. y -> x e. z ) -> + ( ( x e. z -> x e. y ) -> y = z ) ) $= + ( wel wb weq wi wex wn wal axextnd wa dfbi2 imbi1i impexp bitri exbii df-ex + mpbi ) ABDZACDZEZBCFZGZAHZTUAGZUATGZUCGGZIAJIZABCKUEUHAHUIUDUHAUDUFUGLZUCGU + HUBUJUCTUAMNUFUGUCOPQUHARPS $. - $( Declare the syntax for well-founded successor. $) - cwsuc $a class wsuc ( R , A , X ) $. + $( ~ ax-rep without distinct variable conditions or defined symbols. + (Contributed by Scott Fenton, 13-Oct-2010.) $) + axrepprim $p |- -. A. x -. ( -. A. y -. A. z ( ph -> z = y ) -> + A. z -. ( ( A. y z e. x -> -. A. x ( A. z x e. y -> + -. A. y ph ) ) -> -. ( -. A. x ( A. z x e. y -> -. A. y ph ) + -> A. y z e. x ) ) ) $= + ( weq wi wal wex wel wa wb wn axrepnd df-ex df-an exbii exnal bitri bibi2i + dfbi1 albii imbi12i mpbi ) ADCEFDGZCHZDBICGZBCIDGZACGZJZBHZKZDGZFZBHZUDLCGL + ZUFUGUHLFZBGLZFUQUFFLFLZDGZFZLBGLZABCDMUNUTBHVAUMUTBUEUOULUSUDCNUKURDUKUFUQ + KURUJUQUFUJUPLZBHUQUIVBBUGUHOPUPBQRSUFUQTRUAUBPUTBNRUC $. - $( Declare the syntax for well-founded limit class. $) - cwlim $a class WLim ( R , A ) $. + $( ~ ax-un without distinct variable conditions or defined symbols. + (Contributed by Scott Fenton, 13-Oct-2010.) $) + axunprim $p |- -. A. x -. A. y ( -. A. x ( y e. x -> -. x e. z ) + -> y e. x ) $= + ( wel wa wex wi wal axunnd df-an exbii exnal bitri imbi1i albii df-ex mpbi + wn ) BADZACDZEZAFZSGZBHZAFZSTRGZAHRZSGZBHZRAHRZABCIUEUIAFUJUDUIAUCUHBUBUGSU + BUFRZAFUGUAUKASTJKUFALMNOKUIAPMQ $. - $( Define the concept of a successor in a well-founded set. (Contributed by - Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) $) - df-wsuc $a |- wsuc ( R , A , X ) = inf ( Pred ( `' R , A , X ) , A , R ) $. + $( ~ ax-pow without distinct variable conditions or defined symbols. + (Contributed by Scott Fenton, 13-Oct-2010.) $) + axpowprim $p |- ( A. x -. A. y ( A. x ( -. A. z -. x e. y -> A. y x e. z ) + -> y e. x ) -> x = y ) $= + ( weq wel wn wal wi wex axpownd df-ex imbi1i albii exbii bitri sylib con4i + ) ABDZABEZFCGFZACEBGZHZAGZBAEZHZBGZFAGZRFSCIZUAHZAGZUDHZBGZAIZUGFZABCJUMUFA + IUNULUFAUKUEBUJUCUDUIUBAUHTUASCKLMLMNUFAKOPQ $. - ${ - $d R x $. $d A x $. - $( Define the class of limit points of a well-founded set. (Contributed by - Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) $) - df-wlim $a |- WLim ( R , A ) = - { x e. A | ( x =/= inf ( A , A , R ) /\ - x = sup ( Pred ( R , A , x ) , A , R ) ) } $. - $} + $( ~ ax-reg without distinct variable conditions or defined symbols. + (Contributed by Scott Fenton, 13-Oct-2010.) $) + axregprim $p |- ( x e. y -> + -. A. x ( x e. y -> -. A. z ( z e. x -> -. z e. y ) ) ) $= + ( wel wn wi wal wa wex axregnd df-an exbii exnal bitri sylib ) ABDZPCADCBDE + FCGZHZAIZPQEFZAGEZABCJSTEZAIUARUBAPQKLTAMNO $. - $( Equality theorem for well-founded successor. (Contributed by Scott - Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) - wsuceq123 $p |- ( ( R = S /\ A = B /\ X = Y ) -> - wsuc ( R , A , X ) = wsuc ( S , B , Y ) ) $= - ( wceq w3a ccnv cpred cwsuc simp1 cnveqd predeq123 syld3an1 simp2 infeq123d - cinf df-wsuc 3eqtr4g ) CDGZABGZEFGZHZACIZEJZACRBDIZFJZBDRACEKBDFKUDUFACUHBD - UEUGGUBUAUCUFUHGUDCDUAUBUCLZMABUEUGEFNOUAUBUCPUIQACESBDFST $. + $( ~ ax-inf without distinct variable conditions or defined symbols. + (New usage is discouraged.) (Contributed by Scott Fenton, + 13-Oct-2010.) $) + axinfprim $p |- -. A. x -. ( y e. z -> -. ( y e. x -> + -. A. y ( y e. x -> -. A. z ( y e. z -> -. z e. x ) ) ) ) $= + ( wel wa wex wi wn axinfnd df-an exbii exnal bitri imbi2i albii anbi2i mpbi + wal df-ex ) BCDZBADZUATCADZEZCFZGZBRZEZGZAFZTUAUATUBHGZCRHZGZBRZHGHZGZHARHZ + ABCIUIUOAFUPUHUOAUGUNTUGUAUMEUNUFUMUAUEULBUDUKUAUDUJHZCFUKUCUQCTUBJKUJCLMNO + PUAUMJMNKUOASMQ $. - $( Equality theorem for well-founded successor. (Contributed by Scott - Fenton, 13-Jun-2018.) $) - wsuceq1 $p |- ( R = S -> wsuc ( R , A , X ) = wsuc ( S , A , X ) ) $= - ( wceq cwsuc eqid wsuceq123 mp3an23 ) BCEAAEDDEABDFACDFEAGDGAABCDDHI $. + $( ~ ax-ac without distinct variable conditions or defined symbols. + (New usage is discouraged.) (Contributed by Scott Fenton, + 26-Oct-2010.) $) + axacprim $p |- -. A. x -. A. y A. z ( A. x -. ( y e. z -> -. z e. w ) -> + -. A. w -. A. y -. ( ( -. A. w ( y e. z -> ( z e. w -> + ( y e. w -> -. w e. x ) ) ) -> y = w ) -> -. ( y = w -> + -. A. w ( y e. z -> ( z e. w -> ( y e. w -> -. w e. x ) + ) ) ) ) ) $= + ( wel wa wal wex wb wi wn axacnd df-an albii annim anbi2i exbii bitri df-ex + weq anass pm4.63 bitr3i 3bitr2i exnal bibi1i dfbi1 imbi12i 2albii mpbi ) BC + EZCDEZFZAGZUMBDEZDAEZFZFZDHZBDTZIZBGZDHZJZCGBGZAHZUKULKJKZAGZUKULUOUPKJZJZJ + ZDGKZUTJUTVLJKJKZBGZKDGKZJZCGBGZKAGKZABCDLVFVQAHVRVEVQAVDVPBCUNVHVCVOUMVGAU + KULMNVCVNDHVOVBVNDVAVMBVAVLUTIVMUSVLUTUSVKKZDHVLURVSDURUKULUQFZFUKVJKZFVSUK + ULUQUAWAVTUKWAULVIKZFVTULVIOWBUQULUOUPUBPUCPUKVJOUDQVKDUERUFVLUTUGRNQVNDSRU + HUIQVQASRUJ $. - $( Equality theorem for well-founded successor. (Contributed by Scott - Fenton, 13-Jun-2018.) $) - wsuceq2 $p |- ( A = B -> wsuc ( R , A , X ) = wsuc ( R , B , X ) ) $= - ( wceq cwsuc eqid wsuceq123 mp3an13 ) CCEABEDDEACDFBCDFECGDGABCCDDHI $. - $( Equality theorem for well-founded successor. (Contributed by Scott - Fenton, 13-Jun-2018.) $) - wsuceq3 $p |- ( X = Y -> wsuc ( R , A , X ) = wsuc ( R , A , Y ) ) $= - ( wceq cwsuc eqid wsuceq123 mp3an12 ) BBEAAECDEABCFABDFEBGAGAABBCDHI $. +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Untangled classes +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) ${ - nfwsuc.1 $e |- F/_ x R $. - nfwsuc.2 $e |- F/_ x A $. - nfwsuc.3 $e |- F/_ x X $. - $( Bound-variable hypothesis builder for well-founded successor. - (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, - 10-Oct-2021.) $) - nfwsuc $p |- F/_ x wsuc ( R , A , X ) $= - ( cwsuc ccnv cpred cinf df-wsuc nfcnv nfpred nfinf nfcxfr ) ABCDHBCIZDJZB - CKBCDLARBCABQDACEMFGNFEOP $. + $d x A $. + $( We call a class "untanged" if all its members are not members of + themselves. The term originates from Isbell (see citation in ~ dfon2 ). + Using this concept, we can avoid a lot of the uses of the Axiom of + Regularity. Here, we prove a series of properties of untanged classes. + First, we prove that an untangled class is not a member of itself. + (Contributed by Scott Fenton, 28-Feb-2011.) $) + untelirr $p |- ( A. x e. A -. x e. x -> -. A e. A ) $= + ( wel wn wral wcel cv wceq eleq1 eleq2 bitrd notbid rspccv pm2.01d ) AACZ + DZABEBBFZPQDABBAGZBHZOQSOBRFQRBRIRBBJKLMN $. $} ${ - $d R x $. $d S x $. $d A x $. $d B x $. - $( Equality theorem for the limit class. (Contributed by Scott Fenton, - 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) - wlimeq12 $p |- ( ( R = S /\ A = B ) -> - WLim ( R , A ) = WLim ( S , B ) ) $= - ( vx wceq wa cv cinf wne cpred csup crab cwlim simpr infeq123d neeq2d weq - simpl df-wlim predeq123 mp3an3 supeq123d eqeq2d anbi12d rabeqbidv 3eqtr4g - equid ) CDFZABFZGZEHZAACIZJZULACULKZACLZFZGZEAMULBBDIZJZULBDULKZBDLZFZGZE - BMACNBDNUKURVDEABUIUJOZUKUNUTUQVCUKUMUSULUKAACBBDVEVEUIUJSZPQUKUPVBULUKUO - ACVABDUIUJEERUOVAFEUHABCDULULUAUBVEVFUCUDUEUFEACTEBDTUG $. + $d x y A $. + $( The union of a class is untangled iff all its members are untangled. + (Contributed by Scott Fenton, 28-Feb-2011.) $) + untuni $p |- ( A. x e. U. A -. x e. x + <-> A. y e. A A. x e. y -. x e. x ) $= + ( cv cuni wcel wel wn wal wral wrex r19.23v albii ralcom4 eluni2 3bitr4ri + wi imbi1i df-ral ralbii 3bitr4i ) ADZCEZFZAAGHZQZAIZABGZUEQZAIZBCJZUEAUCJ + UEABDZJZBCJUIBCJZAIUHBCKZUEQZAIUKUGUNUPAUHUEBCLMUIBACNUFUPAUDUOUEBUBCORMP + UEAUCSUMUJBCUEAULSTUA $. $} - $( Equality theorem for the limit class. (Contributed by Scott Fenton, - 15-Jun-2018.) $) - wlimeq1 $p |- ( R = S -> WLim ( R , A ) = WLim ( S , A ) ) $= - ( wceq cwlim eqid wlimeq12 mpan2 ) BCDAADABEACEDAFAABCGH $. - - $( Equality theorem for the limit class. (Contributed by Scott Fenton, - 15-Jun-2018.) $) - wlimeq2 $p |- ( A = B -> WLim ( R , A ) = WLim ( R , B ) ) $= - ( wceq cwlim eqid wlimeq12 mpan ) CCDABDACEBCEDCFABCCGH $. - ${ - $d R y $. $d A y $. $d x y $. - nfwlim.1 $e |- F/_ x R $. - nfwlim.2 $e |- F/_ x A $. - $( Bound-variable hypothesis builder for the limit class. (Contributed by - Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) - nfwlim $p |- F/_ x WLim ( R , A ) $= - ( vy cwlim cv cinf cpred csup wceq wa crab df-wlim nfcv nfinf nfne nfpred - wne nfsup nfeq2 nfan nfrabw nfcxfr ) ABCGFHZBBCIZTZUFBCUFJZBCKZLZMZFBNFBC - OULAFBUHUKAAUFUGAUFPZABBCEEDQRAUFUJAUIBCABCUFDEUMSEDUAUBUCEUDUE $. + $d x A $. $d x y $. + untsucf.1 $e |- F/_ y A $. + $( If a class is untangled, then so is its successor. (Contributed by + Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, + 11-Dec-2016.) $) + untsucf $p |- ( A. x e. A -. x e. x -> A. y e. suc A -. y e. y ) $= + ( wel wn wral csuc nfv nfralw cv wcel wceq vex elsuc elequ1 elequ2 notbid + wo bitrd rspccv untelirr eleq1 eleq2 syl5ibrcom jaod biimtrid ralrimi ) A + AEZFZACGZBBEZFZBCHZUJBACDUJBIJBKZUNLUOCLZUOCMZSUKUMUOCBNOUKUPUMUQUJUMAUOC + AKUOMZUIULURUIBAEULABAPABBQTRUAUKUMUQCCLZFACUBUQULUSUQULCUOLUSUOCUOUCUOCC + UDTRUEUFUGUH $. $} - ${ - $d R x $. $d A x $. $d X x $. - $( Membership in the limit class. (Contributed by Scott Fenton, - 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) $) - elwlim $p |- ( X e. WLim ( R , A ) <-> ( X e. A /\ - X =/= inf ( A , A , R ) /\ - X = sup ( Pred ( R , A , X ) , A , R ) ) ) $= - ( vx cwlim wcel cinf wne cpred csup wceq wa neeq1 predeq3 supeq1d eqeq12d - w3a cv id anbi12d df-wlim elrab2 3anass bitr4i ) CABEZFCAFZCAABGZHZCABCIZ - ABJZKZLZLUFUHUKQDRZUGHZUMABUMIZABJZKZLULDCAUEUMCKZUNUHUQUKUMCUGMURUMCUPUJ - URSURAUOUIBABUMCNOPTDABUAUBUFUHUKUCUD $. - $} + $( The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) + (Proof shortened by Andrew Salmon, 27-Aug-2011.) $) + unt0 $p |- A. x e. (/) -. x e. x $= + ( wel wn ral0 ) AABCAD $. ${ - $d R x y z $. $d A x y z $. - $( The zero of a well-founded set is a member of that set. (Contributed by - Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) $) - wzel $p |- ( ( R We A /\ R Se A /\ A =/= (/) ) -> - inf ( A , A , R ) e. A ) $= - ( vx vy vz wwe wse c0 wne w3a wor cv wrex wbr wn wral wi wa wcel wal weso - 3ad2ant1 cpred wceq wss simp1 simp2 ssidd simp3 tz6.26 syl22anc wb elpred - cvv vex elv notbii imnan bitr4i pm2.27 ad2antll rspcev ex ad2antrl jctird - breq1 syl5bi com23 alimdv eq0 r19.26 df-ral bitr3i 3imtr4g reximdva infcl - expr mpd ) ABFZABGZAHIZJZCDEAABVSVTABKWAABUAUBWBABCLZUCZHUDZCAMZDLZWCBNZO - ZDAPWCWGBNZELZWGBNZEAMZQZDAPRZCAMWBVSVTAAUEWAWFVSVTWAUFVSVTWAUGWBAUHVSVTW - AUICAABUJUKWBWEWOCAWBWCASZRZWGWDSZOZDTWGASZWIWNRZQZDTZWEWOWQWSXBDWQWTWSXA - WBWPWTWSXAQWSWTWIQZWBWPWTRRZXAWSWTWHRZOXDWRXFWRXFULCAUNBWCWGDUOUMUPUQWTWH - URUSXEXDWIWNWTXDWIQWBWPWTWIUTVAWPWNWBWTWPWJWMWLWJEWCAWKWCWGBVFVBVCVDVEVGV - QVHVIDWDVJWOXADAPXCWIWNDAVKXADAVLVMVNVOVRVP $. + $d x y A $. + $( If there is an untangled element of a class, then the intersection of + the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) $) + untint $p |- ( E. x e. A A. y e. x -. y e. y -> + A. y e. |^| A -. y e. y ) $= + ( wel wn cv wral cint wcel wss wi intss1 ssralv syl rexlimiv ) BBDEZBAFZG + ZPBCHZGZACQCISQJRTKQCLPBSQMNO $. $} ${ - $d A x y z w $. $d ph x y $. $d R x y z w $. $d X x y z w $. - wsuclem.1 $e |- ( ph -> R We A ) $. - wsuclem.2 $e |- ( ph -> R Se A ) $. - wsuclem.3 $e |- ( ph -> X e. V ) $. - wsuclem.4 $e |- ( ph -> E. w e. A X R w ) $. - $( Lemma for the supremum properties of well-founded successor. - (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, - 10-Oct-2021.) $) - wsuclem $p |- ( ph -> E. x e. A - ( A. y e. Pred ( `' R , A , X ) -. y R x /\ - A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) ) $= - ( cv c0 wrex wbr wa wcel cvv ccnv cpred wceq wn wi wwe wse wss wne predss - wral a1i crab dfpred3g syl elexd wb brcnvg ancoms rexbidva bitrid biimpar - rabn0 syl2anc eqnetrd tz6.26 syl22anc breq1 rexrab w3a noel simp2r eleq2d - rexeqdv mtbiri simp3 elpredg mtbid 3expa ralrimiva simp1rl simp1rr adantr - vex expr 3ad2ant1 elpred mpbir2and rspcev 3expia anim12d ancomsd reximdva - syl5bi sylbid mpd ) AFGUAZIUBZGBNZUBZOUCZBWRPZCNZWSGQZUDZCWRUKZWSXCGQZDNZ - XCGQZDWRPZUEZCFUKZRZBFPZAFGUFFGUGWRFUHZWROUIXBJKXOAFWQIUJULAWRENZIWQQZEFU - MZOAIHSZWRXRUCLEFWQHIUNUOAITSZIXPGQZEFPZXROUIZAIHLUPMXTYCYBYCXQEFPXTYBXQE - FVCXTXQYAEFXPFSXTXQYAUQXPIFTGURUSUTVAVBVDVEBFWRGVFVGAXBXABXCIWQQZCFUMZPZX - NAXABWRYEAXSWRYEUCLCFWQHIUNUOVNYFWSIWQQZXARZBFPAXNYDYGXABCFXCWSIWQVHVIAYH - XMBFAWSFSZRZXAYGXMYJXAXFYGXLAYIXAXFAYIXARZRXECWRAYKXCWRSZXEAYKYLVJZXCWTSZ - XDYMYNXCOSXCVKYMWTOXCAYIXAYLVLVMVOYMWSTSZYLYNXDUQYOYMBWDZULAYKYLVPWRTGWSX - CVQVDVRVSVTWEAYIYGXLAYIYGRZRZXKCFYRXCFSZXGXJYRYSXGVJZWSWRSZXGXJYTUUAYIYGY - IYGAYSXGWAYIYGAYSXGWBYTXSUUAYQUQYRYSXSXGAXSYQLWCWFFHWQIWSYPWGUOWHYRYSXGVP - XIXGDWSWRXHWSXCGVHWIVDWJVTWEWKWLWMWNWOWP $. + $d x A $. + $( If ` A ` is well-founded by ` _E ` , then it is untangled. (Contributed + by Scott Fenton, 1-Mar-2011.) $) + efrunt $p |- ( _E Fr A -> A. x e. A -. x e. x ) $= + ( cep wfr cv wcel wn wa wbr frirr epel sylnib ralrimiva ) BCDZAEZOFZGABNO + BFHOOCIPBOCJAOKLM $. $} ${ - wsucex.1 $e |- ( ph -> R Or A ) $. - $( Existence theorem for well-founded successor. (Contributed by Scott - Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) - wsucex $p |- ( ph -> wsuc ( R , A , X ) e. _V ) $= - ( cwsuc ccnv cpred cinf cvv df-wsuc infexd eqeltrid ) ABCDFBCGDHZBCIJBCDK - ABNCELM $. + $d x y A $. + $( A transitive class is untangled iff its elements are. (Contributed by + Scott Fenton, 7-Mar-2011.) $) + untangtr $p |- ( Tr A -> + ( A. x e. A -. x e. x <-> + A. x e. A A. y e. x -. y e. y ) ) $= + ( wtr wel wn wral cv cuni wss df-tr ssralv sylbi weq elequ1 elequ2 notbid + wi bitrd cbvralvw untuni bitri syl6ib untelirr ralimi impbid1 ) CDZAAEZFZ + ACGZBBEZFZBAHZGZACGZUGUJUIACIZGZUOUGUPCJUJUQRCKUIAUPCLMUQULBUPGUOUIULABUP + ABNZUHUKURUHBAEUKABAOABBPSQTBACUAUBUCUNUIACBUMUDUEUF $. $} - ${ - $d R a b c y $. $d A a b c y $. $d X a b c y $. $d ph a b $. - wsuccl.1 $e |- ( ph -> R We A ) $. - wsuccl.2 $e |- ( ph -> R Se A ) $. - wsuccl.3 $e |- ( ph -> X e. V ) $. - wsuccl.4 $e |- ( ph -> E. y e. A X R y ) $. - $( If ` X ` is a set with an ` R ` successor in ` A ` , then its - well-founded successor is a member of ` A ` . (Contributed by Scott - Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) - wsuccl $p |- ( ph -> wsuc ( R , A , X ) e. A ) $= - ( va vb vc cwsuc ccnv cpred cinf df-wsuc wwe wor weso syl infcl eqeltrid - wsuclem ) ACDFNCDOFPZCDQCCDFRAKLMCUFDACDSCDTGCDUAUBAKLMBCDEFGHIJUEUCUD $. - $} + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Extra propositional calculus theorems +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) ${ - $d R a b c y $. $d A a b c y $. $d X a b c y $. $d ph a b c $. - $d Y y $. - wsuclb.1 $e |- ( ph -> R We A ) $. - wsuclb.2 $e |- ( ph -> R Se A ) $. - wsuclb.3 $e |- ( ph -> X e. V ) $. - wsuclb.4 $e |- ( ph -> Y e. A ) $. - wsuclb.5 $e |- ( ph -> X R Y ) $. - $( A well-founded successor is a lower bound on points after ` X ` . - (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, - 10-Oct-2021.) $) - wsuclb $p |- ( ph -> -. Y R wsuc ( R , A , X ) ) $= - ( va vb vc vy wbr wcel wb syl2anc mpbird ccnv cpred cinf cwsuc wn elpredg - brcnvg wwe wor weso syl cv wrex breq2 rspcev wsuclem inflb df-wsuc breq2i - mpd sylnibr ) AFBCUAZEUBZBCUCZCPZFBCEUDZCPAFVCQZVEUEAVGFEVBPZAVHEFCPZKAFB - QZEDQZVHVIRJIFEBDCUGSTAVKVJVGVHRIJBDVBEFUFSTALMNBVCFCABCUHBCUIGBCUJUKALMN - OBCDEGHIAVJVIEOULZCPZOBUMJKVMVIOFBVLFECUNUOSUPUQUTVFVDFCBCEURUSVA $. + 3jaodd.1 $e |- ( ph -> ( ps -> ( ch -> et ) ) ) $. + 3jaodd.2 $e |- ( ph -> ( ps -> ( th -> et ) ) ) $. + 3jaodd.3 $e |- ( ph -> ( ps -> ( ta -> et ) ) ) $. + $( Double deduction form of ~ 3jaoi . (Contributed by Scott Fenton, + 20-Apr-2011.) $) + 3jaodd $p |- ( ph -> ( ps -> ( ( ch \/ th \/ ta ) -> et ) ) ) $= + ( w3o wi com3r 3jaoi com3l ) CDEJABFCABFKKDEABCFGLABDFHLABEFILMN $. $} + $( Closed form of ~ 3ori . (Contributed by Scott Fenton, 20-Apr-2011.) $) + 3orit $p |- ( ( ph \/ ps \/ ch ) <-> ( ( -. ph /\ -. ps ) -> ch ) ) $= + ( w3o wo wn wi wa df-3or df-or ioran imbi1i 3bitri ) ABCDABEZCENFZCGAFBFHZC + GABCINCJOPCABKLM $. + + $( A biconditional in the antecedent is the same as two implications. + (Contributed by Scott Fenton, 12-Dec-2010.) $) + biimpexp $p |- ( ( ( ph <-> ps ) -> ch ) + <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> ch ) ) ) $= + ( wb wi wa dfbi2 imbi1i impexp bitri ) ABDZCEABEZBAEZFZCELMCEEKNCABGHLMCIJ + $. + ${ - $d R x $. $d A x $. - $( The class of limit points is a subclass of the base class. (Contributed - by Scott Fenton, 16-Jun-2018.) $) - wlimss $p |- WLim ( R , A ) C_ A $= - ( vx cv cinf wne cpred csup wceq wa cwlim df-wlim ssrab3 ) CDZAABEFNABNGA - BHIJCAABKCABLM $. + $d A b $. $d C b $. + $( Ordinal less than is equivalent to having an ordinal between them. + (Contributed by Scott Fenton, 8-Aug-2024.) $) + onelssex $p |- ( ( A e. On /\ C e. On ) -> + ( A e. C <-> E. b e. C A C_ b ) ) $= + ( con0 wcel wa cv wss wrex ssid sseq2 rspcev mpan2 ontr2 expcomd rexlimdv + impbid2 ) ADEBDEFZABEZACGZHZCBIZSAAHZUBAJUAUCCABTAAKLMRUASCBRUATBESATBNOP + Q $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Natural operations on ordinals + Misc. Useful Theorems =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $( Declare new constants $) - $c +no $. - - $( Declare the syntax for natural ordinal addition. See ~ df-nadd . $) - cnadd $a class +no $. + $( Two classes are unequal iff their intersection is a proper subset of one + of them. (Contributed by Scott Fenton, 23-Feb-2011.) $) + nepss $p |- ( A =/= B <-> ( ( A i^i B ) C. A \/ ( A i^i B ) C. B ) ) $= + ( wne cin wss wa wo wpss wceq nne neeq1 biimprcd biimtrid orrd jctl necon3i + wn inidm adantl df-pss inss1 inss2 orim12i ineq2 eqtr3di eqtrdi jaoi impbii + syl ineq1 orbi12i bitr4i ) ABCZABDZAEZUNACZFZUNBEZUNBCZFZGZUNAHZUNBHZGUMVAU + MUPUSGVAUMUPUSUPQUNAIZUMUSUNAJVDUSUMUNABKLMNUPUQUSUTUPUOABUAOUSURABUBOUCUIU + QUMUTUPUMUOABUNAABIZAADUNAABAUDARUEPSUSUMURABUNBVEUNBBDBABBUJBRUFPSUGUHVBUQ + VCUTUNATUNBTUKUL $. ${ - $d x y a z w t $. - $( Define natural ordinal addition. This is a commutative form of addition - over the ordinals. (Contributed by Scott Fenton, 26-Aug-2024.) $) - df-nadd $a |- +no = - frecs ( { <. x , y >. | ( x e. ( On X. On ) /\ y e. ( On X. On ) - /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ - ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ - x =/= y ) ) } , ( On X. On ) , - ( z e. _V , a e. _V |-> - |^| { w e. On | ( ( a " ( { ( 1st ` z ) } X. ( 2nd ` z ) ) ) C_ w /\ - ( a " ( ( 1st ` z ) X. { ( 2nd ` z ) } ) ) C_ w ) } ) ) $. + 3ccased.1 $e |- ( ph -> ( ( ch /\ et ) -> ps ) ) $. + 3ccased.2 $e |- ( ph -> ( ( ch /\ ze ) -> ps ) ) $. + 3ccased.3 $e |- ( ph -> ( ( ch /\ si ) -> ps ) ) $. + 3ccased.4 $e |- ( ph -> ( ( th /\ et ) -> ps ) ) $. + 3ccased.5 $e |- ( ph -> ( ( th /\ ze ) -> ps ) ) $. + 3ccased.6 $e |- ( ph -> ( ( th /\ si ) -> ps ) ) $. + 3ccased.7 $e |- ( ph -> ( ( ta /\ et ) -> ps ) ) $. + 3ccased.8 $e |- ( ph -> ( ( ta /\ ze ) -> ps ) ) $. + 3ccased.9 $e |- ( ph -> ( ( ta /\ si ) -> ps ) ) $. + $( Triple disjunction form of ~ ccased . (Contributed by Scott Fenton, + 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) $) + 3ccased $p |- ( ph -> ( ( ( ch \/ th \/ ta ) /\ ( et \/ ze \/ si ) ) -> + ps ) ) $= + ( wa com12 3jaodan w3o wi 3jaoian ) CDEUAFGHUAZRABCUDABUBZDECFUEGHACFRBIS + ACGRBJSACHRBKSTDFUEGHADFRBLSADGRBMSADHRBNSTEFUEGHAEFRBOSAEGRBPSAEHRBQSTUC + S $. $} ${ - on2recs.1 $e |- F = frecs ( { <. x , y >. | - ( x e. ( On X. On ) /\ y e. ( On X. On ) - /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ - ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ - x =/= y ) ) } , ( On X. On ) , G ) $. + $d R x y z $. $d A x y z $. + $( Expansion of the definition of a strict order. (Contributed by Scott + Fenton, 6-Jun-2016.) $) + dfso3 $p |- ( R Or A <-> + A. x e. A A. y e. A A. z e. A + ( -. x R x /\ + ( ( x R y /\ y R z ) -> x R z ) /\ + ( x R y \/ x = y \/ y R x ) ) ) $= + ( cv wbr wn wa wi weq w3o wral w3a wor wcel c0 wne wb ralbii r19.27zv syl + ne0i ralbiia df-3an 2ralbii df-po anbi1i df-so r19.26-2 3bitr4i 3bitr4ri + wpo ) AFZUNEGHZUNBFZEGZUPCFZEGIUNUREGJZIZUQABKUPUNEGLZIZCDMZBDMZADMUTCDMZ + VAIZBDMZADMZUOUSVANZCDMZBDMADMDEOZVDVGADVCVFBDUPDPDQRVCVFSDUPUCUTVACDUAUB + UDTVJVCABDDVIVBCDUOUSVAUETUFDEUMZVABDMADMZIVEBDMADMZVMIVKVHVLVNVMABCDEUGU + HABDEUIVEVAABDDUJUKUL $. + $} - ${ - $d x y $. - $( Show that double recursion over ordinals yields a function over pairs - of ordinals. (Contributed by Scott Fenton, 3-Sep-2024.) $) - on2recsfn $p |- F Fn ( On X. On ) $= - ( con0 cv wcel c1st cfv cep wbr wceq wo c2nd w3a wfr wtru a1i mptru cxp - wne copab wpo wse wfn eqid onfr frxp2 wwe wor weso sopo mp2b poxp2 epse - epweon sexp2 fpr1 mp3an ) FFUAZAGZVAHBGZVAHVBIJZVCIJZKLVDVEMNVBOJZVCOJZ - KLVFVGMNVBVCUBPPABUCZQZVAVHUDZVAVHUEZCVAUFVIRABFFKKVHVHUGZFKQRUHSZVMUIT - VJRABFFKKVHVLFKUDZRFKUJFKUKVNUQFKULFKUMUNSZVOUOTVKRABFFKKVHVLFKUERFUPSZ - VPURTVAVHCDEUSUT $. - $} + $( A binary relation involving unordered triples. (Contributed by Scott + Fenton, 7-Jun-2016.) $) + brtpid1 $p |- A { <. A , B >. , C , D } B $= + ( cop ctp wbr wcel opex tpid1 df-br mpbir ) ABABEZCDFZGMNHMCDABIJABNKL $. - ${ - $d x y $. - $( Calculate the value of the double ordinal recursion operator. - (Contributed by Scott Fenton, 3-Sep-2024.) $) - on2recsov $p |- ( ( A e. On /\ B e. On ) -> - ( A F B ) = ( <. A , B >. G - ( F |` ( ( suc A X. suc B ) \ { <. A , B >. } ) ) ) ) $= - ( con0 wcel co cxp cfv cep wceq w3a cpred csn wtru a1i cun cop c1st wbr - wa cv c2nd wne copab cres csuc cdif df-ov opelxp wfr wpo wse eqid frxp2 - wo onfr mptru wwe wor weso sopo mp2b poxp2 epse sexp2 3pm3.2i fpr2 mpan - epweon sylbir eqtrid predon adantr uneq1d df-suc eqtr4di adantl xpeq12d - xpord2pred difeq1d eqtrd reseq2d oveq2d ) CHIZDHIZUDZCDEJZCDUAZEHHKZAUE - ZWMIBUEZWMIWNUBLZWOUBLZMUCWPWQNUSWNUFLZWOUFLZMUCWRWSNUSWNWOUGOOABUHZWLP - ZUIZFJZWLECUJZDUJZKZWLQZUKZUIZFJWJWKWLELZXCCDEULWJWLWMIZXJXCNZCDHHUMWMW - TUNZWMWTUOZWMWTUPZOXKXLXMXNXOXMRABHHMMWTWTUQZHMUNRUTSZXQURVAXNRABHHMMWT - XPHMUOZRHMVBHMVCXRVMHMVDHMVEVFSZXSVGVAXORABHHMMWTXPHMUPRHVHSZXTVIVAVJWM - WTEFWLGVKVLVNVOWJXBXIWLFWJXAXHEWJXAHMCPZCQZTZHMDPZDQZTZKZXGUKXHABHHMMWT - CDXPWCWJYGXFXGWJYCXDYFXEWJYCCYBTXDWJYACYBWHYACNWICVPVQVRCVSVTWJYFDYETXE - WJYDDYEWIYDDNWHDVPWAVRDVSVTWBWDWEWFWGWE $. - $} + $( A binary relation involving unordered triples. (Contributed by Scott + Fenton, 7-Jun-2016.) $) + brtpid2 $p |- A { C , <. A , B >. , D } B $= + ( cop ctp wbr wcel opex tpid2 df-br mpbir ) ABCABEZDFZGMNHCMDABIJABNKL $. - $} + $( A binary relation involving unordered triples. (Contributed by Scott + Fenton, 7-Jun-2016.) $) + brtpid3 $p |- A { C , D , <. A , B >. } B $= + ( cop ctp wbr wcel opex tpid3 df-br mpbir ) ABCDABEZFZGMNHCDMABIJABNKL $. ${ - $d a b $. $d a c $. $d a d $. $d a ps $. $d a ta $. $d a x $. - $d a y $. $d b c $. $d b ch $. $d b d $. $d b et $. $d b x $. - $d b y $. $d c d $. $d c ph $. $d c th $. $d c x $. $d c y $. - $d d ps $. $d d x $. $d d y $. $d X a $. $d X b $. $d x y $. - $d Y b $. - on2ind.1 $e |- ( a = c -> ( ph <-> ps ) ) $. - on2ind.2 $e |- ( b = d -> ( ps <-> ch ) ) $. - on2ind.3 $e |- ( a = c -> ( th <-> ch ) ) $. - on2ind.4 $e |- ( a = X -> ( ph <-> ta ) ) $. - on2ind.5 $e |- ( b = Y -> ( ta <-> et ) ) $. - on2ind.i $e |- ( ( a e. On /\ b e. On ) -> - ( ( A. c e. a A. d e. b ch /\ - A. c e. a ps /\ - A. d e. b th ) -> ph ) ) $. - $( Double induction over ordinal numbers. (Contributed by Scott Fenton, - 26-Aug-2024.) $) - on2ind $p |- ( ( X e. On /\ Y e. On ) -> et ) $= - ( con0 cep vx vy cv cxp wcel c1st cfv wbr wceq wo c2nd wne w3a copab eqid - onfr wwe wor wpo epweon weso sopo mp2b epse wa cpred predon adantr adantl - wral raleqdv raleqbidv 3anbi123d sylbid xpord2ind ) ABCDEFUAUBSSTTUAUCZSS - UDZUEUBUCZVQUEVPUFUGZVRUFUGZTUHVSVTUIUJVPUKUGZVRUKUGZTUHWAWBUIUJVPVRULUMU - MUAUBUNZGHIJKLWCUOUPSTUQSTURSTUSUTSTVASTVBVCZSVDZUPWDWEMNOPQIUCZSUEZJUCZS - UEZVEZCLSTWHVFZVJZKSTWFVFZVJZBKWMVJZDLWKVJZUMCLWHVJZKWFVJZBKWFVJZDLWHVJZU - MAWJWNWRWOWSWPWTWJWLWQKWMWFWGWMWFUIWIWFVGVHZWJCLWKWHWIWKWHUIWGWHVGVIZVKVL - WJBKWMWFXAVKWJDLWKWHXBVKVMRVNVO $. + $d x V $. $d A y $. $d ps y $. $d x y $. + iota5f.1 $e |- F/ x ph $. + iota5f.2 $e |- F/_ x A $. + iota5f.3 $e |- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) ) $. + $( A method for computing iota. (Contributed by Scott Fenton, + 13-Dec-2017.) $) + iota5f $p |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A ) $= + ( vy wcel wa cv wceq wb wal cio nfel1 nfan wi eqeq2 alrimi bibi2d imbi12d + weq nfeq2 albid iotaval vtoclg adantl mpd ) ADEJZKZBCLZDMZNZCOZBCPZDMZULU + OCAUKCFCDEGQRHUAUKUPURSZABCIUDZNZCOZUQILZMZSUSIDEVCDMZVBUPVDURVEVAUOCCVCD + GUEVEUTUNBVCDUMTUBUFVCDUQTUCBCIUGUHUIUJ $. $} + $( Closed form of ~ ja . Proved using the completeness script. + (Proof modification is discouraged.) (Contributed by Scott Fenton, + 13-Dec-2021.) $) + jath $p |- ( ( -. ph -> ch ) -> ( ( ps -> ch ) -> + ( ( ph -> ps ) -> ch ) ) ) $= + ( wn wi jcn pm2.21 imim2 ax-mp ax-1 pm2.61 ) ADZLCEZBCEZABEZCEZEZEZEZRLCDZR + EZEZSLTMDZEZEZUBLCFUDUAEZUEUBEUCREUFMQGUCRTHIUDUALHIIUAREZUBSECREZUGCQEZUHC + PEZUICOJPQEZUJUIEPNJZPQCHIIQREZUIUHEQMJZQRCHIICRKIZUARLHIIAREZSREABDZREZEZU + PAUQQEZEZUSAUQPEZEZVAAUQODZEZEZVCABFVEVBEZVFVCEVDPEVGOCGVDPUQHIVEVBAHIIVBUT + EZVCVAEUKVHULPQUQHIVBUTAHIIUTUREZVAUSEUMVIUNQRUQHIUTURAHIIURREZUSUPEBREZVJB + UAEZVKBTQEZEZVLBTNDZEZEZVNBCFVPVMEZVQVNEVOQEVRNPGVOQTHIVPVMBHIIVMUAEZVNVLEU + MVSUNQRTHIVMUABHIIUGVLVKEUOUARBHIIBRKIURRAHIIARKII $. + ${ - $d a b $. $d a c $. $d a d $. $d a e $. $d a f $. $d a ps $. - $d a rh $. $d a th $. $d a x $. $d a y $. $d b c $. $d b ch $. - $d b d $. $d b e $. $d b f $. $d b mu $. $d b th $. $d b x $. - $d b y $. $d c d $. $d c e $. $d c f $. $d c la $. $d c th $. - $d c x $. $d c y $. $d ch f $. $d d e $. $d d f $. $d d ph $. - $d d ta $. $d d x $. $d d y $. $d e et $. $d e f $. $d e ps $. - $d e x $. $d e y $. $d e ze $. $d f si $. $d f x $. $d f y $. - $d X a $. $d X b $. $d X c $. $d x y $. $d Y b $. $d Y c $. $d Z c $. - on3ind.1 $e |- ( a = d -> ( ph <-> ps ) ) $. - on3ind.2 $e |- ( b = e -> ( ps <-> ch ) ) $. - on3ind.3 $e |- ( c = f -> ( ch <-> th ) ) $. - on3ind.4 $e |- ( a = d -> ( ta <-> th ) ) $. - on3ind.5 $e |- ( b = e -> ( et <-> ta ) ) $. - on3ind.6 $e |- ( b = e -> ( ze <-> th ) ) $. - on3ind.7 $e |- ( c = f -> ( si <-> ta ) ) $. - on3ind.8 $e |- ( a = X -> ( ph <-> rh ) ) $. - on3ind.9 $e |- ( b = Y -> ( rh <-> mu ) ) $. - on3ind.10 $e |- ( c = Z -> ( mu <-> la ) ) $. - on3ind.i $e |- ( ( a e. On /\ b e. On /\ c e. On ) -> - ( ( ( A. d e. a A. e e. b A. f e. c th /\ - A. d e. a A. e e. b ch /\ - A. d e. a A. f e. c ze ) - /\ - ( A. d e. a ps /\ - A. e e. b A. f e. c ta /\ - A. e e. b si ) /\ - A. f e. c et ) - -> ph ) ) $. - $( Triple induction over ordinals. (Contributed by Scott Fenton, - 4-Sep-2024.) $) - on3ind $p |- ( ( X e. On /\ Y e. On /\ Z e. On ) -> la ) $= - ( vx vy con0 cep cv cxp wcel c1st cfv wbr wceq wo c2nd w3a wne copab eqid - wa onfr wwe wor wpo epweon weso sopo mp2b epse cpred wral predon 3ad2ant1 - 3ad2ant2 3ad2ant3 raleqdv raleqbidv 3anbi123d sylbid xpord3ind ) ABCDEFGH - IJKULUMUNUNUNUOUOUOULUPZUNUNUQUNUQZURUMUPZWKURWJUSUTZUSUTZWLUSUTZUSUTZUOV - AWNWPVBVCWMVDUTZWOVDUTZUOVAWQWRVBVCWJVDUTZWLVDUTZUOVAWSWTVBVCVEWJWLVFVIVE - ULUMVGZLMNOPQRSTXAVHVJUNUOVKUNUOVLUNUOVMVNUNUOVOUNUOVPVQZUNVRZVJXBXCVJXBX - CUAUBUCUDUEUFUGUHUIUJQUPZUNURZRUPZUNURZSUPZUNURZVEZDMUNUOXHVSZVTZLUNUOXFV - SZVTZTUNUOXDVSZVTZCLXMVTZTXOVTZGMXKVTZTXOVTZVEZBTXOVTZEMXKVTZLXMVTZHLXMVT - ZVEZFMXKVTZVEDMXHVTZLXFVTZTXDVTZCLXFVTZTXDVTZGMXHVTZTXDVTZVEZBTXDVTZEMXHV - TZLXFVTZHLXFVTZVEZFMXHVTZVEAXJYAYOYFYTYGUUAXJXPYJXRYLXTYNXJXNYITXOXDXEXGX - OXDVBXIXDWAWBZXJXLYHLXMXFXGXEXMXFVBXIXFWAWCZXJDMXKXHXIXEXKXHVBXGXHWAWDZWE - WFWFXJXQYKTXOXDUUBXJCLXMXFUUCWEWFXJXSYMTXOXDUUBXJGMXKXHUUDWEWFWGXJYBYPYDY - RYEYSXJBTXOXDUUBWEXJYCYQLXMXFUUCXJEMXKXHUUDWEWFXJHLXMXFUUCWEWGXJFMXKXHUUD - WEWGUKWHWI $. + $d a b $. $d a ph $. $d a ps $. $d a x $. $d a y $. $d b ph $. + $d b ps $. $d b x $. $d b y $. $d ph y $. $d ps x $. $d x y $. + $( Cross product of two class abstractions. (Contributed by Scott Fenton, + 19-Aug-2024.) $) + xpab $p |- ( { x | ph } X. { y | ps } ) = { <. x , y >. | ( ph /\ ps ) } $= + ( va vb cab cxp wa copab wcel wsbc wbr wsb df-clab sban sbsbc sbv 3bitr3i + cv relxp relopabv anbi12i anbi1i sbbii anbi2i bitri bitr4i brabsb 3bitr4i + brxp eqid eqbrriv ) EFACGZBDGZHZABIZCDJZUNUOUAUQCDUBETZUNKZFTZUOKZIZUQDVA + LZCUSLZUSVAUPMUSVAURMVCACENZBDFNZIZVEUTVFVBVGAECOBFDOUCVDCENAVGIZCENZVEVH + VDVICEUQDFNADFNZVGIVDVIABDFPUQDFQVKAVGADFRUDSUEVDCEQVJVFVGCENZIVHAVGCEPVL + VGVFVGCERUFUGSUHUSVAUNUOUKUQCDUSVAURURULUIUJUM $. $} ${ - $d a w $. $d a x $. $d a y $. $d a z $. $d w x $. $d w y $. $d w z $. + $d A x $. $d A y $. $d B x $. $d B y $. $d B z $. $d ph x $. $d x y $. $d x z $. $d y z $. - $( Natural addition is a function over pairs of ordinals. (Contributed by - Scott Fenton, 26-Aug-2024.) $) - naddfn $p |- +no Fn ( On X. On ) $= - ( vx vy vz va vw cnadd cvv cv c1st cfv csn c2nd cxp cima wss wa con0 crab - cint cmpo df-nadd on2recsfn ) ABFCDGGDHZCHZIJZKUDLJZMNEHZOUCUEUFKMNUGOPEQ - RSTABCEDUAUB $. + $( A version of ~ ralxp with explicit substitution. (Contributed by Scott + Fenton, 21-Aug-2024.) $) + ralxpes $p |- ( A. x e. ( A X. B ) [. ( 1st ` x ) / y ]. + [. ( 2nd ` x ) / z ]. ph <-> A. y e. A A. z e. B ph ) $= + ( cv c2nd cfv wsbc c1st nfsbc1v nfcv nfsbcw nfv sbcopeq1a ralxpf ) ADBGZH + IZJZCRKIZJABCDEFTCUALTDCUADUAMADSLNABOACDRPQ $. $} + $( Ordered triple membership in a triple cross product. (Contributed by + Scott Fenton, 21-Aug-2024.) $) + ot2elxp $p |- ( <. <. A , B >. , C >. e. ( ( D X. E ) X. F ) <-> + ( A e. D /\ B e. E /\ C e. F ) ) $= + ( cop cxp wcel wa w3a opelxp anbi1i df-3an 3bitr4i ) ABGZDEHZIZCFIZJADIZBEI + ZJZSJPCGQFHITUASKRUBSABDELMPCQFLTUASNO $. + ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d a d $. $d a f $. $d a t $. - $d a x $. $d A x $. $d B b $. $d b c $. $d b d $. $d b f $. $d b t $. - $d b x $. $d B x $. $d c d $. $d c t $. $d c x $. $d d t $. $d d x $. - $d f p $. $d f q $. $d f t $. $d f x $. $d p q $. $d p t $. $d p x $. - $d q t $. $d q x $. $d t x $. - $( Lemma for ordinal addition closure. (Contributed by Scott Fenton, - 26-Aug-2024.) $) - naddcllem $p |- ( ( A e. On /\ B e. On ) -> - ( ( A +no B ) e. On /\ - ( A +no B ) = |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\ - ( +no " ( A X. { B } ) ) C_ x ) } ) ) $= - ( vc vd vt cnadd co con0 wcel cxp cima wss wa wceq imaeq2d sseq1d anbi12d - wral cvv va vb vf vp vq cv csn crab cint oveq1 eleq1d sneq xpeq1d rabbidv - weq xpeq1 inteqd eqeq12d oveq2 xpeq2 xpeq2d w3a simpl ralimi 3ad2ant2 jca - 3ad2ant3 cop csuc cdif cres c1st cfv c2nd cmpo on2recsov adantr opex wfun - df-nadd wfn naddfn fnfun ax-mp vex sucex difexi resfunexg mp2an wn wel wo - xpex word eloni ad2antlr ordirr olcd ianor opelxp xchnxbir sylibr sssucid - sucid snssi xpss12 ssdifsn mpbiran resima2 ad2antrr orcd wrex cuni simprr - syl cun ralbidv ralsn cdm onss syl2an sseqtrrdi funimassov sylancr mpbird - wb fndmi simprl ralbii unssd ssorduni funimaexg sylib sseq2 eqeltrd sneqd - snex unex xpeq12d imaeq1 uniex elon sucelon onsucuni unssad unssbd rspcev - syl12anc onintrab2 op1std op2ndd eqid ovmpog mp3an12i 3eqtrd syl5 on2ind - ex ) UAUFZUBUFZGHZIJZUVAGUUSUGZUUTKZLZAUFZMZGUUSUUTUGZKZLZUVFMZNZAIUHZUIZ - OZNZDUFZUUTGHZIJZUVRGUVQUGZUUTKZLZUVFMZGUVQUVHKZLZUVFMZNZAIUHZUIZOZNZUVQE - UFZGHZIJZUWMGUVTUWLKZLZUVFMZGUVQUWLUGZKZLZUVFMZNZAIUHZUIZOZNZUUSUWLGHZIJZ - UXGGUVCUWLKZLZUVFMZGUUSUWRKZLZUVFMZNZAIUHZUIZOZNZBUUTGHZIJZUXTGBUGZUUTKZL - ZUVFMZGBUVHKZLZUVFMZNZAIUHZUIZOZNBCGHZIJZUYMGUYBCKZLZUVFMZGBCUGZKZLZUVFMZ - NZAIUHZUIZOZNBCUAUBDEUADUOZUVBUVSUVOUWJVUFUVAUVRIUUSUVQUUTGUJZUKVUFUVAUVR - UVNUWIVUGVUFUVMUWHVUFUVLUWGAIVUFUVGUWCUVKUWFVUFUVEUWBUVFVUFUVDUWAGVUFUVCU - VTUUTUUSUVQULZUMPQVUFUVJUWEUVFVUFUVIUWDGUUSUVQUVHUPPQRUNUQURRUBEUOZUVSUWN - UWJUXEVUIUVRUWMIUUTUWLUVQGUSZUKVUIUVRUWMUWIUXDVUJVUIUWHUXCVUIUWGUXBAIVUIU - WCUWQUWFUXAVUIUWBUWPUVFVUIUWAUWOGUUTUWLUVTUTPQVUIUWEUWTUVFVUIUWDUWSGVUIUV - HUWRUVQUUTUWLULVAPQRUNUQURRVUFUXHUWNUXRUXEVUFUXGUWMIUUSUVQUWLGUJZUKVUFUXG - UWMUXQUXDVUKVUFUXPUXCVUFUXOUXBAIVUFUXKUWQUXNUXAVUFUXJUWPUVFVUFUXIUWOGVUFU - VCUVTUWLVUHUMPQVUFUXMUWTUVFVUFUXLUWSGUUSUVQUWRUPPQRUNUQURRUUSBOZUVBUYAUVO - UYLVULUVAUXTIUUSBUUTGUJZUKVULUVAUXTUVNUYKVUMVULUVMUYJVULUVLUYIAIVULUVGUYE - UVKUYHVULUVEUYDUVFVULUVDUYCGVULUVCUYBUUTUUSBULUMPQVULUVJUYGUVFVULUVIUYFGU - USBUVHUPPQRUNUQURRUUTCOZUYAUYNUYLVUEVUNUXTUYMIUUTCBGUSZUKVUNUXTUYMUYKVUDV - UOVUNUYJVUCVUNUYIVUBAIVUNUYEUYQUYHVUAVUNUYDUYPUVFVUNUYCUYOGUUTCUYBUTPQVUN - UYGUYTUVFVUNUYFUYSGVUNUVHUYRBUUTCULVAPQRUNUQURRUXFEUUTSDUUSSZUWKDUUSSZUXS - EUUTSZVBZUVSDUUSSZUXHEUUTSZNZUUSIJZUUTIJZNZUVPVUSVUTVVAVUQVUPVUTVURUWKUVS - DUUSUVSUWJVCVDVEVURVUPVVAVUQUXSUXHEUUTUXHUXRVCVDVGVFVVEVVBUVPVVEVVBNZUVBU - VOVVFUVAUVNIVVFUVAUUSUUTVHZGUUSVIZUUTVIZKZVVGUGZVJZVKZFUCTTUCUFZFUFZVLVMZ - UGZVVOVNVMZKZLZUVFMZVVNVVPVVRUGZKZLZUVFMZNZAIUHZUIZVOZHZVVMUVDLZUVFMZVVMU - VILZUVFMZNZAIUHZUIZUVNVVEUVAVWJOVVBUDUEUUSUUTGVWIUDUEFAUCVTVPVQVVGTJVVMTJ - ZVVFVWQIJVWJVWQOUUSUUTVRGVSZVVLTJVWRGIIKZWAVWSWBVWTGWCWDZVVJVVKVVHVVIUUSU - AWEZWFUUTUBWEZWFWMWGGVVLTWHWIVVFVWQUVNIVVFVWPUVMVVFVWOUVLAIVVFVWLUVGVWNUV - KVVFVWKUVEUVFVVFUVDVVLMZVWKUVEOVVFVVGUVDJZWJZVXDVVFUUSUVCJZWJZUBUBWKZWJZW - LZVXFVVFVXJVXHVVFUUTWNZVXJVVDVXLVVCVVBUUTWOWPUUTWQXOWRVXGVXINVXKVXEVXGVXI - WSUUSUUTUVCUUTWTXAXBVXDUVDVVJMZVXFUVCVVHMZUUTVVIMVXMUUSVVHJVXNUUSVXBXDUUS - VVHXEWDUUTXCUVCVVHUUTVVIXFWIUVDVVJVVGXGXHXBGUVDVVLXIXOQVVFVWMUVJUVFVVFUVI - VVLMZVWMUVJOVVFVVGUVIJZWJZVXOVVFUAUAWKZWJZUUTUVHJZWJZWLZVXQVVFVXSVYAVVFUU - SWNZVXSVVCVYCVVDVVBUUSWOXJUUSWQXOXKVXRVXTNVYBVXPVXRVXTWSUUSUUTUUSUVHWTXAX - BVXOUVIVVJMZVXQUUSVVHMUVHVVIMZVYDUUSXCUUTVVIJVYEUUTVXCXDUUTVVIXEWDUUSVVHU - VHVVIXFWIUVIVVJVVGXGXHXBGUVIVVLXIXOQRUNUQZVVFUVLAIXLZUVNIJVVFUVEUVJXPZXMZ - VIZIJZUVEVYJMZUVJVYJMZVYGVVFVYIIJZVYKVVFVYIWNZVYNVVFVYHIMZVYOVVFUVEUVJIVV - FUVEIMZVVOUWLGHZIJZEUUTSZFUVCSZVVFVVAWUAVVEVUTVVAXNVYTVVAFUUSVXBFUAUOZVYS - UXHEUUTWUBVYRUXGIVVOUUSUWLGUJUKXQXRXBVVFVWSUVDGXSZMVYQWUAYFVXAVVFUVDVWTWU - CVVEUVDVWTMZVVBVVCUVCIMUUTIMWUDVVDUUSIXEUUTXTUVCIUUTIXFYAVQVWTGWBYGZYBFEU - VCUUTIGYCYDYEVVFUVJIMZUVQVVOGHZIJZFUVHSZDUUSSZVVFVUTWUJVVEVUTVVAYHWUIUVSD - UUSWUHUVSFUUTVXCFUBUOWUGUVRIVVOUUTUVQGUSUKXRYIXBVVFVWSUVIWUCMWUFWUJYFVXAV - VFUVIVWTWUCVVEUVIVWTMZVVBVVCUUSIMUVHIMWUKVVDUUSXTUUTIXEUUSIUVHIXFYAVQWUEY - BDFUUSUVHIGYCYDYEYJZVYHYKXOVYIVYHUVEUVJVWSUVDTJUVETJVXAUVCUUTUUSYQVXCWMGU - VDTYLWIVWSUVITJUVJTJVXAUUSUVHVXBUUTYQWMGUVITYLWIYRUUAUUBXBVYIUUCYMVVFUVEU - VJVYJVVFVYPVYHVYJMWULVYHUUDXOZUUEVVFUVEUVJVYJWUMUUFUVLVYLVYMNAVYJIUVFVYJO - UVGVYLUVKVYMUVFVYJUVEYNUVFVYJUVJYNRUUGUUHUVLAUUIYMZYOFUCVVGVVMTTVWHVWQVWI - VVNUVDLZUVFMZVVNUVILZUVFMZNZAIUHZUIIVVOVVGOZVWGWUTWVAVWFWUSAIWVAVWAWUPVWE - WURWVAVVTWUOUVFWVAVVSUVDVVNWVAVVQUVCVVRUUTWVAVVPUUSUUSUUTVVOVXBVXCUUJZYPU - USUUTVVOVXBVXCUUKZYSPQWVAVWDWUQUVFWVAVWCUVIVVNWVAVVPUUSVWBUVHWVBWVAVVRUUT - WVCYPYSPQRUNUQVVNVVMOZWUTVWPWVDWUSVWOAIWVDWUPVWLWURVWNWVDWUOVWKUVFVVNVVMU - VDYTQWVDWUQVWMUVFVVNVVMUVIYTQRUNUQVWIUULUUMUUNVYFUUOZWUNYOWVEVFUURUUPUUQ - $. + ot21st.1 $e |- A e. _V $. + ot21st.2 $e |- B e. _V $. + ot21st.3 $e |- C e. _V $. + $( Extract the first member of an ordered triple. Deduction version. + (Contributed by Scott Fenton, 21-Aug-2024.) $) + ot21std $p |- ( X = <. <. A , B >. , C >. -> ( 1st ` ( 1st ` X ) ) = A ) $= + ( cop wceq c1st cfv opex op1std fveq2d op1st eqtrdi ) DABHZCHIZDJKZJKQJKA + RSQJQCDABLGMNABEFOP $. + + $( Extract the second member of an ordered triple. Deduction version. + (Contributed by Scott Fenton, 21-Aug-2024.) $) + ot22ndd $p |- ( X = <. <. A , B >. , C >. -> ( 2nd ` ( 1st ` X ) ) = B ) $= + ( cop wceq c1st cfv c2nd opex op1std fveq2d op2nd eqtrdi ) DABHZCHIZDJKZL + KRLKBSTRLRCDABMGNOABEFPQ $. $} ${ - $d A x $. $d B x $. - $( Closure law for natural addition. (Contributed by Scott Fenton, - 26-Aug-2024.) $) - naddcl $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) e. On ) $= - ( vx con0 wcel wa cnadd co csn cxp cima cv wss crab cint naddcllem simpld - wceq ) ADEBDEFABGHZDESGAIBJKCLZMGABIJKTMFCDNORCABPQ $. - - $( The value of natural addition. (Contributed by Scott Fenton, - 26-Aug-2024.) $) - naddov $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) = - |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\ - ( +no " ( A X. { B } ) ) C_ x ) } ) $= - ( con0 wcel wa cnadd co csn cxp cima wss crab cint wceq naddcllem simprd - cv ) BDECDEFBCGHZDESGBICJKARZLGBCIJKTLFADMNOABCPQ $. + otthne.1 $e |- A e. _V $. + otthne.2 $e |- B e. _V $. + otthne.3 $e |- C e. _V $. + $( Contrapositive of the ordered triple theorem. (Contributed by Scott + Fenton, 21-Aug-2024.) $) + otthne $p |- ( <. <. A , B >. , C >. =/= <. <. D , E >. , F >. <-> + ( A =/= D \/ B =/= E \/ C =/= F ) ) $= + ( cop wne wo w3o opthne orbi1i opex df-3or 3bitr4i ) ABJZDEJZKZCFKZLADKZB + EKZLZUBLSCJTFJKUCUDUBMUAUEUBABDEGHNOSCTFABPINUCUDUBQR $. + $} - $d A t $. $d A y $. $d A z $. $d B t $. $d B y $. $d B z $. $d t x $. - $d t y $. $d t z $. $d x y $. $d x z $. - $( Alternate expression for natural addition. (Contributed by Scott - Fenton, 26-Aug-2024.) $) - naddov2 $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) = - |^| { x e. On | ( A. y e. B ( A +no y ) e. x /\ - A. z e. A ( z +no B ) e. x ) } ) $= - ( vt con0 wcel wa cnadd co csn cxp cima cv wss crab cint wral wb cdm onss - naddov snssi xpss12 syl2an fndmi sseqtrrdi wfun wfn fnfun funimassov mpan - naddfn ax-mp wceq oveq1 eleq1d ralbidv ralsng adantr bitrd adantl anbi12d - syl oveq2 rabbidv inteqd eqtrd ) DGHZEGHZIZDEJKJDLZEMZNAOZPZJDELZMZNVOPZI - ZAGQZRDBOZJKZVOHZBESZCOZEJKZVOHZCDSZIZAGQZRADEUCVLWAWKVLVTWJAGVLVPWEVSWIV - LVPFOZWBJKZVOHZBESZFVMSZWEVLVNJUAZPZVPWPTZVLVNGGMZWQVJVMGPEGPVNWTPVKDGUDE - UBVMGEGUEUFWTJUNUGZUHJUIZWRWSJWTUJXBUNWTJUKUOZFBVMEVOJULUMVEVJWPWETVKWOWE - FDGWLDUPZWNWDBEXDWMWCVOWLDWBJUQURUSUTVAVBVLVSWFWLJKZVOHZFVQSZCDSZWIVLVRWQ - PZVSXHTZVLVRWTWQVJDGPVQGPVRWTPVKDUBEGUDDGVQGUEUFXAUHXBXIXJXCCFDVQVOJULUMV - EVKXHWITVJVKXGWHCDXFWHFEGWLEUPXEWGVOWLEWFJVFURUTUSVCVBVDVGVHVI $. + ${ + $d A x y z p $. $d B x y z p $. $d C x y z p $. $d D x y z p $. + $( Membership in a triple cross product. (Contributed by Scott Fenton, + 21-Aug-2024.) $) + elxpxp $p |- ( A e. ( ( B X. C ) X. D ) <-> + E. x e. B E. y e. C E. z e. D A = <. <. x , y >. , z >. ) $= + ( vp cxp wcel cv cop wceq wrex elxp2 opeq1 eqeq2d rexbidv rexxp bitri ) D + EFIZGIJDHKZCKZLZMZCGNZHUANDAKBKLZUCLZMZCGNZBFNAENHCDUAGOUFUJHABEFUBUGMZUE + UICGUKUDUHDUBUGUCPQRST $. $} ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d a d $. $d a x $. $d B b $. - $d b c $. $d b d $. $d b x $. $d c d $. $d c x $. $d d x $. - $( Natural addition commutes. (Contributed by Scott Fenton, - 26-Aug-2024.) $) - naddcom $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) = ( B +no A ) ) $= - ( va vb vc vd vx cv cnadd co wceq weq oveq1 oveq2 eqeq12d con0 wcel wa wb - wral w3a crab cint eleq1 ralimi 3ad2ant3 adantl 3ad2ant2 anbi12d biancomd - ralbi syl rabbidv inteqd naddov2 adantr ancoms 3eqtr4d ex on2ind ) CHZDHZ - IJZVBVAIJZKZEHZVBIJZVBVFIJZKZVFFHZIJZVJVFIJZKZVAVJIJZVJVAIJZKZAVBIJZVBAIJ - ZKABIJZBAIJZKABCDEFCELZVCVGVDVHVAVFVBIMVAVFVBINODFLVGVKVHVLVBVJVFINVBVJVF - IMOWAVNVKVOVLVAVFVJIMVAVFVJINOVAAKVCVQVDVRVAAVBIMVAAVBINOVBBKVQVSVRVTVBBA - INVBBAIMOVAPQZVBPQZRZVMFVBTEVATZVIEVATZVPFVBTZUAZVEWDWHRZVNGHZQZFVBTZVGWJ - QZEVATZRZGPUBZUCZVHWJQZEVATZVOWJQZFVBTZRZGPUBZUCZVCVDWIWPXCWIWOXBGPWIWOWS - XAWIWLXAWNWSWHWLXASZWDWGWEXEWFWGWKWTSZFVBTXEVPXFFVBVNVOWJUDUEWKWTFVBUKULU - FUGWHWNWSSZWDWFWEXGWGWFWMWRSZEVATXGVIXHEVAVGVHWJUDUEWMWREVAUKULUHUGUIUJUM - UNWDVCWQKWHGFEVAVBUOUPWDVDXDKZWHWCWBXIGEFVBVAUOUQUPURUSUT $. + $d A x y z $. $d B x y z $. $d C x y z $. $d D x y z $. + $( Version of ~ elrel for triple cross products. (Contributed by Scott + Fenton, 21-Aug-2024.) $) + elxpxpss $p |- ( ( R C_ ( ( B X. C ) X. D ) /\ A e. R ) -> + E. x E. y E. z A = <. <. x , y >. , z >. ) $= + ( cxp wss wcel wa cv cop wceq wex wrex rexex reximi syl ssel2 elxpxp + sylbi ) HEFIGIZJDHKLDUDKZDAMBMNCMNOZCPZBPZAPZHUDDUAUEUFCGQZBFQZAEQZUIABCD + EFGUBULUHAEQUIUKUHAEUKUGBFQUHUJUGBFUFCGRSUGBFRTSUHAERTUCT $. $} ${ - $d A a $. $d a b $. $d a c $. $d a x $. $d b x $. $d c x $. - $( Ordinal zero is the additive identity for natural addition. - (Contributed by Scott Fenton, 26-Aug-2024.) $) - naddid1 $p |- ( A e. On -> ( A +no (/) ) = A ) $= - ( va vb vc vx cv c0 cnadd co wceq oveq1 id eqeq12d con0 wcel wral wa crab - cint adantr weq wss 0elon naddov2 mpan2 ral0 biantrur wel wb eleq1 ralimi - ralbi syl adantl dfss3 bitr4di bitr3id rabbidv inteqd intmin 3eqtrd tfis3 - ex ) BFZGHIZVDJZCFZGHIZVGJZAGHIZAJBCABCUAZVEVHVDVGVDVGGHKVKLMVDAJZVEVJVDA - VDAGHKVLLMVDNOZVICVDPZVFVMVNQZVEVDDFHIEFZOZDGPZVHVPOZCVDPZQZENRZSZVDVPUBZ - ENRZSZVDVMVEWCJZVNVMGNOWGUCEDCVDGUDUETVOWBWEVOWAWDENWAVTVOWDVRVTVQDUFUGVO - VTCEUHZCVDPZWDVNVTWIUIZVMVNVSWHUIZCVDPWJVIWKCVDVHVGVPUJUKVSWHCVDULUMUNCVD - VPUOUPUQURUSVMWFVDJVNEVDNUTTVAVCVB $. + $d A w $. $d A x $. $d A y $. $d A z $. $d B w $. $d B x $. $d B y $. + $d B z $. $d C w $. $d C x $. $d C y $. $d C z $. $d w x $. $d w y $. + $d w z $. $d x y $. $d x z $. $d y z $. + ral3xpf.1 $e |- F/ y ph $. + ral3xpf.2 $e |- F/ z ph $. + ral3xpf.3 $e |- F/ w ph $. + ral3xpf.4 $e |- F/ x ps $. + ral3xpf.5 $e |- ( x = <. <. y , z >. , w >. -> ( ph <-> ps ) ) $. + $( Restricted for all over a triple cross product. (Contributed by Scott + Fenton, 22-Aug-2024.) $) + ralxp3f $p |- ( A. x e. ( ( A X. B ) X. C ) ph <-> + A. y e. A A. z e. B A. w e. C ps ) $= + ( wral cv wi wal wrex ralbii cxp wcel cop wceq df-ral r19.23 bitri elxpxp + imbi1i 3bitr4ri albii ralcom4 opex ceqsal bitr3i 3bitri ) ACGHUAIUAZOCPZU + QUBZAQZCRURDPEPUCZFPZUCZUDZAQZFIOZEHOZDGOZCRZBFIOZEHOZDGOZACUQUEUTVHCVDFI + SZEHSZAQZDGOVNDGSZAQVHUTVNADGJUFVGVODGVGVMAQZEHOVOVFVQEHVDAFILUFTVMAEHKUF + UGTUSVPADEFURGHIUHUIUJUKVIVGCRZDGOVLVGDCGULVRVKDGVRVFCRZEHOVKVFECHULVSVJE + HVSVECRZFIOVJVEFCIULVTBFIABCVCMVAVBUMNUNTUOTUOTUOUP $. $} ${ - $d A c $. $d A d $. $d A w $. $d A x $. $d A z $. $d B c $. $d B d $. - $d B w $. $d B x $. $d B y $. $d B z $. $d C c $. $d c d $. $d c w $. - $d c x $. $d c y $. $d c z $. $d d w $. $d d x $. $d w x $. $d w y $. - $d w z $. $d x y $. $d x z $. - $( Ordinal less-than-or-equal is preserved by natural addition. - (Contributed by Scott Fenton, 7-Sep-2024.) $) - naddssim $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> - ( A C_ B -> ( A +no C ) C_ ( B +no C ) ) ) $= - ( vc vd vw vx vz vy con0 wcel wss cnadd co wi wa cv oveq2 imbi2d wral weq - sseq12d wceq r19.21v imbi2i crab cint wel rspccva ad4ant24 simprrl eleq1d - bitri sylan simplrl adantr simp-4l onelon naddcl syl2anc mp2and ralrimiva - ontr2 simpllr simprrr ssralv sylc jca expr ss2rabdv intss simplll naddov2 - syl simplrr 3sstr4d exp31 a2d ex syl5bi tfis3 com12 3impia ) AJKZBJKZCJKZ - ABLZACMNZBCMNZLZOZWFWDWEPZWKWLWGADQZMNZBWMMNZLZOZOZWLWGAEQZMNZBWSMNZLZOZO - ZWLWKODECDEUAZWQXCWLXEWPXBWGXEWNWTWOXAWMWSAMRWMWSBMRUBSSWMCUCZWQWKWLXFWPW - JWGXFWNWHWOWIWMCAMRWMCBMRUBSSXDEWMTZWLWGXBEWMTZOZOZWMJKZWRXGWLXCEWMTZOXJW - LXCEWMUDXLXIWLWGXBEWMUDUEUMXKWLXIWQXKWLXIWQOXKWLPZWGXHWPXMWGXHWPXMWGPZXHP - ZAFQZMNZGQZKZFWMTZHQWMMNXRKZHATZPZGJUFZUGZBIQZMNZXRKZIWMTZYAHBTZPZGJUFZUG - ZWNWOXOYLYDLYEYMLXOYKYCGJXOXRJKZYKYCXOYNYKPZPZXTYBYPXSFWMYPFDUHZPZXQBXPMN - ZLZYSXRKZXSXHYQYTXNYOXBYTEXPWMEFUAWTXQXAYSWSXPAMRWSXPBMRUBUIUJYPYIYQUUAXO - YNYIYJUKYHUUAIXPWMIFUAYGYSXRYFXPBMRULUIUNYRXQJKZYNYTUUAPXSOYRWDXPJKZUUBYP - WDYQXOWDYOXNWDXHXKWDWEWGUOUPZUPUPYPXKYQUUCXKWLWGXHYOUQWMXPURUNAXPUSUTXOYN - YKYQUOXQYSXRVCUTVAVBYPWGYJYBXMWGXHYOVDXOYNYIYJVEYAHABVFVGVHVIVJYLYDVKVNXO - WDXKWNYEUCUUDXKWLWGXHVLZGFHAWMVMUTXOWEXKWOYMUCXNWEXHXKWDWEWGVOUPUUEGIHBWM - VMUTVPVQVRVSVRVTWAWBWC $. + $d A p $. $d A x $. $d A y $. $d A z $. $d B p $. $d B x $. $d B y $. + $d B z $. $d C p $. $d C x $. $d C y $. $d C z $. $d p ps $. + $d p x $. $d p y $. $d p z $. $d ph x $. $d ph y $. $d ph z $. + $d x y $. $d x z $. $d y z $. + ralxp3.1 $e |- ( p = <. <. x , y >. , z >. -> ( ph <-> ps ) ) $. + $( Restricted for-all over a triple cross product. (Contributed by Scott + Fenton, 21-Aug-2024.) $) + ralxp3 $p |- ( A. p e. ( ( A X. B ) X. C ) ph <-> + A. x e. A A. y e. B A. z e. C ps ) $= + ( nfv ralxp3f ) ABICDEFGHACKADKAEKBIKJL $. $} + $( Equality theorem for substitution of a class for an ordered triple. + (Contributed by Scott Fenton, 22-Aug-2024.) $) + sbcoteq1a $p |- ( A = <. <. x , y >. , z >. -> + ( [. ( 1st ` ( 1st ` A ) ) / x ]. [. ( 2nd ` ( 1st ` A ) ) / y ]. + [. ( 2nd ` A ) / z ]. ph <-> ph ) ) $= + ( cv cop wceq c2nd cfv wsbc c1st opex vex op2ndd eqcomd sbceq1a syl ot22ndd + wb ot21std 3bitrrd ) EBFZCFZGZDFZGHZAADEIJZKZUICELJZIJZKZULBUJLJZKZUGUFUHHA + UITUGUHUFUEUFEUCUDMDNZOPADUHQRUGUDUKHUIULTUGUKUDUCUDUFEBNZCNZUOSPUICUKQRUGU + CUMHULUNTUGUMUCUCUDUFEUPUQUOUAPULBUMQRUB $. + ${ - $d A b $. $d A x $. $d B b $. $d B c $. $d b x $. $d B x $. $d C b $. - $d C c $. $d c x $. $d C x $. - $( Ordinal less-than is preserved by natural addition. (Contributed by - Scott Fenton, 9-Sep-2024.) $) - naddelim $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A e. B -> - ( A +no C ) e. ( B +no C ) ) ) $= - ( vc vx vb con0 wcel w3a cnadd co wa cv wral crab cint wceq oveq1 eleq1d - wi ad2antlr adantld ralrimiva ovex elintrab sylibr naddov2 3adant1 adantr - rspcv eleqtrrd ex ) AGHZBGHZCGHZIZABHZACJKZBCJKZHUPUQLZURBDMJKEMZHDCNZFMZ - CJKZVAHZFBNZLZEGOPZUSUTVGURVAHZTZEGNURVHHUTVJEGUTVAGHZLVFVIVBUQVFVITUPVKV - EVIFABVCAQVDURVAVCACJRSUJUAUBUCVGEURGACJUDUEUFUPUSVHQZUQUNUOVLUMEDFBCUGUH - UIUKUL $. + $d A w $. $d A x $. $d A y $. $d A z $. $d B w $. $d B x $. $d B y $. + $d B z $. $d C w $. $d C x $. $d C y $. $d C z $. $d ph x $. + $d w x $. $d w y $. $d w z $. $d x y $. $d x z $. $d y z $. + $( Restricted for-all over a triple cross product with explicit + substitution. (Contributed by Scott Fenton, 22-Aug-2024.) $) + ralxp3es $p |- ( A. x e. ( ( A X. B ) X. C ) + [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ]. + [. ( 2nd ` x ) / w ]. ph <-> + A. y e. A A. z e. B A. w e. C ph ) $= + ( cv c2nd cfv wsbc c1st nfsbc1v nfcv nfsbcw nfv sbcoteq1a ralxp3f ) AEBIZ + JKZLZDTMKZJKZLZCUCMKZLABCDEFGHUECUFNUEDCUFDUFOUBDUDNPUEECUFEUFOUBEDUDEUDO + AEUANPPABQACDETRS $. $} - $( Ordinal less-than is not affected by natural addition. (Contributed by - Scott Fenton, 9-Sep-2024.) $) - naddel1 $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> - ( A e. B <-> ( A +no C ) e. ( B +no C ) ) ) $= - ( con0 wcel w3a cnadd co naddelim wss naddssim 3com12 ontri1 ancoms 3adant3 - wn wi wb naddcl 3adant1 3imp3i2an 3imtr3d impcon4bid ) ADEZBDEZCDEZFZABEZAC - GHZBCGHZEZABCIUGBAJZUJUIJZUHPZUKPZUEUDUFULUMQBACKLUDUEULUNRZUFUEUDUPBAMNOUD - UEUFUJDEZUIDEUMUORUEUFUQUDBCSTACSUJUIMUAUBUC $. + $( The union of two ordinals is in a third iff both of the first two are. + (Contributed by Scott Fenton, 10-Sep-2024.) $) + onunel $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A u. B ) e. C <-> + ( A e. C /\ B e. C ) ) ) $= + ( con0 wcel w3a wss cun wa wb wceq biimpi eleq1d adantl ontr2 expdimp bitrd + wi word eloni ssequn1 3adant2 pm4.71rd ssequn2 3adant1 wo ordtri2or2 syl2an + pm4.71d 3adant3 mpjaodan ) ADEZBDEZCDEZFZABGZABHZCEZACEZBCEZIZJBAGZUOUPIZUR + UTVAUPURUTJUOUPUQBCUPUQBKABUALMNVCUTUSUOUPUTUSULUNUPUTIUSRUMABCOUBPUCQUOVBI + ZURUSVAVBURUSJUOVBUQACVBUQAKBAUDLMNVDUSUTUOVBUSUTUMUNVBUSIUTRULBACOUEPUIQUL + UMUPVBUFZUNULASBSVEUMATBTABUGUHUJUK $. - $( Ordinal less-than is not affected by natural addition. (Contributed by - Scott Fenton, 9-Sep-2024.) $) - naddel2 $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> - ( A e. B <-> ( C +no A ) e. ( C +no B ) ) ) $= - ( con0 wcel w3a cnadd co naddel1 wceq naddcom 3adant2 3adant1 eleq12d bitrd - ) ADEZBDEZCDEZFZABEACGHZBCGHZECAGHZCBGHZEABCISTUBUAUCPRTUBJQACKLQRUAUCJPBCK - MNO $. + ${ + $d A x $. $d B x y $. $d F x y $. $d ph y $. $d ps x $. + imaeqsex.1 $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $. + $( Substitute a function value into an existential quantifier over an + image. (Contributed by Scott Fenton, 27-Sep-2024.) $) + imaeqsexv $p |- ( ( F Fn A /\ B C_ A ) -> + ( E. x e. ( F " B ) ph <-> E. y e. B ps ) ) $= + ( wfn wss wa cima wrex cv cfv wceq wex wcel df-rex exbii fvelimab rexcom4 + anbi1d exbidv bitrid eqcom anbi1i ceqsexv rexbii r19.41v 3bitr3ri bitrdi + fvex bitri ) GEIFEJKZACGFLZMZDNZGOZCNZPZDFMZAKZCQZBDFMZUQUTUPRZAKZCQUOVDA + CUPSUOVGVCCUOVFVBADEFUTGUAUCUDUEVAAKZCQZDFMVHDFMZCQVEVDVHDCFUBVIBDFVIUTUS + PZAKZCQBVHVLCVAVKAUSUTUFUGTABCUSURGUMHUHUNUIVJVCCVAADFUJTUKUL $. - $( Ordinal less-than-or-equal is not affected by natural addition. - (Contributed by Scott Fenton, 9-Sep-2024.) $) - naddss1 $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> - ( A C_ B <-> ( A +no C ) C_ ( B +no C ) ) ) $= - ( con0 wcel w3a wn cnadd co wss naddel1 3com12 notbid ontri1 3adant3 naddcl - wb 3adant2 3adant1 syl2anc 3bitr4d ) ADEZBDEZCDEZFZBAEZGZBCHIZACHIZEZGZABJZ - UIUHJZUEUFUJUCUBUDUFUJQBACKLMUBUCULUGQUDABNOUEUIDEZUHDEZUMUKQUBUDUNUCACPRUC - UDUOUBBCPSUIUHNTUA $. + $( Substitute a function value into a universal quantifier over an image. + (Contributed by Scott Fenton, 27-Sep-2024.) $) + imaeqsalv $p |- ( ( F Fn A /\ B C_ A ) -> + ( A. x e. ( F " B ) ph <-> A. y e. B ps ) ) $= + ( wfn wss wa wn cima wrex wral cv cfv wceq notbid dfral2 imaeqsexv + 3bitr4g ) GEIFEJKZALZCGFMZNZLBLZDFNZLACUEOBDFOUCUFUHUDUGCDEFGCPDPGQRABHSU + ASACUETBDFTUB $. + $} - $( Ordinal less-than-or-equal is not affected by natural addition. - (Contributed by Scott Fenton, 9-Sep-2024.) $) - naddss2 $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> - ( A C_ B <-> ( C +no A ) C_ ( C +no B ) ) ) $= - ( con0 wcel w3a cnadd co naddss1 wceq naddcom 3adant2 3adant1 sseq12d bitrd - wss ) ADEZBDEZCDEZFZABPACGHZBCGHZPCAGHZCBGHZPABCITUAUCUBUDQSUAUCJRACKLRSUBU - DJQBCKMNO $. + ${ + $d A x $. + $( The union of a finite ordinal is a finite ordinal. (Contributed by + Scott Fenton, 17-Oct-2024.) $) + nnuni $p |- ( A e. _om -> U. A e. _om ) $= + ( vx com wcel c0 wceq cv csuc wrex cuni nn0suc unieq uni0 eqtrdi eqeltrdi + wo peano1 word nnord syl ordunisuc id eqeltrd eleq1d syl5ibrcom rexlimiv + jaoi ) ACDAEFZABGZHZFZBCIZPAJZCDZBAKUHUNULUHUMECUHUMEJEAELMNQOUKUNBCUICDZ + UNUKUJJZCDUOUPUICUOUIRUPUIFUISUIUATUOUBUCUKUMUPCAUJLUDUEUFUGT $. + $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal Numbers + Properties of real and complex numbers =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $( Set up the new syntax for surreal numbers $) - $c No + ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) ) $= + ( cc0 wne cdiv co c2 cexp c1 cif oveq2 oveq1d oveq1 oveq2d eqeq12d ax-1cn + wceq cc ifcli elimne0 sqdivi dedth ) BEFZABGHZIJHZAIJHZBIJHZGHZSAUEBKLZGH + ZIJHZUHUKIJHZGHZSBKBUKSZUGUMUJUOUPUFULIJBUKAGMNUPUIUNUHGBUKIJOPQAUKCUEBKT + DRUABUBUCUD $. + $} - $( Declare the class of all surreal numbers (see ~ df-no ). $) - csur $a class No $. + ${ + $d N x $. $d M x $. + $( The supremum of a finite sequence of integers. (Contributed by Scott + Fenton, 8-Aug-2013.) $) + supfz $p |- ( N e. ( ZZ>= ` M ) -> sup ( ( M ... N ) , ZZ , < ) = N ) $= + ( vx cuz cfv wcel cz cfz co clt wor cr wss zssre ltso mp2 a1i eluzelz wbr + soss eluzfz2 cv wa cle wn elfzle2 adantl wb elfzelz zred eluzelre syl2anr + lenlt mpbid supmax ) BADEFZCGABHIZBJGJKZUPGLMLJKURNOGLJTPQABRABUAUPCUBZUQ + FZUCUSBUDSZBUSJSUEZUTVAUPUSABUFUGUTUSLFBLFVAVBUHUPUTUSUSABUIUJABUKUSBUMUL + UNUO $. - $( Declare the less than relationship over surreal numbers (see - ~ df-slt ). $) - cslt $a class = ` M ) -> inf ( ( M ... N ) , ZZ , < ) = M ) $= + ( vx cuz cfv wcel cz cfz co clt wor wss zssre ltso soss mp2 a1i wbr zred + cr eluzel2 eluzfz1 cv wa cle wn elfzle1 adantl elfzelz lenlt syl2an mpbid + wb infmin ) BADEFZCGABHIZAJGJKZUOGTLTJKUQMNGTJOPQABUAZABUBUOCUCZUPFZUDAUS + UERZUSAJRUFZUTVAUOUSABUGUHUOATFUSTFVAVBUMUTUOAURSUTUSUSABUISAUSUJUKULUN + $. + $} - $( Declare the birthday function for surreal numbers (see ~ df-bday ). $) - cbday $a class bday $. + $( The sequence ` ( 0 ... ( N - 1 ) ) ` is empty iff ` N ` is zero. + (Contributed by Scott Fenton, 16-May-2014.) $) + fz0n $p |- ( N e. NN0 -> ( ( 0 ... ( N - 1 ) ) = (/) <-> N = 0 ) ) $= + ( cn0 wcel c1 cmin co cc0 clt wbr cfz c0 wceq cz wb nn0z sylancr cle bitr3d + 0z cr peano2zm syl fzn cn wo elnn0 wn nnge1 1re wa subge0 0re resubcl lenlt + nnre sylancl mpbid nnne0 neneqd 2falsed cneg oveq1 eqtr4di neg1lt0 eqbrtrdi + df-neg id 2thd jaoi sylbi ) ABCZADEFZGHIZGVLJFKLZAGLZVKGMCVLMCZVMVNNSVKAMCV + PAOAUAUBGVLUCPVKAUDCZVOUEVMVONZAUFVQVRVOVQVMVOVQDAQIZVMUGZAUHVQATCZDTCZVSVT + NAUOUIWAWBUJZGVLQIZVSVTADUKWCGTCVLTCWDVTNULADUMGVLUNPRUPUQVQAGAURUSUTVOVMVO + VOVLDVAZGHVOVLGDEFWEAGDEVBDVFVCVDVEVOVGVHVIVJR $. ${ - $d f a $. - $( Define the class of surreal numbers. The surreal numbers are a proper - class of numbers developed by John H. Conway and introduced by Donald - Knuth in 1975. They form a proper class into which all ordered fields - can be embedded. The approach we take to defining them was first - introduced by Hary Gonshor, and is based on the conception of a "sign - expansion" of a surreal number. We define the surreals as - ordinal-indexed sequences of ` 1o ` and ` 2o ` , analagous to Gonshor's - ` ( - ) ` and ` ( + ) ` . - - After introducing this definition, we will abstract away from it using - axioms that Norman Alling developed in "Foundations of Analysis over - Surreal Number Fields." This is done in an effort to be agnostic - towards the exact implementation of surreals. (Contributed by Scott - Fenton, 9-Jun-2011.) $) - df-no $a |- No = { f | E. a e. On f : a --> { 1o , 2o } } $. + $d F f $. $d A f $. $d B f $. + $( Value of a sequence shifted by ` A ` . (Contributed by Scott Fenton, + 16-Dec-2017.) $) + shftvalg $p |- ( ( F e. V /\ A e. CC /\ B e. CC ) -> + ( ( F shift A ) ` B ) = ( F ` ( B - A ) ) ) $= + ( vf wcel cc cshi co cmin wceq wa cv wi oveq1 fveq1d fveq1 eqeq12d imbi2d + cfv vex shftval vtoclg 3impib ) CDFAGFZBGFZBCAHIZTZBAJIZCTZKZUEUFLZBEMZAH + IZTZUIUMTZKZNULUKNECDUMCKZUQUKULURUOUHUPUJURBUNUGUMCAHOPUIUMCQRSABUMEUAUB + UCUD $. $} ${ - $d f g x y $. - $( Next, we introduce surreal less-than, a comparison relationship over the - surreals by lexicographically ordering them. (Contributed by Scott - Fenton, 9-Jun-2011.) $) - df-slt $a |- . | ( ( f e. No /\ g e. No ) /\ - E. x e. On ( - A. y e. x ( f ` y ) = ( g ` y ) /\ - ( f ` x ) { <. 1o , (/) >. , - <. 1o , 2o >. , - <. (/) , 2o >. } ( g ` x ) ) ) } $. + $d A k $. $d A m $. $d B k $. $d B m $. $d F k $. $d k m $. + $d k ph $. $d M k $. $d m ph $. $d Z k $. + divcnvlin.1 $e |- Z = ( ZZ>= ` M ) $. + divcnvlin.2 $e |- ( ph -> M e. ZZ ) $. + divcnvlin.3 $e |- ( ph -> A e. CC ) $. + divcnvlin.4 $e |- ( ph -> B e. ZZ ) $. + divcnvlin.5 $e |- ( ph -> F e. V ) $. + divcnvlin.6 $e |- ( ( ph /\ k e. Z ) -> + ( F ` k ) = ( ( k + A ) / ( k + B ) ) ) $. + $( Limit of the ratio of two linear functions. (Contributed by Scott + Fenton, 17-Dec-2017.) $) + divcnvlin $p |- ( ph -> F ~~> 1 ) $= + ( vm c1 co caddc cn wcel cli wbr cz cv cmin cdiv cmpt cc0 wa cc nncn zcnd + adantl subcld adantr wne nnne0 divdird dividd oveq1d eqtrd mpteq2dva nnuz + cvv 1zzd divcnv syl 1cnd nnex mptex a1i divcld fmpttd ffvelcdmda cfv wceq + weq oveq2 oveq2d eqid ovex fvmpt eqtr4d climaddc2 eqbrtrd cuz cres resmpt + wss nnssz ax-mp reseq2i eqtr3i 1p0e1 breq12i wb climres mp2an bitri sylib + zex eluzelz eleq2s ppncand zaddcld oveq1 oveq12d 3eqtr4d climshft2 mpbird + 1z id ) AEPUAUBOUCOUDZBCUEQZRQZXMUFQZUGZPUAUBZAOSXPUGZPUHRQZUAUBZXRAXSOSP + XNXMUFQZRQZUGZXTUAAOSXPYCAXMSTZUIZXPXMXMUFQZYBRQYCYFXMXNXMYEXMUJTAXMUKUMZ + AXNUJTZYEABCKACLULUNZUOZYHYEXMUHUPAXMUQUMZURYFYGPYBRYFXMYHYLUSUTVAVBAUHPD + OSYBUGZYDPVDSVCAVEAYIYMUHUAUBYJXNOVFVGAVHYDVDTAOSYCVIVJVKASUJDUDZYMAOSYBU + JYFXNXMYKYHYLVLVMVNYNSTZYNYDVOZPYNYMVOZRQZVPAYOYPPXNYNUFQZRQZYROYNYCYTSYD + ODVQYBYSPRXMYNXNUFVRZVSYDVTPYSRWAWBYOYQYSPROYNYBYSSYMUUAYMVTXNYNUFWAWBVSW + CUMWDWEYAXQPWFVOZWGZPUAUBZXRXSUUCXTPUAXQSWGZXSUUCSUCWIUUEXSVPWJOUCSXPWHWK + SUUBXQVCWLWMWNWOPUCTXQVDTZUUDXRWPXKOUCXPXAVJZPXQPVDWQWRWSWTAPDEXQCFGVDHIJ + LMUUFAUUGVKAYNHTZUIZYNCRQZXNRQZUUJUFQZYNBRQZUUJUFQUUJXQVOZYNEVOUUIUUKUUMU + UJUFUUIYNCBUUHYNUJTAUUHYNYNUCTZYNFWFVOHFYNXBIXCZULUMUUICACUCTUUHLUOZULABU + JTUUHKUOXDUTUUIUUJUCTUUNUULVPUUIYNCUUHUUOAUUPUMUUQXEOUUJXPUULUCXQXMUUJVPZ + XOUUKXMUUJUFXMUUJXNRXFUURXLXGXQVTUUKUUJUFWAWBVGNXHXIXJ $. $} - $( Finally, we introduce the birthday function. This function maps each - surreal to an ordinal. In our implementation, this is the domain of the - sign function. The important properties of this function are established - later. (Contributed by Scott Fenton, 11-Jun-2011.) $) - df-bday $a |- bday = ( x e. No |-> dom x ) $. - ${ - $d A f x $. - $( Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). - (Contributed by Scott Fenton, 11-Jun-2011.) $) - elno $p |- ( A e. No <-> E. x e. On A : x --> { 1o , 2o } ) $= - ( vf csur wcel cvv cv c1o c2o cpr wf con0 wrex elex ancoms rexlimiva wceq - fex feq1 rexbidv df-no elab2g pm5.21nii ) BDEBFEZAGZHIJZBKZALMZBDNUGUDALU - GUELEUDUEUFLBROPUEUFCGZKZALMUHCBDFUIBQUJUGALUEUFUIBSTCAUAUBUC $. + $d A k $. $d B k $. $d F k $. $d F m $. $d k m $. $d k ph $. + $d M k $. $d Z k $. $d Z m $. + climlec3.1 $e |- Z = ( ZZ>= ` M ) $. + climlec3.2 $e |- ( ph -> M e. ZZ ) $. + climlec3.3 $e |- ( ph -> B e. RR ) $. + climlec3.4 $e |- ( ph -> F ~~> A ) $. + climlec3.5 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) $. + climlec3.6 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ B ) $. + $( Comparison of a constant to the limit of a sequence. (Contributed by + Scott Fenton, 5-Jan-2018.) $) + climlec3 $p |- ( ph -> A <_ B ) $= + ( vm cle wbr cneg cfv cc0 wcel cv cmpt renegcld cmin co cli cvv cuz fvexi + 0cnd mptex a1i wa recnd eqid weq fveq2 negeqd simpr fvmptd3 df-neg eqtrdi + climsubc2 breqtrrdi adantr lenegd mpbid breqtrrd climlec2 climrecl mpbird + cr eqeltrd ) ABCOPCQZBQZOPAVNVODNGNUAZERZQZUBZFGHIACJUCAVSSBUDUEVOUFABSDE + VSFUGGHIKAUJVSUGTANGVRGFUHHUIUKULADUAZGTZUMZVTERZLUNWBVTVSRZWCQZSWCUDUEWB + NVTVRWEGVSVLVSUONDUPVQWCVPVTEUQURAWAUSWBWCLUCZUTZWCVAVBVCBVAVDWBWDWEVLWGW + FVMWBVNWEWDOWBWCCOPVNWEOPMWBWCCLACVLTWAJVEVFVGWGVHVIABCABDEFGHIKLVJJVFVK + $. $} + $( Calculate the logarithm of ` _i ` . (Contributed by Scott Fenton, + 13-Apr-2020.) $) + logi $p |- ( log ` _i ) = ( _i x. ( _pi / 2 ) ) $= + ( ci clog cfv cpi c2 co cmul wceq cc wcel cc0 ax-icn clt wbr halfpire pipos + wb pire ax-mp wtru cdiv ce efhalfpi wne crn ine0 cneg cim cle recni lt0neg2 + mulcli mpbi halfpos2 renegcli 0re lttri mp2an cre reim rere eqtr3i breqtrri + cr caddc a1i ltaddposd mpbii picn times2i breqtrrdi ltdivmul2d mpbird mptru + crp 2rp ltleii eqbrtri ellogrn mpbir3an logeftb mp3an mpbir ) ABCADEUAFZGFZ + HZWEUBCAHZUCAIJAKUDWEBUEJZWFWGQLUFWHWEIJDUGZWEUHCZMNWJDUINAWDLWDOUJZULWIWDW + JMWIKMNZKWDMNZWIWDMNKDMNZWLPDVDJZWNWLQRDUKSUMWNWMPWOWNWMQRDUNSUMWIKWDDRUOUP + OUQURWDUSCZWJWDWDIJWPWJHWKWDUTSWDVDJWPWDHOWDVASVBZVCWJWDDUIWQWDDORWDDMNZTWR + DDEGFZMNTDDDVEFZWSMTWNDWTMNPTDDWOTRVFZXAVGVHDVIVJVKTDDEXAXAEVOJTVPVFVLVMVNV + QVRWEVSVTAWEWAWBWC $. + + $( ` _i ` raised to itself is real. (Contributed by Scott Fenton, + 13-Apr-2020.) $) + iexpire $p |- ( _i ^c _i ) e. RR $= + ( ci ccxp co c1 cneg cpi c2 cdiv cmul ce cfv cr clog cc wcel cc0 wne ax-icn + wceq halfpire ine0 cxpef mp3an logi oveq2i recni mulassi ixi oveq1i 3eqtr2i + fveq2i eqtri neg1rr remulcli reefcl ax-mp eqeltri ) AABCZDEZFGHCZICZJKZLURA + AMKZICZJKZVBANOZAPQVFURVESRUARAAUBUCVDVAJVDAAUTICZICAAICZUTICVAVCVGAIUDUEAA + UTRRUTTUFUGVHUSUTIUHUIUJUKULVALOVBLOUSUTUMTUNVAUOUPUQ $. + + $( The binomial coefficent over negative one is zero. (Contributed by Scott + Fenton, 29-May-2020.) $) + bcneg1 $p |- ( N e. NN0 -> ( N _C -u 1 ) = 0 ) $= + ( cn0 wcel c1 cneg cbc co cc0 cfz cfa cfv cmin cmul cdiv cif wceq neg1z cle + cz wbr bcval mpan2 wa clt wn neg1lt0 neg1rr 0re ltnlei mpbi intnanr wb nn0z + 0z elfz mp3an12 syl mtbiri iffalsed eqtrd ) ABCZADEZFGZVBHAIGCZAJKAVBLGJKVB + JKMGNGZHOZHVAVBSCZVCVFPQVBAUAUBVAVDVEHVAVDHVBRTZVBARTZUCZVHVIVBHUDTVHUEUFVB + HUGUHUIUJUKVAASCZVDVJULZAUMVGHSCVKVLQUNVBHAUOUPUQURUSUT $. + + $( The proportion of one bionmial coefficient to another with ` N ` decreased + by 1. (Contributed by Scott Fenton, 23-Jun-2020.) $) + bcm1nt $p |- ( ( N e. NN /\ K e. ( 0 ... ( N - 1 ) ) ) -> + ( N _C K ) = ( ( ( N - 1 ) _C K ) x. ( N / ( N - K ) ) ) ) $= + ( cn wcel cc0 c1 cmin co cfz wa caddc cbc cdiv cmul wceq bcp1n adantl simpl + nncnd oveq1d 1cnd npcand oveq12d oveq2d 3eqtr3d ) BCDZAEBFGHZIHDZJZUGFKHZAL + HZUGALHZUJUJAGHZMHZNHZBALHULBBAGHZMHZNHUHUKUOOUFAUGPQUIUJBALUIBFUIBUFUHRSUI + UAUBZTUIUNUQULNUIUJBUMUPMURUIUJBAGURTUCUDUE $. + ${ - $d A f g x y $. $d B f g x y $. - $( The value of the surreal less than relationship. (Contributed by Scott - Fenton, 14-Jun-2011.) $) - sltval $p |- ( ( A e. No /\ B e. No ) -> - ( A E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ - ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , - <. (/) , 2o >. } ( B ` x ) ) ) ) $= - ( vf vg csur wcel wa cslt wbr cfv wceq wral cop con0 wrex fveq1 anbi12d - cv c1o c0 c2o ctp eleq1 anbi1d eqeq1d ralbidv breq1d anbi2d eqeq2d breq2d - rexbidv df-slt brabg bianabs ) CGHZDGHZIZCDJKBTZCLZUTDLZMZBATZNZVDCLZVDDL - ZUAUBOUAUCOUBUCOUDZKZIZAPQZETZGHZFTZGHZIZUTVLLZUTVNLZMZBVDNZVDVLLZVDVNLZV - HKZIZAPQZIUQVOIZVAVRMZBVDNZVFWBVHKZIZAPQZIUSVKIEFCDGGJVLCMZVPWFWEWKWLVMUQ - VOVLCGUEUFWLWDWJAPWLVTWHWCWIWLVSWGBVDWLVQVAVRUTVLCRUGUHWLWAVFWBVHVDVLCRUI - SUMSVNDMZWFUSWKVKWMVOURUQVNDGUEUJWMWJVJAPWMWHVEWIVIWMWGVCBVDWMVRVBVAUTVND - RUKUHWMWBVGVFVHVDVNDRULSUMSABEFUNUOUP $. + $d N n m j k $. + $( A product identity for binomial coefficents. (Contributed by Scott + Fenton, 23-Jun-2020.) $) + bcprod $p |- ( N e. NN -> + prod_ k e. ( 1 ... ( N - 1 ) ) ( ( N - 1 ) _C k ) = + prod_ k e. ( 1 ... ( N - 1 ) ) ( k ^ ( ( 2 x. k ) - N ) ) ) $= + ( c1 cmin co cfz cbc cprod cmul cexp wceq oveq2d oveq1d adantr prodeq12dv + c2 wcel cn cdiv nncnd vm vn vj cv cc0 caddc oveq1 1m1e0 eqtrdi fz10 oveq2 + c0 eqeq12d weq prod0 eqtr4i wa cfa cfv nncn 1cnd pncand cuz elnnuz biimpi + simpr cn0 nnnn0 elfzelz bccl syl2an nn0cnd fprodm1 bcnn syl fzfid fprodcl + cz mulid1d fz1ssfz0 bcm1nt sylan2 prodeq2dv nnm1nn0 clt wbr elfznn adantl + sseli cc nnred nn0red cr cle elfzle2 ltm1d lelttrd wb simpl nnsub syl2anc + nnre mpbid nnne0d divcld fprodmul fproddiv chash cfn fzfi sylancr hashfz1 + fprodconst eqtr2d fprodfac nnz 1zzd nn0zd fprodrev nncand prodeq1d 3eqtrd + id oveq12d eqtr4d 2nn a1i nnmulcld nnzd peano2nn zsubcld expclzd subsub4d + expm1d eqtr3d 3eqtr4d 2timesd mvrladdd faccl expcld div32d eqtrd ex nnind + ) CUAUDZCDEZFEZUUFAUDZGEZAHZUUGUUHPUUHIEZUUEDEZJEZAHZKULUEUUHGEZAHZULUUHU + UKCDEZJEZAHZKCUBUDZCDEZFEZUVAUUHGEZAHZUVBUUHUUKUUTDEZJEZAHZKZCUUTCUFEZCDE + ZFEZUVJUUHGEZAHZUVKUUHUUKUVIDEZJEZAHZKZCBCDEZFEZUVRUUHGEZAHZUVSUUHUUKBDEZ + JEZAHZKUAUBBUUECKZUUJUUPUUNUUSUWEUUGULUUIUUOAUWEUUGCUEFEULUWEUUFUECFUWEUU + FCCDEUEUUECCDUGUHUIZLUJUIZUWEUUIUUOKUUHUUGQZUWEUUFUEUUHGUWFMNOUWEUUGULUUM + UURAUWGUWEUUMUURKUWHUWEUULUUQUUHJUUECUUKDUKLNOUMUAUBUNZUUJUVDUUNUVGUWIUUG + UVBUUIUVCAUWIUUFUVACFUUEUUTCDUGZLZUWIUUIUVCKUWHUWIUUFUVAUUHGUWJMNOUWIUUGU + VBUUMUVFAUWKUWIUUMUVFKUWHUWIUULUVEUUHJUUEUUTUUKDUKLNOUMUUEUVIKZUUJUVMUUNU + VPUWLUUGUVKUUIUVLAUWLUUFUVJCFUUEUVICDUGZLZUWLUUIUVLKUWHUWLUUFUVJUUHGUWMMN + OUWLUUGUVKUUMUVOAUWNUWLUUMUVOKUWHUWLUULUVNUUHJUUEUVIUUKDUKLNOUMUUEBKZUUJU + WAUUNUWDUWOUUGUVSUUIUVTAUWOUUFUVRCFUUEBCDUGZLZUWOUUIUVTKUWHUWOUUFUVRUUHGU + WPMNOUWOUUGUVSUUMUWCAUWQUWOUUMUWCKUWHUWOUULUWBUUHJUUEBUUKDUKLNOUMUUPCUUSU + UOAUOUURAUOUPUUTRQZUVHUVQUWRUVHUQZUVDUUTUVAJEZUVAURUSZSEZIEZUVGUXBIEZUVMU + VPUWSUVDUVGUXBIUWRUVHVFMUWRUVMUXCKUVHUWRUVMCUUTFEZUUTUUHGEZAHUVBUXFAHZUUT + UUTGEZIEZUXCUWRUVKUXEUVLUXFAUWRUVJUUTCFUWRUUTCUUTUTZUWRVAZVBZLZUWRUVLUXFK + UUHUVKQUWRUVJUUTUUHGUXLMNOUWRUXFUXHACUUTUWRUUTCVCUSQUUTVDVEZUWRUUHUXEQZUQ + ZUXFUWRUUTVGQZUUHVRQZUXFVGQZUXOUUTVHZUUHCUUTVIUUHUUTVJZVKVLUUHUUTUUTGUKVM + UWRUXIUXGCIEUXGUXCUWRUXHCUXGIUWRUXQUXHCKUXTUUTVNVOLUWRUXGUWRUVBUXFAUWRCUV + AVPZUWRUUHUVBQZUQZUXFUWRUXQUXRUXSUYCUXTUUHCUVAVIZUYAVKVLVQVSUWRUXGUVBUVCU + UTUUTUUHDEZSEZIEZAHUVDUVBUYGAHZIEUXCUWRUVBUXFUYHAUYCUWRUUHUEUVAFEZQUXFUYH + KUVBUYJUUHUVAVTWIUUHUUTWAWBWCUWRUVBUVCUYGAUYBUYDUVCUWRUVAVGQZUXRUVCVGQUYC + UUTWDZUYEUUHUVAVJVKVLUYDUUTUYFUWRUUTWJQZUYCUXJNZUYDUYFUYDUUHUUTWEWFZUYFRQ + ZUYDUUHUVAUUTUYDUUHUYCUUHRQZUWRUUHUVAWGWHZWKUYDUVAUWRUYKUYCUYLNWLUWRUUTWM + QUYCUUTXBNZUYCUUHUVAWNWFUWRUUHCUVAWOWHUYDUUTUYSWPWQUYDUYQUWRUYOUYPWRUYRUW + RUYCWSUUHUUTWTXAXCZTZUYDUYFUYTXDZXEXFUWRUYIUXBUVDIUWRUYIUVBUUTAHZUVBUYFAH + ZSEUXBUWRUVBUUTUYFAUYBUYNVUAVUBXGUWRUWTVUCUXAVUDSUWRVUCUUTUVBXHUSZJEZUWTU + WRUVBXIQUYMVUCVUFKCUVAXJUXJUVBUUTAXMXKUWRVUEUVAUUTJUWRUYKVUEUVAKUYLUVAXLV + OLXNUWRUXAUVBUCUDZUCHZUUTUVADEZUVAFEZUYFAHVUDUWRUYKUXAVUHKUYLUVAUCXOVOUWR + VUGUYFUCAUUTCUVAUUTXPZUWRXQUWRUVAUYLXRUWRVUGUVBQZUQVUGVULVUGRQUWRVUGUVAWG + WHTVUGUYFKYCXSUWRVUJUVBUYFAUWRVUICUVAFUWRUUTCUXJUXKXTMYAYBYDYELYBYBYBNUWR + UVPUXDKUVHUWRUVPUXEUVOAHUVBUVOAHZUUTPUUTIEZUVIDEZJEZIEZUXDUWRUVKUXEUVOAUX + MYAUWRUVOVUPACUUTUXNUXPUUHUVNUXPUUHUXOUYQUWRUUHUUTWGWHZTUXPUUHVURXDUXPUUK + UVIUXPUUKUXPPUUHPRQZUXPYFYGVURYHYIUXPUVIUWRUVIRQUXOUUTYJNYIYKYLAUBUNZUUHU + UTUVNVUOJVUTYCVUTUUKVUNUVIDUUHUUTPIUKMYDVMUWRVUQUVGUXASEZUWTIEUXDUWRVUMVV + AVUPUWTIUWRUVBUVFUUHSEZAHUVGUVBUUHAHZSEVUMVVAUWRUVBUVFUUHAUYBUYDUUHUVEUYD + UUHUYRTZUYDUUHUYRXDZUYDUUKUUTUYDUUKUYDPUUHVUSUYDYFYGUYRYHZYIUWRUUTVRQUYCV + UKNYKZYLZVVDVVEXGUWRUVBUVOVVBAUYDUUHUVECDEZJEUVOVVBUYDVVIUVNUUHJUYDUUKUUT + CUYDUUKVVFTUYNUYDVAYMLUYDUUHUVEVVDVVEVVGYNYOWCUWRUXAVVCUVGSUWRUYKUXAVVCKU + YLUVAAXOVOLYPUWRVUOUVAUUTJUWRVUNUUTDEZCDEVUOUVAUWRVUNUUTCUWRVUNUWRPUUTVUS + UWRYFYGUWRYCYHTUXJUXKYMUWRVVJUUTCDUWRVUNUUTUUTUXJUXJUWRUUTUXJYQYRMYOLYDUW + RUVGUXAUWTUWRUVBUVFAUYBVVHVQUWRUXAUWRUYKUXARQUYLUVAYSVOZTUWRUUTUVAUXJUYLY + TUWRUXAVVKXDUUAUUBYBNYPUUCUUD $. $} ${ - $d A x $. - $( The value of the birthday function within the surreals. (Contributed by - Scott Fenton, 14-Jun-2011.) $) - bdayval $p |- ( A e. No -> ( bday ` A ) = dom A ) $= - ( vx csur wcel cdm cvv cbday cfv wceq dmexg cv dmeq df-bday fvmptg mpdan - ) ACDAEZFDAGHPIACJBABKZEPCFGQALBMNO $. + $d N n m k $. $d C n m k $. + $( A column-sum rule for binomial coefficents. (Contributed by Scott + Fenton, 24-Jun-2020.) $) + bccolsum $p |- ( ( N e. NN0 /\ C e. NN0 ) -> + sum_ k e. ( 0 ... N ) ( k _C C ) = ( ( N + 1 ) _C ( C + 1 ) ) ) $= + ( cn0 wcel cc0 cfz co cbc csu c1 caddc wceq wi oveq2 sumeq1d oveq1 oveq1d + eqeq12d imbi2d vm vn cv 0p1e1 eqtrdi weq cz 0nn0 nn0z bccl sylancr nn0cnd + cc 0z fsum1 cn wo elnn0 cfa cfv cmin cmul cdiv cif cle wbr 1red ltaddrp2d + wn nnrp peano2nn nnred ltnled mpbid elfzle2 nsyl iffalsed 1nn0 nnzd bcval + clt bc0k 3eqtr4rd bcnn ax-mp eqtr4i oveq2d 3eqtr4a sylbi eqtrd wa elnn0uz + jaoi cuz biimpi adantr elfznn0 adantl simplr nn0zd syl2anc fsump1 id 1cnd + nn0cn pncand eqcomd oveqan12rd peano2nn0 bcpasc syl2an 3eqtrd a2d nn0ind + exp31 imp ) CDEADEZFCGHZBUCZAIHZBJZCKLHZAKLHZIHZMZXQFUAUCZGHZXTBJZYFKLHZY + CIHZMZNXQFFGHZXTBJZKYCIHZMZNXQFUBUCZGHZXTBJZYPKLHZYCIHZMZNXQFYSGHZXTBJZYS + KLHZYCIHZMZNXQYENUAUBCYFFMZYKYOXQUUGYHYMYJYNUUGYGYLXTBYFFFGOPUUGYIKYCIUUG + YIFKLHZKYFFKLQUDUERSTUAUBUFZYKUUAXQUUIYHYRYJYTUUIYGYQXTBYFYPFGOPUUIYIYSYC + IYFYPKLQRSTYFYSMZYKUUFXQUUJYHUUCYJUUEUUJYGUUBXTBYFYSFGOPUUJYIUUDYCIYFYSKL + QRSTYFCMZYKYEXQUUKYHYAYJYDUUKYGXRXTBYFCFGOPUUKYIYBYCIYFCKLQRSTXQYMFAIHZYN + XQFUGEUULUMEYMUULMUNXQUULXQFDEZAUGEZUULDEUHAUIAFUJUKULXTUULBFXSFAIQUOUKXQ + AUPEZAFMZUQUULYNMZAURUUOUUQUUPUUOYCFKGHEZKUSUTKYCVAHUSUTYCUSUTVBHVCHZFVDZ + FYNUULUUOUURUUSFUUOYCKVEVFZUURUUOKYCWAVFUVAVIUUOKAUUOVGZAVJVHUUOKYCUVBUUO + YCAVKZVLVMVNYCFKVOVPVQUUOKDEZYCUGEZYNUUTMVRUUOYCUVCVSYCKVTUKAWBWCUUPFFIHZ + KKIHZUULYNUVFKUVGUUMUVFKMUHFWDWEUVDUVGKMVRKWDWEWFAFFIOUUPYCKKIUUPYCUUHKAF + KLQUDUEWGWHWMWIWJYPDEZXQUUAUUFUVHXQUUAUUFUVHXQWKZUUAWKUUCYRYSAIHZLHZYTYSY + CKVAHZIHZLHZUUEUVIUUCUVKMUUAUVIXTUVJBFYPUVHYPFWNUTEZXQUVHUVOYPWLWOWPUVIXS + UUBEZWKZXTUVQXSDEZUUNXTDEUVPUVRUVIXSYSWQWRUVQAUVHXQUVPWSWTAXSUJXAULXSYSAI + QXBWPUUAUVIYRYTUVJUVMLUUAXCUVIUVMUVJUVIUVLAYSIUVIAKXQAUMEUVHAXEWRUVIXDXFW + GXGXHUVIUVNUUEMZUUAUVHYSDEUVEUVSXQYPXIXQYCAXIWTYCYSXJXKWPXLXOXMXNXP $. $} - ${ - $d A x $. - $( A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.) $) - nofun $p |- ( A e. No -> Fun A ) $= - ( vx csur wcel cv c1o c2o cpr wf con0 wrex wfun elno ffun rexlimivw sylbi - ) ACDBEZFGHZAIZBJKALZBAMSTBJQRANOP $. - $( The domain of a surreal is an ordinal. (Contributed by Scott Fenton, - 16-Jun-2011.) $) - nodmon $p |- ( A e. No -> dom A e. On ) $= - ( vx csur wcel cv c1o c2o cpr wf con0 wrex cdm elno fdm biimprcd rexlimiv - eleq1d sylbi ) ACDBEZFGHZAIZBJKALZJDZBAMUAUCBJUAUCSJDUAUBSJSTANQOPR $. +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Infinite products +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) - $( The range of a surreal is a subset of the surreal signs. (Contributed - by Scott Fenton, 16-Jun-2011.) $) - norn $p |- ( A e. No -> ran A C_ { 1o , 2o } ) $= - ( vx csur wcel cv c1o c2o cpr con0 wrex crn wss elno frn rexlimivw sylbi - wf ) ACDBEZFGHZAQZBIJAKSLZBAMTUABIRSANOP $. - $} + $( + @{ + @d A w x @. @d B w x @. @d C w x @. @d ph x @. @d R w x @. + @d w z @. @d X w y @. @d x y @. @d x z @. @d X z @. @d Y w x @. + ntrivcvgvaulem1.1 @e |- ( ( ph /\ x e. A ) -> + E. y e. B A. z e. C Y R x ) @. + @( Lemma for ~ ntrivcvgcau . Specialization of a particular form + while avoiding a distinctness requirement between ` x ` and ` X ` . + (Contributed by Scott Fenton, 30-Dec-2017.) @) + ntrivcvgcaulem1 @p |- ( ( ph /\ X e. A ) -> + E. y e. B A. z e. C Y R X ) @= + ( vw cv wbr wral wrex wcel breq2 ralbidv rexbidv ralrimiva weq sylib wceq + cbvralv rspcv mpan9 ) AJLMZHNZDGOZCFPZLEOZIEQJIHNZDGOZCFPZAJBMZHNZDGOZCFP + ZBEOULAUSBEKUAUSUKBLEBLUBZURUJCFUTUQUIDGUPUHJHRSTUEUCUKUOLIEUHIUDZUJUNCFV + AUIUMDGUHIJHRSTUFUG @. + @} - $( A surreal is a function over its birthday. (Contributed by Scott Fenton, - 16-Jun-2011.) $) - nofnbday $p |- ( A e. No -> A Fn ( bday ` A ) ) $= - ( csur wcel wfun cdm cbday cfv wceq wfn nofun bdayval eqcomd df-fn sylanbrc - ) ABCZADAEZAFGZHAQIAJOQPAKLAQMN $. + @{ + ntrivcvgcaulem2.1 @e |- Z = ( ZZ>= ` M ) @. + ntrivcvgcaulem2.2 @e |- ( ph -> K e. Z ) @. + ntrivcvgcaulem2.3 @e |- ( ( ph /\ k e. Z ) -> A e. CC ) @. + ntrivcvgcaulem2.4 @e |- ( ( ph /\ m e. ( ZZ>= ` K ) ) -> + ( abs ` ( prod_ k e. ( K ... m ) A - 1 ) ) < ( 1 / 2 ) ) @. - $( The domain of a surreal has the ordinal property. (Contributed by Scott - Fenton, 16-Jun-2011.) $) - nodmord $p |- ( A e. No -> Ord dom A ) $= - ( csur wcel cdm con0 word nodmon eloni syl ) ABCADZECJFAGJHI $. + @{ + @d K k @. @d k m @. @d k ph @. + @( Lemma for ~ ntrivcvgcau . A product whose absolute difference + from one is less than one half has an absolute value less than two. + (Contributed by Scott Fenton, 30-Dec-2017.) @) + ntrivcvgcaulem2 @p |- ( ( ph /\ m e. ( ZZ>= ` K ) ) -> + ( abs ` prod_ k e. ( K ... m ) A ) < 2 ) @= + ( cfv wcel co c1 caddc cabs c2 a1i cr cv cuz wa cfz cprod cmin clt elfzuz + fzfid cc uztrn2 syl2an syldan adantlr fprodcl ax-1cn npcand fveq2d abscld + eqeltrd subcld abs1 1re eqeltri readdcld 2re abstrid rereccli wbr halflt1 + cdiv 2ne0 lttrd ltadd1dd oveq2i df-2 3brtr4g lelttrd eqbrtrrd ) ADUAZEUBL + ZMZUCZEVTUDNZBCUEZOUFNZOPNZQLZWEQLRUGWCWGWEQWCWEOWCWDBCWCEVTUIACUAZWDMZBU + JMZWBAWJWIGMZWKAEGMWIWAMWLWJIWIEVTUHFWIEGHUKULJUMUNUOZOUJMWCUPSZUQZURWCWH + WFQLZOQLZPNZRWCWGWCWGWEUJWOWMUTUSWCWPWQWCWFWCWEOWMWNVAZUSZWQTMWCWQOTVBVCV + DSVERTMWCVFSWCWFOWSWNVGWCWPOPNOOPNWRRUGWCWPOOWTOTMWCVCSZXAWCWPORVKNZOWTXB + TMWCRVFVLVHSXAKXBOUGVIWCVJSVMVNWQOWPPVBVOVPVQVRVS @. + @} - ${ - $d x A $. - $( An alternative condition for membership in ` No ` . (Contributed by - Scott Fenton, 21-Mar-2012.) $) - elno2 $p |- ( A e. No <-> - ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) ) $= - ( vx csur wcel wfun cdm con0 crn c1o c2o cpr wss w3a nofun nodmon norn wf - 3jca wa sylibr wrex simp2 wfn simpl funfnd anim1i 3impa df-f feq2 syl2anc - cv rspcev elno impbii ) ACDZAEZAFZGDZAHIJKZLZMZUOUPURUTANAOAPRVABUKZUSAQZ - BGUAZUOVAURUQUSAQZVDUPURUTUBVAAUQUCZUTSZVEUPURUTVGUPURSZVFUTVHAUPURUDUEUF - UGUQUSAUHTVCVEBUQGVBUQUSAUIULUJBAUMTUN $. - $} + @{ + @d A m @. @d K k @. @d k m @. @d K m @. @d k ph @. @d m ph @. + @( Lemma for ~ ntrivcvgcau . A product whose absolute difference from + one is less than one half has no zero values. (Contributed + by Scott Fenton, 30-Dec-2017.) @) + ntrivcvgcaulem3 @p |- ( ( ph /\ k e. ( ZZ>= ` K ) ) -> A =/= 0 ) @= + ( cfv wcel cc0 wceq co c1 cmin cabs clt cv cuz wne csb wi wa cfz cprod c2 + wral cdiv wbr fzfid elfzuz uztrn2 syl2an syldan fprodcl adantr ax-1cn a1i + cc subcld abscld cr 2re 2ne0 rereccli ltnsymd halflt1 cneg absnegi df-neg + fveq2i abs1 3eqtr3i breqtrri oveq1 fveq2d breqtrrid nsyl adantlr fprodm1s + cmul simpr oveq2 sylan9eqr eqtrd mtand neqned ralrimiva nfv nfcsb1v nfcv + mul01d nfne weq csbeq1a neeq1d cbvral rsp sylbir syl imp ) ACUAZEUBLZMZBN + UCZACDUAZBUDZNUCZDXFUJZXGXHUEZAXKDXFAXIXFMZUFZXJNXOXJNOZEXIUGPZBCUHZNOZXO + QUIUKPZXRQRPZSLZTULXSXOYBXTXOYAXOXRQAXRVBMXNAXQBCAEXIUMAXEXQMZXEGMZBVBMZA + EGMZXGYDYCIXEEXIUNFXEEGHUOZUPJUQZURUSQVBMXOUTVAVCVDXTVEMXOUIVFVGVHVAKVIXS + XTNQRPZSLZYBTXTQYJTVJQVKZSLQSLYJQQUTVLYKYISQVMVNVOVPVQXSYAYISXRNQRVRVSVTW + AXOXPUFXREXIQRPZUGPZBCUHZXJWDPZNXOXRYOOXPXOBCEXIAXNWEAYCYEXNYHWBWCUSXPXOY + OYNNWDPNXJNYNWDWFXOYNXOYMBCXOEYLUMAXEYMMZYEXNAYPYDYEAYFXGYDYPIXEEYLUNYGUP + JUQWBURWOWGWHWIWJWKXLXHCXFUJXMXHXKCDXFXHDWLCXJNCXIBWMCNWNWPCDWQBXJNCXIBWR + WSWTXHCXFXAXBXCXD @. + @} - $( Another condition for membership in ` No ` . (Contributed by Scott - Fenton, 14-Apr-2012.) $) - elno3 $p |- ( A e. No <-> ( A : dom A --> { 1o , 2o } /\ dom A e. On ) ) $= - ( wfun cdm con0 wcel crn c1o c2o cpr wss w3a wa csur 3anan32 elno2 wfn df-f - wf funfn anbi1i bitr4i 3bitr4i ) ABZACZDEZAFGHIZJZKUCUGLZUELAMEUDUFARZUELUC - UEUGNAOUIUHUEUIAUDPZUGLUHUDUFAQUCUJUGASTUATUB $. + ntrivcvgcaulem4.5 @e |- ( ( ph /\ x e. RR+ ) -> + E. n e. Z A. q e. ( ZZ>= ` n ) + ( abs ` ( prod_ k e. ( n ... q ) A - 1 ) ) < x ) @. - ${ - $d A a x y $. $d B a x y $. - $( Alternate expression for surreal less than. Two surreals obey surreal - less than iff they obey the sign ordering at the first place they - differ. (Contributed by Scott Fenton, 17-Jun-2011.) $) - sltval2 $p |- ( ( A e. No /\ B e. No ) -> - ( A - ( A ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) - { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } - ( B ` |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) - ) ) $= - ( vy vx wcel wa wbr cfv wceq c1o c0 c2o con0 wne fvex mtbiri adantl fveq2 - wn csur cslt cv wral cop ctp wrex crab cint sltval cvv w3o brtp 1n0 eqeq1 - neii fvprc nsyl2 adantr 2on0 3jaoi sylbi sylib onelon expcom syl5 neeq12d - onintrab onnminsb com12 syldc df-ne con2bii syl6ibr ralrimiv jca ex impac - anass raleq breq12d anbi12d rspcev syl wi eqeq12 csuc 1on 0elon necon3bid - suc11 mp2an mpbir df-2o df-1o eqeq12i nemtbir eqcom bitrdi nesymi 3imtr4i - wb elrab biimpri adantlr ssrab2 ne0i onint sylancr nfrab1 nfint nfcv nffv - nfne elrabf simprbi eqeq12d rspccv ad2antlr simpll oninton ontri1 syl2anc - wss mtod mpbird intss1 eqssd syldan sylan2 fveq2d biimpd pm2.43d rexlimiv - expimpd impbid1 bitr4d ) AUAFBUAFGZABUBHDUCZAIZYSBIZJZDEUCZUDZUUCAIZUUCBI - ZKLUEKMUELMUEUFZHZGZENUGZCUCZAIZUUKBIZOZCNUHZUIZAIZUUPBIZUUGHZEDABUJYRUUS - UUJYRUUSUUJYRUUSGZUUPNFZUUBDUUPUDZUUSGZGZUUJUUTUVAUVBGZUUSGUVDYRUUSUVEYRU - USUVEUUTUVAUVBUUSUVAYRUUSUUPUKFZUVAUUSUUQKJZUURLJZGZUVGUURMJZGZUUQLJZUVJG - ZULUVFKLKMLMUUQUURUUPAPUUPBPUMUVIUVFUVKUVMUVGUVFUVHUVGUVLUVFUVGUVLKLJZKLU - NUPZUUQKLUOQUUPAUQURZUSUVGUVFUVJUVPUSUVJUVFUVLUVJUVHUVFUVJUVHMLJMLUTUPUUR - MLUOQUUPBUQURRVAVBUUNCVHVCRZUUTUUBDUUPUUTYSUUPFZYTUUAOZTZUUBUVRUUTYSNFZUV - TUUTUVAUVRUWAUVQUVAUVRUWAUUPYSVDVEVFUWAUVRUVTUUNUVSCYSUUKYSJUULYTUUMUUAUU - KYSASUUKYSBSVGVIVJVKUVSUUBYTUUAVLVMVNVOVPVQVRUVAUVBUUSVSVCUUIUVCEUUPNUUCU - UPJZUUDUVBUUHUUSUUBDUUCUUPVTUWBUUEUUQUUFUURUUGUUCUUPASUUCUUPBSWAWBWCWDVQU - UIUUSENUUCNFZUUDUUHUUSUWCUUDGZUUHUUSUWDUUHUUHUUSWEUWDUUHGZUUHUUSUWEUUEUUQ - UUFUURUUGUWEUUCUUPAUUHUWDUUEUUFOZUWBUUEKJZUUFLJGZUWGUUFMJZGZUUELJUWIGZULU - UEUUFJZTZUUHUWFUWHUWMUWJUWKUWHUWLUVNUVOUUEKUUFLWFQUWJUWLMKJZUWNKWGZLWGZUW - OUWPOZKLOZUNKNFZLNFZUWQUWRXBWHWIUWSUWTGUWOUWPKLKLWKWJWLWMMUWOKUWPWNWOWPWQ - UWJUWLKMJUWNUUEKUUFMWFKMWRWSQUWKUWLLMJMLUTWTUUELUUFMWFQVAKLKMLMUUEUUFUUCA - PUUCBPUMUUEUUFVLXAUWDUWFUUCUUOFZUWBUWCUWFUXAUUDUXAUWCUWFGUUNUWFCUUCNUUKUU - CJUULUUEUUMUUFUUKUUCASUUKUUCBSVGXCXDXEUWDUXAGZUUCUUPUXBUUCUUPYDZUUPUUCFZT - ZUXBUXDUUQUURJZUXBUUQUUROZUXFTUXBUUPUUOFZUXGUXBUUONYDZUUOLOZUXHUUNCNXFZUX - AUXJUWDUUOUUCXGZRUUOXHXIUXHUVAUXGUUNUXGCUUPNCUUOUUNCNXJXKZCNXLCUUQUURCUUP - ACAXLUXMXMCUUPBCBXLUXMXMXNUUKUUPJUULUUQUUMUURUUKUUPASUUKUUPBSVGXOXPWDUUQU - URVLVCUUDUXDUXFWEUWCUXAUUBUXFDUUPUUCYSUUPJYTUUQUUAUURYSUUPASYSUUPBSXQXRXS - YEUXBUWCUVAUXCUXEXBUWCUUDUXAXTUXAUVAUWDUXAUXIUXJUVAUXKUXLUUOYAXIRUUCUUPYB - YCYFUXAUUPUUCYDUWDUUCUUOYGRYHYIYJZYKUWEUUCUUPBUXNYKWAYLVQYMYOYNYPYQ $. - $} + ntrivcvgcaulem4 @p |- ( ( ph /\ x e. RR+ ) -> + E. j e. ( ZZ>= ` K ) A. p e. ( ZZ>= ` j ) + ( abs ` ( prod_ k e. ( ( j + 1 ) ... p ) A - 1 ) ) < ( x / 2 ) ) @= ? @. - $( The function value of a surreal is either a sign or the empty set. - (Contributed by Scott Fenton, 22-Jun-2011.) $) - nofv $p |- ( A e. No -> ( ( A ` X ) = (/) \/ ( A ` X ) = 1o \/ - ( A ` X ) = 2o ) ) $= - ( csur wcel cfv c0 wceq c1o c2o wo w3o cdm wn pm2.1 wi ndmfv a1i wfun crn - wa cpr nofun norn fvelrn ssel syl5com impancom 1oex con0 elexi elpr2 syl6ib - wss 2on syl2anc orim12d mpi 3orass sylibr ) ACDZBAEZFGZVAHGZVAIGZJZJZVBVCVD - KUTBALDZMZVGJVFVGNUTVHVBVGVEVHVBOUTBAPQUTARZASZHIUAZUMZVGVEOAUBAUCVIVLTVGVA - VKDZVEVIVGVLVMVIVGTVAVJDVLVMBAUDVJVKVAUEUFUGVAHIUHIUIUNUJUKULUOUPUQVBVCVDUR - US $. + @{ + @d j p @. @d j ph @. @d j x @. @d K p @. @d p ph @. @d p x @. @d A m @. + @d j k @. @d j m @. @d K k @. @d k m @. @d K m @. @d k p @. @d k ph @. + @d k x @. @d m ph @. - $( ` (/) ` is not a surreal sign. (Contributed by Scott Fenton, - 16-Jun-2011.) $) - nosgnn0 $p |- -. (/) e. { 1o , 2o } $= - ( c0 c1o c2o cpr wcel wceq wo 1n0 nesymi csuc wne necom df-2o neeq2i bitr4i - nsuceq0 mpbi neii pm3.2ni 0ex elpr mtbir ) ABCDEABFZACFZGUCUDBAHIACBJZAKZAC - KZBPUFAUEKUGUEALCUEAMNOQRSABCTUAUB $. + @( Lemma for ~ ntrivcvgcau . Translate final hypothesis into + a Cauchy-like criterion. (Contributed by Scott Fenton, 30-Dec-2017.) @) + ntrivcvgcaulem5 @p |- ( ( ph /\ x e. RR+ ) -> + E. j e. Z A. p e. ( ZZ>= ` j ) + ( abs ` ( prod_ k e. ( K ... p ) A - prod_ k e. ( K ... j ) A ) ) + < x ) @= + ( wcel wa co cv crp c1 caddc cfz cprod cmin cabs cfv c2 cdiv clt wbr wral + cuz wrex ntrivcvgcaulem4 wss eleqtrdi uzss sseqtrrdi adantr sseld adantrd + syl wi w3a cmul fzfid simplll simpll elfzuz uztrn2 syl2an syl2anc fprodcl + cr cc abscld 3adant3 simpl peano2uzs ax-1cn subcld simplr rphalfcld rpred + 2re a1i cc0 cle absge0d simprl eleq1 anbi2d prodeq1d oveq1d fveq2d breq1d + weq oveq2 imbi12d chvarv ntrivcvgcaulem2 simp3 ltmul12ad wceq c0 eluzelre + cin ad2antrl fzdisj cun simprr elfzuzb sylanbrc fzsplit fprodsplit muls1d + ltp1d eqtr4d absmuld eqtr2d simp1r rpcn 2cn 2ne0 divcan2d 3brtr3d anassrs + wne 3expia ralimdva expimpd jcad reximdv2 mpd ) ABUAZUBRZSZDUAZUCUDTZLUAZ + UETZCEUFZUCUGTZUHUIZYRUJUKTZULUMZLUUAUOUIZUNZDHUOUIZUPHUUCUETZCEUFZHUUAUE + TZCEUFZUGTZUHUIZYRULUMZLUUJUNZDJUPABCDEFGHIJKLMNOPQUQYTUUKUUTDUULJYTUUAUU + LRZUUKSUUAJRZUUTYTUVAUVBUUKYTUULJUUAAUULJURYSAUULIUOUIZJAHUVCRUULUVCURAHJ + UVCNMUSIHUTVEMVAVBVCVDYTUVAUUKUUTYTUVASUUIUUSLUUJYTUVAUUCUUJRZUUIUUSVFYTU + VAUVDSZUUIUUSYTUVEUUIVGZUUPUHUIZUUGVHTZUJUUHVHTZUURYRULUVFUVGUJUUGUUHYTUV + EUVGVQRUUIYTUVESZUUPUVJUUOCEUVJHUUAVIUVJEUAZUUORZSAUVKJRZCVRRZAYSUVEUVLVJ + UVJHJRZUVKUULRZUVMUVLUVJAUVOAYSUVEVKZNVEZUVKHUUAVLIUVKHJMVMZVNOVOVPZVSVTU + JVQRUVFWHWIYTUVEUUGVQRUUIUVJUUFUVJUUEUCUVJUUDCEUVJUUBUUCVIUVJUVKUUDRZSAUV + MUVNAYSUVEUWAVJUVJUUBJRZUVKUUBUOUIRUVMUWAUVJUVBUWBYTUVOUVAUVBUVEAUVOYSNVB + UVAUVDWAIUUAHJMVMVNIUUAJMWBVEUVKUUBUUCVLIUVKUUBJMVMVNOVOVPZUCVRRUVJWCWIWD + ZVSVTYTUVEUUHVQRUUIUVJUUHUVJYRAYSUVEWEWFWGVTYTUVEWJUVGWKUMUUIUVJUUPUVTWLV + TYTUVEUVGUJULUMZUUIUVJAUVAUWEUVQYTUVAUVDWMZACEDHIJMNOAFUAZUULRZSZHUWGUETZ + CEUFZUCUGTZUHUIZUCUJUKTZULUMZVFAUVASZUUPUCUGTZUHUIZUWNULUMZVFFDFDWTZUWIUW + PUWOUWSUWTUWHUVAAUWGUUAUULWNWOUWTUWMUWRUWNULUWTUWLUWQUHUWTUWKUUPUCUGUWTUW + JUUOCEUWGUUAHUEXAWPWQWRWSXBPXCXDVOVTYTUVEWJUUGWKUMUUIUVJUUFUWDWLVTYTUVEUU + IXEXFYTUVEUVHUURXGUUIUVJUURUUPUUFVHTZUHUIUVHUVJUUQUXAUHUVJUUQUUPUUEVHTZUU + PUGTUXAUVJUUNUXBUUPUGUVJUUOUUDCUUMEUVJUUAUUBULUMUUOUUDXJXHXGUVJUUAUVAUUAV + QRYTUVDHUUAXIXKXTHUUAUUBUUCXLVEUVJUUAUUMRZUUMUUOUUDXMXGUVJUVAUVDUXCUWFYTU + VAUVDXNUUAHUUCXOXPUUAHUUCXQVEUVJHUUCVIUVJUVKUUMRZSAUVMUVNAYSUVEUXDVJUVJUV + OUVPUVMUXDUVRUVKHUUCVLUVSVNOVOXRWQUVJUUPUUEUVTUWCXSYAWRUVJUUPUUFUVTUWDYBY + CVTUVFYSUVIYRXGAYSUVEUUIYDYSYRUJYRYEUJVRRYSYFWIUJWJYKYSYGWIYHVEYIYLYJYMYN + YOYPYQ @. + @} + @} + $) ${ - nosgnn0i.1 $e |- X e. { 1o , 2o } $. - $( If ` X ` is a surreal sign, then it is not null. (Contributed by Scott - Fenton, 3-Aug-2011.) $) - nosgnn0i $p |- (/) =/= X $= - ( c0 wceq c1o c2o cpr wcel nosgnn0 eleq1 mpbiri mto neir ) CACADZCEFGZHZI - NPAOHBCAOJKLM $. + $d F j $. $d F k $. $d F n $. $d j k $. $d j n $. $d j ph $. + $d k ph $. $d M j $. $d M k $. $d M n $. $d n ph $. $d Z k $. + iprodefisumlem.1 $e |- Z = ( ZZ>= ` M ) $. + iprodefisumlem.2 $e |- ( ph -> M e. ZZ ) $. + iprodefisumlem.3 $e |- ( ph -> F : Z --> CC ) $. + $( Lemma for ~ iprodefisum . (Contributed by Scott Fenton, + 11-Feb-2018.) $) + iprodefisumlem $p |- ( ph -> seq M ( x. , ( exp o. F ) ) = + ( exp o. seq M ( + , F ) ) ) $= + ( vk vj cmul ce caddc cc wcel cfv wceq fvco3 syl wi co vn ccom cseq cv wa + sylan ffvelcdmda efcl eqeltrd prodf ffnd wfn eff ax-mp serf fnfco sylancr + wf ffn cuz c1 fveq2 2fveq3 eqeq12d imbi2d weq uzid eleqtrrdi syl2anc seq1 + cz fveq2d 3eqtr4d a1i oveq1 3ad2ant3 adantl peano2uz adantr oveq2d expcom + w3a eqcomi eleq2s imp ffvelcdmd efadd eqtr4d 3adant3 eqtrd seqp1 3exp a2d + uzind4 impcom eqfnfvd ) AHDJKBUBZCUCZKLBCUCZUBZADMWRAHWQCDEFAHUDZDNZUEZXA + WQOZXABOZKOZMADMBURZXBXDXFPGDMXAKBQUFXCXEMNXFMNADMXABGUGZXEUHRUIUJUKAKMUL + ZDMWSURZWTDULMMKURXIUMMMKUSUNAHBCDEFXHUOZMDKWSUPUQXCXAWROZXAWSOKOZXAWTOZX + BAXLXMPZAXOSZXACUTOZDAIUDZWROZXRWSOKOZPZSACWROZCWSOZKOZPZSZAUAUDZWROZYGWS + OZKOZPZSAYGVALTZWROZYLWSOZKOZPZSXPIUACXAXRCPZYAYEAYQXSYBXTYDXRCWRVBXRCKWS + VCVDVEIUAVFZYAYKAYRXSYHXTYJXRYGWRVBXRYGKWSVCVDVEXRYLPZYAYPAYSXSYMXTYOXRYL + WRVBXRYLKWSVCVDVEIHVFZYAXOAYTXSXLXTXMXRXAWRVBXRXAKWSVCVDVEYFCVKNZACWQOZCB + OZKOZYBYDAXGCDNUUBUUDPGACXQDAUUACXQNFCVGREVHDMCKBQVIAUUAYBUUBPFJWQCVJRAYC + UUCKAUUAYCUUCPFLBCVJRVLVMVNYGXQNZAYKYPUUEAYKYPUUEAYKWBZYHYLWQOZJTZYIYLBOZ + LTZKOZYMYOUUFUUHYJUUGJTZUUKYKUUEUUHUULPAYHYJUUGJVOVPUUEAUULUUKPYKUUEAUEZU + ULYJUUIKOZJTZUUKUUMUUGUUNYJJUUMXGYLDNZUUGUUNPAXGUUEGVQZUUEUUPAUUEYLXQDCYG + VREVHVSZDMYLKBQVIVTUUMYIMNZUUIMNUUKUUOPUUEAUUSAUUSSYGDXQAYGDNUUSADMYGWSXK + UGWADXQEWCWDWEUUMDMYLBUUQUURWFYIUUIWGVIWHWIWJUUEAYMUUHPZYKUUEUUTAJWQCYGWK + VSWIUUEAYOUUKPYKUUMYNUUJKUUEYNUUJPALBCYGWKVSVLWIVMWLWMWNEWDWOAXJXBXNXMPXK + DMXAKWSQUFWHWP $. $} ${ - $d A x y $. $d B x y $. - $( The restriction of a surreal to an ordinal is still a surreal. - (Contributed by Scott Fenton, 4-Sep-2011.) $) - noreson $p |- ( ( A e. No /\ B e. On ) -> ( A |` B ) e. No ) $= - ( vy vx csur wcel con0 wa cv c1o c2o cpr cres wf wrex elno wi onin fresin - cin feq2 rspcev syl2an an32s ex rexlimiva imp sylanb sylibr ) AEFZBGFZHCI - ZJKLZABMZNZCGOZUNEFUJDIZUMANZDGOZUKUPDAPUSUKUPURUKUPQDGUQGFZURHUKUPUTUKUR - UPUTUKHUQBTZGFVAUMUNNZUPURUQBRUQUMABSUOVBCVAGULVAUMUNUAUBUCUDUEUFUGUHCUNP - UI $. + $d F k $. $d k ph $. $d M k $. $d Z k $. $d F j $. $d j k $. + $d Z j $. $d j ph $. $d M j $. + iprodefisum.1 $e |- Z = ( ZZ>= ` M ) $. + iprodefisum.2 $e |- ( ph -> M e. ZZ ) $. + iprodefisum.3 $e |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B ) $. + iprodefisum.4 $e |- ( ( ph /\ k e. Z ) -> B e. CC ) $. + iprodefisum.5 $e |- ( ph -> seq M ( + , F ) e. dom ~~> ) $. + $( Applying the exponential function to an infinite sum yields an infinite + product. (Contributed by Scott Fenton, 11-Feb-2018.) $) + iprodefisum $p |- ( ph -> prod_ k e. Z ( exp ` B ) = + ( exp ` sum_ k e. Z B ) ) $= + ( vj ce cfv cc wcel syl caddc cseq cli cmpt ccom csu cc0 wne isumcl efne0 + cv cmul ccncf co efcn a1i wa wceq fveq2 eqid fvex adantl eqeltrd serf cuz + fvmpt eqcomi eleq2s seqfeq cdm wbr climdm eqbrtrd climcl climcncf cbvmptv + sylib fmptd iprodefisumlem isum fveq2d 3brtr4d wf fvco3 sylan 3eqtrd efcl + iprodn0 ) ABMNZCMLFLUHZDNZUAZUBZEFBCUCZMNZFGHAWKOPWLUDUEABCDEFGHIJKUFWKUG + QAMRWIESZUBRDESZTNZMNUIWJESWLTAOOWOMWMEFGHMOOUJUKPAULUMACWIEFGHACUHZFPZUN + ZWPWINZWPDNZOWQWSWTUOZALWPWHWTFWIWGWPDUPZWIUQWPDURVCZUSZWRWTBOIJUTZUTVAAW + MWNWOTARCWIDEHWPEVBNZPXAAXAWPFXFXCFXFGVDVEUSVFAWNTVGPWNWOTVHZKWNVIVNZVJAX + GWOOPXHWOWNVKQVLAWIEFGHACFWTOWIXELCFWHWTXBVMVOZVPAWKWOMABCDEFGHIJVQVRVSWR + WPWJNZWSMNZWTMNWFAFOWIVTWQXJXKUOXIFOWPMWIWAWBWRWSWTMXDVRWRWTBMIVRWCWRBOPW + FOPJBWDQWE $. $} ${ - $d A a $. $d B a $. - $( If ` A - ( A |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. _V ) ) $= - ( csur wcel wa wbr cfv c1o c0 cop c2o wceq fvex wn fvprc neii eqeq1 eqcom - bitrdi cslt cv wne con0 crab cint ctp cvv sltval2 w3o brtp 1n0 mtbiri syl - con4i adantr 2on0 adantl 3jaoi sylbi syl6bi ) ADEBDEFABUAGCUBZAHVBBHUCCUD - UEUFZAHZVCBHZIJKILKJLKUGGZVCUHEZABCUIVFVDIMZVEJMZFZVHVELMZFZVDJMZVKFZUJVG - IJILJLVDVEVCANVCBNUKVJVGVLVNVHVGVIVGVHVGOZVMVHOVCAPVMVHIJMZIJULQVMVHJIMVP - VDJIRJISTUMUNUOZUPVHVGVKVQUPVKVGVMVGVKVOVIVKOVCBPVIVKLJMZLJUQQVIVKJLMVRVE - JLRJLSTUMUNUOURUSUTVA $. + $d A j $. $d A k $. $d A z $. $d j k $. $d j ph $. $d k ph $. + $d k z $. + iprodgam.1 $e |- ( ph -> A e. ( CC \ ( ZZ \ NN ) ) ) $. + $( An infinite product version of Euler's gamma function. (Contributed by + Scott Fenton, 12-Feb-2018.) $) + iprodgam $p |- ( ph -> ( _G ` A ) = + ( prod_ k e. NN ( ( ( 1 + ( 1 / k ) ) ^c A ) / ( 1 + ( A / k ) ) ) / + A ) ) $= + ( vj cfv ce cn c1 cdiv co caddc cc wcel wceq clog cmul cmin oveq12d eqtrd + vz clgam cgam cv ccxp cprod cz cdif eflgam oveq1 fvoveq1d sumeq2sdv fveq2 + syl csu df-lgam ovex fvmpt fveq2d cmpt nnuz 1zzd weq id oveq2d oveq2 eqid + adantl wa eldifad adantr peano2nn nncnd nncn cc0 wne nnne0 divcld divne0d + nnne0d logcld mulcld 1cnd addcld simpr dmgmdivn0 cseq cli wbr cdm lgamcvg + subcld seqex breldm isumcl dmgmn0 efsub syl2anc iprodefisum dividd oveq1d + divdird crp 1rp nnrpd rpreccld rpaddcld rpcnd rpne0d cxpefd eflog addcomd + a1i eqtr4d prodeq2dv eqtr3d ) ABUBFZGFZBUCFZHIICUDZJKZLKZBUEKZIBXTJKZLKZJ + KZCUFZBJKZABMUGHUHZUHZNZXRXSODBUIUNAXRHBXTILKZXTJKZPFZQKZYDILKZPFZRKZCUOZ + BPFZRKZGFZYHAXQUUAGAYKXQUUAODUABHUAUDZYNQKZUUCXTJKZILKPFZRKZCUOZUUCPFZRKU + UAYJUBUUCBOZUUHYSUUIYTRUUJHUUGYRCUUJUUDYOUUFYQRUUCBYNQUJUUJUUEYDIPLUUCBXT + JUJUKSULUUCBPUMSUACUPYSYTRUQURUNUSAUUBYSGFZYTGFZJKZYHAYSMNYTMNUUBUUMOAYRC + EHBEUDZILKZUUNJKZPFZQKZBUUNJKZILKPFZRKZUTZIHVAAVBZXTHNZXTUVBFYROAEXTUVAYR + HUVBECVCZUURYOUUTYQRUVEUUQYNBQUVEUUPYMPUVEUUOYLUUNXTJUUNXTILUJUVEVDSUSVEU + VEUUSYDIPLUUNXTBJVFUKSUVBVGZYOYQRUQURVHZAUVDVIZYOYQUVHBYNABMNZUVDABMYIDVJ + ZVKZUVHYMUVHYLXTUVHYLUVDYLHNAXTVLVHZVMZUVDXTMNAXTVNVHZUVDXTVOVPAXTVQVHZVR + UVHYLXTUVMUVNUVHYLUVLVTUVOVSWAWBZUVHYPUVHYDIUVHBXTUVKUVNUVOVRZUVHWCZWDZUV + HBXTAYKUVDDVKAUVDWEZWFZWAZWLZALUVBIWGZXQYTLKZWHWIUWDWHWJNABEUVBUVFDWKUWDU + WEWHLUVBIWMXQYTLUQWNUNZWOABUVJABDWPZWAYSYTWQWRAUUKYGUULBJAHYRGFZCUFUUKYGA + YRCUVBIHVAUVCUVGUWCUWFWSAHUWHYFCUVHUWHYOGFZYQGFZJKZYFUVHYOMNYQMNUWHUWKOUV + PUWBYOYQWQWRUVHUWIYCUWJYEJUVHUWIBYBPFZQKZGFYCUVHYOUWMGUVHYNUWLBQUVHYMYBPU + VHYMXTXTJKZYALKYBUVHXTIXTUVNUVRUVNUVOXBUVHUWNIYALUVHXTUVNUVOWTXATUSVEUSUV + HYBBUVHYBUVHIYAIXCNUVHXDXMUVHXTUVHXTUVTXEXFXGZXHUVHYBUWOXIUVKXJXNUVHUWJYP + YEUVHYPMNYPVOVPUWJYPOUVSUWAYPXKWRUVHYDIUVQUVRXLTSTXOXPAUVIBVOVPUULBOUVJUW + GBXKWRSTTXP $. $} - ${ - $d A a x y $. $d B a x y $. $d X a x y $. - $( If the restrictions of two surreals to a given ordinal obey surreal less - than, then so do the two surreals themselves. (Contributed by Scott - Fenton, 4-Sep-2011.) $) - sltres $p |- ( ( A e. No /\ B e. No /\ X e. On ) -> - ( ( A |` X ) A ( A X. { 1o } ) e. No ) $= - ( con0 wcel c1o csn cxp cdm c2o cpr wf csur 1oex prid1 fconst6 c0 wceq snnz - wne dmxp ax-mp feq2i mpbir a1i eleq1i biimpri elno3 sylanbrc ) ABCZADEZFZGZ - DHIZUJJZUKBCZUJKCUMUHUMAULUJJADULDHLMNUKAULUJUIORUKAPDLQAUISTZUAUBUCUNUHUKA - BUOUDUEUJUFUG $. +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Factorial limits +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) ${ - $d A x $. $d B x $. - $( Lemma for ~ nosepon . Consider a case of proper subset domain. - (Contributed by Scott Fenton, 21-Sep-2020.) $) - noseponlem $p |- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> - -. A. x e. On ( A ` x ) = ( B ` x ) ) $= - ( csur wcel cdm w3a cv cfv wne con0 wrex wceq wral wn nodmon 3ad2ant1 syl - c0 fveq2 word nodmord ordirr ndmfv c1o c2o nosgnn0 elno3 simplbi 3ad2ant2 - cpr wf simp3 ffvelcdmd eleq1 syl5ibcom mtoi neqned necomd eqnetrd neeq12d - rspcev syl2anc df-ne rexbii rexnal bitri sylib ) BDEZCDEZBFZCFZEZGZAHZBIZ - VOCIZJZAKLZVPVQMZAKNOZVNVKKEZVKBIZVKCIZJZVSVIVJWBVMBPQVNWCSWDVNVKVKEOZWCS - MVIVJWFVMVIVKUAWFBUBVKUCRQVKBUDRVNWDSVNWDSVNWDSMZSUEUFUKZEZUGVNWDWHEWGWIV - NVLWHVKCVJVIVLWHCULZVMVJWJVLKECUHUIUJVIVJVMUMUNWDSWHUOUPUQURUSUTVRWEAVKKV - OVKMVPWCVQWDVOVKBTVOVKCTVAVBVCVSVTOZAKLWAVRWKAKVPVQVDVEVTAKVFVGVH $. - - $( Given two unequal surreals, the minimal ordinal at which they differ is - an ordinal. (Contributed by Scott Fenton, 21-Sep-2020.) $) - nosepon $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> - |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. On ) $= - ( csur wcel wne w3a cfv con0 wrex wa wn wceq wral cdm word nodmord 3expia - wo 3impia cv crab cint df-ne rexbii notbii dfral2 bitr4i ordtri3or syl2an - w3o 3orass or12 bitri sylib ord noseponlem wi eqcom ralbii sylnibr ancoms - jaod syld con4d wss ordsson ssralv 3syl adantr wb nofun 3ad2ant1 3ad2ant2 - wfun eqfunfv syl2anc mpbir2and syl5bi necon1ad onintrab2 ) BDEZCDEZBCFZGA - UAZBHZWECHZFZAIJZWHAIUBUCIEWBWCWDWIWBWCKZWIBCWILZWFWGMZAINZWJBCMZWKWLLZAI - JZLWMWIWPWHWOAIWFWGUDUEUFWLAIUGUHWBWCWMWNWBWCWMGZWNBOZCOZMZWLAWRNZWBWCWMW - TWJWTWMWJWTLWRWSEZWSWREZSZWMLZWJWTXDWJXBWTXCUKZWTXDSZWBWRPZWSPXFWCBQZCQWR - WSUIUJXFXBWTXCSSXGXBWTXCULXBWTXCUMUNUOUPWJXBXEXCWBWCXBXEABCUQRWCWBXCXEURW - CWBXCXEWCWBXCGWGWFMZAINWMACBUQWLXJAIWFWGUSUTVARVBVCVDVETWBWCWMXAWBWMXAURZ - WCWBXHWRIVFXKXIWRVGWLAWRIVHVIVJTWQBVOZCVOZWNWTXAKVKWBWCXLWMBVLVMWCWBXMWMC - VLVNABCVPVQVRRVSVTTWHAWAUO $. + $d a b $. $d a k $. $d a n $. $d b n $. $d b x $. $d k n $. $d M a $. + $d M b $. $d M k $. $d M n $. $d M x $. + $( Lemma for ~ faclim . Closed form for a particular sequence. + (Contributed by Scott Fenton, 15-Dec-2017.) $) + faclimlem1 $p |- ( M e. NN0 -> + seq 1 ( x. , ( n e. NN |-> ( ( ( 1 + ( M / n ) ) x. ( 1 + ( 1 / n ) ) ) / + ( 1 + ( ( M + 1 ) / n ) ) ) ) ) = + ( x e. NN |-> ( ( M + 1 ) x. ( ( x + 1 ) / ( x + ( M + 1 ) ) ) ) ) ) $= + ( vb wcel cv cmul cn c1 cdiv co caddc cfv wceq oveq1 oveq12d oveq2d oveq2 + 3eqtrd rpcnd va vk cn0 cmpt cseq wral wa wi fveq2 cz 1z seq1 ax-mp eqtrdi + eqeq12d imbi2d weq 1nn eqid ovex fvmpt nn0cn div1d 1div1e1 oveq2i nn0p1nn + nncnd 1cnd addcomd oveq1d cc ax-1cn addcli nnaddcld nnne0d divassd eqtrid + a1i cuz seqp1 nnuz eleq2s adantr adantl peano2nn syl nnrpd simpl rpaddcld + rpdivcld crp cr nn0re nndivred recnd cc0 cle wbr divge0d ge0p1rpd eqeltrd + nn0ge0 1rp rpreccld rpmulcld mulassd rpne0d divcan5d mul12d mulid1d simpr + adddid nn0cnd divcan2d addassd eqtrd eqtr3d divcan1d 3eqtr3rd exp31 nnind + a2d impcom eqtr4d ralrimiva wfn wb seqfn fneq2i mpbir fnmpti eqfnfv mp2an + sylibr ) CUCEZDFZGBHICBFZJKZLKZIIYQJKZLKZGKZICILKZYQJKZLKZJKZUDZIUEZMZYPA + HUUCAFZILKZUUJUUCLKZJKZGKZUDZMZNZDHUFZUUHUUONZYOUUQDHYOYPHEZUGUUIUUCYPILK + ZYPUUCLKZJKZGKZUUPUUTYOUUIUVDNZYOUAFZUUHMZUUCUVFILKZUVFUUCLKZJKZGKZNZUHYO + IUUGMZUUCIILKZIUUCLKZJKZGKZNZUHYOUBFZUUHMZUUCUVSILKZUVSUUCLKZJKZGKZNZUHYO + UWAUUHMZUUCUWAILKZUWAUUCLKZJKZGKZNZUHYOUVEUHUAUBYPUVFINZUVLUVRYOUWLUVGUVM + UVKUVQUWLUVGIUUHMZUVMUVFIUUHUIIUJEZUWMUVMNUKGUUGIULUMUNUWLUVJUVPUUCGUWLUV + HUVNUVIUVOJUVFIILOUVFIUUCLOPQUOUPUAUBUQZUVLUWEYOUWOUVGUVTUVKUWDUVFUVSUUHU + IUWOUVJUWCUUCGUWOUVHUWAUVIUWBJUVFUVSILOUVFUVSUUCLOPQUOUPUVFUWANZUVLUWKYOU + WPUVGUWFUVKUWJUVFUWAUUHUIUWPUVJUWIUUCGUWPUVHUWGUVIUWHJUVFUWAILOUVFUWAUUCL + OPQUOUPUADUQZUVLUVEYOUWQUVGUUIUVKUVDUVFYPUUHUIUWQUVJUVCUUCGUWQUVHUVAUVIUV + BJUVFYPILOUVFYPUUCLOPQUOUPYOUVMICIJKZLKZIIIJKZLKZGKZIUUCIJKZLKZJKZUVQIHEZ + UVMUXENURBIUUFUXEHUUGYQINZUUBUXBUUEUXDJUXGYSUWSUUAUXAGUXGYRUWRILYQICJRQUX + GYTUWTILYQIIJRQPUXGUUDUXCILYQIUUCJRQPUUGUSZUXBUXDJUTVAUMYOUXEICLKZUVNGKZU + VOJKUUCUVNGKZUVOJKUVQYOUXBUXJUXDUVOJYOUWSUXIUXAUVNGYOUWRCILYOCCVBZVCQUXAU + VNNYOUWTIILVDVEVRPYOUXCUUCILYOUUCYOUUCCVFZVGZVCQPYOUXJUXKUVOJYOUXIUUCUVNG + YOICYOVHUXLVIVJVJYOUUCUVNUVOUXNUVNVKEYOIIVLVLVMVRYOUVOYOIUUCUXFYOURVRUXMV + NZVGYOUVOUXOVOVPSVQUVSHEZYOUWEUWKUXPYOUWEUWKUXPYOUGZUWEUGUWFUVTUWAUUGMZGK + ZUWDUXRGKZUWJUXQUWFUXSNZUWEUXPUYAYOUYAUVSIVSMZHGUUGIUVSVTWAWBWCWCUWEUXSUX + TNUXQUVTUWDUXRGOWDUXQUXTUWJNUWEUXQUXTUWDICUWAJKZLKZIIUWAJKZLKZGKZIUUCUWAJ + KZLKZJKZGKZUUCUWCUYJGKZGKUWJUXPUXTUYKNYOUXPUXRUYJUWDGUXPUWAHEZUXRUYJNUVSW + EZBUWAUUFUYJHUUGYQUWANZUUBUYGUUEUYIJUYOYSUYDUUAUYFGUYOYRUYCILYQUWACJRQUYO + YTUYEILYQUWAIJRQPUYOUUDUYHILYQUWAUUCJRQPUXHUYGUYIJUTVAWFQWCUXQUUCUWCUYJUX + QUUCYOUUCHEUXPUXMWDZVGZUXQUWCUXQUWAUWBUXQUWAUXPUYMYOUYNWCZWGZUXQUVSUUCUXQ + UVSUXPYOWHZWGUXQUUCUYPWGZWIZWJZTZUXQUYJUXQUYGUYIUXQUYDUYFUXQUYDUYCILKWKUX + QIUYCUXQVHZUXQUYCUXQCUWAYOCWLEUXPCWMWDZUYRWNZWOZVIUXQUYCVUGUXQCUWAVUFUYSY + OWPCWQWRUXPCXBWDWSWTXAZUXQIUYEIWKEUXQXCVRZUXQUWAUYSXDZWIZXEZUXQIUYHVUJUXQ + UUCUWAVUAUYSWJZWIZWJTXFUXQUYLUWIUUCGUXQUWAUWCUYGGKZGKZUWAUYIGKZJKVUPUYIJK + UWIUYLUXQVUPUYIUWAUXQVUPUXQUWCUYGVUCVUMXETUXQUYIVUOTZUXQUWAUYRVGZUXQUYIVU + OXGZUXQUWAUYRVOZXHUXQVUQUWGVURUWHJUXQVUQUWCUWAUYGGKZGKUWCUWBUYFGKZGKZUWGU + XQUWAUWCUYGVUTVUDUXQUYGVUMTZXIUXQVVCVVDUWCGUXQUWAUYDGKZUYFGKVVCVVDUXQUWAU + YDUYFVUTUXQUYDVUITUXQUYFVULTZXFUXQVVGUWBUYFGUXQVVGUWAIGKZUWAUYCGKZLKUWACL + KZUWBUXQUWAIUYCVUTVUEVUHXLUXQVVIUWAVVJCLUXQUWAVUTXJZUXQCUWAUXQCUXPYOXKXMZ + VUTVVBXNPUXQVVKUVSUXILKUWBUXQUVSICUXQUVSUYTVGVUEVVMXOUXQUXIUUCUVSLUXQICVU + EVVMVIQXPSVJXQQUXQUWCUWBGKZUYFGKZVVEUWGUXQUWCUWBUYFVUDUXQUWBVUBTZVVHXFUXQ + VVOUWAUYFGKVVIUWAUYEGKZLKUWGUXQVVNUWAUYFGUXQUWAUWBVUTVVPUXQUWBVUBXGXRVJUX + QUWAIUYEVUTVUEUXQUYEVUKTXLUXQVVIUWAVVQILVVLUXQIUWAVUEVUTVVBXNPSXQSUXQVURV + VIUWAUYHGKZLKUWHUXQUWAIUYHVUTVUEUXQUYHVUNTXLUXQVVIUWAVVRUUCLVVLUXQUUCUWAU + YQVUTVVBXNPXPPUXQUWCUYGUYIVUDVVFVUSVVAVPXSQSWCSXTYBYAYCUUTUUPUVDNYOAYPUUN + UVDHUUOADUQZUUMUVCUUCGVVSUUKUVAUULUVBJUUJYPILOUUJYPUUCLOPQUUOUSZUUCUVCGUT + VAWDYDYEUUHHYFZUUOHYFUUSUURYGVWAUUHUYBYFZUWNVWBUKGUUGIYHUMHUYBUUHWAYIYJAH + UUNUUOUUCUUMGUTVVTYKDHUUHUUOYLYMYN $. $} ${ - noextend.1 $e |- X e. { 1o , 2o } $. - $( Extending a surreal by one sign value results in a new surreal. - (Contributed by Scott Fenton, 22-Nov-2021.) $) - noextend $p |- ( A e. No -> ( A u. { <. dom A , X >. } ) e. No ) $= - ( csur wcel cdm cop csn cun wfun con0 crn c1o wss cin wceq cvv syl eqtrid - c0 c2o cpr nofun dmexg funsng sylancl elexi dmsnop ineq2i wn word nodmord - ordirr disjsn sylibr funun syl21anc uneq2i df-suc 3eqtr4i nodmon suceloni - csuc dmun eqeltrid rnun rnsnopg uneq2d norn snssi unssd eqsstrd syl3anbrc - mp1i elno2 ) ADEZAAFZBGHZIZJZVSFZKEVSLZMUAUBZNVSDEVPAJVRJZVQVRFZOZTPVTAUC - VPVQQEZBWCEZWDADUDZCVQBQWCUEUFVPWFVQVQHZOZTWEWJVQVQBBWCCUGUHZUIVPVQVQEUJZ - WKTPVPVQUKWMAULVQUMRVQVQUNUOSAVRUPUQVPWAVQVCZKVQWEIVQWJIWAWNWEWJVQWLURAVR - VDVQUSUTVPVQKEWNKEAVAVQVBRVEVPWBALZBHZIZWCVPWBWOVRLZIWQAVRVFVPWRWPWOVPWGW - RWPPWIVQBQVGRVHSVPWOWPWCAVIWHWPWCNVPCBWCVJVNVKVLVSVOVM $. + $d k m $. $d M k $. $d M m $. $d M n $. + $( Lemma for ~ faclim . Show a limit for the inductive step. (Contributed + by Scott Fenton, 15-Dec-2017.) $) + faclimlem2 $p |- ( M e. NN0 -> + seq 1 ( x. , + ( n e. NN |-> ( ( ( 1 + ( M / n ) ) x. ( 1 + ( 1 / n ) ) ) / + ( 1 + ( ( M + 1 ) / n ) ) ) ) ) ~~> ( M + 1 ) ) $= + ( vm vk wcel cmul cn c1 cv cdiv co caddc cmpt cli cvv nnuz nnex mptex a1i + adantl cn0 cseq faclimlem1 1zzd 1cnd nn0p1nn nnzd cfv wceq weq oveq1 eqid + oveq12d ovex fvmpt divcnvlin nncnd cc wa peano2nn nnred nnaddcld nndivred + simpr adantr recnd fmpttd oveq2d eqtr4d climmulc2 mulid1d breqtrd eqbrtrd + ffvelcdmda ) BUAEZFAGHBAIZJKLKHHVPJKLKFKHBHLKZVPJKLKJKMHUBCGVQCIZHLKZVRVQ + LKZJKZFKZMZVQNCABUCVOWCVQHFKVQNVOHVQDCGWAMZWCHOGPVOUDZVOHVQDWDHOGPWEVOUEV + OVQBUFZUGWDOEVOCGWAQRSDIZGEZWGWDUHZWGHLKZWGVQLKZJKZUIVOCWGWAWLGWDCDUJZVSW + JVTWKJVRWGHLUKVRWGVQLUKUMZWDULWJWKJUNUOZTUPVOVQWFUQZWCOEVOCGWBQRSVOGURWGW + DVOCGWAURVOVRGEZUSZWAWRVSVTWRVSWQVSGEVOVRUTTVAWRVRVQVOWQVDVOVQGEWQWFVEVBV + CVFVGVNWHWGWCUHZVQWIFKZUIVOWHWSVQWLFKZWTCWGWBXAGWCWMWAWLVQFWNVHWCULVQWLFU + NUOWHWIWLVQFWOVHVITVJVOVQWPVKVLVM $. + $} - $( Extend a surreal by a sequence of ordinals. (Contributed by Scott - Fenton, 30-Nov-2021.) $) - noextendseq $p |- ( ( A e. No /\ B e. On ) -> - ( A u. ( ( B \ dom A ) X. { X } ) ) e. No ) $= - ( csur wcel con0 wa cdm cun wfun crn wss mp2b cin c0 wceq eqtri adantr wi - cdif csn cxp c1o c2o cpr nofun wfn fnconstg fnfun wne dmxp ineq2i disjdif - snnzg funun mpan2 sylancl uneq2i nodmon undif eleq1a adantl syl5bi ssdif0 - dmun uneq2 un0 eqtrdi eleq1d biimprcd word eloni ordtri2or2 syl2an mpjaod - sylan eqeltrid rnun norn rnxpss snssi ax-mp sstri sylanblc eqsstrid elno2 - wo unss syl3anbrc ) AEFZBGFZHZABAIZUAZCUBZUCZJZKZWRIZGFWRLZUDUEUFZMWREFWK - WSWLWKAKZWQKZWSAUGCXBFZWQWOUHXDDWOCXBUIWOWQUJNXCXDHWNWQIZOZPQWSXGWNWOOPXF - WOWNXEWPPUKXFWOQDCXBUOWOWPULNZUMWNBUNRAWQUPUQURSWMWTWNWOJZGWTWNXFJXIAWQVF - XFWOWNXHUSRWKWNGFZWLXIGFZAUTXJWLHZWNBMZXKBWNMZXMXIBQZXLXKWNBVAWLXOXKTXJBG - XIVBVCVDXNWOPQZXLXKBWNVEXJXPXKTWLXPXKXJXPXIWNGXPXIWNPJWNWOPWNVGWNVHVIVJVK - SVDXJWNVLBVLXMXNWHWLWNVMBVMWNBVNVOVPVQVRWMXAALZWQLZJZXBAWQVSWMXQXBMZXRXBM - XSXBMWKXTWLAVTSXRWPXBWOWPWAXEWPXBMDCXBWBWCWDXQXRXBWIWEWFWRWGWJ $. + $( Lemma for ~ faclim . Algebraic manipulation for the final induction. + (Contributed by Scott Fenton, 15-Dec-2017.) $) + faclimlem3 $p |- ( ( M e. NN0 /\ B e. NN ) -> + ( ( ( 1 + ( 1 / B ) ) ^ ( M + 1 ) ) / ( 1 + ( ( M + 1 ) / B ) ) ) = + ( ( ( ( 1 + ( 1 / B ) ) ^ M ) / ( 1 + ( M / B ) ) ) x. + ( ( ( 1 + ( M / B ) ) x. ( 1 + ( 1 / B ) ) ) / ( 1 + ( ( M + 1 ) / B ) ) ) + ) ) $= + ( cn0 wcel cn c1 cdiv co caddc cexp cmul crp 1rp a1i adantl rpaddcld rpne0d + rpcnd oveq1d rpdivcld wa nnrp rpreccld simpl expp1d cz nn0z rpexpcl syl2anr + 1cnd nn0nndivcl addcomd nn0ge0div ge0p1rpd eqeltrd divcan1d mulassd 3eqtr2d + recnd rpmulcld nn0p1nn nnrpd adantr divassd eqtrd ) BCDZAEDZUAZFFAGHZIHZBFI + HZJHZFVKAGHZIHZGHVJBJHZFBAGHZIHZGHZVQVJKHZKHZVNGHVRVSVNGHKHVHVLVTVNGVHVLVOV + JKHVRVQKHZVJKHVTVHVJBVHVJVHFVIFLDZVHMNZVGVILDVFVGAAUBZUCZOPZRZVFVGUDUEVHWAV + OVJKVHVOVQVHVOVGVJLDBUFDVOLDVFVGFVIWBVGMNWEPBUGVJBUHUIZRVHVQVHVQVPFIHLVHFVP + VHUJVHVPBAUKZUSULVHVPWIBAUMUNUOZRZVHVQWJQUPSVHVRVQVJVHVRVHVOVQWHWJTRZWKWGUQ + URSVHVRVSVNWLVHVSVHVQVJWJWFUTRVHVNVHFVMWCVHVKAVFVKLDVGVFVKBVAVBVCVGALDVFWDO + TPZRVHVNWMQVDVE $. - $d A x y $. $d X x y $. - $( Calculate the place where a surreal and its extension differ. - (Contributed by Scott Fenton, 22-Nov-2021.) $) - noextenddif $p |- ( A e. No -> - |^| { x e. On | ( A ` x ) =/= ( ( A u. { <. dom A , X >. } ) ` x ) } = - dom A ) $= - ( vy wcel cv cfv wne con0 wss c0 wn wceq syl sylibr fveq2 wa wo 3ad2ant1 - csur cdm cop csn cun crab cint nodmon nosgnn0i a1i word nodmord ndmfv wfn - ordirr cin nofun funfn sylib c1o c2o cpr fnsng sylancl disjsn snidg fvun2 - wfun syl112anc fvsng eqtrd 3netr4d neeq12d onintss sylc wi wb eloni eqcom - ordtri2 orbi1i orcom notbii bitrdi ordsseleq notbid ancoms bitr4d syl2anr - bitri w3a simp3 fvun1 eqcomd 3expia sylbird necon1ad ralrimiva weq ralrab - wral ssint eqssd ) BUAFZAGZBHZXEBBUBZCUCUDZUEZHZIZAJUFZUGZXGXDXGJFZXGBHZX - GXIHZIZXMXGKBUHZXDLCXOXPLCIXDCDUIUJXDXGXGFMZXOLNXDXGUKZXSBULZXGUOOZXGBUMO - XDXPXGXHHZCXDBXGUNZXHXGUDZUNZXGYEUPLNZXGYEFZXPYCNXDBVHYDBUQBURUSZXDXNCUTV - AVBZFZYFXRDXGCJYJVCVDZXDXSYGYBXGXGVEPZXDXNYHXRXGJVFOXGYEBXHXGVGVIXDXNYKYC - CNXRDXGCJYJVJVDVKVLXKXQAXGXEXGNXFXOXJXPXEXGBQXEXGXIQVMVNVOXDXGEGZKZEXLXAZ - XGXMKXDYNBHZYNXIHZIZYOVPZEJXAYPXDYTEJXDYNJFZRZYOYQYRUUBYOMZYNXGFZYQYRNZUU - AYNUKZXTUUDUUCVQXDYNVRYAUUFXTRZUUDXGYNFZXGYNNZSZMZUUCUUGUUDYNXGNZUUHSZMUU - KYNXGVTUUMUUJUUMUUIUUHSUUJUULUUIUUHYNXGVSWAUUIUUHWBWJWCWDXTUUFUUCUUKVQXTU - UFRYOUUJXGYNWEWFWGWHWIXDUUAUUDUUEXDUUAUUDWKZYRYQUUNYDYFYGUUDYRYQNXDUUAYDU - UDYITXDUUAYFUUDYLTXDUUAYGUUDYMTXDUUAUUDWLXGYEBXHYNWMVIWNWOWPWQWRXKYSYOEAJ - AEWSXFYQXJYRXEYNBQXEYNXIQVMWTPEXGXLXBPXC $. + ${ + $d A a b m n x $. + faclim.1 $e |- F = + ( n e. NN |-> ( ( ( 1 + ( 1 / n ) ) ^ A ) / ( 1 + ( A / n ) ) ) ) $. + $( An infinite product expression relating to factorials. Originally due + to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) $) + faclim $p |- ( A e. NN0 -> seq 1 ( x. , F ) ~~> ( ! ` A ) ) $= + ( wcel cmul c1 cseq cn cdiv co caddc cexp cfa cfv cli wceq cc0 oveq12d cc + va vm vb vx cn0 cmpt seqeq3 ax-mp wbr oveq2 oveq1 oveq2d mpteq2dv seqeq3d + cv fveq2 fac0 eqtrdi breq12d weq csn cxp 1red nnrecre readdcld recnd nncn + exp0d nnne0 div0d 1p0e1 1div1e1 fconstmpt eqtr4i wtru nnuz 1zzd climprod1 + mpteq2ia mptru eqbrtri wa cvv simpr seqex faclimlem2 adantr elnnuz biimpi + a1i cuz adantl cfz wf crp nnrp rpreccld rpaddcld nn0z rpexpcld nn0nndivcl + cz 1cnd addcomd nn0ge0div ge0p1rpd eqeltrd rpdivcld rpcnd fmpttd ffvelcdm + 1rp elfznn syl2an adantlr mulcl seqcl rpmulcld nn0p1nn rpdivcl faclimlem3 + nnrpd oveq1d eqid fvmpt 3eqtr4d sylan2 prodfmul climmul facp1 breqtrrd ex + ovex nn0ind eqbrtrid ) AUEEFCGHZFBIGGBUOZJKZLKZAMKZGAYQJKZLKZJKZUFZGHZANO + ZPCUUDQYPUUEQDFCUUDGUGUHFBIYSUAUOZMKZGUUGYQJKZLKZJKZUFZGHZUUGNOZPUIFBIYSR + MKZGRYQJKZLKZJKZUFZGHZGPUIFBIYSUBUOZMKZGUVAYQJKZLKZJKZUFZGHZUVANOZPUIZFBI + YSUVAGLKZMKZGUVJYQJKZLKZJKZUFZGHZUVJNOZPUIZUUEUUFPUIUAUBAUUGRQZUUMUUTUUNG + PUVSUULUUSFGUVSBIUUKUURUVSUUHUUOUUJUUQJUUGRYSMUJUVSUUIUUPGLUUGRYQJUKULSUM + UNUVSUUNRNOGUUGRNUPUQURUSUAUBUTZUUMUVGUUNUVHPUVTUULUVFFGUVTBIUUKUVEUVTUUH + UVBUUJUVDJUUGUVAYSMUJUVTUUIUVCGLUUGUVAYQJUKULSUMUNUUGUVANUPUSUUGUVJQZUUMU + VPUUNUVQPUWAUULUVOFGUWABIUUKUVNUWAUUHUVKUUJUVMJUUGUVJYSMUJUWAUUIUVLGLUUGU + VJYQJUKULSUMUNUUGUVJNUPUSUUGAQZUUMUUEUUNUUFPUWBUULUUDFGUWBBIUUKUUCUWBUUHY + TUUJUUBJUUGAYSMUJUWBUUIUUAGLUUGAYQJUKULSUMUNUUGANUPUSUUTFIGVAVBZGHZGPUUSU + WCQUUTUWDQUUSBIGUFUWCBIUURGYQIEZUURGGJKGUWEUUOGUUQGJUWEYSUWEYSUWEGYRUWEVC + YQVDVEVFVHUWEUUQGRLKGUWEUUPRGLUWEYQYQVGYQVIVJULVKURSVLURVSBIGVMVNFUUSUWCG + UGUHUWDGPUIVOGIVPVOVQVRVTWAUVAUEEZUVIUVRUWFUVIWBZUVPUVHUVJFKZUVQPUWGUVHUV + JUAUVGFBIUVDYSFKZUVMJKZUFZGHZUVPGWCIVPUWGVQUWFUVIWDUVPWCEUWGFUVOGWEWJUWFU + WLUVJPUIUVIBUVAWFWGUWFUUGIEZUUGUVGOZTEUVIUWFUWMWBZUCUDFTUVFGUUGUWMUUGGWKO + EZUWFUWMUWPUUGWHWIWLZUWFUCUOZGUUGWMKEZUWRUVFOZTEZUWMUWFITUVFWNUWRIEZUXAUW + SUWFBIUVETUWFUWEWBZUVEUXCUVBUVDUXCYSUVAUXCGYRGWOEUXCXLWJZUWEYRWOEUWFUWEYQ + YQWPZWQWLWRZUWFUVAXBEUWEUVAWSWGWTUXCUVDUVCGLKWOUXCGUVCUXCXCUXCUVCUVAYQXAZ + VFXDUXCUVCUXGUVAYQXEXFXGZXHXIXJUWRUUGXMZITUWRUVFXKXNXOZUWRTEUDUOZTEWBUWRU + XKFKTEUWOUWRUXKXPWLZXQXOUWFUWMUUGUWLOZTEUVIUWOUCUDFTUWKGUUGUWQUWFUWSUWRUW + KOZTEZUWMUWFITUWKWNUXBUXOUWSUWFBIUWJTUXCUWJUXCUWIUVMUXCUVDYSUXHUXFXRUXCGU + VLUXDUWFUVJWOEYQWOEUVLWOEUWEUWFUVJUVAXSYBUXEUVJYQXTXNWRXHXIXJUXIITUWRUWKX + KXNXOZUXLXQXOUWFUWMUUGUVPOUWNUXMFKQUVIUWOUCUVFUWKUVOGUUGUWQUXJUXPUWFUWSUW + RUVOOZUWTUXNFKZQZUWMUWSUWFUXBUXSUXIUWFUXBWBGGUWRJKZLKZUVJMKZGUVJUWRJKZLKZ + JKZUYAUVAMKZGUVAUWRJKZLKZJKZUYHUYAFKZUYDJKZFKZUXQUXRUWRUVAYAUXBUXQUYEQUWF + BUWRUVNUYEIUVOBUCUTZUVKUYBUVMUYDJUYMYSUYAUVJMUYMYRUXTGLYQUWRGJUJULZYCUYMU + VLUYCGLYQUWRUVJJUJULZSUVOYDUYBUYDJYMYEWLUXBUXRUYLQUWFUXBUWTUYIUXNUYKFBUWR + UVEUYIIUVFUYMUVBUYFUVDUYHJUYMYSUYAUVAMUYNYCUYMUVCUYGGLYQUWRUVAJUJULZSUVFY + DUYFUYHJYMYEBUWRUWJUYKIUWKUYMUWIUYJUVMUYDJUYMUVDUYHYSUYAFUYPUYNSUYOSUWKYD + UYJUYDJYMYESWLYFYGXOYHXOYIUWFUVQUWHQUVIUVAYJWGYKYLYNYO $. $} ${ - $d A x $. - $( Extending a surreal with a negative sign results in a smaller surreal. - (Contributed by Scott Fenton, 22-Nov-2021.) $) - noextendlt $p |- ( A e. No -> ( A u. { <. dom A , 1o >. } ) A . } ) ) $= - ( vx csur wcel c2o cop csn wbr cfv con0 c1o c0 wceq wa syl wfn 2on sylibr - sylancl fvex cdm cun cslt cv wne crab cint ctp w3o wn word nodmord ordirr - ndmfv cin wfun nofun funfn sylib nodmon fnsng snidg fvun2 syl112anc fvsng - disjsn eqtrd jca 3mix3d brtp elexi noextenddif fveq2d 3brtr4d wb noextend - prid2 sltval2 mpdan mpbird ) ACDZAAAUAZEFGZUBZUCHZBUDZAIWFWDIUEBJUFUGZAIZ - WGWDIZKLFKEFLEFUHZHZWAWBAIZWBWDIZWHWIWJWAWLKMZWMLMNZWNWMEMZNZWLLMZWPNZUIW - LWMWJHWAWSWOWQWAWRWPWAWBWBDUJZWRWAWBUKWTAULWBUMOZWBAUNOWAWMWBWCIZEWAAWBPZ - WCWBGZPZWBXDUOLMZWBXDDZWMXBMWAAUPXCAUQAURUSWAWBJDZEJDZXEAUTZQWBEJJVASWAWT - XFXAWBWBVFRWAXHXGXJWBJVBOWBXDAWCWBVCVDWAXHXIXBEMXJQWBEJJVESVGVHVIKLKELEWL - WMWBATWBWDTVJRWAWGWBABAEKEEJQVKVQZVLZVMWAWGWBWDXLVMVNWAWDCDWEWKVOAEXKVPAW - DBVRVSVT $. + $d A k x $. + $( An infinite product expression for factorial. (Contributed by Scott + Fenton, 15-Dec-2017.) $) + iprodfac $p |- ( A e. NN0 -> ( ! ` A ) = + prod_ k e. NN ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) ) $= + ( vx cn0 wcel cn c1 cv cdiv co caddc cexp cprod cfa cfv cmpt oveq2 oveq2d + nnuz crp 1zzd facne0 eqid faclim wceq oveq1d oveq12d ovex fvmpt adantl wa + weq 1rp a1i simpr nnrpd rpreccld rpaddcld nn0z adantr rpexpcld nn0nndivcl + cz recnd addcomd nn0ge0div ge0p1rpd eqeltrd rpdivcld rpcnd iprodn0 eqcomd + 1cnd ) ADEZFGGBHZIJZKJZALJZGAVOIJZKJZIJZBMANOZVNWABCFGGCHZIJZKJZALJZGAWCI + JZKJZIJZPZGWBFSVNUAAUBACWJWJUCZUDVOFEZVOWJOWAUEVNCVOWIWAFWJCBULZWFVRWHVTI + WMWEVQALWMWDVPGKWCVOGIQRUFWMWGVSGKWCVOAIQRUGWKVRVTIUHUIUJVNWLUKZWAWNVRVTW + NVQAWNGVPGTEWNUMUNWNVOWNVOVNWLUOUPUQURVNAVCEWLAUSUTVAWNVTVSGKJTWNGVSWNVMW + NVSAVOVBZVDVEWNVSWOAVOVFVGVHVIVJVKVL $. $} ${ - $d A x y $. $d B x y $. $d X x y $. - $( Given ` A ` less than or equal to ` B ` , equal to ` B ` up to ` X ` , - and ` A ( X ) = 2o ` , then ` B ( X ) = 2o ` . (Contributed by Scott - Fenton, 6-Dec-2021.) $) - nolesgn2o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ - ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B - ( B ` X ) = 2o ) $= - ( vy vx csur wcel con0 w3a cres wceq cfv c2o wa wbr wn c0 c1o wo cop cslt - w3o simpl2 nofv syl 3orel3 syl5com cv wral ctp simp13 fveq1 eqcomd adantr - simpr fvresd 3eqtr3d ralrimiva 3ad2ant2 simprr ancld orim12d 3impia 3mix3 - wrex a1d 3mix2 jaoi fvex brtp sylibr raleq fveq2 breq12d anbi12d syl12anc - rspcev wb simp12 simp11 sltval syl2anc mpbird 3expia syld con1d ) AFGZBFG - ZCHGZIZACJZBCJZKZCALZMKZNZBAUAOZPCBLZMKZWJWPNZWSWQWTWSPZWRQKZWRRKZSZWQWTX - BXCWSUBZXAXDWTWHXEWGWHWIWPUCBCUDUEXBXCWSUFUGWJWPXDWQWJWPXDIZWQDUHZBLZXGAL - ZKZDEUHZUIZXKBLZXKALZRQTRMTQMTUJZOZNZEHVEZXFWIXJDCUIZWRWNXOOZXRWGWHWIWPXD - UKWPWJXSXDWMXSWOWMXJDCWMXGCGZNZXGWLLZXGWKLZXHXIWMYCYDKYAWMYDYCXGWKWLULUMU - NYBXGCBWMYAUOZUPYBXGCAYEUPUQURUNUSXFXCWNQKNZXCWONZXBWONZUBZXTXFYHYGSZYIWJ - WPXDYJWTXBYHXCYGWTXBWOWTWOXBWJWMWOUTZVFVAWTXCWOWTWOXCYKVFVAVBVCYHYIYGYHYF - YGVDYGYFYHVGVHUERQRMQMWRWNCBVICAVIVJVKXQXSXTNECHXKCKZXLXSXPXTXJDXKCVLYLXM - WRXNWNXOXKCBVMXKCAVMVNVOVQVPXFWHWGWQXRVRWGWHWIWPXDVSWGWHWIWPXDVTEDBAWAWBW - CWDWEWFVC $. - - $( Given ` A ` less than or equal to ` B ` , equal to ` B ` up to ` X ` , - and ` A ( X ) = 2o ` , then ` ( A |`` suc X ) = ( B |`` suc X ) ` . - (Contributed by Scott Fenton, 6-Dec-2021.) $) - nolesgn2ores $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ - ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B - ( A |` suc X ) = ( B |` suc X ) ) $= - ( vx csur wcel con0 w3a cres wceq cfv c2o wa wn cdm syl wne fvresd wfun - c0 cslt wbr csuc cv wral cin dmres wss simp11 nodmord ndmfv c1o 2on elexi - word prid2 nosgnn0i neeq1 mpbiri neneqd con4i adantl 3ad2ant2 df-ss sylib - ordsucss eqtrid simp12 nolesgn2o eqtr4d eleq2d wo vex elsuc simp2l fveq1d - sylc adantr simpr 3eqtr3d simp2r fveq2 eqeq12d syl5ibrcom jaod syl5bi imp - ex 3eqtr4d sylbid ralrimiv wb nofun funres 3syl eqfunfv syl2anc mpbir2and - ) AEFZBEFZCGFZHZACIZBCIZJZCAKZLJZMZBAUAUBNZHZACUCZIZBXKIZJZXLOZXMOZJZDUDZ - XLKZXRXMKZJZDXOUEZXJXOXKXPXJXOXKAOZUFZXKAXKUGXJXKYCUHZYDXKJXJYCUOZCYCFZYE - XJWSYFWSWTXAXHXIUIZAUJPXHXBYGXIXGYGXEYGXGYGNXFTJZXGNCAUKYIXFLYIXFLQTLQZLU - LLLGUMUNUPUQZXFTLURUSUTPVAVBVCCYCVFVQXKYCVDVEVGZXJXPXKBOZUFZXKBXKUGXJXKYM - UHZYNXKJXJYMUOZCYMFZYOXJWTYPWSWTXAXHXIVHZBUJPXJCBKZLJZYQABCVIZYQYTYQNYSTJ - ZYTNCBUKUUBYSLUUBYSLQYJYKYSTLURUSUTPVAPCYMVFVQXKYMVDVEVGVJXJYADXOXJXRXOFX - RXKFZYAXJXOXKXRYLVKXJUUCYAXJUUCMZXRAKZXRBKZXSXTXJUUCUUEUUFJZUUCXRCFZXRCJZ - VLXJUUGXRCDVMVNXJUUHUUGUUIXJUUHUUGXJUUHMZXRXCKZXRXDKZUUEUUFXJUUKUULJUUHXJ - XRXCXDXBXEXGXIVOVPVRUUJXRCAXJUUHVSZRUUJXRCBUUMRVTWHXJUUGUUIXFYSJXJXFLYSXB - XEXGXIWAUUAVJUUIUUEXFUUFYSXRCAWBXRCBWBWCWDWEWFWGUUDXRXKAXJUUCVSZRUUDXRXKB - UUNRWIWHWJWKXJXLSZXMSZXNXQYBMWLXJWSASUUOYHAWMXKAWNWOXJWTBSUUPYRBWMXKBWNWO - DXLXMWPWQWR $. - - $( Given ` A ` greater than or equal to ` B ` , equal to ` B ` up to - ` X ` , and ` A ( X ) = 1o ` , then ` B ( X ) = 1o ` . (Contributed by - Scott Fenton, 9-Aug-2024.) $) - nogesgn1o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ - ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A - ( B ` X ) = 1o ) $= - ( vy vx csur wcel con0 w3a cres wceq cfv c1o wa wbr wn c0 c2o wo cop cslt - w3o simpl2 nofv syl 3orel2 syl5com cv wral wrex simp13 fveq1 adantr simpr - ctp fvresd 3eqtr3d ralrimiva 3ad2ant2 simp2r simp3 andi sylib 3mix1 3mix2 - jca jaoi fvex brtp sylibr raleq breq12d anbi12d rspcev syl12anc wb simp11 - fveq2 simp12 sltval syl2anc mpbird 3expia syld con1d 3impia ) AFGZBFGZCHG - ZIZACJZBCJZKZCALZMKZNZABUAOZPCBLZMKZWJWPNZWSWQWTWSPZWRQKZWRRKZSZWQWTXBWSX - CUBZXAXDWTWHXEWGWHWIWPUCBCUDUEXBWSXCUFUGWJWPXDWQWJWPXDIZWQDUHZALZXGBLZKZD - EUHZUIZXKALZXKBLZMQTMRTQRTUOZOZNZEHUJZXFWIXJDCUIZWNWRXOOZXRWGWHWIWPXDUKWP - WJXSXDWMXSWOWMXJDCWMXGCGZNZXGWKLZXGWLLZXHXIWMYCYDKYAXGWKWLULUMYBXGCAWMYAU - NZUPYBXGCBYEUPUQURUMUSXFWOXBNZWOXCNZWNQKXCNZUBZXTXFYFYGSZYIXFWOXDNYJXFWOX - DWJWMWOXDUTWJWPXDVAVFWOXBXCVBVCYFYIYGYFYGYHVDYGYFYHVEVGUEMQMRQRWNWRCAVHCB - VHVIVJXQXSXTNECHXKCKZXLXSXPXTXJDXKCVKYKXMWNXNWRXOXKCAVRXKCBVRVLVMVNVOXFWG - WHWQXRVPWGWHWIWPXDVQWGWHWIWPXDVSEDABVTWAWBWCWDWEWF $. - - $( Given ` A ` greater than or equal to ` B ` , equal to ` B ` up to - ` X ` , and ` A ( X ) = 1o ` , then - ` ( A |`` suc X ) = ( B |`` suc X ) ` . (Contributed by Scott Fenton, - 6-Dec-2021.) $) - nogesgn1ores $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ - ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 1o ) /\ -. A - ( A |` suc X ) = ( B |` suc X ) ) $= - ( vx csur wcel w3a cres wceq cfv c1o wa wn cdm cin syl c0 wne fvresd wfun - con0 cslt wbr csuc cv wral dmres wss word simp11 nodmord ndmfv 1n0 necomi - neeq1 mpbiri neneqd con4i adantl 3ad2ant2 ordsucss df-ss eqtrid nogesgn1o - sylc sylib simp12 eqtr4d eleq2d wo vex elsuc simpl2l fveq1d simpr 3eqtr3d - ex simp2r fveq2 eqeq12d syl5ibrcom jaod syl5bi 3eqtr4d sylbid ralrimiv wb - imp nofun funresd eqfunfv syl2anc mpbir2and ) AEFZBEFZCUAFZGZACHZBCHZIZCA - JZKIZLZABUBUCMZGZACUDZHZBXFHZIZXGNZXHNZIZDUEZXGJZXMXHJZIZDXJUFZXEXJXFXKXE - XJXFANZOZXFAXFUGXEXFXRUHZXSXFIXEXRUIZCXRFZXTXEWNYAWNWOWPXCXDUJZAUKPXCWQYB - XDXBYBWTYBXBYBMXAQIZXBMCAULYDXAKYDXAKRQKRZKQUMUNZXAQKUOUPUQPURUSUTCXRVAVE - XFXRVBVFVCZXEXKXFBNZOZXFBXFUGXEXFYHUHZYIXFIXEYHUIZCYHFZYJXEWOYKWNWOWPXCXD - VGZBUKPXECBJZKIZYLABCVDZYLYOYLMYNQIZYOMCBULYQYNKYQYNKRYEYFYNQKUOUPUQPURPC - YHVAVEXFYHVBVFVCVHXEXPDXJXEXMXJFXMXFFZXPXEXJXFXMYGVIXEYRXPXEYRLZXMAJZXMBJ - ZXNXOXEYRYTUUAIZYRXMCFZXMCIZVJXEUUBXMCDVKVLXEUUCUUBUUDXEUUCUUBXEUUCLZXMWR - JXMWSJYTUUAUUEXMWRWSWTXBWQXDUUCVMVNUUEXMCAXEUUCVOZSUUEXMCBUUFSVPVQXEUUBUU - DXAYNIXEXAKYNWQWTXBXDVRYPVHUUDYTXAUUAYNXMCAVSXMCBVSVTWAWBWCWHYSXMXFAXEYRV - OZSYSXMXFBUUGSWDVQWEWFXEXGTXHTXIXLXQLWGXEXFAXEWNATYCAWIPWJXEXFBXEWOBTYMBW - IPWJDXGXHWKWLWM $. + $d a m $. $d a n $. $d k m $. $d k n $. $d M a $. $d m n $. $d M n $. + faclim2.1 $e |- F = + ( n e. NN |-> + ( ( ( ! ` n ) x. ( ( n + 1 ) ^ M ) ) / ( ! ` ( n + M ) ) ) ) $. + $( Another factorial limit due to Euler. (Contributed by Scott Fenton, + 17-Dec-2017.) $) + faclim2 $p |- ( M e. NN0 -> F ~~> 1 ) $= + ( wcel cn cfa cfv c1 caddc co cexp cmul cdiv cli wceq oveq2 oveq12d nncnd + cc0 va vm vk cn0 cmpt wbr oveq2d fveq2d mpteq2dv breq1d weq wtru cvv nnuz + cv 1zzd nnex mptex a1i 1cnd fveq2 oveq1 oveq1d fvoveq1 eqid ovex peano2nn + fvmpt exp0d nnnn0 faccl mulid1d eqtrd addid1d nnne0d dividd 3eqtrd adantl + syl nncn climconst mptru wa simpr nn0p1nn nnzd divcnvlin adantr cc nnnn0d + nnexpcl sylan ancoms nnmulcld nnred nnnn0addcl nndivred fmpttd ffvelcdmda + recnd adantlr nnaddcld simpl expp1d mulassd eqtr4d nn0addcld facp1 nn0cnd + addassd 3eqtr3d divmuldivd 3eqtr4d climmul 1t1e1 breqtrdi nn0ind eqbrtrid + ex ) CUDEBAFAUOZGHZXTIJKZCLKZMKZXTCJKZGHZNKZUEZIODAFYAYBUAUOZLKZMKZXTYIJK + ZGHZNKZUEZIOUFAFYAYBTLKZMKZXTTJKZGHZNKZUEZIOUFZAFYAYBUBUOZLKZMKZXTUUCJKZG + HZNKZUEZIOUFZAFYAYBUUCIJKZLKZMKZXTUUKJKZGHZNKZUEZIOUFZYHIOUFUAUBCYITPZYOU + UAIOUUSAFYNYTUUSYKYQYMYSNUUSYJYPYAMYITYBLQUGUUSYLYRGYITXTJQUHRUIUJUAUBUKZ + YOUUIIOUUTAFYNUUHUUTYKUUEYMUUGNUUTYJUUDYAMYIUUCYBLQUGUUTYLUUFGYIUUCXTJQUH + RUIUJYIUUKPZYOUUQIOUVAAFYNUUPUVAYKUUMYMUUONUVAYJUULYAMYIUUKYBLQUGUVAYLUUN + GYIUUKXTJQUHRUIUJYICPZYOYHIOUVBAFYNYGUVBYKYDYMYFNUVBYJYCYAMYICYBLQUGUVBYL + YEGYICXTJQUHRUIUJUUBULIUBUUAIUMFUNULUPUUAUMEULAFYTUQURUSULUTUUCFEZUUCUUAH + ZIPULUVCUVDUUCGHZUUKTLKZMKZUUCTJKZGHZNKZUVEUVENKIAUUCYTUVJFUUAAUBUKZYQUVG + YSUVINUVKYAUVEYPUVFMXTUUCGVAUVKYBUUKTLXTUUCIJVBVCRXTUUCTGJVDRUUAVEUVGUVIN + VFVHUVCUVGUVEUVIUVENUVCUVGUVEIMKUVEUVCUVFIUVEMUVCUUKUVCUUKUUCVGSVIUGUVCUV + EUVCUVEUVCUUCUDEZUVEFEUUCVJUUCVKVSZSZVLVMUVCUVHUUCGUVCUUCUUCVTVNUHRUVCUVE + UVNUVCUVEUVMVOVPVQVRWAWBUVLUUJUURUVLUUJWCZUUQIIMKIOUVOIIUCUUIAFYBUUNNKZUE + ZUUQIUMFUNUVOUPUVLUUJWDUUQUMEUVOAFUUPUQURUSUVLUVQIOUFUUJUVLIUUKUCUVQIUMFU + NUVLUPUVLUTUVLUUKUUCWEZWFUVQUMEUVLAFUVPUQURUSUCUOZFEZUVSUVQHZUVSIJKZUVSUU + KJKZNKZPUVLAUVSUVPUWDFUVQAUCUKZYBUWBUUNUWCNXTUVSIJVBZXTUVSUUKJVBRUVQVEUWB + UWCNVFVHZVRWGWHUVLUVTUVSUUIHZWIEUUJUVLFWIUVSUUIUVLAFUUHWIUVLXTFEZWCZUUHUW + JUUEUUGUWJUUEUWJYAUUDUWJXTUDEYAFEUWJXTUVLUWIWDZWJXTVKVSUWIUVLUUDFEZUWIYBF + EZUVLUWLXTVGZYBUUCWKWLWMWNWOUWJUUFUDEUUGFEUWJUUFUWIUVLUUFFEXTUUCWPWMWJUUF + VKVSWQWTWRWSXAUVLUVTUWAWIEUUJUVLFWIUVSUVQUVLAFUVPWIUWJUVPUWJYBUUNUWJYBUWI + UWMUVLUWNVRWOUWJXTUUKUWKUVLUUKFEZUWIUVRWHXBWQWTWRWSXAUVLUVTUVSUUQHZUWHUWA + MKZPUUJUVLUVTWCZUVSGHZUWBUUKLKZMKZUWCGHZNKZUWSUWBUUCLKZMKZUVSUUCJKZGHZNKZ + UWDMKZUWPUWQUWRUXCUXEUWBMKZUXGUWCMKZNKUXIUWRUXAUXJUXBUXKNUWRUXAUWSUXDUWBM + KZMKUXJUWRUWTUXLUWSMUWRUWBUUCUWRUWBUVTUWBFEZUVLUVSVGZVRSZUVLUVTXCZXDUGUWR + UWSUXDUWBUWRUWSUWRUVSUDEUWSFEUWRUVSUVLUVTWDZWJZUVSVKVSZSUWRUXDUVTUVLUXDFE + ZUVTUXMUVLUXTUXNUWBUUCWKWLWMZSUXOXEXFUWRUXFIJKZGHZUXGUYBMKZUXBUXKUWRUXFUD + EZUYCUYDPUWRUVSUUCUXRUXPXGZUXFXHVSUWRUYBUWCGUWRUVSUUCIUWRUVSUXQSUWRUUCUXP + XIUWRUTXJZUHUWRUYBUWCUXGMUYGUGXKRUWRUXEUXGUWBUWCUWRUXEUWRUWSUXDUXSUYAWNSU + WRUXGUWRUYEUXGFEUYFUXFVKVSZSUXOUWRUWCUWRUVSUUKUXQUVLUWOUVTUVRWHXBZSUWRUXG + UYHVOUWRUWCUYIVOXLXFUVTUWPUXCPUVLAUVSUUPUXCFUUQUWEUUMUXAUUOUXBNUWEYAUWSUU + LUWTMXTUVSGVAZUWEYBUWBUUKLUWFVCRXTUVSUUKGJVDRUUQVEUXAUXBNVFVHVRUVTUWQUXIP + UVLUVTUWHUXHUWAUWDMAUVSUUHUXHFUUIUWEUUEUXEUUGUXGNUWEYAUWSUUDUXDMUYJUWEYBU + WBUUCLUWFVCRXTUVSUUCGJVDRUUIVEUXEUXGNVFVHUWGRVRXMXAXNXOXPXSXQXR $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal Numbers: Ordering + Greatest common divisor and divisibility =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - ${ - $d x y z $. - $( Lemma for ~ sltso . The sign expansion relationship totally orders the - surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.) $) - sltsolem1 $p |- { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } - Or ( { 1o , 2o } u. { (/) } ) $= - ( vx vy vz c1o c2o c0 cop cv wbr wcel wceq wa w3o eqtr2 brtp expcom 3jaod - mto ex ad2ant2lr ctp wor cpr csn cun wn 1n0 neii con0 wb 0elon csuc df-2o - 1on df-1o eqeq12i suc11 bitrid nemtbir ancoms nsuceq0 eqeq1i 3pm3.2ni vex - mp2an mtbir a1i w3a pm2.21i ad2ant2rl 3mix2 3jaoi imp syl2anb sylibr eltp - wi eqtr3 3mix2d 3mix1d 3mix1 3mix3 biid 3orbi123i issoi df-tp soeq2 ax-mp - 3mix3d mpbi ) DEFUAZDFGDEGFEGUAZUBZDEUCFUDUEZWLUBZABCWKWLAHZWPWLIZUFWPWKJ - ZWQWPDKZWPFKZLZWSWPEKZLZWTXBLZMXAXCXDXADFKZDFUGUHZWPDFNRXCEDKZXGDFUGDUIJZ - FUIJZXGXEUJUNUKXGDULZFULZKXHXILXEEXJDXKUMUOUPDFUQURVEUSZXBWSXGWPEDNUTRXDE - FKZXMXJFDVAEXJFUMVBUSZXBWTXMWPEFNUTRVCDFDEFEWPWPAVDZXOOVFVGWPBHZWLIZXPCHZ - WLIZLZWPXRWLIZVQWRXPWKJZXRWKJVHXTWSXRFKZLZWSXREKZLZWTYELZMZYAXQWSXPFKZLZW - SXPEKZLZWTYKLZMZXPDKZYCLZYOYELZYIYELZMZYHXSDFDEFEWPXPXOBVDZOZDFDEFEXPXRYT - CVDZOYNYSYHYJYSYHVQYLYMYJYPYHYQYRYPYJYHYOYIYHYCWSYOYILZYHUUCXEXFXPDFNRVIZ - VJPYQYJYHYOYIYHYEWSUUDVJPYJYRYHWSYEYHYIYIYFYDYGVKVJSQYLYPYHYQYRYLYPYHYKYO - YHWSYCYKYOLZYHUUEXGXLXPEDNRVIZTSYLYQYHYKYOYHWSYEUUFTSYLYRYHYKYIYHWSYEYKYI - LZYHUUGXMXNXPEFNRVIZTSQYMYPYHYQYRYMYPYHYKYOYHWTYCUUFTSYMYQYHYKYOYHWTYEUUF - TSYMYRYHYKYIYHWTYEUUHTSQVLVMVNDFDEFEWPXRXOUUBOVOVGWRYBLYNWPXPKZYOWTLZYOXB - LZYIXBLZMZMZXQUUIXPWPWLIZMWRWSXBWTMZYOYKYIMZUUNYBWPDEFXOVPXPDEFYTVPUUPUUQ - UUNWSUUQUUNVQXBWTWSYOUUNYKYIWSYOUUNWSYOLUUIYNUUMWPXPDVRVSSWSYKUUNYLYNUUIU - UMYLYJYMVKVTSWSYIUUNYJYNUUIUUMYJYLYMWAVTSQXBYOUUNYKYIYOXBUUNUUKUUMYNUUIUU - KUUJUULVKWIPXBYKUUNXBYKLUUIYNUUMWPXPEVRVSSYIXBUUNUULUUMYNUUIUULUUJUUKWBWI - PQWTYOUUNYKYIYOWTUUNUUJUUMYNUUIUUJUUKUULWAWIPWTYKUUNYMYNUUIUUMYMYJYLWBVTS - WTYIUUNWTYILUUIYNUUMWPXPFVRVSSQVLVMVNXQYNUUIUUIUUOUUMUUAUUIWCDFDEFEXPWPYT - XOOWDVOWEWKWNKWMWOUJDEFWFWKWNWLWGWHWJ $. - $} + $( Swap the second and third arguments of a gcd. (Contributed by Scott + Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) $) + gcd32 $p |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> + ( ( A gcd B ) gcd C ) = ( ( A gcd C ) gcd B ) ) $= + ( cz wcel w3a cgcd co wceq gcdcom 3adant1 oveq2d gcdass 3com23 3eqtr4d ) AD + EZBDEZCDEZFZABCGHZGHACBGHZGHZABGHCGHACGHBGHZSTUAAGQRTUAIPBCJKLCBAMPRQUCUBIB + CAMNO $. - ${ - $d f g x y $. - $( Surreal less than totally orders the surreals. Axiom O of [Alling] - p. 184. (Contributed by Scott Fenton, 9-Jun-2011.) $) - sltso $p |- ( ( A gcd B ) gcd B ) = ( A gcd B + ) ) $= + ( cz wcel wa cgcd cabs cfv wceq gcdass 3anidm23 gcdid oveq2d adantl gcdabs2 + co 3eqtrd ) ACDZBCDZEABFPZBFPZABBFPZFPZABGHZFPZTRSUAUCIBBAJKSUCUEIRSUBUDAFB + LMNBAOQ $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal Numbers: Birthday Function + Properties of relationships =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ - $d x y $. - $( The birthday function maps the surreals onto the ordinals. Axiom B of - [Alling] p. 184. (Shortened proof on 2012-Apr-14, SF). (Contributed by - Scott Fenton, 11-Jun-2011.) $) - bdayfo $p |- bday : No -onto-> On $= - ( vx vy csur con0 cbday wfo wfn crn wceq cdm wcel wral dmexg rgen df-bday - cv cvv mptfng mpbi c1o wrex cab rnmpt csn cxp noxp1o 1oex snnz dmxp ax-mp - wne eqcomi dmeq rspceeqv sylancl nodmon eleq1a syl rexlimiv impbii abbi2i - c0 wi eqtr4i df-fo mpbir2an ) CDEFECGZEHZDIAPZJZQKZACLVGVKACVICMNACVJEAOZ - RSVHBPZVJIZACUAZBUBDABCVJEVLUCVOBDVMDKZVOVPVMTUDZUEZCKVMVRJZIVOVMUFVSVMVQ - VBUKVSVMITUGUHVMVQUIUJULAVRCVJVSVMVIVRUMUNUOVNVPACVICKVJDKVNVPVCVIUPVJDVM - UQURUSUTVAVDCDEVEVF $. + $d A x y $. + dftr6.1 $e |- A e. _V $. + $( A potential definition of transitivity for sets. (Contributed by Scott + Fenton, 18-Mar-2012.) $) + dftr6 $p |- ( Tr A <-> A e. ( _V \ ran ( ( _E o. _E ) \ _E ) ) ) $= + ( vx vy wel cv wcel wa wi wal cep cdif wn cvv wbr wex epeli anbi12i exbii + 3bitri ccom crn wtr elrn brdif vex brco epel notbii 19.41v exanali bitr3i + bitri exnal con2bii dftr2 eldif mpbiran 3bitr4i ) CDEZDFZAGZHZCFZAGZIDJZC + JZAKKUAZKLZUBZGZMZAUCANVJLGZVKVGVKVDAVIOZCPVFMZCPVGMCAVIBUDVNVOCVNVDAVHOZ + VDAKOZMZHVCDPZVEMZHZVOVDAVHKUEVPVSVRVTVPVDVAKOZVAAKOZHZDPVSDVDAKKCUFBUGWD + VCDWBUTWCVBDVDUHVAABQRSUMVQVEVDABQUIRWAVCVTHDPVOVCVTDUJVCVEDUKULTSVFCUNTU + OCDAUPVMANGVLBANVJUQURUS $. $} - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal Numbers: Density -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $( The value of a surreal at its birthday is ` (/) ` . (Contributed by Scott - Fenton, 14-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) $) - fvnobday $p |- ( A e. No -> ( A ` ( bday ` A ) ) = (/) ) $= - ( csur wcel cbday cfv wn c0 wceq bdayval word nodmord ordirr eqneltrd ndmfv - cdm syl ) ABCZADEZAOZCFRAEGHQRSSAIQSJSSCFAKSLPMRANP $. - ${ - $d A x $. $d B x $. - $( Lemma for ~ nosepne . (Contributed by Scott Fenton, 24-Nov-2021.) $) - nosepnelem $p |- ( ( A e. No /\ B e. No /\ A - ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= - ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) $= - ( csur wcel wbr cfv wne con0 wa c1o cop c2o wceq fvex simpl simpr neeq12d - c0 mpbiri cslt cv crab cint ctp sltval2 w3o brtp csuc df-2o df-1o eqeq12i - 1n0 wb 1on 0elon suc11 mp2an bitri necon3bii mpbir necomi 2on elexi prid2 - nosgnn0i 3jaoi sylbi syl6bi 3impia ) BDEZCDEZBCUAFZAUBZBGVNCGHAIUCUDZBGZV - OCGZHZVKVLJVMVPVQKSLKMLSMLUEFZVRBCAUFVSVPKNZVQSNZJZVTVQMNZJZVPSNZWCJZUGVR - KSKMSMVPVQVOBOVOCOUHWBVRWDWFWBVRKSHZUMWBVPKVQSVTWAPVTWAQRTWDVRKMHMKMKHWGU - MMKKSMKNKUIZSUIZNZKSNZMWHKWIUJUKULKIESIEWJWKUNUOUPKSUQURUSUTVAVBWDVPKVQMV - TWCPVTWCQRTWFVRSMHMKMMIVCVDVEVFWFVPSVQMWEWCPWEWCQRTVGVHVIVJ $. + $d A x $. $d B x $. $d R x $. + coep.1 $e |- A e. _V $. + coep.2 $e |- B e. _V $. + $( Composition with the membership relation. (Contributed by Scott Fenton, + 18-Feb-2013.) $) + coep $p |- ( A ( _E o. R ) B <-> E. x e. B A R x ) $= + ( cv wbr cep wex wcel ccom wrex epeli anbi1ci exbii brco df-rex 3bitr4i + wa ) BAGZDHZUACIHZTZAJUACKZUBTZAJBCIDLHUBACMUDUFAUCUEUBUACFNOPABCIDEFQUBA + CRS $. - $( The value of two non-equal surreals at the first place they differ is - different. (Contributed by Scott Fenton, 24-Nov-2021.) $) - nosepne $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> - ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) =/= - ( B ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) ) $= - ( csur wcel wne cv cfv con0 crab cint wa cslt wbr wo wor nosepnelem necom - 3expia fveq2i wb sltso sotrine mpan wi rabbii inteqi neeq12i bitri sylibr - w3a ancoms jaod sylbid 3impia ) BDEZCDEZBCFZAGZBHZUSCHZFZAIJZKZBHZVDCHZFZ - UPUQLZURBCMNZCBMNZOZVGDMPVHURVKUAUBDBCMUCUDVHVIVGVJUPUQVIVGABCQSUQUPVJVGU - EUQUPVJVGUQUPVJUKVAUTFZAIJZKZCHZVNBHZFZVGACBQVGVPVOFVQVEVPVFVOVDVNBVCVMVB - VLAIUTVARUFUGZTVDVNCVRTUHVPVORUIUJSULUMUNUO $. + $( Composition with the converse membership relation. (Contributed by + Scott Fenton, 18-Feb-2013.) $) + coepr $p |- ( A ( R o. `' _E ) B <-> E. x e. A x R B ) $= + ( cv cep ccnv wbr wa wex wcel ccom wrex vex brcnv epeli bitri anbi1i brco + exbii df-rex 3bitr4i ) BAGZHIZJZUECDJZKZALUEBMZUHKZALBCDUFNJUHABOUIUKAUGU + JUHUGUEBHJUJBUEHEAPQUEBERSTUBABCDUFEFUAUHABUCUD $. $} ${ - $d A x $. $d B x $. - $( If the value of a surreal at a separator is ` 1o ` then the surreal is - lesser. (Contributed by Scott Fenton, 7-Dec-2021.) $) - nosep1o $p |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ - ( A ` |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) = 1o ) -> - A + ( ~P A \ { (/) } ) C_ ran ( _E \ ( _E o. `' R ) ) ) $= + ( vx vz vy cv c0 cdif wcel cep wi wal wss wa wbr wrex anbi12i bitri vex + wn cpw csn ccnv ccom crn wne wral wfr eldif velpw necon3bbii wex wel epel + velsn brdif coep brcnv rexbii dfrex2 3bitrri con1bii exbii df-rex 3bitr4i + elrn imbi12i albii dfss2 df-fr 3bitr4ri ) CFZAUAZGUBZHZIZVLJJBUCZUDZHZUEZ + IZKZCLVLAMZVLGUFZNZDFZEFZBOZTDVLUGZEVLPZKZCLVOVTMABUHWBWKCVPWEWAWJVPVLVMI + ZVLVNIZTZNWEVLVMVNUIWLWCWNWDCAUJWMVLGCGUOUKQRWGVLVSOZEULECUMZWINZEULWAWJW + OWQEWOWGVLJOZWGVLVROZTZNWQWGVLJVRUPWRWPWTWICWGUNWIWSWSWGWFVQOZDVLPWHDVLPW + ITDWGVLVQESZCSZUQXAWHDVLWGWFBXBDSURUSWHDVLUTVAVBQRVCEVLVSXCVFWIEVLVDVEVGV + HCVOVTVICEDABVJVK $. $} ${ - $d A x $. $d B x $. - $( If the value of a surreal at a separator is ` 2o ` then the surreal is - greater. (Contributed by Scott Fenton, 7-Dec-2021.) $) - nosep2o $p |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ - ( A ` |^| { x e. On | ( B ` x ) =/= ( A ` x ) } ) = 2o ) -> - B + ( R Po A /\ ( A X. A ) C_ ( R u. ( _I u. `' R ) ) ) ) $= + ( vx vy wor wpo cv wbr wral wa cid cun wcel wi wal wo brun bitr2i 3bitr4i + vex weq w3o cxp ccnv wss df-so cop opelxp ideq brcnv orbi12i orbi2i df-br + 3orass imbi12i 2albii wrel wb relxp ssrel ax-mp r2al anbi2i bitr4i ) ABEA + BFZCGZDGZBHZCDUAZVGVFBHZUBZDAICAIZJVEAAUCZBKBUDZLZLZUEZJCDABUFVQVLVEVFVGU + GZVMMZVRVPMZNZDOCOZVFAMVGAMJZVKNZDOCOVQVLWAWDCDVSWCVTVKVFVGAAUHVKVFVGVPHZ + VTVHVIVJPZPVHVFVGVOHZPVKWEWFWGVHWGVFVGKHZVFVGVNHZPWFVFVGKVNQWHVIWIVJVFVGD + TZUIVFVGBCTWJUJUKRULVHVIVJUNVFVGBVOQSVFVGVPUMRUOUPVMUQVQWBURAAUSCDVMVPUTV + AVKCDAAVBSVCVD $. $} ${ - $d A x $. $d B x $. - $( Lemma for ~ nosepdm . (Contributed by Scott Fenton, 24-Nov-2021.) $) - nosepdmlem $p |- ( ( A e. No /\ B e. No /\ A - |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) ) $= - ( csur wcel wbr cfv wne con0 cdm wo wn wi wa c1o c0 cop c2o wceq syl cslt - w3a cv crab cint cun ctp sltval2 w3o fvex brtp df-3or ndmfv 1oex nosgnn0i - prid1 neeq1 mpbiri neneqd intnanrd ioran sylanbrc adantl orel1 syl5bi 2on - elexi prid2 con4i syl6 ex com23 sylbid 3impia orrd elun sylibr ) BDEZCDEZ - BCUAFZUBZAUCZBGWBCGHAIUDUEZBJZEZWCCJZEZKWCWDWFUFEWAWEWGVRVSVTWELZWGMZVRVS - NZVTWCBGZWCCGZOPQORQPRQUGFZWIBCAUHWMWKOSZWLPSZNZWNWLRSZNZWKPSZWQNZUIZWJWI - OPORPRWKWLWCBUJWCCUJUKWJWHXAWGWJWHXAWGMWJWHNZXAWTWGXAWPWRKZWTKZXBWTWPWRWT - ULXBXCLZXDWTMWHXEWJWHWSXEWCBUMWSWPLWRLXEWSWNWOWSWKOWSWKOHPOHOORUNUPUOWKPO - UQURUSZUTWSWNWQXFUTWPWRVAVBTVCXCWTVDTVEWQWGWSWGWQWGLZWLRXGWOWLRHZWCCUMWOX - HPRHRORRIVFVGVHUOWLPRUQURTUSVIVCVJVKVLVEVMVNVOWCWDWFVPVQ $. - - $( The first place two surreals differ is an element of the larger of their - domains. (Contributed by Scott Fenton, 24-Nov-2021.) $) - nosepdm $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> - |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. ( dom A u. dom B ) ) $= - ( csur wcel wne cv cfv con0 crab cint cdm cun wa wbr wo wor wb nosepdmlem - cslt sltso sotrine 3expa simplr simpll simpr syl3anc necom rabbii 3eltr4g - mpan inteqi uncom jaodan ex sylbid 3impia ) BDEZCDEZBCFZAGZBHZVACHZFZAIJZ - KZBLZCLZMZEZURUSNZUTBCTOZCBTOZPZVJDTQVKUTVNRUADBCTUBUKVKVNVJVKVLVJVMURUSV - LVJABCSUCVKVMNZVCVBFZAIJZKZVHVGMZVFVIVOUSURVMVRVSEURUSVMUDURUSVMUEVKVMUFA - CBSUGVEVQVDVPAIVBVCUHUIULVGVHUMUJUNUOUPUQ $. + $d b ch $. $d e et $. $d a b c d e f g h p q P $. $d a ps $. $d g si $. + $d a b c d e f g h p q x A $. $d p q x ph $. $d a b c d e f g h x rh $. + $d a b c d e f g h p q x B $. $d a b c d e f g h p q x C $. $d d ta $. + $d a b c d e f g h p q x D $. $d a b c d e f g h p q x E $. $d c th $. + $d a b c d e f g h p q x F $. $d a b c d e f g h p q x G $. $d f ze $. + $d a b c d e f g h p q x H $. $d a b c d e f g h p q x S $. + $d a b c d e f g h x Q $. $d a b c d e f g h x X $. + br8.1 $e |- ( a = A -> ( ph <-> ps ) ) $. + br8.2 $e |- ( b = B -> ( ps <-> ch ) ) $. + br8.3 $e |- ( c = C -> ( ch <-> th ) ) $. + br8.4 $e |- ( d = D -> ( th <-> ta ) ) $. + br8.5 $e |- ( e = E -> ( ta <-> et ) ) $. + br8.6 $e |- ( f = F -> ( et <-> ze ) ) $. + br8.7 $e |- ( g = G -> ( ze <-> si ) ) $. + br8.8 $e |- ( h = H -> ( si <-> rh ) ) $. + br8.9 $e |- ( x = X -> P = Q ) $. + br8.10 $e |- R = { <. p , q >. | + E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P E. e e. P + E. f e. P E. g e. P E. h e. P ( p = <. <. a , b >. , <. c , d >. >. /\ + q = <. <. e , f >. , <. g , h >. >. /\ ph ) } $. + $( Substitution for an eight-place predicate. (Contributed by Scott + Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) $) + br8 $p |- ( ( ( X e. S /\ A e. Q /\ B e. Q ) /\ + ( C e. Q /\ D e. Q /\ E e. Q ) /\ + ( F e. Q /\ G e. Q /\ H e. Q ) ) -> + ( <. <. A , B >. , <. C , D >. >. R <. <. E , F >. , <. G , H >. >. <-> + rh ) ) $= + ( cop wbr wceq wrex wcel opex eqeq1 3anbi1d rexbidv 2rexbidv 3anbi2d brab + cv w3a wa wi wb opth vex sylan9bb eqcoms biimp3a a1i rexlimdva rexlimdvva + sylbi simpl11 simpl12 simpl13 simpl21 simpl22 simpl23 simpl31 eqidd simpr + simpl32 simpl33 opeq2d eqeq2d 3anbi23d rspc2ev syl113anc 3anbi13d rspc3ev + opeq1 opeq1d syl31anc rexeqdv rexeqbidv rspcev syl2anc ex impbid bitrid + opeq2 ) KLVDZMNVDZVDZUCUDVDZUEUFVDZVDZQVEYAUJVPZUKVPZVDZULVPZUMVPZVDZVDZV + FZYDSVPZTVPZVDZUAVPZUBVPZVDZVDZVFZAVQZUBOVGZUAOVGZTOVGZSOVGZUMOVGZULOVGZU + KOVGZUJOVGZJRVGZUGRVHZKPVHZLPVHZVQZMPVHZNPVHZUCPVHZVQZUDPVHZUEPVHZUFPVHZV + QZVQZIUIVPZYKVFZUHVPZYSVFZAVQZUBOVGZUAOVGTOVGZSOVGUMOVGZULOVGUKOVGZUJOVGJ + RVGYLUVGAVQZUBOVGZUAOVGTOVGZSOVGUMOVGZULOVGUKOVGZUJOVGJRVGUUJUIUHYAYDQXSX + TVIYBYCVIUVDYAVFZUVLUVQJUJROUVRUVKUVPUKULOOUVRUVJUVOUMSOOUVRUVIUVNTUAOOUV + RUVHUVMUBOUVRUVEYLUVGAUVDYAYKVJVKVLVMVMVMVMUVFYDVFZUVQUUHJUJROUVSUVPUUFUK + ULOOUVSUVOUUDUMSOOUVSUVNUUBTUAOOUVSUVMUUAUBOUVSUVGYTYLAUVFYDYSVJVNVLVMVMV + MVMVCVOUVCUUJIUVCUUHIJUJROUVCJVPZRVHYEOVHVRVRZUUFIUKULOOUWAYFOVHYHOVHVRVR + ZUUDIUMSOOUWBYIOVHYMOVHVRVRZUUBITUAOOUWCYNOVHYPOVHVRVRZUUAIUBOUUAIVSUWDYQ + OVHVRYLYTAIYLAEYTIAEVTZYKYAYKYAVFYGXSVFZYJXTVFZVRUWEYGYJXSXTYEYFVIYHYIVIW + AUWFACUWGEUWFYEKVFZYFLVFZVRACVTYEYFKLUJWBUKWBWAUWHABUWICUNUOWCWIUWGYHMVFZ + YINVFZVRCEVTYHYIMNULWBUMWBWAUWJCDUWKEUPUQWCWIWCWIWDEIVTZYSYDYSYDVFYOYBVFZ + YRYCVFZVRUWLYOYRYBYCYMYNVIYPYQVIWAUWMEGUWNIUWMYMUCVFZYNUDVFZVREGVTYMYNUCU + DSWBTWBWAUWOEFUWPGURUSWCWIUWNYPUEVFZYQUFVFZVRGIVTYPYQUEUFUAWBUBWBWAUWQGHU + WRIUTVAWCWIWCWIWDWCWEWFWGWHWHWHWHUVCIUUJUVCIVRZUUKUUAUBPVGZUAPVGZTPVGZSPV + GZUMPVGZULPVGZUKPVGZUJPVGZUUJUUKUULUUMUURUVBIWJUWSUULUUMUUOYAXSMYIVDZVDZV + FZYTDVQZUBPVGZUAPVGZTPVGZSPVGUMPVGZUXGUUKUULUUMUURUVBIWKUUKUULUUMUURUVBIW + LUUOUUPUUQUUNUVBIWMUWSUUPUUQUUSYAYAVFZYDYBYRVDZVFZGVQZUBPVGUAPVGZUXOUUOUU + PUUQUUNUVBIWNUUOUUPUUQUUNUVBIWOUUSUUTUVAUUNUURIWPUWSUUTUVAUXPYDYDVFZIUXTU + USUUTUVAUUNUURIWSUUSUUTUVAUUNUURIWTUWSYAWQUWSYDWQUVCIWRUXSUXPUYAIVQUXPYDY + BUEYQVDZVDZVFZHVQUAUBUEUFPPUWQUXRUYDGHUXPUWQUXQUYCYDUWQYRUYBYBYPUEYQXHXAX + BUTXCUWRUYDUYAHIUXPUWRUYCYDYDUWRUYBYCYBYQUFUEXRXAXBVAXCXDXEUXMUXTUXPYTEVQ + ZUBPVGUAPVGUXPYDUCYNVDZYRVDZVFZFVQZUBPVGUAPVGUMSTNUCUDPPPUWKUXKUYEUAUBPPU + WKUXJUXPDEYTUWKUXIYAYAUWKUXHXTXSYINMXRXAXBUQXFVMUWOUYEUYIUAUBPPUWOYTUYHEF + UXPUWOYSUYGYDUWOYOUYFYRYMUCYNXHXIXBURXCVMUWPUYIUXSUAUBPPUWPUYHUXRFGUXPUWP + UYGUXQYDUWPUYFYBYRYNUDUCXRXIXBUSXCVMXGXJUXDUXOYAKYFVDZYJVDZVFZYTBVQZUBPVG + ZUAPVGTPVGZSPVGUMPVGYAXSYJVDZVFZYTCVQZUBPVGZUAPVGTPVGZSPVGUMPVGUJUKULKLMP + PPUWHUXBUYOUMSPPUWHUWTUYNTUAPPUWHUUAUYMUBPUWHYLUYLABYTUWHYKUYKYAUWHYGUYJY + JYEKYFXHXIXBUNXFVLVMVMUWIUYOUYTUMSPPUWIUYNUYSTUAPPUWIUYMUYRUBPUWIUYLUYQBC + YTUWIUYKUYPYAUWIUYJXSYJYFLKXRXIXBUOXFVLVMVMUWJUYTUXNUMSPPUWJUYSUXLTUAPPUW + JUYRUXKUBPUWJUYQUXJCDYTUWJUYPUXIYAUWJYJUXHXSYHMYIXHXAXBUPXFVLVMVMXGXJUUIU + XGJUGRUVTUGVFZUUHUXFUJOPVBVUAUUGUXEUKOPVBVUAUUFUXDULOPVBVUAUUEUXCUMOPVBVU + AUUDUXBSOPVBVUAUUCUXATOPVBVUAUUBUWTUAOPVBVUAUUAUBOPVBXKXLXLXLXLXLXLXLXMXN + XOXPXQ $. $} ${ - $d A x $. $d B x $. $d X x $. - $( The values of two surreals at a point less than their separators are - equal. (Contributed by Scott Fenton, 6-Dec-2021.) $) - nosepeq $p |- ( ( ( A e. No /\ B e. No /\ A =/= B ) /\ - X e. |^| { x e. On | ( A ` x ) =/= ( B ` x ) } ) -> - ( A ` X ) = ( B ` X ) ) $= - ( csur wcel wne w3a cv cfv con0 crab cint wa wn wceq nosepon onelon sylan - fveq2 simpr neeq12d onnminsb sylc df-ne con2bii sylibr ) BEFCEFBCGHZDAIZB - JZUICJZGZAKLMZFZNZDBJZDCJZGZOZUPUQPZUODKFZUNUSUHUMKFUNVAABCQUMDRSUHUNUAUL - URADUIDPUJUPUKUQUIDBTUIDCTUBUCUDURUTUPUQUEUFUG $. + $d b ch $. $d e et $. $d a b c d e f p q P $. $d p q x ph $. $d a ps $. + $d a b c d e f p q x A $. $d a b c d e f p q x B $. $d a b c d e f x Q $. + $d a b c d e f p q x C $. $d a b c d e f p q x D $. $d a b c d e f x X $. + $d a b c d e f p q x E $. $d d ta $. $d c th $. $d a b c d e f x ze $. + $d a b c d e f p q x F $. $d a b c d e f p q x S $. + br6.1 $e |- ( a = A -> ( ph <-> ps ) ) $. + br6.2 $e |- ( b = B -> ( ps <-> ch ) ) $. + br6.3 $e |- ( c = C -> ( ch <-> th ) ) $. + br6.4 $e |- ( d = D -> ( th <-> ta ) ) $. + br6.5 $e |- ( e = E -> ( ta <-> et ) ) $. + br6.6 $e |- ( f = F -> ( et <-> ze ) ) $. + br6.7 $e |- ( x = X -> P = Q ) $. + br6.8 $e |- R = { <. p , q >. | + E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P E. e e. P + E. f e. P ( p = <. a , <. b , c >. >. /\ + q = <. d , <. e , f >. >. /\ ph ) } $. + $( Substitution for a six-place predicate. (Contributed by Scott Fenton, + 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) $) + br6 $p |- ( ( X e. S /\ + ( A e. Q /\ B e. Q /\ C e. Q ) /\ + ( D e. Q /\ E e. Q /\ F e. Q ) ) -> + ( <. A , <. B , C >. >. R <. D , <. E , F >. >. <-> ze ) ) $= + ( cop wbr cv wceq w3a wrex wcel opex eqeq1 eqcom 3anbi1d rexbidv 2rexbidv + bitrdi 3anbi2d brab wa wi wb vex opth sylan9bb sylbi rexlimdva rexlimdvva + biimp3a a1i simpl1 simpl2 opeq1 eqeq1d 3anbi23d opeq2d eqid pm3.2i df-3an + opeq2 mpbiran rspc3ev 3ad2antl3 3anbi13d syl2anc rexeqdv rexeqbidv rspcev + ex impbid bitrid ) IJKUPZUPZLSTUPZUPZOUQUDURZUEURZUFURZUPZUPZXEUSZUGURZQU + RZRURZUPZUPZXGUSZAUTZRMVAZQMVAZUGMVAZUFMVAZUEMVAZUDMVAZHPVAZUAPVBZINVBJNV + BKNVBUTZLNVBSNVBTNVBUTZUTZGUCURZXLUSZUBURZXRUSZAUTZRMVAZQMVAUGMVAZUFMVAUE + MVAZUDMVAHPVAXMYOAUTZRMVAZQMVAUGMVAZUFMVAUEMVAZUDMVAHPVAYGUCUBXEXGOIXDVCL + XFVCYLXEUSZYSUUCHUDPMUUDYRUUBUEUFMMUUDYQUUAUGQMMUUDYPYTRMUUDYMXMYOAUUDYMX + EXLUSXMYLXEXLVDXEXLVEVIVFVGVHVHVHYNXGUSZUUCYEHUDPMUUEUUBYCUEUFMMUUEUUAYAU + GQMMUUEYTXTRMUUEYOXSXMAUUEYOXGXRUSXSYNXGXRVDXGXRVEVIVJVGVHVHVHUOVKYKYGGYK + YEGHUDPMYKHURZPVBXHMVBVLVLZYCGUEUFMMUUGXIMVBXJMVBVLVLZYAGUGQMMUUHXNMVBXOM + VBVLVLZXTGRMXTGVMUUIXPMVBVLXMXSAGXMADXSGXMXHIUSZXKXDUSZVLADVNXHXKIXDUDVOX + IXJVCVPUUJABUUKDUHUUKXIJUSZXJKUSZVLBDVNXIXJJKUEVOUFVOVPUULBCUUMDUIUJVQVRV + QVRXSXNLUSZXQXFUSZVLDGVNXNXQLXFUGVOXOXPVCVPUUNDEUUOGUKUUOXOSUSZXPTUSZVLEG + VNXOXPSTQVORVOVPUUPEFUUQGULUMVQVRVQVRVQWAWBVSVTVTVTYKGYGYKGVLZYHXTRNVAZQN + VAZUGNVAZUFNVAZUENVAZUDNVAZYGYHYIYJGWCUURYIXEXEUSZXSDUTZRNVAZQNVAUGNVAZUV + DYHYIYJGWDYJYHGUVHYIUVFGUVELXQUPZXGUSZEUTUVELSXPUPZUPZXGUSZFUTZUGQRLSTNNN + UUNXSUVJDEUVEUUNXRUVIXGXNLXQWEWFUKWGUUPUVJUVMEFUVEUUPUVIUVLXGUUPXQUVKLXOS + XPWEWHWFULWGUUQUVNUVEXGXGUSZGUTZGUUQUVMUVOFGUVEUUQUVLXGXGUUQUVKXFLXPTSWLW + HWFUMWGUVPUVEUVOVLGUVEUVOXEWIXGWIWJUVEUVOGWKWMVIWNWOUVAUVHIXKUPZXEUSZXSBU + TZRNVAZQNVAUGNVAIJXJUPZUPZXEUSZXSCUTZRNVAZQNVAUGNVAUDUEUFIJKNNNUUJUUSUVTU + GQNNUUJXTUVSRNUUJXMUVRABXSUUJXLUVQXEXHIXKWEWFUHWPVGVHUULUVTUWEUGQNNUULUVS + UWDRNUULUVRUWCBCXSUULUVQUWBXEUULXKUWAIXIJXJWEWHWFUIWPVGVHUUMUWEUVGUGQNNUU + MUWDUVFRNUUMUWCUVECDXSUUMUWBXEXEUUMUWAXDIXJKJWLWHWFUJWPVGVHWNWQYFUVDHUAPU + UFUAUSZYEUVCUDMNUNUWFYDUVBUEMNUNUWFYCUVAUFMNUNUWFYBUUTUGMNUNUWFYAUUSQMNUN + UWFXTRMNUNWRWSWSWSWSWSWTWQXAXBXC $. $} ${ - $d A x $. $d B x $. - $( Given two non-equal surreals, their separator is less than or equal to - the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. - (Contributed by Scott Fenton, 6-Dec-2021.) $) - nosepssdm $p |- ( ( A e. No /\ B e. No /\ A =/= B ) -> - |^| { x e. On | ( A ` x ) =/= ( B ` x ) } C_ dom A ) $= - ( csur wcel wne cfv con0 cdm wn wceq wa c0 wi word nodmord 3ad2ant1 ndmfv - wss syl w3a cv crab cint nosepne neneqd ordn2lp sylibr imp nosepeq simpl1 - imnan ordirr 3syl eqeq1d eqcom bitrdi crn wb simpl2 nofun c1o c2o nosgnn0 - wfun cpr norn sseld mtoi funeldmb syl2anc necon2bbid ordtr1 expdimp con3d - 3ad2ant2 sylbid mpd eqtr4d mtand nosepon nodmon ontri1 mpbird ) BDEZCDEZB - CFZUAZAUBZBGWICGFAHUCUDZBIZSZWKWJEZJZWHWMWJBGZWJCGZKWHWOWPABCUEUFWHWMLZWO - MWPWQWJWKEZJZWOMKWHWMWSWHWMWRLJZWMWSNWHWKOZWTWEWFXAWGBPZQWKWJUGTWMWRULUHU - IWJBRTWQWJCIZEZJZWPMKWQWKBGZWKCGZKZXEABCWKUJWQXHXGMKZXEWQXHMXGKXIWQXFMXGW - QXAWKWKEJXFMKWQWEXAWEWFWGWMUKXBTWKUMWKBRUNUOMXGUPUQWQXIWKXCEZJXEWQXJXGMWQ - CVEZMCURZEZJXJXGMFUSWQWFXKWEWFWGWMUTZCVATWQXMMVBVCVFZEVDWQXLXOMWQWFXLXOSX - NCVGTVHVIWKCVJVKVLWQXDXJWHWMXDXJWHXCOZWMXDLXJNWFWEXPWGCPVPWKWJXCVMTVNVOVQ - VQVRWJCRTVSVTWHWJHEWKHEZWLWNUSABCWAWEWFXQWGBWBQWJWKWCVKWD $. + $d a b c d p q x A $. $d a b c d p q x B $. $d b ch $. $d a b c d x Q $. + $d a b c d p q x C $. $d a b c d p q x D $. $d a ps $. $d a b c d x X $. + $d a b c d p q P $. $d a b c d p q x S $. $d a b c d x ta $. $d c th $. + $d p q x ph $. + br4.1 $e |- ( a = A -> ( ph <-> ps ) ) $. + br4.2 $e |- ( b = B -> ( ps <-> ch ) ) $. + br4.3 $e |- ( c = C -> ( ch <-> th ) ) $. + br4.4 $e |- ( d = D -> ( th <-> ta ) ) $. + br4.5 $e |- ( x = X -> P = Q ) $. + br4.6 $e |- R = { <. p , q >. | + E. x e. S E. a e. P E. b e. P E. c e. P E. d e. P + ( p = <. a , b >. /\ q = <. c , d >. /\ ph ) } $. + $( Substitution for a four-place predicate. (Contributed by Scott Fenton, + 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.) $) + br4 $p |- ( ( X e. S /\ + ( A e. Q /\ B e. Q ) /\ + ( C e. Q /\ D e. Q ) ) -> + ( <. A , B >. R <. C , D >. <-> ta ) ) $= + ( cop wbr cv wceq w3a wrex wcel wa eqeq1 3anbi1d rexbidv 2rexbidv 3anbi2d + opex brab wi wb vex opth sylan9bb eqcoms biimp3a a1i rexlimdva rexlimdvva + sylbi simpl1 simpl2l simpl2r simpl3l simpl3r eqidd simpr 3anbi23d rspc2ev + opeq1 eqeq2d opeq2 syl113anc 3anbi13d syl3anc rexeqdv rexeqbidv rspcev ex + syl2anc impbid bitrid ) GHUHZIJUHZMUIWPRUJZSUJZUHZUKZWQTUJZUAUJZUHZUKZAUL + ZUAKUMZTKUMZSKUMZRKUMZFNUMZONUNZGLUNZHLUNZUOZILUNZJLUNZUOZULZEQUJZWTUKZPU + JZXDUKZAULZUAKUMZTKUMSKUMZRKUMFNUMXAYCAULZUAKUMZTKUMSKUMZRKUMFNUMXKQPWPWQ + MGHVAIJVAXTWPUKZYFYIFRNKYJYEYHSTKKYJYDYGUAKYJYAXAYCAXTWPWTUPUQURUSUSYBWQU + KZYIXIFRNKYKYHXGSTKKYKYGXFUAKYKYCXEXAAYBWQXDUPUTURUSUSUGVBXSXKEXSXIEFRNKX + SFUJZNUNWRKUNUOUOZXGESTKKYMWSKUNXBKUNUOUOZXFEUAKXFEVCYNXCKUNUOXAXEAEXAACX + EEACVDZWTWPWTWPUKWRGUKZWSHUKZUOYOWRWSGHRVESVEVFYPABYQCUBUCVGVMVHCEVDZXDWQ + XDWQUKXBIUKZXCJUKZUOYRXBXCIJTVEUAVEVFYSCDYTEUDUEVGVMVHVGVIVJVKVLVLXSEXKXS + EUOZXLXFUALUMZTLUMZSLUMZRLUMZXKXLXOXREVNUUAXMXNWPWPUKZXECULZUALUMTLUMZUUE + XMXNXLXREVOXMXNXLXREVPUUAXPXQUUFWQWQUKZEUUHXPXQXLXOEVQXPXQXLXOEVRUUAWPVSU + UAWQVSXSEVTUUGUUFUUIEULUUFWQIXCUHZUKZDULTUAIJLLYSXEUUKCDUUFYSXDUUJWQXBIXC + WCWDUDWAYTUUKUUIDEUUFYTUUJWQWQXCJIWEWDUEWAWBWFUUCUUHWPGWSUHZUKZXEBULZUALU + MTLUMRSGHLLYPXFUUNTUALLYPXAUUMABXEYPWTUULWPWRGWSWCWDUBWGUSYQUUNUUGTUALLYQ + UUMUUFBCXEYQUULWPWPWSHGWEWDUCWGUSWBWHXJUUEFONYLOUKZXIUUDRKLUFUUOXHUUCSKLU + FUUOXGUUBTKLUFUUOXFUAKLUFWIWJWJWJWKWMWLWNWO $. $} ${ - $d A a $. $d B a $. - $( Lemma for ~ nodense . Show that a particular abstraction is an ordinal. - (Contributed by Scott Fenton, 16-Jun-2011.) $) - nodenselem4 $p |- ( ( ( A e. No /\ B e. No ) /\ A - |^| { a e. On | ( A ` a ) =/= ( B ` a ) } e. On ) $= - ( csur wcel wa cslt wbr wne cv con0 crab cint simpll simplr wn wceq sltso - cfv wor sonr mpan adantr breq2 notbid syl5ibcom necon2ad nosepon syl3anc - imp ) ADEZBDEZFZABGHZFUKULABIZCJZASUPBSICKLMKEUKULUNNUKULUNOUMUNUOUMUNABU - MAAGHZPZABQZUNPUKURULDGTUKURRDAGUAUBUCUSUQUNABAGUDUEUFUGUJCABUHUI $. - $} + $d A x y z $. $d B x y z $. + $( Another distributive law of converse over class composition. + (Contributed by Scott Fenton, 3-May-2014.) $) + cnvco1 $p |- `' ( `' A o. B ) = ( `' B o. A ) $= + ( vx vy vz ccnv ccom relcnv relco cv wbr wa wex cop wcel vex brcnv bicomi + anbi12ci opelco exbii opelcnv bitri 3bitr4i eqrelriiv ) CDAFZBGZFZBFZAGZU + GHUIAIDJZEJZBKZULCJZUFKZLZEMZUNULAKZULUKUIKZLZEMUNUKNZUHOZVAUJOUPUTEUMUSU + OURUSUMULUKBEPZDPZQRULUNAVCCPZQSUAVBUKUNNUGOUQUNUKUGVEVDUBEUKUNUFBVDVETUC + EUNUKUIAVEVDTUDUE $. - ${ - $d A a $. $d B a $. - $( Lemma for ~ nodense . If the birthdays of two distinct surreals are - equal, then the ordinal from ~ nodenselem4 is an element of that - birthday. (Contributed by Scott Fenton, 16-Jun-2011.) $) - nodenselem5 $p |- ( ( ( A e. No /\ B e. No ) /\ - ( ( bday ` A ) = ( bday ` B ) /\ A - |^| { a e. On | ( - A ` a ) =/= ( B ` a ) } e. - ( bday ` A ) ) $= - ( csur wcel wa cbday cfv wceq cslt wbr cv wne con0 cdm cun wn bdayval syl - crab cint simpll simplr wi wor sltso sonr breq2 notbid syl5ibcom necon2ad - mpan adantr adantrl nosepdm syl3anc simprl uneq2d eqtr3di uneq12d 3eqtr3d - imp unidm eleqtrd eleqtrrd ) ADEZBDEZFZAGHZBGHZIZABJKZFZFZCLZAHVOBHMCNTUA - ZAOZVIVNVPVQBOZPZVQVNVFVGABMZVPVSEVFVGVMUBZVFVGVMUCZVHVLVTVKVHVLVTVFVLVTU - DVGVFVLABVFAAJKZQZABIZVLQDJUEVFWDUFDAJUGULWEWCVLABAJUHUIUJUKUMVBUNCABUOUP - VNVIVJPZVIVSVQVNVIVIPWFVIVNVIVJVIVHVKVLUQURVIVCUSVNVIVQVJVRVNVFVIVQIWAARS - ZVNVGVJVRIWBBRSUTWGVAVDWGVE $. + $( Another distributive law of converse over class composition. + (Contributed by Scott Fenton, 3-May-2014.) $) + cnvco2 $p |- `' ( A o. `' B ) = ( B o. `' A ) $= + ( vx vy vz ccnv ccom relcnv relco cv wbr wa wex cop wcel vex brcnv bicomi + anbi12ci opelco exbii opelcnv bitri 3bitr4i eqrelriiv ) CDABFZGZFZBAFZGZU + GHBUIIDJZEJZUFKZULCJZAKZLZEMZUNULUIKZULUKBKZLZEMUNUKNZUHOZVAUJOUPUTEUMUSU + OURUKULBDPZEPZQURUOUNULACPZVDQRSUAVBUKUNNUGOUQUNUKUGVEVCUBEUKUNAUFVCVETUC + EUNUKBUIVEVCTUDUE $. $} ${ - $d A a $. $d B a $. - $( The restriction of a surreal to the abstraction from ~ nodenselem4 is - still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.) $) - nodenselem6 $p |- ( ( ( A e. No /\ B e. No ) /\ - ( ( bday ` A ) = ( bday ` B ) /\ A - ( A |` |^| { a e. On | ( - A ` a ) =/= ( B ` a ) } ) e. No ) $= - ( csur wcel wa cbday cfv wceq cslt wbr cv con0 crab cint cres nodenselem4 - wne simpll adantrl noreson syl2anc ) ADEZBDEZFZAGHBGHIZABJKZFZFUCCLZAHUIB - HRCMNOZMEZAUJPDEUCUDUHSUEUGUKUFABCQTAUJUAUB $. + $d A x y z p $. $d B x y z p $. + $( Quantifier-free definition of membership in a domain. (Contributed by + Scott Fenton, 21-Jan-2017.) $) + eldm3 $p |- ( A e. dom B <-> ( B |` { A } ) =/= (/) ) $= + ( vx vy vp vz cdm wcel cvv csn cres c0 wne wceq cv eleq1 cop wex wa exbii + elex wn snprc reseq2 res0 eqtrdi sylbi necon1ai reseq2d neeq1d dfclel vex + sneq eldm2 n0 wrex elres pm5.32i opeq1 eqeq2d anbi1d bitr3id exbidv rexsn + weq bitri excom 3bitri 3bitr4i vtoclbg pm5.21nii ) ABGZHZAIHZBAJZKZLMZAVL + UAVNVPLVNUBVOLNZVPLNAUCVRVPBLKLVOLBUDBUEUFUGUHCOZVLHZBVSJZKZLMZVMVQCAIVSA + VLPVSANZWBVPLWDWAVOBVSAUMUIUJVSDOZQZBHZDREOZWFNZWHBHZSZERZDRZVTWCWGWLDEWF + BUKTDVSBCULZUNWCWHWBHZERWKDRZERWMEWBUOWOWPEWOWHFOZWEQZNZWRBHZSZDRZFWAUPWP + FDWHBWAUQXBWPFVSWNFCVEZXAWKDXAWSWJSXCWKWSWJWTWHWRBPURXCWSWIWJXCWRWFWHWQVS + WEUSUTVAVBVCVDVFTWKEDVGVHVIVJVK $. $} + $( Quantifier-free definition of membership in a range. (Contributed by + Scott Fenton, 21-Jan-2017.) $) + elrn3 $p |- ( A e. ran B <-> ( B i^i ( _V X. { A } ) ) =/= (/) ) $= + ( crn wcel ccnv cdm csn cres wne cvv cxp cin df-rn eleq2i eldm3 wceq ineq2i + c0 cnvxp cnvin df-res 3eqtr4ri eqeq1i wrel wb cnveq0 ax-mp bitr4i necon3bii + relinxp 3bitri ) ABCZDABEZFZDUMAGZHZRIBJUOKZLZRIULUNABMNAUMOUPRURRUPRPUREZR + PZURRPZUPUSRUMUQEZLUMUOJKZLUSUPVBVCUMJUOSQBUQTUMUOUAUBUCURUDVAUTUEJUOBUJURU + FUGUHUIUK $. + ${ - $d A a $. $d B a $. $d C a $. - $( Lemma for ~ nodense . ` A ` and ` B ` are equal at all elements of the - abstraction. (Contributed by Scott Fenton, 17-Jun-2011.) $) - nodenselem7 $p |- ( ( ( A e. No /\ B e. No ) /\ - ( ( bday ` A ) = ( bday ` B ) /\ A - ( C e. |^| { a e. On | ( A ` a ) =/= ( B ` a ) } - -> ( A ` C ) = ( B ` C ) ) ) $= - ( csur wcel wa cbday cfv wceq cslt wbr cv wne con0 crab cint w3a simpll - wn simplr wor sltso sonr mpan breq2 syl5ibcom necon2ad imp ad2ant2rl 3jca - notbid nosepeq sylan ex ) AEFZBEFZGAHIBHIJZABKLZGZGZCDMZAIVBBINDOPQFZCAIC - BIJZVAUPUQABNZRVCVDVAUPUQVEUPUQUTSUPUQUTUAUPUSVEUQURUPUSVEUPUSABUPAAKLZTZ - ABJZUSTEKUBUPVGUCEAKUDUEVHVFUSABAKUFULUGUHUIUJUKDABCUMUNUO $. + $d R x y z $. $d A x y z $. + + $( The converse of a partial ordering is still a partial ordering. + (Contributed by Scott Fenton, 13-Jun-2018.) $) + pocnv $p |- ( R Po A -> `' R Po A ) $= + ( vx vy vz wpo ccnv cv wcel wa wbr poirr vex brcnv sylnibr wi 3anrev potr + w3a sylan2b anbi12ci 3imtr4g ispod ) ABFZCDEABGZUDCHZAIZJUFUFBKUFUFUEKAUF + BLUFUFBCMZUHNOUDUGDHZAIZEHZAIZSZJUKUIBKZUIUFBKZJZUKUFBKZUFUIUEKZUIUKUEKZJ + UFUKUEKUMUDULUJUGSUPUQPUGUJULQAUKUIUFBRTURUOUSUNUFUIBUHDMZNUIUKBUTEMZNUAU + FUKBUHVANUBUC $. + + $( The converse of a strict ordering is still a strict ordering. + (Contributed by Scott Fenton, 13-Jun-2018.) $) + socnv $p |- ( R Or A -> `' R Or A ) $= + ( wor ccnv cnvso biimpi ) ABCABDCABEF $. $} ${ - $d A a $. $d B a $. - $( Lemma for ~ nodense . Give a condition for surreal less than when two - surreals have the same birthday. (Contributed by Scott Fenton, - 19-Jun-2011.) $) - nodenselem8 $p |- ( ( A e. No /\ B e. No /\ - ( bday ` A ) = ( bday ` B ) ) -> - ( A - ( ( A ` - |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = - 1o /\ - ( B ` - |^| { a e. On | ( A ` a ) =/= ( B ` a ) } ) = - 2o ) ) ) $= - ( csur wcel cbday cfv wceq wbr c1o c2o wa wi 3impia c0 cop wn nosgnn0 crn - fvex w3a cslt cv wne con0 crab cint nodenselem5 exp32 ctp sltval2 3adant3 - wb w3o brtp eleq2 biimpd cpr nofnbday fnfvelrn eleq1 syl5ibcom sylan norn - wfn sseld adantr syld mtoi ex adantl syl9r imp intnand 3ad2antl1 intnanrd - 3orel13 syl2anc com23 syl5bi sylbid mpdd 3mix2 sylibr syl5ibr impbid ) AD - EZBDEZAFGZBFGZHZUAZABUBIZCUCZAGWNBGUDCUEUFUGZAGZJHZWOBGZKHZLZWLWMWOWIEZWT - WGWHWKWMXAMWGWHLZWKWMXAABCUHUINWLWMWPWRJOPJKPOKPUJIZXAWTMZWGWHWMXCUMWKABC - UKULZXCWQWROHZLZWTWPOHZWSLZUNZWLXDJOJKOKWPWRWOATWOBTUOZWLXAXJWTWLXAXJWTMZ - WLXALZXGQXIQXLXMXFWQWLXAXFQZWGWHWKXAXNMWKXAWOWJEZXBXNWKXAXOWIWJWOUPUQWHXO - XNMWGWHXOXNWHXOLZXFOJKURZEZRXPXFOBSZEZXRWHBWJVEZXOXFXTMBUSYAXOLWRXSEXFXTW - JWOBUTWROXSVAVBVCWHXTXRMXOWHXSXQOBVDVFVGVHVIVJVKVLNVMVNXMXHWSWGWHXAXHQWKW - GXALZXHXRRYBXHOASZEZXRWGAWIVEZXAXHYDMAUSYEXALWPYCEXHYDWIWOAUTWPOYCVAVBVCW - GYDXRMXAWGYCXQOAVDVFVGVHVIVOVPXGWTXIVQVRVJVSVTWAWBWTWMWLXCWTXJXCWTXGXIWCX - KWDXEWEWF $. + sotrd.1 $e |- ( ph -> R Or A ) $. + sotrd.2 $e |- ( ph -> X e. A ) $. + sotrd.3 $e |- ( ph -> Y e. A ) $. + sotrd.4 $e |- ( ph -> Z e. A ) $. + sotrd.5 $e |- ( ph -> X R Y ) $. + sotrd.6 $e |- ( ph -> Y R Z ) $. + $( Transitivity law for strict orderings, deduction form. (Contributed by + Scott Fenton, 24-Nov-2021.) $) + sotrd $p |- ( ph -> X R Z ) $= + ( wbr wor wcel wa wi sotr syl13anc mp2and ) ADECMZEFCMZDFCMZKLABCNDBOEBOF + BOUAUBPUCQGHIJBDEFCRST $. $} ${ - $d A a x y $. $d B a x y $. - $( Given two distinct surreals with the same birthday, there is an older - surreal lying between the two of them. Axiom SD of [Alling] p. 184. - (Contributed by Scott Fenton, 16-Jun-2011.) $) - nodense $p |- ( ( ( A e. No /\ B e. No ) /\ - ( ( bday ` A ) = ( bday ` B ) /\ A - E. x e. No ( ( bday ` x ) e. ( bday ` A ) /\ - A + ( X e. |^| ( F " B ) <-> A. y e. B X e. ( F ` y ) ) ) $= + ( vz cima cint wcel cv wi wal wfn wss wa cfv wral elint wceq wrex r19.23v + imbi1d bitr4di albidv ralcom4 eqcom imbi1i albii fvex eleq2 ceqsalv bitri + fvelimab ralbii bitr3i bitrdi bitrid ) EDCHZIJGKZUSJZEUTJZLZGMZDBNCBOPZEA + KZDQZJZACRZGEUSFSVEVDVGUTTZVBLZACRZGMZVIVEVCVLGVEVCVJACUAZVBLVLVEVAVNVBAB + CUTDUNUCVJVBACUBUDUEVMVKGMZACRVIVKAGCUFVOVHACVOUTVGTZVBLZGMVHVKVQGVJVPVBV + GUTUGUHUIVBVHGVGVFDUJUTVGEUKULUMUOUPUQUR $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal Numbers: Full-Eta Property + Properties of functions and mappings =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - The theorems in this section are derived from "A clean way to separate sets - of surreals" by Paolo Lipparini, - ~ https://doi.org/10.48550/arXiv.1712.03500 . - $) - $( Lemma for full-eta properties. The successor of the union of the image of - the birthday function under a set is an ordinal. (Contributed by Scott - Fenton, 20-Aug-2011.) $) - bdayimaon $p |- ( A e. V -> suc U. ( bday " A ) e. On ) $= - ( wcel cbday cima cuni word cvv wa csuc con0 wfun csur wfo bdayfo funimaexg - fofun ax-mp mpan uniexd wss crn imassrn wceq sseqtri ssorduni jctil sucelon - forn elon2 bitr3i sylib ) ABCZDAEZFZGZUOHCZIZUOJKCZUMUQUPUMUNHDLZUMUNHCMKDN - ZUTOMKDQRDABPSTUNKUAUPUNDUBZKDAUCVAVBKUDOMKDUIRUEUNUFRUGURUOKCUSUOUJUOUHUKU - L $. - - $( Lemma for ~ nolt02o . If ` A ( X ) ` is undefined with ` A ` surreal and - ` X ` ordinal, then ` dom A C_ X ` . (Contributed by Scott Fenton, - 6-Dec-2021.) $) - nolt02olem $p |- ( ( A e. No /\ X e. On /\ ( A ` X ) = (/) ) -> - dom A C_ X ) $= - ( csur wcel con0 cfv c0 wceq w3a cdm wss wn c1o c2o cpr nosgnn0 wa 3ad2ant1 - a1i simpl3 crn simpl1 norn wfun nofun fvelrn sylan sseldd eqeltrrd mtand wb - syl nodmon simp2 ontri1 syl2anc mpbird ) ACDZBEDZBAFZGHZIZAJZBKZBVCDZLZVBVE - GMNOZDZVHLVBPSVBVEQZUTGVGURUSVAVETVIAUAZVGUTVIURVJVGKURUSVAVEUBAUCULVBAUDZV - EUTVJDURUSVKVAAUERBAUFUGUHUIUJVBVCEDZUSVDVFUKURUSVLVAAUMRURUSVAUNVCBUOUPUQ - $. + $( A condition for subset trichotomy for functions. (Contributed by Scott + Fenton, 19-Apr-2011.) $) + funpsstri $p |- ( ( Fun H /\ ( F C_ H /\ G C_ H ) /\ + ( dom F C_ dom G \/ dom G C_ dom F ) ) -> + ( F C. G \/ F = G \/ G C. F ) ) $= + ( wfun wss wa cdm wo w3a wpss wceq w3o cres funssres anim12d ssres2 orim12i + wi ex sseq12 wb ancoms orbi12d syl5ib syl6 3imp sspsstri sylib ) CDZACEZBCE + ZFZAGZBGZEZUNUMEZHZIABEZBAEZHZABJABKBAJLUIULUQUTUIULCUMMZAKZCUNMZBKZFZUQUTR + UIUJVBUKVDUIUJVBCANSUIUKVDCBNSOUQVAVCEZVCVAEZHVEUTUOVFUPVGUMUNCPUNUMCPQVEVF + URVGUSVAAVCBTVDVBVGUSUAVCBVAATUBUCUDUEUFABUGUH $. ${ - $d A x y $. $d B x y $. $d X x y $. - $( Given ` A ` less than ` B ` , equal to ` B ` up to ` X ` , and undefined - at ` X ` , then ` B ( X ) = 2o ` . (Contributed by Scott Fenton, - 6-Dec-2021.) $) - nolt02o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ - ( ( A |` X ) = ( B |` X ) /\ A - ( B ` X ) = 2o ) $= - ( vy vx csur wcel con0 wceq cslt wbr wa cfv c0 c2o c1o syl2anc bitri w3o - wn w3a cres simp11 wor sltso sonr mpan syl simp2r syl5ibrcom mtod simpl2l - breq2 wrel cdm wss simpl11 nofun funrel simpl13 simpl3 nolt02olem syl3anc - wfun 3syl relssres simpl12 simpr 3eqtr3d mtand cv wral cop ctp wi wrex wb - simp12 sltval mpbid df-an rexbii rexnal sylib 1oex prid1 nosgnn0i simpll3 - neii simplr eqeq1 anbi1d eqtr2 syl6bi com12 mtoi simplrr fveq2 rspcv sylc - eqeq12d simprl adantr ontri1 mpbird wo onsseleq ancoms csuc df-1o eqeq12i - mto df-2o 0elon 1on suc11 mp2an nemtbir 2on elexi prid2 3pm3.2i fvex brtp - 3oran con2bii fveq1d breq2d mtbii breq12d notbid syl5ibcom intnanr intnan - mpbi fvres 0ex mtbiri jaod 3orrot sylbid mpd expr ralrimiva nofv ecase23d - ) AFGZBFGZCHGZUAZACUBZBCUBZIZABJKZLZCAMZNIZUAZCBMZOIZUUSNIZUUSPIZUURUVAAB - IZUURUVCAAJKZUURUUGUVDTZUUGUUHUUIUUOUUQUCZFJUDUUGUVEUEFAJUFUGUHUURUVDUVCU - UNUUJUUMUUNUUQUIZABAJUMUJUKUURUVALZUUKUULABUUMUUNUUJUUQUVAULUVHAUNZAUOCUP - ZUUKAIUVHUUGAVDUVIUUGUUHUUIUUOUUQUVAUQZAURAUSVEUVHUUGUUIUUQUVJUVKUUGUUHUU - IUUOUUQUVAUTZUUJUUOUUQUVAVAACVBVCACVFQUVHBUNZBUOCUPZUULBIUVHUUHBVDUVMUUGU - UHUUIUUOUUQUVAVGZBURBUSVEUVHUUHUUIUVAUVNUVOUVLUURUVAVHBCVBVCBCVFQVIVJUURU - VBDVKZAMZUVPBMZIZDEVKZVLZUVTAMZUVTBMZPNVMPOVMNOVMVNZKZTZVOZEHVLZUURUWAUWE - LZEHVPZUWHTZUURUUNUWJUVGUURUUGUUHUUNUWJVQUVFUUGUUHUUIUUOUUQVRZEDABVSQVTUW - JUWGTZEHVPUWKUWIUWMEHUWAUWEWAWBUWGEHWCRWDUURUVBLZUWGEHUWNUVTHGZUWAUWFUWNU - WOUWALZLZUVTCUPZUWFUWQUWRCUVTGZTZUWQUWSUUPUUSIZUWQUXANPIZNPPPOWEWFWGZWIZU - WQUUQUVBUXAUXBVOUUJUUOUUQUVBUWPWHZUURUVBUWPWJZUXAUUQUVBLZUXBUXAUXGUVAUVBL - UXBUXAUUQUVAUVBUUPUUSNWKWLUUSNPWMWNWOQWPUWQUWSLUWSUWAUXAUWQUWSVHUWNUWOUWA - UWSWQUVSUXADCUVTUVPCIUVQUUPUVRUUSUVPCAWRUVPCBWRXAWSWTVJUWQUWOUUIUWRUWTVQU - WNUWOUWAXBZUWNUUIUWPUUGUUHUUIUUOUUQUVBUTXCZUVTCXDQXEUWQUWRUVTCGZUVTCIZXFZ - UWFUWQUWOUUIUWRUXLVQUXHUXIUVTCXGQUWQUXJUWFUXKUWQUVTUUKMZUVTUULMZUWDKZTUXJ - UWFUWQUXMUXMUWDKZUXOUXMPIZUXMNIZLZTZUXQUXMOIZLZTZUXRUYALZTZUAZUXPTUXTUYCU - YEUXSUXBUXDUXRUXQUXBUXMNPWMXHXLUYBPOIZUYGNPUXCUYGNXIZPXIZIZUXBPUYHOUYIXJX - MXKNHGPHGUYJUXBVQXNXONPXPXQRXRZUXMPOWMXLUYDNOINOOPOOHXSXTYAWGWIUXMNOWMXLY - BUXPUYFUXPUXSUYBUYDSUYFTPNPONOUXMUXMUVTUUKYCZUYLYDUXSUYBUYDYERYFYOUWQUXMU - XNUXMUWDUWQUVTUUKUULUWNUUMUWPUUMUUNUUJUUQUVBULXCYGYHYIUXJUXOUWEUXJUXMUWBU - XNUWCUWDUVTCAYPUVTCBYPYJYKYLUWQUWFUXKUUPUUSUWDKZTUWQUYMNPUWDKZUXBPNIZLZTZ - UXBUYGLZTZNNIZUYGLZTZUAZUYNTUYQUYSVUBUXBUYOUXDYMUXBUYGUXDYMUYGUYTUYKYNYBU - YNVUCUYNUYPUYRVUASVUCTPNPONONPYQWEYDUYPUYRVUAYERYFYOUWQUUPNUUSPUWDUXEUXFY - JYRUXKUWEUYMUXKUWBUUPUWCUUSUWDUVTCAWRUVTCBWRYJYKUJYSUUAUUBUUCUUDVJUURUVAU - VBUUTSZUUTUVAUVBSZUURUUHVUDUWLBCUUEUHVUDUVBUUTUVASVUEUVAUVBUUTYTUVBUUTUVA - YTRWDUUF $. - - $( Given ` A ` greater than ` B ` , equal to ` B ` up to ` X ` , and - ` B ( X ) ` undefined, then ` A ( X ) = 1o ` . (Contributed by Scott - Fenton, 9-Aug-2024.) $) - nogt01o $p |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ - ( ( A |` X ) = ( B |` X ) /\ A - ( A ` X ) = 1o ) $= - ( vy vx csur wcel con0 w3a wceq cslt wbr wa cfv c0 c1o c2o wn syl syl2anc - cres wor sltso simp11 sonr sylancr simpl2r simpl2l wrel cdm simpl11 nofun - wss wfun funrel simpl13 simpr nolt02olem syl3anc relssres simpl12 3eqtr3d - simpl3 breqtrrd mtand cv wral cop ctp wi simp2r wb simp12 sltval ralinexa - wrex mpbid con2bii sylib 1n0 neii eqtr2 mto csuc word wne eqeltrri onordi - df-2o 1oex sucid nordeq mp2an eqnetri nesymi 2on0 3pm3.2i fvex brtp 3oran - 2on bitri mpbi adantr fveq1d breq2d mtbii fvres breq12d syl5ibcom intnanr - w3o notbid intnan 2oex 0ex simplr simpll3 mtbiri fveq2 syl5ibrcom eqeq12d - wo rspccv ad2antll eqcom syl6ib sylibd mtoi simprl ontri1 onsseleq bitr3d - mpjaod expr ralrimiva nofv 3orcoma ecase23d ) AFGZBFGZCHGZIZACUAZBCUAZJZA - BKLZMZCBNZOJZIZCANZPJZUULOJZUULQJZUUKUUNAAKLZUUKFKUBYTUUPRUCYTUUAUUBUUHUU - JUDZFAKUEUFUUKUUNMZABAKUUFUUGUUCUUJUUNUGUURUUDUUEABUUFUUGUUCUUJUUNUHUURAU - IZAUJCUMZUUDAJUURYTUUSYTUUAUUBUUHUUJUUNUKZYTAUNUUSAULAUOSSUURYTUUBUUNUUTU - VAYTUUAUUBUUHUUJUUNUPZUUKUUNUQACURUSACUTTUURBUIZBUJCUMZUUEBJUURUUAUVCYTUU - AUUBUUHUUJUUNVAZUUABUNUVCBULBUOSSUURUUAUUBUUJUVDUVEUVBUUCUUHUUJUUNVCBCURU - SBCUTTVBVDVEUUKUUODVFZANZUVFBNZJZDEVFZVGZUVJANZUVJBNZPOVHPQVHOQVHVIZLZRZV - JZEHVGZUUKUVKUVOMEHVPZUVRRUUKUUGUVSUUCUUFUUGUUJVKUUKYTUUAUUGUVSVLUUQYTUUA - UUBUUHUUJVMEDABVNTVQUVRUVSUVKUVOEHVOVRVSUUKUUOMZUVQEHUVTUVJHGZUVKUVPUVTUW - AUVKMZMZUVJCGZUVPUVJCJZUWCUVJUUDNZUVJUUENZUVNLZRUWDUVPUWCUWFUWFUVNLZUWHUW - FPJZUWFOJZMZRZUWJUWFQJZMZRZUWKUWNMZRZIZUWIRUWMUWPUWRUWLPOJPOVTWAUWFPOWBWC - UWOPQJQPQPWDZPWIUWTWEPUWTGUWTPWFUWTQUWTHWIXAWGWHPWJWKUWTPWLWMWNZWOUWFPQWB - WCUWQOQJZQOWPWOZUWFOQWBWCWQUWIUWSUWIUWLUWOUWQXLUWSRPOPQOQUWFUWFUVJUUDWRZU - XDWSUWLUWOUWQWTXBVRXCUWCUWFUWGUWFUVNUWCUVJUUDUUEUVTUUFUWBUUFUUGUUCUUJUUOU - HXDXEXFXGUWDUWHUVOUWDUWFUVLUWGUVMUVNUVJCAXHUVJCBXHXIXMXJUWCUVPUWEUULUUIUV - NLZRUWCUXEQOUVNLZQPJZOOJZMZRZUXGUXBMZRZQOJZUXBMZRZIZUXFRUXJUXLUXOUXGUXHQP - UXAWAXKUXBUXGUXCXNUXBUXMUXCXNWQUXFUXPUXFUXIUXKUXNXLUXPRPOPQOQQOXOXPWSUXIU - XKUXNWTXBVRXCUWCUULQUUIOUVNUUKUUOUWBXQZUUCUUHUUJUUOUWBXRZXIXSUWEUVOUXEUWE - UVLUULUVMUUIUVNUVJCAXTUVJCBXTXIXMYAUWCCUVJGZRZUWDUWEYCZUWCUXSUXBUXCUWCUXS - UUIUULJZUXBUWCUXSUULUUIJZUYBUVKUXSUYCVJUVTUWAUVIUYCDCUVJUVFCJUVGUULUVHUUI - UVFCAXTUVFCBXTYBYDYEUULUUIYFYGUWCUUIOUULQUXRUXQYBYHYIUWCUWAUUBUXTUYAVLUVT - UWAUVKYJUVTUUBUWBYTUUAUUBUUHUUJUUOUPXDUWAUUBMUVJCUMUXTUYAUVJCYKUVJCYLYMTV - QYNYOYPVEUUKUUNUUMUUOXLZUUMUUNUUOXLUUKYTUYDUUQACYQSUUNUUMUUOYRVSYS $. + $d F p x y z $. $d G p x y z $. + $( If a class ` F ` is a proper subset of a function ` G ` , then + ` dom F C. dom G ` . (Contributed by Scott Fenton, 20-Apr-2011.) $) + fundmpss $p |- ( Fun G -> ( F C. G -> dom F C. dom G ) ) $= + ( vx vy vz vp wpss wss wn wa wi syl cv wbr wex wcel adantl ex wal exlimdv + wfun cdm pssss dmss a1i cdif c0 wne pssdif n0 wrel funrel reldif cop wceq + sylib elrel eleq1 df-br bitr4di biimpcd 2eximdv difss ssbri eximi simprbi + mpd adantr brdif ssbrd ad2antlr weq dffun2 2sp breq2 biimprd syl6 simplbi + sps expd impel adantlr com23 mpdd mtod jcad eximdv nss vex notbii anbi12i + syld eldm exbii bitri sylibr dfpss3 syl6ibr ) BUAZABGZAUBZBUBZHZXBXAHIZJX + AXBGWSWTXCXDWTXCKWSWTABHXCABUCZABUDLUEWSWTXDWSWTJZCMZDMZBNZDOZXGEMZANZEOZ + IZJZCOZXDXFFMZBAUFZPZFOZXPWTXTWSWTXRUGUHXTABUIFXRUJUPQXFXSXPFXFXSXGXHXRNZ + DOZCOZXPWSXSYCKZWTWSXRUKZYDWSBUKZYEBULBAUMLYEXSYCYEXSJZXQXGXHUNZUOZDOCOYC + CDXQXRUQYGYIYACDXSYIYAKYEYIXSYAYIXSYHXRPYAXQYHXRURXGXHXRUSUTVAQVBVGRLVHXF + YBXOCXFYBXJXNYBXJKXFYAXIDXRBXGXHBAVCVDVEUEXFYAXNDXFYAXNXFYAJZXMXGXHANZYAY + KIZXFYAXIYLXGXHBAVIZVFQYJXLYKEYJXLXGXKBNZYKWTXLYNKWSYAWTABXGXKXEVJVKYJYNX + LYKWSYAYNXLYKKZKZWTWSXIYPYAWSXIYNYOWSXIYNJZDEVLZYOWSYQYRKZESDSZCSZYSWSYFU + UACDEBVMVFYTYSCYSDEVNVSLYRYKXLXHXKXGAVOVPVQVTYAXIYLYMVRWAWBWCWDTWERTWFWGW + LTVGXDXGXBPZXGXAPZIZJZCOXPCXBXAWHUUEXOCUUBXJUUDXNDXGBCWIZWMUUCXMEXGAUUFWM + WJWKWNWOWPRWFXAXBWQWR $. $} + $( Given two functions with equal domains, equality only requires one + direction of the subset relationship. (Contributed by Scott Fenton, + 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.) $) + funsseq $p |- ( ( Fun F /\ Fun G /\ dom F = dom G ) -> + ( F = G <-> F C_ G ) ) $= + ( wfun cdm wceq w3a eqimss wa cres simpl3 reseq2d funssres 3ad2antl2 simpl2 + wss wrel funrel resdm 3syl 3eqtr3d ex impbid2 ) ACZBCZADZBDZEZFZABEZABOZABG + UHUJUIUHUJHZBUEIZBUFIZABUKUEUFBUCUDUGUJJKUDUCUJULAEUGBALMUKUDBPUMBEUCUDUGUJ + NBQBRSTUAUB $. + ${ - $d S g x $. $d U g x $. $d A g x $. - $( Restriction law for surreals. Lemma 2.1.4 of [Lipparini] p. 3. - (Contributed by Scott Fenton, 5-Dec-2021.) $) - noresle $p |- ( ( ( U e. No /\ S e. No ) /\ - ( dom U C_ A /\ dom S C_ A /\ - A. g e. A -. ( S |` suc g ) - -. S ( ( A F B /\ A F C ) -> B = C ) ) $= + ( vx vy vz cv wbr wa wi wal wceq cvv wcel wb breq12 wfun wrel weq 3adant3 + dffun2 w3a 3adant2 anbi12d eqeq12 3adant1 imbi12d spc3gv mp3an simplbiim + ) DUADUBHKZIKZDLZUOJKZDLZMZIJUCZNZJOIOHOZABDLZACDLZMZBCPZNZHIJDUEAQRBQRCQ + RVCVHNEFGVBVHHIJABCQQQUOAPZUPBPZURCPZUFZUTVFVAVGVLUQVDUSVEVIVJUQVDSVKUOAU + PBDTUDVIVKUSVESVJUOAURCDTUGUHVJVKVAVGSVIUPBURCUIUJUKULUMUN $. $} ${ - $d S x y $. - $( A class of surreals has at most one maximum. (Contributed by Scott - Fenton, 5-Dec-2021.) $) - nomaxmo $p |- ( S C_ No -> E* x e. S A. y e. S -. x E* x e. S A. y e. S -. y - E* x E. u e. A ( G e. dom u /\ - A. v e. A ( -. v ( u |` suc G ) = ( v |` suc G ) ) /\ - ( u ` G ) = x ) ) $= - ( vp vy csur cv wcel cslt wbr wn wceq wi cfv wrex wa weq adantl csuc cres - wss cdm w3a wal reeanv breq1 notbid reseq1 eqeq2d imbi12d simprr2 simprll - wmo rspcdva eqcom syl6ib simprl2 simprlr wo simpl sseldd wor sltso soasym - wral syl2anc pm4.62 sylib mpjaod fveq1d simprl1 sucidg syl fvresd 3eqtr3d - mpan simprl3 simprr3 expr rexlimdvva syl5bir eqeq2 3anbi3d rexbidv eleq2d - alrimivv dmeq breq2 eqeq1d ralbidv fveq1 3anbi123d cbvrexvw bitrdi sylibr - mo4 ) DHUCZECIZUDZJZBIZWTKLZMZWTEUAZUBZXCXFUBZNZOZBDVGZEWTPZAIZNZUEZCDQZE - FIZUDZJZXCXQKLZMZXQXFUBZXHNZOZBDVGZEXQPZGIZNZUEZFDQZRZAGSZOZGUFAUFXPAUOWS - YMAGYKXOYIRZFDQCDQWSYLXOYICFDDUGWSYNYLCFDDWSWTDJZXQDJZRZYNYLWSYQYNRZRZXLY - FXMYGYSEXGPEYBPXLYFYSEXGYBYSWTXQKLZMZXGYBNZXQWTKLZMZYSUUAYBXGNZUUBYSYDUUA - UUEOBDWTBCSZYAUUAYCUUEUUFXTYTXCWTXQKUHUIUUFXHXGYBXCWTXFUJUKULYRYEWSXSYEYH - XOYQUMTWSYOYPYNUNZUPYBXGUQURYSXJUUDUUBOBDXQBFSZXEUUDXIUUBUUHXDUUCXCXQWTKU - HUIUUHXHYBXGXCXQXFUJUKULYRXKWSXBXKXNYIYQUSTWSYOYPYNUTZUPYSYTUUDOZUUAUUDVA - YSWTHJZXQHJZUUJYSDHWTWSYRVBZUUGVCYSDHXQUUMUUIVCHKVDUUKUULRUUJVEHKWTXQVFVR - VHYTUUCVIVJVKVLYSEXFWTYSXBEXFJYRXBWSXBXKXNYIYQVMTEXAVNVOZVPYSEXFXQUUNVPVQ - YRXNWSXBXKXNYIYQVSTYRYHWSXSYEYHXOYQVTTVQWAWBWCWHXPYJAGYLXPXBXKXLYGNZUEZCD - QYJYLXOUUPCDYLXNUUOXBXKXMYGXLWDWEWFUUPYICFDCFSZXBXSXKYEUUOYHUUQXAXREWTXQW - IWGUUQXJYDBDUUQXEYAXIYCUUQXDXTWTXQXCKWJUIUUQXGYBXHWTXQXFUJWKULWLUUQXLYFYG - EWTXQWMWKWNWOWPWRWQ $. - - $( In any class of surreals, there is at most one value of the prefix - property. (Contributed by Scott Fenton, 8-Aug-2024.) $) - noinfprefixmo $p |- ( A C_ No -> - E* x E. u e. A ( G e. dom u /\ - A. v e. A ( -. u ( u |` suc G ) = ( v |` suc G ) ) /\ - ( u ` G ) = x ) ) $= - ( vp vy csur cv wcel cslt wbr wn wceq wi cfv wrex wa weq adantl csuc cres - wss cdm w3a wal reeanv breq2 notbid reseq1 eqeq2d imbi12d simprl2 simprlr - wmo rspcdva simprr2 simprll eqcom syl6ib wo simpl sseldd wor sltso soasym - wral mpan syl2anc imor sylib mpjaod fveq1d simprl1 sucidg 3eqtr3d simprl3 - syl fvresd simprr3 expr rexlimdvva syl5bir alrimivv eqeq2 3anbi3d rexbidv - dmeq eleq2d breq1 eqeq1d ralbidv fveq1 3anbi123d cbvrexvw bitrdi sylibr - mo4 ) DHUCZECIZUDZJZWTBIZKLZMZWTEUAZUBZXCXFUBZNZOZBDVGZEWTPZAIZNZUEZCDQZE - FIZUDZJZXQXCKLZMZXQXFUBZXHNZOZBDVGZEXQPZGIZNZUEZFDQZRZAGSZOZGUFAUFXPAUOWS - YMAGYKXOYIRZFDQCDQWSYLXOYICFDDUGWSYNYLCFDDWSWTDJZXQDJZRZYNYLWSYQYNRZRZXLY - FXMYGYSEXGPEYBPXLYFYSEXGYBYSWTXQKLZMZXGYBNZXQWTKLZMZYSXJUUAUUBOBDXQBFSZXE - UUAXIUUBUUEXDYTXCXQWTKUHUIUUEXHYBXGXCXQXFUJUKULYRXKWSXBXKXNYIYQUMTWSYOYPY - NUNZUPYSUUDYBXGNZUUBYSYDUUDUUGOBDWTBCSZYAUUDYCUUGUUHXTUUCXCWTXQKUHUIUUHXH - XGYBXCWTXFUJUKULYRYEWSXSYEYHXOYQUQTWSYOYPYNURZUPYBXGUSUTYSYTUUDOZUUAUUDVA - YSWTHJZXQHJZUUJYSDHWTWSYRVBZUUIVCYSDHXQUUMUUFVCHKVDUUKUULRUUJVEHKWTXQVFVH - VIYTUUDVJVKVLVMYSEXFWTYSXBEXFJYRXBWSXBXKXNYIYQVNTEXAVOVRZVSYSEXFXQUUNVSVP - YRXNWSXBXKXNYIYQVQTYRYHWSXSYEYHXOYQVTTVPWAWBWCWDXPYJAGYLXPXBXKXLYGNZUEZCD - QYJYLXOUUPCDYLXNUUOXBXKXMYGXLWEWFWGUUPYICFDCFSZXBXSXKYEUUOYHUUQXAXREWTXQW - HWIUUQXJYDBDUUQXEYAXIYCUUQXDXTWTXQXCKWJUIUUQXGYBXHWTXQXFUJWKULWLUUQXLYFYG - EWTXQWMWKWNWOWPWRWQ $. - + funbreq.1 $e |- A e. _V $. + funbreq.2 $e |- B e. _V $. + funbreq.3 $e |- C e. _V $. + $( An equality condition for functions. (Contributed by Scott Fenton, + 18-Feb-2013.) $) + funbreq $p |- ( ( Fun F /\ A F B ) -> ( A F C <-> B = C ) ) $= + ( wfun wbr wa wceq fununiq expdimp wi breq2 biimpcd adantl impbid ) DHZAB + DIZJACDIZBCKZSTUAUBABCDEFGLMTUBUANSUBTUABCADOPQR $. $} ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d A d $. $d a e $. - $d A e $. $d A f $. $d a g $. $d A g $. $d a u $. $d A u $. $d a v $. - $d A v $. $d a x $. $d A x $. $d a y $. $d A y $. $d b x $. $d b y $. - $d c d $. $d c e $. $d c f $. $d c g $. $d c u $. $d c v $. $d c x $. - $d c y $. $d d e $. $d d f $. $d d u $. $d d v $. $d d y $. $d e f $. - $d e g $. $d e u $. $d e v $. $d e x $. $d e y $. $d f g $. $d f u $. - $d f v $. $d f x $. $d f y $. $d g u $. $d g v $. $d g x $. $d g y $. - $d u v $. $d u x $. $d u y $. $d v y $. $d x y $. - nosupcbv.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Lemma to change bound variables in a surreal supremum. (Contributed by - Scott Fenton, 9-Aug-2024.) $) - nosupcbv $p |- S = if ( E. a e. A A. b e. A -. a . } ) , - ( c e. { d | E. e e. A ( d e. dom e /\ - A. f e. A ( -. f ( e |` suc d ) = ( f |` suc d ) ) ) } - |-> ( iota a E. e e. A ( c e. dom e /\ - A. f e. A ( -. f ( e |` suc c ) = ( f |` suc c ) ) /\ - ( e ` c ) = a ) ) ) ) $= - ( cv cslt wral wrex cres wceq wbr wn crio cdm c2o cop csn wcel csuc wi wa - cun cab cfv w3a cio cmpt cif breq1 notbid ralbidv breq2 cbvralvw cbvrexvw - bitrdi cbvriotavw dmeqi opeq1i sneqi uneq12i eleq1w suceq reseq2d eqeq12d - weq imbi2d fveqeq2 3anbi123d rexbidv iotabidv eqeq2 3anbi3d eleq2d reseq1 - dmeq eqeq1d imbi12d eqeq2d fveq1 cbviotavw eqtrdi cbvmptv anbi12d mpteq1i - cbvabv eqtri ifbieq12i ) FAOZBOZPUAZUBZBEQZAERZXBAEUCZXDUDZUEUFZUGZULZIWS - DOZUDZUHZCOZXIPUAZUBZXIWSUIZSZXLXOSZTZUJZCEQZUKZDERZBUMZIOZXJUHZXNXIYDUIZ - SZXLYFSZTZUJZCEQZYDXIUNWRTZUOZDERZAUPZUQZURJOZKOZPUAZUBZKEQZJERZUUAJEUCZU - UCUDZUEUFZUGZULZLMOZGOZUDZUHZHOZUUIPUAZUBZUUIUUHUIZSZUULUUOSZTZUJZHEQZUKZ - GERZMUMZLOZUUJUHZUUNUUIUVDUIZSZUULUVFSZTZUJZHEQZUVDUUIUNZYQTZUOZGERZJUPZU - QZURNXCUUBXHYPUUGUVQXBUUAAJEAJVOZXBYQWSPUAZUBZBEQUUAUVRXAUVTBEUVRWTUVSWRY - QWSPUSUTVAUVTYTBKEBKVOUVSYSWSYRYQPVBUTVCVEZVDXDUUCXGUUFXBUUAAJEUWAVFZXFUU - EXEUUDUEXDUUCUWBVGVHVIVJYPLYCUVPUQUVQILYCYOUVPILVOZYOUVDXJUHZXNXIUVFSZXLU - VFSZTZUJZCEQZUVDXIUNZWRTZUOZDERZAUPUVPUWCYNUWMAUWCYMUWLDEUWCYEUWDYKUWIYLU - WKILXJVKUWCYJUWHCEUWCYIUWGXNUWCYGUWEYHUWFUWCYFUVFXIYDUVDVLZVMUWCYFUVFXLUW - NVMVNVPVAYDUVDWRXIVQVRVSVTUWMUVOAJUVRUWMUWDUWIUWJYQTZUOZDERUVOUVRUWLUWPDE - UVRUWKUWOUWDUWIWRYQUWJWAWBVSUWPUVNDGEDGVOZUWDUVEUWIUVKUWOUVMUWQXJUUJUVDXI - UUIWEZWCUWQUWIXLUUIPUAZUBZUVGUWFTZUJZCEQUVKUWQUWHUXBCEUWQXNUWTUWGUXAUWQXM - UWSXIUUIXLPVBUTZUWQUWEUVGUWFXIUUIUVFWDWFWGVAUXBUVJCHECHVOZUWTUUNUXAUVIUXD - UWSUUMXLUULUUIPUSUTZUXDUWFUVHUVGXLUULUVFWDWHWGVCVEUWQUWJUVLYQUVDXIUUIWIWF - VRVDVEWJWKWLLYCUVCUVPYBUVBBMBMVOZYBUUHXJUHZXNXIUUOSZXLUUOSZTZUJZCEQZUKZDE - RUVBUXFYAUXMDEUXFXKUXGXTUXLBMXJVKUXFXSUXKCEUXFXRUXJXNUXFXPUXHXQUXIUXFXOUU - OXIWSUUHVLZVMUXFXOUUOXLUXNVMVNVPVAWMVSUXMUVADGEUWQUXGUUKUXLUUTUWQXJUUJUUH - UWRWCUWQUXLUWTUUPUXITZUJZCEQUUTUWQUXKUXPCEUWQXNUWTUXJUXOUXCUWQUXHUUPUXIXI - UUIUUOWDWFWGVAUXPUUSCHEUXDUWTUUNUXOUURUXEUXDUXIUUQUUPXLUULUUOWDWHWGVCVEWM - VDVEWOWNWPWQWP $. + br1steq.1 $e |- A e. _V $. + br1steq.2 $e |- B e. _V $. + $( Uniqueness condition for the binary relation ` 1st ` . (Contributed by + Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, + 3-May-2015.) $) + br1steq $p |- ( <. A , B >. 1st C <-> C = A ) $= + ( cvv wcel cop c1st wbr wceq wb br1steqg mp2an ) AFGBFGABHCIJCAKLDEABCFFM + N $. + + $( Uniqueness condition for the binary relation ` 2nd ` . (Contributed by + Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, + 3-May-2015.) $) + br2ndeq $p |- ( <. A , B >. 2nd C <-> C = B ) $= + ( cvv wcel cop c2nd wbr wceq wb br2ndeqg mp2an ) AFGBFGABHCIJCBKLDEABCFFM + N $. $} ${ - $d A a b x y z g v u $. - nosupno.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( The next several theorems deal with a surreal "supremum". This surreal - will ultimately be shown to bound ` A ` below and bound the restriction - of any surreal above. We begin by showing that the given expression - actually defines a surreal number. (Contributed by Scott Fenton, - 5-Dec-2021.) $) - nosupno $p |- ( ( A C_ No /\ A e. V ) -> S e. No ) $= - ( va wcel csur cvv wa wrex cres wceq syl con0 cbday vb vz wss elex cv wbr - cslt wn wral crio cdm c2o cop csn cun csuc wi cab cfv w3a cio cmpt iftrue - cif adantr simprl wreu wrmo nomaxmo ad2antrl reu5 sylanbrc riotacl sseldd - simpl c1o 2on elexi prid2 noextend eqeltrd iffalse wfun crn funmpt iotaex - cpr a1i eqid dmmpti word ssel2 nodmon onss sseld adantrd rexlimdva abssdv - wtr wel wal simplr adantlr ontr1 mpand reseq1 onelon sylan suceloni eloni - simpllr ordsucelsuc mpbid onelss sylc resabs1d eqeq12d syl5ib expimpd vex - weq eleq1w suceq reseq2d imbi2d ralbidv anbi12d rexbidv elab bdayfo ax-mp - wb mpan adantl reximi eqeltrrd eqeltrid wex eqeq2 3anbi3d imim2d reximdva - ralimdv jcad anbi2i 3imtr4g alrimivv dftr2 sylibr cima cuni wfo funimaexg - dford5 fofun uniexd ss2abi bdayval wfn fofn fnfvima mp3an1 elssuni sstrid - ssexd elong mpbird rnmpt weu wmo spcev mp3an3 rexcom4 sylib nosupprefixmo - fvex df-eu iota2 eqcom bitrdi simprr3 adantrr norn simprr1 fvelrn syl2anc - nofun rexlimdvaa sylbird sylan2b eqsstrid syl3anbrc pm2.61ian sylan2 - elno2 ) EHKELUCZEMKZFLKEHUDUWPUWQNZFAUEZBUEZUGUFUHBEUIZAEOZUXAAEUJZUXCUKU - LUMUNUOZGUWTDUEZUKZKZCUEZUXEUGUFUHZUXEUWTUPZPZUXHUXJPZQZUQZCEUIZNZDEOZBUR - ZGUEZUXFKZUXIUXEUXSUPZPZUXHUYAPZQZUQZCEUIZUXSUXEUSZUWSQZUTZDEOZAVAZVBZVDZ - LIUXBUWRUYMLKUXBUWRNZUYMUXDLUXBUYMUXDQUWRUXBUXDUYLVCVEUYNUXCLKUXDLKUYNELU - XCUXBUWPUWQVFUYNUXAAEVGZUXCEKUYNUXBUXAAEVHZUYOUXBUWRVOUWPUYPUXBUWQABEVIVJ - UXAAEVKVLUXAAEVMRVNUXCULVPULULSVQVRVSVTRWAUXBUHZUWRNZUYMUYLLUYQUYMUYLQUWR - UXBUXDUYLWBVEUYRUYLWCZUYLUKZSKZUYLWDZVPULWGZUCZUYLLKUYSUYRGUXRUYKWEWHUWRV - UAUYQUWRUYTUXRSGUXRUYKUYLUYJAWFUYLWIZWJUWRUXRSKZUXRWKZUWPVUGUWQUWPUXRSUCU - XRWSZVUGUWPUXQBSUWPUXPUWTSKZDEUWPUXEEKZNZUXGVUIUXOVUKUXFSUWTVUKUXFSKZUXFS - UCVUKUXELKZVULELUXEWLZUXEWMRZUXFWNRWOWPWQWRUWPJUAWTZUAUEZUXRKZNZJUEZUXRKZ - UQZUAXAJXAVUHUWPVVBJUAUWPVUPVUQUXFKZUXIUXEVUQUPZPZUXHVVDPZQZUQZCEUIZNZDEO - ZNVUTUXFKZUXIUXEVUTUPZPZUXHVVMPZQZUQZCEUIZNZDEOZVUSVVAUWPVUPVVKVVTUWPVUPN - ZVVJVVSDEVWAVUJNZVVJVVLVVRVWBVVCVVLVVIVWBVUPVVCVVLUWPVUPVUJXBVWBVULVUPVVC - NVVLUQUWPVUJVULVUPVUOXCZVUTVUQUXFXDRXEWPVWBVVCVVIVVRVWBVVCNZVVHVVQCEVWDVV - GVVPUXIVVGVVEVVMPZVVFVVMPZQVWDVVPVVEVVFVVMXFVWDVWEVVNVWFVVOVWDUXEVVMVVDVW - DVVDSKZVVMVVDKZVVMVVDUCVWDVUQSKZVWGVWBVULVVCVWIVWCUXFVUQXGXHZVUQXIRVWDVUP - VWHUWPVUPVUJVVCXKVWDVUQWKZVUPVWHYLVWDVWIVWKVWJVUQXJRVUTVUQXLRXMVVDVVMXNXO - ZXPVWDUXHVVMVVDVWLXPXQXRUUAUUCXSUUDUUBXSVURVVKVUPUXQVVKBVUQUAXTBUAYAZUXPV - VJDEVWMUXGVVCUXOVVIBUAUXFYBVWMUXNVVHCEVWMUXMVVGUXIVWMUXKVVEUXLVVFVWMUXJVV - DUXEUWTVUQYCZYDVWMUXJVVDUXHVWNYDXQYEYFYGYHYIUUEUXQVVTBVUTJXTBJYAZUXPVVSDE - VWOUXGVVLUXOVVRBJUXFYBVWOUXNVVQCEVWOUXMVVPUXIVWOUXKVVNUXLVVOVWOUXJVVMUXEU - WTVUTYCZYDVWOUXJVVMUXHVWPYDXQYEYFYGYHYIUUFUUGJUAUXRUUHUUIUXRUUNVLVEUWRUXR - MKVUFVUGYLUWRUXRTEUUJZUUKZMUWQVWRMKUWPUWQVWQMTWCZUWQVWQMKLSTUULZVWSYJLSTU - UOYKTEMUUMYMUUPYNUWRUXRUXGDEOZBURZVWRUXQVXABUXPUXGDEUXGUXOVOYOUUQUWPVXBVW - RUCUWQUWPVXABVWRUWPUXGUWTVWRKDEVUKUXFVWRUWTVUKUXFVWQKUXFVWRUCVUKUXETUSZUX - FVWQVUKVUMVXCUXFQVUNUXEUURRTLUUSZUWPVUJVXCVWQKVWTVXDYJLSTUUTYKLETUXEUVAUV - BYPUXFVWQUVCRWOWQWRVEUVDUVEUXRMUVFRUVGYQYNUWPVUDUYQUWQUWPVUBUBUEZUYKQZGUX - ROZUBURVUCGUBUXRUYKUYLVUEUVHUWPVXGUBVUCUWPVXFVXEVUCKZGUXRUXSUXRKUWPUXTUYF - NZDEOZVXFVXHUQUXQVXJBUXSGXTBGYAZUXPVXIDEVXKUXGUXTUXOUYFBGUXFYBVXKUXNUYECE - VXKUXMUYDUXIVXKUXKUYBUXLUYCVXKUXJUYAUXEUWTUXSYCZYDVXKUXJUYAUXHVXLYDXQYEYF - YGYHYIUWPVXJNZVXFUXTUYFUYGVXEQZUTZDEOZVXHVXMVXPUYKVXEQZVXFVXMUYJAUVIZVXPV - XQYLZVXMUYJAYRZUYJAUVJZVXRVXJVXTUWPVXJUYIAYRZDEOVXTVXIVYBDEUXTUYFUYGUYGQZ - VYBUYGWIUYIUXTUYFVYCUTAUYGUXSUXEUVPUWSUYGQUYHVYCUXTUYFUWSUYGUYGYSYTUVKUVL - YOUYIDAEUVMUVNYNUWPVYAVXJACDEUXSUVOVEUYJAUVQVLVXEMKVXRVXSUBXTUYJVXPAVXEMA - UBYAZUYIVXODEVYDUYHVXNUXTUYFUWSVXEUYGYSYTYHUVRYMRUYKVXEUVSUVTUWPVXPVXHUQV - XJUWPVXOVXHDEUWPVUJVXONNZUYGVXEVUCUXTUYFVXNVUJUWPUWAVYEUXEWDZVUCUYGVYEVUM - VYFVUCUCUWPVUJVUMVXOVUNUWBZUXEUWCRVYEUXEWCZUXTUYGVYFKVYEVUMVYHVYGUXEUWGRU - XTUYFVXNVUJUWPUWDUXSUXEUWEUWFVNYPUWHVEUWIUWJWQWRUWKVJUYLUWOUWLWAUWMYQUWN + $d A p x y z $. + $( Definition of domain in terms of ` 1st ` and image. (Contributed by + Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) + (Proof shortened by Peter Mazsa, 2-Oct-2022.) $) + dfdm5 $p |- dom A = ( ( 1st |` ( _V X. _V ) ) " A ) $= + ( vx vy vp vz c1st cvv cv cop wcel wex wbr wa wceq excom vex bitri anbi1i + exbii 3bitr4i cdm cxp cres cima breq1 eleq1 anbi12d br1steq equcom bitrdi + opex ceqsexv opeq1 eleq1d 3bitr3ri ancom anass brresi elvv 19.41vv elima2 + eldm2 eqriv ) BAUAZFGGUBZUCZAUDZBHZCHZIZAJZCKZDHZAJZVMVHVFLZMZDKZVHVDJVHV + GJVMEHZVIIZNZVMVHFLZVNMZMZEKZDKZCKWDCKZDKVLVQWDCDOVKWECWCDKZEKVRVHNZVSAJZ + MZEKWEVKWGWJEWBWJDVSVRVIUKVTWBVSVHFLZWIMWJVTWAWKVNWIVMVSVHFUEVMVSAUFUGWKW + HWIWKVHVRNWHVRVIVHEPCPUHBEUIQRUJULSWCEDOWIVKEVHBPZWHVSVJAVRVHVIUMUNULUOSV + PWFDVPVOVNMZWFVNVOUPVTEKCKZWAMZVNMWNWBMWMWFWNWAVNUQVOWOVNVOVMVEJZWAMWOVEV + MVHFWLURWPWNWAWPVTCKEKWNECVMUSVTECOQRQRVTWBCEUTTQSTCVHAWLVBDVHVFAWLVATVC $. + + $( Definition of range in terms of ` 2nd ` and image. (Contributed by + Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) + (Proof shortened by Peter Mazsa, 2-Oct-2022.) $) + dfrn5 $p |- ran A = ( ( 2nd |` ( _V X. _V ) ) " A ) $= + ( vx vy vp vz c2nd cvv cv cop wcel wex wbr wa wceq excom vex bitri anbi1i + exbii 3bitr4i crn cxp cres cima breq1 eleq1 anbi12d br2ndeq equcom bitrdi + opex ceqsexv opeq2 eleq1d 3bitr3ri ancom anass brresi elvv 19.41vv elima2 + elrn2 eqriv ) BAUAZFGGUBZUCZAUDZCHZBHZIZAJZCKZDHZAJZVMVIVFLZMZDKZVIVDJVIV + GJVMVHEHZIZNZVMVIFLZVNMZMZEKZDKZCKWDCKZDKVLVQWDCDOVKWECWCDKZEKVRVINZVSAJZ + MZEKWEVKWGWJEWBWJDVSVHVRUKVTWBVSVIFLZWIMWJVTWAWKVNWIVMVSVIFUEVMVSAUFUGWKW + HWIWKVIVRNWHVHVRVICPEPUHBEUIQRUJULSWCEDOWIVKEVIBPZWHVSVJAVRVIVHUMUNULUOSV + PWFDVPVOVNMZWFVNVOUPVTEKCKZWAMZVNMWNWBMWMWFWNWAVNUQVOWOVNVOVMVEJZWAMWOVEV + MVIFWLURWPWNWACEVMUSRQRVTWBCEUTTQSTCVIAWLVBDVIVFAWLVATVC $. $} ${ - $d A g $. $d A p $. $d A q $. $d A u $. $d A v $. $d A y $. $d A z $. - $d g u $. $d g v $. $d g y $. $d p q $. $d p u $. $d p v $. $d p y $. - $d p z $. $d q u $. $d q v $. $d q y $. $d q z $. $d u v $. $d u y $. - $d u z $. $d v y $. $d v z $. $d y z $. - nosupdm.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( The domain of the surreal supremum when there is no maximum. The - primary point of this theorem is to change bound variable. (Contributed - by Scott Fenton, 6-Dec-2021.) $) - nosupdm $p |- ( -. E. x e. A A. y e. A -. x - dom S = { z | E. p e. A ( z e. dom p /\ - A. q e. A ( -. q ( p |` suc z ) = ( q |` suc z ) ) ) } ) $= - ( cv cslt wn wral wrex cdm cres wceq wi wbr wcel csuc wa cab cfv w3a cmpt - cio crio c2o cop csn cun cif iffalse eqtrid iotaex eqid dmmpti eqtrdi weq - dmeqd dmeq eleq2d breq1 notbid reseq1 eqeq2d imbi12d breq2 eqeq1d ralbidv - cbvralvw bitrid anbi12d cbvrexvw eleq1w reseq2d eqeq12d imbi2d rexbidv - suceq cbvabv ) ALZBLZMUANBFOZAFPZNZGQZWFELZQZUBZDLZWKMUAZNZWKWFUCZRZWNWQR - ZSZTZDFOZUDZEFPZBUEZCLZJLZQZUBZILZXGMUAZNZXGXFUCZRZXJXMRZSZTZIFOZUDZJFPZC - UEWIWJHXEHLZWLUBWPWKYAUCZRWNYBRSTDFOYAWKUFWESUGEFPZAUIZUHZQXEWIGYEWIGWHWG - AFUJZYFQUKULUMUNZYEUOYEKWHYGYEUPUQVCHXEYDYEYCAURYEUSUTVAXDXTBCXDWFXHUBZXL - XGWQRZXJWQRZSZTZIFOZUDZJFPBCVBZXTXCYNEJFEJVBZWMYHXBYMYPWLXHWFWKXGVDVEXBXJ - WKMUAZNZWRYJSZTZIFOYPYMXAYTDIFDIVBZWPYRWTYSUUAWOYQWNXJWKMVFVGUUAWSYJWRWNX - JWQVHVIVJVNYPYTYLIFYPYRXLYSYKYPYQXKWKXGXJMVKVGYPWRYIYJWKXGWQVHVLVJVMVOVPV - QYOYNXSJFYOYHXIYMXRBCXHVRYOYLXQIFYOYKXPXLYOYIXNYJXOYOWQXMXGWFXFWCZVSYOWQX - MXJUUBVSVTWAVMVPWBVOWDVA $. + $d A z $. $d B z $. $d C z $. $d D z $. + $( Alternate way of saying that an ordered pair is in a composition. + (Contributed by Scott Fenton, 6-May-2018.) $) + opelco3 $p |- ( <. A , B >. e. ( C o. D ) <-> + B e. ( C " ( D " { A } ) ) ) $= + ( vz cop ccom wcel wbr csn cima df-br cvv wa relco wn c0 ima0 wex wb wceq + brrelex12i snprc noel imaeq2 imaeq2d eqtri eqtrdi eleq2d sylbi con4i elex + imaeq2i mtbiri jca wrex df-rex elimasng elvd bitr4di adantr anbi1d exbidv + cv bitr2id brcog elimag adantl 3bitr4d pm5.21nii bitr3i ) ABFCDGZHABVLIZB + CDAJZKZKZHZABVLLVMAMHZBMHZNZVQABVLCDOUBVQVRVSVRVQVRPVNQUAZVQPAUCWAVQBQHBU + DWAVPQBWAVPCDQKZKZQWAVOWBCVNQDUEUFWCCQKQWBQCDRUMCRUGUHUIUNUJUKBVPULUOVTAE + VDZDIZWDBCIZNZESZWFEVOUPZVMVQWIWDVOHZWFNZESVTWHWFEVOUQVTWKWGEVTWJWEWFVRWJ + WETVSVRWJAWDFDHZWEVRWJWLTEDAWDMMURUSAWDDLUTVAVBVCVEEABCDMMVFVSVQWITVREBCV + OMVGVHVIVJVK $. $} ${ - $d A g u v x y $. $d O u y $. - nosupbday.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Birthday bounding law for surreal supremum. (Contributed by Scott - Fenton, 5-Dec-2021.) $) - nosupbday $p |- ( ( ( A C_ No /\ A e. _V ) /\ - ( O e. On /\ ( bday " A ) C_ O ) ) -> - ( bday ` S ) C_ O ) $= - ( csur wss wcel wa cbday cfv cdm wceq adantr syl cv cvv con0 cima nosupno - bdayval cslt wbr wn wral wrex crio csuc c2o cop csn cun cres cab w3a cmpt - cio cif iftrue eqtrid dmeqd 2oex dmsnop uneq2i dmun df-suc 3eqtr4i eqtrdi - wi word simprrl eloni simprll wreu wrmo simpl nomaxmo adantl reu5 adantrr - sylanbrc riotacl sseldd simprrr wfn wfo bdayfo fofn ax-mp fnfvima mp3an2i - eqeltrrd ordsucss sylc eqsstrd iffalse iotaex eqid simplrl ssel2 ad4ant14 - dmmpti simplrr mp3an1 onelss sseld adantrd rexlimdva abssdv pm2.61ian ) E - JKZEUALZMZHUBLZNEUCZHKZMZMZFNOZFPZHYBFJLZYCYDQXQYEYAABCDEFGUAIUDRFUESATZB - TZUFUGUHBEUIZAEUJZYBYDHKYIYBMZYDYHAEUKZPZULZHYIYDYMQYBYIYDYKYLUMUNUOZUPZP - ZYMYIFYOYIFYIYOGYGDTZPZLZCTZYQUFUGUHZYQYGULZUQYTUUBUQQVMCEUIZMZDEUJZBURZG - TZYRLUUAYQUUGULZUQYTUUHUQQVMCEUIUUGYQOYFQUSDEUJZAVAZUTZVBZYOIYIYOUUKVCVDV - EYLYNPZUPYLYLUOZUPYPYMUUMUUNYLYLUMVFVGVHYKYNVIYLVJVKVLRYJHVNZYLHLYMHKYJXR - UUOYIXQXRXTVOHVPSYJYKNOZYLHYJYKJLUUPYLQYJEJYKYIXOXPYAVQZYJYHAEVRZYKELZYIX - QUURYAYIXQMYIYHAEVSZUURYIXQVTXQUUTYIXOUUTXPABEWARWBYHAEWCWEWDYHAEWFSZWGYK - UESYJXSHUUPYIXQXRXTWHNJWIZYJXOUUSUUPXSLJUBNWJUVBWKJUBNWLWMZUUQUVAJENYKWNW - OWGWPYLHWQWRWSYIUHZYBMYDUUFHUVDYDUUFQYBUVDYDUUKPUUFUVDFUUKUVDFUULUUKIYIYO - UUKWTVDVEGUUFUUJUUKUUIAXAUUKXBXFVLRYBUUFHKUVDYBUUEBHYBUUDYGHLZDEYBYQELZMZ - YSUVEUUCUVGYRHYGUVGXRYRHLYRHKXQXRXTUVFXCUVGYQNOZYRHUVGYQJLZUVHYRQXOUVFUVI - XPYAEJYQXDXEYQUESUVGXSHUVHXQXRXTUVFXGXOUVFUVHXSLZXPYAUVBXOUVFUVJUVCJENYQW - NXHXEWGWPHYRXIWRXJXKXLXMWBWSXNWS $. + $d A x $. $d B p x y z $. $d R p x y z $. + $( Quantifier-free expression saying that a class is a member of an image. + (Contributed by Scott Fenton, 8-May-2018.) $) + elima4 $p |- ( A e. ( R " B ) <-> ( R i^i ( B X. { A } ) ) =/= (/) ) $= + ( vx vp vy vz wcel cvv csn cxp cin c0 wne wceq cv wex wa 3bitri exbii xp0 + cima elex xpeq2 eqtrdi ineq2d in0 necon3i snnzb sylibr sneq xpeq2d neeq1d + eleq1 cop elin ancom anbi1i anass 2exbii 19.41vv bitr3i exrot3 weq anbi2d + elxp opex ceqsexv an12 velsn vex opeq2 eleq1d n0 elima3 vtoclbg pm5.21nii + 3bitr4ri ) ACBUBZHZAIHZCBAJZKZLZMNZAVSUCWEWBMNWAWBMWDMWBMOZWDCMLMWFWCMCWF + WCBMKMWBMBUDBUAUEUFCUGUEUHAUIUJDPZVSHZCBWGJZKZLZMNZVTWEDAIWGAVSUNWGAOZWKW + DMWMWJWCCWMWIWBBWGAUKULUFUMEPZWKHZEQZFPZBHZWQWGUOZCHZRZFQZWLWHWPWNWQGPZUO + ZOZWRXCWIHZRZRZGQFQZWNCHZRZEQZXEXGXJRZRZEQZGQZFQZXBWOXKEWOXJWNWJHZRXRXJRX + KWNCWJUPXJXRUQXRXIXJFGWNBWIVFURSTXLXNGQFQZEQXQXSXKEXSXHXJRZGQFQXKXTXNFGXE + XGXJUSUTXHXJFGVAVBTXNEFGVCVBXPXAFXPXGXDCHZRZGQGDVDZWRYARZRZGQXAXOYBGXMYBE + XDWQXCVGXEXJYAXGWNXDCUNVEVHTYBYEGYBWRXFYARRXFYDRYEWRXFYAUSWRXFYAVIXFYCYDG + WGVJURSTYDXAGWGDVKZYCYAWTWRYCXDWSCXCWGWQVLVMVEVHSTSEWKVNFWGCBYFVOVRVPVQ + $. $} + $( The value of the converse of ` 1st ` restricted to a singleton. + (Contributed by Scott Fenton, 2-Jul-2020.) $) + fv1stcnv $p |- ( ( X e. A /\ Y e. V ) -> + ( `' ( 1st |` ( A X. { Y } ) ) ` X ) = <. X , Y >. ) $= + ( wcel wa c1st csn cxp cres ccnv cfv cop wceq wbr wb cvv bitrd mpbird wf1o + snidg anim2i eqid jctir opex brcnvg mpan2 brres adantr opelxp anbi1i anbi2d + br1steqg bitrid wfn 1stconst f1ocnv f1ofn 3syl simpl fnbrfvb syl2an2 ) CAEZ + DBEZFZCGADHZIZJZKZLCDMZNZCVJVIOZVEVLVCDVFEZFZCCNZFZVEVNVOVDVMVCDBUAUBCUCUDV + EVLVJVGEZVJCGOZFZVPVCVLVSPVDVCVLVJCVHOZVSVCVJQEVLVTPCDUECVJAQVHUFUGVGVJCGAU + HRUIVSVNVRFVEVPVQVNVRCDAVFUJUKVEVRVOVNCDCABUMULUNRSVDVIAUOZVCVCVKVLPVDVGAVH + TAVGVITWAADBUPVGAVHUQAVGVIURUSVCVDUTACVJVIVAVBS $. + + $( The value of the converse of ` 2nd ` restricted to a singleton. + (Contributed by Scott Fenton, 2-Jul-2020.) $) + fv2ndcnv $p |- ( ( X e. V /\ Y e. A ) -> + ( `' ( 2nd |` ( { X } X. A ) ) ` Y ) = <. X , Y >. ) $= + ( wcel wa c2nd csn cxp cres ccnv cfv cop wceq wbr wb wf1o cvv adantl bitrd + snidg anim1i eqid jctir wfn 2ndconst adantr f1ocnv f1ofn 3syl sylancom opex + fnbrfvb brcnvg mpan2 brres opelxp anbi1i br2ndeqg anbi2d bitrid mpbird ) CB + EZDAEZFZDGCHZAIZJZKZLCDMZNZCVFEZVDFZDDNZFZVEVMVNVCVLVDCBUAUBDUCUDVEVKDVJVIO + ZVOVCVDVIAUEZVKVPPVEVGAVHQZAVGVIQVQVCVRVDCABUFUGVGAVHUHAVGVIUIUJADVJVIUMUKV + EVPVJDVHOZVOVDVPVSPZVCVDVJREVTCDULDVJARVHUNUOSVEVSVJVGEZVJDGOZFZVOVDVSWCPVC + VGVJDGAUPSWCVMWBFVEVOWAVMWBCDVFAUQURVEWBVNVMCDDBAUSUTVATTTVB $. + ${ - $d A g $. $d A p $. $d A u $. $d A v $. $d A x $. $d A y $. $d G g $. - $d G p $. $d g u $. $d G u $. $d g v $. $d G v $. $d g x $. $d G x $. - $d g y $. $d G y $. $d p u $. $d p v $. $d U p $. $d U u $. $d u v $. - $d U v $. $d u x $. $d U x $. $d u y $. $d v x $. $d v y $. - nosupfv.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( The value of surreal supremum when there is no maximum. (Contributed by - Scott Fenton, 5-Dec-2021.) $) - nosupfv $p |- ( ( -. E. x e. A A. y e. A -. x ( U |` suc G ) = ( v |` suc G ) ) ) ) -> - ( S ` G ) = ( U ` G ) ) $= - ( vp cv cslt wn wral wrex wcel cres wceq w3a wbr csur wss cvv wa cdm csuc - wi cfv cab cio cmpt crio c2o cop csn cun cif iffalse eqtrid fveq1d simp32 - 3ad2ant1 eleq2d breq2 notbid reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev - dmeq 3impb weq cbvrexvw sylibr eleq1 suceq reseq2d eqeq12d imbi2d rexbidv - 3ad2ant3 elabd 3anbi123d iotabidv eqid iotaex fvmpt syl simp1 simp2 simp3 - fveqeq2 eqidd fveq1 syl13anc weu fvex wex wmo eqeq2 3anbi3d mp3an3 reximi - wb spcev rexcom4 sylib nosupprefixmo adantr 3ad2ant2 df-eu sylanbrc iota2 - sylancr mpbid 3eqtrd ) ALZBLZMUANBEOZAEPZNZEUBUCZEUDQZUEZGEQZIGUFZQZCLZGM - UAZNZGIUGZRZYKYNRZSZUHZCEOZTZTZIFUIZIHYADLZUFZQZYKUUCMUAZNZUUCYAUGZRZYKUU - HRZSZUHZCEOZUEZDEPZBUJZHLZUUDQZUUGUUCUUQUGZRZYKUUSRZSZUHZCEOZUUQUUCUIXTSZ - TZDEPZAUKZULZUIZIUUDQZUUGUUCYNRZYPSZUHZCEOZIUUCUIZXTSZTZDEPZAUKZIGUIZYDYG - UUBUVJSYTYDIFUVIYDFYCYBAEUMZUWBUFUNUOUPUQZUVIURUVIJYCUWCUVIUSUTVAVCUUAIUU - PQUVJUVTSUUAUUOUVKUVOUEZDEPZBIYIYDYGYHYJYSVBYTYDUWEYGYTIKLZUFZQZYKUWFMUAZ - NZUWFYNRZYPSZUHZCEOZUEZKEPZUWEYHYJYSUWPUWOYJYSUEKGEUWFGSZUWHYJUWNYSUWQUWG - YIIUWFGVMVDUWQUWMYRCEUWQUWJYMUWLYQUWQUWIYLUWFGYKMVEVFUWQUWKYOYPUWFGYNVGVH - VIVJVKVLVNUWDUWODKEDKVOZUVKUWHUVOUWNUWRUUDUWGIUUCUWFVMVDUWRUVNUWMCEUWRUUG - UWJUVMUWLUWRUUFUWIUUCUWFYKMVEVFUWRUVLUWKYPUUCUWFYNVGVHVIVJVKVPVQZWDYAISZU - UNUWDDEUWTUUEUVKUUMUVOYAIUUDVRUWTUULUVNCEUWTUUKUVMUUGUWTUUIUVLUUJYPUWTUUH - YNUUCYAIVSZVTUWTUUHYNYKUXAVTWAWBVJVKWCWEHIUVHUVTUUPUVIUUQISZUVGUVSAUXBUVF - UVRDEUXBUURUVKUVDUVOUVEUVQUUQIUUDVRUXBUVCUVNCEUXBUVBUVMUUGUXBUUTUVLUVAYPU - XBUUSYNUUCUUQIVSZVTUXBUUSYNYKUXCVTWAWBVJUUQIXTUUCWOWFWCWGUVIWHUVSAWIWJWKU - UAUVKUVOUVPUWASZTZDEPZUVTUWASZYTYDUXFYGYTYHYJYSUWAUWASZUXFYHYJYSWLYHYJYSW - MYHYJYSWNYTUWAWPUXEYJYSUXHTDGEUUCGSZUVKYJUVOYSUXDUXHUXIUUDYIIUUCGVMVDUXIU - VNYRCEUXIUUGYMUVMYQUXIUUFYLUUCGYKMVEVFUXIUVLYOYPUUCGYNVGVHVIVJUXIUVPUWAUW - AIUUCGWQVHWFVLWRWDUUAUWAUDQUVSAWSZUXFUXGXGIGWTUUAUVSAXAZUVSAXBZUXJYTYDUXK - YGYTUWEUXKUWSUWEUVRAXAZDEPUXKUWDUXMDEUVKUVOUVPUVPSZUXMUVPWHUVRUVKUVOUXNTA - UVPIUUCWTXTUVPSUVQUXNUVKUVOXTUVPUVPXCXDXHXEXFUVRDAEXIXJWKWDYGYDUXLYTYEUXL - YFACDEIXKXLXMUVSAXNXOUVSUXFAUWAUDXTUWASZUVRUXEDEUXOUVQUXDUVKUVOXTUWAUVPXC - XDWCXPXQXRXS $. + $d R x y $. $d A x y $. + $( The image is unaffected by intersection with the domain. (Contributed + by Scott Fenton, 17-Dec-2021.) $) + imaindm $p |- ( R " A ) = ( R " ( A i^i dom R ) ) $= + ( vx vy cima cdm cin cv wbr wrex wcel wa vex breldm pm4.71ri rexbii rexin + bitr4i elima 3bitr4i eqriv ) CBAEZBABFZGZEZDHZCHZBIZDAJZUHDUDJZUGUBKUGUEK + UIUFUCKZUHLZDAJUJUHULDAUHUKUFUGBDMCMZNOPUHDAUCQRDUGBAUMSDUGBUDUMSTUA $. $} + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Set induction (or epsilon induction) +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + ${ - $d A a $. $d a g $. $d A g $. $d A p $. $d a u $. $d A u $. $d a v $. - $d A v $. $d a x $. $d A x $. $d a y $. $d A y $. $d G a $. $d G p $. - $d g u $. $d G u $. $d g v $. $d G v $. $d g x $. $d g y $. $d G y $. - $d p u $. $d p v $. $d S a $. $d U a $. $d U p $. $d U u $. $d u v $. - $d U v $. $d u x $. $d U x $. $d u y $. $d v x $. $d v y $. $d x y $. - nosupres.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( A restriction law for surreal supremum when there is no maximum. - (Contributed by Scott Fenton, 5-Dec-2021.) $) - nosupres $p |- ( ( -. E. x e. A A. y e. A -. x ( U |` suc G ) = ( v |` suc G ) ) ) ) -> - ( S |` suc G ) = ( U |` suc G ) ) $= - ( vp cv cslt wral wcel wa cdm cres wceq wi wbr wrex csur wss cvv csuc w3a - va cfv cin dmres word nosupno 3ad2ant2 nodmord syl cab dmeq eleq2d notbid - wn breq2 reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev weq cbvrexvw sylibr - 3impb wb eleq1 suceq reseq2d eqeq12d imbi2d rexbidv elabg mpbird 3ad2ant3 - cio cmpt crio c2o cop csn cun cif iffalse eqtrid dmeqd iotaex eqid dmmpti - eqtrdi 3ad2ant1 eleqtrrd ordsucss df-ss sylib simp2l simp31 sseldd simp32 - sylc eqtr4d simpl1 simpl2 simpl31 sselda con0 nodmon onelon syl2anc eloni - ordsuc imp simpl33 resabs1 syl5ib imim2d ralimdv nosupfv syl113anc fvresd - simpr 3eqtr4d ex sylbid ralrimiv wfun nofun funres 3syl eqfunfv mpbir2and - ) ALZBLZMUAVABENZAEUBZVAZEUCUDZEUEOZPZGEOZIGQZOZCLZGMUAZVAZGIUFZRZUUJUUMR - ZSZTZCENZUGZUGZFUUMRZUUNSZUVAQZUUNQZSZUHLZUVAUIZUVFUUNUIZSZUHUVCNZUUTUVCU - UMUVDUUTUVCUUMFQZUJZUUMFUUMUKUUTUUMUVKUDZUVLUUMSUUTUVKULZIUVKOUVMUUTFUCOZ - UVNUUFUUCUVOUUSABCDEFHUEJUMUNZFUOUPUUTIYTDLZQZOZUUJUVQMUAZVAZUVQYTUFZRZUU - JUWBRZSZTZCENZPZDEUBZBUQZUVKUUSUUCIUWJOZUUFUUSUWKIUVROZUWAUVQUUMRZUUOSZTZ - CENZPZDEUBZUUSIKLZQZOZUUJUWSMUAZVAZUWSUUMRZUUOSZTZCENZPZKEUBZUWRUUGUUIUUR - UXIUXHUUIUURPKGEUWSGSZUXAUUIUXGUURUXJUWTUUHIUWSGURUSUXJUXFUUQCEUXJUXCUULU - XEUUPUXJUXBUUKUWSGUUJMVBUTUXJUXDUUNUUOUWSGUUMVCVDVEVFVGVHVLUWQUXHDKEDKVIZ - UWLUXAUWPUXGUXKUVRUWTIUVQUWSURUSUXKUWOUXFCEUXKUWAUXCUWNUXEUXKUVTUXBUVQUWS - UUJMVBUTUXKUWMUXDUUOUVQUWSUUMVCVDVEVFVGVJVKUUIUUGUWKUWRVMUURUWIUWRBIUUHYT - ISZUWHUWQDEUXLUVSUWLUWGUWPYTIUVRVNUXLUWFUWOCEUXLUWEUWNUWAUXLUWCUWMUWDUUOU - XLUWBUUMUVQYTIVOZVPUXLUWBUUMUUJUXMVPVQVRVFVGVSVTUNWAWBUUCUUFUVKUWJSUUSUUC - UVKHUWJHLZUVROUWAUVQUXNUFZRUUJUXORSTCENUXNUVQUIYSSUGDEUBZAWCZWDZQUWJUUCFU - XRUUCFUUBUUAAEWEZUXSQWFWGWHWIZUXRWJUXRJUUBUXTUXRWKWLWMHUWJUXQUXRUXPAWNUXR - WOWPWQWRWSIUVKWTXGUUMUVKXAXBWLZUUTUVDUUMUUHUJZUUMGUUMUKUUTUUMUUHUDZUYBUUM - SUUTUUHULZUUIUYCUUTGUCOZUYDUUTEUCGUUCUUDUUEUUSXCUUCUUFUUGUUIUURXDXEZGUOUP - UUCUUFUUGUUIUURXFZIUUHWTXGZUUMUUHXAXBWLXHUUTUVIUHUVCUUTUVFUVCOUVFUUMOZUVI - UUTUVCUUMUVFUYAUSUUTUYIUVIUUTUYIPZUVFFUIZUVFGUIZUVGUVHUYJUUCUUFUUGUVFUUHO - UULGUVFUFZRZUUJUYMRZSZTZCENZUYKUYLSUUCUUFUUSUYIXIUUCUUFUUSUYIXJUUGUUIUURU - UCUUFUYIXKUUTUUMUUHUVFUYHXLUYJUYMUUMUDZUURUYRUUTUYIUYSUUTUUMULZUYIUYSTUUT - IULZUYTUUTIXMOZVUAUUTUUHXMOZUUIVUBUUTUYEVUCUYFGXNUPUYGUUHIXOXPIXQUPIXRXBU - VFUUMWTUPXSUUGUUIUURUUCUUFUYIXTUYSUUQUYQCEUYSUUPUYPUULUUPUUNUYMRZUUOUYMRZ - SUYSUYPUUNUUOUYMVCUYSVUDUYNVUEUYOGUYMUUMYAUUJUYMUUMYAVQYBYCYDXGABCDEFGHUV - FJYEYFUYJUVFUUMFUUTUYIYHZYGUYJUVFUUMGVUFYGYIYJYKYLUUTUVAYMZUUNYMZUVBUVEUV - JPVMUUTUVOFYMVUGUVPFYNUUMFYOYPUUTUYEGYMVUHUYFGYNUUMGYOYPUHUVAUUNYQXPYR $. + $d ph y z $. $d x y z $. + setinds.1 $e |- ( A. y e. x [. y / x ]. ph -> ph ) $. + $( Principle of set induction (or ` _E ` -induction). If a property passes + from all elements of ` x ` to ` x ` itself, then it holds for all + ` x ` . (Contributed by Scott Fenton, 10-Mar-2011.) $) + setinds $p |- ph $= + ( vz cv cvv wcel vex cab wss wi wceq setind wsbc wral dfss3 df-sbc ralbii + nfsbc1v nfcv nfralw nfim raleq sbceq1a imbi12d chvarfv sylbir sylbi sylib + mpg eqcomi abeq2i mpbi ) BFZGHABIABGABJZGEFZUPKZUQUPHZLUPGMEEUPNURABUQOZU + SURCFZUPHZCUQPZUTCUQUPQVCABVAOZCUQPZUTVDVBCUQABVARSVDCUOPZALVEUTLBEVEUTBV + DBCUQBUQUAABVATUBABUQTUCUOUQMVFVEAUTVDCUOUQUDABUQUEUFDUGUHUIABUQRUJUKULUM + UN $. $} ${ - nosupbnd1.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - - ${ - $d A g $. $d A h $. $d A p $. $d A u $. $d A v $. $d A x $. - $d A y $. $d g u $. $d g v $. $d g x $. $d g y $. $d h p $. - $d h u $. $d h v $. $d h x $. $d h y $. $d p u $. $d p v $. - $d p x $. $d p y $. $d S h $. $d S p $. $d U h $. $d U p $. - $d u v $. $d U v $. $d u x $. $d u y $. $d v x $. $d v y $. - $d x y $. - $( Lemma for ~ nosupbnd1 . Establish a soft upper bound. (Contributed - by Scott Fenton, 5-Dec-2021.) $) - nosupbnd1lem1 $p |- ( ( -. E. x e. A A. y e. A -. x -. S ( ph <-> ps ) ) $. + setinds2f.3 $e |- ( A. y e. x ps -> ph ) $. + $( ` _E ` induction schema, using implicit substitution. (Contributed by + Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, + 11-Dec-2016.) $) + setinds2f $p |- ph $= + ( cv wsbc wral wsb sbsbc sbiev bitr3i ralbii sylbi setinds ) ACDACDHIZDCH + ZJBDSJARBDSRACDKBACDLABCDEFMNOGPQ $. + $} - ${ - $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. - $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. - $d v x $. $d v y $. $d W v $. $d x y $. - $( Lemma for ~ nosupbnd1 . When there is no maximum, if any member of - ` A ` is a prolongment of ` S ` , then so are all elements of ` A ` - above it. (Contributed by Scott Fenton, 5-Dec-2021.) $) - nosupbnd1lem2 $p |- ( ( -. E. x e. A A. y e. A -. x - ( W |` dom S ) = S ) $= - ( cv cslt wbr wn csur cvv wcel wa cres wceq wral wss cdm w3a simp3rr wi - wrex con0 simp2l simp3rl sseldd simp3ll nosupno 3ad2ant2 nodmon syl3anc - sltres mtod simp3lr breq2d mtbid nosupbnd1lem1 syld3an3 noreson syl2anc - syl wb wor sltso sotrieq2 mpan mpbir2and ) AKBKLMNBEUAAEUGNZEOUBZEPQZRZ - GEQZGFUCZSZFTZRZIEQZIGLMZNZRZRZUDZIVRSZFTZWHFLMZNZFWHLMNZWGWHVSLMZWJWGW - MWCWBWDWAVMVPUEWGIOQZGOQVRUHQZWMWCUFWGEOIVMVNVOWFUIZWBWDWAVMVPUJZUKZWGE - OGWPVQVTWEVMVPULUKWGFOQZWOVPVMWSWFABCDEFHPJUMUNZFUOVFZIGVRUQUPURWGVSFWH - LVQVTWEVMVPUSUTVAVMVPWFWBWLWQABCDEFIHJVBVCWGWHOQZWSWIWKWLRVGZWGWNWOXBWR - XAIVRVDVEWTOLVHXBWSRXCVIOWHFLVJVKVEVL $. - $} + ${ + $d x y $. $d ph y $. $d ps x $. + setinds2.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + setinds2.2 $e |- ( A. y e. x ps -> ph ) $. + $( ` _E ` induction schema, using implicit substitution. (Contributed by + Scott Fenton, 10-Mar-2011.) $) + setinds2 $p |- ph $= + ( nfv setinds2f ) ABCDBCGEFH $. + $} - ${ - $d A g $. $d A p $. $d A q $. $d A u $. $d A v $. $d A x $. - $d A y $. $d A z $. $d g u $. $d g v $. $d g x $. $d g y $. - $d p q $. $d p u $. $d p v $. $d p y $. $d p z $. $d q u $. - $d q v $. $d q x $. $d q y $. $d q z $. $d S p $. $d S q $. - $d S z $. $d U p $. $d U q $. $d u v $. $d u x $. $d u y $. - $d u z $. $d v x $. $d v y $. $d v z $. $d x y $. $d y z $. - $( Lemma for ~ nosupbnd1 . If ` U ` is a prolongment of ` S ` and in - ` A ` , then ` ( U `` dom S ) ` is not ` 2o ` . (Contributed by Scott - Fenton, 6-Dec-2021.) $) - nosupbnd1lem3 $p |- ( ( -. E. x e. A A. y e. A -. x - ( U ` dom S ) =/= 2o ) $= - ( vp vq cv wn csur wcel wa cres wceq c2o adantr vz cslt wbr wss cvv cdm - wral wrex w3a word nosupno 3ad2ant2 nodmord ordirr 3syl csuc wi simpl3l - cfv ndmfv wne c1o con0 2on elexi prid2 nosgnn0i neeq1 mpbiri neneqd syl - con4i adantl simpl2l sseldd simprl nodmon simpl3r simpll1 simpll2 simpr - c0 simpll3 nosupbnd1lem2 syl112anc eqtr4d simplr nolesgn2ores syl321anc - simprr expr ralrimiva eleq2d breq2 notbid reseq1 eqeq1d imbi12d ralbidv - dmeq anbi12d rspcev syl12anc cab nosupdm 3ad2ant1 eleq1 reseq2d eqeq12d - wb suceq imbi2d rexbidv elabg bitrd mpbird mtand neqned ) ALBLUBUCMBEUG - AEUHMZENUDZEUEOZPZGEOZGFUFZQZFRZPZUIZYDGUSZSYHYISRZYDYDOZYHFNOZYDUJYKMY - BXSYLYGABCDEFHUEIUKULZFUMYDUNUOYHYJPZYKYDJLZUFZOZKLZYOUBUCZMZYOYDUPZQZY - RUUAQZRZUQZKEUGZPZJEUHZYNYCYDGUFZOZYRGUBUCZMZGUUAQZUUCRZUQZKEUGZUUHYCYF - XSYBYJURZYJUUJYHUUJYJUUJMYIWBRZYJMYDGUTUURYISUURYISVAWBSVASVBSSVCVDVEVF - VGYIWBSVHVIVJVKVLVMYNUUOKEYNYREOZUULUUNYNUUSUULPZPZGNOYRNOYDVCOZYEYRYDQ - ZRYJUULUUNUVAENGYNXTUUTXTYAXSYGYJVNTZYNYCUUTUUQTVOUVAENYRUVDYNUUSUULVPV - OUVAYLUVBYNYLUUTYHYLYJYMTTFVQZVKUVAYEFUVCYNYFUUTYCYFXSYBYJVRTUVAXSYBYGU - UTUVCFRXSYBYGYJUUTVSXSYBYGYJUUTVTXSYBYGYJUUTWCYNUUTWAABCDEFGHYRIWDWEWFY - HYJUUTWGYNUUSUULWJGYRYDWHWIWKWLUUGUUJUUPPJGEYOGRZYQUUJUUFUUPUVFYPUUIYDY - OGWTWMUVFUUEUUOKEUVFYTUULUUDUUNUVFYSUUKYOGYRUBWNWOUVFUUBUUMUUCYOGUUAWPW - QWRWSXAXBXCYHYKUUHXJYJYHYKYDUALZYPOZYTYOUVGUPZQZYRUVIQZRZUQZKEUGZPZJEUH - ZUAXDZOZUUHXSYBYKUVRXJYGXSYDUVQYDABUACDEFHKJIXEWMXFYHYLUVBUVRUUHXJYMUVE - UVPUUHUAYDVCUVGYDRZUVOUUGJEUVSUVHYQUVNUUFUVGYDYPXGUVSUVMUUEKEUVSUVLUUDY - TUVSUVJUUBUVKUUCUVSUVIUUAYOUVGYDXKZXHUVSUVIUUAYRUVTXHXIXLWSXAXMXNUOXOTX - PXQXR $. - $} - ${ - $d A g $. $d A u $. $d A v $. $d A w $. $d A x $. $d A y $. - $d g u $. $d g v $. $d g x $. $d g y $. $d S w $. $d U u $. - $d u v $. $d u w $. $d U w $. $d u x $. $d u y $. $d v w $. - $d v x $. $d v y $. $d w x $. $d w y $. $d x y $. - $( Lemma for ~ nosupbnd1 . If ` U ` is a prolongment of ` S ` and in - ` A ` , then ` ( U `` dom S ) ` is not undefined. (Contributed by - Scott Fenton, 6-Dec-2021.) $) - nosupbnd1lem4 $p |- ( ( -. E. x e. A A. y e. A -. x - ( U ` dom S ) =/= (/) ) $= - ( vw cv cslt wbr wn wrex csur wcel wa wceq adantr wral wss cvv cdm cres - w3a cfv c2o wne simpl1 simpl2 simprl simpl3 simprr simp2l simp3l sseldd - c0 wi simpl2l wor sltso soasym syl2an2r mpd jca nosupbnd1lem2 syl112anc - mpan nosupbnd1lem3 neneqd imnan sylib nrexdv simpl3l weq breq2 cbvrexvw - expr breq1 rexbidv bitrid dfrex2 ralbii ralnex 3bitri sylibr rspcv sylc - cbvralvw nosupno 3ad2ant2 nodmon simpl3r simpll1 simpll2 simpll3 eqtr4d - con0 syl simplr nolt02o syl321anc ancld reximdva mtand neqned ) AKZBKZL - MZNBEUAZAEONZEPUBZEUCQZRZGEQZGFUDZUEZFSZRZUFZXQGUGZURYAYBURSZGJKZLMZXQY - DUGZUHSZRZJEOZYAYHJEYAYDEQZRYEYGNZUSYHNYAYJYEYKYAYJYERZRZYFUHYMXLXOYJYD - XQUEZFSZYFUHUIXLXOXTYLUJZXLXOXTYLUKZYAYJYEULZYMXLXOXTYJYDGLMNZRZYOYPYQX - LXOXTYLUMYMYJYSYRYMYEYSYAYJYEUNYAGPQZYLYDPQZYEYSUSZYAEPGXLXMXNXTUOXLXOX - PXSUPUQYMEPYDXMXNXLXTYLUTYRUQPLVAUUAUUBRUUCVBPLGYDVCVIZVDVEVFABCDEFGHYD - IVGZVHABCDEFYDHIVJVHVKVSYEYGVLVMVNYAYCRZYEJEOZYIUUFXPDKZYDLMZJEOZDEUAZU - UGXPXSXLXOYCVOZUUFXLUUKXLXOXTYCUJUUKXJBEOZAEUAXKNZAEUAXLUUJUUMDAEUUJUUH - XILMZBEODAVPZUUMUUIUUOJBEYDXIUUHLVQVRUUPUUOXJBEUUHXHXILVTWAWBWJUUMUUNAE - XJBEWCWDXKAEWEWFWGUUJUUGDGEUUHGSUUIYEJEUUHGYDLVTWAWHWIUUFYEYHJEUUFYJRYE - YGUUFYJYEYGUUFYLRZUUAUUBXQWSQZXRYNSYEYCYGUUFUUAYLUUFEPGXMXNXLXTYCUTZUUL - UQZTUUQEPYDUUFXMYLUUSTUUFYJYEULZUQZUUQFPQZUURUUFUVCYLYAUVCYCXOXLUVCXTAB - CDEFHUCIWKWLTTFWMWTUUQXRFYNUUFXSYLXPXSXLXOYCWNTUUQXLXOXTYTYOXLXOXTYCYLW - OXLXOXTYCYLWPXLXOXTYCYLWQUUQYJYSUVAUUQYEYSUUFYJYEUNZUUFUUAYLUUBUUCUUTUV - BUUDVDVEVFUUEVHWRUVDYAYCYLXAGYDXQXBXCVSXDXEVEXFXG $. - $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Ordinal numbers +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) - ${ - $d A a $. $d A g $. $d a p $. $d A p $. $d a u $. $d A u $. - $d a v $. $d A v $. $d A x $. $d a y $. $d A y $. $d a z $. - $d A z $. $d g u $. $d g v $. $d g x $. $d g y $. $d p u $. - $d p v $. $d p y $. $d p z $. $d S a $. $d S p $. $d S z $. - $d U p $. $d u v $. $d u x $. $d u y $. $d u z $. $d U z $. - $d v x $. $d v y $. $d v z $. $d x y $. $d x z $. $d y z $. - $( Lemma for ~ nosupbnd1 . If ` U ` is a prolongment of ` S ` and in - ` A ` , then ` ( U `` dom S ) ` is not ` 1o ` . (Contributed by Scott - Fenton, 6-Dec-2021.) $) - nosupbnd1lem5 $p |- ( ( -. E. x e. A A. y e. A -. x - ( U ` dom S ) =/= 1o ) $= - ( vz vp wn c1o wceq wi csur wcel wa cres c0 va cv cslt wbr cdm cfv wral - wrex wss cvv w3a word nosupno 3ad2ant2 adantl nodmord 3syl csuc simpr3l - wne ordirr adantr ndmfv c2o 1oex prid1 nosgnn0i neeq1 mpbiri neneqd syl - con4i simp2l simp3l sseldd nofun simpl2l simpll simpl3r simpll1 simpll2 - wfun syl2an simpll3 simprl nosupbnd1lem2 syl112anc eqtr4d simplr simprr - ssel2 eqfunressuc syl213anc expr a2d ralimdva impcom anassrs dmeq breq2 - eleq2d notbid reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev syl12anc cab - wb simplr1 nosupdm con0 nodmon eleq1 suceq reseq2d eqeq12d imbi2d elabg - rexbidv bitrd mpbird mtand neqned rexanali wo simpl nofv 3orel2 syl5com - w3o imdistanda simpl1 simpl2 simpl3 simpr nosupbnd1lem4 pm2.21d expimpd - nosupbnd1lem3 jaod syldc anasss rexlimiva imp sylanbr pm2.61ian ) JUBZG - UCUDZLZFUEZUUJUFZMNZOZJEUGZAUBBUBUCUDLBEUGAEUHLZEPUIZEUJQZRZGEQZGUUMSZF - NZRZUKZUUMGUFZMUTZUUQUVFRZUVGMUVIUVGMNZUUMUUMQZUVIFPQZUUMULUVKLUVFUVLUU - QUVAUURUVLUVEABCDEFHUJIUMUNUOZFUPUUMVAUQUVIUVJRZUVKUUMKUBZUEZQZUUJUVOUC - UDZLZUVOUUMURZSZUUJUVTSZNZOZJEUGZRZKEUHZUVNUVBUUMGUEZQZUULGUVTSZUWBNZOZ - JEUGZUWGUVIUVBUVJUVBUVDUURUVAUUQUSVBUVJUWIUVIUWIUVJUWILUVGTNZUVJLUUMGVC - UWNUVGMUWNUVHTMUTZMMVDVEVFVGZUVGTMVHVIVJVKVLZUOUUQUVFUVJUWMUVFUVJRZUUQU - WMUWRUUPUWLJEUWRUUJEQZRUULUUOUWKUWRUWSUULUUOUWKOUWRUWSUULRZUUOUWKUWRUWT - UUORZRZGWBZUUJWBZUVCUUJUUMSZNUWIUUMUUJUEQZUVGUUNNUWKUXBGPQZUXCUWRUXGUXA - UVFUXGUVJUVFEPGUURUUSUUTUVEVMZUURUVAUVBUVDVNVOVBVBGVPVKUXBUUJPQZUXDUWRU - USUWSUXIUXAUUSUUTUURUVEUVJVQUWSUULUUOVREPUUJWKZWCUUJVPVKUXBUVCFUXEUWRUV - DUXAUVBUVDUURUVAUVJVSVBUXBUURUVAUVEUWTUXEFNZUURUVAUVEUVJUXAVTUURUVAUVEU - VJUXAWAUURUVAUVEUVJUXAWDUWRUWTUUOWEABCDEFGHUUJIWFZWGWHUWRUWIUXAUVJUWIUV - FUWQUOVBUXAUXFUWRUUOUXFUWTUXFUUOUXFLUUNTNZUUOLZUUMUUJVCUXMUUNMUXMUUNMUT - UWOUWPUUNTMVHVIVJVKVLUOUOUXBUVGMUUNUVFUVJUXAWIUWRUWTUUOWJWHGUUJUUMWLWMW - NWNWOWPWQWRUWFUWIUWMRKGEUVOGNZUVQUWIUWEUWMUXOUVPUWHUUMUVOGWSXAUXOUWDUWL - JEUXOUVSUULUWCUWKUXOUVRUUKUVOGUUJUCWTXBUXOUWAUWJUWBUVOGUVTXCXDXEXFXGXHX - IUVNUVKUUMUAUBZUVPQZUVSUVOUXPURZSZUUJUXRSZNZOZJEUGZRZKEUHZUAXJZQZUWGUVN - UURUVKUYGXKUURUVAUVEUUQUVJXLUURUUMUYFUUMABUACDEFHJKIXMXAVKUVNUVLUUMXNQU - YGUWGXKUVIUVLUVJUVMVBFXOUYEUWGUAUUMXNUXPUUMNZUYDUWFKEUYHUXQUVQUYCUWEUXP - UUMUVPXPUYHUYBUWDJEUYHUYAUWCUVSUYHUXSUWAUXTUWBUYHUXRUVTUVOUXPUUMXQZXRUY - HUXRUVTUUJUYIXRXSXTXFXGYBYAUQYCYDYEYFUUQLUULUXNRZJEUHZUVFUVHUULUUOJEYGU - YKUVFUVHUYJUVFUVHOZJEUWSUULUXNUYLUVFUWTUXNRUWTUXMUUNVDNZYHZRUVHUVFUWTUX - NUYNUVFUWTRZUXMUUOUYMYMZUXNUYNUYOUXIUYPUVFUUSUWSUXIUWTUXHUWSUULYIUXJWCU - UJUUMYJVKUXMUUOUYMYKYLYNUVFUWTUYNUVHUYOUXMUVHUYMUYOUXMUVHUYOUUNTUYOUURU - VAUWSUXKUUNTUTUURUVAUVEUWTYOZUURUVAUVEUWTYPZUVFUWSUULWEZUYOUURUVAUVEUWT - UXKUYQUYRUURUVAUVEUWTYQUVFUWTYRUXLWGZABCDEFUUJHIYSWGVJYTUYOUYMUVHUYOUUN - VDUYOUURUVAUWSUXKUUNVDUTUYQUYRUYSUYTABCDEFUUJHIUUBWGVJYTUUCUUAUUDUUEUUF - UUGUUHUUI $. - $} + ${ + $d x y z A $. + $( A class of transitive sets is partially ordered by ` _E ` . + (Contributed by Scott Fenton, 15-Oct-2010.) $) + elpotr $p |- ( A. z e. A Tr z -> _E Po A ) $= + ( vx vy wel wa wi wal wral wtr cep wpo alral alimi syl ralcom ralbii epel + cv wbr ralimi bitri sylib dftr2 df-po anbi12i imbi12i elirrv mtbir bitr3i + wn biantrur 2ralbii bitr4i 3imtr4i ) CDEZDAEZFZCAEZGZDHZCHZABIZUTABIZDBIZ + CBIZASZJZABIBKLZVCUTDBIZCBIZABIZVFVBVKABVBVJCHVKVAVJCUTDBMNVJCBMOUAVLVJAB + IZCBIVFVJACBBPVMVECBUTADBBPQUBUCVHVBABCDVGUDQVICSZVNKTZUKZVNDSZKTZVQVGKTZ + FZVNVGKTZGZFZABIZDBICBIVFCDABKUEVDWDCDBBUTWCABUTWBWCVTURWAUSVRUPVSUQDVNRA + VQRUFAVNRUGVPWBVOCCECUHCVNRUIULUJQUMUNUO $. + $} - ${ - $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. - $d g v $. $d g x $. $d g y $. $d U u $. $d u v $. $d U v $. - $d u x $. $d u y $. $d v x $. $d v y $. $d x y $. - $( Lemma for ~ nosupbnd1 . Establish a hard upper bound when there is no - maximum. (Contributed by Scott Fenton, 6-Dec-2021.) $) - nosupbnd1lem6 $p |- ( ( -. E. x e. A A. y e. A -. x ( U |` dom S ) ( Tr A /\ _E Or A ) ) $= + ( word wtr cep wwe wa wor df-ord wfr zfregfr df-we mpbiran anbi2i bitri ) A + BACZADEZFOADGZFAHPQOPADIQAJADKLMN $. - ${ - $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. - $d g v $. $d g x $. $d g y $. $d U u $. $d u v $. $d U v $. - $d u x $. $d U x $. $d u y $. $d U y $. $d v x $. $d v y $. - $d x y $. - $( Bounding law from below for the surreal supremum. Proposition 4.2 of - [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.) $) - nosupbnd1 $p |- ( ( A C_ No /\ A e. _V /\ U e. A ) -> - ( U |` dom S ) - -. ( Z |` suc dom U ) . } ) ) $= - ( vx vq wcel cslt wbr wn wa csur c2o cfv con0 wceq syl c1o c0 vp wral wss - cvv w3a cdm csuc cres cop csn cun simp1l simp3 breq1 rspcv sylc crab cint - cv wne simpl21 simpl1l sseldd simpl23 simp21 sltso sonr mpan breq2 notbid - wor syl5ibcom con2d neqned nosepssdm syl3anc wo wb nosepon nodmon syl2anc - imp onsseleq ctp wrex adantr sylan simpr weq fveq2 neeq12d onnminsb df-ne - onelon con2bii sylibr simplr ontr1 mp2and fvresd eqtr4d ralrimiva sltval2 - wi mpbid breqtrrd raleq breq12d anbi12d rspcev syl12anc sltval mpbird cxp - noreson cin df-res 2on xpsng sylancl ineq1d 3syl eqtrid nofun eqtrd sylib - wfun mpd eleq2d ex sylbid nosgnn0i mpbiri neneqd intnanrd w3o adantl fvex - ndmfv 3orrot incom word nodmord ordirr disjsn xpdisj1 eqtr3d resundir un0 - uneq2d eqcomi 3eqtr4g wrel resdm sssucid resabs1 mp1i 3brtr4d elexi prid2 - funrel noextend sucelon sltres soasym df-suc reseq2i resundi eqtri rabbii - dmres necom inteqi necomd eqsstrid eqsstrrd df-ss biimpar ralrimiv funres - nesym eqfunfv mpbir2and wfn funfn 1oex prid1 neeq1d 3brtr3d brtp ecase23d - 3bitri simprd neeq1 con4i fnressn opeq2d sneqd uneq12d eqnbrtrd jaodan - mpdan ) CBHZCAUSIJKABUBZLZBMUCZBUDHZDMHZUEZEUSZDIJZEBUBZUEZCDIJZDCUFZUGZU - HZCUXONUIZUJZUKZIJKZUXMUXCUXLUXNUXCUXDUXIUXLULZUXEUXIUXLUMUXKUXNECBUXJCDI - UNUOUPUXMUXNLZFUSZCOZUYDDOZUTZFPUQZURZUXOUCZUYAUYCCMHZUXHCDUTZUYJUYCBMCUX - FUXGUXHUXEUXLUXNVAUXCUXDUXIUXLUXNVBVCZUXFUXGUXHUXEUXLUXNVDZUYCCDUXMUXNCDQ - ZKUXMUYOUXNUXMCCIJZKZUYOUXNKUXMUYKUYQUXMBMCUXEUXFUXGUXHUXLVEUYBVCMIVKZUYK - UYQVFMCIVGVHRUYOUYPUXNCDCIVIVJVLVMWBVNZFCDVOVPUYCUYJUYIUXOHZUYIUXOQZVQZUY - AUYCUYIPHZUXOPHZUYJVUBVRUYCUYKUXHUYLVUCUYMUYNUYSFCDVSZVPUYCUYKVUDUYMCVTZR - ZUYIUXOWCWAUYCVUBUYAUYCUYTUYAVUAUYCUYTLZUXTUXQIJZUYAVUHUXTUXOUHZUXQUXOUHZ - IJZVUIVUHCDUXOUHZVUJVUKIVUHCVUMIJZGUSZCOZVUOVUMOZQZGUAUSZUBZVUSCOZVUSVUMO - ZSTUISNUITNUIWDZJZLZUAPWEZVUHVUCVURGUYIUBZUYICOZUYIVUMOZVVCJZVVFVUHUYKUXH - UYLVUCUYCUYKUYTUYMWFZUYCUXHUYTUYNWFZUYCUYLUYTUYSWFVUEVPZVUHVURGUYIVUHVUOU - YIHZLZVUPVUODOZVUQVVOVUPVVPUTZKZVUPVVPQZVVOVUOPHZVVNVVRVUHVUCVVNVVTVVMUYI - VUOWNWGVUHVVNWHZUYGVVQFVUOFGWIUYEVUPUYFVVPUYDVUOCWJUYDVUODWJWKWLZUPVVQVVS - VUPVVPWMWOWPVVOVUOUXODVVOVVNUYTVUOUXOHZVWAUYCUYTVVNWQVVOVUDVVNUYTLVWCXDVU - HVUDVVNUYCVUDUYTVUGWFZWFVUOUYIUXOWRRWSWTXAXBVUHVVHUYIDOZVVIVVCVUHUXNVVHVW - EVVCJZUXMUXNUYTWQVUHUYKUXHUXNVWFVRZVVKVVLCDFXCZWAXEVUHUYIUXODUYCUYTWHWTXF - VVEVVGVVJLUAUYIPVUSUYIQZVUTVVGVVDVVJVURGVUSUYIXGVWIVVAVVHVVBVVIVVCVUSUYIC - WJVUSUYIVUMWJXHXIXJXKVUHUYKVUMMHZVUNVVFVRVVKVUHUXHVUDVWJVVLVWDDUXOXOWAUAG - CVUMXLWAXMVUHVUJCUXOUHZCVUHVWKUXSUXOUHZUKVWKTUKZVUJVWKVUHVWLTVWKVUHVWLUXS - UXOUDXNZXPZTUXSUXOXQVUHUXOUJZNUJZXNZVWNXPZVWOTVUHVWRUXSVWNVUHVUDNPHVWRUXS - QVWDXRUXONPPXSXTYAVUHVWPUXOXPZTQVWSTQVUHVWTUXOVWPXPZTVWPUXOUUAVUHUXOUXOHK - ZVXATQVUHUYKUXOUUBZVXBVVKCUUCZUXOUUDZYBUXOUXOUUEWPYCVWPUXOVWQUDUUFRUUGYCU - UJCUXSUXOUUHVWMVWKVWKUUIUUKUULVUHCYGZCUUMVWKCQVUHUYKVXFVVKCYDZRCUVACUUNYB - YEUXOUXPUCVUKVUMQVUHUXOUUODUXOUXPUUPUUQUURVUHUXTMHZUXQMHZVUDVULVUIXDUYCVX - HUYTUYCUYKVXHUYMCNSNNPXRUUSUUTZUVBZRWFZUYCVXIUYTUYCUXHUXPPHZVXIUYNUYCVUDV - XMVUGUXOUVCYFDUXPXOWAWFZVWDUXTUXQUXOUVDVPYHVUHVXHVXIVUIUYAXDZVXLVXNUYRVXH - VXILVXOVFMIUXTUXQUVEVHWAYHUYCVUALZUXQUXTUXTIVXPUXQVUMDVWPUHZUKZUXTUXQDUXO - VWPUKZUHVXRUXPVXSDUXOUVFUVGDUXOVWPUVHUVIVXPVUMCVXQUXSVXPVUMCQZVUMUFZUXOQZ - VUQVUPQZGVYAUBZVXPVYAUXODUFZXPZUXODUXOUVKVXPUXOVYEUCVYFUXOQVXPUXOUYIVYEUY - CVUAWHZVXPUYIUYFUYEUTZFPUQZURZVYEUYHVYIUYGVYHFPUYEUYFUVLUVJUVMVXPUXHUYKDC - UTVYJVYEUCUYCUXHVUAUYNWFZUYCUYKVUAUYMWFZVXPCDUYCUYLVUAUYSWFUVNFDCVOVPUVOU - VPUXOVYEUVQYFYCZVXPVYCGVYAVXPVUOVYAHVWCVYCVXPVYAUXOVUOVYMYIVXPVWCVYCVXPVW - CLZVUQVVPVUPVYNVUOUXODVXPVWCWHWTVYNVVRVVPVUPQZVYNVVTVVNVVRVXPVUDVWCVVTVXP - UYKVUDVYLVUFRUXOVUOWNWGVXPVVNVWCVXPUYIUXOVUOVYGYIUVRVWBUPVVQVYOVUPVVPUWAW - OWPYEYJYKUVSVXPVUMYGZVXFVXTVYBVYDLVRVXPUXHDYGZVYPVYKDYDZUXODUVTYBVXPUYKVX - FVYLVXGRGVUMCUWBWAUWCVXPVXQUXOUXODOZUIZUJZUXSVXPDVYEUWDZUXOVYEHZVXQWUAQVX - PVYQWUBVXPUXHVYQVYKVYRRDUWEYFVXPVYSNQZWUCVXPUXOCOZTQZWUDVXPWUFWUDLZWUESQZ - VYSTQZLZWUHWUDLZVXPWUHWUIVXPWUESVXPWUESUTTSUTSSNUWFUWGYLVXPWUETSVXPVXCVXB - WUFVXPUYKVXCVYLVXDRVXEUXOCYSYBUWHYMYNZYOVXPWUHWUDWULYOVXPWUEVYSVVCJZWUGWU - JWUKYPZVXPVVHVWEWUEVYSVVCVXPUXNVWFUXMUXNVUAWQVXPUYKUXHVWGVYLVYKVWHWAXEVUA - VVHWUEQUYCUYIUXOCWJYQVUAVWEVYSQUYCUYIUXODWJYQUWIWUMWUJWUKWUGYPWUKWUGWUJYP - WUNSTSNTNWUEVYSUXOCYRUXODYRUWJWUJWUKWUGYTWUKWUGWUJYTUWLYFUWKUWMZWUCWUDWUC - KWUIWUDKUXODYSWUIVYSNWUIVYSNUTTNUTNVXJYLVYSTNUWNYMYNRUWORVYEUXODUWPWAVXPV - YTUXRVXPVYSNUXOWUOUWQUWRYEUWSYCVXPUYKVXHUXTUXTIJKZVYLVXKUYRVXHWUPVFMUXTIV - GVHYBUWTUXAYJYKYHUXB $. + $d x A $. + $( Lemma for ~ dfon2 . (Contributed by Scott Fenton, 28-Feb-2011.) $) + dfon2lem2 $p |- U. { x | ( x C_ A /\ ph /\ ps ) } C_ A $= + ( cv wss w3a cab cpw cuni simp1 ss2abi df-pw sseqtrri sspwuni mpbi ) CEDF + ZABGZCHZDIZFSJDFSQCHTRQCQABKLCDMNSDOP $. $} ${ - nosupbnd2.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - - ${ - $d A a $. $d a g $. $d A g $. $d a p $. $d A p $. $d A q $. - $d a u $. $d A u $. $d a v $. $d A v $. $d a x $. $d A x $. - $d a y $. $d A y $. $d g p $. $d g q $. $d g u $. $d g v $. - $d g x $. $d g y $. $d p q $. $d p u $. $d p v $. $d p x $. - $d p y $. $d q u $. $d q v $. $d q y $. $d S a $. $d S g $. - $d S p $. $d u v $. $d u x $. $d u y $. $d v x $. $d v y $. - $d x y $. $d Z a $. $d Z g $. $d Z p $. $d Z x $. + $d A x z w t $. - $( Bounding law from above for the surreal supremum. Proposition 4.3 of - [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.) $) - nosupbnd2 $p |- ( ( A C_ No /\ A e. _V /\ Z e. No ) -> - ( A. a e. A a -. ( Z |` dom S ) ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) -> + ( Tr A /\ A. z e. A -. z e. z ) ) ) $= + ( vw vt wcel cv wtr wa wi wel wn wral wss w3a wceq sseq1 treq cvv wal cab + wpss cuni untelirr wrex eluni2 raleq 3anbi123d elequ1 elequ2 bitrd notbid + vex elab cbvralvw biimpi 3ad2ant3 sylbi rsp syl rexlimiv dfon2lem2 dfpss2 + mprg dfon2lem1 ssexg mpan psseq1 anbi12d eleq1 imbi12d spcgv imp csuc csn + sylan snssi cun unss df-suc sseq1i sylbb2 sylancr suctr ax-mp mprgbir nfv + untuni nfra1 nf3an nfab nfuni untsucf raleqf cbvabv elab2g biimprd sucexg + nfcv nfsuc syl11 mp3an23 com12 elssuni sucssel syl5 syld mpd syl6 syl5bir + mpan2i mpani mt3i pm3.2i mpbii ex ) CDGZAHZCUCZXSIZJZXSCGZKZAUAZCIZBBLZMZ + BCNZJZXRYEJZEHZCOZYLIZFFLZMZFYLNZPZEUBZUDZCQZYJYKUUAYTYTGZYHUUBMBYTBYTUEB + HZYTGBALZAYSUFYHAUUCYSUGUUDYHAYSXSYSGZYHBXSNZUUDYHKUUEXSCOZYAYPFXSNZPZUUF + YRUUIEXSAUNYLXSQYMUUGYNYAYQUUHYLXSCRYLXSSYPFYLXSUHUIUOUUHUUGUUFYAUUHUUFYP + YHFBXSFHUUCQZYOYGUUJYOBFLYGFBFUJFBBUKULUMUPUQURUSZYHBXSUTVAVBUSVEYKYTCOZU + UAMZUUBYNYQECVCZUULUUMJYTCUCZYKUUBYTCVDYKUUOYTIZUUBYMYQEVFZYKUUOUUPJZYTCG + ZUUBXRYTTGZYEUURUUSKZUULXRUUTUUNYTCDVGVHUUTYEUVAYDUVAAYTTXSYTQZYBUURYCUUS + UVBXTUUOYAUUPXSYTCVIXSYTSVJXSYTCVKVLVMVNVQUUSYTVOZCOZUUBUUSUULYTVPZCOZUVD + UUNYTCVRUULUVFJYTUVEVSZCOUVDYTUVECVTUVCUVGCYTWAWBWCWDUUSUVDUVCYSGZUUBUVDU + USUVHUVDUVCIZYPFUVCNZUUSUVHKUUPUVIUUQYTWEWFYHBYTNZUVJUVKUUFAYSBAYSWIUUKWG + ZBFYTFYSYRFEYMYNYQFYMFWHYNFWHYPFYLWJWKWLWMZWNWFUVCTGZUVDUVIUVJPZUVHUUSUVN + UVHUVOUUCCOZUUCIZYPFUUCNZPZUVOBUVCYSTUUCUVCQUVPUVDUVQUVIUVRUVJUUCUVCCRUUC + UVCSYPFUUCUVCFUUCWTFYTUVMXAWOUIYRUVSEBYLUUCQYMUVPYNUVQYQUVRYLUUCCRYLUUCSY + PFYLUUCUHUIWPWQWRYTCWSXBXCXDUVHUVCYTOUUSUUBUVCYSXEYTYTCXFXGXHXIXJXLXKXMXN + UUAUUPUVKJYJUUPUVKUUQUVLXOUUAUUPYFUVKYIYTCSYHBYTCUHVJXPVAXQ $. $} ${ - $d a b $. $d a c $. $d a d $. $d a g $. $d a u $. $d a v $. $d a x $. - $d a y $. $d B a $. $d B b $. $d b c $. $d B c $. $d b d $. $d B d $. - $d b e $. $d B e $. $d B g $. $d b u $. $d B u $. $d b v $. $d B v $. - $d b x $. $d B x $. $d b y $. $d B y $. $d c d $. $d c e $. $d c g $. - $d c u $. $d c v $. $d c x $. $d c y $. $d d e $. $d d g $. $d d u $. - $d d v $. $d d x $. $d d y $. $d e g $. $d e u $. $d e v $. $d e x $. - $d e y $. $d g u $. $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. - $d u y $. $d v y $. $d x y $. - noinfcbv.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Change bound variables for surreal infimum. (Contributed by Scott - Fenton, 9-Aug-2024.) $) - noinfcbv $p |- T = if ( E. a e. B A. b e. B -. b . } ) , - ( c e. { b | E. d e. B ( b e. dom d /\ - A. e e. B ( -. d ( d |` suc b ) = ( e |` suc b ) ) ) } - |-> ( iota a E. d e. B ( c e. dom d /\ - A. e e. B ( -. d ( d |` suc c ) = ( e |` suc c ) ) /\ - ( d ` c ) = a ) ) ) ) $= - ( cv cslt wral wrex cres wceq wi wbr wn crio cdm c1o cop csn wcel csuc wa - cun cab cfv w3a cio cmpt cif breq2 notbid ralbidv breq1 cbvralvw cbvrexvw - bitrdi cbvriotavw dmeqi opeq1i sneqi uneq12i eleq1w suceq reseq2d eqeq12d - weq imbi2d fveqeq2 3anbi123d rexbidv iotabidv eqeq2 3anbi3d eleq2d reseq1 - dmeq eqeq1d imbi12d eqeq2d fveq1 cbviotavw eqtrdi cbvmptv anbi12d mpteq1i - cbvabv eqtri ifbieq12i ) FBNZANZOUAZUBZBEPZAEQZXAAEUCZXCUDZUEUFZUGZUKZHWQ - DNZUDZUHZXHCNZOUAZUBZXHWQUIZRZXKXNRZSZTZCEPZUJZDEQZBULZHNZXIUHZXMXHYCUIZR - ZXKYERZSZTZCEPZYCXHUMWRSZUNZDEQZAUOZUPZUQJNZINZOUAZUBZJEPZIEQZYTIEUCZUUBU - DZUEUFZUGZUKZKYPLNZUDZUHZUUGGNZOUAZUBZUUGYPUIZRZUUJUUMRZSZTZGEPZUJZLEQZJU - LZKNZUUHUHZUULUUGUVBUIZRZUUJUVDRZSZTZGEPZUVBUUGUMZYQSZUNZLEQZIUOZUPZUQMXB - UUAXGYOUUFUVOXAYTAIEAIVNZXAWQYQOUAZUBZBEPYTUVPWTUVRBEUVPWSUVQWRYQWQOURUSU - TUVRYSBJEBJVNZUVQYRWQYPYQOVAUSVBVDZVCXCUUBXFUUEXAYTAIEUVTVEZXEUUDXDUUCUEX - CUUBUWAVFVGVHVIYOKYBUVNUPUVOHKYBYNUVNHKVNZYNUVBXIUHZXMXHUVDRZXKUVDRZSZTZC - EPZUVBXHUMZWRSZUNZDEQZAUOUVNUWBYMUWLAUWBYLUWKDEUWBYDUWCYJUWHYKUWJHKXIVJUW - BYIUWGCEUWBYHUWFXMUWBYFUWDYGUWEUWBYEUVDXHYCUVBVKZVLUWBYEUVDXKUWMVLVMVOUTY - CUVBWRXHVPVQVRVSUWLUVMAIUVPUWLUWCUWHUWIYQSZUNZDEQUVMUVPUWKUWODEUVPUWJUWNU - WCUWHWRYQUWIVTWAVRUWOUVLDLEDLVNZUWCUVCUWHUVIUWNUVKUWPXIUUHUVBXHUUGWDZWBUW - PUWHUUGXKOUAZUBZUVEUWESZTZCEPUVIUWPUWGUXACEUWPXMUWSUWFUWTUWPXLUWRXHUUGXKO - VAUSZUWPUWDUVEUWEXHUUGUVDWCWEWFUTUXAUVHCGECGVNZUWSUULUWTUVGUXCUWRUUKXKUUJ - UUGOURUSZUXCUWEUVFUVEXKUUJUVDWCWGWFVBVDUWPUWIUVJYQUVBXHUUGWHWEVQVCVDWIWJW - KKYBUVAUVNYAUUTBJUVSYAYPXIUHZXMXHUUMRZXKUUMRZSZTZCEPZUJZDEQUUTUVSXTUXKDEU - VSXJUXEXSUXJBJXIVJUVSXRUXICEUVSXQUXHXMUVSXOUXFXPUXGUVSXNUUMXHWQYPVKZVLUVS - XNUUMXKUXLVLVMVOUTWLVRUXKUUSDLEUWPUXEUUIUXJUURUWPXIUUHYPUWQWBUWPUXJUWSUUN - UXGSZTZCEPUURUWPUXIUXNCEUWPXMUWSUXHUXMUXBUWPUXFUUNUXGXHUUGUUMWCWEWFUTUXNU - UQCGEUXCUWSUULUXMUUPUXDUXCUXGUUOUUNXKUUJUUMWCWGWFVBVDWLVCVDWNWMWOWPWO $. + $d A x y z $. $d B x y z $. + dfon2lem4.1 $e |- A e. _V $. + dfon2lem4.2 $e |- B e. _V $. + $( Lemma for ~ dfon2 . If two sets satisfy the new definition, then one is + a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.) $) + dfon2lem4 $p |- ( ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) /\ + A. y ( ( y C. B /\ Tr y ) -> y e. B ) ) -> + ( A C_ B \/ B C_ A ) ) $= + ( vz cv wpss wtr wa wcel wi wal wceq wo wss wn cvv eleq1 inss1 sseli wral + cin wel dfon2lem3 ax-mp simprd eleq2 bitrd notbid rspccv syl syl5 pm2.01d + adantr elin sylnib simpld trin syl2an inex1 psseq1 anbi12d imbi12d mpan2d + treq spcv adantl anim12d mtod ianor sylib sspss mpbi inss2 orel1 orc syl6 + olc jaoa mp2ani df-ss sseqin2 orbi12i sylibr ) AHZCIZWGJZKZWGCLZMZANZBHZD + IZWNJZKZWNDLZMZBNZKZCDUDZCOZXBDOZPZCDQZDCQZPXAXBCIZRZXBDIZRZPZXEXAXHXJKZR + XLXAXMXBCLZXBDLZKZXAXBXBLZXPXAXQXQXNXAXQRZXBCXBCDUAZUBWMXNXRMZWTWMGGUEZRZ + GCUCZXTWMCJZYCCSLWMYDYCKMEAGCSUFUGZUHYBXRGXBCGHZXBOZYAXQYGYAXBYFLXQYFXBYF + TYFXBXBUIUJUKULUMUPUNUOXBCDUQURXAXHXNXJXOXAXHXBJZXNWMYDDJZYHWTWMYDYCYEUSW + TYIYBGDUCZDSLWTYIYJKMFBGDSUFUGUSCDUTVAZWMXHYHKZXNMZWTWLYMAXBCDEVBZWGXBOZW + JYLWKXNYOWHXHWIYHWGXBCVCWGXBVGVDWGXBCTVEVHUPVFXAXJYHXOYKWTXJYHKZXOMZWMWSY + QBXBYNWNXBOZWQYPWRXOYRWOXJWPYHWNXBDVCWNXBVGVDWNXBDTVEVHVIVFVJVKXHXJVLVMXL + XHXCPZXJXDPZXEXBCQYSXSXBCVNVOXBDQYTCDVPXBDVNVOXIYSXEXKYTXIYSXCXEXHXCVQXCX + DVRVSXKYTXDXEXJXDVQXDXCVTVSWAWBUMXFXCXGXDCDWCDCWDWEWF $. $} ${ - $d B a b x y z g v u $. $d V g z $. - noinfno.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( The next several theorems deal with a surreal "infimum". This surreal - will ultimately be shown to bound ` B ` above and bound the restriction - of any surreal below. We begin by showing that the given expression - actually defines a surreal number. (Contributed by Scott Fenton, - 8-Aug-2024.) $) - noinfno $p |- ( ( B C_ No /\ B e. V ) -> T e. No ) $= - ( va vb csur wcel wa wrex cres wceq syl con0 cbday vz wss cv cslt wn wral - wbr crio cdm c1o cop csn cun csuc wi cab cfv w3a cio iftrue adantr simprl - cmpt cif wreu wrmo nominmo ad2antrl reu5 sylanbrc riotacl sseldd c2o 1oex - simpl noextend eqeltrd iffalse wfun crn cpr funmpt a1i iotaex eqid dmmpti - prid1 word wtr nodmon simprrl onelon syl2anc rexlimdvaa abssdv wel simplr - wal ssel2 adantlr ontr1 mpand adantrd reseq1 sylan sucelon sylib wb eloni - simpllr ordsucelsuc onelss resabs1d eqeq12d syl5ib imim2d ralimdv expimpd - mpbid sylc vex weq eleq1w reseq2d imbi2d ralbidv anbi12d rexbidv elab cvv - suceq bdayfo ax-mp mpan adantl eqeltrrd eqeltrid wex eqeq2 3anbi3d anbi2i - jcad reximdva 3imtr4g alrimivv dftr2 sylibr cima cuni wfo fofun funimaexg - dford5 uniexd bdayval wfn fofn fnfvima mp3an1 adantrr elssuni ssexd elong - mpbird rnmpt rexab weu wmo fvex spcev mp3an3 reximi rexcom4 noinfprefixmo - df-eu iota2 eqcom bitrdi simprr3 norn nofun simprr1 fvelrn sylbird syl5bi - exlimdv eqsstrid elno2 syl3anbrc pm2.61ian ) ELUBZEHMZNZFBUCZAUCZUDUGUEBE - UFZAEOZUWPAEUHZUWRUIUJUKULUMZGUWNDUCZUIZMZUWTCUCZUDUGUEZUWTUWNUNZPZUXCUXE - PZQZUOZCEUFZNZDEOZBUPZGUCZUXAMZUXDUWTUXNUNZPZUXCUXPPZQZUOZCEUFZUXNUWTUQZU - WOQZURZDEOZAUSZVCZVDZLIUWQUWMUYHLMUWQUWMNZUYHUWSLUWQUYHUWSQUWMUWQUWSUYGUT - VAUYIUWRLMUWSLMUYIELUWRUWQUWKUWLVBUYIUWPAEVEZUWREMUYIUWQUWPAEVFZUYJUWQUWM - VOUWKUYKUWQUWLABEVGVHUWPAEVIVJUWPAEVKRVLUWRUJUJVMVNWGVPRVQUWQUEZUWMNZUYHU - YGLUYLUYHUYGQUWMUWQUWSUYGVRVAUYMUYGVSZUYGUIZSMUYGVTZUJVMWAZUBUYGLMUYNUYMG - UXMUYFWBWCUYMUYOUXMSGUXMUYFUYGUYEAWDUYGWEZWFUYMUXMSMZUXMWHZUWKUYTUYLUWLUW - KUXMSUBUXMWIZUYTUWKUXLBSUWKUXKUWNSMZDEUWKUWTEMZUXKNZNZUXASMZUXBVUBVUEUWTL - MZVUFVUEELUWTUWKVUDVOUWKVUCUXKVBVLZUWTWJZRUWKVUCUXBUXJWKZUXAUWNWLWMWNWOUW - KJKWPZKUCZUXMMZNZJUCZUXMMZUOZKWRJWRVUAUWKVUQJKUWKVUKVULUXAMZUXDUWTVULUNZP - ZUXCVUSPZQZUOZCEUFZNZDEOZNVUOUXAMZUXDUWTVUOUNZPZUXCVVHPZQZUOZCEUFZNZDEOZV - UNVUPUWKVUKVVFVVOUWKVUKNZVVEVVNDEVVPVUCNZVVEVVGVVMVVQVURVVGVVDVVQVUKVURVV - GUWKVUKVUCWQVVQVUFVUKVURNVVGUOVVQVUGVUFUWKVUCVUGVUKELUWTWSWTVUIRZVUOVULUX - AXARXBXCVVQVURVVDVVMVVQVURNZVVCVVLCEVVSVVBVVKUXDVVBVUTVVHPZVVAVVHPZQVVSVV - KVUTVVAVVHXDVVSVVTVVIVWAVVJVVSUWTVVHVUSVVSVUSSMZVVHVUSMZVVHVUSUBVVSVULSMZ - VWBVVQVUFVURVWDVVRUXAVULWLXEZVULXFXGVVSVUKVWCUWKVUKVUCVURXJVVSVULWHZVUKVW - CXHVVSVWDVWFVWEVULXIRVUOVULXKRXSVUSVVHXLXTZXMVVSUXCVVHVUSVWGXMXNXOXPXQXRU - UBUUCXRVUMVVFVUKUXLVVFBVULKYABKYBZUXKVVEDEVWHUXBVURUXJVVDBKUXAYCVWHUXIVVC - CEVWHUXHVVBUXDVWHUXFVUTUXGVVAVWHUXEVUSUWTUWNVULYKZYDVWHUXEVUSUXCVWIYDXNYE - YFYGYHYIUUAUXLVVOBVUOJYABJYBZUXKVVNDEVWJUXBVVGUXJVVMBJUXAYCVWJUXIVVLCEVWJ - UXHVVKUXDVWJUXFVVIUXGVVJVWJUXEVVHUWTUWNVUOYKZYDVWJUXEVVHUXCVWKYDXNYEYFYGY - HYIUUDUUEJKUXMUUFUUGUXMUUMVJVHUYMUXMYJMUYSUYTXHUYMUXMTEUUHZUUIZYJUWMVWMYJ - MZUYLUWLVWNUWKUWLVWLYJTVSZUWLVWLYJMLSTUUJZVWOYLLSTUUKYMTEHUULYNUUNYOYOUWK - UXMVWMUBUYLUWLUWKUXLBVWMUWKUXKUWNVWMMDEVUEUXAVWMUWNVUEUXAVWLMUXAVWMUBVUEU - WTTUQZUXAVWLVUEVUGVWQUXAQVUHUWTUUORUWKVUCVWQVWLMZUXKTLUUPZUWKVUCVWRVWPVWS - YLLSTUUQYMLETUWTUURUUSUUTYPUXAVWLUVARVUJVLWNWOVHUVBUXMYJUVCRUVDYQUYMUYPUA - UCZUYFQZGUXMOZUAUPUYQGUAUXMUYFUYGUYRUVEUYMVXBUAUYQVXBUXOUYANZDEOZVXANZGYR - UYMVWTUYQMZUXLVXDVXAGBBGYBZUXKVXCDEVXGUXBUXOUXJUYABGUXAYCVXGUXIUXTCEVXGUX - HUXSUXDVXGUXFUXQUXGUXRVXGUXEUXPUWTUWNUXNYKZYDVXGUXEUXPUXCVXHYDXNYEYFYGYHU - VFUYMVXEVXFGUYMVXDVXAVXFUYMVXDNZVXAUXOUYAUYBVWTQZURZDEOZVXFVXIVXLUYFVWTQZ - VXAVXIUYEAUVGZVXLVXMXHZVXIUYEAYRZUYEAUVHZVXNVXDVXPUYMVXDUYDAYRZDEOVXPVXCV - XRDEUXOUYAUYBUYBQZVXRUYBWEUYDUXOUYAVXSURAUYBUXNUWTUVIUWOUYBQUYCVXSUXOUYAU - WOUYBUYBYSYTUVJUVKUVLUYDDAEUVMXGYOUYMVXQVXDUWKVXQUYLUWLACDEUXNUVNVHVAUYEA - UVOVJVWTYJMVXNVXOUAYAUYEVXLAVWTYJAUAYBZUYDVXKDEVXTUYCVXJUXOUYAUWOVWTUYBYS - YTYHUVPYNRUYFVWTUVQUVRUYMVXLVXFUOZVXDUWKVYAUYLUWLUWKVXKVXFDEUWKVUCVXKNZNZ - UYBVWTUYQUXOUYAVXJVUCUWKUVSVYCUWTVTZUYQUYBVYCVUGVYDUYQUBVYCELUWTUWKVYBVOU - WKVUCVXKVBVLZUWTUVTRVYCUWTVSZUXOUYBVYDMVYCVUGVYFVYEUWTUWARUXOUYAVXJVUCUWK - UWBUXNUWTUWCWMVLYPWNVHVAUWDXRUWFUWEWOUWGUYGUWHUWIVQUWJYQ $. + $d A x y z $. $d B x y z $. + dfon2lem5.1 $e |- A e. _V $. + dfon2lem5.2 $e |- B e. _V $. + $( Lemma for ~ dfon2 . Two sets satisfying the new definition also satisfy + trichotomy with respect to ` e. ` . (Contributed by Scott Fenton, + 25-Feb-2011.) $) + dfon2lem5 $p |- ( ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) /\ + A. y ( ( y C. B /\ Tr y ) -> y e. B ) ) -> + ( A e. B \/ A = B \/ B e. A ) ) $= + ( vz cv wpss wtr wa wcel wi wal wceq wn wo w3o bitri cvv dfon2lem4 dfpss2 + wss eqcom notbii anbi2i orbi12i andir bitr4i orcom dfon2lem3 ax-mp simpld + wel wral psseq1 treq anbi12d eleq1 imbi12d spcv expcomd imp mpan9 orim12d + sylan2 biimtrid syl5bir mpand 3orrot 3orass df-or sylibr ) AHZCIZVNJZKZVN + CLZMZANZBHZDIZWAJZKZWADLZMZBNZKZCDOZPZDCLZCDLZQZMZWLWIWKRZWHCDUCZDCUCZQZW + JWMABCDEFUAWRWJKZCDIZDCIZQZWHWMXBWPWJKZWQWJKZQWSWTXCXAXDCDUBXAWQDCOZPZKXD + DCUBXFWJWQXEWIDCUDUEUFSUGWPWQWJUHUIXBXAWTQWHWMWTXAUJWHXAWKWTWLWGVTDJZXAWK + MZWGXGGGUNPZGDUOZDTLWGXGXJKMFBGDTUKULUMVTXGXHVTXAXGWKVSXAXGKZWKMADFVNDOZV + QXKVRWKXLVOXAVPXGVNDCUPVNDUQURVNDCUSUTVAVBVCVFVTCJZWGWTWLMVTXMXIGCUOZCTLV + TXMXNKMEAGCTUKULUMWGWTXMWLWFWTXMKZWLMBCEWACOZWDXOWEWLXPWBWTWCXMWACDUPWACU + QURWACDUSUTVAVBVDVEVGVHVIWOWIWKWLRZWNWLWIWKVJXQWIWMQWNWIWKWLVKWIWMVLSSVM + $. $} ${ - $d B g $. $d B p q u v y z $. $d g u $. $d g v $. $d g y $. - noinfdm.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Next, we calculate the domain of ` T ` . This is mostly to change bound - variables. (Contributed by Scott Fenton, 8-Aug-2024.) $) - noinfdm $p |- ( -. E. x e. B A. y e. B -. y - dom T = { z | E. p e. B ( z e. dom p /\ A. q e. B ( -. p - ( p |` suc z ) = ( q |` suc z ) ) ) } ) $= - ( cv cslt wn wral wrex cdm cres wceq wi wbr wcel csuc wa cab cfv w3a cmpt - cio crio c1o cop csn cun cif iffalse eqtrid iotaex eqid dmmpti weq eleq1w - dmeqd reseq2d eqeq12d imbi2d ralbidv anbi12d rexbidv eleq2d notbid reseq1 - suceq dmeq breq1 eqeq1d imbi12d breq2 eqeq2d cbvralvw bitrdi cbvabv eqtri - cbvrexvw eqtrdi ) BLZALZMUANBFOZAFPZNZGQHWFELZQZUBZWKDLZMUAZNZWKWFUCZRZWN - WQRZSZTZDFOZUDZEFPZBUEZHLZWLUBWPWKXFUCZRWNXGRSTDFOXFWKUFWGSUGEFPZAUIZUHZQ - ZCLZJLZQZUBZXMILZMUAZNZXMXLUCZRZXPXSRZSZTZIFOZUDZJFPZCUEZWJGXJWJGWIWHAFUJ - ZYHQUKULUMUNZXJUOXJKWIYIXJUPUQVCXKXEYGHXEXIXJXHAURXJUSUTXDYFBCBCVAZXDXLWL - UBZWPWKXSRZWNXSRZSZTZDFOZUDZEFPYFYJXCYQEFYJWMYKXBYPBCWLVBYJXAYODFYJWTYNWP - YJWRYLWSYMYJWQXSWKWFXLVMZVDYJWQXSWNYRVDVEVFVGVHVIYQYEEJFEJVAZYKXOYPYDYSWL - XNXLWKXMVNVJYSYPXMWNMUAZNZXTYMSZTZDFOYDYSYOUUCDFYSWPUUAYNUUBYSWOYTWKXMWNM - VOVKYSYLXTYMWKXMXSVLVPVQVGUUCYCDIFDIVAZUUAXRUUBYBUUDYTXQWNXPXMMVRVKUUDYMY - AXTWNXPXSVLVSVQVTWAVHWDWAWBWCWE $. + $d S x y z w s t $. + $( Lemma for ~ dfon2 . A transitive class of sets satisfying the new + definition satisfies the new definition. (Contributed by Scott Fenton, + 25-Feb-2011.) $) + dfon2lem6 $p |- ( ( Tr S /\ + A. x e. S A. z ( ( z C. x /\ Tr z ) -> z e. x ) ) -> + A. y ( ( y C. S /\ Tr y ) -> y e. S ) ) $= + ( vs vw vt wtr cv wpss wa wel wi wal wral wcel weq wn syl imbi12d wo cdif + wss pssss ssralv impcom adantrr ad2ant2lr psseq2 anbi1d elequ2 albidv imp + rspccv eldifi rspcv psseq1 treq anbi12d elequ1 syl6ib ad2ant2l adantr w3o + cbvalvw vex dfon2lem5 3orrot 3orass bitri eleq1a nsyli adantll orel1 trss + elndif eldifn ssel con3d syl9 adantl imp31 syl9r mpd biimtrid syl5 mp2and + syl5com ex ssrdv dfpss2 spvv expd com23 com3l adantld imp32 syl5bir mpand + syl6 orrd anassrs ralrimiva wrex c0 pssdif r19.2z ad2antrl eleq1w syl5ibr + wne a1i trel syl7 ad2antrr jaod rexlimdv syld alrimiv ) DHZCIZAIZJZYAHZKZ + CALZMZCNZADOZKZBIZDJZYKHZKZYKDPZMBYJYNYOYJYNKZBEQZBELZUAZEDYKUBZOZYOYPYSE + YTYJYNEIZYTPZYSYJYNUUCKZKZYQYRUUEYKUUBUCZYQRZYRUUEFYKUUBUUEFBLZFELZUUEUUH + KZYAFIZJZYDKZCFLZMZCNZGIZUUBJZUUQHZKZGELZMZGNZUUIUUEUUHUUPUUEYHAYKOZUUHUU + PMYIYNUVDXTUUCYIYLUVDYMYLYIUVDYLYKDUCYIUVDMYKDUDYHAYKDUESUFUGUHYHUUPAUUKY + KAFQZYGUUOCUVEYEUUMYFUUNUVEYCUULYDYBUUKYAUIUJAFCUKTULUNSUMUUEUVCUUHYIUUCU + VCXTYNUUCYIUVCUUCYIYAUUBJZYDKZCELZMZCNZUVCUUCUUBDPZYIUVJMUUBDYKUOZYHUVJAU + UBDAEQZYGUVICUVMYEUVGYFUVHUVMYCUVFYDYBUUBYAUIUJAECUKTULUPSZUVIUVBCGCGQZUV + GUUTUVHUVAUVOUVFUURYDUUSYAUUQUUBUQYAUUQURUSCGEUTTVEVAUFVBVCUUPUVCKUUIFEQZ + EFLZVDZUUJUUICGUUKUUBFVFEVFVGUVRUVPUVQUUIUAZUAZUUJUUIUVRUVPUVQUUIVDUVTUUI + UVPUVQVHUVPUVQUUIVIVJUUJUVPRZUVTUUIMUUDUUHUWAYJUUCUUHUWAYNUUCUUHUWAUUCUVP + UUKYTPUUHUUBYTUUKVKUUKYKDVPVLUMVMVMUWAUVTUVSUUJUUIUVPUVSVNUUJUVQRZUVSUUIM + UUDUUHUWBYJYNUUCUUHUWBYMUUCUUHUWBMMYLYMUUHUUKYKUCZUUCUWBYKUUKVOUUCEBLZRUW + CUWBUUBDYKVQUWCUVQUWDUUKYKUUBVRVSWHVTWAWBVMUVQUUIVNSWCWDWEWFWGWIWJUUFUUGK + YKUUBJZUUEYRYKUUBWKYJYNUUCUWEYRMZYIYNUUCUWFMZMXTYIYMUWGYLUUCYIYMUWFUUCYIU + VJYMUWFMUVNUVJUWEYMYRUVJUWEYMYRUVIUWEYMKZYRMCBCBQZUVGUWHUVHYRUWIUVFUWEYDY + MYAYKUUBUQYAYKURUSCBEUTTWLWMWNWTWOWPWAWQWRWSXAXBXCYPUUAYSEYTXDZYOYLUUAUWJ + MZYJYMYLYTXEXKZUWKYKDXFUWLUUAUWJYSEYTXGWISXHYPYSYOEYTYPYSUUCYOYPYQUUCYOMZ + YRYQUWMMYPUUCYOYQUVKUVLBEDXIXJXLXTYRUWMMYIYNUUCUVKXTYRYOUVLXTYRUVKYODYKUU + BXMWMXNXOXPWNXQXRWDWIXS $. $} ${ - $d B g $. $d B p $. $d B q $. $d B u $. $d B v $. $d B x $. $d B y $. - $d B z $. $d g u $. $d g v $. $d g x $. $d g y $. $d O p $. $d O z $. - $d p q $. $d p u $. $d p v $. $d p y $. $d p z $. $d q u $. $d q v $. - $d q y $. $d q z $. $d u v $. $d u x $. $d u y $. $d u z $. $d V g $. - $d V p $. $d v x $. $d v y $. $d v z $. $d V z $. $d x y $. $d y z $. - noinfbday.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Birthday bounding law for surreal infimum. (Contributed by Scott - Fenton, 8-Aug-2024.) $) - noinfbday $p |- ( ( ( B C_ No /\ B e. V ) /\ - ( O e. On /\ ( bday " B ) C_ O ) ) -> ( bday ` T ) C_ O ) $= - ( csur wss wcel wa cbday cdm wceq syl cv cres vz vp con0 cima cfv noinfno - vq bdayval adantr cslt wbr wn wral wrex crio csuc c1o cop csn cun cab w3a - wi cio cmpt cif iftrue eqtrid dmeqd 1oex dmsnop uneq2i dmun df-suc eqtrdi - 3eqtr4i word simprrl eloni simprll wreu wrmo nominmo reu5 sylanbrc sseldd - simpl riotacl simprrr wfn wfo bdayfo fofn ax-mp fnfvima eqeltrrd ordsucss - mp3an2i sylc eqsstrd noinfdm simplrl ssel2 simplrr mp3an1 ordelss syl2anc - ad4ant14 sseld adantrd rexlimdva abssdv adantl pm2.61ian ) EKLZEIMZNZHUCM - ZOEUDZHLZNZNZFOUEZFPZHXQYCYDQZYAXQFKMYEABCDEFGIJUFFUHRUIBSZASZUJUKULBEUMZ - AEUNZYBYDHLYIYBNZYDYHAEUOZPZUPZHYIYDYMQYBYIYDYKYLUQURUSZUTZPZYMYIFYOYIFYI - YOGYFDSZPZMYQCSZUJUKULZYQYFUPZTYSUUATQVCCEUMNDEUNBVAGSZYRMYTYQUUBUPZTYSUU - CTQVCCEUMUUBYQUEYGQVBDEUNAVDVEZVFYOJYIYOUUDVGVHVIYLYNPZUTYLYLUSZUTYPYMUUE - UUFYLYLUQVJVKVLYKYNVMYLVNVPVOUIYJHVQZYLHMYMHLYJXRUUGYIXQXRXTVRHVSZRYJYKOU - EZYLHYJYKKMUUIYLQYJEKYKYIXOXPYAVTZYJYHAEWAZYKEMZYJYIYHAEWBZUUKYIYBWGYJXOU - UMUUJABEWCRYHAEWDWEYHAEWHRZWFYKUHRYJXSHUUIYIXQXRXTWIOKWJZYJXOUULUUIXSMKUC - OWKUUOWLKUCOWMWNZUUJUUNKEOYKWOWRWFWPYLHWQWSWTYIULZYBNYDUASZUBSZPZMZUUSUGS - ZUJUKULUUSUURUPZTUVBUVCTQVCUGEUMZNZUBEUNZUAVAZHUUQYDUVGQYBABUACDEFGUGUBJX - AUIYBUVGHLUUQYBUVFUAHYBUVEUURHMZUBEYBUUSEMZNZUVAUVHUVDUVJUUTHUURUVJUUGUUT - HMUUTHLUVJXRUUGXQXRXTUVIXBUUHRUVJUUSOUEZUUTHUVJUUSKMZUVKUUTQXOUVIUVLXPYAE - KUUSXCXHUUSUHRUVJXSHUVKXQXRXTUVIXDXOUVIUVKXSMZXPYAUUOXOUVIUVMUUPKEOUUSWOX - EXHWFWPHUUTXFXGXIXJXKXLXMWTXNWT $. + $d A x y z w s t u $. $d B x y z w t $. + dfon2lem7.1 $e |- A e. _V $. + $( Lemma for ~ dfon2 . All elements of a new ordinal are new ordinals. + (Contributed by Scott Fenton, 25-Feb-2011.) $) + dfon2lem7 $p |- ( A. x ( ( x C. A /\ Tr x ) -> x e. A ) -> + ( B e. A -> A. y ( ( y C. B /\ Tr y ) -> y e. B ) ) ) $= + ( vw vt vz vu wpss wtr wa wcel wi wal wss wel wral sylbi imbi12d w3a cuni + vs cv cab wceq csuc csn wn weq elequ1 elequ2 bitrd notbid cbvralvw biimpi + ralimi untuni sylibr vex sseq1 treq raleq 3anbi123d elab dfon2lem3 simprd + cvv ax-mp untelirr syl 3ad2ant3 psseq2 anbi1d albidv 3anbi3i abbii unieqi + mprg eleq2i sylnib dfon2lem2 ssexi snss mtbi intnan cun df-suc unss mtbir + sseq1i bitr4i dfon2lem1 suctr wo elsuc wrex eluni2 rspccv anbi12d cbvalvw + nfa1 psseq1 syl6ib rexlimi rexlimiv rgen dfon2lem6 mp2an eleq2 jaoi sucex + mpbiri elssuni sylbir mp3an23 ralbii bitrdi cbvabv sseqtrdi syl5bir mpani + biimtrid mtoi eleq1 spcv mpan2i mtod dfpss2 biimpri mpan nsyl2 mpbii 3syl + a1i rsp ) AUDZCJZYQKZLZYQCMZNZAOZFUDZCPZUUDKZBUDZGUDZJZUUGKZLZBGQZNZBOZGU + UDRZUAZFUEZUBZCUFZUUGHUDZJZUUJLZBHQZNZBOZHCRZDCMUUGDJZUUJLZUUGDMZNZBOZNUU + CUURCJZUUSUUCUVLUURCMZUUCUVMUURUGZUUEUUFUUGIUDZJZUUJLZBIQZNZBOZIUUDRZUAZF + UEZUBZPZUWEUURUWDPZUURUHZUWDPZLZUWHUWFUURUWDMZUWHHHQZUIZHUURRZUWJUIGGQZUI + ZGYQRZUWMAUUQUWPAUUQRUWLHYQRZAUUQRUWMUWPUWQAUUQUWPUWQUWOUWLGHYQGHUJZUWNUW + KUWRUWNHGQUWKGHGUKGHHULUMUNUOUPUQHAUUQURUSYQUUQMZYQCPZYSUUNGYQRZUAZUWPUUP + UXBFYQAUTZFAUJZUUEUWTUUFYSUUOUXAUUDYQCVAZUUDYQVBZUUNGUUDYQVCZVDVEZUXAUWTU + WPYSUUNUWOGYQUUNIIQUIIUUHRZUWOUUNUUHKZUXIUUHVHMUUNUXJUXILNGUTBIUUHVHVFVIV + GIUUHVJVKUQVLSVSUWMUURUURMUWJHUURVJUURUWDUURUUQUWCUUPUWBFUUOUWAUUEUUFUUNU + VTGIUUDGIUJZUUMUVSBUXKUUKUVQUULUVRUXKUUIUVPUUJUUHUVOUUGVMVNGIBULTVOZUOVPV + QVRVTWAVIUURUWDUURCEUUFUUOFCWBZWCZWDWEWFUWEUURUWGWGZUWDPUWIUVNUXOUWDUURWH + ZWKUURUWGUWDWIWLWJUVMUWGCPZUUCUWEUURCUXNWDUUCUURCPZUXQUWEUXMUXRUXQLZUVNCP + ZUUCUWEUXTUXOCPUXSUVNUXOCUXPWKUURUWGCWIWLUXTUWENUUCUXTUVNUCUDZCPZUYAKZYQU + VOJZYSLZAIQZNZAOZIUYARZUAZUCUEZUBZUWDUXTUVNKZUYHIUVNRZUVNUYLPZUURKZUYMUUE + UUOFWMZUURWNVIUYHIUVNUVOUVNMUVOUURMZUVOUURUFZWOUYHUVOUURIUTWPUYRUYHUYSUYR + IAQZAUUQWQZUYHAUVOUUQWRZUYTUYHAUUQUYGAXBUWSUXBUYTUYHNZUXHUXAUWTVUCYSUXAUY + TUVTUYHUUNUVTGUVOYQUXLWSZUVSUYGBABAUJZUVQUYEUVRUYFVUEUVPUYDUUJYSUUGYQUVOX + CUUGYQVBWTBAIUKTXAXDVLSXESUYSUYHYQUURJZYSLZYQUURMZNZAOZUYPUUTUVOJZUUTKZLZ + HIQZNZHOZIUURRVUJUYQVUPIUURUYRVUAVUPVUBUYTVUPAUUQUWSUXBUYTVUPNZUXHUXAUWTV + UQYSUXAUYTUVTVUPVUDUVSVUOBHBHUJZUVQVUMUVRVUNVURUVPVUKUUJVULUUGUUTUVOXCUUG + UUTVBWTBHIUKTXAXDVLSXFSXGIAHUURXHXIUYSUYGVUIAUYSUYEVUGUYFVUHUYSUYDVUFYSUV + OUURYQVMVNUVOUURYQXJTVOXMXKSXGUXTUYMUYNUAZUVNUYKMUYOUYJVUSUCUVNUURUXNXLUY + AUVNUFUYBUXTUYCUYMUYIUYNUYAUVNCVAUYAUVNVBUYHIUYAUVNVCVDVEUVNUYKXNXOXPUYKU + WCUYJUWBUCFUCFUJZUYBUUEUYCUUFUYIUWAUYAUUDCVAUYAUUDVBVUTUYIUYHIUUDRUWAUYHI + UYAUUDVCUYHUVTIUUDUYGUVSABABUJZUYEUVQUYFUVRVVAUYDUVPYSUUJYQUUGUVOXCYQUUGV + BWTABIUKTXAXQXRVDXSVRXTYOYAYBYCYDUUCUVLUYPUVMUYQUUBUVLUYPLZUVMNAUURUXNYQU + URUFZYTVVBUUAUVMVVCYRUVLYSUYPYQUURCXCYQUURVBWTYQUURCYETYFYGYHUXRUUSUIZUVL + UXMUVLUXRVVDLUURCYIYJYKYLUUSUVEHUURRUVFUVEHUURUUTUURMHAQZAUUQWQUVEAUUTUUQ + WRVVEUVEAUUQUWSUWTYSUVEHYQRZUAZVVEUVENZUUPVVGFYQUXCUXDUUEUWTUUFYSUUOVVFUX + EUXFUXDUUOUXAVVFUXGUUNUVEGHYQUWRUUMUVDBUWRUUKUVBUULUVCUWRUUIUVAUUJUUHUUTU + UGVMVNGHBULTVOUOXRVDVEVVFUWTVVHYSUVEHYQYPVLSXFSXGUVEHUURCVCYMUVEUVKHDCUUT + DUFZUVDUVJBVVIUVBUVHUVCUVIVVIUVAUVGUUJUUTDUUGVMVNUUTDUUGXJTVOWSYN $. $} ${ - $d B g $. $d B u $. $d B v $. $d B x $. $d B y $. $d G g $. $d g u $. - $d G u $. $d g v $. $d G v $. $d g x $. $d G x $. $d g y $. $d G y $. - $d U u $. $d u v $. $d U v $. $d u x $. $d U x $. $d u y $. $d v x $. - $d v y $. - noinffv.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( The value of surreal infimum when there is no minimum. (Contributed by - Scott Fenton, 8-Aug-2024.) $) - noinffv $p |- ( ( -. E. x e. B A. y e. B -. y - ( U |` suc G ) = ( v |` suc G ) ) ) ) -> ( T ` G ) = ( U ` G ) ) $= - ( cv cslt wral wrex wcel cres wceq w3a cfv wbr wn csur wss wa cdm csuc wi - cab cio cmpt crio c1o cop csn cun cif iffalse eqtrid fveq1d 3ad2ant1 dmeq - eleq2d breq1 notbid reseq1 eqeq1d imbi12d ralbidv anbi12d rspcev 3ad2ant3 - 3impb simp32 eleq1 suceq reseq2d eqeq12d imbi2d rexbidv elabg syl fveqeq2 - mpbird 3anbi123d iotabidv eqid iotaex fvmpt simp1 simp2 simp3 eqidd fveq1 - wb syl13anc cvv weu fvex wex wmo eqeq2 3anbi3d spcev mp3an3 rexcom4 sylib - reximi simp2l noinfprefixmo df-eu sylanbrc iota2 sylancr mpbid 3eqtrd ) B - LZALZMUAUBBENZAEOZUBZEUCUDZEJPZUEZGEPZIGUFZPZGCLZMUAZUBZGIUGZQZYHYKQZRZUH - ZCENZSZSZIFTZIHXQDLZUFZPZYTYHMUAZUBZYTXQUGZQZYHUUEQZRZUHZCENZUEZDEOZBUIZH - LZUUAPZUUDYTUUNUGZQZYHUUPQZRZUHZCENZUUNYTTXRRZSZDEOZAUJZUKZTZIUUAPZUUDYTY - KQZYMRZUHZCENZIYTTZXRRZSZDEOZAUJZIGTZYAYDYSUVGRYQYAIFUVFYAFXTXSAEULZUVSUF - UMUNUOUPZUVFUQUVFKXTUVTUVFURUSUTVAYRIUUMPZUVGUVQRYRUWAUVHUVLUEZDEOZYQYAUW - CYDYEYGYPUWCUWBYGYPUEDGEYTGRZUVHYGUVLYPUWDUUAYFIYTGVBVCZUWDUVKYOCEUWDUUDY - JUVJYNUWDUUCYIYTGYHMVDVEUWDUVIYLYMYTGYKVFVGVHVIZVJVKVMZVLYRYGUWAUWCWOYAYD - YEYGYPVNUULUWCBIYFXQIRZUUKUWBDEUWHUUBUVHUUJUVLXQIUUAVOUWHUUIUVKCEUWHUUHUV - JUUDUWHUUFUVIUUGYMUWHUUEYKYTXQIVPZVQUWHUUEYKYHUWIVQVRVSVIVJVTWAWBWDHIUVEU - VQUUMUVFUUNIRZUVDUVPAUWJUVCUVODEUWJUUOUVHUVAUVLUVBUVNUUNIUUAVOUWJUUTUVKCE - UWJUUSUVJUUDUWJUUQUVIUURYMUWJUUPYKYTUUNIVPZVQUWJUUPYKYHUWKVQVRVSVIUUNIXRY - TWCWEVTWFUVFWGUVPAWHWIWBYRUVHUVLUVMUVRRZSZDEOZUVQUVRRZYQYAUWNYDYQYEYGYPUV - RUVRRZUWNYEYGYPWJYEYGYPWKYEYGYPWLYQUVRWMUWMYGYPUWPSDGEUWDUVHYGUVLYPUWLUWP - UWEUWFUWDUVMUVRUVRIYTGWNVGWEVKWPVLYRUVRWQPUVPAWRZUWNUWOWOIGWSYRUVPAWTZUVP - AXAZUWQYQYAUWRYDYQUWCUWRUWGUWCUVOAWTZDEOUWRUWBUWTDEUVHUVLUVMUVMRZUWTUVMWG - UVOUVHUVLUXASAUVMIYTWSXRUVMRUVNUXAUVHUVLXRUVMUVMXBXCXDXEXHUVODAEXFXGWBVLY - RYBUWSYAYBYCYQXIACDEIXJWBUVPAXKXLUVPUWNAUVRWQXRUVRRZUVOUWMDEUXBUVNUWLUVHU - VLXRUVRUVMXBXCVTXMXNXOXP $. + $d A x y z w t $. + $( Lemma for ~ dfon2 . The intersection of a nonempty class ` A ` of new + ordinals is itself a new ordinal and is contained within ` A ` + (Contributed by Scott Fenton, 26-Feb-2011.) $) + dfon2lem8 $p |- ( ( A =/= (/) /\ + A. x e. A A. y ( ( y C. x /\ Tr y ) -> y e. x ) ) -> + ( A. z ( ( z C. |^| A /\ Tr z ) -> z e. |^| A ) /\ + |^| A e. A ) ) $= + ( vt vw cv wpss wtr wa wel wi wal wral wcel wn cvv syl sylbi wceq c0 cint + wne vex dfon2lem3 ax-mp simpld ralimi trint adantl cuni dfon2lem7 alrimiv + df-ral 19.21v albii bitr4i impexp 2albii eluni2 biimpi imim1i alimi alcom + wrex wex 19.23v df-rex imbi1i bitri 3imtr4i sylbir intssuni ssralv adantr + wss mpd dfon2lem6 imp simprd untelirr adantlr risset notbii ralnex psseq2 + intex eqcom anbi1d elequ2 imbi12d albidv rspccv intss1 dfpss2 psseq1 treq + anbi12d eleq1 spcgv expd exp4b com45 com23 syl5 mpdd mpid syl7bi ralrimiv + syl5bir ralim biimtrid wb elintg ad2antrr sylibrd mt3d ex ancld mp2and ) + DUAUCZBGZAGZHZYBIZJZBAKZLZBMZADNZJZDUBZIZEGZFGZHYNIJEFKLEMZFYLNZCGZYLHYRI + JYRYLOLCMZYLDOZJZYJYMYAYJYCIZADNYMYIUUBADYIUUBCCKPCYCNZYCQOYIUUBUUCJLAUDZ + BCYCQUEUFUGUHADUIRUJYKYPFDUKZNZYQYJUUFYAYJFAKZYPLZFMZADNZUUFYIUUIADYIUUHF + BEYCYOUUDULUMUHUUJYCDOZUUHLZFMZAMZUUFUUJUUKUUILZAMUUNUUIADUNUUMUUOAUUKUUH + FUOUPUQUUNUUKUUGJZYPLZFMAMZUUFUUQUULAFUUKUUGYPURUSUUGADVEZYPLZFMZYOUUEOZY + PLZFMUURUUFUUTUVCFUVBUUSYPUVBUUSAYODUTVAVBVCUURUUQAMZFMUVAUUQAFVDUVDUUTFU + VDUUPAVFZYPLUUTUUPYPAVGUUSUVEYPUUGADVHVIUQUPVJYPFUUEUNVKVLSRUJYAUUFYQLZYJ + YAYLUUEVPUVFDVMYPFYLUUEVNRVOVQYMYQJYSYKUUAFCEYLVRYKYSYTYKYSYTYKYSJZYTYLYL + OZYAYSUVHPZYJYAYSJZEEKPEYLNZUVIUVJYMUVKYAYSYMUVKJZYAYLQOZYSUVLLDWGZCEYLQU + ESVSZVTEYLWARWBUVGYTPZYLYNOZEDNZUVHUVPYNYLTZPZEDNZUVGUVRUVPUVSEDVEZPUWAYT + UWBEYLDWCWDUVSEDWEUQUVGUVTUVQLZEDNUWAUVRLUVGUWCEDUVTYLYNTZPZUVGYNDOZUVQUV + SUWDYNYLWHWDUVGUWFYMUWEUVQLZYAYSYMYJUVJYMUVKUVOUGWBYKUWFYMUWGLZLYSYKUWFYB + YNHZYEJZBEKZLZBMZUWHYJUWFUWMLYAYIUWMAYNDYCYNTZYHUWLBUWNYFUWJYGUWKUWNYDUWI + YEYCYNYBWFWIAEBWJWKWLWMUJYAUWFUWMUWHLZLYJUWFYLYNVPZYAUWOYNDWNYAUWMUWPUWHY + AUWMUWPUWEYMUVQYAUWMUWPUWEYMUVQLZUWPUWEJYLYNHZYAUWMJZUWQYLYNWOUWSUWRYMUVQ + YAUWMUWRYMJZUVQLZYAUVMUWMUXALUVNUWLUXABYLQYBYLTZUWJUWTUWKUVQUXBUWIUWRYEYM + YBYLYNWPYBYLWQWRYBYLYNWSWKWTSVSXAXJXBXCXDXEVOXFVOXGXHXIUVTUVQEDXKRXLYAUVH + UVRXMZYJYSYAUVMUXCUVNEYLDQXNSXOXPXQXRXSXEXT $. $} ${ - $d a b $. $d a c $. $d a g $. $d a u $. $d a v $. $d a x $. $d a y $. - $d B a $. $d B b $. $d b c $. $d B c $. $d B g $. $d b u $. $d B u $. - $d b v $. $d B v $. $d B x $. $d b y $. $d B y $. $d c u $. $d c v $. - $d c y $. $d G a $. $d G b $. $d G c $. $d g u $. $d g v $. $d G v $. - $d g x $. $d g y $. $d T a $. $d U a $. $d U b $. $d U c $. $d U u $. - $d u v $. $d U v $. $d u x $. $d U x $. $d u y $. $d V a $. $d V g $. - $d v x $. $d v y $. $d x y $. - noinfres.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( The restriction of surreal infimum when there is no minimum. - (Contributed by Scott Fenton, 8-Aug-2024.) $) - noinfres $p |- ( ( -. E. x e. B A. y e. B -. y - ( U |` suc G ) = ( v |` suc G ) ) ) ) -> - ( T |` suc G ) = ( U |` suc G ) ) $= - ( va vb vc cv wral wcel cres wceq syl cslt wbr wn wrex csur wss wa cdm wi - csuc w3a cfv cin dmres word noinfno 3ad2ant2 nodmord simp31 simp32 simp33 - cab dmeq eleq2d breq1 notbid reseq1 eqeq1d imbi12d ralbidv breq2 cbvralvw - weq eqeq2d bitrdi anbi12d rspcev syl12anc wb eleq1 reseq2d eqeq12d imbi2d - suceq rexbidv elabg mpbird noinfdm 3ad2ant1 eleqtrrd ordsucss df-ss sylib - sylc eqtrid con0 simp2l sseldd nodmon eqtr4d simpl1 simpl2 simpl31 sselda - eloni adantr simpl32 onelon syl2anc sucelon simpl33 resabs1 syl5ib imim2d - simpr ralimdv noinffv syl113anc fvresd 3eqtr4d sylbid ralrimiv wfun nofun - ex funresd eqfunfv mpbir2and ) BOAOUAUBUCBEPAEUDUCZEUEUFZEJQZUGZGEQZIGUHZ - QZGCOZUAUBZUCZGIUJZRZYPYSRZSZUIZCEPZUKZUKZFYSRZYTSZUUGUHZYTUHZSZLOZUUGULZ - UULYTULZSZLUUIPZUUFUUIYSUUJUUFUUIYSFUHZUMZYSFYSUNUUFYSUUQUFZUURYSSUUFUUQU - OZIUUQQUUSUUFFUEQZUUTYLYIUVAUUEABCDEFHJKUPUQZFURTUUFIUULMOZUHZQZUVCNOZUAU - BZUCZUVCUULUJZRZUVFUVIRZSZUIZNEPZUGZMEUDZLVBZUUQUUFIUVQQZIUVDQZUVHUVCYSRZ - UVFYSRZSZUIZNEPZUGZMEUDZUUFYMYOUUDUWFYIYLYMYOUUDUSZYIYLYMYOUUDUTZYIYLYMYO - UUDVAUWEYOUUDUGMGEUVCGSZUVSYOUWDUUDUWIUVDYNIUVCGVCVDUWIUWDGUVFUAUBZUCZYTU - WASZUIZNEPUUDUWIUWCUWMNEUWIUVHUWKUWBUWLUWIUVGUWJUVCGUVFUAVEVFUWIUVTYTUWAU - VCGYSVGVHVIVJUWMUUCNCENCVMZUWKYRUWLUUBUWNUWJYQUVFYPGUAVKVFUWNUWAUUAYTUVFY - PYSVGVNVIVLVOVPVQVRUUFYOUVRUWFVSUWHUVPUWFLIYNUULISZUVOUWEMEUWOUVEUVSUVNUW - DUULIUVDVTUWOUVMUWCNEUWOUVLUWBUVHUWOUVJUVTUVKUWAUWOUVIYSUVCUULIWDZWAUWOUV - IYSUVFUWPWAWBWCVJVPWEWFTWGYIYLUUQUVQSUUEABLCDEFHNMKWHWIWJIUUQWKWNYSUUQWLW - MWOZUUFUUJYSYNUMZYSGYSUNUUFYSYNUFZUWRYSSUUFYNUOZYOUWSUUFYNWPQZUWTUUFGUEQZ - UXAUUFEUEGYIYJYKUUEWQUWGWRZGWSZTYNXETUWHIYNWKWNZYSYNWLWMWOWTUUFUUOLUUIUUF - UULUUIQUULYSQZUUOUUFUUIYSUULUWQVDUUFUXFUUOUUFUXFUGZUULFULZUULGULZUUMUUNUX - GYIYLYMUULYNQYRGUVIRZYPUVIRZSZUIZCEPZUXHUXISYIYLUUEUXFXAYIYLUUEUXFXBYMYOU - UDYIYLUXFXCUUFYSYNUULUXEXDUXGUVIYSUFZUUDUXNUXGYSUOZUXFUXOUXGYSWPQZUXPUXGI - WPQZUXQUXGUXAYOUXRUXGUXBUXAUUFUXBUXFUXCXFUXDTYMYOUUDYIYLUXFXGYNIXHXIIXJWM - YSXETUUFUXFXOZUULYSWKWNYMYOUUDYIYLUXFXKUXOUUCUXMCEUXOUUBUXLYRUUBYTUVIRZUU - AUVIRZSUXOUXLYTUUAUVIVGUXOUXTUXJUYAUXKGUVIYSXLYPUVIYSXLWBXMXNXPWNABCDEFGH - UULJKXQXRUXGUULYSFUXSXSUXGUULYSGUXSXSXTYEYAYBUUFUUGYCZYTYCZUUHUUKUUPUGVSU - UFUVAUYBUVBUVAYSFFYDYFTUUFUXBUYCUXCUXBYSGGYDYFTLUUGYTYGXIYH $. + $d A x y z w t u $. + $( Lemma for ~ dfon2 . A class of new ordinals is well-founded by ` _E ` . + (Contributed by Scott Fenton, 3-Mar-2011.) $) + dfon2lem9 $p |- ( A. x e. A A. y ( ( y C. x /\ Tr y ) -> y e. x ) + -> _E Fr A ) $= + ( vz vt vw vu cv wpss wtr wa wel wi wal wral wss wn wrex cep wcel wne wfr + c0 ssralv cint dfon2lem8 simprd intss1 simpld cvv wceq dfon2lem3 untelirr + intex imp syl eleq1 notbid syl5ibcom a1dd trss eqss simplbi2com syl6 con3 + com23 pm2.61d sylanb syldan ralrimiv eleq2 ralbidv rspcev syl2anc syl6com + syl5 expcom impd alrimiv wbr df-fr epel notbii ralbii rexbii imbi2i albii + bitri sylibr ) BHZAHIWJJKBALMBNZACOZDHZCPZWMUCUAZKZEFLZQZEWMOZFWMRZMZDNZC + SUBZWLXADWLWNWOWTWNWLWKAWMOZWOWTMWKAWMCUDWOXDWTWOXDKZWMUEZWMTZEHZXFTZQZEW + MOZWTXEGHZXFIXLJKXLXFTMGNZXGABGWMUFZUGXEXJEWMEDLXFXHPZXEXJXHWMUHWOXDXMXOX + JMZXEXMXGXNUIWOXFUJTZXMXPWMUNXQXMKZXFXHUKZXPXRXSXJXOXRXFXFTZQZXSXJXRAALQA + XFOZYAXRXFJZYBXQXMYCYBKGAXFUJULUOZUGAXFUMUPXSXTXIXFXHXFUQURUSUTXRXOXSQZXJ + XRXOXIXSMYEXJMXRXIXOXSXRXIXHXFPZXOXSMXRYCXIYFMXRYCYBYDUIXFXHVAUPXSXOYFXFX + HVBVCVDVFXIXSVEVDVFVGVHVIVPVJWSXKFXFWMFHZXFUKZWRXJEWMYHWQXIYGXFXHVKURVLVM + VNVQVOVRVSXCWPXHYGSVTZQZEWMOZFWMRZMZDNXBDFECSWAYMXADYLWTWPYKWSFWMYJWREWMY + IWQFXHWBWCWDWEWFWGWHWI $. $} ${ - $d B g $. $d B h $. $d B p $. $d B q $. $d B u $. $d B v $. $d B x $. - $d B y $. $d g u $. $d g v $. $d g x $. $d g y $. $d h p $. $d h q $. - $d h u $. $d h v $. $d h x $. $d h y $. $d p q $. $d p u $. $d p v $. - $d p x $. $d p y $. $d q u $. $d q v $. $d q y $. $d T h $. $d T p $. - $d U h $. $d U p $. $d u v $. $d U v $. $d u x $. $d u y $. $d V g $. - $d V h $. $d V p $. $d v x $. $d v y $. $d x y $. - noinfbnd1.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Lemma for ~ noinfbnd1 . Establish a soft lower bound. (Contributed by - Scott Fenton, 9-Aug-2024.) $) - noinfbnd1lem1 $p |- ( ( -. E. x e. B A. y e. B -. y -. ( U |` dom T ) ( W |` dom T ) = T ) $= - ( cv cslt wbr wn csur wcel wa cres wceq wss cdm w3a simp3rl noinfbnd1lem1 - wral wrex syld3an3 simp3rr con0 wi simp2l simp3ll sseldd noinfno 3ad2ant2 - nodmon syl sltres syl3anc mtod simp3lr breq1d mtbid noreson syl2anc sltso - wb wor sotrieq2 mpan mpbir2and ) BLALMNOBEUFAEUGOZEPUAZEIQZRZGEQZGFUBZSZF - TZRZJEQZGJMNZOZRZRZUCZJVRSZFTZWHFMNOZFWHMNZOZVMVPWFWBWJWBWDWAVMVPUDZABCDE - FJHIKUEUHWGVSWHMNZWKWGWNWCWBWDWAVMVPUIWGGPQJPQZVRUJQZWNWCUKWGEPGVMVNVOWFU - LZVQVTWEVMVPUMUNWGEPJWQWMUNZWGFPQZWPVPVMWSWFABCDEFHIKUOUPZFUQURZGJVRUSUTV - AWGVSFWHMVQVTWEVMVPVBVCVDWGWHPQZWSWIWJWLRVHZWGWOWPXBWRXAJVRVEVFWTPMVIXBWS - RXCVGPWHFMVJVKVFVL $. + $d x y z w t u v $. + $( ` On ` consists of all sets that contain all its transitive proper + subsets. This definition comes from J. R. Isbell, "A Definition of + Ordinal Numbers", American Mathematical Monthly, vol 67 (1960), pp. + 51-52. (Contributed by Scott Fenton, 20-Feb-2011.) $) + dfon2 $p |- On = { x | A. y ( ( y C. x /\ Tr y ) -> y e. x ) } $= + ( vz vw vu vt vv cv cab wpss wtr wa wel wi wal cep wral vex weq imbi12d + con0 word df-on wss wne tz7.7 df-pss bitr4di exbiri com23 impd alrimiv wn + wwe cvv wcel dfon2lem3 simpld wfr w3o dfon2lem7 ralrimiv dfon2lem9 psseq2 + ax-mp anbi1d elequ2 albidv psseq1 anbi12d elequ1 cbvalvw bitrdi dfon2lem5 + treq anim12d syl6 ralrimivv jca syl wbr dfwe2 epel biid 3orbi123i 2ralbii + rspccv anbi2i bitri sylibr df-ord sylanbrc impbii abbii eqtri ) UAAHZUBZA + IBHZWPJZWRKZLBAMZNZBOZAIAUCWQXCAWQXCWQXBBWQWSWTXAWQWTWSXAWQWTXAWSWQWTLXAW + RWPUDWRWPUELWSWPWRUFWRWPUGUHUIUJUKULXCWPKZWPPUNZWQXCXDCCMUMCWPQZWPUOUPXCX + DXFLNARZBCWPUOUQVEURXCWPPUSZCDMZCDSZDCMZUTZDWPQCWPQZLZXEXCEHZFHZJZXOKZLZE + FMZNZEOZFWPQZXNXCYBFWPBEWPXPXGVAVBYCXHXMFEWPVCYCXLCDWPWPYCCAMZDAMZLGHZCHZ + JZYFKZLZGCMZNZGOZWRDHZJZWTLZBDMZNZBOZLXLYCYDYMYEYSYBYMFYGWPFCSZYBXOYGJZXR + LZECMZNZEOYMYTYAUUDEYTXSUUBXTUUCYTXQUUAXRXPYGXOVDVFFCEVGTVHUUDYLEGEGSZUUB + YJUUCYKUUEUUAYHXRYIXOYFYGVIXOYFVOVJEGCVKTVLVMWGYBYSFYNWPFDSZYBXOYNJZXRLZE + DMZNZEOYSUUFYAUUJEUUFXSUUHXTUUIUUFXQUUGXRXPYNXOVDVFFDEVGTVHUUJYREBEBSZUUH + YPUUIYQUUKUUGYOXRWTXOWRYNVIXOWRVOVJEBDVKTVLVMWGVPGBYGYNCRDRVNVQVRVSVTXEXH + YGYNPWAZXJYNYGPWAZUTZDWPQCWPQZLXNCDWPPWBUUOXMXHUUNXLCDWPWPUULXIXJXJUUMXKD + YGWCXJWDCYNWCWEWFWHWIWJWPWKWLWMWNWO $. + $} - $d B z $. $d p z $. $d q z $. $d T q $. $d T z $. $d U q $. $d u z $. - $d v z $. $d y z $. $d q x $. $d V q $. - $( Lemma for ~ noinfbnd1 . If ` U ` is a prolongment of ` T ` and in - ` B ` , then ` ( U `` dom T ) ` is not ` 1o ` . (Contributed by Scott - Fenton, 9-Aug-2024.) $) - noinfbnd1lem3 $p |- ( ( -. E. x e. B A. y e. B -. y - ( U ` dom T ) =/= 1o ) $= - ( vp vq wn csur wcel wa cres wceq c1o adantr vz cv cslt wbr wral wrex wss - cdm w3a cfv word noinfno 3ad2ant2 nodmord ordirr 3syl wi simpl3l c0 ndmfv - wne 1n0 necomi neeq1 mpbiri neneqd syl con4i adantl simpl2l sseldd simprl - csuc nodmon simpl3r simpll1 simpll2 simpll3 simpr noinfbnd1lem2 syl112anc - con0 eqtr4d simplr simprr nogesgn1ores syl321anc expr eleq2d breq1 notbid - ralrimiva dmeq reseq1 eqeq1d imbi12d ralbidv anbi12d syl12anc cab noinfdm - rspcev wb 3ad2ant1 eleq1 suceq reseq2d eqeq12d imbi2d rexbidv elabg bitrd - mpbird mtand neqned ) BUBAUBUCUDMBEUEAEUFMZENUGZEIOZPZGEOZGFUHZQZFRZPZUIZ - YAGUJZSYEYFSRZYAYAOZYEFNOZYAUKYHMXSXPYIYDABCDEFHIJULUMZFUNYAUOUPYEYGPZYHY - AKUBZUHZOZYLLUBZUCUDZMZYLYAVMZQZYOYRQZRZUQZLEUEZPZKEUFZYKXTYAGUHZOZGYOUCU - DZMZGYRQZYTRZUQZLEUEZUUEXTYCXPXSYGURZYGUUGYEUUGYGUUGMYFUSRZYGMYAGUTUUOYFS - UUOYFSVAUSSVASUSVBVCYFUSSVDVEVFVGVHVIYKUULLEYKYOEOZUUIUUKYKUUPUUIPZPZGNOZ - YONOYAWBOZYBYOYAQZRYGUUIUUKYKUUSUUQYKENGXQXRXPYDYGVJZUUNVKTUURENYOYKXQUUQ - UVBTYKUUPUUIVLVKYKUUTUUQYKYIUUTYEYIYGYJTFVNZVGTUURYBFUVAYKYCUUQXTYCXPXSYG - VOTUURXPXSYDUUQUVAFRXPXSYDYGUUQVPXPXSYDYGUUQVQXPXSYDYGUUQVRYKUUQVSABCDEFG - HIYOJVTWAWCYEYGUUQWDYKUUPUUIWEGYOYAWFWGWHWLUUDUUGUUMPKGEYLGRZYNUUGUUCUUMU - VDYMUUFYAYLGWMWIUVDUUBUULLEUVDYQUUIUUAUUKUVDYPUUHYLGYOUCWJWKUVDYSUUJYTYLG - YRWNWOWPWQWRXBWSYEYHUUEXCYGYEYHYAUAUBZYMOZYQYLUVEVMZQZYOUVGQZRZUQZLEUEZPZ - KEUFZUAWTZOZUUEXPXSYHUVPXCYDXPYAUVOYAABUACDEFHLKJXAWIXDYEYIUUTUVPUUEXCYJU - VCUVNUUEUAYAWBUVEYARZUVMUUDKEUVQUVFYNUVLUUCUVEYAYMXEUVQUVKUUBLEUVQUVJUUAY - QUVQUVHYSUVIYTUVQUVGYRYLUVEYAXFZXGUVQUVGYRYOUVRXGXHXIWQWRXJXKUPXLTXMXNXO + ${ + $d F g $. $d I g $. + $( The value of the recursive definition generator at ` (/) ` when the base + value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) + (Revised by Mario Carneiro, 19-Apr-2014.) $) + rdgprc0 $p |- ( -. I e. _V -> ( rec ( F , I ) ` (/) ) = (/) ) $= + ( vg cvv wcel wn c0 crdg cfv cv wceq cdm wlim crn cuni cif cmpt cres eqid + unieqd con0 0elon rdgval ax-mp res0 fveq2i eqtri eqeq1 wb dmeq limeq rneq + syl fveq12d fveq2d ifbieq12d ifbieq2d eleq1d elrab2 iftruei eleq1i biimpi + id dmmpt simplbiim ndmfv nsyl5 eqtrid ) BDEZFGABHZIZGCDCJZGKZBVLLZMZVLNZO + ZVNOZVLIZAIZPZPZQZIZGVKVJGRZWCIZWDGUAEVKWFKUBBGCAUCUDWEGWCVJUEUFUGGWCLZEZ + VIWDGKWHGDEGGKZBGLZMZGNZOZWJOZGIZAIZPZPZDEZVIWBDEWSCGDWGVMWBWRDVMVMWIWAWQ + BVLGGUHVMVOWKVQVTWMWPVMVNWJKVOWKUIVLGUJZVNWJUKUMVMVPWLVLGULTVMVSWOAVMVRWN + VLGVMVCVMVNWJWTTUNUOUPUQURCDWBWCWCSVDUSWSVIWRBDWIBWQGSUTVAVBVEGWCVFVGVH $. - - $d B w $. $d T w $. $d U w $. $d U x $. $d U y $. $d v w $. $d V w $. - $d w x $. $d w y $. - - $( Lemma for ~ noinfbnd1 . If ` U ` is a prolongment of ` T ` and in - ` B ` , then ` ( U `` dom T ) ` is not undefined. (Contributed by Scott - Fenton, 9-Aug-2024.) $) - noinfbnd1lem4 $p |- ( ( -. E. x e. B A. y e. B -. y - ( U ` dom T ) =/= (/) ) $= - ( vw cslt wbr wn wrex csur wcel wa wceq adantr cv wral wss cdm w3a cfv c0 - cres c1o wi simpl1 simpl2 simprl simpl3 simp2l sselda simp3l sseldd sltso - wne soasym mpan syl2anc impr noinfbnd1lem2 syl112anc noinfbnd1lem3 neneqd - wor jca expr imnan sylib nrexdv breq2 rexbidv dfral2 ralnex rexbii sylibr - xchbinxr simpl3l rspcdva cbvrexvw con0 simpl2l noinfno syl nodmon simpll1 - breq1 simpll2 simpll3 simprr mpd simpl3r eqtr4d simplr syl321anc reximdva - nogt01o ancld mtand neqned ) BUAZAUAZLMZNBEUBZAEOZNZEPUCZEIQZRZGEQZGFUDZU - HZFSZRZUEZXOGUFZUGXSXTUGSZKUAZGLMZXOYBUFZUISZRZKEOZXSYFKEXSYBEQZRZYCYENZU - JYFNXSYHYCYJXSYHYCRZRZYDUIYLXJXMYHYBXOUHZFSZYDUIUTXJXMXRYKUKZXJXMXRYKULZX - SYHYCUMZYLXJXMXRYHGYBLMNZRZYNYOYPXJXMXRYKUNYLYHYRYQXSYHYCYRYIYBPQZGPQZYCY - RUJZXSEPYBXJXKXLXRUOZUPXSUUAYHXSEPGUUCXJXMXNXQUQURTPLVIYTUUARUUBUSPLYBGVA - VBZVCVDVJABCDEFGHIYBJVEZVFABCDEFYBHIJVGVFVHVKYCYEVLVMVNXSYARZYCKEOZYGUUFX - EGLMZBEOZUUGUUFXGBEOZUUIAEGXFGSXGUUHBEXFGXELVOVPUUFXJUUJAEUBZXJXMXRYAUKUU - KUUJNZAEOXIUUJAEVQXHUULAEXGBEVRVSWAVTXNXQXJXMYAWBZWCUUHYCBKEXEYBGLWKWDVMU - UFYCYFKEUUFYHRYCYEUUFYHYCYEUUFYKRZYTUUAXOWEQZYMXPSYCYAYEUUNEPYBUUFXKYKXKX - LXJXRYAWFTZUUFYHYCUMZURZUUNEPGUUPUUFXNYKUUMTURZUUNFPQZUUOUUFUUTYKUUFXMUUT - XJXMXRYAULABCDEFHIJWGWHTFWIWHUUNYMFXPUUNXJXMXRYSYNXJXMXRYAYKWJXJXMXRYAYKW - LXJXMXRYAYKWMUUNYHYRUUQUUNYCYRUUFYHYCWNZUUNYTUUAUUBUURUUSUUDVCWOVJUUEVFUU - FXQYKXNXQXJXMYAWPTWQUVAXSYAYKWRYBGXOXAWSVKXBWTWOXCXD $. - - $d a p $. $d a u $. $d a v $. $d a y $. $d a z $. $d B a $. $d T a $. - $d U z $. $d V z $. $d x z $. - - $( Lemma for ~ noinfbnd1 . If ` U ` is a prolongment of ` T ` and in - ` B ` , then ` ( U `` dom T ) ` is not ` 2o ` . (Contributed by Scott - Fenton, 9-Aug-2024.) $) - noinfbnd1lem5 $p |- ( ( -. E. x e. B A. y e. B -. y - ( U ` dom T ) =/= 2o ) $= - ( vz wn c2o wceq csur wcel wa cres syl c0 va vp cv cslt wbr cdm wral wrex - cfv wi wss w3a wne word noinfno 3ad2ant2 adantl nodmord ordirr cab adantr - csuc simpr3l ndmfv 2on0 necomi neeq1 mpbiri neneqd simpl2l simpl3l sseldd - con4i wfun nofun simprll simpl3r simpll1 simpll2 simpll3 simprl syl112anc - noinfbnd1lem2 eqtr4d simplr simprr eqfunressuc syl213anc expr a2d anassrs - ralimdva impcom eleq2d breq1 notbid reseq1 eqeq1d imbi12d ralbidv anbi12d - dmeq rspcev syl12anc wb nodmon eleq1 suceq reseq2d eqeq12d imbi2d rexbidv - elabg mpbird noinfdm 3ad2ant1 mtand neqned rexanali simpr1 simpr2 simplll - con0 c1o simpr3 simpll noinfbnd1lem4 pm2.21d noinfbnd1lem3 w3o wo simpr2l - nofv 3orel3 sylc mpjaod ex anasss rexlimiva imp sylanbr pm2.61ian ) GKUCZ - UDUEZLZFUFZUUCUIZMNZUJZKEUGZBUCAUCUDUELBEUGAEUHLZEOUKZEIPZQZGEPZGUUFRZFNZ - QZULZUUFGUIZMUMZUUJUUSQZUUTMUVBUUTMNZUUFUUFPZUVBUUFUNZUVDLUVBFOPZUVEUUSUV - FUUJUUNUUKUVFUURABCDEFHIJUOUPUQZFURSUUFUSSUVBUVCQZUVDUUFUAUCZUBUCZUFZPZUV - JUUCUDUEZLZUVJUVIVBZRZUUCUVORZNZUJZKEUGZQZUBEUHZUAUTZPZUVHUWDUUFUVKPZUVNU - VJUUFVBZRZUUCUWFRZNZUJZKEUGZQZUBEUHZUVHUUOUUFGUFZPZUUEGUWFRZUWHNZUJZKEUGZ - UWMUVBUUOUVCUUOUUQUUKUUNUUJVCVAUVCUWOUVBUWOUVCUWOLZUUTMUWTUUTTNZUVAUUFGVD - UXAUVATMUMZMTVEVFZUUTTMVGVHSVIVMZUQUUJUUSUVCUWSUUSUVCQZUUJUWSUXEUUIUWRKEU - XEUUCEPZQUUEUUHUWQUXEUXFUUEUUHUWQUJUXEUXFUUEQZUUHUWQUXEUXGUUHQZQZGVNZUUCV - NZUUPUUCUUFRZNUWOUUFUUCUFPZUUTUUGNUWQUXIGOPUXJUXIEOGUXEUULUXHUULUUMUUKUUR - UVCVJVAZUXEUUOUXHUUOUUQUUKUUNUVCVKVAVLGVOSUXIUUCOPZUXKUXIEOUUCUXNUXEUXFUU - EUUHVPVLUUCVOSUXIUUPFUXLUXEUUQUXHUUOUUQUUKUUNUVCVQVAUXIUUKUUNUURUXGUXLFNZ - UUKUUNUURUVCUXHVRUUKUUNUURUVCUXHVSUUKUUNUURUVCUXHVTUXEUXGUUHWAABCDEFGHIUU - CJWCZWBWDUXIUVCUWOUUSUVCUXHWEZUXDSUXIUUHUXMUXEUXGUUHWFZUXMUUHUXMLZUUGMUXT - UUGTNZUUGMUMZUUFUUCVDUYAUYBUXBUXCUUGTMVGVHSVIVMSUXIUUTMUUGUXRUXSWDGUUCUUF - WGWHWIWIWJWLWMWKUWLUWOUWSQUBGEUVJGNZUWEUWOUWKUWSUYCUVKUWNUUFUVJGXBWNUYCUW - JUWRKEUYCUVNUUEUWIUWQUYCUVMUUDUVJGUUCUDWOWPUYCUWGUWPUWHUVJGUWFWQWRWSWTXAX - CXDUVHUUFYCPZUWDUWMXEUVBUYDUVCUVBUVFUYDUVGFXFSVAUWBUWMUAUUFYCUVIUUFNZUWAU - WLUBEUYEUVLUWEUVTUWKUVIUUFUVKXGUYEUVSUWJKEUYEUVRUWIUVNUYEUVPUWGUVQUWHUYEU - VOUWFUVJUVIUUFXHZXIUYEUVOUWFUUCUYFXIXJXKWTXAXLXMSXNUVHUUFUWCUUFUVBUUFUWCN - ZUVCUUSUYGUUJUUKUUNUYGUURABUACDEFHKUBJXOXPUQVAWNXNXQXRUUJLUUEUUHLZQZKEUHZ - UUSUVAUUEUUHKEXSUYJUUSUVAUYIUUSUVAUJZKEUXFUUEUYHUYKUXGUYHQZUUSUVAUYLUUSQZ - UYAUVAUUGYDNZUYMUYAUVAUYMUUGTUYMUUKUUNUXFUXPUUGTUMUYLUUKUUNUURXTZUYLUUKUU - NUURYAZUXFUUEUYHUUSYBZUYMUUKUUNUURUXGUXPUYOUYPUYLUUKUUNUURYEUXGUYHUUSYFUX - QWBZABCDEFUUCHIJYGWBVIYHUYMUYNUVAUYMUUGYDUYMUUKUUNUXFUXPUUGYDUMUYOUYPUYQU - YRABCDEFUUCHIJYIWBVIYHUYMUYHUYAUYNUUHYJZUYAUYNYKUXGUYHUUSWEUYMUXOUYSUYMEO - UUCUULUUMUUKUURUYLYLUYQVLUUCUUFYMSUYAUYNUUHYNYOYPYQYRYSYTUUAUUB $. - - $( Lemma for ~ noinfbnd1 . Establish a hard lower bound when there is no - minimum. (Contributed by Scott Fenton, 9-Aug-2024.) $) - noinfbnd1lem6 $p |- ( ( -. E. x e. B A. y e. B -. y T - T - -. ( U u. { <. dom U , 1o >. } ) rec ( F , I ) = rec ( F , (/) ) ) $= + ( vx vz vy cvv wcel cv crdg cfv c0 wceq con0 wral wi fveq2 eqeq12d imbi2d + weq wb csuc rdgprc0 0ex rdg0 eqtr4di rdgsuc syl5ibr imim2d wlim cres word + r19.21v wss limord ordsson wfn rdgfnon fvreseq mpanl12 3syl cima cuni crn + wn rneq df-ima 3eqtr4g unieqd vex wa rdglim sylbird biimtrid tfinds com12 + mpan ralrimiv eqfnfv mp2an sylibr ) BFGVDZCHZABIZJZWBAKIZJZLZCMNZWCWELZWA + WGCMWBMGWAWGWADHZWCJZWJWEJZLZOZWAKWCJZKWEJZLZOWAEHZWCJZWRWEJZLZOZWAWRUAZW + CJZXCWEJZLZOWAWGODEWBWJKLZWMWQWAXGWKWOWLWPWJKWCPWJKWEPQRDESZWMXAWAXHWKWSW + LWTWJWRWCPWJWRWEPQRWJXCLZWMXFWAXIWKXDWLXEWJXCWCPWJXCWEPQRDCSZWMWGWAXJWKWD + WLWFWJWBWCPWJWBWEPQRWAWOKWPABUBKAUCUDUEWRMGZXAXFWAXAXFXKWSAJZWTAJZLWSWTAP + XKXDXLXEXMBWRAUFKWRAUFQUGUHXBEWJNWAXAEWJNZOWJUIZWNWAXAEWJULXOXNWMWAXOXNWC + WJUJZWEWJUJZLZWMXOWJUKWJMUMZXRXNTZWJUNWJUOWCMUPZWEMUPZXSXTBAUQZKAUQZEMWJW + CWEURUSUTXRWMXOWCWJVAZVBZWEWJVAZVBZLZXRYEYGXRXPVCXQVCYEYGXPXQVEWCWJVFWEWJ + VFVGVHWJFGZXOWMYITDVIYJXOVJWKYFWLYHBWJFAVKKWJFAVKQVPUGVLUHVMVNVOVQYAYBWIW + HTYCYDCMWCWEVRVSVT $. $} ${ - $d B b $. $d b g $. $d B g $. $d b p $. $d B p $. $d B q $. $d B u $. - $d b v $. $d B v $. $d b x $. $d B x $. $d b y $. $d B y $. $d g p $. - $d g q $. $d g u $. $d g v $. $d g x $. $d g y $. $d p q $. $d p u $. - $d p v $. $d p x $. $d p y $. $d q u $. $d q v $. $d q y $. $d T b $. - $d T g $. $d T p $. $d u v $. $d u x $. $d u y $. $d V b $. $d V g $. - $d V p $. $d v x $. $d V x $. $d v y $. $d x y $. $d Z b $. $d Z g $. - $d Z p $. $d Z x $. - noinfbnd2.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Bounding law from below for the surreal infimum. Analagous to - proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, - 9-Aug-2024.) $) - noinfbnd2 $p |- ( ( B C_ No /\ B e. V /\ Z e. No ) -> - ( A. b e. B Z -. T rec ( F , I ) = + U. { f | E. x e. On ( f Fn x /\ + A. y e. x ( f ` y ) = + if ( y = (/) , I , + if ( Lim y , U. ( f " y ) , + ( F ` ( f ` U. y ) ) ) ) ) } ) $= + ( vg vz cv cfv c0 wceq cuni cif wa con0 cvv wcel fveq2d ifbieq2d crdg wfn + wlim cima wral wrex cab rdgeq2 ifeq1 eqeq2d ralbidv anbi2d rexbidv abbidv + vi unieqd eqeq12d cdm crn cmpt crecs cres df-rdg dfrecs3 wb w3a vex resex + eqeq1 wrel relres reldm0 ax-mp bitrdi dmeq limeq syl rneq eqtr4di fveq12d + df-ima id ifbieq12d eqid imaexg uniex fvex fvmpt cin dmres wss onelss imp + ifex 3adant2 fndm 3ad2ant2 sseqtrrd df-ss sylib eqtrid unieq onelon eloni + word csuc ordzsl iftrue eqtr4d sucid fvres ordunisuc 3eqtr4a nsuceq0 neii + w3o iffalsei wn nlimsucg iffalse eqtri 3eqtr4g reseq2 syl5ibrcom rexlimiv + mp2b wne df-lim simp2bi neneqd iffalsed 3jaoi sylbi sylan9eqr mpdan 3expa + 3eqtr4d ralbidva pm5.32da rexbiia abbii unieqi 3eqtri vtoclg ) DUOIZUAZCI + ZAIZUBZBIZUUGJZUUJKLZUUEUUJUCZUUGUUJUDZMZUUJMZUUGJZDJZNZNZLZBUUHUEZOZAPUF + ZCUGZMZLDEUAZUUIUUKUULEUUSNZLZBUUHUEZOZAPUFZCUGZMZLUOEFUUEELZUUFUVGUVFUVN + UUEEDUHUVOUVEUVMUVOUVDUVLCUVOUVCUVKAPUVOUVBUVJUUIUVOUVAUVIBUUHUVOUUTUVHUU + KUULUUEEUUSUIUJUKULUMUNUPUQUUFGQGIZKLZUUEUVPURZUCZUVPUSZMZUVRMZUVPJZDJZNZ + NZUTZVAUUIUUKUUGUUJVBZUWGJZLZBUUHUEZOZAPUFZCUGZMUVFGDUUEVCABCUWGVDUWNUVEU + WMUVDCUWLUVCAPUUHPRZUUIUWKUVBUWOUUIOUWJUVABUUHUWOUUIUUJUUHRZUWJUVAVEUWOUU + IUWPVFZUWIUUTUUKUWQUWIUWHURZKLZUUEUWRUCZUUOUWRMZUWHJZDJZNZNZUUTUWHQRUWIUX + ELUUGUUJCVGZVHGUWHUWFUXEQUWGUVPUWHLZUVQUWSUWEUXDUUEUXGUVQUWHKLZUWSUVPUWHK + VIUWHVJUXHUWSVEUUGUUJVKUWHVLVMVNUXGUVSUWTUWAUWDUUOUXCUXGUVRUWRLUVSUWTVEUV + PUWHVOZUVRUWRVPVQUXGUVTUUNUXGUVTUWHUSUUNUVPUWHVRUUGUUJWAVSUPUXGUWCUXBDUXG + UWBUXAUVPUWHUXGWBUXGUVRUWRUXIUPVTSWCTUWGWDUWSUUEUXDUOVGUWTUUOUXCUUNUUGQRU + UNQRUXFUUGUUJQWEVMWFUXBDWGWNWNWHVMUWQUWRUUJLZUXEUUTLUWQUWRUUJUUGURZWIZUUJ + UUGUUJWJUWQUUJUXKWKUXLUUJLUWQUUJUUHUXKUWOUWPUUJUUHWKZUUIUWOUWPUXMUUHUUJWL + WMWOUUIUWOUXKUUHLUWPUUHUUGWPWQWRUUJUXKWSWTXAUXJUWQUXEUULUUEUUMUUOUUPUWHJZ + DJZNZNZUUTUXJUWSUULUXDUXPUUEUWRUUJKVIUXJUWTUUMUXCUXOUUOUWRUUJVPUXJUXBUXND + UXJUXAUUPUWHUWRUUJXBSSTTUWQUUJXEZUXQUUTLZUWOUWPUXRUUIUWOUWPOUUJPRUXRUUHUU + JXCUUJXDVQWOUXRUULUUJHIZXFZLZHPUFZUUMXPUXSHUUJXGUULUXSUYCUUMUULUXQUUEUUTU + ULUUEUXPXHUULUUEUUSXHXIUYBUXSHPUXTPRZUXSUYBUYAKLZUUEUYAUCZUUOUYAMZUUGUYAV + BZJZDJZNZNZUYEUUEUYFUUOUYGUUGJZDJZNZNZLUYDUYJUYNUYLUYPUYDUYIUYMDUYDUXTUYH + JZUXTUUGJZUYIUYMUXTUYARUYQUYRLUXTHVGZXJUXTUYAUUGXKVMUYDUYGUXTUYHUYDUXTXEU + YGUXTLUXTXDUXTXLVQZSUYDUYGUXTUUGUYTSXMSUYLUYKUYJUYEUUEUYKUYAKUXTXNXOZXQUX + TQRZUYFXRZUYKUYJLUYSUXTQXSZUYFUUOUYJXTYFYAUYPUYOUYNUYEUUEUYOVUAXQVUBVUCUY + OUYNLUYSVUDUYFUUOUYNXTYFYAYBUYBUXQUYLUUTUYPUYBUULUYEUXPUYKUUEUUJUYAKVIZUY + BUUMUYFUXOUYJUUOUUJUYAVPZUYBUXNUYIDUYBUUPUYGUWHUYHUUJUYAUUGYCUUJUYAXBZVTS + TTUYBUULUYEUUSUYOUUEVUEUYBUUMUYFUURUYNUUOVUFUYBUUQUYMDUYBUUPUYGUUGVUGSSTT + UQYDYEUUMUXQUUSUUTUUMUXPUUOUXQUUSUUMUUOUXOXHUUMUULUUEUXPUUMUUJKUUMUXRUUJK + YGUUJUUPLUUJYHYIYJZYKUUMUUOUURXHYQUUMUULUUEUUSVUHYKXIYLYMVQYNYOXAUJYPYRYS + YTUUAUUBUUCUUD $. $} ${ - $d A a $. $d a g $. $d A g $. $d a u $. $d A u $. $d a v $. $d A v $. - $d a x $. $d A x $. $d a y $. $d A y $. $d B b $. $d b g $. $d B g $. - $d B u $. $d b v $. $d B v $. $d b x $. $d B x $. $d b y $. $d B y $. - $d g u $. $d g v $. $d g x $. $d g y $. $d S a $. $d S b $. $d S g $. - $d S x $. $d T a $. $d T b $. $d T g $. $d T x $. $d u v $. $d u x $. - $d u y $. $d v x $. $d v y $. $d W a $. $d W b $. $d W g $. $d W x $. - $d x y $. - nosupinfsep.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - nosupinfsep.2 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Given two sets of surreals, a surreal ` W ` separates them iff its - restriction to the maximum of ` dom S ` and ` dom T ` separates them. - Corollary 4.4 of [Lipparini] p. 7. (Contributed by Scott Fenton, - 9-Aug-2024.) $) - nosupinfsep $p |- ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) - /\ W e. No ) -> ( ( A. a e. A a - ( A. a e. A a . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - noetasuplem.2 $e |- Z = - ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) ) $. - ${ - $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. - $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. - $d v x $. $d v y $. $d x y $. - $( Lemma for ~ noeta . Establish that our final surreal really is a - surreal. (Contributed by Scott Fenton, 6-Dec-2021.) $) - noetasuplem1 $p |- ( ( A C_ No /\ A e. _V /\ B e. _V ) -> Z e. No ) $= - ( csur wss cvv wcel w3a cbday cima cuni c1o cdm csn cxp nosupno 3adant3 - csuc cdif cun con0 bdayimaon 3ad2ant3 1oex noextendseq syl2anc eqeltrid - c2o prid1 ) ELMZENOZFNOZPZIGQFRSUFZGUAUGTUBUCUHZLKVAGLOZVBUIOZVCLOURUSV - DUTABCDEGHNJUDUEUTURVEUSFNUJUKGVBTTUPULUQUMUNUO $. - $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Defined equality axioms +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) - ${ - $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. - $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. - $d v x $. $d v y $. $d x y $. - $( Lemma for ~ noeta . The restriction of ` Z ` to ` dom S ` is ` S ` . - (Contributed by Scott Fenton, 9-Aug-2024.) $) - noetasuplem2 $p |- ( ( A C_ No /\ A e. _V /\ B e. _V ) -> - ( Z |` dom S ) = S ) $= - ( csur cvv wcel cdm cres c0 cun c1o wceq wss w3a cima cuni csuc csn cxp - cbday cdif reseq1i resundir cin dmres wne 1oex snnz dmxp ineq2i disjdif - ax-mp 3eqtri wrel relres reldm0 mpbir uneq2i wfun nosupno 3adant3 nofun - wb syl funrel resdm 3syl uneq1d un0 eqtrdi eqtrid ) ELUAZEMNZFMNZUBZIGO - ZPZGWDPZQRZGWEGUHFUCUDUEZWDUIZSUFZUGZRZWDPWFWKWDPZRWGIWLWDKUJGWKWDUKWMQ - WFWMQTZWMOZQTZWOWDWKOZULWDWIULQWKWDUMWQWIWDWJQUNWQWITSUOUPWIWJUQUTURWDW - HUSVAWMVBWNWPVKWKWDVCWMVDUTVEVFVAWCWGGQRGWCWFGQWCGVGZGVBWFGTWCGLNZWRVTW - AWSWBABCDEGHMJVHVIGVJVLGVMGVNVOVPGVQVRVS $. - $} + $( A version of ~ ax-ext for use with defined equality. (Contributed by + Scott Fenton, 12-Dec-2010.) $) + axextdfeq $p |- E. z ( ( z e. x -> z e. y ) + -> ( ( z e. y -> z e. x ) -> ( x e. w -> y e. w ) ) ) $= + ( wel wb wi wex weq axextnd ax8 imim2i eximii biimpexp exbii mpbi ) CAEZCBE + ZFZADEBDEGZGZCHQRGRQGTGGZCHSABIZGUACCABJUCTSABDKLMUAUBCQRTNOP $. - ${ - $d A g $. $d A u $. $d A v $. $d A x $. $d A y $. $d g u $. - $d g v $. $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. - $d v x $. $d v y $. $d X u $. $d X v $. $d X x $. $d x y $. - $d X y $. - $( Lemma for ~ noeta . ` Z ` is an upper bound for ` A ` . Part of - Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, - 4-Dec-2021.) $) - noetasuplem3 $p |- ( ( ( A C_ No /\ A e. _V /\ B e. _V ) /\ X e. A ) -> - X z e. y ) -> ( w e. x -> w e. y ) ) $= + ( weq wel wi ax6e ax8 equcoms imim12d eximii ) CDEZCAFZCBFZGDAFZDBFZGGCCDHM + PNOQPNGDCDCAIJCDBIKL $. - ${ - $d A a $. $d a b $. $d A b $. $d a g $. $d A g $. $d a u $. - $d A u $. $d a v $. $d A v $. $d a x $. $d A x $. $d a y $. - $d A y $. $d B a $. $d B b $. $d b g $. $d b p $. $d b q $. - $d b x $. $d g u $. $d g v $. $d g x $. $d g y $. $d p q $. - $d S a $. $d S g $. $d S q $. $d u v $. $d u x $. $d u y $. - $d v x $. $d v y $. $d x y $. $d Z p $. $d Z q $. - $( Lemma for ~ noeta . When ` A ` and ` B ` are separated, then ` Z ` is - a lower bound for ` B ` . Part of Theorem 5.1 of [Lipparini] p. 7-8. - (Contributed by Scott Fenton, 7-Dec-2021.) $) - noetasuplem4 $p |- ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) - /\ - A. a e. A A. b e. B a A. b e. B Z ( A. z ( z e. x <-> z e. y ) -> x = y ) ) $= + ( vw weq wal wn wa wel wb nfnae nfan wnfc nfcvf adantr nfcrd adantl nfbid + cv elequ1 wi bibi12d a1i cbvald axextg syl6bir ) CAECFGZCBECFGZHZCAIZCBIZ + JZCFDAIZDBIZJZDFABEUIUOULDCUGUHCCACKCBCKLUIUMUNCUICDASZUGCUPMUHCANOPUICDB + SZUHCUQMUGCBNQPRDCEZUOULJUAUIURUMUJUNUKDCATDCBTUBUCUDABDUEUF $. + + $( ~ axextb with distinctors instead of distinct variable conditions. + (Contributed by Scott Fenton, 13-Dec-2010.) $) + axextbdist $p |- ( ( -. A. z z = x /\ -. A. z z = y ) + -> ( x = y <-> A. z ( z e. x <-> z e. y ) ) ) $= + ( weq wal wn wa wel wb wi axc9 imp nfnae nfan elequ2 alimd syld axextdist + a1i impbid ) CADCEFZCBDCEFZGZABDZCAHCBHIZCEZUCUDUDCEZUFUAUBUDUGJABCKLUCUD + UECUAUBCCACMCBCMNUDUEJUCABCOSPQABCRT $. $} ${ - $d B g $. $d B u $. $d B v $. $d B x $. $d B y $. $d g u $. $d g v $. - $d g x $. $d g y $. $d u v $. $d u x $. $d u y $. $d v x $. $d v y $. $d x y $. - noetainflem.1 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - noetainflem.2 $e |- W = ( T u. - ( ( suc U. ( bday " A ) \ dom T ) X. { 2o } ) ) $. - $( Lemma for ~ noeta . Establish that this particular construction gives a - surreal. (Contributed by Scott Fenton, 9-Aug-2024.) $) - noetainflem1 $p |- ( ( A e. _V /\ B C_ No /\ B e. _V ) -> W e. No ) $= - ( cvv wcel csur wss w3a cbday cima cuni c2o csuc cdm cdif csn cxp noinfno - cun con0 3adant1 bdayimaon 3ad2ant1 c1o 2oex noextendseq syl2anc eqeltrid - prid2 ) ELMZFNOZFLMZPZIGQERSUAZGUBUCTUDUEUGZNKVAGNMZVBUHMZVCNMUSUTVDURABC - DFGHLJUFUIURUSVEUTELUJUKGVBTULTUMUQUNUOUP $. + 19.12b.1 $e |- F/ y ph $. + 19.12b.2 $e |- F/ x ps $. + $( Version of ~ 19.12vv with not-free hypotheses, instead of distinct + variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) + (Revised by Mario Carneiro, 11-Dec-2016.) $) + 19.12b $p |- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) ) $= + ( wi wal wex 19.21 exbii nfal 19.36 albii bitr2i 3bitri ) ABGZDHZCIABDHZG + ZCIACHZSGZQCIZDHZRTCABDEJKASCBCDFLMUDUABGZDHUBUCUEDABCFMNUABDADCELJOP $. + $} - $( Lemma for ~ noeta . The restriction of ` W ` to the domain of ` T ` is - ` T ` . (Contributed by Scott Fenton, 9-Aug-2024.) $) - noetainflem2 $p |- ( ( B C_ No /\ B e. _V ) -> - ( W |` dom T ) = T ) $= - ( csur cvv wcel cdm cres cun eqtri c0 wceq wss wa cima cuni csuc cdif c2o - cbday csn cxp reseq1i resundir wfun wrel noinfno nofun syl resdm 3syl cin - funrel dmres wne 2oex snnz dmxp ax-mp ineq2i disjdif wb relres reldm0 a1i - mpbir uneq12d un0 eqtrdi eqtrid ) FLUAFMNUBZIGOZPZGVTPZUHEUCUDUEZVTUFZUGU - IZUJZVTPZQZGWAGWFQZVTPWHIWIVTKUKGWFVTULRVSWHGSQGVSWBGWGSVSGUMZGUNWBGTVSGL - NWJABCDFGHMJUOGUPUQGVAGURUSWGSTZVSWKWGOZSTZWLVTWFOZUTZSWFVTVBWOVTWDUTSWNW - DVTWESVCWNWDTUGVDVEWDWEVFVGVHVTWCVIRRWGUNWKWMVJWFVTVKWGVLVGVNVMVOGVPVQVR + $( There is always a set not in ` y ` . (Contributed by Scott Fenton, + 13-Dec-2010.) $) + exnel $p |- E. x -. x e. y $= + ( wel wn wex elirrv nfth weq ax8 con3d spime ax-mp ) BBCZDZABCZDZAEBFZNPABN + AQGABHOMABBIJKL $. + + ${ + $d x z $. $d y z $. + $( Distinctors in terms of membership. (NOTE: this only works with + relations where we can prove ~ el and ~ elirrv .) (Contributed by Scott + Fenton, 15-Dec-2010.) $) + distel $p |- ( -. A. y y = x <-> -. A. y -. x e. y ) $= + ( vz weq wal wn wel wex el df-ex nfnae dveel1 nf5d nfnd elequ2 notbid a1i + wb wi cbvald bitrid mpbii elirrv elequ1 mtbii alimi con3i impbii ) BADZBE + ZFZABGZFZBEZFZUKACGZCHZUOACIUQUPFZCEZFUKUOUPCJUKUSUNUKURUMCBBABKZUKUPBUKU + PBUTBACLMNCBDZURUMRSUKVAUPULCBAOPQTPUAUBUJUNUIUMBUIBBGULBUCBABUDUEUFUGUH $. + $} - $d Y v $. $d Y x $. $d Y y $. - $( Lemma for ~ noeta . ` W ` bounds ` B ` below . (Contributed by Scott - Fenton, 9-Aug-2024.) $) - noetainflem3 $p |- ( ( ( A e. _V /\ B C_ No /\ B e. _V ) /\ Y e. B ) -> - W ( z e. x <-> z e. y ) ) $= + ( weq wel wb wex wi wa axextnd elequ2 jctl eximii dfbi2 exbii mpbir ) ABDZC + AECBEFZFZCGQRHZRQHZIZCGUAUBCCABJUATABCKLMSUBCQRNOP $. - $d A a $. $d a b $. $d a g $. $d a p $. $d a q $. $d a x $. $d B a $. - $d B b $. $d b g $. $d b v $. $d b x $. $d b y $. $d p q $. $d T b $. - $d T g $. $d T q $. $d W p $. $d W q $. - $( Lemma for ~ noeta . If ` A ` precedes ` B ` , then ` W ` is greater - than ` A ` . (Contributed by Scott Fenton, 9-Aug-2024.) $) - noetainflem4 $p |- ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) /\ - A. a e. A A. b e. B a A. a e. A a . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - noetalem1.2 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - noetalem1.3 $e |- Z = - ( S u. ( ( suc U. ( bday " B ) \ dom S ) X. { 1o } ) ) $. - noetalem1.4 $e |- W = ( T u. - ( ( suc U. ( bday " A ) \ dom T ) X. { 2o } ) ) $. - $( Lemma for ~ noeta . Either ` S ` or ` T ` satisfies the final - condition. (Contributed by Scott Fenton, 9-Aug-2024.) $) - noetalem1 $p |- ( ( ( ( A C_ No /\ A e. _V ) /\ ( B C_ No /\ B e. _V ) /\ - A. a e. A A. b e. B a - ( ( S e. No /\ ( A. a e. A a A. x ps ) -> ( -. ps -> A. x -. ph ) ) $= + ( wn wal wi axc7 con1i con3 al2imi syl5 ) BDBCEZDZCEZALFZCEADZCENBBCGHOMPCA + LIJK $. + + $( A more general and closed form of ~ hbim . (Contributed by Scott Fenton, + 13-Dec-2010.) $) + hbimtg $p |- ( ( A. x ( ph -> A. x ch ) /\ ( ps -> A. x th ) ) -> + ( ( ch -> ps ) -> A. x ( ph -> th ) ) ) $= + ( wal wi wa wn hbntg pm2.21 alimi syl6 adantr ala1 imim2i adantl jad ) ACEF + GEFZBDEFZGZHCBADGZEFZSCIZUCGUASUDAIZEFUCACEJUEUBEADKLMNUABUCGSTUCBDAEOPQR + $. + + $( A more general and closed form of ~ hbal . (Contributed by Scott Fenton, + 13-Dec-2010.) $) + hbaltg $p |- ( A. x ( ph -> A. y ps ) -> ( A. x ph -> A. y A. x ps ) ) $= + ( wal wi alim ax-11 syl6 ) ABDEZFCEACEJCEBCEDEAJCGBCDHI $. ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a g $. $d A g $. - $d a u $. $d A u $. $d a v $. $d A v $. $d a x $. $d A x $. $d a y $. - $d A y $. $d B a $. $d B b $. $d b c $. $d B c $. $d b g $. $d B g $. - $d B u $. $d b v $. $d B v $. $d b x $. $d B x $. $d b y $. $d B y $. - $d g u $. $d g v $. $d g x $. $d g y $. $d O c $. $d O u $. $d O y $. - $d S a $. $d S b $. $d S c $. $d S g $. $d S x $. $d T a $. $d T b $. - $d T c $. $d T g $. $d T x $. $d u v $. $d u x $. $d u y $. $d v x $. - $d v y $. $d x y $. - noetalem2.1 $e |- S = if ( E. x e. A A. y e. A -. x . } ) , - ( g e. { y | E. u e. A ( y e. dom u /\ - A. v e. A ( -. v ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. A ( g e. dom u /\ - A. v e. A ( -. v ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - noetalem2.2 $e |- T = if ( E. x e. B A. y e. B -. y . } ) , - ( g e. { y | E. u e. B ( y e. dom u /\ - A. v e. B ( -. u ( u |` suc y ) = ( v |` suc y ) ) ) } - |-> ( iota x E. u e. B ( g e. dom u /\ - A. v e. B ( -. u ( u |` suc g ) = ( v |` suc g ) ) /\ - ( u ` g ) = x ) ) ) ) $. - $( Lemma for ~ noeta . The full statement of the theorem with hypotheses - in place. (Contributed by Scott Fenton, 10-Aug-2024.) $) - noetalem2 $p |- ( ( ( ( A C_ No /\ A e. V ) /\ ( B C_ No /\ B e. W ) /\ - A. a e. A A. b e. B a - E. c e. No ( A. a e. A a A. x ps ) $. + $( A more general form of ~ hbn . (Contributed by Scott Fenton, + 13-Dec-2010.) $) + hbng $p |- ( -. ps -> A. x -. ph ) $= + ( wal wi wn hbntg mpg ) ABCEFBGAGCEFCABCHDI $. + + ${ + hbg.2 $e |- ( ch -> A. x th ) $. + $( A more general form of ~ hbim . (Contributed by Scott Fenton, + 13-Dec-2010.) $) + hbimg $p |- ( ( ps -> ch ) -> A. x ( ph -> th ) ) $= + ( wal wi ax-gen hbimtg mp2an ) ABEHIZEHCDEHIBCIADIEHIMEFJGACBDEKL $. + $} $} + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Well-Founded Induction +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + ${ - $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. $d A d $. - $d a e $. $d A e $. $d a f $. $d A f $. $d a g $. $d A g $. $d a h $. - $d A h $. $d a j $. $d A j $. $d a k $. $d A k $. $d a x $. $d A x $. - $d a y $. $d A y $. $d A z $. $d B a $. $d B b $. $d b c $. $d B c $. - $d b d $. $d B d $. $d b e $. $d B e $. $d b f $. $d B f $. $d b g $. - $d B g $. $d B h $. $d b j $. $d B j $. $d b k $. $d B k $. $d b x $. - $d B x $. $d b y $. $d B y $. $d B z $. $d c d $. $d c e $. $d c f $. - $d c g $. $d c h $. $d c j $. $d c k $. $d c x $. $d c y $. $d d e $. - $d d f $. $d d g $. $d d h $. $d d j $. $d d k $. $d d x $. $d e f $. - $d e g $. $d e h $. $d e j $. $d e k $. $d e x $. $d e y $. $d f g $. - $d f h $. $d f j $. $d f x $. $d f y $. $d f z $. $d g h $. $d g j $. - $d g k $. $d g x $. $d g y $. $d g z $. $d h j $. $d h k $. $d h x $. - $d h y $. $d h z $. $d j k $. $d j x $. $d j y $. $d j z $. $d k x $. - $d k y $. $d k z $. $d O b $. $d O d $. $d O z $. $d x y $. $d x z $. - $d y z $. - $( The full-eta axiom for the surreal numbers. This is the single most - important property of the surreals. It says that, given two sets of - surreals such that one comes completely before the other, there is a - surreal lying strictly between the two. Furthermore, if the birthdays - of members of ` A ` and ` B ` are strictly bounded above by ` O ` , then - ` O ` non-strictly bounds the separator. Axiom FE of [Alling] p. 185. - (Contributed by Scott Fenton, 9-Aug-2024.) $) - noeta $p |- ( ( ( ( A C_ No /\ A e. V ) /\ ( B C_ No /\ B e. W ) /\ - A. x e. A A. y e. B x - E. z e. No ( A. x e. A x + ( A. z A. w ( <. z , w >. e. Pred ( R , ( A X. B ) , <. x , y >. ) -> ch ) + -> ph ) ) $. + frpoins3xpg.2 $e |- ( x = z -> ( ph <-> ps ) ) $. + frpoins3xpg.3 $e |- ( y = w -> ( ps <-> ch ) ) $. + frpoins3xpg.4 $e |- ( x = X -> ( ph <-> th ) ) $. + frpoins3xpg.5 $e |- ( y = Y -> ( th <-> ta ) ) $. + $( Special case of founded partial induction over a cross product. + (Contributed by Scott Fenton, 22-Aug-2024.) $) + frpoins3xpg $p |- ( ( ( R Fr ( A X. B ) /\ R Po ( A X. B ) /\ + R Se ( A X. B ) ) /\ ( X e. A /\ Y e. B ) ) -> ta ) $= + ( vq vp wcel wa cxp wfr wpo wse w3a wral c2nd cfv wsbc c1st cop wceq wrex + cv cpred wi elxp2 nfcv nfsbc1v nfralw nfv nfsbcw wal weq sbcbidv cbvsbcvw + nfim sbcbii bitri ralbii impexp cin elin wss predss sseqin2 eleq2i bitr3i + imbi1i albii df-ral 3bitr4ri eleq1 sbcopeq1a imbi12d ralxpf opelxp 2albii + mpbi r2al biimtrid predeq3 raleqdv syl5ibrcom rexlimd rexlimi sylbi fveq2 + 3bitri ex sbceq1d sbceqbid frpoins2g ralxpes sylib rspc2va sylan2 ancoms + ) MJUBNKUBUCZJKUDZLUEXMLUFXMLUGUHZEXNXLAGKUIFJUIZEXNAGUAUQZUJUKZULZFXPUMU + KZULZUAXMUIXOXTAGTUQZUJUKZULZFYAUMUKZULZUATXMLXPXMUBXPFUQZGUQZUNZUOZGKUPZ + FJUPYETXMLXPURZUIZXTUSZFGXPJKUTYJYMFJYLXTFYEFTYKFYKVAYCFYDVBVCXRFXSVBVJYF + JUBZYIYMGKYNGVDYLXTGYEGTYKGYKVAYCGFYDGYDVAAGYBVBVEVCXRGFXSGXSVAAGXQVBVEVJ + YNYGKUBZYIYMUSYNYOUCZYMYIYETXMLYHURZUIZAUSYRHUQZIUQZUNZYQUBZCUSZIVFHVFZYP + AYRCIYBULZHYDULZTYQUIZYAYQUBZUUFUSZTXMUIZUUDYEUUFTYQYEBGYBULZHYDULUUFYCUU + KFHYDFHVGABGYBPVHVIUUKUUEHYDBCGIYBQVIVKVLVMYAXMUBZUUIUSZTVFUUITVFUUJUUGUU + MUUITUUMUULUUHUCZUUFUSUUIUULUUHUUFVNUUNUUHUUFUUNYAXMYQVOZUBUUHYAXMYQVPUUO + YQYAYQXMVQUUOYQUOXMLYHVRYQXMVSWLZVTWAWBWAWCUUITXMWDUUFTYQWDWEUUJUUCIKUIHJ + UIYSJUBYTKUBUCZUUCUSZIVFHVFUUDUUIUUCTHIJKUUHUUFHUUHHVDUUEHYDVBVJUUHUUFIUU + HIVDUUEIHYDIYDVACIYBVBVEVJUUCTVDYAUUAUOUUHUUBUUFCYAUUAYQWFCHIYAWGWHWIUUCH + IJKWMUURUUCHIUURUUAXMUBZUUBUCZCUSZUUCUVAUUSUUCUSUURUUSUUBCVNUUSUUQUUCYSYT + JKWJWBVLUUTUUBCUUTUUAUUOUBUUBUUAXMYQVPUUOYQUUAUUPVTWAWBWAWKXBXBOWNYIYLYRX + TAYIYETYKYQXMLXPYHWOWPAFGXPWGWHWQXCWRWSWTUATVGZXRYCFXSYDXPYAUMXAUVBAGXQYB + XPYAUJXAXDXEXFAUAFGJKXGXHAEDFGMNJKRSXIXJXK $. + $} + + ${ + $d A p $. $d A q $. $d A x $. $d A y $. $d A z $. $d B p $. $d B q $. + $d B x $. $d B y $. $d B z $. $d C p $. $d C q $. $d C x $. $d C y $. + $d C z $. $d et y $. $d p ph $. $d p q $. $d p x $. $d p y $. + $d p z $. $d ph q $. $d q x $. $d q y $. $d q z $. $d R p $. + $d R q $. $d ta x $. $d X x $. $d x y $. $d X y $. $d x z $. + $d X z $. $d Y y $. $d y z $. $d Y z $. $d Z z $. $d z ze $. + $d A t $. $d A u $. $d A w $. $d B t $. $d B u $. $d B w $. $d C t $. + $d C u $. $d C w $. $d ch u $. $d ch y $. $d ph w $. $d ps t $. + $d ps x $. $d q t $. $d q th $. $d q u $. $d q w $. $d R t $. + $d R u $. $d R w $. $d R x $. $d R y $. $d R z $. $d t u $. $d t w $. + $d t x $. $d t y $. $d t z $. $d th x $. $d th y $. $d th z $. + $d u w $. $d u x $. $d u y $. $d u z $. $d w x $. $d w y $. $d w z $. + frpoins3xp3g.1 $e |- ( ( x e. A /\ y e. B /\ z e. C ) -> + ( A. w A. t A. u + ( <. <. w , t >. , u >. e. Pred ( R , ( ( A X. B ) X. C ) , + <. <. x , y >. , z >. ) -> th ) -> ph ) ) $. + frpoins3xp3g.2 $e |- ( x = w -> ( ph <-> ps ) ) $. + frpoins3xp3g.3 $e |- ( y = t -> ( ps <-> ch ) ) $. + frpoins3xp3g.4 $e |- ( z = u -> ( ch <-> th ) ) $. + frpoins3xp3g.5 $e |- ( x = X -> ( ph <-> ta ) ) $. + frpoins3xp3g.6 $e |- ( y = Y -> ( ta <-> et ) ) $. + frpoins3xp3g.7 $e |- ( z = Z -> ( et <-> ze ) ) $. + $( Special case of founded partial recursion over a triple cross product. + (Contributed by Scott Fenton, 22-Aug-2024.) $) + frpoins3xp3g $p |- ( ( ( R Fr ( ( A X. B ) X. C ) + /\ R Po ( ( A X. B ) X. C ) + /\ R Se ( ( A X. B ) X. C ) ) /\ ( X e. A /\ Y e. B /\ Z e. C ) ) + -> ze ) $= + ( vp vq cxp wfr wpo wse w3a wral wcel cv c2nd cfv wsbc c1st cpred sbcbidv + weq cbvsbcvw sbcbii bitri ralbii cop wceq wrex wi elxpxp nfv nfsbc1v nfim + nfcv nfsbcw wa wal impexp cin elin wss predss sseqin2 eleq2i bitr3i albii + mpbi imbi1i r3al sbcoteq1a imbi12d ralxp3f ot2elxp 2albii 3bitr4ri df-ral + eleq1 biimtrid predeq3 raleqdv syl5ibrcom 3expia rexlimd ex rexlimi sylbi + 2fveq3 fveq2 sbceq1d sbceqbid frpoins2g ralxp3es sylib rspc3v mpan9 ) NOU + JPUJZQUKXSQULXSQUMUNZAJPUOIOUOHNUOZRNUPSOUPTPUPUNGXTAJUHUQZURUSZUTZIYBVAU + SZURUSZUTZHYEVAUSZUTZUHXSUOYAYIAJUIUQZURUSZUTZIYJVAUSZURUSZUTZHYMVAUSZUTZ + UHUIXSQYQUIXSQYBVBZUODLYKUTZMYNUTZKYPUTZUIYRUOZYBXSUPZYIYQUUAUIYRYQBJYKUT + ZIYNUTZKYPUTUUAYOUUEHKYPHKVDZYLUUDIYNUUFABJYKUBVCVCVEUUEYTKYPUUECJYKUTZMY + NUTYTUUDUUGIMYNIMVDBCJYKUCVCVEUUGYSMYNCDJLYKUDVEVFVGVFVGVHUUCYBHUQZIUQZVI + JUQZVIZVJZJPVKZIOVKZHNVKUUBYIVLZHIJYBNOPVMUUNUUOHNUUBYIHUUBHVNYGHYHVOVPUU + HNUPZUUMUUOIOUUPIVNUUBYIIUUBIVNYGIHYHIYHVQYDIYFVOVRVPUUPUUIOUPZUUMUUOVLUU + PUUQVSZUULUUOJPUURJVNUUBYIJUUBJVNYGJHYHJYHVQYDJIYFJYFVQAJYCVOVRVRVPUUPUUQ + UUJPUPZUULUUOVLUUPUUQUUSUNZUUOUULUUAUIXSQUUKVBZUOZAVLUVBKUQZMUQZVILUQZVIZ + UVAUPZDVLZLVTZMVTKVTZUUTAYJXSUPZYJUVAUPZUUAVLZVLZUIVTZUVMUIVTUVJUVBUVNUVM + UIUVNUVKUVLVSZUUAVLUVMUVKUVLUUAWAUVPUVLUUAUVPYJXSUVAWBZUPUVLYJXSUVAWCUVQU + VAYJUVAXSWDUVQUVAVJXSQUUKWEUVAXSWFWJZWGWHWKWHWIUVJUVMUIXSUOZUVOUVHLPUOMOU + OKNUOUVCNUPUVDOUPUVEPUPUNZUVHVLZLVTZMVTKVTUVSUVJUVHKMLNOPWLUVMUVHUIKMLNOP + UVLUUAKUVLKVNYTKYPVOVPUVLUUAMUVLMVNYTMKYPMYPVQYSMYNVOVRVPUVLUUALUVLLVNYTL + KYPLYPVQYSLMYNLYNVQDLYKVOVRVRVPUVHUIVNYJUVFVJUVLUVGUUADYJUVFUVAWTDKMLYJWM + WNWOUVIUWBKMUVHUWALUVHUVFXSUPZUVHVLZUWAUVHUWCUVGVSZDVLUWDUWEUVGDUWEUVFUVQ + UPUVGUVFXSUVAWCUVQUVAUVFUVRWGWHWKUWCUVGDWAWHUWCUVTUVHUVCUVDUVENOPWPWKVGWI + WQWRUVMUIXSWSVGUUAUIUVAWSWRUAXAUULUUBUVBYIAUULUUAUIYRUVAXSQYBUUKXBXCAHIJY + BWMWNXDXEXFXGXFXHXIXAUHUIVDZYGYOHYHYPYBYJVAVAXJUWFYDYLIYFYNYBYJURVAXJUWFA + JYCYKYBYJURXKXLXMXMXNAUHHIJNOPXOXPAGEFHIJRSTNOPUEUFUGXQXR $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal numbers - ordering theorems + Ordering Cross Products, Part 2 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $( Declare a new constant for surreal less than or equal. $) - $c <_s $. - - $( Declare the syntax for surreal less than or equal. $) - csle $a class <_s $. - - $( Define the surreal less than or equal predicate. Compare ~ df-le . - (Contributed by Scott Fenton, 8-Dec-2021.) $) - df-sle $a |- <_s = ( ( No X. No ) \ `' -. A - ( ( A A ( A -. B . | ( x e. ( A X. B ) /\ y e. ( A X. B ) + /\ ( ( ( 1st ` x ) R ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ + ( ( 2nd ` x ) S ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ + x =/= y ) ) } $. - $( Surreal less than obeys trichotomy. (Contributed by Scott Fenton, - 16-Jun-2011.) $) - sltlin $p |- ( ( A e. No /\ B e. No ) -> ( A . T <. c , d >. <-> + ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) /\ + ( ( a R c \/ a = c ) /\ ( b S d \/ b = d ) /\ + ( a =/= c \/ b =/= d ) ) ) ) $= + ( cv wcel cfv wbr wceq wo wne w3a cxp c1st c2nd wa cop weq eleq1 opelxp + bitrdi vex op1std breq1d eqeq1d orbi12d op2ndd neeq1 3anbi123d 3anbi13d + opex breq2d eqeq2d neeq2 opthne 3anbi23d brab ) AMZCDUAZNZBMZVGNZVFUBOZ + VIUBOZEPZVKVLQZRZVFUCOZVIUCOZFPZVPVQQZRZVFVISZTZTHMZCNIMZDNUDZVJWCVLEPZ + WCVLQZRZWDVQFPZWDVQQZRZWCWDUEZVISZTZTWEJMZCNKMZDNUDZWCWOEPZHJUFZRZWDWPF + PZIKUFZRZWCWOSWDWPSRZTZTABWLWOWPUEZGWCWDUSWOWPUSVFWLQZVHWEWBWNVJXGVHWLV + GNWEVFWLVGUGWCWDCDUHUIXGVOWHVTWKWAWMXGVMWFVNWGXGVKWCVLEWCWDVFHUJZIUJZUK + ZULXGVKWCVLXJUMUNXGVRWIVSWJXGVPWDVQFWCWDVFXHXIUOZULXGVPWDVQXKUMUNVFWLVI + UPUQURVIXFQZVJWQWNXEWEXLVJXFVGNWQVIXFVGUGWOWPCDUHUIXLWHWTWKXCWMXDXLWFWR + WGWSXLVLWOWCEWOWPVIJUJZKUJZUKZUTXLVLWOWCXOVAUNXLWIXAWJXBXLVQWPWDFWOWPVI + XMXNUOZUTXLVQWPWDXPVAUNXLWMWLXFSXDVIXFWLVBWCWDWOWPXHXIVCUIUQVDLVE $. + $} - $( Trichotomy law for surreal less than. (Contributed by Scott Fenton, - 22-Apr-2012.) $) - slttrieq2 $p |- ( ( A e. No /\ B e. No ) -> - ( A = B <-> ( -. A R Po A ) $. + poxp2.2 $e |- ( ph -> S Po B ) $. + $( Another way of partially ordering a cross product of two classes. + (Contributed by Scott Fenton, 19-Aug-2024.) $) + poxp2 $p |- ( ph -> T Po ( A X. B ) ) $= + ( vp vq vr vs vt wbr wrex wa wo va vb vc vu cxp cv wcel wn cop wceq weq + elxp2 wne w3a equid pm3.2i neorian con2bii simp3 mto xpord2lem mtbir wb + mpbi breq12 anidms mtbiri rexlimivw sylbi adantl 3reeanv rexbii 2rexbii + wi 3anbi123i 3bitr4ri df-3an simpr1l adantr simpr2r jca simpr2l simpr3r + wpo ad2antrr simpr1r potr syl13anc orc syl6 expd syl6bir simprl1 mpjaod + breq1 a1i breq2 equequ2 orbi12d simprr1 simpr3l simprl2 simprr2 simprr3 + syl5ibcom sylib equequ1 neeq1 3anbi123d anbi1d anbi2d orcom ordi equcom + orbi2i bitri anbi12i 3bitri sylanbrc po2nr syl12anc orcnd com12 syl5bir + syl6bi mt3d 3jca syl2ani 3adant3 bitrdi 3adant1 anbi12d 3adant2 imbi12d + ex syl5ibrcom sylan2br anassrs rexlimdvva imp sylan2b ispod ) AUAUBUCDE + UEZHUAUFZUUCUGZUUDUUDHQZUHZAUUEUUDLUFZMUFZUIZUJZMERZLDRZUUGLMUUDDEULZUU + LUUGLDUUKUUGMEUUKUUFUUJUUJHQZUUOUUHDUGZUUIEUGZSZUURUUHUUHFQLLUKZTZUUIUU + IGQMMUKZTZUUHUUHUMUUIUUIUMTZUNZUNZUVEUVDUVDUVCUUSUVASZUVCUHUUSUVALUOMUO + UPUVCUVFUUHUUHUUIUUIUQURVDUUTUVBUVCUSUTUURUURUVDUSUTBCDEFGHLMLMIVAVBUUK + UUFUUOVCUUDUUJUUDUUJHVEVFVGVHVHVIVJUUEUBUFZUUCUGZUCUFZUUCUGZUNZAUUKUVGN + UFZOUFZUIZUJZUVIPUFZUDUFZUIZUJZUNZUDEROERZMERZPDRZNDRLDRZUUDUVGHQZUVGUV + IHQZSZUUDUVIHQZVNZUULUVOOERZUVSUDERZUNZPDRZNDRLDRUUMUWJNDRZUWKPDRZUNUWD + UVKUULUWJUWKLNPDDDVKUWCUWMLNDDUWBUWLPDUUKUVOUVSMOUDEEEVKVLVMUUEUUMUVHUW + NUVJUWOUUNNOUVGDEULPUDUVIDEULVOVPAUWDUWIAUWCUWILNDDAUUPUVLDUGZSZSUWAUWI + PMDEAUWQUVPDUGZUUQSZUWAUWIVNAUWQUWSSZSUVTUWIOUDEEAUWTUVMEUGZUVQEUGZSZUV + TUWIVNZUWTUXCSAUWQUWSUXCUNZUXDUWQUWSUXCVQAUXESZUWIUVTUURUWPUXASZUUHUVLF + QZLNUKZTZUUIUVMGQZMOUKZTZUUHUVLUMZUUIUVMUMZTZUNZUNZUXGUWRUXBSZUVLUVPFQZ + NPUKZTZUVMUVQGQZOUDUKZTZUVLUVPUMUVMUVQUMTZUNZUNZSZUURUXSUUHUVPFQZLPUKZT + ZUUIUVQGQZMUDUKZTZUUHUVPUMUUIUVQUMTZUNZUNZVNUXRUXFUXQUYGUYRUYHUURUXGUXQ + USUXGUXSUYGUSUXFUXQUYGSZUYRUXFUYSSZUURUXSUYQUYTUUPUUQUXFUUPUYSUUPUWPUWS + UXCAVRVSZUXFUUQUYSUWRUUQUWQUXCAVTVSZWAUYTUWRUXBUXFUWRUYSUWRUUQUWQUXCAWB + ZVSZUXFUXBUYSUXAUXBUWQUWSAWCZVSZWAUYTUYLUYOUYPUYTUXTUYLUYAUYTUXHUXTUYLV + NZUXIUYTUXHUXTUYLUYTUXHUXTSZUYJUYLUYTDFWDZUUPUWPUWRVUHUYJVNAVUIUXEUYSJW + EVUAUXFUWPUYSUUPUWPUWSUXCAWFZVSVUDDUUHUVLUVPFWGWHUYJUYKWIZWJWKUXIVUGVNU + YTUXIUXTUYJUYLUUHUVLUVPFWOVUKWLWPUXJUXMUXPUYGUXFWMZWNUYTUXJUYAUYLVULUYA + UXHUYJUXIUYKUVLUVPUUHFWQNPLWRWSXEUYBUYEUYFUXQUXFWTWNUYTUYCUYOUYDUYTUXKU + YCUYOVNZUXLUYTUXKUYCUYOUYTUXKUYCSZUYMUYOUYTEGWDZUUQUXAUXBVUNUYMVNAVUOUX + EUYSKWEVUBUXFUXAUYSUXAUXBUWQUWSAXAZVSVUFEUUIUVMUVQGWGWHUYMUYNWIZWJWKUXL + VUMVNUYTUXLUYCUYMUYOUUIUVMUVQGWOVUQWLWPUXJUXMUXPUYGUXFXBZWNUYTUXMUYDUYO + VURUYDUXKUYMUXLUYNUVMUVQUUIGWQOUDMWRWSXEUYBUYEUYFUXQUXFXCWNUYTUYPUYAUYD + SZUYTUYFVUSUHUYBUYEUYFUXQUXFXDUVLUVPUVMUVQUQXFUYPUHUYKUYNSZUYTVUSUYPVUT + UUHUVPUUIUVQUQURVUTUYTVUSVUTUYTUXFUVPUVLFQZPNUKZTZUVQUVMGQZUDOUKZTZUVPU + VLUMZUVQUVMUMZTZUNZUYGSZSZVUSVUTUYSVVKUXFVUTUXQVVJUYGVUTUXJVVCUXMVVFUXP + VVIUYKUXJVVCVCUYNUYKUXHVVAUXIVVBUUHUVPUVLFWOLPNXGWSVSUYNUXMVVFVCUYKUYNU + XKVVDUXLVVEUUIUVQUVMGWOMUDOXGWSVJVUTUXNVVGUXOVVHUYKUXNVVGVCUYNUUHUVPUVL + XHVSUYNUXOVVHVCUYKUUIUVQUVMXHVJWSXIXJXKVVLUYAUYDVVLVVAUXTSZUYAVVLVVCUYB + VVMUYATZVVCVVFVVIUYGUXFWMUYBUYEUYFVVJUXFWTVVNUYAVVMTUYAVVATZUYAUXTTZSVV + CUYBSVVMUYAXLUYAVVAUXTXMVVOVVCVVPUYBVVOVVAUYATVVCUYAVVAXLUYAVVBVVANPXNX + OXPUYAUXTXLXQXRXSVVLVUIUWRUWPVVMUHAVUIUXEVVKJWEUXFUWRVVKVUCVSUXFUWPVVKV + UJVSDUVPUVLFXTYAYBVVLVVDUYCSZUYDVVLVVFUYEVVQUYDTZVVCVVFVVIUYGUXFXBUYBUY + EUYFVVJUXFXCVVRUYDVVQTUYDVVDTZUYDUYCTZSVVFUYESVVQUYDXLUYDVVDUYCXMVVSVVF + VVTUYEVVSVVDUYDTVVFUYDVVDXLUYDVVEVVDOUDXNXOXPUYDUYCXLXQXRXSVVLVUOUXBUXA + VVQUHAVUOUXEVVKKWEUXFUXBVVKVUEVSUXFUXAVVKVUPVSEUVQUVMGXTYAYBWAYEYCYDYFY + GYGYOYHUVTUWGUYIUWHUYRUVTUWEUXRUWFUYHUVTUWEUUJUVNHQZUXRUUKUVOUWEVWAVCUV + SUUDUUJUVGUVNHVEYIBCDEFGHLMNOIVAYJUVTUWFUVNUVRHQZUYHUVOUVSUWFVWBVCUUKUV + GUVNUVIUVRHVEYKBCDEFGHNOPUDIVAYJYLUVTUWHUUJUVRHQZUYRUUKUVSUWHVWCVCUVOUU + DUUJUVIUVRHVEYMBCDEFGHLMPUDIVAYJYNYPYQYRYSYRYSYSYTUUAUUB $. + $} - $( Trichotomy law for surreals. (Contributed by Scott Fenton, - 23-Nov-2021.) $) - slttrine $p |- ( ( A e. No /\ B e. No ) -> - ( A =/= B <-> ( A R Fr A ) $. + frxp2.2 $e |- ( ph -> S Fr B ) $. + $( Another way of giving a founded order to a cross product of two + classes. (Contributed by Scott Fenton, 19-Aug-2024.) $) + frxp2 $p |- ( ph -> T Fr ( A X. B ) ) $= + ( va vc ve cv c0 wa wn wi wcel vs vq vp vb vd cxp wss wne wbr wral wrex + vf wal wfr cdm dmss ad2antrl dmxpss sstrdi simprr wrel wceq relxp relss + wb mpi reldm0 syl necon3bid mpbid adantr df-fr sylib dmex sseq1 anbi12d + vex neeq1 raleq rexeqbi1dv imbi12d spcv mp2and csn cima crn rnss rnxpss + imassrn sstrid simprl cin imadisj disjsn bitri necon2abii imaex elimasn + ad2antrr cop wel wex elrel sylan weq wo w3a breq1 notbid simplrr adantl + biimpri rspcdva intnanrd opeq1 eleq1d anbi2d 3anass olc biantrurd neirr + orbi1d biorf bitr4di nonconne biorfi bitr4i bitrdi bitr3d bitrid mpbiri + ax-mp andir impcom opeldm simpr neneqd ioran sylanbrc rexlimddv ralbidv + intn3an1d pm2.61dane intn3an3d eleq1 xpord2lem com12 exlimdvv ralrimiva + mpd breq2 rspcev syl2anc ex alrimiv sylibr ) AUAOZDEUFZUGZUUQPUHZQZUBOZ + UCOZHUIZRZUBUUQUJZUCUUQUKZSZUAUMUURHUNAUVHUAAUVAUVGAUVAQZUDOZLOZFUIZRZU + DUUQUOZUJZUVGLUVNUVIUVNDUGZUVNPUHZUVOLUVNUKZUVIUVNUURUOZDUUSUVNUVSUGAUU + TUUQUURUPUQDEURUSUVIUUTUVQAUUSUUTUTUVIUUQPUVNPUVIUUQVAZUUQPVBUVNPVBVEUU + SUVTAUUTUUSUURVAUVTDEVCUUQUURVDVFUQZUUQVGVHVIVJUVIMOZDUGZUWBPUHZQZUVMUD + UWBUJZLUWBUKZSZMUMZUVPUVQQZUVRSZUVIDFUNZUWIAUWLUVAJVKMLUDDFVLVMUWHUWKMU + VNUUQUAVQZVNUWBUVNVBZUWEUWJUWGUVRUWNUWCUVPUWDUVQUWBUVNDVOUWBUVNPVRVPUWF + UVOLUWBUVNUVMUDUWBUVNVSVTWAWBVHWCUVIUVKUVNTZUVOQZQZUEOZUWBGUIZRZUEUUQUV + KWDZWEZUJZUVGMUXBUWQUXBEUGZUXBPUHZUXCMUXBUKZUWQUXBUUQWFZEUUQUXAWIUWQUXG + UURWFZEUVIUXGUXHUGZUWPUUSUXIAUUTUUQUURWGUQVKDEWHUSWJUWQUWOUXEUVIUWOUVOW + KUWOUXBPUXBPVBUVNUXAWLPVBUWORUUQUXAWMUVNUVKWNWOWPVMUWQNOZEUGZUXJPUHZQZU + WTUEUXJUJZMUXJUKZSZNUMZUXDUXEQZUXFSZAUXQUVAUWPAEGUNUXQKNMUEEGVLVMWSUXPU + XSNUXBUUQUXAUWMWQUXJUXBVBZUXMUXRUXOUXFUXTUXKUXDUXLUXEUXJUXBEVOUXJUXBPVR + VPUXNUXCMUXJUXBUWTUEUXJUXBVSVTWAWBVHWCUWQUWBUXBTZUXCQZQZUVKUWBWTZUUQTZU + VBUYDHUIZRZUBUUQUJZUVGUYCUYAUYEUWQUYAUXCWKUUQUVKUWBLVQZMVQWRVMUYCUYGUBU + UQUYCUBUAXAZQZUVBUXJULOZWTZVBZULXBNXBZUYGUYCUVTUYJUYOUVIUVTUWPUYBUWAWSN + ULUVBUUQXCXDUYKUYNUYGNULUYNUYKUYGUYNUYKUYGSUYCUYMUUQTZQZUXJDTUYLETQZUVK + DTUWBETQZUXJUVKFUIZNLXEZXFZUYLUWBGUIZULMXEZXFZUXJUVKUHZUYLUWBUHZXFZXGZX + GZRZSUYQVUIUYRUYSUYQVUIRZUXJUVKVUAUYQVULVUAUYQVULSUYCUVKUYLWTZUUQTZQZVU + CVUGQZRZSVUOVUCVUGVUOUWTVUCRUEUXBUYLUEULXEUWSVUCUWRUYLUWBGXHXIUWQUYAUXC + VUNXJVUNUYLUXBTZUYCVURVUNUUQUVKUYLUYIULVQZWRXLXKXMXNVUAUYQVUOVULVUQVUAU + YPVUNUYCVUAUYMVUMUUQUXJUVKUYLXOXPXQVUAVUIVUPVUIVUBVUEVUHQZQZVUAVUPVUBVU + EVUHXRVUAVUTVVAVUPVUAVUBVUTVUAUYTXSXTVUAVUTVUEVUGQZVUPVUAVUHVUGVUEVUAVU + HUVKUVKUHZVUGXFZVUGVUAVUFVVCVUGUXJUVKUVKVRYBVVCRVUGVVDVEUVKYAVVCVUGYCYL + YDXQVVBVUPVUDVUGQZXFVUPVUCVUDVUGYMVVEVUPUYLUWBYEYFYGYHYIYJXIWAYKYNUYQVU + FQZVUBVUEVUHVVFUYTRZVUARVUBRVVFUVMVVGUDUVNUXJUDNXEUVLUYTUVJUXJUVKFXHXIU + YCUVOUYPVUFUVIUWOUVOUYBXJWSUYQUXJUVNTZVUFUYPVVHUYCUXJUYLUUQNVQVUSYOXKVK + XMVVFUXJUVKUYQVUFYPYQUYTVUAYRYSUUBUUCUUDUYNUYKUYQUYGVUKUYNUYJUYPUYCUVBU + YMUUQUUEXQUYNUYFVUJUYNUYFUYMUYDHUIVUJUVBUYMUYDHXHBCDEFGHNULLMIUUFYHXIWA + YKUUGUUHUUJUUIUVFUYHUCUYDUUQUVCUYDVBZUVEUYGUBUUQVVIUVDUYFUVCUYDUVBHUUKX + IUUAUULUUMYTYTUUNUUOUAUCUBUURHVLUUP $. + $} - $( Surreal less than or equal in terms of less than. (Contributed by Scott - Fenton, 8-Dec-2021.) $) - slenlt $p |- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> -. B + Pred ( T , ( A X. B ) , <. X , Y >. ) = + ( ( ( Pred ( R , A , X ) u. { X } ) X. ( Pred ( S , B , Y ) u. { Y } ) ) + \ { <. X , Y >. } ) ) $= + ( va vb vc vd cpred csn wceq wcel wa wo ve cxp cv cop cun opeq1 predeq3 + cdif syl sneq uneq12d xpeq1d sneqd difeq12d eqeq12d opeq2 xpeq2d predel + wi a1i eldifi predss snssi unssd xpss12 syl2an sseld syl5 wrex wb elxp2 + wss wbr cvv elpred ax-mp opelxpi adantl biantrurd weq wne w3a xpord2lem + opex eldif opelxp elun vex elv velsn orbi12i bitri anbi12i notbii df-ne + elsn opthne 3bitr2i simprl orbi1d simprr 3anbi12d df-3an bitr2di bitrid + wn pm3.22 bitr4di bitr2d bitr3d bibi12d syl5ibrcom rexlimdvva pm5.21ndd + eleq1 biimtrid eqrdv vtocl2ga ) CDUBZGKUCZLUCZUDZOZCEXTOZXTPZUEZDFYAOZY + APZUEZUBZYBPZUHZQXSGHYAUDZOZCEHOZHPZUEZYIUBZYMPZUHZQXSGHIUDZOZYQDFIOZIP + ZUEZUBZUUAPZUHZQKLHICDXTHQZYCYNYLYTUUIYBYMQYCYNQXTHYAUFZXSGYBYMUGUIUUIY + JYRYKYSUUIYFYQYIUUIYDYOYEYPCEXTHUGXTHUJUKULUUIYBYMUUJUMUNUOYAIQZYNUUBYT + UUHUUKYMUUAQYNUUBQYAIHUPZXSGYMUUAUGUIUUKYRUUFYSUUGUUKYIUUEYQUUKYGUUCYHU + UDDFYAIUGYAIUJUKUQUUKYMUUAUULUMUNUOXTCRZYADRZSZUAYCYLUUOUAUCZXSRZUUPYCR + ZUUPYLRZUURUUQUSUUOXSGYBUUPURUTUUSUUPYJRUUOUUQUUPYJYKVAUUOYJXSUUPUUMYFC + VLYIDVLYJXSVLUUNUUMYDYECYDCVLUUMCEXTVBUTXTCVCVDUUNYGYHDYGDVLUUNDFYAVBUT + YADVCVDYFCYIDVEVFVGVHUUQUUPMUCZNUCZUDZQZNDVIMCVIUUOUURUUSVJZMNUUPCDVKUU + OUVCUVDMNCDUUOUUTCRZUVADRZSZSZUVDUVCUVBYCRZUVBYLRZVJUVIUVBXSRZUVBYBGVMZ + SZUVHUVJYBVNRUVIUVMVJXTYAWDXSVNGYBUVBUUTUVAWDZVOVPUVHUVLUVMUVJUVHUVKUVL + UVGUVKUUOUUTUVACDVQVRVSUVLUVGUUOUUTXTEVMZMKVTZTZUVAYAFVMZNLVTZTZUUTXTWA + UVAYAWATZWBZWBZUVHUVJABCDEFGMNKLJWCUVHUVJUWBUWCUVJUVEUVOSZUVPTZUVFUVRSZ + UVSTZSZUWASZUVHUWBUVJUVBYJRZUVBYKRZXFZSUWIUVBYJYKWEUWJUWHUWLUWAUWJUUTYF + RZUVAYIRZSUWHUUTUVAYFYIWFUWMUWEUWNUWGUWMUUTYDRZUUTYERZTUWEUUTYDYEWGUWOU + WDUWPUVPUWOUWDVJKCVNEXTUUTMWHZVOWIMXTWJWKWLUWNUVAYGRZUVAYHRZTUWGUVAYGYH + WGUWRUWFUWSUVSUWRUWFVJLDVNFYAUVANWHZVOWINYAWJWKWLWMWLUWLUVBYBQZXFUVBYBW + AUWAUWKUXAUVBYBUVNWPWNUVBYBWOUUTUVAXTYAUWQUWTWQWRWMWLUVHUWBUWEUWGUWAWBU + WIUVHUVQUWEUVTUWGUWAUVHUVOUWDUVPUVHUVEUVOUUOUVEUVFWSVSWTUVHUVRUWFUVSUVH + UVFUVRUUOUVEUVFXAVSWTXBUWEUWGUWAXCXDXEUVHUWBUVGUUOSZUWBSUWCUVHUXBUWBUUO + UVGXGVSUVGUUOUWBXCXHXIXEXJXEUVCUURUVIUUSUVJUUPUVBYCXOUUPUVBYLXOXKXLXMXP + XNXQXR $. + $} - $( Surreal less than in terms of less than or equal. (Contributed by Scott - Fenton, 8-Dec-2021.) $) - sltnle $p |- ( ( A e. No /\ B e. No ) -> ( A -. B <_s A ) ) $= - ( csur wcel wa csle wbr cslt wn wb slenlt ancoms con2bid ) ACDZBCDZEBAFGZAB - HGZONPQIJBAKLM $. + ${ + $d A a $. $d a b $. $d A b $. $d a p $. $d A p $. $d a ph $. + $d A x $. $d A y $. $d B a $. $d B b $. $d b p $. $d B p $. + $d b ph $. $d B x $. $d B y $. $d p ph $. $d R x $. $d R y $. + $d S x $. $d S y $. $d T a $. $d T b $. $d T p $. $d x y $. + sexp2.1 $e |- ( ph -> R Se A ) $. + sexp2.2 $e |- ( ph -> S Se B ) $. + $( Condition for the relationship in ~ frxp2 to be set-like. + (Contributed by Scott Fenton, 19-Aug-2024.) $) + sexp2 $p |- ( ph -> T Se ( A X. B ) ) $= + ( vp va vb cv cpred cvv wcel wse csn cxp wral cop wceq wrex wa cun cdif + elxp2 xpord2pred adantl setlikespec sylan adantrr vsnex a1i unexd xpexd + ancoms adantrl difexd eqeltrd eleq1d syl5ibrcom biimtrid ralrimiv dfse3 + predeq3 rexlimdvva sylibr ) ADEUAZHLOZPZQRZLVKUBVKHSAVNLVKVLVKRVLMOZNOZ + UCZUDZNEUEMDUEAVNMNVLDEUIAVRVNMNDEAVODRZVPERZUFZUFZVNVRVKHVQPZQRWBWCDFV + OPZVOTZUGZEGVPPZVPTZUGZUAZVQTZUHZQWAWCWLUDABCDEFGHVOVPIUJUKWBWJWKQWBWFW + IQQWBWDWEQQAVSWDQRZVTADFSZVSWMJVSWNWMDFVOULUSUMUNWEQRWBMUOUPUQWBWGWHQQA + VTWGQRZVSAEGSZVTWOKVTWPWOEGVPULUSUMUTWHQRWBNUOUPUQURVAVBVRVMWCQVKHVLVQV + HVCVDVIVEVFLVKHVGVJ $. + $} - $( Surreal less than or equal in terms of less than. (Contributed by Scott - Fenton, 8-Dec-2021.) $) - sleloe $p |- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> - ( A ( ph <-> ps ) ) $. + xpord2ind.8 $e |- ( b = d -> ( ps <-> ch ) ) $. + xpord2ind.9 $e |- ( a = c -> ( th <-> ch ) ) $. + xpord2ind.11 $e |- ( a = X -> ( ph <-> ta ) ) $. + xpord2ind.12 $e |- ( b = Y -> ( ta <-> et ) ) $. + xpord2ind.i $e |- ( ( a e. A /\ b e. B ) -> + ( ( A. c e. Pred ( R , A , a ) A. d e. Pred ( S , B , b ) ch /\ + A. c e. Pred ( R , A , a ) ps /\ + A. d e. Pred ( S , B , b ) th ) -> ph ) ) $. + $( Induction over the cross product ordering. Note that the + substitutions cover all possible cases of membership in the + predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.) $) + xpord2ind $p |- ( ( X e. A /\ Y e. B ) -> et ) $= + ( cxp wfr wpo wse w3a wcel wa wtru a1i frxp2 poxp2 sexp2 3jca mptru cop + cv cpred wi wal wne wo csn cun wral cdif xpord2pred eleq2d imbi1d eldif + wn opelxp wceq opex elsn notbii df-ne vex opthne 3bitr2i anbi12i imbi1i + bitri impexp bitrdi 2albidv r2al bitr4di wss ssun1 ssralv ax-mp syl weq + ralimi predpoirr eleq1w mtbiri necon2ai olcd pm2.27 ralimia ssun2 neeq1 + orbi2d wb equcoms bicomd imbi12d ralsn ralbii sylib orcd orbi1d ralbidv + syl5 sylbid frpoins3xpg mpan ) IJUMZMUNZYKMUOZYKMUPZUQZNIUROJURUSFYOUTY + LYMYNUTGHIJKLMTIKUNUTUAVAJLUNUTUDVAVBUTGHIJKLMTIKUOZUTUBVAJLUOZUTUEVAVC + UTGHIJKLMTIKUPUTUCVAJLUPUTUFVAVDVEVFABCEFPQRSIJMNOPVHZIURQVHZJURUSZRVHZ + SVHZVGZYKMYRYSVGZVIZURZCVJZSVKRVKZUUAYRVLZUUBYSVLZVMZCVJZSJLYSVIZYSVNZV + OZVPZRIKYRVIZYRVNZVOZVPZAYTUUHUUAUUSURUUBUUOURUSZUULVJZSVKRVKUUTYTUUGUV + BRSYTUUGUUCUUSUUOUMZUUDVNZVQZURZCVJZUVBYTUUFUVFCYTUUEUVEUUCGHIJKLMYRYST + VRVSVTUVGUVAUUKUSZCVJUVBUVFUVHCUVFUUCUVCURZUUCUVDURZWBZUSUVHUUCUVCUVDWA + UVIUVAUVKUUKUUAUUBUUSUUOWCUVKUUCUUDWDZWBUUCUUDVLUUKUVJUVLUUCUUDUUAUUBWE + WFWGUUCUUDWHUUAUUBYRYSRWISWIWJWKWLWNWMUVAUUKCWOWNWPWQUULRSUUSUUOWRWSUUT + CSUUMVPZRUUQVPZBRUUQVPZDSUUMVPZUQYTAUUTUVNUVOUVPUUTUULSUUMVPZRUUQVPZUVN + UUTUUPRUUQVPZUVRUUQUUSWTUUTUVSVJUUQUURXAUUPRUUQUUSXBXCZUUPUVQRUUQUUMUUO + WTUUPUVQVJUUMUUNXAUULSUUMUUOXBXCZXFXDUVQUVMRUUQUULCSUUMUUBUUMURZUUKUULC + VJUWBUUJUUIUWBUUBYSSQXEZUWBYSUUMURZYQUWDWBUEJLYSXGXCSQUUMXHXIXJZXKUUKCX + LXDXMXFXDUUTUUIYSYSVLZVMZBVJZRUUQVPZUVOUUTUULSUUNVPZRUUQVPZUWIUUTUVSUWK + UVTUUPUWJRUUQUUNUUOWTUUPUWJVJUUNUUMXNUULSUUNUUOXBXCXFXDUWJUWHRUUQUULUWH + SYSQWIUWCUUKUWGCBUWCUUJUWFUUIUUBYSYSXOXPUWCBCBCXQQSUHXRXSXTYAYBYCUWHBRU + UQUUAUUQURZUWGUWHBVJUWLUUIUWFUWLUUAYRRPXEZUWLYRUUQURZYPUWNWBUBIKYRXGXCR + PUUQXHXIXJYDUWGBXLXDXMXDUUTYRYRVLZUUJVMZDVJZSUUMVPZUVPUUTUVQRUURVPZUWRU + UTUUPRUURVPZUWSUURUUSWTUUTUWTVJUURUUQXNUUPRUURUUSXBXCUUPUVQRUURUWAXFXDU + VQUWRRYRPWIUWMUULUWQSUUMUWMUUKUWPCDUWMUUIUWOUUJUUAYRYRXOYEUWMDCDCXQPRUI + XRXSXTYFYAYCUWQDSUUMUWBUWPUWQDVJUWBUUJUWOUWEXKUWPDXLXDXMXDVEULYGYHUGUHU + JUKYIYJ $. + $} - $( Trichotomy law for surreal less than or equal. (Contributed by Scott - Fenton, 8-Dec-2021.) $) - sletri3 $p |- ( ( A e. No /\ B e. No ) -> - ( A = B <-> ( A <_s B /\ B <_s A ) ) ) $= - ( csur wcel wa wceq cslt wbr wn csle slttrieq2 slenlt ancoms anbi12d bitrdi - wb ancom bitr4d ) ACDZBCDZEZABFABGHIZBAGHIZEZABJHZBAJHZEZABKUAUGUCUBEUDUAUE - UCUFUBABLTSUFUBPBALMNUCUBQOR $. + $} - $( Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) $) - sltletr $p |- ( ( A e. No /\ B e. No /\ C e. No ) -> - ( ( A A . | ( x e. ( ( A X. B ) X. C ) + /\ y e. ( ( A X. B ) X. C ) + /\ ( ( ( ( 1st ` ( 1st ` x ) ) R ( 1st ` ( 1st ` y ) ) \/ + ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` y ) ) ) /\ + ( ( 2nd ` ( 1st ` x ) ) S ( 2nd ` ( 1st ` y ) ) \/ + ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` y ) ) ) /\ + ( ( 2nd ` x ) T ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) ) /\ + x =/= y ) ) } $. - $( Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) $) - slelttr $p |- ( ( A e. No /\ B e. No /\ C e. No ) -> - ( ( A <_s B /\ B A . , c >. U <. <. d , e >. , f >. <-> + ( ( a e. A /\ b e. B /\ c e. C ) /\ ( d e. A /\ e e. B /\ f e. C ) /\ + ( ( ( a R d \/ a = d ) /\ ( b S e \/ b = e ) /\ ( c T f \/ c = f ) ) /\ + ( a =/= d \/ b =/= e \/ c =/= f ) ) ) ) $= + ( wbr wcel wo w3a cv cop cxp weq wne wa w3o c1st cfv wceq c2nd opex vex + ot21std breq1d eqeq1d orbi12d ot22ndd op2ndd 3anbi123d anbi12d 3anbi13d + eleq1 neeq1 breq2d eqeq2d 3anbi23d brab ot2elxp otthne anbi2i 3anbi123i + neeq2 bitri ) LUAZMUAZUBZNUAZUBZOUAZJUAZUBZKUAZUBZIQVSCDUCEUCZRZWDWERZV + OVTFQZLOUDZSZVPWAGQZMJUDZSZVRWCHQZNKUDZSZTZVSWDUEZUFZTZVOCRVPDRVRERTZVT + CRWADRWCERTZWQVOVTUEVPWAUEVRWCUEUGZUFZTAUAZWERZBUAZWERZXEUHUIZUHUIZXGUH + UIZUHUIZFQZXJXLUJZSZXIUKUIZXKUKUIZGQZXPXQUJZSZXEUKUIZXGUKUIZHQZYAYBUJZS + ZTZXEXGUEZUFZTWFXHVOXLFQZVOXLUJZSZVPXQGQZVPXQUJZSZVRYBHQZVRYBUJZSZTZVSX + GUEZUFZTWTABVSWDIVQVRULWBWCULXEVSUJZXFWFYHYTXHXEVSWEVCUUAYFYRYGYSUUAXOY + KXTYNYEYQUUAXMYIXNYJUUAXJVOXLFVOVPVRXELUMZMUMZNUMZUNZUOUUAXJVOXLUUEUPUQ + UUAXRYLXSYMUUAXPVPXQGVOVPVRXEUUBUUCUUDURZUOUUAXPVPXQUUFUPUQUUAYCYOYDYPU + UAYAVRYBHVQVRXEVOVPULUUDUSZUOUUAYAVRYBUUGUPUQUTXEVSXGVDVAVBXGWDUJZXHWGY + TWSWFXGWDWEVCUUHYRWQYSWRUUHYKWJYNWMYQWPUUHYIWHYJWIUUHXLVTVOFVTWAWCXGOUM + ZJUMZKUMZUNZVEUUHXLVTVOUULVFUQUUHYLWKYMWLUUHXQWAVPGVTWAWCXGUUIUUJUUKURZ + VEUUHXQWAVPUUMVFUQUUHYOWNYPWOUUHYBWCVRHWBWCXGVTWAULUUKUSZVEUUHYBWCVRUUN + VFUQUTXGWDVSVMVAVGPVHWFXAWGXBWSXDVOVPVRCDEVIVTWAWCCDEVIWRXCWQVOVPVRVTWA + WCUUBUUCUUDVJVKVLVN $. + $} - $( Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) $) - sletr $p |- ( ( A e. No /\ B e. No /\ C e. No ) -> - ( ( A <_s B /\ B <_s C ) -> A <_s C ) ) $= - ( csur wcel w3a csle wbr cslt wn wa sltletr 3coml expcomd imp con3d expimpd - wi wb slenlt 3adant1 anbi2d 3adant2 3imtr4d ) ADEZBDEZCDEZFZABGHZCBIHZJZKCA - IHZJZUIBCGHZKACGHZUHUIUKUMUHUIKULUJUHUIULUJRUHULUIUJUGUEUFULUIKUJRCABLMNOPQ - UHUNUKUIUFUGUNUKSUEBCTUAUBUEUGUOUMSUFACTUCUD $. + ${ + $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. + $d A d $. $d a e $. $d A e $. $d a f $. $d A f $. $d a g $. + $d A g $. $d a h $. $d A h $. $d a i $. $d A i $. $d a j $. + $d A j $. $d a k $. $d A k $. $d a l $. $d A l $. $d a ph $. + $d A x $. $d A y $. $d B a $. $d B b $. $d b c $. $d B c $. + $d b d $. $d B d $. $d b e $. $d B e $. $d b f $. $d B f $. + $d b g $. $d B g $. $d b h $. $d B h $. $d b i $. $d B i $. + $d b j $. $d B j $. $d b k $. $d B k $. $d b l $. $d B l $. + $d b ph $. $d B x $. $d B y $. $d C a $. $d C b $. $d C c $. + $d c d $. $d C d $. $d c e $. $d C e $. $d c f $. $d C f $. + $d c g $. $d C g $. $d c h $. $d C h $. $d c i $. $d C i $. + $d c j $. $d C j $. $d c k $. $d C k $. $d c l $. $d C l $. + $d c ph $. $d C x $. $d C y $. $d d e $. $d d f $. $d d g $. + $d d h $. $d d i $. $d d j $. $d d k $. $d d l $. $d d ph $. + $d d x $. $d d y $. $d e f $. $d e g $. $d e h $. $d e i $. + $d e j $. $d e k $. $d e l $. $d e ph $. $d e x $. $d e y $. + $d f g $. $d f h $. $d f i $. $d f j $. $d f k $. $d f l $. + $d f ph $. $d f x $. $d f y $. $d g h $. $d g i $. $d g j $. + $d g k $. $d g l $. $d g ph $. $d g x $. $d g y $. $d h i $. + $d h j $. $d h k $. $d h l $. $d h ph $. $d h x $. $d h y $. + $d i j $. $d i k $. $d i l $. $d i ph $. $d i x $. $d i y $. + $d j k $. $d j l $. $d j ph $. $d j x $. $d j y $. $d k l $. + $d k ph $. $d k x $. $d k y $. $d l ph $. $d l x $. $d l y $. + $d R x $. $d R y $. $d S x $. $d S y $. $d T x $. $d T y $. + $d U a $. $d U b $. $d U c $. $d U d $. $d U e $. $d U f $. + $d U g $. $d U h $. $d U i $. $d U j $. $d U k $. $d U l $. + $d x y $. + poxp3.1 $e |- ( ph -> R Po A ) $. + poxp3.2 $e |- ( ph -> S Po B ) $. + poxp3.3 $e |- ( ph -> T Po C ) $. + $( Triple cross product partial ordering. (Contributed by Scott Fenton, + 21-Aug-2024.) $) + poxp3 $p |- ( ph -> U Po ( ( A X. B ) X. C ) ) $= + ( vd ve wbr wrex wa wo va vb vc vf vg vh vi vj vk vl cxp cv wcel wn cop + elxpxp wi w3a weq wne w3o neirr 3pm3.2ni intnan simp3 mto breq12 anidms + wceq wb xpord3lem bitrdi mtbiri rexlimivw a1i rexlimivv sylbi 3anbi123i + adantl 3reeanv rexbii 2rexbii bitr4i simprl1 simprr2 wpo adantr simpl11 + bitri simpr11 simpr21 potr syl13anc orc syl6 expd breq1 syl6bir simp3l1 + ad2antrl mpjaod breq2 equequ2 orbi12d syl5ibcom simpl12 simpr12 simpr22 + ad2antll simp3l2 simpl13 simpr13 simpr23 simp3l3 3jca simpr3r 3orbi123i + df-ne 3ianor sylib con2bii equequ1 equcom ordir sylanbrc po2nr syl12anc + imp orcnd ex orbi2i 3anim123d syl5bir jca 3adant3 3adant1 rexlimdvw a1d + mt3d rexlimdvv anbi12d 3adant2 imbi12d syl5ibrcom biimtrid ispod ) AUAU + BUCDEUKFUKZJUAULZUUGUMZUUHUUHJQZUNZAUUIUUHOULZPULZUOUDULZUOZVIZUDFRZPER + ZODRZUUKOPUDUUHDEFUPZUUQUUKOPDEUUQUUKUQUULDUMZUUMEUMZSUUPUUKUDFUUPUUJUV + AUVBUUNFUMZURZUVDUULUULGQOOUSTUUMUUMHQPPUSTUUNUUNIQUDUDUSTURZUULUULUTZU + UMUUMUTZUUNUUNUTZVAZSZURZUVKUVJUVIUVEUVFUVGUVHUULVBUUMVBUUNVBVCVDUVDUVD + UVJVEVFUUPUUJUUOUUOJQZUVKUUPUUJUVLVJUUHUUOUUHUUOJVGVHBCDEFGHIJPUDOPUDOK + VKVLVMVNVOVPVQVSAUUIUBULZUUGUMZUCULZUUGUMZURZUUHUVMJQZUVMUVOJQZSZUUHUVO + JQZUQZUVQUUPUVMUEULZUFULZUOUGULZUOZVIZUVOUHULZUIULZUOUJULZUOZVIZURZUJFR + ZUGFRUDFRZUIERZUFERPERZUHDRZUEDRODRZAUWBUVQUUSUWGUGFRZUFERZUEDRZUWLUJFR + ZUIERZUHDRZURZUWSUUIUUSUVNUXBUVPUXEUUTUEUFUGUVMDEFUPUHUIUJUVODEFUPVRUWS + UURUXAUXDURZUHDRZUEDRODRUXFUWRUXHOUEDDUWQUXGUHDUWQUUQUWTUXCURZUIERZUFER + PERUXGUWPUXJPUFEEUWOUXIUIEUUPUWGUWLUDUGUJFFFVTWAWBUUQUWTUXCPUFUIEEEVTWI + WAWBUURUXAUXDOUEUHDDDVTWIWCAUWRUWBOUEDDAUWRUWBUQUVAUWCDUMZSAUWQUWBUHDAU + WPUWBPUFEEAUWPUWBUQUVBUWDEUMZSAUWOUWBUIEAUWNUWBUDUGFFAUWNUWBUQUVCUWEFUM + ZSAUWMUWBUJFAUWBUWMUVDUXKUXLUXMURZUULUWCGQZOUEUSZTZUUMUWDHQZPUFUSZTZUUN + UWEIQZUDUGUSZTZURUULUWCUTUUMUWDUTUUNUWEUTVAZSZURZUXNUWHDUMZUWIEUMZUWJFU + MZURZUWCUWHGQZUEUHUSZTZUWDUWIHQZUFUIUSZTZUWEUWJIQZUGUJUSZTZURZUWCUWHUTZ + UWDUWIUTZUWEUWJUTZVAZSZURZSZUVDUYJUULUWHGQZOUHUSZTZUUMUWIHQZPUIUSZTZUUN + UWJIQZUDUJUSZTZURZUULUWHUTZUUMUWIUTZUUNUWJUTZVAZSZURZUQAVUGVVCAVUGSZUVD + UYJVVBUVDUXNUYEVUFAWDUXNUYJVUEUYFAWEVVDVUQVVAVVDVUJVUMVUPVVDUYKVUJUYLVV + DUXOUYKVUJUQZUXPVVDUXOUYKVUJVVDUXOUYKSZVUHVUJVVDDGWFZUVAUXKUYGVVFVUHUQA + VVGVUGLWGZVUGUVAAUVAUVBUVCUXNUYEVUFWHVSVUGUXKAUXKUXLUXMUYJVUEUYFWJVSZVU + GUYGAUYGUYHUYIUXNVUEUYFWKVSZDUULUWCUWHGWLWMVUHVUIWNZWOWPUXPVVEUQVVDUXPU + YKVUHVUJUULUWCUWHGWQVVKWRVOUYFUXQAVUFUXQUXTUYCUYDUVDUXNWSWTZXAVVDUXQUYL + VUJVVLUYLUXOVUHUXPVUIUWCUWHUULGXBUEUHOXCXDXEVUFUYMAUYFUYMUYPUYSVUDUXNUY + JWSXIZXAVVDUYNVUMUYOVVDUXRUYNVUMUQZUXSVVDUXRUYNVUMVVDUXRUYNSZVUKVUMVVDE + HWFZUVBUXLUYHVVOVUKUQAVVPVUGMWGZVUGUVBAUVAUVBUVCUXNUYEVUFXFVSVUGUXLAUXK + UXLUXMUYJVUEUYFXGVSZVUGUYHAUYGUYHUYIUXNVUEUYFXHVSZEUUMUWDUWIHWLWMVUKVUL + WNZWOWPUXSVVNUQVVDUXSUYNVUKVUMUUMUWDUWIHWQVVTWRVOUYFUXTAVUFUXQUXTUYCUYD + UVDUXNXJWTZXAVVDUXTUYOVUMVWAUYOUXRVUKUXSVULUWDUWIUUMHXBUFUIPXCXDXEVUFUY + PAUYFUYMUYPUYSVUDUXNUYJXJXIZXAVVDUYQVUPUYRVVDUYAUYQVUPUQZUYBVVDUYAUYQVU + PVVDUYAUYQSZVUNVUPVVDFIWFZUVCUXMUYIVWDVUNUQAVWEVUGNWGZVUGUVCAUVAUVBUVCU + XNUYEVUFXKVSVUGUXMAUXKUXLUXMUYJVUEUYFXLVSZVUGUYIAUYGUYHUYIUXNVUEUYFXMVS + ZFUUNUWEUWJIWLWMVUNVUOWNZWOWPUYBVWCUQVVDUYBUYQVUNVUPUUNUWEUWJIWQVWIWRVO + UYFUYCAVUFUXQUXTUYCUYDUVDUXNXNWTZXAVVDUYCUYRVUPVWJUYRUYAVUNUYBVUOUWEUWJ + UUNIXBUGUJUDXCXDXEVUFUYSAUYFUYMUYPUYSVUDUXNUYJXNXIZXAXOVVDVVAUYLUYOUYRU + RZVVDVUDVWLUNZVUGVUDAUYTVUDUXNUYJUYFXPVSVUDUYLUNZUYOUNZUYRUNZVAVWMVUAVW + NVUBVWOVUCVWPUWCUWHXRUWDUWIXRUWEUWJXRXQUYLUYOUYRXSWCXTVVAUNVUIVULVUOURZ + VVDVWLVVAVWQVVAVUIUNZVULUNZVUOUNZVAVWQUNVURVWRVUSVWSVUTVWTUULUWHXRUUMUW + IXRUUNUWJXRXQVUIVULVUOXSWCYAVVDVUIUYLVULUYOVUOUYRVVDVUIUYLVVDVUISZUWHUW + CGQZUYKSZUYLVXAVXBUYLTZUYMVXCUYLTVVDVUIVXDVVDUXQVUIVXDVVLVUIUXOVXBUXPUY + LUULUWHUWCGWQVUIUXPUHUEUSUYLOUHUEYBUHUEYCVLXDXEYHVVDUYMVUIVVMWGVXBUYKUY + LYDYEVVDVXCUNZVUIVVDVVGUYGUXKVXEVVHVVJVVIDUWHUWCGYFYGWGYIYJVVDVULUYOVVD + VULSZUWIUWDHQZUYNSZUYOVXFVXGUYOTZUYPVXHUYOTVVDVULVXIVVDUXTVULVXIVWAVULU + XRVXGUXSUYOUUMUWIUWDHWQVULUXSUIUFUSUYOPUIUFYBUIUFYCVLXDXEYHVVDUYPVULVWB + WGVXGUYNUYOYDYEVVDVXHUNZVULVVDVVPUYHUXLVXJVVQVVSVVREUWIUWDHYFYGWGYIYJVV + DVUOUYRVVDVUOSZUWJUWEIQZUYQSZUYRVXKVXLUYRTZUYSVXMUYRTVXKVXLUJUGUSZTZVXN + VVDVUOVXPVVDUYCVUOVXPVWJVUOUYAVXLUYBVXOUUNUWJUWEIWQUDUJUGYBXDXEYHVXOUYR + VXLUJUGYCYKXTVVDUYSVUOVWKWGVXLUYQUYRYDYEVVDVXMUNZVUOVVDVWEUYIUXMVXQVWFV + WHVWGFUWJUWEIYFYGWGYIYJYLYMYSYNXOYJUWMUVTVUGUWAVVCUWMUVRUYFUVSVUFUWMUVR + UUOUWFJQZUYFUUPUWGUVRVXRVJUWLUUHUUOUVMUWFJVGYOBCDEFGHIJUFUGOPUDUEKVKVLU + WMUVSUWFUWKJQZVUFUWGUWLUVSVXSVJUUPUVMUWFUVOUWKJVGYPBCDEFGHIJUIUJUEUFUGU + HKVKVLUUAUWMUWAUUOUWKJQZVVCUUPUWLUWAVXTVJUWGUUHUUOUVOUWKJVGUUBBCDEFGHIJ + UIUJOPUDUHKVKVLUUCUUDYQYRYTYQYRYTYQYRYTUUEYHUUF $. + $} - ${ - slttrd.1 $e |- ( ph -> A e. No ) $. - slttrd.2 $e |- ( ph -> B e. No ) $. - slttrd.3 $e |- ( ph -> C e. No ) $. ${ - slttrd.4 $e |- ( ph -> A B A R Fr A ) $. + frxp3.2 $e |- ( ph -> S Fr B ) $. + frxp3.3 $e |- ( ph -> T Fr C ) $. + $( Give foundedness over a triple cross product. (Contributed by Scott + Fenton, 21-Aug-2024.) $) + frxp3 $p |- ( ph -> U Fr ( ( A X. B ) X. C ) ) $= + ( vg c0 wa wn wi wcel vs vq vp vd va ve vb vf vc vh vi cxp wss wne wral + cv wbr wrex wal wfr cdm dmss ad2antrl dmxpss sstrdi simprr wrel wceq wb + syl simprl relxp relss mpisyl reldm0 necon3bid mpbid df-fr sylib adantr + vex dmex sseq1 neeq1 anbi12d raleq rexeqbi1dv imbi12d spcv csn cima crn + mp2and imassrn rnss rnxpss sstrid imadisj disjsn bitri necon2abii imaex + cin ad2antrr cop elimasn ad3antrrr opex wel wex w3a weq wo adantl breq1 + notbid simplrr sylibr rspcdva simpr ioran sylanbrc intn3an3d pm2.61dane + neneqd opeq1d eleq1d anbi2d orbi1d neirr orel1 olc impbii bitrdi mpbiri + intnanrd ax-mp impcom opeldm rexlimddv simplrl elxpxpss sylan w3o df-ne + con2bii biimpi intnand simplr opeq2 intn3an2d opeq1 3orass bitrid eleq1 + intn3an1d xpord3lem com12 exlimdv exlimdvv mpd ralrimiva ralbidv rspcev + breq2 syl2anc ex alrimiv ) AUAUPZDEULZFULZUMZUVIPUNZQZUBUPZUCUPZJUQZRZU + BUVIUOZUCUVIURZSZUAUSUVKJUTAUWAUAAUVNUVTAUVNQZUDUPZUEUPZGUQZRZUDUVIVAZV + AZUOZUVTUEUWHUWBUWHDUMZUWHPUNZUWIUEUWHURZUWBUWHUVJVAZDUWBUWGUVJUMZUWHUW + MUMUWBUWGUVKVAZUVJUVLUWGUWOUMAUVMUVIUVKVBVCUVJFVDVEZUWGUVJVBVJDEVDVEUWB + UWGPUNZUWKUWBUVMUWQAUVLUVMVFUWBUVIPUWGPUWBUVIVGZUVIPVHUWGPVHZVIUWBUVLUV + KVGUWRAUVLUVMVKUVJFVLUVIUVKVMVNUVIVOVJVPVQUWBUWGPUWHPUWBUWGVGZUWSUWHPVH + VIUWBUWNUVJVGUWTUWPDEVLUWGUVJVMVNUWGVOVJVPVQUWBOUPZDUMZUXAPUNZQZUWFUDUX + AUOZUEUXAURZSZOUSZUWJUWKQZUWLSZAUXHUVNADGUTUXHLOUEUDDGVRVSVTUXGUXJOUWHU + WGUVIUAWAZWBZWBUXAUWHVHZUXDUXIUXFUWLUXMUXBUWJUXCUWKUXAUWHDWCUXAUWHPWDWE + UXEUWIUEUXAUWHUWFUDUXAUWHWFWGWHWIVJWMUWBUWDUWHTZUWIQZQZUFUPZUGUPZHUQZRZ + UFUWGUWDWJZWKZUOZUVTUGUYBUXPUYBEUMZUYBPUNZUYCUGUYBURZUXPUYBUWGWLZEUWGUY + AWNUWBUYGEUMUXOUWBUYGUVJWLZEUWBUWNUYGUYHUMUWPUWGUVJWOVJDEWPVEVTWQUXPUXN + UYEUWBUXNUWIVKUXNUYBPUYBPVHUWHUYAXCPVHUXNRUWGUYAWRUWHUWDWSWTXAVSAUYDUYE + QZUYFSZUVNUXOAUXAEUMZUXCQZUXTUFUXAUOZUGUXAURZSZOUSZUYJAEHUTUYPMOUGUFEHV + RVSUYOUYJOUYBUWGUYAUXLXBUXAUYBVHZUYLUYIUYNUYFUYQUYKUYDUXCUYEUXAUYBEWCUX + AUYBPWDWEUYMUYCUGUXAUYBUXTUFUXAUYBWFWGWHWIVJXDWMUXPUXRUYBTZUYCQZQZUHUPZ + UIUPZIUQZRZUHUVIUWDUXRXEZWJZWKZUOZUVTUIVUGUYTVUGFUMZVUGPUNZVUHUIVUGURZU + WBVUIUXOUYSUWBVUGUVIWLZFUVIVUFWNUWBVULUVKWLZFUVLVULVUMUMAUVMUVIUVKWOVCU + VJFWPVEWQXDUYTVUEUWGTZVUJUYTUYRVUNUXPUYRUYCVKUWGUWDUXRUEWAZUGWAXFVSVUNV + UGPVUGPVHUWGVUFXCPVHVUNRUVIVUFWRUWGVUEWSWTXAVSAVUIVUJQZVUKSZUVNUXOUYSAU + XAFUMZUXCQZVUDUHUXAUOZUIUXAURZSZOUSZVUQAFIUTVVCNOUIUHFIVRVSVVBVUQOVUGUV + IVUFUXKXBUXAVUGVHZVUSVUPVVAVUKVVDVURVUIUXCVUJUXAVUGFWCUXAVUGPWDWEVUTVUH + UIUXAVUGVUDUHUXAVUGWFWGWHWIVJXGWMUYTVUBVUGTZVUHQZQZVUEVUBXEZUVITZUVOVVH + JUQZRZUBUVIUOZUVTVVGVVEVVIUYTVVEVUHVKUVIVUEVUBUWDUXRXHZUIWAXFVSVVGVVKUB + UVIVVGUBUAXIZQZUVOUXAUJUPZXEZUKUPZXEZVHZUKXJZUJXJOXJZVVKVVGUVLVVNVWBUXP + UVLUYSVVFAUVLUVMUXOUUAXDOUJUKUVODEFUVIUUBUUCVVOVWAVVKOUJVVOVVTVVKUKVVTV + VOVVKVVTVVOVVKSVVGVVSUVITZQZUXADTVVPETVVRFTXKZUWDDTUXRETVUBFTXKZUXAUWDG + UQZOUEXLZXMZVVPUXRHUQZUJUGXLZXMZVVRVUBIUQZUKUIXLZXMZXKZUXAUWDUNZVVPUXRU + NZVVRVUBUNZUUDZQZXKZRZSVWDVXAVWEVWFVWDVXARZUXAUWDVWHVWDVXDVWHVWDVXDSVVG + UWDVVPXEZVVRXEZUVITZQZVWPVWRVWSXMZQZRZSZVXHVXKVVPUXRVWKVXHVXKVWKVXLVVGV + UEVVRXEZUVITZQZVWPVWSQZRZSVXOVXQVVRVUBVXOVWNQVWSVWPVWNVWSRZVXOVWNVXRVWS + VWNVVRVUBUUEUUFUUGXNUUHVXOVWSQZVWPVWSVXSVWOVWIVWLVXSVWMRZVWNRVWORVXSVUD + VXTUHVUGVVRUHUKXLVUCVWMVUAVVRVUBIXOXPVXOVUHVWSUYTVVEVUHVXNXQVTVXSVXNVVR + VUGTVVGVXNVWSUUIUVIVUEVVRVVMUKWAZXFXRXSVXSVVRVUBVXOVWSXTYEVWMVWNYAYBYCY + PYDVWKVXHVXOVXKVXQVWKVXGVXNVVGVWKVXFVXMUVIVWKVXEVUEVVRVVPUXRUWDUUJYFYGY + HVWKVXJVXPVWKVXIVWSVWPVWKVXIUXRUXRUNZVWSXMZVWSVWKVWRVYBVWSVVPUXRUXRWDYI + VYCVWSVYBRVYCVWSSUXRYJVYBVWSYKYQVWSVYBYLYMYNYHXPWHYOYRVXHVWRQZVWPVXIVYD + VWLVWIVWOVYDVWJRZVWKRVWLRVYDUXTVYEUFUYBVVPUFUJXLUXSVWJUXQVVPUXRHXOXPVVG + UYCVXGVWRUXPUYRUYCVVFXQXDVYDVXEUWGTZVVPUYBTVXHVYFVWRVXGVYFVVGVXEVVRUVIU + WDVVPXHVYAYSXNVTUWGUWDVVPVUOUJWAZXFXRXSVYDVVPUXRVXHVWRXTYEVWJVWKYAYBUUK + YPYDVWHVWDVXHVXDVXKVWHVWCVXGVVGVWHVVSVXFUVIVWHVVQVXEVVRUXAUWDVVPUULYFYG + YHVWHVXAVXJVWHVWTVXIVWPVWHVWTUWDUWDUNZVXIXMZVXIVWTVWQVXIXMVWHVYIVWQVWRV + WSUUMVWHVWQVYHVXIUXAUWDUWDWDYIUUNVYIVXIVYHRVYIVXISUWDYJVYHVXIYKYQVXIVYH + YLYMYNYHXPWHYOYRVWDVWQQZVWPVWTVYJVWIVWLVWOVYJVWGRZVWHRVWIRVYJUWFVYKUDUW + HUXAUDOXLUWEVWGUWCUXAUWDGXOXPUYTUWIVVFVWCVWQUWBUXNUWIUYSXQXGVWDUXAUWHTZ + VWQVWDVVQUWGTZVYLVWCVYMVVGVVQVVRUVIUXAVVPXHVYAYSXNUXAVVPUWGOWAVYGYSVJVT + XSVYJUXAUWDVWDVWQXTYEVWGVWHYAYBUUPYPYDYCVVTVVOVWDVVKVXCVVTVVNVWCVVGUVOV + VSUVIUUOYHVVTVVJVXBVVTVVJVVSVVHJUQVXBUVOVVSVVHJXOBCDEFGHIJUGUIOUJUKUEKU + UQYNXPWHYOUURUUSUUTUVAUVBUVSVVLUCVVHUVIUVPVVHVHZUVRVVKUBUVIVYNUVQVVJUVP + VVHUVOJUVEXPUVCUVDUVFYTYTYTUVGUVHUAUCUBUVKJVRXR $. $} ${ - sltletrd.4 $e |- ( ph -> A B <_s C ) $. - $( Surreal less than is transitive. (Contributed by Scott Fenton, - 8-Dec-2021.) $) - sltletrd $p |- ( ph -> A + Pred ( U , ( ( A X. B ) X. C ) , <. <. X , Y >. , Z >. ) = + ( ( ( ( Pred ( R , A , X ) u. { X } ) X. + ( Pred ( S , B , Y ) u. { Y } ) ) X. + ( Pred ( T , C , Z ) u. { Z } ) ) \ + { <. <. X , Y >. , Z >. } ) ) $= + ( cxp cop cpred csn wceq wcel wa va vb vc vq vd ve vf cv cun cdif opeq1 + opeq1d predeq3 syl sneq uneq12d xpeq1d sneqd difeq12d eqeq12d opeq2 w3a + xpeq2d cin wbr wrex wb elxpxp wi df-3an weq wo wne wn eldif opelxp elun + w3o cvv vex elpred velsn orbi12i bitri anbi12i df-ne otthne bitr3i opex + elv elsn simpr1 biantrurd orbi1d simpr2 simpr3 3anbi123d bitrdi bitr4id + xchnxbir anbi1d pm3.22 bitr2d bitrid breq1 xpord3lem bibi12d syl5ibrcom + eleq1 sylan2br anassrs rexlimdva rexlimdvva biimtrid pm5.32d ax-mp elin + 3bitr4g eqrdv wss predss snssi unssd 3ad2ant1 3ad2ant2 syl2anc 3ad2ant3 + a1i xpss12 ssdifssd sseqin2 sylib eqtrd vtocl3ga ) CDNZENZIUAUHZUBUHZOZ + UCUHZOZPZCFYQPZYQQZUIZDGYRPZYRQZUIZNZEHYTPZYTQZUIZNZUUAQZUJZRYPIJYROZYT + OZPZCFJPZJQZUIZUUHNZUULNZUUQQZUJZRYPIJKOZYTOZPZUVADGKPZKQZUIZNZUULNZUVG + QZUJZRYPIUVFLOZPZUVLEHLPZLQZUIZNZUVPQZUJZRUAUBUCJKLCDEYQJRZUUBUURUUOUVE + UWDUUAUUQRUUBUURRUWDYSUUPYTYQJYRUKULZYPIUUAUUQUMUNUWDUUMUVCUUNUVDUWDUUI + UVBUULUWDUUEUVAUUHUWDUUCUUSUUDUUTCFYQJUMYQJUOUPUQUQUWDUUAUUQUWEURUSUTYR + KRZUURUVHUVEUVOUWFUUQUVGRUURUVHRUWFUUPUVFYTYRKJVAULZYPIUUQUVGUMUNUWFUVC + UVMUVDUVNUWFUVBUVLUULUWFUUHUVKUVAUWFUUFUVIUUGUVJDGYRKUMYRKUOUPVCUQUWFUU + QUVGUWGURUSUTYTLRZUVHUVQUVOUWCUWHUVGUVPRUVHUVQRYTLUVFVAZYPIUVGUVPUMUNUW + HUVMUWAUVNUWBUWHUULUVTUVLUWHUUJUVRUUKUVSEHYTLUMYTLUOUPVCUWHUVGUVPUWIURU + SUTYQCSZYRDSZYTESZVBZUUBYPUUOVDZUUOUWMUDUUBUWNUWMUDUHZYPSZUWOUUAIVEZTZU + WPUWOUUOSZTUWOUUBSZUWOUWNSUWMUWPUWQUWSUWPUWOUEUHZUFUHZOZUGUHZOZRZUGEVFZ + UFDVFUECVFUWMUWQUWSVGZUEUFUGUWOCDEVHUWMUXGUXHUEUFCDUWMUXACSZUXBDSZTZTUX + FUXHUGEUWMUXKUXDESZUXFUXHVIZUXKUXLTUWMUXIUXJUXLVBZUXMUXIUXJUXLVJUWMUXNT + ZUXHUXFUXNUWMUXAYQFVEZUEUAVKZVLZUXBYRGVEZUFUBVKZVLZUXDYTHVEZUGUCVKZVLZV + BZUXAYQVMUXBYRVMUXDYTVMVRZTZVBZUXEUUOSZVGUYHUXNUWMTZUYGTZUXOUYIUXNUWMUY + GVJUXOUYIUYGUYKUXOUYIUXIUXPTZUXQVLZUXJUXSTZUXTVLZTZUXLUYBTZUYCVLZTZUYFT + ZUYGUYIUXEUUMSZUXEUUNSZVNZTUYTUXEUUMUUNVOVUAUYSVUCUYFVUAUXCUUISZUXDUULS + ZTUYSUXCUXDUUIUULVPVUDUYPVUEUYRVUDUXAUUESZUXBUUHSZTUYPUXAUXBUUEUUHVPVUF + UYMVUGUYOVUFUXAUUCSZUXAUUDSZVLUYMUXAUUCUUDVQVUHUYLVUIUXQVUHUYLVGUACVSFY + QUXAUEVTZWAWJUEYQWBWCWDVUGUXBUUFSZUXBUUGSZVLUYOUXBUUFUUGVQVUKUYNVULUXTV + UKUYNVGUBDVSGYRUXBUFVTZWAWJUFYRWBWCWDWEWDVUEUXDUUJSZUXDUUKSZVLUYRUXDUUJ + UUKVQVUNUYQVUOUYCVUNUYQVGUCEVSHYTUXDUGVTZWAWJUGYTWBWCWDWEWDUXEUUARZUYFV + UBVUQVNUXEUUAVMUYFUXEUUAWFUXAUXBUXDYQYRYTVUJVUMVUPWGWHUXEUUAUXCUXDWIWKW + TWEWDUXOUYEUYSUYFUXOUYEUYMUYOUYRVBUYSUXOUXRUYMUYAUYOUYDUYRUXOUXPUYLUXQU + XOUXIUXPUWMUXIUXJUXLWLWMWNUXOUXSUYNUXTUXOUXJUXSUWMUXIUXJUXLWOWMWNUXOUYB + UYQUYCUXOUXLUYBUWMUXIUXJUXLWPWMWNWQUYMUYOUYRVJWRXAWSUXOUYJUYGUWMUXNXBWM + XCXDUXFUWQUYHUWSUYIUXFUWQUXEUUAIVEUYHUWOUXEUUAIXEABCDEFGHIUBUCUEUFUGUAM + XFWRUWOUXEUUOXIXGXHXJXKXLXMXNXOUUAVSSUWTUWRVGYSYTWIYPVSIUUAUWOUDVTWAXPU + WOYPUUOXQXRXSUWMUUOYPXTUWNUUORUWMUUMYPUUNUWMUUIYOXTZUULEXTZUUMYPXTUWMUU + ECXTZUUHDXTZVURUWJUWKVUTUWLUWJUUCUUDCUUCCXTUWJCFYQYAYHYQCYBYCYDUWKUWJVV + AUWLUWKUUFUUGDUUFDXTUWKDGYRYAYHYRDYBYCYEUUECUUHDYIYFUWLUWJVUSUWKUWLUUJU + UKEUUJEXTUWLEHYTYAYHYTEYBYCYGUUIYOUULEYIYFYJUUOYPYKYLYMYN $. $} ${ - slelttrd.4 $e |- ( ph -> A <_s B ) $. - slelttrd.5 $e |- ( ph -> B A R Se A ) $. + sexp3.2 $e |- ( ph -> S Se B ) $. + sexp3.3 $e |- ( ph -> T Se C ) $. + $( Show that the triple order is set-like. (Contributed by Scott Fenton, + 21-Aug-2024.) $) + sexp3 $p |- ( ph -> U Se ( ( A X. B ) X. C ) ) $= + ( va vb vc cpred cvv wcel vp cxp cv wral wse cop wceq wrex elxpxp wa wi + w3a df-3an csn cdif xpord3pred adantl simpr1 adantr setlikespec syl2anc + cun vsnex a1i unexd simpr2 simpr3 difexd eqeltrd predeq3 eleq1d biimprd + xpexd com12 syl sylan2br anassrs rexlimdva rexlimdvva biimtrid ralrimiv + dfse3 sylibr ) ADEUBFUBZJUAUCZRZSTZUAWDUDWDJUEAWGUAWDWEWDTWEOUCZPUCZUFQ + UCZUFZUGZQFUHZPEUHODUHAWGOPQWEDEFUIAWMWGOPDEAWHDTZWIETZUJZUJWLWGQFAWPWJ + FTZWLWGUKZWPWQUJAWNWOWQULZWRWNWOWQUMAWSUJZWDJWKRZSTZWRWTXADGWHRZWHUNZVB + ZEHWIRZWIUNZVBZUBZFIWJRZWJUNZVBZUBZWKUNZUOZSWSXAXOUGABCDEFGHIJWHWIWJKUP + UQWTXMXNSWTXIXLSSWTXEXHSSWTXCXDSSWTWNDGUEZXCSTAWNWOWQURAXPWSLUSDGWHUTVA + XDSTWTOVCVDVEWTXFXGSSWTWOEHUEZXFSTAWNWOWQVFAXQWSMUSEHWIUTVAXGSTWTPVCVDV + EVMWTXJXKSSWTWQFIUEZXJSTAWNWOWQVGAXRWSNUSFIWJUTVAXKSTWTQVCVDVEVMVHVIWLX + BWGWLWGXBWLWFXASWDJWEWKVJVKVLVNVOVPVQVRVSVTWAUAWDJWBWC $. $} ${ - sletrd.4 $e |- ( ph -> A <_s B ) $. - sletrd.5 $e |- ( ph -> B <_s C ) $. - $( Surreal less than or equal is transitive. (Contributed by Scott - Fenton, 8-Dec-2021.) $) - sletrd $p |- ( ph -> A <_s C ) $= - ( csle wbr csur wcel wa wi sletr syl3anc mp2and ) ABCJKZCDJKZBDJKZHIABL - MCLMDLMSTNUAOEFGBCDPQR $. + $d A a $. $d a b $. $d A b $. $d a c $. $d A c $. $d a d $. + $d A d $. $d a e $. $d A e $. $d a f $. $d A f $. $d a ps $. + $d a rh $. $d a th $. $d A x $. $d A y $. $d B a $. $d B b $. + $d b c $. $d B c $. $d b ch $. $d b d $. $d B d $. $d b e $. + $d B e $. $d b f $. $d B f $. $d b mu $. $d b th $. $d B x $. + $d B y $. $d C a $. $d C b $. $d C c $. $d c d $. $d C d $. + $d c e $. $d C e $. $d c f $. $d C f $. $d c la $. $d c th $. + $d C x $. $d C y $. $d ch f $. $d d e $. $d d f $. $d d ph $. + $d d ta $. $d e et $. $d e f $. $d e ps $. $d e ze $. $d f si $. + $d R d $. $d R e $. $d R f $. $d R x $. $d R y $. $d S d $. + $d S e $. $d S f $. $d S x $. $d S y $. $d T d $. $d T e $. + $d T f $. $d T x $. $d T y $. $d U a $. $d U b $. $d U c $. + $d U d $. $d U e $. $d U f $. $d X a $. $d X b $. $d X c $. + $d x y $. $d Y b $. $d Y c $. $d Z c $. + xpord3ind.1 $e |- R Fr A $. + xpord3ind.2 $e |- R Po A $. + xpord3ind.3 $e |- R Se A $. + xpord3ind.4 $e |- S Fr B $. + xpord3ind.5 $e |- S Po B $. + xpord3ind.6 $e |- S Se B $. + xpord3ind.7 $e |- T Fr C $. + xpord3ind.8 $e |- T Po C $. + xpord3ind.9 $e |- T Se C $. + xpord3ind.10 $e |- ( a = d -> ( ph <-> ps ) ) $. + xpord3ind.11 $e |- ( b = e -> ( ps <-> ch ) ) $. + xpord3ind.12 $e |- ( c = f -> ( ch <-> th ) ) $. + xpord3ind.13 $e |- ( a = d -> ( ta <-> th ) ) $. + xpord3ind.14 $e |- ( b = e -> ( et <-> ta ) ) $. + xpord3ind.15 $e |- ( b = e -> ( ze <-> th ) ) $. + xpord3ind.16 $e |- ( c = f -> ( si <-> ta ) ) $. + xpord3ind.17 $e |- ( a = X -> ( ph <-> rh ) ) $. + xpord3ind.18 $e |- ( b = Y -> ( rh <-> mu ) ) $. + xpord3ind.19 $e |- ( c = Z -> ( mu <-> la ) ) $. + xpord3ind.i $e |- ( ( a e. A /\ b e. B /\ c e. C ) -> + ( ( ( A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b ) + A. f e. Pred ( T , C , c ) th /\ + A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b ) ch /\ + A. d e. Pred ( R , A , a ) A. f e. Pred ( T , C , c ) ze ) + /\ + ( A. d e. Pred ( R , A , a ) ps /\ + A. e e. Pred ( S , B , b ) A. f e. Pred ( T , C , c ) ta /\ + A. e e. Pred ( S , B , b ) si ) /\ + A. f e. Pred ( T , C , c ) et ) + -> ph ) ) $. + $( Induction over the triple cross product ordering. Note that the + substitutions cover all possible cases of membership in the + predecessor class. (Contributed by Scott Fenton, 4-Sep-2024.) $) + xpord3ind $p |- ( ( X e. A /\ Y e. B /\ Z e. C ) -> la ) $= + ( cxp wfr wpo wse w3a wcel wtru a1i frxp3 mptru poxp3 sexp3 3pm3.2i cop + cv cpred wal wne w3o csn cun wral cdif xpord3pred eleq2d imbi1d eldifsn + wi ot2elxp vex otthne anbi12i bitri imbi1i impexp bitr2i bitr4di albidv + wa 2albidv wss ssun1 ssralv ax-mp ralimi syl wn predpoirr eleq1w mtbiri + weq necon2ai 3mix3d pm2.27 ralimia ssun2 biidd 3orbi123d bicomd equcoms + neeq1 wb imbi12d ralsn sylib 3mix2d ralbidv ralbii 3jca 2ralbidv equcom + r3al 2ralimi 3syl 2ralbii bicom 3imtr3i 3mix1d syl5 sylbid frpoins3xp3g + mpan ) NOVKPVKZTVLZUUMTVMZUUMTVNZVOUCNVPUDOVPUEPVPVOKUUNUUOUUPUUNVQLMNO + PQRSTUJNQVLVQUKVRORVLVQUNVRPSVLVQUQVRVSVTUUOVQLMNOPQRSTUJNQVMZVQULVRORV + MZVQUOVRPSVMZVQURVRWAVTUUPVQLMNOPQRSTUJNQVNVQUMVRORVNVQUPVRPSVNVQUSVRWB + VTWCABCDIJKUFUGUHUIUBUANOPTUCUDUEUFWEZNVPUGWEZOVPUHWEZPVPVOZUIWEZUAWEZW + DUBWEZWDZUUMTUUTUVAWDUVBWDZWFZVPZDWRZUBWGZUAWGUIWGZUVDUUTWHZUVEUVAWHZUV + FUVBWHZWIZDWRZUBPSUVBWFZUVBWJZWKZWLZUAORUVAWFZUVAWJZWKZWLZUINQUUTWFZUUT + WJZWKZWLZAUVCUVMUVDUWIVPUVEUWEVPUVFUWAVPVOZUVRWRZUBWGZUAWGUIWGUWJUVCUVL + UWMUIUAUVCUVKUWLUBUVCUVKUVGUWIUWEVKUWAVKZUVHWJWMZVPZDWRZUWLUVCUVJUWPDUV + CUVIUWOUVGLMNOPQRSTUUTUVAUVBUJWNWOWPUWQUWKUVQXIZDWRUWLUWPUWRDUWPUVGUWNV + PZUVGUVHWHZXIUWRUVGUWNUVHWQUWSUWKUWTUVQUVDUVEUVFUWIUWEUWAWSUVDUVEUVFUUT + UVAUVBUIWTUAWTUBWTXAXBXCXDUWKUVQDXEXFXGXHXJUVRUIUAUBUWIUWEUWAUUBXGUWJDU + BUVSWLZUAUWCWLUIUWGWLZCUAUWCWLZUIUWGWLZGUBUVSWLZUIUWGWLZVOZBUIUWGWLZEUB + UVSWLZUAUWCWLZHUAUWCWLZVOZFUBUVSWLZVOUVCAUWJUXGUXLUXMUWJUXBUXDUXFUWJUWF + UIUWGWLZUVRUBUVSWLZUAUWCWLZUIUWGWLUXBUWGUWIXKUWJUXNWRUWGUWHXLUWFUIUWGUW + IXMXNZUWFUXPUIUWGUWFUWBUAUWCWLZUXPUWCUWEXKUWFUXRWRUWCUWDXLUWBUAUWCUWEXM + XNZUWBUXOUAUWCUVSUWAXKUWBUXOWRUVSUVTXLUVRUBUVSUWAXMXNZXOXPZXOUXOUXAUIUA + UWGUWCUVRDUBUVSUVFUVSVPZUVQUVRDWRUYBUVPUVNUVOUYBUVFUVBUBUHYAZUYBUVBUVSV + PZUUSUYDXQURPSUVBXRXNUBUHUVSXSXTYBZYCUVQDYDXPYEUUCUUDUWJUVNUVOUVBUVBWHZ + WIZCWRZUAUWCWLZUIUWGWLZUXDUWJUVRUBUVTWLZUAUWCWLZUIUWGWLZUYJUWJUXNUYMUXQ + UWFUYLUIUWGUWFUXRUYLUXSUWBUYKUAUWCUVTUWAXKUWBUYKWRUVTUVSYFUVRUBUVTUWAXM + XNZXOXPZXOXPUYKUYHUIUAUWGUWCUVRUYHUBUVBUHWTZUYCUVQUYGDCUYCUVNUVNUVOUVOU + VPUYFUYCUVNYGUYCUVOYGZUVFUVBUVBYKZYHDCYLUHUBUHUBYAZCDVBYIYJYMYNZUUEYOUY + IUXCUIUWGUYHCUAUWCUVEUWCVPZUYGUYHCWRVUAUVOUVNUYFVUAUVEUVAUAUGYAZVUAUVAU + WCVPZUURVUCXQUOORUVAXRXNUAUGUWCXSXTYBZYPUYGCYDXPYEXOXPUWJUVNUVAUVAWHZUV + PWIZGWRZUBUVSWLZUIUWGWLZUXFUWJUXOUAUWDWLZUIUWGWLZVUIUWJUXNVUKUXQUWFVUJU + IUWGUWFUWBUAUWDWLZVUJUWDUWEXKUWFVULWRUWDUWCYFUWBUAUWDUWEXMXNZUWBUXOUAUW + DUXTXOXPZXOXPVUJVUHUIUWGUXOVUHUAUVAUGWTZVUBUVRVUGUBUVSVUBUVQVUFDGVUBUVN + UVNUVOVUEUVPUVPVUBUVNYGZUVEUVAUVAYKZVUBUVPYGZYHUGUAYAZGDYLVUBDGYLVEUGUA + UUAGDUUFUUGYMYQYNYRYOVUHUXEUIUWGVUGGUBUVSUYBVUFVUGGWRUYBUVPUVNVUEUYEYCV + UFGYDXPYEXOXPYSUWJUXHUXJUXKUWJUVNVUEUYFWIZBWRZUIUWGWLZUXHUWJUYKUAUWDWLZ + UIUWGWLZVVBUWJUXNVVDUXQUWFVVCUIUWGUWFVULVVCVUMUWBUYKUAUWDUYNXOXPXOXPVVC + VVAUIUWGVVCUYHUAUWDWLVVAUYKUYHUAUWDUYTYRUYHVVAUAUVAVUOVUBUYGVUTCBVUBUVN + UVNUVOVUEUYFUYFVUPVUQVUBUYFYGYHCBYLUGUAVUSBCVAYIYJYMYNXCYRYOVVABUIUWGUV + DUWGVPZVUTVVABWRVVEUVNVUEUYFVVEUVDUUTUIUFYAZVVEUUTUWGVPZUUQVVGXQULNQUUT + XRXNUIUFUWGXSXTYBUUHVUTBYDXPYEXPUWJUUTUUTWHZUVOUVPWIZEWRZUBUVSWLZUAUWCW + LZUXJUWJUXPUIUWHWLZVVLUWJUWFUIUWHWLZVVMUWHUWIXKUWJVVNWRUWHUWGYFUWFUIUWH + UWIXMXNZUWFUXPUIUWHUYAXOXPUXPVVLUIUUTUFWTZVVFUVRVVJUAUBUWCUVSVVFUVQVVID + EVVFUVNVVHUVOUVOUVPUVPUVDUUTUUTYKVVFUVOYGVVFUVPYGYHVVFEDEDYLUFUIVCYJYIY + MZYTYNYOVVKUXIUAUWCVVJEUBUVSUYBVVIVVJEWRUYBUVPVVHUVOUYEYCVVIEYDXPYEXOXP + UWJVVHUVOUYFWIZHWRZUAUWCWLZUXKUWJUYLUIUWHWLZVVTUWJVVNVWAVVOUWFUYLUIUWHU + YOXOXPVWAVVJUBUVTWLZUAUWCWLZVVTUYLVWCUIUUTVVPVVFUVRVVJUAUBUWCUVTVVQYTYN + VWBVVSUAUWCVVJVVSUBUVBUYPUYCVVIVVREHUYCVVHVVHUVOUVOUVPUYFUYCVVHYGUYQUYR + YHEHYLUHUBUYSHEVFYIYJYMYNYRXCYOVVSHUAUWCVUAVVRVVSHWRVUAUVOVVHUYFVUDYPVV + RHYDXPYEXPYSUWJVVHVUEUVPWIZFWRZUBUVSWLZUXMUWJVUJUIUWHWLZVWFUWJVVNVWGVVO + UWFVUJUIUWHVUNXOXPVWGVVKUAUWDWLZVWFVUJVWHUIUUTVVPVVFUVRVVJUAUBUWDUVSVVQ + YTYNVVKVWFUAUVAVUOVUBVVJVWEUBUVSVUBVVIVWDEFVUBVVHVVHUVOVUEUVPUVPVUBVVHY + GVUQVURYHVUBFEFEYLUGUAVDYJYIYMYQYNXCYOVWEFUBUVSUYBVWDVWEFWRUYBUVPVVHVUE + UYEYCVWDFYDXPYEXPYSVJUUIUUJUTVAVBVGVHVIUUKUUL $. $} $} - $( Surreal less than or equal is reflexive. Theorem 0(iii) of [Conway] - p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) $) - slerflex $p |- ( A e. No -> A <_s A ) $= - ( csur wcel csle wbr cslt wn sltirr wb slenlt anidms mpbird ) ABCZAADEZAAFE - GZAHMNOIAAJKL $. - - $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal numbers - birthday theorems + Well-founded zero, successor, and limits =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) - $( The birthday function is a function. (Contributed by Scott Fenton, - 14-Jun-2011.) $) - bdayfun $p |- Fun bday $= - ( csur con0 cbday wfo wfun bdayfo fofun ax-mp ) ABCDCEFABCGH $. - - $( The birthday function is a function over ` No ` . (Contributed by Scott - Fenton, 30-Jun-2011.) $) - bdayfn $p |- bday Fn No $= - ( csur con0 cbday wfo wfn bdayfo fofn ax-mp ) ABCDCAEFABCGH $. - - $( The birthday function's domain is ` No ` . (Contributed by Scott Fenton, - 14-Jun-2011.) $) - bdaydm $p |- dom bday = No $= - ( csur con0 cbday wfo wf bdayfo fof ax-mp fdmi ) ABCABCDABCEFABCGHI $. + $c wsuc WLim $. - $( The birthday function's range is ` On ` . (Contributed by Scott Fenton, - 14-Jun-2011.) $) - bdayrn $p |- ran bday = On $= - ( csur con0 cbday wfo crn wceq bdayfo forn ax-mp ) ABCDCEBFGABCHI $. + $( Declare the syntax for well-founded successor. $) + cwsuc $a class wsuc ( R , A , X ) $. - $( The value of the birthday function is always an ordinal. (Contributed by - Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, - 8-Dec-2021.) $) - bdayelon $p |- ( bday ` A ) e. On $= - ( csur con0 cbday wfo wf bdayfo fof ax-mp 0elon f0cli ) BCADBCDEBCDFGBCDHIJ - K $. + $( Declare the syntax for well-founded limit class. $) + cwlim $a class WLim ( R , A ) $. - ${ - $d A w x y z $. $d X w x y z $. $d Y w y z $. - $( Lemma for ~ nocvxmin . Given two birthday-minimal elements of a convex - class of surreals, they are not comparable. (Contributed by Scott - Fenton, 30-Jun-2011.) $) - nocvxminlem $p |- ( ( A C_ No /\ A. x e. A A. y e. A - A. z e. No ( ( x z e. A ) ) -> - ( ( ( X e. A /\ Y e. A ) /\ - ( ( bday ` X ) = |^| ( bday " A ) /\ - ( bday ` Y ) = |^| ( bday " A ) ) ) -> - -. X z e. A ) ) -> - E! w e. A ( bday ` w ) = |^| ( bday " A ) ) $= - ( vt c0 wne csur wss cv cslt wbr wa wcel wi wral cbday cfv imp w3a bdayrn - cima cint wceq wrex weq wreu con0 crn imassrn sseqtri wex cdm bdaydm wfun - sseq2i bdayfun funfvima2 mpan sylbir elex2 syl6 exlimdv n0 3imtr4g impcom - onint sylancr wb wfn bdayfn fvelimab adantl mpbid 3adant3 wn ssel anim12d - ad2ant2r nocvxminlem ancom anbi12i syl5bi slttrieq2 biimpar exp32 3adant1 - syl12anc ralrimivv fveqeq2 reu4 sylanbrc ) EGHZEIJZAKZCKZLMWQBKLMNWQEOPCI - QBEQAEQZUADKZRSREUCZUDZUEZDEUFZXBFKZRSXAUEZNZDFUGZPZFEQDEQZXBDEUHWNWOXCWR - WNWONZXAWTOZXCXJWTUIJWTGHZXKWTRUJUIREUKUBULWOWNXLWOWPEOZAUMWSWTODUMZWNXLW - OXMXNAWOXMWPRSZWTOZXNWOERUNZJZXMXPPZXQIEUOUQRUPXRXSUREWPRUSUTVADXOWTVBVCV - DAEVEDWTVEVFVGWTVHVIWOXKXCVJZWNRIVKWOXTVLDIEXARVMUTVNVOVPWOWRXIWNWOWRNZXH - DFEEYAWSEOZXDEOZNZXFXGYAYDXFNZNWSIOZXDIOZNZWSXDLMVQZXDWSLMVQZXGWOYDYHWRXF - WOYDYHWOYBYFYCYGEIWSVREIXDVRVSTVTYAYEYIABCEWSXDWATYAYEYJYEYCYBNZXEXBNZNYA - YJYDYKXFYLYBYCWBXBXEWBWCABCEXDWSWAWDTYHXGYIYJNWSXDWEWFWIWGWJWHXBXEDFEWSXD - XARWKWLWM $. + $d R x $. $d A x $. + $( Define the class of limit points of a well-founded set. (Contributed by + Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) $) + df-wlim $a |- WLim ( R , A ) = + { x e. A | ( x =/= inf ( A , A , R ) /\ + x = sup ( Pred ( R , A , x ) , A , R ) ) } $. $} - $( The surreal numbers are a proper class. (Contributed by Scott Fenton, - 16-Jun-2011.) $) - noprc $p |- -. No e. _V $= - ( csur cvv wcel con0 onprc cbday wfo bdayfo focdmex mpi mto ) ABCZDBCZELADF - GMHADBFIJK $. - + $( Equality theorem for well-founded successor. (Contributed by Scott + Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) + wsuceq123 $p |- ( ( R = S /\ A = B /\ X = Y ) -> + wsuc ( R , A , X ) = wsuc ( S , B , Y ) ) $= + ( wceq w3a ccnv cpred cwsuc simp1 cnveqd predeq123 syld3an1 simp2 infeq123d + cinf df-wsuc 3eqtr4g ) CDGZABGZEFGZHZACIZEJZACRBDIZFJZBDRACEKBDFKUDUFACUHBD + UEUGGUBUAUCUFUHGUDCDUAUBUCLZMABUEUGEFNOUAUBUCPUIQACESBDFST $. -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal numbers: Conway cuts -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) + $( Equality theorem for well-founded successor. (Contributed by Scott + Fenton, 13-Jun-2018.) $) + wsuceq1 $p |- ( R = S -> wsuc ( R , A , X ) = wsuc ( S , A , X ) ) $= + ( wceq cwsuc eqid wsuceq123 mp3an23 ) BCEAAEDDEABDFACDFEAGDGAABCDDHI $. - $( Declare new constant symbols. $) - $c < wsuc ( R , A , X ) = wsuc ( R , B , X ) ) $= + ( wceq cwsuc eqid wsuceq123 mp3an13 ) CCEABEDDEACDFBCDFECGDGABCCDDHI $. - $( Declare the syntax for surreal set less than. $) - csslt $a class <. | ( a C_ No /\ b C_ No /\ - A. x e. a A. y e. b x wsuc ( R , A , X ) = wsuc ( R , A , Y ) ) $= + ( wceq cwsuc eqid wsuceq123 mp3an12 ) BBEAAECDEABCFABDFEBGAGAABBCDHI $. - $( Declare the syntax for the surreal cut operator. $) - cscut $a class |s $. ${ - $d a b x y $. - $( Define the cut operator on surreal numbers. This operator, which Conway - takes as the primitive operator over surreals, picks the surreal lying - between two sets of surreals of minimal birthday. Definition from - [Gonshor] p. 7. (Contributed by Scott Fenton, 7-Dec-2021.) $) - df-scut $a |- |s = ( a e. ~P No , b e. ( < - ( iota_ x e. { y e. No | ( a < - E. z e. No ( A. x e. A x + WLim ( R , A ) = WLim ( S , B ) ) $= + ( vx wceq wa cv cinf wne cpred csup crab cwlim simpr infeq123d neeq2d weq + simpl df-wlim predeq123 mp3an3 supeq123d eqeq2d anbi12d rabeqbidv 3eqtr4g + equid ) CDFZABFZGZEHZAACIZJZULACULKZACLZFZGZEAMULBBDIZJZULBDULKZBDLZFZGZE + BMACNBDNUKURVDEABUIUJOZUKUNUTUQVCUKUMUSULUKAACBBDVEVEUIUJSZPQUKUPVBULUKUO + ACVABDUIUJEERUOVAFEUHABCDULULUAUBVEVFUCUDUEUFEACTEBDTUG $. $} - ${ - $d A a b x y $. $d B a b x y $. - $( Binary relation form of the surreal set less-than relation. - (Contributed by Scott Fenton, 8-Dec-2021.) $) - brsslt $p |- ( A < ( ( A e. _V /\ B e. _V ) /\ - ( A C_ No /\ B C_ No /\ A. x e. A A. y e. B x A e. _V ) $= - ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simpll sylbi - cv ) ABEFAGHZBGHZIAJKBJKCRDRLFDBMCAMNZISCDABOSTUAPQ $. - - $( The second argument of surreal set less than exists. (Contributed by - Scott Fenton, 8-Dec-2021.) $) - ssltex2 $p |- ( A < B e. _V ) $= - ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simplr sylbi - cv ) ABEFAGHZBGHZIAJKBJKCRDRLFDBMCAMNZITCDABOSTUAPQ $. - - $( The first argument of surreal set is a set of surreals. (Contributed by - Scott Fenton, 8-Dec-2021.) $) - ssltss1 $p |- ( A < A C_ No ) $= - ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simpr1 sylbi - cv ) ABEFAGHBGHIZAJKZBJKZCRDRLFDBMCAMZNITCDABOSTUAUBPQ $. + $( Equality theorem for the limit class. (Contributed by Scott Fenton, + 15-Jun-2018.) $) + wlimeq1 $p |- ( R = S -> WLim ( R , A ) = WLim ( S , A ) ) $= + ( wceq cwlim eqid wlimeq12 mpan2 ) BCDAADABEACEDAFAABCGH $. - $( The second argument of surreal set is a set of surreals. (Contributed - by Scott Fenton, 8-Dec-2021.) $) - ssltss2 $p |- ( A < B C_ No ) $= - ( vx vy csslt wbr cvv wcel wa csur wss cslt wral w3a brsslt simpr2 sylbi - cv ) ABEFAGHBGHIZAJKZBJKZCRDRLFDBMCAMZNIUACDABOSTUAUBPQ $. + $( Equality theorem for the limit class. (Contributed by Scott Fenton, + 15-Jun-2018.) $) + wlimeq2 $p |- ( A = B -> WLim ( R , A ) = WLim ( R , B ) ) $= + ( wceq cwlim eqid wlimeq12 mpan ) CCDABDACEBCEDCFABCCGH $. - $( The separation property of surreal set less than. (Contributed by Scott - Fenton, 8-Dec-2021.) $) - ssltsep $p |- ( A < A. x e. A A. y e. B x A e. V ) $. - ssltd.2 $e |- ( ph -> B e. W ) $. - ssltd.3 $e |- ( ph -> A C_ No ) $. - ssltd.4 $e |- ( ph -> B C_ No ) $. - ssltd.5 $e |- ( ( ph /\ x e. A /\ y e. B ) -> x A < ( X e. A /\ + X =/= inf ( A , A , R ) /\ + X = sup ( Pred ( R , A , X ) , A , R ) ) ) $= + ( vx cwlim wcel cinf wne cpred csup wceq wa neeq1 predeq3 supeq1d eqeq12d + w3a cv id anbi12d df-wlim elrab2 3anass bitr4i ) CABEZFCAFZCAABGZHZCABCIZ + ABJZKZLZLUFUHUKQDRZUGHZUMABUMIZABJZKZLULDCAUEUMCKZUNUHUQUKUMCUGMURUMCUPUJ + URSURAUOUIBABUMCNOPTDABUAUBUFUHUKUCUD $. $} ${ - $d A x y $. $d B x y $. $d X x y $. $d Y x y $. - $( Two elements of separated sets obey less than. (Contributed by Scott - Fenton, 20-Aug-2024.) $) - ssltsepc $p |- ( ( A < X + inf ( A , A , R ) e. A ) $= + ( vx vy vz wwe wse c0 wne w3a wor cv wrex wbr wn wral wi wa wcel wal weso + 3ad2ant1 cpred wceq wss simp1 simp2 ssidd simp3 tz6.26 syl22anc wb elpred + cvv vex elv notbii imnan bitr4i pm2.27 ad2antll rspcev ex ad2antrl jctird + breq1 biimtrid com23 alimdv eq0 r19.26 df-ral bitr3i 3imtr4g reximdva mpd + expr infcl ) ABFZABGZAHIZJZCDEAABVSVTABKWAABUAUBWBABCLZUCZHUDZCAMZDLZWCBN + ZOZDAPWCWGBNZELZWGBNZEAMZQZDAPRZCAMWBVSVTAAUEWAWFVSVTWAUFVSVTWAUGWBAUHVSV + TWAUICAABUJUKWBWEWOCAWBWCASZRZWGWDSZOZDTWGASZWIWNRZQZDTZWEWOWQWSXBDWQWTWS + XAWBWPWTWSXAQWSWTWIQZWBWPWTRRZXAWSWTWHRZOXDWRXFWRXFULCAUNBWCWGDUOUMUPUQWT + WHURUSXEXDWIWNWTXDWIQWBWPWTWIUTVAWPWNWBWTWPWJWMWLWJEWCAWKWCWGBVFVBVCVDVEV + GVQVHVIDWDVJWOXADAPXCWIWNDAVKXADAVLVMVNVOVPVR $. $} ${ - ssltsepcd.1 $e |- ( ph -> A < X e. A ) $. - ssltsepcd.3 $e |- ( ph -> Y e. B ) $. - $( Two elements of separated sets obey less than. Deduction form of - ~ ssltsepc . (Contributed by Scott Fenton, 25-Sep-2024.) $) - ssltsepcd $p |- ( ph -> X R We A ) $. + wsuclem.2 $e |- ( ph -> R Se A ) $. + wsuclem.3 $e |- ( ph -> X e. V ) $. + wsuclem.4 $e |- ( ph -> E. w e. A X R w ) $. + $( Lemma for the supremum properties of well-founded successor. + (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, + 10-Oct-2021.) $) + wsuclem $p |- ( ph -> E. x e. A + ( A. y e. Pred ( `' R , A , X ) -. y R x /\ + A. y e. A ( x R y -> E. z e. Pred ( `' R , A , X ) z R y ) ) ) $= + ( cv c0 wrex wbr wa wcel cvv ccnv cpred wceq wn wi wwe wse wss wne predss + wral a1i crab dfpred3g syl elexd wb brcnvg ancoms rexbidva bitrid biimpar + rabn0 syl2anc eqnetrd tz6.26 syl22anc breq1 rexrab w3a noel simp2r eleq2d + rexeqdv mtbiri simp3 elpredg mtbid 3expa ralrimiva simp1rl simp1rr adantr + vex expr 3ad2ant1 elpred mpbir2and rspcev 3expia anim12d ancomsd reximdva + biimtrid sylbid mpd ) AFGUAZIUBZGBNZUBZOUCZBWRPZCNZWSGQZUDZCWRUKZWSXCGQZD + NZXCGQZDWRPZUEZCFUKZRZBFPZAFGUFFGUGWRFUHZWROUIXBJKXOAFWQIUJULAWRENZIWQQZE + FUMZOAIHSZWRXRUCLEFWQHIUNUOAITSZIXPGQZEFPZXROUIZAIHLUPMXTYCYBYCXQEFPXTYBX + QEFVCXTXQYAEFXPFSXTXQYAUQXPIFTGURUSUTVAVBVDVEBFWRGVFVGAXBXABXCIWQQZCFUMZP + ZXNAXABWRYEAXSWRYEUCLCFWQHIUNUOVNYFWSIWQQZXARZBFPAXNYDYGXABCFXCWSIWQVHVIA + YHXMBFAWSFSZRZXAYGXMYJXAXFYGXLAYIXAXFAYIXARZRXECWRAYKXCWRSZXEAYKYLVJZXCWT + SZXDYMYNXCOSXCVKYMWTOXCAYIXAYLVLVMVOYMWSTSZYLYNXDUQYOYMBWDZULAYKYLVPWRTGW + SXCVQVDVRVSVTWEAYIYGXLAYIYGRZRZXKCFYRXCFSZXGXJYRYSXGVJZWSWRSZXGXJYTUUAYIY + GYIYGAYSXGWAYIYGAYSXGWBYTXSUUAYQUQYRYSXSXGAXSYQLWCWFFHWQIWSYPWGUOWHYRYSXG + VPXIXGDWSWRXHWSXCGVHWIVDWJVTWEWKWLWMWNWOWP $. $} ${ - $d A x y $. $d B x y $. $d C x y $. - $( Relationship between surreal set less than and subset. (Contributed by - Scott Fenton, 9-Dec-2021.) $) - sssslt1 $p |- ( ( A < C < A < R Or A ) $. + $( Existence theorem for well-founded successor. (Contributed by Scott + Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) + wsucex $p |- ( ph -> wsuc ( R , A , X ) e. _V ) $= + ( cwsuc ccnv cpred cinf cvv df-wsuc infexd eqeltrid ) ABCDFBCGDHZBCIJBCDK + ABNCELM $. $} ${ - $d A x y $. - $( The empty set is less than any set of surreals. (Contributed by Scott - Fenton, 8-Dec-2021.) $) - nulsslt $p |- ( A e. ~P No -> (/) < A < R We A ) $. + wsuccl.2 $e |- ( ph -> R Se A ) $. + wsuccl.3 $e |- ( ph -> X e. V ) $. + wsuccl.4 $e |- ( ph -> E. y e. A X R y ) $. + $( If ` X ` is a set with an ` R ` successor in ` A ` , then its + well-founded successor is a member of ` A ` . (Contributed by Scott + Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) $) + wsuccl $p |- ( ph -> wsuc ( R , A , X ) e. A ) $= + ( va vb vc cwsuc ccnv cpred cinf df-wsuc wwe wor weso syl infcl eqeltrid + wsuclem ) ACDFNCDOFPZCDQCCDFRAKLMCUFDACDSCDTGCDUAUBAKLMBCDEFGHIJUEUCUD $. $} ${ - $d A x y p q r $. $d B x y p q r $. - $( Conway's Simplicity Theorem. Given ` A ` preceeding ` B ` , there is a - unique surreal of minimal length separating them. This is a fundamental - property of surreals and will be used (via surreal cuts) to prove many - properties later on. Theorem from [Alling] p. 185. (Contributed by - Scott Fenton, 8-Dec-2021.) $) - conway $p |- ( A < - E! x e. { y e. No | ( A < R We A ) $. + wsuclb.2 $e |- ( ph -> R Se A ) $. + wsuclb.3 $e |- ( ph -> X e. V ) $. + wsuclb.4 $e |- ( ph -> Y e. A ) $. + wsuclb.5 $e |- ( ph -> X R Y ) $. + $( A well-founded successor is a lower bound on points after ` X ` . + (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, + 10-Oct-2021.) $) + wsuclb $p |- ( ph -> -. Y R wsuc ( R , A , X ) ) $= + ( va vb vc vy wbr wcel wb syl2anc mpbird ccnv cpred cinf cwsuc wn elpredg + brcnvg wwe wor weso syl cv wrex breq2 rspcev wsuclem inflb df-wsuc breq2i + mpd sylnibr ) AFBCUAZEUBZBCUCZCPZFBCEUDZCPAFVCQZVEUEAVGFEVBPZAVHEFCPZKAFB + QZEDQZVHVIRJIFEBDCUGSTAVKVJVGVHRIJBDVBEFUFSTALMNBVCFCABCUHBCUIGBCUJUKALMN + OBCDEGHIAVJVIEOULZCPZOBUMJKVMVIOFBVLFECUNUOSUPUQUTVFVDFCBCEURUSVA $. $} ${ - $d A a b x y $. $d B a b x y $. - $( The value of the surreal cut operation. (Contributed by Scott Fenton, - 8-Dec-2021.) $) - scutval $p |- ( A < ( A |s B ) = - ( iota_ x e. { y e. No | ( A < - ( ( A |s B ) e. No /\ A < ( A |s B ) e. No ) $= - ( csslt wbr cscut co csur wcel csn scutcut simp1d ) ABCDABEFZGHALIZCDMBCD - ABJK $. + $d R x $. $d A x $. + $( The class of limit points is a subclass of the base class. (Contributed + by Scott Fenton, 16-Jun-2018.) $) + wlimss $p |- WLim ( R , A ) C_ A $= + ( vx cv cinf wne cpred csup wceq wa cwlim df-wlim ssrab3 ) CDZAABEFNABNGA + BHIJCAABKCABLM $. + $} - ${ - scutcld.1 $e |- ( ph -> A < ( A |s B ) e. No ) $= - ( csslt wbr cscut co csur wcel scutcl syl ) ABCEFBCGHIJDBCKL $. - $} - $( The birthday of the surreal cut is equal to the minimum birthday in the - gap. (Contributed by Scott Fenton, 8-Dec-2021.) $) - scutbday $p |- ( A < - ( bday ` ( A |s B ) ) = - |^| ( bday " { x e. No | ( A < ( ( L |s R ) = X <-> - ( L < ( ( L |s R ) = X <-> - ( L < - ( bday ` X ) C_ ( bday ` y ) ) ) ) ) $= - ( vx vz csslt wbr csur wcel wa wceq csn cbday cv wss wi wral wal bitri co - cscut cfv crab cima cint eqscut eqss wb sneq breq2d breq1d anbi12d elrab3 - w3a adantl biimpar wfn bdayfn ssrab2 fnfvima mp3an12 intss1 3syl biantrud - ssint wrex fvelimab mp2an weq rexrab imbi1i r19.23v anbi1ci impexp ralbii - eqcom 3bitr2i albii df-ral ralcom4 3bitr4i sseq2 ceqsalv bitr3di pm5.32da - fvex imbi2d bitrid df-3an 3bitr4g bitrd ) CBGHZDIJZKZCBUBUADLCDMZGHZWPBGH - ZDNUCZNCEOZMZGHZXABGHZKZEIUDZUEZUFZLZUOZWQWRCAOZMZGHZXKBGHZKZWSXJNUCZPZQZ - AIRZUOZEBCDUGWOWQWRKZXHKXTXRKXIXSWOXTXHXRXHWSXGPZXGWSPZKZWOXTKZXRWSXGUHYD - YAYCXRYDYBYAYDDXEJZWSXFJZYBWOYEXTWNYEXTUIWMXDXTEDIWTDLZXBWQXCWRYGXAWPCGWT - DUJZUKYGXAWPBGYHULUMUNUPUQNIURZXEIPZYEYFUSXDEIUTZIXENDVAVBWSXFVCVDVEYAWSF - OZPZFXFRZXRFWSXFVFYNYLXOLZXNYMQZQZFSZAIRZXRYLXFJZYMQZFSYQAIRZFSYNYSUUAUUB - FUUAXNXOYLLZKZAIVGZYMQUUDYMQZAIRUUBYTUUEYMYTUUCAXEVGZUUEYIYJYTUUGUIUSYKAI - XEYLNVHVIXDXNUUCAEIEAVJZXBXLXCXMUUHXAXKCGWTXJUJZUKUUHXAXKBGUUIULUMVKTVLUU - DYMAIVMUUFYQAIUUFYOXNKZYMQYQUUDUUJYMUUCYOXNXOYLVQVNVLYOXNYMVOTVPVRVSYMFXF - VTYQAFIWAWBYRXQAIYPXQFXOXJNWGYOYMXPXNYLXOWSWCWHWDVPTTWEWIWFWQWRXHWJWQWRXR - WJWKWL $. - $} + $( Declare the syntax for natural ordinal addition. See ~ df-nadd . $) + cnadd $a class +no $. ${ - $d A x y z $. $d B x y z $. $d C x y z $. - $( Transitive law for surreal set less than. (Contributed by Scott Fenton, - 9-Dec-2021.) $) - sslttr $p |- ( ( A < A <. | ( x e. ( On X. On ) /\ y e. ( On X. On ) + /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ + ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ + x =/= y ) ) } , ( On X. On ) , + ( z e. _V , a e. _V |-> + |^| { w e. On | ( ( a " ( { ( 1st ` z ) } X. ( 2nd ` z ) ) ) C_ w /\ + ( a " ( ( 1st ` z ) X. { ( 2nd ` z ) } ) ) C_ w ) } ) ) $. $} ${ - $d A x y $. $d B x y $. $d C x y $. - $( Union law for surreal set less than. (Contributed by Scott Fenton, - 9-Dec-2021.) $) - ssltun1 $p |- ( ( A < ( A u. B ) <. | + ( x e. ( On X. On ) /\ y e. ( On X. On ) + /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\ + ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\ + x =/= y ) ) } , ( On X. On ) , G ) $. - $( Union law for surreal set less than. (Contributed by Scott Fenton, - 9-Dec-2021.) $) - ssltun2 $p |- ( ( A < A < ( ( A u. C ) |s ( B u. D ) ) = ( A |s B ) ) $= - ( vx vy vz csslt wbr cscut csn cv cbday wa csur wceq wcel syl syl2anc wss - w3a cun cfv crab cima cint crio simp1 scutcut simp2d simp2 ssltun1 simp3d - co simp3 ssltun2 c0 wne ovex snnz sslttr mp3an3 scutval wral wrex vex weq - elima sneq breq2d breq1d anbi12d rexrab bitri wi wb simplr bdayfn fnbrfvb - wfn mpan simpll1 scutbday simprl ssun1 sssslt1 mpan2 simprr sssslt2 elrab - sylanbrc ssrab2 fnfvima mp3an12 intss1 eqsstrd sseq2 biimpd com12 sylbird - jca impd rexlimdva syl5bi ralrimiv ssint sylibr simp1d eqssd wreu fveqeq2 - ex conway riota2 mpbid eqcomd eqtr4d ) ABHIZCABJUNZKZHIZXTDHIZUAZACUBZBDU - BZJUNZELZMUCMYDFLZKZHIZYIYEHIZNZFOUDZUEZUFZPZEYMUGZXSYCYDYEHIZYFYQPYCYDXT - HIZXTYEHIZYRYCAXTHIZYAYSYCXSOQZUUAXTBHIZYCXRUUBUUAUUCUAXRYAYBUHABUIRZUJXR - YAYBUKACXTULSZYCUUCYBYTYCUUBUUAUUCUUDUMXRYAYBUOXTBDUPSZYSYTXTUQURYRXSABJU - SUTYDXTYEVAVBSZEFYDYEVCRYCYQXSYCXSMUCZYOPZYQXSPZYCUUHYOYCUUHYGTZEYNVDUUHY - OTYCUUKEYNYGYNQZYDGLZKZHIZUUNYEHIZNZUUMYGMIZNZGOVEZYCUUKUULUURGYMVEUUTGYG - MYMEVFVHYLUUQUURGFOFGVGZYJUUOYKUUPUVAYIUUNYDHYHUUMVIZVJUVAYIUUNYEHUVBVKVL - VMVNYCUUSUUKGOYCUUMOQZNZUUQUURUUKUVDUUQUURUUKVOUVDUUQNZUURUUMMUCZYGPZUUKU - VEUVCUVGUURVPZYCUVCUUQVQZMOVTZUVCUVHVROUUMYGMVSWARUVEUUHUVFTZUVGUUKVOUVEU - UHMAYIHIZYIBHIZNZFOUDZUEZUFZUVFUVEXRUUHUVQPXRYAYBUVCUUQWBFABWCRUVEUVFUVPQ - ZUVQUVFTUVEUUMUVOQZUVRUVEUVCAUUNHIZUUNBHIZNZUVSUVIUVEUVTUWAUVEUUOUVTUVDUU - OUUPWDUUOAYDTUVTACWEYDUUNAWFWGRUVEUUPUWAUVDUUOUUPWHUUPBYETUWABDWEUUNYEBWI - WGRXAUVNUWBFUUMOUVAUVLUVTUVMUWAUVAYIUUNAHUVBVJUVAYIUUNBHUVBVKVLWJWKUVJUVO - OTUVSUVRVRUVNFOWLOUVOMUUMWMWNRUVFUVPWORWPUVGUVKUUKUVGUVKUUKUVFYGUUHWQWRWS - RWTXLXBXCXDXEEUUHYNXFXGYCUUHYNQZYOUUHTYCXSYMQZUWCYCUUBYSYTNZUWDYCUUBUUAUU - CUUDXHYCYSYTUUEUUFXAYLUWEFXSOYHXSPZYJYSYKYTUWFYIXTYDHYHXSVIZVJUWFYIXTYEHU - WGVKVLWJWKZUVJYMOTUWDUWCVRYLFOWLOYMMXSWMWNRUUHYNWORXIYCUWDYPEYMXJZUUIUUJV - PUWHYCYRUWIUUGEFYDYEXMRYPUUIEYMXSYGXSYOMXKXNSXOXPXQ $. - $} + ${ + $d x y $. + $( Calculate the value of the double ordinal recursion operator. + (Contributed by Scott Fenton, 3-Sep-2024.) $) + on2recsov $p |- ( ( A e. On /\ B e. On ) -> + ( A F B ) = ( <. A , B >. G + ( F |` ( ( suc A X. suc B ) \ { <. A , B >. } ) ) ) ) $= + ( con0 wcel co cxp cfv cep wceq w3a cpred csn wtru a1i cun cop c1st wbr + wa cv c2nd wne copab cres csuc cdif df-ov opelxp wfr wpo wse eqid frxp2 + wo onfr mptru wwe wor weso sopo mp2b poxp2 epse sexp2 3pm3.2i fpr2 mpan + epweon sylbir eqtrid predon adantr uneq1d df-suc eqtr4di adantl xpeq12d + xpord2pred difeq1d eqtrd reseq2d oveq2d ) CHIZDHIZUDZCDEJZCDUAZEHHKZAUE + ZWMIBUEZWMIWNUBLZWOUBLZMUCWPWQNUSWNUFLZWOUFLZMUCWRWSNUSWNWOUGOOABUHZWLP + ZUIZFJZWLECUJZDUJZKZWLQZUKZUIZFJWJWKWLELZXCCDEULWJWLWMIZXJXCNZCDHHUMWMW + TUNZWMWTUOZWMWTUPZOXKXLXMXNXOXMRABHHMMWTWTUQZHMUNRUTSZXQURVAXNRABHHMMWT + XPHMUOZRHMVBHMVCXRVMHMVDHMVEVFSZXSVGVAXORABHHMMWTXPHMUPRHVHSZXTVIVAVJWM + WTEFWLGVKVLVNVOWJXBXIWLFWJXAXHEWJXAHMCPZCQZTZHMDPZDQZTZKZXGUKXHABHHMMWT + CDXPWCWJYGXFXGWJYCXDYFXEWJYCCYBTXDWJYACYBWHYACNWICVPVQVRCVSVTWJYFDYETXE + WJYDDYEWIYDDNWHDVPWAVRDVSVTWBWDWEWFWGWE $. + $} - ${ - $d a b c x y $. - $( The domain of the surreal cut operation is all separated surreal sets. - (Contributed by Scott Fenton, 8-Dec-2021.) $) - dmscut $p |- dom |s = < No $= - ( va vb vx vy vz csslt csur cscut wf wfn wceq csn cima cbday wbr mpbir2an - cv wrex wcel vex crn wss wfun cdm cpw cfv wa crab cint crio mpofun dmscut - df-scut df-fn cab rnmpo wi elimasn df-br bitr4i co scutval eqeltrrd sylbi - cop scutcl eleq1a syl adantl rexlimivv abssi eqsstri df-f ) FGHIHFJZHUAZG - UBVNHUCHUDFKABGUEZFAQZLMZCQNUFNVQDQLZFOVSBQZFOUGDGUHZMUIKCWAUJZHCDABUMZUK - ULHFUNPVOEQZWBKZBVRRAVPRZEUOGABEVPVRWBHWCUPWFEGWEWDGSZABVPVRVTVRSZWEWGUQZ - VQVPSWHWBGSZWIWHVQVTFOZWJWHVQVTVEFSWKFVQVTATBTURVQVTFUSUTWKVQVTHVAWBGCDVQ - VTVBVQVTVFVCVDWBGWDVGVHVIVJVKVLFGHVMP $. + $d a b $. $d a c $. $d a d $. $d a ps $. $d a ta $. $d a x $. + $d a y $. $d b c $. $d b ch $. $d b d $. $d b et $. $d b x $. + $d b y $. $d c d $. $d c ph $. $d c th $. $d c x $. $d c y $. + $d d ps $. $d d x $. $d d y $. $d X a $. $d X b $. $d x y $. + $d Y b $. + on2ind.1 $e |- ( a = c -> ( ph <-> ps ) ) $. + on2ind.2 $e |- ( b = d -> ( ps <-> ch ) ) $. + on2ind.3 $e |- ( a = c -> ( th <-> ch ) ) $. + on2ind.4 $e |- ( a = X -> ( ph <-> ta ) ) $. + on2ind.5 $e |- ( b = Y -> ( ta <-> et ) ) $. + on2ind.i $e |- ( ( a e. On /\ b e. On ) -> + ( ( A. c e. a A. d e. b ch /\ + A. c e. a ps /\ + A. d e. b th ) -> ph ) ) $. + $( Double induction over ordinal numbers. (Contributed by Scott Fenton, + 26-Aug-2024.) $) + on2ind $p |- ( ( X e. On /\ Y e. On ) -> et ) $= + ( con0 cep vx vy cv cxp wcel c1st cfv wbr wceq wo c2nd wne w3a copab eqid + onfr wwe wor wpo epweon weso sopo mp2b epse wa cpred predon adantr adantl + wral raleqdv raleqbidv 3anbi123d sylbid xpord2ind ) ABCDEFUAUBSSTTUAUCZSS + UDZUEUBUCZVQUEVPUFUGZVRUFUGZTUHVSVTUIUJVPUKUGZVRUKUGZTUHWAWBUIUJVPVRULUMU + MUAUBUNZGHIJKLWCUOUPSTUQSTURSTUSUTSTVASTVBVCZSVDZUPWDWEMNOPQIUCZSUEZJUCZS + UEZVEZCLSTWHVFZVJZKSTWFVFZVJZBKWMVJZDLWKVJZUMCLWHVJZKWFVJZBKWFVJZDLWHVJZU + MAWJWNWRWOWSWPWTWJWLWQKWMWFWGWMWFUIWIWFVGVHZWJCLWKWHWIWKWHUIWGWHVGVIZVKVL + WJBKWMWFXAVKWJDLWKWHXBVKVMRVNVO $. $} ${ - $d A x y z $. $d B x y z $. $d O x y z $. - $( A restatement of ~ noeta using set less than. (Contributed by Scott - Fenton, 10-Aug-2024.) $) - etasslt $p |- ( ( A < - E. x e. No ( A < ( ph <-> ps ) ) $. + on3ind.2 $e |- ( b = e -> ( ps <-> ch ) ) $. + on3ind.3 $e |- ( c = f -> ( ch <-> th ) ) $. + on3ind.4 $e |- ( a = d -> ( ta <-> th ) ) $. + on3ind.5 $e |- ( b = e -> ( et <-> ta ) ) $. + on3ind.6 $e |- ( b = e -> ( ze <-> th ) ) $. + on3ind.7 $e |- ( c = f -> ( si <-> ta ) ) $. + on3ind.8 $e |- ( a = X -> ( ph <-> rh ) ) $. + on3ind.9 $e |- ( b = Y -> ( rh <-> mu ) ) $. + on3ind.10 $e |- ( c = Z -> ( mu <-> la ) ) $. + on3ind.i $e |- ( ( a e. On /\ b e. On /\ c e. On ) -> + ( ( ( A. d e. a A. e e. b A. f e. c th /\ + A. d e. a A. e e. b ch /\ + A. d e. a A. f e. c ze ) + /\ + ( A. d e. a ps /\ + A. e e. b A. f e. c ta /\ + A. e e. b si ) /\ + A. f e. c et ) + -> ph ) ) $. + $( Triple induction over ordinals. (Contributed by Scott Fenton, + 4-Sep-2024.) $) + on3ind $p |- ( ( X e. On /\ Y e. On /\ Z e. On ) -> la ) $= + ( vx vy con0 cep cv cxp wcel c1st cfv wbr wceq wo c2nd w3a wne copab eqid + wa onfr wwe wor wpo epweon weso sopo mp2b epse cpred wral predon 3ad2ant1 + 3ad2ant2 3ad2ant3 raleqdv raleqbidv 3anbi123d sylbid xpord3ind ) ABCDEFGH + IJKULUMUNUNUNUOUOUOULUPZUNUNUQUNUQZURUMUPZWKURWJUSUTZUSUTZWLUSUTZUSUTZUOV + AWNWPVBVCWMVDUTZWOVDUTZUOVAWQWRVBVCWJVDUTZWLVDUTZUOVAWSWTVBVCVEWJWLVFVIVE + ULUMVGZLMNOPQRSTXAVHVJUNUOVKUNUOVLUNUOVMVNUNUOVOUNUOVPVQZUNVRZVJXBXCVJXBX + CUAUBUCUDUEUFUGUHUIUJQUPZUNURZRUPZUNURZSUPZUNURZVEZDMUNUOXHVSZVTZLUNUOXFV + SZVTZTUNUOXDVSZVTZCLXMVTZTXOVTZGMXKVTZTXOVTZVEZBTXOVTZEMXKVTZLXMVTZHLXMVT + ZVEZFMXKVTZVEDMXHVTZLXFVTZTXDVTZCLXFVTZTXDVTZGMXHVTZTXDVTZVEZBTXDVTZEMXHV + TZLXFVTZHLXFVTZVEZFMXHVTZVEAXJYAYOYFYTYGUUAXJXPYJXRYLXTYNXJXNYITXOXDXEXGX + OXDVBXIXDWAWBZXJXLYHLXMXFXGXEXMXFVBXIXFWAWCZXJDMXKXHXIXEXKXHVBXGXHWAWDZWE + WFWFXJXQYKTXOXDUUBXJCLXMXFUUCWEWFXJXSYMTXOXDUUBXJGMXKXHUUDWEWFWGXJYBYPYDY + RYEYSXJBTXOXDUUBWEXJYCYQLXMXFUUCXJEMXKXHUUDWEWFXJHLXMXFUUCWEWGXJFMXKXHUUD + WEWGUKWHWI $. $} ${ - $d A x y z $. $d B x y z $. - $( A version of ~ etasslt with fewer hypotheses but a weaker upper bound. - (Contributed by Scott Fenton, 10-Dec-2021.) $) - etasslt2 $p |- ( A < - E. x e. No ( A < ( bday ` ( A |s B ) ) C_ O ) $= - ( vx vy csslt wbr con0 wcel cbday cun cima wss w3a cv csn cfv csur wa syl - cscut etasslt crab cint wceq simpl1 scutbday wfn bdayfn ssrab2 weq breq2d - co sneq breq1d anbi12d simprl simprr1 simprr2 jca elrabd fnfvima mp3an12i - intss1 eqsstrd simprr3 sstrd rexlimddv ) ABFGZCHIZJABKLCMZNZADOZPZFGZVNBF - GZVMJQZCMZNZABUAUMJQZCMDRDABCUBVLVMRIZVSSZSZVTVQCWCVTJAEOZPZFGZWEBFGZSZER - UCZLZUDZVQWCVIVTWKUEVIVJVKWBUFEABUGTWCVQWJIZWKVQMJRUHWIRMWCVMWIIWLUIWHERU - JWCWHVOVPSEVMREDUKZWFVOWGVPWMWEVNAFWDVMUNZULWMWEVNBFWNUOUPVLWAVSUQWCVOVPV - OVPVRWAVLURVOVPVRWAVLUSUTVARWIJVMVBVCVQWJVDTVEVOVPVRWAVLVFVGVH $. + $d A a $. $d a b $. $d A b $. $d a c $. $d a d $. $d a f $. $d a t $. + $d a x $. $d A x $. $d B b $. $d b c $. $d b d $. $d b f $. $d b t $. + $d b x $. $d B x $. $d c d $. $d c t $. $d c x $. $d d t $. $d d x $. + $d f p $. $d f q $. $d f t $. $d f x $. $d p q $. $d p t $. $d p x $. + $d q t $. $d q x $. $d t x $. + $( Lemma for ordinal addition closure. (Contributed by Scott Fenton, + 26-Aug-2024.) $) + naddcllem $p |- ( ( A e. On /\ B e. On ) -> + ( ( A +no B ) e. On /\ + ( A +no B ) = |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\ + ( +no " ( A X. { B } ) ) C_ x ) } ) ) $= + ( vc vd vt cnadd co con0 wcel cxp cima wss wa wceq imaeq2d sseq1d anbi12d + wral cvv va vb vf vp vq cv csn crab cint oveq1 eleq1d sneq xpeq1d rabbidv + weq xpeq1 inteqd eqeq12d oveq2 xpeq2 xpeq2d w3a simpl ralimi 3ad2ant2 jca + 3ad2ant3 cop csuc cdif cres c1st cfv c2nd cmpo on2recsov adantr opex wfun + df-nadd wfn naddfn fnfun ax-mp vex sucex difexi resfunexg mp2an wn wel wo + xpex word eloni ad2antlr ordirr olcd ianor opelxp xchnxbir sylibr sssucid + sucid snssi xpss12 ssdifsn mpbiran resima2 ad2antrr orcd wrex cuni simprr + syl cun ralbidv ralsn cdm onss syl2an sseqtrrdi funimassov sylancr mpbird + wb fndmi simprl ralbii unssd ssorduni vsnex funimaexg sylib sseq2 eqeltrd + unex sneqd xpeq12d imaeq1 sucelon onsucuni unssad unssbd rspcev onintrab2 + uniex elon syl12anc op1std op2ndd eqid ovmpog mp3an12i 3eqtrd syl5 on2ind + ex ) UAUFZUBUFZGHZIJZUVAGUUSUGZUUTKZLZAUFZMZGUUSUUTUGZKZLZUVFMZNZAIUHZUIZ + OZNZDUFZUUTGHZIJZUVRGUVQUGZUUTKZLZUVFMZGUVQUVHKZLZUVFMZNZAIUHZUIZOZNZUVQE + UFZGHZIJZUWMGUVTUWLKZLZUVFMZGUVQUWLUGZKZLZUVFMZNZAIUHZUIZOZNZUUSUWLGHZIJZ + UXGGUVCUWLKZLZUVFMZGUUSUWRKZLZUVFMZNZAIUHZUIZOZNZBUUTGHZIJZUXTGBUGZUUTKZL + ZUVFMZGBUVHKZLZUVFMZNZAIUHZUIZOZNBCGHZIJZUYMGUYBCKZLZUVFMZGBCUGZKZLZUVFMZ + NZAIUHZUIZOZNBCUAUBDEUADUOZUVBUVSUVOUWJVUFUVAUVRIUUSUVQUUTGUJZUKVUFUVAUVR + UVNUWIVUGVUFUVMUWHVUFUVLUWGAIVUFUVGUWCUVKUWFVUFUVEUWBUVFVUFUVDUWAGVUFUVCU + VTUUTUUSUVQULZUMPQVUFUVJUWEUVFVUFUVIUWDGUUSUVQUVHUPPQRUNUQURRUBEUOZUVSUWN + UWJUXEVUIUVRUWMIUUTUWLUVQGUSZUKVUIUVRUWMUWIUXDVUJVUIUWHUXCVUIUWGUXBAIVUIU + WCUWQUWFUXAVUIUWBUWPUVFVUIUWAUWOGUUTUWLUVTUTPQVUIUWEUWTUVFVUIUWDUWSGVUIUV + HUWRUVQUUTUWLULVAPQRUNUQURRVUFUXHUWNUXRUXEVUFUXGUWMIUUSUVQUWLGUJZUKVUFUXG + UWMUXQUXDVUKVUFUXPUXCVUFUXOUXBAIVUFUXKUWQUXNUXAVUFUXJUWPUVFVUFUXIUWOGVUFU + VCUVTUWLVUHUMPQVUFUXMUWTUVFVUFUXLUWSGUUSUVQUWRUPPQRUNUQURRUUSBOZUVBUYAUVO + UYLVULUVAUXTIUUSBUUTGUJZUKVULUVAUXTUVNUYKVUMVULUVMUYJVULUVLUYIAIVULUVGUYE + UVKUYHVULUVEUYDUVFVULUVDUYCGVULUVCUYBUUTUUSBULUMPQVULUVJUYGUVFVULUVIUYFGU + USBUVHUPPQRUNUQURRUUTCOZUYAUYNUYLVUEVUNUXTUYMIUUTCBGUSZUKVUNUXTUYMUYKVUDV + UOVUNUYJVUCVUNUYIVUBAIVUNUYEUYQUYHVUAVUNUYDUYPUVFVUNUYCUYOGUUTCUYBUTPQVUN + UYGUYTUVFVUNUYFUYSGVUNUVHUYRBUUTCULVAPQRUNUQURRUXFEUUTSDUUSSZUWKDUUSSZUXS + EUUTSZVBZUVSDUUSSZUXHEUUTSZNZUUSIJZUUTIJZNZUVPVUSVUTVVAVUQVUPVUTVURUWKUVS + DUUSUVSUWJVCVDVEVURVUPVVAVUQUXSUXHEUUTUXHUXRVCVDVGVFVVEVVBUVPVVEVVBNZUVBU + VOVVFUVAUVNIVVFUVAUUSUUTVHZGUUSVIZUUTVIZKZVVGUGZVJZVKZFUCTTUCUFZFUFZVLVMZ + UGZVVOVNVMZKZLZUVFMZVVNVVPVVRUGZKZLZUVFMZNZAIUHZUIZVOZHZVVMUVDLZUVFMZVVMU + VILZUVFMZNZAIUHZUIZUVNVVEUVAVWJOVVBUDUEUUSUUTGVWIUDUEFAUCVTVPVQVVGTJVVMTJ + ZVVFVWQIJVWJVWQOUUSUUTVRGVSZVVLTJVWRGIIKZWAVWSWBVWTGWCWDZVVJVVKVVHVVIUUSU + AWEZWFUUTUBWEZWFWMWGGVVLTWHWIVVFVWQUVNIVVFVWPUVMVVFVWOUVLAIVVFVWLUVGVWNUV + KVVFVWKUVEUVFVVFUVDVVLMZVWKUVEOVVFVVGUVDJZWJZVXDVVFUUSUVCJZWJZUBUBWKZWJZW + LZVXFVVFVXJVXHVVFUUTWNZVXJVVDVXLVVCVVBUUTWOWPUUTWQXOWRVXGVXINVXKVXEVXGVXI + WSUUSUUTUVCUUTWTXAXBVXDUVDVVJMZVXFUVCVVHMZUUTVVIMVXMUUSVVHJVXNUUSVXBXDUUS + VVHXEWDUUTXCUVCVVHUUTVVIXFWIUVDVVJVVGXGXHXBGUVDVVLXIXOQVVFVWMUVJUVFVVFUVI + VVLMZVWMUVJOVVFVVGUVIJZWJZVXOVVFUAUAWKZWJZUUTUVHJZWJZWLZVXQVVFVXSVYAVVFUU + SWNZVXSVVCVYCVVDVVBUUSWOXJUUSWQXOXKVXRVXTNVYBVXPVXRVXTWSUUSUUTUUSUVHWTXAX + BVXOUVIVVJMZVXQUUSVVHMUVHVVIMZVYDUUSXCUUTVVIJVYEUUTVXCXDUUTVVIXEWDUUSVVHU + VHVVIXFWIUVIVVJVVGXGXHXBGUVIVVLXIXOQRUNUQZVVFUVLAIXLZUVNIJVVFUVEUVJXPZXMZ + VIZIJZUVEVYJMZUVJVYJMZVYGVVFVYIIJZVYKVVFVYIWNZVYNVVFVYHIMZVYOVVFUVEUVJIVV + FUVEIMZVVOUWLGHZIJZEUUTSZFUVCSZVVFVVAWUAVVEVUTVVAXNVYTVVAFUUSVXBFUAUOZVYS + UXHEUUTWUBVYRUXGIVVOUUSUWLGUJUKXQXRXBVVFVWSUVDGXSZMVYQWUAYFVXAVVFUVDVWTWU + CVVEUVDVWTMZVVBVVCUVCIMUUTIMWUDVVDUUSIXEUUTXTUVCIUUTIXFYAVQVWTGWBYGZYBFEU + VCUUTIGYCYDYEVVFUVJIMZUVQVVOGHZIJZFUVHSZDUUSSZVVFVUTWUJVVEVUTVVAYHWUIUVSD + UUSWUHUVSFUUTVXCFUBUOWUGUVRIVVOUUTUVQGUSUKXRYIXBVVFVWSUVIWUCMWUFWUJYFVXAV + VFUVIVWTWUCVVEUVIVWTMZVVBVVCUUSIMUVHIMWUKVVDUUSXTUUTIXEUUSIUVHIXFYAVQWUEY + BDFUUSUVHIGYCYDYEYJZVYHYKXOVYIVYHUVEUVJVWSUVDTJUVETJVXAUVCUUTUAYLVXCWMGUV + DTYMWIVWSUVITJUVJTJVXAUUSUVHVXBUBYLWMGUVITYMWIYQUUGUUHXBVYIUUAYNVVFUVEUVJ + VYJVVFVYPVYHVYJMWULVYHUUBXOZUUCVVFUVEUVJVYJWUMUUDUVLVYLVYMNAVYJIUVFVYJOUV + GVYLUVKVYMUVFVYJUVEYOUVFVYJUVJYORUUEUUIUVLAUUFYNZYPFUCVVGVVMTTVWHVWQVWIVV + NUVDLZUVFMZVVNUVILZUVFMZNZAIUHZUIIVVOVVGOZVWGWUTWVAVWFWUSAIWVAVWAWUPVWEWU + RWVAVVTWUOUVFWVAVVSUVDVVNWVAVVQUVCVVRUUTWVAVVPUUSUUSUUTVVOVXBVXCUUJZYRUUS + UUTVVOVXBVXCUUKZYSPQWVAVWDWUQUVFWVAVWCUVIVVNWVAVVPUUSVWBUVHWVBWVAVVRUUTWV + CYRYSPQRUNUQVVNVVMOZWUTVWPWVDWUSVWOAIWVDWUPVWLWURVWNWVDWUOVWKUVFVVNVVMUVD + YTQWVDWUQVWMUVFVVNVVMUVIYTQRUNUQVWIUULUUMUUNVYFUUOZWUNYPWVEVFUURUUPUUQ $. $} ${ - $d A x y $. $d B x y $. - $( An upper bound on the birthday of a surreal cut. (Contributed by Scott - Fenton, 10-Dec-2021.) $) - scutbdaybnd2 $p |- ( A < - ( bday ` ( A |s B ) ) C_ suc U. ( bday " ( A u. B ) ) ) $= - ( vx vy csslt wbr csn cbday cfv cun cima cuni csuc wss w3a csur wrex wcel - cv wa co etasslt2 crab cint wceq scutbday adantr wfn bdayfn ssrab2 simprl - cscut simprr1 simprr2 jca weq sneq breq2d breq1d anbi12d sylanbrc fnfvima - elrab mp3an12i intss1 syl eqsstrd simprr3 sstrd rexlimdvaa mpd ) ABEFZACS - ZGZEFZVNBEFZVMHIZHABJKLMZNZOZCPQABULUAHIZVRNZCABUBVLVTWBCPVLVMPRZVTTZTZWA - VQVRWEWAHADSZGZEFZWGBEFZTZDPUCZKZUDZVQVLWAWMUEWDDABUFUGWEVQWLRZWMVQNHPUHW - KPNWEVMWKRZWNUIWJDPUJWEWCVOVPTZWOVLWCVTUKWEVOVPVOVPVSWCVLUMVOVPVSWCVLUNUO - WJWPDVMPDCUPZWHVOWIVPWQWGVNAEWFVMUQZURWQWGVNBEWRUSUTVCVAPWKHVMVBVDVQWLVEV - FVGVOVPVSWCVLVHVIVJVK $. - $} + $d A x $. $d B x $. + $( Closure law for natural addition. (Contributed by Scott Fenton, + 26-Aug-2024.) $) + naddcl $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) e. On ) $= + ( vx con0 wcel wa cnadd co csn cxp cima cv wss crab cint naddcllem simpld + wceq ) ADEBDEFABGHZDESGAIBJKCLZMGABIJKTMFCDNORCABPQ $. - ${ - $( An upper bound on the birthday of a surreal cut when it is a limit - birthday. (Contributed by Scott Fenton, 7-Aug-2024.) $) - scutbdaybnd2lim $p |- ( ( A < - ( bday ` ( A |s B ) ) C_ U. ( bday " ( A u. B ) ) ) $= - ( csslt wbr cscut co cbday cfv wlim wa cun cima cuni wss wcel adantr word - cvv wb con0 csuc wne scutbdaybnd2 wceq wfun bdayfun ssltex1 ssltex2 unexg - wn syl2anc funimaexg sylancr uniexd nlimsucg wi limeq biimpcd adantl mtod - syl neqned bdayelon onordi crn imassrn bdayrn sseqtri ssorduni ax-mp mpbi - ordsuc ordelssne mp2an sylanbrc a1i ordsssuc sylancl mpbird ) ABCDZABEFZG - HZIZJZWBGABKZLZMZNZWBWGUAZOZWDWBWINZWBWIUBZWJVTWKWCABUCPWDWBWIWDWBWIUDZWI - IZWDWGROZWNUJVTWOWCVTWFRVTGUEWEROZWFROUFVTAROBROWPABUGABUHABRRUIUKGWERULU - MUNPWGRUOVAWCWMWNUPVTWMWCWNWBWIUQURUSUTVBWBQWIQZWJWKWLJSWBWAVCZVDWGQZWQWF - TNWSWFGVETGWEVFVGVHWFVIVJZWGVLVKWBWIVMVNVOWDWBTOZWSWHWJSXAWDWRVPWTWBWGVQV - RVS $. - $} + $( The value of natural addition. (Contributed by Scott Fenton, + 26-Aug-2024.) $) + naddov $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) = + |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\ + ( +no " ( A X. { B } ) ) C_ x ) } ) $= + ( con0 wcel wa cnadd co csn cxp cima wss crab cint wceq naddcllem simprd + cv ) BDECDEFBCGHZDESGBICJKARZLGBCIJKTLFADMNOABCPQ $. - ${ - $d A x y $. $d B x y $. $d X x y $. - $( If a surreal lies in a gap and is not equal to the cut, its birthday is - greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.) $) - scutbdaylt $p |- ( ( X e. No /\ ( A < ( bday ` ( A |s B ) ) e. ( bday ` X ) ) $= - ( vy vx csur wcel csn csslt wbr wa wne cbday cfv cv wceq syl3anc sylanbrc - wss syl cscut co w3a crab cima cint simp2l simp2r snnzg 3ad2ant1 scutbday - c0 sslttr wfn bdayfn ssrab2 simp1 simp2 sneq breq2d anbi12d elrab fnfvima - breq1d mp3an12i intss1 crio simprl simprr adantr eqeq1d eqcom bitrdi wreu - eqsstrd biimpa wb biimpri conway fveqeq2 riota2 syl2an2r mpbid scutval ex - eqtr4d necon3d 3impia con0 bdayelon onelpss mp2an ) CFGZACHZIJZWNBIJZKZCA - BUAUBZLZUCZWRMNZCMNZSZXAXBLZXAXBGZWTXAMADOZHZIJZXGBIJZKZDFUDZUEZUFZXBWTAB - IJZXAXMPZWTWOWPWNULLZXNWMWOWPWSUGWMWOWPWSUHWMWQXPWSCFUIZUJAWNBUMZQDABUKZT - WTXBXLGZXMXBSMFUNXKFSWTCXKGZXTUOXJDFUPWTWMWQYAWMWQWSUQWMWQWSURXJWQDCFXFCP - ZXHWOXIWPYBXGWNAIXFCUSZUTYBXGWNBIYCVDVAVBZRFXKMCVCVEXBXLVFTVOWMWQWSXDWMWQ - KZXAXBCWRYEXAXBPZCWRPYEYFKZCEOZMNXMPZEXKVGZWRYGXBXMPZCYJPZYEYFYKYEYFXMXBP - YKYEXAXMXBYEXNXOYEWOWPXPXNWMWOWPVHWMWOWPVIWMXPWQXQVJXRQZXSTVKXMXBVLVMVPYE - YAYFYIEXKVNZYKYLVQYAYEYDVRYGXNYNYEXNYFYMVJZEDABVSTYAYNKYKYJCPYLYIYKEXKCYH - CXMMVTWAYJCVLVMWBWCYGXNWRYJPYOEDABWDTWFWEWGWHXAWIGXBWIGXEXCXDKVQWRWJCWJXA - XBWKWLR $. + $d A t $. $d A y $. $d A z $. $d B t $. $d B y $. $d B z $. $d t x $. + $d t y $. $d t z $. $d x y $. $d x z $. + $( Alternate expression for natural addition. (Contributed by Scott + Fenton, 26-Aug-2024.) $) + naddov2 $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) = + |^| { x e. On | ( A. y e. B ( A +no y ) e. x /\ + A. z e. A ( z +no B ) e. x ) } ) $= + ( vt con0 wcel wa cnadd co csn cxp cima cv wss crab cint wral wb cdm onss + naddov snssi xpss12 syl2an fndmi sseqtrrdi wfun wfn fnfun funimassov mpan + naddfn ax-mp wceq oveq1 eleq1d ralbidv ralsng adantr bitrd adantl anbi12d + syl oveq2 rabbidv inteqd eqtrd ) DGHZEGHZIZDEJKJDLZEMZNAOZPZJDELZMZNVOPZI + ZAGQZRDBOZJKZVOHZBESZCOZEJKZVOHZCDSZIZAGQZRADEUCVLWAWKVLVTWJAGVLVPWEVSWIV + LVPFOZWBJKZVOHZBESZFVMSZWEVLVNJUAZPZVPWPTZVLVNGGMZWQVJVMGPEGPVNWTPVKDGUDE + UBVMGEGUEUFWTJUNUGZUHJUIZWRWSJWTUJXBUNWTJUKUOZFBVMEVOJULUMVEVJWPWETVKWOWE + FDGWLDUPZWNWDBEXDWMWCVOWLDWBJUQURUSUTVAVBVLVSWFWLJKZVOHZFVQSZCDSZWIVLVRWQ + PZVSXHTZVLVRWTWQVJDGPVQGPVRWTPVKDUBEGUDDGVQGUEUFXAUHXBXIXJXCCFDVQVOJULUMV + EVKXHWITVJVKXGWHCDXFWHFEGWLEUPXEWGVOWLEWFJVFURUTUSVCVBVDVGVHVI $. $} ${ - $d A a b c d $. $d B a b c d $. $d C a b c d $. $d D a b c d $. - $d X a d $. $d Y a d $. - $( A comparison law for surreals considered as cuts of sets of surreals. - Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem - 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 11-Dec-2021.) $) - slerec $p |- ( ( ( A < - ( X <_s Y <-> ( A. d e. D X ( A +no B ) = ( B +no A ) ) $= + ( va vb vc vd vx cv cnadd co wceq weq oveq1 oveq2 eqeq12d con0 wcel wa wb + wral w3a crab cint eleq1 ralimi 3ad2ant3 adantl 3ad2ant2 anbi12d biancomd + ralbi syl rabbidv inteqd naddov2 adantr ancoms 3eqtr4d ex on2ind ) CHZDHZ + IJZVBVAIJZKZEHZVBIJZVBVFIJZKZVFFHZIJZVJVFIJZKZVAVJIJZVJVAIJZKZAVBIJZVBAIJ + ZKABIJZBAIJZKABCDEFCELZVCVGVDVHVAVFVBIMVAVFVBINODFLVGVKVHVLVBVJVFINVBVJVF + IMOWAVNVKVOVLVAVFVJIMVAVFVJINOVAAKVCVQVDVRVAAVBIMVAAVBINOVBBKVQVSVRVTVBBA + INVBBAIMOVAPQZVBPQZRZVMFVBTEVATZVIEVATZVPFVBTZUAZVEWDWHRZVNGHZQZFVBTZVGWJ + QZEVATZRZGPUBZUCZVHWJQZEVATZVOWJQZFVBTZRZGPUBZUCZVCVDWIWPXCWIWOXBGPWIWOWS + XAWIWLXAWNWSWHWLXASZWDWGWEXEWFWGWKWTSZFVBTXEVPXFFVBVNVOWJUDUEWKWTFVBUKULU + FUGWHWNWSSZWDWFWEXGWGWFWMWRSZEVATXGVIXHEVAVGVHWJUDUEWMWREVAUKULUHUGUIUJUM + UNWDVCWQKWHGFEVAVBUOUPWDVDXDKZWHWCWBXIGEFVBVAUOUQUPURUSUT $. $} ${ - $d A b c $. $d B b c $. $d C b c $. $d D b c $. $d X b c $. - $d Y b c $. - $( A comparison law for surreals considered as cuts of sets of surreals. - (Contributed by Scott Fenton, 11-Dec-2021.) $) - sltrec $p |- ( ( ( A < - ( X ( E. c e. C X <_s c \/ E. b e. B b <_s Y ) ) ) $= - ( csslt wbr wa cscut cslt csle wral wn wb csur wcel syl2anc co wceq cv wo - wrex simplr simpll simprr simprl slerec syl22anc ancom bitrdi csn scutcut - simp1d ad2antlr eqeltrd ad2antrr slenlt wss sselda adantr sltnle ralbidva - ssltss1 ssltss2 anbi12d ralnex anbi12i ioran bitr4i 3bitr3d con4bid ) ABI - JZCDIJZKZEABLUAZUBZFCDLUAZUBZKZKZEFMJZEHUCZNJZHCUEZGUCZFNJZGBUEZUDZWCFENJ - ZWEEMJZHCOZFWHMJZGBOZKZWDPZWKPZWCWLWPWNKZWQWCVPVOWAVSWLWTQVOVPWBUFVOVPWBU - GVQVSWAUHZVQVSWAUIZCDABFEHGUJUKWPWNULUMWCFRSZERSZWLWRQWCFVTRXAVPVTRSZVOWB - VPXECVTUNZIJXFDIJCDUOUPUQURZWCEVRRXBVOVRRSZVPWBVOXHAVRUNZIJXIBIJABUOUPUSU - RZFEUTTWCWQWFPZHCOZWIPZGBOZKZWSWCWNXLWPXNWCWMXKHCWCWECSZKWERSXDWMXKQWCCRW - EVPCRVAVOWBCDVFUQVBWCXDXPXJVCWEEVDTVEWCWOXMGBWCWHBSZKXCWHRSWOXMQWCXCXQXGV - CWCBRWHVOBRVAVPWBABVGUSVBFWHVDTVEVHXOWGPZWJPZKWSXLXRXNXSWFHCVIWIGBVIVJWGW - JVKVLUMVMVN $. + $d A a $. $d a b $. $d a c $. $d a x $. $d b x $. $d c x $. + $( Ordinal zero is the additive identity for natural addition. + (Contributed by Scott Fenton, 26-Aug-2024.) $) + naddid1 $p |- ( A e. On -> ( A +no (/) ) = A ) $= + ( va vb vc vx cv c0 cnadd co wceq oveq1 id eqeq12d con0 wcel wral wa crab + cint adantr weq wss 0elon naddov2 mpan2 ral0 biantrur wel wb eleq1 ralimi + ralbi syl adantl dfss3 bitr4di bitr3id rabbidv inteqd intmin 3eqtrd tfis3 + ex ) BFZGHIZVDJZCFZGHIZVGJZAGHIZAJBCABCUAZVEVHVDVGVDVGGHKVKLMVDAJZVEVJVDA + VDAGHKVLLMVDNOZVICVDPZVFVMVNQZVEVDDFHIEFZOZDGPZVHVPOZCVDPZQZENRZSZVDVPUBZ + ENRZSZVDVMVEWCJZVNVMGNOWGUCEDCVDGUDUETVOWBWEVOWAWDENWAVTVOWDVRVTVQDUFUGVO + VTCEUHZCVDPZWDVNVTWIUIZVMVNVSWHUIZCVDPWJVIWKCVDVHVGVPUJUKVSWHCVDULUMUNCVD + VPUOUPUQURUSVMWFVDJVNEVDNUTTVAVCVB $. $} ${ - $d A x $. $d B x $. - $( If ` A ` preceeds ` B ` , then ` A ` and ` B ` are disjoint. - (Contributed by Scott Fenton, 18-Sep-2024.) $) - ssltdisj $p |- ( A < ( A i^i B ) = (/) ) $= - ( vx csslt wbr cv wcel wn wral c0 wceq wa cslt csur ssltss1 sselda sltirr - cin syl ssltsepc 3expa mtand ralrimiva disj sylibr ) ABDEZCFZBGZHZCAIABRJ - KUFUICAUFUGAGZLZUHUGUGMEZUKUGNGULHUFANUGABOPUGQSUFUJUHULABUGUGTUAUBUCCABU - DUE $. + $d A c $. $d A d $. $d A w $. $d A x $. $d A z $. $d B c $. $d B d $. + $d B w $. $d B x $. $d B y $. $d B z $. $d C c $. $d c d $. $d c w $. + $d c x $. $d c y $. $d c z $. $d d w $. $d d x $. $d w x $. $d w y $. + $d w z $. $d x y $. $d x z $. + $( Ordinal less-than-or-equal is preserved by natural addition. + (Contributed by Scott Fenton, 7-Sep-2024.) $) + naddssim $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> + ( A C_ B -> ( A +no C ) C_ ( B +no C ) ) ) $= + ( vc vd vw vx vz vy con0 wcel wss cnadd co wi wa cv oveq2 imbi2d wral weq + sseq12d wceq r19.21v imbi2i crab cint wel rspccva ad4ant24 simprrl eleq1d + bitri sylan simplrl adantr simp-4l onelon naddcl syl2anc mp2and ralrimiva + ontr2 simpllr simprrr ssralv sylc jca expr ss2rabdv intss simplll naddov2 + syl simplrr 3sstr4d exp31 a2d ex biimtrid tfis3 com12 3impia ) AJKZBJKZCJ + KZABLZACMNZBCMNZLZOZWFWDWEPZWKWLWGADQZMNZBWMMNZLZOZOZWLWGAEQZMNZBWSMNZLZO + ZOZWLWKODECDEUAZWQXCWLXEWPXBWGXEWNWTWOXAWMWSAMRWMWSBMRUBSSWMCUCZWQWKWLXFW + PWJWGXFWNWHWOWIWMCAMRWMCBMRUBSSXDEWMTZWLWGXBEWMTZOZOZWMJKZWRXGWLXCEWMTZOX + JWLXCEWMUDXLXIWLWGXBEWMUDUEUMXKWLXIWQXKWLXIWQOXKWLPZWGXHWPXMWGXHWPXMWGPZX + HPZAFQZMNZGQZKZFWMTZHQWMMNXRKZHATZPZGJUFZUGZBIQZMNZXRKZIWMTZYAHBTZPZGJUFZ + UGZWNWOXOYLYDLYEYMLXOYKYCGJXOXRJKZYKYCXOYNYKPZPZXTYBYPXSFWMYPFDUHZPZXQBXP + MNZLZYSXRKZXSXHYQYTXNYOXBYTEXPWMEFUAWTXQXAYSWSXPAMRWSXPBMRUBUIUJYPYIYQUUA + XOYNYIYJUKYHUUAIXPWMIFUAYGYSXRYFXPBMRULUIUNYRXQJKZYNYTUUAPXSOYRWDXPJKZUUB + YPWDYQXOWDYOXNWDXHXKWDWEWGUOUPZUPUPYPXKYQUUCXKWLWGXHYOUQWMXPURUNAXPUSUTXO + YNYKYQUOXQYSXRVCUTVAVBYPWGYJYBXMWGXHYOVDXOYNYIYJVEYAHABVFVGVHVIVJYLYDVKVN + XOWDXKWNYEUCUUDXKWLWGXHVLZGFHAWMVMUTXOWEXKWOYMUCXNWEXHXKWDWEWGVOUPUUEGIHB + WMVMUTVPVQVRVSVRVTWAWBWC $. $} - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Surreal numbers - zero and one -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - $( Define new constants. $) - $c 0s 1s $. - - $( Declare the class syntax for surreal zero. $) - c0s $a class 0s $. - - $( Declare the class syntax for surreal one. $) - c1s $a class 1s $. - - $( Define surreal zero. This is the simplest cut of surreal number sets. - Definition from [Conway] p. 17. (Contributed by Scott Fenton, - 7-Aug-2024.) $) - df-0s $a |- 0s = ( (/) |s (/) ) $. - - $( Define surreal one. This is the simplest number greater than surreal - zero. Definition from [Conway] p. 18. (Contributed by Scott Fenton, - 7-Aug-2024.) $) - df-1s $a |- 1s = ( { 0s } |s (/) ) $. - - $( Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) $) - 0sno $p |- 0s e. No $= - ( c0s c0 cscut co csur df-0s csslt wbr wcel cpw 0elpw nulssgt ax-mp eqeltri - scutcl ) ABBCDZEFBBGHZPEIBEJIQEKBLMBBOMN $. - - $( Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) $) - 1sno $p |- 1s e. No $= - ( c1s c0s csn c0 cscut csur df-1s csslt wbr wcel 0sno snelpwi ax-mp nulssgt - co cpw scutcl eqeltri ) ABCZDEOZFGSDHIZTFJSFPJZUABFJUBKBFLMSNMSDQMR $. - ${ - $d x y $. - $( Calculate the birthday of surreal zero. (Contributed by Scott Fenton, - 7-Aug-2024.) $) - bday0s $p |- ( bday ` 0s ) = (/) $= - ( vx c0s cbday cfv c0 cv csn csslt wa csur crab cima cint con0 cscut wcel - wbr co nulssgt 3eqtri df-0s fveq2i cpw wceq 0elpw scutbday mp2b eqtri cdm - crn snelpwi nulsslt jca rabeqc bdaydm eqtr4i imaeq2i imadmrn bdayrn inton - syl inteqi ) BCDZCEAFZGZHQZVEEHQZIZAJKZLZMZNMEVCEEORZCDZVKBVLCUAUBEJUCZPE - EHQVMVKUDJUEESAEEUFUGUHVJNVJCCUIZLCUJNVIVOCVIJVOVHAJVDJPVEVNPZVHVDJUKVPVF - VGVEULVESUMVAUNUOUPUQCURUSTVBUTT $. + $d A b $. $d A x $. $d B b $. $d B c $. $d b x $. $d B x $. $d C b $. + $d C c $. $d c x $. $d C x $. + $( Ordinal less-than is preserved by natural addition. (Contributed by + Scott Fenton, 9-Sep-2024.) $) + naddelim $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A e. B -> + ( A +no C ) e. ( B +no C ) ) ) $= + ( vc vx vb con0 wcel w3a cnadd co wa cv wral crab cint wceq oveq1 eleq1d + wi ad2antlr adantld ralrimiva ovex elintrab sylibr naddov2 3adant1 adantr + rspcv eleqtrrd ex ) AGHZBGHZCGHZIZABHZACJKZBCJKZHUPUQLZURBDMJKEMZHDCNZFMZ + CJKZVAHZFBNZLZEGOPZUSUTVGURVAHZTZEGNURVHHUTVJEGUTVAGHZLVFVIVBUQVFVITUPVKV + EVIFABVCAQVDURVAVCACJRSUJUAUBUCVGEURGACJUDUEUFUPUSVHQZUQUNUOVLUMEDFBCUGUH + UIUKUL $. $} - ${ - $d x y $. - $( Surreal zero is less than surreal one. Theorem from [Conway] p. 7. - (Contributed by Scott Fenton, 7-Aug-2024.) $) - 0slt1s $p |- 0s + ( A e. B <-> ( A +no C ) e. ( B +no C ) ) ) $= + ( con0 wcel w3a cnadd co naddelim wss naddssim 3com12 ontri1 ancoms 3adant3 + wn wi wb naddcl 3adant1 3imp3i2an 3imtr3d impcon4bid ) ADEZBDEZCDEZFZABEZAC + GHZBCGHZEZABCIUGBAJZUJUIJZUHPZUKPZUEUDUFULUMQBACKLUDUEULUNRZUFUEUDUPBAMNOUD + UEUFUJDEZUIDEUMUORUEUFUQUDBCSTACSUJUIMUAUBUC $. - ${ - $d X x $. - $( The only surreal with birthday ` (/) ` is ` 0s ` . (Contributed by - Scott Fenton, 8-Aug-2024.) $) - bday0b $p |- ( X e. No -> ( ( bday ` X ) = (/) <-> X = 0s ) ) $= - ( vx csur wcel cbday cfv c0 c0s wa cscut co df-0s csn csslt wbr cv adantr - wceq syl nulssgt wss wi cpw snelpwi nulsslt id 0ss eqsstrdi a1d ralrimivw - wral adantl wb 0elpw ax-mp eqscut2 mpan mpbir3and eqtr2id ex fveq2 bday0s - w3a eqtrdi impbid1 ) ACDZAEFZGRZAHRZVFVHVIVFVHIZHGGJKZALVJVKARZGAMZNOZVMG - NOZGBPZMZNOVQGNOIZVGVPEFZUAZUBZBCUKZVFVNVHVFVMCUCZDZVNACUDZVMUESQVFVOVHVF - WDVOWEVMTSQVJWABCVHWAVFVHVTVRVHVGGVSVHUFVSUGUHUIULUJVFVLVNVOWBVCUMZVHGGNO - ZVFWFGWCDWGCUNGTUOBGGAUPUQQURUSUTVIVGHEFGAHEVAVBVDVE $. - $} + $( Ordinal less-than is not affected by natural addition. (Contributed by + Scott Fenton, 9-Sep-2024.) $) + naddel2 $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> + ( A e. B <-> ( C +no A ) e. ( C +no B ) ) ) $= + ( con0 wcel w3a cnadd co naddel1 wceq naddcom 3adant2 3adant1 eleq12d bitrd + ) ADEZBDEZCDEZFZABEACGHZBCGHZECAGHZCBGHZEABCISTUBUAUCPRTUBJQACKLQRUAUCJPBCK + MNO $. - ${ - $d x y z $. - $( The birthday of surreal one is ordinal one. (Contributed by Scott - Fenton, 8-Aug-2024.) $) - bday1s $p |- ( bday ` 1s ) = 1o $= - ( vx vy vz cbday cfv c0s csn c1o cima csslt wbr csur wcel 0sno ax-mp wceq - c0 wss wral cslt c1s cscut df-1s fveq2i cun cuni csuc cpw snelpwi nulssgt - co scutbdaybnd2 un0 imaeq2i wfn bdayfn fnsnfv mp2an bday0s 3eqtr2i unieqi - sneqi 0ex unisn eqtri suceq df-1o eqtr4i sseqtri cv wa crab wb fnssintima - cint ssrab2 weq sneq breq2d breq1d anbi12d elrab wn sltirr breq2 necon2ai - wne mtbiri bday0b necon3bid syl5ibr word bdayelon onordi ordge1n0 syl6ibr - ssltsep ralsn ralbii elexi breq1 bitri sylib impel adantrr sylbi scutbday - vex mprgbir sseqtrri eqssi ) UADEFGZQUBUKZDEZHUAXMDUCUDXNHXNDXLQUEZIZUFZU - GZHXLQJKZXNXRRXLLUHMZXSFLMZXTNFLUIOXLUJOZXLQULOXRQUGZHXQQPXRYCPXQQGZUFQXP - YDXPDXLIZFDEZGZYDXOXLDXLUMUNDLUOZYAYGYEPUPNLFDUQURYFQUSVBUTVAQVCVDVEXQQVF - OVGVHVIHDXLAVJZGZJKZYJQJKZVKZALVLZIVOZXNHYORZHBVJZDEZRZBYNYHYNLRYPYSBYNSV - MUPYMALVPBLYNHDVNURYQYNMYQLMZXLYQGZJKZUUAQJKZVKZVKYSYMUUDAYQLABVQZYKUUBYL - UUCUUEYJUUAXLJYIYQVRZVSUUEYJUUAQJUUFVTWAWBYTUUBYSUUCYTFYQTKZYSUUBYTUUGYRQ - WGZYSUUGUUHYTYQFWGUUGYQFYQFPUUGFFTKZYAUUIWCNFWDOYQFFTWEWHWFYTYRQYQFYQWIWJ - WKYRWLYSUUHVMYRYQWMWNYRWOOWPUUBYICVJZTKZCUUASZAXLSZUUGACXLUUAWQUUMYIYQTKZ - AXLSUUGUULUUNAXLUUKUUNCYQBXHUUJYQYITWEWRWSUUNUUGAFFLNWTYIFYQTXAWRXBXCXDXE - XFXIXSXNYOPYBAXLQXGOXJXKVE $. - $} + $( Ordinal less-than-or-equal is not affected by natural addition. + (Contributed by Scott Fenton, 9-Sep-2024.) $) + naddss1 $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> + ( A C_ B <-> ( A +no C ) C_ ( B +no C ) ) ) $= + ( con0 wcel w3a wn cnadd co wss naddel1 3com12 notbid ontri1 3adant3 naddcl + wb 3adant2 3adant1 syl2anc 3bitr4d ) ADEZBDEZCDEZFZBAEZGZBCHIZACHIZEZGZABJZ + UIUHJZUEUFUJUCUBUDUFUJQBACKLMUBUCULUGQUDABNOUEUIDEZUHDEZUMUKQUBUDUNUCACPRUC + UDUOUBBCPSUIUHNTUA $. + $( Ordinal less-than-or-equal is not affected by natural addition. + (Contributed by Scott Fenton, 9-Sep-2024.) $) + naddss2 $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> + ( A C_ B <-> ( C +no A ) C_ ( C +no B ) ) ) $= + ( con0 wcel w3a cnadd co naddss1 wceq naddcom 3adant2 3adant1 sseq12d bitrd + wss ) ADEZBDEZCDEZFZABPACGHZBCGHZPCAGHZCBGHZPABCITUAUCUBUDQSUAUCJRACKLRSUBU + DJQBCKMNO $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -538946,7 +539719,7 @@ property of surreals and will be used (via surreal cuts) to prove many df-new $a |- _N = ( x e. On |-> ( ( _M ` x ) \ ( _Old ` x ) ) ) $. ${ - $d x y z $. + $d x y $. $( Define the left options of a surreal. This is the set of surreals that are simpler and less than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.) $) @@ -539214,7 +539987,7 @@ property of surreals and will be used (via surreal cuts) to prove many c0 ) NABNCBZNDBZEFGZNEQNHOQPNIJKQLM $. ${ - $d A a b x l r $. $d B a b x l r $. + $d A a b x $. $d B a b x $. $( If ` A ` is less than or equal to ordinal ` B ` , then the made set of ` A ` is included in the made set of ` B ` . 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$d A a $. $d A b $. $d a e $. $d A e $. $d a f $. $d A f $. $d A p $. - $d A q $. $d A x $. $d A y $. $d B a $. $d B b $. $d b e $. $d B e $. - $d b f $. $d B f $. $d B p $. $d B q $. $d B x $. $d B y $. $d e f $. - $d e p $. $d e q $. $d f p $. $d f q $. $d p q $. $d p x $. $d p y $. - $d q x $. $d q y $. $d R e $. $d R f $. $d R p $. $d R q $. $d S p $. - $d S q $. + $d A a $. $d A b $. $d A x $. $d A y $. $d B a $. $d B b $. $d B x $. + $d B y $. $( Next we calculate the predecessor class of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.) $) noxpordpred $p |- ( ( A e. No /\ B e. No ) -> @@ -540888,10 +541657,10 @@ property of surreals and will be used (via surreal cuts) to prove many 16-Apr-2012.) $) fobigcup $p |- Bigcup : _V -onto-> _V $= ( vx vy cvv cbigcup wfo wfn crn wceq cuni wcel wral uniexg rgen dfbigcup2 - cv mptfng mpbi wrex cab rnmpt vex snex unisnv eqcomi unieq rspceeqv mp2an - csn 2th abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFZDGZCHAOZIZCJZACKUNURACUPCL - MACUQDANZPQUOBOZUQHACRZBSCABCUQDUSTVABCUTCJVABUAUTUHZCJUTVBIZHVAUTUBVCUTB - UCUDAVBCUQVCUTUPVBUEUFUGUIUJUKCCDULUM $. + cv mptfng mpbi wrex cab rnmpt vex csn vsnex unisnv eqcomi unieq mp2an 2th + rspceeqv abbi2i eqtr4i df-fo mpbir2an ) CCDEDCFZDGZCHAOZIZCJZACKUNURACUPC + LMACUQDANZPQUOBOZUQHACRZBSCABCUQDUSTVABCUTCJVABUAUTUBZCJUTVBIZHVABUCVCUTB + UDUEAVBCUQVCUTUPVBUFUIUGUHUJUKCCDULUM $. $} $( ` Bigcup ` is a function over the universal class. (Contributed by Scott @@ -541092,11 +541861,11 @@ property of surreals and will be used (via surreal cuts) to prove many $( Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.) $) elsingles $p |- ( A e. Singletons <-> E. x A = { x } ) $= - ( vy csingles wcel cvv csn wceq wex elex snex eleq1 mpbiri exlimiv exbidv - cv eqeq1 csingle crn vex wbr df-singles eleq2i elrn brsingle exbii 3bitri - vtoclbg pm5.21nii ) BDEZBFEZBAPZGZHZAIZBDJUNUKAUNUKUMFEULKBUMFLMNCPZDEZUP - UMHZAIZUJUOCBFUPBDLUPBHURUNAUPBUMQOUQUPRSZEULUPRUAZAIUSDUTUPUBUCAUPRCTZUD - VAURAULUPATVBUEUFUGUHUI $. + ( vy csingles wcel cvv csn wceq wex elex vsnex eleq1 mpbiri exlimiv eqeq1 + cv exbidv csingle crn vex wbr df-singles eleq2i elrn exbii 3bitri vtoclbg + brsingle pm5.21nii ) BDEZBFEZBAPZGZHZAIZBDJUNUKAUNUKUMFEAKBUMFLMNCPZDEZUP + UMHZAIZUJUOCBFUPBDLUPBHURUNAUPBUMOQUQUPRSZEULUPRUAZAIUSDUTUPUBUCAUPRCTZUD + VAURAULUPATVBUHUEUFUGUI $. $} ${ @@ -541107,11 +541876,11 @@ property of surreals and will be used (via surreal cuts) to prove many fnsingle $p |- Singleton Fn _V $= ( vx vy vz csingle cvv wfn wfun cdm wceq wrel cv wbr wa weq wal mpbir vex ctxp brsingle mpbir2an wi cxp cep cid csymdif crn cdif difss df-singleton - wss df-rel releqi csn eqtr3 syl2anb ax-gen gen2 dffun2 wcel eqv eqid snex - wex breq2 spcev ax-mp eldm mpgbir df-fn ) DEFDGZDHZEIZVJDJZAKZBKZDLZVNCKZ - DLZMBCNZUAZCOZBOAOVMEEUBZEUCRUDERUEUFZUGZJZWEWDWBUJWBWCUHWDUKPDWDUIULPWAA - BVTCVPVOVNUMZIVQWFIVSVRVNVOAQZBQSVNVQWGCQSVOVQWFUNUOUPUQABCDURTVLVNVKUSZA - AVKUTWHVPBVCZVNWFDLZWIWJWFWFIWFVAVNWFWGVNVBZSPVPWJBWFWKVOWFVNDVDVEVFBVNDW + wss df-rel releqi csn eqtr3 syl2anb ax-gen gen2 dffun2 wcel eqv wex vsnex + eqid breq2 spcev ax-mp eldm mpgbir df-fn ) DEFDGZDHZEIZVJDJZAKZBKZDLZVNCK + ZDLZMBCNZUAZCOZBOAOVMEEUBZEUCRUDERUEUFZUGZJZWEWDWBUJWBWCUHWDUKPDWDUIULPWA + ABVTCVPVOVNUMZIVQWFIVSVRVNVOAQZBQSVNVQWGCQSVOVQWFUNUOUPUQABCDURTVLVNVKUSZ + AAVKUTWHVPBVAZVNWFDLZWIWJWFWFIWFVCVNWFWGAVBZSPVPWJBWFWKVOWFVNDVDVEVFBVNDW GVGPVHDEVIT $. $} @@ -541122,10 +541891,10 @@ property of surreals and will be used (via surreal cuts) to prove many Scott Fenton, 13-Apr-2018.) $) fvsingle $p |- ( Singleton ` A ) = { A } $= ( vx cvv wcel csingle cfv csn wceq cv fveq2 sneq eqeq12d wbr vex brsingle - eqid snex mpbir wfn c0 wb fnsingle mp2an vtoclg fvprc snprc biimpi eqtr4d - fnbrfvb wn pm2.61i ) ACDZAEFZAGZHZBIZEFZUPGZHZUOBACUPAHUQUMURUNUPAEJUPAKL - USUPUREMZUTURURHURPUPURBNZUPQORECSUPCDUSUTUAUBVACUPUREUIUCRUDULUJZUMTUNAE - UEVBUNTHAUFUGUHUK $. + eqid vsnex mpbir wfn c0 wb fnsingle fnbrfvb mp2an vtoclg wn biimpi eqtr4d + fvprc snprc pm2.61i ) ACDZAEFZAGZHZBIZEFZUPGZHZUOBACUPAHUQUMURUNUPAEJUPAK + LUSUPUREMZUTURURHURPUPURBNZBQORECSUPCDUSUTUAUBVACUPUREUCUDRUEULUFZUMTUNAE + UIVBUNTHAUJUGUHUK $. $} ${ @@ -541154,15 +541923,15 @@ property of surreals and will be used (via surreal cuts) to prove many Fenton, 19-Feb-2013.) $) dfiota3 $p |- ( iota x ph ) = U. U. ( { { x | ph } } i^i Singletons ) $= ( vy vz vw cab cv csn wceq cuni csingles cin wex wcel ceqsexv bitri exbii - wa weq eqtri cio df-iota wb abeq1 wel exdistr vex sneq eqeq2d eqeq1 eleq2 - snex anbi12d eqcom velsn equcom anbi12ci bitrdi an13 bitr3i excom eluniab - 3bitr4i mpgbir df-sn dfsingles2 ineq12i 19.42v bicomi abbii unieqi eqtr4i - inab ) ABUAABFZCGZHZIZCFZJVNHZKLZJZJABCUBVRWAVRDGZVNIZWBEGZHZIZRZEMZDFZJZ - WAVRWJIVQVOWJNZUCCVQCWJUDCDUEZWGRZEMDMZWLWHRDMVQWKWLWGDEUFVQWMDMZEMZWNVQE - CSZVNWEIZRZEMWPWRVQEVOCUGWQWEVPVNWDVOUHUIOWSWOEWSWFWCWLRZRZDMWOWTWSDWEWDU - LWFWTWEVNIZVOWENZRWSWFWCXBWLXCWBWEVNUJWBWEVOUKUMXBWRXCWQWEVNUNXCCESWQCWDU - OCEUPPUQUROXAWMDWFWCWLUSQUTQUTWMEDVAPWHDVOVBVCVDVTWIVTWCDFZWFEMZDFZLZWIVS - XDKXFDVNVEDEVFVGXGWCXERZDFWIWCXEDVMXHWHDWHXHWCWFEVHVIVJTTVKVLVKT $. + wa weq eqtri cio df-iota wb abeq1 wel exdistr vex sneq eqeq2d vsnex eqeq1 + eleq2 anbi12d eqcom velsn equcom bitrdi an13 bitr3i excom eluniab 3bitr4i + anbi12ci mpgbir df-sn dfsingles2 ineq12i inab 19.42v bicomi unieqi eqtr4i + abbii ) ABUAABFZCGZHZIZCFZJVNHZKLZJZJABCUBVRWAVRDGZVNIZWBEGZHZIZRZEMZDFZJ + ZWAVRWJIVQVOWJNZUCCVQCWJUDCDUEZWGRZEMDMZWLWHRDMVQWKWLWGDEUFVQWMDMZEMZWNVQ + ECSZVNWEIZRZEMWPWRVQEVOCUGWQWEVPVNWDVOUHUIOWSWOEWSWFWCWLRZRZDMWOWTWSDWEEU + JWFWTWEVNIZVOWENZRWSWFWCXBWLXCWBWEVNUKWBWEVOULUMXBWRXCWQWEVNUNXCCESWQCWDU + OCEUPPVCUQOXAWMDWFWCWLURQUSQUSWMEDUTPWHDVOVAVBVDVTWIVTWCDFZWFEMZDFZLZWIVS + XDKXFDVNVEDEVFVGXGWCXERZDFWIWCXEDVHXHWHDWHXHWCWFEVIVJVMTTVKVLVKT $. $} ${ @@ -541458,16 +542227,16 @@ property of surreals and will be used (via surreal cuts) to prove many ( vy vz csingle cvv csingles wcel csn cima cv wex c0 imaeq2d wbr wa exbii wceq 3bitri cimage ccom cxp cin cdm elex vsnid eleq2 mpbiri n0i wn biimpi snprc ima0 eqtrdi nsyl2 syl exlimiv eleq1 sneq eqeq1d exbidv eldm mpbiran - vex brxp elsingles bitri anbi2i brin 19.42v 3bitr4i excom exancom ceqsexv - snex breq2 brco brsingle anbi1i breq1 brimage eqcom vtoclbg pm5.21nii ) B - CUAZFUBZGHUCZUDZUEZIZBGIZCBJZKZALZJZSZAMZBWJUFWQWLAWQWOWNIZWLWQWSWOWPIAUG - WNWPWOUHUIWSWNNSWLWNWOUJWLUKZWNCNKNWTWMNCWTWMNSBUMULOCUNUOUPUQURDLZWJIZCX - AJZKZWPSZAMZWKWRDBGXABWJUSXABSZXEWQAXGXDWNWPXGXCWMCXABUTOVAVBXBXAELZWIPZE - MZXAXHWGPZXHWPSZQZEMZAMZXFEXAWIDVEZVCXJXMAMZEMXOXIXQEXKXAXHWHPZQXKXLAMZQX - IXQXRXSXKXRXHHIZXSXRXAGIXTXPXAXHGHVFVDAXHVGVHVIXAXHWGWHVJXKXLAVKVLRXMEAVM - VHXNXEAXNXLXKQEMXAWPWGPZXEXKXLEVNXKYAEWPWOVPZXHWPXAWGVQVOYAXAXHFPZXHWPWFP - ZQZEMXHXCSZYDQZEMZXEEXAWPWFFXPYBVRYEYGEYCYFYDXAXHXPEVEVSVTRYHXCWPWFPZWPXD - SXEYDYIEXCXAVPZXHXCWPWFWAVOXCWPCYJYBWBWPXDWCTTTRTWDWE $. + brxp elsingles bitri anbi2i brin 19.42v 3bitr4i excom exancom vsnex breq2 + vex ceqsexv brco brsingle anbi1i breq1 brimage eqcom vtoclbg pm5.21nii ) + BCUAZFUBZGHUCZUDZUEZIZBGIZCBJZKZALZJZSZAMZBWJUFWQWLAWQWOWNIZWLWQWSWOWPIAU + GWNWPWOUHUIWSWNNSWLWNWOUJWLUKZWNCNKNWTWMNCWTWMNSBUMULOCUNUOUPUQURDLZWJIZC + XAJZKZWPSZAMZWKWRDBGXABWJUSXABSZXEWQAXGXDWNWPXGXCWMCXABUTOVAVBXBXAELZWIPZ + EMZXAXHWGPZXHWPSZQZEMZAMZXFEXAWIDVPZVCXJXMAMZEMXOXIXQEXKXAXHWHPZQXKXLAMZQ + XIXQXRXSXKXRXHHIZXSXRXAGIXTXPXAXHGHVEVDAXHVFVGVHXAXHWGWHVIXKXLAVJVKRXMEAV + LVGXNXEAXNXLXKQEMXAWPWGPZXEXKXLEVMXKYAEWPAVNZXHWPXAWGVOVQYAXAXHFPZXHWPWFP + ZQZEMXHXCSZYDQZEMZXEEXAWPWFFXPYBVRYEYGEYCYFYDXAXHXPEVPVSVTRYHXCWPWFPZWPXD + SXEYDYIEXCDVNZXHXCWPWFWAVQXCWPCYJYBWBWPXDWCTTTRTWDWE $. $} ${ @@ -541479,14 +542248,14 @@ property of surreals and will be used (via surreal cuts) to prove many ( vx vy vz vw wfun cv wbr wa wceq wal wcel vex brresi csn bitri cop df-br wi anbi12i cfunpart cimage csingle ccom cvv csingles cxp cres wrel relres cin cdm simprbi cima funpartlem anbi1i elimasn bitr4i eleq2 anbi12d velsn - equtr2 syl2anb syl6bi syl5bi exlimiv impl sylanb sylan2 ax-gen df-funpart - wex gen2 funeqi dffun2 mpbir2an ) AUAZFZAAUBUCUDUEUFUGUKULZUHZUIZBGZCGZVT - HZWBDGZVTHZIWCWEJZSZDKCKZBKZAVSUJWIBWHCDWFWDWBWEAHZWGWFWBVSLZWKVSWBWEADMZ - NUMWDAWBOUNZEGZOZJZEVLZWBWCAHZIZWKWGWDWLWSIWTVSWBWCACMZNWLWRWSEWBAUOUPPWR - WSWKWGWQWSWKIZWGSEXBWCWNLZWEWNLZIZWQWGXBWBWCQALZWBWEQALZIXEWSXFWKXGWBWCAR - WBWEARTXCXFXDXGAWBWCBMZXAUQAWBWEXHWMUQTURWQXEWCWPLZWEWPLZIWGWQXCXIXDXJWNW - PWCUSWNWPWEUSUTXIWCWOJWEWOJWGXJCWOVADWOVACDEVBVCVDVEVFVGVHVIVMVJVRVTFWAWJ - IVQVTAVKVNBCDVTVOPVP $. + wex equtr2 syl2anb syl6bi biimtrid exlimiv impl sylanb sylan2 gen2 ax-gen + df-funpart funeqi dffun2 mpbir2an ) AUAZFZAAUBUCUDUEUFUGUKULZUHZUIZBGZCGZ + VTHZWBDGZVTHZIWCWEJZSZDKCKZBKZAVSUJWIBWHCDWFWDWBWEAHZWGWFWBVSLZWKVSWBWEAD + MZNUMWDAWBOUNZEGZOZJZEVBZWBWCAHZIZWKWGWDWLWSIWTVSWBWCACMZNWLWRWSEWBAUOUPP + WRWSWKWGWQWSWKIZWGSEXBWCWNLZWEWNLZIZWQWGXBWBWCQALZWBWEQALZIXEWSXFWKXGWBWC + ARWBWEARTXCXFXDXGAWBWCBMZXAUQAWBWEXHWMUQTURWQXEWCWPLZWEWPLZIWGWQXCXIXDXJW + NWPWCUSWNWPWEUSUTXIWCWOJWEWOJWGXJCWOVADWOVACDEVCVDVEVFVGVHVIVJVKVLVRVTFWA + WJIVQVTAVMVNBCDVTVOPVP $. $} $( The functional part of ` F ` is a subset of ` F ` . (Contributed by Scott @@ -541815,15 +542584,15 @@ hypothesis from its defining theorem (see ~ altopth ), making it more holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.) $) altopth1 $p |- ( A e. V -> ( << A , B >> = << C , D >> -> A = C ) ) $= - ( caltop wceq csn wa wcel altopthsn sneqrg adantrd syl5bi ) ABFCDFGAHCHGZBH - DHGZIAEJZACGZABCDKQORPACELMN $. + ( caltop wceq csn wa wcel altopthsn sneqrg adantrd biimtrid ) ABFCDFGAHCHGZ + BHDHGZIAEJZACGZABCDKQORPACELMN $. $( Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.) $) altopth2 $p |- ( B e. V -> ( << A , B >> = << C , D >> -> B = D ) ) $= - ( caltop wceq csn wa wcel altopthsn sneqrg adantld syl5bi ) ABFCDFGAHCHGZBH - DHGZIBEJZBDGZABCDKQPROBDELMN $. + ( caltop wceq csn wa wcel altopthsn sneqrg adantld biimtrid ) ABFCDFGAHCHGZ + BHDHGZIBEJZBDGZABCDKQPROBDELMN $. $( Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.) $) @@ -541933,14 +542702,14 @@ hypothesis from its defining theorem (see ~ altopth ), making it more $( An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.) $) altxpsspw $p |- ( A XX. B ) C_ ~P ~P ( A u. ~P B ) $= - ( vz vx vy caltxp cpw cun cv wcel wrex wa csn cpr wss snssi syl snex elpw - prex caltop wceq elaltxp df-altop ssun3 adantr elun1 elun2 sylbir anim12i - wi prss sylib prsspw bitri sylanbrc eqeltrid eleq1a rexlimivv sylbi ssriv - vex ) CABFZABGZHZGZGZCIZVCJVHDIZEIZUAZUBZEBKDAKVHVGJZDEABVHUCVLVMDEABVIAJ - ZVJBJZLZVKVGJVLVMUKVPVKVIMZVIVJMZNZNZVGVIVJUDVPVQVEOZVSVEOZVTVGJZVNWAVOVN - VQAOWAVIAPVQAVDUEQUFVPVIVEJZVRVEJZLWBVNWDVOWEVIAVDUGVOVRBOZWEVJBPWFVRVDJW - EVRBVJRZSVRVDAUHUIQUJVIVRVEDVBWGULUMWCVTVFOWAWBLVTVFVQVSTSVQVSVEVIRVIVRTU - NUOUPUQVKVGVHURQUSUTVA $. + ( vz vx vy caltxp cpw cun cv wcel wrex wa csn cpr wss snssi syl elpw prex + vsnex caltop wceq elaltxp wi df-altop ssun3 adantr elun1 elun2 sylbir vex + anim12i sylib prsspw bitri sylanbrc eqeltrid eleq1a rexlimivv sylbi ssriv + prss ) CABFZABGZHZGZGZCIZVCJVHDIZEIZUAZUBZEBKDAKVHVGJZDEABVHUCVLVMDEABVIA + JZVJBJZLZVKVGJVLVMUDVPVKVIMZVIVJMZNZNZVGVIVJUEVPVQVEOZVSVEOZVTVGJZVNWAVOV + NVQAOWAVIAPVQAVDUFQUGVPVIVEJZVRVEJZLWBVNWDVOWEVIAVDUHVOVRBOZWEVJBPWFVRVDJ + WEVRBETZRVRVDAUIUJQULVIVRVEDUKWGVBUMWCVTVFOWAWBLVTVFVQVSSRVQVSVEDTVIVRSUN + UOUPUQVKVGVHURQUSUTVA $. $} $( The alternate Cartesian product of two sets is a set. (Contributed by @@ -543358,20 +544127,20 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and cee cfv coprab df-colinear breqi wb simpr1 opex brcnvg sylancl df-br wceq eleq1 3anbi2d opeq1 breq1 opeq2 3orbi123d anbi12d rexbidv 3anbi3d 3anbi1d eloprabg 3comr adantl simpl simp2 anim2i axdimuniq adantrrl simprrl fveq2 - cvv 3simpa eleq2d syl5ibrcom mpd syl2an exp32 syl7 rexlimdv sylbid syl5bi - ) ABCKZUALAWGGMZHMZUDUEZNZIMZWJNZJMZWJNZOZWHWLWNKZPLZWLWNWHKZPLZWNWHWLKZP - LZQZRZHSUBZIJGUFZUCZLZDSNZAENZBDUDUEZNZCFNZOZRZAXKNZAWGUAXGHGIJUGUHXOXHWG - AXFLZXPXOXJWGVPNXHXQUIXIXJXLXMUJBCUKAWGEVPXFULUMXQWGAKXFNZXOXPWGAXFUNXOXR - AWJNZBWJNZCWJNZOZAWGPLZBCAKZPLZCABKZPLZQZRZHSUBZXPXNXRYJUIZXIXLXMXJYKXEWK - XTWOOZWHBWNKZPLZBWSPLZWNWHBKZPLZQZRZHSUBWKXTYAOZWHWGPLZBCWHKZPLZCYPPLZQZR - ZHSUBYJIJGBCAXKFEWLBUOZXDYSHSUUGWPYLXCYRUUGWMXTWKWOWLBWJUPUQUUGWRYNWTYOXB - YQUUGWQYMWHPWLBWNURTWLBWSPUSUUGXAYPWNPWLBWHUTTVAVBVCWNCUOZYSUUFHSUUHYLYTY - RUUEUUHWOYAWKXTWNCWJUPVDUUHYNUUAYOUUCYQUUDUUHYMWGWHPWNCBUTTUUHWSUUBBPWNCW - HURTWNCYPPUSVAVBVCWHAUOZUUFYIHSUUIYTYBUUEYHUUIWKXSXTYAWHAWJUPVEUUIUUAYCUU - CYEUUDYGWHAWGPUSUUIUUBYDBPWHACUTTUUIYPYFCPWHABURTVAVBVCVFVGVHXOYIXPHSYIYB - XOWISNZXPYBYHVIXOUUJYBXPXOXIXLRZUUJXSXTRZRZXPUUJYBRXNXLXIXJXLXMVJVKYBUULU - UJXSXTYAVQVKUUKUUMRZDWIUOZXPUUKUUJXTUUOXSBWIDVLVMUUNXPUUOXSUUKUUJXSXTVNUU - OXKWJADWIUDVOVRVSVTWAWBWCWDWEWFWEWF $. + cvv 3simpa eleq2d syl5ibrcom syl2an exp32 syl7 rexlimdv sylbid biimtrid + mpd ) ABCKZUALAWGGMZHMZUDUEZNZIMZWJNZJMZWJNZOZWHWLWNKZPLZWLWNWHKZPLZWNWHW + LKZPLZQZRZHSUBZIJGUFZUCZLZDSNZAENZBDUDUEZNZCFNZOZRZAXKNZAWGUAXGHGIJUGUHXO + XHWGAXFLZXPXOXJWGVPNXHXQUIXIXJXLXMUJBCUKAWGEVPXFULUMXQWGAKXFNZXOXPWGAXFUN + XOXRAWJNZBWJNZCWJNZOZAWGPLZBCAKZPLZCABKZPLZQZRZHSUBZXPXNXRYJUIZXIXLXMXJYK + XEWKXTWOOZWHBWNKZPLZBWSPLZWNWHBKZPLZQZRZHSUBWKXTYAOZWHWGPLZBCWHKZPLZCYPPL + ZQZRZHSUBYJIJGBCAXKFEWLBUOZXDYSHSUUGWPYLXCYRUUGWMXTWKWOWLBWJUPUQUUGWRYNWT + YOXBYQUUGWQYMWHPWLBWNURTWLBWSPUSUUGXAYPWNPWLBWHUTTVAVBVCWNCUOZYSUUFHSUUHY + LYTYRUUEUUHWOYAWKXTWNCWJUPVDUUHYNUUAYOUUCYQUUDUUHYMWGWHPWNCBUTTUUHWSUUBBP + WNCWHURTWNCYPPUSVAVBVCWHAUOZUUFYIHSUUIYTYBUUEYHUUIWKXSXTYAWHAWJUPVEUUIUUA + YCUUCYEUUDYGWHAWGPUSUUIUUBYDBPWHACUTTUUIYPYFCPWHABURTVAVBVCVFVGVHXOYIXPHS + YIYBXOWISNZXPYBYHVIXOUUJYBXPXOXIXLRZUUJXSXTRZRZXPUUJYBRXNXLXIXJXLXMVJVKYB + UULUUJXSXTYAVQVKUUKUUMRZDWIUOZXPUUKUUJXTUUOXSBWIDVLVMUUNXPUUOXSUUKUUJXSXT + VNUUOXKWJADWIUDVOVRVSWFVTWAWBWCWDWEWDWE $. $} $( Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] @@ -543519,32 +544288,32 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ( wcel w3a wa cop wbr ccgr cbtwn wrex wb wi 3jca adantr anbi2d cn cee cfv ccolin w3o ccgr3 brcolinear 3adant3 anbi1d simp1 simp3r simp3l jca simp21 cv simp23 axsegcon syl simprlr simprrr an4 simpl1 simpl21 simpl22 simpl3l - simpl3r cgrcomlr syl122anc simpl23 cgrextend syl133anc sylbid syl5bi expr - simpr imp cgrcom simpl2 brcgr3 syl113anc 3imtr4d an32s reximdva mpd exp32 - 3ancoma btwncom simp3 simp22 syl112anc simpll simplr adantl sylcom bitrid - sylan2b ex expdimp sylbird cgrxfr cgr3permute1 biimprd adantld syld 3jaod - syl131anc expd impd ) GUAHZAGUBUCZHZBXJHZCXJHZIZDXJHZFXJHZJZIZABCKZUDLZAB - KZDFKZMLZJAXSNLZBCAKNLZCYANLZUEZYCJAXSKDFEUOZKZKUFLZEXJOZXRXTYGYCXIXNXTYG - PXQABCGUGUHUIXRYGYCYKXRYDYCYKQZYEYFXRYDYCYKXRYDYCJZJZDYINLZDYHKZACKZMLZJZ - EXJOZYKYNXIXPXOJZXKXMJZIZYTXRUUCYMXRXIUUAUUBXIXNXQUJZXRXPXOXIXNXOXPUKXIXN - XOXPULUMXRXKXMXIXKXLXMXQUNZXIXKXLXMXQUPZUMRSEFDACGUQURYNYSYJEXJXRYHXJHZYM - YSYJQXRUUGJZYMJYOYQYPMLZJZYCUUIXSYIMLZIZYSYJUUHYMUUJUULUUHYMUUJJZJYCUUIUU - KUUHYDYCUUJUSUUHYMYOUUIUTUUHUUMUUKUUMYDYOJZYCUUIJZJZUUHUUKYDYCYOUUIVAUUHU - UPUUNBAKFDKMLZUUIJZJZUUKUUHUUOUURUUNUUHYCUUQUUIUUHXIXKXLXOXPYCUUQPXIXNXQU - UGVBZXKXLXMXIXQUUGVCZXKXLXMXIXQUUGVDZXOXPXIXNUUGVEZXOXPXIXNUUGVFZABDFGVGV - HUITUUHXIXLXKXMXPXOUUGUUSUUKQUUTUVBUVAXKXLXMXIXQUUGVIZUVDUVCXRUUGVOZBACFD - YHGVJVKVLVMVPRVNUUHYSUUJPYMUUHYRUUIYOUUHXIXOUUGXKXMYRUUIPUUTUVCUVFUVAUVED - YHACGVQVHTSUUHYJUULPZYMUUHXIXNXOXPUUGUVGUUTXIXNXQUUGVRZUVCUVDUVFABCDFYHGV - SVTZSWAWBWCWDWEXRYEBYQNLZYLXIXNUVJYEPZXQXNXIXLXKXMIUVKXKXLXMWFBACGWGWPUHX - RUVJYCYKXRUVJYCJZJZFYPNLZYIXSMLZJZEXJOZYKXRUVQUVLXRXIXQXLXMUVQUUDXIXNXQWH - ZXIXKXLXMXQWIZUUFEDFBCGUQWJSUVMUVPYJEXJXRUUGUVLUVPYJQUUHUVLUVPYJUUHUVJUVN - JZYCUUKJZJZUULUVLUVPJZYJUUHUWBUUIUULUUHXIXNXOXPUUGUWBUUIQUUTUVHUVCUVDUVFA - BCDFYHGVJVTUWAUUIUULQUVTUWAUUIUULUWAUUIJYCUUIUUKYCUUKUUIWKUWAUUIVOYCUUKUU - IWLRWQWMWNUWCUVTYCUVOJZJUUHUWBUVJYCUVNUVOVAUUHUWDUWAUVTUUHUVOUUKYCUUHXIXP - UUGXLXMUVOUUKPUUTUVDUVFUVBUVEFYHBCGVQVHTTWOUVIWAWRWBWCWDWEWSXRYFYCYKXRYFY - CJZYHYBNLZACBKKDYHFKKUFLZJZEXJOZYKXRXIXKXMXLXQUWEUWIQUUDUUEUUFUVSUVRACBDE - FGWTXFXRUWHYJEXJUUHUWGYJUWFUUHYJUWGUUHXIXNXOXPUUGYJUWGPUUTUVHUVCUVDUVFABC - DFYHGXAVTXBXCWCXDXGXEXHVL $. + simpl3r cgrcomlr syl122anc simpl23 simpr cgrextend syl133anc biimtrid imp + sylbid expr cgrcom simpl2 brcgr3 syl113anc 3imtr4d an32s reximdva 3ancoma + mpd exp32 btwncom sylan2b simp22 syl112anc simpll simplr ex adantl sylcom + bitrid expdimp sylbird cgrxfr syl131anc cgr3permute1 biimprd adantld syld + simp3 expd 3jaod impd ) GUAHZAGUBUCZHZBXJHZCXJHZIZDXJHZFXJHZJZIZABCKZUDLZ + ABKZDFKZMLZJAXSNLZBCAKNLZCYANLZUEZYCJAXSKDFEUOZKZKUFLZEXJOZXRXTYGYCXIXNXT + YGPXQABCGUGUHUIXRYGYCYKXRYDYCYKQZYEYFXRYDYCYKXRYDYCJZJZDYINLZDYHKZACKZMLZ + JZEXJOZYKYNXIXPXOJZXKXMJZIZYTXRUUCYMXRXIUUAUUBXIXNXQUJZXRXPXOXIXNXOXPUKXI + XNXOXPULUMXRXKXMXIXKXLXMXQUNZXIXKXLXMXQUPZUMRSEFDACGUQURYNYSYJEXJXRYHXJHZ + YMYSYJQXRUUGJZYMJYOYQYPMLZJZYCUUIXSYIMLZIZYSYJUUHYMUUJUULUUHYMUUJJZJYCUUI + UUKUUHYDYCUUJUSUUHYMYOUUIUTUUHUUMUUKUUMYDYOJZYCUUIJZJZUUHUUKYDYCYOUUIVAUU + HUUPUUNBAKFDKMLZUUIJZJZUUKUUHUUOUURUUNUUHYCUUQUUIUUHXIXKXLXOXPYCUUQPXIXNX + QUUGVBZXKXLXMXIXQUUGVCZXKXLXMXIXQUUGVDZXOXPXIXNUUGVEZXOXPXIXNUUGVFZABDFGV + GVHUITUUHXIXLXKXMXPXOUUGUUSUUKQUUTUVBUVAXKXLXMXIXQUUGVIZUVDUVCXRUUGVJZBAC + FDYHGVKVLVOVMVNRVPUUHYSUUJPYMUUHYRUUIYOUUHXIXOUUGXKXMYRUUIPUUTUVCUVFUVAUV + EDYHACGVQVHTSUUHYJUULPZYMUUHXIXNXOXPUUGUVGUUTXIXNXQUUGVRZUVCUVDUVFABCDFYH + GVSVTZSWAWBWCWEWFXRYEBYQNLZYLXIXNUVJYEPZXQXNXIXLXKXMIUVKXKXLXMWDBACGWGWHU + HXRUVJYCYKXRUVJYCJZJZFYPNLZYIXSMLZJZEXJOZYKXRUVQUVLXRXIXQXLXMUVQUUDXIXNXQ + XEZXIXKXLXMXQWIZUUFEDFBCGUQWJSUVMUVPYJEXJXRUUGUVLUVPYJQUUHUVLUVPYJUUHUVJU + VNJZYCUUKJZJZUULUVLUVPJZYJUUHUWBUUIUULUUHXIXNXOXPUUGUWBUUIQUUTUVHUVCUVDUV + FABCDFYHGVKVTUWAUUIUULQUVTUWAUUIUULUWAUUIJYCUUIUUKYCUUKUUIWKUWAUUIVJYCUUK + UUIWLRWMWNWOUWCUVTYCUVOJZJUUHUWBUVJYCUVNUVOVAUUHUWDUWAUVTUUHUVOUUKYCUUHXI + XPUUGXLXMUVOUUKPUUTUVDUVFUVBUVEFYHBCGVQVHTTWPUVIWAWQWBWCWEWFWRXRYFYCYKXRY + FYCJZYHYBNLZACBKKDYHFKKUFLZJZEXJOZYKXRXIXKXMXLXQUWEUWIQUUDUUEUUFUVSUVRACB + DEFGWSWTXRUWHYJEXJUUHUWGYJUWFUUHYJUWGUUHXIXNXOXPUUGYJUWGPUUTUVHUVCUVDUVFA + BCDFYHGXAVTXBXCWCXDXFXGXHVO $. $} $( Change some conditions for outer five segment predicate. (Contributed by @@ -544276,25 +545045,25 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and wo wceq simpl1 simpl2l simprl simpl3l simpl2r syl122anc simprr jca sylibr simpl3r weq simplrr simpll simplr simpr btwnconn1lem2 syl2an opeq2 breq2d breq1d anbi12d anbi2d biimpac jca32 btwnconn1lem13 syl5 expdimp mpd exp32 - ex an4s rexlimdvv orcom simprrl adantl syl5ibrcom simprll orim12d syl5bi - ) EUAJZAEUBUCZJZBWPJZKZCWPJZDWPJZKZLZABUDBCUDKBACMZNOBADMZNOKKZKZDAFPZMZN - OZDXHMCDMZQOZKZCAGPZMZNOZCXNMXKQOZKZKZGWPRFWPRZCXENOZDXDNOZUNZXCXTXFXCXMF - WPRZXRGWPRZXTXCWOWQXAXBYDWOWSXBUEZWOWQWRXBUFZWOWSWTXAUGWOWSXBUHZFADCDESUI - XCWOWQWTXBYEYFYGWOWSWTXAUJYHGACCDESUIXMXRFGWPWPUKULUMXGXSYCFGWPWPXGXHWPJZ - XNWPJZKZXSYCXCYKXFXSYCXCYKKZXFXSKZKZCXHUOZDXNUOZUNZYCYNXHAHPZMZNOXHYRMCBM - QOKZXNAIPZMZNOZXNUUAMZDBMZQOZKZKZIWPRHWPRZYQYNYTHWPRZUUGIWPRZKZUUIYLUULYM - YLUUJUUKYLWOWQYIWTWRUUJWOWSXBYKUPZWQWRWOXBYKUQZXCYIYJURZWTXAWOWSYKUSZWQWR - WOXBYKUTZHAXHCBESVAYLWOWQYJXAWRUUKUUMUUNXCYIYJVBWTXAWOWSYKVEZUUQIAXNDBESV - AVCUMYTUUGHIWPWPUKVDYNUUHYQHIWPWPYNYRWPJZUUAWPJZKZUUHYQYLUVAYMUUHYQYLUVAK - ZYMUUHKZKIHVFZYQUVBWOWQWRLZWTXAYILZYJUUSUUTLZLXFXSUUHLUVDUVCUVBUVEUVFUVGY - LUVEUVAYLWOWQWRUUMUUNUUQTUMZYLUVFUVAYLWTXAYIUUPUURUUOTUMZUVBYJUUSUUTXCYIY - JUVAVGZYLUUSUUTURZYLUUSUUTVBTTUVCXFXSUUHXFXSUUHVHXFXSUUHVIYMUUHVJTABCDEUU - AHFGVKVLUVBUVCUVDYQUVCUVDKYMYTXNYSNOZXNYRMZUUEQOZKZKZKZUVBYQUVDUVCUVQUVDU - UHUVPYMUVDUUGUVOYTUVDUUCUVLUUFUVNUVDUUBYSXNNUUAYRAVMVNUVDUUDUVMUUEQUUAYRX - NVMVOVPVQVQVRUVBUVQYQUVBUVEUVFYJUUSKZKKXFXSUVPLYQUVQUVBUVEUVFUVRUVHUVIUVB - YJUUSUVJUVKVCVSUVQXFXSUVPXFXSUVPVHXFXSUVPVIYMUVPVJTABCDEHFGVTVLWEWAWBWCWF - WDWGWCYQYPYOUNYNYCYOYPWHYNYPYAYOYBYNYAYPXPYMXPYLXFXMXPXQWIWJYPXEXOCNDXNAV - MVNWKYNYBYOXJYMXJYLXFXJXLXRWLWJYOXDXIDNCXHAVMVNWKWMWNWCWFWDWGWC $. + an4s rexlimdvv orcom simprrl adantl syl5ibrcom simprll orim12d biimtrid + ex ) EUAJZAEUBUCZJZBWPJZKZCWPJZDWPJZKZLZABUDBCUDKBACMZNOBADMZNOKKZKZDAFPZ + MZNOZDXHMCDMZQOZKZCAGPZMZNOZCXNMXKQOZKZKZGWPRFWPRZCXENOZDXDNOZUNZXCXTXFXC + XMFWPRZXRGWPRZXTXCWOWQXAXBYDWOWSXBUEZWOWQWRXBUFZWOWSWTXAUGWOWSXBUHZFADCDE + SUIXCWOWQWTXBYEYFYGWOWSWTXAUJYHGACCDESUIXMXRFGWPWPUKULUMXGXSYCFGWPWPXGXHW + PJZXNWPJZKZXSYCXCYKXFXSYCXCYKKZXFXSKZKZCXHUOZDXNUOZUNZYCYNXHAHPZMZNOXHYRM + CBMQOKZXNAIPZMZNOZXNUUAMZDBMZQOZKZKZIWPRHWPRZYQYNYTHWPRZUUGIWPRZKZUUIYLUU + LYMYLUUJUUKYLWOWQYIWTWRUUJWOWSXBYKUPZWQWRWOXBYKUQZXCYIYJURZWTXAWOWSYKUSZW + QWRWOXBYKUTZHAXHCBESVAYLWOWQYJXAWRUUKUUMUUNXCYIYJVBWTXAWOWSYKVEZUUQIAXNDB + ESVAVCUMYTUUGHIWPWPUKVDYNUUHYQHIWPWPYNYRWPJZUUAWPJZKZUUHYQYLUVAYMUUHYQYLU + VAKZYMUUHKZKIHVFZYQUVBWOWQWRLZWTXAYILZYJUUSUUTLZLXFXSUUHLUVDUVCUVBUVEUVFU + VGYLUVEUVAYLWOWQWRUUMUUNUUQTUMZYLUVFUVAYLWTXAYIUUPUURUUOTUMZUVBYJUUSUUTXC + YIYJUVAVGZYLUUSUUTURZYLUUSUUTVBTTUVCXFXSUUHXFXSUUHVHXFXSUUHVIYMUUHVJTABCD + EUUAHFGVKVLUVBUVCUVDYQUVCUVDKYMYTXNYSNOZXNYRMZUUEQOZKZKZKZUVBYQUVDUVCUVQU + VDUUHUVPYMUVDUUGUVOYTUVDUUCUVLUUFUVNUVDUUBYSXNNUUAYRAVMVNUVDUUDUVMUUEQUUA + YRXNVMVOVPVQVQVRUVBUVQYQUVBUVEUVFYJUUSKZKKXFXSUVPLYQUVQUVBUVEUVFUVRUVHUVI + UVBYJUUSUVJUVKVCVSUVQXFXSUVPXFXSUVPVHXFXSUVPVIYMUVPVJTABCDEHFGVTVLWNWAWBW + CWEWDWFWCYQYPYOUNYNYCYOYPWGYNYPYAYOYBYNYAYPXPYMXPYLXFXMXPXQWHWIYPXEXOCNDX + NAVMVNWJYNYBYOXJYMXJYLXFXJXLXRWKWIYOXDXIDNCXHAVMVNWJWLWMWCWEWDWFWC $. $} $( Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. @@ -544443,39 +545212,39 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and vq brab vex opth biid 3anbi123i 2rexbii rexbii wi simpl2l ad2antrl eleenn syl simprlr simprll adantl axdimuniq syl22anc fveq2d rexeqdv exbiri eleq1 anassrs bi2anan9 anbi2d opeq12 breq1d breq2d opeq1 adantr sylan9bb imbi1d - wb anbi12d 3imtr4d com12 expd 3impd rexlimdvv rexlimdvva rexlimdva syl5bi - simpl1 simpl2r simpl3l simpl3r eqidd simpr eqeq1d 3anbi23d anbi1d rspc2ev - expr opeq2 syl113anc 3anbi13d syl3anc 3anbi3d rexeqbidv rspcev syl2anc ex - fveq2 impbid bitrid ) BCLZDELZUBMGNZHNZLZYAOZINZJNZLZYBOZANZYIUCMZYEYGYKL - ZUDMZPZAKNZUEUFZQZRZJYQQZIYQQZHYQQZGYQQZKSQZFSTZBFUEUFZTZCUUFTZPZDUUFTZEU - UFTZPZRZYKYBUCMZYADYKLZUDMZPZAUUFQZUANZYEOZUPNZYIOZYRRZJYQQZIYQQHYQQZGYQQ - KSQYFUVBYRRZJYQQZIYQQHYQQZGYQQKSQUUDUAUPYAYBUBBCUGDEUGUUSYAOZUVEUVHKGSYQU - VIUVDUVGHIYQYQUVIUVCUVFJYQUVIUUTYFUVBYRUVIUUTYAYEOYFUUSYAYEUHYAYEUIUJUKUL - UMUMUVAYBOZUVHUUBKGSYQUVJUVGYTHIYQYQUVJUVFYSJYQUVJUVBYJYFYRUVJUVBYBYIOYJU - VAYBYIUHYBYIUIUJUNULUMUMAKUPUAGHIJUOUQUUMUUDUURUUDYCBOZYDCOZPZYGDOZYHEOZP - 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See ~ brofs and (Revised by NM, 17-Jun-2017.) $) hilbert1.2 $p |- ( P =/= Q -> E* x e. LinesEE ( P e. x /\ Q e. x ) ) $= ( vy wne cv wcel wa weq wi clines2 wral wceq simprl simprr simpl linethru - syl3anc ex eleq2w wrmo cline2 co anim12d eqtr3 syl6 syl5bi expd ralrimivv - an4 anbi12d rmo4 sylibr ) BCEZBAFZGZCUOGZHZBDFZGZCUSGZHZHZADIZJZDKLAKLURA - KUAUNVEADKKUNUOKGZUSKGZHZVCVDVHVCHVFURHZVGVBHZHZUNVDVFVGURVBUJUNVKUOBCUBU - CZMZUSVLMZHVDUNVIVMVJVNUNVIVMUNVIHVFURUNVMUNVFURNUNVFUROUNVIPUOBCQRSUNVJV - NUNVJHVGVBUNVNUNVGVBNUNVGVBOUNVJPUSBCQRSUDUOUSVLUEUFUGUHUIURVBADKVDUPUTUQ - VAADBTADCTUKULUM $. + syl3anc ex eleq2w wrmo an4 cline2 co anim12d syl6 biimtrid expd ralrimivv + eqtr3 anbi12d rmo4 sylibr ) BCEZBAFZGZCUOGZHZBDFZGZCUSGZHZHZADIZJZDKLAKLU + RAKUAUNVEADKKUNUOKGZUSKGZHZVCVDVHVCHVFURHZVGVBHZHZUNVDVFVGURVBUBUNVKUOBCU + CUDZMZUSVLMZHVDUNVIVMVJVNUNVIVMUNVIHVFURUNVMUNVFURNUNVFUROUNVIPUOBCQRSUNV + JVNUNVJHVGVBUNVNUNVGVBNUNVGVBOUNVJPUSBCQRSUEUOUSVLUJUFUGUHUIURVBADKVDUPUT + UQVAADBTADCTUKULUM $. $} ${ @@ -545478,7 +546247,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and E. a e. LinesEE ( P e. a /\ Q e. a /\ R e. a ) ) ) @= ( vp vq vn vx wcel wa cop ccolin wbr cv w3a clines2 wrex wne cline2 co wceq cee cfv cn wi ellines df-3an cab fvline eqeq2d cvv eleq2 3anbi123d - breq1 eqid elab4g 3anbi123i bitrdi pm5.32i 3an6 syl5bi adantld expd + breq1 eqid elab4g 3anbi123i bitrdi pm5.32i 3an6 biimtrid adantld expd sylbid sylan2br expr impd anassrs rexlimdva rexlimivv sylbi rexlimiv impbid1 ) BDKCEKLABCMNOZAFPZKZBVQKZCVQKZQZFRS?WAVPFRVQRKGPZHPZTZVQWBWCUAU BZUCZLZHIPZUDUEZSZGWISIUFSWAVPUGZVQIHGUHWJWKIGUFWIWHUFKZWBWIKZLWGWKHWIWLW @@ -545870,11 +546639,11 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and $( The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) $) hfuni $p |- ( A e. Hf -> U. A e. Hf ) $= - ( chf wcel cuni crnk cfv com rankuni wss wtr rankon ontrci df-tr elhf2g ibi - mpbi word wa wi cvv con0 eqeltrri onordi ordom ordtr2 mp2an eqeltrid uniexg - sylancr wb syl mpbird ) ABCZADZBCZUNEFZGCZUMUPAEFZDZGAHZUMUSURIZURGCZUSGCZU - RJVAURAKLURMPUMVBABNOUSQGQVAVBRVCSUSUPUSUAUTUNKUBUCUDUSURGUEUFUIUGUMUNTCUOU - QUJABUHUNTNUKUL $. + ( chf wcel cuni crnk cfv com rankuni wss wtr con0 rankon ax-mp df-tr elhf2g + ontr mpbi ibi word cvv wa wi eqeltrri onordi ordom ordtr2 mp2an eqeltrid wb + sylancr uniexg syl mpbird ) ABCZADZBCZUOEFZGCZUNUQAEFZDZGAHZUNUTUSIZUSGCZUT + GCZUSJZVBUSKCVEALUSPMUSNQUNVCABORUTSGSVBVCUAVDUBUTUQUTKVAUOLUCUDUEUTUSGUFUG + UJUHUNUOTCUPURUIABUKUOTOULUM $. $( The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) $) @@ -545924,7 +546693,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and wrex risset eqeq1 imbi1d albidv fveq2 eqeq2d eleq1 imbi2d imbi12d cbvalv weq bitrdi nn0suc eqcom vex rankeq0 bicomi bitr3i biimprd syl6bi adantrd com23 a1dd a1i jaod mpd bi2.04 albii 19.21v bitri ralbii r19.21v 3imtr4g - a2d alrimdv omsinds sp syl com12 rexlimdv syl5bi elhf2 anbi12i 2ralbii + a2d alrimdv omsinds sp syl com12 rexlimdv biimtrid elhf2 anbi12i 2ralbii imbi1i anbi2i 3imtr4i ssrdv ) ICJZAKZLJZBKZLJZMZWMWOUAUBCJZNZBCOACOZMZDL CWLWMPQRJZWOPQRJZMZWRNZBCOACOZMZDKZPQZRJZXHCJZNXAXHLJZXKNXJEKZXISZERUCXG XKEXIRUDXGXNXKERXMRJZXGXNXKNXOXNXGXKXOXNXGXKNZNZDTZXQFKZXISZXPNZDTZGKZHK @@ -546104,22 +546873,22 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ( vr vs vt cv wbr wa wi wal brin vex brcnv weq breq1 imbi12d spvv breq2 ccnv cin wrel cdm wer wss inss2 relcnv relss mp2 a1i eqidd anbi2i anbi12i wceq bitri anbi1d 2albidv anbi12d imbi1d albidv anbi2d pm3.3 adantrd impd - com23 4syl adantld jcad bitr2i syl6ib syl5bi bicomi anbi12ci biimpi jctil - 3bitr4i alrimiv alrimivv dfer2 syl3anbrc ) BHZCHZAIZWCDHZAIZJZWBWEAIZKZDL - CLZBLZAAUAZUBZUCZWMUDZWOUOEHZFHZWMIZWQWPWMIZKZWRWQGHZWMIZJZWPXAWMIZKZJZGL - ZFLELWOWMUEWNWKWMWLUFWLUCWNAWLUGAUHWMWLUIUJUKWKWOULWKXGEFWKXFGWKXEWTXCWPW - QAIZWQWPAIZJZWQXAAIZXAWQAIZJZJZWKXDWRXJXBXMWRXHWPWQWLIZJZXJWPWQAWLMZXOXIX - HWPWQAENZFNZOZUMUPXBXKWQXAWLIZJXMWQXAAWLMYAXLXKWQXAAXSGNZOUMUPUNWKXNWPXAA - IZXAWPAIZJZXDWKXNYCYDWKWPWCAIZWFJZWPWEAIZKZDLZCLZXHWQWEAIZJZYHKZDLZXHXKJZ - YCKZXNYCKWJYKBEBEPZWIYICDYRWGYGWHYHYRWDYFWFWBWPWCAQUQWBWPWEAQRURSYJYOCFCF - PZYIYNDYSYGYMYHYSYFXHWFYLWCWQWPATWCWQWEAQZUSUTVASYNYQDGDGPZYMYPYHYCUUAYLX - KXHWEXAWQATVBWEXAWPATRSYQXJXMYCYQXHXMYCKXIYQXMXHYCYQXKXHYCKXLYQXHXKYCXHXK - YCVCVFVDVFVDVEVGWKXAWCAIZWFJZXAWEAIZKZDLZCLZXLYLJZUUDKZDLZXLXIJZYDKZXNYDK - WJUUGBGBGPZWIUUECDUUMWGUUCWHUUDUUMWDUUBWFWBXAWCAQUQWBXAWEAQRURSUUFUUJCFYS - UUEUUIDYSUUCUUHUUDYSUUBXLWFYLWCWQXAATYTUSUTVASUUIUULDEDEPZUUHUUKUUDYDUUNY - LXIXLWEWPWQATVBWEWPXAATRSUULXJXMYDUULXIXMYDKXHUULXMXIYDUULXLXIYDKXKXLXIYD - VCVHVFVHVEVGVIXDYCWPXAWLIZJYEWPXAAWLMUUOYDYCWPXAAXRYBOUMVJVKVLWRWSXPXIWQW - PWLIZJWRWSXHUUPXOXIUUPXHWQWPAXSXROVMXTVNXQWQWPAWLMVQVOVPVRVSEFGWOWMVTWA + com23 adantld bitr2i syl6ib biimtrid bicomi anbi12ci 3bitr4i biimpi jctil + 4syl jcad alrimiv alrimivv dfer2 syl3anbrc ) BHZCHZAIZWCDHZAIZJZWBWEAIZKZ + DLCLZBLZAAUAZUBZUCZWMUDZWOUOEHZFHZWMIZWQWPWMIZKZWRWQGHZWMIZJZWPXAWMIZKZJZ + GLZFLELWOWMUEWNWKWMWLUFWLUCWNAWLUGAUHWMWLUIUJUKWKWOULWKXGEFWKXFGWKXEWTXCW + PWQAIZWQWPAIZJZWQXAAIZXAWQAIZJZJZWKXDWRXJXBXMWRXHWPWQWLIZJZXJWPWQAWLMZXOX + IXHWPWQAENZFNZOZUMUPXBXKWQXAWLIZJXMWQXAAWLMYAXLXKWQXAAXSGNZOUMUPUNWKXNWPX + AAIZXAWPAIZJZXDWKXNYCYDWKWPWCAIZWFJZWPWEAIZKZDLZCLZXHWQWEAIZJZYHKZDLZXHXK + JZYCKZXNYCKWJYKBEBEPZWIYICDYRWGYGWHYHYRWDYFWFWBWPWCAQUQWBWPWEAQRURSYJYOCF + CFPZYIYNDYSYGYMYHYSYFXHWFYLWCWQWPATWCWQWEAQZUSUTVASYNYQDGDGPZYMYPYHYCUUAY + LXKXHWEXAWQATVBWEXAWPATRSYQXJXMYCYQXHXMYCKXIYQXMXHYCYQXKXHYCKXLYQXHXKYCXH + XKYCVCVFVDVFVDVEVPWKXAWCAIZWFJZXAWEAIZKZDLZCLZXLYLJZUUDKZDLZXLXIJZYDKZXNY + DKWJUUGBGBGPZWIUUECDUUMWGUUCWHUUDUUMWDUUBWFWBXAWCAQUQWBXAWEAQRURSUUFUUJCF + YSUUEUUIDYSUUCUUHUUDYSUUBXLWFYLWCWQXAATYTUSUTVASUUIUULDEDEPZUUHUUKUUDYDUU + NYLXIXLWEWPWQATVBWEWPXAATRSUULXJXMYDUULXIXMYDKXHUULXMXIYDUULXLXIYDKXKXLXI + YDVCVGVFVGVEVPVQXDYCWPXAWLIZJYEWPXAAWLMUUOYDYCWPXAAXRYBOUMVHVIVJWRWSXPXIW + QWPWLIZJWRWSXHUUPXOXIUUPXHWQWPAXSXROVKXTVLXQWQWPAWLMVMVNVOVRVSEFGWOWMVTWA $. $} @@ -546129,20 +546898,20 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and /\ A <_ B /\ ( C = A \/ ( A < C /\ C < B ) \/ C = B ) ) ) ) $= ( cxr wcel wa cle wbr w3a wceq clt wi simp1 a1i xrleltne biimprd syl5ibrcom wn wne wo cicc co w3o elicc1 xrletr exp5o com23 imp5q df-ne syl5bir adantlr - 3adant3r3 eqcom necon3bbii syl5bi 3exp com12 imp32 3adantr2 adantll anim12d - ex df-or 3orass pm5.6 orcom imbi2i bitri 3bitr4ri syl6ib 3jcad xrleid breq2 - xrltle adantr adantllr adantrd simpr 3jaod exp31 3impd breq1 ancoms adantld - ad3antrrr ad3antlr impbid bitrd ) ADEZBDEZFZCABUAUBECDEZACGHZCBGHZIZWLABGHZ - CAJZACKHZCBKHZFZCBJZUCZIZABCUDWKWOXCWKWOWLWPXBWOWLLWKWLWMWNMNWIWJWLWMWNWPWI - WLWJWMWNWPLLWIWLWJWMWNWPACBUEUFUGUHWKWOWQRZXARZFWTLZXBWKWOXFWKWOFXDWRXEWSWI - WOXDWRLZWJWIWLWMXGWNXDCASZWIWLWMIZWRCAUIXIWRXHACOPUJULUKWJWOXEWSLZWIWJWLWNX - JWMWJWLWNXJWLWJWNXJLWLWJWNXJXEBCSZWLWJWNIZWSXABCCBUMUNXLWSXKCBOPUOUPUQURUSU - TVAVBWQWTXATZTXDXMLZXBXFWQXMVCWQWTXAVDXFXDXAWTTZLXNXDXAWTVEXOXMXDXAWTVFVGVH - VIVJVKWKXCWLWMWNXCWLLWKWLWPXBMNWKWLWPXBWMWKWLWPXBWMLWKWLFZWPFZWQWMWTXAXQWMW - QAAGHZWIXRWJWLWPAVLWECAAGVMQXQWRWMWSWIWLWPWRWMLZWJWIWLFXSWPACVNVOVPVQXQWMXA - WPXPWPVRZCBAGVMQVSVTWAWKWLWPXBWNWKWLWPXBWNLXQWQWNWTXAXQWNWQWPXTCABGWBQXPWTW - NLZWPWJWLYAWIWJWLFWSWNWRWLWJWSWNLCBVNWCWDUTVOXQWNXABBGHZWJYBWIWLWPBVLWFCBBG - WBQVSVTWAVKWGWH $. + 3adant3r3 eqcom necon3bbii biimtrid com12 imp32 3adantr2 adantll anim12d ex + 3exp df-or 3orass pm5.6 orcom imbi2i bitri 3bitr4ri syl6ib xrleid ad3antrrr + 3jcad breq2 xrltle adantr adantllr adantrd simpr 3jaod exp31 ancoms adantld + 3impd breq1 ad3antlr impbid bitrd ) ADEZBDEZFZCABUAUBECDEZACGHZCBGHZIZWLABG + HZCAJZACKHZCBKHZFZCBJZUCZIZABCUDWKWOXCWKWOWLWPXBWOWLLWKWLWMWNMNWIWJWLWMWNWP + WIWLWJWMWNWPLLWIWLWJWMWNWPACBUEUFUGUHWKWOWQRZXARZFWTLZXBWKWOXFWKWOFXDWRXEWS + WIWOXDWRLZWJWIWLWMXGWNXDCASZWIWLWMIZWRCAUIXIWRXHACOPUJULUKWJWOXEWSLZWIWJWLW + NXJWMWJWLWNXJWLWJWNXJLWLWJWNXJXEBCSZWLWJWNIZWSXABCCBUMUNXLWSXKCBOPUOVBUPUQU + RUSUTVAWQWTXATZTXDXMLZXBXFWQXMVCWQWTXAVDXFXDXAWTTZLXNXDXAWTVEXOXMXDXAWTVFVG + VHVIVJVMWKXCWLWMWNXCWLLWKWLWPXBMNWKWLWPXBWMWKWLWPXBWMLWKWLFZWPFZWQWMWTXAXQW + MWQAAGHZWIXRWJWLWPAVKVLCAAGVNQXQWRWMWSWIWLWPWRWMLZWJWIWLFXSWPACVOVPVQVRXQWM + XAWPXPWPVSZCBAGVNQVTWAWDWKWLWPXBWNWKWLWPXBWNLXQWQWNWTXAXQWNWQWPXTCABGWEQXPW + TWNLZWPWJWLYAWIWJWLFWSWNWRWLWJWSWNLCBVOWBWCUSVPXQWNXABBGHZWJYBWIWLWPBVKWFCB + BGWEQVTWAWDVMWGWH $. ${ $d k m n y ph $. $d k m n x ps $. $d x y $. @@ -546160,31 +546929,31 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and breq1 anbi12d spcev syl2anc csdm adantr php3 cdom ssdomg endomtr ad2antrr cvv ax-mp ad2antlr ensym domentr expcom ad2antll syld domnsym con2i nsyli syl5 impr word wb nnord ad2antrl ordtri1 con2bid mpbird jca31 elin anbi1i - bitri exp44 rexlimdv syl5bi com23 mpdd necon2bd imp31 pm2.21d equcomi a1i - jaod expr impd alrimiv jca eximd impancom 3syl ) ACIJCKZFKZLMZANZCOZFPUAZ - QUBZUUNGKZUCZQUDZGUUNJZADKZUUIUEZBNCDUFZRZDURZNZCOZAUUOCICUUNQUUMCFPUULCU - GCPUHUIZCQUHUJUUIISZUUIUUPLMZGPJZAUUORZGUUIUKZUVIUVKGPUUPPSZUVIAUUOUVMUVI - AUSZUUNUUPUVNUVMUVIANZCOZNZUUPUUNSZUVMUVIAUVQUVOUVPUVMUVOCULUMUNUUMUVPFUU - PPFGUFZUULUVOCUVSUUKUVIAUUJUUPUUILUOUPUQUTZVAVBVCVDVEVFVGVHVIUUNVGUEUUOUU - SVJUUNPVGUUMFPVKVLVMGVGUUNVNVOUURUVFGUUNUVRUVQUURUVFRUVTUVMUURUVPUVFUVMUU - RNZUVOUVECUVMUURCUVMCWDCUUQQCUUNUUPUVGCUUPUHVPVQVRUWAUVOUVEUWAUVONZAUVDUW - AUVIAVSUWBUVCDUWBUVABUVBUWBBUVAUVBUWAUVOBUVAUVBRUVAUUTUUIVTZDCUFZWAUWAUVO - BNZNZUVBUUTUUIWBUWFUWCUVBUWDUWFUWCUVBUVMUURUWEUWCWCZUVMUWEUURUWGUVMUWEUUR - UWGRUVMUWENZUWCUUQQUWHUWCUUTISZUUQQUBZUVMUVOUWCUWIRZBUVMUVIUWKAUVMUVINZUV - JUWKUVIGPWEZUVJUVHUWKUVLUVHUWCUWIUWCUVHUVAUWIUUTUUIWFUUIUUTWGWHTWIWJWKWKU - WHUWIUWCUWJUWIUUTHKZLMZHPJUWHUWCUWJRZHUUTUKUWHUWOUWPHPUWHUWNPSZUWOUWCUWJU - WHUWQUWONZUWCNZNZUUQUWNUWTUWQUUIUWNLMZANZCOZNZUWNUUPSZNZUWNUUQSZUWTUWQUXC - UXEUWHUWQUWOUWCWLUWTUWOBUXCUWHUWQUWOUWCWMUVMUVOBUWSWNUXBUWOBNCUUTDWOUVBUX - AUWOABUUIUUTUWNLWPEWQWRWSUWTUXEUUPUWNUEZWCZUWHUWRUWCUXIUWHUWRNZUWCUUTUUIW - TMZUXIUXJUVHUWCUXKRUWHUVHUWRUVMUVOUVHBUVMUVIUVHAUWLUVJUVHUWMUVLVAWKWKXAUV - HUWCUXKUUIUUTXBTWJUXJUXHUUIUUTXCMZUXKUXHUUPUWNXCMZUXJUXLUWNXGSUXHUXMRHWOU - UPUWNXGXDXHUXJUXMUUIUWNXCMZUXLUWEUXMUXNRZUVMUWRUVIUXOABUVIUXMUXNUUIUUPUWN - XETXFXIUWOUXNUXLRUWHUWQUXNUWOUXLUWOUXNUWNUUTLMUXLUUTUWNXJUUIUWNUUTXKWHXLX - MXNXRUXLUXKUUIUUTXOXPXQXNXSUWTUUPXTZUWNXTZUXEUXIYAUVMUXPUWEUWSUUPYBXFUWRU - XQUWHUWCUWQUXQUWOUWNYBXAYCUXPUXQNUXHUXEUUPUWNYDYEWSYFYGUXGUWNUUNSZUXENUXF - UWNUUNUUPYHUXRUXDUXEUUMUXCFUWNPFHUFZUULUXBCUXSUUKUXAAUUJUWNUUILUOUPUQUTYI - YJVAVBYKYLYMYNYOYPTYNYQYRUWDUVBRUWFDCYSYTUUAYMUUBYNUUCUUDUUETUUFUUGVEVDUU - H $. + bitri exp44 rexlimdv biimtrid com23 mpdd necon2bd imp31 pm2.21d equcomi + a1i jaod expr impd alrimiv jca eximd impancom 3syl ) ACIJCKZFKZLMZANZCOZF + PUAZQUBZUUNGKZUCZQUDZGUUNJZADKZUUIUEZBNCDUFZRZDURZNZCOZAUUOCICUUNQUUMCFPU + ULCUGCPUHUIZCQUHUJUUIISZUUIUUPLMZGPJZAUUORZGUUIUKZUVIUVKGPUUPPSZUVIAUUOUV + MUVIAUSZUUNUUPUVNUVMUVIANZCOZNZUUPUUNSZUVMUVIAUVQUVOUVPUVMUVOCULUMUNUUMUV + PFUUPPFGUFZUULUVOCUVSUUKUVIAUUJUUPUUILUOUPUQUTZVAVBVCVDVEVFVGVHVIUUNVGUEU + UOUUSVJUUNPVGUUMFPVKVLVMGVGUUNVNVOUURUVFGUUNUVRUVQUURUVFRUVTUVMUURUVPUVFU + VMUURNZUVOUVECUVMUURCUVMCWDCUUQQCUUNUUPUVGCUUPUHVPVQVRUWAUVOUVEUWAUVONZAU + VDUWAUVIAVSUWBUVCDUWBUVABUVBUWBBUVAUVBUWAUVOBUVAUVBRUVAUUTUUIVTZDCUFZWAUW + AUVOBNZNZUVBUUTUUIWBUWFUWCUVBUWDUWFUWCUVBUVMUURUWEUWCWCZUVMUWEUURUWGUVMUW + EUURUWGRUVMUWENZUWCUUQQUWHUWCUUTISZUUQQUBZUVMUVOUWCUWIRZBUVMUVIUWKAUVMUVI + NZUVJUWKUVIGPWEZUVJUVHUWKUVLUVHUWCUWIUWCUVHUVAUWIUUTUUIWFUUIUUTWGWHTWIWJW + KWKUWHUWIUWCUWJUWIUUTHKZLMZHPJUWHUWCUWJRZHUUTUKUWHUWOUWPHPUWHUWNPSZUWOUWC + UWJUWHUWQUWONZUWCNZNZUUQUWNUWTUWQUUIUWNLMZANZCOZNZUWNUUPSZNZUWNUUQSZUWTUW + QUXCUXEUWHUWQUWOUWCWLUWTUWOBUXCUWHUWQUWOUWCWMUVMUVOBUWSWNUXBUWOBNCUUTDWOU + VBUXAUWOABUUIUUTUWNLWPEWQWRWSUWTUXEUUPUWNUEZWCZUWHUWRUWCUXIUWHUWRNZUWCUUT + UUIWTMZUXIUXJUVHUWCUXKRUWHUVHUWRUVMUVOUVHBUVMUVIUVHAUWLUVJUVHUWMUVLVAWKWK + XAUVHUWCUXKUUIUUTXBTWJUXJUXHUUIUUTXCMZUXKUXHUUPUWNXCMZUXJUXLUWNXGSUXHUXMR + HWOUUPUWNXGXDXHUXJUXMUUIUWNXCMZUXLUWEUXMUXNRZUVMUWRUVIUXOABUVIUXMUXNUUIUU + PUWNXETXFXIUWOUXNUXLRUWHUWQUXNUWOUXLUWOUXNUWNUUTLMUXLUUTUWNXJUUIUWNUUTXKW + HXLXMXNXRUXLUXKUUIUUTXOXPXQXNXSUWTUUPXTZUWNXTZUXEUXIYAUVMUXPUWEUWSUUPYBXF + UWRUXQUWHUWCUWQUXQUWOUWNYBXAYCUXPUXQNUXHUXEUUPUWNYDYEWSYFYGUXGUWNUUNSZUXE + NUXFUWNUUNUUPYHUXRUXDUXEUUMUXCFUWNPFHUFZUULUXBCUXSUUKUXAAUUJUWNUUILUOUPUQ + UTYIYJVAVBYKYLYMYNYOYPTYNYQYRUWDUVBRUWFDCYSYTUUAYMUUBYNUUCUUDUUETUUFUUGVE + VDUUH $. $} ${ @@ -546737,7 +547506,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and fneint $p |- ( A Fne B -> |^| { x e. B | P e. x } C_ |^| { x e. A | P e. x } ) $= ( vy vz cfne wbr cv wcel crab cint wss wral wa eleq2w elrab fnessex 3expb - wrex intminss sstr sylan expl rexlimiv syl syl5bi ralrimiv ssint sylibr + wrex intminss sstr sylan expl rexlimiv syl biimtrid ralrimiv ssint sylibr ex ) BCGHZDAIJZACKLZEIZMZEUMABKZNUNUQLMULUPEUQUOUQJUOBJZDUOJZOZULUPUMUSAU OBAEDPQULUTUPULUTODFIZJZVAUOMZOZFCTZUPULURUSVEFBCDUORSVDUPFCVACJZVBVCUPVF VBOUNVAMVCUPUMVBAVACAFDPUAUNVAUOUBUCUDUEUFUKUGUHEUNUQUIUJ $. @@ -546852,30 +547621,30 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ( vx vy vw vz wceq cfne wbr wss wa wrex cvv wcel wi vt cv cref wex fnerel crab brrelex2i adantl rabexg syl ssrab2 a1i cuni wral eluni bitri fnessex eleq2i 3expia adantll sseq2 rspcev anim2d reximdv syld com23 impd exlimdv - ex syl5bi elunirab syl6ibr ssrdv unissi simpl sseqtrid eqssd 3expb expcom - eqtr2di ad2antll com12 ad2antrl jcad sseq1 rexbidv elrab simpr ralrimivva - reximdv2 mpd wb eqid isfne2 3syl mpbir2and cbvrexvw biimpi ralrimiv isref - jca32 breq2 breq1 anbi12d spcegv sylc simprrl fnebas eqtr3d eqtrdi vuniex - eqeltrrdi uniexb sylibr simprl fness syl3anc fnetr syl2anc impbid ) CDLZA - BMNZEUBZBOZAYCMNZYCAUCNZPZPZEUDZYAYBYIYAYBPZHUBZIUBZOZIAQZHBUFZRSZYOBOZAY - OMNZYOAUCNZPZPZYIYJBRSZYPYBUUBYAABMUEUGUHZYNHBRUIZUJYJYQYRYSYQYJYNHBUKZUL - YJYRCYOUMZLZUAUBZJUBZSZUUIKUBZOZPZJYOQZUAUUKUNKAUNZYJCUUFYJUACUUFYJUUHCSZ - UUHYKSZYNPZHBQZUUHUUFSUUPUUHUUKSZUUKASZPZKUDZYJUUSUUPUUHAUMZSUVCCUVDUUHFU - RKUUHAUOUPYJUVBUUSKYJUUTUVAUUSYJUVAUUTUUSYJUVAUUTUUSTYJUVAPZUUTUUQYKUUKOZ - PZHBQZUUSYBUVAUUTUVHTYAYBUVAUUTUVHHABUUHUUKUQUSUTUVEUVGUURHBUVEUVFYNUUQUV - AUVFYNTYJUVAUVFYNYMUVFIUUKAYLUUKYKVAVBVIUHVCVDVEVIVFVGVHVJYNHUUHBVKVLVMYJ - BUMZUUFCYOBUUEVNYJCDUVIYAYBVOGVTVPVQZYJUUNKUAAUUKYJUVAUUTPZPZUUMJBQZUUNYB - UVKUVMYAYBUVAUUTUVMJABUUHUUKUQVRUTUVLUUMUUMJBYOUVLUUIBSZUUMPZUUIYOSZUUMUV - LUVOUVNUUIYLOZIAQZPUVPUVLUVOUVNUVRUVOUVNTUVLUVNUUMVOULUVAUVOUVRTYJUUTUVOU - VAUVRUULUVAUVRTUVNUUJUVAUULUVRUVQUULIUUKAYLUUKUUIVAVBVSWAWBWCWDYNUVRHUUIB - YKUUILYMUVQIAYKUUIYLWEWFWGVLUVOUUMTUVLUVNUUMWHULWDWJWKWIYJUUBYPYRUUGUUOPW - LUUCUUDKUAJAYORCUUFFUUFWMZWNWOWPYJYSUUGUUKUUIOZJAQZKYOUNZUVJYJUWAKYOUUKYO - SUUKBSZUUKYLOZIAQZPZYJUWAYNUWEHUUKBYKUUKLYMUWDIAYKUUKYLWEWFWGUWFUWATYJUWE - UWAUWCUWEUWAUWDUVTIJAYLUUIUUKVAWQWRUHULVJWSYJUUBYPYSUUGUWBPWLUUCUUDKJYOAR - UUFCUVSFWTWOWPXAYHUUAEYORYCYOLZYDYQYGYTYCYOBWEUWGYEYRYFYSYCYOAMXBYCYOAUCX - CXDXDXEXFVIYAYHYBEYAYHYBYAYHPZYEYCBMNZYBYAYDYEYFXGZUWHUUBYDYCUMZDLUWIUWHU - VIRSUUBUWHUVIUWKRUWHUWKDUVIUWHCUWKDUWHYECUWKLUWJAYCCUWKFUWKWMZXHUJYAYHVOX - IZGXJEXKXLBXMXNYAYDYGXOUWMYCBRUWKDUWLGXPXQAYCBXRXSVIVHXT $. + biimtrid elunirab syl6ibr ssrdv unissi simpl eqtr2di sseqtrid eqssd 3expb + ex expcom ad2antll com12 ad2antrl jcad sseq1 rexbidv elrab simpr reximdv2 + mpd ralrimivva eqid isfne2 mpbir2and cbvrexvw biimpi ralrimiv isref jca32 + wb 3syl breq2 breq1 anbi12d spcegv simprrl fnebas eqtr3d eqtrdi eqeltrrdi + sylc vuniex uniexb sylibr simprl fness syl3anc fnetr syl2anc impbid ) CDL + ZABMNZEUBZBOZAYCMNZYCAUCNZPZPZEUDZYAYBYIYAYBPZHUBZIUBZOZIAQZHBUFZRSZYOBOZ + AYOMNZYOAUCNZPZPZYIYJBRSZYPYBUUBYAABMUEUGUHZYNHBRUIZUJYJYQYRYSYQYJYNHBUKZ + ULYJYRCYOUMZLZUAUBZJUBZSZUUIKUBZOZPZJYOQZUAUUKUNKAUNZYJCUUFYJUACUUFYJUUHC + SZUUHYKSZYNPZHBQZUUHUUFSUUPUUHUUKSZUUKASZPZKUDZYJUUSUUPUUHAUMZSUVCCUVDUUH + FURKUUHAUOUPYJUVBUUSKYJUUTUVAUUSYJUVAUUTUUSYJUVAUUTUUSTYJUVAPZUUTUUQYKUUK + OZPZHBQZUUSYBUVAUUTUVHTYAYBUVAUUTUVHHABUUHUUKUQUSUTUVEUVGUURHBUVEUVFYNUUQ + UVAUVFYNTYJUVAUVFYNYMUVFIUUKAYLUUKYKVAVBVSUHVCVDVEVSVFVGVHVIYNHUUHBVJVKVL + YJBUMZUUFCYOBUUEVMYJCDUVIYAYBVNGVOVPVQZYJUUNKUAAUUKYJUVAUUTPZPZUUMJBQZUUN + YBUVKUVMYAYBUVAUUTUVMJABUUHUUKUQVRUTUVLUUMUUMJBYOUVLUUIBSZUUMPZUUIYOSZUUM + UVLUVOUVNUUIYLOZIAQZPUVPUVLUVOUVNUVRUVOUVNTUVLUVNUUMVNULUVAUVOUVRTYJUUTUV + OUVAUVRUULUVAUVRTUVNUUJUVAUULUVRUVQUULIUUKAYLUUKUUIVAVBVTWAWBWCWDYNUVRHUU + IBYKUUILYMUVQIAYKUUIYLWEWFWGVKUVOUUMTUVLUVNUUMWHULWDWIWJWKYJUUBYPYRUUGUUO + PWTUUCUUDKUAJAYORCUUFFUUFWLZWMXAWNYJYSUUGUUKUUIOZJAQZKYOUNZUVJYJUWAKYOUUK + YOSUUKBSZUUKYLOZIAQZPZYJUWAYNUWEHUUKBYKUUKLYMUWDIAYKUUKYLWEWFWGUWFUWATYJU + WEUWAUWCUWEUWAUWDUVTIJAYLUUIUUKVAWOWPUHULVIWQYJUUBYPYSUUGUWBPWTUUCUUDKJYO + ARUUFCUVSFWRXAWNWSYHUUAEYORYCYOLZYDYQYGYTYCYOBWEUWGYEYRYFYSYCYOAMXBYCYOAU + CXCXDXDXEXKVSYAYHYBEYAYHYBYAYHPZYEYCBMNZYBYAYDYEYFXFZUWHUUBYDYCUMZDLUWIUW + HUVIRSUUBUWHUVIUWKRUWHUWKDUVIUWHCUWKDUWHYECUWKLUWJAYCCUWKFUWKWLZXGUJYAYHV + NXHZGXIEXLXJBXMXNYAYDYGXOUWMYCBRUWKDUWLGXPXQAYCBXRXSVSVHXT $. $} ${ @@ -546890,18 +547659,18 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ( vx vy wceq cref wbr cv wss cfne wa cvv wcel adantl cuni brrelex2i unexg wex cun refrel brrelex1i syl2anc ssun2 ssun1 eqimss2 adantr ssequn2 sylib eqcomd uneq12i uniun eqtr4i fness syl3anc wrex wral wo elun wi ssid sseq2 - a1i rspcev mpan2 refssex ex jaod syl5bi ralrimiv wb isref mpbir2and jca32 - syl breq2 breq1 anbi12d spcegv sylc vex ssex ad2antrl simprl simpl refbas - eqid ad2antll eqtr3d ssref simprrr reftr exlimdv impbid ) CDJZBAKLZBEMZNZ - AXAOLZXAAKLZPZPZEUCZWSWTXGWSWTPZABUDZQRZBXINZAXIOLZXIAKLZPZPZXGXHAQRZBQRZ - XJWTXPWSBAKUEUASWTXQWSBAKUEUFSABQQUBUGZXHXKXLXMXKXHBAUHVGXHXJAXINZCCDUDZJ - ZXLXRXSXHABUIVGXHXTCXHDCNZXTCJWSYBWTDCUJUKDCULUMUNZAXIQCXTFXTATZBTZUDXITC - YDDYEFGUOABUPUQZURUSXHXMYAHMZIMZNZIAUTZHXIVAZYCXHYJHXIYGXIRYGARZYGBRZVBXH - YJYGABVCXHYLYJYMYLYJVDXHYLYGYGNZYJYGVEYIYNIYGAYHYGYGVFVHVIVGWTYMYJVDWSWTY - MYJIBAYGVJVKSVLVMVNXHXJXMYAYKPVOXRHIXIAQXTCYFFVPVSVQVRXFXOEXIQXAXIJZXBXKX - EXNXAXIBVFYOXCXLXDXMXAXIAOVTXAXIAKWAWBWBWCWDVKWSXFWTEWSXFWTWSXFPZBXAKLZXD - WTYPXQXBDXATZJYQXBXQWSXEBXAEWEWFWGWSXBXEWHYPCDYRWSXFWIXECYRJZWSXBXDYSXCXA - AYRCYRWKZFWJSWLWMBXAQDYRGYTWNUSWSXBXCXDWOBXAAWPUGVKWQWR $. + a1i rspcev mpan2 refssex ex jaod biimtrid ralrimiv wb isref syl mpbir2and + jca32 breq2 breq1 anbi12d spcegv sylc vex ssex ad2antrl simprl simpl eqid + refbas ad2antll eqtr3d ssref simprrr reftr exlimdv impbid ) CDJZBAKLZBEMZ + NZAXAOLZXAAKLZPZPZEUCZWSWTXGWSWTPZABUDZQRZBXINZAXIOLZXIAKLZPZPZXGXHAQRZBQ + RZXJWTXPWSBAKUEUASWTXQWSBAKUEUFSABQQUBUGZXHXKXLXMXKXHBAUHVGXHXJAXINZCCDUD + ZJZXLXRXSXHABUIVGXHXTCXHDCNZXTCJWSYBWTDCUJUKDCULUMUNZAXIQCXTFXTATZBTZUDXI + TCYDDYEFGUOABUPUQZURUSXHXMYAHMZIMZNZIAUTZHXIVAZYCXHYJHXIYGXIRYGARZYGBRZVB + XHYJYGABVCXHYLYJYMYLYJVDXHYLYGYGNZYJYGVEYIYNIYGAYHYGYGVFVHVIVGWTYMYJVDWSW + TYMYJIBAYGVJVKSVLVMVNXHXJXMYAYKPVOXRHIXIAQXTCYFFVPVQVRVSXFXOEXIQXAXIJZXBX + KXEXNXAXIBVFYOXCXLXDXMXAXIAOVTXAXIAKWAWBWBWCWDVKWSXFWTEWSXFWTWSXFPZBXAKLZ + XDWTYPXQXBDXATZJYQXBXQWSXEBXAEWEWFWGWSXBXEWHYPCDYRWSXFWIXECYRJZWSXBXDYSXC + XAAYRCYRWJZFWKSWLWMBXAQDYRGYTWNUSWSXBXCXDWOBXAAWPUGVKWQWR $. $} @@ -546930,41 +547699,41 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). 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X /\ v e. ( F ` x ) ) ) -> x e. v ) $. neibastop1.6 $e |- ( ( ph /\ ( x e. X /\ v e. ( F ` x ) ) ) -> @@ -546987,63 +547756,63 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and cres inss1 sseli anassrs sylan2 adantrl simprl wel fvssunirn frnd sylib sspwuni ad2antrr sstrid sselda elpwid adantrr pweq ineq2d eleq2d rspcev eliun syl2anc sylibr wceq sseq1d adantr adantl iunss ralrimiva ralrimiv - iuneq1 syl12anc fnfvelrn elunii sylanbrc inelcm exlimdv syl5bi rexlimdv - expr impr eleq2 ad3antrrr sseldd cdif difss2d csuc rspe ad2ant2l bitrid - simprlr simpll simprll fveq1i snex eqtri eqtrdi pwidg snssd pwexd inss2 - fr0g elpwi sspwd ralrimivw ssralv mpan9 frsucmpt2 eqsstrd expcom finds2 - wf ssexd fvex elpw syl6ibr com12 mpbiran eleqtrrd peano2 sylancr simprr - ffnfv rabeq2i dfss3 velpw rexlimddv rexlimdvaa expimpd raleqbi1dv snidg - ralimdva peano1 mp2an eqeltrri sylancl rexrn ffvelcdmda simprrr simpllr - eluni2 sstrd simprrl elequ1 imbi12d rspcv syl3c exp32 rexlimdva rabssdv - elin 3impia eqsstrid sseq1 anbi12d ) AJPUOZIJUOZJQUPZIGUQZUOZUXOQUPZURZ - GPUSAJSUTZUOZBUQZNVAZJUTZVBZVCVDZBJVEZUXLAUXTJSUPZJCUQZNVAZLUQZUTZVBZVC - VDZLOVIZVFZUSZCSVJZSULUYPCSVGVHASRUOUXTUYGVKUAJSRVLVMVNAUYEBJUYAJUOUYAS - 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See ~ brofs and wne ctop neibastop1 topontop syl adantr eqid neii1 wceq toponuni ad2antrr sylan sseqtrrd cv wrex neii2 wral wi pweq ineq2d neeq1d raleqbi1dv elrab2 weq simprrr sspwd sslin simprrl wb snssg ad3antlr mpbird fveq2 rspcv ssn0 - ineq1d syl6an expr com23 expimpd syl5bi rexlimdv mpd jca ex wex n0 simprl - elin sseqtrd cvv ciun cmpt crdg com cres crn crab wf simpll simplr elpwid - cdif cbviunv iuneq2d eqtrid mpteq2i rdgeq1 ax-mp reseq1i cbvrexvw rexbidv - w3a bitrid cbvrabv neibastop2lem eleqtrd isneip syl2anc mpbir2and exlimdv - impbid ) AGMTZUIZKGUJZJUKULULTZKMUMZGIULZKUNZUOZUPUSZUIZUUBUUDUUJUUBUUDUI - ZUUEUUIUUKKJUQZMUUBJUTTZUUDKUULUMZAUUMUUAAJMURULTZUUMABDEHIJLMNOPQVAZMJVB - VCZVDZUUCJKUULUULVEZVFVJAMUULVGZUUAUUDAUUOUUTUUPMJVHVCZVIVKUUKUUCCVLZUMZU - VBKUMZUIZCJVMZUUIUUBUUMUUDUVFUURUUCCJKVNVJUUKUVEUUICJUVBJTUVBMUNZTZBVLZIU - LZUVBUNZUOZUPUSZBUVBVOZUIUUKUVEUUIVPZUVJHVLZUNZUOZUPUSZBUVPVOUVNHUVBUVGJU - VSUVMBUVPUVBHCWBZUVRUVLUPUVTUVQUVKUVJUVPUVBVQVRVSVTQWAUUKUVHUVNUVOUUKUVHU - IUVEUVNUUIUUKUVHUVEUVNUUIVPUUKUVHUVEUIZUIZUUFUVKUOZUUHUMZUVNUWCUPUSZUUIUW - BUVKUUGUMUWDUWBUVBKUUKUVHUVCUVDWCWDUVKUUGUUFWEVCUWBGUVBTZUVNUWEVPUWBUWFUV - CUUKUVHUVCUVDWFUUAUWFUVCWGAUUDUWAGUVBMWHWIWJUVMUWEBGUVBUVIGVGZUVLUWCUPUWG - UVJUUFUVKUVIGIWKWNVSWLVCUWCUUHWMWOWPWQWRWSWTXAXBXCUUBUUEUUIUUDUUIUAVLZUUH - TZUAXDUUBUUEUIZUUDUAUUHXEUWJUWIUUDUAUWIUWHUUFTZUWHUUGTZUIZUWJUUDUWHUUFUUG - XGUUBUUEUWMUUDUUBUUEUWMUIZUIZUUDUUNGUBVLZTUWPKUMUIUBJVMZUWOKMUULUUBUUEUWM - XFZAUUTUUAUWNUVAVIZXHUWOBCUCDEUBFGDVLZIULZUDVLZUNZUOZUPUSZUDUEXIUFUEVLZUG - MUGVLZIULZUFVLZUNZUOZXJZXJZXKZUWHUJZXLZXMXNZXOUQZVMZDMXPUWHUHHIUXQJKLMUEA - MLTUUAUWNNVIAMUVGUNUPUJYAIXQUUAUWNOVIUWOAUVIMTZEVLZUVJTZUWTUVJTYKUVJUYAUW - TUOUNUOUPUSAUUAUWNXRZPVJQUWOAUXTUYBUIZUVIUYATUYCRVJUWOAUYDUVBIULZUYAUNUOU - PUSCFVLVOFUVJVMUYCSVJAUUAUWNXSZUWRUUBUUEUWKUWLWFUWOUWHKUUBUUEUWKUWLWCXTUX - PUEXIUCUXFBMUVJUCVLZUNZUOZXJZXJZXKZUXOXLZXMUXNUYLVGUXPUYMVGUEXIUXMUYKUFUC - UXFUXLUYJUFUCWBZUXLBMUVJUXJUOZXJUYJUGBMUXKUYOUGBWBUXHUVJUXJUXGUVIIWKWNYBU - YNBMUYOUYIUYNUXJUYHUVJUXIUYGVQVRYCYDYBYEUXOUXNUYLYFYGYHUXSUYEUHVLZUNZUOZU - PUSZUHUXRVMZDCMUXSUXAUYQUOZUPUSZUHUXRVMDCWBZUYTUXEVUBUDUHUXRUDUHWBZUXDVUA - UPVUDUXCUYQUXAUXBUYPVQVRVSYIVUCVUBUYSUHUXRVUCVUAUYRUPVUCUXAUYEUYQUWTUVBIW - KWNVSYJYLYMYNUWOUUMGUULTUUDUUNUWQUIWGAUUMUUAUWNUUQVIUWOGMUULUYFUWSYOGUBJK - UULUUSYPYQYRWPWSYSWSWRYT $. + ineq1d syl6an expr com23 expimpd biimtrid rexlimdv mpd jca ex wex n0 elin + simprl sseqtrd cvv ciun cmpt crdg com cres crn crab cdif wf simpll simplr + w3a elpwid cbviunv iuneq2d eqtrid mpteq2i rdgeq1 reseq1i cbvrexvw rexbidv + ax-mp bitrid cbvrabv neibastop2lem eleqtrd isneip syl2anc exlimdv impbid + mpbir2and ) AGMTZUIZKGUJZJUKULULTZKMUMZGIULZKUNZUOZUPUSZUIZUUBUUDUUJUUBUU + DUIZUUEUUIUUKKJUQZMUUBJUTTZUUDKUULUMZAUUMUUAAJMURULTZUUMABDEHIJLMNOPQVAZM + JVBVCZVDZUUCJKUULUULVEZVFVJAMUULVGZUUAUUDAUUOUUTUUPMJVHVCZVIVKUUKUUCCVLZU + MZUVBKUMZUIZCJVMZUUIUUBUUMUUDUVFUURUUCCJKVNVJUUKUVEUUICJUVBJTUVBMUNZTZBVL + ZIULZUVBUNZUOZUPUSZBUVBVOZUIUUKUVEUUIVPZUVJHVLZUNZUOZUPUSZBUVPVOUVNHUVBUV + GJUVSUVMBUVPUVBHCWBZUVRUVLUPUVTUVQUVKUVJUVPUVBVQVRVSVTQWAUUKUVHUVNUVOUUKU + VHUIUVEUVNUUIUUKUVHUVEUVNUUIVPUUKUVHUVEUIZUIZUUFUVKUOZUUHUMZUVNUWCUPUSZUU + IUWBUVKUUGUMUWDUWBUVBKUUKUVHUVCUVDWCWDUVKUUGUUFWEVCUWBGUVBTZUVNUWEVPUWBUW + FUVCUUKUVHUVCUVDWFUUAUWFUVCWGAUUDUWAGUVBMWHWIWJUVMUWEBGUVBUVIGVGZUVLUWCUP + UWGUVJUUFUVKUVIGIWKWNVSWLVCUWCUUHWMWOWPWQWRWSWTXAXBXCUUBUUEUUIUUDUUIUAVLZ + UUHTZUAXDUUBUUEUIZUUDUAUUHXEUWJUWIUUDUAUWIUWHUUFTZUWHUUGTZUIZUWJUUDUWHUUF + UUGXFUUBUUEUWMUUDUUBUUEUWMUIZUIZUUDUUNGUBVLZTUWPKUMUIUBJVMZUWOKMUULUUBUUE + UWMXGZAUUTUUAUWNUVAVIZXHUWOBCUCDEUBFGDVLZIULZUDVLZUNZUOZUPUSZUDUEXIUFUEVL + ZUGMUGVLZIULZUFVLZUNZUOZXJZXJZXKZUWHUJZXLZXMXNZXOUQZVMZDMXPUWHUHHIUXQJKLM + UEAMLTUUAUWNNVIAMUVGUNUPUJXQIXRUUAUWNOVIUWOAUVIMTZEVLZUVJTZUWTUVJTYAUVJUY + AUWTUOUNUOUPUSAUUAUWNXSZPVJQUWOAUXTUYBUIZUVIUYATUYCRVJUWOAUYDUVBIULZUYAUN + UOUPUSCFVLVOFUVJVMUYCSVJAUUAUWNXTZUWRUUBUUEUWKUWLWFUWOUWHKUUBUUEUWKUWLWCY + BUXPUEXIUCUXFBMUVJUCVLZUNZUOZXJZXJZXKZUXOXLZXMUXNUYLVGUXPUYMVGUEXIUXMUYKU + FUCUXFUXLUYJUFUCWBZUXLBMUVJUXJUOZXJUYJUGBMUXKUYOUGBWBUXHUVJUXJUXGUVIIWKWN + YCUYNBMUYOUYIUYNUXJUYHUVJUXIUYGVQVRYDYEYCYFUXOUXNUYLYGYKYHUXSUYEUHVLZUNZU + OZUPUSZUHUXRVMZDCMUXSUXAUYQUOZUPUSZUHUXRVMDCWBZUYTUXEVUBUDUHUXRUDUHWBZUXD + VUAUPVUDUXCUYQUXAUXBUYPVQVRVSYIVUCVUBUYSUHUXRVUCVUAUYRUPVUCUXAUYEUYQUWTUV + BIWKWNVSYJYLYMYNUWOUUMGUULTUUDUUNUWQUIWGAUUMUUAUWNUUQVIUWOGMUULUYFUWSYOGU + BJKUULUUSYPYQYTWPWSYRWSWRYS $. $( The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, @@ -547201,19 +547970,19 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ( X = U. T /\ A. x e. S T Fne x ) ) ) $= ( wcel cv cuni wceq wral wa cfne wbr c0 eqid syl wss cvv cpw ctg cfv ciin cin wi riin0 unieqd unipw eqtr2di a1i wne wex n0 w3a unieq eqeq2d rspccva - 3adant1 fnebas eqtr4d 3expia exlimdv syl5bi pm2.61dne adantr adantl fnetr - fnemeet1 ex expcom 3expa ralrimdva simprl eqtr3d eqimss2 ad2antrl sspwuni - jcad sylibr breq2 cbvralvw fnetg ralimi sylbi ad2antll ssiin ssind inex1g - pwexg ad2antrr bastg sstrd isfne4 sylanbrc impbid ) GFHZGBIZJZKZBDLZMZEGU - AZCDCIZUBUCZUDZUEZNOZGEJZKZEAIZNOZADLZMZXBXHXJXMXBXHXJXBXHMGXGJZXIXBGXOKZ - XHXBXPDPDPKZXPUFXBXQXOXCJGXQXGXCCXCXEDUGUHGUIUJUKDPULXKDHZAUMXBXPADUNXBXR - XPAWQXAXRXPWQXAXRUOZGXKJZXOXAXRGXTKZWQWTYABXKDWRXKKWSXTGWRXKUPUQURUSXSXGX - KNOZXOXTKBCXKDFGVIZXGXKXOXTXOQZXTQUTRVAVBVCVDVEZVFXHXIXOKZXBEXGXIXOXIQZYD - UTVGVAVJXBXHXLADWQXAXRXHXLUFZXSYBYHYCXHYBXLEXGXKVHVKRVLVMVSXBXNXHXBXNMZYF - EXGUBUCZSXHYIGXIXOXBXJXMVNXBXPXNYEVFVOYIEXGYJYIEXCXFYIXIGSZEXCSXJYKXBXMXI - GVPVQEGVRVTYIEXESZCDLZEXFSXMYMXBXJXMEXDNOZCDLYMXLYNACDXKXDENWAWBYNYLCDEXD - WCWDWEWFCDXEEWGVTWHYIXGTHZXGYJSWQYOXAXNWQXCTHYOGFWJXCXFTWIRWKXGTWLRWMEXGX - IXOYGYDWNWOVJWP $. + 3adant1 fnemeet1 fnebas eqtr4d 3expia biimtrid pm2.61dne adantr adantl ex + exlimdv fnetr expcom 3expa ralrimdva jcad simprl eqimss2 ad2antrl sspwuni + eqtr3d sylibr breq2 cbvralvw fnetg sylbi ad2antll ssiin ssind pwexg bastg + ralimi inex1g ad2antrr sstrd isfne4 sylanbrc impbid ) GFHZGBIZJZKZBDLZMZE + GUAZCDCIZUBUCZUDZUEZNOZGEJZKZEAIZNOZADLZMZXBXHXJXMXBXHXJXBXHMGXGJZXIXBGXO + KZXHXBXPDPDPKZXPUFXBXQXOXCJGXQXGXCCXCXEDUGUHGUIUJUKDPULXKDHZAUMXBXPADUNXB + XRXPAWQXAXRXPWQXAXRUOZGXKJZXOXAXRGXTKZWQWTYABXKDWRXKKWSXTGWRXKUPUQURUSXSX + GXKNOZXOXTKBCXKDFGUTZXGXKXOXTXOQZXTQVARVBVCVIVDVEZVFXHXIXOKZXBEXGXIXOXIQZ + YDVAVGVBVHXBXHXLADWQXAXRXHXLUFZXSYBYHYCXHYBXLEXGXKVJVKRVLVMVNXBXNXHXBXNMZ + YFEXGUBUCZSXHYIGXIXOXBXJXMVOXBXPXNYEVFVSYIEXGYJYIEXCXFYIXIGSZEXCSXJYKXBXM + XIGVPVQEGVRVTYIEXESZCDLZEXFSXMYMXBXJXMEXDNOZCDLYMXLYNACDXKXDENWAWBYNYLCDE + XDWCWJWDWECDXEEWFVTWGYIXGTHZXGYJSWQYOXAXNWQXCTHYOGFWHXCXFTWKRWLXGTWIRWMEX + GXIXOYGYDWNWOVHWP $. $( Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario @@ -547238,19 +548007,19 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= ( X = U. T /\ A. x e. S x Fne T ) ) ) $= ( wcel cuni wceq wral wa c0 cfne wbr adantr eqid syl ex wss cvv cv unisng csn cif eqcomd iftrue unieqd eqeq2d syl5ibrcom wne wex n0 rspccva 3adant1 - w3a unieq fnejoin1 fnebas eqtrd 3expia syl5bi pm2.61dne sylan9eq wi fnetr - exlimdv 3expa ralrimdva jcad ctg cfv simprl eqtr3d elex ad2antrr eqeltrrd - sseq1 uniexb sylibr eltg3i sylancl eqeltrd snssd wn simplrr ralimi unissb - ssid fnetg ifbothda isfne4 sylanbrc impbid ) FEGZFBUAZHZIZBCJZKZCLIZFUCZC - HZUDZDMNZFDHZIZAUAZDMNZACJZKZWSXDXFXIWSXDXFWSXDFXCHZXEWSFXKIZCLWSXLWTFXAH - ZIZWNXNWRWNXMFFEUBUEOWTXKXMFWTXCXAWTXAXBUFUGUHUICLUJXGCGZAUKWSXLACULWSXOX - LAWNWRXOXLWNWRXOUOZFXGHZXKWRXOFXQIZWNWQXRBXGCWOXGIWPXQFWOXGUPUHUMUNXPXGXC - MNZXQXKIBXGCEFUQZXGXCXQXKXQPXKPZURQUSUTVFVAVBZXCDXKXEYAXEPZURVCRWSXDXHACW - NWRXOXDXHVDZXPXSYDXTXSXDXHXGXCDVERQVGVHVIWSXJXDWSXJKZXKXEIXCDVJVKZSZXDYEF - XKXEWSXLXJYBOWSXFXIVLZVMWTXAYFSZXBYFSZYGYEXAXBXAXCYFVQXBXCYFVQYEYIWTYEFYF - YEFXEYFYHYEDTGZDDSXEYFGYEXETGYKYEFXETYHWNFTGWRXJFEVNVOVPDVRVSDWHDDTVTWAWB - WCOYEWTWDZKZXGYFSZACJZYJYMXIYOWSXFXIYLWEXHYNACXGDWIWFQACYFWGVSWJXCDXKXEYA - YCWKWLRWM $. + unieq fnejoin1 fnebas eqtrd 3expia exlimdv biimtrid pm2.61dne sylan9eq wi + w3a fnetr 3expa ralrimdva jcad simprl eqtr3d sseq1 elex ad2antrr eqeltrrd + ctg cfv uniexb sylibr ssid eltg3i sylancl eqeltrd snssd wn simplrr ralimi + fnetg unissb ifbothda isfne4 sylanbrc impbid ) FEGZFBUAZHZIZBCJZKZCLIZFUC + ZCHZUDZDMNZFDHZIZAUAZDMNZACJZKZWSXDXFXIWSXDXFWSXDFXCHZXEWSFXKIZCLWSXLWTFX + AHZIZWNXNWRWNXMFFEUBUEOWTXKXMFWTXCXAWTXAXBUFUGUHUICLUJXGCGZAUKWSXLACULWSX + OXLAWNWRXOXLWNWRXOVEZFXGHZXKWRXOFXQIZWNWQXRBXGCWOXGIWPXQFWOXGUOUHUMUNXPXG + XCMNZXQXKIBXGCEFUPZXGXCXQXKXQPXKPZUQQURUSUTVAVBZXCDXKXEYAXEPZUQVCRWSXDXHA + CWNWRXOXDXHVDZXPXSYDXTXSXDXHXGXCDVFRQVGVHVIWSXJXDWSXJKZXKXEIXCDVPVQZSZXDY + EFXKXEWSXLXJYBOWSXFXIVJZVKWTXAYFSZXBYFSZYGYEXAXBXAXCYFVLXBXCYFVLYEYIWTYEF + YFYEFXEYFYHYEDTGZDDSXEYFGYEXETGYKYEFXETYHWNFTGWRXJFEVMVNVODVRVSDVTDDTWAWB + WCWDOYEWTWEZKZXGYFSZACJZYJYMXIYOWSXFXIYLWFXHYNACXGDWHWGQACYFWIVSWJXCDXKXE + YAYCWKWLRWM $. $} @@ -547368,32 +548137,32 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= -> ran ( tail ` D ) e. ( fBas ` X ) ) $= ( vz vx vy vu vv wcel c0 wa cfv wss cv wrex adantr wi wb wbr cvv cdir wne vw ctail crn cfbas cpw wnel cin wral w3a tailf wex n0 wf wfn ffn fnfvelrn - frnd ex 3syl ne0i syl6 exlimdv syl5bi imp wn wceq tailini n0i syl fvelrnb - nrexdv mtbird df-nel sylibr reeanv dirge 3expb sylan ad2ant2r dirtr exp32 - anbi12d elvd com23 ad2ant2rl anim12d expr impr vex eltail adantrr adantrl - mp3an3 3imtr4d syl6ibr ssrdv sseq1 rspcev syl2anc rexlimddv ineq1 rexbidv - elin sseq2d ineq2 sylan9bb syl5ibcom rexlimdvva sylbid ralrimivv 3jca cdm - syl5bir dmexg eqeltrid isfbas2 mpbir2and ) AUAIZBJUBZKZAUDLZUEZBUFLIZYDBU - GZMZYDJUBZJYDUHZDNZENZFNZUIZMZDYDOZFYDUJEYDUJZUKZXTYGYAXTBYFYCABCULZUSPYB - YHYIYPXTYAYHYAYKBIZEUMXTYHEBUNXTYSYHEXTYSYKYCLZYDIZYHXTBYFYCUOZYCBUPZYSUU - AQYRBYFYCUQZUUCYSUUABYKYCURUTVAYDYTVBVCVDVEVFYBJYDIZVGYIYBUUEYTJVHZEBOZXT - UUGVGYAXTUUFEBXTYSKYKYTIUUFVGYKABCVIYTYKVJVKVMPXTUUEUUGRZYAXTUUBUUCUUHYRU - UDEBJYCVLVAPVNJYDVOVPYBYOEFYDYDXTYKYDIZYLYDIZKZYOQYAXTUUKGNZYCLZYKVHZGBOZ - HNZYCLZYLVHZHBOZKZYOXTUUBUUCUUKUUTRYRUUDUUCUUIUUOUUJUUSGBYKYCVLHBYLYCVLWD - VAUUTUUNUURKZHBOGBOXTYOUUNUURGHBBVQXTUVAYOGHBBXTUULBIZUUPBIZKZKZYJUUMUUQU - IZMZDYDOZUVAYOUVEUULUCNZASZUUPUVIASZKZUVHUCBXTUVBUVCUVLUCBOUCUULUUPABCVRV - SUVEUVIBIZUVLKZKZUVIYCLZYDIZUVPUVFMZUVHXTUVMUVQUVDUVLXTUUCUVMUVQXTUUBUUCY - RUUDVKBUVIYCURVTWAUVOEUVPUVFUVOYKUVPIZYKUUMIZYKUUQIZKZYKUVFIUVOUVIYKASZUU - LYKASZUUPYKASZKZUVSUWBUVEUVMUVLUWCUWFQUVEUVMKUWCUVLUWFUVEUVMUWCUVLUWFQUVE - UVMUWCKKUVJUWDUVKUWEXTUWCUVJUWDQZUVDUVMXTUWCUWGXTUVJUWCUWDXTUVJUWCUWDQQEX - TYKTIZKZUVJUWCUWDUULUVIYKATWBWCWEWFVFWGXTUWCUVKUWEQZUVDUVMXTUWCUWJXTUVKUW - CUWEXTUVKUWCUWEQQEUWIUVKUWCUWEUUPUVIYKATWBWCWEWFVFWGWHWIWFWJXTUVMUVSUWCRZ - UVDUVLXTUVMUWHUWKEWKZUVIYKTABCWLWOWAUVEUWBUWFRUVNUVEUVTUWDUWAUWEXTUVBUVTU - WDRZUVCXTUVBUWHUWMUWLUULYKTABCWLWOWMXTUVCUWAUWERZUVBXTUVCUWHUWNUWLUUPYKTA - BCWLWOWNWDPWPYKUUMUUQXEWQWRUVGUVRDUVPYDYJUVPUVFWSWTXAXBUUNUVHYJYKUUQUIZMZ - DYDOUURYOUUNUVGUWPDYDUUNUVFUWOYJUUMYKUUQXCXFXDUURUWPYNDYDUURUWOYMYJUUQYLY - KXGXFXDXHXIXJXOXKPXLXMYBBTIZYEYGYQKRXTUWQYAXTBAXNTCAUAXPXQPEFDTBYDXRVKXS - $. + frnd ex 3syl ne0i syl6 exlimdv biimtrid imp wn tailini n0i nrexdv fvelrnb + wceq mtbird df-nel sylibr anbi12d reeanv dirge 3expb sylan ad2ant2r dirtr + syl exp32 elvd com23 ad2ant2rl anim12d expr eltail mp3an3 adantrr adantrl + impr vex 3imtr4d elin syl6ibr ssrdv sseq1 rspcev syl2anc rexlimddv sseq2d + ineq1 rexbidv sylan9bb syl5ibcom rexlimdvva syl5bir sylbid ralrimivv 3jca + ineq2 cdm dmexg eqeltrid isfbas2 mpbir2and ) AUAIZBJUBZKZAUDLZUEZBUFLIZYD + BUGZMZYDJUBZJYDUHZDNZENZFNZUIZMZDYDOZFYDUJEYDUJZUKZXTYGYAXTBYFYCABCULZUSP + YBYHYIYPXTYAYHYAYKBIZEUMXTYHEBUNXTYSYHEXTYSYKYCLZYDIZYHXTBYFYCUOZYCBUPZYS + UUAQYRBYFYCUQZUUCYSUUABYKYCURUTVAYDYTVBVCVDVEVFYBJYDIZVGYIYBUUEYTJVLZEBOZ + XTUUGVGYAXTUUFEBXTYSKYKYTIUUFVGYKABCVHYTYKVIWCVJPXTUUEUUGRZYAXTUUBUUCUUHY + RUUDEBJYCVKVAPVMJYDVNVOYBYOEFYDYDXTYKYDIZYLYDIZKZYOQYAXTUUKGNZYCLZYKVLZGB + OZHNZYCLZYLVLZHBOZKZYOXTUUBUUCUUKUUTRYRUUDUUCUUIUUOUUJUUSGBYKYCVKHBYLYCVK + VPVAUUTUUNUURKZHBOGBOXTYOUUNUURGHBBVQXTUVAYOGHBBXTUULBIZUUPBIZKZKZYJUUMUU + QUIZMZDYDOZUVAYOUVEUULUCNZASZUUPUVIASZKZUVHUCBXTUVBUVCUVLUCBOUCUULUUPABCV + RVSUVEUVIBIZUVLKZKZUVIYCLZYDIZUVPUVFMZUVHXTUVMUVQUVDUVLXTUUCUVMUVQXTUUBUU + CYRUUDWCBUVIYCURVTWAUVOEUVPUVFUVOYKUVPIZYKUUMIZYKUUQIZKZYKUVFIUVOUVIYKASZ + UULYKASZUUPYKASZKZUVSUWBUVEUVMUVLUWCUWFQUVEUVMKUWCUVLUWFUVEUVMUWCUVLUWFQU + VEUVMUWCKKUVJUWDUVKUWEXTUWCUVJUWDQZUVDUVMXTUWCUWGXTUVJUWCUWDXTUVJUWCUWDQQ + EXTYKTIZKZUVJUWCUWDUULUVIYKATWBWDWEWFVFWGXTUWCUVKUWEQZUVDUVMXTUWCUWJXTUVK + UWCUWEXTUVKUWCUWEQQEUWIUVKUWCUWEUUPUVIYKATWBWDWEWFVFWGWHWIWFWNXTUVMUVSUWC + RZUVDUVLXTUVMUWHUWKEWOZUVIYKTABCWJWKWAUVEUWBUWFRUVNUVEUVTUWDUWAUWEXTUVBUV + TUWDRZUVCXTUVBUWHUWMUWLUULYKTABCWJWKWLXTUVCUWAUWERZUVBXTUVCUWHUWNUWLUUPYK + TABCWJWKWMVPPWPYKUUMUUQWQWRWSUVGUVRDUVPYDYJUVPUVFWTXAXBXCUUNUVHYJYKUUQUIZ + MZDYDOUURYOUUNUVGUWPDYDUUNUVFUWOYJUUMYKUUQXEXDXFUURUWPYNDYDUURUWOYMYJUUQY + LYKXNXDXFXGXHXIXJXKPXLXMYBBTIZYEYGYQKRXTUWQYAXTBAXOTCAUAXPXQPEFDTBYDXRWCX + S $. $} ${ @@ -547673,7 +548442,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= $( The double nand. $) w3nand $a wff ( ph -/\ ps -/\ ch ) $. - $( The double nand. This definition allows us to express the input of three + $( The double nand. This definition allows to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.) $) df-3nand $a |- ( ( ph -/\ ps -/\ ch ) <-> ( ph -> ( ps -> -. ch ) ) ) $. @@ -547823,12 +548592,12 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= slide 8). (Contributed by Anthony Hart, 14-Aug-2011.) $) arg-ax $p |- ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( ph -/\ ( ps -/\ ch ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ch -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $= - ( wnan wa wi wn df-nan pm4.57 com12 pm3.45 anim12i jaob 3imtr4i syl6 syl5bi - wo biimpri nannan orel2 simpr a1i jad syl pm3.22 con1d ancli mpbir ) ABCEEZ - UJDCEZCDEZADEZEEZEEUJUJUNFGUJUNABCFZGZUKULUMFZGUJUNUKDCFZHZUPUQDCIUPUSCDFZH - ZADFZHZFZUQUPVDURVDHUTVBRZUPURUTVBJUPVEUTURUPCARZCGZVEUTGZVFUPCVFAUOCAHVFCA - CUAKUOCGVFBCUBUCUDKCCGZACGZFUTUTGZVBUTGZFVGVHVIVKVJVLCCDLACDLMCCANUTUTVBNOU - ECDUFPQUGVAULVCUMULVACDISUMVCADISMPQABCTUKULUMTOUHUJUJUNTUI $. + ( wnan wa wi wn df-nan wo pm4.57 com12 pm3.45 anim12i jaob 3imtr4i biimtrid + syl6 biimpri nannan orel2 simpr a1i jad syl pm3.22 con1d ancli mpbir ) ABCE + EZUJDCEZCDEZADEZEEZEEUJUJUNFGUJUNABCFZGZUKULUMFZGUJUNUKDCFZHZUPUQDCIUPUSCDF + ZHZADFZHZFZUQUPVDURVDHUTVBJZUPURUTVBKUPVEUTURUPCAJZCGZVEUTGZVFUPCVFAUOCAHVF + CACUALUOCGVFBCUBUCUDLCCGZACGZFUTUTGZVBUTGZFVGVHVIVKVJVLCCDMACDMNCCAOUTUTVBO + PUECDUFRQUGVAULVCUMULVACDISUMVCADISNRQABCTUKULUMTPUHUJUJUNTUI $. $( @@ -547916,17 +548685,14 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= See ~ negsym1 for more information. (Contributed by Anthony Hart, 13-Sep-2011.) $) amosym1 $p |- ( E* x E* x F. -> E* x ph ) $= - ( wfal wmo wex weu wi moeu wn mofal 19.8a notnotd ax-mp pm2.21i notnoti nex - eunex mto ja sylbi ) CBDZBDUABEZUABFZGABDZUABHUBUCUDUBIZUDUAUEIBJZUAUBUABKL - MNUCUDUCUAIZBEUGBUAUFOPUABQRNST $. + ( wfal wmo mofal a1i moimi ) ACBDZBHABEFG $. $( A symmetry with ` [ x / y ] ` . See ~ negsym1 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) $) subsym1 $p |- ( [ y / x ] [ y / x ] F. -> [ y / x ] ph ) $= - ( wfal wsb weq wi wa wex fal intnan nex dfsb1 mtbir pm2.21i ) DBCEZBCEZABCE - QBCFZPGZRPHZBIZHUASTBPRPRDGZRDHZBIZHUDUBUCBDRJKLKDBCMNKLKPBCMNO $. + ( wfal wsb sbv falim sylbi sbimi ) DBCEZABCJDADBCFAGHI $. $( (End of Anthony Hart's mathbox.) $) $( End $[ set-mbox-ah.mm $] $) @@ -548012,10 +548778,10 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= Chen-Pang He, 1-Nov-2015.) $) ordtoplem $p |- ( Ord A -> ( A =/= U. A -> A e. S ) ) $= ( cuni wne wceq wn word wcel df-ne con0 csuc wo ordeleqon eqcomi id unieq - unon 3eqtr4a ord orim2i sylbi orcomd orduniorsuc onuni eleq1a 3syl syl5bi - wi syl6c ) AADZEAUKFZGZAHZABIZAUKJUNUMAKIZAUKLZFZUOUNULUPUNUPULUNUPAKFZMU - PULMANUSULUPUSKKDZAUKUTKROUSPAKQSUAUBUCTUNULURAUDTUPUKKIUQBIURUOUIAUECUQB - AUFUGUJUH $. + unon 3eqtr4a ord orim2i sylbi orcomd orduniorsuc wi onuni eleq1a biimtrid + 3syl syl6c ) AADZEAUKFZGZAHZABIZAUKJUNUMAKIZAUKLZFZUOUNULUPUNUPULUNUPAKFZ + MUPULMANUSULUPUSKKDZAUKUTKROUSPAKQSUAUBUCTUNULURAUDTUPUKKIUQBIURUOUEAUFCU + QBAUGUIUJUH $. $} $( An ordinal is a topology iff it is not its supremum (union), proven @@ -548068,18 +548834,18 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= ( vx vo vy con0 wcel csuc wel wb wral wi word cv wa wal ordelon wn ancoms wss syl ct0 weq eloni df-ral anim12dan ordsuc sylbi adantr ontri1 onsssuc ex notbi bitr3d adantrr adantrl bibi12d bitrid biimpd syl6an a2d ordelord - ordelss ordsucsssuc syldan mpbid ssneld jcad pm5.21 idd jad alimdv syl5bi - syl6 syld dfcleq suc11 bitr3id sylibd ralrimivva ctopon onsuctopon ist0-2 - wceq cfv mpbird ) AEFZAGZUAFZBCHZDCHZIZCWGJZBDUBZKZDAJBAJZWFALZWOAUCWPWNB - DAAWPBMZAFZDMZAFZNZNZWLCMZWQGZFZXCWSGZFZIZCOZWMWLXCWGFZWKKZCOXBXIWKCWGUDX - BXKXHCXBXKXJXHKXHXBXJWKXHXBWQEFZWSEFZNZXJXCEFZWKXHKWPWRXLWTXMAWQPAWSPUEZW - PXJXOKZXAWPWGLZXQAUFXRXJXOWGXCPUKUGUHXNXONZWKXHWKWIQZWJQZIZXSXHWIWJULXOXN - YBXHIXOXNNXTXEYAXGXOXLXTXEIXMXOXLNXCWQSXTXEXCWQUIXCWQUJUMUNXOXMYAXGIXLXOX - MNXCWSSYAXGXCWSUIXCWSUJUMUOUPRUQURUSUTXBXJXHXHXBXJQZXEQZXGQZNXHXBYCYDYEWP - WRYCYDKWTWPWRNZXDWGXCYFWQASZXDWGSZAWQVBWPWRWQLZYGYHIZAWQVAYIWPYJWQAVCRVDV - EVFUNWPWTYCYEKWRWPWTNZXFWGXCYKWSASZXFWGSZAWSVBWPWTWSLZYLYMIZAWSVAYNWPYOWS - AVCRVDVEVFUOVGXEXGVHVMXBXHVIVJVNVKVLXBXNXIWMIXPXIXDXFWCXNWMCXDXFVOWQWSVPV - QTVRVSTWFWGAVTWDFWHWOIAWABDCWGAWBTWE $. + ordelss ordsucsssuc syldan mpbid ssneld jcad pm5.21 syl6 idd jad biimtrid + syld alimdv wceq dfcleq suc11 bitr3id sylibd ralrimivva ctopon onsuctopon + cfv ist0-2 mpbird ) AEFZAGZUAFZBCHZDCHZIZCWGJZBDUBZKZDAJBAJZWFALZWOAUCWPW + NBDAAWPBMZAFZDMZAFZNZNZWLCMZWQGZFZXCWSGZFZIZCOZWMWLXCWGFZWKKZCOXBXIWKCWGU + DXBXKXHCXBXKXJXHKXHXBXJWKXHXBWQEFZWSEFZNZXJXCEFZWKXHKWPWRXLWTXMAWQPAWSPUE + ZWPXJXOKZXAWPWGLZXQAUFXRXJXOWGXCPUKUGUHXNXONZWKXHWKWIQZWJQZIZXSXHWIWJULXO + XNYBXHIXOXNNXTXEYAXGXOXLXTXEIXMXOXLNXCWQSXTXEXCWQUIXCWQUJUMUNXOXMYAXGIXLX + OXMNXCWSSYAXGXCWSUIXCWSUJUMUOUPRUQURUSUTXBXJXHXHXBXJQZXEQZXGQZNXHXBYCYDYE + WPWRYCYDKWTWPWRNZXDWGXCYFWQASZXDWGSZAWQVBWPWRWQLZYGYHIZAWQVAYIWPYJWQAVCRV + DVEVFUNWPWTYCYEKWRWPWTNZXFWGXCYKWSASZXFWGSZAWSVBWPWTWSLZYLYMIZAWSVAYNWPYO + WSAVCRVDVEVFUOVGXEXGVHVIXBXHVJVKVMVNVLXBXNXIWMIXPXIXDXFVOXNWMCXDXFVPWQWSV + QVRTVSVTTWFWGAWAWCFWHWOIAWBBDCWGAWDTWE $. $} $( An ordinal topology is T_0. (Contributed by Chen-Pang He, 8-Nov-2015.) $) @@ -550820,9 +551586,9 @@ to the modal axiom (D), and in predicate calculus, they assert that the (Proof modification is discouraged.) (New usage is discouraged.) $) bj-gl4 $p |- ( ( A. x ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) -> A. x ( A. x ph /\ ph ) ) -> ( A. x ph -> A. x A. x ph ) ) $= - ( wal wa wi 19.26 simpr a1i anc2ri syl5bi alimi biimpi imim12i simpl syl6 ) - ABCZADZBCZQEZBCZREPPBCZPDZUAPTRUBASBRUBAQPABFZAUBPUBPEAUAPGHIJKRUBUCLMUAPNO - $. + ( wal wa wi 19.26 simpr a1i anc2ri biimtrid alimi biimpi imim12i simpl syl6 + ) ABCZADZBCZQEZBCZREPPBCZPDZUAPTRUBASBRUBAQPABFZAUBPUBPEAUAPGHIJKRUBUCLMUAP + NO $. $( Over minimal calculus, the modal axiom (4) ( ~ hba1 ) and the modal axiom (K) ( ~ ax-4 ) together imply ~ axc4 . (Contributed by BJ, 29-Nov-2020.) @@ -551278,9 +552044,9 @@ could be deduced from it ( ~ exim would have to be proved first, see $( Closed form of ~ nfim and curried (exported) form of ~ nfimt . (Contributed by BJ, 20-Oct-2021.) $) bj-nfimt $p |- ( F/ x ph -> ( F/ x ps -> F/ x ( ph -> ps ) ) ) $= - ( wnf wi wex wal 19.35 id nfrd imim1d syl5bi imim2d 19.38 syl6 syl9 syl6ibr - df-nf ) ACDZBCDZABEZCFZUACGZEUACDSUBACFZBCFZEZTUCUBACGZUEESUFABCHSUDUGUESAC - SIJKLTUFUDBCGZEUCTUEUHUDTBCTIJMABCNOPUACRQ $. + ( wnf wi wex wal 19.35 id nfrd imim1d biimtrid imim2d 19.38 syl6 syl9 df-nf + syl6ibr ) ACDZBCDZABEZCFZUACGZEUACDSUBACFZBCFZEZTUCUBACGZUEESUFABCHSUDUGUES + ACSIJKLTUFUDBCGZEUCTUEUHUDTBCTIJMABCNOPUACQR $. $( A lemma in closed form used to prove ~ bj-cbval in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use ~ 19.35 since only the @@ -551733,14 +552499,15 @@ could be deduced from it ( ~ exim would have to be proved first, see -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) - $( Commutation of universal quantifiers implies commutation of existential - quantifiers. Can be placed in the ~ ax-4 section, soon after ~ 2nexaln , - and used to prove ~ excom . (Contributed by BJ, 29-Nov-2020.) + $( Commutation of two existential quantifiers on a formula is equivalent to + commutation of two universal quantifiers over the same variables on the + negation of that formula. Can be placed in the ~ ax-4 section, soon after + ~ 2nexaln , and used to prove ~ excom . (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) $) - bj-alcomexcom $p |- ( ( A. x A. y -. ph -> A. y A. x -. ph ) -> + bj-alcomexcom $p |- ( ( A. x A. y -. ph -> A. y A. x -. ph ) <-> ( E. y E. x ph -> E. x E. y ph ) ) $= - ( wn wal wi wex 2nexaln imbi12i con4 sylbir ) ADZCEBEZLBECEZFACGBGZDZABGCGZ - DZFQOFPMRNABCHACBHIOQJK $. + ( wex wi wn wal con34b 2nexaln imbi12i bitr2i ) ABDCDZACDBDZEMFZLFZEAFZCGBG + ZPBGCGZELMHNQORABCIACBIJK $. $( Closed form of ~ hbal . When in main part, prove ~ hbal and ~ hbald from it. (Contributed by BJ, 2-May-2019.) $) @@ -552017,7 +552784,7 @@ is proved in modal (T5). The LHS has the advantage of not involving nested quantifiers on the same variable. Its metaweakening is proved from the core axiom schemes in ~ bj-substw . Note that in the LHS, the reverse implication holds by ~ equs4 (or ~ equs4v if a DV condition is added on - ` x , t ` as in ~ bj-ax12 ). + ` x , t ` as in ~ bj-ax12 ), and the forward implication is ~ sbalex . The LHS can be read as saying that if there exists a setvar equal to a given term witnessing ` ph ` , then all setvars equal to that term also @@ -552408,9 +553175,9 @@ quantifiers appear (since they typically require ~ ax-10 to work with), $( Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.) $) bj-nnfim $p |- ( ( F// x ph /\ F// x ps ) -> F// x ( ph -> ps ) ) $= - ( wnnf wa wex wal 19.35 bj-nnfim2 syl5bi bj-nnfim1 19.38 df-bj-nnf sylanbrc - wi syl6 ) ACDBCDEZABOZCFZRORRCGZORCDSACGBCFOQRABCHABCIJQRACFBCGOTABCKABCLPR - CMN $. + ( wnnf wa wi wex wal 19.35 bj-nnfim2 biimtrid bj-nnfim1 19.38 syl6 sylanbrc + df-bj-nnf ) ACDBCDEZABFZCGZRFRRCHZFRCDSACHBCGFQRABCIABCJKQRACGBCHFTABCLABCM + NRCPO $. ${ bj-nnfimd.1 $e |- ( ph -> F// x ps ) $. @@ -552459,10 +553226,10 @@ two more essential steps but fewer total steps (since there are fewer so using ~ bj-nnfim , ~ bj-nnfnt and ~ bj-nnfbi , but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.) $) bj-nnfor $p |- ( ( F// x ph /\ F// x ps ) -> F// x ( ph \/ ps ) ) $= - ( wnnf wa wo wex wi wal df-bj-nnf 19.43 pm3.48 syl5bi 19.33 anim12i syl2anb - syl6 an4s sylibr ) ACDZBCDZEABFZCGZUBHZUBUBCIZHZEZUBCDTACGZAHZAACIZHZEBCGZB - HZBBCIZHZEUGUAACJBCJUIUMUKUOUGUIUMEZUDUKUOEZUFUCUHULFUPUBABCKUHAULBLMUQUBUJ - UNFUEAUJBUNLABCNQORPUBCJS $. + ( wnnf wa wo wex wi df-bj-nnf 19.43 pm3.48 biimtrid 19.33 syl6 anim12i an4s + wal syl2anb sylibr ) ACDZBCDZEABFZCGZUBHZUBUBCQZHZEZUBCDTACGZAHZAACQZHZEBCG + ZBHZBBCQZHZEUGUAACIBCIUIUMUKUOUGUIUMEZUDUKUOEZUFUCUHULFUPUBABCJUHAULBKLUQUB + UJUNFUEAUJBUNKABCMNOPRUBCIS $. ${ bj-nnford.1 $e |- ( ph -> F// x ps ) $. @@ -552472,9 +553239,9 @@ two more essential steps but fewer total steps (since there are fewer (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) $) bj-nnford $p |- ( ph -> F// x ( ps \/ ch ) ) $= - ( wo wex wi wal wnnf 19.43 bj-nnfed orim12d bj-nnfad 19.33 syl6 df-bj-nnf - syl5bi sylanbrc ) ABCGZDHZUAIUAUADJZIUADKUBBDHZCDHZGAUABCDLAUDBUECABDEMAC - DFMNSAUABDJZCDJZGUCABUFCUGABDEOACDFONBCDPQUADRT $. + ( wo wex wi wnnf 19.43 bj-nnfed orim12d biimtrid bj-nnfad 19.33 df-bj-nnf + wal syl6 sylanbrc ) ABCGZDHZUAIUAUADRZIUADJUBBDHZCDHZGAUABCDKAUDBUECABDEL + ACDFLMNAUABDRZCDRZGUCABUFCUGABDEOACDFOMBCDPSUADQT $. $} $( Nonfreeness in both sides implies nonfreeness in the biconditional. @@ -552564,7 +553331,7 @@ two more essential steps but fewer total steps (since there are fewer (Proof modification is discouraged.) $) bj-nnflemee $p |- ( A. x ( E. y ph -> ph ) -> ( E. y E. x ph -> E. x ph ) ) $= - ( wex wi wal excom exim syl5bi ) ABDZCDACDZBDKAEBFJACBGKABHI $. + ( wex wi wal excom exim biimtrid ) ABDZCDACDZBDKAEBFJACBGKABHI $. $( One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. @@ -552626,15 +553393,15 @@ two more essential steps but fewer total steps (since there are fewer (Contributed by BJ, 2-Dec-2023.) $) bj-19.36im $p |- ( F// x ps -> ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) ) ) $= - ( wi wex wal wnnf 19.35 bj-nnfe imim2d syl5bi ) ABDCEACFZBCEZDBCGZLBDABCHNM - BLBCIJK $. + ( wi wex wal wnnf 19.35 bj-nnfe imim2d biimtrid ) ABDCEACFZBCEZDBCGZLBDABCH + NMBLBCIJK $. $( One direction of ~ 19.37 from the same axioms as ~ 19.37imv . (Contributed by BJ, 2-Dec-2023.) $) bj-19.37im $p |- ( F// x ph -> ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) ) ) $= - ( wi wex wal wnnf 19.35 bj-nnfa imim1d syl5bi ) ABDCEACFZBCEZDACGZAMDABCHNA - LMACIJK $. + ( wi wex wal wnnf 19.35 bj-nnfa imim1d biimtrid ) ABDCEACFZBCEZDACGZAMDABCH + NALMACIJK $. $( Closed form of ~ 19.42 from the same axioms as ~ 19.42v . (Contributed by BJ, 2-Dec-2023.) $) @@ -553111,25 +553878,6 @@ universe has at least two objects (see ~ dtru ). (Contributed by BJ, ( cv cab wcel hbab1 nf5i ) CDABEFBABCGH $. $} - ${ - $d w x y z $. - $( Remove dependency on ~ ax-13 from ~ dtru . (Contributed by BJ, - 31-May-2019.) - - TODO: This predates the removal of ~ ax-13 in ~ dtru . But actually, - ~ sn-dtru is better than either, so move it to Main with ~ sn-el (and - determine whether ~ bj-dtru should be kept as ALT or deleted). - - (Proof modification is discouraged.) (New usage is discouraged.) $) - bj-dtru $p |- -. A. x x = y $= - ( vw vz weq wn wex wal wel wa el ax-nul sp eximii exdistrv mpbir2an ax-mp - ax7 con3d spimevw ax9 com12 con3dimp 2eximi equequ2 notbid syl6bi pm2.61i - wi a1d exlimivv exnal mpbi ) ABEZFZAGZUNAHFCDEZFZDGCGZUPACIZADIZFZJZDGCGZ - USVDUTCGVBDGACKVBAHVBDDALVBAMNUTVBCDOPVCURCDUTUQVAUQUTVACDAUAUBUCUDQURUPC - DDBEZURUPUIVEURCBEZFZUPVEUQVFDBCUEUFVGUOACACEUNVFACBRSTUGVEFZUPURVHUOADAD - EUNVEADBRSTUJUHUKQUNAULUM $. - $} - ${ $d x y $. bj-dtrucor2v.1 $e |- ( x = y -> x =/= y ) $. @@ -554963,15 +555711,9 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ). ( c0 wne cv wcel wex n0 mpbi ) BDEAFBGAHCABIJ $. $} - $( A class is disjoint from its singleton. A consequence of regularity. - Shorter proof than ~ bnj521 and does not depend on ~ df-ne . (Contributed - by BJ, 4-Apr-2019.) $) - bj-disjcsn $p |- ( A i^i { A } ) = (/) $= - ( csn cin c0 wceq wcel wn elirr disjsn mpbir ) AABCDEAAFGAHAAIJ $. - $( Disjointness of the singletons containing 0 and 1. This is a consequence - of ~ bj-disjcsn but the present proof does not use regularity. - (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.) $) + of ~ disjcsn but the present proof does not use regularity. (Contributed + by BJ, 4-Apr-2019.) (Proof modification is discouraged.) $) bj-disjsn01 $p |- ( { (/) } i^i { 1o } ) = (/) $= ( c0 c1o wne csn cin wceq 1n0 necomi disjsn2 ax-mp ) ABCADBDEAFBAGHABIJ $. @@ -555078,23 +555820,23 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ). cab eleq2 abbidv eleq1 biimpd eximi bj-eximcom com12 19.3v sbbii csb wsbc ax-rep sbsbc sbceq2g elv bj-csbsn eqeq2i eqtr2 sneqr sylan2b syl2anb gen2 vex nfa1 mo mpbir bj-sbel1 eleq1i df-clab anbi2i eleq1a eqcoms imdistanri - impac impbii exbii snex isseti 19.42v mpbiran2 3bitr4ri bibi2i albii mpbi - mpg dfcleq issetri ax5e ) BCHZAIJZBHZAUBZKHZDLZWOWKWLDIZHZAUBZKHZWOMZDLZW - PWKWQBNZDLXBDBCUAXCXADXCWSWNNZXAXCWRWMAWQBWLUCUDXDWTWOWSWNKUEUFOUGOWTXBWP - MDXBWTDPWPWTWODUHUIEWSEIZWSNZELFIZXEHZXGWSHZQZFPZELZXHGIZWQHZXMXGJZNZEPZR - ZGLZQZFPZELZXLXQXGXENZMFPELZYBGXPDEFGUNYDXQXQFESZRYCMZEPFPYFFEXQXPXPFESZY - CYEXPEUJZXQXPFEYHUKYGXPXMXEJZNZYCYGXMFXEXOULZNZYJYGXPFXEUMZYLXPFEUOYMYLQE - FXEXMXOKUPUQTYKYIXMFXEURUSTXPYJRXOYINYCXMXOYIUTXGXEFVEVAOVBVCVDXQFEXPEVFV - GVHWGYAXKEXTXJFXSXIXHWRAFSZXOWQHZXIXSYNAXGWLULZWQHYOAFWLWQVIYPXOWQAXGURVJ - TWRFAVKXSYOXPRZGLZYOXRYQGXRXNXPRZYQXQXPXNYHVLYSYQXPXNYOXNYOMXOXMXNXOXMNYO - XMWQXOVMUIVNVOYOXPXNXOWQXMVMVPVQTVRYRYOXPGLGXOXGVSVTYOXPGWAWBTWCWDWEVRWFX - FXKEFXEWSWHVRVHWIWGOWODWJO $. + mpg impac impbii exbii vsnex isseti 19.42v mpbiran2 3bitr4ri bibi2i albii + mpbi dfcleq issetri ax5e ) BCHZAIJZBHZAUBZKHZDLZWOWKWLDIZHZAUBZKHZWOMZDLZ + WPWKWQBNZDLXBDBCUAXCXADXCWSWNNZXAXCWRWMAWQBWLUCUDXDWTWOWSWNKUEUFOUGOWTXBW + PMDXBWTDPWPWTWODUHUIEWSEIZWSNZELFIZXEHZXGWSHZQZFPZELZXHGIZWQHZXMXGJZNZEPZ + RZGLZQZFPZELZXLXQXGXENZMFPELZYBGXPDEFGUNYDXQXQFESZRYCMZEPFPYFFEXQXPXPFESZ + YCYEXPEUJZXQXPFEYHUKYGXPXMXEJZNZYCYGXMFXEXOULZNZYJYGXPFXEUMZYLXPFEUOYMYLQ + EFXEXMXOKUPUQTYKYIXMFXEURUSTXPYJRXOYINYCXMXOYIUTXGXEFVEVAOVBVCVDXQFEXPEVF + VGVHVPYAXKEXTXJFXSXIXHWRAFSZXOWQHZXIXSYNAXGWLULZWQHYOAFWLWQVIYPXOWQAXGURV + JTWRFAVKXSYOXPRZGLZYOXRYQGXRXNXPRZYQXQXPXNYHVLYSYQXPXNYOXNYOMXOXMXNXOXMNY + OXMWQXOVMUIVNVOYOXPXNXOWQXMVMVQVRTVSYRYOXPGLGXOFVTWAYOXPGWBWCTWDWEWFVSWGX + FXKEFXEWSWHVSVHWIVPOWODWJO $. $} ${ $d x A $. $d x B $. $( Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.) $) - bj-clex $p |- ( A e. V -> { x | { x } e. ( A " B ) } e. _V ) $= + bj-clexab $p |- ( A e. V -> { x | { x } e. ( A " B ) } e. _V ) $= ( wcel cima cvv cv csn cab imaexg bj-snsetex syl ) BDEBCFZGEAHINEAJGEBCDK ANGLM $. $} @@ -555129,10 +555871,10 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ). (Contributed by BJ, 6-Oct-2018.) $) bj-elsngl $p |- ( A e. sngl B <-> E. x e. B A = { x } ) $= ( vy bj-csngl wcel cv wceq wa wex csn wrex dfclel df-bj-sngl abeq2i exbii - anbi2i r19.42v bicomi 3bitri rexcom4 eqcom snex eqvinc exancom rexbii ) B - CEZFDGZBHZUHUGFZIZDJUIUHAGZKZHZACLZIZDJZBUMHZACLZDBUGMUKUPDUJUOUIUODUGDAC - NOQPUQUIUNIZACLZDJZUTDJZACLZUSUPVADVAUPUIUNACRSPVDVBUTADCUASVCURACURVCURU - MBHUNUIIDJVCBUMUBDUMBULUCUDUNUIDUETSUFTT $. + anbi2i r19.42v bicomi 3bitri rexcom4 eqcom vsnex eqvinc exancom rexbii ) + BCEZFDGZBHZUHUGFZIZDJUIUHAGKZHZACLZIZDJZBULHZACLZDBUGMUKUODUJUNUIUNDUGDAC + NOQPUPUIUMIZACLZDJZUSDJZACLZURUOUTDUTUOUIUMACRSPVCVAUSADCUASVBUQACUQVBUQU + LBHUMUIIDJVBBULUBDULBAUCUDUMUIDUETSUFTT $. $} ${ @@ -555247,9 +555989,9 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ). (Contributed by BJ, 6-Oct-2018.) $) bj-sngltag $p |- ( A e. V -> ( { A } e. sngl B <-> { A } e. tag B ) ) $= ( wcel csn bj-csngl bj-ctag bj-sngltagi c0 cun df-bj-tag eleq2i wo elun idd - wceq elsni cvv wn syl5bi snprc elex pm2.24d syl5bir syl5 jaod impbid2 ) ACD - ZAEZBFZDZUIBGZDZUIBHUMUIUJIEZJZDZUHUKULUOUIBKLUPUKUIUNDZMUHUKUIUJUNNUHUKUKU - QUHUKOUQUIIPZUHUKUIIQURARDZSUHUKAUAUHUSUKACUBUCUDUEUFTTUG $. + wceq elsni cvv wn biimtrid snprc elex pm2.24d syl5bir syl5 jaod impbid2 ) A + CDZAEZBFZDZUIBGZDZUIBHUMUIUJIEZJZDZUHUKULUOUIBKLUPUKUIUNDZMUHUKUIUJUNNUHUKU + KUQUHUKOUQUIIPZUHUKUIIQURARDZSUHUKAUAUHUSUKACUBUCUDUEUFTTUG $. $( Characterization of the elements of ` B ` in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.) $) @@ -555395,8 +556137,8 @@ A comparison of the different definitions of tuples (strangely not mentioning $d x A $. $d x B $. $( Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.) $) bj-projex $p |- ( B e. V -> ( A Proj B ) e. _V ) $= - ( vx wcel bj-cproj cv csn cima cab cvv df-bj-proj bj-clex eqeltrid ) BCEA - BFDGHBAHZIEDJKDABLDBOCMN $. + ( vx wcel bj-cproj cv csn cima cab cvv df-bj-proj bj-clexab eqeltrid ) BC + EABFDGHBAHZIEDJKDABLDBOCMN $. $} ${ @@ -555669,6 +556411,216 @@ sethood hypotheses (compare ~ opth ). (Contributed by BJ, 6-Oct-2018.) $) ZCDZEFACDCBGHBCDCAGHOQENPCABIJKABCLBACLM $. +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Axioms for finite unions +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + + In this section, we introduce the axiom of singleton ~ ax-bj-sn and the axiom + of binary union ~ ax-bj-bun . Both axioms are implied by the standard + axioms of unordered pair ~ ax-pr and of union ~ ax-un (see ~ snex and + ~ unex ). Conversely, the axiom of unordered pair ~ ax-pr is implied by the + axioms of singleton and of binary union, as proved in ~ bj-prexg and + ~ bj-prex . + + The axioms of union ~ ax-un and of powerset ~ ax-pow are independent of these + axioms: consider respectively the class of pseudo-hereditarily sets + of cardinality less than a given singular strong limit cardinal, see + + Greg Oman, _On the axiom of union_, Arch. Math. Logic (2010) 49:283--289 + + (that model does have finite unions), and the class of well-founded + hereditarily countable sets (or hereditarily less than a given uncountable + regular cardinal). See also ~ https://mathoverflow.net/questions/81815 and + ~ https://mathoverflow.net/questions/48365 . + + A proof by finite induction shows that the existence of finite unions is + equivalent to the existence of binary unions and of nullary unions (the + latter being the axiom of the empty set ~ ax-nul ). + + The axiom of binary union is useful in theories without the axioms of union + ~ ax-un and of powerset ~ ax-pow . For instance, the class of well-founded + sets hereditarily of cardinality at most ` n e. NN0 ` with ordinary + membership relation is a model of + { ~ ax-ext , ~ ax-rep , ~ ax-sep , ~ ax-nul , ~ ax-reg } and the axioms of + existence of unordered ` m ` -tuples for all ` m <_ n ` , and in most cases + one would like to rule out such models, hence the need for extra axioms, + typically variants of powersets or unions. + + The axiom of adjunction ~ ax-bj-adj is more widely used, and is an axiom of + General Set Theory. We prove how to retrieve it from binary union and + singleton in ~ bj-adjfrombun and conversely how to prove from adjunction + singleton ( ~ bj-snfromadj ) and unordered pair ( ~ bj-prfromadj ). + +$) + + ${ + $d x y $. $d ph y $. + $( Two ways of stating that the extension of a formula is a set. + (Contributed by BJ, 18-Jan-2025.) + (Proof modification is discouraged.) $) + bj-abex $p |- ( { x | ph } e. _V <-> E. y A. x ( x e. y <-> ph ) ) $= + ( cab cvv wcel cv wceq wex wel wb wal isset nfcv abeq2f exbii bitri ) ABD + ZEFCGZRHZCIBCJAKBLZCICRMTUACABSBSNOPQ $. + $} + + ${ + $d A x y $. + bj-clex.1 $e |- ( x e. A <-> ph ) $. + $( Two ways of stating that a class is a set. (Contributed by BJ, + 18-Jan-2025.) (Proof modification is discouraged.) $) + bj-clex $p |- ( A e. _V <-> E. y A. x ( x e. y <-> ph ) ) $= + ( cvv wcel cv wceq wex wel wb wal isset dfcleq bibi2i albii bitri exbii ) + DFGCHZDIZCJBCKZALZBMZCJCDNUAUDCUAUBBHDGZLZBMUDBTDOUFUCBUEAUBEPQRSR $. + $} + + ${ + $d x y z $. + $( Two ways of stating the axiom of singleton (which is the universal + closure of either side, see ~ ax-bj-sn ). (Contributed by BJ, + 12-Jan-2025.) (Proof modification is discouraged.) $) + bj-axsn $p |- ( { x } e. _V <-> E. y A. z ( z e. y <-> z = x ) ) $= + ( weq cv csn velsn bj-clex ) CADCBAEZFCIGH $. + $( $j usage 'bj-axsn' avoids 'ax-sep' 'ax-nul' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $d x y z $. + $( Axiom of singleton. (Contributed by BJ, 12-Jan-2025.) $) + ax-bj-sn $a |- A. x E. y A. z ( z e. y <-> z = x ) $. + $} + + ${ + $d x y z $. $d x A $. + $( A singleton built on a set is a set. Contrary to ~ bj-snex , this proof + is intuitionistically valid and does not require ~ ax-nul . + (Contributed by NM, 7-Aug-1994.) Extract it from ~ snex and prove it + from ~ ax-bj-sn . (Revised by BJ, 12-Jan-2025.) + (Proof modification is discouraged.) $) + bj-snexg $p |- ( A e. V -> { A } e. _V ) $= + ( vx vz vy csn cvv wcel cv wceq sneq wel weq wal wex ax-bj-sn spi bj-axsn + wb mpbir eqeltrrdi vtocleg ) AFZGHCABCIZAJUCUDFZGUDAKUEGHDELDCMSDNEOZUFCC + EDPQCEDRTUAUB $. + $( $j usage 'bj-snexg' avoids 'ax-sep' 'ax-nul' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $( A singleton is a set. See also ~ snex , ~ snexALT . (Contributed by + NM, 7-Aug-1994.) Prove it from ~ ax-bj-sn . (Revised by BJ, + 12-Jan-2025.) (Proof modification is discouraged.) $) + bj-snex $p |- { A } e. _V $= + ( cvv wcel csn bj-snexg wn c0 wceq snprc biimpi 0ex eqeltrdi pm2.61i ) AB + CZADZBCABENFZOGBPOGHAIJKLM $. + $( $j usage 'bj-snex' avoids 'ax-sep' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $d x z t $. $d y z t $. + $( Two ways of stating the axiom of binary union (which is the universal + closure of either side, see ~ ax-bj-bun ). (Contributed by BJ, + 12-Jan-2025.) (Proof modification is discouraged.) $) + bj-axbun $p |- ( ( x u. y ) e. _V <-> + E. z A. t ( t e. z <-> ( t e. x \/ t e. y ) ) ) $= + ( wel wo cv cun elun bj-clex ) DAEDBEFDCAGZBGZHDGKLIJ $. + $( $j usage 'bj-axbun' avoids 'ax-sep' 'ax-nul' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $d x y z t $. + $( Axiom of binary union. (Contributed by BJ, 12-Jan-2025.) $) + ax-bj-bun $a |- A. x A. y E. z A. t ( t e. z <-> ( t e. x \/ t e. y ) ) $. + $} + + ${ + $d A x y $. $d B x y $. $d V x $. $d W y $. $d x y z t $. + $( Existence of binary unions of sets, proved from ~ ax-bj-bun . + (Contributed by BJ, 12-Jan-2025.) + (Proof modification is discouraged.) $) + bj-unexg $p |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) $= + ( vx vy vt vz wcel cv wceq wex cun cvv elissetv wa wel wal spi exlimiv wo + exdistrv uneq12 wb ax-bj-bun bj-axbun mpbir eqeltrrdi sylbir syl2an ) ACI + EJZAKZELZFJZBKZFLZABMZNIZBDIEACOFBDOUMUPPULUOPZFLZELURULUOEFUBUTUREUSURFU + SUQUKUNMZNUKAUNBUCVANIGHQGEQGFQUAUDGRHLZVBFVBFREEFHGUESSEFHGUFUGUHTTUIUJ + $. + $( $j usage 'bj-unexg' avoids 'ax-sep' 'ax-nul' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $( Existence of unordered pairs formed on sets, proved from ~ ax-bj-sn and + ~ ax-bj-bun . Contrary to ~ bj-prex , this proof is intuitionistically + valid and does not require ~ ax-nul . (Contributed by BJ, 12-Jan-2025.) + (Proof modification is discouraged.) $) + bj-prexg $p |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V ) $= + ( wcel wa cpr csn cun cvv df-pr bj-snexg bj-unexg syl2an eqeltrid ) ACEZB + DEZFABGAHZBHZIZJABKPRJESJETJEQACLBDLRSJJMNO $. + $( $j usage 'bj-prexg' avoids 'ax-sep' 'ax-nul' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $( Existence of unordered pairs proved from ~ ax-bj-sn and ~ ax-bj-bun . + (Contributed by BJ, 12-Jan-2025.) + (Proof modification is discouraged.) $) + bj-prex $p |- { A , B } e. _V $= + ( cpr csn cun cvv df-pr wcel bj-snex bj-unexg mp2an eqeltri ) ABCADZBDZEZ + FABGMFHNFHOFHAIBIMNFFJKL $. + $( $j usage 'bj-prex' avoids 'ax-sep' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $d x z t $. $d y z t $. + $( Two ways of stating the axiom of adjunction (which is the universal + closure of either side). (Contributed by BJ, 12-Jan-2025.) + (Proof modification is discouraged.) $) + bj-axadj $p |- ( ( x u. { y } ) e. _V <-> + E. z A. t ( t e. z <-> ( t e. x \/ t = y ) ) ) $= + ( wel weq wo cv csn cun wcel elun velsn orbi2i bitri bj-clex ) DAEZDBFZGZ + DCAHZBHZIZJZDHZUCKQUDUBKZGSUDTUBLUERQDUAMNOP $. + $( $j usage 'bj-axadj' avoids 'ax-sep' 'ax-nul' 'ax-pr' 'ax-pow'; $) + $} + + ${ + $d x y z t $. + $( Axiom of adjunction. (Contributed by BJ, 19-Jan-2025.) $) + ax-bj-adj $a |- A. x A. y E. z A. t ( t e. z <-> ( t e. x \/ t = y ) ) $. + $} + + ${ + $d x y z t $. $d A y $. + $( Existence of the result of the adjunction (generalized only in the first + term is this suffices for current applications). (Contributed by BJ, + 19-Jan-2025.) (Proof modification is discouraged.) $) + bj-adjg1 $p |- ( A e. V -> ( A u. { x } ) e. _V ) $= + ( vy vt vz cv csn cun cvv wcel wceq uneq1 eleq1d wel weq wo wb wal spi + wex ax-bj-adj bj-axadj mpbir vtoclg ) DGZAGHZIZJKZBUGIZJKDBCUFBLUHUJJUFBU + GMNUIEFOEDOEAPQRESFUAZUKAUKASDDAFEUBTTDAFEUCUDUE $. + $} + + ${ + $d x y z t $. + $( Singleton from adjunction and empty set. (Contributed by BJ, + 19-Jan-2025.) (Proof modification is discouraged.) $) + bj-snfromadj $p |- { x } e. _V $= + ( c0 cv csn cun cvv 0un wcel 0ex bj-adjg1 ax-mp eqeltrri ) BACDZEZMFMGBFH + NFHIABFJKL $. + $} + + ${ + $( Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) + (Proof modification is discouraged.) $) + bj-prfromadj $p |- { x , y } e. _V $= + ( cv cpr csn cun cvv df-pr wcel bj-snfromadj bj-adjg1 ax-mp eqeltri ) ACZ + BCZDNEZOEFZGNOHPGIQGIAJBPGKLM $. + $} + + ${ + $( Adjunction from singleton and binary union. (Contributed by BJ, + 19-Jan-2025.) (Proof modification is discouraged.) $) + bj-adjfrombun $p |- ( x u. { y } ) e. _V $= + ( cv cvv wcel csn cun vex bj-snexg elv bj-unexg mp2an ) ACZDEBCZFZDEZMOGD + EAHPBNDIJMODDKL $. + $} + + $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Set theory: miscellaneous @@ -555739,6 +556691,31 @@ then use transitivity of a congruence relation (equality for ~ pw0 and biimparc impbid2 ) ABDZCEZABFEZABGZABHUCUAUBUCUAAIEZUBUCATGUAUDUCAABUCAJUCK LATCMNUDUBUCABIORPQS $. + ${ + $d x A $. + $( This theorem ~ bj-velpwALT and the next theorem ~ bj-elpwgALT are + alternate proofs of ~ velpw and ~ elpwg respectively, where one proves + first the setvar case and then generalizes using ~ vtoclbg instead of + proving first the general case using ~ elab2g and then specifying. + Here, this results in needing an extra DV condition, a longer combined + proof and use of ~ ax-12 . In other cases, that order is better (e.g., + ~ vsnex proved before ~ snexg ). (Contributed by BJ, 17-Jan-2025.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + bj-velpwALT $p |- ( x e. ~P A <-> x C_ A ) $= + ( cv cpw wcel wss cab df-pw eleq2i abid bitri ) ACZBDZELLBFZAGZENMOLABHIN + AJK $. + $} + + ${ + $d A x $. $d B x $. + $( Alternate proof of ~ elpwg . See comment for ~ bj-velpwALT . + (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + bj-elpwgALT $p |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) $= + ( vx cv cpw wcel wss eleq1 sseq1 bj-velpwALT vtoclbg ) DEZBFZGMBHANGABHDA + CMANIMABJDBKL $. + $} + ${ $d z x $. $d z y $. $( Justification theorem for ~ dfv2 if it were the definition. See also @@ -556142,10 +557119,10 @@ Moore collections ( ~ df-mre ), topologies ( ~ df-top ), pi-systems, rings of bj-restn0 $p |- ( ( X e. V /\ A e. W ) -> ( X =/= (/) -> ( X |`t A ) =/= (/) ) ) $= ( vx vy wcel wa c0 wne cv crest co wex cin wceq wrex n0 wi vex inex1 jctr - isseti eximi df-rex sylibr rexcom4 sylib a1i syl5bi elrest biimprd eximdv - syld syl6ibr ) DBGACGHZDIJZEKZDALMZGZENZUSIJUPUQURFKZAOZPZFDQZENZVAUQVBDG - ZFNZUPVFFDRVHVFSUPVHVDENZFDQZVFVHVGVIHZFNVJVGVKFVGVIEVCVBAFTUAUCUBUDVIFDU - EUFVDFEDUGUHUIUJUPVEUTEUPUTVEFURADBCUKULUMUNEUSRUO $. + isseti eximi df-rex sylibr rexcom4 sylib a1i biimtrid elrest biimprd syld + eximdv syl6ibr ) DBGACGHZDIJZEKZDALMZGZENZUSIJUPUQURFKZAOZPZFDQZENZVAUQVB + DGZFNZUPVFFDRVHVFSUPVHVDENZFDQZVFVHVGVIHZFNVJVGVKFVGVIEVCVBAFTUAUCUBUDVIF + DUEUFVDFEDUGUHUIUJUPVEUTEUPUTVEFURADBCUKULUNUMEUSRUO $. $} $( Alternate version of ~ bj-restn0 . (Contributed by BJ, 27-Apr-2021.) $) @@ -556292,14 +557269,14 @@ LHS singles out the empty intersection (the empty intersection relative A. x e. ~P A ( X i^i |^| x ) e. B ) ) $= ( cpw wss wcel cv cint c0 csn wral wa cin wb wceq df-ss a1i wi eleq1 cdif cvv ssv int0 sseqtrri mpbi eqcomi eleq1i eldifsn sstr2 intss2 elpwi syl11 - wne syl6 impd syl5bi incom eqcom sylbb sylbi syl5 syld ralrimiv ralbi syl - eqeq1i anbi12d biancomd 0elpw inteq ineq2 3syl bj-raldifsn ax-mp bitr4di - ) BDEZFZDCGZAHZIZCGZABEZJKUAZLZMZDWANZCGZAWDLZDJIZNZCGZMZWHAWCLZVRWFWIWLV - RVSWLWEWIVSWLOVRDWKCWKDDWJFWKDPDUBWJDUCUDUEDWJQUFUGUHRVRWBWHOZAWDLWEWIOVR - WOAWDVRVTWDGZWADFZWOWPVTWCGZVTJUNZMVRWQVTWCJUIVRWRWSWQVTBFZVRWSWQSZWRWTVR - VTVQFXAVTBVQUJVTDUKUOVTBULUMUPUQWQWAWGPZVRWOWQWADNZWAPZXBWADQXDWGWAPXBXCW - GWAWADURVGWGWAUSUTVAXBWOSVRWAWGCTRVBVCVDWBWHAWDVEVFVHVIJWCGWNWMOBVJWHWLAW - CJVTJPWAWJPWGWKPWHWLOVTJVKWAWJDVLWGWKCTVMVNVOVP $. + wne syl6 impd biimtrid incom eqeq1i eqcom sylbb sylbi syl5 ralrimiv ralbi + syld anbi12d biancomd 0elpw inteq ineq2 3syl bj-raldifsn ax-mp bitr4di + syl ) BDEZFZDCGZAHZIZCGZABEZJKUAZLZMZDWANZCGZAWDLZDJIZNZCGZMZWHAWCLZVRWFW + IWLVRVSWLWEWIVSWLOVRDWKCWKDDWJFWKDPDUBWJDUCUDUEDWJQUFUGUHRVRWBWHOZAWDLWEW + IOVRWOAWDVRVTWDGZWADFZWOWPVTWCGZVTJUNZMVRWQVTWCJUIVRWRWSWQVTBFZVRWSWQSZWR + WTVRVTVQFXAVTBVQUJVTDUKUOVTBULUMUPUQWQWAWGPZVRWOWQWADNZWAPZXBWADQXDWGWAPX + BXCWGWAWADURUSWGWAUTVAVBXBWOSVRWAWGCTRVCVFVDWBWHAWDVEVPVGVHJWCGWNWMOBVIWH + WLAWCJVTJPWAWJPWGWKPWHWLOVTJVJWAWJDVKWGWKCTVLVMVNVO $. $} ${ @@ -556438,10 +557415,10 @@ variety of applications (for many families of sets, the family of bj-ismooredr2 $p |- ( ph -> A e. Moore_ ) $= ( cv wss wa c0 wne cuni cint cin wcel anassrs intssuni2 wceq dfss sylbi wi wb incom eqeq2i eleq1 biimpd syl adantll mpd ex wn rint0 eleq1a syl2im - nne syl5bi adantr pm2.61d bj-ismooredr ) ABCABFZCGZHZUSIJZCKZUSLZMZCNZVAV - BVFVAVBHVDCNZVFAUTVBVGEOUTVBVGVFTZAUTVBHVDVCGZVHUSCPVIVDVDVCMZQZVHVDVCRVK - VGVFVKVDVEQVGVFUAVJVEVDVDVCUBUCVDVECUDSUESUFUGUHUIAVBUJZVFTUTVLUSIQZAVFUS - IUNAVCCNVMVEVCQVFDVCUSUKVCCVEULUMUOUPUQUR $. + nne biimtrid adantr pm2.61d bj-ismooredr ) ABCABFZCGZHZUSIJZCKZUSLZMZCNZV + AVBVFVAVBHVDCNZVFAUTVBVGEOUTVBVGVFTZAUTVBHVDVCGZVHUSCPVIVDVDVCMZQZVHVDVCR + VKVGVFVKVDVEQVGVFUAVJVEVDVDVCUBUCVDVECUDSUESUFUGUHUIAVBUJZVFTUTVLUSIQZAVF + USIUNAVCCNVMVEVCQVFDVCUSUKVCCVEULUMUOUPUQUR $. $} ${ @@ -558577,7 +559554,8 @@ singleton on a couple (with disjoint domain) at a point in the domain ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` n ) ) ) ` m ) ) ) $= ( ) ? $. - @( The sum of the empty set is the unit. (Contributed by BJ, 9-Jun-2019.) @) + @( The sum of the empty set is the identity. (Contributed by BJ, + 9-Jun-2019.) @) bj-finsum0 $p |- ( A e. CMnd -> ( A FinSum (/) ) = ( 0g ` A ) ) $= ( ) ? $. @@ -558842,15 +559820,15 @@ coordinates of a barycenter of two points in one dimension (complex ( cmul co caddc wceq c1 wa cmin oveq1 cdiv wi pncand pm5.31 sylancl eqtr2 eqcomd syl6 oveq1d eqtr sylan2 subdird mulid2d eqtrd sylan9eqr ex sylan2d mulcld subadd23d subdid oveq2d eqeq2d sylibd subcld pncan2d eqcom mulcomd - 1cnd syl5ib eqeq1d necomd subne0d rdiv biimpd sylbid syl5bi 3syld ) AFDBM - NZECMNZONZPZDEONZQPZRZFBECBSNZMNZONZPZFBSNZWFPZEWIWEUANPZAWDFBEBMNZSNZVSO - NZPZWHAWCDQESNZPZWAWOAWCWBESNZWPPZWRDPZRZWQAWTWCWSUBWCXAUBADEKLUCWBQESTWC - WSWTUDUEXAWPDWRWPDUFUGUHAWAWQRZWOXBAFWPBMNZVSONZWNWQWAVTXDPFXDPWQVRXCVSOD - WPBMTUIFVTXDUJUKAXCWMVSOAXCQBMNZWLSNWMAQEBAVHLGULAXEBWLSABGUMUIUNUIUOUPUQ - AWNWGFAWNBVSWLSNZONWGABWLVSGAEBLGURAECLHURUSAXFWFBOAWFXFAECBLHGUTUGVAUNVB - VCWHWIWGBSNZPAWJFWGBSTAXGWFWIABWFGAEWELACBHGVDZURVEVBVIWJWFWIPZAWKWIWFVFA - XIWEEMNZWIPZWKAWFXJWIAEWELXHVGVJAXKWKAWEEWIXHLAFBIGVDACBHGABCJVKVLVMVNVOV - PVQ $. + 1cnd syl5ib eqeq1d necomd subne0d rdiv biimpd sylbid biimtrid 3syld ) AFD + BMNZECMNZONZPZDEONZQPZRZFBECBSNZMNZONZPZFBSNZWFPZEWIWEUANPZAWDFBEBMNZSNZV + SONZPZWHAWCDQESNZPZWAWOAWCWBESNZWPPZWRDPZRZWQAWTWCWSUBWCXAUBADEKLUCWBQEST + WCWSWTUDUEXAWPDWRWPDUFUGUHAWAWQRZWOXBAFWPBMNZVSONZWNWQWAVTXDPFXDPWQVRXCVS + ODWPBMTUIFVTXDUJUKAXCWMVSOAXCQBMNZWLSNWMAQEBAVHLGULAXEBWLSABGUMUIUNUIUOUP + UQAWNWGFAWNBVSWLSNZONWGABWLVSGAEBLGURAECLHURUSAXFWFBOAWFXFAECBLHGUTUGVAUN + VBVCWHWIWGBSNZPAWJFWGBSTAXGWFWIABWFGAEWELACBHGVDZURVEVBVIWJWFWIPZAWKWIWFV + FAXIWEEMNZWIPZWKAWFXJWIAEWELXHVGVJAXKWKAWEEWIXHLAFBIGVDACBHGABCJVKVLVMVNV + OVPVQ $. $( Barycentric coordinates in one dimension (complex line). In the statement, ` X ` is the barycenter of the two points ` A , B ` with @@ -559090,7 +560068,7 @@ coordinates of a barycenter of two points in one dimension (complex $} ${ - $d A q r s $. + $d A q r $. $( The irrationals are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 19-May-2024.) $) irrdiff $p |- ( A e. RR -> ( -. A e. QQ <-> A. q e. QQ A. r e. QQ @@ -559246,17 +560224,17 @@ coordinates of a barycenter of two points in one dimension (complex (Contributed by ML, 15-Jul-2020.) $) f1omptsnlem $p |- F : A -1-1-onto-> R $= ( vy crn wf1o cv cfv wceq wi wcel wsbc cvv wb ax-mp wa wf1 wf wral eqid - csn snex eqsbc1 mpbir csb sbcel2 csbconstg eleq2i bitri wrex wex abeq2i - df-rex sylbbr 19.23bi sbcth sbcimg sbcan sbcel1v 3imtr3i sylanbr fvmpt2 - mpbi mpan2 fmpti mpdan fvmpt3i eqeqan12d vex sneqbg bitrdi biimpd rgen2 - sneq dff13 mpbir2an f1f1orn cmpt cab rnmptsn rneqi 3eqtr4i f1oeq3 ) CEI - ZEJZCDEJZCDEUAZWIWKCDEUBAKZELZHKZELZMZWLWNMZNZHCUCACUCACDWLUEZEFWLCOZBK - ZWSMZBWSPZWSDOZXCWSWSMZWSUDWSQOZXCXERWLUFZBWSWSQUGSUHWTWTBWSPZXCXDXHWLB - WSCUIZOWTBWSWLCUJXICWLXFXICMXGBWSCQUKSULUMWTXBTZBWSPZXADOZBWSPZXHXCTXDX - JXLNZBWSPZXKXMNZXFXOXGXNBWSQXJXLAXLXBACUNZXJAUOXQBDGUPXBACUQURUSUTSXFXO - XPRXGXJXLBWSQVASVGWTXBBWSVBBWSDVCVDVEVHZVIWRAHCCWTWNCOZTZWPWQXTWPWSWNUE - ZMZWQWTXSWMWSWOYAWTXDWMWSMXRACWSDEFVFVJAWNWSYACEWLWNVRFXGVKVLWLQOYBWQRA - VMWLWNQVNSVOVPVQAHCDEVSVTCDEWASWHDMWIWJRACWSWBZIXQBWCWHDABCWDEYCFWEGWFW + csn vsnex eqsbc1 mpbir sbcel2 csbconstg eleq2i bitri wrex abeq2i df-rex + csb wex sylbbr 19.23bi sbcth sbcimg sbcan sbcel1v 3imtr3i sylanbr mpan2 + mpbi fmpti fvmpt2 mpdan sneq fvmpt3i eqeqan12d vex sneqbg bitrdi biimpd + rgen2 dff13 mpbir2an f1f1orn cmpt cab rnmptsn rneqi 3eqtr4i f1oeq3 ) CE + IZEJZCDEJZCDEUAZWIWKCDEUBAKZELZHKZELZMZWLWNMZNZHCUCACUCACDWLUEZEFWLCOZB + KZWSMZBWSPZWSDOZXCWSWSMZWSUDWSQOZXCXERAUFZBWSWSQUGSUHWTWTBWSPZXCXDXHWLB + WSCUPZOWTBWSWLCUIXICWLXFXICMXGBWSCQUJSUKULWTXBTZBWSPZXADOZBWSPZXHXCTXDX + JXLNZBWSPZXKXMNZXFXOXGXNBWSQXJXLAXLXBACUMZXJAUQXQBDGUNXBACUOURUSUTSXFXO + XPRXGXJXLBWSQVASVGWTXBBWSVBBWSDVCVDVEVFZVHWRAHCCWTWNCOZTZWPWQXTWPWSWNUE + ZMZWQWTXSWMWSWOYAWTXDWMWSMXRACWSDEFVIVJAWNWSYACEWLWNVKFXGVLVMWLQOYBWQRA + VNWLWNQVOSVPVQVRAHCDEVSVTCDEWASWHDMWIWJRACWSWBZIXQBWCWHDABCWDEYCFWEGWFW HDCEWGSVG $. $} @@ -559284,23 +560262,23 @@ coordinates of a barycenter of two points in one dimension (complex wss csn cab crn df-ima cmpt reseq1i resmpt eqtrid rnmptsn eqtrdi unieqd rneqd eleq2d eleq1w eluniab r19.41v df-rex 3bitr2i anbi2d adantr anim2i ancom eleq2 ibi eximi an12 exbii exsimpr syl exlimiv velsn anbi2i sylib - biimparc vtoclga equid wsb eqid snex sbcg eqsbc1 adantl biimpri 19.23bi - expcom sylbir sbcth sbcimg mpbi sbcan nfv nfab1 nfuni nfcri nfim sbcgfi - sylbird 3imtr3i syl2anbr mpan2 syl5bir mpbidi com12 sbimi equsb3 impbii - sbv bitrdi eqrdv eqcomd ) DCUEZFDUAZUBZDXPAXRDXPAJZXRKXSBJZXSUFZLZADUCZ - BUGZUBZKZXSDKZXPXRYEXSXPXQYDXPXQFDUDZUHZYDFDUIXPYIADYAUJZUHYDXPYHYJXPYH - ACYAUJZDUDYJFYKDGUKACDYAULUMUQABDUNUOUMUPURYFYGIJZDKZYGIXSYEIADUSZYLYEK - ZYGYLXSLZMZANZYMYOYGYLYAKZMZANZYRYOYLXTKZYCMZBNZUUAYCBYLUTZUUCUUABUUCYG - YBYSMZMZANZUUAUUCYGYBUUBMZMZANZUUHUUCYCUUBMUUIADUCUUKUUBYCVGYBUUBADVAUU - IADVBVCUUJUUGAUUIUUFYGUUIUUFYBUUIUUFOUUBYBUUBYSYBXTYAYLVHZVDVEVIVFVJPUU - HYBYTMZANUUAUUGUUMAYGYBYSVKVLYBYTAVMPVNVOPYTYQAYSYPYGIXSVPZVQVLVRYQYMAY - PYMYGYNVSVOVNVTXSXSLZYGYFQZAWAYPIAWBUUPIAWBUUOUUPYPUUPIAYGYPYFYPYOYFYGY - PYSYGYOUUNYGYAYALZYSYOQZYAWCYGYGBYARZYBBYARZUURUUQYASKZUUSYGOXSWDZYGBYA + biimparc vtoclga equid wsb eqid vsnex sbcg eqsbc1 adantl biimpri expcom + 19.23bi sylbir sylbird sbcth sbcimg mpbi sbcan nfab1 nfuni nfcri sbcgfi + nfv 3imtr3i syl2anbr mpan2 syl5bir mpbidi com12 sbimi equsb3 sbv impbii + nfim bitrdi eqrdv eqcomd ) DCUEZFDUAZUBZDXPAXRDXPAJZXRKXSBJZXSUFZLZADUC + ZBUGZUBZKZXSDKZXPXRYEXSXPXQYDXPXQFDUDZUHZYDFDUIXPYIADYAUJZUHYDXPYHYJXPY + HACYAUJZDUDYJFYKDGUKACDYAULUMUQABDUNUOUMUPURYFYGIJZDKZYGIXSYEIADUSZYLYE + KZYGYLXSLZMZANZYMYOYGYLYAKZMZANZYRYOYLXTKZYCMZBNZUUAYCBYLUTZUUCUUABUUCY + GYBYSMZMZANZUUAUUCYGYBUUBMZMZANZUUHUUCYCUUBMUUIADUCUUKUUBYCVGYBUUBADVAU + UIADVBVCUUJUUGAUUIUUFYGUUIUUFYBUUIUUFOUUBYBUUBYSYBXTYAYLVHZVDVEVIVFVJPU + UHYBYTMZANUUAUUGUUMAYGYBYSVKVLYBYTAVMPVNVOPYTYQAYSYPYGIXSVPZVQVLVRYQYMA + YPYMYGYNVSVOVNVTXSXSLZYGYFQZAWAYPIAWBUUPIAWBUUOUUPYPUUPIAYGYPYFYPYOYFYG + YPYSYGYOUUNYGYAYALZYSYOQZYAWCYGYGBYARZYBBYARZUURUUQYASKZUUSYGOAWDZYGBYA SWETUVAUUTUUQOUVBBYAYASWFTYGYBMZBYARZUURBYARZUUSUUTMUURUVCUURQZBYARZUVD UVEQZUVAUVGUVBUVFBYASUVCYSUUBYOYBUUBYSOYGUULWGUVCUUBYOQZAUVCANYCUVIYBAD - VBUUBYCYOUUCYOBYOUUDUUEWHWIWJWKWIXBWLTUVAUVGUVHOUVBUVCUURBYASWMTWNYGYBB - YAWOUURBYAUVBYSYOBYSBWPBIYEBYDYCBWQWRWSWTXAXCXDXEXFIAYEUSXGXHXIIAAXJUUP - IAXLXCTXKXMXNXO $. + VBUUBYCYOUUCYOBYOUUDUUEWHWJWIWKWJWLWMTUVAUVGUVHOUVBUVCUURBYASWNTWOYGYBB + YAWPUURBYAUVBYSYOBYSBXABIYEBYDYCBWQWRWSXLWTXBXCXDXEIAYEUSXFXGXHIAAXIUUP + IAXJXBTXKXMXNXO $. $} $d A u x $. $d A x y $. $d B u x $. @@ -559406,42 +560384,42 @@ coordinates of a barycenter of two points in one dimension (complex two. (Contributed by ML, 17-Jul-2020.) $) topdifinffinlem $p |- ( T e. ( TopOn ` A ) -> A e. Fin ) $= ( vu vy cfn wcel wn wceq wex wa w3a wsbc cvv c0 wo eleq1 wb ax-mp nfab1 - ctopon cfv cpw cv csn wrex cab wss nfv nfcv abid df-rex bitri eqid snex + ctopon cfv cpw cv csn wrex cab wss nfcv abid df-rex bitri eqid wi vsnex cdif snelpwi syl5ibr imdistani anim2i 3impb 3anass sylibr mpbiri difinf - wi snfi sylan2 orcd ancoms 3impa syl rabeq2i 3ad2ant2 mpbid sbcimg mpbi - sbcth sbc3an sbcg 3anbi1i eqsbc1 3anbi2i 3bitri 3anbi3i 3imtr3i pm4.71d - mp3an2 ex anbi1d exbidv bitrid anass exbii exsimpr sylbi syl6bi pm5.32i - ancom bitr4i syl6ib syl6 ax5e dissneq sylan nfielex adantr difss elfvex - ssrd difexg elpwg 3syl adantl mpan2 0fin nsyl ad2antrl wne vsnid inelcm - cin wpss disj4 necon2abii pssned neneqd jca pm4.56 difeq2 eleq1d notbid - sylib eqeq1 orbi12d elrab2 biantrurd bitr4id dfin4 inss2 mp2an eqeltrri - ssfi biortn bitr4di ad2antll mtbird expcom nelneq2 eqcom sylnibr syl6an - exellimddv pm2.65da con4i ) BGHZCBUBUCHZUUQIZUURCBUDZJZUUSEUEZFUEZUFZJZ - FBUGZEUHZCUIUURUVAUUSEUVGCUUSEUJUVFEUAECUKUUSUVBUVGHZUVBCHZFKZUVIUUSUVH - UVEUVILZFKZUVJUUSUVHUVDCHZUVELZFKZUVLUUSUVHUVCBHZUVMLZUVELZFKZUVOUVHUVP - UVELZFKZUUSUVSUVHUVFUWAUVFEULUVEFBUMUNUUSUVTUVRFUUSUVPUVQUVEUUSUVPUVMUU - SUVPUVMUUSUVDUVDJZUVPUVMUVDUOUUSAUEZUVDJZUVPMZAUVDNZUVMAUVDNZUUSUWBUVPM - ZUVMUWEUVMVGZAUVDNZUWFUWGVGZUVDOHZUWJUVCUPZUWIAUVDOUWEUWCCHZUVMUWEUWCUU - THZBUWCUQZGHZIZUWCPJZUWCBJZQZQZLZUWNUWEUUSUWDUWOMZUXCUWEUUSUWDUWOLZLZUX - DUUSUWDUVPUXFUWDUVPLUXEUUSUWDUVPUWOUVPUWOUWDUVDUUTHUVCBURUWCUVDUUTRUSUT - VAVBUUSUWDUWOVCVDUUSUWDUWOUXCUWOUUSUWDLZUXCUXGUXBUWOUXGUWRUXAUWDUUSUWCG - HZUWRUWDUXHUVDGHZUVCVHZUWCUVDGRVEBUWCVFVIVJVAVKVLVMUXBACUUTDVNVDUWDUUSU - WNUVMSUVPUWCUVDCRVOVPVSTUWLUWJUWKSUWMUWEUVMAUVDOVQTVRUWFUUSUWBUVPAUVDNZ - MZUWHUWFUUSAUVDNZUWDAUVDNZUXKMUUSUXNUXKMUXLUUSUWDUVPAUVDVTUXMUUSUXNUXKU - WLUXMUUSSUWMUUSAUVDOWATWBUXNUWBUUSUXKUWLUXNUWBSUWMAUVDUVDOWCTWDWEUXKUVP - UUSUWBUWLUXKUVPSUWMUVPAUVDOWATWFUNUWLUWGUVMSUWMUVMAUVDOWATWGWIWJWHWKWLW - MUVSUVPUVNLZFKUVOUVRUXOFUVPUVMUVEWNWOUVPUVNFWPWQWRUVNUVKFUVNUVEUVMLUVKU - VMUVEWTUVEUVIUVMUVBUVDCRWSXAWOXBUVEUVIFWPXCUVIFXDXCXKFEBCUVGUVGUOXEXFUU - 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wne syl5bi jcad syl6ibr rabss2 ax-mp sseli impbid1 eqrd mpoeq3ia 3eqtri + wne biimtrid jcad syl6ibr rabss2 ax-mp sseli impbid1 eqrd mpoeq3ia 3eqtri a1i ) DFGGUBZUCZABGGAHZCHZIJZXCBHZKJZLZCMUDZNZABGGXGCGUDZNEXAABMMXHNZWTUC ZXIFXKWTABCUAUEGMUFZXMXLXIOPPABMMGGXHUGUHUIABGGXHXJXBGQZXEGQZLZCXHXJXPCUJ XGCMUKXGCGUKXPXCXHQZXCXJQZXPXQXCGQZXGLXRXPXQXSXGXQXCMQZXGLZXPXSXGCMULZXPY @@ -559517,13 +560495,13 @@ coordinates of a barycenter of two points in one dimension (complex ( vx vy vz vl cr cxp cpw cico wf wfn crn wss cxr cle clt mpbir cv wa wcel wrex cres rexpssxrxp df-ico ixxf ffn fnssresb mp2b wbr crab cmpo icorempo wb eqid rneqi wral ssrab2 reex elpw2 rgen2w wceq rnmpo abeq2i simpl simpr - r19.29d2r wi eleq1 biimparc a1i rexlimivv syl ex ssrdv ax-mp eqsstri df-f - syl5bi mpbir2an ) EEFZEGZHVSUAZIWAVSJZWAKZVTLWBVSMMFZLZUBWDMGZHIHWDJWBWEU - LABCNOHABCUCUDWDWFHUEWDVSHUFUGPWCABEEAQZCQZNUHWHBQZOUHRZCEUIZUJZKZVTWAWLA - BCWAWAUMUKUNWKVTSZBEUOAEUOZWMVTLWNABEEWNWKELWJCEUPWKEUQURPUSWODWMVTDQZWMS - WPWKUTZBETAETZWOWPVTSZWRDWMABDEEWKWLWLUMVAVBWOWRWSWOWRRZWNWQRZBETAETWSWTW - NWQABEEWOWRVCWOWRVDVEXAWSABEEXAWSVFWGESWIESRWQWSWNWPWKVTVGVHVIVJVKVLVQVMV - NVOVSVTWAVPVR $. + r19.29d2r wi eleq1 biimparc a1i rexlimivv ex biimtrid ssrdv ax-mp eqsstri + syl df-f mpbir2an ) EEFZEGZHVSUAZIWAVSJZWAKZVTLWBVSMMFZLZUBWDMGZHIHWDJWBW + EULABCNOHABCUCUDWDWFHUEWDVSHUFUGPWCABEEAQZCQZNUHWHBQZOUHRZCEUIZUJZKZVTWAW + LABCWAWAUMUKUNWKVTSZBEUOAEUOZWMVTLWNABEEWNWKELWJCEUPWKEUQURPUSWODWMVTDQZW + MSWPWKUTZBETAETZWOWPVTSZWRDWMABDEEWKWLWLUMVAVBWOWRWSWOWRRZWNWQRZBETAETWSW + TWNWQABEEWOWRVCWOWRVDVEXAWSABEEXAWSVFWGESWIESRWQWSWNWPWKVTVGVHVIVJVPVKVLV + MVNVOVSVTWAVQVR $. $} ${ @@ -559739,47 +560717,47 @@ coordinates of a barycenter of two points in one dimension (complex nfv ad2antll anim12i anim2i 3anass sylibr simprl xrltletr sylc simprr jca simpl sylanbrc adantlr adantr sylan2b sylbi expr ssrd sylanl2 eleq2 sseq1 eleqtrrd anbi12d rspcev syl12anc ancom1s c1 caddc peano2re syl2anc2 ltp1d - expl jca32 breq2 breq1 elrab simpll elioopnf simplbda mp2and syl5bi oveq2 - xrltletrd anbi2d imbi12d cuni cvv unirnioo ex sseq2d impl syl2anc intnand - mpbiri nltmnf eliooord nsyl pm2.21d impd ancomsd mpcom 3jaoi ancoms sseq2 - expdimp syl5ibrcom rexlimivv rgen rgenw iooex rnex icoreunrn eqtr3i tgss2 - rexbidv mp2an raleqi bitr4i mpbir ) FUDZUEUFAUEUFGZCHZUAHZIZUVNDHZIZUVQUV - OGZJZDAUGZKZUAUVLUHZCLUHZUWCCLUWBUAUVLUVOUVLIZUVOUBHZUCHZFMZNZUCOUGUBOUGZ - UWBOOUIZLUJZFVJFUWKUKUWEUWJPULUWKUWLFUMUBUCOOUVOFUNUOUWIUWBUBUCOOUWFOIZUW - GOIZJZUWBUWIUVNUWHIZUVRUVQUWHGZJZDAUGZKZUWNUWMUWTUWNUWMUWPUWSUWNUWGLIZUWG - UPNZUWGQNZURUWMUWPJZUWSKZUWGUQUXAUXEUXBUXCUXAUWMUWPUWSUWMUXAUWPUWSUWMUXAJ - ZUWPJZUVNEHZUSRZUXHUWGSRZJZELUTZAIZUVNUXLIZUXLUWHGZUWSUXGUVNLIZUXAJZUXMUX - FUXAUWPUXPUWMUXAVAZUVNUWFUWGVBZVCZEUVNUWGABVDVEUXGUXPUVNUVNUSRZUVNUWGSRZU - XNUWPUXPUXFUXSTUWPUYAUXFUWPUVNUXSVFTUXGUVNOIZUWFUVNSRZUYBUXFUWPUYCUYDUYBV - GZUWMUXAUWNUWPUYEPUXFUWGUXRVKUWFUWGUVNVHVIVLVMUXGUXPUYAUYBVGZUVNUVNUWGVNM - ZIZUXNUXGUXQUYFUYCUYAUYBVGZUYHUXTUXPUYCUYAUYBUVNWAZVOUXQUYHUYIUXPUYCUWNUY - HUYIPUXAUYJUWGWAZUVNUWGUVNVPVQVRVSUXGUYGUXLUVNUXGUXQUYGUXLNUXTEUVNUWGVTVE - WBWCWDUXAUWMUWNUWPUXOUYKUWOUWPJZEUXLUWHUYLEWLUXKELWEEUWHWFUWOUWPUXHUXLIZU - 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syl5ibrcom adantr jaod syl5bi ralimdv2 cab df-sbc sucex sseq2d raleqbi1dv - cbvabv elab2 bitri syl6ibr wlim ciun ssiun2 adantl rdglim2a mpan sseqtrrd - ex ralrimiva eleq2i abid sylibr a1d tfindes rsp syl eleq1 sseq12d imbi12d - wb eleq12 mpbii vtocleg com12 pm2.43b ) ELMZFEMZFDBUAZNZEYDNZOZYBYCYGYCYB - YCYGPZYCYBYHPZPIELYCIUBZEQZYIYKYIPJFEJUBZFQZYKYIYMYKUCZYJLMZYLYJMZYLYDNZY - JYDNZOZPZPYIYOYSJYJUDZYTUUAIKUUAIRUFZIRUUAIRUEUGYJRQUUAUUBYSJYJUHUUAIRUIU - JUKYOUUAYQYJULZYDNZOZJUUCUDZUUAIUUCUFZYOYSUUEJYJUUCYOYTYLUUCMZUUEPUUHYPYL - YJQZUMYOYTUCZUUEYLYJJUNUOUUJYPUUEUUIYOYTYPUUEPYOYSUUEYPYOYRUUDOZYSUUEYOYR - AYRCUPZUQZYRUUDYRUULURYOUUMSMUUDUUMQYRUULYJYDUTAYRCHUSVAABYJAUBZCUQZUUMYD - SABVBZAYJVBZAYRUULAYJYDABDADASUUOVCZGASUUOVDVEUUPVFUUQVGZAYRCUUSVHVIDUURQ - YDUURBUAQGBDUURVMVJUUNYRQZUUNYRCUULUUTVKAYRCVLVNVOVPVQZYQYRUUDVRVSVTWAYOU - UIUUEPYTYOUUEUUIUUKUVAUUIYQYRUUDYLYJYDTWBWCWDWEWFXDWGUUGUUCUUAIWHZMUUFUUA - IUUCWIYQKUBZYDNZOZJUVCUDZUUFKUUCUVBYJIUNWJUVEUUEJUVCUUCUVCUUCQUVDUUDYQUVC - UUCYDTWKWLUUAUVFIKYSUVEJYJUVCYJUVCQYRUVDYQYJUVCYDTWKWLWMZWNWOWPUVCWQZUUAI - 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This is expressed in its simplest form in ~ wl-luk-a1i , that - allows us prepending an arbitrary antecedent to an implication chain. - Using our antecedent swapping theorems described in - ~ wl-impchain-com-n.m , we may then move such a prepended antecedent to - any desired location within all antecedents. The first series of theorems - of this kind adds a single antecedent somewhere to an implication chain. - The appended number in the theorem name indicates its position within all - antecedents, 1 denoting the head position. A second theorem series - extends this idea to multiple additions (TODO). + allows prepending an arbitrary antecedent to an implication chain. Using + our antecedent swapping theorems described in ~ wl-impchain-com-n.m , we + may then move such a prepended antecedent to any desired location within + all antecedents. The first series of theorems of this kind adds a single + antecedent somewhere to an implication chain. The appended number in the + theorem name indicates its position within all antecedents, 1 denoting the + head position. A second theorem series extends this idea to multiple + additions (TODO). Adding antecedents to an implication chain usually weakens their universality. The consequent afterwards dependends on more conditions @@ -563145,40 +564123,40 @@ sides of the biconditional correlate (they are the same), if they exist at fin2solem breq2 rabbidv ancom2s 3orim123d ralrimivva breq1 eqeq1 r19.21bi mpd 3orbi123d ralbidv cbvmptv eqeq2 anasss issod fin2i2 syl22anc elrnmpti syl adantll sylib ssel2 sonr anassrs elrab simplbi2 ad2antlr elint2 eleq2 - bitri eleq2d rspcv simprbi syl6 syl5bi imbi1d syl5ibcom imp syld adantlll - mtod ex ralrimdva reximdva 3syl syldan alrimiv df-fr simpr df-we sylanbrc - weinxp sqxpexg incom inex1g eqeltrid weeq1 spcegv cfin7 cfin5 cfin6 cfin3 - expl ween cfin4 fin23 fin34 fin45 fin56 fin67 fin71num biimpac sylan ) AU - AIZABUBZAUCUDIZAUEIZUVHUVIJZACKZUFZCUGZUVJUVHUVIABAAUHZUIZUFZUVOUVLABUFZU - VRUVLABUJZUVIUVSUVLDKZALZUWAMUKZJUVMEKZBNZULZCUWAOZEUWAUMZPZDUNUVTUVLUWID - UVLUWBUWCUWHUVLUWBJZUWCJZFUWAGKZFKZBNZGUWAUOZUPZURZUQZUWLUWDBNZGUWAUOZQZE - UWAUMZUWHUWKUWRUWQIZUXBUWKUVHUWQAUSZLZUWQMUKZUWQRUBZUXCUVHUVIUWBUWCUTUWJU - 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NN ) $. $( Lemma for ~ poimir , a variant of Sperner's lemma. 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Proposition 7 of [Viaclovsky8] p. 3. (Contributed by Brendan Leahy, 4-Mar-2018.) @@ -568911,42 +569889,42 @@ curry M LIndF ( R freeLMod I ) ) ) $= cuni wne unieq eqtrdi fveq2d ovol0 eqtr2di a1d cle ovolge0 ad2antll wfo uni0 wex csdm cvv wb reldom brrelex1i 0sdomg syl biimparc fodomr unissb sylancom anbi1i r19.26 bitr4i cfn cen wo brdom2 nnenom sdomen2 isfinite - com ax-mp orbi1i bitri ovolfi expcom ovolctb ex syl5bi imdistani ralimi - jaod sylbi ancoms cima foima raleqdv wfn fofn ssid sseq1 fveqeq2 ralima - anbi12d sylancl bitr3d ciun caddc cmpt cseq crn clt fveq2 sseq1d 2fveq3 - csup weq eqeq1d cbvralvw 0re eleq1a anim2i syl2an fofun funiunfv unieqd - wfun eqtrd adantr rspccva mpteq2dva seqeq3d rneqd supeq1d csn fconstmpt - simprd cxp eqtri 0xr cpnf cmul co cc 0cn ser1const mpan mul01d mpteq2ia - nncn seqeq3 cuz cz 1z seqfn nnuz fneq2i dffn5 bitr3i mpbi 3eqtr4i rneqi - eqtr3i 1nn ne0i rnxp mp2b supeq1i wor xrltso supsn mp2an adantl 3brtr3d - sylbid exlimiv ovolcl xrletri3 sylancr mpbir2and expl pm2.61ine renepnf - imp mp1i ovolre neeqtrrd necon2i expr eqimss necon3bi pm2.61d1 ) BHUAIZ - 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UDXSMLVUIXTWAYAWJWLFYBUXIUYTUXAPUYKUXIUYSUWPNUXIUYSUYLUEZUWPUXIUXHYFUYS - VUQPHBUXHYCDHUXHYDVEUXIUYLBUYNYEYGUIYHUYKVUEMPUXIUYKVUEXGDHMXHZSXIZXJZT - XKXOZMUYKTVUDVUTXKUYKVUCVUSUYKVUBVURXGSUYKDHVUAMUYKUYQHRKUYRLUCZVUAMPZU - YJVVBVVCKGUYQHGDXPZUYGVVBUYIVVCVVDUYFUYRLUYEUYQUXHXLXMVVDUYHVUAMUYEUYQN - UXHXNXQXCYIYPYJYKYLYMVVAMYNZTXKXOZMTVUTVVEXKVUTHVVEYQZXJZVVEVUSVVGGHUYE - XGVVGSXIZOZXHZGHMXHVUSVVGGHVVJMUYEHRZVVJUYEMUUAUUBZMMUUCRVVLVVJVVMPUUDM - UYEUUEUUFVVLUYEUYEUUIUUGYGUUHVVIVUSVVKVVGVURPVVIVUSPDHMYOXGVVGVURSUUJWA - VVISUUKOZWQZVVIVVKPZSUULRVVOUUMXGVVGSUUNWAVVOVVIHWQVVPHVVNVVIUUOUUPGHVV - IUUQUURUUSUVBGHMYOUUTUVASHRHQUFVVHVVEPUVCHSUVDHVVEUVEUVFYRUVGTXKUVHMTRZ - VVFMPUVIYSTMXKUVJUVKYRUHUVLUVMWGUVNUVOUWCYBUWQUXBUXFUXGKVAZUXEUWOUWQVVQ - UXATRVVRYSUWPUVPMUXAUVQUVRUOUVSUVTUWAUWPLMUXAUWPLPZMYTUXAVUPMYTUFVVSXSM - UWBUWDVVSUXALNOYTUWPLNXLUWEUHUWFUWGVEUWHUWQUWPLUWPLUWIUWJUWK $. + com ax-mp orbi1i bitri ovolfi expcom ovolctb ex jaod biimtrid imdistani + ralimi sylbi ancoms cima foima raleqdv fofn ssid fveqeq2 anbi12d ralima + wfn sseq1 sylancl bitr3d ciun caddc cmpt cseq crn clt csup fveq2 sseq1d + weq 2fveq3 eqeq1d cbvralvw 0re eleq1a anim2i syl2an wfun fofun funiunfv + unieqd eqtrd adantr rspccva mpteq2dva seqeq3d supeq1d csn cxp fconstmpt + simprd rneqd eqtri 0xr cpnf cmul co cc 0cn ser1const mpan nncn mpteq2ia + mul01d seqeq3 cuz cz seqfn nnuz fneq2i dffn5 bitr3i mpbi eqtr3i 3eqtr4i + rneqi 1nn ne0i rnxp mp2b supeq1i wor xrltso supsn adantl 3brtr3d sylbid + 1z exlimiv imp ovolcl xrletri3 sylancr mpbir2and expl pm2.61ine renepnf + mp2an mp1i ovolre neeqtrrd necon2i expr eqimss necon3bi pm2.61d1 ) BHUA + IZAUBZHUAIZABJZKBUEZLUCZUWPLUFZUWLUWOUWQUWRUWLUWOUWQKZKZMUWPNOZPZUWRUWT + UXBUDBQBQPZUXBUWTUXCUXAQNOMUXCUWPQNUXCUWPQUEQBQUGUQUHUIUJUKULBQUFZUWLUW + SUXBUXDUWLKZUWSKUXBMUXAUMIZUXAMUMIZUWQUXFUXEUWOUWPUNUOUXEHBCUBZUPZCURZU + WMLUCZUWMNOMPZKZABJZUXGUWSUXDUWLQBUSIZUXJUWLUXOUXDUWLBUTRUXOUXDVABHUAVB + VCBUTVDVEVFHBCVGVIUWQUWOUXNUWQUWOKZUXKUWNKZABJZUXNUXPUXKABJZUWOKUXRUWQU + XSUWOABLVHVJUXKUWNABVKVLUXQUXMABUXKUWNUXLUWNUWMVMRZUWMHVNIZVOZUXKUXLUWN + UWMHUSIZUYAVOUYBUWMHVPUYCUXTUYAUYCUWMVTUSIZUXTHVTVNIUYCUYDVAVQHVTUWMVRW + AUWMVSVLWBWCUXKUXTUXLUYAUXTUXKUXLUWMWDWEUXKUYAUXLUWMWFWGWHWIWJWKWLWMUXJ + UXNUXGUXIUXNUXGUDCUXIUXNGUBZUXHOZLUCZUYFNOZMPZKZGHJZUXGUXIUXMAUXHHWNZJZ + UXNUYKUXIUXMAUYLBHBUXHWOZWPUXIUXHHXBZHHUCUYMUYKVAHBUXHWQZHWRUXMUYJAGHHU + XHUWMUYFPUXKUYGUXLUYIUWMUYFLXCUWMUYFMNWSWTXAXDXEUXIUYKUXGUXIUYKKDHDUBZU + XHOZXFZNOZXGDHUYRNOZXHZSXIZXJZTXKXLZUXAMUMUXIUYOEUBZUXHOZLUCZVUGNOZLRZK + ZEHJZUYTVUEUMIUYKUYPUYKVUHVUIMPZKZEHJVULUYJVUNGEHGEXOZUYGVUHUYIVUMVUOUY + FVUGLUYEVUFUXHXMXNVUOUYHVUIMUYEVUFNUXHXPXQWTXRVUNVUKEHVUMVUJVUHMLRZVUMV + UJUDXSMLVUIXTWAYAWKWLFYBUXIUYTUXAPUYKUXIUYSUWPNUXIUYSUYLUEZUWPUXIUXHYCU + YSVUQPHBUXHYDDHUXHYEVEUXIUYLBUYNYFYGUIYHUYKVUEMPUXIUYKVUEXGDHMXHZSXIZXJ + ZTXKXLZMUYKTVUDVUTXKUYKVUCVUSUYKVUBVURXGSUYKDHVUAMUYKUYQHRKUYRLUCZVUAMP + ZUYJVVBVVCKGUYQHGDXOZUYGVVBUYIVVCVVDUYFUYRLUYEUYQUXHXMXNVVDUYHVUAMUYEUY + QNUXHXPXQWTYIYPYJYKYQYLVVAMYMZTXKXLZMTVUTVVEXKVUTHVVEYNZXJZVVEVUSVVGGHU + YEXGVVGSXIZOZXHZGHMXHVUSVVGGHVVJMUYEHRZVVJUYEMUUAUUBZMMUUCRVVLVVJVVMPUU + DMUYEUUEUUFVVLUYEUYEUUGUUIYGUUHVVIVUSVVKVVGVURPVVIVUSPDHMYOXGVVGVURSUUJ + WAVVISUUKOZXBZVVIVVKPZSUULRVVOUVMXGVVGSUUMWAVVOVVIHXBVVPHVVNVVIUUNUUOGH + VVIUUPUUQUURUUSGHMYOUUTUVASHRHQUFVVHVVEPUVBHSUVCHVVEUVDUVEYRUVFTXKUVGMT + RZVVFMPUVHYSTMXKUVIUWCYRUHUVJUVKWGUVLUVNUVOYBUWQUXBUXFUXGKVAZUXEUWOUWQV + VQUXATRVVRYSUWPUVPMUXAUVQUVRUOUVSUVTUWAUWPLMUXAUWPLPZMYTUXAVUPMYTUFVVSX + SMUWBUWDVVSUXALNOYTUWPLNXMUWEUHUWFUWGVEUWHUWQUWPLUWPLUWIUWJUWK $. $} $( Demonstration of ~ ovoliunnfl . (Contributed by Brendan Leahy, @@ -569393,26 +570371,26 @@ curry M LIndF ( R freeLMod I ) ) ) $= ifex cicc citg2 cab itg2val supeq1i wor xrltso simprr eqeltrd rexlimiva cofr abssi supxrcl mp1i eqeq1d ifbieq2d eqeq2d cbvrexvw 0le0 rpge0 i1ff eqbrtrdi rpre addge01d readdcld ifcl iccssxr fss syl3anc mpand ralimdva - xrletr 3imtr4d rexlimdva anim1d reximdva syl5bi ss2abdv simp3r 3ad2ant2 - sseld w3a rexlimdv3a abssdv xrsupss supub syld supxrlub simprrr simplll - mpan wex csn cxp i1f0 2rp ne0ii ffvelcdm elxrge0 sylib simprd ralrimiva - c0 ralrimivw itg10 eqtr2di biantrud fvconst2 sylan9eq mpteq2dva eqbrtrd - r19.2z mnflt spcev simp-4r simpllr ngtmnft biimprd necon1ad xrltle xrre - syl22anc ccnv crn covol cnvimass fssdm eqcomd wfun difpreima cnvimarndm - fdm ffun difeq1i difeq12d dfss4 mpbi wfn fniniseg syl6bi itg10a breqtrd - ffn difeq1d biantrurd eldif bitr4di bitr4d con2bid 0m0e0 ifeq2da simpll - cof 3syl 1ex fconstmpt resubcl 2re cvol i1fima neldifsn i1fima2 remulcl - mblvol 2cnd 2ne0 mulne0d redivcld mulid1d mul01d ifeq12d i1fmulc i1fsub - i1f1 sylan9eqr wo ianor ifbid ifboth sylancl pm2.61d1 jaoi sylbi eleq1w - pm2.61i fvmpt eqtr4id mpteq2ia i1fpos ad2antrl posdifd mblss ltlen 2pos - ovolge0 mulgt0 mpanl12 divgt0d elrpd inidm ofrfval anim2i divcld npcand - breq1 pm2.24d impcom adantll sylanl1 oveq2 ifeq2d eqcoms nfeq2 mpteq2da - mpd nfmpt1 fvmpt2 simplrr ltdiv1d recn 2halvesd pncand eqtr3d rehalfcld - rehalfcl ltsubaddd 3bitr2d itg1mulc fveq2d itg11 mulcomd divassd 3eqtrd - divcan5rd 3eqtr3d itg1sub subsubd 3eqtrrd 3eqtr4d breqtrrd leidd ltnled - divsubdird resubcld biimpar ltled pm2.61dan bicomd bitrdi notbid pm4.53 - iba breq12d simplbda jaod sylbir olcs ltletrd adantllr pm2.61dane an32s - itg1le expr rexlimdvaa expimpd ancomsd exlimdv rexab 3imtr4g eqsupd + xrletr 3imtr4d rexlimdva anim1d reximdva biimtrid sseld simp3r 3ad2ant2 + ss2abdv w3a rexlimdv3a xrsupss supub syld supxrlub mpan simprrr simplll + abssdv wex csn cxp c0 2rp ne0ii ffvelcdm elxrge0 sylib simprd ralrimiva + i1f0 ralrimivw itg10 eqtr2di biantrud fvconst2 sylan9eq mpteq2dva mnflt + r19.2z eqbrtrd simp-4r simpllr ngtmnft biimprd necon1ad xrltle syl22anc + spcev xrre ccnv crn covol cnvimass fssdm fdm eqcomd wfun ffun difpreima + cnvimarndm difeq1i difeq12d mpbi wfn ffn fniniseg syl6bi itg10a breqtrd + dfss4 difeq1d biantrurd eldif bitr4di bitr4d con2bid 3syl 0m0e0 ifeq2da + cof simpll 1ex fconstmpt resubcl 2re cvol i1fima mblvol i1fima2 remulcl + neldifsn 2cnd 2ne0 mulne0d redivcld mulid1d mul01d ifeq12d i1f1 i1fmulc + i1fsub sylan9eqr ianor ifbid ifboth sylancl pm2.61d1 jaoi sylbi pm2.61i + eleq1w fvmpt eqtr4id mpteq2ia ad2antrl posdifd mblss ovolge0 ltlen 2pos + wo i1fpos mulgt0 mpanl12 divgt0d elrpd inidm anim2i breq1 divcld npcand + ofrfval pm2.24d impcom adantll sylanl1 oveq2 ifeq2d eqcoms nfmpt1 nfeq2 + fvmpt2 mpteq2da simplrr ltdiv1d recn 2halvesd pncand rehalfcl rehalfcld + mpd eqtr3d ltsubaddd 3bitr2d itg1mulc mulcomd divassd divcan5rd 3eqtr3d + fveq2d itg11 3eqtrd itg1sub divsubdird subsubd 3eqtrrd 3eqtr4d breqtrrd + resubcld leidd ltnled biimpar ltled pm2.61dan iba bicomd breq12d bitrdi + notbid pm4.53 simplbda jaod sylbir olcs itg1le adantllr pm2.61dane expr + ltletrd an32s rexlimdvaa expimpd ancomsd exlimdv rexab 3imtr4g eqsupd impr ) HIUFUUAJZEUGZEUUBUHUAUIZEKUUKZLZAUIZWVNUJUHZMZNZUAUJUKZULZAUUCZO PUMZFOPUMZAUAEWWCWWCUNUUDWVMWWECHCUIZDUIZUHZIMZIWWHBUIZUOJZUPZUQZEWVOLZ BURULZWVQWWGUJUHZMZNZDWWAULZAUUCZOPUMWWDOFWWTPGUUEWVMUBUCOWWTWWDPOPUUFW @@ -569432,35 +570410,35 @@ curry M LIndF ( R freeLMod I ) ) ) $= OWWFEWVMHOEUGZWXDWYCWVMWVLOVBXUFIUFUVGHWVLOEUVHWEWFVPZWXLWXOWYEUVLUVIUV JUVKWYDCHWXOWYEKWXPESHOHSQZWYDWGVAZXUEXUGWYDWXPWHWVMECHWYEUQMWXDWYCWVMC HWVLEWVMWBWIZWFZWJWYDCHWXLWYEKWVNESHOXUIXUCXUGWXDWVNCHWXLUQMZWVMWYCWXDC - HHWVNWYQWIZVOXUKWJUVMUVNUVOUVPUVQUVRUWAWVMUCUBUDOWWCWXIPWXAWVMWXBUCUIZW + HHWVNWYQWIZVOXUKWJUVMUVNUVOUVPUVQUWAUVRWVMUCUBUDOWWCWXIPWXAWVMWXBUCUIZW XIPLVCUBWWCVMWXIXUNPLZWXIUDUIZPLZUDWWCULZWCUBOVMNUCOULWVMWWBAOWVMWVTWXC - 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RUXTWOXERWXIWVRUYAUYBXVPXWTXVLWCZXWRWVMXVMXVOXXFXUSWVMXVMNZXVOXWTXVLXXG + RUXSWOXERWXIWVRUYBUXTXVPXWTXVLWCZXWRWVMXVMXVOXXFXUSWVMXVMNZXVOXWTXVLXXG XVOXWTNZNZXVLWVNUYCZWVNUYDZXWBXJZXQZUYEUHZIXXIXXNIMZNZXVTXWAXVLWVMXVTXV MXXHXXOXWOXKXXPWXIWVRIPXXGXVOXWTXXOXLXXIWXDXXOWVRIMWVMWXDXVEXXHXLWXDXXO NCXXMWVNWXDXXOXMWXDXXMHVBZXXOWXDHHXXMWVNWVNXXLUYFWYQUYGRWXDXXOXHWXDWWFH XXMXJZQZWXMXXOWXDXXSWXMWXDXUBXXSWXMWCWYQXUBXXSWWFXXJXWBXQZQZWXMXUBXXRXX - TWWFXUBXXRWVNUKZXYBXXTXJZXJZXXTXUBHXYBXXMXYCXUBXYBHHHWVNUYLZUYHXUBXXMXX - JXXKXQZXXTXJZXYCXUBWVNUYIXXMXYGMHHWVNUYMXXKXWBWVNUYJWKZXYFXYBXXTWVNUYKZ - UYNXNUYOXXTXYBVBXYDXXTMWVNXWBUYFXXTXYBUYPUYQXNXOXUBWVNHUYRZXYAWXMWCZHHW - VNVUCZXYJXYAWYIWXMNWXMHIWWFWVNUYSZWYIWXMXHUYTWKXRWKWOVQVUAXEVUBXWQXPXXI + TWWFXUBXXRWVNUKZXYBXXTXJZXJZXXTXUBHXYBXXMXYCXUBXYBHHHWVNUYHZUYIXUBXXMXX + JXXKXQZXXTXJZXYCXUBWVNUYJXXMXYGMHHWVNUYKXXKXWBWVNUYLWKZXYFXYBXXTWVNUYMZ + UYNXNUYOXXTXYBVBXYDXXTMWVNXWBUYFXXTXYBVUCUYPXNXOXUBWVNHUYQZXYAWXMWCZHHW + 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RR ) $. itg2addnc.g2 $e |- ( ph -> G : RR --> ( 0 [,) +oo ) ) $. itg2addnc.g3 $e |- ( ph -> ( S.2 ` G ) e. RR ) $. @@ -572344,127 +573322,127 @@ curry M LIndF ( R freeLMod I ) ) ) $= recn renegcld iccmbl syl2anc covol mblvol sqrtge0d addge0d subcld sqrtcld syl sqcld subnegd breq2d bitr3d mpbid syl3anc eqtrd eqeltrd volf elpreima wb mp2b 0mbl ax-mp eqtri 0re 3expa fveq2d mpteq2dva cvv cdif eldif 3anass - wn wi biantrurd bitrd 3bitr4rd biimpd 3expia impd syl5bi imp biimprd expd - eqtrdi 3impd sylbid 3impia mpancom recnd cioo sylan2 clt cxr oveq1 adantl - fvoveq1 negeqd oveq12d sqrt0 simpr picn oveq2d casin divcld fvmptd oveq1d - cdiv ovexd cxp ccnv wral copab opabssxp eqsstri cabs areacirclem5 absresq - cif abscld absge0d le2sqd 3bitr4d biimpa ovolicc cpnf wf wfn ffn sylanbrc - simp2 ovol0 eqeltri mpbir2an ifclda ralrimiva renegcl simpl iccssre rembl - fvexd elicc2 simp3 absled iffalse iftrue 3adant2 resqcld 3bitr3rd 2timesd - con3d sselda 3eqtr4d 3eqtrd areacirclem3 iblss2 syl3anbrc areaval elioore - dmarea itgeq2dv rexrd rexr elioo2 absltd notbid lenltd bitr4d anim1i eqle - ioossre 3syl 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XWFYUPYUPSIXWFUGOYMVVKVWIVWJZUNZVWKWGYNWGXWFXWTVXMWWTCYOPZYPNZYVRTYVROUHP + ZUPPZUTNZQPZUMPZQPZYUSXWFGWWTXWOYWEWXAXWPWTYVIXWFXWGWWTRZMXWNYWDVXMQYWFXW + NYWDRXWFYWFXWIYVSXWMYWCUMXWGWWTCYPYOYFYWFXWHYVRXWLYWBQXWGWWTCYOYHZYWFXWKY + WAUTYWFXWJYVTTUPYWFXWHYVROUHYWGYTYNWRYJYJYEYNXWFXUHXUIYUCWWTWXAIXXEXXFYUE + WWTCVWLWFXWFVXMYWDQYSYRXWFYWDYURVXMQXWFYWDYURJUMPYURXWFYVSYURYWCJUMXWFYVS + TUQZYPNZYURXWFYVRYWHYPXWFYVAUQYVRYWHXWFCCXYNXYNXYPVWMXWFYVATYVMYIUXDZWRYW + IYVLUQZYURTSIYWIYWKRVWNTVWOWNYVLYUPVWGUXFWOXMXWFYWCYVRJQPJXWFYWBJYVRQXWFY + WBXVEJXWFYWAJUTXWFYWAYVNJXWFYVTTTUPXWFYVTYWHOUHPTXWFYVRYWHOUHYWJYTVWPXMYN + YVOWGWRYKXMYNXWFYVRXWFWWTCXWFWWTXXDYAXYNXYPYQVWHWGYJXWFYURYURSIXWFYUPYVPV + WQUNZVWKWGYNWGYJXWFVXMYUPYURUPPZQPVXMUGQPYUTVXNXWFYWMUGVXMQYWMUGRXWFYWMYU + PYUPUMPUGYUPYUPYVPYVPVXAVWRWOUNYNXWFVXMYUPYURXYOYVQYWLVWSXWFVXMUGXYOUGSIX + WFYMUNVXBVXCWGUWEVTVXDVXCWG $. $} $( (End of Brendan Leahy's mathbox.) $) @@ -572502,17 +573480,17 @@ curry M LIndF ( R freeLMod I ) ) ) $= ( wa wceq wi wrex vw wral cv cio eqidd ancli eqeq2d anbi12d rspcev sylan2 wcel adantl cvv weu wb csb nfcvd csbiegf csbex eqeltrrdi ad2antrl wex wal eqeq1 anbi2d rexbidv spcegv adantr r19.29 pm3.35 eqeq12 syl5ibrcom ancoms - syl mpd an4 expimpd syl5bi ancomsd expdimp rexlimivw sylan an32s alrimivv - imp ex cbvrexvw bitrdi eu4 sylanbrc iota2 syl2anc mpbid ) ACQZHKRZSZFGUBZ - EGUBZJGUKZBQZQZAIHRZQZEGTZADUCZHRZQZEGTZDUDIRZWTXDWRBWSBIIRZQZXDBXJBIUEUF - XCXKEJGEUCZJRZABXBXJLXMHIIMUGUHUIUJZULXAIUMUKZXHDUNZXDXIUOWSXOWRBWSIEJHUP - UMEJHIGWSEIUQMUREJHPUSUTZVAXAXHDVBZXHCUAUCZKRZQZFGTZQXEXSRZSZUAVCDVCXPWTX - RWRWTXDXRXNWSXDXRSZBWSXOYEXQXHXDDIUMXEIRZXGXCEGYFXFXBAXEIHVDVEVFZVGVNVHVO - ULXAYDDUAWRYDWTWRXHYBYCWRXHQWQXGQZEGTYBYCSZWQXGEGVIYHYIEGYHYBYCWQYBXGYCWQ - YBQWPYAQZFGTZXGYCWPYAFGVIYKXGYCYJXGYCSFGWPYAXGYCWPXGYAYCXGYAQWNXFXTQZQWPY - CAXFCXTVPWPWNYLYCWNWPYLYCSWNWPQYCYLWOWNWOVJXEHXSKVKVLVMVQVRVSVTWAWEWBWCWF - WAVNVQVHWDXHYBDUAYCXHAXSHRZQZEGTYBYCXGYNEGYCXFYMAXEXSHVDVEVFYNYAEFGXLFUCR - ZACYMXTNYOHKXSOUGUHWGWHWIWJXHXDDIUMYGWKWLWM $. + syl mpd an4 expimpd biimtrid ancomsd expdimp rexlimivw imp sylan an32s ex + alrimivv cbvrexvw bitrdi eu4 sylanbrc iota2 syl2anc mpbid ) ACQZHKRZSZFGU + BZEGUBZJGUKZBQZQZAIHRZQZEGTZADUCZHRZQZEGTZDUDIRZWTXDWRBWSBIIRZQZXDBXJBIUE + UFXCXKEJGEUCZJRZABXBXJLXMHIIMUGUHUIUJZULXAIUMUKZXHDUNZXDXIUOWSXOWRBWSIEJH + UPUMEJHIGWSEIUQMUREJHPUSUTZVAXAXHDVBZXHCUAUCZKRZQZFGTZQXEXSRZSZUAVCDVCXPW + TXRWRWTXDXRXNWSXDXRSZBWSXOYEXQXHXDDIUMXEIRZXGXCEGYFXFXBAXEIHVDVEVFZVGVNVH + VOULXAYDDUAWRYDWTWRXHYBYCWRXHQWQXGQZEGTYBYCSZWQXGEGVIYHYIEGYHYBYCWQYBXGYC + WQYBQWPYAQZFGTZXGYCWPYAFGVIYKXGYCYJXGYCSFGWPYAXGYCWPXGYAYCXGYAQWNXFXTQZQW + PYCAXFCXTVPWPWNYLYCWNWPYLYCSWNWPQYCYLWOWNWOVJXEHXSKVKVLVMVQVRVSVTWAWBWCWD + WEWAVNVQVHWFXHYBDUAYCXHAXSHRZQZEGTYBYCXGYNEGYCXFYMAXEXSHVDVEVFYNYAEFGXLFU + CRZACYMXTNYOHKXSOUGUHWGWHWIWJXHXDDIUMYGWKWLWM $. $} ${ @@ -572800,17 +573778,17 @@ curry M LIndF ( R freeLMod I ) ) ) $= ( c C_ B /\ A. x e. A E. y e. c ph /\ A. y e. c E. x e. A ph ) ) $= ( vz vw vv wcel cv wsbc wrex wral nfv wa sbceq2a nfcv nfsbc1v cvv wss w3a crab wex rabexg ssrab2 nfre1 rspcev ancoms anim1ci anasss sbcbidv rexbidv - a1i wceq elrab sylibr sylancom nfsbcw nfrex nfrabw cbvrexfw sylib rexlimd - exp31 ralimia cbvrexw bitrdi simprbi rgen 3jca sseq1 rexeqf ralbid raleqf - nfeq2 3anbi123d spcegv imp syl2an ) EFKACHLZMZBILZMZIDNZHEUDZUAKZWGEUBZAC - WGNZBDOZABDNZCWGOZUCZGLZEUBZACWONZBDOZWLCWOOZUCZGUEZACENZBDOZWFHEFUFXCWIW - KWMWIXCWFHEUGUOXBWJBDBLZDKZAWJCEXECPACWGUHXECLZEKZAWJXEXGQZAQZACJLZMZJWGN - ZWJXHAXFWGKZXLXIXGABWDMZIDNZQZXMAXHXPAXEXGXPAXEQXOXGXEAXOXNAIXDDABWDRZUIU - JUKULUJWFXOHXFEWBXFUPZWEXNIDXRWCABWDACWBRUMUNZUQURXKAJXFWGACXJRZUIUSXKAJC - WGJWGSWFCHEWECIDCDSWCCBWDCWDSACWBTUTVACESVBZACXJTAJPXTVCVDVFVEVGWMXCWLCWG - XMXGWLWFWLHXFEXRWFXOWLXSXNAIBDABWDTAIPXQVHVIUQVJVKUOVLWHWNXAWTWNGWGUAWOWG - UPZWPWIWRWKWSWMWOWGEVMYBWQWJBDBWOWGWFBHEWEBIDBDSWCBWDTVABESVBVQACWOWGCWOS - ZYAVNVOWLCWOWGYCYAVPVRVSVTWA $. + a1i wceq elrab sylibr sylancom nfsbcw nfrexw cbvrexfw sylib exp31 rexlimd + nfrabw ralimia cbvrexw bitrdi simprbi rgen 3jca sseq1 nfeq2 rexeqf ralbid + raleqf 3anbi123d spcegv imp syl2an ) EFKACHLZMZBILZMZIDNZHEUDZUAKZWGEUBZA + CWGNZBDOZABDNZCWGOZUCZGLZEUBZACWONZBDOZWLCWOOZUCZGUEZACENZBDOZWFHEFUFXCWI + WKWMWIXCWFHEUGUOXBWJBDBLZDKZAWJCEXECPACWGUHXECLZEKZAWJXEXGQZAQZACJLZMZJWG + NZWJXHAXFWGKZXLXIXGABWDMZIDNZQZXMAXHXPAXEXGXPAXEQXOXGXEAXOXNAIXDDABWDRZUI + UJUKULUJWFXOHXFEWBXFUPZWEXNIDXRWCABWDACWBRUMUNZUQURXKAJXFWGACXJRZUIUSXKAJ + CWGJWGSWFCHEWECIDCDSWCCBWDCWDSACWBTUTVACESVFZACXJTAJPXTVBVCVDVEVGWMXCWLCW + GXMXGWLWFWLHXFEXRWFXOWLXSXNAIBDABWDTAIPXQVHVIUQVJVKUOVLWHWNXAWTWNGWGUAWOW + GUPZWPWIWRWKWSWMWOWGEVMYBWQWJBDBWOWGWFBHEWEBIDBDSWCBWDTVABESVFVNACWOWGCWO + SZYAVOVPWLCWOWGYCYAVQVRVSVTWA $. $} ${ @@ -572916,27 +573894,27 @@ curry M LIndF ( R freeLMod I ) ) ) $= ( vw vu vv cfn wcel cr cv cle wi wral wrex wa ssel2 adantlr wne reex ssex c0 wss w3a wbr cvv indexfi 3expia sylan2 3adant2 rexn0 rexlimivw 3ad2ant2 r19.2z syl ex ad2antrr sstr ancoms fimaxre sylan anasss ancom2s 3ad2antl3 - anassrs syld a1dd 3impd nfv nfcv nfre1 nfralw nfan nfra1 nfrex weq imbi1d - breq1 ralbidv cbvrexvw rsp adantrr ad2ant2r rspccva adantll simplrr letrd - adantr pm2.27 ad2antlr mpid rexlimdva syl5bi ralimdva an32s exp32 ralrimi - syl5 imp reximdai ssrexv exp43 3ad2ant3 mpdd ) EJKZEUDUAZFLUEZUFZCMZDMZNU - GZAOZDFPZCFQBEPZGMZFUEZXOCXQQZBEPZXOBEQCXQPZUFZGJQZXMABEPZOZDFPZCFQZXGXIX - PYCOZXHXIXGFUHKZYHFLUBUCXGYIXPYCXOBCEFUHGUIUJUKULXJYBYGGJXJXQJKZRZYBHMZXK - NUGZHXQPZCXQQZYGYKXRXTYAYOYKXRXTYAYOOOYKXRRZXTYOYAYPXTXQUDUAZYOXJXTYQOZYJ - XRXHXGYRXIXHXTYQXHXTRXSBEQYQXSBEUPXSYQBEXOCXQUMUNUQURUOUSXJYJXRYQYOOZXIXG - YJXRRYSXHXIXRYJYSXIXRYJYSXIXRRZXQLUEZYJYSXRXIUUAXQFLUTVAUUAYJYQYOCHXQVBUJ - VCVDVEVFVGVHVIURVJXJYBYOYGOZOZYJXIXGUUCXHXIXRXTYAUUBXIXRXTYAUUBYTXTYARZRY - OYFCXQQZYGYTXTYOUUEOYAYTXTRZYNYFCXQYTXTCYTCVKXSCBECEVLXOCXQVMVNVOUUFXKXQK - ZYNYFUUFUUGYNRZRYEDFUUFUUHDYTXTDYTDVKXSDBEDEVLXODCXQDXQVLXNDFVPVQVNVOUUHD - VKVOYTUUHXTXLFKZYEOYTUUHRZXTRUUIXMYDUUJUUIXMRZXTYDUUJUUKRZXTYDUULXSABEXSI - MZXLNUGZAOZDFPZIXQQUULBMEKZRZAXOUUPCIXQCIVRZXNUUODFUUSXMUUNAXKUUMXLNVTVSW - AWBUURUUPAIXQUULUUMXQKZUUPAOUUQUUPUUIUUOOZUULUUTRZAUUODFWCUVBUVAUUNAUVBUU - MXKXLUUJUUTUUMLKZUUKYTUUTUVCUUHXIXRUUTUVCXRUUTRXIUUMFKUVCXQFUUMSFLUUMSUKV - GTTUUJXKLKZUUKUUTYTUUGUVDYNXIXRUUGUVDXRUUGRXIXKFKUVDXQFXKSFLXKSUKVGWDUSUU - LXLLKZUUTYTUUIUVEUUHXMXIUUIUVEXRFLXLSTWEWJUUJUUTUUMXKNUGZUUKUUHUUTUVFYTYN - UUTUVFUUGYMUVFHUUMXQYLUUMXKNVTWFWGWGTUUJUUIXMUUTWHWIUUKUVAUUOOZUUJUUTUUIU - VGXMUUIUUOWKWJWLWMWTTWNWOWPXAWQWRWQWSWRXBWDXRUUEYGOXIUUDYFCXQFXCWLVHXDVJX - EWJXFWNVH $. + anassrs syld a1dd 3impd nfv nfcv nfre1 nfralw nfan nfra1 nfrexw weq breq1 + imbi1d ralbidv cbvrexvw rsp adantrr ad2ant2r adantr rspccva adantll letrd + simplrr pm2.27 ad2antlr mpid syl5 rexlimdva biimtrid ralimdva an32s exp32 + imp ralrimi reximdai ssrexv exp43 3ad2ant3 mpdd ) EJKZEUDUAZFLUEZUFZCMZDM + ZNUGZAOZDFPZCFQBEPZGMZFUEZXOCXQQZBEPZXOBEQCXQPZUFZGJQZXMABEPZOZDFPZCFQZXG + XIXPYCOZXHXIXGFUHKZYHFLUBUCXGYIXPYCXOBCEFUHGUIUJUKULXJYBYGGJXJXQJKZRZYBHM + ZXKNUGZHXQPZCXQQZYGYKXRXTYAYOYKXRXTYAYOOOYKXRRZXTYOYAYPXTXQUDUAZYOXJXTYQO + ZYJXRXHXGYRXIXHXTYQXHXTRXSBEQYQXSBEUPXSYQBEXOCXQUMUNUQURUOUSXJYJXRYQYOOZX + IXGYJXRRYSXHXIXRYJYSXIXRYJYSXIXRRZXQLUEZYJYSXRXIUUAXQFLUTVAUUAYJYQYOCHXQV + BUJVCVDVEVFVGVHVIURVJXJYBYOYGOZOZYJXIXGUUCXHXIXRXTYAUUBXIXRXTYAUUBYTXTYAR + ZRYOYFCXQQZYGYTXTYOUUEOYAYTXTRZYNYFCXQYTXTCYTCVKXSCBECEVLXOCXQVMVNVOUUFXK + XQKZYNYFUUFUUGYNRZRYEDFUUFUUHDYTXTDYTDVKXSDBEDEVLXODCXQDXQVLXNDFVPVQVNVOU + UHDVKVOYTUUHXTXLFKZYEOYTUUHRZXTRUUIXMYDUUJUUIXMRZXTYDUUJUUKRZXTYDUULXSABE + XSIMZXLNUGZAOZDFPZIXQQUULBMEKZRZAXOUUPCIXQCIVRZXNUUODFUUSXMUUNAXKUUMXLNVS + VTWAWBUURUUPAIXQUULUUMXQKZUUPAOUUQUUPUUIUUOOZUULUUTRZAUUODFWCUVBUVAUUNAUV + BUUMXKXLUUJUUTUUMLKZUUKYTUUTUVCUUHXIXRUUTUVCXRUUTRXIUUMFKUVCXQFUUMSFLUUMS + UKVGTTUUJXKLKZUUKUUTYTUUGUVDYNXIXRUUGUVDXRUUGRXIXKFKUVDXQFXKSFLXKSUKVGWDU + SUULXLLKZUUTYTUUIUVEUUHXMXIUUIUVEXRFLXLSTWEWFUUJUUTUUMXKNUGZUUKUUHUUTUVFY + TYNUUTUVFUUGYMUVFHUUMXQYLUUMXKNVSWGWHWHTUUJUUIXMUUTWJWIUUKUVAUUOOZUUJUUTU + UIUVGXMUUIUUOWKWFWLWMWNTWOWPWQWTWRWSWRXAWSXBWDXRUUEYGOXIUUDYFCXQFXCWLVHXD + VJXEWFXFWOVH $. $} $( Membership of a product in a finite interval of integers. (Contributed by @@ -572990,80 +573968,80 @@ curry M LIndF ( R freeLMod I ) ) ) $= $( Lemma for ~ sdc . (Contributed by Jeff Madsen, 2-Sep-2009.) $) sdclem2 $p |- ( ph -> E. f ( f : Z --> A /\ A. n e. Z ch ) ) $= ( vm cv cfv cmpt wf cfz wsbc wral wex wcel wrex ffvelcdmda cab eleq2i - co nfcv nfv nfsbc1v nfan nfrex fvex wceq feq1 sbceq1a anbi12d rexbidv - wa elabf bitri sylib weq cdm fdm adantr wi cuz c1 caddc fveq2 mpteq1d - oveq2 eqeq12d imbi2d cz id fveq12d eqid adantl elfz1eq fveq1d ralrimi - fvmpt ex wfn wb fnmpti sylancl a1i cres w3a cvv simpr cmap ovex ssexg - wss syl sylancr eqeq1 reseq1 eqeq2d fzssp1 fzopth simprd resmpt ax-mp - eleqtrdi fvres eluzfz2 syl5ibrcom sylbid mpd eqtrdi bitrd sbcbidv a1d + co wa nfcv nfsbc1v nfan nfrexw fvex wceq feq1 sbceq1a anbi12d rexbidv + nfv elabf bitri sylib weq cdm fdm adantr wi caddc fveq2 oveq2 mpteq1d + cuz c1 eqeq12d imbi2d cz id fveq12d eqid fvmpt adantl elfz1eq ralrimi + fveq1d ex wfn fnmpti sylancl a1i cres w3a cvv simpr wss cmap ovex syl + ssexg sylancr eqeq1 reseq1 eqeq2d fzssp1 eleqtrdi fzopth simprd ax-mp + wb resmpt fvres eluzfz2 syl5ibrcom sylbid mpd eqtrdi bitrd sbcbidv wo fveq2d eqtr2d eqfnfv mpbird 3simpa reximi ss2abi fvexi 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Then there exists an infinite sequence of elements of ` A ` such that the first ` n ` terms of this sequence satisfy ` ps ` for all ` n ` . This - theorem allows us to construct infinite seqeunces where each term - depends on all the previous terms in the sequence. (Contributed by Jeff - Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, - 3-Jun-2014.) $) + theorem allows to construct infinite sequences where each term depends + on all the previous terms in the sequence. (Contributed by Jeff Madsen, + 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.) $) sdc $p |- ( ph -> E. f ( f : Z --> A /\ A. n e. Z ch ) ) $= ( vx vy vj cv cfz co wf wa wrex cab c1 caddc cres wceq w3a cmpo weq oveq2 feq2d anbi12d cbvrexvw abbii mpoeq123i eqidd eqeq1 3anbi2d rexbidv abbidv @@ -573186,118 +574163,118 @@ curry M LIndF ( R freeLMod I ) ) ) $= mpbid ad2ant2l 3ad2antr2 eluzp1p1 fveq2i biimpa adantlr oveq1 eqtrd sylbi eleqtrrdi eqeq2i ltleii eluz2 mpbir3an fzaddel mpanr12 ifnefalse biimparc fzss1 gtneii simprl adantlrr eqeltri rspcva sylan fzssp1 breqtrri mp3an23 - subadd eqcom addcomi syl5bi mp2b lttri syldan adantlrl ralrimiva cbvralvw - an32s jaodan adantllr adantrlr 3adantr1 syl121anc chvarvv syl2anc exlimdv - ovex mptex rexlimdva 3syld rexlimdvaa mpjaodan sylibrd notbii iman bitr4i - an42s syl6ibr nrexdv df-ne wfr difss difexg fri expr syl5bir ssdif0 eqssd - mt3d rabid2 ) FHGUOMLUPZUQURZGJUPZUSZMVXOUTZUIUPZVAZEVBZCKNVXMUQURZVCZVDZ - JVEZLOVIZUIGVCZVXPVXQHVAZEVBZVYBVDZJVEZLOVIZUEFGVYEUIGUUBZVAVYFFGVYLFGVYL - UUCZVFVAZGVYLVGFVYNUJUPZPUPZIVHZVJUJVYMVCZPVYMVIZFVYRPVYMFVYPVYMUOZVYRVJZ - 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UDYWPVAYVCYWPYUDVVKYWPWWEYUDWPMYLXYFVVLVULXMYVCYUNYWNYVCYUNWWEYUMUOYWNYUD + WWEYUMXPWWEYULXUIYBYCYDVVMUYDYEYFYGVUIYVLAQYUEYVMYVLYUEYUDMVAZVXRYVMXQZYV + MYVLYUDXUKUOYUEYXAVAYVLYUMXUKYUDYWLYULWVAUOYUMXUKVGYWMMNUWNYULMXUIVUTVVNX + UGYUNUUOYSUMYUDXWFYXAXUKXWGUMUNXFXWCYWTXWEYVMVXRXWBYUDMYOXWBYUDWPWXNXLYPY + QXYJYWTVXRYVMXYMYUGWXNWIYRYNWOYVLYWTVXRYVMYUNYWTVJXUGYUNYWTYULMXTVHZMYULW + SVHZYXBVJMNWSVHNYULWSVHYXCMWWENWSWWJUAVVHNYWGWTMNYULWWIYWGYWOVVOUVDMYULWW + IYWOYAXGYWTYUNYXBYWTYUNMYUMUOYXBYUDMYUMXPMYULXUIYBYCYDYEYFYGVUIWLXNUYTVVP + VVTVVQUYHVVRVVPVVSYUIXWPUNKXUPUNKXFZYUFXWNPYUHXWOYUDWYAWPXWGXLYPYXDAQYUEX + WMYUDWYAXWGWKWLXNVWAUYGVWBVWCVWDXVQXWHXWJXWLVBZXWQVDJXWGUMXUKXWFMXUIUQVWI + VWEVXOXWGVAZXULXWHXVPYXEXUQXWQXUKGVXOXWGVSYXFVXSXWJXUNXWLYXFVXQXWIVXRMVXO + XWGVTWAYXFDPXUMXWKXUIVXOXWGVTWLWFYXFCXWPKXUPCXUDYXFXWPXUFYXFXUBXWNPXUCXWO + WYDVXOXWGVTYXFAQXUAXWMWYAVXOXWGVTWLXNWMUWFXBUUSVWFYIVWGVWGVVCUYHYKWUJXUSL + XUIOVXMXUIVAZWUIXURJYXGVXPXULWUHXUOVYBXUQYXGVXNXUKGVXOVXMXUIMUQWGWHYXGEXU + NWUGEWWBYXGXUNWWCYXGDPWWAXUMVXMXUIVXOWKWLWMWNYXGCKVYAXUPVXMXUINUQWGXAXBXC + XDVWHYIVWJVWKVVMVWLVWMVWNYKUGVWOWUBWUCWUKWDFVYEWUKUIVYPGXVTVYDWUJLOXVTVYC + WUIJXVTVXTWUHVXPVYBXVTVXSWUGEXWAXIXJXCXRUWAYFVWSYIYJWUEWUBWUCVJVBZVJWUDVY + TYXHVYPGVYLUVOVWPWUBWUCVWQVWRVWTUVNYKVXAVYNVJVYMVFUYMZFVYSVYMVFVXBFGIVXCZ + VYMGVGZYXIVYSVKUFGVYLVXDYXJYXKYXIVYSVYMYTUOZYXJYXKYXIVBVYSGYTUOYXLRGVYLYT + VXEVNPUJGVYMYTIVXFUXHVXGUXOVXHVXIGVYLVXKUUNWXIFWXJVRVXJVYEUIGVXLUYGVYEVYK + UIHGVXRHVAZVYDVYJLOYXMVYCVYIJYXMVXTVYHVXPVYBYXMVXSVYGEVXRHVXQXHXIXJXCXRVV + EVWH $. $} ${ @@ -573324,8 +574301,8 @@ curry M LIndF ( R freeLMod I ) ) ) $= ( vc vd cv cfz co wf wa wral w3a wex wrex wcel wsbc wi wceq eleq1w anbi2d sbceq2a anbi12d imbi1d cfv wsb c1 cmin sbsbc sbhypf bitr3id simprl adantr nfv wfr wo nfsbc1v nfcv nfor nfim sbceq1a rexbidv orbi12d imbi12d chvarfv - nfrex adantlr wbr nfan anbi1d breq2 adantllr fdc anassrs idd dfsbcq sbcie - fvex bitr3di biimpcd adantl anim1d 3anim123d reximdv mpd chvarvv r19.29a + nfrexw adantlr wbr nfan anbi1d breq2 adantllr fdc anassrs idd dfsbcq fvex + sbcie bitr3di biimpcd adantl anim1d 3anim123d reximdv mpd chvarvv r19.29a eximdv ) FGNMUMZUNUOIKUMZUPZHEUQZCLOXOUNUOURZUSZKUTZMPVAZQIFUKUMZIVBZUQZG QYCVCZUQZYBVDFQUMZIVBZUQZGUQZYBVDUKQYCYHVEZYGYKYBYLYEYJYFGYLYDYIFUKQIVFVG GQYCVHVIVJYGXQNXPVKZYCVEZEUQZXSUSZKUTZMPVAZYBFYDYFYRAQULUMZVCZBCDQYSVCZEF @@ -573337,7 +574314,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= PUQZUQZUUOYHJWNZVDFYTUQZUUQUQZUURVDQULUVDUURQUVCUUQQFYTQFQVTUUKWOUUQQVTWO UURQVTWFUULUVAUVDUVBUURUULUUSUVCUUTUUQUULAYTFUUNVGUULYIUUEUUPUUMWPVIYHYSU UOJWQWJUJWKWRWSWTYGYQYAMPYGYPXTKYGXQXQYOXRXSXSYGXQXAYGYNHEYFYNHVDYEYNYFHY - NGQYMVCYFHGQYMYCXBGHQYMNXPXDUCXCXEXFXGXHYGXSXAXIXNXJXKXLUGXM $. + NGQYMVCYFHGQYMYCXBGHQYMNXPXCUCXDXEXFXGXHYGXSXAXIXNXJXKXLUGXM $. $} ${ @@ -573449,14 +574426,14 @@ curry M LIndF ( R freeLMod I ) ) ) $= crab wne eq0 breq2 elrab notbii imnan sylbb2 alimi df-ral cle ssel2 nnred wceq nnre ad2antlr lenlt biimprd syl2anc ralimdva cc0 cfz fzfi cn0 nnnn0d w3a adantr nnnn0 ad3antlr simpr 3jca ex elfz2nn0 syl6ibr imp ssfi sylancr - dfss3 syld syl5 syl5bi necon3bd an32s 3impa ) BEFZBGHZIZCEHZCAJZKLZABUBZM - UCZWFWIWHWMWFWINZWHWMWNWGWLMWLMUODJZWLHZIZDOZWNWGDWLUDWRCWOKLZIZDBPZWNWGW - RWOBHZWTQZDOXAWQXCDWQXBWSNZIXCWPXDWKWSAWOBWJWOCKUEUFUGXBWSUHUIUJWTDBUKRWN - XAWOCULLZDBPZWGWNWTXEDBWNXBNZWOSHZCSHZWTXEQWFXBXHWIWFXBNZWOBEWOUMZUNTWIXI - WFXBCUPUQXHXINXEWTWOCURUSUTVAWNXFWGWNXFNZVBCVCUAZGHBXMFZWGVBCVDXLWOXMHZDB - PZXNWNXFXPWNXEXODBXGXEWOVEHZCVEHZXEVGZXOXGXEXSXGXENXQXRXEXGXQXEWFXBXQWIXJ - WOXKVFTVHWIXRWFXBXECVIVJXGXEVKVLVMWOCVNVOVAVPDBXMVSRXMBVQVRVMVTWAWBWCVPWD - WE $. + dfss3 syld syl5 biimtrid necon3bd an32s 3impa ) BEFZBGHZIZCEHZCAJZKLZABUB + ZMUCZWFWIWHWMWFWINZWHWMWNWGWLMWLMUODJZWLHZIZDOZWNWGDWLUDWRCWOKLZIZDBPZWNW + GWRWOBHZWTQZDOXAWQXCDWQXBWSNZIXCWPXDWKWSAWOBWJWOCKUEUFUGXBWSUHUIUJWTDBUKR + WNXAWOCULLZDBPZWGWNWTXEDBWNXBNZWOSHZCSHZWTXEQWFXBXHWIWFXBNZWOBEWOUMZUNTWI + XIWFXBCUPUQXHXINXEWTWOCURUSUTVAWNXFWGWNXFNZVBCVCUAZGHBXMFZWGVBCVDXLWOXMHZ + DBPZXNWNXFXPWNXEXODBXGXEWOVEHZCVEHZXEVGZXOXGXEXSXGXENXQXRXEXGXQXEWFXBXQWI + XJWOXKVFTVHWIXRWFXBXECVIVJXGXEVKVLVMWOCVNVOVAVPDBXMVSRXMBVQVRVMVTWAWBWCVP + WDWE $. $} @@ -574164,19 +575141,19 @@ curry M LIndF ( R freeLMod I ) ) ) $= ancoms 2rp rpmulcl mpan ad2antll ad2antrr cdiv cc0 rpcn 2cnd 2ne0 a1i w3a cc divcan3 eqcomd syl3anc oveq2d biimpd adantr imp simpr biimpac 2re rpre remulcl sylancr blhalf expr anasss anassrs eqsstrd syldan adantl cxr rpxr - eleq2 blssm syl3an3 3expa an32s eqssd syl2anc ralrimdva rexlimdvva syl5bi - rspceeqv ex rexn0 jcad impbid2 pm5.32i 3bitri ) BCUCGHZCUAUBZIBCUDGHZCAJZ - DJZBUEGZKZLZDMNZACUFZIZXIIXJXQXIIZIXJXPACNZIXHXRXIABCDUGUHXJXQXIUIXJXSXTX - JXSXTXIXQXTXPACUJUOXJXTXQXIXTCEJZFJZXMKZLZFMNECNXJXQXOYDCYAXLXMKZLADEFCMX - KYALXNYECXKYAXLXMUKOXLYBLYEYCCXLYBYAXMULOUMXJYDXQEFCMXJYACHZYBMHZIZIZYDXP - ACYIXKCHZIZYDXPYKYDIZPYBUNKZMHZCXKYMXMKZLXPYIYNYJYDYGYNXJYFPMHYGYNUPPYBUQ - URZUSUTYLCYOYKYDCYAYMPVAKZXMKZLZCYOQYKYDYSYIYDYSRZYJYGYTXJYFYGYDYSYGYCYRC - YGYBYQYAXMYGYBVHHZPVHHZPVBUBZYBYQLYBVCYGVDUUCYGVEVFUUAUUBUUCVGYQYBYBPVIVJ - VKVLOVMUSVNVOYKYSICYRYOYKYSVPYIYJYSYRYOQZYJYSIYIXKYRHZUUDYSYJUUECYRXKWKVQ - YIUUEUUDXJYFYGUUEUUDRZYGXJYFIZYMSHZUUFYGPSHYBSHUUHVRYBVSPYBVTWAUUGUUHUUEU - UDYMBCYAXKWBWCTWDVOTWEWFWGYKYOCQZYDXJYJYHUUIYHXJYJIYNUUIYGYNYFYPWHXJYJYNU - UIYNXJYJYMWIHUUIYMWJBXKYMCWLWMWNTWOVNWPDYMMXNYOCXLYMXKXMULXAWQXBWRWSWTXTX - IRXJXPACXCVFXDXEXFXG $. + eleq2 blssm syl3an3 3expa an32s eqssd rspceeqv syl2anc ralrimdva biimtrid + ex rexlimdvva rexn0 jcad impbid2 pm5.32i 3bitri ) BCUCGHZCUAUBZIBCUDGHZCA + JZDJZBUEGZKZLZDMNZACUFZIZXIIXJXQXIIZIXJXPACNZIXHXRXIABCDUGUHXJXQXIUIXJXSX + TXJXSXTXIXQXTXPACUJUOXJXTXQXIXTCEJZFJZXMKZLZFMNECNXJXQXOYDCYAXLXMKZLADEFC + MXKYALXNYECXKYAXLXMUKOXLYBLYEYCCXLYBYAXMULOUMXJYDXQEFCMXJYACHZYBMHZIZIZYD + XPACYIXKCHZIZYDXPYKYDIZPYBUNKZMHZCXKYMXMKZLXPYIYNYJYDYGYNXJYFPMHYGYNUPPYB + UQURZUSUTYLCYOYKYDCYAYMPVAKZXMKZLZCYOQYKYDYSYIYDYSRZYJYGYTXJYFYGYDYSYGYCY + RCYGYBYQYAXMYGYBVHHZPVHHZPVBUBZYBYQLYBVCYGVDUUCYGVEVFUUAUUBUUCVGYQYBYBPVI + VJVKVLOVMUSVNVOYKYSICYRYOYKYSVPYIYJYSYRYOQZYJYSIYIXKYRHZUUDYSYJUUECYRXKWK + VQYIUUEUUDXJYFYGUUEUUDRZYGXJYFIZYMSHZUUFYGPSHYBSHUUHVRYBVSPYBVTWAUUGUUHUU + EUUDYMBCYAXKWBWCTWDVOTWEWFWGYKYOCQZYDXJYJYHUUIYHXJYJIYNUUIYGYNYFYPWHXJYJY + NUUIYNXJYJYMWIHUUIYMWJBXKYMCWLWMWNTWOVNWPDYMMXNYOCXLYMXKXMULWQWRXAWSXBWTX + TXIRXJXPACXCVFXDXEXFXG $. $( A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 13-Sep-2015.) $) @@ -574281,25 +575258,25 @@ curry M LIndF ( R freeLMod I ) ) ) $= fdmd sylibr ad2antlr metf ad3antrrr sseqtrd dmss syl dmxpid simprl sseldd 3sstr3g simpllr metcl syl3anc ad2antll readdcld metxmet elind blres inss1 rpre rpxr cmin cle wbr leidd recnd pncand breqtrrd blss2 syl33anc eqsstrd - sstrid sseq2d rspcev syl2anc rexbidv syl5ibrcom rexlimdvva expimpd syl5bi - oveq2 expdimp pm2.61dne ex simprr xpss12 resabs1d eqtr4di syl3an1 adantrr - blbnd 3expa bndss eqeltrrd rexlimdvaa impbid ) BDUAJKZADKZLZCEUGJZKZEAFMZ - BUBJZNZOZFPQZYLYNYSYLYNLZYSERERSZYSUCYTUUAPRUDYRFPUSYSUEPUFUHUUAYRFPUUAYR - RYQOYQUIERYQUJUKULYRFPUMUNUOYLYNERUDZYSYNUUBLCEUPJKZEHMZIMZCUBJNZSZIUQQHE - QZLYLYSHCEIURYLUUCUUHYSYLUUCLZUUGYSHIEUQUUIUUDEKZUUEUQKZLZLZYSUUGUUFYQOZF - PQZUUMUUDABNZUUEUTNZPKZUUFAUUQYPNZOZUUOUUMUUPUUEUUMYJUUDDKZYKUUPPKYJYKUUC - UULVAZUUMEDUUDUUMEEVBZTZDDVBZTZEDUUMUVCUVEOUVDUVFOUUMUVCBTZUVEUUCUVCUVGOZ - YLUULUUCUVCUVGVCZUVCSUVHUUCUVICTZUVCUVJBUVCVDZTUVICUVKGVEBUVCVFVGUUCUVCVH - CCEVIVLVJUVCUVGVKVMVNYJUVGUVESYKUUCUULYJUVEPBBDVOVLVPVQUVCUVEVRVSEVTDVTWC - UUIUUJUUKWAZWBZYJYKUUCUULWDZUUDABDWEWFZUUKUUEPKZUUIUUJUUEWMWGZWHZUUMUUFUU - DUUEYPNZEVCZUUSUUMBDUPJKZUUDDEVCKUUEVHKZUUFUVTSUUMYJUWAUVBBDWIZVSZUUMDEUU - DUVMUVLWJUUKUWBUUIUUJUUEWNWGCBUUDUUEDEGWKWFUUMUVTUVSUUSUVSEWLUUMUWAUVAYKU - VPUURUUPUUQUUEWONZWPWQUVSUUSOUWDUVMUVNUVQUVRUUMUUPUUPUWEWPUUMUUPUVOWRUUMU - UPUUEUUMUUPUVOWSUUMUUEUVQWSWTXABUUDAUUEUUQDXBXCXEXDUUNUUTFUUQPYOUUQSYQUUS - UUFYOUUQAYPXNXFXGXHUUGYRUUNFPEUUFYQUJXIXJXKXLXMXOXPXQYLYRYNFPYLYOPKZYRLLZ - BYQYQVBZVDZUVCVDZCYMUWGUWJUVKCUWGBUVCUWHUWGYRYRUVCUWHOYLUWFYRXRZUWKEYQEYQ - XSXHXTGYAUWGUWIYQUGJKZYRUWJYMKYLUWFUWLYRYJYKUWFUWLYJUWAYKUWFUWLUWCYOBDAYD - YBYEYCUWKEUWIYQYFXHYGYHYI $. + sstrid oveq2 sseq2d rspcev syl2anc rexbidv syl5ibrcom rexlimdvva biimtrid + expimpd expdimp pm2.61dne ex simprr xpss12 resabs1d eqtr4di blbnd syl3an1 + 3expa adantrr bndss eqeltrrd rexlimdvaa impbid ) BDUAJKZADKZLZCEUGJZKZEAF + MZBUBJZNZOZFPQZYLYNYSYLYNLZYSERERSZYSUCYTUUAPRUDYRFPUSYSUEPUFUHUUAYRFPUUA + YRRYQOYQUIERYQUJUKULYRFPUMUNUOYLYNERUDZYSYNUUBLCEUPJKZEHMZIMZCUBJNZSZIUQQ + HEQZLYLYSHCEIURYLUUCUUHYSYLUUCLZUUGYSHIEUQUUIUUDEKZUUEUQKZLZLZYSUUGUUFYQO + ZFPQZUUMUUDABNZUUEUTNZPKZUUFAUUQYPNZOZUUOUUMUUPUUEUUMYJUUDDKZYKUUPPKYJYKU + UCUULVAZUUMEDUUDUUMEEVBZTZDDVBZTZEDUUMUVCUVEOUVDUVFOUUMUVCBTZUVEUUCUVCUVG + OZYLUULUUCUVCUVGVCZUVCSUVHUUCUVICTZUVCUVJBUVCVDZTUVICUVKGVEBUVCVFVGUUCUVC + VHCCEVIVLVJUVCUVGVKVMVNYJUVGUVESYKUUCUULYJUVEPBBDVOVLVPVQUVCUVEVRVSEVTDVT + WCUUIUUJUUKWAZWBZYJYKUUCUULWDZUUDABDWEWFZUUKUUEPKZUUIUUJUUEWMWGZWHZUUMUUF + UUDUUEYPNZEVCZUUSUUMBDUPJKZUUDDEVCKUUEVHKZUUFUVTSUUMYJUWAUVBBDWIZVSZUUMDE + UUDUVMUVLWJUUKUWBUUIUUJUUEWNWGCBUUDUUEDEGWKWFUUMUVTUVSUUSUVSEWLUUMUWAUVAY + KUVPUURUUPUUQUUEWONZWPWQUVSUUSOUWDUVMUVNUVQUVRUUMUUPUUPUWEWPUUMUUPUVOWRUU + MUUPUUEUUMUUPUVOWSUUMUUEUVQWSWTXABUUDAUUEUUQDXBXCXEXDUUNUUTFUUQPYOUUQSYQU + 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A compact @@ -575130,20 +576107,20 @@ counterexample is the discrete extended metric (assigning distinct wi ac6sfi adantr elab2g ibi crn frn ad2antrl inss1 sstrdi sspwuni vex ex sylib rnex uniex elpw sylibr wfo wfn dffn4 fofi syldan inss2 unifi ffn syl2anc elind adantlr simplr fnfvelrn sylan adantll elssuni uniss - 3syl sstr2 syl5com ralimdva impr iunss sstrd rspcev nsyl3 syld syl5bi - exlimdv con4d 3impia ) DOPZFADEUAZQZFJPZEJPZADRZXJXLSZXOXMXOTZEBUBZUC - ZQZBHUDZOUIZRZADUEZXPXMTZYDXNTZADUEXQYCYFADXNYCCUBZXSQZBYBRZTZYCTCEJM - YGEUFZYIYCYKYHXTBYBYGEXSUGUHUJLUKULUMXNADUNUOXPYDDYBNUBZUPZEAUBZYLUQZ - UCZQZADUEZSZNURZYEXJYDYTVAXLXJYDYTXTYQABDYBNXRYOUFXSYPEXRYOUSUTVBVMVC - XPYSYENXPYSYEXMFXSQZBYBRZXPYSSZXMUUBTZYJUUDCFJJYGFUFZYIUUBUUEYHUUABYB - YGFXSUGUHUJLVDVEUUCYLVFZUCZYBPZFUUGUCZQZUUBXJYSUUHXLXJYSSZYAOUUGUUKUU - GHQZUUGYAPUUKUUFYAQUULUUKUUFYBYAYMUUFYBQXJYRDYBYLVGVHZYAOVIVJUUFHVKVN - UUGHUUFYLNVLVOVPVQVRUUKUUFOPZUUFOQUUGOPXJYSDUUFYLVSZUUNUUKYLDVTZUUOYM - UUPXJYRDYBYLWFZVHDYLWAVNDUUFYLWBWCUUKUUFYBOUUMYAOWDVJUUFWEWGWHWIUUCFX - KUUIXJXLYSWJXJYSXKUUIQZXLUUKEUUIQZADUEZUURXJYMYRUUTXJYMSZYQUUSADUVAYN - DPZSZYPUUIQZYQUUSUVCYOUUFPZYOUUGQUVDYMUVBUVEXJYMUUPUVBUVEUUQDYNYLWKWL - WMYOUUFWNYOUUGWOWPEYPUUIWQWRWSWTADEUUIXAVRWIXBUUAUUJBUUGYBXRUUGUFXSUU - IFXRUUGUSUTXCWGXDVMXGXEXFXHXI $. + 3syl sstr2 syl5com ralimdva iunss sstrd rspcev nsyl3 exlimdv biimtrid + impr syld con4d 3impia ) DOPZFADEUAZQZFJPZEJPZADRZXJXLSZXOXMXOTZEBUBZ + UCZQZBHUDZOUIZRZADUEZXPXMTZYDXNTZADUEXQYCYFADXNYCCUBZXSQZBYBRZTZYCTCE + JMYGEUFZYIYCYKYHXTBYBYGEXSUGUHUJLUKULUMXNADUNUOXPYDDYBNUBZUPZEAUBZYLU + QZUCZQZADUEZSZNURZYEXJYDYTVAXLXJYDYTXTYQABDYBNXRYOUFXSYPEXRYOUSUTVBVM + VCXPYSYENXPYSYEXMFXSQZBYBRZXPYSSZXMUUBTZYJUUDCFJJYGFUFZYIUUBUUEYHUUAB + YBYGFXSUGUHUJLVDVEUUCYLVFZUCZYBPZFUUGUCZQZUUBXJYSUUHXLXJYSSZYAOUUGUUK + UUGHQZUUGYAPUUKUUFYAQUULUUKUUFYBYAYMUUFYBQXJYRDYBYLVGVHZYAOVIVJUUFHVK + VNUUGHUUFYLNVLVOVPVQVRUUKUUFOPZUUFOQUUGOPXJYSDUUFYLVSZUUNUUKYLDVTZUUO + YMUUPXJYRDYBYLWFZVHDYLWAVNDUUFYLWBWCUUKUUFYBOUUMYAOWDVJUUFWEWGWHWIUUC + FXKUUIXJXLYSWJXJYSXKUUIQZXLUUKEUUIQZADUEZUURXJYMYRUUTXJYMSZYQUUSADUVA + YNDPZSZYPUUIQZYQUUSUVCYOUUFPZYOUUGQUVDYMUVBUVEXJYMUUPUVBUVEUUQDYNYLWK + WLWMYOUUFWNYOUUGWOWPEYPUUIWQWRWSXFADEUUIWTVRWIXAUUAUUJBUUGYBXRUUGUFXS + UUIFXRUUGUSUTXBWGXCVMXDXGXEXHXI $. $} heibor.4 $e |- G = { <. y , n >. | @@ -575189,57 +576166,57 @@ whose corresponding balls have no finite subcover (i.e. in the set ( ( g ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) $= ( vt cuni cv wf cfv c2nd c1 caddc co wbr cin wcel wa wral wex cdom wrex - com cn0 csn cxp ciun cvv nn0ex fvex snex xpex iunex c1st wrel relopabiv - wss cop wceq w3a 1st2nd mpan eleq1d df-br bitr4di heiborlem2 bitrdi ibi - snid opelxp mpbiran2 fveq2 xpeq12d eleq2d rspcev sylan2br eliun 3adant3 - sneq sylibr syl eqeltrd ssriv ssdomg mp2 cen nn0ennn nnenom entri endom - ax-mp vex xpsnen cfn cpw inss2 ffvelcdmda sselid csdm sylancr ralrimiva - ffvelcdm syl2an oveq1 eqtrid oveq2 eqtrd ineq2d cexp cdiv adantl elpwid - cn c2 oveq2d ovex syl2anc adantr syl3anc sylib heiborlem1 sdomdom sylbi - isfinite endomtr iunctb simp1d peano2nn0 iunin2 cbviunv iuneq1d iuneq2d - domtr eqeq2d rspccva cbl fveq2d df-ov eqtr4di inss1 simp2d sseldd ovmpo - cxmet cxr cmet ccmet cmetmet metxmet nnexpcl nnrpd rpreccld rpxrd blssm - 2nn eqsstrd df-ss eqtr3d eqimss2 simp3d inex1 mopnuni sseqtrd sselda id - adantrr unisn uniiun eqtr3i sseqtri mp3an12 eleq1 biimpri syl3an simprr - snfi rexsn 3expb jca32 ex reximdv2 mpd cmopn fvexi uniex breq1 axcc4dom - 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G @@ -575256,30 +576233,30 @@ whose corresponding balls have no finite subcover (i.e. in the set id cmin cif cmpt cseq fveq1i cz 0z seq1 ax-mp eqtri cvv w3a relopabiv 0nn0 brrelex1i syl iftrue eqid fvmptg sylancr eqtrid eqbrtrd wa df-br cin cop c2nd wral df-ov eqtr4di fvex vex op2ndd oveq1d oveq12d eleq1d - ineq12d anbi12d rspccv syl5bi seqp1 nn0uz eleq2s oveq1i 3eqtr4g eqeq1 - oveq1 ifbieq2d peano2nn0 cn wn nn0p1nn nnne0 neneqd iffalse 3syl ovex - cuz eqeltrdi fvmptd3 cc nn0cn ax-1cn pncan 3eqtrd oveq2d eqtrd breq1d - sylancl biimprd adantrd syl9r a2d nn0ind impcom ) GULUMAGKUNZGQUOZABU - PZKUNZUUDQUOZUQAURKUNZURQUOZUQAUKUPZKUNZUUIQUOZUQAUUIUSUTVAZKUNZUULQU - OZUQAUUCUQBUKGUUDURVBZUUFUUHAUUOUUEUUGUUDURQUUDURKVCUUOVFVDVEUUDUUIVB - ZUUFUUKAUUPUUEUUJUUDUUIQUUDUUIKVCUUPVFVDVEUUDUULVBZUUFUUNAUUQUUEUUMUU - DUULQUUDUULKVCUUQVFVDVEUUDGVBZUUFUUCAUURUUEUUBUUDGQUUDGKVCUURVFVDVEAU - UGIURQAUUGURNULNUPZURVBZIUUSUSVGVAZVHZVIZUNZIUUGURLUVCURVJZUNZUVDURKU - VEUJVKURVLUMUVFUVDVBVMLUVCURVNVOVPAURULUMIVQUMZUVDIVBVTAIURQUOUVGUIIU - RQOUPZULUMCUPZUVHPUNUMUVIUVHHVASUMVRCOQUCVSWAWBNURUVBIULVQUVCUUTIUVAW - CUVCWDZWEWFWGUIWHUUIULUMZAUUKUUNAUUKUUJUUILVAZUULQUOZUUJUUIHVAZUVLUUL - HVAZWKZSUMZWIZUVKUUNUUKUUJUUIWLZQUMZAUVRUUJUUIQWJAUUDLUNZUUDWMUNZUSUT - VAZQUOZUUDHUNZUWAUWCHVAZWKZSUMZWIZBQWNUVTUVRUQUHUWIUVRBUVSQUUDUVSVBZU - WDUVMUWHUVQUWJUWAUVLUWCUULQUWJUWAUVSLUNUVLUUDUVSLVCUUJUUILWOWPZUWJUWB - UUIUSUTUUJUUIUUDUUIKWQUKWRWSWTZVDUWJUWGUVPSUWJUWEUVNUWFUVOUWJUWEUVSHU - NUVNUUDUVSHVCUUJUUIHWOWPUWJUWAUVLUWCUULHUWKUWLXAXCXBXDXEWBXFUVKUVMUUN - UVQUVKUUNUVMUVKUUMUVLUULQUVKUUMUUJUULUVCUNZLVAZUVLUVKUULUVEUNZUUIUVEU - NZUWMLVAZUUMUWNUWOUWQVBUUIURYDUNULLUVCURUUIXGXHXIUULKUVEUJVKUUJUWPUWM - LUUIKUVEUJVKXJXKUVKUWMUUIUUJLUVKUWMUULURVBZIUULUSVGVAZVHZUWSUUIUVKNUU - LUVBUWTULUVCVQUVJUUSUULVBUUTUWRUVAUWSIUUSUULURXLUUSUULUSVGXMXNUUIXOUV - KUWTUWSVQUVKUULXPUMZUWRXQUWTUWSVBUUIXRUXAUULURUULXSXTUWRIUWSYAYBZUULU - SVGYCYEYFUXBUVKUUIYGUMUSYGUMUWSUUIVBUUIYHYIUUIUSYJYOYKYLYMYNYPYQYRYSY - TUUA $. + ineq12d anbi12d rspccv biimtrid cuz seqp1 nn0uz eleq2s oveq1i 3eqtr4g + eqeq1 oveq1 ifbieq2d peano2nn0 cn wn nn0p1nn neneqd iffalse 3syl ovex + nnne0 eqeltrdi fvmptd3 nn0cn ax-1cn pncan sylancl 3eqtrd oveq2d eqtrd + cc breq1d biimprd adantrd syl9r a2d nn0ind impcom ) GULUMAGKUNZGQUOZA + BUPZKUNZUUDQUOZUQAURKUNZURQUOZUQAUKUPZKUNZUUIQUOZUQAUUIUSUTVAZKUNZUUL + QUOZUQAUUCUQBUKGUUDURVBZUUFUUHAUUOUUEUUGUUDURQUUDURKVCUUOVFVDVEUUDUUI + VBZUUFUUKAUUPUUEUUJUUDUUIQUUDUUIKVCUUPVFVDVEUUDUULVBZUUFUUNAUUQUUEUUM + UUDUULQUUDUULKVCUUQVFVDVEUUDGVBZUUFUUCAUURUUEUUBUUDGQUUDGKVCUURVFVDVE + AUUGIURQAUUGURNULNUPZURVBZIUUSUSVGVAZVHZVIZUNZIUUGURLUVCURVJZUNZUVDUR + KUVEUJVKURVLUMUVFUVDVBVMLUVCURVNVOVPAURULUMIVQUMZUVDIVBVTAIURQUOUVGUI + IURQOUPZULUMCUPZUVHPUNUMUVIUVHHVASUMVRCOQUCVSWAWBNURUVBIULVQUVCUUTIUV + AWCUVCWDZWEWFWGUIWHUUIULUMZAUUKUUNAUUKUUJUUILVAZUULQUOZUUJUUIHVAZUVLU + ULHVAZWKZSUMZWIZUVKUUNUUKUUJUUIWLZQUMZAUVRUUJUUIQWJAUUDLUNZUUDWMUNZUS + UTVAZQUOZUUDHUNZUWAUWCHVAZWKZSUMZWIZBQWNUVTUVRUQUHUWIUVRBUVSQUUDUVSVB + ZUWDUVMUWHUVQUWJUWAUVLUWCUULQUWJUWAUVSLUNUVLUUDUVSLVCUUJUUILWOWPZUWJU + WBUUIUSUTUUJUUIUUDUUIKWQUKWRWSWTZVDUWJUWGUVPSUWJUWEUVNUWFUVOUWJUWEUVS + HUNUVNUUDUVSHVCUUJUUIHWOWPUWJUWAUVLUWCUULHUWKUWLXAXCXBXDXEWBXFUVKUVMU + UNUVQUVKUUNUVMUVKUUMUVLUULQUVKUUMUUJUULUVCUNZLVAZUVLUVKUULUVEUNZUUIUV + EUNZUWMLVAZUUMUWNUWOUWQVBUUIURXGUNULLUVCURUUIXHXIXJUULKUVEUJVKUUJUWPU + WMLUUIKUVEUJVKXKXLUVKUWMUUIUUJLUVKUWMUULURVBZIUULUSVGVAZVHZUWSUUIUVKN + UULUVBUWTULUVCVQUVJUUSUULVBUUTUWRUVAUWSIUUSUULURXMUUSUULUSVGXNXOUUIXP + UVKUWTUWSVQUVKUULXQUMZUWRXRUWTUWSVBUUIXSUXAUULURUULYDXTUWRIUWSYAYBZUU + LUSVGYCYEYFUXBUVKUUIYNUMUSYNUMUWSUUIVBUUIYGYHUUIUSYIYJYKYLYMYOYPYQYRY + SYTUUA $. heibor.12 $e |- M = ( n e. NN |-> <. 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WWRVUOVJYHUPZVUKVURWWQWXAWWRWXGWRYMVURVWMWXAVWNWXCVIVJVUOUYGWMWWQVURV + UKWXGWRYMVJVUAUYHWNUYIZXBYRXIVURVWQWWBVWRUOVURVJUNYHUPZWWRVDUPZVWQWWB + VURVWMWXJVWQWRZVWNVWMWXAWXBWXKWXCWXDWVRWXEWXFWXKYFYOUNVUOVJYPYQXLVIVU + RWXIVJWWRVUKVDWXIVJWRVURUYJUYKWXHXJYRXIUYLWKUYNIVUJVUNVULVUPUAUYOUYMW + BVUIVUEVUJVULXEZVUDUQVUMVUIVUCWXLVUDVUHVUCWXLWRZAVUHWVPWXMVUAUYPOVUBO + UMZJUQZVBVJWXNVCUPZVDUPZXEZWXLUSTWXNVUBWRZWXOVUJWXQVULWXNVUBJXFWXSWXP + VUKVBVDWXNVUBVJVCXAXBYSULVUJVULYTUYQVIWKUYRVUJVULVUDXGXHVUHVUGVUQWRAV + UHVUGVUNVUPXEZVUDUQVUQVUHVUFWXTVUDOVUAWXRWXTUSTOMWTZWXOVUNWXQVUPWXNVU + AJXFWYAWXPVUOVBVDWXNVUAVJVCXAXBYSULVUNVUPYTUYQUYRVUNVUPVUDXGXHWKUYSUY + T $. $( Lemma for ~ heibor . Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff @@ -576980,7 +577957,7 @@ the next (since the empty set has a finite subcover, the ${ $d g h x y z G $. $d g h x y z H $. $d g h x y z X $. isring.1 $e |- X = ran G $. - $( The predicate "is a (unital) ring." Definition of ring with unit in + $( The predicate "is a (unital) ring." Definition of "ring with unit" in [Schechter] p. 187. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) $) @@ -577092,7 +578069,7 @@ the next (since the empty set has a finite subcover, the KFSJFSAFSVLAJKCDEFGHIUEUFVCAJFFUGUHVDVKJBFURBPZVCVJAFVPUTVGVBVIVPUSVFURBU RBUQEUIVPUJZTVPVAVHURBURBUQEUKVQTULUMUNUO $. - $( The unit element of a ring is unique. (Contributed by NM, 4-Apr-2009.) + $( The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) $) rngoideu $p |- ( R e. RingOps -> @@ -577462,8 +578439,9 @@ the next (since the empty set has a finite subcover, the uridm.2 $e |- X = ran ( 1st ` R ) $. ${ uridm.3 $e |- U = ( GId ` H ) $. - $( The unit of a ring is an identity element for the multiplication. - (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.) $) + $( The unity element of a ring is an identity element for the + multiplication. (Contributed by FL, 18-Feb-2010.) + (New usage is discouraged.) $) rngoidmlem $p |- ( ( R e. RingOps /\ A e. X ) -> ( ( U H A ) = A /\ ( A H U ) = A ) ) $= ( crngo wcel co wceq wa wi crn cmndo cmagm cexid eqid ex mndomgmid 3syl @@ -577473,8 +578451,9 @@ the next (since the empty set has a finite subcover, the ZVIUSVJNZGVKVIMEVCLZVLEVHVCUJVMUSVJVMUSMZUTVDVAVNEVCAVMUSUOUKULTUMUNUPU QUR $. - $( The unit of a ring is an identity element for the multiplication. - (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.) $) + $( The unity element of a ring is an identity element for the + multiplication. (Contributed by FL, 18-Apr-2010.) + (New usage is discouraged.) $) rngolidm $p |- ( ( R e. RingOps /\ A e. X ) -> ( U H A ) = A ) $= ( crngo wcel wa co wceq rngoidmlem simpld ) BIJAEJKCADLAMACDLAMABCDEFGH NO $. @@ -577482,8 +578461,9 @@ the next (since the empty set has a finite subcover, the ${ uridm2.2 $e |- U = ( GId ` H ) $. - $( The unit of a ring is an identity element for the multiplication. - (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.) $) + $( The unity element of a ring is an identity element for the + multiplication. (Contributed by FL, 18-Apr-2010.) + (New usage is discouraged.) $) rngoridm $p |- ( ( R e. RingOps /\ A e. X ) -> ( A H U ) = A ) $= ( crngo wcel wa co wceq rngoidmlem simprd ) BIJAEJKCADLAMACDLAMABCDEFGH NO $. @@ -577494,8 +578474,8 @@ the next (since the empty set has a finite subcover, the ring1cl.1 $e |- X = ran ( 1st ` R ) $. ring1cl.2 $e |- H = ( 2nd ` R ) $. ring1cl.3 $e |- U = ( GId ` H ) $. - $( The unit of a ring belongs to the base set. (Contributed by FL, - 12-Feb-2010.) (New usage is discouraged.) $) + $( The unity element of a ring belongs to the base set. (Contributed by + FL, 12-Feb-2010.) (New usage is discouraged.) $) rngo1cl $p |- ( R e. RingOps -> U e. X ) $= ( crngo wcel c2nd cfv crn cmagm cexid wa cmndo syl eqid cgi wceq rngomndo eleq1i mndoismgmOLD mndoisexid sylbi elin sylibr fveq2i eqtri iorlid c1st @@ -577513,7 +578493,7 @@ the next (since the empty set has a finite subcover, the uznzr.4 $e |- U = ( GId ` H ) $. uznzr.5 $e |- X = ran G $. $( Obsolete as of 23-Jan-2020. Use ~ 0ring01eqbi instead. In a unital - ring the zero equals the unity iff the ring is the zero ring. + ring the zero equals the ring unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.) $) rngoueqz $p |- ( R e. RingOps -> ( X ~~ 1o <-> U = Z ) ) $= ( vx wcel c1o wceq wi wa syl ex wral cen wbr rngo0cl csn en1eqsn crn c1st @@ -577635,7 +578615,7 @@ the next (since the empty set has a finite subcover, the zerdivempx.4 $e |- X = ran G $. zerdivempx.5 $e |- U = ( GId ` H ) $. $d A a $. $d B a $. $d H a $. $d R a $. $d X a $. $d Z a $. - $( In a unitary ring a left invertible element is not a zero divisor. See + $( In a unital ring a left invertible element is not a zero divisor. See also ~ ringinvnzdiv . (Contributed by Jeff Madsen, 18-Apr-2010.) $) zerdivemp1x $p |- ( ( R e. RingOps /\ A e. X /\ E. a e. X ( a H A ) = U ) -> ( B e. X -> ( ( A H B ) = Z -> B = Z ) ) ) $= @@ -577751,9 +578731,9 @@ the next (since the empty set has a finite subcover, the dvrunz.3 $e |- X = ran G $. dvrunz.4 $e |- Z = ( GId ` G ) $. dvrunz.5 $e |- U = ( GId ` H ) $. - $( In a division ring the unit is different from the zero. (Contributed by - FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) - (New usage is discouraged.) $) + $( In a division ring the ring unit is different from the zero. + (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, + 15-Dec-2013.) (New usage is discouraged.) $) dvrunz $p |- ( R e. DivRingOps -> U =/= Z ) $= ( cdrng wcel cop csn wn wne wceq wb syl cgi fvexi zrdivrng c1o crngo cdif cen wbr cxp cres cgr drngoi simpld rngoueqz rngosn6 biimpd syl6bi pm2.43a @@ -578523,7 +579503,7 @@ the next (since the empty set has a finite subcover, the $( A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) $) - fldcrng $p |- ( K e. Fld -> K e. CRingOps ) $= + fldcrngo $p |- ( K e. Fld -> K e. CRingOps ) $= ( cdrng wcel ccm2 crngo cfld ccring c2nd cfv c1st crn cgi csn cdif cxp cres wa cgr eqid drngoi simpld anim1i df-fld elin2 iscrngo 3imtr4i ) ABCZADCZQAE CZUHQAFCAGCUGUIUHUGUIAHIZAJIZKZUKLIZMNZUNOPRCAUKUJULUMUKSUJSULSUMSTUAUBABDF @@ -578531,9 +579511,9 @@ the next (since the empty set has a finite subcover, the $( The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.) $) isfld2 $p |- ( K e. Fld <-> ( K e. DivRingOps /\ K e. CRingOps ) ) $= - ( cfld wcel cdrng ccring wa flddivrng fldcrng jca crngo iscrngo simprbi cin - ccm2 elin biimpri df-fld eleqtrrdi sylan2 impbii ) ABCZADCZAECZFUAUBUCAGAHI - UCUBANCZUAUCAJCUDAKLUBUDFZADNMZBAUFCUEADNOPQRST $. + ( cfld wcel cdrng ccring wa flddivrng fldcrngo jca ccm2 iscrngo simprbi cin + crngo elin biimpri df-fld eleqtrrdi sylan2 impbii ) ABCZADCZAECZFUAUBUCAGAH + IUCUBAJCZUAUCANCUDAKLUBUDFZADJMZBAUFCUEADJOPQRST $. ${ $d R w x y z $. $d S w x y z $. $d X w x y z $. $d Y w x y z $. @@ -578796,15 +579776,15 @@ the next (since the empty set has a finite subcover, the 0idl $p |- ( R e. RingOps -> { Z } e. ( Idl ` R ) ) $= ( vx vy vz wcel cfv cv co wral wa eqid wceq ovex elsn sylibr eleq1d crngo csn cidl crn wss c2nd rngo0cl snssd fvexi snid velsn rngo0rid mpdan oveq2 - cgi a1i syl5ibrcom syl5bi ralrimiv rngorz ralrimiva oveq1 ralbidv anbi12d - rngolz jca isidl mpbir3and ) AUAIZCUBZAUCJIVJBUDZUECVJIZFKZGKZBLZVJIZGVJM - ZHKZVMAUFJZLZVJIZVMVRVSLZVJIZNZHVKMZNZFVJMVICVKABVKCDVKOZEUGZUHVLVICCBUOE - UIUJUPVIWFFVJVMVJIVMCPZVIWFFCUKVIWFWICVNBLZVJIZGVJMZVRCVSLZVJIZCVRVSLZVJI - ZNZHVKMZNVIWLWRVIWKGVJVNVJIVNCPZVIWKGCUKVIWKWSCCBLZVJIZVIWTCPZXAVICVKIXBW - HCABVKCDWGEULUMWTCCCBQRSWSWJWTVJVNCCBUNTUQURUSVIWQHVKVIVRVKINZWNWPXCWMCPW - NVRABVSVKCEWGDVSOZUTWMCVRCVSQRSXCWOCPWPVRABVSVKCEWGDXDVEWOCCVRVSQRSVFVAVF - WIVQWLWEWRWIVPWKGVJWIVOWJVJVMCVNBVBTVCWIWDWQHVKWIWAWNWCWPWIVTWMVJVMCVRVSU - NTWIWBWOVJVMCVRVSVBTVDVCVDUQURUSFGHABVSVJVKCDXDWGEVGVH $. + cgi a1i syl5ibrcom biimtrid ralrimiv rngorz jca ralrimiva ralbidv anbi12d + rngolz oveq1 isidl mpbir3and ) AUAIZCUBZAUCJIVJBUDZUECVJIZFKZGKZBLZVJIZGV + JMZHKZVMAUFJZLZVJIZVMVRVSLZVJIZNZHVKMZNZFVJMVICVKABVKCDVKOZEUGZUHVLVICCBU + OEUIUJUPVIWFFVJVMVJIVMCPZVIWFFCUKVIWFWICVNBLZVJIZGVJMZVRCVSLZVJIZCVRVSLZV + JIZNZHVKMZNVIWLWRVIWKGVJVNVJIVNCPZVIWKGCUKVIWKWSCCBLZVJIZVIWTCPZXAVICVKIX + BWHCABVKCDWGEULUMWTCCCBQRSWSWJWTVJVNCCBUNTUQURUSVIWQHVKVIVRVKINZWNWPXCWMC + PWNVRABVSVKCEWGDVSOZUTWMCVRCVSQRSXCWOCPWPVRABVSVKCEWGDXDVEWOCCVRVSQRSVAVB + VAWIVQWLWEWRWIVPWKGVJWIVOWJVJVMCVNBVFTVCWIWDWQHVKWIWAWNWCWPWIVTWMVJVMCVRV + SUNTWIWBWOVJVMCVRVSVFTVDVCVDUQURUSFGHABVSVJVKCDXDWGEVGVH $. $} ${ @@ -578877,21 +579857,21 @@ the next (since the empty set has a finite subcover, the ( vx vy vz vi wcel cfv cv co wral wa eqid sylan2 anassrs ralrimiva sylibr wss elint2 ex crngo wne cidl w3a cint c1st crn cgi c2nd intssuni 3ad2ant2 cuni ssel2 idlss 3adant2 unissb sstrd idl0cl vex r19.26 idladdcl ralimdva - c0 fvex wi ovex syl6ibr syl5bir expdimp ralrimiv idllmulcl anass1rs an32s - syl5bi an4s imp idlrmulcl jca wb isidl 3ad2ant1 mpbir3and ) BUAGZAVCUBZAB - UCHZRZUDZAUEZWEGZWHBUFHZUGZRZWJUHHZWHGZCIZDIZWJJZWHGZDWHKZEIZWOBUIHZJZWHG - ZWOWTXAJZWHGZLZEWKKZLZCWHKZWGWHAULZWKWDWCWHXJRWFAUJUKWGFIZWKRZFAKZXJWKRWC - WFXMWDWCWFLZXLFAWCWFXKAGZXLWFXOLZWCXKWEGZXLAWEXKUMZBWJXKWKWJMZWKMZUNNOPUO - FAWKUPQUQWCWFWNWDXNWMXKGZFAKWNXNYAFAWCWFXOYAXPWCXQYAXRBWJXKWMXSWMMZURNOPF - WMAWJUHVDSQUOWCWFXIWDXNXHCWHWOWHGWOXKGZFAKZXNXHFWOACUSSXNYDXHXNYDLZWSXGYE - WRDWHWPWHGWPXKGZFAKZYEWRFWPADUSSXNYDYGWRYDYGLYCYFLZFAKZXNWRYCYFFAUTXNYIWQ - XKGZFAKWRXNYHYJFAWCWFXOYHYJVEZXPWCXQYKXRWCXQLZYHYJWOWPBWJXKXSVATNOVBFWQAW - OWPWJVFSVGVHVIVNVJYEXFEWKXNWTWKGZYDXFXNYMLZYDLZXCXEYOXBXKGZFAKZXCYNYDYQYN - YCYPFAXNYMXOYCYPVEZWCYMWFXOYRXPWCYMLZXQYRXRWCXQYMYRYLYMLZYCYPYLYCYMYPWOWT - BWJXAXKWKXSXAMZXTVKVLTVMNVOOVBVPFXBAWTWOXAVFSQYOXDXKGZFAKZXEYNYDUUCYNYCUU - BFAXNYMXOYCUUBVEZWCYMWFXOUUDXPYSXQUUDXRWCXQYMUUDYTYCUUBYLYCYMUUBWOWTBWJXA - XKWKXSUUAXTVQVLTVMNVOOVBVPFXDAWOWTXAVFSQVRVMPVRTVNVJUOWCWDWIWLWNXIUDVSWFC - DEBWJXAWHWKWMXSUUAXTYBVTWAWB $. + c0 fvex ovex syl6ibr syl5bir expdimp biimtrid ralrimiv idllmulcl anass1rs + wi an32s an4s imp idlrmulcl jca wb isidl 3ad2ant1 mpbir3and ) BUAGZAVCUBZ + ABUCHZRZUDZAUEZWEGZWHBUFHZUGZRZWJUHHZWHGZCIZDIZWJJZWHGZDWHKZEIZWOBUIHZJZW + HGZWOWTXAJZWHGZLZEWKKZLZCWHKZWGWHAULZWKWDWCWHXJRWFAUJUKWGFIZWKRZFAKZXJWKR + WCWFXMWDWCWFLZXLFAWCWFXKAGZXLWFXOLZWCXKWEGZXLAWEXKUMZBWJXKWKWJMZWKMZUNNOP + UOFAWKUPQUQWCWFWNWDXNWMXKGZFAKWNXNYAFAWCWFXOYAXPWCXQYAXRBWJXKWMXSWMMZURNO + PFWMAWJUHVDSQUOWCWFXIWDXNXHCWHWOWHGWOXKGZFAKZXNXHFWOACUSSXNYDXHXNYDLZWSXG + YEWRDWHWPWHGWPXKGZFAKZYEWRFWPADUSSXNYDYGWRYDYGLYCYFLZFAKZXNWRYCYFFAUTXNYI + WQXKGZFAKWRXNYHYJFAWCWFXOYHYJVMZXPWCXQYKXRWCXQLZYHYJWOWPBWJXKXSVATNOVBFWQ + AWOWPWJVESVFVGVHVIVJYEXFEWKXNWTWKGZYDXFXNYMLZYDLZXCXEYOXBXKGZFAKZXCYNYDYQ + YNYCYPFAXNYMXOYCYPVMZWCYMWFXOYRXPWCYMLZXQYRXRWCXQYMYRYLYMLZYCYPYLYCYMYPWO + WTBWJXAXKWKXSXAMZXTVKVLTVNNVOOVBVPFXBAWTWOXAVESQYOXDXKGZFAKZXEYNYDUUCYNYC + UUBFAXNYMXOYCUUBVMZWCYMWFXOUUDXPYSXQUUDXRWCXQYMUUDYTYCUUBYLYCYMUUBWOWTBWJ + XAXKWKXSUUAXTVQVLTVNNVOOVBVPFXDAWOWTXAVESQVRVNPVRTVIVJUOWCWDWIWLWNXIUDVSW + FCDEBWJXAWHWKWMXSUUAXTYBVTWAWB $. $} $( The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, @@ -578915,32 +579895,32 @@ the next (since the empty set has a finite subcover, the sylibr 3ad2antr2 wrex idl0cl r19.2z an12s eluni2 3adantr3 weq sseq1 sseq2 sylan2b orbi12d ralbidv adantr ad2antlr ad2antrl adantll idladdcl ancom2s rspcv ssel2 ancoms expr an42s anasss simprl elunii syl2anc anassrs jaodan - ad2antrr syldan rexlimdvaa syl5bi ralrimiv 3adantr1 idllmulcl exp43 com23 - imp41 sylanl2 simplrr idlrmulcl jca ralrimiva wb isidl mpbir3and ) BUCIZA - UAUBZABUDJZKZCLZDLZKZXLXKKZUEZDAMZCAMZUFZNZAUGZXIIZXTBUHJZUIZKZYBUJJZXTIZ - ELZFLZYBUKZXTIZFXTMZGLZYGBULJZUKZXTIZYGYLYMUKZXTIZNZGYCMZNZEXTMZXGXHXJYDX - QXGXJNZXKYCKZCAMZYDXJXGXKXIIZCAMZUUDCAXIUMZXGUUFUUDXGUUEUUCCAXGUUEUUCBYBX - KYCYBOZYCOZUNUOUPPVICAYCUQURUSXGXHXJYFXQXGXHXJNNYEXKIZCAUTZYFXHXGXJUUKUUB - XHUUJCAMZUUKXJXGUUFUULUUGXGUUFUULXGUUEUUJCAXGUUEUUJBYBXKYEUUHYEOZVAUOUPPV - IUUJCAVBQVCCYEAVDURVEXSYTEXTYGXTIEHRZHAUTXSYTHYGAVDXSUUNYTHAXSHLZAIZUUNNZ - NYKYSXGUUQXRYKXGUUQNZXJXQYKXHUURXJXQYKUURXJNZXQUUOXLKZXLUUOKZUEZDAMZYKUUS - 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Fld <-> ( K e. CRingOps /\ U =/= Z /\ ( Idl ` K ) = { { Z } , X } ) ) $= - ( vy vx vz wcel wceq syl cv wrex wa cfld wne cidl cfv csn cpr w3a fldcrng - ccring cdrng flddivrng dvrunz divrngidl 3jca crngo co cdif wral crngorngo - 3ad2ant1 simp2 crab crn c1st rneqi eqtri rngo1cl ad2antrr cigen eldif wss - wn snssi igenss sylan2 snss biimpri eleq2 syl5ibcom con3dimp sylan anasss - vex sylan2b adantlr wo eldifi snssd igenidl imp an32s ovex elpr sylib ord - mpd sylanl1 eqtr3d eleqtrd eqeq1 rexbidv elrab simprd eqcom rexbii sylibr - prnc ralrimiva 3adant2 jca32 isdrngo3 simp1 isfld2 sylanbrc impbii ) DUAO - ZDUIOZAFUBZDUCUDZFUEZEUFZPZUGZXPXQXRYBDUHXPDUJOZXRDUKZDABCEFGHIJKULQXPYDY - BYEDBCEFGHIJUMQUNYCYDXQXPYCDUOOZXRLRMRZCUPZAPZLESZMEXTUQZURZTTYDYCYFXRYLX - QXRYFYBDUSZUTXQXRYBVAXQYBYLXRXQYBTZYJMYKYNYGYKOZTZAYHPZLESZYJYPAEOZYRYPAN - RZYHPZLESZNEVBZOYSYRTYPAEUUCXQYSYBYOXQYFYSYMDACEEBVCDVDUDZVCIBUUDGVEVFHKV - GQVHYPDYGUEZVIUPZEUUCXQYFYBYOUUFEPZYMYFYBTYOTZUUFXTPZVLZUUGYFYOUUJYBYOYFY - GEOZYGXTOZVLZTUUJYGEXTVJYFUUKUUMUUJYFUUKTUUEUUFVKZUUMUUJUUKYFUUEEVKZUUNYG - EVMDUUEBEGIVNVOUUNUUIUULUUNYGUUFOZUUIUULUUPUUNYGUUFMWCVPVQUUFXTYGVRVSVTWA - WBWDWEUUHUUIUUGUUHUUFYAOZUUIUUGWFYFYOYBUUQYFYOTZYBUUQUURUUFXSOZYBUUQYOYFU - UOUUSYOYGEYGEXTWGZWHDUUEBEGIWIVOXSYAUUFVRVSWJWKUUFXTEDUUEVIWLWMWNWOWPWQXQ - YOUUFUUCPZYBYOXQUUKUVAUUTNLYGDBCEGHIXGVOWEWRWSUUBYRNAEYTAPUUAYQLEYTAYHWTX - AXBWNXCYIYQLEYHAXDXEXFXHXIXJMLDABCEFGHJIKXKXFXQXRYBXLDXMXNXO $. + ( vy vx vz wcel wceq syl cv wrex wa cfld ccring wne cidl cfv csn fldcrngo + cpr w3a cdrng flddivrng dvrunz divrngidl 3jca crngo co cdif wral 3ad2ant1 + crngorngo simp2 crab crn c1st rneqi eqtri rngo1cl ad2antrr cigen wn eldif + wss snssi igenss sylan2 vex biimpri eleq2 syl5ibcom con3dimp sylan anasss + snss sylan2b adantlr wo eldifi snssd igenidl imp an32s ovex sylib ord mpd + elpr sylanl1 prnc eqtr3d eleqtrd eqeq1 rexbidv elrab simprd rexbii sylibr + eqcom ralrimiva 3adant2 jca32 isdrngo3 simp1 isfld2 sylanbrc impbii ) DUA + OZDUBOZAFUCZDUDUEZFUFZEUHZPZUIZXPXQXRYBDUGXPDUJOZXRDUKZDABCEFGHIJKULQXPYD + YBYEDBCEFGHIJUMQUNYCYDXQXPYCDUOOZXRLRMRZCUPZAPZLESZMEXTUQZURZTTYDYCYFXRYL + XQXRYFYBDUTZUSXQXRYBVAXQYBYLXRXQYBTZYJMYKYNYGYKOZTZAYHPZLESZYJYPAEOZYRYPA + NRZYHPZLESZNEVBZOYSYRTYPAEUUCXQYSYBYOXQYFYSYMDACEEBVCDVDUEZVCIBUUDGVEVFHK + VGQVHYPDYGUFZVIUPZEUUCXQYFYBYOUUFEPZYMYFYBTYOTZUUFXTPZVJZUUGYFYOUUJYBYOYF + YGEOZYGXTOZVJZTUUJYGEXTVKYFUUKUUMUUJYFUUKTUUEUUFVLZUUMUUJUUKYFUUEEVLZUUNY + GEVMDUUEBEGIVNVOUUNUUIUULUUNYGUUFOZUUIUULUUPUUNYGUUFMVPWCVQUUFXTYGVRVSVTW + AWBWDWEUUHUUIUUGUUHUUFYAOZUUIUUGWFYFYOYBUUQYFYOTZYBUUQUURUUFXSOZYBUUQYOYF + UUOUUSYOYGEYGEXTWGZWHDUUEBEGIWIVOXSYAUUFVRVSWJWKUUFXTEDUUEVIWLWPWMWNWOWQX + QYOUUFUUCPZYBYOXQUUKUVAUUTNLYGDBCEGHIWRVOWEWSWTUUBYRNAEYTAPUUAYQLEYTAYHXA + XBXCWMXDYIYQLEYHAXGXEXFXHXIXJMLDABCEFGHJIKXKXFXQXRYBXLDXMXNXO $. $} ${ @@ -580725,7 +581705,7 @@ A collection of theorems for commuting equalities (or #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $) - $( Contents: ~ https://us.metamath.org/mpeuni/mmtheorems.html#dtl:20.21 $) + $( Contents: ~ https://us.metamath.org/mpeuni/mmtheorems.html#dtl:20.22 $) $( @@ -580778,8 +581758,8 @@ A collection of theorems for commuting equalities (or quotients $) $c ErALTV $. $( Equivalence relation on its domain quotient predicate $) - $c MembErs $. $( The class of membership equivalence relations $) - $c MembEr $. $( Membership equivalence relation predicate $) + $c CoMembErs $. $( The class of comember equivalence relations $) + $c CoMembEr $. $( Comember equivalence relation predicate $) $c Funss $. $( The class of all function sets (used only once) $) $c FunsALTV $. $( The class of functions, i.e., function relations $) @@ -580793,6 +581773,14 @@ A collection of theorems for commuting equalities (or relation $) $c ElDisj $. $( Disjoint elementhood predicate $) + $c AntisymRel $. $( Antisymmetric relation predicate $) + + $c Parts $. $( The class of all partitions, i.e., partition relations $) + $c Part $. $( Partition predicate $) + + $c MembParts $. $( The class of member partition relations $) + $c MembPart $. $( Member partition predicate $) + $( Extend the definition of a class to include the range Cartesian product class. $) cxrn $a class ( A |X. B ) $. @@ -580886,14 +581874,14 @@ A collection of theorems for commuting equalities (or domain quotient ` A ` .) $) werALTV $a wff R ErALTV A $. - $( Extend the definition of a class to include the membership equivalence + $( Extend the definition of a class to include the comember equivalence relations class. $) - cmembers $a class MembErs $. - $( Extend the definition of a wff to include the membership equivalence - relation predicate. (Read: the membership equivalence relation on ` A ` , - or, the restricted elementhood equivalence relation on its domain quotient + ccomembers $a class CoMembErs $. + $( Extend the definition of a wff to include the comember equivalence + relation predicate. (Read: the comember equivalence relation on ` A ` , + or, the restricted coelement equivalence relation on its domain quotient ` A ` .) $) - wmember $a wff MembEr A $. + wcomember $a wff CoMembEr A $. $( Extend the definition of a class to include the function set class. $) cfunss $a class Funss $. @@ -580914,13 +581902,33 @@ A collection of theorems for commuting equalities (or wdisjALTV $a wff Disj R $. $( Extend the definition of a class to include the disjoint elements class, - i.e., the disjoint elementhood relations class. $) + i.e., the disjoint element relations class. $) celdisjs $a class ElDisjs $. - $( Extend the definition of a wff to include the disjoint elementhood - predicate, i.e., the disjoint elementhood relation predicate. (Read: the - elements of ` A ` are disjoint.) $) + $( Extend the definition of a wff to include the disjoint element predicate, + i.e., the disjoint element relation predicate. (Read: the elements of + ` A ` are disjoint.) $) weldisj $a wff ElDisj A $. + $( Extend the definition of a wff to include the antisymmetry relation + predicate. (Read: ` R ` is an antisymmetric relation.) $) + wantisymrel $a wff AntisymRel R $. + + $( Extend the definition of a class to include the partitions class, i.e., + the partition relations class. $) + cparts $a class Parts $. + $( Extend the definition of a wff to include the partition predicate, i.e., + the partition relation predicate. (Read: ` A ` is a partition by + ` R ` .) $) + wpart $a wff R Part A $. + + $( Extend the definition of a class to include the member partitions class, + i.e., the member partition relations class. $) + cmembparts $a class MembParts $. + $( Extend the definition of a wff to include the member partition predicate, + i.e., the member partition relation predicate. (Read: ` A ` is a member + partition.) $) + wmembpart $a wff MembPart A $. + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -581017,12 +582025,49 @@ A collection of theorems for commuting equalities (or ( wa w3a biantrur 3anass 3bitr4i ) BCFZAKFCABCGAKDHBCEHABCIJ $. $} - $( Substitution of equal classes into elementhood relation. (Contributed by + ${ + bianbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $. + bianbi.2 $e |- ( ps <-> th ) $. + $( Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, + 31-Jul-2023.) $) + bianbi $p |- ( ph <-> ( th /\ ch ) ) $= + ( wa anbi1i bitri ) ABCGDCGEBDCFHI $. + $} + + ${ + bianim.1 $e |- ( ph <-> ( ps /\ ch ) ) $. + bianim.2 $e |- ( ch -> ( ps <-> th ) ) $. + $( Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, + 31-Jul-2023.) $) + bianim $p |- ( ph <-> ( th /\ ch ) ) $= + ( wa pm5.32ri bitri ) ABCGDCGECBDFHI $. + $} + + ${ + biorfd.1 $e |- ( ph -> -. ps ) $. + $( A wff is equivalent to its disjunction with falsehood, deduction form. + (Contributed by Peter Mazsa, 22-Aug-2023.) $) + biorfd $p |- ( ph -> ( ch <-> ( ps \/ ch ) ) ) $= + ( wn wo wb biorf syl ) ABECBCFGDBCHI $. + $} + + $( Substitution of equal classes in binary relation. (Contributed by Peter + Mazsa, 14-Jun-2024.) $) + eqbrtr $p |- ( ( A = B /\ B R C ) -> A R C ) $= + ( wceq wbr breq1 biimpar ) ABEACDFBCDFABCDGH $. + + $( Substitution of equal classes in a binary relation. (Contributed by Peter + Mazsa, 14-Jun-2024.) $) + eqbrb $p |- ( ( A = B /\ A R C ) <-> ( A = B /\ B R C ) ) $= + ( wceq wbr wa simpl eqbrtr jca eqcom anbi1i 3imtr3i impbii ) ABEZACDFZGZOBC + DFZGZBAEZPGZTRGQSUATRTPHBACDIJTOPBAKZLTORUBLMSOPORHABCDIJN $. + + $( Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 22-Jul-2017.) $) eqeltr $p |- ( ( A = B /\ B e. C ) -> A e. C ) $= ( wceq wcel eleq1 biimpar ) ABDACEBCEABCFG $. - $( Substitution of equal classes into elementhood relation. (Contributed by + $( Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 17-Jul-2019.) $) eqelb $p |- ( ( A = B /\ A e. C ) <-> ( A = B /\ B e. C ) ) $= ( wceq wcel wa simpl eqeltr jca eqcom anbi1i 3imtr3i impbii ) ABDZACEZFZNBC @@ -581036,6 +582081,62 @@ A collection of theorems for commuting equalities (or ( wceq wb eqeq12 sylan2 ) ABCGDEGBDGCEGHFBCDEIJ $. $} + $( One-to-one relationship between the successor operation and the singleton. + (Contributed by Peter Mazsa, 31-Dec-2024.) $) + suceqsneq $p |- ( A e. V -> ( suc A = suc B <-> { A } = { B } ) ) $= + ( wcel csuc wceq csn suc11reg sneqbg bitr4id ) ACDAEBEFABFAGBGFABHABCIJ $. + + $( Absorption of union with a singleton by difference. (Contributed by Peter + Mazsa, 24-Jul-2024.) $) + sucdifsn2 $p |- ( ( A u. { A } ) \ { A } ) = A $= + ( csn cin c0 wceq cun cdif disjcsn undif5 ax-mp ) AABZCDEAKFKGAEAHAKIJ $. + + $( The difference between the successor and the singleton of a class is the + class. (Contributed by Peter Mazsa, 20-Sep-2024.) $) + sucdifsn $p |- ( suc A \ { A } ) = A $= + ( csuc csn cdif cun df-suc difeq1i sucdifsn2 eqtri ) ABZACZDAKEZKDAJLKAFGAH + I $. + + $( The restriction to a disjoint is the empty class. (Contributed by Peter + Mazsa, 24-Jul-2024.) $) + disjresin $p |- ( ( A i^i B ) = (/) -> ( R |` ( A i^i B ) ) = (/) ) $= + ( cin c0 wceq cres reseq2 res0 eqtrdi ) ABDZEFCKGCEGEKECHCIJ $. + + $( The intersection of restrictions to disjoint is the empty class. + (Contributed by Peter Mazsa, 24-Jul-2024.) $) + disjresdisj $p |- ( ( A i^i B ) = (/) -> + ( ( R |` A ) i^i ( R |` B ) ) = (/) ) $= + ( cin c0 wceq cres resindi disjresin eqtr3id ) ABDZEFCAGCBGDCKGECABHABCIJ + $. + + $( The difference between restrictions to disjoint is the first restriction. + (Contributed by Peter Mazsa, 24-Jul-2024.) $) + disjresdif $p |- ( ( A i^i B ) = (/) -> + ( ( R |` A ) \ ( R |` B ) ) = ( R |` A ) ) $= + ( cin c0 wceq cres cdif disjresdisj disjdif2 syl ) ABDEFCAGZCBGZDEFLMHLFABC + ILMJK $. + + $( Lemma for ~ ressucdifsn2 . (Contributed by Peter Mazsa, 24-Jul-2024.) $) + disjresundif $p |- ( ( A i^i B ) = (/) -> + ( ( R |` ( A u. B ) ) \ ( R |` B ) ) = ( R |` A ) ) $= + ( cin c0 wceq cun cres cdif resundi difeq1i difun2 eqtri disjresdif eqtrid + ) ABDEFCABGHZCBHZIZCAHZQIZSRSQGZQITPUAQCABJKSQLMABCNO $. + + $( The difference between restrictions to the successor and the singleton of + a class is the restriction to the class, see ~ ressucdifsn . (Contributed + by Peter Mazsa, 24-Jul-2024.) $) + ressucdifsn2 $p |- ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) = + ( R |` A ) $= + ( csn cin c0 wceq cun cres cdif disjcsn disjresundif ax-mp ) AACZDEFBAMGHBM + HIBAHFAJAMBKL $. + + $( The difference between restrictions to the successor and the singleton of + a class is the restriction to the class. (Contributed by Peter Mazsa, + 20-Sep-2024.) $) + ressucdifsn $p |- ( ( R |` suc A ) \ ( R |` { A } ) ) = ( R |` A ) $= + ( csuc cres csn cdif cun df-suc reseq2i difeq1i ressucdifsn2 eqtri ) BACZDZ + BAEZDZFBAOGZDZPFBADNRPMQBAHIJABKL $. + $( Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.) $) inres2 $p |- ( ( R |` A ) i^i S ) = ( ( R i^i S ) |` A ) $= @@ -581052,6 +582153,23 @@ A collection of theorems for commuting equalities (or nexmo1 $p |- ( -. E. x ph -> E* x ph ) $= ( wex wn weu wi wmo pm2.21 moeu sylibr ) ABCZDKABEZFABGKLHABIJ $. + $( Restricted universal quantification over intersection. (Contributed by + Peter Mazsa, 8-Sep-2023.) $) + ralin $p |- ( A. x e. ( A i^i B ) ph <-> + A. x e. A ( x e. B -> ph ) ) $= + ( cv wcel wi cin wa elin imbi1i impexp bitri ralbii2 ) ABEZDFZAGZBCDHZCORFZ + AGOCFZPIZAGTQGSUAAOCDJKTPALMN $. + + ${ + $d A y $. $d x y $. + $( Double restricted universal quantification, special case. (Contributed + by Peter Mazsa, 17-Jun-2020.) $) + r2alan $p |- ( A. x A. y ( ( ( x e. A /\ y e. B ) /\ ph ) -> ps ) <-> + A. x e. A A. y e. B ( ph -> ps ) ) $= + ( cv wcel wa wi wal wral impexp 2albii r2al bitr4i ) CGEHDGFHIZAIBJZDKCKQ + ABJZJZDKCKSDFLCELRTCDQABMNSCDEFOP $. + $} + ${ 3ralbii.1 $e |- ( ph <-> ps ) $. $( Inference adding three restricted universal quantifiers to both sides of @@ -581135,6 +582253,13 @@ A collection of theorems for commuting equalities (or UIMLZUQUKURULUTUPUJCUNUOUHUIUFZUGUTUPUJDVAUGUAUBUC $. $} + $( Binary relation on the converse of an intersection with a Cartesian + product. (Contributed by Peter Mazsa, 27-Jul-2019.) $) + br1cnvinxp $p |- ( C `' ( R i^i ( A X. B ) ) D <-> + ( ( C e. B /\ D e. A ) /\ D R C ) ) $= + ( cxp cin ccnv wbr wcel wa relinxp relbrcnv brinxp2 ancom anbi1i 3bitri ) C + DEABFGZHIDCRIDAJZCBJZKZDCEIZKTSKZUBKCDRABELMABDCENUAUCUBSTOPQ $. + $( Elementhood in a converse ` R ` -coset when ` R ` is a relation. (Contributed by Peter Mazsa, 9-Dec-2018.) $) releleccnv $p |- ( Rel R -> ( A e. [ B ] `' R <-> A R B ) ) $= @@ -581227,6 +582352,14 @@ A collection of theorems for commuting equalities (or AUDJKHZUEUFUIABCDUDRLUFUEBCUCDAMNOUKUGUEIUHIUGUEUHPUJAUDBCDQUGUEUHSUGUEUHTU AUB $. + $( Binary relation on the converse of a restriction. (Contributed by Peter + Mazsa, 27-Jul-2019.) $) + br1cnvres $p |- ( B e. V -> + ( B `' ( R |` A ) C <-> ( C e. A /\ C R B ) ) ) $= + ( cres ccnv wbr cvv cxp cin wcel wa df-res cnveqi breqi wb br1cnvinxp anass + elex bitri baib syl bitrid ) BCDAFZGZHBCDAIJKZGZHZBELZCALZCBDHZMZBCUFUHUEUG + DANOPUJBILZUIUMQBETUIUNUMUIUNUKMULMUNUMMAIBCDRUNUKULSUAUBUCUD $. + ${ $d A y $. $d B y $. $d R y $. $( Elementhood in the domain of a restriction. (Contributed by Peter @@ -581238,6 +582371,37 @@ A collection of theorems for commuting equalities (or BCUCDOPQRUFUGASTUA $. $} + ${ + $d A x $. $d B x $. $d R x $. $d V x $. + $( Element of the range of a restriction. (Contributed by Peter Mazsa, + 26-Dec-2018.) $) + elrnres $p |- ( B e. V -> ( B e. ran ( R |` A ) <-> E. x e. A x R B ) ) $= + ( wcel cres crn cv wbr wex wrex elrng brres exbidv bitrd df-rex bitr4di + wa ) CEFZCDBGZHFZAIZBFUCCDJZSZAKZUDABLTUBUCCUAJZAKUFACUAEMTUGUEABUCCDENOP + UDABQR $. + $} + + ${ + $d A y $. $d B y $. $d R y $. + $( Element of the domain of a restriction to a singleton. (Contributed by + Peter Mazsa, 12-Jun-2024.) $) + eldmressnALTV $p |- ( B e. V -> + ( B e. dom ( R |` { A } ) <-> ( B = A /\ A e. dom R ) ) ) $= + ( vy wcel csn cres cdm wceq wa wbr wex eldmres elsng eldmg bicomd anbi12d + cv bitrd eqelb bitrdi ) BDFZBCAGZHIFZBAJZBCIZFZKZUFAUGFKUCUEBUDFZBESCLEMZ + KUIEUDBCDNUCUJUFUKUHBADOUCUHUKEBCDPQRTBAUGUAUB $. + $} + + ${ + $d A x $. $d B x $. $d R x $. $d W x $. + $( Element of the range of a restriction to a singleton. (Contributed by + Peter Mazsa, 12-Jun-2024.) $) + elrnressn $p |- ( ( A e. V /\ B e. W ) -> + ( B e. ran ( R |` { A } ) <-> A R B ) ) $= + ( vx wcel csn cres crn cv wbr wrex elrnres breq1 rexsng sylan9bbr ) BEGBC + AHZIJGFKZBCLZFRMADGABCLZFRBCENTUAFADSABCOPQ $. + $} + ${ $d A y $. $d R y $. $d V y $. $( Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.) $) @@ -581321,6 +582485,13 @@ A collection of theorems for commuting equalities (or ( [ A ] `' _E = [ B ] `' _E <-> A = B ) ) $= ( wcel cep ccnv cec eccnvep eqeqan12d ) ACEBDEAFGZHABKHBACIBDIJ $. + $( Property of the epsilon relation. (Contributed by Peter Mazsa, + 27-Apr-2020.) $) + disjeccnvep $p |- ( ( A e. V /\ B e. W ) -> + ( ( [ A ] `' _E i^i [ B ] `' _E ) = (/) <-> ( A i^i B ) = (/) ) ) $= + ( wcel wa cep ccnv cec cin c0 eccnvep ineqan12d eqeq1d ) ACEZBDEZFAGHZIZBQI + ZJABJKOPRASBACLBDLMN $. + $( The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019.) $) eccnvepres2 $p |- ( B e. A -> [ B ] ( `' _E |` A ) = B ) $= @@ -581590,6 +582761,18 @@ equivalence of domain elementhood (equivalence is not necessary as HUFUEUHPBUCUDQRSTUBUA $. $} + ${ + $d A x y $. $d B x y $. $d R x y $. + $( Two ways to say that an intersection of the identity relation with a + Cartesian product is a subclass. (Contributed by Peter Mazsa, + 12-Dec-2023.) $) + ref5 $p |- ( ( _I i^i ( A X. B ) ) C_ R <-> A. x e. ( A i^i B ) x R x ) $= + ( vy cv wceq wbr wral wcel cid cxp cin wss equcom imbi1i ralbii ceqsralbv + wi breq2 bitr3i idinxpss ralin 3bitr4i ) AFZEFZGZUEUFDHZSZECIZABIUECJUEUE + DHZSZABIKBCLMDNUKABCMIUJULABUJUFUEGZUHSZECIULUNUIECUMUGUHEAOPQUHUKEUECUFU + EUEDTRUAQAEBCDUBUKABCUCUD $. + $} + ${ $d x y $. $d A y $. $( Two ways to say that an intersection with a Cartesian product is a @@ -581847,7 +583030,7 @@ relation with an intersection with the same Cartesian product (see also ${ eqres.1 $e |- R = ( S |` C ) $. $( Converting a class constant definition by restriction (like ~ df-ers or - ~~? df-parts ) into a binary relation. (Contributed by Peter Mazsa, + ~ df-parts ) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.) $) eqres $p |- ( B e. V -> ( A R B <-> ( A e. C /\ A S B ) ) ) $= ( wbr cres wcel wa breqi brres bitrid ) ABDHABECIZHBFJACJABEHKABDOGLCABEF @@ -581894,18 +583077,18 @@ relation with an intersection with the same Cartesian product (see also relation with Cartesian product of the domain and range of the class. (Contributed by Peter Mazsa, 22-Jul-2019.) $) iss2 $p |- ( A C_ _I <-> A = ( _I i^i ( dom A X. ran A ) ) ) $= - ( vx vy cid wss cdm crn cxp wceq cv cop wcel wb wal wa vex jca2 wi syl5bi - wrel cin ssel opeldm opelrn jcad anandi syl6ibr wbr df-br ideq bitr3i wex - eldm2 eleq1d biimprcd sylcom exlimdv imbi2d syl5ibcom imp adantrd ex impd - opeq2 impbid opelinxp biancomi bitr4di alrimivv relss mpi relinxp sylancl - reli eqrel mpbird inss1 sseq1 mpbiri impbii ) ADEZADAFZAGZHZUAZIZWAWFBJZC - JZKZALZWIWELZMZCNBNZWAWLBCWAWJWIDLZWGWBLZWHWCLZOZOZWKWAWJWRWAWJWNWOOZWNWP - OZOWRWAWJWSWTWAWJWNWOADWIUBZWGWHABPZCPZUCQWAWJWNWPXAWGWHAXBXCUDQUEWNWOWPU - FUGWAWNWQWJWNWGWHIZWAWQWJRZWNWGWHDUHXDWGWHDUIWGWHXCUJUKZWAXDXEWAXDOWOWJWP - WAXDWOWJRZWAWOWGWGKZALZRXDXGWOWJCULWAXICWGAXBUMWAWJXICWAWJWNXIXAWNXDWJXIX - FXDXIWJXDXHWIAWGWHWGVDUNZUOSUPUQSXDXIWJWOXJURUSUTVAVBSVCVEWKWNWQWBWCWGWHD - VFVGVHVIWAATZWETWFWMMWADTXKVNADVJVKWBWCDVLBCAWEVOVMVPWFWAWEDEDWDVQAWEDVRV - SVT $. + ( vx vy cid wss cdm crn cxp wceq cv cop wcel wb wal wa jca2 biimtrid wrel + vex wi cin ssel opeldm opelrn jcad anandi syl6ibr df-br ideq bitr3i eldm2 + wbr wex opeq2 eleq1d biimprcd sylcom exlimdv imbi2d syl5ibcom imp adantrd + ex impd impbid opelinxp biancomi bitr4di alrimivv relss mpi relinxp eqrel + reli sylancl mpbird inss1 sseq1 mpbiri impbii ) ADEZADAFZAGZHZUAZIZWAWFBJ + ZCJZKZALZWIWELZMZCNBNZWAWLBCWAWJWIDLZWGWBLZWHWCLZOZOZWKWAWJWRWAWJWNWOOZWN + WPOZOWRWAWJWSWTWAWJWNWOADWIUBZWGWHABSZCSZUCPWAWJWNWPXAWGWHAXBXCUDPUEWNWOW + PUFUGWAWNWQWJWNWGWHIZWAWQWJTZWNWGWHDULXDWGWHDUHWGWHXCUIUJZWAXDXEWAXDOWOWJ + WPWAXDWOWJTZWAWOWGWGKZALZTXDXGWOWJCUMWAXICWGAXBUKWAWJXICWAWJWNXIXAWNXDWJX + IXFXDXIWJXDXHWIAWGWHWGUNUOZUPQUQURQXDXIWJWOXJUSUTVAVBVCQVDVEWKWNWQWBWCWGW + HDVFVGVHVIWAARZWERWFWMMWADRXKVNADVJVKWBWCDVLBCAWEVMVOVPWFWAWEDEDWDVQAWEDV + RVSVT $. $} ${ @@ -581935,6 +583118,33 @@ relation with an intersection with the same Cartesian product (see also ( ccnv crn cres cdm df-rn reseq2i wrel wceq relcnv dfrel5 mpbi eqtri ) ABZA CZDNNEZDZNOPNAFGNHQNIAJNKLM $. + $( Subset of restriction, special case. (Contributed by Peter Mazsa, + 10-Apr-2023.) $) + relssinxpdmrn $p |- ( Rel R -> + ( R C_ ( S i^i ( dom R X. ran R ) ) <-> R C_ S ) ) $= + ( wrel wss cdm crn cxp wa cin relssdmrn biantrud ssin bitr2di ) ACZABDZOAAE + AFGZDZHABPIDNQOAJKABPLM $. + + $( Two ways to say that a relation is a subclass. (Contributed by Peter + Mazsa, 11-Apr-2023.) $) + cnvref4 $p |- ( Rel R -> + ( ( R i^i ( dom R X. ran R ) ) C_ ( S i^i ( dom R X. ran R ) ) <-> + R C_ S ) ) $= + ( wrel cdm crn cxp cin wceq dfrel6 biimpi dmeqd rneqd xpeq12d ineq2d sseq2d + wss wb relxp relin2 relssinxpdmrn mp2b sseq1d bitrid bitr3d ) ACZAADZAEZFZG + ZBUIDZUIEZFZGZPZUIBUHGZPABPZUEUMUOUIUEULUHBUEUJUFUKUGUEUIAUEUIAHAIJZKUEUIAU + QLMNOUNUIBPZUEUPUHCUICUNURQUFUGRAUHSUIBTUAUEUIABUQUBUCUD $. + + ${ + $d R x y $. + $( Two ways to say that a relation is a subclass of the identity relation. + (Contributed by Peter Mazsa, 26-Jun-2019.) $) + cnvref5 $p |- ( Rel R -> ( R C_ _I <-> A. x A. y ( x R y -> x = y ) ) ) $= + ( wrel cid wss cv wbr wi wal ssrel3 wb cvv ideqg elv imbi2i 2albii bitrdi + wceq ) CDCEFAGZBGZCHZTUAEHZIZBJAJUBTUASZIZBJAJABCEKUDUFABUCUEUBUCUELBTUAM + NOPQR $. + $} + ${ $d A x $. $d B x $. $d R x $. $d V x $. $d W x $. $( Two ways of saying that the coset of ` A ` and the coset of ` B ` have @@ -582066,6 +583276,24 @@ relation with an intersection with the same Cartesian product (see also APZUFTABCUEDRUFUHUIACDSUAUB $. $} + $( Uniqueness is equivalent to non-existence or unique existence. Alternate + definition of the at-most-one quantifier, in terms of the existential + quantifier and the unique existential quantifier. (Contributed by Peter + Mazsa, 19-Nov-2024.) $) + moeu2 $p |- ( E* x ph <-> ( -. E. x ph \/ E! x ph ) ) $= + ( wmo wex weu wi wn wo moeu imor bitri ) ABCABDZABEZFLGMHABILMJK $. + + $( "At most one" picks a variable value, eliminating an existential + quantifier. The proof begins with references *2.21 ( ~ pm2.21 ) and + *14.26 ( ~ eupickbi ) from [WhiteheadRussell] p. 104 and p. 183. + (Contributed by Peter Mazsa, 18-Nov-2024.) + (Proof modification is discouraged.) $) + mopickr $p |- ( ( E* x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) ) $= + ( wmo wa wex wi exancom wn weu wo moeu2 19.8a con3i pm2.21 syl a1d eupickbi + wal sp syl6bi jaoi sylbi biimtrid imp ) BCDZABECFZBAGZUGBAECFZUFUHABCHUFBCF + ZIZBCJZKUIUHGZBCLUKUMULUKUHUIUKBIUHBUJBCMNBAOPQULUIUHCSUHBACRUHCTUAUBUCUDUE + $. + $( Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018.) $) moantr $p |- ( E* x ps -> @@ -582134,6 +583362,13 @@ relation with an intersection with the same Cartesian product (see also ZBJDCBDAKPBOCLMN $. $} + $( Intersection with a converse, binary relation. (Contributed by Peter + Mazsa, 24-Mar-2024.) $) + brcnvin $p |- ( ( A e. V /\ B e. W ) -> + ( A ( R i^i `' S ) B <-> ( A R B /\ B S A ) ) ) $= + ( ccnv cin wbr wa wcel brin brcnvg anbi2d bitrid ) ABCDGZHIABCIZABPIZJAEKBF + KJZQBADIZJABCPLSRTQABEFDMNO $. + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -582315,6 +583550,76 @@ relation with an intersection with the same Cartesian product (see also DUE $. $} + ${ + $d A u v $. $d R u v $. $d V u $. + $( Double restricted quantification over the union of a set and its + singleton. (Contributed by Peter Mazsa, 22-Aug-2023.) $) + disjressuc2 $p |- ( A e. V -> + ( A. u e. ( A u. { A } ) A. v e. ( A u. { A } ) + ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) <-> + ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ + A. u e. A ( [ u ] R i^i [ A ] R ) = (/) ) ) ) $= + ( wcel cv wceq cec c0 wo wral wa eqeq1 eceq1 ineq1d eqeq1d orbi12d anbi2i + cin csn cun eqeq2 ineq2d 2ralunsn eqid biantru bitr4di eqcom bitrdi incom + orci w3a eqtrdi cbvralvw biimpi pm4.71i 3anass df-3an elneq neneqd biorfd + 3bitr2ri ralbiia ) CEFZBGZAGZHZVFDIZVGDIZTZJHZKZACCUAUBZLBVNLZVMACLBCLZVF + CHZVICDIZTZJHZKZBCLZMZVPVTBCLZMVEVOWCCVGHZVRVJTZJHZKZACLZMZWCVEVOWCWICCHZ + VRVRTZJHZKZMZMWJVMWAWHWNBACCEVQVHWEVLWGVFCVGNVQVKWFJVQVIVRVJVFCDOZPQRVGCH + ZVHVQVLVTVGCVFUCWQVKVSJWQVJVRVIVGCDOUDQRVQVQWKVTWMVFCCNVQVSWLJVQVIVRVRWPP + QRUEWIWOWCWNWIWKWMCUFULUGSUHWCVPWBWIMZMVPWBWIUMWJWBWRVPWBWIWBWIWAWHBACVHV + QWEVTWGVHVQWQWEVFVGCNVGCUIUJVHVSWFJVHVSVJVRTWFVHVIVJVRVFVGDOPVJVRUKUNQRUO + UPUQSVPWBWIURVPWBWIUSVCUJWDWBVPVTWABCVFCFZVQVTWSVFCVFCUTVAVBVDSUH $. + $} + + ${ + $d A y z $. $d B y z $. $d R y z $. $d S y z $. $d V y z $. + $d W y z $. + $( Two ways of saying that ` ( R |X. S ) ` -cosets are disjoint. + (Contributed by Peter Mazsa, 19-Jun-2020.) (Revised by Peter Mazsa, + 21-Aug-2023.) $) + disjecxrn $p |- ( ( A e. V /\ B e. W ) -> + ( ( [ A ] ( R |X. S ) i^i [ B ] ( R |X. S ) ) = (/) <-> + ( ( [ A ] R i^i [ B ] R ) = (/) \/ + ( [ A ] S i^i [ B ] S ) = (/) ) ) ) $= + ( vy vz wcel wa cec cin c0 wceq wne cv wbr wex copab bitrdi wo cxrn ecxrn + ineqan12d inopab eqtrdi an4 opabbii neeq1d opabn0 exdistrv ecinn0 anbi12d + wn bitr4d neanior necon4abid ) AEIZBFIZJZACKBCKLZMNADKBDKLZMNUAZACDUBZKZB + VDKZLZMUTVGMOZVAMOZVBMOZJZVCUNUTVHAGPZCQZBVLCQZJZGRZAHPZDQZBVQDQZJZHRZJZV + KUTVHVOVTJZHRGRZWBUTVHWCGHSZMOWDUTVGWEMUTVGVMVRJZVNVSJZJZGHSZWEUTVGWFGHSZ + WGGHSZLWIURUSVEWJVFWKGHACDEUCGHBCDFUCUDWFWGGHUEUFWHWCGHVMVRVNVSUGUHUFUIWC + GHUJTVOVTGHUKTUTVIVPVJWAGABCEFULHABDEFULUMUOVAMVBMUPTUQ $. + $} + + $( Two ways of saying that cosets are disjoint, special case of ~ disjecxrn . + (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, + 25-Aug-2023.) $) + disjecxrncnvep $p |- ( ( A e. V /\ B e. W ) -> + ( ( [ A ] ( R |X. `' _E ) i^i [ B ] ( R |X. `' _E ) ) = (/) <-> + ( ( A i^i B ) = (/) \/ ( [ A ] R i^i [ B ] R ) = (/) ) ) ) $= + ( wcel wa cep ccnv cxrn cec cin c0 wceq disjecxrn bitrdi disjeccnvep orbi1d + wo orcom bitrd ) ADFBEFGZACHIZJZKBUDKLMNZAUCKBUCKLMNZACKBCKLMNZSZABLMNZUGSU + BUEUGUFSUHABCUCDEOUGUFTPUBUFUIUGABDEQRUA $. + + ${ + $d A u v $. $d R u v $. $d V u $. + $( Double restricted quantification over the union of a set and its + singleton. (Contributed by Peter Mazsa, 22-Aug-2023.) $) + disjsuc2 $p |- ( A e. V -> + ( A. u e. ( A u. { A } ) A. v e. ( A u. { A } ) + ( u = v \/ + ( [ u ] ( R |X. `' _E ) i^i [ v ] ( R |X. `' _E ) ) = (/) ) <-> + ( A. u e. A A. v e. A + ( u = v \/ + ( [ u ] ( R |X. `' _E ) i^i [ v ] ( R |X. `' _E ) ) = (/) ) /\ + A. u e. A + ( ( u i^i A ) = (/) \/ + ( [ u ] R i^i [ A ] R ) = (/) ) ) ) ) $= + ( wcel cv wceq cep ccnv cxrn cec cin c0 wo csn cun wral wa disjressuc2 wb + cvv disjecxrncnvep el2v1 ralbidv anbi2d bitrd ) CEFZBGZAGZHUIDIJKZLZUJUKL + MNHOZACCPQZRBUNRUMACRBCRZULCUKLMNHZBCRZSUOUICMNHUIDLCDLMNHOZBCRZSABCUKETU + HUQUSUOUHUPURBCUHUPURUABUICDUBEUCUDUEUFUG $. + $} + ${ $d A u x y z $. $d B u x y z $. $d C u x y z $. $d R u x y z $. $d S u x y z $. @@ -582526,7 +583831,7 @@ relation with an intersection with the same Cartesian product (see also This concept simplifies theorems relating partition and equivalence: the left side of these theorems relate to ` R ` , the right side relate to - ` ,~ R ` (see e.g. ~~? pet ). Without the definition of ` ,~ R ` we + ` ,~ R ` (see e.g. ~ pet ). Without the definition of ` ,~ R ` we should have to relate the right side of these theorems to a composition of a converse (cf. ~ dfcoss3 ) or to the range of a range Cartesian product of classes (cf. ~ dfcoss4 ), which would make the theorems @@ -582579,6 +583884,43 @@ range of a range Cartesian product ( ~ dfcoss4 ), but neither of these ZCFAGPDFAGHBICDJAAKLCDBAMCDBAANO $. $} + ${ + $d R u x y $. + $( Class of cosets by the converse of ` R ` (Contributed by Peter Mazsa, + 17-Jun-2020.) $) + cosscnv $p |- ,~ `' R = { <. x , y >. | E. u ( x R u /\ y R u ) } $= + ( ccnv ccoss cv wbr wa wex copab df-coss wb cvv brcnvg el2v anbi12i exbii + opabbii eqtri ) DEZFCGZAGZUAHZUBBGZUAHZIZCJZABKUCUBDHZUEUBDHZIZCJZABKABCU + ALUHULABUGUKCUDUIUFUJUDUIMCAUBUCNNDOPUFUJMCBUBUENNDOPQRST $. + $} + + ${ + $d A u v x $. $d R u v x $. + $( Class of cosets by the converse of a restriction. (Contributed by Peter + Mazsa, 8-Jun-2020.) $) + coss1cnvres $p |- ,~ `' ( R |` A ) = + { <. u , v >. | + ( ( u e. A /\ v e. A ) /\ E. x ( u R x /\ v R x ) ) } $= + ( cres ccnv ccoss cv wbr wex copab wcel df-coss cvv br1cnvres elv anbi12i + wa wb an4 bitr4i exbii 19.42v bitri opabbii eqtri ) EDFGZHAIZCIZUHJZUIBIZ + UHJZSZAKZCBLUJDMZULDMZSZUJUIEJZULUIEJZSZAKSZCBLCBAUHNUOVBCBUOURVASZAKVBUN + VCAUNUPUSSZUQUTSZSVCUKVDUMVEUKVDTADUIUJEOPQUMVETADUIULEOPQRUPUQUSUTUAUBUC + URVAAUDUEUFUG $. + $} + + ${ + $d A u v x $. + $( Special case of ~ coss1cnvres . (Contributed by Peter Mazsa, + 8-Jun-2020.) $) + coss2cnvepres $p |- ,~ `' ( `' _E |` A ) = + { <. u , v >. | + ( ( u e. A /\ v e. A ) /\ E. x ( x e. u /\ x e. v ) ) } $= + ( cep ccnv cres ccoss cv wcel wa wbr wex copab coss1cnvres wb cvv brcnvep + elv anbi12i exbii anbi2i opabbii eqtri ) EFZDGFHCIZDJBIZDJKZUFAIZUELZUGUI + UELZKZAMZKZCBNUHUIUFJZUIUGJZKZAMZKZCBNABCDUEOUNUSCBUMURUHULUQAUJUOUKUPUJU + OPCUFUIQRSUKUPPBUGUIQRSTUAUBUCUD $. + $} + $( If ` A ` is a set then the class of cosets by ` A ` is a set. (Contributed by Peter Mazsa, 4-Jan-2019.) $) cossex $p |- ( A e. V -> ,~ A e. _V ) $= @@ -582787,6 +584129,64 @@ range of a range Cartesian product ( ~ dfcoss4 ), but neither of these CDEBJKLAMZCELSDELIABNCSEOZHDTHIABNABCDEFGPABCDEEFGQR $. $} + $( Binary relation on a restriction to a singleton. (Contributed by Peter + Mazsa, 11-Jun-2024.) $) + brressn $p |- ( ( B e. V /\ C e. W ) -> + ( B ( R |` { A } ) C <-> ( B = A /\ B R C ) ) ) $= + ( wcel wa csn cres wbr wceq wb brres adantl elsng adantr anbi1d bitrd ) BEG + ZCFGZHZBCDAIZJKZBUCGZBCDKZHZBALZUFHUAUDUGMTUCBCDFNOUBUEUHUFTUEUHMUABAEPQRS + $. + + ${ + $d A a u $. $d R a u $. + $( A class ' R ' restricted to the singleton of the class ' A ' is the + ordered pair class abstraction of the class ' A ' and the sets in + relation ' R ' to ' A ' (and not in relation to the singleton ' { A + } ' ). (Contributed by Peter Mazsa, 16-Jun-2024.) $) + ressn2 $p |- ( R |` { A } ) = { <. a , u >. | ( a = A /\ A R u ) } $= + ( csn cres cv wcel wbr copab wceq dfres2 velsn anbi1i eqbrb bitri opabbii + wa eqtri ) CBEZFDGZTHZUAAGZCIZRZDAJUABKZBUCCIRZDAJDATCLUEUGDAUEUFUDRUGUBU + FUDDBMNUABUCCOPQS $. + $} + + ${ + $d A x $. $d V x $. + $( Any class ' R ' restricted to the singleton of the set ' A ' (see + ~ ressn2 ) is reflexive, see also ~ refrelressn . (Contributed by Peter + Mazsa, 12-Jun-2024.) $) + refressn $p |- ( A e. V -> + A. x e. ( dom ( R |` { A } ) i^i ran ( R |` { A } ) ) + x ( R |` { A } ) x ) $= + ( wcel cv csn cres wbr cdm crn cin wceq wa wi elin wb cvv adantr sylbi + eldmressnALTV elv simplbi a1i elrnressn elvd biimpd adantld eqcomd breq1d + biimtrid mpbidi jcad brressn el2v syl6ibr ralrimiv ) BDEZAFZUSCBGHZIZAUTJ + ZUTKZLZURUSVDEZUSBMZUSUSCIZNZVAURVEVFVGVEVFOURVEUSVBEZUSVCEZNZVFUSVBVCPZV + IVFVJVIVFBCJEZVIVFVMNQABUSCRUAUBUCZSTUDVEBUSCIZVGURVEVKURVOVLURVJVOVIURVJ + VOURVJVOQABUSCDRUEUFUGUHUKVEVKVOVGQZVLVIVPVJVIBUSUSCVIUSBVNUIUJSTULUMVAVH + QAABUSUSCRRUNUOUPUQ $. + $} + + $( Every class ' R ' restricted to the singleton of the class ' A ' (see + ~ ressn2 ) is antisymmetric. (Contributed by Peter Mazsa, + 11-Jun-2024.) $) + antisymressn $p |- + A. x A. y ( ( x ( R |` { A } ) y /\ y ( R |` { A } ) x ) -> x = y ) $= + ( cv csn cres wbr wa wceq wi wb cvv brressn el2v simplbi eqtr3 syl2an gen2 + ) AEZBEZDCFGZHZUATUBHZITUAJZKABUCTCJZUACJZUEUDUCUFTUADHZUCUFUHILABCTUADMMNO + PUDUGUATDHZUDUGUIILBACUATDMMNOPTUACQRS $. + + $( Any class ' R ' restricted to the singleton of the class ' A ' (see + ~ ressn2 ) is transitive, see also ~ trrelressn . (Contributed by Peter + Mazsa, 16-Jun-2024.) $) + trressn $p |- + A. x A. y A. z ( ( x ( R |` { A } ) y /\ y ( R |` { A } ) z ) -> + x ( R |` { A } ) z ) $= + ( cv csn cres wbr wa wi wal wceq eqbrb anbi12i 3imtr4i wb cvv brressn el2v + an3 gen2 ax-gen ) AFZBFZEDGHZIZUECFZUFIZJZUDUHUFIZKZCLBLAULBCUDDMZUDUEEIJZU + EDMZUEUHEIJZJZUMUDUHEIJZUJUKUMDUEEIZJZUODUHEIZJZJUMVAJUQURUMUSUOVAUAUNUTUPV + BUDDUEENUEDUHENOUDDUHENPUGUNUIUPUGUNQABDUDUEERRSTUIUPQBCDUEUHERRSTOUKURQACD + UDUHERRSTPUBUC $. + ${ $d A x $. $d B x $. $d R x $. $d V x $. $d W x $. $( ` A ` and ` B ` are cosets by relation ` R ` : a binary relation. @@ -582817,9 +584217,9 @@ range of a range Cartesian product ( ~ dfcoss4 ), but neither of these ${ $d A u $. $d B u $. $d C u $. $d D u $. $d E u $. $d R u $. $d S u $. $d V u $. $d W u $. $d X u $. $d Y u $. - $( ` <. B , C >. ` and ` <. D , E >. ` are cosets by an intersection with a - restriction: a binary relation. (Contributed by Peter Mazsa, - 8-Jun-2021.) $) + $( ` <. B , C >. ` and ` <. D , E >. ` are cosets by an range Cartesian + product with a restriction: a binary relation. (Contributed by Peter + Mazsa, 8-Jun-2021.) $) br1cossxrnres $p |- ( ( ( B e. V /\ C e. W ) /\ ( D e. X /\ E e. Y ) ) -> ( <. B , C >. ,~ ( R |X. ( S |` A ) ) <. D , E >. <-> @@ -583589,6 +584989,16 @@ when we switch from (general) sets to relations in ~ dfrefrels2 , UCCQTUA $. $} + ${ + $d R x $. + $( Alternate definition of the reflexive relation predicate. (Contributed + by Peter Mazsa, 12-Dec-2023.) $) + dfrefrel5 $p |- ( RefRel R <-> + ( A. x e. ( dom R i^i ran R ) x R x /\ Rel R ) ) $= + ( wrefrel cid cdm crn cxp cin wss wrel cv wbr wral dfrefrel2 ref5 bianbi + ) BCDBEZBFZGHBIBJAKZSBLAQRHMBNAQRBOP $. + $} + ${ $d R r $. $( Element of the class of reflexive relations. (Contributed by Peter @@ -583638,6 +585048,15 @@ when we switch from (general) sets to relations in ~ dfrefrels2 , ( ccoss wrefrel cid cdm crn cxp cin wss wrel wa refrelcoss2 dfrefrel2 mpbir ) ABZCDOEOFGHOIOJKALOMN $. + ${ + $d A x $. $d R x $. $d V x $. + $( Any class ' R ' restricted to the singleton of the set ' A ' (see + ~ ressn2 ) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.) $) + refrelressn $p |- ( A e. V -> RefRel ( R |` { A } ) ) $= + ( vx wcel cv csn cres wbr cdm crn cin wral wrel refressn relres dfrefrel5 + wrefrel sylanblrc ) ACEDFZTBAGZHZIDUBJUBKLMUBNUBRDABCOBUAPDUBQS $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -583721,6 +585140,22 @@ and disjoints ( ~ df-disjs , ~ df-disjALTV ). FLCQUEUGUFABUBUCCSTUA $. $} + $( Alternate definition of the converse reflexive relation predicate. + (Contributed by Peter Mazsa, 25-May-2024.) $) + dfcnvrefrel4 $p |- ( CnvRefRel R <-> ( R C_ _I /\ Rel R ) ) $= + ( wcnvrefrel cdm crn cxp cin cid wss wrel df-cnvrefrel cnvref4 bianim ) ABA + ACADEZFGMFHAIAGHAJAGKL $. + + ${ + $d R x y $. + $( Alternate definition of the converse reflexive relation predicate. + (Contributed by Peter Mazsa, 25-May-2024.) $) + dfcnvrefrel5 $p |- ( CnvRefRel R <-> + ( A. x A. y ( x R y -> x = y ) /\ Rel R ) ) $= + ( wcnvrefrel cid wss wrel cv wbr wceq wi wal dfcnvrefrel4 cnvref5 bianim + ) CDCEFCGAHZBHZCIPQJKBLALCMABCNO $. + $} + ${ $d R r $. $( Element of the class of converse reflexive relations. (Contributed by @@ -584237,6 +585672,15 @@ class of symmetric relations as well (like the elements of the class of ( wceq ccom wrel wa wtrrel coideq id sseq12d releq anbi12d dftrrel2 3bitr4g wss ) ABCZAADZAOZAEZFBBDZBOZBEZFAGBGPRUASUBPQTABABHPIJABKLAMBMN $. + ${ + $d A x y z $. $d R x y z $. + $( Any class ' R ' restricted to the singleton of the class ' A ' (see + ~ ressn2 ) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024.) $) + trrelressn $p |- TrRel ( R |` { A } ) $= + ( vx vy vz csn cres wtrrel cv wbr wa wi wal wrel relres dftrrel3 mpbir2an + trressn ) BAFZGZHCIZDIZTJUBEIZTJKUAUCTJLEMDMCMTNCDEABRBSOCDETPQ $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -584553,10 +585997,10 @@ class of symmetric relations as well (like the elements of the class of ( [ A ] R = [ B ] R \/ ( [ A ] R i^i [ B ] R ) = (/) ) ) $= ( vx weqvrel cec cin c0 wceq wn wcel wbr adantl cvv wb ecexr syl elecALTV sylancl mpbid cv wex wa simpl elinel1 vex elinel2 eqvreltr4d eqvrelthi ex - neq0 exlimdv syl5bi orrd orcomd ) CEZACFZBCFZGZHIZUQURIZUPUTVAUTJDUAZUSKZ - DUBUPVADUSUKUPVCVADUPVCVAUPVCUCZABCUPVCUDZVDAVBBCVEVDVBUQKZAVBCLZVCVFUPVB - UQURUEMZVDANKZVBNKZVFVGOVDVFVIVHVBACPQDUFZAVBCNNRSTVDVBURKZBVBCLZVCVLUPVB - UQURUGMZVDBNKZVJVLVMOVDVLVOVNVBBCPQVKBVBCNNRSTUHUIUJULUMUNUO $. + neq0 exlimdv biimtrid orrd orcomd ) CEZACFZBCFZGZHIZUQURIZUPUTVAUTJDUAZUS + KZDUBUPVADUSUKUPVCVADUPVCVAUPVCUCZABCUPVCUDZVDAVBBCVEVDVBUQKZAVBCLZVCVFUP + VBUQURUEMZVDANKZVBNKZVFVGOVDVFVIVHVBACPQDUFZAVBCNNRSTVDVBURKZBVBCLZVCVLUP + VBUQURUGMZVDBNKZVJVLVMOVDVLVOVNVBBCPQVKBVBCNNRSTUHUIUJULUMUNUO $. $} ${ @@ -584902,13 +586346,19 @@ reflexive relation (see ~ dfrefrel3 ) in equivalence relation. n0eldmqseq $p |- ( ( dom R /. R ) = A -> -. (/) e. A ) $= ( cdm cqs wceq c0 wcel n0eldmqs eleq2 mtbii ) BCBDZAEFKGFAGBHKAFIJ $. + $( Implication of that the empty set is not an element of a class. + (Contributed by Peter Mazsa, 30-Dec-2024.) $) + n0elim $p |- ( -. (/) e. A -> + ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A ) $= + ( c0 wcel cep ccnv cres cdm cqs wceq n0el2 biimpi qseq1d qsresid qsid eqtri + wn eqtrdi ) BACPZDEZAFZGZTHATHZARUAATRUAAIAJKLUBASHAASMANOQ $. + $( Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021.) $) n0el3 $p |- ( -. (/) e. A <-> ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A ) $= - ( c0 wcel cep ccnv cres cdm cqs wceq n0el2 biimpi qseq1d qsresid qsid eqtri - wn eqtrdi n0eldmqseq impbii ) BACPZDEZAFZGZUBHZAITUDAUBHZATUCAUBTUCAIAJKLUE - AUAHAAUAMANOQAUBRS $. + ( c0 wcel wn cep ccnv cres cdm cqs wceq n0elim n0eldmqseq impbii ) BACDEFAG + ZHNIAJAKANLM $. $( The domain quotient binary relation of the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the @@ -584951,7 +586401,7 @@ reflexive relation (see ~ dfrefrel3 ) in equivalence relation. ( wcel wrel cdm cqs wceq crn cuni wi wa unieq wb unidmqseq imp syl5ib ex ) BCDZBEZBFBGZAHZBIAJZHZKUBUAJUCHZSTLUDUAAMSTUEUDNUCBCOPQR $. - $( Lemma for ~ erim2 . (Contributed by Peter Mazsa, 29-Dec-2021.) $) + $( Lemma for ~ erimeq2 . (Contributed by Peter Mazsa, 29-Dec-2021.) $) dmqseqim2 $p |- ( R e. V -> ( Rel R -> ( ( dom R /. R ) = A -> ( B e. ran R <-> B e. U. A ) ) ) ) $= ( wcel wrel cdm cqs wceq crn cuni wb dmqseqim eleq2 syl8 ) CDECFCGCHAICJZAK @@ -585033,14 +586483,13 @@ reflexive relation (see ~ dfrefrel3 ) in equivalence relation. component ( ` dom R = A ` ) of the present ~ df-er . While we acknowledge the need of a domain component, the present ~ df-er definition does not utilize the results revealed by the new theorems in the - Partition-Equivalence Theorem part below (like ~~? pets and ~~? pet ). - From those theorems follows that the natural domain of equivalence - relations is + Partition-Equivalence Theorem part below (like ~ pets and ~ pet ). From + those theorems follows that the natural domain of equivalence relations is not ` R Domain A ` (i.e. ` dom R = A ` see ~ brdomaing ), but ` R DomainQss A ` (i.e. ` ( dom R /. R ) = A ` , see ~ brdmqss ), see - ~ erim vs. ~ prter3 . + ~ erimeq vs. ~ prter3 . While I'm sure we need both equivalence relation ~ df-eqvrels and equivalence relation on domain quotient ~ df-ers , I'm not sure whether we @@ -585059,21 +586508,21 @@ reflexive relation (see ~ dfrefrel3 ) in equivalence relation. ~ brerser . (Contributed by Peter Mazsa, 12-Aug-2021.) $) df-erALTV $a |- ( R ErALTV A <-> ( EqvRel R /\ R DomainQs A ) ) $. - $( Define the class of membership equivalence relations on their domain + $( Define the class of comember equivalence relations on their domain quotients. (Contributed by Peter Mazsa, 28-Nov-2022.) (Revised by Peter Mazsa, 24-Jul-2023.) $) - df-members $a |- MembErs = { a | ,~ ( `' _E |` a ) Ers a } $. + df-comembers $a |- CoMembErs = { a | ,~ ( `' _E |` a ) Ers a } $. - $( Define the membership equivalence relation on the class ` A ` (or, the - restricted elementhood equivalence relation on its domain quotient - ` A ` .) Alternate definitions are ~ dfmember2 and ~ dfmember3 . + $( Define the comember equivalence relation on the class ` A ` (or, the + restricted coelement equivalence relation on its domain quotient ` A ` .) + Alternate definitions are ~ dfcomember2 and ~ dfcomember3 . Later on, in an application of set theory I make a distinction between the default elementhood concept and a special membership concept: membership equivalence relation will be an integral part of that membership concept. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 28-Nov-2022.) $) - df-member $a |- ( MembEr A <-> ,~ ( `' _E |` A ) ErALTV A ) $. + df-comember $a |- ( CoMembEr A <-> ,~ ( `' _E |` A ) ErALTV A ) $. $( Binary equivalence relation with natural domain, see the comment of ~ df-ers . (Contributed by Peter Mazsa, 23-Jul-2021.) $) @@ -585110,28 +586559,30 @@ reflexive relation (see ~ dfrefrel3 ) in equivalence relation. ( wceq werALTV wb erALTVeq1 syl ) ACDFBCGBDGHEBCDIJ $. $} - $( Alternate definition of the membership equivalence relation. (Contributed + $( Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.) $) - dfmember $p |- ( MembEr A <-> ~ A ErALTV A ) $= - ( wmember cep ccnv cres werALTV ccoels df-member df-coels erALTVeq1i bitr4i - ccoss ) ABACDAELZFAAGZFAHANMAIJK $. + dfcomember $p |- ( CoMembEr A <-> ~ A ErALTV A ) $= + ( wcomember ccnv cres werALTV ccoels df-comember df-coels erALTVeq1i bitr4i + cep ccoss ) ABAKCADLZEAAFZEAGANMAHIJ $. - $( Alternate definition of the membership equivalence relation. (Contributed + $( Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.) $) - dfmember2 $p |- ( MembEr A <-> ( EqvRel ~ A /\ ( dom ~ A /. ~ A ) = A ) ) $= - ( wmember ccoels werALTV weqvrel cdm cqs wceq wa dfmember dferALTV2 bitri ) - ABAACZDMEMFMGAHIAJAMKL $. + dfcomember2 $p |- ( CoMembEr A <-> + ( EqvRel ~ A /\ ( dom ~ A /. ~ A ) = A ) ) $= + ( wcomember ccoels werALTV weqvrel cdm cqs wceq dfcomember dferALTV2 bitri + wa ) ABAACZDMEMFMGAHLAIAMJK $. - $( Alternate definition of the membership equivalence relation. (Contributed + $( Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.) $) - dfmember3 $p |- ( MembEr A <-> ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) ) $= - ( wmember ccoels weqvrel cdm cqs wceq wa wcoeleqvrel dfmember2 dfcoeleqvrel - cuni bicomi dmqscoelseq anbi12i bitri ) ABACZDZQEQFAGZHAIZALQFAGZHAJRTSUATR - AKMANOP $. - - $( Two ways to express membership equivalence relation on its domain - quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter - Mazsa, 17-Jul-2023.) $) + dfcomember3 $p |- ( CoMembEr A <-> + ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) ) $= + ( wcomember ccoels weqvrel cdm wceq wa wcoeleqvrel dfcomember2 dfcoeleqvrel + cqs cuni bicomi dmqscoelseq anbi12i bitri ) ABACZDZQEQKAFZGAHZALQKAFZGAIRTS + UATRAJMANOP $. + + $( Two ways to express comember equivalence relation on its domain quotient. + (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, + 17-Jul-2023.) $) eqvreldmqs $p |- ( ( EqvRel ,~ ( `' _E |` A ) /\ ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) <-> ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) ) $= @@ -585139,6 +586590,14 @@ reflexive relation (see ~ dfrefrel3 ) in equivalence relation. cuni bicomi dmqs1cosscnvepreseq anbi12i ) BCADEZFZAGZQHQIAJAMAKIAJSRALNAOP $. + $( Two ways to express comember equivalence relation on its domain quotient. + (Contributed by Peter Mazsa, 30-Dec-2024.) $) + eqvreldmqs2 $p |- ( ( EqvRel ,~ ( `' _E |` A ) /\ + ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) <-> + ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) ) $= + ( cep ccnv cres ccoss weqvrel ccoels cdm cqs wceq df-coels eqvreleqi bicomi + cuni dmqs1cosscnvepreseq anbi12i ) BCADEZFZAGZFZQHQIAJANSIAJTRSQAKLMAOP $. + $( Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when ` A ` and ` R ` are sets. (Contributed by Peter Mazsa, 25-Aug-2021.) $) @@ -585152,10 +586611,10 @@ reflexive relation (see ~ dfrefrel3 ) in equivalence relation. $d A u x y $. $d R u x y $. $d V x y $. $( Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is ~ prter3 in a - more convenient form , see also ~ erim ). (Contributed by Rodolfo + more convenient form , see also ~ erimeq ). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.) $) - erim2 $p |- ( R e. V -> + erimeq2 $p |- ( R e. V -> ( ( EqvRel R /\ ( dom R /. R ) = A ) -> ~ A = R ) ) $= ( vx vy vu wcel wceq wa wrel ad2antrl cv wbr wrex wb simpll eleq2d bitrdi cvv el2v weqvrel cdm cqs ccoels relcoels a1i eqvrelrel brcoels cec simprl @@ -585174,11 +586633,11 @@ coelements on the domain is equal to the relation (this is ~ prter3 in a $( Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is the most - convenient form of ~ prter3 and ~ erim2 ). (Contributed by Peter Mazsa, + convenient form of ~ prter3 and ~ erimeq2 ). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.) $) - erim $p |- ( R e. V -> ( R ErALTV A -> ~ A = R ) ) $= - ( werALTV weqvrel cdm cqs wceq wa wcel ccoels dferALTV2 erim2 syl5bi ) ABDB - EBFBGAHIBCJAKBHABLABCMN $. + erimeq $p |- ( R e. V -> ( R ErALTV A -> ~ A = R ) ) $= + ( werALTV weqvrel cdm cqs wceq wa wcel ccoels dferALTV2 erimeq2 biimtrid ) + ABDBEBFBGAHIBCJAKBHABLABCMN $. $( @@ -585436,22 +586895,22 @@ instead of simply using converse functions (cf. ~ dfdisjALTV ). ~ dfdisjALTV5 . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-disjALTV $a |- ( Disj R <-> ( CnvRefRel ,~ `' R /\ Rel R ) ) $. - $( Define the disjoint elementhood relations class, i.e., the disjoint - elements class. The element of the disjoint elements class and the - disjoint elementhood predicate are the same, that is + $( Define the disjoint element relations class, i.e., the disjoint elements + class. The element of the disjoint elements class and the disjoint + elementhood predicate are the same, that is ` ( A e. ElDisjs <-> ElDisj A ) ` when ` A ` is a set, see ~ eleldisjseldisj . (Contributed by Peter Mazsa, 28-Nov-2022.) $) df-eldisjs $a |- ElDisjs = { a | ( `' _E |` a ) e. Disjs } $. - $( Define the disjoint elementhood relation predicate, i.e., the disjoint + $( Define the disjoint element relation predicate, i.e., the disjoint elementhood predicate. Read: the elements of ` A ` are disjoint. The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is ` ( A e. ElDisjs <-> ElDisj A ) ` when ` A ` is a set, see ~ eleldisjseldisj . As of now, disjoint elementhood is defined as "partition" in set.mm : - compare ~ df-prt with ~ dfeldisj5 . See also the comments of ~~? - dfmembpart2 and of ~~? df-parts . (Contributed by Peter Mazsa, + compare ~ df-prt with ~ dfeldisj5 . See also the comments of + ~ dfmembpart2 and of ~ df-parts . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-eldisj $a |- ( ElDisj A <-> Disj ( `' _E |` A ) ) $. @@ -585719,8 +587178,8 @@ instead of simply using converse functions (cf. ~ dfdisjALTV ). ( wceq wdisjALTV wb disjeq syl ) ABCEBFCFGDBCHI $. $} - $( Lemma for the equality theorem for partition ~~? parteq1 . (Contributed - by Peter Mazsa, 5-Oct-2021.) $) + $( Lemma for the equality theorem for partition ~ parteq1 . (Contributed by + Peter Mazsa, 5-Oct-2021.) $) disjdmqseqeq1 $p |- ( R = S -> ( ( Disj R /\ ( dom R /. R ) = A ) <-> ( Disj S /\ ( dom S /. S ) = A ) ) ) $= @@ -585771,6 +587230,27 @@ instead of simply using converse functions (cf. ~ dfdisjALTV ). ( wceq weldisj wb eldisjeq syl ) ABCEBFCFGDBCHI $. $} + ${ + $d A u v x $. $d R u v x $. + $( Disjoint restriction. (Contributed by Peter Mazsa, 25-Aug-2023.) $) + disjres $p |- ( Rel R -> + ( Disj ( R |` A ) <-> + A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) ) ) $= + ( vx wrel cres wdisjALTV cv wbr wrmo wal wceq cec cin c0 wral wmo relres + wo dfdisjALTV4 mpbiran2 wcel wa wb cvv brres elv mobii df-rmo albii bitri + bitr4i id inecmo bitr4id ) DFDCGZHZBIZEIZDJZBCKZELZUSAIZMZUSDNVDDNOPMTACQ + BCQURUSUTUQJZBRZELZVCURVHUQFDCSEBUQUAUBVGVBEVGUSCUCVAUDZBRVBVFVIBVFVIUEEC + USUTDUFUGUHUIVABCUJUMUKULBAECUSVDDVEUNUOUP $. + $} + + $( Two forms of disjoint elements when the empty set is not an element of the + class. (Contributed by Peter Mazsa, 31-Dec-2024.) $) + eldisjn0elb $p |- ( ( ElDisj A /\ -. (/) e. A ) <-> + ( Disj ( `' _E |` A ) /\ + ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A ) ) $= + ( weldisj cep ccnv cres wdisjALTV c0 wcel wn cdm cqs wceq df-eldisj anbi12i + n0el3 ) ABCDAEZFGAHIPJPKALAMAON $. + $( Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) $) @@ -585778,6 +587258,18 @@ instead of simply using converse functions (cf. ~ dfdisjALTV ). ( cxrn wdisjALTV ccnv ccoss cid wss cin wrel xrnrel dfdisjALTV2 1cosscnvxrn mpbiran2 sseq1i bitri ) ABCZDZQEFZGHZAEFBEFIZGHRTQJABKQLNSUAGABMOP $. + ${ + $d A u v $. $d R u v $. $d S u v $. + $( Disjoint range Cartesian product. (Contributed by Peter Mazsa, + 25-Aug-2023.) $) + disjxrnres5 $p |- ( Disj ( R |X. ( S |` A ) ) <-> + A. u e. A A. v e. A + ( u = v \/ ( [ u ] ( R |X. S ) i^i [ v ] ( R |X. S ) ) = (/) ) ) $= + ( cres cxrn wdisjALTV cv wceq cec cin c0 wral xrnres2 disjeqi wrel xrnrel + wo wb disjres ax-mp bitr3i ) DECFGZHDEGZCFZHZBIZAIZJUHUEKUIUEKLMJSACNBCNZ + UFUDCDEOPUEQUGUJTDERABCUEUAUBUC $. + $} + $( Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) disjorimxrn $p |- ( ( Disj R \/ Disj S ) -> Disj ( R |X. S ) ) $= @@ -585842,6 +587334,885 @@ instead of simply using converse functions (cf. ~ dfdisjALTV ). disjALTVxrnidres $p |- Disj ( R |X. ( _I |` A ) ) $= ( cid wdisjALTV cres cxrn disjALTVid disjimxrnres ax-mp ) CDBCAEFDGABCHI $. + ${ + $d A u v $. $d R u v $. $d V u $. + $( Disjoint range Cartesian product, special case. (Contributed by Peter + Mazsa, 25-Aug-2023.) $) + disjsuc $p |- ( A e. V -> + ( Disj ( R |X. ( `' _E |` suc A ) ) <-> + ( Disj ( R |X. ( `' _E |` A ) ) /\ + A. u e. A + ( ( u i^i A ) = (/) \/ + ( [ u ] R i^i [ A ] R ) = (/) ) ) ) ) $= + ( vv wcel cv wceq cep ccnv cxrn cec c0 wo wral cres wdisjALTV disjxrnres5 + cin wa csn cun csuc disjsuc2 df-suc reseq2i xrneq2i disjeqi bitri 3bitr4g + anbi1i ) BDFAGZEGZHULCIJZKZLUMUOLSMHNZEBBUAUBZOAUQOZUPEBOABOZULBSMHULCLBC + LSMHNABOZTCUNBUCZPZKZQZCUNBPKQZUTTEABCDUDVDCUNUQPZKZQURVCVGVBVFCVAUQUNBUE + UFUGUHEAUQCUNRUIVEUSUTEABCUNRUKUJ $. + $} + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Antisymmetry +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $( Define the antisymmetric relation predicate. (Read: ` R ` is an + antisymmetric relation.) (Contributed by Peter Mazsa, 24-Jun-2024.) $) + df-antisymrel $a |- ( AntisymRel R <-> + ( CnvRefRel ( R i^i `' R ) /\ Rel R ) ) $. + + $( Alternate definition of the antisymmetric relation predicate. + (Contributed by Peter Mazsa, 24-Jun-2024.) $) + dfantisymrel4 $p |- ( AntisymRel R <-> + ( ( R i^i `' R ) C_ _I /\ Rel R ) ) $= + ( wantisymrel ccnv cin wcnvrefrel cid wss df-antisymrel relcnv relin2 ax-mp + wrel dfcnvrefrel4 mpbiran2 bianbi ) ABAACZDZEZALQFGZAHRSQLZPLTAIAPJKQMNO $. + + ${ + $d R x y $. + $( Alternate definition of the antisymmetric relation predicate. + (Contributed by Peter Mazsa, 24-Jun-2024.) $) + dfantisymrel5 $p |- ( AntisymRel R <-> + ( A. x A. y ( ( x R y /\ y R x ) -> x = y ) /\ Rel R ) ) $= + ( wantisymrel ccnv cin wcnvrefrel wrel cv wa wceq wi df-antisymrel relcnv + wbr wal relin2 ax-mp dfcnvrefrel5 cvv mpbiran2 brcnvin el2v imbi1i 2albii + wb bitri bianbi ) CDCCEZFZGZCHAIZBIZCOUMULCOJZULUMKZLZBPAPZCMUKULUMUJOZUO + LZBPAPZUQUKUTUJHZUIHVACNCUIQRABUJSUAUSUPABURUNUOURUNUFABULUMCCTTUBUCUDUEU + GUH $. + $} + + ${ + $d A x y $. $d R x y $. + $( (Contributed by Peter Mazsa, 25-Jun-2024.) $) + antisymrelres $p |- ( AntisymRel ( R |` A ) <-> + A. x e. A A. y e. A ( ( x R y /\ y R x ) -> x = y ) ) $= + ( cres wantisymrel cv wbr wa wceq wi wal wcel wral wrel relres wb cvv elv + brres dfantisymrel5 mpbiran2 anbi12i bitri imbi1i 2albii r2alan 3bitri + an4 ) DCEZFZAGZBGZUJHZUMULUJHZIZULUMJZKZBLALZULCMZUMCMZIULUMDHZUMULDHZIZI + ZUQKZBLALVDUQKBCNACNUKUSUJODCPABUJUAUBURVFABUPVEUQUPUTVBIZVAVCIZIVEUNVGUO + VHUNVGQBCULUMDRTSUOVHQACUMULDRTSUCUTVBVAVCUIUDUEUFVDUQABCCUGUH $. + $} + + ${ + $d A x y $. $d R x y $. + $( (Contributed by Peter Mazsa, 29-Jun-2024.) $) + antisymrelressn $p |- AntisymRel ( R |` { A } ) $= + ( vx vy csn cres wantisymrel cv wbr wa wceq wi antisymressn dfantisymrel5 + wal wrel relres mpbir2an ) BAEZFZGCHZDHZTIUBUATIJUAUBKLDOCOTPCDABMBSQCDTN + R $. + $} + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Partitions: disjoints on domain quotients +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $( Define the class of all partitions, cf. the comment of ~ df-disjs . + Partitions are disjoints on domain quotients (or: domain quotients + restricted to disjoints). + + This is a more general meaning of partition than we we are familiar with: + the conventional meaning of partition (e.g. partition ` A ` of ` X ` , + [Halmos] p. 28: "A partition of ` X ` is a disjoint collection ` A ` of + non-empty subsets of ` X ` whose union is ` X ` ", or Definition 35, + [Suppes] p. 83., cf. https://oeis.org/A000110 ) is what we call membership + partition here, cf. ~ dfmembpart2 . + + The binary partitions relation and the partition predicate are the same, + that is, ` ( R Parts A <-> R Part A ) ` if ` A ` and ` R ` are sets, cf. + ~ brpartspart . (Contributed by Peter Mazsa, 26-Jun-2021.) $) + df-parts $a |- Parts = ( DomainQss |` Disjs ) $. + + $( Define the partition predicate (read: ` A ` is a partition by ` R ` ). + Alternative definition is ~ dfpart2 . The binary partition and the + partition predicate are the same if ` A ` and ` R ` are sets, cf. + ~ brpartspart . (Contributed by Peter Mazsa, 12-Aug-2021.) $) + df-part $a |- ( R Part A <-> ( Disj R /\ R DomainQs A ) ) $. + + $( Define the class of member partition relations on their domain quotients. + (Contributed by Peter Mazsa, 26-Jun-2021.) $) + df-membparts $a |- MembParts = { a | ( `' _E |` a ) Parts a } $. + + $( Define the member partition predicate, or the disjoint restricted element + relation on its domain quotient predicate. (Read: ` A ` is a member + partition.) A alternative definition is ~ dfmembpart2 . + + Member partition is the conventional meaning of partition (see the notes + of ~ df-parts and ~ dfmembpart2 ), we generalize the concept in ~ df-parts + and ~ df-part . + + Member partition and comember equivalence are the same by ~ mpet . + (Contributed by Peter Mazsa, 26-Jun-2021.) $) + df-membpart $a |- ( MembPart A <-> ( `' _E |` A ) Part A ) $. + + $( Alternate definition of the partition predicate. (Contributed by Peter + Mazsa, 5-Sep-2021.) $) + dfpart2 $p |- ( R Part A <-> ( Disj R /\ ( dom R /. R ) = A ) ) $= + ( wpart wdisjALTV wdmqs wa cdm cqs wceq df-part df-dmqs anbi2i bitri ) ABCB + DZABEZFNBGBHAIZFABJOPNABKLM $. + + $( Alternate definition of the conventional membership case of partition. + Partition ` A ` of ` X ` , [Halmos] p. 28: "A partition of ` X ` is a + disjoint collection ` A ` of non-empty subsets of ` X ` whose union is + ` X ` ", or Definition 35, [Suppes] p. 83., cf. https://oeis.org/A000110 . + (Contributed by Peter Mazsa, 14-Aug-2021.) $) + dfmembpart2 $p |- ( MembPart A <-> ( ElDisj A /\ -. (/) e. A ) ) $= + ( wmembpart cep ccnv cres wpart wdisjALTV wdmqs wa weldisj wcel df-membpart + c0 wn df-part df-eldisj bicomi cnvepresdmqs anbi12i 3bitri ) ABACDAEZFUAGZA + UAHZIAJZMAKNZIALAUAOUBUDUCUEUDUBAPQARST $. + + $( Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.) $) + brparts $p |- ( A e. V -> + ( R Parts A <-> ( R e. Disjs /\ R DomainQss A ) ) ) $= + ( cdisjs cparts cdmqss df-parts eqres ) BADEFCGH $. + + $( Binary partitions relation. (Contributed by Peter Mazsa, 30-Dec-2021.) $) + brparts2 $p |- ( ( A e. V /\ R e. W ) -> + ( R Parts A <-> ( R e. Disjs /\ ( dom R /. R ) = A ) ) ) $= + ( wcel wa cparts wbr cdisjs cdmqss cdm cqs wb brparts adantr brdmqss anbi2d + wceq bitrd ) ACEZBDEZFZBAGHZBIEZBAJHZFZUDBKBLARZFTUCUFMUAABCNOUBUEUGUDABCDP + QS $. + + $( Binary partition and the partition predicate are the same if ` A ` and + ` R ` are sets. (Contributed by Peter Mazsa, 5-Sep-2021.) $) + brpartspart $p |- ( ( A e. V /\ R e. W ) -> ( R Parts A <-> R Part A ) ) $= + ( wcel wa cdisjs cdmqss wbr wdisjALTV wdmqs cparts wpart eldisjsdisj adantl + wb brdmqssqs anbi12d brparts adantr df-part a1i 3bitr4d ) ACEZBDEZFZBGEZBAH + IZFZBJZABKZFZBALIZABMZUFUGUJUHUKUEUGUJPUDBDNOABCDQRUDUMUIPUEABCSTUNULPUFABU + AUBUC $. + + $( Equality theorem for partition. (Contributed by Peter Mazsa, + 5-Oct-2021.) $) + parteq1 $p |- ( R = S -> ( R Part A <-> S Part A ) ) $= + ( wceq wdisjALTV cdm cqs wa wpart disjdmqseqeq1 dfpart2 3bitr4g ) BCDBEBFBG + ADHCECFCGADHABIACIABCJABKACKL $. + + $( Equality theorem for partition. (Contributed by Peter Mazsa, + 25-Jul-2024.) $) + parteq2 $p |- ( A = B -> ( R Part A <-> R Part B ) ) $= + ( wceq wdisjALTV cdm cqs wa wpart eqeq2 anbi2d dfpart2 3bitr4g ) ABDZCEZCFC + GZADZHOPBDZHACIBCINQROABPJKACLBCLM $. + + $( Equality theorem for partition. (Contributed by Peter Mazsa, + 25-Jul-2024.) $) + parteq12 $p |- ( ( R = S /\ A = B ) -> ( R Part A <-> S Part B ) ) $= + ( wceq wpart parteq1 parteq2 sylan9bb ) CDEACFADFABEBDFACDGABDHI $. + + ${ + parteq1i.1 $e |- R = S $. + $( Equality theorem for partition, inference version. (Contributed by + Peter Mazsa, 5-Oct-2021.) $) + parteq1i $p |- ( R Part A <-> S Part A ) $= + ( wceq wpart wb parteq1 ax-mp ) BCEABFACFGDABCHI $. + $} + + ${ + parteq1d.1 $e |- ( ph -> R = S ) $. + $( Equality theorem for partition, deduction version. (Contributed by + Peter Mazsa, 5-Oct-2021.) $) + parteq1d $p |- ( ph -> ( R Part A <-> S Part A ) ) $= + ( wceq wpart wb parteq1 syl ) ACDFBCGBDGHEBCDIJ $. + $} + + $( Property of the partition. (Contributed by Peter Mazsa, 24-Jul-2024.) $) + partsuc2 $p |- ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part + ( ( A u. { A } ) \ { A } ) <-> + ( R |` A ) Part A ) $= + ( csn cun cres cdif wceq wpart wb ressucdifsn2 sucdifsn2 parteq12 mp2an ) B + AACZDZEBNEFZBAEZGONFZAGRPHAQHIABJAKRAPQLM $. + + $( Property of the partition. (Contributed by Peter Mazsa, 20-Sep-2024.) $) + partsuc $p |- + ( ( ( R |` suc A ) \ ( R |` { A } ) ) Part ( suc A \ { A } ) <-> + ( R |` A ) Part A ) $= + ( csuc cres csn cdif wceq wpart wb ressucdifsn sucdifsn parteq12 mp2an ) BA + CZDBAEZDFZBADZGNOFZAGRPHAQHIABJAKRAPQLM $. + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Partition-Equivalence Theorems +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + ${ + $d R u x y z $. + $( The "Divide et Aequivalere" Theorem: every disjoint relation generates + equivalent cosets by the relation: generalization of the former + ~ prter1 , cf. ~ eldisjim . (Contributed by Peter Mazsa, 3-May-2019.) + (Revised by Peter Mazsa, 17-Sep-2021.) $) + disjim $p |- ( Disj R -> EqvRel ,~ R ) $= + ( vx vy vz vu wdisjALTV cv ccoss wbr wal weqvrel wrel dfdisjALTV4 simplbi + wa wi wmo trcoss syl eqvrelcoss3 sylibr ) AFZBGZCGZAHZIUDDGZUEIOUCUFUEIPD + JCJBJZUEKUBEGUDAIEQCJZUGUBUHALCEAMNBCDEARSBCDATUA $. + $} + + ${ + disjimi.1 $e |- Disj R $. + $( Every disjoint relation generates equivalent cosets by the relation, + inference version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) + disjimi $p |- EqvRel ,~ R $= + ( wdisjALTV ccoss weqvrel disjim ax-mp ) ACADEBAFG $. + $} + + ${ + detlem.1 $e |- Disj R $. + $( If a relation is disjoint, then it is equivalent to the equivalent + cosets of the relation, inference version. (Contributed by Peter Mazsa, + 30-Sep-2021.) $) + detlem $p |- ( Disj R <-> EqvRel ,~ R ) $= + ( wdisjALTV ccoss weqvrel disjim a1i impbii ) ACZADEZAFIJBGH $. + $} + + $( If the elements of ` A ` are disjoint, then it has equivalent coelements + (former ~ prter1 ). Special case of ~ disjim . (Contributed by Rodolfo + Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015) (Revised + by Peter Mazsa, 8-Feb-2018.) ( Revised by Peter Mazsa, 23-Sep-2021.) $) + eldisjim $p |- ( ElDisj A -> CoElEqvRel A ) $= + ( cep ccnv wdisjALTV ccoss weqvrel weldisj wcoeleqvrel disjim df-coeleqvrel + cres df-eldisj 3imtr4i ) BCAKZDNEFAGAHNIALAJM $. + + $( Alternate form of ~ eldisjim . (Contributed by Peter Mazsa, + 30-Dec-2024.) $) + eldisjim2 $p |- ( ElDisj A -> EqvRel ~ A ) $= + ( cep ccnv wdisjALTV ccoss weqvrel weldisj ccoels disjim df-eldisj df-coels + cres eqvreleqi 3imtr4i ) BCALZDOEZFAGAHZFOIAJQPAKMN $. + + $( The null class is an equivalence relation. (Contributed by Peter Mazsa, + 31-Dec-2021.) $) + eqvrel0 $p |- EqvRel (/) $= + ( c0 ccoss weqvrel disjALTV0 disjimi coss0 eqvreleqi mpbi ) ABZCACADEIAFGH + $. + + $( The cosets by the null class are in equivalence relation if and only if + the null class is disjoint (which it is, see ~ disjALTV0 ). (Contributed + by Peter Mazsa, 31-Dec-2021.) $) + det0 $p |- ( Disj (/) <-> EqvRel ,~ (/) ) $= + ( c0 disjALTV0 detlem ) ABC $. + + $( The cosets by the null class are in equivalence relation. (Contributed by + Peter Mazsa, 31-Dec-2024.) $) + eqvrelcoss0 $p |- EqvRel ,~ (/) $= + ( c0 disjALTV0 disjimi ) ABC $. + + $( The identity relation is an equivalence relation. (Contributed by Peter + Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 31-Dec-2021.) $) + eqvrelid $p |- EqvRel _I $= + ( cid ccoss weqvrel disjALTVid disjimi cossid eqvreleqi mpbi ) ABZCACADEIAF + GH $. + + $( The cosets by a restricted identity relation is an equivalence relation. + (Contributed by Peter Mazsa, 31-Dec-2021.) $) + eqvrel1cossidres $p |- EqvRel ,~ ( _I |` A ) $= + ( cid cres disjALTVidres disjimi ) BACADE $. + + $( The cosets by an intersection with a restricted identity relation are in + equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) + eqvrel1cossinidres $p |- EqvRel ,~ ( R i^i ( _I |` A ) ) $= + ( cid cres cin disjALTVinidres disjimi ) BCADEABFG $. + + $( The cosets by a range Cartesian product with a restricted identity + relation are in equivalence relation. (Contributed by Peter Mazsa, + 31-Dec-2021.) $) + eqvrel1cossxrnidres $p |- EqvRel ,~ ( R |X. ( _I |` A ) ) $= + ( cid cres cxrn disjALTVxrnidres disjimi ) BCADEABFG $. + + $( The cosets by the identity relation are in equivalence relation if and + only if the identity relation is disjoint. (Contributed by Peter Mazsa, + 31-Dec-2021.) $) + detid $p |- ( Disj _I <-> EqvRel ,~ _I ) $= + ( cid disjALTVid detlem ) ABC $. + + $( The cosets by the identity class are in equivalence relation. + (Contributed by Peter Mazsa, 31-Dec-2024.) $) + eqvrelcossid $p |- EqvRel ,~ _I $= + ( cid disjALTVid disjimi ) ABC $. + + $( The cosets by the restricted identity relation are in equivalence relation + if and only if the restricted identity relation is disjoint. (Contributed + by Peter Mazsa, 31-Dec-2021.) $) + detidres $p |- ( Disj ( _I |` A ) <-> EqvRel ,~ ( _I |` A ) ) $= + ( cid cres disjALTVidres detlem ) BACADE $. + + $( The cosets by the intersection with the restricted identity relation are + in equivalence relation if and only if the intersection with the + restricted identity relation is disjoint. (Contributed by Peter Mazsa, + 31-Dec-2021.) $) + detinidres $p |- ( Disj ( R i^i ( _I |` A ) ) <-> + EqvRel ,~ ( R i^i ( _I |` A ) ) ) $= + ( cid cres cin disjALTVinidres detlem ) BCADEABFG $. + + $( The cosets by the range Cartesian product with the restricted identity + relation are in equivalence relation if and only if the range Cartesian + product with the restricted identity relation is disjoint. (Contributed + by Peter Mazsa, 31-Dec-2021.) $) + detxrnidres $p |- ( Disj ( R |X. ( _I |` A ) ) <-> + EqvRel ,~ ( R |X. ( _I |` A ) ) ) $= + ( cid cres cxrn disjALTVxrnidres detlem ) BCADEABFG $. + + ${ + $d R x y $. + $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjlem17 , + (general version of the former ~ prtlem14 ). (Contributed by Peter + Mazsa, 10-Sep-2021.) $) + disjlem14 $p |- ( Disj R -> + ( ( x e. dom R /\ y e. dom R ) -> + ( ( A e. [ x ] R /\ A e. [ y ] R ) -> + [ x ] R = [ y ] R ) ) ) $= + ( wdisjALTV cv cdm wcel wa wceq cec cin c0 wo wi wral dfdisjALTV5 simplbi + wrel rsp2 syl eceq1 a1d elin nel02 pm2.21d syl5bir jaoi syl6 ) DEZAFZDGZH + BFZULHIZUKUMJZUKDKZUMDKZLZMJZNZCUPHCUQHIZUPUQJZOZUJUTBULPAULPZUNUTOUJVDDS + BADQRUTABULULTUAUOVCUSUOVBVAUKUMDUBUCVACURHZUSVBCUPUQUDUSVEVBURCUEUFUGUHU + I $. + $} + + ${ + $d A y $. $d B y $. $d R x y $. + $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjlem18 , + (general version of the former ~ prtlem17 ). (Contributed by Peter + Mazsa, 10-Sep-2021.) $) + disjlem17 $p |- ( Disj R -> + ( ( x e. dom R /\ A e. [ x ] R ) -> + ( E. y e. dom R ( A e. [ y ] R /\ B e. [ y ] R ) -> + B e. [ x ] R ) ) ) $= + ( wdisjALTV cv cdm wcel cec wa wrex wi df-rex an32 wceq disjlem14 biimprd + wex eleq2 syl8 exp4a impd syl5bir expd imp5a imp4b exlimdv biimtrid ex ) + EFZAGZEHZIZCULEJZIZKZCBGZEJZIZDUSIZKZBUMLZDUOIZMVCURUMIZVBKZBSUKUQKZVDVBB + UMNVGVFVDBUKUQVEVBVDUKUQVEUTVAVDUKUQVEUTVAVDMZMZUQVEKUNVEKZUPKUKVIUNVEUPO + UKVJUPVIUKVJUPUTVHUKVJUPUTKUOUSPZVHABCEQVKVDVAUOUSDTRUAUBUCUDUEUFUGUHUIUJ + $. + $} + + ${ + $d A x y $. $d B x y $. $d R x y $. $d V x y $. $d W x y $. + $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjlem19 , + (general version of the former ~ prtlem18 ). (Contributed by Peter + Mazsa, 16-Sep-2021.) $) + disjlem18 $p |- ( ( A e. V /\ B e. W ) -> + ( Disj R -> + ( ( x e. dom R /\ A e. [ x ] R ) -> + ( B e. [ x ] R <-> A ,~ R B ) ) ) ) $= + ( vy wcel wa wdisjALTV cv cec wb wi wrex adantl relbrcoss impel sylibrd + ex cdm ccoss wbr rspe expr wrel disjrel adantr disjlem17 imbi1d impbidd ) + BEHCFHIZDJZAKZDUAZHZBUNDLZHZIZCUQHZBCDUBUCZMNULUMIZUSUTVAVBUSUTVANVBUSIUT + URUTIZAUOOZVAUSUTVDNVBUPURUTVDVCAUOUDUEPVBVAVDMZUSULDUFZVEUMABCDEFQDUGZRU + HSTVBUSBGKDLZHCVHHIGUOOZUTNZVAUTNUMUSVJNULAGBCDUIPVBVAVIUTULVFVAVIMUMGBCD + EFQVGRUJSUKT $. + $} + + ${ + $d A x z $. $d R x z $. $d V x z $. + $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjdmqs , + (general version of the former ~ prtlem19 ). (Contributed by Peter + Mazsa, 16-Sep-2021.) $) + disjlem19 $p |- ( A e. V -> + ( Disj R -> + ( ( x e. dom R /\ A e. [ x ] R ) -> + [ x ] R = [ A ] ,~ R ) ) ) $= + ( vz wcel wdisjALTV cv cdm cec wa ccoss wceq wbr wb wi cvv disjlem18 elvd + imp31 elecALTV ad2antrr bitr4d eqrdv exp31 ) BDFZCGZAHZCIFBUHCJZFKZUIBCLZ + JZMUFUGKUJKZEUIULUMEHZUIFZBUNUKNZUNULFZUFUGUJUOUPOZUFUGUJURPPEABUNCDQRSTU + FUQUPOZUGUJUFUSEBUNUKDQUASUBUCUDUE $. + $} + + ${ + $d R u v x $. + $( Lemma for ~ disjdmqseq via ~ disjdmqs . (Contributed by Peter Mazsa, + 16-Sep-2021.) $) + disjdmqsss $p |- ( Disj R -> ( dom R /. R ) C_ ( dom ,~ R /. ,~ R ) ) $= + ( vv vx vu wdisjALTV cdm cqs cv wcel cec wceq wrex wa wb cvv elv syl wral + wi reximi ccoss wrel disjrel releldmqs disjlem19 ralrimivv 2r19.29 sylbid + ex eqtr ancoms syl6 releldmqscoss sylibrd ssrdv ) AEZBAFZAGZAUAZFUSGZUPBH + ZURIZVACHZUSJZKZCDHZAJZLZDUQLZVAUTIZUPVBVGVDKZVAVGKZMZCVGLZDUQLZVIUPVBVLC + VGLDUQLZVOUPAUBZVBVPNZAUCZVQVRSBCDVAAOUDPQUPVKCVGRDUQRZVPVOSUPVKDCUQVGUPV + FUQIVCVGIMVKSSCDVCAOUEPUFVTVPVOVKVLDCUQVGUGUIQUHVNVHDUQVMVECVGVLVKVEVAVGV + DUJUKTTULUPVQVJVINZVSVQWASBCDVAAOUMPQUNUO $. + $} + + ${ + $d R u v x $. + $( Lemma for ~ disjdmqseq via ~ disjdmqs . (Contributed by Peter Mazsa, + 16-Sep-2021.) $) + disjdmqscossss $p |- ( Disj R -> + ( dom ,~ R /. ,~ R ) C_ ( dom R /. R ) ) $= + ( vv vu vx cv cdm cqs wcel cab cec wceq wrex cvv elv syl wral reximi syl6 + wa wi wdisjALTV wrel wb disjrel releldmqscoss disjlem19 ralrimivv 2r19.29 + ccoss ex sylbid eqtr3 wex df-rex 19.41v bitri simprbi eqcom rexbii syl6ib + ss2abdv abid1 df-qs 3sstr4g ) AUAZBEZAUIZFVGGZHZBIVFCEZAJZKZCAFZLZBIVHVMA + GVEVIVNBVEVIVKVFKZCVMLZVNVEVIVODVKLZCVMLZVPVEVIVKDEZVGJZKZVFVTKZSZDVKLZCV + MLZVRVEVIWBDVKLCVMLZWEVEAUBZVIWFUCZAUDWGWHTBDCVFAMUENOVEWADVKPCVMPZWFWETV + EWACDVMVKVEVJVMHVSVKHZSWATTDCVSAMUFNUGWIWFWEWAWBCDVMVKUHUJOUKWDVQCVMWCVOD + VKVKVFVTULQQRVQVOCVMVQWJDUMZVOVQWJVOSDUMWKVOSVODVKUNWJVODUOUPUQQRVOVLCVMV + KVFURUSUTVABVHVBCBVMAVCVD $. + $} + + $( If a relation is disjoint, its domain quotient is equal to the domain + quotient of the cosets by it. Lemma for ~ partim2 and ~ petlem via + ~ disjdmqseq . (Contributed by Peter Mazsa, 16-Sep-2021.) $) + disjdmqs $p |- ( Disj R -> ( dom R /. R ) = ( dom ,~ R /. ,~ R ) ) $= + ( wdisjALTV cdm cqs ccoss disjdmqsss disjdmqscossss eqssd ) ABACADAEZCIDAFA + GH $. + + $( If a relation is disjoint, its domain quotient is equal to a class if and + only if the domain quotient of the cosets by it is equal to the class. + General version of ~ eldisjn0el (which is the closest theorem to the + former ~ prter2 ). Lemma for ~ partim2 and ~ petlem . (Contributed by + Peter Mazsa, 16-Sep-2021.) $) + disjdmqseq $p |- ( Disj R -> + ( ( dom R /. R ) = A <-> ( dom ,~ R /. ,~ R ) = A ) ) $= + ( wdisjALTV cdm cqs ccoss disjdmqs eqeq1d ) BCBDBEBFZDIEABGH $. + + $( Special case of ~ disjdmqseq (perhaps this is the closest theorem to the + former ~ prter2 ). (Contributed by Peter Mazsa, 26-Sep-2021.) $) + eldisjn0el $p |- ( ElDisj A -> + ( -. (/) e. A <-> ( U. A /. ~ A ) = A ) ) $= + ( cep ccnv cres wdisjALTV cdm cqs wceq ccoss wb weldisj c0 wcel cuni ccoels + wn disjdmqseq df-eldisj n0el3 dmqs1cosscnvepreseq bicomi bibi12i 3imtr4i ) + BCADZEUDFUDGAHZUDIZFUFGAHZJAKLAMPZANAOGAHZJAUDQARUHUEUIUGASUGUIATUAUBUC $. + + $( Disjoint relation on its natural domain implies an equivalence relation on + the cosets of the relation, on its natural domain, cf. ~ partim . Lemma + for ~ petlem . (Contributed by Peter Mazsa, 17-Sep-2021.) $) + partim2 $p |- ( ( Disj R /\ ( dom R /. R ) = A ) -> + ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) $= + ( wdisjALTV cdm cqs wceq ccoss weqvrel disjim adantr disjdmqseq biimpa jca + wa ) BCZBDBEAFZNBGZHZQDQEAFZORPBIJOPSABKLM $. + + $( Partition implies equivalence relation by the cosets of the relation on + its natural domain, cf. ~ partim2 . (Contributed by Peter Mazsa, + 17-Sep-2021.) $) + partim $p |- ( R Part A -> ,~ R ErALTV A ) $= + ( wdisjALTV cdm cqs wceq wa ccoss weqvrel werALTV partim2 dfpart2 dferALTV2 + wpart 3imtr4i ) BCBDBEAFGBHZIPDPEAFGABNAPJABKABLAPMO $. + + $( Partition implies that the class of coelements on the natural domain is + equal to the class of cosets of the relation, cf. ~ erimeq . (Contributed + by Peter Mazsa, 25-Dec-2024.) $) + partimeq $p |- ( R e. V -> ( R Part A -> ~ A = ,~ R ) ) $= + ( wcel ccoss cvv wpart werALTV ccoels wceq cossex partim erimeq syl2im ) BC + DBEZFDABGAOHAIOJBCKABLAOFMN $. + + ${ + $d A u $. $d B u $. $d V u $. + $( Special case of ~ disjlem19 (together with ~ membpartlem19 , this is + former ~ prtlem19 ). (Contributed by Peter Mazsa, 21-Oct-2021.) $) + eldisjlem19 $p |- ( B e. V -> + ( ElDisj A -> + ( ( u e. dom ( `' _E |` A ) /\ B e. u ) -> + u = [ B ] ~ A ) ) ) $= + ( wcel weldisj cv cep ccnv cres cdm wa ccoels cec wceq wi ccoss wdisjALTV + df-eldisj disjlem19 biimtrid expdimp wb eccnvepres3 eleq2d eqeq1d imbi12d + imp adantl mpbid df-coels eceq2i eqeq2i syl6ibr expimpd ex ) CDEZBFZAGZHI + BJZKEZCUSEZLUSCBMZNZOZPUQURLZVAVBVEVFVALZVBUSCUTQZNZOZVEVGCUSUTNZEZVKVIOZ + PZVBVJPZVFVAVLVMUQURVAVLLVMPZURUTRUQVPBSACUTDTUAUHUBVAVNVOUCVFVAVLVBVMVJV + AVKUSCBUSUDZUEVAVKUSVIVQUFUGUIUJVDVIUSVCVHCBUKULUMUNUOUP $. + $} + + ${ + $d A u $. $d B u $. $d V u $. + $( Together with ~ disjlem19 , this is former ~ prtlem19 . (Contributed by + Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) + (Revised by Peter Mazsa, 21-Oct-2021.) $) + membpartlem19 $p |- ( B e. V -> + ( MembPart A -> + ( ( u e. A /\ B e. u ) -> + u = [ B ] ~ A ) ) ) $= + ( wcel wmembpart cv wa ccoels cec wceq wi weldisj c0 dfmembpart2 cep ccnv + wn cres cdm n0el2 biimpi ad2antll eleq2d eldisjlem19 adantrd expd sylbird + imp sylan2b impd ex ) CDEZBFZAGZBEZCUOEZHUOCBIJKZLUMUNHUPUQURUNUMBMZNBERZ + HZUPUQURLZLBOUMVAHZUPUOPQBSTZEZVBVCVDBUOUTVDBKZUMUSUTVFBUAUBUCUDVCVEUQURU + MVAVEUQHURLZUMUSVGUTABCDUEUFUIUGUHUJUKUL $. + $} + + ${ + petlem.1 $e |- ( ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) -> + Disj R ) $. + $( If you can prove that the equivalence of cosets on their natural domain + implies disjointness (e.g. ~ eqvrelqseqdisj5 ), or converse function + (cf. ~ dfdisjALTV ), then disjointness, and equivalence of cosets, both + on their natural domain, are equivalent. Lemma for the Partition + Equivalence Theorem ~ pet2 . (Contributed by Peter Mazsa, + 18-Sep-2021.) $) + petlem $p |- ( ( Disj R /\ ( dom R /. R ) = A ) <-> + ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) $= + ( wdisjALTV cdm wceq wa ccoss weqvrel partim2 disjdmqseq pm5.32i sylanbrc + cqs simpr impbii ) BDZBEBNAFZGZBHZIZTETNAFZGZABJUCQUBSCUAUBOQRUBABKLMP $. + $} + + ${ + petlemi.1 $e |- Disj R $. + $( If you can prove disjointness (e.g. ~ disjALTV0 , ~ disjALTVid , + ~ disjALTVidres , ~ disjALTVxrnidres , search for theorems containing + the ' |- Disj ' string), or the same with converse function (cf. + ~ dfdisjALTV ), then disjointness, and equivalence of cosets, both on + their natural domain, are equivalent. (Contributed by Peter Mazsa, + 18-Sep-2021.) $) + petlemi $p |- ( ( Disj R /\ ( dom R /. R ) = A ) <-> + ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) $= + ( wdisjALTV ccoss weqvrel cdm cqs wceq wa a1i petlem ) ABBDBEZFMGMHAIJCKL + $. + $} + + $( Class ` A ` is a partition by the null class if and only if the cosets by + the null class are in equivalence relation on it. (Contributed by Peter + Mazsa, 31-Dec-2021.) $) + pet02 $p |- ( ( Disj (/) /\ ( dom (/) /. (/) ) = A ) <-> + ( EqvRel ,~ (/) /\ ( dom ,~ (/) /. ,~ (/) ) = A ) ) $= + ( c0 disjALTV0 petlemi ) ABCD $. + + $( Class ` A ` is a partition by the null class if and only if the cosets by + the null class are in equivalence relation on it. (Contributed by Peter + Mazsa, 31-Dec-2021.) $) + pet0 $p |- ( (/) Part A <-> ,~ (/) ErALTV A ) $= + ( c0 wdisjALTV cdm wceq ccoss weqvrel wpart werALTV pet02 dfpart2 dferALTV2 + cqs wa 3bitr4i ) BCBDBMAENBFZGPDPMAENABHAPIAJABKAPLO $. + + $( Class ` A ` is a partition by the identity class if and only if the cosets + by the identity class are in equivalence relation on it. (Contributed by + Peter Mazsa, 31-Dec-2021.) $) + petid2 $p |- ( ( Disj _I /\ ( dom _I /. _I ) = A ) <-> + ( EqvRel ,~ _I /\ ( dom ,~ _I /. ,~ _I ) = A ) ) $= + ( cid disjALTVid petlemi ) ABCD $. + + $( A class is a partition by the identity class if and only if the cosets by + the identity class are in equivalence relation on it. (Contributed by + Peter Mazsa, 31-Dec-2021.) $) + petid $p |- ( _I Part A <-> ,~ _I ErALTV A ) $= + ( cid wdisjALTV cdm cqs wceq ccoss weqvrel werALTV petid2 dfpart2 dferALTV2 + wa wpart 3bitr4i ) BCBDBEAFMBGZHPDPEAFMABNAPIAJABKAPLO $. + + $( Class ` A ` is a partition by the identity class restricted to it if and + only if the cosets by the restricted identity class are in equivalence + relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) + petidres2 $p |- ( ( Disj ( _I |` A ) /\ + ( dom ( _I |` A ) /. ( _I |` A ) ) = A ) <-> + ( EqvRel ,~ ( _I |` A ) /\ + ( dom ,~ ( _I |` A ) /. ,~ ( _I |` A ) ) = A ) ) $= + ( cid cres disjALTVidres petlemi ) ABACADE $. + + $( A class is a partition by identity class restricted to it if and only if + the cosets by the restricted identity class are in equivalence relation on + it, cf. ~ eqvrel1cossidres . (Contributed by Peter Mazsa, + 31-Dec-2021.) $) + petidres $p |- ( ( _I |` A ) Part A <-> ,~ ( _I |` A ) ErALTV A ) $= + ( cid cres wdisjALTV cdm wceq ccoss weqvrel wpart werALTV petidres2 dfpart2 + cqs wa dferALTV2 3bitr4i ) BACZDQEQMAFNQGZHRERMAFNAQIARJAKAQLAROP $. + + $( Class ` A ` is a partition by an intersection with the identity class + restricted to it if and only if the cosets by the intersection are in + equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) + petinidres2 $p |- ( ( Disj ( R i^i ( _I |` A ) ) /\ + ( dom ( R i^i ( _I |` A ) ) /. + ( R i^i ( _I |` A ) ) ) = A ) <-> + ( EqvRel ,~ ( R i^i ( _I |` A ) ) /\ + ( dom ,~ ( R i^i ( _I |` A ) ) /. + ,~ ( R i^i ( _I |` A ) ) ) = A ) ) $= + ( cid cres cin disjALTVinidres petlemi ) ABCADEABFG $. + + $( A class is a partition by an intersection with the identity class + restricted to it if and only if the cosets by the intersection are in + equivalence relation on it. Cf. ~ br1cossinidres , ~ disjALTVinidres and + ~ eqvrel1cossinidres . (Contributed by Peter Mazsa, 31-Dec-2021.) $) + petinidres $p |- ( ( R i^i ( _I |` A ) ) Part A <-> + ,~ ( R i^i ( _I |` A ) ) ErALTV A ) $= + ( cid cres cin wdisjALTV cdm cqs wa ccoss weqvrel wpart werALTV petinidres2 + wceq dfpart2 dferALTV2 3bitr4i ) BCADEZFSGSHAOISJZKTGTHAOIASLATMABNASPATQR + $. + + $( Class ` A ` is a partition by a range Cartesian product with the identity + class restricted to it if and only if the cosets by the range Cartesian + product are in equivalence relation on it. (Contributed by Peter Mazsa, + 31-Dec-2021.) $) + petxrnidres2 $p |- ( ( Disj ( R |X. ( _I |` A ) ) /\ + ( dom ( R |X. ( _I |` A ) ) /. + ( R |X. ( _I |` A ) ) ) = A ) <-> + ( EqvRel ,~ ( R |X. ( _I |` A ) ) /\ + ( dom ,~ ( R |X. ( _I |` A ) ) /. + ,~ ( R |X. ( _I |` A ) ) ) = A ) ) $= + ( cid cres cxrn disjALTVxrnidres petlemi ) ABCADEABFG $. + + $( A class is a partition by a range Cartesian product with the identity + class restricted to it if and only if the cosets by the range Cartesian + product are in equivalence relation on it. Cf. ~ br1cossxrnidres , + ~ disjALTVxrnidres and ~ eqvrel1cossxrnidres . (Contributed by Peter + Mazsa, 31-Dec-2021.) $) + petxrnidres $p |- ( ( R |X. ( _I |` A ) ) Part A <-> + ,~ ( R |X. ( _I |` A ) ) ErALTV A ) $= + ( cid cres cxrn wdisjALTV cdm wceq ccoss weqvrel wpart werALTV petxrnidres2 + cqs wa dfpart2 dferALTV2 3bitr4i ) BCADEZFSGSNAHOSIZJTGTNAHOASKATLABMASPATQ + R $. + + ${ + $d A y $. $d R x y $. + $( The elements of the quotient set of an equivalence relation are disjoint + (cf. ~ eqvreldisj2 , ~ eqvreldisj3 ). (Contributed by Mario Carneiro, + 10-Dec-2016.) (Revised by Peter Mazsa, 3-Dec-2024.) $) + eqvreldisj1 $p |- ( EqvRel R -> + A. x e. ( A /. R ) A. y e. ( A /. R ) + ( x = y \/ ( x i^i y ) = (/) ) ) $= + ( weqvrel cv wceq cin c0 wo cqs simpl simprl simprr qsdisjALTV ralrimivva + wcel wa ) DEZAFZBFZGTUAHIGJABCDKZUBSTUBQZUAUBQZRZRCTUADSUELSUCUDMSUCUDNOP + $. + $} + + ${ + $d A x y $. $d R x y $. + $( The elements of the quotient set of an equivalence relation are disjoint + (cf. ~ eqvreldisj3 ). (Contributed by Mario Carneiro, 10-Dec-2016.) + (Revised by Peter Mazsa, 19-Sep-2021.) $) + eqvreldisj2 $p |- ( EqvRel R -> ElDisj ( A /. R ) ) $= + ( vx vy weqvrel cv wceq cin cqs wral weldisj eqvreldisj1 dfeldisj5 sylibr + c0 wo ) BECFZDFZGQRHOGPDABIZJCSJSKCDABLDCSMN $. + $} + + $( The elements of the quotient set of an equivalence relation are disjoint + (cf. ~ qsdisj2 ). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised + by Peter Mazsa, 20-Jun-2019.) (Revised by Peter Mazsa, 19-Sep-2021.) $) + eqvreldisj3 $p |- ( EqvRel R -> Disj ( `' _E |` ( A /. R ) ) ) $= + ( weqvrel cqs weldisj cep ccnv cres wdisjALTV eqvreldisj2 df-eldisj sylib ) + BCABDZEFGMHIABJMKL $. + + $( Intersection with the converse epsilon relation restricted to the quotient + set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, + 30-May-2020.) (Revised by Peter Mazsa, 31-Dec-2021.) $) + eqvreldisj4 $p |- ( EqvRel R -> Disj ( S i^i ( `' _E |` ( B /. R ) ) ) ) $= + ( weqvrel cep ccnv cqs cres wdisjALTV cin eqvreldisj3 disjimin syl ) BDEFAB + GHZICNJIABKCNLM $. + + $( Range Cartesian product with converse epsilon relation restricted to the + quotient set of an equivalence relation is disjoint. (Contributed by + Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) + eqvreldisj5 $p |- ( EqvRel R -> Disj ( S |X. ( `' _E |` ( B /. R ) ) ) ) $= + ( weqvrel cep ccnv cqs cres wdisjALTV cxrn eqvreldisj3 disjimxrn syl ) BDEF + ABGHZICNJIABKCNLM $. + + $( Implication of ~ eqvreldisj2 , lemma for The Main Theorem of Equivalences + ~ mainer . (Contributed by Peter Mazsa, 23-Sep-2021.) $) + eqvrelqseqdisj2 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> ElDisj A ) $= + ( weqvrel cqs wceq wa weldisj eqvreldisj2 adantr wb eldisjeq adantl mpbid ) + CDZBCEZAFZGPHZAHZORQBCIJQRSKOPALMN $. + + $( Implication of ~ eqvrelqseqdisj2 and ~ n0eldmqseq , see comment of + ~ fences . (Contributed by Peter Mazsa, 30-Dec-2024.) $) + fences3 $p |- ( ( EqvRel R /\ ( dom R /. R ) = A ) -> + ( ElDisj A /\ -. (/) e. A ) ) $= + ( weqvrel cdm cqs wceq wa weldisj c0 wcel eqvrelqseqdisj2 n0eldmqseq adantl + wn jca ) BCZBDZBEAFZGAHIAJNZAQBKRSPABLMO $. + + $( Implication of ~ eqvreldisj3 , lemma for the Member Partition Equivalence + Theorem ~ mpet3 . (Contributed by Peter Mazsa, 27-Oct-2020.) (Revised by + Peter Mazsa, 24-Sep-2021.) $) + eqvrelqseqdisj3 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> + Disj ( `' _E |` A ) ) $= + ( weqvrel cqs wceq wa cep ccnv cres wdisjALTV eqvreldisj3 adantr wb disjeqd + reseq2 adantl mpbid ) CDZBCEZAFZGHIZTJZKZUBAJZKZSUDUABCLMUAUDUFNSUAUCUETAUB + POQR $. + + $( Lemma for ~ petincnvepres2 . (Contributed by Peter Mazsa, + 31-Dec-2021.) $) + eqvrelqseqdisj4 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> + Disj ( S i^i ( `' _E |` A ) ) ) $= + ( weqvrel cqs wceq cep ccnv cres wdisjALTV cin eqvrelqseqdisj3 disjimin syl + wa ) CEBCFAGPHIAJZKDQLKABCMDQNO $. + + $( Lemma for the Partition-Equivalence Theorem ~ pet2 . (Contributed by + Peter Mazsa, 15-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) + eqvrelqseqdisj5 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> + Disj ( S |X. ( `' _E |` A ) ) ) $= + ( weqvrel cqs wceq wa cep ccnv cres wdisjALTV eqvrelqseqdisj3 disjimxrn syl + cxrn ) CEBCFAGHIJAKZLDQPLABCMDQNO $. + + $( The Main Theorem of Equivalences: every equivalence relation implies + equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.) $) + mainer $p |- ( R ErALTV A -> CoMembEr A ) $= + ( weqvrel cdm cqs wceq wa wcoeleqvrel cuni ccoels werALTV wcomember weldisj + eqvrelqseqdisj2 eldisjim syl c0 wcel wn n0eldmqseq adantl wb eldisjn0el jca + mpbid dferALTV2 dfcomember3 3imtr4i ) BCZBDZBEAFZGZAHZAIAJEAFZGABKALULUMUNU + LAMZUMAUJBNZAOPULQARSZUNUKUQUIABTUAULUOUQUNUBUPAUCPUEUDABUFAUGUH $. + + $( Partition with general ` R ` (in addition to the member partition cf. + ~ mpet and ~ mpet2 ) implies equivalent comembers. (Contributed by Peter + Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.) $) + partimcomember $p |- ( R Part A -> CoMembEr A ) $= + ( wpart ccoss werALTV wcomember partim mainer syl ) ABCABDZEAFABGAJHI $. + + $( Member Partition-Equivalence Theorem. Together with ~ mpet ~ mpet2 , + mostly in its conventional ~ cpet and ~ cpet2 form, this is what we used + to think of as the partition equivalence theorem (but cf. ~ pet2 with + general ` R ` ). (Contributed by Peter Mazsa, 4-May-2018.) (Revised by + Peter Mazsa, 26-Sep-2021.) $) + mpet3 $p |- ( ( ElDisj A /\ -. (/) e. A ) <-> + ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) ) $= + ( weldisj c0 wcel wn cep ccnv cres wdisjALTV cdm cqs wceq ccoss wcoeleqvrel + wa weqvrel cuni ccoels eldisjn0elb eqvrelqseqdisj3 petlem eqvreldmqs 3bitri + ) ABCADEOFGAHZIUDJUDKALOUDMZPUEJZUEKALOANAQARKALOASAUDAUFUETUAAUBUC $. + + $( The conventional form of the Member Partition-Equivalence Theorem. In the + conventional case there is no (general) disjoint and no (general) + partition concept: mathematicians have called disjoint or partition what + we call element disjoint or member partition, see also ~ cpet . Together + with ~ cpet , ~ mpet ~ mpet2 , this is what we used to think of as the + partition equivalence theorem (but cf. ~ pet2 with general ` R ` ). + (Contributed by Peter Mazsa, 30-Dec-2024.) $) + cpet2 $p |- ( ( ElDisj A /\ -. (/) e. A ) <-> + ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) ) $= + ( weldisj c0 wcel wn wa cep ccnv cres wdisjALTV cdm cqs wceq weqvrel ccoels + ccoss cuni eldisjn0elb eqvrelqseqdisj3 petlem eqvreldmqs2 3bitri ) ABCADEFG + HAIZJUCKUCLAMFUCPZNUDKZUDLAMFAOZNAQUFLAMFARAUCAUEUDSTAUAUB $. + + $( The conventional form of Member Partition-Equivalence Theorem. In the + conventional case there is no (general) disjoint and no (general) + partition concept: mathematicians have been calling disjoint or partition + what we call element disjoint or member partition, see also ~ cpet2 . Cf. + ~ mpet , ~ mpet2 and ~ mpet3 for unconventional forms of Member + Partition-Equivalence Theorem. Cf. ~ pet and ~ pet2 for + Partition-Equivalence Theorem with general ` R ` . (Contributed by Peter + Mazsa, 31-Dec-2024.) $) + cpet $p |- ( MembPart A <-> + ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) ) $= + ( wmembpart weldisj c0 wcel wn wa ccoels weqvrel cuni cqs dfmembpart2 cpet2 + wceq bitri ) ABACDAEFGAHZIAJPKANGALAMO $. + + $( Member Partition-Equivalence Theorem in almost its shortest possible form, + cf. the 0-ary version ~ mpets . Member partition and comember equivalence + relation are the same (or: each element of ` A ` have equivalent comembers + if and only if ` A ` is a member partition). Together with ~ mpet2 , + ~ mpet3 , and with the conventional ~ cpet and ~ cpet2 , this is what we + used to think of as the partition equivalence theorem (but cf. ~ pet2 with + general ` R ` ). (Contributed by Peter Mazsa, 24-Sep-2021.) $) + mpet $p |- ( MembPart A <-> CoMembEr A ) $= + ( weldisj c0 wcel wn wcoeleqvrel cuni ccoels wceq wmembpart wcomember mpet3 + wa cqs dfmembpart2 dfcomember3 3bitr4i ) ABCADEMAFAGAHNAIMAJAKALAOAPQ $. + + $( Member Partition-Equivalence Theorem in a shorter form. Together with + ~ mpet ~ mpet3 , mostly in its conventional ~ cpet and ~ cpet2 form, this + is what we used to think of as the partition equivalence theorem (but cf. + ~ pet2 with general ` R ` ). (Contributed by Peter Mazsa, + 24-Sep-2021.) $) + mpet2 $p |- ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) $= + ( wmembpart wcomember ccnv cres wpart ccoss werALTV df-membpart df-comember + cep mpet 3bitr3i ) ABACAKDAEZFANGHALAIAJM $. + + $( Member Partition-Equivalence Theorem with binary relations, cf. ~ mpet2 . + (Contributed by Peter Mazsa, 24-Sep-2021.) $) + mpets2 $p |- ( A e. V -> + ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) ) $= + ( wcel cep ccnv cres cparts wbr ccoss wb wpart werALTV mpet2 cvv cnvepresex + cers brpartspart mpdan 1cosscnvepresex brerser bibi12d mpbiri ) ABCZDEAFZAG + HZUDIZAPHZJAUDKZAUFLZJAMUCUEUHUGUIUCUDNCUEUHJABOAUDBNQRUCUFNCUGUIJABSAUFBNT + RUAUB $. + + $( Member Partition-Equivalence Theorem in its shortest possible form: it + shows that member partitions and comember equivalence relations are + literally the same. Cf. ~ pet , the Partition-Equivalence Theorem, with + general ` R ` . (Contributed by Peter Mazsa, 31-Dec-2024.) $) + mpets $p |- MembParts = CoMembErs $= + ( va cep ccnv cv cres cparts wbr ccoss cers cmembparts ccomembers wb mpets2 + cab cvv elv abbii df-membparts df-comembers 3eqtr4i ) BCADZEZUAFGZANUBHUAIG + ZANJKUCUDAUCUDLAUAOMPQARAST $. + + $( Partition with general ` R ` also imply member partition. (Contributed by + Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.) $) + mainpart $p |- ( R Part A -> MembPart A ) $= + ( wpart wcomember wmembpart partimcomember mpet sylibr ) ABCADAEABFAGH $. + + $( The Theorem of Fences by Equivalences: all conceivable equivalence + relations (besides the comember equivalence relation cf. ~ mpet ) generate + a partition of the members. (Contributed by Peter Mazsa, 26-Sep-2021.) $) + fences $p |- ( R ErALTV A -> MembPart A ) $= + ( werALTV wcomember wmembpart mainer mpet sylibr ) ABCADAEABFAGH $. + + $( The Theorem of Fences by Equivalences: all conceivable equivalence + relations (besides the comember equivalence relation cf. ~ mpet3 ) + generate a partition of the members, it alo means that + ` ( R ErALTV A -> ElDisj A ) ` and that + ` ( R ErALTV A -> -. (/) e. A ) ` . (Contributed by Peter Mazsa, + 15-Oct-2021.) $) + fences2 $p |- ( R ErALTV A -> ( ElDisj A /\ -. (/) e. A ) ) $= + ( werALTV wmembpart weldisj c0 wcel wn wa fences dfmembpart2 sylib ) ABCADA + EFAGHIABJAKL $. + + $( The Main Theorem of Equivalences: every equivalence relation implies + equivalent comembers. (Contributed by Peter Mazsa, 15-Oct-2021.) $) + mainer2 $p |- ( R ErALTV A -> ( CoElEqvRel A /\ -. (/) e. A ) ) $= + ( werALTV weldisj c0 wcel wn wa wcoeleqvrel fences2 eldisjim anim1i syl ) A + BCADZEAFGZHAIZOHABJNPOAKLM $. + + $( Every equivalence relation implies equivalent coelements. (Contributed by + Peter Mazsa, 20-Oct-2021.) $) + mainerim $p |- ( R ErALTV A -> CoElEqvRel A ) $= + ( werALTV wcoeleqvrel c0 wcel wn mainer2 simpld ) ABCADEAFGABHI $. + + $( A partition-equivalence theorem with intersection and general ` R ` . + (Contributed by Peter Mazsa, 31-Dec-2021.) $) + petincnvepres2 $p |- ( ( Disj ( R i^i ( `' _E |` A ) ) /\ + ( dom ( R i^i ( `' _E |` A ) ) /. + ( R i^i ( `' _E |` A ) ) ) = A ) <-> + ( EqvRel ,~ ( R i^i ( `' _E |` A ) ) /\ + ( dom ,~ ( R i^i ( `' _E |` A ) ) /. + ,~ ( R i^i ( `' _E |` A ) ) ) = A ) ) $= + ( cep ccnv cres cin ccoss cdm eqvrelqseqdisj4 petlem ) ABCDAEFZAKGZHLBIJ $. + + $( The shortest form of a partition-equivalence theorem with intersection and + general ` R ` . Cf. ~ br1cossincnvepres . Cf. ~ pet . (Contributed by + Peter Mazsa, 23-Sep-2021.) $) + petincnvepres $p |- ( ( R i^i ( `' _E |` A ) ) Part A <-> + ,~ ( R i^i ( `' _E |` A ) ) ErALTV A ) $= + ( cep ccnv cres cin wdisjALTV cdm cqs wa ccoss weqvrel wpart petincnvepres2 + wceq werALTV dfpart2 dferALTV2 3bitr4i ) BCDAEFZGTHTIAOJTKZLUAHUAIAOJATMAUA + PABNATQAUARS $. + + $( Partition-Equivalence Theorem, with general ` R ` . This theorem + (together with ~ pet and ~ pets ) is the main result of my investigation + into set theory, see the comment of ~ pet . (Contributed by Peter Mazsa, + 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.) $) + pet2 $p |- ( ( Disj ( R |X. ( `' _E |` A ) ) /\ + ( dom ( R |X. ( `' _E |` A ) ) /. + ( R |X. ( `' _E |` A ) ) ) = A ) <-> + ( EqvRel ,~ ( R |X. ( `' _E |` A ) ) /\ + ( dom ,~ ( R |X. ( `' _E |` A ) ) /. + ,~ ( R |X. ( `' _E |` A ) ) ) = A ) ) $= + ( cep ccnv cres cxrn ccoss cdm eqvrelqseqdisj5 petlem ) ABCDAEFZAKGZHLBIJ + $. + + $( Partition-Equivalence Theorem with general ` R ` while preserving the + restricted converse epsilon relation of ~ mpet2 (as opposed to + ~ petincnvepres ). A class is a partition by a range Cartesian product + with general ` R ` and the restricted converse element class if and only + if the cosets by the range Cartesian product are in an equivalence + relation on it. Cf. ~ br1cossxrncnvepres . + + This theorem (together with ~ pets and ~ pet2 ) is the main result of my + investigation into set theory. It is no more general than the + conventional Member Partition-Equivalence Theorem ~ mpet , ~ mpet2 and + ~ mpet3 (because you cannot set ` R ` in this theorem in such a way that + you get ~ mpet2 ), i.e., it is not the hypothetical General + Partition-Equivalence Theorem gpet ` |- ( R Part A <-> ,~ R ErALTV A ) ` , + but this one has a general part that ~ mpet2 lacks: ` R ` , which is + sufficient for my future application of set theory, for my purpose outside + of set theory. (Contributed by Peter Mazsa, 23-Sep-2021.) $) + pet $p |- ( ( R |X. ( `' _E |` A ) ) Part A <-> + ,~ ( R |X. ( `' _E |` A ) ) ErALTV A ) $= + ( cep ccnv cres cxrn wdisjALTV cdm wceq wa ccoss weqvrel wpart werALTV pet2 + cqs dfpart2 dferALTV2 3bitr4i ) BCDAEFZGTHTPAIJTKZLUAHUAPAIJATMAUANABOATQAU + ARS $. + + $( Partition-Equivalence Theorem with general ` R ` , with binary relations. + This theorem (together with ~ pet and ~ pet2 ) is the main result of my + investigation into set theory, cf. the comment of ~ pet . (Contributed by + Peter Mazsa, 23-Sep-2021.) $) + pets $p |- ( ( A e. V /\ R e. W ) -> + ( ( R |X. ( `' _E |` A ) ) Parts A <-> + ,~ ( R |X. ( `' _E |` A ) ) Ers A ) ) $= + ( wcel wa cep ccnv cres cxrn cparts wbr ccoss cers wb wpart werALTV pet cvv + syldan xrncnvepresex brpartspart 1cossxrncnvepresex brerser bibi12d mpbiri + ) ACEZBDEZFZBGHAIJZAKLZUJMZANLZOAUJPZAULQZOABRUIUKUNUMUOUGUHUJSEUKUNOABCDUA + AUJCSUBTUGUHULSEUMUOOABCDUCAULCSUDTUEUF $. + $( (End of Peter Mazsa's mathbox.) $) $( End $[ set-mbox-pm.mm $] $) @@ -586085,10 +588456,10 @@ instead of simply using converse functions (cf. ~ dfdisjALTV ). -> ( E. y e. A ( z e. y /\ w e. y ) -> w e. x ) ) ) $= ( wprt cv wcel wel wa wrex wi wex df-rex an32 weq prtlem14 elequ2 biimprd - syl8 exp4a impd syl5bir expd imp5a imp4b exlimdv syl5bi ex ) EFZAGEHZCAIZ - JZCBIZDBIZJZBEKZDAIZLUQBGEHZUPJZBMUJUMJZURUPBENVAUTURBUJUMUSUPURUJUMUSUNU - OURUJUMUSUNUOURLZLZUMUSJUKUSJZULJUJVCUKUSULOUJVDULVCUJVDULUNVBUJVDULUNJAB - PZVBABCEQVEURUOABDRSTUAUBUCUDUEUFUGUHUI $. + syl8 exp4a impd syl5bir expd imp5a imp4b exlimdv biimtrid ex ) EFZAGEHZCA + IZJZCBIZDBIZJZBEKZDAIZLUQBGEHZUPJZBMUJUMJZURUPBENVAUTURBUJUMUSUPURUJUMUSU + NUOURUJUMUSUNUOURLZLZUMUSJUKUSJZULJUJVCUKUSULOUJVDULVCUJVDULUNVBUJVDULUNJ + ABPZVBABCEQVEURUOABDRSTUAUBUCUDUEUFUGUHUI $. $} ${ @@ -586135,19 +588506,19 @@ instead of simply using converse functions (cf. ~ dfdisjALTV ). prter2 $p |- ( Prt A -> ( U. A /. .~ ) = ( A \ { (/) } ) ) $= ( vp vv vz c0 cv wcel wa wceq wrex wex bitri df-rex wral wi wprt cuni cqs csn cdif wne cec rexcom4 r19.41v exbii rexbii elqs eluni2 anbi1i 3bitr4ri - vex prtlem19 ralrimivv 2r19.29 ex syl5bi eqtr3 reximi syl6 19.41v simprbi - syl risset syl6ibr prtlem400 nelelne mp1i jcad eldifsn neldifsn n0el mpbi + vex prtlem19 ralrimivv 2r19.29 ex syl biimtrid reximi syl6 19.41v simprbi + eqtr3 risset syl6ibr wn prtlem400 nelelne mp1i jcad eldifsn neldifsn n0el rspec eldifi jca ancomsd elunii jca2r cvv prtlem11 elv imp eximdv 3imtr3g - wn 19.9v syl5 impbid eqrdv ) DUAZGDUBZEUCZDJUDZUEZWOGKZWQLZWTWSLZWOXAWTDL - ZWTJUFZMXBWOXAXCXDWOXAHKZWTNZHDOZXCWOXAXFIXEOZHDOZXGWOXAXEIKZEUGZNZWTXKNZ - MZIXEOZHDOZXIXAXMIXEOZHDOZWOXPXJXELZXMMZIPZHDOZXSHDOZXMMZIPZXRXAYBXTHDOZI - PYEXTHIDUHYFYDIXSXMHDUIUJQXQYAHDXMIXERUKXAXMIWPOZYEIWPWTEGUPULYGXJWPLZXMM - ZIPYEXMIWPRYIYDIYHYCXMHXJDUMUNUJQQUOWOXLIXESHDSZXRXPTWOXLHIDXEABIHCDEFUQU - RYJXRXPXLXMHIDXEUSUTVGVAXOXHHDXNXFIXEXEWTXKVBVCVCVDXHXFHDXHXSIPZXFXHXSXFM - IPYKXFMXFIXERXSXFIVEQVFVCVDHWTDVHVIJWQLWJXAXDTWOABCDEFVJJWQWTVKVLVMWTDJVN - VIXBXJWTLZIPZXCMZWOXAXBYMXCYMGWSJWSLWJYMGWSSJDVOGIWSVPVQVRWTDWRVSVTWOYLXC - MZIPXAIPYNXAWOYOXAIWOYOYIXAWOYOXMYHWOXCYLXMABIGCDEFUQWAXJWTDWBWCYHXMXAYHX - MXATTGWPWTXJWDEWEWFWGVDWHYLXCIVEXAIWKWIWLWMWN $. + mpbi 19.9v syl5 impbid eqrdv ) DUAZGDUBZEUCZDJUDZUEZWOGKZWQLZWTWSLZWOXAWT + DLZWTJUFZMXBWOXAXCXDWOXAHKZWTNZHDOZXCWOXAXFIXEOZHDOZXGWOXAXEIKZEUGZNZWTXK + NZMZIXEOZHDOZXIXAXMIXEOZHDOZWOXPXJXELZXMMZIPZHDOZXSHDOZXMMZIPZXRXAYBXTHDO + ZIPYEXTHIDUHYFYDIXSXMHDUIUJQXQYAHDXMIXERUKXAXMIWPOZYEIWPWTEGUPULYGXJWPLZX + MMZIPYEXMIWPRYIYDIYHYCXMHXJDUMUNUJQQUOWOXLIXESHDSZXRXPTWOXLHIDXEABIHCDEFU + QURYJXRXPXLXMHIDXEUSUTVAVBXOXHHDXNXFIXEXEWTXKVGVCVCVDXHXFHDXHXSIPZXFXHXSX + FMIPYKXFMXFIXERXSXFIVEQVFVCVDHWTDVHVIJWQLVJXAXDTWOABCDEFVKJWQWTVLVMVNWTDJ + VOVIXBXJWTLZIPZXCMZWOXAXBYMXCYMGWSJWSLVJYMGWSSJDVPGIWSVQWJVRWTDWRVSVTWOYL + XCMZIPXAIPYNXAWOYOXAIWOYOYIXAWOYOXMYHWOXCYLXMABIGCDEFUQWAXJWTDWBWCYHXMXAY + HXMXATTGWPWTXJWDEWEWFWGVDWHYLXCIVEXAIWKWIWLWMWN $. $( For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, @@ -587031,9 +589402,9 @@ This axiom is _logically_ redundant in the (logically complete) (New usage is discouraged.) $) ax12indn $p |- ( -. A. x x = y -> ( x = y -> ( -. ph -> A. x ( x = y -> -. ph ) ) ) ) $= - ( weq wal wn wi wa 19.8a exanali hbn1 con3 syl6 com23 alrimdh syl5bi syl5 - wex expd ) BCEZBFGZUAAGZUAUCHZBFZUAUCIZUFBSZUBUEUFBJUGUAAHZBFZGZUBUEUAABK - UBUJUDBUABLUHBLUBUAUJUCUBUAAUIHUJUCHDAUIMNOPQRT $. + ( weq wal wn wi wa wex 19.8a exanali hbn1 con3 syl6 alrimdh biimtrid syl5 + com23 expd ) BCEZBFGZUAAGZUAUCHZBFZUAUCIZUFBJZUBUEUFBKUGUAAHZBFZGZUBUEUAA + BLUBUJUDBUABMUHBMUBUAUJUCUBUAAUIHUJUCHDAUINOSPQRT $. ${ ax12indi.2 $e |- ( -. A. x x = y -> @@ -587918,11 +590289,11 @@ inference form (e.g., nonzero. ( ~ atne0 analog.) (Contributed by NM, 25-Aug-2014.) $) lsatn0 $p |- ( ph -> U =/= { .0. } ) $= ( vv cv csn cfv wceq wne wcel clmod wb eqid wa clspn cbs cdif wrex islsat - syl mpbid wi eldifsn lspsneq0 sylan biimpd necon3d expimpd neeq1 biimprcd - syl5bi syl6 rexlimdv mpd ) ACJKZLDUAMZMZNZJDUBMZELZUCZUDZCVFOZACBPZVHIADQ - PZVJVHRHJBCVBVEDQEVESZVBSZFGUEUFUGAVDVIJVGAVAVGPZVCVFOZVDVIUHVNVAVEPZVAEO - ZTAVOVAVEEUIAVPVQVOAVPTZVCVFVAEVRVCVFNZVAENZAVKVPVSVTRHVBVEDVAEVLFVMUJUKU - LUMUNUQVDVIVOCVCVFUOUPURUSUT $. + syl mpbid wi eldifsn lspsneq0 sylan biimpd necon3d expimpd biimtrid neeq1 + biimprcd syl6 rexlimdv mpd ) ACJKZLDUAMZMZNZJDUBMZELZUCZUDZCVFOZACBPZVHIA + DQPZVJVHRHJBCVBVEDQEVESZVBSZFGUEUFUGAVDVIJVGAVAVGPZVCVFOZVDVIUHVNVAVEPZVA + EOZTAVOVAVEEUIAVPVQVOAVPTZVCVFVAEVRVCVFNZVAENZAVKVPVSVTRHVBVEDVAEVLFVMUJU + KULUMUNUOVDVIVOCVCVFUPUQURUSUT $. $} ${ @@ -587983,14 +590354,14 @@ inference form (e.g., lsatcmp $p |- ( ph -> ( T C_ U <-> T = U ) ) $= ( vv csn cfv wceq wss wb wcel clmod eqid wa ad2antrr cv cbs c0g cdif wrex clspn clvec lveclmod syl islsat mpbid wi eldifsn lsatn0 clss wo lsatlssel - wne simplrl simpr lspsnat syl31anc ord necon1ad mpd eqimss impbid1 syl5bi - ex sseq2 eqeq2 bibi12d biimprcd syl6 rexlimdv ) ADJUAZKEUFLZLZMZJEUBLZEUC - LZKZUDZUEZCDNZCDMZOZADBPZWDIAEQPZWHWDOAEUGPZWIGEUHUIZJBDVQVTEQWAVTRZVQRZW - ARZFUJUIUKAVSWGJWCAVPWCPZCVRNZCVRMZOZVSWGULWOVPVTPZVPWAURZSZAWRVPVTWAUMAX - AWRAXASZWPWQXBWPWQXBWPSZCWBURZWQAXDXAWPABCEWAWNFWKHUNTXCWQCWBXCWQCWBMZXCW - JCEUOLZPZWSWPWQXEUPAWJXAWPGTAXGXAWPABXFCEXFRZFWKHUQTAWSWTWPUSXBWPUTXFCVQV - TEVPWAWLWNXHWMVAVBVCVDVEVICVRVFVGVIVHVSWGWRVSWEWPWFWQDVRCVJDVRCVKVLVMVNVO - VE $. + wne simplrl simpr lspsnat syl31anc ord necon1ad mpd eqimss biimtrid sseq2 + ex impbid1 eqeq2 bibi12d biimprcd syl6 rexlimdv ) ADJUAZKEUFLZLZMZJEUBLZE + UCLZKZUDZUEZCDNZCDMZOZADBPZWDIAEQPZWHWDOAEUGPZWIGEUHUIZJBDVQVTEQWAVTRZVQR + ZWARZFUJUIUKAVSWGJWCAVPWCPZCVRNZCVRMZOZVSWGULWOVPVTPZVPWAURZSZAWRVPVTWAUM + AXAWRAXASZWPWQXBWPWQXBWPSZCWBURZWQAXDXAWPABCEWAWNFWKHUNTXCWQCWBXCWQCWBMZX + CWJCEUOLZPZWSWPWQXEUPAWJXAWPGTAXGXAWPABXFCEXFRZFWKHUQTAWSWTWPUSXBWPUTXFCV + QVTEVPWAWLWNXHWMVAVBVCVDVEVICVRVFVJVIVGVSWGWRVSWEWPWFWQDVRCVHDVRCVKVLVMVN + VOVE $. $} ${ @@ -588305,13 +590676,13 @@ inference form (e.g., ( wcel wss wn wa wral crab cuni cfv uniss wceq syl eqid wpss clmod w3a cv wrex dfpss3 simprbi wi ss2rab iman ralbii bitr2i clspn cbs simpl1 lsatlss rabss2 4syl simpl2 unimax lssss eqsstrd sstrd adantl lspss syl3anc simpl3 - lssats syl2anc 3sstr4d ex syl5bi con3dimp dfrex2 sylibr sylan2 ) CDUAZEUB - IZCBIZDBIZUCZDCJZKZFUDZDJZWDCJZKLZFAUEZVQCDJWCCDUFUGWAWCLWGKZFAMZKWHWAWJW - BWJWEFANZWFFANZJZWAWBWMWEWFUHZFAMWJWEWFFAUIWNWIFAWEWFUJUKULWAWMWBWAWMLZWK - OZEUMPZPZWLOZWQPZDCWOVRWSEUNPZJWPWSJZWRWTJVRVSVTWMUOZWOWSWFFBNZOZXAWOVRAB - JWLXDJWSXEJXCABEGHUPWFFABUQWLXDQURWOXECXAWOVSXECRVRVSVTWMUSZFCBUTSWOVSCXA - JXFBCXAEXATZGVASVBVCWMXBWAWKWLQVDWPWSWQXAEXGWQTZVEVFWOVRVTDWRRXCVRVSVTWMV - GFABDWQEGXHHVHVIWOVRVSCWTRXCXFFABCWQEGXHHVHVIVJVKVLVMWGFAVNVOVP $. + lssats syl2anc 3sstr4d ex biimtrid con3dimp dfrex2 sylibr sylan2 ) CDUAZE + UBIZCBIZDBIZUCZDCJZKZFUDZDJZWDCJZKLZFAUEZVQCDJWCCDUFUGWAWCLWGKZFAMZKWHWAW + JWBWJWEFANZWFFANZJZWAWBWMWEWFUHZFAMWJWEWFFAUIWNWIFAWEWFUJUKULWAWMWBWAWMLZ + WKOZEUMPZPZWLOZWQPZDCWOVRWSEUNPZJWPWSJZWRWTJVRVSVTWMUOZWOWSWFFBNZOZXAWOVR + ABJWLXDJWSXEJXCABEGHUPWFFABUQWLXDQURWOXECXAWOVSXECRVRVSVTWMUSZFCBUTSWOVSC + XAJXFBCXAEXATZGVASVBVCWMXBWAWKWLQVDWPWSWQXAEXGWQTZVEVFWOVRVTDWRRXCVRVSVTW + MVGFABDWQEGXHHVHVIWOVRVSCWTRXCXFFABCWQEGXHHVHVIVJVKVLVMWGFAVNVOVP $. $} ${ @@ -589048,7 +591419,7 @@ inference form (e.g., l1cvat.n $e |- ( ph -> Q =/= R ) $. l1cvat.l $e |- ( ph -> U C V ) $. l1cvat.m $e |- ( ph -> -. Q C_ U ) $. - $( Create an atom under an element covered by the lattice unit. Part of + $( Create an atom under an element covered by the lattice unity. Part of proof of Lemma B in [Crawley] p. 112. ( ~ 1cvrat analog.) (Contributed by NM, 11-Jan-2015.) $) l1cvat $p |- ( ph -> ( ( Q .(+) R ) i^i U ) e. A ) $= @@ -589350,18 +591721,18 @@ Functionals and kernels of a left vector space (or module) -> E. x e. V ( G ` x ) = .1. ) $= ( vz clvec wcel cfv wceq eqid syl3anc csn cxp wne w3a cv wrex wa wral nne wn ralbii wf wfn wi cbs lflf ffnd fconstfv simplbi2 syl c0g fvexi fconst2 - syl6ib syl5bi necon3ad dfrex2 3impia cinvr cvsca co clmod simp1l lveclmod - cdr lvecdrng simp1r simp2 lflcl simp3 drnginvrcl lmodvscl cmulr syl112anc - syl6ibr lflmul drnginvrl eqtrd fveqeq2 rspcev syl2anc rexlimdv3a 3adant3 - mpd ) GOPZEDPZEFHUAZUBZUCZUDNUEZEQZHUCZNFUFZAUEZEQCRZAFUFZWOWPWSXCWOWPUGZ - WSXBUJZNFUHZUJXCXGXIEWRXIXAHRZNFUHZXGEWRRZXHXJNFXAHUIUKXGXKFWQEULZXLXGEFU - MZXKXMUNXGFBUOQZEBDEXOFGOIXOSZLMUPUQXMXNXKNFHEURUSUTFHEHBVAJVBVCVDVEVFXBN - FVGWEVHWOWPXCXFUNWSXGXBXFNFXGWTFPZXBUDZXABVIQZQZWTGVJQZVKZFPZYBEQZCRZXFXR - GVLPZXTXOPZXQYCXRWOYFWOWPXQXBVMZGVNUTZXRBVOPZXAXOPZXBYGXRWOYJYHBGIVPUTZXR - WOWPXQYKYHWOWPXQXBVQZXGXQXBVRZBDEXOFGWTOIXPLMVSTZXGXQXBVTZXOBXSXAHXPJXSSZ - WATZYNXTYABXOFGWTLIYASZXPWBTXRYDXTXABWCQZVKZCXRYFWPYGXQYDUUARYIYMYRYNBXTY - AYTDEXOFGWTIXPYTSZLYSMWFWDXRYJYKXBUUACRYLYOYPXOBYTCXSXAHXPJUUBKYQWGTWHXEY - EAYBFXDYBCEWIWJWKWLWMWN $. + syl6ib biimtrid necon3ad dfrex2 syl6ibr 3impia cinvr cvsca clmod lveclmod + co simp1l cdr lvecdrng simp1r simp2 lflcl simp3 drnginvrcl lmodvscl cmulr + lflmul syl112anc drnginvrl fveqeq2 rspcev syl2anc rexlimdv3a 3adant3 mpd + eqtrd ) GOPZEDPZEFHUAZUBZUCZUDNUEZEQZHUCZNFUFZAUEZEQCRZAFUFZWOWPWSXCWOWPU + GZWSXBUJZNFUHZUJXCXGXIEWRXIXAHRZNFUHZXGEWRRZXHXJNFXAHUIUKXGXKFWQEULZXLXGE + FUMZXKXMUNXGFBUOQZEBDEXOFGOIXOSZLMUPUQXMXNXKNFHEURUSUTFHEHBVAJVBVCVDVEVFX + BNFVGVHVIWOWPXCXFUNWSXGXBXFNFXGWTFPZXBUDZXABVJQZQZWTGVKQZVNZFPZYBEQZCRZXF + XRGVLPZXTXOPZXQYCXRWOYFWOWPXQXBVOZGVMUTZXRBVPPZXAXOPZXBYGXRWOYJYHBGIVQUTZ + XRWOWPXQYKYHWOWPXQXBVRZXGXQXBVSZBDEXOFGWTOIXPLMVTTZXGXQXBWAZXOBXSXAHXPJXS + SZWBTZYNXTYABXOFGWTLIYASZXPWCTXRYDXTXABWDQZVNZCXRYFWPYGXQYDUUARYIYMYRYNBX + TYAYTDEXOFGWTIXPYTSZLYSMWEWFXRYJYKXBUUACRYLYOYPXOBYTCXSXAHXPJUUBKYQWGTWNX + EYEAYBFXDYBCEWHWIWJWKWLWM $. $} ${ @@ -590161,30 +592532,30 @@ Functionals and kernels of a left vector space (or module) mpbid rexcom4 ovex oveq2 eqeq2d ceqsexv rexbii bitr3i bitrdi bitrid sylib rexbidv rexcom oveq1 cbvrexvw w3a c0g ccntz simp11l lshpdisj cabl lmodabl cin ablcntzd simp12 simp2 simp1rl 3ad2ant1 lspsneli simp1rr simp13 eqtr3d - 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In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in - [Holland95] p. 218. This allows us to reuse our existing collection of + [Holland95] p. 218. This allows to reuse our existing collection of left vector space theorems. The restriction on ` oF ( +g `` v ) ` allows it to be a set; see ~ ofmres . Note the operation reversal in - the scalar product to allow us to use the original scalar ring instead - of the ` oppR ` ring, for convenience. (Contributed by NM, + the scalar product to allow to use the original scalar ring instead of + the ` oppR ` ring, for convenience. (Contributed by NM, 18-Oct-2014.) $) df-ldual $a |- LDual = ( v e. _V |-> ( { <. ( Base ` ndx ) , ( LFnl ` v ) >. , @@ -590588,10 +592959,10 @@ Functionals and kernels of a left vector space (or module) vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in - [Holland95] p. 218. This allows us to reuse our existing collection of + [Holland95] p. 218. This allows to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar - product to allow us to use the original scalar ring instead of the - ` oppR ` ring, for convenience. (Contributed by NM, 18-Oct-2014.) $) + product to allow to use the original scalar ring instead of the ` oppR ` + ring, for convenience. (Contributed by NM, 18-Oct-2014.) $) ldualset $p |- ( ph -> D = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. ( Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) ) $= @@ -591304,14 +593675,14 @@ Functionals and kernels of a left vector space (or module) ( cfv wcel clmod adantr eqid vv cin co cv wa elin cbs csca c0g wceq lkrcl simprl syl3anc cplusg ldualvaddval lkrf0 simprr oveq12d cgrp crg lmodring syl ringgrp grpidcl grplid syl2anc2 3eqtrd wb ldualvaddcl ellkr mpbir2and - syl2anc ex syl5bi ssrdv ) AUAEGPZFGPZUBZEFCUCZGPZUAUDZVRQWAVPQZWAVQQZUEZA - WAVTQZWAVPVQUFAWDWEAWDUEZWEWAHUGPZQZWAVSPZHUHPZUIPZUJZWFHRQZEDQZWBWHAWMWD - MSZAWNWDNSZAWBWCULZDEGWGHWARWGTZIJUKUMZWFWIWAEPZWAFPZWJUNPZUCWKWKXBUCZWKW - FBXBCWJDEFWGHWAWRWJTZXBTZIKLWOWPAFDQZWDOSZWSUOWFWTWKXAWKXBWFWMWNWBWTWKUJW - OWPWQWJDEGHWARWKXDWKTZIJUPUMWFWMXFWCXAWKUJWOXGAWBWCUQWJDFGHWARWKXDXHIJUPU - MURAXCWKUJZWDAWJUSQZWKWJUGPZQXIAWJUTQZXJAWMXLMWJHXDVAVBWJVCVBXKWJWKXKTZXH - VDXKXBWJWKWKXMXEXHVEVFSVGWFWMVSDQZWEWHWLUEVHWOAXNWDABCDEFHIKLMNOVISWJDVSG - WGHWARWKWRXDXHIJVJVLVKVMVNVO $. + syl2anc ex biimtrid ssrdv ) AUAEGPZFGPZUBZEFCUCZGPZUAUDZVRQWAVPQZWAVQQZUE + ZAWAVTQZWAVPVQUFAWDWEAWDUEZWEWAHUGPZQZWAVSPZHUHPZUIPZUJZWFHRQZEDQZWBWHAWM + WDMSZAWNWDNSZAWBWCULZDEGWGHWARWGTZIJUKUMZWFWIWAEPZWAFPZWJUNPZUCWKWKXBUCZW + KWFBXBCWJDEFWGHWAWRWJTZXBTZIKLWOWPAFDQZWDOSZWSUOWFWTWKXAWKXBWFWMWNWBWTWKU + JWOWPWQWJDEGHWARWKXDWKTZIJUPUMWFWMXFWCXAWKUJWOXGAWBWCUQWJDFGHWARWKXDXHIJU + PUMURAXCWKUJZWDAWJUSQZWKWJUGPZQXIAWJUTQZXJAWMXLMWJHXDVAVBWJVCVBXKWJWKXKTZ + XHVDXKXBWJWKWKXMXEXHVEVFSVGWFWMVSDQZWEWHWLUEVHWOAXNWDABCDEFHIKLMNOVISWJDV + SGWGHWARWKWRXDXHIJVJVLVKVMVNVO $. $} ${ @@ -591638,7 +594009,7 @@ Functionals and kernels of a left vector space (or module) op01dm.b $e |- B = ( Base ` K ) $. op01dm.u $e |- U = ( lub ` K ) $. op01dm.g $e |- G = ( glb ` K ) $. - $( Conditions necessary for zero and unit elements to exist. (Contributed + $( Conditions necessary for zero and unity elements to exist. (Contributed by NM, 14-Sep-2018.) $) op01dm $p |- ( K e. OP -> ( B e. dom U /\ B e. dom G ) ) $= ( vx vy wcel cdm w3a cv cfv wceq wbr co wral wa eqid cops cpo coc cple wi @@ -591661,7 +594032,7 @@ Functionals and kernels of a left vector space (or module) ${ op1cl.b $e |- B = ( Base ` K ) $. op1cl.u $e |- .1. = ( 1. ` K ) $. - $( An orthoposet has a unit element. ( ~ helch analog.) (Contributed by + $( An orthoposet has a unity element. ( ~ helch analog.) (Contributed by NM, 22-Oct-2011.) $) op1cl $p |- ( K e. OP -> .1. e. B ) $= ( cops wcel club cfv eqid p1val id cdm cglb op01dm simpld lubcl eqeltrd ) @@ -591731,14 +594102,14 @@ Functionals and kernels of a left vector space (or module) ople1.b $e |- B = ( Base ` K ) $. ople1.l $e |- .<_ = ( le ` K ) $. ople1.u $e |- .1. = ( 1. ` K ) $. - $( Any element is less than the orthoposet unit. ( ~ chss analog.) + $( Any element is less than the orthoposet unity. ( ~ chss analog.) (Contributed by NM, 23-Oct-2011.) $) ople1 $p |- ( ( K e. OP /\ X e. B ) -> X .<_ .1. ) $= ( cops wcel wa club cfv eqid simpl simpr cdm cglb op01dm simpld adantr ple1 ) CIJZEAJZKACLMZBCDIEFUENZGHUCUDOUCUDPUCAUEQJZUDUCUGACRMZQJAUEUHCFUF UHNSTUAUB $. - $( If the orthoposet unit is less than or equal to an element, the element + $( If the orthoposet unity is less than or equal to an element, the element equals the unit. ( ~ chle0 analog.) (Contributed by NM, 5-Dec-2011.) $) op1le $p |- ( ( K e. OP /\ X e. B ) -> ( .1. .<_ X <-> X = .1. ) ) $= @@ -591751,7 +594122,7 @@ Functionals and kernels of a left vector space (or module) $d x y z K $. $d x z .1. $. glb0.g $e |- G = ( glb ` K ) $. glb0.u $e |- .1. = ( 1. ` K ) $. - $( The greatest lower bound of the empty set is the unit element. + $( The greatest lower bound of the empty set is the unity element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.) $) glb0N $p |- ( K e. OP -> ( G ` (/) ) = .1. ) $= ( vx vy vz cops wcel c0 cfv cv cple wbr wral wi wa eqid ral0 crio biid id @@ -591843,7 +594214,7 @@ Functionals and kernels of a left vector space (or module) opoc1.z $e |- .0. = ( 0. ` K ) $. opoc1.u $e |- .1. = ( 1. ` K ) $. opoc1.o $e |- ._|_ = ( oc ` K ) $. - $( Orthocomplement of orthoposet unit. (Contributed by NM, + $( Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012.) $) opoc1 $p |- ( K e. OP -> ( ._|_ ` .1. ) = .0. ) $= ( cops wcel cfv cple wbr wceq cbs eqid op0cl opoccl mpdan ople1 wb mpbird @@ -592763,7 +595134,8 @@ X C ( ._|_ ` Y ) ) ) $= ncvr1.b $e |- B = ( Base ` K ) $. ncvr1.u $e |- .1. = ( 1. ` K ) $. ncvr1.c $e |- C = ( -. .1. C X ) $= ( cops wcel wa wbr cplt cfv cple eqid ad2antrr simplr simpr syl31anc mt2d ople1 wn cpo opposet op1cl pltnle ex simpll cvrlt mtand ) DIJZEAJZKZCEBLZ @@ -594201,11 +596573,51 @@ X C ( ._|_ ` Y ) ) ) $= glbcon.o $e |- ._|_ = ( oc ` K ) $. $( De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume ` HL ` for convenience. (Contributed - by NM, 17-Jan-2012.) (New usage is discouraged.) $) + by NM, 17-Jan-2012.) New ~ df-riota . (Revised by SN, 3-Jan-2025.) + (New usage is discouraged.) $) glbconN $p |- ( ( K e. HL /\ S C_ B ) -> ( G ` S ) = ( ._|_ ` ( U ` { x e. B | ( ._|_ ` x ) e. S } ) ) ) $= ( vz vu wcel cfv wceq wbr wral wi wa vy vw vv vt wss chlt cv crab sseqin2 cin biimpi dfin5 eqtr3di fveq2d cple crio eqid biid id ssrab2 glbval cops + a1i wreu hlop ccla cdm hlclat clatglbcl2 syl2anc glbeu breq1 breq2 imbi2d + ralbidv anbi12d riotaocN ad2antrr opoccl sylancom wrex opococ rspceeqv wb + eqcomd fveq2 eleq1 imbi12d adantl ralxfrd simpr simplr oplecon3b ralbidva + syl3anc bitr4d ralrab weq eleq1d 3bitr4g ad3antrrr riotabidva simpl mpan2 + lubval eqtr4d 3eqtrd sylan9eqr ) CBUEZFUFNZCEOAUGZCNZABUHZEOZXKGOZCNZABUH + ZDOZGOZXICXMEXIBCUJZCXMXIXTCPCBUIUKABCULUMUNXJXNUAUGZLUGZFUOOZQZLXMRZUBUG + ZYBYCQZLXMRZYFYAYCQZSZUBBRZTZUABUPZUCUGZGOZYBYCQZLXMRZYHYFYOYCQZSZUBBRZTZ + UCBUPZGOZXSXJYLUALUBBXMEFYCUFHYCUQZJYLURZXJUSZXMBUEZXJXLABUTVCZVAXJFVBNZY + LUABVDYMUUCPFVEZXJYLUALUBBXMEFYCUFHUUDJUUEUUFXJFVFNUUGXMEVGNFVHUUHBXMEFHJ + VIVJVKYLUUAUAUCBFGHKYAYOPZYEYQYKYTUUKYDYPLXMYAYOYBYCVLVOUUKYJYSUBBUUKYIYR + YHYAYOYFYCVMVNVOVPVQVJXJUUBXRGXJUUBMUGZYNYCQZMXQRZUULUDUGZYCQZMXQRZYNUUOY + CQZSZUDBRZTZUCBUPZXRXJUUAUVAUCBXJYNBNZTZYQUUNYTUUTUVDYBCNZYPSZLBRZUULGOZC + NZUUMSZMBRZYQUUNUVDUVGUVIYOUVHYCQZSZMBRUVKUVDUVFUVMLMUVHBBUVDUULBNZUUIUVH + BNZXJUUIUVCUVNUUJVRZBFGUULHKVSZVTUVDYBBNZTZYBGOZBNZYBUVTGOZPZYBUVHPZMBWAZ + UVDUVRUUIUWAXJUUIUVCUVRUUJVRZBFGYBHKVSZVTUVSUWBYBUVDUVRUUIUWBYBPZUWFBFGYB + HKWBZVTWEMUVTBUVHUWBYBUULUVTGWFWCZVJUWDUVFUVMWDUVDUWDUVEUVIYPUVLYBUVHCWGZ + YBUVHYOYCVMWHWIWJUVDUVJUVMMBUVDUVNTZUUMUVLUVIUWLUUIUVNUVCUUMUVLWDUVPUVDUV + NWKXJUVCUVNWLBFYCGUULYNHUUDKWMWOVNWNWPXLUVEYPLABXKYBCWGZWQXPUVIUUMMABAMWR + XOUVHCXKUULGWFWSZWQWTUVDYTUUOGOZYBYCQZLXMRZUWOYOYCQZSZUDBRUUTUVDYSUWSUBUD + UWOBBUVDUUOBNZUUIUWOBNXJUUIUVCUWTUUJVRZBFGUUOHKVSVTUVDYFBNZTZYFGOZBNZYFUX + DGOZPYFUWOPZUDBWAUVDUXBUUIUXEXJUUIUVCUXBUUJVRZBFGYFHKVSVTUXCUXFYFUVDUXBUU + IUXFYFPUXHBFGYFHKWBVTWEUDUXDBUWOUXFYFUUOUXDGWFWCVJUXGYSUWSWDUVDUXGYHUWQYR + UWRUXGYGUWPLXMYFUWOYBYCVLVOYFUWOYOYCVLWHWIWJUVDUUSUWSUDBUVDUWTTZUUQUWQUUR + UWRUXIUVIUUPSZMBRZUVEUWPSZLBRZUUQUWQUXIUXKUVIUWOUVHYCQZSZMBRUXMUXIUXJUXOM + BUXIUVNTZUUPUXNUVIUXPUUIUVNUWTUUPUXNWDXJUUIUVCUWTUVNUUJXAZUXIUVNWKUVDUWTU + VNWLBFYCGUULUUOHUUDKWMWOVNWNUXIUXLUXOLMUVHBBUXIUVNUUIUVOUXQUVQVTUXIUVRTZU + WAUWCUWEUXIUVRUUIUWAXJUUIUVCUWTUVRUUJXAZUWGVTUXRUWBYBUXIUVRUUIUWHUXSUWIVT + WEUWJVJUWDUXLUXOWDUXIUWDUVEUVIUWPUXNUWKYBUVHUWOYCVMWHWIWJWPXPUVIUUPMABUWN + WQXLUVEUWPLABUWMWQWTUXIUUIUVCUWTUURUWRWDUXAXJUVCUWTWLUVDUWTWKBFYCGYNUUOHU + UDKWMWOWHWNWPVPXBXJXQBUEZXRUVBPXPABUTXJUXTTUVAUCMUDBXQDFYCUFHUUDIUVAURXJU + XTXCXJUXTWKXEXDXFUNXGXH $. + + $( Obsolete version of ~ glbconN as of 3-Jan-2025. (Contributed by NM, + 17-Jan-2012.) (New usage is discouraged.) + (Proof modification is discouraged.) $) + glbconNOLD $p |- ( ( K e. HL /\ S C_ B ) + -> ( G ` S ) = ( ._|_ ` ( U ` { x e. B | ( ._|_ ` x ) e. S } ) ) ) $= + ( vz vu wcel cfv wceq wbr wral wi wa vy vw vv vt wss chlt cv crab sseqin2 + cin biimpi dfin5 eqtr3di fveq2d cple crio eqid biid id ssrab2 glbval cops wreu hlop ccla hlclat clatglbcl sylancl eqeltrrd fvexi riotaclbBAD sylibr a1i breq1 ralbidv breq2 imbi2d anbi12d riotaocN syl2anc ad2antrr sylancom cbs opoccl wrex opococ eqcomd fveq2 rspceeqv eleq1 imbi12d adantl ralxfrd @@ -594247,18 +596659,18 @@ X C ( ._|_ ` Y ) ) ) $= -> ( G ` { x | E. i e. I x = S } ) = ( ._|_ ` ( U ` { x | E. i e. I x = ( ._|_ ` S ) } ) ) ) $= ( vy wcel wa wceq wrex cab cfv chlt wral crab wss vex eqeq1 rexbidv nfra1 - cv elab nfv wi rsp eleq1a syl6 rexlimd syl5bi ssrdv glbconN sylan2 rabbii - fvex df-rab eqtri nfan wb rspa cops hlop opoccl sylan syl opcon2b syl3an1 - pm4.71rd 3expa eqcom bitr3di bitrd anassrs rexbida r19.42v bitr2di abbidv - pm5.32da cbvabv eqtrdi eqtrid fveq2d eqtrd ) HUAOZCBOZEGUBZPZAUIZCQZEGRZA - SZFTZNUIZITZWROZNBUCZDTZITZWOCITZQZEGRZASZDTZITWMWKWRBUDWSXEQWMNWRBWTWROW - TCQZEGRZWMWTBOZWQXLAWTNUEWOWTQWPXKEGWOWTCUFUGUJWMXKXMEGWLEGUHZXMEUKWMEUIG - OZWLXKXMULWLEGUMCBWTUNUOUPUQURNBWRDFHIJKLMUSUTWNXDXJIWNXCXIDWNXCXMXACQZEG - RZPZNSZXIXCXQNBUCXSXBXQNBWQXQAXAWTIVBWOXAQWPXPEGWOXACUFUGUJVAXQNBVCVDWNXS - WTXFQZEGRZNSXIWNXRYANWNYAXMXPPZEGRXRWNXTYBEGWKWMEWKEUKXNVEWKWMXOXTYBVFZWM - XOPWKWLYCWLEGVGWKWLPZXTXMXTPYBYDXTXMYDXFBOZXTXMULWKHVHOZWLYEHVIZBHICJMVJV - KXFBWTUNVLVOYDXMXTXPYDXMPCXAQZXTXPWKWLXMYHXTVFZWKYFWLXMYIYGBHICWTJMVMVNVP - CXAVQVRWEVSUTVTWAXMXPEGWBWCWDYAXHNAWTWOQXTXGEGWTWOXFUFUGWFWGWHWIWIWJ $. + cv elab nfv wi rsp eleq1a syl6 rexlimd biimtrid ssrdv glbconN sylan2 fvex + rabbii df-rab eqtri nfan rspa cops hlop opoccl sylan syl pm4.71rd opcon2b + syl3an1 3expa eqcom bitr3di pm5.32da bitrd anassrs rexbida r19.42v abbidv + wb bitr2di cbvabv eqtrdi eqtrid fveq2d eqtrd ) HUAOZCBOZEGUBZPZAUIZCQZEGR + ZASZFTZNUIZITZWROZNBUCZDTZITZWOCITZQZEGRZASZDTZITWMWKWRBUDWSXEQWMNWRBWTWR + OWTCQZEGRZWMWTBOZWQXLAWTNUEWOWTQWPXKEGWOWTCUFUGUJWMXKXMEGWLEGUHZXMEUKWMEU + IGOZWLXKXMULWLEGUMCBWTUNUOUPUQURNBWRDFHIJKLMUSUTWNXDXJIWNXCXIDWNXCXMXACQZ + EGRZPZNSZXIXCXQNBUCXSXBXQNBWQXQAXAWTIVAWOXAQWPXPEGWOXACUFUGUJVBXQNBVCVDWN + XSWTXFQZEGRZNSXIWNXRYANWNYAXMXPPZEGRXRWNXTYBEGWKWMEWKEUKXNVEWKWMXOXTYBWDZ + WMXOPWKWLYCWLEGVFWKWLPZXTXMXTPYBYDXTXMYDXFBOZXTXMULWKHVGOZWLYEHVHZBHICJMV + IVJXFBWTUNVKVLYDXMXTXPYDXMPCXAQZXTXPWKWLXMYHXTWDZWKYFWLXMYIYGBHICWTJMVMVN + VOCXAVPVQVRVSUTVTWAXMXPEGWBWEWCYAXHNAWTWOQXTXGEGWTWOXFUFUGWFWGWHWIWIWJ $. $} ${ @@ -595346,36 +597758,36 @@ X C ( ._|_ ` Y ) ) ) $= cfv eqid cvrat3 3expd imp4c latjcl latmle1 imp44 simplr2 3jca jca latmcom wn atnle eqeq1d bitrd latmcl latmlem2 breqtrd syl5ibcom cops hlop syl2anc breq2 ople0 sylibd sylbid imp adantrr latmle2 latjcom hlexch3 3expia jcad - 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(Contributed by NM, 28-Jan-2012.) $) cvrat42 $p |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) @@ -596002,8 +598414,9 @@ X C ( ._|_ ` Y ) ) ) $= 1cvrjat.u $e |- .1. = ( 1. ` K ) $. 1cvrjat.c $e |- C = ( ( X .\/ P ) = .1. ) $= ( wcel wbr cfv wceq wb chlt w3a wn wa co coc cp0 simprr cvr1 adantr mpbid @@ -596029,7 +598442,7 @@ X C ( ._|_ ` Y ) ) ) $= 1cvrat.u $e |- .1. = ( 1. ` K ) $. 1cvrat.c $e |- C = ( ( X e. P <-> E. a e. A X = { a } ) ) $= - ( vx wcel cv csn wceq wrex cab pointsetN eleq2d cvv snex eleq1 rexlimivw - mpbiri eqeq1 rexbidv elab3 bitrdi ) DBJZECJEIKZFKZLZMZFANZIOZJEUJMZFANZUG - CUMEABCDIFGHPQULUOIERUNERJZFAUNUPUJRJUISEUJRTUBUAUHEMUKUNFAUHEUJUCUDUEUF - $. + ( vx wcel cv csn wceq wrex cab pointsetN eleq2d cvv vsnex eleq1 rexlimivw + mpbiri eqeq1 rexbidv elab3 bitrdi ) DBJZECJEIKZFKLZMZFANZIOZJEUIMZFANZUGC + ULEABCDIFGHPQUKUNIERUMERJZFAUMUOUIRJFSEUIRTUBUAUHEMUJUMFAUHEUIUCUDUEUF $. $( The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) $) @@ -601607,7 +604019,7 @@ Cambridge University Press (1988). Unlike them, we do not assume that pmap1.u $e |- .1. = ( 1. ` K ) $. pmap1.a $e |- A = ( Atoms ` K ) $. pmap1.m $e |- M = ( pmap ` K ) $. - $( Value of the projective map of a Hilbert lattice at lattice unit. Part + $( Value of the projective map of a Hilbert lattice at lattice unity. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.) $) pmap1N $p |- ( K e. OP -> ( M ` .1. ) = A ) $= @@ -602811,19 +605223,19 @@ their common point (atom). (Contributed by NM, 2-May-2013.) ( chlt wss c0 wne co wa wn wceq wo nne oveq1d syl5ibrcom wcel w3a orbi12i ianor bitri 3adant3r1 padd02 syldan 3ad2antr2 eqtr4d oveq1 eqeq12d eqimss paddssat syl6 padd01 3ad2antr1 sspadd1 simpl simpr3 syl3anc sstrd eqsstrd - 3adant3r3 oveq2 sseq1d jaod 3ad2antr3 oveq2d syl5bi 3impia ) CIUAZDAJZEAJ - ZFAJZUBZDKLZEFBMZKLZNZEKLZFKLZNZNOZDVRBMZDEBMZFBMZJZWDDKPZVRKPZQZEKPZFKPZ - QZQZVLVPNZWHWDVTOZWCOZQWOVTWCUDWQWKWRWNWQVQOZVSOZQWKVQVSUDWSWIWTWJDKRVRKR - UCUEWRWAOZWBOZQWNWAWBUDXAWLXBWMEKRFKRUCUEUCUEWPWKWHWNWPWIWHWJWPWIWEWGPZWH - WPXCWIKVRBMZKEBMZFBMZPWPXDVRXFVLVPVRAJZXDVRPVLVNVOXGVMAIBCEFGHUNUFAIBCVRG - HUGUHWPXEEFBVLVMVNXEEPVOAIBCEGHUGUISUJWIWEXDWGXFDKVRBUKWIWFXEFBDKEBUKSULT - WEWGUMZUOWPWHWJDKBMZWGJWPXIDWGVLVNVMXIDPVOAIBCDGHUPUQZWPDWFWGVLVMVNDWFJVO - AIBCDEGHURVDWPVLWFAJZVOWFWGJVLVPUSVLVMVNXKVOAIBCDEGHUNVDZVLVMVNVOUTAIBCWF - FGHURVAVBVCWJWEXIWGVRKDBVEVFTVGWPWNXCWHWPWLXCWMWPXCWLDKFBMZBMZXIFBMZPWPXN - DFBMXOWPXMFDBVLVMVOXMFPVNAIBCFGHUGVHVIWPXIDFBXJSUJWLWEXNWGXOWLVRXMDBEKFBU - KVIWLWFXIFBEKDBVESULTWPXCWMDEKBMZBMZWFKBMZPWPXQWFXRWPXPEDBVLVMVNXPEPVOAIB - CEGHUPUIVIVLVPXKXRWFPXLAIBCWFGHUPUHUJWMWEXQWGXRWMVRXPDBFKEBVEVIFKWFBVEULT - VGXHUOVGVJVK $. + 3adant3r3 oveq2 sseq1d jaod 3ad2antr3 oveq2d biimtrid 3impia ) CIUAZDAJZE + AJZFAJZUBZDKLZEFBMZKLZNZEKLZFKLZNZNOZDVRBMZDEBMZFBMZJZWDDKPZVRKPZQZEKPZFK + PZQZQZVLVPNZWHWDVTOZWCOZQWOVTWCUDWQWKWRWNWQVQOZVSOZQWKVQVSUDWSWIWTWJDKRVR + KRUCUEWRWAOZWBOZQWNWAWBUDXAWLXBWMEKRFKRUCUEUCUEWPWKWHWNWPWIWHWJWPWIWEWGPZ + WHWPXCWIKVRBMZKEBMZFBMZPWPXDVRXFVLVPVRAJZXDVRPVLVNVOXGVMAIBCEFGHUNUFAIBCV + RGHUGUHWPXEEFBVLVMVNXEEPVOAIBCEGHUGUISUJWIWEXDWGXFDKVRBUKWIWFXEFBDKEBUKSU + LTWEWGUMZUOWPWHWJDKBMZWGJWPXIDWGVLVNVMXIDPVOAIBCDGHUPUQZWPDWFWGVLVMVNDWFJ + VOAIBCDEGHURVDWPVLWFAJZVOWFWGJVLVPUSVLVMVNXKVOAIBCDEGHUNVDZVLVMVNVOUTAIBC + WFFGHURVAVBVCWJWEXIWGVRKDBVEVFTVGWPWNXCWHWPWLXCWMWPXCWLDKFBMZBMZXIFBMZPWP + XNDFBMXOWPXMFDBVLVMVOXMFPVNAIBCFGHUGVHVIWPXIDFBXJSUJWLWEXNWGXOWLVRXMDBEKF + BUKVIWLWFXIFBEKDBVESULTWPXCWMDEKBMZBMZWFKBMZPWPXQWFXRWPXPEDBVLVMVNXPEPVOA + IBCEGHUPUIVIVLVPXKXRWFPXLAIBCWFGHUPUHUJWMWEXQWGXRWMVRXPDBFKEBVEVIFKWFBVEU + LTVGXHUOVGVJVK $. $( Lemma for ~ paddass . Combine ~ paddasslem16 and ~ paddasslem17 . (Contributed by NM, 12-Jan-2012.) $) @@ -602977,19 +605389,19 @@ their common point (atom). (Contributed by NM, 2-May-2013.) ineq1d syl simp21 padd01 sylan9eqr inss1 sspadd1 sstrid wne elin wbr wrex cv wi clat hllatd simprl elpaddn0 syl31anc simpl23 simpl3 simpr2l simpr2r wb simpr1 simpr3 pmodlem1 syl333anc 3exp2 rexlimdvv adantld adantrl exp32 - imp sylbid com34 imp4b syl5bi ssrdv pm2.61da2ne ) EUBQZGARZHARZICQZUCZGIR - ZUCZGHBUDZIUEZGHIUEZBUDZRGSHSXGGSUFZTZXIXJXKXMXHHIXMXHSHBUDZHXMGSHBXGXLUG - UHXMXAXCXNHUFXAXEXFXLUIZXBXCXDXAXFXLUKZAUBBEHLNUJULUMVAXMXAXJARZXBXJXKRXO - XMXCXQXPHIAUNZVBXBXCXDXAXFXLUOAUBBEXJGLNUPUQURXGHSUFZTZXIGIUEZXKXTXHGIXSX - GXHGSBUDZGHSGBUSXGXAXBYBGUFXAXEXFUTXAXBXCXDXFVCAUBBEGLNVDULVEVAXTYAGXKGIV - FXTXAXBXQGXKRXAXEXFXSUIXBXCXDXAXFXSUOXTXCXQXBXCXDXAXFXSUKXRVBAUBBEGXJLNVG - UQVHURXGGSVIHSVITZTZUAXIXKUAVMZXIQYEXHQZYEIQZTYDYEXKQZYEXHIVJXGYCYFYGYHXG - YCYGYFYHXGYCYGYFYHVNXGYCYGTZTZYFYEAQZYEOVMZPVMZDUDFVKZPHVLOGVLZTZYHYJEVOQ - XBXCYCYFYPWDYJEXAXEXFYIUIVPXBXCXDXAXFYIUOXBXCXDXAXFYIUKXGYCYGVQABYEDEFGHP - OJKLNVRVSXGYGYPYHVNYCXGYGTZYOYHYKYQYNYHOPGHXGYGYLGQZYMHQZTZYNYHVNVNXGYGYT - YNYHXGYGYTYNUCZTXAXBXCXDXFYGYRYSYNYHXAXEXFUUAUIXBXCXDXAXFUUAUOXBXCXDXAXFU - UAUKXBXCXDXAXFUUAVTXAXEXFUUAWAXGYGYTYNWEYRYSYGYNXGWBYRYSYGYNXGWCXGYGYTYNW - FABCDEFGHIPOUAJKLMNWGWHWIWNWJWKWLWOWMWPWQWRWSWT $. + imp sylbid com34 imp4b biimtrid ssrdv pm2.61da2ne ) EUBQZGARZHARZICQZUCZG + IRZUCZGHBUDZIUEZGHIUEZBUDZRGSHSXGGSUFZTZXIXJXKXMXHHIXMXHSHBUDZHXMGSHBXGXL + UGUHXMXAXCXNHUFXAXEXFXLUIZXBXCXDXAXFXLUKZAUBBEHLNUJULUMVAXMXAXJARZXBXJXKR + XOXMXCXQXPHIAUNZVBXBXCXDXAXFXLUOAUBBEXJGLNUPUQURXGHSUFZTZXIGIUEZXKXTXHGIX + SXGXHGSBUDZGHSGBUSXGXAXBYBGUFXAXEXFUTXAXBXCXDXFVCAUBBEGLNVDULVEVAXTYAGXKG + IVFXTXAXBXQGXKRXAXEXFXSUIXBXCXDXAXFXSUOXTXCXQXBXCXDXAXFXSUKXRVBAUBBEGXJLN + VGUQVHURXGGSVIHSVITZTZUAXIXKUAVMZXIQYEXHQZYEIQZTYDYEXKQZYEXHIVJXGYCYFYGYH + XGYCYGYFYHXGYCYGYFYHVNXGYCYGTZTZYFYEAQZYEOVMZPVMZDUDFVKZPHVLOGVLZTZYHYJEV + OQXBXCYCYFYPWDYJEXAXEXFYIUIVPXBXCXDXAXFYIUOXBXCXDXAXFYIUKXGYCYGVQABYEDEFG + HPOJKLNVRVSXGYGYPYHVNYCXGYGTZYOYHYKYQYNYHOPGHXGYGYLGQZYMHQZTZYNYHVNVNXGYG + YTYNYHXGYGYTYNUCZTXAXBXCXDXFYGYRYSYNYHXAXEXFUUAUIXBXCXDXAXFUUAUOXBXCXDXAX + FUUAUKXBXCXDXAXFUUAVTXAXEXFUUAWAXGYGYTYNWEYRYSYGYNXGWBYRYSYGYNXGWCXGYGYTY + NWFABCDEFGHIPOUAJKLMNWGWHWIWNWJWKWLWOWMWPWQWRWSWT $. $} ${ @@ -603936,11 +606348,11 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ). .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) $= ( wcel co wn chlt wbr w3a wa wi wo dalawlem13 3expib 3exp dalawlem10 jaod - ianor syl5bi 3imp 3impib ) IUARZBCHSZDHSLRZBEHSCFHSKSZUQJUBTUSCDHSZJUBTUS - DBHSZJUBTUCZUDTZUSDGHSJUBZUCBARCARDARUCZEARFARGARUCZUQEFHSKSUTFGHSKSVAGEH - SKSHSJUBZUPVCVDVEVFUDVGUEZVCURTZVBTZUFUPVDVHUEZURVBULUPVIVKVJUPVIVDVHUPVI - VDUCVEVFVGABCDEFGHIJKLMNOPQUGUHUIUPVJVDVHUPVJVDUCVEVFVGABCDEFGHIJKMNOPUJU - HUIUKUMUNUO $. + ianor biimtrid 3imp 3impib ) IUARZBCHSZDHSLRZBEHSCFHSKSZUQJUBTUSCDHSZJUBT + USDBHSZJUBTUCZUDTZUSDGHSJUBZUCBARCARDARUCZEARFARGARUCZUQEFHSKSUTFGHSKSVAG + EHSKSHSJUBZUPVCVDVEVFUDVGUEZVCURTZVBTZUFUPVDVHUEZURVBULUPVIVKVJUPVIVDVHUP + VIVDUCVEVFVGABCDEFGHIJKLMNOPQUGUHUIUPVJVDVHUPVJVDUCVEVFVGABCDEFGHIJKMNOPU + JUHUIUKUMUNUO $. $( Lemma for ~ dalaw . Swap variable triples ` P Q R ` and ` S T U ` in ~ dalawlem14 , to obtain the elimination of the remaining conditions in @@ -604057,16 +606469,16 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ). 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( ( X C_ Y /\ ( ._|_ ` ( ._|_ ` Y ) ) = Y ) -> ( ( ._|_ ` ( ( ._|_ ` X ) i^i Y ) ) i^i Y ) = ( ._|_ ` ( ._|_ ` X ) ) ) ) $= ( wcel wss w3a cfv wceq cin eqid 2polvalN wa co syl syl2anc syl3anc cpmap - chlt club eqcom 3adant2 eqeq2d biimpd syl5bi coc cmee coml cbs wbr simpl1 - cple hloml ccla hlclat simpl2 atssbase sstrdi clatlubcl simpl3 3jca lubss - simprl omllaw4 sylc fveq2d polval2N simprr ineq12d opoccl pmapmeet eqtr4d - cops hlop clat hllatd latmcl polpmapN eqtrd 3eqtr4d ex sylan2d ) BUBHZDAI - ZEAIZJZECKCKZELZEEBUCKZKZBUAKZKZLZDEIZDCKZEMZCKZEMZWRCKZLZWKEWJLZWIWPWJEU - DWIXDWPWIWJWOEWFWHWJWOLWGAWLBWNCEWLNZFWNNZGOUEUFUGUHWIWQWPPZXCWIXGPZDWLKZ - BUIKZKZWMBUJKZQZXJKZWMXLQZWNKZXIWNKZXAXBXHXOXIWNXHBUKHZXIBULKZHZWMXSHZJXI - WMBUOKZUMZXOXILXHXRXTYAXHWFXRWFWGWHXGUNZBUPRXHBUQHZDXSIXTXHWFYEYDBURRZXHD - AXSWFWGWHXGUSZAXSBXSNZFUTZVAXSDWLBYHXEVBSZXHYEEXSIZYAYFXHEAXSWFWGWHXGVCYI - VAZXSEWLBYHXEVBSZVDXHYEYKWQYCYFYLWIWQWPVFXSDEWLBYBYHYBNZXEVETXSBYBXLXJXIW - MYHYNXLNZXJNZVGVHVIXHXAXNWNKZWOMZXPXHWTYQEWOXHWTXMWNKZCKZYQXHWSYSCXHWSXKW - NKZWOMZYSXHWRUUAEWOXHWFWGWRUUALYDYGACWLBWNXJDXEYPFXFGVJSWIWQWPVKZVLXHWFXK - XSHZYAYSUUBLYDXHBVPHZXTUUDXHWFUUEYDBVQRZYJXSBXJXIYHYPVMSZYMAXSWNBXLXKWMYH - YOFXFVNTVOVIXHWFXMXSHZYTYQLYDXHBVRHUUDYAUUHXHBYDVSUUGYMXSBXLXKWMYHYOVTTZX - SCBWNXJXMYHYPXFGWASWBUUCVLXHWFXNXSHZYAXPYRLYDXHUUEUUHUUJUUFUUIXSBXJXMYHYP - VMSYMAXSWNBXLXNWMYHYOFXFVNTVOXHWFWGXBXQLYDYGAWLBWNCDXEFXFGOSWCWDWE $. + chlt club eqcom 3adant2 eqeq2d biimpd biimtrid coc cmee coml cbs cple wbr + simpl1 hloml ccla hlclat simpl2 sstrdi clatlubcl simpl3 3jca simprl lubss + atssbase omllaw4 sylc fveq2d polval2N simprr ineq12d cops opoccl pmapmeet + hlop eqtr4d clat hllatd latmcl polpmapN eqtrd 3eqtr4d ex sylan2d ) BUBHZD + AIZEAIZJZECKCKZELZEEBUCKZKZBUAKZKZLZDEIZDCKZEMZCKZEMZWRCKZLZWKEWJLZWIWPWJ + EUDWIXDWPWIWJWOEWFWHWJWOLWGAWLBWNCEWLNZFWNNZGOUEUFUGUHWIWQWPPZXCWIXGPZDWL + KZBUIKZKZWMBUJKZQZXJKZWMXLQZWNKZXIWNKZXAXBXHXOXIWNXHBUKHZXIBULKZHZWMXSHZJ + XIWMBUMKZUNZXOXILXHXRXTYAXHWFXRWFWGWHXGUOZBUPRXHBUQHZDXSIXTXHWFYEYDBURRZX + HDAXSWFWGWHXGUSZAXSBXSNZFVFZUTXSDWLBYHXEVASZXHYEEXSIZYAYFXHEAXSWFWGWHXGVB + YIUTZXSEWLBYHXEVASZVCXHYEYKWQYCYFYLWIWQWPVDXSDEWLBYBYHYBNZXEVETXSBYBXLXJX + IWMYHYNXLNZXJNZVGVHVIXHXAXNWNKZWOMZXPXHWTYQEWOXHWTXMWNKZCKZYQXHWSYSCXHWSX + KWNKZWOMZYSXHWRUUAEWOXHWFWGWRUUALYDYGACWLBWNXJDXEYPFXFGVJSWIWQWPVKZVLXHWF + XKXSHZYAYSUUBLYDXHBVMHZXTUUDXHWFUUEYDBVPRZYJXSBXJXIYHYPVNSZYMAXSWNBXLXKWM + YHYOFXFVOTVQVIXHWFXMXSHZYTYQLYDXHBVRHUUDYAUUHXHBYDVSUUGYMXSBXLXKWMYHYOVTT + ZXSCBWNXJXMYHYPXFGWASWBUUCVLXHWFXNXSHZYAXPYRLYDXHUUEUUHUUJUUFUUIXSBXJXMYH + YPVNSYMAXSWNBXLXNWMYHYOFXFVOTVQXHWFWGXBXQLYDYGAWLBWNCDXEFXFGOSWCWDWE $. $( Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) $) @@ -605122,9 +607534,9 @@ X C_ ( U i^i M ) ) $= /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ -. p e. ( X .+ Y ) ) -> ( Y i^i M ) = (/) ) $= ( vq chlt wcel wss w3a cfv c0 wne cv co wn cin wceq wa wex n0 osumcllem7N - 3expia exlimdv syl5bi necon1bd 3impia ) FUBUCJAUDKAUDUEZJKIUFUDJUGUHLUIZA - UCUEZVDJKCUJUCZUKKHULZUGUMVCVEUNZVFVGUGVGUGUHUAUIVGUCZUAUOVHVFUAVGUPVHVIV - FUAVCVEVIVFABCDEFGHIJKUALMNOPQRSTUQURUSUTVAVB $. + 3expia exlimdv biimtrid necon1bd 3impia ) FUBUCJAUDKAUDUEZJKIUFUDJUGUHLUI + ZAUCUEZVDJKCUJUCZUKKHULZUGUMVCVEUNZVFVGUGVGUGUHUAUIVGUCZUAUOVHVFUAVGUPVHV + IVFUAVCVEVIVFABCDEFGHIJKUALMNOPQRSTUQURUSUTVAVB $. $( Lemma for ~ osumclN . (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) $) @@ -605309,9 +607721,9 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . pexmidlem5N $p |- ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( ( ._|_ ` X ) i^i M ) = (/) ) $= ( vq wcel cv c0 wne chlt wss w3a cfv co wn cin wceq wa wex n0 pexmidlem4N - expr exlimdv syl5bi necon1bd impr ) DUAQHAUBIRZAQUCZHSTZURHHGUDZBUEQZUFVA - FUGZSUHUSUTUIZVBVCSVCSTPRVCQZPUJVDVBPVCUKVDVEVBPUSUTVEVBABCDEFGHPIJKLMNOU - LUMUNUOUPUQ $. + expr exlimdv biimtrid necon1bd impr ) DUAQHAUBIRZAQUCZHSTZURHHGUDZBUEQZUF + VAFUGZSUHUSUTUIZVBVCSVCSTPRVCQZPUJVDVBPVCUKVDVEVBPUSUTVEVBABCDEFGHPIJKLMN + OULUMUNUOUPUQ $. $( Lemma for ~ pexmidN . (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.) $) @@ -605622,8 +608034,8 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . lhp1cvr.u $e |- .1. = ( 1. ` K ) $. lhp1cvr.c $e |- C = ( W C .1. ) $= ( wcel cbs cfv wbr eqid islhp simplbda ) EAJFDJFEKLZJFCBMAQBCDEFQNGHIOP $. @@ -605875,8 +608287,8 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . lhp2at.a $e |- A = ( Atoms ` K ) $. lhp2at.h $e |- H = ( LHyp ` K ) $. $( There exist at least two different atoms under a co-atom. This allows - us to create a line under the co-atom. TODO: is this needed? - (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.) $) + to create a line under the co-atom. TODO: is this needed? (Contributed + by NM, 1-Jun-2012.) (New usage is discouraged.) $) lhpex2leN $p |- ( ( K e. HL /\ W e. H ) -> E. p e. A E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) ) $= ( chlt wcel wa cv wbr wne w3a wrex simprr lhpexle1 adantr jca necom bitri @@ -605972,7 +608384,7 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . lhpjat.a $e |- A = ( Atoms ` K ) $. lhpjat.h $e |- H = ( LHyp ` K ) $. $( The join of a co-atom (hyperplane) and an atom not under it is the - lattice unit. (Contributed by NM, 18-May-2012.) $) + lattice unity. (Contributed by NM, 18-May-2012.) $) lhpjat1 $p |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = .1. ) $= ( chlt wcel wa wbr wn cfv eqid cbs ccvr co simpll lhpbase ad2antlr simprl @@ -605981,7 +608393,7 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . BCDFHKVBTZMUIUJUPUQURUKAUTVBBCEFGHVDIJKVELULUM $. $( The join of a co-atom (hyperplane) and an atom not under it is the - lattice unit. (Contributed by NM, 4-Jun-2012.) $) + lattice unity. (Contributed by NM, 4-Jun-2012.) $) lhpjat2 $p |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = .1. ) $= ( chlt wcel wa wbr wn co clat cbs cfv wceq hllat ad2antrr atbase ad2antrl @@ -605999,7 +608411,7 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . lhpj1.u $e |- .1. = ( 1. ` K ) $. lhpj1.h $e |- H = ( LHyp ` K ) $. $( The join of a co-atom (hyperplane) and an element not under it is the - lattice unit. (Contributed by NM, 7-Dec-2012.) $) + lattice unity. (Contributed by NM, 7-Dec-2012.) $) lhpj1 $p |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( W .\/ X ) = .1. ) $= ( vp wcel wa wbr wn co wceq chlt cv catm cfv wrex wb simpll simpr lhpbase @@ -606144,7 +608556,7 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . lhpmat.z $e |- .0. = ( 0. ` K ) $. lhpmat.a $e |- A = ( Atoms ` K ) $. lhpmat.h $e |- H = ( LHyp ` K ) $. - $( An element covered by the lattice unit, when conjoined with an atom not + $( An element covered by the lattice unity, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.) $) lhpmat $p |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = .0. ) $= @@ -606153,7 +608565,7 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . SHTZUQURUSUAVADUBOZURGDUCUDZOZUSVBUEUOVCUPUTDUFUGUQURUSUHUPVEUOUTVDCDGVDU IZMUJUKAVDBDEFGHVFIJKLULUMUN $. - $( An element covered by the lattice unit, when conjoined with an atom, + $( An element covered by the lattice unity, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.) $) lhpmatb $p |- ( ( ( K e. HL /\ W e. H ) /\ P e. A ) @@ -614686,12 +617098,12 @@ then f(s) ` <_ ` f_s(t) ` \/ ` v. TODO: FIX COMMENT. (Contributed by /\ ( ( s e. A /\ -. s .<_ W ) /\ ( s .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( N .\/ ( X ./\ W ) ) ) $= ( chlt wcel wa wbr wn w3a wne cv co wceq cfv cvv wi fvexi anass wral crio - eqid cdleme31fv1 adantl cdleme32fvcl adantrr riotasvd syl5bi mpan2 3impia - cbs ) QUSUTUBNUTVAIEUTIUBRVBVCVAJEUTJUBRVBVCVAVDZUCFUTZIJVEUCUBRVBVCVAZVA - ZUDVFZEUTZWJUBRVBVCZVAWJUCUBSVGZPVGUCVHZVAZUCMVIZTWMPVGZVHZWFWIVAZFVJUTZW - OWRVKFQWEUEVLWOWKWLWNVAZVAWSWTVAWRWKWLWNVMWSXACUDFEWQWPVJWIWPXACVFWQVHVKU - DEVNCFVOZVHWFACEFXBIJMPRSTUAUBUCUDUQURXBVPVQVRWFWGWPFUTWHABCDEFGHIJKLMNOP - QRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURVSVTWAWBWCWD $. + cbs eqid cdleme31fv1 adantl cdleme32fvcl adantrr riotasvd biimtrid 3impia + mpan2 ) QUSUTUBNUTVAIEUTIUBRVBVCVAJEUTJUBRVBVCVAVDZUCFUTZIJVEUCUBRVBVCVAZ + VAZUDVFZEUTZWJUBRVBVCZVAWJUCUBSVGZPVGUCVHZVAZUCMVIZTWMPVGZVHZWFWIVAZFVJUT + ZWOWRVKFQVPUEVLWOWKWLWNVAZVAWSWTVAWRWKWLWNVMWSXACUDFEWQWPVJWIWPXACVFWQVHV + KUDEVNCFVOZVHWFACEFXBIJMPRSTUAUBUCUDUQURXBVQVRVSWFWGWPFUTWHABCDEFGHIJKLMN + OPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURVTWAWBWCWEWD $. $d s t x z Y $. $( Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, @@ -620222,23 +622634,23 @@ says that (our) ~ cdlemg10 "implies (2)" (of p. 116). No details are cfv simp21l simp23 simp32 cdlemg31d syl133anc simp31 nbrne2 necomd simp33 simp21r jca syl2anc 4atex3 df-3an a1i simp12l simp13l cdlemg31a syl122anc wi simpl simp11l hlatlej2 syl3anc clat cbs hllatd eqid atbase syl hlatjcl - wb latjle12 syl13anc mpbi2and adantr adantl lattr mpan2d anim12d reximdva - syl5bi anim2d mpd ) KUEUFZOIUFZUGZDCUFZDOLUHUIZUGZECUFZEOLUHUIZUGZUJZBUKZ - CUFZXOOLUHZUGZNCUFZHGUFZUJZDEULZXOHFUTULZPUKZOLUHUIDYDJUMEYDJUMUNUGPCUOZU - JZUJZAUKZOLUHUIZYHNULZYHXOULZYHNXOJUMZLUHZUJZUGZACUOZYIYJYHDXOJUMZLUHZUGZ - UGZACUOYGXGXJXMXSNOLUHUIZUGYBXPNXOULZUGYEYPXGXJXMYAYFUPZXGXJXMYAYFUQZXGXJ - XMYAYFURZYGXSUUAXNXRXSXTYFUSZYGXGXJXMXRXTYCXSUUAUUCUUDUUEYGXPXQXPXQXSXTXN - YFVAZXPXQXSXTXNYFVJZVKXNXRXSXTYFVBZXNYAYBYCYEVCUUFBCDEFGHIJKLMNOQRSTUAUBU - CUDVDVEZVKXNYAYBYCYEVFYGXPUUBUUGYGXQUUAUUBUUHUUJXQUUAUGXONXONOLVGVHVLVKXN - YAYBYCYEVIACDENXOIJKLOPQRTUAVMVEYGYOYTACYGYHCUFZUGZYNYSYIYNYJYKUGZYMUGUUL - YSYJYKYMVNUULUUMYJYMYRUUMYJVTUULYJYKWAVOUULYMYLYQLUHZYRYGUUNUUKYGNYQLUHZX - OYQLUHZUUNYGXGXHXKXPXTUUOUUCXHXIXGXMYAYFVPZXKXLXGXJYAYFVQUUGUUIBCDEFGHIJK - LMNOQRSTUAUBUCUDVRVSYGXEXHXPUUPXEXFXJXMYAYFWBZUUQUUGCDXOJKLQRTWCWDYGKWEUF - ZNKWFUTZUFZXOUUTUFZYQUUTUFZUUOUUPUGUUNWLYGKUURWGZYGXSUVAUUFCUUTNKUUTWHZTW - IWJYGXPUVBUUGCUUTXOKUVETWIWJYGXEXHXPUVCUURUUQUUGCUUTJKDXOUVERTWKWDZUUTJKL - NXOYQUVEQRWMWNWOWPUULUUSYHUUTUFZYLUUTUFZUVCYMUUNUGYRVTYGUUSUUKUVDWPUUKUVG - YGCUUTYHKUVETWIWQYGUVHUUKYGXEXSXPUVHUURUUFUUGCUUTJKNXOUVERTWKWDWPYGUVCUUK - UVFWPUUTKLYHYLYQUVEQWRWNWSWTXBXCXAXD $. + wb latjle12 syl13anc mpbi2and adantr adantl lattr mpan2d anim12d biimtrid + anim2d reximdva mpd ) KUEUFZOIUFZUGZDCUFZDOLUHUIZUGZECUFZEOLUHUIZUGZUJZBU + KZCUFZXOOLUHZUGZNCUFZHGUFZUJZDEULZXOHFUTULZPUKZOLUHUIDYDJUMEYDJUMUNUGPCUO + ZUJZUJZAUKZOLUHUIZYHNULZYHXOULZYHNXOJUMZLUHZUJZUGZACUOZYIYJYHDXOJUMZLUHZU + GZUGZACUOYGXGXJXMXSNOLUHUIZUGYBXPNXOULZUGYEYPXGXJXMYAYFUPZXGXJXMYAYFUQZXG + XJXMYAYFURZYGXSUUAXNXRXSXTYFUSZYGXGXJXMXRXTYCXSUUAUUCUUDUUEYGXPXQXPXQXSXT + XNYFVAZXPXQXSXTXNYFVJZVKXNXRXSXTYFVBZXNYAYBYCYEVCUUFBCDEFGHIJKLMNOQRSTUAU + BUCUDVDVEZVKXNYAYBYCYEVFYGXPUUBUUGYGXQUUAUUBUUHUUJXQUUAUGXONXONOLVGVHVLVK + XNYAYBYCYEVIACDENXOIJKLOPQRTUAVMVEYGYOYTACYGYHCUFZUGZYNYSYIYNYJYKUGZYMUGU + ULYSYJYKYMVNUULUUMYJYMYRUUMYJVTUULYJYKWAVOUULYMYLYQLUHZYRYGUUNUUKYGNYQLUH + ZXOYQLUHZUUNYGXGXHXKXPXTUUOUUCXHXIXGXMYAYFVPZXKXLXGXJYAYFVQUUGUUIBCDEFGHI + JKLMNOQRSTUAUBUCUDVRVSYGXEXHXPUUPXEXFXJXMYAYFWBZUUQUUGCDXOJKLQRTWCWDYGKWE + UFZNKWFUTZUFZXOUUTUFZYQUUTUFZUUOUUPUGUUNWLYGKUURWGZYGXSUVAUUFCUUTNKUUTWHZ + TWIWJYGXPUVBUUGCUUTXOKUVETWIWJYGXEXHXPUVCUURUUQUUGCUUTJKDXOUVERTWKWDZUUTJ + KLNXOYQUVEQRWMWNWOWPUULUUSYHUUTUFZYLUUTUFZUVCYMUUNUGYRVTYGUUSUUKUVDWPUUKU + VGYGCUUTYHKUVETWIWQYGUVHUUKYGXEXSXPUVHUURUUFUUGCUUTJKNXOUVERTWKWDWPYGUVCU + UKUVFWPUUTKLYHYLYQUVEQWRWNWSWTXAXBXCXD $. $( TODO: Fix comment. (Contributed by NM, 30-May-2013.) $) cdlemg33c0 $p |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) @@ -621567,12 +623979,12 @@ says that (our) ~ cdlemg10 "implies (2)" (of p. 116). No details are -> U = V ) $= ( wcel wa cfv wceq wi wral tendoid chlt cv cid wne adantrr adantrl eqtr4d cres fveq2 eqeq12d syl5ibrcom ralrimivw r19.26 wo wb exmidne pm5.5 bitr3i - jaob ax-mp ralbii tendoeq1 3expia syl5bi mpand 3impia ) GUANIFNOZCENZHENZ - OZDUBZUCAUHZUDZVKCPZVKHPZQZRZDBSZCHQZVGVJOZVKVLQZVPRZDBSZVRVSVTWBDBVTVPWA - VLCPZVLHPZQVTWDVLWEVGVHWDVLQVIACEFGIJKMTUEVGVIWEVLQVHAHEFGIJKMTUFUGWAVNWD - VOWEVKVLCUIVKVLHUIUJUKULWCVROZVPDBSZVTVSWFWBVQOZDBSWGWBVQDBUMWHVPDBWHWAVM - UNZVPRZVPWAVPVMUSWIWJVPUOVKVLUPWIVPUQUTURVAURVGVJWGVSBCDEFGHIKLMVBVCVDVEV - F $. + jaob ax-mp ralbii tendoeq1 3expia biimtrid mpand 3impia ) GUANIFNOZCENZHE + NZOZDUBZUCAUHZUDZVKCPZVKHPZQZRZDBSZCHQZVGVJOZVKVLQZVPRZDBSZVRVSVTWBDBVTVP + WAVLCPZVLHPZQVTWDVLWEVGVHWDVLQVIACEFGIJKMTUEVGVIWEVLQVHAHEFGIJKMTUFUGWAVN + WDVOWEVKVLCUIVKVLHUIUJUKULWCVROZVPDBSZVTVSWFWBVQOZDBSWGWBVQDBUMWHVPDBWHWA + VMUNZVPRZVPWAVPVMUSWIWJVPUOVKVLUPWIVPUQUTURVAURVGVJWGVSBCDEFGHIKLMVBVCVDV + EVF $. $} ${ @@ -626302,7 +628714,7 @@ one of its values (at a non-identity translation) is the identity erng1.e $e |- E = ( ( TEndo ` K ) ` W ) $. erng1.d $e |- D = ( ( EDRing ` K ) ` W ) $. erng1.r $e |- ( ( K e. HL /\ W e. H ) -> D e. Ring ) $. - $( Value of the endomorphism division ring unit. (Contributed by NM, + $( Value of the endomorphism division ring unity. (Contributed by NM, 12-Oct-2013.) $) erng1lem $p |- ( ( K e. HL /\ W e. H ) -> ( 1r ` D ) = ( _I |` T ) ) $= ( vu chlt wcel wa cfv co wceq eqid ccom cid cres cbs cmulr wral tendoidcl @@ -626474,7 +628886,7 @@ one of its values (at a non-identity translation) is the identity erng1r.t $e |- T = ( ( LTrn ` K ) ` W ) $. erng1r.d $e |- D = ( ( EDRing ` K ) ` W ) $. erng1r.r $e |- .1. = ( 1r ` D ) $. - $( The division ring unit of an endomorphism ring. (Contributed by NM, + $( The division ring unity of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) $) erng1r $p |- ( ( K e. HL /\ W e. H ) -> .1. = ( _I |` T ) ) $= ( vf chlt wcel wa cid cres cbs cfv wceq eqid c0g wne w3a ctendo tendoidcl @@ -627269,11 +629681,11 @@ are all translations (for a fiducial co-atom ` W ` ). (Contributed by diaelrnN $p |- ( ( ( K e. V /\ W e. H ) /\ S e. ran I ) -> S C_ T ) $= ( vx vy wcel wa crn wss cv cfv wbr eqid wceq cple cbs crab wrex wfn diafn - wb fvelrnb syl wi breq1 elrab diass ex sseq1 biimpcd syl6 syl5bi rexlimdv - sylbid imp ) EFMGCMNZADOMZABPZVCVDKQZDRZAUAZKLQZGEUBRZSZLEUCRZUDZUEZVEVCD - VMUFVDVNUHLVLCDEVJFGVLTZVJTZHJUGKVMADUIUJVCVHVEKVMVFVMMVFVLMVFGVJSZNZVCVH - VEUKZVKVQLVFVLVIVFGVJULUMVCVRVGBPZVSVCVRVTVLBCDEVJFGVFVOVPHIJUNUOVHVTVEVG - ABUPUQURUSUTVAVB $. + wb fvelrnb syl wi breq1 elrab diass ex sseq1 biimpcd syl6 biimtrid sylbid + rexlimdv imp ) EFMGCMNZADOMZABPZVCVDKQZDRZAUAZKLQZGEUBRZSZLEUCRZUDZUEZVEV + CDVMUFVDVNUHLVLCDEVJFGVLTZVJTZHJUGKVMADUIUJVCVHVEKVMVFVMMVFVLMVFGVJSZNZVC + VHVEUKZVKVQLVFVLVIVFGVJULUMVCVRVGBPZVSVCVRVTVLBCDEVJFGVFVOVPHIJUNUOVHVTVE + VGABUPUQURUSVAUTVB $. $} ${ @@ -630021,7 +632433,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, $( Isomorphism C has domain of lattice atoms that are not less than or equal to the fiducial co-atom ` w ` . The value is a one-dimensional subspace generated by the pair consisting of the ` iota_ ` vector below - and the endomorphism ring unit. Definition of phi(q) in [Crawley] + and the endomorphism ring unity. Definition of phi(q) in [Crawley] p. 121. Note that we use the fixed atom ` ( ( oc ` k ) ` w ) ` to represent the p in their "Choose an atom p..." on line 21. (Contributed by NM, 15-Dec-2013.) $) @@ -632490,7 +634902,7 @@ of phi(x) is independent of the atom q." (Contributed by NM, dih1.i $e |- I = ( ( DIsoH ` K ) ` W ) $. dih1.u $e |- U = ( ( DVecH ` K ) ` W ) $. dih1.v $e |- V = ( Base ` U ) $. - $( The value of isomorphism H at the lattice unit is the set of all + $( The value of isomorphism H at the lattice unity is the set of all vectors. (Contributed by NM, 13-Mar-2014.) $) dih1 $p |- ( ( K e. HL /\ W e. H ) -> ( I ` .1. ) = V ) $= ( vg chlt wcel wa cfv wceq eqid wbr vf vs wrel dihvalrel cltrn ctendo cxp @@ -632637,35 +635049,35 @@ of phi(x) is independent of the atom q." 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TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) @@ -632681,32 +635093,32 @@ of phi(x) is independent of the atom q." 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(Contributed by NM, @@ -632759,26 +635171,26 @@ of phi(x) is independent of the atom q." (Contributed by NM, clat eqeq1 rexbidv elrab sylanbrc eleqtrrdi clatglble mp3an2 latmle1 wral lattrd eqeq2d cbvrexvw bitrdi elrab2 wi simp3 simp13 breq2 rspcva simp11l weq 3ad2ant1 simp12 simp112 simp11r wb clatleglb mpbird latlem12 syl13anc - simp113 mpbi2and 3expia biimprcd syl6 rexlimdv syl5bi simp2 mp3an3 isglbd - expimpd ralrimiv ) HUERZKGRZUFZDCUGZDFUHZKISZUNZUAUBCDFEFUHZHILMOYEUBTZDR - ZUFZCHIYFYGKJUIZYGLMYIHXSXTYBYDYHUJZUKZYEYFCRZYHYEHUORZECUGZYMYEXSYNXSXTY - BYDULHUMZUPZEBTZATZKJUIZUQZADURZBCUSZCQUUBBCUTVAZCEFHLOVBVCZVDYIHVPRZYGCR - ZKCRZYJCRZYLYIDCYGYAYBYDYHVFYEYHVEZVGZYIXTUUHXSXTYBYDYHVHCGHKLPVIZUPZCHJY - GKLNVJVKZUUKYIYNYJERZYFYJISZYIXSYNYKYPUPYIYJUUCEYIUUIYJYTUQZADURZYJUUCRUU - NYIYHYJYJUQUURUUJYIYJVLAYGDYTYJYJYSYGKJVMVNVOUUBUURBYJCYRYJUQUUAUUQADYRYJ - YTVQVRVSVTQWAYNYOUUOUUPUUDCEFHIYJLMOWBWCVOYIUUFUUGUUHYJYGISYLUUKUUMCHIJYG - KLMNWDVKWFYEUATZCRZUUSYGISZUBDWEZUNZUUSYFISZUUSUCTZISZUCEWEZUVCUVFUCEUVEE - RUVECRZUVEUDTZKJUIZUQZUDDURZUFUVCUVFUUBUVLBUVECEBUCWQZUUBUVEYTUQZADURUVLU - VMUUAUVNADYRUVEYTVQVRUVNUVKAUDDAUDWQYTUVJUVEYSUVIKJVMWGWHWIQWJUVCUVHUVLUV - FUVCUVHUFZUVKUVFUDDUVOUVIDRZUUSUVJISZUVKUVFWKUVCUVHUVPUVQUVCUVHUVPUNZUUSU - VIISZUUSKISZUVQUVRUVPUVBUVSUVCUVHUVPWLZYEUUTUVBUVHUVPWMZUVAUVSUBUVIDYGUVI - UUSIWNWOVOUVRCHIUUSYCKLMUVRHUVCUVHXSUVPXSXTYBYDUUTUVBWPZWRZUKZYEUUTUVBUVH - UVPWSZUVRYNYBYCCRUVRXSYNUWDYPUPZYAYBYDUUTUVBUVHUVPWTZCDFHLOVBVOUVRXTUUHUV - CUVHXTUVPXSXTYBYDUUTUVBXAWRUULUPZUVRUUSYCISZUVBUWBUVRYNUUTYBUWJUVBXBUWGUW - FUWHUBCDFHIUUSLMOXCVKXDYAYBYDUUTUVBUVHUVPXGWFUVRUUFUUTUVICRUUHUVSUVTUFUVQ - XBUWEUWFUVRDCUVIUWHUWAVGUWICHIJUUSUVIKLMNXEXFXHXIUVKUVFUVQUVEUVJUUSIWNXJX - KXLXQXMXRUVCYNUUTUVDUVGXBZUVCXSYNUWCYPUPYEUUTUVBXNYNUUTYOUWKUUDUCCEFHIUUS - LMOXCXOVOXDYQYAYBYDXNUUEXP $. + simp113 mpbi2and 3expia biimprcd rexlimdv expimpd biimtrid ralrimiv simp2 + syl6 mp3an3 isglbd ) HUERZKGRZUFZDCUGZDFUHZKISZUNZUAUBCDFEFUHZHILMOYEUBTZ + DRZUFZCHIYFYGKJUIZYGLMYIHXSXTYBYDYHUJZUKZYEYFCRZYHYEHUORZECUGZYMYEXSYNXSX + TYBYDULHUMZUPZEBTZATZKJUIZUQZADURZBCUSZCQUUBBCUTVAZCEFHLOVBVCZVDYIHVPRZYG + CRZKCRZYJCRZYLYIDCYGYAYBYDYHVFYEYHVEZVGZYIXTUUHXSXTYBYDYHVHCGHKLPVIZUPZCH + JYGKLNVJVKZUUKYIYNYJERZYFYJISZYIXSYNYKYPUPYIYJUUCEYIUUIYJYTUQZADURZYJUUCR + UUNYIYHYJYJUQUURUUJYIYJVLAYGDYTYJYJYSYGKJVMVNVOUUBUURBYJCYRYJUQUUAUUQADYR + YJYTVQVRVSVTQWAYNYOUUOUUPUUDCEFHIYJLMOWBWCVOYIUUFUUGUUHYJYGISYLUUKUUMCHIJ + YGKLMNWDVKWFYEUATZCRZUUSYGISZUBDWEZUNZUUSYFISZUUSUCTZISZUCEWEZUVCUVFUCEUV + EERUVECRZUVEUDTZKJUIZUQZUDDURZUFUVCUVFUUBUVLBUVECEBUCWQZUUBUVEYTUQZADURUV + LUVMUUAUVNADYRUVEYTVQVRUVNUVKAUDDAUDWQYTUVJUVEYSUVIKJVMWGWHWIQWJUVCUVHUVL + UVFUVCUVHUFZUVKUVFUDDUVOUVIDRZUUSUVJISZUVKUVFWKUVCUVHUVPUVQUVCUVHUVPUNZUU + SUVIISZUUSKISZUVQUVRUVPUVBUVSUVCUVHUVPWLZYEUUTUVBUVHUVPWMZUVAUVSUBUVIDYGU + VIUUSIWNWOVOUVRCHIUUSYCKLMUVRHUVCUVHXSUVPXSXTYBYDUUTUVBWPZWRZUKZYEUUTUVBU + VHUVPWSZUVRYNYBYCCRUVRXSYNUWDYPUPZYAYBYDUUTUVBUVHUVPWTZCDFHLOVBVOUVRXTUUH + UVCUVHXTUVPXSXTYBYDUUTUVBXAWRUULUPZUVRUUSYCISZUVBUWBUVRYNUUTYBUWJUVBXBUWG + UWFUWHUBCDFHIUUSLMOXCVKXDYAYBYDUUTUVBUVHUVPXGWFUVRUUFUUTUVICRUUHUVSUVTUFU + VQXBUWEUWFUVRDCUVIUWHUWAVGUWICHIJUUSUVIKLMNXEXFXHXIUVKUVFUVQUVEUVJUUSIWNX + JXPXKXLXMXNUVCYNUUTUVDUVGXBZUVCXSYNUWCYPUPYEUUTUVBXOYNUUTYOUWKUUDUCCEFHIU + USLMOXCXQVOXDYQYAYBYDXOUUEXR $. $d u v .<_ $. $d v B $. $d u v G $. $d u v H $. $d u v K $. dihglblem.i $e |- J = ( ( DIsoB ` K ) ` W ) $. @@ -633886,8 +636298,8 @@ of phi(x) is independent of the atom q." (Contributed by NM, $( The value of isomorphism H at the fiducial atom ` P ` is determined by the vector ` <. 0 , S >. ` (the zero translation ~ ltrnid and a nonzero member of the endomorphism ring). In particular, ` S ` can be replaced - with the ring unit ` ( _I |`` T ) ` . (Contributed by NM, 26-Aug-2014.) - (New usage is discouraged.) $) + with the ring unity ` ( _I |`` T ) ` . (Contributed by NM, + 26-Aug-2014.) (New usage is discouraged.) $) dihpN $p |- ( ph -> ( I ` P ) = ( N ` { <. ( _I |` B ) , S >. } ) ) $= ( vg vs clsa cfv cid cres cop c0g eqid dvhlvec dihat cv wceq crio wcel wa @@ -634341,16 +636753,16 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= dihrnss dochval syldan cple clat ad2antrr hlclat ssrab2 clatglbcl sylancl hllat ccla dihcnvcl ssid dihcnvid2 sseqtrrid fveq2 sseq2d elrab clatglble a1i sylanbrc syl3anc wral adantr sseq1d simpll simpr dihord bitr3d biimpd - wb expimpd syl5bi ralrimiv clatleglb mpbird latasymd fveq2d eqtrd ) CUANZ - FANZOZGBUBNZOZGDPZGLUCZBPZQZLCUDPZUEZCUFPZPZEPZBPZGBUGPZEPZBPWKWLGFCUHPPZ - UDPZQWNXCRXFABCXGFGIXFSZJXGSZUILWRXFWTABCDEXGFGUAWRSZWTSZHIJXHXIKUJUKWMXB - XEBWMXAXDEWMWRCCULPZXAXDXJXLSZWICUMNWJWLCUSUNWMCUTNZWSWRQZXAWRNWIXNWJWLCU - OUNZWQLWRUPZWRWSWTCXJXKUQURWRABCFGXJIJVAZWMXNXOXDWSNZXAXDXLTXPXOWMXQVIZWM - XDWRNZGXDBPZQZXSXRWMGGYBGVBABCFGIJVCZVDWQYCLXDWRWOXDRWPYBGWOXDBVEVFVGVJWR - WSWTCXLXDXJXMXKVHVKWMXDXAXLTZXDMUCZXLTZMWSVLZWMYGMWSYFWSNYFWRNZGYFBPZQZOW - MYGWQYKLYFWRWOYFRWPYJGWOYFBVEVFVGWMYIYKYGWMYIOZYKYGYLYBYJQZYKYGYLYBGYJWMY - BGRYIYDVMVNYLWKYAYIYMYGVTWKWLYIVOWMYAYIXRVMWMYIVPWRABCXLFXDYFXJXMIJVQVKVR - VSWAWBWCWMXNYAXOYEYHVTXPXRXTMWRWSWTCXLXDXJXMXKWDVKWEWFWGWGWH $. + wb expimpd biimtrid ralrimiv clatleglb mpbird latasymd fveq2d eqtrd ) CUA + NZFANZOZGBUBNZOZGDPZGLUCZBPZQZLCUDPZUEZCUFPZPZEPZBPZGBUGPZEPZBPWKWLGFCUHP + PZUDPZQWNXCRXFABCXGFGIXFSZJXGSZUILWRXFWTABCDEXGFGUAWRSZWTSZHIJXHXIKUJUKWM + XBXEBWMXAXDEWMWRCCULPZXAXDXJXLSZWICUMNWJWLCUSUNWMCUTNZWSWRQZXAWRNWIXNWJWL + CUOUNZWQLWRUPZWRWSWTCXJXKUQURWRABCFGXJIJVAZWMXNXOXDWSNZXAXDXLTXPXOWMXQVIZ + WMXDWRNZGXDBPZQZXSXRWMGGYBGVBABCFGIJVCZVDWQYCLXDWRWOXDRWPYBGWOXDBVEVFVGVJ + WRWSWTCXLXDXJXMXKVHVKWMXDXAXLTZXDMUCZXLTZMWSVLZWMYGMWSYFWSNYFWRNZGYFBPZQZ + OWMYGWQYKLYFWRWOYFRWPYJGWOYFBVEVFVGWMYIYKYGWMYIOZYKYGYLYBYJQZYKYGYLYBGYJW + MYBGRYIYDVMVNYLWKYAYIYMYGVTWKWLYIVOWMYAYIXRVMWMYIVPWRABCXLFXDYFXJXMIJVQVK + VRVSWAWBWCWMXNYAXOYEYHVTXPXRXTMWRWSWTCXLXDXJXMXKWDVKWEWFWGWGWH $. $} ${ @@ -634859,14 +637271,14 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= dochkrshp $p |- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) e. Y ) ) $= ( cfv wne wcel wceq wa simpr chlt adantr 2fveq3 dochoc1 sylan9eqr necon3d - eqtr4d ex wn df-ne dvhlvec lkrshpor orcomd ord syl5bi syld imp dochshpncl - mpbid necon1d necon3ad jcad dochlkr sylibrd clmod dvhlmod lshpne impbid ) - ADGUAZHUAHUAZIUBZVPKUCZAVQVPVOUDZVOKUCZUEVRAVQVSVTAVPVOVPIAVPVOUBZVPIUDZA - WAUEZWAWBAWAUFWCBEFHIJVOKLMNOPAFUGUCJEUCUEWASUHAWAVTAWAVOIUBZVTAVOIVPVOAV - OIUDZVSAWEUEVPIVOWEAVPIHUAHUAIVOIHHUIABEFHIJLNMOSUJUKZAWEUFUMUNULWDWEUOZA - VTVOIUPAWEVTAVTWEACDKGIBOPQRABEFJLNSUQTURUSUTZVAVBVCVDVEUNVFAVQWGVTAWEVPI - AWEWBWFUNVGWHVBVHABCDEFGHJKLMNQPRSTVIVJAVRVQAVRUEVPKIBOPABVKUCVRABEFJLNSV - LUHAVRUFVMUNVN $. + eqtr4d ex wn df-ne dvhlvec lkrshpor orcomd biimtrid syld dochshpncl mpbid + ord imp necon1d necon3ad jcad dochlkr sylibrd clmod dvhlmod lshpne impbid + ) ADGUAZHUAHUAZIUBZVPKUCZAVQVPVOUDZVOKUCZUEVRAVQVSVTAVPVOVPIAVPVOUBZVPIUD + ZAWAUEZWAWBAWAUFWCBEFHIJVOKLMNOPAFUGUCJEUCUEWASUHAWAVTAWAVOIUBZVTAVOIVPVO + AVOIUDZVSAWEUEVPIVOWEAVPIHUAHUAIVOIHHUIABEFHIJLNMOSUJUKZAWEUFUMUNULWDWEUO + ZAVTVOIUPAWEVTAVTWEACDKGIBOPQRABEFJLNSUQTURUSVDZUTVAVEVBVCUNVFAVQWGVTAWEV + PIAWEWBWFUNVGWHVAVHABCDEFGHJKLMNQPRSTVIVJAVRVQAVRUEVPKIBOPABVKUCVRABEFJLN + SVLUHAVRUFVMUNVN $. $} ${ @@ -637333,24 +639745,24 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane). \/ E. x e. ( V \ { .0. } ) G = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) ) ) ) $= ( wcel cfv wceq cv co csn wrex crio cmpt cdif wo wa wn wne df-ne cur eqid - chlt ad2antrr biimpa orcomd ord syl5bi imp dochkr1 dvhlmod lkrssv dochssv - lcfl2 syl2anc ssdifd ad3antrrr simprl sseldd simprr lcfl6lem reximssdv ex - wss syl5bir orrd olc syl5ibr w3a adantr eldifi adantl snssd dochcl dochoc - cdih crn 3adant3 simp3 fveq2d simpr dochsnkr2 eqtrd 3ad2ant1 lcfl1 mpbird - 3eqtr4d rexlimdv3a jaod impbid ) ANEUPZNQUQZSURZNDSDUSCUSLUSBUSZIUTFUTURC - YDVAZRUQZVBLGVCVDZURZBSUAVAZVEZVBZVFZAYAYLAYAVGZYCYKYCVHZYBSVIZYMYKYBSVJZ - YMYOYKYMYOVGZYDNUQHVKUQZURZYHBYJYBRUQZYIVEZYQBHJYRMNOPQRSTUAUBUCUDUEUHUJY - RVLZUKULAPVMUPTOUPVGZYAYOUNVNANMUPZYAYOUOVNYMYOYTRUQZSVIZYOYNYMUUFYPYMYCU - UFYMUUFYCAYAUUFYCVFZAEJKMNOPQRSTUBUCUDUEUKULUMUNUOWDZVOVPVQVRVSVTYQYDUUAU - PZYSVGZVGZUUAYJYDAUUAYJWNYAYOUUJAYTSYIAUUCYBSWNYTSWNUNAMNQSJUEUKULAJOPTUB - UDUNWAUOWBJOPRSTYBUBUDUEUCWCWEWFWGYQUUIYSWHZWIUUKCDFGHIJYRLMNOPQRSTYDUAUB - UCUDUEUFUGUHUUBUIUJUKULAUUCYAYOUUJUNWGAUUDYAYOUUJUOWGUULYQUUIYSWJWKWLWMWO - WPWMAYCYAYKYCYAAUUGYCUUFWQUUHWRAYHYABYJAYDYJUPZYHWSZYAUUEYBURUUNYFRUQZRUQ - ZYFUUEYBAUUMUUPYFURZYHAUUMVGZUUCYFTPXFUQUQZXGUPZUUQAUUCUUMUNWTZUURUUCYESW - NUUTUVAUURYDSUUMYDSUPAYDSYIXAXBXCJOUUSPRSTYEUBUUSVLZUDUEUCXDWEOUUSPRTYFUB - UVBUCXEWEXHUUNYTUUORUUNYBYFRUUNYBYGQUQZYFUUNNYGQAUUMYHXIXJAUUMUVCYFURYHUU - RCDHFGIJLYGOPQRSTYDUAUBUCUDUEUJUFUGULUHUIYGVLUVAAUUMXKXLXHXMZXJXJUVDXQUUN - EKMNQRUMAUUMUUDYHUOXNXOXPXRXSXT $. + chlt ad2antrr lcfl2 biimpa orcomd ord biimtrid imp dochkr1 dvhlmod lkrssv + dochssv syl2anc ssdifd ad3antrrr simprl sseldd lcfl6lem reximssdv syl5bir + wss simprr ex orrd olc syl5ibr w3a cdih adantr eldifi adantl snssd dochcl + dochoc 3adant3 simp3 fveq2d simpr dochsnkr2 eqtrd 3eqtr4d 3ad2ant1 mpbird + crn lcfl1 rexlimdv3a jaod impbid ) ANEUPZNQUQZSURZNDSDUSCUSLUSBUSZIUTFUTU + RCYDVAZRUQZVBLGVCVDZURZBSUAVAZVEZVBZVFZAYAYLAYAVGZYCYKYCVHZYBSVIZYMYKYBSV + JZYMYOYKYMYOVGZYDNUQHVKUQZURZYHBYJYBRUQZYIVEZYQBHJYRMNOPQRSTUAUBUCUDUEUHU + JYRVLZUKULAPVMUPTOUPVGZYAYOUNVNANMUPZYAYOUOVNYMYOYTRUQZSVIZYOYNYMUUFYPYMY + CUUFYMUUFYCAYAUUFYCVFZAEJKMNOPQRSTUBUCUDUEUKULUMUNUOVOZVPVQVRVSVTWAYQYDUU + AUPZYSVGZVGZUUAYJYDAUUAYJWMYAYOUUJAYTSYIAUUCYBSWMYTSWMUNAMNQSJUEUKULAJOPT + UBUDUNWBUOWCJOPRSTYBUBUDUEUCWDWEWFWGYQUUIYSWHZWIUUKCDFGHIJYRLMNOPQRSTYDUA + UBUCUDUEUFUGUHUUBUIUJUKULAUUCYAYOUUJUNWGAUUDYAYOUUJUOWGUULYQUUIYSWNWJWKWO + WLWPWOAYCYAYKYCYAAUUGYCUUFWQUUHWRAYHYABYJAYDYJUPZYHWSZYAUUEYBURUUNYFRUQZR + UQZYFUUEYBAUUMUUPYFURZYHAUUMVGZUUCYFTPWTUQUQZXPUPZUUQAUUCUUMUNXAZUURUUCYE + SWMUUTUVAUURYDSUUMYDSUPAYDSYIXBXCXDJOUUSPRSTYEUBUUSVLZUDUEUCXEWEOUUSPRTYF + UBUVBUCXFWEXGUUNYTUUORUUNYBYFRUUNYBYGQUQZYFUUNNYGQAUUMYHXHXIAUUMUVCYFURYH + UURCDHFGIJLYGOPQRSTYDUAUBUCUDUEUJUFUGULUHUIYGVLUVAAUUMXJXKXGXLZXIXIUVDXMU + UNEKMNQRUMAUUMUUDYHUOXNXQXOXRXSXT $. $d k l u v w y z $. $d l u x y z .+ $. $d y G $. $d y ph $. $d l z S $. $d l u y z ._|_ $. $d y z .0. $. $d l u x y R $. $d l u x y z .x. $. @@ -638348,19 +640760,18 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane). ( vx vk cv cfv ciun wcel wrex eliun csn wi eleq2i wa chlt cbs adantr eqid wss clmod dvhlmod clss lduallmod ldualelvbase lspsncl syl2anc lssel sylan wceq ldualvbase eleqtrd cvsca co csca wb lspsnel ldualsbase rexeqdv bitrd - lkrssv biimpa clvec dvhlvec lkrss2N mpbird dochss syl3anc sseld ex syl5bi - rexlimdv lspsnid eleqtrrdi 2fveq3 eleq2d rspcev impbid bitrid eqtrid - eqrdv ) ACFDFUHZKUIZMUIZUJZHKUIZMUIZUDAUFXGXIUFUHZXGUKXJXFUKZFDULZAXJXIUK - ZFXJDXFUMAXLXMAXKXMFDXDDUKXDHUNLUIZUKZAXKXMUOZDXNXDUEUPAXOXPAXOUQZXFXIXJX - QJURUKNIUKUQZXEEUSUIZVBXHXEVBZXFXIVBAXRXOUBUTXQGXDKXSEXSVAZRSAEVCUKXOAEIJ - NOQUBVDZUTXQXDBUSUIZGAXNBVEUIZUKZXOXDYCUKABVCUKZHYCUKZYEABETYBVFZABGHYCEV - CRTYCVAZYBUCVGZYDLYCBHYIYDVAZUAVHVIYDXNYCBXDYIYKVJVKAYCGVLXOABGYCEVCRTYIY - BVMUTVNZWCXQXTXDUGUHHBVOUIZVPVLZUGEVQUIZUSUIZULZAXOYQAXOYNUGBVQUIZUSUIZUL - ZYQAYFYGXOYTVRYHYJYMXDUGYRYSLYCBHYRVAZYSVAZYIYMVAZUAVSVIAYNUGYSYPABYRYOYS - YPVCEYOVAZYPVAZTUUAUUBYBVTWAWBWDXQBYPYOYMGHXDKEUGUUDUUERSTUUCAEWEUKXOAEIJ - NOQUBWFUTAHGUKXOUCUTYLWGWHEIJMXSNXHXEOQYAPWIWJWKWLWMWNAXMXLAHDUKXMXLAHXND - 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wi wa mapdordlem1a simprl idd embantd ex sylbid ralimdv2 syl5bi dvhlmod - lssatle mapdordlem1 simprbi adantl adantr simplbi dochlkr simpld simprd - com23 wceq mpbid dochshpsat wex clvec dvhlvec simpr dochsatshp lshpkrex - wrex syl2anc df-rex sylib simprr fveq2d crn clmod lsatssv dochcl dochoc - cdih eqtrd eqeltrd sylanbrc dih1dimat eqtr2d jca eximdv sylibr wb sseq1 - mpd imbi12d ralxfrd bitr2d sylibd simplr sstr ancoms a1i mpand ss2rabdv - impbid bitrd ) APMUNZQMUNZUOULUPZLUNZNUNZPUOZULCUQZYOQUOZULCUQZUOZPQUOZ - AYKYQYLYSACDPFULGHIKLMNOURRSTUGUIUEUAUBUCUKUSACDQFULGHIKLMNOURRSTUGUIUE - UAUBUDUKUSUTAYTUUAAYTYPYRVFZULEVAZUUAYTUUBULCVAAUUCYPYRULCVBAUUBUUBULCE - AYMEVCZYMCVCZUUBVFZUUBAUUDUUEYONUNZJVCZVGZUUFUUBVFZACEFGHIYMKLNFVDUNZOJ - RUESUUKVEZUHUGUIUJUKUBVHAUUIUUJAUUIVGZUUEUUBUUBAUUEUUHVIUUMUUBVJVKVLVMW - FVNVOAUUAUMUPZPUOZUUNQUOZVFZUMBVAUUCABDPQFUMTUFAFIKORSUBVPZUCUDVQAUUQUU - BUMULYOBEAUUDVGZUUGYNWGZYOBVCUUSUUTYNJVCZUUSUUHUUTUVAVGUUDUUHAUUDYMHVCZ - UUHEGHYMLNJUJVRZVSVTUUSFHYMIKLNOJRUESUGUHUIAKURVCOIVCVGZUUDUBWAZUUDUVBA - 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Property (b) of @@ -643001,7 +645412,7 @@ equations of part (5) in [Baer] p. 46, conjoined to share commonality in @{ @d k w @. - @( Define the "e" (unit) vectore. This is the fixed unit vector + @( Define the "e" (unit) vector. This is the fixed unit vector ` <. 0 , 1 >. ` whose span is the lattice isomorphism map of the fiducial atom ` P = ( ( oc ` K ) ` W ) ` (see ~ dihpN ). Its purpose is to provide a fixed reference vector (any other fixed vector would also @@ -643778,12 +646189,12 @@ first fundamental theorem of projective geometry (see ~ mapdpg ). @) mapdh8b $p |- ( ph -> ( I ` <. w , E , T >. ) = ( I ` <. Y , G , T >. ) ) $= ( cv cpr cfv wcel dvhlvec eldifad lspindp5 prcom fveq2i eleq2i wa clvec - csn adantr cdif wne simpr lspexch ex syl5bi mtod mapdh8a ) ABDEFGHIJLKM - NOPQRSTUAUCCVIZUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVCVDVEAUCWKHVJZSVK - ZVLZHUCWKVJSVKVLZAHSTIUCWKUBUGUJAIMPUAUEUFURVMZAUCTUDWAZVBVNAWKTWQVCVNZ - AHTWQVEVNZVGVHVOWNUCHWKVJZSVKZVLZAWOWMXAUCWLWTSWKHVPVQVRAXBWOAXBVSSTIUC - HUDWKUGUIUJAIVTVLXBWPWBAUCTWQWCVLXBVBWBAHTVLXBWSWBAWKTVLXBWRWBAUCWASVKW - KWASVKWDXBVFWBAXBWEWFWGWHWIWJ $. + csn adantr cdif wne simpr lspexch ex biimtrid mtod mapdh8a ) ABDEFGHIJL + KMNOPQRSTUAUCCVIZUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVCVDVEAUCWKHVJZS + VKZVLZHUCWKVJSVKVLZAHSTIUCWKUBUGUJAIMPUAUEUFURVMZAUCTUDWAZVBVNAWKTWQVCV + NZAHTWQVEVNZVGVHVOWNUCHWKVJZSVKZVLZAWOWMXAUCWLWTSWKHVPVQVRAXBWOAXBVSSTI + UCHUDWKUGUIUJAIVTVLXBWPWBAUCTWQWCVLXBVBWBAHTVLXBWSWBAWKTVLXBWRWBAUCWASV + KWKWASVKWDXBVFWBAXBWEWFWGWHWIWJ $. $} ${ @@ -644004,39 +646415,39 @@ first fundamental theorem of projective geometry (see ~ mapdpg ). @) clmod lspprcl simplr simpr lssneln0 clvec dvhlvec lspindpi jca ex mapdhcl reximdva mpd co eqidd simprl mapdheq mpbid simpld ancld weq eleq1w fveq2d sneq neeq1d anbi12d oteq1 oteq3 oteq2d eqtrd reusv3 syl wo ioran xchnxbir - 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(Contributed by NM, 14-May-2015.) (New usage is discouraged.) $) @@ -646284,15 +648695,15 @@ fixed reference functional determined by this vector (corresponding to <-> A. y e. V ( S ` ( F .x. y ) ) = ( G .xb ( S ` y ) ) ) ) $= ( co cfv wceq cv csn cdif wral cun hdmap14lem12 wa wcel velsn lcdlmod c0g clmod eqid lmodvs0 syl2anc hdmapval0 oveq2d fveq2d eqtrd 3eqtr4rd - dvhlmod oveq2 fveq2 eqeq12d syl5ibrcom syl5bi ralrimiv ralunb bitr4di - biantrud lmod0vcl difsnid 3syl raleqdv 3bitrd ) ALRJUPHUQMRHUQIUPURLB - USZJUPZHUQZMWNHUQZIUPZURZBPSUTZVAZVBZWSBXAWTVCZVBZWSBPVBABCDEFGHIJKLM - NOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOVDAXBXBWSBWTVBZVEXDAXEXBAWSBWTWN - WTVFWNSURZAWSBSVGAWSXFLSJUPZHUQZMSHUQZIUPZURAMEVIUQZIUPZXKXJXHAEVJVFM - CVFXLXKURAENOQTUFUIVHUOIFCEMXKUKUGULXKVKZVLVMAXIXKMIAEXKHKNOQSTUAUMUF - XMUHUIVNZVOAXHXIXKAXGSHAKVJVFZLDVFXGSURAKNOQTUAUIVSZUJJGDKLSUDUCUEUMV - LVMVPXNVQVRXFWPXHWRXJXFWOXGHWNSLJVTVPXFWQXIMIWNSHWAVOWBWCWDWEWHWSBXAW - TWFWGAWSBXCPAXOSPVFXCPURXPPKSUBUMWIPSWJWKWLWM $. + dvhlmod oveq2 fveq2 eqeq12d biimtrid ralrimiv biantrud ralunb bitr4di + syl5ibrcom lmod0vcl difsnid 3syl raleqdv 3bitrd ) ALRJUPHUQMRHUQIUPUR + LBUSZJUPZHUQZMWNHUQZIUPZURZBPSUTZVAZVBZWSBXAWTVCZVBZWSBPVBABCDEFGHIJK + LMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOVDAXBXBWSBWTVBZVEXDAXEXBAWSBWT + WNWTVFWNSURZAWSBSVGAWSXFLSJUPZHUQZMSHUQZIUPZURAMEVIUQZIUPZXKXJXHAEVJV + FMCVFXLXKURAENOQTUFUIVHUOIFCEMXKUKUGULXKVKZVLVMAXIXKMIAEXKHKNOQSTUAUM + UFXMUHUIVNZVOAXHXIXKAXGSHAKVJVFZLDVFXGSURAKNOQTUAUIVSZUJJGDKLSUDUCUEU + MVLVMVPXNVQVRXFWPXHWRXJXFWOXGHWNSLJVTVPXFWQXIMIWNSHWAVOWBWHWCWDWEWSBX + AWTWFWGAWSBXCPAXOSPVFXCPURXPPKSUBUMWIPSWJWKWLWM $. $} $( Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, @@ -651245,6 +653656,161 @@ fixed reference functional determined by this vector (corresponding to IEXKFXPVNAXGBWQHCIEXKFXPVOVPVQ $. $} + ${ + $d A a b $. $d F a b $. $d a b ph $. + fldhmf1.1 $e |- ( ph -> K e. Field ) $. + fldhmf1.2 $e |- ( ph -> L e. Field ) $. + fldhmf1.3 $e |- ( ph -> F e. ( K RingHom L ) ) $. + fldhmf1.4 $e |- A = ( Base ` K ) $. + fldhmf1.5 $e |- B = ( Base ` L ) $. + $( A field homomorphism is injective. This follows immediately from the + definition of the ring homomorphism that sends the multiplicative + identity to the multiplicative identity. (Contributed by metakunt, + 7-Jan-2025.) $) + fldhmf1 $p |- ( ph -> F : A -1-1-> B ) $= + ( wne cfv wa co wcel syl wceq eqid syl2anc va vb wf wral wf1 crh rhmf c0g + cv wi cur cminusg cplusg cinvr cmulr cghm ad4antr rhmghm simp-4r cgrp cdr + cfield isfld simpld drnggrp simpllr grpinvcl ghmlin syl3anc ghminv oveq2d + ccrg sylib simpr oveq1d crg ad3antrrr drngring ringgrpd ffvelcdmd grprinv + adantr eqtrd grpcl grpinvinv simplr necomd eqnetrd wb grpinvid2 necon3bid + cui w3a jca drngunit mpbird rhmunitinv elrhmunit unitinvcl biimpd eqeltrd + mpd ringlz eqcomd simprd crngringd unitcl eqcomi eleqtrdi rhmmul unitrinv + cbs fveq2d rhm1 3eqtrd drngunz neneqd pm2.65da neqned ex ralrimiva dff14a + mpbid sylibr ) ABCDUCZUAUIZUBUIZLZYFDMZYGDMZLZUJZUBBUDZUABUDZNBCDUEAYEYNA + DEFUFOPZYEIBCEFDJKUGZQAYMUABAYFBPZNZYLUBBYRYGBPZNZYHYKYTYHNZYIYJUUAYIYJRZ + FUHMZFUKMZRUUAUUBNZUUCYFYGEULMZMZEUMMZOZDMZUUIEUNMZMZDMZFUOMZOZUUIUULEUOM + ZOZDMZUUDUUEUUOUUCUUEUUOUUCUUMUUNOZUUCUUEUUJUUCUUMUUNUUEUUJYIUUGDMZFUMMZO + ZUUCUUEDEFUPOPZYQUUGBPZUUJUVBRUUEYOUVCAYOYQYSYHUUBIUQZEFDURQZAYQYSYHUUBUS + ZUUEEUTPZYSUVDUUEEVAPZUVHAUVIYQYSYHUUBAUVIEVLPZAEVBPUVIUVJNGEVCVMZVDUQZEV + EQZYRYSYHUUBVFZBEUUFYGJUUFSZVGTZUUHUVAEFYFDUUGBJUUHSZUVASZVHVIUUEUVBYIYJF + ULMZMZUVAOZUUCUUEUUTUVTYIUVAUUEUVCYSUUTUVTRUVFUVNBEFDUUFUVSYGJUVOUVSSZVJT + VKUUEUWAYJUVTUVAOZUUCUUEYIYJUVTUVAUUAUUBVNVOUUEFUTPYJCPUWCUUCRUUEFUUEFVAP + ZFVPPZUUAUWDUUBUUAUWDFVLPZUUAFVBPZUWDUWFNAUWGYQYSYHHVQFVCVMVDZWBZFVRQZVSU + UEBCYGDUUEYOYEUVEYPQUVNVTCUVAFUVSYJUUCKUVRUUCSZUWBWATWCWCWCVOUUEUWEUUMCPZ + NUUSUUCRUUEUWEUWLUWJUUEUUMUUJFUNMZMZCUUEYOUUIEWLMZPZUUMUWNRUVEUUEUWPUUIBP + ZUUIEUHMZLZNZUUEUWQUWSUUEUVHYQUVDUWQUVMUVGUVPBUUHEYFUUGJUVQWDVIZUUEUUGUUF + MZYFLZUWSUUEUXBYGYFUUEUVHYSUXBYGRUVMUVNBEUUFYGJUVOWETUUEYFYGYTYHUUBWFWGWH + UUEUVHUVDYQUXCUWSWIUVMUVPUVGUVHUVDYQWMUXBYFUUIUWRBUUHEUUFUUGYFUWRJUVQUWRS + ZUVOWJWKVIYCWNUUEUVIUWPUWTWIUVLBEUWOUUIUWRJUWOSZUXDWOQWPZUUIEFDWQTUUEUWNC + PZUWNUUCLZUUEUWNFWLMZPZUXGUXHNZUUEUWEUUJUXIPZUXJUWJUUEYOUWPUXLUVEUXFUUIEF + DWRTFUXIUWMUUJUXISZUWMSWSTUUEUXJUXKUUEUWDUXJUXKWIUWICFUXIUWNUUCKUXMUWKWOQ + WTXBVDXAWNCFUUNUUMUUCKUUNSZUWKXCQWCXDUUEUURUUOUUEYOUWQUULBPZUURUUORUVEUXA + UUEUULUWOPZUXOUUEEVPPZUWPUXPAUXQYQYSYHUUBAEAUVIUVJUVKXEXFUQUXFEUWOUUKUUIU + XEUUKSZWSTUXPUULEXLMZBUXSEUWOUULUXSSUXEXGBUXSJXHXIQUUIUULEFUUPUUNDBJUUPSZ + UXNXJVIXDUUEUUREUKMZDMZUUDUUEUUQUYADUUEUXQUWPUUQUYARUUEUVIUXQUVLEVRQUXFEU + UPUWOUYAUUKUUIUXEUXRUXTUYASZXKTXMUUEYOUYBUUDRUVEEFUYADUUDUYCUUDSZXNQWCXOU + UEUUCUUDUUAUUCUUDLUUBUUAUUDUUCUUAUWDUUDUUCLUWHFUUDUUCUWKUYDXPQWGWBXQXRXSX + TYAYAWNUAUBBCDYBYD $. + $} + + ${ + $d N a k l $. $d P a k l $. $d a ph $. + aks6d1c2p1.1 $e |- ( ph -> N e. NN ) $. + aks6d1c2p1.2 $e |- ( ph -> P e. Prime ) $. + aks6d1c2p1.3 $e |- ( ph -> P || N ) $. + aks6d1c2p1.4 $e |- E = ( k e. NN0 , l e. NN0 |-> + ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $. + $( In the AKS-theorem the subset defined by ` E ` takes values in the + positive integers. (Contributed by metakunt, 7-Jan-2025.) $) + aks6d1c2p1 $p |- ( ph -> E : ( NN0 X. NN0 ) --> NN ) $= + ( va cn0 cv cfv cexp co cmul cn wcel syl cxp c1st cdiv c2nd cprime adantr + wa prmnn simpr xp1st nnexpcld cdvds wbr wb nndivdvds mpbid xp2nd nnmulcld + jca cmpo cmpt cop wceq vex op1std oveq2d op2ndd mpompt eqcomi eqtri fmptd + oveq12d ) AKLLUAZBKMZUBNZOPZEBUCPZVNUDNZOPZQPZRDAVNVMSZUGZVPVSWBBVOABRSZW + AABUESWCHBUHTZUFWBWAVOLSAWAUIZVNLLUJTUKWBVQVRAVQRSZWAABEULUMZWFIAERSZWCUG + WGWFUNAWHWCGWDUSEBUOTUPUFWBWAVRLSWEVNLLUQTUKURDCFLLBCMZOPZVQFMZOPZQPZUTZK + VMVTVAZJWOWNCFKLLVTWMVNWIWKVBVCZVPWJVSWLQWPVOWIBOWIWKVNCVDZFVDZVEVFWPVRWK + VQOWIWKVNWQWRVGVFVLVHVIVJVK $. + $} + + ${ + $d E a b c d $. $d N k l $. $d N p $. $d P k l $. $d P p $. $d Q p $. + $d a b c d k l ph $. $d a b c d p ph $. + aks6d1c2p2.1 $e |- ( ph -> N e. NN ) $. + aks6d1c2p2.2 $e |- ( ph -> P e. Prime ) $. + aks6d1c2p2.3 $e |- ( ph -> P || N ) $. + aks6d1c2p2.4 $e |- E = ( k e. NN0 , l e. NN0 |-> + ( ( P ^ k ) x. ( ( N / P ) ^ l ) ) ) $. + aks6d1c2p2.5 $e |- ( ph -> Q e. Prime ) $. + aks6d1c2p2.6 $e |- ( ph -> Q || N ) $. + aks6d1c2p2.7 $e |- ( ph -> P =/= Q ) $. + $( Injective condition for countability argument assuming that ` N ` is not + a prime power. (Contributed by metakunt, 7-Jan-2025.) $) + aks6d1c2p2 $p |- ( ph -> E : ( NN0 X. NN0 ) -1-1-> NN ) $= + ( cn0 co wceq wa wcel syl va vb vc vd vp cxp cn wf cv wral wf1 aks6d1c2p1 + wi wn wne cexp cdiv cmul neneq orcd simpr neneqd olcd jaoi anim1ci adantl + wo neqne pm2.61dan impbii orcom bitri ianor bicomi cpc cprime wrex oveq1d + ad5antr neeq12d caddc cc0 0cnd cdvds wbr prmnn jca nndivdvds mpbid adantr + ad2antrr simp-4r nnexpcld pccld nn0cnd simplr simp-5l nncnd nnne0d eqcomd + wb divcan2d breq2d cz nnzd euclemma syl3anc biimpd mpd c1 necom mpbi 1red + imbi2i mpbird ad4antr simpllr nn0zd 3jca pcexp 3netr3d prmdvdsexpr zexpcl + w3a pceq0 expne0d pcmul rspcedvd pcidlem simprl oveq2d biidd nnnn0d bitrd + nnmulcld a1i simprr oveq12d ovmpod ralrimiva prmgt1 ltned sylib dvdsprime + clt necomd pm4.56 syl2anc mtbird pcelnn mulcan2d necon3bid cq nnq divne0d + orcnd addneintrd con3d divcld simp-5r addneintr2d eqidd jaodan necon3abid + mtod pc11 notbid rexnal necon3bbid rexbidv sylan2br ex con4d f1opr sylibr + cmpo ) 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(Contributed by metakunt, 8-Jun-2024.) $) 5bc2eq10 $p |- ( 5 _C 2 ) = ; 1 0 $= @@ -652016,10 +654582,10 @@ fixed reference functional determined by this vector (corresponding to $d B a j k l $. $d B o s $. $d B j l r $. $d B j l w $. $d F b k $. $d K a f j l x y $. $d K b f x y $. $d K a g i k $. $d K o s $. $d K j q $. $d K j l r $. $d K j l w $. $d N a f j l $. $d N b f $. - $d N a g i k $. $d a c g i k $. $d a d f j l x y $. $d a c i k l ph $. - $d b c g i k $. $d b d f x y $. $d b c i k ph $. $d c j k l ph $. - $d d g i k $. $d d o ph s $. $d d j q $. $d d j l ph r $. - $d d j l ph w $. $d k ph s $. $d ph x y $. + $d N a g i k $. $d a d f j l x y $. $d a i k l ph $. $d b g i k $. + $d b d f x y $. $d b i k ph $. $d j k l ph $. $d d g i k $. + $d d o ph s $. $d d j q $. $d d j l ph r $. $d d j l ph w $. + $d k ph s $. $d ph x y $. sticksstones12a.1 $e |- ( ph -> N e. NN0 ) $. sticksstones12a.2 $e |- ( ph -> K e. NN ) $. sticksstones12a.3 $e |- F = ( a e. A |-> ( j e. ( 1 ... K ) |-> @@ -652266,14 +654832,12 @@ fixed reference functional determined by this vector (corresponding to ${ $d A a c j k l $. $d A b c k $. $d A a j k l x y $. $d B a d i k l $. - $d B b d k $. $d B a d j k l $. $d B d o s $. $d B d j l r $. - $d B d j l w $. $d F b c k $. $d F b d k $. $d G c $. $d G d $. - $d K a f j l x y $. $d K b k $. $d K a g i k $. $d K o s $. $d K j q $. - $d K j l r $. $d K j l w $. $d N a f j l $. $d N b k $. $d N a g i k $. - $d a c i k l ph $. $d b c k ph $. $d c g i k $. $d d f j l x y $. - $d d g i k $. $d d i k l ph $. $d d j q $. $d j k l ph x y $. - $d k ph s $. $d o ph s $. $d ph r $. $d ph w $. $d B b d i k $. - $d K b f x y $. $d N b f $. $d b c g i k $. $d b c i k ph $. + $d B b d k $. $d B a d j k l $. $d F b c k $. $d F b d k $. $d G c $. + $d G d $. $d K a f j l x y $. $d K b k $. $d K a g i k $. + $d N a f j l $. $d N b k $. $d N a g i k $. $d a c i k l ph $. + $d b c k ph $. $d c g i k $. $d d f j l x y $. $d d g i k $. + $d d i k l ph $. $d j k l ph x y $. $d B b d i k $. $d K b f x y $. + $d N b f $. $d b c g i k $. $d b c i k ph $. sticksstones12.1 $e |- ( ph -> N e. NN0 ) $. sticksstones12.2 $e |- ( ph -> K e. NN ) $. sticksstones12.3 $e |- F = ( a e. A |-> ( j e. ( 1 ... K ) |-> @@ -654209,30 +656773,15 @@ D Fn ( I ... ( M - 1 ) ) /\ ${ $d w x y z $. - $( A version of ~ el with an inner existential quantifier on ` x ` , which - only uses ~ ax-mp to ~ ax-4 , ~ ax-6 , and ~ ax-pr . (Contributed by - SN, 23-Dec-2024.) $) - sn-el $p |- E. y E. x x e. y $= - ( vz weq wo wel wi wal wex ax-pr ax6ev pm2.07 eximii exim mpi ) ACDZPEZAB - FZGAHZRAIZBCCBAJSQAITPQAACKPLMQRANOM $. - - $( A version of ~ elALT2 with an inner existential quantifier on ` x ` , - which avoids ~ ax-8 . (Contributed by SN, 18-Sep-2023.) + $( Alternate proof of ~ exel , avoiding ~ ax-pr but requiring ~ ax-5 , + ~ ax-9 , and ~ ax-pow . This is similar to how ~ elALT2 uses ~ ax-pow + instead of ~ ax-pr compared to ~ el . (Contributed by SN, 18-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.) $) - sn-elALT2 $p |- E. y E. x x e. y $= + sn-exelALT $p |- E. y E. x x e. y $= ( vw vz wel wi wal wex ax-pow weq ax6ev ax9v1 alrimiv eximii exim mpi ) C AECDEFZCGZABEZFAGZSAHZBDBACITRAHUAADJZRAADKUBQCADCLMNRSAOPN $. - $( $j usage 'sn-elALT2' avoids 'ax-8'; $) - - $( ~ dtru without ~ ax-8 . (Contributed by SN, 21-Sep-2023.) $) - sn-dtru $p |- -. A. x x = y $= - ( vw vz weq wn wex wal wel wa sn-el ax-nul exdistrv mpbir2an ax9v1 eximdv - wi con3d ax7v1 spimevw df-ex syl6ib imnan sylib con2i 2eximi equeuclr a1d - syl6 pm2.61i exlimivv mp2b exnal mpbi ) ABEZFZAGZUOAHFACIZAGZADIZFAHZJZDG - CGZCDEZFZDGCGUQVCUSCGVADGACKDALUSVACDMNVBVECDVDVBVDUSVAFZQVBFVDUSUTAGVFVD - URUTACDAOPUTAUAUBUSVAUCUDUEUFVEUQCDDBEZVEUQQVGVECBEZFZUQVGVHVDDCBUGRVIUPA - CACEUOVHACBSRTUIVGFZUQVEVJUPADADEUOVGADBSRTUHUJUKULUOAUMUN $. - $( $j usage 'sn-dtru' avoids 'ax-8'; $) + $( $j usage 'sn-exelALT' avoids 'ax-7' 'ax-8' 'ax-ext' 'ax-sep' 'ax-nul' + 'ax-pr'; $) $} ${ @@ -654374,7 +656923,6 @@ D Fn ( I ... ( M - 1 ) ) /\ $} ${ - $d F x $. $d S x $. fnimasnd.1 $e |- ( ph -> F Fn A ) $. fnimasnd.2 $e |- ( ph -> S e. A ) $. $( The image of a function by a singleton whose element is in the domain of @@ -654526,6 +657074,27 @@ D Fn ( I ... ( M - 1 ) ) /\ =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) + ${ + ressbasssg.r $e |- R = ( W |`s A ) $. + ressbasssg.b $e |- B = ( Base ` W ) $. + $( The base set of a restriction to ` A ` is a subset of ` A ` and the base + set ` B ` of the original structure. (Contributed by SN, + 10-Jan-2025.) $) + ressbasssg $p |- ( Base ` R ) C_ ( A i^i B ) $= + ( cvv wcel cbs cfv cin wss ressbas ssid eqsstrrdi wn c0 cress reldmress + co ovprc2 eqtrid fveq2d base0 0ss eqsstrri eqsstrdi pm2.61i ) AGHZCIJZABK + ZLUIUJUKUKABCGDEFMUKNOUIPZUJQIJZUKULCQIULCDARTQEDARSUAUBUCUMQUKUDUKUEUFUG + UH $. + $} + + ${ + ressbasss2.r $e |- R = ( W |`s A ) $. + $( The base set of a restriction to ` A ` is a subset of ` A ` . + (Contributed by SN, 10-Jan-2025.) $) + ressbasss2 $p |- ( Base ` R ) C_ A $= + ( cbs cfv cin eqid ressbasssg inss1 sstri ) BEFACEFZGAALBCDLHIALJK $. + $} + ${ nelsubginvcld.g $e |- ( ph -> G e. Grp ) $. nelsubginvcld.s $e |- ( ph -> S e. ( SubGrp ` G ) ) $. @@ -654933,7 +657502,7 @@ D Fn ( I ... ( M - 1 ) ) /\ ringlidmd.u $e |- .1. = ( 1r ` R ) $. ringlidmd.r $e |- ( ph -> R e. Ring ) $. ringlidmd.x $e |- ( ph -> X e. B ) $. - $( The unit element of a ring is a left multiplicative identity. + $( The unity element of a ring is a left multiplicative identity. (Contributed by SN, 14-Aug-2024.) $) ringlidmd $p |- ( ph -> ( .1. .x. X ) = X ) $= ( crg wcel co wceq ringlidm syl2anc ) ACLMFBMEFDNFOJKBCDEFGHIPQ $. @@ -654945,7 +657514,7 @@ D Fn ( I ... ( M - 1 ) ) /\ ringridmd.u $e |- .1. = ( 1r ` R ) $. ringridmd.r $e |- ( ph -> R e. Ring ) $. ringridmd.x $e |- ( ph -> X e. B ) $. - $( The unit element of a ring is a right multiplicative identity. + $( The unity element of a ring is a right multiplicative identity. (Contributed by SN, 14-Aug-2024.) $) ringridmd $p |- ( ph -> ( X .x. .1. ) = X ) $= ( crg wcel co wceq ringridm syl2anc ) ACLMFBMFEDNFOJKBCDEFGHIPQ $. @@ -654963,16 +657532,91 @@ D Fn ( I ... ( M - 1 ) ) /\ $} ${ - drngringd.1 $e |- ( ph -> R e. DivRing ) $. - $( A division ring is a ring. (Contributed by SN, 16-May-2024.) $) - drngringd $p |- ( ph -> R e. Ring ) $= - ( cdr wcel crg drngring syl ) ABDEBFECBGH $. + $d C a b x y $. $d F a b $. $d M a b $. $d ph a b x y $. + rncrhmcl.c $e |- C = ( N |`s ran F ) $. + rncrhmcl.h $e |- ( ph -> F e. ( M RingHom N ) ) $. + rncrhmcl.m $e |- ( ph -> M e. CRing ) $. + $( The range of a commutative ring homomorphism is a commutative ring. + (Contributed by SN, 10-Jan-2025.) $) + rncrhmcl $p |- ( ph -> C e. CRing ) $= + ( vx vy va vb wcel cv cmulr cfv co wceq wa eqid crg cbs wral ccrg crh crn + csubrg rnrhmsubrg subrgring 3syl ressbasss2 sseli anim12i wfn wrex wf syl + rhmf ffnd simpl fvelrnb biimpa syl2an simpr adantr simplrl simprl crngcom + ad3antrrr syl3anc fveq2d rhmmul 3eqtr3d cvv rnexg ressmulr simplrr simprr + oveq123d rexlimddv sylan2 ralrimivva iscrng2 sylanbrc ) ABUAMZINZJNZBOPZQ + ZWGWFWHQZRZJBUBPZUCIWLUCBUDMACDEUEQZMZCUFZEUGPMWEGCDEUHWOEBFUIUJAWKIJWLWL + WFWLMZWGWLMZSAWFWOMZWGWOMZSZWKWPWRWQWSWLWOWFWOBEFUKZULWLWOWGXAULUMAWTSZKN + ZCPZWFRZWKKDUBPZACXFUNZWRXEKXFUOZWTAXFEUBPZCAWNXFXICUPGXFXIDECXFTZXITURUQ + USZWRWSUTXGWRXHKXFWFCVAVBVCXBXCXFMZXESZSZLNZCPZWGRZWKLXFXBXQLXFUOZXMAXGWS + XRWTXKWRWSVDXGWSXRLXFWGCVAVBVCVEXNXOXFMZXQSZSZXDXPEOPZQZXPXDYBQZWIWJYAXCX + ODOPZQZCPZXOXCYEQZCPZYCYDYAYFYHCYADUDMZXLXSYFYHRAYJWTXMXTHVIXBXLXEXTVFZXN + XSXQVGZXFDYEXCXOXJYETZVHVJVKYAWNXLXSYGYCRAWNWTXMXTGVIZYKYLXCXODEYEYBCXFXJ + YMYBTZVLVJYAWNXSXLYIYDRYNYLYKXOXCDEYEYBCXFXJYMYOVLVJVMYAXDWFXPWGYBWHAYBWH + RZWTXMXTAWNWOVNMYPGCWMVOWOEBYBVNFYOVPUJVIZXBXLXEXTVQZXNXSXQVRZVSYAXPWGXDW + FYBWHYQYSYRVSVMVTVTWAWBIJWLBWHWLTWHTWCWD $. + $} + + $( The converse of a ring isomorphism is a ring isomorphism. (Contributed by + SN, 10-Jan-2025.) $) + rimcnv $p |- ( F e. ( R RingIso S ) -> `' F e. ( S RingIso R ) ) $= + ( crh co wcel ccnv wa crs cbs cfv wf wceq eqid rhmf wrel frel dfrel2 isrim0 + sylib syl id eqeltrd anim1ci 3imtr4i ) CABDEZFZCGZBADEFZHUIUHGZUFFZHCABIEFU + HBAIEFUGUKUIUGUJCUFUGAJKZBJKZCLZUJCMZULUMABCULNUMNOUNCPUOULUMCQCRTUAUGUBUCU + DABCSBAUHSUE $. + + $( The composition of ring isomorphisms is a ring isomorphism. (Contributed + by SN, 17-Jan-2025.) $) + rimco $p |- ( ( F e. ( S RingIso T ) /\ G e. ( R RingIso S ) ) -> + ( F o. G ) e. ( R RingIso T ) ) $= + ( crs co wcel ccom crh ccnv isrim0 rhmco cnvco ancoms eqeltrid anim12i an4s + wa syl2anb sylibr ) DBCFGHZEABFGHZSDEIZACJGHZUDKZCAJGZHZSZUDACFGHUBDBCJGHZD + KZCBJGHZSEABJGHZEKZBAJGHZSUIUCBCDLABELUJUMULUOUIUJUMSUEULUOSZUHABCDEMUPUFUN + UKIZUGDENUOULUQUGHCBAUNUKMOPQRTACUDLUA $. + + $( Prove isomorphic by an explicit isomorphism. (Contributed by SN, + 10-Jan-2025.) $) + brrici $p |- ( F e. ( R RingIso S ) -> R ~=r S ) $= + ( crs co wcel c0 wne cric wbr ne0i brric sylibr ) CABDEZFNGHABIJNCKABLM $. - $( A division ring is a group. (Contributed by SN, 16-May-2024.) $) - drnggrpd $p |- ( ph -> R e. Grp ) $= - ( drngringd ringgrpd ) ABABCDE $. + ${ + $d R f $. $d S f $. + $( Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.) $) + ricsym $p |- ( R ~=r S -> S ~=r R ) $= + ( vf cric wbr crs co c0 wne brric wcel wex ccnv rimcnv brrici syl exlimiv + cv n0 sylbi ) ABDEABFGZHIZBADEZABJUBCRZUAKZCLUCCUASUEUCCUEUDMZBAFGKUCABUD + NBAUFOPQTT $. $} + ${ + $d R f g $. $d S f g $. $d T f g $. + $( Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.) $) + rictr $p |- ( ( R ~=r S /\ S ~=r T ) -> R ~=r T ) $= + ( vf vg cric wbr crs co c0 wne brric cv wcel wex n0 exdistrv ccom syl2anb + wa rimco brrici syl ancoms exlimivv sylbir ) ABFGABHIZJKZBCHIZJKZACFGZBCF + GABLBCLUHDMZUGNZDOZEMZUINZEOZUKUJDUGPEUIPUNUQTUMUPTZEODOUKUMUPDEQURUKDEUP + UMUKUPUMTUOULRZACHINUKABCUOULUAACUSUBUCUDUEUFSS $. + $} + + ${ + $d R f $. $d S f $. + $( Ring isomorphism preserves (multiplicative) commutativity. (Contributed + by SN, 10-Jan-2025.) $) + riccrng1 $p |- ( ( R ~=r S /\ R e. CRing ) -> S e. CRing ) $= + ( vf cric wbr cv crs co wcel wex ccrg c0 wne wceq cbs cfv eqid cvv adantr + cress brric n0 bitri wi crn wf1o wfo rimf1o f1ofo forn 3syl oveq2d rimrcl + wa ressid simpl2im eqtr2d rimrhm simpr rncrhmcl eqeltrd ex exlimiv sylanb + crh imp ) ABDEZCFZABGHZIZCJZAKIZBKIZVGVILMVKABUACVIUBUCVKVLVMVJVLVMUDCVJV + LVMVJVLUNZBBVHUEZTHZKVJBVPNVLVJVPBBOPZTHZBVJVOVQBTVJAOPZVQVHUFVSVQVHUGVOV + QNVSVQABVHVSQVQQZUHVSVQVHUIVSVQVHUJUKULVJARIBRIVRBNABVHUMVQBRVTUOUPUQSVNV + PVHABVPQVJVHABVEHIVLABVHURSVJVLUSUTVAVBVCVFVD $. + $} + + $( A ring is commutative if and only if an isomorphic ring is commutative. + (Contributed by SN, 10-Jan-2025.) $) + riccrng $p |- ( R ~=r S -> ( R e. CRing <-> S e. CRing ) ) $= + ( cric wbr ccrg wcel riccrng1 ricsym sylan impbida ) ABCDZAEFZBEFZABGKBACDM + LABHBAGIJ $. + ${ drnginvrcld.b $e |- B = ( Base ` R ) $. drnginvrcld.0 $e |- .0. = ( 0g ` R ) $. @@ -655071,12 +657715,10 @@ D Fn ( I ... ( M - 1 ) ) /\ $} ${ - fldcrngd.1 $e |- ( ph -> R e. Field ) $. - $( A field is a commutative ring. EDITORIAL: Shortens ~ recrng . Also - ~ recrng should be named resrng. Also ~ fldcrng is misnamed. - (Contributed by SN, 23-Nov-2024.) $) - fldcrngd $p |- ( ph -> R e. CRing ) $= - ( cfield wcel ccrg cdr isfld simprbi syl ) ABDEZBFEZCKBGELBHIJ $. + flddrngd.1 $e |- ( ph -> R e. Field ) $. + $( A field is a division ring. (Contributed by SN, 17-Jan-2025.) $) + flddrngd $p |- ( ph -> R e. DivRing ) $= + ( cfield wcel cdr ccrg isfld simplbi syl ) ABDEZBFEZCKLBGEBHIJ $. $} ${ @@ -655088,7 +657730,7 @@ D Fn ( I ... ( M - 1 ) ) /\ $( A vector space is a group. (Contributed by SN, 28-May-2023.) $) lvecgrp $p |- ( W e. LVec -> W e. Grp ) $= - ( clvec wcel clmod cgrp lveclmod lmodgrp syl ) ABCADCAECAFAGH $. + ( clvec wcel lveclmod lmodgrpd ) ABCAADE $. ${ lveclmodd.1 $e |- ( ph -> W e. LVec ) $. @@ -655659,90 +658301,90 @@ proof usage (see ~ mplelbas and ~ mpladd ). Note that hypothesis 0 is wrex syl simprl mapfvd ad4antr adantrl fvoveq1 csn cn0 ad2antrl ad3antrrr simprr simplrr wo adantllr c0 eqnetrrd hasheq0 sylib eleq1w imbi12d caddc cbs euhash1 bitri wreu c0g eubidv df-reu crio eqidd simplr riota2 3bitr4g - necom biimpd necon1bd ifeqda riotacl elmapi ffvelcdmd eqeq2 sylbid syl5bi - cof cgrp grpidcl ifcld fmpttd mpbird oveq1 anbi12d cdif mpbir2and simpllr - nnnn0d eqcomd w3a hashdifsnp1 syl31anc eldifsn iftrue 2thd iffalse eqeq1d - ovexd biorf orcom bitrd pm2.61i necon3abid neanior anass equequ1 ifbieq2d - olc anbi2d pm5.32da anbi1d 3bitr4d bitr2d bitrid eqrdv fveq2d eqtr3d offn - inidm ofval simplrl anassrs grplid grprid ifeq12d ovif12 ifid 3eqtr4g jca - eqcomi rspcedvd crab suppvalfn peano2nn nnne0d necon3bid mp1i mpbid rabn0 - ad3antlr reximddv rspccva adantll adantlrr adantrr equequ2 rexlimdvva mpd - rexcom ovrspc2v exp32 ralrimiv cbvralvw nnindd ralcom biidd biimpcd eleq1 - ralimi rspcv syl5com com23 sylbird an32s adantlr cxp ffn fnsuppeq0 biimpa - ceqsralv sylan2b cfn fsuppimpd hashcl elnn0 mpjaodan anasss ) AHDJUKZJKUL - UMZJGUNZAVVEUOZVVFUOZJKUPUQZURUSZVAUNZVVGVVKUTVBZVVHVVLVVGVVFAVVLVVEVVGAV - VLUOZVVEVVGVVNVVEJDHVCUQZUNZVVGAVVPVVEVDVVLADHJVEIDVEUNZADFUUBNVFZVGRVHVI - AVVLVVPVVGVJAVVPVVLVVGAUBVKZKUPUQZURUSZVAUNZVVSGUNZVJZUBVVOVPZVVPVVLVVGVJ - ZAUCVKZVWAVBZVWCVJZUCVAVPZUBVVOVPZVWEAVWIUBVVOVPZUCVAVPVWKAVWLUCVAAUDVKZV - WAVBZVWCVJZUBVVOVPVLVWAVBZVWCVJZUBVVOVPUEVKZVWAVBZVWCVJZUBVVOVPZVWRVLUUAU - QZVWAVBZVWCVJZUBVVOVPZVWLUDUEVWGVWMVLVBZVWOVWQUBVVOVXFVWNVWPVWCVWMVLVWAVM - VNVOVWMVWRVBZVWOVWTUBVVOVXGVWNVWSVWCVWMVWRVWAVMVNVOVWMVXBVBZVWOVXDUBVVOVX - HVWNVXCVWCVWMVXBVWAVMVNVOVWMVWGVBZVWOVWIUBVVOVXIVWNVWHVWCVWMVWGVWAVMVNVOA - VWQUBVVOVWPUFVKZVVTUNZUFVQZAVVSVVOUNZUOZVWCVWPVWAVLVBZVXLVLVWAVRVVTVEUNVX - OVXLVDVVSKUPVSVVTVEUFUUCVTUUDVXNVXLVXJVVSUSZKWAZUFHUUEZVWCVXNVXLVXJHUNVXQ - UOZUFVQVXRVXNVXKVXSUFVXNVVSHWBZHIUNZKVEUNZVXKVXSVDVXMVXTAVVSDHWCZWDAVYAVX - MRVIVYBVXNKFUUFOVFZVGVXJVVSIVEHKWEWFUUGVXQUFHUUHWGVXNVXRVWCVXNVXRUOZVVSBH - 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$} ${ @@ -656871,28 +659513,28 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove co syl c1 oveq12d eqtrd cn0 cn wo wi elnn0 expgcd wa cabs cfv 0exp cz nnnn0 nnexpcld nnzd gcd0id nnred 0red nngt0d ltled absidd oveq1 nnz 3ad2ant2 nnre 3eqtrrd nngt0 3eqtrd 1z ax-mp eqcomi simp1 simp3 nncnd exp0d 0exp0e1 oveq2d - gcd1 a1i 3eqtr4a syl5bi nn0expcld nn0gcdid0 3eqtr4rd eqtr4di gcd0val eqtr4i - jaod nncn oveq1i oveq12i 3eqtr4i ccase syl2anb 3impia ) AUADZBUADZCUADZABEP - ZCFPZACFPZBCFPZEPZGZWOAUBDZAHGZUCBUBDZBHGZUCWQXCUDZWPAUEBUEXDXFXEXGXHXDXFWQ - XCABCUFIWQCUBDZCHGZUCZXEXFUGZXCCUEZXLXIXCXJXEXFXIXCXEXFXIJZXAHCFPZXAEPZWSXB - XNXPHXAEPZXAUHUIZXAXNXOHXAEXIXEXOHGZXFCUJZKLXNXAUKDXQXRGXNXAXNBCXEXFXIMXIXE - WQXFCULZKUMZUNXAUOQXNXAXNXAYBUPZXNHXAXNUQYCXNXAYBURUSUTVEXNWRBCFXNWRHBEPZBU - HUIZBXEXFWRYDGXIAHBEVANXNBUKDZYDYEGZXFXEYFXIBVBVCBUOZQXFXEYEBGZXIXFBBVDZXFH - BXFUQYJBVFUSUTZVCVGLXNWTXOXAEXEXFWTXOGXIAHCFVANLOIXEXFXJXCXEXFXJJZRRREPZWSX - 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linear (1-dimensional) relation. (Contributed by BJ RQWGVRSWHVRWLVRVSVTAFWIWDLAWOFWIUJUABCDFGHIJKNOPQRSWAVNWBWC $. $} + ${ + $d W l x y $. $d K l x y $. $d B x y $. + prjspnssbas.p $e |- P = ( N PrjSpn K ) $. + prjspnssbas.w $e |- W = ( K freeLMod ( 0 ... N ) ) $. + prjspnssbas.b $e |- B = ( ( Base ` W ) \ { ( 0g ` W ) } ) $. + prjspnssbas.n $e |- ( ph -> N e. NN0 ) $. + prjspnssbas.k $e |- ( ph -> K e. DivRing ) $. + $( A projective point spans a subset of the (nonzero) affine points. + (Contributed by SN, 17-Jan-2025.) $) + prjspnssbas $p |- ( ph -> P C_ ~P B ) $= + ( vx vy vl cv wcel wa cfv co eqid cvsca wceq cbs wrex cqs cpw cprjspn cn0 + copab cdr prjspnval2 syl2anc eqtrid prjspner qsss eqsstrd ) ACBLOZBPMOZBP + QUQNOURFUARZSUBNDUCRZUDQLMUIZUEZBUFACEDUGSZVBGAEUHPDUJPVCVBUBJKLMBVAUTUSD + EFNVATZHIUTTZUSTZUKULUMABVAALMBVAUTUSDEFNVDHIVEVFKUNUOUP $. + + prjspnn0.a $e |- ( ph -> A e. P ) $. + $( A projective point is nonempty. (Contributed by SN, 17-Jan-2025.) $) + prjspnn0 $p |- ( ph -> A =/= (/) ) $= + ( vx vy vl cv wcel wceq eqid wa cvsca cfv co cbs copab cdm cqs c0 wne wer + wrex prjspner erdm syl cprjspn cn0 cdr prjspnval2 syl2anc eqtrid eleqtrd + elqsn0 ) ANQZCROQZCRUAVDPQVEGUBUCZUDSPEUEUCZULUANOUFZUGCSZBCVHUHZRBUIUJAC + VHUKVIANOCVHVGVFEFGPVHTZIJVGTZVFTZLUMCVHUNUOABDVJMADFEUPUDZVJHAFUQREURRVN + VJSKLNOCVHVGVFEFGPVKIJVLVMUSUTVAVBCBVHVCUT $. + $} + ${ 0prjspnlem.b $e |- B = ( ( Base ` W ) \ { ( 0g ` W ) } ) $. 0prjspnlem.w $e |- W = ( K freeLMod ( 0 ... 0 ) ) $. @@ -658828,8 +661495,7 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples $} ${ - $d K a k p $. $d N a k p $. $d P a p $. $d ph a p $. $d N h f n g $. - $d K f n $. $d Y f $. $d .0. p $. + $d K k p $. $d N k p $. $d P p $. $d ph p $. $d N h $. $d .0. p $. prjcrv0.y $e |- Y = ( ( 0 ... N ) mPoly K ) $. prjcrv0.0 $e |- .0. = ( 0g ` Y ) $. prjcrv0.p $e |- P = ( N PrjSpn K ) $. @@ -658838,33 +661504,20 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples $( The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.) $) prjcrv0 $p |- ( ph -> ( ( N PrjCrv K ) ` .0. ) = P ) $= - ( va co cfv cc0 cv wceq eqid wcel cvv vp vn vf vg vh vk cprjcrv cevl cima - cfz c0g csn crab cmhp wfun crn cuni cn0 csupp ccnfld cress cgsu ccnv cmap - cn cfn wss cbs cmpt funmpt cfield mhpfval funeqd mpbiri fldcrngd crnggrpd - ovexd cxp mpl0 0nn0 a1i mhp0cl eqeltrd elunirn2 syl2anc prjcrvval wa ccrg - adantr evl0 imaeq1d cin c0 wne cfrlm cdif ciun wrex cprjspn eleq2i biimpi - clspn adantl cprjsp cdr isfld simplbi syl prjspnval clvec prjspval2 eqtrd - frlmlvec eleqtrd eliun sylib cfsupp wbr frlmbas ssrab2 eqsstrrdi ad2antrr - eldifi adantrr sseldd wel velsn anbi2i lveclmodd lspsnid wn eldifn eldifd - clmod eleq2 syl5ibrcom impr sylan2b elind ne0d rexlimddv xpima2 rabeqcda - ) AFDCUGMNFODUJMZCUHMZNZUAPZUIZCUKNZULZQZUABUMBABUUEFUUDCUNMZCDUUIUAUULRZ - UUERZIUUIRZJKAUULUOZFOUULNZSFUULUPUQSAUUPUBURUCPUUIUSMUTURVAMUDPVBMUBPQUD - UEPVCVEUIVFSUEURUUDVDMUMZUMVGUCEVHNZUMZVIZUOUBURUUTVJAUULUVAAUUSUURECUCUD - UEUBUULUUDTVKUUIUUMGUUSRUUOUURRZAODUJVQZKVLVMVNAFUURUUJVRUUQAUURECUEUUDUU - ITFGUVBUUOHUVCACACKVOZVPZVSAUURCUEUULUUDOTUUIUUMUUOUVBUVCUVEOURSAVTWAWBWC - OFUULWDWEWFAUUKUABAUUGBSZWGZUUHCVHNZUUDVDMZUUJVRZUUGUIZUUJUVGUUFUVJUUGUVG - UVHUUECUUDUUITEFUUNUVHRZGUUOHUVGODUJVQACWHSZUVFUVDWIWJWKUVGUVIUUGWLZWMWNZ - UVKUUJQUVGUUGLPZULCUUDWOMZXBNZNZUVQUKNZULZWPZULZSZUVOLUVQVHNZUWAWPZUVGUUG - LUWFUWCWQZSUWDLUWFWRUVGUUGDCWSMZUWGUVFUUGUWHSZAUVFUWIBUWHUUGIWTXAXCUVGUWH - UVQXDNZUWGUVGDURSZCXESZUWHUWJQAUWKUVFJWIAUWLUVFACVKSZUWLKUWMUWLUVMCXFXGXH - ZWICDXIWEUVGUVQXJSZUWJUWGQAUWOUVFAUWLUUDTSZUWOUWNUVCCUVQUUDTUVQRZXMWEZWIL - UWFUVRUVQUVTUVTRUWFRUVRRZXKXHXLXNLUUGUWFUWCXOXPUVGUVPUWFSZUWDWGZWGZUVNUVP - UXBUVIUUGUVPUXBUWEUVIUVPAUWEUVIVGUVFUXAAUWEUFPUUIXQXRZUFUVIUMZUVIAUWMUWPU - XDUWEQKUVCUXDCUFUVQUUDUVHVKTUUIUWQUVLUUOUXDRXSWEUXCUFUVIXTYAYBUVGUWTUVPUW - ESZUWDUWTUXEUVGUVPUWEUWAYCXCZYDYEUXAUVGUWTUUGUWBQZWGLUAYFZUWDUXGUWTUAUWBY - GYHUVGUWTUXGUXHUVGUWTWGZUXHUXGUVPUWBSUXIUVPUVSUWAUXIUVQYNSZUXEUVPUVSSAUXJ - UVFUWTAUVQUWRYIYBUXFUVRUWEUVQUVPUWERUWSYJWEUWTUVPUWASYKUVGUVPUWEUWAYLXCYM - UUGUWBUVPYOYPYQYRYSYTUUAUVIUUJUUGUUBXHXLUUCXL $. + ( vp vh vk co cfv wceq eqid wcel cvv cprjcrv cc0 cfz cevl cv cima c0g csn + crab cmhp crn cuni fvssunirn ccnv cn cfn cn0 cmap ovexd fldcrngd crnggrpd + cxp mpl0 mhp0cl eqeltrd sselid prjcrvval cbs ccrg adantr evl0 imaeq1d cin + wa wne wss cpw cfrlm cdif flddrngd prjspnssbas cfsupp wbr frlmbas syl2anc + c0 cfield ssrab2 eqsstrrdi ssdifssd sspwd sstrd sselda elpwid sseqin2 cdr + sylib simpr prjspnn0 eqnetrd xpima2 syl eqtrd rabeqcda ) AFDCUAOPFUBDUCOZ + CUDOZPZLUEZUFZCUGPZUHZQZLBUIBABXFFXECUJOZCDXJLXMRZXFRZIXJRZJKADXMPZXMUKUL + FXMDUMAFMUEUNUOUFUPSMUQXEUROUIZXKVBXQAXRECMXEXJTFGXRRZXPHAUBDUCUSZACACKUT + ZVAZVCAXRCMXMXEDTXJXNXPXSXTYBJVDVEVFVGAXLLBAXHBSZVNZXICVHPZXEUROZXKVBZXHU + FZXKYDXGYGXHYDYEXFCXEXJTEFXOYERZGXPHYDUBDUCUSACVISYCYAVJVKVLYDYFXHVMZWFVO + YHXKQYDYJXHWFYDXHYFVPYJXHQYDXHYFABYFVQZXHABCXEVROZVHPZYLUGPUHZVSZVQYKAYOB + CDYLIYLRZYORZJACKVTZWAAYOYFAYMYFYNAYMNUEXJWBWCZNYFUIZYFACWGSXETSYTYMQKXTY + TCNYLXEYEWGTXJYPYIXPYTRWDWEYSNYFWHWIWJWKWLWMWNXHYFWOWQYDXHYOBCDYLIYPYQADU + QSYCJVJACWPSYCYRVJAYCWRWSWTYFXKXHXAXBXCXDXC $. $} @@ -659297,7 +661950,6 @@ evaluates to zero (the "zero set"). (In other words, scalar multiples $} ${ - $d ph s $. $d ph t $. flt4lem4.a $e |- ( ph -> A e. NN ) $. flt4lem4.b $e |- ( ph -> B e. NN ) $. flt4lem4.c $e |- ( ph -> C e. NN ) $. @@ -659620,62 +662272,62 @@ evaluates to zero (the "zero set"). 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(In other words, scalar multiples caddc c1 cmin c2 cdiv cneg cc0 wne cn wn 3nn a1i cn0 nnexpcld pm2.18i nnq mp1i mpan2 qmulcl 1nn ax-mp qsubcl qsqcl mpancom 3cubeslem2 neqned qdivcl syl3anc qnegcl syl 3cubeslem4 oveq1 oveq1d eqeq2d oveq2d rspc3ev syl31anc - id wi w3a wa 3anass simprl syl2an2r simprr eleq1a syl5bi rexlimdv3d mptru - wtru impbii ) AEFZABGZHIJZCGZHIJZSJZDGZHIJZSJZKZDELCELBELZWJHHIJZAHIJZMJZ - TUAJZXAAUBIJZMJZHUBIJZAMJZSJZHSJZUCJZEFZXCUDZXHSJZTSJZXJUCJZEFZXIXJUCJZEF - ZAXKHIJZXPHIJZSJZXRHIJZSJZKZWTWJXDEFZXJEFZXJUEUFZXLWJXCEFZTEFZYFWJXAEFZXB - EFZYIXAUGFZYKWJYMYMUHZHHHUGFZYNUIUJHUKFZYNNUJULUMXAUNUOZWJYPYLNAHOUPXAXBU - QPZTUGFYJURTUNUSZXCTUTQWJXIEFZHEFZYGWJXFEFZXHEFZYTWJYKXEEFUUBYQAVAXAXEUQP - XGEFZWJUUCUUAUUDWJYOUUAUIHUNUSZHVAUOXGAUQVBZXFXHRPZUUEXIHRQZWJXJUEWJAWJVP - ZVCVDZXDXJVEVFWJXOEFZYGYHXQWJXNEFZYJUUKWJXMEFZUUCUULWJYIUUMYRXCVGVHUUFXMX - HRPYSXNTRQUUHUUJXOXJVEVFWJYTYGYHXSUUGUUHUUJXIXJVEVFWJAUUIVIWSYEAXTWNSJZWQ - SJZKAYBWQSJZKBCDXKXPXREEEWKXKKZWRUUOAUUQWOUUNWQSUUQWLXTWNSWKXKHIVJVKVKVLW - MXPKZUUOUUPAUURUUNYBWQSUURWNYAXTSWMXPHIVJVMVKVLWPXRKZUUPYDAUUSWQYCYBSWPXR - HIVJVMVLVNVOWTWJVQWHWSWJBCDEEEWKEFZWMEFZWPEFZVRUUTUVAUVBVSZVSZWHWSWJVQZUU - TUVAUVBVTUVDUVEVQWHUVDWREFZUVEUVDWOEFZWQEFZUVFUUTWLEFZUVCWNEFZUVGUUTYPUVI - NWKHOUPUVDUVAYPUVJUUTUVAUVBWANWMHOQWLWNRWBUVDUVBYPUVHUUTUVAUVBWCNWPHOQWOW - QRPWREAWDVHUJWEWFWGWI $. + id wi wtru w3a wa 3anass simprl syl2an2r simprr biimtrid rexlimdv3d mptru + eleq1a impbii ) AEFZABGZHIJZCGZHIJZSJZDGZHIJZSJZKZDELCELBELZWJHHIJZAHIJZM + JZTUAJZXAAUBIJZMJZHUBIJZAMJZSJZHSJZUCJZEFZXCUDZXHSJZTSJZXJUCJZEFZXIXJUCJZ + EFZAXKHIJZXPHIJZSJZXRHIJZSJZKZWTWJXDEFZXJEFZXJUEUFZXLWJXCEFZTEFZYFWJXAEFZ + XBEFZYIXAUGFZYKWJYMYMUHZHHHUGFZYNUIUJHUKFZYNNUJULUMXAUNUOZWJYPYLNAHOUPXAX + BUQPZTUGFYJURTUNUSZXCTUTQWJXIEFZHEFZYGWJXFEFZXHEFZYTWJYKXEEFUUBYQAVAXAXEU + QPXGEFZWJUUCUUAUUDWJYOUUAUIHUNUSZHVAUOXGAUQVBZXFXHRPZUUEXIHRQZWJXJUEWJAWJ + VPZVCVDZXDXJVEVFWJXOEFZYGYHXQWJXNEFZYJUUKWJXMEFZUUCUULWJYIUUMYRXCVGVHUUFX + MXHRPYSXNTRQUUHUUJXOXJVEVFWJYTYGYHXSUUGUUHUUJXIXJVEVFWJAUUIVIWSYEAXTWNSJZ + WQSJZKAYBWQSJZKBCDXKXPXREEEWKXKKZWRUUOAUUQWOUUNWQSUUQWLXTWNSWKXKHIVJVKVKV + LWMXPKZUUOUUPAUURUUNYBWQSUURWNYAXTSWMXPHIVJVMVKVLWPXRKZUUPYDAUUSWQYCYBSWP + XRHIVJVMVLVNVOWTWJVQVRWSWJBCDEEEWKEFZWMEFZWPEFZVSUUTUVAUVBVTZVTZVRWSWJVQZ + UUTUVAUVBWAUVDUVEVQVRUVDWREFZUVEUVDWOEFZWQEFZUVFUUTWLEFZUVCWNEFZUVGUUTYPU + VINWKHOUPUVDUVAYPUVJUUTUVAUVBWBNWMHOQWLWNRWCUVDUVBYPUVHUUTUVAUVBWDNWPHOQW + OWQRPWREAWHVHUJWEWFWGWI $. $} $( (End of Igor Ieskov's mathbox.) $) @@ -661516,17 +664168,17 @@ family of sets (implicit). (Contributed by Stefan O'Rear, mzpincl $p |- ( V e. _V -> ( mzPoly ` V ) e. ( mzPolyCld ` V ) ) $= ( vf vg va wcel cfv cz cmap co cv wral cof simpr simplr syl2anc ralrimiva wa ovex elint2 sylibr cvv cmzp cmzpcl cint mzpval wss csn cmpt caddc cmul - cxp mzpclall intss1 syl mzpcl1 snex xpex mzpcl2 jca wi vex mzpcl34 3expib - mptex ralimia r19.26 3imtr3i syl2anb anbi12i a1i ralrimivv elmzpcl mpbird - jca32 eqeltrd ) AUAEZAUBFAUCFZUDZVQAUEVPVRVQEVRGGAHIZHIZUFZVSBJZUGZUKZVRE - ZBGKZCVSWBCJZFZUHZVREZBAKZQZWBWGUILZIZVREZWBWGUJLZIZVREZQZCVRKBVRKZQQVPWA - WLWTVPVTVQEWAAULVTVQUMUNVPWFWKVPWEBGVPWBGEZQZWDDJZEZDVQKWEXBXDDVQXBXCVQEZ - QXEXAXDXBXEMVPXAXENXCWBAUOOPDWDVQVSWCGAHRZWBUPUQSTPVPWJBAVPWBAEZQZWIXCEZD - VQKWJXHXIDVQXHXEQXEXGXIXHXEMVPXGXENXCCWBAUROPDWIVQCVSWHXFVDSTPUSVPWSBCVRV - RWBVREZWGVREZQZWSUTVPXLWNXCEZDVQKZWQXCEZDVQKZQZWSXJWBXCEZDVQKZWGXCEZDVQKZ - XQXKDWBVQBVASDWGVQCVASXRXTQZDVQKXMXOQZDVQKXSYAQXQYBYCDVQXEXRXTYCXCWBWGAVB - VCVEXRXTDVQVFXMXODVQVFVGVHWOXNWRXPDWNVQWBWGWMRSDWQVQWBWGWPRSVITVJVKVNCVRB - CBBAVLVMVO $. + cxp mzpclall intss1 syl mzpcl1 vsnex xpex mzpcl2 mptex jca wi vex mzpcl34 + 3expib ralimia r19.26 3imtr3i syl2anb anbi12i a1i ralrimivv jca32 elmzpcl + mpbird eqeltrd ) AUAEZAUBFAUCFZUDZVQAUEVPVRVQEVRGGAHIZHIZUFZVSBJZUGZUKZVR + EZBGKZCVSWBCJZFZUHZVREZBAKZQZWBWGUILZIZVREZWBWGUJLZIZVREZQZCVRKBVRKZQQVPW + AWLWTVPVTVQEWAAULVTVQUMUNVPWFWKVPWEBGVPWBGEZQZWDDJZEZDVQKWEXBXDDVQXBXCVQE + ZQXEXAXDXBXEMVPXAXENXCWBAUOOPDWDVQVSWCGAHRZBUPUQSTPVPWJBAVPWBAEZQZWIXCEZD + VQKWJXHXIDVQXHXEQXEXGXIXHXEMVPXGXENXCCWBAUROPDWIVQCVSWHXFUSSTPUTVPWSBCVRV + RWBVREZWGVREZQZWSVAVPXLWNXCEZDVQKZWQXCEZDVQKZQZWSXJWBXCEZDVQKZWGXCEZDVQKZ + XQXKDWBVQBVBSDWGVQCVBSXRXTQZDVQKXMXOQZDVQKXSYAQXQYBYCDVQXEXRXTYCXCWBWGAVC + VDVEXRXTDVQVFXMXODVQVFVGVHWOXNWRXPDWNVQWBWGWMRSDWQVQWBWGWPRSVITVJVKVLCVRB + CBBAVMVNVO $. $} $( Constant functions are polynomial. See also ~ mzpconstmpt . (Contributed @@ -661811,78 +664463,78 @@ theorem and the several that follow ( ~ mzpaddmpt , ~ mzpmulmpt , vh vi vj vk vl wss cres wtru tru csn cxp caddc cof cmul 0fin cvv mzpconst 0ex 0ss a1i fconstmpt simpr elmapssres sylancl vex fvconst2 syl mpteq2dva eqtr4id fveq1 mpteq2dv eqeq2d rspcev syl12anc fveq2 reseq2 fveq2d anbi12d - mpan sseq1 rexeqbidv sylancr adantl snfi snex vsnid mzpproj mp2an cbvmptv - snssi simpl snssd syl2anc eqid fvmpt fvres ax-mp eqtr2di eqtrid wf w3a wi - fvex simplll simprll unfi unex ssun1 simpllr mzpresrename syl3anc simprlr - cun mzpaddmpt simplr simprr wfn ovex mzpf ffn 3syl ofmpteq reseq1 oveq12d - ssun2 resabs1 fveq2i oveq12i eqtrd eqeq1d rexbidv eqeq1 2rexbidv cbvrexvw - weq bitrdi unssd elmapi fssres syl2anr zex elmap sylibr adantlrr adantrrr - mzpmulmpt simplrr mpbird r19.40 exp32 rexlimdvv ex rexlimivv imp ad2ant2l - simprrr 3adant1 simpld simprd mzpindd eqeq2i anbi2i 2rexbii sylib ) ABHIZ - JZCKZBUJZAGLBUDMZGKZUVKUKZDKZIZNZOZPZDUVKHIZQCRQZUVLAEUVMEKZUVKUKZUVPIZNZ - 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(Contributed by Stefan O'Rear, wn c1 cdioph cmap cab cmzp cuz eldiophb wf1 cid cz ccom wf simp-5r simprr ad2antrr simprl f1f syl mzprename syl3anc w3a diophrw eqcomd fveq1 eqeq1d cmpt anbi2d rexbidv rspceeqv syl2anc simplll simplrl simplrr eldioph2lem2 - abbidv syl22anc sylibr r19.29a eqeq1 syl5ibrcom rexlimdvva adantld syl5bi - rexv simpr simpllr eldioph2 syl121anc adantr eqeltrd rexlimdva2 impbid - wex ) EUALZDMLZNZDUBLUFZUGEUCOZDUDZNZNZCEUHPZLZCBQZAQZXDRSZXKFQZPZUESZNZA - UADUIOZTZBUJZSZFDUKPZTZXIWTCXJGQZXDRSYCHQZPUESNGUAUGIQZUCOZUIOTBUJZSZHYFU - KPZTIEULPZTZNXGYBGBCIEHUMXGYKYBWTXGYHYBIHYJYIXGYEYJLZYDYILZNZNZYBYHYGXSSZ - FYATZYOYFDJQZUNZYRXDRUOXDRSZNZYQJMYOYRMLZNZUUANZKUPDUIOKQYRUQYDPVLZYALZYG - XLXKUUEPZUESZNZAXQTZBUJZSZYQUUDXAYMYFDYRURZUUFWTXAXFYNUUBUUAUSZYOYMUUBUUA - XGYLYMUTVAUUDYSUUMUUCYSYTVBZYFDYRVCVDKYRYDYFDVEVFUUDXAYSYTUULUUNUUOUUCYSY - TUTXAYSYTVGUUKYGYDDYFYRXDBAGKVHVIVFFUUEYAXSUUKYGXMUUESZXRUUJBUUPXPUUIAXQU - UPXOUUHXLUUPXNUUGUEXKXMUUEVJVKVMVNWAVOVPYOUUAJWSZUUAJMTYOWTXCXEYLUUQWTXAX - FYNVQXBXCXEYNVRXBXCXEYNVSXGYLYMVBYEDEJVTWBUUAJWJWCWDYHXTYPFYACYGXSWEVNWFW - GWHWIXGXTXIFYAXGXMYALZNZXTNCXSXHUUSXTWKUUSXSXHLZXTUUSWTXAXEUURUUTWTXAXFUU - RVQWTXAXFUURWLXBXCXEUURVSXGUURWKABXMDEWMWNWOWPWQWR $. + abbidv wex syl22anc rexv sylibr r19.29a eqeq1 syl5ibrcom adantld biimtrid + rexlimdvva simpllr eldioph2 syl121anc adantr eqeltrd rexlimdva2 impbid + simpr ) EUALZDMLZNZDUBLUFZUGEUCOZDUDZNZNZCEUHPZLZCBQZAQZXDRSZXKFQZPZUESZN + ZAUADUIOZTZBUJZSZFDUKPZTZXIWTCXJGQZXDRSYCHQZPUESNGUAUGIQZUCOZUIOTBUJZSZHY + FUKPZTIEULPZTZNXGYBGBCIEHUMXGYKYBWTXGYHYBIHYJYIXGYEYJLZYDYILZNZNZYBYHYGXS + SZFYATZYOYFDJQZUNZYRXDRUOXDRSZNZYQJMYOYRMLZNZUUANZKUPDUIOKQYRUQYDPVLZYALZ + YGXLXKUUEPZUESZNZAXQTZBUJZSZYQUUDXAYMYFDYRURZUUFWTXAXFYNUUBUUAUSZYOYMUUBU + UAXGYLYMUTVAUUDYSUUMUUCYSYTVBZYFDYRVCVDKYRYDYFDVEVFUUDXAYSYTUULUUNUUOUUCY + SYTUTXAYSYTVGUUKYGYDDYFYRXDBAGKVHVIVFFUUEYAXSUUKYGXMUUESZXRUUJBUUPXPUUIAX + QUUPXOUUHXLUUPXNUUGUEXKXMUUEVJVKVMVNWAVOVPYOUUAJWBZUUAJMTYOWTXCXEYLUUQWTX + AXFYNVQXBXCXEYNVRXBXCXEYNVSXGYLYMVBYEDEJVTWCUUAJWDWEWFYHXTYPFYACYGXSWGVNW + HWKWIWJXGXTXIFYAXGXMYALZNZXTNCXSXHUUSXTWSUUSXSXHLZXTUUSWTXAXEUURUUTWTXAXF + UURVQWTXAXFUURWLXBXCXEUURVSXGUURWSABXMDEWMWNWOWPWQWR $. $} ${ @@ -662456,12 +665108,12 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, eldiophss $p |- ( A e. ( Dioph ` B ) -> A C_ ( NN0 ^m ( 1 ... B ) ) ) $= ( vb vc va vd cdioph cfv wcel cn0 cv c1 co wceq wa cn cmap wrex wss simpr cfz cres cc0 cab cmzp eldioph3b vex anbi1d rexbidv elab elfznn elmapssres - eqeq1 ssriv mpan2 ad2antlr eqeltrd adantrd rexlimdva syl5bi ssrdv eqsstrd - ex adantr r19.29an sylbi ) ABGHIBJIZACKZDKZLBUAMZUBZNZVIEKZHUCNZOZDJPQMZR - ZCUDZNZEPUEHZROAJVJQMZSZDCABEUFVGVSWBEVTVGVMVTIOZVSOAVRWAWCVSTWCVRWASVSWC - FVRWAFKZVRIWDVKNZVNOZDVPRZWCWDWAIZVQWGCWDFUGVHWDNZVOWFDVPWIVLWEVNVHWDVKUM - UHUIUJWCWFWHDVPWCVIVPIZOZWEWHVNWKWEWHWKWEOWDVKWAWKWETWJVKWAIZWCWEWJVJPSWL - EVJPVMBUKUNVIJPVJULUOUPUQVCURUSUTVAVDVBVEVF $. + eqeq1 ssriv ad2antlr eqeltrd ex adantrd rexlimdva biimtrid adantr eqsstrd + mpan2 ssrdv r19.29an sylbi ) ABGHIBJIZACKZDKZLBUAMZUBZNZVIEKZHUCNZOZDJPQM + ZRZCUDZNZEPUEHZROAJVJQMZSZDCABEUFVGVSWBEVTVGVMVTIOZVSOAVRWAWCVSTWCVRWASVS + WCFVRWAFKZVRIWDVKNZVNOZDVPRZWCWDWAIZVQWGCWDFUGVHWDNZVOWFDVPWIVLWEVNVHWDVK + UMUHUIUJWCWFWHDVPWCVIVPIZOZWEWHVNWKWEWHWKWEOWDVKWAWKWETWJVKWAIZWCWEWJVJPS + WLEVJPVMBUKUNVIJPVJULVCUOUPUQURUSUTVDVAVBVEVF $. $} @@ -662618,9 +665270,9 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, (New usage is discouraged.) (Proof modification is discouraged.) $) sbcrexgOLD $p |- ( A e. V -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) ) $= - ( vz wrex wsb wsbc dfsbcq2 wceq rexbidv nfcv nfs1v nfrex weq sbequ12 sbie - cv vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGTDLUCUECEABGDKMUBUDB - GUCBCEBENABGOPBGQAUCCEABGRMSUA $. + ( vz wrex wsb wsbc dfsbcq2 cv wceq rexbidv nfcv nfs1v nfrexw sbequ12 sbie + weq vtoclbg ) ACEHZBGIABGIZCEHZUBBDJABDJZCEHGDFUBBGDKGLDMUCUECEABGDKNUBUD + BGUCBCEBEOABGPQBGTAUCCEABGRNSUA $. $} ${ @@ -662734,30 +665386,30 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, eqcomd adantl sbceqbid biimpd fveq1 reseq1 eqeq2d anbi12d rspcev syl12anc impr rexlimdva2 wf caddc nn0p1nn eqeltrid elfz1end sylib ffvelcdm syl2anr elmapi cn adantr simprr mapfzcons1cl ad2antlr eqeltrd wb sbcbidv ad2antll - simprl mpbird anbi2d impbid bitrid abbidv eqtrid nfv nfsbc1v nfsbcw nfrex - nfcv weq sbceq1a rexbidv cbvrexw bitrdi cbvrabw rexrab abbii 3eqtr4g fvex - vex resex sylan9bb sbc2ie rabbii rexeqi eqtrdi cuz simpl cz nn0z peano2uz - uzid 3syl diophrex ) HOPZAFOUAGUBUCZUDUCZUEZGUFUGPZQZBDORZEOUAHUBUCZUDUCZ - UEZLVAZMVAZUUDUHZSZMYTRZLUIZHUFUGZYQUUFUULSUUAYQUUFUUJMBEFVAZUUDUHZTZDGUU - NUGZTZFYSUEZRZLUIZUULYQBEUUGTZDUUHTZMORZLUUEUEZBEUUITZDGUUHUGZTZUUJQZMYSR - ZLUIZUUFUVAYQUVEUUGUUEPZUVDQZLUIUVKUVDLUUEUJYQUVMUVJLUVMUVLUVBDNVAZTZQZNO - RZYQUVJUVMUVLUVONORZQUVQUVDUVRUVLUVCUVOMNOUVBDUUHUVNUKULUMUVLUVONOUNUOYQU - VQUVJYQUVPUVJNOYQUVNOPZQZUVPQUUGGUVNUPUQURZYSPZBEUWAUUDUHZTZDGUWAUGZTZUUG - UWCSZUVJUVTUVLUWBUVOUVTUVLQZYQUVLUVSUWBYQUVSUVLUSUVTUVLUTZYQUVSUVLVBZUUGO - UVNGHIVCVDVEUVTUVLUVOUWFUWHUVOUWFUWHUVBUWDDUVNUWEUWHUWEUVNUWHUVLUVSUWEUVN - SUWIUWJUUGOUVNGHIVFVGVJUWHBEUUGUWCUWHUWCUUGUVLUWCUUGSUVTUUGOUVNGHIVHVKVJZ - VIVLVMVTUVTUVLUWGUVOUWKVEUVIUWFUWGQMUWAYSUUHUWASZUVHUWFUUJUWGUWLUVFUWDDUV - GUWEGUUHUWAVNUWLBEUUIUWCUUHUWAUUDVOZVIVLUWLUUIUWCUUGUWMVPVQVRVSWAYQUVIUVQ - MYSYQUUHYSPZQZUVIQZUVGOPZUVLUVBDUVGTZUVQUWOUWQUVIUWNYROUUHWBGYRPZUWQYQUUH - OYRWJYQGWKPUWSYQGHUAWCUCZWKIHWDWEGWFWGYROGUUHWHWIWLUWPUUGUUIUUEUWOUVHUUJW - MUWNUUIUUEPYQUVIUUHOGHIWNWOWPUWPUWRUVHUWOUVHUUJWTUUJUWRUVHWQUWOUVHUUJUVBU - VFDUVGBEUUGUUIUKWRWSXAUVPUVLUWRQNUVGOUVNUVGSUVOUWRUVLUVBDUVNUVGUKXBVRVSWA - XCXDXEXFUUCUVDELUUEEUUEXKLUUEXKUUCLXGUVCEMOEOXKUVBEDUUHEUUHXKBEUUGXHXIXJE - LXLZUUCUVBDORUVDUXABUVBDOBEUUGXMXNUVBUVCDMOUVBMXGUVBDUUHXHUVBDUUHXMXOXPXQ - UUTUVJLUURUVHUUJMFYSFMXLZUUPUVFDUUQUVGGUUNUUHVNUXBBEUUOUUIUUNUUHUUDVOVIVL - XRXSXTUUTUUKLUUJMUUSYTUURAFYSBADEUUQUUOGUUNYAUUNUUDFYBYCDVAUUQSBCEVAUUOSA - JKYDYEYFYGXSYHWLUUBYQGHYIUGZPZUUAUULUUMPYQUUAYJYQUXDUUAYQGUWTUXCIYQHYKPHU - XCPUWTUXCPHYLHYNHHYMYOWEWLYQUUAUTMLYTGHYPVDWP $. + simprl mpbird anbi2d impbid bitrid abbidv eqtrid nfcv nfv nfsbc1v sbceq1a + nfsbcw nfrexw weq rexbidv cbvrexw bitrdi cbvrabw rexrab abbii 3eqtr4g vex + fvex resex sylan9bb sbc2ie rabbii rexeqi eqtrdi cuz cz nn0z uzid peano2uz + simpl 3syl diophrex ) HOPZAFOUAGUBUCZUDUCZUEZGUFUGPZQZBDORZEOUAHUBUCZUDUC + ZUEZLVAZMVAZUUDUHZSZMYTRZLUIZHUFUGZYQUUFUULSUUAYQUUFUUJMBEFVAZUUDUHZTZDGU + UNUGZTZFYSUEZRZLUIZUULYQBEUUGTZDUUHTZMORZLUUEUEZBEUUITZDGUUHUGZTZUUJQZMYS + RZLUIZUUFUVAYQUVEUUGUUEPZUVDQZLUIUVKUVDLUUEUJYQUVMUVJLUVMUVLUVBDNVAZTZQZN + ORZYQUVJUVMUVLUVONORZQUVQUVDUVRUVLUVCUVOMNOUVBDUUHUVNUKULUMUVLUVONOUNUOYQ + UVQUVJYQUVPUVJNOYQUVNOPZQZUVPQUUGGUVNUPUQURZYSPZBEUWAUUDUHZTZDGUWAUGZTZUU + GUWCSZUVJUVTUVLUWBUVOUVTUVLQZYQUVLUVSUWBYQUVSUVLUSUVTUVLUTZYQUVSUVLVBZUUG + OUVNGHIVCVDVEUVTUVLUVOUWFUWHUVOUWFUWHUVBUWDDUVNUWEUWHUWEUVNUWHUVLUVSUWEUV + NSUWIUWJUUGOUVNGHIVFVGVJUWHBEUUGUWCUWHUWCUUGUVLUWCUUGSUVTUUGOUVNGHIVHVKVJ + ZVIVLVMVTUVTUVLUWGUVOUWKVEUVIUWFUWGQMUWAYSUUHUWASZUVHUWFUUJUWGUWLUVFUWDDU + VGUWEGUUHUWAVNUWLBEUUIUWCUUHUWAUUDVOZVIVLUWLUUIUWCUUGUWMVPVQVRVSWAYQUVIUV + QMYSYQUUHYSPZQZUVIQZUVGOPZUVLUVBDUVGTZUVQUWOUWQUVIUWNYROUUHWBGYRPZUWQYQUU + HOYRWJYQGWKPUWSYQGHUAWCUCZWKIHWDWEGWFWGYROGUUHWHWIWLUWPUUGUUIUUEUWOUVHUUJ + WMUWNUUIUUEPYQUVIUUHOGHIWNWOWPUWPUWRUVHUWOUVHUUJWTUUJUWRUVHWQUWOUVHUUJUVB + UVFDUVGBEUUGUUIUKWRWSXAUVPUVLUWRQNUVGOUVNUVGSUVOUWRUVLUVBDUVNUVGUKXBVRVSW + AXCXDXEXFUUCUVDELUUEEUUEXGLUUEXGUUCLXHUVCEMOEOXGUVBEDUUHEUUHXGBEUUGXIXKXL + ELXMZUUCUVBDORUVDUXABUVBDOBEUUGXJXNUVBUVCDMOUVBMXHUVBDUUHXIUVBDUUHXJXOXPX + QUUTUVJLUURUVHUUJMFYSFMXMZUUPUVFDUUQUVGGUUNUUHVNUXBBEUUOUUIUUNUUHUUDVOVIV + LXRXSXTUUTUUKLUUJMUUSYTUURAFYSBADEUUQUUOGUUNYBUUNUUDFYAYCDVAUUQSBCEVAUUOS + AJKYDYEYFYGXSYHWLUUBYQGHYIUGZPZUUAUULUUMPYQUUAYNYQUXDUUAYQGUWTUXCIYQHYJPH + UXCPUWTUXCPHYKHYLHHYMYOWEWLYQUUAUTMLYTGHYPVDWP $. $} ${ @@ -662774,11 +665426,11 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. ph } e. ( Dioph ` M ) ) -> { u e. ( NN0 ^m ( 1 ... N ) ) | E. v e. NN0 ph } e. ( Dioph ` N ) ) $= ( vb va cn0 wcel cv cfv wsbc c1 cfz co crab wrex nfcv cres cmap cdioph wa - nfv nfsbc1v nfrex wceq sbceq1a cbvrexw rexbidv bitrid cbvrabw rexrabdioph - dfsbcq sbcbidv eqeltrid ) FJKABEDLZMZNZCUROFPQZUAZNZDJOEPQUBQREUCMKUDABJS - ZCJVAUBQZRABHLZNZCILZNZHJSZIVERFUCMVDVJCIVECVETIVETVDIUEVICHJCJTVGCVHUFUG - VDVGHJSCLVHUHZVJAVGBHJAHUEABVFUFABVFUIUJVKVGVIHJVGCVHUIUKULUMVCVIUTCVHNHI - DEFGVFUSUHVGUTCVHABVFUSUOUPUTCVHVBUOUNUQ $. + nfsbc1v nfrexw wceq sbceq1a cbvrexw rexbidv bitrid cbvrabw dfsbcq sbcbidv + nfv rexrabdioph eqeltrid ) FJKABEDLZMZNZCUROFPQZUAZNZDJOEPQUBQREUCMKUDABJ + SZCJVAUBQZRABHLZNZCILZNZHJSZIVERFUCMVDVJCIVECVETIVETVDIUOVICHJCJTVGCVHUEU + FVDVGHJSCLVHUGZVJAVGBHJAHUOABVFUEABVFUHUIVKVGVIHJVGCVHUHUJUKULVCVIUTCVHNH + IDEFGVFUSUGVGUTCVHABVFUSUMUNUTCVHVBUMUPUQ $. rexfrabdioph.2 $e |- L = ( M + 1 ) $. $( Diophantine set builder for existential quantifier, explicit @@ -662980,16 +665632,16 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, ( mzPoly ` ( 1 ... N ) ) ) -> { t e. ( NN0 ^m ( 1 ... N ) ) | A e. NN0 } e. ( Dioph ` N ) ) $= ( vb va vc cn0 wcel cz c1 cfz co cmap cmpt cmzp cfv crab cv wceq nfcv csb - wa wrex cdioph risset rabbii a1i nfv nfcsb1v nfeq2 csbeq1a eqeq2d rexbidv - nfrex cbvrabw eqtrdi caddc cres peano2nn0 adantr ovex cn nn0p1nn elfz1end - cvv sylib mzpproj sylancr eqid rabdiophlem2 eqrabdioph csbeq1 rexrabdioph - syl3anc eqeq1 syldan eqeltrd ) CGHZAIJCKLZMLBNVSOPHZUBZBGHZAGVSMLZQZDRZAE - RZBUAZSZDGUCZEWCQZCUDPZWAWDWEBSZDGUCZAWCQZWJWDWNSWAWBWMAWCDBGUEUFUGWMWIAE - WCAWCTEWCTWMEUHWHADGAGTAWEWGAWFBUIUJUNARWFSZWLWHDGWOBWGWEAWFBUKULUMUOUPVR - VTCJUQLZFRZPZAWQVSURZBUAZSZFGJWPKLZMLQWPUDPHZWJWKHWAWPGHZFIXBMLZWRNXBOPZH - ZFXEWTNXFHXCVRXDVTCUSUTWAXBVEHWPXBHZXGJWPKVAVRXHVTVRWPVBHXHCVCWPVDVFUTFXB - WPVGVHAFBWPCWPVIZVJFWRWTWPVKVNXAWHWRWGSDEFWPCXIWEWRWGVOWFWSSWGWTWRAWFWSBV - LULVMVPVQ $. + wa wrex cdioph risset rabbii a1i nfv nfcsb1v nfeq2 nfrexw csbeq1a rexbidv + eqeq2d cbvrabw eqtrdi caddc cres peano2nn0 adantr cvv cn nn0p1nn elfz1end + ovex sylib mzpproj sylancr eqid rabdiophlem2 eqrabdioph eqeq1 rexrabdioph + syl3anc csbeq1 syldan eqeltrd ) CGHZAIJCKLZMLBNVSOPHZUBZBGHZAGVSMLZQZDRZA + ERZBUAZSZDGUCZEWCQZCUDPZWAWDWEBSZDGUCZAWCQZWJWDWNSWAWBWMAWCDBGUEUFUGWMWIA + EWCAWCTEWCTWMEUHWHADGAGTAWEWGAWFBUIUJUKARWFSZWLWHDGWOBWGWEAWFBULUNUMUOUPV + RVTCJUQLZFRZPZAWQVSURZBUAZSZFGJWPKLZMLQWPUDPHZWJWKHWAWPGHZFIXBMLZWRNXBOPZ + HZFXEWTNXFHXCVRXDVTCUSUTWAXBVAHWPXBHZXGJWPKVEVRXHVTVRWPVBHXHCVCWPVDVFUTFX + BWPVGVHAFBWPCWPVIZVJFWRWTWPVKVNXAWHWRWGSDEFWPCXIWEWRWGVLWFWSSWGWTWRAWFWSB + VOUNVMVPVQ $. $} ${ @@ -663093,28 +665745,29 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, ( vb va vc cn0 wcel cz c1 co cmap cmpt cfv crab cv cmul wceq nfcv cfz w3a cmzp cdvds wbr cneg wo wrex cdioph wral rabdiophlem1 wa wb divides eqeq1d oveq1 rexzrexnn0 bitrdi ralimi r19.26 3imtr3i syl2an 3adant1 csb nfv nfov - rabbi nfcsb1v nfeq nfor nfrex csbeq1a oveq2d eqeq12d orbi12d rexbidv cres - cbvrabw caddc simp1 peano2nn0 3ad2ant1 cvv ovex cn nn0p1nn elfz1end sylib - mzpproj sylancr rabdiophlem2 mzpmulmpt syl2anc 3adant3 3adant2 eqrabdioph - adantr eqid syl3anc mzpnegmpt syl orrabdioph negeq oveq1d csbeq1 eqeltrid - rexrabdioph eqeltrd ) DHIZAJKDUALZMLZBNXJUCOZIZAXKCNXLIZUBZBCUDUEZAHXJMLZ - PZEQZBRLZCSZXSUFZBRLZCSZUGZEHUHZAXQPZDUIOZXMXNXRYGSZXIXMBJIZAXQUJZCJIZAXQ - UJZYIXNABDUKACDUKYJYLULZAXQUJXPYFUMZAXQUJYKYMULYIYNYOAXQYNXPFQZBRLZCSZFJU - HYFFBCUNYRYAYDFEYPXSSYQXTCYPXSBRUPUOYPYBSYQYCCYPYBBRUPUOUQURUSYJYLAXQUTXP - YFAXQVGVAVBVCXOYGXSAYPBVDZRLZAYPCVDZSZYBYSRLZUUASZUGZEHUHZFXQPZYHYFUUFAFX - QAXQTFXQTYFFVEUUEAEHAHTUUBUUDAAYTUUAAXSYSRAXSTARTZAYPBVHZVFAYPCVHZVIAUUCU - UAAYBYSRAYBTUUHUUIVFUUJVIVJVKAQYPSZYEUUEEHUUKYAUUBYDUUDUUKXTYTCUUAUUKBYSX - SRAYPBVLZVMAYPCVLZVNUUKYCUUCCUUAUUKBYSYBRUULVMUUMVNVOVPVRXOXIDKVSLZGQZOZA - UUOXJVQZBVDZRLZAUUQCVDZSZUUPUFZUURRLZUUTSZUGZGHKUUNUALZMLZPUUNUIOZIZUUGYH - IXIXMXNVTXOUVAGUVGPUVHIZUVDGUVGPUVHIZUVIXOUUNHIZGJUVFMLZUUSNUVFUCOZIZGUVM - UUTNUVNIZUVJXIXMUVLXNDWAWBZXIXMUVOXNXIXMULZGUVMUUPNUVNIZGUVMUURNUVNIZUVOX - IUVSXMXIUVFWCIUUNUVFIZUVSKUUNUAWDXIUUNWEIUWADWFUUNWGWHGUVFUUNWIWJWQZAGBUU - NDUUNWRZWKZGUUPUURUVFWLWMWNXIXNUVPXMAGCUUNDUWCWKWOZGUUSUUTUUNWPWSXOUVLGUV - MUVCNUVNIZUVPUVKUVQXIXMUWFXNUVRGUVMUVBNUVNIZUVTUWFUVRUVSUWGUWBGUUPUVFWTXA - UWDGUVBUURUVFWLWMWNUWEGUVCUUTUUNWPWSUVAUVDGUUNXBWMUVEUUEUUPYSRLZUUASZUVBY - SRLZUUASZUGEFGUUNDUWCXSUUPSZUUBUWIUUDUWKUWLYTUWHUUAXSUUPYSRUPUOUWLUUCUWJU - UAUWLYBUVBYSRXSUUPXCXDUOVOYPUUQSZUWIUVAUWKUVDUWMUWHUUSUUAUUTUWMYSUURUUPRA - YPUUQBXEZVMAYPUUQCXEZVNUWMUWJUVCUUAUUTUWMYSUURUVBRUWNVMUWOVNVOXGWMXFXH $. + rabbi nfcsb1v nfrexw csbeq1a oveq2d eqeq12d orbi12d rexbidv cbvrabw caddc + nfeq nfor cres simp1 peano2nn0 3ad2ant1 cvv ovex nn0p1nn elfz1end mzpproj + cn sylib sylancr adantr rabdiophlem2 mzpmulmpt syl2anc 3adant3 eqrabdioph + eqid 3adant2 syl3anc mzpnegmpt orrabdioph negeq oveq1d csbeq1 rexrabdioph + syl eqeltrid eqeltrd ) DHIZAJKDUALZMLZBNXJUCOZIZAXKCNXLIZUBZBCUDUEZAHXJML + ZPZEQZBRLZCSZXSUFZBRLZCSZUGZEHUHZAXQPZDUIOZXMXNXRYGSZXIXMBJIZAXQUJZCJIZAX + QUJZYIXNABDUKACDUKYJYLULZAXQUJXPYFUMZAXQUJYKYMULYIYNYOAXQYNXPFQZBRLZCSZFJ + UHYFFBCUNYRYAYDFEYPXSSYQXTCYPXSBRUPUOYPYBSYQYCCYPYBBRUPUOUQURUSYJYLAXQUTX + PYFAXQVGVAVBVCXOYGXSAYPBVDZRLZAYPCVDZSZYBYSRLZUUASZUGZEHUHZFXQPZYHYFUUFAF + XQAXQTFXQTYFFVEUUEAEHAHTUUBUUDAAYTUUAAXSYSRAXSTARTZAYPBVHZVFAYPCVHZVQAUUC + UUAAYBYSRAYBTUUHUUIVFUUJVQVRVIAQYPSZYEUUEEHUUKYAUUBYDUUDUUKXTYTCUUAUUKBYS + XSRAYPBVJZVKAYPCVJZVLUUKYCUUCCUUAUUKBYSYBRUULVKUUMVLVMVNVOXOXIDKVPLZGQZOZ + AUUOXJVSZBVDZRLZAUUQCVDZSZUUPUFZUURRLZUUTSZUGZGHKUUNUALZMLZPUUNUIOZIZUUGY + HIXIXMXNVTXOUVAGUVGPUVHIZUVDGUVGPUVHIZUVIXOUUNHIZGJUVFMLZUUSNUVFUCOZIZGUV + MUUTNUVNIZUVJXIXMUVLXNDWAWBZXIXMUVOXNXIXMULZGUVMUUPNUVNIZGUVMUURNUVNIZUVO + XIUVSXMXIUVFWCIUUNUVFIZUVSKUUNUAWDXIUUNWHIUWADWEUUNWFWIGUVFUUNWGWJWKZAGBU + UNDUUNWQZWLZGUUPUURUVFWMWNWOXIXNUVPXMAGCUUNDUWCWLWRZGUUSUUTUUNWPWSXOUVLGU + VMUVCNUVNIZUVPUVKUVQXIXMUWFXNUVRGUVMUVBNUVNIZUVTUWFUVRUVSUWGUWBGUUPUVFWTX + FUWDGUVBUURUVFWMWNWOUWEGUVCUUTUUNWPWSUVAUVDGUUNXAWNUVEUUEUUPYSRLZUUASZUVB + YSRLZUUASZUGEFGUUNDUWCXSUUPSZUUBUWIUUDUWKUWLYTUWHUUAXSUUPYSRUPUOUWLUUCUWJ + UUAUWLYBUVBYSRXSUUPXBXCUOVMYPUUQSZUWIUVAUWKUVDUWMUWHUUSUUAUUTUWMYSUURUUPR + AYPUUQBXDZVKAYPUUQCXDZVLUWMUWJUVCUUAUUTUWMYSUURUVBRUWNVKUWOVLVMXEWNXGXH + $. $} @@ -663195,36 +665848,36 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, res0 oveq2i eleqtrdi mp3an3 sselid adantll coeq1 eqid fvex fvmpt eqtr4d ffn eqeq1d rexbidva bitrd rabbidva simplll id fun syl21anc feq1i ad3antlr unex sylib mzprename eldioph4i eqeltrd eleq2 rabbidv syl5ibrcom rexlimdva - eleq1d expimpd syl5bi impcom 3impb ) ADUFGHZCIHZJDUGKZJCUGKZBUDZEUEZBLZAH - ZEIUVHMKZUHZCUFGZHZUVFUVINZUVEUVPUVEDIHZAUAUEZFUEZOZUBUEZGPQZFIRSUIZMKZUJ - ZUAIUVGMKZUHZQZUBUWDUVGOZUKGZUJZNUVQUVPFUAADUWDUBRULHZUWDULHUMRSULUNUOZUW - DUPHZUQUPHZURUWDUQUSUTUWOUWPVAUWDRPJVBKZVMGZUIZUQUSSUWRRSJVMGUWRVCUWQJVMV - DVEVFVGPRHUWSUQUSUTVHPVIUOVJUWDUQVKUOVNZSRVLZVOUVQUVRUWLUVPUVQUVRNZUWIUVP - UBUWKUXBUWBUWKHZNZUVPUWIUVKUWHHZEUVMUHZUVOHUXDUXFUVJUVTOZUCRUWDUVHOZMKZUC - UEZBVPUWDVQZOZLZUWBGZVRZGZPQZFUWEUJZEUVMUHZUVOUXDUXEUXREUVMUXDUVJUVMHZNZU - XEUVKUVTOZUWBGZPQZFUWEUJZUXRUYAUVKUWGHZUXEUYEVAUYAUXTUVIUYFUXDUXTVSUVFUVI - UVRUXCUXTVTUVJIUVHBUVGJDUGWAWBWCUWFUYEUAUVKUWGUVSUVKQZUWCUYDFUWEUYGUWAUYB - PUWBUVSUVKUVTWDWEWFWGWHUYAUYDUXQFUWEUYAUVTUWEHZNZUYCUXPPUYIUYCUXGUXLLZUWB - GZUXPUYIUYBUYJUWBUYIUVIUXTUYHUYBUYJQUVFUVIUVRUXCUXTUYHWIUXDUXTUYHWJUYAUYH - VSUVIUXTUYHWOZUYJUXGBLZUXGUXKLZOUYBUXGBUXKWKUYLUYMUVKUYNUVTUYLUYMUVKTOZUV - KUYLUYMUVKUVTBLZOUYOUVJUVTBWLUYLUYPTUVKUYLUWDIUVTUDZUVIUWDUVHWPZTQZUYPTQU - YHUVIUYQUXTUVTIUWDWMZWNUVIUXTUYHWQUYSUYLUYRUVHUWDWPZTUWDUVHWRUVHSWSSUWDWP - TQZVUATQZCWTSRXAZUVHSUWDXBXCZXDXGUVTBUWDIUVGUVHXEXFXHXIUVKXJXSUYLUYNTUVTO - ZUVTUYLUYNUVJUXKLZUVTUXKLZOVUFUVJUVTUXKWLUYLVUGTVUHUVTUYLUVHIUVJUDZVUGTQZ - UXTUVIVUIUYHUVJIUVHWMXKVUIUWDUWDUXKUDZVUCVUJUWDUWDUXKXLVUKUWDXMUWDUWDUXKX - NUOZVUEUVJUXKUVHIUWDUWDXEXOWHUYHUVIVUHUVTQUXTUYHVUHUVTUWDVQZUVTUVTUWDXPUY - HUYQUVTUWDXQVUMUVTQUYTUWDIUVTYTUWDUVTXRXTXIWNYAXIVUFUVTTOUVTTUVTYBUVTXJXD - XSYAYCXFYDUYIUXGUXIHZUXPUYKQUXTUYHVUNUXDUXTUYHNIUXHMKZUXIUXGUWMIRWSVUOUXI - WSUMYEIRUXHULYFXCUXTUYHUVJVUAVQZUVTVUAVQZQZUXGVUOHVUPTVUQVUPUVJTVQTVUATUV - JVUEYGUVJYIXDVUQUVTTVQTVUATUVTVUEYGUVTYIXDVFUXTUYHVURWOUXGIUVHUWDOZMKVUOU - VHUWDIUVJUVTYHVUSUXHIMUVHUWDYBYJYKYLYMYNUCUXGUXNUYKUXIUXOUXJUXGQUXMUYJUWB - UXJUXGUXLYOYDUXOYPUYJUWBYQYRWHYSUUAUUBUUCUUDUXDUVFUXOUXHUKGHZUXSUVOHUVFUV - IUVRUXCUUEUXDUXHULHZUXCUWJUXHUXLUDZVUTVVAUXDUWDUVHUWNJCUGWAUUKXGUXBUXCVSU - VIVVBUVFUVRUXCUVIUWJUXHUXKBOZUDZVVBUVIVUKUVIUWDUVGWPZTQZVVDVUKUVIVULXGUVI - UUFVVFUVIVVEUVGUWDWPZTUWDUVGWRUVGSWSVUBVVGTQDWTVUDUVGSUWDXBXCXDXGUWDUVGUW - DUVHUXKBUUGUUHUWJUXHVVCUXLUXKBYBUUIUULUUJUCUXLUWBUWJUXHUUMXFFEUXOCUWDUWNU - WTUXAUUNWCUUOUWIUVNUXFUVOUWIUVLUXEEUVMAUWHUVKUUPUUQUUTUURUUSUVAUVBUVCUVD - $. + eleq1d expimpd biimtrid impcom 3impb ) ADUFGHZCIHZJDUGKZJCUGKZBUDZEUEZBLZ + AHZEIUVHMKZUHZCUFGZHZUVFUVINZUVEUVPUVEDIHZAUAUEZFUEZOZUBUEZGPQZFIRSUIZMKZ + UJZUAIUVGMKZUHZQZUBUWDUVGOZUKGZUJZNUVQUVPFUAADUWDUBRULHZUWDULHUMRSULUNUOZ + UWDUPHZUQUPHZURUWDUQUSUTUWOUWPVAUWDRPJVBKZVMGZUIZUQUSSUWRRSJVMGUWRVCUWQJV + MVDVEVFVGPRHUWSUQUSUTVHPVIUOVJUWDUQVKUOVNZSRVLZVOUVQUVRUWLUVPUVQUVRNZUWIU + VPUBUWKUXBUWBUWKHZNZUVPUWIUVKUWHHZEUVMUHZUVOHUXDUXFUVJUVTOZUCRUWDUVHOZMKZ + UCUEZBVPUWDVQZOZLZUWBGZVRZGZPQZFUWEUJZEUVMUHZUVOUXDUXEUXREUVMUXDUVJUVMHZN + ZUXEUVKUVTOZUWBGZPQZFUWEUJZUXRUYAUVKUWGHZUXEUYEVAUYAUXTUVIUYFUXDUXTVSUVFU + VIUVRUXCUXTVTUVJIUVHBUVGJDUGWAWBWCUWFUYEUAUVKUWGUVSUVKQZUWCUYDFUWEUYGUWAU + YBPUWBUVSUVKUVTWDWEWFWGWHUYAUYDUXQFUWEUYAUVTUWEHZNZUYCUXPPUYIUYCUXGUXLLZU + WBGZUXPUYIUYBUYJUWBUYIUVIUXTUYHUYBUYJQUVFUVIUVRUXCUXTUYHWIUXDUXTUYHWJUYAU + YHVSUVIUXTUYHWOZUYJUXGBLZUXGUXKLZOUYBUXGBUXKWKUYLUYMUVKUYNUVTUYLUYMUVKTOZ + UVKUYLUYMUVKUVTBLZOUYOUVJUVTBWLUYLUYPTUVKUYLUWDIUVTUDZUVIUWDUVHWPZTQZUYPT + QUYHUVIUYQUXTUVTIUWDWMZWNUVIUXTUYHWQUYSUYLUYRUVHUWDWPZTUWDUVHWRUVHSWSSUWD + WPTQZVUATQZCWTSRXAZUVHSUWDXBXCZXDXGUVTBUWDIUVGUVHXEXFXHXIUVKXJXSUYLUYNTUV + TOZUVTUYLUYNUVJUXKLZUVTUXKLZOVUFUVJUVTUXKWLUYLVUGTVUHUVTUYLUVHIUVJUDZVUGT + QZUXTUVIVUIUYHUVJIUVHWMXKVUIUWDUWDUXKUDZVUCVUJUWDUWDUXKXLVUKUWDXMUWDUWDUX + KXNUOZVUEUVJUXKUVHIUWDUWDXEXOWHUYHUVIVUHUVTQUXTUYHVUHUVTUWDVQZUVTUVTUWDXP + UYHUYQUVTUWDXQVUMUVTQUYTUWDIUVTYTUWDUVTXRXTXIWNYAXIVUFUVTTOUVTTUVTYBUVTXJ + XDXSYAYCXFYDUYIUXGUXIHZUXPUYKQUXTUYHVUNUXDUXTUYHNIUXHMKZUXIUXGUWMIRWSVUOU + XIWSUMYEIRUXHULYFXCUXTUYHUVJVUAVQZUVTVUAVQZQZUXGVUOHVUPTVUQVUPUVJTVQTVUAT + UVJVUEYGUVJYIXDVUQUVTTVQTVUATUVTVUEYGUVTYIXDVFUXTUYHVURWOUXGIUVHUWDOZMKVU + OUVHUWDIUVJUVTYHVUSUXHIMUVHUWDYBYJYKYLYMYNUCUXGUXNUYKUXIUXOUXJUXGQUXMUYJU + WBUXJUXGUXLYOYDUXOYPUYJUWBYQYRWHYSUUAUUBUUCUUDUXDUVFUXOUXHUKGHZUXSUVOHUVF + UVIUVRUXCUUEUXDUXHULHZUXCUWJUXHUXLUDZVUTVVAUXDUWDUVHUWNJCUGWAUUKXGUXBUXCV + SUVIVVBUVFUVRUXCUVIUWJUXHUXKBOZUDZVVBUVIVUKUVIUWDUVGWPZTQZVVDVUKUVIVULXGU + VIUUFVVFUVIVVEUVGUWDWPZTUWDUVGWRUVGSWSVUBVVGTQDWTVUDUVGSUWDXBXCXDXGUWDUVG + UWDUVHUXKBUUGUUHUWJUXHVVCUXLUXKBYBUUIUULUUJUCUXLUWBUWJUXHUUMXFFEUXOCUWDUW + NUWTUXAUUNWCUUOUWIUVNUXFUVOUWIUVLUXEEUVMAUWHUVKUUPUUQUUTUURUUSUVAUVBUVCUV + D $. $} ${ @@ -663348,10 +666001,10 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, ctbnfien $p |- ( ( ( X ~~ _om /\ Y ~~ _om ) /\ ( A C_ X /\ -. A e. Fin ) ) -> A ~~ Y ) $= ( com cen wbr wa wss cfn wcel wn csdm isfinite notbii wo cdom cvv brrelex1i - wi relen ssdomg syl domen2 sylibd imp brdom2 sylib adantlr ord syl5bi enen2 - impr wb ad2antlr mpbird ) BDEFZCDEFZGZABHZAIJZKZGZGACEFZADEFZURUSVAVDVAADLF - ZKURUSGZVDUTVEAMNVFVEVDUPUSVEVDOZUQUPUSGADPFZVGUPUSVHUPUSABPFZVHUPBQJUSVISB - DETRABQUAUBBDAUCUDUEADUFUGUHUIUJULUQVCVDUMUPVBCDAUKUNUO $. + wi relen ssdomg syl domen2 sylibd imp brdom2 sylib adantlr biimtrid impr wb + ord enen2 ad2antlr mpbird ) BDEFZCDEFZGZABHZAIJZKZGZGACEFZADEFZURUSVAVDVAAD + LFZKURUSGZVDUTVEAMNVFVEVDUPUSVEVDOZUQUPUSGADPFZVGUPUSVHUPUSABPFZVHUPBQJUSVI + SBDETRABQUAUBBDAUCUDUEADUFUGUHULUIUJUQVCVDUKUPVBCDAUMUNUO $. ${ $d A x y $. $d ph x y $. $d B x y $. $d D y $. @@ -663387,36 +666040,36 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, ( va vb vc vd cr wss wcel wn wa cv clt wbr wrex crp wral wceq c0 wne cmin w3a co cabs cfv cfn wi crab ccnv eqeq1 rexbidv elrab simp-4l simpr sseldd csup recnd simp-4r subcld cc0 simprr ad2antrr nelneq syl2anc cc wb subeq0 - necon3abid mpbird absrpcld eleq1 syl5ibrcom rexlimdva syl5bi ssrdv adantr - expimpd cab abrexfi rabssab ssfi sylancl adantl wex simplrl n0 sylib eqid - abscld fvoveq1 rspceeqv mpan2 sylanbrc ne0d exlimddv a1i wor gtso fisupcl - ssrab2 mpan syl3anc cle cinf mp2 fisupg elrabi brcnv notbii lenlt biimprd - vex adantll sylan2 ralimdva adantrd reximdva lbinfle df-inf eqcomi breq1i - soss mpd sylibr sselid lenltd mpbid ralrimiva breq2 notbid ralbidv rspcev - ralnex rexbii rexnal bitri ex 3impa con2d imp ) CIJZDIKZCUAUBZDCKLZMZUDZB - NZDUCUEZUFUGZANZOPZBCQZARSZCUHKZLUUHUUPUUOUUCUUDUUGUUPUUOLZUIUUCUUDMZUUGM - ZUUPUUQUUSUUPMZUUMLZBCSZARQZUUQUUTENZFNZDUCUEZUFUGZTZFCQZEIUJZIOUKZURZRKU - UKUVLOPZLZBCSZUVCUUTUVJRUVLUUSUVJRJUUPUUSGUVJRGNZUVJKZUVPIKZUVPUVGTZFCQZM - UUSUVPRKZUVIUVTEUVPIUVDUVPTUVHUVSFCUVDUVPUVGULUMUNUUSUVRUVTUWAUUSUVRMZUVS - UWAFCUWBUVECKZMZUWAUVSUVGRKUWDUVFUWDUVEDUWDUVEUWDCIUVEUUCUUDUUGUVRUWCUOUW - BUWCUPZUQUSZUWDDUUCUUDUUGUVRUWCUTUSZVAUWDUVFVBUBZUVEDTZLZUWDUWCUUFUWJUWEU - USUUFUVRUWCUURUUEUUFVCVDUVEDCVEVFUWDUVEVGKZDVGKZUWHUWJVHUWFUWGUWKUWLMUWIU - VFVBUVEDVIVJVFVKVLUVPUVGRVMVNVOVSVPVQVRUUTUVJUHKZUVJUAUBZUVJIJZUVLUVJKZUU - PUWMUUSUUPUVIEVTZUHKUVJUWQJUWMFECUVGWAUVIEIWBUWQUVJWCWDWEZUUTUUICKZUWNBUU - TUUEUWSBWFUURUUEUUFUUPWGBCWHWIUUTUWSMZUVJUUKUWTUUKIKUUKUVGTZFCQZUUKUVJKZU - WTUUJUWTUUIDUWTUUIUWTCIUUIUUCUUDUUGUUPUWSUOUUTUWSUPUQUSUWTDUUCUUDUUGUUPUW - SUTUSVAWKZUWSUXBUUTUWSUUKUUKTUXBUUKWJFUUICUVGUUKUUKUVEUUIDUFUCWLWMWNWEUVI - UXBEUUKIUVDUUKTUVHUXAFCUVDUUKUVGULUMUNWOZWPWQZUWOUUTUVIEIXBZWRIUVKWSZUWMU - WNUWOUDUWPWTIUVJUVKXAXCXDZUQUUTUVNBCUWTUVLUUKXEPZUVNUWTUVJIOXFZUUKXEPZUXJ - UWTUWOUVPHNZXEPZHUVJSZGUVJQZUXCUXLUWOUWTUXGWRUUTUXPUWSUUTUVPUXMUVKPZLZHUV - JSZUXMUVPUVKPUXMUULUVKPAUVJQUIHUVJSZMZGUVJQZUXPUUTUVJUVKWSZUWMUWNUYBUYCUU - TUWOUXHUYCUXGWTUVJIUVKYDXGWRUWRUXFGHAUVJUVKXHXDUUTUYAUXOGUVJUVQUUTUVRUYAU - XOUIUVIEUVPIXIUUTUVRMZUXSUXOUXTUYDUXRUXNHUVJUXMUVJKUYDUXMIKZUXRUXNUIZUVIE - UXMIXIUVRUYEUYFUUTUXRUXMUVPOPZLZUVRUYEMZUXNUXQUYGUVPUXMOGXNHXNXJXKUYIUXNU - YHUVPUXMXLXMVPXOXPXQXRXPXSYEVRUXEGHUUKUVJXTXDUVLUXKUUKXEUXKUVLUVJIOYAYBYC - YFUWTUVLUUKUUTUVLIKUWSUUTUVJIUVLUXGUXIYGVRUXDYHYIYJUVBUVOAUVLRUULUVLTZUVA - UVNBCUYJUUMUVMUULUVLUUKOYKYLYMYNVFUVCUUNLZARQUUQUVBUYKARUUMBCYOYPUUNARYQY - RWIYSYTUUAUUB $. + necon3abid mpbird absrpcld eleq1 syl5ibrcom expimpd biimtrid ssrdv adantr + rexlimdva cab abrexfi rabssab ssfi sylancl adantl simplrl n0 sylib abscld + wex eqid fvoveq1 rspceeqv mpan2 sylanbrc ne0d exlimddv ssrab2 a1i fisupcl + wor gtso mpan syl3anc cle cinf mp2 fisupg elrabi vex brcnv notbii biimprd + soss lenlt adantll sylan2 ralimdva adantrd reximdva lbinfle df-inf eqcomi + mpd breq1i sylibr sselid lenltd mpbid notbid ralbidv rspcev ralnex rexbii + ralrimiva breq2 rexnal bitri ex 3impa con2d imp ) CIJZDIKZCUAUBZDCKLZMZUD + ZBNZDUCUEZUFUGZANZOPZBCQZARSZCUHKZLUUHUUPUUOUUCUUDUUGUUPUUOLZUIUUCUUDMZUU + GMZUUPUUQUUSUUPMZUUMLZBCSZARQZUUQUUTENZFNZDUCUEZUFUGZTZFCQZEIUJZIOUKZURZR + KUUKUVLOPZLZBCSZUVCUUTUVJRUVLUUSUVJRJUUPUUSGUVJRGNZUVJKZUVPIKZUVPUVGTZFCQ + ZMUUSUVPRKZUVIUVTEUVPIUVDUVPTUVHUVSFCUVDUVPUVGULUMUNUUSUVRUVTUWAUUSUVRMZU + VSUWAFCUWBUVECKZMZUWAUVSUVGRKUWDUVFUWDUVEDUWDUVEUWDCIUVEUUCUUDUUGUVRUWCUO + UWBUWCUPZUQUSZUWDDUUCUUDUUGUVRUWCUTUSZVAUWDUVFVBUBZUVEDTZLZUWDUWCUUFUWJUW + EUUSUUFUVRUWCUURUUEUUFVCVDUVEDCVEVFUWDUVEVGKZDVGKZUWHUWJVHUWFUWGUWKUWLMUW + IUVFVBUVEDVIVJVFVKVLUVPUVGRVMVNVSVOVPVQVRUUTUVJUHKZUVJUAUBZUVJIJZUVLUVJKZ + UUPUWMUUSUUPUVIEVTZUHKUVJUWQJUWMFECUVGWAUVIEIWBUWQUVJWCWDWEZUUTUUICKZUWNB + UUTUUEUWSBWJUURUUEUUFUUPWFBCWGWHUUTUWSMZUVJUUKUWTUUKIKUUKUVGTZFCQZUUKUVJK + ZUWTUUJUWTUUIDUWTUUIUWTCIUUIUUCUUDUUGUUPUWSUOUUTUWSUPUQUSUWTDUUCUUDUUGUUP + UWSUTUSVAWIZUWSUXBUUTUWSUUKUUKTUXBUUKWKFUUICUVGUUKUUKUVEUUIDUFUCWLWMWNWEU + VIUXBEUUKIUVDUUKTUVHUXAFCUVDUUKUVGULUMUNWOZWPWQZUWOUUTUVIEIWRZWSIUVKXAZUW + MUWNUWOUDUWPXBIUVJUVKWTXCXDZUQUUTUVNBCUWTUVLUUKXEPZUVNUWTUVJIOXFZUUKXEPZU + XJUWTUWOUVPHNZXEPZHUVJSZGUVJQZUXCUXLUWOUWTUXGWSUUTUXPUWSUUTUVPUXMUVKPZLZH + UVJSZUXMUVPUVKPUXMUULUVKPAUVJQUIHUVJSZMZGUVJQZUXPUUTUVJUVKXAZUWMUWNUYBUYC + UUTUWOUXHUYCUXGXBUVJIUVKXNXGWSUWRUXFGHAUVJUVKXHXDUUTUYAUXOGUVJUVQUUTUVRUY + AUXOUIUVIEUVPIXIUUTUVRMZUXSUXOUXTUYDUXRUXNHUVJUXMUVJKUYDUXMIKZUXRUXNUIZUV + IEUXMIXIUVRUYEUYFUUTUXRUXMUVPOPZLZUVRUYEMZUXNUXQUYGUVPUXMOGXJHXJXKXLUYIUX + NUYHUVPUXMXOXMVPXPXQXRXSXQXTYDVRUXEGHUUKUVJYAXDUVLUXKUUKXEUXKUVLUVJIOYBYC + YEYFUWTUVLUUKUUTUVLIKUWSUUTUVJIUVLUXGUXIYGVRUXDYHYIYOUVBUVOAUVLRUULUVLTZU + VAUVNBCUYJUUMUVMUULUVLUUKOYPYJYKYLVFUVCUUNLZARQUUQUVBUYKARUUMBCYMYNUUNARY + QYRWHYSYTUUAUUB $. $( A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in ~ rencldnfilem using @@ -663747,26 +666400,26 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, xpex simplr pellexlem1 syl31anc cdiv simprrr qeqnumdivden oveq1d fveq2d wb breq1d mpbid pellexlem2 jca32 ex breq2 fvoveq1 fveq2 breq12d anbi12d elrab fvex eleq1 anbi1d oveq1 neeq1d anbi2d oveq2d ssrab2 sselid simprr - opelopab 3imtr4g opth oveq12d 3eqtr4d syl5bi opeq12d impbid1 dom2d mpi - ) DEFZDUDGZHFUEZIZBJZEFZCJZEFZIZXMKLMZDXOKLMZUFMZNMZOUAZYAPGZUGKXJUFMUH - MZQRZIZIZBCUIZUJFOAJZQRZYIXJNMPGZYISGZKUKZLMZQRZIZAHULZYHUMRYHEEUNEEUOU - OVHYFBCEEUPUQXLUBUCYQYHUBJZURGZYRSGZUSZUCJZURGZUUBSGZUSZUJXLYRHFZOYRQRZ - YRXJNMZPGZYTYMLMZQRZIZIZYSEFZYTEFZIZYSKLMZDYTKLMZUFMZNMZOUAZUUTPGZYDQRZ - IZIZYRYQFZUUAYHFXLUUMUVEXLUUMIZUUPUVAUVCUVGUUNUUOUVGUUFUUGUUNXLUUFUULUT - ZXLUUFUUGUUKVAYRVBVCZUVGUUFUUOUVHYRVDVEZVFUVGXIUUNUUOXKUVAXIXKUUMVGZUVI - UVJXIXKUUMVIYSYTDVJVKUVGXIUUNUUOYSYTVLMZXJNMZPGZUUJQRZUVCUVKUVIUVJUVGUU - KUVOXLUUFUUGUUKVMUVGUUFUUKUVOVQUVHUUFUUIUVNUUJQUUFUUHUVMPUUFYRUVLXJNYRV - NZVOVPVRVEVSYSYTDVTVKWAWBYPUULAYRHYIYRTZYJUUGYOUUKYIYROQWCUVQYKUUIYNUUJ - QYIYRXJPNWDUVQYLYTYMLYIYRSWEVOWFWGWHYGUUNXPIZUUQXTNMZOUAZUVSPGZYDQRZIZI - UVEBCYSYTYRURWIZYRSWIZXMYSTZXQUVRYFUWCUWFXNUUNXPXMYSEWJWKUWFYBUVTYEUWBU - WFYAUVSOUWFXRUUQXTNXMYSKLWLVOZWMUWFYCUWAYDQUWFYAUVSPUWGVPVRWGWGXOYTTZUV - RUUPUWCUVDUWHXPUUOUUNXOYTEWJWNUWHUVTUVAUWBUVCUWHUVSUUTOUWHXTUUSUUQNUWHX - SUURDUFXOYTKLWLWOWOZWMUWHUWAUVBYDQUWHUVSUUTPUWIVPVRWGWGWSWTXLUVFUUBYQFZ - IZUUAUUETZYRUUBTZVQZXLUWKIZUUFUUBHFZUWNUWOYQHYRYPAHWPZXLUVFUWJUTWQUWOYQ - HUUBUWQXLUVFUWJWRWQUUFUWPIZUWLUWMUWLYSUUCTZYTUUDTZIZUWRUWMYSYTUUCUUDUWD - UWEXAUWRUXAUWMUWRUXAIZUVLUUCUUDVLMZYRUUBUXBYSUUCYTUUDVLUWRUWSUWTUTUWRUW - SUWTWRXBUXBUUFYRUVLTUUFUWPUXAVGUVPVEUXBUWPUUBUXCTUUFUWPUXAVIUUBVNVEXCWB - XDUWMYSUUCYTUUDYRUUBURWEYRUUBSWEXEXFVCWBXGXH $. + opelopab 3imtr4g opth oveq12d 3eqtr4d biimtrid opeq12d impbid1 dom2d + mpi ) DEFZDUDGZHFUEZIZBJZEFZCJZEFZIZXMKLMZDXOKLMZUFMZNMZOUAZYAPGZUGKXJU + FMUHMZQRZIZIZBCUIZUJFOAJZQRZYIXJNMPGZYISGZKUKZLMZQRZIZAHULZYHUMRYHEEUNE + EUOUOVHYFBCEEUPUQXLUBUCYQYHUBJZURGZYRSGZUSZUCJZURGZUUBSGZUSZUJXLYRHFZOY + RQRZYRXJNMZPGZYTYMLMZQRZIZIZYSEFZYTEFZIZYSKLMZDYTKLMZUFMZNMZOUAZUUTPGZY + DQRZIZIZYRYQFZUUAYHFXLUUMUVEXLUUMIZUUPUVAUVCUVGUUNUUOUVGUUFUUGUUNXLUUFU + ULUTZXLUUFUUGUUKVAYRVBVCZUVGUUFUUOUVHYRVDVEZVFUVGXIUUNUUOXKUVAXIXKUUMVG + ZUVIUVJXIXKUUMVIYSYTDVJVKUVGXIUUNUUOYSYTVLMZXJNMZPGZUUJQRZUVCUVKUVIUVJU + VGUUKUVOXLUUFUUGUUKVMUVGUUFUUKUVOVQUVHUUFUUIUVNUUJQUUFUUHUVMPUUFYRUVLXJ + NYRVNZVOVPVRVEVSYSYTDVTVKWAWBYPUULAYRHYIYRTZYJUUGYOUUKYIYROQWCUVQYKUUIY + NUUJQYIYRXJPNWDUVQYLYTYMLYIYRSWEVOWFWGWHYGUUNXPIZUUQXTNMZOUAZUVSPGZYDQR + ZIZIUVEBCYSYTYRURWIZYRSWIZXMYSTZXQUVRYFUWCUWFXNUUNXPXMYSEWJWKUWFYBUVTYE + UWBUWFYAUVSOUWFXRUUQXTNXMYSKLWLVOZWMUWFYCUWAYDQUWFYAUVSPUWGVPVRWGWGXOYT + TZUVRUUPUWCUVDUWHXPUUOUUNXOYTEWJWNUWHUVTUVAUWBUVCUWHUVSUUTOUWHXTUUSUUQN + UWHXSUURDUFXOYTKLWLWOWOZWMUWHUWAUVBYDQUWHUVSUUTPUWIVPVRWGWGWSWTXLUVFUUB + YQFZIZUUAUUETZYRUUBTZVQZXLUWKIZUUFUUBHFZUWNUWOYQHYRYPAHWPZXLUVFUWJUTWQU + WOYQHUUBUWQXLUVFUWJWRWQUUFUWPIZUWLUWMUWLYSUUCTZYTUUDTZIZUWRUWMYSYTUUCUU + DUWDUWEXAUWRUXAUWMUWRUXAIZUVLUUCUUDVLMZYRUUBUXBYSUUCYTUUDVLUWRUWSUWTUTU + WRUWSUWTWRXBUXBUUFYRUVLTUUFUWPUXAVGUVPVEUXBUWPUUBUXCTUUFUWPUXAVIUUBVNVE + XCWBXDUWMYSUUCYTUUDYRUUBURWEYRUUBSWEXEXFVCWBXGXH $. $} ${ @@ -663804,44 +666457,44 @@ to be empty ( ` ( 1 ... 0 ) ` ). 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We are not explicitly defining the ZZ ) -> ( A ^ B ) e. ( Pell14QR ` D ) ) $= ( cn csquarenn cdif wcel cpell14qr cfv co cn0 wa simplll pell14qrexpclnn0 cexp simpllr simpr syl3anc cc recnd cz cr cneg wo c1 cdiv wceq pell14qrre - elznn0 ad2antrr simplr expneg2 pell14qrreccl syl2anc jaodan syl5bi 3impia - eqeltrd expl ) CDEFGZACHIZGZBUAGZABOJZVAGZVCBUBGZBKGZBUCZKGZUDZLUTVBLZVEB - UIVKVFVJVEVKVFLZVGVEVIVLVGLUTVBVGVEUTVBVFVGMUTVBVFVGPVLVGQABCNRVLVILZVDUE - AVHOJZUFJZVAVMASGZBSGVIVDVOUGVKVPVFVIVKAACUHTUJVMBVKVFVIUKTVLVIQZABULRVMU - TVNVAGZVOVAGUTVBVFVIMZVMUTVBVIVRVSUTVBVFVIPVQAVHCNRVNCUMUNURUOUSUPUQ $. + elznn0 ad2antrr simplr expneg2 pell14qrreccl syl2anc jaodan expl biimtrid + eqeltrd 3impia ) CDEFGZACHIZGZBUAGZABOJZVAGZVCBUBGZBKGZBUCZKGZUDZLUTVBLZV + EBUIVKVFVJVEVKVFLZVGVEVIVLVGLUTVBVGVEUTVBVFVGMUTVBVFVGPVLVGQABCNRVLVILZVD + UEAVHOJZUFJZVAVMASGZBSGVIVDVOUGVKVPVFVIVKAACUHTUJVMBVKVFVIUKTVLVIQZABULRV + MUTVNVAGZVOVAGUTVBVFVIMZVMUTVBVIVRVSUTVBVFVIPVQAVHCNRVNCUMUNURUOUPUQUS $. $( First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.) $) @@ -664578,8 +667231,8 @@ trivial solution (1,0). We are not explicitly defining the VSYAQJUVMYJFBXTUUKUUBYQWRWCWSXN $. $( First-quadrant Pell solutions are bounded away from 1. (This particular - bound allows us to prove exact values for the fundamental solution - later.) (Contributed by Stefan O'Rear, 18-Sep-2014.) $) + bound allows to prove exact values for the fundamental solution later.) + (Contributed by Stefan O'Rear, 18-Sep-2014.) $) pell1qrgap $p |- ( ( D e. ( NN \ []NN ) /\ A e. ( Pell1QR ` D ) /\ 1 < A ) -> ( ( sqrt ` ( D + 1 ) ) + ( sqrt ` D ) ) <_ A ) $= ( va vb cn csquarenn wcel cfv c1 clt wbr caddc co csqrt cle cmul wceq cn0 @@ -664703,12 +667356,12 @@ trivial solution (1,0). We are not explicitly defining the ( va vb vc cn wcel cfv c1 cv clt wbr wss cle wral wrex pell14qrre sylancr cr 1re wa csquarenn cdif cpellfund cpell14qr crab cinf pellfundval c0 wne ssrab2 ssrdv sstrid cpell1qr pell1qrss14 pellqrex ssrexv sylc rabn0 breq2 - ex sylibr elrab wi ltle expimpd syl5bi ralrimiv wceq breq1 ralbidv rspcev - infrecl syl3anc eqeltrd ) AEUAUBFZAUCGHBIZJKZBAUDGZUEZRJUFZRBAUGVOVSRLVSU - HUIZCIZDIZMKZDVSNZCROZVTRFVOVSVRRVQBVRUJVOBVRRVOVPVRFVPRFVPAPUTUKULVOVQBV - ROZWAVOAUMGZVRLVQBWHOWGAUNBAUOVQBWHVRUPUQVQBVRURVAVOHRFZHWCMKZDVSNZWFSVOW - JDVSWCVSFWCVRFZHWCJKZTVOWJVQWMBWCVRVPWCHJUSVBVOWLWMWJVOWLTWIWCRFWMWJVCSWC - APHWCVDQVEVFVGWEWKCHRWBHVHWDWJDVSWBHWCMVIVJVKQCDVSVLVMVN $. + ex sylibr elrab ltle expimpd biimtrid ralrimiv wceq breq1 ralbidv infrecl + wi rspcev syl3anc eqeltrd ) AEUAUBFZAUCGHBIZJKZBAUDGZUEZRJUFZRBAUGVOVSRLV + SUHUIZCIZDIZMKZDVSNZCROZVTRFVOVSVRRVQBVRUJVOBVRRVOVPVRFVPRFVPAPUTUKULVOVQ + BVROZWAVOAUMGZVRLVQBWHOWGAUNBAUOVQBWHVRUPUQVQBVRURVAVOHRFZHWCMKZDVSNZWFSV + OWJDVSWCVSFWCVRFZHWCJKZTVOWJVQWMBWCVRVPWCHJUSVBVOWLWMWJVOWLTWIWCRFWMWJVKS + WCAPHWCVCQVDVEVFWEWKCHRWBHVGWDWJDVSWBHWCMVHVIVLQCDVSVJVMVN $. $( Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.) $) @@ -664717,13 +667370,13 @@ trivial solution (1,0). We are not explicitly defining the ( va vb cn csquarenn wcel c1 caddc co csqrt cfv cv clt wbr cle wrex nnrpd cr wss rpsqrtcld cdif cpell14qr crab cinf cpellfund wne ssrab2 pell14qrre c0 wral ex ssrdv sstrid cpell1qr pell1qrss14 pellqrex ssrexv rabn0 sylibr - sylc eldifi peano2nnd rpred readdcld wa breq2 pell14qrgap 3expib ralrimiv - elrab syl5bi infmrgelbi syl31anc pellfundval breqtrrd ) ADEUAFZAGHIZJKZAJ - KZHIZGBLZMNZBAUBKZUCZRMUDZAUEKOVPWDRSWDUIUFZVTRFVTCLZONZCWDUJVTWEONVPWDWC - RWBBWCUGVPBWCRVPWAWCFWARFWAAUHUKULUMVPWBBWCPZWFVPAUNKZWCSWBBWJPWIAUOBAUPW - BBWJWCUQUTWBBWCURUSVPVRVSVPVRVPVQVPVQVPAADEVAZVBQTVCVPVSVPAVPAWKQTVCVDVPW - HCWDWGWDFWGWCFZGWGMNZVEVPWHWBWMBWGWCWAWGGMVFVJVPWLWMWHWGAVGVHVKVICWDVTVLV - MBAVNVO $. + sylc eldifi peano2nnd rpred readdcld wa breq2 pell14qrgap 3expib biimtrid + elrab ralrimiv infmrgelbi syl31anc pellfundval breqtrrd ) ADEUAFZAGHIZJKZ + AJKZHIZGBLZMNZBAUBKZUCZRMUDZAUEKOVPWDRSWDUIUFZVTRFVTCLZONZCWDUJVTWEONVPWD + WCRWBBWCUGVPBWCRVPWAWCFWARFWAAUHUKULUMVPWBBWCPZWFVPAUNKZWCSWBBWJPWIAUOBAU + PWBBWJWCUQUTWBBWCURUSVPVRVSVPVRVPVQVPVQVPAADEVAZVBQTVCVPVSVPAVPAWKQTVCVDV + PWHCWDWGWDFWGWCFZGWGMNZVEVPWHWBWMBWGWCWAWGGMVFVJVPWLWMWHWGAVGVHVIVKCWDVTV + LVMBAVNVO $. $( Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) $) @@ -664745,13 +667398,14 @@ trivial solution (1,0). We are not explicitly defining the -> ( PellFund ` D ) <_ A ) $= ( va vb vc vd wcel cfv c1 clt wbr cv cr cle wceq 3ad2ant1 wral pell14qrre 1re wa cn csquarenn cdif cpell14qr cpellfund crab cinf pellfundval ssrab2 - w3a wss wrex ex ssrdv sstrid breq2 elrab wi ltle sylancr expimpd ralrimiv - syl5bi breq1 ralbidv rspcev simp2 simp3 sylanbrc infrelb syl3anc eqbrtrd - ) BUAUBUCGZABUDHZGZIAJKZUJZBUEHZICLZJKZCVNUFZMJUGZANVMVOVRWBOVPCBUHPVQWAM - UKZDLZELZNKZEWAQZDMULZAWAGZWBANKVMVOWCVPVMWAVNMVTCVNUIVMFVNMVMFLZVNGWJMGW - JBRUMUNUOPVQIMGZIWENKZEWAQZWHSVMVOWMVPVMWLEWAWEWAGWEVNGZIWEJKZTVMWLVTWOCW - EVNVSWEIJUPUQVMWNWOWLVMWNTWKWEMGWOWLURSWEBRIWEUSUTVAVCVBPWGWMDIMWDIOWFWLE - WAWDIWENVDVEVFUTVQVOVPWIVMVOVPVGVMVOVPVHVTVPCAVNVSAIJUPUQVIDEAWAVJVKVL $. + w3a wss wrex ex ssrdv sstrid breq2 elrab wi ltle sylancr expimpd biimtrid + ralrimiv breq1 ralbidv rspcev simp2 sylanbrc infrelb syl3anc eqbrtrd + simp3 ) BUAUBUCGZABUDHZGZIAJKZUJZBUEHZICLZJKZCVNUFZMJUGZANVMVOVRWBOVPCBUH + PVQWAMUKZDLZELZNKZEWAQZDMULZAWAGZWBANKVMVOWCVPVMWAVNMVTCVNUIVMFVNMVMFLZVN + GWJMGWJBRUMUNUOPVQIMGZIWENKZEWAQZWHSVMVOWMVPVMWLEWAWEWAGWEVNGZIWEJKZTVMWL + VTWOCWEVNVSWEIJUPUQVMWNWOWLVMWNTWKWEMGWOWLURSWEBRIWEUSUTVAVBVCPWGWMDIMWDI + OWFWLEWAWDIWENVDVEVFUTVQVOVPWIVMVOVPVGVMVOVPVLVTVPCAVNVSAIJUPUQVHDEAWAVIV + JVK $. ${ $d x D $. $d x A $. @@ -664766,18 +667420,18 @@ trivial solution (1,0). We are not explicitly defining the cinf wceq simp3 eqbrtrrd pellfundre eqeltrrd simp2 mpbid wne pell14qrre c0 ssrab2 ex ssrdv sstrid pell1qrss14 pellqrex ssrexv sylc rabn0 sylibr infmrgelbi syl3anc mtod rexnal breq2 elrab simprl simpl1 syl2anc simprr - 1red ltled jca wb elpell1qr2 mpbird sylan2b adantrr sselid simpr syl5bi - syl a1i imp pellfundlb adantr sseldd simpl2 reximssdv ) CUAUBUCEZBFEZCU - DGZBHIZJZBAKZLIZMZXBXELIZXEBHIZNACUEGZODKZHIZDCUFGZUGZXDXFAXNUHZMXGAXNP - XDXOBXNFHUJZLIZXDXPBHIXQMXDXBXPBHWTXAXBXPUKXCDCUIQZWTXAXCULUMXDXPBXDXBX - PFXRWTXAXBFEXCCUNQUOWTXAXCUPZRUQXDXNFSZXNUTURZXAXOXQTXDXNXMFXLDXMVAZWTX - AXMFSZXCWTDXMFWTXKXMEXKFEXKCUSVBVCQZVDXDXLDXMPZYAXDXJXMSZXLDXJPZYEWTXAY - FXCCVEQWTXAYGXCDCVFQXLDXJXMVGVHXLDXMVIVJXSXTYAXAJXOXQAXNBVKVBVLVMXFAXNV - NVJXDXEXNEZXEXJEZXGYHXDXEXMEZOXEHIZNZYIXLYKDXEXMXKXEOHVOVPZXDYLNZYIYJOX - ELIZNZYNYJYOXDYJYKVQZYNOXEYNWAYNWTYJXEFEWTXAXCYLVRZYQXECUSVSXDYJYKVTWBW - CYNWTYIYPWDYRXECWEWLWFWGWHXDYHXGNZNZXHXIYTWTYJYKXHWTXAXCYSVRYTXNXMXEYBX - DYHXGVQWIZXDYHYKXGXDYHYKYHYLXDYKYMYLYKTXDYJYKWJWMWKWNWHXECWOVLYTXIXGXDY - HXGVTYTXEBYTXMFXEXDYCYSYDWPUUAWQWTXAXCYSWRRWFWCWS $. + 1red ltled jca wb elpell1qr2 syl mpbird sylan2b adantrr sselid biimtrid + simpr a1i imp pellfundlb adantr sseldd simpl2 reximssdv ) CUAUBUCEZBFEZ + CUDGZBHIZJZBAKZLIZMZXBXELIZXEBHIZNACUEGZODKZHIZDCUFGZUGZXDXFAXNUHZMXGAX + NPXDXOBXNFHUJZLIZXDXPBHIXQMXDXBXPBHWTXAXBXPUKXCDCUIQZWTXAXCULUMXDXPBXDX + BXPFXRWTXAXBFEXCCUNQUOWTXAXCUPZRUQXDXNFSZXNUTURZXAXOXQTXDXNXMFXLDXMVAZW + TXAXMFSZXCWTDXMFWTXKXMEXKFEXKCUSVBVCQZVDXDXLDXMPZYAXDXJXMSZXLDXJPZYEWTX + AYFXCCVEQWTXAYGXCDCVFQXLDXJXMVGVHXLDXMVIVJXSXTYAXAJXOXQAXNBVKVBVLVMXFAX + NVNVJXDXEXNEZXEXJEZXGYHXDXEXMEZOXEHIZNZYIXLYKDXEXMXKXEOHVOVPZXDYLNZYIYJ + OXELIZNZYNYJYOXDYJYKVQZYNOXEYNWAYNWTYJXEFEWTXAXCYLVRZYQXECUSVSXDYJYKVTW + BWCYNWTYIYPWDYRXECWEWFWGWHWIXDYHXGNZNZXHXIYTWTYJYKXHWTXAXCYSVRYTXNXMXEY + BXDYHXGVQWJZXDYHYKXGXDYHYKYHYLXDYKYMYLYKTXDYJYKWLWMWKWNWIXECWOVLYTXIXGX + DYHXGVTYTXEBYTXMFXEXDYCYSYDWPUUAWQWTXAXCYSWRRWGWCWS $. $} $( The fundamental solution as an infimum is itself a solution, showing @@ -666707,13 +669361,13 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by cneg c2 cuz cr wo elznn0 wi 3expia adantr simplll simprl simprr nn0z adantl cfv simplr recnd znegclb syl mpbird zmulcld zaddcld syl121anc cmin ad2antrl simpr mulneg1d oveq2d ad2antll mulcld addcld pncand 3eqtrd breq2d bitr2d ex - negsubd jaod expimpd syl5bi 3impia ) AUAUBUNEZCFEZDFEZGZBFEZACHIZADHIZJKZWF - ADBCLIZMIZHIJKZNZWEBUCEZBOEZBTZOEZUDZGWAWDGZWLBUEWRWMWQWLWRWMGZWNWLWPWRWNWL - UFWMWAWDWNWLABCDPUGUHWSWPWLWSWPGZWKWFAWJWOCLIZMIZHIZJKZWHWTWAWBWJFEWPWKXDNW - AWDWMWPUIWRWBWMWPWAWBWCUJQZWTDWIWRWCWMWPWAWBWCUKQWTBCWTWEWOFEZWPXFWSWOULUMW - TBREWEXFNWTBWRWMWPUOUPZBUQURUSXEUTVAWSWPVEAWOCWJPVBWTXCWGWFJWTXBDAHWTXBWJWI - TZMIWJWIVCIDWTXAXHWJMWTBCXGWRCREZWMWPWBXIWAWCCSVDQZVFVGWTWJWIWTDWIWRDREZWMW - PWCXKWAWBDSVHQZWTBCXGXJVIZVJXMVPWTDWIXLXMVKVLVGVMVNVOVQVRVSVT $. + negsubd jaod expimpd biimtrid 3impia ) AUAUBUNEZCFEZDFEZGZBFEZACHIZADHIZJKZ + WFADBCLIZMIZHIJKZNZWEBUCEZBOEZBTZOEZUDZGWAWDGZWLBUEWRWMWQWLWRWMGZWNWLWPWRWN + WLUFWMWAWDWNWLABCDPUGUHWSWPWLWSWPGZWKWFAWJWOCLIZMIZHIZJKZWHWTWAWBWJFEWPWKXD + NWAWDWMWPUIWRWBWMWPWAWBWCUJQZWTDWIWRWCWMWPWAWBWCUKQWTBCWTWEWOFEZWPXFWSWOULU + MWTBREWEXFNWTBWRWMWPUOUPZBUQURUSXEUTVAWSWPVEAWOCWJPVBWTXCWGWFJWTXBDAHWTXBWJ + WITZMIWJWIVCIDWTXAXHWJMWTBCXGWRCREZWMWPWBXIWAWCCSVDQZVFVGWTWJWIWTDWIWRDREZW + MWPWCXKWAWBDSVHQZWTBCXGXJVIZVJXMVPWTDWIXLXMVKVLVGVMVNVOVQVRVSVT $. $( Lemma 2.19 of [JonesMatijasevic] p. 696. Transfer divisibility constraints between Y-values and their indices. (Contributed by Stefan @@ -668475,46 +671129,46 @@ group element in (1,2), contradicting ~ pell14qrgapw . 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EIWNWOWPUVLWMYRYSYRYSUVMYTUVNUVOYT $. + eqidd elrabd elind wi elin elrab anbi2i bitri weq rspcdva biimtrid mpd + ex ne0d imbi1i impexp ralbii2 simplrl fveq2 breq1d simplrr simpr breq2d + wo simprrr mtbird ad3antrrr simprr eleq1w anbi12d imbi12d simplr mp2and + eqtr2d csbeq1d breqd mtbid expr imnan ioran sylanbrc fnwe2val ralrimiva + sylnibr rspcev rexlimdv rexlimdvaa sylbid ) AUBUFZUCUFZGUGZUHZUBKEUIZLU + FZUJZUKZUCUWBULZNUFZMUFZIUGZUHZNUWAUKZMUWAULZAUWBUMUNZFGUOZUWBFUPUWBUQU + TZUWDAEFUVTURUVTUSUWKREFUVTVAUVTUWALVBZVCVDAFGVEUWLSFGVFVGAUWBUVTVHFUVT + UWAVIAEFUVTRVJVKAUVTVLZUWAVMZUQUTUWMAUWPUWAUQAUWPUWAUWOVMZUWAUWOUWAVNAU + WAUWOUPUWQUWAVOAUWAEUWOTAEFUVTRWDVPUWAUWOVQVRVSUAVTUWBUQUWPUQUVTUWAWAWB + WCUCUBFUWBUMGWEWFAUWDUVQKWGZUDUFZKWGZGUGZUHZUCUWAUKZUDUWAULZUWJAUWDUVPU + WSUVTUWAUIZWGZGUGZUHZUBUWBUKZUDUWAULZUXDUWDUWCUCUXEVHZULZAUXJUWCUCUWBUX + KUVTUWAWHZWIAUXEUWAWJZUXLUXJWKAUVTEWJUWAEUPZUXNAEFUVTRXGTEUWAUVTWLWMZUW + CUXIUCUDUWAUXEUVQUXFVOZUVSUXHUBUWBUXQUVRUXGUVQUXFUVPGWNWOWPWQVGWRAUXIUX + CUDUWAAUDLWSZWTZUXIUVQUXEWGZUXFGUGZUHZUCUWAUKZUXCAUXIUYCWKUXRUXIUXHUBUX + KUKZAUYCUXHUBUWBUXKUXMXAAUXNUYDUYCWKUXPUXHUYBUBUCUWAUXEUVPUXTVOUXGUYAUV + PUXTUXFGXBWOXCVGWRXDUXSUYBUXBUCUWAUXSUCLWSZWTZUYAUXAUYFUXTUWRUXFUWTGUYF + UXTUVQKUWAUIZWGZUWRUYFUVQUXEUYGAUXEUYGVOUXRUYEAKUWAETXEXFZXHUYEUYHUWRVO + UXSUVQUWAKXIXJXKUYFUXFUWSUYGWGZUWTUYFUWSUXEUYGUYIXHUXRUYJUWTVOAUYEUWSUW + AKXIXLXKXMWOXNXOXPXOAUXCUWJUDUWAAUXRUXCWTZWTZUEUFZUVPDUWTHXQZUGZUHZUEUW + ACUFZKWGUWTVOZCEXRZVMZUKZUBUYTULZUWJUYLUYTUMUNZUYSUYNUOZUYTUYSUPZUYTUQU + TVUBVUCUYLUWAUYSUWNXSXTAUXRVUDUXCAUXRUWSEUNZVUDAUWAEUWSTYAZAVUFWTUYSUYN + VEVUDABCDEGHIJKUDOPQYBUYSUYNVFVGYCYDVUEUYLUWAUYSYEXTUYLUYTUWSUYLUWAUYSU + WSAUXRUXCYFUYLUYRUWTUWTVOCUWSEUYQUWSUWTKYGAUXRVUFUXCVUGYDUYLUWTYHYIYJUU + AUBUEUYSUYTUMUYNWEWFUYLVUAUWJUBUYTUVPUYTUNZUBLWSZUVPEUNZUVPKWGZUWTVOZWT + ZWTZUYLVUAUWJYKZVUHVUIUVPUYSUNZWTVUNUVPUWAUYSYLVUPVUMVUIUYRVULCUVPEUYQU + VPUWTKYGYMYNYOUYLVUNVUOVUAUYMEUNZUYMKWGUWTVOZWTZUYPYKZUEUWAUKZUYLVUNWTZ + UWJUYPVUTUEUYTUWAUYMUYTUNZUYPYKUELWSZVUSWTZUYPYKVVDVUTYKVVCVVEUYPVVCVVD + UYMUYSUNZWTVVEUYMUWAUYSYLVVFVUSVVDUYRVURCUYMEUYQUYMUWTKYGYMYNYOUUBVVDVU + SUYPUUCYOUUDVVBVVAUWJVVBVVAWTZVUIUWEUVPIUGZUHZNUWAUKZUWJUYLVUIVUMVVAUUE + VVGVVINUWAVVGNLWSZWTZUWEKWGZVUKGUGZVVMVUKVOZUWEUVPDVVMHXQZUGZWTZUUKZVVH + VVLVVNUHVVRUHZVVSUHVVLVVNVVMUWTGUGZVVLUXBVWAUHUCUWAUWEUCNYPZUXAVWAVWBUW + RVVMUWTGUVQUWEKUUFUUGWOVVBUXCVVAVVKAUXRUXCVUNUUHXFVVGVVKUUIYQVVLVUKUWTV + VMGVVBVULVVAVVKUYLVUIVUJVULUULZXFUUJUUMVVLVVOVVQUHZYKVVTVVGVVKVVOVWDVVG + VVKVVOWTZWTZUWEUVPUYNUGZVVQVWFUWEEUNZVVMUWTVOZVWGUHZVVGVVKVWHVVOVVGUWAE + UWEAUXOUYKVUNVVATUUNYAYDVWFVVMVUKUWTVVGVVKVVOUUOZVVBVULVVAVWEVWCXFZXKVW + FVUTVWHVWIWTZVWJYKUEUWAUWEUENYPZVUSVWMUYPVWJVWNVUQVWHVURVWIUENEUUPUYMUW + EUWTKYGUUQVWNUYOVWGUYMUWEUVPUYNXBWOUURVVBVVAVWEUUSVVGVVKVVOYFYQUUTVWFUY + NVVPUWEUVPVWFDUWTVVMHVWFVVMVUKUWTVWKVWLUVAUVBUVCUVDUVEVVOVVQUVFVRVVNVVR + UVGUVHBCDGHIJKNUBOPUVIUVKUVJUWIVVJMUVPUWAMUBYPZUWHVVINUWAVWOUWGVVHUWFUV + PUWEIWNWOWPUVLWMYTYRYTYRUVMYSUVNUVOYS $. $} ${ @@ -669391,47 +672045,47 @@ element in each topology (which need not be in the closed set - 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if it 24-Jan-2015.) $) pwslnmlem1 $p |- ( W e. LNoeM -> Y e. LNoeM ) $= ( vx clnm wcel cbs cfv cv csn cxp cmpt clmhm co crn wceq clmod cvv eqid - lnmlmod snex pwsdiaglmhm sylancl wf1o wfo pwssnf1o elvd f1ofo forn lnmepi - id 3syl syl3anc ) BFGZEBHIZAJZKZEJKLMZBCNOGZUOUSPCHIZQZCFGUOBRGURSGUTBUAU - QUBEUPBUSURSCDUPTZUSTZUCUDUOULUOUPVAUSUEZUPVAUSUFVBUOVEAEUPVABUSUQFSCDVCV + lnmlmod vsnex pwsdiaglmhm sylancl id wf1o pwssnf1o elvd f1ofo forn lnmepi + wfo 3syl syl3anc ) BFGZEBHIZAJZKZEJKLMZBCNOGZUOUSPCHIZQZCFGUOBRGURSGUTBUA + AUBEUPBUSURSCDUPTZUSTZUCUDUOUEUOUPVAUSUFZUPVAUSULVBUOVEAEUPVABUSUQFSCDVCV DVATZUGUHUPVAUSUIUPVAUSUJUMVABCUSVFUKUN $. $} @@ -669511,14 +672165,14 @@ element in each topology (which need not be in the closed set - if it Stefan O'Rear, 24-Jan-2015.) $) pwslnm $p |- ( ( W e. LNoeM /\ I e. Fin ) -> Y e. 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HUVFUVGUVEUVHUVIYTUVJ $. + lidlsubcl lidlacl eqeltrrd rexlimdvaa biimtrid expr 3exp a2d nn0ind rsp + syl6com com23 imp rexlimdv eqelssd ) AUAGHOAUAUFZHQZUOZUVKDUGRZRZUBUFZS + UHZUBUIUJZUVKGQZUVMUVOUIUKULZUMZQZUVRUVMUVKCUNRZQZUWBAHUWCUVKAHFQZHUWCU + PZNUWCHFCUWCUQZJURTZUSZUWCUVNCDUVKUVNUQZIUWGUTTUWBUVOUIQZUVOUVTQZVAUVRU + VOUIUVTVBUWKUVRUWLVCUIUPUWKUVQUBVCUJZUVRVDUWKUVOVEQUWMUVOVFUVOUBVGTUVQU + BVCUIVIVHUWLUVOUKVJZUVRUVOUKVKUWNUVRUKUVPSUHZUBUIUJZVLUIQUKVLSUHZUWPVMV + NUWOUWQUBVLUIUVPVLUKSVOVPVQUWNUVQUWOUBUIUVOUKUVPSVRVSWATVTWBTUVMUVQUVSU + BUIAUVLUVPUIQZUVQUVSWCZWCAUWRUVLUWSUWRAUWSUAHWDZUVLUWSWCAUVOUCUFZSUHZUV + SWCZUAHWDZWCAUVOVLSUHZUVSWCZUAHWDZWCAUWTWCZAUDUFZUVNRZUVPWEWFWGZSUHZUXI + GQZWCZUDHWDZWCUXHUCUBUVPUXAVLVJZUXDUXGAUXPUXCUXFUAHUXPUXBUXEUVSUXAVLUVO + SVOWHWIWJUCUBWKZUXDUWTAUXQUXCUWSUAHUXQUXBUVQUVSUXAUVPUVOSVOWHWIWJZUXAUX + KVJZUXDUXOAUXSUXDUVOUXKSUHZUVSWCZUAHWDUXOUXSUXCUYAUAHUXSUXBUXTUVSUXAUXK + UVOSVOWHWIUYAUXNUAUDHUAUDWKZUXTUXLUVSUXMUYBUVOUXJUXKSUVKUXIUVNWLWMUVKUX + IGWNWOWPWQWJUXRAUXFUAHUVMUXEUVKCWRRZVJZUVSUVMDWSQZUWDUXEUYDWTAUYEUVLLXA + UWIUWCUVNCDUVKUYCUWJIUYCUQZUWGXBXCAUYDUVSWCZUVLAUYCGQZUYGACWSQZGFQZUYHA + UYEUYILCDIXETZMCFGUYCJUYFXDXCUYCGUVKXFTXAXGXHUWRAUWTUXOUWRAUWTUXOUWRAUW + TXIZUXNUDHUYLUXIHQZUOZUXLUXJUVPXOUHZUXMUYNUXJUWAQZUVPXJQUXLUYOWTUYNUXIU + WCQZUYPUYLHUWCUXIAUWRUWFUWTUWHXKUSUWCUVNCDUXIUWJIUWGUTTUYNUVPUWRAUWTUYM + XLXMUXJUVPXNXCUYLUYMUYOUXMUYLUYMUYOUOZUOZUVPUXIXPRZRZUEUFZUVNRZUVPXOUHZ + UXAUVPVUBXPRZRZVJZUOZUEGUJZUCXQZQZUXMUYSVUHUEHUJZUCXQZVUJVUAUYLVUMVUJUP + ZUYRUWRAVUNUWTUWRAUOZUVPHERZRZUVPGERZRZVUMVUJAUWRBUFZVUPRZVUTVURRZUPZBU + IWDVUQVUSUPZPVVCVVDBUVPUIBUBWKVVAVUQVVBVUSVUTUVPVUPWLVUTUVPVURWLXRXSXTV + UOUYEUWEUWRVUQVUMVJAUYEUWRLYAZAUWEUWRNYAUWRAYBZUVNCDEFUCUEHWSUVPIJKUWJY + CYDVUOUYEUYJUWRVUSVUJVJVVEAUYJUWRMYAVVFUVNCDEFUCUEGWSUVPIJKUWJYCYDYEYFX + AUYRVUAVUMQZUYLUYRVUDVUAVUFVJZUOZUEHUJZVVGUYRUYMUYOVUAVUAVJZVVJUYMUYOYB + UYMUYOYGUYRVUAYHVVIUYOVVKUOUEUXIHUEUDWKZVUDUYOVVHVVKVVLVUCUXJUVPXOVUBUX + IUVNWLWMVVLVUFVUAVUAVVLUVPVUEUYTVUBUXIXPWLYIYJYKVPYQVULVVJUCVUAUVPUYTYL + ZUXAVUAVJZVUHVVIUEHVVNVUGVVHVUDUXAVUAVUFYMYNZVSYOUUAYAYPVUKVVIUEGUJZUYS + UXMVUIVVPUCVUAVVMVVNVUHVVIUEGVVOVSYOUYSVVIUXMUEGUYSVUBGQZVVIUOZUOZUXIVU + BCUUBRZWGZVUBCUUCRZWGZUXIGVVSCUUGQZUYQVUBUWCQVWCUXIVJVVSUYIVWDVVSAUYIUW + RAUWTUYRVVRUUDZUYKTZCUUETVVSHUWCUXIVVSAUWFVWEUWHTUYLUYMUYOVVRUUHZYPZVVS + GUWCVUBVVSAGUWCUPZVWEAUYJVWIMUWCGFCUWGJURTTUYSVVQVVIUUFZYPZUWCVWBCVVTUX + IVUBUWGVWBUQZVVTUQZUUIYDVVSUYIUYJVWAGQZVVQVWCGQVWFUYLUYJUYRVVRAUWRUYJUW + TMXKYRVVSVWAUVNRZUVPSUHZVWNVVSUYTUWCVUEUVNCDUXIVUBUVPVVTUWJIUWGVWMUWRAU + WTUYRVVRUUJVVSAUYEVWELTVWHUYLUYMUYOVVRUUKVWKUYSVVQVUDVVHUULUYTUQVUEUQUY + SVVQVUDVVHUUMUUNVVSVWAHQZUWTVWPVWNWCZVVSUYIUWEUYMVUBHQVWQVWFVVSAUWEVWEN + TVWGVVSGHVUBUYLGHUPZUYRVVRAUWRVWSUWTOXKYRVWJYPCFHVVTUXIVUBJVWMUUPYSUWRA + UWTUYRVVRUUOUWSVWRUAVWAHUVKVWAVJZUVQVWPUVSVWNVWTUVOVWOUVPSUVKVWAUVNWLWM + UVKVWAGWNWOXSXCYTVWJVWBCFGVWAVUBJVWLUUQYSUURUUSUUTYTUVAXGXHUVBUVCUVDUWS + UAHUVEUVFUVGUVHUVIYTUVJ $. $} ${ @@ -672118,6 +674772,24 @@ is in the span of P(i)(X), so there is an R-linear combination of $( (End of Stefan O'Rear's mathbox.) $) $( End $[ set-mbox-so.mm $] $) +$( Begin $[ set-mbox-np.mm $] $) +$( Skip $[ set-main.mm $] $) +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Noam Pasman +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + + ${ + r1sssucd.1 $e |- ( ph -> A e. On ) $. + $( Deductive form of ~ r1sssuc . (Contributed by Noam Pasman, + 19-Jan-2025.) $) + r1sssucd $p |- ( ph -> ( R1 ` A ) C_ ( R1 ` suc A ) ) $= + ( con0 wcel cr1 cfv csuc wss r1sssuc syl ) ABDEBFZGBHLGICBJK $. + $} + +$( (End of Noam Pasman's mathbox.) $) +$( End $[ set-mbox-np.mm $] $) $( Begin $[ set-mbox-jpp.mm $] $) $( Skip $[ set-main.mm $] $) @@ -672406,6 +675078,984 @@ is in the span of P(i)(X), so there is an R-linear combination of $) +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Natural addition of Cantor normal forms +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + + $( The product of a limit ordinal with any nonzero ordinal is a limit + ordinal. (Contributed by RP, 8-Jan-2025.) $) + omlimcl2 $p |- ( ( ( A e. On /\ ( B e. C /\ Lim B ) ) /\ (/) e. A ) + -> Lim ( B .o A ) ) $= + ( con0 wcel wlim wa c0 cuni wceq comu co csuc word ad2antrr adantl ad3antlr + ex syl simpr wne ne0i id w3a df-lim biimpri syl2an3an limelon simpll anim1i + eloni 0ellim omlimcl syl21anc syld coa onuni anim12ci jca oalimcl wb oveq2d + omcl omsuc eqtrd limeq mpbird wo orduniorsuc mpjaod ) ADEZBCEZBFZGZGZHAEZGZ + AAIZJZBAKLZFZAVRMZJZVQVSAFZWAVQVSWDVQANZAHUAZVSVSWDVKWEVNVPAUKOZVPWFVOAHUBP + VSUCWDWEWFVSUDAUEUFUGRVQWDWAVQWDGBDEZVKWDGHBEZWAVNWHVKVPWDBCUHZQVQVKWDVKVNV + PUIUJVNWIVKVPWDVMWIVLBULPQBADUMUNRUOVQWCWAVQWCGZWABVRKLZBUPLZFZWKWLDEZVNGZW + NVOWPVPWCVOWOVNVOWHVRDEZGZWOVKWQVNWHAUQWJURZBVRVCSVKVNTUSOWLBCUTSWKVTWMJWAW + NVAWKVTBWBKLZWMWKAWBBKVQWCTVBWKWRWTWMJVOWRVPWCWSOBVRVDSVEVTWMVFSVGRVQWEVSWC + VHWGAVISVJ $. + + ${ + $d A x $. $d B x $. $d C x $. + $( If ` C ` is between ` A ` (inclusive) and ` ( A +o B ) ` (exclusive), + there is an ordinal which equals ` C ` when summed to ` A ` . This is a + slightly different statement than ~ oawordex or ~ oawordeu . + (Contributed by RP, 7-Jan-2025.) $) + oawordex2 $p |- ( ( ( A e. On /\ B e. On ) + /\ ( A C_ C /\ C e. ( A +o B ) ) ) -> + E. x e. B ( A +o x ) = C ) $= + ( con0 wcel wa wss coa co cv wceq wrex simprl wb simpll oacl simpr simprr + adantr onelon syl2an oawordex syl2anc mpbid eqeltrd simpllr oaord syl3anc + mpbird reximssdv ) BEFZCEFZGZBDHZDBCIJZFZGZGZBAKZIJZDLZVBACEUSUOVBAEMZUNU + OUQNUSULDEFZUOVCOULUMURPZUNUPEFUQVDURBCQUOUQRUPDUAUBABDUCUDUEUSUTEFZVBGZG + ZUTCFZVAUPFZVHVADUPUSVFVBSZUSUQVGUNUOUQSTUFVHVFUMULVIVJOUSVFVBNULUMURVGUG + USULVGVETUTCBUHUIUJVKUK $. + $} + + ${ + $d A x $. $d B x $. + $( If an ordinal, ` B ` , is in a half-open interval between some ` A ` and + the next limit ordinal, ` B ` is the sum of the ` A ` and some natural + number. This weakens the antecedent of ~ nnawordex . (Contributed by + RP, 7-Jan-2025.) $) + nnawordexg $p |- ( ( A e. On /\ A C_ B /\ B e. ( A +o _om ) ) + -> E. x e. _om ( A +o x ) = B ) $= + ( con0 wcel wss com coa co w3a wa cv wceq wrex simp1 omelon a1i oawordex2 + 3simpc syl21anc ) BDEZBCFZCBGHIEZJZUAGDEZUBUCKBALHICMAGNUAUBUCOUEUDPQUAUB + UCSABGCRT $. + $} + + $( Closure law for ordinal successor. (Contributed by RP, 8-Jan-2025.) $) + succlg $p |- ( ( A e. B /\ + ( B = (/) \/ ( B = ( _om .o C ) /\ C e. ( On \ 1o ) ) ) ) -> + suc A e. B ) $= + ( wcel c0 wceq com comu co con0 c1o cdif wa wo csuc eleq2 wlim cvv ad2antll + wb noel pm2.21i syl6bi com12 simpl eldifi omex limom pm3.2i ondif1 omlimcl2 + a1i simprbi syl21anc limeq ad2antrl mpbird limsuc syl mpbid ex jaod imp ) A + BDZBEFZBGCHIZFZCJKLDZMZNAOBDZVDVEVJVIVEVDVJVEVDAEDZVJBEAPVKVJAUAUBUCUDVDVIV + JVDVIMZVDVJVDVIUEVLBQZVDVJTVLVMVFQZVLCJDZGRDZGQZMZECDZVNVHVOVDVGCJKUFSVRVLV + PVQUGUHUIULVHVSVDVGVHVOVSCUJUMSCGRUKUNVGVMVNTVDVHBVFUOUPUQBAURUSUTVAVBVC $. + + ${ + $d A x y $. + $( A limit ordinal is either the proper class of ordinals or some nonzero + product with omega. (Contributed by RP, 8-Jan-2025.) $) + dflim5 $p |- ( Lim A <-> ( A = On \/ + E. x e. ( On \ 1o ) A = ( _om .o x ) ) ) $= + ( vy wlim con0 wceq wa wo com co syl limeq coa c0 omelon wi adantr adantl + wcel wn cv comu c1o cdif wrex word limord ordeleqon biimpi pm4.71ri andir + orcomd bitri limon mpbiri pm4.71i orbi1i wne w3a simpl a1i id peano1 3jca + ne0ii omeulem1 3syl biimprd simplr wb on0eln0 necon1bd imp jca oveq2d om0 + nnlim mp1i eqtrd oveq1d nna0r mtbird ex cuni csuc cvv ovex nlimsucg nnord + orduniorsuc w3o 3ianor df-lim xchnxbir sylib pm2.24d nne a1i13 pm2.21 mpd + 3jaod orim1d ord omcl sylancr nnon onuni oasuc syl2anc jaod con2d syl6ibr + anor syl9 com13 3imp simp2 anim12i ondif1 sylibr simpr simpl3 oa0 3eqtr3d + mpdan 3exp expdimp rexlimdv expimpd reximdv2 eldifi limom pm3.2i omlimcl2 + eqeltrd mpanl2 sylbi mpbird rexlimiva impbii orbi2i 3bitr2i ) BDZBEFZUUCG + ZBESZUUCGZHZUUDUUGHUUDBIAUAZUBJZFZAEUCUDZUEZHUUCUUDUUFHZUUCGUUHUUCUUNUUCB + UFZUUNBUGUUOUUFUUDUUOUUFUUDHBUHUIULKUJUUDUUFUUCUKUMUUDUUEUUGUUDUUCUUDUUCE + DUNBELUOUPUQUUGUUMUUDUUGUUMUUGUUJCUAZMJZBFZCIUEZAEUEZUUMUUGUUFIESZUUFINUR + ZUSUUTUUFUUCUTUUFUVAUUFUVBUVAUUFOVAUUFVBUVBUUFNIVCVEVAVDACIBVFVGUUGUUSUUK + AEUULUUGUUIESZUUSUUIUULSZUUKGZUUGUVCGUURUVECIUUGUVCUUPISZUURUVEPUUGUVCUVF + GZUURUVEUUGUVGUURUSZNUUISZUUPNFZGZUVEUUGUVGUURUVKUUCUVGUURUVKPPUUFUURUVGU + UCUVKUURUUCUUQDZUVGUVKUURUVLUUCUUQBLVHUVGUVLUVITZUVJTZHZTUVKUVGUVOUVLUVGU + VMUVLTZUVNUVGUVMUVPUVGUVMGZUVLUUPDZUVQUVFUVRTZUVCUVFUVMVIZUUPVQZKUVQUUINF + ZUVFGZUUQUUPFUVLUVRVJUVQUWBUVFUVGUVMUWBUVCUVMUWBPUVFUVCUVIUUINUVCUVIUUINU + RUUIVKVHVLQVMUVTVNUWCUUQNUUPMJZUUPUWCUUJNUUPMUWCUUJINUBJZNUWCUUINIUBUWBUV + FUTVOUVAUWENFUWCOIVPVRVSVTUVFUWDUUPFUWBUUPWARVSUUQUUPLVGWBWCUVGUVNUVPUVGU + VNGZUVLUUJUUPWDZMJZWEZDZUWHWFSUWJTUWFUUJUWGMWGUWHWFWHVRUWFUUQUWIFUVLUWJVJ + UWFUUQUUJUWGWEZMJZUWIUWFUUPUWKUUJMUVGUVNUUPUWKFZUVFUVNUWMPUVCUVFUVJUWMUVF + UUPUWGFZUWMHZUVJUWMHUVFUUPUFZUWOUUPWIZUUPWJKUVFUWNUVJUWMUVFUWPTZUUPNURZTZ + UWNTZWKZUWNUVJPZUVFUVSUXBUWAUWPUWSUWNUSUXBUVRUWPUWSUWNWLUUPWMWNWOUVFUWRUX + CUWTUXAUVFUWPUXCUWQWPUVFUWTUWNUVJUWTUVJUUPNWQUIWRUXAUXCPUVFUWNUVJWSVAXAWT + XBWTXCRVMVOUWFUUJESZUWGESZUWLUWIFUWFUVAUVCUXDOUVGUVCUVNUVCUVFUTZQIUUIXDZX + EUVGUXEUVNUVFUXEUVCUVFUUPESUXEUUPXFUUPXGKRQUUJUWGXHXIVSUUQUWILKWBWCXJXKUV + IUVJXMXLXNXORXPUVHUVKGZUVDUUKUXHUVCUVIGZUVDUVHUVCUVKUVIUVHUVGUVCUUGUVGUUR + XQZUXFKUVIUVJUTXRUUIXSZXTUXHUUQUUJNMJZBUUJUVKUUQUXLFUVHUVKUUPNUUJMUVIUVJY + AVORUUGUVGUURUVKYBUVHUXLUUJFZUVKUVHUVGUXDUXMUXJUVGUVAUVCUXDOUXFUXGXEUUJYC + VGQYDVNYEYFYGYHYIYJWTUUKUUGAUULUVEUUFUUCUVEBUUJEUVDUUKYAUVDUXDUUKUVDUVAUV + CUXDOUUIEUCYKUXGXEQYOUVEUUCUUJDZUVDUXNUUKUVDUXIUXNUXKUVCUVAIDZGUVIUXNUVAU + XOOYLYMUUIIEYNYPYQQUUKUUCUXNVJUVDBUUJLRYRVNYSYTUUAUUB $. + $} + + ${ + $d A x $. $d B x $. $d C x $. $d D x $. + $( Closure law for ordinal addition. Here we show that ordinal addition is + closed within the empty set or any ordinal power of omega. (Contributed + by RP, 5-Jan-2025.) $) + oacl2g $p |- ( ( ( A e. C /\ B e. C ) /\ + ( C = (/) \/ ( C = ( _om ^o D ) /\ D e. On ) ) ) -> + ( A +o B ) e. C ) $= + ( vx wcel wa c0 wceq com co con0 coa adantr simpl simpr syl wss eleqtrd + wi wo eleq2 noel pm2.21i syl6bi com12 cv ciun omelon jctil eqeltrd adantl + coe oecl onelon expcom jcai oaordi sylc oveq1 eliuni syl2an2r eqcomd ssid + eqsstrdi oaabs2 syl21anc iunssd wrex peano1 oen0 oveq1d oa0r eqtrd sseq2d + sylancl ssidd rspcedvd ssiun eqssd ex jaod imp ) ACFZBCFZGZCHIZCJDUMKZIZD + LFZGZUAABMKZCFZWFWGWMWKWDWGWMTWEWGWDWMWGWDAHFZWMCHAUBWNWMAUCUDUEUFNWFWKWM + WFWKGZWLECEUGZCMKZUHZCWFWDWKWLACMKZFZWLWRFWDWEOWOCLFZALFZGWEWTWOXAXBWKXAW + FWKCWHLWIWJOZWKJLFZWJGZWHLFWKWJXDWIWJPUIUJZJDUNQUKZULWFXAXBTZWKWDXHWEXAWD + XBCAUOUPNNUQWFWEWKWDWEPNBCAURUSEAWQWSCWLWPACMUTVAVBWKWRCIWFWKWRCWKECWQCWK + WPCFZGZWQCCXJWPWHFXAWHCRZWQCIXJWPCWHWKXIPWKWIXIXCNSWKXAXIXGNWKXKXIWKWHCCW + KCWHXCVCZCVDZVENWPCDVFVGXMVEVHWKCWQRZECVICWRRWKXNCCREHCWKHWHCWKXEHJFHWHFX + FVJJDVKVPXLSWKWPHIZGZWQCCXPWQHCMKZCXPWPHCMWKXOPVLWKXQCIZXOWKXAXRXGCVMQNVN + VOWKCVQVRECWQCVSQVTULSWAWBWC $. + $} + + ${ + $d A w x y z $. $d B w x y z $. $d C w x y z $. + $( Ordinal multiplication by a larger ordinal is absorbed when the larger + ordinal is either 2 or ` _om ` raised to some power of ` _om ` . + (Contributed by RP, 12-Jan-2025.) $) + omabs2 $p |- ( ( ( A e. B /\ (/) e. A ) /\ + ( B = (/) \/ B = 2o \/ + ( B = ( _om ^o ( _om ^o C ) ) /\ C e. On ) ) ) + -> ( A .o B ) = B ) $= + ( wcel c0 wa wceq c2o com coe co con0 comu c1o syl wss omelon adantr coa + wi vw vx vy vz w3o eleq2 noel pm2.21i syl6bi com12 cpr wo elpri oveq1 2on + impd om1r ax-mp eqtrdi a1d jaoi df2o3 eleq2s imp oveq2 id eqeq12d simpllr + a1i simpr oecl mpan adantl ad2antlr peano1 sylancl wb mpbird cv wrex cdif + simpl sylancr eqeltrd onelon simplr ondif1 1onn jctil 0elon ad2antrr omcl + cun 1on ad5ant12 w3a oaword biimpd mp3an2ani csuc df-1o ordsucss eqsstrid + word eloni omwordi mpd syl2anc syl3anc sstrd oa0 simplrl eleqtrd ad6antlr + 3syl eqtrd ontr2 oeord syl2an2r ordom mpsyl oveq2d oveq1d sylib eqsstrrid + ex 3jca oeoa mp3an2i omwordri ssidd syl21anc eqtr3d mp3an13 sylcom oeword + mp2an mpjaodan syl5 rexlimdva anbi1d 3imtr4d jctl oen0 omabs syl22anc weu + cotp syl2anr sylanbrc ondif2 mpbir2an oeeu wex euex sylancom omsson ssdif + 0ss mpi sseli eldifi nnon simp-4r oawordri 3sstr3d simp-7l ad4antr mp2and + sylbi om1 simp-5r unssd wne onelpss sseqtrrd oe1 oe0 el1o oveq12d 3eqtr3d + omsuc pm3.2i expd elirr 3syld on0eqel mpjaod oewordi eqcomd simpr3 simprr + 3sstr4d simp1 anim12i sylan pm4.24 oacl oaass simpr1 oaabs2 3eqtrd uneq2d + suceloni sseqtrd oneluni eqtr4i 2a1i jca2 oveq2i eqtri ssun1 simp2 sstrid + eqsstri jca syl6ci bitrd biimprd sseq1i syl6ibr jcai biimpa oaabs ssequn1 + mpbid 3eqtr2d onun2 eqsstrrd eqssd ad5antr mp3and exlimdv ordtri2or 3jaod + ) ABDZEADZFZBEGZBHGZBIICJKZJKZGZCLDZFZUEABMKZBGZUYRUYSVUGUYTVUEUYSUYRVUGU + YSUYPUYQVUGUYSUYPAEDZUYQVUGTZBEAUFVUHVUIAUGUHUIUPUJUYTUYRVUGUYTAHDZUYQFZA + HMKZHGZUYRVUGVUKVUMTUYTVUJUYQVUMUYQVUMTZAENUKZHAVUODAEGZANGZULVUNAENUMVUP + VUNVUQVUPUYQEEDZVUMAEEUFVURVUMEUGUHUIVUQVUMUYQVUQVULNHMKZHANHMUNHLDVUSHGU + OHUQURUSUTVAOVBVCVDVIUYTUYPVUJUYQBHAUFUUAUYTVUFVULBHBHAMVEUYTVFVGUUBUJUYR + VUEVUGUYRVUEFZAIDZVUGIAPZVUTVVAFZVUGAVUBMKZVUBGZVVCVVAUYQVUALDZEVUADZVVEV + UTVVAVJUYPUYQVUEVVAVHVUEVVFUYRVVAVUDVVFVUCILDZVUDVVFQICVKVLZVMZVNVUEVVGUY + 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VUEVVLUYRWWJVVMVNVRYFUYKUYLYFYSYTYTYTUYMYSXGVUTAXDZWVFVVAVVBULVUTVWMXUFVW + RAXEOXTAIUYNVPYRYFUYOVD $. + $} + + ${ + $d A x $. $d B x $. $d C x $. $d D x $. + $( Closure law for ordinal multiplication. (Contributed by RP, + 12-Jan-2025.) $) + omcl2 $p |- ( ( ( A e. C /\ B e. C ) /\ + ( C = (/) \/ ( C = ( _om ^o ( _om ^o D ) ) /\ D e. On ) ) ) -> + ( A .o B ) e. C ) $= + ( vx wcel wa c0 wceq com co con0 comu wi omelon syl adantl simpr wss c1o + coe eleq2 noel pm2.21i syl6bi com12 adantr simpl oecl mpan eqeltrd simpll + wo jctil onelon syl2an2 on0eqel oveq1 om0r sylan9eqr peano1 oen0 eleqtrrd + sylancl ex w3a cv ciun simp1 ad2antrr syl2anc jcai simpl3 simpl2 syl21anc + omordi imp eliuni syl2an2r oveq1d eqtrd 0ss a1i eqsstrd c2o w3o id 3mix3d + adantll omabs2 ssidd sylan mpjaodan iunssd wrex cdif 1onn ondif2 mpbir2an + oeordi mpd ax-mp eqcomi 3eltr4d om1r sseq2d rspcedvd ssiun eleqtrd 3expia + oe0 eqssd com23 jaod ) ACFZBCFZGZCHIZCJJDUAKZUAKZIZDLFZGZUMABMKZCFZXQXRYE + YCXOXRYENXPXRXOYEXRXOAHFZYECHAUBYFYEAUCUDUEUFUGXQYCYEXQYCGZAHIZHAFZUMZYEY + GALFZYJYCCLFZXQXOYKYCCXTLYAYBUHZYBXTLFZYAYBJLFZXSLFZGZYNYBYPYOYOYBYPOJDUI + ZUJOUNZJXSUIPQUKZXOXPYCULCAUOZUPAUQPYGYHYEYIYGYHYEYGYHGYDHCYHYGYDHBMKZHAH + BMURYGBLFZUUBHIYCYLXQXPUUCYTXQXPYCXOXPRUGCBUOUPBUSPUTYGHCFZYHYCUUDXQYCHXT + CYBHXTFZYAYBYQHJFZUUEYSVAJXSVBVDQYMVCQUGUKVEXQYCYIYENXQYIYCYEXOXPYIYCYENX + OXPYIVFZYCYEUUGYCGZYDECEVGZCMKZVHZCUUGXOYCYDACMKZFZYDUUKFXOXPYIVIZUUHYLYK + GZYIXPUUMUUHYLYKYCYLUUGYTQUUHYLYKUUHYLGYLXOYKUUHYLRUUGXOYCYLUUNVJUUAVKVEV + LXOXPYIYCVMXOXPYIYCVNUUOYIGXPUUMBCAVPVQVOEAUUJUULCYDUUIACMURVRVSYCUUKCIUU + GYCUUKCYCECUUJCYCUUICFZGZUUIHIZUUJCSHUUIFZUUQUURGZUUJHCUUTUUJHCMKZHUUTUUI + HCMUUQUURRVTYCUVAHIZUUPUURYCYLUVBYTCUSPVJWAHCSUUTCWBWCWDUUQUUSGZUUJCCUVCU + UPUUSGZXRCWEIZYCWFUUJCIUUPUUSUVDYCUVDWGWIUVCYCXRUVEYCUUPUUSULWHUUICDWJVKU + VCCWKWDUUQUUILFZUURUUSUMYCYLUUPUVFYTCUUIUOWLUUIUQPWMWNYCCUUJSZECWOCUUKSYC + UVGCCSETCYCJHUAKZXTTCYCHXSFZUVHXTFZYCYOYBGZUUFUVIYCYBYOYAYBROUNZVAJDVBVDY + CYPJLWEWPFZUVIUVJNYCUVKYPUVLYRPUVMYOTJFOWQJWRWSHXSJWTVDXATUVHIYCUVHTYOUVH + TIOJXKXBXCWCYMXDYCUUITIZGUUJCCUVNYCUUJTCMKZCUUITCMURYCYLUVOCIYTCXEPUTXFYC + CWKXGECUUJCXHPXLQXIVEXJXMVQXNXAVEXNVQ $. + $} + + $( Closure law for ordinal multiplication. (Contributed by RP, + 14-Jan-2025.) $) + omcl3g $p |- ( ( ( A e. C /\ B e. C ) /\ + ( C e. 3o \/ ( C = ( _om ^o ( _om ^o D ) ) /\ D e. On ) ) ) -> + ( A .o B ) e. C ) $= + ( wcel wa com coe co wceq con0 wo comu c1o c2o df2o3 oveq12 eqeltrdi eleq2 + c0 c3o w3o ctp eltpi df-3o csn cun uneq1i df-suc df-tp 3eqtr4i eqtri eleq2s + csuc cpr orc omcl2 sylan2 ex el1o 0elon om0 ax-mp 0lt1o eqeltri syl2anb a1i + anbi12d 3imtr4d com12 elpri 0ex prid1 3eltr4i 1on om0r 1oex prid2 om1 ccase + wi syl2an 3jaod syl5 olc jaod imp ) ACEZBCEZFZCUAEZCGGDHIHIJDKEFZLABMIZCEZW + JWKWNWLWKCTJZCNJZCOJZUBZWJWNWRCTNOUCZUACTNOUDUAOUNZWSUEOOUFZUGTNUOZXAUGWTWS + OXBXAPUHOUITNOUJUKULUMWJWOWNWPWQWJWOWNWOWJWOCGGCHIHIJCKEFZLWNWOXCUPABCCUQUR + USWPWJWNWPANEZBNEZFZWMNEZWJWNXFXGWAWPXDATJZBTJZXGXEAUTBUTXHXIFZWMTTMIZNATBT + MQZXKTNTKEXKTJVATVBVCZVDVERVFVGWPWHXDWIXECNASCNBSVHCNWMSVIVJWQWJWNWQAOEZBOE + ZFZWMOEZWJWNXPXQWAWQXNXHANJZLZXIBNJZLZXQXOXSAXBOATNVKPUMYABXBOBTNVKPUMXHXIX + RXTXQXJWMXKOXLTXBXKOTNVLVMZXMPVNRXRXIFWMNTMIZOANBTMQTXBYCOYBNKEZYCTJVONVBVC + PVNRXHXTFWMTNMIZOATBNMQTXBYEOYBYDYETJVONVPVCPVNRXRXTFWMNNMIZOANBNMQNXBYFOTN + VQVRYDYFNJVONVSVCPVNRVTWBVGWQWHXNWIXOCOASCOBSVHCOWMSVIVJWCWDWJWLWNWLWJWOWLL + WNWLWOWEABCDUQURUSWFWG $. + + ${ + $d A c f g $. $d B c f g $. $d C c d f g $. $d D c d f g $. + $d E c d f g $. $d F c f g $. $d V c f g $. $d W c f g $. + $( Addition operator for functions from sets into ordinals results in a + function from the intersection of sets into an ordinal. (Contributed by + RP, 5-Jan-2025.) $) + ofoafg $p |- ( ( ( A e. V /\ B e. W /\ C = ( A i^i B ) ) /\ + ( D e. On /\ E e. On /\ F = U_ d e. D ( d +o E ) ) ) + -> ( oF +o |` ( ( D ^m A ) X. ( E ^m B ) ) ) + : ( ( D ^m A ) X. ( E ^m B ) ) --> ( F ^m C ) ) $= + ( vf wcel wceq con0 coa co wa wf wfn adantl ad2antrr vg vc cin w3a cv cof + ciun cmap wral cxp cres wb simp1 elmapg syl2anr simp2 adantr crn wss ffnd + simpl simpr eqid offn simp3 fneq2d mpbird cfv fresin inss1 eqsstrdi sylib + sseqin2 feq2d mpbid ffvelcdmda ad3antlr onelon syl2anc inss2 imp syl21anc + oaordi oveq1 eliuni reseq2d oveq12d eqtr4d fveq1d cvv fnssresd jca inex1g + ofres syl eqeltrd anim1i fnfvof syl2an2r eqtrd 3eltr4d ralrimiva fnfvrnss + sylan2 expcom jcai adantlr oacl iunon syldan 3adant3 elmapd bitrdi sylbid + df-f expr ralrimiv ex ofmres fmpo ) AGKZBHKZCABUCZLZUDZDMKZEMKZFIDIUEZENO + ZUGZLZUDZPZJUEZUAUEZNUFZOZFCUHOZKZUAEBUHOZUIZJDAUHOZUIUUBYTUJZYRYPUUCUKZQ + YMUUAJUUBYMYNUUBKZADYNQZUUAYLYFYAUUEUUFULYEYFYGYKUMZYAYBYDUMZDAYNMGUNUOYM + UUFUUAYMUUFPZYSUAYTUUIYOYTKZBEYOQZYSYMUUJUUKULZUUFYLYGYBUULYEYFYGYKUPZYAY + BYDUPZEBYOMHUNUOUQYMUUFUUKYSYMUUFUUKPZPZYSYQCRZYQURFUSZPZUUPUUQUURUUPUUQY + QYCRZUUPABNYCYNYOGHUUOYNARYMUUOADYNUUFUUKVAUTSZUUOYOBRYMUUOBEYOUUFUUKVBUT + SZYEYAYLUUOUUHTZYEYBYLUUOUUNTZYCVCZVDYEUUQUUTULYLUUOYECYCYQYAYBYDVEZVFTVG + UUQUUPUURUUPUUQUBUEZYQVHZFKZUBCUIUURUUPUVIUBCUUPUVGCKZPZUVGYNCUKZVHZUVGYO + CUKZVHZNOZYJUVHFUVKUVMDKZUVPUVMENOZKZUVPYJKUUPCDUVGUVLUUPACUCZDUVLQZCDUVL + QUUOUWAYMUUFUWAUUKADYNCVIUQSUUPUVTCDUVLYEUVTCLZYLUUOYECAUSZUWBYECYCAUVFAB + VJVKZCAVMVLTVNVOVPZUVKYGUVMMKZUVOEKZUVSYLYGYEUUOUVJUUMVQUVKYFUVQUWFYLYFYE + UUOUVJUUGVQUWEDUVMVRVSUUPCEUVGUVNUUPBCUCZEUVNQZCEUVNQUUOUWIYMUUKUWIUUFBEY + OCVISSUUPUWHCEUVNYEUWHCLZYLUUOYECBUSZUWJYECYCBUVFABVTVKZCBVMVLTVNVOVPYGUW + FPUWGUVSUVOEUVMWCWAWBIUVMYIUVRDUVPYHUVMENWDWEVSUVKUVHUVGUVLUVNYPOZVHZUVPU + UPUVHUWNLUVJUUPUVGYQUWMUUPYQYNYCUKZYOYCUKZYPOZUWMUUPABYCNYNYOGHUVAUVBUVCU + VDUVEWNYEUWMUWQLYLUUOYEUVLUWOUVNUWPYPYECYCYNUVFWFYECYCYOUVFWFWGTWHWIUQUUP + UVLCRZUVNCRZPUVJCWJKZUVJPUWNUVPLUUPUWRUWSUUPACYNUVAYEUWCYLUUOUWDTWKUUPBCY + OUVBYEUWKYLUUOUWLTWKWLUUPUWTUVJYEUWTYLUUOYECYCWJUVFYEYAYCWJKUUHABGWMWOWPT + ZWQCNUVLUVNWJUVGWRWSWTYLYKYEUUOUVJYFYGYKVEZVQXAXBUBCFYQXCXDXEXFUUPYSCFYQQ + UUSUUPFCYQMWJYMFMKZUUOYLUXCYEYLFYJMUXBYFYGYJMKZYKYFYGYIMKZIDUIUXDYFYGPZUX + EIDUXFYHDKZPYHMKZYGUXEYFUXGUXHYGDYHVRXGUXFYGUXGYFYGVBUQYHEXHVSXBIDYIMXIXJ + XKWPSUQUXAXLCFYQXOXMVGXPXNXQXRXNXQJUAUUBYTYQYRUUDUUBYTNJUAXSXTVL $. + $} + + ${ + $d C x $. $d D x $. $d E x $. + $( Addition operator for functions from sets into power of omega results in + a function from the intersection of sets to that power of omega. + (Contributed by RP, 5-Jan-2025.) $) + ofoaf $p |- ( ( ( A e. V /\ B e. W /\ C = ( A i^i B ) ) /\ + ( D e. On /\ E = ( _om ^o D ) ) ) + -> ( oF +o |` ( ( E ^m A ) X. ( E ^m B ) ) ) + : ( ( E ^m A ) X. ( E ^m B ) ) --> ( E ^m C ) ) $= + ( vx con0 wcel com co wceq wa w3a coa cmap omelon wss c0 coe cin ciun cxp + cv cof cres wf simpr simpl oecl sylancr eqeltrd wrex jctil peano1 sylancl + oen0 eleqtrrd oveq1 sseq2d adantl oa0r syl ssid eqsstrrdi rspcedvd eleq2d + wb ssiun biimpa adantr oaabs2 syl21anc eqsstrdi iunssd 3jca ofoafg sylan2 + eqssd ) DIJZEKDUALZMZNZAFJBGJCABUBMOEIJZWEEHEHUEZEPLZUCZMZOEAQLEBQLUDZECQ + LPUFWJUGUHWDWEWEWIWDEWBIWAWCUIZWDKIJZWAWBIJRWAWCUJZKDUKULUMZWNWDEWHWDEWGS + ZHEUNEWHSWDWOETEPLZSZHTEWDTWBEWDWLWANTKJTWBJWDWAWLWMRUOUPKDURUQWKUSWFTMZW + OWQVIWDWRWGWPEWFTEPUTVAVBWDEWPWPWDWEWPEMWNEVCVDWPVEVFVGHEWGEVJVDWDHEWGEWD + WFEJZNZWGEEWTWFWBJZWEWBESWGEMWDWSXAWDEWBWFWKVHVKWDWEWSWNVLWTWBEEWDWCWSWKV + LEVEZVFWFEDVMVNXBVOVPVTVQABCEEEFGHVRVS $. + $} + + ${ + $d A a f h z $. $d B a h $. $d C a f h z $. $d V a h $. + $( Addition operator for functions from a set into a power of omega is an + onto binary operator. (Contributed by RP, 5-Jan-2025.) $) + ofoafo $p |- ( ( A e. V /\ ( B e. On /\ C = ( _om ^o B ) ) ) + -> ( oF +o |` ( ( C ^m A ) X. ( C ^m A ) ) ) + : ( ( C ^m A ) X. ( C ^m A ) ) -onto-> ( C ^m A ) ) $= + ( vh vf vz wcel con0 com co wceq wa coa wf cv c0 syl adantl adantr va coe + cmap cxp cof cres wrex wral wfo cin w3a inidm eqcomi a1i 3jca ofoaf sylan + id simpr csn omelon simpl jca oen0 sylancl eleqtrrd fconst6g oecl sylancr + peano1 eqeltrd elmapd mpbird ovres wfn elmapi ffnd elmapfn anim12i anim1i + offn cfv fnfvof syl2an2r fvconst2g oveq2d onss ad2antrl ffvelcdmda sseldd + wss oa0 3eqtrd eqfnfvd eqtr2d expr jcai oveq2 rspceeqv weq eqeq2d rexbidv + oveq1 rspcev ralrimiva foov sylanbrc ) ADHZBIHZCJBUBKZLZMZMZCAUCKZXNUDZXN + NUEZXOUFZOZEPZFPZGPZXQKZLZGXNUGZFXNUGZEXNUHXOXNXQUIXHXHXHAAAUJZLZUKXLXRXH + XHXHYGXHURZYHYGXHYFAAULZUMUNUOAAABCDDUPUQXMYEEXNXMXSXNHZMZYJXSXSYAXQKZLZG + XNUGZMYEYKYJYNXMYJUSYKAQUTUDZXNHZXSXSYOXQKZLZMYNYKYPYRXMYPYJXMYPACYOOZXLY + SXHXLQCHYSXLQXJCXLJIHZXIMQJHZQXJHXLYTXIYTXLVAUNXIXKVBZVCVJJBVDVEXIXKUSZVF + AQCVGRSXMCAYOIDXLCIHZXHXLCXJIUUCXLYTXIXJIHVAUUBJBVHVIVKSZXHXLVBZVLVMTXMYJ + YPYRXMYJYPMZMZYQXSYOXPKZXSUUGYQUUILXMXSYOXNXNXPVNSUUHUAAUUIXSUUHAANAXSYOD + DUUGXSAVOZXMUUGACXSYJACXSOZYPXSCAVPZTVQSZUUGYOAVOZXMUUGACYOYPYSYJYOCAVPSV + QSXMXHUUGUUFTZUUOYIWAUUMUUHUAPZAHZMZUUPUUIWBZUUPXSWBZUUPYOWBZNKZUUTQNKZUU + TUUHUUJUUNMZUUQXHUUQMUUSUVBLUUGUVDXMYJUUJYPUUNXSCAVRYOCAVRVSSUUHXHUUQUUOV + TANXSYODUUPWCWDUURUVAQUUTNUURUUAUUQUVAQLVJUUHUUQUSAQUUPJWEVIWFUURUUTIHUVC + UUTLUURCIUUTUURUUDCIWKUUHUUDUUQXMUUDUUGUUETTCWGRUUHACUUPXSYJUUKXMYPUULWHW + IWJUUTWLRWMWNWOWPWQGYOXNYLYQXSYAYOXSXQWRWSRVCYDYNFXSXNFEWTZYCYMGXNUVEYBYL + XSXTXSYAXQXCXAXBXDRXEFGEXNXNXNXQXFXG $. + $} + + $( Closure law for component wise addition of ordinal-yielding functions. + (Contributed by RP, 5-Jan-2025.) $) + ofoacl $p |- ( ( ( A e. V /\ ( B e. On /\ C = ( _om ^o B ) ) ) /\ + ( F e. ( C ^m A ) /\ G e. ( C ^m A ) ) ) -> + ( F oF +o G ) e. ( C ^m A ) ) $= + ( wcel con0 com coe co wceq wa cmap coa cof cxp cres ovres adantl cin wf id + w3a inidm a1i eqcomd 3jca ofoaf sylan fovcdmda eqeltrrd ) AFGZBHGCIBJKLMZMZ + DCANKZGEUPGMZMDEOPZUPUPQZRZKZDEURKZUPUQVAVBLUODEUPUPURSTUODEUPUPUPUTUMUMUMA + AAUAZLZUDUNUSUPUTUBUMUMUMVDUMUCZVEUMVCAVCALUMAUEUFUGUHAAABCFFUIUJUKUL $. + + ${ + $d A a $. $d F a $. $d V a $. + $( Identity law for component wise addition of ordinal-yielding functions. + (Contributed by RP, 5-Jan-2025.) $) + ofoaid1 $p |- ( ( ( A e. V /\ B e. On ) /\ F e. ( B ^m A ) ) + -> ( F oF +o ( A X. { (/) } ) ) = F ) $= + ( va wcel con0 wa co wf c0 wfn coa wceq wss syl df-f com peano1 cfv impel + cmap csn cxp cof simpll wi onss sstr expcom anim2d 3imtr4g elmapi adantll + crn fnconstg mp1i w3a simp2 ffnd simp3 simp1 inidm offn jca adantr fnfvof + cv simpr syl12anc fvconst2g sylancr oveq2d ffvelcdmda oa0 eqfnfvd syl3anc + 3eqtrd ) ADFZBGFZHCBAUBIFZHZVSAGCJZAKUCUDZALZCWDMUEIZCNVSVTWAUFVTWAWCVSVT + ABCJZWCWAVTCALZCUOZBOZHWHWIGOZHWGWCVTWJWKWHVTBGOZWJWKUGBUHWJWLWKWIBGUIUJP + UKABCQAGCQULCBAUMUAUNKRFZWEWBSAKRUPUQVSWCWEURZEAWFCWNAAMACWDDDWNAGCVSWCWE + USZUTZVSWCWEVAZVSWCWEVBZWRAVCVDWPWNEVHZAFZHZWSWFTZWSCTZWSWDTZMIZXCKMIZXCX + AWHWEHZVSWTXBXENWNXGWTWNWHWEWPWQVEVFWNVSWTWRVFWNWTVIZAMCWDDWSVGVJXAXDKXCM + XAWMWTXDKNSXHAKWSRVKVLVMXAXCGFXFXCNWNAGWSCWOVNXCVOPVRVPVQ $. + $} + + ${ + $d A a $. $d F a $. $d V a $. + $( Identity law for component wise addition of ordinal-yielding functions. + (Contributed by RP, 5-Jan-2025.) $) + ofoaid2 $p |- ( ( ( A e. V /\ B e. On ) /\ F e. ( B ^m A ) ) + -> ( ( A X. { (/) } ) oF +o F ) = F ) $= + ( va wcel con0 wa co wf c0 wfn coa wceq wss syl df-f com peano1 cfv impel + cmap csn cxp cof simpll wi onss sstr expcom anim2d 3imtr4g elmapi adantll + crn fnconstg mp1i w3a simp3 simp2 ffnd simp1 inidm offn jca adantr fnfvof + cv simpr syl12anc fvconst2g sylancr oveq1d ffvelcdmda oa0r 3eqtrd eqfnfvd + syl3anc ) ADFZBGFZHCBAUBIFZHZVSAGCJZAKUCUDZALZWDCMUEIZCNVSVTWAUFVTWAWCVSV + TABCJZWCWAVTCALZCUOZBOZHWHWIGOZHWGWCVTWJWKWHVTBGOZWJWKUGBUHWJWLWKWIBGUIUJ + PUKABCQAGCQULCBAUMUAUNKRFZWEWBSAKRUPUQVSWCWEURZEAWFCWNAAMAWDCDDVSWCWEUSZW + NAGCVSWCWEUTZVAZVSWCWEVBZWRAVCVDWQWNEVHZAFZHZWSWFTZWSWDTZWSCTZMIZKXDMIZXD + XAWEWHHZVSWTXBXENWNXGWTWNWEWHWOWQVEVFWNVSWTWRVFWNWTVIZAMWDCDWSVGVJXAXCKXD + MXAWMWTXCKNSXHAKWSRVKVLVMXAXDGFXFXDNWNAGWSCWPVNXDVOPVPVQVR $. + $} + + ${ + $d A a $. $d B a $. $d F a $. $d G a $. $d H a $. $d V a $. + $( Component-wise addition of ordinal-yielding functions is associative. + (Contributed by RP, 5-Jan-2025.) $) + ofoaass $p |- ( ( ( A e. V /\ B e. On ) /\ + ( F e. ( B ^m A ) /\ G e. ( B ^m A ) /\ H e. ( B ^m A ) ) ) + -> ( ( F oF +o G ) oF +o H ) = ( F oF +o ( G oF +o H ) ) ) $= + ( wcel con0 wa co coa wfn elmapfn adantl offn cfv wceq wf elmapi fnfvof + va cmap w3a 3ad2ant1 3ad2ant2 simpll inidm 3ad2ant3 cv simpllr ffvelcdmda + cof onelon syl2anc oaass syl3anc adantr anim1i syl21anc oveq1d jca oveq2d + syl2an2r 3eqtr4d eqfnfvd ) AFGZBHGZIZCBAUBJZGZDVIGZEVIGZUCZIZUAACDKULZJZE + VOJZCDEVOJZVOJZVNAAKAVPEFFVNAAKACDFFVMCALZVHVJVKVTVLCBAMUDNZVMDALZVHVKVJW + BVLDBAMUENZVFVGVMUFZWDAUGZOZVMEALZVHVLVJWGVKEBAMUHNZWDWDWEOVNAAKACVRFFWAV + NAAKADEFFWCWHWDWDWEOZWDWDWEOVNUAUIZAGZIZWJVPPZWJEPZKJZWJCPZWJVRPZKJZWJVQP + ZWJVSPZWLWPWJDPZKJZWNKJZWPXAWNKJZKJZWOWRWLWPHGZXAHGZWNHGZXCXEQWLVGWPBGXFV + FVGVMWKUJZVNABWJCVMABCRZVHVJVKXJVLCBASUDNUKBWPUMUNWLVGXABGXGXIVNABWJDVMAB + DRZVHVKVJXKVLDBASUENUKBXAUMUNWLVGWNBGXHXIVNABWJEVMABERZVHVLVJXLVKEBASUHNU + KBWNUMUNWPXAWNUOUPWLWMXBWNKWLVTWBVFWKIZWMXBQVNVTWKWAUQVNWBWKWCUQVNVFWKWDU + RZAKCDFWJTUSUTWLWQXDWPKVNWBWGIWKXMWQXDQVNWBWGWCWHVAXNAKDEFWJTVCVBVDVNVPAL + ZWGIWKXMWSWOQVNXOWGWFWHVAXNAKVPEFWJTVCVNVTVRALZIWKXMWTWRQVNVTXPWAWIVAXNAK + CVRFWJTVCVDVE $. + $} + + ${ + $d A a $. $d F a $. $d G a $. $d V a $. + $( Component-wise addition of natural numnber-yielding functions commutes. + (Contributed by RP, 5-Jan-2025.) $) + ofoacom $p |- ( ( A e. V /\ ( F e. ( _om ^m A ) /\ G e. ( _om ^m A ) ) ) -> + ( F oF +o G ) = ( G oF +o F ) ) $= + ( va wcel com co wa coa wfn elmapfn ad2antrl ad2antll offn wceq wf elmapi + cfv ffvelcdmda cmap cof simpl inidm cv nnacom syl2anc jca anim1i syl2an2r + fnfvof 3eqtr4d eqfnfvd ) ADFZBGAUAHZFZCUOFZIZIZEABCJUBZHZCBUTHZUSAAJABCDD + UPBAKZUNUQBGALMZUQCAKZUNUPCGALNZUNURUCZVGAUDZOUSAAJACBDDVFVDVGVGVHOUSEUEZ + AFZIZVIBSZVICSZJHZVMVLJHZVIVASZVIVBSZVKVLGFVMGFVNVOPUSAGVIBUPAGBQUNUQBGAR + MTUSAGVICUQAGCQUNUPCGARNTVLVMUFUGUSVCVEIVJUNVJIZVPVNPUSVCVEVDVFUHUSUNVJVG + UIZAJBCDVIUKUJUSVEVCIVJVRVQVOPUSVEVCVFVDUHVSAJCBDVIUKUJULUM $. + $} + + ${ + $d S f g $. $d X f g x $. + $( Addition operator for Cantor normal forms is a function into Cantor + normal forms. (Contributed by RP, 2-Jan-2025.) $) + naddcnff $p |- ( ( X e. On /\ S = dom ( _om CNF X ) ) + -> ( oF +o |` ( S X. S ) ) : ( S X. S ) --> S ) $= + ( vf vg vx con0 wcel com co wceq wa cv coa wral wf c0 simpl adantr ffnd + ex ccnf cdm cof cxp cres cfsupp wbr simpr eleq2d omelon a1i cantnfs bitrd + eqid anim12i anassrs wfn crn wss simprl simprr inidm offn simplrl simplrr + wb cfv simpll syl22anc ffvelcdmda nnacl syl2anc eqeltrd ralrimiv fnfvrnss + fnfvof jcai df-f sylibr syl wfun csupp cfn ffun adantl cun simp-4l peano1 + fsuppunfi 0elon oa0 mp1i suppofssd ssfid ovexd isfsupp mpbir2and ad2antrr + cvv sylancl mpbird sylbid ofmres fmpo sylib ) BFGZAHBUAIUBZJZKZCLZDLZMUCZ + IZAGZDANZCANAAUDZAXLXPUEZOXIXOCAXIXJAGZBHXJOZXJPUFUGZKZXOXIXRXJXGGYAXIAXG + XJXFXHUHZUIXIHBXGXJXGUNZHFGXIUJUKZXFXHQZULUMXIYAXOXIYAKZXNDAYFXKAGZBHXKOZ + XKPUFUGZKZXNXIYGYJVFYAXIYGXKXGGYJXIAXGXKYBUIXIHBXGXKYCYDYEULUMRYFYJXNYFYJ + KZXNBHXMOZXMPUFUGZKZYKYLYMYKXFXSYHKZKZYLXIYAYJYPXIXFYAYJKYOYEYAXSYJYHXSXT + QYHYIQUOUOUPYPXMBUQZXMURHUSZKYLYPYQYRYPBBMBXJXKFFYPBHXJXFXSYHUTZSYPBHXKXF + XSYHVAZSXFYOQZUUABVBVCYPYQYRYPYQKYQELZXMVGZHGZEBNZYRYPYQUHYPUUEYQYPUUDEBY + PUUBBGZUUDYPUUFKZUUCUUBXJVGZUUBXKVGZMIZHUUGXJBUQXKBUQXFUUFUUCUUJJUUGBHXJX + FXSYHUUFVDSUUGBHXKXFXSYHUUFVESXFYOUUFVHYPUUFUHBMXJXKFUUBVPVIUUGUUHHGUUIHG + UUJHGYPBHUUBXJYSVJYPBHUUBXKYTVJUUHUUIVKVLVMTVNREBHXMVOVLTVQBHXMVRVSVTYKYL + YMYKYLKZYMXMWAZXMPWBIZWCGZYLUULYKBHXMWDWEUUKXJPWBIXKPWBIWFUUMUUKXJXKPYKXT + YLXIXSXTYJVERYFYHYIYLVEWIUUKBHXJXKFMPXFXHYAYJYLWGPHGUUKWHUKYKXSYLXIXSXTYJ + VDRYFYHYIYLVDPFGZPPMIPJUUKWJPWKWLWMWNUUKXMWSGUUOYMUULUUNKVFUUKXJXKXLWOWJX + MWSFPWPWTWQTVQXIXNYNVFYAYJXIXNXMXGGYNXIAXGXMYBUIXIHBXGXMYCYDYEULUMWRXATXB + VNTXBVNCDAAXMAXQAAMCDXCXDXE $. + $} + + $( Addition operator for Cantor normal forms is a function. (Contributed by + RP, 2-Jan-2025.) $) + naddcnffn $p |- ( ( X e. On /\ S = dom ( _om CNF X ) ) + -> ( oF +o |` ( S X. S ) ) Fn ( S X. S ) ) $= + ( con0 wcel com ccnf co cdm wceq wa cxp coa cof cres naddcnff ffnd ) BCDAEB + FGHIJAAKZALMQNABOP $. + + ${ + $d S f g z $. $d S f x $. $d X f z $. $d X f x $. + $( Addition of Cantor normal forms is a function onto Cantor normal forms. + (Contributed by RP, 2-Jan-2025.) $) + naddcnffo $p |- ( ( X e. On /\ S = dom ( _om CNF X ) ) + -> ( oF +o |` ( S X. S ) ) : ( S X. S ) -onto-> S ) $= + ( vf vg vz con0 wcel com co wceq wa wf cv wrex simpr c0 peano1 mp1i simpl + coa ccnf cdm cxp cof cres wral wfo naddcnff csn cfsupp fconst6g fczfsuppd + vx wbr a1i eleq2d eqid omelon bitrd mpbir2and adantr adantl ovresd biimpd + cantnfs syl56 imp ffnd wfn fnconstg inidm offn simplll syl22anc fvconst2g + cfv fnfvof syl2anc oveq2d ffvelcdmda nnon 3syl 3eqtrd eqfnfvd eqtr2d expr + oa0 oveq2 rspceeqv syl weq oveq1 eqeq2d rexbidv rspcev ralrimiva sylanbrc + jcai foov ) BFGZAHBUAIUBZJZKZAAUCZATUDZXDUEZLCMZDMZEMZXFIZJZEANZDANZCAUFX + DAXFUGABUHXCXMCAXCXGAGZKZXNXGXGXIXFIZJZEANZXMXCXNOXOBPUIUCZAGZXGXGXSXFIZJ + ZKXRXOXTYBXCXTXNXCXTBHXSLZXSPUJUNZPHGZYCXCQBPHUKRXCBFHPWTXBSZYEXCQUOULXCX + TXSXAGYCYDKXCAXAXSWTXBOZUPXCHBXAXSXAUQZHFGXCURUOZYFVEUSUTVAXCXNXTYBXCXNXT + KZKZYAXGXSXEIZXGYKXGXSXEAYJXNXCXNXTSZVBYJXTXCXNXTOVBVCYKUMBYLXGYKBBTBXGXS + FFYKBHXGXCYJBHXGLZYJXNXCYNXGPUJUNZKZYNYMXCXNYPXCXNXGXAGYPXCAXAXGYGUPXCHBX + AXGYHYIYFVEUSVDYNYOSVFVGZVHZYEXSBVIZYKQBPHVJZRXCWTYJYFVAZUUABVKVLYRYKUMMZ + BGZKZUUBYLVPZUUBXGVPZUUBXSVPZTIZUUFPTIZUUFUUDXGBVIZYSWTUUCUUEUUHJYKUUJUUC + YRVAYEYSUUDQYTRWTXBYJUUCVMYKUUCOZBTXGXSFUUBVQVNUUDUUGPUUFTUUDYEUUCUUGPJYE + UUDQUOUUKBPUUBHVOVRVSUUDUUFHGUUFFGUUIUUFJYKBHUUBXGYQVTUUFWAUUFWGWBWCWDWEW + FWREXSAXPYAXGXIXSXGXFWHWIWJXLXRDXGADCWKZXKXQEAUULXJXPXGXHXGXIXFWLWMWNWOVR + WPDECAAAXFWSWQ $. + $} + + $( Closure law for component-wise ordinal addition of Cantor normal forms. + (Contributed by RP, 2-Jan-2025.) $) + naddcnfcl $p |- ( ( ( X e. On /\ S = dom ( _om CNF X ) ) /\ + ( F e. S /\ G e. S ) ) -> ( F oF +o G ) e. S ) $= + ( con0 wcel com ccnf co cdm wceq coa cof cxp ovres adantl naddcnff fovcdmda + wa cres eqeltrrd ) DEFAGDHIJKSZBAFCAFSZSBCLMZAANTZIZBCUDIZAUCUFUGKUBBCAAUDO + PUBBCAAAUEADQRUA $. + + ${ + $d F x $. $d G x $. $d S x $. $d X x $. + $( Component-wise ordinal addition of Cantor normal forms commutes. + (Contributed by RP, 2-Jan-2025.) $) + naddcnfcom $p |- ( ( ( X e. On /\ S = dom ( _om CNF X ) ) /\ + ( F e. S /\ G e. S ) ) -> ( F oF +o G ) = ( G oF +o F ) ) $= + ( vx con0 wcel com co wceq wa coa wf c0 cfsupp wbr simpr eleq2d simpl cfv + ccnf cdm cof eqid omelon a1i cantnfs bitrd syl6bi impel ffnd simpll inidm + offn ffvelcdmda nnacom syl2anc wfn adantr simplll fnfvof syl22anc 3eqtr4d + cv eqfnfvd ) DFGZAHDUAIUBZJZKZBAGZCAGZKZKZEDBCLUCZIZCBVNIZVMDDLDBCFFVMDHB + VIVJDHBMZVLVIVJVQBNOPZKZVQVIVJBVGGVSVIAVGBVFVHQZRVIHDVGBVGUDZHFGVIUEUFZVF + VHSZUGUHVQVRSUIVJVKSUJZUKZVMDHCVIVKDHCMZVLVIVKWFCNOPZKZWFVIVKCVGGWHVIAVGC + VTRVIHDVGCWAWBWCUGUHWFWGSUIVJVKQUJZUKZVFVHVLULZWKDUMZUNVMDDLDCBFFWJWEWKWK + WLUNVMEVDZDGZKZWMBTZWMCTZLIZWQWPLIZWMVOTZWMVPTZWOWPHGWQHGWRWSJVMDHWMBWDUO + VMDHWMCWIUOWPWQUPUQWOBDURZCDURZVFWNWTWRJVMXBWNWEUSZVMXCWNWJUSZVFVHVLWNUTZ + VMWNQZDLBCFWMVAVBWOXCXBVFWNXAWSJXEXDXFXGDLCBFWMVAVBVCVE $. + $} + + ${ + $d F x $. $d S x $. $d X x $. + $( Identity law for component-wise ordinal addition of Cantor normal forms. + (Contributed by RP, 3-Jan-2025.) $) + naddcnfid1 $p |- ( ( ( X e. On /\ S = dom ( _om CNF X ) ) /\ F e. S ) -> + ( F oF +o ( X X. { (/) } ) ) = F ) $= + ( vx con0 wcel com co wceq wa c0 coa wf cfsupp wbr peano1 mp1i a1i adantr + cfv ccnf cdm csn cxp fconst6g simpl fczfsuppd simpr eleq2d omelon cantnfs + cof eqid bitrd mpbir2and wfn simprbda ffnd simplll offn cv simp-4l fnfvof + inidm syl22anc fvconst2g sylancr oveq2d ffvelcdmda nna0 syl 3eqtrd mpidan + eqfnfvd ) CEFZAGCUAHUBZIZJZBAFZCKUCUDZAFZBVTLULHZBIVRWACGVTMZVTKNOZKGFZWC + VRPCKGUEZQVRCEGKVOVQUFZWEVRPRUGVRWAVTVPFWCWDJVRAVPVTVOVQUHZUIVRGCVPVTVPUM + ZGEFVRUJRZWGUKUNUOVRVSJZWAJZDCWBBWLCCLCBVTEEWKBCUPZWAWKCGBVRVSCGBMZBKNOZV + RVSBVPFWNWOJVRAVPBWHUIVRGCVPBWIWJWGUKUNUQZURSZWEVTCUPZWLPWECGVTWFURZQVOVQ + VSWAUSZWTCVDUTWQWLDVAZCFZJZXAWBTZXABTZXAVTTZLHZXEKLHZXEXCWMWRVOXBXDXGIWLW + MXBWQSWEWRXCPWSQVOVQVSWAXBVBWLXBUHZCLBVTEXAVCVEXCXFKXELXCWEXBXFKIPXICKXAG + VFVGVHXCXEGFXHXEIWLCGXABWKWNWAWPSVIXEVJVKVLVNVM $. + $} + + $( Identity law for component-wise ordinal addition of Cantor normal forms. + (Contributed by RP, 3-Jan-2025.) $) + naddcnfid2 $p |- ( ( ( X e. On /\ S = dom ( _om CNF X ) ) /\ F e. S ) -> + ( ( X X. { (/) } ) oF +o F ) = F ) $= + ( con0 wcel com ccnf co cdm wceq wa c0 csn cxp coa cof wf cfsupp peano1 a1i + wbr fconst6g mp1i simpl fczfsuppd simpr eleq2d eqid cantnfs bitrd mpbir2and + omelon naddcnfcom ex mpand imp naddcnfid1 eqtrd ) CDEZAFCGHIZJZKZBAEZKCLMNZ + BOPZHZBVDVEHZBVBVCVFVGJZVBVDAEZVCVHVBVICFVDQZVDLRUAZLFEZVJVBSCLFUBUCVBCDFLU + SVAUDZVLVBSTUEVBVIVDUTEVJVKKVBAUTVDUSVAUFUGVBFCUTVDUTUHFDEVBULTVMUIUJUKVBVI + VCKVHAVDBCUMUNUOUPABCUQUR $. + + ${ + $d F x $. $d G x $. $d H x $. $d S x $. $d X x $. + $( Component-wise addition of Cantor normal forms is associative. + (Contributed by RP, 3-Jan-2025.) $) + naddcnfass $p |- ( ( ( X e. On /\ S = dom ( _om CNF X ) ) /\ + ( F e. S /\ G e. S /\ H e. S ) ) -> + ( ( F oF +o G ) oF +o H ) = ( F oF +o ( G oF +o H ) ) ) $= + ( con0 wcel com co wceq coa wfn simpl syl6bi impel adantr offn cfv fnfvof + wa vx ccnf cdm w3a cof wf cfsupp wbr simpr eleq2d eqid omelon a1i cantnfs + c0 bitrd ffnd simp1 simp2 inidm simp3 cv ffvelcdmda nnaass syl3anc anim1i + syl21anc oveq1d oveq2d 3eqtr4d eqfnfvd ) EFGZAHEUBIUCZJZTZBAGZCAGZDAGZUDZ + TZUAEBCKUEZIZDWAIZBCDWAIZWAIZVTEEKEWBDFFVTEEKEBCFFVOVPBELZVSVOVPEHBUFZBUO + UGUHZTZWFVOVPBVMGWIVOAVMBVLVNUIZUJVOHEVMBVMUKZHFGVOULUMZVLVNMZUNUPZWIEHBW + GWHMZUQNVPVQVRURZOZVOVQCELZVSVOVQEHCUFZCUOUGUHZTZWRVOVQCVMGXAVOAVMCWJUJVO + HEVMCWKWLWMUNUPZXAEHCWSWTMZUQNVPVQVRUSZOZVOVLVSWMPZXFEUTZQZVOVRDELZVSVOVR + EHDUFZDUOUGUHZTZXIVOVRDVMGXLVOAVMDWJUJVOHEVMDWKWLWMUNUPZXLEHDXJXKMZUQNVPV + QVRVAZOZXFXFXGQVTEEKEBWDFFWQVTEEKECDFFXEXPXFXFXGQZXFXFXGQVTUAVBZEGZTZXRWB + RZXRDRZKIZXRBRZXRWDRZKIZXRWCRZXRWERZXTYDXRCRZKIZYBKIZYDYIYBKIZKIZYCYFXTYD + HGYIHGYBHGYKYMJVTEHXRBVOVPWGVSVOVPWIWGWNWONWPOVCVTEHXRCVOVQWSVSVOVQXAWSXB + XCNXDOVCVTEHXRDVOVRXJVSVOVRXLXJXMXNNXOOVCYDYIYBVDVEXTYAYJYBKXTWFWRVLXSTZY + AYJJVTWFXSWQPZVTWRXSXEPZVTVLXSXFVFZEKBCFXRSVGVHXTYEYLYDKXTWRXIYNYEYLJYPVT + XIXSXPPZYQEKCDFXRSVGVIVJXTWBELZXIYNYGYCJVTYSXSXHPYRYQEKWBDFXRSVGXTWFWDELZ + YNYHYFJYOVTYTXSXQPYQEKBWDFXRSVGVJVK $. + $} + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Surreal Contributions +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + + ${ + abeqabi.a $e |- A = { x | ps } $. + $( Generalized condition for a class abstraction to be equal to some class. + (Contributed by RP, 2-Sep-2024.) $) + abeqabi $p |- ( { x | ph } = A <-> A. x ( ph <-> ps ) ) $= + ( cab wceq wb wal eqeq2i abbi bitr4i ) ACFZDGMBCFZGABHCIDNMEJABCKL $. + $} + + ${ + $d x Y $. $d x Z $. + $( Condition for a class abstraction to be a pair. (Contributed by RP, + 25-Aug-2024.) $) + abpr $p |- ( { x | ph } = { Y , Z } <-> + A. x ( ph <-> ( x = Y \/ x = Z ) ) ) $= + ( cv wceq wo cpr dfpr2 abeqabi ) ABEZCFKDFGBCDHBCDIJ $. + $} + + ${ + $d x X $. $d x Y $. $d x Z $. + $( Condition for a class abstraction to be a triple. (Contributed by RP, + 25-Aug-2024.) $) + abtp $p |- ( { x | ph } = { X , Y , Z } <-> + A. x ( ph <-> ( x = X \/ x = Y \/ x = Z ) ) ) $= + ( cv wceq w3o ctp dftp2 abeqabi ) ABFZCGLDGLEGHBCDEIBCDEJK $. + $} + + ${ + $d o O $. $d x o y $. $d ph o $. $d ps x y $. $d ch o $. + ralopabb.o $e |- O = { <. x , y >. | ph } $. + ralopabb.p $e |- ( o = <. x , y >. -> ( ps <-> ch ) ) $. + $( Restricted universal quantification over an ordered-pair class + abstraction. (Contributed by RP, 25-Sep-2024.) $) + ralopabb $p |- ( A. o e. O ps <-> A. x A. y ( ph -> ch ) ) $= + ( wral wi wal wn wex wrex 2nalexn wa cv cop wceq notbid rexopabb 3bitr2ri + annim 2exbii bitri rexnal con4bii ) BFGJZACKZELDLZUKMUJMZENDNZBMZFGOZUIMU + JDEPUOACMZQZENDNUMAUNUPDEFGHFRDRERSTBCIUAUBUQULDEACUDUEUFBFGUGUCUH $. + $} + + ${ + $d x y A $. $d x B y $. $d ch x y $. + bropabg.xA $e |- ( x = A -> ( ph <-> ps ) ) $. + bropabg.yB $e |- ( y = B -> ( ps <-> ch ) ) $. + bropabg.R $e |- R = { <. x , y >. | ph } $. + $( Equivalence for two classes related by an ordered-pair class + abstraction. A generalization of ~ brsslt . (Contributed by RP, + 26-Sep-2024.) $) + bropabg $p |- ( A R B <-> ( ( A e. _V /\ B e. _V ) /\ ch ) ) $= + ( wbr cvv wcel wa bropaex12 brabg biadanii ) FGHLFMNGMNOCADEFGHKPABCDEFGM + MHIJKQR $. + $} + + ${ + fpwfvss.f $e |- F : C --> ~P B $. + $( Functions into a powerset always have values which are subsets. This is + dependant on our convention when the argument is not part of the domain. + (Contributed by RP, 13-Sep-2024.) $) + fpwfvss $p |- ( F ` A ) C_ B $= + ( wcel cfv wss cpw ffvelcdmi elpwid wn cdm wceq fdmi eleq2i ndmfv sylnbir + c0 0ss eqsstrdi pm2.61i ) ACFZADGZBHUCUDBCBIZADEJKUCLUDSBUCADMZFUDSNUFCAC + UEDEOPADQRBTUAUB $. + $} + + $( A class that strictly dominates any set is not empty. (Suggested by SN, + 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.) $) + sdomne0 $p |- ( B ~< A -> A =/= (/) ) $= + ( csdm wbr c0 wne wcel wi relsdom brrelex1i wceq breq1 biimpd 0sdomg sdomtr + cvv a1i ex syl6bir syl pm2.61dne wb brrelex2i ibi syl6 pm2.43i ) BACDZAEFZU + GUGEACDZUHUGBPGZUGUIHZBACIJUJUKBEBEKZUKHUJULUGUIBEACLMQUJBEFEBCDZUKBPNUMUGU + IEBAORSUATUIUHUIAPGUIUHUBEACIUCAPNTUDUEUF $. + + ${ + sdomne0d.a $e |- ( ph -> B ~< A ) $. + sdomne0d.b $e |- ( ph -> B e. V ) $. + $( A class that strictly dominates any set is not empty. (Contributed by + RP, 3-Sep-2024.) $) + sdomne0d $p |- ( ph -> A =/= (/) ) $= + ( csdm wbr c0 wne wcel wi wceq breq1 biimpd a1i 0sdomg sdomtr syl cvv ibi + ex syl6bir pm2.61dne wb relsdom brrelex2i syl6 mpd ) ACBGHZBIJZEAUJIBGHZU + KACDKZUJULLZFUMUNCICIMZUNLUMUOUJULCIBGNOPUMCIJICGHZUNCDQUPUJULICBRUBUCUDS + ULUKULBTKULUKUEIBGUFUGBTQSUAUHUI $. + $} + + ${ + $d ph x $. $d A x $. $d B x $. $d R x $. $d O x $. + safesnsupfiss.small $e |- ( ph -> ( O = (/) \/ O = 1o ) ) $. + safesnsupfiss.finite $e |- ( ph -> B e. Fin ) $. + safesnsupfiss.subset $e |- ( ph -> B C_ A ) $. + safesnsupfiss.ordered $e |- ( ph -> R Or A ) $. + $( If ` B ` is a finite subset of ordered class ` A ` , we can safely + create a small subset with the same largest element and upper bound, if + any. (Contributed by RP, 1-Sep-2024.) $) + safesnsupfiss $p |- ( ph -> + if ( O ~< B , { sup ( B , A , R ) } , B ) C_ B ) $= + ( vx wcel wa wo wceq simpr c0 adantr con0 c1o eleq1 csdm wbr csup csn cif + cv elif elsni wor cfn wne wss 0elon mpbiri 1on jaoi syl sdomne0d syl13anc + wn fisupcl eqeltrd ex syl5 expimpd wi a1i jaod biimtrid ssrdv ) AJECUAUBZ + CBDUCZUDZCUEZCJUFZVNKVKVOVMKZLZVKUTZVOCKZLZMAVSVKVOVMCUGAVQVSVTAVKVPVSVPV + OVLNZAVKLZVSVOVLUHWBWAVSWBWALVOVLCWBWAOWBVLCKZWAWBBDUIZCUJKZCPUKCBULZWCAW + DVKIQAWEVKGQWBCERAVKOAERKZVKAEPNZESNZMWGFWHWGWIWHWGPRKUMEPRTUNWIWGSRKUOES + RTUNUPUQQURAWFVKHQBCDVAUSQVBVCVDVEVTVSVFAVRVSOVGVHVIVJ $. + $} + + ${ + $d ph x $. + safesnsupfiub.small $e |- ( ph -> ( O = (/) \/ O = 1o ) ) $. + safesnsupfiub.finite $e |- ( ph -> B e. Fin ) $. + safesnsupfiub.subset $e |- ( ph -> B C_ A ) $. + safesnsupfiub.ordered $e |- ( ph -> R Or A ) $. + safesnsupfiub.ub $e |- ( ph -> A. x e. B A. y e. C x R y ) $. + $( If ` B ` is a finite subset of ordered class ` A ` , we can safely + create a small subset with the same largest element and upper bound, if + any. (Contributed by RP, 1-Sep-2024.) $) + safesnsupfiub $p |- ( ph -> + A. x e. if ( O ~< B , { sup ( B , A , R ) } , B ) + A. y e. C + x R y ) $= + ( cv wbr wral csdm csup csn wcel cif safesnsupfiss sseld imim1d ralimdv2 + mpd ) ABNZCNGOCFPZBEPUHBHEQOEDGRSEUAZPMAUHUHBEUIAUGUITUGETUHAUIEUGADEGHIJ + KLUBUCUDUEUF $. + $} + + ${ + safesnsupfidom1o.small $e |- ( ph -> ( O = (/) \/ O = 1o ) ) $. + safesnsupfidom1o.finite $e |- ( ph -> B e. Fin ) $. + $( If ` B ` is a finite subset of ordered class ` A ` , we can safely + create a small subset with the same largest element and upper bound, if + any. (Contributed by RP, 1-Sep-2024.) $) + safesnsupfidom1o $p |- ( ph -> + if ( O ~< B , { sup ( B , A , R ) } , B ) ~<_ 1o ) $= + ( wbr c1o cdom wa wceq adantl cvv wcel con0 1on ax-mp c0 wi csdm csup csn + cif iftrue ensn1g domrefg endomtr sylancl wn snprc snex eqeng sylbi 0domg + cen pm2.61i eqbrtrdi iffalse cfn wo wb 0elon eleq1 mpbiri fidomtri sylan2 + jaoi breq2 domtr mpan2 syl6bi biimpd sylbird syl2anc eqbrtrd pm2.61dan + imp ) AECUAHZVSCBDUBZUCZCUDZIJHAVSKWBWAIJVSWBWALAVSWACUEMVTNOZWAIJHZWCWAI + UPHIIJHZWDVTNUFIPOZWEQIPUGRWAIIUHUIWCUJZWASUPHZSIJHZWDWGWASLZWHVTUKWANOWJ + WHTVTULWASNUMRUNWFWIQIPUORZWASIUHUIUQURAVSUJZKWBCIJWLWBCLAVSWACUSMAWLCIJH + ZACUTOZESLZEILZVAZWLWMTGFWNWQKWLCEJHZWMWQWNEPOZWRWLVBWOWSWPWOWSSPOVCESPVD + VEWPWSWFQEIPVDVEVHCEPVFVGWQWRWMTZWNWOWTWPWOWRCSJHZWMESCJVIXAWIWMWKCSIVJVK + VLWPWRWMEICJVIVMVHMVNVOVRVPVQ $. + $} + + ${ + $d A x y $. $d B x y $. $d O x y $. $d R x y $. $d ph x $. + safesnsupfilb.small $e |- ( ph -> ( O = (/) \/ O = 1o ) ) $. + safesnsupfilb.finite $e |- ( ph -> B e. Fin ) $. + safesnsupfilb.subset $e |- ( ph -> B C_ A ) $. + safesnsupfilb.ordered $e |- ( ph -> R Or A ) $. + $( If ` B ` is a finite subset of ordered class ` A ` , we can safely + create a small subset with the same largest element and upper bound, if + any. (Contributed by RP, 3-Sep-2024.) $) + safesnsupfilb $p |- ( ph -> + A. x e. ( B \ if ( O ~< B , { sup ( B , A , R ) } , B ) ) + A. y e. if ( O ~< B , { sup ( B , A , R ) } , B ) + x R y ) $= + ( wbr wral cdif wa wcel wceq wo c0 con0 csdm cv csup csn cif wne ad2antrr + wi wor wss cfn simpr eqidd supgtoreq df-or orcom imbi1i 3bitr4i ralrimiva + wn df-ne sylib iftrue difeq2d adantl raleqdv iftrued w3a adantr c1o 0elon + wb eleq1 mpbiri 1on jaoi syl sdomne0d fisupcl syl2an2r breq2 ralsng bitrd + 3jca ralbidv raldifsnb bitr4di mpbird ral0 iffalse difid eqtrdi pm2.61dan + ) AGEUALZBUBZCUBZFLZCWNEDFUCZUDZEUEZMZBEWTNZMZAWNOZXCWOWRUFZWOWRFLZUHZBEM + ZXDXGBEXDWOEPZOZXFWOWRQZRZXGXJDEWOFWRADFUIZWNXIKUGAEDUJZWNXIJUGAEUKPZWNXI + IUGXDXIULXJWRUMUNXKXFRXKUTZXFUHXLXGXKXFUOXFXKUPXEXPXFWOWRVAUQURVBUSXDXCXA + BEWSNZMZXHXDXABXBXQWNXBXQQAWNWTWSEWNWSEVCVDVEVFXDXRXFBXQMXHXDXAXFBXQXDXAW + QCWSMZXFXDWQCWTWSXDWNWSEAWNULZVGVFXDWREPZXSXFVLAXMWNXOESUFZXNVHYAKXDXOYBX + NAXOWNIVIXDEGTXTXDGSQZGVJQZRZGTPZAYEWNHVIYCYFYDYCYFSTPVKGSTVMVNYDYFVJTPVO + GVJTVMVNVPVQVRAXNWNJVIWDDEFVSVTWQXFCWREWPWRWOFWAWBVQWCWEXFBEWRWFWGWCWHAWN + UTZOZXCXABSMXABWIYHXABXBSYHXBEENSYHWTEEYGWTEQAWNWSEWJVEVDEWKWLVFVNWM $. + $} + + ${ + isoeq145.1 $e |- ( ph -> F = G ) $. + isoeq145.4 $e |- ( ph -> A = C ) $. + isoeq145.5 $e |- ( ph -> B = D ) $. + $( Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.) $) + isoeq145d $p |- ( ph -> ( F Isom R , S ( A , B ) + <-> G Isom R , S ( C , D ) ) ) $= + ( wiso wceq wb isoeq1 syl isoeq4 isoeq5 3bitrd ) ABCFGHMZBCFGIMZDCFGIMZDE + FGIMZAHINUAUBOJBCFGIHPQABDNUBUCOKBCDFGIRQACENUCUDOLDCEFGISQT $. + $} + + ${ + resisoeq45.4 $e |- ( ph -> A = C ) $. + resisoeq45.5 $e |- ( ph -> B = D ) $. + $( Equality deduction for equally restricted isometries. (Contributed by + RP, 14-Jan-2025.) $) + resisoeq45d $p |- ( ph -> ( ( F |` A ) Isom R , S ( A , B ) + <-> ( F |` C ) Isom R , S ( C , D ) ) ) $= + ( cres reseq2d isoeq145d ) ABCDEFGHBKHDKABDHILIJM $. + $} + + $( An equivalence between identically restricted order-reversing + self-isometries. (Contributed by RP, 30-Sep-2024.) $) + negslem1 $p |- ( A = B -> + ( ( F |` A ) Isom R , `' R ( A , A ) <-> + ( F |` B ) Isom R , `' R ( B , B ) ) ) $= + ( wceq ccnv id resisoeq45d ) ABEZAABBCCFDIGZJH $. + + ${ + $d x A $. $d x F $. + $( Equivalence to saying the converse of an involution is the function + itself. (Contributed by RP, 13-Oct-2024.) $) + nvocnvb $p |- ( ( F Fn A /\ `' F = F ) <-> + ( F : A -1-1-onto-> A /\ + A. x e. A ( F ` ( F ` x ) ) = x ) ) $= + ( wfn ccnv wceq wa wf1o cv cfv wral nvof1o fveq1 ad2antlr f1ocnvfv1 sylan + wcel eqtr3d ralrimiva jca wf f1of ffn adantr nvocnv impbii ) CBDZCEZCFZGZ + BBCHZAIZCJZCJZULFZABKZGUJUKUPBCLZUJUOABUJULBQZGUMUHJZUNULUIUSUNFUGURUMUHC + MNUJUKURUSULFUQBBULCOPRSTUKBBCUAZUPUJBBCUBUTUPGUGUIUTUGUPBBCUCUDABCUETPUF + $. + $} + + ${ + $d A a b x y $. $d B a b x y $. $d R a b $. $d S a b $. + $( The following is probably most useful when ` R Or S ` . $) + nla0001.defsslt $e |- .< = { <. a , b >. | ( a C_ S /\ b C_ S /\ + A. x e. a A. y e. b x R y ) } $. + $( Binary relation form of a relation, ` .< ` , which has been extended + from relation ` R ` to subsets of class ` S ` . Usually, we will assume + ` R Or S ` . Definition in [Alling], p. 2. Generalization of + ~ brsslt . (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by + RP, 28-Nov-2023.) $) + rp-brsslt $p |- ( A .< B <-> ( ( A e. _V /\ B e. _V ) /\ ( A C_ S /\ B C_ + S /\ A. x e. A A. y e. B x R y ) ) ) $= + ( cv wss wbr wral w3a wceq sseq1 raleq 3anbi13d ralbidv 3anbi23d bropabg + ) HKZFLZIKZFLZAKBKEMZBUENZAUCNZOCFLZUFUHACNZOUJDFLZUGBDNZACNZOHICDGUCCPUD + UJUIUKUFUCCFQUHAUCCRSUEDPZUFULUKUNUJUEDFQUOUHUMACUGBUEDRTUAJUB $. + + ${ + nla0001.set $e |- ( ph -> A e. _V ) $. + nla0002.sset $e |- ( ph -> A C_ S ) $. + $( Extending a linear order to subsets, the empty set is less than any + subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.) $) + nla0002 $p |- ( ph -> (/) .< A ) $= + ( c0 cvv wcel wss cv wbr wral a1i w3a 0ex 0ss ral0 rp-brsslt syl21anbrc + 3jca ) AMNOZDNOMFPZDFPZBQCQERCDSZBMSZUAMDGRUHAUBTKAUIUJULUIAFUCTLULAUKB + UDTUGBCMDEFGHIJUEUF $. + + $( Extending a linear order to subsets, the empty set is greater than any + subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.) $) + nla0003 $p |- ( ph -> A .< (/) ) $= + ( cvv wcel c0 wss cv wbr wral a1i w3a 0ex 0ss ral0 mpbi 3jca syl21anbrc + ralcom rp-brsslt ) ADMNOMNZDFPZOFPZBQCQERZCOSBDSZUADOGRKUJAUBTAUKULUNLU + LAFUCTUNAUMBDSZCOSUNUOCUDUMCBODUHUETUFBCDOEFGHIJUIUG $. + $} + + $( Extending a linear order to subsets, the empty set is less than itself. + Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.) $) + nla0001 $p |- ( ph -> (/) .< (/) ) $= + ( c0 cvv wcel 0ex a1i wss 0ss nla0002 ) ABCJDEFGHIJKLAMNJEOAEPNQ $. + $} + + ${ + $d A x $. + + $( ` B ` is called a non-limit ordinal if it is not a limit ordinal. + (Contributed by RP, 27-Sep-2023.) + + Alling, Norman L. "Fundamentals of Analysis Over Surreal Numbers + Fields." The Rocky Mountain Journal of Mathematics 19, no. 3 (1989): + 565-73. http://www.jstor.org/stable/44237243. $) + faosnf0.11b $p |- ( ( Ord A /\ -. Lim A /\ A =/= (/) ) -> + E. x e. On A = suc x ) $= + ( word wlim wn c0 wne w3a wa cv csuc wceq con0 wi 3ancomb df-3an wo df-ne + wrex anbi2i imbi1i pm5.6 iman 3bitrri dflim3 xchnxbir 3bitri pm3.35 sylbi + ) BCZBDZEZBFGZHZUJUMIZUOBAJKLAMSZNZIZUPUNUJUMULHUOULIURUJULUMOUJUMULPULUQ + UOUJBFLZUPQZEIZUQUKUQUJUSEZIZUPNUJUTNVAEUOVCUPUMVBUJBFRTUAUJUSUPUBUJUTUCU + DABUEUFTUGUOUPUHUI $. + $} + + ${ + $d f x $. + $( A surreal number, in the functional sign expansion representation, is a + function which maps from an ordinal into a set of two possible signs. + (Contributed by RP, 12-Jan-2025.) $) + dfno2 $p |- No = { f e. ~P ( On X. { 1o , 2o } ) | + ( Fun f /\ dom f e. On ) } $= + ( vx cv c1o c2o cpr wf con0 wrex cab cxp cpw wcel wfun cdm wa adantl wceq + wss simpl csur crab fssxp onss adantr xpss1 syl sstrd sylibr ffun eqeltrd + velpw fdm jca32 rexlimiva simprr wb feq2 funssxp simplbi syl2anr rspcedvd + elpwi impbii abbii df-no df-rab 3eqtr4i ) BCZDEFZACZGZBHIZAJVKHVJKZLZMZVK + NZVKOZHMZPZPZAJUAVTAVOUBVMWAAVMWAVLWABHVIHMZVLPZVPVQVSWCVKVNSZVPWCVKVIVJK + ZVNVLVKWESWBVIVJVKUCQWCVIHSZWEVNSWBWFVLVIUDUEVIHVJUFUGUHAVNULUIVLVQWBVIVJ + VKUJQWCVRVIHVLVRVIRWBVIVJVKUMQWBVLTUKUNUOWAVLVRVJVKGZBVRHVPVQVSUPVIVRRVLW + GUQWAVIVRVJVKURQVTVQWDWGVPVQVSTVKVNVCVQWDPWGVRHSHVJVKUSUTVAVBVDVEABVFVTAV + OVGVH $. + $} + + $( Every ordinal maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + onnog $p |- ( ( A e. On /\ B e. { 1o , 2o } ) + -> ( A X. { B } ) e. No ) $= + ( con0 wcel c1o c2o cpr csn cxp csur fconst6g adantl w3a wfun cdm crn simp3 + wf wss ffund c0 wceq simp2 snnzg dmxp eqcomd 3syl simp1 eqeltrrd frnd elno2 + wne syl3anbrc mpd3an3 ) ACDZBEFGZDZAUPABHZIZRZUSJDZUQUTUOABUPKLUOUQUTMZUSNU + SOZCDUSPUPSVAVBAUPUSUOUQUTQZTVBAVCCVBUQURUAULZAVCUBUOUQUTUCBUPUDVEVCAAURUEU + FUGUOUQUTUHUIVBAUPUSVDUJUSUKUMUN $. + + $( Every ordinal maps to a surreal number of that birthday. (Contributed by + RP, 21-Sep-2023.) $) + onnobdayg $p |- ( ( A e. On /\ B e. { 1o , 2o } ) + -> ( bday ` ( A X. { B } ) ) = A ) $= + ( con0 wcel c1o c2o cpr wa csn cxp cbday cfv cdm csur wceq onnog bdayval c0 + syl wne simpr snnzg dmxp 3syl eqtrd ) ACDZBEFGZDZHZABIZJZKLZUKMZAUIUKNDULUM + OABPUKQSUIUHUJRTUMAOUFUHUABUGUBAUJUCUDUE $. + + $( Bounds formed from the birthday are surreal numbers. (Contributed by RP, + 21-Sep-2023.) $) + bdaybndex $p |- ( ( A e. No /\ B = ( bday ` A ) /\ C e. { 1o , 2o } ) + -> ( B X. { C } ) e. No ) $= + ( csur wcel cbday cfv wceq con0 c1o c2o cpr csn cxp wa simpr bdayval adantr + cdm eqtrd nodmon eqeltrd onnog stoic3 ) ADEZBAFGZHZBIECJKLEBCMNDEUEUGOZBASZ + IUHBUFUIUEUGPUEUFUIHUGAQRTUEUIIEUGAUARUBBCUCUD $. + + $( Bounds formed from the birthday have the same birthday. (Contributed by + RP, 30-Sep-2023.) $) + bdaybndbday $p |- ( ( A e. No /\ B = ( bday ` A ) /\ C e. { 1o , 2o } ) + -> ( bday ` ( B X. { C } ) ) = ( bday ` A ) ) $= + ( csur wcel cbday cfv wceq c1o c2o cpr w3a csn cxp cdm bdaybndex bdayval c0 + syl wne simp3 snnzg dmxp 3syl simp2 3eqtrd ) ADEZBAFGZHZCIJKZEZLZBCMZNZFGZU + NOZBUHULUNDEUOUPHABCPUNQSULUKUMRTUPBHUGUIUKUACUJUBBUMUCUDUGUIUKUEUF $. + + $( Every ordinal maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + onno $p |- ( A e. On -> ( A X. { 2o } ) e. No ) $= + ( con0 wcel c2o c1o cpr csn cxp csur 2oex prid2 onnog mpan2 ) ABCDEDFCADGHI + CEDJKADLM $. + + ${ + onnoi.on $e |- A e. On $. + $( Every ordinal maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + onnoi $p |- ( A X. { 2o } ) e. No $= + ( con0 wcel c2o csn cxp csur onno ax-mp ) ACDAEFGHDBAIJ $. + $} + + $( Ordinal zero maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + 0no $p |- (/) e. No $= + ( c0 c2o csn cxp csur 0xp 0elon onnoi eqeltrri ) ABCZDAEJFAGHI $. + + $( Ordinal one maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + 1no $p |- ( 1o X. { 2o } ) e. No $= + ( c1o 1on onnoi ) ABC $. + + $( Ordinal two maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + 2no $p |- ( 2o X. { 2o } ) e. No $= + ( c2o 2on onnoi ) ABC $. + + $( Ordinal three maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + 3no $p |- ( 3o X. { 2o } ) e. No $= + ( c3o 3on onnoi ) ABC $. + + $( Ordinal four maps to a surreal number. (Contributed by RP, + 21-Sep-2023.) $) + 4no $p |- ( 4o X. { 2o } ) e. No $= + ( c4o 4on onnoi ) ABC $. + + ${ + fnimafnex.f $e |- F Fn B $. + $( The functional image of a function value exists. (Contributed by RP, + 31-Oct-2024.) $) + fnimafnex $p |- ( F " ( G ` A ) ) e. _V $= + ( wfun cfv cvv wcel cima wfn fnfun ax-mp fvex funimaexg mp2an ) CFZADGZHI + CRJHICBKQEBCLMADNCRHOP $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Short Studies @@ -672415,12 +676065,9 @@ is in the span of P(i)(X), so there is an R-linear combination of $( A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.) $) nlimsuc $p |- ( A e. On -> -. Lim suc A ) $= ( con0 wcel csuc word c0 cuni wceq wlim sucidg wn eloni ordirr eleq2 notbid - w3a syl syl5ibrcom mt2d cun neqned wtr ordtr wss df-tr ssequn1 bitri df-suc - csn unieqi uniun eqtri unisng uneq2d eqtrid eqeq1d mpbid neeqtrrd intn3an3d - bitr4id neneqd dflim2 sylnibr ) ABCZADZEZFVECZVEVEGZHZPVEIVDVIVFVGVDVEVHVDV - EAVHVDVEAVDVEAHZAVECZABJVDVKKVJAACZKZVDAEZVMALZAMQVJVKVLVEAANORSUAVDAUBZVHA - HZVDVNVPVOAUCQVDVPAGZATZAHZVQVPVRAUDVTAUEVRAUFUGVDVHVSAVDVHVRAUIZGZTZVSVHAW - ATZGWCVEWDAUHUJAWAUKULVDWBAVRABUMUNUOUPUTUQURVAUSVEVBVC $. + w3a syl syl5ibrcom mt2d neqned onunisuc neeqtrrd neneqd intn3an3d sylnibr + dflim2 ) ABCZADZEZFUHCZUHUHGZHZPUHIUGULUIUJUGUHUKUGUHAUKUGUHAUGUHAHZAUHCZAB + JUGUNKUMAACZKZUGAEUPALAMQUMUNUOUHAANORSTAUAUBUCUDUHUFUE $. $( $j usage 'nlimsuc' avoids 'ax-un'; $) $( 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) @@ -672693,8 +676340,7 @@ is in the span of P(i)(X), so there is an R-linear combination of RP, 15-Apr-2020.) $) ifpbi23 $p |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( if- ( ta , ph , ch ) <-> if- ( ta , ps , th ) ) ) $= - ( wb wa wi wn wif simpl imbi2d simpr anbi12d dfifp2 3bitr4g ) ABFZCDFZGZEAH - ZEIZCHZGEBHZUADHZGEACJEBDJSTUCUBUDSABEQRKLSCDUAQRMLNEACOEBDOP $. + ( wb wa simpl simpr ifpbi23d ) ABFZCDFZGEACBDKLHKLIJ $. $( Restatement of ~ biid . (Contributed by RP, 25-Apr-2020.) $) ifpbiidcor $p |- if- ( ph , ph , -. ph ) $= @@ -673019,10 +676665,10 @@ is in the span of P(i)(X), so there is an R-linear combination of rp-fakeimass $p |- ( ( ph \/ ch ) <-> ( ( ( ph -> ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) ) $= - ( wo wi wb wn conax1 pm2.21d ax-1 ja ax-2 com3r impbid2 2thd jaoi jarl orrd - a1d simplim orcd a1i bija impbii ) ACDZABEZCEZABCEZEZFZAUJCAUGUIUFCUIUFGZUH - AUKBCABHISCUHACBJSZKUIUFACABCLMNCUGUICUFJULOPUGUIUEUGUEUIUGACABCQRSUIGZUEEU - GGUMACAUHTUAUBUCUD $. + ( wo wi wb wn pm2.521g a1d ax-1 ja ax-2 impbid2 2thd jaoi jarl orrd simplim + com3r orcd a1i bija impbii ) ACDZABEZCEZABCEZEZFZAUICAUFUHUECUHUEGUGAABCHIC + UGACBJIZKUHUEACABCLSMCUFUHCUEJUJNOUFUHUDUFUDUHUFACABCPQIUHGZUDEUFGUKACAUGRT + UAUBUC $. $( A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.) $) @@ -673082,14 +676728,13 @@ is in the span of P(i)(X), so there is an R-linear combination of with all the natural numbers between 1 and some ` n e. NN0 ` . (Contributed by RP, 3-Mar-2020.) $) rp-isfinite5 $p |- ( A e. Fin <-> E. n e. NN0 ( 1 ... n ) ~~ A ) $= - ( cfn wcel c1 cv cfz co cen wbr cn0 wrex chash cfv cvv oveq2 mpsyl sylibr - wa wceq wex fvex hashcl isfinite4 biimpi jca breq1d anbi12d spcegv df-rex - eleq1 hasheni eqcomd hashfz1 ovex eqtr eqeng syl5 entr sylancom rexlimiva - syl2anr impbii ) ACDZEBFZGHZAIJZBKLZVDVEKDZVGSZBUAZVHAMNZODVDVLKDZEVLGHZA - IJZSZVKAMUBVDVMVOAUCVDVOAUDZUEUFVJVPBVLOVEVLTZVIVMVGVOVEVLKUKVRVFVNAIVEVL - EGPUGUHUIQVGBKUJRVGVDBKVJVOVDVIVGVNVFIJZVOVGVLVFMNZTZVTVETZVSVIVGVTVLVFAU - LUMVEUNVNODZWAWBSVLVETZVSEVLGUOVLVTVEUPWDVNVFTWCVSVLVEEGPVNVFOUQURQVBVNVF - AUSUTVQRVAVC $. + ( cfn wcel c1 cv cfz co cen wbr cn0 wrex wa wex chash cfv wceq sylibr cvv + oveq2 hashcl isfinite4 biimpi breq1d anbi12d spcedv df-rex hasheni eqcomd + jca eleq1 hashfz1 ovex eqtr eqeng syl5 mpsyl syl2anr entr sylancom impbii + rexlimiva ) ACDZEBFZGHZAIJZBKLZVCVDKDZVFMZBNVGVCVIAOPZKDZEVJGHZAIJZMBKVJA + UAZVCVKVMVNVCVMAUBZUCUJVDVJQZVHVKVFVMVDVJKUKVPVEVLAIVDVJEGTUDUEUFVFBKUGRV + FVCBKVIVMVCVHVFVLVEIJZVMVFVJVEOPZQZVRVDQZVQVHVFVRVJVEAUHUIVDULVLSDZVSVTMV + JVDQZVQEVJGUMVJVRVDUNWBVLVEQWAVQVJVDEGTVLVESUOUPUQURVLVEAUSUTVORVBVA $. $} ${ @@ -673105,16 +676750,16 @@ is in the span of P(i)(X), so there is an R-linear combination of andi orbi12i sylbb2 simp2 oveq2 fz10 eqtrdi simp3 eqbrtrrd simp1 pm2.21dd en0 syl3an2 jaoi simprr jca chash cfv clt nngt0 hash0 a1i hashfz1 3brtr4d csdm nnnn0 fzfi hashsdom mp2an sylib anim1i sdomentr sdomnen exbii 19.42v - wb 3bitr2ri ) ACDZAEFZXAGZXBHZXAGZIZXBJBUAZKUBZALMZBNUCZIXAXBXDIZXAGXFXKX - AXBUDUEXBXDXAUFOXCXBXEXJXCXBXBXAUGXBXAECDZXBXAUHUMECAUIUJUNUKXEXDXGPDZXIG - ZBQZGZXJXAXOXDXAXIBPUCXOABULXIBPUOOUPXJXGNDZXIGZBQXDXNGZBQXPXIBNUOXSXRBXS - XRXSXQXIXSEALMZHZXQXIRZYAXGSFZXIRZIZXQXSYAXRGZYAYCXIGZGZIZYEXSYAXRYGIZGYI - XDYAXNYJXBXTXBAELMZXTAVOZAEUQURUSXNXQYCIZXIGYJXMYMXIXGUTVAXQYCXIUFOVBYAXR - YGVDOYBYFYDYHYAXQXIVCYAYCXIVCVEVFYBXQYDYAXQXIVGYCYAXHEFZXIXQYCXHJSKUBEXGS - JKVHVIVJYAYNXIRZXTXQYOXHEALYAYNXIVGYAYNXIVKVLYAYNXIVMVNVPVQTXDXMXIVRVSXRX - DXNXREXHWHMZXIGZXDXQYPXIXQEVTWAZXHVTWAZWBMZYPXQSXGYRYSWBXGWCYRSFXQWDWEXQX - MYSXGFXGWIZXGWFTWGXLXHCDYTYPWSUMJXGWJEXHWKWLWMWNYQYAXDYQEAWHMYAEXHAWOEAWP - TXTXBXTYKXBEAUQYLOUSWMTXQXMXIUUAWNVSUKWQXDXNBWRWTOVEO $. + wb en0r 3bitr2ri ) ACDZAEFZXBGZXCHZXBGZIZXCJBUAZKUBZALMZBNUCZIXBXCXEIZXBG + XGXLXBXCUDUEXCXEXBUFOXDXCXFXKXDXCXCXBUGXCXBECDZXCXBUHUMECAUIUJUNUKXFXEXHP + DZXJGZBQZGZXKXBXPXEXBXJBPUCXPABULXJBPUOOUPXKXHNDZXJGZBQXEXOGZBQXQXJBNUOXT + XSBXTXSXTXRXJXTEALMZHZXRXJRZYBXHSFZXJRZIZXRXTYBXSGZYBYDXJGZGZIZYFXTYBXSYH + IZGYJXEYBXOYKXCYAXCAELMYAAVOAEUQURUSXOXRYDIZXJGYKXNYLXJXHUTVAXRYDXJUFOVBY + BXSYHVDOYCYGYEYIYBXRXJVCYBYDXJVCVEVFYCXRYEYBXRXJVGYDYBXIEFZXJXRYDXIJSKUBE + XHSJKVHVIVJYBYMXJRZYAXRYNXIEALYBYMXJVGYBYMXJVKVLYBYMXJVMVNVPVQTXEXNXJVRVS + XSXEXOXSEXIWHMZXJGZXEXRYOXJXREVTWAZXIVTWAZWBMZYOXRSXHYQYRWBXHWCYQSFXRWDWE + XRXNYRXHFXHWIZXHWFTWGXMXICDYSYOWSUMJXHWJEXIWKWLWMWNYPYBXEYPEAWHMYBEXIAWOE + AWPTYAXCAWTUSWMTXRXNXJYTWNVSUKWQXEXOBWRXAOVEO $. $} @@ -673192,12 +676837,12 @@ is in the span of P(i)(X), so there is an R-linear combination of E. x e. On A = suc x ) $= ( word wlim wn c0 wne w3a cv csuc wceq con0 wa wi 3ancomb df-3an wo df-ne wrex anbi2i imbi1i pm5.6 iman 3bitrri dflim3 xchnxbir 3bitri pm3.35 sylbi - wcel eloni ordsuc sylib nlimsucg nsuceq0 a1i ordeq limeq notbid 3anbi123d - 3jca neeq1 syl5ibrcom rexlimiv impbii ) BCZBDZEZBFGZHZBAIZJZKZALSZVJVFVIM - ZVOVNNZMZVNVJVFVIVHHVOVHMVQVFVHVIOVFVIVHPVHVPVOVFBFKZVNQZEMZVPVGVPVFVREZM - ZVNNVFVSNVTEVOWBVNVIWAVFBFRTUAVFVRVNUBVFVSUCUDABUEUFTUGVOVNUHUIVMVJALVKLU - JZVJVMVLCZVLDZEZVLFGZHWCWDWFWGWCVKCWDVKUKVKULUMVKLUNWGWCVKUOUPVAVMVFWDVHW - FVIWGBVLUQVMVGWEBVLURUSBVLFVBUTVCVDVE $. + wcel eloni ordsuc sylib nlimsuc nsuceq0 a1i 3jca ordeq limeq notbid neeq1 + 3anbi123d syl5ibrcom rexlimiv impbii ) BCZBDZEZBFGZHZBAIZJZKZALSZVJVFVIMZ + VOVNNZMZVNVJVFVIVHHVOVHMVQVFVHVIOVFVIVHPVHVPVOVFBFKZVNQZEMZVPVGVPVFVREZMZ + VNNVFVSNVTEVOWBVNVIWAVFBFRTUAVFVRVNUBVFVSUCUDABUEUFTUGVOVNUHUIVMVJALVKLUJ + ZVJVMVLCZVLDZEZVLFGZHWCWDWFWGWCVKCWDVKUKVKULUMVKUNWGWCVKUOUPUQVMVFWDVHWFV + IWGBVLURVMVGWEBVLUSUTBVLFVAVBVCVDVE $. $} $( May move to main section after ~ en1 . $) @@ -673252,8 +676897,8 @@ is in the span of P(i)(X), so there is an R-linear combination of $( A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.) $) ensucne0 $p |- ( A ~~ suc B -> A =/= (/) ) $= - ( csuc cen wbr wceq nsuceq0 ensymb en0 bitri nemtbir breq1 mtbiri necon2ai - c0 ) ABCZDEZAOAOFQOPDEZRPOBGRPODEPOFOPHPIJKAOPDLMN $. + ( csuc cen wbr c0 wceq nsuceq0 en0r nemtbir breq1 mtbiri necon2ai ) ABCZDEZ + AFAFGOFNDEZPNFBHNIJAFNDKLM $. $( A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) @@ -673524,9 +677169,9 @@ is in the span of P(i)(X), so there is an R-linear combination of HUEABGMNUEUCBUGHOZUDUEBAIUHABPBAQSTUAS $. ${ - sur0020.a $e |- ( ph -> A e. V ) $. - sur0020.b $e |- ( ph -> B e. W ) $. - sur0020.aneb $e |- ( ph -> A =/= B ) $. + pren2d.a $e |- ( ph -> A e. V ) $. + pren2d.b $e |- ( ph -> B e. W ) $. + pren2d.aneb $e |- ( ph -> A =/= B ) $. $( A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023.) $) pren2d $p |- ( ph -> { A , B } ~~ 2o ) $= @@ -673783,16 +677428,6 @@ of all sets ( ~ inex1g ). $( AFTER 14085 elintg $) - ${ - $d y ph $. $d x y A $. - $( Two ways of saying a set is an element of the intersection of a class. - (Contributed by RP, 13-Aug-2020.) $) - elintabg $p |- ( A e. V -> ( A e. |^| { x | ph } - <-> A. x ( ph -> A e. x ) ) ) $= - ( vy wcel cab cint cv wral wi wal elintg eleq2w ralab2 bitrdi ) CDFCABGZH - FCEIFZEQJACBIFZKBLECQDMARSEBEBCNOP $. - $} - ${ $d x A $. $( Two ways of saying a set is an element of the intersection of a class @@ -674375,22 +678010,21 @@ of all sets ( ~ inex1g ). e. _V $= ( cid cdm crn cun cres wss cvv wcel ssun1 uneq2i wceq ssequn1 mpbi 3eqtri wa ssun2 cv cab cint dmun dmresi rnun uneq12i unidm eqtri reseq2i eqsstri - rnresi wex elexi dmexg rnexg unexg syl2anc resiexd unex dmeq rneq uneq12d - ax-mp reseq2d id sseq12d cleq2lem spcev intexab sylib mp2an ) BBEBFZBGZHZ - IZHZJZEVQFZVQGZHZIZVQJZBAUAZJEWDFZWDGZHZIZWDJZSZAUBUCKLZBVPMWBVPVQWAVOEWA - VOVOHVOVSVOVTVOVSVMVPFZHVMVOHZVOBVPUDWLVOVMVOUENVMVOJWMVOOVMVNMVMVOPQRVTV - NVPGZHVNVOHZVOBVPUFWNVOVNVOULNVNVOJWOVOOVNVMTVNVOPQRUGVOUHUIUJVPBTUKVRWCS - ZWJAUMWKWJWPAVQBVPBCDUNBCLZVPKLDWQVOKWQVMKLVNKLVOKLBCUOBCUPVMVNKKUQURUSVD - UTWIWCWDVQBWDVQOZWHWBWDVQWRWGWAEWRWEVSWFVTWDVQVAWDVQVBVCVEWRVFVGVHVIWJAVJ - VKVL $. + rnresi wex elexi dmexg rnexg unexd resiexd unex dmeq rneq uneq12d reseq2d + ax-mp id sseq12d cleq2lem spcev intexab sylib mp2an ) BBEBFZBGZHZIZHZJZEV + PFZVPGZHZIZVPJZBAUAZJEWCFZWCGZHZIZWCJZSZAUBUCKLZBVOMWAVOVPVTVNEVTVNVNHVNV + RVNVSVNVRVLVOFZHVLVNHZVNBVOUDWKVNVLVNUENVLVNJWLVNOVLVMMVLVNPQRVSVMVOGZHVM + VNHZVNBVOUFWMVNVMVNULNVMVNJWNVNOVMVLTVMVNPQRUGVNUHUIUJVOBTUKVQWBSZWIAUMWJ + WIWOAVPBVOBCDUNBCLZVOKLDWPVNKWPVLVMKKBCUOBCUPUQURVDUSWHWBWCVPBWCVPOZWGWAW + CVPWQWFVTEWQWDVRWEVSWCVPUTWCVPVAVBVCWQVEVFVGVHWIAVIVJVK $. $} $( Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.) $) rtrclexlem $p |- ( R e. V -> ( R u. ( ( dom R u. ran R ) X. ( dom R u. ran R ) ) ) e. _V ) $= - ( wcel cdm crn cun cxp cvv dmexg rnexg unexg syl2anc sqxpexg syl mpdan ) AB - CZADZAEZFZSGZHCZATFHCPSHCZUAPQHCRHCUBABIABJQRHHKLSHMNATBHKO $. + ( wcel cdm crn cun cxp cvv dmexg rnexg unexd sqxpexg syl unexg mpdan ) ABCZ + ADZAEZFZSGZHCZATFHCPSHCUAPQRHHABIABJKSHLMATBHNO $. ${ $d x A $. @@ -674425,15 +678059,15 @@ of all sets ( ~ inex1g ). trclubgNEW $p |- ( ph -> |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } C_ ( R u. ( dom R X. ran R ) ) ) $= ( cv ccom wss cxp cun cvv wcel syl wceq a1i 3sstr3g ssequn1 sylib eqsstrd - ccnv eqsstrid cdm crn dmexd rnexg xpexd syl2anc coeq12d sseq12d cnvssrndm - unexg id ssun1 coundi cnvss coss2 cocnvcnv2 cnvxp coundir coss1 cocnvcnv1 - coeq2i coeq1i xptrrel ssun2 sstri mp1i clublem ) ABEZVHFZVHGCCUAZCUBZHZIZ - VMFZVMGZBCVMACJKZVLJKVMJKDAVJVKJJACJDUCAVPVKJKDCJUDLUECVLJJUJUFVHVMMZVIVN - VHVMVQVHVMVHVMVQUKZVRUGVRUHCVMGACVLULNCSZVKVJHZGZVOACUIWAVNVMCFZVMVLFZIZV - MVMCVLUMWAWDWCVMWAWBWCGWDWCMWAVMVSSZFZVMVTSZFZWBWCWAWEWGGZWFWHGVSVTUNZWEW - GVMUOLVMCUPWGVLVMVKVJUQZVAOWBWCPQWAWCCVLFZVLVLFZIZVMCVLVLURWAWNWMVMWAWLWM - GWNWMMWAWEVLFZWGVLFZWLWMWAWIWOWPGWJWEWGVLUSLCVLUTWGVLVLWKVBOWLWMPQWMVMGWA - WMVLVMVJVKVCVLCVDVENRTRTVFVG $. + ccnv eqsstrid cdm dmexd rnexg xpexd unexd coeq12d sseq12d ssun1 cnvssrndm + crn id coundi cnvss coss2 cocnvcnv2 cnvxp coeq2i coundir cocnvcnv1 coeq1i + coss1 xptrrel ssun2 sstri mp1i clublem ) ABEZVGFZVGGCCUAZCUJZHZIZVLFZVLGZ + BCVLACVKJJDAVIVJJJACJDUBACJKVJJKDCJUCLUDUEVGVLMZVHVMVGVLVOVGVLVGVLVOUKZVP + UFVPUGCVLGACVKUHNCSZVJVIHZGZVNACUIVSVMVLCFZVLVKFZIZVLVLCVKULVSWBWAVLVSVTW + AGWBWAMVSVLVQSZFZVLVRSZFZVTWAVSWCWEGZWDWFGVQVRUMZWCWEVLUNLVLCUOWEVKVLVJVI + UPZUQOVTWAPQVSWACVKFZVKVKFZIZVLCVKVKURVSWLWKVLVSWJWKGWLWKMVSWCVKFZWEVKFZW + JWKVSWGWMWNGWHWCWEVKVALCVKUSWEVKVKWIUTOWJWKPQWKVLGVSWKVKVLVIVJVBVKCVCVDNR + TRTVEVF $. $} ${ @@ -674576,20 +678210,20 @@ of all sets ( ~ inex1g ). ssun1 cv cdif wi cnvresid cnvnonrel cnv0 eqtr4i dmeqi 3eqtr4i 0ss ssequn2 rnssi mpbi rnun 3eqtr4ri rneqi dmss ax-mp uneq12i equncomi reseq2i eqtr2i dmun cnvss eqsstrid sstrdi rneq uneq12d reseq2d id sseq12d syl5ibr adantl - a1i dmexg rnexg unexg syl2anc resiexd mpdan wa dmresi eqimssi unssi ssun2 - rnresi pm3.2i unss ssres2 sylbi ssun4 mp2b clcnvlem ) DCEZFAUAZGZWOHZIZJZ - WOKZFBUAZGZXAHZIZJZXAKZFDFDGZDHZIZJZIZGZXKHZIZJZXKKZABXKDWOXALZDDLLUBZIZM - ZXFWTUCWNXFWTXTFXSGZXSHZIZJZXSKXFYDXQXSXFYDXELZXQYEXEYDXDUDXDYCFXDYBYAXBY - BXCYAXQHZXRHZIZYFYBXBYGYFKYHYFMYGNHZYFXRLZGNLZGYGYIYJYKYJNYKDUEUFUGZUHXRO - NOUINXQXQUJZULPYGYFUKUMXQXRUNXAQUOXQGZXRGZIZYNYAXCYOYNKYPYNMYONGZYNYJHYKH - YOYQYJYKYLUPXRQNQUINXQKYQYNKYMNXQUQURPYOYNUKUMXQXRVCXAOUOUSUTVAVBXEXAVDVE - XQXRTVFXTWSYDWOXSXTWRYCFXTWPYAWQYBWOXSRWOXSVGVHVIXTVJVKVLVMXAWOLZMZWTXFUC - WNWTXFYSFYRGZYRHZIZJZYRKWTUUCWSLZYRUUDWSUUCWRUDWRUUBFWRUUAYTWPUUAWQYTWOQW - OOUSUTVAVBWSWOVDVEYSXEUUCXAYRYSXDUUBFYSXBYTXCUUAXAYRRXAYRVGVHVIYSVJVKVLVM - WOXKMZWSXOWOXKUUEWRXNFUUEWPXLWQXMWOXKRWOXKVGVHVIUUEVJVKDXKKWNDXJTVNWNXJSE - XKSEWNXISWNXGSEXHSEXISEDCVODCVPXGXHSSVQVRVSDXJCSVQVTXPWNXLXIKZXMXIKZWAZXO - XJKZXPUUFUUGXLXGXJGZIXIDXJVCXGUUJXIXGXHTUUJXIXIWBWCWDPXMXHXJHZIXIDXJUNXHU - UKXIXHXGWEUUKXIXIWFWCWDPWGUUHXNXIKUUIXLXMXIWHXNXIFWIWJXOXJDWKWLVNWM $. + a1i dmexg rnexg unexd resiexd unexg mpdan wa resdmss unssi rnresi eqimssi + ssun2 pm3.2i unss ssres2 sylbi ssun4 mp2b clcnvlem ) DCEZFAUAZGZWOHZIZJZW + OKZFBUAZGZXAHZIZJZXAKZFDFDGZDHZIZJZIZGZXKHZIZJZXKKZABXKDWOXALZDDLLUBZIZMZ + XFWTUCWNXFWTXTFXSGZXSHZIZJZXSKXFYDXQXSXFYDXELZXQYEXEYDXDUDXDYCFXDYBYAXBYB + XCYAXQHZXRHZIZYFYBXBYGYFKYHYFMYGNHZYFXRLZGNLZGYGYIYJYKYJNYKDUEUFUGZUHXRON + OUINXQXQUJZULPYGYFUKUMXQXRUNXAQUOXQGZXRGZIZYNYAXCYOYNKYPYNMYONGZYNYJHYKHY + OYQYJYKYLUPXRQNQUINXQKYQYNKYMNXQUQURPYOYNUKUMXQXRVCXAOUOUSUTVAVBXEXAVDVEX + QXRTVFXTWSYDWOXSXTWRYCFXTWPYAWQYBWOXSRWOXSVGVHVIXTVJVKVLVMXAWOLZMZWTXFUCW + NWTXFYSFYRGZYRHZIZJZYRKWTUUCWSLZYRUUDWSUUCWRUDWRUUBFWRUUAYTWPUUAWQYTWOQWO + OUSUTVAVBWSWOVDVEYSXEUUCXAYRYSXDUUBFYSXBYTXCUUAXAYRRXAYRVGVHVIYSVJVKVLVMW + OXKMZWSXOWOXKUUEWRXNFUUEWPXLWQXMWOXKRWOXKVGVHVIUUEVJVKDXKKWNDXJTVNWNXJSEX + KSEWNXISWNXGXHSSDCVODCVPVQVRDXJCSVSVTXPWNXLXIKZXMXIKZWAZXOXJKZXPUUFUUGXLX + GXJGZIXIDXJVCXGUUJXIXGXHTFXIWBWCPXMXHXJHZIXIDXJUNXHUUKXIXHXGWFUUKXIXIWDWE + WCPWGUUHXNXIKUUIXLXMXIWHXNXIFWIWJXOXJDWKWLVNWM $. $} ${ @@ -674650,23 +678284,23 @@ of all sets ( ~ inex1g ). /\ ( y o. y ) C_ y ) ) } ) $= ( vz cvv cid cdm crn cun cres wss ccom cab cint cmpt wcel a1i wceq 3eqtri cv wa crtcl w3a df-rtrcl ancom anbi2i abbii inteqi mpteq2i rtrclexi dmexg - vex rnexg unexg syl2anc resiexg mp2b unex trclexi simpr cotrintab resiexd - dmex rnex mp2an dmtrcl ax-mp dmun dmresi uneq2i ssun1 ssequn1 mpbi rntrcl - eqtri rnun rnresi ssun2 uneq12i unidm reseq2i ssmin sstri eqsstri coeq12d - simprl id sseq12d dmeq rneq uneq12d reseq2d mptrcllem df-3an bitri anbi1i - unss bitr2i eqtr4i ) UAADEASZFZWSGZHZIZCSZJZWSXDJZXDXDKZXDJZUBZCLZMZNZADW - SBSZJZEXMFZXMGZHZIZXMJZXMXMKZXMJZTZTZBLZMZNZACUCYFADXNYAXSTZTZBLZMZNADWSX - CHZXDJZXHTZCLZMZNXLADYEYJYDYIYCYHBYBYGXNXSYAUDUEUFUGUHYAXHYOYOKZYOJZEYOFZ - YOGZHZIZYOJZYJYJKZYJJZABCDYJDOWSDOZBWSDAUKZUIPYODOUUECYKDWSXCUUFUUEXBDOZX - CDOZUUFUUEWTDOZXADOZUUGWSDUJWSDULWTXADDUMZUNXBDUOUPUQURPYQUUEYMCYLXHUSUTP - UUBUUEUUAXCYOYTXBEYTXBXBHXBYRXBYSXBYRYKFZXBYKDOZYRUULQWSXCUUFUUIUUJUUHWSU - UFVBWSUUFVCUUIUUJTXBDUUKVAVDUQZCDYKVEVFUULWTXCFZHWTXBHZXBWSXCVGUUOXBWTXBV - HVIWTXBJUUPXBQWTXAVJWTXBVKVLRVNYSYKGZXBUUMYSUUQQUUNCDYKVMVFUUQXAXCGZHXAXB - HZXBWSXCVOUURXBXAXBVPVIXAXBJUUSXBQXAWTVQXAXBVKVLRVNVRXBVSVNVTXCYKYOXCWSVQ - XHCYKWAWBWCPUUDUUEYHBXNYAXSWEUTPXMYOQZXTYPXMYOUUTXMYOXMYOUUTWFZUVAWDUVAWG - UUTXRUUAXMYOUUTXQYTEUUTXOYRXPYSXMYOWHXMYOWIWJWKUVAWGXDYJQZXGUUCXDYJUVBXDY - JXDYJUVBWFZUVCWDUVCWGWLADYOXKYNXJYMXICXIXEXFTZXHTYMXEXFXHWMUVDYLXHUVDXFXE - TYLXEXFUDWSXCXDWPWNWOWQUFUGUHRWR $. + vex rnexg unexd resiexg mp2b unex trclexi simpr cotrintab dmex rnex unexg + resiexd mp2an dmtrcl ax-mp dmun dmresi uneq2i ssun1 ssequn1 rntrcl rnresi + mpbi eqtri rnun ssun2 uneq12i unidm reseq2i ssmin sstri eqsstri simprl id + coeq12d sseq12d dmeq uneq12d reseq2d mptrcllem df-3an bitri anbi1i bitr2i + rneq unss eqtr4i ) UAADEASZFZWSGZHZIZCSZJZWSXDJZXDXDKZXDJZUBZCLZMZNZADWSB + SZJZEXMFZXMGZHZIZXMJZXMXMKZXMJZTZTZBLZMZNZACUCYFADXNYAXSTZTZBLZMZNADWSXCH + ZXDJZXHTZCLZMZNXLADYEYJYDYIYCYHBYBYGXNXSYAUDUEUFUGUHYAXHYOYOKZYOJZEYOFZYO + GZHZIZYOJZYJYJKZYJJZABCDYJDOWSDOZBWSDAUKZUIPYODOUUECYKDWSXCUUFUUEXBDOXCDO + ZUUFUUEWTXADDWSDUJWSDULUMXBDUNUOUPUQPYQUUEYMCYLXHURUSPUUBUUEUUAXCYOYTXBEY + TXBXBHXBYRXBYSXBYRYKFZXBYKDOZYRUUHQWSXCUUFWTDOZXADOZUUGWSUUFUTWSUUFVAUUJU + UKTXBDWTXADDVBVCVDUPZCDYKVEVFUUHWTXCFZHWTXBHZXBWSXCVGUUMXBWTXBVHVIWTXBJUU + NXBQWTXAVJWTXBVKVNRVOYSYKGZXBUUIYSUUOQUULCDYKVLVFUUOXAXCGZHXAXBHZXBWSXCVP + UUPXBXAXBVMVIXAXBJUUQXBQXAWTVQXAXBVKVNRVOVRXBVSVOVTXCYKYOXCWSVQXHCYKWAWBW + CPUUDUUEYHBXNYAXSWDUSPXMYOQZXTYPXMYOUURXMYOXMYOUURWEZUUSWFUUSWGUURXRUUAXM + YOUURXQYTEUURXOYRXPYSXMYOWHXMYOWPWIWJUUSWGXDYJQZXGUUCXDYJUUTXDYJXDYJUUTWE + ZUVAWFUVAWGWKADYOXKYNXJYMXICXIXEXFTZXHTYMXEXFXHWLUVBYLXHUVBXFXETYLXEXFUDW + SXCXDWQWMWNWOUFUGUHRWR $. $} @@ -674729,9 +678363,9 @@ of all sets ( ~ inex1g ). ~ sqrtcval . (Contributed by RP, 11-May-2024.) $) reabsifneg $p |- ( A e. RR -> ( abs ` A ) = if ( A < 0 , -u A , A ) ) $= ( cr wcel cc0 clt wbr cneg cif cabs cfv wa cle wceq wi ltle mpan2 imdistani - 0re absnid eqcomd syl wn 0red id lenltd bicomd pm5.32i absid sylbi ifeqda ) - ABCZADEFZAGZAHAIJZUKULUMAUNUKULKZUNUMUOUKADLFZKUNUMMUKULUPUKDBCULUPNRADOPQA - SUATUKULUBZKZUNAURUKDALFZKUNAMUKUQUSUKUSUQUKDAUKUCUKUDUEUFUGAUHUITUJT $. + 0re absnid eqcomd syl wn 0red id lenltd bicomd absid sylbida ifeqda ) ABCZA + DEFZAGZAHAIJZUJUKULAUMUJUKKZUMULUNUJADLFZKUMULMUJUKUOUJDBCUKUONRADOPQASUATU + JUKUBZKUMAUJUPDALFZUMAMUJUQUPUJDAUJUCUJUDUEUFAUGUHTUIT $. $( Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.) $) @@ -674826,71 +678460,71 @@ of all sets ( ~ inex1g ). ( if ( ( Im ` A ) < 0 , -u 1 , 1 ) x. ( sqrt ` ( ( ( abs ` A ) - ( Re ` A ) ) / 2 ) ) ) ) ) ) $= ( cc wcel cfv caddc co c2 cdiv csqrt ci cc0 c1 cmul recnd a1i cexp wceq cle - syl eqtrd cabs cre cim clt wbr cneg cif cmin sqrtcvallem5 ax-icn neg1rr 1re - cr ifcli sqrtcvallem3 remulcld mulcld addcld id syl2anc abscl recl readdcld - binom2 rehalfcld sqsqrtd sqmuld i2 ovif neg1sqe1 ifeq12 mp2an ifid resubcld - sq1 3eqtri oveq12d mulid2d 3eqtrd mulm1d negsubd pnncand eqtr4d oveq1d 2cnd - 2timesd wne 2ne0 divsubdird divcan3d 3eqtr3d mul12d mulassd sqrtcvallem4 wb + wb eqtrd cabs cre cim clt wbr cneg cmin sqrtcvallem5 ax-icn cr neg1rr ifcli + cif 1re sqrtcvallem3 remulcld mulcld addcld id binom2 syl2anc recl readdcld + abscl rehalfcld sqsqrtd sqmuld ovif neg1sqe1 sq1 ifeq12 mp2an ifid resubcld + i2 3eqtri oveq12d mulid2d 3eqtrd mulm1d negsubd pnncand 2timesd eqtr4d 2cnd + oveq1d wne 2ne0 divsubdird divcan3d 3eqtr3d mul12d mulassd sqrtcvallem4 syl halfnneg2 mpbird crp 2rp sqrtdivd sqrtcvallem2 resqrtcld 2re 0le2 necon3bii sqrt00 mpbir divmuldivd resqcld imcl absvalsq2 mvrladdd subsq eqtr3d fveq2d absred reabsifneg sqrtmuld 3eqtr3rd remsqsqrt oveq2d renegcld ifcld divassd - ovif12 neg1mulneg1e1 adantr wn ifeqda eqtrid divcan2d add32d replim 3eqtr4d - sqcld sqrtge0d eqeq1d 3bitrd eqtrdi mpbid crred breqtrrd wi wa reim addcomd - cnsqrt00 half0 addeq0 wo olc eqcom sqeqor addid0 sqeq0 3bitr3d syl5ib ancld - sylbid w3a simp2 simp1 negidd 2cn div0i sqrt0 simp3 ltnrd eqnbrtrd iffalsed - 0red subnegd absge0 addid2d rered le0neg2d eqbrtrd 3expib syld sqrtcvallem1 - ixi eqsqrtd eqcomd ) ABCZAUADZAUBDZEFZGHFZIDZJAUCDZKUDUEZLUFZLUGZUWEUWFUHFZ - GHFZIDZMFZMFZEFZAIDUWDUWSAUWDUWIUWRUWDUWIAUIZNZUWDJUWQJBCUWDUJOZUWDUWQUWDUW - MUWPUWMUMCUWDUWKUWLLUMUKULUNOZAUOZUPZNZUQZURZUWDUSUWDUWSGPFZUWIGPFZGUWIUWRM - FZMFZEFUWRGPFZEFZAUWDUWIBCUWRBCUXIUXNQUXAUXGUWIUWRVDUTUWDUXJUXMEFZUXLEFUWFJ - UWJMFZEFUXNAUWDUXOUWFUXLUXPEUWDUXOUWHUWOUFZEFUWHUWOUHFZUWFUWDUXJUWHUXMUXQEU - WDUWHUWDUWHUWDUWGUWDUWEUWFAVAZAVBZVCZVEZNZVFUWDUXMJGPFZUWQGPFZMFUWLUWOMFUXQ - UWDJUWQUXBUXFVGUWDUYDUWLUYEUWOMUYDUWLQUWDVHOUWDUYEUWMGPFZUWPGPFZMFLUWOMFUWO - UWDUWMUWPUWDUWMUXCNZUWDUWPUXDNZVGUWDUYFLUYGUWOMUYFLQUWDUYFUWKUWLGPFZLGPFZUG - 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The right side is slightly more compact than @@ -675128,17 +678762,15 @@ of all sets ( ~ inex1g ). ss2iundf $p |- ( ph -> U_ x e. A B C_ U_ y e. C D ) $= ( vz wss wrex wral ciun cv wcel wa wn wi wal df-ral wceq nfcri nfan simpr eleq1d biimprd wb w3a sseq2d 3expa notbid biimpd imim12d alrimi nfel nfss - ex nfn nfim spcimgft sylc mpid syl5bi con2d dfrex2 syl6ibr ralrimi reximi - mpd ssel r19.37 syl eliun ssrdv ralimi df-iun sseq1i dfss2 ralbii ralcom4 - cab abss nfiun r19.23 albii 3bitrri 3bitri sylibr ) AEGUCZCFUDZBDUEZBDEUF - ZCFGUFZUCZAXCBDJABUGDUHZXCAXHUIZEHUCZXCUAXIXJXBUJZCFUEZUJXCXIXLXJXLCUGZFU - HZXKUKZCULZXIXJUJZXKCFUMXIXPIFUHZXQSXIXMIUNZXOXRXQUKZUKZUKZCULXRXPXTUKXIY - BCAXHCKCBDMUOUPXIXSYAXIXSUIZXRXNXKXQYCXNXRYCXMIFXIXSUQURUSYCXKXQYCXBXJAXH - XSXBXJUTAXHXSVAGHETVBVCVDVEVFVJVGSXOXTCIFXRXQCCIFLPVHXJCCEHNRVIVKVLLVMVNV - OVPVQXBCFVRVSWBVJVTXDEXFUCZBDUEZXGXCYDBDXCUBEXFXCUBUGZEUHZYFGUHZCFUDZYFXF - UHZXCYGYHUKZCFUDYGYIUKXBYKCFEGYFWCWAYGYHCFCUBENUOWDWECYFFGWFVSWGWHXGYGBDU - DZUBWNZXFUCYLYJUKZUBULZYEXEYMXFBUBDEWIWJYLUBXFWOYEYGYJUKZUBULZBDUEYPBDUEZ - UBULYOYDYQBDUBEXFWKWLYPBUBDWMYRYNUBYGYJBDBUBXFCBFGOQWPUOWQWRWSWTXAWE $. + ex nfn nfim spcimgft sylc mpid biimtrid con2d dfrex2 syl6ibr mpd ralrimia + ssel reximi r19.37 syl eliun ssrdv ralimi nfiun iunssf sylibr ) AEGUCZCFU + DZBDUEZBDEUFCFGUFZUCZAWMBDJABUGDUHZUIZEHUCZWMUAWRWSWLUJZCFUEZUJWMWRXAWSXA + CUGZFUHZWTUKZCULZWRWSUJZWTCFUMWRXEIFUHZXFSWRXBIUNZXDXGXFUKZUKZUKZCULXGXEX + IUKWRXKCAWQCKCBDMUOUPWRXHXJWRXHUIZXGXCWTXFXLXCXGXLXBIFWRXHUQURUSXLWTXFXLW + LWSAWQXHWLWSUTAWQXHVAGHETVBVCVDVEVFVJVGSXDXICIFXGXFCCIFLPVHWSCCEHNRVIVKVL + LVMVNVOVPVQWLCFVRVSVTWAWNEWOUCZBDUEWPWMXMBDWMUBEWOWMUBUGZEUHZXNGUHZCFUDZX + NWOUHWMXOXPUKZCFUDXOXQUKWLXRCFEGXNWBWCXOXPCFCUBENUOWDWECXNFGWFVSWGWHBDEWO + CBFGOQWIWJWKWE $. $} ${ @@ -675390,10 +679022,9 @@ Transitive relations (not to be confused with transitive classes). relexpnul $p |- ( ( ( R e. V /\ Rel R ) /\ ( N e. NN0 /\ M e. NN0 ) ) -> ( ( dom ( R ^r N ) i^i ran ( R ^r M ) ) = (/) <-> ( R ^r ( N + M ) ) = (/) ) ) $= - ( crelexp co cdm crn cin c0 wceq ccom wcel wrel wa cn0 caddc coeq0 c1 wi - simprl simprr simpll simplr a1d relexpaddg syl13anc eqeq1d bitr3id ) ACEFZG - ABEFZHIJKUJUKLZJKADMZANZOZCPMZBPMZOZOZACBQFZEFZJKUJUKRUSULVAJUSUPUQUMUTSKZU - NTULVAKUOUPUQUAUOUPUQUBUMUNURUCUSUNVBUMUNURUDUEABCDUFUGUHUI $. + ( crelexp co cdm crn cin c0 wceq ccom wcel wa cn0 caddc coeq0 simplr simprl + wrel simprr relexpaddd eqeq1d bitr3id ) ACEFZGABEFZHIJKUEUFLZJKADMZATZNZCOM + ZBOMZNZNZACBPFEFZJKUEUFQUNUGUOJUNABCUHUIUMRUJUKULSUJUKULUAUBUCUD $. ${ $d n r C N .^ $. @@ -675768,48 +679399,47 @@ over the natural numbers (including zero) is equivalent to the sylib oveq2 dmeqd iunxsn cid cres relexp0g ad2antll dmresi rnresi uneq12d rneqd unidm uneq1d relexpdmg expcom ralrimiv olc ad2antrl inss syl imim1d sseld ralimdv2 mpd iunss sylibr relexprng unssd ssequn2 1ex relexp1g jaoi - ex uneq2d 3impib 3com13 adantr ssel adantl 3adant3 eqtrid wb nn0ex inex1g - ssex eqeltrrid unexg syl2anc mpancom ovex rgenw iunexg 3ad2ant2 relexp0eq - sylancl simp1 mpbid ) BCEZDFGZHIUAZDJZKUBZUCZADBAUDZLUNZMZHLUNZAHUEZDJZIU - EZDJZNZDNZUUPMZHLUNZBHLUNZDUVDOZUURUVFOUVDUULDNZDUVCUULDUULUUSUVANZDJUVCU - UKUVJDHIUFZUGUUSUVADUHUIUJUULDGUVIDOUUKDUKUULDULUMUIUVHUUQUVEHLADUVDUUPUO - UPUQUUNUVEURZUVEUSZNZBURZBUSZNZOZUVFUVGOZUUNUVNAUUTUUPURZMZAUUTUUPUSZMZNZ - AUVBUVTMZAUVBUWBMZNZNZADUVTMZADUWBMZNZNZUVQUVNUWIUWEUWANZNZUWCUWFNZUWJNZN - ZUWMUWONZUWKNZUWLUVLUWNUVMUWPUVLAUVDUVTMAUVCUVTMZUWINZUWNAUVDUUPUTAUVCDUV - TVAUXAUWMUWIUWTUWMUWIUWTUWAUWEAUUTUVBUVTVAVBUJVBVCUVMAUVDUWBMAUVCUWBMZUWJ - NUWPAUVDUUPVDAUVCDUWBVAUXBUWOUWJAUUTUVBUWBVAUJVCVEUWQUWMUWINZUWPNUWSUWNUX - CUWPUWIUWMVFUJUWMUWIUWOUWJVGVHUWRUWHUWKUWRUWAUWENZUWONUWHUWMUXDUWOUWEUWAV - FUJUWAUWEUWCUWFVGVHUJVCUUNUWLUVQUWKNZUVQUUNUWHUVQUWKUUMUUJUUIUWHUVQOZUUMU - UJUUIUXFUUMHDEZIDEZVIZUUJUUIVJZUXFVKZUUMUULKOZTZUXIUULKVLUXMDUUSJZKOZTZDU - VAJZKOZTZVIZTZTUXTUXIUXLUYAUXLUXNUXQNZKOUXOUXRVJUYAUULUYBKUULDUUKJDUVJJUY - BUUKDVMUUKUVJDUVKVNDUUSUVAVOVCVPUXNUXQVQUXOUXRVRVSVTUXTWAUXPUXGUXSUXHUXPU - 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sylbi crn cun cres simpr1 eqtrd oveq1d dmexg rnexg unexg syl2anc relexpiidm - relexp0g simpr2 nn0cnd mul02d eqtr2d jaod syl5bi impcom ) CFGZDFGZHAEGZBCDI - JZKZBLKZCDMUBZNZUCZACOJZDOJZABOJZKZWHWGWOWSNZWGCUDGZCLKZUEWHWTCUFWHXAWTXBWH - DUDGZDLKZUEXAWTNZDUFXCXEXDXAXCWTWOXAXCHZWSWIWKXFWSWNABCDEUGUHUIUIXDXAWTXDXA - HZWIWKWNWSXGWIWKWNWSNXGWIWKHZHZWNWSXIWLWMXIBWJCLIJLXGWIWKUJXIDLCIXDXAXHUKPX - ICXICXDXAXHULUNUMQXIXGWMUOXGXHUPXGWMCLMUBZXAXJUOXDCUQURXGDLCMXDXAUPUSUTRUAV - AVBVCSVDVGWHXBWTWHXBHZWOWSXKWOHZWQVEAVFZAVHZVIZVJZDOJZXPWRXLWPXPDOXLWPALOJZ - XPXLCLAOWHXBWOULZPXLWIXRXPKXKWIWKWNVKZAEVSRZVLVMXLXOTGZWHXQXPKXLWIYBXTWIXMT - GXNTGYBAEVNAEVOXMXNTTVPVQRWHXBWOUKZXODTVRVQXLWRXRXPXLBLAOXLBWJLDIJLXKWIWKWN - VTXLCLDIXSVMXLDXLDYCWAWBQPYAWCQSSWDWEWFWF $. + sylbi crn cun cres simpr1 eqtrd oveq1d dmexg rnexg unexd relexpiidm syl2anc + relexp0g simpr2 nn0cnd mul02d eqtr2d jaod biimtrid impcom ) CFGZDFGZHAEGZBC + DIJZKZBLKZCDMUBZNZUCZACOJZDOJZABOJZKZWHWGWOWSNZWGCUDGZCLKZUEWHWTCUFWHXAWTXB + WHDUDGZDLKZUEXAWTNZDUFXCXEXDXAXCWTWOXAXCHZWSWIWKXFWSWNABCDEUGUHUIUIXDXAWTXD + XAHZWIWKWNWSXGWIWKWNWSNXGWIWKHZHZWNWSXIWLWMXIBWJCLIJLXGWIWKUJXIDLCIXDXAXHUK + PXICXICXDXAXHULUNUMQXIXGWMUOXGXHUPXGWMCLMUBZXAXJUOXDCUQURXGDLCMXDXAUPUSUTRU + AVAVBVCSVDVGWHXBWTWHXBHZWOWSXKWOHZWQVEAVFZAVHZVIZVJZDOJZXPWRXLWPXPDOXLWPALO + JZXPXLCLAOWHXBWOULZPXLWIXRXPKXKWIWKWNVKZAEVSRZVLVMXLXOTGZWHXQXPKXLWIYBXTWIX + MXNTTAEVNAEVOVPRWHXBWOUKZXODTVQVRXLWRXRXPXLBLAOXLBWJLDIJLXKWIWKWNVTXLCLDIXS + VMXLDXLDYCWAWBQPYAWCQSSWDWEWFWF $. ${ $d D j x y $. $d D k x y $. $d D l $. $d D m $. $d N k x $. $d j l $. @@ -676072,22 +679702,22 @@ over the natural numbers (including zero) is equivalent to the -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) $= ( cc0 c1 wcel clt wbr wceq crelexp w3a simp1 oveq2d eqtrd simp2 oveq12d cvv co cpr wa cif wo wi elpri cid cdm crn cun dmresi rnresi uneq12i unidm eqtri - cres reseq2i simp3l relexp0g syl dmexg rnexg unexg syl2anc resiexd 3syl 0re - simp3r breq12d mtbiri iffalsed 3eqtrd 3eqtr4a 3exp relexp1g wn 0lt1 ltnsymi - ltnri 1re mp1i mtbird eqtr4d jaoi ovex mpbiri iftrued 3eqtr4d syl2an impcom - jaod imp ) CFGUAZHZDWMHZUBAEHZBCDIJZCDUCZKZUBZACLTZDLTZABLTZKZWNCFKZCGKZUDZ - DFKZDGKZUDZWTXDUEZWOCFGUFDFGUFXGXJXKXGXHXKXIXEXHXKUEXFXEXHWTXDXEXHWTMZUGUGA - UHZAUIZUJZUPZUHZXPUIZUJZUPZXPXBXCXSXOUGXSXOXOUJXOXQXOXRXOXOUKXOULUMXOUNUOUQ - XLXBXPFLTZXTXLXAXPDFLXLXAAFLTZXPXLCFALXEXHWTNZOXLWPYBXPKXEXHWPWSURZAEUSUTZP - XEXHWTQZRXLWPXPSHYAXTKYDWPXOSWPXMSHXNSHXOSHAEVAAEVBXMXNSSVCVDVEXPSUSVFPXLXC - YBXPXLBFALXLBWRDFXEXHWPWSVHXLWQCDXLWQFFIJFVGVSXLCFDFIYCYFVIVJVKYFVLOYEPVMVN - XFXHWTXDXFXHWTMZXBYBXCYGXAADFLYGXAAGLTZAYGCGALXFXHWTNZOYGWPYHAKZXFXHWPWSURA - EVOZUTPXFXHWTQZRYGBFALYGBWRDFXFXHWPWSVHYGWQCDYGWQGFIJZFGIJZYMVPYGVQFGVGVTVR - WAYGCGDFIYIYLVIWBVKYLVLOWCVNWDXEXIXKUEXFXEXIWTXDXEXIWTMZYBGLTZYBXBXCYBSHYPY - BKYOAFLWEYBSVOWAYOXAYBDGLYOCFALXEXIWTNZOXEXIWTQZRYOBFALYOBWRCFXEXIWPWSVHYOW - QCDYOWQYNVQYOCFDGIYQYRVIWFWGYQVLOWHVNXFXIWTXDXFXIWTMZXBYHXCYSXAADGLYSXAYHAY - SCGALXFXIWTNZOYSWPYJXFXIWPWSURYKUTPXFXIWTQZRYSBGALYSBWRDGXFXIWPWSVHYSWQCDYS - WQGGIJGVTVSYSCGDGIYTUUAVIVJVKUUAVLOWCVNWDWKWLWIWJ $. + cres reseq2i simp3l relexp0g syl dmexg rnexg unexd resiexd simp3r 0re ltnri + 3syl breq12d mtbiri iffalsed 3eqtrd 3eqtr4a 3exp relexp1d 0lt1 ltnsymi mp1i + 1re mtbird eqtr4d jaoi ovex relexp1g mpbiri iftrued 3eqtr4d jaod imp syl2an + wn impcom ) CFGUAZHZDWMHZUBAEHZBCDIJZCDUCZKZUBZACLTZDLTZABLTZKZWNCFKZCGKZUD + ZDFKZDGKZUDZWTXDUEZWOCFGUFDFGUFXGXJXKXGXHXKXIXEXHXKUEXFXEXHWTXDXEXHWTMZUGUG + AUHZAUIZUJZUPZUHZXPUIZUJZUPZXPXBXCXSXOUGXSXOXOUJXOXQXOXRXOXOUKXOULUMXOUNUOU + QXLXBXPFLTZXTXLXAXPDFLXLXAAFLTZXPXLCFALXEXHWTNZOXLWPYBXPKXEXHWPWSURZAEUSUTZ + PXEXHWTQZRXLWPXPSHYAXTKYDWPXOSWPXMXNSSAEVAAEVBVCVDXPSUSVHPXLXCYBXPXLBFALXLB + WRDFXEXHWPWSVEXLWQCDXLWQFFIJFVFVGXLCFDFIYCYFVIVJVKYFVLOYEPVMVNXFXHWTXDXFXHW + TMZXBYBXCYGXAADFLYGXAAGLTZAYGCGALXFXHWTNZOYGAEXFXHWPWSURVOPXFXHWTQZRYGBFALY + GBWRDFXFXHWPWSVEYGWQCDYGWQGFIJZFGIJZYKWKYGVPFGVFVSVQVRYGCGDFIYIYJVIVTVKYJVL + OWAVNWBXEXIXKUEXFXEXIWTXDXEXIWTMZYBGLTZYBXBXCYBSHYNYBKYMAFLWCYBSWDVRYMXAYBD + GLYMCFALXEXIWTNZOXEXIWTQZRYMBFALYMBWRCFXEXIWPWSVEYMWQCDYMWQYLVPYMCFDGIYOYPV + IWEWFYOVLOWGVNXFXIWTXDXFXIWTMZXBYHXCYQXAADGLYQXAYHAYQCGALXFXIWTNZOYQAEXFXIW + PWSURVOPXFXIWTQZRYQBGALYQBWRDGXFXIWPWSVEYQWQCDYQWQGGIJGVSVGYQCGDGIYRYSVIVJV + KYSVLOWAVNWBWHWIWJWL $. $( Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.) $) @@ -676105,7 +679735,7 @@ over the natural numbers (including zero) is equivalent to the ${ $d x y A $. $d x N $. $d x y V $. - $( Absorbtion law for zeroth power of a relation. (Contributed by RP, + $( Absorption law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.) $) relexp0a $p |- ( ( A e. V /\ N e. NN0 ) -> ( ( A ^r N ) ^r 0 ) C_ ( A ^r 0 ) ) $= @@ -676140,25 +679770,25 @@ over the natural numbers (including zero) is equivalent to the orim12i 3ad2antl2 oveq1d eqtrd 3eqtr4d 3exp1 eqcomd oveq12d jaoi 3syl com13 imp simp3 simp2 simp1 nngt0d eqbrtrd iftrued 3eqtrd cid cdm crn cres simpr1 cun simpr2 xpexd dmexg rnexg jca unexg nnnn0d relexpiidm syl2anc simpl2 syl - relexp0g simpl3 ex syld3an3 3exp jaod syl5bi 3ad2ant2 nn0nlt0 breq2d mtbird - wn iffalsed relexp0idm syl3c sylbi 3imp31 impcom ) DEFHUAZEFUBZIZEUEJZFUEJZ - KACJZBGJZABUFUCUGZKZABUHZELMZFLMZXTDLMZIZXOXNXMXSYDNZXOFUDJZFOIZPXNXMYENZNZ - FUIYFYIYGXNEUDJZEOIZPZYFYHEUIZYFYJYHYKXMYJYFYEXMXMXLEIZQZXMXLFIZQZPZDEIZDFI - ZPYJYFYENNZXMYNYPPZYRXKEFUJXMUUBQYRXMYNYPUKULUMYOYSYQYTDXLEUNDXLFUNUQYSUUAY - TYSYJYFXSYDYSYJYFKXSQZXTFLMZXTYBYCYFYSXSUUDXTIZYJYFXSUUEABCFGUOVHUPUUCYAXTF - LYJYSXSYAXTIZYFYJXSUUFABCEGUOZVHZURZUSUUCYCYAXTUUCDEXTLYSYJYFXSRSUUIUTVAVBY - TYJYFXSYDYTYJYFKXSQZYAXTFDLYJYTXSUUFYFUUHURUUJDFYTYJYFXSRVCVDVBVEVFVGYFYKXM - YEYFYKXMDOIZYEYFYKXMKZDXLEOYFYKXMVIUULXKEFUULEOFHYFYKXMVJZUULFYFYKXMVKVLVMV - NUUMVOYFYKUUKKZXSYDUUNXSQZVPXTVQZXTVRZWAZVSZFLMZUUSYBYCUUOUURTJZXOUUTUUSIUU - OXTTJZUUPTJZUUQTJZQUVAUUOABCGUUNXPXQXRVTUUNXPXQXRWBWCZUVBUVCUVDXTTWDXTTWEWF - UUPUUQTTWGVFUUOFYFYKUUKXSRWHUURFTWIWJUUOYAUUSFLUUOYAXTOLMZUUSUUOEOXTLYFYKUU - KXSWKSUUOUVBUVFUUSIUVEXTTWMWLZUTUSUUOYCUVFUUSUUODOXTLYFYKUUKXSWNSUVGUTVAWOW - PWQWRWSYGXNXMYEYGXNXMKZYGYLUUKYEYGXNXMVKZXNYGYLXMXNYLYMULWTUVHDXLFOYGXNXMVI - UVHXKEFUVHXKEOHUAZXNYGUVJXDXMEXAWTUVHFOEHUVIXBXCXEUVIVOYGYJUUKYENYKYGYJUUKX - SYDYGYJUUKKZXSQZYAOLMUVFYBYCUVLYAXTOLUVKXSUUFYJYGXSUUFNUUKUUGWTVHUSUVLFOYAL - YGYJUUKXSRSUVLDOXTLYGYJUUKXSWNSVAVBYGYKUUKXSYDYGYKUUKKZXSQZUVFOLMZUVFYBYCUV - NUVBUVOUVFIUVNABCGUVMXPXQXRVTUVMXPXQXRWBWCXTTXFWLUVNYAUVFFOLUVNEOXTLYGYKUUK - XSWKSYGYKUUKXSRVDUVNDOXTLYGYKUUKXSWNSVAVBWRXGWQVEXHXIXJ $. + relexp0g simpl3 ex syld3an3 3exp biimtrid 3ad2ant2 wn nn0nlt0 breq2d mtbird + jaod iffalsed relexp0idm syl3c sylbi 3imp31 impcom ) DEFHUAZEFUBZIZEUEJZFUE + JZKACJZBGJZABUFUCUGZKZABUHZELMZFLMZXTDLMZIZXOXNXMXSYDNZXOFUDJZFOIZPXNXMYENZ + NZFUIYFYIYGXNEUDJZEOIZPZYFYHEUIZYFYJYHYKXMYJYFYEXMXMXLEIZQZXMXLFIZQZPZDEIZD + FIZPYJYFYENNZXMYNYPPZYRXKEFUJXMUUBQYRXMYNYPUKULUMYOYSYQYTDXLEUNDXLFUNUQYSUU + AYTYSYJYFXSYDYSYJYFKXSQZXTFLMZXTYBYCYFYSXSUUDXTIZYJYFXSUUEABCFGUOVHUPUUCYAX + TFLYJYSXSYAXTIZYFYJXSUUFABCEGUOZVHZURZUSUUCYCYAXTUUCDEXTLYSYJYFXSRSUUIUTVAV + BYTYJYFXSYDYTYJYFKXSQZYAXTFDLYJYTXSUUFYFUUHURUUJDFYTYJYFXSRVCVDVBVEVFVGYFYK + XMYEYFYKXMDOIZYEYFYKXMKZDXLEOYFYKXMVIUULXKEFUULEOFHYFYKXMVJZUULFYFYKXMVKVLV + MVNUUMVOYFYKUUKKZXSYDUUNXSQZVPXTVQZXTVRZWAZVSZFLMZUUSYBYCUUOUURTJZXOUUTUUSI + UUOXTTJZUUPTJZUUQTJZQUVAUUOABCGUUNXPXQXRVTUUNXPXQXRWBWCZUVBUVCUVDXTTWDXTTWE + WFUUPUUQTTWGVFUUOFYFYKUUKXSRWHUURFTWIWJUUOYAUUSFLUUOYAXTOLMZUUSUUOEOXTLYFYK + UUKXSWKSUUOUVBUVFUUSIUVEXTTWMWLZUTUSUUOYCUVFUUSUUODOXTLYFYKUUKXSWNSUVGUTVAW + OWPWQXDWRYGXNXMYEYGXNXMKZYGYLUUKYEYGXNXMVKZXNYGYLXMXNYLYMULWSUVHDXLFOYGXNXM + VIUVHXKEFUVHXKEOHUAZXNYGUVJWTXMEXAWSUVHFOEHUVIXBXCXEUVIVOYGYJUUKYENYKYGYJUU + KXSYDYGYJUUKKZXSQZYAOLMUVFYBYCUVLYAXTOLUVKXSUUFYJYGXSUUFNUUKUUGWSVHUSUVLFOY + ALYGYJUUKXSRSUVLDOXTLYGYJUUKXSWNSVAVBYGYKUUKXSYDYGYKUUKKZXSQZUVFOLMZUVFYBYC + UVNUVBUVOUVFIUVNABCGUVMXPXQXRVTUVMXPXQXRWBWCXTTXFWLUVNYAUVFFOLUVNEOXTLYGYKU + UKXSWKSYGYKUUKXSRVDUVNDOXTLYGYKUUKXSWNSVAVBXDXGWQVEXHXIXJ $. $( The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where ` R ` is a @@ -676172,37 +679802,37 @@ over the natural numbers (including zero) is equivalent to the c2 cid cdm crn wrel relco dfrel2 ax-mp cnvco cnvresid coeq2i coires1 3eqtri cun cnvss resss sstri eqsstrri cnvcnvss a1i simp1 relexp0g relexp1g coeq12d simp2 oveq12d addid2d 3sstr4d cnvexg relexpuzrel syl2anc eluz2nn relexpnndm - 1cnd cvv df-rn ssun2 sstrdi relssres eqtrid eluzge2nn0 relexpcnv 3eqtr4d wb - simp3 3adant1 cnveqb sylancr mpbird coeq1d oveq1d eluzelcn jaod syl5bi jaoi - cc eqsstri addid1d 3adant2 ssun1 coeq2d cin inidm reseq2i 00id 3eqtr4a 3imp - resres ) CUAEZBUAEZADEZACFGZABFGZHZACBIGZFGZJZYBBUBEZBKLZUCYAYCYIUDZBUEYAYJ - YLYKYACUBEZCKLZUCZYJYLUDZYAYOCUEUFZYMYPYNYMYJYCYIYMYJYCMYFYHLZYIABCDUGYFYHU - HZNUIYJBSLZBUMUJUKZEZUCYNYLBULYNYTYLUUBYNYTYCYIYNYTYCMZUNAUOZAUPZVFZOZAHZAY - FYHUUHAJUUCUUHAPZPZAUUHUUHPZPZUUJUUHUQZUULUUHLZUUGAURUUMUUNUUHUSUFUTUULUUIU - UFOZPZUUJUUKUUOJZUULUUPJUUKUUOLUUQUUKUUIUUGPZHUUIUUGHUUOUUGAVAUURUUGUUIUUFV - BZVCUUIUUFVDVEUUKUUOUHUTUUKUUOVGUTUUOUUIJUUPUUJJUUIUUFVHUUOUUIVGUTVIVJAVKVI - 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syl wral vex elab eqid iunrelexpmin1 mpan2 19.21bi syl5bi ralrimiv sylibr - ssint eqssd ovex iunex fvmpt eqtrd mpteq2ia eqtri ) UABHBIZCIZJWOWOKWOJLZ - CUBZUCZMBHANWNAIZOPZQZMBCUDBHWRXAWNHRZWRWNDHANDIZWSOPZQZMZUEZXAXBWRXGXBXG - WQRZWRXGJXBXHWNXGJZXGXGKXGJZXBWNWNSOPZXGWNHUFXBNHRSNRXKXGJUMUGXFWNHEOSNHF - DFHXEENFIZEIZOPZQZDFUNZXEANXLWSOPZQXOXPANXDXQXCXLWSOUHUIAENXQXNWSXMXLOUJU - KULUOZUPUQURXBNSUSUETSVCRXJUTVAXFWNESNHFXRVBUQXBXGHRWOXGTWPXIXJLZVDVEZCVF - XHXSVDWNXFVGXBXTCXTXBWOXGWNVHVIVJWPXSCXGHVKVLVMXGWQVNVOXBXGGIZJZGWQVPXGWR - JXBYBGWQYAWQRWNYAJYAYAKYAJLZXBYBWPYCCYAGVQWOYAWNVHVRXBYCYBVEZGXBNNTYDGVFN - VSXFWNENHGFXRVTWAWBWCWDGXGWQWFWEWGDWNXEXAHXFDBUNANXDWTXCWNWSOUHUIXFVSANWT - UMWNWSOWHWIWJWKWLWM $. + syl wral vex elab eqid iunrelexpmin1 mpan2 biimtrid ralrimiv ssint sylibr + 19.21bi eqssd ovex iunex fvmpt eqtrd mpteq2ia eqtri ) UABHBIZCIZJWOWOKWOJ + LZCUBZUCZMBHANWNAIZOPZQZMBCUDBHWRXAWNHRZWRWNDHANDIZWSOPZQZMZUEZXAXBWRXGXB + XGWQRZWRXGJXBXHWNXGJZXGXGKXGJZXBWNWNSOPZXGWNHUFXBNHRSNRXKXGJUMUGXFWNHEOSN + HFDFHXEENFIZEIZOPZQZDFUNZXEANXLWSOPZQXOXPANXDXQXCXLWSOUHUIAENXQXNWSXMXLOU + JUKULUOZUPUQURXBNSUSUETSVCRXJUTVAXFWNESNHFXRVBUQXBXGHRWOXGTWPXIXJLZVDVEZC + VFXHXSVDWNXFVGXBXTCXTXBWOXGWNVHVIVJWPXSCXGHVKVLVMXGWQVNVOXBXGGIZJZGWQVPXG + WRJXBYBGWQYAWQRWNYAJYAYAKYAJLZXBYBWPYCCYAGVQWOYAWNVHVRXBYCYBVEZGXBNNTYDGV + FNVSXFWNENHGFXRVTWAWFWBWCGXGWQWDWEWGDWNXEXAHXFDBUNANXDWTXCWNWSOUHUIXFVSAN + WTUMWNWSOWHWIWJWKWLWM $. $} ${ @@ -676494,19 +680124,19 @@ over the natural numbers (including zero) is equivalent to the cfv weq iuneq2d oveq2 cbviunv eqtrdi cbvmptv ov2ssiunov2 mp3an23 eqsstrrd c1 relexp1g 1nn0 cuz wceq nn0uz iunrelexpuztr wb wal fvex coeq12d sseq12d wi id 3anbi123d a1i alrimiv elabgt sylancr mpbir3and intss1 syl wral elab - eqid iunrelexpmin2 mpan2 19.21bi syl5bi ralrimiv ssint sylibr eqssd iunex - vex ovex fvmpt eqtrd mpteq2ia eqtri ) UABHUBBIZUCXDUDUEUFZCIZJZXDXFJZXFXF - KZXFJZLZCUGZUHZMBHANXDAIZOTZPZMBCUIBHXMXPXDHQZXMXDDHANDIZXNOTZPZMZUNZXPXQ - XMYBXQYBXLQZXMYBJXQYCXEYBJZXDYBJZYBYBKZYBJZXQXEXDROTZYBXDHUJXQNHQZRNQZYHY - BJUKULYAXDHEORNHFDFHXTENFIZEIZOTZPZDFUOZXTANYKXNOTZPYNYOANXSYPXRYKXNOUMUP - AENYPYMXNYLYKOUQURUSUTZVAVBVCXQXDXDVDOTZYBXDHVEXQYIVDNQYRYBJUKVFYAXDHEOVD - NHFYQVAVBVCXQNRVGUNVHYJYGVIULYAXDERNHFYQVJVBXQYBHQXFYBVHZXKYDYEYGLZVKVPZC - VLYCYTVKXDYAVMXQUUACUUAXQYSXGYDXHYEXJYGXFYBXESXFYBXDSYSXIYFXFYBYSXFYBXFYB - YSVQZUUBVNUUBVOVRVSVTXKYTCYBHWAWBWCYBXLWDWEXQYBGIZJZGXLWFYBXMJXQUUDGXLUUC - XLQXEUUCJZXDUUCJZUUCUUCKZUUCJZLZXQUUDXKUUICUUCGWRCGUOZXGUUEXHUUFXJUUHXFUU - CXESXFUUCXDSUUJXIUUGXFUUCUUJXFUUCXFUUCUUJVQZUUKVNUUKVOVRWGXQUUIUUDVPZGXQN - NVHUULGVLNWHYAXDENHGFYQWIWJWKWLWMGYBXLWNWOWPDXDXTXPHYADBUOANXSXOXRXDXNOUM - UPYAWHANXOUKXDXNOWSWQWTXAXBXC $. + vex eqid iunrelexpmin2 mpan2 19.21bi biimtrid ralrimiv ssint sylibr eqssd + ovex iunex fvmpt eqtrd mpteq2ia eqtri ) UABHUBBIZUCXDUDUEUFZCIZJZXDXFJZXF + XFKZXFJZLZCUGZUHZMBHANXDAIZOTZPZMBCUIBHXMXPXDHQZXMXDDHANDIZXNOTZPZMZUNZXP + XQXMYBXQYBXLQZXMYBJXQYCXEYBJZXDYBJZYBYBKZYBJZXQXEXDROTZYBXDHUJXQNHQZRNQZY + HYBJUKULYAXDHEORNHFDFHXTENFIZEIZOTZPZDFUOZXTANYKXNOTZPYNYOANXSYPXRYKXNOUM + UPAENYPYMXNYLYKOUQURUSUTZVAVBVCXQXDXDVDOTZYBXDHVEXQYIVDNQYRYBJUKVFYAXDHEO + VDNHFYQVAVBVCXQNRVGUNVHYJYGVIULYAXDERNHFYQVJVBXQYBHQXFYBVHZXKYDYEYGLZVKVP + ZCVLYCYTVKXDYAVMXQUUACUUAXQYSXGYDXHYEXJYGXFYBXESXFYBXDSYSXIYFXFYBYSXFYBXF + YBYSVQZUUBVNUUBVOVRVSVTXKYTCYBHWAWBWCYBXLWDWEXQYBGIZJZGXLWFYBXMJXQUUDGXLU + UCXLQXEUUCJZXDUUCJZUUCUUCKZUUCJZLZXQUUDXKUUICUUCGWHCGUOZXGUUEXHUUFXJUUHXF + UUCXESXFUUCXDSUUJXIUUGXFUUCUUJXFUUCXFUUCUUJVQZUUKVNUUKVOVRWGXQUUIUUDVPZGX + QNNVHUULGVLNWIYAXDENHGFYQWJWKWLWMWNGYBXLWOWPWQDXDXTXPHYADBUOANXSXOXRXDXNO + UMUPYAWIANXOUKXDXNOWRWSWTXAXBXC $. $} ${ @@ -677727,12 +681357,12 @@ adopting f( ` ph ` ) as ` if- ( ph , ps , ch ) ` would result in the frege33 $p |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) $= ( wn wi ax-frege28 frege32 ax-mp ) ACZBDZBCZHCDDIJADDHBEABFG $. - $( If as a conseqence of the occurrence of the circumstance ` ph ` , when the - obstacle ` ps ` is removed, ` ch ` takes place, then from the circumstance - that ` ch ` does not take place while ` ph ` occurs the occurrence of the - obstacle ` ps ` can be inferred. Closed form of ~ con1d . Proposition 34 - of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) - (Proof modification is discouraged.) $) + $( If as a consequence of the occurrence of the circumstance ` ph ` , when + the obstacle ` ps ` is removed, ` ch ` takes place, then from the + circumstance that ` ch ` does not take place while ` ph ` occurs the + occurrence of the obstacle ` ps ` can be inferred. Closed form of + ~ con1d . Proposition 34 of [Frege1879] p. 45. (Contributed by RP, + 24-Dec-2019.) (Proof modification is discouraged.) $) frege34 $p |- ( ( ph -> ( -. ps -> ch ) ) -> ( ph -> ( -. ch -> ps ) ) ) $= ( wn wi frege33 frege5 ax-mp ) BDCEZCDBEZEAIEAJEEBCFIJAGH $. @@ -677967,7 +681597,7 @@ the concept (illegal in our notation ) ` ( ph `` ps ) ` with ( wb wi wif frege55cor1a frege9 ax-mp ) BAEZABEZFLACDGBCDGFZFKMFFBAHKLMIJ $. - $( This is the all imporant formula which allows us to apply Frege-style + $( This is the all imporant formula which allows to apply Frege-style definitions and explore their consequences. A closed form of ~ biimpri . Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) $) @@ -679257,7 +682887,7 @@ _Begriffsschrift_ Chapter II with equivalence of classes (where they are -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- ` Fun ``' ``' R ` means the relationship content of procedure ` R ` is - single-valued. The double converse allows us to simply apply this syntax in + single-valued. The double converse allows to simply apply this syntax in place of Frege's even though the original never explicitly limited discussion of propositional statements which vary on two variables to relations. @@ -681001,34 +684631,34 @@ by analogy M(x) would be an 'adherent system'. BJ suggested ( K ` ( s u. t ) ) C_ ( ( K ` s ) u. ( K ` t ) ) $= ( vx cv cun wcel wa c0 wceq cif wo wi ex a1i adantrd adantl id cfv elif wss c3o cpw csn c1o cpr wn wel uneq12 unidm eqtrdi an3 orcd pm2.24 impd - jaao mpdan uneqsn df-3or bitri pm2.21 jaod adantr adantld syl5bi biimpi - w3o elun andi simpl anim1i simpr orim12i sylbi sylan2 olcd sylib expcom - or4 orc snsspr1 eqsstrdi sseld impcom jca olc jaoa anc2li orim12d com12 - jaoi or42 syl6ib 4exmid mpjaod orbi12i sylbbr syl6 ssrdv eqeq1 ifbieq2d - pwuncl prex vex unex ifex fvmpt syl weq uneq12d 3sstr4d rgen2 ) CGZAGZH - ZBUAZXOBUAZXPBUAZHZUCCAUDUEZYBXOYBIZXPYBIZJZXQKUFZLZKUGUHZXQMZXOYFLZYHX - OMZXPYFLZYHXPMZHZXRYAYEFYIYNYEFGZYIIZYJYOYHIZJZYJUIZFCUJZJZNZYLYQJZYLUI - ZFAUJZJZNZNZYOYNIZYPYGYQJZYGUIZYOXQIZJZNZYEUUHYGYOYHXQUBYEYJYLJZYSUUDJZ - NZUUNUUHOZYJUUDJZYLYSJZNZYEUUOUURUUPUUOUUROYEUUOYGUURUUOXQYFYFHYFXOYFXP - YFUKYFULUMUUOUUJUUHYGUUMUUOUUJUUHUUOUUJJZUUBUUGUVBYRUUAYJYLYGYQUNUOUOPY - GUUKUULUUHYGUULUUHOUPUQURUSQUUPUUROYEUUPUUJUUHUUMUUPYGYQUUHYGUUOYJXPKLZ - JZNZXOKLZYLJZNZUUPYQUUHOZYGUUOUVDUVGVIUVHXOXPKUTUUOUVDUVGVAVBUUPUVEUVIU - VGYSUVEUVIOUUDYSUUOUVIUVDYSYJUVIYLYJUVIVCZRYSYJUVIUVCUVJRVDVEUUPYLUVIUV - FUUDYLUVIOYSYLUVIVCSVFVDVGUQUUPUUMUUHUUPUUMJZYRUUCNZUUAUUFNZNUUHUVKUVMU - VLUUMUUPYTUUENZUVMUULUVNUUKUULUVNYOXOXPVJZVHSUUPUVNJUUPYTJZUUPUUEJZNUVM - UUPYTUUEVKUVPUUAUVQUUFUUPYSYTYSUUDVLVMUUPUUDUUEYSUUDVNVMVOVPVQVRYRUUCUU - AUUFWAVSPVDQVDUVAUUROYEUVAUUNYRUUFNZUUAUUCNZNZUUHUUNUVAUVTUUNUUSUVRUUTU - VSUUJUUSUVROZUUMYQUWAYGYQYJUVRUUDYJYQUVRYRUUFWBVTRSUULUWAUUKUULUVNUWAUV - OYTYJUVRUUEUUDYTYJUVRYTYJJZYRUUFUWBYJYQYTYJVNYJYTYQYJXOYHYOYJXOYFYHYJTK - UGWCZWDWEWFWGUOPUUDUUEUVRUUFYRWHVTWIVPSWMUUJUUTUVSOZUUMUUJYLUVSYSYQYLUV - SOYGYLYQUVSUUCUUAWHVTSRUULUWDUUKUULUVNUWDUVOUUTUVNUVSUUTYTUUAUUEUUCYSYT - UUAOYLYSYTUUAUUATPSYLUUEUUCOYSYLUUEYQYLXPYHYOYLXPYFYHYLTUWCWDWEWJVEWKWL - VPSWMWKWLYRUUFUUAUUCWNWOQUUQUVANYEYJYLWPQWQVGUUIYOYKIZYOYMIZNUUHYOYKYMV - JUWEUUBUWFUUGYJYOYHXOUBYLYOYHXPUBWRWSWTXAYEXQYBIXRYILXOXPUDXDDXQDGZYFLZ - YHUWGMZYIYBBUWGXQLZUWHYGUWGXQYHUWGXQYFXBUWJTXCEYGYHXQKUGXEZXOXPCXFZAXFZ - XGXHXIXJYEXSYKXTYMYCXSYKLYDDXOUWIYKYBBDCXKZUWHYJUWGXOYHUWGXOYFXBUWNTXCE - YJYHXOUWKUWLXHXIVEYDXTYMLYCDXPUWIYMYBBDAXKZUWHYLUWGXPYHUWGXPYFXBUWOTXCE - YLYHXPUWKUWMXHXISXLXMXN $. + jaao mpdan uneqsn df-3or bitri pm2.21 jaod adantr adantld biimtrid elun + w3o biimpi andi simpl anim1i simpr orim12i sylbi sylan2 olcd or4 expcom + sylib orc snsspr1 eqsstrdi sseld impcom jca olc jaoa jaoi orim12d com12 + anc2li or42 syl6ib 4exmid mpjaod orbi12i sylbbr syl6 ssrdv pwuncl eqeq1 + ifbieq2d prex vex unex ifex fvmpt syl weq uneq12d 3sstr4d rgen2 ) CGZAG + ZHZBUAZXOBUAZXPBUAZHZUCCAUDUEZYBXOYBIZXPYBIZJZXQKUFZLZKUGUHZXQMZXOYFLZY + HXOMZXPYFLZYHXPMZHZXRYAYEFYIYNYEFGZYIIZYJYOYHIZJZYJUIZFCUJZJZNZYLYQJZYL + UIZFAUJZJZNZNZYOYNIZYPYGYQJZYGUIZYOXQIZJZNZYEUUHYGYOYHXQUBYEYJYLJZYSUUD + JZNZUUNUUHOZYJUUDJZYLYSJZNZYEUUOUURUUPUUOUUROYEUUOYGUURUUOXQYFYFHYFXOYF + XPYFUKYFULUMUUOUUJUUHYGUUMUUOUUJUUHUUOUUJJZUUBUUGUVBYRUUAYJYLYGYQUNUOUO + PYGUUKUULUUHYGUULUUHOUPUQURUSQUUPUUROYEUUPUUJUUHUUMUUPYGYQUUHYGUUOYJXPK + LZJZNZXOKLZYLJZNZUUPYQUUHOZYGUUOUVDUVGVIUVHXOXPKUTUUOUVDUVGVAVBUUPUVEUV + IUVGYSUVEUVIOUUDYSUUOUVIUVDYSYJUVIYLYJUVIVCZRYSYJUVIUVCUVJRVDVEUUPYLUVI + UVFUUDYLUVIOYSYLUVIVCSVFVDVGUQUUPUUMUUHUUPUUMJZYRUUCNZUUAUUFNZNUUHUVKUV + MUVLUUMUUPYTUUENZUVMUULUVNUUKUULUVNYOXOXPVHZVJSUUPUVNJUUPYTJZUUPUUEJZNU + VMUUPYTUUEVKUVPUUAUVQUUFUUPYSYTYSUUDVLVMUUPUUDUUEYSUUDVNVMVOVPVQVRYRUUC + UUAUUFVSWAPVDQVDUVAUUROYEUVAUUNYRUUFNZUUAUUCNZNZUUHUUNUVAUVTUUNUUSUVRUU + TUVSUUJUUSUVROZUUMYQUWAYGYQYJUVRUUDYJYQUVRYRUUFWBVTRSUULUWAUUKUULUVNUWA + UVOYTYJUVRUUEUUDYTYJUVRYTYJJZYRUUFUWBYJYQYTYJVNYJYTYQYJXOYHYOYJXOYFYHYJ + TKUGWCZWDWEWFWGUOPUUDUUEUVRUUFYRWHVTWIVPSWJUUJUUTUVSOZUUMUUJYLUVSYSYQYL + UVSOYGYLYQUVSUUCUUAWHVTSRUULUWDUUKUULUVNUWDUVOUUTUVNUVSUUTYTUUAUUEUUCYS + YTUUAOYLYSYTUUAUUATPSYLUUEUUCOYSYLUUEYQYLXPYHYOYLXPYFYHYLTUWCWDWEWMVEWK + WLVPSWJWKWLYRUUFUUAUUCWNWOQUUQUVANYEYJYLWPQWQVGUUIYOYKIZYOYMIZNUUHYOYKY + MVHUWEUUBUWFUUGYJYOYHXOUBYLYOYHXPUBWRWSWTXAYEXQYBIXRYILXOXPUDXBDXQDGZYF + LZYHUWGMZYIYBBUWGXQLZUWHYGUWGXQYHUWGXQYFXCUWJTXDEYGYHXQKUGXEZXOXPCXFZAX + FZXGXHXIXJYEXSYKXTYMYCXSYKLYDDXOUWIYKYBBDCXKZUWHYJUWGXOYHUWGXOYFXCUWNTX + DEYJYHXOUWKUWLXHXIVEYDXTYMLYCDXPUWIYMYBBDAXKZUWHYLUWGXPYHUWGXPYFXCUWOTX + DEYLYHXPUWKUWMXHXISXLXMXN $. $} ${ @@ -681189,18 +684819,18 @@ Casimir Kuratowski (Kazimierz Kuratowski). ( vc vd cv wss cfv wi wral cun weq sseq1 fveq2 sseq1d imbi12d sseq2 wcel wa cpw sseq2d cbvral2vw ssun1 simprl pwuncl adantl simpl rspc2va syl21anc wceq mpi ssun2 simprr unssd ralrimivva ssequn1 uneq1d uneq1 fveq2d uneq2d - sseq12d uneq2 ancoms unssad adantr sseqtrd ex syl5bi impbii bitri ) CGZDG - ZHZVLBIZVMBIZHZJZDAUAZKCVSKEGZFGZHZVTBIZWABIZHZJZFVSKEVSKZVOVPLZVLVMLZBIZ - HZDVSKCVSKZVRWFVTVMHZWCVPHZJCDEFVSVSCEMZVNWMVQWNVLVTVMNWOVOWCVPVLVTBOZPQD - FMZWMWBWNWEVMWAVTRWQVPWDWCVMWABOZUBQUCWGWLWGWKCDVSVSWGVLVSSZVMVSSZTZTZVOV - PWJXBVLWIHZVOWJHZVLVMUDXBWSWIVSSZWGXCXDJZWGWSWTUEXAXEWGVLVMAUFUGZWGXAUHZW - FXFVLWAHZVOWDHZJEFVLWIVSVSECMZWBXIWEXJVTVLWANXKWCVOWDVTVLBOPQWAWIUKZXIXCX - JXDWAWIVLRXLWDWJVOWAWIBOZUBQUIUJULXBVMWIHZVPWJHZVMVLUMXBWTXEWGXNXOJZWGWSW - TUNXGXHWFXPVMWAHZVPWDHZJEFVMWIVSVSEDMZWBXQWEXRVTVMWANXSWCVPWDVTVMBOPQXLXQ - XNXRXOWAWIVMRXLWDWJVPXMUBQUIUJULUOUPWLWFEFVSVSWBVTWALZWAUKZWLVTVSSWAVSSTZ - TZWEVTWAUQYCYAWEYCYATWCXTBIZWDYCWCYDHYAYCWCWDYDYBWLWCWDLZYDHZWKYFWCVPLZVT - VMLZBIZHCDVTWAVSVSWOWHYGWJYIWOVOWCVPWPURWOWIYHBVLVTVMUSUTVBWQYGYEYIYDWQVP - WDWCWRVAWQYHXTBVMWAVTVCUTVBUIVDVEVFYAYDWDUKYCXTWABOUGVGVHVIUPVJVK $. + sseq12d uneq2 ancoms unssad adantr sseqtrd ex biimtrid impbii bitri ) CGZ + DGZHZVLBIZVMBIZHZJZDAUAZKCVSKEGZFGZHZVTBIZWABIZHZJZFVSKEVSKZVOVPLZVLVMLZB + IZHZDVSKCVSKZVRWFVTVMHZWCVPHZJCDEFVSVSCEMZVNWMVQWNVLVTVMNWOVOWCVPVLVTBOZP + QDFMZWMWBWNWEVMWAVTRWQVPWDWCVMWABOZUBQUCWGWLWGWKCDVSVSWGVLVSSZVMVSSZTZTZV + OVPWJXBVLWIHZVOWJHZVLVMUDXBWSWIVSSZWGXCXDJZWGWSWTUEXAXEWGVLVMAUFUGZWGXAUH + ZWFXFVLWAHZVOWDHZJEFVLWIVSVSECMZWBXIWEXJVTVLWANXKWCVOWDVTVLBOPQWAWIUKZXIX + CXJXDWAWIVLRXLWDWJVOWAWIBOZUBQUIUJULXBVMWIHZVPWJHZVMVLUMXBWTXEWGXNXOJZWGW + SWTUNXGXHWFXPVMWAHZVPWDHZJEFVMWIVSVSEDMZWBXQWEXRVTVMWANXSWCVPWDVTVMBOPQXL + XQXNXRXOWAWIVMRXLWDWJVPXMUBQUIUJULUOUPWLWFEFVSVSWBVTWALZWAUKZWLVTVSSWAVSS + TZTZWEVTWAUQYCYAWEYCYATWCXTBIZWDYCWCYDHYAYCWCWDYDYBWLWCWDLZYDHZWKYFWCVPLZ + VTVMLZBIZHCDVTWAVSVSWOWHYGWJYIWOVOWCVPWPURWOWIYHBVLVTVMUSUTVBWQYGYEYIYDWQ + VPWDWCWRVAWQYHXTBVMWAVTVCUTVBUIVDVEVFYAYDWDUKYCXTWABOUGVGVHVIUPVJVK $. $( Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.) $) @@ -681212,18 +684842,18 @@ Casimir Kuratowski (Kazimierz Kuratowski). wa cpw sseq1 sseq1d cbvral2vw inss1 inss2 elpwi sstrid vex inex2 ad2antll elpw sylibr simprl simpl syl21anc mpi simprr ssind ralrimivva dfss adantl rspc2va ineq1 fveq2d ineq1d sseq12d ineq2 ineq2d ancoms sstrdi eqsstrd ex - adantr syl5bi impbii bitri ) CGZDGZHZVRBIZVSBIZHZJZDAUAZKCWEKEGZFGZHZWFBI - ZWGBIZHZJZFWEKEWEKZVRVSLZBIZWAWBLZHZDWEKCWEKZWDWLWFVSHZWIWBHZJCDEFWEWECEM - ZVTWSWCWTVRWFVSUBXAWAWIWBVRWFBNZUCODFMZWSWHWTWKVSWGWFPXCWBWJWIVSWGBNZQOUD - WMWRWMWQCDWEWEWMVRWERZVSWERZTZTZWOWAWBXHWNVRHZWOWAHZVRVSUEXHWNWERZXEWMXIX - JJZXFXKWMXEXFWNAHXKXFWNVSAVRVSUFZVSAUGUHWNAVSVRDUIUJULUMUKZWMXEXFUNWMXGUO - ZWLXLWNWGHZWOWJHZJZEFWNVRWEWEWFWNSZWHXPWKXQWFWNWGUBXSWIWOWJWFWNBNUCOZFCMZ - XPXIXQXJWGVRWNPYAWJWAWOWGVRBNQOVCUPUQXHWNVSHZWOWBHZXMXHXKXFWMYBYCJZXNWMXE - XFURXOWLYDXREFWNVSWEWEXTFDMZXPYBXQYCWGVSWNPYEWJWBWOWGVSBNQOVCUPUQUSUTWRWL - EFWEWEWHWFWFWGLZSZWRWFWERWGWERTZTZWKWFWGVAYIYGWKYIYGTWIYFBIZWJYGWIYJSYIWF - YFBNVBYIYJWJHYGYIYJWIWJLZWJYHWRYJYKHZWQYLWFVSLZBIZWIWBLZHCDWFWGWEWEXAWOYN - WPYOXAWNYMBVRWFVSVDVEXAWAWIWBXBVFVGXCYNYJYOYKXCYMYFBVSWGWFVHVEXCWBWJWIXDV - IVGVCVJWIWJUFVKVNVLVMVOUTVPVQ $. + adantr biimtrid impbii bitri ) CGZDGZHZVRBIZVSBIZHZJZDAUAZKCWEKEGZFGZHZWF + BIZWGBIZHZJZFWEKEWEKZVRVSLZBIZWAWBLZHZDWEKCWEKZWDWLWFVSHZWIWBHZJCDEFWEWEC + EMZVTWSWCWTVRWFVSUBXAWAWIWBVRWFBNZUCODFMZWSWHWTWKVSWGWFPXCWBWJWIVSWGBNZQO + UDWMWRWMWQCDWEWEWMVRWERZVSWERZTZTZWOWAWBXHWNVRHZWOWAHZVRVSUEXHWNWERZXEWMX + IXJJZXFXKWMXEXFWNAHXKXFWNVSAVRVSUFZVSAUGUHWNAVSVRDUIUJULUMUKZWMXEXFUNWMXG + UOZWLXLWNWGHZWOWJHZJZEFWNVRWEWEWFWNSZWHXPWKXQWFWNWGUBXSWIWOWJWFWNBNUCOZFC + MZXPXIXQXJWGVRWNPYAWJWAWOWGVRBNQOVCUPUQXHWNVSHZWOWBHZXMXHXKXFWMYBYCJZXNWM + XEXFURXOWLYDXREFWNVSWEWEXTFDMZXPYBXQYCWGVSWNPYEWJWBWOWGVSBNQOVCUPUQUSUTWR + WLEFWEWEWHWFWFWGLZSZWRWFWERWGWERTZTZWKWFWGVAYIYGWKYIYGTWIYFBIZWJYGWIYJSYI + WFYFBNVBYIYJWJHYGYIYJWIWJLZWJYHWRYJYKHZWQYLWFVSLZBIZWIWBLZHCDWFWGWEWEXAWO + YNWPYOXAWNYMBVRWFVSVDVEXAWAWIWBXBVFVGXCYNYJYOYKXCYMYFBVSWGWFVHVEXCWBWJWIX + DVIVGVCVJWIWJUFVKVNVLVMVOUTVPVQ $. $} ${ @@ -681916,16 +685546,16 @@ Casimir Kuratowski (Kazimierz Kuratowski). co wf ffvelcdmd ntrneiel crab ntrneifv4 df-rab eqtrdi eleq1d clabel wal wex df-rex bitr4i cvv ibar bibi2d ralbiia wss ssv cdif wn eldif a1i vex mpbiran ntrneinex elpwid sselda con3dimp pm3.14 orcs adantl - sseld 2falsed syl5bi ralrimiv raldifeq bitrid ralv bitr2di rexbidva - ex 3bitrd ) ANEKUAZKUAUBWSNLUAZUBBUCZDUBZEXALUAUBZUDZBUEZWTUBZBCUFZ - XCUGZBDUHZCWTUIZADWSFGHIJKLMNOPQRSADUJZXKEKAKXKXKUKUOUBXKXKKUPADFGH - IJKLMOPQRULKXKXKUMUNTUQURAWSXEWTAWSXCBDUSXEABDEFGHIJKLMOPQRTUTXCBDV - AVBVCXFXGXDUGZBVEZCWTUIZAXJXFCUCZWTUBZXMUDCVFXNXDBCWTVDXMCWTVGVHAXM - XICWTAXPUDZXIXLBVIUHZXMXIXLBDUHXQXRXHXLBDXBXCXDXGXBXCVJVKVLXQXLBDVI - DVIVMXQDVNVRXQXLBVIDVOZXAXSUBZXBVPZXQXLXTXAVIUBYABVSXAVIDVQVTXQYAXL - XQYAUDXGXDXQXGXBXQXODXAXQXODAWTXKXOAWTXKADXKUJZNLALYBDUKUOUBDYBLUPA - DFGHIJKLMOPQRWALYBDUMUNSUQWBWCWBWHWDYAXDVPZXQYAXCVPYCXBXCWEWFWGWIWQ - WJWKWLWMXLBWNWOWPWMWR $. + sseld 2falsed ex biimtrid ralrimiv raldifeq bitrid bitr2di rexbidva + ralv 3bitrd ) ANEKUAZKUAUBWSNLUAZUBBUCZDUBZEXALUAUBZUDZBUEZWTUBZBCU + FZXCUGZBDUHZCWTUIZADWSFGHIJKLMNOPQRSADUJZXKEKAKXKXKUKUOUBXKXKKUPADF + GHIJKLMOPQRULKXKXKUMUNTUQURAWSXEWTAWSXCBDUSXEABDEFGHIJKLMOPQRTUTXCB + DVAVBVCXFXGXDUGZBVEZCWTUIZAXJXFCUCZWTUBZXMUDCVFXNXDBCWTVDXMCWTVGVHA + XMXICWTAXPUDZXIXLBVIUHZXMXIXLBDUHXQXRXHXLBDXBXCXDXGXBXCVJVKVLXQXLBD + VIDVIVMXQDVNVRXQXLBVIDVOZXAXSUBZXBVPZXQXLXTXAVIUBYABVSXAVIDVQVTXQYA + XLXQYAUDXGXDXQXGXBXQXODXAXQXODAWTXKXOAWTXKADXKUJZNLALYBDUKUOUBDYBLU + PADFGHIJKLMOPQRWALYBDUMUNSUQWBWCWBWHWDYAXDVPZXQYAXCVPYCXBXCWEWFWGWI + WJWKWLWMWNXLBWQWOWPWNWR $. $} $} $} @@ -682227,16 +685857,16 @@ base set if and only if the neighborhoods (convergents) of every point s e. ( N ` x ) <-> ( I ` s ) e. ( N ` x ) ) ) ) $= ( wral wcel cvv adantr cv cfv wceq cpw wb wal dfcleq eqcom ralv 3bitr4i wa wss ssv a1i cdif wn vex eldif mpbiran co ntrneiiex elmapi ffvelcdmda - wf syl elpwid sseld con3dimp ffvelcdmd 2falsed syl5bi ralrimiv raldifeq - cmap ex wbr simpr simplr ntrneiel bibi12d ralbidva bitr3d bitrid ralcom - bitrdi ) ALUAZIUBZIUBZWGUCZLCUDZQWFBUAZJUBZRZWGWLRZUEZBCQZLWJQWOLWJQBCQ - AWIWPLWJWIWKWGRZWKWHRZUEZBSQZAWFWJRZUKZWPWGWHUCWSBUFWIWTBWGWHUGWHWGUHWS - BUIUJXBWSBCQWTWPXBWSBCSCSULXBCUMUNXBWSBSCUOZWKXCRZWKCRZUPZXBWSXDWKSRXFB - UQWKSCURUSXBXFWSXBXFUKWQWRXBWQXEXBWGCWKXBWGCAWJWJWFIAIWJWJVNUTRWJWJIVDZ - ACDEFGHIJKMNOPVAIWJWJVBVEZVCZVFVGVHXBWRXEXBWHCWKXBWHCXBWJWJWGIAXGXAXHTX - IVIVFVGVHVJVOVKVLVMXBWSWOBCXBXEUKZWQWMWRWNXJCWFDEFGHIJKWKMNOXBIJHVPZXEA - XKXAPTTZXBXEVQZAXAXEVRVSXJCWGDEFGHIJKWKMNOXLXMXBWGWJRXEXITVSVTWAWBWCWAW - OLBWJCWDWE $. + cmap wf syl sseld con3dimp ffvelcdmd 2falsed biimtrid ralrimiv raldifeq + elpwid ex wbr simpr simplr ntrneiel bibi12d bitr3d bitrid ralcom bitrdi + ralbidva ) ALUAZIUBZIUBZWGUCZLCUDZQWFBUAZJUBZRZWGWLRZUEZBCQZLWJQWOLWJQB + CQAWIWPLWJWIWKWGRZWKWHRZUEZBSQZAWFWJRZUKZWPWGWHUCWSBUFWIWTBWGWHUGWHWGUH + WSBUIUJXBWSBCQWTWPXBWSBCSCSULXBCUMUNXBWSBSCUOZWKXCRZWKCRZUPZXBWSXDWKSRX + FBUQWKSCURUSXBXFWSXBXFUKWQWRXBWQXEXBWGCWKXBWGCAWJWJWFIAIWJWJVDUTRWJWJIV + EZACDEFGHIJKMNOPVAIWJWJVBVFZVCZVNVGVHXBWRXEXBWHCWKXBWHCXBWJWJWGIAXGXAXH + TXIVIVNVGVHVJVOVKVLVMXBWSWOBCXBXEUKZWQWMWRWNXJCWFDEFGHIJKWKMNOXBIJHVPZX + EAXKXAPTTZXBXEVQZAXAXEVRVSXJCWGDEFGHIJKWKMNOXLXMXBWGWJRXEXITVSVTWEWAWBW + EWOLBWJCWCWD $. $} ${ @@ -683825,7 +687455,7 @@ base set if and only if the neighborhoods (convergents) of every point bound variable change is ~ rexlimdvaa . (Contributed by Rohan Ridenour, 3-Aug-2023.) $) rexlimdvaacbv $p |- ( ph -> ( E. x e. A ps -> ch ) ) $= - ( wrex cbvrexv rexlimdvaa syl5bi ) BEGJDFGJACBDEFGHKADCFGILM $. + ( wrex cbvrexv rexlimdvaa biimtrid ) BEGJDFGJACBDEFGHKADCFGILM $. $} ${ @@ -683840,7 +687470,7 @@ base set if and only if the neighborhoods (convergents) of every point condition, which does not require ~ ax-13 . (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.) $) rexlimddvcbvw $p |- ( ph -> ps ) $= - ( wrex cbvrexvw rexlimdvaa syl5bi mpd ) ADEGKZBHPCFGKABDCEFGJLACBFGIMNO + ( wrex cbvrexvw rexlimdvaa biimtrid mpd ) ADEGKZBHPCFGKABDCEFGJLACBFGIMNO $. $} @@ -685515,39 +689145,39 @@ collection and union ( ~ mnuop3d ), from which closure under pairing c0 3syl inex2 elpw mpbir unieq ineq2d ineq1 unieqd sseq12d anbi2d rexbidv vex rspcv ax-mp alrimiv inss2 an12 bicomi anbi2i bitri exbii df-rex eluni 3bitr4i w3a simp1 biimpri 3adant1 sseldd sylib elinel1 elin2d elinel2 cvv - elin elpwpw simprbi jca anim2i eximi sylibr 3expia ralrimiva anim12i sylg - syl5bi elequ2 rexeq imbi12d ralbidv cbvalvw nfv nfa1 nfan elpwi sp biimpi - ssin ex adantr simpl2 simprr simprl 3jca eximdv mpd 3anass bitr4i mpbiran - simp3 imim12d ralimdv imbi1i impexp df-ral syl6ibr anim12d reximdv 3impia - albii dfss2 3com23 syl3an2 3expa sylan2 ralrimia impbii ) AIZJZEBIZKZLZUU - FFIZMZKZUUKUUHJJZKZMZLZNZBEOZFEJZPZUUGELZUUGUUHLZGCUDZCIZUUKQZNZCEOZGDUDZ - 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The derivative of the generalized sum in - ~ binomcxplemnn0 . Part of remark "This convergence allows us to - apply term-by-term differentiation..." in the Wikibooks proof. + ~ binomcxplemnn0 . Part of remark "This convergence allows to apply + term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) $) binomcxplemdvsum $p |- ( ph -> ( CC _D P ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) ) ) $= @@ -689509,10 +693139,10 @@ Deduction Proof (not shown) was minimized. The minimized proof is (New usage is discouraged.) $) onfrALT $p |- _E Fr On $= ( va vy vx con0 cep wfr cv wss c0 wne wa cin wceq wi dfepfr simpr wel wex - wrex expd n0 wn onfrALTlem1 onfrALTlem2 pm2.61 exlimdv syl5bi mpd mpgbir - syl6c ) DEFAGZDHZUKIJZKZUKBGLIMBUKSZNAABDOUNUMUOULUMPUMCAQZCRUNUOCUKUAUNU - PUOCUNUPUKCGLIMZUONUQUBZUONUOUNUPUQUOCBAUCTUNUPURUOCBAUDTUQUOUEUJUFUGUHUI - $. + wrex expd n0 wn onfrALTlem1 onfrALTlem2 pm2.61 syl6c exlimdv biimtrid mpd + mpgbir ) DEFAGZDHZUKIJZKZUKBGLIMBUKSZNAABDOUNUMUOULUMPUMCAQZCRUNUOCUKUAUN + UPUOCUNUPUKCGLIMZUONUQUBZUONUOUNUPUQUOCBAUCTUNUPURUOCBAUDTUQUOUEUFUGUHUIU + J $. $} $( Closed form of right-to-left implication of ~ 19.41 , Theorem 19.41 of @@ -689622,11 +693252,11 @@ Deduction Proof (not shown) was minimized. The minimized proof is E. x E. y ( x = u /\ y = v ) ) $= ( cv wceq wal wn wo wa wex a1d wne biimprcd syl6 eximdv nfnae 19.9 syl6ib wi ax6e2nd ax6e2eq pm2.61i jaoi olc excom neeq1 adantrd simpr neeq2 syl6c - a1i sp necon3ai syl5bi orc pm2.61ine impbii ) AEZBEZFZAGZHZDEZCEZFZIZUSVD - FZUTVEFZJZBKAKZVCVKVFABCDUAZVBVFVKTABCDUBVCVKVFVLLUCUDVKVGTVDVEVFVGVKVFVC - UELVDVEMZVKVCVGVMVKVCBKZVCVKVJAKZBKVMVNVJABUFVMVOVCBVMVOVCAKVCVMVJVCAVMVJ - USUTMZVCVMVJUSVEMZVIVPVMVHVQVIVHVQVMUSVDVEUGNUHVJVITVMVHVIUIULVIVPVQUTVEU - SUJNUKVBUSUTVAAUMUNOPVCAABAQRSPUOVCBABBQRSVCVFUPOUQUR $. + a1i sp necon3ai biimtrid orc pm2.61ine impbii ) AEZBEZFZAGZHZDEZCEZFZIZUS + VDFZUTVEFZJZBKAKZVCVKVFABCDUAZVBVFVKTABCDUBVCVKVFVLLUCUDVKVGTVDVEVFVGVKVF + VCUELVDVEMZVKVCVGVMVKVCBKZVCVKVJAKZBKVMVNVJABUFVMVOVCBVMVOVCAKVCVMVJVCAVM + VJUSUTMZVCVMVJUSVEMZVIVPVMVHVQVIVHVQVMUSVDVEUGNUHVJVITVMVHVIUIULVIVPVQUTV + EUSUJNUKVBUSUTVAAUMUNOPVCAABAQRSPUOVCBABBQRSVCVFUPOUQUR $. $} ${ @@ -689818,7 +693448,7 @@ system of predicate calculus of Chapter IV (Section 19). The because we have already used "sequence" for any succession of objects, where the German is "Folge".) A sequent is a formal expression of the form ` ph , ... , ps ->.. ch , ... , th ` where ` ph , ... , ps ` and - ` ch , ... , th ` are seqences of a finite + ` ch , ... , th ` are sequences of a finite number of 0 or more formulas (substituting Metamath notation for Kleene's notation). The part ` ph , ... , ps ` is the antecedent, and ` ch , ... , th ` the succedent of the sequent @@ -692869,8 +696499,8 @@ Why not create a separate database (setg.mm) of proofs in G1, avoiding translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) $) snssiALT $p |- ( A e. B -> { A } C_ B ) $= - ( vx wcel cv csn wi wal wss wceq velsn eleq1a syl5bi alrimiv dfss2 sylibr - ) ABDZCEZAFZDZRBDZGZCHSBIQUBCTRAJQUACAKABRLMNCSBOP $. + ( vx wcel cv csn wal wss wceq velsn eleq1a biimtrid alrimiv dfss2 sylibr + wi ) ABDZCEZAFZDZRBDZPZCGSBHQUBCTRAIQUACAJABRKLMCSBNO $. $} ${ @@ -698053,11 +701683,6 @@ unification theorem (e.g., the sub-theorem whose assertion is step 5 ABOPQRZOSPTZCDUAZUBZOCDEFGBOUIUKBUIUCUJBCDUJBUDHIUEUFHIJULUGKLMNUH $. $} - $( A member of a union that is not a member of the second class, is a member - of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) - elunnel2 $p |- ( ( A e. ( B u. C ) /\ -. A e. C ) -> A e. B ) $= - ( cun wcel wo elun biimpi orcomd orcanai ) ABCDEZACEZABEZKMLKMLFABCGHIJ $. - ${ adantlllr.1 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $. $( Deduction adding a conjunct to antecedent. (Contributed by Glauco @@ -699244,8 +702869,8 @@ not even needed (it can be any class). (Contributed by Glauco Siliprandi, 23-Oct-2021.) $) iinssf $p |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C ) $= ( vy wss wrex ciin cv wcel wral wb cvv eliin elv wi ssel reximi nfcri syl - r19.36vf syl5bi ssrdv ) CDGZABHZFABCIZDFJZUGKZUHCKZABLZUFUHDKZUIUKMFAUHBC - NOPUFUJULQZABHUKULQUEUMABCDUHRSUJULABAFDETUBUAUCUD $. + r19.36vf biimtrid ssrdv ) CDGZABHZFABCIZDFJZUGKZUHCKZABLZUFUHDKZUIUKMFAUH + BCNOPUFUJULQZABHUKULQUEUMABCDUHRSUJULABAFDETUBUAUCUD $. $} ${ @@ -699798,21 +703423,21 @@ not even needed (it can be any class). (Contributed by Glauco ibi eqtri nfv nfeq2 reean sylibr adantr nfmpt1 nfcxfr nfcri simpll adantl nfrn nfan eqcomd ad2antlr 3eqtrd adantll neneqd pm2.65da adantlr ad2antrl neqned ad2antll ineq12d simpl eqnetrd jca ex reximdv a1d reximdai mpd a1i - rexlimdvv csbeq1 nfin nfne nfrex neeq1 csbid eqtrdi ineq1d neeq1d anbi12d - rexbidv cbvrexw bitri cbvdisj xchnxbir con4i ) BEUAZBKZLZACDLZXIMZAKZGKZN - ZDAXMDUBZUEZONZPZGCQZACQZXJMXKHKZIKZNZYAYBUEZONZPZIXGQHXGQZXTXIHXGYALZYGB - HXGXHYAXHYARSUFYHMYGHIXGYAYBYAYBRZSUCUDUGXKYFXTHIXGXGYAXGUHZYBXGUHZPZYFXT - TTXKYLYFXTYLYFPZYADRZYBXORZPZGCQZACQZXTYLYRYFYLYNACQZYOGCQZPYRYJYSYKYTYJY - SACDYAEXGFUIUPYKYTGCXOYBEXGEACDUJZGCXOUJFAGCDXOGDUKAXMDULZAXMDUMZUNUQUIUP - UOYNYOAGCCYNGURAYBXOUUBUSUTVAVBYMYQXSACYLYFAYJYKAAHXGAEAEUUAFACDVCVDVHZVE - AIXGUUDVEVIYFAURVIYMYQXSTXLCUHYMYPXRGCYFYPXRTYLYFYPXRYFYPPXNXQYCYPXNYEYCY - PPZXLXMUUEXLXMRZYIYPUUFYIYCYPUUFPYADXOYBYNYOUUFVFUUFDXORYPUUCVGYOXOYBRZYN - UUFYOYBXOYOSVJZVKVLVMUUEUUFPYAYBYCYPUUFVFVNVOVRVPYEYPXQYCYEYPPZXPYDOUUIDY - AXOYBYNDYARYEYOYNYADYNSVJVQYOUUGYEYNUUHVSVTYEYPWAWBVMWCWDVGWEWFWGWHWDWIWJ - WHJCAJKZDUBZLZXTXJUULMUUJXMNZUUKXOUEZONZPZGCQZJCQXTJGCUUKXOAUUJXMDWKUCUUQ - XSJACUUPAGCACUKUUMUUOAUUMAURAUUNOAUUKXOAUUJDULZUUBWLAOUKWMVIWNXSJURUUJXLR - ZUUPXRGCUUSUUMXNUUOXQUUJXLXMWOUUSUUNXPOUUSUUKDXOUUSUUKAXLDUBDAUUJXLDWKADW - PWQWRWSWTXAXBXCAJCDUUKJDUKUURAUUJDUMXDXEVAXF $. + rexlimdvv csbeq1 nfrexw neeq1 csbid eqtrdi ineq1d anbi12d rexbidv cbvrexw + nfin nfne neeq1d bitri cbvdisj xchnxbir con4i ) BEUAZBKZLZACDLZXIMZAKZGKZ + NZDAXMDUBZUEZONZPZGCQZACQZXJMXKHKZIKZNZYAYBUEZONZPZIXGQHXGQZXTXIHXGYALZYG + BHXGXHYAXHYARSUFYHMYGHIXGYAYBYAYBRZSUCUDUGXKYFXTHIXGXGYAXGUHZYBXGUHZPZYFX + TTTXKYLYFXTYLYFPZYADRZYBXORZPZGCQZACQZXTYLYRYFYLYNACQZYOGCQZPYRYJYSYKYTYJ + YSACDYAEXGFUIUPYKYTGCXOYBEXGEACDUJZGCXOUJFAGCDXOGDUKAXMDULZAXMDUMZUNUQUIU + PUOYNYOAGCCYNGURAYBXOUUBUSUTVAVBYMYQXSACYLYFAYJYKAAHXGAEAEUUAFACDVCVDVHZV + EAIXGUUDVEVIYFAURVIYMYQXSTXLCUHYMYPXRGCYFYPXRTYLYFYPXRYFYPPXNXQYCYPXNYEYC + YPPZXLXMUUEXLXMRZYIYPUUFYIYCYPUUFPYADXOYBYNYOUUFVFUUFDXORYPUUCVGYOXOYBRZY + NUUFYOYBXOYOSVJZVKVLVMUUEUUFPYAYBYCYPUUFVFVNVOVRVPYEYPXQYCYEYPPZXPYDOUUID + YAXOYBYNDYARYEYOYNYADYNSVJVQYOUUGYEYNUUHVSVTYEYPWAWBVMWCWDVGWEWFWGWHWDWIW + JWHJCAJKZDUBZLZXTXJUULMUUJXMNZUUKXOUEZONZPZGCQZJCQXTJGCUUKXOAUUJXMDWKUCUU + QXSJACUUPAGCACUKUUMUUOAUUMAURAUUNOAUUKXOAUUJDULZUUBWTAOUKXAVIWLXSJURUUJXL + RZUUPXRGCUUSUUMXNUUOXQUUJXLXMWMUUSUUNXPOUUSUUKDXOUUSUUKAXLDUBDAUUJXLDWKAD + WNWOWPXBWQWRWSXCAJCDUUKJDUKUURAUUJDUMXDXEVAXF $. $} ${ @@ -700120,15 +703745,15 @@ not even needed (it can be any class). (Contributed by Glauco (Contributed by Glauco Siliprandi, 24-Dec-2020.) $) mpct $p |- ( ph -> ( A ^m B ) ~<_ _om ) $= ( vy vz cv cmap co com cdom wbr c0 wceq oveq2 breq1d cvv wcel a1i csn cun - ctex syl mapdm0 cfn snfi fict ax-mp eqbrtrd wss cdif cxp cen cin vex snex - vx wa ad2antrr wn eldifn disjsn sylibr ad2antlr mapunen syl31anc mapsnend - adantl simpr endomtr syl2anc xpct ex findcard2d ) ABURHZIJZKLMBNIJZKLMBFH - ZIJZKLMZBVSGHZUAZUBZIJZKLMZBCIJZKLMURFGCVPNOVQVRKLVPNBIPQVPVSOVQVTKLVPVSB - IPQVPWDOVQWEKLVPWDBIPQVPCOVQWGKLVPCBIPQAVRNUAZKLABRSZVRWHOABKLMZWIDBUCUDZ - BRUEUDWHKLMZAWHUFSWLNUGWHUHUITUJAVSCUKZWBCVSULSZUSZUSZWAWFWPWAUSZWEVTBWCI - JZUMZUNMZWSKLMZWFWQVSRSZWCRSZWIVSWCUONOZWTXBWQFUPTXCWQWBUQTAWIWOWAWKUTWOX - DAWAWNXDWMWNWBVSSVAXDWBCVSVBVSWBVCVDVIVEVSWCBRRRVFVGWQWAWRKLMZXAWPWAVJAXE - WOWAAWRBUNMWJXEABWBRRWKWBRSAGUPTVHDWRBKVKVLUTVTWRVMVLWEWSKVKVLVNEVO $. + vx ctex syl mapdm0 cfn snfi fict ax-mp eqbrtrd wss cdif cxp cen cin vsnex + wa vex ad2antrr wn eldifn disjsn sylibr adantl ad2antlr syl31anc mapsnend + mapunen simpr endomtr syl2anc xpct ex findcard2d ) ABUCHZIJZKLMBNIJZKLMBF + HZIJZKLMZBVSGHZUAZUBZIJZKLMZBCIJZKLMUCFGCVPNOVQVRKLVPNBIPQVPVSOVQVTKLVPVS + BIPQVPWDOVQWEKLVPWDBIPQVPCOVQWGKLVPCBIPQAVRNUAZKLABRSZVRWHOABKLMZWIDBUDUE + ZBRUFUEWHKLMZAWHUGSWLNUHWHUIUJTUKAVSCULZWBCVSUMSZURZURZWAWFWPWAURZWEVTBWC + IJZUNZUOMZWSKLMZWFWQVSRSZWCRSZWIVSWCUPNOZWTXBWQFUSTXCWQGUQTAWIWOWAWKUTWOX + DAWAWNXDWMWNWBVSSVAXDWBCVSVBVSWBVCVDVEVFVSWCBRRRVIVGWQWAWRKLMZXAWPWAVJAXE + WOWAAWRBUOMWJXEABWBRRWKWBRSAGUSTVHDWRBKVKVLUTVTWRVMVLWEWSKVKVLVNEVO $. $} ${ @@ -700976,11 +704601,11 @@ not even needed (it can be any class). (Contributed by Glauco rnmptbdlem $p |- ( ph -> ( E. y e. RR A. x e. A B <_ y <-> E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y ) ) $= ( cv cle wbr wral cr wrex wa nfan simpr wcel cmpt crn nfcv nfra1 rnmptbdd - nfrex ex wi nfmpt1 nfrn nfv nfralw breq1 simplr adantlr elrnmpt1d rspcdva - eqid ralrimia a1d reximdai impbid ) AFCKZLMZBENZCOPZDKZVCLMZDBEFUAZUBZNZC - OPZAVFVLAVFQBCDEFAVFBHVEBCOBOUCVDBEUDUFRAVFSUEUGAVKVECOIAVKVEUHVCOTAVKVEA - VKQZVDBEAVKBHVHBDVJBVIBEFUIUJVHBUKULRVMBKETZQZVHVDDVJFVGFVCLUMAVKVNUNVOBE - FVIGVIURVMVNSAVNFGTVKJUOUPUQUSUGUTVAVB $. + nfrexw ex wi nfmpt1 nfrn nfv nfralw simplr eqid adantlr elrnmpt1d rspcdva + breq1 ralrimia a1d reximdai impbid ) AFCKZLMZBENZCOPZDKZVCLMZDBEFUAZUBZNZ + COPZAVFVLAVFQBCDEFAVFBHVEBCOBOUCVDBEUDUFRAVFSUEUGAVKVECOIAVKVEUHVCOTAVKVE + AVKQZVDBEAVKBHVHBDVJBVIBEFUIUJVHBUKULRVMBKETZQZVHVDDVJFVGFVCLURAVKVNUMVOB + EFVIGVIUNVMVNSAVNFGTVKJUOUPUQUSUGUTVAVB $. $} ${ @@ -701198,6 +704823,7 @@ not even needed (it can be any class). (Contributed by Glauco ( wcel wral wf ralrimia fmptff sylib ) ADELZBCMCEFNARBCGJOBCEDFHIKPQ $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Ordering on real numbers - Real and complex numbers basic operations @@ -703481,13 +707107,13 @@ not even needed (it can be any class). (Contributed by Glauco infrnmptle $p |- ( ph -> inf ( ran ( x e. A |-> B ) , RR* , < ) <_ inf ( ran ( x e. A |-> C ) , RR* , < ) ) $= ( vy vz cmpt crn nfv cxr eqid cv wcel cle wbr rnmptssd wa wceq cvv wb vex - wrex elrnmpt ax-mp biimpi adantl nfmpt1 nfrn nfrex simpr elrnmpt1 syl2anc - wi w3a 3adant3 id eqcomd 3ad2ant3 breqtrd breq1 rspcev rexlimd adantr mpd - 3exp infleinf2 ) AJKBCDLZMZBCELZMZAJNAKNABCDOVLFVLPZGUAABCEOVNFVNPZHUAAJQ - ZVORZUBVREUCZBCUGZKQZVRSTZKVMUGZVSWAAVSWAVRUDRVSWAUEJUFBCEVRVNUDVQUHUIUJU - KAWAWDURVSAVTWDBCFWCBKVMBVLBCDULUMWCBNUNABQCRZVTWDAWEVTUSZDVMRZDVRSTZWDAW - EWGVTAWEUBWEDORWGAWEUOGBCDVLOVPUPUQUTWFDEVRSAWEDESTVTIUTVTAEVRUCWEVTVREVT - VAVBVCVDWCWHKDVMWBDVRSVEVFUQVJVGVHVIVK $. + wrex elrnmpt ax-mp biimpi adantl wi nfmpt1 nfrn nfrexw w3a simpr elrnmpt1 + syl2anc 3adant3 id eqcomd 3ad2ant3 breqtrd rspcev 3exp rexlimd adantr mpd + breq1 infleinf2 ) AJKBCDLZMZBCELZMZAJNAKNABCDOVLFVLPZGUAABCEOVNFVNPZHUAAJ + QZVORZUBVREUCZBCUGZKQZVRSTZKVMUGZVSWAAVSWAVRUDRVSWAUEJUFBCEVRVNUDVQUHUIUJ + UKAWAWDULVSAVTWDBCFWCBKVMBVLBCDUMUNWCBNUOABQCRZVTWDAWEVTUPZDVMRZDVRSTZWDA + WEWGVTAWEUBWEDORWGAWEUQGBCDVLOVPURUSUTWFDEVRSAWEDESTVTIUTVTAEVRUCWEVTVREV + TVAVBVCVDWCWHKDVMWBDVRSVJVEUSVFVGVHVIVK $. $} ${ @@ -703539,14 +707165,14 @@ not even needed (it can be any class). (Contributed by Glauco infxrunb3rnmpt $p |- ( ph -> ( A. y e. RR E. x e. A B <_ y <-> inf ( ran ( x e. A |-> B ) , RR* , < ) = -oo ) ) $= ( vz cv cle wbr wrex cr wral cxr wceq nfv wcel wi cmpt crn cinf cmnf nfrn - clt nfmpt1 nfrex w3a wa simpr elrnmpt1 syl2anc 3adant3 simp3 breq1 rspcev - eqid 3exp rexlimd cvv vex elrnmpt ax-mp biimpi biimpcd a1d reximdai com12 - wb syl rexlimi a1i impbid ralbid wss rnmptssd infxrunb3 bitrd ) AECJZKLZB - DMZCNOIJZVTKLZIBDEUAZUBZMZCNOZWFPUFUCUDQZAWBWGCNGAWBWGAWAWGBDFWDBIWFBWEBD - EUGUEWDBRZUHABJDSZWAWGAWKWAUIEWFSZWAWGAWKWLWAAWKUJWKEPSWLAWKUKHBDEWEPWEUR - ZULUMUNAWKWAUOWDWAIEWFWCEVTKUPZUQUMUSUTWGWBTAWDWBIWFWBIRWCWFSZWCEQZBDMZWD - WBTWOWQWCVASWOWQVJIVBBDEWCWEVAWMVCVDVEWDWQWBWDWPWABDWJWDWPWATWKWPWDWAWNVF - VGVHVIVKVLVMVNVOAWFPVPWHWIVJABDEPWEFWMHVQCIWFVRVKVS $. + clt nfmpt1 nfrexw w3a wa simpr eqid elrnmpt1 syl2anc 3adant3 simp3 rspcev + breq1 3exp rexlimd cvv wb vex elrnmpt ax-mp biimpi biimpcd reximdai com12 + a1d syl rexlimi a1i impbid ralbid wss rnmptssd infxrunb3 bitrd ) AECJZKLZ + BDMZCNOIJZVTKLZIBDEUAZUBZMZCNOZWFPUFUCUDQZAWBWGCNGAWBWGAWAWGBDFWDBIWFBWEB + DEUGUEWDBRZUHABJDSZWAWGAWKWAUIEWFSZWAWGAWKWLWAAWKUJWKEPSWLAWKUKHBDEWEPWEU + LZUMUNUOAWKWAUPWDWAIEWFWCEVTKURZUQUNUSUTWGWBTAWDWBIWFWBIRWCWFSZWCEQZBDMZW + DWBTWOWQWCVASWOWQVBIVCBDEWCWEVAWMVDVEVFWDWQWBWDWPWABDWJWDWPWATWKWPWDWAWNV + GVJVHVIVKVLVMVNVOAWFPVPWHWIVBABDEPWEFWMHVQCIWFVRVKVS $. $} ${ @@ -706770,12 +710396,12 @@ closed under the multiplication ( ' X ' ) of a finite number of climinff $p |- ( ph -> F ~~> inf ( ran F , RR , < ) ) $= ( vj cfv cle wbr wi nfcv cr cv wcel wa caddc nfv nfan nffv nfbr nfim wceq c1 co eleq1w anbi2d fvoveq1 breq12d imbi12d chvarfv wral wrex nfci nfralw - fveq2 nfrex wb breq2d cbvralw a1i rexbidv imbi2d climinf ) ABNDEFIJKACUAZ - FUBZUCZVLUKUDULDOZVLDOZPQZRANUAZFUBZUCZVRUKUDULZDOZVRDOZPQZRCNVTWDCAVSCGV - SCUEZUFCWBWCPCWADHCWASUGCPSZCVRDHCVRSUGZUHUIVLVRUJZVNVTVQWDWHVMVSACNFUMUN - WHVOWBVPWCPVLVRUKDUDUOVLVRDVCZUPUQLURABUAZVPPQZCFUSZBTUTZRAWJWCPQZNFUSZBT - UTZRCNAWPCGWOCBTCTSWNCNFCNFWEVACWJWCPCWJSWFWGUHZVBVDUIWHWMWPAWHWLWOBTWLWO - VEWHWKWNCNFWKNUEWQWHVPWCWJPWIVFVGVHVIVJMURVK $. + fveq2 nfrexw wb breq2d cbvralw a1i rexbidv imbi2d climinf ) ABNDEFIJKACUA + ZFUBZUCZVLUKUDULDOZVLDOZPQZRANUAZFUBZUCZVRUKUDULZDOZVRDOZPQZRCNVTWDCAVSCG + VSCUEZUFCWBWCPCWADHCWASUGCPSZCVRDHCVRSUGZUHUIVLVRUJZVNVTVQWDWHVMVSACNFUMU + NWHVOWBVPWCPVLVRUKDUDUOVLVRDVCZUPUQLURABUAZVPPQZCFUSZBTUTZRAWJWCPQZNFUSZB + TUTZRCNAWPCGWOCBTCTSWNCNFCNFWEVACWJWCPCWJSWFWGUHZVBVDUIWHWMWPAWHWLWOBTWLW + OVEWHWKWNCNFWKNUEWQWHVPWCWJPWIVFVGVHVIVJMURVK $. $} ${ @@ -710720,7 +714346,7 @@ closed under the multiplication ( ' X ' ) of a finite number of $. climxrrelem.n $e |- ( ( ph /\ -oo e. CC ) -> D <_ ( abs ` ( -oo - A ) ) ) $. - $( If a seqence ranging over the extended reals converges w.r.t. the + $( If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.) $) @@ -712411,19 +716037,19 @@ reals with any finite set (the extended reals is an example of such a xlimmnfv $p |- ( ph -> ( F ~~>* -oo <-> A. x e. RR E. j e. Z A. k e. ( ZZ>= ` j ) ( F ` k ) <_ x ) ) $= ( vy wbr cv cle wral wrex cr wa wcel cxr cmnf clsxlim cfv cuz cz ad2antrr - wf simplr simpr xlimmnfvlem1 ralrimiva nfcv nfra1 nfrex nfralw nfan nfre1 - nfv adantr clt c1 cmin co w3a 3ad2ant1 uztrn2 3adant1 ffvelcdmd ad5ant134 - simp-4r peano2rem rexrd syl rexr ad4antlr ltm1d xrlelttrd ex ralimdva imp - adantl3r 3impa adantl simpl breq2 ralbidv rexbidv syl2anc adantll reximdd - wceq rspcva xlimmnfvlem2 impbida ) AEUAUBLZDMZEUCZBMZNLZDCMZUDUCZOZCGPZBQ - OZAWORZXCBQXEWRQSZRCDEFWRGAFUESZWOXFHUFIAGTEUGZWOXFJUFAWOXFUHXEXFUIUJUKAX - DRZKCDEFGAXDDADURXCDBQDQULXBDCGDGULWSDXAUMUNUOUPAXDCACURXCCBQCQULXBCGUQUO - UPZAXGXDHUSIAXHXDJUSXIWQKMZUTLZDXAOZCGPKQXIXKQSZRZWQXKVAVBVCZNLZDXAOZXMCG - XIXNCXJXNCURUPXOWTGSZXRXMAXNXSXRXMXDAXNRXSRZXRXMXTXQXLDXAXTWPXASZRZXQXLYB - XQRZWQXPXKAXSYAWQTSXNXQAXSYAVDGTWPEAXSXHYAJVEXSYAWPGSAFWPWTGIVFVGVHVIYCXN - XPTSAXNXSYAXQVJZXNXPXKVKZVLVMXNXKTSAXSYAXQXKVNVOYBXQUIYCXKYDVPVQVRVSVTWAW - BXDXNXRCGPZAXDXNRXPQSZXDYFXNYGXDYEWCXDXNWDXCYFBXPQWRXPWKZXBXRCGYHWSXQDXAW - RXPWQNWEWFWGWLWHWIWJUKWMWN $. + wf simplr simpr xlimmnfvlem1 ralrimiva nfv nfcv nfra1 nfrexw nfralw nfre1 + nfan adantr clt c1 cmin co w3a uztrn2 3adant1 ffvelcdmd ad5ant134 simp-4r + 3ad2ant1 peano2rem rexrd syl rexr ad4antlr xrlelttrd ex ralimdva adantl3r + ltm1d imp 3impa adantl simpl breq2 ralbidv rexbidv rspcva syl2anc adantll + wceq reximdd xlimmnfvlem2 impbida ) AEUAUBLZDMZEUCZBMZNLZDCMZUDUCZOZCGPZB + QOZAWORZXCBQXEWRQSZRCDEFWRGAFUESZWOXFHUFIAGTEUGZWOXFJUFAWOXFUHXEXFUIUJUKA + XDRZKCDEFGAXDDADULXCDBQDQUMXBDCGDGUMWSDXAUNUOUPURAXDCACULXCCBQCQUMXBCGUQU + PURZAXGXDHUSIAXHXDJUSXIWQKMZUTLZDXAOZCGPKQXIXKQSZRZWQXKVAVBVCZNLZDXAOZXMC + GXIXNCXJXNCULURXOWTGSZXRXMAXNXSXRXMXDAXNRXSRZXRXMXTXQXLDXAXTWPXASZRZXQXLY + BXQRZWQXPXKAXSYAWQTSXNXQAXSYAVDGTWPEAXSXHYAJVJXSYAWPGSAFWPWTGIVEVFVGVHYCX + NXPTSAXNXSYAXQVIZXNXPXKVKZVLVMXNXKTSAXSYAXQXKVNVOYBXQUIYCXKYDVTVPVQVRWAVS + WBXDXNXRCGPZAXDXNRXPQSZXDYFXNYGXDYEWCXDXNWDXCYFBXPQWRXPWKZXBXRCGYHWSXQDXA + WRXPWQNWEWFWGWHWIWJWLUKWMWN $. $} ${ @@ -712517,19 +716143,19 @@ reals with any finite set (the extended reals is an example of such a xlimpnfv $p |- ( ph -> ( F ~~>* +oo <-> A. x e. RR E. j e. Z A. k e. ( ZZ>= ` j ) x <_ ( F ` k ) ) ) $= ( vy wbr cv cle wral wrex cr wa wcel cxr cpnf clsxlim cfv cuz cz ad2antrr - wf simplr simpr xlimpnfvlem1 ralrimiva nfcv nfra1 nfrex nfralw nfan nfre1 - nfv adantr clt c1 caddc co simp-4r syl peano2re rexrd w3a 3ad2ant1 uztrn2 - rexr 3adant1 ffvelcdmd ad5ant134 ltp1d xrltletrd ex ralimdva imp adantl3r - 3impa adantl simpl wceq breq1 ralbidv rexbidv rspcva syl2anc xlimpnfvlem2 - adantll reximdd impbida ) AEUAUBLZBMZDMZEUCZNLZDCMZUDUCZOZCGPZBQOZAWNRZXB - BQXDWOQSZRCDEFWOGAFUESZWNXEHUFIAGTEUGZWNXEJUFAWNXEUHXDXEUIUJUKAXCRZKCDEFG - AXCDADURXBDBQDQULXADCGDGULWRDWTUMUNUOUPAXCCACURXBCBQCQULXACGUQUOUPZAXFXCH - USIAXGXCJUSXHKMZWQUTLZDWTOZCGPKQXHXJQSZRZXJVAVBVCZWQNLZDWTOZXLCGXHXMCXIXM - CURUPXNWSGSZXQXLAXMXRXQXLXCAXMRXRRZXQXLXSXPXKDWTXSWPWTSZRZXPXKYAXPRZXJXOW - QYBXMXJTSAXMXRXTXPVDZXJVKVEYBXMXOTSYCXMXOXJVFZVGVEAXRXTWQTSXMXPAXRXTVHGTW - PEAXRXGXTJVIXRXTWPGSAFWPWSGIVJVLVMVNYBXJYCVOYAXPUIVPVQVRVSVTWAXCXMXQCGPZA - XCXMRXOQSZXCYEXMYFXCYDWBXCXMWCXBYEBXOQWOXOWDZXAXQCGYGWRXPDWTWOXOWQNWEWFWG - WHWIWKWLUKWJWM $. + wf simplr simpr xlimpnfvlem1 ralrimiva nfv nfcv nfra1 nfrexw nfralw nfre1 + nfan adantr clt c1 caddc co simp-4r rexr syl peano2re w3a 3ad2ant1 uztrn2 + rexrd 3adant1 ffvelcdmd ad5ant134 ltp1d xrltletrd ralimdva adantl3r 3impa + ex adantl simpl wceq breq1 ralbidv rexbidv rspcva syl2anc adantll reximdd + imp xlimpnfvlem2 impbida ) AEUAUBLZBMZDMZEUCZNLZDCMZUDUCZOZCGPZBQOZAWNRZX + BBQXDWOQSZRCDEFWOGAFUESZWNXEHUFIAGTEUGZWNXEJUFAWNXEUHXDXEUIUJUKAXCRZKCDEF + GAXCDADULXBDBQDQUMXADCGDGUMWRDWTUNUOUPURAXCCACULXBCBQCQUMXACGUQUPURZAXFXC + HUSIAXGXCJUSXHKMZWQUTLZDWTOZCGPKQXHXJQSZRZXJVAVBVCZWQNLZDWTOZXLCGXHXMCXIX + MCULURXNWSGSZXQXLAXMXRXQXLXCAXMRXRRZXQXLXSXPXKDWTXSWPWTSZRZXPXKYAXPRZXJXO + WQYBXMXJTSAXMXRXTXPVDZXJVEVFYBXMXOTSYCXMXOXJVGZVKVFAXRXTWQTSXMXPAXRXTVHGT + WPEAXRXGXTJVIXRXTWPGSAFWPWSGIVJVLVMVNYBXJYCVOYAXPUIVPVTVQWKVRVSXCXMXQCGPZ + AXCXMRXOQSZXCYEXMYFXCYDWAXCXMWBXBYEBXOQWOXOWCZXAXQCGYGWRXPDWTWOXOWQNWDWEW + FWGWHWIWJUKWLWM $. $} ${ @@ -713197,37 +716823,37 @@ distinct definitions for the same symbol (limit of a sequence). wss simp1rl ad2antrr simplr fvmpt2 eleq1w imbi12d chvarvv fvmptd3 3adant2 anbi2d eleq1d oveq12d fveq2d syl3anc simpld ssrab3 sseli nnncan2d eqbrtrd adantl syl1111anc oveq2 fveq2 oveq2d simpll3 ex ralrimiva 3exp ralrimivva - 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KLURUSVJVLVNUTVAEVELVBTGCVFEVCTVFQVKGVD $. $} - $( From compex differentiation to real differentiation. (Contributed by + $( From complex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) dvcnre $p |- ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) ) $= @@ -721010,28 +724636,28 @@ used to represent the final q_n in the paper (the one with n large $= ( vm vi vu vy cv cn wcel c1 cfz co cc0 cfv clt wbr wrex cdif wral wex cle wf wa wceq w3a cfn wss cuni stoweidlem50 crab cmpt nfcv nfra1 nfan nfrabw - nfv nfcxfr nfrab1 nfeq2 nfrex nfss nfdif nf3an nfre1 eqid cvv ctop cmptop - ccn ccmp syl cioo crn ctg retop eqeltri cnfex sylancl ssexd adantr simpr1 - sseqtrdi simpr2 simpr3 wne stoweidlem35 exlimddv cdiv csu cmul nfralw nff - c0 simprl simprrl simprrr caddc 3adant1r adantlr elssuni sseqtrrdi sseldd - cr stoweidlem44 ex exlimdvv mpd ) AUPUTZVAVBZVCUUAVDVEZGRUTZVOZVFDUTZUQUT - UUDVGVGVHVIZUQUUCVJZDHIVKZVLZVPZVPZRVMUPVMZVFUUFSUTZVGZVNVIUUOVCVNVIVPDHV - LPUUNVGVFVQVFUUOVHVIDUUIVLVRSEVJZAURUTZVSVBZUUQOVTZUUIUUQWAZVTZVRZUUMURAB - 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+743306,24 @@ those for the more general case of a piecewise smooth function (see eqtri com cdom wbr cdif wo c2 0red cr 2re unitssre id eleqtrdi sselid cxr cle rexrd 1xr iccgelb syl3anc iccleub 1le2 letrd eliccd eleqtrrdi snelpwi 1re syl cfn snfi fict orc breq1 difeq2 breq1d orbi12d elrab2 sylibr elind - jca wceq eqcomi eleqtrd adantr sseldd ralrimiva elint2 snidg eleq2 rspcev - snex syl2anc eluni2 rgen dfss3 eqssi salexct mptru inss1 salexct2 pm2.65i - wtru sseli eqneltri ) GOPZGUDZGPZGUEUUCUFUUAUUBCGUUBCUUBFUDZCGFGDHUGZQZHO - UHZUIZFNFUUGPZUUHFQUUIFOPZDFQZULUUJUUKFCUJZOLCUMPUULOPCRSUKUNZUMKRSUKUOUP - UMCUQURUPDEFVEZFMEFUSUTVAUUFUUKHFOUUEFDVBVCVDFUUGVFURUTVGUUDUULUDCFUULLVH - CVIVQVJCUUBQUAUGZUUBPZUACVKUUPUACUUOCPZUUOUBUGZPZUBGVLZUUPUUQUUOVMZGPUUOU - VAPZUUTUUQUVAUUHGUUQUVAUCUGZPZUCUUGVKUVAUUHPUUQUVDUCUUGUUQUVCUUGPZULDUVCU - VAUVEDUVCQZUUQUVEUVCOPZUVFUVEUVGUVFULUUFUVFHUVCOUUEUVCDVBVCVNVOVPUUQUVADP - UVEUUQUVAUUNDUUQEFUVAUUQUVABUJZPZUVAVRVSVTZBUVAWAZVRVSVTZWBZULUVAEPUUQUVI - 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Similar to sge0gerp $p |- ( ph -> A <_ ( sum^ ` F ) ) $= ( vy cv csumge0 cxr cle nfv wcel co cxad cpw cfn cin cres cfv crn clt wss cmpt csup wral wa simpr cc0 cpnf cicc wf adantr elinel1 elpwi syl fssresd - adantl sge0xrcl ralrimiva eqid rnmptss crp wbr wrex nfmpt1 nfrn nfrex w3a - cvv id fvexd elrnmpt1 syl2anc 3ad2ant2 simp3 wceq oveq1 breq2d rspce 3exp - rexlimd mpd supxrge sge0sup eqcomd breqtrd ) ADCGUAZUBUCZECMZUDZNUEZUIZUF - ZOUGUJZENUEZPABLWSDABQAWQORZCWNUKWSOUHAXBCWNAWOWNRZULZWPWNWOAXCUMXDGUNUOU - PSZWOEAGXEEUQXCIURXCWOGUHZAXCWOWMRXFWOWMUBUSWOGUTVAVCVBVDVECWNWQOWRWRVFZV - GVAJABMZVHRULZDWQXHTSZPVIZCWNVJDLMZXHTSZPVIZLWSVJZKXIXKXOCWNXICQXNCLWSCWR - CWNWQVKVLXNCQVMXIXCXKXOXIXCXKVNWQWSRZXKXOXCXIXPXKXCXCWQVORXPXCVPXCWPNVQCW - NWQWRVOXGVRVSVTXIXCXKWAXNXKLWQWSXKLQXLWQWBXMXJDPXLWQXHTWCWDWEVSWFWGWHWIAX - AWTACEFGHIWJWKWL $. + adantl sge0xrcl ralrimiva eqid rnmptss crp wbr wrex nfmpt1 nfrexw w3a cvv + nfrn fvexd elrnmpt1 syl2anc 3ad2ant2 simp3 wceq oveq1 breq2d 3exp rexlimd + id rspce mpd supxrge sge0sup eqcomd breqtrd ) ADCGUAZUBUCZECMZUDZNUEZUIZU + FZOUGUJZENUEZPABLWSDABQAWQORZCWNUKWSOUHAXBCWNAWOWNRZULZWPWNWOAXCUMXDGUNUO + UPSZWOEAGXEEUQXCIURXCWOGUHZAXCWOWMRXFWOWMUBUSWOGUTVAVCVBVDVECWNWQOWRWRVFZ + VGVAJABMZVHRULZDWQXHTSZPVIZCWNVJDLMZXHTSZPVIZLWSVJZKXIXKXOCWNXICQXNCLWSCW + RCWNWQVKVOXNCQVLXIXCXKXOXIXCXKVMWQWSRZXKXOXCXIXPXKXCXCWQVNRXPXCWFXCWPNVPC + WNWQWRVNXGVQVRVSXIXCXKVTXNXKLWQWSXKLQXLWQWAXMXJDPXLWQXHTWBWCWGVRWDWEWHWIA + XAWTACEFGHIWJWKWL $. $} ${ @@ -740987,69 +744613,69 @@ the supremum (in the real numbers) of finite subsums. 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Similar to Siliprandi, 11-Oct-2020.) $) sge0xp $p |- ( ph -> ( sum^ ` ( j e. A |-> ( sum^ ` ( k e. B |-> C ) ) ) ) = ( sum^ ` ( z e. ( A X. B ) |-> D ) ) ) $= - ( cmpt csumge0 cfv cvv wcel vi cv csn cxp ciun snex xpexd adantr disjsnxp - a1i wdisj cc0 cpnf cicc co wa cop wceq wrex wb elsnxp ax-mp biimpi adantl - vex nfv nfan wi w3a 3ad2ant3 3expa 3adant3 eqeltrd 3exp rexlimd mpd 3impa - sge0iunmpt iunxpconst eqcomi mpteq1d fveq2d simpr eqid projf1o eqidd opex - opeq2 fvmptd adantlr sge0f1o eqcomd mpteq2da 3eqtr4rd ) ABGCGUBZUCZDUDZUE - ZFPZQRGCBWQFPQRZPZQRBCDUDZFPZQRGCHDEPQRZPZQRAGCWQFBISMAWQSTWOCTZAWPDSJWPS - TAWOUFUJNUGUHGCWQUKACDGUIUJAXFBUBZWQTZFULUMUNUOZTZAXFUPZXHUPZXGWOHUBZUQZU - RZHDUSZXJXHXPXKXHXPWOSTXHXPUTGVEHDSWOXGVAVBVCVDXLXOXJHDXKXHHAXFHKXFHVFVGZ + ( cmpt csumge0 cfv cvv wcel vi cv csn cxp vsnex a1i xpexd adantr disjsnxp + ciun wdisj cc0 cpnf cicc co wa cop wceq wb vex elsnxp ax-mp biimpi adantl + wrex nfv nfan wi w3a 3ad2ant3 3adant3 eqeltrd 3exp rexlimd mpd sge0iunmpt + 3expa 3impa iunxpconst eqcomi mpteq1d simpr eqid projf1o eqidd opeq2 opex + fveq2d fvmptd adantlr sge0f1o eqcomd mpteq2da 3eqtr4rd ) ABGCGUBZUCZDUDZU + JZFPZQRGCBWQFPQRZPZQRBCDUDZFPZQRGCHDEPQRZPZQRAGCWQFBISMAWQSTWOCTZAWPDSJWP + STAGUEUFNUGUHGCWQUKACDGUIUFAXFBUBZWQTZFULUMUNUOZTZAXFUPZXHUPZXGWOHUBZUQZU + RZHDVEZXJXHXPXKXHXPWOSTXHXPUSGUTHDSWOXGVAVBVCVDXLXOXJHDXKXHHAXFHKXFHVFVGZ XHHVFVGXJHVFXKXMDTZXOXJVHVHXHXKXRXOXJXKXRXOVIFEXIXOXKFEURXRLVJXKXREXITZXO - AXFXRXSOVKVLVMVNUHVOVPZVQVRAXCWSQABXBWRFXBWRURAWRXBGCDVSVTUJWAWBAXEXAQAGC - XDWTAGVFXKWTXDXKWQFDEBHUADWOUAUBZUQZPZXNJXKBVFXQLADJTXFNUHXKUAWODYCCAXFWC - YCWDWEAXRXMYCRXNURXFAXRUPZUAXMYBXNDYCSYDYCWFYAXMURYBXNURYDYAXMWOWHVDAXRWC - XNSTYDWOXMWGUJWIWJXTWKWLWMWBWN $. + AXFXRXSOVQVKVLVMUHVNVOZVRVPAXCWSQABXBWRFXBWRURAWRXBGCDVSVTUFWAWHAXEXAQAGC + XDWTAGVFXKWTXDXKWQFDEBHUADWOUAUBZUQZPZXNJXKBVFXQLADJTXFNUHXKUAWODYCCAXFWB + YCWCWDAXRXMYCRXNURXFAXRUPZUAXMYBXNDYCSYDYCWEYAXMURYBXNURYDYAXMWOWFVDAXRWB + XNSTYDWOXMWGUFWIWJXTWKWLWMWHWN $. $} ${ @@ -742343,10 +745969,10 @@ the supremum (in the real numbers) of finite subsums. Similar to cuz nfan adantl cpnf cico cr rge0ssre simpll elpwinss adantr simpr sseldd cc0 adantll sselid fsumreclf rnmptssd supxrcl syl elfzuz eleqtrrdi syldan rexrd fzfid adantlr wrex cle wbr wb vex elrnmpt ax-mp biimpi w3a 3ad2ant1 - cz 3ad2ant2 3adant3 uzfissfz nfmpt1 nfrn nfrex wi id elrnmpt1 simplr nfcv - sumex nfsum1 ad4ant14 simplll 0xr pnfxr icogelb syl3anc fsumlessf eqbrtrd - nfeq 3adant2 breq2 rspcev 3exp rexlimd rexlimdv suplesup2 wral ssriv ovex - mpd elpw mpbir fzfi elini sumeq1 rspceeqv elrnmptd 2a1i ralrimiva dfss3 + cz 3ad2ant2 3adant3 uzfissfz nfmpt1 nfrn nfrexw wi id sumex elrnmpt1 nfcv + simplr nfsum1 nfeq ad4ant14 simplll 0xr icogelb syl3anc fsumlessf eqbrtrd + pnfxr 3adant2 breq2 rspcev 3exp rexlimd mpd rexlimdv suplesup2 wral ssriv + ovex elpw mpbir fzfi elini sumeq1 rspceeqv elrnmptd 2a1i ralrimiva dfss3 imp sylibr supxrss xrletrid eqtrd ) ACFBUHUBUCKFUDZUEUFZKUGZBCUIZUHZUJZMU KULZDFEDUGZUMUNZBCUIZUHZUJZMUKULZACKFBNGAFEVAUCZNFUURUOAIOEVAUPUQJURAUUKU UQAUUJMPZUUKMQAKUUFUUHMUUIAKRUUIUSZAUUGUUFQZSZUUHUVBUUGBCAUVACGUVACRVBUVA @@ -742359,15 +745985,15 @@ the supremum (in the real numbers) of finite subsums. Similar to UWFUWIAUWEUWIKUUFAUVAUWEUWIAUVAUWEWNZUUGUUMPZDFWFUWIUWLUUGDEFAUVAEWPQUWEH WOIUVAAUVKUWEUVLWQAUVAUVCUWEUVDWRWSUWLUWMUWIDFUWLDRUWHDUAUUPDUUODFUUNWTXA UWHDRXBAUWEUVOUWMUWIXCXCUVAAUWESZUVOUWMUWIUWNUVOUWMWNUUNUUPQZUWCUUNWGWHZU - WIUVOUWNUWOUWMUVOUVOUUNNQZUWOUVOXDUWQUVOUUMBCXHODFUUNUUONUVNXETWQUWNUWMUW - PUVOUWNUWMSZUWCUUHUUNWGAUWEUWMXFUWRUUMBUUGCUWNUWMCAUWECGCUWCUUHCUWCXGUUGB - CCUUGXGXIXRVBUWMCRVBUWREUULWDAUVQUVRUWEUWMUWAXJUWRUVQSAUVIVMBWGWHZAUWEUWM - UVQXKUVQUVIUWRUVSVCUVTVMMQZVDMQZUVJUWSUWTUVTXLOUXAUVTXMOJVMVDBXNXOTUWNUWM - VKXPXQXSUWHUWPUAUUNUUPUWGUUNUWCWGXTYATYBXSYCYIYBYDYTWBYEAUUPUUJPZUUSUUQUU - KWGWHAUWDLUUPYFUXBAUWDLUUPAUWCUUPQZSUWCUUNUOZDFWFZUWDUXCUXEAUXCUXEUWJUXCU + WIUVOUWNUWOUWMUVOUVOUUNNQZUWOUVOXDUWQUVOUUMBCXEODFUUNUUONUVNXFTWQUWNUWMUW + PUVOUWNUWMSZUWCUUHUUNWGAUWEUWMXHUWRUUMBUUGCUWNUWMCAUWECGCUWCUUHCUWCXGUUGB + CCUUGXGXIXJVBUWMCRVBUWREUULWDAUVQUVRUWEUWMUWAXKUWRUVQSAUVIVMBWGWHZAUWEUWM + UVQXLUVQUVIUWRUVSVCUVTVMMQZVDMQZUVJUWSUWTUVTXMOUXAUVTXROJVMVDBXNXOTUWNUWM + VKXPXQXSUWHUWPUAUUNUUPUWGUUNUWCWGXTYATYBXSYCYDYBYEYTWBYFAUUPUUJPZUUSUUQUU + KWGWHAUWDLUUPYGUXBAUWDLUUPAUWCUUPQZSUWCUUNUOZDFWFZUWDUXCUXEAUXCUXEUWJUXCU XEWIUWKDFUUNUWCUUONUVNWKWLWMVCAUXEUWDXCUXCAUXDUWDDFUXDUWDXCAUVOUXDKUUFUUH - UWCUUINUUTUXDUUMUUFQZUXDUWFUXFUXDUUMUUEUEUUMUUEQUUMFPCUUMFUVSYGUUMFEUULUM - YHYJYKEUULYLYMOUXDXDKUUMUUFUUHUUNUWCUUGUUMBCYNYOTUWJUXDUWKOYPYQYDVJYIYRLU + UWCUUINUUTUXDUUMUUFQZUXDUWFUXFUXDUUMUUEUEUUMUUEQUUMFPCUUMFUVSYHUUMFEUULUM + YIYJYKEUULYLYMOUXDXDKUUMUUFUUHUUNUWCUUGUUMBCYNYOTUWJUXDUWKOYPYQYEVJYDYRLU UPUUJYSUUAUVMUUPUUJUUBTUUCUUD $. $} @@ -743328,44 +746954,44 @@ nondecreasing measurable sets (with bounded measure) then the measure of bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.) $) meaiuninc3v $p |- ( ph -> S ~~>* ( M ` U_ n e. Z ( E ` n ) ) ) $= ( vx vj cfv cr wa wcel adantr vk cv cle wbr wral wrex ciun clsxlim cz cxr - wf cdm cmea eqid ffvelcdmda meaxrcl fmptd nfv nfcv nfra1 nfrex nfan caddc - c1 co wss adantlr simpr meaiunincf climxlim2 wn clt wb wceq 2fveq3 breq2d - cbvrexvw a1i rexr ad2antlr xrltnled rexbidva bitrd ralbidva rexnal ralbii - ralnex mpbird syldan cpnf cuz simp-4r syl simp-4l uztrn2 ad4ant24 syl2anc - bitri wi eleq1w anbi2d eleq1d imbi12d chvarvv ad5ant13 simplr w3a 3adant3 - 3ad2ant1 simp1 3adant1 simp3 simpll uzssd3 elfzouz adantl adantll fvoveq1 - cfzo sseldd fveq2 sseq12d 3adantl3 ssinc meassle fvexd breqtrrd ad5ant135 - cvv fvmpt2 xrltletrd xrltled ralrimiva ex reximdva ralimdva cmpt ad4ant13 - imp nfmpt1 nfcxfr xlimpnf rspa nfralw ad3antlr dmmeasal com cdom saliuncl - nfre1 uzct ad3antrrr ssiun2s exp31 rexlimd mpd ralrimia xrpnf pm2.61dan ) - ACUBZDPZEPZNUBZUCUDZCGUEZNQUFZBCGUVAUGZEPZUHUDZAUVFRZUVHBFGAFUISUVFITZJAG - UJBUKUVFACGUVBUJBAUUTGSZRZUVAEULZEAEUMSZUVLHTUVNUNZAGUVNUUTDKUOZUPZMUQZTU - 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$d A x $. $d D a $. $d F a $. $d S a $. $d a x $. + $d A x $. smfpimltxr.x $e |- F/_ x F $. smfpimltxr.s $e |- ( ph -> S e. SAlg ) $. smfpimltxr.f $e |- ( ph -> F e. ( SMblFn ` S ) ) $. @@ -752024,7 +755650,7 @@ its dimensional volume (the product of its length in each dimension, $} ${ - $d A x y $. $d B y $. $d R x y $. + $d A x $. $d R x $. smfpimltxrmpt.x $e |- F/ x ph $. smfpimltxrmpt.s $e |- ( ph -> S e. SAlg ) $. smfpimltxrmpt.b $e |- ( ( ph /\ x e. A ) -> B e. V ) $. @@ -752947,7 +756573,7 @@ convergence can be decidedly irregular (Remark 121G of [Fremlin1] $} ${ - $d A x y $. $d B y $. $d L x y $. + $d A x $. $d L x $. smfpimgtxrmpt.x $e |- F/ x ph $. smfpimgtxrmpt.s $e |- ( ph -> S e. SAlg ) $. smfpimgtxrmpt.b $e |- ( ( ph /\ x e. A ) -> B e. V ) $. @@ -753942,19 +757568,19 @@ that every function in the sequence can have a different (partial) (Contributed by Glauco Siliprandi, 23-Oct-2021.) $) smfsup $p |- ( ph -> G e. ( SMblFn ` S ) ) $= ( vm nfcv vw vz cv cfv cle wbr wral cr wrex cdm ciin crab nffv nfdm nfiin - nfv nfbr nfralw nfrex wceq fveq2 cbviin a1i breq1d ralbidv fveq1d cbvralw - dmeqd wb bitrd rexbidv breq2 cbvrexvw cbvrabcsfw cmpt crn clt csup nfrab1 - eqtri nfcxfr nfmpt nfrn nfsup mpteq2dv cbvmpt eqtrd rneqd supeq1d cbvmptf - smfsuplem3 ) AUAUBDESGHIJMNOPDBUCZFUCZGUDZUDZCUCZUEUFZFJUGZCUHUIZBFJWNUJZ - UKZULZUAUCZSUCZGUDZUDZUBUCZUEUFZSJUGZUBUHUIZUASJXEUJZUKZULQWSXJBUAXAXLUAX - ATSBJXKBJTZBXEBXDGLBXDTUMZUNUOWSUAUPXIBUBUHBUHTZXHBSJXMBXFXGUEBXCXEXNBXCT - UMZBUETBXGTUQURUSXAXLUTWLXCUTZFSJWTXKSWTTFXEFXDGKFXDTUMZUNWMXDUTZWNXEWMXD - GVAZVHVBVCXQWSXFWPUEUFZSJUGZCUHUIZXJXQWRYBCUHXQWRXCWNUDZWPUEUFZFJUGZYBXQW - QYEFJXQWOYDWPUEWLXCWNVAZVDVEYFYBVIXQYEYAFSJYESUPFXFWPUEFXCXEXRFXCTUMZFUET - FWPTUQXSYDXFWPUEXSXCWNXEXTVFZVDVGVCVJVKYCXJVIXQYBXICUBUHWPXGUTYAXHSJWPXGX - FUEVLVEVMVCVJVNVTHBDFJWOVOZVPZUHVQVRZVOUADSJXFVOZVPZUHVQVRZVORBUADYLYOBDX - BQWSBXAVSWAUADTUAYLTBYNUHVQBYMBSJXFXMXPWBWCXOBVQTWDXQUHYKYNVQXQYJYMXQYJFJ - YDVOZYMXQFJWOYDYGWEYPYMUTXQFSJYDXFSYDTYHYIWFVCWGWHWIWJVTWK $. + nfv nfbr nfralw nfrexw wceq fveq2 dmeqd cbviin a1i breq1d ralbidv cbvralw + wb fveq1d bitrd rexbidv breq2 cbvrexvw cbvrabcsfw eqtri cmpt crn clt csup + nfrab1 nfcxfr nfmpt nfrn nfsup mpteq2dv cbvmpt supeq1d cbvmptf smfsuplem3 + eqtrd rneqd ) AUAUBDESGHIJMNOPDBUCZFUCZGUDZUDZCUCZUEUFZFJUGZCUHUIZBFJWNUJ + ZUKZULZUAUCZSUCZGUDZUDZUBUCZUEUFZSJUGZUBUHUIZUASJXEUJZUKZULQWSXJBUAXAXLUA + XATSBJXKBJTZBXEBXDGLBXDTUMZUNUOWSUAUPXIBUBUHBUHTZXHBSJXMBXFXGUEBXCXEXNBXC + TUMZBUETBXGTUQURUSXAXLUTWLXCUTZFSJWTXKSWTTFXEFXDGKFXDTUMZUNWMXDUTZWNXEWMX + DGVAZVBVCVDXQWSXFWPUEUFZSJUGZCUHUIZXJXQWRYBCUHXQWRXCWNUDZWPUEUFZFJUGZYBXQ + WQYEFJXQWOYDWPUEWLXCWNVAZVEVFYFYBVHXQYEYAFSJYESUPFXFWPUEFXCXEXRFXCTUMZFUE + TFWPTUQXSYDXFWPUEXSXCWNXEXTVIZVEVGVDVJVKYCXJVHXQYBXICUBUHWPXGUTYAXHSJWPXG + XFUEVLVFVMVDVJVNVOHBDFJWOVPZVQZUHVRVSZVPUADSJXFVPZVQZUHVRVSZVPRBUADYLYOBD + XBQWSBXAVTWAUADTUAYLTBYNUHVRBYMBSJXFXMXPWBWCXOBVRTWDXQUHYKYNVRXQYJYMXQYJF + JYDVPZYMXQFJWOYDYGWEYPYMUTXQFSJYDXFSYDTYHYIWFVDWJWKWGWHVOWI $. $} ${ @@ -753980,8 +757606,8 @@ that every function in the sequence can have a different (partial) cv cr wa eqidd fvmpt2d dmeqd nfcv nfcri nfan 3expa dmmptdf eqtr2d iineq2d eqid nfmpt1 nfmpt nffv nfiin rabeqf syl nfv nfii1 wb simpll simpr eliinid nfdm adantll eqtrd adantlr eleqtrd w3a fveq1d 3adant3 simp3 fvmpt4 breq1d - syl2anc syl3anc ralbida rexbid rabbida eqtrid nfra1 nfrex nfrabw rabidim1 - nfcxfr eleq2s sylan mpteq2da supeq1d mpteq12da fmptd2f smfsup eqeltrd + syl2anc syl3anc ralbida rexbid eqtrid nfra1 nfrexw nfrabw nfcxfr rabidim1 + rabbida eleq2s sylan mpteq2da supeq1d mpteq12da fmptd2f smfsup eqeltrd rneqd ) AIBBURZHURZHLBDEUCZUCZUDZUDZCURZUEUFZHLUGZCUSUHZBHLXSUIZUJZUKZHLX TUCZULZUSUMUNZUCZGUOUDZAIBFHLEUCZULZUSUMUNZUCYKUBABFYOYGYJNAFEYAUEUFZHLUG ZCUSUHZBHLDUJZUKZYGUAAYTYRBYFUKZYGAYSYFUPYTUUAUPAHLDYEMAXPLUQZUTZYEXQUIZD @@ -753990,10 +757616,10 @@ that every function in the sequence can have a different (partial) YDBYFNAXOYFUQZUTZYQYCCUSAUULCOUULCVRVFUUMYPYBHLAUULHMHBYFHLYEVSVEVFUUMUUB UTZAUUBUUHYPYBVTAUULUUBWAUUMUUBWBUUNXOYEDUULUUBXOYEUQAHXOLYEWCWEAUUBYEDUP UULUUCYEUUDDUUFUUJWFWGWHAUUBUUHWIZEXTYAUEUUOXTXOXQUDZEAUUBXTUUPUPUUHUUCXO - XSXQUUEWJWKUUOUUHUUIUUPEUPAUUBUUHWLSBDEKWMWOVIZWNWPWQWRWSWFWTAXOFUQZUTZUS - YNYIUMUUSYMYHUUSHLEXTAUURHMHBFHFYTUAYRHBYSYQHCUSHUSVDYPHLXAXBHLDVSXCXEVEV + XSXQUUEWJWKUUOUUHUUIUUPEUPAUUBUUHWLSBDEKWMWOVIZWNWPWQWRXEWFWSAXOFUQZUTZUS + YNYIUMUUSYMYHUUSHLEXTAUURHMHBFHFYTUAYRHBYSYQHCUSHUSVDYPHLWTXAHLDVSXBXCVEV FUUSUUBUTAUUBUUHEXTUPAUURUUBWAUUSUUBWBUURUUBUUHAUURXOYSUQZUUBUUHUUTXOYTFY - RBYSXDUAXFHXOLDWCXGWEUUQWPXHXNXIXJWTABCYGGHXRYKJLHLXQVLUUKPQRAHLXQYLMTXKY + RBYSXDUAXFHXOLDWCXGWEUUQWPXHXNXIXJWSABCYGGHXRYKJLHLXQVLUUKPQRAHLXQYLMTXKY GVKYKVKXLXM $. $} @@ -754111,19 +757737,19 @@ that every function in the sequence can have a different (partial) (Contributed by Glauco Siliprandi, 23-Oct-2021.) $) smfinf $p |- ( ph -> G e. ( SMblFn ` S ) ) $= ( vm nfcv vw vz cv cfv cle wbr wral cr wrex cdm ciin crab nffv nfdm nfiin - nfv nfbr nfralw nfrex wceq fveq2 cbviin a1i breq2d ralbidv fveq1d cbvralw - dmeqd wb bitrd rexbidv breq1 cbvrexvw cbvrabcsfw cmpt crn clt cinf nfrab1 - eqtri nfcxfr nfmpt nfrn nfinf mpteq2dv cbvmpt eqtrd rneqd infeq1d cbvmptf - smfinflem ) AUAUBDESGHIJMNOPDCUCZBUCZFUCZGUDZUDZUEUFZFJUGZCUHUIZBFJWOUJZU - KZULZUBUCZUAUCZSUCZGUDZUDZUEUFZSJUGZUBUHUIZUASJXFUJZUKZULQWSXJBUAXAXLUAXA - TSBJXKBJTZBXFBXEGLBXETUMZUNUOWSUAUPXIBUBUHBUHTZXHBSJXMBXCXGUEBXCTBUETBXDX - FXNBXDTUMZUQURUSXAXLUTWMXDUTZFSJWTXKSWTTFXFFXEGKFXETUMZUNWNXEUTZWOXFWNXEG - VAZVHVBVCXQWSWLXGUEUFZSJUGZCUHUIZXJXQWRYBCUHXQWRWLXDWOUDZUEUFZFJUGZYBXQWQ - YEFJXQWPYDWLUEWMXDWOVAZVDVEYFYBVIXQYEYAFSJYESUPFWLXGUEFWLTFUETFXDXFXRFXDT - UMZUQXSYDXGWLUEXSXDWOXFXTVFZVDVGVCVJVKYCXJVIXQYBXICUBUHWLXCUTYAXHSJWLXCXG - UEVLVEVMVCVJVNVTHBDFJWPVOZVPZUHVQVRZVOUADSJXGVOZVPZUHVQVRZVORBUADYLYOBDXB - QWSBXAVSWAUADTUAYLTBYNUHVQBYMBSJXGXMXPWBWCXOBVQTWDXQUHYKYNVQXQYJYMXQYJFJY - DVOZYMXQFJWPYDYGWEYPYMUTXQFSJYDXGSYDTYHYIWFVCWGWHWIWJVTWK $. + nfv nfbr nfralw nfrexw wceq fveq2 dmeqd cbviin a1i breq2d ralbidv cbvralw + wb fveq1d bitrd rexbidv breq1 cbvrexvw cbvrabcsfw eqtri cmpt crn clt cinf + nfrab1 nfcxfr nfmpt nfinf mpteq2dv cbvmpt eqtrd infeq1d cbvmptf smfinflem + nfrn rneqd ) AUAUBDESGHIJMNOPDCUCZBUCZFUCZGUDZUDZUEUFZFJUGZCUHUIZBFJWOUJZ + UKZULZUBUCZUAUCZSUCZGUDZUDZUEUFZSJUGZUBUHUIZUASJXFUJZUKZULQWSXJBUAXAXLUAX + ATSBJXKBJTZBXFBXEGLBXETUMZUNUOWSUAUPXIBUBUHBUHTZXHBSJXMBXCXGUEBXCTBUETBXD + XFXNBXDTUMZUQURUSXAXLUTWMXDUTZFSJWTXKSWTTFXFFXEGKFXETUMZUNWNXEUTZWOXFWNXE + GVAZVBVCVDXQWSWLXGUEUFZSJUGZCUHUIZXJXQWRYBCUHXQWRWLXDWOUDZUEUFZFJUGZYBXQW + QYEFJXQWPYDWLUEWMXDWOVAZVEVFYFYBVHXQYEYAFSJYESUPFWLXGUEFWLTFUETFXDXFXRFXD + TUMZUQXSYDXGWLUEXSXDWOXFXTVIZVEVGVDVJVKYCXJVHXQYBXICUBUHWLXCUTYAXHSJWLXCX + GUEVLVFVMVDVJVNVOHBDFJWPVPZVQZUHVRVSZVPUADSJXGVPZVQZUHVRVSZVPRBUADYLYOBDX + BQWSBXAVTWAUADTUAYLTBYNUHVRBYMBSJXGXMXPWBWJXOBVRTWCXQUHYKYNVRXQYJYMXQYJFJ + YDVPZYMXQFJWPYDYGWDYPYMUTXQFSJYDXGSYDTYHYIWEVDWFWKWGWHVOWI $. $} ${ @@ -754150,8 +757776,8 @@ that every function in the sequence can have a different (partial) wal 3expa fvmptelcdm dmmptdf 3eqtrrd iineq2d nfmpt1 nfmpt nffv nfdm nfiin rabeqf syl nfv nfii1 wb simpll eliinid adantll eqcomd adantlr eleqtrd w3a simpr fveq1d 3adant3 fvmpt2 syl2anc eqtr2d breq2d syl3anc ralbida rabbida - simp3 rexbid eqtrd alrimi nfra1 nfrabw nfcxfr eleq2i rabidim1 wi mpteq2da - nfrex biimpi idi rneqd infeq1d ex ralrimi mpteq12f fmptdf smfinf eqeltrd + simp3 rexbid eqtrd alrimi nfrexw nfrabw nfcxfr eleq2i biimpi rabidim1 idi + nfra1 wi mpteq2da rneqd infeq1d ex ralrimi mpteq12f fmptdf smfinf eqeltrd ) AIBCUCZBUCZHUCZHLBDEUDZUDZUEZUEZUFUGZHLUHZCUIUJZBHLYLUKZULZUMZHLYMUDZUN ZUIUOUPZUDZGUQUEZAIBFHLEUDZUNZUIUOUPZUDZUUCIUUHUSAUBURAFYSUSZBVLUUGUUBUSZ BFUHUUHUUCUSAUUIBNAFYGEUFUGZHLUHZCUIUJZBHLDULZUMZYSFUUOUSAUAURAUUOUUMBYRU @@ -754163,10 +757789,10 @@ that every function in the sequence can have a different (partial) HYQDUVHUUQYHYQUTAHYHLYQWIWJAUUQYQDUSUVHUURDYQUVFWKWLWMAUUQUVDWNZEYMYGUFUV LYMYHYJUEZEAUUQYMUVMUSUVDUURYHYLYJUUSWPWQUVLUVDUVEUVMEUSAUUQUVDXESBDEKYJU VCWRWSWTZXAXBXCXFXDXGXGXHAUUJBFNAYHFUTZUUJAUVOVAZUIUUFUUAUOUVPUUEYTUVPHLE - YMAUVOHMHYHFUVJHFUUOUAUUMHBUUNUULHCUIHUIVEUUKHLXIXPHLDWFXJXKVFVGUVPUUQVAA + YMAUVOHMHYHFUVJHFUUOUAUUMHBUUNUULHCUIHUIVEUUKHLXPXIHLDWFXJXKVFVGUVPUUQVAA UUQUVDEYMUSZAUVOUUQWHUVPUUQWOUVOUUQUVDAUVOUUQVAYHUUNUTZUUQUVDUVOUVRUUQUVO - YHUUOUTZUVRUVOUVSFUUOYHUAXLXQUUMBUUNXMWDVJUVOUUQWOHYHLDWIWSWJUVLUVQXNUVNX - RXBXOXSXTYAYBBFUUGYSUUBYCWSXGABCYSGHYKUUCJLHLYJVRUVGPQRAHLYJUUDYKMTYKVHYD + YHUUOUTZUVRUVOUVSFUUOYHUAXLXMUUMBUUNXNWDVJUVOUUQWOHYHLDWIWSWJUVLUVQXQUVNX + OXBXRXSXTYAYBBFUUGYSUUBYCWSXGABCYSGHYKUUCJLHLYJVRUVGPQRAHLYJUUDYKMTYKVHYD YSVHUUCVHYEYF $. $} @@ -754873,8 +758499,6 @@ that every function in the sequence can have a different (partial) $} ${ - $d A y $. $d B y $. $d C y $. $d D y $. $d V x $. $d ph y $. - $d x y $. smfdivdmmbl.1 $e |- F/ x ph $. smfdivdmmbl.2 $e |- F/_ x B $. smfdivdmmbl.3 $e |- ( ph -> S e. SAlg ) $. @@ -755359,6 +758983,242 @@ that every function in the sequence can have a different (partial) $( End $[ set-mbox-ss.mm $] $) +$( Begin $[ set-mbox-et.mm $] $) +$( Skip $[ set-main.mm $] $) +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Ender Ting +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Increasing sequences and subsequences +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + ${ + $( Less-than class is never reflexive. (Contributed by Ender Ting, + 22-Nov-2024.) Prefer to specify theorem domain and then apply ~ ltnri . + (New usage is discouraged.) $) + et-ltneverrefl $p |- -. A < A $= + ( cxr wcel clt wbr wn xrltnr cop opelxp1 con3i ltrelxr sseli nsyl sylnibr + cxp df-br pm2.61i ) ABZCZAADZEZFAGSFZAAHZTCZUAUBUCRROZCZUDUFSAARRIJTUEUCK + LMAATPNQ $. + $} + + ${ + $( Alternative proof that equality is left-Euclidean, using ~ ax7 directly + instead of utility theorems; done for practice. (Contributed by Ender + Ting, 21-Dec-2024.) $) + et-equeucl $p |- ( x = z -> ( y = z -> x = y ) ) $= + ( weq wi equid ax7 com12 ax-mp syl ) ACDZCADZBCDZABDZEAADZKLEAFKOLACAGHIM + LNMCBDZLNEBBDZMPEBFMQPBCBGHILPNCABGHJHJ $. + $} + + ${ + $( The square root of a negative number is not a real number. (Contributed + by Ender Ting, 5-Jan-2025.) $) + et-sqrtnegnre $p |- ( ( A e. RR /\ A < 0 ) -> -. ( sqrt ` A ) e. RR ) $= + ( cr wcel cc0 clt wbr wa cle wi csqrt cfv simpr 0red ltnled biimpd impcom + wn id jcnd ancoms c2 cexp co wceq recn sqsqrtd sqge0 breq2 syl2imc nsyl ) + ABZCZADZEFZGULUMAHZFZIZAJKZUKCZUNULUQQUNULGULUPUNULLULUNUPQZULUNUTULAUMUL + RULMNOPSTULURUAUBUCZAUDZUSUMVAUOFZUPULAAUEUFURUGVBVCUPVAAUMUOUHOUIUJ $. + $} + + ${ + $d B t $. $d T t $. $d k t $. + natlocalincr.1 $e |- A. k e. ( 0 ..^ T ) A. t e. ( 1 ..^ ( T + 1 ) ) ( k < + t -> ( B ` k ) < ( B ` t ) ) $. + $( Global monotonicity on half-open range implies local monotonicity. + Inference form. (Contributed by Ender Ting, 22-Nov-2024.) $) + natlocalincr $p |- A. k e. ( 0 ..^ T ) ( B ` k ) < ( B ` ( k + 1 ) ) $= + ( cv cfv c1 caddc co clt wbr cc0 cfzo wcel wex wi wral ralimi cz wceq rsp + ovex isseti fzoaddel mpan2 0p1e1 oveq1i eleqtrdi eleq1 syl5ibrcom ralimia + 1z imim1d mp2b cr elfzoelz zre ltp1 3syl breq2 ax-2 syl5com breq2d biimpd + fveq2 a2i rspec eximdv mpi ax5e syl rgen ) DFZBGZVNHIJZBGZKLZDMCNJZVNVSOZ + VRAPZVRVTAFZVPUAZAPWAAVPVNHIUCUDVTWCVRAWCVRQZDVSWCVNWBKLZVOWBBGZKLZQZQZDV + SRZWCWGQZDVSRWDDVSRWHAHCHIJZNJZRZDVSRWBWMOZWHQZDVSRWJEWNWPDVSWHAWMUBSWPWI + DVSVTWCWOWHVTWOWCVPWMOVTVPMHIJZWLNJZWMVTHTOVPWROUMVNMCHUEUFWQHWLNUGUHUIWB + VPWMUJUKUNULUOWIWKDVSVTWCWEQWIWKVTWEWCVNVPKLZVTVNTOVNUPOWSVNMCUQVNURVNUSU + TWBVPVNKVAUKWCWEWGVBVCULWKWDDVSWCWGVRWCWGVRWCWFVQVOKWBVPBVFVDVEVGSUOVHVIV + JVRAVKVLVM $. + $} + + ${ + $d B a b k $. $d T a k t $. $d T a b k $. + natglobalincr.1 $e |- A. k e. ( 0 ..^ T ) ( B ` k ) < ( B ` ( k + 1 ) ) $. + natglobalincr.2 $e |- T e. ZZ $. + $( Local monotonicity on half-open integer range implies global + monotonicity. Inference form. (Contributed by Ender Ting, + 23-Nov-2024.) $) + natglobalincr $p |- A. k e. ( 0 ..^ T ) A. t e. ( ( k + 1 ) ... T ) ( B ` k + ) < ( B ` t ) $= + ( cv cfv clt wbr cc0 co wcel cz cle w3a wceq fveq2 cxr syl va vb c1 caddc + cfz wb elfzoelz peano2zd elfz1 sylancl breq2d rspec cop cxp df-br ltrelxr + cfzo sseli sylbi opelxp1 3ad2ant3 opelxp2 0red cr simp1 zre 3syl peano2re + simp21 zred elfzole1 ltp1d wa wi id leltletr mp2and 3ad2ant1 simp22 letrd + 3jca 0zd a1i elfzo mpd3an23 fvoveq1 breq12d vtoclri syl6bir simp3 xrlttrd + simp23 elfzoel2 elfzop1le2 fzindd sylbida rgen2 ) DGZBHZAGZBHZIJZDAKCUQLZ + WRUCUDLZCUELZWRXCMZWTXEMZWTNMXDWTOJWTCOJPZXBXFXDNMCNMZXGXHUFXFWRWRKCUGZUH + ZFWTXDCUIUJXFWSUAGZBHZIJWSXDBHZIJZWSUBGZBHZIJZWSXPUCUDLZBHZIJXBUAUBWTXDCX + LXDQXMXNWSIXLXDBRUKXLXPQXMXQWSIXLXPBRUKXLXSQXMXTWSIXLXSBRUKXLWTQXMXAWSIXL + WTBRUKXODXCEULXFXPNMZXDXPOJZXPCIJZPZXRPZWSXQXTXRXFWSSMZYDXRWSXQUMZSSUNZMZ + YFXRYGIMYIWSXQIUOIYHYGUPURUSZWSXQSSUTTVAXRXFXQSMZYDXRYIYKYJWSXQSSVBTVAYEX + QXTIJZXQXTUMZYHMZXTSMYEKXPOJZYCYLYEKXDXPYEVCYEWRVDMZXDVDMZYEXFWRNMZYPXFYD + XRVEXJWRVFZVGWRVHZTYEXPXFYAYBYCXRVIZVJXFYDKXDOJZXRXFKWROJZWRXDIJZUUBWRKCV + KXFYRUUDXJYRWRYSVLTXFKVDMZYPYQPZUUCUUDVMUUBVNXFYRYPUUFXJYSYPUUEYPYQYPVCYP + VOYTWAVGKWRXDVPTVQVRXFYAYBYCXRVSVTXFYAYBYCXRWLYEYAYOYCVMZYLVNUUAYAUUGXPXC + MZYLYAKNMXIUUHUUGUFYAWBXIYAFWCXPKCWDWEXOYLDXPXCWRXPQWSXQXNXTIWRXPBRWRXPUC + BUDWFWGEWHWITVQZYLYMIMYNXQXTIUOIYHYMUPURUSXQXTSSVBVGXFYDXRWJUUIWKXKWRKCWM + WRKCWNWOWPWQ $. + $} + + $c UpWord $. + + $( Extend class notation to include the set of strictly increasing + sequences. $) + cupword $a class UpWord S $. + + ${ + $d S k w $. + + ${ + $( Strictly increasing sequence is a sequence, adjacent elements of which + increase. (Contributed by Ender Ting, 19-Nov-2024.) $) + df-upword $a |- UpWord S = { w | ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` + w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) } $. + $} + + ${ + $( Empty set is an increasing sequence for every range. (Contributed by + Ender Ting, 19-Nov-2024.) $) + upwordnul $p |- (/) e. UpWord S $= + ( vw vk c0 cv wcel cfv c1 co wbr cc0 chash cmin wceq wi cvv wb ax-mp cz + cle cword caddc clt cfzo wral wa cab cupword wal 0ex elab6g wrd0 eleq1a + fveq2 hash0 eqtrdi oveq1d 0red lem1d eqbrtrd eqeltrdi 1zzd zsubcld fzon + 0z sylancr mpbid rzal syl jca mpgbir df-upword eleqtrri ) DBEZAUAZFZCEZ + VNGVQHUBIVNGUCJZCKVNLGZHMIZUDIZUEZUFZBUGZAUHDWDFZVNDNZWCOZBDPFWEWGBUIQU + JWCBDPUKRWFVPWBDVOFWFVPOAULDVOVNUMRWFWADNZWBWFVTKTJZWHWFVTKHMIKTWFVSKHM + WFVSDLGKVNDLUNUOUPZUQWFKWFURUSUTWFKSFVTSFWIWHQVEWFVSHWFVSKSWJVEVAWFVBVC + KVTVDVFVGVRCWAVHVIVJVKBACVLVM $. + $} + + ${ + $d A w $. + + $( Any increasing sequence is a sequence. (Contributed by Ender Ting, + 19-Nov-2024.) $) + upwordisword $p |- ( A e. UpWord S -> A e. Word S ) $= + ( vw vk cv cword wcel cupword eleq1 cfv c1 caddc clt wbr cc0 chash cmin + co cfzo wral wa df-upword abeq2i simplbi vtoclga ) CEZBFZGZAUGGCABHZUFA + UGIUFUIGUHDEZUFJUJKZLRUFJMNDOUFPJUKQRSRTZUHULUACUICBDUBUCUDUE $. + $} + + ${ + singoutnword.1 $e |- A e. _V $. + $( Singleton with character out of range ` V ` is not a word for that + range. (Contributed by Ender Ting, 21-Nov-2024.) $) + singoutnword $p |- ( -. A e. V -> -. <" A "> e. Word V ) $= + ( cs1 cword wcel cc0 cfv cvv wceq s1fv ax-mp c0 wne s1nz fstwrdne mpan2 + eqeltrrid con3i ) ADZBEFZABFUAAGTHZBAIZFUBAJCAUCKLUATMNUBBFAOBTPQRS $. + $} + + ${ + singoutnupword.1 $e |- A e. _V $. + $( Singleton with character out of range ` S ` is not an increasing + sequence for that range. (Contributed by Ender Ting, 22-Nov-2024.) $) + singoutnupword $p |- ( -. A e. S -> -. <" A "> e. UpWord S ) $= + ( wcel wn cs1 cword cupword singoutnword upwordisword nsyl ) ABDEAFZBGD + LBHDABCILBJK $. + $} + + ${ + $d A w $. + upwordsing.1 $e |- A e. S $. + $( Singleton is an increasing sequence for any compatible range. + (Contributed by Ender Ting, 21-Nov-2024.) $) + upwordsing $p |- <" A "> e. UpWord S $= + ( vw vk cs1 cv cword wcel cfv c1 caddc clt wbr cc0 chash cmin cfzo wceq + co wral wa cab cupword wi wal wb s1cl elab6g mp2b eleq1a c0 fveq2 s1len + oveq1d oveq1i 1m1e0 eqtri eqtrdi oveq2d fzo0 syl jca df-upword eleqtrri + rzal mpgbir ) AFDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBDUCBUDAFDGBHIEGDGJEG + KLTDGJMNEODGPJKQTRTUAUBDUCIDGAFSDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBUEDA + BIAFBHIAFDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBDUCIDGAFSDGBHIEGDGJEGKLTDGJ + MNEODGPJKQTRTUAUBUEDUFUGCABUHDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBDAFBHUI + UJDGAFSDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAABIAFBHIDGAFSDGBHIUECABUHAFBHDG + UKUJDGAFSODGPJKQTRTULSEGDGJEGKLTDGJMNEODGPJKQTRTUADGAFSODGPJKQTRTOORTUL + DGAFSDGPJKQTOORDGAFSDGPJKQTAFPJKQTODGAFSDGPJAFPJKQDGAFPUMUOAFPJKQTKKQTO + AFPJKKQAUNUPUQURUSUTOVAUSEGDGJEGKLTDGJMNEODGPJKQTRTVFVBVCVGDBEVDVE $. + $} + + ${ + $d S t $. + upwordsseti.1 $e |- S e. _V $. + $( Strictly increasing sequences with a set given for range form a set. + (Contributed by Ender Ting, 21-Nov-2024.) $) + upwordsseti $p |- UpWord S e. _V $= + ( vt cupword cword wrdexi cv upwordisword ssriv ssexi ) ADZAEZABFCKLCGA + HIJ $. + $} + + ${ + $d A b k $. $d S b k $. + tworepnotupword.1 $e |- A e. _V $. + $( Concatenation of identical singletons is never an increasing sequence. + (Contributed by Ender Ting, 22-Nov-2024.) $) + tworepnotupword $p |- -. ( <" A "> ++ <" A "> ) e. UpWord S $= + ( vb vk co wcel wn wceq cfv c1 caddc clt wbr cc0 chash cmin cfzo ax-mp + cz cs1 cconcat cupword ovex cv wral wrex wa wex isseti wi 0z ccat2s1len + c0ex c2 oveq1i 2m1e1 eqtri eqeltri breqtrri fzolb mpbir3an eleq1a fveq2 + 1z 0lt1 oveq1d oveq2d eleq2d syl5ibr et-ltneverrefl fveq1 cvv ccat2s1p1 + eqtrdi 1e0p1 fveq2i ccat2s1p2 eqtr3i breq12d mtbiri fvoveq1 biimpd jcad + con3d syl5com eximdv mpi nfre1 rspe exlimi rexnal sylib cword df-upword + syl abeq2i simprbi nsyl eleq1 mtbid vtocle ) AUAZXCUBFZBUCZGZHDXDXCXCUB + UDDUEZXDIZXGXEGZXFXHEUEZXGJZXJKLFXGJZMNZEOXGPJZKQFZRFZUFZXIXHXMHZEXPUGZ + XQHXHXJXPGZXRUHZEUIZXSXHXJOIZEUIYBEOUNUJXHYCYAEXHYCXTXRYCXTXHXJOXDPJZKQ + FZRFZGZOYFGZYCYGUKYHOTGYETGOYEMNULYEKTYEUOKQFKYDUOKQAAUMUPUQURZVEUSOKYE + MVFYIUTOYEVAVBOYFXJVCSXHXPYFXJXHXOYEORXHXNYDKQXGXDPVDVGVHVIVJXHOXGJZOKL + FZXGJZMNZHYCXRXHYMAAMNAVKXHYJAYLAMXHYJOXDJZAOXGXDVLAVMGZYNAICVMAAVNSVOX + HYLYKXDJZAYKXGXDVLKXDJZYPAKYKXDVPVQYOYQAICVMAAVRSVSVOVTWAYCXMYMYCXMYMYC + XKYJXLYLMXJOXGVDXJOKXGLWBVTWCWEWFWDWGWHYAXSEXREXPWIXREXPWJWKWPXMEXPWLWM + XIXGBWNGZXQYRXQUHDXEDBEWOWQWRWSXGXDXEWTXAXB $. + $} + + ${ + $d S a $. $d T a $. + $( There is a finite number of strictly increasing sequences of a given + length over finite alphabet. Trivially holds for invalid lengths + where there're zero matching sequences. (Contributed by Ender Ting, + 5-Jan-2024.) $) + upwrdfi $p |- ( S e. Fin -> { a e. UpWord S | ( # ` a ) = T } e. Fin ) $= + ( cfn wcel cv chash cfv wceq cword crab cupword wrdnfi wss upwordisword + wtru ad2antrl rabss3d mptru ssfi mpan2 syl ) ADZECFZGHBIZCAJZKZUCEZUECA + LZKZUCEZCBAMUHUJUGNZUKULPZUECUIUFUDUIEUDUFEUMUEUDAOQRSUGUJTUAUB $. + $} + $} + + $( + This theorem will show that a given distribution in RR may be satisfied by + a relatively short sequence of value transfers. + ${ + paysd.1 $e |- ( ph -> A e. Fin ) $. + paysd.2 $e |- ( ph -> B : A --> RR ) $. + paysd.3 $e |- ( ph -> sum_ k e. A ( B ` k ) = 0 ) $. + paysd.4 $e |- ( ph -> C = (/) ) $. + paysd $p |- ( ph -> ( C e. Word ( ( A X. A ) X. RR ) /\ ( # ` C ) <_ ( # + ` A ) ) ) $= ( cxp cr cword wcel chash cfv cle wbr c0 wrd0 a1i eqeltrd + cc0 wceq fveq2 hash0 eqtrdi syl cfn cn0 hashcl nn0ge0d eqbrtrd jca ) + ACBBFGFZHZICJZKZBULKZLZMACNZUKEUPUKIAUJOPQAUMRZUNU + OACUPSZUMUQSEURUMUPULKUQCUPULTUAUBUCAUNABUDIUNUEID BUFUCUGUHUI $. + $} + $) + +$( (End of Ender Ting's mathbox.) $) +$( End $[ set-mbox-et.mm $] $) + + $( Begin $[ set-mbox-ju.mm $] $) $( Skip $[ set-main.mm $] $) $( @@ -755437,8 +759297,8 @@ that every function in the sequence can have a different (partial) Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.) $) hirstL-ax3 $p |- ( ( -. ph -> -. ps ) -> ( ( -. ph -> ps ) -> ph ) ) $= - ( wn wi wo pm4.64 pm4.66 pm2.64 com12 sylbi syl5bi ) ACZBDABEZLBCZDZAABFOAN - EZMADABGMPAABHIJK $. + ( wn wi wo pm4.64 pm4.66 pm2.64 com12 sylbi biimtrid ) ACZBDABEZLBCZDZAABFO + ANEZMADABGMPAABHIJK $. $( Recover ~ ax-3 from ~ hirstL-ax3 . (Contributed by Jarvin Udandy, 3-Jul-2015.) (Proof modification is discouraged.) @@ -757727,7 +761587,7 @@ class of all the singletons of (proper) ordered pairs over the elements fsetsnfo $p |- ( S e. V -> F : B -onto-> A ) $= ( vm vn wcel cv wceq wrex cop csn weq opeq2 wf cfv wral wfo fsetsnf eqeq1 vex rexbidv elab2 sneqd eqeq2d cbvrexvw wa simpr cvv cmpt a1i adantl snex - fvmptd eqcomd adantr eqtrd reximdva syl5bi imp ralrimiva dffo3 sylanbrc + fvmptd eqcomd adantr eqtrd reximdva biimtrid imp ralrimiva dffo3 sylanbrc ex ) EGMZDCFUAKNZLNZFUBZOZLDPZKCUCDCFUDABCDEFGHIJUEVKVPKCVKVLCMZVPVQVLEHN ZQZRZOZHDPZVKVPBNZVTOZHDPWBBVLCKUGBKSWDWAHDWCVLVTUFUHIUIWBVLEVMQZRZOZLDPV KVPWAWGHLDHLSZVTWFVLWHVSWEVRVMETUJUKULVKWGVOLDVKVMDMZUMZWGVOWJWGUMVLWFVNW @@ -757801,16 +761661,16 @@ class of all the singletons of (proper) ordered pairs over the elements ( vm vn wa wceq vy wcel wf cv cfv weq wral wf1 cfsetsnfsetf cfsetsnfsetfv wi cmpt ad2ant2r ad2ant2rl eqeq12d cvv wb ralrimiva mpteqb syl simplr idd fvexd rspcimdv csn vex elab2 anbi12i w3a simp3 simp1r fveq2 ralsng mpbird - feq1 wfn anim12i 3ad2ant2 eqfnfv 3exp syl5bi syld sylbid ralrimivva dff13 - ffn imp sylanbrc ) CJUBZKCUBZSZHGIUCQUDZIUEZRUDZIUEZTZQRUFZUKZRHUGQHUGHGI - UHABCDEFGHIJKLMNOPUIWKWRQRHHWKWLHUBZWNHUBZSZSZWPLCKWLUEZULZLCKWNUEZULZTZW - QXBWMXDWOXFWIWSWMXDTWJWTABCDEFGHIJWLKLMNOPUJUMWIWTWOXFTWJWSABCDEFGHIJWNKL - MNOPUJUNUOXBXGXCXETZLCUGZWQXBXCUPUBZLCUGXGXIUQXBXJLCXBLUDZCUBSKWLVCURLCXC - XEUPUSUTXBXIXHWQXBXHXHLKCWIWJXAVAXBXKKTSXHVBVDWKXAXHWQUKZXAKVEZDWLUCZXMDW - NUCZSZWKXLWSXNWTXOXMDAUDZUCZXNAWLHQVFXMDXQWLVOOVGXRXOAWNHRVFXMDXQWNVOOVGV - HWKXPXHWQWKXPXHVIZWQUAUDZWLUEZXTWNUEZTZUAXMUGZXSYDXHWKXPXHVJXSWJYDXHUQWIW - JXPXHVKYCXHUAKCXTKTYAXCYBXEXTKWLVLXTKWNVLUOVMUTVNXSWLXMVPZWNXMVPZSZWQYDUQ - XPWKYGXHXNYEXOYFXMDWLWFXMDWNWFVQVRUAXMWLWNVSUTVNVTWAWGWBWCWCWDQRHGIWEWH + feq1 wfn ffn anim12i 3ad2ant2 eqfnfv 3exp biimtrid syld sylbid ralrimivva + imp dff13 sylanbrc ) CJUBZKCUBZSZHGIUCQUDZIUEZRUDZIUEZTZQRUFZUKZRHUGQHUGH + GIUHABCDEFGHIJKLMNOPUIWKWRQRHHWKWLHUBZWNHUBZSZSZWPLCKWLUEZULZLCKWNUEZULZT + ZWQXBWMXDWOXFWIWSWMXDTWJWTABCDEFGHIJWLKLMNOPUJUMWIWTWOXFTWJWSABCDEFGHIJWN + KLMNOPUJUNUOXBXGXCXETZLCUGZWQXBXCUPUBZLCUGXGXIUQXBXJLCXBLUDZCUBSKWLVCURLC + XCXEUPUSUTXBXIXHWQXBXHXHLKCWIWJXAVAXBXKKTSXHVBVDWKXAXHWQUKZXAKVEZDWLUCZXM + DWNUCZSZWKXLWSXNWTXOXMDAUDZUCZXNAWLHQVFXMDXQWLVOOVGXRXOAWNHRVFXMDXQWNVOOV + GVHWKXPXHWQWKXPXHVIZWQUAUDZWLUEZXTWNUEZTZUAXMUGZXSYDXHWKXPXHVJXSWJYDXHUQW + IWJXPXHVKYCXHUAKCXTKTYAXCYBXEXTKWLVLXTKWNVLUOVMUTVNXSWLXMVPZWNXMVPZSZWQYD + UQXPWKYGXHXNYEXOYFXMDWLVQXMDWNVQVRVSUAXMWLWNVTUTVNWAWBWFWCWDWDWEQRHGIWGWH $. $d A a b y z $. $d B b n x y z $. $d F m n $. $d H b $. $d V y $. @@ -757823,20 +761683,20 @@ class of all the singletons of (proper) ordered pairs over the elements fveq1 adantr eqeq1d ralbidva rexbidv anbi12d elab2 cmpt csn simpllr fmptd eqid snex mptex sylibr wb mpteq2dv eqeq2d adantl simpr eqidd snidg fvmptd ad6antlr eqtr4d ralimdva imp wfn ffn cvv nfv fnmptd jca eqfnfv syl mpbird - ex fvexd rspcedvd simp-4l cfsetsnfsetfv rexbidva rexlimdva expimpd syl5bi - sylan ralrimiv dffo3 sylanbrc ) CJQZKCQZRZHGIUDUASZUBSZIUEZTZUBHUFZUAGUGH - GIUHABCDEFGHIJKLMNOPUIXDXIUAGXEGQCDXEUDZBSZXEUEZMSZTZBCUGZMDUFZRZXDXICDES - ZUDZXKXRUEZXMTZBCUGZMDUFZRXQEXEGUAUJEUAUKZXSXJYCXPCDXRXEULYDYBXOMDYDYAXNB - CYDXKCQZRXTXLXMYDXTXLTYEXKXRXEUMUNUOUPUQURNUSXDXJXPXIXDXJRZXOXIMDYFXMDQZR - ZXOXIYHXORZXIXELCKXFUEZUTZTZUBHUFYIYLXELCKUCKVAZXMUTZUEZUTZTZUBYNHYIYMDYN - UDZYNHQYIUCYMXMDYNYFYGXOUCSZYMQVBYNVDVCYMDASZUDYRAYNHUCYMXMKVEVFYMDYTYNUL - OUSVGXFYNTZYLYQVHYIUUAYKYPXEUUALCYJYOKXFYNUMVIVJVKYIYQXLXKYPUEZTZBCUGZYHX - OUUDYHXNUUCBCYHYERZXNUUCUUEXNRZXLXMUUBUUEXNVLUUFLXKYOXMCYPDUUFYPVMUUFLBUK - ZRZUCKXMXMYMYNDUUHYNVMUUHYSKTRXMVMXCKYMQXBXJYGYEXNUUGKCVNVPUUFYGUUGYFYGYE - XNVBZUNVOUUEYEXNYHYEVLUNUUIVOVQWIVRVSYIXECVTZYPCVTZRZYQUUDVHYHUULXOYFUULY - GYFUUJUUKXJUUJXDCDXEWAVKYFLCYOYPWBYFLWCYFLSCQRKYNWJYPVDWDWEUNUNBCXEYPWFWG - WHWKYIXHYLUBHYIXFHQZRXGYKXEYIXBUUMXGYKTXBXCXJYGXOWLABCDEFGHIJXFKLMNOPWMWR - VJWNWHWIWOWPWQWSUBUAHGIWTXA $. + ex fvexd rspcedvd simp-4l cfsetsnfsetfv sylan rexbidva rexlimdva biimtrid + expimpd ralrimiv dffo3 sylanbrc ) CJQZKCQZRZHGIUDUASZUBSZIUEZTZUBHUFZUAGU + GHGIUHABCDEFGHIJKLMNOPUIXDXIUAGXEGQCDXEUDZBSZXEUEZMSZTZBCUGZMDUFZRZXDXICD + ESZUDZXKXRUEZXMTZBCUGZMDUFZRXQEXEGUAUJEUAUKZXSXJYCXPCDXRXEULYDYBXOMDYDYAX + NBCYDXKCQZRXTXLXMYDXTXLTYEXKXRXEUMUNUOUPUQURNUSXDXJXPXIXDXJRZXOXIMDYFXMDQ + ZRZXOXIYHXORZXIXELCKXFUEZUTZTZUBHUFYIYLXELCKUCKVAZXMUTZUEZUTZTZUBYNHYIYMD + YNUDZYNHQYIUCYMXMDYNYFYGXOUCSZYMQVBYNVDVCYMDASZUDYRAYNHUCYMXMKVEVFYMDYTYN + ULOUSVGXFYNTZYLYQVHYIUUAYKYPXEUUALCYJYOKXFYNUMVIVJVKYIYQXLXKYPUEZTZBCUGZY + HXOUUDYHXNUUCBCYHYERZXNUUCUUEXNRZXLXMUUBUUEXNVLUUFLXKYOXMCYPDUUFYPVMUUFLB + UKZRZUCKXMXMYMYNDUUHYNVMUUHYSKTRXMVMXCKYMQXBXJYGYEXNUUGKCVNVPUUFYGUUGYFYG + YEXNVBZUNVOUUEYEXNYHYEVLUNUUIVOVQWIVRVSYIXECVTZYPCVTZRZYQUUDVHYHUULXOYFUU + LYGYFUUJUUKXJUUJXDCDXEWAVKYFLCYOYPWBYFLWCYFLSCQRKYNWJYPVDWDWEUNUNBCXEYPWF + WGWHWKYIXHYLUBHYIXFHQZRXGYKXEYIXBUUMXGYKTXBXCXJYGXOWLABCDEFGHIJXFKLMNOPWM + WNVJWOWHWIWPWRWQWSUBUAHGIWTXA $. $( The mapping of the class of singleton functions into the class of constant functions is a bijection. (Contributed by AV, 14-Sep-2024.) $) @@ -757932,29 +761792,29 @@ restrictions of the composed functions (to their minimum domains). fcoresf1 $p |- ( ph -> ( X : P -1-1-> E /\ Y : E -1-1-> D ) ) $= ( wceq wi vx vy va vb wf1 wf cv cfv wral wfo fcoreslem3 fof syl ccom wa dff13 wcel fcoresf1lem adantrr adantrl eqeq12d imbi1d a1i imim1d sylbid - fveq2 ralimdvva adantld syl5bi mpd sylanbrc cres crn cin inss2 eqsstrdi - fssresd feq1i sylibr wrex fcoreslem2 eqcomd eleq2d wfn wb fvelrnb bitrd - fofn anbi12d fveqeq2 eqeq1 imbi12d eqeq2d equequ2 rspc2v adantl syld ex - imim2d impl eqeqan12rd eqeq12 ancoms syl5ibcom expd rexlimdva ralrimivv - com23 impd jca ) AFGJUEZGEKUEZAFGJUFZUAUGZJUHZUBUGZJUHZSZXNXPSZTZUBFUIU - AFUIZXKAFGJUJZXMABCDFGHJLMNOUKZFGJULUMAFEIHUNZUEZYARYEFEYDUFZXNYDUHZXPY - DUHZSZXSTZUBFUIUAFUIZUOZAYAUAUBFEYDUPZAYKYAYFAYJXTUAUBFFAXNFUQZXPFUQZUO - UOZYJXOKUHZXQKUHZSZXSTXTYPYIYSXSYPYGYQYHYRAYNYGYQSYOABCDEFGHIJKXNLMNOPQ - URUSAYOYHYRSYNABCDEFGHIJKXPLMNOPQURUTVAVBYPXRYSXSXRYSTYPXOXQKVFVCVDVEVG - VHVIVJUAUBFGJUPVKAGEKUFZXNKUHZXPKUHZSZXSTZUBGUIUAGUIXLAGEIGVLZUFYTADEGI - PAGHVMZDVNZDGUUGSAMVCUUFDVOVPVQGEKUUEQVRVSAUUDUAUBGGAXNGUQZXPGUQZUOUCUG - ZJUHZXNSZUCFVTZUDUGZJUHZXPSZUDFVTZUOUUDAUUHUUMUUIUUQAUUHXNJVMZUQZUUMAGU - URXNAUURGABCDFGHJLMNOWAWBZWCAJFWDZUUSUUMWEAYBUVAYCFGJWHUMZUCFXNJWFUMWGA - UUIXPUURUQZUUQAGUURXPUUTWCAUVAUVCUUQWEUVBUDFXPJWFUMWGWIAUUMUUQUUDAUULUU - QUUDTUCFAUUJFUQZUOZUUQUULUUDUVEUUPUULUUDTUDFUVEUUNFUQZUOZUUPUULUUDUVGUU - KKUHZUUOKUHZSZUUKUUOSZTZUUPUULUOZUUDAUVDUVFUVLAYEUVDUVFUOZUVLTZRYEYLAUV - OYMAYKUVOYFAUVNYKUVLAUVNYKUVLTAUVNUOZYKUUJYDUHZUUNYDUHZSZUUJUUNSZTZUVLU - VNYKUWATAYJUWAUVQYHSZUUJXPSZTUAUBUUJUUNFFXNUUJSYIUWBXSUWCXNUUJYHYDWJXNU - UJXPWKWLXPUUNSZUWBUVSUWCUVTUWDYHUVRUVQXPUUNYDVFWMUBUDUCWNWLWOWPUVPUWAUV - JUVTTUVLUVPUVSUVJUVTUVPUVQUVHUVRUVIAUVDUVQUVHSUVFABCDEFGHIJKUUJLMNOPQUR - USAUVFUVRUVISUVDABCDEFGHIJKUUNLMNOPQURUTVAVBUVPUVTUVKUVJUVTUVKTUVPUUJUU - NJVFVCWSVEWQWRXHVHVIVJWTUVMUVJUUCUVKXSUULUUPUVHUUAUVIUUBUUKXNKVFUUOXPKV - FXAUULUUPUVKXSWEUUKXNUUOXPXBXCWLXDXEXFXHXFXIVEXGUAUBGEKUPVKXJ $. + fveq2 ralimdvva adantld biimtrid mpd sylanbrc cres crn eqsstrdi fssresd + cin inss2 feq1i sylibr wrex fcoreslem2 eqcomd eleq2d fofn fvelrnb bitrd + wfn wb anbi12d fveqeq2 imbi12d eqeq2d equequ2 rspc2v adantl imim2d syld + eqeq1 com23 impl eqeqan12rd eqeq12 ancoms syl5ibcom expd rexlimdva impd + ex ralrimivv jca ) AFGJUEZGEKUEZAFGJUFZUAUGZJUHZUBUGZJUHZSZXNXPSZTZUBFU + IUAFUIZXKAFGJUJZXMABCDFGHJLMNOUKZFGJULUMAFEIHUNZUEZYARYEFEYDUFZXNYDUHZX + PYDUHZSZXSTZUBFUIUAFUIZUOZAYAUAUBFEYDUPZAYKYAYFAYJXTUAUBFFAXNFUQZXPFUQZ + UOUOZYJXOKUHZXQKUHZSZXSTXTYPYIYSXSYPYGYQYHYRAYNYGYQSYOABCDEFGHIJKXNLMNO + PQURUSAYOYHYRSYNABCDEFGHIJKXPLMNOPQURUTVAVBYPXRYSXSXRYSTYPXOXQKVFVCVDVE + VGVHVIVJUAUBFGJUPVKAGEKUFZXNKUHZXPKUHZSZXSTZUBGUIUAGUIXLAGEIGVLZUFYTADE + GIPAGHVMZDVPZDGUUGSAMVCUUFDVQVNVOGEKUUEQVRVSAUUDUAUBGGAXNGUQZXPGUQZUOUC + UGZJUHZXNSZUCFVTZUDUGZJUHZXPSZUDFVTZUOUUDAUUHUUMUUIUUQAUUHXNJVMZUQZUUMA + GUURXNAUURGABCDFGHJLMNOWAWBZWCAJFWGZUUSUUMWHAYBUVAYCFGJWDUMZUCFXNJWEUMW + FAUUIXPUURUQZUUQAGUURXPUUTWCAUVAUVCUUQWHUVBUDFXPJWEUMWFWIAUUMUUQUUDAUUL + UUQUUDTUCFAUUJFUQZUOZUUQUULUUDUVEUUPUULUUDTUDFUVEUUNFUQZUOZUUPUULUUDUVG + UUKKUHZUUOKUHZSZUUKUUOSZTZUUPUULUOZUUDAUVDUVFUVLAYEUVDUVFUOZUVLTZRYEYLA + UVOYMAYKUVOYFAUVNYKUVLAUVNYKUVLTAUVNUOZYKUUJYDUHZUUNYDUHZSZUUJUUNSZTZUV + LUVNYKUWATAYJUWAUVQYHSZUUJXPSZTUAUBUUJUUNFFXNUUJSYIUWBXSUWCXNUUJYHYDWJX + NUUJXPWRWKXPUUNSZUWBUVSUWCUVTUWDYHUVRUVQXPUUNYDVFWLUBUDUCWMWKWNWOUVPUWA + UVJUVTTUVLUVPUVSUVJUVTUVPUVQUVHUVRUVIAUVDUVQUVHSUVFABCDEFGHIJKUUJLMNOPQ + URUSAUVFUVRUVISUVDABCDEFGHIJKUUNLMNOPQURUTVAVBUVPUVTUVKUVJUVTUVKTUVPUUJ + UUNJVFVCWPVEWQXHWSVHVIVJWTUVMUVJUUCUVKXSUULUUPUVHUUAUVIUUBUUKXNKVFUUOXP + KVFXAUULUUPUVKXSWHUUKXNUUOXPXBXCWKXDXEXFWSXFXGVEXIUAUBGEKUPVKXJ $. $} $( A composition is injective iff the restrictions of its components to the @@ -758339,8 +762199,8 @@ Restricted quantification (extension) using implicit substitution, analogous to ~ cbvrex2v . (Contributed by Alexander van der Vekens, 2-Jul-2017.) $) cbvrex2 $p |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps ) $= - ( wrex nfcv nfrex weq cbvrexw rexbidv rexbii bitri ) AEIPZDHPCEIPZFHPBGIP - ZFHPUDUEDFHAFEIFIQJRCDEIDIQKRDFSACEINUATUEUFFHCBEGILMOTUBUC $. + ( wrex nfcv nfrexw weq cbvrexw rexbidv rexbii bitri ) AEIPZDHPCEIPZFHPBGI + PZFHPUDUEDFHAFEIFIQJRCDEIDIQKRDFSACEINUATUEUFFHCBEGILMOTUBUC $. $} $( Example for a theorem about a restricted universal quantification in which @@ -758402,17 +762262,17 @@ the restricting class depends on (actually is) the bound variable: All <-> A = B ) ) $= ( vy wcel wa cv wceq wo wi eqeq1 orbi12d eqeq2 imbi12d rspcv wn com12 ex wreu weq wral wrex reu8 simprlr ioran eqid pm2.24i simplbiim eqtr2 ancoms - syl expimpd ja syld simprll adantr sylbi jaoi impd rexlimdva syl5bi reueq - a1d biimpi adantl wb orbi1d oridm bitrdi reubidv mpbird impbid ) BDFZCDFZ - GZAHZBIZVQCIZJZADTZBCIZWAVTEHZBIZWCCIZJZAEUAZKZEDUBZGZADUCVPWBVTWFAEDWGVR - WDVSWEVQWCBLVQWCCLMUDVPWJWBADVPVQDFZGZVTWIWBVTWLWIWBKZVRWLWMKVSVRWLWMVRWL - GZWICBIZCCIZJZVSKZWBWNVOWIWRKVRVNVOWKUEWHWRECDWEWFWQWGVSWEWDWOWEWPWCCBLWC - CCLMWCCVQNOPULWRWNWBWQVSWNWBKZWQQWOQWPQWSWOWPUFWPWSCUGUHUIVSVRWLWBVSVRGWB - WLVRVSWBVQBCUJZUKVDUMUNRUOSVSWLWMVSWLGZWIBBIZWBJZVRKZWBXAVNWIXDKVSVNVOWKU - PWHXDEBDWDWFXCWGVRWDWDXBWEWBWCBBLWCBCLMWCBVQNOPULXDXAWBXCVRXAWBKZXCQXBQZW - BQZGXEXBWBUFXFXEXGXBXEBUGUHUQURVRVSWLWBVRVSGWBWLWTVDUMUNRUOSUSRUTVAVBVPWB - WAVPWBGZWAVSADTZVPXIWBVOXIVNVOXIADCVCVEVFUQXHVTVSADXHVTVSVSJVSXHVRVSVSWBV - RVSVGVPBCVQNVFVHVSVIVJVKVLSVM $. + syl a1d expimpd ja syld simprll adantr sylbi jaoi impd rexlimdva biimtrid + reueq biimpi adantl wb orbi1d oridm bitrdi reubidv mpbird impbid ) BDFZCD + FZGZAHZBIZVQCIZJZADTZBCIZWAVTEHZBIZWCCIZJZAEUAZKZEDUBZGZADUCVPWBVTWFAEDWG + VRWDVSWEVQWCBLVQWCCLMUDVPWJWBADVPVQDFZGZVTWIWBVTWLWIWBKZVRWLWMKVSVRWLWMVR + WLGZWICBIZCCIZJZVSKZWBWNVOWIWRKVRVNVOWKUEWHWRECDWEWFWQWGVSWEWDWOWEWPWCCBL + WCCCLMWCCVQNOPULWRWNWBWQVSWNWBKZWQQWOQWPQWSWOWPUFWPWSCUGUHUIVSVRWLWBVSVRG + WBWLVRVSWBVQBCUJZUKUMUNUORUPSVSWLWMVSWLGZWIBBIZWBJZVRKZWBXAVNWIXDKVSVNVOW + KUQWHXDEBDWDWFXCWGVRWDWDXBWEWBWCBBLWCBCLMWCBVQNOPULXDXAWBXCVRXAWBKZXCQXBQ + ZWBQZGXEXBWBUFXFXEXGXBXEBUGUHURUSVRVSWLWBVRVSGWBWLWTUMUNUORUPSUTRVAVBVCVP + WBWAVPWBGZWAVSADTZVPXIWBVOXIVNVOXIADCVDVEVFURXHVTVSADXHVTVSVSJVSXHVRVSVSW + BVRVSVGVPBCVQNVFVHVSVIVJVKVLSVM $. $} @@ -758493,12 +762353,12 @@ Analogs to Existential uniqueness (double quantification) ( et -> ( b = y /\ ( ps -> a = x ) ) ) ) ) $= ( vu wreu weq wi wral wa wrex reubii imbi1d ralbidv anbi12d rexbidv bitri reu8 wsb cv wcel nfv nfs1v nfcv nfim nfan sbequ12 equequ1 imbi12d cbvrexw - nfralw imbi1i ralbii anbi2i r19.41 bitr4i r19.28v simplr equequ2 ad2antrl - nfrex anbi2d adantl imp sbievw bicomi sbbii sbco2vv pm3.35 equcomd sylbir - rspc ex com12 ad2antlr simplrr ad2antrr wb sbequ sbbidv imbi2d vex sbc2ie - wsbc a1i biimprd adantld sbsbc sylibr pm2.27 ax7 syl6 syld ralimdva exp31 - syl5bi com24 imp41 jca rspcedvd sbcom2 sbf 3bitri anbi12i rexbii impancom - syl exp32 jcad expimpd ralrimivv syl5 expd impd reximdva reximia sylbi + nfralw imbi1i ralbii anbi2i nfrexw r19.41 bitr4i r19.28v equequ2 ad2antrl + simplr anbi2d rspc adantl imp sbievw bicomi sbco2vv pm3.35 equcomd sylbir + sbbii ex com12 ad2antlr simplrr ad2antrr wb sbbidv imbi2d wsbc vex sbc2ie + sbequ a1i biimprd adantld sbsbc sylibr pm2.27 biimtrid syl6 syld ralimdva + ax7 exp31 com24 imp41 jca rspcedvd sbcom2 sbf 3bitri anbi12i syl impancom + rexbii exp32 jcad expimpd ralrimivv syl5 expd impd reximdva reximia sylbi mpd ) AIMUDZHLUDZACIJUEZUFZJMUGZUHZIMUIZEDYSUFZJMUGZUHZIMUIZHKUEZUFZKLUGZ UHZHLUIZAFOIUEZBNHUEZUFZUHUFZOMUGNLUGZUHZIMUIZHLUIYRUUCHLUDUULYQUUCHLACIJ MRUPUJUUCUUGHKLUUHUUBUUFIMUUHAEUUAUUEPUUHYTUUDJMUUHCDYSQUKULUMUNUPUOUUKUU @@ -758506,31 +762366,31 @@ Analogs to Existential uniqueness (double quantification) USUUKUUCUVHUHUVJUUJUVHUUCUUIUVGKLUUGUVFUUHUUFUVEIUCMUUFUCUTUUTUVDIEIUCVAU VCIJMIMVBZUVAUVBIDIUCVAUVBIUTVCVIVDZIUCUEZEUUTUUEUVDEIUCVEUVNUUDUVCJMUVND UVAYSUVBDIUCVEIUCJVFVGULUMVHVJVKVLUUBUVHIMUVGIKLILVBUVFUUHIUVEIUCMUVLUVMV - SUUHIUTVCVIVMVNUVKUVIUURIMUVKIURMUSUHZUUBUVHUURUVOAUUAUVHUURUFUVOAUHZUUAU - VHUURUUAUVHUHUUAUVGUHZKLUGZUVPUURUUAUVGKLVOUVPUVRUURUVPUVRUHZAUUQUVOAUVRV - PUVSUUPNOLMUVPNURZLUSZOURZMUSZUHZUVRUUPUVPUWDUHZUVRUUAUUTKNUQZUVAKNUQZUVB + MUUHIUTVCVIVNVOUVKUVIUURIMUVKIURMUSUHZUUBUVHUURUVOAUUAUVHUURUFUVOAUHZUUAU + VHUURUUAUVHUHUUAUVGUHZKLUGZUVPUURUUAUVGKLVPUVPUVRUURUVPUVRUHZAUUQUVOAUVRV + SUVSUUPNOLMUVPNURZLUSZOURZMUSZUHZUVRUUPUVPUWDUHZUVRUUAUUTKNUQZUVAKNUQZUVB UFZJMUGZUHZUCMUIZHNUEZUFZUHZUUPUWAUVRUWNUFUVPUWCUVQUWNKUVTLUUAUWMKUUAKUTU - WKUWLKUWJKUCMKMVBZUWFUWIKUUTKNVAUWHKJMUWOUWGUVBKUVAKNVAUVBKUTVCVIVDVSUWLK + WKUWLKUWJKUCMKMVBZUWFUWIKUUTKNVAUWHKJMUWOUWGUVBKUVAKNVAUVBKUTVCVIVDVMUWLK UTVCVDKNUEZUVGUWMUUAUWPUVFUWKUUHUWLUWPUVEUWJUCMUWPUUTUWFUVDUWIUUTKNVEUWPU - VCUWHJMUWPUVAUWGUVBUVAKNVEUKULUMUNKNHVQVGVTWJVRUWEUUAUWMUUPUWEUUAUHZCJOUQ + VCUWHJMUWPUVAUWGUVBUVAKNVEUKULUMUNKNHVQVGVTWAVRUWEUUAUWMUUPUWEUUAUHZCJOUQ ZIOUEZUFZUWMUUPUFZUWEUUAUWTUWDUUAUWTUFZUVPUWCUXBUWAYTUWTJUWBMUWRUWSJCJOVA - UWSJUTVCJOUECUWRYSUWSCJOVEJOIVQVGWJWAWAWBUWTAIOUQZUWSUFZUWQUXAUWRUXCUWSUW - 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sylbi nrexdv syl5bi ralrimiv oteq3 cbviunv eleq2i notbii ralbii disj olcd - ex sneqd pm2.61i ralrimivva oteq1 iuneq2d disjor ) ADJZEGUAZFCEKZAFKZLZMZ - NZFCGKZAWJLZMZNZUBUCOZUDZGBPEBPEBWMUEWGWSEGBBWHWGWIBJZWNBJZQZQZWSUFWHWSXC - WHWRUGUHWHRZXCWSXDXCQZWRWHXEHKZWQJZRZHWMPZWRXEXFICWNAIKZLZMZNZJZRZHWMPXIX - EXOHWMXFWMJXFWLJZFCUIXEXOFXFCWLUJXEXPXOFCXEWJCJZQZXPQZXFXLJZICUIXNXSXTICX - SXJCJZQZXFXKOZXTYBYCRZWKXKOZRZXSYFYAXRYFXPXDXCXQYFXQXCXDYFXQXCXDYFUFXQXCQ - ZYEWHYGYEWHAAOZFIUAZURZWHYGWTWGXQYEYJUKXCWTXQWGWTXAULUMXQWGXBULXQXCUNWIAW - JWNBAXJDCUOUPWHYHYIUQUSUTVTVAVBSSXSYDYFUKZYAXPYKXRXPXFWKOZYKHWKVCYLYCYEXF - WKXKVDVEVIUMSVFHXKVCVGVJIXFCXLUJVGVHVKVLXHXOHWMXGXNWQXMXFFICWPXLYIWOXKWJX - JWNAVMWAVNVOVPVQTHWMWQVRTVSVTWBWCBWMWQEGWHFCWLWPWHWKWOWIWNAWJWDWAWEWFT $. + ex sylbi nrexdv biimtrid ralrimiv oteq3 cbviunv eleq2i notbii ralbii disj + sneqd olcd pm2.61i ralrimivva oteq1 iuneq2d disjor ) ADJZEGUAZFCEKZAFKZLZ + MZNZFCGKZAWJLZMZNZUBUCOZUDZGBPEBPEBWMUEWGWSEGBBWHWGWIBJZWNBJZQZQZWSUFWHWS + XCWHWRUGUHWHRZXCWSXDXCQZWRWHXEHKZWQJZRZHWMPZWRXEXFICWNAIKZLZMZNZJZRZHWMPX + IXEXOHWMXFWMJXFWLJZFCUIXEXOFXFCWLUJXEXPXOFCXEWJCJZQZXPQZXFXLJZICUIXNXSXTI + CXSXJCJZQZXFXKOZXTYBYCRZWKXKOZRZXSYFYAXRYFXPXDXCXQYFXQXCXDYFXQXCXDYFUFXQX + CQZYEWHYGYEWHAAOZFIUAZURZWHYGWTWGXQYEYJUKXCWTXQWGWTXAULUMXQWGXBULXQXCUNWI + AWJWNBAXJDCUOUPWHYHYIUQUSUTVIVAVBSSXSYDYFUKZYAXPYKXRXPXFWKOZYKHWKVCYLYCYE + XFWKXKVDVEVJUMSVFHXKVCVGVKIXFCXLUJVGVHVLVMXHXOHWMXGXNWQXMXFFICWPXLYIWOXKW + JXJWNAVNVTVOVPVQVRTHWMWQVSTWAVIWBWCBWMWQEGWHFCWLWPWHWKWOWIWNAWJWDVTWEWFT + $. $} @@ -760631,15 +764492,15 @@ Negated membership (alternative) (Contributed by AV, 31-Jul-2022.) $) f1oresf1o2 $p |- ( ph -> ( F |` D ) : D -1-1-onto-> { y e. B | ch } ) $= ( cv wceq wrex wcel wa syl adantr wi ex cfv w3a wf1o f1of sselda ffvelcdm - wf jca 3adant3 wb eleq1 3ad2ant3 mpbid eqcom biimpd com23 3imp rexlimdv3a - syl5bi wfo f1ofo foelcdmi sylan nfv nfre1 nfim rspe expcom eqcoms sylbird - adantl rexlimd syld impd impbid f1oresf1o ) ABCDEFGHIJACLZHUAZDLZMZCGNZVS - FOZBPZAVTWCCGAVQGOZVTUBZWBBWEVRFOZWBWEEFHUGZVQEOZPZWFAWDWIVTAWDPWGWHAWGWD - AEFHUCZWGIEFHUDQRAGEVQJUEUHUIEFVQHUFQVTAWFWBUJWDVRVSFUKULUMAWDVTBAVTWDBVT - VSVRMZAWDBSZVRVSUNZAWKWLAWKPZWDBKUOTUSUPUQUHURAWBBWAAWBVTCENZBWASZAWBWOAE - FHUTZWBWOAWJWQIEFHVAQCEFHVSVBVCTAVTWPCEACVDBWACBCVDVTCGVEVFAWHVTWPSVTWKAW - HPWPWMAWKWPSWHAWKWPWNBWDWAKWKWDWASZAWRVRVSWDVTWAVTCGVGVHVIVKVJTRUSTVLVMVN - VOVP $. + wf 3adant3 wb eleq1 3ad2ant3 mpbid eqcom biimpd biimtrid com23 rexlimdv3a + jca 3imp wfo f1ofo foelcdmi sylan nfre1 nfim expcom eqcoms adantl sylbird + nfv rspe rexlimd syld impd impbid f1oresf1o ) ABCDEFGHIJACLZHUAZDLZMZCGNZ + VSFOZBPZAVTWCCGAVQGOZVTUBZWBBWEVRFOZWBWEEFHUGZVQEOZPZWFAWDWIVTAWDPWGWHAWG + WDAEFHUCZWGIEFHUDQRAGEVQJUEURUHEFVQHUFQVTAWFWBUIWDVRVSFUJUKULAWDVTBAVTWDB + VTVSVRMZAWDBSZVRVSUMZAWKWLAWKPZWDBKUNTUOUPUSURUQAWBBWAAWBVTCENZBWASZAWBWO + AEFHUTZWBWOAWJWQIEFHVAQCEFHVSVBVCTAVTWPCEACVJBWACBCVJVTCGVDVEAWHVTWPSVTWK + AWHPWPWMAWKWPSWHAWKWPWNBWDWAKWKWDWASZAWRVRVSWDVTWAVTCGVKVFVGVHVITRUOTVLVM + VNVOVP $. $} @@ -761129,17 +764990,17 @@ Infinity and the extended real number system (cont.) - extension cfzo wn cuz cfv cn elfzo0 nnre 3ad2ant2 nn0re resubcl 3ad2ant1 lenlt bicomd syl2anc biimpa nnz zsubcl syl2an impancom imp subge0 mpbird elnn0z sylanbrc nn0z ltle simplr1 nn0sub mpbid elnn0uz sylib ltsub1d wceq nncn nn0cn breq2d - cc nncan biimpd sylbid ex com3l 3impia impcom 3jca exp31 3adant2 sylbi 3imp - syl5bi elfzo2 sylibr ) ADCUAEZFZBWMFZBCAGEZHIUBZJBWPGEZDUCUDFZAKFZWRAHIZJZW - RDAUAEFWNWOWQXBWNALFZCUEFZACHIZJWOWQXBMZMZACUFXCXEXGXDWOBLFZXDBCHIZJZXCXENZ - XFBCUFXKXJWQXBXKXJNZWQNZWSWTXAXMWRLFZWSXMWPBOIZXNXLWQXOXLWPPFZBPFZWQXOQXJCP - FZAPFZXPXKXDXHXRXICUGZUHZXCXSXEAUIZRZCAUJZSXJXQXKXHXDXQXIBUIZUKTXPXQNXOWQWP - BULUMUNUOXMWPLFZXHXOXNQXLYFWQXLWPKFZDWPOIZYFXJCKFZWTYGXKXDXHYIXICUPUHXCWTXE - AVERZCAUQSXLYHACOIZXKXJYKXCXJXEYKXCXSXRXEYKMXJYBYAACVFURUSUTXJXRXSYHYKQXKYA - YCCAVASVBWPVCVDRXHXDXIXKWQVGWPBVHUNVIWRVJVKXLWTWQXKWTXJYJRRXLXAWQXJXKXAXHXD - XIXKXAMXKXHXDNZXIXAXCYLXIXAMZMXEXCYLYMXCYLNZXIWRCWPGEZHIZXAYNBCWPYLXQXCXHXQ - XDYERTYLXRXCXDXRXHXTTZTYLXRXSXPXCYQYBYDSVLYNYPXAYNYOAWRHYLCVQFZAVQFYOAVMXCX - DYRXHCVNTAVOCAVRSVPVSVTWARWBWCWDRWEWFWJWGWHWIWRDAWKWL $. + cc nncan biimpd sylbid com3l 3impia impcom 3jca exp31 biimtrid 3adant2 3imp + ex sylbi elfzo2 sylibr ) ADCUAEZFZBWMFZBCAGEZHIUBZJBWPGEZDUCUDFZAKFZWRAHIZJ + ZWRDAUAEFWNWOWQXBWNALFZCUEFZACHIZJWOWQXBMZMZACUFXCXEXGXDWOBLFZXDBCHIZJZXCXE + NZXFBCUFXKXJWQXBXKXJNZWQNZWSWTXAXMWRLFZWSXMWPBOIZXNXLWQXOXLWPPFZBPFZWQXOQXJ + CPFZAPFZXPXKXDXHXRXICUGZUHZXCXSXEAUIZRZCAUJZSXJXQXKXHXDXQXIBUIZUKTXPXQNXOWQ + WPBULUMUNUOXMWPLFZXHXOXNQXLYFWQXLWPKFZDWPOIZYFXJCKFZWTYGXKXDXHYIXICUPUHXCWT + XEAVERZCAUQSXLYHACOIZXKXJYKXCXJXEYKXCXSXRXEYKMXJYBYAACVFURUSUTXJXRXSYHYKQXK + YAYCCAVASVBWPVCVDRXHXDXIXKWQVGWPBVHUNVIWRVJVKXLWTWQXKWTXJYJRRXLXAWQXJXKXAXH + XDXIXKXAMXKXHXDNZXIXAXCYLXIXAMZMXEXCYLYMXCYLNZXIWRCWPGEZHIZXAYNBCWPYLXQXCXH + XQXDYERTYLXRXCXDXRXHXTTZTYLXRXSXPXCYQYBYDSVLYNYPXAYNYOAWRHYLCVQFZAVQFYOAVMX + CXDYRXHCVNTAVOCAVRSVPVSVTWIRWAWBWCRWDWEWFWGWJWHWRDAWKWL $. $( A half-open integer range can represent an ordered pair, analogous to ~ fzopth . (Contributed by Alexander van der Vekens, 1-Jul-2018.) $) @@ -762023,46 +765884,46 @@ all preimages of values of a function and the range of the function (see cxr simpr ciccp cc0 cfz cn0 nnnn0 nn0fz0 sylib iccpartxr cpnf cmnf w3o cz elfzoelz ad2antll caddc cuz elfzo2 cle eluzelz peano2zd 3ad2ant1 simp2 wb elxr zltp1le sylan biimp3a eluz2 syl3anbrc sylbi eqcomd eleq1d com12 1red - biimpcd elfz2 zre syl3anc expcomd adantrd 3adant2 3ad2ant3 syl5bi 3adant3 - letr wne anim12ci 3adant1 ltlen nesym anbi2i bitr2di expd adantld elfzelz - biimpd 1zzd elfzel2 3jca 3ad2ant2 elfzo mpbird 3exp impcom iccpartipre ex - jca pm2.61i cmin cmap 1eluzge0 fzoss1 mp1i elfzoel2 fzoval eleq2d elfzouz - wss sseld sylbid breq2 sseldd iccpartimp simprd smonoord ltpnf elfzubelfz - syl5ibr elfzofz iccpartgtprec eqcoms c2 nnne0 biimpri nn0n0n1ge2 ige2m1fz - df-ne nltmnf pm2.21dd 3jaoi impl ralrimiva mpcom expcom mpd ) ADUBGZCUCZB - HZDBHZIJZCKDUDLZUEZEDKUFZAUVEUVKMZMUVLUVKAUVEUVLUVKUVGKBHZIJZCUGUEUVOCUHU - VLUVIUVOCUVJUGUVLUVJKKUDLUGDKKUDUIKUJUKUVLUVHUVNUVGIDKBULUMUNUOUPAUVLUQZU - 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(Contributed by AV, 12-Jul-2020.) $) @@ -762073,39 +765934,39 @@ all preimages of values of a function and the range of the function (see cuz fveq2i fveq2 eqcomd eleq1d biimpcd ad3antrrr wne cn ciccp elfz2nn0 cz cle w3a elfzo2 simpl1 simpr2 nn0ge0 0red eluzelre syl3anc expcomd syl5com zre lelttr 3impia 3ad2ant2 imp elnnz sylanbrc nn0re expd exp31 com34 3imp - com35 expdcom 3imp1 elfzo0 syl3anbrc syl5bi sylbi impcom simpr fzo1fzo0n0 - ex iccpartipre exp32 expdimp pm2.61dne cmin cmap ad3antlr elfzoelz fzoval - eleq2d wss elfzouz2 fzoss2 sseld sylbid iccpartimp simprd smonoord lbfzo0 - ralrimiva sylibr 3jca wb breq1 mpbid 1nn0 a1i nnnn0 nnge1 eqeltrid pnfnlt - pm2.21dd mnflt ralbidv mpbird 3jaoi mpcom expcom pm2.61i ) DGUDZAHBIZCUBZ - BIZJKZCGDUELZUFZMUUOUVAAUUOUVAUUSCUCUFUUSCUGUUOUUSCUUTUCUUOUUTGGUELUCDGGU - EUHGURUIUJUKULAUUOUMZUVAUUPUNNZAUVBOZUVAAUVCUVBABHDEFADUONZHHDUPLZNADEUQD - USPUTQUVCUUPRNZUUPVAUDZUUPVBUDZVCUVDUVAMZUUPVDUVGUVJUVHUVIUVGUVDUVAUVGUVD - OZUUSCUUTUVKUUQUUTNZOZUABHUUQUVMVEUVLUUQHGVFLZVJIZNUVKUVLUUQGVJIZUVOUUQGD - 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Corresponds to ~ fourierdlem11 in GS's mathbox. @@ -762134,12 +765995,12 @@ all preimages of values of a function and the range of the function (see ( vk cv cfv clt wbr cc0 cfzo co wcel c1 cun cz wceq syl csn caddc w3a 0zd cn nnz nngt0 3jca fzopred 0p1e1 a1i oveq1d uneq2d eqtrd eleq2d wo elun wi elsni wa fveq2 adantr iccpartlt adantl eqbrtrd ex wral breq1d iccpartiltu - weq rspccv syl11 jaoi com12 syl5bi sylbid ralrimiv ) ACHZBIZDBIZJKZCLDMNZ - AVRWBOVRLUAZPDMNZQZOZWAAWBWEVRAWBWCLPUBNZDMNZQZWEALROZDROZLDJKZUCZWBWISAD - UEOZWMEWNWJWKWLWNUDDUFDUGUHTLDUITAWHWDWCAWGPDMWGPSAUJUKULUMUNUOWFVRWCOZVR - WDOZUPZAWAVRWCWDUQWQAWAWOAWAURZWPWOVRLSZWRVRLUSWSAWAWSAUTVSLBIZVTJWSVSWTS - AVRLBVAVBAWTVTJKWSABDEFVCVDVEVFTGHZBIZVTJKZGWDVGWPWAAXCWAGVRWDGCVJXBVSVTJ - XAVRBVAVHVKABGDEFVIVLVMVNVOVPVQ $. + weq rspccv syl11 jaoi com12 biimtrid sylbid ralrimiv ) ACHZBIZDBIZJKZCLDM + NZAVRWBOVRLUAZPDMNZQZOZWAAWBWEVRAWBWCLPUBNZDMNZQZWEALROZDROZLDJKZUCZWBWIS + ADUEOZWMEWNWJWKWLWNUDDUFDUGUHTLDUITAWHWDWCAWGPDMWGPSAUJUKULUMUNUOWFVRWCOZ + VRWDOZUPZAWAVRWCWDUQWQAWAWOAWAURZWPWOVRLSZWRVRLUSWSAWAWSAUTVSLBIZVTJWSVSW + TSAVRLBVAVBAWTVTJKWSABDEFVCVDVEVFTGHZBIZVTJKZGWDVGWPWAAXCWAGVRWDGCVJXBVSV + TJXAVRBVAVHVKABGDEFVIVLVMVNVOVPVQ $. $( If there is a partition, then all intermediate points and the upper bound are strictly greater than the lower bound. 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Q c = { x , y } ) ) $= ( wcel cv wral wceq wa wi imp weq com12 cxp wss wbr w3a cpr wrex cspr cfv - wb cop df-br simpl adantl opelxp sylib prelspr syl2an2r ex syl5bi 3adant3 - ssel wo vex preq12b breq12 biimpd adantr ancomsd 3ad2ant3 com23 bi2anan9r - rsp2 eleq1 ancoms imbi12d mpbid expimpd jaoi ralrimivva crab eleq2i eqeq1 - imbi1d 2ralbidv elrab bitri sylanbrc eqidd rspcev syl2anc prsprel mpanr12 - syl6bi preq1 eqeq2d breq1 preq2 breq2 rspc2v imp4c mpcom sylanb rexlimiva - cvv a1d impbid ) EFLZDEEUAZUBZAMZBMZDUCZXKXJDUCZUIZBENAENZUDZXLJMZXJXKUEZ - OZJCUFZXPXLXTXPXLPZXRCLZXRXROZXTYAXREUGUHZLZXRHMZIMZUEZOZYFYGDUCZQZIENHEN - ZYBXPXLYEXGXIXLYEQXOXLXJXKUJZDLZXGXIPZYEXJXKDUKYOYNYEYOXGYNXJELZXKELZPZYE - XGXIULYOYNPYMXHLZYRYOYNYSXIYNYSQXGDXHYMVAUMRXJXKEEUNUOEFXJXKUPUQURUSUTRYA - YKHIEEYIAHSBISPZAISZBHSZPZVBZYAYFELZYGELZPZPZYJXJXKYFYGAVCZBVCZHVCIVCVDUU - DUUHYJYTUUHYJQUUCUUHYTYJYAYTYJQZUUGXLUUKXPYTXLYJYTXLYJXJYFXKYGDVEVFTUMVGT - UUCYAUUGYJUUCYAPYQYPPZXMQZUUGYJQZYAUUMUUCXPXLUUMXPUULXLXMXOXGUULXLXMQZQXI - 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elpw eqcom syl5bi ancoms eqssd ex syl2anb sylbid rgen2 dff13 mpbir2an ) - CDEUACDEUBMOZEPZNOZEPZQZMNUCZUDZNCUEMCUEABCDEFGHIJKLUFVPMNCCVJCRZVLCRZS - VNIOAOBOUGQZIVJUIABUHZVSIVLUIABUHZQZVOVQVRVKVTVMWAABCDEFVJGHIJKLUJABCDE - FVLGHIJKLUJUKVQVJFULPZTZVLWCTZWBVOUDVRVQVJWCUMZRWDCWFVJJUNVJWCMUOUSUPVR - VLWFRWECWFVLJUNVLWCNUOUSUPWDWESZWBVOWGWBSVJVLWGWBVJVLTABFMNIUQURWGWBVLV - JTZWEWDWBWHUDWBWAVTQWEWDSWHVTWAUTABFNMIUQVAVBURVCVDVEVFVGMNCDEVHVI $. + elpw eqcom biimtrid ancoms eqssd ex syl2anb sylbid rgen2 dff13 mpbir2an + ) CDEUACDEUBMOZEPZNOZEPZQZMNUCZUDZNCUEMCUEABCDEFGHIJKLUFVPMNCCVJCRZVLCR + ZSVNIOAOBOUGQZIVJUIABUHZVSIVLUIABUHZQZVOVQVRVKVTVMWAABCDEFVJGHIJKLUJABC + DEFVLGHIJKLUJUKVQVJFULPZTZVLWCTZWBVOUDVRVQVJWCUMZRWDCWFVJJUNVJWCMUOUSUP + VRVLWFRWECWFVLJUNVLWCNUOUSUPWDWESZWBVOWGWBSVJVLWGWBVJVLTABFMNIUQURWGWBV + LVJTZWEWDWBWHUDWBWAVTQWEWDSWHVTWAUTABFNMIUQVAVBURVCVDVEVFVGMNCDEVHVI $. $d a b c f q t x y $. $d r t $. $d F f t $. $d P f t $. $d R f p $. $d R t $. $d V f q c t x y $. $d W a b c f t x y $. @@ -763356,19 +767217,20 @@ valid cases ( ` (/) ` is the last symbol) and invalid cases ( ` (/) ` cpr copab cxp cpw wbr breq bibi12d 2ralbidv elrab2 cspr sprsymrelfolem1 crab eqid eleqtrri rexeq opabbidv eqeq2d adantl wss velpw wrel cvv xpss wb sstr2 mpi df-rel sylibr dfrel4v nfv nfra1 nfan sprsymrelfolem2 3expa - nfra2w opabbid biimpd ex com23 syl5bi expcom sylbi rspcedvd sprsymrelfv - mpd imp31 impcom rexbidva mpbird ralrimiva dffo3 sylanbrc ) FGUDZCDEUEZ - NPZOPZEUFZQZOCUGZNDRCDEUHXDXCABCDEFHIJKLMUIUJXCXINDXCXEDUDZSZXIXEJPAPZB - PZUKQZJXFUGZABULZQZOCUGZXJXCXRXJXEFFUMZUNZUDZXLXMXEUOZXMXLXEUOZVNZBFRZA - FRZSZXCXRTXLXMHPZUOZXMXLYHUOZVNZBFRAFRYFHXEXTDYHXEQZYKYDABFFYLYIYBYJYCX - LXMYHXEUPXMXLYHXEUPUQURLUSYGXCXRYGXCSZXQXEXNJUAPUBPZUCPZUKQYNYOXEUOTUCF - RUBFRUAFUTUFZVBZUGZABULZQZOYQCYQCUDYMYQYPUNCYQXEFUAUBUCYQVCZVAKVDUJXFYQ - QZXQYTVNYMUUBXPYSXEUUBXOYRABXNJXFYQVEVFVGVHYAYFXCYTYAXEXSVIZYFXCYTTTNXS - VJUUCXCYFYTXCUUCYFYTTZXCUUCSZXEVKZUUDUUCUUFXCUUCXEVLVLUMZVIZUUFUUCXSUUG - VIUUHFFVMXEXSUUGVOVPXEVQVRVHUUFXEYBABULZQZUUEUUDABXEVSUUEYFUUJYTUUEYFUU - JYTTUUEYFSZUUJYTUUKUUIYSXEUUKYBYRABUUEYFAUUEAVTYEAFWAWBUUEYFBUUEBVTYDAB - FFWEWBXCUUCYFYBYRVNABYQXEFGUAUBUCJUUAWCWDWFVGWGWHWIWJWOWKWIWLWPWMWHWLWQ - XKXHXQOCXKXFCUDZSXGXPXEUULXGXPQXKABCDEFXFHIJKLMWNVHVGWRWSWTONCDEXAXB $. + nfra2w opabbid biimpd ex com23 biimtrid mpd expcom sylbi imp31 rspcedvd + impcom sprsymrelfv rexbidva mpbird ralrimiva dffo3 sylanbrc ) FGUDZCDEU + EZNPZOPZEUFZQZOCUGZNDRCDEUHXDXCABCDEFHIJKLMUIUJXCXINDXCXEDUDZSZXIXEJPAP + ZBPZUKQZJXFUGZABULZQZOCUGZXJXCXRXJXEFFUMZUNZUDZXLXMXEUOZXMXLXEUOZVNZBFR + ZAFRZSZXCXRTXLXMHPZUOZXMXLYHUOZVNZBFRAFRYFHXEXTDYHXEQZYKYDABFFYLYIYBYJY + CXLXMYHXEUPXMXLYHXEUPUQURLUSYGXCXRYGXCSZXQXEXNJUAPUBPZUCPZUKQYNYOXEUOTU + CFRUBFRUAFUTUFZVBZUGZABULZQZOYQCYQCUDYMYQYPUNCYQXEFUAUBUCYQVCZVAKVDUJXF + YQQZXQYTVNYMUUBXPYSXEUUBXOYRABXNJXFYQVEVFVGVHYAYFXCYTYAXEXSVIZYFXCYTTTN + XSVJUUCXCYFYTXCUUCYFYTTZXCUUCSZXEVKZUUDUUCUUFXCUUCXEVLVLUMZVIZUUFUUCXSU + UGVIUUHFFVMXEXSUUGVOVPXEVQVRVHUUFXEYBABULZQZUUEUUDABXEVSUUEYFUUJYTUUEYF + UUJYTTUUEYFSZUUJYTUUKUUIYSXEUUKYBYRABUUEYFAUUEAVTYEAFWAWBUUEYFBUUEBVTYD + ABFFWEWBXCUUCYFYBYRVNABYQXEFGUAUBUCJUUAWCWDWFVGWGWHWIWJWKWLWIWMWNWOWHWM + WPXKXHXQOCXKXFCUDZSXGXPXEUULXGXPQXKABCDEFXFHIJKLMWQVHVGWRWSWTONCDEXAXB + $. $( The mapping ` F ` is a bijection between the subsets of the set of pairs over a fixed set ` V ` into the symmetric relations ` R ` on the @@ -763470,7 +767332,7 @@ valid cases ( ` (/) ` is the last symbol) and invalid cases ( ` (/) ` ( va vb wcel wa cinf csup cop cv wrex wbr cif ifcl wor cxp cin cpr prpair wceq wne simpll simplrl simplrr infsupprpr syl13anc df-br sylib w3a infpr simprr 3adant1 eqeltrd suppr jca 3expb adantr opelxp sylibr infeq1 supeq1 - elind wb opeq12d eleq1d ad2antrl mpbird rexlimdvva syl5bi imp eleqtrrdi + elind wb opeq12d eleq1d ad2antrl mpbird rexlimdvva biimtrid imp eleqtrrdi ex ) DBUAZEAKZLEDBMZEDBNZOZBDDUBZUCZCVSVTWCWEKZVTEIPZJPZUDZUFZWGWHUGZLZJD QIDQVSWFFADEIJHUEVSWLWFIJDDVSWGDKZWHDKZLZLZWLWFWPWLLZWFWIDBMZWIDBNZOZWEKZ WQBWDWTWQWRWSBRZWTBKWQVSWMWNWKXBVSWOWLUHVSWMWNWLUIVSWMWNWLUJWPWJWKUQDWGWH @@ -763508,49 +767370,49 @@ valid cases ( ` (/) ` is the last symbol) and invalid cases ( ` (/) ` sotric simplbiim impcom eqtrd anbi12d adantrr simpl simprll simprlr orc adantr syl3anc eqneqall adantld ancomd sylan2 anbi1d ancoms 3jaoi mpcom olc sylbid ad2antrl imp a1d ancom2s expdimp vex preq12b syl6ibr imbi12d - eqeqan12rd eqeq12 com12 mpbird rexlimdvva com13 syl5bi sylbi 3imp31 ) E - BUDZFALZGALZUEZGCUFZFCUFZMGEBUGZGEBUHZUJZFEBUGZFEBUHZUJZMZGFMZYJYKYOYLY - RYJHGHUKZEBUGZUUAEBUHZUJZYOACULKUUAGMUUBYMUUCYNEUUAGBUMEUUAGBUNUOYGYHYI - UPYOULLYJYMYNVBUQURYJHFUUDYRACULKUUAFMUUBYPUUCYQEUUAFBUMEUUAFBUNUOYGYHY - IUSYRULLYJYPYQVBUQURUTYIYHYGYSYTNZYIGUAUKZUBUKZVAZMZUUFUUGVCZOZUBEVPUAE - 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(Contributed by AV, 11-Aug-2021.) $) @@ -766500,19 +770362,19 @@ become clearer and sometimes also shorter (see, for example, ~ divgcdoddALTV va vb vz ceven oddz zaddcl syl2an eqeq1 rexbidv dfodd6 elrab2 ex ad3antlr adantr peano2zd wb oveq2 eqeq2d adantl oveq12 2cnd anim1i ancoms syl 1cnd mulcl sylan add4d simpl simpr adddid oveq1d addcl 1p1e2 eqtr4i a1i oveq2d - 2t1e2 3eqtr4rd eqtrd rspcedvd rexlimdva2 expimpd syl5bi dfeven4 sylanbrc - sylbi ) AFGZBFGZHABIJZKGZWKLCMZNJZOZCKPZWKUEGWIAKGZBKGZWLWJAUFBUFABUGUHWI - WJWPWIWQALDMZNJZQIJZOZDKPZHZWJWPRUBMZXAOZDKPXCUBAKFXEAOXFXBDKXEAXAUIUJUBD - UKULWJWRBLEMZNJZQIJZOZEKPZHZXDWPUCMZXIOZEKPXKUCBKFXMBOXNXJEKXMBXIUIUJUCEU - KULWQXCXLWPRZWQXBXODKWQWSKGZHZXBHZWRXKWPXRWRHZXJWPEKXSXGKGZHZXJHZWOWKLWSX - GIJZQIJZNJZOZCYDKYBYCYAYCKGZXJXSXTYGXPXTYGRWQXBWRXPXTYGWSXGUGUMUNSUOUPWMY - DOZWOYFUQYBYHWNYEWKWMYDLNURUSUTYBWKXAXIIJZYEYAXJWKYIOZXBXJYJRXQWRXTXBXJYJ - AXABXIIVAUMUNSYAYIYEOZXJXSXTYKXPXTYKRWQXBWRXPXTYKXPWSTGZXGTGZYKXTWSUAXGUA - YLYMHZYIWTXHIJZQQIJZIJZYEYNWTQXHQYNLTGZYLHZWTTGYMYLYSYMYRYLYMVBVCVDLWSVGV - EYNVFZYLYRYMXHTGYLVBLXGVGVHYTVIYNLYCNJZLQNJZIJYOUUBIJYEYQYNUUAYOUUBIYNLWS - XGYNVBZYLYMVJYLYMVKVLVMYNLYCQUUCWSXGVNYTVLYNYPUUBYOIYPUUBOYNYPLUUBVOVSVPV - QVRVTWAUHUMUNSUOWAWBWCWDWCSWEWHSUDMZWNOZCKPWPUDWKKUEUUDWKOUUEWOCKUUDWKWNU - IUJUDCWFULWG $. + 2t1e2 3eqtr4rd eqtrd rspcedvd rexlimdva2 expimpd biimtrid sylbi sylanbrc + dfeven4 ) AFGZBFGZHABIJZKGZWKLCMZNJZOZCKPZWKUEGWIAKGZBKGZWLWJAUFBUFABUGUH + WIWJWPWIWQALDMZNJZQIJZOZDKPZHZWJWPRUBMZXAOZDKPXCUBAKFXEAOXFXBDKXEAXAUIUJU + BDUKULWJWRBLEMZNJZQIJZOZEKPZHZXDWPUCMZXIOZEKPXKUCBKFXMBOXNXJEKXMBXIUIUJUC + EUKULWQXCXLWPRZWQXBXODKWQWSKGZHZXBHZWRXKWPXRWRHZXJWPEKXSXGKGZHZXJHZWOWKLW + SXGIJZQIJZNJZOZCYDKYBYCYAYCKGZXJXSXTYGXPXTYGRWQXBWRXPXTYGWSXGUGUMUNSUOUPW + MYDOZWOYFUQYBYHWNYEWKWMYDLNURUSUTYBWKXAXIIJZYEYAXJWKYIOZXBXJYJRXQWRXTXBXJ + YJAXABXIIVAUMUNSYAYIYEOZXJXSXTYKXPXTYKRWQXBWRXPXTYKXPWSTGZXGTGZYKXTWSUAXG + UAYLYMHZYIWTXHIJZQQIJZIJZYEYNWTQXHQYNLTGZYLHZWTTGYMYLYSYMYRYLYMVBVCVDLWSV + GVEYNVFZYLYRYMXHTGYLVBLXGVGVHYTVIYNLYCNJZLQNJZIJYOUUBIJYEYQYNUUAYOUUBIYNL + WSXGYNVBZYLYMVJYLYMVKVLVMYNLYCQUUCWSXGVNYTVLYNYPUUBYOIYPUUBOYNYPLUUBVOVSV + PVQVRVTWAUHUMUNSUOWAWBWCWDWCSWEWFSUDMZWNOZCKPWPUDWKKUEUUDWKOUUEWOCKUUDWKW + NUIUJUDCWHULWG $. $( The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.) $) @@ -766520,18 +770382,18 @@ become clearer and sometimes also shorter (see, for example, ~ divgcdoddALTV ( vn vi vj codd wcel wa caddc co cz c2 cv cmul c1 wceq wrex wi imp cc zcn va vb vz ceven evenz zaddcl syl2an eqeq1 rexbidv dfodd6 elrab2 dfeven4 ex oddz ad3antlr adantr oveq2 oveq1d eqeq2d adantl oveq12 2cnd mulcld ancoms - 1cnd mulcl add32d adddid eqcomd eqtrd rspcedvd rexlimdva2 r19.29an syl5bi - wb expimpd sylbi sylanbrc ) AFGZBUEGZHABIJZKGZWBLCMZNJZOIJZPZCKQZWBFGVTAK - GZBKGZWCWAAUOBUFABUGUHVTWAWHVTWIALDMZNJZOIJZPZDKQZHZWAWHRUBMZWMPZDKQWOUBA - KFWQAPWRWNDKWQAWMUIUJUBDUKULWAWJBLEMZNJZPZEKQZHZWPWHUCMZWTPZEKQXBUCBKUEXD - BPXEXAEKXDBWTUIUJUCEUMULWIWNXCWHRDKWIWKKGZHZWNHZWJXBWHXHWJHZXAWHEKXIWSKGZ - HZXAHZWGWBLWKWSIJZNJZOIJZPZCXMKXKXMKGZXAXIXJXQXFXJXQRWIWNWJXFXJXQWKWSUGUN - UPSUQWDXMPZWGXPVPXLXRWFXOWBXRWEXNOIWDXMLNURUSUTVAXLWBWMWTIJZXOXKXAWBXSPZW - NXAXTRXGWJXJWNXAXTAWMBWTIVBUNUPSXKXSXOPZXAXIXJYAXFXJYARWIWNWJXFXJYAXFXJHZ - XSWLWTIJZOIJXOYBWLOWTXJXFWLTGXJXFHZLWKYDVCXFWKTGZXJWKUAZVAVDVEYBVFXFLTGWS - TGZWTTGXJXFVCWSUAZLWSVGUHVHYBYCXNOIYBXNYCYBLWKWSYBVCXFYEXJYFUQXJYGXFYHVAV - IVJUSVKUNUPSUQVKVLVMVQVNVOVRSUDMZWFPZCKQWHUDWBKFYIWBPYJWGCKYIWBWFUIUJUDCU - KULVS $. + wb mulcl add32d adddid eqcomd eqtrd rspcedvd rexlimdva2 r19.29an biimtrid + 1cnd expimpd sylbi sylanbrc ) AFGZBUEGZHABIJZKGZWBLCMZNJZOIJZPZCKQZWBFGVT + AKGZBKGZWCWAAUOBUFABUGUHVTWAWHVTWIALDMZNJZOIJZPZDKQZHZWAWHRUBMZWMPZDKQWOU + BAKFWQAPWRWNDKWQAWMUIUJUBDUKULWAWJBLEMZNJZPZEKQZHZWPWHUCMZWTPZEKQXBUCBKUE + XDBPXEXAEKXDBWTUIUJUCEUMULWIWNXCWHRDKWIWKKGZHZWNHZWJXBWHXHWJHZXAWHEKXIWSK + GZHZXAHZWGWBLWKWSIJZNJZOIJZPZCXMKXKXMKGZXAXIXJXQXFXJXQRWIWNWJXFXJXQWKWSUG + UNUPSUQWDXMPZWGXPVFXLXRWFXOWBXRWEXNOIWDXMLNURUSUTVAXLWBWMWTIJZXOXKXAWBXSP + ZWNXAXTRXGWJXJWNXAXTAWMBWTIVBUNUPSXKXSXOPZXAXIXJYAXFXJYARWIWNWJXFXJYAXFXJ + HZXSWLWTIJZOIJXOYBWLOWTXJXFWLTGXJXFHZLWKYDVCXFWKTGZXJWKUAZVAVDVEYBVPXFLTG + WSTGZWTTGXJXFVCWSUAZLWSVGUHVHYBYCXNOIYBXNYCYBLWKWSYBVCXFYEXJYFUQXJYGXFYHV + AVIVJUSVKUNUPSUQVKVLVMVQVNVOVRSUDMZWFPZCKQWHUDWBKFYIWBPYJWGCKYIWBWFUIUJUD + CUKULVS $. $} $( The difference of two odds is even. (Contributed by Scott Fenton, @@ -766824,14 +770686,14 @@ become clearer and sometimes also shorter (see, for example, ~ divgcdoddALTV ( wcel cprime w3a caddc co wceq c2 wo wi wa codd adantl cc zcnd prmz cz w3o ceven olc a1d wn cmin wne df-ne csn cdif eldifsn oddprmALTV emoo expcom syl sylbir ex syl5bir com23 3ad2ant3 impcom 3adant3 3simpa 3ad2ant2 eqcom evenz - adantr zaddcl syl2an subadd2d biimprd syl5bi 3impia odd2prm2 syl3anc df-3or - orcd pm2.61i sylibr ) DUBEZAFEZBFEZCFEZGZDABHIZCHIZJZGZAKJZBKJZLZCKJZLZWIWJ - WLUAWLWHWMMWLWMWHWLWKUCUDWLUEZWHWMWNWHNZWKWLWODCUFIZOEZWAWBNZWPWEJZWKWHWNWQ - VTWDWNWQMZWGWDVTWTWCWAVTWTMWBWCWNVTWQWNCKUGZWCVTWQMZCKUHWCXAXBWCXANCFKUIUJE - ZXBCFKUKXCCOEZXBCULVTXDWQDCUMUNUOUPUQURUSUTVAVBVAWHWRWNWDVTWRWGWAWBWCVCVDPW - HWSWNVTWDWGWSWGWFDJZVTWDNZWSDWFVEXFWSXEXFDCWEVTDQEWDVTDDVFRVGWDCQEZVTWCWAXG - WBWCCCSRUTPWDWEQEZVTWAWBXHWCWRWEWAATEBTEWETEWBASBSABVHVIRVBPVJVKVLVMPABWPVN - VOVQUQVRWIWJWLVPVS $. + adantr zaddcl syl2an subadd2d biimprd biimtrid 3impia odd2prm2 syl3anc orcd + pm2.61i df-3or sylibr ) DUBEZAFEZBFEZCFEZGZDABHIZCHIZJZGZAKJZBKJZLZCKJZLZWI + WJWLUAWLWHWMMWLWMWHWLWKUCUDWLUEZWHWMWNWHNZWKWLWODCUFIZOEZWAWBNZWPWEJZWKWHWN + WQVTWDWNWQMZWGWDVTWTWCWAVTWTMWBWCWNVTWQWNCKUGZWCVTWQMZCKUHWCXAXBWCXANCFKUIU + JEZXBCFKUKXCCOEZXBCULVTXDWQDCUMUNUOUPUQURUSUTVAVBVAWHWRWNWDVTWRWGWAWBWCVCVD + PWHWSWNVTWDWGWSWGWFDJZVTWDNZWSDWFVEXFWSXEXFDCWEVTDQEWDVTDDVFRVGWDCQEZVTWCWA + XGWBWCCCSRUTPWDWEQEZVTWAWBXHWCWRWEWAATEBTEWETEWBASBSABVHVIRVBPVJVKVLVMPABWP + VNVOVPUQVQWIWJWLVRVS $. ${ $d N p q $. $d P p q $. $d Q p q $. $d R p q $. @@ -767305,18 +771167,18 @@ pseudoprimes shall be composite (positive) integers, they must be ( c2 cfv wcel w3a co cmo wceq wa c4 c1 wb 3ad2ant1 adantr cr wbr clt 2re c3 a1i cfppr cuz codd cprime wnel cexp cmin cn fpprel mp1i eluz4eluz2 fppr2odd 2nn adantl simpr2 3jca fpprwppr crp cc0 eluz4nn nnrpd 0le2 cz eluz2 zlem1lt - cle wi 4z mpan 4m1e3 breq1i 3re zre 2lt3 simpr lttrd ex syl5bi sylbid sylbi - modid syl22anc sylan9eq jca pm2.43i ge2nprmge4 3adant2 simp3 eluz2nn eqcomd - 3imp syl eqeq2d biimpa cgcd gcd2odd1 3ad2ant2 fpprwpprb mpbir2and impbii ) - ABUACDZABUBCDZAUCDZAUDUEZEZBAUFFAGFZBHZIZXAXHXAXAAJUBCDZXDBAKUGFUFFAGFKHZEZ - XHBUHDZXAXKLXAUMBAUIUJXAXKXHXAXKIZXEXGXMXBXCXDXKXBXAXIXDXBXJAUKMUNXAXCXKAUL - NXAXIXDXJUOUPXAXKXFBAGFZBBAUQXIXDXNBHZXJXIBODZAURDZUSBVFPZBAQPZXOXPXIRTXIAA - UTVAXRXIVBTXIJVCDZAVCDZJAVFPZEXSJAVDXTYAYBXSYAYBXSVGVGXTYAYBJKUGFZAQPZXSXTY - AYBYDLVHJAVEVIYDSAQPZYAXSYCSAQVJVKYAYEXSYAYEIZBSAXPYFRTSODYFVLTYAAODYEAVMNB - SQPYFVNTYAYEVOVPVQVRVSTWKVTZBAWAZWBMWCWDVQVSWEXHXAXIXDIZXLXFXNHZIZXEYIXGXEX - IXDXBXDXIXCAWFWGZXBXCXDWHWDNXHXLYJXLXHUMTXEXGYJXEBXNXFXEXNBXEXPXQXRXSXOXPXE - RTXBXCXQXDXBAAWIVAMXRXEVBTXEXIXSYLYGWLYHWBWJWMWNWDXHABWOFKHZXAYIYKILXEYMXGX - CXBYMXDAWPWQNBAWRWLWSWT $. + cle wi 4z mpan 4m1e3 breq1i 3re zre 2lt3 simpr lttrd ex biimtrid 3imp sylbi + sylbid modid syl22anc sylan9eq jca pm2.43i ge2nprmge4 3adant2 simp3 eluz2nn + syl eqcomd eqeq2d biimpa cgcd gcd2odd1 3ad2ant2 fpprwpprb mpbir2and impbii + ) ABUACDZABUBCDZAUCDZAUDUEZEZBAUFFAGFZBHZIZXAXHXAXAAJUBCDZXDBAKUGFUFFAGFKHZ + EZXHBUHDZXAXKLXAUMBAUIUJXAXKXHXAXKIZXEXGXMXBXCXDXKXBXAXIXDXBXJAUKMUNXAXCXKA + ULNXAXIXDXJUOUPXAXKXFBAGFZBBAUQXIXDXNBHZXJXIBODZAURDZUSBVFPZBAQPZXOXPXIRTXI + AAUTVAXRXIVBTXIJVCDZAVCDZJAVFPZEXSJAVDXTYAYBXSYAYBXSVGVGXTYAYBJKUGFZAQPZXSX + TYAYBYDLVHJAVEVIYDSAQPZYAXSYCSAQVJVKYAYEXSYAYEIZBSAXPYFRTSODYFVLTYAAODYEAVM + NBSQPYFVNTYAYEVOVPVQVRWATVSVTZBAWBZWCMWDWEVQWAWFXHXAXIXDIZXLXFXNHZIZXEYIXGX + EXIXDXBXDXIXCAWGWHZXBXCXDWIWENXHXLYJXLXHUMTXEXGYJXEBXNXFXEXNBXEXPXQXRXSXOXP + XERTXBXCXQXDXBAAWJVAMXRXEVBTXEXIXSYLYGWKYHWCWLWMWNWEXHABWOFKHZXAYIYKILXEYMX + GXCXBYMXDAWPWQNBAWRWKWSWT $. $( Fermat's little theorem with base 8 reversed is not generally true: There is an integer ` p ` (for example 9, see ~ 9fppr8 ) so that " ` p ` is @@ -767693,17 +771555,17 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or ( vp vq wcel w3a cprime wa c5 clt wbr wi c3 ancoms cz cle cr c6 a1i imp ex cgbe ceven cv codd caddc wceq wrex isgbe cuz cfv oddprmuzge3 eluz2 zre co 3re pm3.2i pm3.22 le2add sylancr ancomsd 3p3e6 breq1i 5lt6 5re readdcl - 6re ltletr syl3anc mpani syl5bi syld syl2an adantl com23 exp4b 3imp com13 - 3adant1 sylbi an4s 3adant3 impcom wb breq2 3ad2ant3 mpbird rexlimdvv ) AU - ADAUBDZBUCZUDDZCUCZUDDZAWIWKUEUNZUFZEZCFUGBFUGZGHAIJZACBUHWHWPWQWHWOWQBCF - FWIFDZWKFDZGZWOWQKKWHWTWOWQWTWOGWQHWMIJZWOWTXAWJWLWTXAKWNWJWLGWTXAWJWRWLW - SXAWJWRGWILUIUJZDZWKXBDZXAWLWSGWRWJXCWIUKMWSWLXDWKUKMXCXDXAXCLNDZWINDZLWI - OJZEZXDXAKLWIULXDXEWKNDZLWKOJZEZXHXALWKULXFXGXKXAKZXEXFXGXLXKXGXFXAXEXIXJ - XGXFXAKZKXEXIXJXGXMXEXIGXFXJXGGZXAXIXFXNXAKZKXEXIXFXOXIWKPDZWIPDZXOXFWKUM - WIUMXPXQGZXNLLUEUNZWMOJZXAXRXGXJXTXRLPDZYAGXQXPGXGXJGXTKYAYAUOUOUPXPXQUQL - LWIWKURUSUTXTQWMOJZXRXAXSQWMOVAVBXRHQIJZYBXAVCXRHPDZQPDZWMPDZYCYBGXAKYDXR - VDRYEXRVFRXQXPYFWIWKVEMHQWMVGVHVIVJVKVLTVMVNVOVPVQSVRVJVSSVLVTTWAWBWOWQXA - WCZWTWNWJYGWLAWMHIWDWEVMWFTRWGSVS $. + ltletr syl3anc mpani biimtrid syld syl2an adantl com23 exp4b 3imp 3adant1 + 6re com13 sylbi an4s 3adant3 impcom wb breq2 3ad2ant3 mpbird rexlimdvv ) + AUADAUBDZBUCZUDDZCUCZUDDZAWIWKUEUNZUFZEZCFUGBFUGZGHAIJZACBUHWHWPWQWHWOWQB + CFFWIFDZWKFDZGZWOWQKKWHWTWOWQWTWOGWQHWMIJZWOWTXAWJWLWTXAKWNWJWLGWTXAWJWRW + LWSXAWJWRGWILUIUJZDZWKXBDZXAWLWSGWRWJXCWIUKMWSWLXDWKUKMXCXDXAXCLNDZWINDZL + WIOJZEZXDXAKLWIULXDXEWKNDZLWKOJZEZXHXALWKULXFXGXKXAKZXEXFXGXLXKXGXFXAXEXI + XJXGXFXAKZKXEXIXJXGXMXEXIGXFXJXGGZXAXIXFXNXAKZKXEXIXFXOXIWKPDZWIPDZXOXFWK + UMWIUMXPXQGZXNLLUEUNZWMOJZXAXRXGXJXTXRLPDZYAGXQXPGXGXJGXTKYAYAUOUOUPXPXQU + QLLWIWKURUSUTXTQWMOJZXRXAXSQWMOVAVBXRHQIJZYBXAVCXRHPDZQPDZWMPDZYCYBGXAKYD + XRVDRYEXRVQRXQXPYFWIWKVEMHQWMVFVGVHVIVJVKTVLVMVNVOVRSVPVIVSSVKVTTWAWBWOWQ + XAWCZWTWNWJYGWLAWMHIWDWEVLWFTRWGSVS $. $( Any weak odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.) $) @@ -767711,8 +771573,8 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or ( vp vq vr wcel caddc co cprime wa c5 wbr wi c2 cz anim12i cr 3ad2ant2 c4 cle c6 cgbow codd wceq wrex clt isgbow w3a cuz cfv prmuz2 eluz2 sylib zre cv 2re pm3.2i jctil simp3 le2add sylc 2p2e4 breq1i zaddcl zred adantr 4re - simpr 4p2e6 5lt6 5re a1i zaddcld ltletr syl3anc mpani syl5bi expcom com12 - 6re imp exp31 breq2 syl5ibrcom syl2an rexlimdva adantl rexlimdvva sylbi + simpr 4p2e6 5lt6 5re a1i 6re zaddcld ltletr syl3anc mpani biimtrid expcom + com12 imp exp31 breq2 syl5ibrcom syl2an rexlimdva adantl rexlimdvva sylbi mpd ) AUAEAUBEZABUNZCUNZFGZDUNZFGZUCZDHUDZCHUDBHUDZIJAUEKZADCBUFWJWRWSWJW QWSBCHHWKHEZWLHEZIZWQWSLWJXBWPWSDHXBMNEZWKNEZMWKSKZUGZXCWLNEZMWLSKZUGZIZX CWNNEZMWNSKZUGZWPWSLWNHEZWTXFXAXIWTWKMUHUIZEXFWKUJMWKUKULXAWLXOEXIWLUJMWL @@ -767723,8 +771585,8 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or ZWMPEZWNPEZIZIYFXLIYKYIYPYMYHYNXMYOYGYNYFYGWMWKWLVCZVDVEXKXCYOXLWNUMQOYLX TVFUOUPUQYHYFXMXLYGYFVGXCXKXLURORMWMWNUSUTYHXMYKXPLZYGXMYRLYFXMYGYRXKXCYG YRLXLYGXKYRYKTWOSKZYGXKIZXPYJTWOSVHVBYTJTUEKZYSXPVIYTJPEZTPEZWOPEUUAYSIXP - LUUBYTVJVKUUCYTVSVKYTWOYTWMWNYGWMNEXKYQVEYGXKVGVLVDJTWOVMVNVOVPVQQVRVEVTW - IWAVPVQQVRQVTWIVTAWOJUEWBWCWDWEWFWGVTWH $. + LUUBYTVJVKUUCYTVLVKYTWOYTWMWNYGWMNEXKYQVEYGXKVGVMVDJTWOVNVOVPVQVRQVSVEVTW + IWAVQVRQVSQVTWIVTAWOJUEWBWCWDWEWFWGVTWH $. $( Any weak odd Goldbach number is greater than or equal to 7. Because of ~ 7gbow , this bound is strict. (Contributed by AV, 20-Jul-2020.) $) @@ -767872,28 +771734,28 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or cv cgbe ceven wral c5 cgbow cz oddz c1 wb 5nn nnzi zltp1le mpan wo breq1i 5p1e6 cr 6re a1i zre leloed bitrid 6nn 6p1e7 7re cmin simpr 3odd omoeALTV jctir breq2 eleq1 imbi12d rspcv 3syl 4p3e7 eqcomi 4re 3re ltaddsub biimpd - w3a syl3anc syl5bi impcom adantr pm2.27 syl wrex isgbe 3prm zcn 3cn npcan - cc eqcomd oveq2 eqcoms sylan9eq rspcedeq2vd oveq1 eqeq2d rexbidv 3ad2ant3 - syl5ib com12 ad4antlr reximdva jctild isgbow syl6ibr 3syld ex com23 7gbow - adantld mpbii a1d sylbid 6even evennodd pm2.21d ax-mp syl6bir com24 mpcom - jaoi ralrimiva ) FBUAZGHZYJUBIZJZBUCUDZUEAUAZGHZYOUFIZJZAKYOKIZYNYRYOUGIZ - YSYNYRJYOUHYTYPYNYSYQYTYPUEUILMZYONHZYNYSYQJZJZUEUGIYTYPUUBUJUEUKULUEYOUM - UNYTUUBOYOGHZOYOPZUOZUUDUUBOYONHYTUUGUUAOYONUQUPYTOYOOURIYTUSUTYOVAZVBVCU - UGYTUUDUUEYTUUDJZUUFYTUUEUUDYTUUEOUILMZYONHZUUDOUGIYTUUEUUKUJOVDULOYOUMUN - YTUUKQYOGHZQYOPZUOZUUDUUKQYONHYTUUNUUJQYONVEUPYTQYOQURIYTVFUTUUHVBVCUUNYT - UUDUULUUIUUMUULYTUUDUULYTRZYSYNYQUUOYSYNYQJUUOYSRZYNFYOSVGMZGHZUUQUBIZJZU - USYQUUPYSSKIZRUUQUCIZYNUUTJUUPYSUVAUUOYSVHZVIVKYOSVJYMUUTBUUQUCYJUUQPYKUU - RYLUUSYJUUQFGVLYJUUQUBVMVNVOVPUUPUURUUTUUSJUUOUURYSYTUULUURUULFSLMZYOGHZY - TUURQUVDYOGUVDQVQVRUPYTFURIZSURIZYOURIZUVEUURJUVFYTVSUTUVGYTVTUTUUHUVFUVG - UVHWCUVEUURFSYOWAWBWDWEWFWGUURUUSWHWIUUSUVBCUAZKIZDUAZKIZUUQUVIUVKLMZPZWC - ZDTWJZCTWJZRUUPYQUUQDCWKUUPUVQYQUVBUUPUVQYSYOUVMEUAZLMZPZETWJZDTWJZCTWJZR - YQUUPUVQUWCYSUUPUVPUWBCTUUPUVITIZRUVOUWADTYTUVOUWAJUULYSUWDUVKTIUVOYTUWAU - VNUVJYTUWAJUVLYTYOUUQUVRLMZPZETWJUVNUWAYTESTYOUWESTIYTWLUTYTUVRSPYOUUQSLM - ZUWEYTYOWPIZSWPIZRZYOUWGPYTUWHUWIYOWMWNVKUWJUWGYOYOSWOWQWIUWGUWEPSUVRSUVR - UUQLWRWSWTXAUVNUWFUVTETUVNUWEUVSYOUUQUVMUVRLXBXCXDXFXEXGXHXIXIUVCXJYOEDCX - KXLXQWEXMXNXOXNUUMUUDYTUUMUUCYNUUMYQYSUUMQUFIYQXPQYOUFVMXRXSXSXSYHXGXTXTX - GUUFUUDYTUUFUUCYNUUFYSOKIZYQOYOKVMOUCIZUWKYQJYAUWLUWKYQOYBYCYDYEXSXSYHXGX - TXTYFYGWFYI $. + w3a syl3anc biimtrid impcom adantr pm2.27 syl wrex isgbe cc zcn 3cn npcan + 3prm eqcomd oveq2 eqcoms sylan9eq rspcedeq2vd oveq1 eqeq2d rexbidv syl5ib + 3ad2ant3 com12 ad4antlr reximdva jctild syl6ibr adantld 3syld com23 7gbow + isgbow ex mpbii a1d jaoi 6even evennodd pm2.21d ax-mp syl6bir com24 mpcom + sylbid ralrimiva ) FBUAZGHZYJUBIZJZBUCUDZUEAUAZGHZYOUFIZJZAKYOKIZYNYRYOUG + IZYSYNYRJYOUHYTYPYNYSYQYTYPUEUILMZYONHZYNYSYQJZJZUEUGIYTYPUUBUJUEUKULUEYO + UMUNYTUUBOYOGHZOYOPZUOZUUDUUBOYONHYTUUGUUAOYONUQUPYTOYOOURIYTUSUTYOVAZVBV + CUUGYTUUDUUEYTUUDJZUUFYTUUEUUDYTUUEOUILMZYONHZUUDOUGIYTUUEUUKUJOVDULOYOUM + UNYTUUKQYOGHZQYOPZUOZUUDUUKQYONHYTUUNUUJQYONVEUPYTQYOQURIYTVFUTUUHVBVCUUN + YTUUDUULUUIUUMUULYTUUDUULYTRZYSYNYQUUOYSYNYQJUUOYSRZYNFYOSVGMZGHZUUQUBIZJ + ZUUSYQUUPYSSKIZRUUQUCIZYNUUTJUUPYSUVAUUOYSVHZVIVKYOSVJYMUUTBUUQUCYJUUQPYK + UURYLUUSYJUUQFGVLYJUUQUBVMVNVOVPUUPUURUUTUUSJUUOUURYSYTUULUURUULFSLMZYOGH + ZYTUURQUVDYOGUVDQVQVRUPYTFURIZSURIZYOURIZUVEUURJUVFYTVSUTUVGYTVTUTUUHUVFU + VGUVHWCUVEUURFSYOWAWBWDWEWFWGUURUUSWHWIUUSUVBCUAZKIZDUAZKIZUUQUVIUVKLMZPZ + WCZDTWJZCTWJZRUUPYQUUQDCWKUUPUVQYQUVBUUPUVQYSYOUVMEUAZLMZPZETWJZDTWJZCTWJ + ZRYQUUPUVQUWCYSUUPUVPUWBCTUUPUVITIZRUVOUWADTYTUVOUWAJUULYSUWDUVKTIUVOYTUW + AUVNUVJYTUWAJUVLYTYOUUQUVRLMZPZETWJUVNUWAYTESTYOUWESTIYTWPUTYTUVRSPYOUUQS + LMZUWEYTYOWLIZSWLIZRZYOUWGPYTUWHUWIYOWMWNVKUWJUWGYOYOSWOWQWIUWGUWEPSUVRSU + VRUUQLWRWSWTXAUVNUWFUVTETUVNUWEUVSYOUUQUVMUVRLXBXCXDXEXFXGXHXIXIUVCXJYOED + CXPXKXLWEXMXQXNXQUUMUUDYTUUMUUCYNUUMYQYSUUMQUFIYQXOQYOUFVMXRXSXSXSXTXGYHY + HXGUUFUUDYTUUFUUCYNUUFYSOKIZYQOYOKVMOUCIZUWKYQJYAUWLUWKYQOYBYCYDYEXSXSXTX + GYHYHYFYGWFYI $. $( If the strong binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, @@ -767906,18 +771768,18 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or 3re imp syl isgbe 3prm wb 3anbi3d oveq2 eqeq2d anbi12d adantl simp1 simp2 w3a 3jca cc zcnd ad3antrrr 3cn prmz zaddcl syl2an adantll subadd2d biimpa cz eqcomd 3ad2antr3 jca rspcedvd ex reximdva jctild isgbo syl6ibr adantld - syl5bi 3syld com12 expd ralrimiv ) FBGZHIZXGUAJZKZBUBUCZUDAGZHIZXLUEJZKAL - XKXLLJZXMXNXOXMMZXKXNXPXKFXLNUFOZHIZXQUAJZKZXSXNXPXONLJZMXQUBJZXKXTKXPXOY - AXOXMUGZUHUIXLNUJXJXTBXQUBXGXQPXHXRXIXSXGXQFHUKXGXQUAUNULUOUMXPXRXTXSKXOX - MXRXMFNQOZXLHIZXOXRYDUDXLHUPUQXOYEXRXOFNXLFURJXOUSRNURJXOVFRXOXLXLUTZVAVB - VCVDVGXRXSVEVHXSYBCGZLJZDGZLJZXQYGYIQOZPZVSZDSTZCSTZMXPXNXQDCVIXPYOXNYBXP - YOXOYHYJEGZLJZVSZXLYKYPQOZPZMZESTZDSTZCSTZMXNXPYOUUDXOXPYNUUCCSXPYGSJZMZY - MUUBDSUUFYISJZMZYMUUBUUHYMMZUUAYHYJYAVSZXLYKNQOZPZMZENSNSJUUIVJRYPNPZUUAU - UMVKUUIUUNYRUUJYTUULUUNYQYAYHYJYPNLUNVLUUNYSUUKXLYPNYKQVMVNVOVPUUIUUJUULY - MUUJUUHYMYHYJYAYHYJYLVQYHYJYLVRYAYMUHRVTVPUUHYHYLUULYJUUHYLMUUKXLUUHYLUUK - XLPUUHXLNYKXOXLWAJXMUUEUUGXOXLYFWBWCNWAJUUHWDRUUEUUGYKWAJXPUUEUUGMYKUUEYG - WKJYIWKJYKWKJUUGYGWEYIWEYGYIWFWGWBWHWIWJWLWMWNWOWPWQWQYCWRXLEDCWSWTXAXBXC - XDXEXF $. + biimtrid 3syld com12 expd ralrimiv ) FBGZHIZXGUAJZKZBUBUCZUDAGZHIZXLUEJZK + ALXKXLLJZXMXNXOXMMZXKXNXPXKFXLNUFOZHIZXQUAJZKZXSXNXPXONLJZMXQUBJZXKXTKXPX + OYAXOXMUGZUHUIXLNUJXJXTBXQUBXGXQPXHXRXIXSXGXQFHUKXGXQUAUNULUOUMXPXRXTXSKX + OXMXRXMFNQOZXLHIZXOXRYDUDXLHUPUQXOYEXRXOFNXLFURJXOUSRNURJXOVFRXOXLXLUTZVA + VBVCVDVGXRXSVEVHXSYBCGZLJZDGZLJZXQYGYIQOZPZVSZDSTZCSTZMXPXNXQDCVIXPYOXNYB + XPYOXOYHYJEGZLJZVSZXLYKYPQOZPZMZESTZDSTZCSTZMXNXPYOUUDXOXPYNUUCCSXPYGSJZM + ZYMUUBDSUUFYISJZMZYMUUBUUHYMMZUUAYHYJYAVSZXLYKNQOZPZMZENSNSJUUIVJRYPNPZUU + AUUMVKUUIUUNYRUUJYTUULUUNYQYAYHYJYPNLUNVLUUNYSUUKXLYPNYKQVMVNVOVPUUIUUJUU + LYMUUJUUHYMYHYJYAYHYJYLVQYHYJYLVRYAYMUHRVTVPUUHYHYLUULYJUUHYLMUUKXLUUHYLU + UKXLPUUHXLNYKXOXLWAJXMUUEUUGXOXLYFWBWCNWAJUUHWDRUUEUUGYKWAJXPUUEUUGMYKUUE + YGWKJYIWKJYKWKJUUGYGWEYIWEYGYIWFWGWBWHWIWJWLWMWNWOWPWQWQYCWRXLEDCWSWTXAXB + XCXDXEXF $. $} $( Lemma 1 for ~ sbgoldbalt : If an even number greater than 4 is the sum of @@ -767958,23 +771820,23 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or ( c4 clt wbr wcel wi c2 caddc co wceq cprime wrex cle c3 cr a1i wa codd cv cgbe ceven c1 cz wb evenz zltp1le sylancr 2p1e3 breq1i 3re zred leloed 2z wo 3z 3p1e4 4re pm3.35 w3a isgbe simp3 reximdva imp sylbi a1d ex com23 - syl 2prm 2p2e4 eqcomi mp3an eqeq1 2rexbidv mpbii jaoi com12 sylbid syl5bi - rspceov wn 3odd eleq1 oddneven pm2.21d 2lt4 2re lttr syl3anc mpani simpll - simpr anim1i adantr df-3an sylibr sbgoldbaltlem2 sylanbrc syl6ibr embantd - sylc jca impbid ralbiia ) DAUAZEFZXGUBGZHZIXGEFZXGCUAZBUAZJKZLZBMNZCMNZHZ - AUCXGUCGZXJXRXSXKXJXQXSXKIUDJKZXGOFZXJXQHZXSIUEGXGUEGZXKYAUFUOXGUGZIXGUHU - IYAPXGOFZXSYBXTPXGOUJUKXSYEPXGEFZPXGLZUPZYBXSPXGPQGXSULRXSXGYDUMZUNYHXSYB - YFXSYBHZYGXSYFYBXSYFPUDJKZXGOFZYBXSPUEGYCYFYLUFUQYDPXGUHUIYLDXGOFZXSYBYKD - XGOURUKXSYMXHDXGLZUPZYBXSDXGDQGZXSUSRZYIUNYOXSYBXHYJYNXHXJXSXQXHXJXSXQHZX - HXJSXIYRXHXIUTXIXQXSXIXSXLTGZXMTGZXOVAZBMNZCMNZSZXQXGBCVBZXSUUCXQXSUUBXPC - MXSXLMGZSZUUAXOBMUUAXOHUUGXMMGZSYSYTXOVCRVDVDVEVFVGVJVHVIYNYBXSYNXQXJYNDX - NLZBMNCMNZXQIMGZUUKDIIJKZLUUJVKVKUULDVLVMCBMMIIDJWBVNYNUUIXOCBMMDXGXNVOVP - VQVGVGVRVSVTWAVTVSYGXSYBYGXGTGZXSWCYGPTGUUMWDPXGTWEVQXGWFVJWGVRVSVTWAVTVI - XSXHXRXIXSXHXRXIHXSXHSZXKXQXIXSXHXKXSIDEFZXHXKWHXSIQGZYPXGQGUUOXHSXKHUUPX - SWIRYQYIIDXGWJWKWLVEUUNXQUUDXIUUNXQUUDUUNXQSXSUUCXSXHXQWMUUNXQUUCUUNXPUUB - CMUUNUUFSZXOUUABMUUQUUHSZXOUUAUURXOSZYSYTSZXOUUAUUSUUFUUHSZXSXHXOVAZUUTUU - RUVAXOUUQUUFUUHUUNUUFWNWOWPUUSUUNXOSUVBUURUUNXOUUNUUFUUHWMWOXSXHXOWQWRXLX - MXGWSXCUURXOWNYSYTXOWQWTVHVDVDVEXDVHUUEXAXBVHVIXEXF $. + 2prm 2p2e4 eqcomi rspceov mp3an eqeq1 2rexbidv mpbii jaoi sylbid biimtrid + syl com12 wn 3odd eleq1 oddneven pm2.21d 2lt4 2re lttr mpani simpll simpr + syl3anc anim1i adantr df-3an sbgoldbaltlem2 sylc sylanbrc syl6ibr embantd + sylibr jca impbid ralbiia ) DAUAZEFZXGUBGZHZIXGEFZXGCUAZBUAZJKZLZBMNZCMNZ + HZAUCXGUCGZXJXRXSXKXJXQXSXKIUDJKZXGOFZXJXQHZXSIUEGXGUEGZXKYAUFUOXGUGZIXGU + HUIYAPXGOFZXSYBXTPXGOUJUKXSYEPXGEFZPXGLZUPZYBXSPXGPQGXSULRXSXGYDUMZUNYHXS + YBYFXSYBHZYGXSYFYBXSYFPUDJKZXGOFZYBXSPUEGYCYFYLUFUQYDPXGUHUIYLDXGOFZXSYBY + KDXGOURUKXSYMXHDXGLZUPZYBXSDXGDQGZXSUSRZYIUNYOXSYBXHYJYNXHXJXSXQXHXJXSXQH + ZXHXJSXIYRXHXIUTXIXQXSXIXSXLTGZXMTGZXOVAZBMNZCMNZSZXQXGBCVBZXSUUCXQXSUUBX + PCMXSXLMGZSZUUAXOBMUUAXOHUUGXMMGZSYSYTXOVCRVDVDVEVFVGWAVHVIYNYBXSYNXQXJYN + DXNLZBMNCMNZXQIMGZUUKDIIJKZLUUJVJVJUULDVKVLCBMMIIDJVMVNYNUUIXOCBMMDXGXNVO + VPVQVGVGVRWBVSVTVSWBYGXSYBYGXGTGZXSWCYGPTGUUMWDPXGTWEVQXGWFWAWGVRWBVSVTVS + VIXSXHXRXIXSXHXRXIHXSXHSZXKXQXIXSXHXKXSIDEFZXHXKWHXSIQGZYPXGQGUUOXHSXKHUU + PXSWIRYQYIIDXGWJWNWKVEUUNXQUUDXIUUNXQUUDUUNXQSXSUUCXSXHXQWLUUNXQUUCUUNXPU + UBCMUUNUUFSZXOUUABMUUQUUHSZXOUUAUURXOSZYSYTSZXOUUAUUSUUFUUHSZXSXHXOVAZUUT + UURUVAXOUUQUUFUUHUUNUUFWMWOWPUUSUUNXOSUVBUURUUNXOUUNUUFUUHWLWOXSXHXOWQXCX + LXMXGWRWSUURXOWMYSYTXOWQWTVHVDVDVEXDVHUUEXAXBVHVIXEXF $. $( If the strong binary Goldbach conjecture is valid, the binary Goldbach conjecture is valid. (Contributed by AV, 23-Dec-2021.) $) @@ -768023,15 +771885,15 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or ( vm c4 cv clt wbr cgbe wcel wi ceven wral caddc co cprime wrex c6 c5 cuz wceq cfv breq2 eleq1w imbi12d cbvralvw cz cle w3a eluz2 wo sgoldbeven3prm codd zeoALTV expdcom cgbow sbgoldbwt wa rspa c1 df-6 breq1i 5nn nnzi oddz - wb zltp1le sylancr biimprd syl5bi imp isgbow simprbi a1i embantd ex com23 - adantl mpd syl com13 jaoi 3adant1 sylbi impcom ralrimiva ) FAGZHIZWHJKZLZ - AMNFEGZHIZWLJKZLZEMNZWHDGCGOPBGOPUBBQRCQRDQRZASUAUCZNWKWOAEMWHWLUBWIWMWJW - NWHWLFHUDAEJUEUFUGWPWQAWRWHWRKZWPWQWSSUHKZWHUHKZSWHUIIZUJWPWQLZSWHUKXAXBX - CWTXAXBXCXAWHMKZWHUNKZULXBXCLZWHUOXDXFXEWPXDXBWQEWHBCDUMUPWPXBXEWQWPTWHHI - ZWHUQKZLZAUNNZXBXEWQLLAEURXJXEXBWQXJXEXBWQLZXJXEUSXIXKXIAUNUTXEXIXKLXJXEX - BXIWQXEXBXIWQLXEXBUSZXGXHWQXEXBXGXBTVAOPZWHUIIZXEXGSXMWHUIVBVCXEXGXNXETUH - KXAXGXNVGTVDVEWHVFTWHVHVIVJVKVLXHWQLXLXHXEWQWHBCDVMVNVOVPVQVRVSVTVQVRWAWB - WCWAVLWDWEWFWGWE $. + wb zltp1le sylancr biimprd biimtrid imp isgbow simprbi a1i embantd adantl + ex com23 mpd syl com13 jaoi 3adant1 sylbi impcom ralrimiva ) FAGZHIZWHJKZ + LZAMNFEGZHIZWLJKZLZEMNZWHDGCGOPBGOPUBBQRCQRDQRZASUAUCZNWKWOAEMWHWLUBWIWMW + JWNWHWLFHUDAEJUEUFUGWPWQAWRWHWRKZWPWQWSSUHKZWHUHKZSWHUIIZUJWPWQLZSWHUKXAX + BXCWTXAXBXCXAWHMKZWHUNKZULXBXCLZWHUOXDXFXEWPXDXBWQEWHBCDUMUPWPXBXEWQWPTWH + HIZWHUQKZLZAUNNZXBXEWQLLAEURXJXEXBWQXJXEXBWQLZXJXEUSXIXKXIAUNUTXEXIXKLXJX + EXBXIWQXEXBXIWQLXEXBUSZXGXHWQXEXBXGXBTVAOPZWHUIIZXEXGSXMWHUIVBVCXEXGXNXET + UHKXAXGXNVGTVDVEWHVFTWHVHVIVJVKVLXHWQLXLXHXEWQWHBCDVMVNVOVPVRVSVQVTVRVSWA + WBWCWAVLWDWEWFWGWE $. $d n p q r x y $. $( If the modern version of the original formulation of the Goldbach @@ -768046,21 +771908,21 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or cfv clt ceven nfra1 eqeq1 rexbidv 2rexbidv cbvralvw cz 6nn nnzi a1i evenz 2z zaddcld adantr cmin c4 4cn 2cn 4p2e6 mvrraddi 2p2e4 2evenALTV evenltle eqcomi mp3an2 eqbrtrrid eqbrtrid cr w3a wb 6re 2re zred 3jca lesubadd syl - mpbid eluz2 syl3anbrc rspcv syl5bi nfre1 nfcv nfrex simplrl simplrr simpr - nfv simp-4l mogoldbblem oveq1 eqeq2d oveq2 sylibr syl3anc rexlimdva2 expr - cbvrex2vw rexlimd ex syldc expd ralrimi ) AHZDHZCHZIJZBHZIJZKZBLMZCLMDLMZ - ANUAUCZUBZOXHUDPZXHXKKZCLMZDLMZQAUEXPAXQUFXRXHUERZXSYBYCXSSZXRXHOIJZXMKZB - LMZCLMZDLMZYBXREHZXMKZBLMZCLMDLMZEXQUBZYDYIXPYMAEXQXHYJKZXOYLDCLLYOXNYKBL - XHYJXMUGUHUIUJYDYEXQRZYNYIQYDNUKRZYEUKRZNYETPZYPYQYDNULUMUNYCYRXSYCXHOXHU - OZOUKRYCUPUNUQURYDNOUSJZXHTPZYSYDUUAUTXHTNUTOVAVBUTOIJNVCVHVDYDUTOOIJZXHT - VEYCOUERXSUUCXHTPVFOXHVGVIVJVKYDNVLRZOVLRZXHVLRZVMZUUBYSVNYCUUGXSYCUUDUUE - UUFUUDYCVOUNUUEYCVPUNYCXHYTVQVRURNOXHVSVTWANYEWBWCYMYIEYEXQYJYEKZYLYGDCLL - UUHYKYFBLYJYEXMUGUHUIWDVTWEYDYHYBDLYDDWLYADLWFYDXILRZYHYBQYDUUISZYGYBCLUU - JCWLYACDLCLWGXTCLWFWHYDUUIXJLRZYGYBQYDUUIUUKSZSZYFYBBLUUMXLLRZSZYFSUUIUUK - UUNVMZYCYFYBUUOUUPYFUUOUUIUUKUUNYDUUIUUKUUNWIYDUUIUUKUUNWJUUMUUNWKVRURYCX - SUULUUNYFWMUUOYFWKUUPYCYFVMXHFHZGHZIJZKZGLMFLMYBXIXJXLXHGFWNXTUUTXHUUQXJI - JZKDCFGLLXIUUQKXKUVAXHXIUUQXJIWOWPXJUURKUVAUUSXHXJUURUUQIWQWPXBWRWSWTXAXC - 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3m1e2 mpan biimprd syl5bi wn lenltd pm2.21 syl6bi syldc eqcom biimpi 2a1d - jaoi com12 sylbid imp 2lt3 breq1 mpbiri impbid1 sylbi orbi1d bitrd biimpd - 3adant1 2prm eleq1 nnsum3primesprm syl 3prm nnsum3primes4 anbi2d 2rexbidv - eqeq1 5prm cgbe 6gbe nnsum3primesgbe 7prm 8gbe ) CEUAUBFZCGHIZUDCEJZCKJZL - ZCMJZLZCNJZLZCOJZLZCPJZLZCGJZLZDUEZKHIZCUFUUCUCUGZBUEAUEUBBUHZJZUDZAQUUEU - IUGZUJDUKUJZYHYIUUBYHYICGRIZUUALZUUBYHCGECULZGUMFYHUNSUOYHUULUUBYHUUKYTUU - AYHUUKCPRIZYSLZYTYHCPHIZCPUFUPUGZRIZUUOUUKYHCUQFZPUQFUUPUURTECURZPUTUSCPV - AVBYHCPUUMPUMFYHVCSUOUURUUKTYHUUQGCRVDVESVFYHUUNYRYSYHUUNCORIZYQLZYRYHCOH - IZCOUFUPUGZRIZUVBUUNYHUUSOUQFUVCUVETUUTOVGUSCOVAVBYHCOUUMOUMFYHVHSUOUVEUU - NTYHUVDPCRVIVESVFYHUVAYPYQYHUVACNRIZYOLZYPYHCNHIZCNUFUPUGZRIZUVGUVAYHUUSN - UQFUVHUVJTUUTNVJUSCNVAVBYHCNUUMNUMFYHVKSUOUVJUVATYHUVIOCRVLVESVFYHUVFYNYO - YHUVFCMRIZYMLZYNYHCMHIZCMUFUPUGZRIZUVLUVFYHUUSMUQFUVMUVOTUUTVMCMVAVBYHCMU - UMMUMFYHVNSUOUVOUVFTYHUVNNCRVOVESVFYHUVKYLYMYHUVKCKRIZYKLZYLYHCKHIZCKUFUP - UGZRIZUVQUVKYHUUSKUQFZUVRUVTTUUTVPCKVAVBYHCKUUMKUMFZYHVQSUOUVTUVKTYHUVSMC - 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(Contributed by AV, 24-Jul-2020.) (Proof shortened @@ -768415,39 +772277,39 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or feq2i prmex ovex elmapg mp1i mpbird fveq1 sumeq2dv eqeq2d wo velsn orbi2i elun 3bitri wne cr mtbiri adantld ex eqcomd mpd elfz2 3lt4 ltnle necon2ad 3re mpbii breq1 eqcoms 3ad2ant3 sylbi fvunsn ffvelcdm ancoms prmz eqeltrd - zcnd fveq2 cdm fdm eleq2 fsnunfv sylan9eq eqeltrdi com12 syl5bi sumeq12dv - fsumm1 biimpa oveq1d oveq2d eluzelcn npcand eqtrd 3eqtrrd rspcedvd expcom - jaoi elmapi syl11 rexlimdv rexlimdva2 ) UDCUEZUFEUWBUGFGCUAUHZDUQUIHZFZDU - JFZIZDJKLMZBUEZAUEZHZBUKZNZAOUWHULMZUMZUWCUWGIZDUBUEZPQMNZUBUGUMZUWOUWCUW - GUWSCUBDUNURUWPUWRUWOUBUGUWPUWQUGFZIZUWRIZDPUOMZJPLMZUWIUCUEZHZBUKZNZUCOU - XDULMZUMZUWOUXBUWCUXCUSUIHFZUXCUAFZUXJUWCUWGUWTUWRUPUWGUXKUWCUWTUWRUWEUXK - UWFUWEUSRFZPRFZDUSPQMZUIHZFZUXKUXMUWEUSUTVASUXNUWEVBSUWEUXQUWDUXPDUQUXOUI - UXOUQVCVDVEVFVJPUSDVGVHTVIUXBUWFPUAFZIZUXLUXAUXSUWRUWPUXSUWTUWPUXRUWFUWGU - XRUWFIUWCUWEUXRUWFUXRUWEVKSVLVMVNTTDPVOVPUWCUXKUXLIUXJUCBCUXCVQURVRUWGUXJ - 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(Contributed by AV, @@ -768468,42 +772330,42 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or 3eqtri sylibr prmex elmapg mpbird fveq1 sumeq2sdv velsn orbi2i 3bitri wne elun elfz2 3lt4 ltnle mpbii eqcoms necon2ad 3ad2ant3 fvunsn ffvelcdm prmz breq1 ancoms zcnd eqeltrd fveq2 cdm eleq2 fsnunfv sylan9eq eqeltrdi com12 - jaoi syl5bi fsumm1 sumeq12dv biimpa oveq1d oveq2d eluzelcn npcand 3eqtrrd - fdm eqtrd rspcedvd expcom elmapi syl11 rexlimdv mpd evengpoap3 r19.29a ) - UDCUEZUFEUWRUAFUGCUHUIZDGUSUJZUKHFZDULFZIZDGJUMKZBUEZAUEZHZBUNZLZAMUXDUOK - ZUPZUWSUXCIZDUBUEZNOKLZUXKUBUAUXLUXMUAFZIZUXNIZDNUQKZGNUMKZUXEUCUEZHZBUNZ - LZUCMUXSUOKZUPZUXKUXQUWSUXRPUKHFZUXRUHFZUYEUWSUXCUXOUXNURUXCUYFUWSUXOUXNU - XAUYFUXBUXAPQFZNQFZDPNOKZUKHFZUYFUYHUXAPUTVARZUYIUXAVLRZUXAUYJQFDQFZUYJDS - EZUYKUXAPNUYLUYMVBUWTDVCUXAUWTQFZUYNUWTDSEZVDUYOUWTDVEUYNUYQUYOUYPUYNUYQU - YOUYQPJOKZDSEZUYNUYOUYRUWTDSVFVGUYNUYJUYRUFEZUYSUYOGGUJUWTUYJUYRUFGGUSVHV - HVIWEVJVKVFVMUYNUYJVNFUYRVNFDVNFUYTUYSIUYOUGUYNPNPVNFUYNVORZNVNFZUYNVPRVQ - 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elfzo2 caddc df-9 breq2i eluz2nn 8nn jctir adantr nnleltp1 biimprd syl5bi - syl 3impia jca sylbi nnsum4primesle9 a1d ceven 4nn a1i oveq2 oveq2d breq1 - sumeq1d eqeq2d anbi12d rexeqbidv adantl 4re leidi nnsum4primeseven impcom - r19.42v sylanbrc rspcedvd 3nn 3re 3lt4 6nn 6re 6lt9 eluzuzle mp2an anim1i - ex nnsum4primesodd mpan9 eluzelz zeoALTV mpjaodan jaoi ralrimiva ) UECFZU - AGYGUBHUFCUCUDZEFZIJGZDFZKYILMZBFAFUGZBUHZUIZNZAUJYLUKMZOZEPOZDQULUGZYKYT - HZYHYSUUAYKQRUMMZHZYKRULUGZHZUNZYHYSUFZUUAYKUUBUUDUOZHZUUFRYTHZUUAUUIUPUU - JQUQHRUQHZQRJGURRUSUTQRVAVCVBVDQRVEVFUUJYTUUHYKQRVGVHVIYKUUBUUDVJVKUUCUUG - UUEUUCYSYHUUCUUAYKVLJGZNZYSUUCUUAUUKYKRUAGZVMZUUMYKQRVOUUOUUAUULUUAUUKUUN - VNUUAUUKUUNUULUUNYKVLKVPMZUAGZUUAUUKNZUULRUUPYKUAVQVRUURUULUUQUURYKPHZVLP - HZNZUULUUQUPUUAUVAUUKUUAUUSUUTYKVSVTWAWBYKVLWCWFWDWEWGWHWIABYKEWJWFWKUUEY - KWLHZUUGYKUCHZUUEUVBNZYHYSUVDYHNZYRIIJGZYKKILMZYMBUHZUIZNZAUJUVGUKMZOZEIP - IPHUVEWMWNYIIUIZYRUVLUPUVEUVMYPUVJAYQUVKUVMYLUVGUJUKYIIKLWOZWPUVMYJUVFYOU - VIYIIIJWQUVMYNUVHYKUVMYLUVGYMBUVNWRWSWTXAXBUVEUVFUVIAUVKOZUVLUVFUVEIXCXDW - 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(Contributed by AV, @@ -768558,29 +772420,29 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or ( vo c4 cv clt wbr wcel wi c3 cle co wa cn c2 c9 c6 cgbe ceven c1 cfz cfv wral csu wceq cprime cmap wrex cuz cfzo wo cun wb cz 2z 9nn nnzi 2re 2lt9 9re ltleii eluz2 mpbir3an fzouzsplit eleq2d ax-mp elun bitri c8 w3a simp1 - elfzo2 caddc df-9 breq2i eluz2nn 8nn jctir adantr nnleltp1 biimprd syl5bi - syl 3impia jca sylbi nnsum3primesle9 a1d codd weq breq2 eleq1w imbi12d cr - rspcv 4re eluzelre 3jca adantl eluzle 4lt9 jctil ltletr sylc pm2.27 syl5d - a1i ex impcom nnsum3primesgbe syl6 c5 cgbow 3nn oveq2 oveq2d breq1 eqeq2d - sumeq1d anbi12d rexeqbidv 3re leidi 6nn 6re 6lt9 eluzuzle nnsum4primesodd - mp2an anim1i mpan9 r19.42v sylanbrc rspcedvd expcom sbgoldbwt syl11 jaoi - eluzelz zeoALTV mpjaodan ralrimiva ) GCHZIJZUUFUAKZLZCUBUFZEHZMNJZDHZUCUU - KUDOZBHAHUEZBUGZUHZPZAUIUUNUJOZUKZEQUKZDRULUEZUUMUVBKZUUJUVAUVCUUMRSUMOZK - 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+772456,27 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or 9nn 1nn ax-1cn addassi 1p1e2 oveq2i eqtri 3eqtri 2nn wn 2p1e3 lenltd pm2.21 2cn syl6bi eleq1 c6 6even epee eqeltrri evennodd ax-mp pm2.21i syl6bir jaoi 6p6e12 sylbid 11gbo mpbii 2a1d 5p5e10 5odd opoeALTV 9gbo 8even imp 3ad2ant3 - c5 syl5bi 3impia ) ABCZDAEFZADGUAUBZUCHCZAUDCZYFAUECZDAIFZAYEEFZUFZYCYDUGZY - GDUECYEUECYFYKJDUHUIYEYEGUAUJUPUKULZUIDYEAUMUNYKYLYGYJYHYLYGUOYIYLYJYGYCYDY - JYGUOZYCYDKAEFZKAUQZURZYNYCDLCZALCZYDYQJDUSMAUTZYRYSUGZYDDGNHZAIFZKAIFZYQDA - VAUUCUUDJUUAUUBKAIVBVCOYRKPCZAPCUUDYQJYSUUEYRVDOAVEKAVJVFVGVHYQYCYNYOYCYNUO - ZYPYCYOYNYCYOQAEFZQAUQZURZYNYCYOKGNHZAIFZQAIFZUUIYCKLCYSYOUUKJKVIMYTKAVAVHU - UKUULJYCUUJQAIVKVCOYCQAQPCYCVLOYCAYTVMZVNVGUUIYCYNUUGUUFUUHYCUUGYNYCUUGGVOU - BZAEFZUUNAUQZURZYNYCUUGQGNHZAIFZUUNAIFZUUQYCQLCYSUUGUUSJQWCMYTQAVAVHUUSUUTJ - YCUURUUNAIVPVCOYCUUNAUUNPCYCVQOUUMVNVGUUQYCYNUUOUUFUUPYCUUOYNYCUUOGGUBZAEFZ - 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cc0 simp1 ralimi cn0 cn c3 cuz nnm1nn0 3ad2ant1 sylbi eluzge3nn - elfzo1 a1i a1d cz elfzo2 cr eluzelre adantr 1red resubcld zre adantl lttr - ltm1d mpand 3impia 3jcad syl imp elfzo0 sylibr fveq2 eleq1d eldifi expcom - syl6 com13 mpcom ad2antrr wb 3anbi3d eqeq2d oddprmALTV ad3antrrr anim12ci - oveq2 3simpa df-3an cc oddz zcnd com23 reximdva syld prmz oveq1 sylan9req - npcand exp31 impcom jca rspcedvd ex exp41 com25 syl6com ancoms 3impib mpd - com15 imp31 ) AFUDBUEUFUGZUHZIUIUGZUHZIFEUJZFUDUKUFEUJULUFUGIUVBUMUFUNUOU - PUHZIFUDUMUFZEUJZUMUFZUQUGZUVFHURUPZUNUVFURUPZUSZLUTZUIUGZKUTZUIUGZJUTZUI - UGZUSZIUVKUVMUKUFZUVOUKUFZVAZUHZJVBVCZKVBVCZLVBVCZUVAAUUTUURUVCUVJVDAUURU - UTVEUUSUUTVFAUURUUTVLABUVFCDEFGHIMNOPQRSTUAUVFVGVHVIAUURUUTUVJUWDVDZAUNDU - TZURUPZUWFHURUPZUHZUWFVJUGZVDZDUQVKZUURUUTUWEVDVDOUVJUWLUURUUTAUWDUVGUVHU - VIUWLUURUUTAUWDVDVDVDZVDUVGUWLUVHUVIUHZUWMUVGUWLUVIUVHUHZUVFVJUGZVDZUWNUW - MVDUWLUNUCUTZURUPZUWRHURUPZUHZUWRVJUGZVDZUCUQVKUVGUWQUWKUXCDUCUQUWFUWRVAZ - 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elfzo2 eluzelre adantr ltm1d 1red resubcld zre adantl + cr lttr mpand 3impia 3jcad syl imp elfzo0 sylibr fveq2 eleq1d eldifi syl6 + expcom com13 mpcom ad2antrr wb 3anbi3d eqeq2d oddprmALTV ad3antrrr 3simpa + oveq2 anim12ci df-3an cc oddz zcnd com23 reximdva syld npcand oveq1 exp31 + prmz sylan9req impcom rspcedvd ex exp41 com25 syl6com ancoms 3impib com15 + jca mpd imp31 ) AFUDBUEUFUGZUHZIUIUGZUHZIFEUJZFUDUKUFEUJULUFUGIUVBUMUFUNU + OUPUHZIFUDUMUFZEUJZUMUFZUQUGZUVFHURUPZUNUVFURUPZUSZLUTZUIUGZKUTZUIUGZJUTZ + UIUGZUSZIUVKUVMUKUFZUVOUKUFZVAZUHZJVBVCZKVBVCZLVBVCZUVAAUUTUURUVCUVJVDAUU + RUUTVEUUSUUTVFAUURUUTVLABUVFCDEFGHIMNOPQRSTUAUVFVGVHVIAUURUUTUVJUWDVDZAUN + DUTZURUPZUWFHURUPZUHZUWFVJUGZVDZDUQVKZUURUUTUWEVDVDOUVJUWLUURUUTAUWDUVGUV + HUVIUWLUURUUTAUWDVDVDVDZVDUVGUWLUVHUVIUHZUWMUVGUWLUVIUVHUHZUVFVJUGZVDZUWN + UWMVDUWLUNUCUTZURUPZUWRHURUPZUHZUWRVJUGZVDZUCUQVKUVGUWQUWKUXCDUCUQUWFUWRV + AZUWIUXAUWJUXBUXDUWGUWSUWHUWTUWFUWRUNURVMUWFUWRHURVNVOUWFUWRVJVPVQVRUXCUW + 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X ) $. @@ -769353,17 +773215,17 @@ an isomorphism in the category of graphs (something like "GraphIsom ( vc vd ve wf cv cfv wceq wi wral isomuspgrlem2b wcel wa isomuspgrlem2a wf1 cima wb syl imaeq2 fveq2 eqeq12d rspcv ad2antrl imp eqcomd ad2antll mpidan wss wf1o f1of1 cuspgr cupgr uspgrupgr cuhgr upgruhgr cedg eleq2i - cvtx cpw edguhgr elpwi sseqtrrdi ex syl5bi anim12d 3syl syl2an2r biimpd - f1imaeq sylbid ralrimivva dff13 sylanbrc ) AEHGUFUCUGZGUHZUDUGZGUHZUIZW - OWQUIZUJZUDEUKUCEUKEHGUPABCDEFGHIJLMNOPQRSTUAULAXAUCUDEEAWOEUMZWQEUMZUN - ZUNZWSFWOUQZFWQUQZUIZWTAXDFUEUGZUQZXIGUHZUIZUEEUKZWSXHURAFKUMXMUBBCDUEE - FGHIJKNOPQRUOUSXEXMUNZWPXFWRXGXNXFWPXEXMXFWPUIZXBXMXOUJAXCXLXOUEWOEXIWO - UIXJXFXKWPXIWOFUTXIWOGVAVBVCVDVEVFXNXGWRXEXMXGWRUIZXCXMXPUJAXBXLXPUEWQE - XIWQUIXJXGXKWRXIWQFUTXIWQGVAVBVCVGVEVFVBVHXEXHWTAIJFUPZXDWOIVIZWQIVIZUN - ZXHWTURAIJFVJXQTIJFVKUSAXDXTACVLUMCVMUMZXDXTUJZSCVNYACVOUMZYBCVPYCXBXRX - CXSXBWOCVQUHZUMZYCXREYDWOPVRYCYEXRYCYEUNWOCVSUHZVTZUMZXRWOCWAYHWOYFIWOY - FWBNWCUSWDWEXCWQYDUMZYCXSEYDWQPVRYCYIXSYCYIUNWQYGUMZXSWQCWAYJWQYFIWQYFW - BNWCUSWDWEWFUSWGVEIJWOWQFWJWHWIWKWLUCUDEHGWMWN $. + cvtx cpw edguhgr elpwi sseqtrrdi biimtrid anim12d 3syl f1imaeq syl2an2r + ex biimpd sylbid ralrimivva dff13 sylanbrc ) AEHGUFUCUGZGUHZUDUGZGUHZUI + ZWOWQUIZUJZUDEUKUCEUKEHGUPABCDEFGHIJLMNOPQRSTUAULAXAUCUDEEAWOEUMZWQEUMZ + UNZUNZWSFWOUQZFWQUQZUIZWTAXDFUEUGZUQZXIGUHZUIZUEEUKZWSXHURAFKUMXMUBBCDU + EEFGHIJKNOPQRUOUSXEXMUNZWPXFWRXGXNXFWPXEXMXFWPUIZXBXMXOUJAXCXLXOUEWOEXI + WOUIXJXFXKWPXIWOFUTXIWOGVAVBVCVDVEVFXNXGWRXEXMXGWRUIZXCXMXPUJAXBXLXPUEW + QEXIWQUIXJXGXKWRXIWQFUTXIWQGVAVBVCVGVEVFVBVHXEXHWTAIJFUPZXDWOIVIZWQIVIZ + UNZXHWTURAIJFVJXQTIJFVKUSAXDXTACVLUMCVMUMZXDXTUJZSCVNYACVOUMZYBCVPYCXBX + RXCXSXBWOCVQUHZUMZYCXREYDWOPVRYCYEXRYCYEUNWOCVSUHZVTZUMZXRWOCWAYHWOYFIW + OYFWBNWCUSWIWDXCWQYDUMZYCXSEYDWQPVRYCYIXSYCYIUNWQYGUMZXSWQCWAYJWQYFIWQY + FWBNWCUSWIWDWEUSWFVEIJWOWQFWGWHWJWKWLUCUDEHGWMWN $. $d B c d $. $d E b m n $. $d F a m n $. $d F y $. $d G y $. $d K e m n $. $d V m n $. $d W c d m n $. $d X y $. $d ph e m n $. @@ -769763,19 +773625,19 @@ it allows only walks consisting of not proper hyperedges (i.e. edges cpr cwlks cupgr cupwlks wlkv ciedg cdm cword chash cfz cvtx wf cv c1 cfzo caddc csn wss wif iswlk simpr1 simpr2 wn df-ifp dfsn2 preq2 eqtrid eqeq2d wo biimpa a1d cedg simpr simpl upgredginwlk syl2anr imp wne simprr simplr - df-ne fvexd 3jca sylbir adantl upgredgpr syl31anc eqcomd mpd com12 syl5bi - id jaoi ralimdva ex com23 3impia sylbid impd wb isupwlk mpbird mpid ) BAC - UAEFZCUBGZBACUCEFZXCXDCHGBHGAHGIZXEABCUDXCXDXFXEXCXDJZXFJXEBCUEEZUFUGGZKB - UHEZUILCUJEZAUKZDULZBEXHEZXMAEZXMUMUOLZAEZTZMZDKXJUNLZNZIZXFXGYBXFXCXDYBX - FXCXIXLXOXQMZXNXOUPZMZXRXNUQZURZDXTNZIZXDYBOAHDBCXHXKHHXKPZXHPZUSXFXDYIYB - XFXDYIYBXFXDJZYIJXIXLYAYLXIXLYHUTYLXIXLYHVAYIYLYAXIXLYHYLYAOXIXLJZYLYHYAY - MYLYHYAOYMYLJZYGXSDXTYGYCYEJZYCVBZYFJZVHZYNXMXTGZJZXSYCYEYFVCYRYTXSYOYTXS - OYQYOXSYTYCYEXSYCYDXRXNYCYDXOXOTXRXOVDXOXQXOVEVFVGVIVJYTYQXSYTXNCVKEZGZYQ - XSOYNYSUUBYLXDXIYSUUBOYMXFXDVLXIXLVMUUABCXHXMYKUUAPZVNVOVPYTUUBYQXSYTUUBJ - ZYQJZXRXNUUEXDUUBYFXOHGZXQHGZXOXQVQZIZXRXNMUUDXDYQYTXDUUBYNXDYSYMXFXDVRQQ - QYTUUBYQVSUUDYPYFVRYQUUIUUDYPUUIYFYPUUHUUIXOXQVTUUHUUFUUGUUHUUHXMAWAUUHXP - AWAUUHWKWBWCQWDXOXQXNHUUACXKHYJUUCWEWFWGRWHWIWLWIWJWMWNWOWPSWBRWOWQWRSXFX - EYBWSXGAHDBCXHXKHHYJYKWTWDXARXBS $. + df-ne fvexd id sylbir adantl upgredgpr syl31anc eqcomd mpd com12 biimtrid + 3jca jaoi ralimdva ex com23 3impia sylbid impd wb isupwlk mpbird mpid ) B + ACUAEFZCUBGZBACUCEFZXCXDCHGBHGAHGIZXEABCUDXCXDXFXEXCXDJZXFJXEBCUEEZUFUGGZ + KBUHEZUILCUJEZAUKZDULZBEXHEZXMAEZXMUMUOLZAEZTZMZDKXJUNLZNZIZXFXGYBXFXCXDY + BXFXCXIXLXOXQMZXNXOUPZMZXRXNUQZURZDXTNZIZXDYBOAHDBCXHXKHHXKPZXHPZUSXFXDYI + YBXFXDYIYBXFXDJZYIJXIXLYAYLXIXLYHUTYLXIXLYHVAYIYLYAXIXLYHYLYAOXIXLJZYLYHY + AYMYLYHYAOYMYLJZYGXSDXTYGYCYEJZYCVBZYFJZVHZYNXMXTGZJZXSYCYEYFVCYRYTXSYOYT + XSOYQYOXSYTYCYEXSYCYDXRXNYCYDXOXOTXRXOVDXOXQXOVEVFVGVIVJYTYQXSYTXNCVKEZGZ + YQXSOYNYSUUBYLXDXIYSUUBOYMXFXDVLXIXLVMUUABCXHXMYKUUAPZVNVOVPYTUUBYQXSYTUU + BJZYQJZXRXNUUEXDUUBYFXOHGZXQHGZXOXQVQZIZXRXNMUUDXDYQYTXDUUBYNXDYSYMXFXDVR + QQQYTUUBYQVSUUDYPYFVRYQUUIUUDYPUUIYFYPUUHUUIXOXQVTUUHUUFUUGUUHUUHXMAWAUUH + XPAWAUUHWBWKWCQWDXOXQXNHUUACXKHYJUUCWEWFWGRWHWIWLWIWJWMWNWOWPSWKRWOWQWRSX + FXEYBWSXGAHDBCXHXKHHYJYKWTWDXARXBS $. $} $( In a pseudograph, the definitions for a walk and a simple walk are @@ -770155,13 +774017,13 @@ their group (addition) operations are equal for all pairs of elements of mgmpropd $p |- ( ph -> ( K e. Mgm <-> L e. Mgm ) ) $= ( va cv cplusg cfv co cbs wcel wral eleq2d eqid cmgm wa wceq simpl eqcomd biimpcd adantr impcom biimpd adantld imp syl12anc eleq1d 2ralbidva eqtr3d - wi raleqbidv bitrd c0 wne wb wex n0 ismgmn0 syl6bi exlimdv syl5bi 3bitr4d - mpd ) ABLZCLZEMNZOZEPNZQZCVNRBVNRZVJVKFMNZOZFPNZQZCVSRZBVSRZEUAQZFUAQZAVP - VRVNQZCVNRZBVNRWBAVOWEBCVNVNAVJVNQZVKVNQZUBZUBZVMVRVNWJAVJDQZVKDQZVMVRUCA - WIUDWIAWKWGAWKUPWHAWGWKAVNDVJADVNGUEZSUFUGUHAWIWLAWHWLWGAWHWLAVNDVKWMSUIU - JUKJULUMUNAWFWABVNVSADVNVSGHUOZAWEVTCVNVSWNAVNVSVRWNSUQUQURADUSUTZWCVPVAZ - IWOKLZDQZKVBZAWPKDVCZAWRWPKAWRWQVNQWPADVNWQGSBCWQVNEVLVNTVLTVDVEVFVGVIAWO - WDWBVAZIWOWSAXAWTAWRXAKAWRWQVSQXAADVSWQHSBCWQVSFVQVSTVQTVDVEVFVGVIVH $. + wi raleqbidv bitrd c0 wne wex ismgmn0 syl6bi exlimdv biimtrid mpd 3bitr4d + wb n0 ) ABLZCLZEMNZOZEPNZQZCVNRBVNRZVJVKFMNZOZFPNZQZCVSRZBVSRZEUAQZFUAQZA + VPVRVNQZCVNRZBVNRWBAVOWEBCVNVNAVJVNQZVKVNQZUBZUBZVMVRVNWJAVJDQZVKDQZVMVRU + CAWIUDWIAWKWGAWKUPWHAWGWKAVNDVJADVNGUEZSUFUGUHAWIWLAWHWLWGAWHWLAVNDVKWMSU + IUJUKJULUMUNAWFWABVNVSADVNVSGHUOZAWEVTCVNVSWNAVNVSVRWNSUQUQURADUSUTZWCVPV + HZIWOKLZDQZKVAZAWPKDVIZAWRWPKAWRWQVNQWPADVNWQGSBCWQVNEVLVNTVLTVBVCVDVEVFA + WOWDWBVHZIWOWSAXAWTAWRXAKAWRWQVSQXAADVSWQHSBCWQVSFVQVSTVQTVBVCVDVEVFVG $. $} ${ @@ -770698,14 +774560,14 @@ their group (addition) operations are equal for all pairs of elements of ( va vc wrex wral wcel wa simpr wceq wn adantr cv cmpo co cmnd wnel chash wne cfv clt wbr cvv cbs fvexi a1i simpl hashgt12el2 syl3anc rexbii rexnal c1 df-ne bitri eqidd weq ovmpod eqtr3d ex ralimdva rexlimdva con3d bicomi - ralbii ralnex 3bitr3i syl6ib syl5bi syl5 mp2and cplusg eqcomi isnmnd syl - ) AKUAZLUAZBCDDEUBZUCZWDUGZLDMZKDNZFUDUEAEDOZUTDUFUHUIUJZWIIJWJWKPZEWDUGZ - LDMZAWIWLDUKOZWKWJWNWOWLDFULGUMUNWJWKQWJWKUOEDUKLUPUQWNEWDRZLDNZSZAWIWNWP - SZLDMWRWMWSLDEWDVAURWPLDUSVBAWRWFWDRZLDNZKDMZSZWIAXBWQAXAWQKDAWCDOZPZWTWP - LDXEWDDOZPZWTWPXGWTPWFEWDXGWFERWTXGBCWCWDDDEEWEDXGWEVCXGBKVDCLVDPPEVCXEXD - XFAXDQTXEXFQXEWJXFAWJXDITTVETXGWTQVFVGVHVIVJXASZKDNWTSZLDMZKDNXCWIXHXJKDX - JXHWTLDUSVKVLXAKDVMXJWHKDXIWGLDWGXIWFWDVAVKURVLVNVOVPVQVRLKDFWEGFVSUHWEHV - TWAWB $. + ralbii ralnex 3bitr3i syl6ib biimtrid syl5 mp2and cplusg eqcomi isnmnd + syl ) AKUAZLUAZBCDDEUBZUCZWDUGZLDMZKDNZFUDUEAEDOZUTDUFUHUIUJZWIIJWJWKPZEW + DUGZLDMZAWIWLDUKOZWKWJWNWOWLDFULGUMUNWJWKQWJWKUOEDUKLUPUQWNEWDRZLDNZSZAWI + WNWPSZLDMWRWMWSLDEWDVAURWPLDUSVBAWRWFWDRZLDNZKDMZSZWIAXBWQAXAWQKDAWCDOZPZ + WTWPLDXEWDDOZPZWTWPXGWTPWFEWDXGWFERWTXGBCWCWDDDEEWEDXGWEVCXGBKVDCLVDPPEVC + XEXDXFAXDQTXEXFQXEWJXFAWJXDITTVETXGWTQVFVGVHVIVJXASZKDNWTSZLDMZKDNXCWIXHX + JKDXJXHWTLDUSVKVLXAKDVMXJWHKDXIWGLDWGXIWFWDVAVKURVLVNVOVPVQVRLKDFWEGFVSUH + WEHVTWAWB $. $} ${ @@ -771358,10 +775220,10 @@ Similar results are obtained for an associative operation (defining 16-Jan-2020.) $) mgm2mgm $p |- ( M e. MgmALT <-> M e. Mgm ) $= ( vx vy cmgm2 wcel cmgm cplusg cfv cbs ccllaw wbr eqid ismgmALT cv co cvv - wral wb fvex iscllaw biimprd syl5bi sylbid pm2.43i mgmplusgiopALT mpbird - mp2an ismgm impbii ) ADEZAFEZUJUKUJUJAGHZAIHZJKZUKUMADULUMLZULLZMUNBNCNUL - OUMECUMQBUMQZUJUKULPEUMPEUNUQRAGSAISBCUMPPULTUGUJUKUQBCUMADULUOUPUHUAUBUC - UDUKUJUNAUEUMAFULUOUPMUFUI $. + wral wb fvex iscllaw mp2an biimprd biimtrid sylbid pm2.43i mgmplusgiopALT + ismgm mpbird impbii ) ADEZAFEZUJUKUJUJAGHZAIHZJKZUKUMADULUMLZULLZMUNBNCNU + LOUMECUMQBUMQZUJUKULPEUMPEUNUQRAGSAISBCUMPPULTUAUJUKUQBCUMADULUOUPUGUBUCU + DUEUKUJUNAUFUMAFULUOUPMUHUI $. $} ${ @@ -771496,8 +775358,8 @@ Nonzero rings (extension) ( crg cnzr cdif wcel csn wceq 0ringdif simprbi ) BFGHIBFIACJKABCDELM $. 0ring1eq0.1 $e |- .1. = ( 1r ` R ) $. - $( In a zero ring, a ring which is not a nonzero ring, the unit equals the - zero element. (Contributed by AV, 17-Apr-2020.) $) + $( In a zero ring, a ring which is not a nonzero ring, the ring unity + equals the zero element. (Contributed by AV, 17-Apr-2020.) $) 0ring1eq0 $p |- ( R e. ( Ring \ NzRing ) -> .1. = .0. ) $= ( crg cnzr cdif wcel wn wa wceq eldif cbs cfv chash c1 0ringnnzr eqid imp 0ring01eq eqcomd ex sylbird sylbi ) BHIJKBHKZBIKLZMCDNZBHIOUHUIUJUHUIBPQZ @@ -771618,10 +775480,10 @@ Nonzero rings (extension) ringrng $p |- ( R e. Ring -> R e. Rng ) $= ( vx vy vz cabl wcel crg crng ringabl cgrp cmgp cmnd cv cplusg co wceq wa cfv wral eqid cmulr cbs isring csgrp simpl mndsgrp 3ad2ant2 adantl simpr3 - w3a isrng syl3anbrc ex syl5bi mpcom ) AEFZAGFZAHFZAIUQAJFZAKRZLFZBMZCMZDM - ZANRZOAUARZOVBVCVFOVBVDVFOZVEOPVBVCVEOVDVFOVGVCVDVFOVEOPQDAUBRZSCVHSBVHSZ - UJZUPURBCDVHVEAVFUTVHTZUTTZVETZVFTZUCUPVJURUPVJQUPUTUDFZVIURUPVJUEVJVOUPV - AUSVOVIUTUFUGUHUPUSVAVIUIBCDVHVEAVFUTVKVLVMVNUKULUMUNUO $. + w3a isrng syl3anbrc ex biimtrid mpcom ) AEFZAGFZAHFZAIUQAJFZAKRZLFZBMZCMZ + DMZANRZOAUARZOVBVCVFOVBVDVFOZVEOPVBVCVEOVDVFOVGVCVDVFOVEOPQDAUBRZSCVHSBVH + SZUJZUPURBCDVHVEAVFUTVHTZUTTZVETZVFTZUCUPVJURUPVJQUPUTUDFZVIURUPVJUEVJVOU + PVAUSVOVIUTUFUGUHUPUSVAVIUIBCDVHVEAVFUTVKVLVMVNUKULUMUNUO $. $( The unital rings are (non-unital) rings. (Contributed by AV, 20-Mar-2020.) $) @@ -771653,7 +775515,7 @@ Nonzero rings (extension) rngdi.b $e |- B = ( Base ` R ) $. rngdi.p $e |- .+ = ( +g ` R ) $. rngdi.t $e |- .x. = ( .r ` R ) $. - $( Distributive law for the multiplication operation of a nonunital ring + $( Distributive law for the multiplication operation of a non-unital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.) $) rngdir $p |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) $= @@ -771673,7 +775535,7 @@ Nonzero rings (extension) ${ rngcl.b $e |- B = ( Base ` R ) $. rngcl.t $e |- .x. = ( .r ` R ) $. - $( Closure of the multiplication operation of a nonunital ring. + $( Closure of the multiplication operation of a non-unital ring. (Contributed by AV, 17-Apr-2020.) $) rngcl $p |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B ) $= ( crng wcel cmgp cfv cmgm csgrp eqid rngmgp sgrpmgm syl mgpbas mgpplusg @@ -771681,7 +775543,7 @@ Nonzero rings (extension) FFRBCUDUFGSUAUB $. rnglz.z $e |- .0. = ( 0g ` R ) $. - $( The zero of a nonunital ring is a left-absorbing element. (Contributed + $( The zero of a non-unital ring is a left-absorbing element. (Contributed by AV, 17-Apr-2020.) $) rnglz $p |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = .0. ) $= ( crng wcel wa co cplusg cfv wceq cgrp cabl syl adantr syl2anc rngabl w3a @@ -772040,8 +775902,8 @@ Nonzero rings (extension) GVHABFVRVHVRFDULPZDVRFWATHVRTZUMUNUQUOUPVGVLURKVMURKZVSVNKVHCVLVLTZUSVHVJ WCVPDVMVMTZUSSABVLVMVSVRBCVLWDGUTDVRVMWEWBVAVSTVBUHVCVDCDEVLVMWDWEVEVF $. - $( The constant mapping to zero is a nonunital ring homomorphism from any - nonunital ring to the zero ring. (Contributed by AV, 17-Apr-2020.) $) + $( The constant mapping to zero is a non-unital ring homomorphism from any + non-unital ring to the zero ring. (Contributed by AV, 17-Apr-2020.) $) c0rnghm $p |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> H e. ( S RngHomo T ) ) $= ( crng wcel crg cnzr wa co cmgp cfv cgrp syl eqid cdif cghm wss ringssrng @@ -772071,20 +775933,20 @@ Nonzero rings (extension) cvv wb eqid mndidcl fmptd cmpt a1i eqidd vsnid eleq2 mpbird adantl fvmptd weq simpr oveq12d mndlid mpdan mgmcl syl3anc wi syl6bi mpd fveq2d 3eqtrrd elsni eqtr2d raleqdv raleqbidv fvoveq1 fveq2 oveq1d eqeq12d oveq2 2ralsng - id oveq2d el2v bitrdi jca ex exlimdv syl5bi 3impia ismgmhm sylanbrc ) CUA - MZDUBMZBUGNUCOZUDZXFCUBMZPBCUENZEUHZJQZKQZDUFNZRENZXLENZXMENZCUFNZRZOZKBS - ZJBSZPZEDCUIRMXHXIXFXEXFXIXFPXGXEXIXFCUJUKULUMXEXFXGYCXGBLQZUNZOZLUOZXEXF - PZYCBUSMXGYGUTBDUEGUPBUSLUQURYHYFYCLYHYFYCYHYFPZXKYBYIABFXJEYIFXJMZAQBMYH - YJYFXEYJXFXJCFXJVAZHVBZTTZTIVCYIYBYDYDXNRZENZYDENZYPXRRZOZYIYPFOZYRYIAYDF - FBEXJEABFVDOYIIVEYIALVLPFVFYFYDBMZYHYFYTYDYEMZUUAYFLVGVEBYEYDVHVIVJZYMVKY - IYSPZYQFFXRRZFYOUUCYPFYPFXRYIYSVMZUUEVNYIUUDFOZYSYHUUFYFXEUUFXFXEYJUUFYLX - JXRCFFYKXRVAZHVOVPTTTUUCYOYPFYIYOYPOYSYIYNYDEYIYTYNYDOZUUBYIYTPZYNBMZUUHU - UIXFYTYTUUJYIXFYTYHXFYFXEXFVMTTYIYTVMZUUKBDYDYDXNGXNVAZVQVRYIUUJUUHVSZYTY - FUUMYHYFUUJYNYEMUUHBYEYNVHYNYDWDVTVJTWAVPWBTUUEWEWCVPYIYBXTKYESZJYESZYRYF - YBUUOUTYHYFYAUUNJBYEYFWNZYFXTKBYEUUPWFWGVJUUOYRUTLLXTYDXMXNRZENZYPXQXRRZO - YRJKYDYDUSUSJLVLZXOUURXSUUSXLYDXMEXNWHUUTXPYPXQXRXLYDEWIWJWKKLVLZUURYOUUS - YQUVAUUQYNEXMYDYDXNWLWBUVAXQYPYPXRXMYDEWIWOWKWMWPWQVIWRWSWTXAXBJKBXJXNXRD - CEGYKUULUUGXCXD $. + id oveq2d el2v bitrdi jca ex exlimdv biimtrid 3impia ismgmhm sylanbrc ) C + UAMZDUBMZBUGNUCOZUDZXFCUBMZPBCUENZEUHZJQZKQZDUFNZRENZXLENZXMENZCUFNZRZOZK + BSZJBSZPZEDCUIRMXHXIXFXEXFXIXFPXGXEXIXFCUJUKULUMXEXFXGYCXGBLQZUNZOZLUOZXE + XFPZYCBUSMXGYGUTBDUEGUPBUSLUQURYHYFYCLYHYFYCYHYFPZXKYBYIABFXJEYIFXJMZAQBM + YHYJYFXEYJXFXJCFXJVAZHVBZTTZTIVCYIYBYDYDXNRZENZYDENZYPXRRZOZYIYPFOZYRYIAY + DFFBEXJEABFVDOYIIVEYIALVLPFVFYFYDBMZYHYFYTYDYEMZUUAYFLVGVEBYEYDVHVIVJZYMV + KYIYSPZYQFFXRRZFYOUUCYPFYPFXRYIYSVMZUUEVNYIUUDFOZYSYHUUFYFXEUUFXFXEYJUUFY + LXJXRCFFYKXRVAZHVOVPTTTUUCYOYPFYIYOYPOYSYIYNYDEYIYTYNYDOZUUBYIYTPZYNBMZUU + HUUIXFYTYTUUJYIXFYTYHXFYFXEXFVMTTYIYTVMZUUKBDYDYDXNGXNVAZVQVRYIUUJUUHVSZY + TYFUUMYHYFUUJYNYEMUUHBYEYNVHYNYDWDVTVJTWAVPWBTUUEWEWCVPYIYBXTKYESZJYESZYR + YFYBUUOUTYHYFYAUUNJBYEYFWNZYFXTKBYEUUPWFWGVJUUOYRUTLLXTYDXMXNRZENZYPXQXRR + ZOYRJKYDYDUSUSJLVLZXOUURXSUUSXLYDXMEXNWHUUTXPYPXQXRXLYDEWIWJWKKLVLZUURYOU + USYQUVAUUQYNEXMYDYDXNWLWBUVAXQYPYPXRXMYDEWIWOWKWMWPWQVIWRWSWTXAXBJKBXJXNX + RDCEGYKUULUUGXCXD $. ${ $d Z x $. @@ -772114,8 +775976,8 @@ Nonzero rings (extension) CUEUHUIUJUK $. $} - $( The constant mapping to zero is a nonunital ring homomorphism from the - zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.) $) + $( The constant mapping to zero is a non-unital ring homomorphism from the + zero ring to any non-unital ring. (Contributed by AV, 17-Apr-2020.) $) zrrnghm $p |- ( ( S e. Rng /\ T e. ( Ring \ NzRing ) ) -> H e. ( T RngHomo S ) ) $= ( va vc wcel wa co cfv wceq wral syl adantr eqid crng crg cnzr cdif cmulr @@ -773529,7 +777391,7 @@ underlying set (base set) and the morphisms (non-unital ring zrinitorngc.c $e |- C = ( RngCat ` U ) $. zrinitorngc.z $e |- ( ph -> Z e. ( Ring \ NzRing ) ) $. zrinitorngc.e $e |- ( ph -> Z e. U ) $. - $( The zero ring is an initial object in the category of nonunital rings. + $( The zero ring is an initial object in the category of non-unital rings. (Contributed by AV, 18-Apr-2020.) $) zrinitorngc $p |- ( ph -> Z e. ( InitO ` C ) ) $= ( vh vr vx cfv wcel wa wceq crng eqid eleq2d adantr va cinito cv chom weu @@ -773552,7 +777414,7 @@ underlying set (base set) and the morphisms (non-unital ring UUKWQXTWLNUUKUUIWBWOVSWPWRWSWTTXAXBWSXNYDJYAYBXJYAXMXCXDWIXEAXPBJXLEKYMYT AYSBXFNFBCDGXGWIUUAXIXH $. - $( The zero ring is a terminal object in the category of nonunital rings. + $( The zero ring is a terminal object in the category of non-unital rings. (Contributed by AV, 17-Apr-2020.) $) zrtermorngc $p |- ( ph -> Z e. ( TermO ` C ) ) $= ( vh vr vx cfv wcel cv wa wceq crng eqid adantr va ctermo chom co weu cbs @@ -774102,24 +777964,22 @@ subcategory subset of the mappings between base sets of unital rings (in ( co wa wcel wceq wbr csect cfv crh ccom cid cbs cres crs ccnv ringccat ccat syl isinv w3a ringcsect df-3an bitrdi 3ancoma bitri anbi12d anandi eqid wf1o simplrl adantl wf rhmf anim12i ad2antlr simpr ad2antrl fcof1o - jca32 eqcom anbi2i sylib anass sylanbrc wb syl2anc anbi1d adantr mpbird - isrim rimrhm isrim0 eleq1 eqcoms anbi2d sylan9bbr syl6bi expdimp impcom - com12 coeq1 ad2antll rimf1o f1ococnv1 eqtrd jca31 biimpcd coeq2 impbida - wi f1ococnv2 3bitrd ) AEFIJGQUAEFIJCUBUCZQUAZFEJIXHQUAZRZEIJUDQSZFJIUDQ - ZSZRZFEUEZUFIUGUCZUHZTZRZXORZXTEFUEZUFJUGUCZUHZTZRZRZEIJUIQSZFEUJZTZRZA - BCXHEFGIJLPADHSCULSMCDHKUKUMNOXHVCZUNAXKXTXOYERZRYGAXIXTXJYMAXIXLXNXSUO - XTABCXHDXQEFHIJKLMNOXQVCZYLUPXLXNXSUQURAXJXNXLYEUOZYMABCXHDYCFEHJIKLMON - YCVCZYLUPYOXLXNYEUOYMXNXLYEUSXLXNYEUQUTURVAXTXOYEVBURAYGYKAYGRZYKXLXQYC - EVDZRZYJRZYQXLYRYJRZYTYGXLAXTXLXNYFVEVFYQXQYCEVGZYCXQFVGZRZYEXSRRZUUAYG - UUEAYGUUDYEXSXOUUDXTYFXLUUBXNUUCXQYCIJEYNYPVHYCXQJIFYPYNVHVIVJYFYEYAXTY - EVKVFXTXSYAYEXOXSVKVLVNVFUUEYRYIFTZRUUAXQYCEFVMUUFYJYRYIFVOVPVQUMXLYRYJ - VRVSAYKYTVTYGAYHYSYJAIBSZJBSZYHYSVTNOXQYCIJEBBYNYPWEWAWBWCWDAYKRZXTXOYF - UUIXLXNXSYHXLAYJXQYCIJEYNYPWFVLYKAXNYHYJAXNYJARZYHXNUUJYHXOXNAYHXLYIXMS - ZRZYJXOAUUGUUHYHUULVTNOIJEBBWGWAZYJUUKXNXLUUKXNVTYIFYIFXMWHWIWJWKXLXNVK - WLWOWMWNUUIXPYIEUEZXRYJXPUUNTAYHFYIEWPWQUUIYRUUNXRTYHYRAYJXQYCIJEYNYPWR - VLZXQYCEWSUMWTZXAUUIXOUULYKAUULYHAUULXEYJAYHUULUUMXBWCWNUUIXNUUKXLYJXNU - UKVTAYHFYIXMWHWQWJWDZUUIXOXSYEUUQUUPUUIYBEYIUEZYDYJYBUURTAYHFYIEXCWQUUI - YRUURYDTUUOXQYCEXFUMWTXAXAXDXG $. + jca32 eqcom anbi2i sylib anass sylanbrc a1i anbi1d adantr mpbird rimrhm + wb isrim isrim0 simprbi eleq1 syl5ibrcom coeq1 ad2antll f1ococnv1 eqtrd + imp rimf1o jca31 biimpi anbi2d coeq2 f1ococnv2 impbida 3bitrd ) AEFIJGQ + UAEFIJCUBUCZQUAZFEJIXDQUAZRZEIJUDQSZFJIUDQZSZRZFEUEZUFIUGUCZUHZTZRZXKRZ + XPEFUEZUFJUGUCZUHZTZRZRZEIJUIQSZFEUJZTZRZABCXDEFGIJLPADHSCULSMCDHKUKUMN + OXDVCZUNAXGXPXKYARZRYCAXEXPXFYIAXEXHXJXOUOXPABCXDDXMEFHIJKLMNOXMVCZYHUP + XHXJXOUQURAXFXJXHYAUOZYIABCXDDXSFEHJIKLMONXSVCZYHUPYKXHXJYAUOYIXJXHYAUS + XHXJYAUQUTURVAXPXKYAVBURAYCYGAYCRZYGXHXMXSEVDZRZYFRZYMXHYNYFRZYPYCXHAXP + XHXJYBVEVFYMXMXSEVGZXSXMFVGZRZYAXORRZYQYCUUAAYCYTYAXOXKYTXPYBXHYRXJYSXM + XSIJEYJYLVHXSXMJIFYLYJVHVIVJYBYAXQXPYAVKVFXPXOXQYAXKXOVKVLVNVFUUAYNYEFT + ZRYQXMXSEFVMUUBYFYNYEFVOVPVQUMXHYNYFVRVSAYGYPWEYCAYDYOYFYDYOWEAXMXSIJEY + JYLWFVTWAWBWCAYGRZXPXKYBUUCXHXJXOYDXHAYFIJEWDVLYGXJAYDYFXJYDXJYFYEXISZY + DXHUUDIJEWGZWHFYEXIWIZWJWOVFUUCXLYEEUEZXNYFXLUUGTAYDFYEEWKWLUUCYNUUGXNT + YDYNAYFXMXSIJEYJYLWPVLZXMXSEWMUMWNZWQUUCXKXHUUDRZYDUUJAYFYDUUJUUEWRVLUU + CXJUUDXHYFXJUUDWEAYDUUFWLWSWCZUUCXKXOYAUUKUUIUUCXREYEUEZXTYFXRUULTAYDFY + EEWTWLUUCYNUULXTTUUHXMXSEXAUMWNWQWQXBXC $. $} ${ @@ -774522,25 +778382,23 @@ the category of sets which sends each ring to its underlying set (base ( F ( X N Y ) G <-> ( F e. ( X RingIso Y ) /\ G = `' F ) ) ) $= ( co wa wcel wceq wbr csect cfv crh ccom cid cbs cres ccnv ringccatALTV crs ccat syl eqid isinv w3a ringcsectALTV df-3an bitrdi 3ancoma anbi12d - bitri anandi wf1o simplrl adantl wf anim12i ad2antlr simpr ad2antrl jca - fcof1o eqcom anbi2i sylib anass sylanbrc wb anbi1d adantr mpbird rimrhm - rhmf isrim isrim0 eleq1 eqcoms anbi2d syl6bi com12 expdimp impcom coeq1 - sylan9bbr ad2antll rimf1o f1ococnv1 eqtrd jca31 biimpcd coeq2 f1ococnv2 - wi impbida 3bitrd ) AEFIJGQUAEFIJCUBUCZQUAZFEJIXGQUAZRZEIJUDQSZFJIUDQZS - ZRZFEUEZUFIUGUCZUHZTZRZXNRZXSEFUEZUFJUGUCZUHZTZRZRZEIJUKQSZFEUIZTZRZABC - XGEFGIJLPADHSCULSMCDHKUJUMNOXGUNZUOAXJXSXNYDRZRYFAXHXSXIYLAXHXKXMXRUPXS - ABCXGDXPEFHIJKLMNOXPUNZYKUQXKXMXRURUSAXIXMXKYDUPZYLABCXGDYBFEHJIKLMONYB - UNZYKUQYNXKXMYDUPYLXMXKYDUTXKXMYDURVBUSVAXSXNYDVCUSAYFYJAYFRZYJXKXPYBEV - DZRZYIRZYPXKYQYIRZYSYFXKAXSXKXMYEVEVFYPXPYBEVGZYBXPFVGZRZYDXRRZRZYTYFUU - EAYFUUCUUDXNUUCXSYEXKUUAXMUUBXPYBIJEYMYOWDYBXPJIFYOYMWDVHVIYFYDXRYEYDXT - XSYDVJVFXSXRXTYDXNXRVJVKVLVLVFUUEYQYHFTZRYTXPYBEFVMUUFYIYQYHFVNVOVPUMXK - YQYIVQVRAYJYSVSYFAYGYRYIAIBSZJBSZRZYGYRVSAUUGUUHNOVLZXPYBIJEBBYMYOWEUMV - TWAWBAYJRZXSXNYEUUKXKXMXRYGXKAYIXPYBIJEYMYOWCVKYJAXMYGYIAXMYIARZYGXMUUL - YGXNXMAYGXKYHXLSZRZYIXNAUUIYGUUNVSUUJIJEBBWFUMZYIUUMXMXKUUMXMVSYHFYHFXL - WGWHWIWOXKXMVJWJWKWLWMUUKXOYHEUEZXQYIXOUUPTAYGFYHEWNWPUUKYQUUPXQTYGYQAY - IXPYBIJEYMYOWQVKZXPYBEWRUMWSZWTUUKXNUUNYJAUUNYGAUUNXDYIAYGUUNUUOXAWAWMU - UKXMUUMXKYIXMUUMVSAYGFYHXLWGWPWIWBZUUKXNXRYDUUSUURUUKYAEYHUEZYCYIYAUUTT - AYGFYHEXBWPUUKYQUUTYCTUUQXPYBEXCUMWSWTWTXEXF $. + bitri anandi wf1o simplrl wf rhmf anim12i ad2antlr simpr ad2antrl jca32 + adantl fcof1o eqcom anbi2i sylib anass sylanbrc isrim a1i anbi1d adantr + mpbird rimrhm isrim0 simprbi eleq1 syl5ibrcom imp coeq1 ad2antll rimf1o + wb f1ococnv1 eqtrd jca31 biimpi anbi2d coeq2 f1ococnv2 impbida 3bitrd ) + AEFIJGQUAEFIJCUBUCZQUAZFEJIXDQUAZRZEIJUDQSZFJIUDQZSZRZFEUEZUFIUGUCZUHZT + ZRZXKRZXPEFUEZUFJUGUCZUHZTZRZRZEIJUKQSZFEUIZTZRZABCXDEFGIJLPADHSCULSMCD + HKUJUMNOXDUNZUOAXGXPXKYARZRYCAXEXPXFYIAXEXHXJXOUPXPABCXDDXMEFHIJKLMNOXM + UNZYHUQXHXJXOURUSAXFXJXHYAUPZYIABCXDDXSFEHJIKLMONXSUNZYHUQYKXHXJYAUPYIX + JXHYAUTXHXJYAURVBUSVAXPXKYAVCUSAYCYGAYCRZYGXHXMXSEVDZRZYFRZYMXHYNYFRZYP + YCXHAXPXHXJYBVEVMYMXMXSEVFZXSXMFVFZRZYAXORRZYQYCUUAAYCYTYAXOXKYTXPYBXHY + RXJYSXMXSIJEYJYLVGXSXMJIFYLYJVGVHVIYBYAXQXPYAVJVMXPXOXQYAXKXOVJVKVLVMUU + AYNYEFTZRYQXMXSEFVNUUBYFYNYEFVOVPVQUMXHYNYFVRVSAYGYPWNYCAYDYOYFYDYOWNAX + MXSIJEYJYLVTWAWBWCWDAYGRZXPXKYBUUCXHXJXOYDXHAYFIJEWEVKYGXJAYDYFXJYDXJYF + YEXISZYDXHUUDIJEWFZWGFYEXIWHZWIWJVMUUCXLYEEUEZXNYFXLUUGTAYDFYEEWKWLUUCY + NUUGXNTYDYNAYFXMXSIJEYJYLWMVKZXMXSEWOUMWPZWQUUCXKXHUUDRZYDUUJAYFYDUUJUU + EWRVKUUCXJUUDXHYFXJUUDWNAYDUUFWLWSWDZUUCXKXOYAUUKUUIUUCXREYEUEZXTYFXRUU + LTAYDFYEEWTWLUUCYNUULXTTUUHXMXSEXAUMWPWQWQXBXC $. $} ${ @@ -775440,10 +779298,10 @@ Basic algebraic structures (extension) domains, i.e. the first base class may depend on the second base class), analogous to ~ mpoexxg . (Contributed by AV, 30-Mar-2019.) $) mpoexxg2 $p |- ( ( B e. R /\ A. y e. B A e. S ) -> F e. _V ) $= - ( wcel wral wa wfun cdm cvv mpofun cv csn cxp sylancr ciun dmmpossx2 snex - wss xpexg mpan2 ralimi iunexg sylan2 ssexg funex ) DFJZCGJZBDKZLZHMHNZOJZ - HOJABCDEHIPUOUPBDCBQZRZSZUAZUDVAOJZUQABCDEHIUBUNULUTOJZBDKVBUMVCBDUMUSOJV - CURUCCUSGOUEUFUGBDUTFOUHUIUPVAOUJTOHUKT $. + ( wcel wral wa wfun cdm cvv mpofun cv csn cxp sylancr wss dmmpossx2 vsnex + ciun xpexg mpan2 ralimi iunexg sylan2 ssexg funex ) DFJZCGJZBDKZLZHMHNZOJ + ZHOJABCDEHIPUOUPBDCBQRZSZUDZUAUTOJZUQABCDEHIUBUNULUSOJZBDKVAUMVBBDUMUROJV + BBUCCURGOUEUFUGBDUSFOUHUIUPUTOUJTOHUKT $. $} ${ @@ -775593,19 +779451,19 @@ Basic algebraic structures (extension) prmz zcnd biimpd sylbid prmuz2 nprm sylan2 eleq1 notbid pm2.24 com12 syl6bi wn com3l mpcom jaoi prmnn nnred mul02lem2 wne elnnne0 eqneqall eqcoms com23 elnnz lt0neg1 nngt0d simpr anim12ci orcd simprl mul2lt0bi mpbird wb nn0nlt0 - breq1 nnnn0 pm2.21d syldc sylbird adantld syl5bi imp 3impib ) CUADZAEDZBEDZ - CAFGZBHZABHZIZWSCUBDZCUCDZCUDZUEDZJZKWTXAJZXEIZCUFXFXLXJXFCUEDZCLHZKXLCUGXM - XLXNXMCUHHZCUIUJUKZDZKXLCULXOXLXQXOXKXEXOXKJZXCUHAFGZBHZXDXRXBXSBXOXBXSHXKC - UHAFUMMNXKXTXDIXOXKXTXDXKXSABWTXSAHXAWTAWTAAUOUPUNMNUQOURPXQXKXEXBEDZVGZXQX - KJZXEXKXQAXPDZYBWTYDXAAUSMCAUTVAXCYBYCXDXCYBXAVGZYCXDIXCYAXAXBBEVBVCYCYEXDX - KYEXDIZXQXAYFWTXAXDVDOOVEVFVHVIPVJTXNXCXKXDXNXCLAFGZBHZXKXDIXNXBYGBCLAFUMNX - KYHXDXKYHLBHZXDXKYGLBWTYGLHZXAWTAUCDZYJWTAAVKZVLZAVMQMNXAYIXDIZWTXABUEDZYNB - VKZYOBUBDZBLVNZJYNBVOYRYNYQYIYRXDYRXDIBLXDBLVPVQVEOTQOURVEVFVRVJTXGXIXLXIXH - UADZLXHRSZJXGXLXHVSXGYTXLYSXGYTCLRSZXLCVTXGUUAXLXKXGUUAJZXBLRSZXEXKUUBUUCXK - UUBJZUUCUUALARSZJZLCRSALRSJZKUUDUUFUUGXKUUEUUBUUAWTUUEXAWTAYLWAMXGUUAWBWCWD - UUDCAXKXGUUAWEXKYKUUBWTYKXAYMMMWFWGPXKXCUUCXDXKXCUUCXDIXKXCJUUCBLRSZXDXCUUC - UUHWHXKXBBLRWJOXKUUHXDIZXCXAUUIWTXAYOUUIYPYOYQUUIBWKYQUUHXDBWIWLQQOMURPVRWM - PWNWOWPWQVJTWR $. + breq1 nnnn0 pm2.21d syldc sylbird adantld biimtrid imp 3impib ) CUADZAEDZBE + DZCAFGZBHZABHZIZWSCUBDZCUCDZCUDZUEDZJZKWTXAJZXEIZCUFXFXLXJXFCUEDZCLHZKXLCUG + XMXLXNXMCUHHZCUIUJUKZDZKXLCULXOXLXQXOXKXEXOXKJZXCUHAFGZBHZXDXRXBXSBXOXBXSHX + KCUHAFUMMNXKXTXDIXOXKXTXDXKXSABWTXSAHXAWTAWTAAUOUPUNMNUQOURPXQXKXEXBEDZVGZX + QXKJZXEXKXQAXPDZYBWTYDXAAUSMCAUTVAXCYBYCXDXCYBXAVGZYCXDIXCYAXAXBBEVBVCYCYEX + DXKYEXDIZXQXAYFWTXAXDVDOOVEVFVHVIPVJTXNXCXKXDXNXCLAFGZBHZXKXDIXNXBYGBCLAFUM + NXKYHXDXKYHLBHZXDXKYGLBWTYGLHZXAWTAUCDZYJWTAAVKZVLZAVMQMNXAYIXDIZWTXABUEDZY + NBVKZYOBUBDZBLVNZJYNBVOYRYNYQYIYRXDYRXDIBLXDBLVPVQVEOTQOURVEVFVRVJTXGXIXLXI + XHUADZLXHRSZJXGXLXHVSXGYTXLYSXGYTCLRSZXLCVTXGUUAXLXKXGUUAJZXBLRSZXEXKUUBUUC + XKUUBJZUUCUUALARSZJZLCRSALRSJZKUUDUUFUUGXKUUEUUBUUAWTUUEXAWTAYLWAMXGUUAWBWC + WDUUDCAXKXGUUAWEXKYKUUBWTYKXAYMMMWFWGPXKXCUUCXDXKXCUUCXDIXKXCJUUCBLRSZXDXCU + UCUUHWHXKXBBLRWJOXKUUHXDIZXCXAUUIWTXAYOUUIYPYOYQUUIBWKYQUUHXDBWIWLQQOMURPVR + WMPWNWOWPWQVJTWR $. $( 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.) $) 2t6m3t4e0 $p |- ( ( 2 x. 6 ) - ( 3 x. 4 ) ) = 0 $= @@ -776084,22 +779942,22 @@ a function into a (ring theoretic) domain equals the support of the ( vx vv wcel wa co cfv crab wn wceq eqid eqtrd cdm cvv cmnd cv cplusg cof cmap c0g wne wo csupp cun ioran nne anbi12i bitri elmapfn ad2antrl adantr wfn ad2antll simplr inidm simplrl simplrr ofval an32s cbs mndidcl ad4antr - ancli mndlid syl sylibr ex con4d ss2rabdv wfun offun ovexd fvexd suppval1 - syl5bi syl3anc cmpt offvalfv dmeqd ovex dmmpti eqtrdi rabeqdv elmapfun id - wf elmapi fdm rabeq 3syl simprr fdmd uneq12d unrab 3sstr4d ) DUAJZEFJZKZA - CEUELZJZBXEJZKZKZHUBZABDUCMZUDZLZMZDUFMZUGZHENZXJAMZXOUGZXJBMZXOUGZUHZHEN - ZXMXOUILZAXOUILZBXOUILZUJZXIXPYBHEXIXJEJZKZYBXPYBOZXRXOPZXTXOPZKZYIXPOZYJ - XSOZYAOZKYMXSYAUKYOYKYPYLXRXOULXTXOULUMUNYIYMYNYIYMKZXNXOPYNYQXNXOXOXKLZX - OXIYMYHXNYRPXIYMKEEXOXOXKEABFFXJXIAEURZYMXFYSXDXGACEUOUPZUQXIBEURZYMXGUUA - XDXFBCEUOUSZUQXIXCYMXBXCXHUTZUQZUUDEVAXIYKYLYHVBXIYKYLYHVCVDVEYQXBXODVFMZ - 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LMod /\ ( ( # ` E ) = 0 \/ ( # ` E ) = 1 ) /\ S e. ~P B ) -> S linIndS M ) $= ( vf vv vx wcel cfv wceq wi wa cvv wb adantr c0g clmod chash cc0 clininds - c1 wo cpw wbr wne lmodsn0 cbs fvexi hasheq0 ax-mp eqneqall syl5bi syl csn - c0 com12 crg lmodring eqid 0ring sylan cv cfsupp clinc co wral cmap simpr - adantl wf snex jctil elmapg cmpt cxp fconst2 fconstmpt eqeq2i bitri eqidd - fvex fvexd fvmptd ralrimiva a1d breq1 oveq1 anbi12d fveq1 ralbidv imbi12d - eqeq1d syl5ibrcom sylbid ralrimiv mpbird simpl ancomd islininds mpbir2and - raleqdv mpancom expcom jaoi expd 3imp ) EUALZDUBMZUCNZXLUENZUFZCAUGZLZCEU - DUHZXOXKXQXROXOXKXQXRXMXKXQPZXROXNXSXMXRXKXMXROZXQXKDUSUIZXTDBEGHUJXMDUSN - ZYAXRDQLXMYBRDBUKHULDQUMUNYBYAXRXRDUSUOUTUPUQSUTXSXNXRDBTMZURZNZXSXNPZXRX - SBVALZXNYEXKYGXQBEGVBSDBYCHYCVCZVDVEYEYFPZXRXQIVFZYCVGUHZYJCEVHMZVIZETMZN - ZPZJVFZYJMZYCNZJCVJZOZIDCVKVIZVJZYFXQYEXSXQXNXKXQVLSZVMYIUUCUUAIYDCVKVIZV - JZYIUUAIUUEYIYJUUELZCYDYJVNZUUAYIYDQLZXQPZUUGUUHRYFUUJYEYFXQUUIUUDYCVOVPV - MYDCYJQXPVQUQUUHYJKCYCVRZNZYIUUAUUHYJCYDVSZNUULCYCYJBTWEVTUUMUUKYJKCYCWAW - 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c0g fvexi fdmfifsupp pm2.27 syl5bi ralimdva snlindsntorlem syld impbid wb - syl fveqeq2 ralsng bicomd imbi2d ralbidv biantrurd 3bitrd bitrid snex + c0g fdmfifsupp pm2.27 syl biimtrid ralimdva snlindsntorlem syld impbid wb + fvexi fveqeq2 ralsng bicomd imbi2d ralbidv biantrurd 3bitrd bitrid snex islininds sylancr bitr4d ) EUCQZFAQZRZIUDZFDUEZHUFZICGUGUHZUIZFUGZAUJQZPU DZGUKUMZUUNUULEULUNUEZHSZRZUAUDZUUNUNGSZUAUULUIZTZPCUULUOUEZUIZRZUULEUPUM ZUUKUUHHSZUUGGSZTZICUIZUUFUVEUUKUVGUQZIUUJUIUVJUUIUVKIUUJUUHHURUSUVGICGUT @@ -778306,7 +782164,7 @@ there exists a regular element r of the ring (an element that is neither VLFDUEZHSZUWSUVJRUVMUXEUWMHDUWAUVLFOXBXCUVIUXFUVMTIUVLCUUGUVLSZUVGUXFUVHU VMUXGUUHUXEHUUGUVLFDXDVTUUGUVLGXEXFXIXGXHXJXKXLXMXNXMXOXHUUFUVOUUQUVMTZPU VCUIUVJUUFUVNUXHPUVCUVNUUOUXHTZUUFUVQRZUXHUUOUUQUVMXPUXJUUOUXIUXHTUXJUULC - UUNXQGUVQUWQUUFUWRVNUULXRQUXJFXSXTGXQQUXJGBYAMYBXTYCUUOUXHYDYKYEYFABCDPEF + UUNXQGUVQUWQUUFUWRVNUULXRQUXJFXSXTGXQQUXJGBYAMYKXTYBUUOUXHYCYDYEYFABCDPEF GHIJKLMNOYGYHYIUUFUVNUVBPUVCUUFUVMUVAUURUUFUVAUVMUUEUVAUVMYJUUDUUTUVMUAFA UUSFGUUNYLYMVNYNYOYPUUFUUMUVDUUEUUMUUDFAVOVNYQYRYSUUFUULXQQUUDUVFUVEYJFYT UWKUAABUULPCEXQUCGHJNKLMUUAUUBUUC $. @@ -778345,39 +782203,39 @@ there exists a regular element r of the ring (an element that is neither c0g simp1 anim12ci simp2 cgrp lmodfgrp simpl grpinvcl lincvalpr syl112anc syl2an simpll adantl 3jca jca simprr ldepsprlem sylc eqtrd wo wn lmodring crg eqcom csn 01eq0ring sneq eqeq2d eleq2 elsni anim1i ancomd lmodvs1 syl - 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V /\ Z e. W ) /\ Z e/ ran F ) -> ( F supp Z ) = dom F ) $= ( vx wfun wcel w3a crn wnel wa csupp co cv cfv wne cdm crab wceq suppval1 - adantr wn df-nel fvelrn 3ad2antl1 eleq1 eqcoms syl5ibrcom necon3bd syl5bi - wral wb impancom ralrimiv rabid2 sylibr eqtr4d ) AFZABGZDCGZHZDAIZJZKZADL - MZENZAOZDPZEAQZRZVIVAVEVJSVCEBCADTUAVDVHEVIUKVIVJSVDVHEVIVAVFVIGZVCVHVCDV - BGZUBVAVKKZVHDVBUCVMVLVGDVMVLVGDSVGVBGZURUSVKVNUTVFAUDUEVLVNULDVGDVGVBUFU - GUHUIUJUMUNVHEVIUOUPUQ $. + adantr wral wn df-nel fvelrn 3ad2antl1 eleq1 syl5ibrcom necon3bd biimtrid + wb eqcoms impancom ralrimiv rabid2 sylibr eqtr4d ) AFZABGZDCGZHZDAIZJZKZA + DLMZENZAOZDPZEAQZRZVIVAVEVJSVCEBCADTUAVDVHEVIUBVIVJSVDVHEVIVAVFVIGZVCVHVC + DVBGZUCVAVKKZVHDVBUDVMVLVGDVMVLVGDSVGVBGZURUSVKVNUTVFAUEUFVLVNUKDVGDVGVBU + GULUHUIUJUMUNVHEVIUOUPUQ $. $} $( An integer greater than 1 is a complex number not equal to 0 or 1. @@ -780296,13 +784154,13 @@ Logarithm to an arbitrary base (extension) ( wcel cblen cfv c1 wceq cc0 wo c2 co caddc wbr clt wa crp a1i sylbid fveq2 jaoi eqtrdi cn0 cn elnn0 clogb cfl blennn eqeq1d cle cmin wne 2rp nnrp 1ne2 wi cr necomi relogbcl syl3anc flcld zcnd 1cnd addlsub 1m1e0 eqeq2d cz wb 0z - flbi sylancl 3bitrd 0p1e1 breq2i anbi2i nnlog2ge0lt1 biimpar olcd ex syl5bi - orc a1d sylbi blen0 blen1 impbid1 ) AUABZACDZEFZAGFZAEFZHZWEAUBBZWHHWGWJUNZ - AUCWKWLWHWKWGIAUDJZUEDZEKJZEFZWJWKWFWOEAUFUGWKWPGWMUHLZWMGEKJZMLZNZWJWKWPWN - EEUIJZFWNGFZWTWKWNEEWKWNWKWMWKIOBZAOBIEUJZWMUOBZXCWKUKPAULXDWKEIUMUPPIAUQUR - ZUSUTWKVAZXGVBWKXAGWNXAGFWKVCPVDWKXEGVEBXBWTVFXFVGWMGVHVIVJWTWQWMEMLZNZWKWJ - WSXHWQWREWMMVKVLVMWKXIWJWKXINWIWHWKWIXIAVNVOVPVQVRQQWHWJWGWHWIVSVTSWAWHWGWI - WHWFGCDEAGCRWBTWIWFECDEAECRWCTSWD $. + flbi sylancl 3bitrd breq2i anbi2i nnlog2ge0lt1 biimpar olcd ex biimtrid orc + 0p1e1 a1d sylbi blen0 blen1 impbid1 ) AUABZACDZEFZAGFZAEFZHZWEAUBBZWHHWGWJU + NZAUCWKWLWHWKWGIAUDJZUEDZEKJZEFZWJWKWFWOEAUFUGWKWPGWMUHLZWMGEKJZMLZNZWJWKWP + WNEEUIJZFWNGFZWTWKWNEEWKWNWKWMWKIOBZAOBIEUJZWMUOBZXCWKUKPAULXDWKEIUMUPPIAUQ + URZUSUTWKVAZXGVBWKXAGWNXAGFWKVCPVDWKXEGVEBXBWTVFXFVGWMGVHVIVJWTWQWMEMLZNZWK + WJWSXHWQWREWMMVSVKVLWKXIWJWKXINWIWHWKWIXIAVMVNVOVPVQQQWHWJWGWHWIVRVTSWAWHWG + WIWHWFGCDEAGCRWBTWIWFECDEAECRWCTSWD $. $( The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, @@ -780530,17 +784388,17 @@ last digit of the integer part (for ` b = ` 10 the first digit before cuz cn clogb cfl caddc ccxp cexp nnre 3ad2ant2 eluzelre eluz2nn nn0ge0d syl nnnn0 crp nnrp relogbzcl sylan2 3adant3 recxpcld 3ad2ant3 cpr cdif csn wceq leidd eluz2cnn0n1 nncn nnne0 eldifsn sylanbrc cxplogb syl2an breqtrrd eluz2 - wne cz flltp1 ltletr syl3anc mpand ex com23 3impia com12 syl5bi wb eluz2gt1 - zre cxplt syl12anc mpbid lelttrd eluzelcn eluz2n0 eluzelz cxpexpz breq2d - jca ) ADUAEFZCUBFZBACUCGZUDEHUEGZUAEFZIZCABUFGZJKZCABUGGZJKZXECAXBUFGZXFXAW - TCLFXDCUHZUIXEAXBWTXAALFZXDDAUJZMZWTXANAOKZXDWTAUBFZXOAUKXPAAUNULUMMZWTXAXB - LFZXDXAWTCUOFXRCUPACUQURZUSZUTXEABXNXQXDWTBLFZXAXCBUJVAZUTWTXACXJOKXDWTXAPZ - CCXJOXACCOKWTXACXKVFQWTARNHVBVCFCRNVDVCFZXJCVEXAAVGXACRFCNVPYDCVHCVICRNVJVK - ACVLVMVNUSXEXBBJKZXJXFJKZWTXAXDYEXDXCVQFZBVQFZXCBOKZIZYCYEXCBVOYJYCYEYGYHYI - YCYESYGYHPZYCYIYEYKYCYIYESYKYCPZXBXCJKZYIYEYLXRYMYCXRYKXSQZXBVRUMYLXRXCLFZY - AYMYIPYESYNYKYOYCYGYOYHXCWITTYKYAYCYHYAYGBWIQTXBXCBVSVTWAWBWCWDWEWFWDXEXLHA - JKZPZXRYAYEYFWGWTXAYQXDWTXLYPXMAWHWSMXTYBAXBBWJWKWLWMXEARFZANVPZYHXGXIWGWTX - AYRXDDAWNMWTXAYSXDAWOMXDWTYHXAXCBWPVAYRYSYHIXFXHCJABWQWRVTWL $. + wne cz flltp1 zre ltletr syl3anc mpand com23 3impia com12 biimtrid eluz2gt1 + ex jca cxplt syl12anc mpbid lelttrd eluzelcn eluz2n0 eluzelz cxpexpz breq2d + wb ) ADUAEFZCUBFZBACUCGZUDEHUEGZUAEFZIZCABUFGZJKZCABUGGZJKZXECAXBUFGZXFXAWT + CLFXDCUHZUIXEAXBWTXAALFZXDDAUJZMZWTXANAOKZXDWTAUBFZXOAUKXPAAUNULUMMZWTXAXBL + FZXDXAWTCUOFXRCUPACUQURZUSZUTXEABXNXQXDWTBLFZXAXCBUJVAZUTWTXACXJOKXDWTXAPZC + CXJOXACCOKWTXACXKVFQWTARNHVBVCFCRNVDVCFZXJCVEXAAVGXACRFCNVPYDCVHCVICRNVJVKA + CVLVMVNUSXEXBBJKZXJXFJKZWTXAXDYEXDXCVQFZBVQFZXCBOKZIZYCYEXCBVOYJYCYEYGYHYIY + CYESYGYHPZYCYIYEYKYCYIYESYKYCPZXBXCJKZYIYEYLXRYMYCXRYKXSQZXBVRUMYLXRXCLFZYA + YMYIPYESYNYKYOYCYGYOYHXCVSTTYKYAYCYHYAYGBVSQTXBXCBVTWAWBWHWCWDWEWFWDXEXLHAJ + KZPZXRYAYEYFWSWTXAYQXDWTXLYPXMAWGWIMXTYBAXBBWJWKWLWMXEARFZANVPZYHXGXIWSWTXA + YRXDDAWNMWTXAYSXDAWOMXDWTYHXAXCBWPVAYRYSYHIXFXHCJABWQWRWAWL $. $( The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.) $) @@ -780610,30 +784468,30 @@ last digit of the integer part (for ` b = ` 10 the first digit before elrege0 sylibr syl3anc nn0cn 3ad2ant3 3ad2ant2 subeq0ad mpbird oveq2d exp0d nn0digval eqtrd 1zzd flid eluz2gt1 1mod eqtr2d wn simprl1 adantl zsubcld wi clt nn0re sublt0d biimprd expnegico01 ico01fl0 crp nnrpd 0mod eluzelz lenlt - impcom bicomd biimpd 3simpc nn0sub mpbid zexpcl expm1d pm4.56 syl5bi znnsub - wo axlttri expdimp nnm1nn0 eqeltrd reexpclzd mod0 pm2.61ian ifeqda 3eqtr4d - ) AUAUBDEZBUCEZCUCEZUDZACFGZABFGUEGZHDZAIGZACBUFGZFGZHDZAIGZBYPAUGDGZBCTZJK - UHYOYRUUBAIYOYQUUAHYOUUAYQYOAUOEZAKUIZLZCMEZBMEZLZUUAYQTYLYMUUHYNYLUUFUUGUA - AUJZYLAAUKZULZUMNYMYNUUKYLYMYNLZUUJUUIYMUUJYNUUIBUNZCUNZUPZUQURACBUSUTVAVBV - CYOAVDEZYMYPKVEVGGEZUUDYSTYLYMUUSYNUUMNYLYMYNVFYOYPVHEZKYPVIOZLZUUTYLYNUVCY - MYLYNLZUVAUVBYLAVHEZYNUVAUAAVJZACVKVLUVDACYLUVEYNUVFPYLYNVMYLKAVIOYNYLAAVNV - OPVPUMVQYPVRVSAYPBWHVTYOUUEJKUUCYOUUELZUUCJAIGZJUVGUUBJAIUVGUUBJHDZJUVGUUAJ - HUVGUUAAKFGZJUVGYTKAFUVGYTKTZCBTZUVGBCYOUUEVMVAYOUVKUVLQUUEYOCBYNYLCUOEYMCW - AWBYMYLBUOEYNBWAWCWDPWEWFYOUVJJTZUUEYLYMUVMYNYLAUULWGNPWIVBUVGJMEUVIJTUVGWJ - JWKRWIVCYOUVHJTZUUEYLYMUVNYNYLUVEJAWTOUVNUVFAWLAWMUTNPWNYOUUEWOZLZUUCKCBWTO - ZUVPUUCKTUVQUVPLZUUCKAIGZKUVRUUBKAIUVRUUAKJVGGEZUUBKTUVRYLYTMEZYTKWTOZUVTYL - YMYNUVOUVQWPYOUWAUVQUVOYMYNUWAYLUUOCBYNUUIYMUUQWQYMUUJYNUUPPWRURZSUVPUVQUWB - YOUVQUWBWSUVOYOUWBUVQYOCBYNYLCVHEZYMCXAZWBYMYLBVHEZYNBXAZWCXBXCPXKAYTXDVTUU - AXERVCYOUVSKTZUVQUVOYLYMUWHYNYLAXFEZUWHYLAUUMXGZAXHRNSWIUVQWOZUVPLZUUCUUAAI - GZKUWLUUBUUAAIUWLUUAMEZUUBUUATUWLAMEZYTUCEZUWNYOUWOUWKUVOYLYMUWOYNUAAXINSZU - WLBCVIOZUWPUVPUWKUWRYOUWKUWRWSZUVOYMYNUWSYLUUOUWKUWRUUOUWFUWDLZUWKUWRQYMUWF - YNUWDUWGUWEUPZUWTUWRUWKBCXJXLRXMURPXKUWLUUOUWRUWPQYOUUOUWKUVOYLYMYNXNSBCXOR - XPAYTXQUTUUAWKRVCUWLUWMKTZUUAAUEGZMEZUWLUXCAYTJUFGZFGZMYOUXCUXFTUWKUVOYOUXF - UXCYOAYTYLYMUUFYNUULNYLYMUUGYNUUNNZUWCXRVASUWLUWOUXEUCEZUXFMEUWQUWLYTVDEZUX - HUWLBCWTOZUXIUVPUWKUXJYOUVOUWKUXJUVOUWKLUUEUVQYBWOZYOUXJUUEUVQXSYOUXJUXKYOU - WTUXJUXKQYMYNUWTYLUXAURBCYCRXCXTYDXKUWLUUJUUILZUXJUXIQYOUXLUWKUVOYMYNUXLYLU - URURSBCYARXPYTYERAUXEXQUTYFYOUXBUXDQZUWKUVOYOUUAVHEUWIUXMYOAYTYLYMUVEYNUVFN - UXGUWCYGYLYMUWIYNUWJNUUAAYHUTSWEWIYIVAYJYK $. + impcom bicomd biimpd 3simpc nn0sub zexpcl expm1d wo pm4.56 axlttri biimtrid + mpbid expdimp znnsub nnm1nn0 eqeltrd reexpclzd pm2.61ian ifeqda 3eqtr4d + mod0 ) AUAUBDEZBUCEZCUCEZUDZACFGZABFGUEGZHDZAIGZACBUFGZFGZHDZAIGZBYPAUGDGZB + CTZJKUHYOYRUUBAIYOYQUUAHYOUUAYQYOAUOEZAKUIZLZCMEZBMEZLZUUAYQTYLYMUUHYNYLUUF + UUGUAAUJZYLAAUKZULZUMNYMYNUUKYLYMYNLZUUJUUIYMUUJYNUUIBUNZCUNZUPZUQURACBUSUT + VAVBVCYOAVDEZYMYPKVEVGGEZUUDYSTYLYMUUSYNUUMNYLYMYNVFYOYPVHEZKYPVIOZLZUUTYLY + NUVCYMYLYNLZUVAUVBYLAVHEZYNUVAUAAVJZACVKVLUVDACYLUVEYNUVFPYLYNVMYLKAVIOYNYL + AAVNVOPVPUMVQYPVRVSAYPBWHVTYOUUEJKUUCYOUUELZUUCJAIGZJUVGUUBJAIUVGUUBJHDZJUV + GUUAJHUVGUUAAKFGZJUVGYTKAFUVGYTKTZCBTZUVGBCYOUUEVMVAYOUVKUVLQUUEYOCBYNYLCUO + EYMCWAWBYMYLBUOEYNBWAWCWDPWEWFYOUVJJTZUUEYLYMUVMYNYLAUULWGNPWIVBUVGJMEUVIJT + UVGWJJWKRWIVCYOUVHJTZUUEYLYMUVNYNYLUVEJAWTOUVNUVFAWLAWMUTNPWNYOUUEWOZLZUUCK + CBWTOZUVPUUCKTUVQUVPLZUUCKAIGZKUVRUUBKAIUVRUUAKJVGGEZUUBKTUVRYLYTMEZYTKWTOZ + UVTYLYMYNUVOUVQWPYOUWAUVQUVOYMYNUWAYLUUOCBYNUUIYMUUQWQYMUUJYNUUPPWRURZSUVPU + VQUWBYOUVQUWBWSUVOYOUWBUVQYOCBYNYLCVHEZYMCXAZWBYMYLBVHEZYNBXAZWCXBXCPXKAYTX + DVTUUAXERVCYOUVSKTZUVQUVOYLYMUWHYNYLAXFEZUWHYLAUUMXGZAXHRNSWIUVQWOZUVPLZUUC + UUAAIGZKUWLUUBUUAAIUWLUUAMEZUUBUUATUWLAMEZYTUCEZUWNYOUWOUWKUVOYLYMUWOYNUAAX + INSZUWLBCVIOZUWPUVPUWKUWRYOUWKUWRWSZUVOYMYNUWSYLUUOUWKUWRUUOUWFUWDLZUWKUWRQ + YMUWFYNUWDUWGUWEUPZUWTUWRUWKBCXJXLRXMURPXKUWLUUOUWRUWPQYOUUOUWKUVOYLYMYNXNS + BCXORYBAYTXPUTUUAWKRVCUWLUWMKTZUUAAUEGZMEZUWLUXCAYTJUFGZFGZMYOUXCUXFTUWKUVO + YOUXFUXCYOAYTYLYMUUFYNUULNYLYMUUGYNUUNNZUWCXQVASUWLUWOUXEUCEZUXFMEUWQUWLYTV + DEZUXHUWLBCWTOZUXIUVPUWKUXJYOUVOUWKUXJUVOUWKLUUEUVQXRWOZYOUXJUUEUVQXSYOUXJU + XKYOUWTUXJUXKQYMYNUWTYLUXAURBCXTRXCYAYCXKUWLUUJUUILZUXJUXIQYOUXLUWKUVOYMYNU + XLYLUURURSBCYDRYBYTYERAUXEXPUTYFYOUXBUXDQZUWKUVOYOUUAVHEUWIUXMYOAYTYLYMUVEY + NUVFNUXGUWCYGYLYMUWIYNUWJNUUAAYKUTSWEWIYHVAYIYJ $. $( All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.) $) dig1 $p |- ( ( B e. ( ZZ>= ` 2 ) /\ K e. 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(Contributed by AV, @@ -780976,20 +784834,20 @@ last digit of the integer part (for ` b = ` 10 the first digit before cmin expimpd wb eqcom a1i nncn 1cnd addlsub 1m1e0 eqeq2d 3bitrd csn oveq1 oveq2d 0p1e1 oveq2i fzo01 eqtri sumeq1d cc 0cn cz 2nn dig0 mp2an 2cn exp0 0z ax-mp oveq12d cr 1re mul02lem2 sumsn eqtr2di syl6bi adantl fveq2 blen0 - eqeq1d adantr mpbird a1d jaoi sylbi com12 ralrimiv ex syl5bi ) CEZFGZAEZH - ZXKIXMJKZBEZXKLUAGZKZLXPUBKZMKZBNZHZOZCPUCDEZFGXMHZYDXOXPYDXQKZXSMKZBNZHZ - OZDPUCZXMQRZXLXMSUDKZHZXKIYMJKZXTBNZHZOZCPUCZYCYJCDPCDUEZXNYEYBYIXKYDXMFU - FYTXKYDYAYHYTUGYTXOXTYGBYTXRYFXSMXKYDXPXQUHUIUMUJUKULYLYKYSYLYKUNZYRCPXKP - RZUUAYRUUBXKQRZXKIHZUOUUAYROZXKUPUUCUUEUUDUUCXKLUQKQRZUUEXKSVBKLUQKPRZUUC - UUFUNYLYKYRDABCURVCUUCUUGUNYLYKYRDABCUSVCXKUTVAUUDYLYKYRUUDYLUNZYRYKUUHYR - SYMHZIYOXPIXQKZXSMKZBNZHZOZYLUUNUUDYLUUIXMIHZUUMYLUUIYMSHZXMSSVBKZHUUOUUI - UUPVDYLSYMVEVFYLXMSSXMVGYLVHZUURVIYLUUQIXMUUQIHYLVJVFVKVLUUOUULIVMZUUKBNZ - IUUOYOUUSUUKBUUOYOIISUDKZJKZUUSUUOYMUVAIJXMISUDVNVOUVBISJKUUSUVASIJVPVQVR - VSTVTIWARZUVCUUTIHWBWBUUKIBIWAXPIHZUUKISMKZIUVDUUJIXSSMUVDUUJIIXQKZIXPIIX - QVNLQRIWCRUVFIHWDWILIWEWFTUVDXSLIUBKZSXPILUBUHLWARUVGSHWGLWHWJTWKSWLRUVEI - HWMSWNWJTWOWFWPWQWRUUDYRUUNVDYLUUDYNUUIYQUUMUUDXLSYMUUDXLIFGSXKIFWSWTTXAU - UDXKIYPUULUUDUGUUDYOXTUUKBUUDXRUUJXSMXKIXPXQUHUIUMUJUKXBXCXDVCXEXFXGXHXIX - J $. + eqeq1d adantr mpbird a1d jaoi sylbi com12 ralrimiv ex biimtrid ) CEZFGZAE + ZHZXKIXMJKZBEZXKLUAGZKZLXPUBKZMKZBNZHZOZCPUCDEZFGXMHZYDXOXPYDXQKZXSMKZBNZ + HZOZDPUCZXMQRZXLXMSUDKZHZXKIYMJKZXTBNZHZOZCPUCZYCYJCDPCDUEZXNYEYBYIXKYDXM + FUFYTXKYDYAYHYTUGYTXOXTYGBYTXRYFXSMXKYDXPXQUHUIUMUJUKULYLYKYSYLYKUNZYRCPX + KPRZUUAYRUUBXKQRZXKIHZUOUUAYROZXKUPUUCUUEUUDUUCXKLUQKQRZUUEXKSVBKLUQKPRZU + UCUUFUNYLYKYRDABCURVCUUCUUGUNYLYKYRDABCUSVCXKUTVAUUDYLYKYRUUDYLUNZYRYKUUH + YRSYMHZIYOXPIXQKZXSMKZBNZHZOZYLUUNUUDYLUUIXMIHZUUMYLUUIYMSHZXMSSVBKZHUUOU + UIUUPVDYLSYMVEVFYLXMSSXMVGYLVHZUURVIYLUUQIXMUUQIHYLVJVFVKVLUUOUULIVMZUUKB + NZIUUOYOUUSUUKBUUOYOIISUDKZJKZUUSUUOYMUVAIJXMISUDVNVOUVBISJKUUSUVASIJVPVQ + VRVSTVTIWARZUVCUUTIHWBWBUUKIBIWAXPIHZUUKISMKZIUVDUUJIXSSMUVDUUJIIXQKZIXPI + IXQVNLQRIWCRUVFIHWDWILIWEWFTUVDXSLIUBKZSXPILUBUHLWARUVGSHWGLWHWJTWKSWLRUV + EIHWMSWNWJTWOWFWPWQWRUUDYRUUNVDYLUUDYNUUIYQUUMUUDXLSYMUUDXLIFGSXKIFWSWTTX + AUUDXKIYPUULUUDUGUUDYOXTUUKBUUDXRUUJXSMXKIXPXQUHUIUMUJUKXBXCXDVCXEXFXGXHX + IXJ $. $d L a k x y $. $( Lemma 2 for ~ nn0sumshdig . (Contributed by AV, 7-Jun-2020.) $) @@ -782351,28 +786209,28 @@ Elementary geometry (extension) ( vx cr wcel wa c1 cfv wceq c2 wb eqeq1d adantl vy wo cop cpr cmap eleq2i wi co oveq2i bitri wf elmapi wrex wne 1ne2 1ex 2ex fprb ax-mp fveq1 fvpr1 cv vex eqtrdi fvpr2 anbi12d opeq2 adantr preq12d eqeq2d biimpcd sylbid ex - rexlimdvva syl5bi syl5 imp orim12d ad2antlr mpbird expl prelrrx2 ad2antrr - elprg elpri eleq1 cvv simpl a1i fvpr1g mp3an2i simpr fvpr2g jca orcd olcd - jaod impbid ) AKLZBKLZMZEKLZFKLZMZMZGCLZNGOZAPZQGOZBPZMZXGEPZXIFPZMZUBZMZ - GNAUCZQBUCZUDZNEUCZQFUCZUDZUDLZXEXFXOYCXEXFMZXOMYCGXSPZGYBPZUBZYDXOYGYDXK - YEXNYFXEXFXKYEUGZXFGKNQUDZUEUHZLZXEYHXFGKDUEUHZLYKCYLGIUFYLYJGDYIKUEHUIUF - UJZYKYIKGUKZXEYHGKYIULZYNGNJVBZUCZQUAVBZUCZUDZPZUAKUMJKUMZXEYHNQUNZYNUUBR - UOJUANQKGUPUQURUSZXEUUAYHJUAKKXEYPKLYRKLMMZUUAYHUUEUUAMZXKYPAPZYRBPZMZYEU - UAXKUUIRUUEUUAXHUUGXJUUHUUAXGYPAUUAXGNYTOZYPNGYTUTUUCUUJYPPUONQYPYRUPJVCV - AUSVDZSUUAXIYRBUUAXIQYTOZYRQGYTUTUUCUULYRPUONQYPYRUQUAVCVEUSVDZSVFTUUAUUI - YEUGUUEUUIUUAYEUUIYTXSGUUIYQXQYSXRUUGYQXQPUUHYPANVGVHUUHYSXRPUUGYRBQVGTVI - VJVKTVLVMVNVOVPVOVQXEXFXNYFUGZXFYKXEUUNYMYKYNXEUUNYOYNUUBXEUUNUUDXEUUAUUN - JUAKKUUEUUAUUNUUFXNYPEPZYRFPZMZYFUUAXNUUQRUUEUUAXLUUOXMUUPUUAXGYPEUUKSUUA - XIYRFUUMSVFTUUAUUQYFUGUUEUUQUUAYFUUQYTYBGUUQYQXTYSYAUUOYQXTPUUPYPENVGVHUU - PYSYAPUUOYRFQVGTVIVJVKTVLVMVNVOVPVOVQVRVQXFYCYGRXEXOGXSYBCWDVSVTWAYCYGXEX - PGXSYBWEXEYEXPYFXEYEXPXEYEMZXFXOUURXFXSCLZXAUUSXDYEABCDHIWBWCYEXFUUSRXEGX - SCWFTVTUURXKXNUURXKNXSOZAPZQXSOZBPZMZXAUVDXDYEXAUVAUVCNWGLZXAWSUUCUVAUPWS - WTWHUUCXAUOWIZNQABWGKWJWKQWGLZXAWTUUCUVCUQWSWTWLUVFNQABWGKWMWKWNWCYEXKUVD - RXEYEXHUVAXJUVCYEXGUUTANGXSUTSYEXIUVBBQGXSUTSVFTVTWOWNVMXEYFXPXEYFMZXFXOU - VHXFYBCLZXDUVIXAYFEFCDHIWBVSYFXFUVIRXEGYBCWFTVTUVHXNXKUVHXNNYBOZEPZQYBOZF - PZMZXDUVNXAYFXDUVKUVMUVEXDXBUUCUVKUPXBXCWHUUCXDUOWIZNQEFWGKWJWKUVGXDXCUUC - UVMUQXBXCWLUVONQEFWGKWMWKWNVSYFXNUVNRXEYFXLUVKXMUVMYFXGUVJENGYBUTSYFXIUVL - FQGYBUTSVFTVTWPWNVMWQVPWR $. + rexlimdvva biimtrid syl5 imp orim12d elprg ad2antlr mpbird elpri prelrrx2 + expl ad2antrr eleq1 cvv simpl a1i fvpr1g mp3an2i simpr jca orcd olcd jaod + fvpr2g impbid ) AKLZBKLZMZEKLZFKLZMZMZGCLZNGOZAPZQGOZBPZMZXGEPZXIFPZMZUBZ + MZGNAUCZQBUCZUDZNEUCZQFUCZUDZUDLZXEXFXOYCXEXFMZXOMYCGXSPZGYBPZUBZYDXOYGYD + XKYEXNYFXEXFXKYEUGZXFGKNQUDZUEUHZLZXEYHXFGKDUEUHZLYKCYLGIUFYLYJGDYIKUEHUI + UFUJZYKYIKGUKZXEYHGKYIULZYNGNJVBZUCZQUAVBZUCZUDZPZUAKUMJKUMZXEYHNQUNZYNUU + BRUOJUANQKGUPUQURUSZXEUUAYHJUAKKXEYPKLYRKLMMZUUAYHUUEUUAMZXKYPAPZYRBPZMZY + EUUAXKUUIRUUEUUAXHUUGXJUUHUUAXGYPAUUAXGNYTOZYPNGYTUTUUCUUJYPPUONQYPYRUPJV + CVAUSVDZSUUAXIYRBUUAXIQYTOZYRQGYTUTUUCUULYRPUONQYPYRUQUAVCVEUSVDZSVFTUUAU + UIYEUGUUEUUIUUAYEUUIYTXSGUUIYQXQYSXRUUGYQXQPUUHYPANVGVHUUHYSXRPUUGYRBQVGT + VIVJVKTVLVMVNVOVPVOVQXEXFXNYFUGZXFYKXEUUNYMYKYNXEUUNYOYNUUBXEUUNUUDXEUUAU + UNJUAKKUUEUUAUUNUUFXNYPEPZYRFPZMZYFUUAXNUUQRUUEUUAXLUUOXMUUPUUAXGYPEUUKSU + UAXIYRFUUMSVFTUUAUUQYFUGUUEUUQUUAYFUUQYTYBGUUQYQXTYSYAUUOYQXTPUUPYPENVGVH + UUPYSYAPUUOYRFQVGTVIVJVKTVLVMVNVOVPVOVQVRVQXFYCYGRXEXOGXSYBCVSVTWAWDYCYGX + EXPGXSYBWBXEYEXPYFXEYEXPXEYEMZXFXOUURXFXSCLZXAUUSXDYEABCDHIWCWEYEXFUUSRXE + GXSCWFTWAUURXKXNUURXKNXSOZAPZQXSOZBPZMZXAUVDXDYEXAUVAUVCNWGLZXAWSUUCUVAUP + WSWTWHUUCXAUOWIZNQABWGKWJWKQWGLZXAWTUUCUVCUQWSWTWLUVFNQABWGKWQWKWMWEYEXKU + VDRXEYEXHUVAXJUVCYEXGUUTANGXSUTSYEXIUVBBQGXSUTSVFTWAWNWMVMXEYFXPXEYFMZXFX + OUVHXFYBCLZXDUVIXAYFEFCDHIWCVTYFXFUVIRXEGYBCWFTWAUVHXNXKUVHXNNYBOZEPZQYBO + ZFPZMZXDUVNXAYFXDUVKUVMUVEXDXBUUCUVKUPXBXCWHUUCXDUOWIZNQEFWGKWJWKUVGXDXCU + UCUVMUQXBXCWLUVONQEFWGKWQWKWMVTYFXNUVNRXEYFXLUVKXMUVMYFXGUVJENGYBUTSYFXIU + VLFQGYBUTSVFTWAWOWMVMWPVPWR $. $} ${ @@ -782824,61 +786682,61 @@ left module (or any extended structure having a base set, an addition, + ( t x. ( y ` i ) ) ) ) ) $= ( wcel cr c1 co cmin cmul caddc wceq cc0 oveq1d cdiv cn cfz cmap csn cdif cv w3a wa cfv wral cicc wrex cico cioc oveq2 oveq1 oveq12d eqeq2d ralbidv - cbvrexvw wss wi unitssre ssrexv mp1i syl5bi cle wbr clt cxr wb 0re elico2 - cneg 1xr mp2an simp1 1red resubcld 1cnd wne ltne 3adant2 subne0d redivcld - recnd sylbi ad2antlr renegcld adantl wf elmapi 3ad2ant2 ffvelcdmda eldifi - eqcom ad2antrr 3ad2ant3 mulcld subcld subadd2d bitr4id divmuld divsubdird - divrec2d div23d eqtrd bitrid divcld mulneg1d eqcomd oveq2d reccld negsubd - syl subnegd cc muldivdir syl112anc mulid1d npcand 3eqtr2d biimpd ralimdva - 3eqtr3d sylbid imp rspcedvd rexlimdva2 0xr elioc2 gt0ne0 3adant3 rereccld - 1re negsubdi2d eqtr3d eqeq1d a1i 3bitr3d subaddd addcld divmul2d comraddd - elioc1 anim1i divdird dividd 3jaod ) GUAJZAUFZKLGUBMZUCMZJZBUFZUUMUUKUDZU - EJZUGZHUFZUUMJZUHZDUFZUUSUIZLEUFZNMZUVBUUKUIZOMZUVDUVBUUOUIZOMZPMZQZDUULU - JZERLUKMZULZUVCLCUFZNMZUVFOMZUVOUVHOMZPMZQZDUULUJZCKULZUVFLIUFZNMZUVCOMZU - 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(Contributed by AV, 15-Feb-2023.) $) eenglngeehlnmlem2 $p |- ( ( ( N e. NN /\ x e. ( RR ^m ( 1 ... N ) ) @@ -783328,35 +787186,35 @@ between the two points ("point-slope form"), sometimes also written as oveq2d fvpr2g bibi12d notbid rspcev sylancr expcom notnotb cneg cdiv 1red 2ex sylbir renegcld simprl redivcld syl2anc ax-1ne0 neii mpbir mulid1d cc 2th negcld divcan2d negidd eqtrd simprr eqeq12d mtbiri ovex ex nne bicomi - 1re oveq1 ax-1cn mul02i id eqeqan12d a1d syl2anbr jaoi3 orcoms imp syl5bi - sylbi rexnal syl6ib con4d df-3an syl6ibr ) AUAIZBUAIZCUAIZUBZAJFUCZUDZKLZ - BMYTUDZKLZNLZCOZUUAPOZQZFDUGZAPUEZBPOZUFZCPOZUFZUUJUUKUUMUBYSUUNUUIYSUUNR - ZUUHRZFDUHZUUIRUUOUULRZUUMRZUIZYSUUQUULUUMUJUUTYSUUQUUSUURYSUUQUKZUUSUVAU - URUUSCPUEZUVACPULYSUVBUUQYSUVBUFZJPUMMPUMUNZDIZAPKLZBPKLZNLZCOZPPOZQZRZUU - QPUAIZUVMUVEUTUTPPDEGHUOUPUVCUVKUUMUVJQZUVCUUMUVJRZQZUVNRUVBUVPYSUVBUUMUV - OUUMUVBUVOUVOCPUQURUVJUUMPVAZUSVBVCUUMUVJVDVEUVCUVIUUMUVJYSUVIUUMQUVBYSUV - IPCOUUMYSUVHPCYSUVHPPNLZPYSUVFPUVGPNYSAYSAYPYQYRVFZVGZVHYSBYSBYPYQYRVIZVG - ZVHVJVKSVLPCVMVNVOVPVQUUPUVLFUVDDYTUVDOZUUHUVKUWCUUFUVIUUGUVJUWCUUEUVHCUW - 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UYPXAXDXEXFVPXOYDYEYGXKUULLDYHYIYJYK $. + ax-1rid orcoms rexnal syl6ib biimtrid con4d ) AUAUBZBUAUBZBUCUDZUEZCUAU + BZUFZIUAUBZUEZAUCTZICBUGUHZTZUEZAUILUJZURZUKUHZBULUUDURZUKUHZUMUHZCTZUU + GITZUNZLDUOZUUCUPZAUCUDZIUUAUDZUQZYSUUMUPZUUNYTUPZUUBUPZUQUUQYTUUBUSUUO + UUSUUPUUTAUCUTIUUAUTVAVBYSUUQUULUPZLDVCZUURUUQYSUVBUUPUUOYSUVBVJZUUPUVC + UUOUUPYSUVBUUPYSUEZUIUCVDULUUAVDZVEZDUBZAUIUVFURZUKUHZBULUVFURZUKUHZUMU + HZCTZUVJITZUNZUPZUVBUVDUCUAUBZUUAUAUBZUVGUVDVKYSUVRUUPYSCBYQYPYRYLYOYPV + FZVGZYQYMYRYOYLYMYPYMYNVHZVIZVGYQYNYRYLYMYNYPVLZVGZVMVNUCUUADGMOVOVPUVD + UVOCCTZUUAITZUNZUUPUWGUPZYSUUPUWEUWFUPZUNUWHUUPUWEUWIUUPUWIUWEUUPUUAIUU + PIUUAUUPWAVQVRVSUWIUWEVJUUPUWICVTWBWCUWEUWFWDWEVGUVDUVMUWEUVNUWFUVDUVLC + CUVDUVLUCCUMUHZCYSUVLUWJTUUPYSUVIUCUVKCUMYSUVIAUCUKUHZUCYSUVHUCAUKYSUVQ + UVHUCTYSVKUCUUAUAWFWGWHYQUWKUCTZYRYLYOUWLYPYLAAWIZWJWKVGWLYSUVKBUUAUKUH + ZCYSUVJUUABUKYSUUAWMUBZUVJUUATYSCBUGWNZUCUUAWMWOWGZWHYSCBYQCWPUBZYRYQCU + VSWQZVGYQBWPUBZYRYOYLUWTYPYOBUWAWQVIVGZUWDWRWLWSVNYSUWJCTZUUPYQUXBYRYQC + UWSWTVGVNWLXAYSUVNUWFUNUUPYSUVJUUAIUWQXAVNXDXBUVAUVPLUVFDUUDUVFTZUULUVO + UXCUUJUVMUUKUVNUXCUUIUVLCUXCUUFUVIUUHUVKUMUXCUUEUVHAUKUIUUDUVFXCWHUXCUU + GUVJBUKULUUDUVFXCZWHWSXAUXCUUGUVJIUXDXAXDXEXFVPXOUUPUPUUBUUOUVCIUUAXGUU + BUUOUEZYSUVBUXEYSUEZUIUIVDUVEVEZDUBZAUIUXGURZUKUHZBULUXGURZUKUHZUMUHZCT + ZUXKITZUNZUPZUVBYSUXHUXEYSUIUAUBZUVRUEZUXHYQUXSYRYQUXRUVRYQXHYQCBUVSUWB + UWCVMXIVGUIUUADGMOVOWGVNUXFUXQACUMUHZCTZUWFUNZUPZUXFUYBYTUWFUNZUXEUYDUP + ZYSUXEYTUWIUNUYEUXEYTUWIUUOYTUWIVJUUBYTUUOUWIUWIAUCXJXKVNUUBUWIYTVJZUUO + UYFUUAIUWFYTXLXMVGWCYTUWFWDWEVGUXFUYAYTUWFUXFUYACAUMUHZCTZYTUXFUXTUYGCU + XFACUXFAYLYOYPYRUXEXNWQUXFCYSYPUXEUVTVNWQXPXAUXFUWRAWPUBZUEZUYHYTUNYSUY + JUXEYQUYJYRYLYPUYJYOYLUYIYPUWRUWMCWIXQXRVGVNCAXSWGXTYAXBYSUXQUYCUNUXEYS + UXPUYBYSUXNUYAUXOUWFYSUXMUXTCYSUXJAUXLCUMYSUXJAUIUKUHZAYSUXIUIAUKYSUIWM + UBZUXIUITUYLYSYBWBUIUUAWMWFWGWHYQUYKATZYRYLYOUYMYPAYFWKVGWLYSUXLUWNCYSU + XKUUABUKYSUWOUXKUUATUWPUIUUAWMWOWGZWHYSCBYSCUVTWQUXAUWDWRWLWSXAYSUXKUUA + IUYNXAXDXEVNYCUVAUXQLUXGDUUDUXGTZUULUXPUYOUUJUXNUUKUXOUYOUUIUXMCUYOUUFU + XJUUHUXLUMUYOUUEUXIAUKUIUUDUXGXCWHUYOUUGUXKBUKULUUDUXGXCZWHWSXAUYOUUGUX + KIUYPXAXDXEXFVPXOYDYEYGXKUULLDYHYIYJYK $. $( Example for a horizontal line ` G ` passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.) $) @@ -783912,37 +787770,37 @@ holds even for degenerate lines ( ` A = B = 0 ` ). 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Solutions of the quadratic equations for the coordinates of the intersection points of a horizontal line and a @@ -783962,28 +787820,28 @@ holds even for degenerate lines ( ` A = B = 0 ` ). (Contributed by subcld eqeltrid sqrtcld addid2d sq0i simp2 sqvald eqtrid simplrr divcan5d oveq2d eqcomd eqeq2d biimpd subid1d oveq12d eqtr2d biimpa jctird mulneg2d cc simp1rr negcld df-neg eqtr4di divnegd 3eqtr4d 3ad2ant1 simp1l3 simp1l2 - 3ad2ant2 addid1d orim12d syl5bi expimpd syld ) AUAKZBUAKZCUAKZUBZALMZBLUC - ZNZNZFUDKZLDUEUFZNZGUAKHUAKNZUBZGUGUHOHUGUHOPOFUGUHOZMAGQOBHQOPOCMNHCBROZ - MZGDUIUJZBROZUKZMZGYBMZULZNGACQOZBYAQOZPOZEROZMZHBCQOZAYAQOZUMOZEROZMZNZG - YGYHUMOZEROZMZHYLYMPOZEROZMZNZULZABCDEFGHIJUNXQXTYFUUEYFYEYDULXQXTNZUUEYD - YEUOUUFYEYQYDUUDUUFYEYKYPUUFYEYKUUFYBYJGUUFYJYBXQYJYBMZXTXLXOUUGXPXLXONZY - JYHEROZYBUUHYIYHERUUHYILYHPOYHUUHYGLYHPUUHYGLCQOZLXLYGUUJMZXOXIUUKXHXJALC - QUPUQSUUHCUUHCXEXFXGXKXOUSURZUTTZVAUUHYHUUHBYAUUHBXEXFXGXKXOVBURZUUHDUUHD - XREQOZCUGUHOZUMOZWIJUUHUUOUUPUUHXREUUHFXOFWIKZXLXOFXMFUAKXNFVCSURZVDVEXLE - WIKZXOXHUUTXKXEXFUUTXGXEXFNEABEIVFURVGSZSVHUUHCUULVEVIVJVKZVHVLTVAUUHUUIY - HBBQOZROYBUUHEUVCYHRXLEUVCMZXOXLEAUGUHOZBUGUHOZPOZUVCIXLUVGLUVFPOZUVCXLUV - ELUVFPXIUVELMXHXJAVMUQVAXHUVHUVCMXKXHUVHUVFUVCXHUVFXHBXHBXEXFXGVNURZVEVLX - HBUVIVOTSTVPZSZVSUUHYABBUVBUUNUUNXHXIXJXOVQZUVLVRTTVGSVTWAWBXQXTYPXQXSYOH - XQYOYLUVCROZXSXLXOYOUVMMXPUUHYNYLEUVCRUUHYNYLLUMOYLUUHYMLYLUMUUHYMLYAQOZL - XLYMUVNMZXOXIUVOXHXJALYAQUPUQZSUUHYAUVBUTTVSUUHYLUUHBCUUNUULVHWCTUVKWDVGX - QCBBXLXOCWIKXPUULVGXLXOBWIKXPUUNVGZUVQXIXJXHXOXPWJZUVRVRWEWAWFWGUUFYDYTUU - CUUFYDYTUUFYCYSGUUFYSYCXQYSYCMZXTXLXOUVSXPUUHYHUKZEROZYAUKZBROZYSYCUUHUWA - BUWBQOZUVCROUWCUUHUVTUWDEUVCRUUHUWDUVTUUHBYAUUNUVBWHVTUVKWDUUHUWBBBUUHYAU - VBWKUUNUUNUVLUVLVRTUUHYRUVTERUUHYRLYHUMOUVTUUHYGLYHUMUUMVAYHWLWMVAUUHYABU - VBUUNUVLWNWOVGSVTWAWBXQXTUUCXQXSUUBHXQUUBUVMXSXQUUAYLEUVCRXQUUAYLLPOYLXQY - MLYLPXQYMUVNLXLXOUVOXPUVPWPXQYAXQDXQDUUQWIJXQUUOUUPXQXREXQFXOXLUURXPUUSWS - VEXLXOUUTXPUVAWPVHXQCXQCXEXFXGXKXOXPWQURZVEVIVJVKUTTVSXQYLXQBCXQBXEXFXGXK - XOXPWRURZUWEVHWTTXLXOUVDXPUVJWPWDXQCBBUWEUWFUWFUVRUVRVRWEWAWFWGXAXBXCXD + 3ad2ant2 addid1d orim12d biimtrid expimpd syld ) AUAKZBUAKZCUAKZUBZALMZBL + UCZNZNZFUDKZLDUEUFZNZGUAKHUAKNZUBZGUGUHOHUGUHOPOFUGUHOZMAGQOBHQOPOCMNHCBR + OZMZGDUIUJZBROZUKZMZGYBMZULZNGACQOZBYAQOZPOZEROZMZHBCQOZAYAQOZUMOZEROZMZN + ZGYGYHUMOZEROZMZHYLYMPOZEROZMZNZULZABCDEFGHIJUNXQXTYFUUEYFYEYDULXQXTNZUUE + YDYEUOUUFYEYQYDUUDUUFYEYKYPUUFYEYKUUFYBYJGUUFYJYBXQYJYBMZXTXLXOUUGXPXLXON + ZYJYHEROZYBUUHYIYHERUUHYILYHPOYHUUHYGLYHPUUHYGLCQOZLXLYGUUJMZXOXIUUKXHXJA + LCQUPUQSUUHCUUHCXEXFXGXKXOUSURZUTTZVAUUHYHUUHBYAUUHBXEXFXGXKXOVBURZUUHDUU + HDXREQOZCUGUHOZUMOZWIJUUHUUOUUPUUHXREUUHFXOFWIKZXLXOFXMFUAKXNFVCSURZVDVEX + LEWIKZXOXHUUTXKXEXFUUTXGXEXFNEABEIVFURVGSZSVHUUHCUULVEVIVJVKZVHVLTVAUUHUU + IYHBBQOZROYBUUHEUVCYHRXLEUVCMZXOXLEAUGUHOZBUGUHOZPOZUVCIXLUVGLUVFPOZUVCXL + UVELUVFPXIUVELMXHXJAVMUQVAXHUVHUVCMXKXHUVHUVFUVCXHUVFXHBXHBXEXFXGVNURZVEV + LXHBUVIVOTSTVPZSZVSUUHYABBUVBUUNUUNXHXIXJXOVQZUVLVRTTVGSVTWAWBXQXTYPXQXSY + OHXQYOYLUVCROZXSXLXOYOUVMMXPUUHYNYLEUVCRUUHYNYLLUMOYLUUHYMLYLUMUUHYMLYAQO + ZLXLYMUVNMZXOXIUVOXHXJALYAQUPUQZSUUHYAUVBUTTVSUUHYLUUHBCUUNUULVHWCTUVKWDV + GXQCBBXLXOCWIKXPUULVGXLXOBWIKXPUUNVGZUVQXIXJXHXOXPWJZUVRVRWEWAWFWGUUFYDYT + UUCUUFYDYTUUFYCYSGUUFYSYCXQYSYCMZXTXLXOUVSXPUUHYHUKZEROZYAUKZBROZYSYCUUHU + WABUWBQOZUVCROUWCUUHUVTUWDEUVCRUUHUWDUVTUUHBYAUUNUVBWHVTUVKWDUUHUWBBBUUHY + AUVBWKUUNUUNUVLUVLVRTUUHYRUVTERUUHYRLYHUMOUVTUUHYGLYHUMUUMVAYHWLWMVAUUHYA + BUVBUUNUVLWNWOVGSVTWAWBXQXTUUCXQXSUUBHXQUUBUVMXSXQUUAYLEUVCRXQUUAYLLPOYLX + QYMLYLPXQYMUVNLXLXOUVOXPUVPWPXQYAXQDXQDUUQWIJXQUUOUUPXQXREXQFXOXLUURXPUUS + WSVEXLXOUUTXPUVAWPVHXQCXQCXEXFXGXKXOXPWQURZVEVIVJVKUTTVSXQYLXQBCXQBXEXFXG + XKXOXPWRURZUWEVHWTTXLXOUVDXPUVJWPWDXQCBBUWEUWFUWFUVRUVRVRWEWAWFWGXAXBXCXD $. $( Lemma for ~ itsclc0 . Solutions of the quadratic equations for the @@ -784142,18 +788000,18 @@ coordinates of the intersection points of a (nondegenerate) line and a crab cmul cdiv cmin cpnf cico wb rprege0 elrege0 sylibr adantr 3ad2ant3 csqrt eqid 2sphere0 eleq2d syl oveq2d oveq12d eqeq1d elrab2 a1i anbi12d fveq1 wi oveq1d elrab 3simpa simpl3 rrx2pxel rrx2pyel jca itsclc0xyqsol - adantl syl3anc expcomd expimpd com23 adantld syl5bi impd sylbid ) AUCUD - BUCUDCUCUDUEZAUFUGBUFUGUHZGUIUDZUFDUJUKZULZUEZLMGHUMZUDZLKUDZULLUNNUOZU - PZUQURUMZUQXLUPZUQURUMZUSUMZGUQURUMZUTZNEVAZUDZLEUDZAUNLUPZVBUMZBUQLUPZ - VBUMZUSUMZCUTZULZULYCACVBUMZBDVMUPZVBUMZUSUMFVCUMUTYEBCVBUMZAYKVBUMZVDU - MFVCUMUTULYCYJYLVDUMFVCUMUTYEYMYNUSUMFVCUMUTULUHZXHXJYAXKYIXHGUFVEVFUMU - DZXJYAVGXGXCYPXDXEYPXFXEGUCUDUFGUJUKULYPGVHGVIVJVKVLYPXIXTLXTEGHIJMNOPQ - RSXTVNVOVPVQXKYIVGXHAXMVBUMZBXOVBUMZUSUMZCUTYHNLEKXLLUTZYSYGCYTYQYDYRYF - USYTXMYCAVBUNXLLWDZVRYTXOYEBVBUQXLLWDZVRVSVTUBWAWBWCXHYAYIYOYAYBYCUQURU - MZYEUQURUMZUSUMZXRUTZULXHYIYOWEZXSUUFNLEYTXQUUEXRYTXNUUCXPUUDUSYTXMYCUQ - URUUAWFYTXOYEUQURUUBWFVSVTWGXHUUFUUGYBXHYIUUFYOXHYBYHUUFYOWEXHYBULZUUFY - HYOUUHXCXDULZXGYCUCUDZYEUCUDZULZUUFYHULYOWEXHUUIYBXCXDXGWHVKXCXDXGYBWIY - BUULXHYBUUJUUKEJLOQWJEJLOQWKWLWNABCDFGYCYETUAWMWOWPWQWRWSWTXAXB $. + adantl syl3anc expcomd expimpd com23 adantld biimtrid impd sylbid ) AUC + UDBUCUDCUCUDUEZAUFUGBUFUGUHZGUIUDZUFDUJUKZULZUEZLMGHUMZUDZLKUDZULLUNNUO + ZUPZUQURUMZUQXLUPZUQURUMZUSUMZGUQURUMZUTZNEVAZUDZLEUDZAUNLUPZVBUMZBUQLU + PZVBUMZUSUMZCUTZULZULYCACVBUMZBDVMUPZVBUMZUSUMFVCUMUTYEBCVBUMZAYKVBUMZV + DUMFVCUMUTULYCYJYLVDUMFVCUMUTYEYMYNUSUMFVCUMUTULUHZXHXJYAXKYIXHGUFVEVFU + MUDZXJYAVGXGXCYPXDXEYPXFXEGUCUDUFGUJUKULYPGVHGVIVJVKVLYPXIXTLXTEGHIJMNO + PQRSXTVNVOVPVQXKYIVGXHAXMVBUMZBXOVBUMZUSUMZCUTYHNLEKXLLUTZYSYGCYTYQYDYR + YFUSYTXMYCAVBUNXLLWDZVRYTXOYEBVBUQXLLWDZVRVSVTUBWAWBWCXHYAYIYOYAYBYCUQU + RUMZYEUQURUMZUSUMZXRUTZULXHYIYOWEZXSUUFNLEYTXQUUEXRYTXNUUCXPUUDUSYTXMYC + UQURUUAWFYTXOYEUQURUUBWFVSVTWGXHUUFUUGYBXHYIUUFYOXHYBYHUUFYOWEXHYBULZUU + FYHYOUUHXCXDULZXGYCUCUDZYEUCUDZULZUUFYHULYOWEXHUUIYBXCXDXGWHVKXCXDXGYBW + IYBUULXHYBUUJUUKEJLOQWJEJLOQWKWLWNABCDFGYCYETUAWMWOWPWQWRWSWTXAXB $. $( The intersection points of a (nondegenerate) line through two points and a circle around the origin. 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See ~ ralbidc for a more $) ${ - $d A x $. $d B x $. $d X x $. fvconst0ci.1 $e |- B e. _V $. fvconst0ci.2 $e |- Y = ( ( A X. { B } ) ` X ) $. $( A constant function's value is either the constant or the empty set. @@ -785411,7 +789268,6 @@ additional condition (deduction form). See ~ ralbidc for a more $} ${ - $d A x $. $d B x $. $d X x $. fvconstdomi.1 $e |- B e. _V $. $( A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, @@ -785634,10 +789490,10 @@ redundant here (see ~ neircl ), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.) $) opnneilv $p |- ( ph -> ( E. x e. ( ( nei ` J ) ` S ) ps -> E. y e. J ( S C_ y /\ ch ) ) ) $= - ( cnei cfv wrex cv wcel wa wex wss df-rex ctop syl5bi neii2 r19.41dv expl - sylan anass expimpd anim2d reximdv syld exlimdv ) BDFGJKKZLDMZUKNZBOZDPAF - EMZQZCOZEGLZBDUKRAUNURDAUNUPUOULQZOZBOZEGLZURAUMBVBAUMOUTBEGAGSNUMUTEGLHF - EGULUAUDUBUCAVAUQEGVAUPUSBOZOAUQUPUSBUEAVCCUPAUSBCIUFUGTUHUIUJT $. + ( cnei cfv wrex cv wcel wa wex wss df-rex ctop biimtrid neii2 sylan anass + r19.41dv expl expimpd anim2d reximdv syld exlimdv ) BDFGJKKZLDMZUKNZBOZDP + AFEMZQZCOZEGLZBDUKRAUNURDAUNUPUOULQZOZBOZEGLZURAUMBVBAUMOUTBEGAGSNUMUTEGL + HFEGULUAUBUDUEAVAUQEGVAUPUSBOZOAUQUPUSBUCAVCCUPAUSBCIUFUGTUHUIUJT $. opnneil.3 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $. $( A variant of ~ opnneilv . (Contributed by Zhi Wang, 31-Aug-2024.) $) @@ -786894,8 +790750,8 @@ have GLB (expanded version). (Contributed by Zhi Wang, $} ${ - $d F t v w x y z $. $d I t v w y z $. $d S t v w x y z $. - $d G v w y z $. $d T t v w y z $. $d ph t v w y z $. + $d F v w x y z $. $d I v w y z $. $d S v w x y z $. $d G v w y z $. + $d T v w y z $. $d ph v w y z $. ipoglb.g $e |- ( ph -> G = ( glb ` I ) ) $. ipoglbdm.t $e |- ( ph -> T = U. { x e. F | x C_ |^| S } ) $. $( The domain of the GLB of the inclusion poset. (Contributed by Zhi @@ -788619,8 +792475,8 @@ structure becomes the object here (see ~ mndtcbasval ) , instead of just ~ ax-13 . See ~ nfiundg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) $) nfiund $p |- ( ph -> F/_ y U_ x e. A B ) $= - ( vz ciun cv wcel wrex cab df-iun nfv nfcrd nfrexd nfabdw nfcxfrd ) ACBDE - JIKELZBDMZINBIDEOAUBCIAIPAUACBDFGACIEHQRST $. + ( vz ciun cv wcel wrex cab df-iun nfv nfcrd nfrexdw nfabdw nfcxfrd ) ACBD + EJIKELZBDMZINBIDEOAUBCIAIPAUACBDFGACIEHQRST $. $( $j usage 'nfiund' avoids 'ax-13'; $) $} @@ -788634,8 +792490,8 @@ structure becomes the object here (see ~ mndtcbasval ) , instead of just a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.) $) nfiundg $p |- ( ph -> F/_ y U_ x e. A B ) $= - ( vz ciun cv wcel wrex cab df-iun nfv nfcrd nfrexdg nfabd nfcxfrd ) ACBDE - JIKELZBDMZINBIDEOAUBCIAIPAUACBDFGACIEHQRST $. + ( vz ciun cv wcel wrex cab df-iun nfv nfcrd nfrexd nfabd nfcxfrd ) ACBDEJ + IKELZBDMZINBIDEOAUBCIAIPAUACBDFGACIEHQRST $. $} $( ${ @@ -788661,10 +792517,10 @@ structure becomes the object here (see ~ mndtcbasval ) , instead of just class. (Contributed by Emmett Weisz, 3-Nov-2019.) $) iunord $p |- ( A. x e. A Ord B -> Ord U_ x e. A B ) $= ( vy word wral ciun wtr con0 wss ordtr ralimi triun wcel wrex eliun nfra1 - syl cv nfv wi rsp ordelon ex syl6 rexlimd syl5bi ssrdv ordon trssord 3exp - mpii sylc ) CEZABFZABCGZHZUPIJZUPEZUOCHZABFUQUNUTABCKLABCMRUODUPIDSZUPNVA - CNZABOUOVAINZAVABCPUOVBVCABUNABQVCATUOASBNUNVBVCUAUNABUBUNVBVCCVAUCUDUEUF - UGUHUQURIEZUSUIUQURVDUSUPIUJUKULUM $. + syl cv nfv wi ordelon syl6 rexlimd biimtrid ssrdv ordon trssord 3exp mpii + rsp ex sylc ) CEZABFZABCGZHZUPIJZUPEZUOCHZABFUQUNUTABCKLABCMRUODUPIDSZUPN + VACNZABOUOVAINZAVABCPUOVBVCABUNABQVCATUOASBNUNVBVCUAUNABUKUNVBVCCVAUBULUC + UDUEUFUQURIEZUSUGUQURVDUSUPIUHUIUJUM $. $} ${ @@ -788706,9 +792562,9 @@ structure becomes the object here (see ~ mndtcbasval ) , instead of just $( Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) $) tfis2d $p |- ( ph -> ( x e. On -> ps ) ) $= - ( cv con0 wcel wi weq wb com12 pm5.74d wral r19.21v a2d syl5bi tfis2 ) DH - ZIJZABABKZACKZDEDELZABCAUEBCMFNOUDEUAPACEUAPZKUBUCACEUAQUBAUFBAUBUFBKGNRS - TN $. + ( cv con0 wcel wi weq wb com12 pm5.74d wral r19.21v a2d biimtrid tfis2 ) + DHZIJZABABKZACKZDEDELZABCAUEBCMFNOUDEUAPACEUAPZKUBUCACEUAQUBAUFBAUBUFBKGN + RSTN $. $} ${ @@ -791450,191 +795306,4 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler, $( End $[ set-mbox-kz.mm $] $) -$( Begin $[ set-mbox-et.mm $] $) -$( Skip $[ set-main.mm $] $) -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Ender Ting -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# -$) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Increasing sequences and subsequences -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -$) - - ${ - $( Less-than class is never reflexive. (Contributed by Ender Ting, - 22-Nov-2024.) Prefer to specify theorem domain and then apply ~ ltnri . - (New usage is discouraged.) $) - et-ltneverrefl $p |- -. A < A $= - ( cxr wcel clt wbr wn xrltnr cop opelxp1 con3i ltrelxr sseli nsyl sylnibr - cxp df-br pm2.61i ) ABZCZAADZEZFAGSFZAAHZTCZUAUBUCRROZCZUDUFSAARRIJTUEUCK - LMAATPNQ $. - $} - - ${ - $d B t $. $d T t $. $d k t $. - natlocalincr.1 $e |- A. k e. ( 0 ..^ T ) A. t e. ( 1 ..^ ( T + 1 ) ) ( k < - t -> ( B ` k ) < ( B ` t ) ) $. - $( Global monotonicity on half-open range implies local monotonicity. - (Contributed by Ender Ting, 22-Nov-2024.) $) - natlocalincr $p |- A. k e. ( 0 ..^ T ) ( B ` k ) < ( B ` ( k + 1 ) ) $= - ( cv cfv c1 caddc co clt wbr cc0 cfzo wcel wex wi wral ralimi cz wceq rsp - ovex isseti fzoaddel mpan2 0p1e1 oveq1i eleqtrdi eleq1 syl5ibrcom ralimia - 1z imim1d mp2b cr elfzoelz zre ltp1 3syl breq2 ax-2 syl5com breq2d biimpd - fveq2 a2i rspec eximdv mpi ax5e syl rgen ) DFZBGZVNHIJZBGZKLZDMCNJZVNVSOZ - VRAPZVRVTAFZVPUAZAPWAAVPVNHIUCUDVTWCVRAWCVRQZDVSWCVNWBKLZVOWBBGZKLZQZQZDV - SRZWCWGQZDVSRWDDVSRWHAHCHIJZNJZRZDVSRWBWMOZWHQZDVSRWJEWNWPDVSWHAWMUBSWPWI - DVSVTWCWOWHVTWOWCVPWMOVTVPMHIJZWLNJZWMVTHTOVPWROUMVNMCHUEUFWQHWLNUGUHUIWB - VPWMUJUKUNULUOWIWKDVSVTWCWEQWIWKVTWEWCVNVPKLZVTVNTOVNUPOWSVNMCUQVNURVNUSU - TWBVPVNKVAUKWCWEWGVBVCULWKWDDVSWCWGVRWCWGVRWCWFVQVOKWBVPBVFVDVEVGSUOVHVIV - JVRAVKVLVM $. - $} - - ${ - $d B a b k $. $d T a k t $. $d T a b k $. - natglobalincr.1 $e |- A. k e. ( 0 ..^ T ) ( B ` k ) < ( B ` ( k + 1 ) ) $. - natglobalincr.2 $e |- T e. ZZ $. - $( Local monotonicity on half-open integer range implies global - monotonicity. (Contributed by Ender Ting, 23-Nov-2024.) $) - natglobalincr $p |- A. k e. ( 0 ..^ T ) A. t e. ( ( k + 1 ) ... T ) ( B ` k - ) < ( B ` t ) $= - ( cv cfv clt wbr cc0 co wcel cz cle w3a wceq fveq2 cxr syl va vb c1 caddc - cfz wb elfzoelz peano2zd elfz1 sylancl breq2d rspec cop cxp df-br ltrelxr - cfzo sseli sylbi opelxp1 3ad2ant3 opelxp2 0red cr simp1 zre 3syl peano2re - simp21 zred elfzole1 ltp1d wa wi id leltletr mp2and 3ad2ant1 simp22 letrd - 3jca 0zd a1i elfzo mpd3an23 fvoveq1 breq12d vtoclri syl6bir simp3 xrlttrd - simp23 elfzoel2 elfzop1le2 fzindd sylbida rgen2 ) DGZBHZAGZBHZIJZDAKCUQLZ - WRUCUDLZCUELZWRXCMZWTXEMZWTNMXDWTOJWTCOJPZXBXFXDNMCNMZXGXHUFXFWRWRKCUGZUH - ZFWTXDCUIUJXFWSUAGZBHZIJWSXDBHZIJZWSUBGZBHZIJZWSXPUCUDLZBHZIJXBUAUBWTXDCX - LXDQXMXNWSIXLXDBRUKXLXPQXMXQWSIXLXPBRUKXLXSQXMXTWSIXLXSBRUKXLWTQXMXAWSIXL - WTBRUKXODXCEULXFXPNMZXDXPOJZXPCIJZPZXRPZWSXQXTXRXFWSSMZYDXRWSXQUMZSSUNZMZ - YFXRYGIMYIWSXQIUOIYHYGUPURUSZWSXQSSUTTVAXRXFXQSMZYDXRYIYKYJWSXQSSVBTVAYEX - QXTIJZXQXTUMZYHMZXTSMYEKXPOJZYCYLYEKXDXPYEVCYEWRVDMZXDVDMZYEXFWRNMZYPXFYD - XRVEXJWRVFZVGWRVHZTYEXPXFYAYBYCXRVIZVJXFYDKXDOJZXRXFKWROJZWRXDIJZUUBWRKCV - KXFYRUUDXJYRWRYSVLTXFKVDMZYPYQPZUUCUUDVMUUBVNXFYRYPUUFXJYSYPUUEYPYQYPVCYP - VOYTWAVGKWRXDVPTVQVRXFYAYBYCXRVSVTXFYAYBYCXRWLYEYAYOYCVMZYLVNUUAYAUUGXPXC - MZYLYAKNMXIUUHUUGUFYAWBXIYAFWCXPKCWDWEXOYLDXPXCWRXPQWSXQXNXTIWRXPBRWRXPUC - BUDWFWGEWHWITVQZYLYMIMYNXQXTIUOIYHYMUPURUSXQXTSSVBVGXFYDXRWJUUIWKXKWRKCWM - WRKCWNWOWPWQ $. - $} - - $c UpWord $. - - $( Extend class notation to include the set of strictly increasing - sequences. $) - cupword $a class UpWord S $. - - ${ - $d S k w $. - - ${ - $( Strictly increasing sequence is a sequence, adjacent elements of which - increase. (Contributed by Ender Ting, 19-Nov-2024.) $) - df-upword $a |- UpWord S = { w | ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` - w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) } $. - $} - - ${ - $( Empty set is an increasing sequence for every range. (Contributed by - Ender Ting, 19-Nov-2024.) $) - upwordnul $p |- (/) e. UpWord S $= - ( vw vk c0 cv wcel cfv c1 co wbr cc0 chash cmin wceq wi cvv wb ax-mp cz - cle cword caddc clt cfzo wral wa cab cupword wal 0ex elab6g wrd0 eleq1a - fveq2 hash0 eqtrdi oveq1d 0red lem1d eqbrtrd eqeltrdi 1zzd zsubcld fzon - 0z sylancr mpbid rzal syl jca mpgbir df-upword eleqtrri ) DBEZAUAZFZCEZ - VNGVQHUBIVNGUCJZCKVNLGZHMIZUDIZUEZUFZBUGZAUHDWDFZVNDNZWCOZBDPFWEWGBUIQU - JWCBDPUKRWFVPWBDVOFWFVPOAULDVOVNUMRWFWADNZWBWFVTKTJZWHWFVTKHMIKTWFVSKHM - WFVSDLGKVNDLUNUOUPZUQWFKWFURUSUTWFKSFVTSFWIWHQVEWFVSHWFVSKSWJVEVAWFVBVC - KVTVDVFVGVRCWAVHVIVJVKBACVLVM $. - $} - - ${ - $d A w $. - - $( Any increasing sequence is a sequence. (Contributed by Ender Ting, - 19-Nov-2024.) $) - upwordisword $p |- ( A e. UpWord S -> A e. Word S ) $= - ( vw vk cv cword wcel cupword eleq1 cfv c1 caddc clt wbr cc0 chash cmin - co cfzo wral wa df-upword abeq2i simplbi vtoclga ) CEZBFZGZAUGGCABHZUFA - UGIUFUIGUHDEZUFJUJKZLRUFJMNDOUFPJUKQRSRTZUHULUACUICBDUBUCUDUE $. - $} - - ${ - singoutnword.1 $e |- A e. _V $. - $( Singleton with character out of range ` V ` is not a word for that - range. (Contributed by Ender Ting, 21-Nov-2024.) $) - singoutnword $p |- ( -. A e. V -> -. <" A "> e. Word V ) $= - ( cs1 cword wcel cc0 cfv cvv wceq s1fv ax-mp c0 wne s1nz fstwrdne mpan2 - eqeltrrid con3i ) ADZBEFZABFUAAGTHZBAIZFUBAJCAUCKLUATMNUBBFAOBTPQRS $. - $} - - ${ - singoutnupword.1 $e |- A e. _V $. - $( Singleton with character out of range ` S ` is not an increasing - sequence for that range. (Contributed by Ender Ting, 22-Nov-2024.) $) - singoutnupword $p |- ( -. A e. S -> -. <" A "> e. UpWord S ) $= - ( wcel wn cs1 cword cupword singoutnword upwordisword nsyl ) ABDEAFZBGD - LBHDABCILBJK $. - $} - - ${ - $d A w $. - upwordsing.1 $e |- A e. S $. - $( Singleton is an increasing sequence for any compatible range. - (Contributed by Ender Ting, 21-Nov-2024.) $) - upwordsing $p |- <" A "> e. UpWord S $= - ( vw vk cs1 cv cword wcel cfv c1 caddc clt wbr cc0 chash cmin cfzo wceq - co wral wa cab cupword wi wal wb s1cl elab6g mp2b eleq1a c0 fveq2 s1len - oveq1d oveq1i 1m1e0 eqtri eqtrdi oveq2d fzo0 syl jca df-upword eleqtrri - rzal mpgbir ) AFDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBDUCBUDAFDGBHIEGDGJEG - KLTDGJMNEODGPJKQTRTUAUBDUCIDGAFSDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBUEDA - BIAFBHIAFDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBDUCIDGAFSDGBHIEGDGJEGKLTDGJ - MNEODGPJKQTRTUAUBUEDUFUGCABUHDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAUBDAFBHUI - UJDGAFSDGBHIEGDGJEGKLTDGJMNEODGPJKQTRTUAABIAFBHIDGAFSDGBHIUECABUHAFBHDG - UKUJDGAFSODGPJKQTRTULSEGDGJEGKLTDGJMNEODGPJKQTRTUADGAFSODGPJKQTRTOORTUL - DGAFSDGPJKQTOORDGAFSDGPJKQTAFPJKQTODGAFSDGPJAFPJKQDGAFPUMUOAFPJKQTKKQTO - AFPJKKQAUNUPUQURUSUTOVAUSEGDGJEGKLTDGJMNEODGPJKQTRTVFVBVCVGDBEVDVE $. - $} - - ${ - $d S t $. - upwordsseti.1 $e |- S e. _V $. - $( Strictly increasing sequences with a set given for range form a set. - (Contributed by Ender Ting, 21-Nov-2024.) $) - upwordsseti $p |- UpWord S e. _V $= - ( vt cupword cword wrdexi cv upwordisword ssriv ssexi ) ADZAEZABFCKLCGA - HIJ $. - $} - - ${ - $d A b k $. $d S b k $. - tworepnotupword.1 $e |- A e. _V $. - $( Word of two matching characters is never an increasing sequence. - (Contributed by Ender Ting, 22-Nov-2024.) $) - tworepnotupword $p |- -. ( <" A "> ++ <" A "> ) e. UpWord S $= - ( vb vk co wcel wn wceq cfv c1 caddc clt wbr cc0 chash cmin cfzo ax-mp - cz cs1 cconcat cupword ovex cv wral wrex wa wex isseti wi 0z ccat2s1len - c0ex c2 oveq1i 2m1e1 eqtri eqeltri breqtrri fzolb mpbir3an eleq1a fveq2 - 1z 0lt1 oveq1d oveq2d eleq2d syl5ibr et-ltneverrefl fveq1 cvv ccat2s1p1 - eqtrdi 1e0p1 fveq2i ccat2s1p2 eqtr3i breq12d mtbiri fvoveq1 biimpd jcad - con3d syl5com eximdv mpi nfre1 rspe exlimi rexnal sylib cword df-upword - syl abeq2i simprbi nsyl eleq1 mtbid vtocle ) AUAZXCUBFZBUCZGZHDXDXCXCUB - UDDUEZXDIZXGXEGZXFXHEUEZXGJZXJKLFXGJZMNZEOXGPJZKQFZRFZUFZXIXHXMHZEXPUGZ - XQHXHXJXPGZXRUHZEUIZXSXHXJOIZEUIYBEOUNUJXHYCYAEXHYCXTXRYCXTXHXJOXDPJZKQ - FZRFZGZOYFGZYCYGUKYHOTGYETGOYEMNULYEKTYEUOKQFKYDUOKQAAUMUPUQURZVEUSOKYE - MVFYIUTOYEVAVBOYFXJVCSXHXPYFXJXHXOYEORXHXNYDKQXGXDPVDVGVHVIVJXHOXGJZOKL - FZXGJZMNZHYCXRXHYMAAMNAVKXHYJAYLAMXHYJOXDJZAOXGXDVLAVMGZYNAICVMAAVNSVOX - HYLYKXDJZAYKXGXDVLKXDJZYPAKYKXDVPVQYOYQAICVMAAVRSVSVOVTWAYCXMYMYCXMYMYC - XKYJXLYLMXJOXGVDXJOKXGLWBVTWCWEWFWDWGWHYAXSEXREXPWIXREXPWJWKWPXMEXPWLWM - XIXGBWNGZXQYRXQUHDXEDBEWOWQWRWSXGXDXEWTXAXB $. - $} - $} - -$( (End of Ender Ting's mathbox.) $) -$( End $[ set-mbox-et.mm $] $) - $( End $[ set-mbox.mm $] $)