diff --git a/changes-set.txt b/changes-set.txt index c94e1c9e8..1a90fa438 100644 --- a/changes-set.txt +++ b/changes-set.txt @@ -89,6 +89,15 @@ DONE: Date Old New Notes 27-Jan-25 --- --- Moved surreal cofinality theorems from SF's mathbox to main set.mm +26-Jan-25 an2anr [same] Moved from PM's mathbox to main set.mm +26-Jan-25 ismhmd [same] Moved from SN's mathbox to main set.mm +26-Jan-25 ringcld [same] Moved from SN's mathbox to main set.mm +26-Jan-25 ablcmnd [same] Moved from SN's mathbox to main set.mm +26-Jan-25 ringabld [same] Moved from SN's mathbox to main set.mm +26-Jan-25 ringcmnd [same] Moved from SN's mathbox to main set.mm +26-Jan-25 pwsexpg [same] Moved from SN's mathbox to main set.mm +26-Jan-25 drnginvrcld [same] Moved from SN's mathbox to main set.mm +26-Jan-25 pwspjmhmmgpd [same] Moved from SN's mathbox to main set.mm 24-Jan-25 rextru [same] Moved from ZW's mathbox to main set.mm 19-Jan-25 --- --- Moved surreal cut and option theorems from SF's mathbox to main set.mm diff --git a/constr.mm b/constr.mm new file mode 100644 index 000000000..cdc52d6ba --- /dev/null +++ b/constr.mm @@ -0,0 +1,144 @@ +$[ set.mm $] + +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Constructible numbers +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + This section defines the set of constructible points as complex numbers which + can be drawn starting from two points (we take ` 0 ` and ` 1 ` ), and taking + intersections of circles and lines. + + We initially define two sets: + + + We then proceed by showing that those two sets are equal. + + This construction is useful for proving the impossibility of doubling the + cube ( ~ imp2cube ), and of angle trisection ( ~ imp3ang ) +$) + + $( All algebraic numbers admit a minimal polynomial. $) + $( minplyeu $p |- ( ph -> E! p e. ( Monic1p ` E ) A. q e. ( Monic1p ` E ) ( -> p = q ) ) $) + + $( Use infval $) + $( minplyf $) + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Degree of a field extension +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + ${ + fldextdgirred.b $e |- B = ( Base ` F ) $. + fldextdgirred.0 $e |- .0. = ( 0g ` F ) $. + fldextdgirred.d $e |- D = ( deg1 ` F ) $. + fldextdgirred.o $e |- O = ( F evalSub1 E ) $. + fldextdgirred.m $e |- M = ( Poly1 ` F ) $. + fldextdgirred.k $e |- K = ( F |`s E ) $. + fldextdgirred.l $e |- L = ( F |`s ( F fldGen ( E u. { Z } ) ) ) $. + fldextdgirred.f $e |- ( ph -> F e. Field ) $. + fldextdgirred.e $e |- ( ph -> E e. ( SubDRing ` F ) ) $. + fldextdgirred.p $e |- ( ph -> P e. dom O ) $. + fldextdgirred.1 $e |- ( ph -> P e. ( Irred ` M ) ) $. + fldextdgirred.z $e |- ( ph -> Z e. ( B \ E ) ) $. + fldextdgirred.2 $e |- ( ph -> ( ( O ` P ) ` Z ) = .0. ) $. + $( If ` Z ` is the root of an irreducible polynomial ` P ` over a field + ` K ` , then the degree of the field extension ` [ K [ Z ] : K ] ` is + the degree of the polynomial ` P ` . $) + fldextdgirred $p |- ( ph -> ( L [:] K ) = ( D ` P ) ) $= ? $. + $} + + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Quadratic field extensions +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $c /FldExt2 $. + + $( Extend class notation with the quadratic field extension relation. $) + cfldext2 $a class /FldExt2 $. + + ${ + $d e f $. + $( Define the quadratic field extension relation. Quadratic field + extensions are field extensions of degree 2. $) + df-fldext2 $a |- /FldExt2 = { <. e , f >. | ( e /FldExt f /\ ( e [:] f ) = 2 ) } $. + $} + + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Chain +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $c Chain $. + + $( Extend class notation with the (finite) chain builder function. $) + cchn $a class Chain $. + + ${ + $d r c n $. + $( Define the (finite) chain builder function. A chain is defined to be a + sequence of objects, whereas each object is in a given relation with the + next one. $) + df-chn $a |- Chain = ( r e. _V |-> { c e. Word dom r | + A. n e. ( dom c \ { 0 } ) ( c ` ( n - 1 ) ) r ( c ` n ) } ) $. + $} + + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Constructible points +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $c Constr $. + + $( Extend class notation with the set of geometrically constructible points. $) + cconstr $a class Constr $. + + $( Define the set of geometrically constructible points, by recursively adding + the line-line, line-circle and circle-circle intersections constructions + using points in a previous iteration. $) + df-constr $a |- Constr = ( rec ( ( s e. _V |-> { x e. CC | + ( E. a e. s E. b e. s E. c e. s E. d e. s E. t e. RR E. r e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ x = ( c + ( r x. ( d - c ) ) ) /\ -. ( Im ` ( ( * ` ( b - a ) ) x. ( d - c ) ) ) = 0 ) + \/ E. a e. s E. b e. s E. c e. s E. e e. s E. f e. s E. t e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ ( abs ` ( x - c ) ) = ( abs ` ( e - f ) ) ) + \/ E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s E. t e. RR ( -. a = d /\ ( abs ` ( x - a ) ) = ( abs ` ( b - c ) ) /\ ( abs ` ( x - d ) ) = ( abs ` ( e - f ) ) ) ) + } ) , { 0 , 1 } ) " _om ) $. + + ${ + constrtow2.b $e |- B = ( Base ` F ) $. + constrtow2.q $e |- Q = ( CCfld |`s QQ ) $. + constrtow2.a $e |- ( ph -> A e. Constr ) $. + constrtow2.f $e |- ( ph -> F e. Field ) $. + constrtow2.1 $e |- ( ph -> A e. B ) $. + $( If an algebraically constructible point ` A ` is in a field ` F ` , then + there is a tower of quadratic field extensions from ` QQ ` to ` F ` . $) + constrtow2 $p |- ( ph -> E. t e. ( Chain ` `' /FldExt2 ) ( ( t ` 0 ) = Q + /\ ( lastS ` t ) = F ) ) $= ? $. + $} + + +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Impossible constructions +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $( Impossibility of doubling the cube. This is expressed by stating that the + cubic root of 2, which is the side length of a cube of volume ` 2 ` , is + not constructible. $) + imp2cube $p |- -. ( 2 ^c ( 1 / 3 ) ) e. Constr $= ? $. + + $( Impossibility of trisecting angles. This is expressed by stating that the + cosine of an angle of ` ( _pi / 9 ) ` which would be the third of the + constructible angle ` ( _pi / 3 ) ` , is not constructible. $) + imp3ang $p |- -. ( cos ` ( _pi / 9 ) ) e. Constr $= ? $. diff --git a/iset.mm b/iset.mm index f7e836f62..b5fd86a01 100644 --- a/iset.mm +++ b/iset.mm @@ -148174,21 +148174,60 @@ since the target of the homomorphism (operator ` O ` in our model) need $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - 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 ~ https://us.metamath.org/mpeuni/df-unit.html in set.mm), and we have + "the ring unity is a unit", see ~ https://us.metamath.org/mpeuni/1unit.html . + $) + $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 ~ dfur2g , which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under @@ -148228,9 +148267,9 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This $( --.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Semirings --.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $c SRing $. @@ -148373,7 +148412,7 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This AVKDVJVCNVGVACVCDVJVPSVAVJVFDVCVACVCEFVPSUNTUO $. $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 -> @@ -148426,7 +148465,7 @@ 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 ) $= @@ -148462,14 +148501,14 @@ of the multiplicative monoid ( ~ df-mgp ) of a ring-like structure. This VKRVERVDRUHUMUPVBVJUBUQUPUSVGVAVIUPURVFEUPDVDEECVEBCVCIVMGUIZBDVCIVMHUJZU PEUKZSTUPUTVHEUPEEDVDCVEVNVPVOSTULUNUO $. - $( 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 ) $= @@ -148784,7 +148823,7 @@ 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 @@ -148945,8 +148984,8 @@ the additive structure must be abelian (see ~ ringcom ), care must be 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.) $) + $( 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 cv co wceq wa wral wreu cfv eqid syl oveqd eqeq1d cmgp ringmgp @@ -148980,7 +149019,7 @@ the additive structure must be abelian (see ~ ringcom ), care must be ${ 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 ) $= @@ -149014,13 +149053,13 @@ the additive structure must be abelian (see ~ ringcom ), care must be BEVCVJQVDQVCQUGUHUQUSVFVAVHUQURVEEUQDVCEECVDUOCVDMUPBCVBIVKGUIRZUODVCMUPB DVBIVKHUJRZUQEUKZSTUQUTVGEUQEEDVCCVDVLVNVMSTUMUN $. - $( 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 @@ -149075,8 +149114,8 @@ 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 @@ -149318,7 +149357,7 @@ 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. ) ) $= @@ -149331,7 +149370,7 @@ 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. ) ) $= @@ -149761,6 +149800,229 @@ the additive structure must be abelian (see ~ ringcom ), care must be $} +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Divisibility +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= +$) + + $( Introduce new constant symbols. $) + $c ||r $. $( Ring divisibility relation $) + $c Unit $. $( Units in a ring $) + $c Irred $. $( Ring irreducibles $) + + $( Ring divisibility relation. $) + cdsr $a class ||r $. + + $( Units in a ring. $) + cui $a class Unit $. + + $( Ring irreducibles. $) + cir $a class Irred $. + + ${ + $d b w x y z $. + $( Define the (right) divisibility relation in a ring. Access to the left + divisibility relation is available through + ` ( ||r `` ( oppR `` R ) ) ` . (Contributed by Mario Carneiro, + 1-Dec-2014.) $) + df-dvdsr $a |- ||r = ( w e. _V |-> { <. x , y >. | ( x e. ( Base ` w ) /\ + E. z e. ( Base ` w ) ( z ( .r ` w ) x ) = y ) } ) $. + + $( Define the set of units in a ring, that is, all elements with a left and + right multiplicative inverse. (Contributed by Mario Carneiro, + 1-Dec-2014.) $) + df-unit $a |- Unit = ( w e. _V |-> + ( `' ( ( ||r ` w ) i^i ( ||r ` ( oppR ` w ) ) ) " { ( 1r ` w ) } ) ) $. + + $( Define the set of irreducible elements in a ring. (Contributed by Mario + Carneiro, 4-Dec-2014.) $) + df-irred $a |- Irred = ( w e. _V |-> + [_ ( ( Base ` w ) \ ( Unit ` w ) ) / b ]_ + { z e. b | A. x e. b A. y e. b ( x ( .r ` w ) y ) =/= z } ) $. + $} + + ${ + $d w x y z R $. + $( The divides relation is a relation. (Contributed by Mario Carneiro, + 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.) $) + reldvdsrsrg $p |- ( R e. SRing -> Rel ( ||r ` R ) ) $= + ( vx vz vy vw csrg wcel cdsr cfv cvv cxp cbs cv cmulr co wceq copab fveq2 + wrex wa wrel df-dvdsr eleq2d oveqd eqeq1d rexeqbidv anbi12d opabbidv elex + wss wfn basfn funfvex funfni sylancr xpexg syl2anc simplll simplr simpllr + simpr eqid srgcl eqeltrrd rexlimdva2 imdistanda ssopab2dv df-xp sseqtrrdi + syl3anc ssexd fvmptd3 eqsstrd xpss sstrdi df-rel sylibr ) AFGZAHIZJJKZUJV + SUAVRVSALIZWAKZVTVRVSBMZWAGZCMZWCANIZOZDMZPZCWASZTZBDQZWBVREAWCEMZLIZGZWE + WCWMNIZOZWHPZCWNSZTZBDQWLJHJBDCEUBWMAPZWTWKBDXAWOWDWSWJXAWNWAWCWMALRZUCXA + WRWICWNWAXBXAWQWGWHXAWPWFWEWCWMANRUDUEUFUGUHAFUIZVRWLWBJVRWAJGZXDWBJGVRLJ + UKAJGXDULXCXDJALALUMUNUOZXEWAWAJJUPUQVRWLWDWHWAGZTZBDQWBVRWKXGBDVRWDWJXFV + RWDTZWIXFCWAXHWEWAGZTZWITZWGWHWAXJWIVAXKVRXIWDWGWAGVRWDXIWIURXHXIWIUSVRWD + XIWIUTWAAWFWEWCWAVBWFVBVCVJVDVEVFVGBDWAWAVHVIZVKVLXLVMWAWAVNVOVSVPVQ $. + $} + + ${ + $d x y .|| $. $d r x y z B $. $d x y z X $. $d x y z Y $. $d x y Z $. + $d r x y z R $. $d r x y z .x. $. $d x y z ph $. + dvdsrvald.1 $e |- ( ph -> B = ( Base ` R ) ) $. + dvdsrvald.2 $e |- ( ph -> .|| = ( ||r ` R ) ) $. + dvdsrvald.r $e |- ( ph -> R e. SRing ) $. + ${ + dvdsrvald.3 $e |- ( ph -> .x. = ( .r ` R ) ) $. + $( Value of the divides relation. (Contributed by Mario Carneiro, + 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.) $) + dvdsrvald $p |- ( ph -> .|| = { <. x , y >. | + ( x e. B /\ E. z e. B ( z .x. x ) = y ) } ) $= + ( cfv cv cbs wcel wceq wa copab cvv vr cdsr cmulr df-dvdsr fveq2 eleq2d + wrex oveqd eqeq1d rexeqbidv anbi12d opabbidv csrg elexd cxp wfn funfvex + co basfn funfni sylancr xpexg syl2anc simprr ad2antrr simprl eqid srgcl + simplr syl3anc eqeltrrd rexlimdvaa imdistanda ssopab2dv df-xp sseqtrrdi + ssexd fvmptd3 3eqtr4d ) AGUBMBNZGOMZPZDNZVTGUCMZURZCNZQZDWAUGZRZBCSZFVT + EPZWCVTHURZWFQZDEUGZRZBCSAUAGVTUANZOMZPZWCVTWPUCMZURZWFQZDWQUGZRZBCSWJT + UBTBCDUAUDWPGQZXCWIBCXDWRWBXBWHXDWQWAVTWPGOUEZUFXDXAWGDWQWAXEXDWTWEWFXD + WSWDWCVTWPGUCUEUHUIUJUKULAGUMKUNZAWJWAWAUOZTAWATPZXHXGTPAOTUPGTPXHUSXFX + HTGOGOUQUTVAZXIWAWATTVBVCAWJWBWFWAPZRZBCSXGAWIXKBCAWBWHXJAWBRZWGXJDWAXL + WCWAPZWGRZRZWEWFWAXLXMWGVDXOGUMPZXMWBWEWAPAXPWBXNKVEXLXMWGVFAWBXNVIWAGW + DWCVTWAVGWDVGVHVJVKVLVMVNBCWAWAVOVPVQVRJAWOWIBCAWKWBWNWHAEWAVTIUFAWMWGD + EWAIAWLWEWFAHWDWCVTLUHUIUJUKULVS $. + + $( Value of the divides relation. (Contributed by Mario Carneiro, + 1-Dec-2014.) $) + dvdsrd $p |- ( ph + -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) ) $= + ( vx vy cvv wcel wa cv co wceq wbr wrex wrel cfv reldvdsrsrg syl releqd + cdsr csrg mpbird brrelex12 sylan simplr elexd simprr cmulr cbs ad2antrr + simprl eleqtrd eqid srgcl syl3anc oveqd 3eltr4d eqeltrrd jca rexlimdvaa + ex expimpd wb copab dvdsrvald adantr breqd simpl eleq1d eqeq12d rexbidv + oveq2d simpr anbi12d brabga adantl bitrd pm5.21ndd ) AGOPZHOPZQZGHDUAZG + CPZBRZGFSZHTZBCUBZQZAWJWIADUCZWJWIAWQEUHUDZUCZAEUIPZWSKEUEUFADWRJUGUJGH + DUKULVIAWKWOWIAWKQZWNWIBCXAWLCPZWNQZQZWGWHXDGCAWKXCUMZUNXDHCXDWMHCXAXBW + NUOXDWLGEUPUDZSZEUQUDZWMCXDWTWLXHPGXHPXGXHPAWTWKXCKURXDWLCXHXAXBWNUSACX + HTWKXCIURZUTXDGCXHXEXIUTXHEXFWLGXHVAXFVAVBVCXDFXFWLGAFXFTWKXCLURVDXIVEV + FUNVGVHVJAWIWJWPVKAWIQZWJGHMRZCPZWLXKFSZNRZTZBCUBZQZMNVLZUAZWPXJDXRGHAD + XRTWIAMNBCDEFIJKLVMVNVOWIXSWPVKAXQWPMNGHXROOXKGTZXNHTZQZXLWKXPWOYBXKGCX + TYAVPZVQYBXOWNBCYBXMWMXNHYBXKGWLFYCVTXTYAWAVRVSWBXRVAWCWDWEVIWF $. + + ${ + dvdsr2d.x $e |- ( ph -> X e. B ) $. + $( Value of the divides relation. (Contributed by Mario Carneiro, + 1-Dec-2014.) $) + dvdsr2d $p |- ( ph -> ( X .|| Y <-> E. z e. B ( z .x. X ) = Y ) ) $= + ( wbr wcel cv co wceq wrex dvdsrd mpbirand ) AGHDNGCOBPGFQHRBCSMABCDE + FGHIJKLTUA $. + + dvdsrmuld.y $e |- ( ph -> Y e. B ) $. + $( A left-multiple of ` X ` is divisible by ` X ` . (Contributed by + Mario Carneiro, 1-Dec-2014.) $) + dvdsrmuld $p |- ( ph -> X .|| ( Y .x. X ) ) $= + ( vz co wbr wcel cv wceq wrex eqid oveq1 eqeq1d rspcev sylancl dvdsrd + mpbir2and ) AFGFEOZCPFBQNRZFEOZUHSZNBTZLAGBQUHUHSZULMUHUAUKUMNGBUIGSU + JUHUHUIGFEUBUCUDUEANBCDEFUHHIJKUFUG $. + $} + $} + + ${ + dvdsrcld.d $e |- ( ph -> X .|| Y ) $. + $( Closure of a dividing element. (Contributed by Mario Carneiro, + 5-Dec-2014.) $) + dvdsrcld $p |- ( ph -> X e. B ) $= + ( vz wcel cv cmulr cfv co wceq wrex wbr wa eqidd dvdsrd mpbid simpld ) + AEBLZKMEDNOZPFQKBRZAEFCSUEUGTJAKBCDUFEFGHIAUFUAUBUCUD $. + $} + $} + + ${ + $d x y .|| $. $d r x y z B $. $d x y z X $. $d x y z Y $. $d x y Z $. + $d r x y z R $. $d r x y z .x. $. + dvdsr.1 $e |- B = ( Base ` R ) $. + dvdsr.2 $e |- .|| = ( ||r ` R ) $. + $( Closure of a dividing element. (Contributed by Mario Carneiro, + 5-Dec-2014.) $) + dvdsrcl2 $p |- ( ( R e. Ring /\ X .|| Y ) -> Y e. B ) $= + ( vx crg wcel wbr wa cv cmulr cfv co wceq wrex cbs a1i cdsr ringsrg eqidd + dvdsrd pm5.32i eqid ringcl 3expa an32s eleq1 syl5ibcom rexlimdva sylbi + impr ) CIJZDEBKZLUODAJZHMZDCNOZPZEQZHARZLZLEAJZUOUPVCUOHABCUSDEACSOQUOFTB + CUAOQUOGTCUBUOUSUCUDUEUOUQVBVDUOUQLZVAVDHAVEURAJZLUTAJZVAVDUOVFUQVGUOVFUQ + VGACUSURDFUSUFUGUHUIUTEAUJUKULUNUM $. + + $( An element in a (unital) ring divides itself. (Contributed by Mario + Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) $) + dvdsrid $p |- ( ( R e. Ring /\ X e. B ) -> X .|| X ) $= + ( crg wcel wa cur cfv cmulr co cbs wceq a1i cdsr csrg adantr eqid ringsrg + eqidd simpr ringidcl dvdsrmuld ringlidm breqtrd ) CGHZDAHZIZDCJKZDCLKZMDB + UJABCULDUKACNKOUJEPBCQKOUJFPUHCRHUICUASUJULUBUHUIUCUHUKAHUIACUKEUKTZUDSUE + ACULUKDEULTUMUFUG $. + + $( Divisibility is transitive. (Contributed by Mario Carneiro, + 1-Dec-2014.) $) + dvdsrtr $p |- ( ( R e. Ring /\ Y .|| Z /\ Z .|| X ) -> Y .|| X ) $= + ( vy vx wcel wbr wa cv cfv co wceq wrex a1i eqidd crg cdsr ringsrg dvdsrd + cmulr cbs anbi12d an4 bitrdi reeanv ad2antrr simplrl simpll simprr simprl + csrg eqid ringcl syl3anc dvdsrmuld ringass syl13anc oveq2 sylan9eq breq2d + breqtrd id syl5ibcom rexlimdvva syl5bir expimpd sylbid 3impib ) CUAKZEFBL + ZFDBLZEDBLZVNVOVPMZEAKZFAKZMZINZECUEOZPZFQZIARZJNZFWCPZDQZJARZMZMZVQVNVRV + SWFMZVTWJMZMWLVNVOWMVPWNVNIABCWCEFACUFOQZVNGSZBCUBOQZVNHSZCUCZVNWCTZUDVNJ + ABCWCFDWPWRWSWTUDUGVSWFVTWJUHUIVNWAWKVQWKWEWIMZJARIARVNWAMZVQWEWIIJAAUJXB + XAVQIJAAXBWBAKZWGAKZMZMZEWGWDWCPZBLXAVQXFEWGWBWCPZEWCPZXGBXFABCWCEXHWOXFG + SWQXFHSVNCUPKWAXEWSUKXFWCTVNVSVTXEULZXFVNXDXCXHAKVNWAXEUMZXBXCXDUNZXBXCXD + UOZACWCWGWBGWCUQZURUSUTXFVNXDXCVSXIXGQXKXLXMXJACWCWGWBEGXNVAVBVFXAXGDEBWE + WIXGWHDWDFWGWCVCWIVGVDVEVHVIVJVKVLVM $. + + ${ + dvdsrmul1.3 $e |- .x. = ( .r ` R ) $. + $( The divisibility relation is preserved under right-multiplication. + (Contributed by Mario Carneiro, 1-Dec-2014.) $) + dvdsrmul1 $p |- ( ( R e. Ring /\ Z e. B /\ X .|| Y ) -> + ( X .x. Z ) .|| ( Y .x. Z ) ) $= + ( vx crg wcel wbr co wa cv wceq cfv a1i wrex cbs cdsr csrg adantr cmulr + ringsrg dvdsrd simplll syl simplr simpllr ringcl syl3anc simpr syl13anc + dvdsrmuld ringass breqtrrd oveq1 breq2d syl5ibcom expimpd sylbid 3impia + rexlimdva ) CLMZGAMZEFBNZEGDOZFGDOZBNZVGVHPZVIEAMZKQZEDOZFRZKAUAZPVLVMK + ABCDEFACUBSRZVMHTBCUCSRZVMITVGCUDMZVHCUGZUEDCUFSRZVMJTUHVMVNVRVLVMVNPZV + QVLKAWDVOAMZPZVJVPGDOZBNVQVLWFVJVOVJDOZWGBWFABCDVJVOVSWFHTVTWFITWFVGWAV + GVHVNWEUIZWBUJWCWFJTWFVGVNVHVJAMWIVMVNWEUKZVGVHVNWEULZACDEGHJUMUNWDWEUO + ZUQWFVGWEVNVHWGWHRWIWLWJWKACDVOEGHJURUPUSVQWGVKVJBVPFGDUTVAVBVFVCVDVE + $. + $} + + dvdsrneg.5 $e |- N = ( invg ` R ) $. + $( An element divides its negative. (Contributed by Mario Carneiro, + 1-Dec-2014.) $) + dvdsrneg $p |- ( ( R e. Ring /\ X e. B ) -> X .|| ( N ` X ) ) $= + ( crg wcel wa cur cfv cmulr co cbs wceq a1i adantr eqid cdsr csrg ringsrg + eqidd simpr cgrp ringgrp ringidcl grpinvcl syl2anc simpl ringnegl breqtrd + dvdsrmuld ) CIJZEAJZKZECLMZDMZECNMZOEDMBUQABCUTEUSACPMQUQFRBCUAMQUQGRUOCU + BJUPCUCSUQUTUDUOUPUEZUOUSAJZUPUOCUFJURAJVBCUGACURFURTZUHACDURFHUIUJSUNUQA + CUTURDEFUTTVCHUOUPUKVAULUM $. + $} + + ${ + $d B x $. $d R x $. $d X x $. $d .0. x $. + dvdsr0.b $e |- B = ( Base ` R ) $. + dvdsr0.d $e |- .|| = ( ||r ` R ) $. + dvdsr0.z $e |- .0. = ( 0g ` R ) $. + $( In a ring, zero is divisible by all elements. ("Zero divisor" as a term + has a somewhat different meaning.) (Contributed by Stefan O'Rear, + 29-Mar-2015.) $) + dvdsr01 $p |- ( ( R e. Ring /\ X e. B ) -> X .|| .0. ) $= + ( vx crg wcel wa wbr cv cmulr cfv co wceq wrex a1i ring0cl oveq1 syl2an2r + eqid ringlz eqeq1d rspcev cbs cdsr csrg ringsrg adantr eqidd simpr mpbird + dvdsr2d ) CJKZDAKZLZDEBMINZDCOPZQZERZIASZUQEAKUREDVAQZERZVDACEFHUAACVADEF + VAUDHUEVCVFIEAUTERVBVEEUTEDVAUBUFUGUCUSIABCVADEACUHPRUSFTBCUIPRUSGTUQCUJK + URCUKULUSVAUMUQURUNUPUO $. + + $d B w $. $d .0. w $. + $( Only zero is divisible by zero. (Contributed by Stefan O'Rear, + 29-Mar-2015.) $) + dvdsr02 $p |- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> X = .0. ) ) $= + ( vx vw crg wcel wa cv cfv wceq wrex a1i adantr wb wbr cmulr co cdsr csrg + cbs ringsrg ring0cl dvdsr2d ringrz eqeq1d eqcom bitrdi rexbidva wex elex2 + eqid r19.9rmv 3syl bitr4d bitrd ) CKLZDALZMZEDBUAINZECUBOZUCZDPZIAQZDEPZV + DIABCVFEDACUFOPVDFRBCUDOPVDGRVBCUELVCCUGSVFVFPVDVFUQZRVBEALZVCACEFHUHZSUI + VBVIVJTVCVBVIVJIAQZVJVBVHVJIAVBVEALMZVHEDPVJVOVGEDACVFVEEFVKHUJUKEDULUMUN + VBVLJNALJUOVJVNTVMJEAUPVJIJAURUSUTSVA $. + $} + + $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# The complex numbers as an algebraic extensible structure @@ -167013,6 +167275,16 @@ Norman Megill (2007) section 1.1.3. Megill then states, "A number of htmldef "oppR" as "oppr"; althtmldef "oppR" as "oppr"; latexdef "oppR" as "\mathrm{opp}_r"; +htmldef "||r" as " ||r"; + althtmldef "||r" as "∥r"; + latexdef "||r" as "\mathrel{\parallel_\mathrm{r}}"; +htmldef "Unit" as "Unit"; + althtmldef "Unit" as "Unit"; + latexdef "Unit" as "\mathrm{Unit}"; +htmldef "Irred" as "Irred"; + althtmldef "Irred" as "Irred"; + latexdef "Irred" as "\mathrm{Irred}"; htmldef "numer" as "numer"; althtmldef "numer" as "numer"; latexdef "numer" as "\mathrm{numer}"; diff --git a/mmil.raw.html b/mmil.raw.html index c7974ced0..b01bf26b0 100644 --- a/mmil.raw.html +++ b/mmil.raw.html @@ -10898,6 +10898,36 @@ may be possible once SubGrp is defined + + reldvdsr + ~ reldvdsrsrg + + + + dvdsrval + ~ dvdsrvald + + + + dvdsr + ~ dvdsrd + + + + dvdsr2 + ~ dvdsr2d + + + + dvdsrmul + ~ dvdsrmuld + + + + dvdsrcl + ~ dvdsrcld + + df-drng none diff --git a/set.mm b/set.mm index 9b06e6110..ede5aec72 100644 --- a/set.mm +++ b/set.mm @@ -5780,6 +5780,12 @@ use disjunction (although this is not required since definitions are ( wa anbi2d biancomd ) ADBFCDABCDEGH $. $} + $( Double commutation in conjunction. (Contributed by Peter Mazsa, + 27-Jun-2019.) $) + an2anr $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> + ( ( ps /\ ph ) /\ ( th /\ ch ) ) ) $= + ( wa ancom anbi12i ) ABEBAECDEDCEABFCDFG $. + $( Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) $) pm4.38 $p |- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) -> @@ -227478,6 +227484,27 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is SJKXEIVOVTVGWEWIWRWTWAWBWCWD $. $} + ${ + $d ph x y $. $d B x y $. $d F x y $. $d S x y $. $d T x y $. + ismhmd.b $e |- B = ( Base ` S ) $. + ismhmd.c $e |- C = ( Base ` T ) $. + ismhmd.p $e |- .+ = ( +g ` S ) $. + ismhmd.q $e |- .+^ = ( +g ` T ) $. + ismhmd.0 $e |- .0. = ( 0g ` S ) $. + ismhmd.z $e |- Z = ( 0g ` T ) $. + ismhmd.s $e |- ( ph -> S e. Mnd ) $. + ismhmd.t $e |- ( ph -> T e. Mnd ) $. + ismhmd.f $e |- ( ph -> F : B --> C ) $. + ismhmd.a $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> + ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) $. + ismhmd.h $e |- ( ph -> ( F ` .0. ) = Z ) $. + $( Deduction version of ~ ismhm . (Contributed by SN, 27-Jul-2024.) $) + ismhmd $p |- ( ph -> F e. ( S MndHom T ) ) $= + ( cmnd wcel wf cv cfv wceq wral w3a cmhm ralrimivva 3jca ismhm syl21anbrc + co ) AHUDUEIUDUEDEJUFZBUGZCUGZFUQJUHUSJUHUTJUHGUQUIZCDUJBDUJZKJUHLUIZUKJH + IULUQUESTAURVBVCUAAVABCDDUBUMUCUNBCDEFGHIJLKMNOPQRUOUP $. + $} + ${ $d f s t x y B $. $d x y F $. $d x y S $. $d x y T $. $( Reverse closure of a monoid homomorphism. (Contributed by Mario @@ -243658,6 +243685,14 @@ an extension of the previous (inserting an element and its inverse at ablcmn $p |- ( G e. Abel -> G e. CMnd ) $= ( cabl wcel cgrp ccmn isabl simprbi ) ABCADCAECAFG $. + ${ + ablcmnd.1 $e |- ( ph -> G e. Abel ) $. + $( An Abelian group is a commutative monoid. (Contributed by SN, + 1-Jun-2024.) $) + ablcmnd $p |- ( ph -> G e. CMnd ) $= + ( cabl wcel ccmn ablcmn syl ) ABDEBFECBGH $. + $} + ${ $d g x y B $. $d g x y G $. $d g .+ $. iscmn.b $e |- B = ( Base ` G ) $. @@ -252166,6 +252201,18 @@ the additive structure must be abelian (see ~ ringcom ), care must be HFUCDEUEUHGUDUAUB $. $} + ${ + ringcld.b $e |- B = ( Base ` R ) $. + ringcld.t $e |- .x. = ( .r ` R ) $. + ringcld.r $e |- ( ph -> R e. Ring ) $. + ringcld.x $e |- ( ph -> X e. B ) $. + ringcld.y $e |- ( ph -> Y e. B ) $. + $( Closure of the multiplication operation of a ring. (Contributed by SN, + 29-Jul-2024.) $) + ringcld $p |- ( ph -> ( X .x. Y ) e. B ) $= + ( crg wcel co ringcl syl3anc ) ACLMEBMFBMEFDNBMIJKBCDEFGHOP $. + $} + ${ ringdi.b $e |- B = ( Base ` R ) $. ringdi.p $e |- .+ = ( +g ` R ) $. @@ -252348,6 +252395,17 @@ the additive structure must be abelian (see ~ ringcom ), care must be ringcmn $p |- ( R e. Ring -> R e. CMnd ) $= ( crg wcel cabl ccmn ringabl ablcmn syl ) ABCADCAECAFAGH $. + ${ + ringabld.1 $e |- ( ph -> R e. Ring ) $. + $( A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.) $) + ringabld $p |- ( ph -> R e. Abel ) $= + ( crg wcel cabl ringabl syl ) ABDEBFECBGH $. + + $( A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.) $) + ringcmnd $p |- ( ph -> R e. CMnd ) $= + ( ringabld ablcmnd ) ABABCDE $. + $} + ${ $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 ) ) $. @@ -253081,6 +253139,65 @@ the additive structure must be abelian (see ~ ringcom ), care must be FWQWTXBWBWFHWLULXDVCWFLWOULXNVCVTSTWAWC $. $} + ${ + $d A a b x $. $d B a b x $. $d M a b $. $d R x $. $d T a b $. + $d Y x $. $d ph a b x $. + pwspjmhmmgpd.y $e |- Y = ( R ^s I ) $. + pwspjmhmmgpd.b $e |- B = ( Base ` Y ) $. + pwspjmhmmgpd.m $e |- M = ( mulGrp ` Y ) $. + pwspjmhmmgpd.t $e |- T = ( mulGrp ` R ) $. + pwspjmhmmgpd.r $e |- ( ph -> R e. Ring ) $. + pwspjmhmmgpd.i $e |- ( ph -> I e. V ) $. + pwspjmhmmgpd.a $e |- ( ph -> A e. I ) $. + $( The projection given by ~ pwspjmhm is also a monoid homomorphism between + the respective multiplicative groups. (Contributed by SN, + 30-Jul-2024.) $) + pwspjmhmmgpd $p |- ( ph -> ( x e. B |-> ( x ` A ) ) e. ( M MndHom T ) ) $= + ( cfv wcel wceq va vb cbs cmulr cv cmpt cur mgpbas mgpplusg ringidval crg + eqid cmnd pwsring syl2anc ringmgp syl wa adantr pwselbas ffvelcdmd fmpttd + simpr co cof simprl simprr pwsmulrval fveq1d ffnd inidm eqidd ofval eqtrd + mpidan ringcl syl3an1 3expb fveq1 fvex fvmpt oveq12d 3eqtr4d csn ringidcl + cxp 3syl pws1 fvconst2 3eqtr2d ismhmd ) AUAUBDEUCRZJUDRZEUDRZHFBDCBUEZRZU + FZJUGRZEUGRZDJHMLUHWLEFNWLULZUHJWMHMWMULZUIEWNFNWNULZUIJWRHMWRULZUJEWSFNW + SULZUJAJUKSZHUMSAEUKSZGISZXEOPEGIJKUNUOZJHMUPUQAXFFUMSOEFNUPUQABDWPWLAWOD + SZURZGWLCWOXJWLEGDUKWOJIKWTLAXFXIOUSAXGXIPUSAXIVCUTACGSZXIQUSVAVBAUAUEZDS + ZUBUEZDSZURZURZCXLXNWMVDZRZCXLRZCXNRZWNVDZXRWQRZXLWQRZXNWQRZWNVDXQXSCXLXN + WNVEVDZRZYBXQCXRYFXQDEWMWNXLXNGUKIJKLAXFXPOUSZAXGXPPUSZAXMXOVFZAXMXOVGZXB + XAVHVIAXPXKYGYBTQXQGGXTYAWNGXLXNIICXQGWLXLXQWLEGDUKXLJIKWTLYHYIYJUTVJXQGW + LXNXQWLEGDUKXNJIKWTLYHYIYKUTVJYIYIGVKXQXKURZXTVLYLYAVLVMVOVNXQXRDSZYCXSTA + XMXOYMAXEXMXOYMXHDJWMXLXNLXAVPVQVRBXRWPXSDWQCWOXRVSWQULZCXRVTWAUQXQYDXTYE + YAWNXQXMYDXTTYJBXLWPXTDWQCWOXLVSYNCXLVTWAUQXQXOYEYATYKBXNWPYADWQCWOXNVSYN + CXNVTWAUQWBWCAWRWQRZCWRRZCGWSWDWFZRZWSAXEWRDSYOYPTXHDJWRLXCWEBWRWPYPDWQCW + OWRVSYNCWRVTWAWGACYQWRAXFXGYQWRTOPEWSGIJKXDWHUOVIAXKYRWSTQGWSCEUGVTWIUQWJ + WK $. + $} + + ${ + $d A x $. $d B x $. $d N x $. $d R x $. $d X x $. $d Y x $. + $d .xb x $. $d ph x $. + pwsexpg.y $e |- Y = ( R ^s I ) $. + pwsexpg.b $e |- B = ( Base ` Y ) $. + pwsexpg.m $e |- M = ( mulGrp ` Y ) $. + pwsexpg.t $e |- T = ( mulGrp ` R ) $. + pwsexpg.s $e |- .xb = ( .g ` M ) $. + pwsexpg.g $e |- .x. = ( .g ` T ) $. + pwsexpg.r $e |- ( ph -> R e. Ring ) $. + pwsexpg.i $e |- ( ph -> I e. V ) $. + pwsexpg.n $e |- ( ph -> N e. NN0 ) $. + pwsexpg.x $e |- ( ph -> X e. B ) $. + pwsexpg.a $e |- ( ph -> A e. I ) $. + $( Value of a group exponentiation in a structure power. Compare + ~ pwsmulg . (Contributed by SN, 30-Jul-2024.) $) + pwsexpg $p |- ( ph -> ( ( N .xb X ) ` A ) = ( N .x. ( X ` A ) ) ) $= + ( vx cfv cmpt cmhm wcel cn0 wceq pwspjmhmmgpd mgpbas mhmmulg syl3anc cmnd + co crg pwsring syl2anc ringmgp syl mulgnn0cl fveq1 eqid fvex fvmpt oveq2d + cv 3eqtr3d ) AJLEUQZUECBUEVIZUFZUGZUFZJLVNUFZGUQZBVKUFZJBLUFZGUQAVNIFUHUQ + UIJUJUIZLCUIZVOVQUKAUEBCDFHIKMNOPQTUAUDULUBUCCEGVNIFJLCMIPOUMZRSUNUOAVKCU + IZVOVRUKAIUPUIZVTWAWCAMURUIZWDADURUIHKUIWETUADHKMNUSUTMIPVAVBUBUCCEIJLWBR + VCUOUEVKVMVRCVNBVLVKVDVNVEZBVKVFVGVBAVPVSJGAWAVPVSUKUCUELVMVSCVNBVLLVDWFB + LVFVGVBVHVJ $. + $} + ${ $d p q u v .+ $. $d a b p q u v w x y z ph $. $d a b p q u v w x y z U $. $d p q u x .1. $. $d p q u v w B $. $d a b p q u x y z F $. $d p q R $. @@ -255144,6 +255261,14 @@ nonzero elements form a group under multiplication (from which it ex 3imtr3d 3impib simprd ) BIJZDAJZDEKZLDCMZAJZUJEKZUGUHUIUKULNZUGDBOMZJZ UJUNJZUHUINUMUGBPJZUOUPQBRUQUOUPBUNCDUNSZHUAUCUBABUNDEFURGTABUNUJEFURGTUD UEUF $. + + drnginvrcld.r $e |- ( ph -> R e. DivRing ) $. + drnginvrcld.x $e |- ( ph -> X e. B ) $. + drnginvrcld.1 $e |- ( ph -> X =/= .0. ) $. + $( Closure of the multiplicative inverse in a division ring. ( ~ reccld + analog). (Contributed by SN, 14-Aug-2024.) $) + drnginvrcld $p |- ( ph -> ( I ` X ) e. B ) $= + ( cdr wcel wne cfv drnginvrcl syl3anc ) ACMNEBNEFOEDPBNJKLBCDEFGHIQR $. $} ${ @@ -340460,15 +340585,15 @@ Hilbert space (in the algebraic sense, meaning that all algebraically ( vg vh ccom crn wss caddc c1 cfv co cle wa cr wcel vn cioo cuni cabs cxr cv cmin cseq clt csup covol c2 cdiv wbr cxp cin cn cmap cun simpld simprd wrex crp rphalfcld eqid ovolgelb syl3anc reeanv w3a cmpt 3ad2ant1 simp3ll - simp2l simp3lr simp2r simp3rl simp3rr ovolunlem1 rexlimdvv syl5bir mp2and - cif 3exp ) ABUBHUFZJKUCLZMUDUGJZWDJNUHZKUEUIUJBUKOZDULUMPZMPQUNZRZHQSSUOU - PUQURPZVBZCUBIUFZJKUCLZMWFWNJNUHZKUEUIUJCUKOZWIMPQUNZRZIWLVBZBCUSUKOWHWQM - PDMPQUNZABSLZWHSTZWIVCTZWMAXBXCEUTAXBXCEVAADGVDZBWIWGHWGVEZVFVGACSLZWQSTZ - XDWTAXGXHFUTAXGXHFVAXECWIWPIWPVEZVFVGWMWTRWKWSRZIWLVBHWLVBAXAWKWSHIWLWLVH - AXJXAHIWLWLAWDWLTZWNWLTZRZXJXAAXMXJVIBCDWGWPMWFUAUQUAUFZULUMPZUQTXOWNOXNN - MPULUMPWDOWBVJZJNUHZUAWDWNXPAXMXBXCRXJEVKAXMXGXHRXJFVKAXMDVCTXJGVKXFXIXQV - 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rspccva sylan simprlr - simprll eleq2d cbvrabv ovolicc2lem4 anassrs syl5bir expimpd rexlimddv + simprll eleq2d cbvrabv ovolicc2lem4 anassrs biimtrrid expimpd rexlimddv csn ) ADUAUGZUHZEDUIUJFUKULUMUNUOUSZUAHADHUPZUHUVMUAHUQADEURUJZUVODQADU LUHEULUHDEUOUSZDUVPUHZADKUTAELUTMDEVAVBZVCUADHVDVEAUVLHUHZUVMVFZVFZGGUB UGZVGZCUGZJVHZIVHZVIVHZEUOUSZUWHEVJZUWEUWCVHZUHZCGVMZVFZUBVKZUVNAUWOUWA @@ -341602,10 +341727,10 @@ Hilbert space (in the algebraic sense, meaning that all algebraically ( vol* ` B ) = ( ( vol* ` ( B i^i A ) ) + ( vol* ` ( B \ A ) ) ) ) $= ( vx cvol cdm wcel cr wss covol cfv cin cdif caddc co wceq cpw reex elpw2 wi fveq2d cv wral ismbl fveq2 eleq1d ineq1 difeq1 oveq12d eqeq12d imbi12d - rspccv simplbiim syl5bir 3imp ) ADEFZBGHZBIJZGFZUQBAKZIJZBALZIJZMNZOZUPBG - PZFZUOURVDSZBGQRUOAGHCUAZIJZGFZVIVHAKZIJZVHALZIJZMNZOZSZCVEUBVFVGSCAUCVQV - GCBVEVHBOZVJURVPVDVRVIUQGVHBIUDZUEVRVIUQVOVCVSVRVLUTVNVBMVRVKUSIVHBAUFTVR - VMVAIVHBAUGTUHUIUJUKULUMUN $. + rspccv simplbiim biimtrrid 3imp ) 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(Contributed by w3o cxp cle cr wf wfn wb dyadf ovelrn anbi12d mp2b reeanv bitr4i ad2antrl ffn nn0re ad2antll dyaddisjlem ancom anbi12i sylanb wo orcom incom eqeq1i orbi12i df-3or 3bitr4i sylib lecasei simpl fveq2d simpr sseq12d 3orbi123d - wbr ineq12d eqeq1d syl5ibrcom rexlimdvva syl5bir rexlimivv sylbi ) CEUAZK - ZDWIKZLZCGUBZHUBZEUCZMZHNOZDIUBZJUBZEUCZMZJNOZLZIPOGPOZCQRZDQRZUDZXFXEUDZ - CSRZDSRZUEZTMZUFZWLWQGPOZXBIPOZLZXDPNUGZUHUIUIUGUEZEUJEXQUKZWLXPULABEFUMX - QXREUTXSWJXNWKXOGHPNCEUNIJPNDEUNUOUPWQXBGIPPUQURXCXMGIPPXCWPXALZJNOHNOWMP - KZWRPKZLZXMWPXAHJNNUQYCXTXMHJNNYCWNNKZWSNKZLZLZXMXTWOQRZWTQRZUDZYIYHUDZWO - SRZWTSRZUEZTMZUFZYGYPWNWSYDWNUIKYCYEWNVAUSYEWSUIKYCYDWSVAVBABWMWRWNWSEFVC - YGWSWNUHWAZLYKYJYMYLUEZTMZUFZYPYGYBYALZYEYDLZLYQYTYCUUAYFUUBYAYBVDYDYEVDV - EABWRWMWSWNEFVCVFYKYJVGZYSVGYJYKVGZYOVGYTYPUUCUUDYSYOYKYJVHYRYNTYMYLVIVJV - KYKYJYSVLYJYKYOVLVMVNVOXTXGYJXHYKXLYOXTXEYHXFYIXTCWOQWPXAVPZVQZXTDWTQWPXA - VRZVQZVSXTXFYIXEYHUUHUUFVSXTXKYNTXTXIYLXJYMXTCWOSUUEVQXTDWTSUUGVQWBWCVTWD - WEWFWGWH $. + wbr ineq12d eqeq1d syl5ibrcom rexlimdvva biimtrrid rexlimivv sylbi ) CEUA + ZKZDWIKZLZCGUBZHUBZEUCZMZHNOZDIUBZJUBZEUCZMZJNOZLZIPOGPOZCQRZDQRZUDZXFXEU + DZCSRZDSRZUEZTMZUFZWLWQGPOZXBIPOZLZXDPNUGZUHUIUIUGUEZEUJEXQUKZWLXPULABEFU + MXQXREUTXSWJXNWKXOGHPNCEUNIJPNDEUNUOUPWQXBGIPPUQURXCXMGIPPXCWPXALZJNOHNOW + MPKZWRPKZLZXMWPXAHJNNUQYCXTXMHJNNYCWNNKZWSNKZLZLZXMXTWOQRZWTQRZUDZYIYHUDZ + WOSRZWTSRZUEZTMZUFZYGYPWNWSYDWNUIKYCYEWNVAUSYEWSUIKYCYDWSVAVBABWMWRWNWSEF + VCYGWSWNUHWAZLYKYJYMYLUEZTMZUFZYPYGYBYALZYEYDLZLYQYTYCUUAYFUUBYAYBVDYDYEV + DVEABWRWMWSWNEFVCVFYKYJVGZYSVGYJYKVGZYOVGYTYPUUCUUDYSYOYKYJVHYRYNTYMYLVIV + JVKYKYJYSVLYJYKYOVLVMVNVOXTXGYJXHYKXLYOXTXEYHXFYIXTCWOQWPXAVPZVQZXTDWTQWP + XAVRZVQZVSXTXFYIXEYHUUHUUFVSXTXKYNTXTXIYLXJYMXTCWOSUUEVQXTDWTSUUGVQWBWCVT + WDWEWFWGWH $. ${ dyadmax.2 $e |- ( ph -> A e. ZZ ) $. @@ -343498,27 +343624,27 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cz inss2 sstri sstrdi eleq2 rexima sylancr crab c0 ssrab2 adantr sstrid wral wne simprl ssid fveq2 sseq2d rspcev sylancl sylibr dyadmax syl2anc rabn0 elrab simprlr simplrr sseldd simprll imbi1i impexp bitri ad2antll - sstr2 ancrd imim1d syl5bir imim2d biimtrid ralimdv2 impr sseq1d equequ1 - imbi12d ralbidv elrab2 sylanbrc wfun cdm ffun ssrab3 sseqtrrdi ad2antrr - fdmi funfvima2 mpd elunii exp32 rexlimdv rexlimdvaa sylbid ssrdv imass2 - uniss mp1i eqssd ) ANFUBZUCZNHUBZUCZALYLYNLOZYLPYOUAOZPZUAYKUDZAYOYNPZU - AYOYKUEAYRYOUIOZNUFZPZUIFUDZYSANUGUGUHZUJZFUUDQYRUUCUKUUDUGULZNUMZUUEUN - UUDUUFNUOUPAFGUQZUUDKUUHURUSUSUHZUTZUUDVEVAUHZUUJGUMUUHUUJQBCGIVBUUKUUJ - GVCUPUUJUUIUUDURUUIVFVDVGVGVHZYQUUBUAUIUUDFNYPUUAYOVIVJVKAUUBYSUIFAYTFP - ZUUBRZRZMOZNUFZEOZNUFZQZMESZTZEUUAYONUFZQZLFVLZVQZMUVEUDZYSUUOUVEUUHQUV - EVMVRZUVGUUOUVEFUUHUVDLFVNAFUUHQUUNKVOVPUUOUVDLFUDZUVHUUOUUMUUAUUAQZUVI - AUUMUUBVSUUAVTUVDUVJLYTFLUISUVCUUAUUAYOYTNWAWBWCWDUVDLFWHWEBCMEUVEGIWFW - GUUOUVFYSMUVEUUPUVEPUUPFPZUUAUUQQZRZUUOUVFYSTUVDUVLLUUPFLMSUVCUUQUUAYOU - UPNWAWBWIUUOUVMUVFYSUUOUVMUVFRZRZYOUUQPUUQYMPZYSUVOUUAUUQYOUUOUVKUVLUVF - WJAUUMUUBUVNWKWLUVOUUPHPZUVPUVOUVKUVBEFVQZUVQUUOUVKUVLUVFWMUUOUVMUVFUVR - UUOUVMRZUVBUVBEUVEFUURUVEPZUVBTZUURFPZUUAUUSQZUVBTZTZUVSUWBUVBTUWAUWBUW - CRZUVBTUWEUVTUWFUVBUVDUWCLUURFLESUVCUUSUUAYOUURNWAWBWIWNUWBUWCUVBWOWPUV - SUWDUVBUWBUWDUWCUUTRZUVATUVSUVBUWCUUTUVAWOUVSUUTUWGUVAUVSUUTUWCUVLUUTUW - CTUUOUVKUUAUUQUUSWRWQWSWTXAXBXCXDXEDOZNUFZUUSQZDESZTZEFVQZUVRDUUPFHDMSZ - UWLUVBEFUWNUWJUUTUWKUVAUWNUWIUUQUUSUWHUUPNWAXFDMEXGXHXIJXJXKUVONXLZHNXM - ZQZUVQUVPTUUGUWOUNUUDUUFNXNUPAUWQUUNUVNAHUUDUWPAHFUUDUWMDFHJXOZUULVPUUD - UUFNUNXRXPXQHUUPNXSVKXTYOUUQYMYAWGYBXCYCXTYDYEXCYFYMYKQZYNYLQAHFQUWSUWR - HFNYGUPYMYKYHYIYJ $. + sstr2 ancrd imim1d imim2d biimtrid ralimdv2 impr sseq1d equequ1 imbi12d + biimtrrid ralbidv elrab2 sylanbrc wfun cdm ffun fdmi sseqtrrdi ad2antrr + ssrab3 funfvima2 mpd elunii exp32 rexlimdv rexlimdvaa sylbid ssrdv mp1i + imass2 uniss eqssd ) ANFUBZUCZNHUBZUCZALYLYNLOZYLPYOUAOZPZUAYKUDZAYOYNP + ZUAYOYKUEAYRYOUIOZNUFZPZUIFUDZYSANUGUGUHZUJZFUUDQYRUUCUKUUDUGULZNUMZUUE + UNUUDUUFNUOUPAFGUQZUUDKUUHURUSUSUHZUTZUUDVEVAUHZUUJGUMUUHUUJQBCGIVBUUKU + UJGVCUPUUJUUIUUDURUUIVFVDVGVGVHZYQUUBUAUIUUDFNYPUUAYOVIVJVKAUUBYSUIFAYT + FPZUUBRZRZMOZNUFZEOZNUFZQZMESZTZEUUAYONUFZQZLFVLZVQZMUVEUDZYSUUOUVEUUHQ + UVEVMVRZUVGUUOUVEFUUHUVDLFVNAFUUHQUUNKVOVPUUOUVDLFUDZUVHUUOUUMUUAUUAQZU + VIAUUMUUBVSUUAVTUVDUVJLYTFLUISUVCUUAUUAYOYTNWAWBWCWDUVDLFWHWEBCMEUVEGIW + FWGUUOUVFYSMUVEUUPUVEPUUPFPZUUAUUQQZRZUUOUVFYSTUVDUVLLUUPFLMSUVCUUQUUAY + OUUPNWAWBWIUUOUVMUVFYSUUOUVMUVFRZRZYOUUQPUUQYMPZYSUVOUUAUUQYOUUOUVKUVLU + VFWJAUUMUUBUVNWKWLUVOUUPHPZUVPUVOUVKUVBEFVQZUVQUUOUVKUVLUVFWMUUOUVMUVFU + VRUUOUVMRZUVBUVBEUVEFUURUVEPZUVBTZUURFPZUUAUUSQZUVBTZTZUVSUWBUVBTUWAUWB + UWCRZUVBTUWEUVTUWFUVBUVDUWCLUURFLESUVCUUSUUAYOUURNWAWBWIWNUWBUWCUVBWOWP + UVSUWDUVBUWBUWDUWCUUTRZUVATUVSUVBUWCUUTUVAWOUVSUUTUWGUVAUVSUUTUWCUVLUUT + UWCTUUOUVKUUAUUQUUSWRWQWSWTXHXAXBXCXDDOZNUFZUUSQZDESZTZEFVQZUVRDUUPFHDM + SZUWLUVBEFUWNUWJUUTUWKUVAUWNUWIUUQUUSUWHUUPNWAXEDMEXFXGXIJXJXKUVONXLZHN + XMZQZUVQUVPTUUGUWOUNUUDUUFNXNUPAUWQUUNUVNAHUUDUWPAHFUUDUWMDFHJXRZUULVPU + UDUUFNUNXOXPXQHUUPNXSVKXTYOUUQYMYAWGYBXBYCXTYDYEXBYFYMYKQZYNYLQAHFQUWSU + WRHFNYHUPYMYKYIYGYJ $. $( Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.) $) @@ -345685,24 +345811,24 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Mario Carneiro, 19-Jun-2014.) $) @@ -346274,18 +346400,18 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by @@ -348038,12 +348165,12 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Stefan O'Rear, 29-Mar-2015.) (New usage is discouraged.) @@ -357047,30 +357174,30 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cn wfun mptexd funmpt cres difss resmpt ax-mp eqsstrri mptexg funsssuppss resss mp3an12i fsuppsssupp gsumsubmcl cmnd ringmnd simprl ffvelcdmd fveq2 syl22anc gsumsn syl3anc sylib elnn0 orel2 sylc eqeltrd nn0nnaddcl syl2anc - wo simprr nnne0d eqnetrd expr syl5bir rexlimdva necon4bd wfn wb ffnd 0nn0 - fnconstg eqfnfv c0ex fvconst2 eqeq2d ralbiia bitrdi sylibrd syl fconstmpt - psrbag0 oveq2i gsumz mpan eqtrid eqtrd fveqeq2 syl5ibrcom impbid ) EFLZGA - LZUAZGDMZNOZGENUBUCZOZUVEUVGJUGZGMZNOZJEUDZUVIUVEUVMUVFNUVMUHUVLUHZJEUEUV - EUVFNUFZUVLJEUIUVEUVNUVOJEUVNUVKNUFZUVEUVJELZUAUVOUVKNUJZUVEUVQUVPUVOUVEU - VQUVPUAZUAZUVFPKEUVJUBZUKZKUGZGMZULZQRZPKUWAUWDULQRZUMRZNUVTUVFPGQRZPKEUW - DULZQRUWHUVDUVFUWIOUVCUVSBGPBUGZQRZUWIADUWKGPQUNIPGQUOUSUPUVTGUWJPQUVEGUW - JOUVSUVEKESGACGEFHUQZURUTZVAUVTEVJUWBUWAUMKPFUWDNVBVCVDPVELZPVFLUVTVGPVHV - IZUVCUVDUVSVKUVTUWCELZUAUWDUVTESUWCGUVEESGVTZUVSUWMUTZVLVMUVTGUWJNVNUWNUV - EGNVNVOZUVSUVDUVCUWTACEFGHVPVQUTVRZUWBUWAVSZWAOUVTUXBUWAUWBVSWAUWBUWAWBUW - AEWCWDWEUVQEUWBUWAWFZOUVEUVPUVQUXCEEUVJWGWHWLWIWJUVTUWHUVTUWFSLUWGXBLUWHX - BLUVTUWBSUWEPTNVCUWPUVCUWBTLUVDUVSEUWAFWKZWMSPWNMLUVTWOWEUVTKUWBUWDSUVTUW - RUWQUWDSLUWCUWBLUWSUWCEUWAWPESUWCGWQWRWSUVTUWETLZUWEXCZUWJNVNVOUWENWTRUWJ - NWTRXAZUWENVNVOUVCUXEUVDUVSUVCKUWBUWDTUXDXDWMUXFUVTKUWBUWDXEWEUXAUWJXCUWE - UWJXAUVTUWJTLZUXGKEUWDXEUWEUWJUWBXFZUWJUWBEXAUXIUWEOEUWAXGKEUWBUWDXHXIUWJ - UWBXMXJUVCUXHUVDUVSKEUWDFXKWMUWEUWJTNXLXNUWJUWETNXOYBXPUVTUWGUVKXBUVTPXQL - ZUVQUVKVJLUWGUVKOUWOUXJUVTVGPXRZVIUVEUVQUVPXSZUVTUVKUVTESUVJGUWSUXLXTZVMU - WDVJUVKKPUVJEVBUWCUVJGYAYCYDUVTUVNUVKXBLZUVLYLZUXNUVTUVPUVNUVEUVQUVPYMUVR - YEUVTUVKSLUXOUXMUVKYFYEUVLUXNYGYHYIUWFUWGYJYKYNYOYPYQYRYQYSUVEUVIUVKUVJUV - HMZOZJEUDZUVMUVEGEYTUVHEYTZUVIUXRUUAUVEESGUWMUUBNSLUXSUVEUUCENSUUDVIJEGUV - HUUEYKUXQUVLJEUVQUXPNUVKENUVJUUFUUGUUHUUIUUJUUKUVEUVGUVIUVHDMZNOUVEUXTPUV - HQRZNUVEUVHALZUXTUYAOUVCUYBUVDACEFHUUNUTBUVHUWLUYAADUWKUVHPQUNIPUVHQUOUSU - ULUVEUYAPJENULZQRZNUVHUYCPQJENUUMUUOUVCUYDNOZUVDUXJUVCUYEUWOUXJVGUXKXIEJP - FNVCUUPUUQUTUURUUSGUVHNDUUTUVAUVB $. + wo simprr nnne0d eqnetrd expr biimtrrid rexlimdva necon4bd wfn wb ralbiia + ffnd 0nn0 fnconstg eqfnfv c0ex fvconst2 eqeq2d bitrdi sylibrd psrbag0 syl + fconstmpt oveq2i gsumz mpan eqtrid eqtrd fveqeq2 syl5ibrcom impbid ) EFLZ + GALZUAZGDMZNOZGENUBUCZOZUVEUVGJUGZGMZNOZJEUDZUVIUVEUVMUVFNUVMUHUVLUHZJEUE + UVEUVFNUFZUVLJEUIUVEUVNUVOJEUVNUVKNUFZUVEUVJELZUAUVOUVKNUJZUVEUVQUVPUVOUV + EUVQUVPUAZUAZUVFPKEUVJUBZUKZKUGZGMZULZQRZPKUWAUWDULQRZUMRZNUVTUVFPGQRZPKE + UWDULZQRUWHUVDUVFUWIOUVCUVSBGPBUGZQRZUWIADUWKGPQUNIPGQUOUSUPUVTGUWJPQUVEG + UWJOUVSUVEKESGACGEFHUQZURUTZVAUVTEVJUWBUWAUMKPFUWDNVBVCVDPVELZPVFLUVTVGPV + HVIZUVCUVDUVSVKUVTUWCELZUAUWDUVTESUWCGUVEESGVTZUVSUWMUTZVLVMUVTGUWJNVNUWN + UVEGNVNVOZUVSUVDUVCUWTACEFGHVPVQUTVRZUWBUWAVSZWAOUVTUXBUWAUWBVSWAUWBUWAWB + UWAEWCWDWEUVQEUWBUWAWFZOUVEUVPUVQUXCEEUVJWGWHWLWIWJUVTUWHUVTUWFSLUWGXBLUW + HXBLUVTUWBSUWEPTNVCUWPUVCUWBTLUVDUVSEUWAFWKZWMSPWNMLUVTWOWEUVTKUWBUWDSUVT + UWRUWQUWDSLUWCUWBLUWSUWCEUWAWPESUWCGWQWRWSUVTUWETLZUWEXCZUWJNVNVOUWENWTRU + WJNWTRXAZUWENVNVOUVCUXEUVDUVSUVCKUWBUWDTUXDXDWMUXFUVTKUWBUWDXEWEUXAUWJXCU + WEUWJXAUVTUWJTLZUXGKEUWDXEUWEUWJUWBXFZUWJUWBEXAUXIUWEOEUWAXGKEUWBUWDXHXIU + WJUWBXMXJUVCUXHUVDUVSKEUWDFXKWMUWEUWJTNXLXNUWJUWETNXOYBXPUVTUWGUVKXBUVTPX + QLZUVQUVKVJLUWGUVKOUWOUXJUVTVGPXRZVIUVEUVQUVPXSZUVTUVKUVTESUVJGUWSUXLXTZV + MUWDVJUVKKPUVJEVBUWCUVJGYAYCYDUVTUVNUVKXBLZUVLYLZUXNUVTUVPUVNUVEUVQUVPYMU + VRYEUVTUVKSLUXOUXMUVKYFYEUVLUXNYGYHYIUWFUWGYJYKYNYOYPYQYRYQYSUVEUVIUVKUVJ + UVHMZOZJEUDZUVMUVEGEYTUVHEYTZUVIUXRUUAUVEESGUWMUUCNSLUXSUVEUUDENSUUEVIJEG + UVHUUFYKUXQUVLJEUVQUXPNUVKENUVJUUGUUHUUIUUBUUJUUKUVEUVGUVIUVHDMZNOUVEUXTP + UVHQRZNUVEUVHALZUXTUYAOUVCUYBUVDACEFHUULUTBUVHUWLUYAADUWKUVHPQUNIPUVHQUOU + SUUMUVEUYAPJENULZQRZNUVHUYCPQJENUUNUUOUVCUYDNOZUVDUXJUVCUYEUWOUXJVGUXKXIE + JPFNVCUUPUUQUTUURUUSGUVHNDUUTUVAUVB $. $} ${ @@ -358226,16 +358353,16 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vx vy cdomn wcel cnzr cv cfv co wceq wral syl wne cn0 eqid crg syl3anc wa cmulr c0g wo wi domnnzr ply1nz wn neanior cdg1 caddc domnring ad2antrr cbs crlreg simplrl simprl cco1 simpll deg1ldgdomn simplrr simprr deg1mul2 - deg1nn0cl nn0addcld eqeltrd wb ply1ring ringcl deg1nn0clb syl2anc syl5bir - mpbird ex necon4bd ralrimivva isdomn sylanbrc ) BFGZAHGZDIZEIZAUAJZKZAUBJ - ZLVTWDLWAWDLUCZUDZEAUMJZMDWGMAFGVRBHGVSBUEABCUFNVRWFDEWGWGVRVTWGGZWAWGGZT - ZTZWEWCWDWEUGVTWDOZWAWDOZTZWKWCWDOZVTWDWAWDUHWKWNWOWKWNTZWOWCBUIJZJZPGZWP - WRVTWQJZWAWQJZUJKPWPWGWQABWBBUNJZVTWAWDWQQZCXBQZWGQZWBQZWDQZVRBRGZWJWNBUK - ZULZVRWHWIWNUOZWKWLWMUPZWPVRWHWLWTVTUQJZJXBGVRWJWNURXKXLXMWGWQABXBVTWDXCC - XGXEXDXMQUSSVRWHWIWNUTZWKWLWMVAZVBWPWTXAWPXHWHWLWTPGXJXKXLWGWQABVTWDXCCXG - XEVCSWPXHWIWMXAPGXJXNXOWGWQABWAWDXCCXGXEVCSVDVEWPXHWCWGGZWOWSVFXJWPARGZWH - WIXPVRXQWJWNVRXHXQXIABCVGNULXKXNWGAWBVTWAXEXFVHSWGWQABWCWDXCCXGXEVIVJVLVM - VKVNVODEWGAWBWDXEXFXGVPVQ $. + deg1nn0cl nn0addcld eqeltrd wb ringcl deg1nn0clb syl2anc mpbird biimtrrid + ply1ring ex necon4bd ralrimivva isdomn sylanbrc ) BFGZAHGZDIZEIZAUAJZKZAU + BJZLVTWDLWAWDLUCZUDZEAUMJZMDWGMAFGVRBHGVSBUEABCUFNVRWFDEWGWGVRVTWGGZWAWGG + ZTZTZWEWCWDWEUGVTWDOZWAWDOZTZWKWCWDOZVTWDWAWDUHWKWNWOWKWNTZWOWCBUIJZJZPGZ + WPWRVTWQJZWAWQJZUJKPWPWGWQABWBBUNJZVTWAWDWQQZCXBQZWGQZWBQZWDQZVRBRGZWJWNB + UKZULZVRWHWIWNUOZWKWLWMUPZWPVRWHWLWTVTUQJZJXBGVRWJWNURXKXLXMWGWQABXBVTWDX + CCXGXEXDXMQUSSVRWHWIWNUTZWKWLWMVAZVBWPWTXAWPXHWHWLWTPGXJXKXLWGWQABVTWDXCC + XGXEVCSWPXHWIWMXAPGXJXNXOWGWQABWAWDXCCXGXEVCSVDVEWPXHWCWGGZWOWSVFXJWPARGZ + WHWIXPVRXQWJWNVRXHXQXIABCVLNULXKXNWGAWBVTWAXEXFVGSWGWQABWCWDXCCXGXEVHVIVJ + VMVKVNVODEWGAWBWDXEXFXGVPVQ $. $( The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.) $) @@ -359265,16 +359392,16 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cv wral w3a cdomn isidom simplbi simprbi domnnzr syl simpl eldifsn adantl wne fta1g ralrimiva 3jca simp1 cmulr co wo wi cbs simp2 wn df-ne cv1 eqid cvsca simpll1 simplrl simplrr simprl simprr simpll3 cnveqd imaeq1d fveq2d - fveq2 breq12d rspccv fta1blem expr syl5bir orrd ex isdomn sylanbrc impbii - ralrimivva ) DUAQZDUBQZDUCQZEUKZFRZUDZGUEZUFZUGRZXCBRZUHUIZEAHUEUJZULZUMZ - WTXAXBXLWTXADUNQZDUOZUPWTXNXBWTXAXNXOUQDURUSWTXJEXKWTXCXKQZSABCDXCFGHIJKL - MNWTXPUTXPXCAQZWTXPXQXCHVCZXCAHVAZUPVBXPXRWTXPXQXRXSUQVBVDVEVFXMXAXNWTXAX - BXLVGXMXBOUKZPUKZDVHRZVIGTZXTGTZYAGTZVJZVKZPDVLRZULOYHULXNXAXBXLVMXMYGOPY - HYHXMXTYHQZYAYHQZSZSZYCYFYLYCSZYDYEYDVNXTGVCZYMYEXTGVOYLYCYNYEYLYCYNSZSZA - BCDCVRRZYBYHXTYAFGDVPRZHIJKLMNYHVQZYBVQZYRVQYQVQXAXBXLYKYOVSXMYIYJYOVTXMY - IYJYOWAYLYCYNWBYLYCYNWCYPXLXTYRYQVIZXKQUUAFRZUDZXFUFZUGRZUUABRZUHUIZVKXAX - BXLYKYOWDXJUUGEUUAXKXCUUATZXHUUEXIUUFUHUUHXGUUDUGUUHXEUUCXFUUHXDUUBXCUUAF - WHWEWFWGXCUUABWHWIWJUSWKWLWMWNWOWSOPYHDYBGYSYTMWPWQXOWQWR $. + breq12d rspccv fta1blem expr biimtrrid orrd ex ralrimivva isdomn sylanbrc + fveq2 impbii ) DUAQZDUBQZDUCQZEUKZFRZUDZGUEZUFZUGRZXCBRZUHUIZEAHUEUJZULZU + MZWTXAXBXLWTXADUNQZDUOZUPWTXNXBWTXAXNXOUQDURUSWTXJEXKWTXCXKQZSABCDXCFGHIJ + KLMNWTXPUTXPXCAQZWTXPXQXCHVCZXCAHVAZUPVBXPXRWTXPXQXRXSUQVBVDVEVFXMXAXNWTX + AXBXLVGXMXBOUKZPUKZDVHRZVIGTZXTGTZYAGTZVJZVKZPDVLRZULOYHULXNXAXBXLVMXMYGO + PYHYHXMXTYHQZYAYHQZSZSZYCYFYLYCSZYDYEYDVNXTGVCZYMYEXTGVOYLYCYNYEYLYCYNSZS + ZABCDCVRRZYBYHXTYAFGDVPRZHIJKLMNYHVQZYBVQZYRVQYQVQXAXBXLYKYOVSXMYIYJYOVTX + MYIYJYOWAYLYCYNWBYLYCYNWCYPXLXTYRYQVIZXKQUUAFRZUDZXFUFZUGRZUUABRZUHUIZVKX + AXBXLYKYOWDXJUUGEUUAXKXCUUATZXHUUEXIUUFUHUUHXGUUDUGUUHXEUUCXFUUHXDUUBXCUU + AFWRWEWFWGXCUUABWRWHWIUSWJWKWLWMWNWOOPYHDYBGYSYTMWPWQXOWQWS $. $} ${ @@ -360258,19 +360385,19 @@ or are almost disjoint (the interiors are disjoint). 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S /\ y e. S ) ) -> ( x x. y ) e. S ) $. $( The product of two polynomials is a polynomial. (Contributed by Mario @@ -360280,19 +360407,19 @@ or are almost disjoint (the interiors are disjoint). 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S ) $. $( The difference of two polynomials is a polynomial. (Contributed by @@ -360428,24 +360555,24 @@ or are almost disjoint (the interiors are disjoint). 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This is one of the primary interests of the Möbius function as an arithmetic @@ -382559,21 +382686,21 @@ particular proof approach is due to Cauchy (1821). 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(Contributed by Mario Carneiro, 28-Apr-2016.) $) @@ -384606,21 +384733,21 @@ particular proof approach is due to Cauchy (1821). 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B ) $. @@ -384675,32 +384802,32 @@ particular proof approach is due to Cauchy (1821). 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YTVWRUYKUYSVWPVUDUYKVWKVXBUYJUVJUYIUBQZRVXBUYIUVJVXBVXGVXHVXJUVDZVXLYSV - XBVWBVXHVXFVXSUYIVUBUPUQZVYBRSVXLVYCVXKVYAVXBVXIVYDVXJUXPUYIYRXBUVJUYIV - UBYEYFVEYQVWPVUDUYGUYTUVAYTUVBYNUVCUVEUVFUVGUVHUVIYT $. + simprd expcom a2d biimtrrid prmind mpcom ) AFOGPQZUBQZRSZFGTQZGFTQZPQZR + UCZFRUDQZOUEQZGRUDQZOUEQZPQZUFQZSZAUVKFGUBQZRAFUKUGZOUKUGZGUKUGZFOUBQZR + SUVKUWDSHUWFAUHUIJAUWHOFUBQZRAFULUGZOULUGZUWHUWISAFHUJZUMFOUNUOAOFUPUQU + RZUWIRSZIAOUSUGZUWJUWMUWNUTVAUWLOFVBVCVDVEFOGVFVGLVEUWEAUVLUWCVHZHAEVIZ + UVJUBQZRSZUWQGTQZGUWQTQZPQZUVPUWQRUDQZOUEQZUVTPQZUFQZSZVHZVHAUWSRGTQZGR + TQZPQZUVPVJUVTPQZUFQZSZVHZVHACVIZUVJUBQZRSZUXPGTQZGUXPTQZPQZUVPUXPRUDQZ + OUEQZUVTPQZUFQZSZVHZVHZAUAVIZUVJUBQZRSZUYIGTQZGUYITQZPQZUVPUYIRUDQZOUEQ + ZUVTPQZUFQZSZVHZVHZAUXPUYIPQZUVJUBQZRSZVUBGTQZGVUBTQZPQZUVPVUBRUDQZOUEQ + ZUVTPQZUFQZSZVHZVHZAUWPVHECUAFUWQRSZUXHUXOAVUOUXGUXNUWSVUOUXBUXKUXFUXMV + UOUWTUXIUXAUXJPUWQRGTVKUWQRGTVLVMVUOUXEUXLUVPUFVUOUXDVJUVTPVUOUXDVJOUEQ + VJVUOUXCVJOUEVUOUXCRRUDQVJUWQRRUDVKVQVNVOOVPVRVSVNVOVTWAWBWBECWCZUXHUYG + AVUPUWSUXRUXGUYFVUPUWRUXQRUWQUXPUVJUBVKWDVUPUXBUYAUXFUYEVUPUWTUXSUXAUXT + PUWQUXPGTVKUWQUXPGTVLVMVUPUXEUYDUVPUFVUPUXDUYCUVTPVUPUXCUYBOUEUWQUXPRUD + VKVOVOVTWAWEWBEUAWCZUXHUYTAVUQUWSUYKUXGUYSVUQUWRUYJRUWQUYIUVJUBVKWDVUQU + XBUYNUXFUYRVUQUWTUYLUXAUYMPUWQUYIGTVKUWQUYIGTVLVMVUQUXEUYQUVPUFVUQUXDUY + PUVTPVUQUXCUYOOUEUWQUYIRUDVKVOVOVTWAWEWBUWQVUBSZUXHVUMAVURUWSVUDUXGVULV + URUWRVUCRUWQVUBUVJUBVKWDVURUXBVUGUXFVUKVURUWTVUEUXAVUFPUWQVUBGTVKUWQVUB + GTVLVMVURUXEVUJUVPUFVURUXDVUIUVTPVURUXCVUHOUEUWQVUBRUDVKVOVOVTWAWEWBUWQ + FSZUXHUWPAVUSUWSUVLUXGUWCVUSUWRUVKRUWQFUVJUBVKWDVUSUXBUVOUXFUWBVUSUWTUV + MUXAUVNPUWQFGTVKUWQFGTVLVMVUSUXEUWAUVPUFVUSUXDUVRUVTPVUSUXCUVQOUEUWQFRU + DVKVOVOVTWAWEWBAUXNUWSARRPQZUVPVJUFQZUXKUXMVUTRVVAWFUVPWGUGVVARSWIUVPWH + WJWKAUXIRUXJRPAUXIROUFQZGTQZRVVBRGTWLWMRULUGZRVJWNZWOAGULUGZRGUBQRSZVVC + RSVVDVVEXAWPWQAGJUJZAVVFVVGVVHGWRXBRGWSWTXCAUXJGVVBTQZRVVBRGTWLXDAVVFRU + KUGZGRUBQRSZVVIRSVVHVVJAXEUIAVVFVVKVVHGXFXBGRXGXHXCVMAUXLVJUVPUFAUVTAUV + SAUVSAUWGUVSXIUGJGXJXBXKXMXLVTXNYCAUWQUSUGZUXHAVVLUWSUXGAVVLUWSWOZUWQUS + OYDXOUGZUWQGUBQZRSZWOUXGAVVMWOZVVNVVPVVQVVLUWQOWNZVVNAVVLUWSXPZVVQUWQOU + BQRSZVVRVVQUWQULUGZUWKUVJULUGZUWSOUVJUPUQZVVTVVLVWAAUWSUWQXQXRUWKVVQUMU + IVVQUWKVVFVWBUMVVQGAUWGVVMJXSZUJZOGXTZVCAVVLUWSYAZVVQUWKVVFVWCUMVWEOGYB + ZVCUWQOUVJYEYFZVVQVVLUWOVVTVVRUTVVSVAUWQOYGUOVDUWQUSOYHYIVVQUWRVVORVVQU + WQUKUGZUWFUWGVVTUWRVVOSVVLVWJAUWSUWQYJXRUWFVVQUHUIVWDVWIUWQOGVFVGVWGYKY + LMUUAYNUUNUYHVUAWOAUYGUYTWOZVHUXPOUUBUUCZUGZUYIVWLUGZWOZVUNAUYGUYTUUDVW + OAVWKVUMAVWOVWKVUMVHAVWOWOZVUDVWKVULVWPVUDVWKVULVWPVUDVWKWOZWOZUXPUYIVU + BGVWRUXPUYIVWRVWMUXPUKUGAVWMVWNVWQUUEUXPYMXBZVWRVWNUYIUKUGAVWMVWNVWQUUF + UYIYMXBZUUGVWPVUDOVUBUPUQZURVWKVWPVUDWOZVXAORUPUQZUUHVXBVXAOVUCUPUQZVXC + VXBVXAVWCVXDVXBUWKVVFVWCUMAVVFVWOVUDVVHYOZVWHVCUWKVXBVUBULUGZVWBVXAVWCW + OVXDVHUMVXBUXPULUGZUYIULUGZWOZVXFVWOVXIAVUDVWMVXGVWNVXHOUXPYPOUYIYPUUIU + UJZUXPUYIXTXBZVXBUWKVVFVWBUMVXEVWFVCZOVUBUVJUUKWTUULVXBVUCROUPVWPVUDUUO + ZUUMUUPUUQYQAUWGVWOVWQJYOAOGUPUQURVWOVWQKYOVWPVUDVUBGUBQRSZVWKVXBVXFVVF + VWBVUDGUVJUPUQZVXNVXKVXEVXLVXMVXBUWKVVFVXOUMVXEOGYRVCVUBGUVJYEYFYQVWSVW + TVWRVUBUURVWRUXRUYFVWPVUDUXRVWKVXBUXQUVJUXPUBQZRVXBUXPUVJVXBVXGVXHVXJUU + SZVXLYSVXBVWBVXGVXFUVJVUBUBQZRSZUXPVUBUPUQZVXPRSVXLVXQVXKVXBVXRVUCRVXBU + VJVUBVXLVXKYSVXMVEZVXBVXIVXTVXJUXPUYIYBXBUVJUXPVUBYEYFVEYQVWPVUDUYGUYTU + UTYTVWRUYKUYSVWPVUDUYKVWKVXBUYJUVJUYIUBQZRVXBUYIUVJVXBVXGVXHVXJUVDZVXLY + SVXBVWBVXHVXFVXSUYIVUBUPUQZVYBRSVXLVYCVXKVYAVXBVXIVYDVXJUXPUYIYRXBUVJUY + IVUBYEYFVEYQVWPVUDUYGUYTUVAYTUVBYNUVCUVEUVFUVGUVHUVIYT $. $} $( Extend ~ lgsquad to coprime odd integers (the domain of the Jacobi @@ -389223,12 +389350,12 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). 2sqlem5 $p |- ( ph -> N e. S ) $= ( vp vq vx vy cv co cz wrex wcel wa cexp caddc wceq cmul 2sqlem2 reeanv c2 sylib cprime simplrr simprlr simplrl simprll simprrr simprrl 2sqlem4 - cn ad2antrr expr rexlimdvva syl5bir mp2and ) ACKOZUGUAPLOZUGUAPUBPUCZLQ - RZKQRZECUDPZMOZUGUAPNOZUGUAPUBPUCZNQRZMQRZEDSZACDSVGJKLBCDFUEUHAVHDSVMI - MNBVHDFUEUHVGVMTVFVLTZMQRKQRAVNVFVLKMQQUFAVOVNKMQQVOVEVKTZNQRLQRAVCQSZV - IQSZTZTZVNVEVKLNQQUFVTVPVNLNQQVTVDQSZVJQSZTZVPVNVTWCVPTZTBVIVJVCVDCDEFA - EUQSVSWDGURACUISVSWDHURAVQVRWDUJVTWAWBVPUKAVQVRWDULVTWAWBVPUMVTWCVEVKUN - VTWCVEVKUOUPUSUTVAUTVAVB $. + cn ad2antrr expr rexlimdvva biimtrrid mp2and ) ACKOZUGUAPLOZUGUAPUBPUCZ + LQRZKQRZECUDPZMOZUGUAPNOZUGUAPUBPUCZNQRZMQRZEDSZACDSVGJKLBCDFUEUHAVHDSV + MIMNBVHDFUEUHVGVMTVFVLTZMQRKQRAVNVFVLKMQQUFAVOVNKMQQVOVEVKTZNQRLQRAVCQS + ZVIQSZTZTZVNVEVKLNQQUFVTVPVNLNQQVTVDQSZVJQSZTZVPVNVTWCVPTZTBVIVJVCVDCDE + FAEUQSVSWDGURACUISVSWDHURAVQVRWDUJVTWAWBVPUKAVQVRWDULVTWAWBVPUMVTWCVEVK + UNVTWCVEVKUOUPUSUTVAUTVAVB $. $} ${ @@ -389563,14 +389690,14 @@ minus one divided by eight (` ( 2 /L N ) ` = -1^(((N^2)-1)/8) ). ( va vb wcel cv c2 cexp co caddc wceq cz wrex c1 wa rexlimdvva 1z eqtrdi cmul cprime c4 cmo wo wn df-ne cgcd prmz ad3antrrr simplrr bezout syl2anc wne simplll simpllr simplr simprll simprlr simprr 2sqblem mpd ex impancom - expr syl5bir orrd oveq1 oveq1d eqeq2d oveq2d 1p1e2 rspc2ev mp3an12 adantl - sq1 2sq jaodan impbida ) CUAFZCAGZHIJZBGZHIJZKJZLZBMNAMNZCHLZCUBUCJOLZUDV - SWFPZWGWHWGUECHUMZWIWHCHUFVSWJWFWHVSWJPZWEWHABMMWKVTMFZWBMFZPZPZWEWHWOWEP - ZCWBUGJCDGZTJWBEGZTJKJLZEMNDMNZWHWPCMFZWMWTVSXAWJWNWECUHUIWKWLWMWEUJDECWB - UKULWPWSWHDEMMWPWQMFZWRMFZPZWSWHWPXDWSPZPWQWRCVTWBWKWNWEXEUNWKWNWEXEUOWOW - EXEUPWPXBXCWSUQWPXBXCWSURWPXDWSUSUTVDQVAVBQVCVEVFVSWGWFWHWGWFVSOMFZXFWGWF - RRWEWGCOWCKJZLABOOMMVTOLZWDXGCXHWAOWCKXHWAOHIJZOVTOHIVGVOSVHVIWBOLZXGHCXJ - XGOOKJHXJWCOOKXJWCXIOWBOHIVGVOSVJVKSVIVLVMVNABCVPVQVR $. + expr biimtrrid orrd oveq1 sq1 oveq1d eqeq2d oveq2d rspc2ev mp3an12 adantl + 1p1e2 2sq jaodan impbida ) CUAFZCAGZHIJZBGZHIJZKJZLZBMNAMNZCHLZCUBUCJOLZU + DVSWFPZWGWHWGUECHUMZWIWHCHUFVSWJWFWHVSWJPZWEWHABMMWKVTMFZWBMFZPZPZWEWHWOW + EPZCWBUGJCDGZTJWBEGZTJKJLZEMNDMNZWHWPCMFZWMWTVSXAWJWNWECUHUIWKWLWMWEUJDEC + WBUKULWPWSWHDEMMWPWQMFZWRMFZPZWSWHWPXDWSPZPWQWRCVTWBWKWNWEXEUNWKWNWEXEUOW + OWEXEUPWPXBXCWSUQWPXBXCWSURWPXDWSUSUTVDQVAVBQVCVEVFVSWGWFWHWGWFVSOMFZXFWG + WFRRWEWGCOWCKJZLABOOMMVTOLZWDXGCXHWAOWCKXHWAOHIJZOVTOHIVGVHSVIVJWBOLZXGHC + XJXGOOKJHXJWCOOKXJWCXIOWBOHIVGVHSVKVOSVJVLVMVNABCVPVQVR $. $} $( ` 2 ` is the sum of squares of two nonnegative integers iff the two @@ -397844,31 +397971,31 @@ a multiplicative function (but not completely multiplicative). wcel w3a cr simpld rpred simprd pntrlog2bnd syl2anc reeanv c4 cdc simpl adantr rpaddcl syl2an ltaddrp jca adantrr simprlr eqid wss cxr ad2antrl rpxr rpre ltaddrp2d ltled iooss1 simprrl ssralv ralimdv sylc simprrr - pntleme expr rexlimdvva syl5bir mp2and ) ACURZDURZUSUTZVAROVBVCZVDVCZUU - GVBVCZNURUUFVBVCZUSUTZVLZGURZKVEUUOVFVCVGVEZOVHUTZGUUGUUKVIVCZVJZVLZDVM - VKZCUOURZVNVOVCZVJZNQVNVPVCZVJZUOVMVKZUUGKVEZVGVEUUGVQVEZVBVCVRUVIVFVCV - AUUGSVFVCVSVEVTVCUUGUPURZVFVCKVEVGVEUVJVQVEVBVCUPWAVBVCWBVCUUGVFVCUQURZ - VHUTDVAVNVOVCVJZUQVMVKZFURZKVEUVNVFVCVGVELPLWCWDVCVBVCWBVCVHUTFEURVNVPV - CVJEVMVKZAUUHVARMURZVBVCZVDVCZUUGVBVCZUULUSUTZVLZUUPUVPVHUTZGUUGUVSVIVC - ZVJZVLZDVMVKZCBURZVNVOVCZVJZNIUVPVFVCZWHVEZVNVPVCZVJZBVMVKZUVGMWEVAVOVC - ZOUVPOWRZUWNUVACUWHVJZNUVEVJZBVMVKUVGUWPUWMUWRBVMUWPUWIUWQNUWLUVEUWPUWK - QVNVPUWPUWKIOVFVCZWHVEQUWPUWJUWSWHUVPOIVFWFWGULWIWJUWPUWFUVACUWHUWPUWEU - UTDVMUWPUWAUUNUWDUUSUWPUVTUUMUUHUWPUVSUUKUULUSUWPUVRUUJUUGVBUWPUVQUUIVA - VDUVPORVBWFWKWJZWLWSUWPUWBUUQGUWCUURUWPUVSUUKUUGVIUWTWKUVPOUUPVHWMWNWOW - PWQWNWPUWRUVFBUOVMUWGUVBWRZUWQUVDNUVEUXAUVACUWHUVCUWGUVBVNVOWTXAWQXBXCU - HAOUWOXHZVAQUSUTZLOWBVCZVMXHZAOVMXHQVMXHUXBUXCUXEXIAHIJKLOPQRTUAUBUDUEU - FUGUIUJUKULXDXEXFXGASXJXHZVASVHUTZUVMASASVMXHZUXGUMXKZXLZAUXHUXGUMXMDSK - 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( cfv wcel wceq wi c1 cmul fveq2 eqeq12d cq vn vy vz cv c2 cn eluz2nn cuz co imbi2d cc0 wne ax-1ne0 qrng1 qrng0 abv1z sylancl eqtr4d cprime expcom wa jcab oveq12 adantr cz eluzelz ad2antrl syl ad2antll qrngbas - cvv cmulr qex ccnfld cnfldmul ressmulr abvmul syl3anc syl5ibr syl5bir - zq ax-mp a2d prmind impcom sylan2 ostthlem1 ) ABCUADEGHIJUAUDZUEUHLZM - AWHUFMZWHDLZWHELZNZWHUGWJAWMAFUDZDLZWNELZNZOAPDLZPELZNZOAUBUDZDLZXAEL - ZNZOZAUCUDZDLZXFELZNZOZAXAXFQUIZDLZXKELZNZOZAWMOFUBUCWHWNPNZWQWTAXPWO - WRWPWSWNPDRWNPERSUJWNXANZWQXDAXQWOXBWPXCWNXADRWNXAERSUJWNXFNZWQXIAXRW - OXGWPXHWNXFDRWNXFERSUJWNXKNZWQXNAXSWOXLWPXMWNXKDRWNXKERSUJWNWHNZWQWMA - XTWOWKWPWLWNWHDRWNWHERSUJAWRPWSADBMZPUKULZWRPNIUMBCPDUKHCGUNZCGUOZUPU - QAEBMZYBWSPNJUMBCPEUKHYCYDUPUQURAWNUSMWQKUTXEXJVAAXDXIVAZOXAWIMZXFWIM - ZVAZXOAXDXIVBYIAYFXNAYIYFXNOYFXNAYIVAZXBXGQUIZXCXHQUIZNXBXCXGXHQVCYJX - LYKXMYLYJYAXATMZXFTMZXLYKNAYAYIIVDYJXAVEMZYMYGYOAYHUEXAVFVGXAWAVHZYJX - FVEMZYNYHYQAYGUEXFVFVIXFWAVHZBTCQDXAXFHCGVJZTVKMQCVLLNVMTVNCQVKGVOVPW - 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(Contributed by Scott Fenton, 8-Aug-2024.) $) @@ -399889,8 +400017,8 @@ a multiplicative function (but not completely multiplicative). 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 + syl simprr3 expr rexlimdvva biimtrrid alrimivv eqeq2 3anbi3d rexbidv dmeq + fvresd eleq2d breq1 eqeq1d ralbidv fveq1 3anbi123d cbvrexvw bitrdi sylibr mo4 ) DHUCZECIZUDZJZWTBIZKLZMZWTEUAZUBZXCXFUBZNZOZBDVGZEWTPZAIZNZUEZCDQZE FIZUDZJZXQXCKLZMZXQXFUBZXHNZOZBDVGZEXQPZGIZNZUEZFDQZRZAGSZOZGUFAUFXPAUOWS YMAGYKXOYIRZFDQCDQWSYLXOYICFDDUGWSYNYLCFDDWSWTDJZXQDJZRZYNYLWSYQYNRZRZXLY @@ -399899,10 +400027,10 @@ a multiplicative function (but not completely multiplicative). 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In contrast to the definition in Aksoy et ( vx wcel cfv wa wceq cc0 cfzo co wral wi imp ex eqid com12 adantr cuspgr cwlks c1st chash w3a cv c2nd cfz wb 3anan32 a1i wlkeq 3expa 3adant1 caddc cpr ciedg fzofzp1 adantl fveq2 eqeq12d impancom ralrimiv fvoveq1 cbvralvw - c1 rspcdv sylibr wss fzossfz ssralv mp1i r19.26 ralimdv syl5bir expd syld - preq12 mpd cdm cword cvtx cupgr uspgrupgr upgrwlkcompim syl oveq2 raleqdv - 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 biimtrid ralimdva 3syld expimpd pm4.71d 3bitr4d ) DUAGZBDUBHZ - GZCYIGZIZEBUCHZUDHZJZUEZECUCHZUDHZJZAUFZYMHZYTYQHZJZAKELMZNZYTBUGHZHZYTCU - GHZHZJZAKEUHMZNZUEZYSUULIZUUEIZBCJZUUNUUMUUOUIYPYSUUEUULUJUKYLYOUUPUUMUIZ - YHYJYKYOUUQABCDEULUMUNYPUUNUUEYPYSUULUUEYPYSIZUULUUGYTVFUOMZUUFHZUPZUUIUU - SUUHHZUPZJZAUUDNZUUBDUQHZHZUUAUVFHZJZAUUDNZUUEUURUULUVEUURUULIZUUTUVBJZAU - 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In contrast to the definition in Aksoy et ( vk cfv co cc0 wcel wrex cvv w3a wa wceq wi adantr cwlkson wbr chash wne cv crn cvtx cwlks eqid wlkonprop cpr wss caddc cfzo fveq2 fv0p1e1 preq12d c1 sseq1d rexbidv wlkvtxiedg cn0 wlkcl cn elnnne0 simplbi2 lbfzo0 syl6ibr - wral syl imp rspcdva fvex prss eleq1 ax-1 syl6bi impd syl5bir reximdv mpd - adantl ex 3adant3 3ad2ant3 ) ECABFUAJKUBZEUCJZLUDZADUEZMZDGUFZNZWFFOMAFUG - JZMBWMMPZEOMCOMQZECFUHJUBZLCJZARZWGCJBRZPZPWHWLSZABCEFWMWMUIUJWTWNXAWOWPW - RXAWSWPWRQZWHWLXBWHQZWQURCJZUKZWIULZDWKNZWLXCIUEZCJZXHURUMKCJZUKZWIULZDWK - NZXGILWGUNKZLXHLRZXLXFDWKXOXKXEWIXOXIWQXJXDXHLCUOCXHUPUQUSUTXBXMIXNVIZWHW - PXPWRCDIEFGHVATTXBWHLXNMZWPWHXQSZWRWPWGVBMZXRCEFVCXSWHWGVDMZXQXTXSWHWGVEV - FWGVGVHVJTVKVLXBXGWLSWHXBXFWJDWKXFWQWIMZXDWIMZQXBWJWQXDWILCVMURCVMVNXBYAY - BWJWRYAYBWJSZSWPWRYAWJYCWQAWIVOWJYBVPVQWBVRVSVTTWAWCWDWEVJVK $. + wral syl imp rspcdva fvex prss eleq1 ax-1 syl6bi adantl biimtrrid reximdv + impd mpd ex 3adant3 3ad2ant3 ) ECABFUAJKUBZEUCJZLUDZADUEZMZDGUFZNZWFFOMAF + UGJZMBWMMPZEOMCOMQZECFUHJUBZLCJZARZWGCJBRZPZPWHWLSZABCEFWMWMUIUJWTWNXAWOW + PWRXAWSWPWRQZWHWLXBWHQZWQURCJZUKZWIULZDWKNZWLXCIUEZCJZXHURUMKCJZUKZWIULZD + WKNZXGILWGUNKZLXHLRZXLXFDWKXOXKXEWIXOXIWQXJXDXHLCUOCXHUPUQUSUTXBXMIXNVIZW + HWPXPWRCDIEFGHVATTXBWHLXNMZWPWHXQSZWRWPWGVBMZXRCEFVCXSWHWGVDMZXQXTXSWHWGV + EVFWGVGVHVJTVKVLXBXGWLSWHXBXFWJDWKXFWQWIMZXDWIMZQXBWJWQXDWILCVMURCVMVNXBY + AYBWJWRYAYBWJSZSWPWRYAWJYCWQAWIVOWJYBVPVQVRWAVSVTTWBWCWDWEVJVK $. $} $( The length of a walk between two different vertices is not 0 (i.e. is at @@ -427696,13 +427824,13 @@ segment of the walk (of length ` N ` ) forms a walk on the subgraph ( P ` k ) e. ( I ` ( F ` k ) ) ) ) $= ( cfv wcel c1 co wi wral caddc cpr wss wa fvex chash cn cmin cfzo fzo0end cv cc0 wceq fveq2 fvoveq1 preq12d 2fveq3 sseq12d rspcv syl prss cc npcan1 - nncn fveq2d eleq1d biimpd adantld syl5bir syld syl5com simpl a1i ralimdva - sylbir mpd jca ) ADUAJZUBKZVMBJZVMLUCMZDJEJZKZNCUFZBJZVSDJEJZKZCUGVMUDMZO - ZAVTVSLPMZBJZQZWARZCWCOZVNVRIVNWIVPBJZVPLPMZBJZQZVQRZVRVNVPWCKWIWNNVMUEWH - WNCVPWCVSVPUHZWGWMWAVQWOVTWJWFWLVSVPBUIVSVPLBPUJUKVSVPEDULUMUNUOWNWJVQKZW - LVQKZSVNVRWJWLVQVPBTWKBTUPVNWQVRWPVNWQVRVNWLVOVQVNWKVMBVNVMUQKWKVMUHVMUSV - MURUOUTVAVBVCVDVEVFAWIWDIAWHWBCWCWHWBNAVSWCKSWHWBWFWAKZSWBVTWFWAVSBTWEBTU - PWBWRVGVJVHVIVKVL $. + nncn fveq2d eleq1d biimpd adantld biimtrrid syld syl5com simpl sylbir a1i + ralimdva mpd jca ) ADUAJZUBKZVMBJZVMLUCMZDJEJZKZNCUFZBJZVSDJEJZKZCUGVMUDM + ZOZAVTVSLPMZBJZQZWARZCWCOZVNVRIVNWIVPBJZVPLPMZBJZQZVQRZVRVNVPWCKWIWNNVMUE + WHWNCVPWCVSVPUHZWGWMWAVQWOVTWJWFWLVSVPBUIVSVPLBPUJUKVSVPEDULUMUNUOWNWJVQK + ZWLVQKZSVNVRWJWLVQVPBTWKBTUPVNWQVRWPVNWQVRVNWLVOVQVNWKVMBVNVMUQKWKVMUHVMU + SVMURUOUTVAVBVCVDVEVFAWIWDIAWHWBCWCWHWBNAVSWCKSWHWBWFWAKZSWBVTWFWAVSBTWEB + TUPWBWRVGVHVIVJVKVL $. $( Lemma 3 for ~ wlkd . (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.) $) @@ -429037,18 +429165,19 @@ where f enumerates the (indices of the) different edges, and p pthdlem2 $p |- ( ph -> ( ( P " { 0 , R } ) i^i ( P " ( 1 ..^ R ) ) ) = (/) ) $= ( cfv cc0 wceq cima c1 cfzo co c0 wcel cvv cn0 cfz chash cpr cin wn cword - cn wi lencl wne df-ne elnnne0 simplbi2 syl5bir 3syl wnel eqid pthdlem2lem - wa wo orci mp3an3 olci wf wb cmin wrdffz syl adantr oveq2i sylibr nnm1nn0 - feq2i eqeltrid adantl fvinim0ffz syl2anc mpbir2and ex oveq1 eqtrid oveq2d - syld cle wbr caddc c2 1p1e2 breqtrri 0re 1re lesubadd2i mpbir cz peano2zm - 0le2 1z 0z ax-mp fzon mp2an mpbi eqtrdi imaeq2d ima0 ineq2d in0 pm2.61d2 - ) ABUAIZJKZBJCUBLZBMCNOZLZUCZPKZAXIUDZXHUFQZXNABRUEQZXHSQZXOXPUGFRBUHXOXH - JUIZXRXPXHJUJXPXRXSXHUKULUMUNAXPXNAXPURZXNJBIXLUOZCBIXLUOZAXPJJKZJCKZUSYA - YCYDJUPUTABCDEJFGHUQVAAXPCJKZCCKZUSYBYFYECUPVBABCDECFGHUQVAXTJCTOZRBVCZCS - QZXNYAYBURVDXTJXHMVEOZTOZRBVCZYHAYLXPAXQYLFRBVFVGVHYGYKRBCYJJTGVIVLVJXPYI - AXPCYJSGXHVKVMVNBCRVOVPVQVRWBXIXMXJPUCPXIXLPXJXIXLBPLPXIXKPBXIXKMJMVEOZNO - 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(Contributed by Alexander van der Vekens, @@ -433640,10 +433769,10 @@ that the word does not contain the terminating vertex p(n) of the walk, -> ( W cyclShift N ) e. ( ClWWalks ` G ) ) $= ( chash cfv wceq cclwwlk wcel cc0 cfz co wa ccsh oveq2 cvtx cword c0 simprl cvv wne eqid clwwlkbp simp2d cshwn adantr sylan9eq eqeltrd wn cfzo wi df-ne - syl fzofzim expcom syl5bir adantl impcom clwwisshclwws syl2anc pm2.61ian ) - BCDEZFZCAGEZHZBIVAJKHZLZCBMKZVCHZVBVFLVGCVCVBVFVGCVAMKZCBVACMNVDVICFZVEVDCA - OEZPHZVJVDASHVLCQTAVKCVKUAUBUCVKCUDULUEUFVBVDVERUGVBUHZVFLVDBIVAUIKHZVHVMVD - VERVFVMVNVEVMVNUJVDVMBVATZVEVNBVAUKVOVEVNBVAUMUNUOUPUQABCURUSUT $. + syl fzofzim expcom biimtrrid adantl impcom clwwisshclwws syl2anc pm2.61ian + ) BCDEZFZCAGEZHZBIVAJKHZLZCBMKZVCHZVBVFLVGCVCVBVFVGCVAMKZCBVACMNVDVICFZVEVD + CAOEZPHZVJVDASHVLCQTAVKCVKUAUBUCVKCUDULUEUFVBVDVERUGVBUHZVFLVDBIVAUIKHZVHVM + VDVERVFVMVNVEVMVNUJVDVMBVATZVEVNBVAUKVOVEVNBVAUMUNUOUPUQABCURUSUT $. ${ erclwwlk.r $e |- .~ = { <. u , w >. | ( u e. ( ClWWalks ` G ) @@ -435241,10 +435370,10 @@ prime number equals this length (in an undirected simple graph). clwwlknon1sn $p |- ( X e. V -> ( ( X C 1 ) = { <" X "> } <-> { X } e. E ) ) $= ( wcel c1 co cs1 csn wceq wn wnel df-nel wa c0 ex adantl cword s1cli snnz - clwwlknon1nloop cvv elexi nesymi mtbiri syl syl5bir clwwlknon1loop impbid - eqeq1 con4d ) EDIZEJAKZELZMZNZEMZBIZUPVBUTVBOVABPZUPUTOZVABQUPVCVDUPVCRUQ - SNZVDVCVEUPABCDEFGHUEUAVEUTSUSNUSSURURUFUBEUCUGUDUHUQSUSUNUIUJTUKUOUPVBUT - ABCDEFGHULTUM $. + clwwlknon1nloop cvv elexi nesymi eqeq1 syl biimtrrid con4d clwwlknon1loop + mtbiri impbid ) EDIZEJAKZELZMZNZEMZBIZUPVBUTVBOVABPZUPUTOZVABQUPVCVDUPVCR + UQSNZVDVCVEUPABCDEFGHUEUAVEUTSUSNUSSURURUFUBEUCUGUDUHUQSUSUIUNUJTUKULUPVB + UTABCDEFGHUMTUO $. $} $( There is at most one (closed) walk on vertex ` X ` of length ` 1 ` as word @@ -439476,12 +439605,12 @@ third vertex being the middle vertex of a (simple) path/walk of length 2 4-Jan-2022.) $) frgr2wwlkeu $p |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> E! c e. V <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) $= - ( cfrgr wcel wa wne w3a cv cs3 c2 cwwlksnon co wreu cpr syl5bir wb df-3an - cedg cfv eqid frcond2 3impib cusgr cumgr wi frgrusgr 3anan32 umgrwwlks2on - usgrumgr ex 3syl impl reubidva 3adant3 mpbird ) CGHZADHZBDHZIZABJZKAELZBM - ABNCOPPHZEDQZAVERCUBUCZHVEBRVHHIZEDQZUTVCVDVJVCVDIVAVBVDKUTVJVAVBVDUAABVH - CDEFVHUDZUESUFUTVCVGVJTVDUTVCIVFVIEDUTVCVEDHZVFVITZUTCUGHCUHHZVCVLIZVMUIC - UJCUMVOVAVLVBKZVNVMVAVLVBUKVNVPVMAVEBVHCDFVKULUNSUOUPUQURUS $. + ( cfrgr wcel wa wne w3a cv cs3 c2 cwwlksnon co wreu cpr biimtrrid wb cedg + cfv df-3an eqid frcond2 3impib cusgr cumgr frgrusgr usgrumgr umgrwwlks2on + wi 3anan32 ex 3syl impl reubidva 3adant3 mpbird ) CGHZADHZBDHZIZABJZKAELZ + BMABNCOPPHZEDQZAVERCUAUBZHVEBRVHHIZEDQZUTVCVDVJVCVDIVAVBVDKUTVJVAVBVDUCAB + VHCDEFVHUDZUESUFUTVCVGVJTVDUTVCIVFVIEDUTVCVEDHZVFVITZUTCUGHCUHHZVCVLIZVMU + LCUICUJVOVAVLVBKZVNVMVAVLVBUMVNVPVMAVEBVHCDFVKUKUNSUOUPUQURUS $. $( In a friendship graph, there is always a path/walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, @@ -441032,25 +441161,25 @@ number of vertices in a friendship graph is (k(k-1)+1), see c0 cle wb cr nn0re 1re lenlt sylancl cn elnnne0 biimpd sylbir a1d expimpd nnle1eq1 sylbird cvtx fveq2i eqeq1i simpr rusgr1vtx syl2an ex cusgr cxnn0 orcd cv cvtxdg wral eqid rusgrprop0 simp2 hashnncl df-ne nngt1ne1 biimprd - w3a syl5bir syl6bir imp vdgn1frgrv3 3imp3i2an r19.26 wrex r19.2z eqneqall - neeq1 com12 rexlimivw expd com34 3ad2ant3 mpd 3exp com15 com13 syl6 com23 - pm2.61i exp4b simprl simpl ad2antlr anim12ci frgrreggt1 syl31anc olcd 2a1 - exp31 pm2.61ii ) FBUAGZBHIZBJIZUBZCUCKZCUMLZMZAUDKZABUEGZMZYFNNYCOZYFOZYI - YLYFYIYLMZYMYNMZYFYOBUFKZYPYFNYOYKYJMZYHYGMZMYQYOYSYRYOYSYRMYIYSYLYRYGYHU - GYJYKUGUHUIUJABCDUKPYPYQYOYFYNYMYQYOYFNZNZYNBHLZBJLZMZYMUUANBHBJULUUDYQYM - YTUUDYQYMYTNUUDYQMZYMBFIZYTUUEYMBFUNGZUUFYQUUGYMUOZUUDYQBUPKFUPKUUHBUQURB - FUSUTQYQUUDUUGUUFNZYQUUBUUCUUIYQUUBMZUUIUUCUUJBVAKZUUIBVBUUKUUGUUFBVGVCVD - VEVFRVHCSTZFIZUUFYTNUUMYTUUFUUMYOYFUUMYOMYDYEUUMAVITZSTZFIZYKYDYOUUMUUPUU - 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cmul com34 syl5bir mpcom eqeq2d pm2.21 ad2antrr syl5com wo frgrreg mpjaod - exp32 exp4c com3r 3exp impcomd com24 3exp1 3imp21 ) CUAHZAUBHZABUDUEZCUCI - ZJKZUUSLKZUUSUFKZUGZUUSUQHZUUPUUQUURUVCMZMZCUHUVDUUPUUQUURUVCUVCUVDUUPUUQ - NZUUROZUVCMZUVCUVHUIUVCPUUTPZUVAPZUVBPZNUVIUUTUVAUVBUJUVJUVKUVLUVIUVHUVKU - VLUVJUVCUVGUURUVKUVLUVJUVCMMMUVGUVJUVKUVLUURUVCUVDUUPUUQUVJUVKUVLUVEMZMZM - UVDUVJUUQUUPUVNUVJUUSJQZUVDUUQUUPUVNMMUUSJUKUUPUVOUUQUVDUVNUUPUVOUUQUVDUV - NMMZCULQZUUPUVOOZUVPUUPUVOUVQUUPUUSJCULCUAUMUNUOUVDUVRUUQUVQUVNUVDUVOUUPU - UQUVQUVNMMZUVDUVOUUPUVSMZUVDUVOOUUSUPHZUVTUUSURUWAUVKUUQUVQUUPUVMUVKUUSLQ - ZUWAUUQUVQUUPUVMMZMMZUUSLUKUWAUWBUWDUWAUWBOUUSRUSIHZUWDUUSUTUWEUUQUVQUWCU - USRKZUWEUUQUVQNZUWCMUWGUWFUWCUWEUUQUVQUWFUWCMZUUQUVQUWHMMUWEUUPUVQUWFUUQU - VMUUPUWFUVQUUQUVMMZUUPUWFUVQUWIMZUUPUWFOEVAZFVAZQZCUWKUWLVBZKZOZFVCEVCUWJ - CUAEFVDUWPUWJEFUWPUUQUVQUVMUWPUUQUVQUVMMUWPAUBVEZUUQPUWMUWKVFHZUWLVFHZUWM - NAVGIZUWNKZUWQUWOUWMUWRUWSUWMUWRUWMEVHVIUWSUWMFVHVIUWMWBVKUWOUXACUWTUWNDV - JVLUWKUWLAVFVFVMVNAUBVOVPVQVRVSWCSVRVTVIWAWDUWGUUPUWFPZUVMUWEUUQUVQUUPUXB - 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XMSUULVTYTSVTYSUUMWAXRYEVTWAXLYPUUNYTUUO $. + cmul com34 biimtrrid mpcom eqeq2d pm2.21 ad2antrr syl5com wo mpjaod exp32 + frgrreg exp4c com3r 3exp impcomd com24 3exp1 3imp21 ) CUAHZAUBHZABUDUEZCU + CIZJKZUUSLKZUUSUFKZUGZUUSUQHZUUPUUQUURUVCMZMZCUHUVDUUPUUQUURUVCUVCUVDUUPU + UQNZUUROZUVCMZUVCUVHUIUVCPUUTPZUVAPZUVBPZNUVIUUTUVAUVBUJUVJUVKUVLUVIUVHUV + KUVLUVJUVCUVGUURUVKUVLUVJUVCMMMUVGUVJUVKUVLUURUVCUVDUUPUUQUVJUVKUVLUVEMZM + ZMUVDUVJUUQUUPUVNUVJUUSJQZUVDUUQUUPUVNMMUUSJUKUUPUVOUUQUVDUVNUUPUVOUUQUVD + UVNMMZCULQZUUPUVOOZUVPUUPUVOUVQUUPUUSJCULCUAUMUNUOUVDUVRUUQUVQUVNUVDUVOUU + PUUQUVQUVNMMZUVDUVOUUPUVSMZUVDUVOOUUSUPHZUVTUUSURUWAUVKUUQUVQUUPUVMUVKUUS + LQZUWAUUQUVQUUPUVMMZMMZUUSLUKUWAUWBUWDUWAUWBOUUSRUSIHZUWDUUSUTUWEUUQUVQUW + CUUSRKZUWEUUQUVQNZUWCMUWGUWFUWCUWEUUQUVQUWFUWCMZUUQUVQUWHMMUWEUUPUVQUWFUU + QUVMUUPUWFUVQUUQUVMMZUUPUWFUVQUWIMZUUPUWFOEVAZFVAZQZCUWKUWLVBZKZOZFVCEVCU + WJCUAEFVDUWPUWJEFUWPUUQUVQUVMUWPUUQUVQUVMMUWPAUBVEZUUQPUWMUWKVFHZUWLVFHZU + WMNAVGIZUWNKZUWQUWOUWMUWRUWSUWMUWRUWMEVHVIUWSUWMFVHVIUWMWBVKUWOUXACUWTUWN + DVJVLUWKUWLAVFVFVMVNAUBVOVPVQVRVSWCSVRVTVIWAWDUWGUUPUWFPZUVMUWEUUQUVQUUPU + XBUVMMZMUUPUUQUVQUWEUXCUUPUVQUUQUWEUXCMZUUPUVQUUQUXDMUUPUVQOZUVLUWEUXBUUQ + UVEUXEUVLUXBUWEUVFUXEUVLUXBUWEUVFUXEUUQUVLUXBOZUWEOZUVEUXEUUQUURUXGUVCUXE + UUQUURUXGUVCMZUXEUUQUUROZOZBJKZUXHBRKZUXIUXEUXKUXHMZUURUUQUXEUXMMZUURAWEH + ZBWFHZGVAZAWGIZIZBKZGCWHZNUUQUXNMZGUXRABCDUXRWIWJUYAUXOUYBUXPUUQUYAUXNUXG + UYAUXEUXKUUQUVCUWEUYAUXEUXKUUQUVCMMMMUXFUWEUUQUXEUXKUYAUVCUWEUUQUXEUXKUYA + UVCMMMZUWEUUQOUUQLUUSWKUEZOZUXSJQZGCWHZUYCUWEUYDUUQUUSWLWMUYEUYFGCUUQUXQC + HUYDUYFAUXQCDWNWOWPUXKUYAUYGUXEUVCUXKUYAUXSJKZGCWHZUYGUXEUVCMZMUXKUXTUYHG + CBJUXSWQWRUYIUYGUYJUYIUYGOUYHUYFOZGCWHZUYJUYHUYFGCWSUYLWTGCWHZUYJUYKWTGCU + YKUYFPZUYFOUYFUYNOZWTUYHUYNUYFUYNUYHUXSJXAXBXCUYNUYFXDUYOUYFXEXFXGXHUXEUY + MUVCUVQUYMUVCMUUPUVQUYMWTUVCWTGCXOUVCXIXJXKWDXLXMSXNXPXQSXRXKXSWDXTWCYAYA + UXJUUSBBLYBTZYQTZLYCTZKZUXLUXHUXEUXIUYSABCDYDYEUXLUYSUVBUXHUXLUYRUFUUSUXL + UYRRRLYBTZYQTZLYCTZUFUXLUYQVUALYCUXLBRUYPUYTYQUXLWBBRLYBYFYGYHVUBRLYCTUFV + UARLYCVUARLYQTRUYTLRYQYIYJYKYLYMYNYLYOUUAUXGUVBUVCUVLUVBUVCMUXBUWEUVBUVCU + UBUUCWDXNUUDUXEUXIUXKUXLUUEABCDUUHYEUUFUUGYRVRUUIYRXRSVRVTWAUUJYPUUKXMSYS + XRXMSUULVTYTSVTYSUUMWAXRYEVTWAXLYPUUNYTUUO $. $( If a nonempty finite friendship graph is ` K `-regular, then it must have order 1 or 3. Special case of ~ frgrregord013 . (Contributed by @@ -447002,15 +447131,15 @@ which is an Abelian group (i.e. the vectors, with the operation of ( ( A S B ) = Z <-> ( A = 0 \/ B = Z ) ) ) $= ( wcel cc co wceq cc0 wa c1 oveq2 3ad2antl2 3adant2 3eqtr3d cnv w3a wo wn wne df-ne cdiv ad2antlr recid2 oveq1d simpl1 reccl simpl2 simpl3 syl13anc - cmul nvsass nvsid adantr adantlr nvsz anassrs 3adantl3 ex syl5bir orrd wi - sylan2 nv0 oveq1 eqeq1d syl5ibrcom 3adant3 jaod impbid ) DUAJZAKJZBEJZUBZ - ABCLZFMZANMZBFMZUCZVSWAWDVSWAOZWBWCWBUDANUEZWEWCANUFWEWFWCWEWFOPAUGLZVTCL - ZWGFCLZBFWAWHWIMVSWFVTFWGCQUHVSWFWHBMWAVSWFOZWGAUPLZBCLZPBCLZWHBVQVPWFWLW - MMVRVQWFOZWKPBCAUIUJRWJVPWGKJZVQVRWLWHMVPVQVRWFUKVQVPWFWOVRAULZRVPVQVRWFU - MVPVQVRWFUNWGABCDEGHUQUOVSWMBMZWFVPVRWQVQBCDEGHURSUSTUTVSWFWIFMZWAVPVQWFW - RVRVPVQWFWRWNVPWOWRWPWGCDFHIVAVHVBVCUTTVDVEVFVDVSWBWAWCVPVRWBWAVGVQVPVROW - AWBNBCLZFMBCDEFGHIVIWBVTWSFANBCVJVKVLSVPVQWCWAVGVRVPVQOWAWCAFCLZFMACDFHIV - AWCVTWTFBFACQVKVLVMVNVO $. + cmul nvsass adantr adantlr nvsz sylan2 anassrs 3adantl3 ex biimtrrid orrd + nvsid wi nv0 oveq1 eqeq1d syl5ibrcom 3adant3 jaod impbid ) DUAJZAKJZBEJZU + BZABCLZFMZANMZBFMZUCZVSWAWDVSWAOZWBWCWBUDANUEZWEWCANUFWEWFWCWEWFOPAUGLZVT + CLZWGFCLZBFWAWHWIMVSWFVTFWGCQUHVSWFWHBMWAVSWFOZWGAUPLZBCLZPBCLZWHBVQVPWFW + LWMMVRVQWFOZWKPBCAUIUJRWJVPWGKJZVQVRWLWHMVPVQVRWFUKVQVPWFWOVRAULZRVPVQVRW + FUMVPVQVRWFUNWGABCDEGHUQUOVSWMBMZWFVPVRWQVQBCDEGHVGSURTUSVSWFWIFMZWAVPVQW + FWRVRVPVQWFWRWNVPWOWRWPWGCDFHIUTVAVBVCUSTVDVEVFVDVSWBWAWCVPVRWBWAVHVQVPVR + OWAWBNBCLZFMBCDEFGHIVIWBVTWSFANBCVJVKVLSVPVQWCWAVHVRVPVQOWAWCAFCLZFMACDFH + IUTWCVTWTFBFACQVKVLVMVNVO $. $} ${ @@ -451094,37 +451223,37 @@ orthogonal vectors (i.e. whose inner product is 0) is the sum of the fveq1 breq1d cbvralvw ralbidv bitrid cbvrexvw 2fveq3 rexralbidv wa crab breq2 cn cmpt wss crp adantr simpr sylib cbvrabv rabbidv eqtrid cbvmptv cblo ubthlem1 ad3antrrr ad2antrr simplrl simplrr simprl simprr ubthlem2 - wcel expr rexlimdva rexlimdvva mpd ex syl5bir cnmcv cmul cnv ccbn ax-mp - bnnv eqid nvcl mpan remulcl syl2an cba wf adantlr ad2ant2r blof mp3an12 - sselda syl simplr ffvelcdmd cxr cmnf clt simpllr nmogtmnf xrre syl22anc - nmoxr ad2antlr syl2anc nmblolbi cc0 nvge0 jca lemul1a syl31anc ralimdva - letrd brralrspcev syl6an ralrimdva impbid ) ABUHZCUHZUIZHUIZKUHZTUJZCEU - KKULUMZBJUKZYTFIUNUOZUIZLUHZTUJZCEUKZLULUMZUUFUAUHZUBUHZUIZHUIZUUITUJZU - BEUKZLULUMZUAJUKZAUULUUSUUEUABJUUSUUMYTUIZHUIZUUCTUJZCEUKZKULUMUUMYSUPZ - UUEUURUVDLKULUURUVBUUITUJZCEUKUUIUUCUPZUVDUUQUVFUBCEUUNYTUPZUUPUVBUUITU - VHUUOUVAHUUMUUNYTURUQUSUTUVGUVFUVCCEUUIUUCUVBTVHVAVBVCUVEUVCUUDKCULEUVE - UVBUUBUUCTUUMYSHYTVDUSVEVBUTZAUUTUULAUUTVFZUCUHZUUMDUOUDUHZTUJUAJVGUEUH - ZUFVIUUIUUNUIZHUIZUFUHZTUJZUBEUKZLJVGZVJZUIVKZUDVLUMZUCJUMUEVIUMUULUVJB - UCUACUVTDEFUGUEGHIJUDKMNOPQRAEFIVTUOZVKZUUTSVMUVJUUTUUFAUUTVNUVIVOZUFUG - VIUVSUVBUGUHZTUJZCEUKZUAJVGZUVPUWFUPZUVSUVBUVPTUJZCEUKZUAJVGUWIUVRUWLLU - AJUVRUUIYTUIZHUIZUVPTUJZCEUKUUIUUMUPZUWLUVQUWOUBCEUVHUVOUWNUVPTUVHUVNUW - 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althtmldef "fldGen" as "fldGen"; - latexdef "fldGen" as "\mathrm{fldGen}"; +htmldef "fldGen" as " fldGen "; + althtmldef "fldGen" as " fldGen "; + latexdef "fldGen" as "\mathbin{\operatorname{fldGen}}"; +htmldef "AlgNb" as " AlgNb "; + althtmldef "AlgNb" as " AlgNb "; + latexdef "AlgNb" as "\mathbin{\operatorname{AlgNb}}"; +htmldef "minPoly" as " minPoly "; + althtmldef "minPoly" as " minPoly "; + latexdef "minPoly" as "\mathbin{\operatorname{minPoly}}"; htmldef "repr" as "repr"; althtmldef "repr" as "repr"; @@ -459667,13 +459802,13 @@ space independently from the axioms (see comments in ~ ax-hilex ). Note co cin ccauold cmopn clm cdm chlo simpl hlcompl eldm2g adantr mpbid df-br sylancr ctopon cnv cxmet hlnvi cva csm cno df-hba fveq2i eqtr4i mopntopon wi imsxmet mp2b lmcl mpan a1i cres h2hlm breqi cvv brres elv bitri adantl - baib biimprd jcad syl5bir eximdv mpd elin df-rex 3imtr4i h2hcau eleq2s ) - CAFZGHZAIJZCBKLZMLZIUAUBUCZUDZUECWQNZCWRNZOZWMINZWNOZAPZCWSNWOXBCWMQWPUFL - ZUGLZNZAPZXEXBCXGUHNZXIXBBUINWTXJEWTXAUJWPBCXFWPRZXFRZUKUPWTXJXISXAACXGWQ - ULUMUNXBXHXDAXHCWMXGHZXBXDCWMXGUOXBXMXCWNXMXCVHXBXFIUQLNZXMXCBURNWPIUSLNX - NBEUTZWPBIIVAVBQVCQZTLBTLVDBXPTDVEVFZXKVIWPXFIXLVGVJWMCXFIVKVLVMXBWNXMXAW - NXMSWTWNXAXMWNCWMXGWRVNZHZXAXMOZCWMGXRWPBXFDXOXQXKXLVOVPXSXTSAWRCWMXGVQVR - VSVTWBWAWCWDWEWFWGCWQWRWHWNAIWIWJWPBDXOXQXKWKWL $. + baib biimprd jcad biimtrrid eximdv mpd elin df-rex 3imtr4i h2hcau eleq2s + ) CAFZGHZAIJZCBKLZMLZIUAUBUCZUDZUECWQNZCWRNZOZWMINZWNOZAPZCWSNWOXBCWMQWPU + FLZUGLZNZAPZXEXBCXGUHNZXIXBBUINWTXJEWTXAUJWPBCXFWPRZXFRZUKUPWTXJXISXAACXG + WQULUMUNXBXHXDAXHCWMXGHZXBXDCWMXGUOXBXMXCWNXMXCVHXBXFIUQLNZXMXCBURNWPIUSL + NXNBEUTZWPBIIVAVBQVCQZTLBTLVDBXPTDVEVFZXKVIWPXFIXLVGVJWMCXFIVKVLVMXBWNXMX + AWNXMSWTWNXAXMWNCWMXGWRVNZHZXAXMOZCWMGXRWPBXFDXOXQXKXLVOVPXSXTSAWRCWMXGVQ + VRVSVTWBWAWCWDWEWFWGCWQWRWHWNAIWIWJWPBDXOXQXKWKWL $. $} @@ -459870,13 +460005,13 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). ( ( A .h B ) = 0h <-> ( A = 0 \/ B = 0h ) ) ) $= ( cc wcel wa csm co c0v wceq c1 oveq2 ad2antlr adantlr 3eqtr3d hvmul0 ex wi cc0 eqeq1d syl5ibrcom chba wo wn df-ne cdiv cmul recid2 oveq1d reccl simpll - wne simplr ax-hvmulass syl3anc ax-hvmulid syl syl5bir orrd ax-hvmul0 adantl - oveq1 adantr jaod impbid ) ACDZBUADZEZABFGZHIZARIZBHIZUBZVGVIVLVGVIEZVJVKVJ - UCARUKZVMVKARUDVMVNVKVMVNEJAUEGZVHFGZVOHFGZBHVIVPVQIVGVNVHHVOFKLVGVNVPBIVIV - GVNEZVOAUFGZBFGZJBFGZVPBVEVNVTWAIVFVEVNEZVSJBFAUGUHMVRVOCDZVEVFVTVPIVEVNWCV - FAUIZMVEVFVNUJVEVFVNULVOABUMUNVFWABIVEVNBUOLNMVGVNVQHIZVIVEVNWEVFWBWCWEWDVO - OUPMMNPUQURPVGVJVIVKVFVJVIQVEVFVIVJRBFGZHIBUSVJVHWFHARBFVASTUTVEVKVIQVFVEVI - VKAHFGZHIAOVKVHWGHBHAFKSTVBVCVD $. + simplr ax-hvmulass syl3anc ax-hvmulid biimtrrid orrd ax-hvmul0 oveq1 adantl + wne syl adantr jaod impbid ) ACDZBUADZEZABFGZHIZARIZBHIZUBZVGVIVLVGVIEZVJVK + VJUCARUTZVMVKARUDVMVNVKVMVNEJAUEGZVHFGZVOHFGZBHVIVPVQIVGVNVHHVOFKLVGVNVPBIV + IVGVNEZVOAUFGZBFGZJBFGZVPBVEVNVTWAIVFVEVNEZVSJBFAUGUHMVRVOCDZVEVFVTVPIVEVNW + CVFAUIZMVEVFVNUJVEVFVNUKVOABULUMVFWABIVEVNBUNLNMVGVNVQHIZVIVEVNWEVFWBWCWEWD + VOOVAMMNPUOUPPVGVJVIVKVFVJVIQVEVFVIVJRBFGZHIBUQVJVHWFHARBFURSTUSVEVKVIQVFVE + VIVKAHFGZHIAOVKVHWGHBHAFKSTVBVCVD $. $( Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) $) @@ -462895,10 +463030,10 @@ to a member of the subspace (Definition of complete subspace in [Beran] subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.) $) ocnel $p |- ( ( H e. SH /\ A e. ( _|_ ` H ) /\ A =/= 0h ) -> -. A e. H ) $= - ( csh wcel cort cfv c0v wne wn c0h wceq cin elin ocin eleq2d biimpd syl5bir - wa wi expcomd imp elch0 syl6ib necon3ad 3impia ) BCDZABEFZDZAGHABDZIUFUHRZU - IAGUJUIAJDZAGKUFUHUIUKSUFUIUHUKUIUHRABUGLZDZUFUKABUGMUFUMUKUFULJABNOPQTUAAU - BUCUDUE $. + ( csh wcel cort cfv c0v wne wn wa c0h wceq cin elin eleq2d biimpd biimtrrid + wi ocin expcomd imp elch0 syl6ib necon3ad 3impia ) BCDZABEFZDZAGHABDZIUFUHJ + ZUIAGUJUIAKDZAGLUFUHUIUKRUFUIUHUKUIUHJABUGMZDZUFUKABUGNUFUMUKUFULKABSOPQTUA + AUBUCUDUE $. ${ $d x y A $. @@ -462957,16 +463092,16 @@ to a member of the subspace (Definition of complete subspace in [Beran] Mario Carneiro, 15-May-2014.) (New usage is discouraged.) $) pjhthmo $p |- ( ( A e. SH /\ B e. SH /\ ( A i^i B ) = 0H ) -> E* x ( x e. A /\ E. y e. B C = ( x +h y ) ) ) $= - ( vz vw csh wcel cin c0h wceq cv cva co wrex wa wal syl5bir eqeq2d w3a wi - wmo reeanv simpll1 simpll2 simpll3 simplrl simprll simplrr simprlr eqtr3d - an4 simprrl simprrr shuni simpld exp32 rexlimdvv expimpd alrimivv rexbidv - eleq1w oveq1 oveq2 cbvrexvw bitrdi anbi12d mo4 sylibr ) CHIZDHIZCDJKLZUAZ - AMZCIZEVOBMZNOZLZBDPZQZFMZCIZEWBGMZNOZLZGDPZQZQZVOWBLZUBZFRARWAAUCVNWKAFW - IVPWCQZVTWGQZQVNWJVPWCVTWGUMVNWLWMWJWMVSWFQZGDPBDPVNWLQZWJVSWFBGDDUDWOWNW - JBGDDWOVQDIZWDDIZQZWNWJWOWRWNQZQZWJVQWDLZWTVOVQWBWDCDVKVLVMWLWSUEVKVLVMWL - WSUFVKVLVMWLWSUGVNVPWCWSUHWOWPWQWNUIVNVPWCWSUJWOWPWQWNUKWTEVRWEWOWRVSWFUN - WOWRVSWFUOULUPUQURUSSUTSVAWAWHAFWJVPWCVTWGAFCVCWJVTEWBVQNOZLZBDPWGWJVSXCB - DWJVRXBEVOWBVQNVDTVBXCWFBGDXAXBWEEVQWDWBNVETVFVGVHVIVJ $. + ( vz vw csh wcel cin c0h wceq cv cva co wrex wa wal biimtrrid eqeq2d w3a + wi wmo an4 reeanv simpll1 simpll2 simpll3 simplrl simprll simplrr simprlr + simprrl simprrr eqtr3d shuni simpld exp32 rexlimdvv alrimivv eleq1w oveq1 + expimpd rexbidv oveq2 cbvrexvw bitrdi anbi12d mo4 sylibr ) CHIZDHIZCDJKLZ + UAZAMZCIZEVOBMZNOZLZBDPZQZFMZCIZEWBGMZNOZLZGDPZQZQZVOWBLZUBZFRARWAAUCVNWK + AFWIVPWCQZVTWGQZQVNWJVPWCVTWGUDVNWLWMWJWMVSWFQZGDPBDPVNWLQZWJVSWFBGDDUEWO + WNWJBGDDWOVQDIZWDDIZQZWNWJWOWRWNQZQZWJVQWDLZWTVOVQWBWDCDVKVLVMWLWSUFVKVLV + MWLWSUGVKVLVMWLWSUHVNVPWCWSUIWOWPWQWNUJVNVPWCWSUKWOWPWQWNULWTEVRWEWOWRVSW + FUMWOWRVSWFUNUOUPUQURUSSVCSUTWAWHAFWJVPWCVTWGAFCVAWJVTEWBVQNOZLZBDPWGWJVS + XCBDWJVRXBEVOWBVQNVBTVDXCWFBGDXAXBWEEVQWDWBNVETVFVGVHVIVJ $. $} ${ @@ -465200,17 +465335,17 @@ to a member of the subspace (Definition of complete subspace in [Beran] ( vx vz vw vy cspn cfv chba wss csh wcel ax-mp cv wa wel wi wral co mp2an cun spancl shscli shssii spanss2 unss12 shunssi sstri spanss wceq sseqtri cph spanid cva wex wrex shseli bitri wb vex elspani anbi12i r19.26 bitr4i - r19.27v sylanb anim12 syl5bir shaddcl 3expib sylan9r eleq1 biimprd sylan9 - r2ex unss expl ralimia unssi sylibr syl exlimivv sylbi ssriv eqssi ) ABUC - ZIJZAIJZBIJZUNUAZWIWLIJZWLWLKLWHWLLWIWMLWLWJWKAKLZWJMNCAUDOZBKLZWKMNDBUDO - ZUEZUFWHWJWKUCZWLAWJLZBWKLZWHWSLWNWTCAUGOWPXADBUGOAWJBWKUHUBWJWKWOWQUIUJW - HWLUKUBWLMNWMWLULWRWLUOOUMEWLWIEPZWLNZFPZWJNZGPZWKNZQZXBXDXFUPUAZULZQZGUQ - FUQZXBWINZXCXJGWKURFWJURXLFGWJWKXBWOWQUSXJFGWJWKVQUTXKXMFGXKAHPZLZFHRZSZB - XNLZGHRZSZQZXJQZHMTZXMXHYAHMTZXJYCXHXQHMTZXTHMTZQYDXEYEXGYFWNXEYEVACHAXDF - VBVCOWPXGYFVADHBXFGVBVCOVDXQXTHMVEVFYAXJHMVGVHYCWHXNLZEHRZSZHMTZXMYBYIHMX - NMNZYAXJYIYKYAQYGXIXNNZXJYHYAYGXPXSQZYKYLYGXOXRQYAYMABXNVRXOXPXRXSVIVJYKX - PXSYLXDXFXNVKVLVMXJYHYLXBXIXNVNVOVPVSVTWHKLXMYJVAABKCDWAHWHXBEVBVCOWBWCWD - WEWFWG $. + r2ex r19.27v sylanb anim12 biimtrrid shaddcl 3expib sylan9r eleq1 biimprd + unss sylan9 expl ralimia unssi sylibr syl exlimivv sylbi ssriv eqssi ) AB + UCZIJZAIJZBIJZUNUAZWIWLIJZWLWLKLWHWLLWIWMLWLWJWKAKLZWJMNCAUDOZBKLZWKMNDBU + DOZUEZUFWHWJWKUCZWLAWJLZBWKLZWHWSLWNWTCAUGOWPXADBUGOAWJBWKUHUBWJWKWOWQUIU + JWHWLUKUBWLMNWMWLULWRWLUOOUMEWLWIEPZWLNZFPZWJNZGPZWKNZQZXBXDXFUPUAZULZQZG + UQFUQZXBWINZXCXJGWKURFWJURXLFGWJWKXBWOWQUSXJFGWJWKVGUTXKXMFGXKAHPZLZFHRZS + ZBXNLZGHRZSZQZXJQZHMTZXMXHYAHMTZXJYCXHXQHMTZXTHMTZQYDXEYEXGYFWNXEYEVACHAX + DFVBVCOWPXGYFVADHBXFGVBVCOVDXQXTHMVEVFYAXJHMVHVIYCWHXNLZEHRZSZHMTZXMYBYIH + MXNMNZYAXJYIYKYAQYGXIXNNZXJYHYAYGXPXSQZYKYLYGXOXRQYAYMABXNVQXOXPXRXSVJVKY + KXPXSYLXDXFXNVLVMVNXJYHYLXBXIXNVOVPVRVSVTWHKLXMYJVAABKCDWAHWHXBEVBVCOWBWC + WDWEWFWG $. $} $( The span of a union is the subspace sum of spans. (Contributed by NM, @@ -465375,12 +465510,12 @@ equals the join of their closures (double orthocomplements). (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) $) spansni $p |- ( span ` { A } ) = ( _|_ ` ( _|_ ` { A } ) ) $= ( vx vz vy csn cspn cfv cort chba wcel wss snssi spanssoc mp2b cv csm csh - cc wi co wceq wrex wral wa elexi snss shmulcl 3expia ancoms syl5bir eleq1 - imbi2d syl5ibrcom ralrimdva rexlimiv h1de2ci wb vex elspani 3imtr4i ssriv - eqssi ) AFZGHZVDIHIHZAJKZVDJLZVEVFLBAJMZVDNOCVFVECPZDPZAQUAZUBZDSUCVDEPZL - ZVJVNKZTZERUDZVJVFKVJVEKZVMVRDSVKSKZVMVQERVTVNRKZUEZVQVMVOVLVNKZTVOAVNKZW - BWCAVNAJBUFUGWAVTWDWCTWAVTWDWCVKAVNUHUIUJUKVMVPWCVOVJVLVNULUMUNUOUPDVJABU - QVGVHVSVRURBVIEVDVJCUSUTOVAVBVC $. + cc wi co wceq wrex wral elexi snss shmulcl 3expia ancoms biimtrrid imbi2d + wa eleq1 syl5ibrcom ralrimdva rexlimiv h1de2ci wb vex elspani ssriv eqssi + 3imtr4i ) AFZGHZVDIHIHZAJKZVDJLZVEVFLBAJMZVDNOCVFVECPZDPZAQUAZUBZDSUCVDEP + ZLZVJVNKZTZERUDZVJVFKVJVEKZVMVRDSVKSKZVMVQERVTVNRKZULZVQVMVOVLVNKZTVOAVNK + ZWBWCAVNAJBUEUFWAVTWDWCTWAVTWDWCVKAVNUGUHUIUJVMVPWCVOVJVLVNUMUKUNUOUPDVJA + BUQVGVHVSVRURBVIEVDVJCUSUTOVCVAVB $. $( Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) $) @@ -468256,7 +468391,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. $( A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) $) hon0 $p |- ( T : ~H --> ~H -> -. T = (/) ) $= - ( chba wf c0 wceq c0v ax-hv0cl n0ii wfn fn0 wi ffn fndmu syl syl5bir mtoi + ( chba wf c0 wceq c0v ax-hv0cl n0ii wfn fn0 wi ffn fndmu syl biimtrrid mtoi ex ) BBACZADEZBDEZFBGHSADIZRTAJRABIZUATKBBALUBUATBDAMQNOP $. ${ @@ -472649,14 +472784,14 @@ Positive operators (cont.) ( vx cho wcel wa ch0o cleo wbr co cc0 cfv csp chba wral cr hmopre leoppos cle wceq chos cv r19.26 caddc wi addge0 ex syl2an anandirs wf anim12i cva hmopf w3a hosval oveq1d 3expa adantlr adantll simpr ax-his2 syl3anc eqtrd - ffvelcdm sylan breq2d sylibrd ralimdva syl5bir bi2anan9 hmops syl 3imtr4d - wb imp ) ADEZBDEZFZGAHIZGBHIZFZGABUAJZHIZVRKCUBZALZWDMJZSIZCNOZKWDBLZWDMJ - ZSIZCNOZFZKWDWBLZWDMJZSIZCNOZWAWCWMWGWKFZCNOVRWQWGWKCNUCVRWRWPCNVRWDNEZFZ - WRKWFWJUDJZSIZWPVPVQWSWRXBUEZVPWSFWFPEZWJPEZXCVQWSFWDAQWDBQXDXEFWRXBWFWJU - FUGUHUIWTWOXAKSVRNNAUJZNNBUJZFZWSWOXATVPXFVQXGAUMBUMUKXHWSFZWOWEWIULJZWDM - JZXAXFXGWSWOXKTXFXGWSUNWNXJWDMWDABUOUPUQXIWENEZWINEZWSXKXATXFWSXLXGNNWDAV - DURXGWSXMXFNNWDBVDUSXHWSUTWEWIWDVAVBVCVEVFVGVHVIVPVSWHVQVTWLCARCBRVJVRWBD - EWCWQVNABVKCWBRVLVMVO $. + ffvelcdm sylan breq2d sylibrd ralimdva biimtrrid bi2anan9 syl 3imtr4d imp + wb hmops ) ADEZBDEZFZGAHIZGBHIZFZGABUAJZHIZVRKCUBZALZWDMJZSIZCNOZKWDBLZWD + MJZSIZCNOZFZKWDWBLZWDMJZSIZCNOZWAWCWMWGWKFZCNOVRWQWGWKCNUCVRWRWPCNVRWDNEZ + FZWRKWFWJUDJZSIZWPVPVQWSWRXBUEZVPWSFWFPEZWJPEZXCVQWSFWDAQWDBQXDXEFWRXBWFW + JUFUGUHUIWTWOXAKSVRNNAUJZNNBUJZFZWSWOXATVPXFVQXGAUMBUMUKXHWSFZWOWEWIULJZW + DMJZXAXFXGWSWOXKTXFXGWSUNWNXJWDMWDABUOUPUQXIWENEZWINEZWSXKXATXFWSXLXGNNWD + AVDURXGWSXMXFNNWDBVDUSXHWSUTWEWIWDVAVBVCVEVFVGVHVIVPVSWHVQVTWLCARCBRVJVRW + BDEWCWQVNABVOCWBRVKVLVM $. $( The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed @@ -472724,12 +472859,12 @@ Positive operators (cont.) leoptr $p |- ( ( ( S e. HrmOp /\ T e. HrmOp /\ U e. HrmOp ) /\ ( S <_op T /\ T <_op U ) ) -> S <_op U ) $= ( vx cho wcel w3a cleo wbr wa cfv csp co cle chba wral cr hmopre wb leop2 - cv r19.26 wi letr syl3an ralimdva syl5bir 3adant3 3adant1 anbi12d 3adant2 - 3anandirs 3imtr4d imp ) AEFZBEFZCEFZGZABHIZBCHIZJZACHIZURDUAZAKVCLMZVCBKV - CLMZNIZDOPZVEVCCKVCLMZNIZDOPZJZVDVHNIZDOPZVAVBVKVFVIJZDOPURVMVFVIDOUBURVN - VLDOUOUPUQVCOFZVNVLUCZUOVOJVDQFUPVOJVEQFUQVOJVHQFVPVCARVCBRVCCRVDVEVHUDUE - ULUFUGURUSVGUTVJUOUPUSVGSUQDABTUHUPUQUTVJSUODBCTUIUJUOUQVBVMSUPDACTUKUMUN - $. + cv r19.26 letr syl3an 3anandirs biimtrrid 3adant3 3adant1 anbi12d 3adant2 + wi ralimdva 3imtr4d imp ) AEFZBEFZCEFZGZABHIZBCHIZJZACHIZURDUAZAKVCLMZVCB + KVCLMZNIZDOPZVEVCCKVCLMZNIZDOPZJZVDVHNIZDOPZVAVBVKVFVIJZDOPURVMVFVIDOUBUR + VNVLDOUOUPUQVCOFZVNVLUKZUOVOJVDQFUPVOJVEQFUQVOJVHQFVPVCARVCBRVCCRVDVEVHUC + UDUEULUFURUSVGUTVJUOUPUSVGSUQDABTUGUPUQUTVJSUODBCTUHUIUOUQVBVMSUPDACTUJUM + UN $. $( A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) @@ -476640,17 +476775,17 @@ p C_ ( A vH B ) ) /\ A =/= 0H ) ( vc cv chj co wss wa cat wi wral cch wcel wrex cin chjcomi sseq2i anbi2i cdmd wbr ssin bitri mdsymlem5 sseq1 chincl mpan2 chub2 sylancr sstr2 syl5 weq impd a1i com13 adantrr ad2ant2r adantll com12 expd pm2.61d2 rexlimivv - syl6bi imim2d com34 imp4b syl5bir ex ralimdva wb sylancl chrelat3 syl2anc - chjcli chjcl adantr sylibrd com3r ralrimiv dmdbr2 mp2an sylibr ) FKZABLMZ - NZWIEKZDKZLMNZWLANZWMBNZOZOZDPUAEPUAZQZFPRZAJKZNZXBBALMZUBZXBBUBZALMZNZQZ - JSRZBAUFUGZXAXIJSXBSTZXCXAXHXLXCXAXHQXLXCOZXAWIXENZWIXGNZQZFPRZXHXMWTXPFP - XMWIPTZOZWTXPXNWIXBNZWKOZXSWTOXOYAXTWIXDNZOXNWKYBXTWJXDWIABGHUCUDUEWIXBXD - UHUIXSWTXTWKXOXSWTWKXTXOXSWSXTXOQZWKWSXSYCWRXSYCQZEDPPWLPTWMPTOEFURZWRYDQ - ABCDEFJGHIUJYEWRXSYCWRXSOYEYCWQXSYEYCQZWNWOXMYFWPXRWOXLYFXCXTYEWOXLOZXOYE - YGXOQQXTYEWOXLXOYEWOWIANZXLXOQWLWIAUKXLAXGNZYHXOXLASTZXFSTZYIGXLBSTZYKHXB - BULUMZAXFUNUOWIAXGUPUQVIUSUTVAVBVCVDVEVFVGVHVEVJVKVLVMVNVOXLXHXQVPZXCXLXE - STZXGSTZYNXLXDSTYOBAHGVTXBXDULUMXLYKYJYPYMGXFAWAVQFXEXGVRVSWBWCVNWDWEYLYJ - XKXJVPHGJBAWFWGWH $. + syl6bi imim2d com34 imp4b biimtrrid ex ralimdva wb chjcl sylancl chrelat3 + chjcli syl2anc adantr sylibrd com3r ralrimiv dmdbr2 mp2an sylibr ) FKZABL + MZNZWIEKZDKZLMNZWLANZWMBNZOZOZDPUAEPUAZQZFPRZAJKZNZXBBALMZUBZXBBUBZALMZNZ + QZJSRZBAUFUGZXAXIJSXBSTZXCXAXHXLXCXAXHQXLXCOZXAWIXENZWIXGNZQZFPRZXHXMWTXP + FPXMWIPTZOZWTXPXNWIXBNZWKOZXSWTOXOYAXTWIXDNZOXNWKYBXTWJXDWIABGHUCUDUEWIXB + XDUHUIXSWTXTWKXOXSWTWKXTXOXSWSXTXOQZWKWSXSYCWRXSYCQZEDPPWLPTWMPTOEFURZWRY + DQABCDEFJGHIUJYEWRXSYCWRXSOYEYCWQXSYEYCQZWNWOXMYFWPXRWOXLYFXCXTYEWOXLOZXO + YEYGXOQQXTYEWOXLXOYEWOWIANZXLXOQWLWIAUKXLAXGNZYHXOXLASTZXFSTZYIGXLBSTZYKH + XBBULUMZAXFUNUOWIAXGUPUQVIUSUTVAVBVCVDVEVFVGVHVEVJVKVLVMVNVOXLXHXQVPZXCXL + XESTZXGSTZYNXLXDSTYOBAHGVTXBXDULUMXLYKYJYPYMGXFAVQVRFXEXGVSWAWBWCVNWDWEYL + YJXKXJVPHGJBAWFWGWH $. $( Lemma for ~ mdsymi . Lemma 4(i) of [Maeda] p. 168. Note that Maeda's 1965 definition of dual modular pair has reversed arguments compared to @@ -476797,21 +476932,21 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, elin cch spansnj spansnch chjcom sylan2 eqtrd mpan adantr sylibrd expdimp wb com12 ssid sneq fveq2d spansn0 oveq2d shs0i inss1 chub2i ssini chincli c0h eqssi chjcomi chabs1i eqtri mpbiri pm2.61d2 adantrr sumdmdlem shsub2i - eqsstrdi adantl sstrd sseld syl5bir mpand exp32 com34 pm2.18 syl8 pm2.43d - syl5 ssrdv chsleji jctil eqss sylibr ) AUBZCFGZBCFGZHZXQBHZCFGZIZAJUCZBCK - GZXRIZXRYDIZLYDXRMYCYFYEYCUAXRYDYCUAUBZXRNZYGYDNZYHYGUDNZYCYHYIOZYGXRBCDE - UEUFYCYJYHYIUGZYIOYIYCYJYLYHYIYCYJYLYKYCYJYLLZLZYGCYGUHZPUIZKGZNZYHYIYJYR - YCYLYJYPYQYGYJYPUJNCUJNYPYQIYGUKCEULZYPCUOUMYGUNUPUQYRYHLYGYQXRHZNYNYIYGY - QXRVFYNYTYDYGYNYTYQBHZCFGZYDYCYJYTUUBIZYLYCYJLYGQMZUUCYCYJUUDUGZUUCYJUUEL - ZYCUUCUUFYCYPCFGZXRHZUUGBHZCFGZIZUUCUUFYPJNZYCUUKOUUEYJYGQURUULYGQUSYGUTV - AYBUUKAYPJXPYPMZXSUUHYAUUJUUMXQUUGXRXPYPCFVBZRUUMXTUUICFUUMXQUUGBUUNRSVCV - DVEYJUUCUUKVQUUEYJYTUUHUUBUUJYJYQUUGXRCVGNZYJYQUUGMEUUOYJLYQCYPFGZUUGCYGV - HYJUUOYPVGNUUPUUGMYGVICYPVJVKVLVMZRYJUUAUUICFYJYQUUGBUUQRSVCVNVOVRVPUUDUU - CCCICVSZUUDYTCUUBCUUDYTCXRHZCUUDYQCXRUUDYQCWIKGCUUDYPWICKUUDYPQUHZPUIWIUU - DYOUUTPYGQVTWAWBTWCCYSWDTZRUUSCCXRWECCXRUURCBEDWFWGWJTUUDUUBCBHZCFGZCUUDU - UAUVBCFUUDYQCBUVARSUVCCUVBFGCUVBCCBEDWHEWKCBEDWLWMZTVCWNWOWPYMUUBYDIYCYMU - UBCYDYMUUBUVCCYMUUAUVBCFBCYGDEWQSUVDTCBYSBDULWRWSWTXAXBXCXDXEXFYIXGXHXJXI - XKBCDEXLXMYDXRXNXO $. + eqsstrdi adantl sstrd sseld biimtrrid mpand exp32 com34 syl8 syl5 pm2.43d + pm2.18 ssrdv chsleji jctil eqss sylibr ) AUBZCFGZBCFGZHZXQBHZCFGZIZAJUCZB + CKGZXRIZXRYDIZLYDXRMYCYFYEYCUAXRYDYCUAUBZXRNZYGYDNZYHYGUDNZYCYHYIOZYGXRBC + DEUEUFYCYJYHYIUGZYIOYIYCYJYLYHYIYCYJYLYKYCYJYLLZLZYGCYGUHZPUIZKGZNZYHYIYJ + YRYCYLYJYPYQYGYJYPUJNCUJNYPYQIYGUKCEULZYPCUOUMYGUNUPUQYRYHLYGYQXRHZNYNYIY + GYQXRVFYNYTYDYGYNYTYQBHZCFGZYDYCYJYTUUBIZYLYCYJLYGQMZUUCYCYJUUDUGZUUCYJUU + ELZYCUUCUUFYCYPCFGZXRHZUUGBHZCFGZIZUUCUUFYPJNZYCUUKOUUEYJYGQURUULYGQUSYGU + TVAYBUUKAYPJXPYPMZXSUUHYAUUJUUMXQUUGXRXPYPCFVBZRUUMXTUUICFUUMXQUUGBUUNRSV + CVDVEYJUUCUUKVQUUEYJYTUUHUUBUUJYJYQUUGXRCVGNZYJYQUUGMEUUOYJLYQCYPFGZUUGCY + GVHYJUUOYPVGNUUPUUGMYGVICYPVJVKVLVMZRYJUUAUUICFYJYQUUGBUUQRSVCVNVOVRVPUUD + UUCCCICVSZUUDYTCUUBCUUDYTCXRHZCUUDYQCXRUUDYQCWIKGCUUDYPWICKUUDYPQUHZPUIWI + UUDYOUUTPYGQVTWAWBTWCCYSWDTZRUUSCCXRWECCXRUURCBEDWFWGWJTUUDUUBCBHZCFGZCUU + DUUAUVBCFUUDYQCBUVARSUVCCUVBFGCUVBCCBEDWHEWKCBEDWLWMZTVCWNWOWPYMUUBYDIYCY + MUUBCYDYMUUBUVCCYMUUAUVBCFBCYGDEWQSUVDTCBYSBDULWRWSWTXAXBXCXDXEXFYIXJXGXH + XIXKBCDEXLXMYDXRXNXO $. $( The subspace sum of two Hilbert lattice elements is closed iff the elements are a dual modular pair. Theorem 2 of [Holland] p. 1519. @@ -476961,17 +477096,17 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, weq wreu biimpi reeanv eqtr2 cmv wb anim12i hvaddsub4 syl2an an4s adantll shsubcl mp3an1 ancoms eleq1 syl5ibrcom adantl adantr jctild eleq2 bitr3id csh elin ad2antrr sylibd elch0 hvsubeq0 bitrid ad2antlr sylbid rexlimdvva - syl5 syl5bir ralrimivva oveq1 eqeq2d rexbidv oveq2 cbvrexvw bitrdi sylibr - c0v reu4 ) ECDUAJKZCDUBZLMZNEAOZBOZPJZMZBDQZACQZWPEHOZIOZPJZMZIDQZNZAHUEZ - RZHCUCACUCZNWPACUFWIWQWKXFWIWQABCDEFGUDUGWKXEAHCCXCWOXANZIDQBDQWKWLCKZWRC - KZNZNZXDWOXABIDDUHXKXGXDBIDDXGWNWTMZXKWMDKZWSDKZNZNZXDEWNWTUIXPXLWLWRUJJZ - WSWMUJJZMZXDXJXOXLXSUKZWKXHXMXIXNXTXHXMNWLSKZWMSKZNWRSKZWSSKZNXTXIXNNXHYA - XMYBWLCFTZWMDGTULXIYCXNYDWRCFTZWSDGTULWLWMWRWSUMUNUOUPXPXSXQLKZXDXPXSXQCK - ZXQDKZNZYGXJXOXSYJRWKXJXONXSYIYHXOXSYIRXJXOYIXSXRDKZXNXMYKDVGKXNXMYKGWSWM - DUQURUSXQXRDUTVAVBXJYHXOCVGKXHXIYHFWLWRCUQURVCVDUPWKYJYGUKXJXOYJXQWJKWKYG - XQCDVHWJLXQVEVFVIVJXJYGXDUKZWKXOXHYAYCYLXIYEYFYGXQWGMYAYCNXDXQVKWLWRVLVMU - NVNVJVOVQVPVRVSULWPXBAHCXDWPEWRWMPJZMZBDQXBXDWOYNBDXDWNYMEWLWRWMPVTWAWBYN - XABIDBIUEYMWTEWMWSWRPWCWAWDWEWHWF $. + syl5 biimtrrid ralrimivva oveq1 eqeq2d rexbidv oveq2 cbvrexvw bitrdi reu4 + c0v sylibr ) ECDUAJKZCDUBZLMZNEAOZBOZPJZMZBDQZACQZWPEHOZIOZPJZMZIDQZNZAHU + EZRZHCUCACUCZNWPACUFWIWQWKXFWIWQABCDEFGUDUGWKXEAHCCXCWOXANZIDQBDQWKWLCKZW + RCKZNZNZXDWOXABIDDUHXKXGXDBIDDXGWNWTMZXKWMDKZWSDKZNZNZXDEWNWTUIXPXLWLWRUJ + JZWSWMUJJZMZXDXJXOXLXSUKZWKXHXMXIXNXTXHXMNWLSKZWMSKZNWRSKZWSSKZNXTXIXNNXH + YAXMYBWLCFTZWMDGTULXIYCXNYDWRCFTZWSDGTULWLWMWRWSUMUNUOUPXPXSXQLKZXDXPXSXQ + CKZXQDKZNZYGXJXOXSYJRWKXJXONXSYIYHXOXSYIRXJXOYIXSXRDKZXNXMYKDVGKXNXMYKGWS + WMDUQURUSXQXRDUTVAVBXJYHXOCVGKXHXIYHFWLWRCUQURVCVDUPWKYJYGUKXJXOYJXQWJKWK + YGXQCDVHWJLXQVEVFVIVJXJYGXDUKZWKXOXHYAYCYLXIYEYFYGXQWGMYAYCNXDXQVKWLWRVLV + MUNVNVJVOVQVPVRVSULWPXBAHCXDWPEWRWMPJZMZBDQXBXDWOYNBDXDWNYMEWLWRWMPVTWAWB + YNXABIDBIUEYMWTEWMWSWRPWCWAWDWEWFWH $. $} ${ @@ -477451,6 +477586,13 @@ and the expression ( x e. A /\ x e. B ) ` . ( wo w3o df-3or sylbir olcs ) ABFZCDKCFABCGDABCHEIJ $. $} + $( Associative law for four conjunctions with a triple conjunction. + (Contributed by Thierry Arnoux, 21-Jan-2025.) $) + 13an22anass $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ch + /\ th ) ) ) $= + ( wa w3a an2anr an4 bitri an43 3bitr2ri 3an4anass ancom ) ABECDEEZBCEDAEEZB + CDFZAEAPEOCBEADEEZACEDBEEZNBCDAGRCAEBDEEQACDBGCABDHIACDBJKBCDALPAMK $. + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -479018,21 +479160,21 @@ Class abstractions (a.k.a. class builders) leltne syl3an vex nfcsb1v nfcv nfiun nfdif csbeq1a oveq2 iuneq1d difeq12d wi csbief ineq12i cuz cfv simp1 nnuz eleqtrdi simp2 nnzd elfzo2 syl3anbrc simp3 nfcsbw csbhypf equcoms eqcomd ssiun2sf syl ssdifssd ssrind eqsstrid - cz disjdif sylancl 3expia 3adant3 sylbird syl5bir orrd adantl wlogle mpan - sseq0 rgen2 disjors mpbir ) DLACMDNZOUFZBUAZUBZUCHIPZDHNZYFQZDINZYFQZRZSU - DZUGZILUEHLUEYNHILLUHYHLTZYJLTZURZYNUIUHJKPZDJNZYFQZDKNZYFQZRZSUDZUGYNYNH - 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orcom ralbii r19.30 risset biorf sylnbi adantl syl5ibr syl5bir olc ralimi - wrex impbid1 nfv nfcsb1v nfcv nfin nfeq1 csbeq1a cbvralw a1i iunss iunin1 - wss eqeq1i 3bitr3ri bitri nfcvd csbiegf 3bitr4d bitr4d anbi2d clel5 incom - ss0b anass anidm anbi2i ) EFUDZEBUDZUAZJZABEUBUCZCUEZABCUEZHUFZEKZAXLCUGZ - AECUGZLZMKZNZHBOZJZEIUFZKZXOAYACUGZLZMKZNZIBOZJZXKABCUHDLZMKZJZXEXJYHPXGX - EXJXLYAKZXNYCLZMKZNZIBOZXRJZHBOZYGJZYHXEXJYOIXIOZHBOZYFIXIOZJZYSXJYTHXIOX - EUUCAXICHIUIYTUUBHBEFXMYOYFIXIXMYLYBYNYEXLEYAUJXMYMYDMXMXNXOYCAXLECUKQRUL - SUMUNXEUUAYRUUBYGXEYTYQHBYOXRIBEFYAEKZYLXMYNXQYAEXLUPUUDYMXPMUUDYCXOXNAYA - ECUKZUORULUMSXEUUBYGEEKZXOXOLZMKZNZJYGYFUUIIBEFUUDYBUUFYEUUHYAEEUPUUDYDUU - GMUUDYCXOXOUUEUORULUMUUIYGUUFUUHEUQURUSUTVAVBYRXTYGYRYPHBOZXSJXTYPXRHBVCX - KUUJXSABCHIUIVDVEVDTVFXHYHYKYJJZYKXHXTYKYGYJXHXSYJXKXHXSXQHBOZYJXHXSUULXS - XQXMNZHBOZXHUULUUMXRHBXQXMVGVHUUNUULXHUULXMHBVRZNZXQXMHBVIXHUULUUOUULNZUU - PXGUULUUQPZXEXFUUOUURHEBVJUUOUULVKVLVMUUOUULVGTVNVOXQXRHBXQXMVPVQVSXEYJUU - LPXGXECDLZMKZABOZXNDLZMKZHBOZYJUULUVAUVDPXEUUTUVCAHBUUTHVTAUVBMAXNDAXLCWA - 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XFXGXHXJUVBUVCXKYOUVEXLXMXNXOXPXSXQXRXTYAYBDYEYIHIYCYD $. + biimtrrid wlogle mpan rgen2 disjors mpbir ) DLEMUFZACLDNZMUFZBUAZUBZUCHIO + ZDHNZYIPZDINZYIPZQZRUDZUGZIYEUEHYEUEYQHIYEYEUHYKYESZYMYESZURZYQUIUHJKOZDJ + NZYIPZDKNZYIPZQZRUDZUGYQYQHIJKYEJHOZKIOZURZUUAYJUUGYPUUBYKUUDYMUJUUJUUFYO + RUUHUUIUUCYLUUEYNDUUBYKYIUKDUUDYMYIUKULUMUNJIOZKHOZURZUUAYJUUGYPUUMUUAIHO + YJUUBYMUUDYKUJIHUOUPUUMUUFYORUUMUUFYNYLQYOUUKUULUUCYNUUEYLDUUBYMYIUKDUUDY + KYIUKULYNYLUQUSUMUNYETUTUHYEVATEVBZVCVDZVEUHYTURYQVFYRYSYKYMVGVHZVIZYQUHU + UQYJYPYJVMYMYKVJZUUQYPYMYKVKUUQUURYKYMVLVHZYPYRYKTSYSYMTSUUPUUPUUSUURVNYE + TYKUUOVOYETYMUUOVOUUPVPYKYMVQVRYRYSUUSYPWFUUPYRYSUUSYPYRYSUUSVIZYOCLYMMUF + ZBUAZDYMAPZUVBUBZQZUTUVERUDYPUUTYODYKAPZCLYKMUFZBUAZUBZUVDQUVEYLUVIYNUVDD + YKYIUVIHVSDUVFUVHDYKAVTCDUVGBDUVGWAFWBWCDHOZAUVFYHUVHDYKAWDUVJCYGUVGBYFYK + LMWGWEWHWIDYMYIUVDIVSDUVCUVBDYMAVTCDUVABDUVAWAFWBWCDIOZAUVCYHUVBDYMAWDUVK + CYGUVABYFYMLMWGWEWHWIWJUUTUVIUVBUVDUUTUVFUVBUVHUUTYKUVASZUVFUVBUTUUTYKLWK + WLZSYMWRSUUSUVLUUTYKVAUVMUUTYEVAYKUUNYRYSUUSWMWNWOWPUUTYMUUTYEVAYMUUNYRYS + UUSWQWNWSYRYSUUSWTYKLYMXAXBCUVABYKUVFCHOUVFBUVFBUDHCDHCNZABDUVNWAFGXCXDXE + XIXFXGXHXJUVBUVCXKYOUVEXLXMXNXOXPXSXQXRXTYAYBDYEYIHIYCYD $. $} ${ @@ -482726,18 +482868,18 @@ its graph has a given second element (that is, function value). wss 1nn0 fz1ssfz0 sstrdi sselda bcval2 nnnn0d elfzelz adantl bccl syl2anc fzss2 nn0zd eqeltrrd clt cn elfznn 1zzd simpr cle elfzm11 biimpa ltsubnn0 w3a simp3d faccld zmulcld zcnd wne facne0 mulne0d dvdsfac nn0red ltsubrpd - uzid nnrpd prmndvdsfaclt wo ioran euclemma con3d syl5bir syl32anc dvdszzq - biimpd breqtrrd ) AUACZBDADUBEZFEZCZGZAAHIZABUBEZHIZBHIZUCEZUDEZABUEEZJXC - XDXHAXIXIUFWSXBUGZXCXJXIKXCBLAFEZCXJXIUHWSXAXLBWSXADAFEZXLWSAWTUIICZXAXMU - MWSAKCZDMCXNWSAAUJZNZUNADUKULWTDAVDOAUOUPUQBAUROZXCXJXCAMCZBKCZXJMCWSXSXB - WSAXPUSPZXBXTWSBDWTUTVABAVBVCVEVFXCXFXGXCXFXCXEXCXSBMCZBAVGQZXEMCZYAXCBXB - BVHCWSBWTVIVAZUSZXCDKCZXOXBYCXCVJWSXOXBXQPWSXBVKYGXOGZXBGXTDBVLQZYCYHXBXT - YIYCVPBDAVMVNVQRZXSYBGYCYDABVOSRZVRNZXCXGXCBYFVRNZVSXCXFXGXCXFYLVTXCXGYMV - TXCYDXFLWAYKXEWBOXCYBXGLWAYFBWBOWCWSAXDJQZXBWSAVHCAAUIICZYNXPWSXOYOXQAWGO - AAWDVCPXCWSXFKCZXGKCZAXFJQZTZAXGJQZTZAXHJQZTZXKYLYMXCWSYDXEAVGQZYSXKYKXCA - BXCAYAWEXCBYEWHWFWSYDGUUDYSAXEWISRXCWSYBYCUUAXKYFYJWSYBGYCUUAABWISRWSYPYQ - VPZYSUUAGZUUCUUFYRYTWJZTUUEUUCYRYTWKUUEUUBUUGUUEUUBUUGAXFXGWLWQWMWNSWOWPX - RWR $. + uzid nnrpd prmndvdsfaclt wo ioran euclemma biimpd con3d biimtrrid dvdszzq + syl32anc breqtrrd ) AUACZBDADUBEZFEZCZGZAAHIZABUBEZHIZBHIZUCEZUDEZABUEEZJ + XCXDXHAXIXIUFWSXBUGZXCXJXIKXCBLAFEZCXJXIUHWSXAXLBWSXADAFEZXLWSAWTUIICZXAX + MUMWSAKCZDMCXNWSAAUJZNZUNADUKULWTDAVDOAUOUPUQBAUROZXCXJXCAMCZBKCZXJMCWSXS + XBWSAXPUSPZXBXTWSBDWTUTVABAVBVCVEVFXCXFXGXCXFXCXEXCXSBMCZBAVGQZXEMCZYAXCB + XBBVHCWSBWTVIVAZUSZXCDKCZXOXBYCXCVJWSXOXBXQPWSXBVKYGXOGZXBGXTDBVLQZYCYHXB + XTYIYCVPBDAVMVNVQRZXSYBGYCYDABVOSRZVRNZXCXGXCBYFVRNZVSXCXFXGXCXFYLVTXCXGY + MVTXCYDXFLWAYKXEWBOXCYBXGLWAYFBWBOWCWSAXDJQZXBWSAVHCAAUIICZYNXPWSXOYOXQAW + GOAAWDVCPXCWSXFKCZXGKCZAXFJQZTZAXGJQZTZAXHJQZTZXKYLYMXCWSYDXEAVGQZYSXKYKX + CABXCAYAWEXCBYEWHWFWSYDGUUDYSAXEWISRXCWSYBYCUUAXKYFYJWSYBGYCUUAABWISRWSYP + YQVPZYSUUAGZUUCUUFYRYTWJZTUUEUUCYRYTWKUUEUUBUUGUUEUUBUUGAXFXGWLWMWNWOSWQW + PXRWR $. $} $( Numerator and denominator of the negative. (Contributed by Thierry @@ -494225,7 +494367,7 @@ such that every prime ideal contains a prime element (this is a fply1.3 $e |- P = ( Base ` ( Poly1 ` R ) ) $. fply1.4 $e |- ( ph -> F : ( NN0 ^m 1o ) --> B ) $. fply1.5 $e |- ( ph -> F finSupp .0. ) $. - $( Conditions for a function to be an univariate polynomial. (Contributed + $( Conditions for a function to be a univariate polynomial. (Contributed by Thierry Arnoux, 19-Aug-2023.) $) fply1 $p |- ( ph -> F e. P ) $= ( vf c1o co cfv wcel cfn cn0 cmap eqid cmps cbs cfsupp wbr ccnv cima crab @@ -494257,6 +494399,282 @@ such that every prime ideal contains a prime element (this is a PZVEAVQWCWEKMBDECGHJIUQTURUTVFVEAFGBUDVAVB $. $} + ${ + evls1scafv.q $e |- Q = ( S evalSub1 R ) $. + evls1scafv.w $e |- W = ( Poly1 ` U ) $. + evls1scafv.u $e |- U = ( S |`s R ) $. + evls1scafv.b $e |- B = ( Base ` S ) $. + evls1scafv.a $e |- A = ( algSc ` W ) $. + evls1scafv.s $e |- ( ph -> S e. CRing ) $. + evls1scafv.r $e |- ( ph -> R e. ( SubRing ` S ) ) $. + evls1scafv.x $e |- ( ph -> X e. R ) $. + evls1scafv.1 $e |- ( ph -> C e. B ) $. + $( Value of the univariate polynomial evaluation for scalars. (Contributed + by Thierry Arnoux, 21-Jan-2025.) $) + evls1scafv $p |- ( ph -> ( ( Q ` ( A ` X ) ) ` C ) = X ) $= + ( cfv csn cxp evls1sca fveq1d wcel wceq fvconst2g syl2anc eqtrd ) ADJBTET + ZTDCJUAUBZTZJADUJUKABCEFGHIJKLMNOPQRUCUDAJFUEDCUEULJUFRSCJDFUGUHUI $. + $} + + ${ + evls1expd.q $e |- Q = ( S evalSub1 R ) $. + evls1expd.k $e |- K = ( Base ` S ) $. + evls1expd.w $e |- W = ( Poly1 ` U ) $. + evls1expd.u $e |- U = ( S |`s R ) $. + evls1expd.b $e |- B = ( Base ` W ) $. + evls1expd.s $e |- ( ph -> S e. CRing ) $. + evls1expd.r $e |- ( ph -> R e. ( SubRing ` S ) ) $. + evls1expd.1 $e |- ./\ = ( .g ` ( mulGrp ` W ) ) $. + evls1expd.2 $e |- .^ = ( .g ` ( mulGrp ` S ) ) $. + evls1expd.n $e |- ( ph -> N e. NN0 ) $. + evls1expd.m $e |- ( ph -> M e. B ) $. + evls1expd.c $e |- ( ph -> C e. K ) $. + $( Univariate polynomial evaluation builder for an exponential. See also + ~ evl1expd . (Contributed by Thierry Arnoux, 24-Jan-2025.) $) + evls1expd $p |- ( ph + -> ( ( Q ` ( N ./\ M ) ) ` C ) = ( N .^ ( ( Q ` M ) ` C ) ) ) $= + ( co cfv cpws cmgp cmg eqid evls1pw fveq1d cbs cvv crngringd wcel a1i crh + fvexi wf ccrg csubrg evls1rhm syl2anc rhmf syl ffvelcdmd pwsexpg eqtrd ) + ACLJKUFDUGZUGCLJDUGZFIUHUFZUIUGZUJUGZUFZUGLCVLUGHUFACVKVPABDEFGKMUIUGZILM + JNQPVQUKORUASTUCUDULUMACVMUNUGZFVOFUIUGZHIVNLUOVLVMVMUKZVRUKZVNUKVSUKVOUK + UBAFSUPIUOUQAIFUNOUTURUCABVRJDADMVMUSUFUQZBVRDVAAFVBUQEFVCUGUQWBSTIDEFVMG + MNOVTQPVDVEBVRMVMDRWAVFVGUDVHUEVIVJ $. + $} + + ${ + evls1varpwval.q $e |- Q = ( S evalSub1 R ) $. + evls1varpwval.u $e |- U = ( S |`s R ) $. + evls1varpwval.w $e |- W = ( Poly1 ` U ) $. + evls1varpwval.x $e |- X = ( var1 ` U ) $. + evls1varpwval.b $e |- B = ( Base ` S ) $. + evls1varpwval.e $e |- ./\ = ( .g ` ( mulGrp ` W ) ) $. + evls1varpwval.f $e |- .^ = ( .g ` ( mulGrp ` S ) ) $. + evls1varpwval.s $e |- ( ph -> S e. CRing ) $. + evls1varpwval.r $e |- ( ph -> R e. ( SubRing ` S ) ) $. + evls1varpwval.n $e |- ( ph -> N e. NN0 ) $. + evls1varpwval.c $e |- ( ph -> C e. B ) $. + $( Univariate polynomial evaluation for subrings maps the exponentiation of + a variable to the exponentiation of the evaluated variable. See + ~ evl1varpwval . (Contributed by Thierry Arnoux, 24-Jan-2025.) $) + evls1varpwval $p |- ( ph -> ( ( Q ` ( N ./\ X ) ) ` C ) = ( N .^ C ) ) + $= + ( co cfv cbs eqid csubrg wcel crg subrgring vr1cl 3syl evls1expd cid cres + evls1var fveq1d wceq fvresi syl eqtrd oveq2d ) ACJLIUDDUEUEJCLDUEZUEZHUDJ + CHUDAKUFUEZCDEFGHBLIJKMQONVFUGZTUARSUBAEFUHUEUIGUJUILVFUIUAEFGNUKVFKGLPOV + GULUMUCUNAVECJHAVECUOBUPZUEZCACVDVHABDEFGLMPNQTUAUQURACBUIVICUSUCBCUTVAVB + VCVB $. + $} + + ${ + ressply1evl.q $e |- Q = ( S evalSub1 R ) $. + ressply1evl.k $e |- K = ( Base ` S ) $. + ressply1evl.w $e |- W = ( Poly1 ` U ) $. + ressply1evl.u $e |- U = ( S |`s R ) $. + ressply1evl.b $e |- B = ( Base ` W ) $. + + ${ + $d .x. k $. $d .x. x $. $d A i j k $. $d A x $. $d B k $. + $d K k x $. $d M k $. $d Q k x $. $d S k $. $d S x $. $d U i j k $. + $d U x $. $d W i j k $. $d W x $. $d i j k ph $. $d ph x $. + evls1fpws.s $e |- ( ph -> S e. CRing ) $. + evls1fpws.r $e |- ( ph -> R e. ( SubRing ` S ) ) $. + evls1fpws.y $e |- ( ph -> M e. B ) $. + evls1fpws.1 $e |- .x. = ( .r ` S ) $. + evls1fpws.2 $e |- .^ = ( .g ` ( mulGrp ` S ) ) $. + evls1fpws.a $e |- A = ( coe1 ` M ) $. + $( Evaluation of a univariate subring polynomial as a function in a power + series. (Contributed by Thierry Arnoux, 23-Jan-2025.) $) + evls1fpws $p |- ( ph -> ( Q ` M ) = ( x e. K + |-> ( S gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ x ) ) ) ) ) ) $= + ( vj vi cfv cn0 cv cv1 cmgp cmg co cvsca cmpt cgsu crg wcel wceq csubrg + subrgring syl eqid ply1coe syl2anc fveq2d cpws c0g wa csca cbs ply1lmod + clmod adantr coe1fvalcl sylan ply1sca eleqtrd cmnd ply1ring simpr vr1cl + ringmgp mgpbas mulgnn0cl syl3anc lmodvscl ssidd cvv fvexd fveq2 oveq12d + oveq1 clt wbr wral wrex coe1ae0 ad3antrrr eqtrd oveq1d ad4ant13 lmod0vs + wi ex imim2d ralimdva reximdva mpd mptnn0fsuppd evls1gsumadd cascl ccrg + cmulr crh evls1rhm wf ffvelcdmd rhmmul casa subrgcrng ply1assa asclmul1 + asclf cof fvexi a1i rhmf pwsmulrval pwselbas ffnd inidm ad2antrr simplr + wss mpteq2dva nn0ex csupp cfsupp ressbas2 eleqtrrd evls1varpwval offval + subrgss evls1scafv 3eqtr3d oveq2d crngringd sseldd ringcld 3impa 3com23 + ringcmnd 3expb wfun cfn mptexd coe1sfi fsuppimpd cdif csn cxp wfn coe1f + funmpt fvdifsupp subrg0 eqtr4d eldifad ringlz fconstmpt eqtr4di cmnmndd + pws0g suppss2 suppssfifsupp syl32anc pwsgsum 3eqtrd ) AMEUHNJUIJUJZCUHZ + UWAIUKUHZNULUHZUMUHZUNZNUOUHZUNZUPUQUNZEUHZBLGJUIUWBUWABUJZKUNZHUNZUPUQ + UNUPZAMUWIEAIURUSZMDUSZMUWIUTAFGVAUHUSZUWOUAFGIRVBVCZUBCDNIUWGJUWEMUWDU + WCQUWCVDZSUWGVDZUWDVDZUWEVDZUEVEVFVGAUWJGLVHUNZJUIUWHEUHZUPZUQUNUXCJUIB + LUWMUPZUPZUQUNUWNAJDUXCEFGILUINUWHNVIUHZOPQUXHVDZRUXCVDZSTUAAUWAUIUSZVJ + ZNVNUSZUWBNVKUHZVLUHZUSZUWFDUSZUWHDUSAUXMUXKAUWOUXMUWRNIQVMVCZVOUXLUWBI + VLUHZUXOAUWPUXKUWBUXSUSUBCDNIMUXSUWAUESQUXSVDZVPVQZAUXSUXOUTUXKAIUXNVLA + UWOIUXNUTZUWRNIURQVRVCZVGVOVSZUXLUWDVTUSZUXKUWCDUSZUXQUXLNURUSZUYEAUYGU + XKAUWOUYGUWRNIQWAVCZVONUWDUXAWDZVCAUXKWBUXLUWOUYFAUWOUXKUWRVODNIUWCUWSQ + SWCZVCDUWEUWDUWAUWCDNUWDUXASWEZUXBWFWGZUWBUWGUXNUXODNUWFSUXNVDZUWTUXOVD + ZWHWGZAUIWIAUFDUWHUFUJZCUHZUYPUWCUWEUNZUWGUNZJWJUXHUGANVIWKUYOUWAUYPUTU + WBUYQUWFUYRUWGUWAUYPCWLUWAUYPUWCUWEWNWMAUGUJZUYPWOWPZUYQIVIUHZUTZXEZUFU + IWQZUGUIWRZVUAUYSUXHUTZXEZUFUIWQZUGUIWRAUWPVUFUBCDNIUFMVUBUGUESQVUBVDZW + SVCAVUEVUIUGUIAUYTUIUSZVJZVUDVUHUFUIVULUYPUIUSZVJZVUCVUGVUAVUNVUCVUGVUN + VUCVJZUYSUXNVIUHZUYRUWGUNZUXHVUOUYQVUPUYRUWGVUOUYQVUBVUPVUNVUCWBVUOIUXN + VIAUYBVUKVUMVUCUYCWTVGXAXBVUOUXMUYRDUSZVUQUXHUTAUXMVUKVUMVUCUXRWTAVUMVU + RVUKVUCAVUMVJUYEVUMUYFVURAUYEVUMAUYGUYEUYHUYIVCVOAVUMWBAUYFVUMAUWOUYFUW + RUYJVCVODUWEUWDUYPUWCUYKUXBWFWGXCUWGUXNVUPDNUYRUXHSUYMUWTVUPVDUXIXDVFXA + XFXGXHXIXJXKXLAUXEUXGUXCUQAJUIUXDUXFUXLUWBNXMUHZUHZUWFNXOUHZUNZEUHZVUTE + UHZUWFEUHZUXCXOUHZUNZUXDUXFUXLENUXCXPUNUSZVUTDUSUXQVVCVVGUTAVVHUXKAGXNU + SZUWQVVHTUALEFGUXCINOPUXJRQXQVFVOZUXLUXODUWBVUSAUXODVUSXRUXKAVUSDUXNUXO + NVUSVDZUYMUYHUXRUYNSYEVOUYDXSZUYLVUTUWFNUXCVVAVVFEDSVVAVDZVVFVDZXTWGUXL + VVBUWHEUXLNYAUSZUXPUXQVVBUWHUTAVVOUXKAIXNUSZVVOAVVIUWQVVPTUAFGIRYBVFNIQ + YCVCVOUYDUYLVUSUWBUWGVVAUXNUXODNUWFVVKUYMUYNSVVMUWTYDWGVGUXLVVGVVDVVEHY + FUNUXFUXLUXCVLUHZGVVFHVVDVVELXNWJUXCUXJVVQVDZAVVIUXKTVOZLWJUSZUXLLGVLPY + GZYHZUXLDVVQVUTEUXLVVHDVVQEXRVVJDVVQNUXCESVVRYIVCZVVLXSZUXLDVVQUWFEVWCU + YLXSZUCVVNYJUXLBLLUWBUWLHLVVDVVEWJWJUXLLLVVDUXLLGLVVQXNVVDUXCWJUXJPVVRV + VSVWBVWDYKYLUXLLLVVEUXLLGLVVQXNVVEUXCWJUXJPVVRVVSVWBVWEYKYLVWBVWBLYMUXL + UWKLUSZVJZVUSLUWKEFGINUWBOQRPVVKAVVIUXKVWFTYNZAUWQUXKVWFUAYNZUXLUWBFUSV + WFUXLUWBUXSFUYAAFUXSUTZUXKAFLYPZVWJAUWQVWKUAFLGPUUEZVCFLIGRPUUAVCVOUUBZ + VOUXLVWFWBZUUFVWGLUWKEFGIKUWEUWANUWCORQUWSPUXBUDVWHVWIAUXKVWFYOZVWNUUCU + UDXAUUGYQUUHABJLGUWMLUIWJWJUXCUXCVIUHZUXJPVWPVDVVTAVWAYHZUIWJUSZAYRYHZA + GAGTUUIZUUNZAVWFUXKUWMLUSZAUXKVWFVXBAUXKVWFVXBVWGLGHUWBUWLPUCAGURUSZUXK + VWFVWTYNUXLUWBLUSVWFUXLFLUWBUXLUWQVWKAUWQUXKUAVOVWLVCVWMUUJVOVWGGULUHZV + TUSZUXKVWFUWLLUSZAVXEUXKVWFAVXCVXEVWTGVXDVXDVDZWDZVCYNVWOVWNLKVXDUWAUWK + LGVXDVXGPWEUDWFZWGUUKUULUUMUUOAUXGWJUSUXGUUPZVWPWJUSCVUBYSUNZUUQUSUXGVW + PYSUNVXKYPUXGVWPYTWPAJUIUXFWJVWSUURVXJAJUIUXFUVFYHAUXCVIWKACVUBAUWPCVUB + YTWPUBCDNIMVUBUESQVUJUUSVCUUTAUIUXFJWJVXKVWPAUWAUIVXKUVAUSZVJZUXFLGVIUH + ZUVBUVCZVWPVXMUXFBLVXNUPVXOVXMBLUWMVXNVXMVWFVJZUWMVXNUWLHUNZVXNVXPUWBVX + NUWLHVXMUWBVXNUTVWFVXMUWBVUBVXNVXMUICWJWJUWAVUBACUIUVDVXLAUIUXSCAUWPUIU + XSCXRUBCDNIMUXSUESQUXTUVEVCYLVOVWRVXMYRYHVXMIVIWKAVXLWBUVGAVXNVUBUTZVXL + AUWQVXRUAFGIVXNRVXNVDZUVHVCVOUVIVOXBVXPVXCVXFVXQVXNUTAVXCVXLVWFVWTYNZVX + PVXEUXKVWFVXFVXPVXCVXEVXTVXHVCVXPUWAUIVXKAVXLVWFYOUVJVXMVWFWBVXIWGLGHUW + LVXNPUCVXSUVKVFXAYQBLVXNUVLUVMAVXOVWPUTZVXLAGVTUSVVTVYAAGVXAUVNVWQGLWJU + XCVXNUXJVXSUVOVFVOXAVWSUVPVXKUXGWJWJVWPUVQUVRUVSUVTXA $. + $} + + ${ + $d B k m x $. $d E k m $. $d E k x $. $d K f x $. $d K k $. + $d K y $. $d Q k m $. $d Q k x $. $d R f x $. $d S f g x $. + $d S k $. $d U f x $. $d U k $. $d W k $. $d W x $. $d k m ph $. + $d k ph x $. + ressply1evl.e $e |- E = ( eval1 ` S ) $. + ressply1evl.s $e |- ( ph -> S e. CRing ) $. + ressply1evl.r $e |- ( ph -> R e. ( SubRing ` S ) ) $. + $( Evaluation of a univariate subring polynomial is the same as the + evaluation in the bigger ring. (Contributed by Thierry Arnoux, + 23-Jan-2025.) $) + ressply1evl $p |- ( ph -> Q = ( E |` B ) ) $= + ( cfv wcel eqid vm vx vk cres wceq cv wral wa cn0 cco1 cmgp cmg co cmpt + cmulr cgsu cress cpl1 cbs evl1fval1 adantr csubrg crg crngringd subrgid + ccrg syl cps1 ressply1bas2 inss2 eqsstrdi ressid fveq2d sseqtrrd sselda + cin evls1fpws simpr eqtr4d ralrimiva wfn wss cpws crh evl1rhm rhmf 3syl + wb wf ffnd evls1rhm syl2anc fvreseq1 syl21anc mpbird eqcomd ) AGBUDZCAW + QCUEZUAUFZGRZWSCRZUEZUABUGZAXBUABAWSBSZUHZWTUBHEUCUIUCUFZWSUJRZRXFUBUFE + UKRULRZUMEUORZUMUNUPUMUNXAXEUBXGEHUQUMZURRZUSRZGHEXIXJUCXHHWSXKHGEOKUTK + XKTXJTXLTAEVFSZXDPVAZAHEVBRZSZXDAEVCSXPAEPVDHEKVEVGVAABXLWSABEURRZUSRZX + LABFVHRZUSRZXRVPXRABXTEXQDIFXRXSXQTZMLNQXSTXTTXRTZVIXTXRVJVKZAXKXQUSAXJ + EURAXMXJEUEPHEVFKVLVGVMVMVNVOXITZXHTZXGTZVQXEUBXGBCDEXIFUCXHHWSIJKLMNXN + ADXOSZXDQVAAXDVRYDYEYFVQVSVTAGXRWACBWABXRWBWRXCWHAXREHWCUMZUSRZGAXMGXQY + HWDUMSXRYIGWIPHXQEYHGOYAYHTZKWEXRYIXQYHGYBYITZWFWGWJABYICACIYHWDUMSZBYI + CWIAXMYGYLPQHCDEYHFIJKYJMLWKWLBYIIYHCNYKWFVGWJYCUAXRBGCWMWNWOWP $. + $} + + ${ + evls1muld.1 $e |- .X. = ( .r ` W ) $. + evls1muld.2 $e |- .x. = ( .r ` S ) $. + evls1muld.s $e |- ( ph -> S e. CRing ) $. + evls1muld.r $e |- ( ph -> R e. ( SubRing ` S ) ) $. + evls1muld.m $e |- ( ph -> M e. B ) $. + evls1muld.n $e |- ( ph -> N e. B ) $. + evls1muld.c $e |- ( ph -> C e. K ) $. + $( Univariate polynomial evaluation of a product of polynomials. + (Contributed by Thierry Arnoux, 24-Jan-2025.) $) + evls1muld $p |- ( ph -> ( ( Q ` ( M .X. N ) ) ` C ) + = ( ( ( Q ` M ) ` C ) .x. ( ( Q ` N ) ` C ) ) ) $= + ( co ce1 cfv cpl1 cmulr cress wcel wceq eqid ressply1mul syl12anc oveqi + id cvv cbs fvexi ressmulr 3eqtr4g fveq2d fveq1d cres ressply1evl csubrg + crg subrgring ply1ring 3syl ringcld fvresd eqtr2d cps1 cin ressply1bas2 + ax-mp inss2 eqsstrdi sseldd jca evl1muld simprd 3eqtr3d ) ACKLHUFZFUGUH + ZUHZUHCKLFUIUHZUJUHZUFZWHUHZUHZCWGDUHZUHCKDUHZUHZCLDUHZUHZGUFZACWIWMAWG + WLWHAKLMUJUHZUFZKLWJBUKUFZUJUHZUFZWGWLAAKBULLBULXBXEUMAURUCUDABXCFWJEMI + KLWJUNZQPRUBXCUNZUOUPHXAKLSUQWKXDKLBUSULWKXDUMBMUTRVABWJXCWKUSXGWKUNZVB + VSUQVCVDVEACWIWOAWOWGWHBVFZUHWIAWGDXIABDEFIWHJMNOPQRWHUNZUAUBVGZVEAWGBW + HABMHKLRSAEFVHUHULIVIULMVIULUBEFIQVJMIPVKVLUCUDVMVNVOVEAWLWJUTUHZULWNWT + UMAJWJFWKGXLKLWHWQWSCXJXFOXLUNZUAUEAKXLULCKWHUHZUHWQUMABXLKABIVPUHZUTUH + ZXLVQXLABXPFWJEMIXLXOXFQPRUBXOUNXPUNXMVRXPXLVTWAZUCWBACXNWPAWPKXIUHXNAK + DXIXKVEAKBWHUCVNVOVEWCALXLULCLWHUHZUHWSUMABXLLXQUDWBACXRWRAWRLXIUHXRALD + XIXKVEALBWHUDVNVOVEWCXHTWDWEWF $. + $} + $} + + ${ + $d H f $. $d H h $. $d R f h $. $d h m $. + ressdeg1.h $e |- H = ( R |`s T ) $. + ressdeg1.d $e |- D = ( deg1 ` R ) $. + ressdeg1.u $e |- U = ( Poly1 ` H ) $. + ressdeg1.b $e |- B = ( Base ` U ) $. + ressdeg1.p $e |- ( ph -> P e. B ) $. + ressdeg1.t $e |- ( ph -> T e. ( SubRing ` R ) ) $. + $( The degree of a univariate polynomial in a structure restriction. + (Contributed by Thierry Arnoux, 20-Jan-2025.) $) + ressdeg1 $p |- ( ph -> ( D ` P ) = ( ( deg1 ` H ) ` P ) ) $= + ( cfv csupp cxr clt wcel eqid cco1 co csup cdg1 csubrg wceq subrg0 oveq2d + c0g syl supeq1d cpl1 cbs cps1 ressply1bas2 eleqtrd elin2d deg1val 3eqtr4d + cin ) ADUAOZEUIOZPUBZQRUCZVAHUIOZPUBZQRUCZDCOZDHUDOZOZAQVCVFRAVBVEVAPAFEU + EOSVBVEUFNFEHVBIVBTZUGUJUHUKADEULOZUMOZSVHVDUFAHUNOZUMOZVMDADBVOVMUTMABVO + EVLFGHVMVNVLTZIKLNVNTVOTVMTZUOUPUQVAVMCVLEDVBJVPVQVKVATZURUJADBSVJVGUFMVA + BVIGHDVEVITKLVETVRURUJUS $. + $} + + ${ + ply1ascl0.w $e |- W = ( Poly1 ` R ) $. + ply1ascl0.a $e |- A = ( algSc ` W ) $. + ply1ascl0.o $e |- O = ( 0g ` R ) $. + ply1ascl0.1 $e |- .0. = ( 0g ` W ) $. + ply1ascl0.r $e |- ( ph -> R e. Ring ) $. + $( The zero scalar as a polynomial. (Contributed by Thierry Arnoux, + 20-Jan-2025.) $) + ply1ascl0 $p |- ( ph -> ( A ` O ) = .0. ) $= + ( cascl cfv c0g csca crg wcel syl fveq2d eqid wceq ply1sca clmod ply1lmod + eqtrid ply1ring ascl0 eqtrd fveq1i 3eqtr4g ) ADELMZMZENMZDBMFAULEOMZNMZUK + MUMADUOUKADCNMUOIACUNNACPQZCUNUAKECPGUBRSUESAUKUNEUKTUNTAUPEUCQKECGUDRAUP + EPQKECGUFRUGUHDBUKHUIJUJ $. + $} + + ${ + ressply.1 $e |- S = ( Poly1 ` R ) $. + ressply.2 $e |- H = ( R |`s T ) $. + ressply.3 $e |- U = ( Poly1 ` H ) $. + ressply.4 $e |- B = ( Base ` U ) $. + ressply.5 $e |- ( ph -> T e. ( SubRing ` R ) ) $. + ${ + ressply10g.6 $e |- Z = ( 0g ` S ) $. + $( A restricted polynomial algebra has the same group identity (zero + polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.) $) + ressply10g $p |- ( ph -> Z = ( 0g ` U ) ) $= + ( c0g cfv cascl eqid wcel syl subrg1ascl fveq1d crg subrgring ply1ascl0 + cres csubrg wceq subrg0 csubg subrgsubg subg0cl eqeltrrd fvresd 3eqtr3d + 3syl fveq2d subrgrcl 3eqtr2rd ) AFOPZGOPZDQPZPZCOPZVBPHAVAFQPZPVAVBEUFZ + PUTVCAVAVEVFAVBVEDCEFGIVBRZJKMVERZUAUBAVEGVAFUTKVHVARUTRAECUGPSZGUCSMEC + GJUDTUEAVAEVBAVDVAEAVIVDVAUHMECGVDJVDRZUITZAVIECUJPSVDESMECUKECVDVJULUP + UMUNUOAVDVAVBVKUQAVBCVDDHIVGVJNAVICUCSMECURTUEUS $. + $} + + ${ + $d B p $. $d M p $. $d N p $. $d p ph $. + ressply1mon1p.m $e |- M = ( Monic1p ` R ) $. + ressply1mon1p.n $e |- N = ( Monic1p ` H ) $. + $( The monic polynomials of a restricted polynomial algebra. + (Contributed by Thierry Arnoux, 21-Jan-2025.) $) + ressply1mon1p $p |- ( ph -> N = ( B i^i M ) ) $= + ( wcel cfv wa eqid vp cin cv c0g wne cdg1 cco1 cur wceq w3a cbs ismon1p + anbi2i cress co ressply1bas ressbasss eqsstrdi sseld anbi1d 13an22anass + pm4.71d wb ressply10g neeq2d adantr simpr csubrg ressdeg1 fveq2d subrg1 + bitr4di syl eqeq12d anbi12d pm5.32da 3anass bitr3d bitr2id elin 3bitr4g + eqrdv ) AUAIBHUBZAUAUCZBQZWDFUDRZUEZWDGUFRZRZWDUGRZRZGUHRZUIZUJZWEWDHQZ + SZWDIQWDWCQWPWEWDDUKRZQZWDDUDRZUEZWDCUFRZRZWJRZCUHRZUIZUJZSZAWNWOXFWEWQ + XADCXDWDHWSJWQTZWSTZXATZOXDTZULUMAWEWTXESZSZXGWNAXMWEWRSZXLSXGAWEXNXLAW + EWRABWQWDABDBUNUOZUKRWQABXOCDEFGJKLMNXOTZUPBWQXODXPXHUQURUSVBUTWEWRWTXE + VAVLAXMWEWGWMSZSWNAWEXLXQAWESZWTWGXEWMAWTWGVCWEAWSWFWDABCDEFGWSJKLMNXIV + DVEVFXRXCWKXDWLXRXBWIWJXRBXAWDCEFGKXJLMAWEVGAECVHRQZWENVFVIVJAXDWLUIZWE + AXSXTNECGXDKXKVKVMVFVNVOVPWEWGWMVQVLVRVSBWHFGWLWDIWFLMWFTWHTPWLTULWDBHV + TWAWB $. + $} + $} + ${ $d P n $. $d R n $. ply1chr.1 $e |- P = ( Poly1 ` R ) $. @@ -495940,6 +496358,131 @@ such that every prime ideal contains a prime element (this is a $} +$( +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- + Algebraic numbers and Minimal polynomials +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- +$) + + $c AlgNb minPoly $. + + $( Extend class notation with the algebraic number builder function. $) + calgnb $a class AlgNb $. + + $( Extend class notation with the minimal polynomial builder function. $) + cminply $a class minPoly $. + + ${ + $d e f p $. + $( Define the algebraic number builder function. This definition is + similar to ~ df-aa . (Contributed by Thierry Arnoux, 19-Jan-2025.) $) + df-algnb $a |- AlgNb = ( e e. _V , f e. _V |-> U_ p e. ( dom ( e evalSub1 f + ) \ { ( 0g ` ( Poly1 ` e ) ) } ) ( `' ( ( e evalSub1 f ) ` p ) " { ( 0g + ` e ) } ) ) $. + $} + + ${ + $d e f p x $. + $( Define the minimal polynomial builder function . (Contributed by + Thierry Arnoux, 19-Jan-2025.) $) + df-minply $a |- minPoly = ( e e. _V , f e. _V |-> ( x e. ( e AlgNb f ) |-> + inf ( ( `' ( ( e evalSub1 f ) ` p ) " { ( 0g ` e ) } ) , ( Monic1p ` e ) + , ( ( deg1 ` e ) oR < ( deg1 ` e ) ) ) ) ) $. + $} + + ${ + $d .0. e f $. $d E e f p $. $d F e f p $. $d O e f p $. $d Z e f p $. + algnbval.o $e |- O = ( E evalSub1 F ) $. + algnbval.z $e |- Z = ( 0g ` ( Poly1 ` E ) ) $. + algnbval.1 $e |- .0. = ( 0g ` E ) $. + algnbval.2 $e |- ( ph -> E e. Field ) $. + algnbval.3 $e |- ( ph -> F e. ( SubDRing ` E ) ) $. + $( The algebraic numbers over a field ` F ` within a field ` E ` . That + is, the numbers ` X ` which are roots of nonzero polynomials ` p ( X ) ` + with coefficients in ` ( Base `` F ) ` . This is expressed by the idiom + ` ( ``' ( O `` p ) " { .0. } ) ` , which can be translated into + ` { x e. ( Base `` E ) | ( ( O `` p ) `` x ) = .0. } ` by ~ fniniseg2 . + (Contributed by Thierry Arnoux, 26-Jan-2025.) $) + algnbval $p |- ( ph -> ( E AlgNb F ) + = U_ p e. ( dom O \ { Z } ) ( `' ( O ` p ) " { .0. } ) ) $= + ( ve vf cvv wcel csn cfv ces1 c0g cdif cv ccnv cima ciun calgnb co cfield + cdm wceq elexd csdrg wral ovexi dmex difexi fvex cnvex imaex rgenw iunexg + a1i sylancl cpl1 wa oveq12 eqtr4di dmeqd simpl fveq2d sneqd fveq1d cnveqd + difeq12d imaeq12d iuneq12d df-algnb ovmpoga syl3anc ) ABOPCOPGDUIZFQZUAZG + UBZDRZUCZEQZUDZUEZOPZBCUFUGWHUJABUHKUKACBULRLUKAWBOPZWGOPZGWBUMWIWJAVTWAD + DBCSHUNUOUPVBWKGWBWEWFWDWCDUQURUSUTGWBWGOOVAVCMNBCOOGMUBZNUBZSUGZUIZWLVDR + ZTRZQZUAZWCWNRZUCZWLTRZQZUDZUEWHUFOWLBUJZWMCUJZVEZGWSWBXDWGXGWOVTWRWAXGWN + DXGWNBCSUGDWLBWMCSVFHVGZVHXGWQFXGWQBVDRZTRFXGWPXITXGWLBVDXEXFVIZVJVJIVGVK + VNXGXAWEXCWFXGWTWDXGWCWNDXHVLVMXGXBEXGXBBTREXGWLBTXJVJJVGVKVOVPMNGVQVRVS + $. + + ${ + $d B p $. $d E p $. $d F p $. $d O p $. $d X p $. $d Z p $. + $d p ph $. + isalgnb.b $e |- B = ( Base ` E ) $. + $( Property for an element ` X ` of a field ` E ` to be algebraic over a + subfield ` F ` . (Contributed by Thierry Arnoux, 26-Jan-2025.) $) + isalgnb $p |- ( ph -> ( X e. ( E AlgNb F ) <-> ( X e. B + /\ E. p e. ( dom O \ { Z } ) ( ( O ` p ) ` X ) = .0. ) ) ) $= + ( co wcel cfv wa eqid calgnb wceq cdm cdif wrex ccnv cima ciun algnbval + cv csn eleq2d eliun bitrdi wf wfn wb cpws cbs cfield adantr fvexi cress + cvv a1i cpl1 crh ccrg csubrg fldcrngd csdrg cdr issdrg simp2bi evls1rhm + syl syl2anc rhmf simpr eldifad fdmd eleqtrd ffvelcdmd pwselbas fniniseg + ffn 3syl rexbidva bitrd r19.42v ) AFCDUAPZQZFBQZFIUJZERZRGUBZSZIEUCZHUK + ZUDZUEZWMWPIWTUESAWLFWOUFGUKUGZQZIWTUEZXAAWLFIWTXBUHZQXDAWKXEFACDEGHIJK + LMNUIULIFWTXBUMUNAXCWQIWTAWNWTQZSZBBWOUOWOBUPXCWQUQXGBCBCBURPZUSRZUTWOX + HVDXHTZOXITZACUTQXFMVABVDQXGBCUSOVBVEXGCDVCPZVFRZUSRZXIWNEAXNXIEUOZXFAE + XMXHVGPQZXOACVHQDCVIRQZXPACMVJADCVKRQZXQNXRCVLQXQXLVLQCDVMVNVPBEDCXHXLX + MJOXJXLTXMTVOVQXNXIXMXHEXNTXKVRVPZVAXGWNWRXNXGWNWRWSAXFVSVTAWRXNUBXFAXN + XIEXSWAVAWBWCWDBBWOWFBGFWOWEWGWHWIWMWPIWTWJUN $. + $} + + ${ + $d .0. p q $. $d E p q $. $d F p q $. $d O p q $. $d X p q $. + $d Z p q $. $d p ph q $. + minplyeulem.x $e |- ( ph -> X e. ( E AlgNb F ) ) $. + $( An algebraic number ` X ` over ` F ` is a root of some monic + polynomial ` p ` with coefficients in ` F ` . (Contributed by Thierry + Arnoux, 26-Jan-2025.) $) + minplyeulem $p |- ( ph -> E. p e. ( Monic1p ` E ) ( ( O ` p ) ` X ) = .0. + ) $= + ( cfv wceq wcel eqid ad2antrr syl vq cv cmn1 wrex cdm csn cdif wa cress + co cdg1 cco1 cinvr cpl1 cascl cmulr cbs cin csdrg csubrg issdrg simp2bi + cdr ressply1mon1p inss2 eqsstrdi crg cuc1p simp3bi drngringd c0g simplr + wne eldifad cpws crh wf ccrg fldcrngd evls1rhm syl2anc eleqtrd eldifsni + rhmf fdmd ad2antlr ressply10g neeqtrd drnguc1p syl3anc uc1pmon1p sseldd + simpr fveq2d fveq1d eqeq1d csca casa fldsdrgfld ply1assa assaring clmod + cfield crngringd asclf deg1nn0cl coe1fvalcl deg1ldg drnginvrcld ply1sca + ply1lmod cn0 ffvelcdmd calgnb isalgnb mpbid simpld evls1muld oveq2d cvv + fvexd pwselbas ringrz 3eqtrd rspcedvd simprd r19.29a ) AEUAUBZDOOZFPZEH + UBZDOZOZFPZHBUCOZUDUADUEZGUFZUGZAYHYRQZUHZYJUHZYNEYHBCUIUJZUKOZOZYHULOZ + OZUUBUMOZOZUUBUNOZUOOZOZYHUUIUPOZUJZDOZOZFPHUUMYOUUAUUBUCOZYOUUMUUAUUPU + UIUQOZYOURYOUUAUUQBBUNOZCUUIUUBYOUUPUURRZUUBRZUUIRZUUQRZUUACBUSOQZCBUTO + QZAUVCYSYJMSUVCBVCQZUVDUUBVCQZBCVAZVBZTZYORUUPRZVDUUQYOVEVFUUAUUBVGQZYH + UUBVHOZQZUUMUUPQUUAUUBAUVFYSYJAUVCUVFMUVCUVEUVDUVFUVGVITSZVJZUUAUVFYHUU + QQZYHUUIVKOZVMZUVMUVNUUAYHYPUUQUUAYHYPYQAYSYJVLVNAYPUUQPYSYJAUUQBBUQOZV + OUJZUQOZDADUUIUVTVPUJQZUUQUWADVQZABVRQZUVDUWBABLVSZAUVCUVDMUVHTZUVSDCBU + VTUUBUUIIUVSRZUVTRZUUTUVAVTWAZUUQUWAUUIUVTDUVBUWARZWDZTWESWBZUUAYHGUVQY + SYHGVMAYJYHYPGWCWFAGUVQPYSYJAUUQBUURCUUIUUBGUUSUUTUVAUVBUWFJWGSWHZUUQUV + LUUIUUBYHUVQUVAUVBUVQRZUVLRZWIWJUUJUVLUUCUUIUUBUULUUGUUPYHUWOUVJUVAUULR + ZUUJRZUUCRZUUGRZWKWAWLUUAYKUUMPZUHZYMUUOFUXAEYLUUNUXAYKUUMDUUAUWTWMWNWO + WPUUAUUOEUUKDOZOZYIBUPOZUJUXCFUXDUJZFUUAUUQEDCBUXDUULUUBUVSUUKYHUUIIUWG + UVAUUTUVBUWPUXDRZAUWDYSYJUWESUVIUUAUUIWQOZUQOZUUQUUHUUJUUAUUJUUQUXGUXHU + UIUWQUXGRUUAUUIWRQZUUIVGQAUXIYSYJAUUBVRQUXIAUUBABXCQZUVCUUBXCQZLMCBWSWA + ZVSZUUIUUBUVAWTTSUUIXATAUUIXBQZYSYJAUVKUXNAUUBUXMXDUUIUUBUVAXKTSUXHRUVB + XEUUAUUHUUBUQOZUXHUUAUXOUUBUUGUUFUUBVKOZUXORZUXPRZUWSUVNUUAUVPUUDXLQZUU + FUXOQUWLUUAUVKUVPUVRUXSUVOUWLUWMUUQUUCUUIUUBYHUVQUWRUVAUWNUVBXFWJUUEUUQ + UUIUUBYHUXOUUDUUERZUVBUVAUXQXGWAUUAUVKUVPUVRUUFUXPVMUVOUWLUWMUUEUUQUUCU + UIUUBYHUXPUVQUWRUVAUWNUVBUXRUXTXHWJXIAUXOUXHPYSYJAUUBUXGUQAUXKUUBUXGPUX + LUUIUUBXCUVAXJTWNSWBXMZUWLAEUVSQZYSYJAUYBYJUAYRUDZAEBCXNUJQUYBUYCUHNAUV + SBCDEFGUAIJKLMUWGXOXPZXQSZXRUUAYIFUXCUXDYTYJWMXSUUABVGQZUXCUVSQUXEFPAUY + FYSYJABUWEXDSUUAUVSUVSEUXBUUAUVSBUVSUWAXCUXBUVTXTUWHUWGUWJAUXJYSYJLSUUA + BUQYAUUAUUQUWAUUKDUUAUWBUWCAUWBYSYJUWISUWKTUYAXMYBUYEXMUVSBUXDUXCFUWGUX + FKYCWAYDYEAUYBUYCUYDYFYG $. + $} + $} + + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Matrices @@ -511850,18 +512393,18 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry wb mpbird adantr wn cuz wss 1zzd zsubcld cn nnaddcl mp2an a1i nnred syl zred nnzd zlem1lt syl2anc mpbid ltled eluz fzss2 crab cr cinf wral wrex sseld rabid wi ballotlemsup wor ltso id inflb ballotlemi breq2d sylibrd - notbid syl5bir syland imp biid sylnib anassrs pm2.65da nrexdv ) BKFUDUE - ZEUFZBGUGZUGUHUIZEUJBHUGZUJUKULZUMULZXOXPYAUEZURZXRXPXSUNUOZYCYDXRYCYDX - PXTUPUOZYBYEXOXPUJXTUQUSYBXPUTUEXSUTUEZYDYEVHXOXPUJXTVAXOXSUJIJVBULZXOX - SUJYGUMULZUEZXSXQUGUHUIABCDEFGHIJKLMNOPQRSTVCVDZVEZXPXSVFVGVIVJXOYBXRYD - VKZXOYBXRURZURYDYDXOYMYLXOYBXPYHUEZXRYLXOYAYHXPXOYGXTVLUGUEZYAYHVMXOYOX - TYGUPUOZXOXTYGXOXTXOXSUJYKXOVNVOZWBXOYGYGVPUEZXOIVPUEJVPUEYRMNIJVQVRVSZ - VTXOXSYGUPUOZXTYGUNUOZXOYIYTYJXSUJYGUQWAXOYFYGUTUEZYTUUAVHYKXOYGYSWCZXS - YGWDWEWFWGXOXTUTUEUUBYOYPVHYQUUCXTYGWHWEVIXTUJYGWIWAWOYNXRURXPXREYHWJZU - EZXOYLXREYHWPXOUUEXPUUDWKUNWLZUNUOZVKZYLXOUAUFZUBUFZUNUOVKUAUUDWMUUJUUI - UNUOUCUFUUIUNUOUCUUDWNWQUAWKWMURUBWKWNZUUEUUHWQAUCUBUABCDEFGHIJKLMNOPQR - STWRUUKUBUAUCWKUUDXPUNWKUNWSUUKWTVSUUKXAXBWAXOYDUUGXOXSUUFXPUNABCDEFGHI - JKLMNOPQRSTXCXDXFXEXGXHXIYDXJXKXLXMXN $. + notbid biimtrrid syland imp biid sylnib anassrs pm2.65da nrexdv ) BKFUD + UEZEUFZBGUGZUGUHUIZEUJBHUGZUJUKULZUMULZXOXPYAUEZURZXRXPXSUNUOZYCYDXRYCY + DXPXTUPUOZYBYEXOXPUJXTUQUSYBXPUTUEXSUTUEZYDYEVHXOXPUJXTVAXOXSUJIJVBULZX + OXSUJYGUMULZUEZXSXQUGUHUIABCDEFGHIJKLMNOPQRSTVCVDZVEZXPXSVFVGVIVJXOYBXR + YDVKZXOYBXRURZURYDYDXOYMYLXOYBXPYHUEZXRYLXOYAYHXPXOYGXTVLUGUEZYAYHVMXOY + OXTYGUPUOZXOXTYGXOXTXOXSUJYKXOVNVOZWBXOYGYGVPUEZXOIVPUEJVPUEYRMNIJVQVRV + SZVTXOXSYGUPUOZXTYGUNUOZXOYIYTYJXSUJYGUQWAXOYFYGUTUEZYTUUAVHYKXOYGYSWCZ + XSYGWDWEWFWGXOXTUTUEUUBYOYPVHYQUUCXTYGWHWEVIXTUJYGWIWAWOYNXRURXPXREYHWJ + ZUEZXOYLXREYHWPXOUUEXPUUDWKUNWLZUNUOZVKZYLXOUAUFZUBUFZUNUOVKUAUUDWMUUJU + UIUNUOUCUFUUIUNUOUCUUDWNWQUAWKWMURUBWKWNZUUEUUHWQAUCUBUABCDEFGHIJKLMNOP + QRSTWRUUKUBUAUCWKUUDXPUNWKUNWSUUKWTVSUUKXAXBWAXOYDUUGXOXSUUFXPUNABCDEFG + HIJKLMNOPQRSTXCXDXFXEXGXHXIYDXJXKXLXMXN $. $( If the first vote is for B, the vote on the first tie is for A. (Contributed by Thierry Arnoux, 1-Dec-2016.) $) @@ -518208,11 +518751,11 @@ conditions of the Five Segment Axiom ( ~ axtg5seg ). See ~ df-ofs . (New usage is discouraged.) $) bnj23 $p |- ( A. z e. B -. z R y -> A. w e. A ( w R y -> [. w / x ]. ph ) ) $= - ( cv wbr wn wral wsbc wi wcel wa wb cvv syl5bir sbcng crab eleq2i elrabsf - elv nfcv bitri weq breq1 notbid rspccv expdimp con4d ralrimiva ) DJZCJZHK - ZLZDGMZEJZUPHKZABUTNZOEFUSUTFPZQZVBVAVBLZALZBUTNZVDVALZVGVEREABUTSUAUEUSV - CVGVHVCVGQZUTGPZUSVHVJUTVFBFUBZPVIGVKUTIUCVFBUTFBFUFUDUGURVHDUTGDEUHUQVAU - OUTUPHUIUJUKTULTUMUN $. + ( cv wbr wn wral wsbc wi wcel wa wb cvv biimtrrid elv crab eleq2i elrabsf + sbcng nfcv bitri weq breq1 notbid rspccv expdimp con4d ralrimiva ) DJZCJZ + HKZLZDGMZEJZUPHKZABUTNZOEFUSUTFPZQZVBVAVBLZALZBUTNZVDVALZVGVEREABUTSUEUAU + SVCVGVHVCVGQZUTGPZUSVHVJUTVFBFUBZPVIGVKUTIUCVFBUTFBFUFUDUGURVHDUTGDEUHUQV + AUOUTUPHUIUJUKTULTUMUN $. $} ${ @@ -520777,31 +521320,31 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a ancom csn cdif eldifi eleq2s nnord ordtr1 imp syl3an3 wral rsp mpd bnj446 mpan9 pm3.35 sylan2b iuneq1 bnj658 simp3bi bnj240 anim12i anim2i biancomi simpl simp3 simpl1 bnj589 bnj590 syldan simpr 3eqtr4d ex syl bnj593 19.9v - 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(Contributed by ( wcel wceq cin c0 wne wa wral cfv w3a wn df-ne wi cv csn cdif cres crest co chmeo cuni ccnv cima wss cvmsi simp3d simprd simpl ralimi sneq difeq2d ineq1 eqeq1d raleqbidv rspccva sylan necom eldifsn biimpri sylan2b rspccv - syl ineq2 syl2im expd 3impia syl5bir orrd ) GHFUANZCGNZDGNZUBZCDOZCDPZQOZ - WEUCCDRZWDWGCDUDWAWBWCWHWGUEWAWBSZWCWHWGWICAUFZPZQOZAGCUGZUHZTZWCWHSDWNNZ - WGWABUFZWJPZQOZAGWQUGZUHZTZBGTZWBWOWAXBJWQUIEWQUJUKKHUJUKULUKNZSZBGTZXCWA - GUMJUNHUOOZXFWAHKNGEUPGQRSXGXFSABEFGHIJKLMUQURUSXEXBBGXBXDUTVAVNXBWOBCGWQ - COZWSWLAXAWNXHWTWMGWQCVBVCXHWRWKQWQCWJVDVEVFVGVHWHWCDCRZWPCDVIWPWCXISDGCV - JVKVLWLWGADWNWJDOWKWFQWJDCVOVEVMVPVQVRVSVT $. + syl ineq2 syl2im expd 3impia biimtrrid orrd ) GHFUANZCGNZDGNZUBZCDOZCDPZQ + OZWEUCCDRZWDWGCDUDWAWBWCWHWGUEWAWBSZWCWHWGWICAUFZPZQOZAGCUGZUHZTZWCWHSDWN + NZWGWABUFZWJPZQOZAGWQUGZUHZTZBGTZWBWOWAXBJWQUIEWQUJUKKHUJUKULUKNZSZBGTZXC + WAGUMJUNHUOOZXFWAHKNGEUPGQRSXGXFSABEFGHIJKLMUQURUSXEXBBGXBXDUTVAVNXBWOBCG + WQCOZWSWLAXAWNXHWTWMGWQCVBVCXHWRWKQWQCWJVDVEVFVGVHWHWCDCRZWPCDVIWPWCXISDG + CVJVKVLWLWGADWNWJDOWKWFQWJDCVOVEVMVPVQVRVSVT $. $( Every element of an even covering of ` U ` is homeomorphic to ` U ` via ` F ` . (Contributed by Mario Carneiro, 13-Feb-2015.) $) @@ -530051,45 +530594,45 @@ property of an acyclic graph (see also ~ acycgr0v ). 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(Contributed by Mario Carneiro, @@ -532620,38 +533163,38 @@ codes is an increasing chain (with respect to inclusion). 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Show the cut properties of surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) $) @@ -545099,49 +545642,49 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). 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See ~ brofs and <. A , <. B , C >. >. Cgr3 <. G , <. H , I >. >. ) -> <. D , <. E , F >. >. Cgr3 <. G , <. H , I >. >. ) ) $= ( wcel w3a wa cop ccgr wbr ccgr3 wi axcgrtr syl133anc cn cee cfv 3an6 simpl - simpr11 simpr12 simpr21 simpr22 simpr31 simpr32 simpr13 3anim123d wb brcgr3 - simpr23 simpr33 syl5bir 3adant3r3 3adant3r2 anbi12d 3adant3r1 3imtr4d ) JUA - KZAJUBUCZKZBVEKZCVEKZLZDVEKZEVEKZFVEKZLZGVEKZHVEKZIVEKZLZLZMZABNZDENZOPZACN - ZDFNZOPZBCNZEFNZOPZLZVTGHNZOPZWCGINZOPZWFHINZOPZLZMZWAWJOPZWDWLOPZWGWNOPZLZ - AWFNZDWGNZQPZXBGWNNZQPZMXCXEQPZWQWBWKMZWEWMMZWHWOMZLVSXAWBWKWEWMWHWOUDVSXHW - RXIWSXJWTVSVDVFVGVJVKVNVOXHWRRVDVRUEZVFVGVHVMVQVDUFZVFVGVHVMVQVDUGZVJVKVLVI - VQVDUHZVJVKVLVIVQVDUIZVNVOVPVIVMVDUJZVNVOVPVIVMVDUKZABDEGHJSTVSVDVFVHVJVLVN - VPXIWSRXKXLVFVGVHVMVQVDULZXNVJVKVLVIVQVDUPZXPVNVOVPVIVMVDUQZACDFGIJSTVSVDVG - VHVKVLVOVPXJWTRXKXMXRXOXSXQXTBCEFHIJSTUMURVSXDWIXFWPVDVIVMXDWIUNVQABCDEFJUO - USVDVIVQXFWPUNVMABCGHIJUOUTVAVDVMVQXGXAUNVIDEFGHIJUOVBVC $. + simpr11 simpr12 simpr21 simpr22 simpr31 simpr32 simpr13 3anim123d biimtrrid + simpr23 simpr33 wb brcgr3 3adant3r3 3adant3r2 anbi12d 3adant3r1 3imtr4d ) J + UAKZAJUBUCZKZBVEKZCVEKZLZDVEKZEVEKZFVEKZLZGVEKZHVEKZIVEKZLZLZMZABNZDENZOPZA + CNZDFNZOPZBCNZEFNZOPZLZVTGHNZOPZWCGINZOPZWFHINZOPZLZMZWAWJOPZWDWLOPZWGWNOPZ + LZAWFNZDWGNZQPZXBGWNNZQPZMXCXEQPZWQWBWKMZWEWMMZWHWOMZLVSXAWBWKWEWMWHWOUDVSX + HWRXIWSXJWTVSVDVFVGVJVKVNVOXHWRRVDVRUEZVFVGVHVMVQVDUFZVFVGVHVMVQVDUGZVJVKVL + VIVQVDUHZVJVKVLVIVQVDUIZVNVOVPVIVMVDUJZVNVOVPVIVMVDUKZABDEGHJSTVSVDVFVHVJVL + VNVPXIWSRXKXLVFVGVHVMVQVDULZXNVJVKVLVIVQVDUOZXPVNVOVPVIVMVDUPZACDFGIJSTVSVD + VGVHVKVLVOVPXJWTRXKXMXRXOXSXQXTBCEFHIJSTUMUNVSXDWIXFWPVDVIVMXDWIUQVQABCDEFJ + URUSVDVIVQXFWPUQVMABCGHIJURUTVAVDVMVQXGXAUQVIDEFGHIJURVBVC $. $( Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) $) @@ -546344,30 +546887,30 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and jca imp simpll1 axsegcon syl122anc reeanv sylanbrc ad2antrr simprl simprr simpr simp-4l simplrl simprrr simpllr 3jca simplrr simp1ll btwnconn1lem12 simp1lr simpl1r simpll2 simpl3l simpl3r jca32 syl2an an4s rexlimddv exp32 - rexlimdvv mpd expr syl5bir orrd ) EUFKZAEUBUCZKZBXCKZLZCXCKZDXCKZGMZXCKZL - ZHMZXCKZFMZXCKZNZNZNZABUDZBCUDZNZBACOPQBADOPQNZNZDAXIOPQZDXIOCDOZRQZNZCAX - LOPQZCXLOZYERQZNZNZXIAXNOZPQXIXNOCBORQNZXLYMPQXLXNODBORQNZNZLZNZCXIUGZDXL - UGZYSUHCXIUDZYRYTCXIUIXRYQUUAYTXRYQUUANZNUAMZCXIOPQUUCDXLOPQNZYTUAXCXRUUB - UUDUAXCUJZXRUUBCXLAOPQZDXIAOPQZNZUUEUUBYHYDNXRUUHUUBYHYDYQYHUUAYHYJYGYCYP - UKSYQYDUUAYDYFYKYCYPULSVHXRYHUUFYDUUGXRXBXGXDXMYHUUFUMXBXDXEXQUNZXGXHXJXP - XFUOZXBXDXEXQUPZXFXKXMXOUQZCAXLEURUSXRXBXHXDXJYDUUGUMUUIXGXHXJXPXFUTZUUKX - GXHXJXPXFVAZDAXIEURUSVBVCXRXBXMXJXDXGXHUUHUUEVDUUIUULUUNUUKUUJUUMUAXLXIAC - DEVEVFVGVIXRUUCXCKZUUBUUDYTXRUUONZUUBUUDNZNZCXIIMZOPQCUUSOYIRQNZCXLJMZOPQ - CUVAOCUUCORQNZNZJXCUJIXCUJZYTUUPUVDUUQUUPUUTIXCUJZUVBJXCUJZUVDUUPXBXJXGXG - XMUVEXBXDXEXQUUOVJZXRXJUUOUUNSXRXGUUOUUJSZUVHXRXMUUOUULSZIXICCXLEVKVLUUPX - BXMXGXGUUOUVFUVGUVIUVHUVHXRUUOVRJXLCCUUCEVKVLUUTUVBIJXCXCVMVNSUURUVCYTIJX - CXCUURUUSXCKZUVAXCKZNZUVCYTUUPUVLUUQUVCYTUUPUVLNZUUQUVCNZNUVAUUSUEMZOPQUV - AUVOOUVAUUSORQNZYTUEXCUVMUVPUEXCUJZUVNUVMXBUVJUVKUVKUVJUVQXRXBUUOUVLUUIVO - UUPUVJUVKVPZUUPUVJUVKVQZUVSUVRUEUUSUVAUVAUUSEVKVLSUVMUVOXCKZUVNUVPYTUVMUV - TNZXFXKXMXOUUOLZNZUVJUVTUVKLZLXSXTUUALZYBNZYLYPLZUUDUUTUVBUVPLZNNYTUVNUVP - NZUWAXFUWCUWDXFXQUUOUVLUVTVSUWAXKUWBUUPXKUVLUVTXFXKXPUUOVTVOUWAXMXOUUOXRX - MUUOUVLUVTUULTXRXOUUOUVLUVTXFXKXMXOWATXRUUOUVLUVTWBWCVHUWAUVJUVTUVKUUPUVJ - UVKUVTVTUVMUVTVRUUPUVJUVKUVTWDWCWCUWIUWGUUDUWHUWIUWFYLYPUWIUWEYBUWIXSXTUU - AUVNXSUVPYQXSUUAUUDUVCXSXTYBYLYPWETSUVNXTUVPYQXTUUAUUDUVCXSXTYBYLYPWGTSUV - NUUAUVPYQUUAUUDUVCWBSWCUUBYBUUDUVCUVPYAYBYLYPUUAWHTVHUUQYLUVCUVPYCYLYPUUA - UUDWIVOUWIYNYOUUBYNUUDUVCUVPYNYOYCYLUUAWJTUUBYOUUDUVCUVPYNYOYCYLUUAWKTVHW - CUUBUUDUVCUVPWBUWIUUTUVBUVPUUQUUTUVBUVPVTUUQUUTUVBUVPWDUVNUVPVRWCWLABCDUU - SUVOUVAUUCEFGHWFWMWNWOWNWPWQWRWNWOWSWTXA $. + rexlimdvv mpd expr biimtrrid orrd ) EUFKZAEUBUCZKZBXCKZLZCXCKZDXCKZGMZXCK + ZLZHMZXCKZFMZXCKZNZNZNZABUDZBCUDZNZBACOPQBADOPQNZNZDAXIOPQZDXIOCDOZRQZNZC + AXLOPQZCXLOZYERQZNZNZXIAXNOZPQXIXNOCBORQNZXLYMPQXLXNODBORQNZNZLZNZCXIUGZD + XLUGZYSUHCXIUDZYRYTCXIUIXRYQUUAYTXRYQUUANZNUAMZCXIOPQUUCDXLOPQNZYTUAXCXRU + UBUUDUAXCUJZXRUUBCXLAOPQZDXIAOPQZNZUUEUUBYHYDNXRUUHUUBYHYDYQYHUUAYHYJYGYC + YPUKSYQYDUUAYDYFYKYCYPULSVHXRYHUUFYDUUGXRXBXGXDXMYHUUFUMXBXDXEXQUNZXGXHXJ + XPXFUOZXBXDXEXQUPZXFXKXMXOUQZCAXLEURUSXRXBXHXDXJYDUUGUMUUIXGXHXJXPXFUTZUU + KXGXHXJXPXFVAZDAXIEURUSVBVCXRXBXMXJXDXGXHUUHUUEVDUUIUULUUNUUKUUJUUMUAXLXI + ACDEVEVFVGVIXRUUCXCKZUUBUUDYTXRUUONZUUBUUDNZNZCXIIMZOPQCUUSOYIRQNZCXLJMZO + PQCUVAOCUUCORQNZNZJXCUJIXCUJZYTUUPUVDUUQUUPUUTIXCUJZUVBJXCUJZUVDUUPXBXJXG + XGXMUVEXBXDXEXQUUOVJZXRXJUUOUUNSXRXGUUOUUJSZUVHXRXMUUOUULSZIXICCXLEVKVLUU + PXBXMXGXGUUOUVFUVGUVIUVHUVHXRUUOVRJXLCCUUCEVKVLUUTUVBIJXCXCVMVNSUURUVCYTI + JXCXCUURUUSXCKZUVAXCKZNZUVCYTUUPUVLUUQUVCYTUUPUVLNZUUQUVCNZNUVAUUSUEMZOPQ + UVAUVOOUVAUUSORQNZYTUEXCUVMUVPUEXCUJZUVNUVMXBUVJUVKUVKUVJUVQXRXBUUOUVLUUI + VOUUPUVJUVKVPZUUPUVJUVKVQZUVSUVRUEUUSUVAUVAUUSEVKVLSUVMUVOXCKZUVNUVPYTUVM + UVTNZXFXKXMXOUUOLZNZUVJUVTUVKLZLXSXTUUALZYBNZYLYPLZUUDUUTUVBUVPLZNNYTUVNU + VPNZUWAXFUWCUWDXFXQUUOUVLUVTVSUWAXKUWBUUPXKUVLUVTXFXKXPUUOVTVOUWAXMXOUUOX + RXMUUOUVLUVTUULTXRXOUUOUVLUVTXFXKXMXOWATXRUUOUVLUVTWBWCVHUWAUVJUVTUVKUUPU + VJUVKUVTVTUVMUVTVRUUPUVJUVKUVTWDWCWCUWIUWGUUDUWHUWIUWFYLYPUWIUWEYBUWIXSXT + UUAUVNXSUVPYQXSUUAUUDUVCXSXTYBYLYPWETSUVNXTUVPYQXTUUAUUDUVCXSXTYBYLYPWGTS + UVNUUAUVPYQUUAUUDUVCWBSWCUUBYBUUDUVCUVPYAYBYLYPUUAWHTVHUUQYLUVCUVPYCYLYPU + UAUUDWIVOUWIYNYOUUBYNUUDUVCUVPYNYOYCYLUUAWJTUUBYOUUDUVCUVPYNYOYCYLUUAWKTV + HWCUUBUUDUVCUVPWBUWIUUTUVBUVPUUQUUTUVBUVPVTUUQUUTUVBUVPWDUVNUVPVRWCWLABCD + UUSUVOUVAUUCEFGHWFWMWNWOWNWPWQWRWNWOWSWTXA $. $} ${ @@ -546672,15 +547215,15 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ( ( <. A , B >. Cgr <. E , F >. /\ <. C , D >. Cgr <. G , H >. ) -> ( <. A , B >. Seg<_ <. C , D >. <-> <. E , F >. Seg<_ <. G , H >. ) ) ) $= - ( wcel w3a cop ccgr wbr wa csegle df-3an seglecgr12im syl5bir expd cn wi wb - simp11 simp12 simp13 simp23 simp31 cgrcom syl122anc simp21 simp22 syl333anc - cee cfv simp32 simp33 anbi12d sylbid impbidd ) IUAJZAIUNUOZJZBVBJZKZCVBJZDV - BJZEVBJZKZFVBJZGVBJZHVBJZKZKZABLZEFLZMNZCDLZGHLZMNZOZVOVRPNZVPVSPNZVNWAWBWC - WAWBOVQVTWBKVNWCVQVTWBQABCDEFGHIRSTVNWAVPVOMNZVSVRMNZOZWCWBUBVNVQWDVTWEVNVA - VCVDVHVJVQWDUCVAVCVDVIVMUDZVAVCVDVIVMUEZVAVCVDVIVMUFZVEVFVGVHVMUGZVEVIVJVKV - LUHZABEFIUIUJVNVAVFVGVKVLVTWEUCWGVEVFVGVHVMUKZVEVFVGVHVMULZVEVIVJVKVLUPZVEV - IVJVKVLUQZCDGHIUIUJURVNWFWCWBWFWCOWDWEWCKZVNWBWDWEWCQVNVAVHVJVKVLVCVDVFVGWP - WBUBWGWJWKWNWOWHWIWLWMEFGHABCDIRUMSTUSUT $. + ( wcel w3a cop ccgr wbr wa csegle df-3an seglecgr12im biimtrrid expd cn cee + cfv wi wb simp11 simp12 simp13 simp23 simp31 cgrcom syl122anc simp21 simp22 + simp32 simp33 anbi12d syl333anc sylbid impbidd ) IUAJZAIUBUCZJZBVBJZKZCVBJZ + DVBJZEVBJZKZFVBJZGVBJZHVBJZKZKZABLZEFLZMNZCDLZGHLZMNZOZVOVRPNZVPVSPNZVNWAWB + WCWAWBOVQVTWBKVNWCVQVTWBQABCDEFGHIRSTVNWAVPVOMNZVSVRMNZOZWCWBUDVNVQWDVTWEVN + VAVCVDVHVJVQWDUEVAVCVDVIVMUFZVAVCVDVIVMUGZVAVCVDVIVMUHZVEVFVGVHVMUIZVEVIVJV + KVLUJZABEFIUKULVNVAVFVGVKVLVTWEUEWGVEVFVGVHVMUMZVEVFVGVHVMUNZVEVIVJVKVLUOZV + EVIVJVKVLUPZCDGHIUKULUQVNWFWCWBWFWCOWDWEWCKZVNWBWDWEWCQVNVAVHVJVKVLVCVDVFVG + WPWBUDWGWJWKWNWOWHWIWLWMEFGHABCDIRURSTUSUT $. ${ $d A y $. $d B y $. $d N y $. @@ -548238,21 +548781,21 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and elicc3 $p |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <-> ( C e. RR* /\ 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 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 $. + wn wne wo cicc co elicc1 xrletr exp5o com23 imp5q df-ne biimtrrid 3adant3r3 + adantlr eqcom necon3bbii biimtrid 3exp com12 imp32 3adantr2 adantll anim12d + ex df-or 3orass pm5.6 orcom imbi2i bitri 3bitr4ri syl6ib 3jcad xrleid breq2 + w3o ad3antrrr xrltle adantr adantllr adantrd simpr 3jaod exp31 3impd ancoms + breq1 adantld ad3antlr impbid bitrd ) ADEZBDEZFZCABUAUBECDEZACGHZCBGHZIZWLA + BGHZCAJZACKHZCBKHZFZCBJZVMZIZABCUCWKWOXCWKWOWLWPXBWOWLLWKWLWMWNMNWIWJWLWMWN + WPWIWLWJWMWNWPLLWIWLWJWMWNWPACBUDUEUFUGWKWOWQRZXARZFWTLZXBWKWOXFWKWOFXDWRXE + WSWIWOXDWRLZWJWIWLWMXGWNXDCASZWIWLWMIZWRCAUHXIWRXHACOPUIUJUKWJWOXEWSLZWIWJW + LWNXJWMWJWLWNXJWLWJWNXJLWLWJWNXJXEBCSZWLWJWNIZWSXABCCBULUMXLWSXKCBOPUNUOUPU + QURUSUTVAWQWTXATZTXDXMLZXBXFWQXMVBWQWTXAVCXFXDXAWTTZLXNXDXAWTVDXOXMXDXAWTVE + VFVGVHVIVJWKXCWLWMWNXCWLLWKWLWPXBMNWKWLWPXBWMWKWLWPXBWMLWKWLFZWPFZWQWMWTXAX + QWMWQAAGHZWIXRWJWLWPAVKVNCAAGVLQXQWRWMWSWIWLWPWRWMLZWJWIWLFXSWPACVOVPVQVRXQ + WMXAWPXPWPVSZCBAGVLQVTWAWBWKWLWPXBWNWKWLWPXBWNLXQWQWNWTXAXQWNWQWPXTCABGWDQX + PWTWNLZWPWJWLYAWIWJWLFWSWNWRWLWJWSWNLCBVOWCWEUSVPXQWNXABBGHZWJYBWIWLWPBVKWF + CBBGWDQVTWAWBVJWGWH $. ${ $d k m n y ph $. $d k m n x ps $. $d x y $. @@ -549043,38 +549586,38 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ineq2d neeq1d elrab2 simprbi simprr sylc ssn0 syl2anc rexlimdvaa biimtrid rsp ralrimiv sylanbrc ex alrimiv anbi12i an4 bitri inss1 elpwi sstrid vex inex1 ssralv ax-mp inss2 anim12i r19.26 wex exdistrv simprl sselid elpwid - n0 simplll ad2antrr simplr sseldd syl13anc exlimdvv syl5bir ralimdva syl5 - ss2in impr sylan2b ralrimivva wb mpbi ssexd uniexb istopg mpbir2and pwidg - cvv a1i csn cdif ffvelcdmda eldifi df-ss eldifsni eqnetrd ralrimiva eqssd - 3syl istopon ) AGUBOZIGUCZUDGIUEUFOAUUANUGZGPZUUCUCZGOZUHZNUIZUUCUAUGZQZG - OZUAGRNGRZAUUGNAUUDUUFAUUDSZUUEIUKZOZBUGZFUFZUUEUKZQZTULZBUUERZUUFUUMUUEI - PZUUOUUMUUCUUNPUVBUUMUUCGUUNAUUDUJZGUUQEUGZUKZQZTULZBUVDRZEUUNUMUUNMUVHEU - UNUNUOZUPUUCIUQVCUUEINURUSUTUUMUUTBUUEUUPUUEOUUPUUIOZUAUUCVAUUMUUTUAUUPUU - CVBUUMUVJUUTUAUUCUUMUUIUUCOZUVJSSZUUQUUIUKZQZUUSPZUVNTULZUUTUVLUVMUURPUVO - UVLUUIUUEUVKUUIUUEPUUMUVJUUIUUCVDVEVFUVMUURUUQVGVHUVLUVPBUUIRZUVJUVPUVLUU - IGOZUVQUUMUVKUVRUVJUUMUUCGUUIUVCVIVJUVRUUIUUNOZUVQUVHUVQEUUIUUNGUVGUVPBUV - 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X /\ v e. ( F ` x ) ) ) -> x e. v ) $. neibastop1.6 $e |- ( ( ph /\ ( x e. X /\ v e. ( F ` x ) ) ) -> @@ -549482,28 +550025,28 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= 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 $. + ineq1 ineq2 sylan9bb syl5ibcom rexlimdvva biimtrrid sylbid ralrimivv 3jca + rexbidv cdm dmexg eqeltrid isfbas2 mpbir2and ) AUAIZBJUBZKZAUDLZUEZBUFLIZ + YDBUGZMZYDJUBZJYDUHZDNZENZFNZUIZMZDYDOZFYDUJEYDUJZUKZXTYGYAXTBYFYCABCULZU + SPYBYHYIYPXTYAYHYAYKBIZEUMXTYHEBUNXTYSYHEXTYSYKYCLZYDIZYHXTBYFYCUOZYCBUPZ + YSUUAQYRBYFYCUQZUUCYSUUABYKYCURUTVAYDYTVBVCVDVEVFYBJYDIZVGYIYBUUEYTJVLZEB + OZXTUUGVGYAXTUUFEBXTYSKYKYTIUUFVGYKABCVHYTYKVIWCVJPXTUUEUUGRZYAXTUUBUUCUU + HYRUUDEBJYCVKVAPVMJYDVNVOYBYOEFYDYDXTYKYDIZYLYDIZKZYOQYAXTUUKGNZYCLZYKVLZ + GBOZHNZYCLZYLVLZHBOZKZYOXTUUBUUCUUKUUTRYRUUDUUCUUIUUOUUJUUSGBYKYCVKHBYLYC + VKVPVAUUTUUNUURKZHBOGBOXTYOUUNUURGHBBVQXTUVAYOGHBBXTUULBIZUUPBIZKZKZYJUUM + UUQUIZMZDYDOZUVAYOUVEUULUCNZASZUUPUVIASZKZUVHUCBXTUVBUVCUVLUCBOUCUULUUPAB + CVRVSUVEUVIBIZUVLKZKZUVIYCLZYDIZUVPUVFMZUVHXTUVMUVQUVDUVLXTUUCUVMUVQXTUUB + UUCYRUUDWCBUVIYCURVTWAUVOEUVPUVFUVOYKUVPIZYKUUMIZYKUUQIZKZYKUVFIUVOUVIYKA + SZUULYKASZUUPYKASZKZUVSUWBUVEUVMUVLUWCUWFQUVEUVMKUWCUVLUWFUVEUVMUWCUVLUWF + QUVEUVMUWCKKUVJUWDUVKUWEXTUWCUVJUWDQZUVDUVMXTUWCUWGXTUVJUWCUWDXTUVJUWCUWD + QQEXTYKTIZKZUVJUWCUWDUULUVIYKATWBWDWEWFVFWGXTUWCUVKUWEQZUVDUVMXTUWCUWJXTU + VKUWCUWEXTUVKUWCUWEQQEUWIUVKUWCUWEUUPUVIYKATWBWDWEWFVFWGWHWIWFWNXTUVMUVSU + WCRZUVDUVLXTUVMUWHUWKEWOZUVIYKTABCWJWKWAUVEUWBUWFRUVNUVEUVTUWDUWAUWEXTUVB + UVTUWDRZUVCXTUVBUWHUWMUWLUULYKTABCWJWKWLXTUVCUWAUWERZUVBXTUVCUWHUWNUWLUUP + YKTABCWJWKWMVPPWPYKUUMUUQWQWRWSUVGUVRDUVPYDYJUVPUVFWTXAXBXCUUNUVHYJYKUUQU + IZMZDYDOUURYOUUNUVGUWPDYDUUNUVFUWOYJUUMYKUUQXEXDXNUURUWPYNDYDUURUWOYMYJUU + QYLYKXFXDXNXGXHXIXJXKPXLXMYBBTIZYEYGYQKRXTUWQYAXTBAXOTCAUAXPXQPEFDTBYDXRW + CXS $. $} ${ @@ -555534,8 +556077,8 @@ References are made to the second edition (1927, reprinted 1963) of $( Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) $) bj-sblem2 $p |- ( A. x ( ph -> ( ch -> ps ) ) -> ( ( E. x ph -> ch ) -> A. x ( ph -> ps ) ) ) $= - ( wex wi wal 19.23v ax-2 al2imi syl5bir ) ADECFACFZDGACBFFZDGABFZDGACDHML - NDACBIJK $. + ( wex wi wal 19.23v ax-2 al2imi biimtrrid ) ADECFACFZDGACBFFZDGABFZDGACDH + MLNDACBIJK $. $( Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) $) bj-sblem $p |- ( A. x ( ph -> ( ps <-> ch ) ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ch ) ) ) $= @@ -557330,9 +557873,9 @@ 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 biimtrid snprc elex pm2.24d syl5bir syl5 jaod impbid2 ) A - CDZAEZBFZDZUIBGZDZUIBHUMUIUJIEZJZDZUHUKULUOUIBKLUPUKUIUNDZMUHUKUIUJUNNUHUKU - KUQUHUKOUQUIIPZUHUKUIIQURARDZSUHUKAUAUHUSUKACUBUCUDUEUFTTUG $. + wceq elsni cvv wn biimtrid snprc elex pm2.24d biimtrrid syl5 jaod impbid2 ) + ACDZAEZBFZDZUIBGZDZUIBHUMUIUJIEZJZDZUHUKULUOUIBKLUPUKUIUNDZMUHUKUIUJUNNUHUK + UKUQUHUKOUQUIIPZUHUKUIIQURARDZSUHUKAUAUHUSUKACUBUCUDUEUFTTUG $. $( Characterization of the elements of ` B ` in terms of elements of its tagged version. (Contributed by BJ, 6-Oct-2018.) $) @@ -559182,11 +559725,11 @@ variety of applications (for many families of sets, the family of copsex2d $p |- ( ph -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) ) $= ( wceq wex wa syl cv cop wb wcel elisset exdistrv wnf nfe1 nfbid bj-nfexd - a1i 19.9d opeq12 copsexgw bicomd eqcoms adantl bitrd ex bj-exlimd syl5bir - mp2and ) ADUAZFQZDRZEUAZGQZERZFGUBZVCVFUBZQZBSZERZDRZCUCZAFHUDVENDFHUETAG - IUDVHOEGIUETVEVHSVDVGSZERZDRAVOVDVGDEUFAAVQVOVODJVOADAVNCDVNDUGAVMDUHUKLU - IULAAVPVOVOEKVOAEAVNCEAVMEDJVMEUGAVLEUHUKUJMUIULAVPVOAVPSVNBCVPVNBUCZAVPV - JVIQVRVCVFFGUMVRVIVJVKBVNBDEVIUNUOUPTUQPURUSUTUTVAVB $. + a1i 19.9d opeq12 copsexgw bicomd eqcoms adantl bj-exlimd biimtrrid mp2and + bitrd ex ) ADUAZFQZDRZEUAZGQZERZFGUBZVCVFUBZQZBSZERZDRZCUCZAFHUDVENDFHUET + AGIUDVHOEGIUETVEVHSVDVGSZERZDRAVOVDVGDEUFAAVQVOVODJVOADAVNCDVNDUGAVMDUHUK + LUIULAAVPVOVOEKVOAEAVNCEAVMEDJVMEUGAVLEUHUKUJMUIULAVPVOAVPSVNBCVPVNBUCZAV + PVJVIQVRVCVFFGUMVRVIVJVKBVNBDEVIUNUOUPTUQPVAVBURURUSUT $. $} ${ @@ -561605,21 +562148,21 @@ coordinates of a barycenter of two points in one dimension (complex ancom eleq2 ibi eximi an12 exbii exsimpr syl exlimiv velsn anbi2i sylib 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 - VBUUBYCYOUUCYOBYOUUDUUEWHWJWIWKWJWLWMTUVAUVGUVHOUVBUVCUURBYASWNTWOYGYBB - YAWPUURBYAUVBYSYOBYSBXABIYEBYDYCBWQWRWSXLWTXBXCXDXEIAYEUSXFXGXHIAAXIUUP - IAXJXBTXKXMXNXO $. + nfv nfim 3imtr3i syl2anbr mpan2 biimtrrid mpbidi com12 sbimi equsb3 sbv + impbii bitrdi eqrdv eqcomd ) DCUEZFDUAZUBZDXPAXRDXPAJZXRKXSBJZXSUFZLZAD + UCZBUGZUBZKZXSDKZXPXRYEXSXPXQYDXPXQFDUDZUHZYDFDUIXPYIADYAUJZUHYDXPYHYJX + PYHACYAUJZDUDYJFYKDGUKACDYAULUMUQABDUNUOUMUPURYFYGIJZDKZYGIXSYEIADUSZYL + YEKZYGYLXSLZMZANZYMYOYGYLYAKZMZANZYRYOYLXTKZYCMZBNZUUAYCBYLUTZUUCUUABUU + CYGYBYSMZMZANZUUAUUCYGYBUUBMZMZANZUUHUUCYCUUBMUUIADUCUUKUUBYCVGYBUUBADV + AUUIADVBVCUUJUUGAUUIUUFYGUUIUUFYBUUIUUFOUUBYBUUBYSYBXTYAYLVHZVDVEVIVFVJ + PUUHYBYTMZANUUAUUGUUMAYGYBYSVKVLYBYTAVMPVNVOPYTYQAYSYPYGIXSVPZVQVLVRYQY + MAYPYMYGYNVSVOVNVTXSXSLZYGYFQZAWAYPIAWBUUPIAWBUUOUUPYPUUPIAYGYPYFYPYOYF + YGYPYSYGYOUUNYGYAYALZYSYOQZYAWCYGYGBYARZYBBYARZUURUUQYASKZUUSYGOAWDZYGB + YASWETUVAUUTUUQOUVBBYAYASWFTYGYBMZBYARZUURBYARZUUSUUTMUURUVCUURQZBYARZU + VDUVEQZUVAUVGUVBUVFBYASUVCYSUUBYOYBUUBYSOYGUULWGUVCUUBYOQZAUVCANYCUVIYB + ADVBUUBYCYOUUCYOBYOUUDUUEWHWJWIWKWJWLWMTUVAUVGUVHOUVBUVCUURBYASWNTWOYGY + BBYAWPUURBYAUVBYSYOBYSBXABIYEBYDYCBWQWRWSXBWTXCXDXEXFIAYEUSXGXHXIIAAXJU + UPIAXKXCTXLXMXNXO $. $} $d A u x $. $d A x y $. $d B u x $. @@ -562717,26 +563260,26 @@ coordinates of a barycenter of two points in one dimension (complex vo cxp wss eleq2 anbi12d anass crdg nfv cuni c1st cmpo nfmpo2 nfcxfr nfcv cif nfrdg nffv nfeq2 nfn notbid anbi2d opeq2 rdgeq2 fveq1d eqeq2d imbi12d nfim syl vex csuc opex rdg0 a1i con0 nnon fveq2 sylan9eq finxpreclem5 imp - 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(Contributed by Brendan Leahy, @@ -569073,183 +569616,183 @@ curry M LIndF ( R freeLMod I ) ) ) $= con3i nn0cni mulid2i 3eqtri breqtrrdi fsumdvds cuz nncnd npcan1 nnm1nn0 cc nn0zd uzid peano2uz 3syl eqeltrrd fzss2 ssralv wnel raldifb biimparc nnel velsn cle nnred ltm1d nn0red ltnled mpbid elfzle2 syl5ibrcom con2d - clt nsyl imp syl5 expdimp an32s biimtrid idd jad ralimdaa syl5bir con3d - dfrex2 ralimdva syld ss2rabdv hashssdif ciun df-ne ralbii ralnex nnnn0d - 3imtr4g nn0uz eleqtrdi eqeltrd fzsplit2 nnzd fzsn uneq2d raleqdv ralunb - difss ssrexv ralimi biantrurd bitr4id nn0fz0 nfre1 nfralw eldifsn diffi - ad3antrrr ssrab2 ssdomg mp2 hashdifsn mpan 1cnd addsubd hashfz0 3eqtr4d - sylan9eqr hashen cmpt biimpac rabid simplbi2com impancom ancrd reximdv2 - simplbi pm4.71rd df-mpt nfrab1 nfcri cbvopab1 eqtr4i breqi df-br 3bitri - opabidw bitr4di rexbidva nfmpt nfbr cbvrexfw bitrdi sylibd ralimia eqid - breq1 nfel1 eleq1d elrabf simprbi fmpti jctil fodomfi fisseneq mp3an12i - endomtr sbth 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NN ) $. $( Lemma for ~ poimir , a variant of Sperner's lemma. 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$. $( Alternate proof of ~ itg2add using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable @@ -573555,120 +574098,120 @@ curry M LIndF ( R freeLMod I ) ) ) $= inmbl finiunmbl eqeltrd ltsubadd syl3an1 elioomnf ftc1anclem1 itg2addnc ismbf2d breqtrd itg2cl anandis rpaddcld rpxrd xrlelttr anim12i anandirs mpand syld addsub4 replimd oveq1d subdi mp3an3an 3eqtr4rd rpcn 2halvesd - breq12d sylibd reximdvva syl5bir mp2and ) AKUBUCZUDZCTCUEZFUCZVYGIUFZUG - UHZUIUFZVYGGUEZUFZUJUKZULUFZUMZUNUFZKUUAUOUKZUPUQZGUUBURZUSZCTVYJUTUFZV - YGHUEZUFZUJUKZULUFZUMZUNUFZVYRUPUQZHVYTUSZCTVYJVYMVAWUDVBUKZVCUKZUJUKZU - LUFZUMZUNUFZKUPUQZHVYTUSGVYTUSZVYEAVYRUBUCZWUAKUUCZABCDEFGIJVYRLMNOPQRS - VDVEVYFWUJCTVYHVYGUAFWAVAUOUKZUAUEZIUFZVBUKZUMZUFZUGUHZUIUFZWUDUJUKULUF - ZUMZUNUFZVYRUPUQZHVYTUSZVYEAWUSWVMWUTABCDEFHWVEBDEVFUKCDBUEZVGUKWVFUUHU - 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ralrimiv idllmulcl anass1rs + ovex an32s an4s imp idlrmulcl jca wb isidl 3ad2ant1 mpbir3and ) BUAGZAVCU + BZABUCHZRZUDZAUEZWEGZWHBUFHZUGZRZWJUHHZWHGZCIZDIZWJJZWHGZDWHKZEIZWOBUIHZJ + ZWHGZWOWTXAJZWHGZLZEWKKZLZCWHKZWGWHAULZWKWDWCWHXJRWFAUJUKWGFIZWKRZFAKZXJW + KRWCWFXMWDWCWFLZXLFAWCWFXKAGZXLWFXOLZWCXKWEGZXLAWEXKUMZBWJXKWKWJMZWKMZUNN + OPUOFAWKUPQUQWCWFWNWDXNWMXKGZFAKWNXNYAFAWCWFXOYAXPWCXQYAXRBWJXKWMXSWMMZUR + NOPFWMAWJUHVDSQUOWCWFXIWDXNXHCWHWOWHGWOXKGZFAKZXNXHFWOACUSSXNYDXHXNYDLZWS + XGYEWRDWHWPWHGWPXKGZFAKZYEWRFWPADUSSXNYDYGWRYDYGLYCYFLZFAKZXNWRYCYFFAUTXN + YIWQXKGZFAKWRXNYHYJFAWCWFXOYHYJVEZXPWCXQYKXRWCXQLZYHYJWOWPBWJXKXSVATNOVBF + WQAWOWPWJVMSVFVGVHVIVJYEXFEWKXNWTWKGZYDXFXNYMLZYDLZXCXEYOXBXKGZFAKZXCYNYD + YQYNYCYPFAXNYMXOYCYPVEZWCYMWFXOYRXPWCYMLZXQYRXRWCXQYMYRYLYMLZYCYPYLYCYMYP + WOWTBWJXAXKWKXSXAMZXTVKVLTVNNVOOVBVPFXBAWTWOXAVMSQYOXDXKGZFAKZXEYNYDUUCYN + YCUUBFAXNYMXOYCUUBVEZWCYMWFXOUUDXPYSXQUUDXRWCXQYMUUDYTYCUUBYLYCYMUUBWOWTB + 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(Contributed by Jeff Madsen, @@ -581836,31 +582379,31 @@ the next (since the empty set has a finite subcover, the wrex crab cab df-rab eqtrdi abeq2d anbi12d adantr reeanv anbi2i an4 bitri prnc crngm4 3com23 3expa adantllr syl3an1 3expb idllmulcl sylanl1 anassrs rngocl syldan eqeltrrd oveq12 eleq1d syl5ibrcom rexlimdvva adantld sylbid - syl5bir ralrimivv ex igenss snss sylibr ssel syl5com orim12d imim12d syld - vex ralrimdvva adantrd imdistand df-3an 3imtr4g ispridl2 impbid ) BUAOZAB - UBUCOZABUDUCZOZAEUEZFUFZGUFZDPZAOZYFAOZYGAOZUGZQZGERFERZUHZYAYBYDYEKUFZLU - FZDPZAOZLMUFZRZKNUFZRZUUBASZYTASZUGZQZMYCRNYCRZUHZYOYABUIOZYBUUIUJBUKZKLA - BCDENMHIJULUMYAYDYETZUUHTUULYNTUUIYOYAUULUUHYNYAYDUUHYNQZYEYAYDUUMYAYDTZU - UHYMFGEEUUNYFEOZYGEOZTZTZUUHYSLBYGUTZUNPZRZKBYFUTZUNPZRZUVCASZUUTASZUGZQZ - YMYAUUQUUHUVHQZYDYAUUQTZUVCYCOZUUTYCOZUVIYAUUOUVKUUPYAUUJUVBESZUVKUUOUUKY - FEUOZBUVBCEHJUPUQURYAUUPUVLUUOYAUUJUUSESZUVLUUPUUKYGEUOZBUUSCEHJUPUQUSUUG - UVHUUAKUVCRZUVEUUEUGZQNMUVCUUTYCYCUUBUVCVAZUUCUVQUUFUVRUUAKUUBUVCVBUVSUUD - UVEUUEUUBUVCAVCVDVEYTUUTVAZUVQUVDUVRUVGUVTUUAUVAKUVCYSLYTUUTVBVFUVTUUEUVF - UVEYTUUTAVCVGVEVHVIVJUURYIUVDUVGYLUURYIUVDUURYITZYSKLUVCUUTUWAYPUVCOZYQUU - TOZTZYPEOZYPUUBYFDPZVAZNEVKZTZYQEOZYQYTYGDPZVAZMEVKZTZTZYSUURUWDUWOUJZYIY - AUUQUWPYDUVJUWBUWIUWCUWNYAUUOUWBUWIUJUUPYAUUOTZUWIKUVCUWQUVCUWHKEVLUWIKVM - KNYFBCDEHIJWCUWHKEVNVOVPURYAUUPUWCUWNUJUUOYAUUPTZUWNLUUTUWRUUTUWMLEVLUWNL - VMLMYGBCDEHIJWCUWMLEVNVOVPUSVQVJVRUWOUWEUWJTZUWGUWLTZMEVKNEVKZTZUWAYSUXBU - WSUWHUWMTZTUWOUXAUXCUWSUWGUWLNMEEVSVTUWEUWJUWHUWMWAWBUWAUXAYSUWSUWAUWTYSN - MEEUWAUUBEOZYTEOZTZTZYSUWTUWFUWKDPZAOUXGUUBYTDPZYHDPZUXHAUURUXFUXJUXHVAZY - IYAUUQUXFUXKYDYAUUQUXFUXKYAUXFUUQUXKUUBYTYFYGBCDEHIJWDWEWFWGVJUUNYIUXFUXJ - AOZUUQUUNYITUXFUXIEOZUXLUUNUXFUXMYIYAUXFUXMYDYAUXDUXEUXMYAUUJUXDUXEUXMUUK - UUBYTBCDEHIJWMWHWIVJVJUUNYIUXMUXLYAUUJYDYIUXMTUXLUUKYHUXIBCDAEHIJWJWKWLWN - WGWOUWTYRUXHAYPUWFYQUWKDWPWQWRWSWTXBXAXCXDYAUUQUVGYLQYDUVJUVEYJUVFYKUVJYF - UVCOZUVEYJYAUUOUXNUUPUWQUVBUVCSZUXNYAUUJUVMUXOUUOUUKUVNBUVBCEHJXEUQYFUVCF - XMXFXGURUVCAYFXHXIUVJYGUUTOZUVFYKYAUUPUXPUUOUWRUUSUUTSZUXPYAUUJUVOUXQUUPU - UKUVPBUUSCEHJXEUQYGUUTGXMXFXGUSUUTAYGXHXIXJVJXKXLXNXDXOXPYDYEUUHXQYDYEYNX - QXRXAYAUUJYOYBQUUKUUJYOYBABCDEFGHIJXSXDUMXT $. + biimtrrid ralrimivv ex igenss snss sylibr ssel syl5com orim12d ralrimdvva + vex imim12d syld adantrd imdistand df-3an 3imtr4g ispridl2 impbid ) BUAOZ + ABUBUCOZABUDUCZOZAEUEZFUFZGUFZDPZAOZYFAOZYGAOZUGZQZGERFERZUHZYAYBYDYEKUFZ + LUFZDPZAOZLMUFZRZKNUFZRZUUBASZYTASZUGZQZMYCRNYCRZUHZYOYABUIOZYBUUIUJBUKZK + LABCDENMHIJULUMYAYDYETZUUHTUULYNTUUIYOYAUULUUHYNYAYDUUHYNQZYEYAYDUUMYAYDT + ZUUHYMFGEEUUNYFEOZYGEOZTZTZUUHYSLBYGUTZUNPZRZKBYFUTZUNPZRZUVCASZUUTASZUGZ + QZYMYAUUQUUHUVHQZYDYAUUQTZUVCYCOZUUTYCOZUVIYAUUOUVKUUPYAUUJUVBESZUVKUUOUU + KYFEUOZBUVBCEHJUPUQURYAUUPUVLUUOYAUUJUUSESZUVLUUPUUKYGEUOZBUUSCEHJUPUQUSU + UGUVHUUAKUVCRZUVEUUEUGZQNMUVCUUTYCYCUUBUVCVAZUUCUVQUUFUVRUUAKUUBUVCVBUVSU + UDUVEUUEUUBUVCAVCVDVEYTUUTVAZUVQUVDUVRUVGUVTUUAUVAKUVCYSLYTUUTVBVFUVTUUEU + VFUVEYTUUTAVCVGVEVHVIVJUURYIUVDUVGYLUURYIUVDUURYITZYSKLUVCUUTUWAYPUVCOZYQ + UUTOZTZYPEOZYPUUBYFDPZVAZNEVKZTZYQEOZYQYTYGDPZVAZMEVKZTZTZYSUURUWDUWOUJZY + IYAUUQUWPYDUVJUWBUWIUWCUWNYAUUOUWBUWIUJUUPYAUUOTZUWIKUVCUWQUVCUWHKEVLUWIK + VMKNYFBCDEHIJWCUWHKEVNVOVPURYAUUPUWCUWNUJUUOYAUUPTZUWNLUUTUWRUUTUWMLEVLUW + NLVMLMYGBCDEHIJWCUWMLEVNVOVPUSVQVJVRUWOUWEUWJTZUWGUWLTZMEVKNEVKZTZUWAYSUX + BUWSUWHUWMTZTUWOUXAUXCUWSUWGUWLNMEEVSVTUWEUWJUWHUWMWAWBUWAUXAYSUWSUWAUWTY + SNMEEUWAUUBEOZYTEOZTZTZYSUWTUWFUWKDPZAOUXGUUBYTDPZYHDPZUXHAUURUXFUXJUXHVA + ZYIYAUUQUXFUXKYDYAUUQUXFUXKYAUXFUUQUXKUUBYTYFYGBCDEHIJWDWEWFWGVJUUNYIUXFU + XJAOZUUQUUNYITUXFUXIEOZUXLUUNUXFUXMYIYAUXFUXMYDYAUXDUXEUXMYAUUJUXDUXEUXMU + UKUUBYTBCDEHIJWMWHWIVJVJUUNYIUXMUXLYAUUJYDYIUXMTUXLUUKYHUXIBCDAEHIJWJWKWL + WNWGWOUWTYRUXHAYPUWFYQUWKDWPWQWRWSWTXBXAXCXDYAUUQUVGYLQYDUVJUVEYJUVFYKUVJ + YFUVCOZUVEYJYAUUOUXNUUPUWQUVBUVCSZUXNYAUUJUVMUXOUUOUUKUVNBUVBCEHJXEUQYFUV + CFXLXFXGURUVCAYFXHXIUVJYGUUTOZUVFYKYAUUPUXPUUOUWRUUSUUTSZUXPYAUUJUVOUXQUU + PUUKUVPBUUSCEHJXEUQYGUUTGXLXFXGUSUUTAYGXHXIXJVJXMXNXKXDXOXPYDYEUUHXQYDYEY + NXQXRXAYAUUJYOYBQUUKUUJYOYBABCDEFGHIJXSXDUMXT $. $d A a b $. $d B b $. $( Property of a prime ideal in a commutative ring. (Contributed by Jeff @@ -583356,12 +583899,6 @@ A collection of theorems for commuting equalities (or ( cv cvv wcel el3v3 elvd ) ABCACFGHBDEIJ $. $} - $( Double commutation in conjunction. (Contributed by Peter Mazsa, - 27-Jun-2019.) $) - an2anr $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> - ( ( ps /\ ph ) /\ ( th /\ ch ) ) ) $= - ( wa ancom anbi12i ) ABEBAECDEDCEABFCDFG $. - $( Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.) $) anan $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ( ( ph /\ th ) /\ ta ) ) <-> @@ -589017,10 +589554,10 @@ the null class is disjoint (which it is, see ~ disjALTV0 ). (Contributed ( ( 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 $. + wrel rsp2 syl eceq1 a1d elin nel02 pm2.21d biimtrrid jaoi syl6 ) DEZAFZDG + ZHBFZULHIZUKUMJZUKDKZUMDKZLZMJZNZCUPHCUQHIZUPUQJZOZUJUTBULPAULPZUNUTOUJVD + DSBADQRUTABULULTUAUOVCUSUOVBVAUKUMDUBUCVACURHZUSVBCUPUQUDUSVEVBURCUEUFUGU + HUI $. $} ${ @@ -589033,11 +589570,11 @@ the null class is disjoint (which it is, see ~ disjALTV0 ). (Contributed ( 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 - $. + wex eleq2 syl8 exp4a impd biimtrrid expd imp5a imp4b exlimdv biimtrid ex + ) EFZAGZEHZIZCULEJZIZKZCBGZEJZIZDUSIZKZBUMLZDUOIZMVCURUMIZVBKZBSUKUQKZVDV + BBUMNVGVFVDBUKUQVEVBVDUKUQVEUTVAVDUKUQVEUTVAVDMZMZUQVEKUNVEKZUPKUKVIUNVEU + POUKVJUPVIUKVJUPUTVHUKVJUPUTKUOUSPZVHABCEQVKVDVAUOUSDTRUAUBUCUDUEUFUGUHUI + UJ $. $} ${ @@ -589785,9 +590322,9 @@ This theorem (together with ~ pet and ~ pet2 ) is the main result of my prtlem14 $p |- ( Prt A -> ( ( x e. A /\ y e. A ) -> ( ( w e. x /\ w e. y ) -> x = y ) ) ) $= ( wprt cv wcel wa wceq cin c0 wo wi wral df-prt rsp2 sylbi elin wn wal sp - eq0 pm2.21d syl5bir jao1i syl6 ) DEZAFZDGBFZDGHZUHUIIZUHUIJZKIZLZCFZUHGUO - UIGHZUKMUGUNBDNADNUJUNMABDOUNABDDPQUKUMUPUPUOULGZUMUKUOUHUIRUMUQUKUMUQSZC - TURCULUBURCUAQUCUDUEUF $. + eq0 pm2.21d biimtrrid jao1i syl6 ) DEZAFZDGBFZDGHZUHUIIZUHUIJZKIZLZCFZUHG + UOUIGHZUKMUGUNBDNADNUJUNMABDOUNABDDPQUKUMUPUPUOULGZUMUKUOUHUIRUMUQUKUMUQS + ZCTURCULUBURCUAQUCUDUEUF $. $( Lemma for ~ prter1 and ~ prtex . (Contributed by Rodolfo Medina, 13-Oct-2010.) $) @@ -589809,10 +590346,10 @@ This theorem (together with ~ pet and ~ pet2 ) is the main result of my -> ( 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 biimtrid ex ) EFZAGEHZCA - IZJZCBIZDBIZJZBEKZDAIZLUQBGEHZUPJZBMUJUMJZURUPBENVAUTURBUJUMUSUPURUJUMUSU - NUOURUJUMUSUNUOURLZLZUMUSJUKUSJZULJUJVCUKUSULOUJVDULVCUJVDULUNVBUJVDULUNJ - ABPZVBABCEQVEURUOABDRSTUAUBUCUDUEUFUGUHUI $. + syl8 exp4a impd biimtrrid expd imp5a imp4b exlimdv biimtrid ex ) EFZAGEHZ + CAIZJZCBIZDBIZJZBEKZDAIZLUQBGEHZUPJZBMUJUMJZURUPBENVAUTURBUJUMUSUPURUJUMU + SUNUOURUJUMUSUNUOURLZLZUMUSJUKUSJZULJUJVCUKUSULOUJVDULVCUJVDULUNVBUJVDULU + NJABPZVBABCEQVEURUOABDRSTUAUBUCUDUEUFUGUHUI $. $} ${ @@ -591964,14 +592501,14 @@ inference form (e.g., lpssat $p |- ( ph -> E. q e. A ( q C_ U /\ -. q C_ T ) ) $= ( wss wn syl crab cuni cfv adantr cv wral wrex wpss dfpss3 simprbi ralbii wa iman ss2rab clspn clmod wcel cbs lsatlss rabss2 uniss 4syl wceq unimax - wi eqid lssss eqsstrd sstrd adantl syl3anc lssats syl2anc 3sstr4d syl5bir - lspss ex mtod dfrex2 sylibr ) AGUAZENZVQDNZOUHZOZGBUBZOVTGBUCAWBEDNZADEUD - ZWCOZMWDDENWEDEUEUFPWBVRVSVAZGBUBZAWCWFWAGBVRVSUIUGWGVRGBQZVSGBQZNZAWCVRV - SGBUJAWJWCAWJUHZWHRZFUKSZSZWIRZWMSZEDWKFULUMZWOFUNSZNZWLWONZWNWPNAWQWJJTA - WSWJAWOVSGCQZRZWRAWQBCNWIXANWOXBNJBCFHIUOVSGBCUPWIXAUQURAXBDWRADCUMZXBDUS - KGDCUTPAXCDWRNKCDWRFWRVBZHVCPVDVETWJWTAWHWIUQVFWLWOWMWRFXDWMVBZVLVGAEWNUS - ZWJAWQECUMXFJLGBCEWMFHXEIVHVITADWPUSZWJAWQXCXGJKGBCDWMFHXEIVHVITVJVMVKVKV - NVTGBVOVP $. + wi eqid lssss eqsstrd sstrd adantl lspss syl3anc lssats syl2anc biimtrrid + 3sstr4d ex mtod dfrex2 sylibr ) AGUAZENZVQDNZOUHZOZGBUBZOVTGBUCAWBEDNZADE + UDZWCOZMWDDENWEDEUEUFPWBVRVSVAZGBUBZAWCWFWAGBVRVSUIUGWGVRGBQZVSGBQZNZAWCV + RVSGBUJAWJWCAWJUHZWHRZFUKSZSZWIRZWMSZEDWKFULUMZWOFUNSZNZWLWONZWNWPNAWQWJJ + TAWSWJAWOVSGCQZRZWRAWQBCNWIXANWOXBNJBCFHIUOVSGBCUPWIXAUQURAXBDWRADCUMZXBD + USKGDCUTPAXCDWRNKCDWRFWRVBZHVCPVDVETWJWTAWHWIUQVFWLWOWMWRFXDWMVBZVGVHAEWN + USZWJAWQECUMXFJLGBCEWMFHXEIVIVJTADWPUSZWJAWQXCXGJKGBCDWMFHXEIVIVJTVLVMVKV + KVNVTGBVOVP $. $} ${ @@ -592009,12 +592546,12 @@ inference form (e.g., -> ( T C_ U <-> A. p e. A ( p C_ T -> p C_ U ) ) ) $= ( wss crab cuni cfv wcel adantr uniss wceq cv wral expcom ralrimivw clspn wi sstr ss2rab wa clmod cbs lsatlss rabss2 4syl unimax eqid lssss eqsstrd - syl sstrd adantl lspss syl3anc ex syl2anc sseq12d sylibrd syl5bir impbid2 - lssats ) ADEMZGUAZDMZVLEMZUFZGBUBZVKVOGBVMVKVNVLDEUGUCUDVPVMGBNZVNGBNZMZA - VKVMVNGBUHAVSVQOZFUEPZPZVROZWAPZMZVKAVSWEAVSUIFUJQZWCFUKPZMZVTWCMZWEAWFVS - JRAWHVSAWCVNGCNZOZWGAWFBCMVRWJMWCWKMJBCFHIULVNGBCUMVRWJSUNAWKEWGAECQZWKET - LGECUOUSAWLEWGMLCEWGFWGUPZHUQUSURUTRVSWIAVQVRSVAVTWCWAWGFWMWAUPZVBVCVDADW - BEWDAWFDCQDWBTJKGBCDWAFHWNIVJVEAWFWLEWDTJLGBCEWAFHWNIVJVEVFVGVHVI $. + syl sstrd adantl syl3anc lssats syl2anc sseq12d sylibrd biimtrrid impbid2 + lspss ex ) ADEMZGUAZDMZVLEMZUFZGBUBZVKVOGBVMVKVNVLDEUGUCUDVPVMGBNZVNGBNZM + ZAVKVMVNGBUHAVSVQOZFUEPZPZVROZWAPZMZVKAVSWEAVSUIFUJQZWCFUKPZMZVTWCMZWEAWF + VSJRAWHVSAWCVNGCNZOZWGAWFBCMVRWJMWCWKMJBCFHIULVNGBCUMVRWJSUNAWKEWGAECQZWK + ETLGECUOUSAWLEWGMLCEWGFWGUPZHUQUSURUTRVSWIAVQVRSVAVTWCWAWGFWMWAUPZVIVBVJA + DWBEWDAWFDCQDWBTJKGBCDWAFHWNIVCVDAWFWLEWDTJLGBCEWAFHWNIVCVDVEVFVGVH $. $} ${ @@ -594053,19 +594590,19 @@ Functionals and kernels of a left vector space (or module) ( vr vs vz cv wcel w3a wa cfv co wceq wrex cmulr cplusg adantr simpr2 csn clvec lshpkrlem3 simpr3 lveclmod simpr1 lmodvscl syl3anc lmodvacl 3reeanv clmod syl wi simp1l simp1r1 simp1r2 simp1r3 simp2ll simp2lr simp2r simp31 - jca simp32 simp33 lshpkrlem5 syl333anc expdimp rexlimdv rexlimdvva mp3and - 3exp syl5bir ) AUAUTZNVAZEUTZPVAZDUTZPVAZVBZVCZXFUQUTZXFLVDZTIVEGVEVFZUQJ - VGZXHURUTZXHLVDZTIVEGVEVFZURJVGZXDXFIVEZXHGVEZUSUTZYALVDZTIVEGVEVFZUSJVGZ - YCXDXMFVHVDVEXQFVIVDVEVFZXKBCUQFGHIJKLMNOPQXFSTUBUCUDUEUFAQVMVAZXJUGVJZAJ - MVAXJUHVJZATPVAXJUIVJZAXEXGXIVKZAJTVLOVDHVEPVFXJUKVJZULUMUNUOUPVNXKBCURFG - HIJKLMNOPQXHSTUBUCUDUEUFYHYIYJAXEXGXIVOZYLULUMUNUOUPVNXKBCUSFGHIJKLMNOPQY - ASTUBUCUDUEUFYHYIYJXKQWBVAZXTPVAZXIYAPVAXKYGYNYHQVPWCZXKYNXEXGYOYPAXEXGXI - VQYKXDIFNPQXFUBULUNUMVRVSYMGPQXTXHUBUCVTVSYLULUMUNUOUPVNXOXSYEVBXNXRYDVBZ - USJVGZURJVGUQJVGXKYFXNXRYDUQURUSJJJWAXKYRYFUQURJJXKXLJVAZXPJVAZVCZVCYQYFU - SJXKUUAYBJVAZYQYFWDXKUUAUUBVCZYQYFXKUUCYQVBZAXEXGXIYSYTUUBVCXNXRYDYFAXJUU - CYQWEXEXGXIAUUCYQWFXEXGXIAUUCYQWGXEXGXIAUUCYQWHYSYTUUBXKYQWIUUDYTUUBYSYTU - UBXKYQWJXKUUAUUBYQWKWMXKUUCXNXRYDWLXKUUCXNXRYDWNXKUUCXNXRYDWOABCUSDEFGHIJ - KLMNOPQTSTURUQUAUBUCUDUEUFUGUHUIUIUKULUMUNUOUPWPWQXBWRWSWTXCXA $. + jca simp32 simp33 lshpkrlem5 syl333anc 3exp rexlimdv rexlimdvva biimtrrid + expdimp mp3and ) AUAUTZNVAZEUTZPVAZDUTZPVAZVBZVCZXFUQUTZXFLVDZTIVEGVEVFZU + QJVGZXHURUTZXHLVDZTIVEGVEVFZURJVGZXDXFIVEZXHGVEZUSUTZYALVDZTIVEGVEVFZUSJV + GZYCXDXMFVHVDVEXQFVIVDVEVFZXKBCUQFGHIJKLMNOPQXFSTUBUCUDUEUFAQVMVAZXJUGVJZ + AJMVAXJUHVJZATPVAXJUIVJZAXEXGXIVKZAJTVLOVDHVEPVFXJUKVJZULUMUNUOUPVNXKBCUR + FGHIJKLMNOPQXHSTUBUCUDUEUFYHYIYJAXEXGXIVOZYLULUMUNUOUPVNXKBCUSFGHIJKLMNOP + QYASTUBUCUDUEUFYHYIYJXKQWBVAZXTPVAZXIYAPVAXKYGYNYHQVPWCZXKYNXEXGYOYPAXEXG + XIVQYKXDIFNPQXFUBULUNUMVRVSYMGPQXTXHUBUCVTVSYLULUMUNUOUPVNXOXSYEVBXNXRYDV + BZUSJVGZURJVGUQJVGXKYFXNXRYDUQURUSJJJWAXKYRYFUQURJJXKXLJVAZXPJVAZVCZVCYQY + FUSJXKUUAYBJVAZYQYFWDXKUUAUUBVCZYQYFXKUUCYQVBZAXEXGXIYSYTUUBVCXNXRYDYFAXJ + UUCYQWEXEXGXIAUUCYQWFXEXGXIAUUCYQWGXEXGXIAUUCYQWHYSYTUUBXKYQWIUUDYTUUBYSY + TUUBXKYQWJXKUUAUUBYQWKWMXKUUCXNXRYDWLXKUUCXNXRYDWNXKUUCXNXRYDWOABCUSDEFGH + IJKLMNOPQTSTURUQUAUBUCUDUEUFUGUHUIUIUKULUMUNUOUPWPWQWRXBWSWTXAXC $. $} ${ @@ -596477,7 +597014,7 @@ X C ( ._|_ ` Y ) ) ) $= cvrnbtwn $p |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> -. ( X .< Z /\ Z .< Y ) ) $= ( vz wcel w3a wbr wa wn cv wrex wb cvrval 3adant3r3 wi ralnex breq2 breq1 - wral anbi12d notbid rspcv syl5bir adantld 3ad2ant3 adantl sylbid 3impia + wral anbi12d notbid rspcv biimtrrid adantld 3ad2ant3 adantl sylbid 3impia wceq ) EAMZFBMZGBMZHBMZNZFGCOZFHDOZHGDOZPZQZURVBPVCFGDOZFLRZDOZVIGDOZPZLB SQZPZVGURUSUTVCVNTVALABCDEFGIJKUAUBVBVNVGUCZURVAUSVOUTVAVMVGVHVMVLQZLBUGV AVGVLLBUDVPVGLHBVIHUQZVLVFVQVJVDVKVEVIHFDUEVIHGDUFUHUIUJUKULUMUNUOUP $. @@ -597103,27 +597640,27 @@ X C ( ._|_ ` Y ) ) ) $= ( vx wcel wa wbr cfv wceq wss adantr wb coml ccla cal w3a cv crab cmee co coc cp0 simpl2 ssrab2 atssbase rabss2 ax-mp lubss mp3an23 atlpos 3ad2ant3 syl cpo simpl simpr lubid sylan breqtrd wrex wn breq1 elrab simpll2 sstri - wi lubel mp3an3 sylancom ex syl5bir expdimp simpll3 eqid atn0 clat simpl3 - atllat atbase adantl clatlubcl sylancl cops simpl1 omlop syl2anc latlem12 - wne opoccl syl13anc opnoncon breq2d ople0 syl2an 3bitrd expr necon3ad mpd - biimpa syld imnan sylib simplr mtbid nrexdv latmcl syl3anc atlex necon1bd - omllaw3 mp2and ) EUAMZEUBMZEUCMZUDZGCMZNZAUEZGFOZABUFZDPZGFOZGYHEUIPZPZEU - GPZUHZEUJPZQZYHGQZYDYHYFACUFZDPZGFYDXTYHYRFOZXSXTYAYCUKZXTYQCRYGYQRZYSYFA - CULBCRUUABCEHKUMZYFABCUNUOCYGYQDEFHIJUPUQUTYBEVAMZYCYRGQYAXSUUCXTEURUSUUC - YCNACDEFGHIJUUCYCVBUUCYCVCVDVEVFYDLUEZYMFOZLBVGZVHYOYDUUELBYDUUDBMZNZUUDG - FOZUUDYKFOZNZUUEUUHUUIUUJVHZVMUUKVHUUHUUIUUDYHFOZUULYDUUGUUIUUMUUGUUINUUD - YGMZYDUUMYFUUIAUUDBYEUUDGFVIVJYDUUNUUMYDUUNXTUUMXSXTYAYCUUNVKXTUUNYGCRZUU - MYGBCYFABULUUBVLZCYGDEFUUDHIJVNVOVPVQVRVSUUHUUMUULUUHUUMNZUUDYNWOZUULUUHU - URUUMYDUUGYAUURXSXTYAYCUUGVTBUUDEYNYNWAZKWBVPSUUQUUJUUDYNUUHUUMUUJUUDYNQZ - UUHUUMUUJNZUUTUUHUVAUUDYHYKYLUHZFOZUUDYNFOZUUTUUHEWCMZUUDCMZYHCMZYKCMZUVA - UVCTYDUVEUUGYDYAUVEXSXTYAYCWDEWEUTZSZUUGUVFYDBCUUDEHKWFZWGZYDUVGUUGYDXTUU - OUVGYTUUPCYGDEHJWHWIZSYDUVHUUGYDEWJMZUVGUVHYDXSUVNXSXTYAYCWKZEWLUTZUVMCEY - JYHHYJWAZWPWMZSZCEFYLUUDYHYKHIYLWAZWNWQYDUVCUVDTUUGYDUVBYNUUDFYDUVNUVGUVB - YNQUVPUVMCEYLYJYHYNHUVQUVTUUSWRWMWSSYDUVNUVFUVDUUTTUUGUVPUVKCEFUUDYNHIUUS - WTXAXBXFXCXDXEVQXGUUIUUJXHXIUUHUVEUVFYCUVHUUKUUETUVJUVLYBYCUUGXJUVSCEFYLU - UDGYKHIUVTWNWQXKXLYDUUFYMYNYDYMYNWOZUUFYDUWANYAYMCMZUWAUUFXSXTYAYCUWAVTYD - UWBUWAYDUVEYCUVHUWBUVIYBYCVCZUVRCEYLGYKHUVTXMXNSYDUWAVCLBCEFYMYNHIUUSKXOX - NVQXPXEYDXSUVGYCYIYONYPVMUVOUVMUWCCEFYLYJYHGYNHIUVTUVQUUSXQXNXR $. + lubel mp3an3 sylancom biimtrrid expdimp wne simpll3 eqid atn0 clat simpl3 + wi ex atllat atbase adantl clatlubcl sylancl simpl1 omlop opoccl latlem12 + cops syl2anc syl13anc opnoncon breq2d ople0 syl2an 3bitrd biimpa necon3ad + expr syld imnan sylib simplr mtbid nrexdv latmcl syl3anc necon1bd omllaw3 + mpd atlex mp2and ) EUAMZEUBMZEUCMZUDZGCMZNZAUEZGFOZABUFZDPZGFOZGYHEUIPZPZ + EUGPZUHZEUJPZQZYHGQZYDYHYFACUFZDPZGFYDXTYHYRFOZXSXTYAYCUKZXTYQCRYGYQRZYSY + FACULBCRUUABCEHKUMZYFABCUNUOCYGYQDEFHIJUPUQUTYBEVAMZYCYRGQYAXSUUCXTEURUSU + UCYCNACDEFGHIJUUCYCVBUUCYCVCVDVEVFYDLUEZYMFOZLBVGZVHYOYDUUELBYDUUDBMZNZUU + DGFOZUUDYKFOZNZUUEUUHUUIUUJVHZWDUUKVHUUHUUIUUDYHFOZUULYDUUGUUIUUMUUGUUINU + UDYGMZYDUUMYFUUIAUUDBYEUUDGFVIVJYDUUNUUMYDUUNXTUUMXSXTYAYCUUNVKXTUUNYGCRZ + UUMYGBCYFABULUUBVLZCYGDEFUUDHIJVMVNVOWEVPVQUUHUUMUULUUHUUMNZUUDYNVRZUULUU + HUURUUMYDUUGYAUURXSXTYAYCUUGVSBUUDEYNYNVTZKWAVOSUUQUUJUUDYNUUHUUMUUJUUDYN + QZUUHUUMUUJNZUUTUUHUVAUUDYHYKYLUHZFOZUUDYNFOZUUTUUHEWBMZUUDCMZYHCMZYKCMZU + VAUVCTYDUVEUUGYDYAUVEXSXTYAYCWCEWFUTZSZUUGUVFYDBCUUDEHKWGZWHZYDUVGUUGYDXT + UUOUVGYTUUPCYGDEHJWIWJZSYDUVHUUGYDEWOMZUVGUVHYDXSUVNXSXTYAYCWKZEWLUTZUVMC + EYJYHHYJVTZWMWPZSZCEFYLUUDYHYKHIYLVTZWNWQYDUVCUVDTUUGYDUVBYNUUDFYDUVNUVGU + VBYNQUVPUVMCEYLYJYHYNHUVQUVTUUSWRWPWSSYDUVNUVFUVDUUTTUUGUVPUVKCEFUUDYNHIU + USWTXAXBXCXEXDXPWEXFUUIUUJXGXHUUHUVEUVFYCUVHUUKUUETUVJUVLYBYCUUGXIUVSCEFY + LUUDGYKHIUVTWNWQXJXKYDUUFYMYNYDYMYNVRZUUFYDUWANYAYMCMZUWAUUFXSXTYAYCUWAVS + YDUWBUWAYDUVEYCUVHUWBUVIYBYCVCZUVRCEYLGYKHUVTXLXMSYDUWAVCLBCEFYMYNHIUUSKX + QXMWEXNXPYDXSUVGYCYIYONYPWDUVOUVMUWCCEFYLYJYHGYNHIUVTUVQUUSXOXMXR $. $} ${ @@ -597139,12 +597676,13 @@ X C ( ._|_ ` Y ) ) ) $= ( wcel w3a wbr wi wa crab wss cfv wceq atlatmstc coml ccla cal cv simpl13 wral cpo atlpos syl atbase adantl simpl2 postr syl13anc expcomd ralrimdva simpl3 ss2rab club simpl12 ssrab2 atssbase sstri lubss mp3an2 sylancom ex - eqid 3adant3 3adant2 breq12d sylibd syl5bir impbid ) CUAKZCUBKZCUCKZLZEBK - ZFBKZLZEFDMZGUDZEDMZWCFDMZNZGAUFZWAWBWFGAWAWCAKZOZWDWBWEWICUGKZWCBKZVSVTW - DWBOWENWIVQWJVOVPVQVSVTWHUECUHUIWHWKWAABWCCHJUJUKVRVSVTWHULVRVSVTWHUQBCDW - CEFHIUMUNUOUPWGWDGAPZWEGAPZQZWAWBWDWEGAURWAWNWLCUSRZRZWMWORZDMZWBWAWNWRWA - WNVPWRVOVPVQVSVTWNUTVPWMBQWNWRWMABWEGAVAABCHJVBVCBWLWMWOCDHIWOVHZVDVEVFVG - WAWPEWQFDVRVSWPESVTGABWOCDEHIWSJTVIVRVTWQFSVSGABWOCDFHIWSJTVJVKVLVMVN $. + eqid 3adant3 3adant2 breq12d sylibd biimtrrid impbid ) CUAKZCUBKZCUCKZLZE + BKZFBKZLZEFDMZGUDZEDMZWCFDMZNZGAUFZWAWBWFGAWAWCAKZOZWDWBWEWICUGKZWCBKZVSV + TWDWBOWENWIVQWJVOVPVQVSVTWHUECUHUIWHWKWAABWCCHJUJUKVRVSVTWHULVRVSVTWHUQBC + DWCEFHIUMUNUOUPWGWDGAPZWEGAPZQZWAWBWDWEGAURWAWNWLCUSRZRZWMWORZDMZWBWAWNWR + WAWNVPWRVOVPVQVSVTWNUTVPWMBQWNWRWMABWEGAVAABCHJVBVCBWLWMWOCDHIWOVHZVDVEVF + VGWAWPEWQFDVRVSWPESVTGABWOCDEHIWSJTVIVRVTWQFSVSGABWOCDFHIWSJTVJVKVLVMVN + $. $} ${ @@ -597160,11 +597698,11 @@ X C ( ._|_ ` Y ) ) ) $= e. B ) -> ( X .< Y -> E. p e. A ( -. p .<_ X /\ p .<_ Y ) ) ) $= ( coml wcel w3a wbr wn wa wi wral ccla cal cv cpo simp13 atlpos pltnle ex - wrex syld3an1 iman ancom xchbinx ralbii wb atlatle 3com23 biimprd syl5bir - syl con3d dfrex2 syl6ibr syld ) DMNZDUANZDUBNZOZFBNZGBNZOZFGCPZGFEPZQZHUC - ZFEPZQZVOGEPZRZHAUIZDUDNZVIVHVJVLVNSVKVGWAVEVFVGVIVJUEDUFUTWAVIVJOVLVNBCD - EFGIJKUGUHUJVKVNVSQZHATZQVTVKWCVMWCVRVPSZHATZVKVMWDWBHAWDVRVQRVSVRVPUKVRV - QULUMUNVKVMWEVHVJVIVMWEUOABDEGFHIJLUPUQURUSVAVSHAVBVCVD $. + wrex syl syld3an1 iman xchbinx ralbii wb atlatle 3com23 biimprd biimtrrid + ancom con3d dfrex2 syl6ibr syld ) DMNZDUANZDUBNZOZFBNZGBNZOZFGCPZGFEPZQZH + UCZFEPZQZVOGEPZRZHAUIZDUDNZVIVHVJVLVNSVKVGWAVEVFVGVIVJUEDUFUJWAVIVJOVLVNB + CDEFGIJKUGUHUKVKVNVSQZHATZQVTVKWCVMWCVRVPSZHATZVKVMWDWBHAWDVRVQRVSVRVPULV + RVQUTUMUNVKVMWEVHVJVIVMWEUOABDEGFHIJLUPUQURUSVAVSHAVBVCVD $. $} ${ @@ -601109,26 +601647,26 @@ projective geometry (where it is either given as an axiom or stated as wne col cbs cfv simpl1 hlol syl simpr1 simpr2 eqid hlatjcl simpl3 meetat2 oveq1 hlatjidm syl2anc sylan9eqr oveq1d clat hllatd atbase latmcom eleq1d eqtrd eqeq1d orbi12d mpbird adantlr wn df-ne clln simpll1 simpll2 simpll3 - oveq2d llni2 syl31anc simplr2 simpr3 2llnmat syl32anc 3exp2 imp31 syl5bir - wi orrd orcomd pm2.61dane syl13anc oveq2 olj01 mpjaodan ) GUANZBANZCANZUB - ZDANZEANZEIOZPZBCFQZDEFQZUGZUBZRZXDXGXHHQZANZXLIOZPZXEXKXDRXBXCXDXIXOXBXJ - XDUCXCXFXIXBXDUDXKXDUEXCXFXIXBXDUFXBXCXDXIUBZRZXOBCXQBCOZRZXOXHCHQZANZXTI - OZPZXQYCXRXQGUHNZXHGUIUJZNZXAYCXQWSYDWSWTXAXPUKZGULZUMZXQWSXCXDYFYGXBXCXD - XIUNXBXCXDXIUOZAYEFGDEYEUPZJMUQSZWSWTXAXPURZAYECGHXHIYKKLMUSSTXSXMYAXNYBX - SXLXTAXSXLCXHHQZXTXSXGCXHHXRXQXGCCFQZCBCCFUTXQWSXAYOCOYGYMAFGCJMVAVBVCVDX - QYNXTOZXRXQGVENCYENZYFYPXQGYGVFXQXAYQYMAYECGYKMVGUMYLYEGHCXHYKKVHSTVJZVIX - SXLXTIYRVKVLVMXQBCUGZRZXODEXQDEOZXOYSXQUUARZXOXGEHQZANZUUCIOZPZXQUUFUUAXQ - YDXGYENZXDUUFYIXBUUGXPAYEFGBCYKJMUQZTYJAYEEGHXGIYKKLMUSSTUUBXMUUDXNUUEUUB - XLUUCAUUBXHEXGHUUAXQXHEEFQZEDEEFUTXQWSXDUUIEOYGYJAFGEJMVAVBVCWAZVIUUBXLUU - CIUUJVKVLVMVNYTDEUGZRZXNXMUULXNXMXNVOXLIUGZUULXMXLIVPXQYSUUKUUMXMWKXQYSUU - KUUMXMXQYSUUKUUMUBZRZWSXGGVQUJZNZXHUUPNZXIUUMXMWSWTXAXPUUNVRZUUOWSWTXAYSU - UQUUSWSWTXAXPUUNVSWSWTXAXPUUNVTXQYSUUKUUMUNABCFGUUPJMUUPUPZWBWCUUOWSXCXDU - UKUURUUSXCXDXIXBUUNUDXCXDXIXBUUNWDXQYSUUKUUMUOADEFGUUPJMUUTWBWCXCXDXIXBUU - NUFXQYSUUKUUMWEAGHUUPXGXHIKLMUUTWFWGWHWIWJWLWMWNWNWOXKXERZXOXGDHQZANZUVBI - OZPZXKUVEXEXKYDUUGXCUVEXKWSYDWSWTXAXJUKYHUMZXBUUGXJUUHTXBXCXFXIUNZAYEDGHX - GIYKKLMUSSTUVAXMUVCXNUVDUVAXLUVBAUVAXHDXGHXEXKXHDIFQZDEIDFWPXKYDDYENZUVHD - OUVFXKXCUVIUVGAYEDGYKMVGUMYEFGDIYKJLWQVBVCWAZVIUVAXLUVBIUVJVKVLVMXBXCXFXI - UOWR $. + oveq2d wi llni2 syl31anc simplr2 simpr3 2llnmat syl32anc 3exp2 imp31 orrd + biimtrrid orcomd pm2.61dane syl13anc oveq2 olj01 mpjaodan ) GUANZBANZCANZ + UBZDANZEANZEIOZPZBCFQZDEFQZUGZUBZRZXDXGXHHQZANZXLIOZPZXEXKXDRXBXCXDXIXOXB + XJXDUCXCXFXIXBXDUDXKXDUEXCXFXIXBXDUFXBXCXDXIUBZRZXOBCXQBCOZRZXOXHCHQZANZX + TIOZPZXQYCXRXQGUHNZXHGUIUJZNZXAYCXQWSYDWSWTXAXPUKZGULZUMZXQWSXCXDYFYGXBXC + XDXIUNXBXCXDXIUOZAYEFGDEYEUPZJMUQSZWSWTXAXPURZAYECGHXHIYKKLMUSSTXSXMYAXNY + BXSXLXTAXSXLCXHHQZXTXSXGCXHHXRXQXGCCFQZCBCCFUTXQWSXAYOCOYGYMAFGCJMVAVBVCV + DXQYNXTOZXRXQGVENCYENZYFYPXQGYGVFXQXAYQYMAYECGYKMVGUMYLYEGHCXHYKKVHSTVJZV + IXSXLXTIYRVKVLVMXQBCUGZRZXODEXQDEOZXOYSXQUUARZXOXGEHQZANZUUCIOZPZXQUUFUUA + XQYDXGYENZXDUUFYIXBUUGXPAYEFGBCYKJMUQZTYJAYEEGHXGIYKKLMUSSTUUBXMUUDXNUUEU + UBXLUUCAUUBXHEXGHUUAXQXHEEFQZEDEEFUTXQWSXDUUIEOYGYJAFGEJMVAVBVCWAZVIUUBXL + UUCIUUJVKVLVMVNYTDEUGZRZXNXMUULXNXMXNVOXLIUGZUULXMXLIVPXQYSUUKUUMXMWBXQYS + UUKUUMXMXQYSUUKUUMUBZRZWSXGGVQUJZNZXHUUPNZXIUUMXMWSWTXAXPUUNVRZUUOWSWTXAY + SUUQUUSWSWTXAXPUUNVSWSWTXAXPUUNVTXQYSUUKUUMUNABCFGUUPJMUUPUPZWCWDUUOWSXCX + DUUKUURUUSXCXDXIXBUUNUDXCXDXIXBUUNWEXQYSUUKUUMUOADEFGUUPJMUUTWCWDXCXDXIXB + UUNUFXQYSUUKUUMWFAGHUUPXGXHIKLMUUTWGWHWIWJWLWKWMWNWNWOXKXERZXOXGDHQZANZUV + BIOZPZXKUVEXEXKYDUUGXCUVEXKWSYDWSWTXAXJUKYHUMZXBUUGXJUUHTXBXCXFXIUNZAYEDG + HXGIYKKLMUSSTUVAXMUVCXNUVDUVAXLUVBAUVAXHDXGHXEXKXHDIFQZDEIDFWPXKYDDYENZUV + HDOUVFXKXCUVIUVGAYEDGYKMVGUMYEFGDIYKJLWQVBVCWAZVIUVAXLUVBIUVJVKVLVMXBXCXF + XIUOWR $. $( The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.) $) @@ -612123,11 +612661,11 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . -> ( ( R ` F ) e. A \/ ( R ` F ) = .0. ) ) $= ( vp wcel wa cfv wceq wn wne chlt cv cple wbr wrex eqid lhpexnle ad2antrr df-ne simplll simpr simpllr simplr adantr trl0 syl112anc ex necon3d trlat - mpd rexlimddv syl5bir orrd orcomd ) FUAOGEOPZDCOZPZDBQZHRZVHAOZVGVIVJVISV - HHTZVGVJVHHUIVGVKVJVGVKPZNUBZGFUCQZUDSZVJNAVEVONAUEVFVKAEFVNGNVNUFZJKUGUH - VLVMAOVOPZPZVEVQVFVMDQZVMTZVJVEVFVKVQUJZVLVQUKVEVFVKVQULZVRVKVTVGVKVQUMVR - VSVMVHHVRVSVMRZVIVRWCPVEVQVFWCVIVRVEWCWAUNVLVQWCUMVRVFWCWBUNVRWCUKAVMBCDE - FVNGHVPIJKLMUOUPUQURUTAVMBCDEFVNGVPJKLMUSUPVAUQVBVCVD $. + mpd rexlimddv biimtrrid orrd orcomd ) FUAOGEOPZDCOZPZDBQZHRZVHAOZVGVIVJVI + SVHHTZVGVJVHHUIVGVKVJVGVKPZNUBZGFUCQZUDSZVJNAVEVONAUEVFVKAEFVNGNVNUFZJKUG + UHVLVMAOVOPZPZVEVQVFVMDQZVMTZVJVEVFVKVQUJZVLVQUKVEVFVKVQULZVRVKVTVGVKVQUM + VRVSVMVHHVRVSVMRZVIVRWCPVEVQVFWCVIVRVEWCWAUNVLVQWCUMVRVFWCWBUNVRWCUKAVMBC + DEFVNGHVPIJKLMUOUPUQURUTAVMBCDEFVNGVPJKLMUSUPVAUQVBVCVD $. $( The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.) $) @@ -613096,18 +613634,18 @@ removing it shorten (and not lengthen) proofs using it? 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(Contributed by NM, wi simpl11 simpl13 syl2anc clat simp11l adantr hllatd simp11r lhpbase syl wb latmcl syl3anc latmle2 dibelval3 syl12anc anbi12d reeanv simpll1 simpr simplr dihord10 3exp2 oveq12 eqeq2d imbi1d imbi2d biimprd com23 impr syl6 - com12 rexlimdvv syl5bir sylbid exp32 ralrimiv simp11 simp2l simp2r trlord - mpd syl122anc mpbird ) QVGVHZUBMVHZVIZFAVHFUBRVJVKVIZTAVHTUBRVJVKVIZVLZUC - BVHZUDBVHZVIZFOVMUCUBSVNZNVMEVNTOVMZUDUBSVNZNVMZEVNVOZVLZUUDUUFRVJZVBVPZG - VMZUUDRVJZUULUUFRVJZWJZVBHVQZUUIUUOVBHUUIUUKHVHZUUMUUNUUIUUQUUMVIZVIZUUKU - AVRZVCVPZVDVPZDVNZVSZVDUUGVTVCUUEVTZUUNUUSYTUUAUUBUUHUUQUUMUVEYTUUCUUHUUR - WAUUAUUBYTUUHUURWBUUAUUBYTUUHUURWCZYTUUCUUHUURWDUUIUUQUUMWEUUIUUQUUMWFVCV - DABCDEFGHIVBJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAWGWHUUSU - VDUUNVCVDUUEUUGUUSUVAUUEVHZUVBUUGVHZVIUVALVEVPZVMUVIVRZVSZVEKVTZUVBVFVPZU - AVRZVSZUVMGVMUUFRVJZVIZVFHVTZVIZUVDUUNWJZUUSUVGUVLUVHUVRUUSYQYSUVGUVLXAYQ - YRYSUUCUUHUURWKZYQYRYSUUCUUHUURWLACTHJKLMOQRVGUBUVAVEUFUIUJURUOUSULVAWIWM - UUSYQUUFBVHZUUFUBRVJZUVHUVRXAUWAUUSQWNVHZUUBUBBVHZUWBUUSQUUIYOUURYOYPYRYS - UUCUUHWOZWPWQZUVFUUSYPUWEUUIYPUURYOYPYRYSUUCUUHWRZWPBMQUBUEUJWSZWTZBQSUDU - BUEUHXBZXCUUSUWDUUBUWEUWCUWGUVFUWJBQRSUDUBUEUFUHXDZXCBGHVFJMNQRVGUBUUFUVB - UAUEUFUJUOUPUQUKXEXFXGUVSUVKUVQVIZVFHVTVEKVTUUSUVTUVKUVQVEVFKHXHUUSUWMUVT - VEVFKHUUSUVIKVHUVMHVHVIZUVPUUTUVJUVNDVNZVSZUUNWJZWJZUWMUVTWJUUSUWNUVPUWPU - UNUUSUWNUVPUWPVLZVIYTUURUWSUUNYTUUCUUHUURUWSXIUUIUURUWSXKUUSUWSXJABCDEFGH - IVBVFJKLMNOPQRSTUAUBUCUDVEUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAXLXCXMUWMUWRU - VTUVKUVOUVPUWRUVTWJUVKUVOVIZUWRUVPUVTUWTUVPUVTWJUWRUWTUVTUWQUVPUWTUVDUWPU - UNUWTUVCUWOUUTUVAUVJUVBUVNDXNXOXPXQXRXSXTYBYAYCYDYEYCYLYFYGUUIYQUUDBVHZUU - DUBRVJZUWBUWCUUJUUPXAYQYRYSUUCUUHYHUUIUWDUUAUWEUXAUUIQUWFWQZYTUUAUUBUUHYI - ZUUIYPUWEUWHUWIWTZBQSUCUBUEUHXBXCUUIUWDUUAUWEUXBUXCUXDUXEBQRSUCUBUEUFUHXD - XCUUIUWDUUBUWEUWBUXCYTUUAUUBUUHYJZUXEUWKXCUUIUWDUUBUWEUWCUXCUXFUXEUWLXCAB - GHVBMQRUBUUDUUFUEUFUIUJUOUPYKYMYN $. + com12 rexlimdvv biimtrrid sylbid mpd ralrimiv simp11 simp2l simp2r trlord + exp32 syl122anc mpbird ) QVGVHZUBMVHZVIZFAVHFUBRVJVKVIZTAVHTUBRVJVKVIZVLZ + UCBVHZUDBVHZVIZFOVMUCUBSVNZNVMEVNTOVMZUDUBSVNZNVMZEVNVOZVLZUUDUUFRVJZVBVP + ZGVMZUUDRVJZUULUUFRVJZWJZVBHVQZUUIUUOVBHUUIUUKHVHZUUMUUNUUIUUQUUMVIZVIZUU + KUAVRZVCVPZVDVPZDVNZVSZVDUUGVTVCUUEVTZUUNUUSYTUUAUUBUUHUUQUUMUVEYTUUCUUHU + URWAUUAUUBYTUUHUURWBUUAUUBYTUUHUURWCZYTUUCUUHUURWDUUIUUQUUMWEUUIUUQUUMWFV + CVDABCDEFGHIVBJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAWGWHUU + SUVDUUNVCVDUUEUUGUUSUVAUUEVHZUVBUUGVHZVIUVALVEVPZVMUVIVRZVSZVEKVTZUVBVFVP + ZUAVRZVSZUVMGVMUUFRVJZVIZVFHVTZVIZUVDUUNWJZUUSUVGUVLUVHUVRUUSYQYSUVGUVLXA + YQYRYSUUCUUHUURWKZYQYRYSUUCUUHUURWLACTHJKLMOQRVGUBUVAVEUFUIUJURUOUSULVAWI + WMUUSYQUUFBVHZUUFUBRVJZUVHUVRXAUWAUUSQWNVHZUUBUBBVHZUWBUUSQUUIYOUURYOYPYR + YSUUCUUHWOZWPWQZUVFUUSYPUWEUUIYPUURYOYPYRYSUUCUUHWRZWPBMQUBUEUJWSZWTZBQSU + DUBUEUHXBZXCUUSUWDUUBUWEUWCUWGUVFUWJBQRSUDUBUEUFUHXDZXCBGHVFJMNQRVGUBUUFU + VBUAUEUFUJUOUPUQUKXEXFXGUVSUVKUVQVIZVFHVTVEKVTUUSUVTUVKUVQVEVFKHXHUUSUWMU + VTVEVFKHUUSUVIKVHUVMHVHVIZUVPUUTUVJUVNDVNZVSZUUNWJZWJZUWMUVTWJUUSUWNUVPUW + PUUNUUSUWNUVPUWPVLZVIYTUURUWSUUNYTUUCUUHUURUWSXIUUIUURUWSXKUUSUWSXJABCDEF + GHIVBVFJKLMNOPQRSTUAUBUCUDVEUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAXLXCXMUWMUW + RUVTUVKUVOUVPUWRUVTWJUVKUVOVIZUWRUVPUVTUWTUVPUVTWJUWRUWTUVTUWQUVPUWTUVDUW + PUUNUWTUVCUWOUUTUVAUVJUVBUVNDXNXOXPXQXRXSXTYBYAYCYDYEYCYFYLYGUUIYQUUDBVHZ + UUDUBRVJZUWBUWCUUJUUPXAYQYRYSUUCUUHYHUUIUWDUUAUWEUXAUUIQUWFWQZYTUUAUUBUUH + YIZUUIYPUWEUWHUWIWTZBQSUCUBUEUHXBXCUUIUWDUUAUWEUXBUXCUXDUXEBQRSUCUBUEUFUH + XDXCUUIUWDUUBUWEUWBUXCYTUUAUUBUUHYJZUXEUWKXCUUIUWDUUBUWEUWCUXCUXFUXEUWLXC + ABGHVBMQRUBUUDUUFUEUFUIUJUOUPYKYMYN $. $( Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.) $) @@ -637325,17 +637863,17 @@ of phi(x) is independent of the atom q." (Contributed by NM, ( vq vr chlt wcel wa wbr wn co w3a wrex cfv cin wceq simp1 simp3ll simp3r cv simp2 lhpmcvr6N syl112anc simp3l simp2l simp1l latmcom syl3anc eqbrtrd clat hllatd reeanv simp11 simp12 3ad2ant1 simp3l1 simp3r1 simp3l3 simp3r3 - simp2r simp13r 3jca dihmeetlem19N syl33anc 3exp rexlimdvv syl5bir mp2and - jca ) HUEUFZKEUFZUGZLBUFZLKIUHUIZUGZMBUFZMKIUHUIZUGZLMJUJZKIUHZUGZUKZUCUS - ZKIUHUIZXBMIUHUIZXBLIUHZUKZUCAULZUDUSZKIUHUIZXHLIUHUIZXHMIUHZUKZUDAULZWRF - UMLFUMMFUMUNUOZXAWKWNWOWSXGWKWNWTUPZWKWNWTUTWOWPWSWKWNUQZWKWNWQWSURZABEGH - IJKLMUCNOQRSPVAVBXAWKWQWLMLJUJZKIUHXMXOWKWNWQWSVCWKWLWMWTVDZXAXRWRKIXAHVI - UFWOWLXRWRUOXAHWIWJWNWTVEVJXPXSBHJMLNRVFVGXQVHABEGHIJKMLUDNOQRSPVAVBXGXMU - GXFXLUGZUDAULUCAULXAXNXFXLUCUDAAVKXAXTXNUCUDAAXAXBAUFZXHAUFZUGZXTXNXAYCXT - UKZWKWNWOYAXCUGYBXIUGXEXKWSUKXNWKWNWTYCXTVLWKWNWTYCXTVMXAYCWOXTXPVNYDYAXC - XAYAYBXTVDXCXDXEXLXAYCVOWHYDYBXIXAYAYBXTVSXIXJXKXFXAYCVPWHYDXEXKWSXCXDXEX - LXAYCVQXIXJXKXFXAYCVRWQWSWKWNYCXTVTWAABCDEFGHIJKLMUDUCNOPQRSTUAUBWBWCWDWE - WFWG $. + jca simp2r simp13r 3jca dihmeetlem19N syl33anc rexlimdvv biimtrrid mp2and + 3exp ) HUEUFZKEUFZUGZLBUFZLKIUHUIZUGZMBUFZMKIUHUIZUGZLMJUJZKIUHZUGZUKZUCU + SZKIUHUIZXBMIUHUIZXBLIUHZUKZUCAULZUDUSZKIUHUIZXHLIUHUIZXHMIUHZUKZUDAULZWR + FUMLFUMMFUMUNUOZXAWKWNWOWSXGWKWNWTUPZWKWNWTUTWOWPWSWKWNUQZWKWNWQWSURZABEG + HIJKLMUCNOQRSPVAVBXAWKWQWLMLJUJZKIUHXMXOWKWNWQWSVCWKWLWMWTVDZXAXRWRKIXAHV + IUFWOWLXRWRUOXAHWIWJWNWTVEVJXPXSBHJMLNRVFVGXQVHABEGHIJKMLUDNOQRSPVAVBXGXM + UGXFXLUGZUDAULUCAULXAXNXFXLUCUDAAVKXAXTXNUCUDAAXAXBAUFZXHAUFZUGZXTXNXAYCX + TUKZWKWNWOYAXCUGYBXIUGXEXKWSUKXNWKWNWTYCXTVLWKWNWTYCXTVMXAYCWOXTXPVNYDYAX + CXAYAYBXTVDXCXDXEXLXAYCVOVSYDYBXIXAYAYBXTVTXIXJXKXFXAYCVPVSYDXEXKWSXCXDXE + XLXAYCVQXIXJXKXFXAYCVRWQWSWKWNYCXTWAWBABCDEFGHIJKLMUDUCNOPQRSTUAUBWCWDWHW + EWFWG $. $} ${ @@ -638673,11 +639211,11 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= by NM, 1-Jan-2015.) $) dochkrshp4 $p |- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V \/ ( L ` G ) = V ) ) ) $= - ( cfv wceq wne wo wa df-ne dochkrshp3 biimprd expdimp syl5bir orrd orcomd - wn ex simpl syl6bi dochoc1 2fveq3 id eqeq12d syl5ibrcom jaod impbid ) ADG - SZHSHSZVBTZVCIUAZVBITZUBZAVDVGAVDUCZVFVEVHVFVEVFUKVBIUAZVHVEVBIUDAVDVIVEA - VEVDVIUCZABCDEFGHIJKLMNOPQRUEZUFUGUHUIUJULAVEVDVFAVEVJVDVKVDVIUMUNAVDVFIH - SHSZITABEFHIJKMLNQUOVFVCVLVBIVBIHHUPVFUQURUSUTVA $. + ( cfv wceq wne wo wa wn df-ne dochkrshp3 biimprd expdimp biimtrrid orcomd + orrd ex simpl syl6bi dochoc1 2fveq3 id eqeq12d syl5ibrcom jaod impbid ) A + DGSZHSHSZVBTZVCIUAZVBITZUBZAVDVGAVDUCZVFVEVHVFVEVFUDVBIUAZVHVEVBIUEAVDVIV + EAVEVDVIUCZABCDEFGHIJKLMNOPQRUFZUGUHUIUKUJULAVEVDVFAVEVJVDVKVDVIUMUNAVDVF + IHSHSZITABEFHIJKMLNQUOVFVCVLVBIVBIHHUPVFUQURUSUTVA $. $} ${ @@ -641099,23 +641637,23 @@ orthocomplement of its kernel (when its kernel is a closed hyperplane). ( 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 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 $. + dochssv syl2anc ssdifd ad3antrrr simprl sseldd simprr reximssdv biimtrrid + wss lcfl6lem ex orrd olc syl5ibr w3a cdih crn adantr eldifi adantl dochcl + snssd dochoc 3adant3 simp3 fveq2d simpr dochsnkr2 3eqtr4d 3ad2ant1 mpbird + eqtrd lcfl1 rexlimdv3a jaod impbid ) ANEUPZNQUQZSURZNDSDUSCUSLUSBUSZIUTFU + TURCYDVAZRUQZVBLGVCVDZURZBSUAVAZVEZVBZVFZAYAYLAYAVGZYCYKYCVHZYBSVIZYMYKYB + SVJZYMYOYKYMYOVGZYDNUQHVKUQZURZYHBYJYBRUQZYIVEZYQBHJYRMNOPQRSTUAUBUCUDUEU + HUJYRVLZUKULAPVMUPTOUPVGZYAYOUNVNANMUPZYAYOUOVNYMYOYTRUQZSVIZYOYNYMUUFYPY + MYCUUFYMUUFYCAYAUUFYCVFZAEJKMNOPQRSTUBUCUDUEUKULUMUNUOVOZVPVQVRVSVTWAYQYD + UUAUPZYSVGZVGZUUAYJYDAUUAYJWMYAYOUUJAYTSYIAUUCYBSWMYTSWMUNAMNQSJUEUKULAJO + PTUBUDUNWBUOWCJOPRSTYBUBUDUEUCWDWEWFWGYQUUIYSWHZWIUUKCDFGHIJYRLMNOPQRSTYD + UAUBUCUDUEUFUGUHUUBUIUJUKULAUUCYAYOUUJUNWGAUUDYAYOUUJUOWGUULYQUUIYSWJWNWK + WOWLWPWOAYCYAYKYCYAAUUGYCUUFWQUUHWRAYHYABYJAYDYJUPZYHWSZYAUUEYBURUUNYFRUQ + ZRUQZYFUUEYBAUUMUUPYFURZYHAUUMVGZUUCYFTPWTUQUQZXAUPZUUQAUUCUUMUNXBZUURUUC + YESWMUUTUVAUURYDSUUMYDSUPAYDSYIXCXDXFJOUUSPRSTYEUBUUSVLZUDUEUCXEWEOUUSPRT + YFUBUVBUCXGWEXHUUNYTUUORUUNYBYFRUUNYBYGQUQZYFUUNNYGQAUUMYHXIXJAUUMUVCYFUR + YHUURCDHFGIJLYGOPQRSTYDUAUBUCUDUEUJUFUGULUHUIYGVLUVAAUUMXKXLXHXPZXJXJUVDX + MUUNEKMNQRUMAUUMUUDYHUOXNXQXOXRXSXT $. $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. $. @@ -658794,47 +659332,6 @@ D Fn ( I ... ( M - 1 ) ) /\ YPIQUAUDUOVULUPUJUMVUOYSABEJHUVDUVJFYPORUBUEUOVUNUPUKUNVUOYSWRYN $. $} - ${ - $d ph x y $. $d B x y $. $d F x y $. $d S x y $. $d T x y $. - ismhmd.b $e |- B = ( Base ` S ) $. - ismhmd.c $e |- C = ( Base ` T ) $. - ismhmd.p $e |- .+ = ( +g ` S ) $. - ismhmd.q $e |- .+^ = ( +g ` T ) $. - ismhmd.0 $e |- .0. = ( 0g ` S ) $. - ismhmd.z $e |- Z = ( 0g ` T ) $. - ismhmd.s $e |- ( ph -> S e. Mnd ) $. - ismhmd.t $e |- ( ph -> T e. Mnd ) $. - ismhmd.f $e |- ( ph -> F : B --> C ) $. - ismhmd.a $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> - ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) $. - ismhmd.h $e |- ( ph -> ( F ` .0. ) = Z ) $. - $( Deduction version of ~ ismhm . (Contributed by SN, 27-Jul-2024.) $) - ismhmd $p |- ( ph -> F e. ( S MndHom T ) ) $= - ( cmnd wcel wf cv cfv wceq wral w3a cmhm ralrimivva 3jca ismhm syl21anbrc - co ) AHUDUEIUDUEDEJUFZBUGZCUGZFUQJUHUSJUHUTJUHGUQUIZCDUJBDUJZKJUHLUIZUKJH - IULUQUESTAURVBVCUAAVABCDDUBUMUCUNBCDEFGHIJLKMNOPQRUOUP $. - $} - - ${ - ablcmnd.1 $e |- ( ph -> G e. Abel ) $. - $( An Abelian group is a commutative monoid. (Contributed by SN, - 1-Jun-2024.) $) - ablcmnd $p |- ( ph -> G e. CMnd ) $= - ( cabl wcel ccmn ablcmn syl ) ABDEBFECBGH $. - $} - - ${ - ringcld.b $e |- B = ( Base ` R ) $. - ringcld.t $e |- .x. = ( .r ` R ) $. - ringcld.r $e |- ( ph -> R e. Ring ) $. - ringcld.x $e |- ( ph -> X e. B ) $. - ringcld.y $e |- ( ph -> Y e. B ) $. - $( Closure of the multiplication operation of a ring. (Contributed by SN, - 29-Jul-2024.) $) - ringcld $p |- ( ph -> ( X .x. Y ) e. B ) $= - ( crg wcel co ringcl syl3anc ) ACLMEBMFBMEFDNBMIJKBCDEFGHOP $. - $} - ${ ringassd.b $e |- B = ( Base ` R ) $. ringassd.t $e |- .x. = ( .r ` R ) $. @@ -658873,17 +659370,6 @@ D Fn ( I ... ( M - 1 ) ) /\ ( crg wcel co wceq ringridm syl2anc ) ACLMFBMFEDNFOJKBCDEFGHIPQ $. $} - ${ - ringabld.1 $e |- ( ph -> R e. Ring ) $. - $( A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.) $) - ringabld $p |- ( ph -> R e. Abel ) $= - ( crg wcel cabl ringabl syl ) ABDEBFECBGH $. - - $( A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.) $) - ringcmnd $p |- ( ph -> R e. CMnd ) $= - ( ringabld ablcmnd ) ABABCDE $. - $} - ${ $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 ) $. @@ -658971,17 +659457,12 @@ D Fn ( I ... ( M - 1 ) ) /\ LABHBAGIJ $. ${ - drnginvrcld.b $e |- B = ( Base ` R ) $. - drnginvrcld.0 $e |- .0. = ( 0g ` R ) $. - drnginvrcld.i $e |- I = ( invr ` R ) $. - drnginvrcld.r $e |- ( ph -> R e. DivRing ) $. - drnginvrcld.x $e |- ( ph -> X e. B ) $. - drnginvrcld.1 $e |- ( ph -> X =/= .0. ) $. - $( Closure of the multiplicative inverse in a division ring. ( ~ reccld - analog). (Contributed by SN, 14-Aug-2024.) $) - drnginvrcld $p |- ( ph -> ( I ` X ) e. B ) $= - ( cdr wcel wne cfv drnginvrcl syl3anc ) ACMNEBNEFOEDPBNJKLBCDEFGHIQR $. - + drnginvrn0d.b $e |- B = ( Base ` R ) $. + drnginvrn0d.0 $e |- .0. = ( 0g ` R ) $. + drnginvrn0d.i $e |- I = ( invr ` R ) $. + drnginvrn0d.r $e |- ( ph -> R e. DivRing ) $. + drnginvrn0d.x $e |- ( ph -> X e. B ) $. + drnginvrn0d.1 $e |- ( ph -> X =/= .0. ) $. $( A multiplicative inverse in a division ring is nonzero. ( ~ recne0d analog). (Contributed by SN, 14-Aug-2024.) $) drnginvrn0d $p |- ( ph -> ( I ` X ) =/= .0. ) $= @@ -659215,65 +659696,6 @@ D Fn ( I ... ( M - 1 ) ) /\ JKLSTUA $. $} - ${ - $d A a b x $. $d B a b x $. $d M a b $. $d R x $. $d T a b $. - $d Y x $. $d ph a b x $. - pwspjmhmmgpd.y $e |- Y = ( R ^s I ) $. - pwspjmhmmgpd.b $e |- B = ( Base ` Y ) $. - pwspjmhmmgpd.m $e |- M = ( mulGrp ` Y ) $. - pwspjmhmmgpd.t $e |- T = ( mulGrp ` R ) $. - pwspjmhmmgpd.r $e |- ( ph -> R e. Ring ) $. - pwspjmhmmgpd.i $e |- ( ph -> I e. V ) $. - pwspjmhmmgpd.a $e |- ( ph -> A e. I ) $. - $( The projection given by ~ pwspjmhm is also a monoid homomorphism between - the respective multiplicative groups. (Contributed by SN, - 30-Jul-2024.) $) - pwspjmhmmgpd $p |- ( ph -> ( x e. B |-> ( x ` A ) ) e. ( M MndHom T ) ) $= - ( cfv wcel wceq va vb cbs cmulr cv cmpt cur mgpbas mgpplusg ringidval crg - eqid cmnd pwsring syl2anc ringmgp syl wa adantr pwselbas ffvelcdmd fmpttd - simpr co cof simprl simprr pwsmulrval fveq1d ffnd inidm eqidd ofval eqtrd - mpidan ringcl syl3an1 3expb fveq1 fvex fvmpt oveq12d 3eqtr4d csn ringidcl - cxp 3syl pws1 fvconst2 3eqtr2d ismhmd ) AUAUBDEUCRZJUDRZEUDRZHFBDCBUEZRZU - FZJUGRZEUGRZDJHMLUHWLEFNWLULZUHJWMHMWMULZUIEWNFNWNULZUIJWRHMWRULZUJEWSFNW - SULZUJAJUKSZHUMSAEUKSZGISZXEOPEGIJKUNUOZJHMUPUQAXFFUMSOEFNUPUQABDWPWLAWOD - SZURZGWLCWOXJWLEGDUKWOJIKWTLAXFXIOUSAXGXIPUSAXIVCUTACGSZXIQUSVAVBAUAUEZDS - ZUBUEZDSZURZURZCXLXNWMVDZRZCXLRZCXNRZWNVDZXRWQRZXLWQRZXNWQRZWNVDXQXSCXLXN - WNVEVDZRZYBXQCXRYFXQDEWMWNXLXNGUKIJKLAXFXPOUSZAXGXPPUSZAXMXOVFZAXMXOVGZXB - XAVHVIAXPXKYGYBTQXQGGXTYAWNGXLXNIICXQGWLXLXQWLEGDUKXLJIKWTLYHYIYJUTVJXQGW - LXNXQWLEGDUKXNJIKWTLYHYIYKUTVJYIYIGVKXQXKURZXTVLYLYAVLVMVOVNXQXRDSZYCXSTA - XMXOYMAXEXMXOYMXHDJWMXLXNLXAVPVQVRBXRWPXSDWQCWOXRVSWQULZCXRVTWAUQXQYDXTYE - YAWNXQXMYDXTTYJBXLWPXTDWQCWOXLVSYNCXLVTWAUQXQXOYEYATYKBXNWPYADWQCWOXNVSYN - CXNVTWAUQWBWCAWRWQRZCWRRZCGWSWDWFZRZWSAXEWRDSYOYPTXHDJWRLXCWEBWRWPYPDWQCW - OWRVSYNCWRVTWAWGACYQWRAXFXGYQWRTOPEWSGIJKXDWHUOVIAXKYRWSTQGWSCEUGVTWIUQWJ - WK $. - $} - - ${ - $d A x $. $d B x $. $d N x $. $d R x $. $d X x $. $d Y x $. - $d .xb x $. $d ph x $. - pwsexpg.y $e |- Y = ( R ^s I ) $. - pwsexpg.b $e |- B = ( Base ` Y ) $. - pwsexpg.m $e |- M = ( mulGrp ` Y ) $. - pwsexpg.t $e |- T = ( mulGrp ` R ) $. - pwsexpg.s $e |- .xb = ( .g ` M ) $. - pwsexpg.g $e |- .x. = ( .g ` T ) $. - pwsexpg.r $e |- ( ph -> R e. Ring ) $. - pwsexpg.i $e |- ( ph -> I e. V ) $. - pwsexpg.n $e |- ( ph -> N e. NN0 ) $. - pwsexpg.x $e |- ( ph -> X e. B ) $. - pwsexpg.a $e |- ( ph -> A e. I ) $. - $( Value of a group exponentiation in a structure power. Compare - ~ pwsmulg . (Contributed by SN, 30-Jul-2024.) $) - pwsexpg $p |- ( ph -> ( ( N .xb X ) ` A ) = ( N .x. ( X ` A ) ) ) $= - ( vx cfv cmpt cmhm wcel cn0 wceq pwspjmhmmgpd mgpbas mhmmulg syl3anc cmnd - co crg pwsring syl2anc ringmgp syl mulgnn0cl fveq1 eqid fvex fvmpt oveq2d - cv 3eqtr3d ) AJLEUQZUECBUEVIZUFZUGZUFZJLVNUFZGUQZBVKUFZJBLUFZGUQAVNIFUHUQ - UIJUJUIZLCUIZVOVQUKAUEBCDFHIKMNOPQTUAUDULUBUCCEGVNIFJLCMIPOUMZRSUNUOAVKCU - IZVOVRUKAIUPUIZVTWAWCAMURUIZWDADURUIHKUIWETUADHKMNUSUTMIPVAVBUBUCCEIJLWBR - VCUOUEVKVMVRCVNBVLVKVDVNVEZBVKVFVGVBAVPVSJGAWAVPVSUKUCUELVMVSCVNBVLLVDWFB - LVFVGVBVHVJ $. - $} - ${ $d B x $. $d I a x y $. $d J a x y $. $d M a x $. $d R a $. $d T a $. $d U a $. $d Y a y $. $d ph a x y $. @@ -661745,11 +662167,11 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove remulid2 $p |- ( A e. RR -> ( 1 x. A ) = A ) $= ( vx cr wcel cc0 wceq c1 co wn wne df-ne wa ax-rrecex simpll recnd simprl cmul cv mulassd simprr oveq1d remulinvcom ax-1rid eqtrd 3eqtr3d rexlimddv - oveq2d syl ex syl5bir 1re remul01 mp1i oveq2 id 3eqtr4d pm2.61d2 ) ACDZAE - FZGAQHZAFZUSIAEJZURVAAEKURVBVAURVBLZABRZQHZGFZVABCBAMVCVDCDZVFLZLZVEAQHAV - DAQHZQHZUTAVIAVDAVIAURVBVHNZOZVIVDVCVGVFPZOVMSVIVEGAQVCVGVFTZUAVIVKAGQHZA - VIVJGAQVIAVDVLVNVOUBUGVIURVPAFVLAUCUHUDUEUFUIUJUSGEQHZEUTAGCDVQEFUSUKGULU - MAEGQUNUSUOUPUQ $. + oveq2d syl ex biimtrrid 1re remul01 mp1i oveq2 id 3eqtr4d pm2.61d2 ) ACDZ + AEFZGAQHZAFZUSIAEJZURVAAEKURVBVAURVBLZABRZQHZGFZVABCBAMVCVDCDZVFLZLZVEAQH + AVDAQHZQHZUTAVIAVDAVIAURVBVHNZOZVIVDVCVGVFPZOVMSVIVEGAQVCVGVFTZUAVIVKAGQH + ZAVIVJGAQVIAVDVLVNVOUBUGVIURVPAFVLAUCUHUDUEUFUIUJUSGEQHZEUTAGCDVQEFUSUKGU + LUMAEGQUNUSUOUPUQ $. $( $j usage 'remulid2' avoids 'ax-mulcom'; $) $} @@ -666371,54 +666793,54 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, resundir simprlr rspcev ex simpr difss resabs1d resabs1 anbi1d rexlimdva2 rspc2ev impbid nn0ssz mapss mp2an sseli sumsqeq0 oveq12d eqid ovex eqeq1d jca fvmpt bitr4d rexbidva bitrd bitr3id abbidv simpl fzssuz pm3.2i simprr - uzssz mzpaddmpt eldioph2 eqeltrd ineq12 eleq1d syl5ibrcom syl5bir anabsi5 - sstri sylbid ) ACUFFZGZBUWIGZABUGZUWIGZUWJCHGZUWJUWKIZUWMUHACUIUWNUWOADUJ - ZUAUJZJCUKKZLZMZUWQUBUJZFZNMZIZUAHOCJULKZUMFZUNZPKZQZDUOZMZUBUXGUPFZQZBUW - PUCUJZUWRLZMZUXNUDUJZFZNMZIZUCHRPKZQZDUOZMZUDRUPFZQZIZUWMUWNUWJUXMUWKUYFU - WNUWNUXGUQGZUXGURGZUSUWRUXGUTZUWJUXMVAUWNVBZOUQGZUYHUWNVCOUXFUQVDVEUWNUYI - VFURGZVNUWNCOGZUXGVFVGVHUYIUYMVACVIZCVJUXGVFVKVLVMUWNUWRUXGRUGZUXGCVOZUXG - RVPVQZUADAUXGCUBVRVSUWNUWNRUQGZRURGUSZUWRRUTZUWKUYFVAUYKUYSUWNVTWEJOGZUYT - UWNWAJRWBWCVEVUAUWNUBUWRRUXACWDWFZWEUCDBRCUDVRVSWGUYGUXKUYDIZUDUYEQUBUXLQ - UWNUWMUXKUYDUBUDUXLUYEWHUWNVUDUWMUBUDUXLUYEUWNUXAUXLGZUXQUYEGZIZIZUWMVUDU - XJUYCUGZUWIGVUHVUIUWPUEUJZUWRLZMZVUJEOOPKZEUJZUXGLZUXAFZSWIKZVUNRLZUXQFZS - 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(Contributed by Stefan O'Rear, fveq2 ovex fvmpt eqeq1d simplrl mzpf ffvelcdmd zcnd simplrr bitr2d anbi2d wf mul0ord bitr3id rexbidva abbidv eqtrid simpl a1i simprl feqmptd simprr eqeltrrd mzpmulmpt syl2anc eldioph2 syl3anc eqeltrd syl5ibrcom rexlimdvva - uneq12 eleq1d syl5bir sylbid anabsi5 ) ACUAIZJZBXRJZABUBZXRJZXSCKJZXSXTLZ - YBUCACUDYCYDADMEMZUECUFNZUGOZYEFMZIZPOZLZEKQUHNZRZDUIZOZFQUJIZRZBYGYEGMZI - ZPOZLZEYLRZDUIZOZGYPRZLZYBYCYCQUKJZLZQULJUMZYFQUSZLZYDUUFUNYCUUGUOUPUUIUU - JUESJUUIUQUEQURUTVAFYFQYHCVBVCZVDUUHUUKLXSYQXTUUEEDAQCFVEEDBQCGVEVFVGUUFY - OUUDLZGYPRFYPRYCYBYOUUDFGYPYPVHYCUUMYBFGYPYPYCYHYPJZYRYPJZLZLZYBUUMYNUUCU - BZXRJUUQUURYGYEHSQUHNZHMZYHIZUUTYRIZVINZVJZIZPOZLZEYLRZDUIZXRUUQUURYMUUBV - KZDUIUVIYMUUBDVLUUQUVJUVHDUVJYKUUAVKZEYLRUUQUVHYKUUAEYLVMUUQUVKUVGEYLUVKY - GYJYTVKZLUUQYEYLJZLZUVGYGYJYTVNUVNUVLUVFYGUVNUVFYIYSVINZPOUVLUVNUVEUVOPUV - NYEUUSJZUVEUVOOUVMUVPUUQYLUUSYESUKJKSUSYLUUSUSVOVPKSQUKVQVRVSVTZHYEUVCUVO - UUSUVDUUTYEOUVAYIUVBYSVIUUTYEYHWCUUTYEYRWCWAUVDWBYIYSVIWDWETWFUVNYIYSUVNY - IUVNUUSSYEYHUVNUUNUUSSYHWNZYCUUNUUOUVMWGYHQWHZTUVQWIWJUVNYSUVNUUSSYEYRUVN - UUOUUSSYRWNZYCUUNUUOUVMWKYRQWHZTUVQWIWJWOWLWMWPWQWPWRWSUUQYCUUGUUJLZUVDYP - JZUVIXRJYCUUPWTUWBUUQUUGUUJUOUULVDXAUUQHUUSUVAVJZYPJHUUSUVBVJZYPJUWCUUQYH - UWDYPUUQHUUSSYHUUQUUNUVRYCUUNUUOXBZUVSTXCUWFXEUUQYRUWEYPUUQHUUSSYRUUQUUOU - VTYCUUNUUOXDZUWATXCUWGXEHUVAUVBQXFXGEDUVDQCXHXIXJUUMYAUURXRAYNBUUCXMXNXKX - LXOXPTXQ $. + uneq12 eleq1d biimtrrid sylbid anabsi5 ) ACUAIZJZBXRJZABUBZXRJZXSCKJZXSXT + LZYBUCACUDYCYDADMEMZUECUFNZUGOZYEFMZIZPOZLZEKQUHNZRZDUIZOZFQUJIZRZBYGYEGM + ZIZPOZLZEYLRZDUIZOZGYPRZLZYBYCYCQUKJZLZQULJUMZYFQUSZLZYDUUFUNYCUUGUOUPUUI + UUJUESJUUIUQUEQURUTVAFYFQYHCVBVCZVDUUHUUKLXSYQXTUUEEDAQCFVEEDBQCGVEVFVGUU + FYOUUDLZGYPRFYPRYCYBYOUUDFGYPYPVHYCUUMYBFGYPYPYCYHYPJZYRYPJZLZLZYBUUMYNUU + CUBZXRJUUQUURYGYEHSQUHNZHMZYHIZUUTYRIZVINZVJZIZPOZLZEYLRZDUIZXRUUQUURYMUU + BVKZDUIUVIYMUUBDVLUUQUVJUVHDUVJYKUUAVKZEYLRUUQUVHYKUUAEYLVMUUQUVKUVGEYLUV + KYGYJYTVKZLUUQYEYLJZLZUVGYGYJYTVNUVNUVLUVFYGUVNUVFYIYSVINZPOUVLUVNUVEUVOP + UVNYEUUSJZUVEUVOOUVMUVPUUQYLUUSYESUKJKSUSYLUUSUSVOVPKSQUKVQVRVSVTZHYEUVCU + VOUUSUVDUUTYEOUVAYIUVBYSVIUUTYEYHWCUUTYEYRWCWAUVDWBYIYSVIWDWETWFUVNYIYSUV + NYIUVNUUSSYEYHUVNUUNUUSSYHWNZYCUUNUUOUVMWGYHQWHZTUVQWIWJUVNYSUVNUUSSYEYRU + VNUUOUUSSYRWNZYCUUNUUOUVMWKYRQWHZTUVQWIWJWOWLWMWPWQWPWRWSUUQYCUUGUUJLZUVD + YPJZUVIXRJYCUUPWTUWBUUQUUGUUJUOUULVDXAUUQHUUSUVAVJZYPJHUUSUVBVJZYPJUWCUUQ + YHUWDYPUUQHUUSSYHUUQUUNUVRYCUUNUUOXBZUVSTXCUWFXEUUQYRUWEYPUUQHUUSSYRUUQUU + OUVTYCUUNUUOXDZUWATXCUWGXEHUVAUVBQXFXGEDUVDQCXHXIXJUUMYAUURXRAYNBUUCXMXNX + KXLXOXPTXQ $. $} ${ @@ -690430,11 +690852,11 @@ collection and union ( ~ mnuop3d ), from which closure under pairing inaex $p |- ( A e. On -> E. x e. Inacc A e. x ) $= ( con0 wcel cv cina csuc cdif cint wceq wa wss cwina inawina winaon ssriv syl onmindif c0 cvv mpan adantr simpr eleqtrrd difss sstri inaprc neli wi - wne ssdif0 sucexg ssexg expcom syl5bir mtoi onint sylancr eldifad rspcime - neqned ) BCDZBAEZDAFBGZHZIZFVBVCVFJZKBVFVCVBBVFDZVGFCLVBVHAFCVCFDVCMDVCCD - VCNVCOQPZFBRUAUBVBVGUCUDVBVFFVDVBVECLVESUJVFVEDVEFCFVDUEVIUFVBVESVBVESJZF - TDZFTUGUHVJFVDLZVBVKFVDUKVBVDTDZVLVKUIBCULVLVMVKFVDTUMUNQUOUPVAVEUQURUSUT - $. + ssdif0 sucexg ssexg expcom biimtrrid neqned onint sylancr eldifad rspcime + wne mtoi ) BCDZBAEZDAFBGZHZIZFVBVCVFJZKBVFVCVBBVFDZVGFCLVBVHAFCVCFDVCMDVC + CDVCNVCOQPZFBRUAUBVBVGUCUDVBVFFVDVBVECLVESUTVFVEDVEFCFVDUEVIUFVBVESVBVESJ + ZFTDZFTUGUHVJFVDLZVBVKFVDUJVBVDTDZVLVKUIBCUKVLVMVKFVDTULUMQUNVAUOVEUPUQUR + US $. $} ${ @@ -692504,9 +692926,9 @@ collection and union ( ~ mnuop3d ), from which closure under pairing Salmon, 24-May-2011.) $) pm10.57 $p |- ( A. x ( ph -> ( ps \/ ch ) ) -> ( A. x ( ph -> ps ) \/ E. x ( ph /\ ch ) ) ) $= - ( wo wi wal wa wex wn alnex imnan pm2.53 con1d imim3i syl5bir al2imi orrd ) - ABCEZFZDGZABFZDGZACHZDIZUAUEUCUEJUDJZDGUAUCUDDKTUFUBDUFACJZFTUBACLSUGBASBCB - CMNOPQPNR $. + ( wo wi wal wa wex wn alnex imnan pm2.53 con1d imim3i biimtrrid al2imi orrd + ) ABCEZFZDGZABFZDGZACHZDIZUAUEUCUEJUDJZDGUAUCUDDKTUFUBDUFACJZFTUBACLSUGBASB + CBCMNOPQPNR $. $( @@ -702737,46 +703159,46 @@ unification theorem (e.g., the sub-theorem whose assertion is step 5 syl vy vz vg vu wne cfv wi csn cun fneq2 raleq exbidv 0ex fneq1 fn0 mpbir eqid ceqsexv2d ral0 exan cfn wn cop wf biimpi ad2antrl vex simpllr fsnunf dffn2 syl121anc sylibr simprr nfra1 nfan simpr nelne2 necomd fvunsn neeq1 - nfv fveq2 eleq1d eleq2w bitrd imbi12d rspcv syl5bir syl3c eqeltrd simp-4l - cbvralvw w3a elsni 3ad2ant2 simp1 eqtrd simp3 pm2.21ddne syl3anc wo sylib - elun mpjaodan ralrimi syl21anc eximdv snex unex fveq1 ralbidv spcev eximi - imbi2d syl6 ax5e imp an32s cbvexvw neq0 exdistrv simprrl simprrr ad5ant12 - simpl simplrl fndmd 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ZWHXCWCWTTXEWTWCUPWCWSGIUQURXDWGGIWCFWRWGWRWFHISZWCFWGWQWFLHILHUKDBWPWERL + HGUSVAUTFWCXFWGUBWDXFWGWDWFHIUJVBVCVDVEVFVGVHVIVBVJVGVKVHVIVFVGVG $. $d ch c $. $d et c $. $( Implication of a double restricted existential uniqueness in terms of @@ -766299,15 +766721,15 @@ Infinity and the extended real number system (cont.) - extension wbr ex cuz cfv unidm eqcomi fzsn sneq oveq1d uneq12d clt wn cle zre peano2z lep1d zred lenltd mpbid wb fzonlt0 mpancom uneq2d un0 3eqtr4a eluzelz syl11 eqtrdi fzisfzounsn adantl w3a eluz2 simpl1 simpl2 wne nesym cr ltlen syl2an - biimprd exp4b 3imp syl5bir imp 3jca sylbi impcom fzopred syl eqtrd pm2.61i - ) ABCZBAUAUBDZABEFZAGZAHIFZBJFZKZBGZKZCZLBMDZWJWSWKWTWJWSWTWJNWQWQWQKZWLWRX - AWQWQUCUDWJWTWLBBEFWQABBEOBUEPWJWTWRWQBHIFZBJFZKZWQKXAWJWPXDWQWJWMWQWOXCABU - FWJWNXBBJABHIOUGUHQWTXDWQWQWTXDWQRKWQWTXCRWQWTXBBUISUJZXCRCZWTBXBUKSXEWTBBU - LZUNWTBXBXGWTXBBUMZUOUPUQXBMDWTXEXFURXHXBBUSUTUQVAWQVBVFQPVCTABVDVEWJUJZWKW - SXIWKNZWLABJFZWQKZWRWKWLXLCXIABVGVHXJXKWPWQXJAMDZWTABUISZVIZXKWPCWKXIXOWKXM - WTABUKSZVIZXIXOLABVJXQXIXOXQXINXMWTXNXMWTXPXIVKXMWTXPXIVLXQXIXNXIBAVMZXQXNB - AVNXMWTXPXRXNLXMWTXPXRXNXMWTNXNXPXRNZXMAVODBVODXNXSURWTAULXGABVPVQVRVSVTWAW - BWCTWDWEABWFWGQWHTWI $. + biimprd exp4b 3imp biimtrrid imp 3jca sylbi impcom fzopred eqtrd pm2.61i + syl ) ABCZBAUAUBDZABEFZAGZAHIFZBJFZKZBGZKZCZLBMDZWJWSWKWTWJWSWTWJNWQWQWQKZW + LWRXAWQWQUCUDWJWTWLBBEFWQABBEOBUEPWJWTWRWQBHIFZBJFZKZWQKXAWJWPXDWQWJWMWQWOX + CABUFWJWNXBBJABHIOUGUHQWTXDWQWQWTXDWQRKWQWTXCRWQWTXBBUISUJZXCRCZWTBXBUKSXEW + TBBULZUNWTBXBXGWTXBBUMZUOUPUQXBMDWTXEXFURXHXBBUSUTUQVAWQVBVFQPVCTABVDVEWJUJ + ZWKWSXIWKNZWLABJFZWQKZWRWKWLXLCXIABVGVHXJXKWPWQXJAMDZWTABUISZVIZXKWPCWKXIXO + WKXMWTABUKSZVIZXIXOLABVJXQXIXOXQXINXMWTXNXMWTXPXIVKXMWTXPXIVLXQXIXNXIBAVMZX + QXNBAVNXMWTXPXRXNLXMWTXPXRXNXMWTNXNXPXRNZXMAVODBVODXNXSURWTAULXGABVPVQVRVSV + TWAWBWCTWDWEABWFWIQWGTWH $. $( Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020.) $)