diff --git a/discouraged b/discouraged index 4291629ea8..382134ee7f 100644 --- a/discouraged +++ b/discouraged @@ -5988,7 +5988,7 @@ "erngdvlem1-rN" is used by "erngdvlem3-rN". "erngdvlem1-rN" is used by "erngdvlem4-rN". "erngdvlem3-rN" is used by "erngdvlem4-rN". -"erngdvlem3-rN" is used by "erngrng-rN". +"erngdvlem3-rN" is used by "erngring-rN". "erngdvlem4-rN" is used by "erngdv-rN". "erngfmul-rN" is used by "erngdvlem3-rN". "erngfmul-rN" is used by "erngdvlem4-rN". @@ -6414,7 +6414,7 @@ "grpomuldivass" is used by "grpopncan". "grpomuldivass" is used by "nvaddsubass". "grpon0" is used by "0ngrp". -"grpon0" is used by "rngon0". +"grpon0" is used by "rngone0". "grpon0" is used by "vcoprnelem". "grponnncan2" is used by "nvnnncan2". "grponpcan" is used by "ablonnncan". @@ -12215,7 +12215,7 @@ "rngogrpo" is used by "rngokerinj". "rngogrpo" is used by "rngolcan". "rngogrpo" is used by "rngolz". -"rngogrpo" is used by "rngon0". +"rngogrpo" is used by "rngone0". "rngogrpo" is used by "rngonegcl". "rngogrpo" is used by "rngonegmn1l". "rngogrpo" is used by "rngonegmn1r". @@ -12251,7 +12251,7 @@ "rngomndo" is used by "isdrngo2". "rngomndo" is used by "rngo1cl". "rngomndo" is used by "rngoidmlem". -"rngon0" is used by "rngoueqz". +"rngone0" is used by "rngoueqz". "rngopidOLD" is used by "exidcl". "rngopidOLD" is used by "exidresid". "rngopidOLD" is used by "isexid2". @@ -15739,7 +15739,7 @@ New usage of "erngfset-rN" is discouraged (1 uses). New usage of "erngmul-rN" is discouraged (2 uses). New usage of "erngplus-rN" is discouraged (1 uses). New usage of "erngplus2-rN" is discouraged (0 uses). -New usage of "erngrng-rN" is discouraged (0 uses). +New usage of "erngring-rN" is discouraged (0 uses). New usage of "erngset-rN" is discouraged (3 uses). New usage of "eu1OLD" is discouraged (0 uses). New usage of "eu2OLD" is discouraged (0 uses). @@ -17860,7 +17860,7 @@ New usage of "rngolcan" is discouraged (0 uses). New usage of "rngolidm" is discouraged (7 uses). New usage of "rngolz" is discouraged (5 uses). New usage of "rngomndo" is discouraged (3 uses). -New usage of "rngon0" is discouraged (1 uses). +New usage of "rngone0" is discouraged (1 uses). New usage of "rngopidOLD" is discouraged (4 uses). New usage of "rngorcan" is discouraged (0 uses). New usage of "rngoridm" is discouraged (3 uses). diff --git a/set.mm b/set.mm index 6144811e38..b89f0b76e3 100644 --- a/set.mm +++ b/set.mm @@ -95,6 +95,220 @@ proposed syl imtri (analogous to *bitr*, sstri, etc.) DONE: Date Old New Notes +09-Feb-20 df-rng df-ring Rename Ring label fragment rng -> ring +09-Feb-20 isrng.b isring.b +09-Feb-20 isrng.g isring.g +09-Feb-20 isrng.p isring.p +09-Feb-20 isrng.t isring.t +09-Feb-20 isrng isring +09-Feb-20 rnggrp ringgrp +09-Feb-20 rngmgp.g ringmgp.g +09-Feb-20 rngmgp ringmgp +09-Feb-20 rngmnd ringmnd +09-Feb-20 crngrng crngring +09-Feb-20 rngi.b ringi.b +09-Feb-20 rngi.p ringi.p +09-Feb-20 rngi.t ringi.t +09-Feb-20 rngi ringi +09-Feb-20 rngcl.b ringcl.b +09-Feb-20 rngcl.t ringcl.t +09-Feb-20 rngcl ringcl +09-Feb-20 rngass ringass +09-Feb-20 rngideu ringideu +09-Feb-20 rngdi.b ringdi.b +09-Feb-20 rngdi.p ringdi.p +09-Feb-20 rngdi.t ringdi.t +09-Feb-20 rngdi ringdi +09-Feb-20 rngdir ringdir +09-Feb-20 rngidcl.b ringidcl.b +09-Feb-20 rngidcl.u ringidcl.u +09-Feb-20 rngidcl ringidcl +09-Feb-20 rng0cl.b ring0cl.b +09-Feb-20 rng0cl.z ring0cl.z +09-Feb-20 rng0cl ring0cl +09-Feb-20 rngidmlem ringidmlem +09-Feb-20 rnglidm ringlidm +09-Feb-20 rngridm ringridm +09-Feb-20 isrngid isringid +09-Feb-20 rngidss.g ringidss.g +09-Feb-20 rngidss.b ringidss.b +09-Feb-20 rngidss.u ringidss.u +09-Feb-20 rngidss ringidss +09-Feb-20 rngacl.b ringacl.b +09-Feb-20 rngacl.p ringacl.p +09-Feb-20 rngacl ringacl +09-Feb-20 rngcom ringcom +09-Feb-20 rngabl ringabl +09-Feb-20 rngcmn ringcmn +09-Feb-20 rngpropd.1 ringpropd.1 +09-Feb-20 rngpropd.2 ringpropd.2 +09-Feb-20 rngpropd.3 ringpropd.3 +09-Feb-20 rngpropd.4 ringpropd.4 +09-Feb-20 rngpropd ringpropd +09-Feb-20 rngprop.b ringprop.b +09-Feb-20 rngprop.p ringprop.p +09-Feb-20 rngprop.m ringprop.m +09-Feb-20 rngprop ringprop +09-Feb-20 isrngd.b isringd.b +09-Feb-20 isrngd.p isringd.p +09-Feb-20 isrngd.t isringd.t +09-Feb-20 isrngd.g isringd.g +09-Feb-20 isrngd.c isringd.c +09-Feb-20 isrngd.a isringd.a +09-Feb-20 isrngd.d isringd.d +09-Feb-20 isrngd.e isringd.e +09-Feb-20 isrngd.u isringd.u +09-Feb-20 isrngd.i isringd.i +09-Feb-20 isrngd.h isringd.h +09-Feb-20 isrngd isringd +09-Feb-20 rnglz ringlz +09-Feb-20 rngrz ringrz +09-Feb-20 rngsrg ringsrg +09-Feb-20 rng1eq0.b ring1eq0.b +09-Feb-20 rng1eq0.u ring1eq0.u +09-Feb-20 rng1eq0.z ring1eq0.z +09-Feb-20 rng1eq0 ring1eq0 +09-Feb-20 rng1ne0.b ring1ne0.b +09-Feb-20 rng1ne0.u ring1ne0.u +09-Feb-20 rng1ne0.z ring1ne0.z +09-Feb-20 rng1ne0 ring1ne0 +09-Feb-20 rngnegl.b ringnegl.b +09-Feb-20 rngnegl.t ringnegl.t +09-Feb-20 rngnegl.u ringnegl.u +09-Feb-20 rngnegl.n ringnegl.n +09-Feb-20 rngnegl.r ringnegl.r +09-Feb-20 rngnegl.x ringnegl.x +09-Feb-20 rngnegl ringnegl +09-Feb-20 rngneglmul.b ringneglmul.b +09-Feb-20 rngneglmul.t ringneglmul.t +09-Feb-20 rngneglmul.n ringneglmul.n +09-Feb-20 rngneglmul.r ringneglmul.r +09-Feb-20 rngneglmul.x ringneglmul.x +09-Feb-20 rngneglmul.y ringneglmul.y +09-Feb-20 rngmneg1 ringmneg1 +09-Feb-20 rngmneg2 ringmneg2 +09-Feb-20 rngm2neg ringm2neg +09-Feb-20 rngsubdi.b ringsubdi.b +09-Feb-20 rngsubdi.t ringsubdi.t +09-Feb-20 rngsubdi.m ringsubdi.m +09-Feb-20 rngsubdi.r ringsubdi.r +09-Feb-20 rngsubdi.x ringsubdi.x +09-Feb-20 rngsubdi.y ringsubdi.y +09-Feb-20 rngsubdi.z ringsubdi.z +09-Feb-20 rngsubdi ringsubdi +09-Feb-20 rng1.m ring1.m +09-Feb-20 rng1 ring1 +09-Feb-20 rngn0 ringn0 +09-Feb-20 rnglghm.b ringlghm.b +09-Feb-20 rnglghm.t ringlghm.t +09-Feb-20 rnglghm ringlghm +09-Feb-20 rngrghm ringrghm +09-Feb-20 prdsrngd.y prdsringd.y +09-Feb-20 prdsrngd.i prdsringd.i +09-Feb-20 prdsrngd.s prdsringd.s +09-Feb-20 prdsrngd.r prdsringd.r +09-Feb-20 prdsrngd prdsringd +09-Feb-20 pwsrng.y pwsring.y +09-Feb-20 pwsrng pwsring +09-Feb-20 imasrng.u imasring.u +09-Feb-20 imasrng.v imasring.v +09-Feb-20 imasrng.p imasring.p +09-Feb-20 imasrng.t imasring.t +09-Feb-20 imasrng.o imasring.o +09-Feb-20 imasrng.f imasring.f +09-Feb-20 imasrng.e1 imasring.e1 +09-Feb-20 imasrng.e2 imasring.e2 +09-Feb-20 imasrng.r imasring.r +09-Feb-20 imasrng imasring +09-Feb-20 qusrng2.u qusring2.u +09-Feb-20 qusrng2.v qusring2.v +09-Feb-20 qusrng2.p qusring2.p +09-Feb-20 qusrng2.t qusring2.t +09-Feb-20 qusrng2.o qusring2.o +09-Feb-20 qusrng2.r qusring2.r +09-Feb-20 qusrng2.e1 qusring2.e1 +09-Feb-20 qusrng2.e2 qusring2.e2 +09-Feb-20 qusrng2.x qusring2.x +09-Feb-20 qusrng2 qusring2 +09-Feb-20 opprrng opprring +09-Feb-20 opprrngb opprringb +09-Feb-20 rnginvcl.3 ringinvcl.3 +09-Feb-20 rnginvcl ringinvcl +09-Feb-20 rnginvdv.b ringinvdv.b +09-Feb-20 rnginvdv.u ringinvdv.u +09-Feb-20 rnginvdv.d ringinvdv.d +09-Feb-20 rnginvdv.o ringinvdv.o +09-Feb-20 rnginvdv.i ringinvdv.i +09-Feb-20 rnginvdv ringinvdv +09-Feb-20 subrgrng.1 subrgring.1 +09-Feb-20 subrgrng subrgring +09-Feb-20 srngrng srngring +09-Feb-20 lmodrng.1 lmodring.1 +09-Feb-20 lmodrng lmodring +09-Feb-20 qusrng.u qusring.u +09-Feb-20 qusrng.i qusring.i +09-Feb-20 qusrng qusring +09-Feb-20 lpirrng lpirring +09-Feb-20 nzrrng nzrring +09-Feb-20 rngelnzr.z ringelnzr.z +09-Feb-20 rngelnzr.b ringelnzr.b +09-Feb-20 rngelnzr ringelnzr +09-Feb-20 0rngnnzr 0ringnnzr +09-Feb-20 0rng.b 0ring.b +09-Feb-20 0rng.0 0ring.0 +09-Feb-20 0rng 0ring +09-Feb-20 0rng01eq.1 0ring01eq.1 +09-Feb-20 0rng01eq 0ring01eq +09-Feb-20 01eq0rng 01eq0ring +09-Feb-20 0rng01eqbi 0ring01eqbi +09-Feb-20 rng1zr.b ring1zr.b +09-Feb-20 rng1zr.p ring1zr.p +09-Feb-20 rng1zr.t ring1zr.t +09-Feb-20 rng1zr ring1zr +09-Feb-20 domnrng domnring +09-Feb-20 assarng assaring +09-Feb-20 psrrng.s psrring.s +09-Feb-20 psrrng.i psrring.i +09-Feb-20 psrrng.r psrring.r +09-Feb-20 psrrng psrring +09-Feb-20 mplrng mplring +09-Feb-20 opsrrng.o opsrring.o +09-Feb-20 opsrrng.i opsrring.i +09-Feb-20 opsrrng.r opsrring.r +09-Feb-20 opsrrng.t opsrring.t +09-Feb-20 opsrrng opsrring +09-Feb-20 psr1rng.s psr1ring.s +09-Feb-20 psr1rng psr1ring +09-Feb-20 ply1rng.p ply1ring.p +09-Feb-20 ply1rng ply1ring +09-Feb-20 cnrng cnring +09-Feb-20 zringrng zringring +09-Feb-20 rngvcl.b ringvcl.b +09-Feb-20 rngvcl.t ringvcl.t +09-Feb-20 rngvcl ringvcl +09-Feb-20 matrng matring +09-Feb-20 pmatrng.p pmatring.p +09-Feb-20 pmatrng.c pmatring.c +09-Feb-20 pmatrng pmatring +09-Feb-20 trgrng trgring +09-Feb-20 tdrgrng tdrgring +09-Feb-20 nrgrng nrgring +09-Feb-20 clmrng clmring +09-Feb-20 rngn0.1 rngone0.1 +09-Feb-20 rngn0.2 rngone0.2 +09-Feb-20 rngon0 rngone0 +09-Feb-20 rnginvval.b ringinvval.b +09-Feb-20 rnginvval.p ringinvval.p +09-Feb-20 rnginvval.o ringinvval.o +09-Feb-20 rnginvval.n ringinvval.n +09-Feb-20 rnginvval.u ringinvval.u +09-Feb-20 rnginvval ringinvval +09-Feb-20 orngrng orngring +09-Feb-20 lnrrng lnrring +09-Feb-20 mendrng mendring +09-Feb-20 ringrng0 ringrng0 +09-Feb-20 erngrng eringring +09-Feb-20 erngrng-rN erngring-rN 08-Feb-20 xrsmgm [same] moved from AV's mathbox to main set.mm 08-Feb-20 xrsnsgrp [same] moved from AV's mathbox to main set.mm 08-Feb-20 xrsmgmdifsgrp [same] moved from AV's mathbox to main set.mm @@ -202457,7 +202671,7 @@ The definition of magmas (` Mgm `, see ~ df-mgm ) concentrates on the closure $} $( The internal operation for a set is the trivial operation iff the set is a - singleton. Formerly part of proof of ~ rng1zr . (Contributed by FL, + singleton. Formerly part of proof of ~ ring1zr . (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.) $) intopsn $p |- ( ( .o. : ( B X. B ) --> B /\ Z e. B ) -> ( B = { Z } <-> .o. = { <. <. Z , Z >. , Z >. } ) ) $= @@ -202900,8 +203114,8 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is $( Obsolete version of ~ mndcl as of 8-Feb-2020. Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario - Carneiro, 6-Jan-2015.) (Revised by AV, 6-Feb-2020.) (New usage is - discouraged.) (Proof modification is discouraged.) $) + Carneiro, 6-Jan-2015.) (Revised by AV, 6-Feb-2020.) + (New usage is discouraged.) (Proof modification is discouraged.) $) mndclOLD $p |- ( ( G e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B ) $= ( vx vy vz ve wcel co cv wceq wral wa wi oveq1 eleq1d cmnd oveq1d eqeq12d wrex ismnd ralbidv anbi12d oveq2 oveq2d rspc2va simpld expcom expd adantr @@ -202914,8 +203128,8 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is $( Obsolete version of ~ mndass as of 8-Feb-2020. A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Revised by AV, - 6-Feb-2020.) (New usage is discouraged.) (Proof modification is - discouraged.) $) + 6-Feb-2020.) (New usage is discouraged.) + (Proof modification is discouraged.) $) mndassOLD $p |- ( ( G e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) ) $= ( vx vy vz ve wcel co wceq cv wral wa wi oveq1 cmnd w3a wrex ismnd eleq1d @@ -205202,7 +205416,7 @@ commutative group is an Abelian group (see ~ df-abl ). Subgroups can group is the set of complex numbers ` CC ` over the group operation ` + ` (addition), as proven in ~ cnaddablx ; an Abelian group is a group as proven in ~ ablgrp . Other structures include groups, including - unital rings ( ~ df-rng ) and fields ( ~ df-field ). (Contributed by + unital rings ( ~ df-ring ) and fields ( ~ df-field ). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) $) df-grp $a |- Grp = { g e. Mnd | A. a e. ( Base ` g ) E. m e. ( Base ` g ) ( m ( +g ` g ) a ) = ( 0g ` g ) } $. @@ -205348,7 +205562,7 @@ with the same (relevant) properties is also a group. (Contributed by $( Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring ` R ` is a group before we know that it is also a ring. (Theorem - ~ rnggrp , on the other hand, requires that we know in advance that + ~ ringgrp , on the other hand, requires that we know in advance that ` R ` is a ring.) (Contributed by NM, 11-Oct-2013.) $) grpss $p |- ( G e. Grp <-> R e. Grp ) $= ( cgrp wcel cbs baseid cnx cfv cop cplusg cpr opex eleqtrri strss plusgid @@ -219222,7 +219436,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements explicit structure indices. The actual structure is dependent on how ` Base ` and ` +g ` is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see - ~ cnrng . (Contributed by NM, 20-Oct-2012.) + ~ cnring . (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.) $) cnaddabl $p |- G e. Abel $= ( vx vy vz cc caddc cv cneg cc0 cvv wcel cbs cfv wceq cnex grpbase addcom @@ -226488,7 +226702,7 @@ factorization into prime power factors (even if the exponents are average ring, or even for a field, but it will be a monoid, and ~ unitgrp shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication - operation of a ring is a monoid" ( ~ rngmgp ) or "the multiplicative + operation of a ring is a monoid" ( ~ ringmgp ) or "the multiplicative identity" in terms of the identity of a monoid ( ~ df-1r ). (Contributed by Mario Carneiro, 21-Dec-2014.) $) df-mgp $a |- mulGrp = ( w e. _V |-> @@ -227055,7 +227269,7 @@ factorization into prime power factors (even if the exponents are srglmhm.b $e |- B = ( Base ` R ) $. srglmhm.t $e |- .x. = ( .r ` R ) $. $( Left-multiplication in a semiring by a fixed element of the ring is a - monoid homomorphism, analogous to ~ rnglghm . (Contributed by AV, + monoid homomorphism, analogous to ~ ringlghm . (Contributed by AV, 23-Aug-2019.) $) srglmhm $p |- ( ( R e. SRing /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) e. ( R MndHom R ) ) $= @@ -227073,7 +227287,7 @@ factorization into prime power factors (even if the exponents are BBWEWECCWAWMWMFFXGXGXKXKVLVM $. $( Right-multiplication in a semiring by a fixed element of the ring is a - monoid homomorphism, analogous to ~ rngrghm . (Contributed by AV, + monoid homomorphism, analogous to ~ ringrghm . (Contributed by AV, 23-Aug-2019.) $) srgrmhm $p |- ( ( R e. SRing /\ X e. B ) -> ( x e. B |-> ( x .x. X ) ) e. ( R MndHom R ) ) $= @@ -227413,11 +227627,11 @@ would be the integrable nonnegative (resp. positive) bounded functions on structure, and where the addition is left- and right-distributive for the multiplication. Definition of a ring in [BourbakiAlg1] p. 92 or of a ring with identity in [Roman] p. 19. So that the additive structure - must be abelian (see ~ rngcom ), care must be taken that in the case of + must be abelian (see ~ ringcom ), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.) $) - df-rng $a |- Ring = { f e. Grp | ( ( mulGrp ` f ) e. Mnd /\ + df-ring $a |- Ring = { f e. Grp | ( ( mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. A. x e. r A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) @@ -227431,48 +227645,48 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of ${ $d b p r t x y z B $. $d b p r t x y z .+ $. $d b p r t x y z R $. $d r G $. $d b p r t x y z .x. $. - isrng.b $e |- B = ( Base ` R ) $. - isrng.g $e |- G = ( mulGrp ` R ) $. - isrng.p $e |- .+ = ( +g ` R ) $. - isrng.t $e |- .x. = ( .r ` R ) $. + isring.b $e |- B = ( Base ` R ) $. + isring.g $e |- G = ( mulGrp ` R ) $. + isring.p $e |- .+ = ( +g ` R ) $. + isring.t $e |- .x. = ( .r ` R ) $. $( The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) $) - isrng $p |- ( R e. Ring <-> ( R e. Grp /\ G e. Mnd + isring $p |- ( R e. Ring <-> ( R e. Grp /\ G e. Mnd /\ A. x e. B A. y e. B A. z e. B ( ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) /\ ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) ) ) ) $= ( wcel cv co wceq wa wral cfv cvv vr vp vt vb crg cgrp cmnd w3a cmgp wsbc cmulr cplusg cbs fveq2 syl6eqr eleq1d fvex a1i simpl fveq2d simpllr simpr - simpll eqidd simplr oveqd eqeq12d anbi12d raleqbidv sbcied2 df-rng elrab2 - oveq123d 3anass bitr4i ) FUEMFUFMZHUGMZANZBNZCNZEOZGOZVRVSGOZVRVTGOZEOZPZ - VRVSEOZVTGOZWDVSVTGOZEOZPZQZCDRZBDRZADRZQZQVPVQWOUHUANZUISZUGMZVRVSVTUBNZ - OZUCNZOZVRVSXBOZVRVTXBOZWTOZPZVRVSWTOZVTXBOZXEVSVTXBOZWTOZPZQZCUDNZRZBXNR - ZAXNRZUCWQUKSZUJZUBWQULSZUJZUDWQUMSZUJZQWPUAFUFUEWQFPZWSVQYCWOYDWRHUGYDWR - FUISHWQFUIUNJUOUPYDYAWOUDYBDTYBTMYDWQUMUQURYDYBFUMSDWQFUMUNIUOYDXNDPZQZXS - WOUBXTETXTTMYFWQULUQURYFXTFULSEYFWQFULYDYEUSUTKUOYFWTEPZQZXQWOUCXRGTXRTMY - HWQUKUQURYHXRFUKSGYHWQFUKYDYEYGVCUTLUOYHXBGPZQZXPWNAXNDYDYEYGYIVAZYJXOWMB - XNDYKYJXMWLCXNDYKYJXGWFXLWKYJXCWBXFWEYJVRVRXAWAXBGYHYIVBZYJVRVDYJWTEVSVTY - FYGYIVEZVFVMYJXDWCXEWDWTEYMYJXBGVRVSYLVFYJXBGVRVTYLVFZVMVGYJXIWHXKWJYJXHW - GVTVTXBGYLYJWTEVRVSYMVFYJVTVDVMYJXEWDXJWIWTEYMYNYJXBGVSVTYLVFVMVGVHVIVIVI - VJVJVJVHABCUCUAUDUBVKVLVPVQWOVNVO $. + simpll eqidd simplr oveqd oveq123d eqeq12d anbi12d sbcied2 df-ring elrab2 + raleqbidv 3anass bitr4i ) FUEMFUFMZHUGMZANZBNZCNZEOZGOZVRVSGOZVRVTGOZEOZP + ZVRVSEOZVTGOZWDVSVTGOZEOZPZQZCDRZBDRZADRZQZQVPVQWOUHUANZUISZUGMZVRVSVTUBN + ZOZUCNZOZVRVSXBOZVRVTXBOZWTOZPZVRVSWTOZVTXBOZXEVSVTXBOZWTOZPZQZCUDNZRZBXN + RZAXNRZUCWQUKSZUJZUBWQULSZUJZUDWQUMSZUJZQWPUAFUFUEWQFPZWSVQYCWOYDWRHUGYDW + RFUISHWQFUIUNJUOUPYDYAWOUDYBDTYBTMYDWQUMUQURYDYBFUMSDWQFUMUNIUOYDXNDPZQZX + SWOUBXTETXTTMYFWQULUQURYFXTFULSEYFWQFULYDYEUSUTKUOYFWTEPZQZXQWOUCXRGTXRTM + YHWQUKUQURYHXRFUKSGYHWQFUKYDYEYGVCUTLUOYHXBGPZQZXPWNAXNDYDYEYGYIVAZYJXOWM + BXNDYKYJXMWLCXNDYKYJXGWFXLWKYJXCWBXFWEYJVRVRXAWAXBGYHYIVBZYJVRVDYJWTEVSVT + YFYGYIVEZVFVGYJXDWCXEWDWTEYMYJXBGVRVSYLVFYJXBGVRVTYLVFZVGVHYJXIWHXKWJYJXH + WGVTVTXBGYLYJWTEVRVSYMVFYJVTVDVGYJXEWDXJWIWTEYMYNYJXBGVSVTYLVFVGVHVIVMVMV + MVJVJVJVIABCUCUAUDUBVKVLVPVQWOVNVO $. $} ${ $d r G $. $d r x y z R $. $( A ring is a group. (Contributed by NM, 15-Sep-2011.) $) - rnggrp $p |- ( R e. Ring -> R e. Grp ) $= + ringgrp $p |- ( R e. Ring -> R e. Grp ) $= ( vx vy vz crg wcel cgrp cmgp cfv cmnd cv cplusg cmulr wceq cbs wral eqid - co wa isrng simp1bi ) AEFAGFAHIZJFBKZCKZDKZALIZRAMIZRUCUDUGRUCUEUGRZUFRNU - CUDUFRUEUGRUHUDUEUGRUFRNSDAOIZPCUIPBUIPBCDUIUFAUGUBUIQUBQUFQUGQTUA $. + co wa isring simp1bi ) AEFAGFAHIZJFBKZCKZDKZALIZRAMIZRUCUDUGRUCUEUGRZUFRN + UCUDUFRUEUGRUHUDUEUGRUFRNSDAOIZPCUIPBUIPBCDUIUFAUGUBUIQUBQUFQUGQTUA $. - rngmgp.g $e |- G = ( mulGrp ` R ) $. + ringmgp.g $e |- G = ( mulGrp ` R ) $. $( A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.) $) - rngmgp $p |- ( R e. Ring -> G e. Mnd ) $= + ringmgp $p |- ( R e. Ring -> G e. Mnd ) $= ( vx vy vz crg wcel cgrp cmnd cv cplusg cfv co cmulr wceq cbs wral eqid - wa isrng simp2bi ) AGHAIHBJHDKZEKZFKZALMZNAOMZNUCUDUGNUCUEUGNZUFNPUCUDUFN - UEUGNUHUDUEUGNUFNPTFAQMZREUIRDUIRDEFUIUFAUGBUISCUFSUGSUAUB $. + wa isring simp2bi ) AGHAIHBJHDKZEKZFKZALMZNAOMZNUCUDUGNUCUEUGNZUFNPUCUDUF + NUEUGNUHUDUEUGNUFNPTFAQMZREUIRDUIRDEFUIUFAUGBUISCUFSUGSUAUB $. $( A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) $) @@ -227488,53 +227702,53 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.) $) - rngmnd $p |- ( R e. Ring -> R e. Mnd ) $= - ( crg wcel cgrp cmnd rnggrp grpmnd syl ) ABCADCAECAFAGH $. + ringmnd $p |- ( R e. Ring -> R e. Mnd ) $= + ( crg wcel cgrp cmnd ringgrp grpmnd syl ) ABCADCAECAFAGH $. $( A ring is a magma. (Contributed by AV, 31-Jan-2020.) $) ringmgm $p |- ( R e. Ring -> R e. Mgm ) $= - ( crg wcel cgrp cmgm rnggrp grpmgm syl ) ABCADCAECAFAGH $. + ( crg wcel cgrp cmgm ringgrp grpmgm syl ) ABCADCAECAFAGH $. $( A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) $) - crngrng $p |- ( R e. CRing -> R e. Ring ) $= + crngring $p |- ( R e. CRing -> R e. Ring ) $= ( ccrg wcel crg cmgp cfv ccmn eqid iscrng simplbi ) ABCADCAEFZGCAKKHIJ $. $( Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.) $) mgpf $p |- ( mulGrp |` Ring ) : Ring --> Mnd $= ( va crg cmnd cmgp cres wf wfn cv cfv wcel wral cvv wss fnmgp fnssres mp2an - ssv fvres eqid rngmgp eqeltrd rgen ffnfv mpbir2an ) BCDBEZFUEBGZAHZUEIZCJZA - BKDLGBLMUFNBQLBDOPUIABUGBJUHUGDIZCUGBDRUGUJUJSTUAUBABCUEUCUD $. + ssv fvres eqid ringmgp eqeltrd rgen ffnfv mpbir2an ) BCDBEZFUEBGZAHZUEIZCJZ + ABKDLGBLMUFNBQLBDOPUIABUGBJUHUGDIZCUGBDRUGUJUJSTUAUBABCUEUCUD $. ${ $d x y z B $. $d x y z R $. $d x y z .x. $. $d x y z X $. $d x y z Y $. $d x y z .+ $. $d x y z Z $. - rngi.b $e |- B = ( Base ` R ) $. - rngi.p $e |- .+ = ( +g ` R ) $. - rngi.t $e |- .x. = ( .r ` R ) $. + ringi.b $e |- B = ( Base ` R ) $. + ringi.p $e |- .+ = ( +g ` R ) $. + ringi.t $e |- .x. = ( .r ` R ) $. $( Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) - rngi $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) + ringi $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) /\ ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) ) $= - ( vx vy vz wcel wa co wceq cv wral rsp crg w3a cgrp cmgp cfv cmnd simp3bi - eqid isrng adantr simpr1 simpr2 simpr3 simpld caovdig simprd caovdirg jca - sylc ) CUANZEANFANGANUBOEFGBPDPEFDPEGDPZBPQEFBPGDPVAFGDPBPQUTKLMEFGABDBAU - TKRZANZLRZANZMRZANZUBZOZVBVDVFBPDPVBVDDPVBVFDPZBPQZVBVDBPVFDPVJVDVFDPBPQZ - VIVKVLOZMASZVGVMVIVNLASZVEVNVIVOKASZVCVOUTVPVHUTCUCNCUDUEZUFNVPKLMABCDVQH - VQUHIJUIUGUJUTVCVEVGUKVOKATUSUTVCVEVGULVNLATUSUTVCVEVGUMVMMATUSZUNUOUTKLM - EFGABDBAVIVKVLVRUPUQUR $. + ( vx vy vz wcel wa co wceq cv wral rsp crg w3a cgrp cmgp cmnd eqid isring + cfv simp3bi adantr simpr1 sylc simpr2 simpr3 simpld caovdig caovdirg jca + simprd ) CUANZEANFANGANUBOEFGBPDPEFDPEGDPZBPQEFBPGDPVAFGDPBPQUTKLMEFGABDB + AUTKRZANZLRZANZMRZANZUBZOZVBVDVFBPDPVBVDDPVBVFDPZBPQZVBVDBPVFDPVJVDVFDPBP + QZVIVKVLOZMASZVGVMVIVNLASZVEVNVIVOKASZVCVOUTVPVHUTCUCNCUDUHZUENVPKLMABCDV + QHVQUFIJUGUIUJUTVCVEVGUKVOKATULUTVCVEVGUMVNLATULUTVCVEVGUNVMMATULZUOUPUTK + LMEFGABDBAVIVKVLVRUSUQUR $. $} ${ - rngcl.b $e |- B = ( Base ` R ) $. - rngcl.t $e |- .x. = ( .r ` R ) $. + ringcl.b $e |- B = ( Base ` R ) $. + ringcl.t $e |- .x. = ( .r ` R ) $. $( Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) - rngcl $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B ) $= - ( crg wcel cmgp cfv cmnd co eqid rngmgp mgpbas mgpplusg mndcl syl3an1 ) B - HIBJKZLIDAIEAIDECMAIBTTNZOACTDEABTUAFPBCTUAGQRS $. + ringcl $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B ) $= + ( crg wcel cmgp cfv cmnd co eqid ringmgp mgpbas mgpplusg mndcl syl3an1 ) + BHIBJKZLIDAIEAIDECMAIBTTNZOACTDEABTUAFPBCTUAGQRS $. $( A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) $) @@ -227548,68 +227762,68 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of monoid. (Contributed by Mario Carneiro, 15-Jun-2015.) $) iscrng2 $p |- ( R e. CRing <-> ( R e. Ring /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) ) $= - ( ccrg wcel crg cmgp cfv ccmn wa cv co wceq wral eqid iscrng rngmgp iscmn - cmnd wb mgpbas mgpplusg baib syl pm5.32i bitri ) DHIDJIZDKLZMIZNUKAOZBOZE - PUOUNEPQBCRACRZNDULULSZTUKUMUPUKULUCIZUMUPUDDULUQUAUMURUPABCEULCDULUQFUED - EULUQGUFUBUGUHUIUJ $. + ( ccrg wcel crg cmgp cfv ccmn wa cv co wceq wral eqid iscrng cmnd ringmgp + wb mgpbas mgpplusg iscmn baib syl pm5.32i bitri ) DHIDJIZDKLZMIZNUKAOZBOZ + EPUOUNEPQBCRACRZNDULULSZTUKUMUPUKULUAIZUMUPUCDULUQUBUMURUPABCEULCDULUQFUD + DEULUQGUEUFUGUHUIUJ $. $( Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) - rngass $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> + ringass $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) ) $= - ( crg wcel cmgp cfv cmnd w3a co wceq eqid rngmgp mgpbas mgpplusg mndass + ( crg wcel cmgp cfv cmnd w3a co wceq eqid ringmgp mgpbas mgpplusg mndass sylan ) BIJBKLZMJDAJEAJFAJNDECOFCODEFCOCOPBUCUCQZRACUCDEFABUCUDGSBCUCUDHT UAUB $. $d u x B $. $d u x R $. $d u x .x. $. $( The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) $) - rngideu $p |- ( R e. Ring -> + ringideu $p |- ( R e. Ring -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) ) $= - ( crg wcel cmgp cfv cmnd cv co wceq wa wral wreu eqid rngmgp mgpplusg syl - mgpbas mndideu ) DHIDJKZLIBMZAMZENUGOUGUFENUGOPACQBCRDUEUESZTABCEUECDUEUH - FUCDEUEUHGUAUDUB $. + ( crg wcel cmgp cfv cmnd cv co wceq wa wral wreu eqid ringmgp mndideu syl + mgpbas mgpplusg ) DHIDJKZLIBMZAMZENUGOUGUFENUGOPACQBCRDUEUESZTABCEUECDUEU + HFUCDEUEUHGUDUAUB $. $} ${ - rngdi.b $e |- B = ( Base ` R ) $. - rngdi.p $e |- .+ = ( +g ` R ) $. - rngdi.t $e |- .x. = ( .r ` R ) $. + ringdi.b $e |- B = ( Base ` R ) $. + ringdi.p $e |- .+ = ( +g ` R ) $. + ringdi.t $e |- .x. = ( .r ` R ) $. $( Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) $) - rngdi $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> + ringdi $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) ) $= - ( crg wcel w3a wa co wceq rngi simpld ) CKLEALFALGALMNEFGBODOEFDOEGDOZBOP - EFBOGDOSFGDOBOPABCDEFGHIJQR $. + ( crg wcel w3a wa co wceq ringi simpld ) CKLEALFALGALMNEFGBODOEFDOEGDOZBO + PEFBOGDOSFGDOBOPABCDEFGHIJQR $. $( Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) $) - rngdir $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> + ringdir $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) ) $= - ( crg wcel w3a wa co wceq rngi simprd ) CKLEALFALGALMNEFGBODOEFDOEGDOZBOP - EFBOGDOSFGDOBOPABCDEFGHIJQR $. + ( crg wcel w3a wa co wceq ringi simprd ) CKLEALFALGALMNEFGBODOEFDOEGDOZBO + PEFBOGDOSFGDOBOPABCDEFGHIJQR $. $} ${ - rngidcl.b $e |- B = ( Base ` R ) $. - rngidcl.u $e |- .1. = ( 1r ` R ) $. + ringidcl.b $e |- B = ( Base ` R ) $. + ringidcl.u $e |- .1. = ( 1r ` R ) $. $( The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) $) - rngidcl $p |- ( R e. Ring -> .1. e. B ) $= - ( crg wcel cmgp cfv cmnd eqid rngmgp mgpbas rngidval mndidcl syl ) BFGBHI - ZJGCAGBQQKZLAQCABQRDMBCQRENOP $. + ringidcl $p |- ( R e. Ring -> .1. e. B ) $= + ( crg wcel cmgp cfv cmnd eqid ringmgp mgpbas rngidval mndidcl syl ) BFGBH + IZJGCAGBQQKZLAQCABQRDMBCQRENOP $. $} ${ - rng0cl.b $e |- B = ( Base ` R ) $. - rng0cl.z $e |- .0. = ( 0g ` R ) $. + ring0cl.b $e |- B = ( Base ` R ) $. + ring0cl.z $e |- .0. = ( 0g ` R ) $. $( The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) $) - rng0cl $p |- ( R e. Ring -> .0. e. B ) $= - ( crg wcel cgrp rnggrp grpidcl syl ) BFGBHGCAGBIABCDEJK $. + ring0cl $p |- ( R e. Ring -> .0. e. B ) $= + ( crg wcel cgrp ringgrp grpidcl syl ) BFGBHGCAGBIABCDEJK $. $} ${ @@ -227617,197 +227831,199 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of rngidm.b $e |- B = ( Base ` R ) $. rngidm.t $e |- .x. = ( .r ` R ) $. rngidm.u $e |- .1. = ( 1r ` R ) $. - $( Lemma for ~ rnglidm and ~ rngridm . (Contributed by NM, 15-Sep-2011.) - (Revised by Mario Carneiro, 27-Dec-2014.) $) - rngidmlem $p |- ( ( R e. Ring /\ X e. B ) + $( Lemma for ~ ringlidm and ~ ringridm . (Contributed by NM, + 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) $) + ringidmlem $p |- ( ( R e. Ring /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) ) $= - ( crg wcel cmgp cfv cmnd co wceq wa eqid rngmgp mgpbas mgpplusg rngidval + ( crg wcel cmgp cfv cmnd co wceq wa eqid ringmgp mgpbas mgpplusg rngidval mndlrid sylan ) BIJBKLZMJEAJDECNEOEDCNEOPBUDUDQZRACUDEDABUDUEFSBCUDUEGTBD UDUEHUAUBUC $. $( The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) $) - rnglidm $p |- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X ) $= - ( crg wcel wa co wceq rngidmlem simpld ) BIJEAJKDECLEMEDCLEMABCDEFGHNO $. + ringlidm $p |- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X ) $= + ( crg wcel wa co wceq ringidmlem simpld ) BIJEAJKDECLEMEDCLEMABCDEFGHNO + $. $( The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) $) - rngridm $p |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X ) $= - ( crg wcel wa co wceq rngidmlem simprd ) BIJEAJKDECLEMEDCLEMABCDEFGHNO $. + ringridm $p |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X ) $= + ( crg wcel wa co wceq ringidmlem simprd ) BIJEAJKDECLEMEDCLEMABCDEFGHNO + $. $( Properties showing that an element ` I ` is the unity element of a ring. (Contributed by NM, 7-Aug-2013.) $) - isrngid $p |- ( R e. Ring + isringid $p |- ( R e. Ring -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> .1. = I ) ) $= ( vy crg wcel cmgp cfv eqid mgpbas rngidval cv co wceq mgpplusg wral wreu - wa wrex rngideu reurex syl ismgmid ) CKLZABDFJCMNZEBCUKUKOZGPCEUKULIQCDUK - ULHUAUJJRZARZDSUNTUNUMDSUNTUDABUBZJBUCUOJBUEAJBCDGHUFUOJBUGUHUI $. + wa wrex ringideu reurex syl ismgmid ) CKLZABDFJCMNZEBCUKUKOZGPCEUKULIQCDU + KULHUAUJJRZARZDSUNTUNUMDSUNTUDABUBZJBUCUOJBUEAJBCDGHUFUOJBUGUHUI $. $} ${ $d y A $. $d y B $. $d y M $. $d y .1. $. $d y R $. - rngidss.g $e |- M = ( ( mulGrp ` R ) |`s A ) $. - rngidss.b $e |- B = ( Base ` R ) $. - rngidss.u $e |- .1. = ( 1r ` R ) $. + ringidss.g $e |- M = ( ( mulGrp ` R ) |`s A ) $. + ringidss.b $e |- B = ( Base ` R ) $. + ringidss.u $e |- .1. = ( 1r ` R ) $. $( A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) $) - rngidss $p |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> + ringidss $p |- ( ( R e. Ring /\ A C_ B /\ .1. e. A ) -> .1. = ( 0g ` M ) ) $= ( vy wcel cbs cfv eqid wceq co cvv oveqd 3ad2antl1 eqtr3d syldan crg cmgp wss w3a cplusg c0g simp3 mgpbas ressbas2 3ad2ant2 eleqtrd cv simp2 sselda - eqsstr3d wa cmulr fvex syl6eqel mgpplusg ressplusg adantr rnglidm rngridm - syl ismgmid2 ) CUAJZABUCZDAJZUDZIEKLZEUELZDEEUFLZVKMVMMVLMVJDAVKVGVHVIUGV - HVGAVKNVIABECUBLZFBCVNVNMZGUHUIUJZUKVJIULZVKJZVQBJZDVQVLOZVQNVJVKBVQVJVKA - BVPVGVHVIUMUOUNZVJVSUPZDVQCUQLZOZVTVQWBWCVLDVQVJWCVLNZVSVJAPJWEVJAVKPVPEK - URUSAWCVNEPFCWCVNVOWCMZUTVAVEVBZQVGVHVSWDVQNVIBCWCDVQGWFHVCRSTVJVRVSVQDVL - OZVQNWAWBVQDWCOZWHVQWBWCVLVQDWGQVGVHVSWIVQNVIBCWCDVQGWFHVDRSTVF $. + eqsstr3d wa fvex syl6eqel mgpplusg ressplusg syl adantr ringlidm ringridm + cmulr ismgmid2 ) CUAJZABUCZDAJZUDZIEKLZEUELZDEEUFLZVKMVMMVLMVJDAVKVGVHVIU + GVHVGAVKNVIABECUBLZFBCVNVNMZGUHUIUJZUKVJIULZVKJZVQBJZDVQVLOZVQNVJVKBVQVJV + KABVPVGVHVIUMUOUNZVJVSUPZDVQCVELZOZVTVQWBWCVLDVQVJWCVLNZVSVJAPJWEVJAVKPVP + EKUQURAWCVNEPFCWCVNVOWCMZUSUTVAVBZQVGVHVSWDVQNVIBCWCDVQGWFHVCRSTVJVRVSVQD + VLOZVQNWAWBVQDWCOZWHVQWBWCVLVQDWGQVGVHVSWIVQNVIBCWCDVQGWFHVDRSTVF $. $} ${ - rngacl.b $e |- B = ( Base ` R ) $. - rngacl.p $e |- .+ = ( +g ` R ) $. + ringacl.b $e |- B = ( Base ` R ) $. + ringacl.p $e |- .+ = ( +g ` R ) $. $( Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.) $) - rngacl $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B ) $= - ( crg wcel cgrp co rnggrp grpcl syl3an1 ) CHICJIDAIEAIDEBKAICLABCDEFGMN + ringacl $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B ) $= + ( crg wcel cgrp co ringgrp grpcl syl3an1 ) CHICJIDAIEAIDEBKAICLABCDEFGMN $. $( Commutativity of the additive group of a ring. (See also ~ lmodcom .) (Contributed by Gérard Lang, 4-Dec-2014.) $) - rngcom $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> + ringcom $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) ) $= - ( wcel co wceq cfv rngacl syl3anc syl13anc rngdir rnglidm syl2anc oveq12d - eqid grpass crg w3a cur cmulr simp1 rngidcl syl simp2 simp3 rngdi 3eqtr3d - eqtr3d eqtrd cgrp rnggrp 3eqtr4d wb grprcan mpbid 3com23 grplcan ) CUAHZD - AHZEAHZUBZDDEBIZBIZDEDBIZBIZJZVFVHJZVEDDBIZEBIZVFDBIZVGVIVEVMEBIZVNEBIZJZ - VMVNJZVEVLEEBIZBIZVFVFBIZVOVPVECUCKZWBBIZDCUDKZIZWCEWDIZBIZWBVFWDIZWHBIZV - TWAVEWCVFWDIZWGWIVEVBWCAHZVCVDWJWGJVBVCVDUEZVEVBWBAHZWMWKWLVEVBWMWLACWBFW - BSZUFUGZWOABCWBWBFGLMVBVCVDUHZVBVCVDUIZABCWDWCDEFGWDSZUJNVEVBWMWMVFAHZWJW - IJWLWOWOABCDEFGLZABCWDWBWBVFFGWRONULVEWEVLWFVSBVEWEWBDWDIZXABIZVLVEVBWMWM - VCWEXBJWLWOWOWPABCWDWBWBDFGWRONVEXADXADBVEVBVCXADJWLWPACWDWBDFWRWNPQZXCRU - MVEWFWBEWDIZXDBIZVSVEVBWMWMVDWFXEJWLWOWOWQABCWDWBWBEFGWRONVEXDEXDEBVEVBVD - XDEJWLWQACWDWBEFWRWNPQZXFRUMRVEWHVFWHVFBVEVBWSWHVFJWLWTACWDWBVFFWRWNPQZXG - RUKVECUNHZVLAHZVDVDVOVTJVEVBXHWLCUOUGZVEVBVCVCXIWLWPWPABCDDFGLMZWQWQABCVL - EEFGTNVEXHWSVCVDVPWAJXJWTWPWQABCVFDEFGTNUPVEXHVMAHZVNAHZVDVQVRUQXJVEVBXIV - DXLWLXKWQABCVLEFGLMVEVBWSVCXMWLWTWPABCVFDFGLMWQABCVMVNEFGURNUSVEXHVCVCVDV - MVGJXJWPWPWQABCDDEFGTNVEXHVCVDVCVNVIJXJWPWQWPABCDEDFGTNUKVEXHWSVHAHZVCVJV - KUQXJWTVBVDVCXNABCEDFGLUTWPABCVFVHDFGVANUS $. + ( wcel co wceq cfv eqid ringacl syl3anc syl13anc ringdir ringlidm syl2anc + oveq12d grpass crg w3a cur cmulr simp1 ringidcl simp2 simp3 ringdi eqtr3d + syl eqtrd 3eqtr3d cgrp ringgrp 3eqtr4d wb grprcan mpbid 3com23 grplcan ) + CUAHZDAHZEAHZUBZDDEBIZBIZDEDBIZBIZJZVFVHJZVEDDBIZEBIZVFDBIZVGVIVEVMEBIZVN + EBIZJZVMVNJZVEVLEEBIZBIZVFVFBIZVOVPVECUCKZWBBIZDCUDKZIZWCEWDIZBIZWBVFWDIZ + WHBIZVTWAVEWCVFWDIZWGWIVEVBWCAHZVCVDWJWGJVBVCVDUEZVEVBWBAHZWMWKWLVEVBWMWL + ACWBFWBLZUFUKZWOABCWBWBFGMNVBVCVDUGZVBVCVDUHZABCWDWCDEFGWDLZUIOVEVBWMWMVF + AHZWJWIJWLWOWOABCDEFGMZABCWDWBWBVFFGWRPOUJVEWEVLWFVSBVEWEWBDWDIZXABIZVLVE + VBWMWMVCWEXBJWLWOWOWPABCWDWBWBDFGWRPOVEXADXADBVEVBVCXADJWLWPACWDWBDFWRWNQ + RZXCSULVEWFWBEWDIZXDBIZVSVEVBWMWMVDWFXEJWLWOWOWQABCWDWBWBEFGWRPOVEXDEXDEB + VEVBVDXDEJWLWQACWDWBEFWRWNQRZXFSULSVEWHVFWHVFBVEVBWSWHVFJWLWTACWDWBVFFWRW + NQRZXGSUMVECUNHZVLAHZVDVDVOVTJVEVBXHWLCUOUKZVEVBVCVCXIWLWPWPABCDDFGMNZWQW + QABCVLEEFGTOVEXHWSVCVDVPWAJXJWTWPWQABCVFDEFGTOUPVEXHVMAHZVNAHZVDVQVRUQXJV + EVBXIVDXLWLXKWQABCVLEFGMNVEVBWSVCXMWLWTWPABCVFDFGMNWQABCVMVNEFGUROUSVEXHV + CVCVDVMVGJXJWPWPWQABCDDEFGTOVEXHVCVDVCVNVIJXJWPWQWPABCDEDFGTOUMVEXHWSVHAH + ZVCVJVKUQXJWTVBVDVCXNABCEDFGMUTWPABCVFVHDFGVAOUS $. $} ${ $d x y R $. $( A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.) $) - rngabl $p |- ( R e. Ring -> R e. Abel ) $= - ( vx vy crg wcel cbs cfv cplusg eqidd rnggrp cv eqid rngcom isabld ) ADEZ - BCAFGZAHGZAOPIOQIAJPQABKCKPLQLMN $. + ringabl $p |- ( R e. Ring -> R e. Abel ) $= + ( vx vy crg wcel cbs cfv cplusg eqidd ringgrp cv eqid ringcom isabld ) AD + EZBCAFGZAHGZAOPIOQIAJPQABKCKPLQLMN $. $} $( A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) $) - rngcmn $p |- ( R e. Ring -> R e. CMnd ) $= - ( crg wcel cabl ccmn rngabl ablcmn syl ) ABCADCAECAFAGH $. + ringcmn $p |- ( R e. Ring -> R e. CMnd ) $= + ( crg wcel cabl ccmn ringabl ablcmn syl ) ABCADCAECAFAGH $. ${ $d s u v w x y B $. $d s u v w x y K $. $d s u v w x y ph $. $d s u v w x y L $. - rngpropd.1 $e |- ( ph -> B = ( Base ` K ) ) $. - rngpropd.2 $e |- ( ph -> B = ( Base ` L ) ) $. - rngpropd.3 $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> + ringpropd.1 $e |- ( ph -> B = ( Base ` K ) ) $. + ringpropd.2 $e |- ( ph -> B = ( Base ` L ) ) $. + ringpropd.3 $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) ) $. - rngpropd.4 $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> + ringpropd.4 $e |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) ) $. $( If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.) $) - rngpropd $p |- ( ph -> ( K e. Ring <-> L e. Ring ) ) $= + ringpropd $p |- ( ph -> ( K e. Ring <-> L e. Ring ) ) $= ( vu vv vw wcel cfv co wceq wa wral proplem cgrp cmgp cv cplusg cmulr cbs cmnd w3a crg simpll simprll simplrl simprlr ad2antrr eleqtrd simprr grpcl eqid syl3anc eleqtrrd syl12anc oveq2d eqtrd simplrr mgpbas mgpplusg mndcl oveq12d eqeq12d oveq1d anbi12d anassrs 2ralbidva adantr raleqdv raleqbidv wb ralbidva 3bitr3d pm5.32da df-3an 3bitr4g grppropd syl6eq oveqi 3eqtr3g - mndpropd 3anbi12d bitrd isrng ) AEUANZEUBOZUGNZKUCZLUCZMUCZEUDOZPZEUEOZPZ - WNWOWSPZWNWPWSPZWQPZQZWNWOWQPZWPWSPZXBWOWPWSPZWQPZQZRZMEUFOZSZLXKSZKXKSZU - HZFUANZFUBOZUGNZWNWOWPFUDOZPZFUEOZPZWNWOYAPZWNWPYAPZXSPZQZWNWOXSPZWPYAPZY - DWOWPYAPZXSPZQZRZMFUFOZSZLYMSZKYMSZUHZEUINFUINAXOWKWMYPUHZYQAWKWMRZXNRYSY - PRXOYRAYSXNYPAYSRZXJMDSZLDSZKDSYLMDSZLDSZKDSXNYPYTUUAUUCKLDDYTWNDNZWODNZR - ZRXJYLMDYTUUGWPDNZXJYLVQYTUUGUUHRZRZXDYFXIYKUUJWTYBXCYEUUJWTWNWRYAPZYBUUJ - AUUEWRDNWTUUKQAYSUUIUJZYTUUEUUFUUHUKZUUJWRXKDUUJWKWOXKNZWPXKNZWRXKNAWKWMU - UIULZUUJWODXKYTUUEUUFUUHUMZADXKQZYSUUIGUNZUOZUUJWPDXKYTUUGUUHUPZUUSUOZXKW - QEWOWPXKURZWQURZUQUSUUSUTABCDDWSYAWNWRJTVAUUJWRXTWNYAUUJAUUFUUHWRXTQUULUU - QUVAABCDDWQXSWOWPITVAVBVCUUJXCXAXBXSPZYEUUJAXADNXBDNZXCUVEQUULUUJXAXKDUUJ - WMWNXKNZUUNXAXKNAWKWMUUIVDZUUJWNDXKUUMUUSUOZUUTXKWSWLWNWOXKEWLWLURZUVCVEZ - EWSWLUVJWSURZVFZVGUSUUSUTUUJXBXKDUUJWMUVGUUOXBXKNUVHUVIUVBXKWSWLWNWPUVKUV - MVGUSUUSUTZABCDDWQXSXAXBITVAUUJXAYCXBYDXSUUJAUUEUUFXAYCQUULUUMUUQABCDDWSY - AWNWOJTVAUUJAUUEUUHXBYDQUULUUMUVAABCDDWSYAWNWPJTVAZVHVCVIUUJXFYHXHYJUUJXF - XEWPYAPZYHUUJAXEDNUUHXFUVPQUULUUJXEXKDUUJWKUVGUUNXEXKNUUPUVIUUTXKWQEWNWOU - VCUVDUQUSUUSUTUVAABCDDWSYAXEWPJTVAUUJXEYGWPYAUUJAUUEUUFXEYGQUULUUMUUQABCD - DWQXSWNWOITVAVJVCUUJXHXBXGXSPZYJUUJAUVFXGDNXHUVQQUULUVNUUJXGXKDUUJWMUUNUU - OXGXKNUVHUUTUVBXKWSWLWOWPUVKUVMVGUSUUSUTABCDDWQXSXBXGITVAUUJXBYDXGYIXSUVO - UUJAUUFUUHXGYIQUULUUQUVAABCDDWSYAWOWPJTVAVHVCVIVKVLVRVMYTUUBXMKDXKAUURYSG - VNZYTUUAXLLDXKUVRYTXJMDXKUVRVOVPVPYTUUDYOKDYMADYMQYSHVNZYTUUCYNLDYMUVSYTY - LMDYMUVSVOVPVPVSVTWKWMXNWAWKWMYPWAWBAWKXPWMXRYPABCDEFGHIWCABCDWLXQADXKWLU - FOGUVKWDADYMXQUFOHYMFXQXQURZYMURZVEWDABUCZDNCUCZDNRRUWBUWCWSPUWBUWCYAPUWB - UWCWLUDOZPUWBUWCXQUDOZPJWSUWDUWBUWCUVMWEYAUWEUWBUWCFYAXQUVTYAURZVFWEWFWGW - HWIKLMXKWQEWSWLUVCUVJUVDUVLWJKLMYMXSFYAXQUWAUVTXSURUWFWJWB $. + mndpropd 3anbi12d bitrd isring ) AEUANZEUBOZUGNZKUCZLUCZMUCZEUDOZPZEUEOZP + ZWNWOWSPZWNWPWSPZWQPZQZWNWOWQPZWPWSPZXBWOWPWSPZWQPZQZRZMEUFOZSZLXKSZKXKSZ + UHZFUANZFUBOZUGNZWNWOWPFUDOZPZFUEOZPZWNWOYAPZWNWPYAPZXSPZQZWNWOXSPZWPYAPZ + YDWOWPYAPZXSPZQZRZMFUFOZSZLYMSZKYMSZUHZEUINFUINAXOWKWMYPUHZYQAWKWMRZXNRYS + YPRXOYRAYSXNYPAYSRZXJMDSZLDSZKDSYLMDSZLDSZKDSXNYPYTUUAUUCKLDDYTWNDNZWODNZ + RZRXJYLMDYTUUGWPDNZXJYLVQYTUUGUUHRZRZXDYFXIYKUUJWTYBXCYEUUJWTWNWRYAPZYBUU + JAUUEWRDNWTUUKQAYSUUIUJZYTUUEUUFUUHUKZUUJWRXKDUUJWKWOXKNZWPXKNZWRXKNAWKWM + UUIULZUUJWODXKYTUUEUUFUUHUMZADXKQZYSUUIGUNZUOZUUJWPDXKYTUUGUUHUPZUUSUOZXK + WQEWOWPXKURZWQURZUQUSUUSUTABCDDWSYAWNWRJTVAUUJWRXTWNYAUUJAUUFUUHWRXTQUULU + UQUVAABCDDWQXSWOWPITVAVBVCUUJXCXAXBXSPZYEUUJAXADNXBDNZXCUVEQUULUUJXAXKDUU + JWMWNXKNZUUNXAXKNAWKWMUUIVDZUUJWNDXKUUMUUSUOZUUTXKWSWLWNWOXKEWLWLURZUVCVE + ZEWSWLUVJWSURZVFZVGUSUUSUTUUJXBXKDUUJWMUVGUUOXBXKNUVHUVIUVBXKWSWLWNWPUVKU + VMVGUSUUSUTZABCDDWQXSXAXBITVAUUJXAYCXBYDXSUUJAUUEUUFXAYCQUULUUMUUQABCDDWS + YAWNWOJTVAUUJAUUEUUHXBYDQUULUUMUVAABCDDWSYAWNWPJTVAZVHVCVIUUJXFYHXHYJUUJX + FXEWPYAPZYHUUJAXEDNUUHXFUVPQUULUUJXEXKDUUJWKUVGUUNXEXKNUUPUVIUUTXKWQEWNWO + UVCUVDUQUSUUSUTUVAABCDDWSYAXEWPJTVAUUJXEYGWPYAUUJAUUEUUFXEYGQUULUUMUUQABC + DDWQXSWNWOITVAVJVCUUJXHXBXGXSPZYJUUJAUVFXGDNXHUVQQUULUVNUUJXGXKDUUJWMUUNU + UOXGXKNUVHUUTUVBXKWSWLWOWPUVKUVMVGUSUUSUTABCDDWQXSXBXGITVAUUJXBYDXGYIXSUV + OUUJAUUFUUHXGYIQUULUUQUVAABCDDWSYAWOWPJTVAVHVCVIVKVLVRVMYTUUBXMKDXKAUURYS + GVNZYTUUAXLLDXKUVRYTXJMDXKUVRVOVPVPYTUUDYOKDYMADYMQYSHVNZYTUUCYNLDYMUVSYT + YLMDYMUVSVOVPVPVSVTWKWMXNWAWKWMYPWAWBAWKXPWMXRYPABCDEFGHIWCABCDWLXQADXKWL + UFOGUVKWDADYMXQUFOHYMFXQXQURZYMURZVEWDABUCZDNCUCZDNRRUWBUWCWSPUWBUWCYAPUW + BUWCWLUDOZPUWBUWCXQUDOZPJWSUWDUWBUWCUVMWEYAUWEUWBUWCFYAXQUVTYAURZVFWEWFWG + WHWIKLMXKWQEWSWLUVCUVJUVDUVLWJKLMYMXSFYAXQUWAUVTXSURUWFWJWB $. $( If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) $) crngpropd $p |- ( ph -> ( K e. CRing <-> L e. CRing ) ) $= - ( crg wcel cmgp cfv ccmn wa ccrg cbs eqid co rngpropd mgpbas syl6eq cmulr - cv cplusg mgpplusg oveqi 3eqtr3g cmnpropd anbi12d iscrng 3bitr4g ) AEKLZE - MNZOLZPFKLZFMNZOLZPEQLFQLAUNUQUPUSABCDEFGHIJUAABCDUOURADERNZUORNGUTEUOUOS - ZUTSUBUCADFRNZURRNHVBFURURSZVBSUBUCABUEZDLCUEZDLPPVDVEEUDNZTVDVEFUDNZTVDV - EUOUFNZTVDVEURUFNZTJVFVHVDVEEVFUOVAVFSUGUHVGVIVDVEFVGURVCVGSUGUHUIUJUKEUO - VAULFURVCULUM $. + ( crg wcel cmgp cfv ccmn wa ccrg cbs eqid co ringpropd mgpbas cv mgpplusg + syl6eq cmulr cplusg oveqi 3eqtr3g cmnpropd anbi12d iscrng 3bitr4g ) AEKLZ + EMNZOLZPFKLZFMNZOLZPEQLFQLAUNUQUPUSABCDEFGHIJUAABCDUOURADERNZUORNGUTEUOUO + SZUTSUBUEADFRNZURRNHVBFURURSZVBSUBUEABUCZDLCUCZDLPPVDVEEUFNZTVDVEFUFNZTVD + VEUOUGNZTVDVEURUGNZTJVFVHVDVEEVFUOVAVFSUDUHVGVIVDVEFVGURVCVGSUDUHUIUJUKEU + OVAULFURVCULUM $. $} ${ $d x y K $. $d x y L $. - rngprop.b $e |- ( Base ` K ) = ( Base ` L ) $. - rngprop.p $e |- ( +g ` K ) = ( +g ` L ) $. - rngprop.m $e |- ( .r ` K ) = ( .r ` L ) $. + ringprop.b $e |- ( Base ` K ) = ( Base ` L ) $. + ringprop.p $e |- ( +g ` K ) = ( +g ` L ) $. + ringprop.m $e |- ( .r ` K ) = ( .r ` L ) $. $( If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) $) - rngprop $p |- ( K e. Ring <-> L e. Ring ) $= + ringprop $p |- ( K e. Ring <-> L e. Ring ) $= ( vx vy crg wcel wtru cbs cfv wceq a1i cv cplusg co wa oveqi cmulr eqidd - wb rngpropd trud ) AHIBHIUBJFGAKLZABJUEUAUEBKLMJCNFOZGOZAPLZQUFUGBPLZQMJU - FUEIUGUEIRRZUHUIUFUGDSNUFUGATLZQUFUGBTLZQMUJUKULUFUGESNUCUD $. + wb ringpropd trud ) AHIBHIUBJFGAKLZABJUEUAUEBKLMJCNFOZGOZAPLZQUFUGBPLZQMJ + UFUEIUGUEIRRZUHUIUFUGDSNUFUGATLZQUFUGBTLZQMUJUKULUFUGESNUCUD $. $} ${ $d u x .1. $. $d u x y z B $. $d u x y z ph $. $d u x y z R $. $d u .x. $. - isrngd.b $e |- ( ph -> B = ( Base ` R ) ) $. - isrngd.p $e |- ( ph -> .+ = ( +g ` R ) ) $. - isrngd.t $e |- ( ph -> .x. = ( .r ` R ) ) $. - isrngd.g $e |- ( ph -> R e. Grp ) $. - isrngd.c $e |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B ) $. - isrngd.a $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> + isringd.b $e |- ( ph -> B = ( Base ` R ) ) $. + isringd.p $e |- ( ph -> .+ = ( +g ` R ) ) $. + isringd.t $e |- ( ph -> .x. = ( .r ` R ) ) $. + isringd.g $e |- ( ph -> R e. Grp ) $. + isringd.c $e |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B ) $. + isringd.a $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) ) $. - isrngd.d $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> + isringd.d $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) ) $. - isrngd.e $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> + isringd.e $e |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) ) $. - isrngd.u $e |- ( ph -> .1. e. B ) $. - isrngd.i $e |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x ) $. - isrngd.h $e |- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x ) $. + isringd.u $e |- ( ph -> .1. e. B ) $. + isringd.i $e |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x ) $. + isringd.h $e |- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x ) $. $( Properties that determine a ring. (Contributed by NM, 2-Aug-2013.) $) - isrngd $p |- ( ph -> R e. Ring ) $= + isringd $p |- ( ph -> R e. Ring ) $= ( cgrp wcel cmgp cfv cmnd cv cplusg co cmulr wceq wa cbs wral eqid mgpbas crg syl6eq mgpplusg ismndd eleq2d 3anbi123d biimpar adantr eqidd proplem3 - w3a oveq123d 3eqtr3d jca syldan ralrimivvva isrng syl3anbrc ) AGUAUBGUCUD - ZUEUBBUFZCUFZDUFZGUGUDZUHZGUIUDZUHZVOVPVTUHZVOVQVTUHZVRUHZUJZVOVPVRUHZVQV - TUHZWCVPVQVTUHZVRUHZUJZUKZDGULUDZUMCWLUMBWLUMGUPUBMABCDEHVNIAEWLVNULUDJWL - GVNVNUNZWLUNZUOUQAHVTVNUGUDLGVTVNWMVTUNZURUQNORSTUSAWKBCDWLWLWLAVOWLUBZVP - WLUBZVQWLUBZVFZVOEUBZVPEUBZVQEUBZVFZWKAXCWSAWTWPXAWQXBWRAEWLVOJUTAEWLVPJU - TAEWLVQJUTVAVBAXCUKZWEWJXDVOVPVQFUHZHUHVOVPHUHZVOVQHUHZFUHWAWDPXDVOVOXEVS - HVTAHVTUJXCLVCZXDVOVDAXCCDFVRKVEVGXDXFWBXGWCFVRAFVRUJXCKVCZAXCBCHVTLVEAXC - BDHVTLVEZVGVHXDVOVPFUHZVQHUHXGVPVQHUHZFUHWGWIQXDXKWFVQVQHVTXHAXCBCFVRKVEX - DVQVDVGXDXGWCXLWHFVRXIXJAXCCDHVTLVEVGVHVIVJVKBCDWLVRGVTVNWNWMVRUNWOVLVM + w3a oveq123d 3eqtr3d jca syldan ralrimivvva isring syl3anbrc ) AGUAUBGUCU + DZUEUBBUFZCUFZDUFZGUGUDZUHZGUIUDZUHZVOVPVTUHZVOVQVTUHZVRUHZUJZVOVPVRUHZVQ + VTUHZWCVPVQVTUHZVRUHZUJZUKZDGULUDZUMCWLUMBWLUMGUPUBMABCDEHVNIAEWLVNULUDJW + LGVNVNUNZWLUNZUOUQAHVTVNUGUDLGVTVNWMVTUNZURUQNORSTUSAWKBCDWLWLWLAVOWLUBZV + PWLUBZVQWLUBZVFZVOEUBZVPEUBZVQEUBZVFZWKAXCWSAWTWPXAWQXBWRAEWLVOJUTAEWLVPJ + UTAEWLVQJUTVAVBAXCUKZWEWJXDVOVPVQFUHZHUHVOVPHUHZVOVQHUHZFUHWAWDPXDVOVOXEV + SHVTAHVTUJXCLVCZXDVOVDAXCCDFVRKVEVGXDXFWBXGWCFVRAFVRUJXCKVCZAXCBCHVTLVEAX + CBDHVTLVEZVGVHXDVOVPFUHZVQHUHXGVPVQHUHZFUHWGWIQXDXKWFVQVQHVTXHAXCBCFVRKVE + XDVQVDVGXDXGWCXLWHFVRXIXJAXCCDHVTLVEVGVHVIVJVKBCDWLVRGVTVNWNWMVRUNWOVLVM $. iscrngd.c $e |- ( ( ph /\ x e. B /\ y e. B ) -> @@ -227815,10 +228031,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.) $) iscrngd $p |- ( ph -> R e. CRing ) $= - ( crg wcel cmgp ccmn ccrg isrngd eqid mgpbas syl6eq cmulr cplusg mgpplusg - cfv cbs ismndd iscmnd iscrng sylanbrc ) AGUBUCGUDUNZUEUCGUFUCABCDEFGHIJKL - MNOPQRSTUGABCEHUTAEGUOUNZUTUOUNJVAGUTUTUHZVAUHUIUJZAHGUKUNZUTULUNLGVDUTVB - VDUHUMUJZABCDEHUTIVCVENORSTUPUAUQGUTVBURUS $. + ( crg wcel cmgp cfv ccmn ccrg isringd cbs eqid mgpbas syl6eq cmulr cplusg + mgpplusg ismndd iscmnd iscrng sylanbrc ) AGUBUCGUDUEZUFUCGUGUCABCDEFGHIJK + LMNOPQRSTUHABCEHUTAEGUIUEZUTUIUEJVAGUTUTUJZVAUJUKULZAHGUMUEZUTUNUELGVDUTV + BVDUJUOULZABCDEHUTIVCVENORSTUPUAUQGUTVBURUS $. $} ${ @@ -227827,176 +228043,176 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of rngz.z $e |- .0. = ( 0g ` R ) $. $( The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) $) - rnglz $p |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. ) $= - ( crg wcel wa co cplusg cfv wceq cgrp rnggrp grpidcl syl adantr mpdan w3a - eqid grplid oveq1d simpr rngdir syldan simpl rngcl syl3anc grprid syl2anc - 3jca eqcomd 3eqtr3d wb grplcan syl13anc mpbid ) BIJZDAJZKZEDCLZVDBMNZLZVD - EVELZOZVDEOZVCEEVELZDCLZVDVFVGVCVJEDCVAVJEOZVBVABPJZVLBQZVMEAJZVLABEFHRZA - VEBEEFVEUCZHUDUASTUEVAVBVOVOVBUBVKVFOVCVOVOVBVAVOVBVAVMVOVNVPSTZVRVAVBUFZ - UNAVEBCEEDFVQGUGUHVCVMVDAJZVDVGOVAVMVBVNTZVCVAVOVBVTVAVBUIVRVSABCEDFGUJUK - ZVMVTKVGVDAVEBVDEFVQHULUOUMUPVCVMVTVOVTVHVIUQWAWBVRWBAVEBVDEVDFVQURUSUT - $. + ringlz $p |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. ) $= + ( crg wcel wa co cplusg cfv wceq cgrp ringgrp grpidcl syl adantr eqid w3a + grplid mpdan oveq1d simpr 3jca ringdir syldan simpl ringcl syl3anc grprid + eqcomd syl2anc 3eqtr3d wb grplcan syl13anc mpbid ) BIJZDAJZKZEDCLZVDBMNZL + ZVDEVELZOZVDEOZVCEEVELZDCLZVDVFVGVCVJEDCVAVJEOZVBVABPJZVLBQZVMEAJZVLABEFH + RZAVEBEEFVEUAZHUCUDSTUEVAVBVOVOVBUBVKVFOVCVOVOVBVAVOVBVAVMVOVNVPSTZVRVAVB + UFZUGAVEBCEEDFVQGUHUIVCVMVDAJZVDVGOVAVMVBVNTZVCVAVOVBVTVAVBUJVRVSABCEDFGU + KULZVMVTKVGVDAVEBVDEFVQHUMUNUOUPVCVMVTVOVTVHVIUQWAWBVRWBAVEBVDEVDFVQURUSU + T $. $( The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) $) - rngrz $p |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. ) $= - ( crg wcel wa co cplusg cfv wceq cgrp rnggrp grplid syl adantr eqid mpdan - grpidcl oveq2d w3a simpr 3jca rngdi syldan mpd3an3 eqcomd syl2anc 3eqtr3d - rngcl wb grprcan syl13anc mpbid ) BIJZDAJZKZDECLZVBBMNZLZEVBVCLZOZVBEOZVA - DEEVCLZCLZVBVDVEVAVHEDCUSVHEOZUTUSBPJZVJBQZVKEAJZVJABEFHUCZAVCBEEFVCUAZHR - UBSTUDUSUTUTVMVMUEVIVDOVAUTVMVMUSUTUFUSVMUTUSVKVMVLVNSTZVPUGAVCBCDEEFVOGU - HUIVAVKVBAJZVBVEOUSVKUTVLTZUSUTVMVQVPABCDEFGUNUJZVKVQKVEVBAVCBVBEFVOHRUKU - LUMVAVKVQVMVQVFVGUOVRVSVPVSAVCBVBEVBFVOUPUQUR $. + ringrz $p |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. ) $= + ( crg wcel wa co cplusg cfv wceq cgrp ringgrp grplid syl adantr mpdan w3a + grpidcl eqid oveq2d simpr ringdi syldan ringcl mpd3an3 syl2anc 3eqtr3d wb + 3jca eqcomd grprcan syl13anc mpbid ) BIJZDAJZKZDECLZVBBMNZLZEVBVCLZOZVBEO + ZVADEEVCLZCLZVBVDVEVAVHEDCUSVHEOZUTUSBPJZVJBQZVKEAJZVJABEFHUCZAVCBEEFVCUD + ZHRUASTUEUSUTUTVMVMUBVIVDOVAUTVMVMUSUTUFUSVMUTUSVKVMVLVNSTZVPUNAVCBCDEEFV + OGUGUHVAVKVBAJZVBVEOUSVKUTVLTZUSUTVMVQVPABCDEFGUIUJZVKVQKVEVBAVCBVBEFVOHR + UOUKULVAVKVQVMVQVFVGUMVRVSVPVSAVCBVBEVBFVOUPUQUR $. $} ${ $d x y z R $. $( Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.) $) - rngsrg $p |- ( R e. Ring -> R e. SRing ) $= + ringsrg $p |- ( R e. Ring -> R e. SRing ) $= ( vx vy vz crg wcel ccmn cmgp cfv cmnd cv cplusg co cmulr wceq wa cbs w3a - wral eqid c0g csrg rngcmn cgrp isrng biimpi simp2d simp3d rnglz rngrz jca - ralrimiva r19.26 sylanbrc 3jca issrg sylibr ) AEFZAGFZAHIZJFZBKZCKZDKZALI - ZMANIZMVBVCVFMVBVDVFMZVEMOVBVCVEMVDVFMVGVCVDVFMVEMOPDAQIZSCVHSZAUAIZVBVFM - VJOZVBVJVFMVJOZPZPBVHSZRAUBFURUSVAVNAUCURAUDFZVAVIBVHSZURVOVAVPRBCDVHVEAV - FUTVHTZUTTZVETZVFTZUEUFZUGURVPVMBVHSVNURVOVAVPWAUHURVMBVHURVBVHFPVKVLVHAV - FVBVJVQVTVJTZUIVHAVFVBVJVQVTWBUJUKULVIVMBVHUMUNUOBCDVHVEAVFUTVJVQVRVSVTWB - UPUQ $. + wral eqid c0g csrg ringcmn cgrp isring biimpi simp2d simp3d ringlz ringrz + jca ralrimiva r19.26 sylanbrc 3jca issrg sylibr ) AEFZAGFZAHIZJFZBKZCKZDK + ZALIZMANIZMVBVCVFMVBVDVFMZVEMOVBVCVEMVDVFMVGVCVDVFMVEMOPDAQIZSCVHSZAUAIZV + BVFMVJOZVBVJVFMVJOZPZPBVHSZRAUBFURUSVAVNAUCURAUDFZVAVIBVHSZURVOVAVPRBCDVH + VEAVFUTVHTZUTTZVETZVFTZUEUFZUGURVPVMBVHSVNURVOVAVPWAUHURVMBVHURVBVHFPVKVL + VHAVFVBVJVQVTVJTZUIVHAVFVBVJVQVTWBUJUKULVIVMBVHUMUNUOBCDVHVEAVFUTVJVQVRVS + VTWBUPUQ $. $} ${ - rng1eq0.b $e |- B = ( Base ` R ) $. - rng1eq0.u $e |- .1. = ( 1r ` R ) $. - rng1eq0.z $e |- .0. = ( 0g ` R ) $. + ring1eq0.b $e |- B = ( Base ` R ) $. + ring1eq0.u $e |- .1. = ( 1r ` R ) $. + ring1eq0.z $e |- .0. = ( 0g ` R ) $. $( If one and zero are equal, then any two elements of a ring are equal. Alternatively, every ring has one distinct from zero except the zero ring containing the single element ` { 0 } ` . (Contributed by Mario Carneiro, 10-Sep-2014.) $) - rng1eq0 $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> + ring1eq0 $p |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( .1. = .0. -> X = Y ) ) $= - ( crg wcel w3a wceq wa co oveq1d rnglz syl2anc eqtr4d rnglidm cmulr simpr - cfv simpl1 simpl2 eqid simpl3 3eqtr3d ex ) BJKZDAKZEAKZLZCFMZDEMUMUNNZCDB - UAUCZOZCEUPOZDEUOUQFDUPOZURUOCFDUPUMUNUBZPUOURFEUPOZUSUOCFEUPUTPUOUSFVAUO - UJUKUSFMUJUKULUNUDZUJUKULUNUEZABUPDFGUPUFZIQRUOUJULVAFMVBUJUKULUNUGZABUPE - FGVDIQRSSSUOUJUKUQDMVBVCABUPCDGVDHTRUOUJULUREMVBVEABUPCEGVDHTRUHUI $. + ( crg wcel w3a wceq wa co oveq1d ringlz syl2anc eqtr4d ringlidm cmulr cfv + simpr simpl1 simpl2 eqid simpl3 3eqtr3d ex ) BJKZDAKZEAKZLZCFMZDEMUMUNNZC + DBUAUBZOZCEUPOZDEUOUQFDUPOZURUOCFDUPUMUNUCZPUOURFEUPOZUSUOCFEUPUTPUOUSFVA + UOUJUKUSFMUJUKULUNUDZUJUKULUNUEZABUPDFGUPUFZIQRUOUJULVAFMVBUJUKULUNUGZABU + PEFGVDIQRSSSUOUJUKUQDMVBVCABUPCDGVDHTRUOUJULUREMVBVEABUPCEGVDHTRUHUI $. $} ${ $d B x y $. $d R x y $. $d .1. x y $. $d .0. x y $. - rng1ne0.b $e |- B = ( Base ` R ) $. - rng1ne0.u $e |- .1. = ( 1r ` R ) $. - rng1ne0.z $e |- .0. = ( 0g ` R ) $. + ring1ne0.b $e |- B = ( Base ` R ) $. + ring1ne0.u $e |- .1. = ( 1r ` R ) $. + ring1ne0.z $e |- .0. = ( 0g ` R ) $. $( If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.) $) - rng1ne0 $p |- ( ( R e. Ring /\ 1 < ( # ` B ) ) -> .1. =/= .0. ) $= + ring1ne0 $p |- ( ( R e. Ring /\ 1 < ( # ` B ) ) -> .1. =/= .0. ) $= ( vx vy cv wne wrex crg wcel c1 cfv wa cvv cbs wi clt wbr fvex hashgt12el - chash eqeltri mpan adantl w3a rng1eq0 3expib adantr com3l rexlimivv mpcom - necon3d ) HJZIJZKZIALHALZBMNZOAUEPUAUBZQZCDKZVBUTVAARNVBUTABSPREBSUCUFARH - IUDUGUHUSVCVDTHIAAVCUQANZURANZQZUSVDVAVGUSVDTZTVBVAVEVFVHVAVEVFUICDUQURAB - CUQURDEFGUJUPUKULUMUNUO $. + chash eqeltri mpan adantl w3a necon3d 3expib adantr com3l rexlimivv mpcom + ring1eq0 ) HJZIJZKZIALHALZBMNZOAUEPUAUBZQZCDKZVBUTVAARNVBUTABSPREBSUCUFAR + HIUDUGUHUSVCVDTHIAAVCUQANZURANZQZUSVDVAVGUSVDTZTVBVAVEVFVHVAVEVFUICDUQURA + BCUQURDEFGUPUJUKULUMUNUO $. $} ${ - rngnegl.b $e |- B = ( Base ` R ) $. - rngnegl.t $e |- .x. = ( .r ` R ) $. - rngnegl.u $e |- .1. = ( 1r ` R ) $. - rngnegl.n $e |- N = ( invg ` R ) $. - rngnegl.r $e |- ( ph -> R e. Ring ) $. - rngnegl.x $e |- ( ph -> X e. B ) $. + ringnegl.b $e |- B = ( Base ` R ) $. + ringnegl.t $e |- .x. = ( .r ` R ) $. + ringnegl.u $e |- .1. = ( 1r ` R ) $. + ringnegl.n $e |- N = ( invg ` R ) $. + ringnegl.r $e |- ( ph -> R e. Ring ) $. + ringnegl.x $e |- ( ph -> X e. B ) $. $( Negation in a ring is the same as left multiplication by -1. ( ~ rngonegmn1l analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) - rngnegl $p |- ( ph -> ( ( N ` .1. ) .x. X ) = ( N ` X ) ) $= - ( cfv co wceq wcel syl syl2anc eqid cplusg c0g rngidcl cgrp rnggrp rngdir - crg grpinvcl syl13anc grprinv oveq1d rnglz eqtrd rnglidm 3eqtr3rd syl3anc - wb rngcl grpinvid1 mpbird eqcomd ) AGFNZEFNZGDOZAVBVDPZGVDCUANZOZCUBNZPZA - EVCVFOZGDOZEGDOZVDVFOZVHVGACUGQZEBQZVCBQZGBQZVKVMPLAVNVOLBCEHJUCRZACUDQZV - OVPAVNVSLCUERZVRBCFEHKUHSZMBVFCDEVCGHVFTZIUFUIAVKVHGDOZVHAVJVHGDAVSVOVJVH - PVTVRBVFCFEVHHWBVHTZKUJSUKAVNVQWCVHPLMBCDGVHHIWDULSUMAVLGVDVFAVNVQVLGPLMB - CDEGHIJUNSUKUOAVSVQVDBQZVEVIUQVTMAVNVPVQWELWAMBCDVCGHIURUPBVFCFGVDVHHWBWD - KUSUPUTVA $. + ringnegl $p |- ( ph -> ( ( N ` .1. ) .x. X ) = ( N ` X ) ) $= + ( cfv co wceq wcel syl syl2anc eqid cplusg c0g crg ringidcl cgrp grpinvcl + ringgrp ringdir syl13anc grprinv oveq1d ringlz eqtrd ringlidm 3eqtr3rd wb + ringcl syl3anc grpinvid1 mpbird eqcomd ) AGFNZEFNZGDOZAVBVDPZGVDCUANZOZCU + BNZPZAEVCVFOZGDOZEGDOZVDVFOZVHVGACUCQZEBQZVCBQZGBQZVKVMPLAVNVOLBCEHJUDRZA + CUEQZVOVPAVNVSLCUGRZVRBCFEHKUFSZMBVFCDEVCGHVFTZIUHUIAVKVHGDOZVHAVJVHGDAVS + VOVJVHPVTVRBVFCFEVHHWBVHTZKUJSUKAVNVQWCVHPLMBCDGVHHIWDULSUMAVLGVDVFAVNVQV + LGPLMBCDEGHIJUNSUKUOAVSVQVDBQZVEVIUPVTMAVNVPVQWELWAMBCDVCGHIUQURBVFCFGVDV + HHWBWDKUSURUTVA $. $( Negation in a ring is the same as right multiplication by -1. ( ~ rngonegmn1r analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) rngnegr $p |- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) ) $= - ( cfv co wceq wcel syl syl2anc eqid cplusg c0g cgrp rnggrp grpinvcl rngdi - crg rngidcl syl13anc grplinv oveq2d rngrz eqtrd rngridm 3eqtr3rd wb rngcl - syl3anc grpinvid2 mpbird eqcomd ) AGFNZGEFNZDOZAVBVDPZVDGCUANZOZCUBNZPZAG - VCEVFOZDOZVDGEDOZVFOZVHVGACUGQZGBQZVCBQZEBQZVKVMPLMACUCQZVQVPAVNVRLCUDRZA - VNVQLBCEHJUHRZBCFEHKUESZVTBVFCDGVCEHVFTZIUFUIAVKGVHDOZVHAVJVHGDAVRVQVJVHP - VSVTBVFCFEVHHWBVHTZKUJSUKAVNVOWCVHPLMBCDGVHHIWDULSUMAVLGVDVFAVNVOVLGPLMBC - DEGHIJUNSUKUOAVRVOVDBQZVEVIUPVSMAVNVOVPWELMWABCDGVCHIUQURBVFCFGVDVHHWBWDK - USURUTVA $. - $} - - ${ - rngneglmul.b $e |- B = ( Base ` R ) $. - rngneglmul.t $e |- .x. = ( .r ` R ) $. - rngneglmul.n $e |- N = ( invg ` R ) $. - rngneglmul.r $e |- ( ph -> R e. Ring ) $. - rngneglmul.x $e |- ( ph -> X e. B ) $. - rngneglmul.y $e |- ( ph -> Y e. B ) $. + ( cfv co wceq wcel syl syl2anc eqid cplusg c0g crg cgrp ringidcl grpinvcl + ringgrp ringdi syl13anc grplinv oveq2d ringrz ringridm 3eqtr3rd wb ringcl + eqtrd syl3anc grpinvid2 mpbird eqcomd ) AGFNZGEFNZDOZAVBVDPZVDGCUANZOZCUB + NZPZAGVCEVFOZDOZVDGEDOZVFOZVHVGACUCQZGBQZVCBQZEBQZVKVMPLMACUDQZVQVPAVNVRL + CUGRZAVNVQLBCEHJUERZBCFEHKUFSZVTBVFCDGVCEHVFTZIUHUIAVKGVHDOZVHAVJVHGDAVRV + QVJVHPVSVTBVFCFEVHHWBVHTZKUJSUKAVNVOWCVHPLMBCDGVHHIWDULSUQAVLGVDVFAVNVOVL + GPLMBCDEGHIJUMSUKUNAVRVOVDBQZVEVIUOVSMAVNVOVPWELMWABCDGVCHIUPURBVFCFGVDVH + HWBWDKUSURUTVA $. + $} + + ${ + ringneglmul.b $e |- B = ( Base ` R ) $. + ringneglmul.t $e |- .x. = ( .r ` R ) $. + ringneglmul.n $e |- N = ( invg ` R ) $. + ringneglmul.r $e |- ( ph -> R e. Ring ) $. + ringneglmul.x $e |- ( ph -> X e. B ) $. + ringneglmul.y $e |- ( ph -> Y e. B ) $. $( Negation of a product in a ring. ( ~ mulneg1 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) - rngmneg1 $p |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) ) $= - ( cur cfv co crg wcel syl rngnegl wceq cgrp eqid rngidcl grpinvcl syl2anc - rnggrp rngass syl13anc oveq1d rngcl syl3anc 3eqtr3d ) ACNOZEOZFDPZGDPZUOF - GDPZDPZFEOZGDPUREOACQRZUOBRZFBRZGBRZUQUSUAKACUBRZUNBRZVBAVAVEKCUGSAVAVFKB - CUNHUNUCZUDSBCEUNHJUEUFLMBCDUOFGHIUHUIAUPUTGDABCDUNEFHIVGJKLTUJABCDUNEURH - IVGJKAVAVCVDURBRKLMBCDFGHIUKULTUM $. + ringmneg1 $p |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) ) $= + ( cur cfv co crg wcel syl ringnegl wceq ringgrp ringidcl grpinvcl syl2anc + cgrp eqid ringass syl13anc oveq1d ringcl syl3anc 3eqtr3d ) ACNOZEOZFDPZGD + PZUOFGDPZDPZFEOZGDPUREOACQRZUOBRZFBRZGBRZUQUSUAKACUFRZUNBRZVBAVAVEKCUBSAV + AVFKBCUNHUNUGZUCSBCEUNHJUDUELMBCDUOFGHIUHUIAUPUTGDABCDUNEFHIVGJKLTUJABCDU + NEURHIVGJKAVAVCVDURBRKLMBCDFGHIUKULTUM $. $( Negation of a product in a ring. ( ~ mulneg2 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) - rngmneg2 $p |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) ) $= - ( co cur cfv crg wcel syl rngnegr wceq cgrp eqid rngidcl grpinvcl syl2anc - rnggrp rngass syl13anc rngcl syl3anc oveq2d 3eqtr3rd ) AFGDNZCOPZEPZDNZFG - UPDNZDNZUNEPFGEPZDNACQRZFBRZGBRZUPBRZUQUSUAKLMACUBRZUOBRZVDAVAVEKCUGSAVAV - FKBCUOHUOUCZUDSBCEUOHJUEUFBCDFGUPHIUHUIABCDUOEUNHIVGJKAVAVBVCUNBRKLMBCDFG - HIUJUKTAURUTFDABCDUOEGHIVGJKMTULUM $. + ringmneg2 $p |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) ) $= + ( co cur cfv crg wcel syl rngnegr wceq cgrp ringgrp eqid ringidcl syl2anc + grpinvcl ringass syl13anc ringcl syl3anc oveq2d 3eqtr3rd ) AFGDNZCOPZEPZD + NZFGUPDNZDNZUNEPFGEPZDNACQRZFBRZGBRZUPBRZUQUSUAKLMACUBRZUOBRZVDAVAVEKCUCS + AVAVFKBCUOHUOUDZUESBCEUOHJUGUFBCDFGUPHIUHUIABCDUOEUNHIVGJKAVAVBVCUNBRKLMB + CDFGHIUJUKTAURUTFDABCDUOEGHIVGJKMTULUM $. $( Double negation of a product in a ring. ( ~ mul2neg analog.) (Contributed by Mario Carneiro, 4-Dec-2014.) $) - rngm2neg $p |- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) ) $= - ( cfv co cgrp wcel crg rnggrp syl2anc syl grpinvcl rngmneg1 rngmneg2 wceq - fveq2d rngcl syl3anc grpinvinv 3eqtrd ) AFENGENZDOFUKDOZENFGDOZENZENZUMAB - CDEFUKHIJKLACPQZGBQZUKBQACRQZUPKCSUAZMBCEGHJUBTUCAULUNEABCDEFGHIJKLMUDUFA - UPUMBQZUOUMUEUSAURFBQUQUTKLMBCDFGHIUGUHBCEUMHJUITUJ $. + ringm2neg $p |- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) ) $= + ( cfv co cgrp wcel crg ringgrp syl2anc syl grpinvcl ringmneg1 fveq2d wceq + ringmneg2 ringcl syl3anc grpinvinv 3eqtrd ) AFENGENZDOFUKDOZENFGDOZENZENZ + UMABCDEFUKHIJKLACPQZGBQZUKBQACRQZUPKCSUAZMBCEGHJUBTUCAULUNEABCDEFGHIJKLMU + FUDAUPUMBQZUOUMUEUSAURFBQUQUTKLMBCDFGHIUGUHBCEUMHJUITUJ $. $} ${ - rngsubdi.b $e |- B = ( Base ` R ) $. - rngsubdi.t $e |- .x. = ( .r ` R ) $. - rngsubdi.m $e |- .- = ( -g ` R ) $. - rngsubdi.r $e |- ( ph -> R e. Ring ) $. - rngsubdi.x $e |- ( ph -> X e. B ) $. - rngsubdi.y $e |- ( ph -> Y e. B ) $. - rngsubdi.z $e |- ( ph -> Z e. B ) $. + ringsubdi.b $e |- B = ( Base ` R ) $. + ringsubdi.t $e |- .x. = ( .r ` R ) $. + ringsubdi.m $e |- .- = ( -g ` R ) $. + ringsubdi.r $e |- ( ph -> R e. Ring ) $. + ringsubdi.x $e |- ( ph -> X e. B ) $. + ringsubdi.y $e |- ( ph -> Y e. B ) $. + ringsubdi.z $e |- ( ph -> Z e. B ) $. $( Ring multiplication distributes over subtraction. ( ~ subdi analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) - rngsubdi $p |- ( ph -> + ringsubdi $p |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) ) $= - ( cfv co wcel wceq syl2anc cminusg cplusg cgrp rnggrp eqid grpinvcl rngdi - crg syl syl13anc rngmneg2 oveq2d eqtrd grpsubval rngcl syl3anc 3eqtr4d ) - AFGHCUAPZPZCUBPZQZDQZFGDQZFHDQZURPZUTQZFGHEQZDQVCVDEQZAVBVCFUSDQZUTQZVFAC - UHRZFBRZGBRZUSBRZVBVJSLMNACUCRZHBRZVNAVKVOLCUDUIOBCURHIURUEZUFTBUTCDFGUSI - UTUEZJUGUJAVIVEVCUTABCDURFHIJVQLMOUKULUMAVGVAFDAVMVPVGVASNOBUTCUREGHIVRVQ - KUNTULAVCBRZVDBRZVHVFSAVKVLVMVSLMNBCDFGIJUOUPAVKVLVPVTLMOBCDFHIJUOUPBUTCU - REVCVDIVRVQKUNTUQ $. + ( cfv co wcel wceq syl2anc cminusg cplusg crg ringgrp syl grpinvcl ringdi + cgrp syl13anc ringmneg2 oveq2d eqtrd grpsubval ringcl syl3anc 3eqtr4d + eqid ) AFGHCUAPZPZCUBPZQZDQZFGDQZFHDQZURPZUTQZFGHEQZDQVCVDEQZAVBVCFUSDQZU + TQZVFACUCRZFBRZGBRZUSBRZVBVJSLMNACUHRZHBRZVNAVKVOLCUDUEOBCURHIURUQZUFTBUT + CDFGUSIUTUQZJUGUIAVIVEVCUTABCDURFHIJVQLMOUJUKULAVGVAFDAVMVPVGVASNOBUTCURE + GHIVRVQKUMTUKAVCBRZVDBRZVHVFSAVKVLVMVSLMNBCDFGIJUNUOAVKVLVPVTLMOBCDFHIJUN + UOBUTCUREVCVDIVRVQKUMTUP $. $( Ring multiplication distributes over subtraction. ( ~ subdir analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) $) rngsubdir $p |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) ) $= - ( cfv co wcel wceq syl2anc cminusg cplusg crg cgrp rnggrp syl eqid rngdir - grpinvcl syl13anc rngmneg1 oveq2d eqtrd grpsubval oveq1d syl3anc 3eqtr4d - rngcl ) AFGCUAPZPZCUBPZQZHDQZFHDQZGHDQZUSPZVAQZFGEQZHDQVDVEEQZAVCVDUTHDQZ - VAQZVGACUCRZFBRZUTBRZHBRZVCVKSLMACUDRZGBRZVNAVLVPLCUEUFNBCUSGIUSUGZUITOBV - ACDFUTHIVAUGZJUHUJAVJVFVDVAABCDUSGHIJVRLNOUKULUMAVHVBHDAVMVQVHVBSMNBVACUS - EFGIVSVRKUNTUOAVDBRZVEBRZVIVGSAVLVMVOVTLMOBCDFHIJURUPAVLVQVOWALNOBCDGHIJU - RUPBVACUSEVDVEIVSVRKUNTUQ $. + ( cfv co wcel wceq syl2anc cminusg crg cgrp ringgrp eqid grpinvcl ringdir + cplusg syl ringmneg1 oveq2d eqtrd grpsubval oveq1d ringcl syl3anc 3eqtr4d + syl13anc ) AFGCUAPZPZCUHPZQZHDQZFHDQZGHDQZUSPZVAQZFGEQZHDQVDVEEQZAVCVDUTH + DQZVAQZVGACUBRZFBRZUTBRZHBRZVCVKSLMACUCRZGBRZVNAVLVPLCUDUINBCUSGIUSUEZUFT + OBVACDFUTHIVAUEZJUGURAVJVFVDVAABCDUSGHIJVRLNOUJUKULAVHVBHDAVMVQVHVBSMNBVA + CUSEFGIVSVRKUMTUNAVDBRZVEBRZVIVGSAVLVMVOVTLMOBCDFHIJUOUPAVLVQVOWALNOBCDGH + IJUOUPBVACUSEVDVEIVSVRKUMTUQ $. $} ${ @@ -228010,96 +228226,97 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of mulgass2 $p |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( ( N .x. X ) .X. Y ) = ( N .x. ( X .X. Y ) ) ) $= ( wcel co wceq cc0 oveq1 oveq1d eqeq12d cfv adantr syl3anc vx vy cz wi cv - crg cneg c1 caddc w3a c0g eqid rnglz 3adant3 simp3 mulg0 syl rngcl 3com23 - 3eqtr4d cn0 wa cplusg cgrp simpl1 rnggrp nn0z adantl mulgp1 mulgcl simpl2 - 3ad2ant1 rngdir syl13anc eqtrd syl5ibr ex cn cminusg nnz mulgneg rngmneg1 - fveq2 zindd 3exp com24 3imp2 ) BUFKZEUCKZFAKZGAKZEFCLZGDLZEFGDLZCLZMZWHWK - WJWIWPWHWKWJWIWPUDUAUEZFCLZGDLZWQWNCLZMNFCLZGDLZNWNCLZMUBUEZFCLZGDLZXDWNC - LZMZXDUGZFCLZGDLZXIWNCLZMZXDUHUILZFCLZGDLZXNWNCLZMZWPWHWKWJUJZUAUBEWQNMZW - SXBWTXCXTWRXAGDWQNFCOPWQNWNCOQWQXDMZWSXFWTXGYAWRXEGDWQXDFCOPWQXDWNCOQWQXN - MZWSXPWTXQYBWRXOGDWQXNFCOPWQXNWNCOQWQXIMZWSXKWTXLYCWRXJGDWQXIFCOPWQXIWNCO - QWQEMZWSWMWTWOYDWRWLGDWQEFCOPWQEWNCOQXSBUKRZGDLZYEXBXCWHWKYFYEMWJABDGYEHJ - YEULZUMUNXSXAYEGDXSWJXAYEMWHWKWJUOZACBFYEHYGIUPUQPXSWNAKZXCYEMWHWJWKYIABD - FGHJURUSZACBWNYEHYGIUPUQUTXSXDVAKZXHXRUDXHXRXSYKVBZXFWNBVCRZLZXGWNYMLZMXF - XGWNYMOYLXPYNXQYOYLXPXEFYMLZGDLZYNYLXOYPGDYLBVDKZXDUCKZWJXOYPMYLWHYRWHWKW - JYKVEZBVFZUQZYKYSXSXDVGVHZXSWJYKYHSZAYMCBXDFHIYMULZVITPYLWHXEAKZWJWKYQYNM - YTYLYRYSWJUUFXSYRYKWHWKYRWJUUAVLZSUUCUUDACBXDFHIVJZTUUDWHWKWJYKVKAYMBDXEF - GHUUEJVMVNVOYLYRYSYIXQYOMUUBUUCXSYIYKYJSAYMCBXDWNHIUUEVITQVPVQXSXDVRKZXHX - MUDXHXMXSUUIVBZXFBVSRZRZXGUUKRZMXFXGUUKWCUUJXKUULXLUUMUUJXKXEUUKRZGDLUULU - UJXJUUNGDUUJYRYSWJXJUUNMXSYRUUIUUGSZUUIYSXSXDVTVHZXSWJUUIYHSZACBUUKXDFHIU - UKULZWATPUUJABDUUKXEGHJUURWHWKWJUUIVEUUJYRYSWJUUFUUOUUPUUQUUHTWHWKWJUUIVK - WBVOUUJYRYSYIXLUUMMUUOUUPXSYIUUIYJSACBUUKXDWNHIUURWATQVPVQWDWEWFWG $. + crg cneg caddc w3a c0g eqid ringlz 3adant3 simp3 mulg0 syl ringcl 3eqtr4d + c1 3com23 cn0 wa cplusg cgrp simpl1 ringgrp adantl mulgp1 3ad2ant1 mulgcl + nn0z simpl2 ringdir syl13anc eqtrd syl5ibr ex cminusg fveq2 nnz ringmneg1 + cn mulgneg zindd 3exp com24 3imp2 ) BUFKZEUCKZFAKZGAKZEFCLZGDLZEFGDLZCLZM + ZWHWKWJWIWPWHWKWJWIWPUDUAUEZFCLZGDLZWQWNCLZMNFCLZGDLZNWNCLZMUBUEZFCLZGDLZ + XDWNCLZMZXDUGZFCLZGDLZXIWNCLZMZXDUSUHLZFCLZGDLZXNWNCLZMZWPWHWKWJUIZUAUBEW + QNMZWSXBWTXCXTWRXAGDWQNFCOPWQNWNCOQWQXDMZWSXFWTXGYAWRXEGDWQXDFCOPWQXDWNCO + QWQXNMZWSXPWTXQYBWRXOGDWQXNFCOPWQXNWNCOQWQXIMZWSXKWTXLYCWRXJGDWQXIFCOPWQX + IWNCOQWQEMZWSWMWTWOYDWRWLGDWQEFCOPWQEWNCOQXSBUJRZGDLZYEXBXCWHWKYFYEMWJABD + GYEHJYEUKZULUMXSXAYEGDXSWJXAYEMWHWKWJUNZACBFYEHYGIUOUPPXSWNAKZXCYEMWHWJWK + YIABDFGHJUQUTZACBWNYEHYGIUOUPURXSXDVAKZXHXRUDXHXRXSYKVBZXFWNBVCRZLZXGWNYM + LZMXFXGWNYMOYLXPYNXQYOYLXPXEFYMLZGDLZYNYLXOYPGDYLBVDKZXDUCKZWJXOYPMYLWHYR + WHWKWJYKVEZBVFZUPZYKYSXSXDVKVGZXSWJYKYHSZAYMCBXDFHIYMUKZVHTPYLWHXEAKZWJWK + YQYNMYTYLYRYSWJUUFXSYRYKWHWKYRWJUUAVIZSUUCUUDACBXDFHIVJZTUUDWHWKWJYKVLAYM + BDXEFGHUUEJVMVNVOYLYRYSYIXQYOMUUBUUCXSYIYKYJSAYMCBXDWNHIUUEVHTQVPVQXSXDWB + KZXHXMUDXHXMXSUUIVBZXFBVRRZRZXGUUKRZMXFXGUUKVSUUJXKUULXLUUMUUJXKXEUUKRZGD + LUULUUJXJUUNGDUUJYRYSWJXJUUNMXSYRUUIUUGSZUUIYSXSXDVTVGZXSWJUUIYHSZACBUUKX + DFHIUUKUKZWCTPUUJABDUUKXEGHJUURWHWKWJUUIVEUUJYRYSWJUUFUUOUUPUUQUUHTWHWKWJ + UUIVLWAVOUUJYRYSYIXLUUMMUUOUUPXSYIUUIYJSACBUUKXDWNHIUURWCTQVPVQWDWEWFWG + $. $} ${ $d Z a b c $. $d M a b c $. - rng1.m $e |- M = { <. ( Base ` ndx ) , { Z } >. , + ring1.m $e |- M = { <. ( Base ` ndx ) , { Z } >. , <. ( +g ` ndx ) , { <. <. Z , Z >. , Z >. } >. , <. ( .r ` ndx ) , { <. <. Z , Z >. , Z >. } >. } $. $( The (smallest) structure representing a _zero ring_. (Contributed by AV, 28-Apr-2019.) $) - rng1 $p |- ( Z e. V -> M e. Ring ) $= + ring1 $p |- ( Z e. V -> M e. Ring ) $= ( va vb vc wcel cfv cop co wceq wa wral ax-mp eqtrd oveq2d oveq1d eqeq12d cvv cgrp cmgp cmnd cv csn w3a crg cnx cbs cplusg eqid grp1 rngbase eqcomi snex rngplusg eqcomd grppropstr sylibr mnd1 mgpbas grpbase eqtri mgpplusg cpr cmulr rngmulr grpplusg mndprop df-ov a1i opex jca fvsng oveq12d oveq1 - id syl anbi12d ralbidv ralsng oveq2 bitrd mpbird 3jca isrng ) CBHZAUAHZAU - BIZUCHZEUDZFUDZGUDZCCJZCJZUEZKZWPKZWKWLWPKZWKWMWPKZWPKZLZWSWMWPKZWTWQWPKZ - LZMZGCUEZNZFXGNZEXGNZUFAUGHWGWHWJXJWGUHUIIXGJUHUJIWPJVEZUAHWHCXKBXKUKZULX - GWPAXKXGAUIIZXGTHZXGXMLCUOZXGWPAWPTDUMOZUNWPTHZAUJIZWPLWOUOZXQWPXRXGWPAWP - TDUPZUQOXLURUSWGXKUCHWJCXKBXLUTWIXKWIUIIZXGXKUIIZXGYAXGAWIWIUKZXPVAUNXNXG - YBLXOXGWPXKTXLVBOVCWIUJIZWPXKUJIZYDAVFIZWPYFYDAYFWIYCYFUKVDUNWPYFXQWPYFLX - SXGWPAWPTDVGOZUNVCXQWPYELXSXGWPXKTXLVHOVCVIUSWGXJCCCWPKZWPKZYHYHWPKZLZYHC - WPKZYJLZMZWGYKYMWGYICYJWGYIYHCWGYHCCWPWGYHWNWPIZCYHYOLWGCCWPVJVKZWGWNTHZW - GMYOCLWGYQWGYQWGCCVLVKWGVQVMWNCTBVNVRZPZQYSPWGYJCWGYJYOCWGYJYHYOWGYHCYHCW - PYSYSVOYPPYRPUQZPWGYLCYJWGYLYHCWGYHCCWPYSRYSPYTPVMWGXJCWQWPKZCWLWPKZCWMWP - KZWPKZLZUUBWMWPKZUUCWQWPKZLZMZGXGNZFXGNZYNXIUUKECBWKCLZXHUUJFXGUULXFUUIGX - GUULXBUUEXEUUHUULWRUUAXAUUDWKCWQWPVPUULWSUUBWTUUCWPWKCWLWPVPZWKCWMWPVPZVO - SUULXCUUFXDUUGUULWSUUBWMWPUUMRUULWTUUCWQWPUUNRSVSVTVTWAWGUUKCUUCWPKZYHUUC - WPKZLZYHWMWPKZUUCUUCWPKZLZMZGXGNZYNUUJUVBFCBWLCLZUUIUVAGXGUVCUUEUUQUUHUUT - UVCUUAUUOUUDUUPUVCWQUUCCWPWLCWMWPVPZQUVCUUBYHUUCWPWLCCWPWBZRSUVCUUFUURUUG - UUSUVCUUBYHWMWPUVERUVCWQUUCUUCWPUVDQSVSVTWAUVAYNGCBWMCLZUUQYKUUTYMUVFUUOY - IUUPYJUVFUUCYHCWPWMCCWPWBZQUVFUUCYHYHWPUVGQSUVFUURYLUUSYJWMCYHWPWBUVFUUCY - HUUCYHWPUVGUVGVOSVSWAWCWCWDWEEFGXGWPAWPWIXPYCXQWPXRLXSXTOYGWFUS $. + id syl anbi12d ralbidv ralsng oveq2 bitrd mpbird 3jca isring ) CBHZAUAHZA + UBIZUCHZEUDZFUDZGUDZCCJZCJZUEZKZWPKZWKWLWPKZWKWMWPKZWPKZLZWSWMWPKZWTWQWPK + ZLZMZGCUEZNZFXGNZEXGNZUFAUGHWGWHWJXJWGUHUIIXGJUHUJIWPJVEZUAHWHCXKBXKUKZUL + XGWPAXKXGAUIIZXGTHZXGXMLCUOZXGWPAWPTDUMOZUNWPTHZAUJIZWPLWOUOZXQWPXRXGWPAW + PTDUPZUQOXLURUSWGXKUCHWJCXKBXLUTWIXKWIUIIZXGXKUIIZXGYAXGAWIWIUKZXPVAUNXNX + GYBLXOXGWPXKTXLVBOVCWIUJIZWPXKUJIZYDAVFIZWPYFYDAYFWIYCYFUKVDUNWPYFXQWPYFL + XSXGWPAWPTDVGOZUNVCXQWPYELXSXGWPXKTXLVHOVCVIUSWGXJCCCWPKZWPKZYHYHWPKZLZYH + CWPKZYJLZMZWGYKYMWGYICYJWGYIYHCWGYHCCWPWGYHWNWPIZCYHYOLWGCCWPVJVKZWGWNTHZ + WGMYOCLWGYQWGYQWGCCVLVKWGVQVMWNCTBVNVRZPZQYSPWGYJCWGYJYOCWGYJYHYOWGYHCYHC + WPYSYSVOYPPYRPUQZPWGYLCYJWGYLYHCWGYHCCWPYSRYSPYTPVMWGXJCWQWPKZCWLWPKZCWMW + PKZWPKZLZUUBWMWPKZUUCWQWPKZLZMZGXGNZFXGNZYNXIUUKECBWKCLZXHUUJFXGUULXFUUIG + XGUULXBUUEXEUUHUULWRUUAXAUUDWKCWQWPVPUULWSUUBWTUUCWPWKCWLWPVPZWKCWMWPVPZV + OSUULXCUUFXDUUGUULWSUUBWMWPUUMRUULWTUUCWQWPUUNRSVSVTVTWAWGUUKCUUCWPKZYHUU + CWPKZLZYHWMWPKZUUCUUCWPKZLZMZGXGNZYNUUJUVBFCBWLCLZUUIUVAGXGUVCUUEUUQUUHUU + TUVCUUAUUOUUDUUPUVCWQUUCCWPWLCWMWPVPZQUVCUUBYHUUCWPWLCCWPWBZRSUVCUUFUURUU + GUUSUVCUUBYHWMWPUVERUVCWQUUCUUCWPUVDQSVSVTWAUVAYNGCBWMCLZUUQYKUUTYMUVFUUO + YIUUPYJUVFUUCYHCWPWMCCWPWBZQUVFUUCYHYHWPUVGQSUVFUURYLUUSYJWMCYHWPWBUVFUUC + YHUUCYHWPUVGUVGVOSVSWAWCWCWDWEEFGXGWPAWPWIXPYCXQWPXRLXSXTOYGWFUS $. $} $( Rings exist. (Contributed by AV, 29-Apr-2019.) $) - rngn0 $p |- Ring =/= (/) $= - ( vz cv cvv wcel cnx cbs cfv csn cop cplusg cmulr ctp crg wne vex eqid rng1 - c0 ne0i mp2b ) ABZCDEFGUAHIEJGUAUAIUAIHZIEKGUBILZMDMRNAOUCCUAUCPQMUCST $. + ringn0 $p |- Ring =/= (/) $= + ( vz cv cvv wcel cnx cbs cfv csn cop cplusg cmulr ctp crg c0 wne eqid ring1 + vex ne0i mp2b ) ABZCDEFGUAHIEJGUAUAIUAIHZIEKGUBILZMDMNOARUCCUAUCPQMUCST $. ${ $d x y z B $. $d x y z R $. $d x y z .x. $. $d x y z X $. - rnglghm.b $e |- B = ( Base ` R ) $. - rnglghm.t $e |- .x. = ( .r ` R ) $. + ringlghm.b $e |- B = ( Base ` R ) $. + ringlghm.t $e |- .x. = ( .r ` R ) $. $( Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.) $) - rnglghm $p |- ( ( R e. Ring /\ X e. B ) -> + ringlghm $p |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) e. ( R GrpHom R ) ) $= ( vy vz crg wcel wa cfv cv co eqid wceq oveq2 ovex fvmpt cplusg cmpt cgrp - rnggrp adantr rngcl 3expa fmptd w3a 3anass rngdi sylan2br anassrs adantlr - rngacl 3expb syl oveqan12d adantl 3eqtr4d isghmd ) CJKZEBKZLZHICUAMZVECCA - BEANZDOZUBZBBFFVEPZVIVBCUCKVCCUDUEZVJVDABVGBVHVBVCVFBKVGBKBCDEVFFGUFUGVHP - ZUHVDHNZBKZINZBKZLZLZEVLVNVEOZDOZEVLDOZEVNDOZVEOZVRVHMZVLVHMZVNVHMZVEOZVB - VCVPVSWBQZVCVPLVBVCVMVOUIWGVCVMVOUJBVECDEVLVNFVIGUKULUMVQVRBKZWCVSQVBVPWH - VCVBVMVOWHBVECVLVNFVIUOUPUNAVRVGVSBVHVFVREDRVKEVRDSTUQVPWFWBQVDVMVOWDVTWE - WAVEAVLVGVTBVHVFVLEDRVKEVLDSTAVNVGWABVHVFVNEDRVKEVNDSTURUSUTVA $. + ringgrp adantr ringcl 3expa fmptd w3a 3anass ringdi anassrs ringacl 3expb + sylan2br adantlr syl oveqan12d adantl 3eqtr4d isghmd ) CJKZEBKZLZHICUAMZV + ECCABEANZDOZUBZBBFFVEPZVIVBCUCKVCCUDUEZVJVDABVGBVHVBVCVFBKVGBKBCDEVFFGUFU + GVHPZUHVDHNZBKZINZBKZLZLZEVLVNVEOZDOZEVLDOZEVNDOZVEOZVRVHMZVLVHMZVNVHMZVE + OZVBVCVPVSWBQZVCVPLVBVCVMVOUIWGVCVMVOUJBVECDEVLVNFVIGUKUOULVQVRBKZWCVSQVB + VPWHVCVBVMVOWHBVECVLVNFVIUMUNUPAVRVGVSBVHVFVREDRVKEVRDSTUQVPWFWBQVDVMVOWD + VTWEWAVEAVLVGVTBVHVFVLEDRVKEVLDSTAVNVGWABVHVFVNEDRVKEVNDSTURUSUTVA $. $( Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.) $) - rngrghm $p |- ( ( R e. Ring /\ X e. B ) -> + ringrghm $p |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( x .x. X ) ) e. ( R GrpHom R ) ) $= ( vy vz crg wcel wa cfv cv co eqid wceq oveq1 ovex fvmpt cplusg cmpt cgrp - rnggrp adantr rngcl 3expa an32s fmptd w3a df-3an rngdir sylan2br anass1rs - rngacl 3expb adantlr syl oveqan12d adantl 3eqtr4d isghmd ) CJKZEBKZLZHICU - AMZVFCCABANZEDOZUBZBBFFVFPZVJVCCUCKVDCUDUEZVKVEABVHBVIVCVGBKZVDVHBKZVCVLV - DVMBCDVGEFGUFUGUHVIPZUIVEHNZBKZINZBKZLZLZVOVQVFOZEDOZVOEDOZVQEDOZVFOZWAVI - MZVOVIMZVQVIMZVFOZVCVSVDWBWEQZVSVDLVCVPVRVDUJWJVPVRVDUKBVFCDVOVQEFVJGULUM - UNVTWABKZWFWBQVCVSWKVDVCVPVRWKBVFCVOVQFVJUOUPUQAWAVHWBBVIVGWAEDRVNWAEDSTU - RVSWIWEQVEVPVRWGWCWHWDVFAVOVHWCBVIVGVOEDRVNVOEDSTAVQVHWDBVIVGVQEDRVNVQEDS - TUSUTVAVB $. + ringgrp adantr ringcl 3expa an32s fmptd ringdir sylan2br anass1rs ringacl + w3a df-3an 3expb adantlr syl oveqan12d adantl 3eqtr4d isghmd ) CJKZEBKZLZ + HICUAMZVFCCABANZEDOZUBZBBFFVFPZVJVCCUCKVDCUDUEZVKVEABVHBVIVCVGBKZVDVHBKZV + CVLVDVMBCDVGEFGUFUGUHVIPZUIVEHNZBKZINZBKZLZLZVOVQVFOZEDOZVOEDOZVQEDOZVFOZ + WAVIMZVOVIMZVQVIMZVFOZVCVSVDWBWEQZVSVDLVCVPVRVDUNWJVPVRVDUOBVFCDVOVQEFVJG + UJUKULVTWABKZWFWBQVCVSWKVDVCVPVRWKBVFCVOVQFVJUMUPUQAWAVHWBBVIVGWAEDRVNWAE + DSTURVSWIWEQVEVPVRWGWCWHWDVFAVOVHWCBVIVGVOEDRVNVOEDSTAVQVHWDBVIVGVQEDRVNV + QEDSTUSUTVAVB $. $} ${ @@ -228118,19 +228335,19 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) $) gsummulc1 $p |- ( ph -> ( R gsum ( k e. A |-> ( X .x. Y ) ) ) = ( ( R gsum ( k e. A |-> X ) ) .x. Y ) ) $= - ( vx cv co cmpt cgsu crg wcel ccmn rngcmn syl cmnd rngmnd rngrghm syl2anc - cghm cmhm ghmmhm oveq1 gsummhm2 ) AUABCUAUBZJFUCZIJFUCGEGBIUDUEUCZJFUCEEH - IKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUOUCUGZVDEEUPUCUGAVCJCUGV - EPRUACEFJLOUMUNEEVDUQUJSTUTIJFURUTVBJFURUS $. + ( vx cv co cmpt cgsu crg wcel ccmn ringcmn syl cmnd ringmnd cghm ringrghm + cmhm syl2anc ghmmhm oveq1 gsummhm2 ) AUABCUAUBZJFUCZIJFUCGEGBIUDUEUCZJFUC + EEHIKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUMUCUGZVDEEUOUCUGAVCJC + UGVEPRUACEFJLOUNUPEEVDUQUJSTUTIJFURUTVBJFURUS $. $( A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) $) gsummulc2 $p |- ( ph -> ( R gsum ( k e. A |-> ( Y .x. X ) ) ) = ( Y .x. ( R gsum ( k e. A |-> X ) ) ) ) $= - ( vx cv co cmpt cgsu crg wcel ccmn rngcmn syl cmnd rngmnd rnglghm syl2anc - cghm cmhm ghmmhm oveq2 gsummhm2 ) AUABCJUAUBZFUCZJIFUCGJEGBIUDUEUCZFUCEEH - IKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUOUCUGZVDEEUPUCUGAVCJCUGV - EPRUACEFJLOUMUNEEVDUQUJSTUTIJFURUTVBJFURUS $. + ( vx cv co cmpt cgsu crg wcel ccmn ringcmn syl cmnd ringmnd cghm ringlghm + cmhm syl2anc ghmmhm oveq2 gsummhm2 ) AUABCJUAUBZFUCZJIFUCGJEGBIUDUEUCZFUC + EEHIKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUMUCUGZVDEEUOUCUGAVCJC + UGVEPRUACEFJLOUNUPEEVDUQUJSTUTIJFURUTVBJFURUS $. $} ${ @@ -228152,10 +228369,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of (Proof modification is discouraged.) $) gsummulc1OLD $p |- ( ph -> ( R gsum ( k e. A |-> ( X .x. Y ) ) ) = ( ( R gsum ( k e. A |-> X ) ) .x. Y ) ) $= - ( vx cv co cmpt cgsu crg wcel ccmn rngcmn syl cmnd rngmnd rngrghm syl2anc - cghm cmhm ghmmhm oveq1 gsummhm2OLD ) AUABCUAUBZJFUCZIJFUCGEGBIUDUEUCZJFUC - EEHIKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUOUCUGZVDEEUPUCUGAVCJC - UGVEPRUACEFJLOUMUNEEVDUQUJSTUTIJFURUTVBJFURUS $. + ( vx cv co cmpt cgsu crg wcel ccmn ringcmn syl cmnd ringmnd cghm ringrghm + cmhm syl2anc ghmmhm oveq1 gsummhm2OLD ) AUABCUAUBZJFUCZIJFUCGEGBIUDUEUCZJ + FUCEEHIKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUMUCUGZVDEEUOUCUGAV + CJCUGVEPRUACEFJLOUNUPEEVDUQUJSTUTIJFURUTVBJFURUS $. $( A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of ~ gsummulc1 as of @@ -228163,10 +228380,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of (Proof modification is discouraged.) $) gsummulc2OLD $p |- ( ph -> ( R gsum ( k e. A |-> ( Y .x. X ) ) ) = ( Y .x. ( R gsum ( k e. A |-> X ) ) ) ) $= - ( vx cv co cmpt cgsu crg wcel ccmn rngcmn syl cmnd rngmnd rnglghm syl2anc - cghm cmhm ghmmhm oveq2 gsummhm2OLD ) AUABCJUAUBZFUCZJIFUCGJEGBIUDUEUCZFUC - EEHIKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUOUCUGZVDEEUPUCUGAVCJC - UGVEPRUACEFJLOUMUNEEVDUQUJSTUTIJFURUTVBJFURUS $. + ( vx cv co cmpt cgsu crg wcel ccmn ringcmn syl cmnd ringmnd cghm ringlghm + cmhm syl2anc ghmmhm oveq2 gsummhm2OLD ) AUABCJUAUBZFUCZJIFUCGJEGBIUDUEUCZ + FUCEEHIKLMAEUFUGZEUHUGPEUIUJAVCEUKUGPEULUJQAUACVAUDZEEUMUCUGZVDEEUOUCUGAV + CJCUGVEPRUACEFJLOUNUPEEVDUQUJSTUTIJFURUTVBJFURUS $. $} ${ @@ -228185,16 +228402,16 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of gsummgp0 $p |- ( ph -> ( G gsum ( n e. N |-> A ) ) = .0. ) $= ( wceq co wcel adantr cmpt cgsu cv wa csn cdif cun cmulr difsnid ad2antrl cfv eqcomd mpteq1d oveq2d cbs eqid mgpbas mgpplusg ccmn crngmgp syl diffi - ccrg cfn simpl eldifi syl2an simprl neldifsnd cmnd crngrng rngmnd mndidcl - crg wb eleq1 ad2antll mpbird weq adantlr gsumunsnd oveq2 sylan2 ralrimiva - 3syl gsummptcl rngrz syl2anc eqtrd 3eqtrd rexlimddv ) ACIQZGFHBUAZUBRZIQE - HPAEUCZHSZWLUDZUDZWNGFHWOUEZUFZWSUGZBUAZUBRGFWTBUAUBRZCDUHUKZRZIWRWMXBGUB - WRFHXABWPHXAQAWLWPXAHHWOUIULUJUMUNWRWTDUOUKZXDFGWOHBCXFDGJXFUPZUQZDXDGJXD - UPZURAGUSSZWQADVCSZXJLDGJUTVAZTAWTVDSZWQAHVDSXMMHWSVBVAZTWRAFUCZHSZBXFSZX - OWTSZAWQVEXOHWSVFZNVGAWPWLVHWRWOHVIWRCXFSZIXFSZAYAWQADVNSZDVJSYAAXKYBLDVK - VAZDVLXFDIXGKVMWETWLXTYAVOAWPCIXFVPVQVRAFEVSBCQWQOVTWAWRXEXCIXDRZIWLXEYDQ - AWPCIXCXDWBVQWRYBXCXFSZYDIQAYBWQYCTAYEWQAXFFGWTBXHXLXNAXQFWTXRAXPXQXSNWCW - DWFTXFDXDXCIXGXIKWGWHWIWJWK $. + ccrg cfn eldifi syl2an simprl neldifsnd crg cmnd crngring ringmnd mndidcl + simpl 3syl wb eleq1 ad2antll mpbird weq adantlr gsumunsnd oveq2 ralrimiva + sylan2 gsummptcl ringrz syl2anc eqtrd 3eqtrd rexlimddv ) ACIQZGFHBUAZUBRZ + IQEHPAEUCZHSZWLUDZUDZWNGFHWOUEZUFZWSUGZBUAZUBRGFWTBUAUBRZCDUHUKZRZIWRWMXB + GUBWRFHXABWPHXAQAWLWPXAHHWOUIULUJUMUNWRWTDUOUKZXDFGWOHBCXFDGJXFUPZUQZDXDG + JXDUPZURAGUSSZWQADVCSZXJLDGJUTVAZTAWTVDSZWQAHVDSXMMHWSVBVAZTWRAFUCZHSZBXF + SZXOWTSZAWQVNXOHWSVEZNVFAWPWLVGWRWOHVHWRCXFSZIXFSZAYAWQADVISZDVJSYAAXKYBL + DVKVAZDVLXFDIXGKVMVOTWLXTYAVPAWPCIXFVQVRVSAFEVTBCQWQOWAWBWRXEXCIXDRZIWLXE + YDQAWPCIXCXDWCVRWRYBXCXFSZYDIQAYBWQYCTAYEWQAXFFGWTBXHXLXNAXQFWTXRAXPXQXSN + WEWDWFTXFDXDXCIXGXIKWGWHWIWJWK $. $} ${ @@ -228222,38 +228439,38 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of ( ( R gsum ( x e. I |-> X ) ) .x. ( R gsum ( y e. J |-> Y ) ) ) = ( R gsum ( x e. I , y e. J |-> ( X .x. Y ) ) ) ) $= ( vi vj co cmpt2 cgsu cmpt cfv ccnv cvv csn cdif cima cxp crg wcel ccmn - cv rngcmn syl adantr wa wf eqid fmptd simpl ffvelrn simpr rngcl syl3anc - syl2an cfn xpfi syl2anc wbr wn wceq wo ianor brxp xchnxbir simprl eldif - biimpri sylan wss ssid a1i suppssrOLD syldan oveq1d rnglz simprr oveq2d - eqtrd rngrz jaodan sylan2b anasss gsum2d2 nffvmpt1 nfcv fveq2 oveqan12d - nfov weq cbvmpt2 simp2 3adant3 fvmpt2 3adant2 oveq12d mpt2eq3dva syl5eq - w3a simp3 nfmpt mpteq2dv cbvmpt syl6eq mpteq2dva 3eqtr3d cplusg adantlr - 3expa gsummulc2OLD gsumclOLD gsummulc1OLD 3eqtrrd ) AEBCGHKLFUFZUGZUHUF - ZEBGECHYLUIZUHUFZUIZUHUFZEBGKECHLUIZUHUFZFUFZUIZUHUFEBGKUIZUHUFYTFUFAEU - DUEGHUDUTZUUCUJZUEUTZYSUJZFUFZUGZUHUFEUDGEUEHUUHUIZUHUFZUIZUHUFYNYRAGDH - UUCUKULMUMUNZUOZYSUKUUMUOZUPZUDUEEIJUUHMNPAEUQURZEUSURSEVAVBZQAHJURZUUD - GURZRVCAUUTUUFHURZVDZVDZUUQUUEDURZUUGDURZUUHDURAUUQUVBSVCZAGDUUCVEZUUTU - VDUVBABGKDUUCTUUCVFZVGZUUTUVAVHGDUUDUUCVIVMZAHDYSVEZUVAUVEUVBACHLDYSUAY - SVFZVGZUUTUVAVJHDUUFYSVIVMZDEFUUEUUGNOVKVLAUUNVNURUUOVNURZUUPVNURUBUCUU - NUUOVOVPAUVBUUDUUFUUPVQZVRZUUHMVSZUVQUVCUUDUUNURZVRZUUFUUOURZVRZVTZUVRU - VSUWAVDUWCUVPUVSUWAWAUUDUUFUUNUUOWBWCUVCUVTUVRUWBUVCUVTVDZUUHMUUGFUFZMU - WDUUEMUUGFUVCUVTUUDGUUNUNURZUUEMVSUVCUUTUVTUWFAUUTUVAWDUWFUUTUVTVDUUDGU - UNWEWFWGUVCGDUUCUUNUUDMAUVGUVBUVIVCUUNUUNWHUVCUUNWIWJWKWLWMUVCUWEMVSZUV - TUVCUUQUVEUWGUVFUVNDEFUUGMNOPWNVPVCWQUVCUWBVDZUUHUUEMFUFZMUWHUUGMUUEFUV - CUWBUUFHUUOUNURZUUGMVSUVCUVAUWBUWJAUUTUVAWOUWJUVAUWBVDUUFHUUOWEWFWGUVCH - DYSUUOUUFMAUVKUVBUVMVCUUOUUOWHUVCUUOWIWJWKWLWPUVCUWIMVSZUWBUVCUUQUVDUWK - UVFUVJDEFUUEMNOPWRVPVCWQWSWTXAXBAUUIYMEUHAUUIBCGHBUTZUUCUJZCUTZYSUJZFUF - ZUGYMUDUEBCGHUUHUWPBUUEUUGFBGKUUDXCBFXDBUUGXDXGZCUUEUUGFCUUEXDCFXDZCHLU - UFXCZXGUDUWPXDUEUWPXDZUDBXHZUECXHZUUEUWMUUGUWOFUUDUWLUUCXEZUUFUWNYSXEZX - FXIABCGHUWPYLAUWLGURZUWNHURZXQZUWMKUWOLFUXGUXEKDURZUWMKVSAUXEUXFXJAUXEU - XHUXFTXKBGKDUUCUVHXLVPUXGUXFLDURZUWOLVSAUXEUXFXRAUXFUXIUXEUAXMCHLDYSUVL - XLVPXNZXOXPWPAUULYQEUHAUULBGECHUWPUIZUHUFZUIYQUDBGUUKUXLBEUUJUHBEXDBUHX - DBUEHUUHBHXDUWQXSXGUDUXLXDUXAUUJUXKEUHUXAUUJUEHUWMUUGFUFZUIUXKUXAUEHUUH - UXMUXAUUEUWMUUGFUXCWMXTUECHUXMUWPCUWMUUGFCUWMXDUWRUWSXGUWTUXBUUGUWOUWMF - UXDWPYAYBWPYAABGUXLYPAUXEVDZUXKYOEUHUXNCHUWPYLAUXEUXFUWPYLVSUXJYGYCWPYC - XPWPYDAYQUUBEUHABGYPUUAUXNHDEYEUJZEFCJLKMNPUXOVFZOAUUQUXESVCAUUSUXERVCT - AUXFUXIUXEUAYFAUVOUXEUCVCYHYCWPAGDUXOEFBIKYTMNPUXPOSQAHDYSEJMNPUURRUVMU - CYITUBYJYK $. + cv ringcmn syl adantr wa eqid fmptd simpl ffvelrn syl2an ringcl syl3anc + wf simpr cfn xpfi syl2anc wn wceq wo ianor brxp xchnxbir simprl biimpri + wbr eldif sylan wss suppssrOLD syldan oveq1d ringlz eqtrd simprr oveq2d + ssid a1i ringrz jaodan sylan2b gsum2d2 nffvmpt1 nfcv nfov weq oveqan12d + anasss fveq2 cbvmpt2 w3a simp2 3adant3 simp3 3adant2 oveq12d mpt2eq3dva + fvmpt2 syl5eq nfmpt mpteq2dv cbvmpt syl6eq 3expa 3eqtr3d cplusg adantlr + mpteq2dva gsummulc2OLD gsumclOLD gsummulc1OLD 3eqtrrd ) AEBCGHKLFUFZUGZ + UHUFZEBGECHYLUIZUHUFZUIZUHUFZEBGKECHLUIZUHUFZFUFZUIZUHUFEBGKUIZUHUFYTFU + FAEUDUEGHUDUTZUUCUJZUEUTZYSUJZFUFZUGZUHUFEUDGEUEHUUHUIZUHUFZUIZUHUFYNYR + AGDHUUCUKULMUMUNZUOZYSUKUUMUOZUPZUDUEEIJUUHMNPAEUQURZEUSURSEVAVBZQAHJUR + ZUUDGURZRVCAUUTUUFHURZVDZVDZUUQUUEDURZUUGDURZUUHDURAUUQUVBSVCZAGDUUCVLZ + UUTUVDUVBABGKDUUCTUUCVEZVFZUUTUVAVGGDUUDUUCVHVIZAHDYSVLZUVAUVEUVBACHLDY + SUAYSVEZVFZUUTUVAVMHDUUFYSVHVIZDEFUUEUUGNOVJVKAUUNVNURUUOVNURZUUPVNURUB + UCUUNUUOVOVPAUVBUUDUUFUUPWEZVQZUUHMVRZUVQUVCUUDUUNURZVQZUUFUUOURZVQZVSZ + UVRUVSUWAVDUWCUVPUVSUWAVTUUDUUFUUNUUOWAWBUVCUVTUVRUWBUVCUVTVDZUUHMUUGFU + FZMUWDUUEMUUGFUVCUVTUUDGUUNUNURZUUEMVRUVCUUTUVTUWFAUUTUVAWCUWFUUTUVTVDU + UDGUUNWFWDWGUVCGDUUCUUNUUDMAUVGUVBUVIVCUUNUUNWHUVCUUNWPWQWIWJWKUVCUWEMV + RZUVTUVCUUQUVEUWGUVFUVNDEFUUGMNOPWLVPVCWMUVCUWBVDZUUHUUEMFUFZMUWHUUGMUU + EFUVCUWBUUFHUUOUNURZUUGMVRUVCUVAUWBUWJAUUTUVAWNUWJUVAUWBVDUUFHUUOWFWDWG + UVCHDYSUUOUUFMAUVKUVBUVMVCUUOUUOWHUVCUUOWPWQWIWJWOUVCUWIMVRZUWBUVCUUQUV + DUWKUVFUVJDEFUUEMNOPWRVPVCWMWSWTXGXAAUUIYMEUHAUUIBCGHBUTZUUCUJZCUTZYSUJ + ZFUFZUGYMUDUEBCGHUUHUWPBUUEUUGFBGKUUDXBBFXCBUUGXCXDZCUUEUUGFCUUEXCCFXCZ + CHLUUFXBZXDUDUWPXCUEUWPXCZUDBXEZUECXEZUUEUWMUUGUWOFUUDUWLUUCXHZUUFUWNYS + XHZXFXIABCGHUWPYLAUWLGURZUWNHURZXJZUWMKUWOLFUXGUXEKDURZUWMKVRAUXEUXFXKA + UXEUXHUXFTXLBGKDUUCUVHXQVPUXGUXFLDURZUWOLVRAUXEUXFXMAUXFUXIUXEUAXNCHLDY + SUVLXQVPXOZXPXRWOAUULYQEUHAUULBGECHUWPUIZUHUFZUIYQUDBGUUKUXLBEUUJUHBEXC + BUHXCBUEHUUHBHXCUWQXSXDUDUXLXCUXAUUJUXKEUHUXAUUJUEHUWMUUGFUFZUIUXKUXAUE + HUUHUXMUXAUUEUWMUUGFUXCWKXTUECHUXMUWPCUWMUUGFCUWMXCUWRUWSXDUWTUXBUUGUWO + UWMFUXDWOYAYBWOYAABGUXLYPAUXEVDZUXKYOEUHUXNCHUWPYLAUXEUXFUWPYLVRUXJYCYG + WOYGXRWOYDAYQUUBEUHABGYPUUAUXNHDEYEUJZEFCJLKMNPUXOVEZOAUUQUXESVCAUUSUXE + RVCTAUXFUXIUXEUAYFAUVOUXEUCVCYHYGWOAGDUXOEFBIKYTMNPUXPOSQAHDYSEJMNPUURR + UVMUCYITUBYJYK $. $} gsumdixp.xf $e |- ( ph -> ( x e. I |-> X ) finSupp .0. ) $. @@ -228264,39 +228481,39 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of gsumdixp $p |- ( ph -> ( ( R gsum ( x e. I |-> X ) ) .x. ( R gsum ( y e. J |-> Y ) ) ) = ( R gsum ( x e. I , y e. J |-> ( X .x. Y ) ) ) ) $= - ( vi vj co cmpt2 cgsu cmpt cv cfv csupp cxp crg wcel rngcmn syl adantr wa - ccmn wf eqid fmptd simpl ffvelrn syl2an simpr rngcl syl3anc cfn fsuppimpd - xpfi syl2anc wbr wn wceq wo ianor brxp xchnxbir cdif simprl eldif biimpri - sylan cvv wss ssid a1i c0g fvex eqeltri suppssr syldan oveq1d rnglz eqtrd - simprr oveq2d rngrz jaodan sylan2b anasss gsum2d2 nffvmpt1 nfcv weq fveq2 - nfov oveqan12d cbvmpt2 w3a simp2 3adant3 simp3 3adant2 oveq12d mpt2eq3dva - fvmpt2 syl5eq nfmpt mpteq2dv cbvmpt syl6eq 3expa mpteq2dva 3eqtr3d cplusg - adantlr cfsupp gsummulc2 gsumcl gsummulc1 3eqtrrd ) AEBCGHKLFUFZUGZUHUFZE - BGECHYOUIZUHUFZUIZUHUFZEBGKECHLUIZUHUFZFUFZUIZUHUFEBGKUIZUHUFUUCFUFAEUDUE - GHUDUJZUUFUKZUEUJZUUBUKZFUFZUGZUHUFEUDGEUEHUUKUIZUHUFZUIZUHUFYQUUAAGDHUUF - MULUFZUUBMULUFZUMZUDUEEIJUUKMNPAEUNUOZEUTUOSEUPUQZQAHJUOZUUGGUOZRURAUVBUU - IHUOZUSZUSZUUSUUHDUOZUUJDUOZUUKDUOAUUSUVDSURZAGDUUFVAZUVBUVFUVDABGKDUUFTU - UFVBZVCZUVBUVCVDGDUUGUUFVEVFZAHDUUBVAZUVCUVGUVDACHLDUUBUAUUBVBZVCZUVBUVCV - GHDUUIUUBVEVFZDEFUUHUUJNOVHVIAUUPVJUOUUQVJUOUURVJUOAUUFMUBVKAUUBMUCVKUUPU - UQVLVMAUVDUUGUUIUURVNZVOZUUKMVPZUVRUVEUUGUUPUOZVOZUUIUUQUOZVOZVQZUVSUVTUW - BUSUWDUVQUVTUWBVRUUGUUIUUPUUQVSVTUVEUWAUVSUWCUVEUWAUSZUUKMUUJFUFZMUWEUUHM - UUJFUVEUWAUUGGUUPWAUOZUUHMVPUVEUVBUWAUWGAUVBUVCWBUWGUVBUWAUSUUGGUUPWCWDWE - UVEGDWFUUFIUUPUUGMAUVIUVDUVKURUUPUUPWGUVEUUPWHWIAGIUOUVDQURMWFUOUVEMEWJUK - WFPEWJWKWLWIZWMWNWOUVEUWFMVPZUWAUVEUUSUVGUWIUVHUVPDEFUUJMNOPWPVMURWQUVEUW - CUSZUUKUUHMFUFZMUWJUUJMUUHFUVEUWCUUIHUUQWAUOZUUJMVPUVEUVCUWCUWLAUVBUVCWRU - WLUVCUWCUSUUIHUUQWCWDWEUVEHDWFUUBJUUQUUIMAUVMUVDUVOURUUQUUQWGUVEUUQWHWIAU - VAUVDRURUWHWMWNWSUVEUWKMVPZUWCUVEUUSUVFUWMUVHUVLDEFUUHMNOPWTVMURWQXAXBXCX - DAUULYPEUHAUULBCGHBUJZUUFUKZCUJZUUBUKZFUFZUGYPUDUEBCGHUUKUWRBUUHUUJFBGKUU - GXEBFXFBUUJXFXIZCUUHUUJFCUUHXFCFXFZCHLUUIXEZXIUDUWRXFUEUWRXFZUDBXGZUECXGZ - UUHUWOUUJUWQFUUGUWNUUFXHZUUIUWPUUBXHZXJXKABCGHUWRYOAUWNGUOZUWPHUOZXLZUWOK - UWQLFUXIUXGKDUOZUWOKVPAUXGUXHXMAUXGUXJUXHTXNBGKDUUFUVJXSVMUXIUXHLDUOZUWQL - VPAUXGUXHXOAUXHUXKUXGUAXPCHLDUUBUVNXSVMXQZXRXTWSAUUOYTEUHAUUOBGECHUWRUIZU - HUFZUIYTUDBGUUNUXNBEUUMUHBEXFBUHXFBUEHUUKBHXFUWSYAXIUDUXNXFUXCUUMUXMEUHUX - CUUMUEHUWOUUJFUFZUIUXMUXCUEHUUKUXOUXCUUHUWOUUJFUXEWOYBUECHUXOUWRCUWOUUJFC - UWOXFUWTUXAXIUXBUXDUUJUWQUWOFUXFWSYCYDWSYCABGUXNYSAUXGUSZUXMYREUHUXPCHUWR - YOAUXGUXHUWRYOVPUXLYEYFWSYFXTWSYGAYTUUEEUHABGYSUUDUXPHDEYHUKZEFCJLKMNPUXQ - VBZOAUUSUXGSURAUVAUXGRURTAUXHUXKUXGUAYIAUUBMYJVNUXGUCURYKYFWSAGDUXQEFBIKU - UCMNPUXROSQAHDUUBEJMNPUUTRUVOUCYLTUBYMYN $. + ( vi vj co cmpt2 cgsu cmpt cfv csupp cxp crg wcel ccmn ringcmn syl adantr + cv wa eqid fmptd simpl ffvelrn syl2an simpr ringcl syl3anc fsuppimpd xpfi + wf cfn syl2anc wbr wn wceq wo ianor brxp xchnxbir cdif simprl eldif sylan + biimpri cvv wss ssid a1i fvex eqeltri suppssr syldan oveq1d ringlz simprr + c0g eqtrd oveq2d ringrz jaodan sylan2b anasss gsum2d2 nffvmpt1 nfcv fveq2 + nfov weq oveqan12d cbvmpt2 w3a simp2 3adant3 fvmpt2 simp3 3adant2 oveq12d + mpt2eq3dva syl5eq mpteq2dv cbvmpt syl6eq mpteq2dva 3eqtr3d cplusg adantlr + nfmpt 3expa cfsupp gsummulc2 gsumcl gsummulc1 3eqtrrd ) AEBCGHKLFUFZUGZUH + UFZEBGECHYOUIZUHUFZUIZUHUFZEBGKECHLUIZUHUFZFUFZUIZUHUFEBGKUIZUHUFUUCFUFAE + UDUEGHUDUSZUUFUJZUEUSZUUBUJZFUFZUGZUHUFEUDGEUEHUUKUIZUHUFZUIZUHUFYQUUAAGD + HUUFMUKUFZUUBMUKUFZULZUDUEEIJUUKMNPAEUMUNZEUOUNSEUPUQZQAHJUNZUUGGUNZRURAU + VBUUIHUNZUTZUTZUUSUUHDUNZUUJDUNZUUKDUNAUUSUVDSURZAGDUUFVKZUVBUVFUVDABGKDU + UFTUUFVAZVBZUVBUVCVCGDUUGUUFVDVEZAHDUUBVKZUVCUVGUVDACHLDUUBUAUUBVAZVBZUVB + UVCVFHDUUIUUBVDVEZDEFUUHUUJNOVGVHAUUPVLUNUUQVLUNUURVLUNAUUFMUBVIAUUBMUCVI + UUPUUQVJVMAUVDUUGUUIUURVNZVOZUUKMVPZUVRUVEUUGUUPUNZVOZUUIUUQUNZVOZVQZUVSU + VTUWBUTUWDUVQUVTUWBVRUUGUUIUUPUUQVSVTUVEUWAUVSUWCUVEUWAUTZUUKMUUJFUFZMUWE + UUHMUUJFUVEUWAUUGGUUPWAUNZUUHMVPUVEUVBUWAUWGAUVBUVCWBUWGUVBUWAUTUUGGUUPWC + WEWDUVEGDWFUUFIUUPUUGMAUVIUVDUVKURUUPUUPWGUVEUUPWHWIAGIUNUVDQURMWFUNUVEME + WQUJWFPEWQWJWKWIZWLWMWNUVEUWFMVPZUWAUVEUUSUVGUWIUVHUVPDEFUUJMNOPWOVMURWRU + VEUWCUTZUUKUUHMFUFZMUWJUUJMUUHFUVEUWCUUIHUUQWAUNZUUJMVPUVEUVCUWCUWLAUVBUV + CWPUWLUVCUWCUTUUIHUUQWCWEWDUVEHDWFUUBJUUQUUIMAUVMUVDUVOURUUQUUQWGUVEUUQWH + WIAUVAUVDRURUWHWLWMWSUVEUWKMVPZUWCUVEUUSUVFUWMUVHUVLDEFUUHMNOPWTVMURWRXAX + BXCXDAUULYPEUHAUULBCGHBUSZUUFUJZCUSZUUBUJZFUFZUGYPUDUEBCGHUUKUWRBUUHUUJFB + GKUUGXEBFXFBUUJXFXHZCUUHUUJFCUUHXFCFXFZCHLUUIXEZXHUDUWRXFUEUWRXFZUDBXIZUE + CXIZUUHUWOUUJUWQFUUGUWNUUFXGZUUIUWPUUBXGZXJXKABCGHUWRYOAUWNGUNZUWPHUNZXLZ + UWOKUWQLFUXIUXGKDUNZUWOKVPAUXGUXHXMAUXGUXJUXHTXNBGKDUUFUVJXOVMUXIUXHLDUNZ + UWQLVPAUXGUXHXPAUXHUXKUXGUAXQCHLDUUBUVNXOVMXRZXSXTWSAUUOYTEUHAUUOBGECHUWR + UIZUHUFZUIYTUDBGUUNUXNBEUUMUHBEXFBUHXFBUEHUUKBHXFUWSYHXHUDUXNXFUXCUUMUXME + UHUXCUUMUEHUWOUUJFUFZUIUXMUXCUEHUUKUXOUXCUUHUWOUUJFUXEWNYAUECHUXOUWRCUWOU + UJFCUWOXFUWTUXAXHUXBUXDUUJUWQUWOFUXFWSYBYCWSYBABGUXNYSAUXGUTZUXMYREUHUXPC + HUWRYOAUXGUXHUWRYOVPUXLYIYDWSYDXTWSYEAYTUUEEUHABGYSUUDUXPHDEYFUJZEFCJLKMN + PUXQVAZOAUUSUXGSURAUVAUXGRURTAUXHUXKUXGUAYGAUUBMYJVNUXGUCURYKYDWSAGDUXQEF + BIKUUCMNPUXROSQAHDUUBEJMNPUUTRUVOUCYLTUBYMYN $. $} ${ @@ -228343,56 +228560,56 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of by Mario Carneiro, 11-Mar-2015.) $) prdsmulrcl $p |- ( ph -> ( F .x. G ) e. B ) $= ( wcel vx co cv cfv cmulr cmpt crg wf wfn ffn syl prdsmulrval cbs wral wa - ffvelrnda adantr simpr prdsbasprj eqid rngcl syl3anc ralrimiva prdsbasmpt - mpbird eqeltrd ) AFGEUBUAHUAUCZFUDZVGGUDZVGCUDZUEUDZUBZUFZBAUABCDEFGHIJKL - MOPAHUGCUHCHUIZQHUGCUJUKZRSNULAVMBTVLVJUMUDZTZUAHUNAVQUAHAVGHTZUOZVJUGTVH - VPTVIVPTVQAHUGVGCQUPVSBCDFHVGIJKLMADITVROUQZAHJTVRPUQZAVNVRVOUQZAFBTVRRUQ - AVRURZUSVSBCDGHVGIJKLMVTWAWBAGBTVRSUQWCUSVPVJVKVHVIVPUTVKUTVAVBVCAUABCDVL - HIJKLMOPVOVDVEVF $. + ffvelrnda adantr simpr prdsbasprj eqid ringcl ralrimiva prdsbasmpt mpbird + syl3anc eqeltrd ) AFGEUBUAHUAUCZFUDZVGGUDZVGCUDZUEUDZUBZUFZBAUABCDEFGHIJK + LMOPAHUGCUHCHUIZQHUGCUJUKZRSNULAVMBTVLVJUMUDZTZUAHUNAVQUAHAVGHTZUOZVJUGTV + HVPTVIVPTVQAHUGVGCQUPVSBCDFHVGIJKLMADITVROUQZAHJTVRPUQZAVNVRVOUQZAFBTVRRU + QAVRURZUSVSBCDGHVGIJKLMVTWAWBAGBTVRSUQWCUSVPVJVKVHVIVPUTVKUTVAVEVBAUABCDV + LHIJKLMOPVOVCVDVF $. $} ${ $d w I $. $d w x y z ph $. $d w x y R $. $d w x y S $. $d w x y z Y $. $d w V $. $d w W $. - prdsrngd.y $e |- Y = ( S Xs_ R ) $. - prdsrngd.i $e |- ( ph -> I e. W ) $. - prdsrngd.s $e |- ( ph -> S e. V ) $. - prdsrngd.r $e |- ( ph -> R : I --> Ring ) $. + prdsringd.y $e |- Y = ( S Xs_ R ) $. + prdsringd.i $e |- ( ph -> I e. W ) $. + prdsringd.s $e |- ( ph -> S e. V ) $. + prdsringd.r $e |- ( ph -> R : I --> Ring ) $. $( A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.) $) - prdsrngd $p |- ( ph -> Y e. Ring ) $= + prdsringd $p |- ( ph -> Y e. Ring ) $= ( vx vw wcel cfv cmnd co crg eqid adantr vy vz cgrp cmgp cv cmulr wceq wa - cplusg cbs wral wf wss rnggrp ssriv sylancl prdsgrpd ccom cprds cres mgpf - fss fco2 sylancr prdsmndd wfn ffn prdsmgp simpld simprd proplem3 mndpropd - eqidd syl mpbird w3a cmpt ffvelrnda simplr1 simpr prdsbasprj simpr2 rngdi - simpr3 prdsplusgfval oveq2d prdsmulrfval oveq12d 3eqtr4d mpteq2dva simpr1 - rngmnd prdsplusgcl prdsmulrval prdsmulrcl prdsplusgval rngdir ralrimivvva - syl13anc oveq1d jca isrng syl3anbrc ) AGUCNGUDOZPNZLUEZUAUEZUBUEZGUIOZQZG - UFOZQZXFXGXKQZXFXHXKQZXIQZUGZXFXGXIQZXHXKQZXNXGXHXKQZXIQZUGZUHZUBGUJOZUKU - AYCUKLYCUKGRNABCDEFGHIJADRBULZRUCUMDUCBULKLRUCXFUNUODRUCBVBUPUQAXECUDBURZ - USQZPNAYECDEFYFYFSZIJARPUDRUTULYDDPYEULVAKDRPUDBVCVDVEALUAXDUJOZXDYFAYHVM - AYHYFUJOUGZXDUIOZYFUIOZUGZABCDXDFEGYFHXDSZYGIJAYDBDVFZKDRBVGVNZVHZVIAXFYH - NXGYHNUHLUAYJYKAYIYLYPVJVKVLVOAYBLUAUBYCYCYCAXFYCNZXGYCNZXHYCNZVPZUHZXPYA - UUAMDMUEZXFOZUUBXJOZUUBBOZUFOZQZVQMDUUBXMOZUUBXNOZUUEUIOZQZVQXLXOUUAMDUUG - UUKUUAUUBDNZUHZUUCUUBXGOZUUBXHOZUUJQZUUFQZUUCUUNUUFQZUUCUUOUUFQZUUJQZUUGU - UKUUMUUERNZUUCUUEUJOZNZUUNUVBNZUUOUVBNZUUQUUTUGUUADRUUBBAYDYTKTZVRZUUMYCB - CXFDUUBEFGHYCSZUUACENZUULAUVIYTJTZTZUUADFNZUULAUVLYTITZTZUUAYNUULAYNYTYOT - ZTZYQYRYSAUULVSZUUAUULVTZWAZUUMYCBCXGDUUBEFGHUVHUVKUVNUVPUUAYRUULAYQYRYSW - BZTZUVRWAZUUMYCBCXHDUUBEFGHUVHUVKUVNUVPUUAYSUULAYQYRYSWDZTZUVRWAZUVBUUJUU - EUUFUUCUUNUUOUVBSZUUJSZUUFSZWCWSUUMUUDUUPUUCUUFUUMYCXIBCXGXHDUUBEFGHUVHUV - KUVNUVPUWAUWDXISZUVRWEWFUUMUUHUURUUIUUSUUJUUMYCBCXKXFXGDUUBEFGHUVHUVKUVNU - VPUVQUWAXKSZUVRWGUUMYCBCXKXFXHDUUBEFGHUVHUVKUVNUVPUVQUWDUWJUVRWGZWHWIWJUU - AMYCBCXKXFXJDEFGHUVHUVJUVMUVOAYQYRYSWKZUUAYCXIBCXGXHDEFGHUVHUWIUVJUVMADPB - ULZYTAYDRPUMUWMKLRPXFWLUODRPBVBUPTZUVTUWCWMUWJWNUUAMYCXIBCXMXNDEFGHUVHUVJ - UVMUVOUUAYCBCXKXFXGDEFGHUVHUWJUVJUVMUVFUWLUVTWOUUAYCBCXKXFXHDEFGHUVHUWJUV - JUVMUVFUWLUWCWOZUWIWPWIUUAMDUUBXQOZUUOUUFQZVQMDUUIUUBXSOZUUJQZVQXRXTUUAMD - UWQUWSUUMUUCUUNUUJQZUUOUUFQZUUSUUNUUOUUFQZUUJQZUWQUWSUUMUVAUVCUVDUVEUXAUX - CUGUVGUVSUWBUWEUVBUUJUUEUUFUUCUUNUUOUWFUWGUWHWQWSUUMUWPUWTUUOUUFUUMYCXIBC - XFXGDUUBEFGHUVHUVKUVNUVPUVQUWAUWIUVRWEWTUUMUUIUUSUWRUXBUUJUWKUUMYCBCXKXGX - HDUUBEFGHUVHUVKUVNUVPUWAUWDUWJUVRWGWHWIWJUUAMYCBCXKXQXHDEFGHUVHUVJUVMUVOU - UAYCXIBCXFXGDEFGHUVHUWIUVJUVMUWNUWLUVTWMUWCUWJWNUUAMYCXIBCXNXSDEFGHUVHUVJ - UVMUVOUWOUUAYCBCXKXGXHDEFGHUVHUWJUVJUVMUVFUVTUWCWOUWIWPWIXAWRLUAUBYCXIGXK - XDUVHYMUWIUWJXBXC $. + cplusg cbs wral wf wss ringgrp ssriv fss sylancl prdsgrpd ccom cprds cres + mgpf fco2 sylancr prdsmndd eqidd wfn ffn prdsmgp simpld proplem3 mndpropd + syl simprd mpbird ffvelrnda simplr1 simpr prdsbasprj simpr2 simpr3 ringdi + cmpt syl13anc prdsplusgfval oveq2d prdsmulrfval oveq12d 3eqtr4d mpteq2dva + w3a simpr1 ringmnd prdsplusgcl prdsmulrval prdsmulrcl prdsplusgval oveq1d + ringdir jca ralrimivvva isring syl3anbrc ) AGUCNGUDOZPNZLUEZUAUEZUBUEZGUI + OZQZGUFOZQZXFXGXKQZXFXHXKQZXIQZUGZXFXGXIQZXHXKQZXNXGXHXKQZXIQZUGZUHZUBGUJ + OZUKUAYCUKLYCUKGRNABCDEFGHIJADRBULZRUCUMDUCBULKLRUCXFUNUODRUCBUPUQURAXECU + DBUSZUTQZPNAYECDEFYFYFSZIJARPUDRVAULYDDPYEULVBKDRPUDBVCVDVEALUAXDUJOZXDYF + AYHVFAYHYFUJOUGZXDUIOZYFUIOZUGZABCDXDFEGYFHXDSZYGIJAYDBDVGZKDRBVHVMZVIZVJ + AXFYHNXGYHNUHLUAYJYKAYIYLYPVNVKVLVOAYBLUAUBYCYCYCAXFYCNZXGYCNZXHYCNZWKZUH + ZXPYAUUAMDMUEZXFOZUUBXJOZUUBBOZUFOZQZWCMDUUBXMOZUUBXNOZUUEUIOZQZWCXLXOUUA + MDUUGUUKUUAUUBDNZUHZUUCUUBXGOZUUBXHOZUUJQZUUFQZUUCUUNUUFQZUUCUUOUUFQZUUJQ + ZUUGUUKUUMUUERNZUUCUUEUJOZNZUUNUVBNZUUOUVBNZUUQUUTUGUUADRUUBBAYDYTKTZVPZU + UMYCBCXFDUUBEFGHYCSZUUACENZUULAUVIYTJTZTZUUADFNZUULAUVLYTITZTZUUAYNUULAYN + YTYOTZTZYQYRYSAUULVQZUUAUULVRZVSZUUMYCBCXGDUUBEFGHUVHUVKUVNUVPUUAYRUULAYQ + YRYSVTZTZUVRVSZUUMYCBCXHDUUBEFGHUVHUVKUVNUVPUUAYSUULAYQYRYSWAZTZUVRVSZUVB + UUJUUEUUFUUCUUNUUOUVBSZUUJSZUUFSZWBWDUUMUUDUUPUUCUUFUUMYCXIBCXGXHDUUBEFGH + UVHUVKUVNUVPUWAUWDXISZUVRWEWFUUMUUHUURUUIUUSUUJUUMYCBCXKXFXGDUUBEFGHUVHUV + KUVNUVPUVQUWAXKSZUVRWGUUMYCBCXKXFXHDUUBEFGHUVHUVKUVNUVPUVQUWDUWJUVRWGZWHW + IWJUUAMYCBCXKXFXJDEFGHUVHUVJUVMUVOAYQYRYSWLZUUAYCXIBCXGXHDEFGHUVHUWIUVJUV + MADPBULZYTAYDRPUMUWMKLRPXFWMUODRPBUPUQTZUVTUWCWNUWJWOUUAMYCXIBCXMXNDEFGHU + VHUVJUVMUVOUUAYCBCXKXFXGDEFGHUVHUWJUVJUVMUVFUWLUVTWPUUAYCBCXKXFXHDEFGHUVH + UWJUVJUVMUVFUWLUWCWPZUWIWQWIUUAMDUUBXQOZUUOUUFQZWCMDUUIUUBXSOZUUJQZWCXRXT + UUAMDUWQUWSUUMUUCUUNUUJQZUUOUUFQZUUSUUNUUOUUFQZUUJQZUWQUWSUUMUVAUVCUVDUVE + UXAUXCUGUVGUVSUWBUWEUVBUUJUUEUUFUUCUUNUUOUWFUWGUWHWSWDUUMUWPUWTUUOUUFUUMY + CXIBCXFXGDUUBEFGHUVHUVKUVNUVPUVQUWAUWIUVRWEWRUUMUUIUUSUWRUXBUUJUWKUUMYCBC + XKXGXHDUUBEFGHUVHUVKUVNUVPUWAUWDUWJUVRWGWHWIWJUUAMYCBCXKXQXHDEFGHUVHUVJUV + MUVOUUAYCXIBCXFXGDEFGHUVHUWIUVJUVMUWNUWLUVTWNUWCUWJWOUUAMYCXIBCXNXSDEFGHU + VHUVJUVMUVOUWOUUAYCBCXKXGXHDEFGHUVHUWJUVJUVMUVFUVTUWCWPUWIWQWIWTXALUAUBYC + XIGXKXDUVHYMUWIUWJXBXC $. $} ${ @@ -228406,15 +228623,15 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.) $) prdscrngd $p |- ( ph -> Y e. CRing ) $= - ( vx vy crg wcel cmgp cfv ccmn ccrg wf wss crngrng ssriv sylancl prdsrngd - cv fss ccom cprds co eqid cres wfn wral cvv fnmgp ssv fnssres mp2an fvres - crngmgp eqeltrd rgen ffnfv mpbir2an fco2 sylancr prdscmnd cbs wceq cplusg - ffn syl prdsmgp simpld wa simprd proplem3 cmnpropd mpbird iscrng sylanbrc - eqidd ) AGNOGPQZROZGSOABCDEFGHIJADSBTZSNUADNBTKLSNLUFZUBUCDSNBUGUDUEAWECP - BUHZUIUJZROAWHCDEFWIWIUKZIJASRPSULZTZWFDRWHTWLWKSUMZWGWKQZROZLSUNPUOUMSUO - UAWMUPSUQUOSPURUSWOLSWGSOWNWGPQZRWGSPUTWGWPWPUKVAVBVCLSRWKVDVEKDSRPBVFVGV - HALMWDVIQZWDWIAWQWCAWQWIVIQVJZWDVKQZWIVKQZVJZABCDWDFEGWIHWDUKZWJIJAWFBDUM - KDSBVLVMVNZVOAWGWQOMUFWQOVPLMWSWTAWRXAXCVQVRVSVTGWDXBWAWB $. + ( vx vy crg wcel cmgp cfv ccmn ccrg wf wss crngring fss sylancl prdsringd + cv ssriv ccom cprds co eqid cres wfn wral cvv fnmgp fnssres mp2an crngmgp + fvres eqeltrd rgen ffnfv mpbir2an fco2 sylancr prdscmnd eqidd wceq cplusg + ssv cbs ffn syl simpld wa simprd proplem3 cmnpropd mpbird iscrng sylanbrc + prdsmgp ) AGNOGPQZROZGSOABCDEFGHIJADSBTZSNUADNBTKLSNLUFZUBUGDSNBUCUDUEAWE + CPBUHZUIUJZROAWHCDEFWIWIUKZIJASRPSULZTZWFDRWHTWLWKSUMZWGWKQZROZLSUNPUOUMS + UOUAWMUPSVKUOSPUQURWOLSWGSOWNWGPQZRWGSPUTWGWPWPUKUSVAVBLSRWKVCVDKDSRPBVEV + FVGALMWDVLQZWDWIAWQVHAWQWIVLQVIZWDVJQZWIVJQZVIZABCDWDFEGWIHWDUKZWJIJAWFBD + UMKDSBVMVNWCZVOAWGWQOMUFWQOVPLMWSWTAWRXAXCVQVRVSVTGWDXBWAWB $. $} ${ @@ -228436,13 +228653,13 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $} ${ - pwsrng.y $e |- Y = ( R ^s I ) $. + pwsring.y $e |- Y = ( R ^s I ) $. $( A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.) $) - pwsrng $p |- ( ( R e. Ring /\ I e. V ) -> Y e. Ring ) $= + pwsring $p |- ( ( R e. Ring /\ I e. V ) -> Y e. Ring ) $= ( crg wcel wa csca cfv csn cxp cprds co eqid pwsval cvv simpr fvex a1i wf - fconst6g adantr prdsrngd eqeltrd ) AFGZBCGZHZDAIJZBAKLZMNZFAUIBFCDEUIOPUH - UJUIBQCUKUKOUFUGRUIQGUHAISTUFBFUJUAUGBAFUBUCUDUE $. + fconst6g adantr prdsringd eqeltrd ) AFGZBCGZHZDAIJZBAKLZMNZFAUIBFCDEUIOPU + HUJUIBQCUKUKOUFUGRUIQGUHAISTUFBFUJUAUGBAFUBUCUDUE $. $} ${ @@ -228500,120 +228717,120 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $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 $. $d a b p q u v x y z V $. $d p q u v .x. $. - imasrng.u $e |- ( ph -> U = ( F "s R ) ) $. - imasrng.v $e |- ( ph -> V = ( Base ` R ) ) $. - imasrng.p $e |- .+ = ( +g ` R ) $. - imasrng.t $e |- .x. = ( .r ` R ) $. - imasrng.o $e |- .1. = ( 1r ` R ) $. - imasrng.f $e |- ( ph -> F : V -onto-> B ) $. - imasrng.e1 $e |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) + imasring.u $e |- ( ph -> U = ( F "s R ) ) $. + imasring.v $e |- ( ph -> V = ( Base ` R ) ) $. + imasring.p $e |- .+ = ( +g ` R ) $. + imasring.t $e |- .x. = ( .r ` R ) $. + imasring.o $e |- .1. = ( 1r ` R ) $. + imasring.f $e |- ( ph -> F : V -onto-> B ) $. + imasring.e1 $e |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) ) $. - imasrng.e2 $e |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) + imasring.e2 $e |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) ) $. - imasrng.r $e |- ( ph -> R e. Ring ) $. + imasring.r $e |- ( ph -> R e. Ring ) $. $( The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) $) - imasrng $p |- ( ph -> ( U e. Ring /\ ( F ` .1. ) = ( 1r ` U ) ) ) $= + imasring $p |- ( ph -> ( U e. Ring /\ ( F ` .1. ) = ( 1r ` U ) ) ) $= ( vu vv vw vx vy vz crg wcel cfv cur wceq cplusg cmulr imasbas eqidd cgrp - c0g a1i rnggrp syl eqid imasgrp simpld cxp wf cv co wa cbs adantr eleqtrd - simprl rngcl syl3anc eleqtrrd caovclg imasmulf fovrn syl3an1 w3a wrex crn - simprr wfo forn eleq2d 3anbi123d wfn wb fvelrnb bitr3d 3reeanv syl6bbr wi - fofn simp2 3ad2ant1 3adant3r3 simp3 simpr3 rngass syl13anc simpl 3adantr3 - fveq2d imasmulval simpr1 3adantr1 3eqtr4d oveq1d 3adant3r1 oveq2d oveq12d - simp1 eqeq12d syl5ibcom 3exp2 imp32 rexlimdv rexlimdvva sylbid imp rngacl - rngdi 3adantr2 imasaddval 3adant3r2 rngdir rngidcl ffvelrnd simpr rnglidm - fof biimpa syl2anc eqtrd oveq2 rexlimdva mpd3an3 rngridm isrngd wral jcad - id oveq1 sylbird ralrimiv isrngid mpbi2and eqcomd jca ) AFUIUJZGHUKZFULUK - ZUMAUCUDUEBFUNUKZFFUOUKZUUOABDFHIUINOSUBUPZAUUQUQAUURUQAFURUJDUSUKZHUKFUS - UKUMABCDFHIUUTJKLMNOCDUNUKUMAPUTSTADUIUJZDURUJUBDVAVBUUTVCVDVEABBVFBUURVG - 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qusring2.t $e |- .x. = ( .r ` R ) $. + qusring2.o $e |- .1. = ( 1r ` R ) $. + qusring2.r $e |- ( ph -> .~ Er V ) $. + qusring2.e1 $e |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) ) $. - qusrng2.e2 $e |- ( ph -> + qusring2.e2 $e |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) ) $. - qusrng2.x $e |- ( ph -> R e. Ring ) $. + qusring2.x $e |- ( ph -> R e. Ring ) $. $( The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) $) - qusrng2 $p |- ( ph -> ( U e. Ring /\ [ .1. ] .~ = ( 1r ` U ) ) ) $= + qusring2 $p |- ( ph -> ( U e. Ring /\ [ .1. ] .~ = ( 1r ` U ) ) ) $= ( vu vx vy crg wcel cec cur cfv wceq wa cv cmpt cqs cvv eqid wer cbs fvex - syl6eqel erex sylc qusval quslem co adantr simprl eleqtrd simprr eleqtrrd - rngacl syl3anc ercpbl rngcl imasrng divsfval eqcomd eqeq1d anbi2d mpbird - ) AFUEUFZGCUGZFUHUIZUJZUKWAGUBHUBULCUGUMZUIZWCUJZUKAHCUNBDEFGWEHIJKLAUBCD - FWEHUOUEMNWEUPZAHCUQHUOUFCUOUFRAHDURUIZUONDURUSUTZHCUOVAVBZUAVCNOPQAUBCDF - WEHUOUEMNWHWKUAVDAUBKULZLULZJULZIULZBCWEHUCUDRWJWHAUCULZHUFZUDULZHUFZUKZU - KZWPWRBVEZWIHXADUEUFZWPWIUFZWRWIUFZXBWIUFAXCWTUAVFZXAWPHWIAWQWSVGAHWIUJWT - NVFZVHZXAWRHWIAWQWSVIXGVHZWIBDWPWRWIUPZOVKVLXGVJSVMAUBWLWMWNWOECWEHUCUDRW - JWHXAWPWREVEZWIHXAXCXDXEXKWIUFXFXHXIWIDEWPWRXJPVNVLXGVJTVMUAVOAWDWGWAAWBW - FWCAWFWBAUBGCWEHRWJWHVPVQVRVSVT $. + syl6eqel erex sylc qusval quslem co adantr simprl eleqtrd ringacl syl3anc + simprr eleqtrrd ercpbl ringcl imasring divsfval eqcomd eqeq1d anbi2d + mpbird ) AFUEUFZGCUGZFUHUIZUJZUKWAGUBHUBULCUGUMZUIZWCUJZUKAHCUNBDEFGWEHIJ + KLAUBCDFWEHUOUEMNWEUPZAHCUQHUOUFCUOUFRAHDURUIZUONDURUSUTZHCUOVAVBZUAVCNOP + QAUBCDFWEHUOUEMNWHWKUAVDAUBKULZLULZJULZIULZBCWEHUCUDRWJWHAUCULZHUFZUDULZH + UFZUKZUKZWPWRBVEZWIHXADUEUFZWPWIUFZWRWIUFZXBWIUFAXCWTUAVFZXAWPHWIAWQWSVGA + HWIUJWTNVFZVHZXAWRHWIAWQWSVKXGVHZWIBDWPWRWIUPZOVIVJXGVLSVMAUBWLWMWNWOECWE + HUCUDRWJWHXAWPWREVEZWIHXAXCXDXEXKWIUFXFXHXIWIDEWPWRXJPVNVJXGVLTVMUAVOAWDW + GWAAWBWFWCAWFWBAUBGCWEHRWJWHVPVQVRVSVT $. $} ${ @@ -228632,11 +228849,11 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of crngbinom $p |- ( ( ( R e. CRing /\ N e. NN0 ) /\ ( A e. S /\ B e. S ) ) -> ( N .^ ( A .+ B ) ) = ( R gsum ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ A ) .X. ( k .^ B ) ) ) ) ) ) $= - ( wcel wa co ccrg cn0 csrg ccmn w3a cc0 cfz cv cbc cmin cmpt cgsu crngrng - wceq crg rngsrg syl adantr crngmgp simpr 3jca csrgbinom sylan ) DUARZKUBR - ZSZDUCRZJUDRZVEUEAERBERSKABCTITDHUFKUGTKHUHZUITKVIUJTAITVIBITGTFTUKULTUNV - FVGVHVEVDVGVEVDDUORVGDUMDUPUQURVDVHVEDJPUSURVDVEUTVAABCDEFGHIJKLMNOPQVBVC - $. + ( wcel wa co ccrg cn0 csrg ccmn w3a cc0 cfz cv cbc cmin cmpt crg crngring + cgsu wceq ringsrg syl adantr crngmgp simpr 3jca csrgbinom sylan ) DUARZKU + BRZSZDUCRZJUDRZVEUEAERBERSKABCTITDHUFKUGTKHUHZUITKVIUJTAITVIBITGTFTUKUNTU + OVFVGVHVEVDVGVEVDDULRVGDUMDUPUQURVDVHVEDJPUSURVDVEUTVAABCDEFGHIJKLMNOPQVB + VC $. $} $( @@ -228732,33 +228949,33 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) $) - opprrng $p |- ( R e. Ring -> O e. Ring ) $= + opprring $p |- ( R e. Ring -> O e. Ring ) $= ( vx vy vz wcel cbs cfv cplusg cmulr wceq eqid a1i cv co opprmul syl13anc - cgrp 3eqtr4g crg cur opprbas oppradd eqidd rnggrp grpprop sylib w3a rngcl - 3com23 syl5eqel wa simpl simpr3 simpr2 simpr1 rngass eqcomd oveq1i oveq2i - eqtri rngdir oveq12i rngdi rngidcl rngridm syl5eq rnglidm isrngd ) AUAGZD - EFAHIZAJIZBBKIZAUBIZVLBHILVKVLABCVLMZUCZNVMBJILVKVMABCVMMZUDZNVKVNUEVKASG - BSGAUFABVQVSUGUHVKDOZVLGZEOZVLGZUIVTWBVNPZWBVTAKIZPZVLVLAVNWEBVTWBVPWEMZC - VNMZQZVKWCWAWFVLGVLAWEWBVTVPWGUJUKULVKWAWCFOZVLGZUIZUMZWJWFWEPZWJWBWEPZVT - WEPZWDWJVNPZVTWBWJVNPZVNPZWMWPWNWMVKWKWCWAWPWNLVKWLUNZVKWAWCWKUOZVKWAWCWK - UPZVKWAWCWKUQZVLAWEWJWBVTVPWGURRUSWQWFWJVNPWNWDWFWJVNWIUTVLAVNWEBWFWJVPWG - CWHQVBWSVTWOVNPWPWRWOVTVNVLAVNWEBWBWJVPWGCWHQZVAVLAVNWEBVTWOVPWGCWHQVBTWM - WBWJVMPZVTWEPZWFWJVTWEPZVMPZVTXEVNPWDVTWJVNPZVMPWMVKWCWKWAXFXHLWTXBXAXCVL - VMAWEWBWJVTVPVRWGVCRVLAVNWEBVTXEVPWGCWHQWDWFXIXGVMWIVLAVNWEBVTWJVPWGCWHQZ - VDTWMWJVTWBVMPZWEPZXGWOVMPZXKWJVNPXIWRVMPWMVKWKWAWCXLXMLWTXAXCXBVLVMAWEWJ - VTWBVPVRWGVERVLAVNWEBXKWJVPWGCWHQXIXGWRWOVMXJXDVDTVLAVOVPVOMZVFVKWAUMZVOV - TVNPVTVOWEPVTVLAVNWEBVOVTVPWGCWHQVLAWEVOVTVPWGXNVGVHXOVTVOVNPVOVTWEPVTVLA - VNWEBVTVOVPWGCWHQVLAWEVOVTVPWGXNVIVHVJ $. - - $( Bidirectional form of ~ opprrng . (Contributed by Mario Carneiro, + cgrp 3eqtr4g crg cur opprbas oppradd eqidd ringgrp grpprop sylib syl5eqel + w3a ringcl 3com23 simpl simpr3 simpr2 simpr1 ringass eqcomd oveq1i oveq2i + wa eqtri ringdir oveq12i ringdi ringidcl ringridm syl5eq ringlidm isringd + ) AUAGZDEFAHIZAJIZBBKIZAUBIZVLBHILVKVLABCVLMZUCZNVMBJILVKVMABCVMMZUDZNVKV + NUEVKASGBSGAUFABVQVSUGUHVKDOZVLGZEOZVLGZUJVTWBVNPZWBVTAKIZPZVLVLAVNWEBVTW + BVPWEMZCVNMZQZVKWCWAWFVLGVLAWEWBVTVPWGUKULUIVKWAWCFOZVLGZUJZVAZWJWFWEPZWJ + WBWEPZVTWEPZWDWJVNPZVTWBWJVNPZVNPZWMWPWNWMVKWKWCWAWPWNLVKWLUMZVKWAWCWKUNZ + VKWAWCWKUOZVKWAWCWKUPZVLAWEWJWBVTVPWGUQRURWQWFWJVNPWNWDWFWJVNWIUSVLAVNWEB + WFWJVPWGCWHQVBWSVTWOVNPWPWRWOVTVNVLAVNWEBWBWJVPWGCWHQZUTVLAVNWEBVTWOVPWGC + WHQVBTWMWBWJVMPZVTWEPZWFWJVTWEPZVMPZVTXEVNPWDVTWJVNPZVMPWMVKWCWKWAXFXHLWT + XBXAXCVLVMAWEWBWJVTVPVRWGVCRVLAVNWEBVTXEVPWGCWHQWDWFXIXGVMWIVLAVNWEBVTWJV + PWGCWHQZVDTWMWJVTWBVMPZWEPZXGWOVMPZXKWJVNPXIWRVMPWMVKWKWAWCXLXMLWTXAXCXBV + LVMAWEWJVTWBVPVRWGVERVLAVNWEBXKWJVPWGCWHQXIXGWRWOVMXJXDVDTVLAVOVPVOMZVFVK + WAVAZVOVTVNPVTVOWEPVTVLAVNWEBVOVTVPWGCWHQVLAWEVOVTVPWGXNVGVHXOVTVOVNPVOVT + WEPVTVLAVNWEBVTVOVPWGCWHQVLAWEVOVTVPWGXNVIVHVJ $. + + $( Bidirectional form of ~ opprring . (Contributed by Mario Carneiro, 6-Dec-2014.) $) - opprrngb $p |- ( R e. Ring <-> O e. Ring ) $= - ( vx vy crg wcel opprrng cfv eqid wtru cbs opprbas a1i cv cplusg co cmulr - wceq wa coppr wb eqidd oppradd opprmul eqtr2i rngpropd trud sylibr impbii - oveqi ) AFGZBFGZABCHUMBUAIZFGZULBUNUNJZHULUOUBKDEALIZAUNKUQUCUQUNLISKUQBU - NUPUQABCUQJZMZMNDOZEOZAPIZQUTVAUNPIZQSKUTUQGVAUQGTTZVBVCUTVAVBBUNUPVBABCV - BJUDUDUKNUTVAARIZQZUTVAUNRIZQZSVDVHVAUTBRIZQVFUQBVGVIUNUTVAUSVIJZUPVGJUEU - QAVIVEBVAUTURVEJCVJUEUFNUGUHUIUJ $. + opprringb $p |- ( R e. Ring <-> O e. Ring ) $= + ( vx vy crg wcel opprring cfv eqid wtru cbs wceq opprbas a1i cv cplusg co + wa cmulr coppr eqidd oppradd oveqi opprmul eqtr2i ringpropd sylibr impbii + wb trud ) AFGZBFGZABCHUMBUAIZFGZULBUNUNJZHULUOUJKDEALIZAUNKUQUBUQUNLIMKUQ + BUNUPUQABCUQJZNZNODPZEPZAQIZRUTVAUNQIZRMKUTUQGVAUQGSSZVBVCUTVAVBBUNUPVBAB + CVBJUCUCUDOUTVAATIZRZUTVAUNTIZRZMVDVHVAUTBTIZRVFUQBVGVIUNUTVAUSVIJZUPVGJU + EUQAVIVEBVAUTURVEJCVJUEUFOUGUKUHUI $. ${ oppr0.2 $e |- .0. = ( 0g ` R ) $. @@ -228815,15 +229032,15 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of (Contributed by Mario Carneiro, 14-Jun-2015.) $) mulgass3 $p |- ( ( R e. Ring /\ ( N e. ZZ /\ X e. B /\ Y e. B ) ) -> ( X .X. ( N .x. Y ) ) = ( N .x. ( X .X. Y ) ) ) $= - ( vx vy wcel wa cfv co wceq eqid cvv a1i crg w3a coppr cmg opprrng adantr - cz cmulr simpr1 simpr3 simpr2 opprbas mulgass2 opprmul oveq2i 3eqtr3g cbs - syl13anc wss ssv cplusg ovex oppradd oveqi mulgpropd oveqd oveq2d 3eqtr4d - cv ) BUAMZEUGMZFAMZGAMZUBZNZFEGBUCOZUDOZPZDPZEFGDPZVQPZFEGCPZDPEVTCPVOVRF - VPUHOZPZEGFWCPZVQPZVSWAVOVPUAMZVKVMVLWDWFQVJWGVNBVPVPRZUEUFVJVKVLVMUIVJVK - VLVMUJVJVKVLVMUKAVPVQWCEGFABVPWHHULZVQRZWCRZUMURABWCDVPVRFHJWHWKUNWEVTEVQ - ABWCDVPGFHJWHWKUNUOUPVOWBVRFDVOCVQEGVOKLACVQBVPSIWJABUQOQVOHTAVPUQOQVOWIT - ASUSVOAUTTKVIZLVIZBVAOZPZSMVOWLSMWMSMNNZWLWMWNVBTWOWLWMVPVAOZPQWPWNWQWLWM - WNBVPWHWNRVCVDTVEZVFVGVOCVQEVTWRVFVH $. + ( vx vy wcel wa cfv co wceq eqid cvv a1i crg w3a coppr cmg cmulr opprring + adantr simpr1 simpr3 simpr2 opprbas mulgass2 syl13anc opprmul 3eqtr3g cbs + cz oveq2i wss cv cplusg ovex oppradd oveqi mulgpropd oveqd oveq2d 3eqtr4d + ssv ) BUAMZEUQMZFAMZGAMZUBZNZFEGBUCOZUDOZPZDPZEFGDPZVQPZFEGCPZDPEVTCPVOVR + FVPUEOZPZEGFWCPZVQPZVSWAVOVPUAMZVKVMVLWDWFQVJWGVNBVPVPRZUFUGVJVKVLVMUHVJV + KVLVMUIVJVKVLVMUJAVPVQWCEGFABVPWHHUKZVQRZWCRZULUMABWCDVPVRFHJWHWKUNWEVTEV + QABWCDVPGFHJWHWKUNURUOVOWBVRFDVOCVQEGVOKLACVQBVPSIWJABUPOQVOHTAVPUPOQVOWI + TASUSVOAVITKUTZLUTZBVAOZPZSMVOWLSMWMSMNNZWLWMWNVBTWOWLWMVPVAOZPQWPWNWQWLW + MWNBVPWHWNRVCVDTVEZVFVGVOCVQEVTWRVFVH $. $} $( @@ -228934,31 +229151,31 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.) $) dvdsrcl2 $p |- ( ( R e. Ring /\ X .|| Y ) -> Y e. B ) $= - ( vx wbr crg wcel cv cmulr cfv co wceq wrex wa eqid dvdsr rngcl syl5ibcom - 3expa an32s eleq1 rexlimdva impr sylan2b ) DEBICJKZDAKZHLZDCMNZOZEPZHAQZR - EAKZHABCULDEFGULSZTUIUJUOUPUIUJRZUNUPHAURUKAKZRUMAKZUNUPUIUSUJUTUIUSUJUTA - CULUKDFUQUAUCUDUMEAUEUBUFUGUH $. + ( vx wbr crg wcel cv cmulr cfv co wceq wrex wa eqid dvdsr 3expa syl5ibcom + ringcl an32s eleq1 rexlimdva impr sylan2b ) DEBICJKZDAKZHLZDCMNZOZEPZHAQZ + REAKZHABCULDEFGULSZTUIUJUOUPUIUJRZUNUPHAURUKAKZRUMAKZUNUPUIUSUJUTUIUSUJUT + ACULUKDFUQUCUAUDUMEAUEUBUFUGUH $. $( 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 wbr eqid rngidcl dvdsrmul syl2anr rnglidm - id breqtrd ) CGHZDAHZIDCJKZDCLKZMZDBUCUCUDAHDUFBNUBUCTACUDEUDOZPABCUEDUDE - FUEOZQRACUEUDDEUHUGSUA $. + ( crg wcel wa cur cfv cmulr co id eqid ringidcl dvdsrmul syl2anr ringlidm + wbr breqtrd ) CGHZDAHZIDCJKZDCLKZMZDBUCUCUDAHDUFBTUBUCNACUDEUDOZPABCUEDUD + EFUEOZQRACUEUDDEUHUGSUA $. $( Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.) $) dvdsrtr $p |- ( ( R e. Ring /\ Y .|| Z /\ Z .|| X ) -> Y .|| X ) $= ( vy vx crg wcel wbr wa cv cmulr co wceq wrex dvdsr cfv anbi12i an4 bitri - reeanv simplrl simpll simprr simprl rngcl syl3anc dvdsrmul syl2anc rngass - eqid syl13anc breqtrd oveq2 id sylan9eq breq2d rexlimdvva syl5bir expimpd - syl5ibcom syl5bi 3impib ) CKLZEFBMZFDBMZEDBMZVIVJNZEALZFALZNZIOZECPUAZQZF - RZIASZJOZFVQQZDRZJASZNZNZVHVKVLVMVTNZVNWDNZNWFVIWGVJWHIABCVQEFGHVQUOZTJAB - CVQFDGHWITUBVMVTVNWDUCUDVHVOWEVKWEVSWCNZJASIASVHVONZVKVSWCIJAAUEWKWJVKIJA - AWKVPALZWAALZNZNZEWAVRVQQZBMWJVKWOEWAVPVQQZEVQQZWPBWOVMWQALZEWRBMVHVMVNWN - UFZWOVHWMWLWSVHVOWNUGZWKWLWMUHZWKWLWMUIZACVQWAVPGWIUJUKABCVQEWQGHWIULUMWO - VHWMWLVMWRWPRXAXBXCWTACVQWAVPEGWIUNUPUQWJWPDEBVSWCWPWBDVRFWAVQURWCUSUTVAV - EVBVCVDVFVG $. + eqid reeanv simplrl simpll simprr simprl syl3anc dvdsrmul syl2anc ringass + ringcl syl13anc breqtrd oveq2 breq2d syl5ibcom rexlimdvva syl5bir expimpd + id sylan9eq syl5bi 3impib ) CKLZEFBMZFDBMZEDBMZVIVJNZEALZFALZNZIOZECPUAZQ + ZFRZIASZJOZFVQQZDRZJASZNZNZVHVKVLVMVTNZVNWDNZNWFVIWGVJWHIABCVQEFGHVQUEZTJ + ABCVQFDGHWITUBVMVTVNWDUCUDVHVOWEVKWEVSWCNZJASIASVHVONZVKVSWCIJAAUFWKWJVKI + JAAWKVPALZWAALZNZNZEWAVRVQQZBMWJVKWOEWAVPVQQZEVQQZWPBWOVMWQALZEWRBMVHVMVN + WNUGZWOVHWMWLWSVHVOWNUHZWKWLWMUIZWKWLWMUJZACVQWAVPGWIUOUKABCVQEWQGHWIULUM + WOVHWMWLVMWRWPRXAXBXCWTACVQWAVPEGWIUNUPUQWJWPDEBVSWCWPWBDVRFWAVQURWCVDVEU + SUTVAVBVCVFVG $. ${ dvdsrmul1.3 $e |- .x. = ( .r ` R ) $. @@ -228966,23 +229183,23 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of (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 cv wceq wrex wa dvdsr simplll simplr simpllr rngcl - dvdsrmul sylancom simpr rngass syl13anc breqtrrd oveq1 breq2d syl5ibcom - syl3anc rexlimdva expimpd syl5bi 3impia ) CLMZGAMZEFBNZEGDOZFGDOZBNZVAE - AMZKPZEDOZFQZKARZSUSUTSZVDKABCDEFHIJTVJVEVIVDVJVESZVHVDKAVKVFAMZSZVBVGG - DOZBNVHVDVMVBVFVBDOZVNBVKVLVBAMZVBVOBNVMUSVEUTVPUSUTVEVLUAZVJVEVLUBZUSU - TVEVLUCZACDEGHJUDUNABCDVBVFHIJUEUFVMUSVLVEUTVNVOQVQVKVLUGVRVSACDVFEGHJU - HUIUJVHVNVCVBBVGFGDUKULUMUOUPUQUR $. + ( vx crg wcel wbr co cv wceq wrex wa dvdsr simplll simplr simpllr simpr + ringcl syl3anc dvdsrmul sylancom ringass syl13anc breqtrrd oveq1 breq2d + syl5ibcom rexlimdva expimpd syl5bi 3impia ) CLMZGAMZEFBNZEGDOZFGDOZBNZV + AEAMZKPZEDOZFQZKARZSUSUTSZVDKABCDEFHIJTVJVEVIVDVJVESZVHVDKAVKVFAMZSZVBV + GGDOZBNVHVDVMVBVFVBDOZVNBVKVLVBAMZVBVOBNVMUSVEUTVPUSUTVEVLUAZVJVEVLUBZU + SUTVEVLUCZACDEGHJUEUFABCDVBVFHIJUGUHVMUSVLVEUTVNVOQVQVKVLUDVRVSACDVFEGH + JUIUJUKVHVNVCVBBVGFGDULUMUNUOUPUQUR $. $} 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 wbr id cgrp rnggrp eqid grpinvcl dvdsrmul - rngidcl syl2anc syl2anr simpl simpr rngnegl breqtrd ) CIJZEAJZKZECLMZDMZE - CNMZOZEDMBUKUKUNAJZEUPBPUJUKQUJCRJUMAJUQCSACUMFUMTZUCACDUMFHUAUDABCUOEUNF - GUOTZUBUEULACUOUMDEFUSURHUJUKUFUJUKUGUHUI $. + ( crg wcel wa cur cfv cmulr co wbr id cgrp ringgrp eqid ringidcl grpinvcl + syl2anc dvdsrmul syl2anr simpl simpr ringnegl breqtrd ) CIJZEAJZKZECLMZDM + ZECNMZOZEDMBUKUKUNAJZEUPBPUJUKQUJCRJUMAJUQCSACUMFUMTZUAACDUMFHUBUCABCUOEU + NFGUOTZUDUEULACUOUMDEFUSURHUJUKUFUJUKUGUHUI $. $} ${ @@ -228994,19 +229211,19 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of has a somewhat different meaning, see ~ df-rlreg .) (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 rng0cl adantr rnglz eqeq1d - eqid oveq1 rspcev syl2anc wb dvdsr2 adantl mpbird ) CJKZDAKZLZDEBMZINZDCO - PZQZERZIASZUNEAKZEDUQQZERZUTULVAUMACEFHTUAACUQDEFUQUDZHUBUSVCIEAUPERURVBE - UPEDUQUEUCUFUGUMUOUTUHULIABCUQDEFGVDUIUJUK $. + ( vx crg wcel wa wbr cv cmulr cfv co wceq wrex ring0cl adantr eqid ringlz + oveq1 eqeq1d rspcev syl2anc wb dvdsr2 adantl mpbird ) CJKZDAKZLZDEBMZINZD + COPZQZERZIASZUNEAKZEDUQQZERZUTULVAUMACEFHTUAACUQDEFUQUBZHUCUSVCIEAUPERURV + BEUPEDUQUDUEUFUGUMUOUTUHULIABCUQDEFGVDUIUJUK $. $( 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 crg wcel wa wbr cv cmulr cfv wceq wrex wb adantr rng0cl dvdsr2 rngrz - co eqid syl eqeq1d eqcom syl6bb rexbidva cgrp c0 wne rnggrp r19.9rzv 3syl - grpbn0 bitr4d bitrd ) CJKZDAKZLZEDBMZINZECOPZUDZDQZIARZDEQZVBEAKZVCVHSUTV - JVAACEFHUATIABCVEEDFGVEUEZUBUFUTVHVISVAUTVHVIIARZVIUTVGVIIAUTVDAKLZVGEDQV - IVMVFEDACVEVDEFVKHUCUGEDUHUIUJUTCUKKAULUMVIVLSCUNACFUQVIIAUOUPURTUS $. + ( vx crg wcel wa wbr cv cmulr cfv wceq wrex wb adantr ring0cl eqid dvdsr2 + co syl ringrz eqeq1d eqcom syl6bb rexbidva c0 wne ringgrp grpbn0 r19.9rzv + cgrp 3syl bitr4d bitrd ) CJKZDAKZLZEDBMZINZECOPZUDZDQZIARZDEQZVBEAKZVCVHS + UTVJVAACEFHUATIABCVEEDFGVEUBZUCUEUTVHVISVAUTVHVIIARZVIUTVGVIIAUTVDAKLZVGE + DQVIVMVFEDACVEVDEFVKHUFUGEDUHUIUJUTCUPKAUKULVIVLSCUMACFUNVIIAUOUQURTUS $. $} ${ @@ -229037,10 +229254,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.) $) 1unit $p |- ( R e. Ring -> .1. e. U ) $= - ( crg wcel cdsr cfv wbr coppr cbs rngidcl dvdsrid opprrng opprbas syl2anc - eqid mpdan isunit sylanbrc ) AFGZCCAHIZJZCCAKIZHIZJZCBGUBCALIZGZUDUHACUHR - ZEMZUHUCACUJUCRZNSUBUEFGUIUGAUEUERZOUKUHUFUECUHAUEUMUJPUFRZNQUCAUEBCUFCDE - ULUMUNTUA $. + ( crg wcel cdsr cfv wbr coppr cbs ringidcl dvdsrid mpdan opprring opprbas + eqid syl2anc isunit sylanbrc ) AFGZCCAHIZJZCCAKIZHIZJZCBGUBCALIZGZUDUHACU + HRZEMZUHUCACUJUCRZNOUBUEFGUIUGAUEUERZPUKUHUFUECUHAUEUMUJQUFRZNSUCAUEBCUFC + DEULUMUNTUA $. $} ${ @@ -229099,10 +229316,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.) $) dvdsunit $p |- ( ( R e. CRing /\ Y .|| X /\ X e. U ) -> Y e. U ) $= - ( ccrg wcel wbr wa cur cfv crg wi crngrng eqid wb crngunit adantr dvdsrtr - cbs 3expia sylan 3imtr4d 3impia ) BHIZEDAJZDCIZECIZUGUHKDBLMZAJZEUKAJZUIU - JUGBNIZUHULUMOBPUNUHULUMBUBMZABUKEDUOQGUAUCUDUGUIULRUHABCUKDFUKQZGSTUGUJU - MRUHABCUKEFUPGSTUEUF $. + ( ccrg wcel wbr wa cur cfv crg wi crngring eqid wb crngunit adantr 3expia + cbs dvdsrtr sylan 3imtr4d 3impia ) BHIZEDAJZDCIZECIZUGUHKDBLMZAJZEUKAJZUI + UJUGBNIZUHULUMOBPUNUHULUMBUBMZABUKEDUOQGUCUAUDUGUIULRUHABCUKDFUKQZGSTUGUJ + UMRUHABCUKEFUPGSTUEUF $. $} ${ @@ -229114,17 +229331,17 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of unitmulcl $p |- ( ( R e. Ring /\ X e. U /\ Y e. U ) -> ( X .x. Y ) e. U ) $= ( crg wcel co cfv cdsr wbr eqid unitcl syl wa isunit sylib syl3anc wceq - w3a cur coppr simp1 cbs simp3 dvdsrmul1 rnglidm syl2anc breqtrd dvdsrtr - simpld opprrng opprmul simprd opprbas rngridm syl5eq syl5eqbrr sylanbrc - simp2 cmulr ) AHIZDCIZECIZUBZDEBJZAUCKZALKZMZVHVIAUDKZLKZMZVHCIVGVDVHEV - JMEVIVJMZVKVDVEVFUEZVGVHVIEBJZEVJVGVDEAUFKZIZDVIVJMZVHVQVJMVPVGVFVSVDVE - VFUGZVRACEVRNZFOPZVGVTDVIVMMZVGVEVTWDQVDVEVFVBZVJAVLCVIVMDFVINZVJNZVLNZ - VMNZRSZUMVRVJABDVIEWBWGGUHTVGVDVSVQEUAVPWCVRABVIEWBGWFUIUJUKVGVOEVIVMMZ - VGVFVOWKQWAVJAVLCVIVMEFWFWGWHWIRSZUMVRVJAVIVHEWBWGULTVGVLHIZVHDVMMWDVNV - GVDWMVPAVLWHUNPZVGVHEDVLVCKZJZDVMVRAWOBVLEDWBGWHWONZUOVGWPVIDWOJZDVMVGW - MDVRIZWKWPWRVMMWNVGVEWSWEVRACDWBFOPZVGVOWKWLUPVRVMVLWOEVIDVRAVLWHWBUQZW - IWQUHTVGWRDVIBJZDVRAWOBVLVIDWBGWHWQUOVGVDWSXBDUAVPWTVRABVIDWBGWFURUJUSU - KUTVGVTWDWJUPVRVMVLVIVHDXAWIULTVJAVLCVIVMVHFWFWGWHWIRVA $. + w3a cur coppr simp1 cbs simp3 simpld dvdsrmul1 ringlidm syl2anc breqtrd + simp2 dvdsrtr opprring opprmul simprd opprbas ringridm syl5eq syl5eqbrr + cmulr sylanbrc ) AHIZDCIZECIZUBZDEBJZAUCKZALKZMZVHVIAUDKZLKZMZVHCIVGVDV + HEVJMEVIVJMZVKVDVEVFUEZVGVHVIEBJZEVJVGVDEAUFKZIZDVIVJMZVHVQVJMVPVGVFVSV + DVEVFUGZVRACEVRNZFOPZVGVTDVIVMMZVGVEVTWDQVDVEVFUMZVJAVLCVIVMDFVINZVJNZV + LNZVMNZRSZUHVRVJABDVIEWBWGGUITVGVDVSVQEUAVPWCVRABVIEWBGWFUJUKULVGVOEVIV + MMZVGVFVOWKQWAVJAVLCVIVMEFWFWGWHWIRSZUHVRVJAVIVHEWBWGUNTVGVLHIZVHDVMMWD + VNVGVDWMVPAVLWHUOPZVGVHEDVLVBKZJZDVMVRAWOBVLEDWBGWHWONZUPVGWPVIDWOJZDVM + VGWMDVRIZWKWPWRVMMWNVGVEWSWEVRACDWBFOPZVGVOWKWLUQVRVMVLWOEVIDVRAVLWHWBU + RZWIWQUITVGWRDVIBJZDVRAWOBVLVIDWBGWHWQUPVGVDWSXBDUAVPWTVRABVIDWBGWFUSUK + UTULVAVGVTWDWJUQVRVMVLVIVHDXAWIUNTVJAVLCVIVMVHFWFWGWHWIRVC $. unitmulclb.1 $e |- B = ( Base ` R ) $. $( Reversal of ~ unitmulcl in a commutative ring. (Contributed by Mario @@ -229132,12 +229349,12 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of unitmulclb $p |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( ( X .x. Y ) e. U <-> ( X e. U /\ Y e. U ) ) ) $= ( ccrg wcel w3a co wa wbr wi dvdsrmul syl2anc dvdsunit 3expia cfv simp1 - cdsr simp2 simp3 eqid crngcom breqtrrd jcad crg crngrng 3ad2ant1 3expib - unitmulcl syl impbid ) BJKZEAKZFAKZLZEFCMZDKZEDKZFDKZNZUTVBVCVDUTUQEVAB - UCUAZOZVBVCPUQURUSUBZUTEFECMZVAVFUTURUSEVIVFOUQURUSUDZUQURUSUEZAVFBCEFI - VFUFZHQRABCEFIHUGUHUQVGVBVCVFBDVAEGVLSTRUTUQFVAVFOZVBVDPVHUTUSURVMVKVJA - VFBCFEIVLHQRUQVMVBVDVFBDVAFGVLSTRUIUTBUJKZVEVBPUQURVNUSBUKULVNVCVDVBBCD - EFGHUNUMUOUP $. + cdsr simp2 simp3 eqid crngcom breqtrrd jcad crngring 3ad2ant1 unitmulcl + crg 3expib syl impbid ) BJKZEAKZFAKZLZEFCMZDKZEDKZFDKZNZUTVBVCVDUTUQEVA + BUCUAZOZVBVCPUQURUSUBZUTEFECMZVAVFUTURUSEVIVFOUQURUSUDZUQURUSUEZAVFBCEF + IVFUFZHQRABCEFIHUGUHUQVGVBVCVFBDVAEGVLSTRUTUQFVAVFOZVBVDPVHUTUSURVMVKVJ + AVFBCFEIVLHQRUQVMVBVDVFBDVAFGVLSTRUIUTBUMKZVEVBPUQURVNUSBUJUKVNVCVDVBBC + DEFGHULUNUOUP $. $} $d x y z G $. $d m x y z R $. $d m x y z U $. @@ -229153,40 +229370,40 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of unitgrp $p |- ( R e. Ring -> G e. Grp ) $= ( vy vm wcel cfv cbs wceq cvv eqid cv co unitcl wa wbr wrex syl2anc vx vz crg cmulr cur unitgrpbas cplusg fvex eqeltri cmgp mgpplusg ressplusg mp1i - a1i unitmulcl w3a 3anim123i rngass sylan2 1unit rnglidm cdsr coppr isunit - simpr sylib wb adantl dvdsr2 syl opprbas anbi12d reeanv ad2antrr dvdsrmul - simplll syl13anc simprrl opprmul simprrr syl5eqr 3eqtr3d rngridm sylanbrc - simprl simplr oveq1d oveq2d 3brtr3d syl5eq breqtrd jca rexlimdvaa expimpd - reximdv2 syl5bir sylbid mpd isgrpde ) AUCHZUAFUBBAUDIZCAUEIZBCJIZKWTABCDE - UFZUNBLHXACUGIKWTBXCLXDCJUHUIBXAAUJIZCLEAXAXEXEMXAMZUKULUMAXABUANZFNZDXFU - OXGBHZXHBHZUBNZBHZUPWTXGAJIZHZXHXMHZXKXMHZUPXGXHXAOXKXAOXGXHXKXAOXAOKXIXN - XJXOXLXPXMABXGXMMZDPZXMABXHXQDPXMABXKXQDPUQXMAXAXGXHXKXQXFURUSABXBDXBMZUT - XIWTXNXBXGXAOXGKXRXMAXAXBXGXQXFXSVAUSWTXIQZXGXBAVBIZRZXGXBAVCIZVBIZRZQZXH - XGXAOZXBKZFBSZXTXIYFWTXIVEYAAYCBXBYDXGDXSYAMZYCMZYDMZVDVFXTYFYHFXMSZGNZXG - YCUDIZOZXBKZGXMSZQZYIXTYBYMYEYRXTXNYBYMVGXIXNWTXRVHZFXMYAAXAXGXBXQYJXFVIV - JXTXNYEYRVGYTGXMYDYCYOXGXBXMAYCYKXQVKZYLYOMZVIVJVLYSYHYQQZGXMSZFXMSXTYIYH - YQFGXMXMVMXTUUDYHFXMBXTXOUUDXJYHQZXTXOQZUUCUUEGXMUUFYNXMHZUUCQZQZXJYHUUIX - HXBYARXHXBYDRXJUUIYNXGYNXAOZXHXBYAUUIUUGXNYNUUJYARUUFUUGUUCWEZXTXNXOUUHYT - VNZXMYAAXAYNXGXQYJXFVOTUUIXBYNXAOZXHXBXAOZYNXHUUIYGYNXAOZXHUUJXAOZUUMUUNU - UIWTXOXNUUGUUOUUPKWTXIXOUUHVPZXTXOUUHWFZUULUUKXMAXAXHXGYNXQXFURVQUUIYGXBY - NXAUUFUUGYHYQVRZWGUUIUUJXBXHXAUUIUUJYPXBXMAYOXAYCYNXGXQXFYKUUBVSUUFUUGYHY - QVTWAZWHWBUUIWTUUGUUMYNKUUQUUKXMAXAXBYNXQXFXSVATUUIWTXOUUNXHKUUQUURXMAXAX - BXHXQXFXSWCTWBUUTWIUUIXHXGXHYOOZXBYDUUIXOXNXHUVAYDRUURUULXMYDYCYOXHXGUUAY - LUUBVOTUUIUVAYGXBXMAYOXAYCXGXHXQXFYKUUBVSUUSWJWKYAAYCBXBYDXHDXSYJYKYLVDWD - UUSWLWMWNWOWPWQWRWS $. + a1i unitmulcl w3a 3anim123i sylan2 1unit ringlidm cdsr coppr simpr isunit + ringass sylib wb adantl dvdsr2 syl opprbas anbi12d reeanv simprl ad2antrr + dvdsrmul simplll simplr syl13anc simprrl opprmul simprrr syl5eqr ringridm + oveq1d oveq2d 3eqtr3d 3brtr3d syl5eq breqtrd sylanbrc rexlimdvaa reximdv2 + jca expimpd syl5bir sylbid mpd isgrpde ) AUCHZUAFUBBAUDIZCAUEIZBCJIZKWTAB + CDEUFZUNBLHXACUGIKWTBXCLXDCJUHUIBXAAUJIZCLEAXAXEXEMXAMZUKULUMAXABUANZFNZD + XFUOXGBHZXHBHZUBNZBHZUPWTXGAJIZHZXHXMHZXKXMHZUPXGXHXAOXKXAOXGXHXKXAOXAOKX + IXNXJXOXLXPXMABXGXMMZDPZXMABXHXQDPXMABXKXQDPUQXMAXAXGXHXKXQXFVEURABXBDXBM + ZUSXIWTXNXBXGXAOXGKXRXMAXAXBXGXQXFXSUTURWTXIQZXGXBAVAIZRZXGXBAVBIZVAIZRZQ + ZXHXGXAOZXBKZFBSZXTXIYFWTXIVCYAAYCBXBYDXGDXSYAMZYCMZYDMZVDVFXTYFYHFXMSZGN + ZXGYCUDIZOZXBKZGXMSZQZYIXTYBYMYEYRXTXNYBYMVGXIXNWTXRVHZFXMYAAXAXGXBXQYJXF + VIVJXTXNYEYRVGYTGXMYDYCYOXGXBXMAYCYKXQVKZYLYOMZVIVJVLYSYHYQQZGXMSZFXMSXTY + IYHYQFGXMXMVMXTUUDYHFXMBXTXOUUDXJYHQZXTXOQZUUCUUEGXMUUFYNXMHZUUCQZQZXJYHU + UIXHXBYARXHXBYDRXJUUIYNXGYNXAOZXHXBYAUUIUUGXNYNUUJYARUUFUUGUUCVNZXTXNXOUU + HYTVOZXMYAAXAYNXGXQYJXFVPTUUIXBYNXAOZXHXBXAOZYNXHUUIYGYNXAOZXHUUJXAOZUUMU + UNUUIWTXOXNUUGUUOUUPKWTXIXOUUHVQZXTXOUUHVRZUULUUKXMAXAXHXGYNXQXFVEVSUUIYG + XBYNXAUUFUUGYHYQVTZWEUUIUUJXBXHXAUUIUUJYPXBXMAYOXAYCYNXGXQXFYKUUBWAUUFUUG + YHYQWBWCZWFWGUUIWTUUGUUMYNKUUQUUKXMAXAXBYNXQXFXSUTTUUIWTXOUUNXHKUUQUURXMA + XAXBXHXQXFXSWDTWGUUTWHUUIXHXGXHYOOZXBYDUUIXOXNXHUVAYDRUURUULXMYDYCYOXHXGU + UAYLUUBVPTUUIUVAYGXBXMAYOXAYCXGXHXQXFYKUUBWAUUSWIWJYAAYCBXBYDXHDXSYJYKYLV + DWKUUSWNWLWOWMWPWQWRWS $. $( The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.) $) unitabl $p |- ( R e. CRing -> G e. Abel ) $= - ( ccrg wcel cgrp ccmn cabl crg crngrng unitgrp syl cmgp cmnd eqid crngmgp - cfv grpmnd subcmn syl2anc isabl sylanbrc ) AFGZCHGZCIGZCJGUEAKGUFALABCDEM - NZUEAOSZIGCPGZUGAUIUIQRUEUFUJUHCTNBUICEUAUBCUCUD $. + ( ccrg wcel cgrp ccmn cabl crg crngring unitgrp syl cmgp cfv cmnd crngmgp + eqid grpmnd subcmn syl2anc isabl sylanbrc ) AFGZCHGZCIGZCJGUEAKGUFALABCDE + MNZUEAOPZIGCQGZUGAUIUISRUEUFUJUHCTNBUICEUAUBCUCUD $. unitgrp.3 $e |- .1. = ( 1r ` R ) $. $( The identity of the multiplicative group is ` 1r ` . (Contributed by Mario Carneiro, 2-Dec-2014.) $) unitgrpid $p |- ( R e. Ring -> .1. = ( 0g ` G ) ) $= - ( crg wcel c0g cfv wceq 1unit cbs wss eqid unitss rngidss mp3an2 mpdan ) + ( crg wcel c0g cfv wceq 1unit cbs wss eqid unitss ringidss mp3an2 mpdan ) AHIZCBIZCDJKLZABCEGMUABANKZOUBUCUDABUDPZEQBUDACDFUEGRST $. $} @@ -229198,10 +229415,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of ring. (Contributed by Mario Carneiro, 18-Jun-2015.) $) unitsubm $p |- ( R e. Ring -> U e. ( SubMnd ` M ) ) $= ( crg wcel csubmnd cfv cbs wss cur cress cmnd eqid unitss a1i 1unit syl - co cgrp cmgp oveq1i unitgrp grpmnd w3a wb rngmgp mgpbas issubm2 mpbir3and - rngidval ) AFGZBCHIGZBAJIZKZALIZBGZCBMTZNGZUPUMUOABUOOZDPQABUQDUQOZRUMUSU - AGUTABUSDCAUBIBMEUCUDUSUESUMCNGUNUPURUTUFUGACEUHUOBUSCUQUOACEVAUIAUQCEVBU - LUSOUJSUK $. + co cgrp cmgp oveq1i unitgrp grpmnd w3a wb ringmgp mgpbas rngidval issubm2 + mpbir3and ) AFGZBCHIGZBAJIZKZALIZBGZCBMTZNGZUPUMUOABUOOZDPQABUQDUQOZRUMUS + UAGUTABUSDCAUBIBMEUCUDUSUESUMCNGUNUPURUTUFUGACEUHUOBUSCUQUOACEVAUIAUQCEVB + UJUSOUKSUL $. $} $c invr $. @@ -229246,10 +229463,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $. ${ - rnginvcl.3 $e |- B = ( Base ` R ) $. + ringinvcl.3 $e |- B = ( Base ` R ) $. $( The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.) $) - rnginvcl $p |- ( ( R e. Ring /\ X e. U ) -> ( I ` X ) e. B ) $= + ringinvcl $p |- ( ( R e. Ring /\ X e. U ) -> ( I ` X ) e. B ) $= ( crg wcel wa cfv unitinvcl unitcl syl ) BIJECJKEDLZCJPAJBCDEFGMABCPHFN O $. $} @@ -229281,9 +229498,9 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.) $) 1rinv $p |- ( R e. Ring -> ( I ` .1. ) = .1. ) $= - ( crg wcel cfv cmulr co cbs wceq cui eqid 1unit rnginvcl rnglidm unitrinv - mpdan eqtr3d ) AFGZBBCHZAIHZJZUBBUAUBAKHZGZUDUBLUABAMHZGZUFAUGBUGNZEOZUEA - UGCBUIDUENZPSUEAUCBUBUKUCNZEQSUAUHUDBLUJAUCUGBCBUIDULERST $. + ( crg wcel cfv cmulr cbs wceq cui 1unit ringinvcl mpdan ringlidm unitrinv + co eqid eqtr3d ) AFGZBBCHZAIHZRZUBBUAUBAJHZGZUDUBKUABALHZGZUFAUGBUGSZEMZU + EAUGCBUIDUESZNOUEAUCBUBUKUCSZEPOUAUHUDBKUJAUCUGBCBUIDULEQOT $. $} ${ @@ -229293,7 +229510,7 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( The additive identity is a unit if and only if ` 1 = 0 ` , i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.) $) 0unit $p |- ( R e. Ring -> ( .0. e. U <-> .1. = .0. ) ) $= - ( crg wcel wceq wa cinvr cfv cmulr co eqid unitrinv cbs rnginvcl rnglz + ( crg wcel wceq wa cinvr cfv cmulr co eqid unitrinv cbs ringinvcl ringlz syldan eqtr3d simpr 1unit adantr eqeltrrd impbida ) AHIZDBIZCDJZUHUIKDDAL MZMZANMZOZCDAUMBCUKDEUKPZUMPZGQUHUIULARMZIUNDJUQABUKDEUOUQPZSUQAUMULDURUP FTUAUBUHUJKCDBUHUJUCUHCBIUJABCEGUDUEUFUG $. @@ -229307,11 +229524,11 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of 4-Dec-2014.) $) unitnegcl $p |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. U ) $= ( crg wcel wa cfv cur cdsr wbr eqid syl2an breqtrd isunit dvdsrtr syl3anc - dvdsrneg coppr simpl cbs cgrp rnggrp unitcl grpinvcl wceq grpinvinv simpr - syldan simpld opprrng adantr opprbas opprneg syl2anc simprd sylanbrc + dvdsrneg coppr simpl cbs cgrp unitcl grpinvcl syldan wceq grpinvinv simpr + ringgrp simpld opprring adantr opprbas opprneg syl2anc simprd sylanbrc sylib ) AGHZDBHZIZDCJZAKJZALJZMZVDVEAUAJZLJZMZVDBHVCVAVDDVFMDVEVFMZVGVAVB - UBVCVDVDCJZDVFVAVBVDAUCJZHZVDVLVFMVAAUDHZDVMHZVNVBAUEZVMABDVMNZEUFZVMACDV - RFUGOZVMVFACVDVRVFNZFTUKVAVOVPVLDUHVBVQVSVMACDVRFUIOZPVCVKDVEVIMZVCVBVKWC + UBVCVDVDCJZDVFVAVBVDAUCJZHZVDVLVFMVAAUDHZDVMHZVNVBAUKZVMABDVMNZEUEZVMACDV + RFUFOZVMVFACVDVRVFNZFTUGVAVOVPVLDUHVBVQVSVMACDVRFUIOZPVCVKDVEVIMZVCVBVKWC IVAVBUJVFAVHBVEVIDEVENZWAVHNZVINZQUTZULVMVFAVEVDDVRWARSVCVHGHZVDDVIMWCVJV AWHVBAVHWEUMUNZVCVDVLDVIVCWHVNVDVLVIMWIVTVMVIVHCVDVMAVHWEVRUOZWFACVHWEFUP TUQWBPVCVKWCWGURVMVIVHVEVDDWJWFRSVFAVHBVEVIVDEWDWAWEWFQUS $. @@ -229365,10 +229582,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.) $) dvrcl $p |- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( X ./ Y ) e. B ) $= - ( crg wcel w3a co cinvr cfv cmulr wceq eqid dvrval 3adant1 rnginvcl rngcl - 3adant2 syld3an3 eqeltrd ) CJKZEAKZFDKZLEFBMZEFCNOZOZCPOZMZAUGUHUIUMQUFAB - CULDUJEFGULRZHUJRZISTUFUGUHUKAKZUMAKUFUHUPUGACDUJFHUOGUAUCACULEUKGUNUBUDU - E $. + ( crg wcel w3a co cinvr cfv cmulr wceq eqid dvrval 3adant1 3adant2 ringcl + ringinvcl syld3an3 eqeltrd ) CJKZEAKZFDKZLEFBMZEFCNOZOZCPOZMZAUGUHUIUMQUF + ABCULDUJEFGULRZHUJRZISTUFUGUHUKAKZUMAKUFUHUPUGACDUJFHUOGUCUAACULEUKGUNUBU + DUE $. $} ${ @@ -229398,7 +229615,7 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( A cancellation law for division. ( ~ div1 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.) $) dvr1 $p |- ( ( R e. Ring /\ X e. B ) -> ( X ./ .1. ) = X ) $= - ( crg wcel wa co cinvr cfv cmulr cui wceq id eqid 1unit syl2anr rngridm + ( crg wcel wa co cinvr cfv cmulr cui wceq id eqid 1unit syl2anr ringridm dvrval 1rinv adantr oveq2d 3eqtrd ) CIJZEAJZKZEDBLZEDCMNZNZCONZLZEDUNLEUI UIDCPNZJUKUOQUHUIRCUPDUPSZHTABCUNUPULEDFUNSZUQULSZGUCUAUJUMDEUNUHUMDQUICD ULUSHUDUEUFACUNDEFURHUBUG $. @@ -229414,11 +229631,11 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of dvrass $p |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .x. Y ) ./ Z ) = ( X .x. ( Y ./ Z ) ) ) $= ( crg wcel w3a co cfv wceq syl2anc dvrval cinvr simpr1 simpr2 simpr3 eqid - wa simpl rnginvcl rngass syl13anc rngcl 3adant3r3 oveq2d 3eqtr4d ) CMNZFA - NZGANZHENZOZUFZFGDPZHCUAQZQZDPZFGVCDPZDPZVAHBPZFGHBPZDPUTUOUPUQVCANZVDVFR - UOUSUGZUOUPUQURUBUOUPUQURUCZUTUOURVIVJUOUPUQURUDZACEVBHJVBUEZIUHSACDFGVCI - LUIUJUTVAANZURVGVDRUOUPUQVNURACDFGILUKULVLABCDEVBVAHILJVMKTSUTVHVEFDUTUQU - RVHVERVKVLABCDEVBGHILJVMKTSUMUN $. + wa simpl ringinvcl ringass syl13anc ringcl 3adant3r3 oveq2d 3eqtr4d ) CMN + ZFANZGANZHENZOZUFZFGDPZHCUAQZQZDPZFGVCDPZDPZVAHBPZFGHBPZDPUTUOUPUQVCANZVD + VFRUOUSUGZUOUPUQURUBUOUPUQURUCZUTUOURVIVJUOUPUQURUDZACEVBHJVBUEZIUHSACDFG + VCILUIUJUTVAANZURVGVDRUOUPUQVNURACDFGILUKULVLABCDEVBVAHILJVMKTSUTVHVEFDUT + UQURVHVERVKVLABCDEVBGHILJVMKTSUMUN $. $( A cancellation law for division. ( ~ divcan1 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, @@ -229426,11 +229643,11 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of dvrcan1 $p |- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( ( X ./ Y ) .x. Y ) = X ) $= ( crg wcel w3a co cfv wceq eqid 3adant2 eqtrd cinvr dvrval 3adant1 oveq1d - simp1 simp2 rnginvcl unitcl 3ad2ant3 rngass syl13anc cur unitlinv rngridm - oveq2d 3adant3 ) CLMZFAMZGEMZNZFGBOZGDOFGCUAPZPZDOZGDOZFUTVAVDGDURUSVAVDQ - UQABCDEVBFGHKIVBRZJUBUCUDUTVEFVCGDOZDOZFUTUQURVCAMZGAMZVEVHQUQURUSUEUQURU - SUFUQUSVIURACEVBGIVFHUGSUSUQVJURACEGHIUHUIACDFVCGHKUJUKUTVHFCULPZDOZFUTVG - VKFDUQUSVGVKQURCDEVKVBGIVFKVKRZUMSUOUQURVLFQUSACDVKFHKVMUNUPTTT $. + simp1 ringinvcl unitcl 3ad2ant3 ringass syl13anc unitlinv oveq2d ringridm + simp2 cur 3adant3 ) CLMZFAMZGEMZNZFGBOZGDOFGCUAPZPZDOZGDOZFUTVAVDGDURUSVA + VDQUQABCDEVBFGHKIVBRZJUBUCUDUTVEFVCGDOZDOZFUTUQURVCAMZGAMZVEVHQUQURUSUEUQ + URUSUNUQUSVIURACEVBGIVFHUFSUSUQVJURACEGHIUGUHACDFVCGHKUIUJUTVHFCUOPZDOZFU + TVGVKFDUQUSVGVKQURCDEVKVBGIVFKVKRZUKSULUQURVLFQUSACDVKFHKVMUMUPTTT $. $( A cancellation law for division. ( ~ divcan3 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, @@ -229438,10 +229655,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of dvrcan3 $p |- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( ( X .x. Y ) ./ Y ) = X ) $= ( crg wcel w3a co cur cfv wceq simp1 simp2 unitcl 3ad2ant3 simp3 syl13anc - dvrass eqid dvrid 3adant2 oveq2d rngridm 3adant3 3eqtrd ) CLMZFAMZGEMZNZF - GDOGBOZFGGBOZDOZFCPQZDOZFUPUMUNGAMZUOUQUSRUMUNUOSUMUNUOTUOUMVBUNACEGHIUAU - BUMUNUOUCABCDEFGGHIJKUEUDUPURUTFDUMUOURUTRUNBCEUTGIJUTUFZUGUHUIUMUNVAFRUO - ACDUTFHKVCUJUKUL $. + dvrass eqid dvrid 3adant2 oveq2d ringridm 3adant3 3eqtrd ) CLMZFAMZGEMZNZ + FGDOGBOZFGGBOZDOZFCPQZDOZFUPUMUNGAMZUOUQUSRUMUNUOSUMUNUOTUOUMVBUNACEGHIUA + UBUMUNUOUCABCDEFGGHIJKUEUDUPURUTFDUMUOURUTRUNBCEUTGIJUTUFZUGUHUIUMUNVAFRU + OACDUTFHKVCUJUKUL $. $} ${ @@ -229454,24 +229671,24 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of dvreq1 $p |- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( ( X ./ Y ) = .1. <-> X = Y ) ) $= ( crg wcel w3a co wceq cmulr cfv oveq1 3adant2 eqid dvrcan1 unitcl sylan2 - rnglidm eqeq12d syl5ib dvrid eqeq1d syl5ibrcom impbid ) CLMZFAMZGDMZNZFGB - OZEPZFGPZUQUPGCQRZOZEGUSOZPUOURUPEGUSSUOUTFVAGABCUSDFGHIJUSUAZUBULUNVAGPZ - UMUNULGAMVCACDGHIUCACUSEGHVBKUEUDTUFUGUOUQURGGBOZEPZULUNVEUMBCDEGIJKUHTUR - UPVDEFGGBSUIUJUK $. + ringlidm eqeq12d syl5ib dvrid eqeq1d syl5ibrcom impbid ) CLMZFAMZGDMZNZFG + BOZEPZFGPZUQUPGCQRZOZEGUSOZPUOURUPEGUSSUOUTFVAGABCUSDFGHIJUSUAZUBULUNVAGP + ZUMUNULGAMVCACDGHIUCACUSEGHVBKUEUDTUFUGUOUQURGGBOZEPZULUNVEUMBCDEGIJKUHTU + RUPVDEFGGBSUIUJUK $. $} ${ - rnginvdv.b $e |- B = ( Base ` R ) $. - rnginvdv.u $e |- U = ( Unit ` R ) $. - rnginvdv.d $e |- ./ = ( /r ` R ) $. - rnginvdv.o $e |- .1. = ( 1r ` R ) $. - rnginvdv.i $e |- I = ( invr ` R ) $. + ringinvdv.b $e |- B = ( Base ` R ) $. + ringinvdv.u $e |- U = ( Unit ` R ) $. + ringinvdv.d $e |- ./ = ( /r ` R ) $. + ringinvdv.o $e |- .1. = ( 1r ` R ) $. + ringinvdv.i $e |- I = ( invr ` R ) $. $( Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.) $) - rnginvdv $p |- ( ( R e. Ring /\ X e. U ) -> ( I ` X ) = ( .1. ./ X ) ) $= - ( crg wcel wa co cfv cmulr wceq rngidcl eqid dvrval sylan rnginvcl syldan - rnglidm eqtr2d ) CMNZGDNZOEGBPZEGFQZCRQZPZUKUHEANUIUJUMSACEHKTABCULDFEGHU - LUAZILJUBUCUHUIUKANUMUKSACDFGILHUDACULEUKHUNKUFUEUG $. + ringinvdv $p |- ( ( R e. Ring /\ X e. U ) -> ( I ` X ) = ( .1. ./ X ) ) $= + ( crg wcel wa co cfv cmulr wceq ringidcl dvrval ringinvcl ringlidm syldan + eqid sylan eqtr2d ) CMNZGDNZOEGBPZEGFQZCRQZPZUKUHEANUIUJUMSACEHKTABCULDFE + GHULUEZILJUAUFUHUIUKANUMUKSACDFGILHUBACULEUKHUNKUCUDUG $. $} ${ @@ -229610,13 +229827,13 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of Carneiro, 4-Dec-2014.) $) irredn0 $p |- ( ( R e. Ring /\ X e. I ) -> X =/= .0. ) $= ( vx vy crg wcel wa wn cfv cv co wceq wrex eqid adantr eqeq1d wne cmulr - cui cbs cdif rng0cl anim1i eldif sylibr rnglz mpdan oveq1 oveq2 rspc2ev - wo syl3anc ex wb isnirred syl mpbird simpr eleq1 syl5ibcom necon3bd mpd - orrd ) AIJZCBJZKZDBJZLZCDUAVHVLVIVHVLDAUCMZJZGNZHNZAUBMZOZDPZHAUDMZVMUE - ZQGWAQZUOZVHVNWBVHVNLZWBVHWDKZDWAJZWFDDVQOZDPZWBWEDVTJZWDKWFVHWIWDVTADV - TRZFUFZUGDVTVMUHUIZWLVHWHWDVHWIWHWKVTAVQDDWJVQRZFUJUKSVSWHDVPVQOZDPGHDD - WAWAVODPVRWNDVODVPVQULTVPDPWNWGDVPDDVQUMTUNUPUQVGVHWIVLWCURWKGHVTAVQVMB - WADWJVMREWARWMUSUTVASVJVKCDVJVICDPVKVHVIVBCDBVCVDVEVF $. + cui cbs wo ring0cl anim1i eldif sylibr ringlz mpdan oveq1 oveq2 rspc2ev + cdif syl3anc ex orrd isnirred syl mpbird simpr eleq1 syl5ibcom necon3bd + wb mpd ) AIJZCBJZKZDBJZLZCDUAVHVLVIVHVLDAUCMZJZGNZHNZAUBMZOZDPZHAUDMZVM + UOZQGWAQZUEZVHVNWBVHVNLZWBVHWDKZDWAJZWFDDVQOZDPZWBWEDVTJZWDKWFVHWIWDVTA + DVTRZFUFZUGDVTVMUHUIZWLVHWHWDVHWIWHWKVTAVQDDWJVQRZFUJUKSVSWHDVPVQOZDPGH + DDWAWAVODPVRWNDVODVPVQULTVPDPWNWGDVPDDVQUMTUNUPUQURVHWIVLWCVFWKGHVTAVQV + MBWADWJVMREWARWMUSUTVASVJVKCDVJVICDPVKVHVIVBCDBVCVDVEVG $. $} ${ @@ -229660,26 +229877,26 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of simp2 cdvr simp1 simp3 unitdvcl 3com23 syl2anc irredcl 3ad2ant2 dvrcan3 wi 3expia eleq1d sylibd ad2antrr eldifi ad2antrl dvrcl eldifn unitmulcl wa dvrcan1 eldifd simprr oveq1d ad2antlr dvrass syl13anc 3eqtr3d eqeq1d - oveq2 rspcev rexlimdvaa reximdva orim12d unitcl 3ad2ant3 rngcl isnirred - mtod wb syl 3imtr4d mt4d ) AUAMZEDMZFCMZUBZWLEFBNZDMZWKWLWMUGWNWOCMZJOZ - KOZBNZWOPZKAUCUDZCUEZQZJXCQZUFZECMZWRLOZBNZEPZLXCQZJXCQZUFZWPRZWLRZWNWQ - XGXEXLWNWQWOFAUHUDZNZCMZXGWNWKWMWQXRUQWKWLWMUIZWKWLWMUJZWKWMWQXRWKWQWMX - RXPACWOFHXPSZUKULURUMWNXQECWNWKEXBMZWMXQEPZXSWLWKYBWMXBADEGXBSZUNUOZXTX - BXPABCEFYDHYAIUPTZUSUTWNXDXKJXCWNWRXCMZVGZXAXKKXCYHWSXCMZXAVGZVGZWSFXPN - ZXCMWRYLBNZEPZXKYKYLXBCYKWKWSXBMZWMYLXBMWNWKYGYJXSVAZYIYOYHXAWSXBCVBVCZ - WNWMYGYJXTVAZXBXPACWSFYDHYAVDTYKYLCMZWSCMZYIYTRYHXAWSXBCVEVCYKYSYLFBNZC - MZYTYKWKWMYSUUBUQYPYRWKWMYSUUBWKYSWMUUBABCYLFHIVFULURUMYKUUAWSCYKWKYOWM - UUAWSPYPYQYRXBXPABCWSFYDHYAIVHTUSUTWFVIYKWTFXPNZXQYMEYKWTWOFXPYHYIXAVJV - KYKWKWRXBMZYOWMUUCYMPYPYGUUDWNYJWRXBCVBVLYQYRXBXPABCWRWSFYDHYAIVMVNWNYC - YGYJYFVAVOXJYNLYLXCXHYLPXIYMEXHYLWRBVQVPVRUMVSVTWAWNWOXBMZXNXFWGWNWKYBF - XBMZUUEXSYEWMWKUUFWLXBACFYDHWBWCXBABEFYDIWDTJKXBABCDXCWOYDHGXCSZIWEWHWN - YBXOXMWGYEJLXBABCDXCEYDHGUUGIWEWHWIWJ $. + mtod oveq2 rspcev rexlimdvaa reximdva orim12d wb unitcl 3ad2ant3 ringcl + isnirred syl 3imtr4d mt4d ) AUAMZEDMZFCMZUBZWLEFBNZDMZWKWLWMUGWNWOCMZJO + ZKOZBNZWOPZKAUCUDZCUEZQZJXCQZUFZECMZWRLOZBNZEPZLXCQZJXCQZUFZWPRZWLRZWNW + QXGXEXLWNWQWOFAUHUDZNZCMZXGWNWKWMWQXRUQWKWLWMUIZWKWLWMUJZWKWMWQXRWKWQWM + XRXPACWOFHXPSZUKULURUMWNXQECWNWKEXBMZWMXQEPZXSWLWKYBWMXBADEGXBSZUNUOZXT + XBXPABCEFYDHYAIUPTZUSUTWNXDXKJXCWNWRXCMZVGZXAXKKXCYHWSXCMZXAVGZVGZWSFXP + NZXCMWRYLBNZEPZXKYKYLXBCYKWKWSXBMZWMYLXBMWNWKYGYJXSVAZYIYOYHXAWSXBCVBVC + ZWNWMYGYJXTVAZXBXPACWSFYDHYAVDTYKYLCMZWSCMZYIYTRYHXAWSXBCVEVCYKYSYLFBNZ + CMZYTYKWKWMYSUUBUQYPYRWKWMYSUUBWKYSWMUUBABCYLFHIVFULURUMYKUUAWSCYKWKYOW + MUUAWSPYPYQYRXBXPABCWSFYDHYAIVHTUSUTVQVIYKWTFXPNZXQYMEYKWTWOFXPYHYIXAVJ + VKYKWKWRXBMZYOWMUUCYMPYPYGUUDWNYJWRXBCVBVLYQYRXBXPABCWRWSFYDHYAIVMVNWNY + CYGYJYFVAVOXJYNLYLXCXHYLPXIYMEXHYLWRBVRVPVSUMVTWAWBWNWOXBMZXNXFWCWNWKYB + FXBMZUUEXSYEWMWKUUFWLXBACFYDHWDWEXBABEFYDIWFTJKXBABCDXCWOYDHGXCSZIWGWHW + NYBXOXMWCYEJLXBABCDXCEYDHGUUGIWGWHWIWJ $. $( The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) $) irredlmul $p |- ( ( R e. Ring /\ X e. U /\ Y e. I ) -> ( X .x. Y ) e. I ) $= - ( crg wcel w3a co coppr cfv cmulr cbs eqid opprmul opprrng opprirred + ( crg wcel w3a co coppr cfv cmulr cbs eqid opprmul opprring opprirred opprunit irredrmul syl3an1 3com23 syl5eqelr ) AJKZECKZFDKZLEFBMFEANOZPO ZMZDAQOZAUKBUJFEUMRIUJRZUKRZSUGUIUHULDKZUGUJJKUIUHUPAUJUNTUJUKCDFEAUJDU NGUAAUJCHUNUBUOUCUDUEUF $. @@ -229717,9 +229934,9 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of Carneiro, 4-Dec-2014.) $) irrednegb $p |- ( ( R e. Ring /\ X e. B ) -> ( X e. I <-> ( N ` X ) e. I ) ) $= - ( crg wcel cfv irredneg adantlr wceq cgrp rnggrp grpinvinv sylan adantr - wa eqeltrrd impbida ) BIJZEAJZTZECJZEDKZCJZUCUFUHUDBCDEFGLMUEUHTUGDKZEC - UEUIENZUHUCBOJUDUJBPABDEHGQRSUCUHUICJUDBCDUGFGLMUAUB $. + ( crg wcel wa irredneg adantlr wceq cgrp ringgrp grpinvinv sylan adantr + cfv eqeltrrd impbida ) BIJZEAJZKZECJZEDTZCJZUCUFUHUDBCDEFGLMUEUHKUGDTZE + CUEUIENZUHUCBOJUDUJBPABDEHGQRSUCUHUICJUDBCDUGFGLMUAUB $. $} $} @@ -229763,24 +229980,24 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of dfrhm2 $p |- RingHom = ( r e. Ring , s e. Ring |-> ( ( r GrpHom s ) i^i ( ( mulGrp ` r ) MndHom ( mulGrp ` s ) ) ) ) $= ( vv vw vf vx vy crg cv cbs cfv wceq co wa wral crab wcel cab wb eqid crh - cur cplusg cmulr cmap csb cmpt2 cghm cmgp cmhm df-rnghom wf rnggrp isghm3 - cin cgrp syl2an abbi2dv df-rab fvex elmap anbi1i abbii eqtri syl6eqr cmnd - rngmgp mgpbas mgpplusg rngidval ismhm baib 3anass bitr4i ineq12d r19.26-2 - w3a ancom anass 3bitri a1i rabbiia oveq12 ancoms raleqbi1dv adantr anbi2d - raleq rabeqbidv csbie2 inrab 3eqtr4i syl6reqr mpt2eq3ia ) UABAHHCBIZJKZDA - IZJKZWOUBKZEIZKWQUBKZLZFIZGIZWOUCKZMWTKXCWTKZXDWTKZWQUCKZMLZXCXDWOUDKZMWT - KXFXGWQUDKZMLZNZGCIZOZFXNOZNZEDIZXNUEMZPZUFUFZUGBAHHWOWQUHMZWOUIKZWQUIKZU - JMZUOZUGFGDCEABUKBAHHYAYFWOHQZWQHQZNZYFXIGWPOFWPOZEWRWPUEMZPZXLGWPOFWPOZX - BNZEYKPZUOZYAYIYBYLYEYOYIYBWPWRWTULZYJNZERZYLYIYREYBYGWOUPQWQUPQWTYBQYRSY - HWOUMWQUMGFXEXHWOWQWTWPWRWPTZWRTZXETXHTUNUQURYLWTYKQZYJNZERYSYJEYKUSUUCYR - EUUBYQYJWRWPWTWQJUTZWOJUTZVAZVBVCVDVEYIYEYQYMXBVQZERZYOYIUUGEYEYGYCVFQZYD - VFQZWTYEQZUUGSYHWOYCYCTZVGWQYDYDTZVGUUKUUIUUJNUUGFGWPWRXJXKYCYDWTXAWSWPWO - YCUULYTVHWRWQYDUUMUUAVHWOXJYCUULXJTVIWQXKYDUUMXKTVIWOWSYCUULWSTVJWQXAYDUU - MXATVJVKVLUQURYOUUBYNNZERUUHYNEYKUSUUNUUGEUUNYQYNNUUGUUBYQYNUUFVBYQYMXBVM - VNVCVDVEVOXBXMGWPOZFWPOZNZEYKPZYJYNNZEYKPYAYPUUQUUSEYKUUQUUSSUUBUUQUUPXBN - YJYMNZXBNUUSXBUUPVRUUPUUTXBXIXLFGWPWPVPVBYJYMXBVSVTWAWBCDWPWRXTUURUUEUUDX - NWPLZXRWRLZNZXQUUQEXSYKUVBUVAXSYKLXRWRXNWPUEWCWDUVCXPUUPXBUVAXPUUPSUVBXOU - UOFXNWPXMGXNWPWHWEWFWGWIWJYJYNEYKWKWLWMWNVD $. + cur cplusg cmulr cmap csb cmpt2 cghm cmgp cmhm cin df-rnghom cgrp ringgrp + wf isghm3 syl2an abbi2dv df-rab fvex elmap anbi1i abbii eqtri syl6eqr w3a + cmnd ringmgp mgpbas mgpplusg rngidval baib 3anass bitr4i ineq12d r19.26-2 + ismhm ancom anass 3bitri a1i oveq12 ancoms raleq raleqbi1dv adantr anbi2d + rabbiia rabeqbidv csbie2 inrab 3eqtr4i syl6reqr mpt2eq3ia ) UABAHHCBIZJKZ + DAIZJKZWOUBKZEIZKWQUBKZLZFIZGIZWOUCKZMWTKXCWTKZXDWTKZWQUCKZMLZXCXDWOUDKZM + WTKXFXGWQUDKZMLZNZGCIZOZFXNOZNZEDIZXNUEMZPZUFUFZUGBAHHWOWQUHMZWOUIKZWQUIK + ZUJMZUKZUGFGDCEABULBAHHYAYFWOHQZWQHQZNZYFXIGWPOFWPOZEWRWPUEMZPZXLGWPOFWPO + ZXBNZEYKPZUKZYAYIYBYLYEYOYIYBWPWRWTUOZYJNZERZYLYIYREYBYGWOUMQWQUMQWTYBQYR + SYHWOUNWQUNGFXEXHWOWQWTWPWRWPTZWRTZXETXHTUPUQURYLWTYKQZYJNZERYSYJEYKUSUUC + YREUUBYQYJWRWPWTWQJUTZWOJUTZVAZVBVCVDVEYIYEYQYMXBVFZERZYOYIUUGEYEYGYCVGQZ + YDVGQZWTYEQZUUGSYHWOYCYCTZVHWQYDYDTZVHUUKUUIUUJNUUGFGWPWRXJXKYCYDWTXAWSWP + WOYCUULYTVIWRWQYDUUMUUAVIWOXJYCUULXJTVJWQXKYDUUMXKTVJWOWSYCUULWSTVKWQXAYD + UUMXATVKVQVLUQURYOUUBYNNZERUUHYNEYKUSUUNUUGEUUNYQYNNUUGUUBYQYNUUFVBYQYMXB + VMVNVCVDVEVOXBXMGWPOZFWPOZNZEYKPZYJYNNZEYKPYAYPUUQUUSEYKUUQUUSSUUBUUQUUPX + BNYJYMNZXBNUUSXBUUPVRUUPUUTXBXIXLFGWPWPVPVBYJYMXBVSVTWAWHCDWPWRXTUURUUEUU + DXNWPLZXRWRLZNZXQUUQEXSYKUVBUVAXSYKLXRWRXNWPUEWBWCUVCXPUUPXBUVAXPUUPSUVBX + OUUOFXNWPXMGXNWPWDWEWFWGWIWJYJYNEYKWKWLWMWNVD $. $} $( Define the ring isomorphism relation, analogous to ~ df-gic : Two @@ -229895,13 +230112,13 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of 13-Jun-2015.) $) isrhm2d $p |- ( ph -> F e. ( R RingHom S ) ) $= ( crg wcel wa cghm co cmgp cfv cmhm crh jca cmnd cbs wf cv wceq c0g w3a - wral eqid rngmgp syl ralrimivva rngidval fveq2i 3eqtr3g mgpbas mgpplusg - ghmf 3jca ismhm sylanbrc isrhm ) AEUBUCZFUBUCZUDJEFUEUFUCZJEUGUHZFUGUHZ - UIUFUCZUDJEFUJUFUCAVNVOQRUKAVPVSUAAVQULUCZVRULUCZUDDFUMUHZJUNZBUOZCUOZG - UFJUHWDJUHWEJUHHUFUPZCDUSBDUSZVQUQUHZJUHZVRUQUHZUPZURVSAVTWAAVNVTQEVQVQ - UTZVAVBAVOWARFVRVRUTZVAVBUKAWCWGWKAVPWCUAEFJDWBLWBUTZVIVBAWFBCDDTVCAIJU - HKWIWJSIWHJEIVQWLMVDVEFKVRWMNVDVFVJBCDWBGHVQVRJWJWHDEVQWLLVGWBFVRWMWNVG - EGVQWLOVHFHVRWMPVHWHUTWJUTVKVLUKEFJVQVRWLWMVMVL $. + wral eqid ringmgp ghmf ralrimivva rngidval fveq2i 3eqtr3g 3jca mgpplusg + syl mgpbas ismhm sylanbrc isrhm ) AEUBUCZFUBUCZUDJEFUEUFUCZJEUGUHZFUGUH + ZUIUFUCZUDJEFUJUFUCAVNVOQRUKAVPVSUAAVQULUCZVRULUCZUDDFUMUHZJUNZBUOZCUOZ + GUFJUHWDJUHWEJUHHUFUPZCDUSBDUSZVQUQUHZJUHZVRUQUHZUPZURVSAVTWAAVNVTQEVQV + QUTZVAVIAVOWARFVRVRUTZVAVIUKAWCWGWKAVPWCUAEFJDWBLWBUTZVBVIAWFBCDDTVCAIJ + UHKWIWJSIWHJEIVQWLMVDVEFKVRWMNVDVFVGBCDWBGHVQVRJWJWHDEVQWLLVJWBFVRWMWNV + JEGVQWLOVHFHVRWMPVHWHUTWJUTVKVLUKEFJVQVRWLWMVMVL $. $} isrhmd.c $e |- C = ( Base ` S ) $. @@ -229913,8 +230130,8 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.) $) isrhmd $p |- ( ph -> F e. ( R RingHom S ) ) $= - ( crg wcel cgrp rnggrp syl isghmd isrhm2d ) ABCDHIJKLMNOPQRSTUAUBUCABCFGH - IMDEOUDUEUFAHUIUJHUKUJTHULUMAIUIUJIUKUJUAIULUMUGUHUNUO $. + ( crg wcel cgrp ringgrp syl isghmd isrhm2d ) ABCDHIJKLMNOPQRSTUAUBUCABCFG + HIMDEOUDUEUFAHUIUJHUKUJTHULUMAIUIUJIUKUJUAIULUMUGUHUNUO $. $} ${ @@ -229999,20 +230216,20 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) $) pwsco1rhm $p |- ( ph -> ( g e. C |-> ( g o. F ) ) e. ( Z RingHom Y ) ) $= - ( wcel eqid vx vy crg wa cv ccom cmpt cghm co cmgp cfv crh pwsrng syl2anc - cmhm jca cmnd rngmnd syl pwsco1mhm cgrp wceq rnggrp ghmmhmb eleqtrrd cpws - cbs rngmgp cmap pwsbas syl6eqr mgpbas eqtr3d mpteq1d cplusg pwsmgp simpld - eqidd simprd proplem3 mhmpropd 3eltr4d isrhm sylanbrc ) AKUCSZJUCSZUDFDFU - EGUFZUGZKJUHUIZSZWHKUJUKZJUJUKZUOUIZSZUDWHKJULUISAWEWFAEUCSZCISZWEOQECIKM - UMUNZAWOBHSZWFOPEBHJLUMUNZUPAWJWNAWHKJUOUIZWIABCDEFGHIJKLMNAWOEUQSZOEURUS - ZPQRUTAKVASZJVASZWIWTVBAWEXCWQKVCUSAWFXDWSJVCUSKJVDUNVEAFEUJUKZCVFUIZVGUK - ZWGUGXFXEBVFUIZUOUIWHWMABCXGXEFGHIXHXFXHTZXFTZXGTZAWOXEUQSZOEXEXETZVHUSZP - QRUTAFDXGWGAEVGUKZCVIUIZDXGAXPKVGUKZDAXAWPXPXQVBXBQXOECUQIKMXOTZVJUNNVKAX - LWPXPXGVBXNQXOXECUQIXFXJXOEXEXMXRVLVJUNVMVNAUAUBWKVGUKZWLVGUKZWKWLXFXHAXS - VRAXTVRAXSXGVBZWKVOUKZXFVOUKZVBZAWOWPYAYDUDOQXSXGYBYCECXEWKUCIKXFMXMXJWKT - ZXSTXKYBTYCTVPUNZVQAXTXHVGUKZVBZWLVOUKZXHVOUKZVBZAWOWRYHYKUDOPXTYGYIYJEBX - EWLUCHJXHLXMXIWLTZXTTYGTYITYJTVPUNZVQAUAUEZXSSUBUEZXSSUDUAUBYBYCAYAYDYFVS - VTAYNXTSYOXTSUDUAUBYIYJAYHYKYMVSVTWAWBUPKJWHWKWLYEYLWCWD $. + ( wcel eqid vx vy crg wa ccom cmpt cghm cmgp cfv cmhm crh pwsring syl2anc + cv jca cmnd ringmnd syl pwsco1mhm cgrp wceq ringgrp ghmmhmb eleqtrrd cpws + cbs ringmgp cmap pwsbas syl6eqr mgpbas eqtr3d mpteq1d eqidd cplusg pwsmgp + co simpld simprd proplem3 mhmpropd 3eltr4d isrhm sylanbrc ) AKUCSZJUCSZUD + FDFUNGUEZUFZKJUGVQZSZWHKUHUIZJUHUIZUJVQZSZUDWHKJUKVQSAWEWFAEUCSZCISZWEOQE + CIKMULUMZAWOBHSZWFOPEBHJLULUMZUOAWJWNAWHKJUJVQZWIABCDEFGHIJKLMNAWOEUPSZOE + UQURZPQRUSAKUTSZJUTSZWIWTVAAWEXCWQKVBURAWFXDWSJVBURKJVCUMVDAFEUHUIZCVEVQZ + VFUIZWGUFXFXEBVEVQZUJVQWHWMABCXGXEFGHIXHXFXHTZXFTZXGTZAWOXEUPSZOEXEXETZVG + URZPQRUSAFDXGWGAEVFUIZCVHVQZDXGAXPKVFUIZDAXAWPXPXQVAXBQXOECUPIKMXOTZVIUMN + VJAXLWPXPXGVAXNQXOXECUPIXFXJXOEXEXMXRVKVIUMVLVMAUAUBWKVFUIZWLVFUIZWKWLXFX + HAXSVNAXTVNAXSXGVAZWKVOUIZXFVOUIZVAZAWOWPYAYDUDOQXSXGYBYCECXEWKUCIKXFMXMX + JWKTZXSTXKYBTYCTVPUMZVRAXTXHVFUIZVAZWLVOUIZXHVOUIZVAZAWOWRYHYKUDOPXTYGYIY + JEBXEWLUCHJXHLXMXIWLTZXTTYGTYITYJTVPUMZVRAUAUNZXSSUBUNZXSSUDUAUBYBYCAYAYD + YFVSVTAYNXTSYOXTSUDUAUBYIYJAYHYKYMVSVTWAWBUOKJWHWKWLYEYLWCWD $. $} ${ @@ -230026,21 +230243,21 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) $) pwsco2rhm $p |- ( ph -> ( g e. B |-> ( F o. g ) ) e. ( Y RingHom Z ) ) $= - ( wcel co cfv syl eqid vx vy crg wa cv ccom cmpt cghm cmgp rhmrcl1 pwsrng - cmhm crh syl2anc rhmrcl2 rhmghm ghmmhm pwsco2mhm cgrp wceq rnggrp ghmmhmb - jca eleqtrrd cpws rhmmhm cmap pwsbas syl6eqr rngmgp mgpbas eqtr3d mpteq1d - cmnd eqidd cplusg pwsmgp simpld simprd proplem3 mhmpropd 3eltr4d sylanbrc - cbs isrhm ) AIUCPZJUCPZUDFCGFUEUFZUGZIJUHQZPZWIIUIRZJUIRZULQZPZUDWIIJUMQP - AWFWGADUCPZBHPZWFAGDEUMQPZWPODEGUJSZNDBHIKUKUNZAEUCPZWQWGAWRXAODEGUOSZNEB - HJLUKUNZVCAWKWOAWIIJULQZWJABCDEFGHIJKLMNAGDEUHQPZGDEULQPAWRXEODEGUPSDEGUQ - SURAIUSPZJUSPZWJXDUTAWFXFWTIVASAWGXGXCJVASIJVBUNVDAFDUIRZBVEQZWDRZWHUGXIE - UIRZBVEQZULQWIWNABXJXHXKFGHXIXLXITZXLTZXJTZNAWRGXHXKULQPODEGXHXKXHTZXKTZV - FSURAFCXJWHADWDRZBVGQZCXJAXSIWDRZCAWPWQXSXTUTWSNXRDBUCHIKXRTZVHUNMVIAXHVN - PZWQXSXJUTAWPYBWSDXHXPVJSNXRXHBVNHXIXMXRDXHXPYAVKVHUNVLVMAUAUBWLWDRZWMWDR - ZWLWMXIXLAYCVOAYDVOAYCXJUTZWLVPRZXIVPRZUTZAWPWQYEYHUDWSNYCXJYFYGDBXHWLUCH - IXIKXPXMWLTZYCTXOYFTYGTVQUNZVRAYDXLWDRZUTZWMVPRZXLVPRZUTZAXAWQYLYOUDXBNYD - YKYMYNEBXKWMUCHJXLLXQXNWMTZYDTYKTYMTYNTVQUNZVRAUAUEZYCPUBUEZYCPUDUAUBYFYG - AYEYHYJVSVTAYRYDPYSYDPUDUAUBYMYNAYLYOYQVSVTWAWBVCIJWIWLWMYIYPWEWC $. + ( wcel co cfv syl eqid vx vy crg ccom cmpt cghm cmgp cmhm rhmrcl1 pwsring + crh syl2anc rhmrcl2 jca rhmghm ghmmhm pwsco2mhm cgrp wceq ringgrp ghmmhmb + wa cv eleqtrrd cpws rhmmhm cmap pwsbas syl6eqr cmnd ringmgp mgpbas eqtr3d + mpteq1d eqidd cplusg pwsmgp simpld simprd proplem3 mhmpropd 3eltr4d isrhm + cbs sylanbrc ) AIUCPZJUCPZVBFCGFVCUDZUEZIJUFQZPZWIIUGRZJUGRZUHQZPZVBWIIJU + KQPAWFWGADUCPZBHPZWFAGDEUKQPZWPODEGUISZNDBHIKUJULZAEUCPZWQWGAWRXAODEGUMSZ + NEBHJLUJULZUNAWKWOAWIIJUHQZWJABCDEFGHIJKLMNAGDEUFQPZGDEUHQPAWRXEODEGUOSDE + GUPSUQAIURPZJURPZWJXDUSAWFXFWTIUTSAWGXGXCJUTSIJVAULVDAFDUGRZBVEQZWDRZWHUE + XIEUGRZBVEQZUHQWIWNABXJXHXKFGHXIXLXITZXLTZXJTZNAWRGXHXKUHQPODEGXHXKXHTZXK + TZVFSUQAFCXJWHADWDRZBVGQZCXJAXSIWDRZCAWPWQXSXTUSWSNXRDBUCHIKXRTZVHULMVIAX + HVJPZWQXSXJUSAWPYBWSDXHXPVKSNXRXHBVJHXIXMXRDXHXPYAVLVHULVMVNAUAUBWLWDRZWM + WDRZWLWMXIXLAYCVOAYDVOAYCXJUSZWLVPRZXIVPRZUSZAWPWQYEYHVBWSNYCXJYFYGDBXHWL + UCHIXIKXPXMWLTZYCTXOYFTYGTVQULZVRAYDXLWDRZUSZWMVPRZXLVPRZUSZAXAWQYLYOVBXB + NYDYKYMYNEBXKWMUCHJXLLXQXNWMTZYDTYKTYMTYNTVQULZVRAUAVCZYCPUBVCZYCPVBUAUBY + FYGAYEYHYJVSVTAYRYDPYSYDPVBUAUBYMYNAYLYOYQVSVTWAWBUNIJWIWLWMYIYPWCWE $. $} ${ @@ -230054,10 +230271,10 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of f1rhm0to0 $p |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = N <-> X = .0. ) ) $= ( crh co wcel wf1 cfv wceq syl 3ad2ant1 cghm rhmghm ghmid eqeq2d wi simp2 - w3a simp3 crg rhmrcl1 rng0cl f1veqaeq syl12anc sylbird fveq2 sylan9eqr ex - impbid ) ECDMNOZABEPZGAOZUGZGEQZFRZGHRZVBVDVCHEQZRZVEVBVFFVCUSUTVFFRZVAUS - ECDUANOVHCDEUBCDEHFLKUCSTZUDVBUTVAHAOZVGVEUEUSUTVAUFUSUTVAUHUSUTVJVAUSCUI - OVJCDEUJACHILUKSTABGHEULUMUNVBVEVDVEVBVCVFFGHEUOVIUPUQUR $. + w3a simp3 crg rhmrcl1 ring0cl f1veqaeq syl12anc fveq2 sylan9eqr ex impbid + sylbird ) ECDMNOZABEPZGAOZUGZGEQZFRZGHRZVBVDVCHEQZRZVEVBVFFVCUSUTVFFRZVAU + SECDUANOVHCDEUBCDEHFLKUCSTZUDVBUTVAHAOZVGVEUEUSUTVAUFUSUTVAUHUSUTVJVAUSCU + IOVJCDEUJACHILUKSTABGHEULUMURVBVEVDVEVBVCVFFGHEUNVIUOUPUQ $. ${ $d A x $. $d B x $. $d F x $. $d N x $. $d R x $. $d S x $. @@ -230102,20 +230319,20 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of ( vx vy co wcel wceq wa cfv wb adantr crh wf1 ccnv csn cima cv simpl f1fn wfn adantl elpreima syl biimpa simpld simprd elsnc sylib wi w3a f1rhm0to0 fvex biimpd 3expa imp syl21anc ex elsn syl6ibr ssrdv wss crg cgrp rhmrcl1 - rnggrp grpidcl 3syl cghm rhmghm ghmid sylibr wf ffn mpbir2and snssd eqssd - rhmf wral simpr2l simpr2r simpr3 eqid ghmeqker syl31anc eleqtrd grpsubeq0 - csg simpr1 ovex syl3anc mpbid 3anassrs ralrimivva dff13 sylanbrc impbida + ringgrp grpidcl 3syl cghm rhmghm ghmid sylibr wf rhmf ffn mpbir2and snssd + eqssd wral csg simpr2l simpr2r simpr3 eqid ghmeqker syl31anc eleqtrd ovex + simpr1 grpsubeq0 syl3anc mpbid 3anassrs ralrimivva dff13 sylanbrc impbida ) ECDUANOZABEUBZEUCGUDZUEZFUDZPZXFXGQZXIXJXLLXIXJXLLUFZXIOZXMFPZXMXJOXLXN XOXLXNQZXLXMAOZXMERZGPZXOXLXNUGXPXQXRXHOZXLXNXQXTQZXLEAUIZXNYASXGYBXFABEU HUJAXMXHEUKULUMZUNXPXTXSXPXQXTYCUOXRGXMEVAUPUQXLXQQXSXOXFXGXQXSXOURXFXGXQ USXSXOABCDEGXMFHIKJUTVBVCVDVEVFLFVGVHVIXFXJXIVJXGXFFXIXFFXIOZFAOZFERZXHOZ XFCVKOZCVLOZYECDEVMZCVNZACFHJVOVPXFYFGPZYGXFECDVQNOZYLCDEVRZCDEFGJKVSULYF - GFEVAUPVTXFABEWAZYBYDYEYGQSABCDEHIWFZABEWBAFXHEUKVPWCWDTWEXFXKQZYOXRMUFZE + GFEVAUPVTXFABEWAZYBYDYEYGQSABCDEHIWBZABEWCAFXHEUKVPWDWETWFXFXKQZYOXRMUFZE RPZXMYRPZURZMAWGLAWGXGXFYOXKYPTYQUUALMAAYQXQYRAOZQZQYSYTXFXKUUCYSYTXFXKUU - CYSUSZQZXMYRCWPRZNZFPZYTUUEUUGXJOUUHUUEUUGXIXJUUEYMXQUUBYSUUGXIOZXFYMUUDY - NTXQUUBXKYSXFWHZXQUUBXKYSXFWIZXFXKUUCYSWJYMXQUUBUSYSUUIACDXMEXIUUFYRGHKXI - WKUUFWKZWLUMWMXFXKUUCYSWQWNUUGFXMYRUUFWRUPUQUUEYIXQUUBUUHYTSUUEYHYIXFYHUU - DYJTYKULUUJUUKACUUFXMYRFHJUULWOWSWTXAVFXBLMABEXCXDXE $. + CYSUSZQZXMYRCWHRZNZFPZYTUUEUUGXJOUUHUUEUUGXIXJUUEYMXQUUBYSUUGXIOZXFYMUUDY + NTXQUUBXKYSXFWIZXQUUBXKYSXFWJZXFXKUUCYSWKYMXQUUBUSYSUUIACDXMEXIUUFYRGHKXI + WLUUFWLZWMUMWNXFXKUUCYSWQWOUUGFXMYRUUFWPUPUQUUEYIXQUUBUUHYTSUUEYHYIXFYHUU + DYJTYKULUUJUUKACUUFXMYRFHJUULWRWSWTXAVFXBLMABEXCXDXE $. $} ${ @@ -230220,7 +230437,7 @@ must be abelian (see ~ rngcom ), care must be taken that in the case of $( A division ring is a group. (Contributed by NM, 8-Sep-2011.) $) drnggrp $p |- ( R e. DivRing -> R e. Grp ) $= - ( cdr wcel crg cgrp drngrng rnggrp syl ) ABCADCAECAFAGH $. + ( cdr wcel crg cgrp drngrng ringgrp syl ) ABCADCAECAFAGH $. $( A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) $) @@ -230240,28 +230457,28 @@ nonzero elements form a group under multiplication (from which it ( vx wcel cfv wceq wa cgrp eqid cress co adantr syl2anc wbr cvv cdif cmgp cdr crg cui csn isdrng adantl syl6eqr unitgrp eqeltrrd cv wne unitcl cdvr oveq2 c0g cmulr wss cbs difss mgpbas ressbas2 ax-mp grpidcl eldifsn sylib - ad2antlr simprd eldifad simpr dvrcan1 syl3anc dvrcl rngrz 3netr4d necon3i - simpll syl sylanbrc ssrdv cur cdsr coppr cminusg eldifi grpinvcl dvdsrmul - ex adantll cplusg fvex eqeltri mgpplusg ressplusg grplinv rngidcl rnglidm - difexg mp2b mpdan 1unit grpid mpbid eqtrd breqtrd opprbas opprmul grprinv - wb sseldd syl5eq isunit eqssd impbida pm5.32i bitri ) BUCIBUDIZBUEJZADUFZ - UAZKZLZXRCMIZLZABXSDEXSNZFUGXRYBYDXRYBYDYCBUBJZXSOPZCMYCYHYGYAOPZCYBYHYIK - XRXSYAYGOUPUHGUIXRYHMIYBBXSYHYFYHNUJQUKYEXSYAYEHXSYAYEHULZXSIZYJYAIZYEYKL - ZYJAIZYJDUMZYLYKYNYEABXSYJEYFUNUHYMCUQJZYJBUOJZPZYJBURJZPZYRDYSPZUMYOYMYP - DYTUUAYMYPAIZYPDUMZYMYPYAIZUUBUUCLYDUUDXRYKYACYPYAAUSYACUTJKAXTVAYAACYGGA - BYGYGNZEVBVCVDZYPNZVEVHZYPADVFVGVIYMXRUUBYKYTYPKXRYDYKVRZYMYPAXTUUHVJZYEY - KVKZAYQBYSXSYPYJEYFYQNZYSNZVLVMYMXRYRAIZUUADKUUIYMXRUUBYKUUNUUIUUJUUKAYQB - XSYPYJEYFUULVNVMABYSYRDEUUMFVORVPYJDYTUUAYJDYRYSUPVQVSYJADVFVTWIWAZYEHYAX - SYEYLYKYEYLLZYJBWBJZBWCJZSYJUUQBWDJZWCJZSYKUUPYJYJCWEJZJZYJYSPZUUQUURUUPY - NUVBAIZYJUVCUURSYLYNYEYJAXTWFUHZUUPUVBAXTYDYLUVBYAIXRYACUVAYJUUFUVANZWGWJ - VJZAUURBYSYJUVBEUURNZUUMWHRUUPUVCYPUUQYDYLUVCYPKXRYAYSCUVAYJYPUUFATIYATIY - SCWKJKABUTJTEBUTWLWMAXTTWSYAYSYGCTGBYSYGUUEUUMWNWOWTZUUGUVFWPWJYEYPUUQKZY - LYEUUQUUQYSPUUQKZUVJXRUVKYDXRUUQAIUVKABUUQEUUQNZWQABYSUUQUUQEUUMUVLWRXAQY - EYDUUQYAIUVKUVJXJXRYDVKYEXSYAUUQUUOXRUUQXSIYDBXSUUQYFUVLXBQXKYAYSCUUQYPUU - FUVIUUGXCRXDQZXEXFUUPYJUVBYJUUSURJZPZUUQUUTUUPYNUVDYJUVOUUTSUVEUVGAUUTUUS - UVNYJUVBABUUSUUSNZEXGUUTNZUVNNZWHRUUPUVOYJUVBYSPZUUQABUVNYSUUSUVBYJEUUMUV - PUVRXHUUPUVSYPUUQYDYLUVSYPKXRYAYSCUVAYJYPUUFUVIUUGUVFXIWJUVMXEXLXFUURBUUS - XSUUQUUTYJYFUVLUVHUVPUVQXMVTWIWAXNXOXPXQ $. + ad2antlr simprd simpll eldifad simpr dvrcan1 syl3anc dvrcl ringrz 3netr4d + necon3i syl sylanbrc ssrdv cur cdsr coppr cminusg eldifi grpinvcl adantll + ex dvdsrmul cplusg fvex eqeltri mgpplusg ressplusg mp2b ringidcl ringlidm + difexg grplinv mpdan wb 1unit grpid mpbid breqtrd opprbas opprmul grprinv + sseldd eqtrd syl5eq isunit eqssd impbida pm5.32i bitri ) BUCIBUDIZBUEJZAD + UFZUAZKZLZXRCMIZLZABXSDEXSNZFUGXRYBYDXRYBYDYCBUBJZXSOPZCMYCYHYGYAOPZCYBYH + YIKXRXSYAYGOUPUHGUIXRYHMIYBBXSYHYFYHNUJQUKYEXSYAYEHXSYAYEHULZXSIZYJYAIZYE + YKLZYJAIZYJDUMZYLYKYNYEABXSYJEYFUNUHYMCUQJZYJBUOJZPZYJBURJZPZYRDYSPZUMYOY + MYPDYTUUAYMYPAIZYPDUMZYMYPYAIZUUBUUCLYDUUDXRYKYACYPYAAUSYACUTJKAXTVAYAACY + GGABYGYGNZEVBVCVDZYPNZVEVHZYPADVFVGVIYMXRUUBYKYTYPKXRYDYKVJZYMYPAXTUUHVKZ + YEYKVLZAYQBYSXSYPYJEYFYQNZYSNZVMVNYMXRYRAIZUUADKUUIYMXRUUBYKUUNUUIUUJUUKA + YQBXSYPYJEYFUULVOVNABYSYRDEUUMFVPRVQYJDYTUUAYJDYRYSUPVRVSYJADVFVTWIWAZYEH + YAXSYEYLYKYEYLLZYJBWBJZBWCJZSYJUUQBWDJZWCJZSYKUUPYJYJCWEJZJZYJYSPZUUQUURU + UPYNUVBAIZYJUVCUURSYLYNYEYJAXTWFUHZUUPUVBAXTYDYLUVBYAIXRYACUVAYJUUFUVANZW + GWHVKZAUURBYSYJUVBEUURNZUUMWJRUUPUVCYPUUQYDYLUVCYPKXRYAYSCUVAYJYPUUFATIYA + TIYSCWKJKABUTJTEBUTWLWMAXTTWSYAYSYGCTGBYSYGUUEUUMWNWOWPZUUGUVFWTWHYEYPUUQ + KZYLYEUUQUUQYSPUUQKZUVJXRUVKYDXRUUQAIUVKABUUQEUUQNZWQABYSUUQUUQEUUMUVLWRX + AQYEYDUUQYAIUVKUVJXBXRYDVLYEXSYAUUQUUOXRUUQXSIYDBXSUUQYFUVLXCQXJYAYSCUUQY + PUUFUVIUUGXDRXEQZXKXFUUPYJUVBYJUUSURJZPZUUQUUTUUPYNUVDYJUVOUUTSUVEUVGAUUT + UUSUVNYJUVBABUUSUUSNZEXGUUTNZUVNNZWJRUUPUVOYJUVBYSPZUUQABUVNYSUUSUVBYJEUU + MUVPUVRXHUUPUVSYPUUQYDYLUVSYPKXRYAYSCUVAYJYPUUFUVIUUGUVFXIWHUVMXKXLXFUURB + UUSXSUUQUUTYJYFUVLUVHUVPUVQXMVTWIWAXNXOXPXQ $. $} ${ @@ -230274,12 +230491,12 @@ nonzero elements form a group under multiplication (from which it 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.) $) drngprop $p |- ( K e. DivRing <-> L e. DivRing ) $= ( vx vy crg wcel cui cfv cbs c0g csn cdif wceq wa a1i co eqid eqidd cmulr - cv oveqi unitpropd cplusg grpidpropd sneqd difeq2d eqeq12d pm5.32i anbi1i - cdr rngprop bitri isdrng 3bitr4i ) AHIZAJKZALKZAMKZNZOZPZQZBHIZBJKZUTBMKZ - NZOZPZQZAUMIBUMIVEURVKQVLURVDVKURUSVGVCVJURFGUTABURUTUAZUTBLKPURCRZFUCZGU - CZAUBKZSVOVPBUBKZSPURVOUTIVPUTIQQZVQVRVOVPEUDRUEURVBVIUTURVAVHURFGUTABVMV - NVOVPAUFKZSVOVPBUFKZSPVSVTWAVOVPDUDRUGUHUIUJUKURVFVKABCDEUNULUOUTAUSVAUTT - USTVATUPUTBVGVHCVGTVHTUPUQ $. + cdr cv oveqi unitpropd cplusg grpidpropd difeq2d eqeq12d pm5.32i ringprop + sneqd anbi1i bitri isdrng 3bitr4i ) AHIZAJKZALKZAMKZNZOZPZQZBHIZBJKZUTBMK + ZNZOZPZQZAUCIBUCIVEURVKQVLURVDVKURUSVGVCVJURFGUTABURUTUAZUTBLKPURCRZFUDZG + UDZAUBKZSVOVPBUBKZSPURVOUTIVPUTIQQZVQVRVOVPEUERUFURVBVIUTURVAVHURFGUTABVM + VNVOVPAUGKZSVOVPBUGKZSPVSVTWAVOVPDUERUHUMUIUJUKURVFVKABCDEULUNUOUTAUSVAUT + TUSTVATUPUTBVGVHCVGTVHTUPUQ $. $} ${ @@ -230358,9 +230575,9 @@ nonzero elements form a group under multiplication (from which it analog.) (Contributed by NM, 19-Apr-2014.) $) drnginvrcl $p |- ( ( R e. DivRing /\ X e. B /\ X =/= .0. ) -> ( I ` X ) e. B ) $= - ( cdr wcel wne cfv wa cui eqid drngunit crg wi drngrng rnginvcl ex 3impib - syl sylbird ) BIJZDAJZDEKZDCLAJZUEUFUGMDBNLZJZUHABUIDEFUIOZGPUEBQJZUJUHRB - SULUJUHABUICDUKHFTUAUCUDUB $. + ( cdr wcel wne cfv wa cui eqid drngunit crg wi drngrng ringinvcl sylbird + ex syl 3impib ) BIJZDAJZDEKZDCLAJZUEUFUGMDBNLZJZUHABUIDEFUIOZGPUEBQJZUJUH + RBSULUJUHABUICDUKHFTUBUCUAUD $. $( The multiplicative inverse in a division ring is nonzero. ( ~ recne0 analog.) (Contributed by NM, 19-Apr-2014.) $) @@ -230407,16 +230624,16 @@ nonzero elements form a group under multiplication (from which it drngmul0or $p |- ( ph -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) ) $= ( co wceq wa cfv wcel adantr syl2anc wo wn wne df-ne cinvr oveq2 ad2antlr - cur cdr simpr eqid drnginvrl syl3anc oveq1d crg drngrng drnginvrcl rngass - syl13anc rnglidm 3eqtr3d adantlr rngrz ex syl5bir orrd rnglz oveq1 eqeq1d - syl syl5ibrcom jaod impbid ) AEFDNZGOZEGOZFGOZUAZAVOVRAVOPZVPVQVPUBEGUCZV - SVQEGUDVSVTVQVSVTPZECUEQZQZVNDNZWCGDNZFGVOWDWEOAVTVNGWCDUFUGAVTWDFOVOAVTP - ZWCEDNZFDNZCUHQZFDNZWDFWFWGWIFDWFCUIRZEBRZVTWGWIOAWKVTKSZAWLVTLSZAVTUJZBC - DWIWBEGHIJWIUKZWBUKZULUMUNWFCUORZWCBRZWLFBRZWHWDOAWRVTAWKWRKCUPVJZSWFWKWL - VTWSWMWNWOBCWBEGHIWQUQUMZWNAWTVTMSBCDWCEFHJURUSAWJFOZVTAWRWTXCXAMBCDWIFHJ - WPUTTSVAVBWAWRWSWEGOVSWRVTAWRVOXASSAVTWSVOXBVBBCDWCGHJIVCTVAVDVEVFVDAVPVO - VQAVOVPGFDNZGOZAWRWTXEXAMBCDFGHJIVGTVPVNXDGEGFDVHVIVKAVOVQEGDNZGOZAWRWLXG - XALBCDEGHJIVCTVQVNXFGFGEDUFVIVKVLVM $. + cur cdr simpr drnginvrl syl3anc oveq1d crg drngrng syl drnginvrcl ringass + eqid syl13anc ringlidm 3eqtr3d adantlr ringrz syl5bir ringlz oveq1 eqeq1d + ex orrd syl5ibrcom jaod impbid ) AEFDNZGOZEGOZFGOZUAZAVOVRAVOPZVPVQVPUBEG + UCZVSVQEGUDVSVTVQVSVTPZECUEQZQZVNDNZWCGDNZFGVOWDWEOAVTVNGWCDUFUGAVTWDFOVO + AVTPZWCEDNZFDNZCUHQZFDNZWDFWFWGWIFDWFCUIRZEBRZVTWGWIOAWKVTKSZAWLVTLSZAVTU + JZBCDWIWBEGHIJWIUSZWBUSZUKULUMWFCUNRZWCBRZWLFBRZWHWDOAWRVTAWKWRKCUOUPZSWF + WKWLVTWSWMWNWOBCWBEGHIWQUQULZWNAWTVTMSBCDWCEFHJURUTAWJFOZVTAWRWTXCXAMBCDW + IFHJWPVATSVBVCWAWRWSWEGOVSWRVTAWRVOXASSAVTWSVOXBVCBCDWCGHJIVDTVBVIVEVJVIA + VPVOVQAVOVPGFDNZGOZAWRWTXEXAMBCDFGHJIVFTVPVNXDGEGFDVGVHVKAVOVQEGDNZGOZAWR + WLXGXALBCDEGHJIVDTVQVNXFGFGEDUFVHVKVLVM $. $( A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.) $) @@ -230440,9 +230657,9 @@ nonzero elements form a group under multiplication (from which it $( The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.) $) opprdrng $p |- ( R e. DivRing <-> O e. DivRing ) $= - ( crg wcel cui cfv cbs c0g csn cdif wceq cdr opprrngb anbi1i eqid opprbas - wa isdrng opprunit oppr0 3bitr4i ) ADEZAFGZAHGZAIGZJKLZRBDEZUGRAMEBMEUCUH - UGABCNOUEAUDUFUEPZUDPZUFPZSUEBUDUFUEABCUIQABUDUJCTABUFCUKUASUB $. + ( crg wcel cui cfv cbs c0g csn cdif wceq cdr opprringb anbi1i eqid isdrng + wa opprbas opprunit oppr0 3bitr4i ) ADEZAFGZAHGZAIGZJKLZRBDEZUGRAMEBMEUCU + HUGABCNOUEAUDUFUEPZUDPZUFPZQUEBUDUFUEABCUISABUDUJCTABUFCUKUAQUB $. $} ${ @@ -230467,28 +230684,28 @@ nonzero elements form a group under multiplication (from which it isdrngd $p |- ( ph -> R e. DivRing ) $= ( wcel vz crg cmgp cfv cbs c0g csn cdif cress co cgrp wa cdr wceq difss wss syl5sseq eqid mgpbas ressbas2 syl cmulr cplusg fvex syl6eqel difexg - cvv mgpplusg ressplusg 3syl eqtrd wne eldifsn w3a syl3an1 3expib eleq2d - rngcl anbi12d oveqd eleq12d 3imtr4d 3adant2r 3adant3r sylanbrc syl3an3b - cv 3impib syl3an2b eldifi 3anim123i wi rngass ex eqidd oveq123d eqeq12d - 3anbi123d imp sylan2 cur rngidcl 3eltr4d rnglidm eqeq1d adantrr sylan2b - isgrpd sneqd difeq12d oveq2d eleq1d anbi2d mpbi2and isdrng2 sylibr ) AE - UBTZEUCUDZEUEUDZEUFUDZUGZUHZUIUJZUKTZULZEUMTAXQXRDIUGZUHZUIUJZUKTZYENAB - CUAYGFYHHGAYGXSUPYGYHUEUDUNADYGXSDYFUOJUQYGXSYHXRYHURZXSEXRXRURZXSURZUS - UTVAAFEVBUDZYHVCUDZKADVGTYGVGTYMYNUNADXSVGJEUEVDVEDYFVGVFYGYMXRYHVGYJEY - MXRYKYMURZVHVIVJVKBWGZYGTZAYPDTZYPIVLZULZCWGZYGTZYPUUAFUJZYGTZYPDIVMZUU - BAYTUUADTZUUAIVLZULZUUDUUADIVMAYTUUHVNUUCDTZUUCIVLUUDAYTUUFUUIUUGAYRUUF - UUIYSAYRUUFUUIAYPXSTZUUAXSTZULYPUUAYMUJZXSTZYRUUFULUUIAUUJUUKUUMAXQUUJU - UKUUMNXSEYMYPUUAYLYOVRVOVPAYRUUJUUFUUKADXSYPJVQZADXSUUAJVQZVSAUUCUULDXS - AFYMYPUUAKVTZJWAWBWHWCWDOUUCDIVMWEWFWIYQUUBUAWGZYGTZVNAYRUUFUUQDTZVNZUU - CUUQFUJZYPUUAUUQFUJZFUJZUNZYQYRUUBUUFUURUUSYPDYFWJUUADYFWJUUQDYFWJWKAUU - TUVDAUUJUUKUUQXSTZVNZUULUUQYMUJZYPUUAUUQYMUJZYMUJZUNZUUTUVDAXQUVFUVJWLN - XQUVFUVJXSEYMYPUUAUUQYLYOWMWNVAAYRUUJUUFUUKUUSUVEUUNUUOADXSUUQJVQWRAUVA - UVGUVCUVIAUUCUULUUQUUQFYMKUUPAUUQWOWPAYPYPUVBUVHFYMKAYPWOZAFYMUUAUUQKVT - WPWQWBWSWTAGDTGIVLGYGTAEXAUDZXSGDAXQUVLXSTNXSEUVLYLUVLURZXBVAMJXCPGDIVM - WEYQAYTGYPFUJZYPUNZUUEAYRUVOYSAYRUVOAUUJUVLYPYMUJZYPUNZYRUVOAXQUUJUVQWL - NXQUUJUVQXSEYMUVLYPYLYOUVMXDWNVAUUNAUVNUVPYPAGUVLYPYPFYMKMUVKWPXEWBWSXF - XGYQAYTHYGTZUUEAYTULHDTHIVLUVRQRHDIVMWEXGYQAYTHYPFUJGUNUUESXGXHAYIYDXQA - YHYCUKAYGYBXRUIADXSYFYAJAIXTLXIXJXKXLXMXNXSEYCXTYLXTURYCURXOXP $. + cvv mgpplusg ressplusg 3syl eqtrd wne eldifsn w3a ringcl syl3an1 3expib + eleq2d anbi12d oveqd eleq12d 3impib 3adant2r 3adant3r sylanbrc syl3an3b + cv 3imtr4d syl3an2b eldifi 3anim123i wi ringass 3anbi123d eqidd eqeq12d + ex oveq123d imp sylan2 ringidcl 3eltr4d ringlidm eqeq1d adantrr sylan2b + cur isgrpd sneqd difeq12d oveq2d eleq1d anbi2d mpbi2and isdrng2 sylibr + ) AEUBTZEUCUDZEUEUDZEUFUDZUGZUHZUIUJZUKTZULZEUMTAXQXRDIUGZUHZUIUJZUKTZY + ENABCUAYGFYHHGAYGXSUPYGYHUEUDUNADYGXSDYFUOJUQYGXSYHXRYHURZXSEXRXRURZXSU + RZUSUTVAAFEVBUDZYHVCUDZKADVGTYGVGTYMYNUNADXSVGJEUEVDVEDYFVGVFYGYMXRYHVG + YJEYMXRYKYMURZVHVIVJVKBWGZYGTZAYPDTZYPIVLZULZCWGZYGTZYPUUAFUJZYGTZYPDIV + MZUUBAYTUUADTZUUAIVLZULZUUDUUADIVMAYTUUHVNUUCDTZUUCIVLUUDAYTUUFUUIUUGAY + RUUFUUIYSAYRUUFUUIAYPXSTZUUAXSTZULYPUUAYMUJZXSTZYRUUFULUUIAUUJUUKUUMAXQ + UUJUUKUUMNXSEYMYPUUAYLYOVOVPVQAYRUUJUUFUUKADXSYPJVRZADXSUUAJVRZVSAUUCUU + LDXSAFYMYPUUAKVTZJWAWHWBWCWDOUUCDIVMWEWFWIYQUUBUAWGZYGTZVNAYRUUFUUQDTZV + NZUUCUUQFUJZYPUUAUUQFUJZFUJZUNZYQYRUUBUUFUURUUSYPDYFWJUUADYFWJUUQDYFWJW + KAUUTUVDAUUJUUKUUQXSTZVNZUULUUQYMUJZYPUUAUUQYMUJZYMUJZUNZUUTUVDAXQUVFUV + JWLNXQUVFUVJXSEYMYPUUAUUQYLYOWMWQVAAYRUUJUUFUUKUUSUVEUUNUUOADXSUUQJVRWN + AUVAUVGUVCUVIAUUCUULUUQUUQFYMKUUPAUUQWOWRAYPYPUVBUVHFYMKAYPWOZAFYMUUAUU + QKVTWRWPWHWSWTAGDTGIVLGYGTAEXGUDZXSGDAXQUVLXSTNXSEUVLYLUVLURZXAVAMJXBPG + DIVMWEYQAYTGYPFUJZYPUNZUUEAYRUVOYSAYRUVOAUUJUVLYPYMUJZYPUNZYRUVOAXQUUJU + VQWLNXQUUJUVQXSEYMUVLYPYLYOUVMXCWQVAUUNAUVNUVPYPAGUVLYPYPFYMKMUVKWRXDWH + WSXEXFYQAYTHYGTZUUEAYTULHDTHIVLUVRQRHDIVMWEXFYQAYTHYPFUJGUNUUESXFXHAYIY + DXQAYHYCUKAYGYBXRUIADXSYFYAJAIXTLXIXJXKXLXMXNXSEYCXTYLXTURYCURXOXP $. $} ${ @@ -230500,18 +230717,18 @@ nonzero elements form a group under multiplication (from which it (Contributed by NM, 10-Aug-2013.) $) isdrngrd $p |- ( ph -> R e. DivRing ) $= ( cfv vz coppr cdr wcel cmulr cbs eqid opprbas syl6eq eqidd oppr0 oppr1 - c0g cur crg opprrng syl cv wne wa w3a co wi eleq1 neeq1 anbi12d 3anbi2d - wceq oveq1 neeq1d imbi12d 3anbi3d oveq2 3ad2ant1 oveqd opprmul eqnetrrd - syl6eqr 3com23 chvarv adantr eqtr3d syl5eq isdrngd opprdrng sylibr ) AE - UBTZUCUDEUCUDABUADWGWGUETZGHIADEUFTZWGUFTJWIEWGWGUGZWIUGZUHUIAWHUJAIEUM - TZWGUMTLEWGWLWJWLUGUKUIAGEUNTZWGUNTMEWMWGWJWMUGULUIAEUOUDWGUOUDNEWGWJUP - UQACURZDUDZWNIUSZUTZUAURZDUDZWRIUSZUTZVAZWNWRWHVBZIUSZVCZABURZDUDZXFIUS - ZUTZXAVAZXFWRWHVBZIUSZVCCBWNXFVHZXBXJXDXLXMWQXIAXAXMWOXGWPXHWNXFDVDWNXF - IVEVFVGXMXCXKIWNXFWRWHVIVJVKAWQXIVAZWNXFWHVBZIUSZVCXEBUAXFWRVHZXNXBXPXD - XQXIXAAWQXQXGWSXHWTXFWRDVDXFWRIVEVFVLXQXOXCIXFWRWNWHVMVJVKAXIWQXPAXIWQV - AZXFWNFVBZXOIXRXSXFWNEUETZVBXOXRFXTXFWNAXIFXTVHZWQKVNVOWIEWHXTWGWNXFWKX - TUGZWJWHUGZVPVROVQVSVTVTPQRAXIUTZHXFWHVBXFHXTVBZGWIEWHXTWGHXFWKYBWJYCVP - YDXFHFVBYEGYDFXTXFHAYAXIKWAVOSWBWCWDEWGWJWEWF $. + c0g cur crg opprring syl cv wne wa w3a co wi wceq eleq1 anbi12d 3anbi2d + neeq1 oveq1 neeq1d imbi12d 3anbi3d oveq2 3ad2ant1 oveqd opprmul syl6eqr + eqnetrrd 3com23 chvarv adantr eqtr3d syl5eq isdrngd opprdrng sylibr ) A + EUBTZUCUDEUCUDABUADWGWGUETZGHIADEUFTZWGUFTJWIEWGWGUGZWIUGZUHUIAWHUJAIEU + MTZWGUMTLEWGWLWJWLUGUKUIAGEUNTZWGUNTMEWMWGWJWMUGULUIAEUOUDWGUOUDNEWGWJU + PUQACURZDUDZWNIUSZUTZUAURZDUDZWRIUSZUTZVAZWNWRWHVBZIUSZVCZABURZDUDZXFIU + SZUTZXAVAZXFWRWHVBZIUSZVCCBWNXFVDZXBXJXDXLXMWQXIAXAXMWOXGWPXHWNXFDVEWNX + FIVHVFVGXMXCXKIWNXFWRWHVIVJVKAWQXIVAZWNXFWHVBZIUSZVCXEBUAXFWRVDZXNXBXPX + DXQXIXAAWQXQXGWSXHWTXFWRDVEXFWRIVHVFVLXQXOXCIXFWRWNWHVMVJVKAXIWQXPAXIWQ + VAZXFWNFVBZXOIXRXSXFWNEUETZVBXOXRFXTXFWNAXIFXTVDZWQKVNVOWIEWHXTWGWNXFWK + XTUGZWJWHUGZVPVQOVRVSVTVTPQRAXIUTZHXFWHVBXFHXTVBZGWIEWHXTWGHXFWKYBWJYCV + PYDXFHFVBYEGYDFXTXFHAYAXIKWAVOSWBWCWDEWGWJWEWF $. $} $} @@ -230529,12 +230746,12 @@ nonzero elements form a group under multiplication (from which it 27-Jun-2015.) $) drngpropd $p |- ( ph -> ( K e. DivRing <-> L e. DivRing ) ) $= ( crg wcel cui cfv cbs c0g wceq wa adantr eqid csn cdif cdr eqtr3d cplusg - unitpropd cv co adantlr grpidpropd sneqd difeq12d eqeq12d pm5.32da anbi1d - rngpropd bitrd isdrng 3bitr4g ) AEKLZEMNZEONZEPNZUAZUBZQZRZFKLZFMNZFONZFP - NZUAZUBZQZRZEUCLFUCLAVGUTVNRVOAUTVFVNAUTRZVAVIVEVMAVAVIQUTABCDEFGHJUFSVPV - BVJVDVLAVBVJQUTADVBVJGHUDSVPVCVKVPBCDEFADVBQUTGSADVJQUTHSABUGZDLCUGZDLRVQ - VREUENUHVQVRFUENUHQUTIUIUJUKULUMUNAUTVHVNABCDEFGHIJUPUOUQVBEVAVCVBTVATVCT - URVJFVIVKVJTVITVKTURUS $. + unitpropd cv adantlr grpidpropd sneqd difeq12d eqeq12d pm5.32da ringpropd + co anbi1d bitrd isdrng 3bitr4g ) AEKLZEMNZEONZEPNZUAZUBZQZRZFKLZFMNZFONZF + PNZUAZUBZQZRZEUCLFUCLAVGUTVNRVOAUTVFVNAUTRZVAVIVEVMAVAVIQUTABCDEFGHJUFSVP + VBVJVDVLAVBVJQUTADVBVJGHUDSVPVCVKVPBCDEFADVBQUTGSADVJQUTHSABUGZDLCUGZDLRV + QVREUENUOVQVRFUENUOQUTIUHUIUJUKULUMAUTVHVNABCDEFGHIJUNUPUQVBEVAVCVBTVATVC + TURVJFVIVKVJTVITVKTURUS $. $( If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, @@ -230614,26 +230831,26 @@ nonzero elements form a group under multiplication (from which it $( Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) $) subrgid $p |- ( R e. Ring -> B e. ( SubRing ` R ) ) $= - ( crg wcel cress co wa wss cur cfv csubrg ressid id eqeltrd ancli rngidcl - eqid ssid jctil issubrg sylanbrc ) BDEZUCBAFGZDEZHAAIZBJKZAEZHABLKEUCUEUC - UDBDABDCMUCNOPUCUHUFABUGCUGRZQASTAABUGCUIUAUB $. + ( crg wcel cress co wa wss cur cfv csubrg ressid id eqeltrd eqid ringidcl + ancli ssid jctil issubrg sylanbrc ) BDEZUCBAFGZDEZHAAIZBJKZAEZHABLKEUCUEU + CUDBDABDCMUCNORUCUHUFABUGCUGPZQASTAABUGCUIUAUB $. $} ${ - subrgrng.1 $e |- S = ( R |`s A ) $. + subrgring.1 $e |- S = ( R |`s A ) $. $( A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) - subrgrng $p |- ( A e. ( SubRing ` R ) -> S e. Ring ) $= + subrgring $p |- ( A e. ( SubRing ` R ) -> S e. Ring ) $= ( csubrg cfv wcel cress co crg wa cbs wss cur eqid issubrg simplbi simprd syl5eqel ) ABEFGZCBAHIZJDTBJGZUAJGZTUBUCKABLFZMBNFZAGKAUDBUEUDOUEOPQRS $. $( A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.) $) subrgcrng $p |- ( ( R e. CRing /\ A e. ( SubRing ` R ) ) -> S e. CRing ) $= - ( ccrg wcel csubrg cfv wa cmgp ccmn subrgrng adantl cress co eqid mgpress - crg cmnd crngmgp adantr rngmgp syl eqeltrd subcmn syl2anc eqeltrrd iscrng - sylanbrc ) BEFZABGHZFZIZCRFZCJHZKFCEFULUNUJABCDLMZUMBJHZANOZUOKABCUQEUKDU - QPZQZUMUQKFZURSFURKFUJVAULBUQUSTUAUMURUOSUTUMUNUOSFUPCUOUOPZUBUCUDAUQURUR - PUEUFUGCUOVBUHUI $. + ( ccrg wcel csubrg cfv wa crg cmgp ccmn subrgring adantl cress co mgpress + eqid cmnd crngmgp adantr ringmgp eqeltrd subcmn syl2anc eqeltrrd sylanbrc + syl iscrng ) BEFZABGHZFZIZCJFZCKHZLFCEFULUNUJABCDMNZUMBKHZAOPZUOLABCUQEUK + DUQRZQZUMUQLFZURSFURLFUJVAULBUQUSTUAUMURUOSUTUMUNUOSFUPCUOUORZUBUHUCAUQUR + URRUDUEUFCUOVBUIUG $. $} $( Reverse closure for a subring predicate. (Contributed by Mario Carneiro, @@ -230644,9 +230861,9 @@ nonzero elements form a group under multiplication (from which it $( A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.) $) subrgsubg $p |- ( A e. ( SubRing ` R ) -> A e. ( SubGrp ` R ) ) $= - ( csubrg cfv wcel cgrp cbs wss cress csubg crg subrgrcl rnggrp eqid subrgss - co syl subrgrng issubg syl3anbrc ) ABCDEZBFEZABGDZHBAIPZFEZABJDEUABKEUBABLB - MQAUCBUCNZOUAUDKEUEABUDUDNRUDMQUCABUFST $. + ( csubrg cfv wcel cgrp cbs wss cress co csubg crg subrgrcl ringgrp syl eqid + subrgss subrgring issubg syl3anbrc ) ABCDEZBFEZABGDZHBAIJZFEZABKDEUABLEUBAB + MBNOAUCBUCPZQUAUDLEUEABUDUDPRUDNOUCABUFST $. ${ subrg0.1 $e |- S = ( R |`s A ) $. @@ -230685,12 +230902,12 @@ nonzero elements form a group under multiplication (from which it subrg1 $p |- ( A e. ( SubRing ` R ) -> .1. = ( 1r ` S ) ) $= ( vx cfv wcel cur cbs cmulr co wceq wa eqid crg oveqd eqeq1d syldan sylan csubrg cv wral subrg1cl subrgbas eleqtrd subrgss eqsstr3d sselda subrgrcl - rngidmlem ressmulr anbi12d biimpa ralrimiva subrgrng isrngid syl mpbi2and - wb syl6reqr ) ABUBHZIZCJHZBJHZDVDVFCKHZIZVFGUCZCLHZMZVINZVIVFVJMZVINZOZGV - GUDZVEVFNZVDVFAVGABVFVFPZUEABCEUFZUGVDVOGVGVDVIVGIVIBKHZIZVOVDVGVTVIVDVGA - VTVSAVTBVTPZUHUIUJVDWAVFVIBLHZMZVINZVIVFWCMZVINZOZVOVDBQIWAWHABUKVTBWCVFV - IWBWCPZVRULUAVDWHVOVDWEVLWGVNVDWDVKVIVDWCVJVFVIABCWCVCEWIUMZRSVDWFVMVIVDW - CVJVIVFWJRSUNUOTTUPVDCQIVHVPOVQVAABCEUQGVGCVJVEVFVGPVJPVEPURUSUTFVB $. + ringidmlem ressmulr biimpa ralrimiva subrgring isringid mpbi2and syl6reqr + anbi12d wb syl ) ABUBHZIZCJHZBJHZDVDVFCKHZIZVFGUCZCLHZMZVINZVIVFVJMZVINZO + ZGVGUDZVEVFNZVDVFAVGABVFVFPZUEABCEUFZUGVDVOGVGVDVIVGIVIBKHZIZVOVDVGVTVIVD + VGAVTVSAVTBVTPZUHUIUJVDWAVFVIBLHZMZVINZVIVFWCMZVINZOZVOVDBQIWAWHABUKVTBWC + VFVIWBWCPZVRULUAVDWHVOVDWEVLWGVNVDWDVKVIVDWCVJVFVIABCWCVCEWIUMZRSVDWFVMVI + VDWCVJVIVFWJRSUTUNTTUOVDCQIVHVPOVQVAABCEUPGVGCVJVEVFVGPVJPVEPUQVBURFUS $. $} ${ @@ -230709,11 +230926,11 @@ nonzero elements form a group under multiplication (from which it Carneiro, 2-Dec-2014.) $) subrgmcl $p |- ( ( A e. ( SubRing ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A ) $= - ( csubrg cfv wcel w3a cress cmulr cbs eqid subrgrng 3ad2ant1 wceq eleqtrd - co crg simp2 subrgbas simp3 rngcl syl3anc ressmulr oveqd 3eltr4d ) ABGHZI - ZDAIZEAIZJZDEBAKSZLHZSZUNMHZDECSAUMUNTIZDUQIEUQIUPUQIUJUKURULABUNUNNZOPUM - DAUQUJUKULUAUJUKAUQQULABUNUSUBPZRUMEAUQUJUKULUCUTRUQUNUODEUQNUONUDUEUMCUO - DEUJUKCUOQULABUNCUIUSFUFPUGUTUH $. + ( csubrg cfv wcel w3a cress co cmulr cbs crg eqid subrgring 3ad2ant1 wceq + eleqtrd simp2 subrgbas simp3 ringcl syl3anc ressmulr oveqd 3eltr4d ) ABGH + ZIZDAIZEAIZJZDEBAKLZMHZLZUNNHZDECLAUMUNOIZDUQIEUQIUPUQIUJUKURULABUNUNPZQR + UMDAUQUJUKULUAUJUKAUQSULABUNUSUBRZTUMEAUQUJUKULUCUTTUQUNUODEUQPUOPUDUEUMC + UODEUJUKCUOSULABUNCUIUSFUFRUGUTUH $. $} ${ @@ -230722,11 +230939,11 @@ nonzero elements form a group under multiplication (from which it Mario Carneiro, 15-Jun-2015.) $) subrgsubm $p |- ( A e. ( SubRing ` R ) -> A e. ( SubMnd ` M ) ) $= ( csubrg cfv wcel csubmnd cbs wss cur cress co cmnd eqid subrgss subrg1cl - cmgp crg rngmgp wceq subrgrcl mgpress mpancom subrgrng syl eqeltrd w3a wb - mgpbas rngidval issubm2 3syl mpbir3and ) ABEFZGZACHFGZABIFZJZBKFZAGZCALMZ - NGZAURBUROZPABUTUTOZQUPVBBALMZRFZNBSGZUPVBVGUAABUBZABVFCSUOVFOZDUCUDUPVFS - GVGNGABVFVJUEVFVGVGOTUFUGUPVHCNGUQUSVAVCUHUIVIBCDTURAVBCUTURBCDVDUJBUTCDV - EUKVBOULUMUN $. + cmgp crg ringmgp subrgrcl mgpress mpancom subrgring syl eqeltrd wb mgpbas + wceq w3a rngidval issubm2 3syl mpbir3and ) ABEFZGZACHFGZABIFZJZBKFZAGZCAL + MZNGZAURBUROZPABUTUTOZQUPVBBALMZRFZNBSGZUPVBVGUIABUAZABVFCSUOVFOZDUBUCUPV + FSGVGNGABVFVJUDVFVGVGOTUEUFUPVHCNGUQUSVAVCUJUGVIBCDTURAVBCUTURBCDVDUHBUTC + DVEUKVBOULUMUN $. $} ${ @@ -230758,17 +230975,17 @@ nonzero elements form a group under multiplication (from which it subrguss $p |- ( A e. ( SubRing ` R ) -> V C_ U ) $= ( vx cfv wcel wa cur cdsr wbr eqid wceq adantr cmulr co coppr simpr sylib csubrg cv isunit simpld subrg1 breqtrrd wss subrgdvds ssbrd mpd cinvr cbs - subrgbas subrgss eqsstr3d unitcl adantl sseldd crg subrgrng sylan opprbas - rnginvcl dvdsrmul syl2anc opprmul unitrinv ressmulr oveqd 3eqtr4d breqtrd - syl5eq sylanbrc ex ssrdv ) ABUDJZKZIEDVTIUEZEKZWADKZVTWBLZWABMJZBNJZOZWAW - EBUAJZNJZOWCWDWAWECNJZOWGWDWACMJZWEWJWDWAWKWJOZWAWKCUAJZNJZOZWDWBWLWOLVTW - BUBWJCWMEWKWNWAHWKPZWJPZWMPWNPUFUCUGVTWEWKQWBABCWEFWEPZUHRZUIWDWJWFWAWEVT - WJWFUJWBAWFBCWJFWFPZWQUKRULUMWDWAWACUNJZJZWAWHSJZTZWEWIWDWABUOJZKXBXEKWAX - DWIOWDCUOJZXEWAWDXFAXEVTAXFQWBABCFUPRVTAXEUJWBAXEBXEPZUQRURZWBWAXFKVTXFCE - WAXFPZHUSUTVAWDXFXEXBXHVTCVBKZWBXBXFKABCFVCZXFCEXAWAHXAPZXIVFVDVAXEWIWHXC - WAXBXEBWHWHPZXGVEWIPZXCPZVGVHWDXDWAXBBSJZTZWEXEBXCXPWHXBWAXGXPPZXMXOVIWDW - AXBCSJZTZWKXQWEVTXJWBXTWKQXKCXSEWKXAWAHXLXSPWPVJVDWDXPXSWAXBVTXPXSQWBABCX - PVSFXRVKRVLWSVMVOVNWFBWHDWEWIWAGWRWTXMXNUFVPVQVR $. + subrgbas subrgss eqsstr3d unitcl adantl subrgring ringinvcl sylan opprbas + sseldd crg dvdsrmul syl2anc opprmul unitrinv oveqd 3eqtr4d syl5eq breqtrd + ressmulr sylanbrc ex ssrdv ) ABUDJZKZIEDVTIUEZEKZWADKZVTWBLZWABMJZBNJZOZW + AWEBUAJZNJZOWCWDWAWECNJZOWGWDWACMJZWEWJWDWAWKWJOZWAWKCUAJZNJZOZWDWBWLWOLV + TWBUBWJCWMEWKWNWAHWKPZWJPZWMPWNPUFUCUGVTWEWKQWBABCWEFWEPZUHRZUIWDWJWFWAWE + VTWJWFUJWBAWFBCWJFWFPZWQUKRULUMWDWAWACUNJZJZWAWHSJZTZWEWIWDWABUOJZKXBXEKW + AXDWIOWDCUOJZXEWAWDXFAXEVTAXFQWBABCFUPRVTAXEUJWBAXEBXEPZUQRURZWBWAXFKVTXF + CEWAXFPZHUSUTVEWDXFXEXBXHVTCVFKZWBXBXFKABCFVAZXFCEXAWAHXAPZXIVBVCVEXEWIWH + XCWAXBXEBWHWHPZXGVDWIPZXCPZVGVHWDXDWAXBBSJZTZWEXEBXCXPWHXBWAXGXPPZXMXOVIW + DWAXBCSJZTZWKXQWEVTXJWBXTWKQXKCXSEWKXAWAHXLXSPWPVJVCWDXPXSWAXBVTXPXSQWBAB + CXPVSFXRVORVKWSVLVMVNWFBWHDWEWIWAGWRWTXMXNUFVPVQVR $. $} ${ @@ -230781,17 +230998,17 @@ nonzero elements form a group under multiplication (from which it subrginv $p |- ( ( A e. ( SubRing ` R ) /\ X e. U ) -> ( I ` X ) = ( J ` X ) ) $= ( cfv wcel cur cmulr co wceq adantr eqid sylan csubrg wa crg cbs subrgrcl - subrgbas subrgss eqsstr3d subrgrng rnginvcl sseldd unitcl adantl subrguss - wss sselda syldan rngass syl13anc unitlinv ressmulr subrg1 3eqtr4d oveq1d - cui oveqd unitrinv oveq2d 3eqtr3d rnglidm syl2anc rngridm ) ABUALZMZGDMZU - BZBNLZGELZBOLZPZGFLZVQVSPZVRWAVPWAGVSPZVRVSPZWAGVRVSPZVSPZVTWBVPBUCMZWABU - DLZMZGWHMVRWHMZWDWFQVNWGVOABUEZRZVPCUDLZWHWAVNWMWHUOVOVNWMAWHABCHUFAWHBWH - SZUGUHRZVNCUCMZVOWAWMMABCHUIZWMCDFGJKWMSZUJTUKZVPWMWHGWOVOGWMMVNWMCDGWRJU - LUMUKVNVOGBVELZMZWJVNDWTGABCWTDHWTSZJUNUPZVNWGXAWJWKWHBWTEGXBIWNUJTUQZWHB - VSWAGVRWNVSSZURUSVPWCVQVRVSVPWAGCOLZPZCNLZWCVQVNWPVOXGXHQWQCXFDXHFGJKXFSX - HSUTTVPVSXFWAGVNVSXFQVOABCVSVMHXEVARVFVNVQXHQVOABCVQHVQSZVBRVCVDVPWEVQWAV - SVNVOXAWEVQQZXCVNWGXAXJWKBVSWTVQEGXBIXEXIVGTUQVHVIVPWGWJVTVRQWLXDWHBVSVQV - RWNXEXIVJVKVPWGWIWBWAQWLWSWHBVSVQWAWNXEXIVLVKVI $. + wss subrgbas subrgss eqsstr3d subrgring sseldd unitcl adantl cui subrguss + ringinvcl sselda syldan ringass syl13anc unitlinv ressmulr subrg1 3eqtr4d + oveqd oveq1d unitrinv oveq2d 3eqtr3d ringlidm syl2anc ringridm ) ABUALZMZ + GDMZUBZBNLZGELZBOLZPZGFLZVQVSPZVRWAVPWAGVSPZVRVSPZWAGVRVSPZVSPZVTWBVPBUCM + ZWABUDLZMZGWHMVRWHMZWDWFQVNWGVOABUEZRZVPCUDLZWHWAVNWMWHUFVOVNWMAWHABCHUGA + WHBWHSZUHUIRZVNCUCMZVOWAWMMABCHUJZWMCDFGJKWMSZUPTUKZVPWMWHGWOVOGWMMVNWMCD + GWRJULUMUKVNVOGBUNLZMZWJVNDWTGABCWTDHWTSZJUOUQZVNWGXAWJWKWHBWTEGXBIWNUPTU + RZWHBVSWAGVRWNVSSZUSUTVPWCVQVRVSVPWAGCOLZPZCNLZWCVQVNWPVOXGXHQWQCXFDXHFGJ + KXFSXHSVATVPVSXFWAGVNVSXFQVOABCVSVMHXEVBRVEVNVQXHQVOABCVQHVQSZVCRVDVFVPWE + VQWAVSVNVOXAWEVQQZXCVNWGXAXJWKBVSWTVQEGXBIXEXIVGTURVHVIVPWGWJVTVRQWLXDWHB + VSVQVRWNXEXIVJVKVPWGWIWBWAQWLWSWHBVSVQWAWNXEXIVLVKVI $. $} ${ @@ -230827,20 +231044,20 @@ nonzero elements form a group under multiplication (from which it subrgunit $p |- ( A e. ( SubRing ` R ) -> ( X e. V <-> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) ) $= ( cfv wcel eqid wceq adantr wbr cmulr co syl2anc csubrg w3a wa subrguss - sselda unitcl adantl subrgbas eleqtrrd cinvr subrgrng rnginvcl subrginv - cbs crg sylan 3eltr4d 3jca cdsr simpr2 eleqtrd simpr3 dvdsrmul subrgrcl - cur coppr simpr1 unitlinv ressmulr oveqd subrg1 3eqtr3d breqtrd opprbas - opprmul unitrinv syl5eq isunit sylanbrc impbida ) ABUALZMZGFMZGDMZGAMZG - ELZAMZUBZWBWCUCZWDWEWGWBFDGABCDFHIJUDUEWIGCUNLZAWCGWJMZWBWJCFGWJNZJUFUG - WBAWJOZWCABCHUHZPZUIWIGCUJLZLZWJWFAWBCUOMWCWQWJMABCHUKWJCFWPGJWPNZWLULU - PABCFEWPGHKJWRUMWOUQURWBWHUCZGCVELZCUSLZQGWTCVFLZUSLZQWCWSGWFGCRLZSZWTX - AWSWKWFWJMZGXEXAQWSGAWJWBWDWEWGUTWBWMWHWNPZVAZWSWFAWJWBWDWEWGVBXGVAZWJX - ACXDGWFWLXANZXDNZVCTWSWFGBRLZSZBVELZXEWTWSBUOMZWDXMXNOWBXOWHABVDPZWBWDW - EWGVGZBXLDXNEGIKXLNZXNNZVHTWSXLXDWFGWBXLXDOWHABCXLWAHXRVIPZVJWBXNWTOWHA - BCXNHXSVKPZVLVMWSGWFGXBRLZSZWTXCWSWKXFGYCXCQXHXIWJXCXBYBGWFWJCXBXBNZWLV - NXCNZYBNZVCTWSYCGWFXDSZWTWJCYBXDXBWFGWLXKYDYFVOWSGWFXLSZXNYGWTWSXOWDYHX - NOXPXQBXLDXNEGIKXRXSVPTWSXLXDGWFXTVJYAVLVQVMXACXBFWTXCGJWTNXJYDYEVRVSVT - $. + sselda cbs unitcl subrgbas eleqtrrd cinvr crg subrgring ringinvcl sylan + adantl subrginv 3eltr4d 3jca cur coppr simpr2 eleqtrd dvdsrmul subrgrcl + simpr3 simpr1 unitlinv ressmulr subrg1 3eqtr3d breqtrd opprbas unitrinv + cdsr oveqd opprmul syl5eq isunit sylanbrc impbida ) ABUALZMZGFMZGDMZGAM + ZGELZAMZUBZWBWCUCZWDWEWGWBFDGABCDFHIJUDUEWIGCUFLZAWCGWJMZWBWJCFGWJNZJUG + UOWBAWJOZWCABCHUHZPZUIWIGCUJLZLZWJWFAWBCUKMWCWQWJMABCHULWJCFWPGJWPNZWLU + MUNABCFEWPGHKJWRUPWOUQURWBWHUCZGCUSLZCVNLZQGWTCUTLZVNLZQWCWSGWFGCRLZSZW + TXAWSWKWFWJMZGXEXAQWSGAWJWBWDWEWGVAWBWMWHWNPZVBZWSWFAWJWBWDWEWGVEXGVBZW + JXACXDGWFWLXANZXDNZVCTWSWFGBRLZSZBUSLZXEWTWSBUKMZWDXMXNOWBXOWHABVDPZWBW + DWEWGVFZBXLDXNEGIKXLNZXNNZVGTWSXLXDWFGWBXLXDOWHABCXLWAHXRVHPZVOWBXNWTOW + HABCXNHXSVIPZVJVKWSGWFGXBRLZSZWTXCWSWKXFGYCXCQXHXIWJXCXBYBGWFWJCXBXBNZW + LVLXCNZYBNZVCTWSYCGWFXDSZWTWJCYBXDXBWFGWLXKYDYFVPWSGWFXLSZXNYGWTWSXOWDY + HXNOXPXQBXLDXNEGIKXRXSVMTWSXLXDGWFXTVOYAVJVQVKXACXBFWTXCGJWTNXJYDYEVRVS + VT $. $} subrgugrp.4 $e |- G = ( ( mulGrp ` R ) |`s U ) $. @@ -230848,16 +231065,16 @@ nonzero elements form a group under multiplication (from which it ring. (Contributed by Mario Carneiro, 4-Dec-2014.) $) subrgugrp $p |- ( A e. ( SubRing ` R ) -> V e. ( SubGrp ` G ) ) $= ( vx vy cfv wcel cv cmulr co wral eqid cvv csubrg csubg c0 cinvr subrguss - wss wne crg cur subrgrng 1unit ne0i 3syl w3a wceq ressmulr 3ad2ant1 oveqd - wa unitmulcl syl3an1 eqeltrd 3expa ralrimiva subrginv unitinvcl sylan jca - cgrp wb subrgrcl unitgrp unitgrpbas cplusg cui fvex eqeltri cmgp mgpplusg - ressplusg ax-mp invrfval issubg2 mpbir3and ) ABUAMZNZFEUBMNZFDUFZFUCUGZKO - ZLOZBPMZQZFNZLFRZWJBUDMZMZFNZUSZKFRZABCDFGHIUEWFCUHNZCUIMZFNWIABCGUJZCFXB - IXBSUKFXBULUMWFWSKFWFWJFNZUSZWOWRXEWNLFWFXDWKFNZWNWFXDXFUNZWMWJWKCPMZQZFX - GWLXHWJWKWFXDWLXHUOXFABCWLWEGWLSZUPUQURWFXAXDXFXIFNXCCXHFWJWKIXHSUTVAVBVC - VDXEWQWJCUDMZMZFABCFWPXKWJGWPSZIXKSZVEWFXAXDXLFNXCCFXKWJIXNVFVGVBVHVDWFBU - HNEVINWGWHWIWTUNVJABVKBDEHJVLKLDWLFEWPBDEHJVMDTNWLEVNMUODBVOMTHBVOVPVQDWL - BVRMZETJBWLXOXOSXJVSVTWABDEWPHJXMWBWCUMWD $. + wss wne wa crg cur subrgring 1unit ne0i 3syl wceq ressmulr 3ad2ant1 oveqd + w3a unitmulcl syl3an1 eqeltrd 3expa ralrimiva subrginv unitinvcl jca cgrp + wb subrgrcl unitgrp unitgrpbas cplusg cui fvex eqeltri mgpplusg ressplusg + sylan cmgp ax-mp invrfval issubg2 mpbir3and ) ABUAMZNZFEUBMNZFDUFZFUCUGZK + OZLOZBPMZQZFNZLFRZWJBUDMZMZFNZUHZKFRZABCDFGHIUEWFCUINZCUJMZFNWIABCGUKZCFX + BIXBSULFXBUMUNWFWSKFWFWJFNZUHZWOWRXEWNLFWFXDWKFNZWNWFXDXFUSZWMWJWKCPMZQZF + XGWLXHWJWKWFXDWLXHUOXFABCWLWEGWLSZUPUQURWFXAXDXFXIFNXCCXHFWJWKIXHSUTVAVBV + CVDXEWQWJCUDMZMZFABCFWPXKWJGWPSZIXKSZVEWFXAXDXLFNXCCFXKWJIXNVFVSVBVGVDWFB + UINEVHNWGWHWIWTUSVIABVJBDEHJVKKLDWLFEWPBDEHJVLDTNWLEVMMUODBVNMTHBVNVOVPDW + LBVTMZETJBWLXOXOSXJVQVRWABDEWPHJXMWBWCUNWD $. $} ${ @@ -230873,19 +231090,19 @@ nonzero elements form a group under multiplication (from which it ( wcel cfv cv co w3a wa wceq syl adantlr syldan vu vv vw crg csubrg csubg wral subrgsubg subrg1cl subrgmcl 3expb ralrimivva 3jca cress simpl cplusg wss cbs simpr1 eqid subgbas ressplusg cmulr ressmulr subggrp simpr3 oveq1 - cgrp eleq1d oveq2 rspc2v syl5com 3impib subgss sseld 3anim123d imp rngass - rngdi rngdir simpr2 rnglidm rngridm isrngd jca issubrg sylanbrc impbid2 - ex ) EUDKZCEUELKZCEUFLZKZGCKZAMZBMZFNZCKZBCUGACUGZOZWKWMWNWSCEUHCEGIUIWKW - RABCCWKWOCKWPCKWRCEFWOWPJUJUKULUMWJWTWKWJWTPZWJECUNNZUDKZPCDUQZWNPWKXAWJX - CWJWTUOXAUAUBUCCEUPLZXBFGXAWMCXBURLQWJWMWNWSUSZCEXBXBUTZVARXAWMXEXBUPLQXF - CXEEXBWLXGXEUTZVBRXAWMFXBVCLQXFCEXBFWLXGJVDRXAWMXBVHKXFCEXBXGVERXAUAMZCKZ - UBMZCKZXIXKFNZCKZXAWSXJXLPXNWJWMWNWSVFWRXNXIWPFNZCKABXIXKCCWOXIQWQXOCWOXI - WPFVGVIWPXKQXOXMCWPXKXIFVJVIVKVLVMXAXJXLUCMZCKZOZXIDKZXKDKZXPDKZOZXMXPFNX - IXKXPFNZFNQZXAXRYBXAXJXSXLXTXQYAXACDXIXAWMXDXFDCEHVNRZVOZXACDXKYEVOXACDXP - YEVOVPVQZWJYBYDWTDEFXIXKXPHJVRSTXAXRYBXIXKXPXENFNXMXIXPFNZXENQZYGWJYBYIWT - DXEEFXIXKXPHXHJVSSTXAXRYBXIXKXENXPFNYHYCXENQZYGWJYBYJWTDXEEFXIXKXPHXHJVTS - TWJWMWNWSWAZXAXJXSGXIFNXIQZXAXJXSYFVQZWJXSYLWTDEFGXIHJIWBSTXAXJXSXIGFNXIQ - ZYMWJXSYNWTDEFGXIHJIWCSTWDWEXAXDWNYEYKWECDEGHIWFWGWIWH $. + cgrp eleq1d oveq2 rspc2v syl5com 3impib subgss sseld 3anim123d imp ringdi + ringass ringdir simpr2 ringlidm ringridm isringd issubrg sylanbrc impbid2 + jca ex ) EUDKZCEUELKZCEUFLZKZGCKZAMZBMZFNZCKZBCUGACUGZOZWKWMWNWSCEUHCEGIU + IWKWRABCCWKWOCKWPCKWRCEFWOWPJUJUKULUMWJWTWKWJWTPZWJECUNNZUDKZPCDUQZWNPWKX + AWJXCWJWTUOXAUAUBUCCEUPLZXBFGXAWMCXBURLQWJWMWNWSUSZCEXBXBUTZVARXAWMXEXBUP + LQXFCXEEXBWLXGXEUTZVBRXAWMFXBVCLQXFCEXBFWLXGJVDRXAWMXBVHKXFCEXBXGVERXAUAM + ZCKZUBMZCKZXIXKFNZCKZXAWSXJXLPXNWJWMWNWSVFWRXNXIWPFNZCKABXIXKCCWOXIQWQXOC + WOXIWPFVGVIWPXKQXOXMCWPXKXIFVJVIVKVLVMXAXJXLUCMZCKZOZXIDKZXKDKZXPDKZOZXMX + PFNXIXKXPFNZFNQZXAXRYBXAXJXSXLXTXQYAXACDXIXAWMXDXFDCEHVNRZVOZXACDXKYEVOXA + CDXPYEVOVPVQZWJYBYDWTDEFXIXKXPHJVSSTXAXRYBXIXKXPXENFNXMXIXPFNZXENQZYGWJYB + YIWTDXEEFXIXKXPHXHJVRSTXAXRYBXIXKXENXPFNYHYCXENQZYGWJYBYJWTDXEEFXIXKXPHXH + JVTSTWJWMWNWSWAZXAXJXSGXIFNXIQZXAXJXSYFVQZWJXSYLWTDEFGXIHJIWBSTXAXJXSXIGF + NXIQZYMWJXSYNWTDEFGXIHJIWCSTWDWHXAXDWNYEYKWHCDEGHIWEWFWIWG $. $} ${ @@ -230895,13 +231112,13 @@ nonzero elements form a group under multiplication (from which it Carneiro, 6-Dec-2014.) $) opprsubrg $p |- ( SubRing ` R ) = ( SubRing ` O ) $= ( vx vy vz csubrg cfv cv wcel crg subrgrcl csubg cmulr co wral w3a a1i wb - eqid opprrngb sylibr cur opprsubg eleq2d ralcom cbs opprmul eleq1i bitr4i - 2ralbii 3anbi13d issubrg2 opprbas oppr1 sylbi 3bitr4d pm5.21nii eqriv - wceq ) DAGHZBGHZDIZVAJZAKJZVCVBJZVCALVFBKJZVEVCBLABCUAZUBVEVCAMHZJZAUCHZV - CJZEIZFIZANHZOZVCJZFVCPEVCPZQVCBMHZJZVLVNVMBNHZOZVCJZEVCPFVCPZQZVDVFVEVJV - TVRWDVLVEVIVSVCVIVSUTVEABCUDRUEVRWDSVEVRVQEVCPFVCPWDVQEFVCVCUFWCVQFEVCVCW - BVPVCAUGHZAWAVOBVNVMWFTZVOTZCWATZUHUIUKUJRULEFVCWFAVOVKWGVKTZWHUMVEVGVFWE - SVHFEVCWFBWAVKWFABCWGUNAVKBCWJUOWIUMUPUQURUS $. + eqid opprringb sylibr cur wceq opprsubg eleq2d ralcom cbs opprmul 2ralbii + eleq1i bitr4i 3anbi13d issubrg2 opprbas oppr1 sylbi 3bitr4d pm5.21nii + eqriv ) DAGHZBGHZDIZVAJZAKJZVCVBJZVCALVFBKJZVEVCBLABCUAZUBVEVCAMHZJZAUCHZ + VCJZEIZFIZANHZOZVCJZFVCPEVCPZQVCBMHZJZVLVNVMBNHZOZVCJZEVCPFVCPZQZVDVFVEVJ + VTVRWDVLVEVIVSVCVIVSUDVEABCUERUFVRWDSVEVRVQEVCPFVCPWDVQEFVCVCUGWCVQFEVCVC + WBVPVCAUHHZAWAVOBVNVMWFTZVOTZCWATZUIUKUJULRUMEFVCWFAVOVKWGVKTZWHUNVEVGVFW + ESVHFEVCWFBWAVKWFABCWGUOAVKBCWJUPWIUNUQURUSUT $. $} ${ @@ -230955,24 +231172,24 @@ nonzero elements form a group under multiplication (from which it issubdrg $p |- ( ( R e. DivRing /\ A e. ( SubRing ` R ) ) -> ( S e. DivRing <-> A. x e. ( A \ { .0. } ) ( I ` x ) e. A ) ) $= ( cdr wcel cfv wa cdif syl wceq wb eqid ad2antlr wss csubrg cv wral cinvr - csn cbs crg cui simpllr subrgrng c0g simpr eldifsn sylib subrgbas eleqtrd - simpld simprd subrg0 neeqtrd drngunit mpbir2and rnginvcl syl2anc subrginv - wne 3eltr4d ralrimiva cin subrguss isdrng simprbi ad2antrr sseqtrd unitss - syl5sseqr ssind subrgss difin2 sseqtr4d simprl sseldd subrgunit mpbir3and - simprr w3a expr ralimdva dfss3 sylibr eqssd sneqd difeq12d eqtrd sylanbrc - imp impbida ) CJKZBCUALKZMZDJKZAUBZELZBKZABFUEZNZUCZWTXAMZXDAXFXHXBXFKZMZ - XBDUDLZLZDUFLZXCBXJDUGKZXBDUHLZKZXLXMKXJWSXNWRWSXAXIUIZBCDGUJZOXJXPXBXMKZ - XBDUKLZVFZXJXBBXMXJXBBKZXBFVFZXJXIYBYCMZXHXIULXBBFUMZUNZUQXJWSBXMPZXQBCDG - UOZOZUPXJXBFXTXJYBYCYFURXJWSFXTPZXQBCDFGHUSZOUTXAXPXSYAMQWTXIXMDXOXBXTXMR - ZXORZXTRZVASVBZXMDXOXKXBYMXKRZYLVCVDXJWSXPXCXLPXQYOBCDXOEXKXBGIYMYPVEVDYI - VGVHWTXGMZXNXOXMXTUEZNZPXAWSXNWRXGXRSYQXOXFYSYQXOXFYQXOCUFLZXENZBVIZXFYQX - OUUABYQXOCUHLZUUAWSXOUUCTWRXGBCDUUCXOGUUCRZYMVJSWRUUCUUAPZWSXGWRCUGKUUEYT - CUUCFYTRZUUDHVKVLVMVNYQXMXOBXMDXOYLYMVOWSYGWRXGYHSZVPVQYQBYTTZXFUUBPWSUUH - WRXGBYTCUUFVRZSBXEYTVSOVTYQXPAXFUCZXFXOTWTXGUUJWTXDXPAXFWTXIXDXPWTXIXDMZM - ZXPXBUUCKZYBXDUULUUMXBYTKZYCUULBYTXBWSUUHWRUUKUUISUULYBYCUULXIYDWTXIXDWAY - EUNZUQZWBUULYBYCUUOURWRUUMUUNYCMQWSUUKYTCUUCXBFUUFUUDHVAVMVBUUPWTXIXDWEWS - XPUUMYBXDWFQWRUUKBCDUUCEXOXBGUUDYMIWCSWDWGWHWPAXFXOWIWJWKYQBXMXEYRUUGYQFX - TWSYJWRXGYKSWLWMWNXMDXOXTYLYMYNVKWOWQ $. + csn cbs crg cui simpllr subrgring c0g simpr eldifsn sylib simpld subrgbas + eleqtrd simprd subrg0 neeqtrd drngunit mpbir2and syl2anc subrginv 3eltr4d + wne ringinvcl ralrimiva subrguss isdrng simprbi ad2antrr unitss syl5sseqr + cin sseqtrd ssind subrgss difin2 simprl sseldd simprr subrgunit mpbir3and + sseqtr4d w3a expr ralimdva imp dfss3 sylibr eqssd sneqd difeq12d sylanbrc + eqtrd impbida ) CJKZBCUALKZMZDJKZAUBZELZBKZABFUEZNZUCZWTXAMZXDAXFXHXBXFKZ + MZXBDUDLZLZDUFLZXCBXJDUGKZXBDUHLZKZXLXMKXJWSXNWRWSXAXIUIZBCDGUJZOXJXPXBXM + KZXBDUKLZVFZXJXBBXMXJXBBKZXBFVFZXJXIYBYCMZXHXIULXBBFUMZUNZUOXJWSBXMPZXQBC + DGUPZOZUQXJXBFXTXJYBYCYFURXJWSFXTPZXQBCDFGHUSZOUTXAXPXSYAMQWTXIXMDXOXBXTX + MRZXORZXTRZVASVBZXMDXOXKXBYMXKRZYLVGVCXJWSXPXCXLPXQYOBCDXOEXKXBGIYMYPVDVC + YIVEVHWTXGMZXNXOXMXTUEZNZPXAWSXNWRXGXRSYQXOXFYSYQXOXFYQXOCUFLZXENZBVOZXFY + QXOUUABYQXOCUHLZUUAWSXOUUCTWRXGBCDUUCXOGUUCRZYMVISWRUUCUUAPZWSXGWRCUGKUUE + YTCUUCFYTRZUUDHVJVKVLVPYQXMXOBXMDXOYLYMVMWSYGWRXGYHSZVNVQYQBYTTZXFUUBPWSU + UHWRXGBYTCUUFVRZSBXEYTVSOWEYQXPAXFUCZXFXOTWTXGUUJWTXDXPAXFWTXIXDXPWTXIXDM + ZMZXPXBUUCKZYBXDUULUUMXBYTKZYCUULBYTXBWSUUHWRUUKUUISUULYBYCUULXIYDWTXIXDV + TYEUNZUOZWAUULYBYCUUOURWRUUMUUNYCMQWSUUKYTCUUCXBFUUFUUDHVAVLVBUUPWTXIXDWB + WSXPUUMYBXDWFQWRUUKBCDUUCEXOXBGUUDYMIWCSWDWGWHWIAXFXOWJWKWLYQBXMXEYRUUGYQ + FXTWSYJWRXGYKSWMWNWPXMDXOXTYLYMYNVJWOWQ $. $} ${ @@ -230982,16 +231199,16 @@ nonzero elements form a group under multiplication (from which it 4-Dec-2014.) $) subsubrg $p |- ( A e. ( SubRing ` R ) -> ( B e. ( SubRing ` S ) <-> ( B e. ( SubRing ` R ) /\ B C_ A ) ) ) $= - ( csubrg cfv wcel wss wa crg cress co cbs adantr wceq adantl subrgrng jca - eqid cur subrgrcl subrgss subrgbas sseqtr4d oveq1i syl5eq syldan eqeltrrd - ressabs subrg1 subrg1cl eqeltrd issubrg sylanbrc adantrl ad2antrl sseqtrd - sstrd simprr impbida ) ACFGZHZBDFGHZBVBHZBAIZJZVCVDJZVEVFVHCKHZCBLMZKHZJB - CNGZIZCUAGZBHZJVEVHVIVKVCVIVDACUBOVHDBLMZVJKVCVDVFVPVJPZVHBDNGZAVDBVRIZVC - BVRDVRTZUCQVCAVRPZVDACDEUDZOUEZVCVFJVPCALMZBLMVJDWDBLEUFABCVBUJUGZUHVDVPK - HZVCBDVPVPTRQUISVHVMVOVHBAVLWCVCAVLIVDAVLCVLTZUCOUSVHVNDUAGZBVCVNWHPZVDAC - DVNEVNTZUKZOVDWHBHZVCBDWHWHTZULQUMSBVLCVNWGWJUNUOWCSVCVGJZDKHZWFJVSWLJVDW - NWOWFVCWOVGACDEROWNVPVJKVCVFVQVEWEUPVEVKVCVFBCVJVJTRUQUMSWNVSWLWNBAVRVCVE - VFUTVCWAVGWBOURWNVNWHBVCWIVGWKOVEVOVCVFBCVNWJULUQUISBVRDWHVTWMUNUOVA $. + ( csubrg cfv wcel wss crg cress cbs adantr wceq eqid adantl subrgring jca + wa co cur subrgrcl subrgss subrgbas sseqtr4d oveq1i ressabs syl5eq syldan + eqeltrrd subrg1 subrg1cl eqeltrd issubrg sylanbrc adantrl ad2antrl simprr + sstrd sseqtrd impbida ) ACFGZHZBDFGHZBVBHZBAIZSZVCVDSZVEVFVHCJHZCBKTZJHZS + BCLGZIZCUAGZBHZSVEVHVIVKVCVIVDACUBMVHDBKTZVJJVCVDVFVPVJNZVHBDLGZAVDBVRIZV + CBVRDVROZUCPVCAVRNZVDACDEUDZMUEZVCVFSVPCAKTZBKTVJDWDBKEUFABCVBUGUHZUIVDVP + JHZVCBDVPVPOQPUJRVHVMVOVHBAVLWCVCAVLIVDAVLCVLOZUCMUSVHVNDUAGZBVCVNWHNZVDA + CDVNEVNOZUKZMVDWHBHZVCBDWHWHOZULPUMRBVLCVNWGWJUNUOWCRVCVGSZDJHZWFSVSWLSVD + WNWOWFVCWOVGACDEQMWNVPVJJVCVFVQVEWEUPVEVKVCVFBCVJVJOQUQUMRWNVSWLWNBAVRVCV + EVFURVCWAVGWBMUTWNVNWHBVCWIVGWKMVEVOVCVFBCVNWJULUQUJRBVRDWHVTWMUNUOVA $. $( The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.) $) @@ -231010,11 +231227,11 @@ nonzero elements form a group under multiplication (from which it issubrg3 $p |- ( R e. Ring -> ( S e. ( SubRing ` R ) <-> ( S e. ( SubGrp ` R ) /\ S e. ( SubMnd ` M ) ) ) ) $= ( vx vy crg wcel csubrg cfv csubg cur cv cmulr wral wa eqid 3anass syl6bb - w3a co csubmnd cbs issubrg2 wss wb rngmgp subgss mgpbas rngidval mgpplusg - cmnd issubm baibd syl2an pm5.32da bitr4d ) AGHZBAIJHZBAKJHZALJZBHZEMFMANJ - ZUABHFBOEBOZPZPZUTBCUBJHZPURUSUTVBVDTVFEFBAUCJZAVCVAVHQZVAQZVCQZUDUTVBVDR - SURUTVGVEURCULHZBVHUEZVGVEUFUTACDUGVHBAVIUHVLVGVMVEVLVGVMVBVDTVMVEPEFVHVC - BCVAVHACDVIUIAVACDVJUJAVCCDVKUKUMVMVBVDRSUNUOUPUQ $. + w3a csubmnd cbs issubrg2 cmnd wss ringmgp subgss mgpbas rngidval mgpplusg + co wb issubm baibd syl2an pm5.32da bitr4d ) AGHZBAIJHZBAKJHZALJZBHZEMFMAN + JZUKBHFBOEBOZPZPZUTBCUAJHZPURUSUTVBVDTVFEFBAUBJZAVCVAVHQZVAQZVCQZUCUTVBVD + RSURUTVGVEURCUDHZBVHUEZVGVEULUTACDUFVHBAVIUGVLVGVMVEVLVGVMVBVDTVMVEPEFVHV + CBCVAVHACDVIUHAVACDVJUIAVCCDVKUJUMVMVBVDRSUNUOUPUQ $. $} ${ @@ -231024,11 +231241,11 @@ nonzero elements form a group under multiplication (from which it resrhm $p |- ( ( F e. ( S RingHom T ) /\ X e. ( SubRing ` S ) ) -> ( F |` X ) e. ( U RingHom T ) ) $= ( crh co wcel csubrg cfv wa crg cres cghm cmgp cmhm rhmrcl2 syl2an eqid - subrgrng anim12ci rhmghm subrgsubg resghm csubmnd rhmmhm subrgsubm resmhm - csubg cress wceq rhmrcl1 mgpress sylan oveq1d eleqtrd jca isrhm sylanbrc + subrgring anim12ci csubg rhmghm subrgsubg resghm csubmnd rhmmhm subrgsubm + cress resmhm wceq rhmrcl1 mgpress sylan oveq1d eleqtrd jca isrhm sylanbrc ) DABGHIZEAJKZIZLZCMIZBMIZLDENZCBOHIZVGCPKZBPKZQHZIZLVGCBGHIVAVFVCVEABDRE - ACFUAUBVDVHVLVADABOHIEAUJKIVHVCABDUCEAUDABCDEFUESVDVGAPKZEUKHZVJQHZVKVADV - MVJQHIEVMUFKIVGVOIVCABDVMVJVMTZVJTZUGEAVMVPUHVMVJVNDEVNTUISVDVNVIVJQVAAMI + ACFUAUBVDVHVLVADABOHIEAUCKIVHVCABDUDEAUEABCDEFUFSVDVGAPKZEUJHZVJQHZVKVADV + MVJQHIEVMUGKIVGVOIVCABDVMVJVMTZVJTZUHEAVMVPUIVMVJVNDEVNTUKSVDVNVIVJQVAAMI VCVNVIULABDUMEACVMMVBFVPUNUOUPUQURCBVGVIVJVITVQUSUT $. $} @@ -231066,30 +231283,30 @@ nonzero elements form a group under multiplication (from which it ( Z ` S ) e. ( SubRing ` R ) ) $= ( vx vz wcel wa cfv co wral wceq syl2anc ralrimiva wb sseldi vy crg csubg wss csubrg csubmnd c0 wne cv cplusg cminusg mgpbas cntzssv a1i c0g simpll - cmulr ssel2 adantll eqid rnglz rngrz eqtr4d rng0cl adantr mgpplusg cntzel - simpr mpbird syl simpl2 cntzi simpl3 oveq12d simpl1l simp1r sselda rngdir - w3a syl13anc rngdi 3eqtr4d simp1l simp2 simp3 rngacl syl3anc 3expa fveq2d - ne0i simplr rngmneg1 rngmneg2 cgrp rnggrp ad2antrr grpinvcl jca mpbir3and - simplll issubg2 cmnd rngmgp cntzsubm sylan issubrg3 mpbir2and ) BUBKZCAUD - ZLZCEMZBUEMKZXKBUCMKZXKDUFMKZXJXMXKAUDZXKUGUHZIUIZUAUIZBUJMZNZXKKZUAXKOZX - QBUKMZMZXKKZLZIXKOZXOXJACDEABDGFULZHUMZUNXJBUOMZXKKZXPXJYKYJJUIZBUQMZNZYL - YJYMNZPZJCOZXJYPJCXJYLCKZLZYNYJYOYSXHYLAKZYNYJPXHXIYRUPZXIYRYTXHCAYLURUSZ - ABYMYLYJFYMUTZYJUTZVAQYSXHYTYOYJPUUAUUBABYMYLYJFUUCUUDVBQVCRXJXIYJAKZYKYQ - SXHXIVHXHUUEXIABYJFUUDVDVEJAYMCDYJEYHBYMDGUUCVFZHVGQVIXKYJWJVJXJYFIXKXJXQ - XKKZLZYBYEUUHYAUAXKXJUUGXRXKKZYAXJUUGUUIVSZYAXTYLYMNZYLXTYMNZPZJCOZUUJUUM - JCUUJYRLZXQYLYMNZXRYLYMNZXSNZYLXQYMNZYLXRYMNZXSNZUUKUULUUOUUPUUSUUQUUTXSU - UOUUGYRUUPUUSPZXJUUGUUIYRVKZUUJYRVHZYMCDXQYLEUUFHVLZQUUOUUIYRUUQUUTPXJUUG - UUIYRVMZUVDYMCDXRYLEUUFHVLQVNUUOXHXQAKZXRAKZYTUUKUURPXHXIUUGUUIYRVOZUUOXK - AXQYIUVCTZUUOXKAXRYIUVFTZUUJCAYLXHXIUUGUUIVPZVQZAXSBYMXQXRYLFXSUTZUUCVRVT - UUOXHYTUVGUVHUULUVAPUVIUVMUVJUVKAXSBYMYLXQXRFUVNUUCWAVTWBRUUJXIXTAKZYAUUN - SUVLUUJXHUVGUVHUVOXHXIUUGUUIWCUUJXKAXQYIXJUUGUUIWDTUUJXKAXRYIXJUUGUUIWETA - XSBXQXRFUVNWFWGJAYMCDXTEYHUUFHVGQVIWHRUUHYEYDYLYMNZYLYDYMNZPZJCOZUUHUVRJC - UUHYRLZUUPYCMUUSYCMUVPUVQUVTUUPUUSYCUUGYRUVBXJUVEUSWIUVTABYMYCXQYLFUUCYCU - TZXHXIUUGYRWTZUVTXKAXQYIXJUUGYRWKTZUUHCAYLXHXIUUGWKZVQZWLUVTABYMYCYLXQFUU - CUWAUWBUWEUWCWMWBRUUHXIYDAKZYEUVSSUWDUUHBWNKZUVGUWFXHUWGXIUUGBWOZWPUUHXKA - XQYIXJUUGVHTABYCXQFUWAWQQJAYMCDYDEYHUUFHVGQVIWRRXJUWGXMXOXPYGVSSXHUWGXIUW - HVEIUAAXSXKBYCFUVNUWAXAVJWSXHDXBKXIXNBDGXCACDEYHHXDXEXHXLXMXNLSXIBXKDGXFV - EXG $. + cmulr ssel2 adantll eqid ringlz ringrz eqtr4d simpr ring0cl adantr cntzel + mgpplusg mpbird syl w3a simpl2 cntzi simpl3 oveq12d simpl1l simp1r sselda + ringdir syl13anc ringdi 3eqtr4d simp1l simp2 simp3 ringacl syl3anc fveq2d + ne0i 3expa simplll ringmneg1 ringmneg2 cgrp ringgrp ad2antrr grpinvcl jca + simplr issubg2 mpbir3and cmnd ringmgp cntzsubm sylan issubrg3 mpbir2and ) + BUBKZCAUDZLZCEMZBUEMKZXKBUCMKZXKDUFMKZXJXMXKAUDZXKUGUHZIUIZUAUIZBUJMZNZXK + KZUAXKOZXQBUKMZMZXKKZLZIXKOZXOXJACDEABDGFULZHUMZUNXJBUOMZXKKZXPXJYKYJJUIZ + BUQMZNZYLYJYMNZPZJCOZXJYPJCXJYLCKZLZYNYJYOYSXHYLAKZYNYJPXHXIYRUPZXIYRYTXH + CAYLURUSZABYMYLYJFYMUTZYJUTZVAQYSXHYTYOYJPUUAUUBABYMYLYJFUUCUUDVBQVCRXJXI + YJAKZYKYQSXHXIVDXHUUEXIABYJFUUDVEVFJAYMCDYJEYHBYMDGUUCVHZHVGQVIXKYJWIVJXJ + YFIXKXJXQXKKZLZYBYEUUHYAUAXKXJUUGXRXKKZYAXJUUGUUIVKZYAXTYLYMNZYLXTYMNZPZJ + COZUUJUUMJCUUJYRLZXQYLYMNZXRYLYMNZXSNZYLXQYMNZYLXRYMNZXSNZUUKUULUUOUUPUUS + UUQUUTXSUUOUUGYRUUPUUSPZXJUUGUUIYRVLZUUJYRVDZYMCDXQYLEUUFHVMZQUUOUUIYRUUQ + UUTPXJUUGUUIYRVNZUVDYMCDXRYLEUUFHVMQVOUUOXHXQAKZXRAKZYTUUKUURPXHXIUUGUUIY + RVPZUUOXKAXQYIUVCTZUUOXKAXRYIUVFTZUUJCAYLXHXIUUGUUIVQZVRZAXSBYMXQXRYLFXSU + TZUUCVSVTUUOXHYTUVGUVHUULUVAPUVIUVMUVJUVKAXSBYMYLXQXRFUVNUUCWAVTWBRUUJXIX + TAKZYAUUNSUVLUUJXHUVGUVHUVOXHXIUUGUUIWCUUJXKAXQYIXJUUGUUIWDTUUJXKAXRYIXJU + UGUUIWETAXSBXQXRFUVNWFWGJAYMCDXTEYHUUFHVGQVIWJRUUHYEYDYLYMNZYLYDYMNZPZJCO + ZUUHUVRJCUUHYRLZUUPYCMUUSYCMUVPUVQUVTUUPUUSYCUUGYRUVBXJUVEUSWHUVTABYMYCXQ + YLFUUCYCUTZXHXIUUGYRWKZUVTXKAXQYIXJUUGYRWSTZUUHCAYLXHXIUUGWSZVRZWLUVTABYM + YCYLXQFUUCUWAUWBUWEUWCWMWBRUUHXIYDAKZYEUVSSUWDUUHBWNKZUVGUWFXHUWGXIUUGBWO + ZWPUUHXKAXQYIXJUUGVDTABYCXQFUWAWQQJAYMCDYDEYHUUFHVGQVIWRRXJUWGXMXOXPYGVKS + XHUWGXIUWHVFIUAAXSXKBYCFUVNUWAWTVJXAXHDXBKXIXNBDGXCACDEYHHXDXEXHXLXMXNLSX + IBXKDGXFVFXG $. $} ${ @@ -231102,14 +231319,15 @@ nonzero elements form a group under multiplication (from which it Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) $) pwsdiagrhm $p |- ( ( R e. Ring /\ I e. W ) -> F e. ( R RingHom Y ) ) $= - ( vy vz crg wcel wa co cfv eqid cbs eqidd cghm cmgp cmhm crh simpl pwsrng - jca cgrp rnggrp pwsdiagghm cpws cmnd rngmgp mgpbas pwsdiagmhm wceq cplusg - sylan pwsmgp simpld cv simprd proplem3 mhmpropd eleqtrrd isrhm sylanbrc ) - CMNZEFNZOZVHGMNZODCGUAPNZDCUBQZGUBQZUCPZNZODCGUDPNVJVHVKVHVIUECEFGHUFUGVJ - VLVPVHCUHNVIVLCUIABCDEFGHIJUJURVJDVMVMEUKPZUCPZVOVHVMULNVIDVRNCVMVMRZUMAB - VMDEFVQVQRZBCVMVSIUNJUOURVJKLVMSQZVNSQZVMVNVMVQVJWATZVJWBTWCVJWBVQSQZUPZV - NUQQZVQUQQZUPZWBWDWFWGCEVMVNMFGVQHVSVTVNRZWBRWDRWFRWGRUSZUTVJKVAZWANLVAZW - ANOOWKWLVMUQQPTVJWKWBNWLWBNOKLWFWGVJWEWHWJVBVCVDVEUGCGDVMVNVSWIVFVG $. + ( vy vz crg wcel wa co cfv eqid cbs eqidd cghm cmgp crh simpl pwsring jca + cmhm cgrp ringgrp pwsdiagghm sylan cpws cmnd ringmgp mgpbas cplusg pwsmgp + pwsdiagmhm wceq simpld simprd proplem3 mhmpropd eleqtrrd isrhm sylanbrc + cv ) CMNZEFNZOZVHGMNZODCGUAPNZDCUBQZGUBQZUGPZNZODCGUCPNVJVHVKVHVIUDCEFGHU + EUFVJVLVPVHCUHNVIVLCUIABCDEFGHIJUJUKVJDVMVMEULPZUGPZVOVHVMUMNVIDVRNCVMVMR + ZUNABVMDEFVQVQRZBCVMVSIUOJURUKVJKLVMSQZVNSQZVMVNVMVQVJWATZVJWBTWCVJWBVQSQ + ZUSZVNUPQZVQUPQZUSZWBWDWFWGCEVMVNMFGVQHVSVTVNRZWBRWDRWFRWGRUQZUTVJKVGZWAN + LVGZWANOOWKWLVMUPQPTVJWKWBNWLWBNOKLWFWGVJWEWHWJVAVBVCVDUFCGDVMVNVSWIVEVF + $. $} ${ @@ -231125,19 +231343,19 @@ nonzero elements form a group under multiplication (from which it the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.) $) subrgpropd $p |- ( ph -> ( SubRing ` K ) = ( SubRing ` L ) ) $= - ( cfv crg wcel co wa cbs cvv wceq eqid ax-mp vs csubrg cress wss rngpropd - cv cur cin ineq2d vex ressbas syl6eq cplusg inss2 sseli anim12i ressplusg - oveqi 3eqtr3g sylan2 cmulr ressmulr anbi12d eqtr3d sseq2d issubrg 3bitr4g - rngidpropd eleq1d eqrdv ) AUAEUBKZFUBKZAELMZEUAUFZUCNZLMZOZVNEPKZUDZEUGKZ - VNMZOZOFLMZFVNUCNZLMZOZVNFPKZUDZFUGKZVNMZOZOVNVKMVNVLMAVQWFWBWKAVMWCVPWEA - BCDEFGHIJUEABCVNDUHZVOWDAWLVNVRUHZVOPKZADVRVNGUIVNQMZWMWNRUAUJZVNVRVOQEVO - SZVRSZUKTULAWLVNWGUHZWDPKZADWGVNHUIWOWSWTRWPVNWGWDQFWDSZWGSZUKTULBUFZWLMZ - CUFZWLMZOZAXCDMZXEDMZOZXCXEVOUMKZNZXCXEWDUMKZNZRXDXHXFXIWLDXCVNDUNZUOWLDX - EXOUOUPZAXJOZXCXEEUMKZNXCXEFUMKZNXLXNIXRXKXCXEWOXRXKRWPVNXREVOQWQXRSUQTUR - XSXMXCXEWOXSXMRWPVNXSFWDQXAXSSUQTURUSUTXGAXJXCXEVOVAKZNZXCXEWDVAKZNZRXPXQ - XCXEEVAKZNXCXEFVAKZNYAYCJYDXTXCXEWOYDXTRWPVNEVOYDQWQYDSVBTURYEYBXCXEWOYEY - BRWPVNFWDYEQXAYESVBTURUSUTUEVCAVSWHWAWJAVRWGVNADVRWGGHVDVEAVTWIVNABCDEFGH - JVHVIVCVCVNVREVTWRVTSVFVNWGFWIXBWISVFVGVJ $. + ( cfv crg wcel co wa cbs cvv wceq eqid ax-mp vs csubrg cv cress ringpropd + wss cur cin ineq2d vex ressbas syl6eq inss2 sseli anim12i ressplusg oveqi + cplusg 3eqtr3g sylan2 cmulr ressmulr anbi12d eqtr3d sseq2d eleq1d issubrg + rngidpropd 3bitr4g eqrdv ) AUAEUBKZFUBKZAELMZEUAUCZUDNZLMZOZVNEPKZUFZEUGK + ZVNMZOZOFLMZFVNUDNZLMZOZVNFPKZUFZFUGKZVNMZOZOVNVKMVNVLMAVQWFWBWKAVMWCVPWE + ABCDEFGHIJUEABCVNDUHZVOWDAWLVNVRUHZVOPKZADVRVNGUIVNQMZWMWNRUAUJZVNVRVOQEV + OSZVRSZUKTULAWLVNWGUHZWDPKZADWGVNHUIWOWSWTRWPVNWGWDQFWDSZWGSZUKTULBUCZWLM + ZCUCZWLMZOZAXCDMZXEDMZOZXCXEVOURKZNZXCXEWDURKZNZRXDXHXFXIWLDXCVNDUMZUNWLD + XEXOUNUOZAXJOZXCXEEURKZNXCXEFURKZNXLXNIXRXKXCXEWOXRXKRWPVNXREVOQWQXRSUPTU + QXSXMXCXEWOXSXMRWPVNXSFWDQXAXSSUPTUQUSUTXGAXJXCXEVOVAKZNZXCXEWDVAKZNZRXPX + QXCXEEVAKZNXCXEFVAKZNYAYCJYDXTXCXEWOYDXTRWPVNEVOYDQWQYDSVBTUQYEYBXCXEWOYE + YBRWPVNFWDYEQXAYESVBTUQUSUTUEVCAVSWHWAWJAVRWGVNADVRWGGHVDVEAVTWIVNABCDEFG + HJVHVFVCVCVNVREVTWRVTSVGVNWGFWIXBWISVGVIVJ $. $} ${ @@ -231158,17 +231376,17 @@ nonzero elements form a group under multiplication (from which it $( Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.) $) rhmpropd $p |- ( ph -> ( J RingHom K ) = ( L RingHom M ) ) $= - ( co wcel cfv vf crh crg wa cv cghm cmgp rngpropd anbi12d ghmpropd eleq2d - cmhm cbs mgpbas syl6eq cmulr cplusg mgpplusg oveqi 3eqtr3g mhmpropd isrhm - eqid 3bitr4g eqrdv ) AUAFGUBRZHIUBRZAFUCSZGUCSZUDZUAUEZFGUFRZSZVKFUGTZGUG - TZULRZSZUDZUDHUCSZIUCSZUDZVKHIUFRZSZVKHUGTZIUGTZULRZSZUDZUDVKVFSVKVGSAVJW - AVRWHAVHVSVIVTABCDFHJLNPUHABCEGIKMOQUHUIAVMWCVQWGAVLWBVKABCDEFGHIJKLMNOUJ - UKAVPWFVKABCDEVNVOWDWEADFUMTZVNUMTJWIFVNVNVCZWIVCUNUOAEGUMTZVOUMTKWKGVOVO - VCZWKVCUNUOADHUMTZWDUMTLWMHWDWDVCZWMVCUNUOAEIUMTZWEUMTMWOIWEWEVCZWOVCUNUO - ABUEZDSCUEZDSUDUDWQWRFUPTZRWQWRHUPTZRWQWRVNUQTZRWQWRWDUQTZRPWSXAWQWRFWSVN - WJWSVCURUSWTXBWQWRHWTWDWNWTVCURUSUTAWQESWRESUDUDWQWRGUPTZRWQWRIUPTZRWQWRV - OUQTZRWQWRWEUQTZRQXCXEWQWRGXCVOWLXCVCURUSXDXFWQWRIXDWEWPXDVCURUSUTVAUKUIU - IFGVKVNVOWJWLVBHIVKWDWEWNWPVBVDVE $. + ( co wcel cfv vf crh crg cghm cmgp cmhm ringpropd anbi12d ghmpropd eleq2d + wa cv cbs eqid mgpbas syl6eq cmulr cplusg mgpplusg oveqi 3eqtr3g mhmpropd + isrhm 3bitr4g eqrdv ) AUAFGUBRZHIUBRZAFUCSZGUCSZUKZUAULZFGUDRZSZVKFUETZGU + ETZUFRZSZUKZUKHUCSZIUCSZUKZVKHIUDRZSZVKHUETZIUETZUFRZSZUKZUKVKVFSVKVGSAVJ + WAVRWHAVHVSVIVTABCDFHJLNPUGABCEGIKMOQUGUHAVMWCVQWGAVLWBVKABCDEFGHIJKLMNOU + IUJAVPWFVKABCDEVNVOWDWEADFUMTZVNUMTJWIFVNVNUNZWIUNUOUPAEGUMTZVOUMTKWKGVOV + OUNZWKUNUOUPADHUMTZWDUMTLWMHWDWDUNZWMUNUOUPAEIUMTZWEUMTMWOIWEWEUNZWOUNUOU + PABULZDSCULZDSUKUKWQWRFUQTZRWQWRHUQTZRWQWRVNURTZRWQWRWDURTZRPWSXAWQWRFWSV + NWJWSUNUSUTWTXBWQWRHWTWDWNWTUNUSUTVAAWQESWRESUKUKWQWRGUQTZRWQWRIUQTZRWQWR + VOURTZRWQWRWEURTZRQXCXEWQWRGXCVOWLXCUNUSUTXDXFWQWRIXDWEWPXDUNUSUTVAVBUJUH + UHFGVKVNVOWJWLVCHIVKWDWEWNWPVCVDVE $. $} $( @@ -231263,41 +231481,41 @@ Absolute value (abstract algebra) syl5breqr adantr fveq2 breq2d syl5ibrcom wne w3a clt simp1 simp2 3ad2ant1 cr eleqtrrd simp3 neeqtrrd syl3anc wi 0re 3adant3 ltle sylancr mpd 3expia pm2.61dne elrege0 ralrimiva ffnfv gt0ne0d necon4d eqeq1d impbid oveq1 crg - sylanbrc eqid rnglz syl2anc sylan9eqr oveq1d ffvelrnd recnd eqtrd 3eqtr4d - mul02d oveq2 rngrz oveq2d mul01d simpl1 oveqd simpl2 simprl simpl3 simprr - syl122anc pm2.61da2ne cgrp rnggrp grplid addge02d eqbrtrd grprid addge01d - cc eqbrtrrd jca ralrimiv isabv mpbir2and ) AIGUBUCZDAIUUAUDZGUEUCZUFUGUHU - IZIUJZBUKZIUCZUFULZUUFGUMUCZULZUNZUUFCUKZGUOUCZUIZIUCZUUGUULIUCZURUIZULZU - UFUULGUPUCZUIZIUCZUUGUUPUQUIZUSUTZVAZCUUCVBZVAZBUUCVBZAIUUCVCZUUGUUDUDZBU - UCVBUUEAUUCWCIUJZUVHAEWCIUJUVJQAEUUCWCILVDVEZUUCWCIVFVGAUVIBUUCAUUFUUCUDZ - VAZUUGWCUDZUFUUGUSUTZUVIAUUCWCUUFIUVKVHZUVMUVOUUFUUIUVMUVOUUJUFUUIIUCZUSU - TZAUVRUVLAUFUFUVQUSVIAJIUCUVQUFAJUUIIOVJRVKZVLVMUUJUUGUVQUFUSUUFUUIIVNZVO - VPAUVLUUFUUIVQZUVOAUVLUWAVRZUFUUGVSUTZUVOUWBAUUFEUDZUUFJVQZUWCAUVLUWAVTUW - BUUFUUCEAUVLUWAWAAUVLEUUCULZUWALWBWDUWBUUFUUIJAUVLUWAWEAUVLJUUIULZUWAOWBW - FSWGZUWBUFWCUDUVNUWCUVOWHWIAUVLUVNUWAUVPWJUFUUGWKWLWMWNWOUUGWPXEWQBUUCUUD - IWRXEAUVFBUUCUVMUUKUVEUVMUUHUUJUVMUUFUUIUUGUFAUVLUWAUUGUFVQUWBUUGUWHWSWNW - TUVMUUHUUJUVQUFULZAUWIUVLUVSVMUUJUUGUVQUFUVTXAVPXBUVMUVDCUUCAUVLUULUUCUDZ - 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(Contributed by Mario Carneiro, 8-Sep-2014.) $) abv0 $p |- ( F e. A -> ( F ` .0. ) = 0 ) $= - ( wcel cbs cfv cc0 wceq crg abvrcl eqid rng0cl syl abveq0 mpbiri mpdan + ( wcel cbs cfv cc0 wceq crg abvrcl eqid ring0cl syl abveq0 mpbiri mpdan wa ) CAGZDBHIZGZDCIJKZUABLGUCABCEMUBBDUBNZFOPUAUCTUDDDKDNAUBBCDDEUEFQRS $. $} @@ -231418,13 +231636,13 @@ Absolute value (abstract algebra) (Contributed by Mario Carneiro, 8-Sep-2014.) $) abv1z $p |- ( ( F e. A /\ .1. =/= .0. ) -> ( F ` .1. ) = 1 ) $= ( wcel wne wa cfv co cdiv c1 eqid adantr syl3anc wceq eqtr3d cmul cbs - cr crg abvrcl rngidcl syl abvcl mpdan recnd cc0 simpl abvne0 divcan3d - simpr cmulr rnglidm syl2anc fveq2d abvmul oveq1d dividd ) DAIZCEJZKZC - DLZVFUAMZVFNMZVFOVEVFVFVEVFVCVFUCIZVDVCCBUBLZIZVIVCBUDIZVKABDFUEZVJBC - VJPZGUFUGZAVJBDCFVNUHUIQUJZVPVEVCVKVDVFUKJVCVDULZVCVKVDVOQZVCVDUOAVJB - DCEFVNHUMRZUNVEVFVFNMVHOVEVFVGVFNVECCBUPLZMZDLZVFVGVEWACDVEVLVKWACSVC - VLVDVMQVRVJBVTCCVNVTPZGUQURUSVEVCVKVKWBVGSVQVRVRAVJBVTDCCFVNWCUTRTVAV - EVFVPVSVBTT $. + cr crg abvrcl ringidcl syl abvcl mpdan recnd cc0 simpl simpr divcan3d + abvne0 cmulr ringlidm syl2anc fveq2d abvmul oveq1d dividd ) DAIZCEJZK + ZCDLZVFUAMZVFNMZVFOVEVFVFVEVFVCVFUCIZVDVCCBUBLZIZVIVCBUDIZVKABDFUEZVJ + BCVJPZGUFUGZAVJBDCFVNUHUIQUJZVPVEVCVKVDVFUKJVCVDULZVCVKVDVOQZVCVDUMAV + JBDCEFVNHUORZUNVEVFVFNMVHOVEVFVGVFNVECCBUPLZMZDLZVFVGVEWACDVEVLVKWACS + VCVLVDVMQVRVJBVTCCVNVTPZGUQURUSVEVCVKVKWBVGSVQVRVRAVJBVTDCCFVNWCUTRTV + AVEVFVPVSVBTT $. $} $( The absolute value of one is one in a division ring. (Contributed by @@ -231442,22 +231660,22 @@ Absolute value (abstract algebra) abvneg $p |- ( ( F e. A /\ X e. B ) -> ( F ` ( N ` X ) ) = ( F ` X ) ) $= ( wcel wa cfv wceq adantr eqid fveq2d c1 cmul co syl2anc cur c0g crg wi - abvrcl cgrp rnggrp syl grpinvcl sylan simpr rng1eq0 syl3anc imp c2 cexp - wne cmulr cr rngidcl abvcl mpdan recnd sqvald mpd3an23 rngmneg2 rngnegl - abvmul grpinvinv 3eqtrd 3eqtr2d abv1z sq1 syl6eqr cc0 cle wbr wb abvge0 - eqtrd 1re 0le1 sq11 mpanr12 biimpa syldan adantlr oveq1d eqtr3d mulid2d - simpl pm2.61dane ) DAJZFBJZKZFELZDLZFDLZMCUALZCUBLZWOWSWTMZKWPFDWOXAWPF - MZWOCUCJZWPBJZWNXAXBUDWMXCWNACDGUEZNZWMCUFJZWNXDWMXCXGXECUGUHZBCEFHIUIU - JWMWNUKZBCWSWPFWTHWSOZWTOZULUMUNPWOWSWTUQZKZQWRRSZWQWRXMWSELZDLZWRRSZXN - WQXMXPQWRRWMXLXPQMZWNWMXLXPUOUPSZQUOUPSZMZXRWMXLKZXSQXTYBXSWSDLZQWMXSYC - MXLWMXSXPXPRSZXOXOCURLZSZDLZYCWMXPWMXPWMXOBJZXPUSJZWMXGWSBJZYHXHWMXCYJX - EBCWSHXJUTUHZBCEWSHIUITZABCDXOGHVAVBZVCVDWMYHYHYGYDMYLYLABCYEDXOXOGHYEO - ZVHVEWMYFWSDWMYFXOWSYESZELXOELZWSWMBCYEEXOWSHYNIXEYLYKVFWMYOXOEWMBCYEWS - EWSHYNXJIXEYKVGPWMXGYJYPWSMXHYKBCEWSHIVITVJPVKNACWSDWTGXJXKVLVTVMVNWMYA - XRWMYIVOXPVPVQZYAXRVRZYMWMYHYQYLABCDXOGHVSVBYIYQKQUSJVOQVPVQYRWAWBXPQWC - WDTWEWFWGWHWOXQWQMXLWOXOFYESZDLZXQWQWOWMYHWNYTXQMWMWNWKWMYHWNYLNXIABCYE - DXOFGHYNVHUMWOYSWPDWOBCYEWSEFHYNXJIXFXIVGPWINWIWOXNWRMXLWOWRWOWRABCDFGH - VAVCWJNWIWL $. + abvrcl cgrp ringgrp syl grpinvcl sylan ring1eq0 syl3anc imp wne c2 cexp + simpr cmulr ringidcl abvcl mpdan recnd sqvald abvmul mpd3an23 ringmneg2 + cr ringnegl grpinvinv 3eqtrd 3eqtr2d abv1z eqtrd sq1 syl6eqr cc0 cle wb + wbr abvge0 0le1 sq11 mpanr12 biimpa syldan adantlr oveq1d simpl mulid2d + 1re eqtr3d pm2.61dane ) DAJZFBJZKZFELZDLZFDLZMCUALZCUBLZWOWSWTMZKWPFDWO + XAWPFMZWOCUCJZWPBJZWNXAXBUDWMXCWNACDGUEZNZWMCUFJZWNXDWMXCXGXECUGUHZBCEF + HIUIUJWMWNUQZBCWSWPFWTHWSOZWTOZUKULUMPWOWSWTUNZKZQWRRSZWQWRXMWSELZDLZWR + RSZXNWQXMXPQWRRWMXLXPQMZWNWMXLXPUOUPSZQUOUPSZMZXRWMXLKZXSQXTYBXSWSDLZQW + MXSYCMXLWMXSXPXPRSZXOXOCURLZSZDLZYCWMXPWMXPWMXOBJZXPVGJZWMXGWSBJZYHXHWM + XCYJXEBCWSHXJUSUHZBCEWSHIUITZABCDXOGHUTVAZVBVCWMYHYHYGYDMYLYLABCYEDXOXO + GHYEOZVDVEWMYFWSDWMYFXOWSYESZELXOELZWSWMBCYEEXOWSHYNIXEYLYKVFWMYOXOEWMB + CYEWSEWSHYNXJIXEYKVHPWMXGYJYPWSMXHYKBCEWSHIVITVJPVKNACWSDWTGXJXKVLVMVNV + OWMYAXRWMYIVPXPVQVSZYAXRVRZYMWMYHYQYLABCDXOGHVTVAYIYQKQVGJVPQVQVSYRWJWA + XPQWBWCTWDWEWFWGWOXQWQMXLWOXOFYESZDLZXQWQWOWMYHWNYTXQMWMWNWHWMYHWNYLNXI + ABCYEDXOFGHYNVDULWOYSWPDWOBCYEWSEFHYNXJIXFXIVHPWKNWKWOXNWRMXLWOWRWOWRAB + CDFGHUTVBWINWKWL $. $} ${ @@ -231467,12 +231685,12 @@ Absolute value (abstract algebra) abvsubtri $p |- ( ( F e. A /\ X e. B /\ Y e. B ) -> ( F ` ( X .- Y ) ) <_ ( ( F ` X ) + ( F ` Y ) ) ) $= ( wcel w3a co cfv cminusg cplusg caddc cle wceq eqid 3adant1 fveq2d wbr - grpsubval cgrp crg abvrcl 3ad2ant1 rnggrp simp3 grpinvcl syl2anc abvtri - syl syld3an3 abvneg 3adant2 oveq2d breqtrd eqbrtrd ) DAKZFBKZGBKZLZFGEM - ZDNFGCONZNZCPNZMZDNZFDNZGDNZQMZRVDVEVIDVBVCVEVISVABVHCVFEFGIVHTZVFTZJUD - UAUBVDVJVKVGDNZQMZVMRVAVBVCVGBKZVJVQRUCVDCUEKZVCVRVDCUFKZVSVAVBVTVCACDH - UGUHCUIUNVAVBVCUJBCVFGIVOUKULABVHCDFVGHIVNUMUOVDVPVLVKQVAVCVPVLSVBABCDV - FGHIVOUPUQURUSUT $. + grpsubval crg abvrcl 3ad2ant1 ringgrp syl simp3 grpinvcl syl2anc abvtri + cgrp syld3an3 abvneg 3adant2 oveq2d breqtrd eqbrtrd ) DAKZFBKZGBKZLZFGE + MZDNFGCONZNZCPNZMZDNZFDNZGDNZQMZRVDVEVIDVBVCVEVISVABVHCVFEFGIVHTZVFTZJU + DUAUBVDVJVKVGDNZQMZVMRVAVBVCVGBKZVJVQRUCVDCUNKZVCVRVDCUEKZVSVAVBVTVCACD + HUFUGCUHUIVAVBVCUJBCVFGIVOUKULABVHCDFVGHIVNUMUOVDVPVLVKQVAVCVPVLSVBABCD + VFGHIVOUPUQURUSUT $. $} abvrec.z $e |- .0. = ( 0g ` R ) $. @@ -231539,22 +231757,22 @@ Absolute value (abstract algebra) abvres $p |- ( ( F e. A /\ C e. ( SubRing ` R ) ) -> ( F |` C ) e. B ) $= ( wcel cfv wa wceq adantl eqid syl cc0 fvres syl3anc co vx vy csubrg cres cplusg cmulr c0g cabv a1i cbs subrgbas ressplusg ressmulr csubg subrgsubg - subg0 crg subrgrng cr wf wss abvf subrgss fssres syl2an subg0cl 3syl abv0 - adantr eqtrd cv wne w3a clt simp1l sselda 3adant3 simp3 3ad2ant2 breqtrrd - wbr abvgt0 cmul simp1r simp2l sseldd simp3l abvmul subrgmcl oveq12d caddc - 3eqtr4d cle abvtri subrgacl 3brtr4d isabvd ) FAJZCDUCKZJZLZUAUBBCDUEKZEDU - FKZFCUDZDUGKZBEUHKMXAIUIWTCEUJKMWRCDEHUKNWTXBEUEKMWRCXBDEWSHXBOZULNWTXCEU - FKMWRCDEXCWSHXCOZUMNXACDUNKJZXEEUGKMWTXHWRCDUONZCDEXEHXEOZUPPWTEUQJWRCDEH - URNWRDUJKZUSFUTCXKVAZCUSXDUTWTAXKDFGXKOZVBCXKDXMVCZXKUSCFVDVEXAXEXDKZXEFK - ZQXAXHXECJXOXPMXICDXEXJVFXECFRVGWRXPQMWTADFXEGXJVHVIVJXAUAVKZCJZXQXEVLZVM - ZQXQFKZXQXDKZVNXTWRXQXKJZXSQYAVNWAWRWTXRXSVOXAXRYCXSXACXKXQWTXLWRXNNVPVQX - AXRXSVRAXKDFXQXEGXMXJWBSXRXAYBYAMZXSXQCFRZVSVTXAXRXSLZUBVKZCJZYGXEVLZLZVM - ZXQYGXCTZFKZYAYGFKZWCTZYLXDKZYBYGXDKZWCTYKWRYCYGXKJZYMYOMWRWTYFYJVOZYKCXK - XQYKWTXLWRWTYFYJWDZXNPZXAXRXSYJWEZWFZYKCXKYGUUAXAYFYHYIWGZWFZAXKDXCFXQYGG - XMXGWHSYKYLCJZYPYMMYKWTXRYHUUFYTUUBUUDCDXCXQYGXGWISYLCFRPYKYBYAYQYNWCYKXR - YDUUBYEPZYKYHYQYNMUUDYGCFRPZWJWLYKXQYGXBTZFKZYAYNWKTZUUIXDKZYBYQWKTWMYKWR - YCYRUUJUUKWMWAYSUUCUUEAXKXBDFXQYGGXMXFWNSYKUUICJZUULUUJMYKWTXRYHUUMYTUUBU - UDCXBDXQYGXFWOSUUICFRPYKYBYAYQYNWKUUGUUHWJWPWQ $. + subg0 crg subrgring cr wss abvf subrgss fssres syl2an subg0cl 3syl adantr + wf abv0 eqtrd wne w3a clt wbr simp1l sselda 3adant3 simp3 abvgt0 3ad2ant2 + breqtrrd cmul simp1r simp2l sseldd simp3l abvmul subrgmcl oveq12d 3eqtr4d + cv caddc cle abvtri subrgacl 3brtr4d isabvd ) FAJZCDUCKZJZLZUAUBBCDUEKZED + UFKZFCUDZDUGKZBEUHKMXAIUIWTCEUJKMWRCDEHUKNWTXBEUEKMWRCXBDEWSHXBOZULNWTXCE + UFKMWRCDEXCWSHXCOZUMNXACDUNKJZXEEUGKMWTXHWRCDUONZCDEXEHXEOZUPPWTEUQJWRCDE + HURNWRDUJKZUSFVHCXKUTZCUSXDVHWTAXKDFGXKOZVACXKDXMVBZXKUSCFVCVDXAXEXDKZXEF + KZQXAXHXECJXOXPMXICDXEXJVEXECFRVFWRXPQMWTADFXEGXJVIVGVJXAUAWKZCJZXQXEVKZV + LZQXQFKZXQXDKZVMXTWRXQXKJZXSQYAVMVNWRWTXRXSVOXAXRYCXSXACXKXQWTXLWRXNNVPVQ + XAXRXSVRAXKDFXQXEGXMXJVSSXRXAYBYAMZXSXQCFRZVTWAXAXRXSLZUBWKZCJZYGXEVKZLZV + LZXQYGXCTZFKZYAYGFKZWBTZYLXDKZYBYGXDKZWBTYKWRYCYGXKJZYMYOMWRWTYFYJVOZYKCX + KXQYKWTXLWRWTYFYJWCZXNPZXAXRXSYJWDZWEZYKCXKYGUUAXAYFYHYIWFZWEZAXKDXCFXQYG + GXMXGWGSYKYLCJZYPYMMYKWTXRYHUUFYTUUBUUDCDXCXQYGXGWHSYLCFRPYKYBYAYQYNWBYKX + RYDUUBYEPZYKYHYQYNMUUDYGCFRPZWIWJYKXQYGXBTZFKZYAYNWLTZUUIXDKZYBYQWLTWMYKW + RYCYRUUJUUKWMVNYSUUCUUEAXKXBDFXQYGGXMXFWNSYKUUICJZUULUUJMYKWTXRYHUUMYTUUB + UUDCXBDXQYGXFWOSUUICFRPYKYBYAYQYNWLUUGUUHWIWPWQ $. $} ${ @@ -231573,24 +231791,25 @@ Absolute value (abstract algebra) 6-May-2015.) $) abvtrivd $p |- ( ph -> F e. A ) $= ( wceq cc0 c1 cplusg cfv cabv a1i cbs eqidd cmulr c0g cv cif cr wcel wa - 0re 1re keepel fmptd crg rng0cl iftrue c0ex fvmpt 3syl wne w3a clt 0lt1 - eqeq1 ifbid 1ex ifex ifnefalse sylan9eq 3adant1 syl5breqr cmul co 1t1e1 - eqcomi 3ad2ant1 simp2l simp3l rngcl syl3anc syl wn neneqd iffalse eqtrd - simp2r simp3r oveq12d 3eqtr4a caddc cle c2 breq1 0le2 1le2 keephyp df-2 - wbr breqtri cgrp rnggrp eqid grpcl 3brtr4d isabvd ) ACDEFGUAUBZGHIJEGUC - UBRAKUDFGUEUBRALUDAXJUFHGUGUBRAOUDJGUHUBRAMUDPABFBUIZJRZSTUJZUKIXMUKULA - XKFULUMXLSTUKUNUOUPUDNUQAGURULZJFULJIUBSRPFGJLMUSBJXMSFIXLSTUTNVAVBVCAC - UIZFULZXOJVDZVESTXOIUBZVFVGXPXQXRTRAXPXQXRXOJRZSTUJZTBXOXMXTFIXKXORXLXS - STXKXOJVHVINXSSTVAVJVKVBZXOJSTVLVMVNVOAXPXQUMZDUIZFULZYCJVDZUMZVEZTTTVP - VQZXOYCHVQZIUBZXRYCIUBZVPVQYHTVRVSYGYJYIJRZSTUJZTYGYIFULZYJYMRYGXNXPYDY - NAYBXNYFPVTAXPXQYFWAZAYBYDYEWBZFGHXOYCLOWCWDBYIXMYMFIXKYIRXLYLSTXKYIJVH - VINYLSTVAVJVKVBWEYGYLWFYMTRYGYIJQWGYLSTWHWEWIYGXRTYKTVPYGXRXTTYGXPXRXTR - YOYAWEYGXSWFXTTRYGXOJAXPXQYFWJWGXSSTWHWEWIZYGYKYCJRZSTUJZTYGYDYKYSRYPBY - CXMYSFIXKYCRXLYRSTXKYCJVHVINYRSTVAVJVKVBWEYGYRWFYSTRYGYCJAYBYDYEWKWGYRS - TWHWEWIZWLWMYGXOYCXJVQZJRZSTUJZTTWNVQZUUAIUBZXRYKWNVQWOUUCUUDWOXBYGUUCW - PUUDWOUUBSWPWOXBTWPWOXBUUCWPWOXBSTSUUCWPWOWQTUUCWPWOWQWRWSWTXAXCUDYGUUA - FULZUUEUUCRYGGXDULZXPYDUUFAYBUUGYFAXNUUGPGXEWEVTYOYPFXJGXOYCLXJXFXGWDBU - UAXMUUCFIXKUUARXLUUBSTXKUUAJVHVINUUBSTVAVJVKVBWEYGXRTYKTWNYQYTWLXHXI $. + 0re 1re keepel fmptd crg ring0cl c0ex fvmpt 3syl wne w3a clt 0lt1 eqeq1 + iftrue ifbid 1ex ifex ifnefalse sylan9eq 3adant1 syl5breqr 1t1e1 eqcomi + cmul co 3ad2ant1 simp2l simp3l ringcl syl3anc syl neneqd iffalse simp2r + wn eqtrd simp3r oveq12d 3eqtr4a caddc cle wbr c2 0le2 1le2 keephyp df-2 + breq1 breqtri cgrp ringgrp eqid grpcl 3brtr4d isabvd ) ACDEFGUAUBZGHIJE + GUCUBRAKUDFGUEUBRALUDAXJUFHGUGUBRAOUDJGUHUBRAMUDPABFBUIZJRZSTUJZUKIXMUK + ULAXKFULUMXLSTUKUNUOUPUDNUQAGURULZJFULJIUBSRPFGJLMUSBJXMSFIXLSTVHNUTVAV + BACUIZFULZXOJVCZVDSTXOIUBZVEVFXPXQXRTRAXPXQXRXOJRZSTUJZTBXOXMXTFIXKXORX + LXSSTXKXOJVGVINXSSTUTVJVKVAZXOJSTVLVMVNVOAXPXQUMZDUIZFULZYCJVCZUMZVDZTT + TVRVSZXOYCHVSZIUBZXRYCIUBZVRVSYHTVPVQYGYJYIJRZSTUJZTYGYIFULZYJYMRYGXNXP + YDYNAYBXNYFPVTAXPXQYFWAZAYBYDYEWBZFGHXOYCLOWCWDBYIXMYMFIXKYIRXLYLSTXKYI + JVGVINYLSTUTVJVKVAWEYGYLWIYMTRYGYIJQWFYLSTWGWEWJYGXRTYKTVRYGXRXTTYGXPXR + XTRYOYAWEYGXSWIXTTRYGXOJAXPXQYFWHWFXSSTWGWEWJZYGYKYCJRZSTUJZTYGYDYKYSRY + PBYCXMYSFIXKYCRXLYRSTXKYCJVGVINYRSTUTVJVKVAWEYGYRWIYSTRYGYCJAYBYDYEWKWF + YRSTWGWEWJZWLWMYGXOYCXJVSZJRZSTUJZTTWNVSZUUAIUBZXRYKWNVSWOUUCUUDWOWPYGU + UCWQUUDWOUUBSWQWOWPTWQWOWPUUCWQWOWPSTSUUCWQWOXBTUUCWQWOXBWRWSWTXAXCUDYG + UUAFULZUUEUUCRYGGXDULZXPYDUUFAYBUUGYFAXNUUGPGXEWEVTYOYPFXJGXOYCLXJXFXGW + DBUUAXMUUCFIXKUUARXLUUBSTXKUUAJVGVINUUBSTUTVJVKVAWEYGXRTYKTWNYQYTWLXHXI + $. $} $( The trivial absolute value. (This theorem is true as long as ` R ` is a @@ -231618,18 +231837,18 @@ Absolute value (abstract algebra) 4-Oct-2015.) $) abvpropd $p |- ( ph -> ( AbsVal ` K ) = ( AbsVal ` L ) ) $= ( cfv wcel co cv wceq wb wa wral anbi12d eqid vf cabv crg cbs cc0 cpnf wf - cico c0g cmulr cmul cplusg caddc cle wbr rngpropd eqtr3d feq2d grpidpropd - adantr eqeq2d bibi2d fveq2d eqeq1d breq1d anassrs ralbidva raleqdv anbi2d - raleqbidv 3bitr3d abvrcl isabv biadan2 3bitr4g eqrdv ) AUAEUBKZFUBKZAEUCL - ZEUDKZUEUFUHMZUANZUGZBNZWBKZUEOZWDEUIKZOZPZWDCNZEUJKZMZWBKZWEWJWBKZUKMZOZ - WDWJEULKZMZWBKZWEWNUMMZUNUOZQZCVTRZQZBVTRZQZQFUCLZFUDKZWAWBUGZWFWDFUIKZOZ - PZWDWJFUJKZMZWBKZWOOZWDWJFULKZMZWBKZWTUNUOZQZCXHRZQZBXHRZQZQWBVQLZWBVRLZA - VSXGXFYEABCDEFGHIJUPAWCXIXEYDAVTXHWAWBADVTXHGHUQURAWIXBCDRZQZBDRXLYACDRZQ - ZBDRXEYDAYIYKBDAWDDLZQZWIXLYHYJYMWHXKWFYMWGXJWDAWGXJOYLABCDEFGHIUSUTVAVBY - MXBYACDAYLWJDLZXBYAPAYLYNQQZWPXPXAXTYOWMXOWOYOWLXNWBJVCVDYOWSXSWTUNYOWRXR - WBIVCVESVFVGSVGAYIXDBDVTGAYHXCWIAXBCDVTGVHVIVJAYKYCBDXHHAYJYBXLAYACDXHHVH - VIVJVKSSYFVSXFVQEWBVQTZVLBCVQVTWQEWKWBWGYPVTTWQTWKTWGTVMVNYGXGYEVRFWBVRTZ - VLBCVRXHXQFXMWBXJYQXHTXQTXMTXJTVMVNVOVP $. + cico c0g cmulr cmul cplusg caddc ringpropd eqtr3d feq2d grpidpropd adantr + cle eqeq2d bibi2d fveq2d eqeq1d breq1d anassrs ralbidva raleqdv raleqbidv + wbr anbi2d 3bitr3d abvrcl isabv biadan2 3bitr4g eqrdv ) AUAEUBKZFUBKZAEUC + LZEUDKZUEUFUHMZUANZUGZBNZWBKZUEOZWDEUIKZOZPZWDCNZEUJKZMZWBKZWEWJWBKZUKMZO + ZWDWJEULKZMZWBKZWEWNUMMZUSVIZQZCVTRZQZBVTRZQZQFUCLZFUDKZWAWBUGZWFWDFUIKZO + ZPZWDWJFUJKZMZWBKZWOOZWDWJFULKZMZWBKZWTUSVIZQZCXHRZQZBXHRZQZQWBVQLZWBVRLZ + AVSXGXFYEABCDEFGHIJUNAWCXIXEYDAVTXHWAWBADVTXHGHUOUPAWIXBCDRZQZBDRXLYACDRZ + QZBDRXEYDAYIYKBDAWDDLZQZWIXLYHYJYMWHXKWFYMWGXJWDAWGXJOYLABCDEFGHIUQURUTVA + YMXBYACDAYLWJDLZXBYAPAYLYNQQZWPXPXAXTYOWMXOWOYOWLXNWBJVBVCYOWSXSWTUSYOWRX + RWBIVBVDSVEVFSVFAYIXDBDVTGAYHXCWIAXBCDVTGVGVJVHAYKYCBDXHHAYJYBXLAYACDXHHV + GVJVHVKSSYFVSXFVQEWBVQTZVLBCVQVTWQEWKWBWGYPVTTWQTWKTWGTVMVNYGXGYEVRFWBVRT + ZVLBCVRXHXQFXMWBXJYQXHTXQTXMTXJTVMVNVOVP $. $} $( @@ -231719,7 +231938,7 @@ Absolute value (abstract algebra) $} $( A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.) $) - srngrng $p |- ( R e. *Ring -> R e. Ring ) $= + srngring $p |- ( R e. *Ring -> R e. Ring ) $= ( csr wcel cstf cfv coppr crh co crg eqid srngrhm rhmrcl1 syl ) ABCADEZAAFE ZGHCAICANOOJNJKAONLM $. @@ -231766,11 +231985,11 @@ Absolute value (abstract algebra) srngadd $p |- ( ( R e. *Ring /\ X e. B /\ Y e. B ) -> ( .* ` ( X .+ Y ) ) = ( ( .* ` X ) .+ ( .* ` Y ) ) ) $= ( csr wcel w3a co cstf cfv wceq eqid syl syl3an1 stafval coppr cghm crh - srngrhm rhmghm oppradd ghmlin srngrng 3ad2ant2 3ad2ant3 oveq12d 3eqtr3d - crg rngacl ) CJKZEAKZFAKZLZEFBMZCNOZOZEUTOZFUTOZBMZUSDOZEDOZFDOZBMUOUTC - CUAOZUBMKZUPUQVAVDPUOUTCVHUCMKVICUTVHVHQZUTQZUDCVHUTUERBBCVHEUTFAHIBCVH - VJIUFUGSURUSAKZVAVEPUOCUMKUPUQVLCUHABCEFHIUNSUSACUTDHGVKTRURVBVFVCVGBUP - UOVBVFPUQEACUTDHGVKTUIUQUOVCVGPUPFACUTDHGVKTUJUKUL $. + srngrhm rhmghm oppradd ghmlin srngring ringacl 3ad2ant2 oveq12d 3eqtr3d + crg 3ad2ant3 ) CJKZEAKZFAKZLZEFBMZCNOZOZEUTOZFUTOZBMZUSDOZEDOZFDOZBMUOU + TCCUAOZUBMKZUPUQVAVDPUOUTCVHUCMKVICUTVHVHQZUTQZUDCVHUTUERBBCVHEUTFAHIBC + VHVJIUFUGSURUSAKZVAVEPUOCUMKUPUQVLCUHABCEFHIUISUSACUTDHGVKTRURVBVFVCVGB + UPUOVBVFPUQEACUTDHGVKTUJUQUOVCVGPUPFACUTDHGVKTUNUKUL $. $} srngmul.t $e |- .x. = ( .r ` R ) $. @@ -231780,11 +231999,11 @@ Absolute value (abstract algebra) srngmul $p |- ( ( R e. *Ring /\ X e. B /\ Y e. B ) -> ( .* ` ( X .x. Y ) ) = ( ( .* ` Y ) .x. ( .* ` X ) ) ) $= ( csr wcel w3a co cstf cfv coppr wceq eqid syl3an1 stafval srngrhm rhmmul - cmulr crh opprmul crg srngrng rngcl syl 3ad2ant3 3ad2ant2 oveq12d 3eqtr3d - syl6eq ) BJKZEAKZFAKZLZEFCMZBNOZOZFUTOZEUTOZCMZUSDOZFDOZEDOZCMURVAVCVBBPO - ZUCOZMZVDUOUTBVHUDMKUPUQVAVJQBUTVHVHRZUTRZUAEFBVHCVIUTAHIVIRZUBSABVICVHVC - VBHIVKVMUEUNURUSAKZVAVEQUOBUFKUPUQVNBUGABCEFHIUHSUSABUTDHGVLTUIURVBVFVCVG - CUQUOVBVFQUPFABUTDHGVLTUJUPUOVCVGQUQEABUTDHGVLTUKULUM $. + cmulr crh opprmul syl6eq crg srngring ringcl syl 3ad2ant3 oveq12d 3eqtr3d + 3ad2ant2 ) BJKZEAKZFAKZLZEFCMZBNOZOZFUTOZEUTOZCMZUSDOZFDOZEDOZCMURVAVCVBB + POZUCOZMZVDUOUTBVHUDMKUPUQVAVJQBUTVHVHRZUTRZUAEFBVHCVIUTAHIVIRZUBSABVICVH + VCVBHIVKVMUEUFURUSAKZVAVEQUOBUGKUPUQVNBUHABCEFHIUISUSABUTDHGVLTUJURVBVFVC + VGCUQUOVBVFQUPFABUTDHGVLTUKUPUOVCVGQUQEABUTDHGVLTUNULUM $. $} ${ @@ -231796,10 +232015,10 @@ Absolute value (abstract algebra) redundant, as can be seen from the proof of ~ issrngd .) (Contributed by Mario Carneiro, 6-Oct-2015.) $) srng1 $p |- ( R e. *Ring -> ( .* ` .1. ) = .1. ) $= - ( csr wcel cstf cfv crg cbs wceq srngrng eqid rngidcl stafval 3syl crh co - coppr srngrhm oppr1 rhm1 syl eqtr3d ) AFGZBAHIZIZBCIZBUFAJGBAKIZGUHUILAMU - JABUJNZEOBUJAUGCUKDUGNZPQUFUGAATIZRSGUHBLAUGUMUMNZULUAAUMBUGBEABUMUNEUBUC - UDUE $. + ( csr wcel cstf cfv crg cbs wceq srngring eqid ringidcl stafval coppr crh + 3syl co srngrhm oppr1 rhm1 syl eqtr3d ) AFGZBAHIZIZBCIZBUFAJGBAKIZGUHUILA + MUJABUJNZEOBUJAUGCUKDUGNZPSUFUGAAQIZRTGUHBLAUGUMUMNZULUAAUMBUGBEABUMUNEUB + UCUDUE $. $} ${ @@ -231808,10 +232027,10 @@ Absolute value (abstract algebra) $( The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.) $) srng0 $p |- ( R e. *Ring -> ( .* ` .0. ) = .0. ) $= - ( csr wcel cstf cfv crg cgrp cbs wceq srngrng rnggrp eqid grpidcl stafval - 4syl co coppr crh cghm srngrhm rhmghm oppr0 ghmid 3syl eqtr3d ) AFGZCAHIZ - IZCBIZCUJAJGAKGCALIZGULUMMANAOUNACUNPZEQCUNAUKBUODUKPZRSUJUKAAUAIZUBTGUKA - UQUCTGULCMAUKUQUQPZUPUDAUQUKUEAUQUKCCEAUQCUREUFUGUHUI $. + ( csr wcel cstf cfv crg cgrp cbs wceq srngring ringgrp grpidcl stafval co + eqid 4syl coppr crh cghm srngrhm rhmghm oppr0 ghmid 3syl eqtr3d ) AFGZCAH + IZIZCBIZCUJAJGAKGCALIZGULUMMANAOUNACUNSZEPCUNAUKBUODUKSZQTUJUKAAUAIZUBRGU + KAUQUCRGULCMAUKUQUQSZUPUDAUQUKUEAUQUKCCEAUQCUREUFUGUHUI $. $} ${ @@ -231839,42 +232058,42 @@ Absolute value (abstract algebra) 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.) $) issrngd $p |- ( ph -> R e. *Ring ) $= ( cfv co wceq cstf coppr crh wcel ccnv csr cbs cplusg cmulr cur oppr1 crg - eqid opprrng syl cstv cv wral rngidcl wa ex eleq2d fveq1d fveq12d 3imtr3d - eqeq1d eqcomd ralrimiva id fveq2 fveq2d eqeq12d rspcv sylc oveq1d eleq12d - imp ralrimiv eleq1d 3expib anbi12d oveqd oveq123d ralrimivv oveq1 rspc2va - oveq2d oveq2 eqtr4d rnglidm syl2anc 3eqtr3d stafval 3eqtr4d opprmul rngcl - syl21anc syl6eqr 3expb sylan adantl opprbas oppradd staffval fmptd rngacl - oveqan12d isrhmd cmpt wf1o wf fmpt sylibr r19.21bi rspccva adantrl eqeq2d - syl5ibrcom adantrr impbid f1ocnv2d simprd syl6reqr issrng sylanbrc ) AEUA - RZEEUBRZUCSUDYFYFUEZTEUFUDABCEUGRZYIEUHRZYJEYGEUIRZYGUIRZEUJRZYFYMYIUMZYM - UMZEYMYGYGUMZYOUKYKUMZYLUMZMAEULUDZYGULUDMEYGYPUNUOAYMEUPRZRZUUAYTRZYMYFR - ZYMAYMUUAYKSZUUDYTRZUUAUUBAUUDUUBUUAYKSZUUEAYMUUBUUAYKAYMYIUDZBUQZUUHYTRZ - YTRZTZBYIURZYMUUBTZAYSUUGMYIEYMYNYOUSUOZAUUKBYIAUUHYIUDZUTUUJUUHAUUOUUJUU - HTZAUUHHUDZUUHGRZGRZUUHTZUUOUUPAUUQUUTQVAAHYIUUHIVBZAUUSUUJUUHAUURUUIGYTL - AUUHGYTLVCZVDVFVEVQVGZVHZUUKUUMBYMYIUUHYMTZUUHYMUUJUUBUVEVIUVEUUIUUAYTUUH - YMYTVJZVKVLVMVNZVOAUUGUUAYIUDZUUHCUQZYKSZYTRZUVIYTRZUUIYKSZTZCYIURBYIURUU - EUUFTZUUNAUUGUUIYIUDZBYIURUVHUUNAUVPBYIAUUQUURHUDZUUOUVPAUUQUVQNVAUVAAUUR - UUIHYIUVBIVPVEZVRUVPUVHBYMYIUVEUUIUUAYIUVFVSVMVNZAUVNBCYIYIAUUQUVIHUDZUTZ - UUHUVIFSZGRZUVIGRZUURFSZTZUUOUVIYIUDZUTZUVNAUUQUVTUWFPVTAUUQUUOUVTUWGUVAA - HYIUVIIVBWAZAUWCUVKUWEUVMAUWBUVJGYTLAFYKUUHUVIKWBVDAUWDUVLUURUUIFYKKAUVIG - YTLVCZUVBWCVLVEZWDUVNUVOYMUVIYKSZYTRZUVLUUAYKSZTBCYMUUAYIYIUVEUVKUWMUVMUW - NUVEUVJUWLYTUUHYMUVIYKWEVKUVEUUIUUAUVLYKUVFWGVLUVIUUATZUWMUUEUWNUUFUWOUWL - UUDYTUVIUUAYMYKWHVKUWOUVLUUBUUAYKUVIUUAYTVJVOVLWFWQWIAYSUVHUUDUUATMUVSYIE - YKYMUUAYNYQYOWJWKZAUUDUUAYTUWPVKWLAUUGUUCUUATUUNYMYIEYFYTYNYTUMZYFUMZWMUO - UVGWNAUWHUTZUVKUUIUVLYLSZUVJYFRZUUHYFRZUVIYFRZYLSZUWSUVKUVMUWTAUWHUVNUWKV - QYIEYLYKYGUUIUVLYNYQYPYRWOWRUWSUVJYIUDZUXAUVKTAYSUWHUXEMYSUUOUWGUXEYIEYKU - UHUVIYNYQWPWSWTUVJYIEYFYTYNUWQUWRWMUOUWHUXDUWTTAUUOUWGUXBUUIUXCUVLYLUUHYI - EYFYTYNUWQUWRWMZUVIYIEYFYTYNUWQUWRWMZXGXAWNYIEYGYPYNXBYJUMZYJEYGYPUXHXCAB - YIUUIYIYFAUUOUVPUVRVQZBYIEYFYTYNUWQUWRXDZXEZUWSUUHUVIYJSZYTRZUUIUVLYJSZUX - LYFRZUXBUXCYJSZAUWHUXMUXNTZAUWAUUHUVIDSZGRZUURUWDDSZTZUWHUXQAUUQUVTUYAOVT - UWIAUXSUXMUXTUXNAUXRUXLGYTLADYJUUHUVIJWBVDAUURUUIUWDUVLDYJJUVBUWJWCVLVEVQ - UWSUXLYIUDZUXOUXMTAYSUWHUYBMYSUUOUWGUYBYIYJEUUHUVIYNUXHXFWSWTUXLYIEYFYTYN - UWQUWRWMUOUWHUXPUXNTAUUOUWGUXBUUIUXCUVLYJUXFUXGXGXAWNXHAYHCYIUVLXIZYFAYIY - IYFXJYHUYCTABCYIYIUUIUVLYFUXJUXIAUVLYIUDZCYIAYIYIYFXKUYDCYIURUXKCYIYIUVLY - FCYIEYFYTYNUWQUWRXDZXLXMXNUWSUUHUVLTZUVIUUITZUWSUYGUYFUVIUVLYTRZTZAUWGUYI - UUOAUULUWGUYIUVDUUKUYIBUVIYIUUHUVITZUUHUVIUUJUYHUYJVIUYJUUIUVLYTUUHUVIYTV - JVKVLXOWTXPUYFUUIUYHUVIUUHUVLYTVJXQXRUWSUYFUYGUUKAUUOUUKUWGUVCXSUYGUVLUUJ - UUHUVIUUIYTVJXQXRXTYAYBUYEYCEYFYGYPUWRYDYE $. + eqid opprring syl cstv wral ringidcl eleq2d fveq1d fveq12d eqeq1d 3imtr3d + cv wa ex imp eqcomd ralrimiva id fveq2 fveq2d eqeq12d sylc oveq1d eleq12d + rspcv ralrimiv eleq1d 3expib anbi12d oveqd oveq123d ralrimivv oveq1 oveq2 + rspc2va syl21anc ringlidm syl2anc 3eqtr3d stafval 3eqtr4d opprmul syl6eqr + oveq2d eqtr4d ringcl 3expb sylan oveqan12d opprbas oppradd staffval fmptd + adantl ringacl isrhmd cmpt wf1o wf sylibr r19.21bi rspccva adantrl eqeq2d + fmpt syl5ibrcom adantrr impbid f1ocnv2d simprd syl6reqr issrng sylanbrc ) + AEUARZEEUBRZUCSUDYFYFUEZTEUFUDABCEUGRZYIEUHRZYJEYGEUIRZYGUIRZEUJRZYFYMYIU + MZYMUMZEYMYGYGUMZYOUKYKUMZYLUMZMAEULUDZYGULUDMEYGYPUNUOAYMEUPRZRZUUAYTRZY + MYFRZYMAYMUUAYKSZUUDYTRZUUAUUBAUUDUUBUUAYKSZUUEAYMUUBUUAYKAYMYIUDZBVDZUUH + YTRZYTRZTZBYIUQZYMUUBTZAYSUUGMYIEYMYNYOURUOZAUUKBYIAUUHYIUDZVEUUJUUHAUUOU + UJUUHTZAUUHHUDZUUHGRZGRZUUHTZUUOUUPAUUQUUTQVFAHYIUUHIUSZAUUSUUJUUHAUURUUI + GYTLAUUHGYTLUTZVAVBVCVGVHZVIZUUKUUMBYMYIUUHYMTZUUHYMUUJUUBUVEVJUVEUUIUUAY + TUUHYMYTVKZVLVMVQVNZVOAUUGUUAYIUDZUUHCVDZYKSZYTRZUVIYTRZUUIYKSZTZCYIUQBYI + UQUUEUUFTZUUNAUUGUUIYIUDZBYIUQUVHUUNAUVPBYIAUUQUURHUDZUUOUVPAUUQUVQNVFUVA + AUURUUIHYIUVBIVPVCZVRUVPUVHBYMYIUVEUUIUUAYIUVFVSVQVNZAUVNBCYIYIAUUQUVIHUD + ZVEZUUHUVIFSZGRZUVIGRZUURFSZTZUUOUVIYIUDZVEZUVNAUUQUVTUWFPVTAUUQUUOUVTUWG + UVAAHYIUVIIUSWAZAUWCUVKUWEUVMAUWBUVJGYTLAFYKUUHUVIKWBVAAUWDUVLUURUUIFYKKA + UVIGYTLUTZUVBWCVMVCZWDUVNUVOYMUVIYKSZYTRZUVLUUAYKSZTBCYMUUAYIYIUVEUVKUWMU + VMUWNUVEUVJUWLYTUUHYMUVIYKWEVLUVEUUIUUAUVLYKUVFWPVMUVIUUATZUWMUUEUWNUUFUW + OUWLUUDYTUVIUUAYMYKWFVLUWOUVLUUBUUAYKUVIUUAYTVKVOVMWGWHWQAYSUVHUUDUUATMUV + SYIEYKYMUUAYNYQYOWIWJZAUUDUUAYTUWPVLWKAUUGUUCUUATUUNYMYIEYFYTYNYTUMZYFUMZ + WLUOUVGWMAUWHVEZUVKUUIUVLYLSZUVJYFRZUUHYFRZUVIYFRZYLSZUWSUVKUVMUWTAUWHUVN + UWKVGYIEYLYKYGUUIUVLYNYQYPYRWNWOUWSUVJYIUDZUXAUVKTAYSUWHUXEMYSUUOUWGUXEYI + EYKUUHUVIYNYQWRWSWTUVJYIEYFYTYNUWQUWRWLUOUWHUXDUWTTAUUOUWGUXBUUIUXCUVLYLU + UHYIEYFYTYNUWQUWRWLZUVIYIEYFYTYNUWQUWRWLZXAXFWMYIEYGYPYNXBYJUMZYJEYGYPUXH + XCABYIUUIYIYFAUUOUVPUVRVGZBYIEYFYTYNUWQUWRXDZXEZUWSUUHUVIYJSZYTRZUUIUVLYJ + SZUXLYFRZUXBUXCYJSZAUWHUXMUXNTZAUWAUUHUVIDSZGRZUURUWDDSZTZUWHUXQAUUQUVTUY + AOVTUWIAUXSUXMUXTUXNAUXRUXLGYTLADYJUUHUVIJWBVAAUURUUIUWDUVLDYJJUVBUWJWCVM + VCVGUWSUXLYIUDZUXOUXMTAYSUWHUYBMYSUUOUWGUYBYIYJEUUHUVIYNUXHXGWSWTUXLYIEYF + YTYNUWQUWRWLUOUWHUXPUXNTAUUOUWGUXBUUIUXCUVLYJUXFUXGXAXFWMXHAYHCYIUVLXIZYF + AYIYIYFXJYHUYCTABCYIYIUUIUVLYFUXJUXIAUVLYIUDZCYIAYIYIYFXKUYDCYIUQUXKCYIYI + UVLYFCYIEYFYTYNUWQUWRXDZXQXLXMUWSUUHUVLTZUVIUUITZUWSUYGUYFUVIUVLYTRZTZAUW + GUYIUUOAUULUWGUYIUVDUUKUYIBUVIYIUUHUVITZUUHUVIUUJUYHUYJVJUYJUUIUVLYTUUHUV + IYTVKVLVMXNWTXOUYFUUIUYHUVIUUHUVLYTVKXPXRUWSUYFUYGUUKAUUOUUKUWGUVCXSUYGUV + LUUJUUHUVIUUIYTVKXPXRXTYAYBUYEYCEYFYGYPUWRYDYE $. $} ${ @@ -231887,18 +232106,18 @@ Absolute value (abstract algebra) identity. (Contributed by Thierry Arnoux, 19-Apr-2019.) $) idsrngd $p |- ( ph -> R e. *Ring ) $= ( cfv wceq wcel cv wa simpr fveq2d eqeq12d rspcdv mpd co va vb cplusg cbs - cmulr a1i cstv ccrg crg crngrng syl wral ralrimiva adantr eqeltrd 3adant2 - eqidd cgrp rnggrp eqid grpcl syl3an1 3adant3 oveq12d eqtr4d crngcom rngcl - w3a 3eqtr4d eqtrd issrngd ) AUAUBDUCJZDDUEJZECCDUDJKAFUFAVLUQAVMUQEDUGJKA - GUFADUHLZDUILZHDUJUKZAUAMZCLZNZVQEJZVQCVSBMZEJZWAKZBCULZVTVQKZAWDVRAWCBCI - UMZUNZVSWCWEBVQCAVROZVSWAVQKZNZWBVTWAVQWJWAVQEVSWIOZPWKQRSZWHUOZAVRUBMZCL - ZVHZVQWNVLTZEJZWQVTWNEJZVLTWPWDWRWQKZAWOWDVRAWDWOWFUNZUPZWPWCWTBWQCADURLZ - VRWOWQCLAVOXCVPDUSUKCVLDVQWNFVLUTVAVBWPWAWQKZNZWBWRWAWQXEWAWQEWPXDOZPXFQR - SWPVTVQWSWNVLAVRWEWOWLVCZAWOWSWNKZVRAWONZWDXHXAXIWCXHBWNCAWOOXIWAWNKZNZWB - WSWAWNXKWAWNEXIXJOZPXLQRSUPZVDVEWPVQWNVMTZWNVQVMTZXNEJZWSVTVMTAVNVRWOXNXO - KHCDVMVQWNFVMUTZVFVBWPWDXPXNKZXBWPWCXRBXNCAVOVRWOXNCLVPCDVMVQWNFXQVGVBWPW - AXNKZNZWBXPWAXNXTWAXNEWPXSOZPYAQRSWPWSWNVTVQVMXMXGVDVIVSVTEJZVTVQVSWDYBVT - KZWGVSWCYCBVTCWMVSWAVTKZNZWBYBWAVTYEWAVTEVSYDOZPYFQRSWLVJVK $. + cmulr a1i eqidd cstv ccrg crg crngring syl wral ralrimiva eqeltrd 3adant2 + adantr w3a cgrp ringgrp eqid grpcl syl3an1 3adant3 oveq12d eqtr4d crngcom + ringcl 3eqtr4d eqtrd issrngd ) AUAUBDUCJZDDUEJZECCDUDJKAFUFAVLUGAVMUGEDUH + JKAGUFADUILZDUJLZHDUKULZAUAMZCLZNZVQEJZVQCVSBMZEJZWAKZBCUMZVTVQKZAWDVRAWC + BCIUNZUQZVSWCWEBVQCAVROZVSWAVQKZNZWBVTWAVQWJWAVQEVSWIOZPWKQRSZWHUOZAVRUBM + ZCLZURZVQWNVLTZEJZWQVTWNEJZVLTWPWDWRWQKZAWOWDVRAWDWOWFUQZUPZWPWCWTBWQCADU + SLZVRWOWQCLAVOXCVPDUTULCVLDVQWNFVLVAVBVCWPWAWQKZNZWBWRWAWQXEWAWQEWPXDOZPX + FQRSWPVTVQWSWNVLAVRWEWOWLVDZAWOWSWNKZVRAWONZWDXHXAXIWCXHBWNCAWOOXIWAWNKZN + ZWBWSWAWNXKWAWNEXIXJOZPXLQRSUPZVEVFWPVQWNVMTZWNVQVMTZXNEJZWSVTVMTAVNVRWOX + NXOKHCDVMVQWNFVMVAZVGVCWPWDXPXNKZXBWPWCXRBXNCAVOVRWOXNCLVPCDVMVQWNFXQVHVC + WPWAXNKZNZWBXPWAXNXTWAXNEWPXSOZPYAQRSWPWSWNVTVQVMXMXGVEVIVSVTEJZVTVQVSWDY + BVTKZWGVSWCYCBVTCWMVSWAVTKZNZWBYBWAVTYEWAVTEVSYDOZPYFQRSWLVJVK $. $} @@ -232121,10 +232340,10 @@ Absolute value (abstract algebra) NUJUINQUFUCJZUHUINUHQRRCUKSDUKSBUFOJZSEURSDCUMUOUIUPUQUFURUKABEUKTUMTUITU FTURTUOTUPTUQTUDUE $. - lmodrng.1 $e |- F = ( Scalar ` W ) $. + lmodring.1 $e |- F = ( Scalar ` W ) $. $( The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) $) - lmodrng $p |- ( W e. LMod -> F e. Ring ) $= + lmodring $p |- ( W e. LMod -> F e. Ring ) $= ( vr vw vx vq clmod wcel cgrp crg cv cfv co cbs cplusg wceq wa wral eqid cvsca w3a cmulr cur islmod simp2bi ) BHIBJIAKIDLZELZBUAMZNZBOMZIUGUHFLZBP MZNUINUJUGULUINUMNQGLZUGAPMZNUHUINUNUHUINUJUMNQUBUNUGAUCMZNUHUINUNUJUINQA @@ -232135,7 +232354,7 @@ Absolute value (abstract algebra) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmodfgrp $p |- ( W e. LMod -> F e. Grp ) $= - ( clmod wcel crg cgrp lmodrng rnggrp syl ) BDEAFEAGEABCHAIJ $. + ( clmod wcel crg cgrp lmodring ringgrp syl ) BDEAFEAGEABCHAIJ $. $} ${ @@ -232164,8 +232383,8 @@ Absolute value (abstract algebra) $( Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmodmcl $p |- ( ( W e. LMod /\ X e. K /\ Y e. K ) -> ( X .x. Y ) e. K ) $= - ( clmod wcel crg co lmodrng rngcl syl3an1 ) DJKBLKECKFCKEFAMCKBDGNCBAEFHI - OP $. + ( clmod wcel crg co lmodring ringcl syl3an1 ) DJKBLKECKFCKEFAMCKBDGNCBAEF + HIOP $. $} ${ @@ -232329,7 +232548,7 @@ Absolute value (abstract algebra) (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmod0cl $p |- ( W e. LMod -> .0. e. K ) $= - ( clmod wcel crg lmodrng rng0cl syl ) CHIAJIDBIACEKBADFGLM $. + ( clmod wcel crg lmodring ring0cl syl ) CHIAJIDBIACEKBADFGLM $. $} ${ @@ -232340,7 +232559,7 @@ Absolute value (abstract algebra) (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmod1cl $p |- ( W e. LMod -> .1. e. K ) $= - ( clmod wcel crg lmodrng rngidcl syl ) DHIBJIACIBDEKCBAFGLM $. + ( clmod wcel crg lmodring ringidcl syl ) DHIBJIACIBDEKCBAFGLM $. $} ${ @@ -232404,13 +232623,13 @@ Absolute value (abstract algebra) p. 51. ( ~ ax-hvmul0 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmod0vs $p |- ( ( W e. LMod /\ X e. V ) -> ( O .x. X ) = .0. ) $= - ( clmod wcel co cplusg cfv wceq eqid syl wa cbs crg lmodrng adantr rng0cl - simpl simpr lmodvsdir syl13anc cgrp rnggrp grplid syl2anc oveq1d lmodvscl - eqtr3d wb syl3anc lmod0vid syldan mpbid eqcomd ) EMNZFDNZUAZGCFAOZVFVGVGE - PQZOZVGRZGVGRZVFCCBPQZOZFAOZVIVGVFVDCBUBQZNZVPVEVNVIRVDVEUGZVFBUCNZVPVDVR - VEBEIUDUEZVOBCVOSZKUFTZWAVDVEUHZVHVLCCABVODEFHVHSZIJVTVLSZUIUJVFVMCFAVFBU - KNZVPVMCRVFVRWEVSBULTWAVOVLBCCVTWDKUMUNUOUQVDVEVGDNZVJVKURVFVDVPVEWFVQWAW - BCABVODEFHIJVTUPUSVHDEVGGHWCLUTVAVBVC $. + ( clmod wcel co cplusg cfv wceq eqid syl wa cbs simpl crg lmodring adantr + ring0cl simpr syl13anc cgrp ringgrp grplid syl2anc oveq1d eqtr3d lmodvscl + lmodvsdir wb syl3anc lmod0vid syldan mpbid eqcomd ) EMNZFDNZUAZGCFAOZVFVG + VGEPQZOZVGRZGVGRZVFCCBPQZOZFAOZVIVGVFVDCBUBQZNZVPVEVNVIRVDVEUCZVFBUDNZVPV + DVRVEBEIUEUFZVOBCVOSZKUGTZWAVDVEUHZVHVLCCABVODEFHVHSZIJVTVLSZUQUIVFVMCFAV + FBUJNZVPVMCRVFVRWEVSBUKTWAVOVLBCCVTWDKULUMUNUOVDVEVGDNZVJVKURVFVDVPVEWFVQ + WAWBCABVODEFHIJVTUPUSVHDEVGGHWCLUTVAVBVC $. $} ${ @@ -232422,12 +232641,12 @@ Absolute value (abstract algebra) [Kreyszig] p. 51. ( ~ hvmul0 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) $) lmodvs0 $p |- ( ( W e. LMod /\ X e. K ) -> ( X .x. .0. ) = .0. ) $= - ( clmod wcel wa c0g cfv cmulr co wceq eqid adantr crg lmodrng rngrz sylan - oveq1d cbs simpl simpr rng0cl syl lmod0vcl syl13anc lmod0vs syldan oveq2d - lmodvsass eqtrd 3eqtr3d ) DKLZECLZMZEBNOZBPOZQZFAQZVBFAQZEFAQZFVAVDVBFAUS - BUALZUTVDVBRBDGUBZCBVCEVBIVCSZVBSZUCUDUEVAVEEVFAQZVGVAUSUTVBCLZFDUFOZLZVE - VLRUSUTUGUSUTUHVAVHVMUSVHUTVITCBVBIVKUIUJUSVOUTVNDFVNSZJUKTZEVBAVCBCVNDFV - PGHIVJUPULVAVFFEAUSUTVOVFFRVQABVBVNDFFVPGHVKJUMUNZUOUQVRUR $. + ( clmod wcel wa c0g cfv cmulr co wceq eqid adantr crg ringrz sylan oveq1d + lmodring cbs simpl ring0cl syl lmod0vcl lmodvsass syl13anc lmod0vs syldan + simpr oveq2d eqtrd 3eqtr3d ) DKLZECLZMZEBNOZBPOZQZFAQZVBFAQZEFAQZFVAVDVBF + AUSBUALZUTVDVBRBDGUEZCBVCEVBIVCSZVBSZUBUCUDVAVEEVFAQZVGVAUSUTVBCLZFDUFOZL + ZVEVLRUSUTUGUSUTUOVAVHVMUSVHUTVITCBVBIVKUHUIUSVOUTVNDFVNSZJUJTZEVBAVCBCVN + DFVPGHIVJUKULVAVFFEAUSUTVOVFFRVQABVBVNDFFVPGHVKJUMUNZUPUQVRUR $. $} ${ @@ -232446,22 +232665,22 @@ Absolute value (abstract algebra) ( wcel co wceq oveq1 vx vy cn0 w3a clmod wi wa cv c1 caddc oveq1d eqeq12d cc0 imbi2d weq c0g simpr adantr eqid lmod0vs syl2anc simpl mulg0 lmodvscl cfv syl syl3anc 3eqtr4rd cplusg cmnd cgrp lmodgrp grpmnd adantl mulgnn0p1 - lmodrng rngmnd mulgnn0cl lmodvsdir syl13anc eqcomd eqtr3d sylan9eqr eqtrd - crg simprll exp31 a2d nn0ind exp4c com12 3imp impcom ) AFQZGUCQZJHQZUDIUE - QZGAJBRZDRZGACRZJBRZSZWNWOWPWQXBUFZWOWNWPXCUFWOWNWPWQXBWNWPUGZWQUGZUAUHZW - RDRZXFACRZJBRZSZUFXEUMWRDRZUMACRZJBRZSZUFXEUBUHZWRDRZXOACRZJBRZSZUFXEXOUI - UJRZWRDRZXTACRZJBRZSZUFXEXBUFUAUBGXFUMSZXJXNXEYEXGXKXIXMXFUMWRDTYEXHXLJBX - FUMACTUKULUNUAUBUOZXJXSXEYFXGXPXIXRXFXOWRDTYFXHXQJBXFXOACTUKULUNXFXTSZXJY - DXEYGXGYAXIYCXFXTWRDTYGXHYBJBXFXTACTUKULUNXFGSZXJXBXEYHXGWSXIXAXFGWRDTYHX - HWTJBXFGACTUKULUNXEEUPVEZJBRZIUPVEZXMXKXEWQWPYJYKSXDWQUQZXDWPWQWNWPUQURZB - EYIHIJYKKLMYIUSZYKUSZUTVAXEXLYIJBXEWNXLYISXDWNWQWNWPVBURZFCEAYINYNPVCVFUK - XEWRHQZXKYKSXEWQWNWPYQYLYPYMABEFHIJKLMNVDVGZHDIWRYKKYOOVCVFVHXOUCQZXEXSYD - YSXEXSYDYSXEUGZXSUGYAXPWRIVIVEZRZYCYTYAUUBSZXSYTIVJQZYSYQUUCXEUUDYSWQUUDX - DWQIVKQUUDIVLIVMVFVNVNYSXEVBZXEYQYSYRVNHUUADIXOWRKOUUAUSZVOVGURXSYTUUBXRW - RUUARZYCXPXRWRUUATYTXQAEVIVEZRZJBRZUUGYCYTWQXQFQZWNWPUUJUUGSXEWQYSYLVNYTE - VJQZYSWNUUKXEUULYSWQUULXDWQEWEQUULEILVPEVQVFVNVNZUUEYSWNWPWQWFZFCEXOANPVR - VGUUNXEWPYSYMVNUUAUUHXQABEFHIJKUUFLMNUUHUSZVSVTYTUUIYBJBYTYBUUIYTUULYSWNY - BUUISUUMUUEUUNFUUHCEXOANPUUOVOVGWAUKWBWCWDWGWHWIWJWKWLWM $. + crg lmodring ringmnd mulgnn0cl lmodvsdir syl13anc eqcomd eqtr3d sylan9eqr + simprll eqtrd exp31 a2d nn0ind exp4c com12 3imp impcom ) AFQZGUCQZJHQZUDI + UEQZGAJBRZDRZGACRZJBRZSZWNWOWPWQXBUFZWOWNWPXCUFWOWNWPWQXBWNWPUGZWQUGZUAUH + ZWRDRZXFACRZJBRZSZUFXEUMWRDRZUMACRZJBRZSZUFXEUBUHZWRDRZXOACRZJBRZSZUFXEXO + UIUJRZWRDRZXTACRZJBRZSZUFXEXBUFUAUBGXFUMSZXJXNXEYEXGXKXIXMXFUMWRDTYEXHXLJ + BXFUMACTUKULUNUAUBUOZXJXSXEYFXGXPXIXRXFXOWRDTYFXHXQJBXFXOACTUKULUNXFXTSZX + JYDXEYGXGYAXIYCXFXTWRDTYGXHYBJBXFXTACTUKULUNXFGSZXJXBXEYHXGWSXIXAXFGWRDTY + HXHWTJBXFGACTUKULUNXEEUPVEZJBRZIUPVEZXMXKXEWQWPYJYKSXDWQUQZXDWPWQWNWPUQUR + ZBEYIHIJYKKLMYIUSZYKUSZUTVAXEXLYIJBXEWNXLYISXDWNWQWNWPVBURZFCEAYINYNPVCVF + UKXEWRHQZXKYKSXEWQWNWPYQYLYPYMABEFHIJKLMNVDVGZHDIWRYKKYOOVCVFVHXOUCQZXEXS + YDYSXEXSYDYSXEUGZXSUGYAXPWRIVIVEZRZYCYTYAUUBSZXSYTIVJQZYSYQUUCXEUUDYSWQUU + DXDWQIVKQUUDIVLIVMVFVNVNYSXEVBZXEYQYSYRVNHUUADIXOWRKOUUAUSZVOVGURXSYTUUBX + RWRUUARZYCXPXRWRUUATYTXQAEVIVEZRZJBRZUUGYCYTWQXQFQZWNWPUUJUUGSXEWQYSYLVNY + TEVJQZYSWNUUKXEUULYSWQUULXDWQEVPQUULEILVQEVRVFVNVNZUUEYSWNWPWQWEZFCEXOANP + VSVGUUNXEWPYSYMVNUUAUUHXQABEFHIJKUUFLMNUUHUSZVTWAYTUUIYBJBYTYBUUIYTUULYSW + NYBUUISUUMUUEUUNFUUHCEXOANPUUOVOVGWBUKWCWDWFWGWHWIWJWKWLWM $. $} ${ @@ -232582,12 +232801,13 @@ Absolute value (abstract algebra) $( Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) $) lmodvsneg $p |- ( ph -> ( N ` ( R .x. X ) ) = ( ( M ` R ) .x. X ) ) $= - ( wcel cur cfv cmulr clmod wceq cgrp crg lmodrng syl rnggrp eqid grpinvcl - co rngidcl syl2anc lmodvsass syl13anc rngnegl lmodvscl lmodvneg1 3eqtr3rd - oveq1d syl3anc ) AEUAUBZGUBZCEUCUBZUMZJDUMZVECJDUMZDUMZCGUBZJDUMVIHUBZAIU - DTZVEFTZCFTZJBTZVHVJUEQAEUFTZVDFTZVNAEUGTZVQAVMVSQEILUHUIZEUJUIAVSVRVTFEV - DOVDUKZUNUIFEGVDOPULUOSRVECDVFEFBIJKLMOVFUKZUPUQAVGVKJDAFEVFVDGCOWBWAPVTS - URVBAVMVIBTZVJVLUEQAVMVOVPWCQSRCDEFBIJKLMOUSVCDVDEGHBIVIKNLMWAPUTUOVA $. + ( wcel cur cfv cmulr co clmod wceq cgrp crg lmodring syl ringgrp ringidcl + eqid grpinvcl syl2anc lmodvsass syl13anc ringnegl oveq1d lmodvscl syl3anc + lmodvneg1 3eqtr3rd ) AEUAUBZGUBZCEUCUBZUDZJDUDZVECJDUDZDUDZCGUBZJDUDVIHUB + ZAIUETZVEFTZCFTZJBTZVHVJUFQAEUGTZVDFTZVNAEUHTZVQAVMVSQEILUIUJZEUKUJAVSVRV + TFEVDOVDUMZULUJFEGVDOPUNUOSRVECDVFEFBIJKLMOVFUMZUPUQAVGVKJDAFEVFVDGCOWBWA + PVTSURUSAVMVIBTZVJVLUFQAVMVOVPWCQSRCDEFBIJKLMOUTVADVDEGHBIVIKNLMWAPVBUOVC + $. $} ${ @@ -232745,12 +232965,12 @@ Absolute value (abstract algebra) lmodsubvs $p |- ( ph -> ( X .- ( A .x. Y ) ) = ( X .+ ( ( N ` A ) .x. Y ) ) ) $= ( co cur cfv clmod wcel wceq lmodvscl syl3anc eqid lmodvsubval2 cmulr crg - cgrp lmodrng syl rnggrp rngidcl grpinvcl syl2anc lmodvsass rngnegl oveq1d - syl13anc eqtr3d oveq2d eqtrd ) AKBLDUDZGUDZKEUEUFZHUFZVJDUDZCUDZKBHUFZLDU - DZCUDAJUGUHZKIUHVJIUHZVKVOUITUBAVRBFUHZLIUHZVSTUAUCBDEFIJLMQPRUJUKKVJCDVL - EGHIJMNOQPSVLULZUMUKAVNVQKCAVMBEUNUFZUDZLDUDZVNVQAVRVMFUHZVTWAWEVNUITAEUP - UHZVLFUHZWFAEUOUHZWGAVRWITEJQUQURZEUSURAWIWHWJFEVLRWBUTURFEHVLRSVAVBUAUCV - MBDWCEFIJLMQPRWCULZVCVFAWDVPLDAFEWCVLHBRWKWBSWJUAVDVEVGVHVI $. + cgrp lmodring ringgrp ringidcl grpinvcl syl2anc lmodvsass syl13anc oveq1d + syl ringnegl eqtr3d oveq2d eqtrd ) AKBLDUDZGUDZKEUEUFZHUFZVJDUDZCUDZKBHUF + ZLDUDZCUDAJUGUHZKIUHVJIUHZVKVOUITUBAVRBFUHZLIUHZVSTUAUCBDEFIJLMQPRUJUKKVJ + CDVLEGHIJMNOQPSVLULZUMUKAVNVQKCAVMBEUNUFZUDZLDUDZVNVQAVRVMFUHZVTWAWEVNUIT + AEUPUHZVLFUHZWFAEUOUHZWGAVRWITEJQUQVEZEURVEAWIWHWJFEVLRWBUSVEFEHVLRSUTVAU + AUCVMBDWCEFIJLMQPRWCULZVBVCAWDVPLDAFEWCVLHBRWKWBSWJUAVFVDVGVHVI $. $} ${ @@ -232769,17 +232989,17 @@ Absolute value (abstract algebra) lmodsubdi $p |- ( ph -> ( A .x. ( X .- Y ) ) = ( ( A .x. X ) .- ( A .x. Y ) ) ) $= ( co cur cminusg cplusg clmod wcel wceq lmodvsubval2 syl3anc oveq2d cmulr - cfv crg lmodrng syl rngnegr rngnegl eqtr4d oveq1d rnggrp rngidcl grpinvcl - eqid cgrp syl2anc lmodvsass syl13anc 3eqtr3d lmodvscl lmodvsdi 3eqtr4rd ) - ABIJFTZCTBIDUAUKZDUBUKZUKZJCTZHUCUKZTZCTZBICTZBJCTZFTZAVKVQBCAHUDUEZIGUEZ - JGUEZVKVQUFPRSIJVPCVLDFVMGHKVPVBZOMLVMVBZVLVBZUGUHUIAVSBVOCTZVPTZVSVNVTCT - ZVPTZVRWAAWHWJVSVPABVNDUJUKZTZJCTZVNBWLTZJCTZWHWJAWMWOJCAWMBVMUKWOAEDWLVL - VMBNWLVBZWGWFAWBDULUEZPDHMUMUNZQUOAEDWLVLVMBNWQWGWFWSQUPUQURAWBBEUEZVNEUE - ZWDWNWHUFPQADVCUEZVLEUEZXAAWRXBWSDUSUNAWRXCWSEDVLNWGUTUNEDVMVLNWFVAVDZSBV - NCWLDEGHJKMLNWQVEVFAWBXAWTWDWPWJUFPXDQSVNBCWLDEGHJKMLNWQVEVFVGUIAWBWTWCVO - GUEZVRWIUFPQRAWBXAWDXEPXDSVNCDEGHJKMLNVHUHVPBCDEGHIVOKWEMLNVIVFAWBVSGUEZV - TGUEZWAWKUFPAWBWTWCXFPQRBCDEGHIKMLNVHUHAWBWTWDXGPQSBCDEGHJKMLNVHUHVSVTVPC - VLDFVMGHKWEOMLWFWGUGUHVJUQ $. + cfv eqid crg lmodring syl rngnegr ringnegl eqtr4d oveq1d ringgrp ringidcl + grpinvcl syl2anc lmodvsass syl13anc 3eqtr3d lmodvscl lmodvsdi 3eqtr4rd + cgrp ) ABIJFTZCTBIDUAUKZDUBUKZUKZJCTZHUCUKZTZCTZBICTZBJCTZFTZAVKVQBCAHUDU + EZIGUEZJGUEZVKVQUFPRSIJVPCVLDFVMGHKVPULZOMLVMULZVLULZUGUHUIAVSBVOCTZVPTZV + SVNVTCTZVPTZVRWAAWHWJVSVPABVNDUJUKZTZJCTZVNBWLTZJCTZWHWJAWMWOJCAWMBVMUKWO + AEDWLVLVMBNWLULZWGWFAWBDUMUEZPDHMUNUOZQUPAEDWLVLVMBNWQWGWFWSQUQURUSAWBBEU + EZVNEUEZWDWNWHUFPQADVJUEZVLEUEZXAAWRXBWSDUTUOAWRXCWSEDVLNWGVAUOEDVMVLNWFV + BVCZSBVNCWLDEGHJKMLNWQVDVEAWBXAWTWDWPWJUFPXDQSVNBCWLDEGHJKMLNWQVDVEVFUIAW + BWTWCVOGUEZVRWIUFPQRAWBXAWDXEPXDSVNCDEGHJKMLNVGUHVPBCDEGHIVOKWEMLNVHVEAWB + VSGUEZVTGUEZWAWKUFPAWBWTWCXFPQRBCDEGHIKMLNVGUHAWBWTWDXGPQSBCDEGHJKMLNVGUH + VSVTVPCVLDFVMGHKWEOMLWFWGUGUHVIUR $. $} ${ @@ -232797,17 +233017,17 @@ Absolute value (abstract algebra) analog.) (Contributed by NM, 2-Jul-2014.) $) lmodsubdir $p |- ( ph -> ( ( A S B ) .x. X ) = ( ( A .x. X ) .- ( B .x. X ) ) ) $= - ( cminusg cfv cplusg cur clmod wcel wceq cgrp crg lmodrng syl rnggrp eqid - grpinvcl syl2anc lmodvsdir syl13anc cmulr rngnegl oveq1d lmodvsass eqtr3d - co rngidcl oveq2d eqtrd grpsubval lmodvscl syl3anc lmodvsubval2 3eqtr4d ) - ABCFUBUCZUCZFUDUCZVDZKEVDZBKEVDZFUEUCZVMUCZCKEVDZEVDZJUDUCZVDZBCDVDZKEVDV - RWAHVDZAVQVRVNKEVDZWCVDZWDAJUFUGZBGUGZVNGUGZKIUGZVQWHUHRSAFUIUGZCGUGZWKAF - UJUGZWMAWIWORFJNUKULZFUMULZTGFVMCOVMUNZUOUPUAWCVOBVNEFGIJKLWCUNZNMOVOUNZU - QURAWGWBVRWCAVTCFUSUCZVDZKEVDZWGWBAXBVNKEAGFXAVSVMCOXAUNZVSUNZWRWPTUTVAAW - IVTGUGZWNWLXCWBUHRAWMVSGUGZXFWQAWOXGWPGFVSOXEVEULGFVMVSOWRUOUPTUAVTCEXAFG - IJKLNMOXDVBURVCVFVGAWEVPKEAWJWNWEVPUHSTGVOFVMDBCOWTWRQVHUPVAAWIVRIUGZWAIU - GZWFWDUHRAWIWJWLXHRSUABEFGIJKLNMOVIVJAWIWNWLXIRTUACEFGIJKLNMOVIVJVRWAWCEV - SFHVMIJLWSPNMWRXEVKVJVL $. + ( cminusg cfv cplusg co cur clmod wcel wceq cgrp crg lmodring syl ringgrp + eqid grpinvcl lmodvsdir syl13anc cmulr ringnegl oveq1d ringidcl lmodvsass + syl2anc eqtr3d oveq2d grpsubval lmodvscl syl3anc lmodvsubval2 3eqtr4d + eqtrd ) ABCFUBUCZUCZFUDUCZUEZKEUEZBKEUEZFUFUCZVMUCZCKEUEZEUEZJUDUCZUEZBCD + UEZKEUEVRWAHUEZAVQVRVNKEUEZWCUEZWDAJUGUHZBGUHZVNGUHZKIUHZVQWHUIRSAFUJUHZC + GUHZWKAFUKUHZWMAWIWORFJNULUMZFUNUMZTGFVMCOVMUOZUPVDUAWCVOBVNEFGIJKLWCUOZN + MOVOUOZUQURAWGWBVRWCAVTCFUSUCZUEZKEUEZWGWBAXBVNKEAGFXAVSVMCOXAUOZVSUOZWRW + PTUTVAAWIVTGUHZWNWLXCWBUIRAWMVSGUHZXFWQAWOXGWPGFVSOXEVBUMGFVMVSOWRUPVDTUA + VTCEXAFGIJKLNMOXDVCURVEVFVLAWEVPKEAWJWNWEVPUISTGVOFVMDBCOWTWRQVGVDVAAWIVR + IUHZWAIUHZWFWDUIRAWIWJWLXHRSUABEFGIJKLNMOVHVIAWIWNWLXIRTUACEFGIJKLNMOVHVI + VRWAWCEVSFHVMIJLWSPNMWRXEVJVIVK $. $} ${ @@ -232875,56 +233095,56 @@ Absolute value (abstract algebra) lmodprop2d $p |- ( ph -> ( K e. LMod <-> L e. LMod ) ) $= ( wcel vr vw vz vq cgrp crg cv cvsca cfv co wral w3a clmod wi lmodgrp a1i cbs cplusg wceq cur wa eqid islmod simp2bi simplr simprl ad2antrr eleqtrd - cmulr simprr lmodvscl syl3anc eleqtrrd ralrimivva 3jcad grppropd rngpropd - ex syl5ibr adantlr 3eltr4d adantr simpll simprlr simprrr proplem syl12anc - wb eleq1d simplr1 simprrl grpcl simplr3 proplem2 syl21anc oveq12d eqeq12d - oveq2d eqtrd simplr2 simprll rngacl ad2ant2r oveq1d 3anbi123d rngidcl syl - rngcl rngidpropd eqeq1d anbi12d anassrs 2ralbidva eleq2d anbi1d raleqbidv - 3anbi1d 3bitr3d 3bitr4g pm5.21ndd ) AHUETZFUFTZBUGZCUGZHUHUIZUJZDTZCDUKBE - UKZULZHUMTZIUMTZAYJYAYBYHYJYAUNAHUOUPYJYBUNAYJYAYBUAUGZUBUGZYEUJZHUQUIZTZ - YLYMUCUGZHURUIZUJZYEUJZYNYLYQYEUJZYRUJZUSZUDUGZYLFURUIZUJZYMYEUJZUUDYMYEU - JZYNYRUJZUSZULZUUDYLFVIUIZUJZYMYEUJZUUDYNYEUJZUSZFUTUIZYMYEUJZYMUSZVAZVAZ - UBYOUKZUCYOUKZUAFUQUIZUKZUDUVDUKZUCUBYRUUEYEUULUUQFUVDYOHUAUDYOVBZYRVBZYE - VBZLUVDVBZUUEVBZUULVBZUUQVBZVCZVDUPAYJYHAYJVAZYGBCEDUVOYCETZYDDTZVAZVAZYF - YODUVSYJYCUVDTYDYOTYFYOTAYJUVRVEUVSYCEUVDUVOUVPUVQVFAEUVDUSZYJUVRNVGVHUVS - 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LMod /\ U e. S /\ X e. U ) -> ( N ` X ) e. U ) $= ( clmod wcel w3a csca cfv cur cminusg cvsca co wceq cbs eqid syl wa lssel - lmodvneg1 sylan2 3impb simp1 simp2 cgrp lmodrng 3ad2ant1 rngidcl grpinvcl - crg rnggrp syl2anc simp3 lssvscl syl22anc eqeltrrd ) DHIZBAIZEBIZJZDKLZML - ZVDNLZLZEDOLZPZECLZBUTVAVBVIVJQZVAVBUAUTEDRLZIVKABVLDEVLSZFUBVHVEVDVFCVLD - EVMGVDSZVHSZVESZVFSZUCUDUEVCUTVAVGVDRLZIZVBVIBIUTVAVBUFUTVAVBUGVCVDUHIZVE - VRIZVSVCVDUMIZVTUTVAWBVBVDDVNUIUJZVDUNTVCWBWAWCVRVDVEVRSZVPUKTVRVDVFVEWDV - QULUOUTVAVBUPVRAVHBVDDVGEVNVOWDFUQURUS $. + lmodvneg1 sylan2 3impb simp1 simp2 crg lmodring 3ad2ant1 ringgrp ringidcl + cgrp grpinvcl syl2anc simp3 lssvscl syl22anc eqeltrrd ) DHIZBAIZEBIZJZDKL + ZMLZVDNLZLZEDOLZPZECLZBUTVAVBVIVJQZVAVBUAUTEDRLZIVKABVLDEVLSZFUBVHVEVDVFC + VLDEVMGVDSZVHSZVESZVFSZUCUDUEVCUTVAVGVDRLZIZVBVIBIUTVAVBUFUTVAVBUGVCVDUMI + ZVEVRIZVSVCVDUHIZVTUTVAWBVBVDDVNUIUJZVDUKTVCWBWAWCVRVDVEVRSZVPULTVRVDVFVE + WDVQUNUOUTVAVBUPVRAVHBVDDVGEVNVOWDFUQURUS $. $} ${ @@ -233436,26 +233656,26 @@ Absolute value (abstract algebra) islss3 $p |- ( W e. LMod -> ( U e. S <-> ( U C_ V /\ X e. LMod ) ) ) $= ( vx wcel wa adantl cfv cbs wceq eqid eqidd syl co cvv va clmod wss lssss vb csca cplusg cvsca cmulr cur ressbas2 sylan2 ressplusg resssca ressvsca - crg lmodrng adantr csubg cgrp lsssubg subggrp cv lssvscl 3impb w3a simpll - simpr1 ad2antlr simpr2 sseldd simpr3 lmodvsdi lmodvsdir lmodvsass lmodvs1 - syl13anc sselda adantlr syldan islmodd simprl fvex syl6eqel clss eqsstr3d - jca eqcomd a1i c0 lmodgrp ad2antll grpbn0 lss1 lsscl sylan islssd eqeltrd - wne impbida ) DUBJZBAJZBCUCZEUBJZKZXAXBKZXCXDXBXCXAABCDGHUDZLZXFIUAUEDUFM - ZNMZDUGMZXIUGMZDUHMZXIUIMZXIUJMZXIBEXBXAXCBENMZOZXGXCXQXABCEDFGUKZLULXBXK - EUGMZOZXABXKDEAFXKPZUMLXBXIEUFMZOZXABXIDEAFXIPZUNLXBXMEUHMZOZXABXMDEAFXMP - ZUOLXFXJQXFXLQXFXNQXFXOQXAXIUPJXBXIDYDUQURXFBDUSMJEUTJZABDHVABDEFVBRXFIVC - ZXJJZUAVCZBJZYIYKXMSZBJXJAXMBXIDYIYKYDYGXJPZHVDVEXFYJYLUEVCZBJZVFZKZXAYJY - KCJYOCJZYIYKYOXKSXMSYMYIYOXMSZXKSOXAXBYQVGXFYJYLYPVHYRBCYKXBXCXAYQXGVIZXF - YJYLYPVJVKYRBCYOUUAXFYJYLYPVLVKXKYIXMXIXJCDYKYOGYAYDYGYNVMVQXFYJYKXJJZYPV - 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(Contributed by Mario Carneiro, 11-Jan-2015.) $) pwslmod $p |- ( ( R e. LMod /\ I e. V ) -> Y e. LMod ) $= - ( vx clmod wcel wa csca cfv csn cxp cprds eqid pwsval crg lmodrng adantr + ( vx clmod wcel wa csca cfv csn cxp cprds eqid pwsval crg lmodring adantr co simpr wf fconst6g cv wceq fvconst2g adantlr fveq2d prdslmodd eqeltrd ) AGHZBCHZIZDAJKZBALMZNTZGAUNBGCDEUNOZPUMFUOUNBCUPUPOUKUNQHULUNAUQRSUKULUAU KBGUOUBULBAGUCSUMFUDZBHZIURUOKZAJUKUSUTAUEULBAURGUFUGUHUIUJ $. @@ -234503,13 +234723,13 @@ Absolute value (abstract algebra) O'Rear, 28-Feb-2015.) $) lmodvsinv $p |- ( ( W e. LMod /\ R e. K /\ X e. B ) -> ( ( M ` R ) .x. X ) = ( N ` ( R .x. X ) ) ) $= - ( wcel cfv co wceq syl clmod w3a cur cmulr simp1 cgrp crg 3ad2ant1 rnggrp - lmodrng eqid rngidcl grpinvcl syl2anc simp2 simp3 syl13anc rngnegl oveq1d - lmodvsass lmodvscl lmodvneg1 3eqtr3d ) HUAPZBEPZIAPZUBZDUCQZFQZBDUDQZRZIC - RZVIBICRZCRZBFQZICRVMGQZVGVDVIEPZVEVFVLVNSVDVEVFUEZVGDUFPZVHEPZVQVGDUGPZV - SVDVEWAVFDHKUJUHZDUITVGWAVTWBEDVHOVHUKZULTEDFVHONUMUNVDVEVFUOZVDVEVFUPVIB - CVJDEAHIJKLOVJUKZUTUQVGVKVOICVGEDVJVHFBOWEWCNWBWDURUSVGVDVMAPVNVPSVRBCDEA - HIJKLOVACVHDFGAHVMJMKLWCNVBUNVC $. + ( wcel cfv co wceq syl clmod w3a cur cmulr cgrp lmodring 3ad2ant1 ringgrp + simp1 crg ringidcl grpinvcl syl2anc simp2 simp3 lmodvsass syl13anc oveq1d + eqid ringnegl lmodvscl lmodvneg1 3eqtr3d ) HUAPZBEPZIAPZUBZDUCQZFQZBDUDQZ + RZICRZVIBICRZCRZBFQZICRVMGQZVGVDVIEPZVEVFVLVNSVDVEVFUIZVGDUEPZVHEPZVQVGDU + JPZVSVDVEWAVFDHKUFUGZDUHTVGWAVTWBEDVHOVHUSZUKTEDFVHONULUMVDVEVFUNZVDVEVFU + OVIBCVJDEAHIJKLOVJUSZUPUQVGVKVOICVGEDVJVHFBOWEWCNWBWDUTURVGVDVMAPVNVPSVRB + CDEAHIJKLOVACVHDFGAHVMJMKLWCNVBUMVC $. $} ${ @@ -234552,39 +234772,39 @@ Absolute value (abstract algebra) ( F ` ( ( x .x. y ) .+ z ) ) = ( ( x .X. ( F ` y ) ) .+^ ( F ` z ) ) ) ) ) $= ( clmod wcel wa clmhm co wf wceq cv cfv wral w3a lmhmf lmhmsca cghm lmghm adantr lmhmlmod1 simpr1 simpr2 syl3anc simpr3 ghmlin lmhmlin oveq1d eqtrd - lmodvscl 3adant3r3 ralrimivvva 3jca adantl lmodgrp anim12i wi cur lmodrng - cgrp crg ad2antrr eqid rngidcl oveq1 fveq2d eqeq12d 2ralbidv 3syl simplll - rspcv simprl lmodvs1 syl2anc simplrr simpllr ffvelrnd eqtr3d sylibd exp32 - simplrl 2ralbidva 3imp2 jca isghm c0g ghmid ad3antrrr grpidcl oveq2 fveq2 - sylanbrc syl oveq2d simprr grprid simplr3 simplr2 syl6eqr simplr1 anassrs - cbs eleqtrrd ralimdva 3exp2 com45 mpd wb islmhm3 mpbir3and impbida ) HUEU - FZIUEUFZUGZMHIUHUIUFZDEMUJZONUKZAULZBULZJUIZCULZFUIZMUMZYRYSMUMZKUIZUUAMU - MZGUIZUKZCDUNZBDUNZALUNZUOZYOUULYNYOYPYQUUKDEHIMPQUPHIMNORSUQYOUUHABCLDDY - OYRLUFZYSDUFZUUADUFZUOZUGZUUCYTMUMZUUFGUIZUUGUUQMHIURUIUFZYTDUFZUUOUUCUUS - UKYOUUTUUPHIMUSUTUUQYLUUMUUNUVAYOYLUUPHIMVAUTYOUUMUUNUUOVBYOUUMUUNUUOVCYR - JNLDHYSPRUCTVJZVDYOUUMUUNUUOVEFGHIYTMUUADPUAUBVFVDUUQUURUUEUUFGYOUUMUUNUU - 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LMod /\ .1. =/= .0. ) /\ B e. J /\ E e. B ) -> -. E e. ( N ` ( B \ { E } ) ) ) $= ( clmod wcel wne cfv cbs eqid syl2anc wa w3a cvsca co csn cdif wceq simp2 - simp1l simp3 lbsel lmodvs1 wn lmodrng rngidcl 3syl simp1r lbsind syl22anc - crg eqneltrrd ) GNOZBHPZUAZAEOZCAOZUBZBCGUCQZUDZCACUEUFFQZVGVBCGRQZOZVICU - GVBVCVEVFUIZVGVEVFVLVDVEVFUHZVDVEVFUJZACEVKGVKSZIUKTVHBDVKGCVPKVHSZLULTVG - VEVFBDRQZOZVCVIVJOUMVNVOVGVBDUTOVSVMDGKUNVRDBVRSZLUOUPVBVCVEVFUQBAVHCDEVR - FVKGHVPIJKVQVTMURUSVA $. + simp1l simp3 lmodvs1 wn crg lmodring ringidcl 3syl simp1r lbsind syl22anc + lbsel eqneltrrd ) GNOZBHPZUAZAEOZCAOZUBZBCGUCQZUDZCACUEUFFQZVGVBCGRQZOZVI + CUGVBVCVEVFUIZVGVEVFVLVDVEVFUHZVDVEVFUJZACEVKGVKSZIUTTVHBDVKGCVPKVHSZLUKT + VGVEVFBDRQZOZVCVIVJOULVNVOVGVBDUMOVSVMDGKUNVRDBVRSZLUOUPVBVCVEVFUQBAVHCDE + VRFVKGHVPIJKVQVTMURUSVA $. $d x y .0. $. $d x y B $. $d x y C $. $d x y J $. $d x N $. $d x V $. $d x y .1. $. $d x y W $. @@ -236158,28 +236378,28 @@ division ring is an Abelian group (vectors) together with a division ring lspsneq $p |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E. k e. ( K \ { .0. } ) X = ( k .x. Y ) ) ) $= ( vj csn cfv wceq cv co cdif wrex wa c0g cur wne clmod crg clvec lveclmod - wcel syl lmodrng eqid rngidcl 3syl cdr lvecdrng drngunz sylanbrc ad2antrr - eldifsn lmod0vcl lmodvs1 syl2anc adantl adantr lspsneq0b biimpar 3eqtr4rd - oveq2 simpr eqeq2d rspcev wss eqimss clss lspsncl lspsnel5 mpbird lspsnel - oveq1 wb mpbid simprl biimpd necon3d eqnetrrd lvecvsn0 simpld ex reximdv2 - imp jca mpd pm2.61dane wi eldifi eldifsni lspsnvs syl121anc fveq2d eqeq1d - sneq biimprcd syl6 rexlimdv impbid weq cbvrexv syl6bb ) AIUBZFUCZJUBFUCZU - DZIUAUEZJCUFZUDZUAEKUBZUGZUHZIDUEZJCUFZUDZDYFUHAYAYGAYAYGAYAUIZYGJHUJUCZY - KJYLUDZUIZBUKUCZYFUQZIYOJCUFZUDZYGAYPYAYMAYOEUQZYOKULZYPAHUMUQZBUNUQYSAHU - OUQZUUARHUPZURZBHMUSEBYONYOUTZVAVBAUUBBVCUQYTRBHMVDBYOKOUUEVEVBYOEKVHVFVG - YNYOYLCUFZYLYQIAUUFYLUDZYAYMAUUAYLGUQZUUGUUDAUUBUUAUUHRUUCGHYLLYLUTZVIVBC - YOBGHYLLMPUUEVJVKVGYMYQUUFUDYKJYLYOCVQVLYKIYLUDZYMYKFGHIJYLLUUIQAUUAYAUUD - VMZAIGUQYASVMZAJGUQZYATVMZAYAVRVNZVOVPYDYRUAYOYFYBYOUDYCYQIYBYOJCWHVSVTVK - YKJYLULZUIZYDUAEUHZYGYKUURUUPYKIXTUQZUURYKUUSXSXTWAZYAUUTAXSXTWBVLYKHWCUC - 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( SubRing ` W ) -> A e. LMod ) $= ( vx vy vz cfv wcel cplusg co wceq eqid cgrp cv wa adantr sseldi syl13anc w3a csubrg cbs cin cmulr cur cress subrgss srabase sraaddg srasca sravsca - csra a1i ressbas ressplusg ressmulr subrg1 subrgrng subrgrcl rnggrp eqidd - crg syl proplem3 grppropd mpbid 3ad2ant1 inss2 sseli 3ad2ant2 simp3 rngcl - syl3anc simpr1 simpr2 simpr3 rngdi rngdir rngass rnglidm sylan islmodd ) - BCUAHZIZEFGBCUBHZUCZCJHZWGCUDHZWHCUEHZCBUFKZWEAWDABCABCULHHLWDDUMZBWECWEM - ZUGZUHZWDABCWKWMUIZWDABCWKWMUJWDABCWKWMUKBWEWJWCCWJMZWLUNBWGCWJWCWPWGMZUO - BCWJWHWCWPWHMZUPBCWJWIWPWIMZUQBCWJWPURWDCNIZANIWDCVBIZWTBCUSZCUTVCWDEFWEC - AWDWEVAWNWDEOZWEIZFOZWEIZPEFWGAJHWOVDVEVFWDXCWFIZXFTXAXDXFXCXEWHKZWEIWDXG - XAXFXBVGXGWDXDXFWFWEXCBWEVHZVIVJWDXGXFVKWECWHXCXEWLWRVLVMWDXGXFGOZWEIZTZP - ZXAXDXFXKXCXEXJWGKWHKXHXCXJWHKZWGKLWDXAXLXBQXMWFWEXCXIWDXGXFXKVNRWDXGXFXK - VOWDXGXFXKVPWEWGCWHXCXEXJWLWQWRVQSWDXGXEWFIZXKTZPZXAXDXFXKXCXEWGKXJWHKXNX - EXJWHKZWGKLWDXAXPXBQZXQWFWEXCXIWDXGXOXKVNRZXQWFWEXEXIWDXGXOXKVORZWDXGXOXK - VPZWEWGCWHXCXEXJWLWQWRVRSXQXAXDXFXKXHXJWHKXCXRWHKLXSXTYAYBWECWHXCXEXJWLWR - VSSWDXAXDWIXCWHKXCLXBWECWHWIXCWLWRWSVTWAWB $. + csra a1i ressbas ressplusg ressmulr subrg1 subrgring crg subrgrcl ringgrp + eqidd proplem3 grppropd mpbid 3ad2ant1 inss2 sseli 3ad2ant2 simp3 syl3anc + ringcl simpr1 simpr2 simpr3 ringdi ringdir ringass ringlidm sylan islmodd + syl ) BCUAHZIZEFGBCUBHZUCZCJHZWGCUDHZWHCUEHZCBUFKZWEAWDABCABCULHHLWDDUMZB + WECWEMZUGZUHZWDABCWKWMUIZWDABCWKWMUJWDABCWKWMUKBWEWJWCCWJMZWLUNBWGCWJWCWP + WGMZUOBCWJWHWCWPWHMZUPBCWJWIWPWIMZUQBCWJWPURWDCNIZANIWDCUSIZWTBCUTZCVAWBW + DEFWECAWDWEVBWNWDEOZWEIZFOZWEIZPEFWGAJHWOVCVDVEWDXCWFIZXFTXAXDXFXCXEWHKZW + EIWDXGXAXFXBVFXGWDXDXFWFWEXCBWEVGZVHVIWDXGXFVJWECWHXCXEWLWRVLVKWDXGXFGOZW + EIZTZPZXAXDXFXKXCXEXJWGKWHKXHXCXJWHKZWGKLWDXAXLXBQXMWFWEXCXIWDXGXFXKVMRWD + XGXFXKVNWDXGXFXKVOWEWGCWHXCXEXJWLWQWRVPSWDXGXEWFIZXKTZPZXAXDXFXKXCXEWGKXJ + WHKXNXEXJWHKZWGKLWDXAXPXBQZXQWFWEXCXIWDXGXOXKVMRZXQWFWEXEXIWDXGXOXKVNRZWD + XGXOXKVOZWEWGCWHXCXEXJWLWQWRVQSXQXAXDXFXKXHXJWHKXCXRWHKLXSXTYAYBWECWHXCXE + XJWLWRVRSWDXAXDWIXCWHKXCLXBWECWHWIXCWLWRWSVSVTWA $. $} ${ @@ -237597,12 +237817,12 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $= $( Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) $) issubrngd2 $p |- ( ph -> D e. ( SubRing ` I ) ) $= - ( csubrg cfv wcel csubg cur cv cmulr co wral crg cgrp rnggrp syl eqeltrrd - issubgrpd2 wa proplem3 3expb ralrimivva w3a cbs eqid issubrg2 mpbir3and - wb ) ADIUCUDUEZDIUFUDUEZIUGUDZDUEZBUHZCUHZIUIUDZUJZDUEZCDUKBDUKZABCDEFIJK - LMNOPQAIULUEZIUMUEUBIUNUOUQAHVJDRTUPAVPBCDDAVLDUEZVMDUEZURZURVLVMGUJZVODA - WABCGVNSUSAVSVTWBDUEUAUTUPVAAVRVHVIVKVQVBVGUBBCDIVCUDZIVNVJWCVDVJVDVNVDVE - UOVF $. + ( csubrg cfv wcel csubg cur cv cmulr wral crg cgrp ringgrp syl issubgrpd2 + co eqeltrrd wa proplem3 3expb ralrimivva w3a cbs eqid issubrg2 mpbir3and + wb ) ADIUCUDUEZDIUFUDUEZIUGUDZDUEZBUHZCUHZIUIUDZUPZDUEZCDUJBDUJZABCDEFIJK + LMNOPQAIUKUEZIULUEUBIUMUNUOAHVJDRTUQAVPBCDDAVLDUEZVMDUEZURZURVLVMGUPZVODA + WABCGVNSUSAVSVTWBDUEUAUTUQVAAVRVHVIVKVQVBVGUBBCDIVCUDZIVNVJWCVDVJVDVNVDVE + UNVF $. $} ${ @@ -237832,11 +238052,11 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $= lidlsubg $p |- ( ( R e. Ring /\ I e. U ) -> I e. ( SubGrp ` R ) ) $= ( vx vy crg wcel wa csubg cfv cbs wss c0 wne cv wral eqid syl ralrimiva cplusg cminusg lidlss adantl lidl0cl ne0i lidlacl anassrs lidlnegcl 3expa - co c0g jca cgrp w3a wb rnggrp adantr issubg2 mpbir3and ) AGHZCBHZIZCAJKHZ - CALKZMZCNOZEPZFPZAUAKZUKCHZFCQZVHAUBKZKCHZIZECQZVBVFVAVECBAVERZDUCUDVCAUL - KZCHVGABCVRDVRRUECVRUFSVCVOECVCVHCHZIZVLVNVTVKFCVCVSVICHVKVJABCVHVIDVJRZU - GUHTVAVBVSVNABCVMVHDVMRZUIUJUMTVCAUNHZVDVFVGVPUOUPVAWCVBAUQUREFVEVJCAVMVQ - WAWBUSSUT $. + co c0g jca cgrp w3a wb ringgrp adantr issubg2 mpbir3and ) AGHZCBHZIZCAJKH + ZCALKZMZCNOZEPZFPZAUAKZUKCHZFCQZVHAUBKZKCHZIZECQZVBVFVAVECBAVERZDUCUDVCAU + LKZCHVGABCVRDVRRUECVRUFSVCVOECVCVHCHZIZVLVNVTVKFCVCVSVICHVKVJABCVHVIDVJRZ + UGUHTVAVBVSVNABCVMVHDVMRZUIUJUMTVCAUNHZVDVFVGVPUOUPVAWCVBAUQUREFVEVJCAVMV + QWAWBUSSUT $. ${ lidlsubcl.m $e |- .- = ( -g ` R ) $. @@ -237874,11 +238094,11 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $= $( An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) $) lidl1el $p |- ( ( R e. Ring /\ I e. U ) -> ( .1. e. I <-> I = B ) ) $= - ( va crg wcel wa wceq wss lidlssOLD adantr cv cmulr cfv co eqid rngridm - ad2ant2rl lidlmcl ancom2s eqeltrrd expr ssrdv eqssd ex eleq2 syl5ibrcom - rngidcl impbid ) BJKZECKZLZDEKZEAMZUQURUSUQURLZEAUQEANURAECJBGFOPUTIAEU - QURIQZAKZVAEKUQURVBLLVADBRSZTZVAEUOVBVDVAMUPURABVCDVAGVCUAZHUBUCUQVBURV - DEKABVCCEVADFGVEUDUEUFUGUHUIUJUQURUSDAKZUOVFUPABDGHUMPEADUKULUN $. + ( va crg wcel wa wceq wss lidlssOLD adantr cv cmulr cfv co eqid lidlmcl + ringridm ancom2s eqeltrrd expr ssrdv eqssd ex ringidcl eleq2 syl5ibrcom + ad2ant2rl impbid ) BJKZECKZLZDEKZEAMZUQURUSUQURLZEAUQEANURAECJBGFOPUTIA + EUQURIQZAKZVAEKUQURVBLLVADBRSZTZVAEUOVBVDVAMUPURABVCDVAGVCUAZHUCUMUQVBU + RVDEKABVCCEVADFGVEUBUDUEUFUGUHUIUQURUSDAKZUOVFUPABDGHUJPEADUKULUN $. $} $} @@ -237941,7 +238161,7 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $= $( The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) $) rsp1 $p |- ( R e. Ring -> ( K ` { .1. } ) = B ) $= - ( crg wcel csn cfv wceq wss rngidcl snssd rspssid mpdan cur cvv fvex + ( crg wcel csn cfv wceq wss ringidcl snssd rspssid mpdan cur cvv fvex eqeltri snss sylibr clidl wb eqid rspcl lidl1el mpbid ) BHIZCCJZDKZIZ ULALZUJUKULMZUMUJUKAMZUOUJCAABCFGNOZABUKDEFPQCULCBRKSGBRTUAUBUCUJULBU DKZIZUMUNUEUJUPUSUQABURUKDEFURUFZUGQABURCULUTFGUHQUI $. @@ -238093,31 +238313,31 @@ left ideal which is also a right ideal (or a left ideal over the opposite 2idlcpbl $p |- ( ( R e. Ring /\ S e. I ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) ) $= ( wcel wbr co cfv eqid syl2anc crg csg simpll w3a csubg wer clidl 2idlval - wa coppr elin2 simplbi ad2antlr lidlsubg eqger syl simprl cabl wss rngabl - ersym ad2antrr lidlss eqgabl mpbid simp2d simprr simp1d syl3anc cgrp wceq - rngcl rnggrp grpnnncan2 syl13anc rngsubdi lidlmcl syl22anc eqeltrrd cmulr - wb simp3d opprmul rngsubdir syl5eq opprrng simprbi lidlsubcl mpbir3and ex - opprbas ) EUAOZFIOZUIZACHPZBDHPZUIZABGQZCDGQZHPZWNWQUIZWTWRJOZWSJOZWSWREU - BRZQZFOZXAWLAJOZBJOZXBWLWMWQUCZXACJOZXGACXDQZFOZXACAHPZXJXGXLUDZXAACHJXAF - EUEROZJHUFXAWLFEUGRZOZXOXIWMXQWLWQWMXQFEUJRZUGRZOZFXPXSIEIXPXSXRXPSZXRSZX - SSZMUHUKZULUMZEXPFYAUNTHEJFKLUOUPWNWOWPUQVAXAEUROZFJUSZXMXNWAWLYFWMWQEUTV - BZXAXQYGYEJFXPEKYAVCUPZCAHFEXDJKXDSZLVDTVEZVFZXAXHDJOZDBXDQZFOZXAWPXHYMYO - UDZWNWOWPVGXAYFYGWPYPWAYHYIBDHFEXDJKYJLVDTVEZVHZJEGABKNVLVIZXAWLXJYMXCXIX - AXJXGXLYKVHZXAXHYMYOYQVFZJEGCDKNVLVIZXAWSCBGQZXDQZWRUUCXDQZXDQZXEFXAEVJOZ - XCXBUUCJOZUUFXEVKWLUUGWMWQEVMVBUUBYSXAWLXJXHUUHXIYTYRJEGCBKNVLVIJEXDWSWRU - UCKYJVNVOXAWLXQUUDFOUUEFOUUFFOXIYEXACYNGQZUUDFXAJEGXDCDBKNYJXIYTUUAYRVPXA - WLXQXJYOUUIFOXIYEYTXAXHYMYOYQWBJEGXPFCYNYAKNVQVRVSXABXKXRVTRZQZUUEFXAUUKX - KBGQUUEJEUUJGXRBXKKNYBUUJSZWCXAJEGXDACBKNYJXIYLYTYRWDWEXAXRUAOZXTXHXLUUKF - OWLUUMWMWQEXRYBWFVBWMXTWLWQWMXQXTYDWGUMYRXAXJXGXLYKWBJXRUUJXSFBXKYCJEXRYB - KWKUULVQVRVSEXPFXDUUDUUEYAYJWHVRVSXAYFYGWTXBXCXFUDWAYHYIWRWSHFEXDJKYJLVDT - WIWJ $. + coppr elin2 simplbi ad2antlr lidlsubg eqger syl simprl ersym cabl ringabl + wa wss wb ad2antrr lidlss eqgabl simp2d simprr simp1d ringcl syl3anc cgrp + mpbid wceq ringgrp grpnnncan2 syl13anc ringsubdi simp3d syl22anc eqeltrrd + lidlmcl cmulr opprmul rngsubdir syl5eq opprring simprbi opprbas lidlsubcl + mpbir3and ex ) EUAOZFIOZUTZACHPZBDHPZUTZABGQZCDGQZHPZWNWQUTZWTWRJOZWSJOZW + SWREUBRZQZFOZXAWLAJOZBJOZXBWLWMWQUCZXACJOZXGACXDQZFOZXACAHPZXJXGXLUDZXAAC + HJXAFEUEROZJHUFXAWLFEUGRZOZXOXIWMXQWLWQWMXQFEUIRZUGRZOZFXPXSIEIXPXSXRXPSZ + XRSZXSSZMUHUJZUKULZEXPFYAUMTHEJFKLUNUOWNWOWPUPUQXAEUROZFJVAZXMXNVBWLYFWMW + QEUSVCZXAXQYGYEJFXPEKYAVDUOZCAHFEXDJKXDSZLVETVLZVFZXAXHDJOZDBXDQZFOZXAWPX + HYMYOUDZWNWOWPVGXAYFYGWPYPVBYHYIBDHFEXDJKYJLVETVLZVHZJEGABKNVIVJZXAWLXJYM + XCXIXAXJXGXLYKVHZXAXHYMYOYQVFZJEGCDKNVIVJZXAWSCBGQZXDQZWRUUCXDQZXDQZXEFXA + EVKOZXCXBUUCJOZUUFXEVMWLUUGWMWQEVNVCUUBYSXAWLXJXHUUHXIYTYRJEGCBKNVIVJJEXD + WSWRUUCKYJVOVPXAWLXQUUDFOUUEFOUUFFOXIYEXACYNGQZUUDFXAJEGXDCDBKNYJXIYTUUAY + RVQXAWLXQXJYOUUIFOXIYEYTXAXHYMYOYQVRJEGXPFCYNYAKNWAVSVTXABXKXRWBRZQZUUEFX + AUUKXKBGQUUEJEUUJGXRBXKKNYBUUJSZWCXAJEGXDACBKNYJXIYLYTYRWDWEXAXRUAOZXTXHX + LUUKFOWLUUMWMWQEXRYBWFVCWMXTWLWQWMXQXTYDWGULYRXAXJXGXLYKVRJXRUUJXSFBXKYCJ + EXRYBKWHUULWAVSVTEXPFXDUUDUUEYAYJWIVSVTXAYFYGWTXBXCXFUDVBYHYIWRWSHFEXDJKY + JLVETWJWK $. $} ${ $d y z F $. $d a b c d x y z I $. $d c d .1. $. $d a b c d x y z R $. $d a b c d x y z S $. $d a b c d x y z U $. $d a b c d x y z X $. - qusrng.u $e |- U = ( R /s ( R ~QG S ) ) $. - qusrng.i $e |- I = ( 2Ideal ` R ) $. + qusring.u $e |- U = ( R /s ( R ~QG S ) ) $. + qusring.i $e |- I = ( 2Ideal ` R ) $. ${ qus1.o $e |- .1. = ( 1r ` R ) $. $( The multiplicative identity of the quotient ring. (Contributed by @@ -238126,18 +238346,18 @@ left ideal which is also a right ideal (or a left ideal over the opposite ( U e. Ring /\ [ .1. ] ( R ~QG S ) = ( 1r ` U ) ) ) $= ( vd vc va vb wcel cfv co wceq eqid syl cv wbr crg cplusg cqg cmulr cbs wa cqus a1i csubg wer clidl coppr 2idlval elin2 simplbi lidlsubg sylan2 - eqger cnsg wi cabl rngabl adantr ablnsg eleqtrrd eqgcpbl 2idlcpbl simpl - qusrng2 ) AUAMZBEMZUFZAUBNZABUCOZAAUDNZCDAUENZIJKLCAVNUGOPVLFUHVPVPPVLV - PQZUHVMQZVOQZHVLBAUINZMZVPVNUJVKVJBAUKNZMZWAVKWCBAULNZUKNZMBWBWEEAEWBWE - WDWBQZWDQWEQGUMUNUOAWBBWFUPUQZVNAVPBVQVNQZURRVLBAUSNZMKSZJSZVNTLSZISZVN - TUFWJWLVMOWKWMVMOVNTUTVLBVTWIWGVLAVAMZWIVTPVJWNVKAVBVCAVDRVEWJWLWKWMVMV - NAVPBVQWHVRVFRWJWLWKWMABVOVNEVPVQWHGVSVGVJVKVHVI $. + eqger cnsg cabl ringabl adantr eleqtrrd eqgcpbl 2idlcpbl simpl qusring2 + wi ablnsg ) AUAMZBEMZUFZAUBNZABUCOZAAUDNZCDAUENZIJKLCAVNUGOPVLFUHVPVPPV + LVPQZUHVMQZVOQZHVLBAUINZMZVPVNUJVKVJBAUKNZMZWAVKWCBAULNZUKNZMBWBWEEAEWB + WEWDWBQZWDQWEQGUMUNUOAWBBWFUPUQZVNAVPBVQVNQZURRVLBAUSNZMKSZJSZVNTLSZISZ + VNTUFWJWLVMOWKWMVMOVNTVHVLBVTWIWGVLAUTMZWIVTPVJWNVKAVAVBAVIRVCWJWLWKWMV + MVNAVPBVQWHVRVDRWJWLWKWMABVOVNEVPVQWHGVSVEVJVKVFVG $. $} $( If ` S ` is a two-sided ideal in ` R ` , then ` U = R / S ` is a ring, called the quotient ring of ` R ` by ` S ` . (Contributed by Mario Carneiro, 14-Jun-2015.) $) - qusrng $p |- ( ( R e. Ring /\ S e. I ) -> U e. Ring ) $= + qusring $p |- ( ( R e. Ring /\ S e. I ) -> U e. Ring ) $= ( crg wcel wa cur cfv cqg co cec wceq eqid qus1 simpld ) AGHBDHICGHAJKZAB LMNCJKOABCSDEFSPQR $. @@ -238148,19 +238368,19 @@ left ideal which is also a right ideal (or a left ideal over the opposite ` R / S ` . (Contributed by Mario Carneiro, 15-Jun-2015.) $) qusrhm $p |- ( ( R e. Ring /\ S e. I ) -> F e. ( R RingHom U ) ) $= ( wcel cfv eqid co cec a1i divsfval wceq cv vy vz vd vc va vb cmulr simpl - crg wa cur qusrng cqg csubg wer clidl coppr 2idlval elin2 lidlsubg sylan2 - simplbi eqger syl cvv cbs fvex eqeltri qus1 simprd eqtrd cqus rngcl 3expb - 2idlcpbl adantlr caovclg qusmulval adantr oveq12d 3eqtr4rd cnsg cghm cabl - rngabl ablnsg eleqtrrd qusghm isrhm2d ) BUILZCFLZUJZUAUBGBDBUGMZDUGMZBUKM - ZEDUKMZJWONZWPNWMNZWNNZWJWKUHZBCDFHIULWLWOEMWOBCUMOZPZWPWLAWOXAEGWLCBUNMZ - LZGXAUOZWKWJCBUPMZLZXDWKXGCBUQMZUPMZLCXFXIFBFXFXIXHXFNZXHNXINIURUSVBBXFCX - JUTVAZXABGCJXANZVCVDZGVELZWLGBVFMZVEJBVFVGVHZQKRWLDUILXBWPSBCDWOFHIWQVIVJ - VKWLUATZGLZUBTZGLZUJZUJZXQXAPZXSXAPZWNOZXQXSWMOZXAPZXQEMZXSEMZWNOYFEMWLXR - XTYEYGSWLXABWNWMDGXQXSUIUCUDUEUFDBXAVLOSWLHQGXOSWLJQXMWTUETUFTUDTZUCTZBCW - MXAFGJXLIWRVOWLUAUBYJYKGGGWMWJYAYFGLZWKWJXRXTYLGBWMXQXSJWRVMVNVPVQWRWSVRV - NYBYHYCYIYDWNYBAXQXAEGWLXEYAXMVSZXNYBXPQZKRYBAXSXAEGYMYNKRVTYBAYFXAEGYMYN - KRWAWLCBWBMZLEBDWCOLWLCXCYOXKWLBWDLZYOXCSWJYPWKBWEVSBWFVDWGAEBDGCJHKWHVDW - I $. + crg wa cur qusring cqg csubg wer clidl coppr 2idlval elin2 simplbi sylan2 + lidlsubg eqger syl cvv cbs fvex eqeltri qus1 simprd eqtrd 2idlcpbl ringcl + cqus 3expb adantlr caovclg qusmulval adantr oveq12d 3eqtr4rd cnsg ringabl + cghm cabl ablnsg eleqtrrd qusghm isrhm2d ) BUILZCFLZUJZUAUBGBDBUGMZDUGMZB + UKMZEDUKMZJWONZWPNWMNZWNNZWJWKUHZBCDFHIULWLWOEMWOBCUMOZPZWPWLAWOXAEGWLCBU + NMZLZGXAUOZWKWJCBUPMZLZXDWKXGCBUQMZUPMZLCXFXIFBFXFXIXHXFNZXHNXINIURUSUTBX + FCXJVBVAZXABGCJXANZVCVDZGVELZWLGBVFMZVEJBVFVGVHZQKRWLDUILXBWPSBCDWOFHIWQV + IVJVKWLUATZGLZUBTZGLZUJZUJZXQXAPZXSXAPZWNOZXQXSWMOZXAPZXQEMZXSEMZWNOYFEMW + LXRXTYEYGSWLXABWNWMDGXQXSUIUCUDUEUFDBXAVNOSWLHQGXOSWLJQXMWTUETUFTUDTZUCTZ + BCWMXAFGJXLIWRVLWLUAUBYJYKGGGWMWJYAYFGLZWKWJXRXTYLGBWMXQXSJWRVMVOVPVQWRWS + VRVOYBYHYCYIYDWNYBAXQXAEGWLXEYAXMVSZXNYBXPQZKRYBAXSXAEGYMYNKRVTYBAYFXAEGY + MYNKRWAWLCBWBMZLEBDWDOLWLCXCYOXKWLBWELZYOXCSWJYPWKBWCVSBWFVDWGAEBDGCJHKWH + VDWI $. $} ${ @@ -238196,23 +238416,23 @@ left ideal which is also a right ideal (or a left ideal over the opposite (Contributed by Mario Carneiro, 15-Jun-2015.) $) quscrng $p |- ( ( R e. CRing /\ S e. I ) -> U e. CRing ) $= ( vx vy vd vc va wcel wa crg cv cfv co wceq eqid cec vu vv ccrg cmulr cbs - vb c2idl crngrng adantr simpr crng2idl eleqtrd qusrng syl2anc cqg cqs cvv - wral cqus eqidd ovex qusbas eleq2d anbi12d oveq2 eqeq12d crngcom 3adant1r - a1i oveq1 3expa eceq1d csubg wer lidlsubg sylan eqger syl wbr wi 2idlcpbl - rngcl 3expb qusmulval 3eqtr4rd ectocld sylbird ralrimivv iscrng2 sylanbrc - an32s expl ) AUCLZBDLZMZCNLZGOZHOZCUDPZQZWRWQWSQZRZHCUEPZURGXCURCUCLWOANL - ZBAUGPZLZWPWMXDWNAUHZUIZWOBDXEWMWNUJWMDXERWNADFUKUIULZABCXEEXESZUMUNWOXBG - HXCXCWOWQXCLZWRXCLZMWQAUEPZABUOQZUPZLZWRXOLZMXBWOXPXKXQXLWOXOXCWQWOXNACXM - UQNCAXNUSQRWOEVIZWOXMUTZXNUQLWOABUOVAVIXHVBZVCWOXOXCWRXTVCVDWOXPXQXBWQUAO - ZXNTZWSQZYBWQWSQZRZXBWOXPMUAWRXMXNXOXOSZYBWRRYCWTYDXAYBWRWQWSVEYBWRWQWSVJ - VFWOYAXMLZXPYEUBOZXNTZYBWSQZYBYIWSQZRYEWOYGMZUBWQXMXNXOYFYIWQRYJYCYKYDYIW - QYBWSVJYIWQYBWSVEVFYLYHXMLZMZYAYHAUDPZQZXNTZYHYAYOQZXNTZYKYJYNYPYRXNWOYGY - MYPYRRZWMYGYMYTWNXMAYOYAYHXMSZYOSZVGVHVKVLWOYGYMYKYQRWOXNAWSYOCXMYAYHNIJK - UFXRXSWOBAVMPLZXMXNVNWMXDWNUUCXGADBFVOVPXNAXMBUUAXNSZVQVRZXHWOXDXFKOZJOZX - NVSUFOZIOZXNVSMUUFUUHYOQUUGUUIYOQZXNVSVTXHXIUUFUUHUUGUUIABYOXNXEXMUUAUUDX - JUUBWAUNZWOXDUUGXMLZUUIXMLZMUUJXMLZXHXDUULUUMUUNXMAYOUUGUUIUUAUUBWBWCVPZU - UBWSSZWDVKWOYMYGYJYSRZWOYMYGUUQWOXNAWSYOCXMYHYANIJKUFXRXSUUEXHUUKUUOUUBUU - PWDVKWKWEWFWKWFWLWGWHGHXCCWSXCSUUPWIWJ $. + wral c2idl crngring adantr simpr crng2idl eleqtrd qusring syl2anc cqg cqs + cvv cqus a1i eqidd ovex qusbas eleq2d anbi12d oveq2 oveq1 eqeq12d crngcom + vb 3adant1r 3expa eceq1d csubg wer lidlsubg sylan eqger syl wbr wi ringcl + 2idlcpbl 3expb qusmulval an32s 3eqtr4rd ectocld sylbird ralrimivv iscrng2 + expl sylanbrc ) AUCLZBDLZMZCNLZGOZHOZCUDPZQZWRWQWSQZRZHCUEPZUFGXCUFCUCLWO + ANLZBAUGPZLZWPWMXDWNAUHZUIZWOBDXEWMWNUJWMDXERWNADFUKUIULZABCXEEXESZUMUNWO + XBGHXCXCWOWQXCLZWRXCLZMWQAUEPZABUOQZUPZLZWRXOLZMXBWOXPXKXQXLWOXOXCWQWOXNA + CXMUQNCAXNURQRWOEUSZWOXMUTZXNUQLWOABUOVAUSXHVBZVCWOXOXCWRXTVCVDWOXPXQXBWQ + UAOZXNTZWSQZYBWQWSQZRZXBWOXPMUAWRXMXNXOXOSZYBWRRYCWTYDXAYBWRWQWSVEYBWRWQW + SVFVGWOYAXMLZXPYEUBOZXNTZYBWSQZYBYIWSQZRYEWOYGMZUBWQXMXNXOYFYIWQRYJYCYKYD + YIWQYBWSVFYIWQYBWSVEVGYLYHXMLZMZYAYHAUDPZQZXNTZYHYAYOQZXNTZYKYJYNYPYRXNWO + YGYMYPYRRZWMYGYMYTWNXMAYOYAYHXMSZYOSZVHVJVKVLWOYGYMYKYQRWOXNAWSYOCXMYAYHN + IJKVIXRXSWOBAVMPLZXMXNVNWMXDWNUUCXGADBFVOVPXNAXMBUUAXNSZVQVRZXHWOXDXFKOZJ + OZXNVSVIOZIOZXNVSMUUFUUHYOQUUGUUIYOQZXNVSVTXHXIUUFUUHUUGUUIABYOXNXEXMUUAU + UDXJUUBWBUNZWOXDUUGXMLZUUIXMLZMUUJXMLZXHXDUULUUMUUNXMAYOUUGUUIUUAUUBWAWCV + PZUUBWSSZWDVKWOYMYGYJYSRZWOYMYGUUQWOXNAWSYOCXMYHYANIJKVIXRXSUUEXHUUKUUOUU + BUUPWDVKWEWFWGWEWGWKWHWIGHXCCWSXCSUUPWJWL $. $} $( @@ -238275,7 +238495,7 @@ left ideal which is also a right ideal (or a left ideal over the opposite $( The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.) $) lpi0 $p |- ( R e. Ring -> { .0. } e. P ) $= - ( vg crg wcel csn crsp cfv wceq cbs wrex eqid rng0cl rsp0 eqcomd sneq + ( vg crg wcel csn crsp cfv wceq cbs wrex eqid ring0cl rsp0 eqcomd sneq cv fveq2d eqeq2d rspcev syl2anc islpidl mpbird ) BGHZCIZAHUHFTZIZBJKZKZ LZFBMKZNZUGCUNHUHUHUKKZLZUOUNBCUNOZEPUGUPUHBUKCUKOZEQRUMUQFCUNUICLZULUP UHUTUJUHUKUICSUAUBUCUDUNABFUHUKDUSURUEUF $. @@ -238286,7 +238506,7 @@ left ideal which is also a right ideal (or a left ideal over the opposite $( The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.) $) lpi1 $p |- ( R e. Ring -> B e. P ) $= - ( vg crg wcel csn crsp cfv wceq wrex cur eqid rngidcl rsp1 eqcomd sneq + ( vg crg wcel csn crsp cfv wceq wrex cur eqid ringidcl rsp1 eqcomd sneq cv fveq2d eqeq2d rspcev syl2anc islpidl mpbird ) CGHZABHAFTZIZCJKZKZLZF AMZUGCNKZAHAUNIZUJKZLZUMACUNEUNOZPUGUPAACUNUJUJOZEURQRULUQFUNAUHUNLZUKU PAUTUIUOUJUHUNSUAUBUCUDABCFAUJDUSEUEUF $. @@ -238321,7 +238541,7 @@ left ideal which is also a right ideal (or a left ideal over the opposite $( Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.) $) - lpirrng $p |- ( R e. LPIR -> R e. Ring ) $= + lpirring $p |- ( R e. LPIR -> R e. Ring ) $= ( clpir wcel crg clidl cfv clpidl wceq eqid islpir simplbi ) ABCADCAEFZAGFZ HMALMILIJK $. @@ -238427,7 +238647,7 @@ left ideal which is also a right ideal (or a left ideal over the opposite $} $( A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) $) - nzrrng $p |- ( R e. NzRing -> R e. Ring ) $= + nzrring $p |- ( R e. NzRing -> R e. Ring ) $= ( cnzr wcel crg cur cfv c0g wne eqid isnzr simplbi ) ABCADCAEFZAGFZHALMLIMI JK $. @@ -238444,15 +238664,15 @@ left ideal which is also a right ideal (or a left ideal over the opposite elements. (Contributed by Stefan O'Rear, 24-Feb-2015.) $) isnzr2 $p |- ( R e. NzRing <-> ( R e. Ring /\ 2o ~<_ B ) ) $= ( vx vy wcel cfv wne wa c2o cdom wbr eqid c1o cv wceq wrex adantr cvv wb - cnzr crg cur isnzr csdm wn rngidcl rng0cl simpr df-ne neeq1 syl5bbr neeq2 - c0g rspc2ev syl3anc ex wi rng1eq0 necon3bd rexlimdvva impbid fvex eqeltri - 3expb cbs 1sdom ax-mp syl6bbr csuc 1onn sucdom df-2o breq1i bitr4i syl6bb - com pm5.32i bitri ) BUAFBUBFZBUCGZBUNGZHZIZVTJAKLZIBWAWBWAMZWBMZUDVTWCWEV - TWCNAUELZWEVTWCDOZEOZPZUFZEAQDAQZWHVTWCWMVTWCWMWDWAAFZWBAFZWCWMVTWNWCABWA - CWFUGRVTWOWCABWBCWGUHRVTWCUIWLWCWAWJHZDEWAWBAAWLWIWJHWIWAPWPWIWJUJWIWAWJU - KULWJWBWAUMUOUPUQVTWLWCDEAAVTWIAFZWJAFZIIWKWAWBVTWQWRWAWBPWKURABWAWIWJWBC - WFWGUSVEUTVAVBASFWHWMTABVFGSCBVFVCVDDEASVGVHVIWHNVJZAKLZWENVQFWHWTTVKNAVL - VHJWSAKVMVNVOVPVRVS $. + cnzr crg cur c0g isnzr csdm wn ringidcl ring0cl simpr df-ne neeq1 syl5bbr + neeq2 rspc2ev syl3anc ex wi ring1eq0 3expb necon3bd rexlimdvva impbid cbs + fvex eqeltri 1sdom ax-mp syl6bbr csuc com 1onn sucdom df-2o breq1i bitr4i + syl6bb pm5.32i bitri ) BUAFBUBFZBUCGZBUDGZHZIZVTJAKLZIBWAWBWAMZWBMZUEVTWC + WEVTWCNAUFLZWEVTWCDOZEOZPZUGZEAQDAQZWHVTWCWMVTWCWMWDWAAFZWBAFZWCWMVTWNWCA + BWACWFUHRVTWOWCABWBCWGUIRVTWCUJWLWCWAWJHZDEWAWBAAWLWIWJHWIWAPWPWIWJUKWIWA + WJULUMWJWBWAUNUOUPUQVTWLWCDEAAVTWIAFZWJAFZIIWKWAWBVTWQWRWAWBPWKURABWAWIWJ + WBCWFWGUSUTVAVBVCASFWHWMTABVDGSCBVDVEVFDEASVGVHVIWHNVJZAKLZWENVKFWHWTTVLN + AVMVHJWSAKVNVOVPVQVRVS $. $} @@ -238462,13 +238682,14 @@ left ideal which is also a right ideal (or a left ideal over the opposite elements. Analogous to ~ isnzr2 . (Contributed by AV, 14-Apr-2019.) $) isnzr2hash $p |- ( R e. NzRing <-> ( R e. Ring /\ 1 < ( # ` B ) ) ) $= ( wcel cur cfv c0g wa c1 chash clt wbr eqid cxr cvv hashxrcl mp1i fvex c2 - cbs cnzr crg wne isnzr rngidcl rng0cl cpr 1re rexri a1i prex eqeltri 1lt2 - wceq hashprg biimpa syl5breqr wss cle simpl prss hashss sylancr xrltletrd - wi sylib ex syl2anc imdistani rng1ne0 jca impbii bitri ) BUADBUBDZBEFZBGF - ZUCZHZVNIAJFZKLZHZBVOVPVOMZVPMZUDVRWAVNVQVTVNVOADZVPADZVQVTVEABVOCWBUEABV - PCWCUFWDWEHZVQVTWFVQHZIVOVPUGZJFZVSINDWGIUHUIUJWHODWINDWGVOVPUKWHOPQAODZV - SNDWGABTFOCBTRULZAOPQWGISWIKUMWFVQWISUNVOVPAUOUPUQWGWJWHAURZWIVSUSLWKWGWF - WLWFVQUTVOVPABERBGRVAVFAWHOVBVCVDVGVHVIWAVNVQVNVTUTABVOVPCWBWCVJVKVLVM $. + cbs cnzr crg wne isnzr wi ringidcl ring0cl cpr 1re rexri a1i prex eqeltri + 1lt2 wceq hashprg biimpa syl5breqr wss simpl prss sylib sylancr xrltletrd + cle hashss ex syl2anc imdistani ring1ne0 jca impbii bitri ) BUADBUBDZBEFZ + BGFZUCZHZVNIAJFZKLZHZBVOVPVOMZVPMZUDVRWAVNVQVTVNVOADZVPADZVQVTUEABVOCWBUF + ABVPCWCUGWDWEHZVQVTWFVQHZIVOVPUHZJFZVSINDWGIUIUJUKWHODWINDWGVOVPULWHOPQAO + DZVSNDWGABTFOCBTRUMZAOPQWGISWIKUNWFVQWISUOVOVPAUPUQURWGWJWHAUSZWIVSVELWKW + GWFWLWFVQUTVOVPABERBGRVAVBAWHOVFVCVDVGVHVIWAVNVQVNVTUTABVOVPCWBWCVJVKVLVM + $. $} ${ @@ -238477,21 +238698,21 @@ left ideal which is also a right ideal (or a left ideal over the opposite $( The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.) $) opprnzr $p |- ( R e. NzRing -> O e. NzRing ) $= - ( cnzr wcel crg c2o cbs cfv wbr nzrrng opprrng syl isnzr2 simprbi opprbas - cdom eqid sylanbrc ) ADEZBFEZGAHIZQJZBDETAFEZUAAKABCLMTUDUCUBAUBRZNOUBBUB - ABCUEPNS $. + ( cnzr wcel crg c2o cbs cfv cdom wbr nzrring opprring eqid isnzr2 simprbi + syl opprbas sylanbrc ) ADEZBFEZGAHIZJKZBDETAFEZUAALABCMQTUDUCUBAUBNZOPUBB + UBABCUEROS $. $} ${ - rngelnzr.z $e |- .0. = ( 0g ` R ) $. - rngelnzr.b $e |- B = ( Base ` R ) $. + ringelnzr.z $e |- .0. = ( 0g ` R ) $. + ringelnzr.b $e |- B = ( Base ` R ) $. $( A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.) $) - rngelnzr $p |- ( ( R e. Ring /\ X e. ( B \ { .0. } ) ) -> R e. NzRing ) $= + ringelnzr $p |- ( ( R e. Ring /\ X e. ( B \ { .0. } ) ) -> R e. NzRing ) $= ( crg wcel csn cdif wa cur cfv wne cnzr simpl eldifsni adantl wceq wi mpd - eldifi rng0cl adantr eqid rng1eq0 syl3anc necon3d isnzr sylanbrc ) BGHZCA - DIZJHZKZUKBLMZDNZBOHUKUMPZUNCDNZUPUMURUKCADQRUNUODCDUNUKCAHZDAHZUODSCDSTU - QUMUSUKCAULUBRUKUTUMABDFEUCUDABUOCDDFUOUEZEUFUGUHUABUODVAEUIUJ $. + eldifi ring0cl adantr eqid ring1eq0 syl3anc necon3d isnzr sylanbrc ) BGHZ + CADIZJHZKZUKBLMZDNZBOHUKUMPZUNCDNZUPUMURUKCADQRUNUODCDUNUKCAHZDAHZUODSCDS + TUQUMUSUKCAULUBRUKUTUMABDFEUCUDABUOCDDFUOUEZEUFUGUHUABUODVAEUIUJ $. $} ${ @@ -238500,10 +238721,10 @@ left ideal which is also a right ideal (or a left ideal over the opposite $( A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.) $) nzrunit $p |- ( ( R e. NzRing /\ A e. U ) -> A =/= .0. ) $= - ( cnzr wcel wne wn wceq cur cfv eqid nzrnz crg wb nzrrng 0unit necon3bbid - syl mpbird eleq1 notbid syl5ibrcom necon2ad imp ) BGHZACHZADIUHUIADUHUIJA - DKZDCHZJZUHULBLMZDIZBUMDUMNZFOUHBPHZULUNQBRUPUKUMDBCUMDEFUOSTUAUBUJUIUKAD - CUCUDUEUFUG $. + ( cnzr wcel wne wn wceq cur cfv eqid nzrnz crg nzrring 0unit necon3bbid + wb syl mpbird eleq1 notbid syl5ibrcom necon2ad imp ) BGHZACHZADIUHUIADUHU + IJADKZDCHZJZUHULBLMZDIZBUMDUMNZFOUHBPHZULUNTBQUPUKUMDBCUMDEFUORSUAUBUJUIU + KADCUCUDUEUFUG $. $} ${ @@ -238512,71 +238733,71 @@ left ideal which is also a right ideal (or a left ideal over the opposite 15-Jun-2015.) $) subrgnzr $p |- ( ( R e. NzRing /\ A e. ( SubRing ` R ) ) -> S e. NzRing ) $= - ( cnzr wcel csubrg cfv wa crg cur c0g wne subrgrng adantl eqid nzrnz wceq - adantr subrg1 subrg0 3netr3d isnzr sylanbrc ) BEFZABGHFZIZCJFZCKHZCLHZMCE - FUFUHUEABCDNOUGBKHZBLHZUIUJUEUKULMUFBUKULUKPZULPZQSUFUKUIRUEABCUKDUMTOUFU - LUJRUEABCULDUNUAOUBCUIUJUIPUJPUCUD $. + ( cnzr wcel csubrg cfv crg cur c0g wne subrgring adantl eqid nzrnz adantr + wa wceq subrg1 subrg0 3netr3d isnzr sylanbrc ) BEFZABGHFZRZCIFZCJHZCKHZLC + EFUFUHUEABCDMNUGBJHZBKHZUIUJUEUKULLUFBUKULUKOZULOZPQUFUKUISUEABCUKDUMTNUF + ULUJSUEABCULDUNUANUBCUIUJUIOUJOUCUD $. $} $( A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.) $) - 0rngnnzr $p |- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 + 0ringnnzr $p |- ( R e. Ring -> ( ( # ` ( Base ` R ) ) = 1 <-> -. R e. NzRing ) ) $= ( wcel cbs cfv c1 wceq clt wbr wa wn 1re ex wi wb cvv c0 syl a1i cc0 com12 crg chash cnzr ltnri breq2 mtbiri adantl intnand wo pm2.21 cle cxr hashxrcl - ianor fvex ax-mp rexri xrlenlt mp2an bicomi wne cgrp rnggrp grpbn0 simpr cn - eqid cfn 1nn0 hashbnd syl3anc hashcl hasheq0 biimpd necon3d impcom sylanbrc - cn0 elnnne0 adantr mpcom nnle1eq1 mpbid sylbi jaoi impbid isnzr2hash notbii - syl6bb ) AUABZACDZUBDZEFZWJEWLGHZIZJZAUCBZJWJWMWPWJWMWPWJWMIWNWJWMWNJZWJWMW - NEEGHEKUDWLEEGUEUFUGUHLWPWJWMWPWJJZWRUIWJWMMZWJWNUNWSWTWRWJWMUJWRWLEUKHZWTX - AWRWLULBZEULBXAWRNWKOBZXBACUOZWKOUMUPEKUQWLEURUSUTWJXAWMWJWKPVAZXAWMMWJAVBB - XEAVCWKAWKVGZVDQXEXAWMXEXAIZXAWMXEXAVEZXGWLVFBZXAWMNWKVHBZXGXIXGXCEVRBZXAXJ - XCXGXDRXKXGVIRXHWKEOVJVKXJWLVRBZXGXIMWKVLXGXLXIXEXLXIMXAXEXLXIXEXLIXLWLSVAZ - XIXEXLVEXLXEXMXLWLSWKPXLWLSFZWKPFZXLXCXNXONXCXLXDRWKOVMQVNVOVPWLVSVQLVTTQWA - WLWBQWCLQTWDWEWDTWFWOWQWQWOWKAXFWGUTWHWI $. - - ${ - 0rng.b $e |- B = ( Base ` R ) $. - 0rng.0 $e |- .0. = ( 0g ` R ) $. + ianor fvex ax-mp rexri xrlenlt mp2an bicomi wne cgrp ringgrp eqid grpbn0 cn + simpr cfn 1nn0 hashbnd syl3anc hashcl hasheq0 biimpd necon3d impcom elnnne0 + sylanbrc adantr mpcom nnle1eq1 mpbid sylbi impbid isnzr2hash notbii syl6bb + cn0 jaoi ) AUABZACDZUBDZEFZWJEWLGHZIZJZAUCBZJWJWMWPWJWMWPWJWMIWNWJWMWNJZWJW + MWNEEGHEKUDWLEEGUEUFUGUHLWPWJWMWPWJJZWRUIWJWMMZWJWNUNWSWTWRWJWMUJWRWLEUKHZW + TXAWRWLULBZEULBXAWRNWKOBZXBACUOZWKOUMUPEKUQWLEURUSUTWJXAWMWJWKPVAZXAWMMWJAV + BBXEAVCWKAWKVDZVEQXEXAWMXEXAIZXAWMXEXAVGZXGWLVFBZXAWMNWKVHBZXGXIXGXCEWHBZXA + XJXCXGXDRXKXGVIRXHWKEOVJVKXJWLWHBZXGXIMWKVLXGXLXIXEXLXIMXAXEXLXIXEXLIXLWLSV + AZXIXEXLVGXLXEXMXLWLSWKPXLWLSFZWKPFZXLXCXNXONXCXLXDRWKOVMQVNVOVPWLVQVRLVSTQ + VTWLWAQWBLQTWCWIWCTWDWOWQWQWOWKAXFWEUTWFWG $. + + ${ + 0ring.b $e |- B = ( Base ` R ) $. + 0ring.0 $e |- .0. = ( 0g ` R ) $. $( If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, ~ https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019.) $) - 0rng $p |- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> B = { .0. } ) $= - ( crg wcel chash cfv c1 wceq csn wi rng0cl c1o cen wbr cvv wb cbs eqeltri - fvex hashen1 ax-mp en1eqsn ex syl5bi syl imp ) BFGZAHIJKZACLKZUJCAGZUKULM - ABCDENUKAOPQZUMULARGUKUNSABTIRDBTUBUAARUCUDUMUNULCAUEUFUGUHUI $. + 0ring $p |- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> B = { .0. } ) $= + ( crg wcel chash cfv c1 wceq csn wi ring0cl c1o cen wbr cvv wb cbs syl5bi + fvex eqeltri hashen1 ax-mp en1eqsn ex syl imp ) BFGZAHIJKZACLKZUJCAGZUKUL + MABCDENUKAOPQZUMULARGUKUNSABTIRDBTUBUCARUDUEUMUNULCAUFUGUAUHUI $. - 0rng01eq.1 $e |- .1. = ( 1r ` R ) $. + 0ring01eq.1 $e |- .1. = ( 1r ` R ) $. $( In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019.) $) - 0rng01eq $p |- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> .0. = .1. ) $= - ( crg wcel chash cfv c1 wceq wa csn 0rng wi rngidcl eleq2 elsni syl5com + 0ring01eq $p |- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> .0. = .1. ) $= + ( crg wcel chash cfv c1 wceq wa csn 0ring wi ringidcl eleq2 elsni syl5com eqcomd syl6bi adantr mpd ) BHIZAJKLMZNADOZMZDCMZABDEFPUFUIUJQUGUFCAIZUIUJ ABCEGRUIUKCUHIZUJAUHCSULCDCDTUBUCUAUDUE $. $( If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) $) - 01eq0rng $p |- ( ( R e. Ring /\ .0. = .1. ) -> B = { .0. } ) $= + 01eq0ring $p |- ( ( R e. Ring /\ .0. = .1. ) -> B = { .0. } ) $= ( crg wcel wceq chash cfv c1 wne wi cvv cbs ax-mp c0 eqneqall csn cc0 clt - wbr w3o fvex eqeltri hashv01gt1 rng0cl wb hasheq0 ne0i syl5com syl5bi syl - com12 a1d wa rng1ne0 necomd ex a1i com13 3jaoi necon4d imp 0rng syldan ) - BHIZDCJZAKLZMJZADUAJVIVJVLVIVKMDCVKUBJZVLMVKUCUDZUEZVIVKMNZDCNZOZOZAPIZVO - ABQLPEBQUFUGZAPUHRVMVSVLVNVIVMVRVIDAIZVMVROABDEFUIVMASJZWBVRVTVMWCUJWAAPU - KRWBASNWCVRADULVRASTUMUNUOUPVLVRVIVQVKMTUQVPVIVNVQVIVNVQOOVPVIVNVQVIVNURC - DABCDEGFUSUTVAVBVCVDRVEVFABDEFVGVH $. + wbr w3o fvex eqeltri hashv01gt1 ring0cl hasheq0 ne0i syl5com syl5bi com12 + wb syl a1d wa ring1ne0 necomd ex a1i com13 3jaoi necon4d imp 0ring syldan + ) BHIZDCJZAKLZMJZADUAJVIVJVLVIVKMDCVKUBJZVLMVKUCUDZUEZVIVKMNZDCNZOZOZAPIZ + VOABQLPEBQUFUGZAPUHRVMVSVLVNVIVMVRVIDAIZVMVROABDEFUIVMASJZWBVRVTVMWCUOWAA + PUJRWBASNWCVRADUKVRASTULUMUPUNVLVRVIVQVKMTUQVPVIVNVQVIVNVQOOVPVIVNVQVIVNU + RCDABCDEGFUSUTVAVBVCVDRVEVFABDEFVGVH $. $( In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by AV, 23-Jan-2020.) $) - 0rng01eqbi $p |- ( R e. Ring -> ( B ~~ 1o <-> .1. = .0. ) ) $= - ( wcel chash cfv c1 wceq cvv cbs fvex eqeltri mp1i wa ex c0g crg 0rng01eq - c1o cen wbr wb hashen1 eqcomd eqcom csn 01eq0rng fveq2 hashsng syl syl5bi - eqtrd impbid bitr3d ) BUAHZAIJZKLZAUCUDUEZCDLZAMHVAVBUFUSABNJMEBNOPAMUGQU - SVAVCUSVAVCUSVARDCABCDEFGUBUHSVCDCLZUSVACDUIUSVDVAUSVDRADUJZLZVAABCDEFGUK - VFUTVEIJZKAVEIULDMHVGKLVFDBTJMFBTOPDMUMQUPUNSUOUQUR $. + 0ring01eqbi $p |- ( R e. Ring -> ( B ~~ 1o <-> .1. = .0. ) ) $= + ( wcel chash cfv c1 wceq cvv cbs fvex eqeltri mp1i wa ex c0g wb 0ring01eq + crg c1o cen wbr hashen1 eqcomd eqcom 01eq0ring fveq2 hashsng eqtrd syl5bi + csn syl impbid bitr3d ) BUCHZAIJZKLZAUDUEUFZCDLZAMHVAVBUAUSABNJMEBNOPAMUG + QUSVAVCUSVAVCUSVARDCABCDEFGUBUHSVCDCLZUSVACDUIUSVDVAUSVDRADUOZLZVAABCDEFG + UJVFUTVEIJZKAVEIUKDMHVGKLVFDBTJMFBTOPDMULQUMUPSUNUQUR $. $} ${ @@ -238588,26 +238809,26 @@ left ideal which is also a right ideal (or a left ideal over the opposite ring. (Contributed by AV, 29-Apr-2019.) $) rng1nnzr $p |- ( Z e. V -> M e/ NzRing ) $= ( wcel cnzr wn wnel cbs cfv chash c1 wceq csn cvv snex cop rngbase eqcomd - mp1i fveq2d hashsng eqtrd crg wb rng1 0rngnnzr syl mpbid df-nel sylibr ) - CBEZAFEGZAFHULAIJZKJZLMZUMULUOCNZKJLULUNUQKULUQUNUQOEUQUNMULCPUQCCQCQNZAU - RODRTSUACBUBUCULAUDEUPUMUEABCDUFAUGUHUIAFUJUK $. + mp1i fveq2d hashsng eqtrd crg wb ring1 0ringnnzr syl mpbid df-nel sylibr + ) CBEZAFEGZAFHULAIJZKJZLMZUMULUOCNZKJLULUNUQKULUQUNUQOEUQUNMULCPUQCCQCQNZ + AURODRTSUACBUBUCULAUDEUPUMUEABCDUFAUGUHUIAFUJUK $. $} ${ - rng1zr.b $e |- B = ( Base ` R ) $. - rng1zr.p $e |- .+ = ( +g ` R ) $. - rng1zr.t $e |- .* = ( .r ` R ) $. + ring1zr.b $e |- B = ( Base ` R ) $. + ring1zr.p $e |- .+ = ( +g ` R ) $. + ring1zr.t $e |- .* = ( .r ` R ) $. $( The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption ` R e. Ring ` could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.) $) - rng1zr $p |- ( ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) + ring1zr $p |- ( ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) ) $= - ( crg wcel csrg cxp wfn csn wceq cop wa wb rngsrg srg1zr syl3anl1 ) CIJCK - JBAALZMDUBMEAJAENOBEEPEPNZODUCOQRCSABCDEFGHTUA $. + ( crg wcel csrg cxp wfn csn wceq cop wa wb ringsrg srg1zr syl3anl1 ) CIJC + KJBAALZMDUBMEAJAENOBEEPEPNZODUCOQRCSABCDEFGHTUA $. $( The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption @@ -238619,8 +238840,8 @@ zero ring (at least if its operations are internal binary operations). /\ Z e. B ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) ) $= ( crg wcel cxp wfn w3a wa c1o cen wbr csn wceq cop en1eqsnbi adantl bitrd - wb rng1zr ) CIJBAAKZLDUFLMZEAJZNAOPQZAERSZBEETETRZSDUKSNUHUIUJUDUGEAUAUBA - BCDEFGHUEUC $. + wb ring1zr ) CIJBAAKZLDUFLMZEAJZNAOPQZAERSZBEETETRZSDUKSNUHUIUJUDUGEAUAUB + ABCDEFGHUEUC $. ringen1zr.0 $e |- Z = ( 0g ` R ) $. $( The only unital ring with one element is the zero ring (at least if its @@ -238632,7 +238853,7 @@ zero ring (at least if its operations are internal binary operations). ringen1zr $p |- ( ( R e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) ) $= - ( crg wcel cxp wfn w3a c1o cen wbr cop csn wceq rng0cl 3ad2ant1 rngen1zr + ( crg wcel cxp wfn w3a c1o cen wbr cop csn wceq ring0cl 3ad2ant1 rngen1zr wa wb mpdan ) CJKZBAALZMZDUHMZNEAKZAOPQBEERERSZTDULTUDUEUGUIUKUJACEFIUAUB ABCDEFGHUCUF $. $} @@ -238739,7 +238960,7 @@ zero ring (at least if its operations are internal binary operations). rrgeq0 $p |- ( ( R e. Ring /\ X e. E /\ Y e. B ) -> ( ( X .x. Y ) = .0. <-> Y = .0. ) ) $= ( vx vy crg wcel w3a co wceq wi cv rrgeq0i 3adant1 simp1 wral crab rrgval - ssrab2 eqsstri simp2 sseldi rngrz syl2anc oveq2 eqeq1d syl5ibrcom impbid + ssrab2 eqsstri simp2 sseldi ringrz syl2anc oveq2 eqeq1d syl5ibrcom impbid ) BNOZEDOZFAOZPZEFCQZGRZFGRZURUSVBVCSUQABCDEFGHIJKUAUBUTVBVCEGCQZGRZUTUQE AOVEUQURUSUCUTDAEDLTMTZCQGRVFGRSMAUDZLAUEALMABCDGHIJKUFVGLAUGUHUQURUSUIUJ ABCEGIJKUKULVCVAVDGFGECUMUNUOUP $. @@ -238807,15 +239028,15 @@ zero ring (at least if its operations are internal binary operations). 22-Mar-2015.) $) unitrrg $p |- ( R e. Ring -> U C_ E ) $= ( vx vy crg wcel cv wa cbs cfv cmulr co c0g wceq wi eqid adantr cinvr cur - unitcl adantl oveq2 unitlinv oveq1d simpll rnginvcl simpr rngass syl13anc - wral rnglidm adantlr 3eqtr3d rngrz syl2anc eqeq12d syl5ib ralrimiva isrrg - sylanbrc ex ssrdv ) AHIZFBCVFFJZBIZVGCIZVFVHKZVGALMZIZVGGJZANMZOZAPMZQZVM - VPQZRZGVKUMVIVHVLVFVKABVGVKSZEUCUDZVJVSGVKVQVGAUAMZMZVOVNOZWCVPVNOZQVJVMV - KIZKZVRVOVPWCVNUEWGWDVMWEVPWGWCVGVNOZVMVNOZAUBMZVMVNOZWDVMWGWHWJVMVNVJWHW - JQWFAVNBWJWBVGEWBSZVNSZWJSZUFTUGWGVFWCVKIZVLWFWIWDQVFVHWFUHZVJWOWFVKABWBV - GEWLVTUITZVJVLWFWATVJWFUJVKAVNWCVGVMVTWMUKULVFWFWKVMQVHVKAVNWJVMVTWMWNUNU - OUPWGVFWOWEVPQWPWQVKAVNWCVPVTWMVPSZUQURUSUTVAGVKAVNCVGVPDVTWMWRVBVCVDVE - $. + wral unitcl adantl oveq2 unitlinv oveq1d ringinvcl simpr ringass syl13anc + simpll ringlidm adantlr 3eqtr3d ringrz syl2anc eqeq12d ralrimiva sylanbrc + syl5ib isrrg ex ssrdv ) AHIZFBCVFFJZBIZVGCIZVFVHKZVGALMZIZVGGJZANMZOZAPMZ + QZVMVPQZRZGVKUCVIVHVLVFVKABVGVKSZEUDUEZVJVSGVKVQVGAUAMZMZVOVNOZWCVPVNOZQV + JVMVKIZKZVRVOVPWCVNUFWGWDVMWEVPWGWCVGVNOZVMVNOZAUBMZVMVNOZWDVMWGWHWJVMVNV + JWHWJQWFAVNBWJWBVGEWBSZVNSZWJSZUGTUHWGVFWCVKIZVLWFWIWDQVFVHWFUMZVJWOWFVKA + BWBVGEWLVTUITZVJVLWFWATVJWFUJVKAVNWCVGVMVTWMUKULVFWFWKVMQVHVKAVNWJVMVTWMW + NUNUOUPWGVFWOWEVPQWPWQVKAVNWCVPVTWMVPSZUQURUSVBUTGVKAVNCVGVPDVTWMWRVCVAVD + VE $. $} ${ @@ -238848,8 +239069,8 @@ zero ring (at least if its operations are internal binary operations). SQ $. $( A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.) $) - domnrng $p |- ( R e. Domn -> R e. Ring ) $= - ( cdomn wcel cnzr crg domnnzr nzrrng syl ) ABCADCAECAFAGH $. + domnring $p |- ( R e. Domn -> R e. Ring ) $= + ( cdomn wcel cnzr crg domnnzr nzrring syl ) ABCADCAECAFAGH $. domneq0.b $e |- B = ( Base ` R ) $. domneq0.t $e |- .x. = ( .r ` R ) $. @@ -238860,13 +239081,13 @@ zero ring (at least if its operations are internal binary operations). ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) ) $= ( vx vy wcel co wceq wo cv wi wral eqeq1d syl2anc cdomn w3a 3simpc isdomn cnzr simprbi 3ad2ant1 oveq1 eqeq1 orbi1d imbi12d oveq2 orbi2d rspc2va crg - wa domnrng simp3 rnglz syl5ibrcom simp2 rngrz jaod impbid ) BUALZDALZEALZ - UBZDECMZFNZDFNZEFNZOZVHVFVGUPJPZKPZCMZFNZVNFNZVOFNZOZQZKARJARZVJVMQZVEVFV - GUCVEVFWBVGVEBUELWBJKABCFGHIUDUFUGWAWCDVOCMZFNZVKVSOZQJKDEAAVNDNZVQWEVTWF - WGVPWDFVNDVOCUHSWGVRVKVSVNDFUIUJUKVOENZWEVJWFVMWHWDVIFVOEDCULSWHVSVLVKVOE - FUIUMUKUNTVHVKVJVLVHVJVKFECMZFNZVHBUOLZVGWJVEVFWKVGBUQUGZVEVFVGURABCEFGHI - USTVKVIWIFDFECUHSUTVHVJVLDFCMZFNZVHWKVFWNWLVEVFVGVAABCDFGHIVBTVLVIWMFEFDC - ULSUTVCVD $. + wa domnring simp3 ringlz syl5ibrcom simp2 ringrz jaod impbid ) BUALZDALZE + ALZUBZDECMZFNZDFNZEFNZOZVHVFVGUPJPZKPZCMZFNZVNFNZVOFNZOZQZKARJARZVJVMQZVE + VFVGUCVEVFWBVGVEBUELWBJKABCFGHIUDUFUGWAWCDVOCMZFNZVKVSOZQJKDEAAVNDNZVQWEV + TWFWGVPWDFVNDVOCUHSWGVRVKVSVNDFUIUJUKVOENZWEVJWFVMWHWDVIFVOEDCULSWHVSVLVK + VOEFUIUMUKUNTVHVKVJVLVHVJVKFECMZFNZVHBUOLZVGWJVEVFWKVGBUQUGZVEVFVGURABCEF + GHIUSTVKVIWIFDFECUHSUTVHVJVLDFCMZFNZVHWKVFWNWLVEVFVGVAABCDFGHIVBTVLVIWMFE + FDCULSUTVCVD $. $( In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.) $) @@ -238927,14 +239148,14 @@ zero ring (at least if its operations are internal binary operations). Carneiro, 6-May-2015.) $) abvn0b $p |- ( R e. Domn <-> ( R e. NzRing /\ A =/= (/) ) ) $= ( vx vy vz cdomn wcel cnzr c0 wne wa domnnzr cbs cfv c0g wceq eqid wral - cv cc0 c1 cif cmpt cmulr domnrng domnmuln0 abvtrivd ne0i syl jca co wo wi - wex n0 wn neanior abvdom 3expib syl5bi expdimp syl5bir ralrimivva exlimiv - an4 necon4bd sylbi anim2i isdomn sylibr impbii ) BGHZBIHZAJKZLZVMVNVOBMVM - DBNOZDTZBPOZQUAUBUCUDZAHVOVMDEFAVQBBUEOZVTVSCVQRZVSRZVTRWARZBUFVQBWAETZFT - ZVSWBWDWCUGUHAVTUIUJUKVPVNWEWFWAULZVSQWEVSQWFVSQUMZUNZFVQSEVQSZLVMVOWJVNV - OVRAHZDUOWJDAUPWKWJDWKWIEFVQVQWKWEVQHZWFVQHZLZLZWHWGVSWHUQWEVSKZWFVSKZLZW - OWGVSKZWEVSWFVSURWKWNWRWSWNWRLWLWPLZWMWQLZLWKWSWLWMWPWQVFWKWTXAWSAVQBWAVR - WEWFVSCWBWCWDUSUTVAVBVCVGVDVEVHVIEFVQBWAVSWBWDWCVJVKVL $. + cv cc0 c1 cif cmpt cmulr domnring domnmuln0 abvtrivd syl jca co wo wi wex + ne0i neanior an4 abvdom 3expib syl5bi expdimp syl5bir necon4bd ralrimivva + n0 wn exlimiv sylbi anim2i isdomn sylibr impbii ) BGHZBIHZAJKZLZVMVNVOBMV + MDBNOZDTZBPOZQUAUBUCUDZAHVOVMDEFAVQBBUEOZVTVSCVQRZVSRZVTRWARZBUFVQBWAETZF + TZVSWBWDWCUGUHAVTUOUIUJVPVNWEWFWAUKZVSQWEVSQWFVSQULZUMZFVQSEVQSZLVMVOWJVN + VOVRAHZDUNWJDAVEWKWJDWKWIEFVQVQWKWEVQHZWFVQHZLZLZWHWGVSWHVFWEVSKZWFVSKZLZ + WOWGVSKZWEVSWFVSUPWKWNWRWSWNWRLWLWPLZWMWQLZLWKWSWLWMWPWQUQWKWTXAWSAVQBWAV + RWEWFVSCWBWCWDURUSUTVAVBVCVDVGVHVIEFVQBWAVSWBWDWCVJVKVL $. $} $( A division ring is a domain. (Contributed by Mario Carneiro, @@ -238975,35 +239196,35 @@ zero ring (at least if its operations are internal binary operations). ( wcel vy ccnv cfv co wbr csn eldifad wf1o wf wf1 cv wceq wral wne cdif wi wa eldifsni syl ad2antrr oveq1 ovex fvmpt adantl eqeq1d cdomn adantr wo wb simpr domneq0 syl3anc bitrd biimpa ord necon1ad ex ralrimiva cghm - mpd cmpt crg domnrng rngrghm syl2anc syl5eqel ghmf1 mpbird cen f1finf1o - cfn enrefg f1ocnv f1of 3syl rngidcl ffvelrnd dvdsrmul cbvmptv f1ocnvfv2 - mpbid eqtri eqtr3d breqtrd ) ACHIUBZUCZCGUDZHEACDTZXFDTZCXGEUEACDJUFZRU - GZADDHXEADDIUHZDDXEUHDDXEUIADDIUJZXLAXMUAUKZIUCZJULZXNJULZUPZUADUMZAXRU - ADAXNDTZUQZXPXQYAXPUQZCJUNZXQAYCXTXPACDXJUOTYCRCDJURUSUTYBXQCJYBXQCJULZ - YAXPXQYDVHZYAXPXNCGUDZJULZYEYAXOYFJXTXOYFULABXNBUKZCGUDZYFDIYHXNCGVAZSX - NCGVBVCVDVEYAFVFTZXTXHYGYEVIAYKXTPVGAXTVJAXHXTXKVGDFGXNCJKOLVKVLVMVNVOV - PVTVQVRAIFFVSUDZTXMXSVIAIBDYIWAZYLSAFWBTZXHYMYLTAYKYNPFWCUSZXKBDFGCKOWD - WEWFUAFFJIDDJKKLLWGUSWHADDWIUEZDWKTZXMXLVIAYQYPQDWKWLUSQDDIWJWEXAZDDIWM - DDXEWNWOAYNHDTZYODFHKMWPUSZWQZDEFGCXFKNOWRWEAXFIUCZXGHAXIUUBXGULUUAUAXF - YFXGDIXNXFCGVAIYMUADYFWASBUADYIYFYJWSXBXFCGVBVCUSAXLYSUUBHULYRYTDDHIWTW - EXCXD $. + mpd cmpt crg domnring ringrghm syl2anc syl5eqel ghmf1 mpbird cen enrefg + cfn f1finf1o mpbid f1ocnv f1of ringidcl ffvelrnd dvdsrmul cbvmptv eqtri + 3syl f1ocnvfv2 eqtr3d breqtrd ) ACHIUBZUCZCGUDZHEACDTZXFDTZCXGEUEACDJUF + ZRUGZADDHXEADDIUHZDDXEUHDDXEUIADDIUJZXLAXMUAUKZIUCZJULZXNJULZUPZUADUMZA + XRUADAXNDTZUQZXPXQYAXPUQZCJUNZXQAYCXTXPACDXJUOTYCRCDJURUSUTYBXQCJYBXQCJ + ULZYAXPXQYDVHZYAXPXNCGUDZJULZYEYAXOYFJXTXOYFULABXNBUKZCGUDZYFDIYHXNCGVA + ZSXNCGVBVCVDVEYAFVFTZXTXHYGYEVIAYKXTPVGAXTVJAXHXTXKVGDFGXNCJKOLVKVLVMVN + VOVPVTVQVRAIFFVSUDZTXMXSVIAIBDYIWAZYLSAFWBTZXHYMYLTAYKYNPFWCUSZXKBDFGCK + OWDWEWFUAFFJIDDJKKLLWGUSWHADDWIUEZDWKTZXMXLVIAYQYPQDWKWJUSQDDIWLWEWMZDD + IWNDDXEWOXAAYNHDTZYODFHKMWPUSZWQZDEFGCXFKNOWRWEAXFIUCZXGHAXIUUBXGULUUAU + AXFYFXGDIXNXFCGVAIYMUADYFWASBUADYIYFYJWSWTXFCGVBVCUSAXLYSUUBHULYRYTDDHI + XBWEXCXD $. $} $( A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.) $) fidomndrng $p |- ( B e. Fin -> ( R e. Domn <-> R e. DivRing ) ) $= ( vx vy wcel cdomn wa cfv wceq adantl wss eqid syl wb cv cdsr wbr cmulr - co cfn cdr crg cui c0g csn cdif domnrng cin c0 cur cnzr wne domnnzr nzrnz + co cfn cdr crg cui c0g csn cdif domnring cin c0 wn cur cnzr domnnzr nzrnz neneqd 0unit mtbird disjsn sylibr unitss reldisj ax-mp sylib coppr simplr - wn cmpt simpll simpr fidomndrnglem opprbas oppr0 opprdomn isunit sylanbrc - oppr1 ex ssrdv eqssd isdrng drngdomn impbid1 ) AUAFZBGFZBUBFZWDWEWFWDWEHZ - BUCFZBUDIZABUEIZUFZUGZJWFWEWHWDBUHKZWGWIWLWGWIWKUIUJJZWIWLLZWGWJWIFZVGWNW - GWPBUKIZWJJZWGWQWJWGBULFZWQWJUMWEWSWDBUNKBWQWJWQMZWJMZUONUPWGWHWPWROWMBWI - WQWJWIMZXAWTUQNURWIWJUSUTWIALWNWOOABWICXBVAWIWKAVBVCVDWGDWLWIWGDPZWLFZXCW - IFZWGXDHZXCWQBQIZRXCWQBVEIZQIZRXEXFEXCAXGBBSIZWQEAEPZXCXJTVHZWJCXAWTXGMZX - JMWDWEXDVFZWDWEXDVIZWGXDVJZXLMVKXFEXCAXIXHXHSIZWQEAXKXCXQTVHZWJABXHXHMZCV - LBXHWJXSXAVMBWQXHXSWTVQXIMZXQMXFWEXHGFXNBXHXSVNNXOXPXRMVKXGBXHWIWQXIXCXBW - TXMXSXTVOVPVRVSVTABWIWJCXBXAWAVPVRBWBWC $. + wne cmpt simpll simpr fidomndrnglem opprbas oppr0 oppr1 opprdomn sylanbrc + isunit ex ssrdv eqssd isdrng drngdomn impbid1 ) AUAFZBGFZBUBFZWDWEWFWDWEH + ZBUCFZBUDIZABUEIZUFZUGZJWFWEWHWDBUHKZWGWIWLWGWIWKUIUJJZWIWLLZWGWJWIFZUKWN + WGWPBULIZWJJZWGWQWJWGBUMFZWQWJVGWEWSWDBUNKBWQWJWQMZWJMZUONUPWGWHWPWROWMBW + IWQWJWIMZXAWTUQNURWIWJUSUTWIALWNWOOABWICXBVAWIWKAVBVCVDWGDWLWIWGDPZWLFZXC + WIFZWGXDHZXCWQBQIZRXCWQBVEIZQIZRXEXFEXCAXGBBSIZWQEAEPZXCXJTVHZWJCXAWTXGMZ + XJMWDWEXDVFZWDWEXDVIZWGXDVJZXLMVKXFEXCAXIXHXHSIZWQEAXKXCXQTVHZWJABXHXHMZC + VLBXHWJXSXAVMBWQXHXSWTVNXIMZXQMXFWEXHGFXNBXHXSVONXOXPXRMVKXGBXHWIWQXIXCXB + WTXMXSXTVQVPVRVSVTABWIWJCXBXAWAVPVRBWBWC $. $( A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.) $) @@ -239140,7 +239361,7 @@ zero ring (at least if its operations are internal binary operations). $( An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.) $) - assarng $p |- ( W e. AssAlg -> W e. Ring ) $= + assaring $p |- ( W e. AssAlg -> W e. Ring ) $= ( vz vx vy casa wcel clmod crg csca cfv ccrg w3a cv cvsca cmulr wceq wral co cbs eqid wa isassa simplbi simp2d ) AEFZAGFZAHFZAIJZKFZUEUFUGUILBMZCMZ ANJZRDMZAOJZRUJUKUMUNRULRZPUKUJUMULRUNRUOPUADASJZQCUPQBUHSJZQCDUQULUNUHUP @@ -239170,15 +239391,15 @@ zero ring (at least if its operations are internal binary operations). ( ( A .x. X ) .X. ( C .x. Y ) ) = ( ( C .* A ) .x. ( X .X. Y ) ) ) $= ( wcel wa co casa wceq simp1 simpr 3ad2ant2 clmod assalmod simpl lmodvscl w3a syl3an 3ad2ant3 assaassr syl13anc assaass eqcomd lmodvsass oveq1d crg - 3ad2ant1 assasca crngrng adantr adantl rngcl syl3anc 3adant3 eqtrd 3eqtrd - ccrg syl ) IUARZABRZCBRZSZJHRZKHRZSZUJZAJDTZCKDTETZCVTKETDTZCVTDTZKETZCAG - TZJKETDTZVSVLVNVTHRZVQWAWBUBVLVOVRUCZVOVLVNVRVMVNUDZUEZVLIUFRZVOVMVRVPWGI - UGZVMVNUHZVPVQUHZADFBHIJLMPNUIUKZVRVLVQVOVPVQUDULZCBDEFHIVTKLMNPQUMUNVSVL - VNWGVQWBWDUBWHWJWOWPVLVNWGVQUJSWDWBCBDEFHIVTKLMNPQUOUPUNVSWDWEJDTZKETZWFV - SWKVNVMVPWDWRUBVLVOWKVRWLUTWJVOVLVMVRWMUEVRVLVPVOWNULZWKVNVMVPUJSZWCWQKEW - TWQWCCADGFBHIJLMPNOUQUPURUNVSVLWEBRZVPVQWRWFUBWHVLVOXAVRVLVOSFUSRZVNVMXAV - LXBVOVLFVJRXBFIMVAFVBVKVCVOVNVLWIVDVOVMVLWMVDBFGCANOVEVFVGWSWPWEBDEFHIJKL - MNPQUOUNVHVI $. + 3ad2ant1 ccrg assasca crngring adantr adantl ringcl syl3anc 3adant3 eqtrd + syl 3eqtrd ) IUARZABRZCBRZSZJHRZKHRZSZUJZAJDTZCKDTETZCVTKETDTZCVTDTZKETZC + AGTZJKETDTZVSVLVNVTHRZVQWAWBUBVLVOVRUCZVOVLVNVRVMVNUDZUEZVLIUFRZVOVMVRVPW + GIUGZVMVNUHZVPVQUHZADFBHIJLMPNUIUKZVRVLVQVOVPVQUDULZCBDEFHIVTKLMNPQUMUNVS + VLVNWGVQWBWDUBWHWJWOWPVLVNWGVQUJSWDWBCBDEFHIVTKLMNPQUOUPUNVSWDWEJDTZKETZW + FVSWKVNVMVPWDWRUBVLVOWKVRWLUTWJVOVLVMVRWMUEVRVLVPVOWNULZWKVNVMVPUJSZWCWQK + EWTWQWCCADGFBHIJLMPNOUQUPURUNVSVLWEBRZVPVQWRWFUBWHVLVOXAVRVLVOSFUSRZVNVMX + AVLXBVOVLFVARXBFIMVBFVCVJVDVOVNVLWIVEVOVMVLWMVEBFGCANOVFVGVHWSWPWEBDEFHIJ + KLMNPQUOUNVIVK $. $} ${ @@ -239223,20 +239444,20 @@ zero ring (at least if its operations are internal binary operations). issubassa $p |- ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) -> ( S e. AssAlg <-> ( A e. ( SubRing ` W ) /\ A e. L ) ) ) $= ( wcel cfv wa crg co jca wceq ad2antrl eqid cv vy vz vx casa csubrg cress - wss w3a simpl1 assarng syl adantl syl5eqelr simpl3 issubrg sylanbrc clmod - simpl2 assalmod wb islss3 3syl mpbir2and cbs cvsca cmulr subrgss ressbas2 - csca resssca eqidd ressvsca ressmulr lsslmod syl2an subrgrng ccrg assasca - simpr adantr simpll simpr1 simpr2 sseldd simpr3 assaass syl13anc assaassr - isassad 3ad2antl1 impbida ) FUDKZCAKZAEUGZUHZBUDKZAFUELZKZADKZMZWOWPMZWRW - SXAFNKZFAUFOZNKZMWNWMMWRXAXBXDXAWLXBWLWMWNWPUIZFUJUKXAXCBNGWPBNKZWOBUJULU - MPXAWNWMWLWMWNWPUNZWLWMWNWPURPAEFCIJUOUPXAWSWNBUQKZXGWPXHWOBUSULXAWLFUQKZ - WSWNXHMUTXEFUSZDAEFBGIHVAVBVCPWLWMWTWPWNWLWTMZUAUBFVILZVDLZFVELZFVFLZXLAB - UCXKWNABVDLQWRWNWLWSAEFIVGRZAEBFGIVHUKWRXLBVILQWLWSAXLFBWQGXLSZVJRXKXMVKW - RXNBVELQWLWSAXNFBWQGXNSZVLRWRXOBVFLQWLWSAFBXOWQGXOSZVMRWLXIWSXHWTXJWRWSVS - DAFBGHVNVOWRXFWLWSAFBGVPRWLXLVQKWTXLFXQVRVTXKUCTZXMKZUATZAKZUBTZAKZUHZMZW - LYAYBEKZYDEKZXTYBXNOYDXOOXTYBYDXOOXNOZQWLWTYFWAZXKYAYCYEWBZYGAEYBXKWNYFXP - VTZXKYAYCYEWCWDZYGAEYDYMXKYAYCYEWEWDZXTXMXNXOXLEFYBYDIXQXMSZXRXSWFWGYGWLY - AYHYIYBXTYDXNOXOOYJQYKYLYNYOXTXMXNXOXLEFYBYDIXQYPXRXSWHWGWIWJWK $. + wss w3a simpl1 assaring syl adantl syl5eqelr simpl3 simpl2 sylanbrc clmod + issubrg assalmod wb islss3 3syl mpbir2and csca cbs cvsca subrgss ressbas2 + cmulr resssca eqidd ressvsca ressmulr simpr lsslmod syl2an subrgring ccrg + assasca adantr simpll simpr1 simpr2 sseldd simpr3 assaass isassad impbida + syl13anc assaassr 3ad2antl1 ) FUDKZCAKZAEUGZUHZBUDKZAFUELZKZADKZMZWOWPMZW + RWSXAFNKZFAUFOZNKZMWNWMMWRXAXBXDXAWLXBWLWMWNWPUIZFUJUKXAXCBNGWPBNKZWOBUJU + LUMPXAWNWMWLWMWNWPUNZWLWMWNWPUOPAEFCIJURUPXAWSWNBUQKZXGWPXHWOBUSULXAWLFUQ + KZWSWNXHMUTXEFUSZDAEFBGIHVAVBVCPWLWMWTWPWNWLWTMZUAUBFVDLZVELZFVFLZFVILZXL + ABUCXKWNABVELQWRWNWLWSAEFIVGRZAEBFGIVHUKWRXLBVDLQWLWSAXLFBWQGXLSZVJRXKXMV + KWRXNBVFLQWLWSAXNFBWQGXNSZVLRWRXOBVILQWLWSAFBXOWQGXOSZVMRWLXIWSXHWTXJWRWS + VNDAFBGHVOVPWRXFWLWSAFBGVQRWLXLVRKWTXLFXQVSVTXKUCTZXMKZUATZAKZUBTZAKZUHZM + ZWLYAYBEKZYDEKZXTYBXNOYDXOOXTYBYDXOOXNOZQWLWTYFWAZXKYAYCYEWBZYGAEYBXKWNYF + XPVTZXKYAYCYEWCWDZYGAEYDYMXKYAYCYEWEWDZXTXMXNXOXLEFYBYDIXQXMSZXRXSWFWIYGW + LYAYHYIYBXTYDXNOXOOYJQYKYLYNYOXTXMXNXOXLEFYBYDIXQYPXRXSWJWIWGWKWH $. $} ${ @@ -239248,23 +239469,23 @@ zero ring (at least if its operations are internal binary operations). A e. AssAlg ) $= ( vy vz vx wcel cfv wa cmulr co cbs wceq eqid adantl crg adantr cv cplusg ccrg csubrg cress a1i wss subrgss srabase srasca subrgbas sravsca sramulr - clmod sralmod crngrng eqidd sraaddg proplem3 rngpropd mpbid subrgcrng w3a - csra simpr1 sseldd simpr2 simpr3 rngass syl13anc cmgp ccmn crngmgp mgpbas - ad2antrr mgpplusg cmn12 isassad ) CUAHZBCUBIHZJZEFBCKIZVTCBUCLZCMIZAGVSAB - CABCVBIINVSDUDZVRBWBUEZVQBWBCWBOZUFPZUGZVSABCWCWFUHVRBWAMINVQBCWAWAOZUIPV - SABCWCWFUJVSABCWCWFUKZVRAULHVQABCDUMPVSCQHZAQHVQWJVRCUNRZVSGEWBCAVSWBUOWG - VSGSZWBHZESZWBHZJZGECTIATIVSABCWCWFUPUQVSWPGEVTAKIWIUQURUSBCWAWHUTVSWLBHZ - WOFSZWBHZVAZJZWJWMWOWSWLWNVTLWRVTLWLWNWRVTLVTLZNVSWJWTWKRXABWBWLVSWDWTWFR - VSWQWOWSVCVDZVSWQWOWSVEZVSWQWOWSVFZWBCVTWLWNWRWEVTOZVGVHXACVIIZVJHZWOWMWS - WNWLWRVTLVTLXBNVQXHVRWTCXGXGOZVKVMXDXCXEWBVTXGWNWLWRWBCXGXIWEVLCVTXGXIXFV - NVOVHVP $. + csra clmod sralmod crngring eqidd sraaddg proplem3 ringpropd mpbid simpr1 + subrgcrng w3a sseldd simpr2 simpr3 ringass syl13anc cmgp crngmgp ad2antrr + ccmn mgpbas mgpplusg cmn12 isassad ) CUAHZBCUBIHZJZEFBCKIZVTCBUCLZCMIZAGV + SABCABCULIINVSDUDZVRBWBUEZVQBWBCWBOZUFPZUGZVSABCWCWFUHVRBWAMINVQBCWAWAOZU + IPVSABCWCWFUJVSABCWCWFUKZVRAUMHVQABCDUNPVSCQHZAQHVQWJVRCUORZVSGEWBCAVSWBU + PWGVSGSZWBHZESZWBHZJZGECTIATIVSABCWCWFUQURVSWPGEVTAKIWIURUSUTBCWAWHVBVSWL + BHZWOFSZWBHZVCZJZWJWMWOWSWLWNVTLWRVTLWLWNWRVTLVTLZNVSWJWTWKRXABWBWLVSWDWT + WFRVSWQWOWSVAVDZVSWQWOWSVEZVSWQWOWSVFZWBCVTWLWNWRWEVTOZVGVHXACVIIZVLHZWOW + MWSWNWLWRVTLVTLXBNVQXHVRWTCXGXGOZVJVKXDXCXEWBVTXGWNWLWRWBCXGXIWEVMCVTXGXI + XFVNVOVHVP $. $} $( The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.) $) rlmassa $p |- ( R e. CRing -> ( ringLMod ` R ) e. AssAlg ) $= - ( ccrg wcel cbs cfv csubrg crglmod casa crg crngrng eqid subrgid syl rlmval - sraassa mpdan ) ABCZADEZAFECZAGEZHCQAICSAJRARKLMTRAANOP $. + ( ccrg wcel cbs cfv csubrg crglmod casa crg crngring subrgid rlmval sraassa + eqid syl mpdan ) ABCZADEZAFECZAGEZHCQAICSAJRARNKOTRAALMP $. ${ $d r w x y z K $. $d r w x y z L $. $d r w x y z P $. $d r w x y z ph $. @@ -239284,33 +239505,33 @@ zero ring (at least if its operations are internal binary operations). associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) $) assapropd $p |- ( ph -> ( K e. AssAlg <-> L e. AssAlg ) ) $= - ( wcel co wceq wral vr vz vw clmod crg wa casa assalmod assarng lmodpropd - wi jca a1i syl5ibr rngpropd jcad wb csca cfv ccrg w3a cv cvsca cbs eqtr3d - cmulr eleq1d 3anbi123d adantr simpll simplrl simprl fveq2d syl5eq eleqtrd - syl simprrl eqid lmodvscl syl3anc eleqtrrd simprrr proplem syl12anc eqtrd - oveq1d simplrr rngcl eqeq12d anbi12d anassrs 2ralbidva ralbidva raleqbidv - oveq2d raleqdv 3bitr3d isassa 3bitr4g ex pm5.21ndd ) AGUDQZGUEQZUFZGUGQZH - UGQZXEXDUKAXEXBXCGUHGUIULUMAXFXBXCXFXBAHUDQZHUHABCDEFGHIJKMNOPUJZUNXFXCAH - UEQZHUIABCDGHIJKLUOZUNUPAXDXEXFUQAXDUFZXBXCGURUSZUTQZVAZUAVBZUBVBZGVCUSZR - ZUCVBZGVFUSZRZXOXPXSXTRZXQRZSZXPXOXSXQRZXTRZYCSZUFZUCGVDUSZTZUBYITZUAXLVD - USZTZUFXGXIHURUSZUTQZVAZXOXPHVCUSZRZXSHVFUSZRZXOXPXSYSRZYQRZSZXPXOXSYQRZY - SRZUUBSZUFZUCHVDUSZTZUBUUHTZUAYNVDUSZTZUFXEXFXKXNYPYMUULAXNYPUQXDAXBXGXCX - IXMYOXHXJAXLYNUTAFXLYNMNVEVGVHVIXKYHUCDTZUBDTZUAETUUGUCDTZUBDTZUAETYMUULX - KUUNUUPUAEXKXOEQZUFYHUUGUBUCDDXKUUQXPDQZXSDQZUFZYHUUGUQXKUUQUUTUFZUFZYDUU - CYGUUFUVBYAYTYCUUBUVBYAXRXSYSRZYTUVBAXRDQUUSYAUVCSAXDUVAVJZUVBXRYIDUVBXBX - OYLQZXPYIQZXRYIQAXBXCUVAVKZUVBXOEYLXKUUQUUTVLZUVBAEYLSZUVDAEFVDUSZYLOAFXL - VDMVMVNZVPVOZUVBXPDYIXKUUQUURUUSVQZUVBADYISZUVDIVPZVOZXOXQXLYLYIGXPYIVRZX - LVRZXQVRZYLVRZVSVTUVOWAXKUUQUURUUSWBZABCDDXTYSXRXSLWCWDUVBXRYRXSYSUVBAUUQ - UURXRYRSUVDUVHUVMABCEDXQYQXOXPPWCWDWFWEUVBYCXOYBYQRZUUBUVBAUUQYBDQYCUWBSU - VDUVHUVBYBYIDUVBXCUVFXSYIQZYBYIQAXBXCUVAWGUVPUVBXSDYIUWAUVOVOZYIGXTXPXSUV - QXTVRZWHVTUVOWAABCEDXQYQXOYBPWCWDUVBYBUUAXOYQUVBAUURUUSYBUUASUVDUVMUWAABC - DDXTYSXPXSLWCWDWOWEZWIUVBYFUUEYCUUBUVBYFXPYEYSRZUUEUVBAUURYEDQYFUWGSUVDUV - MUVBYEYIDUVBXBUVEUWCYEYIQUVGUVLUWDXOXQXLYLYIGXSUVQUVRUVSUVTVSVTUVOWAABCDD - XTYSXPYELWCWDUVBYEUUDXPYSUVBAUUQUUSYEUUDSUVDUVHUWAABCEDXQYQXOXSPWCWDWOWEU - WFWIWJWKWLWMXKUUNYKUAEYLAUVIXDUVKVIXKUUMYJUBDYIAUVNXDIVIZXKYHUCDYIUWHWPWN - WNXKUUPUUJUAEUUKAEUUKSXDAEUVJUUKOAFYNVDNVMVNVIXKUUOUUIUBDUUHADUUHSXDJVIZX - KUUGUCDUUHUWIWPWNWNWQWJUBUCYLXQXTXLYIGUAUVQUVRUVTUVSUWEWRUBUCUUKYQYSYNUUH - HUAUUHVRYNVRUUKVRYQVRYSVRWRWSWTXA $. + ( wcel co wceq wral vr vz vw clmod wa casa wi assalmod assaring lmodpropd + crg jca a1i syl5ibr ringpropd jcad wb csca cfv ccrg cv cvsca cmulr eqtr3d + w3a cbs eleq1d adantr simpll simplrl simprl fveq2d syl5eq eleqtrd simprrl + 3anbi123d syl eqid lmodvscl syl3anc eleqtrrd simprrr proplem oveq1d eqtrd + syl12anc simplrr ringcl oveq2d eqeq12d anbi12d anassrs 2ralbidva ralbidva + raleqdv raleqbidv 3bitr3d isassa 3bitr4g ex pm5.21ndd ) AGUDQZGUKQZUEZGUF + QZHUFQZXEXDUGAXEXBXCGUHGUIULUMAXFXBXCXFXBAHUDQZHUHABCDEFGHIJKMNOPUJZUNXFX + CAHUKQZHUIABCDGHIJKLUOZUNUPAXDXEXFUQAXDUEZXBXCGURUSZUTQZVEZUAVAZUBVAZGVBU + SZRZUCVAZGVCUSZRZXOXPXSXTRZXQRZSZXPXOXSXQRZXTRZYCSZUEZUCGVFUSZTZUBYITZUAX + LVFUSZTZUEXGXIHURUSZUTQZVEZXOXPHVBUSZRZXSHVCUSZRZXOXPXSYSRZYQRZSZXPXOXSYQ + RZYSRZUUBSZUEZUCHVFUSZTZUBUUHTZUAYNVFUSZTZUEXEXFXKXNYPYMUULAXNYPUQXDAXBXG + XCXIXMYOXHXJAXLYNUTAFXLYNMNVDVGVPVHXKYHUCDTZUBDTZUAETUUGUCDTZUBDTZUAETYMU + ULXKUUNUUPUAEXKXOEQZUEYHUUGUBUCDDXKUUQXPDQZXSDQZUEZYHUUGUQXKUUQUUTUEZUEZY + DUUCYGUUFUVBYAYTYCUUBUVBYAXRXSYSRZYTUVBAXRDQUUSYAUVCSAXDUVAVIZUVBXRYIDUVB + XBXOYLQZXPYIQZXRYIQAXBXCUVAVJZUVBXOEYLXKUUQUUTVKZUVBAEYLSZUVDAEFVFUSZYLOA + FXLVFMVLVMZVQVNZUVBXPDYIXKUUQUURUUSVOZUVBADYISZUVDIVQZVNZXOXQXLYLYIGXPYIV + RZXLVRZXQVRZYLVRZVSVTUVOWAXKUUQUURUUSWBZABCDDXTYSXRXSLWCWFUVBXRYRXSYSUVBA + UUQUURXRYRSUVDUVHUVMABCEDXQYQXOXPPWCWFWDWEUVBYCXOYBYQRZUUBUVBAUUQYBDQYCUW + BSUVDUVHUVBYBYIDUVBXCUVFXSYIQZYBYIQAXBXCUVAWGUVPUVBXSDYIUWAUVOVNZYIGXTXPX + SUVQXTVRZWHVTUVOWAABCEDXQYQXOYBPWCWFUVBYBUUAXOYQUVBAUURUUSYBUUASUVDUVMUWA + ABCDDXTYSXPXSLWCWFWIWEZWJUVBYFUUEYCUUBUVBYFXPYEYSRZUUEUVBAUURYEDQYFUWGSUV + DUVMUVBYEYIDUVBXBUVEUWCYEYIQUVGUVLUWDXOXQXLYLYIGXSUVQUVRUVSUVTVSVTUVOWAAB + CDDXTYSXPYELWCWFUVBYEUUDXPYSUVBAUUQUUSYEUUDSUVDUVHUWAABCEDXQYQXOXSPWCWFWI + WEUWFWJWKWLWMWNXKUUNYKUAEYLAUVIXDUVKVHXKUUMYJUBDYIAUVNXDIVHZXKYHUCDYIUWHW + OWPWPXKUUPUUJUAEUUKAEUUKSXDAEUVJUUKOAFYNVFNVLVMVHXKUUOUUIUBDUUHADUUHSXDJV + HZXKUUGUCDUUHUWIWOWPWPWQWKUBUCYLXQXTXLYIGUAUVQUVRUVTUVSUWEWRUBUCUUKYQYSYN + UUHHUAUUHVRYNVRUUKVRYQVRYSVRWRWSWTXA $. $} ${ @@ -239326,15 +239547,15 @@ zero ring (at least if its operations are internal binary operations). ( vs vw wcel wss cfv cv csubrg crab cint wceq cbs casa wa cpw cmpt casp cin clss fveq2 syl6eqr pweqd ineq12d rabeq inteqd mpteq12dv df-asp fvex syl cvv eqeltri pwex mptex fvmpt syl5eq fveq1d adantr simpr sylibr wrex - elpw2 crg assarng clmod assalmod lss1 elind sseq2 rspcev sylan intexrab - subrgid sylib sseq1 rabbidv eqid fvmptg syl2anc eqtrd ) FUALZCEMZUBZCBN - ZCJEUCZJOZAOZMZAFPNZDUFZQZRZUDZNZCWNMZAWQQZRZWHWKXASWIWHCBWTWHBFUENWTGK - FJKOZTNZUCZWOAXEPNZXEUGNZUFZQZRZUDWTUAUEXEFSZJXGXLWLWSXMXFEXMXFFTNZEXEF - TUHHUIUJXMXKWRXMXJWQSXKWRSXMXHWPXIDXEFPUHXMXIFUGNDXEFUGUHIUIUKWOAXJWQUL - UQUMUNKAJUOJWLWSEEXNURHFTUPUSZUTVAVBVCVDVEWJCWLLZXDURLZXAXDSWJWIXPWHWIV - FCEXOVIVGWJXBAWQVHZXQWHEWQLWIXRWHWPDEWHFVJLEWPLFVKEFHVTUQWHFVLLEDLFVMDE - FHIVNUQVOXBWIAEWQWNECVPVQVRXBAWQVSWAJCWSXDWLURWTWMCSZWRXCXSWOXBAWQWMCWN - WBWCUMWTWDWEWFWG $. + elpw2 crg assaring subrgid clmod assalmod lss1 elind sseq2 rspcev sylan + intexrab sylib sseq1 rabbidv eqid fvmptg syl2anc eqtrd ) FUALZCEMZUBZCB + NZCJEUCZJOZAOZMZAFPNZDUFZQZRZUDZNZCWNMZAWQQZRZWHWKXASWIWHCBWTWHBFUENWTG + KFJKOZTNZUCZWOAXEPNZXEUGNZUFZQZRZUDWTUAUEXEFSZJXGXLWLWSXMXFEXMXFFTNZEXE + FTUHHUIUJXMXKWRXMXJWQSXKWRSXMXHWPXIDXEFPUHXMXIFUGNDXEFUGUHIUIUKWOAXJWQU + LUQUMUNKAJUOJWLWSEEXNURHFTUPUSZUTVAVBVCVDVEWJCWLLZXDURLZXAXDSWJWIXPWHWI + VFCEXOVIVGWJXBAWQVHZXQWHEWQLWIXRWHWPDEWHFVJLEWPLFVKEFHVLUQWHFVMLEDLFVND + EFHIVOUQVPXBWIAEWQWNECVQVRVSXBAWQVTWAJCWSXDWLURWTWMCSZWRXCXSWOXBAWQWMCW + NWBWCUMWTWDWEWFWG $. $( The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.) $) @@ -239431,24 +239652,24 @@ zero ring (at least if its operations are internal binary operations). $( The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.) $) asclf $p |- ( ph -> A : K --> B ) $= - ( vx cv cur cfv cvsca wcel adantr eqid co wa clmod crg rngidcl lmodvscl - simpr syl syl3anc asclfval fmptd ) AMEMNZFOPZFQPZUAZCBAULERZUBFUCRZUPUM - CRZUOCRAUQUPJSAUPUGAURUPAFUDRURICFUMLUMTZUEUHSULUNDECFUMLHUNTZKUFUIMBUN - UMDEFGHKUTUSUJUK $. + ( vx cv cur cfv cvsca wcel adantr eqid co clmod simpr ringidcl lmodvscl + wa crg syl syl3anc asclfval fmptd ) AMEMNZFOPZFQPZUAZCBAULERZUFFUBRZUPU + MCRZUOCRAUQUPJSAUPUCAURUPAFUGRURICFUMLUMTZUDUHSULUNDECFUMLHUNTZKUEUIMBU + NUMDEFGHKUTUSUJUK $. $} $( The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.) $) asclghm $p |- ( ph -> A e. ( F GrpHom W ) ) $= - ( vx vy cplusg cfv cbs eqid crg wcel syl co wceq asclval cgrp clmod asclf - lmodrng rnggrp cv cur cvsca adantr simprl simprr lmodvsdir syl13anc grpcl - wa rngidcl 3expb sylan oveqan12d adantl 3eqtr4d isghmd ) AIJCKLZDKLZCDBCM - LZDMLZVENZVFNZVCNZVDNZACOPZCUAPZADUBPZVKHCDFUDQCUEQZADOPZDUAPGDUEQABVFCVE - DEFGHVGVHUCAIUFZVEPZJUFZVEPZUOZUOZVPVRVCRZDUGLZDUHLZRZVPWCWDRZVRWCWDRZVDR - ZWBBLZVPBLZVRBLZVDRZWAVMVQVSWCVFPZWEWHSAVMVTHUIAVQVSUJAVQVSUKAWMVTAVOWMGV - FDWCVHWCNZUPQUIVDVCVPVRWDCVEVFDWCVHVJFWDNZVGVIULUMWAWBVEPZWIWESAVLVTWPVNV - LVQVSWPVEVCCVPVRVGVIUNUQURBWDWCCVEDWBEFVGWOWNTQVTWLWHSAVQVSWJWFWKWGVDBWDW - CCVEDVPEFVGWOWNTBWDWCCVEDVREFVGWOWNTUSUTVAVB $. + ( vx vy cplusg cfv cbs eqid crg wcel syl co wceq asclval lmodring ringgrp + clmod asclf cv cur cvsca adantr simprl simprr ringidcl lmodvsdir syl13anc + cgrp wa grpcl 3expb sylan oveqan12d adantl 3eqtr4d isghmd ) AIJCKLZDKLZCD + BCMLZDMLZVENZVFNZVCNZVDNZACOPZCUNPZADUCPZVKHCDFUAQCUBQZADOPZDUNPGDUBQABVF + CVEDEFGHVGVHUDAIUEZVEPZJUEZVEPZUOZUOZVPVRVCRZDUFLZDUGLZRZVPWCWDRZVRWCWDRZ + VDRZWBBLZVPBLZVRBLZVDRZWAVMVQVSWCVFPZWEWHSAVMVTHUHAVQVSUIAVQVSUJAWMVTAVOW + MGVFDWCVHWCNZUKQUHVDVCVPVRWDCVEVFDWCVHVJFWDNZVGVIULUMWAWBVEPZWIWESAVLVTWP + VNVLVQVSWPVEVCCVPVRVGVIUPUQURBWDWCCVEDWBEFVGWOWNTQVTWLWHSAVQVSWJWFWKWGVDB + WDWCCVEDVPEFVGWOWNTBWDWCCVEDVREFVGWOWNTUSUTVAVB $. $} ${ @@ -239462,23 +239683,23 @@ zero ring (at least if its operations are internal binary operations). operation. (Contributed by Mario Carneiro, 9-Mar-2015.) $) asclmul1 $p |- ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( ( A ` R ) .X. X ) = ( R .x. X ) ) $= - ( casa wcel cfv co wceq w3a cur eqid asclval 3ad2ant2 simp1 simp2 assarng - oveq1d crg 3ad2ant1 rngidcl simp3 assaass syl13anc rnglidm syl2anc oveq2d - syl 3eqtrd ) HPQZBFQZIGQZUAZBARZIDSBHUBRZCSZIDSZBVFIDSZCSZBICSVDVEVGIDVBV - AVEVGTVCACVFEFHBJKLOVFUCZUDUEUIVDVAVBVFGQZVCVHVJTVAVBVCUFVAVBVCUGVDHUJQZV - LVAVBVMVCHUHUKZGHVFMVKULUSVAVBVCUMZBFCDEGHVFIMKLONUNUOVDVIIBCVDVMVCVIITVN - VOGHDVFIMNVKUPUQURUT $. + ( casa wcel cfv co wceq w3a cur eqid 3ad2ant2 oveq1d simp1 simp2 assaring + asclval crg 3ad2ant1 ringidcl syl simp3 assaass syl13anc ringlidm syl2anc + oveq2d 3eqtrd ) HPQZBFQZIGQZUAZBARZIDSBHUBRZCSZIDSZBVFIDSZCSZBICSVDVEVGID + VBVAVEVGTVCACVFEFHBJKLOVFUCZUIUDUEVDVAVBVFGQZVCVHVJTVAVBVCUFVAVBVCUGVDHUJ + QZVLVAVBVMVCHUHUKZGHVFMVKULUMVAVBVCUNZBFCDEGHVFIMKLONUOUPVDVIIBCVDVMVCVII + TVNVOGHDVFIMNVKUQURUSUT $. $( Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.) $) asclmul2 $p |- ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( X .X. ( A ` R ) ) = ( R .x. X ) ) $= ( wcel cfv co wceq oveq2d casa w3a cur asclval 3ad2ant2 simp1 simp2 simp3 - crg assarng 3ad2ant1 rngidcl syl assaassr syl13anc rngridm syl2anc 3eqtrd - eqid ) HUAPZBFPZIGPZUBZIBAQZDRIBHUCQZCRZDRZBIVEDRZCRZBICRVCVDVFIDVAUTVDVF - SVBACVEEFHBJKLOVEUSZUDUETVCUTVAVBVEGPZVGVISUTVAVBUFUTVAVBUGUTVAVBUHZVCHUI - PZVKUTVAVMVBHUJUKZGHVEMVJULUMBFCDEGHIVEMKLONUNUOVCVHIBCVCVMVBVHISVNVLGHDV - EIMNVJUPUQTUR $. + eqid crg assaring 3ad2ant1 ringidcl syl assaassr syl13anc ringridm 3eqtrd + syl2anc ) HUAPZBFPZIGPZUBZIBAQZDRIBHUCQZCRZDRZBIVEDRZCRZBICRVCVDVFIDVAUTV + DVFSVBACVEEFHBJKLOVEUIZUDUETVCUTVAVBVEGPZVGVISUTVAVBUFUTVAVBUGUTVAVBUHZVC + HUJPZVKUTVAVMVBHUKULZGHVEMVJUMUNBFCDEGHIVEMKLONUOUPVCVHIBCVCVMVBVHISVNVLG + HDVEIMNVJUQUSTUR $. $} ${ @@ -239504,21 +239725,21 @@ zero ring (at least if its operations are internal binary operations). Carneiro, 8-Mar-2015.) $) asclrhm $p |- ( W e. AssAlg -> A e. ( F RingHom W ) ) $= ( vx vy wcel cbs cfv cmulr cur eqid syl co asclval adantr oveq2d syl13anc - wceq casa ccrg crg assasca crngrng assarng cvsca rngidcl assalmod lmodvs1 - 3syl clmod syl2anc eqtrd cv rnglidm simpl simprl lmodvscl syl3anc assaass - wa simprr assaassr lmodvsass 3eqtr4rd rngcl 3expb oveq12d 3eqtr4d asclghm - sylan isrhm2d ) CUAHZFGBIJZBCBKJZCKJZBLJZACLJZVOMZVRMZVSMZVPMZVQMZVNBUBHB - UCHZBCEUDBUENZCUFZVNVRAJZVRVSCUGJZOZVSVNWEVRVOHWHWJTWFVOBVRVTWAUHAWIVSBVO - CVRDEVTWIMZWBPUKVNCULHZVSCIJZHZWJVSTCUIZVNCUCHZWNWGWMCVSWMMZWBUHNZWIVRBWM - CVSWQEWKWAUJUMUNVNFUOZVOHZGUOZVOHZVBZVBZWSXAVPOZVSWIOZWSVSWIOZXAVSWIOZVQO - ZXEAJZWSAJZXAAJZVQOXDWSXAVSVSVQOZWIOZWIOZWSXHWIOZXIXFXDXNXHWSWIXDXMVSXAWI - VNXMVSTZXCVNWPWNXQWGWRWMCVQVSVSWQWDWBUPUMQRRXDXIWSVSXHVQOZWIOZXOXDVNWTWNX - HWMHZXIXSTVNXCUQZVNWTXBURZVNWNXCWRQZXDWLXBWNXTVNWLXCWOQZVNWTXBVCZYCXAWIBV - OWMCVSWQEWKVTUSUTWSVOWIVQBWMCVSXHWQEVTWKWDVASXDXRXNWSWIXDVNXBWNWNXRXNTYAY - EYCYCXAVOWIVQBWMCVSVSWQEVTWKWDVDSRUNXDWLWTXBWNXFXPTYDYBYEYCWSXAWIVPBVOWMC - VSWQEWKVTWCVESVFXDXEVOHZXJXFTVNWEXCYFWFWEWTXBYFVOBVPWSXAVTWCVGVHVLAWIVSBV - OCXEDEVTWKWBPNXDXKXGXLXHVQXDWTXKXGTYBAWIVSBVOCWSDEVTWKWBPNXDXBXLXHTYEAWIV - SBVOCXADEVTWKWBPNVIVJVNABCDEWGWOVKVM $. + wceq casa ccrg crg assasca crngring assaring cvsca ringidcl 3syl assalmod + clmod lmodvs1 syl2anc eqtrd cv wa ringlidm simprl simprr lmodvscl syl3anc + simpl assaass assaassr lmodvsass 3eqtr4rd oveq12d 3eqtr4d asclghm isrhm2d + ringcl 3expb sylan ) CUAHZFGBIJZBCBKJZCKJZBLJZACLJZVOMZVRMZVSMZVPMZVQMZVN + BUBHBUCHZBCEUDBUENZCUFZVNVRAJZVRVSCUGJZOZVSVNWEVRVOHWHWJTWFVOBVRVTWAUHAWI + VSBVOCVRDEVTWIMZWBPUIVNCUKHZVSCIJZHZWJVSTCUJZVNCUCHZWNWGWMCVSWMMZWBUHNZWI + VRBWMCVSWQEWKWAULUMUNVNFUOZVOHZGUOZVOHZUPZUPZWSXAVPOZVSWIOZWSVSWIOZXAVSWI + OZVQOZXEAJZWSAJZXAAJZVQOXDWSXAVSVSVQOZWIOZWIOZWSXHWIOZXIXFXDXNXHWSWIXDXMV + SXAWIVNXMVSTZXCVNWPWNXQWGWRWMCVQVSVSWQWDWBUQUMQRRXDXIWSVSXHVQOZWIOZXOXDVN + WTWNXHWMHZXIXSTVNXCVBZVNWTXBURZVNWNXCWRQZXDWLXBWNXTVNWLXCWOQZVNWTXBUSZYCX + AWIBVOWMCVSWQEWKVTUTVAWSVOWIVQBWMCVSXHWQEVTWKWDVCSXDXRXNWSWIXDVNXBWNWNXRX + NTYAYEYCYCXAVOWIVQBWMCVSVSWQEVTWKWDVDSRUNXDWLWTXBWNXFXPTYDYBYEYCWSXAWIVPB + VOWMCVSWQEWKVTWCVESVFXDXEVOHZXJXFTVNWEXCYFWFWEWTXBYFVOBVPWSXAVTWCVKVLVMAW + IVSBVOCXEDEVTWKWBPNXDXKXGXLXHVQXDWTXKXGTYBAWIVSBVOCWSDEVTWKWBPNXDXBXLXHTY + EAWIVSBVOCXADEVTWKWBPNVGVHVNABCDEWGWOVIVJ $. $} ${ @@ -239530,10 +239751,10 @@ zero ring (at least if its operations are internal binary operations). identity. (Contributed by Mario Carneiro, 9-Mar-2015.) $) rnascl $p |- ( W e. AssAlg -> ran A = ( N ` { .1. } ) ) $= ( vx vy casa wcel csn cfv cv cvsca co wceq csca cbs eqid cab crn assalmod - wrex clmod crg assarng rngidcl syl lspsn syl2anc asclfval rnmpt syl6reqr - ) DJKZBLCMZHNINBDOMZPZQIDRMZSMZUDHUAZAUBUODUEKBDSMZKZUPVAQDUCUODUFKVCDUGV - BDBVBTZFUHUIHUQIUSUTCVBDBUSTZUTTZVDUQTZGUJUKIHUTURAIAUQBUSUTDEVEVFVGFULUM - UN $. + wrex clmod crg assaring ringidcl lspsn syl2anc asclfval rnmpt syl6reqr + syl ) DJKZBLCMZHNINBDOMZPZQIDRMZSMZUDHUAZAUBUODUEKBDSMZKZUPVAQDUCUODUFKVC + DUGVBDBVBTZFUHUNHUQIUSUTCVBDBUSTZUTTZVDUQTZGUIUJIHUTURAIAUQBUSUTDEVEVFVGF + UKULUM $. $} ${ @@ -239613,14 +239834,14 @@ zero ring (at least if its operations are internal binary operations). ( R ` ( ran C u. S ) ) ) $= ( vx wcel wss wa cfv crab cint cab eqid adantr casa cv csubrg cin crn cun clss wceq elin anbi1i anass issubassa2 anbi1d unss syl6bb pm5.32da syl5bb - bitri abbidv df-rab 3eqtr4g inteqd cmre crg assarng subrgmre syl csca cbs - aspval wf assalmod asclf frn simpr unssd mrcval syl2anc 3eqtr4d ) FUALZDE - MZNZDKUBZMZKFUCOZFUGOZUDZPZQBUEZDUFZWCMZKWEPZQZDAOWJCOZWBWHWLWBWCWGLZWDNZ - KRZWCWELZWKNZKRZWHWLVTWQWTUHWAVTWPWSKWPWRWCWFLZWDNZNZVTWSWPWRXANZWDNXCWOX - DWDWCWEWFUIUJWRXAWDUKURVTWRXBWKVTWRNZXBWIWCMZWDNWKXEXAXFWDBWCWFFHWFSZULUM - WIDWCUNUOUPUQUSTWDKWGUTWKKWEUTVAVBKADWFEFGJXGVJWBWEEVCOLZWJEMWNWMUHVTXHWA - VTFVDLXHFVEZEFJVFVGTWBWIDEVTWIEMZWAVTFVHOZVIOZEBVKXJVTBEXKXLFHXKSXIFVLXLS - JVMXLEBVNVGTVTWAVOVPWEWJCEKIVQVRVS $. + bitri abbidv df-rab 3eqtr4g inteqd aspval cmre crg assaring subrgmre csca + syl cbs wf assalmod asclf frn simpr unssd mrcval syl2anc 3eqtr4d ) FUALZD + EMZNZDKUBZMZKFUCOZFUGOZUDZPZQBUEZDUFZWCMZKWEPZQZDAOWJCOZWBWHWLWBWCWGLZWDN + ZKRZWCWELZWKNZKRZWHWLVTWQWTUHWAVTWPWSKWPWRWCWFLZWDNZNZVTWSWPWRXANZWDNXCWO + XDWDWCWEWFUIUJWRXAWDUKURVTWRXBWKVTWRNZXBWIWCMZWDNWKXEXAXFWDBWCWFFHWFSZULU + MWIDWCUNUOUPUQUSTWDKWGUTWKKWEUTVAVBKADWFEFGJXGVCWBWEEVDOLZWJEMWNWMUHVTXHW + AVTFVELXHFVFZEFJVGVITWBWIDEVTWIEMZWAVTFVHOZVJOZEBVKXJVTBEXKXLFHXKSXIFVLXL + SJVMXLEBVNVITVTWAVOVPWEWJCEKIVQVRVS $. $} ${ @@ -239636,13 +239857,13 @@ zero ring (at least if its operations are internal binary operations). 26-Aug-2019.) $) assamulgscmlem1 $p |- ( ( ( A e. B /\ X e. V ) /\ W e. AssAlg ) -> ( 0 E ( A .x. X ) ) = ( ( 0 .^ A ) .x. ( 0 E X ) ) ) $= - ( wcel wa casa cur cfv co cc0 wceq clmod assalmod crg assarng rngidcl syl - eqid lmodvs1 eqcomd syl2anc adantl simpll simplr lmodvscl mgpbas rngidval - syl3anc mulg0 oveq12d 3eqtr4d ) ABTZKITZUAZJUBTZUAZJUCUDZFUCUDZVMCUEZUFAK - CUEZDUEZUFAEUEZUFKDUEZCUEVKVMVOUGZVJVKJUHTZVMITZVTJUIZVKJUJTWBJUKIJVMLVMU - NZULUMWAWBUAVOVMCVNFIJVMLMOVNUNZUOUPUQURVLVPITZVQVMUGVLWAVHVIWFVKWAVJWCUR - VHVIVKUSZVHVIVKUTZACFBIJKLMONVAVDIDHVPVMIJHRLVBZJVMHRWDVCZSVEUMVLVRVNVSVM - CVLVHVRVNUGWGBEGAVNBFGPNVBFVNGPWEVCQVEUMVLVIVSVMUGWHIDHKVMWIWJSVEUMVFVG + ( wcel wa casa cur cfv cc0 wceq clmod assalmod crg assaring eqid ringidcl + syl lmodvs1 eqcomd syl2anc adantl simpll simplr lmodvscl syl3anc rngidval + co mgpbas mulg0 oveq12d 3eqtr4d ) ABTZKITZUAZJUBTZUAZJUCUDZFUCUDZVMCVCZUE + AKCVCZDVCZUEAEVCZUEKDVCZCVCVKVMVOUFZVJVKJUGTZVMITZVTJUHZVKJUITWBJUJIJVMLV + MUKZULUMWAWBUAVOVMCVNFIJVMLMOVNUKZUNUOUPUQVLVPITZVQVMUFVLWAVHVIWFVKWAVJWC + UQVHVIVKURZVHVIVKUSZACFBIJKLMONUTVAIDHVPVMIJHRLVDZJVMHRWDVBZSVEUMVLVRVNVS + VMCVLVHVRVNUFWGBEGAVNBFGPNVDFVNGPWEVBQVEUMVLVIVSVMUFWHIDHKVMWIWJSVEUMVFVG $. $( Lemma for ~ assamulgscm (induction step). (Contributed by AV, @@ -239651,28 +239872,28 @@ zero ring (at least if its operations are internal binary operations). /\ W e. AssAlg ) -> ( ( y E ( A .x. X ) ) = ( ( y .^ A ) .x. ( y E X ) ) -> ( ( y + 1 ) E ( A .x. X ) ) = ( ( ( y + 1 ) .^ A ) .x. ( ( y + 1 ) E X ) ) ) ) ) $= - ( cv cn0 wcel wa casa co wceq caddc cmulr cfv cmnd crg assarng rngmgp syl - adantl adantr simpll clmod assalmod simplr lmodvscl syl3anc eqid mgpplusg - c1 mgpbas mulgnn0p1 oveq1 csca cbs simprr ccrg assasca crngrng 3syl simpl - wi a1i fveq2i syl6eq eleq2d biimpcd imp eqcomi mulgnn0cl simprlr syl13anc - assaass oveq2d eqcomd peano2nn0 w3a lmodvsass 3eqtrd simprll fveq2d oveqd - assaassr eqtrd oveq1d sylan9eqr exp31 ) AUAZUBUCZBCUCZLJUCZUDZKUEUCZUDZXD - BLDUFZEUFZXDBFUFZXDLEUFZDUFZUGZXDVFUHUFZXKEUFZXQBFUFZXQLEUFZDUFZUGXEXJUDZ - XPUDZXRXLXKKUIUJZUFZYAYCIUKUCZXEXKJUCZXRYEUGYBYFXPXJYFXEXIYFXHXIKULUCYFKU - MKISUNUOUPUPZUQXEXJXPURYBYGXPXJYGXEXJKUSUCZXFXGYGXIYIXHKUTUPZXFXGXIURXFXG - XIVABDGCJKLMNPOVBVCUPZUQJYDEIXDXKJKISMVGZTKYDISYDVDZVEZVHVCXPYBYEXOXKYDUF - ZYAXLXOXKYDVIYBYOXMXNXKYDUFZDUFZXMBKVJUJZUIUJZUFZXTDUFZYAYBXIXMYRVKUJZUCZ - XNJUCZYGYOYQUGXEXHXIVLZYBHUKUCZXEBUUBUCZUUCXJUUFXEXIUUFXHXIGVMUCGULUCUUFG - KNVNGVOGHQUNVPUPUPZXEXJVQZXJUUGXEXHXIUUGXFXIUUGVRXGXIXFUUGXICUUBBXICGVKUJ - ZUUBCUUJUGXIOVSGYRVKNVTWAWBWCUQWDUPZUUBFHXDBUUBGHQYRGVKGYRNWEVTVGRWFVCZYB - YFXEXGUUDYHUUIXEXFXGXIWGZJEIXDLYLTWFVCZYKXMUUBDYDYRJKXNXKMYRVDZUUBVDZPYMW - IWHYBYQXMBXNLYDUFZDUFZDUFXMBXTDUFZDUFZUUAYBYPUURXMDYBXIUUGUUDXGYPUURUGUUE - UUKUUNUUMBUUBDYDYRJKXNLMUUOUUPPYMWSWHWJYBUURUUSXMDYBUUQXTBDYBXTUUQYBYFXEX - GXTUUQUGYHUUIUUMJYDEIXDLYLTYNVHVCWKWJWJYBYIUUCUUGXTJUCZUUTUUAUGXJYIXEYJUP - UULUUKYBYFXQUBUCZXGUVAYHXEUVBXJXDWLUQUUMJEIXQLYLTWFVCYIUUCUUGUVAWMUDUUAUU - TXMBDYSYRUUBJKXTMUUOPUUPYSVDWNWKWHWOYBYTXSXTDYBXSYTYBXSXMBGUIUJZUFZYTYBUU - FXEXFXSUVDUGUUHUUIXEXFXGXIWPCUVCFHXDBCGHQOVGRGUVCHQUVCVDVEVHVCYBUVCYSXMBY - BGYRUIGYRUGYBNVSWQWRWTWKXAWOXBWTXC $. + ( cv cn0 wcel wa casa co wceq c1 caddc cmulr cfv crg assaring ringmgp syl + cmnd adantl adantr simpll clmod assalmod simplr lmodvscl syl3anc mgpplusg + mgpbas eqid mulgnn0p1 oveq1 csca cbs simprr ccrg assasca crngring 3syl wi + simpl a1i fveq2i syl6eq eleq2d biimpcd mulgnn0cl simprlr assaass syl13anc + imp eqcomi assaassr oveq2d eqcomd peano2nn0 lmodvsass 3eqtrd fveq2d oveqd + w3a simprll eqtrd oveq1d sylan9eqr exp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d A x y $. $d B x y $. $d E x y $. $d N x y $. $d V x y $. $d W x y $. $d X x y $. $d .x. x y $. $d .^ x y $. @@ -240417,21 +240638,21 @@ zero ring (at least if its operations are internal binary operations). Mario Carneiro, 29-Dec-2014.) $) psrmulcllem $p |- ( ph -> ( X .x. Y ) e. B ) $= ( co wcel cvv vk vx vy cv cle cofr wbr crab cfv cmin cof cmulr cmpt cbs - cgsu cmap cfn c0g eqid crg ccmn adantr rngcmn syl cmps reldmpsr elbasov - wf wa simpld psrbaglefi sylan ad2antrr psrelbas simpr breq1 elrab sylib - ffvelrnd simplr psrbagf syl2anc simprd psrbagcon syl13anc rngcl syl3anc - cn0 fmptd fvex a1i fdmfifsupp gsumcl ccnv cima ovex rabex eqeltri elmap - cn sylibr psrmulfval psrbas 3eltr4d ) AUACDUBUCUDZUAUDZUEUFZUGZUCCUHZUB - UDZIUIZXFXJUJUKRZJUIZDULUIZRZUMZUORZUMZDUNUIZCUPRZIJFRBACXSXRVHXRXTSAUA - CXQXSXRAXFCSZVIZXIXSXPDUQDURUIZXSUSZYCUSYBDUTSZDVASAYEYANVBDVCVDAHTSZYA - XIUQSAYFDTSZAIBSYFYGVIOIBEVEHDVFKLVGVDVJZUCCGXFHTQVKVLZYBUBXIXOXSXPYBXJ - XISZVIZYEXKXSSXMXSSXOXSSAYEYAYJNVMYKCXSXJIACXSIVHYAYJABCDEGHXSIKYDQLOVN - VMYKXJCSZXJXFXGUGZYKYJYLYMVIYBYJVOXHYMUCXJCXEXJXFXGVPVQVRZVJZVSYKCXSXLJ - ACXSJVHYAYJABCDEGHXSJKYDQLPVNVMYKXLCSZXLXFXGUGZYKYFYAHWHXJVHZYMYPYQVIAY - FYAYJYHVMZAYAYJVTYKYFYLYRYSYOCGXJHTQWAWBYKYLYMYNWCCGXFXJHTQWDWEVJVSXSDX - NXKXMYDXNUSZWFWGXPUSWIZYBXIXSXPTYCUUAYIYCTSYBDURWJWKWLWMXRUSWIXSCXRDUNW - JCGUDWNWTWOUQSZGWHHUPRZUHTQUUBGUUCWHHUPWPWQWRWSXAAUBUCBCDEFXNGUAIJHKLYT - MQOPXBABCDEGHXSTKYDQLYHXCXD $. + cgsu cmap wf cfn c0g eqid crg ccmn adantr ringcmn cmps reldmpsr elbasov + wa simpld psrbaglefi sylan ad2antrr psrelbas simpr breq1 elrab ffvelrnd + syl cn0 simplr psrbagf syl2anc simprd psrbagcon syl13anc ringcl syl3anc + sylib fmptd fvex a1i fdmfifsupp gsumcl ccnv cn cima rabex eqeltri elmap + ovex sylibr psrmulfval psrbas 3eltr4d ) AUACDUBUCUDZUAUDZUEUFZUGZUCCUHZ + UBUDZIUIZXFXJUJUKRZJUIZDULUIZRZUMZUORZUMZDUNUIZCUPRZIJFRBACXSXRUQXRXTSA + UACXQXSXRAXFCSZVHZXIXSXPDURDUSUIZXSUTZYCUTYBDVASZDVBSAYEYANVCDVDVRAHTSZ + YAXIURSAYFDTSZAIBSYFYGVHOIBEVEHDVFKLVGVRVIZUCCGXFHTQVJVKZYBUBXIXOXSXPYB + XJXISZVHZYEXKXSSXMXSSXOXSSAYEYAYJNVLYKCXSXJIACXSIUQYAYJABCDEGHXSIKYDQLO + VMVLYKXJCSZXJXFXGUGZYKYJYLYMVHYBYJVNXHYMUCXJCXEXJXFXGVOVPWHZVIZVQYKCXSX + LJACXSJUQYAYJABCDEGHXSJKYDQLPVMVLYKXLCSZXLXFXGUGZYKYFYAHVSXJUQZYMYPYQVH + AYFYAYJYHVLZAYAYJVTYKYFYLYRYSYOCGXJHTQWAWBYKYLYMYNWCCGXFXJHTQWDWEVIVQXS + DXNXKXMYDXNUTZWFWGXPUTWIZYBXIXSXPTYCUUAYIYCTSYBDUSWJWKWLWMXRUTWIXSCXRDU + NWJCGUDWNWOWPURSZGVSHUPRZUHTQUUBGUUCVSHUPWTWQWRWSXAAUBUCBCDEFXNGUAIJHKL + YTMQOPXBABCDEGHXSTKYDQLYHXCXD $. $} $( Closure of the power series multiplication operation. (Contributed by @@ -240525,13 +240746,13 @@ zero ring (at least if its operations are internal binary operations). (Contributed by Mario Carneiro, 29-Dec-2014.) $) psrvscacl $p |- ( ph -> ( X .x. F ) e. B ) $= ( vf wcel co cvv vx vy cv ccnv cn cima cfn cn0 cmap csn cxp cmulr cfv cof - crab wf crg eqid rngcl 3expb sylan fconst6g syl psrelbas ovex rabex inidm - wa a1i off fvex eqeltri elmap sylibr psrvsca cmps reldmpsr elbasov simpld - cbs psrbas 3eltr4d ) AQUCUDUEUFUGRZQUHGUISZUOZIUJUKZFCULUMZUNSZHWEUISZIFE - SBAWEHWHUPWHWIRAUAUBWEWEWEWGHHHWFFTTACUQRZUAUCZHRZUBUCZHRZVHWKWMWGSHRZNWJ - WLWNWOHCWGWKWMLWGURZUSUTVAAIHRWEHWFUPOWEIHVBVCABWECDQGHFJLWEURZMPVDWETRAW - CQWDUHGUIVEVFZVIZWSWEVGVJHWEWHHCVTUMTLCVTVKVLWRVMVNABWECDEWGQFGHIJKLMWPWQ - OPVOABWECDQGHTJLWQMAGTRZCTRZAFBRWTXAVHPFBDVPGCVQJMVRVCVSWAWB $. + crab wf crg wa eqid ringcl 3expb sylan fconst6g psrelbas ovex rabex inidm + syl a1i off cbs fvex eqeltri elmap sylibr psrvsca reldmpsr elbasov simpld + cmps psrbas 3eltr4d ) AQUCUDUEUFUGRZQUHGUISZUOZIUJUKZFCULUMZUNSZHWEUISZIF + ESBAWEHWHUPWHWIRAUAUBWEWEWEWGHHHWFFTTACUQRZUAUCZHRZUBUCZHRZURWKWMWGSHRZNW + JWLWNWOHCWGWKWMLWGUSZUTVAVBAIHRWEHWFUPOWEIHVCVHABWECDQGHFJLWEUSZMPVDWETRA + WCQWDUHGUIVEVFZVIZWSWEVGVJHWEWHHCVKUMTLCVKVLVMWRVNVOABWECDEWGQFGHIJKLMWPW + QOPVPABWECDQGHTJLWQMAGTRZCTRZAFBRWTXAURPFBDVTGCVQJMVRVHVSWAWB $. $} ${ @@ -240655,48 +240876,48 @@ zero ring (at least if its operations are internal binary operations). $d j k n r s t u v w x y z ph $. $d g h j k r w x y V $. $d k x y .x. $. $d f g h j k n x Z $. $d r s t x y z S $. $d x y .1. $. $d j k x .X. $. $d f g h j k n u v w x Y $. - psrrng.s $e |- S = ( I mPwSer R ) $. - psrrng.i $e |- ( ph -> I e. V ) $. - psrrng.r $e |- ( ph -> R e. Ring ) $. + psrring.s $e |- S = ( I mPwSer R ) $. + psrring.i $e |- ( ph -> I e. V ) $. + psrring.r $e |- ( ph -> R e. Ring ) $. $( The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.) $) psrlmod $p |- ( ph -> S e. LMod ) $= ( vf vr cfv eqidd wcel syl cv eqid co csn cvv psrvsca vx vy vz cbs cplusg - vs cvsca cmulr cur crg psrsca cgrp rnggrp psrgrp w3a 3ad2ant1 simp2 simp3 - vt psrvscacl wa ccnv cima cfn cn0 cmap crab cxp cof ovex rabex a1i simpr1 - cn fconst6g simpr2 psrelbas simpr3 wceq adantr rngdi caofdi psradd oveq2d - wf sylan oveq12d 3eqtr4d psraddcl 3adant3r3 rngdir caofdir ofc12 3eqtr2rd - oveq1d rngacl syl3anc rngass caofass rngcl rngidcl simpr rnglidm caofid0l - eqtrd islmodd ) AUAUBUCBUDKZCUEKZBUEKZCUGKZBUHKZBUIKZBCUDKZCAXMLAXHLABCDE - UJFGHUKAXJLAXGLAXILAXKLAXLLHABCDEFGABUJMZBULMZHBUMNZUNAUAOZXGMZUBOZXMMZUO - XMBCXJXSDXGXQFXJPZXGPZXMPZAXRXNXTHUPAXRXTUQAXRXTURUTZAXRXTUCOZXMMZUOZVAZI - OVBVNVCVDMZIVEDVFQZVGZXQRVHZXSYEXHQZXKVIZQZXQXSXJQZXQYEXJQZXIVIZQZXQYMXJQ - YPYQXHQYHYLXSYEYRQZYNQYLXSYNQZYLYEYNQZYRQYOYSYHJUFUSYKXIXGXKYLXSYEXGXISYK - SMZYHYIIYJVEDVFVJVKZVLYHXRYKXGYLWEZAXRXTYFVMZYKXQXGVOZNYHXMYKBCIDXGXSFYBY - KPZYCAXRXTYFVPZVQYHXMYKBCIDXGYEFYBUUHYCAXRXTYFVRZVQYHXNJOZXGMZUFOZXGMUSOZ - XGMUOZUUKUUMUUNXIQXKQUUKUUMXKQZUUKUUNXKQZXIQVSAXNYGHVTZXGXIBXKUUKUUMUUNYB - 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( NN0 ^m I ) | ( `' f " NN ) e. Fin } $. @@ -240708,12 +240929,12 @@ zero ring (at least if its operations are internal binary operations). $( The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.) $) psr1cl $p |- ( ph -> U e. B ) $= - ( cbs cfv cmap co wf wcel cv cc0 csn cxp wceq cif crg eqid rngidcl ifcl - rng0cl syl2anc syl adantr fmptd fvex ccnv cn cima cfn cn0 crab cvv ovex - rabex eqeltri elmap sylibr psrbas eleqtrrd ) AGEUAUBZDUCUDZCADVQGUEGVRU - FABDBUGZJUHUIUJUKZHLULZVQGAWAVQUFZVSDUFAEUMUFZWBOWCHVQUFLVQUFWBVQEHVQUN - ZRUOVQELWDQUQVTHLVQUPURUSUTSVAVQDGEUAVBDIUGVCVDVEVFUFZIVGJUCUDZVHVIPWEI - WFVGJUCVJVKVLVMVNACDEFIJVQKMWDPTNVOVP $. + ( cbs cfv cmap co wf wcel cv cc0 csn cxp wceq cif eqid ringidcl ring0cl + crg ifcl syl2anc syl adantr fmptd fvex ccnv cima cfn cn0 crab cvv rabex + cn ovex eqeltri elmap sylibr psrbas eleqtrrd ) AGEUAUBZDUCUDZCADVQGUEGV + RUFABDBUGZJUHUIUJUKZHLULZVQGAWAVQUFZVSDUFAEUPUFZWBOWCHVQUFLVQUFWBVQEHVQ + UMZRUNVQELWDQUOVTHLVQUQURUSUTSVAVQDGEUAVBDIUGVCVJVDVEUFZIVFJUCUDZVGVHPW + EIWFVFJUCVKVIVLVMVNACDEFIJVQKMWDPTNVOVP $. psrlidm.t $e |- .x. = ( .r ` S ) $. psrlidm.x $e |- ( ph -> X e. B ) $. @@ -240725,46 +240946,46 @@ zero ring (at least if its operations are internal binary operations). cv wa cle cofr wbr crab cmin cof cmpt cgsu cc0 csn cxp adantr psrmulval cmulr simpr cres wss wceq fconstmpt fczpsrbag syl5eqel wral cn0 psrbagf sylan ffvelrnda nn0ge0d ralrimiva 0nn0 fconst6 mp1i inidm a1i fvconst2g - eqidd ofrfval mpbird breq1 elrab sylanbrc resmpt oveq2d cvv ccmn rngcmn - snssd crg ccnv cima cfn cmap ovex rab2ex ad2antrr simpld syldan syl2anc - sylib simprd psrbagcon syl13anc ffvelrnd rngcl syl3anc fmptd cif sylan2 - cn cdif cur fvex eqeltri c0g fvmpt wn adantl eqtrd fveq2d oveq12d eqeq1 - eldifi ifbid ifex eldifn elsn sylnib iffalse oveq1d rnglz suppss2 csupp - wfun cfsupp rabex2 mptrabex funmpt suppssfifsupp syl32anc rngmnd iftrue - snfi gsumres nn0cn subid1d caofid0r rnglidm eqeltrd fveq2 oveq2 3eqtr3d - cmnd gsumsn 3eqtrd eqfnfvd ) AUEDHMGUHZMADEUIUJZUVPUKUVPDULACDEFJKUVQUV - POUVQUMZRUBACEFGKHMOUBUCQABCDEFHIJKLNOPQRSTUAUBUNZUDUOUPDUVQUVPUQURADUV - QMUKZMDULACDEFJKUVQMOUVRRUBUDUPZDUVQMUQURAUEUTZDUSZVAZUWBUVPUJEUFUGUTZU - 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(Contributed by Mario Carneiro, 29-Dec-2014.) 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(Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by AV, @@ -240824,47 +241045,47 @@ zero ring (at least if its operations are internal binary operations). 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(Contributed by Mario Carneiro, 29-Dec-2014.) Obsolete version of @@ -240874,48 +241095,48 @@ zero ring (at least if its operations are internal binary operations). 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Z ) ) ) $= ( vx vk vg vj vh vn vz co cbs cfv wf wfn psrmulcl psrelbas ffn syl cv eqid wcel wa cle cofr wbr crab cmin cmulr cmpt cgsu adantr simpr ccmn - cof crg rngcmn ad2antrr breq1 elrab sylib simpld ad3antrrr cn0 simplr - ffvelrnd psrbagf syl2anc simprd psrbagcon syl13anc rngcl syl3anc wceq - anasss fveq2 fveq2d oveq12d oveq2d psrass1lem psrmulval oveq1d cplusg - cfn c0g psrbaglefi cvv fvex a1i fsuppmptdm gsummulc1 rngass ffvelrnda - oveq2 sylan cc nn0cn nnncan2 syl3an mpteq2dva feqmptd offval2 3eqtr4d - ovex eqtr4d 3eqtr2d wfun w3a csupp wss cfsupp ccnv cn cima cmap mptex - rab2ex funmpt 3pm3.2i cdm suppssdm dmmptss syl12anc gsummulc2 eqfnfvd - sstri suppssfifsupp ) AUBCJKFUIZLFUIZJKLFUIZFUIZACDUJUKZUUGULUUGCUMAB - CDEGHUUJUUGMUUJUSZPRABDEFHUUFLMRQOABDEFHJKMRQOSTUNZUAUNUOCUUJUUGUPUQA - CUUJUUIULUUICUMABCDEGHUUJUUIMUUKPRABDEFHJUUHMRQOSABDEFHKLMRQOTUAUNZUN - UOCUUJUUIUPUQAUBURZCUTZVAZDUCUDURZUUNVBVCZVDZUDCVEZUCURZUUFUKZUUNUVAV - 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(Contributed by Mario Carneiro, @@ -241047,38 +241268,38 @@ zero ring (at least if its operations are internal binary operations). cmulr wa wceq eqid psradd fveq1d ad2antrr ssrab2 simpr sseldi cvv cbs wf wfn psrelbas ffn syl ccnv cima cfn cn0 cmap ovex rabex eqeltri a1i inidm eqidd ofval mpdan eqtrd oveq1d crg ffvelrnd psrbagconcl syl3anc - simplr rngdir syl13anc mpteq2dva psrbaglefi sylan rngcl eqtr4d oveq2d - cn offval2 c0g ccmn adantr rngcmn fmptd wfun wss cfsupp rabexg mptexg - csupp funmpt fvex cdm suppssdm dmmptss suppssfifsupp syl32anc gsumadd - sstri cgrp rnggrp psraddcl psrmulfval psrmulcl 3eqtr4d ) AUDCEUEUFUGU - DUGZUHUIUJZUFCUKZUEUGZKLDULZUMZYMYPUNUOULZMUMZEUTUMZULZUPZUQULZUPUDCE - UEYOYPKUMZYTUUAULZUPZUQULZEUEYOYPLUMZYTUUAULZUPZUQULZEURUMZULZUPZYQMG - ULKMGULZLMGULZDULZAUDCUUDUUNAYMCUSZVAZUUDEUUGUUKUUMUOZULZUQULUUNUUTUU - CUVBEUQUUTUUCUEYOUUFUUJUUMULZUPUVBUUTUEYOUUBUVCUUTYPYOUSZVAZUUBUUEUUI - UUMULZYTUUAULZUVCUVEYRUVFYTUUAUVEYRYPKLUVAULZUMZUVFAYRUVIVBUUSUVDAYPY - QUVHABUUMDEFIKLNSUUMVCZUCTUAVDVEVFUVEYPCUSZUVIUVFVBUVEYOCYPYNUFCVGZUU - TUVDVHZVIZUVECCUUEUUIUUMCKLVJVJYPUVECEVKUMZKVLZKCVMAUVPUUSUVDABCEFHIU - VOKNUVOVCZQSTVNVFZCUVOKVOVPUVECUVOLVLZLCVMAUVSUUSUVDABCEFHIUVOLNUVQQS - UAVNVFZCUVOLVOVPCVJUSZUVECHUGVQXEVRVSUSZHVTIWAULZUKVJQUWBHUWCVTIWAWBW - CWDZWEZUWECWFUVEUVKVAZUUEWGUWFUUIWGWHWIWJWKUVEEWLUSZUUEUVOUSZUUIUVOUS - ZYTUVOUSZUVGUVCVBAUWGUUSUVDPVFZUVECUVOYPKUVRUVNWMZUVECUVOYPLUVTUVNWMZ - UVECUVOYSMACUVOMVLUUSUVDABCEFHIUVOMNUVQQSUBVNVFUVEYOCYSUVLUVEIJUSZUUS - UVDYSYOUSAUWNUUSUVDOVFAUUSUVDWPUVMUFCYOHYMIJYPQYOVCWNWOVIWMZUVOUUMEUU - AUUEUUIYTUVQUVJUUAVCZWQWRWJWSUUTUEYOUUFUUJUUMUUGUUKVSUVOUVOAUWNUUSYOV - SUSZOUFCHYMIJQWTXAZUVEUWGUWHUWJUUFUVOUSUWKUWLUWOUVOEUUAUUEYTUVQUWPXBW - OZUVEUWGUWIUWJUUJUVOUSUWKUWMUWOUVOEUUAUUIYTUVQUWPXBWOZUUTUUGWGUUTUUKW - GXFXCXDUUTYOUVOUUMUUGEUUKVSEXGUMZUVQUXAVCUVJUUTUWGEXHUSAUWGUUSPXIEXJV - PUWRUUTUEYOUUFUVOUUGUWSUUGVCZXKUUTUEYOUUJUVOUUKUWTUUKVCZXKUUTUUGVJUSZ - UUGXLZUXAVJUSZUWQUUGUXAXQULZYOXMZUUGUXAXNUJUUTYOVJUSZUXDUUTUWAUXIUWAU - UTUWDWEYNUFCVJXOVPZUEYOUUFVJXPVPUXEUUTUEYOUUFXRWEUXFUUTEXGXSWEZUWRUXH - UUTUXGUUGXTYOUUGUXAYAUEYOUUFUUGUXBYBYFWEYOUUGVJVJUXAYCYDUUTUUKVJUSZUU - KXLZUXFUWQUUKUXAXQULZYOXMZUUKUXAXNUJUUTUXIUXLUXJUEYOUUJVJXPVPUXMUUTUE - YOUUJXRWEUXKUWRUXOUUTUXNUUKXTYOUUKUXAYAUEYOUUJUUKUXCYBYFWEYOUUKVJVJUX - AYCYDYEWJWSAUEUFBCEFGUUAHUDYQMINSUWPRQABDEFIKLNSUCAUWGEYGUSPEYHVPTUAY - IUBYJAUURUUPUUQUVAULUUOABUUMDEFIUUPUUQNSUVJUCABEFGIKMNSRPTUBYKABEFGIL - MNSRPUAUBYKVDAUDCUUHUULUUMUUPUUQVJVJVJUWAAUWDWEUUHVJUSUUTEUUGUQWBWEUU - LVJUSUUTEUUKUQWBWEAUEUFBCEFGUUAHUDKMINSUWPRQTUBYJAUEUFBCEFGUUAHUDLMIN - SUWPRQUAUBYJXFWJYL $. + cn simplr ringdir syl13anc mpteq2dva psrbaglefi ringcl offval2 eqtr4d + sylan oveq2d c0g ccmn adantr ringcmn fmptd csupp cfsupp rabexg mptexg + wfun wss funmpt fvex cdm suppssdm dmmptss sstri suppssfifsupp gsumadd + syl32anc cgrp ringgrp psraddcl psrmulfval psrmulcl 3eqtr4d ) AUDCEUEU + FUGUDUGZUHUIUJZUFCUKZUEUGZKLDULZUMZYMYPUNUOULZMUMZEUTUMZULZUPZUQULZUP + UDCEUEYOYPKUMZYTUUAULZUPZUQULZEUEYOYPLUMZYTUUAULZUPZUQULZEURUMZULZUPZ + YQMGULKMGULZLMGULZDULZAUDCUUDUUNAYMCUSZVAZUUDEUUGUUKUUMUOZULZUQULUUNU + UTUUCUVBEUQUUTUUCUEYOUUFUUJUUMULZUPUVBUUTUEYOUUBUVCUUTYPYOUSZVAZUUBUU + EUUIUUMULZYTUUAULZUVCUVEYRUVFYTUUAUVEYRYPKLUVAULZUMZUVFAYRUVIVBUUSUVD + AYPYQUVHABUUMDEFIKLNSUUMVCZUCTUAVDVEVFUVEYPCUSZUVIUVFVBUVEYOCYPYNUFCV + GZUUTUVDVHZVIZUVECCUUEUUIUUMCKLVJVJYPUVECEVKUMZKVLZKCVMAUVPUUSUVDABCE + FHIUVOKNUVOVCZQSTVNVFZCUVOKVOVPUVECUVOLVLZLCVMAUVSUUSUVDABCEFHIUVOLNU + VQQSUAVNVFZCUVOLVOVPCVJUSZUVECHUGVQWPVRVSUSZHVTIWAULZUKVJQUWBHUWCVTIW + AWBWCWDZWEZUWECWFUVEUVKVAZUUEWGUWFUUIWGWHWIWJWKUVEEWLUSZUUEUVOUSZUUIU + VOUSZYTUVOUSZUVGUVCVBAUWGUUSUVDPVFZUVECUVOYPKUVRUVNWMZUVECUVOYPLUVTUV + NWMZUVECUVOYSMACUVOMVLUUSUVDABCEFHIUVOMNUVQQSUBVNVFUVEYOCYSUVLUVEIJUS + ZUUSUVDYSYOUSAUWNUUSUVDOVFAUUSUVDWQUVMUFCYOHYMIJYPQYOVCWNWOVIWMZUVOUU + MEUUAUUEUUIYTUVQUVJUUAVCZWRWSWJWTUUTUEYOUUFUUJUUMUUGUUKVSUVOUVOAUWNUU + SYOVSUSZOUFCHYMIJQXAXEZUVEUWGUWHUWJUUFUVOUSUWKUWLUWOUVOEUUAUUEYTUVQUW + PXBWOZUVEUWGUWIUWJUUJUVOUSUWKUWMUWOUVOEUUAUUIYTUVQUWPXBWOZUUTUUGWGUUT + UUKWGXCXDXFUUTYOUVOUUMUUGEUUKVSEXGUMZUVQUXAVCUVJUUTUWGEXHUSAUWGUUSPXI + EXJVPUWRUUTUEYOUUFUVOUUGUWSUUGVCZXKUUTUEYOUUJUVOUUKUWTUUKVCZXKUUTUUGV + JUSZUUGXPZUXAVJUSZUWQUUGUXAXLULZYOXQZUUGUXAXMUJUUTYOVJUSZUXDUUTUWAUXI + UWAUUTUWDWEYNUFCVJXNVPZUEYOUUFVJXOVPUXEUUTUEYOUUFXRWEUXFUUTEXGXSWEZUW + RUXHUUTUXGUUGXTYOUUGUXAYAUEYOUUFUUGUXBYBYCWEYOUUGVJVJUXAYDYFUUTUUKVJU + SZUUKXPZUXFUWQUUKUXAXLULZYOXQZUUKUXAXMUJUUTUXIUXLUXJUEYOUUJVJXOVPUXMU + UTUEYOUUJXRWEUXKUWRUXOUUTUXNUUKXTYOUUKUXAYAUEYOUUJUUKUXCYBYCWEYOUUKVJ + VJUXAYDYFYEWJWTAUEUFBCEFGUUAHUDYQMINSUWPRQABDEFIKLNSUCAUWGEYGUSPEYHVP + TUAYIUBYJAUURUUPUUQUVAULUUOABUUMDEFIUUPUUQNSUVJUCABEFGIKMNSRPTUBYKABE + FGILMNSRPUAUBYKVDAUDCUUHUULUUMUUPUUQVJVJVJUWAAUWDWEUUHVJUSUUTEUUGUQWB + WEUULVJUSUUTEUUKUQWBWEAUEUFBCEFGUUAHUDKMINSUWPRQTUBYJAUEUFBCEFGUUAHUD + LMINSUWPRQUAUBYJXCWJYL $. $} ${ @@ -241093,29 +241314,29 @@ zero ring (at least if its operations are internal binary operations). = ( A .x. ( X .X. Y ) ) ) $= ( vk vx vy cv cle cofr wbr crab cfv cmin cof cmulr cmpt cgsu wcel cbs co adantr syl6eleq ad2antrr ssrab2 simpr sseldi psrvscaval oveq1d crg - eqid wceq psrelbas ffvelrnd simplr psrbagconcl syl3anc syl13anc eqtrd - wa rngass mpteq2dva oveq2d cplusg cfn c0g psrbaglefi sylan rngcl wfun - cvv w3a csupp wss cfsupp ccnv cn cima cn0 cmap rabex2 mptrabex funmpt - ovex fvex 3pm3.2i a1i suppssdm sstri suppssfifsupp syl12anc gsummulc2 - cdm dmmptss psrvscacl psrmulfval csn psrmulcl psrvsca offval2 3eqtr4d - cxp fconstmpt ) AUFDEUGUHUIUFUIZUJUKULZUHDUMZUGUIZBMGVBZUNZYEYHUOUPVB - ZNUNZEUQUNZVBZURZUSVBZURUFDBEUGYGYHMUNZYLYMVBZURZUSVBZYMVBZURZYINHVBB - MNHVBZGVBZAUFDYPUUAAYEDUTZWAZYPEUGYGBYRYMVBZURZUSVBUUAUUFYOUUHEUSUUFU - GYGYNUUGUUFYHYGUTZWAZYNBYQYMVBZYLYMVBZUUGUUJYJUUKYLYMUUJCDEFGYMIMJEVA - UNZBYHOUDUUMVLZTYMVLZRUUFBUUMUTZUUIUUFBKUUMABKUTUUEUEVCZUCVDZVCZAMCUT - UUEUUIUAVEZUUJYGDYHYFUHDVFZUUFUUIVGZVHZVIVJUUJEVKUTZUUPYQUUMUTZYLUUMU - TZUULUUGVMAUVDUUEUUIQVEZUUSUUJDUUMYHMUUJCDEFIJUUMMOUUNRTUUTVNUVCVOZUU - JDUUMYKNUUJCDEFIJUUMNOUUNRTANCUTUUEUUIUBVEVNUUJYGDYKUVAUUJJLUTZUUEUUI - YKYGUTAUVIUUEUUIPVEAUUEUUIVPUVBUHDYGIYEJLYHRYGVLVQVRVHVOZUUMEYMBYQYLU - UNUUOWBVSVTWCWDUUFYGUUMEWEUNZEYMUGWFYRBEWGUNZUUNUVLVLUVKVLUUOAUVDUUEQ - VCAUVIUUEYGWFUTZPUHDIYEJLRWHWIZUURUUJUVDUVEUVFYRUUMUTUVGUVHUVJUUMEYMY - QYLUUNUUOWJVRUUFYSWLUTZYSWKZUVLWLUTZWMZUVMYSUVLWNVBZYGWOZYSUVLWPULUVR - UUFUVOUVPUVQYFUGUHDYRWLIUIWQWRWSWFUTIWTJXAVBDWLRWTJXAXEXBZXCUGYGYRXDE - WGXFXGXHUVNUVTUUFUVSYSXNYGYSUVLXIUGYGYRYSYSVLXOXJXHYGYSWLWLUVLXKXLXMV - TWCAUGUHCDEFHYMIUFYINJOTUUOSRACEFGMJKBOUDUCTQUEUAXPUBXQAUUDDBXRYCZUUC - YMUPVBUUBACDEFGYMIUUCJKBOUDUCTUUORUEACEFHJMNOTSQUAUBXSXTAUFDBYTYMUWBU - UCWLKWLDWLUTAUWAXHUUQYTWLUTUUFEYSUSXEXHUWBUFDBURVMAUFDBYDXHAUGUHCDEFH - YMIUFMNJOTUUOSRUAUBXQYAVTYB $. + wa eqid psrelbas ffvelrnd simplr psrbagconcl syl3anc ringass syl13anc + wceq eqtrd mpteq2dva oveq2d cplusg cfn c0g psrbaglefi ringcl cvv wfun + sylan w3a csupp wss cfsupp ccnv cn cima cn0 cmap ovex rabex2 mptrabex + funmpt fvex 3pm3.2i a1i suppssdm dmmptss sstri suppssfifsupp syl12anc + cdm gsummulc2 psrvscacl psrmulfval csn cxp psrmulcl psrvsca fconstmpt + offval2 3eqtr4d ) AUFDEUGUHUIUFUIZUJUKULZUHDUMZUGUIZBMGVBZUNZYEYHUOUP + VBZNUNZEUQUNZVBZURZUSVBZURUFDBEUGYGYHMUNZYLYMVBZURZUSVBZYMVBZURZYINHV + BBMNHVBZGVBZAUFDYPUUAAYEDUTZVLZYPEUGYGBYRYMVBZURZUSVBUUAUUFYOUUHEUSUU + FUGYGYNUUGUUFYHYGUTZVLZYNBYQYMVBZYLYMVBZUUGUUJYJUUKYLYMUUJCDEFGYMIMJE + VAUNZBYHOUDUUMVMZTYMVMZRUUFBUUMUTZUUIUUFBKUUMABKUTUUEUEVCZUCVDZVCZAMC + UTUUEUUIUAVEZUUJYGDYHYFUHDVFZUUFUUIVGZVHZVIVJUUJEVKUTZUUPYQUUMUTZYLUU + MUTZUULUUGWAAUVDUUEUUIQVEZUUSUUJDUUMYHMUUJCDEFIJUUMMOUUNRTUUTVNUVCVOZ + UUJDUUMYKNUUJCDEFIJUUMNOUUNRTANCUTUUEUUIUBVEVNUUJYGDYKUVAUUJJLUTZUUEU + UIYKYGUTAUVIUUEUUIPVEAUUEUUIVPUVBUHDYGIYEJLYHRYGVMVQVRVHVOZUUMEYMBYQY + LUUNUUOVSVTWBWCWDUUFYGUUMEWEUNZEYMUGWFYRBEWGUNZUUNUVLVMUVKVMUUOAUVDUU + EQVCAUVIUUEYGWFUTZPUHDIYEJLRWHWLZUURUUJUVDUVEUVFYRUUMUTUVGUVHUVJUUMEY + MYQYLUUNUUOWIVRUUFYSWJUTZYSWKZUVLWJUTZWMZUVMYSUVLWNVBZYGWOZYSUVLWPULU + VRUUFUVOUVPUVQYFUGUHDYRWJIUIWQWRWSWFUTIWTJXAVBDWJRWTJXAXBXCZXDUGYGYRX + EEWGXFXGXHUVNUVTUUFUVSYSXNYGYSUVLXIUGYGYRYSYSVMXJXKXHYGYSWJWJUVLXLXMX + OWBWCAUGUHCDEFHYMIUFYINJOTUUOSRACEFGMJKBOUDUCTQUEUAXPUBXQAUUDDBXRXSZU + UCYMUPVBUUBACDEFGYMIUUCJKBOUDUCTUUORUEACEFHJMNOTSQUAUBXTYAAUFDBYTYMUW + BUUCWJKWJDWJUTAUWAXHUUQYTWJUTUUFEYSUSXBXHUWBUFDBURWAAUFDBYBXHAUGUHCDE + FHYMIUFMNJOTUUOSRUAUBXQYCWBYD $. $} psrcom.c $e |- ( ph -> R e. CRing ) $. @@ -241123,43 +241344,43 @@ zero ring (at least if its operations are internal binary operations). Carneiro, 7-Jan-2015.) $) psrcom $p |- ( ph -> ( X .X. Y ) = ( Y .X. X ) ) $= ( vx vk vg vj vz cv cle cofr wbr crab cfv cmin cof cmulr cmpt cgsu wcel - co wa ccom cbs cfn c0g eqid ccmn crg rngcmn syl adantr psrbaglefi sylan - ad2antrr wf psrelbas simpr breq1 elrab sylib simpld ffvelrnd cn0 simplr - psrbagf syl2anc simprd psrbagcon syl13anc rngcl syl3anc fmptd cvv csupp - wfun wss cfsupp ccnv cn cima cmap ovex eqeltri a1i rabexg mptexg funmpt - rabex fvex suppssdm dmmptss suppssfifsupp syl32anc psrbagconf1o gsumf1o - cdm sstri wf1o psrbagconcl eqidd wceq fveq2 oveq2 fveq2d oveq12d fmptco - ffvelrnda cc nn0cn syl2an mpteq2dva feqmptd offval2 3eqtr4d oveq2d ccrg - nncan crngcom eqtrd psrmulfval ) AUACDUBUCUFZUAUFZUGUHZUIZUCCUJZUBUFZJU - KZYTUUDULUMZURZKUKZDUNUKZURZUOZUPURZUOUACDUDUUCUDUFZKUKZYTUUMUUFURZJUKZ - UUIURZUOZUPURZUOJKFURKJFURAUACUULUUSAYTCUQZUSZUULDUUKUDUUCUUOUOZUTZUPUR - UUSUVAUUCDVAUKZUUCUUKDUVBVBDVCUKZUVDVDZUVEVDADVEUQZUUTADVFUQZUVGNDVGVHV - IAHIUQZUUTUUCVBUQZMUCCGYTHIOVJVKZUVAUBUUCUUJUVDUUKUVAUUDUUCUQZUSZUVHUUE - UVDUQUUHUVDUQUUJUVDUQAUVHUUTUVLNVLUVMCUVDUUDJACUVDJVMZUUTUVLABCDEGHUVDJ - 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XIWAUQZUXJWAUQZUXLUXJXSZUXBHWAUXHYTUVAHWAYTVMZUXAAUVIUUTUXQMCGYTHIOWCVK - VIZYEZUXBHWAUXHUUMUXBUVIUUMCUQZHWAUUMVMZUXCUXBUXTUUMYTUUAUIZUXBUXAUXTUY - BUSUXEUUBUYBUCUUMCYSUUMYTUUAVPVQVRZVSZCGUUMHIOWCWDZYEZUXNUXIYFUQUXJYFUQ - UXPUXOUXIYGUXJYGUXIUXJYOYHWDYIUXBUEHUXIUXKULYTUUOIWAWKUXCUXSUXKWKUQUXMU - XIUXJULWTXBUXBUEHWAYTUXRYJZUXBUEHUXIUXJULYTUUMIWAWAUXCUXSUYFUYGUXBUEHWA - UUMUYEYJZYKYKUYHYLYBYMUXBDYNUQZUUPUVDUQUUNUVDUQUXGUUQXSAUYIUUTUXATVLUXB - CUVDUUOJAUVNUUTUXAUVOVLUXBUUOCUQZUUOYTUUAUIZUXBUVIUUTUYAUYBUYJUYKUSUXCU - XDUYEUXBUXTUYBUYCWECGYTUUMHIOWFWGVSVTUXBCUVDUUMKAUVTUUTUXAUWAVLUYDVTUVD - DUUIUUPUUNUVFUWFYPWIYQYIYQYMYQYIAUBUCBCDEFUUIGUAJKHLQUWFPORSYRAUDUCBCDE - FUUIGUAKJHLQUWFPOSRYRYL $. + co wa ccom cbs cfn c0g eqid ccmn crg ringcmn adantr psrbaglefi ad2antrr + syl sylan wf psrelbas simpr breq1 elrab sylib simpld cn0 simplr psrbagf + ffvelrnd syl2anc simprd psrbagcon syl13anc syl3anc fmptd cvv wfun csupp + ringcl cfsupp ccnv cn cima cmap ovex rabex eqeltri rabexg mptexg funmpt + wss a1i fvex suppssdm dmmptss sstri suppssfifsupp syl32anc psrbagconf1o + cdm wf1o gsumf1o psrbagconcl eqidd fveq2 oveq2 fveq2d oveq12d ffvelrnda + wceq fmptco nn0cn nncan syl2an mpteq2dva feqmptd offval2 3eqtr4d oveq2d + cc ccrg crngcom eqtrd psrmulfval ) AUACDUBUCUFZUAUFZUGUHZUIZUCCUJZUBUFZ + JUKZYTUUDULUMZURZKUKZDUNUKZURZUOZUPURZUOUACDUDUUCUDUFZKUKZYTUUMUUFURZJU + KZUUIURZUOZUPURZUOJKFURKJFURAUACUULUUSAYTCUQZUSZUULDUUKUDUUCUUOUOZUTZUP + URUUSUVAUUCDVAUKZUUCUUKDUVBVBDVCUKZUVDVDZUVEVDADVEUQZUUTADVFUQZUVGNDVGV + KVHAHIUQZUUTUUCVBUQZMUCCGYTHIOVIVLZUVAUBUUCUUJUVDUUKUVAUUDUUCUQZUSZUVHU + UEUVDUQUUHUVDUQUUJUVDUQAUVHUUTUVLNVJUVMCUVDUUDJACUVDJVMZUUTUVLABCDEGHUV + DJLUVFOQRVNZVJUVMUUDCUQZUUDYTUUAUIZUVMUVLUVPUVQUSUVAUVLVOUUBUVQUCUUDCYS + UUDYTUUAVPVQVRZVSZWCUVMCUVDUUGKACUVDKVMZUUTUVLABCDEGHUVDKLUVFOQSVNZVJUV + MUUGCUQZUUGYTUUAUIZUVMUVIUUTHVTUUDVMZUVQUWBUWCUSAUVIUUTUVLMVJZAUUTUVLWA + UVMUVIUVPUWDUWEUVSCGUUDHIOWBWDUVMUVPUVQUVRWECGYTUUDHIOWFWGVSWCUVDDUUIUU + EUUHUVFUUIVDZWMWHUUKVDZWIUVAUUKWJUQZUUKWKZUVEWJUQZUVJUUKUVEWLURZUUCXEZU + UKUVEWNUIUVAUUCWJUQZUWHUVACWJUQZUWMUWNUVACGUFWOWPWQVBUQZGVTHWRURZUJWJOU + WOGUWPVTHWRWSWTXAXFUUBUCCWJXBVKUBUUCUUJWJXCVKUWIUVAUBUUCUUJXDXFUWJUVADV + CXGXFUVKUWLUVAUWKUUKXNUUCUUKUVEXHUBUUCUUJUUKUWGXIXJXFUUCUUKWJWJUVEXKXLA + UVIUUTUUCUUCUVBXOMUDUCCUUCGYTHIOUUCVDZXMVLXPUVAUVCUURDUPUVAUVCUDUUCUUPY + TUUOUUFURZKUKZUUIURZUOUURUVAUDUBUUCUUCUUOUUJUWTUVBUUKUVAUUMUUCUQZUSZUVI + UUTUXAUUOUUCUQAUVIUUTUXAMVJZAUUTUXAWAZUVAUXAVOZUCCUUCGYTHIUUMOUWQXQWHUV + AUVBXRUVAUUKXRUUDUUOYDZUUEUUPUUHUWSUUIUUDUUOJXSUXFUUGUWRKUUDUUOYTUUFXTY + AYBYEUVAUDUUCUWTUUQUXBUWTUUPUUNUUIURZUUQUXBUWSUUNUUPUUIUXBUWRUUMKUXBUEH + UEUFZYTUKZUXIUXHUUMUKZULURZULURZUOUEHUXJUOUWRUUMUXBUEHUXLUXJUXBUXHHUQUS + ZUXIVTUQZUXJVTUQZUXLUXJYDZUXBHVTUXHYTUVAHVTYTVMZUXAAUVIUUTUXQMCGYTHIOWB + VLVHZYCZUXBHVTUXHUUMUXBUVIUUMCUQZHVTUUMVMZUXCUXBUXTUUMYTUUAUIZUXBUXAUXT + UYBUSUXEUUBUYBUCUUMCYSUUMYTUUAVPVQVRZVSZCGUUMHIOWBWDZYCZUXNUXIYNUQUXJYN + UQUXPUXOUXIYFUXJYFUXIUXJYGYHWDYIUXBUEHUXIUXKULYTUUOIVTWJUXCUXSUXKWJUQUX + MUXIUXJULWSXFUXBUEHVTYTUXRYJZUXBUEHUXIUXJULYTUUMIVTVTUXCUXSUYFUYGUXBUEH + VTUUMUYEYJZYKYKUYHYLYAYMUXBDYOUQZUUPUVDUQUUNUVDUQUXGUUQYDAUYIUUTUXATVJU + XBCUVDUUOJAUVNUUTUXAUVOVJUXBUUOCUQZUUOYTUUAUIZUXBUVIUUTUYAUYBUYJUYKUSUX + CUXDUYEUXBUXTUYBUYCWECGYTUUMHIOWFWGVSWCUXBCUVDUUMKAUVTUUTUXAUWAVJUYDWCU + VDDUUIUUPUUNUVFUWFYPWHYQYIYQYMYQYIAUBUCBCDEFUUIGUAJKHLQUWFPORSYRAUDUCBC + DEFUUIGUAKJHLQUWFPOSRYRYL $. psrass.k $e |- K = ( Base ` R ) $. psrass.n $e |- .x. = ( .s ` S ) $. @@ -241171,48 +241392,48 @@ zero ring (at least if its operations are internal binary operations). ( vk vx vy vu vv vw co wceq psrass23l cv cle cofr wbr crab cfv cmin cof cmulr cmpt cgsu wcel wa cbs eqid adantr syl6eleq ad2antrr ssrab2 simplr simpr psrbagconcl sseldi psrvscaval oveq2d psrelbas ffvelrnd ccrg 3expb - syl3anc crngcom sylan crg w3a rngass caov12d eqtrd mpteq2dva cplusg cfn - c0g psrbaglefi rngcl cvv wfun csupp wss cfsupp ccnv cn cima cmap rabex2 - cn0 ovex mptrabex funmpt 3pm3.2i a1i cdm suppssdm dmmptss suppssfifsupp - fvex sstri syl12anc gsummulc2 psrvscacl psrmulfval csn psrmulcl psrvsca - cxp fconstmpt offval2 3eqtr4d jca ) ABMGUMNHUMBMNHUMZGUMZUNMBNGUMZHUMZY - NUNABCDEFGHIJKLMNOPQRSTUAUBUDUEUFUOAUGDEUHUIUPUGUPZUQURUSZUIDUTZUHUPZMV - AZYQYTVBVCUMZYOVAZEVDVAZUMZVEZVFUMZVEUGDBEUHYSUUAUUBNVAZUUDUMZVEZVFUMZU - UDUMZVEZYPYNAUGDUUGUULAYQDVGZVHZUUGEUHYSBUUIUUDUMZVEZVFUMUULUUOUUFUUQEV - FUUOUHYSUUEUUPUUOYTYSVGZVHZUUEUUABUUHUUDUMZUUDUMUUPUUSUUCUUTUUAUUDUUSCD - EFGUUDINJEVIVAZBUUBOUEUVAVJZTUUDVJZRUUOBUVAVGUURUUOBKUVAABKVGUUNUFVKZUD - VLZVKZANCVGUUNUURUBVMZUUSYSDUUBYRUIDVNZUUSJLVGZUUNUURUUBYSVGAUVIUUNUURP - VMAUUNUURVOUUOUURVPZUIDYSIYQJLYTRYSVJVQWEVRZVSVTUUSUJUKULUUABUUHUVAUUDU - USDUVAYTMUUSCDEFIJUVAMOUVBRTAMCVGUUNUURUAVMWAUUSYSDYTUVHUVJVRWBZUVFUUSD - UVAUUBNUUSCDEFIJUVANOUVBRTUVGWAUVKWBZUUSEWCVGZUJUPZUVAVGZUKUPZUVAVGZVHU - VOUVQUUDUMZUVQUVOUUDUMUNZAUVNUUNUURUCVMUVNUVPUVRUVTUVAEUUDUVOUVQUVBUVCW - FWDWGUUSEWHVGZUVPUVRULUPZUVAVGWIUVSUWBUUDUMUVOUVQUWBUUDUMUUDUMUNAUWAUUN - UURQVMZUVAEUUDUVOUVQUWBUVBUVCWJWGWKWLWMVTUUOYSUVAEWNVAZEUUDUHWOUUIBEWPV - AZUVBUWEVJUWDVJUVCAUWAUUNQVKAUVIUUNYSWOVGZPUIDIYQJLRWQWGZUVEUUSUWAUUAUV - AVGUUHUVAVGUUIUVAVGUWCUVLUVMUVAEUUDUUAUUHUVBUVCWRWEUUOUUJWSVGZUUJWTZUWE - WSVGZWIZUWFUUJUWEXAUMZYSXBZUUJUWEXCUSUWKUUOUWHUWIUWJYRUHUIDUUIWSIUPXDXE - XFWOVGIXIJXGUMDWSRXIJXGXJXHZXKUHYSUUIXLEWPXSXMXNUWGUWMUUOUWLUUJXOYSUUJU - WEXPUHYSUUIUUJUUJVJXQXTXNYSUUJWSWSUWEXRYAYBWLWMAUHUICDEFHUUDIUGMYOJOTUV - CSRUAACEFGNJKBOUEUDTQUFUBYCYDAYNDBYEYHZYMUUDVCUMUUMACDEFGUUDIYMJKBOUEUD - TUVCRUFACEFHJMNOTSQUAUBYFYGAUGDBUUKUUDUWOYMWSKWSDWSVGAUWNXNUVDUUKWSVGUU - OEUUJVFXJXNUWOUGDBVEUNAUGDBYIXNAUHUICDEFHUUDIUGMNJOTUVCSRUAUBYDYJWLYKYL - $. + syl3anc crngcom sylan crg w3a ringass eqtrd mpteq2dva cplusg psrbaglefi + caov12d cfn c0g ringcl cvv wfun csupp wss cfsupp ccnv cn cima cmap ovex + cn0 rabex2 mptrabex fvex 3pm3.2i a1i cdm suppssdm dmmptss suppssfifsupp + funmpt syl12anc gsummulc2 psrvscacl psrmulfval csn cxp psrmulcl psrvsca + sstri fconstmpt offval2 3eqtr4d jca ) ABMGUMNHUMBMNHUMZGUMZUNMBNGUMZHUM + ZYNUNABCDEFGHIJKLMNOPQRSTUAUBUDUEUFUOAUGDEUHUIUPUGUPZUQURUSZUIDUTZUHUPZ + MVAZYQYTVBVCUMZYOVAZEVDVAZUMZVEZVFUMZVEUGDBEUHYSUUAUUBNVAZUUDUMZVEZVFUM + ZUUDUMZVEZYPYNAUGDUUGUULAYQDVGZVHZUUGEUHYSBUUIUUDUMZVEZVFUMUULUUOUUFUUQ + EVFUUOUHYSUUEUUPUUOYTYSVGZVHZUUEUUABUUHUUDUMZUUDUMUUPUUSUUCUUTUUAUUDUUS + CDEFGUUDINJEVIVAZBUUBOUEUVAVJZTUUDVJZRUUOBUVAVGUURUUOBKUVAABKVGUUNUFVKZ + UDVLZVKZANCVGUUNUURUBVMZUUSYSDUUBYRUIDVNZUUSJLVGZUUNUURUUBYSVGAUVIUUNUU + RPVMAUUNUURVOUUOUURVPZUIDYSIYQJLYTRYSVJVQWEVRZVSVTUUSUJUKULUUABUUHUVAUU + DUUSDUVAYTMUUSCDEFIJUVAMOUVBRTAMCVGUUNUURUAVMWAUUSYSDYTUVHUVJVRWBZUVFUU + SDUVAUUBNUUSCDEFIJUVANOUVBRTUVGWAUVKWBZUUSEWCVGZUJUPZUVAVGZUKUPZUVAVGZV + HUVOUVQUUDUMZUVQUVOUUDUMUNZAUVNUUNUURUCVMUVNUVPUVRUVTUVAEUUDUVOUVQUVBUV + CWFWDWGUUSEWHVGZUVPUVRULUPZUVAVGWIUVSUWBUUDUMUVOUVQUWBUUDUMUUDUMUNAUWAU + UNUURQVMZUVAEUUDUVOUVQUWBUVBUVCWJWGWOWKWLVTUUOYSUVAEWMVAZEUUDUHWPUUIBEW + QVAZUVBUWEVJUWDVJUVCAUWAUUNQVKAUVIUUNYSWPVGZPUIDIYQJLRWNWGZUVEUUSUWAUUA + UVAVGUUHUVAVGUUIUVAVGUWCUVLUVMUVAEUUDUUAUUHUVBUVCWRWEUUOUUJWSVGZUUJWTZU + WEWSVGZWIZUWFUUJUWEXAUMZYSXBZUUJUWEXCUSUWKUUOUWHUWIUWJYRUHUIDUUIWSIUPXD + XEXFWPVGIXIJXGUMDWSRXIJXGXHXJZXKUHYSUUIXSEWQXLXMXNUWGUWMUUOUWLUUJXOYSUU + JUWEXPUHYSUUIUUJUUJVJXQYHXNYSUUJWSWSUWEXRXTYAWKWLAUHUICDEFHUUDIUGMYOJOT + UVCSRUAACEFGNJKBOUEUDTQUFUBYBYCAYNDBYDYEZYMUUDVCUMUUMACDEFGUUDIYMJKBOUE + UDTUVCRUFACEFHJMNOTSQUAUBYFYGAUGDBUUKUUDUWOYMWSKWSDWSVGAUWNXNUVDUUKWSVG + UUOEUUJVFXHXNUWOUGDBVEUNAUGDBYIXNAUHUICDEFHUUDIUGMNJOTUVCSRUAUBYCYJWKYK + YL $. $} $( The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.) $) - psrrng $p |- ( ph -> S e. Ring ) $= + psrring $p |- ( ph -> S e. Ring ) $= ( vx vy vz vr vf cfv cv wcel eqidd w3a eqid adantr cplusg cmulr ccnv cima - cbs cn cfn cn0 cmap co crab cc0 csn cxp wceq cur c0g cif cmpt cgrp rnggrp - crg syl psrgrp 3ad2ant1 simp2 simp3 psrmulcl simpr1 simpr2 simpr3 psrass1 - wa psrdi psrdir psr1cl simpr psrlidm psrridm isrngd ) AIJKCUENZCUANZCCUBN - ZLMOUCUFUDUGPMUHDUIUJUKZLODULUMUNUOBUPNZBUQNZURUSZAWAQAWBQAWCQABCDEFGABVB - PZBUTPHBVAVCVDAIOZWAPZJOZWAPZRWABCWCDWIWKFWASZWCSZAWJWHWLHVEAWJWLVFAWJWLV - GVHAWJWLKOZWAPZRZVMZWAWDBCWCMDEWIWKWOFADEPZWQGTZAWHWQHTZWDSZWNWMAWJWLWPVI - ZAWJWLWPVJZAWJWLWPVKZVLWRWAWDWBBCWCMDEWIWKWOFWTXAXBWNWMXCXDXEWBSZVNWRWAWD - WBBCWCMDEWIWKWOFWTXAXBWNWMXCXDXEXFVOALWAWDBCWGWEMDEWFFGHXBWFSZWESZWGSZWMV - PAWJVMZLWAWDBCWCWGWEMDEWIWFFAWSWJGTZAWHWJHTZXBXGXHXIWMWNAWJVQZVRXJLWAWDBC - WCWGWEMDEWIWFFXKXLXBXGXHXIWMWNXMVSVT $. + cbs cn cfn cn0 cmap co crab cc0 csn cxp wceq cur c0g cif cmpt crg ringgrp + cgrp psrgrp 3ad2ant1 simp2 simp3 psrmulcl wa simpr1 simpr2 simpr3 psrass1 + syl psrdi psrdir psr1cl simpr psrlidm psrridm isringd ) AIJKCUENZCUANZCCU + BNZLMOUCUFUDUGPMUHDUIUJUKZLODULUMUNUOBUPNZBUQNZURUSZAWAQAWBQAWCQABCDEFGAB + UTPZBVBPHBVAVMVCAIOZWAPZJOZWAPZRWABCWCDWIWKFWASZWCSZAWJWHWLHVDAWJWLVEAWJW + LVFVGAWJWLKOZWAPZRZVHZWAWDBCWCMDEWIWKWOFADEPZWQGTZAWHWQHTZWDSZWNWMAWJWLWP + VIZAWJWLWPVJZAWJWLWPVKZVLWRWAWDWBBCWCMDEWIWKWOFWTXAXBWNWMXCXDXEWBSZVNWRWA + WDWBBCWCMDEWIWKWOFWTXAXBWNWMXCXDXEXFVOALWAWDBCWGWEMDEWFFGHXBWFSZWESZWGSZW + MVPAWJVHZLWAWDBCWCWGWEMDEWIWFFAWSWJGTZAWHWJHTZXBXGXHXIWMWNAWJVQZVRXJLWAWD + BCWCWGWEMDEWIWFFXKXLXBXGXHXIWMWNXMVSVT $. psr1.d $e |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } $. psr1.z $e |- .0. = ( 0g ` R ) $. @@ -241223,11 +241444,11 @@ zero ring (at least if its operations are internal binary operations). psr1 $p |- ( ph -> U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) ) ) $= ( vy wcel cv cc0 csn cxp wceq cif cmpt cbs cmulr co wa wral psr1cl adantr - cfv eqid crg psrlidm psrridm jca ralrimiva wb psrrng isrngid syl mpbi2and - simpr ) ABCBUAIUBUCUDUEGKUFUGZEUHUOZTZVHSUAZEUIUOZUJVKUEZVKVHVLUJVKUEZUKZ - SVIULZFVHUEZABVICDEVHGHIJKLMNOPQVHUPZVIUPZUMAVOSVIAVKVITZUKZVMVNWABVICDEV - LVHGHIJVKKLAIJTVTMUNZADUQTVTNUNZOPQVRVSVLUPZAVTVGZURWABVICDEVLVHGHIJVKKLW - BWCOPQVRVSWDWEUSUTVAAEUQTVJVPUKVQVBADEIJLMNVCSVIEVLFVHVSWDRVDVEVF $. + cfv eqid crg simpr psrlidm psrridm ralrimiva wb psrring isringid mpbi2and + jca syl ) ABCBUAIUBUCUDUEGKUFUGZEUHUOZTZVHSUAZEUIUOZUJVKUEZVKVHVLUJVKUEZU + KZSVIULZFVHUEZABVICDEVHGHIJKLMNOPQVHUPZVIUPZUMAVOSVIAVKVITZUKZVMVNWABVICD + EVLVHGHIJVKKLAIJTVTMUNZADUQTVTNUNZOPQVRVSVLUPZAVTURZUSWABVICDEVLVHGHIJVKK + LWBWCOPQVRVSWDWEUTVFVAAEUQTVJVPUKVQVBADEIJLMNVCSVIEVLFVHVSWDRVDVGVE $. $} ${ @@ -241238,24 +241459,24 @@ zero ring (at least if its operations are internal binary operations). $( The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.) $) psrcrng $p |- ( ph -> S e. CRing ) $= - ( vx vy vf crg wcel cfv ccrg syl cbs eqid cv 3ad2ant1 cmgp crngrng psrrng - ccmn wceq mgpbas a1i cplusg mgpplusg cmnd rngmgp w3a ccnv cn cima cfn cn0 - cmulr cmap co crab simp2 simp3 psrcom iscmnd iscrng sylanbrc ) ACLMZCUANZ - UDMCOMABCDEFGABOMZBLMZHBUBPZUCZAIJCQNZCURNZVIVNVIQNUEAVNCVIVIRZVNRZUFUGVO - VIUHNUEACVOVIVPVORZUIUGAVHVIUJMVMCVIVPUKPAISZVNMZJSZVNMZULVNKSUMUNUOUPMKU - QDUSUTVAZBCVOKDEVSWAFAVTDEMWBGTAVTVKWBVLTWCRVRVQAVTWBVBAVTWBVCAVTVJWBHTVD - VECVIVPVFVG $. + ( vx vy vf crg wcel cfv ccrg syl cbs eqid cv 3ad2ant1 cmgp crngring cmulr + ccmn psrring wceq mgpbas a1i cplusg mgpplusg cmnd ringmgp w3a ccnv cn cfn + cima cn0 cmap co crab simp2 simp3 psrcom iscmnd iscrng sylanbrc ) ACLMZCU + ANZUDMCOMABCDEFGABOMZBLMZHBUBPZUEZAIJCQNZCUCNZVIVNVIQNUFAVNCVIVIRZVNRZUGU + HVOVIUINUFACVOVIVPVORZUJUHAVHVIUKMVMCVIVPULPAISZVNMZJSZVNMZUMVNKSUNUOUQUP + MKURDUSUTVAZBCVOKDEVSWAFAVTDEMWBGTAVTVKWBVLTWCRVRVQAVTWBVBAVTWBVCAVTVJWBH + TVDVECVIVPVFVG $. $( The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) $) psrassa $p |- ( ph -> S e. AssAlg ) $= ( vy vz vx vf cbs cfv eqidd wcel cv co adantr eqid cvsca cmulr psrsca crg - ccrg crngrng syl psrlmod psrrng w3a wa wceq ccnv cima cfn cn0 cmap simpr2 - cn crab simpr3 simpr1 psrass23 simpld simprd isassad ) AIJBMNZCUANZCUBNZB - CMNZCKAVJOABCDEUEFGHUCAVGOAVHOAVIOABCDEFGABUEPZBUDPZHBUFUGZUHABCDEFGVMUIH - AKQZVGPZIQZVJPZJQZVJPZUJZUKZVNVPVHRVRVIRVNVPVRVIRVHRZULZVPVNVRVHRVIRWBULZ - WAVNVJLQUMUSUNUOPLUPDUQRUTZBCVHVILDVGEVPVRFADEPVTGSAVLVTVMSWETVITVJTAVOVQ - VSURAVOVQVSVAAVKVTHSVGTVHTAVOVQVSVBVCZVDWAWCWDWFVEVF $. + ccrg crngring syl psrlmod psrring w3a wa wceq ccnv cima cfn cn0 cmap crab + cn simpr2 simpr3 simpr1 psrass23 simpld simprd isassad ) AIJBMNZCUANZCUBN + ZBCMNZCKAVJOABCDEUEFGHUCAVGOAVHOAVIOABCDEFGABUEPZBUDPZHBUFUGZUHABCDEFGVMU + IHAKQZVGPZIQZVJPZJQZVJPZUJZUKZVNVPVHRVRVIRVNVPVRVIRVHRZULZVPVNVRVHRVIRWBU + LZWAVNVJLQUMUSUNUOPLUPDUQRURZBCVHVILDVGEVPVRFADEPVTGSAVLVTVMSWETVITVJTAVO + VQVSUTAVOVQVSVAAVKVTHSVGTVHTAVOVQVSVBVCZVDWAWCWDWFVEVF $. $} ${ @@ -241352,19 +241573,19 @@ zero ring (at least if its operations are internal binary operations). subrgpsr $p |- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B e. ( SubRing ` S ) ) $= ( vx vf wcel cfv crg cbs adantl eqid vy csubrg wa cress co simpl subrgrcl - wss cur subrgrng wceq a1i simpr resspsrbas resspsradd resspsrmul rngpropd - psrrng cv mpbid jca ressbasss syl6eqss ccnv cn cima cfn cn0 cmap crab cc0 - wf csn cxp c0g cif cmpt subrg1cl csubg subrgsubg subg0cl syl ifcl syl2anc - subrgbas eleqtrd adantr fmptd psr1 feq1d mpbird rabex elmap sylibr psrbas - fvex ovex eleqtrrd issubrg sylanbrc ) GHOZDBUBPOZUCZCQOZCAUDUEZQOZUCACRPZ - UHZCUIPZAOZUCACUBPOXCXDXFXCBCGHIXAXBUFZXBBQOXADBUGSZURXCEQOXFXCFEGHKXKXBF - QOXADBFJUJSURXCMUAAEXEAERPUKXCLULXCAXEBCDEFGIJKLXETZXAXBUMZUNZXCAXEBCDEFG - MUSZUAUSZIJKLXMXNUOXCAXEBCDEFGXPXQIJKLXMXNUPUQUTVAXCXHXJXCAXERPXGXOAXGXEC - XMXGTZVBVCXCXIFRPZNUSVDVEVFVGOZNVHGVIUEZVJZVIUEZAXCYBXSXIVLZXIYCOXCYDYBXS - MYBXPGVKVMVNUKZBUIPZBVOPZVPZVQZVLXCMYBYHXSYIXCYHXSOXPYBOXCYHDXSXBYHDOZXAX - BYFDOYGDOZYJDBYFYFTZVRXBDBVSPOYKDBVTDBYGYGTZWAWBYEYFYGDWCWDSXBDXSUKXADBFJ - WESWFWGYITWHXCYBXSXIYIXCMYBBCXIYFNGHYGIXKXLYBTZYMYLXITZWIWJWKXSYBXIFRWPXT - NYAVHGVIWQWLWMWNXCAYBFENGXSHKXSTYNLXKWOWRVAAXGCXIXRYOWSWT $. + wss cur psrring subrgring wceq a1i simpr resspsrbas resspsradd resspsrmul + cv ringpropd mpbid jca ressbasss syl6eqss ccnv cima cfn cn0 cmap crab cc0 + cn csn cxp c0g cif cmpt subrg1cl csubg subrgsubg subg0cl syl ifcl syl2anc + wf subrgbas eleqtrd adantr fmptd psr1 feq1d mpbird fvex ovex rabex sylibr + elmap psrbas eleqtrrd issubrg sylanbrc ) GHOZDBUBPOZUCZCQOZCAUDUEZQOZUCAC + RPZUHZCUIPZAOZUCACUBPOXCXDXFXCBCGHIXAXBUFZXBBQOXADBUGSZUJXCEQOXFXCFEGHKXK + XBFQOXADBFJUKSUJXCMUAAEXEAERPULXCLUMXCAXEBCDEFGIJKLXETZXAXBUNZUOZXCAXEBCD + EFGMURZUAURZIJKLXMXNUPXCAXEBCDEFGXPXQIJKLXMXNUQUSUTVAXCXHXJXCAXERPXGXOAXG + XECXMXGTZVBVCXCXIFRPZNURVDVKVEVFOZNVGGVHUEZVIZVHUEZAXCYBXSXIWDZXIYCOXCYDY + BXSMYBXPGVJVLVMULZBUIPZBVNPZVOZVPZWDXCMYBYHXSYIXCYHXSOXPYBOXCYHDXSXBYHDOZ + XAXBYFDOYGDOZYJDBYFYFTZVQXBDBVRPOYKDBVSDBYGYGTZVTWAYEYFYGDWBWCSXBDXSULXAD + BFJWESWFWGYITWHXCYBXSXIYIXCMYBBCXIYFNGHYGIXKXLYBTZYMYLXITZWIWJWKXSYBXIFRW + LXTNYAVGGVHWMWNWPWOXCAYBFENGXSHKXSTYNLXKWQWRVAAXGCXIXRYOWSWT $. $} ${ @@ -241453,13 +241674,13 @@ zero ring (at least if its operations are internal binary operations). ` i ` . (Contributed by Mario Carneiro, 29-Dec-2014.) $) mvrf $p |- ( ph -> V : I --> B ) $= ( vx vf vh vy wf cv wcel eqid ccnv cn cima cfn cn0 cmap crab wceq cc0 cif - co c1 cmpt cur cfv c0g wa cbs crg rngidcl syl ifcl syl2anc ad2antrr fmptd - rng0cl fvex ovex rabex elmap sylibr psrbas adantr eleqtrrd mvrfval mpbird - feq1d ) AEBFQEBMENORUAUBUCUDSZOUEEUFUKZUGZNRZPEPRMRZUHULUIUJUMUHZCUNUOZCU - PUOZUJZUMZUMZQAMEWGBWHAWBESZUQZWGCURUOZVTUFUKZBWJVTWKWGQWGWLSWJNVTWFWKWGA - WFWKSZWIWAVTSAWDWKSZWEWKSZWMACUSSZWNLWKCWDWKTZWDTZUTVAAWPWOLWKCWEWQWETZVF - VAWCWDWEWKVBVCVDWGTVEWKVTWGCURVGVROVSUEEUFVHVIVJVKABWLUHWIABVTCDOEWKGHWQV - TTZJKVLVMVNWHTVEAEBFWHAMPVTCWDNOEFGUSWEIWTWSWRKLVOVQVP $. + co c1 cmpt cur cfv c0g cbs crg ringidcl syl ring0cl ifcl syl2anc ad2antrr + wa fmptd fvex ovex rabex elmap sylibr psrbas adantr eleqtrrd feq1d mpbird + mvrfval ) AEBFQEBMENORUAUBUCUDSZOUEEUFUKZUGZNRZPEPRMRZUHULUIUJUMUHZCUNUOZ + CUPUOZUJZUMZUMZQAMEWGBWHAWBESZVEZWGCUQUOZVTUFUKZBWJVTWKWGQWGWLSWJNVTWFWKW + GAWFWKSZWIWAVTSAWDWKSZWEWKSZWMACURSZWNLWKCWDWKTZWDTZUSUTAWPWOLWKCWEWQWETZ + VAUTWCWDWEWKVBVCVDWGTVFWKVTWGCUQVGVROVSUEEUFVHVIVJVKABWLUHWIABVTCDOEWKGHW + QVTTZJKVLVMVNWHTVFAEBFWHAMPVTCWDNOEFGURWEIWTWSWRKLVQVOVP $. ${ mvrf1.z $e |- .0. = ( 0g ` R ) $. @@ -241696,31 +241917,31 @@ zero ring (at least if its operations are internal binary operations). 16-Jul-2019.) $) mpllsslem $p |- ( ph -> U e. ( LSubSp ` S ) ) $= ( vu vv vw cbs cfv cplusg clss cvsca crg psrsca eqidd wceq a1i csubg wcel - vk wss cgrp rnggrp syl mplsubglem subgss c0g c0 eqid subg0cl ne0i 3syl cv - wne w3a wa co adantr csupp simprl simprr eleq2d oveq1 eleq1d elrab syl6bb - crab mpbid simpld psrvscacl cvv wi wal ovex wral simprd alrimiv ralrimiva - expr sseq2 imbi1d albidv rspcv sylc psrelbas cdif cmulr eldifi psrvscaval - adantl ssid ccnv cn cima cfn cn0 cmap rabex2 eqeltri suppssr oveq2d rngrz - syl2anc 3eqtrd suppss sseq1 eleq1 imbi12d spcgv mpbir2and 3adantr3 simpr3 - fvex syl3c subgcl syl3anc islssd ) AUEGUHUIZHUJUIZHUKUIZHULUIZIGEHUFUGAGH - LMUMOSUDUNAYRUOEHUHUIUPAPUQAYSUOAUUAUOAYTUOAIHURUIUSZIEVAABCDEFGHIJKLMNOP - QRSTUAUBUCAGUMUSZGVBUSUDGVCVDVEZEIHPVFVDAUUBHVGUIZIUSIVHVNUUDIHUUEUUEVIVJ - IUUEVKVLAUEVMZYRUSZUFVMZIUSZUGVMZIUSZVOZVPUUBUUFUUHUUAVQZIUSZUUKUUMUUJYSV - QIUSAUUBUULUUDVRAUUGUUIUUNUUKAUUGUUIVPZVPZUUNUUMEUSZUUMNVSVQZDUSZUUPEGHUU - AUUHLYRUUFOUUAVIZYRVIZPAUUCUUOUDVRZAUUGUUIVTZUUPUUHEUSZUUHNVSVQZDUSZUUPUU - IUVDUVFVPZAUUGUUIWAUUPUUIUUHKVMZNVSVQZDUSZKEWGZUSUVGUUPIUVKUUHAIUVKUPUUOU - CVRZWBUVJUVFKUUHEUVHUUHUPUVIUVEDUVHUUHNVSWCWDWEWFWHZWIZWJZUUPUURWKUSZCVMZ - UVEVAZUVQDUSZWLZCWMZUURUVEVAZUUSUVPUUPUUMNVSWNUQUUPUVFUVQBVMZVAZUVSWLZCWM - ZBDWOZUWAUUPUVDUVFUVMWPAUWGUUOAUWFBDAUWCDUSZVPUWECAUWHUWDUVSUBWSWQWRVRUWF - UWABUVEDUWCUVEUPZUWEUVTCUWIUWDUVRUVSUWCUVEUVQWTXAXBXCXDUUPFYRUTUUMUVENUUP - EFGHJLYRUUMOUVARPUVOXEUUPUTVMZFUVEXFUSZVPZUWJUUMUIUUFUWJUUHUIZGXGUIZVQUUF - NUWNVQZNUWLEFGHUUAUWNJUUHLYRUUFUWJOUUTUVAPUWNVIZRUUPUUGUWKUVCVRUUPUVDUWKU - VNVRUWKUWJFUSUUPUWJFUVEXHXJXIUWLUWMNUUFUWNUUPFYRWKUUHWKUVEUWJNUUPEFGHJLYR - UUHOUVARPUVNXEUVEUVEVAUUPUVEXKUQFWKUSUUPJVMXLXMXNXOUSJXPLXQVQFWKRXPLXQWNX - RUQNWKUSUUPNGVGUIWKQGVGYMXSUQXTYAUUPUWONUPZUWKUUPUUCUUGUWQUVBUVCYRGUWNUUF - NUVAUWPQYBYCVRYDYEUVTUWBUUSWLCUURWKUVQUURUPUVRUWBUVSUUSUVQUURUVEYFUVQUURD - YGYHYIYNUUPUUNUUMUVKUSUUQUUSVPUUPIUVKUUMUVLWBUVJUUSKUUMEUVHUUMUPUVIUURDUV - HUUMNVSWCWDWEWFYJYKAUUGUUIUUKYLYSIHUUMUUJYSVIYOYPYQ $. + vk wss cgrp ringgrp syl mplsubglem subgss c0g c0 wne eqid subg0cl ne0i cv + 3syl w3a wa co adantr csupp simprl simprr crab eleq2d oveq1 eleq1d syl6bb + elrab simpld psrvscacl cvv wi wal ovex wral simprd expr alrimiv ralrimiva + mpbid sseq2 imbi1d albidv rspcv sylc psrelbas cdif eldifi psrvscaval ssid + cmulr adantl ccnv cn cima cfn cn0 cmap rabex2 fvex eqeltri suppssr oveq2d + ringrz syl2anc 3eqtrd suppss sseq1 eleq1 imbi12d spcgv mpbir2and 3adantr3 + syl3c simpr3 subgcl syl3anc islssd ) AUEGUHUIZHUJUIZHUKUIZHULUIZIGEHUFUGA + GHLMUMOSUDUNAYRUOEHUHUIUPAPUQAYSUOAUUAUOAYTUOAIHURUIUSZIEVAABCDEFGHIJKLMN + OPQRSTUAUBUCAGUMUSZGVBUSUDGVCVDVEZEIHPVFVDAUUBHVGUIZIUSIVHVIUUDIHUUEUUEVJ + VKIUUEVLVNAUEVMZYRUSZUFVMZIUSZUGVMZIUSZVOZVPUUBUUFUUHUUAVQZIUSZUUKUUMUUJY + SVQIUSAUUBUULUUDVRAUUGUUIUUNUUKAUUGUUIVPZVPZUUNUUMEUSZUUMNVSVQZDUSZUUPEGH + UUAUUHLYRUUFOUUAVJZYRVJZPAUUCUUOUDVRZAUUGUUIVTZUUPUUHEUSZUUHNVSVQZDUSZUUP + UUIUVDUVFVPZAUUGUUIWAUUPUUIUUHKVMZNVSVQZDUSZKEWBZUSUVGUUPIUVKUUHAIUVKUPUU + OUCVRZWCUVJUVFKUUHEUVHUUHUPUVIUVEDUVHUUHNVSWDWEWGWFWSZWHZWIZUUPUURWJUSZCV + MZUVEVAZUVQDUSZWKZCWLZUURUVEVAZUUSUVPUUPUUMNVSWMUQUUPUVFUVQBVMZVAZUVSWKZC + WLZBDWNZUWAUUPUVDUVFUVMWOAUWGUUOAUWFBDAUWCDUSZVPUWECAUWHUWDUVSUBWPWQWRVRU + WFUWABUVEDUWCUVEUPZUWEUVTCUWIUWDUVRUVSUWCUVEUVQWTXAXBXCXDUUPFYRUTUUMUVENU + UPEFGHJLYRUUMOUVARPUVOXEUUPUTVMZFUVEXFUSZVPZUWJUUMUIUUFUWJUUHUIZGXJUIZVQU + UFNUWNVQZNUWLEFGHUUAUWNJUUHLYRUUFUWJOUUTUVAPUWNVJZRUUPUUGUWKUVCVRUUPUVDUW + KUVNVRUWKUWJFUSUUPUWJFUVEXGXKXHUWLUWMNUUFUWNUUPFYRWJUUHWJUVEUWJNUUPEFGHJL + YRUUHOUVARPUVNXEUVEUVEVAUUPUVEXIUQFWJUSUUPJVMXLXMXNXOUSJXPLXQVQFWJRXPLXQW + MXRUQNWJUSUUPNGVGUIWJQGVGXSXTUQYAYBUUPUWONUPZUWKUUPUUCUUGUWQUVBUVCYRGUWNU + UFNUVAUWPQYCYDVRYEYFUVTUWBUUSWKCUURWJUVQUURUPUVRUWBUVSUUSUVQUURUVEYGUVQUU + RDYHYIYJYMUUPUUNUUMUVKUSUUQUUSVPUUPIUVKUUMUVLWCUVJUUSKUUMEUVHUUMUPUVIUURD + UVHUUMNVSWDWEWGWFYKYLAUUGUUIUUKYNYSIHUUMUUJYSVJYOYPYQ $. $} ${ @@ -241813,30 +242034,30 @@ zero ring (at least if its operations are internal binary operations). (Proof modification is discouraged.) $) mpllsslemOLD $p |- ( ph -> U e. ( LSubSp ` S ) ) $= ( vu vv vw cbs cfv cplusg clss cvsca crg psrsca eqidd wceq a1i csubg wcel - vk wss cgrp rnggrp syl mplsubglemOLD subgss c0g c0 eqid subg0cl ne0i 3syl - wne cv w3a wa co adantr ccnv cvv csn cdif cima simprl simprr eleq2d cnveq - imaeq1d eleq1d elrab syl6bb mpbid simpld psrvscacl wi wal simprd psrelbas - crab cmulr eldifi adantl psrvscaval ssid suppssrOLD oveq2d syl2anc 3eqtrd - rngrz suppssOLD ssexd wral expr alrimiv ralrimiva sseq2 imbi1d rspcv sylc - albidv sseq1 eleq1 imbi12d spcgv mpbir2and 3adantr3 simpr3 subgcl syl3anc - syl3c islssd ) AUEGUHUIZHUJUIZHUKUIZHULUIZIGEHUFUGAGHLMUMOSUDUNAYLUOEHUHU - IUPAPUQAYMUOAYOUOAYNUOAIHURUIUSZIEVAABCDEFGHIJKLMNOPQRSTUAUBUCAGUMUSZGVBU - SUDGVCVDVEZEIHPVFVDAYPHVGUIZIUSIVHVMYRIHYSYSVIVJIYSVKVLAUEVNZYLUSZUFVNZIU - SZUGVNZIUSZVOZVPYPYTUUBYOVQZIUSZUUEUUGUUDYMVQIUSAYPUUFYRVRAUUAUUCUUHUUEAU - UAUUCVPZVPZUUHUUGEUSZUUGVSZVTNWAWBZWCZDUSZUUJEGHYOUUBLYLYTOYOVIZYLVIZPAYQ - UUIUDVRZAUUAUUCWDZUUJUUBEUSZUUBVSZUUMWCZDUSZUUJUUCUUTUVCVPZAUUAUUCWEUUJUU - 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(Contributed by Mario Carneiro, 9-Jan-2015.) $) mplsubrg $p |- ( ph -> U e. ( SubRing ` S ) ) $= - ( vx vk vf cfv wcel co eqid cvv vy csubrg csubg cur cmulr wral crg rnggrp - cgrp syl mplsubg cbs c0g cfsupp wbr psrrng rngidcl ccnv cima cfn cn0 cmap - cv crab cc0 csn cxp wceq cif cmpt psr1 wfun w3a csupp wss mptrabex funmpt - cn ovex fvex 3pm3.2i a1i snfi cdif wa wne eldifsni adantl ifnefalse rabex + ( vx vk vf cfv wcel co eqid cvv vy csubrg csubg cur cv cmulr wral ringgrp + crg cgrp syl mplsubg cbs c0g cfsupp wbr psrring ringidcl ccnv cn cima cfn + cn0 cmap crab cc0 csn cxp wceq cif cmpt psr1 wfun w3a csupp ovex mptrabex + wss funmpt fvex 3pm3.2i a1i snfi cdif wne eldifsni adantl ifnefalse rabex suppss2 suppssfifsupp syl12anc eqbrtrd mplelbas sylanbrc caddc cof adantr - simprl simprr mplsubrglem ralrimivva wb issubrg2 mpbir3and ) AEDUBPQZEDUC - PQZDUDPZEQZMVCZUAVCZDUEPZREQZUAEUFMEUFZABCDEFGHIJKACUGQZCUIQLCUHUJUKAXIDU - LPZQZXICUMPZUNUOXJADUGQZXRACDFGHKLUPZXQDXIXQSZXISZUQUJAXINOVCURVRUSUTQZOV - AFVBRZVDZNVCZFVEVFVGZVHCUDPZXSVIZVJZXSUNANYFCDXIYIOFGXSHKLYFSZXSSZYISYCVK - AYKTQZYKVLZXSTQZVMZYHVFZUTQZYKXSVNRYRVOYKXSUNUOYQAYNYOYPYDNOYEYJTVAFVBVSZ - VPNYFYJVQCUMVTWAWBYSAYHWCWBAYFYJNTYRXSAYGYFYRWDQZWEYGYHWFZYJXSVHUUAUUBAYG - YFYHWGWHYGYHYIXSWIUJYFTQAYDOYEYTWJWBWKYRYKTTXSWLWMWNXQBCDEFXIXSIHYBYMJWOW - PAXNMUAEEAXKEQZXLEQZWEZWEWQWRXKXSVNRXLXSVNRVGUSZYFBCDCUEPZEOFGXKXLXSHIJAF - GQUUEKWSAXPUUELWSYLYMUUFSUUGSAUUCUUDWTAUUCUUDXAXBXCAXTXGXHXJXOVMXDYAMUAEX - QDXMXIYBYCXMSXEUJXF $. + wa simprl simprr mplsubrglem ralrimivva wb issubrg2 mpbir3and ) AEDUBPQZE + DUCPQZDUDPZEQZMUEZUAUEZDUFPZREQZUAEUGMEUGZABCDEFGHIJKACUIQZCUJQLCUHUKULAX + IDUMPZQZXICUNPZUOUPXJADUIQZXRACDFGHKLUQZXQDXIXQSZXISZURUKAXINOUEUSUTVAVBQ + ZOVCFVDRZVEZNUEZFVFVGVHZVICUDPZXSVJZVKZXSUOANYFCDXIYIOFGXSHKLYFSZXSSZYISY + CVLAYKTQZYKVMZXSTQZVNZYHVGZVBQZYKXSVORYRVRYKXSUOUPYQAYNYOYPYDNOYEYJTVCFVD + VPZVQNYFYJVSCUNVTWAWBYSAYHWCWBAYFYJNTYRXSAYGYFYRWDQZWSYGYHWEZYJXSVIUUAUUB + AYGYFYHWFWGYGYHYIXSWHUKYFTQAYDOYEYTWIWBWJYRYKTTXSWKWLWMXQBCDEFXIXSIHYBYMJ + WNWOAXNMUAEEAXKEQZXLEQZWSZWSWPWQXKXSVORXLXSVORVHVAZYFBCDCUFPZEOFGXKXLXSHI + JAFGQUUEKWRAXPUUELWRYLYMUUFSUUGSAUUCUUDWTAUUCUUDXAXBXCAXTXGXHXJXOVNXDYAMU + AEXQDXMXIYBYCXMSXEUKXF $. $} ${ @@ -242179,23 +242400,23 @@ zero ring (at least if its operations are internal binary operations). $( The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.) $) - mplrng $p |- ( ( I e. V /\ R e. Ring ) -> P e. Ring ) $= - ( wcel crg wa cbs cfv cmps co csubrg eqid simpl mplsubrg mplval2 subrgrng - simpr syl ) CDFZBGFZHZAIJZCBKLZMJFAGFUCABUEUDCDUENZEUDNZUAUBOUAUBSPUDUEAA - BUEUDCEUFUGQRT $. + mplring $p |- ( ( I e. V /\ R e. Ring ) -> P e. Ring ) $= + ( wcel crg wa cbs cfv cmps co eqid simpl simpr mplsubrg mplval2 subrgring + csubrg syl ) CDFZBGFZHZAIJZCBKLZSJFAGFUCABUEUDCDUEMZEUDMZUAUBNUAUBOPUDUEA + ABUEUDCEUFUGQRT $. $( The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.) $) mplcrng $p |- ( ( I e. V /\ R e. CRing ) -> P e. CRing ) $= ( wcel ccrg wa cmps co cbs cfv csubrg eqid simpl simpr psrcrng crg adantl - crngrng mplsubrg mplval2 subrgcrng syl2anc ) CDFZBGFZHZCBIJZGFAKLZUHMLFAG - FUGBUHCDUHNZUEUFOZUEUFPQUGABUHUICDUJEUINZUKUFBRFUEBTSUAUIUHAABUHUICEUJULU - BUCUD $. + crngring mplsubrg mplval2 subrgcrng syl2anc ) CDFZBGFZHZCBIJZGFAKLZUHMLFA + GFUGBUHCDUHNZUEUFOZUEUFPQUGABUHUICDUJEUINZUKUFBRFUEBTSUAUIUHAABUHUICEUJUL + UBUCUD $. $( The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.) $) mplassa $p |- ( ( I e. V /\ R e. CRing ) -> P e. AssAlg ) $= - ( wcel ccrg wa casa cbs cfv cmps csubrg clss eqid simpl crg crngrng syl + ( wcel ccrg wa casa cbs cfv cmps csubrg clss eqid simpl crg crngring syl co adantl mplsubrg mpllss cur wb simpr psrassa subrg1cl subrgss issubassa wss mplval2 syl3anc mpbir2and ) CDFZBGFZHZAIFZAJKZCBLTZMKFZUSUTNKZFZUQABU TUSCDUTOZEUSOZUOUPPZUPBQFUOBRUAZUBZUQABUTUSCDVDEVEVFVGUCUQUTIFUTUDKZUSFZU @@ -242318,11 +242539,11 @@ series in the subring which are also polynomials (in the parent algebra. (Contributed by Mario Carneiro, 4-Jul-2015.) $) subrgmvrf $p |- ( ph -> V : I --> B ) $= ( vx cfv wcel eqid syl wfn cv wral wf cmps co cbs csubrg crg subrgrcl ffn - mvrf wa cmvr wceq subrgmvr fveq1d adantr subrgrng simpr eqeltrd ralrimiva - mvrcl ffnfv sylanbrc ) AHGUAZPUBZHQZBRZPGUCGBHUDAGGCUEUFZUGQZHUDVFAVKCVJG - HIVJSJVKSKADCUHQRZCUIRLDCUJTULGVKHUKTAVIPGAVGGRZUMZVHVGGFUNUFZQZBAVHVPUOV - MAVGHVOACDFGHIJKLMUPUQURVNBEFGVOIVGNVOSOAGIRVMKURAFUIRZVMAVLVQLDCFMUSTURA - VMUTVCVAVBPGBHVDVE $. + mvrf wa cmvr wceq subrgmvr fveq1d subrgring simpr mvrcl eqeltrd ralrimiva + adantr ffnfv sylanbrc ) AHGUAZPUBZHQZBRZPGUCGBHUDAGGCUEUFZUGQZHUDVFAVKCVJ + GHIVJSJVKSKADCUHQRZCUIRLDCUJTULGVKHUKTAVIPGAVGGRZUMZVHVGGFUNUFZQZBAVHVPUO + VMAVGHVOACDFGHIJKLMUPUQVCVNBEFGVOIVGNVOSOAGIRVMKVCAFUIRZVMAVLVQLDCFMURTVC + AVMUSUTVAVBPGBHVDVE $. $} ${ @@ -242340,17 +242561,17 @@ series in the subring which are also polynomials (in the parent $( A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) $) mplmon $p |- ( ph -> ( y e. D |-> if ( y = X , .1. , .0. ) ) e. B ) $= - ( cv wceq cif cmpt cmps co cbs cfv wcel cfsupp wbr cmap wf rngidcl rng0cl - crg eqid ifcld syl adantr fmptd fvex ccnv cn cima cfn cn0 cvv ovex rabex2 - elmap sylibr psrbas eleqtrrd wfun w3a csn csupp wss mptex eqeltri 3pm3.2i - funmpt c0g a1i snfi cdif wa wn wne eldifsni adantl neneqd iffalse suppss2 - suppssfifsupp syl12anc mplelbas sylanbrc ) ABDBUAZKUBZGLUCZUDZIFUEUFZUGUH - ZUIXCLUJUKZXCCUIAXCFUGUHZDULUFZXEADXGXCUMXCXHUIABDXBXGXCAXBXGUIZWTDUIAFUP - UIZXISXJXAGLXGXGFGXGUQZPUNXGFLXKOUOURUSUTXCUQVAXGDXCFUGVBHUAVCVDVEVFUIHVG - IULUFDVHQVGIULVIVJZVKVLAXEDFXDHIXGJXDUQZXKQXEUQZRVMVNAXCVHUIZXCVOZLVHUIZV - PZKVQZVFUIZXCLVRUFXSVSXFXRAXOXPXQBDXBXLVTBDXBWCLFWDUHVHOFWDVBWAWBWEXTAKWF - WEADXBBVHXSLAWTDXSWGUIZWHZXAWIXBLUBYBWTKYAWTKWJAWTDKWKWLWMXAGLWNUSDVHUIAX - LWEWOXSXCVHVHLWPWQXEEFXDCIXCLMXMXNONWRWS $. + ( cv wceq cif cmpt cmps co cbs cfv wcel cfsupp wbr cmap crg eqid ringidcl + wf ring0cl ifcld syl adantr fmptd fvex ccnv cima cfn cn0 cvv rabex2 elmap + cn ovex sylibr psrbas eleqtrrd w3a csn csupp wss mptex funmpt c0g eqeltri + wfun 3pm3.2i a1i snfi cdif wa wn wne eldifsni adantl neneqd suppssfifsupp + iffalse suppss2 syl12anc mplelbas sylanbrc ) ABDBUAZKUBZGLUCZUDZIFUEUFZUG + UHZUIXCLUJUKZXCCUIAXCFUGUHZDULUFZXEADXGXCUPXCXHUIABDXBXGXCAXBXGUIZWTDUIAF + UMUIZXISXJXAGLXGXGFGXGUNZPUOXGFLXKOUQURUSUTXCUNVAXGDXCFUGVBHUAVCVJVDVEUIH + VFIULUFDVGQVFIULVKVHZVIVLAXEDFXDHIXGJXDUNZXKQXEUNZRVMVNAXCVGUIZXCWCZLVGUI + ZVOZKVPZVEUIZXCLVQUFXSVRXFXRAXOXPXQBDXBXLVSBDXBVTLFWAUHVGOFWAVBWBWDWEXTAK + WFWEADXBBVGXSLAWTDXSWGUIZWHZXAWIXBLUBYBWTKYAWTKWJAWTDKWKWLWMXAGLWOUSDVGUI + AXLWEWPXSXCVGVGLWNWQXEEFXDCIXCLMXMXNONWRWS $. mplmonmul.t $e |- .x. = ( .r ` P ) $. mplmonmul.x $e |- ( ph -> Y e. D ) $. @@ -242365,64 +242586,64 @@ series in the subring which are also polynomials (in the parent ( y e. D |-> if ( y = ( X oF + Y ) , .1. , .0. ) ) ) $= ( vk vj vx vz cv wceq cif cmpt cle cofr wbr crab cfv cmin cof cmulr caddc co cgsu eqid mplmon mplmul eqeq1 ifbid cbvmptv wcel wa csn cres wss simpr - snssd resmpt syl oveq2d cmnd cbs crg ad2antrr rngmnd cur cvv fvex eqeltri - iftrue fvmpt ssrab2 simplr psrbagconcl syl3anc sseldi c0g oveq12d rngidcl - ifex rng0cl ifcld rnglidm syl2anc wral cn0 wb wf psrbagf ffvelrnda adantr - adantlr cc nn0cn subadd syl3an eqcom syl6bb ralbidva mpteqb ovex a1i mprg - 3bitr4g feqmptd offval2 eqeq12d 3bitr4d 3eqtrd eqeltrrd eqeltrd wn mplelf - c0 ffvelrnd iffalse cfn cdif cmap suppss2 sylan2 fveq2 oveq2 fveq2d gsum0 - gsumsn cin disjsn wfn rngcl fmptd ffn fnresdisj biimpa sylan2br cr nn0red - 3syl nn0addge1 ralrimiva ofrfval2 mpbird breq1 elrab breq2 rabbidv eleq2d - sylanbrc syl5ibrcom con3dimp pm2.61dan ccmn rngcmn psrbaglefi sylan ssdif - 3eqtr4a ax-mp sseli wne eldifsni adantl neneqd ccnv cn cima rabex2 oveq1d - suppssr eldifi rnglz eqtrd rabex w3a 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(Contributed by Mario Carneiro, 7-Jan-2015.) 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D |-> if ( y = Y , .1. , .0. ) ) = ( G gsum ( k e. I |-> ( ( Y ` k ) .^ ( V ` k ) ) ) ) ) $= - ( vx ccrg wcel crg crngrng syl cv cfv cplusg co wceq wa mplcrng syl2anc - cbs adantr simprr mvrcl simprl cmulr mgpplusg eqcomi crngcom ralrimivva - eqid syl3anc mplcoe5 ) AUFBCDEFGHIJKLMNOPQRSTUAUBUCAEUGUHZEUIUHZUDEUJUK - ZUEABULZLUMZUFULZLUMZJUNUMZUOVSVQVTUOUPZUFBKKAVRKUHZVPKUHZUQZUQZDUGUHZV - QDUTUMZUHVSWGUHWAAWFWDAKMUHZVMWFTUDDEKMPURUSVAWEWGDEKLMVPPUCWGVJZAWHWDT - VAZAVNWDVOVAZAWBWCVBVCWEWGDEKLMVRPUCWIWJWKAWBWCVDVCWGDVTVQVSWIDVEUMZVTD - WLJUAWLVJVFVGVHVKVIVL $. + ( vx ccrg wcel crg crngring syl cv cfv cplusg co wa cbs mplcrng syl2anc + wceq adantr eqid simprr mvrcl simprl mgpplusg eqcomi crngcom ralrimivva + cmulr syl3anc mplcoe5 ) AUFBCDEFGHIJKLMNOPQRSTUAUBUCAEUGUHZEUIUHZUDEUJU + KZUEABULZLUMZUFULZLUMZJUNUMZUOVSVQVTUOUTZUFBKKAVRKUHZVPKUHZUPZUPZDUGUHZ + VQDUQUMZUHVSWGUHWAAWFWDAKMUHZVMWFTUDDEKMPURUSVAWEWGDEKLMVPPUCWGVBZAWHWD + TVAZAVNWDVOVAZAWBWCVCVDWEWGDEKLMVRPUCWIWJWKAWBWCVEVDWGDVTVQVSWIDVJUMZVT + DWLJUAWLVBVFVGVHVKVIVL $. $( Decompose a monomial into a finite product of powers of variables. (The assumption that ` R ` is a commutative ring is not strictly @@ -242796,75 +243017,75 @@ series in the subring which are also polynomials (in the parent c0 eqid eqeq12d imbi12d imbi2d ccrg caddc adantr ffvelrnda 0nn0 sylancl weq ifcl fmptd suppss2OLD ssfi syl2anc eqeltrrd mpbir2and syl5ss nn0cnd eqidd oveq12d simpr 3eqtr4d wo eqtrd mvrcl mulgnn0cl syl3anc fveq2 a2d - cur cun noel mtbiri fconstmpt syl6eqr rngidval gsum0 crg crngrng eqcomd - mpt0 mpl1 a1d ssun1 sstr2 ax-mp imim1i cmulr oveq1 cof cbs eldifn ssun2 - simprll simprr vex snss sylibr ffvelrnd snfi eldifsni sylancr mplmonmul - wne neneqd mplcoe3 offval2 addid2d elsni simprlr ad2antrr fveq2d sseldi - eqneltrd addid1d elsn sylnib biorf orcom bitri syl6rbbr 3eqtr3rd mgpbas - elun mgpplusg ccmn mplcrng crngmgp sselda cmnmnd adantlr syldan syl5ibr - cmnd gsumunsn expr syl5 expcom findcard2s mpcom mpd resmpt oveq1d mulg0 - eldifi sylan2 gsumresOLD 3eqtr2d ) ABCBUJZNUKZFOULZUMBCUXTUFKUFUJZNUNZU - 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(Contributed by Mario Carneiro, 10-Jan-2015.) $) mplbas2 $p |- ( ph -> ( A ` ran V ) = ( Base ` P ) ) $= ( cfv wcel eqid syl co cvv vx vk vf vy vz vu vv crn cbs casa wss mplbasss - psrassa a1i wf wfn cv wral ccrg crg crngrng ffn wa adantr simpr ralrimiva - mvrf mvrcl ffnfv sylanbrc frn aspss syl3anc csubrg clss wceq mpllss aspid - mplsubrg sseqtrd ccnv cn cima cfn cn0 cmap crab cur c0g cif cvsca mplcoe1 - cmpt cgsu cabl mplrng syl2anc rngabl ovex rabex csubg aspsubrg wb mplval2 - syl6ss subsubrg mpbir2and subrgsubg clmod mpllmod ad2antrr asplss psrlmod - csca lsslss mplelf ffvelrnda mplsca fveq2d cmgp ccmn ad3antrrr fmptd wfun - eleqtrd csupp cfsupp wbr funmpt fvex cdif cc0 adantl suppssr oveq1d eqtrd - eldifi suppss2 suppssfifsupp eqeltrd cmg mplcoe2 rngidval mplcrng crngmgp - csubmnd subrgsubm simplll psrbag biimpa simpld aspssid sylan sseldd cmulr - fnfvelrn mgpbas mgpplusg syl3an1 mulgnn0subcl mptexg simprd elrabi elmapi - subrgmcl subrg1cl frnnn0supp eqimss c0ex sylanl2 mulg0 gsumsubmcl lssvscl - syl32anc syl22anc mptrabex 3pm3.2i mplelbas simprbi fsuppimpd ssid mplmon - w3a lmod0vs sylan2 syl12anc gsumsubgcl ex ssrdv eqssd ) AGUHZBOZCUIOZAUWL - UWMBOZUWMAEUJPZUWMEUIOZUKZUWKUWMUKZUWLUWNUKADEFHJMNUMZUWQAUWPCDEUWMFIJUWM - QZUWPQZULZUNAFUWMGUOZUWRAGFUPZUAUQZGOUWMPZUAFURUXCAFUWPGUOUXDAUWPDEFGHJKU - XAMADUSPZDUTPZNDVARZVGFUWPGVBRZAUXFUAFAUXEFPZVCUWMCDFGHUXEIKUWTAFHPZUXKMV - DAUXHUXKUXIVDAUXKVEVHVFUAFUWMGVIVJFUWMGVKRZBUWMUWKUWPELUXAVLVMAUWOUWMEVNO - ZPZUWMEVOOZPZUWNUWMVPUWSACDEUWMFHJIUWTMUXIVSZACDEUWMFHJIUWTMUXIVQZBUWMUXP - UWPELUXAUXPQZVRVMVTZAUAUWMUWLAUXEUWMPZUXEUWLPAUYBVCZUXECUBUCUQWAWBWCWDPZU - CWEFWFSZWGZUBUQZUXEOZUDUYFUDUQUYGVPDWHOZDWIOZWJWMZCWKOZSZWMZWNSUWLUYCUDUW - MUYFCDUYLUYIUCUBFHUXEUYJIUYFQZUYJQZUYIQZAUXLUYBMVDZUWTUYLQZAUXHUYBUXIVDZA - UYBVEZWLUYCUYFUWLUYNCTCWIOZVUBQZACWOPZUYBACUTPZVUDAUXLUXHVUEMUXICDFHIWPWQ - CWRRVDUYFTPUYCUYDUCUYEWEFWFWSZWTUNZAUWLCXAOPZUYBAUWLCVNOPZVUHAVUIUWLUXNPZ - UWLUWMUKZAUWOUWKUWPUKZVUJUWSAUWKUWMUWPUXMUXBXEZBUWKUWPELUXAXBWQUYAAUXOVUI - VUJVUKVCXCUXRUWMUWLECCDEUWMFIJUWTXDZXFRXGZUWLCXHRVDUYCUBUYFUYMUWLUYNUYCUY - 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psrassa a1i wf wfn cv wral ccrg crg crngrng mvrf wa simpr mvrcl ralrimiva - ffn ffnfv sylanbrc aspss syl3anc csubrg clss wceq mplsubrg mpllss sseqtrd - frn aspid ccnv cn cima cfn cn0 cmap co crab cur c0g cif cmpt cgsu mplcoe1 - cvsca cabl mplrng syl2anc rngabl ovex rabex csubg syl6ss aspsubrg mplval2 - cvv wb subsubrg mpbir2and subrgsubg clmod mpllmod ad2antrr asplss psrlmod - csca lsslss mplelf ffvelrnda mplsca fveq2d eleqtrd cmgp cmg rngidval ccmn - mplcoe2 mplcrng crngmgp csubmnd fmptd csn suppssrOLD oveq1d eldifi adantl - cdif cc0 eqtrd suppss2OLD ssfi eqeltrd subrgsubm simplll psrbag ad3antrrr - biimpa simpld aspssid fnfvelrn sylan sseldd cmulr mgpbas mgpplusg syl3an1 - subrgmcl subrg1cl mulgnn0subcl simprd nn0suppOLD 3syl mulg0 gsumsubmclOLD - eqimss lssvscl syl22anc mplelbasOLD simprbi ssid mplmon lmod0vs sylan2 ex - gsumsubgclOLD ssrdv eqssd ) AGUHZBOZCUIOZAUVQUVRBOZUVRAEUJPZUVREUIOZQZUVP - UVRQZUVQUVSQADEFHJMNULZUWBAUWACDEUVRFIJUVRRZUWARZUKZUMAFUVRGUNZUWCAGFUOZU - AUPZGOUVRPZUAFUQUWHAFUWAGUNUWIAUWADEFGHJKUWFMADURPZDUSPZNDUTSZVAFUWAGVFSZ - 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D |-> if ( y = K , X , .0. ) ) ) $= ( cv wceq cif cmpt csn cxp cmulr cfv cof cbs eqid mplmon mplvsca cvv wcel co ccnv cima cfn cn0 cmap crab ovex rabex eqeltri a1i adantr cur fvex c0g - cn wa ifex fconstmpt eqidd offval2 oveq2 eqeq1d crg rngridm iftrue eqcomd - syl2anc sylan9eq wn rngrz iffalse ifbothda mpteq2dv 3eqtrd ) AMBDBUEZKUFZ - HNUGZUHZGUTDMUIUJZWRFUKULZUMUTBDMWQWTUTZUHBDWPMNUGZUHAEUNULZDEFGWTIWRJCMO - PTXCUOZWTUOZQUDABXCDEFHIJLKNOXDSRQUAUBUCUPUQABDMWQWTWSWRURCURDURUSADIUEVA - VOVBVCUSZIVDJVEUTZVFURQXFIXGVDJVEVGVHVIVJAMCUSZWODUSZUDVKWQURUSAXIVPWPHNH - FVLULURRFVLVMVINFVNULURSFVNVMVIVQVJWSBDMUHUFABDMVRVJAWRVSVTABDXAXBWPMHWTU - TZXBUFMNWTUTZXBUFXAXBUFAHNHWQUFXJXAXBHWQMWTWAWBNWQUFXKXAXBNWQMWTWAWBAWPXJ - MXBAFWCUSZXHXJMUFUBUDCFWTHMTXERWDWGWPXBMWPMNWEWFWHAWPWIZXKNXBAXLXHXKNUFUB - UDCFWTMNTXESWJWGXMXBNWPMNWKWFWHWLWMWN $. + cn wa ifex fconstmpt eqidd offval2 oveq2 eqeq1d crg syl2anc iftrue eqcomd + ringridm sylan9eq wn ringrz iffalse ifbothda mpteq2dv 3eqtrd ) AMBDBUEZKU + FZHNUGZUHZGUTDMUIUJZWRFUKULZUMUTBDMWQWTUTZUHBDWPMNUGZUHAEUNULZDEFGWTIWRJC + MOPTXCUOZWTUOZQUDABXCDEFHIJLKNOXDSRQUAUBUCUPUQABDMWQWTWSWRURCURDURUSADIUE + VAVOVBVCUSZIVDJVEUTZVFURQXFIXGVDJVEVGVHVIVJAMCUSZWODUSZUDVKWQURUSAXIVPWPH + NHFVLULURRFVLVMVINFVNULURSFVNVMVIVQVJWSBDMUHUFABDMVRVJAWRVSVTABDXAXBWPMHW + TUTZXBUFMNWTUTZXBUFXAXBUFAHNHWQUFXJXAXBHWQMWTWAWBNWQUFXKXAXBNWQMWTWAWBAWP + XJMXBAFWCUSZXHXJMUFUBUDCFWTHMTXERWGWDWPXBMWPMNWEWFWHAWPWIZXKNXBAXLXHXKNUF + UBUDCFWTMNTXESWJWDXMXBNWPMNWKWFWHWLWMWN $. $} ${ @@ -243456,9 +243677,9 @@ series in the subring which are also polynomials (in the parent $( The scalar injection is a function into the polynomial algebra. (Contributed by Stefan O'Rear, 9-Mar-2015.) $) mplasclf $p |- ( ph -> A : K --> B ) $= - ( wf cfv cbs wcel crg eqid csca mplrng mpllmod asclf mplsca fveq2d syl5eq - wa syl2anc feq2d mpbird ) AGCBODUAPZQPZCBOZAFHRZESRZUNMNUOUPUHBCULUMDLULT - DEFHIUBDEFHIUCUMTJUDUIAGUMCBAGEQPUMKAEULQADEFHSIMNUEUFUGUJUK $. + ( wf cfv cbs wcel crg eqid wa mplring mpllmod asclf syl2anc mplsca fveq2d + csca syl5eq feq2d mpbird ) AGCBODUHPZQPZCBOZAFHRZESRZUNMNUOUPUABCULUMDLUL + TDEFHIUBDEFHIUCUMTJUDUEAGUMCBAGEQPUMKAEULQADEFHSIMNUFUGUIUJUK $. $} ${ @@ -243480,15 +243701,15 @@ series in the subring which are also polynomials (in the parent vf cvv cress ovex eqeltri a1i mplsca fveq2d eqtrd fneq2d mpbiri wss crg co subrgrcl subrgss fnssres syl2anc cv wa fvres adantl ccnv cn cima cfn cn0 cmap crab cc0 csn cxp c0g cmpt subrg0 ifeq2d adantr mpteq2dv sselda - cif mplascl subrgrng eleq2d biimpa 3eqtr4d eqtr2d eqfnfvd ) AUAFCBFUCZA - CFUDCGUERZUFRZUDCWTXAGQWTSXASUGAFXACAFHUFRZXAAFEUHRTZFXBUIPFEHMUJUKZAHW - TUFAGHIJUMNOHUMTAHEFUNVEUMMEFUNUOUPUQURUSUTVAVBABEUFRZUDZFXEVCZWSFUDAXF - BDUERZUFRZUDBXHXIDLXHSXISUGAXEXIBAEXHUFADEIJVDKOAXCEVDTZPFEVFUKZURUSVAV - BAXCXGPFXEEXESZVGUKZXEFBVHVIAUAVJZFTZVKZXNWSRZXNBRZXNCRZXOXQXRUIAXNFBVL - VMXPUBULVJVNVOVPVQTULVRIVSVEVTZUBVJIWAWBWCUIZXNEWDRZWKZWEUBXTYAXNHWDRZW - KZWEXRXSXPUBXTYCYEAYCYEUIXOAYAYBYDXNAXCYBYDUIPFEHYBMYBSZWFUKWGWHWIXPUBB - XEXTDEULIJXNYBKXTSZYFXLLAIJTXOOWHZAXJXOXKWHAFXEXNXMWJWLXPUBCXBXTGHULIJX - NYDNYGYDSXBSQYHAHVDTZXOAXCYIPFEHMWMUKWHAXOXNXBTAFXBXNXDWNWOWLWPWQWR $. + cif mplascl subrgring eleq2d biimpa 3eqtr4d eqtr2d eqfnfvd ) AUAFCBFUCZ + ACFUDCGUERZUFRZUDCWTXAGQWTSXASUGAFXACAFHUFRZXAAFEUHRTZFXBUIPFEHMUJUKZAH + WTUFAGHIJUMNOHUMTAHEFUNVEUMMEFUNUOUPUQURUSUTVAVBABEUFRZUDZFXEVCZWSFUDAX + FBDUERZUFRZUDBXHXIDLXHSXISUGAXEXIBAEXHUFADEIJVDKOAXCEVDTZPFEVFUKZURUSVA + VBAXCXGPFXEEXESZVGUKZXEFBVHVIAUAVJZFTZVKZXNWSRZXNBRZXNCRZXOXQXRUIAXNFBV + LVMXPUBULVJVNVOVPVQTULVRIVSVEVTZUBVJIWAWBWCUIZXNEWDRZWKZWEUBXTYAXNHWDRZ + WKZWEXRXSXPUBXTYCYEAYCYEUIXOAYAYBYDXNAXCYBYDUIPFEHYBMYBSZWFUKWGWHWIXPUB + BXEXTDEULIJXNYBKXTSZYFXLLAIJTXOOWHZAXJXOXKWHAFXEXNXMWJWLXPUBCXBXTGHULIJ + XNYDNYGYDSXBSQYHAHVDTZXOAXCYIPFEHMWMUKWHAXOXNXBTAFXBXNXDWNWOWLWPWQWR $. $} subrgasclcl.b $e |- B = ( Base ` U ) $. @@ -243500,18 +243721,19 @@ series in the subring which are also polynomials (in the parent subrgasclcl $p |- ( ph -> ( ( A ` X ) e. B <-> X e. T ) ) $= ( vf vx cfv wcel wa cbs cc0 csn cxp cv ccnv cn cima cfn cmap co crab wceq cn0 c0g cif wral adantr eqid psrbag0 syl cmpt wf cmps csubrg crg subrgrcl - mplascl wss subrgrng mplsubrg sselda eqeltrrd psrelbas fmpt sylibr iftrue - subrgss eleq1d rspcv sylc subrgbas eleqtrrd cascl cres subrgascl sylan9eq - fveq1d fvres csca mplrng mpllmod asclf syl2anc mplsca fveq2d eqtrd eleq2d - biimpa ffvelrnd impbida ) ALBUDZCUEZLFUEZAXIUFZLHUGUDZFXKIUHUIUJZUBUKULUM - UNUOUEUBUTIUPUQURZUEZUCUKXMUSZLEVAUDZVBZXLUEZUCXNVCZLXLUEZXKIKUEZXOAYBXIQ - VDXNUBIKXNVEZVFVGXKXNXLUCXNXRVHZVIXTXKIHVJUQZUGUDZXNHYEUBIXLYDYEVEZXLVEYC - YFVEZXKXHYDYFAXHYDUSXIAUCBJXNDEUBIKLXQMYCXQVETNQAFEVKUDUEZEVLUERFEVMVGUAV - NVDACYFXHACYEVKUDUECYFVOAGHYECIKYGPSQAYIHVLUEZRFEHOVPVGZVQCYFYEYHWDVGVRVS - VTUCXNXLXRYDYDVEWAWBXSYAUCXMXNXPXRLXLXPLXQWCWEWFWGAFXLUSZXIAYIYLRFEHOWHVG - ZVDWIAXJUFZLGWJUDZUDZXHCAXJYPLBFWKZUDXHALYOYQABYODEFGHIKMNOPQRYOVEZWLWNLF - BWOWMYNGWPUDZUGUDZCLYOAYTCYOVIZXJAYBYJUUAQYKYBYJUFYOCYSYTGYRYSVEGHIKPWQGH - IKPWRYTVESWSWTVDAXJLYTUEAFYTLAFXLYTYMAHYSUGAGHIKVLPQYKXAXBXCXDXEXFVSXG $. + mplascl subrgring mplsubrg subrgss sselda eqeltrrd psrelbas sylibr iftrue + wss fmpt eleq1d rspcv sylc subrgbas eleqtrrd cascl subrgascl fveq1d fvres + cres csca mplring mpllmod asclf syl2anc mplsca fveq2d eqtrd eleq2d biimpa + sylan9eq ffvelrnd impbida ) ALBUDZCUEZLFUEZAXIUFZLHUGUDZFXKIUHUIUJZUBUKUL + UMUNUOUEUBUTIUPUQURZUEZUCUKXMUSZLEVAUDZVBZXLUEZUCXNVCZLXLUEZXKIKUEZXOAYBX + IQVDXNUBIKXNVEZVFVGXKXNXLUCXNXRVHZVIXTXKIHVJUQZUGUDZXNHYEUBIXLYDYEVEZXLVE + YCYFVEZXKXHYDYFAXHYDUSXIAUCBJXNDEUBIKLXQMYCXQVETNQAFEVKUDUEZEVLUERFEVMVGU + AVNVDACYFXHACYEVKUDUECYFWCAGHYECIKYGPSQAYIHVLUEZRFEHOVOVGZVPCYFYEYHVQVGVR + VSVTUCXNXLXRYDYDVEWDWAXSYAUCXMXNXPXRLXLXPLXQWBWEWFWGAFXLUSZXIAYIYLRFEHOWH + VGZVDWIAXJUFZLGWJUDZUDZXHCAXJYPLBFWNZUDXHALYOYQABYODEFGHIKMNOPQRYOVEZWKWL + LFBWMXEYNGWOUDZUGUDZCLYOAYTCYOVIZXJAYBYJUUAQYKYBYJUFYOCYSYTGYRYSVEGHIKPWP + GHIKPWQYTVESWRWSVDAXJLYTUEAFYTLAFXLYTYMAHYSUGAGHIKVLPQYKWTXAXBXCXDXFVSXG + $. $} ${ @@ -243554,23 +243776,23 @@ series in the subring which are also polynomials (in the parent ( y e. D |-> if ( y = Y , G , .0. ) ) ) = ( y e. D |-> if ( y = ( X oF + Y ) , ( F .x. G ) , .0. ) ) ) $= ( cv wceq cur cfv cif cmpt cvsca co caddc cof casa wcel csca ccrg mplassa - cbs syl2anc mplsca fveq2d syl5eq eleqtrd eqid crg crngrng mplmon assalmod + cbs syl2anc mplsca fveq2d syl5eq eleqtrd crg crngring syl mplmon assalmod clmod lmodvscl syl3anc assaass syl13anc assaassr oveq2d cmulr psrbagaddcl - syl mplmonmul lmodvsass syl5req oveqd oveq1d 3eqtr2d 3eqtrd mplmon2 rngcl - oveq12d 3eqtr3d ) AJBDBUIZNUJZFUKULZPUMUNZEUOULZUPZKBDWPOUJZWRPUMUNZWTUPZ - GUPZJKHUPZBDWPNOUQURUPZUJZWRPUMUNZWTUPZBDWQJPUMUNZBDXBKPUMUNZGUPBDXHXFPUM - UNAXEJWSXDGUPZWTUPZJKWSXCGUPZWTUPZWTUPZXJAEUSUTZJEVAULZVDULZUTZWSEVDULZUT - ZXDYBUTZXEXNUJALMUTZFVBUTZXRUAUBEFLMQVCVEZAJCXTUGACFVDULXTTAFXSVDAEFLMVBQ - UAUBVFZVGVHZVIZABYBDEFWRILMNPQYBVJZSWRVJZRUAAYFFVKUTZUBFVLWDZUEVMZAEVOUTZ - KXTUTZXCYBUTZYDAXRYPYGEVNWDZAKCXTUHYIVIZABYBDEFWRILMOPQYKSYLRUAYNUFVMZKWT - XSXTYBEXCYKXSVJZWTVJZXTVJZVPVQJXTWTGXSYBEWSXDYKUUBUUDUUCUCVRVSAXMXPJWTAXR - YQYCYRXMXPUJYGYTYOUUAKXTWTGXSYBEWSXCYKUUBUUDUUCUCVTVSWAAXQJKXIWTUPZWTUPZJ - KXSWBULZUPZXIWTUPZXJAXPUUEJWTAXOXIKWTABYBDEFGWRILMNOPQYKSYLRUAYNUEUCUFWEW - AWAAYPYAYQXIYBUTUUIUUFUJYSYJYTABYBDEFWRILMXGPQYKSYLRUAYNAYENDUTODUTXGDUTU - AUEUFDINOLMRWCVQZVMJKWTUUGXSXTYBEXIYKUUBUUCUUDUUGVJWFVSAUUHXFXIWTAUUGHJKA - HFWBULUUGUDAFXSWBYHVGWGWHWIWJWKAXAXKXDXLGABCDEFWTWRILNMJPQUUCRYLSTUAYNUEU - GWLABCDEFWTWRILOMKPQUUCRYLSTUAYNUFUHWLWNABCDEFWTWRILXGMXFPQUUCRYLSTUAYNUU - JAYMJCUTKCUTXFCUTYNUGUHCFHJKTUDWMVQWLWO $. + eqid mplmonmul syl5req oveqd oveq1d 3eqtr2d 3eqtrd mplmon2 oveq12d ringcl + lmodvsass 3eqtr3d ) AJBDBUIZNUJZFUKULZPUMUNZEUOULZUPZKBDWPOUJZWRPUMUNZWTU + PZGUPZJKHUPZBDWPNOUQURUPZUJZWRPUMUNZWTUPZBDWQJPUMUNZBDXBKPUMUNZGUPBDXHXFP + UMUNAXEJWSXDGUPZWTUPZJKWSXCGUPZWTUPZWTUPZXJAEUSUTZJEVAULZVDULZUTZWSEVDULZ + UTZXDYBUTZXEXNUJALMUTZFVBUTZXRUAUBEFLMQVCVEZAJCXTUGACFVDULXTTAFXSVDAEFLMV + BQUAUBVFZVGVHZVIZABYBDEFWRILMNPQYBWDZSWRWDZRUAAYFFVJUTZUBFVKVLZUEVMZAEVOU + TZKXTUTZXCYBUTZYDAXRYPYGEVNVLZAKCXTUHYIVIZABYBDEFWRILMOPQYKSYLRUAYNUFVMZK + WTXSXTYBEXCYKXSWDZWTWDZXTWDZVPVQJXTWTGXSYBEWSXDYKUUBUUDUUCUCVRVSAXMXPJWTA + XRYQYCYRXMXPUJYGYTYOUUAKXTWTGXSYBEWSXCYKUUBUUDUUCUCVTVSWAAXQJKXIWTUPZWTUP + ZJKXSWBULZUPZXIWTUPZXJAXPUUEJWTAXOXIKWTABYBDEFGWRILMNOPQYKSYLRUAYNUEUCUFW + EWAWAAYPYAYQXIYBUTUUIUUFUJYSYJYTABYBDEFWRILMXGPQYKSYLRUAYNAYENDUTODUTXGDU + TUAUEUFDINOLMRWCVQZVMJKWTUUGXSXTYBEXIYKUUBUUCUUDUUGWDWNVSAUUHXFXIWTAUUGHJ + KAHFWBULUUGUDAFXSWBYHVGWFWGWHWIWJAXAXKXDXLGABCDEFWTWRILNMJPQUUCRYLSTUAYNU + EUGWKABCDEFWTWRILOMKPQUUCRYLSTUAYNUFUHWKWLABCDEFWTWRILXGMXFPQUUCRYLSTUAYN + UUJAYMJCUTKCUTXFCUTYNUGUHCFHJKTUDWMVQWKWO $. $} ${ @@ -243596,54 +243818,54 @@ series in the subring which are also polynomials (in the parent needed. (Contributed by Stefan O'Rear, 11-Mar-2015.) $) mplind $p |- ( ph -> X e. H ) $= ( vz vw cin inss1 crn cmps co casp cfv casa wcel cbs wss cvv eqid psrassa - inss2 csubrg ccrg crg crngrng syl mplsubrg subrgss syl5ss wfn cv wf mvrf2 - wral ffn ralrimiva fnfvrnss syl2anc ssind aspss syl3anc mplbas2 clss wceq - frn syl6eqr csubg cur c0 wne cminusg wa a1i csca crh mplassa asclrhm rhm1 - mplsca eqeltrrd rngidcl fveq2d syl5eq eleqtrrd fveq2 eleq1d assarng elind - rspcva ne0i sseli anim12i caovclg sylan2 cgrp clmod lmodgrp adantr simprl - assalmod sseldi simprr grpcl anassrs simpr simpl cghm rhmghm grpinvcl jca - rngnegl w3a wb mpbir3and ralrimivva simprbda ghminv eqtrd rnggrp syl12anc - issubg2 rngcl issubrg2 mplval2 subsubrg cvsca asclmul1 lmodvscl mpbir2and - mpllss syldan islss4 lsslss syl21anc aspid 3sstr3d sseldd ) AIDUKZIMIDULZ - ADUVBMALUMZJGUNUOZUPUQZUQZUVBUVFUQZDUVBAUVEURUSZUVBUVEUTUQZVAUVDUVBVAUVGU - VHVAAGUVEJVBUVEVCZUGUHVDZAUVBDUVJIDVEZADUVEVFUQZUSZDUVJVAANGUVEDJVBUVKQUA - UGAGVGUSZGVHUSUHGVIVJZVKZDUVJUVEUVJVCZVLVJVMAUVDIDALJVNZBVOZLUQIUSZBJVRUV - DIVAAJDLVPZUVTADNGJLVBQPUAUGUVQVQZJDLVSVJAUWBBJUEVTBJILWAWBAUWCUVDDVAUWDJ - 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V ) $. evlslem4.j $e |- ( ph -> J e. W ) $. @@ -243758,23 +243980,24 @@ polynomial algebra (with coefficients in ` R ` ) to a function from cv wceq oveqan12d cbvmpt2 cop cvv wcel wa vex eqop2 adantl mpt2mpt eqtr4i sylbi simp2 3adant3 eqid fvmpt2 syl2anc simp3 3adant2 mpt2eq3dva syl5reqr w3a oveq12d oveq1d cdif wo cun difxp eleq2i elun bitri xp1st wss ssid a1i - fmptd c0g eqeltri suppssr sylan2 crg adantr wf xp2nd ffvelrn syl2an rnglz - fvex eqtrd oveq2d rngrz jaodan sylan2b xpexg suppss2 eqsstrd ) ABCGHKLFUE - ZUFZMUGUEUBGHUHZUBUQZUIUJZBGKUKZUJZXRULUJZCHLUKZUJZFUEZUKZMUGUEXTMUGUEZYC - MUGUEZUHZAXPYFMUGAYFBCGHBUQZXTUJZCUQZYCUJZFUEZUFZXPYOUCUDGHUCUQZXTUJZUDUQ - ZYCUJZFUEZUFYFBCUCUDGHYNYTUCYNUMUDYNUMBYQYSFBGKYPUNBFUMBYSUMUOCYQYSFCYQUM - CFUMCHLYRUNUOYJYPURYLYRURYKYQYMYSFYJYPXTUPYLYRYCUPUSUTUCUDUBGHYEYTXRYPYRV - AURXRVBVBUHVCZXSYPURZYBYRURZVDZVDYEYTURZXRYPYRUCVEUDVEVFUUDUUEUUAUUBUUCYA - YQYDYSFXSYPXTUPYBYRYCUPUSVGVJVHVIABCGHYNXOAYJGVCZYLHVCZVTZYKKYMLFUUHUUFKD - VCZYKKURAUUFUUGVKAUUFUUIUUGRVLBGKDXTXTVMZVNVOUUHUUGLDVCZYMLURAUUFUUGVPAUU - GUUKUUFSVQCHLDYCYCVMZVNVOWAVRVSWBAXQYEUBVBYIMXRXQYIWCZVCZAXRGYGWCZHUHZVCZ - XRGHYHWCZUHZVCZWDZYEMURZUUNXRUUPUUSWEZVCUVAUUMUVCXRYGYHGHWFWGXRUUPUUSWHWI - AUUQUVBUUTAUUQVDZYEMYDFUEZMUVDYAMYDFUUQAXSUUOVCYAMURXRUUOHWJAGDVBXTIYGXSM - ABGKDXTRUUJWNZYGYGWKAYGWLWMTMVBVCAMEWOUJVBOEWOXFWPWMZWQWRWBUVDEWSVCZYDDVC - ZUVEMURAUVHUUQQWTAHDYCXAYBHVCUVIUUQACHLDYCSUULWNZXRUUOHXBHDYBYCXCXDDEFYDM - NPOXEVOXGAUUTVDZYEYAMFUEZMUVKYDMYAFUUTAYBUURVCYDMURXRGUURXBAHDVBYCJYHYBMU - VJYHYHWKAYHWLWMUAUVGWQWRXHUVKUVHYADVCZUVLMURAUVHUUTQWTAGDXTXAXSGVCUVMUUTU - VFXRGUURWJGDXSXTXCXDDEFYAMNPOXIVOXGXJXKAGIVCHJVCXQVBVCTUAGHIJXLVOXMXN $. + fmptd c0g fvex eqeltri suppssr sylan2 crg adantr wf ffvelrn syl2an ringlz + xp2nd eqtrd oveq2d ringrz jaodan sylan2b xpexg suppss2 eqsstrd ) ABCGHKLF + UEZUFZMUGUEUBGHUHZUBUQZUIUJZBGKUKZUJZXRULUJZCHLUKZUJZFUEZUKZMUGUEXTMUGUEZ + YCMUGUEZUHZAXPYFMUGAYFBCGHBUQZXTUJZCUQZYCUJZFUEZUFZXPYOUCUDGHUCUQZXTUJZUD + UQZYCUJZFUEZUFYFBCUCUDGHYNYTUCYNUMUDYNUMBYQYSFBGKYPUNBFUMBYSUMUOCYQYSFCYQ + UMCFUMCHLYRUNUOYJYPURYLYRURYKYQYMYSFYJYPXTUPYLYRYCUPUSUTUCUDUBGHYEYTXRYPY + RVAURXRVBVBUHVCZXSYPURZYBYRURZVDZVDYEYTURZXRYPYRUCVEUDVEVFUUDUUEUUAUUBUUC + YAYQYDYSFXSYPXTUPYBYRYCUPUSVGVJVHVIABCGHYNXOAYJGVCZYLHVCZVTZYKKYMLFUUHUUF + KDVCZYKKURAUUFUUGVKAUUFUUIUUGRVLBGKDXTXTVMZVNVOUUHUUGLDVCZYMLURAUUFUUGVPA + UUGUUKUUFSVQCHLDYCYCVMZVNVOWAVRVSWBAXQYEUBVBYIMXRXQYIWCZVCZAXRGYGWCZHUHZV + CZXRGHYHWCZUHZVCZWDZYEMURZUUNXRUUPUUSWEZVCUVAUUMUVCXRYGYHGHWFWGXRUUPUUSWH + WIAUUQUVBUUTAUUQVDZYEMYDFUEZMUVDYAMYDFUUQAXSUUOVCYAMURXRUUOHWJAGDVBXTIYGX + SMABGKDXTRUUJWNZYGYGWKAYGWLWMTMVBVCAMEWOUJVBOEWOWPWQWMZWRWSWBUVDEWTVCZYDD + VCZUVEMURAUVHUUQQXAAHDYCXBYBHVCUVIUUQACHLDYCSUULWNZXRUUOHXFHDYBYCXCXDDEFY + DMNPOXEVOXGAUUTVDZYEYAMFUEZMUVKYDMYAFUUTAYBUURVCYDMURXRGUURXFAHDVBYCJYHYB + MUVJYHYHWKAYHWLWMUAUVGWRWSXHUVKUVHYADVCZUVLMURAUVHUUTQXAAGDXTXBXSGVCUVMUU + TUVFXRGUURWJGDXSXTXCXDDEFYAMNPOXIVOXGXJXKAGIVCHJVCXQVBVCTUAGHIJXLVOXMXN + $. $} ${ @@ -243869,64 +244092,64 @@ polynomial algebra (with coefficients in ` R ` ) to a function from evlslem2 $p |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( E ` ( x ( .r ` P ) y ) ) = ( ( E ` x ) .x. ( E ` y ) ) ) $= ( vz cv wcel wa weq cfv cif cmpt cgsu co cmulr ccom cmpt2 cvv c0g eqid cn - ccnv cima cfn cn0 cmap ovex rabex2 a1i crg ccrg crngrng syl mplrng adantr - syl2anc cbs ad2antrr simprl mplelf ffvelrnda mplmon2cl simprr cfsupp wral - simpr wbr wfun w3a csupp wss mptex funmpt fvex 3pm3.2i mplelsfi fsuppimpd - cdif ssid eqeltri suppssr ifeq1d ifid syl6eq mpteq2dv wceq csn cxp rnggrp - cgrp mpl0 fconstmpt eqtr4d suppss2 suppssfifsupp syl12anc ralrimiva fveq1 - breq1d cbvralv sylib adantrr fveq2 adantrl gsumdixp fveq2d gsummhm oveq2d - wf eqtrd eqidd fmptco oveq12d ffvelrnd suppssfv 3eqtr4d mplcoe4 fmptd cof - r19.21bi equequ2 ifbieq1d syl5eqbr ccmn rngcmn cmnd rngmnd xpex cmhm cghm - cbvmptv ghmmhm rngcl syl3anc ralrimivva fmpt2 mpt2fun xpfi evlslem4 caddc - mpt2ex mplmon2mul anassrs 3impb mpt2eq3dva feqmptd fmpt2co ghmid 3eqtr2d - ghmf ) ABUHZDUIZCUHZDUIZUJZUJZFLEMEMLUKZLUHZUVMULZPUMZUNZUNZUOUPZFKEMEMKU - KZKUHZUVOULZPUMZUNZUNZUOUPZFUQULZUPZNULZHNUWDURZUOUPZHNUWKURZUOUPZIUPZUVM - 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D |-> ( ( F ` ( Y ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) finSupp ( 0g ` S ) ) ) $= - ( vx cv cfv cof co cgsu cmpt wf c0g cfsupp wbr wcel wa crg ccrg crngrng - syl adantr crh rhmf mplelf ffvelrnda ffvelrnd mgpbas eqid crngmgp simpr - ccmn cvv psrbagev2 rngcl syl3anc fmptd wfun csupp cfn wss ccnv cima cn0 - cmap ovex a1i rabexd mptexg funmpt fvex mplelsfi fsuppimpd wceq feqmptd - cn oveq1d eqimss2 cghm rhmghm ghmid 3syl suppssfv rnglz sylan suppssov1 - suppssfifsupp syl32anc jca ) ADCTDTURZRUSZMUSZHYBNLUTVAVBVAZIVAZVCZVDYG - GVEUSZVFVGZATDYFCYGAYBDVHZVIZGVJVHZYDCVHYECVHYFCVHAYLYJAGVKVHZYLUMGVLVM - ZVNYKPCYCMAPCMVDZYJAMFGVOVAVHZYOUNPCFGMUDUCVPVMVNADPYBRABDEFJOPRUAUDUBU - EUPVQZVRVSYKYBCDHLJNOHVEUSZUECGHUFUCVTUGYRWAAHWDVHZYJAYMYSUMGHUFWBVMVNA - YJWCAOCNVDYJUOVNAOWEVHYJUKVNWFZCGIYDYEUCUHWGWHYGWAWIAYGWEVHZYGWJZYHWEVH - ZRFVEUSZWKVAZWLVHYGYHWKVAUUEWMYIADWEVHUUAAJURWNXHWOWLVHJWPOWQVAZDWEUEUU - FWEVHAWPOWQWRWSWTTDYFWEXAVMUUBATDYFXBWSUUCAGVEXCWSZARUUDABEFROVKUUDUAUB - UUDWAZUPULXDXEATUQYDYEDCUUEIWEWEYHYHATYCDWEMUUEWEUUDYHAUUETDYCVCZUUDWKV - AZXFUUJUUEWMARUUIUUDWKATDPRYQXGXIUUJUUEXJVMAYPMFGXKVAVHUUDMUSYHXFUNFGMX - LFGMUUDYHUUHYHWAZXMXNYCWEVHYKYBRXCWSUUDWEVHAFVEXCWSXOAYLUQURZCVHYHUULIV - AYHXFYNCGIUULYHUCUHUUKXPXQYDWEVHYKYCMXCWSYTUUGXRUUEYGWEWEYHXSXTYA $. + ( vx cv cfv cof co cgsu cmpt wf c0g cfsupp wbr wcel wa crg crngring syl + ccrg adantr crh rhmf mplelf ffvelrnda ffvelrnd mgpbas eqid ccmn crngmgp + simpr cvv psrbagev2 ringcl syl3anc fmptd wfun csupp cfn wss ccnv cn cn0 + cima cmap ovex a1i rabexd mptexg funmpt fvex mplelsfi fsuppimpd feqmptd + wceq oveq1d eqimss2 cghm rhmghm ghmid 3syl suppssfv sylan suppssfifsupp + ringlz suppssov1 syl32anc jca ) ADCTDTURZRUSZMUSZHYBNLUTVAVBVAZIVAZVCZV + DYGGVEUSZVFVGZATDYFCYGAYBDVHZVIZGVJVHZYDCVHYECVHYFCVHAYLYJAGVMVHZYLUMGV + KVLZVNYKPCYCMAPCMVDZYJAMFGVOVAVHZYOUNPCFGMUDUCVPVLVNADPYBRABDEFJOPRUAUD + UBUEUPVQZVRVSYKYBCDHLJNOHVEUSZUECGHUFUCVTUGYRWAAHWBVHZYJAYMYSUMGHUFWCVL + VNAYJWDAOCNVDYJUOVNAOWEVHYJUKVNWFZCGIYDYEUCUHWGWHYGWAWIAYGWEVHZYGWJZYHW + EVHZRFVEUSZWKVAZWLVHYGYHWKVAUUEWMYIADWEVHUUAAJURWNWOWQWLVHJWPOWRVAZDWEU + EUUFWEVHAWPOWRWSWTXATDYFWEXBVLUUBATDYFXCWTUUCAGVEXDWTZARUUDABEFROVMUUDU + AUBUUDWAZUPULXEXFATUQYDYEDCUUEIWEWEYHYHATYCDWEMUUEWEUUDYHAUUETDYCVCZUUD + WKVAZXHUUJUUEWMARUUIUUDWKATDPRYQXGXIUUJUUEXJVLAYPMFGXKVAVHUUDMUSYHXHUNF + GMXLFGMUUDYHUUHYHWAZXMXNYCWEVHYKYBRXDWTUUDWEVHAFVEXDWTXOAYLUQURZCVHYHUU + LIVAYHXHYNCGIUULYHUCUHUUKXRXPYDWEVHYKYCMXDWTYTUUGXSUUEYGWEWEYHXQXTYA $. $( Lemma for ~ evlseu . Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) Obsolete version of @@ -243984,19 +244207,19 @@ polynomial algebra (with coefficients in ` R ` ) to a function from ( ( F ` ( Y ` b ) ) .x. ( T gsum ( b oF .^ G ) ) ) ) " ( _V \ { ( 0g ` S ) } ) ) e. Fin ) ) $= ( vx cv cfv cof co cgsu cmpt wf ccnv cvv c0g csn cdif cima cfn wcel crg - wa ccrg crngrng syl adantr crh rhmf mplelf ffvelrnda ffvelrnd eqid ccmn - mgpbas crngmgp simpr psrbagev2 rngcl syl3anc fmptd wss mplelsfiOLD wceq - feqmptd cnveqd imaeq1d eqimss2 cghm rhmghm ghmid 3syl suppssfvOLD rnglz - fvex a1i sylan suppssov1OLD ssfi syl2anc jca ) ADCTDTURZRUSZMUSZHXMNLUT - VAVBVAZIVAZVCZVDXRVEVFGVGUSZVHVIVJZVKVLZATDXQCXRAXMDVLZVNZGVMVLZXOCVLXP - CVLXQCVLAYDYBAGVOVLZYDUMGVPVQZVRYCPCXNMAPCMVDZYBAMFGVSVAVLZYGUNPCFGMUDU - CVTVQVRADPXMRABDEFJOPRUAUDUBUEUPWAZWBWCYCXMCDHLJNOHVGUSZUECGHUFUCWFUGYJ - WDAHWEVLZYBAYEYKUMGHUFWGVQVRAYBWHAOCNVDYBUOVRAOVFVLYBUKVRWIZCGIXOXPUCUH - WJWKXRWDWLARVEZVFFVGUSZVHVIZVJZVKVLXTYPWMYAABEFROVOYNUAUBYNWDZUPULWNATU - QXOXPDCYPIVFXSXSATXNDMYPVFYNXSAYPTDXNVCZVEZYOVJZWOYTYPWMAYMYSYOARYRATDP - RYIWPWQWRYTYPWSVQAYHMFGWTVAVLYNMUSXSWOUNFGMXAFGMYNXSYQXSWDZXBXCXNVFVLYC - XMRXFXGXDAYDUQURZCVLXSUUBIVAXSWOYFCGIUUBXSUCUHUUAXEXHXOVFVLYCXNMXFXGYLX - IYPXTXJXKXL $. + ccrg crngring syl adantr crh rhmf mplelf ffvelrnda ffvelrnd mgpbas eqid + wa crngmgp simpr psrbagev2 ringcl syl3anc fmptd wss mplelsfiOLD feqmptd + ccmn wceq cnveqd imaeq1d eqimss2 cghm rhmghm ghmid 3syl a1i suppssfvOLD + fvex ringlz sylan suppssov1OLD ssfi syl2anc jca ) ADCTDTURZRUSZMUSZHXMN + LUTVAVBVAZIVAZVCZVDXRVEVFGVGUSZVHVIVJZVKVLZATDXQCXRAXMDVLZWEZGVMVLZXOCV + LXPCVLXQCVLAYDYBAGVNVLZYDUMGVOVPZVQYCPCXNMAPCMVDZYBAMFGVRVAVLZYGUNPCFGM + UDUCVSVPVQADPXMRABDEFJOPRUAUDUBUEUPVTZWAWBYCXMCDHLJNOHVGUSZUECGHUFUCWCU + GYJWDAHWOVLZYBAYEYKUMGHUFWFVPVQAYBWGAOCNVDYBUOVQAOVFVLYBUKVQWHZCGIXOXPU + CUHWIWJXRWDWKARVEZVFFVGUSZVHVIZVJZVKVLXTYPWLYAABEFROVNYNUAUBYNWDZUPULWM + ATUQXOXPDCYPIVFXSXSATXNDMYPVFYNXSAYPTDXNVCZVEZYOVJZWPYTYPWLAYMYSYOARYRA + TDPRYIWNWQWRYTYPWSVPAYHMFGWTVAVLYNMUSXSWPUNFGMXAFGMYNXSYQXSWDZXBXCXNVFV + LYCXMRXFXDXEAYDUQURZCVLXSUUBIVAXSWPYFCGIUUBXSUCUHUUAXGXHXOVFVLYCXNMXFXD + YLXIYPXTXJXKXL $. $} ${ @@ -244011,48 +244234,48 @@ polynomial algebra (with coefficients in ` R ` ) to a function from (Contributed by Stefan O'Rear, 8-Mar-2015.) $) evlslem3 $p |- ( ph -> ( E ` ( x e. D |-> if ( x = A , H , .0. ) ) ) = ( ( F ` H ) .x. ( T gsum ( A oF .^ G ) ) ) ) $= - ( vy vz cv wceq cif cmpt cfv cof co cgsu wcel cvv crg crngrng mplmon2cl - ccrg syl fveq1 fveq2d oveq1d mpteq2dv oveq2d fvmpt wa simpr c0g eqeltri - ovex fvex ifexg syl2anc adantr eqeq1 ifbid eqid fvmptg mpteq2dva rngmnd - a1i cmnd ccnv cn cima cfn cn0 cmap rabex2 wf rhmf rng0cl ifcld ffvelrnd + ( vy vz cv wceq cif cmpt cfv cof co cgsu wcel cvv ccrg crg crngring syl + mplmon2cl fveq1 fveq2d oveq1d mpteq2dv oveq2d ovex fvmpt simpr c0g fvex + eqeltri a1i ifexg syl2anc adantr eqeq1 ifbid eqid fvmptg mpteq2dva cmnd + wa ringmnd ccnv cn cima cfn cn0 cmap rabex2 rhmf ring0cl ifcld ffvelrnd crh mgpbas ccmn crngmgp cmnmnd ad2antrr simprl simprr mulgnn0cl syl3anc - psrbagf sylan off wfun wb cdif cc0 wfn ffn eldifi adantl ffvelrn syl2an - inidm wn 3eqtrd eqtrd csupp cfsupp ffun psrbag simplbda fnfvof syl22anc - wss wbr eldifn ad2antlr elpreima mpbir2and mtand elnn0 sylib orel1 sylc - wo mulg0 suppss suppssfifsupp syl32anc gsumcl rngcl csn eldifsni neneqd - fmptd iffalse rhmghm ghmid sylan2 rnglz suppss2 gsumpt iftrue oveq12d - 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mplsubrg mplval2 + cascl subsubrg2 fveq2d ressascl syl6reqr rneqd cmre subrgmre 3syl syl3anc + cpw unssd ad2antrr wfn wb rhmf ffn adantr uneq1d assaring eqsstr3d submrc + fveq12d eqtr2d 3eqtr3d simpr rhmeql mrcsscl eqsstrd simprl simprr 3imtr4d + ad2antlr fnreseql fneqeql2 syl5 ralrimivva reseq1 rmo4 sylc reu5 sylanbrc + sylibr rmoim ) AGUEZBUFZHUGZUVGKUFZIUGZUHZGDFUIUJZUKZUVLGUVMULZUVLGUVMUMA + UADUNUOZFUBUCUEUPVMUQURUSUCUTJVAUJVBZUBUEZUAUEUOHUOFVCUOZUVRIUVSVDUOZVEUJ + VFUJFVGUOZUJVHVFUJVHZUVMUSZUWBBUFZHUGZUWBKUFZIUGZVIUVNABUVPCUVQDEFUVSUWAU + CUWBUVTHIJEUNUOZKUAUBLUVPVNZMUWHVNUVQVNUVSVNUVTVNUWAVNOUWBVNPQRSTNVJUWCUW + EUWGUVNUVLUWEUWGUHGUWBUVMUVGUWBUGZUVIUWEUVKUWGUWJUVHUWDHUVGUWBBVKVLUWJUVJ + UWFIUVGUWBKVKVLVOVPVQVRAUVLUVGBVSZKVSZVTZWAZHBUPUFZIKUPUFZVTZUGZWHZGUVMWB + UWRGUVMULZUVOAUWSGUVMAUVLUVGUWKWAZUWOUGZUVGUWLWAZUWPUGZUHZUWRAUVLUXEAUVIU + XBUVKUXDABWCZUVIUXBADWDUOZUNUOZUVPBWRZUXFAJWEUSZEWFUSZUXIPAEWGUSUXKQEWIVR + 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( W RingHom T ) /\ ( ( Q o. A ) = X /\ ( Q o. V ) = Y ) ) ) $= ( vm cvv wcel ccrg csubrg cfv w3a cv ccom wceq crh crab crio evlsval wreu wa co cbs eqid simp1 subrgcrng 3adant1 cmap simp2 pwscrng sylancl csn cxp - ovex cmpt cres wss subrgss 3ad2ant3 resmpt syl syl6eqr crngrng pwsdiagrhm - crg 3ad2ant2 simp3 resrhm syl2anc eqeltrrd wf fvex eqeltri simpl1 sylancr - wb elmapg biimpa simplr ffvelrnd simpl2 pwselbasb mpbird riotacl2 eqeltrd - fmptd evlseu coeq1 eqeq1d anbi12d elrab sylib ) JUEUFZFUGUFZEFUHUIUFZUJZD - UDUKZBULZMUMZXOKULZNUMZUSZUDLGUNUTZUOZUFDYAUFDBULZMUMZDKULZNUMZUSZUSXNDXT - UDYAUPZYBABCDEFGHUDIJKLMNOPQRSTUAUBUCUQXNXTUDYAURYHYBUFXNBGVAUIZLHGUDMNJK - PYIVBZUAQXKXLXMVCXLXMHUGUFXKEFHRVDVEXNXLCJVFUTZUEUFZGUGUFXKXLXMVGCJVFVLZF - YKUEGSVHVIXNACYKAUKZVJVKZVMZEVNZMHGUNUTZXNYQAEYOVMZMXNECVOZYQYSUMXMXKYTXL - ECFTVPVQACEYOVRVSUBVTXNYPFGUNUTUFZXMYQYRUFXNFWCUFZYLUUAXLXKUUBXMFWAWDYMAC - FYPYKUEGSTYPVBWBVIXKXLXMWEFGHYPERWFWGWHXNAJIYKYNIUKZUIZVMZYINXNYNJUFZUSZU - UEYIUFZYKCUUEWIZUUGIYKUUDCUUEUUGUUCYKUFZUSJCYNUUCUUGUUJJCUUCWIZUUGCUEUFXK - UUJUUKWNCFVAUIUETFVAWJWKXKXLXMUUFWLCJUUCUEUEWOWMWPXNUUFUUJWQWRUUEVBXDUUGX - LYLUUHUUIWNXKXLXMUUFWSYMCFYKYIUGUUEGUESTYJWTVIXAUCXDXEXTUDYAXBVSXCXTYGUDD - YAXODUMZXQYDXSYFUULXPYCMXODBXFXGUULXRYENXODKXFXGXHXIXJ $. + ovex cmpt wss subrgss 3ad2ant3 resmpt syl syl6eqr crg crngring pwsdiagrhm + cres 3ad2ant2 simp3 resrhm syl2anc eqeltrrd wf fvex eqeltri simpl1 elmapg + wb sylancr biimpa simplr ffvelrnd simpl2 pwselbasb mpbird evlseu riotacl2 + fmptd eqeltrd coeq1 eqeq1d anbi12d elrab sylib ) JUEUFZFUGUFZEFUHUIUFZUJZ + DUDUKZBULZMUMZXOKULZNUMZUSZUDLGUNUTZUOZUFDYAUFDBULZMUMZDKULZNUMZUSZUSXNDX + TUDYAUPZYBABCDEFGHUDIJKLMNOPQRSTUAUBUCUQXNXTUDYAURYHYBUFXNBGVAUIZLHGUDMNJ + KPYIVBZUAQXKXLXMVCXLXMHUGUFXKEFHRVDVEXNXLCJVFUTZUEUFZGUGUFXKXLXMVGCJVFVLZ + FYKUEGSVHVIXNACYKAUKZVJVKZVMZEWCZMHGUNUTZXNYQAEYOVMZMXNECVNZYQYSUMXMXKYTX + LECFTVOVPACEYOVQVRUBVSXNYPFGUNUTUFZXMYQYRUFXNFVTUFZYLUUAXLXKUUBXMFWAWDYMA + CFYPYKUEGSTYPVBWBVIXKXLXMWEFGHYPERWFWGWHXNAJIYKYNIUKZUIZVMZYINXNYNJUFZUSZ + UUEYIUFZYKCUUEWIZUUGIYKUUDCUUEUUGUUCYKUFZUSJCYNUUCUUGUUJJCUUCWIZUUGCUEUFX + KUUJUUKWNCFVAUIUETFVAWJWKXKXLXMUUFWLCJUUCUEUEWMWOWPXNUUFUUJWQWRUUEVBXDUUG + XLYLUUHUUIWNXKXLXMUUFWSYMCFYKYIUGUUEGUESTYJWTVIXAUCXDXBXTUDYAXCVRXEXTYGUD + DYAXODUMZXQYDXSYFUULXPYCMXODBXFXGUULXRYENXODKXFXGXHXIXJ $. $} ${ @@ -244390,15 +244613,15 @@ polynomial algebra (with coefficients in ` R ` ) to a function from by Stefan O'Rear, 13-Mar-2015.) $) evlssca $p |- ( ph -> ( Q ` ( A ` X ) ) = ( ( B ^m I ) X. { X } ) ) $= ( vx vy ccom cfv cmap co cv csn cxp cmpt wceq cmvr cpws crh wcel cvv ccrg - wa csubrg elex syl eqid evlsval2 syl3anc simprd simpld fveq1d wf subrgrng - cbs crg mplasclf wss subrgss ressbas2 3syl feq2d mpbird fvco3 sneq xpeq2d - syl2anc ovex snex xpex fvmpt 3eqtr3d ) AKDBUCZUDZKUAECHUEUFZUAUGZUHZUIZUJ - ZUDZKBUDDUDZWJKUHZUIZAKWHWNAWHWNUKZDHGULUFZUCUAHUBWJWKUBUGUDUJUJZUKZADJFW - JUMUFZUNUFUOZWSXBURZAHUPUOZFUQUOEFUSUDUOZXDXEURAHIUOXFQHIUTVARSUABCDEFXCG - UBHWTJWNXALMWTVBNXCVBOPWNVBZXAVBVCVDVEVFVGAEJVJUDZBVHZKEUOZWIWPUKAXJGVJUD - ZXIBVHABXIJGHXLIMXIVBXLVBPQAXGGVKUOSEFGNVIVAVLAEXLXIBAXGECVMEXLUKSECFOVNE - CGFNOVOVPVQVRTEXIKDBVSWBAXKWOWRUKTUAKWMWREWNWKKUKWLWQWJWKKVTWAXHWJWQCHUEW - CKWDWEWFVAWG $. + wa csubrg elex syl evlsval2 syl3anc simprd simpld fveq1d cbs wf subrgring + eqid crg mplasclf subrgss ressbas2 3syl feq2d mpbird fvco3 syl2anc xpeq2d + wss sneq ovex snex xpex fvmpt 3eqtr3d ) AKDBUCZUDZKUAECHUEUFZUAUGZUHZUIZU + JZUDZKBUDDUDZWJKUHZUIZAKWHWNAWHWNUKZDHGULUFZUCUAHUBWJWKUBUGUDUJUJZUKZADJF + WJUMUFZUNUFUOZWSXBURZAHUPUOZFUQUOEFUSUDUOZXDXEURAHIUOXFQHIUTVARSUABCDEFXC + GUBHWTJWNXALMWTVJNXCVJOPWNVJZXAVJVBVCVDVEVFAEJVGUDZBVHZKEUOZWIWPUKAXJGVGU + DZXIBVHABXIJGHXLIMXIVJXLVJPQAXGGVKUOSEFGNVIVAVLAEXLXIBAXGECWAEXLUKSECFOVM + ECGFNOVNVOVPVQTEXIKDBVRVSAXKWOWRUKTUAKWMWREWNWKKUKWLWQWJWKKWBVTXHWJWQCHUE + WCKWDWEWFVAWG $. $} ${ @@ -244416,14 +244639,14 @@ polynomial algebra (with coefficients in ` R ` ) to a function from evlsvar $p |- ( ph -> ( Q ` ( V ` X ) ) = ( g e. ( B ^m I ) |-> ( g ` X ) ) ) $= ( cfv vx ccom cmap co cmpt cmpl cascl csn cxp wceq cpws crh wcel cvv ccrg - cv wa csubrg elex syl eqid evlsval2 syl3anc simprd fveq1d wfn wf subrgrng - cbs crg mvrf2 ffn fvco2 syl2anc fveq2 mpteq2dv ovex mptex fvmpt 3eqtr3d ) - AKCIUBZTZKUAHGBHUCUDZUAUPZGUPZTZUEZUEZTZKITCTZGWCKWETZUEZAKWAWHACHFUFUDZU - GTZUBUADWCWDUHUIUEZUJZWAWHUJZACWMEWCUKUDZULUDUMZWPWQUQZAHUNUMZEUOUMDEURTU - MZWSWTUQAHJUMXAPHJUSUTQRUAWNBCDEWRFGHIWMWOWHLWMVAZMNWRVAOWNVAWOVAWHVAZVBV - CVDVDVEAIHVFZKHUMZWBWJUJAHWMVITZIVGXEAXGWMFHIJXCMXGVAPAXBFVJUMRDEFNVHUTVK - HXGIVLUTSHCIKVMVNAXFWIWLUJSUAKWGWLHWHWDKUJGWCWFWKWDKWEVOVPXDGWCWKBHUCVQVR - VSUTVT $. + cv wa csubrg elex syl evlsval2 syl3anc simprd fveq1d wfn cbs wf subrgring + eqid crg mvrf2 ffn fvco2 syl2anc fveq2 mpteq2dv ovex mptex fvmpt 3eqtr3d + ) AKCIUBZTZKUAHGBHUCUDZUAUPZGUPZTZUEZUEZTZKITCTZGWCKWETZUEZAKWAWHACHFUFUD + ZUGTZUBUADWCWDUHUIUEZUJZWAWHUJZACWMEWCUKUDZULUDUMZWPWQUQZAHUNUMZEUOUMDEUR + TUMZWSWTUQAHJUMXAPHJUSUTQRUAWNBCDEWRFGHIWMWOWHLWMVIZMNWRVIOWNVIWOVIWHVIZV + AVBVCVCVDAIHVEZKHUMZWBWJUJAHWMVFTZIVGXEAXGWMFHIJXCMXGVIPAXBFVJUMRDEFNVHUT + VKHXGIVLUTSHCIKVMVNAXFWIWLUJSUAKWGWLHWHWDKUJGWCWFWKWDKWEVOVPXDGWCWKBHUCVQ + VRVSUTVT $. $} ${ @@ -244444,10 +244667,10 @@ polynomial algebra (with coefficients in ` R ` ) to a function from $( The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) $) evlrhm $p |- ( ( I e. V /\ R e. CRing ) -> Q e. ( W RingHom T ) ) $= - ( wcel ccrg wa cress co cmpl crh adantl eqid cfv crg crngrng subrgid wceq - csubrg syl evlval evlsrhm mpd3an3 ressid oveq2d syl6eqr oveq1d eleqtrd ) - EFLZCMLZNZBECAOPZQPZDRPZGDRPUPUQACUFUALZBVALURCUBLZVBUQVCUPCUCSACIUDUGABA - CDUSEFUTABCEHIUHUTTUSTKIUIUJURUTGDRURUTECQPGURUSCEQUQUSCUEUPACMIUKSULJUMU + ( wcel ccrg wa cress co cmpl crh adantl eqid cfv crg crngring subrgid syl + csubrg evlval evlsrhm mpd3an3 wceq ressid oveq2d syl6eqr oveq1d eleqtrd ) + EFLZCMLZNZBECAOPZQPZDRPZGDRPUPUQACUFUALZBVALURCUBLZVBUQVCUPCUCSACIUDUEABA + CDUSEFUTABCEHIUGUTTUSTKIUHUIURUTGDRURUTECQPGURUSCEQUQUSCUJUPACMIUKSULJUMU NUO $. $} @@ -244469,12 +244692,12 @@ polynomial algebra (with coefficients in ` R ` ) to a function from 12-Sep-2019.) $) evlsscasrng $p |- ( ph -> ( Q ` ( A ` X ) ) = ( O ` ( C ` X ) ) ) $= ( cfv ces co cmap csn cxp cress cmpl cascl ccrg wcel ressid eqcomd oveq2d - wceq syl syl5eq fveq2d fveq1d eqid crg csubrg crngrng subrgid wss subrgss - 3syl sseldd evlssca eqtrd evlval a1i 3eqtr4rd ) ANDUGZCJHUHUIUGZUGZCJUJUI - NUKULZVTKUGNBUGFUGAWBNJHCUMUIZUNUIZUOUGZUGZWAUGWCAVTWGWAANDWFADEUOUGWFUBA - EWEUOAEJHUNUIWESAHWDJUNAHUPUQZHWDVAUDWHWDHCHUPTURUSVBUTVCVDVCVEVDAWFCWACH - WDJLWENWAVFWEVFWDVFTWFVFUCUDAWHHVGUQCHVHUGZUQUDHVICHTVJVMAGCNAGWIUQGCVKUE - GCHTVLVBUFVNVOVPAVTKWAKWAVAACKHJPTVQVRVEABCFGHIJLMNOQRTUAUCUDUEUFVOVS $. + wceq syl syl5eq fveq2d fveq1d eqid crg csubrg crngring subrgid wss sseldd + 3syl subrgss evlssca eqtrd evlval a1i 3eqtr4rd ) ANDUGZCJHUHUIUGZUGZCJUJU + INUKULZVTKUGNBUGFUGAWBNJHCUMUIZUNUIZUOUGZUGZWAUGWCAVTWGWAANDWFADEUOUGWFUB + AEWEUOAEJHUNUIWESAHWDJUNAHUPUQZHWDVAUDWHWDHCHUPTURUSVBUTVCVDVCVEVDAWFCWAC + HWDJLWENWAVFWEVFWDVFTWFVFUCUDAWHHVGUQCHVHUGZUQUDHVICHTVJVMAGCNAGWIUQGCVKU + EGCHTVNVBUFVLVOVPAVTKWAKWAVAACKHJPTVQVRVEABCFGHIJLMNOQRTUAUCUDUEUFVOVS $. $} ${ @@ -244488,10 +244711,10 @@ polynomial algebra (with coefficients in ` R ` ) to a function from $( Simple polynomial evaluation maps scalars to constant functions. (Contributed by AV, 12-Sep-2019.) $) evlsca $p |- ( ph -> ( Q ` ( A ` X ) ) = ( ( B ^m I ) X. { X } ) ) $= - ( co cfv eqid wcel cress cmpl cascl ces cmap csn cxp ccrg crngrng subrgid - crg csubrg 3syl evlsscasrng evlssca eqtr3d ) AIFECUAQZUBQZUCRZRCFEUDQRZRI - BRDRCFUEQIUFUGAUSCBHUTCEUQFDGURIUTSZJURSZUQSZKLUSSZMNOAEUHTEUKTCEULRTOEUI - CELUJUMZPUNAUSCUTCEUQFGURIVAVBVCLVDNOVEPUOUP $. + ( co cfv eqid wcel cress cmpl cascl ces cmap csn cxp ccrg csubrg crngring + crg subrgid 3syl evlsscasrng evlssca eqtr3d ) AIFECUAQZUBQZUCRZRCFEUDQRZR + IBRDRCFUEQIUFUGAUSCBHUTCEUQFDGURIUTSZJURSZUQSZKLUSSZMNOAEUHTEUKTCEUIRTOEU + JCELULUMZPUNAUSCUTCEUQFGURIVAVBVCLVDNOVEPUOUP $. $} ${ @@ -244510,12 +244733,12 @@ polynomial algebra (with coefficients in ` R ` ) to a function from itself. (Contributed by AV, 12-Sep-2019.) $) evlsvarsrng $p |- ( ph -> ( Q ` ( V ` X ) ) = ( O ` ( V ` X ) ) ) $= ( vg cfv cmap co cv cmpt evlsvar ces cmvr wceq evlval a1i fveq1d subrgmvr - eqid ccrg wcel ressid syl eqcomd oveq2d 3eqtr2d fveq2d crg csubrg crngrng - cress subrgid 3syl 3eqtrrd eqtrd ) AKJUBZDUBUACHUCUDKUAUEUBUFZVLIUBZACDEF - GUAHJBKLNOPQRSTUGAVNVLCHFUHUDUBZUBKHFCVGUDZUIUDZUBZVOUBVMAVLIVOIVOUJACIFH - MPUKULUMAVLVRVOAKJVQAJHGUIUDZHFUIUDZVQJVSUJANULAFEGHVTBVTUOQSOUNAFVPHUIAV - PFAFUPUQZVPFUJRCFUPPURUSUTVAVBUMVCACVOCFVPUAHVQBKVOUOVQUOVPUOPQRAWAFVDUQC - FVEUBUQRFVFCFPVHVITUGVJVK $. + cress eqid ccrg wcel ressid syl eqcomd oveq2d 3eqtr2d fveq2d crg crngring + csubrg subrgid 3syl 3eqtrrd eqtrd ) AKJUBZDUBUACHUCUDKUAUEUBUFZVLIUBZACDE + FGUAHJBKLNOPQRSTUGAVNVLCHFUHUDUBZUBKHFCUOUDZUIUDZUBZVOUBVMAVLIVOIVOUJACIF + HMPUKULUMAVLVRVOAKJVQAJHGUIUDZHFUIUDZVQJVSUJANULAFEGHVTBVTUPQSOUNAFVPHUIA + VPFAFUQURZVPFUJRCFUQPUSUTVAVBVCUMVDACVOCFVPUAHVQBKVOUPVQUPVPUPPQRAWAFVEUR + CFVGUBURRFVFCFPVHVITUGVJVK $. $} ${ @@ -244530,11 +244753,11 @@ polynomial algebra (with coefficients in ` R ` ) to a function from (Contributed by AV, 12-Sep-2019.) $) evlvar $p |- ( ph -> ( Q ` ( V ` X ) ) = ( g e. ( B ^m I ) |-> ( g ` X ) ) ) $= - ( cress co cfv eqid wcel cmvr ces cmap cv cmpt crg csubrg crngrng subrgid - ccrg 3syl evlsvarsrng evlsvar subrgmvr fveq1d eqcomd fveq2d 3eqtr3rd ) AI - FDBPQZUAQZRZBFDUBQRZRVACREBFUCQIEUDRUEIGRZCRAHBVBBDUSFCUTIVBSZJUTSZUSSZLM - NADUJTDUFTBDUGRTNDUHBDLUIUKZOULABVBBDUSEFUTHIVDVEVFLMNVGOUMAVAVCCAVCVAAIG - UTADBUSFGHKMVGVFUNUOUPUQUR $. + ( cress co cfv eqid wcel cmvr ces cmap cmpt ccrg crg csubrg crngring 3syl + cv subrgid evlsvarsrng evlsvar subrgmvr fveq1d eqcomd fveq2d 3eqtr3rd ) A + IFDBPQZUAQZRZBFDUBQRZRVACREBFUCQIEUJRUDIGRZCRAHBVBBDUSFCUTIVBSZJUTSZUSSZL + MNADUETDUFTBDUGRTNDUHBDLUKUIZOULABVBBDUSEFUTHIVDVEVFLMNVGOUMAVAVCCAVCVAAI + GUTADBUSFGHKMVGVFUNUOUPUQUR $. $} ${ @@ -244549,15 +244772,15 @@ polynomial algebra (with coefficients in ` R ` ) to a function from Mario Carneiro, 19-Mar-2015.) $) mpfconst $p |- ( ph -> ( ( B ^m I ) X. { X } ) e. Q ) $= ( co cfv cress eqid cbs wcel cmap csn cxp ces crn cmpl evlssca wfn cpws - cascl crh wf ccrg csubrg evlsrhm syl3anc rhmf ffn 3syl crg subrgrng syl - csca wa mplrng mpllmod asclf syl2anc wss wceq subrgss ressbas2 cvv ovex - a1i mplsca fveq2d eqtrd eleqtrd ffvelrnd fnfvelrn eqeltrrd syl6eleqr ) - ABFUAOZHUBUCZDFEUDOPZUEZCAHFEDQOZUFOZUJPZPZWFPZWEWGAWJBWFDEWHFGWIHWFRZW - IRZWHRZIWJRZKLMNUGAWFWISPZUHZWKWQTWLWGTAWFWIEWDUIOZUKOTZWQWSSPZWFULWRAF - GTZEUMTDEUNPTZWTKLMBWFDEWSWHFGWIWMWNWOWSRIUOUPWQXAWIWSWFWQRZXARUQWQXAWF - URUSAWIVCPZSPZWQHWJAXBWHUTTZXFWQWJULKAXCXGMDEWHWOVAVBXBXGVDWJWQXEXFWIWP - XERWIWHFGWNVEWIWHFGWNVFXFRXDVGVHAHDXFNADWHSPZXFAXCDBVIDXHVJMDBEIVKDBWHE - WOIVLUSAWHXESAWIWHFGVMWNKWHVMTAEDQVNVOVPVQVRVSVTWQWKWFWAVHWBJWC $. + crh wf ccrg csubrg evlsrhm syl3anc rhmf ffn 3syl csca crg subrgring syl + cascl wa mplring mpllmod syl2anc wss wceq subrgss ressbas2 cvv ovex a1i + asclf mplsca fveq2d eqtrd eleqtrd ffvelrnd fnfvelrn eqeltrrd syl6eleqr + ) ABFUAOZHUBUCZDFEUDOPZUEZCAHFEDQOZUFOZVCPZPZWFPZWEWGAWJBWFDEWHFGWIHWFR + ZWIRZWHRZIWJRZKLMNUGAWFWISPZUHZWKWQTWLWGTAWFWIEWDUIOZUJOTZWQWSSPZWFUKWR + AFGTZEULTDEUMPTZWTKLMBWFDEWSWHFGWIWMWNWOWSRIUNUOWQXAWIWSWFWQRZXARUPWQXA + WFUQURAWIUSPZSPZWQHWJAXBWHUTTZXFWQWJUKKAXCXGMDEWHWOVAVBXBXGVDWJWQXEXFWI + WPXERWIWHFGWNVEWIWHFGWNVFXFRXDVOVGAHDXFNADWHSPZXFAXCDBVHDXHVIMDBEIVJDBW + HEWOIVKURAWHXESAWIWHFGVLWNKWHVLTAEDQVMVNVPVQVRVSVTWQWKWFWAVGWBJWC $. $} $d B f $. $d I f $. $d J f $. $d R f $. $d S f $. @@ -244566,12 +244789,12 @@ polynomial algebra (with coefficients in ` R ` ) to a function from Mario Carneiro, 20-Mar-2015.) $) mpfproj $p |- ( ph -> ( f e. ( B ^m I ) |-> ( f ` J ) ) e. Q ) $= ( co cfv eqid cbs wcel cress cmvr ces cmap cmpt evlsvar crn cmpl wfn cpws - cv crh wf ccrg csubrg evlsrhm syl3anc rhmf ffn 3syl crg subrgrng fnfvelrn - syl mvrcl syl2anc syl6eleqr eqeltrrd ) AHGEDUAPZUBPZQZDGEUCPQZQZFBGUDPZHF - UKQUECABVLDEVIFGVJIHVLRZVJRZVIRZJLMNOUFAVMVLUGZCAVLGVIUHPZSQZUIZVKVTTVMVR - TAVLVSEVNUJPZULPTZVTWBSQZVLUMWAAGITEUNTDEUOQTZWCLMNBVLDEWBVIGIVSVOVSRZVQW - BRJUPUQVTWDVSWBVLVTRZWDRURVTWDVLUSUTAVTVSVIGVJIHWFVPWGLAWEVIVATNDEVIVQVBV - DOVEVTVKVLVCVFKVGVH $. + cv crh wf ccrg csubrg evlsrhm syl3anc rhmf ffn crg subrgring syl fnfvelrn + 3syl mvrcl syl2anc syl6eleqr eqeltrrd ) AHGEDUAPZUBPZQZDGEUCPQZQZFBGUDPZH + FUKQUECABVLDEVIFGVJIHVLRZVJRZVIRZJLMNOUFAVMVLUGZCAVLGVIUHPZSQZUIZVKVTTVMV + RTAVLVSEVNUJPZULPTZVTWBSQZVLUMWAAGITEUNTDEUOQTZWCLMNBVLDEWBVIGIVSVOVSRZVQ + WBRJUPUQVTWDVSWBVLVTRZWDRURVTWDVLUSVDAVTVSVIGVJIHWFVPWGLAWEVIUTTNDEVIVQVA + VBOVEVTVKVLVCVFKVGVH $. $} ${ @@ -244581,12 +244804,12 @@ polynomial algebra (with coefficients in ` R ` ) to a function from mpfsubrg $p |- ( ( I e. _V /\ S e. CRing /\ R e. ( SubRing ` S ) ) -> Q e. ( SubRing ` ( S ^s ( ( Base ` S ) ^m I ) ) ) ) $= ( cvv wcel ccrg csubrg cfv w3a ces co cress cmpl cbs cima cmap eqid crg - cpws crn crh wf wfn wceq evlsrhm rhmf fnima 4syl syl6reqr subrgrng mplrng - ffn sylan2 3adant2 subrgid syl rhmima syl2anc eqeltrd ) DFGZCHGZBCIJGZKZA - BDCLMJZDCBNMZOMZPJZQZCCPJZDRMUAMZIJZVEVJVFUBZAVEVFVHVLUCMGZVIVLPJZVFUDVFV - IUEVJVNUFVKVFBCVLVGDFVHVFSVHSZVGSZVLSVKSUGZVIVPVHVLVFVISZVPSUHVIVPVFUNVIV - FUIUJEUKVEVOVIVHIJGZVJVMGVSVEVHTGZWAVBVDWBVCVDVBVGTGWBBCVGVRULVHVGDFVQUMU - OUPVIVHVTUQURVFVHVLVIUSUTVA $. + cpws crn crh wf wfn wceq evlsrhm rhmf ffn 4syl syl6reqr subrgring mplring + fnima sylan2 3adant2 subrgid syl rhmima syl2anc eqeltrd ) DFGZCHGZBCIJGZK + ZABDCLMJZDCBNMZOMZPJZQZCCPJZDRMUAMZIJZVEVJVFUBZAVEVFVHVLUCMGZVIVLPJZVFUDV + FVIUEVJVNUFVKVFBCVLVGDFVHVFSVHSZVGSZVLSVKSUGZVIVPVHVLVFVISZVPSUHVIVPVFUIV + IVFUNUJEUKVEVOVIVHIJGZVJVMGVSVEVHTGZWAVBVDWBVCVDVBVGTGWBBCVGVRULVHVGDFVQU + MUOUPVIVHVTUQURVFVHVLVIUSUTVA $. ${ mpff.b $e |- B = ( Base ` S ) $. @@ -244662,63 +244885,63 @@ polynomial algebra (with coefficients in ` R ` ) to a function from wa crg elpreima simpld syl3anc cof ovex ffvelrnd eqtrd fnfvelrn syl6eleqr fvimacnvi jca wi fvex eleq1 vex elab syl5bbr anbi12d oveq12 eleq1d vtocl2 imbi12d syl12anc eqeltrd mpbir2and adantlr cmgp mgpplusg eleq2d ralrimiva - csn cxp wral cbvralv sylib r19.21bi simpr cmpt simp1d simp2d crngrng cghm - simp3d subrgcrng mplrng simprl simprr rngacl rhmghm a1i pwsplusgval simpl - ghmlin bi2anan9 anbi2d rngcl cmhm rhmmhm mgpbas mhmlin pwsmulrval mplassa - csca casa asclrhm mplsca fveq2d wss subrgss ressbas2 biimpar evlssca snex - biimpa xpex sneq xpeq2d syldan mvrcl evlsvar mptex sylibr mpteq2dv mplind - fveq2 syl5ibcom rexlimdva mpd elabg ) AKBJUTZVAZIAUQVBZOTPVCVDVEZVEZKVFZU - QTPOVGVDZVHVDZVIVEZVNZUWMAKUWOVJZVAZUXAAKNUXBUPUDVKAUWOUWTVLZUXCUXAVMAUWT - PLTVOVDZVPVDZVIVEZUWOVQZUXDAUWOUWSUXFVRVDVAZUXHATVSVAZPVTVAZOPWAVEVAZWBZU - XIAKNVAZUXMUPNOPTKUDWCWDZLUWOOPUXFUWRTVSUWSUWOWEZUWSWEZUWRWEZUXFWEZUAWFWD - ZUWTUXGUWSUXFUWOUWTWEZUXGWEZWHWDZUWTUXGUWOWIWDZUQUWTKUWOWGWDWJAUWQUWMUQUW - TAUWNUWTVAZXAZUWPUWLVAZUWQUWMUYFUWOWKZUWNUWOWLUWLWMZVAUYGAUYHUYEAUXHUYHUY - CUWTUXGUWOWNWDZWOUYFURUSUWTUWSWPVEZUWSWQVEZUWRUWSWRVEZUYITUWRVIVEZTUWRWSV - DZUWNUWSUYNWEUYOWEZUXQUYLWEZUYMWEZUYKWEZUYAAURVBZUYIVAZUSVBZUYIVAZXAZUYTV - UBUYLVDZUYIVAZUYEAVUDXAZVUFVUEUWTVAZVUEUWOVEZUWLVAZVUGUWSXBVAZUYTUWTVAZVU - 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(Contributed by Mario Carneiro, 9-Feb-2015.) $) ply1crng $p |- ( R e. CRing -> P e. CRing ) $= - ( ccrg wcel cps1 cfv c1o cmpl co cbs csubrg eqid psr1crng ply1bas crngrng - crg ply1subrg syl syl5eqelr ply1val subrgcrng syl2anc ) BDEZBFGZDEHBIJKGZ - UELGZEADEBUEUEMZNUDUFAKGZUGABUEUICUHUIMZOUDBQEUIUGEBPABUEUICUHUJRSTUFUEAA - BUECUHUAUBUC $. + ( ccrg wcel cps1 cfv c1o cmpl co cbs csubrg psr1crng ply1bas crg crngring + eqid ply1subrg syl syl5eqelr ply1val subrgcrng syl2anc ) BDEZBFGZDEHBIJKG + ZUELGZEADEBUEUEQZMUDUFAKGZUGABUEUICUHUIQZNUDBOEUIUGEBPABUEUICUHUJRSTUFUEA + ABUECUHUAUBUC $. $( The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.) $) ply1assa $p |- ( R e. CRing -> P e. AssAlg ) $= - ( ccrg wcel casa cbs cfv cps1 clss crg crngrng eqid ply1subrg syl ply1lss - csubrg cur co cress wss wa psr1assa subrg1cl subrgss cmpl ply1val ply1bas - wb c1o oveq2i eqtr4i issubassa syl3anc mpbir2and ) BDEZAFEZAGHZBIHZQHEZUR - USJHZEZUPBKEZUTBLZABUSURCUSMZURMZNOZUPVCVBVDABUSURCVEVFPOUPUSFEUSRHZUREZU - RUSGHZUAZUQUTVBUBUIBUSVEUCUPUTVIVGURUSVHVHMZUDOUPUTVKVGURVJUSVJMZUEOURAVH - VAVJUSAUSUJBUFSGHZTSUSURTSABUSCVEUGURVNUSTABUSURCVEVFUHUKULVAMVMVLUMUNUO + ( ccrg wcel casa cbs cfv cps1 csubrg clss crg crngring eqid ply1subrg syl + ply1lss cur co cress wss wa wb psr1assa subrg1cl subrgss c1o cmpl ply1val + ply1bas oveq2i eqtr4i issubassa syl3anc mpbir2and ) BDEZAFEZAGHZBIHZJHEZU + RUSKHZEZUPBLEZUTBMZABUSURCUSNZURNZOPZUPVCVBVDABUSURCVEVFQPUPUSFEUSRHZUREZ + URUSGHZUAZUQUTVBUBUCBUSVEUDUPUTVIVGURUSVHVHNZUEPUPUTVKVGURVJUSVJNZUFPURAV + HVAVJUSAUSUGBUHSGHZTSUSURTSABUSCVEUIURVNUSTABUSURCVEVFUJUKULVANVMVLUMUNUO $. $} @@ -245458,16 +245681,16 @@ series in the subring which are also polynomials (in the parent gsumply1subr $p |- ( ph -> ( S gsum F ) = ( U gsum F ) ) $= ( cfv wcel cvv vt vs co cress csubrg csubmnd subrgply1 subrgsubg subgsubm cgsu syl 3syl eqid gsumsubm wf fex syl2anc ovex a1i cpl1 fvex eqeltri cbs - csubg oveq2i ressply1bas eqcomd crg cmgm subrgrng ringmgm cv cplusg simpl - wa wceq eleq2d biimpcd adantr impcom adantl ressply1add syl12anc wfun crn - wi ffun wss frn sseqtrd gsummgmpropd eqtrd ) AEHUJUCECUDUCZHUJUCGHUJUCABC - HEWMJPAFDUERSZCEUERSZCEUFRSZOCDEFGIKLMNUGZWOCEVDRSWPCEUHCEUIUKULQWMUMZUNA - UAHWMGTTTUBABCHUOZBJSHTSQPBCJHUPUQWMTSAECUDURUSGTSAGIUTRTMIUTVAVBUSAGVCRZ - WMVCRZAWTWMDEFGIKLMWTUMOCWTEUDNVEVFVGAWNWMVHSZWMVISOWNWOXBWQCEWMWRVJUKWMV - KULAUBVLZXASZUAVLZXASZVOZVOZXCXEGVMRUCZXCXEWMVMRUCZXHAXCCSZXECSZXIXJVPAXG - VNXGAXKXDAXKWFXFAXDXKAXACXCACXAACWMDEFGIKLMNOWRVFZVGZVQVRVSVTXGAXLXFAXLWF - XDAXFXLAXACXEXNVQVRWAVTACWMDEFGIXCXEKLMNOWRWBWCVGAWSHWDQBCHWGUKAHWEZCXAAW - SXOCWHQBCHWIUKXMWJWKWL $. + csubg oveq2i ressply1bas eqcomd crg cmgm subrgring ringmgm cv cplusg wceq + wa simpl wi eleq2d biimpcd adantr impcom adantl ressply1add syl12anc wfun + ffun crn wss frn sseqtrd gsummgmpropd eqtrd ) AEHUJUCECUDUCZHUJUCGHUJUCAB + CHEWMJPAFDUERSZCEUERSZCEUFRSZOCDEFGIKLMNUGZWOCEVDRSWPCEUHCEUIUKULQWMUMZUN + AUAHWMGTTTUBABCHUOZBJSHTSQPBCJHUPUQWMTSAECUDURUSGTSAGIUTRTMIUTVAVBUSAGVCR + ZWMVCRZAWTWMDEFGIKLMWTUMOCWTEUDNVEVFVGAWNWMVHSZWMVISOWNWOXBWQCEWMWRVJUKWM + VKULAUBVLZXASZUAVLZXASZVOZVOZXCXEGVMRUCZXCXEWMVMRUCZXHAXCCSZXECSZXIXJVNAX + GVPXGAXKXDAXKVQXFAXDXKAXACXCACXAACWMDEFGIKLMNOWRVFZVGZVRVSVTWAXGAXLXFAXLV + QXDAXFXLAXACXEXNVRVSWBWAACWMDEFGIXCXEKLMNOWRWCWDVGAWSHWEQBCHWFUKAHWGZCXAA + WSXOCWHQBCHWIUKXMWJWKWL $. $} ${ @@ -245540,41 +245763,41 @@ series in the subring which are also polynomials (in the parent ( F .xb G ) = ( G .x. F ) ) $= ( wcel co cfv cvv vb va ve vd vc vf crg w3a cv ccnv cn cima cfn cmap crab cn0 cle cofr wbr cmin cof cmulr cmpt cgsu ccom cbs c0g eqid ccmn 3ad2ant1 - wa rngcmn adantr ovex rabex a1i simpll1 wf psrelbas elrabi ffvelrn syl2an - simp3 simp2 ad2antrr ssrab2 cmps reldmpsr strov2rcl 3ad2ant3 simplr simpr - psrbagconcl sseldi ffvelrnd rngcl fmptd wfun csupp wss cfsupp mptexg mp1i - syl3anc funmpt fvex psrbaglefi sylan suppssdm dmmptss sstri suppssfifsupp - syl32anc wf1o psrbagconf1o gsumf1o eqidd wceq fveq2 fveq2d oveq12d fmptco - cdm oveq2 psrbagf cc nn0cn nncan adantl caonncan oveq2d opprmul mpteq2dva - syl6eqr eqtrd coppr cplusg psrmulfval fveq2i eleqtrd id eqeltri gsumpropd - mptex opprbas oppradd 3eqtrd psrbaspropd eqtri 3eqtr4g 3eqtr4rd ) BUGQZFA - QZGAQZUHZUAUBUIUJUKULUMQZUBUPHUNRZUOZBUCUDUIUAUIZUQURUSZUDUURUOZUCUIZGSZU - USUVBUTVAZRZFSZBVBSZRZVCZVDRZVCUAUURCUEUVAUEUIZFSZUUSUVKUVDRZGSZCVBSZRZVC - ZVDRZVCGFERFGDRUUOUAUURUVJUVRUUOUUSUURQZVKZUVJBUVIUEUVAUVMVCZVEZVDRBUVQVD - RZUVRUVTUVABVFSZUVAUVIBUWATBVGSZUWDVHZUWEVHUUOBVIQZUVSUULUUMUWGUUNBVLVJVM - 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UVKFUXLUCUFUUSUVKUPHUTTUXMUVTHUPUUSVRZUXKUUOUWQUVSUXOUWRUURUBUUSHTUWNYEXH - VMUVTUWQUVKUURQHUPUVKVRUXKUUOUWQUVSUWRVMUUTUDUVKUURVTUURUBUVKHTUWNYEWBUVB - UPQZUFUIZUPQZVKUVBUVBUXQUTRUTRUXQXRZUXLUXPUVBYFQUXQYFQUXSUXRUVBYGUXQYGUVB - UXQYHWBYIYJXTYKUWDBUVOUVGCUVLUVNUWFUWTLUVOVHZYLYNYMYOYKUUOUWCUVRXRZUVSUUL - UUMUYAUUNUULUVQBCTUGTUVQTQUULUEUVAUVPUWIUUDVPUULUUACTQUULCBYPSTLBYPXFUUBV - PUWDCVFSXRZUULUWDBCLUWFUUEZVPBYQSZCYQSXRUULUYDBCLUYDVHUUFVPUUCVJVMUUGYMUU - OUCUDAUURBIEUVGUBUAGFHKPUWTNUWNUWOUWPYRUUOUEUDJVFSZUURCJDUVOUBUAFGHMUYEVH - UXTOUWNUUOFAUYEUWPUUOHBWGRZVFSZHCWGRZVFSAUYEUUOBCHUYBUUOUYCVPUUHAIVFSUYGP - IUYFVFKYSUUIJUYHVFMYSUUJZYTUUOGAUYEUWOUYIYTYRUUK $. + wa ringcmn adantr ovex rabex a1i simpll1 wf simp3 psrelbas elrabi ffvelrn + syl2an simp2 ad2antrr ssrab2 cmps reldmpsr strov2rcl 3ad2ant3 psrbagconcl + simplr simpr syl3anc sseldi ffvelrnd ringcl fmptd csupp wss cfsupp mptexg + wfun mp1i funmpt fvex psrbaglefi sylan cdm suppssdm dmmptss suppssfifsupp + sstri syl32anc wf1o psrbagconf1o gsumf1o eqidd fveq2 oveq2 fveq2d oveq12d + wceq fmptco psrbagf cc nn0cn nncan adantl caonncan oveq2d opprmul syl6eqr + mpteq2dva eqtrd coppr cplusg psrmulfval fveq2i eleqtrd id eqeltri opprbas + mptex oppradd gsumpropd 3eqtrd psrbaspropd eqtri 3eqtr4g 3eqtr4rd ) BUGQZ + FAQZGAQZUHZUAUBUIUJUKULUMQZUBUPHUNRZUOZBUCUDUIUAUIZUQURUSZUDUURUOZUCUIZGS + ZUUSUVBUTVAZRZFSZBVBSZRZVCZVDRZVCUAUURCUEUVAUEUIZFSZUUSUVKUVDRZGSZCVBSZRZ + VCZVDRZVCGFERFGDRUUOUAUURUVJUVRUUOUUSUURQZVKZUVJBUVIUEUVAUVMVCZVEZVDRBUVQ + VDRZUVRUVTUVABVFSZUVAUVIBUWATBVGSZUWDVHZUWEVHUUOBVIQZUVSUULUUMUWGUUNBVLVJ + VMUVATQZUVTUUTUDUURUUPUBUUQUPHUNVNVOVOZVPUVTUCUVAUVHUWDUVIUVTUVBUVAQZVKZU + ULUVCUWDQZUVFUWDQUVHUWDQUULUUMUUNUVSUWJVQUVTUURUWDGVRZUVBUURQUWLUWJUUOUWM + UVSUUOAUURBIUBHUWDGKUWFUURVHZPUULUUMUUNVSZVTVMUUTUDUVBUURWAUURUWDUVBGWBWC + UWKUURUWDUVEFUUOUURUWDFVRUVSUWJUUOAUURBIUBHUWDFKUWFUWNPUULUUMUUNWDZVTWEUW + KUVAUURUVEUUTUDUURWFUWKHTQZUVSUWJUVEUVAQUUOUWQUVSUWJUUNUULUWQUUMABIWGHGKP + WHWIWJZWEUUOUVSUWJWLUVTUWJWMUDUURUVAUBUUSHTUVBUWNUVAVHZWKWNWOWPUWDBUVGUVC + UVFUWFUVGVHZWQWNUVIVHZWRUVTUVITQZUVIXCZUWETQZUVAUMQZUVIUWEWSRZUVAWTZUVIUW + EXAUSUWHUXBUVTUWIUCUVAUVHTXBXDUXCUVTUCUVAUVHXEVPUXDUVTBVGXFVPUUOUWQUVSUXE + UWRUDUURUBUUSHTUWNXGXHUXGUVTUXFUVIXIUVAUVIUWEXJUCUVAUVHUVIUXAXKXMVPUVAUVI + TTUWEXLXNUUOUWQUVSUVAUVAUWAXOUWRUEUDUURUVAUBUUSHTUWNUWSXPXHXQUVTUWBUVQBVD + UVTUWBUEUVAUVNUUSUVMUVDRZFSZUVGRZVCUVQUVTUEUCUVAUVAUVMUVHUXJUWAUVIUVTUVKU + VAQZVKZUWQUVSUXKUVMUVAQUUOUWQUVSUXKUWRWEZUUOUVSUXKWLUVTUXKWMUDUURUVAUBUUS + HTUVKUWNUWSWKWNUVTUWAXRUVTUVIXRUVBUVMYCZUVCUVNUVFUXIUVGUVBUVMGXSUXNUVEUXH + FUVBUVMUUSUVDXTYAYBYDUVTUEUVAUXJUVPUXLUXJUVNUVLUVGRUVPUXLUXIUVLUVNUVGUXLU + XHUVKFUXLUCUFUUSUVKUPHUTTUXMUVTHUPUUSVRZUXKUUOUWQUVSUXOUWRUURUBUUSHTUWNYE + XHVMUVTUWQUVKUURQHUPUVKVRUXKUUOUWQUVSUWRVMUUTUDUVKUURWAUURUBUVKHTUWNYEWCU + VBUPQZUFUIZUPQZVKUVBUVBUXQUTRUTRUXQYCZUXLUXPUVBYFQUXQYFQUXSUXRUVBYGUXQYGU + VBUXQYHWCYIYJYAYKUWDBUVOUVGCUVLUVNUWFUWTLUVOVHZYLYMYNYOYKUUOUWCUVRYCZUVSU + ULUUMUYAUUNUULUVQBCTUGTUVQTQUULUEUVAUVPUWIUUDVPUULUUACTQUULCBYPSTLBYPXFUU + BVPUWDCVFSYCZUULUWDBCLUWFUUCZVPBYQSZCYQSYCUULUYDBCLUYDVHUUEVPUUFVJVMUUGYN + UUOUCUDAUURBIEUVGUBUAGFHKPUWTNUWNUWOUWPYRUUOUEUDJVFSZUURCJDUVOUBUAFGHMUYE + VHUXTOUWNUUOFAUYEUWPUUOHBWGRZVFSZHCWGRZVFSAUYEUUOBCHUYBUUOUYCVPUUHAIVFSUY + GPIUYFVFKYSUUIJUYHVFMYSUUJZYTUUOGAUYEUWOUYIYTYRUUK $. $} ${ @@ -245653,17 +245876,17 @@ series in the subring which are also polynomials (in the parent ${ $d ph x y $. $d I x y $. $d O x y $. $d R x y $. - opsrrng.o $e |- O = ( ( I ordPwSer R ) ` T ) $. - opsrrng.i $e |- ( ph -> I e. V ) $. - opsrrng.r $e |- ( ph -> R e. Ring ) $. - opsrrng.t $e |- ( ph -> T C_ ( I X. I ) ) $. + opsrring.o $e |- O = ( ( I ordPwSer R ) ` T ) $. + opsrring.i $e |- ( ph -> I e. V ) $. + opsrring.r $e |- ( ph -> R e. Ring ) $. + opsrring.t $e |- ( ph -> T C_ ( I X. I ) ) $. $( Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) $) - opsrrng $p |- ( ph -> O e. Ring ) $= - ( vx vy cmps crg wcel cfv cv cplusg proplem3 cmulr co eqid psrrng opsrbas - cbs eqidd wa opsrplusg opsrmulr rngpropd mpbid ) ADBMUAZNOENOABULDFULUBZH - IUCAKLULUEPZULEAUNUFABULCDEUMGJUDAKQUNOLQUNOUGZKLULRPERPABULCDEUMGJUHSAUO - KLULTPETPABULCDEUMGJUISUJUK $. + opsrring $p |- ( ph -> O e. Ring ) $= + ( vx vy cmps crg wcel cfv cv cplusg proplem3 cmulr eqid psrring cbs eqidd + co opsrbas wa opsrplusg opsrmulr ringpropd mpbid ) ADBMUEZNOENOABULDFULUA + ZHIUBAKLULUCPZULEAUNUDABULCDEUMGJUFAKQUNOLQUNOUGZKLULRPERPABULCDEUMGJUHSA + UOKLULTPETPABULCDEUMGJUISUJUK $. $( Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.) $) @@ -245676,22 +245899,22 @@ series in the subring which are also polynomials (in the parent $} ${ - psr1rng.s $e |- S = ( PwSer1 ` R ) $. + psr1ring.s $e |- S = ( PwSer1 ` R ) $. $( Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) $) - psr1rng $p |- ( R e. Ring -> S e. Ring ) $= - ( crg wcel c0 c1o con0 psr1val 1on a1i id cxp wss 0ss opsrrng ) ADEZAFGBH - ABCIGHEQJKQLFGGMZNQROKP $. + psr1ring $p |- ( R e. Ring -> S e. Ring ) $= + ( crg wcel c0 c1o con0 psr1val 1on a1i id cxp wss 0ss opsrring ) ADEZAFGB + HABCIGHEQJKQLFGGMZNQROKP $. $} ${ - ply1rng.p $e |- P = ( Poly1 ` R ) $. + ply1ring.p $e |- P = ( Poly1 ` R ) $. $( Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) $) - ply1rng $p |- ( R e. Ring -> P e. Ring ) $= + ply1ring $p |- ( R e. Ring -> P e. Ring ) $= ( crg wcel c1o cmpl co cbs cfv cps1 csubrg eqid ply1bas ply1subrg ply1val - syl5eqelr subrgrng syl ) BDEZFBGHIJZBKJZLJZEADETUAAIJZUCABUBUDCUBMZUDMZNA - BUBUDCUEUFOQUAUBAABUBCUEPRS $. + syl5eqelr subrgring syl ) BDEZFBGHIJZBKJZLJZEADETUAAIJZUCABUBUDCUBMZUDMZN + ABUBUDCUEUFOQUAUBAABUBCUEPRS $. $} ${ @@ -245863,12 +246086,12 @@ series in the subring which are also polynomials (in the parent coe1z $p |- ( R e. Ring -> ( coe1 ` .0. ) = ( NN0 X. { Y } ) ) $= ( va vb vc crg wcel cn0 c1o cv cxp cmpt con0 eqid wceq csn ccom cco1 cmap cfv co wa wf fconst6g adantl nn0ex 1on elexi elmap sylibr cmpl psr1baslem - eqidd ply1mpl0 a1i rnggrp mpl0 fconstmpt syl6eq fmptco cbs ply1rng rng0cl - coe1fval2 3syl 3eqtr4d ) BKLZDHMNHOZUAPZQZUBZHMCQZDUCUEZMCUAZPZVLHIMMNUDU - FZVNCCVODVLVMMLZUGNMVNUHZVNWALWBWCVLNVMMUIUJMNVNUKNRULUMUNUOVLVOURVLDWAVS - PIWACQVLWANBUPUFZBJNCRDWDSZJUQGABWDDWEEFUSNRLVLULUTBVAVBIWACVCVDIOVNTCURV - EVLAKLDAVFUEZLVRVPTABEVGWFADWFSZFVHHVRWFABDVOVRSWGEVOSVIVJVTVQTVLHMCVCUTV - K $. + eqidd ply1mpl0 a1i ringgrp fconstmpt syl6eq fmptco cbs ply1ring coe1fval2 + mpl0 ring0cl 3syl 3eqtr4d ) BKLZDHMNHOZUAPZQZUBZHMCQZDUCUEZMCUAZPZVLHIMMN + UDUFZVNCCVODVLVMMLZUGNMVNUHZVNWALWBWCVLNVMMUIUJMNVNUKNRULUMUNUOVLVOURVLDW + AVSPIWACQVLWANBUPUFZBJNCRDWDSZJUQGABWDDWEEFUSNRLVLULUTBVAVHIWACVBVCIOVNTC + URVDVLAKLDAVEUEZLVRVPTABEVFWFADWFSZFVIHVRWFABDVOVRSWGEVOSVGVJVTVQTVLHMCVB + UTVK $. $} ${ @@ -245884,14 +246107,14 @@ series in the subring which are also polynomials (in the parent ( va wcel co cn0 c1o ccom cfv eqid cvv crg w3a csn cxp cmpt cof cco1 cmpl cv cps1 ply1bas ply1plusg simp2 simp3 mpladd coeq1d cmap wfn cbs ply1basf wf ffn syl 3ad2ant2 3ad2ant3 wf1o c0 df1o2 nn0ex mapsnf1o3 f1of mp1i ovex - 0ex a1i inidm ofco eqtrd ply1rng rngacl syl3an1 coe1fval2 oveq12d 3eqtr4d - wceq ) DUAMZEAMZFAMZUBZEFCNZLOPLUIUCUDUEZQZEWKQZFWKQZBUFZNZWJUGRZEUGRZFUG - RZWONWIWLEFWONZWKQWPWIWJWTWKWIAPDUHNZBCDPEFXASZGDDUJRZAHXCSIUKKCDXAGHXBJU - LWFWGWHUMWFWGWHUNUOUPWIOPUQNZXDXDOBEFWKTTTWGWFEXDURZWHWGXDDUSRZEVAXEAGDEX - FHIXFSZUTXDXFEVBVCVDWHWFFXDURZWGWHXDXFFVAXHAGDFXFHIXGUTXDXFFVBVCVEOXDWKVF - OXDWKVAWILOPWKVGVHVIVNWKSZVJOXDWKVKVLXDTMWIOPUQVMVOZXJOTMWIVIVOXDVPVQVRWI - WJAMZWQWLWEWFGUAMWGWHXKGDHVSACGEFIJVTWALWQAGDWJWKWQSIHXIWBVCWIWRWMWSWNWOW - GWFWRWMWEWHLWRAGDEWKWRSIHXIWBVDWHWFWSWNWEWGLWSAGDFWKWSSIHXIWBVEWCWD $. + 0ex a1i inidm ofco eqtrd wceq ply1ring ringacl syl3an1 coe1fval2 oveq12d + 3eqtr4d ) DUAMZEAMZFAMZUBZEFCNZLOPLUIUCUDUEZQZEWKQZFWKQZBUFZNZWJUGRZEUGRZ + FUGRZWONWIWLEFWONZWKQWPWIWJWTWKWIAPDUHNZBCDPEFXASZGDDUJRZAHXCSIUKKCDXAGHX + BJULWFWGWHUMWFWGWHUNUOUPWIOPUQNZXDXDOBEFWKTTTWGWFEXDURZWHWGXDDUSRZEVAXEAG + DEXFHIXFSZUTXDXFEVBVCVDWHWFFXDURZWGWHXDXFFVAXHAGDFXFHIXGUTXDXFFVBVCVEOXDW + KVFOXDWKVAWILOPWKVGVHVIVNWKSZVJOXDWKVKVLXDTMWIOPUQVMVOZXJOTMWIVIVOXDVPVQV + RWIWJAMZWQWLVSWFGUAMWGWHXKGDHVTACGEFIJWAWBLWQAGDWJWKWQSIHXIWCVCWIWRWMWSWN + WOWGWFWRWMVSWHLWRAGDEWKWRSIHXIWCVDWHWFWSWNVSWGLWSAGDFWKWSSIHXIWCVEWDWE $. $( A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.) $) @@ -245916,17 +246139,17 @@ series in the subring which are also polynomials (in the parent coe1subfv $p |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` ( F .- G ) ) ` X ) = ( ( ( coe1 ` F ) ` X ) N ( ( coe1 ` G ) ` X ) ) ) $= - ( wcel cn0 cco1 cfv co wceq adantr eqid crg wa cplusg simpl1 cgrp ply1rng - w3a rnggrp syl grpsubcl syl3an1 simpl3 coe1addfv syl31anc 3ad2ant1 simpl2 - simpr grpnpcan syl3anc fveq2d fveq1d eqtr3d wb wf coe1f 3ad2ant2 3ad2ant3 - cbs ffvelrnda grpsubadd syl13anc mpbird eqcomd ) BUAMZCAMZDAMZUGZGNMZUBZG - COPZPZGDOPZPZFQZGCDEQZOPZPZVSWDWGRZWGWCBUCPZQZWARZVSGWEDHUCPZQZOPZPZWJWAV - SVNWEAMZVPVRWOWJRVNVOVPVRUDVQWPVRVNHUEMZVOVPWPVNHUAMWQHBIUFHUHUIZAHECDJKU - JUKZSVNVOVPVRULZVQVRUQAWIWLBWEDGHIJWLTZWITZUMUNVSGWNVTVSWMCOVSWQVOVPWMCRV - QWQVRVNVOWQVPWRUOSVNVOVPVRUPWTAWLHECDJXAKURUSUTVAVBVSBUEMZWABVHPZMWCXDMWG - XDMWHWKVCVQXCVRVNVOXCVPBUHUOSVQNXDGVTVOVNNXDVTVDVPVTAHBCXDVTTJIXDTZVEVFVI - VQNXDGWBVPVNNXDWBVDVOWBAHBDXDWBTJIXEVEVGVIVQNXDGWFVQWPNXDWFVDWSWFAHBWEXDW - FTJIXEVEUIVIXDWIBFWAWCWGXEXBLVJVKVLVM $. + ( wcel cn0 cco1 cfv co wceq adantr eqid crg w3a wa cplusg simpl1 ply1ring + cgrp ringgrp syl grpsubcl simpl3 simpr coe1addfv syl31anc 3ad2ant1 simpl2 + syl3an1 grpnpcan syl3anc fveq2d fveq1d eqtr3d wb coe1f 3ad2ant2 ffvelrnda + cbs wf 3ad2ant3 grpsubadd syl13anc mpbird eqcomd ) BUAMZCAMZDAMZUBZGNMZUC + ZGCOPZPZGDOPZPZFQZGCDEQZOPZPZVSWDWGRZWGWCBUDPZQZWARZVSGWEDHUDPZQZOPZPZWJW + AVSVNWEAMZVPVRWOWJRVNVOVPVRUEVQWPVRVNHUGMZVOVPWPVNHUAMWQHBIUFHUHUIZAHECDJ + KUJUQZSVNVOVPVRUKZVQVRULAWIWLBWEDGHIJWLTZWITZUMUNVSGWNVTVSWMCOVSWQVOVPWMC + RVQWQVRVNVOWQVPWRUOSVNVOVPVRUPWTAWLHECDJXAKURUSUTVAVBVSBUGMZWABVGPZMWCXDM + WGXDMWHWKVCVQXCVRVNVOXCVPBUHUOSVQNXDGVTVOVNNXDVTVHVPVTAHBCXDVTTJIXDTZVDVE + VFVQNXDGWBVPVNNXDWBVHVOWBAHBDXDWBTJIXEVDVIVFVQNXDGWFVQWPNXDWFVHWSWFAHBWEX + DWFTJIXEVDUIVFXDWIBFWAWCWGXEXBLVJVKVLVM $. $} ${ @@ -245980,39 +246203,39 @@ series in the subring which are also polynomials (in the parent ( vc wcel co cn0 c1o cfv eqid vd vb va crg w3a csn cxp cmpt ccom cle cofr cv wbr cmap crab cmin cof cgsu cco1 cc0 cfz fconst6g nn0ex con0 1on elexi wf elmap sylibr adantl eqidd cmps psr1bas2 psr1mulr psr1baslem psrmulfval - 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(Contributed by AV, 14-Aug-2019.) $) ply1moncl $p |- ( ( R e. Ring /\ D e. NN0 ) -> ( D .^ X ) e. B ) $= - ( crg wcel cn0 wa cmnd co ply1rng adantr rngmgp syl simpr vr1cl mulgnn0cl - mgpbas syl3anc ) DMNZBONZPFQNZUIGANZBGERANUHUJUIUHCMNUJCDHSCFJUAUBTUHUIUC - UHUKUIACDGIHLUDTAEFBGACFJLUFKUEUG $. + ( crg wcel cn0 wa cmnd co ply1ring adantr ringmgp syl simpr vr1cl syl3anc + mgpbas mulgnn0cl ) DMNZBONZPFQNZUIGANZBGERANUHUJUIUHCMNUJCDHSCFJUAUBTUHUI + UCUHUKUIACDGIHLUDTAEFBGACFJLUFKUGUE $. $} ${ @@ -246130,12 +246353,12 @@ series in the subring which are also polynomials (in the parent 27-Mar-2015.) $) coe1tmfv2 $p |- ( ph -> ( ( coe1 ` ( C .x. ( D .^ X ) ) ) ` F ) = .0. ) $= - ( vx co cco1 cfv cn0 cv wceq cif cmpt wcel coe1tm syl3anc fveq1d rng0cl - crg syl ifcl syl2anc eqeq1 ifbid fvmptg wn necomd neneqd iffalse 3eqtrd - eqid ) AHBCKGUFFUFUGUHZUHHUEUIUEUJZCUKZBLULZUMZUHZHCUKZBLULZLAHVLVPAEUS - UNZBIUNZCUIUNVLVPUKTUAUBUEBCDEFGIJKLMNOPQRSUOUPUQAHUIUNVSIUNZVQVSUKUCAW - ALIUNZWBUAAVTWCTIELNMURUTVRBLIVAVBUEHVOVSUIIVPVMHUKVNVRBLVMHCVCVDVPVKVE - VBAVRVFVSLUKAHCACHUDVGVHVRBLVIUTVJ $. + ( vx co cco1 cfv cn0 cv wceq cif cmpt crg coe1tm syl3anc fveq1d ring0cl + wcel syl ifcl syl2anc eqeq1 eqid fvmptg wn necomd neneqd iffalse 3eqtrd + ifbid ) AHBCKGUFFUFUGUHZUHHUEUIUEUJZCUKZBLULZUMZUHZHCUKZBLULZLAHVLVPAEU + NUSZBIUSZCUIUSVLVPUKTUAUBUEBCDEFGIJKLMNOPQRSUOUPUQAHUIUSVSIUSZVQVSUKUCA + WALIUSZWBUAAVTWCTIELNMURUTVRBLIVAVBUEHVOVSUIIVPVMHUKVNVRBLVMHCVCVKVPVDV + EVBAVRVFVSLUKAHCACHUDVGVHVRBLVIUTVJ $. $} coe1tmmul.b $e |- B = ( Base ` P ) $. @@ -246151,43 +246374,43 @@ series in the subring which are also polynomials (in the parent ( x e. NN0 |-> if ( D <_ x , ( ( ( coe1 ` A ) ` ( x - D ) ) .X. C ) , .0. ) ) ) $= ( vy co cco1 cfv cn0 cc0 cfz cmin cmpt cgsu cle wbr cif crg wcel ply1tmcl - cv wceq syl3anc coe1mul wa eqeq2 cvv cmnd adantr rngmnd syl a1i simprr wb - ovex simprl nn0sub syl2anc mpbid nn0ge0d cr nn0re ad2antrl subge02d fznn0 - nn0red mpbir2and ad2antrr wf coe1f elfznn0 adantl ffvelrnd fznn0sub rngcl - eqid fmptd csn cdif eldifi wne eldifsn simplrl nn0cnd nncand eqcomd oveq2 - cc eqeq2d syl5ibrcom necon3d impr sylan2b coe1tmfv2 oveq2d sylan2 suppss2 - rngrz eqtrd gsumpt fveq2 fveq2d oveq12d fvmpt coe1tmfv1 3eqtrd anassrs wn - ad2antll ad2antlr clt ltnled mpbird mpteq2dva lelttrd gtned gsumz sylancl - ifbothda ) ACEFOLULJULZIULUMUNZBUOHUKUPBVGZUQULZUKVGZCUMUNZUNZUUHUUJURULZ - UUFUMUNZUNZKULZUSZUTULZUSZBUOFUUHVAVBZUUHFURULZUUKUNZEKULZPVCZUSAHVDVEZCD - VEZUUFDVEZUUGUUSVHUHUGAUVEEMVEZFUOVEZUVGUHUIUJDEFGHJLMNORSTUAUBUCUDVFVIZU - KDHIKBCUUFGSUEUFUDVJVIABUOUURUVDUUTUURUVCVHZUURPVHUURUVDVHAUUHUOVEZVKZUVC - PUVCUVDUURVLPUVDUURVLAUVLUUTUVKAUVLUUTVKZVKZUURUVAUUQUNZUVBUUHUVAURULZUUN - 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(Contributed by Stefan O'Rear, 29-Mar-2015.) $) @@ -246195,33 +246418,33 @@ series in the subring which are also polynomials (in the parent ( x e. NN0 |-> if ( D <_ x , ( C .X. ( ( coe1 ` A ) ` ( x - D ) ) ) , .0. ) ) ) $= ( vy co cco1 cfv cn0 cc0 cfz cmin cmpt cgsu cle wbr cif crg wcel ply1tmcl - cv wceq syl3anc coe1mul eqeq2 cvv cmnd ad2antrr rngmnd syl ovex a1i simpr - wa wb fznn0 ad2antlr mpbir2and eqid coe1f adantr elfznn0 ffvelrn fznn0sub - wf syl2an rngcl fmptd csn cdif ad3antrrr eldifi adantl eldifsni coe1tmfv2 - wne necomd oveq1d rnglz syl2anc sylan2 adantlr eqtrd suppss2 gsumpt fveq2 - oveq2 fveq2d oveq12d fvmpt coe1tmfv1 wn elfzle2 breq1 syl5ibrcom necon3bd - imp an32s mpteq2dva oveq2d gsumz ifbothda ) AEFOLULJULZCIULUMUNZBUOHUKUPB - VGZUQULZUKVGZYIUMUNZUNZYKYMURULZCUMUNZUNZKULZUSZUTULZUSZBUOFYKVAVBZEYKFUR - ULZYQUNZKULZPVCZUSAHVDVEZYIDVEZCDVEZYJUUBVHUHAUUHEMVEZFUOVEZUUIUHUIUJDEFG - HJLMNORSTUAUBUCUDVFVIZUGUKDHIKBYICGSUEUFUDVJVIABUOUUAUUGUUCUUAUUFVHUUAPVH - UUAUUGVHAYKUOVEZVTZUUFPUUFUUGUUAVKPUUGUUAVKUUOUUCVTZUUAFYTUNZUUFUUPYLMYTH - VLFPRQUUPUUHHVMVEZAUUHUUNUUCUHVNHVOZVPYLVLVEZUUPUPYKUQVQZVRZUUPFYLVEZUULU - UCAUULUUNUUCUJVNUUOUUCVSUUNUVCUULUUCVTWAAUUCFYKWBWCWDZUUOYLMYTWKUUCUUOUKY - LYSMYTUUOYMYLVEZVTZUUHYOMVEZYRMVEZYSMVEAUUHUUNUVEUHVNZUUOUOMYNWKZYMUOVEZU - VGUVEAUVJUUNAUUIUVJUUMYNDGHYIMYNWEUDSRWFVPWGYMYKWHZUOMYMYNWIWLUUOUOMYQWKZ - YPUOVEUVHUVEAUVMUUNAUUJUVMUGYQDGHCMYQWEUDSRWFVPWGYMUPYKWJUOMYPYQWIWLZMHKY - OYRRUFWMVIYTWEZWNWGUUPYLYSUKVLFWOZPUUPYMYLUVPWPVEZVTZYSPYRKULZPUVRYOPYRKU - VREFGHJLYMMNOPQRSTUAUBUCAUUHUUNUUCUVQUHWQAUUKUUNUUCUVQUIWQAUULUUNUUCUVQUJ - WQUVQUVKUUPUVQUVEUVKYMYLUVPWRZUVLVPWSUVQFYMXBZUUPUVQYMFYMYLFWTXCWSXAXDUUO - UVQUVSPVHZUUCUVQUUOUVEUWBUVTUVFUUHUVHUWBUVIUVNMHKYRPRUFQXEXFZXGXHXIUVBXJX - KUUPUUQFYNUNZUUEKULZUUFUUPUVCUUQUWEVHUVDUKFYSUWEYLYTYMFVHZYOUWDYRUUEKYMFY - NXLUWFYPUUDYQYMFYKURXMXNXOUVOUWDUUEKVQXPVPUUPUWDEUUEKAUWDEVHZUUNUUCAUUHUU - KUULUWGUHUIUJEFGHJLMNOPQRSTUAUBUCXQVIVNXDXIXIUUOUUCXRZVTZUUAHUKYLPUSZUTUL - ZPUWIYTUWJHUTUWIUKYLYSPUWIUVEVTZYSUVSPUWLYOPYRKUWLEFGHJLYMMNOPQRSTUAUBUCA - UUHUUNUWHUVEUHWQAUUKUUNUWHUVEUIWQAUULUUNUWHUVEUJWQUVEUVKUWIUVLWSUUOUVEUWH - UWAUVFUWHUWAUVFUUCFYMUVFUUCFYMVHYMYKVAVBZUVEUWMUUOYMUPYKXSWSFYMYKVAXTYAYB - YCYDXAXDUUOUVEUWBUWHUWCXHXIYEYFUWIUURUUTUWKPVHAUURUUNUWHAUUHUURUHUUSVPVNU - UTUWIUVAVRYLUKHVLPQYGXFXIYHYEXI $. + cv wceq syl3anc coe1mul wa eqeq2 cvv cmnd ad2antrr ringmnd syl ovex simpr + wb fznn0 ad2antlr mpbir2and wf eqid coe1f adantr elfznn0 ffvelrn fznn0sub + a1i syl2an ringcl csn cdif ad3antrrr eldifi adantl wne eldifsni coe1tmfv2 + fmptd necomd oveq1d ringlz syl2anc sylan2 eqtrd gsumpt fveq2 oveq2 fveq2d + adantlr suppss2 oveq12d fvmpt coe1tmfv1 elfzle2 breq1 syl5ibrcom necon3bd + wn imp an32s mpteq2dva oveq2d gsumz ifbothda ) AEFOLULJULZCIULUMUNZBUOHUK + UPBVGZUQULZUKVGZYIUMUNZUNZYKYMURULZCUMUNZUNZKULZUSZUTULZUSZBUOFYKVAVBZEYK + FURULZYQUNZKULZPVCZUSAHVDVEZYIDVEZCDVEZYJUUBVHUHAUUHEMVEZFUOVEZUUIUHUIUJD + EFGHJLMNORSTUAUBUCUDVFVIZUGUKDHIKBYICGSUEUFUDVJVIABUOUUAUUGUUCUUAUUFVHUUA + PVHUUAUUGVHAYKUOVEZVKZUUFPUUFUUGUUAVLPUUGUUAVLUUOUUCVKZUUAFYTUNZUUFUUPYLM + YTHVMFPRQUUPUUHHVNVEZAUUHUUNUUCUHVOHVPZVQYLVMVEZUUPUPYKUQVRZWKZUUPFYLVEZU + ULUUCAUULUUNUUCUJVOUUOUUCVSUUNUVCUULUUCVKVTAUUCFYKWAWBWCZUUOYLMYTWDUUCUUO + UKYLYSMYTUUOYMYLVEZVKZUUHYOMVEZYRMVEZYSMVEAUUHUUNUVEUHVOZUUOUOMYNWDZYMUOV + EZUVGUVEAUVJUUNAUUIUVJUUMYNDGHYIMYNWEUDSRWFVQWGYMYKWHZUOMYMYNWIWLUUOUOMYQ + WDZYPUOVEUVHUVEAUVMUUNAUUJUVMUGYQDGHCMYQWEUDSRWFVQWGYMUPYKWJUOMYPYQWIWLZM + HKYOYRRUFWMVIYTWEZXBWGUUPYLYSUKVMFWNZPUUPYMYLUVPWOVEZVKZYSPYRKULZPUVRYOPY + RKUVREFGHJLYMMNOPQRSTUAUBUCAUUHUUNUUCUVQUHWPAUUKUUNUUCUVQUIWPAUULUUNUUCUV + QUJWPUVQUVKUUPUVQUVEUVKYMYLUVPWQZUVLVQWRUVQFYMWSZUUPUVQYMFYMYLFWTXCWRXAXD + UUOUVQUVSPVHZUUCUVQUUOUVEUWBUVTUVFUUHUVHUWBUVIUVNMHKYRPRUFQXEXFZXGXMXHUVB + XNXIUUPUUQFYNUNZUUEKULZUUFUUPUVCUUQUWEVHUVDUKFYSUWEYLYTYMFVHZYOUWDYRUUEKY + MFYNXJUWFYPUUDYQYMFYKURXKXLXOUVOUWDUUEKVRXPVQUUPUWDEUUEKAUWDEVHZUUNUUCAUU + HUUKUULUWGUHUIUJEFGHJLMNOPQRSTUAUBUCXQVIVOXDXHXHUUOUUCYBZVKZUUAHUKYLPUSZU + TULZPUWIYTUWJHUTUWIUKYLYSPUWIUVEVKZYSUVSPUWLYOPYRKUWLEFGHJLYMMNOPQRSTUAUB + UCAUUHUUNUWHUVEUHWPAUUKUUNUWHUVEUIWPAUULUUNUWHUVEUJWPUVEUVKUWIUVLWRUUOUVE + UWHUWAUVFUWHUWAUVFUUCFYMUVFUUCFYMVHYMYKVAVBZUVEUWMUUOYMUPYKXRWRFYMYKVAXSX + TYAYCYDXAXDUUOUVEUWBUWHUWCXMXHYEYFUWIUURUUTUWKPVHAUURUUNUWHAUUHUURUHUUSVQ + VOUUTUWIUVAWKYLUKHVMPQYGXFXHYHYEXH $. coe1tmmul2fv.y $e |- ( ph -> Y e. NN0 ) $. $( Function value of a right-multiplication by a term in the shifted @@ -246258,18 +246481,18 @@ series in the subring which are also polynomials (in the parent ( x e. NN0 |-> if ( D <_ x , ( ( coe1 ` A ) ` ( x - D ) ) , .0. ) ) ) $= ( cur cfv co cvsca cco1 cn0 cle wbr cmin cmulr cif cmpt cbs eqid crg wcel - rngidcl syl coe1tmmul csca wceq ply1sca fveq2d oveq1d clmod ply1lmod cmnd - cv ply1rng rngmgp vr1cl mgpbas mulgnn0cl syl3anc lmodvs1 syl2anc eqtrd wa - ad2antrr wf coe1f simplr simpr nn0sub2 ffvelrnd rnglidm ifeq1da mpteq2dva - 3syl 3eqtr3d ) AGUCUDZEKIUEZFUFUDZUEZCHUEZUGUDBUHEBVJZUIUJZWMWREUKUEZCUGU - DZUDZGULUDZUEZLUMZUNWNCHUEZUGUDBUHWSXBLUMZUNABCDWMEFGHWOXCIGUOUDZJKLMXHUP - ZNOWOUPZPQRSXCUPZUATAGUQURZWMXHURTXHGWMXIWMUPZUSUTUBVAAWQXFUGAWPWNCHAWPFV - BUDZUCUDZWNWOUEZWNAWMXOWNWOAGXNUCAXLGXNVCTFGUQNVDUTVEVFAFVGURZWNDURZXPWNV - CAXLXQTFGNVHUTAJVIURZEUHURZKDURZXRAXLFUQURXSTFGNVKFJPVLWKUBAXLYATDFGKONRV - MUTDIJEKDFJPRVNQVOVPWOXOXNDFWNRXNUPXJXOUPVQVRVSVFVEABUHXEXGAWRUHURZVTZWSX - DXBLYCWSVTZXLXBXHURXDXBVCAXLYBWSTWAYDUHXHWTXAAUHXHXAWBZYBWSACDURYEUAXADFG - CXHXAUPRNXIWCUTWAYDXTYBWSWTUHURAXTYBWSUBWAAYBWSWDYCWSWEEWRWFVPWGXHGXCWMXB - XIXKXMWHVRWIWJWL $. + cv ringidcl coe1tmmul csca wceq ply1sca fveq2d oveq1d clmod ply1lmod cmnd + syl ply1ring ringmgp vr1cl mgpbas mulgnn0cl syl3anc lmodvs1 syl2anc eqtrd + 3syl wa ad2antrr wf coe1f simplr simpr nn0sub2 ffvelrnd ifeq1da mpteq2dva + ringlidm 3eqtr3d ) AGUCUDZEKIUEZFUFUDZUEZCHUEZUGUDBUHEBUSZUIUJZWMWREUKUEZ + CUGUDZUDZGULUDZUEZLUMZUNWNCHUEZUGUDBUHWSXBLUMZUNABCDWMEFGHWOXCIGUOUDZJKLM + XHUPZNOWOUPZPQRSXCUPZUATAGUQURZWMXHURTXHGWMXIWMUPZUTVJUBVAAWQXFUGAWPWNCHA + WPFVBUDZUCUDZWNWOUEZWNAWMXOWNWOAGXNUCAXLGXNVCTFGUQNVDVJVEVFAFVGURZWNDURZX + PWNVCAXLXQTFGNVHVJAJVIURZEUHURZKDURZXRAXLFUQURXSTFGNVKFJPVLVTUBAXLYATDFGK + ONRVMVJDIJEKDFJPRVNQVOVPWOXOXNDFWNRXNUPXJXOUPVQVRVSVFVEABUHXEXGAWRUHURZWA + ZWSXDXBLYCWSWAZXLXBXHURXDXBVCAXLYBWSTWBYDUHXHWTXAAUHXHXAWCZYBWSACDURYEUAX + ADFGCXHXAUPRNXIWDVJWBYDXTYBWSWTUHURAXTYBWSUBWBAYBWSWEYCWSWFEWRWGVPWHXHGXC + WMXBXIXKXMWKVRWIWJWL $. coe1pwmulfv.y $e |- ( ph -> Y e. NN0 ) $. $( Function value of a right-multiplication by a variable power in the @@ -246374,7 +246597,7 @@ series in the subring which are also polynomials (in the parent $( A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) $) ply1sclf $p |- ( R e. Ring -> A : K --> B ) $= - ( crg wcel cid cfv ply1sca2 ply1rng ply1lmod cbs c1 df-base strfvi + ( crg wcel cid cfv ply1sca2 ply1ring ply1lmod cbs c1 df-base strfvi asclf ) DJKABDLMECGCDFNCDFOCDFPDQRESHTIUA $. $( The value of the algebra scalars function for (univariate) @@ -246423,22 +246646,22 @@ series in the subring which are also polynomials (in the parent $( The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.) $) ply1scl0 $p |- ( R e. Ring -> ( A ` .0. ) = Y ) $= - ( crg wcel cfv cur cvsca co c0g cbs wceq eqid syl cid rng0cl c1 df-base - ply1sca2 strfvi asclval fveq2d syl6reqr oveq1d ply1lmod ply1rng rngidcl - fvi clmod lmod0vs syl2anc 3eqtrd ) CJKZEALZEBMLZBNLZOZCUALZPLZVAVBOZDUS - ECQLZKUTVCRVGCEVGSZHUBAVBVAVDVGBEGBCFUEZCQUCVGUDVHUFVBSZVASZUGTUSEVEVAV - BUSVECPLEUSVDCPCJUNUHHUIUJUSBUOKVABQLZKZVFDRBCFUKUSBJKVMBCFULVLBVAVLSZV - KUMTVBVDVEVLBVADVNVIVJVESIUPUQUR $. + ( crg wcel cfv cur cvsca co c0g cbs wceq eqid syl cid ring0cl c1 strfvi + ply1sca2 df-base asclval fveq2d syl6reqr oveq1d clmod ply1lmod ply1ring + fvi ringidcl lmod0vs syl2anc 3eqtrd ) CJKZEALZEBMLZBNLZOZCUALZPLZVAVBOZ + DUSECQLZKUTVCRVGCEVGSZHUBAVBVAVDVGBEGBCFUEZCQUCVGUFVHUDVBSZVASZUGTUSEVE + VAVBUSVECPLEUSVDCPCJUNUHHUIUJUSBUKKVABQLZKZVFDRBCFULUSBJKVMBCFUMVLBVAVL + SZVKUOTVBVDVEVLBVADVNVIVJVESIUPUQUR $. ply1scln0.k $e |- K = ( Base ` R ) $. $( Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.) $) ply1scln0 $p |- ( ( R e. Ring /\ X e. K /\ X =/= .0. ) -> ( A ` X ) =/= Y ) $= - ( crg wcel wne w3a cfv wa wceq adantr cbs wf1 wb ply1sclf1 simpr rng0cl - eqid f1fveq syl12anc biimpd necon3d 3impia ply1scl0 3ad2ant1 neeqtrd ) + ( crg wcel wne w3a cfv wa wceq adantr cbs wf1 wb eqid ply1sclf1 ring0cl + simpr f1fveq syl12anc biimpd necon3d 3impia ply1scl0 3ad2ant1 neeqtrd ) CMNZEDNZEGOZPEAQZGAQZFUPUQURUSUTOUPUQRZUSUTEGVAUSUTSZEGSZVADBUAQZAUBZUQ - GDNZVBVCUCUPVEUQAVDBCDHILVDUGUDTUPUQUEUPVFUQDCGLJUFTDVDEGAUHUIUJUKULUPU + GDNZVBVCUCUPVEUQAVDBCDHILVDUDUETUPUQUGUPVFUQDCGLJUFTDVDEGAUHUIUJUKULUPU QUTFSURABCFGHIJKUMUNUO $. $} @@ -246448,12 +246671,12 @@ series in the subring which are also polynomials (in the parent $( The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.) $) ply1scl1 $p |- ( R e. Ring -> ( A ` .1. ) = N ) $= - ( crg wcel cfv cvsca co cur cbs wceq eqid rngidcl syl c1 df-base strfvi - cid ply1sca2 asclval fvi fveq2d syl6eqr oveq1d ply1lmod ply1rng lmodvs1 - clmod syl2anc 3eqtr2d ) CJKZDALZDEBMLZNZCUDLZOLZEUSNZEUQDCPLZKURUTQVDCD - VDRZHSAUSEVAVDBDGBCFUEZCPUAVDUBVEUCUSRZIUFTUQVBDEUSUQVBCOLDUQVACOCJUGUH - HUIUJUQBUNKEBPLZKZVCEQBCFUKUQBJKVIBCFULVHBEVHRZISTUSVBVAVHBEVJVFVGVBRUM - UOUP $. + ( crg wcel cfv cvsca co cur cbs wceq eqid ringidcl syl ply1sca2 df-base + cid c1 strfvi asclval fvi fveq2d syl6eqr oveq1d clmod ply1lmod ply1ring + lmodvs1 syl2anc 3eqtr2d ) CJKZDALZDEBMLZNZCUCLZOLZEUSNZEUQDCPLZKURUTQVD + CDVDRZHSAUSEVAVDBDGBCFUAZCPUDVDUBVEUEUSRZIUFTUQVBDEUSUQVBCOLDUQVACOCJUG + UHHUIUJUQBUKKEBPLZKZVCEQBCFULUQBJKVIBCFUMVHBEVHRZISTUSVBVAVHBEVJVFVGVBR + UNUOUP $. $} $} @@ -246465,12 +246688,12 @@ series in the subring which are also polynomials (in the parent $( The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.) $) ply1idvr1 $p |- ( R e. Ring -> ( 0 .^ X ) = ( 1r ` P ) ) $= - ( crg wcel cc0 co cur cfv cascl cvsca cbs wceq eqid rngidcl mpdan ply1sca - csca ply1scltm fveq2d oveq1d clmod ply1lmod cn0 ply1moncl lmodvs1 syl2anc - 0nn0 mpan2 3eqtrrd ply1scl1 eqtrd ) BJKZLECMZBNOZAPOZOZANOZUSVCVAUTAQOZMZ - AUDOZNOZUTVEMZUTUSVABROZKVCVFSVJBVAVJTZVATZUAVBABVECVAVJDEVKFGVETZHIVBTZU - EUBUSVAVHUTVEUSBVGNABJFUCUFUGUSAUHKUTAROZKZVIUTSABFUIUSLUJKVPUNVOLABCDEFG - HIVOTZUKUOVEVHVGVOAUTVQVGTVMVHTULUMUPVBABVAVDFVNVLVDTUQUR $. + ( crg wcel cc0 co cur cfv cascl cvsca cbs wceq eqid csca ringidcl ply1sca + ply1scltm mpdan fveq2d oveq1d clmod ply1lmod 0nn0 ply1moncl mpan2 lmodvs1 + cn0 syl2anc 3eqtrrd ply1scl1 eqtrd ) BJKZLECMZBNOZAPOZOZANOZUSVCVAUTAQOZM + ZAUAOZNOZUTVEMZUTUSVABROZKVCVFSVJBVAVJTZVATZUBVBABVECVAVJDEVKFGVETZHIVBTZ + UDUEUSVAVHUTVEUSBVGNABJFUCUFUGUSAUHKUTAROZKZVIUTSABFUIUSLUNKVPUJVOLABCDEF + GHIVOTZUKULVEVHVGVOAUTVQVGTVMVHTUMUOUPVBABVAVDFVNVLVDTUQUR $. $} ${ @@ -246489,33 +246712,33 @@ series in the subring which are also polynomials (in the parent ( wcel wa cfv wceq cn co adantr wi vn vk vs crg cv cco1 wral cc0 cfz cmin cmulr cmpt cgsu cn0 cvv coe1mul 3expb oveq2 oveq1 fveq2d oveq2d mpteq12dv eqid weq adantl nnnn0 ovex a1i fvmptd r19.26 nncn subid1d sylan9eqr simpl - eqeltrd fveq2 eqeq1d rspcva ex syl elfznn0 coe1fvalcl syl2an rngrz expcom - cbs syl2anc com23 syld com12 expd com24 com13 wn biimpri anim12ci elnnne0 - wne df-ne sylibr eleq2i biimpi fznn0sub rnglz com14 imp pm2.61i mpteq2dva - a1dd syl5bi cmnd rngmnd gsumz eqtrd ralrimiva cbvralv ) CUDMZEAMZFAMZNZNZ - HUEZEUFOZOZGPZYBFUFOZOZGPZNHQUGZYBEFDRUFOZOZGPZHQUGZYAYINZUAUEZYJOZGPZUAQ - UGYMYNYQUAQYNYOQMZNZYPCUBUHYOUIRZUBUEZYCOZYOUUAUJRZYFOZCUKOZRZULZUMRZGYSU - CYOCUBUHUCUEZUIRZUUBUUIUUAUJRZYFOZUUERZULZUMRZUUHUNYJUOYNYJUCUNUUOULPZYRY - AUUPYIXQXRXSUUPUBACDUUEUCEFBILUUEVCZJUPUQSSUCUAVDZUUOUUHPYSUURUUNUUGCUMUU - RUBUUJUUMYTUUFUUIYOUHUIURUURUULUUDUUBUUEUURUUKUUCYFUUIYOUUAUJUSUTVAVBVAVE - YRYOUNMYNYOVFVEUUHUOMYSCUUGUMVGVHVIYSUUHCUBYTGULZUMRZGYSUUGUUSCUMYSUBYTUU - FGYSUUAYTMZUUFGPZYNYRUVAUVBTZYNYRUVAUVBYAYIYRUVANZUVBTZYIYEHQUGZYHHQUGZNZ - YAUVEYEYHHQVJUUAUHPZYAUVHUVETTUVHYAUVIUVEUVGYAUVIUVETTUVFUVGUVDUVIYAUVBUV - GUVDUVIYAUVBTZUVDUVINZUVGUVJUVKUVGUUDGPZUVJUVKUUCQMZUVGUVLTUVKUUCYOQUVIUV - DUUCYOUHUJRZYOUUAUHYOUJURYRUVNYOPUVAYRYOYOVKVLSVMUVDYRUVIYRUVAVNSVOUVMUVG - 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Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) finSupp ( 0g ` P ) ) $= ( wcel cfv cn0 crg wa csca cvv cbs cv co eqid clmod ply1lmod adantr nn0ex - a1i cmnd ply1rng rngmgp syl ad2antrr simpr vr1cl mgpbas mulgnn0cl syl3anc - wf coe1f adantl c0g cfsupp wbr coe1sfi ply1sca eqcomd fveq2d mptscmfsuppd - wceq breqtrrd ) DUARZHBRZUBZABCCUCSZEFUDTDUESZFUFZJGUGZMVTUHNVQCUIRVRCDKU - JUKTUDRVSULUMVSWBTRZUBIUNRZWDJBRZWCBRVQWEVRWDVQCUARWECDKUOCIOUPUQURVSWDUS - VQWFVRWDBCDJLKMUTURBGIWBJBCIOMVAPVBVCVRTWAAVDVQABCDHWAQMKWAUHVEVFVSADVGSZ - VTVGSVHVRAWGVHVIVQABCDHWGQMKWGUHVJVFVSVTDVGVQVTDVOVRVQDVTCDUAKVKVLUKVMVPV - N $. + a1i ply1ring ringmgp syl ad2antrr simpr vr1cl mgpbas mulgnn0cl syl3anc wf + cmnd coe1f adantl c0g cfsupp wbr coe1sfi wceq ply1sca eqcomd mptscmfsuppd + fveq2d breqtrrd ) DUARZHBRZUBZABCCUCSZEFUDTDUESZFUFZJGUGZMVTUHNVQCUIRVRCD + KUJUKTUDRVSULUMVSWBTRZUBIVDRZWDJBRZWCBRVQWEVRWDVQCUARWECDKUNCIOUOUPUQVSWD + URVQWFVRWDBCDJLKMUSUQBGIWBJBCIOMUTPVAVBVRTWAAVCVQABCDHWAQMKWAUHVEVFVSADVG + SZVTVGSVHVRAWGVHVIVQABCDHWGQMKWGUHVJVFVSVTDVGVQVTDVKVRVQDVTCDUAKVLVMUKVOV + PVN $. $} ${ @@ -246562,46 +246785,46 @@ series in the subring which are also polynomials (in the parent cmpt cgsu c0 com eqid psr1baslem 1onn cps1 ply1bas ply1vsca simpl mplcoe1 a1i simpr fvcoe1 adantll cmgp cmvr cmg simpll cplusg csn wral eqidd fveq2 oveq1d oveq2d eqeq12d ralsn sylibr ralbidv raleqi raleqbii mplcoe5 mpteq1 - 0ex df1o2 ax-mp oveq2i cmnd cvv mplrng rngmgp syl ad2antrr cbs mgpbas wss - mpan ssv ovex cmulr ply1mulr mgpplusg eqtr3i oveqi mulgpropd oveqd adantr - ply1rng wf elmapi ffvelrn sylancl adantl mulgnn0cl syl3anc eqeltrd vr1val - 0lt1o vr1cl syl6eqr oveq12d gsumsn syl5eq mpteq2dva ccom nn0ex mptex cpl1 - 3eqtrd fvex eqeltri clmod ply1plusg ply1mpl0 ccmn mpllmod sylancr lmodcmn - gsumpropd csca ply1lmod ffvelrnda ply1sca eqcomd fveq2d eleqtrrd lmodvscl - coe1f fmptd ply1coefsupp wf1o mapsnf1o2 gsumf1o oveq1 fmptco 3eqtrrd ) DU - FRZHBRZUGZHUHDUISZUAUJUHUKSZUAULZHTZUBUVIUBULZUVJUMDUNTZDUOTZUPUQZESZUQZU - RSUVHUAUVIUSUVJTZATZUVRJGSZESZUQZURSZCFUJFULZATZUWDJGSZESZUQZURSZUVGUBBUV - IUVHDEUVMUCUAUHUTHUVNUVHVAZUCVBZUVNVAZUVMVAZUHUTRZUVGVCVICDDVDTZBKUWOVAMV - EZDUVHECKUWJNVFUVEUVFVGZUVEUVFVJVHUVGUVQUWBUVHURUVGUAUVIUVPUWAUVGUVJUVIRZ - UGZUVKUVSUVOUVTEUVFUWRUVKUVSUMUVEAHBUVJQVKVLUWSUVOUVHVMTZUDUHUDULZUVJTZUX - AUHDVNSZTZUWTVOTZSZUQZURSZUVRJUXESZUVTUWSUEUBUVIUVHDUVMUCUDUXEUWTUHUXCUTU - VJUVNUWJUWKUWLUWMUWNUWSVCVIUWTVAZUXEVAZUXCVAUVEUVFUWRVPUVGUWRVJUWSUVLUXCT - 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CRing /\ K e. B ) -> K = ( P gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) ) $= ( wcel cfv va vb vd vc ccrg wa c1o cmpl co cn0 cmap wceq cur c0g cif cmpt - cv cgsu c0 com eqid psr1baslem 1onn a1i cps1 ply1bas ply1vsca crg crngrng - adantr simpr mplcoe1 fvcoe1 adantll cmgp cmvr simpll mplcoe2 df1o2 mpteq1 - cmg csn ax-mp oveq2i cmnd cvv ccmn mplcrng mpan crngmgp syl cmnmnd mgpbas - 0ex cbs wss ssv cplusg ovex cmulr ply1mulr mgpplusg oveqi mulgpropd oveqd - eqtr3i ply1crng rngmgp 3syl elmapi 0lt1o ffvelrn sylancl adantl mulgnn0cl - vr1cl syl3anc eqeltrd fveq2 vr1val syl6eqr gsumsn syl5eq 3eqtrd mpteq2dva - wf oveq12d oveq2d ccom nn0ex mptex cpl1 eqeltri ply1plusg gsumpropd csupp - fvex cfsupp wbr eqidd clmod mpllmod sylancr lmodcmn coe1f ffvelrnda simpl - csca mplsca mp2an lmodvscl fmptd w3a cfn funmpt 3pm3.2i coe1sfi fsuppimpd - wfun feqmptd eqcomd oveq1d syl6eqss lmod0vs sylan suppssov1 suppssfifsupp - ssid syl12anc wf1o mapsnf1o2 gsumf1o oveq1 fmptco 3eqtrrd ) DUESZHBSZUFZH - UGDUHUIZUAUJUGUKUIZUAUQZHTZUBUVTUBUQZUWAULDUMTZDUNTZUOUPZEUIZUPZURUIUVSUA - 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NN ( ( coe1 ` M ) ` n ) = .0. ) ) $= ( wcel wa cfv wceq adantr cc0 vk crg cv wrex cco1 cn wral simpl cply1coe0 anim1i syl wb fveq2 fveq1d eqeq1d ralbidv adantl mpbird rexlimdva w3a cn0 - ex simpr csca eqid ply1rng ply1lmod asclf 0nn0 coe1fvalcl sylancl ply1sca - cbs eqcomd fveq2d eleqtrrd ffvelrnd 3jca cif cvv cmpt coe1scl syl2anc weq - wf wn nnne0 neneqd eqeq1 notbid iffalse nnnn0 c0g fvex eqeltri a1i fvmptd - eqtrd ralimdva imp ply1sclid jca csn cun df-n0 raleqi c0ex eqeq12d syl5bb - ralunsn eqcoe1ply1eq sylc eqeq2d rspcedv mpd impbid ) DUBOZGBOZPZGIUCZAQZ - RZIFUDZEUCZGUEQZQZHRZEUFUGZXSYBYHIFXSXTFOZPZYBYHYJYBPZYHYDYAUEQZQZHRZEUFU - GZYKXQYIPZYOYJYPYBXSXQYIXQXRUHZUJSABCDXTEFHJKLMNUIUKYBYHYOULYJYBYGYNEUFYB - YFYMHYBYDYEYLGYAUEUMUNUOUPUQURVBUSXSYHYCXSYHPZGTYEQZAQZRZYCYRXQXRYTBOZUTZ - YFYDYTUEQZQZRZEVAUGZUUAXSUUCYHXSXQXRUUBYQXQXRVCZXSCVDQZVMQZBYSAXQUUJBAWEX - RXQABUUIUUJCNUUIVECDLVFCDLVGUUJVEMVHSXSYSDVMQZUUJXSXRTVAOZYSUUKOUUHVIYEBC - DGUUKTYEVEZMLUUKVEVJVKXQUUJUUKRXRXQUUIDVMXQDUUICDUBLVLVNVOSVPVQVRSYRUUGUU - FEUFUGZYSTUUDQZRZPZYRUUNUUPXSYHUUNXSYGUUFEUFXSYDUFOZPZYGUUFUUSYGPYFHUUEUU - SYGVCUUSHUUERYGUUSUUEHUUSUAYDUAUCZTRZYSHVSZHVAUUDVTXSUUDUAVAUVBWARZUURXSX - QYSFOZUVCYQXSXRUULUVDUUHVIYEBCDGFTUUMMLJVJVKZUAACDFYSHLNJKWBWCSUUSUAEWDZP - ZUVAWFZUVBHRUVGUVHYDTRZWFZUUSUVJUVFUURUVJXSUURYDTYDWGWHUQSUVFUVHUVJULUUSU - VFUVAUVIUUTYDTWIWJUQURUVAYSHWKUKUURYDVAOXSYDWLUQHVTOUUSHDWMQVTKDWMWNWOWPW - QVNSWRVBWSWTXSUUPYHXSXQUVDUUPYQUVEACDFYSLNJXAWCSXBUUGUUFEUFTXCXDZUGZYRUUQ - UUFEVAUVKXEXFYRTVTOZUVLUUQULUVMYRXGWPUUFUUPEUFTVTUVIYFYSUUEUUOYDTYEUMYDTU - UDUMXHXJUKXIURYEBUUDCDEGYTLMUUMUUDVEXKXLYRYBUUAIYSFXSUVDYHUVESXTYSRZYBUUA - ULYRUVNYAYTGXTYSAUMXMUQXNXOVBXP $. + ex simpr csca cbs wf eqid ply1ring ply1lmod asclf 0nn0 coe1fvalcl sylancl + ply1sca eqcomd fveq2d eleqtrrd ffvelrnd 3jca cif cvv cmpt coe1scl syl2anc + weq wn nnne0 neneqd eqeq1 notbid iffalse nnnn0 c0g fvex eqeltri a1i eqtrd + fvmptd ralimdva imp ply1sclid jca csn df-n0 raleqi eqeq12d ralunsn syl5bb + cun c0ex eqcoe1ply1eq sylc eqeq2d rspcedv mpd impbid ) DUBOZGBOZPZGIUCZAQ + ZRZIFUDZEUCZGUEQZQZHRZEUFUGZXSYBYHIFXSXTFOZPZYBYHYJYBPZYHYDYAUEQZQZHRZEUF + UGZYKXQYIPZYOYJYPYBXSXQYIXQXRUHZUJSABCDXTEFHJKLMNUIUKYBYHYOULYJYBYGYNEUFY + BYFYMHYBYDYEYLGYAUEUMUNUOUPUQURVBUSXSYHYCXSYHPZGTYEQZAQZRZYCYRXQXRYTBOZUT + ZYFYDYTUEQZQZRZEVAUGZUUAXSUUCYHXSXQXRUUBYQXQXRVCZXSCVDQZVEQZBYSAXQUUJBAVF + XRXQABUUIUUJCNUUIVGCDLVHCDLVIUUJVGMVJSXSYSDVEQZUUJXSXRTVAOZYSUUKOUUHVKYEB + CDGUUKTYEVGZMLUUKVGVLVMXQUUJUUKRXRXQUUIDVEXQDUUICDUBLVNVOVPSVQVRVSSYRUUGU + UFEUFUGZYSTUUDQZRZPZYRUUNUUPXSYHUUNXSYGUUFEUFXSYDUFOZPZYGUUFUUSYGPYFHUUEU + USYGVCUUSHUUERYGUUSUUEHUUSUAYDUAUCZTRZYSHVTZHVAUUDWAXSUUDUAVAUVBWBRZUURXS + XQYSFOZUVCYQXSXRUULUVDUUHVKYEBCDGFTUUMMLJVLVMZUAACDFYSHLNJKWCWDSUUSUAEWEZ + PZUVAWFZUVBHRUVGUVHYDTRZWFZUUSUVJUVFUURUVJXSUURYDTYDWGWHUQSUVFUVHUVJULUUS + UVFUVAUVIUUTYDTWIWJUQURUVAYSHWKUKUURYDVAOXSYDWLUQHWAOUUSHDWMQWAKDWMWNWOWP + WRVOSWQVBWSWTXSUUPYHXSXQUVDUUPYQUVEACDFYSLNJXAWDSXBUUGUUFEUFTXCXIZUGZYRUU + QUUFEVAUVKXDXEYRTWAOZUVLUUQULUVMYRXJWPUUFUUPEUFTWAUVIYFYSUUEUUOYDTYEUMYDT + UUDUMXFXGUKXHURYEBUUDCDEGYTLMUUMUUDVGXKXLYRYBUUAIYSFXSUVDYHUVESXTYSRZYBUU + AULYRUVNYAYTGXTYSAUMXMUQXNXOVBXP $. $} ${ @@ -246763,33 +246986,33 @@ series in the subring which are also polynomials (in the parent -> ( ( coe1 ` ( P gsum ( x e. ( m u. { a } ) |-> M ) ) ) ` K ) = ( R gsum ( x e. ( m u. { a } ) |-> ( ( coe1 ` M ) ` K ) ) ) ) ) ) $= ( vy wcel cmpt cgsu co cco1 cfv cv cfn wn w3a wral wi csn cun wa ralunb - wceq csb cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cvv eqid crg ply1rng - syl rngcmn 3ad2ant3 ad2antrr simpll1 rspcsbela expcom adantl adantr imp - ccmn vex simpll2 ssnid mpan csbeq1 gsumunsn syl5eq eqcomi oveq1i syl6eq - a1i fveq2d fveq1d cn0 gsummptcl coe1addfv syl31anc eqtrd oveq1 sylan9eq - simplr cbs csbfv12 csbfv2g ax-mp csbconstg fveq12i eqtri coe1f ffvelrnd - wf syl5eqel nffv csbhypf syl6req exp31 com23 ex a2d imp4b syl5bi ) FUAZ - UBOZIUAZXOOUCZAUDZHCOZBXOUEZGDBXOHPZQRZSTTZEBXOGHSTZTZPZQRZUKZUFZXTBXOX - QUGZUHZUEZGDBYLHPZQRZSTZTZEBYLYFPZQRZUKZUFYMYAXTBYKUEZUIXSYJUIYTXTBXOYK - UJXSYJYAUUAYTXSYAYIUUAYTUFZXSYAYIUUBUFXSYAUIZUUAYIYTUUCUUAYIYTUUCUUAUIZ - YIUIYQYHGBXQHULZSTZTZEUMTZRZYSUUDYIYQYDUUGUUHRZUUIUUDYQGYCUUEDUMTZRZSTZ - TZUUJUUDGYPUUMUUDYOUULSUUDYODNXOBNUAZHULZPZQRZUUEUUKRZUULUUDYODNYLUUPPZ - QRUUSYNUUTDQBNYLHUUPNHUNZBUUOHUOZBUUOHUPZUQURUUDXOCUUKNDXQUSUUPUUEKUUKU - TZXSDVMOZYAUUAAXPUVEXRADVAOZUVEAEVAOZUVFLDEJVBVCDVDVCVEVFZXPXRAYAUUAVGZ - UUDUUOXOOZUUPCOZUUCUVJUVKUFZUUAYAUVLXSUVJYAUVKBUUOXOHCVHVIVJVKVLZXQUSOU - UDIVNWDZXPXRAYAUUAVOZUUAUUECOZUUCXQYKOUUAUVPIVPBXQYKHCVHVQVJZBUUOXQHVRV - SVTUURYCUUEUUKUUQYBDQYBUUQBNXOHUUPUVAUVBUVCUQWAURWBWCWEWFUUDUVGYCCOUVPG - WGOZUUNUUJUKXSUVGYAUUAAXPUVGXRLVEVFUUDCBDXOHKUVHUVIXSYAUUAWNWHUVQXSUVRY - AUUAAXPUVRXRMVEZVFZCUUHUUKEYCUUEGDJKUVDUUHUTZWIWJWKYDYHUUGUUHWLWMUUDUUI - YSUKYIUUDYSENXOBUUOYFULZPZQRZUUGUUHRZUUIUUDYSENYLUWBPZQRUWEYRUWFEQBNYLY - FUWBNYFUNZBUUOYFUOZBUUOYFUPZUQURUUDXOEWOTZUUHNEXQUSUWBUUGUWJUTZUWAXSEVM - OZYAUUAAXPUWLXRAUVGUWLLEVDVCVEVFUVIUUDUVJUIZUWBGUUPSTZTZUWJUWBBUUOGULZB - UUOYEULZTUWOBUUOGYEWPUWPGUWQUWNUUOUSOZUWQUWNUKNVNZBUUOHUSSWQWRUWRUWPGUK - UWSBUUOGUSWSWRWTXAUWMWGUWJGUWNUWMUVKWGUWJUWNXDUVMUWNCDEUUPUWJUWNUTKJUWK - XBVCUUCUVRUUAUVJXSUVRYAUVSVKVFXCXEUVNUVOUUDWGUWJGUUFUUDUVPWGUWJUUFXDUVQ - UUFCDEUUEUWJUUFUTKJUWKXBVCUVTXCBNXQYFUUGBXQUNBGUUFBUUESBSUNBXQHUOXFBGUN - XFBUAXQUKZGYEUUFUWTHUUESBXQHUPWEWFXGVSVTUWDYHUUGUUHUWCYGEQYGUWCBNXOYFUW - BUWGUWHUWIUQWAURWBXHVKWKXIXJXKXLXMXNXK $. + wceq csb cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cvv crg ply1ring syl + eqid ccmn ringcmn 3ad2ant3 ad2antrr simpll1 rspcsbela expcom adantl imp + adantr vex a1i simpll2 mpan csbeq1 gsumunsn syl5eq eqcomi oveq1i syl6eq + ssnid fveq2d fveq1d cn0 simplr gsummptcl coe1addfv syl31anc eqtrd oveq1 + sylan9eq cbs csbfv12 csbfv2g ax-mp csbconstg eqtri wf ffvelrnd syl5eqel + fveq12i coe1f nffv csbhypf syl6req exp31 com23 ex a2d imp4b syl5bi ) FU + AZUBOZIUAZXOOUCZAUDZHCOZBXOUEZGDBXOHPZQRZSTTZEBXOGHSTZTZPZQRZUKZUFZXTBX + OXQUGZUHZUEZGDBYLHPZQRZSTZTZEBYLYFPZQRZUKZUFYMYAXTBYKUEZUIXSYJUIYTXTBXO + YKUJXSYJYAUUAYTXSYAYIUUAYTUFZXSYAYIUUBUFXSYAUIZUUAYIYTUUCUUAYIYTUUCUUAU + IZYIUIYQYHGBXQHULZSTZTZEUMTZRZYSUUDYIYQYDUUGUUHRZUUIUUDYQGYCUUEDUMTZRZS + TZTZUUJUUDGYPUUMUUDYOUULSUUDYODNXOBNUAZHULZPZQRZUUEUUKRZUULUUDYODNYLUUP + PZQRUUSYNUUTDQBNYLHUUPNHUNZBUUOHUOZBUUOHUPZUQURUUDXOCUUKNDXQUSUUPUUEKUU + KVCZXSDVDOZYAUUAAXPUVEXRADUTOZUVEAEUTOZUVFLDEJVAVBDVEVBVFVGZXPXRAYAUUAV + HZUUDUUOXOOZUUPCOZUUCUVJUVKUFZUUAYAUVLXSUVJYAUVKBUUOXOHCVIVJVKVMVLZXQUS + OUUDIVNVOZXPXRAYAUUAVPZUUAUUECOZUUCXQYKOUUAUVPIWDBXQYKHCVIVQVKZBUUOXQHV + RVSVTUURYCUUEUUKUUQYBDQYBUUQBNXOHUUPUVAUVBUVCUQWAURWBWCWEWFUUDUVGYCCOUV + PGWGOZUUNUUJUKXSUVGYAUUAAXPUVGXRLVFVGUUDCBDXOHKUVHUVIXSYAUUAWHWIUVQXSUV + RYAUUAAXPUVRXRMVFZVGZCUUHUUKEYCUUEGDJKUVDUUHVCZWJWKWLYDYHUUGUUHWMWNUUDU + UIYSUKYIUUDYSENXOBUUOYFULZPZQRZUUGUUHRZUUIUUDYSENYLUWBPZQRUWEYRUWFEQBNY + LYFUWBNYFUNZBUUOYFUOZBUUOYFUPZUQURUUDXOEWOTZUUHNEXQUSUWBUUGUWJVCZUWAXSE + VDOZYAUUAAXPUWLXRAUVGUWLLEVEVBVFVGUVIUUDUVJUIZUWBGUUPSTZTZUWJUWBBUUOGUL + ZBUUOYEULZTUWOBUUOGYEWPUWPGUWQUWNUUOUSOZUWQUWNUKNVNZBUUOHUSSWQWRUWRUWPG + UKUWSBUUOGUSWSWRXDWTUWMWGUWJGUWNUWMUVKWGUWJUWNXAUVMUWNCDEUUPUWJUWNVCKJU + WKXEVBUUCUVRUUAUVJXSUVRYAUVSVMVGXBXCUVNUVOUUDWGUWJGUUFUUDUVPWGUWJUUFXAU + VQUUFCDEUUEUWJUUFVCKJUWKXEVBUVTXBBNXQYFUUGBXQUNBGUUFBUUESBSUNBXQHUOXFBG + UNXFBUAXQUKZGYEUUFUWTHUUESBXQHUPWEWFXGVSVTUWDYHUUGUUHUWCYGEQYGUWCBNXOYF + UWBUWGUWHUWIUQWAURWBXHVMWLXIXJXKXLXMXNXK $. $} $d B a m n $. $d B x $. $d K x $. $d K a m n $. $d N n x $. $d m y $. @@ -246838,14 +247061,14 @@ series in the subring which are also polynomials (in the parent (Contributed by AV, 8-Oct-2019.) $) gsumsmonply1 $p |- ( ph -> ( P gsum ( k e. NN0 |-> ( A .* ( k .^ X ) ) ) ) e. B ) $= - ( cn0 cv co cmpt cvv c0g cfv eqid crg wcel ccmn ply1rng rngcmn 3syl nn0ex - a1i r19.21bi w3a 3ad2ant1 simp3 simp2 cmgp ply1tmcl syl3anc mpd3an3 fmptd - clmod ply1lmod syl csca wceq ply1sca ply1moncl sylan mptscmfsupp0 gsumcl - ) AUBCFUBBFUCZJGUDZHUDZUEZDUFDUGUHZMWBUIZAEUJUKZDUJUKDULUKPDELUMDUNUOUBUF - UKAUPUQZAFUBVTCWAAVRUBUKZBIUKZVTCUKZAWGFUBTURZAWFWGUSWDWGWFWHAWFWDWGPUTAW - FWGVAAWFWGVBCBVRDEHGIDVCUHZJQLNRWJUIZOMVDVEVFWAUIVGAIUBDEBFHCUFVSWBKWEAWD - DVHUKPDELVIVJAWDEDVKUHVLPDEUJLVMVJMWIAWDWFVSCUKPCVRDEGWJJLNWKOMVNVOWCSRUA - VPVQ $. + ( cn0 cv co cmpt cvv c0g cfv eqid crg wcel ccmn ply1ring ringcmn 3syl a1i + nn0ex r19.21bi w3a 3ad2ant1 simp3 simp2 cmgp ply1tmcl syl3anc fmptd clmod + mpd3an3 ply1lmod csca wceq ply1sca ply1moncl sylan mptscmfsupp0 gsumcl + syl ) AUBCFUBBFUCZJGUDZHUDZUEZDUFDUGUHZMWBUIZAEUJUKZDUJUKDULUKPDELUMDUNUO + UBUFUKAUQUPZAFUBVTCWAAVRUBUKZBIUKZVTCUKZAWGFUBTURZAWFWGUSWDWGWFWHAWFWDWGP + UTAWFWGVAAWFWGVBCBVRDEHGIDVCUHZJQLNRWJUIZOMVDVEVHWAUIVFAIUBDEBFHCUFVSWBKW + EAWDDVGUKPDELVIVQAWDEDVJUHVKPDEUJLVLVQMWIAWDWFVSCUKPCVRDEGWJJLNWKOMVMVNWC + SRUAVOVP $. $d A n s x $. $d K n $. $d L k n s x $. $d P k n s $. $d R k n $. $d X n s $. $d .0. k s n x $. $d .^ k n s x $. $d .* n s $. @@ -246860,65 +247083,65 @@ series in the subring which are also polynomials (in the parent co cmap wcel cvv wf r19.21bi eqid fmptd cbs fvex eqeltri a1i nn0ex elmapg wb sylancl mpbird c0g fsuppmapnn0ub mpd wa simpr rspcsbela syl2anc fvmpts ad2antrr eqeq1d imbi2d ralimdva cc0 cfz nfv nfcv nfcsb1v nfeq1 nfim nfral - 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CRing /\ N e. NN0 /\ A e. B ) -> ( N .^ ( X .+ A ) ) = ( P gsum ( k e. ( 0 ... N ) |-> ( ( N _C k ) .x. ( ( ( N - k ) .^ A ) .X. ( k .^ X ) ) ) ) ) ) $= - ( ccrg wcel cn0 w3a co cc0 cfz cbc cmin cmpt cgsu ccmn wceq crg crngrng - cv ply1rng rngcmn 3syl 3ad2ant1 vr1cl syl cmncom syl3anc oveq2d cbs cfv - simp3 ply1crng simp2 eleq2i biimpi 3ad2ant3 syl6eleq crngbinom syl22anc - eqid eqtrd ) EUAUBZKUCUBZABUBZUDZKLADUEZIUEKALDUEZIUEZCHUFKUGUEKHUPZUHU - EKWFUIUEAIUEWFLIUEGUEFUEUJUKUEZWBWCWDKIWBCULUBZLBUBZWAWCWDUMVSVTWHWAVSE - UNUBZCUNUBWHEUOZCEMUQCURUSUTVSVTWIWAVSWJWIWKBCELNMTVAVBZUTVSVTWAVHBDCLA - TOVCVDVEWBCUAUBZVTACVFVGZUBZLWNUBZWEWGUMVSVTWMWACEMVIUTVSVTWAVJWAVSWOVT - WAWOBWNATVKVLVMVSVTWPWAVSLBWNWLTVNUTALDCWNFGHIJKWNVQPQORSVOVPVR $. + ( ccrg wcel cn0 w3a co cc0 cfz cv cbc cmin cmpt cgsu ccmn wceq crngring + crg ply1ring ringcmn 3ad2ant1 vr1cl syl simp3 cmncom syl3anc oveq2d cbs + 3syl cfv ply1crng simp2 eleq2i biimpi 3ad2ant3 syl6eleq crngbinom eqtrd + eqid syl22anc ) EUAUBZKUCUBZABUBZUDZKLADUEZIUEKALDUEZIUEZCHUFKUGUEKHUHZ + UIUEKWFUJUEAIUEWFLIUEGUEFUEUKULUEZWBWCWDKIWBCUMUBZLBUBZWAWCWDUNVSVTWHWA + VSEUPUBZCUPUBWHEUOZCEMUQCURVGUSVSVTWIWAVSWJWIWKBCELNMTUTVAZUSVSVTWAVBBD + CLATOVCVDVEWBCUAUBZVTACVFVHZUBZLWNUBZWEWGUNVSVTWMWACEMVIUSVSVTWAVJWAVSW + OVTWAWOBWNATVKVLVMVSVTWPWAVSLBWNWLTVNUSALDCWNFGHIJKWNVQPQORSVOVRVP $. $} $d S k $. @@ -247010,30 +247233,30 @@ series in the subring which are also polynomials (in the parent |-> ( ( N _C k ) .x. ( ( S ` ( ( N - k ) E A ) ) .X. ( k .^ X ) ) ) ) ) ) $= ( ccrg wcel cn0 w3a cfv co cc0 cfz cv cbc cmin cmpt cgsu cbs wceq wf csca - eqid crg crngrng ply1rng syl 3ad2ant1 clmod ply1lmod asclf ply1sca fveq2d - syl5eq feq2d mpbird simp3 ffvelrnd lply1binom syld3an3 cur cvsca cmgp cmg - wa casa ply1assa adantr fznn0sub adantl eleq2d biimpa 3adant2 assamulgscm - rngidcl syl13anc eqcomd oveqd cmnd rngmgp mgpbas rngidval mulgnn0z syl2an - oveq12d asclval oveq2d syl6eleq mulgnn0cl syl3anc syl6eq eleqtrrd 3eqtr4d - eqtrd simpr oveq1d mpteq2dva ) DUGUHZNUIUHZAMUHZUJZNOAEUKZCULJULZBHUMNUNU - LZNHUOZUPULZNYFUQULZYCJULZYFOJULZGULZFULZURZUSULZBHYEYGYHAIULZEUKZYJGULZF - ULZURZUSULXSXTYAYCBUTUKZUHYDYNVAYBMYTAEYBMYTEVBBVCUKZUTUKZYTEVBYBEYTUUAUU - BBUDUUAVDZXSXTBVEUHZYAXSDVEUHZUUDDVFZBDPVGVHZVIXSXTBVJUHZYAXSUUEUUHUUFBDP - VKVHVIUUBVDZYTVDZVLYBMUUBYTEYBMDUTUKZUUBUCYBDUUAUTXSXTDUUAVAZYABDUGPVMZVI - ZVNVOVPVQXSXTYAVRVSYCYTBCDFGHJKNOPQRSTUAUBUUJVTWAYBYMYSBUSYBHYEYLYRYBYFYE - UHZWFZYKYQYGFUUPYIYPYJGUUPYHABWBUKZBWCUKZULZJULZYOUUQUURULZYIYPUUPUUTYHAU - UAWDUKZWEUKZULZYHUUQJULZUURULZUVAUUPBWGUHZYHUIUHZAUUBUHZUUQYTUHZUUTUVFVAY - BUVGUUOXSXTUVGYABDPWHVIWIUUOUVHYBYFUMNWJZWKZYBUVIUUOXSYAUVIXTXSYAUVIXSMUU - BAXSMUUKUUBUCXSDUUAUTUUMVNVOWLWMWNWIZYBUVJUUOXSXTUVJYAXSUUDUVJUUGYTBUUQUU - JUUQVDZWPVHVIWIAUUBUURJUVCUUAUVBKYHYTBUUQUUJUUCUUIUURVDZUVBVDUVCVDUAUBWOW - QUUPUVDYOUVEUUQUURUUPUVCIYHAUUPIUVCYBIUVCVAZUUOXSXTUVPYAXSILWEUKUVCUFXSLU - VBWEXSLDWDUKUVBUEXSDUUAWDUUMVNVOVNVOVIWIWRWSYBKWTUHZUVHUVEUUQVAUUOXSXTUVQ - YAXSUUDUVQUUGBKUAXAVHVIUVKYTJKYHUUQYTBKUAUUJXBUBBUUQKUAUVNXCXDXEXFXOUUPYC - UUSYHJUUPUVIYCUUSVAUVMEUURUUQUUAUUBBAUDUUCUUIUVOUVNXGVHXHUUPYOUUBUHYPUVAV - AUUPYOLUTUKZUUBUUPLWTUHZUVHAUVRUHZYOUVRUHYBUVSUUOXSXTUVSYAXSUUEUVSUUFDLUE - XAVHVIWIUVLYBUVTUUOXSYAUVTXTXSYAWFAMUVRXSYAXPMDLUEUCXBXIWNWIUVRILYHAUVRVD - UFXJXKUUPUUBUUKUVRUUPUUADUTUUPDUUAYBUULUUOUUNWIWRVNUUKDLUEUUKVDXBXLXMEUUR - UUQUUAUUBBYOUDUUCUUIUVOUVNXGVHXNXQXHXRXHXO $. + eqid crg crngring ply1ring syl 3ad2ant1 clmod asclf ply1sca fveq2d syl5eq + ply1lmod feq2d mpbird simp3 ffvelrnd lply1binom syld3an3 wa cur cvsca cmg + cmgp casa ply1assa adantr fznn0sub adantl eleq2d biimpa ringidcl syl13anc + 3adant2 assamulgscm eqcomd oveqd ringmgp mgpbas rngidval mulgnn0z oveq12d + cmnd syl2an eqtrd asclval oveq2d simpr syl6eleq mulgnn0cl syl6eq eleqtrrd + syl3anc 3eqtr4d oveq1d mpteq2dva ) DUGUHZNUIUHZAMUHZUJZNOAEUKZCULJULZBHUM + NUNULZNHUOZUPULZNYFUQULZYCJULZYFOJULZGULZFULZURZUSULZBHYEYGYHAIULZEUKZYJG + ULZFULZURZUSULXSXTYAYCBUTUKZUHYDYNVAYBMYTAEYBMYTEVBBVCUKZUTUKZYTEVBYBEYTU + UAUUBBUDUUAVDZXSXTBVEUHZYAXSDVEUHZUUDDVFZBDPVGVHZVIXSXTBVJUHZYAXSUUEUUHUU + FBDPVOVHVIUUBVDZYTVDZVKYBMUUBYTEYBMDUTUKZUUBUCYBDUUAUTXSXTDUUAVAZYABDUGPV + LZVIZVMVNVPVQXSXTYAVRVSYCYTBCDFGHJKNOPQRSTUAUBUUJVTWAYBYMYSBUSYBHYEYLYRYB + YFYEUHZWBZYKYQYGFUUPYIYPYJGUUPYHABWCUKZBWDUKZULZJULZYOUUQUURULZYIYPUUPUUT + YHAUUAWFUKZWEUKZULZYHUUQJULZUURULZUVAUUPBWGUHZYHUIUHZAUUBUHZUUQYTUHZUUTUV + FVAYBUVGUUOXSXTUVGYABDPWHVIWIUUOUVHYBYFUMNWJZWKZYBUVIUUOXSYAUVIXTXSYAUVIX + SMUUBAXSMUUKUUBUCXSDUUAUTUUMVMVNWLWMWPWIZYBUVJUUOXSXTUVJYAXSUUDUVJUUGYTBU + UQUUJUUQVDZWNVHVIWIAUUBUURJUVCUUAUVBKYHYTBUUQUUJUUCUUIUURVDZUVBVDUVCVDUAU + BWQWOUUPUVDYOUVEUUQUURUUPUVCIYHAUUPIUVCYBIUVCVAZUUOXSXTUVPYAXSILWEUKUVCUF + XSLUVBWEXSLDWFUKUVBUEXSDUUAWFUUMVMVNVMVNVIWIWRWSYBKXEUHZUVHUVEUUQVAUUOXSX + TUVQYAXSUUDUVQUUGBKUAWTVHVIUVKYTJKYHUUQYTBKUAUUJXAUBBUUQKUAUVNXBXCXFXDXGU + UPYCUUSYHJUUPUVIYCUUSVAUVMEUURUUQUUAUUBBAUDUUCUUIUVOUVNXHVHXIUUPYOUUBUHYP + UVAVAUUPYOLUTUKZUUBUUPLXEUHZUVHAUVRUHZYOUVRUHYBUVSUUOXSXTUVSYAXSUUEUVSUUF + DLUEWTVHVIWIUVLYBUVTUUOXSYAUVTXTXSYAWBAMUVRXSYAXJMDLUEUCXAXKWPWIUVRILYHAU + VRVDUFXLXOUUPUUBUUKUVRUUPUUADUTUUPDUUAYBUULUUOUUNWIWRVMUUKDLUEUUKVDXAXMXN + EUURUUQUUAUUBBYOUDUUCUUIUVOUVNXHVHXPXQXIXRXIXG $. $} $( @@ -247160,11 +247383,11 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evls1rhmlem $p |- ( R e. CRing -> F e. ( ( R ^s ( B ^m 1o ) ) RingHom T ) ) $= ( ccrg wcel c1o cmap co cbs cfv cv cmpt cvv eqid cpws csn cxp ccom pwsbas - crh wceq ovex mpan2 mpteq1d syl5eq crngrng fvex eqeltri a1i wf1o wf df1o2 - c0 0ex mapsnf1o3 f1of mp1i pwsco1rhm eqeltrd ) DJKZFADCLMNZUANZOPZAQBCLBQ - UBUCRZUDZRZVHEUFNVFFACVGMNZVKRVLIVFAVMVIVKVFVGSKZVMVIUGCLMUHZCDVGJSVHVHTZ - GUEUIUJUKVFCVGVIDAVJSSEVHHVPVITDULCSKVFCDOPSGDOUMUNZUOVNVFVOUOCVGVJUPCVGV - JUQVFBCLVJUSURVQUTVJTVACVGVJVBVCVDVE $. + crh wceq ovex mpan2 mpteq1d syl5eq crngring fvex eqeltri wf1o wf c0 df1o2 + a1i 0ex mapsnf1o3 f1of mp1i pwsco1rhm eqeltrd ) DJKZFADCLMNZUANZOPZAQBCLB + QUBUCRZUDZRZVHEUFNVFFACVGMNZVKRVLIVFAVMVIVKVFVGSKZVMVIUGCLMUHZCDVGJSVHVHT + ZGUEUIUJUKVFCVGVIDAVJSSEVHHVPVITDULCSKVFCDOPSGDOUMUNZUSVNVFVOUSCVGVJUOCVG + VJUPVFBCLVJUQURVQUTVJTVACVGVJVBVCVDVE $. $} ${ @@ -247205,27 +247428,28 @@ univariate polynomial evaluation map function for a (sub)ring is a ring (Contributed by AV, 8-Sep-2019.) $) evls1sca $p |- ( ph -> ( Q ` ( A ` X ) ) = ( B X. { X } ) ) $= ( cfv wcel cvv vx vy vz c1o cmap co cv csn cxp cmpt ccom ces cmpl cpws wf - cbs wceq crh con0 ccrg csubrg 1on eqid evlsrhm syl3anc rhmf csca subrgrng - a1i syl crg ply1rng clmod ply1lmod asclf wss subrgss ply1sca fveq2d eqtrd - ressbas2 cps1 ply1bas eqcomd feq23d ffvelrnd fvco3 syl2anc cascl ply1ascl - mpbird syl6eq fveq1d evlssca eqidd adantl sseldd fconst6g wb fvex eqeltri - coeq1 ovex elmapg snex mptexg coexg fvmptd pm3.2i fconstmpt fmptco 3eqtrd - xpex wa cpw elpwg evls1fval 3eqtr4d ) AIBRZUACCUDUEUFZUEUFZUAUGZUBCUDUBUG - ZUHUIZUJZUKZUJZEUDFULUFZRZUKZRZUBCIUJZXSDRCIUHZUIZAYKXSYIRZYGRZXTYMUIZYGR - ZYLAUDGUMUFZUPRZFXTUNUFZUPRZYIUOZXSYTSYKYPUQAYIYSUUAURUFSZUUCAUDUSSZFUTSZ - EFVARZSZUUDUUEAVBVIZOPCYIEFUUAGUDUSYSYIVCZYSVCZLUUAVCMVDVEYTUUBYSUUAYIYTV - CUUBVCVFVJAEYTIBAEYTBUOHVGRZUPRZHUPRZBUOABUUNUULUUMHNUULVCAGVKSZHVKSAUUHU - UOPEFGLVHVJZHGKVLVJAUUOHVMSUUPHGKVNVJUUMVCUUNVCZVOAEYTUUMUUNBAEGUPRZUUMAE - CVPZEUURUQAUUHUUSPECFMVQZVJZECGFLMWAVJAGUULUPAUUOGUULUQUUPHGVKKVRVJVSVTAU - UNYTUUNYTUQAHGGWBRZUUNKUVBVCUUQWCVIWDWEWKQWFYTUUBXSYGYIWGWHAYOYQYGAYOIYSW - IRZRZYIRYQAXSUVDYIAIBUVCABHWIRZUVCBUVEUQANVIUVEHGKUVEVCWJWLWMVSAUVCCYIEFG - UDUSYSIUUJUUKLMUVCVCUUIOPQWNVTVSAYRYQYEUKZYLAUAYQYFUVFYAYGTAYGWOYBYQUQYFU - VFUQAYBYQYEXBWPAYQYASZXTCYQUOZAICSUVHAECIUVAQWQXTICWRVJACTSZXTTSZUVGUVHWS - UVIACFUPRTMFUPWTXAZVIZUVJACUDUEXCZVICXTYQTTXDWHWKAYQTSZYETSZUVFTSUVNAXTYM - UVMIXEXMVIAUVIUVOUVLUBCYDTXFVJYQYETTXGWHXHAUBUCCXTYDIIYEYQAYCCSZXNZYDXTSZ - UDCYDUOZUVPUVSAUDYCCWRWPUVQUVIUUEXNZUVRUVSWSUVTUVQUVIUUEUVKVBXIVICUDYDTUS - XDVJWKAYEWOYQUCXTIUJUQAUCXTIXJVIUCUGYDUQIWOXKVTXLAXSDYJAUUFECXOSZDYJUQOAU - UHUWAPUUHUWAUUSUUTECUUGXPWKVJUAUBCDEFYHUTJYHVCMXQWHWMYNYLUQAUBCIXJVIXR $. + cbs wceq crh con0 ccrg csubrg 1on a1i eqid evlsrhm syl3anc rhmf subrgring + syl crg ply1ring clmod ply1lmod asclf wss subrgss ressbas2 ply1sca fveq2d + csca eqtrd cps1 ply1bas eqcomd feq23d mpbird ffvelrnd fvco3 syl2anc cascl + ply1ascl syl6eq fveq1d evlssca eqidd coeq1 adantl sseldd fconst6g wb fvex + eqeltri ovex elmapg snex xpex mptexg coexg fvmptd pm3.2i fconstmpt fmptco + wa 3eqtrd cpw elpwg evls1fval 3eqtr4d ) AIBRZUACCUDUEUFZUEUFZUAUGZUBCUDUB + UGZUHUIZUJZUKZUJZEUDFULUFZRZUKZRZUBCIUJZXSDRCIUHZUIZAYKXSYIRZYGRZXTYMUIZY + GRZYLAUDGUMUFZUPRZFXTUNUFZUPRZYIUOZXSYTSYKYPUQAYIYSUUAURUFSZUUCAUDUSSZFUT + SZEFVARZSZUUDUUEAVBVCZOPCYIEFUUAGUDUSYSYIVDZYSVDZLUUAVDMVEVFYTUUBYSUUAYIY + TVDUUBVDVGVIAEYTIBAEYTBUOHVTRZUPRZHUPRZBUOABUUNUULUUMHNUULVDAGVJSZHVJSAUU + HUUOPEFGLVHVIZHGKVKVIAUUOHVLSUUPHGKVMVIUUMVDUUNVDZVNAEYTUUMUUNBAEGUPRZUUM + AECVOZEUURUQAUUHUUSPECFMVPZVIZECGFLMVQVIAGUULUPAUUOGUULUQUUPHGVJKVRVIVSWA + AUUNYTUUNYTUQAHGGWBRZUUNKUVBVDUUQWCVCWDWEWFQWGYTUUBXSYGYIWHWIAYOYQYGAYOIY + SWJRZRZYIRYQAXSUVDYIAIBUVCABHWJRZUVCBUVEUQANVCUVEHGKUVEVDWKWLWMVSAUVCCYIE + FGUDUSYSIUUJUUKLMUVCVDUUIOPQWNWAVSAYRYQYEUKZYLAUAYQYFUVFYAYGTAYGWOYBYQUQY + FUVFUQAYBYQYEWPWQAYQYASZXTCYQUOZAICSUVHAECIUVAQWRXTICWSVIACTSZXTTSZUVGUVH + WTUVIACFUPRTMFUPXAXBZVCZUVJACUDUEXCZVCCXTYQTTXDWIWFAYQTSZYETSZUVFTSUVNAXT + YMUVMIXEXFVCAUVIUVOUVLUBCYDTXGVIYQYETTXHWIXIAUBUCCXTYDIIYEYQAYCCSZXMZYDXT + SZUDCYDUOZUVPUVSAUDYCCWSWQUVQUVIUUEXMZUVRUVSWTUVTUVQUVIUUEUVKVBXJVCCUDYDT + USXDVIWFAYEWOYQUCXTIUJUQAUCXTIXKVCUCUGYDUQIWOXLWAXNAXSDYJAUUFECXOSZDYJUQO + AUUHUWAPUUHUWAUUSUUTECUUGXPWFVIUAUBCDEFYHUTJYHVDMXQWIWMYNYLUQAUBCIXKVCXR + $. $} ${ @@ -247246,13 +247470,13 @@ univariate polynomial evaluation map function for a (sub)ring is a ring sums. (Contributed by AV, 14-Sep-2019.) $) evls1gsumadd $p |- ( ph -> ( Q ` ( W gsum ( x e. N |-> Y ) ) ) = ( P gsum ( x e. N |-> ( Q ` Y ) ) ) ) $= - ( cfv cmpt cgsu cvv crg wcel ccmn csubrg subrgrng syl ply1rng rngcmn 3syl - co wa cmnd ccrg crngrng cbs eqeltri a1i jca pwsrng rngmnd cn0 nn0ex ssexd - fvex crh cghm cmhm evls1rhm syl2anc rhmghm ghmmhm gsummptmhm eqcomd ) ADB - JLEUFUGUHUSKBJLUGUHUSEUFABJCLKDEUIMTQAHUJUKZKUJUKKULUKAFGUMUFUKZWCUBFGHRU - NUOKHPUPKUQURAGUJUKZIUIUKZUTDUJUKDVAUKAWEWFAGVBUKZWEUAGVCUOWFAIGVDUFUIOGV - DVMVEVFVGGIUIDSVHDVIURAJVJUIVJUIUKAVKVFUDVLAEKDVNUSUKZEKDVOUSUKEKDVPUSUKA - WGWDWHUAUBIEFGDHKNOSRPVQVRKDEVSKDEVTURUCUEWAWB $. + ( cfv cmpt cgsu co cvv crg wcel ccmn csubrg subrgring ply1ring ringcmn wa + syl 3syl cmnd ccrg crngring cbs eqeltri a1i jca pwsring ringmnd cn0 nn0ex + fvex ssexd crh cghm cmhm evls1rhm syl2anc rhmghm ghmmhm gsummptmhm eqcomd + ) ADBJLEUFUGUHUIKBJLUGUHUIEUFABJCLKDEUJMTQAHUKULZKUKULKUMULAFGUNUFULZWCUB + FGHRUOUSKHPUPKUQUTAGUKULZIUJULZURDUKULDVAULAWEWFAGVBULZWEUAGVCUSWFAIGVDUF + UJOGVDVLVEVFVGGIUJDSVHDVIUTAJVJUJVJUJULAVKVFUDVMAEKDVNUIULZEKDVOUIULEKDVP + UIULAWGWDWHUAUBIEFGDHKNOSRPVQVRKDEVSKDEVTUTUCUEWAWB $. $} ${ @@ -247276,12 +247500,12 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evls1gsummul $p |- ( ph -> ( Q ` ( G gsum ( x e. N |-> Y ) ) ) = ( H gsum ( x e. N |-> ( Q ` Y ) ) ) ) $= ( cfv cmpt cgsu co cvv mgpbas rngidval ccrg wcel csubrg subrgcrng syl2anc - ccmn ply1crng crngmgp 3syl crg wa cmnd crngrng syl cbs fvex eqeltri jctir - pwsrng rngmgp cn0 nn0ex a1i ssexd cmhm evls1rhm rhmmhm gsummptmhm eqcomd - crh ) AKBMOEUJUKULUMJBMOUKULUMEUJABMCOJKEUNICNJSUDUONIJSTUPAHUQURZNUQURJV - BURAGUQURZFGUSUJURZWGUEUFFGHUAUTVANHRVCNJSVDVEAGVFURZLUNURZVGDVFURKVHURAW - JWKAWHWJUEGVIVJLGVKUJUNQGVKVLVMVNGLUNDUBVODKUCVPVEAMVQUNVQUNURAVRVSUHVTAE - NDWFUMURZEJKWAUMURAWHWIWLUEUFLEFGDHNPQUBUARWBVANDEJKSUCWCVJUGUIWDWE $. + ccmn ply1crng crngmgp 3syl crg cmnd crngring syl cbs fvex eqeltri pwsring + wa jctir ringmgp cn0 nn0ex a1i crh cmhm evls1rhm rhmmhm gsummptmhm eqcomd + ssexd ) AKBMOEUJUKULUMJBMOUKULUMEUJABMCOJKEUNICNJSUDUONIJSTUPAHUQURZNUQUR + JVBURAGUQURZFGUSUJURZWGUEUFFGHUAUTVANHRVCNJSVDVEAGVFURZLUNURZVNDVFURKVGUR + AWJWKAWHWJUEGVHVILGVJUJUNQGVJVKVLVOGLUNDUBVMDKUCVPVEAMVQUNVQUNURAVRVSUHWF + AENDVTUMURZEJKWAUMURAWHWIWLUEUFLEFGDHNPQUBUARWBVANDEJKSUCWCVIUGUIWDWE $. $} ${ @@ -247303,10 +247527,10 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evls1varpw $p |- ( ph -> ( Q ` ( N .^ X ) ) = ( N ( .g ` ( mulGrp ` ( S ^s B ) ) ) ( Q ` X ) ) ) $= ( cpws co cmgp cfv cmhm wcel cn0 cbs cmg wceq ccrg csubrg wa crh jca eqid - evls1rhm rhmmhm 3syl crg subrgrng vr1cl mgpbas mhmmulg syl3anc ) ACHEBUBU - CZUDUEZUFUCUGZIUHUGKJUIUEZUGZIKGUCCUEIKCUEVHUJUEZUCUKAEULUGZDEUMUEUGZUNCJ - VGUOUCUGVIAVMVNSTUPBCDEVGFJLQVGUQMNURJVGCHVHOVHUQUSUTUAAVNFVAUGVKTDEFMVBV - JJFKPNVJUQZVCUTVJGVLCHVHIKVJJHOVOVDRVLUQVEVF $. + evls1rhm rhmmhm 3syl crg subrgring vr1cl mgpbas mhmmulg syl3anc ) ACHEBUB + UCZUDUEZUFUCUGZIUHUGKJUIUEZUGZIKGUCCUEIKCUEVHUJUEZUCUKAEULUGZDEUMUEUGZUNC + JVGUOUCUGVIAVMVNSTUPBCDEVGFJLQVGUQMNURJVGCHVHOVHUQUSUTUAAVNFVAUGVKTDEFMVB + VJJFKPNVJUQZVCUTVJGVLCHVHIKVJJHOVOVDRVLUQVEVF $. $} ${ @@ -247337,13 +247561,13 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ( y e. B |-> ( 1o X. { y } ) ) ) ) $= ( vx wcel cfv c1o cmap co ccom ccrg csn cxp cmpt evl1fval fveq1i cpws cbs wa cv wceq crh con0 1on simpl eqid evlrhm sylancr rhmf syl fvco3 sylancom - wf syl5eq ffvelrn crg cvv crngrng adantr ovex sylancl eleqtrrd coeq1 fvex - pwsbas eqeltri mptex coex fvmpt eqtrd ) EUAOZBFOZUIZBHPZBDPZNCCQRSZRSZNUJ - ZACQAUJUBUCZUDZTZUDZPZWEWJTZWCWDBWLDTZPZWMBHWONACDEHIJKUEUFWAWBFEWFUGSZUH - PZDVCZWPWMUKWCDGWQULSOZWSWCQUMOWAWTUNWAWBUOCDEWQQUMGJKLWQUPZUQURFWRGWQDMW - RUPUSUTZFWRBWLDVAVBVDWCWEWGOWMWNUKWCWEWRWGWAWBWSWEWROXBFWRBDVEVBWCEVFOZWF - VGOWGWRUKWAXCWBEVHVICQRVJCEWFVFVGWQXAKVOVKVLNWEWKWNWGWLWHWEWJVMWLUPWEWJBD - VNACWICEUHPVGKEUHVNVPVQVRVSUTVT $. + syl5eq ffvelrn crg cvv crngring adantr ovex pwsbas sylancl eleqtrrd coeq1 + wf fvex eqeltri mptex coex fvmpt eqtrd ) EUAOZBFOZUIZBHPZBDPZNCCQRSZRSZNU + JZACQAUJUBUCZUDZTZUDZPZWEWJTZWCWDBWLDTZPZWMBHWONACDEHIJKUEUFWAWBFEWFUGSZU + HPZDVNZWPWMUKWCDGWQULSOZWSWCQUMOWAWTUNWAWBUOCDEWQQUMGJKLWQUPZUQURFWRGWQDM + WRUPUSUTZFWRBWLDVAVBVCWCWEWGOWMWNUKWCWEWRWGWAWBWSWEWROXBFWRBDVDVBWCEVEOZW + FVFOWGWRUKWAXCWBEVGVHCQRVICEWFVEVFWQXAKVJVKVLNWEWKWNWGWLWHWEWJVMWLUPWEWJB + DVOACWICEUHPVFKEUHVOVPVQVRVSUTVT $. $} ${ @@ -247414,18 +247638,18 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evl1sca $p |- ( ( R e. CRing /\ X e. B ) -> ( O ` ( A ` X ) ) = ( B X. { X } ) ) $= ( vy vx wcel cfv c1o co cxp cmpt wceq eqid ccrg wa cevl csn ccom cmap cbs - cv crg crngrng adantr ply1sclf ffvelrn sylancom cmpl cps1 ply1bas evl1val - syl syldan cress cascl ressid oveq2d fveq2d ply1ascl syl6reqr fveq1d con0 - wf evlval 1on a1i simpl csubrg subrgid simpr evlssca eqtrd coeq1d wral c0 - wf1o df1o2 cvv fvex eqeltri 0ex mapsnf1o3 f1of mp1i fmpt sylibr fconstmpt - eqidd fmptcof syl6eqr 3eqtrd ) DUAMZFBMZUBZFANZENZXBODUCPZNZKBOKUHUDQZRZU - EZBOUFPZFUDZQZXGUEZBXJQZWSWTXBCUGNZMZXCXHSWSWTBXNAVJZXOXADUIMZXPWSXQWTDUJ - UKZAXNCDBHJIXNTZULUSBXNFAUMUNKXBBXDDXNODUOPZEGXDTZIXTTCDDUPNZXNHYBTXSUQUR - UTXAXEXKXGXAXEFODBVAPZUOPZVBNZNZXDNXKXAXBYFXDXAFAYEXAYEXTVBNAXAYDXTVBXAYC - DOUOWSYCDSWTBDUAIVCUKVDVEACDHJVFVGVHVEXAYEBXDBDYCOVIYDFBXDDOYAIVKYDTYCTIY - ETOVIMXAVLVMWSWTVNXAXQBDVONMXRBDIVPUSWSWTVQVRVSVTXAXLKBFRXMXAKLBXIXFFFXGX - KXABXIXGVJZXFXIMKBWABXIXGWCYGXAKBOXGWBWDBDUGNWEIDUGWFWGWHXGTZWIBXIXGWJWKK - BXIXFXGYHWLWMXAXGWOXKLXIFRSXALXIFWNVMLUHXFSFWOWPKBFWNWQWR $. + cv wf crg crngring adantr ply1sclf syl ffvelrn sylancom cmpl cps1 ply1bas + evl1val syldan cress ressid oveq2d fveq2d ply1ascl syl6reqr fveq1d evlval + cascl con0 1on a1i simpl csubrg subrgid simpr evlssca coeq1d wral wf1o c0 + eqtrd df1o2 cvv fvex eqeltri 0ex mapsnf1o3 f1of mp1i fmpt eqidd fconstmpt + sylibr fmptcof syl6eqr 3eqtrd ) DUAMZFBMZUBZFANZENZXBODUCPZNZKBOKUHUDQZRZ + UEZBOUFPZFUDZQZXGUEZBXJQZWSWTXBCUGNZMZXCXHSWSWTBXNAUIZXOXADUJMZXPWSXQWTDU + KULZAXNCDBHJIXNTZUMUNBXNFAUOUPKXBBXDDXNODUQPZEGXDTZIXTTCDDURNZXNHYBTXSUSU + TVAXAXEXKXGXAXEFODBVBPZUQPZVJNZNZXDNXKXAXBYFXDXAFAYEXAYEXTVJNAXAYDXTVJXAY + CDOUQWSYCDSWTBDUAIVCULVDVEACDHJVFVGVHVEXAYEBXDBDYCOVKYDFBXDDOYAIVIYDTYCTI + YETOVKMXAVLVMWSWTVNXAXQBDVONMXRBDIVPUNWSWTVQVRWCVSXAXLKBFRXMXAKLBXIXFFFXG + XKXABXIXGUIZXFXIMKBVTBXIXGWAYGXAKBOXGWBWDBDUGNWEIDUGWFWGWHXGTZWIBXIXGWJWK + KBXIXFXGYHWLWOXAXGWMXKLXIFRSXALXIFWNVMLUHXFSFWMWPKBFWNWQWR $. evl1scad.u $e |- U = ( Base ` P ) $. evl1scad.1 $e |- ( ph -> R e. CRing ) $. @@ -247435,10 +247659,10 @@ univariate polynomial evaluation map function for a (sub)ring is a ring Carneiro, 4-Jul-2015.) $) evl1scad $p |- ( ph -> ( ( A ` X ) e. U /\ ( ( O ` ( A ` X ) ) ` Y ) = X ) ) $= - ( cfv wcel wceq ccrg crg wf crngrng ply1sclf ffvelrnd csn evl1sca syl2anc - 3syl cxp fveq1d fvconst2g eqtrd jca ) AHBRZFSIUPGRZRZHTACFHBAEUASZEUBSCFB - UCOEUDBFDECKMLNUEUJPUFAURICHUGUKZRZHAIUQUTAUSHCSZUQUTTOPBCDEGHJKLMUHUIULA - VBICSVAHTPQCHICUMUIUNUO $. + ( cfv wcel wceq crg wf crngring ply1sclf ffvelrnd csn cxp evl1sca syl2anc + ccrg 3syl fveq1d fvconst2g eqtrd jca ) AHBRZFSIUPGRZRZHTACFHBAEUJSZEUASCF + BUBOEUCBFDECKMLNUDUKPUEAURICHUFUGZRZHAIUQUTAUSHCSZUQUTTOPBCDEGHJKLMUHUIUL + AVBICSVAHTPQCHICUMUIUNUO $. $} ${ @@ -247450,16 +247674,16 @@ univariate polynomial evaluation map function for a (sub)ring is a ring (Contributed by Mario Carneiro, 12-Jun-2015.) $) evl1var $p |- ( R e. CRing -> ( O ` X ) = ( _I |` B ) ) $= ( vy vz ccrg wcel cfv c1o co cv ccom c0 cbs eqid cmvr cevl cmpt cmap ccnv - csn cxp cid cres cpl1 wceq crg crngrng vr1cl syl cmpl ply1bas mpdan df1o2 - cps1 evl1val cvv fvex eqeltri mapsncnv coeq2i ressid oveq2d fveq1d vr1val - 0ex cress syl6eqr fveq2d con0 evlval 1on a1i csubrg subrgid 0lt1o evlsvar - id eqtr3d coeq1d syl5eqr wf1o mapsnf1o2 f1ococnv2 mp1i 3eqtrd ) BJKZDCLZD - MBUANZLZHAMHOUEUFUBZPZIAMUCNZQIOLUBZWRUDZPZUGAUHZWKDBUILZRLZKZWLWPUJWKBUK - KZXDBULZXCXBBDFXBSZXCSZUMUNHDAWMBXCMBUONZCEWMSZGXISXBBBUSLZXCXGXKSXHUPUTU - QWKWPWNWSPWTWSWOWNIHAMWRQURABRLVAGBRVBVCZVJWRSZVDVEWKWNWRWSWKQMBAVKNZTNZL - ZWMLWNWRWKXPDWMWKXPQMBTNZLDWKQXOXQWKXNBMTABJGVFVGVHBDFVIVLVMWKAWMABXNIMXO - VNQAWMBMXJGVOXOSXNSGMVNKWKVPVQWKWBWKXEABVRLKXFABGVSUNQMKWKVTVQWAWCWDWEWQA - WRWFWTXAUJWKIAMWRQURXLVJXMWGWQAWRWHWIWJ $. + csn cxp cid cres cpl1 wceq crg crngring vr1cl syl cmpl cps1 ply1bas mpdan + evl1val df1o2 cvv fvex eqeltri mapsncnv coeq2i cress ressid oveq2d fveq1d + 0ex vr1val syl6eqr fveq2d con0 evlval 1on id csubrg subrgid 0lt1o evlsvar + a1i eqtr3d coeq1d syl5eqr wf1o mapsnf1o2 f1ococnv2 mp1i 3eqtrd ) BJKZDCLZ + DMBUANZLZHAMHOUEUFUBZPZIAMUCNZQIOLUBZWRUDZPZUGAUHZWKDBUILZRLZKZWLWPUJWKBU + KKZXDBULZXCXBBDFXBSZXCSZUMUNHDAWMBXCMBUONZCEWMSZGXISXBBBUPLZXCXGXKSXHUQUS + URWKWPWNWSPWTWSWOWNIHAMWRQUTABRLVAGBRVBVCZVJWRSZVDVEWKWNWRWSWKQMBAVFNZTNZ + LZWMLWNWRWKXPDWMWKXPQMBTNZLDWKQXOXQWKXNBMTABJGVGVHVIBDFVKVLVMWKAWMABXNIMX + OVNQAWMBMXJGVOXOSXNSGMVNKWKVPWBWKVQWKXEABVRLKXFABGVSUNQMKWKVTWBWAWCWDWEWQ + AWRWFWTXAUJWKIAMWRQUTXLVJXMWGWQAWRWHWIWJ $. evl1vard.p $e |- P = ( Poly1 ` R ) $. evl1vard.u $e |- U = ( Base ` P ) $. @@ -247468,9 +247692,9 @@ univariate polynomial evaluation map function for a (sub)ring is a ring $( Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.) $) evl1vard $p |- ( ph -> ( X e. U /\ ( ( O ` X ) ` Y ) = Y ) ) $= - ( wcel cfv wceq ccrg syl crg crngrng vr1cl 3syl cid evl1var fveq1d fvresi - cres eqtrd jca ) AGEPZHGFQZQZHRADSPZDUAPULNDUBECDGJLMUCUDAUNHUEBUIZQZHAHU - MUPAUOUMUPRNBDFGIJKUFTUGAHBPUQHROBHUHTUJUK $. + ( wcel cfv wceq ccrg syl crg crngring 3syl cid cres evl1var fveq1d fvresi + vr1cl eqtrd jca ) AGEPZHGFQZQZHRADSPZDUAPULNDUBECDGJLMUIUCAUNHUDBUEZQZHAH + UMUPAUOUMUPRNBDFGIJKUFTUGAHBPUQHROBHUHTUJUK $. $} ${ @@ -247487,15 +247711,15 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ( cfv eqid c1o co c0 wcel cbs vy cv1 ce1 cid subrgvr1 syl6reqr fveq2d ces cres cv csn cxp cmpt ccom cevl cmvr con0 1on a1i 0lt1o evlsvarsrng vr1val fveq1d syl5eq 3eqtr4d coeq1d ccrg csubrg cress cmpl wceq cpl1 cps1 fveq2i - subrgmvr ply1bas eqcomi subrgvr1cl evls1val syl3anc crngrng vr1cl evl1val - crg 3syl syl2anc evl1var syl 3eqtrd ) AGCNEUBNZCNZWJEUCNZNZUDBUIZAGWJCAWJ - FUBNGAEDFWJWJOZMJUEIUFUGAWJDPEUHQZNZNZUABPUAUJUKULUMZUNZWJPEUOQZNZWSUNZWK - WMAWRXBWSARPFUPQZNZWQNXEXANWRXBAUQBWQDEFPXAXDRWQOXAOZXDOJKPUQSAURUSZLMRPS - AUTUSVAAWJXEWQAWJRPEUPQZNXEEWJWOVBARXHXDAEDFPXHUQXHOXGMJVOVCVDZUGAWJXEXAX - IUGVEVFAEVGSZDEVHNSWJPEDVIQZVJQZTNZSWKWTVKLMAXMEDFVLNZFWJWOMJXNOXNTNZXMXK - VLNZXKXKVMNZXOXPOXQOXNXPTFXKVLJVNVNVPVQVRUAWJBCDEWPXMXLHWPOKXLOXMOVSVTAXJ - WJPEVJQZTNZSZWMXCVKLAXJEWDSXTLEWAXSEVLNZEWJWOYAOZYATNZXSYAEEVMNZYCYBYDOYC - OVPVQWBWEUAWJBXAEXSXRWLWLOZXFKXROXSOWCWFVEAXJWMWNVKLBEWLWJYEWOKWGWHWI $. + subrgmvr ply1bas eqcomi subrgvr1cl evls1val syl3anc crngring 3syl evl1val + crg vr1cl syl2anc evl1var syl 3eqtrd ) AGCNEUBNZCNZWJEUCNZNZUDBUIZAGWJCAW + JFUBNGAEDFWJWJOZMJUEIUFUGAWJDPEUHQZNZNZUABPUAUJUKULUMZUNZWJPEUOQZNZWSUNZW + KWMAWRXBWSARPFUPQZNZWQNXEXANWRXBAUQBWQDEFPXAXDRWQOXAOZXDOJKPUQSAURUSZLMRP + SAUTUSVAAWJXEWQAWJRPEUPQZNXEEWJWOVBARXHXDAEDFPXHUQXHOXGMJVOVCVDZUGAWJXEXA + XIUGVEVFAEVGSZDEVHNSWJPEDVIQZVJQZTNZSWKWTVKLMAXMEDFVLNZFWJWOMJXNOXNTNZXMX + KVLNZXKXKVMNZXOXPOXQOXNXPTFXKVLJVNVNVPVQVRUAWJBCDEWPXMXLHWPOKXLOXMOVSVTAX + JWJPEVJQZTNZSZWMXCVKLAXJEWDSXTLEWAXSEVLNZEWJWOYAOZYATNZXSYAEEVMNZYCYBYDOY + COVPVQWEWBUAWJBXAEXSXRWLWLOZXFKXROXSOWCWFVEAXJWMWNVKLBEWLWJYEWOKWGWHWI $. $} ${ @@ -247515,12 +247739,12 @@ univariate polynomial evaluation map function for a (sub)ring is a ring 13-Sep-2019.) $) evls1scasrng $p |- ( ph -> ( Q ` ( A ` X ) ) = ( O ` ( C ` X ) ) ) $= ( cfv ces1 co csn cxp cress cpl1 cascl ccrg wcel ressid eqcomd syl fveq2d - wceq syl5eq fveq1d crg csubrg crngrng subrgid wss subrgss sseldd evls1sca - eqid 3syl eqtrd evl1fval1 a1i 3eqtr4rd ) ALDUDZHCUEUFZUDZCLUGUHZVOJUDLBUD - FUDAVQLHCUIUFZUJUDZUKUDZUDZVPUDVRAVOWBVPALDWAADEUKUDWATAEVTUKAEHUJUDVTQAH - VSUJAHULUMZHVSURUAWCVSHCHULRUNUOUPUQUSUQUSUTUQAWACVPCHVSVTLVPVIVTVIVSVIRW - AVIUAAWCHVAUMCHVBUDZUMUAHVCCHRVDVJAGCLAGWDUMGCVEUBGCHRVFUPUCVGVHVKAVOJVPJ - VPURACJHNRVLVMUTABCFGHIKLMOPRSUAUBUCVHVN $. + wceq syl5eq fveq1d eqid crg csubrg crngring subrgid 3syl subrgss evls1sca + wss sseldd eqtrd evl1fval1 a1i 3eqtr4rd ) ALDUDZHCUEUFZUDZCLUGUHZVOJUDLBU + DFUDAVQLHCUIUFZUJUDZUKUDZUDZVPUDVRAVOWBVPALDWAADEUKUDWATAEVTUKAEHUJUDVTQA + HVSUJAHULUMZHVSURUAWCVSHCHULRUNUOUPUQUSUQUSUTUQAWACVPCHVSVTLVPVAVTVAVSVAR + WAVAUAAWCHVBUMCHVCUDZUMUAHVDCHRVEVFAGCLAGWDUMGCVIUBGCHRVGUPUCVJVHVKAVOJVP + JVPURACJHNRVLVMUTABCFGHIKLMOPRSUAUBUCVHVN $. $} ${ @@ -247537,11 +247761,11 @@ univariate polynomial evaluation map function for a (sub)ring is a ring the ring itself. (Contributed by AV, 12-Sep-2019.) $) evls1varsrng $p |- ( ph -> ( Q ` V ) = ( O ` V ) ) $= ( cfv cv1 wceq eqid wcel cid cres evls1var ces1 co cress evl1fval1 fveq1d - a1i subrgvr1 ccrg ressid syl eqcomd fveq2d 3eqtr2d csubrg crngrng subrgid - crg 3syl 3eqtrrd eqtrd ) AHCPUABUBZHGPZABCDEFHIKLMNOUCAVEHEBUDUEZPEBUFUEZ - QPZVFPVDAHGVFGVFRABGEJMUGUIUHAHVHVFAHFQPZEQPZVHHVIRAKUIAEDFVJVJSOLUJAEVGQ - AVGEAEUKTZVGERNBEUKMULUMUNUOUPUOABVFBEVGVHVFSVHSVGSMNAVKEUTTBEUQPTNEURBEM - USVAUCVBVC $. + a1i subrgvr1 ressid syl eqcomd fveq2d 3eqtr2d crg csubrg crngring subrgid + ccrg 3syl 3eqtrrd eqtrd ) AHCPUABUBZHGPZABCDEFHIKLMNOUCAVEHEBUDUEZPEBUFUE + ZQPZVFPVDAHGVFGVFRABGEJMUGUIUHAHVHVFAHFQPZEQPZVHHVIRAKUIAEDFVJVJSOLUJAEVG + QAVGEAEUTTZVGERNBEUTMUKULUMUNUOUNABVFBEVGVHVFSVHSVGSMNAVKEUPTBEUQPTNEURBE + MUSVAUCVBVC $. $} ${ @@ -247585,18 +247809,18 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evl1subd $p |- ( ph -> ( ( M .- N ) e. U /\ ( ( O ` ( M .- N ) ) ` Y ) = ( V D W ) ) ) $= ( co wcel cfv wceq cgrp cpws cghm crh ccrg evl1rhm syl rhmghm ghmgrp1 - eqid simpld grpsubcl syl3anc cof csg ghmsub cvv cbs crngrng 3syl fvex - crg rnggrp eqeltri a1i rhmf ffvelrnd pwssub syl22anc eqtrd fveq1d wfn - wf pwselbas ffn fnfvof simprd oveq12d 3eqtrd jca ) AGIHUDZFUEZMWHJUFZ - UFZKLCUDZUGADUHUEZGFUEZIFUEZWIAJDEBUIUDZUJUDUEZWMAJDWPUKUDUEZWQAEULUE - ZWRRBDEWPJNOWPUQZPUMUNZDWPJUOUNZDWPJUPUNAWNMGJUFZUFZKUGZTURZAWOMIJUFZ - UFZLUGZUAURZFDHGIQUBUSUTAWKMXCXGCVAUDZUFZXDXHCUDZWLAMWJXKAWJXCXGWPVBU - FZUDZXKAWQWNWOWJXOUGXBXFXJFDWPGJHXNIQUBXNUQZVCUTAEUHUEZBVDUEZXCWPVEUF - ZUEXGXSUEXOXKUGAWSEVIUEXQREVFEVJVGXRABEVEUFVDPEVEVHVKVLZAFXSGJAWRFXSJ - VTXAFXSDWPJQXSUQZVMUNZXFVNZAFXSIJYBXJVNZXSEXCXGBCXNVDWPWTYAUCXPVOVPVQ - VRAXCBVSZXGBVSZXRMBUEXLXMUGABBXCVTYEABEBXSULXCWPVDWTPYARXTYCWABBXCWBU - NABBXGVTYFABEBXSULXGWPVDWTPYARXTYDWABBXGWBUNXTSBCXCXGVDMWCVPAXDKXHLCA - WNXETWDAWOXIUAWDWEWFWG $. + eqid simpld grpsubcl syl3anc cof csg ghmsub cvv cbs crg crngring 3syl + ringgrp fvex eqeltri a1i wf ffvelrnd pwssub syl22anc eqtrd fveq1d wfn + rhmf pwselbas ffn fnfvof simprd oveq12d 3eqtrd jca ) AGIHUDZFUEZMWHJU + FZUFZKLCUDZUGADUHUEZGFUEZIFUEZWIAJDEBUIUDZUJUDUEZWMAJDWPUKUDUEZWQAEUL + UEZWRRBDEWPJNOWPUQZPUMUNZDWPJUOUNZDWPJUPUNAWNMGJUFZUFZKUGZTURZAWOMIJU + FZUFZLUGZUAURZFDHGIQUBUSUTAWKMXCXGCVAUDZUFZXDXHCUDZWLAMWJXKAWJXCXGWPV + BUFZUDZXKAWQWNWOWJXOUGXBXFXJFDWPGJHXNIQUBXNUQZVCUTAEUHUEZBVDUEZXCWPVE + UFZUEXGXSUEXOXKUGAWSEVFUEXQREVGEVIVHXRABEVEUFVDPEVEVJVKVLZAFXSGJAWRFX + SJVMXAFXSDWPJQXSUQZVTUNZXFVNZAFXSIJYBXJVNZXSEXCXGBCXNVDWPWTYAUCXPVOVP + VQVRAXCBVSZXGBVSZXRMBUEXLXMUGABBXCVMYEABEBXSULXCWPVDWTPYARXTYCWABBXCW + BUNABBXGVMYFABEBXSULXGWPVDWTPYARXTYDWABBXGWBUNXTSBCXCXGVDMWCVPAXDKXHL + CAWNXETWDAWOXIUAWDWEWFWG $. $} ${ @@ -247606,17 +247830,17 @@ univariate polynomial evaluation map function for a (sub)ring is a ring (Contributed by Mario Carneiro, 4-Jul-2015.) $) evl1muld $p |- ( ph -> ( ( M .xb N ) e. U /\ ( ( O ` ( M .xb N ) ) ` Y ) = ( V .x. W ) ) ) $= - ( co wcel cfv wceq crg cpws crh ccrg evl1rhm syl rhmrcl1 simpld rngcl - eqid syl3anc cof cmulr rhmmul cbs cvv fvex eqeltri wf rhmf pwsmulrval - a1i ffvelrnd eqtrd fveq1d wfn pwselbas fnfvof syl22anc simprd oveq12d - ffn 3eqtrd jca ) AHIEUDZGUEZMWBJUFZUFZKLFUDZUGACUHUEZHGUEZIGUEZWCAJCD - BUIUDZUJUDUEZWGADUKUEWKRBCDWJJNOWJUQZPULUMZCWJJUNUMAWHMHJUFZUFZKUGZTU - OZAWIMIJUFZUFZLUGZUAUOZGCEHIQUBUPURAWEMWNWRFUSUDZUFZWOWSFUDZWFAMWDXBA - WDWNWRWJUTUFZUDZXBAWKWHWIWDXFUGWMWQXAHICWJEXEJGQUBXEUQZVAURAWJVBUFZDX - EFWNWRBUKVCWJWLXHUQZRBVCUEZABDVBUFVCPDVBVDVEVIZAGXHHJAWKGXHJVFWMGXHCW - JJQXIVGUMZWQVJZAGXHIJXLXAVJZUCXGVHVKVLAWNBVMZWRBVMZXJMBUEXCXDUGABBWNV - FXOABDBXHUKWNWJVCWLPXIRXKXMVNBBWNVSUMABBWRVFXPABDBXHUKWRWJVCWLPXIRXKX - NVNBBWRVSUMXKSBFWNWRVCMVOVPAWOKWSLFAWHWPTVQAWIWTUAVQVRVTWA $. + ( co wcel cfv wceq crg cpws crh ccrg eqid evl1rhm syl rhmrcl1 syl3anc + simpld ringcl cof cmulr rhmmul cbs cvv fvex eqeltri a1i rhmf ffvelrnd + wf pwsmulrval eqtrd fveq1d wfn pwselbas fnfvof syl22anc simprd 3eqtrd + ffn oveq12d jca ) AHIEUDZGUEZMWBJUFZUFZKLFUDZUGACUHUEZHGUEZIGUEZWCAJC + DBUIUDZUJUDUEZWGADUKUEWKRBCDWJJNOWJULZPUMUNZCWJJUOUNAWHMHJUFZUFZKUGZT + UQZAWIMIJUFZUFZLUGZUAUQZGCEHIQUBURUPAWEMWNWRFUSUDZUFZWOWSFUDZWFAMWDXB + AWDWNWRWJUTUFZUDZXBAWKWHWIWDXFUGWMWQXAHICWJEXEJGQUBXEULZVAUPAWJVBUFZD + XEFWNWRBUKVCWJWLXHULZRBVCUEZABDVBUFVCPDVBVDVEVFZAGXHHJAWKGXHJVIWMGXHC + WJJQXIVGUNZWQVHZAGXHIJXLXAVHZUCXGVJVKVLAWNBVMZWRBVMZXJMBUEXCXDUGABBWN + VIXOABDBXHUKWNWJVCWLPXIRXKXMVNBBWNVSUNABBWRVIXPABDBXHUKWRWJVCWLPXIRXK + XNVNBBWRVSUNXKSBFWNWRVCMVOVPAWOKWSLFAWHWPTVQAWIWTUAVQVTVRWA $. $} $} @@ -247646,22 +247870,22 @@ univariate polynomial evaluation map function for a (sub)ring is a ring Mario Carneiro, 12-Jun-2015.) $) evl1expd $p |- ( ph -> ( ( N .xb M ) e. U /\ ( ( O ` ( N .xb M ) ) ` Y ) = ( N .^ V ) ) ) $= - ( vx vy wcel cfv wceq cmgp cmnd cn0 crg ccrg crngrng syl ply1rng rngmgp - eqid 3syl simpld mgpbas mulgnn0cl syl3anc cpws cmg cmhm evl1rhm mhmmulg - crh rhmmhm cbs cvv eqidd cplusg fvex eqeltri pwsmgp sylancl wss ssv a1i - co wa ovex simprd proplem3 mulgpropd oveqd eqtrd fveq1d ffvelrnd syl5eq - cv wf rhmf eleqtrd pwsmulg syl23anc oveq2d jca ) AIHEWAZFUEZLWTJUFZUFZI - KGWAZUGACUHUFZUIUEZIUJUEZHFUEZXAADUKUEZCUKUEXFADULUEZXIQDUMUNZCDNUOCXEX - EUQZUPURUBAXHLHJUFZUFZKUGZSUSZFEXEIHFCXEXLPUTZTVAVBAXCLIXMDUHUFZBVCWAZV - DUFZWAZUFZXDALXBYAAXBIXMDBVCWAZUHUFZVDUFZWAZYAAJXEYDVEWAUEZXGXHXBYFUGAJ - CYCVHWAUEZYGAXJYHQBCDYCJMNYCUQZOVFUNZCYCJXEYDXLYDUQZVIUNUBXPFEYEJXEYDIH - XQTYEUQZVGVBAYEXTIXMAUCUDYDVJUFZYEXTYDXSVKYLXTUQZAYMVLAYMXSVJUFZUGZYDVM - UFZXSVMUFZUGZAXJBVKUEZYPYSWBQBDVJUFVKODVJVNVOZYMYOYQYRDBXRYDULVKYCXSYIX - RUQZXSUQZYKYMUQYOUQZYQUQYRUQVPVQZUSZYMVKVRAYMVSVTUCWLZUDWLZYQWAVKUEAUUG - VKUEUUHVKUEWBZWBUUGUUHYQWCVTAUUIUCUDYQYRAYPYSUUEWDWEWFWGWHWIAYBIXNGWAZX - DAXRUIUEZYTXGXMYOUELBUEYBUUJUGAXIUUKXKDXRUUBUPUNYTAUUAVTUBAXMYCVJUFZYOA - FUULHJAYHFUULJWMYJFUULCYCJPUULUQZWNUNXPWJAUULYMYOUULYCYDYKUUMUTUUFWKWOR - LYOXRXTGBIVKXMXSUUCUUDYNUAWPWQAXNKIGAXHXOSWDWRWHWHWS $. + ( vx vy wcel cfv wceq cmgp cmnd cn0 crg ccrg crngring syl ply1ring eqid + co ringmgp 3syl simpld mgpbas mulgnn0cl syl3anc cpws cmg evl1rhm rhmmhm + cmhm crh mhmmulg cbs cvv eqidd cplusg wa eqeltri pwsmgp sylancl wss ssv + fvex cv ovex simprd proplem3 mulgpropd oveqd eqtrd fveq1d rhmf ffvelrnd + a1i wf syl5eq eleqtrd pwsmulg syl23anc oveq2d jca ) AIHEUQZFUEZLWTJUFZU + FZIKGUQZUGACUHUFZUIUEZIUJUEZHFUEZXAADUKUEZCUKUEXFADULUEZXIQDUMUNZCDNUOC + XEXEUPZURUSUBAXHLHJUFZUFZKUGZSUTZFEXEIHFCXEXLPVAZTVBVCAXCLIXMDUHUFZBVDU + QZVEUFZUQZUFZXDALXBYAAXBIXMDBVDUQZUHUFZVEUFZUQZYAAJXEYDVHUQUEZXGXHXBYFU + GAJCYCVIUQUEZYGAXJYHQBCDYCJMNYCUPZOVFUNZCYCJXEYDXLYDUPZVGUNUBXPFEYEJXEY + DIHXQTYEUPZVJVCAYEXTIXMAUCUDYDVKUFZYEXTYDXSVLYLXTUPZAYMVMAYMXSVKUFZUGZY + DVNUFZXSVNUFZUGZAXJBVLUEZYPYSVOQBDVKUFVLODVKWAVPZYMYOYQYRDBXRYDULVLYCXS + YIXRUPZXSUPZYKYMUPYOUPZYQUPYRUPVQVRZUTZYMVLVSAYMVTWLUCWBZUDWBZYQUQVLUEA + UUGVLUEUUHVLUEVOZVOUUGUUHYQWCWLAUUIUCUDYQYRAYPYSUUEWDWEWFWGWHWIAYBIXNGU + QZXDAXRUIUEZYTXGXMYOUELBUEYBUUJUGAXIUUKXKDXRUUBURUNYTAUUAWLUBAXMYCVKUFZ + YOAFUULHJAYHFUULJWMYJFUULCYCJPUULUPZWJUNXPWKAUULYMYOUULYCYDYKUUMVAUUFWN + WORLYOXRXTGBIVLXMXSUUCUUDYNUAWPWQAXNKIGAXHXOSWDWRWHWHWS $. $} $} @@ -247672,31 +247896,31 @@ univariate polynomial evaluation map function for a (sub)ring is a ring 12-Jun-2015.) $) pf1const $p |- ( ( R e. CRing /\ X e. B ) -> ( B X. { X } ) e. Q ) $= ( ccrg wcel wa csn cxp ce1 cfv crn cpl1 eqid cbs co wf adantr evl1sca wfn - cascl cpws crh evl1rhm rhmf ffn crg crngrng ply1sclf syl ffvelrn sylancom - 3syl fnfvelrn syl2anc eqeltrrd syl6eleqr ) CGHZDAHZIZADJKZCLMZNZBVBDCOMZU - CMZMZVDMZVCVEVGAVFCVDDVDPZVFPZEVGPZUAVBVDVFQMZUBZVHVMHZVIVEHVBVDVFCAUDRZU - ERHZVMVPQMZVDSVNUTVQVAAVFCVPVDVJVKVPPEUFTVMVRVFVPVDVMPZVRPUGVMVRVDUHUOUTV - AAVMVGSZVOVBCUIHZVTUTWAVACUJTVGVMVFCAVKVLEVSUKULAVMDVGUMUNVMVHVDUPUQURFUS - $. + cpws crh evl1rhm rhmf ffn 3syl crg crngring ply1sclf syl ffvelrn sylancom + cascl fnfvelrn syl2anc eqeltrrd syl6eleqr ) CGHZDAHZIZADJKZCLMZNZBVBDCOMZ + UOMZMZVDMZVCVEVGAVFCVDDVDPZVFPZEVGPZUAVBVDVFQMZUBZVHVMHZVIVEHVBVDVFCAUCRZ + UDRHZVMVPQMZVDSVNUTVQVAAVFCVPVDVJVKVPPEUETVMVRVFVPVDVMPZVRPUFVMVRVDUGUHUT + VAAVMVGSZVOVBCUIHZVTUTWAVACUJTVGVMVFCAVKVLEVSUKULAVMDVGUMUNVMVHVDUPUQURFU + S $. $( The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.) $) pf1id $p |- ( R e. CRing -> ( _I |` B ) e. Q ) $= ( ccrg wcel cid cres ce1 cfv crn cv1 eqid evl1var cpl1 cbs wfn cpws co wf - crh evl1rhm rhmf ffn 3syl crg crngrng vr1cl syl fnfvelrn syl2anc eqeltrrd - syl6eleqr ) CFGZHAIZCJKZLZBUOCMKZUQKZUPURACUQUSUQNZUSNZDOUOUQCPKZQKZRZUSV - DGZUTURGUOUQVCCASTZUBTGVDVGQKZUQUAVEAVCCVGUQVAVCNZVGNDUCVDVHVCVGUQVDNZVHN - UDVDVHUQUEUFUOCUGGVFCUHVDVCCUSVBVIVJUIUJVDUSUQUKULUMEUN $. + crh evl1rhm rhmf ffn crg crngring syl fnfvelrn syl2anc eqeltrrd syl6eleqr + 3syl vr1cl ) CFGZHAIZCJKZLZBUOCMKZUQKZUPURACUQUSUQNZUSNZDOUOUQCPKZQKZRZUS + VDGZUTURGUOUQVCCASTZUBTGVDVGQKZUQUAVEAVCCVGUQVAVCNZVGNDUCVDVHVCVGUQVDNZVH + NUDVDVHUQUEUMUOCUFGVFCUGVDVCCUSVBVIVJUNUHVDUSUQUIUJUKEUL $. $( Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) $) pf1subrg $p |- ( R e. CRing -> Q e. ( SubRing ` ( R ^s B ) ) ) $= ( ccrg wcel ce1 cfv cpl1 cbs cima cpws co csubrg crn crh wf wfn eqid wceq - evl1rhm rhmf ffn fnima 4syl syl6eqr casa crg ply1assa assarng 3syl rhmima - subrgid syl2anc eqeltrrd ) CFGZCHIZCJIZKIZLZBCAMNZOIZUQVAURPZBUQURUSVBQNG - ZUTVBKIZURRURUTSVAVDUAAUSCVBURURTUSTZVBTDUBZUTVFUSVBURUTTZVFTUCUTVFURUDUT - URUEUFEUGUQVEUTUSOIGZVAVCGVHUQUSUHGUSUIGVJUSCVGUJUSUKUTUSVIUNULURUSVBUTUM - UOUP $. + evl1rhm rhmf ffn fnima 4syl syl6eqr casa ply1assa assaring subrgid rhmima + crg 3syl syl2anc eqeltrrd ) CFGZCHIZCJIZKIZLZBCAMNZOIZUQVAURPZBUQURUSVBQN + GZUTVBKIZURRURUTSVAVDUAAUSCVBURURTUSTZVBTDUBZUTVFUSVBURUTTZVFTUCUTVFURUDU + TURUEUFEUGUQVEUTUSOIGZVAVCGVHUQUSUHGUSUMGVJUSCVGUIUSUJUTUSVIUKUNURUSVBUTU + LUOUP $. $} ${ @@ -247830,53 +248054,53 @@ univariate polynomial evaluation map function for a (sub)ring is a ring fveq2d coeq1d sylibr eqeltrrd cevl crn ces evlval rneqi ad2antrl ad2antll an4 anbi2d mapsnf1o3 f1of inidm cpl1 ce1 pf1rcl csca casa mplassa sylancr ply1ascl asclrhm mplsca oveq1d eleqtrrd ffvelrnda evl1val syl2anc evl1sca - 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( m u. { a } ) |-> M ) ) ) ` Y ) = ( R gsum ( x e. ( m u. { a } ) |-> ( ( O ` M ) ` Y ) ) ) ) ) ) $= ( vy wcel cfv cv cfn wn w3a wral cmpt cgsu co wceq wi csn cun wa ralunb - csb cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cvv eqid ccmn crg crngrng - ccrg syl ply1rng rngcmn 3ad2ant3 simpll1 rspcsbela expcom adantl adantr - ad2antrr imp vex a1i simpll2 ssnid csbeq1 gsumunsn syl5eq eqcomi oveq1i + csb cplusg nfcv nfcsb1v csbeq1a cbvmpt oveq2i cvv eqid crg crngring syl + ccmn ply1ring ringcmn 3ad2ant3 ad2antrr simpll1 rspcsbela expcom adantl + ccrg adantr imp vex a1i ssnid mpan csbeq1 gsumunsn syl5eq eqcomi oveq1i eqtrd fveq2d fveq1d simplr gsummptcl eqidd jca evl1addd simprd sylan9eq - mpan oveq1 csbfv12 csbfv2g ax-mp csbconstg fveq12i fveval1fvcl syl5eqel - eqtri nffv csbhypf syl6req exp31 com23 ex a2d imp4b syl5bi ) GUAZUBSZKU - AZXQSUCZAUDZHFSZBXQUEZJDBXQHUFZUGUHZITTZEBXQJHITZTZUFZUGUHZUIZUJZYBBXQX - SUKZULZUEZJDBYNHUFZUGUHZITZTZEBYNYHUFZUGUHZUIZUJYOYCYBBYMUEZUMYAYLUMUUB - YBBXQYMUNYAYLYCUUCUUBYAYCYKUUCUUBUJZYAYCYKUUDUJYAYCUMZUUCYKUUBUUEUUCYKU - UBUUEUUCUMZYKUMYSYJJBXSHUOZITZTZEUPTZUHZUUAUUFYKYSYFUUIUUJUHZUUKUUFYSJY - EUUGDUPTZUHZITZTZUULUUFJYRUUOUUFYQUUNIUUFYQDRXQBRUAZHUOZUFZUGUHZUUGUUMU - HZUUNUUFYQDRYNUURUFZUGUHUVAYPUVBDUGBRYNHUURRHUQZBUUQHURZBUUQHUSZUTVAUUF - XQFUUMRDXSVBUURUUGOUUMVCZYADVDSZYCUUCAXRUVGXTADVESZUVGAEVESZUVHAEVGSZUV - IPEVFVHZDEMVIVHDVJVHVKVQZXRXTAYCUUCVLZUUFUUQXQSZUURFSZUUEUVNUVOUJZUUCYC - UVPYAUVNYCUVOBUUQXQHFVMVNVOVPVRZXSVBSUUFKVSVTZXRXTAYCUUCWAZUUCUUGFSZUUE - XSYMSUUCUVTKWBBXSYMHFVMWRVOZBUUQXSHWCWDWEUVAUUNUIUUFUUTYEUUGUUMUUSYDDUG - YDUUSBRXQHUURUVCUVDUVEUTWFVAWGVTWHWIWJUUFUUNFSUUPUULUIUUFCDUUJUUMEFYEUU - GIYFUUIJLMNOYAUVJYCUUCAXRUVJXTPVKVQZYAJCSZYCUUCAXRUWCXTQVKVQZUUFYEFSYFY - FUIUUFFBDXQHOUVLUVMYAYCUUCWKWLUUFYFWMWNUUFUVTUUIUUIUIUWAUUFUUIWMWNUVFUU - JVCZWOWPWHYFYJUUIUUJWSWQUUFUUKUUAUIYKUUFUUAERXQBUUQYHUOZUFZUGUHZUUIUUJU - HZUUKUUFUUAERYNUWFUFZUGUHUWIYTUWJEUGBRYNYHUWFRYHUQZBUUQYHURZBUUQYHUSZUT - VAUUFXQCUUJREXSVBUWFUUINUWEYAEVDSZYCUUCAXRUWNXTAUVIUWNUVKEVJVHVKVQUVMUU - FUVNUMZUWFJUURITZTZCUWFBUUQJUOZBUUQYGUOZTUWQBUUQJYGWTUWRJUWSUWPUUQVBSZU - WSUWPUIRVSZBUUQHVBIXAXBUWTUWRJUIUXABUUQJVBXCXBXDXGUWOCDEFUURIJLMNOUUFUV - JUVNUWBVPUUFUWCUVNUWDVPUVQXEXFUVRUVSUUFCDEFUUGIJLMNOUWBUWDUWAXEBRXSYHUU - IBXSUQBJUUHBUUGIBIUQBXSHURXHBJUQXHBUAXSUIZJYGUUHUXBHUUGIBXSHUSWIWJXIWDW - EUWHYJUUIUUJUWGYIEUGYIUWGBRXQYHUWFUWKUWLUWMUTWFVAWGXJVPWHXKXLXMXNXOXPXM - $. + simpll2 oveq1 csbfv12 csbfv2g ax-mp csbconstg fveq12i eqtri fveval1fvcl + syl5eqel nffv csbhypf syl6req exp31 com23 ex a2d imp4b syl5bi ) GUAZUBS + ZKUAZXQSUCZAUDZHFSZBXQUEZJDBXQHUFZUGUHZITTZEBXQJHITZTZUFZUGUHZUIZUJZYBB + XQXSUKZULZUEZJDBYNHUFZUGUHZITZTZEBYNYHUFZUGUHZUIZUJYOYCYBBYMUEZUMYAYLUM + UUBYBBXQYMUNYAYLYCUUCUUBYAYCYKUUCUUBUJZYAYCYKUUDUJYAYCUMZUUCYKUUBUUEUUC + YKUUBUUEUUCUMZYKUMYSYJJBXSHUOZITZTZEUPTZUHZUUAUUFYKYSYFUUIUUJUHZUUKUUFY + SJYEUUGDUPTZUHZITZTZUULUUFJYRUUOUUFYQUUNIUUFYQDRXQBRUAZHUOZUFZUGUHZUUGU + UMUHZUUNUUFYQDRYNUURUFZUGUHUVAYPUVBDUGBRYNHUURRHUQZBUUQHURZBUUQHUSZUTVA + UUFXQFUUMRDXSVBUURUUGOUUMVCZYADVGSZYCUUCAXRUVGXTADVDSZUVGAEVDSZUVHAEVPS + ZUVIPEVEVFZDEMVHVFDVIVFVJVKZXRXTAYCUUCVLZUUFUUQXQSZUURFSZUUEUVNUVOUJZUU + CYCUVPYAUVNYCUVOBUUQXQHFVMVNVOVQVRZXSVBSUUFKVSVTZXRXTAYCUUCWRZUUCUUGFSZ + UUEXSYMSUUCUVTKWABXSYMHFVMWBVOZBUUQXSHWCWDWEUVAUUNUIUUFUUTYEUUGUUMUUSYD + DUGYDUUSBRXQHUURUVCUVDUVEUTWFVAWGVTWHWIWJUUFUUNFSUUPUULUIUUFCDUUJUUMEFY + EUUGIYFUUIJLMNOYAUVJYCUUCAXRUVJXTPVJVKZYAJCSZYCUUCAXRUWCXTQVJVKZUUFYEFS + YFYFUIUUFFBDXQHOUVLUVMYAYCUUCWKWLUUFYFWMWNUUFUVTUUIUUIUIUWAUUFUUIWMWNUV + FUUJVCZWOWPWHYFYJUUIUUJWSWQUUFUUKUUAUIYKUUFUUAERXQBUUQYHUOZUFZUGUHZUUIU + UJUHZUUKUUFUUAERYNUWFUFZUGUHUWIYTUWJEUGBRYNYHUWFRYHUQZBUUQYHURZBUUQYHUS + ZUTVAUUFXQCUUJREXSVBUWFUUINUWEYAEVGSZYCUUCAXRUWNXTAUVIUWNUVKEVIVFVJVKUV + MUUFUVNUMZUWFJUURITZTZCUWFBUUQJUOZBUUQYGUOZTUWQBUUQJYGWTUWRJUWSUWPUUQVB + SZUWSUWPUIRVSZBUUQHVBIXAXBUWTUWRJUIUXABUUQJVBXCXBXDXEUWOCDEFUURIJLMNOUU + FUVJUVNUWBVQUUFUWCUVNUWDVQUVQXFXGUVRUVSUUFCDEFUUGIJLMNOUWBUWDUWAXFBRXSY + HUUIBXSUQBJUUHBUUGIBIUQBXSHURXHBJUQXHBUAXSUIZJYGUUHUXBHUUGIBXSHUSWIWJXI + WDWEUWHYJUUIUUJUWGYIEUGYIUWGBRXQYHUWFUWKUWLUWMUTWFVAWGXJVQWHXKXLXMXNXOX + PXM $. $} evl1gsumd.m $e |- ( ph -> A. x e. N M e. U ) $. @@ -247941,25 +248165,25 @@ univariate polynomial evaluation map function for a (sub)ring is a ring = ( R gsum ( x e. N |-> ( ( O ` M ) ` Y ) ) ) ) $= ( cgsu cfv vn vm va wcel wral cmpt co wceq cfn wi cv wa c0 csn cun anbi2d raleq mpteq1 oveq2d fveq2d fveq1d eqeq12d imbi12d cascl mpt0 oveq2i gsum0 - c0g eqid eqtri fveq2i crg ccrg crngrng ply1scl0 eqcomd syl5eq cgrp rnggrp - syl grpidcl evl1scad simprd syl6eqr adantr evl1gsumdlem 3expia a2d impexp - eqtrd wn 3imtr4g findcard2s expd mpcom mpd ) AGFUDZBHUEZJDBHGUFZSUGZITZTZ - EBHJGITTZUFZSUGZUHZQHUIUDZAWRXFUJRXGAWRXFAWQBUAUKZUEZULZJDBXHGUFZSUGZITZT - ZEBXHXCUFZSUGZUHZUJAWQBUMUEZULZJDBUMGUFZSUGZITZTZEBUMXCUFZSUGZUHZUJAWQBUB - UKZUEZULZJDBYGGUFZSUGZITZTZEBYGXCUFZSUGZUHZUJZAWQBYGUCUKZUNUOZUEZULZJDBYS - GUFZSUGZITZTZEBYSXCUFZSUGZUHZUJZAWRULZXFUJUAUBUCHXHUMUHZXJXSXQYFUUKXIXRAW - QBXHUMUQUPUUKXNYCXPYEUUKJXMYBUUKXLYAIUUKXKXTDSBXHUMGURUSUTVAUUKXOYDESBXHU - MXCURUSVBVCXHYGUHZXJYIXQYPUULXIYHAWQBXHYGUQUPUULXNYMXPYOUULJXMYLUULXLYKIU - ULXKYJDSBXHYGGURUSUTVAUULXOYNESBXHYGXCURUSVBVCXHYSUHZXJUUAXQUUHUUMXIYTAWQ - BXHYSUQUPUUMXNUUEXPUUGUUMJXMUUDUUMXLUUCIUUMXKUUBDSBXHYSGURUSUTVAUUMXOUUFE - SBXHYSXCURUSVBVCXHHUHZXJUUJXQXFUUNXIWRAWQBXHHUQUPUUNXNXBXPXEUUNJXMXAUUNXL - WTIUUNXKWSDSBXHHGURUSUTVAUUNXOXDESBXHHXCURUSVBVCAYFXRAYCEVHTZYEAYCJUUODVD - TZTZITZTZUUOAJYBUURAYBDVHTZITUURYAUUTIYADUMSUGUUTXTUMDSBGVEVFDUUTUUTVIZVG - VJVKAUUTUUQIAUUQUUTAEVLUDZUUQUUTUHAEVMUDUVBOEVNVTZUUPDEUUTUUOLUUPVIZUUOVI - ZUVAVOVTVPUTVQVAAUUQFUDUUSUUOUHAUUPCDEFIUUOJKLMUVDNOAEVRUDZUUOCUDAUVBUVFU - VCEVSVTCEUUOMUVEWAVTPWBWCWJYEEUMSUGUUOYDUMESBXCVEVFEUUOUVEVGVJWDWEYGUIUDZ - YRYGUDWKZULZAYHYPUJZUJAYTUUHUJZUJYQUUIUVIAUVJUVKUVGUVHAUVJUVKUJABCDEFUBGI - JUCKLMNOPWFWGWHAYHYPWIAYTUUHWIWLWMWNWOWP $. + c0g eqid fveq2i crg ccrg crngring syl ply1scl0 eqcomd syl5eq cgrp ringgrp + eqtri grpidcl evl1scad simprd eqtrd syl6eqr adantr wn evl1gsumdlem 3expia + a2d impexp 3imtr4g findcard2s expd mpcom mpd ) AGFUDZBHUEZJDBHGUFZSUGZITZ + TZEBHJGITTZUFZSUGZUHZQHUIUDZAWRXFUJRXGAWRXFAWQBUAUKZUEZULZJDBXHGUFZSUGZIT + ZTZEBXHXCUFZSUGZUHZUJAWQBUMUEZULZJDBUMGUFZSUGZITZTZEBUMXCUFZSUGZUHZUJAWQB + UBUKZUEZULZJDBYGGUFZSUGZITZTZEBYGXCUFZSUGZUHZUJZAWQBYGUCUKZUNUOZUEZULZJDB + YSGUFZSUGZITZTZEBYSXCUFZSUGZUHZUJZAWRULZXFUJUAUBUCHXHUMUHZXJXSXQYFUUKXIXR + AWQBXHUMUQUPUUKXNYCXPYEUUKJXMYBUUKXLYAIUUKXKXTDSBXHUMGURUSUTVAUUKXOYDESBX + HUMXCURUSVBVCXHYGUHZXJYIXQYPUULXIYHAWQBXHYGUQUPUULXNYMXPYOUULJXMYLUULXLYK + IUULXKYJDSBXHYGGURUSUTVAUULXOYNESBXHYGXCURUSVBVCXHYSUHZXJUUAXQUUHUUMXIYTA + WQBXHYSUQUPUUMXNUUEXPUUGUUMJXMUUDUUMXLUUCIUUMXKUUBDSBXHYSGURUSUTVAUUMXOUU + FESBXHYSXCURUSVBVCXHHUHZXJUUJXQXFUUNXIWRAWQBXHHUQUPUUNXNXBXPXEUUNJXMXAUUN + XLWTIUUNXKWSDSBXHHGURUSUTVAUUNXOXDESBXHHXCURUSVBVCAYFXRAYCEVHTZYEAYCJUUOD + VDTZTZITZTZUUOAJYBUURAYBDVHTZITUURYAUUTIYADUMSUGUUTXTUMDSBGVEVFDUUTUUTVIZ + VGVTVJAUUTUUQIAUUQUUTAEVKUDZUUQUUTUHAEVLUDUVBOEVMVNZUUPDEUUTUUOLUUPVIZUUO + VIZUVAVOVNVPUTVQVAAUUQFUDUUSUUOUHAUUPCDEFIUUOJKLMUVDNOAEVRUDZUUOCUDAUVBUV + FUVCEVSVNCEUUOMUVEWAVNPWBWCWDYEEUMSUGUUOYDUMESBXCVEVFEUUOUVEVGVTWEWFYGUIU + DZYRYGUDWGZULZAYHYPUJZUJAYTUUHUJZUJYQUUIUVIAUVJUVKUVGUVHAUVJUVKUJABCDEFUB + GIJUCKLMNOPWHWIWJAYHYPWKAYTUUHWKWLWMWNWOWP $. $} ${ @@ -247982,14 +248206,14 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evl1gsumadd $p |- ( ph -> ( Q ` ( W gsum ( x e. N |-> Y ) ) ) = ( P gsum ( x e. N |-> ( Q ` Y ) ) ) ) $= ( cmpt cgsu co cfv ces1 wceq evl1fval1 a1i fveq1d cpl1 ccrg wcel ressid - syl eqcomd fveq2d syl5eq oveq1d cbs c0g eqid crg csubrg crngrng subrgid - cress 3syl cv adantr eleqtrd cfsupp syl6eqr breqtrrd evls1gsumadd eqtrd - wa mpteq2dv oveq2d 3eqtrd ) AIBHJUBZUCUDZEUEWBFGUFUDZUEZDBHJWCUEZUBZUCU - DZDBHJEUEZUBZUCUDAWBEWCEWCUGAGEFLMUHUIZUJAWDFGVGUDZUKUEZWAUCUDZWCUEWGAW - BWMWCAIWLWAUCAIFUKUEWLNAFWKUKAWKFAFULUMZWKFUGQGFULMUNUOUPUQURZUSUQABWLU - TUEZDWCGFWKGHWLJWLVAUEZWCVBMWLVBWQVBWKVBOWPVBQAWNFVCUMGFVDUEUMQFVEGFMVF - VHABVIHUMZVQZJCWPRWSCIUTUEWPPWSIWLUTAIWLUGWRWOVJUQURVKSAWAKWQVLUAAWQIVA - UEKAWLIVAAIWLWOUPUQTVMVNVOVPAWFWIDUCABHWEWHAWHWEAJEWCWJUJUPVRVSVT $. + cress syl eqcomd fveq2d syl5eq oveq1d cbs c0g eqid crg crngring subrgid + csubrg 3syl cv wa adantr eleqtrd syl6eqr breqtrrd evls1gsumadd mpteq2dv + cfsupp eqtrd oveq2d 3eqtrd ) AIBHJUBZUCUDZEUEWBFGUFUDZUEZDBHJWCUEZUBZUC + UDZDBHJEUEZUBZUCUDAWBEWCEWCUGAGEFLMUHUIZUJAWDFGUOUDZUKUEZWAUCUDZWCUEWGA + WBWMWCAIWLWAUCAIFUKUEWLNAFWKUKAWKFAFULUMZWKFUGQGFULMUNUPUQURUSZUTURABWL + VAUEZDWCGFWKGHWLJWLVBUEZWCVCMWLVCWQVCWKVCOWPVCQAWNFVDUMGFVGUEUMQFVEGFMV + FVHABVIHUMZVJZJCWPRWSCIVAUEWPPWSIWLVAAIWLUGWRWOVKURUSVLSAWAKWQVQUAAWQIV + BUEKAWLIVBAIWLWOUQURTVMVNVOVRAWFWIDUCABHWEWHAWHWEAJEWCWJUJUQVPVSVT $. $} ${ @@ -248013,11 +248237,12 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evl1gsummul $p |- ( ph -> ( Q ` ( G gsum ( x e. N |-> Y ) ) ) = ( H gsum ( x e. N |-> ( Q ` Y ) ) ) ) $= ( cfv cmpt cgsu co cvv mgpbas rngidval ccrg wcel ccmn ply1crng crngmgp wa - 3syl crg cmnd crngrng syl cbs eqeltri jctir pwsrng rngmgp cn0 nn0ex ssexd - fvex a1i crh cmhm evl1rhm rhmmhm gsummptmhm eqcomd ) AIBKMEUFUGUHUIHBKMUG - UHUIEUFABKCMHIEUJGCLHUCRUKLGHUCUBULAFUMUNZLUMUNHUOUNSLFPUPLHUCUQUSAFUTUNZ - JUJUNZURDUTUNIVAUNAWAWBAVTWASFVBVCJFVDUFUJOFVDVLVEVFFJUJDQVGDIUDVHUSAKVIU - JVIUJUNAVJVMUAVKAVTELDVNUIUNEHIVOUIUNSJLFDENPQOVPLDEHIUCUDVQUSTUEVRVS $. + 3syl crg cmnd crngring syl cbs fvex eqeltri jctir pwsring ringmgp cn0 a1i + nn0ex ssexd crh cmhm evl1rhm rhmmhm gsummptmhm eqcomd ) AIBKMEUFUGUHUIHBK + MUGUHUIEUFABKCMHIEUJGCLHUCRUKLGHUCUBULAFUMUNZLUMUNHUOUNSLFPUPLHUCUQUSAFUT + UNZJUJUNZURDUTUNIVAUNAWAWBAVTWASFVBVCJFVDUFUJOFVDVEVFVGFJUJDQVHDIUDVIUSAK + VJUJVJUJUNAVLVKUAVMAVTELDVNUIUNEHIVOUIUNSJLFDENPQOVPLDEHIUCUDVQUSTUEVRVS + $. $} ${ @@ -248039,12 +248264,12 @@ univariate polynomial evaluation map function for a (sub)ring is a ring = ( N ( .g ` ( mulGrp ` ( R ^s B ) ) ) ( Q ` X ) ) ) $= ( co cfv cmgp cress cv1 cpl1 cmg ces1 cpws wceq evl1fval1 a1i fveq2i ccrg eqtri wcel ressid eqcomd fveq2d syl5eq eqidd oveq123d fveq12d eqid csubrg - syl crg crngrng subrgid 3syl evls1varpw eqcomi oveq2d 3eqtrd ) AGIERZCSGD - BUARZUBSZVMUCSZTSZUDSZRZDBUERZSGVNVSSZDBUFRTSUDSZRGICSZWARAVLVRCVSCVSUGAB - CDJNUHZUIAGGIVNEVQAEDUCSZTSZUDSZVQEFUDSWFOFWEUDFHTSWELHWDTKUJULUJULAWEVPU - DAWDVOTADVMUCAVMDADUKUMZVMDUGPBDUKNUNVCUOZUPUPUPUQAGURAIDUBSVNMADVMUBWHUP - UQZUSUTABVSBDVMVQVPGVOVNVSVAVMVAVOVAVPVAVNVANVQVAPAWGDVDUMBDVBSUMPDVEBDNV - FVGQVHAVTWBGWAAVNIVSCVSCUGACVSWCVIUIAIVNWIUOUTVJVK $. + syl crg crngring subrgid 3syl evls1varpw eqcomi oveq2d 3eqtrd ) AGIERZCSG + DBUARZUBSZVMUCSZTSZUDSZRZDBUERZSGVNVSSZDBUFRTSUDSZRGICSZWARAVLVRCVSCVSUGA + BCDJNUHZUIAGGIVNEVQAEDUCSZTSZUDSZVQEFUDSWFOFWEUDFHTSWELHWDTKUJULUJULAWEVP + UDAWDVOTADVMUCAVMDADUKUMZVMDUGPBDUKNUNVCUOZUPUPUPUQAGURAIDUBSVNMADVMUBWHU + PUQZUSUTABVSBDVMVQVPGVOVNVSVAVMVAVOVAVPVAVNVANVQVAPAWGDVDUMBDVBSUMPDVEBDN + VFVGQVHAVTWBGWAAVNIVSCVSCUGACVSWCVIUIAIVNWIUOUTVJVK $. ${ evl1varpwval.c $e |- ( ph -> C e. B ) $. @@ -248073,19 +248298,19 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evl1scvarpw $p |- ( ph -> ( Q ` ( A .X. ( N .^ X ) ) ) = ( ( B X. { A } ) .xb ( N F ( Q ` X ) ) ) ) $= ( co cfv cascl cmulr csn cxp casa wcel csca wceq ccrg ply1assa syl6eleq - cbs syl ply1sca eqcomd fveq2d eleqtrrd cmnd cn0 crg crngrng rngmgp eqid - ply1rng vr1cl mgpbas mulgnn0cl syl3anc asclmul1 crh evl1rhm clmod asclf - ply1lmod ffvelrnd rhmmul evl1sca syl2anc cpws cmgp cmg evl1varpw fveq2i - eqtri a1i oveqd eqtrd oveq12d 3eqtrd ) ABMOIUJZHUJZDUKBNULUKZUKZXANUMUK - ZUJZDUKZXDDUKZXADUKZGUJZCBUNUOZMODUKZJUJZGUJAXBXFDAXFXBANUPUQZBNURUKZVC - UKZUQXANVCUKZUQZXFXBUSAEUTUQZXNUBNEQVAVDABEVCUKZXPABCXTUETVBAXOEVCAXSXO - EUSUBXSEXONEUTQVEVFVDVGVHZAKVIUQZMVJUQOXQUQZXRANVKUQZYBAEVKUQZYDAXSYEUB - EVLVDZNEQVOVDZNKRVMVDUCAYEYCYFXQNEOSQXQVNZVPVDXQIKMOXQNKRYHVQUAVRVSZXCB - HXEXOXPXQNXAXCVNZXOVNZXPVNZYHXEVNZUDVTVSVFVGADNFWAUJUQZXDXQUQXRXGXJUSAX - SYNUBCNEFDPQUFTWBVDAXPXQBXCAXCXQXOXPNYJYKYGAYENWCUQYFNEQWEVDYLYHWDYAWFY - IXDXANFXEGDXQYHYMUGWGVSAXHXKXIXMGAXSBCUQXHXKUSUBUEXCCNEDBPQTYJWHWIAXIMX - LECWJUJZWKUKZWLUKZUJXMACDEIKMNOPQRSTUAUBUCWMAYQJMXLAJYQJYQUSAJLWLUKYQUI - LYPWLLFWKUKYPUHFYOWKUFWNWOWNWOWPVFWQWRWSWT $. + cbs syl ply1sca eqcomd eleqtrrd cmnd cn0 crngring ply1ring ringmgp eqid + fveq2d crg vr1cl mgpbas mulgnn0cl syl3anc asclmul1 crh evl1rhm ply1lmod + clmod asclf ffvelrnd rhmmul evl1sca syl2anc cpws evl1varpw fveq2i eqtri + cmgp cmg a1i oveqd eqtrd oveq12d 3eqtrd ) ABMOIUJZHUJZDUKBNULUKZUKZXANU + MUKZUJZDUKZXDDUKZXADUKZGUJZCBUNUOZMODUKZJUJZGUJAXBXFDAXFXBANUPUQZBNURUK + ZVCUKZUQXANVCUKZUQZXFXBUSAEUTUQZXNUBNEQVAVDABEVCUKZXPABCXTUETVBAXOEVCAX + SXOEUSUBXSEXONEUTQVEVFVDVNVGZAKVHUQZMVIUQOXQUQZXRANVOUQZYBAEVOUQZYDAXSY + EUBEVJVDZNEQVKVDZNKRVLVDUCAYEYCYFXQNEOSQXQVMZVPVDXQIKMOXQNKRYHVQUAVRVSZ + XCBHXEXOXPXQNXAXCVMZXOVMZXPVMZYHXEVMZUDVTVSVFVNADNFWAUJUQZXDXQUQXRXGXJU + SAXSYNUBCNEFDPQUFTWBVDAXPXQBXCAXCXQXOXPNYJYKYGAYENWDUQYFNEQWCVDYLYHWEYA + WFYIXDXANFXEGDXQYHYMUGWGVSAXHXKXIXMGAXSBCUQXHXKUSUBUEXCCNEDBPQTYJWHWIAX + IMXLECWJUJZWNUKZWOUKZUJXMACDEIKMNOPQRSTUAUBUCWKAYQJMXLAJYQJYQUSAJLWOUKY + QUILYPWOLFWNUKYPUHFYOWNUFWLWMWLWMWPVFWQWRWSWT $. $} ${ @@ -248098,12 +248323,12 @@ univariate polynomial evaluation map function for a (sub)ring is a ring the evaluated variable. (Contributed by AV, 18-Sep-2019.) $) evl1scvarpwval $p |- ( ph -> ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) = ( A .x. ( N E C ) ) ) $= - ( co cbs cfv wcel wceq eqid cmnd cn0 crg ccrg crngrng syl ply1rng vr1cl - rngmgp mgpbas mulgnn0cl syl3anc evl1varpwval jca evl1vsd simprd ) ABMOJ - UJZHUJZNUKULZUMDVMEULULBMDIUJZGUJUNACNFHGVNVLBEVODPQTVNUOZUBUFAVLVNUMZD - VLEULULVOUNAKUPUMZMUQUMOVNUMZVQANURUMZVRAFURUMZVTAFUSUMWAUBFUTVAZNFQVBV - ANKRVDVAUCAWAVSWBVNNFOSQVPVCVAVNJKMOVNNKRVPVEUAVFVGACDEFIJKLMNOPQRSTUAU - BUCUFUGUHVHVIUEUDUIVJVK $. + ( co cbs cfv wcel wceq eqid cmnd cn0 crg ccrg crngring ply1ring ringmgp + syl vr1cl mgpbas mulgnn0cl syl3anc evl1varpwval jca evl1vsd simprd ) AB + MOJUJZHUJZNUKULZUMDVMEULULBMDIUJZGUJUNACNFHGVNVLBEVODPQTVNUOZUBUFAVLVNU + MZDVLEULULVOUNAKUPUMZMUQUMOVNUMZVQANURUMZVRAFURUMZVTAFUSUMWAUBFUTVCZNFQ + VAVCNKRVBVCUCAWAVSWBVNNFOSQVPVDVCVNJKMOVNNKRVPVEUAVFVGACDEFIJKLMNOPQRST + UAUBUCUFUGUHVHVIUEUDUIVJVK $. $} $} @@ -248134,17 +248359,17 @@ univariate polynomial evaluation map function for a (sub)ring is a ring evl1gsummon $p |- ( ph -> ( ( Q ` ( W gsum ( x e. M |-> ( A .X. ( N .^ X ) ) ) ) ) ` C ) = ( R gsum ( x e. M |-> ( A .x. ( N E C ) ) ) ) ) $= - ( co cmpt cgsu cfv cpws eqid cv wcel clmod csca cbs ccrg crngrng ply1lmod - wa crg syl adantr r19.21bi wceq ply1sca fveq2d syl5eq eleqtrd cn0 ply1rng - cmnd rngmgp vr1cl mgpbas mulgnn0cl lmodvscl evl1gsumaddval evl1scvarpwval - syl3anc mpteq2dva oveq2d eqtrd ) AEQBOCPRKUPZIUPZUQURUPFUSUSGBOEWOFUSUSZU - QZURUPGBOCPEJUPHUPZUQZURUPABDEGNUTUPZFGNOQWOSTUAWTVAUBUJABVBOVCZVJZQVDVCZ - CQVEUSZVFUSZVCWNDVCZWODVCAXCXAAGVKVCZXCAGVGVCZXGUJGVHVLZQGUAVIVLVMXBCNXEA - CNVCBOUKVNZANXEVOXAANGVFUSXETAGXDVFAXHGXDVOUJQGVGUAVPVLVQVRVMVSXBLWBVCZPV - TVCZRDVCZXFAXKXAAQVKVCZXKAXGXNXIQGUAWAVLQLUFWCVLVMAXLBOUNVNZXBXGXMAXGXAXI - VMDQGRUCUAUBWDVLDKLPRDQLUFUBWEUGWFWJCIXDXEDQWNUBXDVAUHXEVAWGWJULUMUOWHAWQ - WSGURABOWPWRXBCNEFGHIJKLMPQRSUAUFUCTUGAXHXAUJVMXOUHXJAENVCXAUOVMUDUEUIWIW - KWLWM $. + ( co cmpt cgsu cfv cpws eqid cv wcel clmod csca cbs crg ccrg crngring syl + ply1lmod adantr r19.21bi wceq ply1sca fveq2d syl5eq eleqtrd cmnd ply1ring + cn0 ringmgp vr1cl mgpbas mulgnn0cl lmodvscl evl1gsumaddval evl1scvarpwval + wa syl3anc mpteq2dva oveq2d eqtrd ) AEQBOCPRKUPZIUPZUQURUPFUSUSGBOEWOFUSU + SZUQZURUPGBOCPEJUPHUPZUQZURUPABDEGNUTUPZFGNOQWOSTUAWTVAUBUJABVBOVCZWIZQVD + VCZCQVEUSZVFUSZVCWNDVCZWODVCAXCXAAGVGVCZXCAGVHVCZXGUJGVIVJZQGUAVKVJVLXBCN + XEACNVCBOUKVMZANXEVNXAANGVFUSXETAGXDVFAXHGXDVNUJQGVHUAVOVJVPVQVLVRXBLVSVC + ZPWAVCZRDVCZXFAXKXAAQVGVCZXKAXGXNXIQGUAVTVJQLUFWBVJVLAXLBOUNVMZXBXGXMAXGX + AXIVLDQGRUCUAUBWCVJDKLPRDQLUFUBWDUGWEWJCIXDXEDQWNUBXDVAUHXEVAWFWJULUMUOWG + AWQWSGURABOWPWRXBCNEFGHIJKLMPQRSUAUFUCTUGAXHXAUJVLXOUHXJAENVCXAUOVLUDUEUI + WHWKWLWM $. $} $( @@ -248461,8 +248686,8 @@ univariate polynomial evaluation map function for a (sub)ring is a ring $( The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) - cnrng $p |- CCfld e. Ring $= - ( ccnfld ccrg wcel crg cncrng crngrng ax-mp ) ABCADCEAFG $. + cnring $p |- CCfld e. Ring $= + ( ccnfld ccrg wcel crg cncrng crngring ax-mp ) ABCADCEAFG $. $( The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see ~ xrsmgmdifsgrp .) @@ -248478,25 +248703,25 @@ univariate polynomial evaluation map function for a (sub)ring is a ring $( The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) cnfld0 $p |- 0 = ( 0g ` CCfld ) $= - ( ccnfld c0g cfv cc0 caddc co wceq 00id cgrp cc wb crg cnrng rnggrp ax-mp - wcel 0cn cnfldbas cnfldadd eqid grpid mp2an mpbi eqcomi ) ABCZDDDEFDGZUED - GZHAIPZDJPUFUGKALPUHMANOQJEADUERSUETUAUBUCUD $. + ( ccnfld c0g cfv cc0 caddc co wceq 00id cgrp wcel cc wb crg ringgrp ax-mp + cnring 0cn cnfldbas cnfldadd eqid grpid mp2an mpbi eqcomi ) ABCZDDDEFDGZU + EDGZHAIJZDKJUFUGLAMJUHPANOQKEADUERSUETUAUBUCUD $. $( The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) cnfld1 $p |- 1 = ( 1r ` CCfld ) $= ( vx ccnfld cur cfv c1 cc wcel cv cmul wceq wral ax-1cn mulid2 mulid1 jca - co wa rgen pm3.2i crg wb cnrng cnfldbas cnfldmul eqid isrngid mpbi eqcomi - ax-mp ) BCDZEEFGZEAHZIPULJZULEIPULJZQZAFKZQZUJEJZUKUPLUOAFULFGUMUNULMULNO - RSBTGUQURUAUBAFBIUJEUCUDUJUEUFUIUGUH $. + co wa rgen pm3.2i crg cnring cnfldbas cnfldmul eqid isringid ax-mp eqcomi + wb mpbi ) BCDZEEFGZEAHZIPULJZULEIPULJZQZAFKZQZUJEJZUKUPLUOAFULFGUMUNULMUL + NORSBTGUQURUHUAAFBIUJEUBUCUJUDUEUFUIUG $. $( The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) cnfldneg $p |- ( X e. CC -> ( ( invg ` CCfld ) ` X ) = -u X ) $= - ( cc wcel ccnfld cminusg cfv cneg wceq caddc co negid wb negcl cgrp cnrng - cc0 crg rnggrp ax-mp cnfldbas cnfldadd cnfld0 eqid grpinvid1 mp3an1 mpdan - mpbird ) ABCZADEFZFAGZHZAUJIJPHZAKUHUJBCZUKULLZAMDNCZUHUMUNDQCUOODRSBIDUI - AUJPTUAUBUIUCUDUEUFUG $. + ( cc wcel ccnfld cminusg cfv cneg wceq caddc co cc0 negid wb negcl cnring + cgrp crg ringgrp ax-mp cnfldbas cnfld0 eqid grpinvid1 mp3an1 mpdan mpbird + cnfldadd ) ABCZADEFZFAGZHZAUJIJKHZALUHUJBCZUKULMZANDPCZUHUMUNDQCUOODRSBID + UIAUJKTUGUAUIUBUCUDUEUF $. $( The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.) $) @@ -248509,61 +248734,61 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldsub $p |- - = ( -g ` CCfld ) $= ( vx vy cc cv cmin cmpt2 ccnfld cfv wcel caddc cnfldbas eqid wceq wfn ffn co wf ax-mp fnov mpbi csg cminusg cneg cnfldadd grpsubval cnfldneg adantl - wa oveq2d negsub 3eqtrrd mpt2eq3ia cxp subf cgrp crg cnrng rnggrp grpsubf - mp2b 3eqtr4i ) ABCCADZBDZEPZFZABCCVBVCGUAHZPZFZEVFABCCVDVGVBCIZVCCIZUHZVG - VBVCGUBHZHZJPVBVCUCZJPVDCJGVLVFVBVCKUDVLLVFLZUEVKVMVNVBJVJVMVNMVIVCUFUGUI - VBVCUJUKULECCUMZNZEVEMVPCEQVQUNVPCEORABCCESTVFVPNZVFVHMGUOIZVPCVFQVRGUPIV - SUQGURRCGVFKVOUSVPCVFOUTABCCVFSTVA $. + wa oveq2d negsub 3eqtrrd mpt2eq3ia cxp subf cgrp crg ringgrp grpsubf mp2b + cnring 3eqtr4i ) ABCCADZBDZEPZFZABCCVBVCGUAHZPZFZEVFABCCVDVGVBCIZVCCIZUHZ + VGVBVCGUBHZHZJPVBVCUCZJPVDCJGVLVFVBVCKUDVLLVFLZUEVKVMVNVBJVJVMVNMVIVCUFUG + UIVBVCUJUKULECCUMZNZEVEMVPCEQVQUNVPCEORABCCESTVFVPNZVFVHMGUOIZVPCVFQVRGUP + IVSUTGUQRCGVFKVOURVPCVFOUSABCCVFSTVA $. $( The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) $) cndrng $p |- CCfld e. DivRing $= ( vx vy ccnfld cdr wcel wtru cc cmul c1 cv cdiv co cc0 cbs cfv a1i wne wa - wceq adantl cnfldbas cmulr c0g cnfld0 cur cnfld1 crg cnrng mulne0 3adant1 - cnfldmul ax-1ne0 reccl recne0 recid2 isdrngd trud ) CDEFABGCHIIAJZKLZMGCN - OSFUAPHCUBOSFUKPMCUCOSFUDPICUEOSFUFPCUGEFUHPURGEURMQRZBJZGEVAMQRURVAHLMQF - URVAUIUJIMQFULPUTUSGEFURUMTUTUSMQFURUNTUTUSURHLISFURUOTUPUQ $. + wceq adantl cnfldbas cnfldmul c0g cnfld0 cur cnfld1 cnring mulne0 3adant1 + cmulr crg ax-1ne0 reccl recne0 recid2 isdrngd trud ) CDEFABGCHIIAJZKLZMGC + NOSFUAPHCUJOSFUBPMCUCOSFUDPICUEOSFUFPCUKEFUGPURGEURMQRZBJZGEVAMQRURVAHLMQ + FURVAUHUIIMQFULPUTUSGEFURUMTUTUSMQFURUNTUTUSURHLISFURUOTUPUQ $. $( The division operation in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) $) cnflddiv $p |- / = ( /r ` CCfld ) $= - ( vx vy vz cc cc0 cv cmul co wceq cmpt2 ccnfld cfv cdiv wa cnrng cnfldbas - wcel eqid cnfldmul eqtr3d csn cdif crio cdvr cnfld0 cndrng drngui dvrcan1 - cinvr crg mp3an1 oveq1d dvrcl wne simpr eldifsn sylib simpld simprd simpl - divcan4d divval syl3anc dvrval mpt2eq3ia df-div dvrfval 3eqtr4i ) ABDDEUA - UBZBFZCFGHAFZICDUCZJABDVIVKVJKUILZLGHZJMKUDLZABDVIVLVNVKDQZVJVIQZNZVKVJVO - HZVLVNVRVKVJMHZVSVLVRVSVJGHZVJMHVTVSVRWAVKVJMKUJQZVPVQWAVKIODVOKGVIVKVJPD - KEPUEUFUGZVORZSUHUKULVRVSVJWBVPVQVSDQODVOKVIVKVJPWCWDUMUKVRVJDQZVJEUNZVRV - QWEWFNVPVQUOVJDEUPUQZURZVRWEWFWGUSZVATVRVPWEWFVTVLIVPVQUTWHWICVKVJVBVCTDV - OKGVIVMVKVJPSWCVMRZWDVDTVEABCVFABDVOKGVIVMPSWCWJWDVGVH $. + ( vx vy vz cc cc0 cv cmul wceq cmpt2 ccnfld cfv cdiv wcel cnring cnfldbas + co wa eqid cnfldmul eqtr3d csn cdif crio cinvr cdvr cnfld0 cndrng dvrcan1 + crg drngui mp3an1 oveq1d dvrcl simpr eldifsn sylib simpld simprd divcan4d + wne simpl divval syl3anc dvrval mpt2eq3ia df-div dvrfval 3eqtr4i ) ABDDEU + AUBZBFZCFGPAFZHCDUCZIABDVIVKVJJUDKZKGPZILJUEKZABDVIVLVNVKDMZVJVIMZQZVKVJV + OPZVLVNVRVKVJLPZVSVLVRVSVJGPZVJLPVTVSVRWAVKVJLJUIMZVPVQWAVKHNDVOJGVIVKVJO + DJEOUFUGUJZVORZSUHUKULVRVSVJWBVPVQVSDMNDVOJVIVKVJOWCWDUMUKVRVJDMZVJEUTZVR + VQWEWFQVPVQUNVJDEUOUPZUQZVRWEWFWGURZUSTVRVPWEWFVTVLHVPVQVAWHWICVKVJVBVCTD + VOJGVIVMVKVJOSWCVMRZWDVDTVEABCVFABDVOJGVIVMOSWCWJWDVGVH $. $( The multiplicative inverse in the field of complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) $) cnfldinv $p |- ( ( X e. CC /\ X =/= 0 ) -> ( ( invr ` CCfld ) ` X ) = ( 1 / X ) ) $= - ( cc wcel cc0 wne wa csn cdif ccnfld cinvr cfv c1 cdiv wceq eldifsn cnrng - co crg cnfldbas cnfld0 cndrng drngui cnflddiv cnfld1 eqid rnginvdv sylbir - mpan ) ABCADEFABDGHZCZAIJKZKLAMQNZABDOIRCUJULPBMIUILUKASBIDSTUAUBUCUDUKUE - UFUHUG $. + ( cc wcel cc0 wne wa csn cdif ccnfld cinvr c1 cdiv co wceq eldifsn cnring + cfv crg cnfldbas cnfld0 cndrng drngui cnflddiv cnfld1 eqid ringinvdv mpan + sylbir ) ABCADEFABDGHZCZAIJQZQKALMNZABDOIRCUJULPBLIUIKUKASBIDSTUAUBUCUDUK + UEUFUGUH $. $( The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.) $) cnfldmulg $p |- ( ( A e. ZZ /\ B e. CC ) -> ( A ( .g ` CCfld ) B ) = ( A x. B ) ) $= ( vx vy cc wcel ccnfld cfv co cmul wceq cv cneg c1 caddc eqeq12d cnfldbas - cc0 oveq1 eqid cz cmg cnfld0 mulg0 mul02 eqtr4d cn0 wi wa cmnd crg rngmnd - cnrng ax-mp cnfldadd mp3an1 nn0cn adantr 1cnd simpr adddird mulid2 adantl - mulgnn0p1 oveq2d eqtrd syl5ibr expcom cn cminusg fveq2 nncn mulneg1 sylan - mulgnegnn mulcl cnfldneg syl zindd impcom ) BEFZAUAFABGUBHZIZABJIZKZCLZBW - BIZWFBJIZKRBWBIZRBJIZKDLZBWBIZWKBJIZKZWKMZBWBIZWOBJIZKZWKNOIZBWBIZWSBJIZK - ZWEWACDAWFRKWGWIWHWJWFRBWBSWFRBJSPWFWKKWGWLWHWMWFWKBWBSWFWKBJSPWFWSKWGWTW - HXAWFWSBWBSWFWSBJSPWFWOKWGWPWHWQWFWOBWBSWFWOBJSPWFAKWGWCWHWDWFABWBSWFABJS - PWAWIRWJEWBGBRQUCWBTZUDBUEUFWKUGFZWAWNXBUHWNXBXDWAUIZWLBOIZWMBOIZKWLWMBOS - XEWTXFXAXGGUJFZXDWAWTXFKGUKFXHUMGULUNEOWBGWKBQXCUOVDUPXEXAWMNBJIZOIXGXEWK - NBXDWKEFZWAWKUQURXEUSXDWAUTVAXEXIBWMOWAXIBKXDBVBVCVEVFPVGVHWKVIFZWAWNWRUH - WNWRXKWAUIZWLGVJHZHZWMXMHZKWLWMXMVKXLWPXNWQXOEWBGXMWKBQXCXMTVOXLWQWMMZXOX - KXJWAWQXPKWKVLZWKBVMVNXLWMEFZXOXPKXKXJWAXRXQWKBVPVNWMVQVRUFPVGVHVSVT $. + cc0 oveq1 eqid cz cmg cnfld0 mulg0 mul02 eqtr4d cn0 wi wa cmnd crg cnring + ringmnd ax-mp cnfldadd mulgnn0p1 mp3an1 nn0cn adantr simpr adddird mulid2 + 1cnd adantl oveq2d eqtrd syl5ibr expcom cn cminusg mulgnegnn nncn mulneg1 + fveq2 sylan mulcl cnfldneg syl zindd impcom ) BEFZAUAFABGUBHZIZABJIZKZCLZ + BWBIZWFBJIZKRBWBIZRBJIZKDLZBWBIZWKBJIZKZWKMZBWBIZWOBJIZKZWKNOIZBWBIZWSBJI + ZKZWEWACDAWFRKWGWIWHWJWFRBWBSWFRBJSPWFWKKWGWLWHWMWFWKBWBSWFWKBJSPWFWSKWGW + TWHXAWFWSBWBSWFWSBJSPWFWOKWGWPWHWQWFWOBWBSWFWOBJSPWFAKWGWCWHWDWFABWBSWFAB + JSPWAWIRWJEWBGBRQUCWBTZUDBUEUFWKUGFZWAWNXBUHWNXBXDWAUIZWLBOIZWMBOIZKWLWMB + OSXEWTXFXAXGGUJFZXDWAWTXFKGUKFXHULGUMUNEOWBGWKBQXCUOUPUQXEXAWMNBJIZOIXGXE + WKNBXDWKEFZWAWKURUSXEVCXDWAUTVAXEXIBWMOWAXIBKXDBVBVDVEVFPVGVHWKVIFZWAWNWR + UHWNWRXKWAUIZWLGVJHZHZWMXMHZKWLWMXMVNXLWPXNWQXOEWBGXMWKBQXCXMTVKXLWQWMMZX + OXKXJWAWQXPKWKVLZWKBVMVOXLWMEFZXOXPKXKXJWAXRXQWKBVPVOWMVQVRUFPVGVHVSVT $. $( The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, @@ -248572,25 +248797,25 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ( B ( .g ` ( mulGrp ` CCfld ) ) A ) = ( A ^ B ) ) $= ( vx vy cn0 wcel cc ccnfld cfv co cexp wceq wi cc0 c1 oveq1 oveq2 eqeq12d imbi2d cmul cmgp cv caddc eqid cnfldbas mgpbas cnfld1 rngidval mulg0 exp0 - cmg eqtr4d cmnd crg cnrng rngmgp ax-mp cnfldmul mgpplusg mulgnn0p1 mp3an1 - wa ancoms expp1 syl5ibr expcom a2d nn0ind impcom ) BEFAGFZBAHUAIZUKIZJZAB - KJZLZVJCUBZAVLJZAVPKJZLZMVJNAVLJZANKJZLZMVJDUBZAVLJZAWCKJZLZMVJWCOUCJZAVL - JZAWGKJZLZMVJVOMCDBVPNLZVSWBVJWKVQVTVRWAVPNAVLPVPNAKQRSVPWCLZVSWFVJWLVQWD - VRWEVPWCAVLPVPWCAKQRSVPWGLZVSWJVJWMVQWHVRWIVPWGAVLPVPWGAKQRSVPBLZVSVOVJWN - VQVMVRVNVPBAVLPVPBAKQRSVJVTOWAGVLVKAOGHVKVKUDZUEUFZHOVKWOUGUHVLUDZUIAUJUL - WCEFZVJWFWJVJWRWFWJMWFWJVJWRVBZWDATJZWEATJZLWDWEATPWSWHWTWIXAWRVJWHWTLZVK - UMFZWRVJXBHUNFXCUOHVKWOUPUQGTVLVKWCAWPWQHTVKWOURUSUTVAVCAWCVDRVEVFVGVHVI - $. + cmg eqtr4d wa crg cnring ringmgp ax-mp cnfldmul mgpplusg mulgnn0p1 mp3an1 + cmnd ancoms expp1 syl5ibr expcom a2d nn0ind impcom ) BEFAGFZBAHUAIZUKIZJZ + ABKJZLZVJCUBZAVLJZAVPKJZLZMVJNAVLJZANKJZLZMVJDUBZAVLJZAWCKJZLZMVJWCOUCJZA + VLJZAWGKJZLZMVJVOMCDBVPNLZVSWBVJWKVQVTVRWAVPNAVLPVPNAKQRSVPWCLZVSWFVJWLVQ + WDVRWEVPWCAVLPVPWCAKQRSVPWGLZVSWJVJWMVQWHVRWIVPWGAVLPVPWGAKQRSVPBLZVSVOVJ + WNVQVMVRVNVPBAVLPVPBAKQRSVJVTOWAGVLVKAOGHVKVKUDZUEUFZHOVKWOUGUHVLUDZUIAUJ + ULWCEFZVJWFWJVJWRWFWJMWFWJVJWRUMZWDATJZWEATJZLWDWEATPWSWHWTWIXAWRVJWHWTLZ + VKVBFZWRVJXBHUNFXCUOHVKWOUPUQGTVLVKWCAWPWQHTVKWOURUSUTVAVCAWCVDRVEVFVGVHV + I $. $( The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.) $) cnsrng $p |- CCfld e. *Ring $= ( vx ccnfld csr wcel wtru caddc cmul ccj cbs cfv wceq cnfldbas a1i cplusg - vy cc cv adantl co 3adant1 cnfldadd cmulr cnfldmul cstv cnfldcj crg cnrng - cjcl cjadd wa mulcom fveq2d cjmul ancoms eqtrd cjcj issrngd trud ) BCDEAO - FBGHPPBIJKELMFBNJKEUAMGBUBJKEUCMHBUDJKEUEMBUFDEUGMAQZPDZUSHJZPDEUSUHRUTOQ - ZPDZUSVBFSHJVAVBHJZFSKEUSVBUITUTVCUSVBGSZHJZVDVAGSZKEUTVCUJZVFVBUSGSZHJZV - GVHVEVIHUSVBUKULVCUTVJVGKVBUSUMUNUOTUTVAHJUSKEUSUPRUQUR $. + vy cc cv adantl co 3adant1 cnfldadd cnfldmul cstv cnfldcj crg cnring cjcl + cmulr cjadd wa mulcom fveq2d cjmul ancoms eqtrd cjcj issrngd trud ) BCDEA + OFBGHPPBIJKELMFBNJKEUAMGBUHJKEUBMHBUCJKEUDMBUEDEUFMAQZPDZUSHJZPDEUSUGRUTO + QZPDZUSVBFSHJVAVBHJZFSKEUSVBUITUTVCUSVBGSZHJZVDVAGSZKEUTVCUJZVFVBUSGSZHJZ + VGVHVEVIHUSVBUKULVCUTVJVGKVBUSUMUNUOTUTVAHJUSKEUSUPRUQUR $. $} ${ @@ -248768,9 +248993,9 @@ univariate polynomial evaluation map function for a (sub)ring is a ring 18-Jun-2015.) $) cnsubmlem $p |- A e. ( SubMnd ` CCfld ) $= ( ccnfld csubmnd cfv wcel cc wss cc0 cv caddc co wral ssriv rgen2a crg - cmnd w3a wb cnrng rngmnd cnfldbas cnfld0 cnfldadd issubm mp2b mpbir3an - ) CGHIJZCKLZMCJZANBNOPCJZBCQACQZACKDRFUOABCESGTJGUAJULUMUNUPUBUCUDGUEAB - KOCGMUFUGUHUIUJUK $. + cmnd w3a cnring ringmnd cnfldbas cnfld0 cnfldadd issubm mp2b mpbir3an + wb ) CGHIJZCKLZMCJZANBNOPCJZBCQACQZACKDRFUOABCESGTJGUAJULUMUNUPUBUKUCGU + DABKOCGMUEUFUGUHUIUJ $. $} cnsubglem.3 $e |- ( x e. A -> -u x e. A ) $. @@ -248781,10 +249006,10 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnsubglem $p |- A e. ( SubGrp ` CCfld ) $= ( ccnfld csubg cfv wcel cc wss c0 wne cv caddc wral ax-mp co cminusg wa ssriv ne0i ralrimiva cneg wceq cnfldneg syl eqeltrd jca rgen w3a wb crg - cgrp cnrng rnggrp cnfldbas cnfldadd eqid issubg2 mpbir3an ) CIJKLZCMNZC - OPZAQZBQRUACLZBCSZVHIUBKZKZCLZUCZACSZACMEUDDCLVGHCDUETVNACVHCLZVJVMVPVI - BCFUFVPVLVHUGZCVPVHMLVLVQUHEVHUIUJGUKULUMIUQLZVEVFVGVOUNUOIUPLVRURIUSTA - BMRCIVKUTVAVKVBVCTVD $. + cgrp cnring ringgrp cnfldbas cnfldadd eqid issubg2 mpbir3an ) CIJKLZCMN + ZCOPZAQZBQRUACLZBCSZVHIUBKZKZCLZUCZACSZACMEUDDCLVGHCDUETVNACVHCLZVJVMVP + VIBCFUFVPVLVHUGZCVPVHMLVLVQUHEVHUIUJGUKULUMIUQLZVEVFVGVOUNUOIUPLVRURIUS + TABMRCIVKUTVAVKVBVCTVD $. $} cnsubrglem.4 $e |- 1 e. A $. @@ -248793,9 +249018,9 @@ univariate polynomial evaluation map function for a (sub)ring is a ring 4-Dec-2014.) $) cnsubrglem $p |- A e. ( SubRing ` CCfld ) $= ( ccnfld csubrg cfv wcel csubg c1 cv cmul co wral cnsubglem rgen2a crg wb - w3a cnrng cc cnfldbas cnfld1 cnfldmul issubrg2 ax-mp mpbir3an ) CIJKLZCIM - KLZNCLZAOBOPQCLZBCRACRZABCNDEFGSGUOABCHTIUALULUMUNUPUCUBUDABCUEIPNUFUGUHU - IUJUK $. + w3a cnring cc cnfldbas cnfld1 cnfldmul issubrg2 ax-mp mpbir3an ) CIJKLZCI + MKLZNCLZAOBOPQCLZBCRACRZABCNDEFGSGUOABCHTIUALULUMUNUPUCUBUDABCUEIPNUFUGUH + UIUJUK $. cnsubrglem.6 $e |- ( ( x e. A /\ x =/= 0 ) -> ( 1 / x ) e. A ) $. $( Lemma for ~ resubdrg and friends. (Contributed by Mario Carneiro, @@ -248803,12 +249028,12 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnsubdrglem $p |- ( A e. ( SubRing ` CCfld ) /\ ( CCfld |`s A ) e. DivRing ) $= ( ccnfld cfv wcel co cdr cc0 cdif cndrng eqid cnfld0 cc csubrg cnsubrglem - cress cv cinvr csn wral wb issubdrg mp2an cdiv crg wceq cnrng ssriv ssdif - c1 wss ax-mp cnfldbas drngui cnflddiv cnfld1 rnginvdv sylancr wne eldifsn - sseli wa sylbi eqeltrd mprgbir pm3.2i ) CJUAKLZJCUCMZNLZABCDEFGHUBZVPAUDZ - JUEKZKZCLZACOUFZPZJNLVNVPWAAWCUGUHQVQACJVOVSOVORSVSRZUIUJVRWCLZVTUQVRUKMZ - CWEJULLVRTWBPZLVTWFUMUNWCWGVRCTURWCWGURACTDUOCTWBUPUSVHTUKJWGUQVSVRUTTJOU - TSQVAVBVCWDVDVEWEVRCLVROVFVIWFCLVRCOVGIVJVKVLVM $. + cress cv cinvr csn wral wb issubdrg mp2an c1 cdiv wceq cnring ssriv ssdif + crg wss ax-mp sseli cnfldbas drngui cnflddiv cnfld1 ringinvdv sylancr wne + wa eldifsn sylbi eqeltrd mprgbir pm3.2i ) CJUAKLZJCUCMZNLZABCDEFGHUBZVPAU + DZJUEKZKZCLZACOUFZPZJNLVNVPWAAWCUGUHQVQACJVOVSOVORSVSRZUIUJVRWCLZVTUKVRUL + MZCWEJUQLVRTWBPZLVTWFUMUNWCWGVRCTURWCWGURACTDUOCTWBUPUSUTTULJWGUKVSVRVATJ + OVASQVBVCVDWDVEVFWEVRCLVROVGVHWFCLVRCOVIIVJVKVLVM $. $} ${ @@ -248850,11 +249075,11 @@ univariate polynomial evaluation map function for a (sub)ring is a ring absabv $p |- abs e. ( AbsVal ` CCfld ) $= ( vx vy cabs ccnfld cfv wcel wtru caddc cmul cc0 wceq a1i wne wbr 3adant1 cc cv wa co ad2ant2r cabv eqidd cbs cnfldbas cplusg cnfldadd cnfldmul c0g - cmulr cnfld0 crg cnrng cr wf absf abs0 absgt0 biimpa absmul abstri isabvd - clt cle trud ) CDUAEZFGABVEPHDICJGVEUBPDUCEKGUDLHDUEEKGUFLIDUIEKGUGLJDUHE - KGUJLDUKFGULLPUMCUNGUOLJCEJKGUPLAQZPFZVFJMZJVFCEZVBNZGVGVHVJVFUQUROVGVHRZ - BQZPFZVLJMZRZVFVLISCEVIVLCEZISKZGVGVMVQVHVNVFVLUSTOVKVOVFVLHSCEVIVPHSVCNZ - GVGVMVRVHVNVFVLUTTOVAVD $. + cmulr cnfld0 crg cnring cr wf absf clt absgt0 biimpa absmul abstri isabvd + abs0 cle trud ) CDUAEZFGABVEPHDICJGVEUBPDUCEKGUDLHDUEEKGUFLIDUIEKGUGLJDUH + EKGUJLDUKFGULLPUMCUNGUOLJCEJKGVBLAQZPFZVFJMZJVFCEZUPNZGVGVHVJVFUQUROVGVHR + ZBQZPFZVLJMZRZVFVLISCEVIVLCEZISKZGVGVMVQVHVNVFVLUSTOVKVOVFVLHSCEVIVPHSVCN + ZGVGVMVRVHVNVFVLUTTOVAVD $. $( The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) $) @@ -248949,16 +249174,16 @@ univariate polynomial evaluation map function for a (sub)ring is a ring $( Lemma for ~ gzrngunit . (Contributed by Mario Carneiro, 4-Dec-2014.) $) gzrngunitlem $p |- ( A e. ( Unit ` Z ) -> 1 <_ ( abs ` A ) ) $= ( cfv wcel c1 cle wbr c2 cexp co cc0 cgz ccnfld wceq wb gzsubrg ax-mp syl - mp2b cui cabs sq1 cn ax-1ne0 csubrg crg subrgrng eqid csubg c0g subrgsubg - cnfld0 subg0 cur cnfld1 subrg1 0unit nemtbir wn cn0 wo subrgbas gzabssqcl - cbs unitcl elnn0 sylib ord gzcn abscld recnd sqeq0 abs00ad biimpcd sylbid - cc eleq1 syld mt3i nnge1d syl5eqbr cr absge0d wa 1re 0le1 mpanl12 syl2anc - le2sq mpbird ) ABUADZEZFAUBDZGHZFIJKZWNIJKZGHZWMWPFWQGUCWMWQWMWQUDEZLWLEZ - WTFLUEMNUFDEZBUGEWTFLOPQMNBCUHBWLFLWLUIZXAMNUJDELBUKDOQMNULMNBLCUMUNTXAFB - UODOQMNBFCUPUQRURTUSWMWSUTWQLOZWTWMWSXCWMWQVAEZWSXCVBWMAMEZXDMBWLAXAMBVED - OQMNBCVCRXBVFZAVDSWQVGVHVIWMXCWNLOZWTWMWNVQEXCXGPWMWNWMAWMXEAVQEXFAVJSZVK - ZVLWNVMSWMXGALOZWTWMAXHVNXJWMWTALWLVRVOVPVPVSVTWAWBWMWNWCEZLWNGHZWOWRPZXI - WMAXHWDFWCELFGHXKXLWEXMWFWGFWNWJWHWIWK $. + mp2b cui cabs sq1 cn ax-1ne0 csubrg subrgring eqid csubg subrgsubg cnfld0 + crg c0g subg0 cur cnfld1 subrg1 0unit nemtbir wn cn0 cbs unitcl gzabssqcl + wo subrgbas elnn0 sylib ord gzcn abscld recnd sqeq0 abs00ad eleq1 biimpcd + cc sylbid syld mt3i nnge1d syl5eqbr cr absge0d 0le1 le2sq mpanl12 syl2anc + wa 1re mpbird ) ABUADZEZFAUBDZGHZFIJKZWNIJKZGHZWMWPFWQGUCWMWQWMWQUDEZLWLE + ZWTFLUEMNUFDEZBULEWTFLOPQMNBCUGBWLFLWLUHZXAMNUIDELBUMDOQMNUJMNBLCUKUNTXAF + BUODOQMNBFCUPUQRURTUSWMWSUTWQLOZWTWMWSXCWMWQVAEZWSXCVEWMAMEZXDMBWLAXAMBVB + DOQMNBCVFRXBVCZAVDSWQVGVHVIWMXCWNLOZWTWMWNVQEXCXGPWMWNWMAWMXEAVQEXFAVJSZV + KZVLWNVMSWMXGALOZWTWMAXHVNXJWMWTALWLVOVPVRVRVSVTWAWBWMWNWCEZLWNGHZWOWRPZX + IWMAXHWDFWCELFGHXKXLWIXMWJWEFWNWFWGWHWK $. $( The units on ` ZZ [ _i ] ` are the gaussian integers with norm ` 1 ` . (Contributed by Mario Carneiro, 4-Dec-2014.) $) @@ -248967,23 +249192,23 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ( cfv wcel cgz cabs c1 wceq ccnfld gzsubrg ax-mp eqid cle wbr cdiv co cc0 cc a1i cui wa csubrg cbs subrgbas unitcl cinvr subrginv mpan wne gzcn syl 0red 1re abscld 0lt1 gzrngunitlem ltletrd gt0ne0d abs00ad necon3bid mpbid - cr clt cnfldinv syl2anc eqtr3d subrgrng unitinvcl eqeltrrd ax-1cn absdivd - crg breqtrd 1div1e1 eqcomi oveq1i 3brtr4g wb lerec syl22anc mpbird letri3 - abs1 sylancl mpbir2and jca cdif adantr simpr ax-1ne0 eqnetrd fveq2 syl6eq - csn abs0 necon3i eldifsn sylanbrc simpl ccj cmul c2 cexp absvalsqd oveq1d - sq1 cjcld divcan3d 3eqtr2d gzcjcl cnfldbas cnfld0 cndrng drngui subrgunit - eqeltrd w3a syl3anbrc impbii ) ABUADZEZAFEZAGDZHIZUBZYBYCYEFBYAAFJUCDEZFB - UDDIKFJBCUELYAMZUFZYBYEYDHNOZHYDNOZYBYJHHPQZHYDPQZNOZYBHHGDZYDPQZYLYMNYBH - HAPQZGDZYPNYBYQYAEHYRNOYBABUGDZDZYQYAYBAJUGDZDZYTYQYGYBUUBYTIKFJBYAUUAYSA - CUUAMZYHYSMZUHUIYBASEZARUJZUUBYQIZYBYCUUEYIAUKZULZYBYDRUJZUUFYBYDYBRHYDYB - UMHVCEZYBUNTZYBAUUIUOZRHVDOZYBUPTZABCUQZURZUSYBYDRARYBAUUIUTVAVBZAVEZVFVG - BVMEZYBYTYAEYGUUTKFJBCVHLBYAYSAYHUUDVIUIVJYQBCUQULYBHAHSEYBVKTUUIUURVLVNV - OHYOYDPYOHWDVPVQVRYBYDVCEZRYDVDOUUKUUNYJYNVSUUMUUQUULUUOYDHVTWAWBUUPYBUVA - UUKYEYJYKUBVSUUMUNYDHWCWEWFWGYFASRWOWHZEZYCUUBFEZYBYFUUEUUFUVCYCUUEYEUUHW - IZYFUUJUUFYFYDHRYCYEWJZHRUJYFWKTWLARYDRARIYDRGDRARGWMWPWNWQULZASRWRWSYCYE - WTYFUUBAXADZFYFUUBYQAUVHXBQZAPQUVHYFUUEUUFUUGUVEUVGUUSVFYFUVIHAPYFYDXCXDQ - ZUVIHYFAUVEXEYFUVJHXCXDQHYFYDHXCXDUVFXFXGWNVGXFYFUVHAYFAUVEXHUVEUVGXIXJYC - UVHFEYEAXKWIXQYGYBUVCYCUVDXRVSKFJBUVBUUAYAACSJRXLXMXNXOYHUUCXPLXSXT $. + cr clt cnfldinv syl2anc eqtr3d subrgring unitinvcl ax-1cn absdivd breqtrd + crg eqeltrrd 1div1e1 abs1 eqcomi oveq1i 3brtr4g wb syl22anc mpbird letri3 + lerec sylancl mpbir2and jca csn adantr simpr ax-1ne0 eqnetrd fveq2 syl6eq + cdif abs0 necon3i eldifsn sylanbrc simpl ccj cmul c2 absvalsqd oveq1d sq1 + cexp divcan3d 3eqtr2d gzcjcl eqeltrd w3a cnfldbas cnfld0 cndrng subrgunit + cjcld drngui syl3anbrc impbii ) ABUADZEZAFEZAGDZHIZUBZYBYCYEFBYAAFJUCDEZF + BUDDIKFJBCUELYAMZUFZYBYEYDHNOZHYDNOZYBYJHHPQZHYDPQZNOZYBHHGDZYDPQZYLYMNYB + HHAPQZGDZYPNYBYQYAEHYRNOYBABUGDZDZYQYAYBAJUGDZDZYTYQYGYBUUBYTIKFJBYAUUAYS + ACUUAMZYHYSMZUHUIYBASEZARUJZUUBYQIZYBYCUUEYIAUKZULZYBYDRUJZUUFYBYDYBRHYDY + BUMHVCEZYBUNTZYBAUUIUOZRHVDOZYBUPTZABCUQZURZUSYBYDRARYBAUUIUTVAVBZAVEZVFV + GBVMEZYBYTYAEYGUUTKFJBCVHLBYAYSAYHUUDVIUIVNYQBCUQULYBHAHSEYBVJTUUIUURVKVL + VOHYOYDPYOHVPVQVRVSYBYDVCEZRYDVDOUUKUUNYJYNVTUUMUUQUULUUOYDHWDWAWBUUPYBUV + AUUKYEYJYKUBVTUUMUNYDHWCWEWFWGYFASRWHWOZEZYCUUBFEZYBYFUUEUUFUVCYCUUEYEUUH + WIZYFUUJUUFYFYDHRYCYEWJZHRUJYFWKTWLARYDRARIYDRGDRARGWMWPWNWQULZASRWRWSYCY + EWTYFUUBAXADZFYFUUBYQAUVHXBQZAPQUVHYFUUEUUFUUGUVEUVGUUSVFYFUVIHAPYFYDXCXG + QZUVIHYFAUVEXDYFUVJHXCXGQHYFYDHXCXGUVFXEXFWNVGXEYFUVHAYFAUVEXQUVEUVGXHXIY + CUVHFEYEAXJWIXKYGYBUVCYCUVDXLVTKFJBUVBUUAYAACSJRXMXNXOXRYHUUCXPLXSXT $. $} ${ @@ -248996,18 +249221,18 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ( vf vx c0 wceq ccnfld cgsu co csu cfv wcel wa cc0 cc syl vn chash cn cfz cmpt c1 cv wf1o wex wi mpteq1 mpt0 syl6eq oveq2d cnfld0 gsum0 sum0 eqtr4i sumeq1 eqtr4d a1i caddc ccom cseq csupp ccntz cnfldbas cnfldadd eqid cmnd - cfn crg cnrng rngmnd mp1i adantr wf fmptd ccmn rngcmn cntzcmnf simprl wf1 - f1of1 crn cdm suppssdm fdm syl5sseq wfo f1ofo forn 3syl sseqtr4d gsumval3 - simprr sumfc fveq2 ffvelrnda f1of fvco3 fsum syl5eqr expr exlimdv expimpd - sylan wo fz1f1o mpjaod ) ABIJZKDBCUEZLMZBCDNZJZBUBOZUCPZUFXPUDMZBGUGZUHZG - UIZQZXKXOUJAXKXMICDNZXNXKXMKILMZYCXKXLIKLXKXLDICUEIDBICUKDCULUMUNYDRYCKRU - OUPCDUQURUMBICDUSUTVAAXQYAXOAXQQXTXOGAXQXTXOAXQXTQZQZXMXPVBXLXSVCZUFVDOZX - NYFBSVBXLKXSXPVKYGRVEMZRKVFOZVGUOVHYJVIZKVLPZKVJPYFVMKVNVOABVKPZYEEVPABSX - LVQZYEADBCSXLFXLVIVRVPZYFBSXLKYJVGYKYLKVSPYFVMKVTVOYOWAAXQXTWBZYFXTXRBXSW - CAXQXTWPZXRBXSWDTYFXLRVEMZBXSWEZYFXLWFZYRBXLRWGYFYNYTBJYOBSXLWHTWIYFXTXRB - XSWJYSBJYQXRBXSWKXRBXSWLWMWNYIVIWOYFXNBHUGZXLOZHNYHBCHDWQYFBUUBUAUGZXSOZX - LOZHUAXSYGXPUUAUUDXLWRYPYQYFBSUUAXLYOWSYFXRBXSVQZUUCXRPUUCYGOUUEJYFXTUUFY - QXRBXSWTTXRBUUCXLXSXAXGXBXCUTXDXEXFAYMXKYBXHEBGXITXJ $. + cfn crg cnring ringmnd mp1i adantr fmptd ccmn ringcmn cntzcmnf simprl wf1 + wf simprr f1of1 crn cdm suppssdm fdm syl5sseq wfo f1ofo sseqtr4d gsumval3 + forn 3syl sumfc fveq2 ffvelrnda f1of fsum syl5eqr expr exlimdv expimpd wo + fvco3 sylan fz1f1o mpjaod ) ABIJZKDBCUEZLMZBCDNZJZBUBOZUCPZUFXPUDMZBGUGZU + HZGUIZQZXKXOUJAXKXMICDNZXNXKXMKILMZYCXKXLIKLXKXLDICUEIDBICUKDCULUMUNYDRYC + KRUOUPCDUQURUMBICDUSUTVAAXQYAXOAXQQXTXOGAXQXTXOAXQXTQZQZXMXPVBXLXSVCZUFVD + OZXNYFBSVBXLKXSXPVKYGRVEMZRKVFOZVGUOVHYJVIZKVLPZKVJPYFVMKVNVOABVKPZYEEVPA + BSXLWCZYEADBCSXLFXLVIVQVPZYFBSXLKYJVGYKYLKVRPYFVMKVSVOYOVTAXQXTWAZYFXTXRB + XSWBAXQXTWDZXRBXSWETYFXLRVEMZBXSWFZYFXLWGZYRBXLRWHYFYNYTBJYOBSXLWITWJYFXT + XRBXSWKYSBJYQXRBXSWLXRBXSWOWPWMYIVIWNYFXNBHUGZXLOZHNYHBCHDWQYFBUUBUAUGZXS + OZXLOZHUAXSYGXPUUAUUDXLWRYPYQYFBSUUAXLYOWSYFXRBXSWCZUUCXRPUUCYGOUUEJYFXTU + UFYQXRBXSWTTXRBUUCXLXSXGXHXAXBUTXCXDXEAYMXKYBXFEBGXITXJ $. $} ${ @@ -249020,16 +249245,16 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ( vy vz cc wcel cn0 cexp co caddc cfv cmul wceq cc0 ax-mp ccnfld cv wf c1 cmpt wral cmhm expcl eqid fmptd wa expadd 3expb nn0addcl oveq2 ovex fvmpt adantl syl oveqan12d 3eqtr4d ralrimivva 0nn0 exp0 syl5eq cmnd w3a csubmnd - nn0subm submmnd crg rngmgp pm3.2i submbas cnfldbas mgpbas cplusg cnfldadd - cnrng cbs ressplusg cnfldmul mgpplusg cnfld0 subm0 rngidval ismhm mpbiran - c0g cnfld1 syl3anbrc ) BIJZKIAKBAUAZLMZUDZUBZGUAZHUAZNMZWNOZWPWNOZWQWNOZP - MZQZHKUEGKUEZRWNOZUCQZWNDCUFMJZWKAKWMIWNBWLUGWNUHZUIWKXCGHKKWKWPKJZWQKJZU - JZUJZBWRLMZBWPLMZBWQLMZPMZWSXBWKXIXJXMXPQBWPWQUKULXLWRKJZWSXMQXKXQWKWPWQU - MUQAWRWMXMKWNWLWRBLUNXHBWRLUOUPURXKXBXPQWKXIXJWTXNXAXOPAWPWMXNKWNWLWPBLUN - XHBWPLUOUPAWQWMXOKWNWLWQBLUNXHBWQLUOUPUSUQUTVAWKXEBRLMZUCRKJXEXRQVBARWMXR - KWNWLRBLUNXHBRLUOUPSBVCVDXGDVEJZCVEJZUJWOXDXFVFXSXTKTVGOZJZXSVHKDTEVISTVJ - JXTVRTCFVKSVLGHKINPDCWNUCRYBKDVSOQVHKDTEVMSITCFVNVOYBNDVPOQVHKNTDYAEVQVTS - TPCFWAWBYBRDWHOQVHKDTREWCWDSTUCCFWIWEWFWGWJ $. + nn0subm submmnd crg cnring ringmgp pm3.2i cnfldbas mgpbas cplusg cnfldadd + cbs submbas ressplusg cnfldmul mgpplusg c0g cnfld0 subm0 rngidval mpbiran + cnfld1 ismhm syl3anbrc ) BIJZKIAKBAUAZLMZUDZUBZGUAZHUAZNMZWNOZWPWNOZWQWNO + ZPMZQZHKUEGKUEZRWNOZUCQZWNDCUFMJZWKAKWMIWNBWLUGWNUHZUIWKXCGHKKWKWPKJZWQKJ + ZUJZUJZBWRLMZBWPLMZBWQLMZPMZWSXBWKXIXJXMXPQBWPWQUKULXLWRKJZWSXMQXKXQWKWPW + QUMUQAWRWMXMKWNWLWRBLUNXHBWRLUOUPURXKXBXPQWKXIXJWTXNXAXOPAWPWMXNKWNWLWPBL + UNXHBWPLUOUPAWQWMXOKWNWLWQBLUNXHBWQLUOUPUSUQUTVAWKXEBRLMZUCRKJXEXRQVBARWM + XRKWNWLRBLUNXHBRLUOUPSBVCVDXGDVEJZCVEJZUJWOXDXFVFXSXTKTVGOZJZXSVHKDTEVIST + VJJXTVKTCFVLSVMGHKINPDCWNUCRYBKDVROQVHKDTEVSSITCFVNVOYBNDVPOQVHKNTDYAEVQV + TSTPCFWAWBYBRDWCOQVHKDTREWDWESTUCCFWHWFWIWGWJ $. $} ${ @@ -249038,21 +249263,21 @@ univariate polynomial evaluation map function for a (sub)ring is a ring (Contributed by Thierry Arnoux, 1-May-2018.) $) nn0srg $p |- ( CCfld |`s NN0 ) e. SRing $= ( vx vy vz ccnfld cn0 co wcel cfv cmnd caddc cmul wceq wa wral ax-mp eqid - cc0 cvv cc nn0sscn cress csrg ccmn cmgp csubmnd crg cnrng nn0subm submcmn - cv rngcmn mp2an nn0ex mgpress wss c1 w3a nn0mulcl rgen2 3pm3.2i wb rngmgp - cnfldbas mgpbas cnfld1 rngidval cnfldmul mgpplusg issubm submmnd eqeltrri - 1nn0 mpbir simpl nn0cn syl simprl sseldi simprr adddid adddird ralrimivva - jca mul02d mul01d rgen cbs ressbas2 cnfldadd ressplusg cmulr ressmulr c0g - cplusg rngmnd 0nn0 cnfld0 ress0g mp3an issrg mpbir3an ) DEUAFZUBGXBUCGZXB - UDHZIGAUJZBUJZCUJZJFKFXEXFKFZXEXGKFZJFLZXEXFJFXGKFXIXFXGKFJFLZMZCENBENZQX - EKFQLZXEQKFQLZMZMZAENDUCGZEDUEHGXCDUFGZXRUGDUKOZUHEDXBXBPZUIULDUDHZEUAFZX - DIXRERGZYCXDLXTUMEDXBYBUCRYAYBPZUNULEYBUEHGZYCIGYFESUOZUPEGZXHEGZBENAENZU - QZYGYHYJTVLYIABEEXEXFURUSUTYBIGZYFYKVAXSYLUGDYBYEVBOABSKEYBUPSDYBYEVCVDDU - PYBYEVEVFDKYBYEVGVHVIOVMEYCYBYCPVJOVKXQAEXEEGZXMXPYMXLBCEEYMXFEGZXGEGZMZM - ZXJXKYQXEXFXGYQYMXESGYMYPVNXEVOZVPZYQESXFTYMYNYOVQVRZYQESXGTYMYNYOVSVRZVT - YQXEXFXGYSYTUUAWAWCWBYMXNXOYMXEYRWDYMXEYRWEWCWCWFABCEJXBKXDQYGEXBWGHLTESX - BDYAVCWHOXDPYDJXBWNHLUMEJDXBRYAWIWJOYDKXBWKHLUMEDXBKRYAVGWLODIGZQEGYGQXBW - MHLXSUUBUGDWOOWPTESDXBQYAVCWQWRWSWTXA $. + cc0 cvv cc nn0sscn cress csrg ccmn cmgp cv csubmnd cnring ringcmn nn0subm + crg submcmn mp2an nn0ex mgpress wss c1 w3a 1nn0 nn0mulcl rgen2 3pm3.2i wb + ringmgp cnfldbas mgpbas cnfld1 rngidval cnfldmul mgpplusg issubm eqeltrri + mpbir submmnd simpl nn0cn syl simprl sseldi simprr adddird jca ralrimivva + adddid mul02d mul01d rgen cbs ressbas2 cplusg cnfldadd ressplusg ressmulr + cmulr c0g ringmnd 0nn0 cnfld0 ress0g mp3an issrg mpbir3an ) DEUAFZUBGXBUC + GZXBUDHZIGAUEZBUEZCUEZJFKFXEXFKFZXEXGKFZJFLZXEXFJFXGKFXIXFXGKFJFLZMZCENBE + NZQXEKFQLZXEQKFQLZMZMZAENDUCGZEDUFHGXCDUJGZXRUGDUHOZUIEDXBXBPZUKULDUDHZEU + AFZXDIXRERGZYCXDLXTUMEDXBYBUCRYAYBPZUNULEYBUFHGZYCIGYFESUOZUPEGZXHEGZBENA + ENZUQZYGYHYJTURYIABEEXEXFUSUTVAYBIGZYFYKVBXSYLUGDYBYEVCOABSKEYBUPSDYBYEVD + VEDUPYBYEVFVGDKYBYEVHVIVJOVLEYCYBYCPVMOVKXQAEXEEGZXMXPYMXLBCEEYMXFEGZXGEG + ZMZMZXJXKYQXEXFXGYQYMXESGYMYPVNXEVOZVPZYQESXFTYMYNYOVQVRZYQESXGTYMYNYOVSV + RZWCYQXEXFXGYSYTUUAVTWAWBYMXNXOYMXEYRWDYMXEYRWEWAWAWFABCEJXBKXDQYGEXBWGHL + TESXBDYAVDWHOXDPYDJXBWIHLUMEJDXBRYAWJWKOYDKXBWMHLUMEDXBKRYAVHWLODIGZQEGYG + QXBWNHLXSUUBUGDWOOWPTESDXBQYAVDWQWRWSWTXA $. $} ${ @@ -249061,26 +249286,26 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ~ subcmn ). (Contributed by Thierry Arnoux, 6-Sep-2018.) $) rge0srg $p |- ( CCfld |`s ( 0 [,) +oo ) ) e. SRing $= ( vx vy vz ccnfld cc0 cpnf co wcel cfv cmnd caddc cmul wceq wa wral ax-mp - cc cnfldbas c1 cvv cico cress csrg ccmn cmgp csubmnd crg cnrng rngcmn wss - cv cr rge0ssre ax-resscn sstri 0e0icopnf ge0addcl rgen2 3pm3.2i wb rngmnd + cc cnfldbas c1 cvv cico cress csrg ccmn cv csubmnd crg cnring ringcmn wss + cmgp cr rge0ssre ax-resscn sstri 0e0icopnf ge0addcl rgen2 3pm3.2i ringmnd w3a cnfld0 cnfldadd issubm mpbir eqid submcmn mp2an cle wbr clt 1re ltpnf - 0le1 cxr 0re pnfxr elico2 mpbir3an ge0mulcl rngmgp mgpbas cnfld1 rngidval - cnfldmul mp2b submmnd simpll sseldi simplr simpr adddid adddird ralrimiva - mgpplusg jca sseli mul02d mul01d jca32 rgen ressbas2 cnfldex ovex mgpress - cbs cplusg ressplusg cmulr ressmulr c0g ress0g mp3an issrg ) DEFUAGZUBGZU - CHXQUDHZDUEIZXPUBGZJHZAUKZBUKZCUKZKGLGYBYCLGZYBYDLGZKGMZYBYCKGZYDLGYFYCYD - LGKGMZNZCXPOZBXPOZEYBLGEMZYBELGEMZNNZAXPODUDHZXPDUFIHZXRDUGHZYPUHDUIPYQXP - QUJZEXPHZYHXPHZBXPOAXPOZVBZYSYTUUBXPULQUMUNUOZUPUUAABXPXPYBYCUQURUSDJHZYQ - UUCUTYRUUEUHDVAPZABQKXPDERVCVDVEPVFXPDXQXQVGZVHVIXPXSUFIHZYAUUHYSSXPHZYEX - PHZBXPOAXPOZVBZYSUUIUUKUUDUUISULHZESVJVKZSFVLVKZVMVOUUMUUOVMSVNPEULHFVPHU - UIUUMUUNUUOVBUTVQVREFSVSVIVTUUJABXPXPYBYCWAURUSYRXSJHUUHUULUTUHDXSXSVGZWB - ABQLXPXSSQDXSUUPRWCDSXSUUPWDWEDLXSUUPWFWPVEWGVFXPXTXSXTVGWHPYOAXPYBXPHZYL - YMYNUUQYKBXPUUQYCXPHZNZYJCXPUUSYDXPHZNZYGYIUVAYBYCYDUVAXPQYBUUDUUQUURUUTW - IWJZUVAXPQYCUUDUUQUURUUTWKWJZUVAXPQYDUUDUUSUUTWLWJZWMUVAYBYCYDUVBUVCUVDWN - WQWOWOUUQYBXPQYBUUDWRZWSUUQYBUVEWTXAXBABCXPKXQLXTEYSXPXQXGIMUUDXPQXQDUUGR - XCPDTHXPTHZXTXQUEIMXDEFUAXEZXPDXQXSTTUUGUUPXFVIUVFKXQXHIMUVGXPKDXQTUUGVDX - IPUVFLXQXJIMUVGXPDXQLTUUGWFXKPUUEYTYSEXQXLIMUUFUPUUDXPQDXQEUUGRVCXMXNXOVT - $. + wb 0le1 cxr pnfxr elico2 mpbir3an ge0mulcl ringmgp mgpbas cnfld1 rngidval + 0re cnfldmul mgpplusg mp2b submmnd simpll sseldi simplr simpr adddird jca + adddid ralrimiva sseli mul02d mul01d jca32 rgen cbs ressbas2 cnfldex ovex + mgpress cplusg ressplusg cmulr ressmulr c0g ress0g mp3an issrg ) DEFUAGZU + BGZUCHXQUDHZDUKIZXPUBGZJHZAUEZBUEZCUEZKGLGYBYCLGZYBYDLGZKGMZYBYCKGZYDLGYF + YCYDLGKGMZNZCXPOZBXPOZEYBLGEMZYBELGEMZNNZAXPODUDHZXPDUFIHZXRDUGHZYPUHDUIP + YQXPQUJZEXPHZYHXPHZBXPOAXPOZVAZYSYTUUBXPULQUMUNUOZUPUUAABXPXPYBYCUQURUSDJ + HZYQUUCVNYRUUEUHDUTPZABQKXPDERVBVCVDPVEXPDXQXQVFZVGVHXPXSUFIHZYAUUHYSSXPH + ZYEXPHZBXPOAXPOZVAZYSUUIUUKUUDUUISULHZESVIVJZSFVKVJZVLVOUUMUUOVLSVMPEULHF + VPHUUIUUMUUNUUOVAVNWEVQEFSVRVHVSUUJABXPXPYBYCVTURUSYRXSJHUUHUULVNUHDXSXSV + FZWAABQLXPXSSQDXSUUPRWBDSXSUUPWCWDDLXSUUPWFWGVDWHVEXPXTXSXTVFWIPYOAXPYBXP + HZYLYMYNUUQYKBXPUUQYCXPHZNZYJCXPUUSYDXPHZNZYGYIUVAYBYCYDUVAXPQYBUUDUUQUUR + UUTWJWKZUVAXPQYCUUDUUQUURUUTWLWKZUVAXPQYDUUDUUSUUTWMWKZWPUVAYBYCYDUVBUVCU + VDWNWOWQWQUUQYBXPQYBUUDWRZWSUUQYBUVEWTXAXBABCXPKXQLXTEYSXPXQXCIMUUDXPQXQD + UUGRXDPDTHXPTHZXTXQUKIMXEEFUAXFZXPDXQXSTTUUGUUPXGVHUVFKXQXHIMUVGXPKDXQTUU + GVCXIPUVFLXQXJIMUVGXPDXQLTUUGWFXKPUUEYTYSEXQXLIMUUFUPUUDXPQDXQEUUGRVBXMXN + XOVS $. $} $( @@ -249095,7 +249320,7 @@ According to Wikipedia ("Integer", 25-May-2019, characterizes the ring ` Z ` ." In set.mm, there was no explicit definition for the ring of integers until June 2019, but it was denoted by ` ( CCfld |``s ZZ ) ` , the field of complex numbers restricted to the -integers. In ~ zringrng it is shown that this restriction is a ring (it is +integers. In ~ zringring it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in ~ zringlpir ), and ~ zringbas shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition ~ df-zring of the ring of integers. @@ -249123,18 +249348,18 @@ According to Wikipedia ("Integer", 25-May-2019, $( The ring of integers is a ring. (Contributed by AV, 20-May-2019.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 13-Jun-2019.) $) - zringrng $p |- ZZring e. Ring $= - ( zring ccrg wcel crg zringcrng crngrng ax-mp ) ABCADCEAFG $. + zringring $p |- ZZring e. Ring $= + ( zring ccrg wcel crg zringcrng crngring ax-mp ) ABCADCEAFG $. $( The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.) $) zringabl $p |- ZZring e. Abel $= - ( zring crg wcel cabl zringrng rngabl ax-mp ) ABCADCEAFG $. + ( zring crg wcel cabl zringring ringabl ax-mp ) ABCADCEAFG $. $( The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.) $) zringgrp $p |- ZZring e. Grp $= - ( zring crg wcel cgrp zringrng rnggrp ax-mp ) ABCADCEAFG $. + ( zring crg wcel cgrp zringring ringgrp ax-mp ) ABCADCEAFG $. $( The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.) $) @@ -249170,9 +249395,9 @@ According to Wikipedia ("Integer", 25-May-2019, $( The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) $) zring0 $p |- 0 = ( 0g ` ZZring ) $= - ( ccnfld cmnd wcel cc0 cz cc wss zring c0g cfv wceq ccrg crg cncrng crngrng - rngmnd mp2b 0z zsscn df-zring cnfldbas cnfld0 ress0g mp3an ) ABCZDECEFGDHIJ - KALCAMCUENAOAPQRSEFAHDTUAUBUCUD $. + ( ccnfld cmnd wcel cc0 cz cc wss zring c0g cfv wceq ccrg crg cncrng ringmnd + crngring mp2b 0z zsscn df-zring cnfldbas cnfld0 ress0g mp3an ) ABCZDECEFGDH + IJKALCAMCUENAPAOQRSEFAHDTUAUBUCUD $. $( The multiplicative neutral element of the ring of integers (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) $) @@ -249218,9 +249443,9 @@ According to Wikipedia ("Integer", 25-May-2019, Arnoux, 1-Nov-2017.) (New usage is discouraged.) Use ~ zring instead. $) zrng0 $p |- 0 = ( 0g ` Z ) $= - ( ccnfld cmnd wcel cc0 cz wss c0g cfv wceq ccrg crg cncrng crngrng rngmnd - cc mp2b 0z zsscn cnfldbas cnfld0 ress0g mp3an ) CDEZFGEGQHFAIJKCLECMEUENC - OCPRSTGQCAFBUAUBUCUD $. + ( ccnfld cmnd wcel cc0 cz cc wss c0g cfv wceq crg cncrng crngring ringmnd + ccrg mp2b 0z zsscn cnfldbas cnfld0 ress0g mp3an ) CDEZFGEGHIFAJKLCQECMEUE + NCOCPRSTGHCAFBUAUBUCUD $. $( The multiplicative neutral element of the ring of integers (Contributed by Thierry Arnoux, 1-Nov-2017.) (New usage is discouraged.) Use @@ -249271,14 +249496,14 @@ According to Wikipedia ("Integer", 25-May-2019, ( va cc0 wne cn wcel wa wceq eleq1 syl5ibrcom zring ccnfld cz eqid adantr cfv syl cv cin c0 cabs simplr cminusg csubg csubrg zsubrg subrgsubg ax-mp cneg wss zringbas lidlss sselda df-zring subginv sylancr cc zcnd cnfldneg - clidl eqtr3d crg zringrng simpr lidlnegcl syl3anc eqeltrrd wo zred absord - a1i mpjaod nnabscl sylan elind ne0i csn wrex zring0 lidlnz r19.29a ) AEUA - ZFGZBHUBZUCGZEBAWEBIZJZWFJZWEUDSZWGIWHWKBHWLWKWLWEKZWLBIZWLWEULZKZWKWNWMW - IAWIWFUEWLWEBLMWKWNWPWOBIZWJWQWFWJWENUFSZSZWOBWJWEOUFSZSZWSWOWJPOUGSIZWEP - IZXAWSKPOUHSIXBUIPOUJUKABPWEABNVCSZIZBPUMCPBXDNUNXDQZUOTUPZPONWTWRWEUQWTQ - WRQZURUSWJWEUTIXAWOKWJWEXGVAWEVBTVDWJNVEIZXEWIWSBIXIWJVFVNAXEWICRAWIVGNXD - BWRWEXFXHVHVIVJRWLWOBLMWJWMWPVKWFWJWEWJWEXGVLVMRVOWJXCWFWLHIXGWEVPVQVRWGW - LVSTAXIXEBFVTGWFEBWAXIAVFVNCDENXDBFXFWBWCVIWD $. + clidl eqtr3d crg zringring a1i simpr lidlnegcl syl3anc eqeltrrd wo absord + zred mpjaod nnabscl sylan elind ne0i csn wrex zring0 lidlnz r19.29a ) AEU + AZFGZBHUBZUCGZEBAWEBIZJZWFJZWEUDSZWGIWHWKBHWLWKWLWEKZWLBIZWLWEULZKZWKWNWM + WIAWIWFUEWLWEBLMWKWNWPWOBIZWJWQWFWJWENUFSZSZWOBWJWEOUFSZSZWSWOWJPOUGSIZWE + PIZXAWSKPOUHSIXBUIPOUJUKABPWEABNVCSZIZBPUMCPBXDNUNXDQZUOTUPZPONWTWRWEUQWT + QWRQZURUSWJWEUTIXAWOKWJWEXGVAWEVBTVDWJNVEIZXEWIWSBIXIWJVFVGAXEWICRAWIVHNX + DBWRWEXFXHVIVJVKRWLWOBLMWJWMWPVLWFWJWEWJWEXGVNVMRVOWJXCWFWLHIXGWEVPVQVRWG + WLVSTAXIXEBFVTGWFEBWAXIAVFVGCDENXDBFXFWBWCVJWD $. zringlpirlem.g $e |- G = sup ( ( I i^i NN ) , RR , `' < ) $. $( Lemma for ~ zringlpir . A nonzero ideal of integers contains the least @@ -249299,18 +249524,18 @@ According to Wikipedia ("Integer", 25-May-2019, cin ccnv csup cuz c0 nnuz sseqtri zringlpirlem1 infmssuzcl sylancr sseldi wne syl5eqel nnrpd modlt zmodcld nn0red nnred ltnled mpbid cdiv cneg cmin wa cfl nncnd nndivred flcld mulcld negsubd znegcld mulcomd mulneg2d eqtrd - zcnd oveq2d modval 3eqtr4rd crg zringrng zringlpirlem2 zringmulr syl22anc - lidlmcl zringplusg lidlacl eqeltrd adantr simpr elind infmssuzle syl5eqbr - a1i mtand cn0 elnn0 sylib orel1 sylc wb dvdsval3 mpbird ) ABDUAIZDBUBJZUD - UCZAYBKLZUEYDYCUFZYCAYDBYBMIZAYBBUGIZYFUEADUHLZBUILZYGADACNDACOUJPZLZCNUK - ENCYJOULYJUMZUNUOHUPZUQZABACKUSZKBCKURZABYOUHUGUTVAZYOGAYOQVBPZUKZYOVCVJY - QYOLYOKYRYPVDVEZACEFVFYOQVGVHVKVIZVLZDBVMRAYBBAYBADBYMUUAVNZVOABUUAVPVQVR - AYDWBZBYQYBMGUUDYSYBYOLYQYBMIYTUUDCKYBAYBCLYDAYBDDBVSJZWCPZVTZBSJZTJZCADB - UUFSJZVTZTJDUUJWAJZUUIYBADUUJADYMWMABUUFABUUAWDZAUUFAUUEADBYNUUAWEWFZWMZW - GWHAUUHUUKDTAUUHBUUGSJUUKAUUGBAUUGAUUFUUNWIZWMUUMWJABUUFUUMUUOWKWLWNAYHYI - YBUULUCYNUUBDBWORWPAOWQLZYKDCLUUHCLZUUICLUUQAWRXKZEHAUUQYKUUGNLBCLUURUUSE - UUPABCEFGWSNOSYJCUUGBYLULWTXBXATOYJCDUUHYLXCXDXAXEXFAYDXGXHYBYOQXIVHXJXLA - YBXMLYEUUCYBXNXOYDYCXPXQABKLDNLYAYCXRUUAYMBDXSRXT $. + zcnd oveq2d modval 3eqtr4rd crg zringring zringlpirlem2 zringmulr lidlmcl + a1i syl22anc zringplusg lidlacl eqeltrd adantr simpr elind syl5eqbr mtand + infmssuzle cn0 elnn0 sylib orel1 sylc wb dvdsval3 mpbird ) ABDUAIZDBUBJZU + DUCZAYBKLZUEYDYCUFZYCAYDBYBMIZAYBBUGIZYFUEADUHLZBUILZYGADACNDACOUJPZLZCNU + KENCYJOULYJUMZUNUOHUPZUQZABACKUSZKBCKURZABYOUHUGUTVAZYOGAYOQVBPZUKZYOVCVJ + YQYOLYOKYRYPVDVEZACEFVFYOQVGVHVKVIZVLZDBVMRAYBBAYBADBYMUUAVNZVOABUUAVPVQV + RAYDWBZBYQYBMGUUDYSYBYOLYQYBMIYTUUDCKYBAYBCLYDAYBDDBVSJZWCPZVTZBSJZTJZCAD + BUUFSJZVTZTJDUUJWAJZUUIYBADUUJADYMWMABUUFABUUAWDZAUUFAUUEADBYNUUAWEWFZWMZ + WGWHAUUHUUKDTAUUHBUUGSJUUKAUUGBAUUGAUUFUUNWIZWMUUMWJABUUFUUMUUOWKWLWNAYHY + IYBUULUCYNUUBDBWORWPAOWQLZYKDCLUUHCLZUUICLUUQAWRXBZEHAUUQYKUUGNLBCLUURUUS + EUUPABCEFGWSNOSYJCUUGBYLULWTXAXCTOYJCDUUHYLXDXEXCXFXGAYDXHXIYBYOQXLVHXJXK + AYBXMLYEUUCYBXNXOYDYCXPXQABKLDNLYAYCXRUUAYMBDXSRXT $. $} ${ @@ -249318,15 +249543,15 @@ According to Wikipedia ("Integer", 25-May-2019, $( The integers are a principal ideal ring but not a field. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) $) zringlpir $p |- ZZring e. LPIR $= - ( vx vy vz zring clpir wcel crg clidl cfv clpidl wss zringrng cv cdivides - cc0 wa wbr wral simpr eqid csn eleq1 wne wrex cn cin cr clt zringlpirlem2 - ccnv csup simpl simpll simplr zringlpirlem3 ralrimiva wceq ralbidv rspcev - wel breq1 syl2anc wb dvdsrzring lpigen mpan adantr mpbird zring0 pm2.61ne - lpi0 mp1i ssriv islpir2 mpbir2an ) DEFDGFZDHIZDJIZKLAVQVRAMZVQFZVSVRFZOUA - ZVRFZVSWBVSWBVRUBVTVSWBUCZPZWABMZCMZNQZCVSRZBVSUDZWEVSUEUFUGUHUJUKZVSFWKW - GNQZCVSRZWJWEWKVSVTWDULVTWDSWKTZUIWEWLCVSWECAUTZPWKVSWGVTWDWOUMVTWDWOUNWN - WEWOSUOUPWIWMBWKVSWFWKUQWHWLCVSWFWKWGNVAURUSVBVTWAWJVCZWDVPVTWPLBCNVRDVQV - SVQTZVRTZVDVEVFVGVHVPWCVTLVRDOWRVIVKVLVJVMVRDVQWRWQVNVO $. + ( vx vy vz zring clpir crg clidl cfv clpidl wss zringring cv cc0 cdivides + wcel wa wbr wral simpr eqid csn eleq1 wne wrex cn cin clt ccnv csup simpl + cr zringlpirlem2 simpll simplr zringlpirlem3 ralrimiva wceq breq1 ralbidv + rspcev syl2anc wb dvdsrzring lpigen mpan adantr mpbird lpi0 mp1i pm2.61ne + wel zring0 ssriv islpir2 mpbir2an ) DEODFOZDGHZDIHZJKAVQVRALZVQOZVSVROZMU + AZVROZVSWBVSWBVRUBVTVSWBUCZPZWABLZCLZNQZCVSRZBVSUDZWEVSUEUFUKUGUHUIZVSOWK + WGNQZCVSRZWJWEWKVSVTWDUJVTWDSWKTZULWEWLCVSWECAVKZPWKVSWGVTWDWOUMVTWDWOUNW + NWEWOSUOUPWIWMBWKVSWFWKUQWHWLCVSWFWKWGNURUSUTVAVTWAWJVBZWDVPVTWPKBCNVRDVQ + VSVQTZVRTZVCVDVEVFVGVPWCVTKVRDMWRVLVHVIVJVMVRDVQWRWQVNVO $. $( The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.) $) @@ -249350,18 +249575,18 @@ According to Wikipedia ("Integer", 25-May-2019, of ~ zringlpirlem1 as of 9-Jun-2019. (New usage is discouraged.) $) zlpirlem1 $p |- ( ph -> ( I i^i NN ) =/= (/) ) $= ( va cc0 wne cn wcel cfv cz ccnfld zsubrg ax-mp eqid wceq cc adantr cv c0 - cin crg clidl csn csubrg subrgrng a1i c0g cnfld0 subrg0 lidlnz syl3anc wa - wrex cabs cneg simplr eleq1 syl5ibrcom cminusg subrgsubg wss cbs cnfldbas - csubg zsscn ressbas2 lidlssOLD sylancr sselda subginv cnfldneg syl eqtr3d - zcnd simpr lidlnegcl eqeltrrd zred absord mpjaod nnabscl sylan elind ne0i - wo ex rexlimdva mpd ) AGUAZHIZGBUPZBJUCZUBIZACUDKZBCUELZKZBHUFIWNWQAMNUGL - KZWQOMNCDUHPZUIEFGCWRBHWRQZWTHCUJLROMNCHDUKULPUMUNAWMWPGBAWLBKZUOZWMWPXDW - MUOZWLUQLZWOKWPXEBJXFXEXFWLRZXFBKZXFWLURZRZXEXHXGXCAXCWMUSXFWLBUTVAXEXHXJ - XIBKZXDXKWMXDWLCVBLZLZXIBXDWLNVBLZLZXMXIXDMNVGLKZWLMKZXOXMRWTXPOMNVCPABMW - LAWQWSBMVDXAEMBWRUDCMSVDMCVELRVHMSCNDVFVIPXBVJVKVLZMNCXNXLWLDXNQXLQZVMVKX - DWLSKXOXIRXDWLXRVQWLVNVOVPXDWQWSXCXMBKWQXDXAUIAWSXCETAXCVRCWRBXLWLXBXSVSU - NVTTXFXIBUTVAXDXGXJWHWMXDWLXDWLXRWAWBTWCXDXQWMXFJKXRWLWDWEWFWOXFWGVOWIWJW - K $. + wrex cin crg csn csubrg subrgring a1i c0g cnfld0 subrg0 lidlnz syl3anc wa + clidl cabs cneg simplr eleq1 syl5ibrcom cminusg csubg subrgsubg wss zsscn + cbs cnfldbas ressbas2 lidlssOLD sylancr sselda subginv cnfldneg syl simpr + zcnd eqtr3d lidlnegcl eqeltrrd wo zred absord nnabscl sylan elind ne0i ex + mpjaod rexlimdva mpd ) AGUAZHIZGBUCZBJUDZUBIZACUEKZBCUPLZKZBHUFIWNWQAMNUG + LKZWQOMNCDUHPZUIEFGCWRBHWRQZWTHCUJLROMNCHDUKULPUMUNAWMWPGBAWLBKZUOZWMWPXD + WMUOZWLUQLZWOKWPXEBJXFXEXFWLRZXFBKZXFWLURZRZXEXHXGXCAXCWMUSXFWLBUTVAXEXHX + JXIBKZXDXKWMXDWLCVBLZLZXIBXDWLNVBLZLZXMXIXDMNVCLKZWLMKZXOXMRWTXPOMNVDPABM + WLAWQWSBMVEXAEMBWRUECMSVEMCVGLRVFMSCNDVHVIPXBVJVKVLZMNCXNXLWLDXNQXLQZVMVK + XDWLSKXOXIRXDWLXRVQWLVNVOVRXDWQWSXCXMBKWQXDXAUIAWSXCETAXCVPCWRBXLWLXBXSVS + UNVTTXFXIBUTVAXDXGXJWAWMXDWLXDWLXRWBWCTWIXDXQWMXFJKXRWLWDWEWFWOXFWGVOWHWJ + WK $. zlpirlem.g $e |- G = sup ( ( I i^i NN ) , RR , `' < ) $. $( Lemma for ~ zlpir . A nonzero ideal of integers contains the least @@ -249380,25 +249605,25 @@ According to Wikipedia ("Integer", 25-May-2019, (New usage is discouraged.) $) zlpirlem3 $p |- ( ph -> G || X ) $= ( wbr co wceq cn wcel cz cfv ccnfld cmul caddc cdivides cmo cc0 wn wo cle - clt cr crp crg clidl wss csubrg zsubrg subrgrng mp1i cc cnfldbas ressbas2 - cbs zsscn ax-mp eqid lidlssOLD syl2anc sseldd zred cin inss2 ccnv csup c1 - cuz c0 wne nnuz sseqtri zlpirlem1 infmssuzcl sylancr syl5eqel nnrpd modlt - sseldi zmodcld nn0red nnred ltnled mpbid wa cdiv cfl cneg cmin zcnd nncnd - nndivred flcld mulcld negsubd znegcld mulcomd mulneg2d 3eqtr4rd zlpirlem2 - eqtrd oveq2d modval cvv cmulr cnfldmul ressmulr lidlmcl syl22anc cnfldadd - zex cplusg ressplusg lidlacl adantr simpr elind infmssuzle syl5eqbr mtand - eqeltrd cn0 elnn0 sylib orel1 sylc wb dvdsval3 mpbird ) ABDUAKZDBUBLZUCMZ - AYPNOZUDYRYQUEZYQAYRBYPUFKZAYPBUGKZYTUDADUHOZBUIOZUUAADACPDAEUJOZCEUKQZOZ - CPULPRUMQOUUDAUNPREFUOUPZGPCUUEUJEPUQULPEUTQMVAPUQERFURUSVBZUUEVCZVDVEJVF - ZVGZABACNVHZNBCNVIZABUULUHUGVJVKZUULIAUULVLVMQZULZUULVNVOUUNUULOUULNUUOUU - MVPVQZACEFGHVRUULVLVSVTWAWDZWBZDBWCVEAYPBAYPADBUUJUURWEZWFABUURWGWHWIAYRW - JZBUUNYPUFIUVAUUPYPUULOUUNYPUFKUUQUVACNYPAYPCOYRAYPDDBWKLZWLQZWMZBSLZTLZC - ADBUVCSLZWMZTLDUVGWNLZUVFYPADUVGADUUJWOABUVCABUURWPZAUVCAUVBADBUUKUURWQWR - ZWOZWSWTAUVEUVHDTAUVEBUVDSLUVHAUVDBAUVDAUVCUVKXAZWOUVJXBABUVCUVJUVLXCXFXG - AUUBUUCYPUVIMUUKUUSDBXHVEXDAUUDUUFDCOUVECOZUVFCOUUGGJAUUDUUFUVDPOBCOUVNUU - GGUVMABCEFGHIXEPESUUECUVDBUUIUUHPXIOZSEXJQMXPPRESXIFXKXLVBXMXNTEUUECDUVEU - UIUVOTEXQQMXPPTREXIFXOXRVBXSXNYFXTAYRYAYBYPUULVLYCVTYDYEAYPYGOYSUUTYPYHYI - YRYQYJYKABNODPOYOYQYLUURUUJBDYMVEYN $. + clt cr crp crg clidl wss csubrg zsubrg subrgring cc cbs cnfldbas ressbas2 + mp1i zsscn ax-mp eqid lidlssOLD syl2anc sseldd zred cin inss2 ccnv c1 cuz + csup c0 wne nnuz sseqtri zlpirlem1 infmssuzcl syl5eqel sseldi nnrpd modlt + sylancr zmodcld nn0red nnred ltnled mpbid wa cdiv cneg cmin zcnd nndivred + nncnd flcld mulcld negsubd znegcld mulcomd mulneg2d eqtrd oveq2d 3eqtr4rd + cfl modval zlpirlem2 cvv cmulr cnfldmul ressmulr syl22anc cplusg cnfldadd + zex lidlmcl ressplusg lidlacl eqeltrd adantr simpr elind infmssuzle mtand + syl5eqbr cn0 elnn0 sylib orel1 sylc wb dvdsval3 mpbird ) ABDUAKZDBUBLZUCM + ZAYPNOZUDYRYQUEZYQAYRBYPUFKZAYPBUGKZYTUDADUHOZBUIOZUUAADACPDAEUJOZCEUKQZO + ZCPULPRUMQOUUDAUNPREFUOUTZGPCUUEUJEPUPULPEUQQMVAPUPERFURUSVBZUUEVCZVDVEJV + FZVGZABACNVHZNBCNVIZABUULUHUGVJVMZUULIAUULVKVLQZULZUULVNVOUUNUULOUULNUUOU + UMVPVQZACEFGHVRUULVKVSWDVTWAZWBZDBWCVEAYPBAYPADBUUJUURWEZWFABUURWGWHWIAYR + WJZBUUNYPUFIUVAUUPYPUULOUUNYPUFKUUQUVACNYPAYPCOYRAYPDDBWKLZXFQZWLZBSLZTLZ + CADBUVCSLZWLZTLDUVGWMLZUVFYPADUVGADUUJWNABUVCABUURWPZAUVCAUVBADBUUKUURWOW + QZWNZWRWSAUVEUVHDTAUVEBUVDSLUVHAUVDBAUVDAUVCUVKWTZWNUVJXAABUVCUVJUVLXBXCX + DAUUBUUCYPUVIMUUKUUSDBXGVEXEAUUDUUFDCOUVECOZUVFCOUUGGJAUUDUUFUVDPOBCOUVNU + UGGUVMABCEFGHIXHPESUUECUVDBUUIUUHPXIOZSEXJQMXPPRESXIFXKXLVBXQXMTEUUECDUVE + UUIUVOTEXNQMXPPTREXIFXOXRVBXSXMXTYAAYRYBYCYPUULVKYDWDYFYEAYPYGOYSUUTYPYHY + IYRYQYJYKABNODPOYOYQYLUURUUJBDYMVEYN $. $} ${ @@ -249409,15 +249634,15 @@ According to Wikipedia ("Integer", 25-May-2019, 9-Jun-2019. (New usage is discouraged.) $) zlpir $p |- Z e. LPIR $= ( vx vy vz wcel cfv cz ccnfld zsubrg ax-mp cv cc0 cdivides wbr wral simpr - wa eqid wceq clpir crg clidl clpidl wss csubrg subrgrng csn eleq1 wrex cn - wne cin cr clt ccnv simpl zlpirlem2 wel simpll simplr zlpirlem3 ralrimiva - csup breq1 ralbidv rspcev syl2anc wb dvdsrz lpigen mpan adantr mpbird c0g - cnfld0 subrg0 lpi0 mp1i pm2.61ne ssriv islpir2 mpbir2an ) AUAFAUBFZAUCGZA - UDGZUEHIUFGFZWDJHIABUGKZCWEWFCLZWEFZWIWFFZMUHZWFFZWIWLWIWLWFUIWJWIWLULZRZ - WKDLZELZNOZEWIPZDWIUJZWOWIUKUMUNUOUPVDZWIFXAWQNOZEWIPZWTWOXAWIABWJWNUQWJW - NQXASZURWOXBEWIWOECUSZRXAWIWQABWJWNXEUTWJWNXEVAXDWOXEQVBVCWSXCDXAWIWPXATW - RXBEWIWPXAWQNVEVFVGVHWJWKWTVIZWNWDWJXFWHDENWFAWEWIWESZWFSZABVJVKVLVMVNWDW - MWJWHWFAMXHWGMAVOGTJHIAMBVPVQKVRVSVTWAWFAWEXHXGWBWC $. + wa eqid wceq clpir crg clidl clpidl wss csubrg subrgring csn eleq1 wne cn + wrex cin cr clt ccnv csup zlpirlem2 wel simpll simplr zlpirlem3 ralrimiva + simpl breq1 ralbidv rspcev syl2anc dvdsrz lpigen adantr mpbird c0g cnfld0 + wb mpan subrg0 lpi0 mp1i pm2.61ne ssriv islpir2 mpbir2an ) AUAFAUBFZAUCGZ + AUDGZUEHIUFGFZWDJHIABUGKZCWEWFCLZWEFZWIWFFZMUHZWFFZWIWLWIWLWFUIWJWIWLUJZR + ZWKDLZELZNOZEWIPZDWIULZWOWIUKUMUNUOUPUQZWIFXAWQNOZEWIPZWTWOXAWIABWJWNVDWJ + WNQXASZURWOXBEWIWOECUSZRXAWIWQABWJWNXEUTWJWNXEVAXDWOXEQVBVCWSXCDXAWIWPXAT + WRXBEWIWPXAWQNVEVFVGVHWJWKWTVOZWNWDWJXFWHDENWFAWEWIWESZWFSZABVIVJVPVKVLWD + WMWJWHWFAMXHWGMAVMGTJHIAMBVNVQKVRVSVTWAWFAWEXHXGWBWC $. $( The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) Obsolete version of ~ zringcyg as of 9-Jun-2019. @@ -249498,44 +249723,44 @@ According to Wikipedia ("Integer", 25-May-2019, 10-Jun-2019.) $) prmirredlem $p |- ( A e. NN -> ( A e. I <-> A e. Prime ) ) $= ( vy cn wcel cprime wa cfv cdivides wbr c1 wceq co cmul cc0 ad2antrr cabs - cz syl vx c2 cuz cv wo wi wral wne zring crg zringrng zring1 irredn1 mpan - anim2i eluz2b3 sylibr cui cdiv nnz ad2antrl simprr nnne0 dvdsval2 syl3anc - wb mpbid zcnd cc nncn divcan2d simplr eqeltrd zringbas zringmulr irredmul - eqid zringunit baib nnnn0 nn0re nn0ge0 absidd eqeq1d bitrd cr nnre simprl - cn0 nndivred cle nnred nngt0 divge0 syl22anc 1cnd divmuld mulid1d orbi12d - clt 3bitrd expr ralrimiva isprm2 sylanbrc prmz 1nprm prmnn 3syl syl5ibcom - wn id eleq1 adantld syl5bi mtoi simplrl nnne0d simpr simplrr mul02d oveq1 - 3netr4d necon3i absne0d neneqd nn0abscl elnn0 sylib mt3d simprbi dvdsmul1 - ord ad2antlr breqtrd absdvdsb syl2anc eqeq1 abscld recnd breq1 imbi12d ex - rspcv syl3c mulcand fveq2d absmuld 3eqtr3d eqcom syl6bb mpbird ralrimivva - eqeq12d 3bitr2d isirred2 syl3anbrc adantl impbida ) AEFZABFZAGFZUUTUVAHZA - UBUCIFZDUDZAJKZUVELMZUVEAMZUEZUFZDEUGZUVBUVCUUTALUHZHUVDUVAUVLUUTUIUJFUVA - UVLUKUILBACULUMUNUOAUPUQUVCUVJDEUVCUVEEFZUVFUVIUVCUVMUVFHZHZUVEUIURIZFZAU - VEUSNZUVPFZUEZUVIUVOUVESFZUVRSFZUVEUVRONZBFUVTUVMUWAUVCUVFUVEUTVAZUVOUVFU - WBUVCUVMUVFVBUVOUWAUVEPUHZASFZUVFUWBVFUWDUVMUWEUVCUVFUVEVCVAZUUTUWFUVAUVN - AUTQZUVEAVDVEVGZUVOUWCABUVOAUVEUVOAUWHVHZUVMUVEVIFUVCUVFUVEVJVAZUWGVKUUTU - VAUVNVLVMSUIOUVPBUVEUVRCVNUVPVQZVOVPVEUVOUVQUVGUVSUVHUVOUVQUVERIZLMZUVGUV - OUWAUVQUWNVFZUWDUVQUWAUWNUVEVRVSZTUVOUWMUVELUVMUWMUVEMZUVCUVFUVMUVEWIFZUW - QUVEVTUWRUVEUVEWAUVEWBWCTVAWDWEUVOUVSUVRRIZLMZUVHUVOUWBUVSUWTVFUWIUVSUWBU - WTUVRVRVSTUVOUWTUVRLMUVELONZAMUVHUVOUWSUVRLUVOUVRUVOAUVEUUTAWFFZUVAUVNAWG - QZUVCUVMUVFWHZWJUVOUXBPAWKKZUVEWFFPUVEWTKZPUVRWKKUXCUUTUXEUVAUVNUUTAWIFZU - XEAVTZAWBZTQUVOUVEUXDWLUVMUXFUVCUVFUVEWMVAAUVEWNWOWCWDUVOAUVELUWJUWKUVOWP - UWGWQUVOUXAUVEAUVOUVEUWKWRWDXAWEWSVGXBXCDAXDZXEUVBUVAUUTUVBUWFAUVPFZXKUAU - DZUVEONZAMZUXLUVPFZUVQUEZUFZDSUGUASUGUVAAXFZUVBUXKLGFZXGUXKUWFARIZLMZHUVB - UXSAVRUVBUYAUXSUWFUVBUXTGFUYAUXSUVBUXTAGUVBUUTUXGUXTAMZAXHZUXHUXGAAWAUXIW - CXIZUVBXLVMUXTLGXMXJXNXOXPUVBUXQUADSSUVBUXLSFZUWAHZHZUXNUXPUYGUXNHZUXPUXL - 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This @@ -249550,18 +249775,18 @@ According to Wikipedia ("Integer", 25-May-2019, (Revised by AV, 10-Jun-2019.) $) prmirred $p |- ( A e. I <-> ( A e. ZZ /\ ( abs ` A ) e. Prime ) ) $= ( wcel cz cfv cprime zring zringbas wb cn wa cc0 wceq wo wi eleq1d adantl - mpan ccnfld cabs irredcl cn0 cr cneg ax-1 crg wne zringrng zring0 irredn0 - elnn0 necon2bi pm2.21d sylbi prmnn a1i prmirredlem pm5.21ndd nn0re nn0ge0 - jaoi absidd bitr4d cminusg eqid irrednegb csubg csubrg subrgsubg df-zring - zsubrg ax-mp subginv zcn cnfldneg syl eqtr3d bitrd adantr zre cle nn0ge0d - cc nnnn0 le0neg1d mpbird absnidd 3bitr4d adantrl elznn0nn biimpi mpjaodan - wbr biadan2 ) ABDZAEDZAUAFZGDZEHBACIUBWQAUCDZWPWSJZAUDDZAUEZKDZLZWTXAWQWT - WPAGDZWSWTAKDZWPXFWTXGAMNZOWPXGPZAULXGXIXHXGWPUFXHWPXGWPAMHUGDZWPAMUHUIHB - AMCUJUKSUMUNVBUOXFXGPWTAUPUQXGWPXFJPWTABCURUQUSWTWRAGWTAAUTAVAVCQVDRWQXDX - AXBWQXDLZXCBDZXCGDZWPWSXDXLXMJWQXCBCURRWQWPXLJXDWQWPAHVEFZFZBDZXLXJWQWPXP - JUIEHBXNACXNVFZIVGSWQXOXCBWQATVEFZFZXOXCETVHFDZWQXSXONETVIFDXTVLETVJVMETH - XRXNAVKXRVFXQVNSWQAWDDXSXCNAVOAVPVQVRQVSVTXKWRXCGXKAWQXBXDAWAVTZXKAMWBWNM - XCWBWNZXDYBWQXDXCXCWEWCRXKAYAWFWGWHQWIWJWQWTXEOAWKWLWMWO $. + mpan ccnfld cabs irredcl cn0 cr cneg elnn0 ax-1 crg wne zringring irredn0 + zring0 necon2bi pm2.21d jaoi sylbi prmnn a1i prmirredlem pm5.21ndd nn0ge0 + nn0re absidd bitr4d cminusg irrednegb csubg csubrg zsubrg subrgsubg ax-mp + eqid df-zring subginv cc zcn cnfldneg syl eqtr3d bitrd adantr zre cle wbr + nn0ge0d le0neg1d mpbird absnidd 3bitr4d adantrl elznn0nn mpjaodan biadan2 + nnnn0 biimpi ) ABDZAEDZAUAFZGDZEHBACIUBWQAUCDZWPWSJZAUDDZAUEZKDZLZWTXAWQW + TWPAGDZWSWTAKDZWPXFWTXGAMNZOWPXGPZAUFXGXIXHXGWPUGXHWPXGWPAMHUHDZWPAMUIUJH + BAMCULUKSUMUNUOUPXFXGPWTAUQURXGWPXFJPWTABCUSURUTWTWRAGWTAAVBAVAVCQVDRWQXD + XAXBWQXDLZXCBDZXCGDZWPWSXDXLXMJWQXCBCUSRWQWPXLJXDWQWPAHVEFZFZBDZXLXJWQWPX + PJUJEHBXNACXNVLZIVFSWQXOXCBWQATVEFZFZXOXCETVGFDZWQXSXONETVHFDXTVIETVJVKET + HXRXNAVMXRVLXQVNSWQAVODXSXCNAVPAVQVRVSQVTWAXKWRXCGXKAWQXBXDAWBWAZXKAMWCWD + MXCWCWDZXDYBWQXDXCXCWNWERXKAYAWFWGWHQWIWJWQWTXEOAWKWOWLWM $. $} ${ @@ -249573,46 +249798,46 @@ According to Wikipedia ("Integer", 25-May-2019, ~ prmirredlem as of 10-Jun-2019. (New usage is discouraged.) $) prmirredlemOLD $p |- ( A e. NN -> ( A e. I <-> A e. Prime ) ) $= ( vy cn wcel wa cfv cdivides wbr c1 wceq cz co cmul cc0 ad2antrr syl wral - vx cprime c2 cuz cv wo wi wne crg ccnfld csubrg zsubrg subrgrng ax-mp cur - cnfld1 subrg1 irredn1 mpan anim2i eluz2b3 sylibr cui cdiv ad2antrl simprr - nnz wb nnne0 dvdsval2 syl3anc mpbid zcnd nncn divcan2d simplr eqeltrd cbs - subrgbas eqid cvv cmulr zex cnfldmul ressmulr irredmul cabs zrngunit baib - cc cn0 nnnn0 nn0re nn0ge0 absidd eqeq1d bitrd cr nnre simprl nndivred cle - clt nnred nngt0 divge0 syl22anc ax-1cn a1i divmuld mulid1d 3bitrd orbi12d - expr ralrimiva isprm2 sylanbrc wn prmz 1nprm prmnn 3syl syl5ibcom adantld - id eleq1 syl5bi simplrl nnne0d simpr simplrr mul02d 3netr4d oveq1 necon3i - mtoi eqeq1 abscld recnd absne0d neneqd nn0abscl elnn0 sylib mt3d dvdsmul1 - ord simprbi ad2antlr breqtrd absdvdsb syl2anc breq1 imbi12d rspcv mulcand - syl3c fveq2d absmuld 3eqtr3d eqeq12d eqcom syl6bb 3bitr2d mpbird isirred2 - ex ralrimivva syl3anbrc adantl impbida ) AGHZABHZAUCHZUVMUVNIZAUDUEJHZFUF - ZAKLZUVRMNZUVRANZUGZUHZFGUAZUVOUVPUVMAMUIZIUVQUVNUWEUVMCUJHZUVNUWEOUKULJH - ZUWFUMOUKCDUNUOCMBAEUWGMCUPJNUMOUKCMDUQURUOUSUTVAAVBVCUVPUWCFGUVPUVRGHZUV - SUWBUVPUWHUVSIZIZUVRCVDJZHZAUVRVEPZUWKHZUGZUWBUWJUVROHZUWMOHZUVRUWMQPZBHU - WOUWHUWPUVPUVSUVRVHVFZUWJUVSUWQUVPUWHUVSVGUWJUWPUVRRUIZAOHZUVSUWQVIUWSUWH - UWTUVPUVSUVRVJVFZUVMUXAUVNUWIAVHSZUVRAVKVLVMZUWJUWRABUWJAUVRUWJAUXCVNZUWH - UVRWKHUVPUVSUVRVOVFZUXBVPUVMUVNUWIVQVROCQUWKBUVRUWMEUWGOCVSJNUMOUKCDVTUOZ - UWKWAZOWBHQCWCJNWDOUKCQWBDWEWFUOZWGVLUWJUWLUVTUWNUWAUWJUWLUVRWHJZMNZUVTUW - JUWPUWLUXKVIZUWSUWLUWPUXKUVRCDWIWJZTUWJUXJUVRMUWHUXJUVRNZUVPUVSUWHUVRWLHZ - UXNUVRWMUXOUVRUVRWNUVRWOWPTVFWQWRUWJUWNUWMWHJZMNZUWAUWJUWQUWNUXQVIUXDUWNU - WQUXQUWMCDWIWJTUWJUXQUWMMNUVRMQPZANUWAUWJUXPUWMMUWJUWMUWJAUVRUVMAWSHZUVNU - WIAWTSZUVPUWHUVSXAZXBUWJUXSRAXCLZUVRWSHRUVRXDLZRUWMXCLUXTUVMUYBUVNUWIUVMA - WLHZUYBAWMZAWOZTSUWJUVRUYAXEUWHUYCUVPUVSUVRXFVFAUVRXGXHWPWQUWJAUVRMUXEUXF - MWKHZUWJXIXJUXBXKUWJUXRUVRAUWJUVRUXFXLWQXMWRXNVMXOXPFAXQZXRUVOUVNUVMUVOUX - AAUWKHZXSUBUFZUVRQPZANZUYJUWKHZUWLUGZUHZFOUAUBOUAUVNAXTZUVOUYIMUCHZYAUYIU - XAAWHJZMNZIUVOUYQACDWIUVOUYSUYQUXAUVOUYRUCHUYSUYQUVOUYRAUCUVOUVMUYDUYRANZ - AYBZUYEUYDAAWNUYFWPYCZUVOYFVRUYRMUCYGYDYEYHYQUVOUYOUBFOOUVOUYJOHZUWPIZIZU - YLUYNVUEUYLIZUYNUYJWHJZMNZVUGANZUGZVUFVUGGHZUWDVUGAKLZVUJVUFVUKVUGRNZVUFV - UGRVUFUYJVUFUYJUVOVUCUWPUYLYIZVNZVUFUYKRUVRQPZUIUYJRUIVUFARUYKVUPVUFAUVOU - VMVUDUYLVUASYJVUEUYLYKZVUFUVRVUFUVRUVOVUCUWPUYLYLZVNZYMYNUYJRUYKVUPUYJRUV - RQYOYPTUUAZUUBVUFVUKVUMVUFVUGWLHZVUKVUMUGVUFVUCVVAVUNUYJUUCTVUGUUDUUEUUHU - UFUVOUWDVUDUYLUVOUVQUWDUYHUUISVUFUYJAKLZVULVUFUYJUYKAKVUDUYJUYKKLUVOUYLUY - JUVRUUGUUJVUQUUKVUFVUCUXAVVBVULVIVUNUVOUXAVUDUYLUYPSUYJAUULUUMVMUWCVULVUJ - UHFVUGGUVRVUGNZUVSVULUWBVUJUVRVUGAKUUNVVCUVTVUHUWAVUIUVRVUGMYRUVRVUGAYRXN - UUOUUPUURVUFUYMVUHUWLVUIVUFVUCUYMVUHVIVUNUYMVUCVUHUYJCDWIWJTVUFUWLUXKVUGU - XJQPZVUGMQPZNZVUIVUFUWPUXLVURUXMTVUFUXJMVUGVUFUXJVUFUVRVUSYSYTUYGVUFXIXJV - UFVUGVUFUYJVUOYSYTZVUTUUQVUFVVFAVUGNVUIVUFVVDAVVEVUGVUFUYKWHJUYRVVDAVUFUY - KAWHVUQUUSVUFUYJUVRVUOVUSUUTUVOUYTVUDUYLVUBSUVAVUFVUGVVGXLUVBAVUGUVCUVDUV - EXNUVFUVHUVIUBFOCQUWKBAUXGUXHEUXIUVGUVJUVKUVL $. + vx cprime c2 cuz cv wo wi wne ccnfld csubrg zsubrg subrgring ax-mp cnfld1 + crg cur subrg1 irredn1 mpan anim2i eluz2b3 sylibr cui nnz ad2antrl simprr + cdiv wb nnne0 dvdsval2 syl3anc mpbid zcnd cc nncn divcan2d simplr eqeltrd + cbs subrgbas eqid cvv cmulr cnfldmul ressmulr irredmul cabs zrngunit baib + zex cn0 nnnn0 nn0re nn0ge0 absidd eqeq1d bitrd cr simprl nndivred cle clt + nnre nnred divge0 syl22anc ax-1cn a1i divmuld mulid1d 3bitrd orbi12d expr + nngt0 ralrimiva isprm2 sylanbrc wn prmz 1nprm prmnn 3syl id eleq1 adantld + syl5ibcom syl5bi mtoi simplrl nnne0d simpr simplrr mul02d 3netr4d necon3i + oveq1 eqeq1 abscld recnd absne0d neneqd nn0abscl elnn0 sylib mt3d simprbi + ord dvdsmul1 ad2antlr breqtrd absdvdsb syl2anc breq1 imbi12d rspcv fveq2d + syl3c mulcand absmuld 3eqtr3d eqeq12d syl6bb 3bitr2d mpbird ex ralrimivva + eqcom isirred2 syl3anbrc adantl impbida ) AGHZABHZAUCHZUVMUVNIZAUDUEJHZFU + FZAKLZUVRMNZUVRANZUGZUHZFGUAZUVOUVPUVMAMUIZIUVQUVNUWEUVMCUPHZUVNUWEOUJUKJ + HZUWFULOUJCDUMUNCMBAEUWGMCUQJNULOUJCMDUOURUNUSUTVAAVBVCUVPUWCFGUVPUVRGHZU + VSUWBUVPUWHUVSIZIZUVRCVDJZHZAUVRVHPZUWKHZUGZUWBUWJUVROHZUWMOHZUVRUWMQPZBH + UWOUWHUWPUVPUVSUVRVEVFZUWJUVSUWQUVPUWHUVSVGUWJUWPUVRRUIZAOHZUVSUWQVIUWSUW + HUWTUVPUVSUVRVJVFZUVMUXAUVNUWIAVESZUVRAVKVLVMZUWJUWRABUWJAUVRUWJAUXCVNZUW + HUVRVOHUVPUVSUVRVPVFZUXBVQUVMUVNUWIVRVSOCQUWKBUVRUWMEUWGOCVTJNULOUJCDWAUN + ZUWKWBZOWCHQCWDJNWKOUJCQWCDWEWFUNZWGVLUWJUWLUVTUWNUWAUWJUWLUVRWHJZMNZUVTU + WJUWPUWLUXKVIZUWSUWLUWPUXKUVRCDWIWJZTUWJUXJUVRMUWHUXJUVRNZUVPUVSUWHUVRWLH + ZUXNUVRWMUXOUVRUVRWNUVRWOWPTVFWQWRUWJUWNUWMWHJZMNZUWAUWJUWQUWNUXQVIUXDUWN + UWQUXQUWMCDWIWJTUWJUXQUWMMNUVRMQPZANUWAUWJUXPUWMMUWJUWMUWJAUVRUVMAWSHZUVN + UWIAXDSZUVPUWHUVSWTZXAUWJUXSRAXBLZUVRWSHRUVRXCLZRUWMXBLUXTUVMUYBUVNUWIUVM + AWLHZUYBAWMZAWOZTSUWJUVRUYAXEUWHUYCUVPUVSUVRXOVFAUVRXFXGWPWQUWJAUVRMUXEUX + FMVOHZUWJXHXIUXBXJUWJUXRUVRAUWJUVRUXFXKWQXLWRXMVMXNXPFAXQZXRUVOUVNUVMUVOU + XAAUWKHZXSUBUFZUVRQPZANZUYJUWKHZUWLUGZUHZFOUAUBOUAUVNAXTZUVOUYIMUCHZYAUYI + UXAAWHJZMNZIUVOUYQACDWIUVOUYSUYQUXAUVOUYRUCHUYSUYQUVOUYRAUCUVOUVMUYDUYRAN + ZAYBZUYEUYDAAWNUYFWPYCZUVOYDVSUYRMUCYEYGYFYHYIUVOUYOUBFOOUVOUYJOHZUWPIZIZ + UYLUYNVUEUYLIZUYNUYJWHJZMNZVUGANZUGZVUFVUGGHZUWDVUGAKLZVUJVUFVUKVUGRNZVUF + VUGRVUFUYJVUFUYJUVOVUCUWPUYLYJZVNZVUFUYKRUVRQPZUIUYJRUIVUFARUYKVUPVUFAUVO + UVMVUDUYLVUASYKVUEUYLYLZVUFUVRVUFUVRUVOVUCUWPUYLYMZVNZYNYOUYJRUYKVUPUYJRU + VRQYQYPTUUAZUUBVUFVUKVUMVUFVUGWLHZVUKVUMUGVUFVUCVVAVUNUYJUUCTVUGUUDUUEUUH + UUFUVOUWDVUDUYLUVOUVQUWDUYHUUGSVUFUYJAKLZVULVUFUYJUYKAKVUDUYJUYKKLUVOUYLU + YJUVRUUIUUJVUQUUKVUFVUCUXAVVBVULVIVUNUVOUXAVUDUYLUYPSUYJAUULUUMVMUWCVULVU + JUHFVUGGUVRVUGNZUVSVULUWBVUJUVRVUGAKUUNVVCUVTVUHUWAVUIUVRVUGMYRUVRVUGAYRX + MUUOUUPUURVUFUYMVUHUWLVUIVUFVUCUYMVUHVIVUNUYMVUCVUHUYJCDWIWJTVUFUWLUXKVUG + UXJQPZVUGMQPZNZVUIVUFUWPUXLVURUXMTVUFUXJMVUGVUFUXJVUFUVRVUSYSYTUYGVUFXHXI + VUFVUGVUFUYJVUOYSYTZVUTUUSVUFVVFAVUGNVUIVUFVVDAVVEVUGVUFUYKWHJUYRVVDAVUFU + YKAWHVUQUUQVUFUYJUVRVUOVUSUUTUVOUYTVUDUYLVUBSUVAVUFVUGVVGXKUVBAVUGUVHUVCU + VDXMUVEUVFUVGUBFOCQUWKBAUXGUXHEUXIUVIUVJUVKUVL $. $( The positive irreducible elements of ` ZZ ` are the prime numbers. This is an alternative way to define ` Prime ` . (Contributed by Mario @@ -249629,18 +249854,18 @@ According to Wikipedia ("Integer", 25-May-2019, prmirredOLD $p |- ( A e. I <-> ( A e. ZZ /\ ( abs ` A ) e. Prime ) ) $= ( wcel cz cfv cprime ccnfld wceq zsubrg ax-mp wb cn wi mpan eleq1d adantl cc0 cabs csubrg cbs subrgbas irredcl cn0 cr cneg wa wo elnn0 ax-1 crg wne - subrgrng csubg subrgsubg cnfld0 subg0 irredn0 necon2bi pm2.21d jaoi sylbi - c0g prmnn a1i prmirredlemOLD pm5.21ndd nn0re nn0ge0 absidd bitr4d cminusg - eqid irrednegb subginv zcn cnfldneg syl eqtr3d bitrd adantr zre cle nnnn0 - cc wbr nn0ge0d le0neg1d absnidd 3bitr4d adantrl elznn0nn mpjaodan biadan2 - mpbird biimpi ) ABFZAGFZAUAHZIFZGCBAEGJUBHFZGCUCHKLGJCDUDMZUEWTAUFFZWSXBN - ZAUGFZAUHZOFZUIZXEXFWTXEWSAIFZXBXEAOFZWSXKXEXLATKZUJWSXLPZAUKXLXNXMXLWSUL - XMWSXLWSATCUMFZWSATUNXCXOLGJCDUOMZCBATEGJUPHFZTCVEHKXCXQLGJUQMZGJCTDURUSM - UTQVAVBVCVDXKXLPXEAVFVGXLWSXKNPXEABCDEVHVGVIXEXAAIXEAAVJAVKVLRVMSWTXIXFXG - WTXIUIZXHBFZXHIFZWSXBXIXTYANWTXHBCDEVHSWTWSXTNXIWTWSACVNHZHZBFZXTXOWTWSYD - NXPGCBYBAEYBVOZXDVPQWTYCXHBWTAJVNHZHZYCXHXQWTYGYCKXRGJCYFYBADYFVOYEVQQWTA - WGFYGXHKAVRAVSVTWARWBWCXSXAXHIXSAWTXGXIAWDWCZXSATWEWHTXHWEWHZXIYIWTXIXHXH - WFWISXSAYHWJWQWKRWLWMWTXEXJUJAWNWRWOWP $. + subrgring csubg c0g subrgsubg cnfld0 subg0 irredn0 necon2bi pm2.21d sylbi + jaoi prmnn a1i prmirredlemOLD pm5.21ndd nn0ge0 absidd bitr4d cminusg eqid + nn0re irrednegb subginv cc zcn cnfldneg syl eqtr3d bitrd adantr zre nnnn0 + cle wbr nn0ge0d le0neg1d mpbird absnidd 3bitr4d adantrl elznn0nn mpjaodan + biimpi biadan2 ) ABFZAGFZAUAHZIFZGCBAEGJUBHFZGCUCHKLGJCDUDMZUEWTAUFFZWSXB + NZAUGFZAUHZOFZUIZXEXFWTXEWSAIFZXBXEAOFZWSXKXEXLATKZUJWSXLPZAUKXLXNXMXLWSU + LXMWSXLWSATCUMFZWSATUNXCXOLGJCDUOMZCBATEGJUPHFZTCUQHKXCXQLGJURMZGJCTDUSUT + MVAQVBVCVEVDXKXLPXEAVFVGXLWSXKNPXEABCDEVHVGVIXEXAAIXEAAVOAVJVKRVLSWTXIXFX + GWTXIUIZXHBFZXHIFZWSXBXIXTYANWTXHBCDEVHSWTWSXTNXIWTWSACVMHZHZBFZXTXOWTWSY + DNXPGCBYBAEYBVNZXDVPQWTYCXHBWTAJVMHZHZYCXHXQWTYGYCKXRGJCYFYBADYFVNYEVQQWT + AVRFYGXHKAVSAVTWAWBRWCWDXSXAXHIXSAWTXGXIAWEWDZXSATWGWHTXHWGWHZXIYIWTXIXHX + HWFWISXSAYHWJWKWLRWMWNWTXEXJUJAWOWQWPWR $. $} ${ @@ -249655,16 +249880,16 @@ According to Wikipedia ("Integer", 25-May-2019, ( vy vz cc wcel cc0 wa cz cv cexp co cfv cmul wceq ccnfld wne csn cdif wf cmpt caddc wral zring cghm expclzlem 3expa eqid fmptd zaddcl adantl oveq2 expaddz ovex fvmpt syl oveqan12d 3eqtr4d ralrimivva zringgrp crg cnfldbas - cgrp cnrng cnfld0 cndrng drngui cress oveq1i eqtri unitgrp ax-mp zringbas - cmgp pm3.2i wss cbs difss mgpbas ressbas2 zringplusg cvv cui fvex eqeltri - cplusg cnfldmul mgpplusg ressplusg isghm mpbiran sylanbrc ) BIJZBKUAZLZMI - KUBZUCZAMBANZOPZUEZUDZGNZHNZUFPZXDQZXFXDQZXGXDQZRPZSZHMUGGMUGZXDUHCUIPJZW - SAMXCXAXDWQWRXBMJXCXAJBXBUJUKXDULZUMWSXMGHMMWSXFMJZXGMJZLZLZBXHOPZBXFOPZB - XGOPZRPZXIXLBXFXGUQXTXHMJZXIYASXSYEWSXFXGUNUOAXHXCYAMXDXBXHBOUPXPBXHOURUS - UTXSXLYDSWSXQXRXJYBXKYCRAXFXCYBMXDXBXFBOUPXPBXFOURUSAXGXCYCMXDXBXGBOUPXPB - XGOURUSVAUOVBVCXOUHVGJZCVGJZLXEXNLYFYGVDTVEJYGVHTXACITKVFVIVJVKZCDXAVLPTV - RQZXAVLPFDYIXAVLEVMVNVOVPVSHGUFRUHCXDMXAVQXAIVTXACWAQSIWTWBXAICDFITDEVFWC - WDVPWEXAWFJRCWJQSXATWGQWFYHTWGWHWIXARDCWFFTRDEWKWLWMVPWNWOWP $. + cgrp cnring cnfld0 cndrng drngui cress cmgp oveq1i eqtri unitgrp zringbas + ax-mp pm3.2i wss cbs difss mgpbas ressbas2 zringplusg cvv cplusg cui fvex + eqeltri cnfldmul mgpplusg ressplusg isghm mpbiran sylanbrc ) BIJZBKUAZLZM + IKUBZUCZAMBANZOPZUEZUDZGNZHNZUFPZXDQZXFXDQZXGXDQZRPZSZHMUGGMUGZXDUHCUIPJZ + WSAMXCXAXDWQWRXBMJXCXAJBXBUJUKXDULZUMWSXMGHMMWSXFMJZXGMJZLZLZBXHOPZBXFOPZ + BXGOPZRPZXIXLBXFXGUQXTXHMJZXIYASXSYEWSXFXGUNUOAXHXCYAMXDXBXHBOUPXPBXHOURU + SUTXSXLYDSWSXQXRXJYBXKYCRAXFXCYBMXDXBXFBOUPXPBXFOURUSAXGXCYCMXDXBXGBOUPXP + BXGOURUSVAUOVBVCXOUHVGJZCVGJZLXEXNLYFYGVDTVEJYGVHTXACITKVFVIVJVKZCDXAVLPT + VMQZXAVLPFDYIXAVLEVNVOVPVRVSHGUFRUHCXDMXAVQXAIVTXACWAQSIWTWBXAICDFITDEVFW + CWDVRWEXAWFJRCWGQSXATWHQWFYHTWHWIWJXARDCWFFTRDEWKWLWMVRWNWOWP $. $} ${ @@ -249680,18 +249905,18 @@ According to Wikipedia ("Integer", 25-May-2019, ( vy cc wcel wa cz cexp co cfv cmul wceq ccnfld ax-mp vz cc0 wne csn cdif cv cmpt wf caddc wral cghm expclzlem 3expa eqid fmptd zaddcl adantl oveq2 expaddz ovex fvmpt syl oveqan12d 3eqtr4d ralrimivva cgrp csubrg subrgsubg - csubg zsubrg subggrp crg cnrng cnfldbas cnfld0 cndrng drngui cress oveq1i - cmgp eqtri unitgrp pm3.2i cbs subrgbas wss difss mgpbas ressbas2 cnfldadd - cplusg ressplusg cvv cui eqeltri cnfldmul mgpplusg isghm mpbiran sylanbrc - fvex ) BJKZBUBUCZLZMJUBUDZUEZAMBAUFZNOZUGZUHZIUFZUAUFZUIOZXIPZXKXIPZXLXIP - ZQOZRZUAMUJIMUJZXIECUKOKZXDAMXHXFXIXBXCXGMKXHXFKBXGULUMXIUNZUOXDXRIUAMMXD - XKMKZXLMKZLZLZBXMNOZBXKNOZBXLNOZQOZXNXQBXKXLUSYEXMMKZXNYFRYDYJXDXKXLUPUQA - XMXHYFMXIXGXMBNURYABXMNUTVAVBYDXQYIRXDYBYCXOYGXPYHQAXKXHYGMXIXGXKBNURYABX - KNUTVAAXLXHYHMXIXGXLBNURYABXLNUTVAVCUQVDVEXTEVFKZCVFKZLXJXSLYKYLMSVIPKZYK - MSVGPZKZYMVJMSVHTMSEFVKTSVLKYLVMSXFCJSUBVNVOVPVQZCDXFVROSVTPZXFVROHDYQXFV - RGVSWAWBTWCUAIUIQECXIMXFYOMEWDPRVJMSEFWETXFJWFXFCWDPRJXEWGXFJCDHJSDGVNWHW - ITYOUIEWKPRVJMUISEYNFWJWLTXFWMKQCWKPRXFSWNPWMYPSWNXAWOXFQDCWMHSQDGWPWQWLT - WRWSWT $. + csubg zsubrg subggrp crg cnring cnfldbas cnfld0 cndrng drngui cmgp oveq1i + cress eqtri unitgrp pm3.2i subrgbas difss mgpbas ressbas2 cplusg cnfldadd + cbs wss ressplusg cvv cui fvex eqeltri cnfldmul mgpplusg mpbiran sylanbrc + isghm ) BJKZBUBUCZLZMJUBUDZUEZAMBAUFZNOZUGZUHZIUFZUAUFZUIOZXIPZXKXIPZXLXI + PZQOZRZUAMUJIMUJZXIECUKOKZXDAMXHXFXIXBXCXGMKXHXFKBXGULUMXIUNZUOXDXRIUAMMX + DXKMKZXLMKZLZLZBXMNOZBXKNOZBXLNOZQOZXNXQBXKXLUSYEXMMKZXNYFRYDYJXDXKXLUPUQ + AXMXHYFMXIXGXMBNURYABXMNUTVAVBYDXQYIRXDYBYCXOYGXPYHQAXKXHYGMXIXGXKBNURYAB + XKNUTVAAXLXHYHMXIXGXLBNURYABXLNUTVAVCUQVDVEXTEVFKZCVFKZLXJXSLYKYLMSVIPKZY + KMSVGPZKZYMVJMSVHTMSEFVKTSVLKYLVMSXFCJSUBVNVOVPVQZCDXFVTOSVRPZXFVTOHDYQXF + VTGVSWAWBTWCUAIUIQECXIMXFYOMEWJPRVJMSEFWDTXFJWKXFCWJPRJXEWEXFJCDHJSDGVNWF + WGTYOUIEWHPRVJMUISEYNFWIWLTXFWMKQCWHPRXFSWNPWMYPSWNWOWPXFQDCWMHSQDGWQWRWL + TXAWSWT $. $} ${ @@ -249723,18 +249948,18 @@ According to Wikipedia ("Integer", 25-May-2019, 12-Jun-2019.) $) mulgrhm $p |- ( R e. Ring -> F e. ( ZZring RingHom R ) ) $= ( vx wcel cz zring cfv c1 co wceq cv oveq1 ovex fvmpt cmul cmulr zringbas - vy crg zring1 zringmulr zringrng a1i id 1z ax-mp cbs rngidcl mulg1 syl5eq - eqid syl wa cgrp rnggrp adantr simprr mulgcl syl3anc rnglidm syldan simpl - oveq2d simprl mulgass2 syl13anc mulgass 3eqtr4rd zmulcl oveqan12d 3eqtr4d - adantl cghm mulgghm2 syl2anc isrhm2d ) AUEJZIUDKLAUAAUBMZNECUCUFHUGWDUQZL - UEJWCUHUIWCUJWCNEMZNCBOZCNKJWFWGPUKDNDQZCBOZWGKEWHNCBRGNCBSTULWCCAUMMZJZW - GCPWJACWJUQZHUNZWJBACWLFUOURUPWCIQZKJZUDQZKJZUSZUSZWNWPUAOZCBOZWNCBOZWPCB - OZWDOZWTEMZWNEMZWPEMZWDOZWSWNCXCWDOZBOZWNXCBOZXDXAWSXIXCWNBWCWRXCWJJZXIXC - PWSAUTJZWQWKXLWCXMWRAVAZVBZWCWOWQVCZWCWKWRWMVBZWJBAWPCWLFVDVEZWJAWDCXCWLW - EHVFVGVIWSWCWOWKXLXDXJPWCWRVHWCWOWQVJZXQXRWJABWDWNCXCWLFWEVKVLWSXMWOWQWKX - AXKPXOXSXPXQWJBAWNWPCWLFVMVLVNWSWTKJZXEXAPWRXTWCWNWPVOVRDWTWIXAKEWHWTCBRG - WTCBSTURWRXHXDPWCWOWQXFXBXGXCWDDWNWIXBKEWHWNCBRGWNCBSTDWPWIXCKEWHWPCBRGWP - CBSTVPVRVQWCXMWKELAVSOJXNWMWJABCDEFGWLVTWAWB $. + vy crg zring1 zringmulr eqid zringring a1i id 1z ax-mp cbs ringidcl mulg1 + syl syl5eq wa ringgrp adantr simprr mulgcl syl3anc ringlidm syldan oveq2d + simprl mulgass2 syl13anc mulgass 3eqtr4rd zmulcl adantl oveqan12d 3eqtr4d + cgrp simpl cghm mulgghm2 syl2anc isrhm2d ) AUEJZIUDKLAUAAUBMZNECUCUFHUGWD + UHZLUEJWCUIUJWCUKWCNEMZNCBOZCNKJWFWGPULDNDQZCBOZWGKEWHNCBRGNCBSTUMWCCAUNM + ZJZWGCPWJACWJUHZHUOZWJBACWLFUPUQURWCIQZKJZUDQZKJZUSZUSZWNWPUAOZCBOZWNCBOZ + WPCBOZWDOZWTEMZWNEMZWPEMZWDOZWSWNCXCWDOZBOZWNXCBOZXDXAWSXIXCWNBWCWRXCWJJZ + XIXCPWSAVQJZWQWKXLWCXMWRAUTZVAZWCWOWQVBZWCWKWRWMVAZWJBAWPCWLFVCVDZWJAWDCX + CWLWEHVEVFVGWSWCWOWKXLXDXJPWCWRVRWCWOWQVHZXQXRWJABWDWNCXCWLFWEVIVJWSXMWOW + QWKXAXKPXOXSXPXQWJBAWNWPCWLFVKVJVLWSWTKJZXEXAPWRXTWCWNWPVMVNDWTWIXAKEWHWT + CBRGWTCBSTUQWRXHXDPWCWOWQXFXBXGXCWDDWNWIXBKEWHWNCBRGWNCBSTDWPWIXCKEWHWPCB + RGWPCBSTVOVNVPWCXMWKELAVSOJXNWMWJABCDEFGWLVTWAWB $. $( The powers of the element ` 1 ` give the unique ring homomorphism from ` ZZ ` to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) @@ -249767,16 +249992,16 @@ According to Wikipedia ("Integer", 25-May-2019, (New usage is discouraged.) $) mulgghm2OLD $p |- ( ( R e. Grp /\ .1. e. B ) -> F e. ( Z GrpHom R ) ) $= ( vx vy wcel wa cz caddc co cfv wceq cgrp wf cv cplusg wral cghm ccnfld - simpl csubrg zsubrg subrgrng rnggrp mp2b jctil mulgcl 3expa an32s fmptd - crg eqid mulgdir 3exp2 imp42 zaddcl adantl oveq1 ovex fvmpt syl 3eqtr4d - oveqan12d ralrimivva jca cbs subrgbas ax-mp cnfldadd ressplusg sylanbrc - isghm ) BUANZDANZOZGUANZWAOPAFUBZLUCZMUCZQRZFSZWFFSZWGFSZBUDSZRZTZMPUEL - PUEZOFGBUFRNWCWAWDWAWBUHPUGUISZNZGUSNWDUJPUGGHUKGULUMUNWCWEWOWCEPEUCZDC - RZAFWAWRPNZWBWSANZWAWTWBXAACBWRDKIUOUPUQJURWCWNLMPPWCWFPNZWGPNZOZOZWHDC - RZWFDCRZWGDCRZWLRZWIWMWAXDWBXFXITZWAXBXCWBXJWAXBXCWBXJAWLCBWFWGDKIWLUTZ - VAVBVCUQXEWHPNZWIXFTXDXLWCWFWGVDVEEWHWSXFPFWRWHDCVFJWHDCVGVHVIXDWMXITWC - XBXCWJXGWKXHWLEWFWSXGPFWRWFDCVFJWFDCVGVHEWGWSXHPFWRWGDCVFJWGDCVGVHVKVEV - JVLVMMLQWLGBFPAWQPGVNSTUJPUGGHVOVPKWQQGUDSTUJPQUGGWPHVQVRVPXKVTVS $. + simpl csubrg crg zsubrg subrgring ringgrp mp2b jctil mulgcl 3expa an32s + fmptd eqid mulgdir 3exp2 imp42 zaddcl adantl oveq1 ovex fvmpt oveqan12d + syl 3eqtr4d ralrimivva subrgbas ax-mp cnfldadd ressplusg isghm sylanbrc + jca cbs ) BUANZDANZOZGUANZWAOPAFUBZLUCZMUCZQRZFSZWFFSZWGFSZBUDSZRZTZMPU + ELPUEZOFGBUFRNWCWAWDWAWBUHPUGUISZNZGUJNWDUKPUGGHULGUMUNUOWCWEWOWCEPEUCZ + DCRZAFWAWRPNZWBWSANZWAWTWBXAACBWRDKIUPUQURJUSWCWNLMPPWCWFPNZWGPNZOZOZWH + DCRZWFDCRZWGDCRZWLRZWIWMWAXDWBXFXITZWAXBXCWBXJWAXBXCWBXJAWLCBWFWGDKIWLU + TZVAVBVCURXEWHPNZWIXFTXDXLWCWFWGVDVEEWHWSXFPFWRWHDCVFJWHDCVGVHVJXDWMXIT + WCXBXCWJXGWKXHWLEWFWSXGPFWRWFDCVFJWFDCVGVHEWGWSXHPFWRWGDCVFJWGDCVGVHVIV + EVKVLVSMLQWLGBFPAWQPGVTSTUKPUGGHVMVNKWQQGUDSTUKPQUGGWPHVOVPVNXKVQVR $. $} mulgrhmOLD.4 $e |- .1. = ( 1r ` R ) $. @@ -249785,20 +250010,20 @@ According to Wikipedia ("Integer", 25-May-2019, of ~ mulgrhm as of 12-Jun-2019. (New usage is discouraged.) $) mulgrhmOLD $p |- ( R e. Ring -> F e. ( Z RingHom R ) ) $= ( wcel cz cmul cfv c1 ccnfld wceq zsubrg ax-mp co vx crg cmulr csubrg cbs - vy subrgbas cur cnfld1 subrg1 cnfldmul ressmulr eqid subrgrng id 1z oveq1 - mp1i ovex fvmpt rngidcl mulg1 syl syl5eq cgrp rnggrp adantr simprr mulgcl - cv syl3anc rnglidm syldan oveq2d simpl mulgass2 syl13anc mulgass 3eqtr4rd - simprl zmulcl adantl oveqan12d 3eqtr4d cghm mulgghm2OLD syl2anc isrhm2d - wa ) AUBKZUAUFLFAMAUCNZOECLPUDNZKZLFUENQRLPFGUGSWMOFUHNQRLPFOGUIUJSJWMMFU - CNQRLPFMWLGUKULSWKUMZWMFUBKWJRLPFGUNURWJUOWJOENZOCBTZCOLKWOWPQUPDODVJZCBT - ZWPLEWQOCBUQIOCBUSUTSWJCAUENZKZWPCQWSACWSUMZJVAZWSBACXAHVBVCVDWJUAVJZLKZU - FVJZLKZWIZWIZXCXEMTZCBTZXCCBTZXECBTZWKTZXIENZXCENZXEENZWKTZXHXCCXLWKTZBTZ - XCXLBTZXMXJXHXRXLXCBWJXGXLWSKZXRXLQXHAVEKZXFWTYAWJYBXGAVFZVGZWJXDXFVHZWJW - TXGXBVGZWSBAXECXAHVIVKZWSAWKCXLXAWNJVLVMVNXHWJXDWTYAXMXSQWJXGVOWJXDXFVTZY - FYGWSABWKXCCXLXAHWNVPVQXHYBXDXFWTXJXTQYDYHYEYFWSBAXCXECXAHVRVQVSXHXILKZXN - XJQXGYIWJXCXEWAWBDXIWRXJLEWQXICBUQIXICBUSUTVCXGXQXMQWJXDXFXOXKXPXLWKDXCWR - XKLEWQXCCBUQIXCCBUSUTDXEWRXLLEWQXECBUQIXECBUSUTWCWBWDWJYBWTEFAWETKYCXBWSA - BCDEFGHIXAWFWGWH $. + vy subrgbas cur cnfld1 subrg1 cnfldmul ressmulr eqid subrgring mp1i id 1z + cv oveq1 ovex fvmpt ringidcl mulg1 syl5eq wa ringgrp adantr simprr mulgcl + syl syl3anc ringlidm syldan oveq2d simpl simprl mulgass2 syl13anc mulgass + cgrp 3eqtr4rd zmulcl adantl oveqan12d 3eqtr4d mulgghm2OLD syl2anc isrhm2d + cghm ) AUBKZUAUFLFAMAUCNZOECLPUDNZKZLFUENQRLPFGUGSWMOFUHNQRLPFOGUIUJSJWMM + FUCNQRLPFMWLGUKULSWKUMZWMFUBKWJRLPFGUNUOWJUPWJOENZOCBTZCOLKWOWPQUQDODURZC + BTZWPLEWQOCBUSIOCBUTVASWJCAUENZKZWPCQWSACWSUMZJVBZWSBACXAHVCVJVDWJUAURZLK + ZUFURZLKZVEZVEZXCXEMTZCBTZXCCBTZXECBTZWKTZXIENZXCENZXEENZWKTZXHXCCXLWKTZB + TZXCXLBTZXMXJXHXRXLXCBWJXGXLWSKZXRXLQXHAVTKZXFWTYAWJYBXGAVFZVGZWJXDXFVHZW + JWTXGXBVGZWSBAXECXAHVIVKZWSAWKCXLXAWNJVLVMVNXHWJXDWTYAXMXSQWJXGVOWJXDXFVP + ZYFYGWSABWKXCCXLXAHWNVQVRXHYBXDXFWTXJXTQYDYHYEYFWSBAXCXECXAHVSVRWAXHXILKZ + XNXJQXGYIWJXCXEWBWCDXIWRXJLEWQXICBUSIXICBUTVAVJXGXQXMQWJXDXFXOXKXPXLWKDXC + WRXKLEWQXCCBUSIXCCBUTVADXEWRXLLEWQXECBUSIXECBUTVAWDWCWEWJYBWTEFAWITKYCXBW + SABCDEFGHIXAWFWGWH $. $( The powers of the element ` 1 ` give the unique ring homomorphism from ` ZZ ` to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) @@ -250021,10 +250246,10 @@ According to Wikipedia ("Integer", 25-May-2019, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) $) zlmsca $p |- ( G e. V -> ZZring = ( Scalar ` W ) ) $= ( wcel cnx csca cfv zring cop csts co cvsca cmg scaid wne c5 c6 5re crg - 5lt6 ltneii scandx vscandx neeq12i mpbir setsnid wceq zringrng mpan2 eqid - setsid zlmval fveq2d 3eqtr4a ) ABEZAFGHZIJKLZGHZURFMHZANHZJKLZGHICGHVAUTG - UROUQUTPQRPQRSUAUBUQQUTRUCUDUEUFUGUPITEIUSUHUIBIGTAOULUJUPCVBGVAABCDVAUKU - MUNUO $. + 5lt6 ltneii scandx vscandx neeq12i mpbir wceq zringring setsid mpan2 eqid + setsnid zlmval fveq2d 3eqtr4a ) ABEZAFGHZIJKLZGHZURFMHZANHZJKLZGHICGHVAUT + GUROUQUTPQRPQRSUAUBUQQUTRUCUDUEUFULUPITEIUSUGUHBIGTAOUIUJUPCVBGVAABCDVAUK + UMUNUO $. ${ zlmvsca.2 $e |- .x. = ( .g ` G ) $. @@ -250048,25 +250273,25 @@ According to Wikipedia ("Integer", 25-May-2019, zlmlmod $p |- ( G e. Abel <-> W e. LMod ) $= ( vx vy cabl wcel cz cplusg cfv caddc cmul c1 zring cbs wceq eqid a1i cv co vz clmod cmg zlmbas zlmplusg zlmsca cvsca zlmvsca zringplusg zringmulr - zringbas cmulr cur zring1 crg zringrng cgrp ablprop ablgrp mulgcl syl3an1 - sylbi mulgdi w3a mulgdir sylan mulgass mulg1 adantl islmodd sylibr impbii - lmodabl ) AFGZBUBGZVNDEUAHAIJZKAUCJZLMNAOJZBVRBOJPVNVRABCVRQZUDZRVPBIJPVN - VPABCVPQZUEZRAFBCUFVQBUGJPVNVQABCVQQZUHRHNOJPVNUKRKNIJPVNUIRLNULJPVNUJRMN - UMJPVNUNRNUOGVNUPRVNBFGZBUQGABVTWBURZBUSVBVNAUQGZDSZHGZESZVRGWGWIVQTVRGAU - SZVRVQAWGWIVSWCUTVAVRVPVQAWGWIUASZVSWCWAVCVNWFWHWIHGWKVRGVDZWGWIKTWKVQTWG - WKVQTWIWKVQTZVPTPWJVRVPVQAWGWIWKVSWCWAVEVFVNWFWLWGWILTWKVQTWGWMVQTPWJVRVQ - AWGWIWKVSWCVGVFWGVRGMWGVQTWGPVNVRVQAWGVSWCVHVIVJVOWDVNBVMWEVKVL $. + zringbas cmulr cur zring1 crg zringring cgrp ablprop ablgrp sylbi syl3an1 + mulgcl mulgdi mulgdir sylan mulgass adantl islmodd lmodabl sylibr impbii + w3a mulg1 ) AFGZBUBGZVNDEUAHAIJZKAUCJZLMNAOJZBVRBOJPVNVRABCVRQZUDZRVPBIJP + VNVPABCVPQZUEZRAFBCUFVQBUGJPVNVQABCVQQZUHRHNOJPVNUKRKNIJPVNUIRLNULJPVNUJR + MNUMJPVNUNRNUOGVNUPRVNBFGZBUQGABVTWBURZBUSUTVNAUQGZDSZHGZESZVRGWGWIVQTVRG + AUSZVRVQAWGWIVSWCVBVAVRVPVQAWGWIUASZVSWCWAVCVNWFWHWIHGWKVRGVLZWGWIKTWKVQT + WGWKVQTWIWKVQTZVPTPWJVRVPVQAWGWIWKVSWCWAVDVEVNWFWLWGWILTWKVQTWGWMVQTPWJVR + VQAWGWIWKVSWCVFVEWGVRGMWGVQTWGPVNVRVQAWGVSWCVMVGVHVOWDVNBVIWEVJVK $. $( The ` ZZ ` -module operation turns a ring into an associative algebra over ` ZZ ` . (Contributed by Mario Carneiro, 2-Oct-2015.) $) zlmassa $p |- ( G e. Ring <-> W e. AssAlg ) $= ( vy vz vx crg wcel casa cz cmg cfv cmulr zring cbs wceq eqid zlmbas a1i - cv zlmsca zringbas cvsca zlmvsca zlmmulr cabl clmod rngabl zlmlmod cplusg - sylib zlmplusg rngprop biimpi zringcrng mulgass2 mulgass3 isassad assarng - ccrg sylibr impbii ) AGHZBIHZVCDEJAKLZAMLZNAOLZBFVGBOLPVCVGABCVGQZRZSAGBC - UAJNOLPVCUBSVEBUCLPVCVEABCVEQZUDSVFBMLPVCVFABCVFQZUEZSVCAUFHBUGHAUHABCUIU - KVCBGHZABVIAUJLZABCVNQULVLUMZUNNUTHVCUOSVGAVEVFFTZDTZETZVHVJVKUPVGAVEVFVP - VQVRVHVJVKUQURVDVMVCBUSVOVAVB $. + cv zlmsca zringbas cvsca zlmvsca zlmmulr cabl clmod ringabl zlmlmod sylib + cplusg zlmplusg ringprop biimpi ccrg zringcrng mulgass2 mulgass3 assaring + isassad sylibr impbii ) AGHZBIHZVCDEJAKLZAMLZNAOLZBFVGBOLPVCVGABCVGQZRZSA + GBCUAJNOLPVCUBSVEBUCLPVCVEABCVEQZUDSVFBMLPVCVFABCVFQZUEZSVCAUFHBUGHAUHABC + UIUJVCBGHZABVIAUKLZABCVNQULVLUMZUNNUOHVCUPSVGAVEVFFTZDTZETZVHVJVKUQVGAVEV + FVPVQVRVHVJVKURUTVDVMVCBUSVOVAVB $. $} ${ @@ -250088,8 +250313,8 @@ According to Wikipedia ("Integer", 25-May-2019, $( Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.) $) chrcl $p |- ( R e. Ring -> C e. NN0 ) $= - ( crg wcel cur cfv cod cn0 eqid chrval cbs rngidcl odcl syl syl5eqelr ) B - DEZABFGZBHGZGZIABRSSJZRJZCKQRBLGZETIEUCBRUCJZUBMRBSUCUDUANOP $. + ( crg wcel cur cfv cod cn0 eqid chrval cbs ringidcl odcl syl syl5eqelr ) + BDEZABFGZBHGZGZIABRSSJZRJZCKQRBLGZETIEUCBRUCJZUBMRBSUCUDUANOP $. chrid.l $e |- L = ( ZRHom ` R ) $. chrid.z $e |- .0. = ( 0g ` R ) $. @@ -250097,20 +250322,20 @@ According to Wikipedia ("Integer", 25-May-2019, characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.) $) chrid $p |- ( R e. Ring -> ( L ` C ) = .0. ) $= - ( crg wcel cfv cur cmg co cz wceq chrcl nn0zd eqid zrhmulg mpdan cod odid - chrval oveq1i cbs rngidcl syl syl5eqr eqtrd ) BHIZACJZABKJZBLJZMZDUJANIUK - UNOUJAABEPQBUMULCAFUMRZULRZSTUJUNULBUAJZJZULUMMZDURAULUMABULUQUQRZUPEUCUD - UJULBUEJZIUSDOVABULVARZUPUFULUMBUQVADVBUTUOGUBUGUHUI $. + ( crg wcel cfv cur cmg co cz wceq chrcl nn0zd eqid zrhmulg mpdan ringidcl + cod chrval oveq1i cbs odid syl syl5eqr eqtrd ) BHIZACJZABKJZBLJZMZDUJANIU + KUNOUJAABEPQBUMULCAFUMRZULRZSTUJUNULBUBJZJZULUMMZDURAULUMABULUQUQRZUPEUCU + DUJULBUEJZIUSDOVABULVARZUPUAULUMBUQVADVBUTUOGUFUGUHUI $. $( The ` ZZ ` ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.) $) chrdvds $p |- ( ( R e. Ring /\ N e. ZZ ) -> ( C || N <-> ( L ` N ) = .0. ) ) $= ( crg wcel cz wa cdivides wbr cur cfv cmg wceq eqid adantr co chrval cgrp - cod breq1i cbs rnggrp rngidcl simpr oddvds syl3anc syl5bbr zrhmulg eqeq1d - wb bitr4d ) BIJZDKJZLZADMNZDBOPZBQPZUAZERZDCPZERUTVABUDPZPZDMNZUSVDVGADMA - BVAVFVFSZVASZFUBUEUSBUCJZVABUFPZJZURVHVDUOUQVKURBUGTUQVMURVLBVAVLSZVJUHTU - QURUIVAVBBDVFVLEVNVIVBSZHUJUKULUSVEVCEBVBVACDGVOVJUMUNUP $. + cod breq1i wb ringgrp ringidcl simpr oddvds syl3anc syl5bbr eqeq1d bitr4d + cbs zrhmulg ) BIJZDKJZLZADMNZDBOPZBQPZUAZERZDCPZERUTVABUDPZPZDMNZUSVDVGAD + MABVAVFVFSZVASZFUBUEUSBUCJZVABUOPZJZURVHVDUFUQVKURBUGTUQVMURVLBVAVLSZVJUH + TUQURUIVAVBBDVFVLEVNVIVBSZHUJUKULUSVEVCEBVBVACDGVOVJUPUMUN $. $( If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, @@ -250118,12 +250343,12 @@ According to Wikipedia ("Integer", 25-May-2019, chrcong $p |- ( ( R e. Ring /\ M e. ZZ /\ N e. ZZ ) -> ( C || ( M - N ) <-> ( L ` M ) = ( L ` N ) ) ) $= ( crg wcel cz co cdivides wbr cfv wceq eqid 3ad2ant1 zrhmulg w3a cmin cur - cmg cod chrval breq1i cgrp wb rnggrp rngidcl simp2 simp3 odcong syl112anc - cbs syl5bbr 3adant3 3adant2 eqeq12d bitr4d ) BJKZDLKZELKZUAZADEUBMZNOZDBU - CPZBUDPZMZEVHVIMZQZDCPZECPZQVGVHBUEPZPZVFNOZVEVLVPAVFNABVHVOVORZVHRZGUFUG - VEBUHKZVHBUPPZKZVCVDVQVLUIVBVCVTVDBUJSVBVCWBVDWABVHWARZVSUKSVBVCVDULVBVCV - DUMVHVIBDEVOWAFWCVRVIRZIUNUOUQVEVMVJVNVKVBVCVMVJQVDBVIVHCDHWDVSTURVBVDVNV - KQVCBVIVHCEHWDVSTUSUTVA $. + cmg cod chrval breq1i cgrp cbs wb ringgrp ringidcl simp2 odcong syl112anc + simp3 syl5bbr 3adant3 3adant2 eqeq12d bitr4d ) BJKZDLKZELKZUAZADEUBMZNOZD + BUCPZBUDPZMZEVHVIMZQZDCPZECPZQVGVHBUEPZPZVFNOZVEVLVPAVFNABVHVOVORZVHRZGUF + UGVEBUHKZVHBUIPZKZVCVDVQVLUJVBVCVTVDBUKSVBVCWBVDWABVHWARZVSULSVBVCVDUMVBV + CVDUPVHVIBDEVOWAFWCVRVIRZIUNUOUQVEVMVJVNVKVBVCVMVJQVDBVIVHCDHWDVSTURVBVDV + NVKQVCBVIVHCEHWDVSTUSUTVA $. $} $( Nonzero rings are precisely those with characteristic not 1. (Contributed @@ -250156,21 +250381,21 @@ According to Wikipedia ("Integer", 25-May-2019, ( chr ` R ) e. Prime ) ) $= ( vx vy wcel cfv cc0 wceq wa co cdivides wbr cz cn0 syl sylanbrc wb zring eqid chrdvds syl2anc cdomn cchr cprime wn wne df-ne c2 cv cmul wo wi wral - cuz cn c1 csn cdif domnrng chrcl adantr simpr eldifsn dfn2 syl6eleqr cnzr - crg domnnzr nzrrng chrnzr ibi eluz2b3 czrh c0g crh ad2antrr zrhrhm simprl - cmulr simprr zringbas zringmulr rhmmul syl3anc eqeq1d cbs simpll ffvelrnd - rhmf domneq0 bitrd biimpd zmulcl adantl orbi12d 3imtr4d ralrimivva isprm6 - wf ex syl5bir orrd ) AUADZAUBEZFGZXCUCDZXDUDXCFUEZXBXEXCFUFXBXFXEXBXFHZXC - UGUMEDZXCBUHZCUHZUIIZJKZXCXIJKZXCXJJKZUJZUKZCLULBLULXEXGXCUNDXCUOUEZXHXGX - CMFUPUQZUNXGXCMDZXFXCXRDXBXSXFXBAVFDZXSAURZXCAXCRZUSNUTXBXFVAXCMFVBOVCVDX - BXQXFXBAVEDZXQAVGYCXQYCXTYCXQPAVHAVINVJNUTXCVKOXGXPBCLLXGXILDZXJLDZHZHZXK - AVLEZEZAVMEZGZXIYHEZYJGZXJYHEZYJGZUJZXLXOYGYKYPYGYKYLYNAVREZIZYJGZYPYGYIY - RYJYGYHQAVNIDZYDYEYIYRGYGXTYTXBXTXFYFYAVOZAYHYHRZVPNZXGYDYEVQZXGYDYEVSZXI - XJQAUIYQYHLVTWAYQRZWBWCWDYGXBYLAWEEZDYNUUGDYSYPPXBXFYFWFYGLUUGXIYHYGYTLUU - GYHWRUUCLUUGQAYHVTUUGRZWHNZUUDWGYGLUUGXJYHUUIUUEWGUUGAYQYLYNYJUUHUUFYJRZW - IWCWJWKYGXTXKLDZXLYKPUUAYFUUKXGXIXJWLWMXCAYHXKYJYBUUBUUJSTYGXMYMXNYOYGXTY - DXMYMPUUAUUDXCAYHXIYJYBUUBUUJSTYGXTYEXNYOPUUAUUEXCAYHXJYJYBUUBUUJSTWNWOWP - BCXCWQOWSWTXA $. + cuz cn c1 csn cdif crg domnring chrcl adantr simpr eldifsn dfn2 syl6eleqr + cnzr domnnzr nzrring chrnzr ibi eluz2b3 czrh cmulr ad2antrr zrhrhm simprl + c0g crh simprr zringbas zringmulr rhmmul syl3anc eqeq1d cbs rhmf ffvelrnd + simpll wf domneq0 bitrd biimpd zmulcl adantl orbi12d ralrimivva isprm6 ex + 3imtr4d syl5bir orrd ) AUADZAUBEZFGZXCUCDZXDUDXCFUEZXBXEXCFUFXBXFXEXBXFHZ + XCUGUMEDZXCBUHZCUHZUIIZJKZXCXIJKZXCXJJKZUJZUKZCLULBLULXEXGXCUNDXCUOUEZXHX + GXCMFUPUQZUNXGXCMDZXFXCXRDXBXSXFXBAURDZXSAUSZXCAXCRZUTNVAXBXFVBXCMFVCOVDV + EXBXQXFXBAVFDZXQAVGYCXQYCXTYCXQPAVHAVINVJNVAXCVKOXGXPBCLLXGXILDZXJLDZHZHZ + XKAVLEZEZAVQEZGZXIYHEZYJGZXJYHEZYJGZUJZXLXOYGYKYPYGYKYLYNAVMEZIZYJGZYPYGY + IYRYJYGYHQAVRIDZYDYEYIYRGYGXTYTXBXTXFYFYAVNZAYHYHRZVONZXGYDYEVPZXGYDYEVSZ + XIXJQAUIYQYHLVTWAYQRZWBWCWDYGXBYLAWEEZDYNUUGDYSYPPXBXFYFWHYGLUUGXIYHYGYTL + UUGYHWIUUCLUUGQAYHVTUUGRZWFNZUUDWGYGLUUGXJYHUUIUUEWGUUGAYQYLYNYJUUHUUFYJR + ZWJWCWKWLYGXTXKLDZXLYKPUUAYFUUKXGXIXJWMWNXCAYHXKYJYBUUBUUJSTYGXMYMXNYOYGX + TYDXMYMPUUAUUDXCAYHXIYJYBUUBUUJSTYGXTYEXNYOPUUAUUEXCAYHXJYJYBUUBUUJSTWOWS + WPBCXCWQOWRWTXA $. $} ${ @@ -250179,8 +250404,8 @@ According to Wikipedia ("Integer", 25-May-2019, $( The set ` n ZZ ` is an ideal in ` ZZ ` . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) $) znlidl $p |- ( N e. ZZ -> ( S ` { N } ) e. ( LIdeal ` ZZring ) ) $= - ( wcel zring crg csn wss clidl zringrng snssi zringbas eqid rspcl sylancr - cz cfv ) BPDEFDBGZPHRAQEIQZDJBPKPESRACLSMNO $. + ( cz wcel zring crg csn wss cfv clidl zringring snssi zringbas eqid rspcl + sylancr ) BDEFGEBHZDIRAJFKJZELBDMDFSRACNSOPQ $. znval.u $e |- U = ( ZZring /s ( ZZring ~QG ( S ` { N } ) ) ) $. $( The value of the ` Z/nZ ` structure. It is defined as the quotient ring @@ -250207,18 +250432,18 @@ According to Wikipedia ("Integer", 25-May-2019, Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.) $) znval $p |- ( N e. NN0 -> Y = ( U sSet <. ( le ` ndx ) , .<_ >. ) ) $= ( vz cfv co zring cc0 wceq syl6eqr vn vs cn0 wcel czn cnx cple cop csts - vf cv csn crsp cqg cqus czrh cz cfzo cif cres cle ccom csb crg zringrng - ccnv a1i wa cvv ovex simpr fveq2d simpl sneqd fveq12d oveq12d sylan9eqr + vf csn crsp cqg cqus czrh cfzo cif cres cle ccom ccnv csb crg zringring + cv cz a1i wa ovex id simpr fveq2d simpl sneqd fveq12d oveq12d sylan9eqr fvex resex simpll eqeq1d oveq2d ifbieq2d reseq12d coeq1d cnveqd coeq12d - id csbied opeq2d df-zn fvmpt syl5eq ) EUCUDGEUEOBUFUGOZDUHZUIPZJUAENQUB - NUKZWQUAUKZULZWQUMOZOZUNPZUOPZUBUKZWNUJXDUPOZWRRSZUQRWRURPZUSZUTZUJUKZV - AVBZXJVFZVBZVCZUHZUIPZVCZVCWPUCUEWRESZNQXQWPVDQVDUDXRVEVGXRWQQSZVHZUBXC - XPWPVIXCVIUDXTWQXBUOVJVGXTXDXCSZVHZXDBXOWOUIYAXTXDXCBYAWHXTXCQQEULZAOZU - NPZUOPBXTWQQXBYEUOXRXSVKZXTWQQXAYDUNYFXTWSYCWTAXTWTQUMOAXTWQQUMYFVLHTXT - WREXRXSVMVNVOVPVPITVQZYBXNDWNYBUJXIXMDVIXIVIUDYBXEXHXDUPVRVSVGYBXJXISZV - HZXMCVAVBZCVFZVBDYIXKYJXLYKYIXJCVAYHYBXJXICYHWHYBXIBUPOZFUTCYBXEYLXHFYB - XDBUPYGVLYBXHERSZUQREURPZUSFYBXFYMXGYNUQYBWRERXRXSYAVTZWAYBWRERURYOWBWC - LTWDKTVQZWEYIXJCYPWFWGMTWIWJVPWIWINUJUAUBWKBWOUIVJWLWM $. + cvv csbied opeq2d df-zn fvmpt syl5eq ) EUCUDGEUEOBUFUGOZDUHZUIPZJUAENQU + BNVEZWQUAVEZUKZWQULOZOZUMPZUNPZUBVEZWNUJXDUOOZWRRSZVFRWRUPPZUQZURZUJVEZ + USUTZXJVAZUTZVBZUHZUIPZVBZVBWPUCUEWRESZNQXQWPVCQVCUDXRVDVGXRWQQSZVHZUBX + CXPWPWHXCWHUDXTWQXBUNVIVGXTXDXCSZVHZXDBXOWOUIYAXTXDXCBYAVJXTXCQQEUKZAOZ + UMPZUNPBXTWQQXBYEUNXRXSVKZXTWQQXAYDUMYFXTWSYCWTAXTWTQULOAXTWQQULYFVLHTX + TWREXRXSVMVNVOVPVPITVQZYBXNDWNYBUJXIXMDWHXIWHUDYBXEXHXDUOVRVSVGYBXJXISZ + VHZXMCUSUTZCVAZUTDYIXKYJXLYKYIXJCUSYHYBXJXICYHVJYBXIBUOOZFURCYBXEYLXHFY + BXDBUOYGVLYBXHERSZVFREUPPZUQFYBXFYMXGYNVFYBWRERXRXSYAVTZWAYBWRERUPYOWBW + CLTWDKTVQZWEYIXJCYPWFWGMTWIWJVPWIWINUJUAUBWKBWOUIVIWLWM $. $} znle.l $e |- .<_ = ( le ` Y ) $. @@ -250245,9 +250470,9 @@ According to Wikipedia ("Integer", 25-May-2019, Carneiro, 14-Jun-2015.) Obsolete version of ~ znlidl as of 13-Jun-2019. (New usage is discouraged.) $) znlidlOLD $p |- ( N e. ZZ -> ( S ` { N } ) e. ( LIdeal ` Z ) ) $= - ( cz wcel crg csn wss cfv clidl ccnfld csubrg zsubrg subrgrng ax-mp snssi - cbs wceq subrgbas eqid rspcl sylancr ) BFGCHGZBIZFJUFAKCLKZGFMNKGZUEOFMCD - PQBFRFCUGUFAEUHFCSKTOFMCDUAQUGUBUCUD $. + ( cz crg csn wss cfv clidl ccnfld csubrg zsubrg subrgring ax-mp snssi cbs + wcel wceq subrgbas eqid rspcl sylancr ) BFSCGSZBHZFIUFAJCKJZSFLMJSZUENFLC + DOPBFQFCUGUFAEUHFCRJTNFLCDUAPUGUBUCUD $. znvalOLD.u $e |- U = ( Z /s ( Z ~QG ( S ` { N } ) ) ) $. $( The value of the ` Z/nZ ` structure. It is defined as the quotient ring @@ -250328,9 +250553,9 @@ According to Wikipedia ("Integer", 25-May-2019, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) $) znbas $p |- ( N e. NN0 -> ( ZZ /. R ) = ( Base ` Y ) ) $= ( cn0 wcel cz cqs zring cqus co cbs cfv cvv crg a1i cqg wceq zringbas csn - eqidd ovex eqeltri zringrng qusbas oveq2i znbas2 eqtrd ) CHIZJAKLAMNZOPDO - PULALUMJQRULUMUDJLOPUAULUBSAQIULALCUCBPZTNZQGLUNTUEUFSLRIULUGSUHBUMCDEAUO - LMGUIFUJUK $. + eqidd ovex eqeltri zringring qusbas oveq2i znbas2 eqtrd ) CHIZJAKLAMNZOPD + OPULALUMJQRULUMUDJLOPUAULUBSAQIULALCUCBPZTNZQGLUNTUEUFSLRIULUGSUHBUMCDEAU + OLMGUIFUJUK $. $} ${ @@ -250356,13 +250581,13 @@ According to Wikipedia ("Integer", 25-May-2019, classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) $) znzrh2 $p |- ( N e. NN0 -> L = ( x e. ZZ |-> [ x ] .~ ) ) $= - ( wcel czrh cfv cz cec zring cqus co wceq crg cn0 cmpt crh clidl zringrng - cv csn nn0z znlidl syl oveq2i ccrg c2idl zringcrng eqid crng2idl zringbas - cqg ax-mp eceq2 mpteq2i qusrhm sylancr zncrng2 crngrng zrhrhmb 4syl mpbid - wb znzrh eqtr2d syl5eq ) EUAKZDFLMZANAUFZBOZUBZJVMVQPBQRZLMZVNVMVQPVRUCRK - ZVQVSSZVMPTKEUGCMZPUDMZKZVTUEVMENKZWDEUHZCEGUIUJAPWBVRVQWCNBPWBURRZPQHUKZ - PULKWCPUMMSUNPWCWCUOUPUSUQANVPVOWGOZBWGSVPWISHBWGVOUTUSVAVBVCVMWEVRULKVRT - KVTWAVIWFCVREGWHVDVRVEVRVQVSVSUOVFVGVHCVREFGWHIVJVKVL $. + ( wcel czrh cfv cz cec zring cqus co wceq crg cn0 cv cmpt clidl zringring + crh csn nn0z znlidl syl cqg oveq2i ccrg c2idl zringcrng crng2idl zringbas + eqid ax-mp eceq2 mpteq2i qusrhm sylancr wb zncrng2 crngring zrhrhmb mpbid + 4syl znzrh eqtr2d syl5eq ) EUAKZDFLMZANAUBZBOZUCZJVMVQPBQRZLMZVNVMVQPVRUF + RKZVQVSSZVMPTKEUGCMZPUDMZKZVTUEVMENKZWDEUHZCEGUIUJAPWBVRVQWCNBPWBUKRZPQHU + LZPUMKWCPUNMSUOPWCWCURUPUSUQANVPVOWGOZBWGSVPWISHBWGVOUTUSVAVBVCVMWEVRUMKV + RTKVTWAVDWFCVREGWHVEVRVFVRVQVSVSURVGVIVHCVREFGWHIVJVKVL $. $( The ` ZZ ` ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, @@ -250383,11 +250608,11 @@ According to Wikipedia ("Integer", 25-May-2019, (Contributed by Mario Carneiro, 15-Jun-2015.) $) znzrhfo $p |- ( N e. NN0 -> L : ZZ -onto-> B ) $= ( vx wcel cz wfo zring cfv cqg co cvv crg wceq a1i eqid cn0 csn crsp cmpt - cv cec cqs cqus eqidd cbs zringbas zringrng quslem wb znbas syl6eqr foeq3 - ovex syl mpbid znzrh2 foeq1 mpbird ) CUAIZJABKZJAHJHUELCUBLUCMZMZNOZUFUDZ - KZVDJJVHUGZVIKZVJVDHVHLLVHUHOZVIJPQVDVMUIJLUJMRVDUKSVITVHPIVDLVGNURSLQIVD - ULSUMVDVKARVLVJUNVDVKDUJMAVHVFCDVFTZEVHTZUOFUPVKAJVIUQUSUTVDBVIRVEVJUNHVH - VFBCDVNVOEGVAJABVIVBUSVC $. + cv cec cqs cqus eqidd cbs zringbas ovex zringring quslem wb znbas syl6eqr + foeq3 syl mpbid znzrh2 foeq1 mpbird ) CUAIZJABKZJAHJHUELCUBLUCMZMZNOZUFUD + ZKZVDJJVHUGZVIKZVJVDHVHLLVHUHOZVIJPQVDVMUIJLUJMRVDUKSVITVHPIVDLVGNULSLQIV + DUMSUNVDVKARVLVJUOVDVKDUJMAVHVFCDVFTZEVHTZUPFUQVKAJVIURUSUTVDBVIRVEVJUOHV + HVFBCDVNVOEGVAJABVIVBUSVC $. $} ${ @@ -250397,12 +250622,12 @@ According to Wikipedia ("Integer", 25-May-2019, (Contributed by Mario Carneiro, 21-Apr-2016.) $) zncyg $p |- ( N e. NN0 -> Y e. CycGrp ) $= ( vn vx cn0 wcel cgrp cz cv cmg cfv co cmpt crn cbs wceq syl eqid rneqd - wrex ccyg crg ccrg zncrng crngrng rnggrp cur rngidcl czrh zrhval2 znzrhfo - wfo forn eqtr3d oveq2 mpteq2dv eqeq1d rspcev syl2anc iscyg sylanbrc ) AFG - ZBHGZDIDJZEJZBKLZMZNZOZBPLZQZEVKUAZBUBGVCBUCGZVDVCBUDGVNABCUEBUFRZBUGRVCB - UHLZVKGZDIVEVPVGMZNZOZVKQZVMVCVNVQVOVKBVPVKSZVPSZUIRVCBUJLZOZVTVKVCWDVSVC - VNWDVSQVOBVGVPDWDWDSZVGSZWCUKRTVCIVKWDUMWEVKQVKWDABCWBWFULIVKWDUNRUOVLWAE - VPVKVFVPQZVJVTVKWHVIVSWHDIVHVRVFVPVEVGUPUQTURUSUTEVKVGDBWBWGVAVB $. + wrex ccyg crg ccrg crngring ringgrp cur ringidcl czrh zrhval2 wfo znzrhfo + zncrng forn eqtr3d oveq2 mpteq2dv eqeq1d rspcev syl2anc iscyg sylanbrc ) + AFGZBHGZDIDJZEJZBKLZMZNZOZBPLZQZEVKUAZBUBGVCBUCGZVDVCBUDGVNABCUMBUERZBUFR + VCBUGLZVKGZDIVEVPVGMZNZOZVKQZVMVCVNVQVOVKBVPVKSZVPSZUHRVCBUILZOZVTVKVCWDV + SVCVNWDVSQVOBVGVPDWDWDSZVGSZWCUJRTVCIVKWDUKWEVKQVKWDABCWBWFULIVKWDUNRUOVL + WAEVPVKVFVPQZVJVTVKWHVIVSWHDIVHVRVFVPVEVGUPUQTURUSUTEVKVGDBWBWGVAVB $. zndvds.2 $e |- L = ( ZRHom ` Y ) $. $( Express equality of equivalence classes in ` ZZ / n ZZ ` in terms of @@ -250411,19 +250636,19 @@ According to Wikipedia ("Integer", 25-May-2019, ( ( L ` A ) = ( L ` B ) <-> N || ( A - B ) ) ) $= ( vx cfv wceq wcel cz cmin co cdivides zring eqid zringbas sylancr ccnfld cn0 w3a wbr eqcom csn crsp cqg cec znzrhval 3adant2 3adant3 eqeq12d csubg - wer crg clidl zringrng wss nn0z 3ad2ant1 snssd rspcl lidlsubg eqger simp3 - syl erth csg cabl wb zringabl lidlss eqgabl wa simp2 jca biantrurd df-3an - syl6bbr cv cab csubrg zsubrg subrgsubg cnfldsub df-zring subgsub syld3an1 - mp1i eqcomd dvdsrzring rspsn eleq12d ovex breq2 syl6bb 3bitr2d syl5bb - elab ) ACIZBCIZJXAWTJZDUAKZALKZBLKZUBZDABMNZOUCZWTXAUDXFXBBPDUEZPUFIZIZUG - NZUHZAXLUHZJBAXLUCZXHXFXAXMWTXNXCXEXAXMJXDBXLXJCDEXJQZXLQZFGUIUJXCXDWTXNJ - XEAXLXJCDEXPXQFGUIUKULXFBAXLLXFXKPUMIKZLXLUNXFPUOKZXKPUPIZKZXRUQXFXSXILUR - YAUQXFDLXCXDDLKZXEDUSUTZVALPXTXIXJXPRXTQZVBSZPXTXKYDVCSXLPLXKRXQVDVFXCXDX - EVEZVGXFXOXEXDABPVHIZNZXKKZUBZYIXHXFPVIKXKLURZXOYJVJVKXFYAYKYELXKXTPRYDVL - VFBAXLXKPYGLRYGQZXQVMSXFYIXEXDVNZYIVNYJXFYMYIXFXEXDYFXCXDXEVOVPVQXEXDYIVR - VSXFYIXGDHVTZOUCZHWAZKXHXFYHXGXKYPXFXGYHLTUMIKZXDXCXEXGYHJLTWBIKYQXFWCLTW - DWILTPMYGABWEWFYLWGWHWJXFXSYBXKYPJUQYCHLOPDXJRXPWKWLSWMYOXHHXGABMWNYNXGDO - WOWSWPWQWQWR $. + wer crg clidl zringring wss 3ad2ant1 snssd rspcl lidlsubg eqger syl simp3 + nn0z erth csg wb zringabl lidlss eqgabl wa simp2 biantrurd df-3an syl6bbr + cabl jca cv csubrg zsubrg subrgsubg mp1i cnfldsub df-zring subgsub eqcomd + syld3an1 dvdsrzring rspsn eleq12d ovex breq2 elab syl6bb 3bitr2d syl5bb + cab ) ACIZBCIZJXAWTJZDUAKZALKZBLKZUBZDABMNZOUCZWTXAUDXFXBBPDUEZPUFIZIZUGN + ZUHZAXLUHZJBAXLUCZXHXFXAXMWTXNXCXEXAXMJXDBXLXJCDEXJQZXLQZFGUIUJXCXDWTXNJX + EAXLXJCDEXPXQFGUIUKULXFBAXLLXFXKPUMIKZLXLUNXFPUOKZXKPUPIZKZXRUQXFXSXILURY + AUQXFDLXCXDDLKZXEDVFUSZUTLPXTXIXJXPRXTQZVASZPXTXKYDVBSXLPLXKRXQVCVDXCXDXE + VEZVGXFXOXEXDABPVHIZNZXKKZUBZYIXHXFPVRKXKLURZXOYJVIVJXFYAYKYELXKXTPRYDVKV + DBAXLXKPYGLRYGQZXQVLSXFYIXEXDVMZYIVMYJXFYMYIXFXEXDYFXCXDXEVNVSVOXEXDYIVPV + QXFYIXGDHVTZOUCZHWSZKXHXFYHXGXKYPXFXGYHLTUMIKZXDXCXEXGYHJLTWAIKYQXFWBLTWC + WDLTPMYGABWEWFYLWGWIWHXFXSYBXKYPJUQYCHLOPDXJRXPWJWKSWLYOXHHXGABMWMYNXGDOW + NWOWPWQWQWR $. zndvds0.3 $e |- .0. = ( 0g ` Y ) $. $( Special case of ~ zndvds when one argument is zero. (Contributed by @@ -250431,10 +250656,11 @@ According to Wikipedia ("Integer", 25-May-2019, zndvds0 $p |- ( ( N e. NN0 /\ A e. ZZ ) -> ( ( L ` A ) = .0. <-> N || A ) ) $= ( cn0 wcel cz wa cfv cc0 wceq co cdivides wbr zring 3syl wb zndvds mp3an3 - cmin 0z crh cghm ccrg crg zncrng adantr zrhrhm rhmghm zring0 ghmid eqeq2d - crngrng simpr zcnd subid1d breq2d 3bitr3d ) CIJZAKJZLZABMZNBMZOZCANUDPZQR - ZVFEOCAQRVCVDNKJVHVJUAUEANBCDFGUBUCVEVGEVFVEBSDUFPJZBSDUGPJVGEOVEDUHJZDUI - JVKVCVLVDCDFUJUKDUQDBGULTSDBUMSDBNEUNHUOTUPVEVIACQVEAVEAVCVDURUSUTVAVB $. + cmin crh cghm ccrg crg zncrng adantr crngring zrhrhm rhmghm zring0 eqeq2d + 0z ghmid simpr zcnd subid1d breq2d 3bitr3d ) CIJZAKJZLZABMZNBMZOZCANUDPZQ + RZVFEOCAQRVCVDNKJVHVJUAUPANBCDFGUBUCVEVGEVFVEBSDUEPJZBSDUFPJVGEOVEDUGJZDU + HJVKVCVLVDCDFUIUJDUKDBGULTSDBUMSDBNEUNHUQTUOVEVIACQVEAVEAVCVDURUSUTVAVB + $. $} ${ @@ -250451,45 +250677,45 @@ According to Wikipedia ("Integer", 25-May-2019, 13-Jun-2019.) $) znf1o $p |- ( N e. NN0 -> F : W -1-1-onto-> B ) $= ( vx vy wcel cfv wceq cz co cc0 wa wbr wb vz cn0 wf1 wfo wf1o wf weq wral - cv wi czrh cres wss ccrg crg crh zncrng crngrng eqid zrhrhm zringbas rhmf - zring 4syl cfzo cif sseq1 ssid ssriv keephyp eqsstri fssres sylancl feq1i - elfzoelz sylibr cmin cdivides fveq1i fvres ad2antrl syl5eq ad2antll simpl - eqeq12d simprl sseldi simprr zndvds syl3anc cn wo cmo moddvds crp cle clt - elnn0 cr zred nnrp adantr wne nnne0 ifnefalse syl elfzole1 elfzolt2 modid - eleqtrd syl22anc bitr3d breq1d 0nn0 syl6eqel sylan zsubcld 0dvds subeq0ad - id zcnd 3bitrd jaoian sylanb biimpd ralrimivva dff13 sylanbrc crn zmodfzo - wrex ancoms eleqtrrd zre mpbid mpbird fveq2 eqeq2d rspcev syl2anc modabs2 - syl2anr simpr nnnn0 eqcomd iftrue biimpar syl6eleqr rexbiia ralrimiva wfn - eleq2d eqidd znzrhfo fofn eqeq1 rexbidv ralrn 3syl raleqdv dffo3 df-f1o - forn ) CUBLZDABUCZDABUDZDABUEUVDDABUFZJUIZBMZKUIZBMZNZJKUGZUJZKDUHJDUHUVE - UVDDAEUKMZDULZUFZUVGUVDOAUVOUFZDOUMUVQUVDEUNLEUOLUVOVCEUPPLUVRCEFUQEUREUV - OUVOUSZUTOAVCEUVOVAGVBVDDCQNZOQCVEPZVFZOIUVTOOUMUWAOUMUWBOUMOUWAOUWBOVGUW - AUWBOVGOVHJUWAOUVHQCVOVIZVJVKZOADUVOVLVMDABUVPHVNVPZUVDUVNJKDDUVDUVHDLZUV - JDLZRZRZUVLUVMUWIUVLUVHUVOMZUVJUVOMZNZCUVHUVJVQPZVRSZUVMUWIUVIUWJUVKUWKUW - IUVIUVHUVPMZUWJUVHBUVPHVSUWFUWOUWJNUVDUWGUVHDUVOVTWAWBUWIUVKUVJUVPMZUWKUV - JBUVPHVSZUWGUWPUWKNUVDUWFUVJDUVOVTZWCWBWEUWIUVDUVHOLZUVJOLZUWLUWNTUVDUWHW - DUWIDOUVHUWDUVDUWFUWGWFWGZUWIDOUVJUWDUVDUWFUWGWHWGZUVHUVJUVOCEFUVSWIWJUVD - CWKLZUVTWLZUWHUWNUVMTZCWRZUXCUWHUXEUVTUXCUWHRZUVHCWMPZUVJCWMPZNZUWNUVMUXG - UXCUWSUWTUXJUWNTUXCUWHWDUXGDOUVHUWDUXCUWFUWGWFZWGZUXGDOUVJUWDUXCUWFUWGWHZ - WGZUVHUVJCWNWJUXGUXHUVHUXIUVJUXGUVHWSLCWOLZQUVHWPSZUVHCWQSZUXHUVHNUXGUVHU - XLWTUXCUXOUWHCXAZXBZUXGUVHUWALZUXPUXGUVHDUWAUXKUXCDUWANZUWHUXCDUWBUWAIUXC - CQXCUWBUWANCXDCQOUWAXEXFWBZXBZXJZUVHQCXGXFUXGUXTUXQUYDUVHQCXHXFUVHCXIXKUX - GUVJWSLUXOQUVJWPSZUVJCWQSZUXIUVJNUXGUVJUXNWTUXSUXGUVJUWALZUYEUXGUVJDUWAUX - MUYCXJZUVJQCXGXFUXGUYGUYFUYHUVJQCXHXFUVJCXIXKWEXLUVTUWHRZUWNQUWMVRSZUWMQN - ZUVMUYICQUWMVRUVTUWHWDXMUYIUWMOLUYJUYKTUYIUVHUVJUVTUVDUWHUWSUVTCQUBUVTXTX - NXOZUXAXPZUVTUVDUWHUWTUYLUXBXPZXQUWMXRXFUYIUVHUVJUYIUVHUYMYAUYIUVJUYNYAXS - YBYCYDYBYEYFJKDABYGYHUVDUVGUVHUVKNZKDYKZJAUHZUVFUWEUVDUYPJUVOYIZUHZUYQUVD - UYSUAUIZUVOMZUVKNZKDYKZUAOUHZUVDVUCUAOUVDUYTOLZRVUAUWKNZKDYKZVUCUVDUXDVUE - VUGUXFUXCVUEVUGUVTUXCVUERZUYTCWMPZDLVUAVUIUVOMZNZVUGVUHVUIUWADVUEUXCVUIUW - ALUYTCYJYLZUXCUYAVUEUYBXBYMVUHVUJVUAVUHVUJVUANZCVUIUYTVQPVRSZVUHVUICWMPVU - INZVUNVUEUYTWSLUXOVUOUXCUYTYNUXRUYTCUUAUUBVUHUXCVUIOLZVUEVUOVUNTUXCVUEWDV - UHUWAOVUIUWCVULWGZUXCVUEUUCZVUIUYTCWNWJYOVUHUVDVUPVUEVUMVUNTUXCUVDVUECUUD - XBVUQVURVUIUYTUVOCEFUVSWIWJYPUUEVUFVUKKVUIDUVJVUINUWKVUJVUAUVJVUIUVOYQYRY - SYTUVTVUERZUYTDLVUAVUANZVUGVUSUYTUWBDUVTUYTUWBLVUEUVTUWBOUYTUVTOUWAUUFUUL - UUGIUUHVUSVUAUUMVUFVUTKUYTDKUAUGUWKVUAVUAUVJUYTUVOYQYRYSYTYCYDVUBVUFKDUWG - UVKUWKVUAUWGUVKUWPUWKUWQUWRWBYRUUIVPUUJUVDOAUVOUDZUVOOUUKUYSVUDTAUVOCEFGU - VSUUNZOAUVOUUOUYPVUCJUAOUVOUVHVUANUYOVUBKDUVHVUAUVKUUPUUQUURUUSYPUVDUYPJU - YRAUVDVVAUYRANVVBOAUVOUVCXFUUTYOKJDABUVAYHDABUVBYH $. + cv wi czrh cres wss ccrg crg zring crh zncrng crngring eqid zringbas rhmf + zrhrhm 4syl cfzo sseq1 ssid elfzoelz ssriv keephyp eqsstri fssres sylancl + cif feq1i sylibr cmin cdivides fveq1i fvres ad2antrl syl5eq eqeq12d simpl + ad2antll simprl sseldi simprr zndvds syl3anc cn elnn0 cmo moddvds crp cle + wo clt zred nnrp adantr wne nnne0 ifnefalse syl eleqtrd elfzole1 elfzolt2 + cr modid syl22anc bitr3d breq1d id 0nn0 syl6eqel sylan zsubcld 0dvds zcnd + subeq0ad 3bitrd jaoian sylanb biimpd ralrimivva sylanbrc wrex crn zmodfzo + dff13 ancoms eleqtrrd zre mpbid mpbird fveq2 eqeq2d rspcev syl2anc eqcomd + modabs2 syl2anr simpr nnnn0 iftrue eleq2d biimpar syl6eleqr eqidd rexbiia + ralrimiva znzrhfo fofn eqeq1 rexbidv ralrn 3syl forn raleqdv dffo3 df-f1o + wfn ) CUBLZDABUCZDABUDZDABUEUVDDABUFZJUIZBMZKUIZBMZNZJKUGZUJZKDUHJDUHUVEU + VDDAEUKMZDULZUFZUVGUVDOAUVOUFZDOUMUVQUVDEUNLEUOLUVOUPEUQPLUVRCEFUREUSEUVO + UVOUTZVCOAUPEUVOVAGVBVDDCQNZOQCVEPZVNZOIUVTOOUMUWAOUMUWBOUMOUWAOUWBOVFUWA + UWBOVFOVGJUWAOUVHQCVHVIZVJVKZOADUVOVLVMDABUVPHVOVPZUVDUVNJKDDUVDUVHDLZUVJ + DLZRZRZUVLUVMUWIUVLUVHUVOMZUVJUVOMZNZCUVHUVJVQPZVRSZUVMUWIUVIUWJUVKUWKUWI + UVIUVHUVPMZUWJUVHBUVPHVSUWFUWOUWJNUVDUWGUVHDUVOVTWAWBUWIUVKUVJUVPMZUWKUVJ + BUVPHVSZUWGUWPUWKNUVDUWFUVJDUVOVTZWEWBWCUWIUVDUVHOLZUVJOLZUWLUWNTUVDUWHWD + UWIDOUVHUWDUVDUWFUWGWFWGZUWIDOUVJUWDUVDUWFUWGWHWGZUVHUVJUVOCEFUVSWIWJUVDC + WKLZUVTWQZUWHUWNUVMTZCWLZUXCUWHUXEUVTUXCUWHRZUVHCWMPZUVJCWMPZNZUWNUVMUXGU + XCUWSUWTUXJUWNTUXCUWHWDUXGDOUVHUWDUXCUWFUWGWFZWGZUXGDOUVJUWDUXCUWFUWGWHZW + GZUVHUVJCWNWJUXGUXHUVHUXIUVJUXGUVHXILCWOLZQUVHWPSZUVHCWRSZUXHUVHNUXGUVHUX + LWSUXCUXOUWHCWTZXAZUXGUVHUWALZUXPUXGUVHDUWAUXKUXCDUWANZUWHUXCDUWBUWAIUXCC + QXBUWBUWANCXCCQOUWAXDXEWBZXAZXFZUVHQCXGXEUXGUXTUXQUYDUVHQCXHXEUVHCXJXKUXG + UVJXILUXOQUVJWPSZUVJCWRSZUXIUVJNUXGUVJUXNWSUXSUXGUVJUWALZUYEUXGUVJDUWAUXM + UYCXFZUVJQCXGXEUXGUYGUYFUYHUVJQCXHXEUVJCXJXKWCXLUVTUWHRZUWNQUWMVRSZUWMQNZ + UVMUYICQUWMVRUVTUWHWDXMUYIUWMOLUYJUYKTUYIUVHUVJUVTUVDUWHUWSUVTCQUBUVTXNXO + XPZUXAXQZUVTUVDUWHUWTUYLUXBXQZXRUWMXSXEUYIUVHUVJUYIUVHUYMXTUYIUVJUYNXTYAY + BYCYDYBYEYFJKDABYKYGUVDUVGUVHUVKNZKDYHZJAUHZUVFUWEUVDUYPJUVOYIZUHZUYQUVDU + YSUAUIZUVOMZUVKNZKDYHZUAOUHZUVDVUCUAOUVDUYTOLZRVUAUWKNZKDYHZVUCUVDUXDVUEV + UGUXFUXCVUEVUGUVTUXCVUERZUYTCWMPZDLVUAVUIUVOMZNZVUGVUHVUIUWADVUEUXCVUIUWA + LUYTCYJYLZUXCUYAVUEUYBXAYMVUHVUJVUAVUHVUJVUANZCVUIUYTVQPVRSZVUHVUICWMPVUI + NZVUNVUEUYTXILUXOVUOUXCUYTYNUXRUYTCUUBUUCVUHUXCVUIOLZVUEVUOVUNTUXCVUEWDVU + HUWAOVUIUWCVULWGZUXCVUEUUDZVUIUYTCWNWJYOVUHUVDVUPVUEVUMVUNTUXCUVDVUECUUEX + AVUQVURVUIUYTUVOCEFUVSWIWJYPUUAVUFVUKKVUIDUVJVUINUWKVUJVUAUVJVUIUVOYQYRYS + YTUVTVUERZUYTDLVUAVUANZVUGVUSUYTUWBDUVTUYTUWBLVUEUVTUWBOUYTUVTOUWAUUFUUGU + UHIUUIVUSVUAUUJVUFVUTKUYTDKUAUGUWKVUAVUAUVJUYTUVOYQYRYSYTYCYDVUBVUFKDUWGU + VKUWKVUAUWGUVKUWPUWKUWQUWRWBYRUUKVPUULUVDOAUVOUDZUVOOUVCUYSVUDTAUVOCEFGUV + SUUMZOAUVOUUNUYPVUCJUAOUVOUVHVUANUYOVUBKDUVHVUAUVKUUOUUPUUQUURYPUVDUYPJUY + RAUVDVVAUYRANVVBOAUVOUUSXEUUTYOKJDABUVAYGDABUVBYG $. $} ${ @@ -250502,11 +250728,11 @@ According to Wikipedia ("Integer", 25-May-2019, 21-Apr-2016.) (Revised by AV, 13-Jun-2019.) $) zzngim $p |- L e. ( ZZring GrpIso Y ) $= ( zring cgim co wcel cghm cbs cfv wf1o cc0 0nn0 mp2b eqid cres wceq ax-mp - cz crg crh cn0 ccrg zncrng crngrng zrhrhm rhmghm czrh wfo znzrhfo fnresdm - wfn fofn reseq1i eqtr3i cfzo iftruei eqcomi znf1o zringbas isgim mpbir2an - cif ) AEBFGHAEBIGHZTBJKZALZBUAHZAEBUBGHVEMUCHZBUDHVHNMBCUEBUFOBADUGEBAUHO - VIVGNVFAMTBCVFPZATQZABUIKZTQTVFAUJZATUMVKARVIVMNVFAMBCVJDUKSTVFAUNTAULOAV - LTDUOUPMMRZTMMUQGZVDTVNTVOMPURUSUTSTVFEBAVAVJVBVC $. + cz crg crh cn0 ccrg zncrng crngring zrhrhm rhmghm wfo wfn znzrhfo fnresdm + czrh fofn reseq1i eqtr3i cif iftruei eqcomi znf1o zringbas isgim mpbir2an + cfzo ) AEBFGHAEBIGHZTBJKZALZBUAHZAEBUBGHVEMUCHZBUDHVHNMBCUEBUFOBADUGEBAUH + OVIVGNVFAMTBCVFPZATQZABUMKZTQTVFAUIZATUJVKARVIVMNVFAMBCVJDUKSTVFAUNTAULOA + VLTDUOUPMMRZTMMVDGZUQTVNTVOMPURUSUTSTVFEBAVAVJVBVC $. $} ${ @@ -250617,30 +250843,30 @@ According to Wikipedia ("Integer", 25-May-2019, znfld $p |- ( N e. Prime -> Y e. Field ) $= ( vx vy vz vw wcel syl cv cfv co wceq wo c2o wbr c2 wb wa cz cprime cidom cfield ccrg cdomn cn0 prmnn nnnn0 zncrng cnzr cmulr c0g cbs wral crg cdom - cn wi crngrng 4syl chash cle hash2 cuz prmuz2 eluzle eqid znhash breqtrrd - syl5eqbr cfn cvv 2onn nnfi ax-mp fvex hashdom mp2an sylib isnzr2 sylanbrc - com czrh wrex wfo znzrhfo foelrn anim12dan sylan reeanv cdivides euclemma - cmul 3expb zring crh adantr zrhrhm simprl simprr zringmulr rhmmul syl3anc + cn crngring 4syl chash cle hash2 cuz prmuz2 eluzle eqid breqtrrd syl5eqbr + wi znhash cfn cvv com 2onn nnfi ax-mp hashdom mp2an sylib isnzr2 sylanbrc + fvex czrh wrex wfo znzrhfo foelrn anim12dan reeanv cmul cdivides euclemma + sylan 3expb zring crh adantr zrhrhm simprl simprr zringbas rhmmul syl3anc eqeq1d zmulcl zndvds0 syl2an bitr3d syl2anc orbi12d 3bitr4d biimpd oveq12 - zringbas orbi1d orbi2d sylan9bb imbi12d syl5ibrcom rexlimdvva syl5bir imp - eqeq1 syldan ralrimivva isdomn isidom znfi fiidomfld mpbid ) AUAHZBUBHZBU - CHZYKBUDHZBUEHZYLYKAUFHZYNYKAUQHZYPAUGZAUHZIZABCUIZIYKBUJHZDJZEJZBUKKZLZB - ULKZMZUUCUUGMZUUDUUGMZNZURZEBUMKZUNDUUMUNYOYKBUOHZOUUMUPPZUUBYKYQYPYNUUNY - RYSUUABUSUTZYKOVAKZUUMVAKZVBPZUUOYKUUQQUURVBVCYKQAUURVBYKAQVDKHQAVBPAVEQA - VFIYKYQUURAMYRUUMABCUUMVGZVHIVIVJOVKHZUUMVLHUUSUUOROWBHUVAVMOVNVOBUMVPOUU - MVLVQVRVSUUMBUUTVTWAYKUULDEUUMUUMYKUUCUUMHZUUDUUMHZSZUUCFJZBWCKZKZMZFTWDZ - UUDGJZUVFKZMZGTWDZSZUULYKTUUMUVFWEZUVDUVNYKYPUVOYTUUMUVFABCUUTUVFVGZWFIUV - OUVBUVIUVCUVMFTUUMUUCUVFWGGTUUMUUDUVFWGWHWIYKUVNUULUVNUVHUVLSZGTWDFTWDYKU - ULUVHUVLFGTTWJYKUVQUULFGTTYKUVETHZUVJTHZSZSZUULUVQUVGUVKUUELZUUGMZUVGUUGM - ZUVKUUGMZNZURUWAUWCUWFUWAAUVEUVJWMLZWKPZAUVEWKPZAUVJWKPZNZUWCUWFYKUVRUVSU - WHUWKRAUVEUVJWLWNUWAUWGUVFKZUUGMZUWCUWHUWAUWLUWBUUGUWAUVFWOBWPLHZUVRUVSUW - LUWBMUWAUUNUWNYKUUNUVTUUPWQBUVFUVPWRIYKUVRUVSWSZYKUVRUVSWTZUVEUVJWOBWMUUE - UVFTXNXAUUEVGZXBXCXDYKYPUWGTHUWMUWHRUVTYTUVEUVJXEUWGUVFABUUGCUVPUUGVGZXFX - GXHUWAUWDUWIUWEUWJUWAYPUVRUWDUWIRYKYPUVTYTWQZUWOUVEUVFABUUGCUVPUWRXFXIUWA - YPUVSUWEUWJRUWSUWPUVJUVFABUUGCUVPUWRXFXIXJXKXLUVQUUHUWCUUKUWFUVQUUFUWBUUG - UUCUVGUUDUVKUUEXMXDUVHUUKUWDUUJNUVLUWFUVHUUIUWDUUJUUCUVGUUGYCXOUVLUUJUWEU - WDUUDUVKUUGYCXPXQXRXSXTYAYBYDYEDEUUMBUUEUUGUUTUWQUWRYFWABYGWAYKUUMVKHZYLY - MRYKYQUWTYRUUMABCUUTYHIUUMBUUTYIIYJ $. + zringmulr eqeq1 orbi1d sylan9bb imbi12d syl5ibrcom rexlimdvva syl5bir imp + orbi2d syldan ralrimivva isdomn isidom znfi fiidomfld mpbid ) AUAHZBUBHZB + UCHZYKBUDHZBUEHZYLYKAUFHZYNYKAUQHZYPAUGZAUHZIZABCUIZIYKBUJHZDJZEJZBUKKZLZ + BULKZMZUUCUUGMZUUDUUGMZNZVIZEBUMKZUNDUUMUNYOYKBUOHZOUUMUPPZUUBYKYQYPYNUUN + YRYSUUABURUSZYKOUTKZUUMUTKZVAPZUUOYKUUQQUURVAVBYKQAUURVAYKAQVCKHQAVAPAVDQ + AVEIYKYQUURAMYRUUMABCUUMVFZVJIVGVHOVKHZUUMVLHUUSUUOROVMHUVAVNOVOVPBUMWBOU + UMVLVQVRVSUUMBUUTVTWAYKUULDEUUMUUMYKUUCUUMHZUUDUUMHZSZUUCFJZBWCKZKZMZFTWD + ZUUDGJZUVFKZMZGTWDZSZUULYKTUUMUVFWEZUVDUVNYKYPUVOYTUUMUVFABCUUTUVFVFZWFIU + VOUVBUVIUVCUVMFTUUMUUCUVFWGGTUUMUUDUVFWGWHWMYKUVNUULUVNUVHUVLSZGTWDFTWDYK + UULUVHUVLFGTTWIYKUVQUULFGTTYKUVETHZUVJTHZSZSZUULUVQUVGUVKUUELZUUGMZUVGUUG + MZUVKUUGMZNZVIUWAUWCUWFUWAAUVEUVJWJLZWKPZAUVEWKPZAUVJWKPZNZUWCUWFYKUVRUVS + UWHUWKRAUVEUVJWLWNUWAUWGUVFKZUUGMZUWCUWHUWAUWLUWBUUGUWAUVFWOBWPLHZUVRUVSU + WLUWBMUWAUUNUWNYKUUNUVTUUPWQBUVFUVPWRIYKUVRUVSWSZYKUVRUVSWTZUVEUVJWOBWJUU + EUVFTXAXNUUEVFZXBXCXDYKYPUWGTHUWMUWHRUVTYTUVEUVJXEUWGUVFABUUGCUVPUUGVFZXF + XGXHUWAUWDUWIUWEUWJUWAYPUVRUWDUWIRYKYPUVTYTWQZUWOUVEUVFABUUGCUVPUWRXFXIUW + AYPUVSUWEUWJRUWSUWPUVJUVFABUUGCUVPUWRXFXIXJXKXLUVQUUHUWCUUKUWFUVQUUFUWBUU + GUUCUVGUUDUVKUUEXMXDUVHUUKUWDUUJNUVLUWFUVHUUIUWDUUJUUCUVGUUGXOXPUVLUUJUWE + UWDUUDUVKUUGXOYCXQXRXSXTYAYBYDYEDEUUMBUUEUUGUUTUWQUWRYFWABYGWAYKUUMVKHZYL + YMRYKYQUWTYRUUMABCUUTYHIUUMBUUTYIIYJ $. $( The ` Z/nZ ` structure is a domain (and hence a field) precisely when ` n ` is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) $) @@ -250650,33 +250876,33 @@ According to Wikipedia ("Integer", 25-May-2019, hash2 cdom cnzr cdomn ccrg isidom simprbi domnnzr crg eqid isnzr2 cfn cvv adantl c0 csn cpr df2o2 prfi eqeltri fvex hashdom sylibr syl5eqbrr znhash mp2an breqtrd eluz2 syl3anbrc cdiv czrh cmulr cmul cc nncn nnne0 divcan1d - c0g wne fveq2d zring crh ad2antlr domnrng zrhrhm simprr dvdsval2 zringbas - syl3anc mpbid zringmulr rhmmul iddvds cn0 nnnn0 zndvds0 syl2anc mpbird wf - 3eqtr3d rhmf ffvelrnd domneq0 cr nnre nngt0 divgt0d elnnz sylanbrc dvdsle - clt 1red 0lt1 lediv2 syl222anc nnle1eq1 div1d breq1d sylibd sylbid dvdseq - 3bitr3rd expr syl21anc orim12d ralrimiva isprm2 ex cfield fldidom impbid1 - mpd znfld ) AEFZBUAFZAUBFZUUIUUJUUKUUIUUJUCZAGUDHFZDUEZAUFIZUUNJKZUUNAKZU - GZUHZDEUIUUKUULGLFZALFZGAMIUUMUUTUULUJUKUUIUVAUUJAULZUMUULGBUNHZUOHZAMUUL - GNUOHZUVDMUPUULNUVCUQIZUVEUVDMIZUUJUVFUUIUUJBURFZUVFUUJBUSFZUVHUUJBUTFUVI - BVAVBZBVCOUVHBVDFZUVFUVCBUVCVEZVFVBOVINVGFUVCVHFUVGUVFPNVJVJVKZVLVGVMVJUV - MVNVOBUNVPNUVCVHVQWAVRVSUUIUVDAKUUJUVCABCUVLVTUMWBGAWCWDUULUUSDEUULUUNEFZ - UUOUURUULUVNUUOUCZUCZAUUNWETZBWFHZHZBWMHZKZUUNUVRHZUVTKZUGZUURUVPUVSUWBBW - GHZTZUVTKZUWDUVPUVQUUNWHTZUVRHZAUVRHZUWFUVTUVPUWHAUVRUVPAUUNUUIAWIFUUJUVO - AWJQZUVNUUNWIFUULUUOUUNWJRUVNUUNSWNZUULUUOUUNWKRZWLWOUVPUVRWPBWQTFZUVQLFZ - UUNLFZUWIUWFKUVPUVKUWNUVPUVIUVKUUJUVIUUIUVOUVJWRZBWSOBUVRUVRVEZWTOZUVPUUO - UWOUULUVNUUOXAZUVPUWPUWLUVAUUOUWOPUVNUWPUULUUOUUNULRZUWMUUIUVAUUJUVOUVBQZ - UUNAXBXDXEZUXAUVQUUNWPBWHUWEUVRLXCXFUWEVEZXGXDUVPUWJUVTKZAAUFIZUVPUVAUXFU - XBAXHOUVPAXIFZUVAUXEUXFPUUIUXGUUJUVOAXJQZUXBAUVRABUVTCUWRUVTVEZXKXLXMXOUV - PUVIUVSUVCFUWBUVCFUWGUWDPUWQUVPLUVCUVQUVRUVPUWNLUVCUVRXNUWSLUVCWPBUVRXCUV - LXPOZUXCXQUVPLUVCUUNUVRUXJUXAXQUVCBUWEUVSUWBUVTUVLUXDUXIXRXDXEUVPUWAUUPUW - CUUQUVPUWAAUVQUFIZUUPUVPUXGUWOUWAUXKPUXHUXCUVQUVRABUVTCUWRUXIXKXLUVPUXKAU - VQMIZUUPUVPUVAUVQEFZUXKUXLUHUXBUVPUWOSUVQYFIUXMUXCUVPAUUNUUIAXSFZUUJUVOAX - TQZUVNUUNXSFZUULUUOUUNXTRZUUISAYFIZUUJUVOAYAQZUVNSUUNYFIZUULUUOUUNYARZYBU - VQYCYDAUVQYEXLUVPUUNJMIZAJWETZUVQMIZUUPUXLUVPUXPUXTJXSFSJYFIZUXNUXRUYBUYD - PUXQUYAUVPYGUYEUVPYHUKUXOUXSUUNJAYIYJUVNUYBUUPPUULUUOUUNYKRUVPUYCAUVQMUVP - AUWKYLYMYQYNYOUVPUWCAUUNUFIZUUQUVPUXGUWPUWCUYFPUXHUXAUUNUVRABUVTCUWRUXIXK - XLUVPUUNXIFZUXGUUOUYFUUQUHUVNUYGUULUUOUUNXJRUXHUWTUYGUXGUCUUOUYFUUQUUNAYP - YRYSYOYTUUGYRUUADAUUBYDUUCUUKBUUDFUUJABCUUHBUUEOUUF $. + c0g wne fveq2d zring crh ad2antlr domnring zrhrhm simprr dvdsval2 syl3anc + mpbid zringbas zringmulr rhmmul iddvds cn0 zndvds0 syl2anc mpbird 3eqtr3d + nnnn0 wf rhmf ffvelrnd domneq0 clt cr nngt0 divgt0d elnnz sylanbrc dvdsle + nnre 1red 0lt1 lediv2 nnle1eq1 div1d breq1d 3bitr3rd sylibd sylbid dvdseq + syl222anc expr syl21anc orim12d mpd ralrimiva isprm2 cfield znfld fldidom + ex impbid1 ) AEFZBUAFZAUBFZUUIUUJUUKUUIUUJUCZAGUDHFZDUEZAUFIZUUNJKZUUNAKZ + UGZUHZDEUIUUKUULGLFZALFZGAMIUUMUUTUULUJUKUUIUVAUUJAULZUMUULGBUNHZUOHZAMUU + LGNUOHZUVDMUPUULNUVCUQIZUVEUVDMIZUUJUVFUUIUUJBURFZUVFUUJBUSFZUVHUUJBUTFUV + IBVAVBZBVCOUVHBVDFZUVFUVCBUVCVEZVFVBOVINVGFUVCVHFUVGUVFPNVJVJVKZVLVGVMVJU + VMVNVOBUNVPNUVCVHVQWAVRVSUUIUVDAKUUJUVCABCUVLVTUMWBGAWCWDUULUUSDEUULUUNEF + ZUUOUURUULUVNUUOUCZUCZAUUNWETZBWFHZHZBWMHZKZUUNUVRHZUVTKZUGZUURUVPUVSUWBB + WGHZTZUVTKZUWDUVPUVQUUNWHTZUVRHZAUVRHZUWFUVTUVPUWHAUVRUVPAUUNUUIAWIFUUJUV + OAWJQZUVNUUNWIFUULUUOUUNWJRUVNUUNSWNZUULUUOUUNWKRZWLWOUVPUVRWPBWQTFZUVQLF + ZUUNLFZUWIUWFKUVPUVKUWNUVPUVIUVKUUJUVIUUIUVOUVJWRZBWSOBUVRUVRVEZWTOZUVPUU + OUWOUULUVNUUOXAZUVPUWPUWLUVAUUOUWOPUVNUWPUULUUOUUNULRZUWMUUIUVAUUJUVOUVBQ + ZUUNAXBXCXDZUXAUVQUUNWPBWHUWEUVRLXEXFUWEVEZXGXCUVPUWJUVTKZAAUFIZUVPUVAUXF + UXBAXHOUVPAXIFZUVAUXEUXFPUUIUXGUUJUVOAXNQZUXBAUVRABUVTCUWRUVTVEZXJXKXLXMU + VPUVIUVSUVCFUWBUVCFUWGUWDPUWQUVPLUVCUVQUVRUVPUWNLUVCUVRXOUWSLUVCWPBUVRXEU + VLXPOZUXCXQUVPLUVCUUNUVRUXJUXAXQUVCBUWEUVSUWBUVTUVLUXDUXIXRXCXDUVPUWAUUPU + WCUUQUVPUWAAUVQUFIZUUPUVPUXGUWOUWAUXKPUXHUXCUVQUVRABUVTCUWRUXIXJXKUVPUXKA + UVQMIZUUPUVPUVAUVQEFZUXKUXLUHUXBUVPUWOSUVQXSIUXMUXCUVPAUUNUUIAXTFZUUJUVOA + YFQZUVNUUNXTFZUULUUOUUNYFRZUUISAXSIZUUJUVOAYAQZUVNSUUNXSIZUULUUOUUNYARZYB + UVQYCYDAUVQYEXKUVPUUNJMIZAJWETZUVQMIZUUPUXLUVPUXPUXTJXTFSJXSIZUXNUXRUYBUY + DPUXQUYAUVPYGUYEUVPYHUKUXOUXSUUNJAYIYQUVNUYBUUPPUULUUOUUNYJRUVPUYCAUVQMUV + PAUWKYKYLYMYNYOUVPUWCAUUNUFIZUUQUVPUXGUWPUWCUYFPUXHUXAUUNUVRABUVTCUWRUXIX + JXKUVPUUNXIFZUXGUUOUYFUUQUHUVNUYGUULUUOUUNXNRUXHUWTUYGUXGUCUUOUYFUUQUUNAY + PYRYSYOYTUUAYRUUBDAUUCYDUUGUUKBUUDFUUJABCUUEBUUFOUUH $. $} ${ @@ -250687,11 +250913,11 @@ According to Wikipedia ("Integer", 25-May-2019, Stefan O'Rear, 6-Sep-2015.) $) znchr $p |- ( N e. NN0 -> ( chr ` Y ) = N ) $= ( vx cn0 wcel cchr cfv wceq cv cdivides wbr wb wral czrh c0g crg syl eqid - wa ccrg zncrng crngrng nn0z chrdvds syl2an zndvds0 sylan2 bitrd ralrimiva - cz chrcl dvdsext mpancom mpbird ) AEFZBGHZAIZUQDJZKLZAUSKLZMZDENZUPVBDEUP - USEFZTUTUSBOHZHBPHZIZVAUPBQFZUSUKFZUTVGMVDUPBUAFVHABCUBBUCRZUSUDZUQBVEUSV - FUQSZVESZVFSZUEUFVDUPVIVGVAMVKUSVEABVFCVMVNUGUHUIUJUQEFZUPURVCMUPVHVOVJUQ - BVLULRDUQAUMUNUO $. + wa cz ccrg zncrng crngring nn0z chrdvds syl2an zndvds0 sylan2 bitrd chrcl + ralrimiva dvdsext mpancom mpbird ) AEFZBGHZAIZUQDJZKLZAUSKLZMZDENZUPVBDEU + PUSEFZTUTUSBOHZHBPHZIZVAUPBQFZUSUAFZUTVGMVDUPBUBFVHABCUCBUDRZUSUEZUQBVEUS + VFUQSZVESZVFSZUFUGVDUPVIVGVAMVKUSVEABVFCVMVNUHUIUJULUQEFZUPURVCMUPVHVOVJU + QBVLUKRDUQAUMUNUO $. znunit.u $e |- U = ( Unit ` Y ) $. ${ @@ -250703,38 +250929,38 @@ According to Wikipedia ("Integer", 25-May-2019, ( vn wcel cz wa cfv wbr co wceq wrex c1 syl cdivides vx vm cn0 cur cdsr cv cmulr cbs cgcd ccrg wb zncrng adantr crngunit wf wfo znzrhfo ffvelrn eqid fof sylancom dvdsr2 cmul cmin crn forn rexeqdv wfn ffn oveq1 rexrn - eqeq1d 3syl bitr3d zring crh crg zrhrhm simpr simplr zringbas zringmulr - crngrng rhmmul syl3anc zrh1 eqeq12d simpll zmulcld 1zzd zndvds rexbidva - nn0z ad2antrr gcddvds syl2anc simpld wi gcdcld nn0zd adantrr dvdsmultr2 - simprd simprr peano2zm dvdstr mp2and dvdssub2 syl31anc mpbid rexlimdvaa - mpd dvds1 caddc bezout eqeq1 2rexbidv syl5ibcom cneg ad3antrrr dvdsmul1 - zmulcl dvdsnegb zcnd cc zcn mulcomd oveq1d mulcld subnegd eqtr4d oveq2d - ad2antlr negcld nncand eqtrd breqtrrd oveq2 breq2d 3bitrd reximdva syld - syl5ibrcom rexlimdva impbid ) DUCJZAKJZLZACMZBJZUUIEUDMZEUEMZNZUAUFZUUI - EUGMZOZUUKPZUAEUHMZQZADUIOZRPZUUHEUJJZUUJUUMUKUUFUVBUUGDEFULUMZUULEBUUK - UUIGUUKUSZUULUSZUNSUUHUUIUURJZUUMUUSUKUUFUUGKUURCUOZUVFUUHKUURCUPZUVGUU - FUVHUUGUURCDEFUURUSZHUQUMZKUURCUTSZKUURACURVAUAUURUULEUUOUUIUUKUVIUVEUU - OUSZVBSUUHUUSIUFZCMZUUIUUOOZUUKPZIKQZDUVMAVCOZRVDOZTNZIKQZUVAUUHUUQUACV - EZQZUUSUVQUUHUUQUAUWBUURUUHUVHUWBUURPUVJKUURCVFSVGUUHUVGCKVHUWCUVQUKUVK - KUURCVIUUQUVPUAIKCUUNUVNPUUPUVOUUKUUNUVNUUIUUOVJVLVKVMVNUUHUVPUVTIKUUHU - 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(Contributed by Mario @@ -250765,28 +250991,28 @@ According to Wikipedia ("Integer", 25-May-2019, cn0 wa cgcd cdiv cc nncn cc0 simplr nnz wne nnne0 simpr necon3ai syl21anc wn gcdn0cl nncnd nnne0d divcan2d gcddvds simpld simprd nndivdvds dvdsmulc simpll mpbid syl3anc mpd eqbrtrrd cmulr c0g wf fof ffvelrnd rrgeq0i zring - crh crg ccrg zncrng crngrng zrhrhm zringbas rhmmul eqeq1d zmulcld zndvds0 - zringmulr bitr3d 3imtr3d divcan1d mulid1d 1zzd dvdscmulr syl112anc gcdcld - 3brtr4d dvds1 znunit mpbird eleq1 imbi12d syl5ibrcom rexlimdva com23 mpdd - ssrdv wss unitrrg eqssd ) CUDHZBAYHUABAYHUAUEZBHZYIUCUEZDUFIZIZJZUCKUGZYI - AHZYHYJYOYHKDUBIZYLUHZYIYQHYOYJYHCURHZYRCUIZYQYLCDEYQUJZYLUJZUKZLBYQYIYQD - BGUUAULUMUCKYQYIYLUNUOUPYHYOYJYPYHYNYJYPUQZUCKYHYKKHZUSZUUDYNYMBHZYMAHZUQ - UUFUUGUUHUUFUUGUSZUUHYKCUTMZNJZUUIUUJNOPZUUKUUICUUJVAMZUUJQMZUUMNQMZOPZUU - LUUICUUMUUNUUOOUUICYKUUMQMZOPZCUUMOPZUUIUUJUUMQMZCUUQOUUICUUJYHCVBHUUEUUG - CVCRZUUIUUJUUIUUECKHZYKVDJZCVDJZUSZVLZUUJUDHZYHUUEUUGVEZYHUVBUUEUUGCVFRZU - 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( ( SymGrp ` A ) MndHom ( mulGrp ` R ) ) ) $= ( crg wcel cfv zring cmgp cmhm co eqid syl ccnfld c1 cress csubmnd cz wss cc cc0 wa czrh cpsgn csymg ccom cfn crh zrhrhm rhmmhm cpr psgnghm2 ghmmhm - cneg cghm csn cdif csubg cnmsgnsubg subgsubm ax-mp wb cnrng cnfld0 cndrng - cnfldbas drngui unitsubm subsubm mp2b simpli 1z neg1z prssi csubrg zsubrg - mpbi mp2an subrgsubm zringmpg eqcomi mpbir2an cvv wceq zex ressabs oveq1i - eqtr3i resmhm2 sylancl mhmco syl2an ) BCDZBUAEZFGEZBGEZHIDZAUBEZAUCEZWMHI - DZWLWPUDWQWNHIDAUEDZWKWLFBUFIDWOBWLWLJUGFBWLWMWNWMJWNJUHKWSWPWQLGEZMMULZU - IZNIZHIDZXBWMOEDZWRWSWPWQXCUMIDXDAWQXCWPWQJWPJXCJUJWQXCWPUKKXEXBWTOEZDZXB - PQZXGXBRSUNUOZQZXBWTXINIZOEDZXGXJTZXBXKUPEDXLXKXKJZUQXBXKURUSLCDXIXFDXLXM - UTVALXIWTRLSVDVBVCVEWTJZVFXBXIWTXKXNVGVHVOVIMPDXAPDXHVJVKMXAPVLVPZPLVMEDP - XFDXEXGXHTUTVNPLWTXOVQXBPWTWMWTPNIZWMVRVSVGVHVTWQWMXCWPXBXQXBNIZXCWMXBNIP - WADXHXRXCWBWCXPPXBWTWAWDVPXQWMXBNVRWEWFWGWHWQWMWNWLWPWIWJ $. + cneg cghm csn cdif csubg cnmsgnsubg subgsubm ax-mp cnring cnfldbas cnfld0 + wb cndrng drngui unitsubm subsubm mp2b simpli 1z neg1z prssi mp2an csubrg + mpbi zsubrg subrgsubm zringmpg eqcomi mpbir2an cvv wceq zex oveq1i eqtr3i + ressabs resmhm2 sylancl mhmco syl2an ) BCDZBUAEZFGEZBGEZHIDZAUBEZAUCEZWMH + IDZWLWPUDWQWNHIDAUEDZWKWLFBUFIDWOBWLWLJUGFBWLWMWNWMJWNJUHKWSWPWQLGEZMMULZ + UIZNIZHIDZXBWMOEDZWRWSWPWQXCUMIDXDAWQXCWPWQJWPJXCJUJWQXCWPUKKXEXBWTOEZDZX + BPQZXGXBRSUNUOZQZXBWTXINIZOEDZXGXJTZXBXKUPEDXLXKXKJZUQXBXKURUSLCDXIXFDXLX + MVCUTLXIWTRLSVAVBVDVEWTJZVFXBXIWTXKXNVGVHVOVIMPDXAPDXHVJVKMXAPVLVMZPLVNED + PXFDXEXGXHTVCVPPLWTXOVQXBPWTWMWTPNIZWMVRVSVGVHVTWQWMXCWPXBXQXBNIZXCWMXBNI + PWADXHXRXCWBWCXPPXBWTWAWFVMXQWMXBNVRWDWEWGWHWQWMWNWLWPWIWJ $. zrhpsgninv.p $e |- P = ( Base ` ( SymGrp ` N ) ) $. zrhpsgninv.y $e |- Y = ( ZRHom ` R ) $. @@ -251282,13 +251508,13 @@ According to Wikipedia ("Integer", 25-May-2019, zrhpsgnelbas $p |- ( ( R e. Ring /\ N e. Fin /\ Q e. P ) -> ( Y ` ( S ` Q ) ) e. ( Base ` R ) ) $= ( cfv c1 wcel wceq eqid eqeltrd 3ad2ant1 fveq2 eleq1d syl5ibr neg1z wo wi - cneg cpr crg cfn w3a cbs psgnran 3adant1 elpri cur zrh1 rngidcl cmg co cz - zrhmulg mpan2 cgrp rnggrp a1i mulgcl syl3anc jaoi syl mpcom ) BDJZKKUCZUD - LZCUELZEUFLZBALZUGZVHFJZCUHJZLZVLVMVJVKABDEGHUIUJVJVHKMZVHVIMZUAVNVQUBZVH - KVIUKVRVTVSVNVQVRKFJZVPLZVKVLWBVMVKWACULJZVPCWCFIWCNZUMVPCWCVPNZWDUNZOPVR - VOWAVPVHKFQRSVNVQVSVIFJZVPLZVKVLWHVMVKWGVIWCCUOJZUPZVPVKVIUQLZWGWJMTCWIWC - FVIIWINZWDURUSVKCUTLWKWCVPLWJVPLCVAWKVKTVBWFVPWICVIWCWEWLVCVDOPVSVOWGVPVH - VIFQRSVEVFVG $. + cneg cpr crg cfn w3a cbs psgnran 3adant1 cur zrh1 ringidcl cmg co zrhmulg + elpri cz mpan2 cgrp ringgrp a1i mulgcl syl3anc jaoi syl mpcom ) BDJZKKUCZ + UDLZCUELZEUFLZBALZUGZVHFJZCUHJZLZVLVMVJVKABDEGHUIUJVJVHKMZVHVIMZUAVNVQUBZ + VHKVIUQVRVTVSVNVQVRKFJZVPLZVKVLWBVMVKWACUKJZVPCWCFIWCNZULVPCWCVPNZWDUMZOP + VRVOWAVPVHKFQRSVNVQVSVIFJZVPLZVKVLWHVMVKWGVIWCCUNJZUOZVPVKVIURLZWGWJMTCWI + WCFVIIWINZWDUPUSVKCUTLWKWCVPLWJVPLCVAWKVKTVBWFVPWICVIWCWEWLVCVDOPVSVOWGVP + VHVIFQRSVEVFVG $. $( Embedding of permutation signs into a ring results in an element of the ring. (Contributed by AV, 1-Jan-2019.) $) @@ -251571,9 +251797,9 @@ According to Wikipedia ("Integer", 25-May-2019, $( The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) $) re0g $p |- 0 = ( 0g ` RRfld ) $= - ( ccnfld cmnd wcel cc0 cr cc wss crefld c0g cfv wceq ccrg crg cncrng rngmnd - crngrng mp2b 0re ax-resscn df-refld cnfldbas cnfld0 ress0g mp3an ) ABCZDECE - FGDHIJKALCAMCUENAPAOQRSEFAHDTUAUBUCUD $. + ( ccnfld cmnd wcel cc0 cr wss crefld c0g cfv wceq ccrg crg crngring ringmnd + cc cncrng mp2b 0re ax-resscn df-refld cnfldbas cnfld0 ress0g mp3an ) ABCZDE + CEOFDGHIJAKCALCUEPAMANQRSEOAGDTUAUBUCUD $. $( The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.) $) @@ -251646,16 +251872,16 @@ According to Wikipedia ("Integer", 25-May-2019, regsumsupp $p |- ( ( F : I --> RR /\ F finSupp 0 /\ I e. V ) -> ( RRfld gsum F ) = sum_ x e. ( F supp 0 ) ( F ` x ) ) $= ( cr wf cc0 wcel crefld co ccnfld cc cnfldbas wss ax-resscn a1i caddc cvv - cgsu wceq cfsupp wbr w3a csupp cres cv cfv cmpt csu cnfld0 crg ccmn cnrng - mp1i simp3 simp1 fss sylancl ssid simp2 gsumres cnfldadd df-refld cnfldex - rngcmn 0red wa simpr addid2d addid1d jca gsumress eqtr2d cdm suppssdm fdm - syl syl5sseq feqresmpt oveq2d id fsuppimpd 3ad2ant2 simpl1 ffvelrnd recnd - cfn sselda gsumfsum 3eqtrd ) CEBFZBGUAUBZCDHZUCZIBSJZKBBGUDJZUEZSJZKAWPAU - FZBUGZUHZSJWPWTAUIWNWRKBSJWOWNCLBKDWPGMUJKUKHKULHWNUMKVEUNWKWLWMUOZWNWKEL - NZCLBFWKWLWMUPZOCELBUQURWPWPNWNWPUSPWKWLWMUTVAWNACLQEBKIRDGMVBVCKRHWNVDPX - BXCWNOPXDWNVFWNWSLHZVGZGWSQJWSTWSGQJWSTXFWSWNXEVHZVIXFWSXGVJVKVLVMWNWQXAK - SWNACEWPBXDWNBVNZWPCBGVOWNWKXHCTXDCEBVPVQVRZVSVTWNWPWTAWLWKWPWGHWMWLBGWLW - AWBWCWNWSWPHZVGZWTXKCEWSBWKWLWMXJWDWNWPCWSXIWHWEWFWIWJ $. + cgsu wceq cfsupp wbr w3a csupp cres cv cfv cmpt csu cnfld0 cnring ringcmn + crg ccmn mp1i simp3 simp1 fss sylancl ssid simp2 gsumres cnfldadd cnfldex + df-refld 0red wa addid2d addid1d jca gsumress eqtr2d cdm suppssdm fdm syl + simpr syl5sseq feqresmpt oveq2d cfn id fsuppimpd 3ad2ant2 simpl1 ffvelrnd + sselda recnd gsumfsum 3eqtrd ) CEBFZBGUAUBZCDHZUCZIBSJZKBBGUDJZUEZSJZKAWP + AUFZBUGZUHZSJWPWTAUIWNWRKBSJWOWNCLBKDWPGMUJKUMHKUNHWNUKKULUOWKWLWMUPZWNWK + ELNZCLBFWKWLWMUQZOCELBURUSWPWPNWNWPUTPWKWLWMVAVBWNACLQEBKIRDGMVCVEKRHWNVD + PXBXCWNOPXDWNVFWNWSLHZVGZGWSQJWSTWSGQJWSTXFWSWNXEVQZVHXFWSXGVIVJVKVLWNWQX + AKSWNACEWPBXDWNBVMZWPCBGVNWNWKXHCTXDCEBVOVPVRZVSVTWNWPWTAWLWKWPWAHWMWLBGW + LWBWCWDWNWSWPHZVGZWTXKCEWSBWKWLWMXJWEWNWPCWSXIWGWFWHWIWJ $. $} $( @@ -251896,16 +252122,16 @@ According to Wikipedia ("Integer", 25-May-2019, ip2di $p |- ( ph -> ( ( A .+ B ) ., ( C .+ D ) ) = ( ( ( A ., C ) .+^ ( B ., D ) ) .+^ ( ( A ., D ) .+^ ( B ., C ) ) ) ) $= ( cphl wcel wceq clmod phllmod syl lmodvacl syl3anc ipdir syl13anc ipdi - ccmn cbs cfv csr crg phlsrng srngrng rngcmn 4syl eqid ipcl cmncom eqtrd - co oveq12d cmn4 syl122anc 3eqtrd ) ABCFVFDEFVFZIVFZBVKIVFZCVKIVFZGVFZBD - IVFZBEIVFZGVFZCEIVFZCDIVFZGVFZGVFZVPVSGVFVQVTGVFGVFZAKUBUCZBJUCZCJUCZVK - JUCZVLVOUDQRSAKUEUCZDJUCZEJUCZWGAWDWHQKUFUGTUAFJKDENOUHUIBCVKFGHIJKLMNO - PUJUKAVMVRVNWAGAWDWEWIWJVMVRUDQRTUABDEFGHIJKLMNOPULUKAVNVTVSGVFZWAAWDWF - WIWJVNWKUDQSTUACDEFGHIJKLMNOPULUKAHUMUCZVTHUNUOZUCZVSWMUCZWKWAUDAWDHUPU - CHUQUCWLQHKLURHUSHUTVAZAWDWFWIWNQSTCDHIWMJKLMNWMVBZVCUIZAWDWFWJWOQSUACE - HIWMJKLMNWQVCUIZWMGHVTVSWQPVDUIVEVGAWLVPWMUCZVQWMUCZWOWNWBWCUDWPAWDWEWI - WTQRTBDHIWMJKLMNWQVCUIAWDWEWJXAQRUABEHIWMJKLMNWQVCUIWSWRWMGHVTVPVQVSWQP - VHVIVJ $. + co ccmn cbs cfv csr crg phlsrng srngring ringcmn 4syl eqid cmncom eqtrd + ipcl oveq12d cmn4 syl122anc 3eqtrd ) ABCFUMDEFUMZIUMZBVKIUMZCVKIUMZGUMZ + BDIUMZBEIUMZGUMZCEIUMZCDIUMZGUMZGUMZVPVSGUMVQVTGUMGUMZAKUBUCZBJUCZCJUCZ + VKJUCZVLVOUDQRSAKUEUCZDJUCZEJUCZWGAWDWHQKUFUGTUAFJKDENOUHUIBCVKFGHIJKLM + NOPUJUKAVMVRVNWAGAWDWEWIWJVMVRUDQRTUABDEFGHIJKLMNOPULUKAVNVTVSGUMZWAAWD + WFWIWJVNWKUDQSTUACDEFGHIJKLMNOPULUKAHUNUCZVTHUOUPZUCZVSWMUCZWKWAUDAWDHU + QUCHURUCWLQHKLUSHUTHVAVBZAWDWFWIWNQSTCDHIWMJKLMNWMVCZVFUIZAWDWFWJWOQSUA + CEHIWMJKLMNWQVFUIZWMGHVTVSWQPVDUIVEVGAWLVPWMUCZVQWMUCZWOWNWBWCUDWPAWDWE + WIWTQRTBDHIWMJKLMNWQVFUIAWDWEWJXAQRUABEHIWMJKLMNWQVFUIWSWRWMGHVTVPVQVSW + QPVHVIVJ $. $} ${ @@ -251951,19 +252177,19 @@ According to Wikipedia ("Integer", 25-May-2019, Carneiro, 8-Oct-2015.) $) ip2subdi $p |- ( ph -> ( ( A .- B ) ., ( C .- D ) ) = ( ( ( A ., C ) .+ ( B ., D ) ) S ( ( A ., D ) .+ ( B ., C ) ) ) ) $= - ( cbs cfv eqid crg wcel cabl clmod cphl phllmod syl lmodrng rngabl ipcl - co syl3anc ablsubsub4 wceq lmodvsubcl ipsubdir syl13anc ipsubdi oveq12d - oveq1d cgrp rnggrp grpsubcl ablsubsub 3eqtrd rngacl abladdsub 3eqtr4d ) - ABDIUQZBEIUQZGUQZCDIUQZGUQZCEIUQZFUQZVOVPVRFUQZGUQZVTFUQZBCJUQDEJUQZIUQ - ZVOVTFUQWBGUQZAVSWCVTFAHUDUEZFHGVOVPVRWHUFZRQAHUGUHZHUIUHZALUJUHZWJALUK - UHZWLSLULUMZHLMUNUMZHUOUMZAWMBKUHZDKUHZVOWHUHZSTUBBDHIWHKLMNOWIUPURZAWM - WQEKUHZVPWHUHZSTUCBEHIWHKLMNOWIUPURZAWMCKUHZWRVRWHUHZSUAUBCDHIWHKLMNOWI - UPURZUSVFAWFBWEIUQZCWEIUQZGUQZVQVRVTGUQZGUQWAAWMWQXDWEKUHZWFXIUTSTUAAWL - WRXAXKWNUBUCJKLDEOPVAURBCWEGHIJKLMNOPQVBVCAXGVQXHXJGAWMWQWRXAXGVQUTSTUB - UCBDEGHIJKLMNOPQVDVCAWMXDWRXAXHXJUTSUAUBUCCDEGHIJKLMNOPQVDVCVEAWHFHGVQV - RVTWIRQWPAHVGUHZWSXBVQWHUHAWJXLWOHVHUMWTXCWHHGVOVPWIQVIURXFAWMXDXAVTWHU - HZSUAUCCEHIWHKLMNOWIUPURZVJVKAWKWSXMWBWHUHZWGWDUTWPWTXNAWJXBXEXOWOXCXFW - HFHVPVRWIRVLURWHFHGVOVTWBWIRQVMVCVN $. + ( co cbs cfv eqid crg wcel cabl clmod cphl phllmod syl lmodring ringabl + ipcl syl3anc ablsubsub4 oveq1d wceq lmodvsubcl ipsubdir ipsubdi oveq12d + syl13anc ringgrp grpsubcl ablsubsub 3eqtrd ringacl abladdsub 3eqtr4d + cgrp ) ABDIUDZBEIUDZGUDZCDIUDZGUDZCEIUDZFUDZVOVPVRFUDZGUDZVTFUDZBCJUDDE + JUDZIUDZVOVTFUDWBGUDZAVSWCVTFAHUEUFZFHGVOVPVRWHUGZRQAHUHUIZHUJUIZALUKUI + ZWJALULUIZWLSLUMUNZHLMUOUNZHUPUNZAWMBKUIZDKUIZVOWHUIZSTUBBDHIWHKLMNOWIU + QURZAWMWQEKUIZVPWHUIZSTUCBEHIWHKLMNOWIUQURZAWMCKUIZWRVRWHUIZSUAUBCDHIWH + KLMNOWIUQURZUSUTAWFBWEIUDZCWEIUDZGUDZVQVRVTGUDZGUDWAAWMWQXDWEKUIZWFXIVA + STUAAWLWRXAXKWNUBUCJKLDEOPVBURBCWEGHIJKLMNOPQVCVFAXGVQXHXJGAWMWQWRXAXGV + QVASTUBUCBDEGHIJKLMNOPQVDVFAWMXDWRXAXHXJVASUAUBUCCDEGHIJKLMNOPQVDVFVEAW + HFHGVQVRVTWIRQWPAHVNUIZWSXBVQWHUIAWJXLWOHVGUNWTXCWHHGVOVPWIQVHURXFAWMXD + XAVTWHUIZSUAUCCEHIWHKLMNOWIUQURZVIVJAWKWSXMWBWHUIZWGWDVAWPWTXNAWJXBXEXO + WOXCXFWHFHVPVRWIRVKURWHFHGVOVTWBWIRQVLVFVM $. $} ipdir.f $e |- K = ( Base ` F ) $. @@ -252133,42 +252359,42 @@ According to Wikipedia ("Integer", 25-May-2019, fveq2d oveq2 mpteq2dv feq1d rspccva sylan eqidd 3anrot 3anbi123d oveq123d 3exp syl5bbr eqeq12d imbi12d imp31 3exp2 impancom 3imp2 clss lveclmod syl clmod adantr eqid lss1 sylancom ovex fvmpt3i simpr2 oveq2d simpr3 oveq12d - 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In this sense, -linear equations, matrices and determinants are usually regarded as "over a -ring" in this part. +always presumed (the ring is a unital ring, see also ~ df-ring ). In this +sense, linear equations, matrices and determinants are usually regarded as +"over a ring" in this part. $) $( @@ -254163,18 +254389,18 @@ consisting of the same ring regarded as a (left) module over itself, see ( cv wcel w3a cfv co cmpt cfsupp wbr csupp cfn cof wf cmap 3ad2ant1 simp2 wfn frlmbasmap syl2anc elmapi syl ffn simp3 inidm eqidd offval oveq1d cvv wa wfun wss ovex a1i wceq funmpt funeq mpbird frlmbasfsupp crg cdr cfield - ccrg isfld sylib simpld drngrng rng0cl rnglz sylan suppofss1d fsuppsssupp - fsuppimpd syl22anc eqeltrrd wb mptexg c0g fvex eqeltri funisfsupp syl3anc - ) AFUHZLUIZGUHZLUIZUJZBIBUHZXHUKZXMXJUKZEULZUMZOUNUOZXQOUPULZUQUIZXLXHXJE - URZULZOUPULZXSUQXLYBXQOUPXLBIIXNXOEIXHXJMMXLICXHUSZXHIVCXLXHCIUTULZUIZYDX - LIMUIZXIYFAXIYGXKUGVAZAXIXKVBZLDNICMXHPQSVDVEXHCIVFVGZICXHVHVGXLICXJUSZXJ - IVCXLXJYEUIZYKXLYGXKYLYHAXIXKVILDNICMXJPQSVDVEXJCIVFVGZICXJVHVGYHYHIVJXLX - MIUIVOZXNVKYNXOVKVLZVMXLYBVNUIZYBVPZXHOUNUOZYCXHOUPULVQZYCUQUIYPXLXHXJYAV - RVSXLYBXQVTZYQYOYTYQXQVPZUUAYTBIXPWAZVSYBXQWBWCVGXLYGXIYRYHYILDNIMXHOPUBS - WDVEXLBICXHXJMEOYHXLDWEUIZOCUIAXIUUCXKADWFUIZUUCAUUDDWHUIZADWGUIUUDUUEVOU - DDWIWJWKDWLVGVAZCDOQUBWMVGYJYMXLUUCXMCUIOXMEULOVTUUFCDEXMOQRUBWNWOWPYPYQV - OYRYSVOVOYBOXHYBVNOWQWRWSWTXLUUAXQVNUIZOVNUIZXRXTXAUUAXLUUBVSXLYGUUGYHBIX - PMXBVGUUHXLODXCUKVNUBDXCXDXEVSXQVNVNOXFXGWC $. + ccrg isfld sylib simpld drngrng ring0cl ringlz sylan suppofss1d fsuppimpd + fsuppsssupp syl22anc eqeltrrd mptexg c0g fvex eqeltri funisfsupp syl3anc + wb ) AFUHZLUIZGUHZLUIZUJZBIBUHZXHUKZXMXJUKZEULZUMZOUNUOZXQOUPULZUQUIZXLXH + XJEURZULZOUPULZXSUQXLYBXQOUPXLBIIXNXOEIXHXJMMXLICXHUSZXHIVCXLXHCIUTULZUIZ + YDXLIMUIZXIYFAXIYGXKUGVAZAXIXKVBZLDNICMXHPQSVDVEXHCIVFVGZICXHVHVGXLICXJUS + ZXJIVCXLXJYEUIZYKXLYGXKYLYHAXIXKVILDNICMXJPQSVDVEXJCIVFVGZICXJVHVGYHYHIVJ + XLXMIUIVOZXNVKYNXOVKVLZVMXLYBVNUIZYBVPZXHOUNUOZYCXHOUPULVQZYCUQUIYPXLXHXJ + YAVRVSXLYBXQVTZYQYOYTYQXQVPZUUAYTBIXPWAZVSYBXQWBWCVGXLYGXIYRYHYILDNIMXHOP + UBSWDVEXLBICXHXJMEOYHXLDWEUIZOCUIAXIUUCXKADWFUIZUUCAUUDDWHUIZADWGUIUUDUUE + VOUDDWIWJWKDWLVGVAZCDOQUBWMVGYJYMXLUUCXMCUIOXMEULOVTUUFCDEXMOQRUBWNWOWPYP + YQVOYRYSVOVOYBOXHYBVNOWRWQWSWTXLUUAXQVNUIZOVNUIZXRXTXGUUAXLUUBVSXLYGUUGYH + BIXPMXAVGUUHXLODXBUKVNUBDXBXCXDVSXQVNVNOXEXFWC $. $( Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.) $) @@ -254183,84 +254409,84 @@ consisting of the same ring regarded as a (left) module over itself, see vf cdr ccrg cfield wa isfld sylib simpld frlmsca syl2anc cmulr cstv clmod clvec crg drngrng syl frlmlmod eqeltrrd islvec sylanbrc simprd idsrngd cv eqid w3a co cmpt cgsu cof 3ad2ant1 simp2 simp3 frlmipval syl22anc wf cmap - wfn frlmbasmap elmapi ffn inidm offval oveq2d eqtrd ccmn rngcmn ffvelrnda - adantr rngcl syl3anc fmptd frlmphllem cfsupp weq fveq2 cvv simpr syl13anc - fveq2d ovex mpteq2dva syl5eq wfun wbr csupp wi oveq12d cmpt2 fveq1 oveq1d - id mpteq2dv simprl fveq1d simprr ovmpt2d gsumcl eqeltrd simp31 simp33 wss - simp32 oveq1i fneq1i fvmptd rngass funmpt funeqd frlmbasfsupp rngrz sylan - cbvmptv mpbiri anim12i rng0cl suppofss2d fsuppsssupp eqbrtrrd simp1 eleq1 - 3adant2 eqeq1d 3anbi23d eqeq1 imbi12d chvarv gsummptfsadd frlmip syl6reqr - cbvmpt2v syl6eqr eleqtrd lmodvscl frlmplusgval frlmvscaval fvmpt2d rngdir - 3syl 3eqtrd gsummulc2 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(Contributed by Stefan O'Rear, 3-Feb-2015.) $) uvcvvcl2 $p |- ( ph -> ( ( U ` J ) ` K ) e. B ) $= - ( cfv wceq cur crg wcel eqid c0g cif uvcvval syl31anc rngidcl rng0cl ifcl - syl2anc syl eqeltrd ) AGFDOOZGFPZCQOZCUAOZUBZBACRSZEHSFESGESUKUOPKLMNCDUM - EFGRHUNIUMTZUNTZUCUDAUPUOBSZKUPUMBSUNBSUSBCUMJUQUEBCUNJURUFULUMUNBUGUHUIU - J $. + ( cfv wceq cur crg wcel eqid c0g cif uvcvval syl31anc ring0cl syl2anc syl + ringidcl ifcl eqeltrd ) AGFDOOZGFPZCQOZCUAOZUBZBACRSZEHSFESGESUKUOPKLMNCD + UMEFGRHUNIUMTZUNTZUCUDAUPUOBSZKUPUMBSUNBSUSBCUMJUQUHBCUNJURUEULUMUNBUIUFU + GUJ $. $} ${ @@ -254397,29 +254623,29 @@ is used for the (special) unit vectors forming the standard basis of free O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) $) uvcff $p |- ( ( R e. Ring /\ I e. W ) -> U : I --> B ) $= ( vi vj crg wcel wa wf cv cfv eqid cvv a1i wceq cur c0g cif cmpt cbs cmap - cfsupp wbr rngidcl rng0cl ifcld ad3antrrr fmptd fvex elmapg mpan ad2antlr - co wb mpbird wfun csn cfn csupp wss mptexg funmpt snfi cdif wn wne adantl - eldifsni neneqd iffalse syl simplr suppss2 suppssfifsupp frlmelbas adantr - syl32anc mpbir2and uvcfval feq1d ) BLMZDEMZNZDACODAJDKDKPZJPZUAZBUBQZBUCQ - ZUDZUEZUEZOWIJDWPAWQWIWKDMZNZWPAMZWPBUFQZDUGUSMZWPWNUHUIZWSXBDXAWPOZWSKDW - OXAWPWGWOXAMWHWRWJDMWGWLWMWNXAXABWMXARZWMRZUJXABWNXEWNRZUKULUMWPRUNWHXBXD - UTZWGWRXASMWHXHBUFUOXADWPSEUPUQURVAWSWPSMZWPVBZWNSMZWKVCZVDMZWPWNVEUSXLVF - XCWHXIWGWRKDWOEVGURXJWSKDWOVHTXKWSBUCUOTXMWSWKVITWSDWOKEXLWNWSWJDXLVJMZNZ - WLVKWOWNUAXOWJWKXNWJWKVLWSWJDWKVNVMVOWLWMWNVPVQWGWHWRVRVSXLWPSSWNVTWCWIWT - XBXCNUTWRABFDXALEWPWNHXEXGIWAWBWDWQRUNWIDACWQBCWMJKDLEWNGXFXGWEWFVA $. + co cfsupp wbr ringidcl ring0cl ifcld ad3antrrr fmptd fvex elmapg ad2antlr + wb mpan mpbird wfun csn cfn csupp wss mptexg funmpt snfi cdif wn eldifsni + wne adantl neneqd iffalse simplr suppss2 suppssfifsupp syl32anc frlmelbas + syl adantr mpbir2and uvcfval feq1d ) BLMZDEMZNZDACODAJDKDKPZJPZUAZBUBQZBU + CQZUDZUEZUEZOWIJDWPAWQWIWKDMZNZWPAMZWPBUFQZDUGUHMZWPWNUIUJZWSXBDXAWPOZWSK + DWOXAWPWGWOXAMWHWRWJDMWGWLWMWNXAXABWMXARZWMRZUKXABWNXEWNRZULUMUNWPRUOWHXB + XDUSZWGWRXASMWHXHBUFUPXADWPSEUQUTURVAWSWPSMZWPVBZWNSMZWKVCZVDMZWPWNVEUHXL + VFXCWHXIWGWRKDWOEVGURXJWSKDWOVHTXKWSBUCUPTXMWSWKVITWSDWOKEXLWNWSWJDXLVJMZ + NZWLVKWOWNUAXOWJWKXNWJWKVMWSWJDWKVLVNVOWLWMWNVPWBWGWHWRVQVRXLWPSSWNVSVTWI + WTXBXCNUSWRABFDXALEWPWNHXEXGIWAWCWDWQRUOWIDACWQBCWMJKDLEWNGXFXGWEWFVA $. $( In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) $) uvcf1 $p |- ( ( R e. NzRing /\ I e. W ) -> U : I -1-1-> B ) $= - ( vi vj wcel wa cv cfv wceq wral crg wne eqid cnzr wf wi wf1 nzrrng uvcff - sylan cur c0g nzrnz ad3antrrr simpllr simplrl uvcvv1 simplrr simpr necomd - uvcvv0 3netr4d fveq1 necon3i syl ex necon4d ralrimivva dff13 sylanbrc ) B - UALZDELZMZDACUBZJNZCOZKNZCOZPVLVNPUCZKDQJDQDACUDVHBRLZVIVKBUEZABCDEFGHIUF - UGVJVPJKDDVJVLDLZVNDLZMZMZVLVNVMVOWBVLVNSZVMVOSZWBWCMZVLVMOZVLVOOZSWDWEBU - HOZBUIOZWFWGVHWHWISVIWAWCBWHWIWHTZWITZUJUKWEBCWHDVLREGVHVQVIWAWCVRUKZVHVI - WAWCULZVJVSVTWCUMZWJUNWEBCDVNVLREWIGWLWMVJVSVTWCUOWNWEVLVNWBWCUPUQWKURUSV - MVOWFWGVLVMVOUTVAVBVCVDVEJKDACVFVG $. + ( vi vj wcel wa cv cfv wceq wral crg wne eqid cnzr wf nzrring uvcff sylan + wi wf1 cur c0g nzrnz ad3antrrr simpllr simplrl uvcvv1 simpr necomd uvcvv0 + simplrr 3netr4d fveq1 necon3i syl ex necon4d ralrimivva dff13 sylanbrc ) + BUALZDELZMZDACUBZJNZCOZKNZCOZPVLVNPUFZKDQJDQDACUGVHBRLZVIVKBUCZABCDEFGHIU + DUEVJVPJKDDVJVLDLZVNDLZMZMZVLVNVMVOWBVLVNSZVMVOSZWBWCMZVLVMOZVLVOOZSWDWEB + UHOZBUIOZWFWGVHWHWISVIWAWCBWHWIWHTZWITZUJUKWEBCWHDVLREGVHVQVIWAWCVRUKZVHV + IWAWCULZVJVSVTWCUMZWJUNWEBCDVNVLREWIGWLWMVJVSVTWCURWNWEVLVNWBWCUOUPWKUQUS + VMVOWFWGVLVMVOUTVAVBVCVDVEJKDACVFVG $. $} ${ @@ -254435,41 +254661,41 @@ is used for the (special) unit vectors forming the standard basis of free uvcresum $p |- ( ( R e. Ring /\ I e. W /\ X e. B ) -> X = ( Y gsum ( X oF .x. U ) ) ) $= ( vb va wcel cfv co cmpt eqid wceq crg w3a cv cmulr cgsu cof cbs frlmbasf - wf 3adant1 feqmptd c0g cmnd simpl1 rngmnd syl simpl2 simpr wral ffvelrnda - csn cxp uvcff 3adant3 frlmvscafval adantr syl2anc fconstmpt offval2 eqtrd - wa a1i clmod csca frlmlmod frlmsca eleqtrd lmodvscl syl3anc eqeltrrd fmpt - fveq2d sylibr r19.21bi an32s fmptd wne 3ad2ant1 simp2 simp3 uvcvv0 oveq2d - adantlr rngrz suppsssn gsumpt fveq2 fveq1d oveq12d ovex adantl cur uvcvv1 - fvmpt rngridm mpteq2dva eqtr4d simp1 cvv wfun csupp cfn wss cfsupp mptexg - wbr 3ad2ant2 funmpt fvex frlmbasfsupp fsuppimpd cdif eqcomd ssid syl6eqss - suppssr oveq1d eldifi sylan2 lmod0vs 3eqtr3d suppss2 syl32anc frlmgsum - suppssfifsupp ) BUAOZEFOZGAOZUBZGHMENEMUCZGPZNUCZYTDPZPZBUDPZQZRZRZUEQZHG + wf 3adant1 feqmptd wa c0g cmnd simpl1 ringmnd simpl2 simpr wral ffvelrnda + syl csn cxp uvcff 3adant3 frlmvscafval adantr syl2anc fconstmpt a1i eqtrd + offval2 clmod csca frlmlmod frlmsca fveq2d eleqtrd lmodvscl eqeltrrd fmpt + syl3anc sylibr r19.21bi an32s fmptd wne simp2 simp3 uvcvv0 oveq2d adantlr + 3ad2ant1 ringrz suppsssn gsumpt fveq2 fveq1d ovex fvmpt adantl cur uvcvv1 + oveq12d ringridm mpteq2dva eqtr4d simp1 cvv wfun csupp cfn wss cfsupp wbr + mptexg 3ad2ant2 funmpt fvex frlmbasfsupp fsuppimpd eqcomd syl6eqss oveq1d + cdif suppssr eldifi sylan2 lmod0vs 3eqtr3d suppss2 suppssfifsupp syl32anc + ssid frlmgsum ) BUAOZEFOZGAOZUBZGHMENEMUCZGPZNUCZYTDPZPZBUDPZQZRZRZUEQZHG DCUFQZUEQYSGNEBMEUUFRZUEQZRZUUIYSGNEUUBGPZRUUMYSNEBUGPZGYQYREUUOGUIYPABHE - UUOFGJUUOSZKUHUJZUKYSNEUULUUNYSUUBEOZVKZUULUUBUUKPZUUNUUSEUUOUUKBFUUBBULP - ZUUPUVASZUUSYPBUMOYPYQYRUURUNZBUOUPYPYQYRUURUQZYSUURURZUUSMEUUFUUOUUKYSYT - EOZUURUUFUUOOZYSUVFVKZUVGNEUVHEUUOUUGUIZUVGNEUSUVHYQUUGAOUVIYPYQYRUVFUQZU - VHUUAUUCCQZUUGAUVHUVKEUUAVAVBZUUCUUEUFQUUGUVHUUAABCUUEEUUOFUUCHJKUUPUVJYS - EUUOYTGUUQUTZYSEAYTDYPYQEADUIYRABDEFHIJKVCVDZUTZLUUESZVEUVHNEUUAUUDUUEUVL - UUCFUUOUUOUVJUVHUUAUUOOZUURUVMVFUVHEUUOUUBUUCUVHYQUUCAOZEUUOUUCUIUVJUVOAB - HEUUOFUUCJUUPKUHVGZUTUVLNEUUARTUVHNEUUAVHVLUVHNEUUOUUCUVSUKVIVJZUVHHVMOZU - UAHVNPZUGPZOUVRUVKAOYSUWAUVFYPYQUWAYRBHEFJVOVDZVFUVHUUAUUOUWCUVMYSUUOUWCT - UVFYSBUWBUGYPYQBUWBTYRBHEUAFJVPVDZWBVFVQUVOUUACUWBUWCAHUUCKUWBSZLUWCSVRVS - VTZABHEUUOFUUGJUUPKUHVGNEUUOUUFUUGUUGSWAWCWDWEUUKSZWFUUSEUUFMFUUBUVAUUSUV + UUOFGJUUOSZKUHUJZUKYSNEUULUUNYSUUBEOZULZUULUUBUUKPZUUNUUSEUUOUUKBFUUBBUMP + ZUUPUVASZUUSYPBUNOYPYQYRUURUOZBUPVAYPYQYRUURUQZYSUURURZUUSMEUUFUUOUUKYSYT + EOZUURUUFUUOOZYSUVFULZUVGNEUVHEUUOUUGUIZUVGNEUSUVHYQUUGAOUVIYPYQYRUVFUQZU + VHUUAUUCCQZUUGAUVHUVKEUUAVBVCZUUCUUEUFQUUGUVHUUAABCUUEEUUOFUUCHJKUUPUVJYS + EUUOYTGUUQUTZYSEAYTDYPYQEADUIYRABDEFHIJKVDVEZUTZLUUESZVFUVHNEUUAUUDUUEUVL + UUCFUUOUUOUVJUVHUUAUUOOZUURUVMVGUVHEUUOUUBUUCUVHYQUUCAOZEUUOUUCUIUVJUVOAB + HEUUOFUUCJUUPKUHVHZUTUVLNEUUARTUVHNEUUAVIVJUVHNEUUOUUCUVSUKVLVKZUVHHVMOZU + UAHVNPZUGPZOUVRUVKAOYSUWAUVFYPYQUWAYRBHEFJVOVEZVGUVHUUAUUOUWCUVMYSUUOUWCT + UVFYSBUWBUGYPYQBUWBTYRBHEUAFJVPVEZVQVGVRUVOUUACUWBUWCAHUUCKUWBSZLUWCSVSWB + VTZABHEUUOFUUGJUUPKUHVHNEUUOUUFUUGUUGSWAWCWDWEUUKSZWFUUSEUUFMFUUBUVAUUSUV FYTUUBWGZUBZUUFUUAUVAUUEQZUVAUWJUUDUVAUUAUUEUWJBDEYTUUBUAFUVAIUUSUVFYPUWI - UVCWHZUUSUVFYQUWIUVDWHUUSUVFUWIWIUUSUVFUURUWIUVEWHUUSUVFUWIWJUVBWKWLUWJYP - UVQUWKUVATUWLUUSUVFUVQUWIYSUVFUVQUURUVMWMVDUUOBUUEUUAUVAUUPUVPUVBWNVGVJUV + UVCWMZUUSUVFYQUWIUVDWMUUSUVFUWIWHUUSUVFUURUWIUVEWMUUSUVFUWIWIUVBWJWKUWJYP + UVQUWKUVATUWLUUSUVFUVQUWIYSUVFUVQUURUVMWLVEUUOBUUEUUAUVAUUPUVPUVBWNVHVKUV DWOWPUUSUUTUUNUUBUUBDPZPZUUEQZUUNUURUUTUWOTYSMUUBUUFUWOEUUKYTUUBTZUUAUUNU - UDUWNUUEYTUUBGWQUWPUUBUUCUWMYTUUBDWQWRWSUWHUUNUWNUUEWTXDXAUUSUWOUUNBXBPZU - UEQZUUNUUSUWNUWQUUNUUEUUSBDUWQEUUBUAFIUVCUVDUVEUWQSZXCWLUUSYPUUNUUOOUWRUU - NTUVCYSEUUOUUBGUUQUTUUOBUUEUWQUUNUUPUVPUWSXEVGVJVJVJXFXGYSNMABUUFEEFFHHUL - PZJKUWTSZYPYQYRWIZUXBYPYQYRXHUWGYSUUHXIOZUUHXJZUWTXIOZGUVAXKQZXLOUUHUWTXK - QUXFXMUUHUWTXNXPYQYPUXCYRMEUUGFXOXQUXDYSMEUUGXRVLUXEYSHULXSVLYSGUVAYQYRGU - VAXNXPYPABHEFGUVAJUVBKXTUJYAYSEUUGMFUXFUWTYSYTEUXFYBOZVKZUVKUWBULPZUUCCQZ + UDUWNUUEYTUUBGWQUWPUUBUUCUWMYTUUBDWQWRXDUWHUUNUWNUUEWSWTXAUUSUWOUUNBXBPZU + UEQZUUNUUSUWNUWQUUNUUEUUSBDUWQEUUBUAFIUVCUVDUVEUWQSZXCWKUUSYPUUNUUOOUWRUU + NTUVCYSEUUOUUBGUUQUTUUOBUUEUWQUUNUUPUVPUWSXEVHVKVKVKXFXGYSNMABUUFEEFFHHUM + PZJKUWTSZYPYQYRWHZUXBYPYQYRXHUWGYSUUHXIOZUUHXJZUWTXIOZGUVAXKQZXLOUUHUWTXK + QUXFXMUUHUWTXNXOYQYPUXCYRMEUUGFXPXQUXDYSMEUUGXRVJUXEYSHUMXSVJYSGUVAYQYRGU + VAXNXOYPABHEFGUVAJUVBKXTUJYAYSEUUGMFUXFUWTYSYTEUXFYEOZULZUVKUWBUMPZUUCCQZ UUGUWTUXHUUAUXIUUCCYSEUUOXIGFUXFYTUXIUUQYSGUXIXKQUXFUXFYSUXIUVAGXKYSUWBBU - LYSBUWBUWEYCWBWLUXFYDYEUXBUXIXIOYSUWBULXSVLYFYGUXGYSUVFUVKUUGTYTEUXFYHZUV - TYIUXHUWAUVRUXJUWTTYSUWAUXGUWDVFUXGYSUVFUVRUXKUVOYICUWBUXIAHUUCUWTKUWFLUX - ISUXAYJVGYKUXBYLUXFUUHXIXIUWTYOYMYNXGYSUUJUUHHUEYSUUJMEUVKRUUHYSMEUUAUUCC - GDFUUOAUXBUVMUVOYSMEUUOGUUQUKYSMEADUVNUKVIYSMEUVKUUGUVTXFVJWLXG $. + MYSBUWBUWEYBVQWKUXFYNYCUXBUXIXIOYSUWBUMXSVJYFYDUXGYSUVFUVKUUGTYTEUXFYGZUV + TYHUXHUWAUVRUXJUWTTYSUWAUXGUWDVGUXGYSUVFUVRUXKUVOYHCUWBUXIAHUUCUWTKUWFLUX + ISUXAYIVHYJUXBYKUXFUUHXIXIUWTYLYMYOXGYSUUJUUHHUEYSUUJMEUVKRUUHYSMEUUAUUCC + GDFUUOAUXBUVMUVOYSMEUUOGUUQUKYSMEADUVNUKVLYSMEUVKUUGUVTXFVKWKXG $. $} ${ @@ -254515,20 +254741,20 @@ is used for the (special) unit vectors forming the standard basis of free 24-Jun-2019.) $) frlmssuvc2 $p |- ( ph -> -. ( X .x. ( U ` L ) ) e. C ) $= ( cfv co wcel csupp wss wa cv wne wn eldifad cmulr cur csn crg wf uvcff - crab syl2anc ffvelrnd frlmvscaval uvcvv1 oveq2d rngridm 3eqtrd eldifsni - eqid wceq syl eqnetrd fveq2 neeq1d elrab sylanbrc eldifbd nelss wfn cvv - cdif clmod csca frlmlmod frlmsca fveq2d syl5eq eleqtrd lmodvscl syl3anc - cbs frlmbasf ffn c0g fvex eqeltri suppvalfn sseq1d mtbird intnand oveq1 - a1i elrab2 sylnibr ) ANLGUHZFUIZCUJZXJOUKUIZJULZUMXJDUJAXMXKAXMBUNZXJUH - ZOUOZBIVDZJULZALXQUJZLJUJUPXRUPALIUJLXJUHZOUOZXSALIJUFUQZAXTNOAXTNLXIUH - ZEURUHZUINEUSUHZYDUIZNANCEFYDILKMXIHPRSUDANKOUTZUGUQZAICLGAEVAUJZIMUJZI - CGVBUCUDCEGIMHQPRVCVEYBVFZYBTYDVMZVGAYCYENYDAEGYEILVAMQUCUDYBYEVMZVHVIA - YINKUJYFNVNUCYHKEYDYENSYLYMVJVEVKANKYGWEUJNOUOUGNKOVLVOVPXPYABLIXNLVNXO - XTOXNLXJVQVRVSVTALIJUFWALXQJWBVEAXLXQJAXJIWCZYJOWDUJZXLXQVNAIKXJVBZYNAY - JXKYPUDAHWFUJZNHWGUHZWOUHZUJXICUJXKAYIYJYQUCUDEHIMPWHVEANKYSYHAKEWOUHYS - SAEYRWOAYIYJEYRVNUCUDEHIVAMPWIVEWJWKWLYKNFYRYSCHXIRYRVMTYSVMWMWNCEHIKMX - JPSRWPVEIKXJWQVOUDYOAOEWRUHWDUAEWRWSWTXFBXJMWDIOXAWNXBXCXDXNOUKUIZJULXM - BXJCDXNXJVNYTXLJXNXJOUKXEXBUBXGXH $. + crab syl2anc ffvelrnd eqid frlmvscaval uvcvv1 oveq2d wceq ringridm cdif + 3eqtrd eldifsni syl eqnetrd fveq2 neeq1d elrab sylanbrc eldifbd wfn cvv + nelss clmod cbs frlmlmod frlmsca fveq2d syl5eq eleqtrd lmodvscl syl3anc + csca frlmbasf ffn c0g eqeltri a1i suppvalfn sseq1d mtbird intnand oveq1 + fvex elrab2 sylnibr ) ANLGUHZFUIZCUJZXJOUKUIZJULZUMXJDUJAXMXKAXMBUNZXJU + HZOUOZBIVDZJULZALXQUJZLJUJUPXRUPALIUJLXJUHZOUOZXSALIJUFUQZAXTNOAXTNLXIU + HZEURUHZUINEUSUHZYDUIZNANCEFYDILKMXIHPRSUDANKOUTZUGUQZAICLGAEVAUJZIMUJZ + ICGVBUCUDCEGIMHQPRVCVEYBVFZYBTYDVGZVHAYCYENYDAEGYEILVAMQUCUDYBYEVGZVIVJ + AYINKUJYFNVKUCYHKEYDYENSYLYMVLVEVNANKYGVMUJNOUOUGNKOVOVPVQXPYABLIXNLVKX + OXTOXNLXJVRVSVTWAALIJUFWBLXQJWEVEAXLXQJAXJIWCZYJOWDUJZXLXQVKAIKXJVBZYNA + YJXKYPUDAHWFUJZNHWOUHZWGUHZUJXICUJXKAYIYJYQUCUDEHIMPWHVEANKYSYHAKEWGUHY + SSAEYRWGAYIYJEYRVKUCUDEHIVAMPWIVEWJWKWLYKNFYRYSCHXIRYRVGTYSVGWMWNCEHIKM + XJPSRWPVEIKXJWQVPUDYOAOEWRUHWDUAEWRXFWSWTBXJMWDIOXAWNXBXCXDXNOUKUIZJULX + MBXJCDXNXJVKYTXLJXNXJOUKXEXBUBXGXH $. $} $} @@ -254578,20 +254804,20 @@ is used for the (special) unit vectors forming the standard basis of free (Proof modification is discouraged.) $) frlmssuvc2OLD $p |- ( ph -> -. ( X .x. ( U ` L ) ) e. C ) $= ( cfv co wcel ccnv cvv csn cdif cima wss wa cv wne wn eldifad cmulr cur - crab crg wf uvcff syl2anc ffvelrnd eqid frlmvscaval uvcvv1 wceq rngridm - oveq2d 3eqtrd eldifsni syl eqnetrd fveq2 neeq1d elrab eldifbd nelss wfn - sylanbrc clmod csca cbs frlmlmod frlmsca fveq2d syl5eq eleqtrd lmodvscl - syl3anc frlmbasf fnniniseg2OLD 3syl sseq1d mtbird intnand cnveq imaeq1d - ffn elrab2 sylnibr ) ANLGUHZFUIZCUJZXIUKZULOUMZUNZUOZJUPZUQXIDUJAXOXJAX - OBURZXIUHZOUSZBIVDZJUPZALXSUJZLJUJUTXTUTALIUJLXIUHZOUSZYAALIJUFVAZAYBNO - AYBNLXHUHZEVBUHZUINEVCUHZYFUIZNANCEFYFILKMXHHPRSUDANKXLUGVAZAICLGAEVEUJ - ZIMUJZICGVFUCUDCEGIMHQPRVGVHYDVIZYDTYFVJZVKAYEYGNYFAEGYGILVEMQUCUDYDYGV - JZVLVOAYJNKUJYHNVMUCYIKEYFYGNSYMYNVNVHVPANKXLUNUJNOUSUGNKOVQVRVSXRYCBLI - XPLVMXQYBOXPLXIVTWAWBWFALIJUFWCLXSJWDVHAXNXSJAIKXIVFZXIIWEXNXSVMAYKXJYO - UDAHWGUJZNHWHUHZWIUHZUJXHCUJXJAYJYKYPUCUDEHIMPWJVHANKYRYIAKEWIUHYRSAEYQ - WIAYJYKEYQVMUCUDEHIVEMPWKVHWLWMWNYLNFYQYRCHXHRYQVJTYRVJWOWPCEHIKMXIPSRW - QVHIKXIXEBIOXIWRWSWTXAXBXPUKZXMUOZJUPXOBXICDXPXIVMZYTXNJUUAYSXKXMXPXIXC - XDWTUBXFXG $. + crab crg uvcff syl2anc ffvelrnd eqid frlmvscaval uvcvv1 oveq2d ringridm + wf wceq 3eqtrd eldifsni syl eqnetrd fveq2 neeq1d elrab sylanbrc eldifbd + nelss wfn clmod csca cbs frlmlmod frlmsca fveq2d syl5eq eleqtrd syl3anc + lmodvscl frlmbasf ffn fnniniseg2OLD sseq1d mtbird intnand cnveq imaeq1d + 3syl elrab2 sylnibr ) ANLGUHZFUIZCUJZXIUKZULOUMZUNZUOZJUPZUQXIDUJAXOXJA + XOBURZXIUHZOUSZBIVDZJUPZALXSUJZLJUJUTXTUTALIUJLXIUHZOUSZYAALIJUFVAZAYBN + OAYBNLXHUHZEVBUHZUINEVCUHZYFUIZNANCEFYFILKMXHHPRSUDANKXLUGVAZAICLGAEVEU + JZIMUJZICGVNUCUDCEGIMHQPRVFVGYDVHZYDTYFVIZVJAYEYGNYFAEGYGILVEMQUCUDYDYG + VIZVKVLAYJNKUJYHNVOUCYIKEYFYGNSYMYNVMVGVPANKXLUNUJNOUSUGNKOVQVRVSXRYCBL + IXPLVOXQYBOXPLXIVTWAWBWCALIJUFWDLXSJWEVGAXNXSJAIKXIVNZXIIWFXNXSVOAYKXJY + OUDAHWGUJZNHWHUHZWIUHZUJXHCUJXJAYJYKYPUCUDEHIMPWJVGANKYRYIAKEWIUHYRSAEY + QWIAYJYKEYQVOUCUDEHIVEMPWKVGWLWMWNYLNFYQYRCHXHRYQVITYRVIWPWOCEHIKMXIPSR + WQVGIKXIWRBIOXIWSXEWTXAXBXPUKZXMUOZJUPXOBXICDXPXIVOZYTXNJUUAYSXKXMXPXIX + CXDWTUBXFXG $. $} $} @@ -254751,28 +254977,28 @@ is used for the (special) unit vectors forming the standard basis of free syl cdm fdm syl5sseq ralrimiva rabid2 sylibr ssid frlmsslsp wfn ffn fnima mp3an3 3syl fveq2d 3eqtr2rd simpll simplr difssd ssnid ad2antrl syl5eleqr snssi dfss4 sylib frlmsca difeq12d eleq2d biimpar adantrl frlmssuvc2 ccnv - sneqd wfun cnzr rngelnzr syl2anc uvcf1 df-f1 simprbi imadif simprl fnsnfv - wf1 f1fn eqcomd eqtr2d syl3anc eqtrd neleqtrrd ralrimivva wb sneq difeq2d - oveq2 eleq12d notbid ralbidv ralrn cvv w3a cfrlm ovex eqeltri islbs ax-mp - mpbird syl3anbrc ) AUAMZDFMZUBZBUCZCUGNZOZYLCUDNZNZYMPZJUEZKUEZCUFNZUHZYL - YSQZRZYONZMZUIZJCUJNZUGNZUUGUKNZQZRZSZKYLSZYLEMZYKDYMBULZYNYMABDFCHGYMTZU - MZDYMBUNVAYKYMYSAUKNZUOUHZDOZKYMUPZBDUQZYONZYPYKUUTKYMSYMUVAPYKUUTKYMYKYS - YMMZUBDAUGNZYSULZUUTYJUVDUVFYIYMACDUVEFYSGUVETZUUPURUSUVFYSVBUUSDYSUURUTD - UVEYSVCVDVAVEUUTKYMVFVGYIYJDDOUVCUVAPDVHKYMUVAABDDYOFCUURGHYOTZUUPUURTZUV - ATVIVMYKUVBYLYOYKUUOBDVJZUVBYLPZUUQDYMBVKZDBVLZVNVOVPYKUUMYRLUEZBNZYTUHZY - LUVOQZRZYONZMZUIZJUUKSZLDSZYKUWALJDUUKYKUVNDMZYRUUKMZUBZUBZUVSUUSDUVNQZRZ - OKYMUPZUVPUWGKYMUWJAYTBCDUWIUVEUVNFYRUURGHUUPUVGYTTZUVIUWJTZYIYJUWFVQZYIY - JUWFVRZUWGDUWHVSZUWGUVNUWHDUWIRZLVTUWGUWHDOZUWPUWHPUWDUWQYKUWEUVNDWCWAUWH - DWDWEWBYKUWEYRUVEUURQZRZMZUWDYKUWTUWEYKUWSUUKYRYKUVEUUHUWRUUJYKAUUGUGACDU - AFGWFZVOYKUURUUIYKAUUGUKUXAVOWMWGWHWIWJZWKUWGUVSBUWIUQZYONZUWJUWGUVRUXCYO - 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(Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) $) frlmup1 $p |- ( ph -> E e. ( F LMHom T ) ) $= - ( vy vz vw cvsca cfv csca cbs eqid crg clmod lmodrng syl eqeltrd frlmlmod - wcel syl2anc wceq frlmsca eqtr3d cplusg cgrp lmodgrp cv cof co cgsu wa wi - weq eleq1 anbi2d oveq1 oveq2d eleq1d imbi12d ccmn lmodcmn adantr ad2antrr - c0g simprl fveq2d eleqtrd simprr lmodvscl syl3anc wf frlmbasf sylan inidm - off cvv wfun cfn wss cfsupp wbr ovex a1i ffun fvex frlmbasfsupp wb eqcomd - breq2d mpbird fsuppimpd lmod0vs sylancom suppssof1 suppssfifsupp syl32anc - csupp ssid gsumcl chvarv fmptd feq1d adantrr adantrl fvmpt oveq1d wfn ffn - offn oveqd ffvelrnda syl13anc eqtrd simpr fnfvof syl22anc oveq12d 3eqtr4d - ad2antll cmpt feqmptd cmulr eleqtrrd breq1d gsumadd lmodvacl frlmplusgval - 3expb lmodvsdir eqfnfvd ad2antrl isghmd eleq2d biimpar eqbrtrrd gsumvsmul - dffn2 sylib simplrl adantlrl adantlr lmodvsass simplrr mpteq2dva islmhmd + ( vy vz vw cvsca cfv csca cbs eqid crg wcel lmodring syl eqeltrd frlmlmod + clmod syl2anc wceq frlmsca eqtr3d cplusg cgrp lmodgrp cv cof co wa wi weq + cgsu eleq1 anbi2d oveq1 oveq2d eleq1d imbi12d c0g lmodcmn adantr ad2antrr + simprl fveq2d eleqtrd simprr lmodvscl syl3anc wf frlmbasf sylan inidm off + ccmn cvv wfun csupp cfn wss cfsupp wbr ovex a1i ffun fvex frlmbasfsupp wb + eqcomd breq2d fsuppimpd lmod0vs sylancom suppssof1 suppssfifsupp syl32anc + mpbird ssid gsumcl chvarv fmptd feq1d adantrr adantrl oveq1d wfn ffn offn + fvmpt oveqd syl13anc eqtrd simpr fnfvof syl22anc oveq12d 3eqtr4d ad2antll + ffvelrnda cmpt feqmptd cmulr eleqtrrd gsumadd lmodvacl 3expb frlmplusgval + breq1d lmodvsdir eqfnfvd ad2antrl isghmd biimpar eqbrtrrd gsumvsmul dffn2 + eleq2d sylib simplrl adantlrl adantlr lmodvsass simplrr mpteq2dva islmhmd frlmvscaval ) AUBUCJGJUEUFZHIGUGUFZJUGUFZUVFUHUFZDNUVDUIZPUVFUIZUVEUIZUVG - 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I ) $. @@ -254876,25 +255102,25 @@ is used for the (special) unit vectors forming the standard basis of free (Contributed by Stefan O'Rear, 7-Feb-2015.) 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( F LMHom T ) ( m o. U ) = A ) $= ( vx wcel wceq cfv eqid syl2anc syl vy clmod wf w3a cv ccom clmhm co wrex wrmo wreu cbs cvsca cof cgsu cmpt simp1 simp2 csca a1i simp3 frlmup1 ovex - wfn crn wss fnmpti crg lmodrng 3ad2ant1 uvcff ffn frn syl3anc 3ad2ant3 wa - fnco adantr simpr fvco2 simpl1 simpl2 simpl3 frlmup2 eqtrd eqfnfvd eqeq1d - coeq1 rspcev cres ccnv wi wral wfun funcoeqres ex ralrimivw clspn frlmlbs - ffun clbs lbssp lspextmo rmoim sylc reu5 sylanbrc ) DUBOZHIOZHBAUCZUDZFUE - ZEUFZAPZFGDUGUHZUIZXNFXOUJZXNFXOUKXKNGULQZDNUEADUMQZUNUHZUOUHZUPZXOOYBEUF - ZAPZXPXKNAXRBCDXSYBGHIKXRRZMXSRZYBRZXHXIXJUQXHXIXJURZCDUSQPZXKJUTXHXIXJVA - VBXKUAHYCAXKYBXRVDZEHVDZEVEZXRVFZYCHVDYJXKNXRYAYBDXTUOVCYGVGUTXKHXREUCZYK - XKCVHOZXIYNXHXIYOXJCDJVIVJZYHXRCEHIGLKYEVKSZHXREVLZTXKYNYMYQHXREVMTZXRHYB - EVQVNXJXHAHVDXIHBAVLVOXKUAUEZHOZVPZYTYCQZYTEQYBQZYTAQUUBYKUUAUUCUUDPUUBYN - YKXKYNUUAYQVRYRTXKUUAVSZHYBEYTVTSUUBNAXRBCDXSEYBGHIYTKYEMYFYGXHXIXJUUAWAX - HXIXJUUAWBYIUUBJUTXHXIXJUUAWCUUELWDWEWFXNYDFYBXOXLYBPXMYCAXLYBEWHWGWISXKX - NXLYLWJAEWKUFZPZWLZFXOWMZUUGFXOUJZXQXKEWNZUUIXKYNUUKYQHXREWTTUUKUUHFXOUUK - XNUUGXLEAWOWPWQTXKYMYLGWRQZQXRPZUUJYSXKYLGXAQZOZUUMXKYOXIUUOYPYHCEGHUUNIK - LUUNRZWSSYLUUNUULXRGYEUUPUULRZXBTXRGDFUUFUULYLYEUUQXCSXNUUGFXOXDXEXNFXOXF - XG $. + wfn crn wss fnmpti crg lmodring 3ad2ant1 ffn frn fnco syl3anc 3ad2ant3 wa + uvcff adantr simpr fvco2 simpl1 simpl2 simpl3 frlmup2 eqtrd eqfnfvd coeq1 + eqeq1d rspcev cres ccnv wi wral wfun ffun funcoeqres ralrimivw clspn clbs + ex frlmlbs lbssp lspextmo rmoim sylc reu5 sylanbrc ) DUBOZHIOZHBAUCZUDZFU + EZEUFZAPZFGDUGUHZUIZXNFXOUJZXNFXOUKXKNGULQZDNUEADUMQZUNUHZUOUHZUPZXOOYBEU + FZAPZXPXKNAXRBCDXSYBGHIKXRRZMXSRZYBRZXHXIXJUQXHXIXJURZCDUSQPZXKJUTXHXIXJV + AVBXKUAHYCAXKYBXRVDZEHVDZEVEZXRVFZYCHVDYJXKNXRYAYBDXTUOVCYGVGUTXKHXREUCZY + KXKCVHOZXIYNXHXIYOXJCDJVIVJZYHXRCEHIGLKYEVQSZHXREVKZTXKYNYMYQHXREVLTZXRHY + BEVMVNXJXHAHVDXIHBAVKVOXKUAUEZHOZVPZYTYCQZYTEQYBQZYTAQUUBYKUUAUUCUUDPUUBY + NYKXKYNUUAYQVRYRTXKUUAVSZHYBEYTVTSUUBNAXRBCDXSEYBGHIYTKYEMYFYGXHXIXJUUAWA + XHXIXJUUAWBYIUUBJUTXHXIXJUUAWCUUELWDWEWFXNYDFYBXOXLYBPXMYCAXLYBEWGWHWISXK + XNXLYLWJAEWKUFZPZWLZFXOWMZUUGFXOUJZXQXKEWNZUUIXKYNUUKYQHXREWOTUUKUUHFXOUU + KXNUUGXLEAWPWTWQTXKYMYLGWRQZQXRPZUUJYSXKYLGWSQZOZUUMXKYOXIUUOYPYHCEGHUUNI + KLUUNRZXASYLUUNUULXRGYEUUPUULRZXBTXRGDFUUFUULYLYEUUQXCSXNUUGFXOXDXEXNFXOX + FXG $. $} ${ @@ -255260,12 +255486,12 @@ each element of the range ( ~ islindf5 ). (Contributed by Stefan ( K ` ( F " ( dom F \ { E } ) ) ) ) $= ( clmod wcel cnzr wa clindf wbr cdm cfv cbs eqid syl2anc adantl 3ad2ant1 w3a cur cvsca co csn cdif cima wceq simp1l wf simp2 lindff simp3 ffvelrnd - lmodvs1 c0g wne crg nzrrng rngidcl syl nzrnz lindfind syl22anc eqneltrrd - wn ) EHIZDJIZKZBELMZABNZIZUAZDUBOZABOZEUCOZUDZVOBVKAUEUFUGCOZVMVGVOEPOZIV - QVOUHVGVHVJVLUIZVMVKVSABVMVJVGVKVSBUJVIVJVLUKZVTVSBEHVSQZULRVIVJVLUMZUNVP - VNDVSEVOWBGVPQZVNQZUORVMVJVLVNDPOZIZVNDUPOZUQZVQVRIVFWAWCVIVJWGVLVHWGVGVH - DURIWGDUSWFDVNWFQZWEUTVASTVIVJWIVLVHWIVGDVNWHWEWHQZVBSTVNVPABWFDCEWHWDFGW - KWJVCVDVE $. + lmodvs1 c0g wne wn crg nzrring ringidcl nzrnz lindfind syl22anc eqneltrrd + syl ) EHIZDJIZKZBELMZABNZIZUAZDUBOZABOZEUCOZUDZVOBVKAUEUFUGCOZVMVGVOEPOZI + VQVOUHVGVHVJVLUIZVMVKVSABVMVJVGVKVSBUJVIVJVLUKZVTVSBEHVSQZULRVIVJVLUMZUNV + PVNDVSEVOWBGVPQZVNQZUORVMVJVLVNDPOZIZVNDUPOZUQZVQVRIURWAWCVIVJWGVLVHWGVGV + HDUSIWGDUTWFDVNWFQZWEVAVFSTVIVJWIVLVHWIVGDVNWHWEWHQZVBSTVNVPABWFDCEWHWDFG + WKWJVCVDVE $. $( In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. @@ -255719,17 +255945,17 @@ each element of the range ( ~ islindf5 ). (Contributed by Stefan islindf5 $p |- ( ph -> ( A LIndF T <-> E : B -1-1-> C ) ) $= ( vy clindf wbr cv cfv c0g wceq wi wral wf1 cof co cgsu csn cxp cfrlm cbs csca clmod wcel wf wb eqid islindf4 syl3anc wa oveq1 oveq2d adantl eqeq1d - fvmpt crg lmodrng eqeltrd frlm0 syl2anc fveq2d sneqd xpeq2d eqtr3d adantr - ovex syl eqeq2d imbi12d ralbidva eqcomd oveq1d syl6eqr raleqdv clmhm cghm - bitr4d frlmup1 lmghm ghmf1 3syl ) ACGUCUDZUBUEZIUFZGUGUFZUHZWTJUGUFZUHZUI - ZUBDUJZDEIUKZAWSGWTCHULZUMZUNUMZXBUHZWTKGUSUFZUGUFZUOZUPZUHZUIZUBXMKUQUMZ - URUFZUJZXGAGUTVAZKLVAZKECVBWSYAVCRSUAUBEXMHCKXTGLXNXBOXMVDZPXBVDZXNVDXTVD - VEVFAXGXRUBDUJYAAXFXRUBDAWTDVAZVGZXCXLXEXQYGXAXKXBYFXAXKUHABWTGBUEZCXIUMZ - UNUMXKDIYHWTUHYIXJGUNYHWTCXIVHVIQGXJUNWCVLVJVKYGXDXPWTAXDXPUHYFAKFUGUFZUO - ZUPZXDXPAFVMVAYCYLXDUHAFXMVMTAYBXMVMVARXMGYDVNWDVOSFJKLYJMYJVDVPVQAYKXOKA - YJXNAFXMUGTVRVSVTWAWBWEWFWGAXRUBXTDAXTJURUFDAXSJURAXSFKUQUMJAXMFKUQAFXMTW - HWIMWJVRNWJWKWNWNAIJGWLUMVAIJGWMUMVAXHXGVCABCDEFGHIJKLMNOPQRSTUAWOJGIWPUB - JGXBIDEXDNOXDVDYEWQWRWN $. + ovex fvmpt crg lmodring eqeltrd frlm0 syl2anc fveq2d xpeq2d eqtr3d adantr + syl sneqd eqeq2d imbi12d ralbidva eqcomd oveq1d syl6eqr bitr4d clmhm cghm + raleqdv frlmup1 lmghm ghmf1 3syl ) ACGUCUDZUBUEZIUFZGUGUFZUHZWTJUGUFZUHZU + IZUBDUJZDEIUKZAWSGWTCHULZUMZUNUMZXBUHZWTKGUSUFZUGUFZUOZUPZUHZUIZUBXMKUQUM + ZURUFZUJZXGAGUTVAZKLVAZKECVBWSYAVCRSUAUBEXMHCKXTGLXNXBOXMVDZPXBVDZXNVDXTV + DVEVFAXGXRUBDUJYAAXFXRUBDAWTDVAZVGZXCXLXEXQYGXAXKXBYFXAXKUHABWTGBUEZCXIUM + ZUNUMXKDIYHWTUHYIXJGUNYHWTCXIVHVIQGXJUNVLVMVJVKYGXDXPWTAXDXPUHYFAKFUGUFZU + OZUPZXDXPAFVNVAYCYLXDUHAFXMVNTAYBXMVNVARXMGYDVOWCVPSFJKLYJMYJVDVQVRAYKXOK + AYJXNAFXMUGTVSWDVTWAWBWEWFWGAXRUBXTDAXTJURUFDAXSJURAXSFKUQUMJAXMFKUQAFXMT + WHWIMWJVSNWJWNWKWKAIJGWLUMVAIJGWMUMVAXHXGVCABCDEFGHIJKLMNOPQRSTUAWOJGIWPU + BJGXBIDEXDNOXDVDYEWQWRWK $. $} ${ @@ -255793,11 +256019,11 @@ each element of the range ( ~ islindf5 ). (Contributed by Stefan E. k W ~=m ( F freeLMod k ) ) ) $= ( vj wcel c0 wne cv cfrlm co clmic wbr wex vex syl cvv eqid clmod lbslcic n0 wa cen enref mp3an3 weq oveq2 breq2d spcev exlimdv syl5bi clbs lmicsym - ex cfv lmiclcl cuvc crn lmodrng frlmlbs sylancl ne0i lmiclbs sylc exlimiv - crg impbid1 ) DUAHZCIJZDBAKZLMZNOZAPZVKGKZCHZGPVJVOGCUCVJVQVOGVJVQVOVJVQU - DDBVPLMZNOZVOVJVQVPVPUEOVSVPGQZUFVPBVPCDFEUBUGVNVSAVPVTAGUHVMVRDNVLVPBLUI - UJUKRUPULUMVNVKAVNVMDNOVMUNUQZIJZVKDVMUOVNVJWBDVMURVJBVLUSMZUTZWAHZWBVJBV - HHVLSHWEBDFVAAQBWCVMVLWASVMTWCTWATZVBVCWAWDVDRRVMDWACWFEVEVFVGVI $. + ex cfv lmiclcl cuvc crn crg lmodring frlmlbs sylancl ne0i lmiclbs exlimiv + sylc impbid1 ) DUAHZCIJZDBAKZLMZNOZAPZVKGKZCHZGPVJVOGCUCVJVQVOGVJVQVOVJVQ + UDDBVPLMZNOZVOVJVQVPVPUEOVSVPGQZUFVPBVPCDFEUBUGVNVSAVPVTAGUHVMVRDNVLVPBLU + IUJUKRUPULUMVNVKAVNVMDNOVMUNUQZIJZVKDVMUOVNVJWBDVMURVJBVLUSMZUTZWAHZWBVJB + VAHVLSHWEBDFVBAQBWCVMVLWASVMTWCTWATZVCVDWAWDVERRVMDWACWFEVFVHVGVI $. $} ${ @@ -255856,11 +256082,11 @@ each element of the range ( ~ islindf5 ). (Contributed by Stefan frlmisfrlm $p |- ( ( R e. NzRing /\ I e. Y /\ I ~~ J ) -> ( R freeLMod I ) ~=m ( R freeLMod J ) ) $= ( cnzr wcel cen wbr w3a cfrlm co csca cfv clmic clmod cuvc crn eqid 3adant3 - sylan clbs crg nzrrng frlmlmod frlmlbs ensymd uvcendim entr syl2anc lbslcic - simp3 syl3anc wceq frlmsca oveq1d breqtrrd ) AEFZBDFZBCGHZIZABJKZVALMZCJKZA - CJKNUTVAOFZABPKZQZVAUAMZFZCVFGHZVAVCNHUQURVDUSUQAUBFZURVDAUCZAVABDVARZUDTSU - QURVHUSUQVJURVHVKAVEVABVGDVLVERZVGRZUETSUTCBGHBVFGHZVIUTBCUQURUSUKUFUQURVOU - SAVEBDVMUGSCBVFUHUIVFVBCVGVAVBRVNUJULUTAVBCJUQURAVBUMUSAVABEDVLUNSUOUP $. + sylan clbs crg nzrring frlmlmod frlmlbs simp3 uvcendim entr syl2anc lbslcic + ensymd syl3anc wceq frlmsca oveq1d breqtrrd ) AEFZBDFZBCGHZIZABJKZVALMZCJKZ + ACJKNUTVAOFZABPKZQZVAUAMZFZCVFGHZVAVCNHUQURVDUSUQAUBFZURVDAUCZAVABDVARZUDTS + UQURVHUSUQVJURVHVKAVEVABVGDVLVERZVGRZUETSUTCBGHBVFGHZVIUTBCUQURUSUFUKUQURVO + USAVEBDVMUGSCBVFUHUIVFVBCVGVAVBRVNUJULUTAVBCJUQURAVBUMUSAVABEDVLUNSUOUP $. $( Every free module is isomorphic to the free module of "column vectors" of the same dimension over the same (nonzero) ring. (Contributed by AV, @@ -256191,13 +256417,13 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. $} ${ - rngvcl.b $e |- B = ( Base ` R ) $. - rngvcl.t $e |- .x. = ( .r ` R ) $. + ringvcl.b $e |- B = ( Base ` R ) $. + ringvcl.t $e |- .x. = ( .r ` R ) $. $( Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) $) - rngvcl $p |- ( ( R e. Ring /\ X e. ( B ^m I ) /\ Y e. ( B ^m I ) ) -> + ringvcl $p |- ( ( R e. Ring /\ X e. ( B ^m I ) /\ Y e. ( B ^m I ) ) -> ( X oF .x. Y ) e. ( B ^m I ) ) $= - ( crg wcel cmgp cfv cmnd cmap co cof eqid rngmgp mgpbas mgpplusg syl3an1 + ( crg wcel cmgp cfv cmnd cmap co cof eqid ringmgp mgpbas mgpplusg syl3an1 mndvcl ) BIJBKLZMJEADNOZJFUDJEFCPOUDJBUCUCQZRACDUCEFABUCUEGSBCUCUEHTUBUA $. $} @@ -256259,16 +256485,16 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) $) mamucl $p |- ( ph -> ( X F Y ) e. ( B ^m ( M X. P ) ) ) $= ( vi co wcel vk vj cmulr cfv cmpt cgsu cmpt2 cxp cmap eqid mamuval wral - cv crg wa rngcmn syl adantr cfn ad2antrr wf elmapi simplrl simpr fovrnd - simplrr rngcl syl3anc ralrimiva gsummptcl ralrimivva cvv wb cbs eqeltri - ccmn fvex xpfi syl2anc elmapg sylancr fmpt2 syl6rbbr mpbid eqeltrd ) AH - IESRUAFCDUBGRUMZUBUMZHSZWGUAUMZISZDUCUDZSZUEUFSZUGZBFCUHZUISZABCDWKRUBU - AEFGUNHILJWKUJZKMNOPQUKAWMBTZUACULRFULZWNWPTZAWRRUAFCAWFFTZWICTZUOZUOZB - UBDGWLJADVPTZXCADUNTZXEKDUPUQURAGUSTXCNURXDWLBTZUBGXDWGGTZUOZXFWHBTWJBT - XGAXFXCXHKUTXIWFWGBFGHAFGUHZBHVAZXCXHAHBXJUISTXKPHBXJVBUQUTAXAXBXHVCXDX - HVDZVEXIWGWIBGCIAGCUHZBIVAZXCXHAIBXMUISTXNQIBXMVBUQUTXLAXAXBXHVFVEBDWKW - HWJJWQVGVHVIVJVKAWTWOBWNVAZWSABVLTWOUSTZWTXOVMBDVNUDVLJDVNVQVOAFUSTCUST - XPMOFCVRVSBWOWNVLUSVTWARUAFCWMBWNWNUJWBWCWDWE $. + cv crg wa ccmn ringcmn syl adantr cfn ad2antrr wf elmapi simplrl fovrnd + simpr simplrr ringcl syl3anc ralrimiva gsummptcl ralrimivva cvv wb fvex + cbs eqeltri xpfi syl2anc elmapg sylancr fmpt2 syl6rbbr mpbid eqeltrd ) + AHIESRUAFCDUBGRUMZUBUMZHSZWGUAUMZISZDUCUDZSZUEUFSZUGZBFCUHZUISZABCDWKRU + BUAEFGUNHILJWKUJZKMNOPQUKAWMBTZUACULRFULZWNWPTZAWRRUAFCAWFFTZWICTZUOZUO + ZBUBDGWLJADUPTZXCADUNTZXEKDUQURUSAGUTTXCNUSXDWLBTZUBGXDWGGTZUOZXFWHBTWJ + BTXGAXFXCXHKVAXIWFWGBFGHAFGUHZBHVBZXCXHAHBXJUISTXKPHBXJVCURVAAXAXBXHVDX + DXHVFZVEXIWGWIBGCIAGCUHZBIVBZXCXHAIBXMUISTXNQIBXMVCURVAXLAXAXBXHVGVEBDW + KWHWJJWQVHVIVJVKVLAWTWOBWNVBZWSABVMTWOUTTZWTXOVNBDVPUDVMJDVPVOVQAFUTTCU + TTXPMOFCVRVSBWOWNVMUTVTWARUAFCWMBWNWNUJWBWCWDWE $. $} ${ @@ -256289,39 +256515,39 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. $( Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.) $) mamuass $p |- ( ph -> ( ( X F Y ) G Z ) = ( X H ( Y I Z ) ) ) $= - ( vi vk vj vl co wceq cv wral wcel wa cfv cmpt cgsu ccmn crg rngcmn syl - cmulr adantr cfn ad2antrr cxp cmap elmapi simplrl fovrnd adantrl simprr - wf simpr simprl simplrr adantrr eqid rngcl syl3anc mamufv oveq1d cplusg - gsumcom3fi c0g anassrs cvv ovex a1i fvex fsuppmptdm gsummulc1 mpteq2dva - rngass syl13anc oveq2d 3eqtr2d anass1rs gsummulc2 eqtr4d 3eqtr4d mamucl - ralrimivva wfn wb ffn 3syl eqfnov2 syl2anc mpbird ) ALMEULZNFULZLMNHULZ - GULZUMZUHUNZUIUNZXOULZXSXTXQULZUMZUICUOUHIUOZAYCUHUIICAXSIUPZXTCUPZUQZU - QZDUJKXSUJUNZXNULZYIXTNULZDVEURZULZUSZUTULZDUKJXSUKUNZLULZYPXTXPULZYLUL - ZUSZUTULZYAYBYHDUJKDUKJYQYPYIMULZYKYLULZYLULZUSZUTULZUSZUTULDUKJDUJKUUD - USUTULZUSZUTULYOUUAYHKBJUJUKDUUDOADVAUPZYGADVBUPZUUJPDVCVDVFAKVGUPZYGSV - FZAJVGUPZYGRVFZYHYIKUPZYPJUPZUQZUQZUUKYQBUPZUUCBUPZUUDBUPAUUKYGUURPVHZY - HUUQUUTUUPYHUUQUQZXSYPBIJLAIJVIZBLVPZYGUUQALBUVDVJULUPZUVEUALBUVDVKVDVH - 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( ( ( M X. O ) X. { X } ) oF .x. 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matgrp jca grpinvval2 cbs rnggrp 3ad2ant1 eleq2i biimpi 3ad2ant2 df-3an - oveqd simp3 sylanbrc matecl cmpt2 anim1i ancoms mat0op eqidd simp3r a1i - cv fvex ovmpt2d eqcomd oveq1d eqtrd 3eqtr4d ) CUBOZIBOZDFOZEFOZPZUCZDEA - UDQZIAUEQZRZRZDEWNRZDEIRZCUEQZRZDEIHQZRWSGQZWHWNBOZWIPZWIWLWQXASWHWIXEW - LWHWIPZXDWIXFAUFOZXDXFFUGOZWHXGWIXHWHWIXHCUJOABCFIJKUHUIZUKWHWIULACFJUT - UMZBAWNKWNTZUNUOWHWIUPZVAUQABCWODEWTFWNIJKWOTZWTTZURUSWMXBWPDEWHWIXBWPS - ZWLXFXGWIXOXJXLBAWOHIWNKXMMXKVBUMUQVJWMXCCUDQZWSWTRZXAWMCUFOZWSCVCQZOZX - CXQSWHWIXRWLCVDVEWMWJWKIAVCQZOZUCZXTWMWLYBYCWHWIWLVKZWIWHYBWLWIYBBYAIKV - FVGVHWJWKYBVIVLACDEXSIFJXSTZVMUOXSCWTGWSXPYEXNLXPTZVBUMWMXPWRWSWTWMWRXP - WMNUADEFFXPXPWNUJWHWIWNNUAFFXPVNSZWLXFXHWHPZYGWIWHYHWIXHWHXIVOVPACNUAFX - PJYFVQUOUQWMNWADSUAWAESPPXPVRWMWJWKYDUIWHWIWJWKVSXPUJOWMCUDWBVTWCWDWEWF - WG $. + matgrp jca grpinvval2 oveqd cbs ringgrp 3ad2ant1 eleq2i biimpi 3ad2ant2 + simp3 df-3an sylanbrc matecl cmpt2 anim1i ancoms mat0op cv eqidd simp3r + fvex a1i ovmpt2d eqcomd oveq1d eqtrd 3eqtr4d ) CUBOZIBOZDFOZEFOZPZUCZDE + AUDQZIAUEQZRZRZDEWNRZDEIRZCUEQZRZDEIHQZRWSGQZWHWNBOZWIPZWIWLWQXASWHWIXE + WLWHWIPZXDWIXFAUFOZXDXFFUGOZWHXGWIXHWHWIXHCUJOABCFIJKUHUIZUKWHWIULACFJU + TUMZBAWNKWNTZUNUOWHWIUPZVAUQABCWODEWTFWNIJKWOTZWTTZURUSWMXBWPDEWHWIXBWP + SZWLXFXGWIXOXJXLBAWOHIWNKXMMXKVBUMUQVCWMXCCUDQZWSWTRZXAWMCUFOZWSCVDQZOZ + XCXQSWHWIXRWLCVEVFWMWJWKIAVDQZOZUCZXTWMWLYBYCWHWIWLVJZWIWHYBWLWIYBBYAIK + VGVHVIWJWKYBVKVLACDEXSIFJXSTZVMUOXSCWTGWSXPYEXNLXPTZVBUMWMXPWRWSWTWMWRX + PWMNUADEFFXPXPWNUJWHWIWNNUAFFXPVNSZWLXFXHWHPZYGWIWHYHWIXHWHXIVOVPACNUAF + XPJYFVQUOUQWMNVRDSUAVRESPPXPVSWMWJWKYDUIWHWIWJWKVTXPUJOWMCUDWAWBWCWDWEW + FWG $. $} matvscacell.k $e |- K = ( Base ` R ) $. @@ -256968,7 +257194,7 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. The matrix algebra =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The main result of this subsection are the theorems showing that -` ( N Mat R ) ` is a ring (see ~ matrng ) and an associative algebra (see +` ( N Mat R ) ` is a ring (see ~ matring ) and an associative algebra (see ~ matassa ). Additionally, theorems for the identity matrix and transposed matrices are provided. $) @@ -256997,11 +257223,11 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. $( The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.) $) mamumat1cl $p |- ( ph -> I e. ( B ^m ( M X. M ) ) ) $= - ( wcel wral cv cvv cfn cxp cmap co wf weq cif wa crg rngidcl ifcl syl2anc - rng0cl syl adantr ralrimivva fmpt2 sylib cbs cfv fvex eqeltri xpfi elmapg - wb sylancr mpbird ) AGBHHUAZUBUCPZVGBGUDZAEFUEZDIUFZBPZFHQEHQVIAVLEFHHAVL - ERHPFRHPUGACUHPZVLKVMDBPIBPVLBCDJLUIBCIJMULVJDIBUJUKUMUNUOEFHHVKBGNUPUQAB - SPVGTPZVHVIVDBCURUSSJCURUTVAAHTPZVOVNOOHHVBUKBVGGSTVCVEVF $. + ( wcel wral cv cvv cfn cxp cmap co wf weq cif wa ringidcl ring0cl syl2anc + crg ifcl syl adantr ralrimivva fmpt2 sylib wb cbs cfv fvex eqeltri elmapg + xpfi sylancr mpbird ) AGBHHUAZUBUCPZVGBGUDZAEFUEZDIUFZBPZFHQEHQVIAVLEFHHA + VLERHPFRHPUGACUKPZVLKVMDBPIBPVLBCDJLUHBCIJMUIVJDIBULUJUMUNUOEFHHVKBGNUPUQ + ABSPVGTPZVHVIURBCUSUTSJCUSVAVBAHTPZVOVNOOHHVDUJBVGGSTVCVEVF $. $d i j A $. $d i j J $. $d i j .0. $. $d i j .1. $. $( The components of the identity matrix (as operation in maps-to @@ -257025,30 +257251,30 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. 22-Jul-2019.) $) mamulid $p |- ( ph -> ( I F X ) = X ) $= ( vl vk vm co wceq cv wral wcel cmulr cfv cmpt cgsu crg eqid adantr cfn - wa cxp cmap mamumat1cl simprl simprr mamufv cmnd rngmnd syl ad2antrr wf - elmapi simplrl simpr fovrnd simplrr rngcl syl3anc fmptd wne w3a weq cif - 3adant3 simp2 mat1comp wb equcom ifbid eqtrd syl2anc ifnefalse 3ad2ant3 - a1i oveq1d rnglz suppsssn gsumpt oveq2 oveq1 oveq12d ovex fvmpt equequ1 - ad2antrl equequ2 equid iftruei syl6eq cur eqeltri ovmpt2 anidms fovrnda - cvv fvex rnglidm 3eqtrd ralrimivva wfn mamucl ffn eqfnov2 mpbird ) AHKG - UEZKUFZUBUGZUCUGZYCUEZYEYFKUEZUFZUCJUHUBIUHZAYIUBUCIJAYEIUIZYFJUIZURZUR - ZYGCUDIYEUDUGZHUEZYOYFKUEZCUJUKZUEZULZUMUEYEYTUKZYHYNBJCYRUDGYEYFIIUNHK - TMYRUOZACUNUIZYMNUPZAIUQUIYMRUPZUUEAJUQUIYMSUPAHBIIUSZUTUEUIZYMABCDEFHI - LMNOPQRVAZUPAKBIJUSZUTUEZUIZYMUAUPAYKYLVBZAYKYLVCVDYNIBYTCUQYELMPYNUUCC - VEUIUUDCVFVGUUEUULYNUDIYSBYTYNYOIUIZURZUUCYPBUIYQBUIZYSBUIAUUCYMUUMNVHZ - UUNYEYOBIIHAUUFBHVIZYMUUMAUUGUUQUUHHBUUFVJVGVHAYKYLUUMVKZYNUUMVLZVMUUNY - OYFBIJKAUUIBKVIZYMUUMAUUKUUTUAKBUUIVJVGZVHUUSAYKYLUUMVNVMZBCYRYPYQMUUBV - OVPYTUOZVQYNIYSUDUQYELYNUUMYOYEVRZVSZYSLYQYRUEZLUVEYPLYQYRUVEYPUDUBVTZD - LWAZLUVEYKUUMYPUVHUFYNUUMYKUVDUURWBYNUUMUVDWCYKUUMURZYPUBUDVTZDLWAUVHAY - EBCDEFHYOILMNOPQRWDUVIUVJUVGDLUVJUVGWEUVIUBUDWFWLWGWHWIUVDYNUVHLUFUUMYO - YEDLWJWKWHWMYNUUMUVFLUFZUVDUUNUUCUUOUVKUUPUVBBCYRYQLMUUBPWNWIWBWHUUEWOW - PYNUUAYEYEHUEZYHYRUEZDYHYRUEZYHYKUUAUVMUFAYLUDYEYSUVMIYTUVGYPUVLYQYHYRY - OYEYEHWQYOYEYFKWRWSUVCUVLYHYRWTXAXCYNUVLDYHYRYKUVLDUFZAYLYKUVOEFYEYEIIE - FVTZDLWADHUBFVTZDLWAZEUBVTUVPUVQDLEUBFXBWGFUBVTZUVRUBUBVTZDLWADUVSUVQUV - TDLFUBUBXDWGUVTDLUBXEXFXGQDCXHUKXMOCXHXNXIXJXKXCWMYNUUCYHBUIUVNYHUFUUDA - YEYFBIJKUVAXLBCYRDYHMUUBOXOWIXPXPXQAYCUUIXRZKUUIXRZYDYJWEAUUIBYCVIZUWAA - YCUUJUIUWCABJCGIIHKMNTRRSUUHUAXSYCBUUIVJVGUUIBYCXTVGAUUTUWBUVAUUIBKXTVG - UBUCIJYCKYAWIYB $. + wa cxp cmap mamumat1cl simprl simprr mamufv ringmnd syl ad2antrr elmapi + cmnd wf simplrl simpr fovrnd simplrr ringcl syl3anc wne w3a weq 3adant3 + fmptd cif simp2 mat1comp wb equcom a1i ifbid syl2anc ifnefalse 3ad2ant3 + eqtrd oveq1d ringlz suppsssn gsumpt oveq2 oveq12d ovex ad2antrl equequ1 + oveq1 fvmpt equequ2 equid iftruei syl6eq cur fvex eqeltri ovmpt2 anidms + cvv fovrnda ringlidm 3eqtrd ralrimivva wfn mamucl ffn eqfnov2 mpbird ) + AHKGUEZKUFZUBUGZUCUGZYCUEZYEYFKUEZUFZUCJUHUBIUHZAYIUBUCIJAYEIUIZYFJUIZU + RZURZYGCUDIYEUDUGZHUEZYOYFKUEZCUJUKZUEZULZUMUEYEYTUKZYHYNBJCYRUDGYEYFII + UNHKTMYRUOZACUNUIZYMNUPZAIUQUIYMRUPZUUEAJUQUIYMSUPAHBIIUSZUTUEUIZYMABCD + EFHILMNOPQRVAZUPAKBIJUSZUTUEZUIZYMUAUPAYKYLVBZAYKYLVCVDYNIBYTCUQYELMPYN + UUCCVIUIUUDCVEVFUUEUULYNUDIYSBYTYNYOIUIZURZUUCYPBUIYQBUIZYSBUIAUUCYMUUM + NVGZUUNYEYOBIIHAUUFBHVJZYMUUMAUUGUUQUUHHBUUFVHVFVGAYKYLUUMVKZYNUUMVLZVM + UUNYOYFBIJKAUUIBKVJZYMUUMAUUKUUTUAKBUUIVHVFZVGUUSAYKYLUUMVNVMZBCYRYPYQM + UUBVOVPYTUOZWAYNIYSUDUQYELYNUUMYOYEVQZVRZYSLYQYRUEZLUVEYPLYQYRUVEYPUDUB + VSZDLWBZLUVEYKUUMYPUVHUFYNUUMYKUVDUURVTYNUUMUVDWCYKUUMURZYPUBUDVSZDLWBU + VHAYEBCDEFHYOILMNOPQRWDUVIUVJUVGDLUVJUVGWEUVIUBUDWFWGWHWLWIUVDYNUVHLUFU + UMYOYEDLWJWKWLWMYNUUMUVFLUFZUVDUUNUUCUUOUVKUUPUVBBCYRYQLMUUBPWNWIVTWLUU + EWOWPYNUUAYEYEHUEZYHYRUEZDYHYRUEZYHYKUUAUVMUFAYLUDYEYSUVMIYTUVGYPUVLYQY + HYRYOYEYEHWQYOYEYFKXBWRUVCUVLYHYRWSXCWTYNUVLDYHYRYKUVLDUFZAYLYKUVOEFYEY + EIIEFVSZDLWBDHUBFVSZDLWBZEUBVSUVPUVQDLEUBFXAWHFUBVSZUVRUBUBVSZDLWBDUVSU + VQUVTDLFUBUBXDWHUVTDLUBXEXFXGQDCXHUKXMOCXHXIXJXKXLWTWMYNUUCYHBUIUVNYHUF + UUDAYEYFBIJKUVAXNBCYRDYHMUUBOXOWIXPXPXQAYCUUIXRZKUUIXRZYDYJWEAUUIBYCVJZ + UWAAYCUUJUIUWCABJCGIIHKMNTRRSUUHUAXSYCBUUIVHVFUUIBYCXTVFAUUTUWBUVAUUIBK + XTVFUBUCIJYCKYAWIYB $. $} $d k B $. $d k .0. $. $d l m F $. $d k R $. @@ -257059,28 +257285,28 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) $) mamurid $p |- ( ph -> ( X F I ) = X ) $= ( vl vm vk co wceq cv wral wcel wa cmulr cfv cmpt cgsu crg adantr cfn cxp - eqid cmap mamumat1cl simprl simprr mamufv cmnd rngmnd syl ad2antrr elmapi - wf simplrl simpr fovrnd simplrr rngcl syl3anc fmptd wne w3a weq cif simp2 - 3adant3 mat1comp syl2anc 3ad2ant3 eqtrd oveq2d rngrz suppsssn oveq2 oveq1 - ifnefalse gsumpt oveq12d ovex fvmpt ad2antll equequ1 ifbid equequ2 syl6eq - iftruei cur cvv fvex eqeltri ovmpt2 anidms fovrnda rngridm ralrimivva wfn - 3eqtrd wb mamucl ffn eqfnov2 mpbird ) AKHGUEZKUFZUBUGZUCUGZXTUEZYBYCKUEZU - FZUCIUHUBJUHZAYFUBUCJIAYBJUIZYCIUIZUJZUJZYDCUDIYBUDUGZKUEZYLYCHUEZCUKULZU - EZUMZUNUEYCYQULZYEYKBICYOUDGYBYCJIUOKHTMYOUSZACUOUIZYJNUPZAJUQUIYJSUPAIUQ - UIYJRUPZUUBAKBJIURZUTUEZUIZYJUAUPAHBIIURZUTUEUIZYJABCDEFHILMNOPQRVAZUPAYH - YIVBAYHYIVCZVDYKIBYQCUQYCLMPYKYTCVEUIUUACVFVGUUBUUIYKUDIYPBYQYKYLIUIZUJZY - TYMBUIZYNBUIYPBUIAYTYJUUJNVHZUUKYBYLBJIKAUUCBKVJZYJUUJAUUEUUNUAKBUUCVIVGZ - VHAYHYIUUJVKYKUUJVLZVMZUUKYLYCBIIHAUUFBHVJZYJUUJAUUGUURUUHHBUUFVIVGVHUUPA - YHYIUUJVNZVMBCYOYMYNMYSVOVPYQUSZVQYKIYPUDUQYCLYKUUJYLYCVRZVSZYPYMLYOUEZLU - VBYNLYMYOUVBYNUDUCVTZDLWAZLUVBUUJYIYNUVEUFYKUUJUVAWBYKUUJYIUVAUUSWCAYLBCD - EFHYCILMNOPQRWDWEUVAYKUVELUFUUJYLYCDLWMWFWGWHYKUUJUVCLUFZUVAUUKYTUULUVFUU - MUUQBCYOYMLMYSPWIWEWCWGUUBWJWNYKYRYEYCYCHUEZYOUEZYEDYOUEZYEYIYRUVHUFAYHUD - YCYPUVHIYQUVDYMYEYNUVGYOYLYCYBKWKYLYCYCHWLWOUUTYEUVGYOWPWQWRYIUVHUVIUFAYH - YIUVGDYEYOYIUVGDUFEFYCYCIIEFVTZDLWADHUCFVTZDLWAZEUCVTUVJUVKDLEUCFWSWTFUCV - TZUVLUCUCVTZDLWADUVMUVKUVNDLFUCUCXAWTUVNDLYCUSXCXBQDCXDULXEOCXDXFXGXHXIWH - WRYKYTYEBUIUVIYEUFUUAAYBYCBJIKUUOXJBCYODYEMYSOXKWEXNXNXLAXTUUCXMZKUUCXMZY - AYGXOAUUCBXTVJZUVOAXTUUDUIUVQABICGJIKHMNTSRRUAUUHXPXTBUUCVIVGUUCBXTXQVGAU - UNUVPUUOUUCBKXQVGUBUCJIXTKXRWEXS $. + eqid cmap mamumat1cl simprl simprr mamufv cmnd ringmnd ad2antrr wf elmapi + syl simplrl simpr fovrnd simplrr ringcl syl3anc fmptd wne w3a weq 3adant3 + cif simp2 mat1comp ifnefalse 3ad2ant3 eqtrd oveq2d ringrz suppsssn gsumpt + syl2anc oveq2 oveq1 oveq12d fvmpt ad2antll equequ1 equequ2 iftruei syl6eq + ovex ifbid cur cvv fvex eqeltri ovmpt2 anidms fovrnda ringridm ralrimivva + 3eqtrd wfn wb mamucl ffn eqfnov2 mpbird ) AKHGUEZKUFZUBUGZUCUGZXTUEZYBYCK + UEZUFZUCIUHUBJUHZAYFUBUCJIAYBJUIZYCIUIZUJZUJZYDCUDIYBUDUGZKUEZYLYCHUEZCUK + ULZUEZUMZUNUEYCYQULZYEYKBICYOUDGYBYCJIUOKHTMYOUSZACUOUIZYJNUPZAJUQUIYJSUP + AIUQUIYJRUPZUUBAKBJIURZUTUEZUIZYJUAUPAHBIIURZUTUEUIZYJABCDEFHILMNOPQRVAZU + PAYHYIVBAYHYIVCZVDYKIBYQCUQYCLMPYKYTCVEUIUUACVFVJUUBUUIYKUDIYPBYQYKYLIUIZ + UJZYTYMBUIZYNBUIYPBUIAYTYJUUJNVGZUUKYBYLBJIKAUUCBKVHZYJUUJAUUEUUNUAKBUUCV + IVJZVGAYHYIUUJVKYKUUJVLZVMZUUKYLYCBIIHAUUFBHVHZYJUUJAUUGUURUUHHBUUFVIVJVG + UUPAYHYIUUJVNZVMBCYOYMYNMYSVOVPYQUSZVQYKIYPUDUQYCLYKUUJYLYCVRZVSZYPYMLYOU + EZLUVBYNLYMYOUVBYNUDUCVTZDLWBZLUVBUUJYIYNUVEUFYKUUJUVAWCYKUUJYIUVAUUSWAAY + LBCDEFHYCILMNOPQRWDWLUVAYKUVELUFUUJYLYCDLWEWFWGWHYKUUJUVCLUFZUVAUUKYTUULU + VFUUMUUQBCYOYMLMYSPWIWLWAWGUUBWJWKYKYRYEYCYCHUEZYOUEZYEDYOUEZYEYIYRUVHUFA + YHUDYCYPUVHIYQUVDYMYEYNUVGYOYLYCYBKWMYLYCYCHWNWOUUTYEUVGYOXBWPWQYIUVHUVIU + FAYHYIUVGDYEYOYIUVGDUFEFYCYCIIEFVTZDLWBDHUCFVTZDLWBZEUCVTUVJUVKDLEUCFWRXC + FUCVTZUVLUCUCVTZDLWBDUVMUVKUVNDLFUCUCWSXCUVNDLYCUSWTXAQDCXDULXEOCXDXFXGXH + XIWHWQYKYTYEBUIUVIYEUFUUAAYBYCBJIKUUOXJBCYODYEMYSOXKWLXMXMXLAXTUUCXNZKUUC + XNZYAYGXOAUUCBXTVHZUVOAXTUUDUIUVQABICGJIKHMNTSRRUAUUHXPXTBUUCVIVJUUCBXTXQ + VJAUUNUVPUUOUUCBKXQVJUBUCJIXTKXRWLXS $. $} ${ @@ -257090,45 +257316,45 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. $( Existence of the matrix ring, see also the statement "For a given integer n > 0 the set of square n x n matrices form a ring." in [Lang] p. 504. (Contributed by Stefan O'Rear, 5-Sep-2015.) $) - matrng $p |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring ) $= + matring $p |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring ) $= ( vx vy va vb wcel crg wa cfv co eqid cv mamucl eleqtrd matplusg2 syl2anc wceq vz cfn cbs cxp cmap cplusg cmmul weq cur c0g cif cmpt2 matbas2 eqidd cotp matmulr clmod cgrp matlmod lmodgrp syl w3a simp1r simp1l simp2 simp3 simplr simpll simpr1 simpr2 simpr3 mamuass cof adantr oveq2d mamudi simpr - mamudir 3eqtr4d oveq1d simpl mamumat1cl mamulid mamurid isrngd ) CUBIZBJI - ZKZEFUABUCLZCCUDUEMZAUFLZABCCCUOUGMZGHCCGHUHBUILZBUJLZUKULZABWICJDWINZUMZ - WHWKUNABWLCJDWLNZUPWHAUQIAURIABCDUSAUTVAWHEOZWJIZFOZWJIZVBWICBWLCCWSXAWPW - FWGWTXBVCWRWFWGWTXBVDZXCXCWHWTXBVEWHWTXBVFPWHWTXBUAOZWJIZVBZKZWICBWLWLWLW - LCCCWSXAXDWPWFWGXFVGZWFWGXFVHZXIXIXIWHWTXBXEVIZWHWTXBXEVJZWHWTXBXEVKZWRWR - WRWRVLXGWSXAXDBUFLZVMZMZWLMWSXAWLMZWSXDWLMZXNMZWSXAXDWKMZWLMXPXQWKMZXGWIX - MBWLCCCWSXAXDWPXHWRXIXIXIXMNZXJXKXLVRXGXSXOWSWLXGXAAUCLZIZXDYBIXSXOTXGXAW - JYBXKWHWJYBTXFWQVNZQZXGXDWJYBXLYDQAYBXMWKBCXAXDDYBNZWKNZYARSVOXGXPYBIXQYB - IZXTXRTXGXPWJYBXGWICBWLCCWSXAWPXHWRXIXIXIXJXKPYDQXGXQWJYBXGWICBWLCCWSXDWP - XHWRXIXIXIXJXLPYDQZAYBXMWKBCXPXQDYFYGYARSVSXGWSXAXNMZXDWLMXQXAXDWLMZXNMZW - SXAWKMZXDWLMXQYKWKMZXGWIXMBWLCCCWSXAXDWPXHWRXIXIXIYAXJXKXLVPXGYMYJXDWLXGW - SYBIYCYMYJTXGWSWJYBXJYDQYEAYBXMWKBCWSXADYFYGYARSVTXGYHYKYBIYNYLTYIXGYKWJY - BXGWICBWLCCXAXDWPXHWRXIXIXIXKXLPYDQAYBXMWKBCXQYKDYFYGYARSVSWHWIBWMGHWOCWN - WPWFWGVQWMNZWNNZWONZWFWGWAWBWHWTKZWIBWMGHWLWOCCWSWNWPWFWGWTVGZYOYPYQWFWGW - TVHZYTWRWHWTVQZWCYRWIBWMGHWLWOCCWSWNWPYSYOYPYQYTYTWRUUAWDWE $. + mamudir 3eqtr4d oveq1d simpl mamumat1cl mamulid mamurid isringd ) CUBIZBJ + IZKZEFUABUCLZCCUDUEMZAUFLZABCCCUOUGMZGHCCGHUHBUILZBUJLZUKULZABWICJDWINZUM + ZWHWKUNABWLCJDWLNZUPWHAUQIAURIABCDUSAUTVAWHEOZWJIZFOZWJIZVBWICBWLCCWSXAWP + WFWGWTXBVCWRWFWGWTXBVDZXCXCWHWTXBVEWHWTXBVFPWHWTXBUAOZWJIZVBZKZWICBWLWLWL + WLCCCWSXAXDWPWFWGXFVGZWFWGXFVHZXIXIXIWHWTXBXEVIZWHWTXBXEVJZWHWTXBXEVKZWRW + RWRWRVLXGWSXAXDBUFLZVMZMZWLMWSXAWLMZWSXDWLMZXNMZWSXAXDWKMZWLMXPXQWKMZXGWI + XMBWLCCCWSXAXDWPXHWRXIXIXIXMNZXJXKXLVRXGXSXOWSWLXGXAAUCLZIZXDYBIXSXOTXGXA + WJYBXKWHWJYBTXFWQVNZQZXGXDWJYBXLYDQAYBXMWKBCXAXDDYBNZWKNZYARSVOXGXPYBIXQY + BIZXTXRTXGXPWJYBXGWICBWLCCWSXAWPXHWRXIXIXIXJXKPYDQXGXQWJYBXGWICBWLCCWSXDW + PXHWRXIXIXIXJXLPYDQZAYBXMWKBCXPXQDYFYGYARSVSXGWSXAXNMZXDWLMXQXAXDWLMZXNMZ + WSXAWKMZXDWLMXQYKWKMZXGWIXMBWLCCCWSXAXDWPXHWRXIXIXIYAXJXKXLVPXGYMYJXDWLXG + WSYBIYCYMYJTXGWSWJYBXJYDQYEAYBXMWKBCWSXADYFYGYARSVTXGYHYKYBIYNYLTYIXGYKWJ + YBXGWICBWLCCXAXDWPXHWRXIXIXIXKXLPYDQAYBXMWKBCXQYKDYFYGYARSVSWHWIBWMGHWOCW + NWPWFWGVQWMNZWNNZWONZWFWGWAWBWHWTKZWIBWMGHWLWOCCWSWNWPWFWGWTVGZYOYPYQWFWG + WTVHZYTWRWHWTVQZWCYRWIBWMGHWLWOCCWSWNWPYSYOYPYQYTYTWRUUAWDWE $. $( Existence of the matrix algebra, see also the statement "Then Mat_n(R) is an algebra over R" in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.) $) matassa $p |- ( ( N e. Fin /\ R e. CRing ) -> A e. AssAlg ) $= ( vy vz vx wcel ccrg wa cbs cfv co cxp eqid wceq eleqtrd matvsca2 syl2anc - cv cfn cvsca cotp cmmul matbas2 matsca2 eqidd matmulr crg crngrng matlmod - clmod sylan2 matrng simpr w3a csn cmulr cof ad2antlr simpll simpr1 simpr2 - simpr3 mamuvs1 adantr oveq1d mamucl 3eqtr4d simplr mamuvs2 oveq2d isassad - cmap ) CUAHZBIHZJZEFBKLZAUBLZBCCCUCUDMZBVRCCNZVNMZAGABVRCIDVROZUEZABCIDUF - VQVRUGVQVSUGABVTCIDVTOZUHVPVOBUIHZAULHBUJZABCDUKUMVPVOWFAUIHWGABCDUNUMVOV - PUOVQGTZVRHZETZWBHZFTZWBHZUPZJZWAWHUQNZWJBURLZUSZMZWLVTMWPWJWLVTMZWRMZWHW - JVSMZWLVTMWHWTVSMZWOVRBWQVTCCCWHWJWLWCVPWFVOWNWGUTZWEVOVPWNVAZXEXEWQOZVQW - IWKWMVBZVQWIWKWMVCZVQWIWKWMVDZVEWOXBWSWLVTWOWIWJAKLZHXBWSPXGWOWJWBXJXHVQW - BXJPWNWDVFZQAXJWABVSWQVRCWHWJDXJOZWCVSOZXFWAOZRSVGWOWIWTXJHXCXAPXGWOWTWBX - JWOVRCBVTCCWJWLWCXDWEXEXEXEXHXIVHXKQAXJWABVSWQVRCWHWTDXLWCXMXFXNRSZVIWOWJ - WPWLWRMZVTMXAWJWHWLVSMZVTMXCWOVRBWQVTCCCWJWHWLVOVPWNVJWCXFWEXEXEXEXHXGXIV - KWOXQXPWJVTWOWIWLXJHXQXPPXGWOWLWBXJXIXKQAXJWABVSWQVRCWHWLDXLWCXMXFXNRSVLX - OVIVM $. + cv cfn cvsca cotp cmmul cmap matbas2 matsca2 eqidd matmulr clmod crngring + crg matlmod sylan2 matring simpr w3a csn cmulr cof ad2antlr simpll simpr1 + simpr2 simpr3 mamuvs1 adantr oveq1d mamucl 3eqtr4d simplr mamuvs2 isassad + oveq2d ) CUAHZBIHZJZEFBKLZAUBLZBCCCUCUDMZBVRCCNZUEMZAGABVRCIDVROZUFZABCID + UGVQVRUHVQVSUHABVTCIDVTOZUIVPVOBULHZAUJHBUKZABCDUMUNVPVOWFAULHWGABCDUOUNV + OVPUPVQGTZVRHZETZWBHZFTZWBHZUQZJZWAWHURNZWJBUSLZUTZMZWLVTMWPWJWLVTMZWRMZW + HWJVSMZWLVTMWHWTVSMZWOVRBWQVTCCCWHWJWLWCVPWFVOWNWGVAZWEVOVPWNVBZXEXEWQOZV + QWIWKWMVCZVQWIWKWMVDZVQWIWKWMVEZVFWOXBWSWLVTWOWIWJAKLZHXBWSPXGWOWJWBXJXHV + QWBXJPWNWDVGZQAXJWABVSWQVRCWHWJDXJOZWCVSOZXFWAOZRSVHWOWIWTXJHXCXAPXGWOWTW + BXJWOVRCBVTCCWJWLWCXDWEXEXEXEXHXIVIXKQAXJWABVSWQVRCWHWTDXLWCXMXFXNRSZVJWO + WJWPWLWRMZVTMXAWJWHWLVSMZVTMXCWOVRBWQVTCCCWJWHWLVOVPWNVKWCXFWEXEXEXEXHXGX + IVLWOXQXPWJVTWOWIWLXJHXQXPPXGWOWLWBXJXIXKQAXJWABVSWQVRCWHWLDXLWCXMXFXNRSV + NXOVJVM $. $} ${ @@ -257210,14 +257436,14 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. ( i e. N , j e. N |-> if ( i = j , .1. , .0. ) ) ) $= ( vx wcel crg wa cv wceq cbs cfv co eqid cfn cif cmpt2 cmulr wral cur cxp simpr simpl mamumat1cl matbas2 eleqtrd cmmul matmulr adantr simplr simpll - cmap cotp oveqd eleq2d biimpar mamulid eqtr3d mamurid ralrimiva wb matrng - jca isrngid syl mpbi2and ) FUALZBMLZNZDEFFDOEOPCGUBUCZAQRZLZVPKOZAUDRZSZV - SPZVSVPVTSZVSPZNZKVQUEZAUFRZVPPZVOVPBQRZFFUGURSZVQVOWIBCDEVPFGWITZVMVNUHI - JVPTZVMVNUIUJABWIFMHWKUKZULVOWEKVQVOVSVQLZNZWBWDWOVPVSBFFFUSUMSZSWAVSWOWP - VTVPVSVOWPVTPWNABWPFMHWPTZUNUOZUTWOWIBCDEWPVPFFVSGWKVMVNWNUPZIJWLVMVNWNUQ - ZWTWQVOVSWJLWNVOWJVQVSWMVAVBZVCVDWOVSVPWPSWCVSWOWPVTVSVPWRUTWOWIBCDEWPVPF - FVSGWKWSIJWLWTWTWQXAVEVDVIVFVOAMLVRWFNWHVGABFHVHKVQAVTWGVPVQTVTTWGTVJVKVL - $. + cmap oveqd eleq2d biimpar mamulid eqtr3d mamurid jca ralrimiva wb matring + cotp isringid syl mpbi2and ) FUALZBMLZNZDEFFDOEOPCGUBUCZAQRZLZVPKOZAUDRZS + ZVSPZVSVPVTSZVSPZNZKVQUEZAUFRZVPPZVOVPBQRZFFUGURSZVQVOWIBCDEVPFGWITZVMVNU + HIJVPTZVMVNUIUJABWIFMHWKUKZULVOWEKVQVOVSVQLZNZWBWDWOVPVSBFFFVIUMSZSWAVSWO + WPVTVPVSVOWPVTPWNABWPFMHWPTZUNUOZUSWOWIBCDEWPVPFFVSGWKVMVNWNUPZIJWLVMVNWN + UQZWTWQVOVSWJLWNVOWJVQVSWMUTVAZVBVCWOVSVPWPSWCVSWOWPVTVSVPWRUSWOWIBCDEWPV + PFFVSGWKWSIJWLWTWTWQXAVDVCVEVFVOAMLVRWFNWHVGABFHVHKVQAVTWGVPVQTVTTWGTVJVK + VL $. $d I i j $. $d J i j $. $d ph i j $. mat1ov.n $e |- ( ph -> N e. Fin ) $. @@ -257241,8 +257467,8 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. mat1bas.i $e |- .1. = ( 1r ` ( N Mat R ) ) $. $( The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.) $) mat1bas $p |- ( ( R e. Ring /\ N e. Fin ) -> .1. e. B ) $= - ( crg wcel cfn wa cmat co eqid matrng ancoms cbs cfv fveq2i eqtri rngidcl - syl ) CIJZEKJZLECMNZIJZDBJUEUDUGUFCEUFOPQBUFDBARSUFRSGAUFRFTUAHUBUC $. + ( crg wcel cfn wa cmat co eqid matring ancoms cbs cfv fveq2i ringidcl syl + eqtri ) CIJZEKJZLECMNZIJZDBJUEUDUGUFCEUFOPQBUFDBARSUFRSGAUFRFTUCHUAUB $. $} ${ @@ -257257,16 +257483,16 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. matsc $p |- ( ( N e. Fin /\ R e. Ring /\ L e. K ) -> ( L .x. ( 1r ` A ) ) = ( i e. N , j e. N |-> if ( i = j , L , .0. ) ) ) $= ( cfn wcel cfv co cmpt2 wceq eqid crg w3a cur cxp csn cmulr cof weq simp3 - cif cbs 3simpa matrng rngidcl 3syl matvsca2 syl2anc cvv simp1 simp13 fvex - wa cv c0g eqeltri ifex a1i fconstmpt2 mat1 3adant3 offval22 oveq2 rngridm - ifsb 3adant1 rngrz ifeq12d syl5eq mpt2eq3dv 3eqtrd ) HNOZBUAOZGFOZUBZGAUC - PZCQZHHUDZGUEUDZWEBUFPZUGQZDEHHGDEUHZBUCPZIUJZWIQZRDEHHWKGIUJZRWDWCWEAUKP - ZOZWFWJSWAWBWCUIWDWAWBVBAUAOWQWAWBWCULABHJUMWPAWEWPTZWETUNUOAWPWGBCWIFHGW - EJWRKLWITZWGTUPUQWDDEHHGWMWIWHWENNFURWAWBWCUSZWTWAWBWCDVCHOZEVCHOZUTWMURO - WDXAXBUBWKWLIBUCVAIBVDPURMBVDVAVEVFVGWHDEHHGRSWDDEHHGVHVGWAWBWEDEHHWMRSWC - ABWLDEHIJWLTZMVIVJVKWDDEHHWNWOWDWNWKGWLWIQZGIWIQZUJWOWKWLIWNXDXEWMWLGWIVL - WMIGWIVLVNWDWKXDGXEIWBWCXDGSWAFBWIWLGKWSXCVMVOWBWCXEISWAFBWIGIKWSMVPVOVQV - RVSVT $. + cif cbs wa 3simpa matring ringidcl 3syl matvsca2 syl2anc cvv simp1 simp13 + fvex c0g eqeltri ifex a1i fconstmpt2 mat1 3adant3 offval22 oveq2 ringridm + cv ifsb 3adant1 ringrz ifeq12d syl5eq mpt2eq3dv 3eqtrd ) HNOZBUAOZGFOZUBZ + GAUCPZCQZHHUDZGUEUDZWEBUFPZUGQZDEHHGDEUHZBUCPZIUJZWIQZRDEHHWKGIUJZRWDWCWE + AUKPZOZWFWJSWAWBWCUIWDWAWBULAUAOWQWAWBWCUMABHJUNWPAWEWPTZWETUOUPAWPWGBCWI + FHGWEJWRKLWITZWGTUQURWDDEHHGWMWIWHWENNFUSWAWBWCUTZWTWAWBWCDVMHOZEVMHOZVAW + MUSOWDXAXBUBWKWLIBUCVBIBVCPUSMBVCVBVDVEVFWHDEHHGRSWDDEHHGVGVFWAWBWEDEHHWM + RSWCABWLDEHIJWLTZMVHVIVJWDDEHHWNWOWDWNWKGWLWIQZGIWIQZUJWOWKWLIWNXDXEWMWLG + WIVKWMIGWIVKVNWDWKXDGXEIWBWCXDGSWAFBWIWLGKWSXCVLVOWBWCXEISWAFBWIGIKWSMVPV + OVQVRVSVT $. $} ${ @@ -257447,17 +257673,17 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. = ( U gsum ( r e. N |-> ( r M r ) ) ) ) $= ( wceq cfv co ccrg wcel cid cres w3a ccom cv cmpt cgsu fveq2 fveq1 oveq1d mpteq2dv oveq2d oveq12d 3ad2ant3 cur cfn cvv matrcl simpld csymg c0g czrh - cpsgn coeq12i a1i eqid symgid adantl fveq12d cmhm crngrng cmgp zrhpsgnmhm - wa crg oveq2i syl6eleqr sylan rngidval mhm0 eqtrd fvresi mpteq2dva sylan2 - syl cbs adantr matgsumcl rnglidm syl2anc 3adant3 ) DUAUBZHBUBZCUCIUDZRZUE - CJEUFZSZGKIKUGZCSZWTHTZUHZUITZFTZWPWRSZGKIWTWPSZWTHTZUHZUITZFTZGKIWTWTHTZ - UHZUITZWQWNXEXKRWOWQWSXFXDXJFCWPWRUJWQXCXIGUIWQKIXBXHWQXAXGWTHWTCWPUKULUM - UNUOUPWNWOXKXNRWQWNWOVPZXKDUQSZXNFTZXNWOWNIURUBZXKXQRWOXRDUSUBABDIHLMUTVA - WNXRVPZXFXPXJXNFXSXFIVBSZVCSZDVDSZIVESZUFZSZXPXSWPYAWRYDWRYDRXSJYBEYCOPVF - VGXRWPYARWNIXTURXTVHVIVJVKXSYDXTGVLTZUBZYEXPRWNDVQUBZXRYGDVMZYHXRVPYDXTDV - NSZVLTYFIDVOGYJXTVLNVRVSVTXTGYDXPYAYAVHDXPGNXPVHZWAWBWGWCXSXIXMGUIXSKIXHX - LXSWTIUBZVPXGWTWTHYLXGWTRXSIWTWDVJULWEUNUOWFXOYHXNDWHSZUBXQXNRWNYHWOYIWIA - BDGHIKLMNWJYMDFXPXNYMVHQYKWKWLWCWMWC $. + cpsgn coeq12i a1i eqid symgid adantl fveq12d cmhm crg crngring zrhpsgnmhm + wa cmgp oveq2i syl6eleqr sylan rngidval syl eqtrd fvresi mpteq2dva sylan2 + mhm0 cbs adantr matgsumcl ringlidm syl2anc 3adant3 ) DUAUBZHBUBZCUCIUDZRZ + UECJEUFZSZGKIKUGZCSZWTHTZUHZUITZFTZWPWRSZGKIWTWPSZWTHTZUHZUITZFTZGKIWTWTH + TZUHZUITZWQWNXEXKRWOWQWSXFXDXJFCWPWRUJWQXCXIGUIWQKIXBXHWQXAXGWTHWTCWPUKUL + UMUNUOUPWNWOXKXNRWQWNWOVPZXKDUQSZXNFTZXNWOWNIURUBZXKXQRWOXRDUSUBABDIHLMUT + VAWNXRVPZXFXPXJXNFXSXFIVBSZVCSZDVDSZIVESZUFZSZXPXSWPYAWRYDWRYDRXSJYBEYCOP + VFVGXRWPYARWNIXTURXTVHVIVJVKXSYDXTGVLTZUBZYEXPRWNDVMUBZXRYGDVNZYHXRVPYDXT + DVQSZVLTYFIDVOGYJXTVLNVRVSVTXTGYDXPYAYAVHDXPGNXPVHZWAWGWBWCXSXIXMGUIXSKIX + HXLXSWTIUBZVPXGWTWTHYLXGWTRXSIWTWDVJULWEUNUOWFXOYHXNDWHSZUBXQXNRWNYHWOYIW + IABDGHIKLMNWJYMDFXPXNYMVHQYKWKWLWCWMWC $. $} ${ @@ -257497,26 +257723,26 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. madetsmelbas $p |- ( ( R e. CRing /\ M e. B /\ Q e. P ) -> ( ( ( Y o. S ) ` Q ) ( .r ` R ) ( G gsum ( n e. N |-> ( ( Q ` n ) M n ) ) ) ) e. ( Base ` R ) ) $= - ( wcel cfv co ccrg w3a crg ccom cbs cv cmpt cgsu crngrng 3ad2ant1 cfn cvv - cmulr matrcl simpld simp3 zrhcopsgnelbas syl3anc eqid mgpbas ccmn crngmgp - 3ad2ant2 wral simp2 matepmcl gsummptcl rngcl ) EUARZIBRZDCRZUBZEUCRZDKFUD - SZEUESZRZHGJGUFZDSVQITZUGUHTZVORVNVSEUMSZTVORVIVJVMVKEUIUJZVLVMJUKRZVKVPW - AVJVIWBVKVJWBEULRABEJIOPUNUOVCZVIVJVKUPZCDEFJKLMNUQURVLVOGHJVRVOEHQVOUSZU - TVIVJHVARVKEHQVBUJWCVLVMVKVJVRVORGJVDWAWDVIVJVKVEABCDEGIJOPLVFURVGVOEVTVN - VSWEVTUSVHUR $. + ( wcel cfv co ccrg w3a crg ccom cbs cmpt cgsu cmulr crngring 3ad2ant1 cfn + cv cvv matrcl simpld 3ad2ant2 simp3 zrhcopsgnelbas syl3anc mgpbas crngmgp + eqid ccmn wral simp2 matepmcl gsummptcl ringcl ) EUARZIBRZDCRZUBZEUCRZDKF + UDSZEUESZRZHGJGULZDSVQITZUFUGTZVORVNVSEUHSZTVORVIVJVMVKEUIUJZVLVMJUKRZVKV + PWAVJVIWBVKVJWBEUMRABEJIOPUNUOUPZVIVJVKUQZCDEFJKLMNURUSVLVOGHJVRVOEHQVOVB + ZUTVIVJHVCRVKEHQVAUJWCVLVMVKVJVRVORGJVDWAWDVIVJVKVEABCDEGIJOPLVFUSVGVOEVT + VNVSWEVTVBVHUS $. $( A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.) $) madetsmelbas2 $p |- ( ( R e. CRing /\ M e. B /\ Q e. P ) -> ( ( ( Y o. S ) ` Q ) ( .r ` R ) ( G gsum ( n e. N |-> ( n M ( Q ` n ) ) ) ) ) e. ( Base ` R ) ) $= - ( wcel cfv co ccrg w3a crg ccom cbs cv cmpt cgsu crngrng 3ad2ant1 cfn cvv - cmulr matrcl simpld simp3 zrhcopsgnelbas syl3anc eqid mgpbas ccmn crngmgp - 3ad2ant2 wral simp2 matepm2cl gsummptcl rngcl ) EUARZIBRZDCRZUBZEUCRZDKFU - DSZEUESZRZHGJGUFZVQDSITZUGUHTZVORVNVSEUMSZTVORVIVJVMVKEUIUJZVLVMJUKRZVKVP - WAVJVIWBVKVJWBEULRABEJIOPUNUOVCZVIVJVKUPZCDEFJKLMNUQURVLVOGHJVRVOEHQVOUSZ - UTVIVJHVARVKEHQVBUJWCVLVMVKVJVRVORGJVDWAWDVIVJVKVEABCDEGIJOPLVFURVGVOEVTV - NVSWEVTUSVHUR $. + ( wcel cfv co ccrg w3a crg ccom cbs cmpt cgsu cmulr crngring 3ad2ant1 cfn + cv cvv matrcl simpld 3ad2ant2 simp3 zrhcopsgnelbas syl3anc mgpbas crngmgp + eqid ccmn wral simp2 matepm2cl gsummptcl ringcl ) EUARZIBRZDCRZUBZEUCRZDK + FUDSZEUESZRZHGJGULZVQDSITZUFUGTZVORVNVSEUHSZTVORVIVJVMVKEUIUJZVLVMJUKRZVK + VPWAVJVIWBVKVJWBEUMRABEJIOPUNUOUPZVIVJVKUQZCDEFJKLMNURUSVLVOGHJVRVOEHQVOV + BZUTVIVJHVCRVKEHQVAUJWCVLVMVKVJVRVORGJVDWAWDVIVJVKVEABCDEGIJOPLVFUSVGVOEV + TVNVSWEVTVBVHUS $. $} $( @@ -257547,18 +257773,18 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. $( The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) $) mat0dim0 $p |- ( R e. Ring -> ( 0g ` A ) = (/) ) $= - ( crg wcel c0g cfv cbs wceq cgrp cfn 0fin matrng mpan rnggrp eqid grpidcl - c0 3syl csn cmat co fveq2i mat0dimbas0 syl5eq eleq2d elsni syl6bi mpd ) B - DEZAFGZAHGZEZUKRIZUJADEZAJEUMRKEUJUOLABRCMNAOULAUKULPUKPQSUJUMUKRTZEUNUJU - LUPUKUJULRBUAUBZHGUPAUQHCUCBDUDUEUFUKRUGUHUI $. + ( crg wcel c0g cfv cbs c0 wceq cgrp cfn 0fin matring mpan ringgrp grpidcl + eqid 3syl csn cmat co fveq2i mat0dimbas0 syl5eq eleq2d elsni syl6bi mpd ) + BDEZAFGZAHGZEZUKIJZUJADEZAKEUMILEUJUOMABICNOAPULAUKULRUKRQSUJUMUKITZEUNUJ + ULUPUKUJULIBUAUBZHGUPAUQHCUCBDUDUEUFUKIUGUHUI $. $( The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) $) mat0dimid $p |- ( R e. Ring -> ( 1r ` A ) = (/) ) $= - ( crg wcel cur cfv cbs c0 wceq cfn 0fin matrng mpan eqid rngidcl syl cmat - csn co fveq2i mat0dimbas0 syl5eq eleq2d elsni syl6bi mpd ) BDEZAFGZAHGZEZ - UIIJZUHADEZUKIKEUHUMLABICMNUJAUIUJOUIOPQUHUKUIISZEULUHUJUNUIUHUJIBRTZHGUN - AUOHCUABDUBUCUDUIIUEUFUG $. + ( crg wcel cur cfv cbs c0 wceq cfn 0fin matring mpan eqid ringidcl syl co + csn cmat fveq2i mat0dimbas0 syl5eq eleq2d elsni syl6bi mpd ) BDEZAFGZAHGZ + EZUIIJZUHADEZUKIKEUHUMLABICMNUJAUIUJOUIOPQUHUKUIISZEULUHUJUNUIUHUJIBTRZHG + UNAUOHCUABDUBUCUDUIIUEUFUG $. $( The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) $) @@ -257577,14 +257803,14 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. commutative ring. (Contributed by AV, 10-Aug-2019.) $) mat0dimcrng $p |- ( R e. Ring -> A e. CRing ) $= ( vx vy crg wcel cv cfv co wceq cbs wral c0 cvv 0ex oveq1 oveq2 eqeq12d - wb ccrg cfn 0fin matrng mpan cmat csn mat0dimbas0 wi eqcomi fveq2i eqeq1i - cmulr wa eqidd ralbidv ralsng ax-mp bitri sylibr raleq bitrd adantr sylbi - mpbird ex mpcom eqid iscrng2 sylanbrc ) BFGZAFGZDHZEHZAUMIZJZVNVMVOJZKZEA - LIZMZDVSMZAUAGNUBGVKVLUCABNCUDUENBUFJZLIZNUGZKZVKWABFUHWEVSWDKZVKWAUIWCVS - WDWBALAWBCUJUKULWFVKWAWFVKUNZWAVREWDMZDWDMZWGNNVOJZWJKZWIWGWJUOWINVNVOJZV - NNVOJZKZEWDMZWKNOGZWIWOTPWHWODNOVMNKZVRWNEWDWQVPWLVQWMVMNVNVOQVMNVNVORSUP - UQURWPWOWKTPWNWKENOVNNKWLWJWMWJVNNNVORVNNNVOQSUQURUSUTWFWAWITVKWFWAVTDWDM - WIVTDVSWDVAWFVTWHDWDVREVSWDVAUPVBVCVEVFVDVGDEVSAVOVSVHVOVHVIVJ $. + wb cmulr ccrg cfn 0fin matring mpan cmat csn mat0dimbas0 wi eqcomi fveq2i + eqeq1i wa eqidd ralsng ax-mp bitri sylibr raleq bitrd adantr mpbird sylbi + ralbidv ex mpcom eqid iscrng2 sylanbrc ) BFGZAFGZDHZEHZAUAIZJZVNVMVOJZKZE + ALIZMZDVSMZAUBGNUCGVKVLUDABNCUEUFNBUGJZLIZNUHZKZVKWABFUIWEVSWDKZVKWAUJWCV + SWDWBALAWBCUKULUMWFVKWAWFVKUNZWAVREWDMZDWDMZWGNNVOJZWJKZWIWGWJUOWINVNVOJZ + VNNVOJZKZEWDMZWKNOGZWIWOTPWHWODNOVMNKZVRWNEWDWQVPWLVQWMVMNVNVOQVMNVNVORSV + EUPUQWPWOWKTPWNWKENOVNNKWLWJWMWJVNNNVORVNNNVOQSUPUQURUSWFWAWITVKWFWAVTDWD + MWIVTDVSWDUTWFVTWHDWDVREVSWDUTVEVAVBVCVFVDVGDEVSAVOVSVHVOVHVIVJ $. $} ${ @@ -257664,21 +257890,21 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. co wfn opex eqeltri anim2i ancomd fnsng xpsng fneq1d mpbird sneqi syl6eqr a1i anidms adantr xpeq1d sylan snex inidm cv wi elsni fveq2 eqcomi opeq1i syl6eq eqtrd fveq1d fvsng sylan9eq ex impcom offval cbs simprl w3a df-3an - simpr sylibr mat1dimbas eqid matvsca2 syl2anc 3anass biimpri rngcl fmptsn - adantlr sylancr 3eqtr4d ) CUBLZDFLZMZGBLZHBLZMZMZDNZXEUCZGNZUCZEHONZCUDPZ - UEUHZUAENZGHXJUHZUFZGXIAUGPZUHZEXMONZXDUAXLXLGHXJXLXHXIQQXDXHXLUIXLXGUCZX - LUIZXDXSEGOZNZXLUIZXCYBWTXCEQLZXAMZYBXCXAYCXBYCXAYCXBEDDOZQKDDUJZUKZUTULU - MZEGQBUNRSXDXLXRYAXCXRYATZWTXCYDYIYHEGQBUORSUPUQXDXLXHXRXDXFXLXGWTXFXLTZX - CWSYJWRWSYJWSWSMXFYENZXLDDFFUOZEYEKURUSVASVBVCUPUQXCXIXLUIZWTXAYCXBYMYCXA - YGUTZEHQBUNVDSXLQLXDEVEUTZYOXLVFUAVGZXLLZXDYPXHPZGTZYQYPETZXDYSVHYPEVIZYT - XDYSYTXDYREXHPZGYPEXHVJXDUUBEYAPZGXDEXHYAXDXHYKXGUCZYAXDXFYKXGWTXFYKTZXCW - SUUEWRWSUUEYLVASVBVCXCUUDYATZWTXCYEQLZXAMZUUFXCXAUUGXBUUGXAUUGXBYFUTULUMU - UHUUDYEGOZNYAYEGQBUOUUIXTYEEGEYEKVKVLURVMRSVNVOXCUUCGTZWTXCYDUUJYHEGQBVPR - SVNVQVRRVSYQXDYPXIPZHTZYQYTXDUULVHUUAYTXDUULYTXDUUKEXIPZHYPEXIVJXCUUMHTZW - TXAYCXBUUNYNEHQBVPVDSVQVRRVSVTXDXAXIAWAPZLZXPXKTWTXAXBWBXDWRWSXBWCZUUPXDW - TXBMUUQXCXBWTXAXBWEULWRWSXBWDWFABCDEFHIJKWGRAUUOXFCXOXJBXEGXIIUUOWHJXOWHX - JWHZXFWHWIWJXDYCXMBLZXQXNTYGXDWRXAXBWCZUUSWRXCUUTWSUUTWRXCMWRXAXBWKWLWOBC - XJGHJUURWMRUAEXMQBWNWPWQ $. + simpr sylibr mat1dimbas eqid syl2anc 3anass biimpri adantlr ringcl fmptsn + matvsca2 sylancr 3eqtr4d ) CUBLZDFLZMZGBLZHBLZMZMZDNZXEUCZGNZUCZEHONZCUDP + ZUEUHZUAENZGHXJUHZUFZGXIAUGPZUHZEXMONZXDUAXLXLGHXJXLXHXIQQXDXHXLUIXLXGUCZ + XLUIZXDXSEGOZNZXLUIZXCYBWTXCEQLZXAMZYBXCXAYCXBYCXAYCXBEDDOZQKDDUJZUKZUTUL + UMZEGQBUNRSXDXLXRYAXCXRYATZWTXCYDYIYHEGQBUORSUPUQXDXLXHXRXDXFXLXGWTXFXLTZ + XCWSYJWRWSYJWSWSMXFYENZXLDDFFUOZEYEKURUSVASVBVCUPUQXCXIXLUIZWTXAYCXBYMYCX + AYGUTZEHQBUNVDSXLQLXDEVEUTZYOXLVFUAVGZXLLZXDYPXHPZGTZYQYPETZXDYSVHYPEVIZY + TXDYSYTXDYREXHPZGYPEXHVJXDUUBEYAPZGXDEXHYAXDXHYKXGUCZYAXDXFYKXGWTXFYKTZXC + WSUUEWRWSUUEYLVASVBVCXCUUDYATZWTXCYEQLZXAMZUUFXCXAUUGXBUUGXAUUGXBYFUTULUM + UUHUUDYEGOZNYAYEGQBUOUUIXTYEEGEYEKVKVLURVMRSVNVOXCUUCGTZWTXCYDUUJYHEGQBVP + RSVNVQVRRVSYQXDYPXIPZHTZYQYTXDUULVHUUAYTXDUULYTXDUUKEXIPZHYPEXIVJXCUUMHTZ + WTXAYCXBUUNYNEHQBVPVDSVQVRRVSVTXDXAXIAWAPZLZXPXKTWTXAXBWBXDWRWSXBWCZUUPXD + WTXBMUUQXCXBWTXAXBWEULWRWSXBWDWFABCDEFHIJKWGRAUUOXFCXOXJBXEGXIIUUOWHJXOWH + XJWHZXFWHWOWIXDYCXMBLZXQXNTYGXDWRXAXBWCZUUSWRXCUUTWSUUTWRXCMWRXAXBWJWKWLB + CXJGHJUURWMRUAEXMQBWNWPWQ $. $d B k y $. $d E k $. $d O k x y $. $d R k $. $d V k $. $d X k y $. $d Y k y $. @@ -257690,10 +257916,10 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. ( vk wcel cop csn co wceq a1i adantr cvv vx vy crg wa cmulr cotp cmmul cv cfv cmpt cgsu cmpt2 cfn snfi simpl eqid matmulr eqcomd syl2anc oveqd cmap cxp wss wf1o opex f1osng syl2an f1of syl opeq1d sneqd simpr feq12d mpbird - wf xpsng snssi adantl fss wb fvex eqeltri snex xpex elmapg mamuval rngcmn - cbs ccmn wral df-ov opeq1i sneqi fveq1i eqtri anim2i ancomd fvsng eqeltrd - syl5eq sylan rngcl syl3anc oveq12d eqcomi eleq12d ralsng gsummptcl oveq1d - oveq2 oveq1 mpteq2dv oveq2d mpt2sn rngmnd gsumsn opeq12d opeq2d 3eqtrd + wf xpsng snssi adantl fss wb cbs fvex eqeltri snex elmapg mamuval ringcmn + xpex ccmn wral df-ov opeq1i sneqi fveq1i eqtri anim2i ancomd fvsng syl5eq + eqeltrd sylan ringcl syl3anc oveq2 oveq1 oveq12d eqcomi eleq12d gsummptcl + ralsng oveq1d mpteq2dv oveq2d mpt2sn ringmnd gsumsn opeq12d opeq2d 3eqtrd cmnd ) CUCMZDFMZUDZGBMZHBMZUDZUDZEGNZOZEHNZOZAUEUIZPYIYKCDOZYMYMUFUGPZPUA UBYMYMCLYMUAUHZLUHZYIPZYPUBUHZYKPZCUEUIZPZUJZUKPZULZEGHYTPZNZOZYGYLYNYIYK YCYLYNQZYFYCYMUMMZYAUUHUUIYCDUNZRYAYBUOZUUIYAUDYNYLACYNYMUCIYNUPZUQURUSSU @@ -257701,8 +257927,8 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. ZVAPZMZUUPBYIVOZYGUUPGOZYIVOZUUTBVCZUUSYGUVADDNZOZUUTUVCGNZOZVOZYGUVDUUTU VFVDZUVGYCUVCTMZYDUVHYFUVIYCDDVEZRZYDYEUOZUVCGTBVFVGUVDUUTUVFVHVIYGUUPUVD UUTYIUVFYGYHUVEYGEUVCGEUVCQYGKRZVJVKYCUUPUVDQZYFYCYBYBUVNYAYBVLZUVODDFFVP - USSZVMVNYFUVBYCYDUVBYEGBVQSVRUUPUUTBYIVSUSYGBTMZUUPTMZUURUUSVTUVQYGBCWHUI - ZTJCWHWAWBRZUVRYGYMYMDWCZUWAWDRZBUUPYITTWEUSVNYGYKUUQMZUUPBYKVOZYGUUPHOZY + USSZVMVNYFUVBYCYDUVBYEGBVQSVRUUPUUTBYIVSUSYGBTMZUUPTMZUURUUSVTUVQYGBCWAUI + ZTJCWAWBWCRZUVRYGYMYMDWDZUWAWHRZBUUPYITTWEUSVNYGYKUUQMZUUPBYKVOZYGUUPHOZY KVOZUWEBVCZUWDYGUWFUVDUWEUVCHNZOZVOZYGUVDUWEUWIVDZUWJYCUVIYEUWKYFUVKYDYEV LZUVCHTBVFVGUVDUWEUWIVHVIYGUUPUVDUWEYKUWIYGYJUWHYGEUVCHUVMVJVKUVPVMVNYFUW GYCYEUWGYDHBVQVRVRUUPUWEBYKVSUSYGUVQUVRUWCUWDVTUVTUWBBUUPYKTTWEUSVNWFYGUU @@ -257710,14 +257936,14 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. UUDUWSQYCYBYFUVOSZUXDYGUVSLCYMUWOUVSUPYCCWIMZYFYAUXEYBCWGSSUUOYGUWOUVSMZL YMWJZUXBBMZYGYAUWTBMUXABMUXHUUNYGUWTGBYGUWTUVCUVFUIZGUWTUVCYIUIUXIDDYIWKU VCYIUVFYHUVEEUVCGKWLWMWNWOYFUXIGQZYCYFUVIYDUDUXJYFYDUVIYEUVIYDUVIYEUVJRWP - WQUVCGTBWRVIVRWTZYFYDYCUVLVRWSYGUXAHBYGUXAUVCUWIUIZHUXAUVCYKUIUXLDDYKWKUV - CYKUWIYJUWHEUVCHKWLWMWNWOYFUXLHQZYCYDUVIYEUXMUVIYDUVJRUVCHTBWRXAVRWTZYFYE - YCUWLVRWSBCYTUWTUXAJUUMXBXCZYCUXGUXHVTZYFYBUXPYAUXFUXHLDFYPDQZUWOUXBUVSBU - XQUWMUWTUWNUXAYTYPDDYIXJYPDDYKXKXDZUVSBQUXQBUVSJXERXFXGVRSVNXHUAUBDDUUCCL + WQUVCGTBWRVIVRWSZYFYDYCUVLVRWTYGUXAHBYGUXAUVCUWIUIZHUXAUVCYKUIUXLDDYKWKUV + CYKUWIYJUWHEUVCHKWLWMWNWOYFUXLHQZYCYDUVIYEUXMUVIYDUVJRUVCHTBWRXAVRWSZYFYE + YCUWLVRWTBCYTUWTUXAJUUMXBXCZYCUXGUXHVTZYFYBUXPYAUXFUXHLDFYPDQZUWOUXBUVSBU + XQUWMUWTUWNUXAYTYPDDYIXDYPDDYKXEXFZUVSBQUXQBUVSJXGRXHXJVRSVNXIUAUBDDUUCCL YMUWMYSYTPZUJZUKPUVSUWQUUDFFUUDUPYODQZUUBUXTCUKUYALYMUUAUXSUYAYQUWMYSYTYO - DYPYIXKXIXLXMYRDQZUXTUWPCUKUYBLYMUXSUWOUYBYSUWNUWMYTYRDYPYKXJXMXLXMXNXCYG - UWRUXCYGUVCEUWQUXBUVCEQYGEUVCKXERYGCXTMZYBUXHUWQUXBQYCUYCYFYAUYCYBCXOSSUX - DUXOUWOBUXBLCDFJUXRXPXCXQVKYGUXCUUFYGUXBUUEEYGUWTGUXAHYTUXKUXNXDXRVKXSXS + DYPYIXEXKXLXMYRDQZUXTUWPCUKUYBLYMUXSUWOUYBYSUWNUWMYTYRDYPYKXDXMXLXMXNXCYG + UWRUXCYGUVCEUWQUXBUVCEQYGEUVCKXGRYGCXTMZYBUXHUWQUXBQYCUYCYFYAUYCYBCXOSSUX + DUXOUWOBUXBLCDFJUXRXPXCXQVKYGUXCUUFYGUXBUUEEYGUWTGUXAHYTUXKUXNXFXRVKXSXS $. $d A a b x y $. $d B a b $. $d E a b $. $d O a b $. $d R a b $. @@ -257726,19 +257952,19 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. commutative ring. (Contributed by AV, 16-Aug-2019.) $) mat1dimcrng $p |- ( ( R e. CRing /\ E e. V ) -> A e. CRing ) $= ( vx vy va vb wcel wa cv co wceq csn cop ccrg crg cmulr cfv cbs wral snfi - cfn crngrng adantr matrng sylancr wb mat1dimelbas anbi12d sylan wi simpll - wrex simprl simprr eqid crngcom syl3anc opeq2d sneqd anim1i pm3.22 syl2an - mat1dimmul 3eqtr4d expr imp oveq12 ex ad2antlr expcom rexlimdva ralrimivv - impd sylbid iscrng2 sylanbrc ) CUANZDFNZOZAUBNZJPZKPZAUCUDZQZWIWHWJQZRZKA - UEUDZUFJWNUFAUANWFDSZUHNCUBNZWGDUGWDWPWECUIZUJACWOGUKULWFWMJKWNWNWFWHWNNZ - WIWNNZOZWHELPZTSZRZLBUSZWIEMPZTSZRZMBUSZOZWMWDWPWEWTXIUMWQWPWEOZWRXDWSXHA - BCDWHEFLGHIUNABCDWIEFMGHIUNUOUPWFXDXHWMWFXCXHWMUQZLBWFXABNZOZXCXKXMXCOZXG - WMMBXNXEBNZOZXGWMXPXGOXBXFWJQZXFXBWJQZWKWLXPXQXRRZXGXNXOXSXMXOXSUQXCWFXLX - OXSWFXLXOOZOZEXAXECUCUDZQZTZSZEXEXAYBQZTZSZXQXRYAYDYGYAYCYFEYAWDXLXOYCYFR - WDWEXTURWFXLXOUTWFXLXOVABCYBXAXEHYBVBVCVDVEVFWFXJXTXQYERWDWPWEWQVGZABCDEF - XAXEGHIVJUPWFXJXOXLOXRYHRXTYIXLXOVHABCDEFXEXAGHIVJVIVKVLUJVMUJXPXGWKXQRZX - CXGYJUQXMXOXCXGYJWHXBWIXFWJVNVOVPVMXPXGWLXRRZXCXGYKUQXMXOXGXCYKWIXFWHXBWJ - VNVQVPVMVKVOVRVOVRVTWAVSJKWNAWJWNVBWJVBWBWC $. + cfn crngring adantr matring sylancr wrex wb mat1dimelbas anbi12d sylan wi + simpll simprl simprr eqid crngcom syl3anc opeq2d anim1i mat1dimmul pm3.22 + sneqd syl2an 3eqtr4d expr oveq12 ex ad2antlr expcom rexlimdva impd sylbid + imp ralrimivv iscrng2 sylanbrc ) CUANZDFNZOZAUBNZJPZKPZAUCUDZQZWIWHWJQZRZ + KAUEUDZUFJWNUFAUANWFDSZUHNCUBNZWGDUGWDWPWECUIZUJACWOGUKULWFWMJKWNWNWFWHWN + NZWIWNNZOZWHELPZTSZRZLBUMZWIEMPZTSZRZMBUMZOZWMWDWPWEWTXIUNWQWPWEOZWRXDWSX + HABCDWHEFLGHIUOABCDWIEFMGHIUOUPUQWFXDXHWMWFXCXHWMURZLBWFXABNZOZXCXKXMXCOZ + XGWMMBXNXEBNZOZXGWMXPXGOXBXFWJQZXFXBWJQZWKWLXPXQXRRZXGXNXOXSXMXOXSURXCWFX + LXOXSWFXLXOOZOZEXAXECUCUDZQZTZSZEXEXAYBQZTZSZXQXRYAYDYGYAYCYFEYAWDXLXOYCY + FRWDWEXTUSWFXLXOUTWFXLXOVABCYBXAXEHYBVBVCVDVEVIWFXJXTXQYERWDWPWEWQVFZABCD + EFXAXEGHIVGUQWFXJXOXLOXRYHRXTYIXLXOVHABCDEFXEXAGHIVGVJVKVLUJVTUJXPXGWKXQR + ZXCXGYJURXMXOXCXGYJWHXBWIXFWJVMVNVOVTXPXGWLXRRZXCXGYKURXMXOXGXCYKWIXFWHXB + WJVMVPVOVTVKVNVQVNVQVRVSWAJKWNAWJWNVBWJVBWBWC $. $} ${ @@ -257798,22 +258024,22 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. additive group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) $) mat1ghm $p |- ( ( R e. Ring /\ E e. V ) -> F e. ( R GrpHom A ) ) $= - ( wcel cfv adantr co wceq syl3anc vw vy vi vj crg cplusg eqid cgrp rnggrp - wa csn cfn snfi simpl matgrp sylancr mat1f wral simpr adantl mat1rhmelval - cv oveq12d mat1rhmcl snidg matplusgcell syl21anc rngacl 3eqtr4rd wb oveq1 - jca eqeq12d oveq2 2ralsng syl2anc mpbird matrng eqmat isghmd ) DUEOZEIOZU - JZUAUBDUFPZBUFPZDBFGCJLWDUGZWEUGZWADUHOWBDUIQWCEUKZULOZWABUHOEUMZWAWBUNZB - DWHKUOUPABCDEFGHIJKLMNUQWCUAVBZGOZUBVBZGOZUJZUJZWLWNWDRZFPZWLFPZWNFPZWERZ - SZUCVBZUDVBZWSRZXDXEXBRZSZUDWHURUCWHURZWQXIEEWSRZEEXBRZSZWQEEWTRZEEXARZWD - RZWRXKXJWQXMWLXNWNWDWQWAWBWMXMWLSWCWAWPWKQZWCWBWPWAWBUSZQZWPWMWCWMWOUNUTZ - ABCDEFGHIWLJKLMNVATWQWAWBWOXNWNSXPXRWPWOWCWMWOUSUTZABCDEFGHIWNJKLMNVATVCW - QWTCOZXACOZEWHOZYCUJZXKXOSWQWAWBWMYAXPXRXSABCDEFGHIWLJKLMNVDTZWQWAWBWOYBX - PXRXTABCDEFGHIWNJKLMNVDTZWCYDWPWBYDWAWBYCYCEIVEZYGVLUTQBCWDWEDEEWHWTXAKLW - GWFVFVGWQWAWBWRGOZXJWRSXPXRWQWAWMWOYHXPXSXTGWDDWLWNJWFVHTZABCDEFGHIWRJKLM - NVATVIWCXIXLVJZWPWCWBWBYJXQXQXHEXEWSRZEXEXBRZSXLUCUDEEIIXDESXFYKXGYLXDEXE - WSVKXDEXEXBVKVMXEESYKXJYLXKXEEEWSVNXEEEXBVNVMVOVPQVQWQWSCOZXBCOZXCXIVJWQW - AWBYHYMXPXRYIABCDEFGHIWRJKLMNVDTWQBUEOZYAYBYNWCYOWPWCWIWAYOWJWKBDWHKVRUPQ - YEYFCWEBWTXALWGVHTBCDUCUDWHWSXBKLVSVPVQVT $. + ( wcel cfv adantr co wceq syl3anc vw vy vi vj wa cplusg eqid cgrp ringgrp + crg csn snfi simpl matgrp sylancr mat1f cv wral simpr adantl mat1rhmelval + cfn oveq12d mat1rhmcl snidg jca matplusgcell syl21anc ringacl 3eqtr4rd wb + oveq1 eqeq12d oveq2 2ralsng syl2anc mpbird matring eqmat isghmd ) DUJOZEI + OZUEZUAUBDUFPZBUFPZDBFGCJLWDUGZWEUGZWADUHOWBDUIQWCEUKZVBOZWABUHOEULZWAWBU + MZBDWHKUNUOABCDEFGHIJKLMNUPWCUAUQZGOZUBUQZGOZUEZUEZWLWNWDRZFPZWLFPZWNFPZW + ERZSZUCUQZUDUQZWSRZXDXEXBRZSZUDWHURUCWHURZWQXIEEWSRZEEXBRZSZWQEEWTRZEEXAR + ZWDRZWRXKXJWQXMWLXNWNWDWQWAWBWMXMWLSWCWAWPWKQZWCWBWPWAWBUSZQZWPWMWCWMWOUM + UTZABCDEFGHIWLJKLMNVATWQWAWBWOXNWNSXPXRWPWOWCWMWOUSUTZABCDEFGHIWNJKLMNVAT + VCWQWTCOZXACOZEWHOZYCUEZXKXOSWQWAWBWMYAXPXRXSABCDEFGHIWLJKLMNVDTZWQWAWBWO + YBXPXRXTABCDEFGHIWNJKLMNVDTZWCYDWPWBYDWAWBYCYCEIVEZYGVFUTQBCWDWEDEEWHWTXA + KLWGWFVGVHWQWAWBWRGOZXJWRSXPXRWQWAWMWOYHXPXSXTGWDDWLWNJWFVITZABCDEFGHIWRJ + KLMNVATVJWCXIXLVKZWPWCWBWBYJXQXQXHEXEWSRZEXEXBRZSXLUCUDEEIIXDESXFYKXGYLXD + EXEWSVLXDEXEXBVLVMXEESYKXJYLXKXEEEWSVNXEEEXBVNVMVOVPQVQWQWSCOZXBCOZXCXIVK + WQWAWBYHYMXPXRYIABCDEFGHIWRJKLMNVDTWQBUJOZYAYBYNWCYOWPWCWIWAYOWJWKBDWHKVR + UOQYEYFCWEBWTXALWGVITBCDUCUDWHWSXBKLVSVPVQVT $. ${ $d B e w $. $d E e $. $d F e x y $. $d K e $. $d M w x y $. @@ -257825,50 +258051,50 @@ as element of (the base set of) ` ( ( 1 ... n ) Mat R ) `. over this ring. (Contributed by AV, 22-Dec-2019.) $) mat1mhm $p |- ( ( R e. Ring /\ E e. V ) -> F e. ( M MndHom N ) ) $= ( wcel co vw vy vi vj ve crg wa cmnd wf cv cmulr cfv wceq wral cur cmhm - w3a rngmgp adantr csn cfn snfi simpl matrng sylancr syl mat1f cmpt cgsu - jca rngmnd simpr simpll cbs eqid snidg simprl mat1rhmcl syl3anc matecld - simprr rngcl oveq2 oveq1 oveq12d adantl gsumsnd mat1rhmelval matmulcell - ad2antlr 3eqtr4rd wb eqeq12d 2ralsng sylancom mpbird syl2anc ralrimivva - eqtrd eqmat cop rngidcl mat1rhmval mpd3an3 mat1dimid eqtr4d 3jca mgpbas - mgpplusg rngidval ismhm sylanbrc ) DUFSZEKSZUGZHUHSZIUHSZUGGCFUIZUAUJZU - BUJZDUKULZTZFULZXSFULZXTFULZBUKULZTZUMZUBGUNUAGUNZDUOULZFULZBUOULZUMZUQ - FHIUPTSXOXPXQXMXPXNDHQURUSXOBUFSZXQXOEUTZVASXMYNEVBXMXNVCBDYOMVDVEZBIRU - RVFVJXOXRYIYMABCDEFGJKLMNOPVGXOYHUAUBGGXOXSGSZXTGSZUGZUGZYHUCUJZUDUJZYC - TZUUAUUBYGTZUMZUDYOUNUCYOUNZYTUUFEEYCTZEEYGTZUMZYTDUEYOEUEUJZYDTZUUJEYE - TZYATZVHVITZYBUUHUUGYTUUNEEYDTZEEYETZYATZYBYTUUMGUUQUEDEKLXODUHSZYSXMUU - RXNDVKUSUSXOXNYSXMXNVLZUSZYTXMUUOGSUUPGSUUQGSXMXNYSVMZYTBBVNULZDEEGYDYO - MLUVBVOZXNEYOSZXMYSEKVPZWJZUVFYTXMXNYQYDUVBSUVAUUTXOYQYRVQZABUVBDEFGJKX - SLMUVCOPVRVSVTYTBUVBDEEGYEYOMLUVCUVFUVFYTXMXNYRYEUVBSUVAUUTXOYQYRWAZABU - VBDEFGJKXTLMUVCOPVRVSVTGDYAUUOUUPLYAVOZWBVSUUJEUMZUUMUUQUMYTUVJUUKUUOUU - LUUPYAUUJEEYDWCUUJEEYEWDWEWFWGYTUUOXSUUPXTYAYTXMXNYQUUOXSUMUVAUUTUVGABC - DEFGJKXSLMNOPWHVSYTXMXNYRUUPXTUMUVAUUTUVHABCDEFGJKXTLMNOPWHVSWEWSYTXMYD - CSZYECSZUGUVDUVDUGZUUHUUNUMUVAYTUVKUVLYTXMXNYQUVKUVAUUTUVGABCDEFGJKXSLM - NOPVRVSZYTXMXNYRUVLUVAUUTUVHABCDEFGJKXTLMNOPVRVSZVJXNUVMXMYSXNUVDUVDUVE - UVEVJWJBCDYAYFUEEEYOYDYEMNUVIYFVOZWIVSYTXMXNYBGSZUUGYBUMUVAUUTYTXMYQYRU - VQUVAUVGUVHGDYAXSXTLUVIWBVSZABCDEFGJKYBLMNOPWHVSWKXOUUFUUIWLZYSXMXNXNUV - SUUSUUEEUUBYCTZEUUBYGTZUMUUIUCUDEEKKUUAEUMUUCUVTUUDUWAUUAEUUBYCWDUUAEUU - BYGWDWMUUBEUMUVTUUGUWAUUHUUBEEYCWCUUBEEYGWCWMWNWOUSWPYTYCCSZYGCSZYHUUFW - LYTXMXNUVQUWBUVAUUTUVRABCDEFGJKYBLMNOPVRVSYTYNUVKUVLUWCXOYNYSYPUSUVNUVO - CBYFYDYENUVPWBVSBCDUCUDYOYCYGMNWTWQWPWRXOYKJYJXAUTZYLXMXNYJGSZYKUWDUMXM - UWEXNGDYJLYJVOZXBUSABCDEFGJKYJLMNOPXCXDBGDEJKMLOXEXFXGUAUBGCYAYFHIFYLYJ - GDHQLXHCBIRNXHDYAHQUVIXIBYFIRUVPXIDYJHQUWFXJBYLIRYLVOXJXKXL $. + w3a ringmgp adantr csn cfn snfi simpl matring sylancr syl jca cmpt cgsu + mat1f ringmnd simpr simpll eqid snidg ad2antlr simprl mat1rhmcl syl3anc + cbs matecld simprr ringcl oveq2 oveq1 oveq12d adantl mat1rhmelval eqtrd + gsumsnd matmulcell 3eqtr4rd wb eqeq12d sylancom mpbird eqmat ralrimivva + 2ralsng syl2anc cop ringidcl mat1rhmval mpd3an3 mat1dimid eqtr4d mgpbas + 3jca mgpplusg rngidval ismhm sylanbrc ) DUFSZEKSZUGZHUHSZIUHSZUGGCFUIZU + AUJZUBUJZDUKULZTZFULZXSFULZXTFULZBUKULZTZUMZUBGUNUAGUNZDUOULZFULZBUOULZ + UMZUQFHIUPTSXOXPXQXMXPXNDHQURUSXOBUFSZXQXOEUTZVASXMYNEVBXMXNVCBDYOMVDVE + ZBIRURVFVGXOXRYIYMABCDEFGJKLMNOPVJXOYHUAUBGGXOXSGSZXTGSZUGZUGZYHUCUJZUD + UJZYCTZUUAUUBYGTZUMZUDYOUNUCYOUNZYTUUFEEYCTZEEYGTZUMZYTDUEYOEUEUJZYDTZU + UJEYETZYATZVHVITZYBUUHUUGYTUUNEEYDTZEEYETZYATZYBYTUUMGUUQUEDEKLXODUHSZY + SXMUURXNDVKUSUSXOXNYSXMXNVLZUSZYTXMUUOGSUUPGSUUQGSXMXNYSVMZYTBBVTULZDEE + GYDYOMLUVBVNZXNEYOSZXMYSEKVOZVPZUVFYTXMXNYQYDUVBSUVAUUTXOYQYRVQZABUVBDE + FGJKXSLMUVCOPVRVSWAYTBUVBDEEGYEYOMLUVCUVFUVFYTXMXNYRYEUVBSUVAUUTXOYQYRW + BZABUVBDEFGJKXTLMUVCOPVRVSWAGDYAUUOUUPLYAVNZWCVSUUJEUMZUUMUUQUMYTUVJUUK + UUOUULUUPYAUUJEEYDWDUUJEEYEWEWFWGWJYTUUOXSUUPXTYAYTXMXNYQUUOXSUMUVAUUTU + VGABCDEFGJKXSLMNOPWHVSYTXMXNYRUUPXTUMUVAUUTUVHABCDEFGJKXTLMNOPWHVSWFWIY + TXMYDCSZYECSZUGUVDUVDUGZUUHUUNUMUVAYTUVKUVLYTXMXNYQUVKUVAUUTUVGABCDEFGJ + KXSLMNOPVRVSZYTXMXNYRUVLUVAUUTUVHABCDEFGJKXTLMNOPVRVSZVGXNUVMXMYSXNUVDU + VDUVEUVEVGVPBCDYAYFUEEEYOYDYEMNUVIYFVNZWKVSYTXMXNYBGSZUUGYBUMUVAUUTYTXM + YQYRUVQUVAUVGUVHGDYAXSXTLUVIWCVSZABCDEFGJKYBLMNOPWHVSWLXOUUFUUIWMZYSXMX + NXNUVSUUSUUEEUUBYCTZEUUBYGTZUMUUIUCUDEEKKUUAEUMUUCUVTUUDUWAUUAEUUBYCWEU + UAEUUBYGWEWNUUBEUMUVTUUGUWAUUHUUBEEYCWDUUBEEYGWDWNWSWOUSWPYTYCCSZYGCSZY + HUUFWMYTXMXNUVQUWBUVAUUTUVRABCDEFGJKYBLMNOPVRVSYTYNUVKUVLUWCXOYNYSYPUSU + VNUVOCBYFYDYENUVPWCVSBCDUCUDYOYCYGMNWQWTWPWRXOYKJYJXAUTZYLXMXNYJGSZYKUW + DUMXMUWEXNGDYJLYJVNZXBUSABCDEFGJKYJLMNOPXCXDBGDEJKMLOXEXFXHUAUBGCYAYFHI + FYLYJGDHQLXGCBIRNXGDYAHQUVIXIBYFIRUVPXIDYJHQUWFXJBYLIRYLVNXJXKXL $. $} $( There is a ring homomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) $) mat1rhm $p |- ( ( R e. Ring /\ E e. V ) -> F e. ( R RingHom A ) ) $= - ( crg wcel wa co cmgp cfv cghm cmhm crh simpl csn cfn snfi matrng sylancr - jca mat1ghm eqid mat1mhm isrhm sylanbrc ) DOPZEIPZQZUPBOPZQFDBUARPZFDSTZB - STZUBRPZQFDBUCRPURUPUSUPUQUDZUREUEZUFPUPUSEUGVDBDVEKUHUIUJURUTVCABCDEFGHI - JKLMNUKABCDEFGVAVBHIJKLMNVAULZVBULZUMUJDBFVAVBVFVGUNUO $. + ( crg wcel wa co cmgp cfv cghm cmhm crh simpl csn cfn matring sylancr jca + snfi mat1ghm eqid mat1mhm isrhm sylanbrc ) DOPZEIPZQZUPBOPZQFDBUARPZFDSTZ + BSTZUBRPZQFDBUCRPURUPUSUPUQUDZUREUEZUFPUPUSEUJVDBDVEKUGUHUIURUTVCABCDEFGH + IJKLMNUKABCDEFGVAVBHIJKLMNVAULZVBULZUMUIDBFVAVBVFVGUNUO $. $( There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) $) mat1rngiso $p |- ( ( R e. Ring /\ E e. V ) -> F e. ( R RingIso A ) ) $= - ( crg wcel wa crs co crh wf1o mat1rhm mat1f1o csn cfn snfi matrng sylancr - wb simpl isrim syldan mpbir2and ) DOPZEIPZQZFDBRSPZFDBTSPZGCFUAZABCDEFGHI - JKLMNUBABCDEFGHIJKLMNUCUNUOBOPZUQURUSQUIUPEUDZUEPUNUTEUFUNUOUJBDVAKUGUHGC - DBFOOJLUKULUM $. + ( crg wcel wa crs co crh wf1o mat1rhm mat1f1o wb csn snfi matring sylancr + cfn simpl isrim syldan mpbir2and ) DOPZEIPZQZFDBRSPZFDBTSPZGCFUAZABCDEFGH + IJKLMNUBABCDEFGHIJKLMNUCUNUOBOPZUQURUSQUDUPEUEZUIPUNUTEUFUNUOUJBDVAKUGUHG + CDBFOOJLUKULUM $. $} ${ @@ -257987,12 +258213,12 @@ that the ring of scalar matrices (over a commutative ring) is a commutative $( The identity matrix is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.) $) dmatid $p |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` A ) e. D ) $= - ( vi vj wcel crg wa cur cfv cv wceq wral cfn wne co wi matrng rngidcl syl - cif simpl adantr simpr adantl mat1ov ifnefalse sylan9eq ralrimivva dmatel - eqid ex mpbir2and ) EUAMZDNMZOZAPQZCMVDBMZKRZLRZUBZVFVGVDUCZFSZUDZLETKETV - CANMVEADEGUEBAVDHVDURZUFUGVCVKKLEEVCVFEMZVGEMZOZOZVHVJVPVHVIVFVGSDPQZFUHF - VPADVDVQVFVGEFGVQURIVCVAVOVAVBUIUJVCVBVOVAVBUKUJVOVMVCVMVNUIULVOVNVCVMVNU - KULVLUMVFVGVQFUNUOUSUPABCDKLVDENFGHIJUQUT $. + ( vi vj wcel crg wa cur cfv cv wceq wral cfn wne co matring eqid ringidcl + wi syl cif simpl adantr simpr adantl mat1ov ifnefalse sylan9eq ralrimivva + ex dmatel mpbir2and ) EUAMZDNMZOZAPQZCMVDBMZKRZLRZUBZVFVGVDUCZFSZUGZLETKE + TVCANMVEADEGUDBAVDHVDUEZUFUHVCVKKLEEVCVFEMZVGEMZOZOZVHVJVPVHVIVFVGSDPQZFU + IFVPADVDVQVFVGEFGVQUEIVCVAVOVAVBUJUKVCVBVOVAVBULUKVOVMVCVMVNUJUMVOVNVCVMV + NULUMVLUNVFVGVQFUOUPURUQABCDKLVDENFGHIJUSUT $. $d I i j $. $d J j $. $d X i j $. $d .0. i j $. $( An extradiagonal entry of a diagonal matrix is equal to zero. @@ -258014,73 +258240,73 @@ that the ring of scalar matrices (over a commutative ring) is a commutative |-> if ( x = y , ( ( x X y ) ( .r ` R ) ( x Y y ) ) , .0. ) ) ) $= ( wcel wa co wceq ad2antlr adantl vk cfn crg cmulr cfv cotp cmmul cv cmpt cgsu cmpt2 cif eqid matmulr adantr eqcomd oveqd cbs simplr simpll dmatmat - cxp cmap imp matbas2i syl adantrr adantrl mamuval cdif cplusg ccmn rngcmn - w3a csn 3ad2ant1 cvv c0g ovex fvex fsuppmptdm simp2 simpr syl3anc simplr3 - a1i matecl rngcl simp3 syl6eleq wi imp32 eqtr ancoms oveq2d adantlr oveq1 - a1d oveq12d gsumdifsnd simprl 3jca eldifi eldifsni necomd dmatelnd oveq1d - wne syl13anc rnglz syl2anc eqtrd mpteq2dva cmnd diffi rngmnd anim12ci jca - gsumz mndlid 3eqtrd iftrue eqtr4d simprr df-ne neeq1 biimpcd sylbir rngrz - wn impcom biimpri pm2.61ian anim2i ancomd iffalse mpt2eq3dva ) GUBOZFUCOZ - PZHEOZIEOZPZPZHICUDUEZQHIFGGGUFUGQZQABGGFUAGAUHZUAUHZHQZUUHBUHZIQZFUDUEZQ - ZUIZUJQZUKABGGUUGUUJRZUUGUUJHQZUUGUUJIQZUULQZJULZUKUUDUUEUUFHIUUDUUFUUEYT - UUFUUERUUCCFUUFGUCKUUFUMZUNUOUPUQUUDFURUEZGFUULAUABUUFGGUCHIUVAUVBUMZUULU - MZYRYSUUCUSZYRYSUUCUTZUVFUVFYTUUAHUVBGGVBVCQZOZUUBYTUUAPHDOZUVHYTUUAUVICD - EFHGUCJKLMNVAVDZCDFUVBHGKUVCLVEVFVGYTUUBIUVGOZUUAYTUUBPIDOZUVKYTUUBUVLCDE - FIGUCJKLMNVAVDCDFUVBIGKUVCLVEVFVHVIUUDABGGUUOUUTUUPUUDUUGGOZUUJGOZVNZUUOU - UTRUUPUVOPZUUOUUSUUTUVPUUOFUAGUUGVOZVJZUUMUIZUJQZUUSFVKUEZQJUUSUWAQZUUSUV - 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(Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.) $) dmatsubcl $p |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( X e. D /\ Y e. D ) ) -> ( X ( -g ` A ) Y ) e. D ) $= - ( vi vj wcel crg wa co wceq adantr cfn csg cfv cv wne wi wral cgrp matrng - rnggrp syl dmatmat imp adantrr adantrl grpsubcl syl3anc simpr matsubgcell - eqid anim12d simpll simprl 3jca simplrl simplrr dmatelnd syl13anc oveq12d - w3a simprr cbs rng0cl jca adantl ad3antrrr 3eqtrd ex ralrimivva wb dmatel - grpsubid mpbir2and ) EUAOZDPOZQZFCOZGCOZQZQZFGAUBUCZRZCOZWLBOZMUDZNUDZUEZ - WOWPWLRZHSZUFZNEUGMEUGZWJAUHOZFBOZGBOZWNWFXBWIWFAPOXBADEIUIAUJUKTWFWGXCWH - WFWGXCABCDFEPHIJKLULZUMUNWFWHXDWGWFWHXDABCDGEPHIJKLULZUMUOBAWKFGJWKUTZUPU - QWJWTMNEEWJWOEOZWPEOZQZQZWQWSXKWQQZWRWOWPFRZWOWPGRZDUBUCZRZHHXORZHXKWRXPS - ZWQXKWEXCXDQZXJXRWJWEXJWFWEWIWDWEURTZTWJXSXJWFWIXSWFWGXCWHXDXEXFVAUMTWJXJ - URABDWKWOWPXOEFGIJXGXOUTZUSUQTXLXMHXNHXOXLWDWEWGVJZXHXIWQXMHSXKYBWQWJYBXJ - WJWDWEWGWDWEWIVBZXTWFWGWHVCVDTTWJXHXIWQVEZWJXHXIWQVFZXKWQURZABCDWOWPEFHIJ - KLVGVHXLWDWEWHVJZXHXIWQXNHSXKYGWQWJYGXJWJWDWEWHYCXTWFWGWHVKVDTTYDYEYFABCD - WOWPEGHIJKLVGVHVIWFXQHSZWIXJWQWFDUHOZHDVLUCZOZQZYHWEYLWDWEYIYKDUJYJDHYJUT - ZKVMVNVOYJDXOHHYMKYAWBUKVPVQVRVSWFWMWNXAQVTWIABCDMNWLEPHIJKLWATWC $. + ( vi vj wcel crg wa co wceq adantr cfn csg cfv cv wi wral matring ringgrp + wne cgrp syl dmatmat imp adantrr adantrl eqid syl3anc anim12d matsubgcell + grpsubcl simpr w3a simpll simprl simplrl simplrr dmatelnd syl13anc simprr + oveq12d cbs ring0cl jca adantl grpsubid ad3antrrr 3eqtrd ex ralrimivva wb + 3jca dmatel mpbir2and ) EUAOZDPOZQZFCOZGCOZQZQZFGAUBUCZRZCOZWLBOZMUDZNUDZ + UIZWOWPWLRZHSZUEZNEUFMEUFZWJAUJOZFBOZGBOZWNWFXBWIWFAPOXBADEIUGAUHUKTWFWGX + CWHWFWGXCABCDFEPHIJKLULZUMUNWFWHXDWGWFWHXDABCDGEPHIJKLULZUMUOBAWKFGJWKUPZ + UTUQWJWTMNEEWJWOEOZWPEOZQZQZWQWSXKWQQZWRWOWPFRZWOWPGRZDUBUCZRZHHXORZHXKWR + XPSZWQXKWEXCXDQZXJXRWJWEXJWFWEWIWDWEVATZTWJXSXJWFWIXSWFWGXCWHXDXEXFURUMTW + JXJVAABDWKWOWPXOEFGIJXGXOUPZUSUQTXLXMHXNHXOXLWDWEWGVBZXHXIWQXMHSXKYBWQWJY + BXJWJWDWEWGWDWEWIVCZXTWFWGWHVDWATTWJXHXIWQVEZWJXHXIWQVFZXKWQVAZABCDWOWPEF + HIJKLVGVHXLWDWEWHVBZXHXIWQXNHSXKYGWQWJYGXJWJWDWEWHYCXTWFWGWHVIWATTYDYEYFA + BCDWOWPEGHIJKLVGVHVJWFXQHSZWIXJWQWFDUJOZHDVKUCZOZQZYHWEYLWDWEYIYKDUHYJDHY + JUPZKVLVMVNYJDXOHHYMKYAVOUKVPVQVRVSWFWMWNXAQVTWIABCDMNWLEPHIJKLWBTWC $. $d A x y $. $d B x y $. $d B z $. $d D z $. $d N z $. $d R z $. $( The set of diagonal matrices is a subgroup of the matrix group/algebra. @@ -258088,10 +258314,10 @@ that the ring of scalar matrices (over a commutative ring) is a commutative dmatsgrp $p |- ( ( R e. Ring /\ N e. Fin ) -> D e. ( SubGrp ` A ) ) $= ( vx vy vz crg wcel wa cfv cv wral ancoms cfn csubg wss c0 wne co dmatmat csg wi ssrdv cur dmatid ne0i syl dmatsubcl ancom1s ralrimivva cgrp w3a wb - matrng rnggrp eqid issubg4 3syl mpbir3and ) DNOZEUAOZPZCAUBQOZCBUCZCUDUEZ - KRZLRZAUHQZUFCOZLCSKCSZVIMCBVHVGMRZCOVRBOUIABCDVRENFGHIJUGTUJVIAUKQZCOZVL - VHVGVTABCDEFGHIJULTCVSUMUNVIVPKLCCVHVGVMCOVNCOPVPABCDEVMVNFGHIJUOUPUQVIAN - OZAUROVJVKVLVQUSUTVHVGWAADEGVATAVBKLBCAVOHVOVCVDVEVF $. + matring ringgrp eqid issubg4 3syl mpbir3and ) DNOZEUAOZPZCAUBQOZCBUCZCUDU + EZKRZLRZAUHQZUFCOZLCSKCSZVIMCBVHVGMRZCOVRBOUIABCDVRENFGHIJUGTUJVIAUKQZCOZ + VLVHVGVTABCDEFGHIJULTCVSUMUNVIVPKLCCVHVGVMCOVNCOPVPABCDEVMVNFGHIJUOUPUQVI + ANOZAUROVJVKVLVQUSUTVHVGWAADEGVATAVBKLBCAVOHVOVCVDVEVF $. $d B m $. $d N m $. $d R m $. $d X m $. $d Y m $. $d .0. i j m x y $. $( The product of two diagonal matrices is a diagonal matrix. (Contributed @@ -258100,31 +258326,31 @@ that the ring of scalar matrices (over a commutative ring) is a commutative -> ( X ( .r ` A ) Y ) e. D ) $= ( vx vy vi vj wcel wa wceq co vm cfn crg cv cmulr cfv cif cmpt2 wral crab wne wi cbs simpll simplr w3a 3ad2ant1 simp2 simp3 dmatmat adantrr matecld - eqid imp adantrl rngcl syl3anc rng0cl adantl adantr ifcld matbas2d eqeq12 - cvv eqidd oveq12d ifbieq1d simplrl simplrr ovex c0g fvex eqeltri ifex a1i - oveq12 ovmpt2d ifnefalse eqtrd ex ralrimivva oveq eqeq1d ralbidv sylanbrc - imbi2d elrab dmatmul dmatval 3eltr4d ) EUBQZDUCQZRZFCQZGCQZRZRZMNEEMUDZNU - DZSZXHXIFTZXHXIGTZDUEUFZTZHUGZUHZOUDZPUDZUKZXQXRUAUDZTZHSZULZPEUIZOEUIZUA - BUJZFGAUEUFTCXGXPBQXSXQXRXPTZHSZULZPEUIZOEUIZXPYFQXGMNABXODDUMUFZEUCIYLVC - ZJXAXBXFUNXAXBXFUOZXGXHEQZXIEQZUPZXJXNHYLYQXBXKYLQXLYLQXNYLQXGYOXBYPYNUQY - QAAUMUFZDXHXIYLFEIYMYRVCZXGYOYPURZXGYOYPUSZXGYOFYRQZYPXCXDUUBXEXCXDUUBAYR - CDFEUCHIYSKLUTVDVAUQVBYQAYRDXHXIYLGEIYMYSYTUUAXGYOGYRQZYPXCXEUUCXDXCXEUUC - AYRCDGEUCHIYSKLUTVDVEUQVBYLDXMXKXLYMXMVCVFVGXGYOHYLQZYPXCUUDXFXBUUDXAYLDH - YMKVHVIVJUQVKVLXGYIOPEEXGXQEQZXREQZRRZXSYHUUGXSRZYGXQXRSZXQXRFTZXQXRGTZXM - TZHUGZHUUHMNXQXREEXOUUMXPVNUUHXPVOXHXQSXIXRSRZXOUUMSUUHUUNXJUUIXNUULHXHXQ - XIXRVMUUNXKUUJXLUUKXMXHXQXIXRFWFXHXQXIXRGWFVPVQVIXGUUEUUFXSVRXGUUEUUFXSVS - UUMVNQUUHUUIUULHUUJUUKXMVTHDWAUFVNKDWAWBWCWDWEWGXSUUMHSUUGXQXRUULHWHVIWIW - JWKYEYKUAXPBXTXPSZYDYJOEUUOYCYIPEUUOYBYHXSUUOYAYGHXQXRXTXPWLWMWPWNWNWQWOM - NABCDEFGHIJKLWRXCCYFSXFABCDOPUAEUCHIJKLWSVJWT $. + imp adantrl ringcl syl3anc ring0cl adantl adantr ifcld matbas2d cvv eqidd + eqid eqeq12 oveq12 oveq12d ifbieq1d simplrl simplrr ovex c0g fvex eqeltri + ifex a1i ovmpt2d ifnefalse eqtrd ex ralrimivva oveq eqeq1d imbi2d ralbidv + elrab sylanbrc dmatmul dmatval 3eltr4d ) EUBQZDUCQZRZFCQZGCQZRZRZMNEEMUDZ + NUDZSZXHXIFTZXHXIGTZDUEUFZTZHUGZUHZOUDZPUDZUKZXQXRUAUDZTZHSZULZPEUIZOEUIZ + UABUJZFGAUEUFTCXGXPBQXSXQXRXPTZHSZULZPEUIZOEUIZXPYFQXGMNABXODDUMUFZEUCIYL + VNZJXAXBXFUNXAXBXFUOZXGXHEQZXIEQZUPZXJXNHYLYQXBXKYLQXLYLQXNYLQXGYOXBYPYNU + QYQAAUMUFZDXHXIYLFEIYMYRVNZXGYOYPURZXGYOYPUSZXGYOFYRQZYPXCXDUUBXEXCXDUUBA + YRCDFEUCHIYSKLUTVCVAUQVBYQAYRDXHXIYLGEIYMYSYTUUAXGYOGYRQZYPXCXEUUCXDXCXEU + UCAYRCDGEUCHIYSKLUTVCVDUQVBYLDXMXKXLYMXMVNVEVFXGYOHYLQZYPXCUUDXFXBUUDXAYL + DHYMKVGVHVIUQVJVKXGYIOPEEXGXQEQZXREQZRRZXSYHUUGXSRZYGXQXRSZXQXRFTZXQXRGTZ + XMTZHUGZHUUHMNXQXREEXOUUMXPVLUUHXPVMXHXQSXIXRSRZXOUUMSUUHUUNXJUUIXNUULHXH + XQXIXRVOUUNXKUUJXLUUKXMXHXQXIXRFVPXHXQXIXRGVPVQVRVHXGUUEUUFXSVSXGUUEUUFXS + VTUUMVLQUUHUUIUULHUUJUUKXMWAHDWBUFVLKDWBWCWDWEWFWGXSUUMHSUUGXQXRUULHWHVHW + IWJWKYEYKUAXPBXTXPSZYDYJOEUUOYCYIPEUUOYBYHXSUUOYAYGHXQXRXTXPWLWMWNWOWOWPW + QMNABCDEFGHIJKLWRXCCYFSXFABCDOPUAEUCHIJKLWSVIWT $. $( The set of diagonal matrices is a subring of the matrix ring/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) $) dmatsrng $p |- ( ( R e. Ring /\ N e. Fin ) -> D e. ( SubRing ` A ) ) $= ( vx vy crg wcel wa cfv cv wral ancoms eqid cfn csubrg csubg cur cmulr co - dmatsgrp dmatid dmatmulcl ralrimivva w3a wb matrng issubrg2 syl mpbir3and - ) DMNZEUANZOZCAUBPNZCAUCPNZAUDPZCNZKQZLQZAUEPZUFCNZLCRKCRZABCDEFGHIJUGURU - QVCABCDEFGHIJUHSURUQVHURUQOVGKLCCABCDEVDVEFGHIJUIUJSUSAMNZUTVAVCVHUKULURU - QVIADEGUMSKLCBAVFVBHVBTVFTUNUOUP $. + dmatsgrp dmatid dmatmulcl ralrimivva w3a matring issubrg2 syl mpbir3and + wb ) DMNZEUANZOZCAUBPNZCAUCPNZAUDPZCNZKQZLQZAUEPZUFCNZLCRKCRZABCDEFGHIJUG + URUQVCABCDEFGHIJUHSURUQVHURUQOVGKLCCABCDEVDVEFGHIJUIUJSUSAMNZUTVAVCVHUKUP + URUQVIADEGULSKLCBAVFVBHVBTVFTUMUNUO $. $d C x y $. $d D a b x y $. $d N a b $. $d R a b $. dmatcrng.c $e |- C = ( A |`s D ) $. @@ -258132,22 +258358,22 @@ that the ring of scalar matrices (over a commutative ring) is a commutative commutative ring . (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) $) dmatcrng $p |- ( ( R e. CRing /\ N e. Fin ) -> C e. CRing ) $= - ( vx vy va vb wcel wa cfv co ccrg cfn crg cv wceq cbs wral csubrg crngrng - cmulr dmatsrng sylan subrgrng syl cif cmpt2 w3a simp1lr simp2 dmatmat imp - eqid simp3 adantrr 3ad2ant1 matecld adantrl crngcom syl3anc ifeq1d anim2i - mpt2eq3dva dmatmul pm3.22 syl2an 3eqtr4d ralrimivva ancoms subrgbas oveqd - wb eqcomd ressmulr eqeq12d raleqbidv mpbird iscrng2 sylanbrc ) EUAQZFUBQZ - RZCUCQZMUDZNUDZCUJSZTZWNWMWOTZUEZNCUFSZUGZMWSUGZCUAQWKDAUHSZQZWLWIEUCQZWJ - XCEUIZABDEFGHIJKUKULZDACLUMUNWKXAWMWNAUJSZTZWNWMXGTZUEZNDUGZMDUGZWJWIXLWJ - WIRZXJMNDDXMWMDQZWNDQZRZRZOPFFOUDZPUDZUEZXRXSWMTZXRXSWNTZEUJSZTZGUOZUPZOP - FFXTYBYAYCTZGUOZUPZXHXIXQOPFFYEYHXQXRFQZXSFQZUQZXTYDYGGYLWIYAEUFSZQYBYMQY - DYGUEWJWIXPYJYKURYLAAUFSZEXRXSYMWMFHYMVBZYNVBZXQYJYKUSZXQYJYKVCZXQYJWMYNQ - ZYKXMXNYSXOXMXNYSAYNDEWMFUAGHYPJKUTVAVDVEVFYLAYNEXRXSYMWNFHYOYPYQYRXQYJWN - YNQZYKXMXOYTXNXMXOYTAYNDEWNFUAGHYPJKUTVAVGVEVFYMEYCYAYBYOYCVBVHVIVJVLXMWJ - XDRZXPXHYFUEWIXDWJXEVKZOPABDEFWMWNGHIJKVMULXMUUAXOXNRXIYIUEXPUUBXNXOVNOPA - BDEFWNWMGHIJKVMVOVPVQVRWKXCXAXLWAXFXCWTXKMWSDXCDWSDACLVSWBZXCWRXJNWSDUUCX - CWPXHWQXIXCWOXGWMWNXCXGWODACXGXBLXGVBWCWBZVTXCWOXGWNWMUUDVTWDWEWEUNWFMNWS - CWOWSVBWOVBWGWH $. + ( vx vy va vb wcel wa cfv co ccrg cfn crg cv cmulr wceq cbs wral crngring + csubrg dmatsrng sylan subrgring syl cif cmpt2 simp1lr simp2 simp3 dmatmat + w3a imp adantrr 3ad2ant1 matecld adantrl crngcom ifeq1d mpt2eq3dva anim2i + syl3anc dmatmul pm3.22 syl2an 3eqtr4d ralrimivva ancoms subrgbas ressmulr + eqid wb eqcomd oveqd eqeq12d raleqbidv mpbird iscrng2 sylanbrc ) EUAQZFUB + QZRZCUCQZMUDZNUDZCUESZTZWNWMWOTZUFZNCUGSZUHZMWSUHZCUAQWKDAUJSZQZWLWIEUCQZ + WJXCEUIZABDEFGHIJKUKULZDACLUMUNWKXAWMWNAUESZTZWNWMXGTZUFZNDUHZMDUHZWJWIXL + WJWIRZXJMNDDXMWMDQZWNDQZRZRZOPFFOUDZPUDZUFZXRXSWMTZXRXSWNTZEUESZTZGUOZUPZ + OPFFXTYBYAYCTZGUOZUPZXHXIXQOPFFYEYHXQXRFQZXSFQZVAZXTYDYGGYLWIYAEUGSZQYBYM + QYDYGUFWJWIXPYJYKUQYLAAUGSZEXRXSYMWMFHYMVTZYNVTZXQYJYKURZXQYJYKUSZXQYJWMY + NQZYKXMXNYSXOXMXNYSAYNDEWMFUAGHYPJKUTVBVCVDVEYLAYNEXRXSYMWNFHYOYPYQYRXQYJ + WNYNQZYKXMXOYTXNXMXOYTAYNDEWNFUAGHYPJKUTVBVFVDVEYMEYCYAYBYOYCVTVGVKVHVIXM + WJXDRZXPXHYFUFWIXDWJXEVJZOPABDEFWMWNGHIJKVLULXMUUAXOXNRXIYIUFXPUUBXNXOVMO + PABDEFWNWMGHIJKVLVNVOVPVQWKXCXAXLWAXFXCWTXKMWSDXCDWSDACLVRWBZXCWRXJNWSDUU + CXCWPXHWQXIXCWOXGWMWNXCXGWODACXGXBLXGVTVSWBZWCXCWOXGWNWMUUDWCWDWEWEUNWFMN + WSCWOWSVTWOVTWGWH $. $} ${ @@ -258164,16 +258390,16 @@ that the ring of scalar matrices (over a commutative ring) is a commutative -> ( C .* M ) e. D ) $= ( vi vj wcel wa co adantr cfn crg cv wne c0g cfv wceq wi wral simprl eqid dmatmat com12 adantl impcom jca matvscl syldan dmatel cmulr simp-4r simpr - wb w3a anim1i 3jca matvscacell syl oveq2 rngrz 3eqtrd ex imim2d ralimdvva - expimpd sylbid impr mpbir2and ) IUAQZEUBQZRZCGQZHDQZRZRZCHFSZDQZWFBQZOUCZ - PUCZUDZWIWJWFSZEUEUFZUGZUHZPIUIOIUIZWAWDWBHBQZRZWHWEWBWQWAWBWCUJWDWAWQWCW - AWQUHWBWAWCWQABDEHIUBWMKLWMUKZNULUMUNUOUPABCEFGIHJKLMUQURWAWBWCWPWAWBRZWC - WQWKWIWJHSZWMUGZUHZPIUIOIUIZRZWPWAWCXEVCWBABDEOPHIUBWMKLWSNUSTWTWQXDWPWTW - QRZXCWOOPIIXFWIIQWJIQRZRZXBWNWKXHXBWNXHXBRZWLCXAEUTUFZSZCWMXJSZWMXIVTWRXG - VDZWLXKUGXHXMXBXHVTWRXGVSVTWBWQXGVAXFWRXGWTWBWQWAWBVBVETXFXGVBVFTABEFXJWI - WJGICHKLJMXJUKZVGVHXBXKXLUGXHXAWMCXJVIUNXHXLWMUGZXBXFXOXGXFVTWBRZXOWTXPWQ - WAVTWBVSVTVBVETGEXJCWMJXNWSVJVHTTVKVLVMVNVOVPVQWAWGWHWPRVCWDABDEOPWFIUBWM - KLWSNUSTVR $. + wb w3a anim1i 3jca matvscacell syl ringrz 3eqtrd imim2d ralimdvva expimpd + oveq2 ex sylbid impr mpbir2and ) IUAQZEUBQZRZCGQZHDQZRZRZCHFSZDQZWFBQZOUC + ZPUCZUDZWIWJWFSZEUEUFZUGZUHZPIUIOIUIZWAWDWBHBQZRZWHWEWBWQWAWBWCUJWDWAWQWC + WAWQUHWBWAWCWQABDEHIUBWMKLWMUKZNULUMUNUOUPABCEFGIHJKLMUQURWAWBWCWPWAWBRZW + CWQWKWIWJHSZWMUGZUHZPIUIOIUIZRZWPWAWCXEVCWBABDEOPHIUBWMKLWSNUSTWTWQXDWPWT + WQRZXCWOOPIIXFWIIQWJIQRZRZXBWNWKXHXBWNXHXBRZWLCXAEUTUFZSZCWMXJSZWMXIVTWRX + GVDZWLXKUGXHXMXBXHVTWRXGVSVTWBWQXGVAXFWRXGWTWBWQWAWBVBVETXFXGVBVFTABEFXJW + IWJGICHKLJMXJUKZVGVHXBXKXLUGXHXAWMCXJVNUNXHXLWMUGZXBXFXOXGXFVTWBRZXOWTXPW + QWAVTWBVSVTVBVETGEXJCWMJXNWSVIVHTTVJVOVKVLVMVPVQWAWGWHWPRVCWDABDEOPWFIUBW + MKLWSNUSTVR $. $} ${ @@ -258232,15 +258458,15 @@ that the ring of scalar matrices (over a commutative ring) is a commutative scmatscmide $p |- ( ( ( N e. Fin /\ R e. Ring /\ C e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( C .* .1. ) J ) = if ( I = J , C , .0. ) ) $= - ( wcel wa co cfv wceq cfn crg w3a cmulr cur cif cbs simpl2 matrng rngidcl - simp3 eqid syl 3adant3 jca adantr simpr matvscacell syl3anc simpl1 simprl - simprr mat1ov oveq2d ovif2 rngridm 3adant1 rngrz ifeq12d syl5eq 3eqtrd ) - IUAPZDUBPZCBPZUCZFIPZHIPZQZQZFHCEGRRZCFHERZDUDSZRZCFHTZDUESZJUFZWBRZWDCJU - FZVSVMVNEAUGSZPZQZVRVTWCTVLVMVNVRUHZVOWKVRVOVNWJVLVMVNUKVLVMWJVNVLVMQAUBP - WJADIKUIWIAEWIULZNUJUMUNUOUPVOVRUQAWIDGWBFHBICEKWMLOWBULZURUSVSWAWFCWBVSA - DEWEFHIJKWEULZMVLVMVNVRUTWLVOVPVQVAVOVPVQVBNVCVDVOWGWHTVRVOWGWDCWEWBRZCJW - BRZUFWHWDCWEJWBVEVOWDWPCWQJVMVNWPCTVLBDWBWECLWNWOVFVGVMVNWQJTVLBDWBCJLWNM - VHVGVIVJUPVK $. + ( wcel wa co cfv wceq cfn crg w3a cmulr cur cif simpl2 simp3 matring eqid + cbs syl 3adant3 jca adantr simpr matvscacell syl3anc simpl1 simprl simprr + ringidcl mat1ov oveq2d ringridm 3adant1 ringrz ifeq12d syl5eq 3eqtrd + ovif2 ) IUAPZDUBPZCBPZUCZFIPZHIPZQZQZFHCEGRRZCFHERZDUDSZRZCFHTZDUESZJUFZW + BRZWDCJUFZVSVMVNEAUKSZPZQZVRVTWCTVLVMVNVRUGZVOWKVRVOVNWJVLVMVNUHVLVMWJVNV + LVMQAUBPWJADIKUIWIAEWIUJZNVBULUMUNUOVOVRUPAWIDGWBFHBICEKWMLOWBUJZUQURVSWA + WFCWBVSADEWEFHIJKWEUJZMVLVMVNVRUSWLVOVPVQUTVOVPVQVANVCVDVOWGWHTVRVOWGWDCW + EWBRZCJWBRZUFWHWDCWEJWBVKVOWDWPCWQJVMVNWPCTVLBDWBWECLWNWOVEVFVMVNWQJTVLBD + WBCJLWNMVGVFVHVIUOVJ $. $d B i j x y $. $d N i j x y $. $d R i j x y $. $d S i j x y $. $d T i j x y $. $d .1. i j x y $. $d .0. i j x y $. $d .* i j x y $. @@ -258257,27 +258483,27 @@ that the ring of scalar matrices (over a commutative ring) is a commutative cif cur cbs eqid dmatid syl5eqel adantr jca dmatscmcl syldan simprr oveqi a1i dmatmul eqtrd simpll simplr 3jca 3ad2ant1 scmatscmide syl2anc oveq12d w3a 3simpc ifeq1d mpt2eq3dva iftrue adantl ifeq1da wral cvv eqeq12 eqcomi - eqidd ifbieq1d ovex c0g fvex eqeltri ifex ovmpt2d rngcl eqtr4d ralrimivva - syl sylan wb rng0cl ifcld matbas2d matrng rngidcl matvscl eqmat mpbird ) - JUESZCUFSZUGZDBSZEBSZUGZUGZDHITZEHITZGTZUAUBJJUAUBUHZUAUIZUBUIZXRTZYBYCXS - TZCUJUKZTZKUPZULZDEFTZHITZXMXPXRJCUMTZSZXSYLSZUGZXTYIUNXQYMYNXMXPXNHYLSZU - GYMXQXNYPXMXNXOUOZXMYPXPXMHAUQUKYLOAAURUKZYLCJKLYRUSZNYLUSZUTVAVBZVCAYRDY - LCIBHJMLYSPYTVDVEXMXPXOYPUGYNXQXOYPXMXNXOVFZUUAVCAYREYLCIBHJMLYSPYTVDVEVC - XMYOUGZXTXRXSAUJUKZTZYIXTUUEUNUUCGUUDXRXSRVGVHUAUBAYRYLCJXRXSKLYSNYTVIVJV - EXQYIUAUBJJYAYADKUPZYAEKUPZYFTZKUPZULZYKXQUAUBJJYHUUIXQYBJSZYCJSZVRZYAYGU - UHKUUMYDUUFYEUUGYFUUMXKXLXNVRZUUKUULUGZYDUUFUNXQUUKUUNUULXQXKXLXNXKXLXPVK - ZXKXLXPVLZYQVMVNXQUUKUULVSZABDCHYBIYCJKLMNOPVOVPUUMXKXLXOVRZUUOYEUUGUNXQU - UKUUSUULXQXKXLXOUUPUUQUUBVMVNUURABECHYBIYCJKLMNOPVOVPVQVTWAXQUUJUAUBJJYAD - EYFTZKUPZULZYKXQUAUBJJUUIUVAUUMYAUUHUUTKYAUUHUUTUNUUMYAUUFDUUGEYFYADKWBYA - EKWBVQWCWDWAXQUVBYKUNZUCUIZUDUIZUVBTZUVDUVEYKTZUNZUDJWEUCJWEZXQUVHUCUDJJX - QUVDJSZUVEJSZUGZUGZUVFUCUDUHZYJKUPZUVGUVMUAUBUVDUVEJJUVAUVOUVBWFUVMUVBWIU - AUCUHUBUDUHUGZUVAUVOUNUVMUVPYAUVNUUTYJKYBUVDYCUVEWGUUTYJUNUVPYFFDEFYFQWHV - GVHWJWCXQUVJUVKUOXQUVJUVKVFUVOWFSUVMUVNYJKDEFWKKCWLUKWFNCWLWMWNWOVHWPXQXK - XLYJBSZVRUVLUVGUVOUNXQXKXLUVQUUPUUQXQXLXNXOVRZUVQXQXLXNXOUUQYQUUBVMZBCFDE - MQWQWTZVMABYJCHUVDIUVEJKLMNOPVOXAWRWSXQUVBYRSYKYRSZUVCUVIXBXQUAUBAYRUVACB - JUFLMYSUUPUUQXQUUKUVABSUULXQYAUUTKBXQUVRUUTBSUVSBCYFDEMYFUSWQWTXMKBSZXPXL - UWBXKBCKMNXCWCVBXDVNXEXMXPUVQHYRSZUGUWAXQUVQUWCUVTXMUWCXPXMAUFSUWCACJLXFY - RAHYSOXGWTVBVCAYRYJCIBJHMLYSPXHVEAYRCUCUDJUVBYKLYSXIVPXJVJVJVJ $. + eqidd ifbieq1d ovex c0g fvex eqeltri ifex ovmpt2d ringcl syl sylan eqtr4d + ralrimivva ring0cl ifcld matbas2d matring ringidcl matvscl eqmat mpbird + wb ) JUESZCUFSZUGZDBSZEBSZUGZUGZDHITZEHITZGTZUAUBJJUAUBUHZUAUIZUBUIZXRTZY + BYCXSTZCUJUKZTZKUPZULZDEFTZHITZXMXPXRJCUMTZSZXSYLSZUGZXTYIUNXQYMYNXMXPXNH + YLSZUGYMXQXNYPXMXNXOUOZXMYPXPXMHAUQUKYLOAAURUKZYLCJKLYRUSZNYLUSZUTVAVBZVC + AYRDYLCIBHJMLYSPYTVDVEXMXPXOYPUGYNXQXOYPXMXNXOVFZUUAVCAYREYLCIBHJMLYSPYTV + DVEVCXMYOUGZXTXRXSAUJUKZTZYIXTUUEUNUUCGUUDXRXSRVGVHUAUBAYRYLCJXRXSKLYSNYT + VIVJVEXQYIUAUBJJYAYADKUPZYAEKUPZYFTZKUPZULZYKXQUAUBJJYHUUIXQYBJSZYCJSZVRZ + YAYGUUHKUUMYDUUFYEUUGYFUUMXKXLXNVRZUUKUULUGZYDUUFUNXQUUKUUNUULXQXKXLXNXKX + LXPVKZXKXLXPVLZYQVMVNXQUUKUULVSZABDCHYBIYCJKLMNOPVOVPUUMXKXLXOVRZUUOYEUUG + UNXQUUKUUSUULXQXKXLXOUUPUUQUUBVMVNUURABECHYBIYCJKLMNOPVOVPVQVTWAXQUUJUAUB + JJYADEYFTZKUPZULZYKXQUAUBJJUUIUVAUUMYAUUHUUTKYAUUHUUTUNUUMYAUUFDUUGEYFYAD + KWBYAEKWBVQWCWDWAXQUVBYKUNZUCUIZUDUIZUVBTZUVDUVEYKTZUNZUDJWEUCJWEZXQUVHUC + UDJJXQUVDJSZUVEJSZUGZUGZUVFUCUDUHZYJKUPZUVGUVMUAUBUVDUVEJJUVAUVOUVBWFUVMU + VBWIUAUCUHUBUDUHUGZUVAUVOUNUVMUVPYAUVNUUTYJKYBUVDYCUVEWGUUTYJUNUVPYFFDEFY + FQWHVGVHWJWCXQUVJUVKUOXQUVJUVKVFUVOWFSUVMUVNYJKDEFWKKCWLUKWFNCWLWMWNWOVHW + PXQXKXLYJBSZVRUVLUVGUVOUNXQXKXLUVQUUPUUQXQXLXNXOVRZUVQXQXLXNXOUUQYQUUBVMZ + BCFDEMQWQWRZVMABYJCHUVDIUVEJKLMNOPVOWSWTXAXQUVBYRSYKYRSZUVCUVIXJXQUAUBAYR + UVACBJUFLMYSUUPUUQXQUUKUVABSUULXQYAUUTKBXQUVRUUTBSUVSBCYFDEMYFUSWQWRXMKBS + ZXPXLUWBXKBCKMNXBWCVBXCVNXDXMXPUVQHYRSZUGUWAXQUVQUWCUVTXMUWCXPXMAUFSUWCAC + JLXEYRAHYSOXFWRVBVCAYRYJCIBJHMLYSPXGVEAYRCUCUDJUVBYKLYSXHVPXIVJVJVJ $. $} ${ @@ -258316,15 +258542,15 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) -> S = { m e. B | E. c e. K A. i e. N A. j e. N ( i m j ) = if ( i = j , c , .0. ) } ) $= ( wcel wa cv wceq cfn crg cur cfv co wrex crab weq cif wral eqid scmatval - cvsca wb simpr adantr simpll matrng rngidcl anim1i ancomd matvscl syl2anc - syl eqmat w3a simplll simpllr scmatscmide sylan eqeq2d 2ralbidva rexbidva - 3jca bitrd rabbidva eqtrd ) IUAQZCUBQZRZDGSZKSZAUCUDZAUMUDZUEZTZKHUFZGBUG - ESZFSZWAUEZEFUHWBJUIZTZFIUJEIUJZKHUFZGBUGABCDWDWCGHIUBKOLMWCUKZWDUKZNULVT - WGWNGBVTWABQZRZWFWMKHWRWBHQZRZWFWJWHWIWEUEZTZFIUJEIUJZWMWTWQWEBQZWFXCUNWR - WQWSVTWQUOUPWTVTWSWCBQZRXDVTWQWSUQWTXEWSWRXEWSVTXEWQVTAUBQXEACILURBAWCMWO - USVDUPUTVAABWBCWDHIWCOLMWPVBVCABCEFIWAWELMVEVCWTXBWLEFIIWTWHIQWIIQRZRXAWK - WJWTVRVSWSVFXFXAWKTWTVRVSWSVRVSWQWSVGVRVSWQWSVHWRWSUOVNAHWBCWCWHWDWIIJLOP - WOWPVIVJVKVLVOVMVPVQ $. + cvsca wb adantr simpll matring ringidcl syl anim1i ancomd matvscl syl2anc + simpr eqmat simplll simpllr 3jca scmatscmide sylan eqeq2d 2ralbidva bitrd + w3a rexbidva rabbidva eqtrd ) IUAQZCUBQZRZDGSZKSZAUCUDZAUMUDZUEZTZKHUFZGB + UGESZFSZWAUEZEFUHWBJUIZTZFIUJEIUJZKHUFZGBUGABCDWDWCGHIUBKOLMWCUKZWDUKZNUL + VTWGWNGBVTWABQZRZWFWMKHWRWBHQZRZWFWJWHWIWEUEZTZFIUJEIUJZWMWTWQWEBQZWFXCUN + WRWQWSVTWQVDUOWTVTWSWCBQZRXDVTWQWSUPWTXEWSWRXEWSVTXEWQVTAUBQXEACILUQBAWCM + WOURUSUOUTVAABWBCWDHIWCOLMWPVBVCABCEFIWAWELMVEVCWTXBWLEFIIWTWHIQWIIQRZRXA + WKWJWTVRVSWSVNXFXAWKTWTVRVSWSVRVSWQWSVFVRVSWQWSVGWRWSVDVHAHWBCWCWHWDWIIJL + OPWOWPVIVJVKVLVMVOVPVQ $. $d I i j $. $d J j $. $d K m $. $d M i j $. $d .0. i j m $. $( Alternate proof of ~ scmate : An entry of an ` N ` x ` N ` scalar @@ -258362,41 +258588,41 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) scmatscm $p |- ( ( ( N e. Fin /\ R e. Ring ) /\ C e. S ) -> E. c e. K A. m e. B ( C .X. m ) = ( c .* m ) ) $= ( vk wcel co vi vj vx vy cfn crg wa cv cur wceq wrex wral eqid scmatscmid - cfv 3expa oveq1 cmulr cmpt cgsu simpr ad4antr simpl adantr matrng rngidcl - csb syl anim1i ancomd matvscl syl2anc matmulcell syl3anc weq cmpt2 df-3an - c0g cif w3a sylibr ad3antrrr matsc oveqd cvv eqidd eqeq12 adantl vex fvex - ifbid ifex a1i ovmpt2d eqtrd oveq1d mpteq2dva oveq2d ovif simp-6r simplrr - ad2antrr matecld rnglz ifeq2d syl5eq cmnd rngmnd wb equcom ifbi ax-mp cbs - mpteq2i eleq2i biimpi ralrimiva gsummpt1n0 3eqtrd csbov2g csbov1g csbvarg - rngcl matvscacell eqtr4d ralrimivva mpbird sylan9eqr ex impancom ralrimiv - eqmat reximdva mpd ) JUESZDUFSZUGZCESZUGZCKUHZAUIUOZHTZUJZKIUKZCGUHZFTZYT - UUEHTZUJZGBULZKIUKYOYPYRUUDABDEHUUAICJUFKLMNUUAUMZOQUNUPYSUUCUUIKIYSYTISZ - UGZUUCUUIUULUUCUGUUHGBUULUUEBSZUUCUUHUULUUMUGZUUCUUHUUCUUNUUFUUBUUEFTZUUG - CUUBUUEFUQUUNUUOUUGUJZUAUHZUBUHZUUOTZUUQUURUUGTZUJZUBJULUAJULZUUNUVAUAUBJ - JUUNUUQJSZUURJSZUGZUGZUUSYTUUQUURUUETZDURUOZTZUUTUVFUUSDRJUUQRUHZUUBTZUVJ - 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(Contributed by AV, 18-Dec-2019.) $) $( The identity matrix is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) $) scmatid $p |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` A ) e. S ) $= - ( vc wcel crg cur cfv co wceq eqid cfn wa cv cvsca cbs matrng rngidcl syl - wrex csca matsca2 eqcomd fveq2d adantl eqeltrd oveq1 eqeq2d clmod matlmod - wb lmodvs1 syl2anc rspcedvd scmatel mpbir2and ) FUANZCONZUBZAPQZDNVIBNZVI - MUCZVIAUDQZRZSZMCUEQZUIVHAONVJACFHUFBAVIIVITZUGUHZVHVNVIAUJQZPQZVIVLRZSZM - VSVOVHVSCPQZVOVHVRCPVHCVRACFOHUKULUMVGWBVONVFVOCWBVOTZWBTUGUNUOVKVSSZVNWA - UTVHWDVMVTVIVKVSVIVLUPUQUNVHVTVIVHAURNVJVTVISACFHUSVQVLVSVRBAVIIVRTVLTZVS - TVAVBULVCABCDVLVIVOVIFOMWCHIVPWELVDVE $. + ( vc wcel crg cur cfv co wceq eqid cfn wa cvsca cbs wrex matring ringidcl + cv syl matsca2 eqcomd fveq2d adantl eqeltrd wb oveq1 eqeq2d clmod matlmod + csca lmodvs1 syl2anc rspcedvd scmatel mpbir2and ) FUANZCONZUBZAPQZDNVIBNZ + VIMUHZVIAUCQZRZSZMCUDQZUEVHAONVJACFHUFBAVIIVITZUGUIZVHVNVIAUTQZPQZVIVLRZS + ZMVSVOVHVSCPQZVOVHVRCPVHCVRACFOHUJUKULVGWBVONVFVOCWBVOTZWBTUGUMUNVKVSSZVN + WAUOVHWDVMVTVIVKVSVIVLUPUQUMVHVTVIVHAURNVJVTVISACFHUSVQVLVSVRBAVIIVRTVLTZ + VSTVAVBUKVCABCDVLVIVOVIFOMWCHIVPWELVDVE $. ${ $d A i j $. $d B c i j m $. $d E c i j $. $d N i j m $. $d R i j m $. @@ -258442,23 +258668,23 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) -> ( X ( +g ` A ) Y ) e. S ) $= ( wcel wa cfv co wceq eqid vc vd ve cfn crg cur cvsca cplusg scmatscmid cv wrex 3expa adantrr wi 3expia oveq12 adantl csca cbs matlmod ad2antrr - clmod matsca2 fveq2d syl5eq eleq2d biimpd adantr imp biimpa rngidcl syl - matrng lmodvsdir syl13anc eqcomd simpll oveqd rnggrp simpr simplr grpcl - syl3anc eqeltrd matvscl syl12anc wb oveq1 eqeq2d eqidd rspcedvd scmatel - cgrp mpbir2and exp32 rexlimdva com23 syld com12 impcom mpd ) FUDOZCUEOZ - PZGDOZHDOZPZPGUAUJZAUFQZAUGQZRZSZUAEUKZGHAUHQZRZDOZXDXEXMXFXBXCXEXMABCD - XJXIEGFUEUALJKXITZXJTZNUIULUMXGXDXMXPUNZXFXDXSUNXEXDXFXSXDXFHUBUJZXIXJR - ZSZUBEUKZXSXBXCXFYCABCDXJXIEHFUEUBLJKXQXRNUIUOXDYBXSUBEXDXTEOZPZXMYBXPY - EXLYBXPUNUAEYEXHEOZPZXLYBXPYGXLYBPZPXOXKYAXNRZDYHXOYISYGGXKHYAXNUPUQYGY - IDOYHYGYIXHXTAURQZUHQZRZXIXJRZDYGYMYIYGAVBOZXHYJUSQZOZXTYOOZXIBOZYMYISX - DYNYDYFACFJUTVAYEYFYPXDYFYPUNYDXDYFYPXDEYOXHXDECUSQYOLXDCYJUSACFUEJVCZV - DVEZVFVGVHVIYEYQYFXDYDYQXDEYOXTYTVFVJVHXDYRYDYFXDAUEOYRACFJVMBAXIKXQVKV - LVAZXNYKXHXTXJYJYOBAXIKXNTYJTXRYOTYKTVNVOVPYGYMDOZYMBOZYMUCUJZXIXJRZSZU - CEUKZYGXDYLEOYRUUCXDYDYFVQYGYLXHXTCUHQZRZEYGYKUUHXHXTYGYJCUHXDYJCSYDYFX - DCYJYSVPVAVDVRYGCWMOZYFYDUUIEOXDUUJYDYFXCUUJXBCVSUQVAYEYFVTXDYDYFWAEUUH - CXHXTLUUHTWBWCWDZUUAABYLCXJEFXILJKXRWEWFYGUUFYMYMSZUCYLEUUKUUDYLSZUUFUU - LWGYGUUMUUEYMYMUUDYLXIXJWHWIUQYGYMWJWKXDUUBUUCUUGPWGYDYFABCDXJXIEYMFUEU - CLJKXQXRNWLVAWNWDVHWDWOWPWQWPWRWSUQWTXA $. + clmod matsca2 fveq2d syl5eq eleq2d biimpd adantr imp biimpa matring syl + ringidcl lmodvsdir syl13anc eqcomd simpll oveqd cgrp simpr simplr grpcl + ringgrp syl3anc eqeltrd matvscl syl12anc wb oveq1 eqeq2d eqidd rspcedvd + scmatel mpbir2and exp32 rexlimdva com23 syld com12 impcom mpd ) FUDOZCU + EOZPZGDOZHDOZPZPGUAUJZAUFQZAUGQZRZSZUAEUKZGHAUHQZRZDOZXDXEXMXFXBXCXEXMA + BCDXJXIEGFUEUALJKXITZXJTZNUIULUMXGXDXMXPUNZXFXDXSUNXEXDXFXSXDXFHUBUJZXI + XJRZSZUBEUKZXSXBXCXFYCABCDXJXIEHFUEUBLJKXQXRNUIUOXDYBXSUBEXDXTEOZPZXMYB + XPYEXLYBXPUNUAEYEXHEOZPZXLYBXPYGXLYBPZPXOXKYAXNRZDYHXOYISYGGXKHYAXNUPUQ + YGYIDOYHYGYIXHXTAURQZUHQZRZXIXJRZDYGYMYIYGAVBOZXHYJUSQZOZXTYOOZXIBOZYMY + ISXDYNYDYFACFJUTVAYEYFYPXDYFYPUNYDXDYFYPXDEYOXHXDECUSQYOLXDCYJUSACFUEJV + CZVDVEZVFVGVHVIYEYQYFXDYDYQXDEYOXTYTVFVJVHXDYRYDYFXDAUEOYRACFJVKBAXIKXQ + VMVLVAZXNYKXHXTXJYJYOBAXIKXNTYJTXRYOTYKTVNVOVPYGYMDOZYMBOZYMUCUJZXIXJRZ + SZUCEUKZYGXDYLEOYRUUCXDYDYFVQYGYLXHXTCUHQZRZEYGYKUUHXHXTYGYJCUHXDYJCSYD + YFXDCYJYSVPVAVDVRYGCVSOZYFYDUUIEOXDUUJYDYFXCUUJXBCWCUQVAYEYFVTXDYDYFWAE + UUHCXHXTLUUHTWBWDWEZUUAABYLCXJEFXILJKXRWFWGYGUUFYMYMSZUCYLEUUKUUDYLSZUU + FUULWHYGUUMUUEYMYMUUDYLXIXJWIWJUQYGYMWKWLXDUUBUUCUUGPWHYDYFABCDXJXIEYMF + UEUCLJKXQXRNWMVAWNWEVHWEWOWPWQWPWRWSUQWTXA $. $( The difference of two scalar matrices is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) $) @@ -258466,23 +258692,23 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) -> ( X ( -g ` A ) Y ) e. S ) $= ( wcel wa cfv co wceq eqid vc vd ve cfn crg cv cur cvsca csg scmatscmid wrex 3expa adantrr 3expia oveq12 adantl csca cbs clmod matlmod ad2antrr - wi matsca2 fveq2d syl5eq eleq2d biimpd adantr imp biimpa matrng rngidcl - syl lmodsubdir eqcomd simpll oveqd rnggrp simpr simplr grpsubcl syl3anc - cgrp eqeltrd matvscl syl12anc wb oveq1 eqidd rspcedvd scmatel mpbir2and - eqeq2d exp32 rexlimdva com23 syld com12 impcom mpd ) FUDOZCUEOZPZGDOZHD - OZPZPGUAUFZAUGQZAUHQZRZSZUAEUKZGHAUIQZRZDOZXCXDXLXEXAXBXDXLABCDXIXHEGFU - EUALJKXHTZXITZNUJULUMXFXCXLXOVBZXEXCXRVBXDXCXEXRXCXEHUBUFZXHXIRZSZUBEUK - ZXRXAXBXEYBABCDXIXHEHFUEUBLJKXPXQNUJUNXCYAXRUBEXCXSEOZPZXLYAXOYDXKYAXOV - BUAEYDXGEOZPZXKYAXOYFXKYAPZPXNXJXTXMRZDYGXNYHSYFGXJHXTXMUOUPYFYHDOYGYFY - HXGXSAUQQZUIQZRZXHXIRZDYFYLYHYFXGXSYJXIYIYIURQZXMBAXHKXQYITYMTXMTYJTXCA - USOYCYEACFJUTVAYDYEXGYMOZXCYEYNVBYCXCYEYNXCEYMXGXCECURQYMLXCCYIURACFUEJ - VCZVDVEZVFVGVHVIYDXSYMOZYEXCYCYQXCEYMXSYPVFVJVHXCXHBOZYCYEXCAUEOYRACFJV - KBAXHKXPVLVMVAZVNVOYFYLDOZYLBOZYLUCUFZXHXIRZSZUCEUKZYFXCYKEOYRUUAXCYCYE - VPYFYKXGXSCUIQZRZEYFYJUUFXGXSYFYICUIXCYICSYCYEXCCYIYOVOVAVDVQYFCWCOZYEY - CUUGEOXCUUHYCYEXBUUHXACVRUPVAYDYEVSXCYCYEVTECUUFXGXSLUUFTWAWBWDZYSABYKC - XIEFXHLJKXQWEWFYFUUDYLYLSZUCYKEUUIUUBYKSZUUDUUJWGYFUUKUUCYLYLUUBYKXHXIW - HWMUPYFYLWIWJXCYTUUAUUEPWGYCYEABCDXIXHEYLFUEUCLJKXPXQNWKVAWLWDVHWDWNWOW - PWOWQWRUPWSWT $. + wi matsca2 fveq2d syl5eq eleq2d biimpd adantr biimpa matring lmodsubdir + imp ringidcl syl eqcomd simpll oveqd cgrp ringgrp simpr simplr grpsubcl + syl3anc eqeltrd matvscl syl12anc wb oveq1 eqeq2d eqidd rspcedvd scmatel + mpbir2and exp32 rexlimdva com23 syld com12 impcom mpd ) FUDOZCUEOZPZGDO + ZHDOZPZPGUAUFZAUGQZAUHQZRZSZUAEUKZGHAUIQZRZDOZXCXDXLXEXAXBXDXLABCDXIXHE + GFUEUALJKXHTZXITZNUJULUMXFXCXLXOVBZXEXCXRVBXDXCXEXRXCXEHUBUFZXHXIRZSZUB + EUKZXRXAXBXEYBABCDXIXHEHFUEUBLJKXPXQNUJUNXCYAXRUBEXCXSEOZPZXLYAXOYDXKYA + XOVBUAEYDXGEOZPZXKYAXOYFXKYAPZPXNXJXTXMRZDYGXNYHSYFGXJHXTXMUOUPYFYHDOYG + YFYHXGXSAUQQZUIQZRZXHXIRZDYFYLYHYFXGXSYJXIYIYIURQZXMBAXHKXQYITYMTXMTYJT + XCAUSOYCYEACFJUTVAYDYEXGYMOZXCYEYNVBYCXCYEYNXCEYMXGXCECURQYMLXCCYIURACF + UEJVCZVDVEZVFVGVHVLYDXSYMOZYEXCYCYQXCEYMXSYPVFVIVHXCXHBOZYCYEXCAUEOYRAC + FJVJBAXHKXPVMVNVAZVKVOYFYLDOZYLBOZYLUCUFZXHXIRZSZUCEUKZYFXCYKEOYRUUAXCY + CYEVPYFYKXGXSCUIQZRZEYFYJUUFXGXSYFYICUIXCYICSYCYEXCCYIYOVOVAVDVQYFCVROZ + YEYCUUGEOXCUUHYCYEXBUUHXACVSUPVAYDYEVTXCYCYEWAECUUFXGXSLUUFTWBWCWDZYSAB + YKCXIEFXHLJKXQWEWFYFUUDYLYLSZUCYKEUUIUUBYKSZUUDUUJWGYFUUKUUCYLYLUUBYKXH + XIWHWIUPYFYLWJWKXCYTUUAUUEPWGYCYEABCDXIXHEYLFUEUCLJKXPXQNWLVAWMWDVHWDWN + WOWPWOWQWRUPWSWT $. $d B c d $. $( The product of two scalar matrices is a scalar matrix. (Contributed @@ -258491,20 +258717,20 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) -> ( X ( .r ` A ) Y ) e. S ) $= ( wcel crg wa co wceq adantr vc vd ve cfn cmulr cfv cur cvsca wrex eqid cv scmatel oveq12 adantll simp-4l simplr anim1i scmatscmiddistr syl2anc - ancomd simpl simpr adantl rngcl syl3anc matrng rngidcl matvscl syl12anc - wi syl wb oveq1 eqeq2d eqidd rspcedvd mpbir2and exp32 imp eqeltrd exp31 - rexlimdva expimpd com23 sylbid imp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wi ancomd simpl adantl ringcl syl3anc matring ringidcl matvscl syl12anc + simpr syl oveq1 eqeq2d eqidd rspcedvd mpbir2and exp32 imp eqeltrd exp31 + wb rexlimdva expimpd com23 sylbid imp32 ) FUDOZCPOZQZGDOZHDOZGHAUEUFZRZ + DOZWIWJGBOZGUAUKZAUGUFZAUHUFZRZSZUAEUIZQZWKWNUTABCDWRWQEGFPUALJKWQUJZWR + UJZNULWIWKXBWNWIWKHBOZHUBUKZWQWRRZSZUBEUIZQXBWNUTZABCDWRWQEHFPUBLJKXCXD + NULWIXEXIXJWIXEQZXHXJUBEXKXFEOZQZXBXHWNXMWOXAXHWNUTZXMWOQZWTXNUAEXOWPEO + ZQZWTXHWNXQWTQZXHQWMWSXGWLRZDWTXHWMXSSXQGWSHXGWLUMUNXRXSDOZXHXQXTWTXQXS + WPXFCUEUFZRZWQWRRZDXQWIXPXLQXSYCSWIXEXLWOXPUOXQXLXPXOXLXPXKXLWOUPUQVAAE + CWPXFYAWLWQWRFIJLMXCXDYAUJZWLUJURUSXOXPYCDOZXMXPYEUTZWOXKXLYFWIXLYFUTXE + WIXLXPYEWIXLXPQZQZYEYCBOZYCUCUKZWQWRRZSZUCEUIZYHWIYBEOZWQBOZYIWIYGVBYHW + HXPXLYNWGWHYGUPYGXPWIXLXPVJVCYGXLWIXLXPVBVCECYAWPXFLYDVDVEZWIYOYGWIAPOY + OACFJVFBAWQKXCVGVKTABYBCWREFWQLJKXDVHVIYHYLYCYCSZUCYBEYPYJYBSZYLYQWAYHY + RYKYCYCYJYBWQWRVLVMVCYHYCVNVOWIYEYIYMQWAYGABCDWRWQEYCFPUCLJKXCXDNULTVPV + QTVRTVRVSTTVSVTWBWCWDWBWCWEWDWEWF $. $} $d A x y $. $d B x y z $. $d N x y z $. $d R x y z $. $d S x y z $. @@ -258512,19 +258738,19 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) $) scmatsgrp $p |- ( ( N e. Fin /\ R e. Ring ) -> S e. ( SubGrp ` A ) ) $= ( vx vy vz wcel crg cfv cv wral cfn wa csubg wss c0 wne co scmatmat ssrdv - csg cur scmatid ne0i syl scmatsubcl ralrimivva cgrp wb matrng rnggrp eqid - w3a issubg4 3syl mpbir3and ) FUAPCQPUBZDAUCRPZDBUDZDUEUFZMSZNSZAUJRZUGDPZ - NDTMDTZVFODBABCDOSFQHILUHUIVFAUKRZDPVIABCDEFGHIJKLULDVOUMUNVFVMMNDDABCDEF - VJVKGHIJKLUOUPVFAQPAUQPVGVHVIVNVBURACFHUSAUTMNBDAVLIVLVAVCVDVE $. + csg cur scmatid ne0i syl scmatsubcl ralrimivva wb matring ringgrp issubg4 + cgrp w3a eqid 3syl mpbir3and ) FUAPCQPUBZDAUCRPZDBUDZDUEUFZMSZNSZAUJRZUGD + PZNDTMDTZVFODBABCDOSFQHILUHUIVFAUKRZDPVIABCDEFGHIJKLULDVOUMUNVFVMMNDDABCD + EFVJVKGHIJKLUOUPVFAQPAVAPVGVHVIVNVBUQACFHURAUSMNBDAVLIVLVCUTVDVE $. $( The set of scalar matrices is a subring of the matrix ring/algebra. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.) $) scmatsrng $p |- ( ( N e. Fin /\ R e. Ring ) -> S e. ( SubRing ` A ) ) $= ( vx vy wcel crg cfv cv wral eqid cfn wa csubrg csubg cur cmulr scmatsgrp - co scmatid scmatmulcl ralrimivva w3a wb matrng issubrg2 syl mpbir3and ) F - UAOCPOUBZDAUCQOZDAUDQOZAUEQZDOZMRZNRZAUFQZUHDOZNDSMDSZABCDEFGHIJKLUGABCDE - FGHIJKLUIURVFMNDDABCDEFVCVDGHIJKLUJUKURAPOUSUTVBVGULUMACFHUNMNDBAVEVAIVAT - VETUOUPUQ $. + co scmatid scmatmulcl ralrimivva w3a wb matring issubrg2 syl mpbir3and ) + FUAOCPOUBZDAUCQOZDAUDQOZAUEQZDOZMRZNRZAUFQZUHDOZNDSMDSZABCDEFGHIJKLUGABCD + EFGHIJKLUIURVFMNDDABCDEFVCVDGHIJKLUJUKURAPOUSUTVBVGULUMACFHUNMNDBAVEVAIVA + TVETUOUPUQ $. $d C x y $. ${ @@ -258534,23 +258760,24 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) commutative ring. (Contributed by AV, 21-Aug-2019.) $) scmatcrng $p |- ( ( N e. Fin /\ R e. CRing ) -> C e. CRing ) $= ( vx vy va vb wcel co cfn ccrg wa crg cv cmulr cfv wceq cbs wral csubrg - crngrng scmatsrng sylan2 subrgrng syl cif cmpt2 w3a simp1lr simp2 simp3 - scmatmat imp adantrr 3ad2ant1 matecld adantrl crngcom ifeq1d mpt2eq3dva - syl3anc cdmat anim2i adantr wi scmatdmat anim12d dmatmul syl2anc ancomd - eqid 3eqtr4d ralrimivva subrgbas eqcomd ressmulr oveqd raleqbidv mpbird - wb eqeq12d iscrng2 sylanbrc ) GUASZDUBSZUCZCUDSZOUEZPUEZCUFUGZTZWTWSXAT - ZUHZPCUIUGZUJZOXEUJZCUBSWQEAUKUGZSZWRWPWODUDSZXIDULZABDEFGHIJKLMUMUNZEA - CNUOUPWQXGWSWTAUFUGZTZWTWSXMTZUHZPEUJZOEUJZWQXPOPEEWQWSESZWTESZUCZUCZQR - GGQUEZRUEZUHZYCYDWSTZYCYDWTTZDUFUGZTZHUQZURZQRGGYEYGYFYHTZHUQZURZXNXOYB - QRGGYJYMYBYCGSZYDGSZUSZYEYIYLHYQWPYFFSYGFSYIYLUHWOWPYAYOYPUTYQAAUIUGZDY - CYDFWSGIKYRWBZYBYOYPVAZYBYOYPVBZYBYOWSYRSZYPWQXSUUBXTWQXSUUBAYRDEWSGUBI - YSMVCVDVEVFVGYQAYRDYCYDFWTGIKYSYTUUAYBYOWTYRSZYPWQXTUUCXSWQXTUUCAYRDEWT - GUBIYSMVCVDVHVFVGFDYHYFYGKYHWBVIVLVJVKYBWOXJUCZWSGDVMTZSZWTUUESZUCZXNYK - UHWQUUDYAWPXJWOXKVNVOZWQYAUUHWQXSUUFXTUUGWPWOXJXSUUFVPXKABUUEDEFWSGHIJK - LMUUEWBZVQUNWPWOXJXTUUGVPXKABUUEDEFWTGHIJKLMUUJVQUNVRVDZQRABUUEDGWSWTHI - JLUUJVSVTYBUUDUUGUUFUCXOYNUHUUIYBUUFUUGUUKWAQRABUUEDGWTWSHIJLUUJVSVTWCW - DWQXIXGXRWKXLXIXFXQOXEEXIEXEEACNWEWFZXIXDXPPXEEUULXIXBXNXCXOXIXAXMWSWTX - IXMXAEACXMXHNXMWBWGWFZWHXIXAXMWTWSUUMWHWLWIWIUPWJOPXECXAXEWBXAWBWMWN $. + crngring scmatsrng subrgring syl cif cmpt2 w3a simp1lr eqid simp2 simp3 + sylan2 scmatmat adantrr 3ad2ant1 matecld adantrl crngcom syl3anc ifeq1d + imp mpt2eq3dva cdmat anim2i adantr wi scmatdmat anim12d dmatmul syl2anc + ancomd 3eqtr4d ralrimivva wb subrgbas eqcomd ressmulr eqeq12d raleqbidv + oveqd mpbird iscrng2 sylanbrc ) GUASZDUBSZUCZCUDSZOUEZPUEZCUFUGZTZWTWSX + ATZUHZPCUIUGZUJZOXEUJZCUBSWQEAUKUGZSZWRWPWODUDSZXIDULZABDEFGHIJKLMUMVCZ + EACNUNUOWQXGWSWTAUFUGZTZWTWSXMTZUHZPEUJZOEUJZWQXPOPEEWQWSESZWTESZUCZUCZ + QRGGQUEZRUEZUHZYCYDWSTZYCYDWTTZDUFUGZTZHUPZUQZQRGGYEYGYFYHTZHUPZUQZXNXO + YBQRGGYJYMYBYCGSZYDGSZURZYEYIYLHYQWPYFFSYGFSYIYLUHWOWPYAYOYPUSYQAAUIUGZ + DYCYDFWSGIKYRUTZYBYOYPVAZYBYOYPVBZYBYOWSYRSZYPWQXSUUBXTWQXSUUBAYRDEWSGU + BIYSMVDVLVEVFVGYQAYRDYCYDFWTGIKYSYTUUAYBYOWTYRSZYPWQXTUUCXSWQXTUUCAYRDE + WTGUBIYSMVDVLVHVFVGFDYHYFYGKYHUTVIVJVKVMYBWOXJUCZWSGDVNTZSZWTUUESZUCZXN + YKUHWQUUDYAWPXJWOXKVOVPZWQYAUUHWQXSUUFXTUUGWPWOXJXSUUFVQXKABUUEDEFWSGHI + JKLMUUEUTZVRVCWPWOXJXTUUGVQXKABUUEDEFWTGHIJKLMUUJVRVCVSVLZQRABUUEDGWSWT + HIJLUUJVTWAYBUUDUUGUUFUCXOYNUHUUIYBUUFUUGUUKWBQRABUUEDGWTWSHIJLUUJVTWAW + CWDWQXIXGXRWEXLXIXFXQOXEEXIEXEEACNWFWGZXIXDXPPXEEUULXIXBXNXCXOXIXAXMWSW + TXIXMXAEACXMXHNXMUTWHWGZWKXIXAXMWTWSUUMWKWIWJWJUOWLOPXECXAXEUTXAUTWMWN + $. $} $d D x $. @@ -258562,7 +258789,7 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) ( vx vy wcel cfv cfn crg wa csubg cbs wss c0 wne cv csg co wral scmatdmat ssrdv wceq dmatsgrp ancoms subgbas eqcomd syl sseqtr4d cur scmatid adantr ne0i wi com12 impcom a1d imp32 eqid subgsub syl3anc scmatsubcl ralrimivva - w3a eqeltrd cgrp csubrg dmatsrng subrgrng rnggrp issubg4 3syl mpbir3and + w3a eqeltrd cgrp csubrg dmatsrng subrgring ringgrp issubg4 3syl mpbir3and wb ) HUASZEUBSZUCZFCUDTSZFCUETZUFZFUGUHZQUIZRUIZCUJTZUKZFSZRFULQFULZWIFDW KWIQFDABDEFGWNHIJKLMNOUMZUNWIDAUDTSZWKDUOWHWGXAABDEHIJKMOUPUQZXADWKDACPUR USUTVAWIAVBTZFSWMABEFGHIJKLMNVCFXCVEUTWIWRQRFFWIWNFSZWOFSZUCZUCZWQWNWOAUJ @@ -258576,12 +258803,12 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) scmatsrng1 $p |- ( ( N e. Fin /\ R e. Ring ) -> S e. ( SubRing ` C ) ) $= ( vx vy wcel cfv cfn crg wa csubrg csubg cv cmulr co wral scmatsgrp1 wceq cur dmatsrng ancoms subrg1 syl eqcomd scmatid eqeltrd ressmulr scmatmulcl - eqid proplem3 ralrimivva w3a wb subrgrng cbs issubrg2 3syl mpbir3and ) HU - ASZEUBSZUCZFCUDTSZFCUETSZCULTZFSZQUFZRUFZCUGTZUHZFSZRFUIQFUIZABCDEFGHIJKL - MNOPUJVNVQAULTZFVNWEVQVNDAUDTZSZWEVQUKVMVLWGABDEHIJKMOUMUNZDACWEPWEVBUOUP - UQABEFGHIJKLMNURUSVNWCQRFFVNVSFSVTFSUCZUCWBVSVTAUGTZUHFVNWIQRWAWJVNWJWAVN - WGWJWAUKWHDACWJWFPWJVBUTUPUQVCABEFGHVSVTIJKLMNVAUSVDVNWGCUBSVOVPVRWDVEVFW - HDACPVGQRFCVHTZCWAVQWKVBVQVBWAVBVIVJVK $. + eqid proplem3 ralrimivva w3a wb subrgring cbs issubrg2 3syl mpbir3and ) H + UASZEUBSZUCZFCUDTSZFCUETSZCULTZFSZQUFZRUFZCUGTZUHZFSZRFUIQFUIZABCDEFGHIJK + LMNOPUJVNVQAULTZFVNWEVQVNDAUDTZSZWEVQUKVMVLWGABDEHIJKMOUMUNZDACWEPWEVBUOU + PUQABEFGHIJKLMNURUSVNWCQRFFVNVSFSVTFSUCZUCWBVSVTAUGTZUHFVNWIQRWAWJVNWJWAV + NWGWJWAUKWHDACWJWFPWJVBUTUPUQVCABEFGHVSVTIJKLMNVAUSVDVNWGCUBSVOVPVRWDVEVF + WHDACPVGQRFCVHTZCWAVQWKVBVQVBWAVBVIVJVK $. $} ${ @@ -258597,20 +258824,21 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) -> ( C .* X ) e. S ) $= ( wcel crg wa co cbs cfv wceq eqid vc ve cfn cv cur wrex wi scmatel oveq2 adantl csca cmulr matlmod ad3antrrr matsca2 fveq2d syl5eq eleq2d ad2antrr - clmod biimpa matrng rngidcl syl lmodvsass syl13anc eqcomd simplll simp-4r - adantr oveqd simpllr eqcomi eleq2i rngcl syl3anc eqeltrd matvscl syl12anc - biimpi oveq1 eqcoms rspcedeq2vd wb mpbir2and rexlimdva com23 sylbid imp32 - ex expimpd ) GUCMZCNMZOZBFMZHDMZBHEPZDMZWNWPWOWRWNWPHAQRZMZHUAUDZAUERZEPZ - SZUACQRZUFZOZWOWRUGAWSCDEXBXEHGNUAXETJWSTZXBTZLKUHWNWOXGWRWNWOXGWRUGWNWOO - ZWTXFWRXJWTOZXDWRUAXEXKXAXEMZOZXDWRXMXDOWQBXCEPZDXDWQXNSXMHXCBEUIUJXMXNDM - XDXMXNBXAAUKRZULRZPZXBEPZDXMXRXNXMAUTMZBXOQRZMZXAXTMZXBWSMZXRXNSWNXSWOWTX - LACGJUMUNXJYAWTXLWNWOYAWNFXTBWNFXEXTIWNCXOQACGNJUOZUPUQURVAUSXKXLYBXKXEXT - XAXKCXOQWNCXOSZWOWTYDUSUPURVAWNYCWOWTXLWNANMYCACGJVBWSAXBXHXIVCVDUNZBXAEX - PXOXTWSAXBXHXOTLXTTXPTVEVFVGXMXRDMZXRWSMZXRUBUDZXBEPZSZUBFUFZXMWNXQFMYCYH - WNWOWTXLVHXMXQBXACULRZPZFXMXPYMBXAXMXOCULXJXOCSWTXLXJCXOWNYEWOYDVJVGUSUPV - KXMWMWOXAFMZYNFMWLWMWOWTXLVIWNWOWTXLVLXLYOXKXLYOXEFXAFXEIVMVNVTUJFCYMBXAI - YMTVOVPVQZYFAWSXQCEFGXBIJXHLVRVSXMUBXQFXRYJYPYIXQSYKXMYKXQYIXQYIXBEWAWBUJ - WCWNYGYHYLOWDWOWTXLAWSCDEXBFXRGNUBIJXHXILKUHUNWEVQVJVQWJWFWKWJWGWHWGWI $. + clmod biimpa matring ringidcl syl lmodvsass syl13anc eqcomd simplll oveqd + adantr simp-4r simpllr eqcomi eleq2i biimpi syl3anc eqeltrd matvscl oveq1 + ringcl syl12anc eqcoms rspcedeq2vd wb mpbir2and ex rexlimdva com23 sylbid + expimpd imp32 ) GUCMZCNMZOZBFMZHDMZBHEPZDMZWNWPWOWRWNWPHAQRZMZHUAUDZAUERZ + EPZSZUACQRZUFZOZWOWRUGAWSCDEXBXEHGNUAXETJWSTZXBTZLKUHWNWOXGWRWNWOXGWRUGWN + WOOZWTXFWRXJWTOZXDWRUAXEXKXAXEMZOZXDWRXMXDOWQBXCEPZDXDWQXNSXMHXCBEUIUJXMX + NDMXDXMXNBXAAUKRZULRZPZXBEPZDXMXRXNXMAUTMZBXOQRZMZXAXTMZXBWSMZXRXNSWNXSWO + WTXLACGJUMUNXJYAWTXLWNWOYAWNFXTBWNFXEXTIWNCXOQACGNJUOZUPUQURVAUSXKXLYBXKX + EXTXAXKCXOQWNCXOSZWOWTYDUSUPURVAWNYCWOWTXLWNANMYCACGJVBWSAXBXHXIVCVDUNZBX + AEXPXOXTWSAXBXHXOTLXTTXPTVEVFVGXMXRDMZXRWSMZXRUBUDZXBEPZSZUBFUFZXMWNXQFMY + CYHWNWOWTXLVHXMXQBXACULRZPZFXMXPYMBXAXMXOCULXJXOCSWTXLXJCXOWNYEWOYDVJVGUS + UPVIXMWMWOXAFMZYNFMWLWMWOWTXLVKWNWOWTXLVLXLYOXKXLYOXEFXAFXEIVMVNVOUJFCYMB + XAIYMTVTVPVQZYFAWSXQCEFGXBIJXHLVRWAXMUBXQFXRYJYPYIXQSYKXMYKXQYIXQYIXBEVSW + BUJWCWNYGYHYLOWDWOWTXLAWSCDEXBFXRGNUBIJXHXILKUHUNWEVQVJVQWFWGWJWFWHWIWHWK + $. $} ${ @@ -258665,12 +258893,12 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) scmatrhmcl $p |- ( ( N e. Fin /\ R e. Ring /\ X e. K ) -> ( F ` X ) e. C ) $= ( vc wcel crg wceq cfn w3a cfv co scmatrhmval 3adant1 cbs cv 3simpa simp3 - wrex wa matrng 3adant3 eqid rngidcl syl matvscl syl12anc wb eqeq2d adantl - oveq1 eqidd rspcedvd scmatel mpbir2and eqeltrd ) IUARZDSRZJHRZUBZJFUCZJEG - UDZCVJVKVMVNTVIABDEFGHISJKLMNOUEUFVLVNCRZVNBUGUCZRZVNQUHZEGUDZTZQHUKZVLVI - VJULVKEVPRZVQVIVJVKUIVIVJVKUJZVLBSRZWBVIVJWDVKBDILUMUNVPBEVPUOZMUPUQBVPJD - GHIEKLWENURUSVLVTVNVNTZQJHWCVRJTZVTWFUTVLWGVSVNVNVRJEGVCVAVBVLVNVDVEVIVJV - OVQWAULUTVKBVPDCGEHVNISQKLWEMNPVFUNVGVH $. + wrex wa matring 3adant3 ringidcl syl matvscl syl12anc oveq1 eqeq2d adantl + eqid wb eqidd rspcedvd scmatel mpbir2and eqeltrd ) IUARZDSRZJHRZUBZJFUCZJ + EGUDZCVJVKVMVNTVIABDEFGHISJKLMNOUEUFVLVNCRZVNBUGUCZRZVNQUHZEGUDZTZQHUKZVL + VIVJULVKEVPRZVQVIVJVKUIVIVJVKUJZVLBSRZWBVIVJWDVKBDILUMUNVPBEVPVBZMUOUPBVP + JDGHIEKLWENUQURVLVTVNVNTZQJHWCVRJTZVTWFVCVLWGVSVNVNVRJEGUSUTVAVLVNVDVEVIV + JVOVQWAULVCVKBVPDCGEHVNISQKLWEMNPVFUNVGVH $. $d C x $. $d N x $. $( There is a function from a ring to any ring of scalar matrices over this @@ -258699,27 +258927,27 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) scmatf1 $p |- ( ( N e. Fin /\ N =/= (/) /\ R e. Ring ) -> F : K -1-1-> C ) $= ( vi wcel wceq wral wa vy vz vj cfn c0 wne crg w3a cfv wf1 scmatf 3adant2 - wf cv wi co wb simpr simpl scmatrhmval syl2an eqeq12d 3adantl2 cbs matrng - eqid rngidcl syl anim12ci matvscl syldan jca eqmat csn cun difsnid eqcomd - cdif adantl raleqdv oveq2 ralunsn c0g anim2i df-3an sylibr id scmatscmide - cif iftruei syl6eq anbi2d 3bitrd ralbidva r19.26 rspn0 adantr com12 sylbi - 3ad2ant2 sylbid ralrimivva dff13 sylanbrc ) IUDQZIUEUFZDUGQZUHZHCFUMZUAUN - ZFUIZUBUNZFUIZRZXJXLRZUOZUBHSUAHSHCFUJXEXGXIXFABCDEFGHIJKLMNOUKULXHXPUAUB - HHXHXJHQZXLHQZTZTZXNXJEGUPZXLEGUPZRZXOXEXGXSXNYCUQXFXEXGTZXSTZXKYAXMYBYDX - GXQXKYARXSXEXGURZXQXRUSZABDEFGHIUGXJJKLMNUTVAYDXGXRXMYBRXSYFXQXRURZABDEFG - HIUGXLJKLMNUTVAVBVCXTYCPUNZUCUNZYAUPZYIYJYBUPZRZUCISZPISZXOXTYABVDUIZQZYB - YPQZTZYCYOUQXEXGXSYSXFYEYQYRYDXSXQEYPQZTYQYDYTXSXQYDBUGQYTBDIKVEYPBEYPVFZ - LVGVHZYGVIBYPXJDGHIEJKUUAMVJVKYDXSXRYTTYRYDYTXSXRUUBYHVIBYPXLDGHIEJKUUAMV - JVKVLVCBYPDPUCIYAYBKUUAVMVHXTYOYMUCIYIVNZVRZSZXOTZPISZXOXEXGXSYOUUGUQXFYE - YNUUFPIYEYIIQZTZYNYMUCUUDUUCVOZSZUUEYIYIYAUPZYIYIYBUPZRZTZUUFUUIYMUCIUUJU - UHIUUJRYEUUHUUJIIYIVPVQVSVTUUHUUKUUOUQYEYMUUNUCUUDYIIYJYIRYKUULYLUUMYJYIY - IYAWAYJYIYIYBWAVBWBVSUUIUUNXOUUEUUIUULXJUUMXLUUIUULYIYIRZXJDWCUIZWIZXJYEX - EXGXQUHZUUHUUHTZUULUURRUUHYEYDXQTUUSXSXQYDYGWDXEXGXQWEWFUUHUUHUUHUUHWGZUV - AVLZBHXJDEYIGYIIUUQKJUUQVFZLMWHVAUUPXJUUQYIVFZWJWKUUIUUMUUPXLUUQWIZXLYEXE - XGXRUHZUUTUUMUVERUUHYEYDXRTUVFXSXRYDYHWDXEXGXRWEWFUVBBHXLDEYIGYIIUUQKJUVC - LMWHVAUUPXLUUQUVDWJWKVBWLWMWNVCUUGXTXOUUGUUEPISZXOPISZTXTXOUOZUUEXOPIWOUV - HUVIUVGXTUVHXOXHUVHXOUOZXSXFXEUVJXGXOPIWPWTWQWRVSWSWRXAXAXAXBUAUBHCFXCXD - $. + wf cv wi co wb simpr scmatrhmval syl2an eqeq12d 3adantl2 cbs matring eqid + simpl ringidcl syl anim12ci matvscl syldan jca eqmat csn cdif cun difsnid + eqcomd adantl raleqdv ralunsn c0g cif anim2i df-3an sylibr id scmatscmide + oveq2 iftruei syl6eq anbi2d 3bitrd ralbidva r19.26 3ad2ant2 adantr sylbid + rspn0 com12 sylbi ralrimivva dff13 sylanbrc ) IUDQZIUEUFZDUGQZUHZHCFUMZUA + UNZFUIZUBUNZFUIZRZXJXLRZUOZUBHSUAHSHCFUJXEXGXIXFABCDEFGHIJKLMNOUKULXHXPUA + UBHHXHXJHQZXLHQZTZTZXNXJEGUPZXLEGUPZRZXOXEXGXSXNYCUQXFXEXGTZXSTZXKYAXMYBY + DXGXQXKYARXSXEXGURZXQXRVFZABDEFGHIUGXJJKLMNUSUTYDXGXRXMYBRXSYFXQXRURZABDE + FGHIUGXLJKLMNUSUTVAVBXTYCPUNZUCUNZYAUPZYIYJYBUPZRZUCISZPISZXOXTYABVCUIZQZ + YBYPQZTZYCYOUQXEXGXSYSXFYEYQYRYDXSXQEYPQZTYQYDYTXSXQYDBUGQYTBDIKVDYPBEYPV + EZLVGVHZYGVIBYPXJDGHIEJKUUAMVJVKYDXSXRYTTYRYDYTXSXRUUBYHVIBYPXLDGHIEJKUUA + MVJVKVLVBBYPDPUCIYAYBKUUAVMVHXTYOYMUCIYIVNZVOZSZXOTZPISZXOXEXGXSYOUUGUQXF + YEYNUUFPIYEYIIQZTZYNYMUCUUDUUCVPZSZUUEYIYIYAUPZYIYIYBUPZRZTZUUFUUIYMUCIUU + JUUHIUUJRYEUUHUUJIIYIVQVRVSVTUUHUUKUUOUQYEYMUUNUCUUDYIIYJYIRYKUULYLUUMYJY + IYIYAWIYJYIYIYBWIVAWAVSUUIUUNXOUUEUUIUULXJUUMXLUUIUULYIYIRZXJDWBUIZWCZXJY + EXEXGXQUHZUUHUUHTZUULUURRUUHYEYDXQTUUSXSXQYDYGWDXEXGXQWEWFUUHUUHUUHUUHWGZ + UVAVLZBHXJDEYIGYIIUUQKJUUQVEZLMWHUTUUPXJUUQYIVEZWJWKUUIUUMUUPXLUUQWCZXLYE + XEXGXRUHZUUTUUMUVERUUHYEYDXRTUVFXSXRYDYHWDXEXGXRWEWFUVBBHXLDEYIGYIIUUQKJU + VCLMWHUTUUPXLUUQUVDWJWKVAWLWMWNVBUUGXTXOUUGUUEPISZXOPISZTXTXOUOZUUEXOPIWO + UVHUVIUVGXTUVHXOXHUVHXOUOZXSXFXEUVJXGXOPIWSWPWQWTVSXAWTWRWRWRXBUAUBHCFXCX + D $. $( There is a bijection between a ring and any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 26-Dec-2019.) $) @@ -258735,25 +258963,25 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) additive group of the ring of scalar matrices over this ring. (Contributed by AV, 22-Dec-2019.) $) scmatghm $p |- ( ( N e. Fin /\ R e. Ring ) -> F e. ( R GrpHom S ) ) $= - ( wcel cfv eqid vy vz cfn crg wa cplusg cbs rnggrp adantl csubg scmatsgrp - cgrp c0g subggrp syl wf scmatf wceq wb scmatstrbas feq3 mpbird cv co csca - matsca2 cscmat ovex eqeltri resssca mp1i eqtrd fveq2d oveqd oveq1d adantr - cvv clmod clss matlmod scmatlss lsslmod syl2anc syl5eq eleq2d adantrd imp - adantld cur scmatid 3eltr4d cvsca ressvsca ax-mp lmodvsdir syl13anc simpr - biimpd a1i w3a anim1i 3anass sylibr rngacl scmatrhmval ad2ant2lr ad2ant2l - oveq12d 3eqtr4d isghmd ) JUCRZDUDRZUEZUAUBDUFSZEUFSZDEGIEUGSZKXPTZXNTZXOT - ZXLDULRXKDUHUIXMCBUJSREULRBBUGSZDCIJDUMSZLXTTZKYATZPUKCBEQUNUOXMIXPGUPZIC - GUPZABCDFGHIJKLMNOPUQXMXPCURYDYEUSBCDEJLPQUTZXPCIGVAUOVBXMUAVCZIRZUBVCZIR - ZUEZUEZYGYIXNVDZFHVDZYGFHVDZYIFHVDZXOVDZYMGSZYGGSZYIGSZXOVDYLYNYGYIEVESZU - FSZVDZFHVDZYQXMYNUUDURYKXMYMUUCFHXMXNUUBYGYIXMDUUAUFXMDBVESZUUABDJUDLVFCV - QRZUUEUUAURXMCJDVGVDVQPJDVGVHVIZCUUEBEVQQUUETVJVKVLZVMVNVOVPYLEVRRZYGUUAU - GSZRZYIUUJRZFXPRZUUDYQURXMUUIYKXMBVRRCBVSSZRUUIBDJLVTBDCJLPWAUUNCBEQUUNTW - BWCVPXMYKUUKXMYHUUKYJXMYHUUKXMIUUJYGXMIDUGSUUJKXMDUUAUGUUHVMWDZWEWRWFWGXM - YKUULXMYJUULYHXMYJUULXMIUUJYIUUOWEWRWHWGXMUUMYKXMBWISZCFXPBXTDCIJYALYBKYC - PWJFUUPURXMMWSYFWKVPXOUUBYGYIHUUAUUJXPEFXQXSUUATUUFHEWLSURUUGCHBEVQQNWMWN - UUJTUUBTWOWPVLYLXLYMIRZYRYNURXMXLYKXKXLWQZVPYLXLYHYJWTZUUQYLXLYKUEUUSXMXL - YKUURXAXLYHYJXBXCIXNDYGYIKXRXDUOABDFGHIJUDYMKLMNOXEWCYLYSYOYTYPXOXLYHYSYO - URXKYJABDFGHIJUDYGKLMNOXEXFXLYJYTYPURXKYHABDFGHIJUDYIKLMNOXEXGXHXIXJ $. + ( wcel cfv eqid vy vz cfn crg wa cplusg cbs cgrp ringgrp adantl csubg c0g + scmatsgrp subggrp wf scmatf wceq wb scmatstrbas feq3 mpbird cv co matsca2 + syl csca cvv cscmat ovex eqeltri resssca eqtrd fveq2d oveqd oveq1d adantr + mp1i clmod clss matlmod scmatlss lsslmod syl2anc syl5eq eleq2d biimpd imp + adantrd adantld cur scmatid a1i 3eltr4d cvsca ressvsca lmodvsdir syl13anc + ax-mp simpr w3a anim1i 3anass sylibr ringacl scmatrhmval ad2ant2l oveq12d + ad2ant2lr 3eqtr4d isghmd ) JUCRZDUDRZUEZUAUBDUFSZEUFSZDEGIEUGSZKXPTZXNTZX + OTZXLDUHRXKDUIUJXMCBUKSREUHRBBUGSZDCIJDULSZLXTTZKYATZPUMCBEQUNVEXMIXPGUOZ + ICGUOZABCDFGHIJKLMNOPUPXMXPCUQYDYEURBCDEJLPQUSZXPCIGUTVEVAXMUAVBZIRZUBVBZ + IRZUEZUEZYGYIXNVCZFHVCZYGFHVCZYIFHVCZXOVCZYMGSZYGGSZYIGSZXOVCYLYNYGYIEVFS + ZUFSZVCZFHVCZYQXMYNUUDUQYKXMYMUUCFHXMXNUUBYGYIXMDUUAUFXMDBVFSZUUABDJUDLVD + CVGRZUUEUUAUQXMCJDVHVCVGPJDVHVIVJZCUUEBEVGQUUETVKVQVLZVMVNVOVPYLEVRRZYGUU + AUGSZRZYIUUJRZFXPRZUUDYQUQXMUUIYKXMBVRRCBVSSZRUUIBDJLVTBDCJLPWAUUNCBEQUUN + TWBWCVPXMYKUUKXMYHUUKYJXMYHUUKXMIUUJYGXMIDUGSUUJKXMDUUAUGUUHVMWDZWEWFWHWG + XMYKUULXMYJUULYHXMYJUULXMIUUJYIUUOWEWFWIWGXMUUMYKXMBWJSZCFXPBXTDCIJYALYBK + YCPWKFUUPUQXMMWLYFWMVPXOUUBYGYIHUUAUUJXPEFXQXSUUATUUFHEWNSUQUUGCHBEVGQNWO + WRUUJTUUBTWPWQVLYLXLYMIRZYRYNUQXMXLYKXKXLWSZVPYLXLYHYJWTZUUQYLXLYKUEUUSXM + XLYKUURXAXLYHYJXBXCIXNDYGYIKXRXDVEABDFGHIJUDYMKLMNOXEWCYLYSYOYTYPXOXLYHYS + YOUQXKYJABDFGHIJUDYGKLMNOXEXHXLYJYTYPUQXKYHABDFGHIJUDYIKLMNOXEXFXGXIXJ $. ${ $d M y z $. $d T y z $. @@ -258764,36 +258992,36 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) ring. (Contributed by AV, 29-Dec-2019.) $) scmatmhm $p |- ( ( N e. Fin /\ R e. Ring ) -> F e. ( M MndHom T ) ) $= ( vy vz cfn wcel crg wa cmnd cbs cfv wf cv cmulr wceq wral cur w3a cmhm - co rngmgp adantl csubrg eqid scmatsrng subrgrng 3syl scmatf scmatstrbas - c0g eqidd feq23d mpbird scmatscmiddistr ressmulr syl adantr oveqd simpr - eqtr3d anim1i 3anass sylibr rngcl scmatrhmval syl2anc ad2ant2lr oveq12d - ad2ant2l 3eqtr4d ralrimivva rngidcl matsca2 fveq2d oveq1d clmod matlmod - csca matrng lmodvs1 eqtrd subrg1 jca31 mgpbas mgpplusg rngidval ismhm - 3jca ) LUDUEZDUFUEZUGZKUHUEZFUHUEZUGJEUIUJZHUKZUBULZUCULZDUMUJZUSZHUJZX - OHUJZXPHUJZEUMUJZUSZUNZUCJUOUBJUOZDUPUJZHUJZEUPUJZUNZUQZUGHKFURUSUEXJXK - XLYJXIXKXHDKTUTVAXJCBVBUJZUEZEUFUEXLBBUIUJZDCJLDVIUJZNYMVCZMYNVCZRVDZCB - ESVEEFUAUTVFXJXNYEYIXJXNJCHUKABCDGHIJLMNOPQRVGXJJXMJCHXJJVJBCDELNRSVHVK - VLXJYDUBUCJJXJXOJUEZXPJUEZUGZUGZXRGIUSZXOGIUSZXPGIUSZYBUSZXSYCUUAUUCUUD - BUMUJZUSUUBUUEBJDXOXPXQUUFGILYNNMYPOPXQVCZUUFVCZVMUUAUUFYBUUCUUDXJUUFYB - UNZYTXJYLUUIYQCBEUUFYKSUUHVNVOVPVQVSUUAXIXRJUEZXSUUBUNXJXIYTXHXIVRZVPUU - AXIYRYSUQZUUJUUAXIYTUGUULXJXIYTUUKVTXIYRYSWAWBJDXQXOXPMUUGWCVOABDGHIJLU - FXRMNOPQWDWEUUAXTUUCYAUUDYBXIYRXTUUCUNXHYSABDGHIJLUFXOMNOPQWDWFXIYSYAUU - DUNXHYRABDGHIJLUFXPMNOPQWDWHWGWIWJXJYGGYHXJYGYFGIUSZGXJXIYFJUEZYGUUMUNU - UKXIUUNXHJDYFMYFVCZWKVAABDGHIJLUFYFMNOPQWDWEXJUUMBWQUJZUPUJZGIUSZGXJYFU - UQGIXJDUUPUPBDLUFNWLWMWNXJBWOUEGYMUEZUURGUNBDLNWPXJBUFUEUUSBDLNWRYMBGYO - OWKVOIUUQUUPYMBGYOUUPVCPUUQVCWSWEWTWTXJYLGYHUNYQCBEGSOXAVOWTXGXBUBUCJXM - XQYBKFHYHYFJDKTMXCXMEFUAXMVCXCDXQKTUUGXDEYBFUAYBVCXDDYFKTUUOXEEYHFUAYHV - CXEXFWB $. + co ringmgp adantl csubrg c0g eqid scmatsrng subrgring 3syl scmatf eqidd + scmatstrbas feq23d mpbird scmatscmiddistr syl adantr oveqd eqtr3d simpr + ressmulr anim1i 3anass sylibr ringcl syl2anc ad2ant2lr ad2ant2l oveq12d + scmatrhmval 3eqtr4d ralrimivva csca matsca2 fveq2d oveq1d clmod matlmod + ringidcl matring eqtrd subrg1 3jca jca31 mgpbas mgpplusg rngidval ismhm + lmodvs1 ) LUDUEZDUFUEZUGZKUHUEZFUHUEZUGJEUIUJZHUKZUBULZUCULZDUMUJZUSZHU + JZXOHUJZXPHUJZEUMUJZUSZUNZUCJUOUBJUOZDUPUJZHUJZEUPUJZUNZUQZUGHKFURUSUEX + JXKXLYJXIXKXHDKTUTVAXJCBVBUJZUEZEUFUEXLBBUIUJZDCJLDVCUJZNYMVDZMYNVDZRVE + ZCBESVFEFUAUTVGXJXNYEYIXJXNJCHUKABCDGHIJLMNOPQRVHXJJXMJCHXJJVIBCDELNRSV + JVKVLXJYDUBUCJJXJXOJUEZXPJUEZUGZUGZXRGIUSZXOGIUSZXPGIUSZYBUSZXSYCUUAUUC + UUDBUMUJZUSUUBUUEBJDXOXPXQUUFGILYNNMYPOPXQVDZUUFVDZVMUUAUUFYBUUCUUDXJUU + FYBUNZYTXJYLUUIYQCBEUUFYKSUUHVSVNVOVPVQUUAXIXRJUEZXSUUBUNXJXIYTXHXIVRZV + OUUAXIYRYSUQZUUJUUAXIYTUGUULXJXIYTUUKVTXIYRYSWAWBJDXQXOXPMUUGWCVNABDGHI + JLUFXRMNOPQWHWDUUAXTUUCYAUUDYBXIYRXTUUCUNXHYSABDGHIJLUFXOMNOPQWHWEXIYSY + AUUDUNXHYRABDGHIJLUFXPMNOPQWHWFWGWIWJXJYGGYHXJYGYFGIUSZGXJXIYFJUEZYGUUM + UNUUKXIUUNXHJDYFMYFVDZWQVAABDGHIJLUFYFMNOPQWHWDXJUUMBWKUJZUPUJZGIUSZGXJ + YFUUQGIXJDUUPUPBDLUFNWLWMWNXJBWOUEGYMUEZUURGUNBDLNWPXJBUFUEUUSBDLNWRYMB + GYOOWQVNIUUQUUPYMBGYOUUPVDPUUQVDXGWDWSWSXJYLGYHUNYQCBEGSOWTVNWSXAXBUBUC + JXMXQYBKFHYHYFJDKTMXCXMEFUAXMVDXCDXQKTUUGXDEYBFUAYBVDXDDYFKTUUOXEEYHFUA + YHVDXEXFWB $. $} $( There is a ring homomorphism from a ring to the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.) $) scmatrhm $p |- ( ( N e. Fin /\ R e. Ring ) -> F e. ( R RingHom S ) ) $= ( wcel wa cfv cfn crg cghm co cmgp cmhm crh csubrg cbs c0g eqid scmatsrng - simpr subrgrng syl jca scmatghm scmatmhm isrhm sylibr ) JUARZDUBRZSZVBEUB - RZSZGDEUCUDRZGDUETZEUETZUFUDRZSZSGDEUGUDRVCVEVJVCVBVDVAVBUMVCCBUHTRVDBBUI - TZDCIJDUJTZLVKUKKVLUKPULCBEQUNUOUPVCVFVIABCDEFGHIJKLMNOPQUQABCDEVHFGHIVGJ - KLMNOPQVGUKZVHUKZURUPUPDEGVGVHVMVNUSUT $. + simpr subrgring syl jca scmatghm scmatmhm isrhm sylibr ) JUARZDUBRZSZVBEU + BRZSZGDEUCUDRZGDUETZEUETZUFUDRZSZSGDEUGUDRVCVEVJVCVBVDVAVBUMVCCBUHTRVDBBU + ITZDCIJDUJTZLVKUKKVLUKPULCBEQUNUOUPVCVFVIABCDEFGHIJKLMNOPQUQABCDEVHFGHIVG + JKLMNOPQVGUKZVHUKZURUPUPDEGVGVHVMVNUSUT $. $( There is a ring isomorphism from a ring to the ring of scalar matrices over this ring with positive dimension. (Contributed by AV, @@ -258801,11 +259029,11 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) scmatrngiso $p |- ( ( N e. Fin /\ N =/= (/) /\ R e. Ring ) -> F e. ( R RingIso S ) ) $= ( wcel crg cfv cfn c0 wne w3a crs crh wf1o scmatrhm 3adant2 scmatf1o wceq - co cbs wb scmatstrbas f1oeq3 syl mpbird wa simp3 csubrg c0g eqid subrgrng - scmatsrng isrim syl2anc mpbir2and ) JUARZJUBUCZDSRZUDZGDEUEULRZGDEUFULRZI - EUMTZGUGZVIVKVNVJABCDEFGHIJKLMNOPQUHUIVLVPICGUGZABCDFGHIJKLMNOPUJVLVOCUKZ - VPVQUNVIVKVRVJBCDEJLPQUOUIVOCIGUPUQURVLVKESRZVMVNVPUSUNVIVJVKUTVIVKVSVJVI - VKUSCBVATRVSBBUMTZDCIJDVBTZLVTVCKWAVCPVECBEQVDUQUIIVODEGSSKVOVCVFVGVH $. + co cbs wb scmatstrbas f1oeq3 syl mpbird wa csubrg c0g scmatsrng subrgring + simp3 eqid isrim syl2anc mpbir2and ) JUARZJUBUCZDSRZUDZGDEUEULRZGDEUFULRZ + IEUMTZGUGZVIVKVNVJABCDEFGHIJKLMNOPQUHUIVLVPICGUGZABCDFGHIJKLMNOPUJVLVOCUK + ZVPVQUNVIVKVRVJBCDEJLPQUOUIVOCIGUPUQURVLVKESRZVMVNVPUSUNVIVJVKVDVIVKVSVJV + IVKUSCBUTTRVSBBUMTZDCIJDVATZLVTVEKWAVEPVBCBEQVCUQUIIVODEGSSKVOVEVFVGVH $. $} ${ @@ -258828,13 +259056,13 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) 21-Dec-2019.) $) mat0scmat $p |- ( R e. Ring -> (/) e. ( (/) ScMat R ) ) $= ( vc crg wcel c0 cscmat co cmat cbs cfv cv cur cvsca wceq wrex csn 0ex wb - eqid eqeq2d snid mat0dimbas0 rngidcl oveq1 adantl mat0dimscm mpdan eqcomd - syl5eleqr rspcedvd mat0dimid oveq2d rexbidv mpbird 0fin scmatel mpbir2and - cfn wa mpan ) ACDZEEAFGZDZEEAHGZIJZDZEBKZVDLJZVDMJZGZNZBAIJZOZVAEEPVEEQUA - ACUBUIVAVMEVGEVIGZNZBVLOVAVOEALJZEVIGZNZBVPVLVLAVPVLSZVPSUCZVGVPNZVOVRRVA - WAVNVQEVGVPEVIUDTUEVAVQEVAVPVLDVQENVTVDAVPVDSZUFUGUHUJVAVKVOBVLVAVJVNEVAV - HEVGVIVDAWBUKULTUMUNEURDVAVCVFVMUSRUOVDVEAVBVIVHVLEECBVSWBVESVHSVISVBSUPU - TUQ $. + eqid eqeq2d mat0dimbas0 syl5eleqr ringidcl oveq1 adantl mat0dimscm eqcomd + snid mpdan rspcedvd mat0dimid oveq2d rexbidv mpbird cfn 0fin scmatel mpan + wa mpbir2and ) ACDZEEAFGZDZEEAHGZIJZDZEBKZVDLJZVDMJZGZNZBAIJZOZVAEEPVEEQU + HACUAUBVAVMEVGEVIGZNZBVLOVAVOEALJZEVIGZNZBVPVLVLAVPVLSZVPSUCZVGVPNZVOVRRV + AWAVNVQEVGVPEVIUDTUEVAVQEVAVPVLDVQENVTVDAVPVDSZUFUIUGUJVAVKVOBVLVAVJVNEVA + VHEVGVIVDAWBUKULTUMUNEUODVAVCVFVMUSRUPVDVEAVBVIVHVLEECBVSWBVESVHSVISVBSUQ + URUT $. $} ${ @@ -258848,19 +259076,19 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) -> ( M e. B -> M e. ( N ScMat R ) ) ) $= ( ve vc wcel cfv wceq co csn cbs wa cop cvv eqid chash c1 cscmat hash1snb crg wi cv wex cmat cvsca wrex simpr wb mat1dimelbas mpan2 cmulr mat1dimid - cur vex a1i sylan2 oveq2d simpl jctir rngidcl mat1dimscm syl12anc rngridm - adantr opeq2d sneqd 3eqtrrd eqtrd ex reximdva sylbid imp cfn snfi scmatel - syl2anc mpbir2and fveq2i eqtri oveq1 fveq2d syl5eq eleq2d imbi12d syl5ibr - exlimiv syl6bi 3imp ) EFKZEUALUBMZCUEKZDBKZDECUCNZKZUFZWNWOEIUGZOZMZIUHWP - WTUFZEFIUDXCXDIWPWTXCDXBCUINZPLZKZDXBCUCNZKZUFWPXGXIWPXGQZXIXGDJUGZXEURLZ - XEUJLZNZMZJCPLZUKZWPXGULWPXGXQWPXGDXAXARZXKRZOZMZJXPUKZXQWPXASKZXGYBUMIUS - ZXEXPCXADXRSJXETZXPTZXRTZUNUOWPYAXOJXPWPXKXPKZQZYAXOYIYAQDXTXNYIYAULYIXTX - NMYAYIXNXKXRCURLZROZXMNZXRXKYJCUPLZNZRZOZXTYIXLYKXKXMYHWPYCXLYKMYCYHYDUTX - EXPCXAXRSYEYFYGUQVAVBYIWPYCQYHYJXPKZYLYPMYIWPYCWPYHVCYDVDWPYHULWPYQYHXPCY - JYFYJTZVEVIXEXPCXAXRSXKYJYEYFYGVFVGYIYOXSYIYNXKXRXPCYMYJXKYFYMTYRVHVJVKVL - VIVMVNVOVPVQXJXBVRKZWPXIXGXQQUMYSXJXAVSUTWPXGVCXEXFCXHXMXLXPDXBUEJYFYEXFT - XLTXMTXHTVTWAWBVNXCWQXGWSXIXCBXFDXCBECUINZPLZXFBAPLUUAHAYTPGWCWDXCYTXEPEX - BCUIWEWFWGWHXCWRXHDEXBCUCWEWHWIWJWKWLWM $. + cur vex a1i sylan2 oveq2d simpl jctir ringidcl adantr mat1dimscm syl12anc + ringridm opeq2d sneqd 3eqtrrd ex reximdva sylbid imp snfi scmatel syl2anc + eqtrd mpbir2and fveq2i eqtri fveq2d syl5eq eleq2d imbi12d syl5ibr exlimiv + cfn oveq1 syl6bi 3imp ) EFKZEUALUBMZCUEKZDBKZDECUCNZKZUFZWNWOEIUGZOZMZIUH + WPWTUFZEFIUDXCXDIWPWTXCDXBCUINZPLZKZDXBCUCNZKZUFWPXGXIWPXGQZXIXGDJUGZXEUR + LZXEUJLZNZMZJCPLZUKZWPXGULWPXGXQWPXGDXAXARZXKRZOZMZJXPUKZXQWPXASKZXGYBUMI + USZXEXPCXADXRSJXETZXPTZXRTZUNUOWPYAXOJXPWPXKXPKZQZYAXOYIYAQDXTXNYIYAULYIX + TXNMYAYIXNXKXRCURLZROZXMNZXRXKYJCUPLZNZRZOZXTYIXLYKXKXMYHWPYCXLYKMYCYHYDU + TXEXPCXAXRSYEYFYGUQVAVBYIWPYCQYHYJXPKZYLYPMYIWPYCWPYHVCYDVDWPYHULWPYQYHXP + CYJYFYJTZVEVFXEXPCXAXRSXKYJYEYFYGVGVHYIYOXSYIYNXKXRXPCYMYJXKYFYMTYRVIVJVK + VLVFVTVMVNVOVPXJXBWJKZWPXIXGXQQUMYSXJXAVQUTWPXGVCXEXFCXHXMXLXPDXBUEJYFYEX + FTXLTXMTXHTVRVSWAVMXCWQXGWSXIXCBXFDXCBECUINZPLZXFBAPLUUAHAYTPGWBWCXCYTXEP + EXBCUIWKWDWEWFXCWRXHDEXBCUCWKWFWGWHWIWLWM $. $} $( @@ -259005,16 +259233,16 @@ seems more natural to use the definition of a free (left) module, avoiding to an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.) $) mavmulcl $p |- ( ph -> ( X .X. Y ) e. ( B ^m N ) ) $= - ( vi co wcel vj cv cfv cmpt cgsu cmap crg mavmulval wral wa rngcmn adantr - ccmn syl cfn ad2antrr cxp cbs wceq matbas2 syl2anc eleqtrrd elmapi fovrnd - wf simpr ffvelrnd rngcl syl3anc ralrimiva gsummptcl cvv wb eqeltri elmapg - fvex sylancr eqid fmpt syl6rbbr mpbid eqeltrd ) AHIFSRGDUAGRUBZUAUBZHSZWD - IUCZESZUDUESZUDZCGUFSZABCDEFRUAGUGHIJKLMNOPQUHAWHCTZRGUIZWIWJTZAWKRGAWCGT - ZUJZCUADGWGLADUMTZWNADUGTZWPNDUKUNULAGUOTZWNOULWOWGCTZUAGWOWDGTZUJZWQWECT - WFCTWSAWQWNWTNUPXAWCWDCGGHAGGUQZCHVEZWNWTAHCXBUFSZTXCAHBURUCZXDPAWRWQXDXE - USONBDCGUGJLUTVAVBHCXBVCUNUPWOWNWTAWNVFULWOWTVFZVDXAGCWDIAGCIVEZWNWTAIWJT - XGQICGVCUNUPXFVGCDEWEWFLMVHVIVJVKVJAWMGCWIVEZWLACVLTWRWMXHVMCDURUCVLLDURV - PVNOCGWIVLUOVOVQRGCWHWIWIVRVSVTWAWB $. + ( vi co wcel vj cv cfv cmpt cgsu cmap crg mavmulval wral ccmn ringcmn syl + wa adantr cfn ad2antrr cxp cbs wceq matbas2 syl2anc eleqtrrd elmapi simpr + wf fovrnd ffvelrnd ringcl syl3anc ralrimiva gsummptcl cvv wb fvex eqeltri + elmapg sylancr eqid fmpt syl6rbbr mpbid eqeltrd ) AHIFSRGDUAGRUBZUAUBZHSZ + WDIUCZESZUDUESZUDZCGUFSZABCDEFRUAGUGHIJKLMNOPQUHAWHCTZRGUIZWIWJTZAWKRGAWC + GTZUMZCUADGWGLADUJTZWNADUGTZWPNDUKULUNAGUOTZWNOUNWOWGCTZUAGWOWDGTZUMZWQWE + CTWFCTWSAWQWNWTNUPXAWCWDCGGHAGGUQZCHVEZWNWTAHCXBUFSZTXCAHBURUCZXDPAWRWQXD + XEUSONBDCGUGJLUTVAVBHCXBVCULUPWOWNWTAWNVDUNWOWTVDZVFXAGCWDIAGCIVEZWNWTAIW + JTXGQICGVCULUPXFVGCDEWEWFLMVHVIVJVKVJAWMGCWIVEZWLACVLTWRWMXHVMCDURUCVLLDU + RVNVOOCGWIVLUOVPVQRGCWHWIWIVRVSVTWAWB $. $} ${ @@ -259033,26 +259261,26 @@ seems more natural to use the definition of a free (left) module, avoiding to ( vi vj cfv co wcel wceq adantr vx vy cur cv cmulr cmpt cgsu crg eqid cfn cbs cmat fveq2i mat1bas syl2anc mavmulval weq c0g cif cmpt2 mat1 proplem3 oveq1d mpteq2dv oveq2d cvv eqidd eqeq12 ifbid adantl simpr fvex a1i ifcld - wa ovmpt2d iftrue wi cmap wf wb eqeltri elmapg ffvelrn syl6bi mpd rnglidm - ex imp fveq2 equcoms 3eqtrd eqtr4d wn rnglz eqcom sylnbi eqcomd pm2.61ian - iffalse eqtrd mpteq2dva rngmnd syl syl6eleq gsummptif1n0 wfn dffn5 bitr4i - cmnd ffn sylibr ) ABUCPZGEQNFDOFNUDZOUDZXMQZXOGPZDUEPZQZUFZUGQZUFNFXNGPZU - FZGABCDXRENOFUHXMGHJIXRUIZKLADUHRZFUJRZXMBUKPZRKLBYGDXMFHYGUIBFDULQUCHUMU - NUOMUPANFYAYBAXNFRZVOZYADOFXNXOUAUBFFUAUBUQZDUCPZDURPZUSZUTZQZXQXRQZUFZUG - QDOFONUQZYBYLUSZUFZUGQYBYIXTYQDUGYIOFXSYPYIXPYOXQXRAYHNOXMYNAYFYEXMYNSLKB - DYKUAUBFYLHYKUIZYLUIZVAUOVBVCVDVEYIYQYTDUGYIOFYPYSYIXOFRZVOZYPNOUQZYKYLUS - ZXQXRQZYSUUDYOUUFXQXRUUDUAUBXNXOFFYMUUFYNVFUUDYNVGUANUQUBOUQVOZYMUUFSUUDU - UHYJUUEYKYLUAUDXNUBUDXOVHVIVJYIYHUUCAYHVKZTYIUUCVKUUDUUEYKYLVFYKVFRUUDDUC - VLVMYLVFRUUDDURVLVMVNVPVCUUEUUDUUGYSSUUEUUDVOZUUGYBYSUUJUUGYKXQXRQZXQYBUU - JUUFYKXQXRUUEUUFYKSUUDUUEYKYLVQTVCUUDUUKXQSZUUEUUDYEXQCRZUULYIYEUUCAYEYHK - TTZYIUUCUUMAUUCUUMVRZYHAGCFVSQRZUUOMAUUPFCGVTZUUOACVFRZYFUUPUUQWAUURACDUK - PZVFIDUKVLWBVMLCFGVFUJWCUOZUUQUUCUUMFCXOGWDWHWEWFTWIZCDXRYKXQIYDUUAWGUOVJ - UUEXQYBSZUUDUVBONXOXNGWJWKTWLUUEYSYBSZUUDUVCONYRYBYLVQWKTWMUUEWNZUUDVOUUG - YLXQXRQZYLYSUVDUUGUVESUUDUVDUUFYLXQXRUUEYKYLWTVCTUUDUVEYLSZUVDUUDYEUUMUVF - UUNUVACDXRXQYLIYDUUBWOUOVJUVDYLYSSUUDUVDYSYLUUEYRYSYLSXNXOWPYRYBYLWTWQWRT - WLWSXAXBVEYIYBOYTDFUJXNYLUUBADXJRZYHAYEUVGKDXCXDTAYFYHLTUUIYTUIAYHYBUUSRZ - AUUPYHUVHVRZMAUUPUUQUVIUUTUUQYHUVHUUQYHVOYBCUUSFCXNGWDIXEWHWEWFWIXFWLXBAG - FXGZYCGSZAUUPUVJMAUUPUUQUVJUUTFCGXKWEWFUVKGYCSUVJYCGWPNFGXHXIXLWL $. + wa ovmpt2d iftrue wi cmap wf wb eqeltri elmapg ffvelrn ex syl6bi ringlidm + mpd imp fveq2 equcoms 3eqtrd eqtr4d wn iffalse ringlz eqcom sylnbi eqcomd + pm2.61ian eqtrd mpteq2dva ringmnd syl syl6eleq gsummptif1n0 wfn ffn dffn5 + cmnd bitr4i sylibr ) ABUCPZGEQNFDOFNUDZOUDZXMQZXOGPZDUEPZQZUFZUGQZUFNFXNG + PZUFZGABCDXRENOFUHXMGHJIXRUIZKLADUHRZFUJRZXMBUKPZRKLBYGDXMFHYGUIBFDULQUCH + UMUNUOMUPANFYAYBAXNFRZVOZYADOFXNXOUAUBFFUAUBUQZDUCPZDURPZUSZUTZQZXQXRQZUF + ZUGQDOFONUQZYBYLUSZUFZUGQYBYIXTYQDUGYIOFXSYPYIXPYOXQXRAYHNOXMYNAYFYEXMYNS + LKBDYKUAUBFYLHYKUIZYLUIZVAUOVBVCVDVEYIYQYTDUGYIOFYPYSYIXOFRZVOZYPNOUQZYKY + LUSZXQXRQZYSUUDYOUUFXQXRUUDUAUBXNXOFFYMUUFYNVFUUDYNVGUANUQUBOUQVOZYMUUFSU + UDUUHYJUUEYKYLUAUDXNUBUDXOVHVIVJYIYHUUCAYHVKZTYIUUCVKUUDUUEYKYLVFYKVFRUUD + DUCVLVMYLVFRUUDDURVLVMVNVPVCUUEUUDUUGYSSUUEUUDVOZUUGYBYSUUJUUGYKXQXRQZXQY + BUUJUUFYKXQXRUUEUUFYKSUUDUUEYKYLVQTVCUUDUUKXQSZUUEUUDYEXQCRZUULYIYEUUCAYE + YHKTTZYIUUCUUMAUUCUUMVRZYHAGCFVSQRZUUOMAUUPFCGVTZUUOACVFRZYFUUPUUQWAUURAC + DUKPZVFIDUKVLWBVMLCFGVFUJWCUOZUUQUUCUUMFCXOGWDWEWFWHTWIZCDXRYKXQIYDUUAWGU + OVJUUEXQYBSZUUDUVBONXOXNGWJWKTWLUUEYSYBSZUUDUVCONYRYBYLVQWKTWMUUEWNZUUDVO + UUGYLXQXRQZYLYSUVDUUGUVESUUDUVDUUFYLXQXRUUEYKYLWOVCTUUDUVEYLSZUVDUUDYEUUM + UVFUUNUVACDXRXQYLIYDUUBWPUOVJUVDYLYSSUUDUVDYSYLUUEYRYSYLSXNXOWQYRYBYLWOWR + WSTWLWTXAXBVEYIYBOYTDFUJXNYLUUBADXJRZYHAYEUVGKDXCXDTAYFYHLTUUIYTUIAYHYBUU + SRZAUUPYHUVHVRZMAUUPUUQUVIUUTUUQYHUVHUUQYHVOYBCUUSFCXNGWDIXEWEWFWHWIXFWLX + BAGFXGZYCGSZAUUPUVJMAUUPUUQUVJUUTFCGXHWFWHUVKGYCSUVJYCGWQNFGXIXKXLWL $. $d j k B $. $d k N $. $d k R $. $d i j k X $. $d k Y $. $d i j k Z $. $d k ph $. $d i j .X. $. $d i k .x. $. @@ -259065,36 +259293,36 @@ seems more natural to use the definition of a free (left) module, avoiding to mavmulass $p |- ( ph -> ( ( X .X. Z ) .x. Y ) = ( X .x. ( Z .x. Y ) ) ) $= ( co vi vj vk cmap wcel wf wfn cmulr cfv eqid cxp cbs cfn matbas2 syl2anc crg wceq eleqtrrd mamucl eleqtrd mavmulcl elmapi ffn 3syl cv wa cmpt cgsu - ccmn rngcmn syl adantr ad2antrr simplr simprr fovrnd simprl wi ffvelrn ex - imp ad2ant2r rngcl syl3anc gsumcom3fi simpr mamufv oveq1d c0g adantlr cvv - cplusg ovex a1i fvex fsuppmptdm gsummulc1 rngass anassrs mpteq2dva oveq2d - syl13anc 3eqtr2d mavmulfv gsummulc2 eqtr4d 3eqtr4d eqfnfvd ) AUAGHJFTZIET - ZHJIETZETZAXJCGUDTZUEGCXJUFXJGUGABCDDUHUIZEGXIIKMLXNUJZNOAXICGGUKZUDTZBUL - UIZACGDFGGHJLNQOOOAHXRXQRAGUMUEZDUPUEZXQXRUQONBDCGUPKLUNUOZURZAJXRXQSYAUR - ZUSYAUTZPVAXJCGVBGCXJVCVDAXLXMUEGCXLUFXLGUGABCDXNEGHXKKMLXONORABCDXNEGJIK - MLXONOSPVAZVAXLCGVBGCXLVCVDAUAVEZGUEZVFZDUBGYFUBVEZXITZYIIUIZXNTZVGZVHTZD - UCGYFUCVEZHTZYOXKUIZXNTZVGZVHTZYFXJUIYFXLUIYHDUBGDUCGYPYOYIJTZYKXNTZXNTZV - GZVHTZVGZVHTDUCGDUBGUUCVGVHTZVGZVHTYNYTYHGCGUBUCDUUCLADVIUEZYGAXTUUINDVJV - KVLAXSYGOVLZUUJYHYIGUEZYOGUEZVFZVFZXTYPCUEZUUBCUEZUUCCUEAXTYGUUMNVMZUUNYF - YOCGGHAXPCHUFZYGUUMAHXQUEZUURYBHCXPVBVKZVMAYGUUMVNYHUUKUULVOZVPZUUNXTUUAC - UEZYKCUEZUUPUUQUUNYOYICGGJAXPCJUFZYGUUMAJXQUEZUVEYCJCXPVBVKZVMUVAYHUUKUUL - VQVPZAUUKUVDYGUULAUUKUVDAIXMUEZGCIUFZUUKUVDVRZPICGVBUVJUUKUVDGCYIIVSVTVDZ - WAZWBZCDXNUUAYKLXOWCZWDCDXNYPUUBLXOWCWDWEYHYMUUFDVHYHUBGYLUUEYHUUKVFZYLDU - CGYPUUAXNTZVGZVHTZYKXNTDUCGUVQYKXNTZVGZVHTUUEUVPYJUVSYKXNUVPCGDXNUCFYFYIG - GUPHJQLXOAXTYGUUKNVMZAXSYGUUKOVMZUWCUWCAUUSYGUUKYBVMAUVFYGUUKYCVMAYGUUKVN - YHUUKWFWGWHUVPGCDWLUIZDXNUCUMUVQYKDWIUIZLUWEUJZUWDUJZXOUWBUWCAUUKUVDYGUVM - WJUVPUULVFZXTUUOUVCUVQCUEYHXTUUKUULAXTYGNVLZVMYHUULUUOUUKYHUULVFZYFYOCGGH - AUURYGUULUUTVMAYGUULVNYHUULWFZVPZWJUWHYOYICGGJYHUVEUUKUULAUVEYGUVGVLZVMUV - PUULWFYHUUKUULVNVPCDXNYPUUALXOWCWDUVPUCGUVRWKWKUVQUWEUVRUJUWCUVQWKUEUWHYP - UUAXNWMWNUWEWKUEZUVPDWIWOZWNWPWQUVPUWAUUDDVHUVPUCGUVTUUCYHUUKUULUVTUUCUQZ - UUNXTUUOUVCUVDUWPUUQUVBUVHUVNCDXNYPUUAYKLXOWRXBWSWTXAXCWTXAYHYSUUHDVHYHUC - GYRUUGUWJYRYPDUBGUUBVGZVHTZXNTUUGUWJYQUWRYPXNUWJBCDXNEUBYOGUPJIKMLXOAXTYG - UULNVMZAXSYGUULOVMZAJXRUEYGUULSVMAUVIYGUULPVMUWKXDXAUWJGCUWDDXNUBUMUUBYPU - WELUWFUWGXOUWSUWTUWLUWJUUKVFZXTUVCUVDUUPYHXTUULUUKUWIVMUXAYOYICGGJYHUVEUU - LUUKUWMVMYHUULUUKVNUWJUUKWFVPUWJUUKUVDAUVKYGUULUVLVMWAUVOWDUWJUBGUWQWKWKU - UBUWEUWQUJUWTUUBWKUEUXAUUAYKXNWMWNUWNUWJUWOWNWPXEXFWTXAXGYHBCDXNEUBYFGUPX - IIKMLXOUWIUUJAXIXRUEYGYDVLAUVIYGPVLAYGWFZXDYHBCDXNEUCYFGUPHXKKMLXOUWIUUJA - HXRUEYGRVLAXKXMUEYGYEVLUXBXDXGXH $. + ringcmn syl adantr ad2antrr simplr simprr fovrnd simprl wi ffvelrn ex imp + ad2ant2r ringcl syl3anc gsumcom3fi simpr mamufv oveq1d cplusg c0g adantlr + ccmn cvv ovex a1i fsuppmptdm gsummulc1 ringass syl13anc anassrs mpteq2dva + fvex oveq2d 3eqtr2d mavmulfv gsummulc2 eqtr4d 3eqtr4d eqfnfvd ) AUAGHJFTZ + IETZHJIETZETZAXJCGUDTZUEGCXJUFXJGUGABCDDUHUIZEGXIIKMLXNUJZNOAXICGGUKZUDTZ + BULUIZACGDFGGHJLNQOOOAHXRXQRAGUMUEZDUPUEZXQXRUQONBDCGUPKLUNUOZURZAJXRXQSY + AURZUSYAUTZPVAXJCGVBGCXJVCVDAXLXMUEGCXLUFXLGUGABCDXNEGHXKKMLXONORABCDXNEG + JIKMLXONOSPVAZVAXLCGVBGCXLVCVDAUAVEZGUEZVFZDUBGYFUBVEZXITZYIIUIZXNTZVGZVH + TZDUCGYFUCVEZHTZYOXKUIZXNTZVGZVHTZYFXJUIYFXLUIYHDUBGDUCGYPYOYIJTZYKXNTZXN + TZVGZVHTZVGZVHTDUCGDUBGUUCVGVHTZVGZVHTYNYTYHGCGUBUCDUUCLADWKUEZYGAXTUUIND + VIVJVKAXSYGOVKZUUJYHYIGUEZYOGUEZVFZVFZXTYPCUEZUUBCUEZUUCCUEAXTYGUUMNVLZUU + NYFYOCGGHAXPCHUFZYGUUMAHXQUEZUURYBHCXPVBVJZVLAYGUUMVMYHUUKUULVNZVOZUUNXTU + UACUEZYKCUEZUUPUUQUUNYOYICGGJAXPCJUFZYGUUMAJXQUEZUVEYCJCXPVBVJZVLUVAYHUUK + UULVPVOZAUUKUVDYGUULAUUKUVDAIXMUEZGCIUFZUUKUVDVQZPICGVBUVJUUKUVDGCYIIVRVS + VDZVTZWAZCDXNUUAYKLXOWBZWCCDXNYPUUBLXOWBWCWDYHYMUUFDVHYHUBGYLUUEYHUUKVFZY + LDUCGYPUUAXNTZVGZVHTZYKXNTDUCGUVQYKXNTZVGZVHTUUEUVPYJUVSYKXNUVPCGDXNUCFYF + YIGGUPHJQLXOAXTYGUUKNVLZAXSYGUUKOVLZUWCUWCAUUSYGUUKYBVLAUVFYGUUKYCVLAYGUU + KVMYHUUKWEWFWGUVPGCDWHUIZDXNUCUMUVQYKDWIUIZLUWEUJZUWDUJZXOUWBUWCAUUKUVDYG + UVMWJUVPUULVFZXTUUOUVCUVQCUEYHXTUUKUULAXTYGNVKZVLYHUULUUOUUKYHUULVFZYFYOC + GGHAUURYGUULUUTVLAYGUULVMYHUULWEZVOZWJUWHYOYICGGJYHUVEUUKUULAUVEYGUVGVKZV + LUVPUULWEYHUUKUULVMVOCDXNYPUUALXOWBWCUVPUCGUVRWLWLUVQUWEUVRUJUWCUVQWLUEUW + HYPUUAXNWMWNUWEWLUEZUVPDWIXAZWNWOWPUVPUWAUUDDVHUVPUCGUVTUUCYHUUKUULUVTUUC + UQZUUNXTUUOUVCUVDUWPUUQUVBUVHUVNCDXNYPUUAYKLXOWQWRWSWTXBXCWTXBYHYSUUHDVHY + HUCGYRUUGUWJYRYPDUBGUUBVGZVHTZXNTUUGUWJYQUWRYPXNUWJBCDXNEUBYOGUPJIKMLXOAX + TYGUULNVLZAXSYGUULOVLZAJXRUEYGUULSVLAUVIYGUULPVLUWKXDXBUWJGCUWDDXNUBUMUUB + YPUWELUWFUWGXOUWSUWTUWLUWJUUKVFZXTUVCUVDUUPYHXTUULUUKUWIVLUXAYOYICGGJYHUV + EUULUUKUWMVLYHUULUUKVMUWJUUKWEVOUWJUUKUVDAUVKYGUULUVLVLVTUVOWCUWJUBGUWQWL + WLUUBUWEUWQUJUWTUUBWLUEUXAUUAYKXNWMWNUWNUWJUWOWNWOXEXFWTXBXGYHBCDXNEUBYFG + UPXIIKMLXOUWIUUJAXIXRUEYGYDVKAUVIYGPVKAYGWEZXDYHBCDXNEUCYFGUPHXKKMLXOUWIU + UJAHXRUEYGRVKAXKXMUEYGYEVKUXBXDXGXH $. $} ${ @@ -259333,14 +259561,14 @@ seems more natural to use the definition of a free (left) module, avoiding to /\ ( K e. N /\ L e. N ) ) -> ( K ( M ( N matRRep R ) S ) L ) e. B ) $= ( vi vj crg wcel cfv co wceq eqid adantr 3ad2ant1 cbs w3a cmarrep c0g cif - wa cv cmpt2 marrepval 3adantl1 matrcl simpld 3ad2ant2 simpl1 simp3 rng0cl - cfn cvv ifcld simp2 eleq2i biimpi matecl syl3anc matbas2d eqeltrd ) CMNZG - BNZDCUAOZNZUBZEHNFHNUFZUFZEFGDHCUCPZPPZKLHHKUGZEQZLUGZFQZDCUDOZUEZVPVRGPZ - UEZUHZBVHVJVLVOWDQVGABVNCDKLEFGHVTIJVNRVTRZUIUJVMKLABWCCVIHMIVIRZJVKHUQNZ - VLVHVGWGVJVHWGCURNABCHGIJUKULUMSVGVHVJVLUNVMVPHNZVRHNZUBZVQWAWBVIVMWHWAVI - NZWIVKWKVLVKVSDVTVIVGVHVJUOVGVHVTVINVJVICVTWFWEUPTUSSTWJWHWIGAUAOZNZWBVIN - VMWHWIUTVMWHWIUOVMWHWMWIVKWMVLVHVGWMVJVHWMBWLGJVAVBUMSTACVPVRVIGHIWFVCVDU - SVEVF $. + wa cmpt2 marrepval 3adantl1 cfn cvv matrcl simpld 3ad2ant2 simpl1 ring0cl + cv simp3 ifcld simp2 eleq2i biimpi matecl syl3anc matbas2d eqeltrd ) CMNZ + GBNZDCUAOZNZUBZEHNFHNUFZUFZEFGDHCUCPZPPZKLHHKUQZEQZLUQZFQZDCUDOZUEZVPVRGP + ZUEZUGZBVHVJVLVOWDQVGABVNCDKLEFGHVTIJVNRVTRZUHUIVMKLABWCCVIHMIVIRZJVKHUJN + ZVLVHVGWGVJVHWGCUKNABCHGIJULUMUNSVGVHVJVLUOVMVPHNZVRHNZUBZVQWAWBVIVMWHWAV + INZWIVKWKVLVKVSDVTVIVGVHVJURVGVHVTVINVJVICVTWFWEUPTUSSTWJWHWIGAUAOZNZWBVI + NVMWHWIUTVMWHWIURVMWHWMWIVKWMVLVHVGWMVJVHWMBWLGJVAVBUNSTACVPVRVIGHIWFVCVD + USVEVF $. $} ${ @@ -259472,14 +259700,14 @@ seems more natural to use the definition of a free (left) module, avoiding to if ( J = L , ( I X L ) , .0. ) ) ) $= ( crg wcel w3a co cmulr cfv wceq cur cif wa simp3 jca ma1repveval syl3an3 simp2 oveq2d ovif2 a1i cbs simp1 3ad2ant3 eleq2i biimpi 3ad2ant1 3ad2ant2 - eqid matecl syl3anc rngridm syl2anc rngrz ifeq12d syl5eq ifeq2d 3eqtrd ) - DUAUBZMBUBZCLUBZIKUBZUCZGKUBZHKUBZJKUBZUCZUCZGJMUDZJHFUDZDUEUFZUDWFHIUGZJ - CUFZHJUGZDUHUFZNUIZUIZWHUDZWIWFWJWHUDZWFWMWHUDZUIZWIWPWKWFNUIZUIWEWGWNWFW - HWDVPVTWCWBUJWGWNUGWDWCWBWAWBWCUKZWAWBWCUOULABCDEFJHIMKLNOPQRSTUMUNUPWOWR - UGWEWIWFWJWMWHUQURWEWIWQWSWPWEWQWKWFWLWHUDZWFNWHUDZUIWSWKWFWLNWHUQWEWKXAW - FXBNWEVPWFDUSUFZUBZXAWFUGVPVTWDUTZWEWAWCMAUSUFZUBZXDWDVPWAVTWAWBWCUTVAWDV - PWCVTWTVAVTVPXGWDVQVRXGVSVQXGBXFMPVBVCVDVEADGJXCMKOXCVFZVGVHZXCDWHWLWFXHW - HVFZWLVFVIVJWEVPXDXBNUGXEXIXCDWHWFNXHXJSVKVJVLVMVNVO $. + eqid matecl syl3anc ringridm syl2anc ringrz ifeq12d syl5eq ifeq2d 3eqtrd + ) DUAUBZMBUBZCLUBZIKUBZUCZGKUBZHKUBZJKUBZUCZUCZGJMUDZJHFUDZDUEUFZUDWFHIUG + ZJCUFZHJUGZDUHUFZNUIZUIZWHUDZWIWFWJWHUDZWFWMWHUDZUIZWIWPWKWFNUIZUIWEWGWNW + FWHWDVPVTWCWBUJWGWNUGWDWCWBWAWBWCUKZWAWBWCUOULABCDEFJHIMKLNOPQRSTUMUNUPWO + WRUGWEWIWFWJWMWHUQURWEWIWQWSWPWEWQWKWFWLWHUDZWFNWHUDZUIWSWKWFWLNWHUQWEWKX + AWFXBNWEVPWFDUSUFZUBZXAWFUGVPVTWDUTZWEWAWCMAUSUFZUBZXDWDVPWAVTWAWBWCUTVAW + DVPWCVTWTVAVTVPXGWDVQVRXGVSVQXGBXFMPVBVCVDVEADGJXCMKOXCVFZVGVHZXCDWHWLWFX + HWHVFZWLVFVIVJWEVPXDXBNUGXEXIXCDWHWFNXHXJSVKVJVLVMVNVO $. $d B l $. $d C l $. $d I l $. $d J l $. $d K l $. $d N l $. $d R l $. $d V l $. $d X l $. $d .0. l $. @@ -259492,18 +259720,18 @@ seems more natural to use the definition of a free (left) module, avoiding to = ( I X J ) ) $= ( crg wcel w3a wne cv co cmulr cfv cmpt cgsu cif wa simp1 adantr 3ad2ant3 wceq simp2 simpr mulmarep1el syl113anc mpteq2dva oveq2d wn biimpi iffalse - df-ne syl mpteq2dv cfn cmnd rngmnd 3ad2ant1 cvv matrcl simpld 3ad2ant2 wb - eqcom ifbi oveq2 adantl ifeq1da eqtrd ax-mp mpteq2i eleq2i matecl syl3anc - cbs eqid gsummptif1n0 3eqtrd ) DUAUBZLBUBZCKUBZIJUBZUCZGJUBZHJUBZHIUDZUCZ - UCZDNJGNUEZLUFZXCHFUFDUGUHZUFZUIZUJUFDNJHIUPZXDXCCUHXEUFZHXCUPZXDMUKZUKZU - IZUJUFDNJXKUIZUJUFGHLUFZXBXGXMDUJXBNJXFXLXBXCJUBZULWMWQWRWSXPXFXLUPXBWMXP - WMWQXAUMUNXBWQXPWMWQXAUQUNXBWRXPXAWMWRWQWRWSWTUMUOZUNXBWSXPXAWMWSWQWRWSWT - UQUOZUNXBXPURABCDEFGHIXCJKLMOPQRSTUSUTVAVBXBXMXNDUJXBNJXLXKXBXHVCZXLXKUPX - AWMXSWQWTWRXSWSWTXSHIVFVDUOUOXHXIXKVEVGVHVBXBXONXNDJVIHMSWMWQDVJUBXADVKVL - WQWMJVIUBZXAWNWOXTWPWNXTDVMUBABDJLOPVNVOVLVPXRNJXKXCHUPZXOMUKZXJYAVQZXKYB - UPHXCVRYCXKYAXDMUKYBXJYAXDMVSYCYAXDXOMYAXDXOUPYCXCHGLVTWAWBWCWDWEXBWRWSLA - WIUHZUBZXODWIUHZUBXQXRWQWMYEXAWNWOYEWPWNYEBYDLPWFVDVLVPADGHYFLJOYFWJWGWHW - KWL $. + df-ne syl mpteq2dv cfn cmnd ringmnd 3ad2ant1 matrcl simpld 3ad2ant2 eqcom + cvv ifbi oveq2 adantl ifeq1da eqtrd ax-mp mpteq2i cbs eleq2i eqid syl3anc + wb matecl gsummptif1n0 3eqtrd ) DUAUBZLBUBZCKUBZIJUBZUCZGJUBZHJUBZHIUDZUC + ZUCZDNJGNUEZLUFZXCHFUFDUGUHZUFZUIZUJUFDNJHIUPZXDXCCUHXEUFZHXCUPZXDMUKZUKZ + UIZUJUFDNJXKUIZUJUFGHLUFZXBXGXMDUJXBNJXFXLXBXCJUBZULWMWQWRWSXPXFXLUPXBWMX + PWMWQXAUMUNXBWQXPWMWQXAUQUNXBWRXPXAWMWRWQWRWSWTUMUOZUNXBWSXPXAWMWSWQWRWSW + TUQUOZUNXBXPURABCDEFGHIXCJKLMOPQRSTUSUTVAVBXBXMXNDUJXBNJXLXKXBXHVCZXLXKUP + XAWMXSWQWTWRXSWSWTXSHIVFVDUOUOXHXIXKVEVGVHVBXBXONXNDJVIHMSWMWQDVJUBXADVKV + LWQWMJVIUBZXAWNWOXTWPWNXTDVQUBABDJLOPVMVNVLVOXRNJXKXCHUPZXOMUKZXJYAWIZXKY + BUPHXCVPYCXKYAXDMUKYBXJYAXDMVRYCYAXDXOMYAXDXOUPYCXCHGLVSVTWAWBWCWDXBWRWSL + AWEUHZUBZXODWEUHZUBXQXRWQWMYEXAWNWOYEWPWNYEBYDLPWFVDVLVOADGHYFLJOYFWGWJWH + WKWL $. $d A l $. $d F l $. $d Z l $. $d .X. l $. mulmarep1gsum2.x $e |- .X. = ( R maVecMul <. N , N >. ) $. @@ -259790,38 +260018,38 @@ determinant function is alternating (see ~ mdetralt ) and normalized ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsum ( x e. N |-> ( x M ( p ` x ) ) ) ) ) ) ) ) $= ( vq vy ccrg wcel wa cfv cv ccom co cmpt cgsu csymg cminusg wceq mdetleib - adantl cbs eqid crg crngrng rngcmn syl adantr cfn matrcl simpld symgbasfi - ccmn ad2antrr cmgp cmhm wf czrh cpsgn coeq12i zrhpsgnmhm syl5eqel syl2anc - cvv mgpbas mhmf ffvelrnda crngmgp cxp cmap simpr matbas2i 3syl symgbasf1o - elmapi wf1o f1of ralrimiva gsummptcl rngcl syl3anc cgrp symggrp grpinvf1o - fovrnd gsummptfif1o feqmptd eqidd fveq2 fveq1 oveq1d mpteq2dv oveq2d ccnv - oveq12d fmptco symginv fveq2d zrhpsgninv ad2antlr fveq1d eqeltrd syl6eleq - eqtrd f1ocnv id f1ocnvfv1 sylan mpteq2dva 3eqtrd ) FUDUEZJCUEZUFZJDUGZFUB - EUBUHZLGUIZUGZIUCKUCUHZYKUGZYNJUJZUKZULUJZHUJZUKZULUJZFYTKUMUGZUNUGZUIZUL - UJFMEMUHZYLUGZIAKAUHZUUGUUEUGZJUJZUKZULUJZHUJZUKZULUJYHYJUUAUOYGUCBCDEFGH - IJKLUBNOPQRSTUAUPUQYIFURUGZEUBYTFUUCEYSUUNUSZYGFVIUEZYHYGFUTUEZUUPFVAZFVB - VCVDYIKVEUEZEVEUEYIUUSFVTUEZYHUUSUUTUFYGBCFKJOPVFUQVGZKEUUBUUBUSZQVHVCYIY - SUUNUEZUBEYIYKEUEZUFZUUQYMUUNUEYRUUNUEUVCYGUUQYHUVDUURVJYIEUUNYKYLYIYLUUB - FVKUGZVLUJZUEZEUUNYLVMYIUUQUUSUVHYGUUQYHUURVDUVAUUQUUSUFYLFVNUGZKVOUGZUIU - VGLUVIGUVJRSVPKFVQVRVSEUUNUUBUVFYLQUUNFUVFUVFUSUUOWAWBVCWCUVEUUNUCIKYPUUN - FIUAUUOWAZYGIVIUEZYHUVDFIUAWDZVJYIUUSUVDUVAVDUVEYPUUNUEUCKUVEYNKUEZUFYOYN - UUNKKJYIKKWEZUUNJVMZUVDUVNYIYHJUUNUVOWFUJUEUVPYGYHWGBCFUUNJKOUUOPWHJUUNUV - OWKWIZVJUVEKKYNYKUVEKKYKWLZKKYKVMUVDUVRYIKEYKUUBUVBQWJUQKKYKWMVCWCUVEUVNW - GXAWNWOUUNFHYMYRUUOTWPWQWNYTUSYIEUUBUUCQUUCUSZYIUUSUUBWRUEUVAKUUBVEUVBWSV - CWTZXBYIUUDUUMFULYIUUDMEUUEUUCUGZYLUGZIUCKYNUWAUGZYNJUJZUKZULUJZHUJZUKUUM - YIMUBEEUWAYSUWGUUCYTYIEEUUEUUCYIEEUUCWLEEUUCVMUVTEEUUCWMVCZWCYIMEEUUCUWHX - CYIYTXDYKUWAUOZYMUWBYRUWFHYKUWAYLXEUWIYQUWEIULUWIUCKYPUWDUWIYOUWCYNJYNYKU - WAXFXGXHXIXKXLYIMEUWGUULYIUUEEUEZUFZUWBUUFUWFUUKHUWKUWBUUEXJZYLUGZUUFUWKU - WAUWLYLUWJUWAUWLUOZYIKEUUEUUBUUCUVBQUVSXMZUQXNUWKUUQUUSUWJUWMUUFUOYGUUQYH - UWJUURVJYIUUSUWJUVAVDZYIUWJWGEFGUUEKLQRSXOWQXTUWKUWFIUWEUUEUIZULUJUUKUWKI - URUGZKUCUWEIUUEKUWDUWRUSYGUVLYHUWJUVMVJUWPUWKUWDUWRUEUCKUWKUVNUFZUWDUUNUW - RUWSUWCYNUUNKKJYIUVPUWJUVNUVQVJUWSUWCYNUWLUGKUWSYNUWAUWLUWJUWNYIUVNUWOXPX - QUWKKKYNUWLUWKKKUUEWLZKKUWLWLKKUWLVMUWJUWTYIKEUUEUUBUVBQWJUQZKKUUEYAKKUWL - WMWIWCXRUWKUVNWGXAUVKXSWNUWEUSUXAXBUWKUWQUUJIULUWKUWQAKUUHUWAUGZUUHJUJZUK - UUJUWKAUCKKUUHUWDUXCUUEUWEUWKKKUUGUUEUWKUWTKKUUEVMUXAKKUUEWMVCZWCUWKAKKUU - EUXDXCUWKUWEXDYNUUHUOZUWCUXBYNUUHJYNUUHUWAXEUXEYBXKXLUWKAKUXCUUIUWKUUGKUE - ZUFZUXBUUGUUHJUXGUXBUUHUWLUGZUUGUXGUUHUWAUWLUWJUWNYIUXFUWOXPXQUWKUWTUXFUX - HUUGUOUXAKKUUGUUEYCYDXTXGYEXTXIXTXKYEXTXIYF $. + adantl cbs eqid ccmn crg crngring ringcmn syl adantr cfn matrcl symgbasfi + cvv simpld ad2antrr cmgp cmhm wf czrh coeq12i zrhpsgnmhm syl5eqel syl2anc + cpsgn mgpbas mhmf ffvelrnda crngmgp cmap simpr matbas2i elmapi symgbasf1o + cxp 3syl wf1o f1of fovrnd ralrimiva gsummptcl ringcl syl3anc cgrp symggrp + grpinvf1o gsummptfif1o feqmptd eqidd fveq2 oveq1d mpteq2dv oveq2d oveq12d + fveq1 fmptco ccnv symginv fveq2d zrhpsgninv eqtrd ad2antlr fveq1d eqeltrd + f1ocnv syl6eleq id f1ocnvfv1 sylan mpteq2dva 3eqtrd ) FUDUEZJCUEZUFZJDUGZ + FUBEUBUHZLGUIZUGZIUCKUCUHZYKUGZYNJUJZUKZULUJZHUJZUKZULUJZFYTKUMUGZUNUGZUI + ZULUJFMEMUHZYLUGZIAKAUHZUUGUUEUGZJUJZUKZULUJZHUJZUKZULUJYHYJUUAUOYGUCBCDE + FGHIJKLUBNOPQRSTUAUPUQYIFURUGZEUBYTFUUCEYSUUNUSZYGFUTUEZYHYGFVAUEZUUPFVBZ + FVCVDVEYIKVFUEZEVFUEYIUUSFVIUEZYHUUSUUTUFYGBCFKJOPVGUQVJZKEUUBUUBUSZQVHVD + YIYSUUNUEZUBEYIYKEUEZUFZUUQYMUUNUEYRUUNUEUVCYGUUQYHUVDUURVKYIEUUNYKYLYIYL + UUBFVLUGZVMUJZUEZEUUNYLVNYIUUQUUSUVHYGUUQYHUURVEUVAUUQUUSUFYLFVOUGZKVTUGZ + UIUVGLUVIGUVJRSVPKFVQVRVSEUUNUUBUVFYLQUUNFUVFUVFUSUUOWAWBVDWCUVEUUNUCIKYP + UUNFIUAUUOWAZYGIUTUEZYHUVDFIUAWDZVKYIUUSUVDUVAVEUVEYPUUNUEUCKUVEYNKUEZUFY + OYNUUNKKJYIKKWJZUUNJVNZUVDUVNYIYHJUUNUVOWEUJUEUVPYGYHWFBCFUUNJKOUUOPWGJUU + NUVOWHWKZVKUVEKKYNYKUVEKKYKWLZKKYKVNUVDUVRYIKEYKUUBUVBQWIUQKKYKWMVDWCUVEU + VNWFWNWOWPUUNFHYMYRUUOTWQWRWOYTUSYIEUUBUUCQUUCUSZYIUUSUUBWSUEUVAKUUBVFUVB + WTVDXAZXBYIUUDUUMFULYIUUDMEUUEUUCUGZYLUGZIUCKYNUWAUGZYNJUJZUKZULUJZHUJZUK + UUMYIMUBEEUWAYSUWGUUCYTYIEEUUEUUCYIEEUUCWLEEUUCVNUVTEEUUCWMVDZWCYIMEEUUCU + WHXCYIYTXDYKUWAUOZYMUWBYRUWFHYKUWAYLXEUWIYQUWEIULUWIUCKYPUWDUWIYOUWCYNJYN + YKUWAXJXFXGXHXIXKYIMEUWGUULYIUUEEUEZUFZUWBUUFUWFUUKHUWKUWBUUEXLZYLUGZUUFU + WKUWAUWLYLUWJUWAUWLUOZYIKEUUEUUBUUCUVBQUVSXMZUQXNUWKUUQUUSUWJUWMUUFUOYGUU + QYHUWJUURVKYIUUSUWJUVAVEZYIUWJWFEFGUUEKLQRSXOWRXPUWKUWFIUWEUUEUIZULUJUUKU + WKIURUGZKUCUWEIUUEKUWDUWRUSYGUVLYHUWJUVMVKUWPUWKUWDUWRUEUCKUWKUVNUFZUWDUU + NUWRUWSUWCYNUUNKKJYIUVPUWJUVNUVQVKUWSUWCYNUWLUGKUWSYNUWAUWLUWJUWNYIUVNUWO + XQXRUWKKKYNUWLUWKKKUUEWLZKKUWLWLKKUWLVNUWJUWTYIKEUUEUUBUVBQWIUQZKKUUEXTKK + UWLWMWKWCXSUWKUVNWFWNUVKYAWOUWEUSUXAXBUWKUWQUUJIULUWKUWQAKUUHUWAUGZUUHJUJ + ZUKUUJUWKAUCKKUUHUWDUXCUUEUWEUWKKKUUGUUEUWKUWTKKUUEVNUXAKKUUEWMVDZWCUWKAK + KUUEUXDXCUWKUWEXDYNUUHUOZUWCUXBYNUUHJYNUUHUWAXEUXEYBXIXKUWKAKUXCUUIUWKUUG + KUEZUFZUXBUUGUUHJUXGUXBUUHUWLUGZUUGUXGUUHUWAUWLUWJUWNYIUXFUWOXQXRUWKUWTUX + FUXHUUGUOUXAKKUUGUUEYCYDXPXFYEXPXHXPXIYEXPXHYF $. $} ${ @@ -259886,23 +260114,23 @@ determinant function is alternating (see ~ mdetralt ) and normalized ( vm vp vx wcel c0 co cbs cfv cv cmpt cgsu csn wceq eqid a1i cvv mpteq2dv oveq2d 3eqtrd crg cmdat cmat csymg czrh cpsgn ccom cmulr cur cop mdetfval cmgp mat0dimbas0 mpteq1d 0ex ovex oveq fmptsng sylancl c0g mpt0 syl6eq wa - gsum0 rngidval eqcomi cfn 0fin zrhcopsgnelbas mp3an2 rngridm syldan eqtrd - mpteq2dva cevpm simpl c1 simpr elsni psgn0fv0 syl eleq2s adantl psgnevpmb - fveq2 symgbas0 wb mpbir2and zrhpsgnevpm syl3anc cmnd rngmnd rngidcl eqidd - gsumsn opeq2d sneqd eqtr3d ) AUAEZFAUBGZBFAUCGZHIZACFUDIZHIZCJZAUEIZFUFIZ - UGIZAULIZDFDJZXEIZXJBJZGZKZLGZAUHIZGZKZLGZKZBFMZXSKZFAUIIZUJZMZWTXTNWSDXA - XBWTXDAXGXPXIBFXFCWTOXAOXBOXDOZXFOZXGOZXPOZXIOZUKPWSBXBYAXSAUAUMUNWSFACXD - XHXIDFXKXJFGZKZLGZXPGZKZLGZUJZMZYBYEWSFQEZYPQEYRYBNYSWSUOPZAYOLUPBFXSYPQQ - XLFNZXRYOALUUACXDXQYNUUAXOYMXHXPUUAXNYLXILUUADFXMYKXKXJXLFUQRSSRSURUSWSYQ - YDWSYPYCFWSYPACXDXHXIUTIZXPGZKZLGACXDXHKZLGZYCWSYOUUDALWSCXDYNUUCWSYMUUBX - HXPWSYMXIFLGUUBWSYLFXILYLFNWSDYKVAPSXIUUBUUBOVDVBSRSWSUUDUUEALWSCXDUUCXHW - SXEXDEZVCZUUCXHYCXPGZXHUUHUUBYCXHXPUUBYCNUUHYCUUBAYCXIYJYCOZVEVFPSWSUUGXH - AHIZEZUUIXHNWSFVGEZUUGUULVHXDXEAXGFXFYFYHYGVIVJUUKAXPYCXHUUKOZYIUUJVKVLVM - VNSWSUUFACXDYCKZLGACYAYCKZLGZYCWSUUEUUOALWSCXDXHYCUUHWSUUMXEFVOIEZXHYCNWS - UUGVPUUMUUHVHPZUUHUURUUGXEXGIZVQNZWSUUGVRUUGUVAWSUVAXEYAXDXEYAEXEFNZUVAXE - FVSUVBUUTFXGIVQXEFXGWEVTVBWAWFWBWCUUHUUMUURUUGUVAVCWGUUSFXDXCXEXGXCOYFYHW - DWAWHAXGYCXEFXFYGYHUUJWIWJVNSWSUUOUUPALWSCXDYAYCXDYANWSWFPUNSWSAWKEYSYCUU - KEUUQYCNAWLYTUUKAYCUUNUUJWMYCUUKYCCAFQUUNUVBYCWNWOWJTTWPWQWRT $. + gsum0 rngidval eqcomi cfn zrhcopsgnelbas mp3an2 ringridm syldan mpteq2dva + 0fin eqtrd cevpm simpl c1 simpr elsni psgn0fv0 syl symgbas0 eleq2s adantl + fveq2 psgnevpmb mpbir2and zrhpsgnevpm syl3anc cmnd ringmnd ringidcl eqidd + wb gsumsn opeq2d sneqd eqtr3d ) AUAEZFAUBGZBFAUCGZHIZACFUDIZHIZCJZAUEIZFU + FIZUGIZAULIZDFDJZXEIZXJBJZGZKZLGZAUHIZGZKZLGZKZBFMZXSKZFAUIIZUJZMZWTXTNWS + DXAXBWTXDAXGXPXIBFXFCWTOXAOXBOXDOZXFOZXGOZXPOZXIOZUKPWSBXBYAXSAUAUMUNWSFA + CXDXHXIDFXKXJFGZKZLGZXPGZKZLGZUJZMZYBYEWSFQEZYPQEYRYBNYSWSUOPZAYOLUPBFXSY + PQQXLFNZXRYOALUUACXDXQYNUUAXOYMXHXPUUAXNYLXILUUADFXMYKXKXJXLFUQRSSRSURUSW + SYQYDWSYPYCFWSYPACXDXHXIUTIZXPGZKZLGACXDXHKZLGZYCWSYOUUDALWSCXDYNUUCWSYMU + UBXHXPWSYMXIFLGUUBWSYLFXILYLFNWSDYKVAPSXIUUBUUBOVDVBSRSWSUUDUUEALWSCXDUUC + XHWSXEXDEZVCZUUCXHYCXPGZXHUUHUUBYCXHXPUUBYCNUUHYCUUBAYCXIYJYCOZVEVFPSWSUU + GXHAHIZEZUUIXHNWSFVGEZUUGUULVMXDXEAXGFXFYFYHYGVHVIUUKAXPYCXHUUKOZYIUUJVJV + KVNVLSWSUUFACXDYCKZLGACYAYCKZLGZYCWSUUEUUOALWSCXDXHYCUUHWSUUMXEFVOIEZXHYC + NWSUUGVPUUMUUHVMPZUUHUURUUGXEXGIZVQNZWSUUGVRUUGUVAWSUVAXEYAXDXEYAEXEFNZUV + AXEFVSUVBUUTFXGIVQXEFXGWEVTVBWAWBWCWDUUHUUMUURUUGUVAVCWNUUSFXDXCXEXGXCOYF + YHWFWAWGAXGYCXEFXFYGYHUUJWHWIVLSWSUUOUUPALWSCXDYAYCXDYANWSWBPUNSWSAWJEYSY + CUUKEUUQYCNAWKYTUUKAYCUUNUUJWLYCUUKYCCAFQUUNUVBYCWMWOWITTWPWQWRT $. $( The determinant for 0-dimensional matrices is a one-to-one function from the singleton containing the empty set onto the singleton containing the @@ -259932,17 +260160,17 @@ determinant function is alternating (see ~ mdetralt ) and normalized 23-Jul-2019.) $) mdetf $p |- ( R e. CRing -> D : B --> K ) $= ( vm vp vc wcel cfv cv co wa syl eqid ccrg csymg cbs czrh cpsgn ccom cmgp - cmpt cgsu cmulr crg crngrng adantr rngcmn cfn cvv matrcl adantl symgbasfi - ccmn simpld ad2antrr cmhm wf zrhpsgnmhm syl2anc mgpbas mhmf ffvelrnda cxp - crngmgp cmap matbas2i ad3antlr elmapi symgbasf fovrnd ralrimiva gsummptcl - simpr rngcl syl3anc mdetfval fmptd ) DUANZKBDLFUBOZUCOZLPZDUDOZFUEOZUFZOZ - DUGOZMFMPZWHOZWNKPZQZUHUIQZDUJOZQZUHUIQECWEWPBNZRZELDWGWTJXBDUKNZDUTNWEXC - XADULZUMZDUNSXBFUONZWGUONXBXFDUPNZXAXFXGRWEABDFWPHIUQURVAZFWGWFWFTZWGTZUS - SXBWTENZLWGXBWHWGNZRZXCWLENWRENXKWEXCXAXLXDVBXBWGEWHWKXBWKWFWMVCQNZWGEWKV - DXBXCXFXNXEXHFDVEVFWGEWFWMWKXJEDWMWMTZJVGZVHSVIXMEMWMFWQXPWEWMUTNXAXLDWMX - OVKVBXBXFXLXHUMXMWQENMFXMWNFNZRZWOWNEFFWPXRWPEFFVJZVLQNZXSEWPVDXAXTWEXLXQ - ABDEWPFHJIVMVNWPEXSVOSXMFFWNWHXLFFWHVDXBFWGWHWFXIXJVPURVIXMXQVTVQVRVSEDWS - WLWRJWSTZWAWBVRVSMABCWGDWJWSWMKFWILGHIXJWITWJTYAXOWCWD $. + cmpt cgsu cmulr crg ccmn crngring adantr ringcmn cfn matrcl adantl simpld + cvv symgbasfi ad2antrr cmhm wf zrhpsgnmhm syl2anc mgpbas mhmf crngmgp cxp + ffvelrnda matbas2i ad3antlr elmapi symgbasf simpr fovrnd ralrimiva ringcl + cmap gsummptcl syl3anc mdetfval fmptd ) DUANZKBDLFUBOZUCOZLPZDUDOZFUEOZUF + ZOZDUGOZMFMPZWHOZWNKPZQZUHUIQZDUJOZQZUHUIQECWEWPBNZRZELDWGWTJXBDUKNZDULNW + EXCXADUMZUNZDUOSXBFUPNZWGUPNXBXFDUTNZXAXFXGRWEABDFWPHIUQURUSZFWGWFWFTZWGT + ZVASXBWTENZLWGXBWHWGNZRZXCWLENWRENXKWEXCXAXLXDVBXBWGEWHWKXBWKWFWMVCQNZWGE + WKVDXBXCXFXNXEXHFDVEVFWGEWFWMWKXJEDWMWMTZJVGZVHSVKXMEMWMFWQXPWEWMULNXAXLD + WMXOVIVBXBXFXLXHUNXMWQENMFXMWNFNZRZWOWNEFFWPXRWPEFFVJZVTQNZXSEWPVDXAXTWEX + LXQABDEWPFHJIVLVMWPEXSVNSXMFFWNWHXLFFWHVDXBFWGWHWFXIXJVOURVKXMXQVPVQVRWAE + DWSWLWRJWSTZVSWBVRWAMABCWGDWJWSWMKFWILGHIXJWITWJTYAXOWCWD $. $( The determinant evaluates to an element of the base ring. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 7-Feb-2019.) $) @@ -259968,40 +260196,41 @@ determinant function is alternating (see ~ mdetralt ) and normalized cdif crab psgnfn c0 adantl rabeq difeq1 dmeqd eleq1d rabsnif restidsing cif cres cxp xpsng anidms syl5req fnsng fnnfpeq0 mpbird syl6eqel iftrue 0fin 3eqtrrd fneq2d mpbiri snid fvco2 fveq1d snidg mp1i eleqtrrd psgnsn - ancli crg crngrng 3ad2ant1 zrh1 3eqtrd simp2l cmnd rngmgp eleq2i biimpi - wb eleq2 simpl 3jca syl2an 3adant1 matecl mgpbas syl2anc gsumsn syl3anc - simpr syl6eleq eqvisset fvsng id rnglidm opeq2d sneqd eqidd rngmnd ) DU - DMZGEUEZNZEHMZUIZFBMZUFZFCOZDUAGUGOZUHOZUAVAZDUJOZGUKOZULZOZDUMOZLGLVAZ - UUTOZUVFFPZVBZQPZDUNOZPZVBZQPZDUBEEUOZUEZUEZEEFPZVBZQPZUVRUUOUUJUUQUVNN - UUNLABCUUSDUVBUVKUVEFGUVAUAIJKUUSRZUVARZUVBRZUVKRZUVERZUPUQUUPUVMUVSDQU - UPUVMUAUVQUVLVBZUVPUVPUVCOZUVELGUVFUVPOZUVFFPZVBZQPZUVKPZUOZUEZUVSUUPUA - UUSUVQUVLUUPUUSUUKUGOZUHOZUVQUUNUUJUUSUWPNZUUOUULUWQUUMUULUURUWOUHGUUKU - GURUSUTZVCUUPUUMUWPUVQNZUUJUULUUMUUOVDZUUKUWPUWOEHUWORZUWPRZUUKRZVEZSVF - 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eqtrd mdetdiaglem iffalse pm2.61dan mpteq2dva oveq2d cvv cmnd crg crngrng - df-ne rngmnd 3ad2ant1 adantr fvex a1i c0g symgid 3ad2ant2 symggrp grpidcl - cgrp eqeltrd mgpbas ccmn crngmgp simpl2 eleq2i biimpi ralrimiva gsummptcl - 3ad2ant3 matecl gsummptif1n0 3eqtrd ex ) DUBQZJUCQZIBQZUDZEUEZFUEZUFXQXRI - RKSUGFJUHEJUHZICTZHGJGUEZYAIRZUIUJRZSXPXSUKZXTDUAJULTZUNTZUAUEZDUMTZJUOTZ - UPTHGJYAYGTYAIRUIUJRDUQTZRZUIZUJRZDUAYFYGURJUSZSZYCKUTZUIZUJRYCYDXOXTYMSX - MXNXOXSVAZGABCYFDYIYJHIJYHUALMNYFVBZYHVBZYIVBZYJVBZOVCVDYDYLYQDUJYDUAYFYK - YPYDYGYFQZUKZYOYKYPSUUDYOUKZYKYCYPUUEXMXOYOYKYCSYDXMUUCYOXMXNXOXSVGVEYDXO - UUCYOYRVEUUDYOVFABYGDYIYJHIJYHGMNOYTUUAUUBVHVIYOYCYPSUUDYOYPYCYOYCKVJVKVL - VQUUDYOVMZUKZYKKYPUUGXPXSUUCYGYNUFZUKYKKSXPXSUUCUUFVNYDXSUUCUUFXPXSVFVEUU - DUUCUUFUUHYDUUCVFUUHUUFYGYNWGVOVPABCYGDYIYJEFGHYFIJKYHLMNOPYSYTUUAUUBVRVI - UUGYPKUUFYPKSUUDYOYCKVSVLVKVQVTWAWBYDYCUAYQDYFWCYNKPXPDWDQZXSXMXNUUIXOXMD - WEQUUIDWFDWHVDWIWJYFWCQYDYEUNWKWLXPYNYFQXSXPYNYEWMTZYFXNXMYNUUJSXOJYEUCYE - VBZWNWOXPYEWRQZUUJYFQXNXMUULXOJYEUCUUKWPWOYFYEUUJYSUUJVBWQVDWSWJYQVBYDDUN - 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CRing /\ N e. Fin ) -> ( D ` I ) = .1. ) $= ( vi vj wcel wa cfv co cbs wceq eqid adantr ccrg cfn chash cmg cv weq c0g - cmgp cif wral id crg crngrng anim1i ancomd matrng rngidcl 3syl syl simplr - simprl simprr mat1ov ralrimivva mdetdiagid sylc rngsrg hashcl srg1expzeq1 - jca32 csrg cn0 syl2an eqtrd ) CUAMZFUBMZNZEBOZFUCOZDCUHOZUDOZPZDVQVQEAQOZ - MZDCQOZMZNNKUEZLUEZEPKLUFDCUGOZUIRZLFUJKFUJVRWBRVQVQWDWFVQUKVQVPCULMZNAUL - MWDVQWKVPVOWKVPCUMZUNUOACFHUPWCAEWCSZIUQURVOWFVPVOWKWFWLWECDWESZJUQUSTVJV - QWJKLFFVQWGFMZWHFMZNZNACEDWGWHFWIHJWISZVOVPWQUTVQWKWQVOWKVPWLTTVQWOWPVAVQ - WOWPVBIVCVDAWCWEBCWAKLVTEFDWIGHWMVTSZWRWNWASZVEVFVOCVKMZVSVLMWBDRVPVOWKXA - WLCVGUSFVHCWADVTVSWSWTJVIVMVN $. + cmgp cif wral crg crngring anim1i ancomd matring ringidcl 3syl syl simplr + id jca32 simprl simprr mat1ov ralrimivva mdetdiagid sylc csrg cn0 ringsrg + hashcl srg1expzeq1 syl2an eqtrd ) CUAMZFUBMZNZEBOZFUCOZDCUHOZUDOZPZDVQVQE + AQOZMZDCQOZMZNNKUEZLUEZEPKLUFDCUGOZUIRZLFUJKFUJVRWBRVQVQWDWFVQUTVQVPCUKMZ + 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UUSUURUXMUXNUULVIZTUWDUYSVWEVWHUVOHUQVUCVWGVWIHEFUVDUVNPQUUMYPAUEUUSUVPWB + WBUVOVWKUVPVLVWLUVOWBUQUWDUVDUVNFUUNYKVWKWBUQAEYTUUOYKUUPUUQXFAVUAUYDUVSU + VLVERSUFBCDUUSEUVBFUVEJIUVAUEMNOUXNUVAVLZUVBVLZQVUEYSVQAUVTUVQKFAVUAUYMUV + TUVQVERUAUFBCDUUSEUVBFUVELIUVAUEMNOUXNVWMVWNQVUEYSVQXEYR $. $} ${ @@ -260324,23 +260553,23 @@ determinant function is alternating (see ~ mdetralt ) and normalized mdetrsca2 $p |- ( ph -> ( D ` ( i e. N , j e. N |-> if ( i = I , ( F .x. X ) , Y ) ) ) = ( F .x. ( D ` ( i e. N , j e. N |-> if ( i = I , X , Y ) ) ) ) ) $= - ( cmat co cbs cfv cv wceq cif eqid ccrg wcel w3a crg crngrng syl 3ad2ant1 - cmpt2 rngcl syl3anc ifcld matbas2d csn cxp cof cres cvv snex snssd sselda - cfn a1i 3adant3 syld3an2 fconstmpt2 eqidd offval22 wa elsni adantr iftrue - mpt2eq3ia oveq2i 3eqtr4g wss ssid resmpt2 sylancl oveq2d 3eqtr4rd cdif wn - eldifsni 3ad2ant2 neneqd iffalse eqtr4d mpt2eq3dva difss mp2an mdetrsca - wne ) AJCUBUCZXBUDUEZBCDHIJEFJJEUFZHUGZGKDUCZLUHZUQZGEFJJXEKLUHZUQZMXBUIZ - XCUIZNOPAEFXBXCXGCIJUJXKNXLQPAXDJUKZFUFJUKZULZXEXFLIXOCUMUKZGIUKZKIUKZXFI - UKAXMXPXNACUJUKXPPCUNUOUPAXMXQXNTUPRICDGKNOURUSSUTVATAEFXBXCXICIJUJXKNXLQ - PXOXEKLIRSUTVAUAAHVBZJVCZGVBVCZEFXSJXIUQZDVDZUCZEFXSJXGUQZYAXJXTVEZYCUCXH - XTVEZAYAEFXSJKUQZYCUCEFXSJXFUQYDYEAEFXSJGKDYAYHVFVJIIXSVFUKAHVGVKQAXDXSUK - ZXQXNTUPAXMYIXNXRAYIXMXNAXSJXDAHJUAVHZVIVLRVMYAEFXSJGUQUGAEFXSJGVNVKAYHVO - VPYBYHYAYCEFXSJXIKYIXNVQZXEXIKUGYIXEXNXDHVRVSZXEKLVTUOWAWBEFXSJXGXFYKXEXG - XFUGYLXEXFLVTUOWAWCAYFYBYAYCAXSJWDZJJWDZYFYBUGYJJWEZEFJJXSJXIWFWGWHAYMYNY - GYEUGYJYOEFJJXSJXGWFWGWIAEFJXSWJZJXGUQZEFYPJXIUQZXHYPJVCZVEZXJYSVEZAEFYPJ - XGXIAXDYPUKZXNULZXEWKZXGXIUGUUCXDHUUBAXDHXAXNXDJHWLWMWNUUDXGLXIXEXFLWOXEK - LWOWPUOWQYPJWDZYNYTYQUGJXSWRZYOEFJJYPJXGWFWSUUEYNUUAYRUGUUFYOEFJJYPJXIWFW - SWCWT $. + ( cmat co cbs cfv wceq cif cmpt2 eqid ccrg wcel w3a crg crngring 3ad2ant1 + syl ringcl syl3anc ifcld matbas2d csn cxp cof cres cvv cfn snex a1i snssd + sselda 3adant3 syld3an2 fconstmpt2 eqidd offval22 adantr iftrue mpt2eq3ia + cv elsni oveq2i 3eqtr4g wss ssid resmpt2 sylancl oveq2d 3eqtr4rd cdif wne + wa wn eldifsni 3ad2ant2 neneqd iffalse eqtr4d mpt2eq3dva difss mdetrsca + mp2an ) AJCUBUCZXBUDUEZBCDHIJEFJJEVSZHUFZGKDUCZLUGZUHZGEFJJXEKLUGZUHZMXBU + IZXCUIZNOPAEFXBXCXGCIJUJXKNXLQPAXDJUKZFVSJUKZULZXEXFLIXOCUMUKZGIUKZKIUKZX + FIUKAXMXPXNACUJUKXPPCUNUPUOAXMXQXNTUORICDGKNOUQURSUSUTTAEFXBXCXICIJUJXKNX + LQPXOXEKLIRSUSUTUAAHVAZJVBZGVAVBZEFXSJXIUHZDVCZUCZEFXSJXGUHZYAXJXTVDZYCUC + XHXTVDZAYAEFXSJKUHZYCUCEFXSJXFUHYDYEAEFXSJGKDYAYHVEVFIIXSVEUKAHVGVHQAXDXS + UKZXQXNTUOAXMYIXNXRAYIXMXNAXSJXDAHJUAVIZVJVKRVLYAEFXSJGUHUFAEFXSJGVMVHAYH + VNVOYBYHYAYCEFXSJXIKYIXNWKZXEXIKUFYIXEXNXDHVTVPZXEKLVQUPVRWAEFXSJXGXFYKXE + XGXFUFYLXEXFLVQUPVRWBAYFYBYAYCAXSJWCZJJWCZYFYBUFYJJWDZEFJJXSJXIWEWFWGAYMY + NYGYEUFYJYOEFJJXSJXGWEWFWHAEFJXSWIZJXGUHZEFYPJXIUHZXHYPJVBZVDZXJYSVDZAEFY + PJXGXIAXDYPUKZXNULZXEWLZXGXIUFUUCXDHUUBAXDHWJXNXDJHWMWNWOUUDXGLXIXEXFLWPX + EKLWPWQUPWRYPJWCZYNYTYQUFJXSWSZYOEFJJYPJXGWEXAUUEYNUUAYRUFUUFYOEFJJYPJXIW + EXAWBWT $. $} ${ @@ -260357,13 +260586,13 @@ determinant function is alternating (see ~ mdetralt ) and normalized (Contributed by SO, 16-Jul-2018.) $) mdetr0 $p |- ( ph -> ( D ` ( i e. N , j e. N |-> if ( i = I , .0. , X ) ) ) = .0. ) $= - ( wceq cfv wcel cv cmulr co cif eqid crg ccrg crngrng syl rng0cl 3ad2ant1 - cmpt2 mdetrsca2 rnglz syl2anc ifeq1d mpt2eq3dv fveq2d cmat wf mdetf ifcld - cbs w3a matbas2d ffvelrnd 3eqtr3d ) ADEHHDUAZFRZJJCUBSZUCZIUDZULZBSJDEHHV - IJIUDZULZBSZVJUCZVPJABCVJDEJFGHJIKLVJUEZNOAVHHTZJGTZEUAHTZACUFTZVTACUGTZW - BNCUHUIZGCJLMUJUIZUKZPWEQUMAVMVOBADEHHVLVNAVIVKJIAWBVTVKJRWDWEGCVJJJLVRMU - NUOUPUQURAWBVPGTVQJRWDAHCUSUCZVCSZGVOBAWCWHGBUTNWGWHBCGHKWGUEZWHUEZLVAUIA - DEWGWHVNCGHUGWILWJONAVSWAVDVIJIGWFPVBVEVFGCVJVPJLVRMUNUOVG $. + ( wceq cfv wcel cv cmulr co cif cmpt2 eqid crg ccrg crngring syl 3ad2ant1 + ring0cl mdetrsca2 ringlz syl2anc ifeq1d mpt2eq3dv fveq2d cbs wf mdetf w3a + cmat ifcld matbas2d ffvelrnd 3eqtr3d ) ADEHHDUAZFRZJJCUBSZUCZIUDZUEZBSJDE + HHVIJIUDZUEZBSZVJUCZVPJABCVJDEJFGHJIKLVJUFZNOAVHHTZJGTZEUAHTZACUGTZVTACUH + TZWBNCUIUJZGCJLMULUJZUKZPWEQUMAVMVOBADEHHVLVNAVIVKJIAWBVTVKJRWDWEGCVJJJLV + RMUNUOUPUQURAWBVPGTVQJRWDAHCVCUCZUSSZGVOBAWCWHGBUTNWGWHBCGHKWGUFZWHUFZLVA + UJADEWGWHVNCGHUHWILWJONAVSWAVBVIJIGWFPVDVEVFGCVJVPJLVRMUNUOVG $. $} ${ @@ -260378,14 +260607,14 @@ determinant function is alternating (see ~ mdetralt ) and normalized mdet0 $p |- ( ( R e. CRing /\ N e. Fin /\ N =/= (/) ) -> ( D ` Z ) = .0. ) $= ( vi vx vy wcel cfv wceq cv adantr cfn c0 wne wex wa n0 cmpt2 weq cif crg - ccrg crngrng anim1i ancomd c0g mat0op syl5eq fveq2d ifid eqcomi mpt2eq3dv - syl a1i cbs eqid simpll simpr cmnd rngmnd mndidcl 3ad2ant1 mdetr0 exlimdv - 3eqtrd ex syl5bi 3impia ) DUKPZEUAPZEUBUCZGCQZFRZVTMSZEPZMUDVRVSUEZWBMEUF - WEWDWBMWEWDWBWEWDUEZWANOEEFUGZCQNOEENMUHZFFUIZUGZCQFWFGWGCWFVSDUJPZUEZGWG - RWEWLWDWEWKVSVRWKVSDULZUMUNTWLGAUOQWGKADNOEFILUPUQVBURWFWGWJCWFNOEEFWIFWI - RWFWIFWHFUSUTVCVAURWFCDNOWCDVDQZEFFHWNVEZLVRVSWDVFWEVSWDVRVSVGTWFNSEPFWNP - ZOSEPWEWPWDWEDVHPZWPVRWQVSVRWKWQWMDVIVBTWNDFWOLVJVBTVKWEWDVGVLVNVOVMVPVQ - $. + ccrg crngring anim1i ancomd c0g mat0op syl5eq syl fveq2d eqcomi mpt2eq3dv + ifid a1i eqid simpll simpr cmnd ringmnd mndidcl 3ad2ant1 mdetr0 3eqtrd ex + cbs exlimdv syl5bi 3impia ) DUKPZEUAPZEUBUCZGCQZFRZVTMSZEPZMUDVRVSUEZWBME + UFWEWDWBMWEWDWBWEWDUEZWANOEEFUGZCQNOEENMUHZFFUIZUGZCQFWFGWGCWFVSDUJPZUEZG + WGRWEWLWDWEWKVSVRWKVSDULZUMUNTWLGAUOQWGKADNOEFILUPUQURUSWFWGWJCWFNOEEFWIF + WIRWFWIFWHFVBUTVCVAUSWFCDNOWCDVNQZEFFHWNVDZLVRVSWDVEWEVSWDVRVSVFTWFNSEPFW + NPZOSEPWEWPWDWEDVGPZWPVRWQVSVRWKWQWMDVHURTWNDFWOLVIURTVJWEWDVFVKVLVMVOVPV + Q $. $} ${ @@ -260405,26 +260634,26 @@ determinant function is alternating (see ~ mdetralt ) and normalized if ( i = I , ( X .+ Y ) , Z ) ) ) = ( ( D ` ( i e. N , j e. N |-> if ( i = I , X , Z ) ) ) .+ ( D ` ( i e. N , j e. N |-> if ( i = I , Y , Z ) ) ) ) ) $= - ( cmat co cbs cfv cv wceq cif eqid ccrg wcel w3a crg crngrng syl 3ad2ant1 - cmpt2 rngacl syl3anc ifcld matbas2d csn cof cxp cres cvv cfn snex a1i wss - snssd simp2 sseldd syld3an2 eqidd offval22 eqcomd adantr iftrue mpt2eq3ia - wa elsni oveq12i 3eqtr4g ssid resmpt2 sylancl oveq12d 3eqtr4d wn eldifsni - cdif neneqd iffalse eqtr4d 3ad2ant2 mpt2eq3dva difss mp2an mdetrlin ) AID - UBUCZXAUDUEZBCDGIEFIIEUFZGUGZJKCUCZLUHZUQZEFIIXDJLUHZUQZEFIIXDKLUHZUQZMXA - UIZXBUIZOPAEFXAXBXFDHIUJXLNXMQPAXCIUKZFUFIUKZULZXDXELHXPDUMUKZJHUKZKHUKZX - EHUKAXNXQXOADUJUKXQPDUNUOUPRSHCDJKNOURUSTUTVAAEFXAXBXHDHIUJXLNXMQPXPXDJLH - RTUTVAAEFXAXBXJDHIUJXLNXMQPXPXDKLHSTUTVAUAAEFGVBZIXFUQZEFXTIXHUQZEFXTIXJU - QZCVCZUCZXGXTIVDZVEZXIYFVEZXKYFVEZYDUCAEFXTIXEUQZEFXTIJUQZEFXTIKUQZYDUCZY - AYEAYMYJAEFXTIJKCYKYLVFVGHHXTVFUKAGVHVIQAXNXCXTUKZXOXRAYNXOULXTIXCAYNXTIV - JZXOAGIUAVKZUPAYNXOVLVMZRVNAXNYNXOXSYQSVNAYKVOAYLVOVPVQEFXTIXFXEYNXOWAZXD - XFXEUGYNXDXOXCGWBVRZXDXELVSUOVTYBYKYCYLYDEFXTIXHJYRXDXHJUGYSXDJLVSUOVTEFX - TIXJKYRXDXJKUGYSXDKLVSUOVTWCWDAYOIIVJZYGYAUGYPIWEZEFIIXTIXFWFWGAYHYBYIYCY - DAYOYTYHYBUGYPUUAEFIIXTIXHWFWGAYOYTYIYCUGYPUUAEFIIXTIXJWFWGWHWIAEFIXTWLZI - XFUQZEFUUBIXHUQZXGUUBIVDZVEZXIUUEVEZAEFUUBIXFXHXCUUBUKZAXFXHUGZXOUUHXDWJZ - UUIUUHXCGXCIGWKWMZUUJXFLXHXDXELWNZXDJLWNWOUOWPWQUUBIVJZYTUUFUUCUGIXTWRZUU - AEFIIUUBIXFWFWSZUUMYTUUGUUDUGUUNUUAEFIIUUBIXHWFWSWDAUUCEFUUBIXJUQZUUFXKUU - EVEZAEFUUBIXFXJUUHAXFXJUGZXOUUHUUJUURUUKUUJXFLXJUULXDKLWNWOUOWPWQUUOUUMYT - UUQUUPUGUUNUUAEFIIUUBIXJWFWSWDWT $. + ( cmat co cbs cfv wceq cif cmpt2 eqid ccrg wcel w3a crg crngring 3ad2ant1 + cv syl ringacl syl3anc ifcld matbas2d csn cof cxp cres cvv cfn snex snssd + a1i simp2 sseldd syld3an2 eqidd offval22 eqcomd wa elsni adantr mpt2eq3ia + iftrue oveq12i 3eqtr4g ssid resmpt2 sylancl oveq12d 3eqtr4d cdif eldifsni + wss wn neneqd iffalse eqtr4d 3ad2ant2 mpt2eq3dva difss mp2an mdetrlin ) A + IDUBUCZXAUDUEZBCDGIEFIIEUPZGUFZJKCUCZLUGZUHZEFIIXDJLUGZUHZEFIIXDKLUGZUHZM + XAUIZXBUIZOPAEFXAXBXFDHIUJXLNXMQPAXCIUKZFUPIUKZULZXDXELHXPDUMUKZJHUKZKHUK + ZXEHUKAXNXQXOADUJUKXQPDUNUQUORSHCDJKNOURUSTUTVAAEFXAXBXHDHIUJXLNXMQPXPXDJ + LHRTUTVAAEFXAXBXJDHIUJXLNXMQPXPXDKLHSTUTVAUAAEFGVBZIXFUHZEFXTIXHUHZEFXTIX + JUHZCVCZUCZXGXTIVDZVEZXIYFVEZXKYFVEZYDUCAEFXTIXEUHZEFXTIJUHZEFXTIKUHZYDUC + ZYAYEAYMYJAEFXTIJKCYKYLVFVGHHXTVFUKAGVHVJQAXNXCXTUKZXOXRAYNXOULXTIXCAYNXT + IWKZXOAGIUAVIZUOAYNXOVKVLZRVMAXNYNXOXSYQSVMAYKVNAYLVNVOVPEFXTIXFXEYNXOVQZ + XDXFXEUFYNXDXOXCGVRVSZXDXELWAUQVTYBYKYCYLYDEFXTIXHJYRXDXHJUFYSXDJLWAUQVTE + FXTIXJKYRXDXJKUFYSXDKLWAUQVTWBWCAYOIIWKZYGYAUFYPIWDZEFIIXTIXFWEWFAYHYBYIY + CYDAYOYTYHYBUFYPUUAEFIIXTIXHWEWFAYOYTYIYCUFYPUUAEFIIXTIXJWEWFWGWHAEFIXTWI + ZIXFUHZEFUUBIXHUHZXGUUBIVDZVEZXIUUEVEZAEFUUBIXFXHXCUUBUKZAXFXHUFZXOUUHXDW + LZUUIUUHXCGXCIGWJWMZUUJXFLXHXDXELWNZXDJLWNWOUQWPWQUUBIWKZYTUUFUUCUFIXTWRZ + UUAEFIIUUBIXFWEWSZUUMYTUUGUUDUFUUNUUAEFIIUUBIXHWEWSWCAUUCEFUUBIXJUHZUUFXK + UUEVEZAEFUUBIXFXJUUHAXFXJUFZXOUUHUUJUURUUKUUJXFLXJUULXDKLWNWOUQWPWQUUOUUM + YTUUQUUPUFUUNUUAEFIIUUBIXJWEWSWCWT $. $} ${ @@ -260446,85 +260675,85 @@ determinant function is alternating (see ~ mdetralt ) and normalized 23-Jul-2018.) $) mdetralt $p |- ( ph -> ( D ` X ) = .0. ) $= ( vp vc vq cfv csymg cbs cv czrh cpsgn ccom cmgp co cmpt cgsu cmulr cevpm - cres cdif cplusg wcel wceq eqid mdetleib syl crg ccmn ccrg crngrng rngcmn - cfn cvv wa matrcl simpld symgbasfi adantr cmhm wf zrhpsgnmhm syl2anc mhmf - mgpbas ffvelrnda cxp cmap matbas2i elmapi ad2antrr wf1o symgbasf1o adantl - crngmgp simpr fovrnd ralrimiva gsummptcl rngcl syl3anc cin c0 disjdif a1i - f1of resmpt ax-mp oveq1d sylan2 eqtrd mpteq2dva syl5eq oveq2d eqidd fveq2 - wss cgrp fmptco eqtr4d ssfi cpr wne wi oveq2 c1 cmul ccnfld sylan oveq12d - weq mpteq2dv fveq1d pmtrprfv syl13anc oveq1 syl5ibrcom 3ad2ant1 pm2.61dne - eqeq12d wn 3eqtrd cun evpmss undif eqcomi gsummptfidmsplitres cminusg cur - zrhpsgnevpm sseli rnglidm difss zrhpsgnodpm eldifi rngnegl rnggrp grpinvf - mpbi feqmptd cabl rngabl difssd gsummptfidminv cpmtr crn c2o cen pr2nelem - prssi pmtrrn pmtrodpm evpmodpmf1o gsummptfif1o anbi2d eleq1d imbi12d cneg - wbr eleq1 symggrp sseldi grpcl cress cghm psgnghm2 prex cnfldmul mgpplusg - symgtrf ressplusg ghmlin psgnpmtr psgnevpm neg1cn syl6eq psgnodpmr chvarv - mulid1i fveq1 symgov fvco3 wral rspcv sylc prcom fveq2i fveq1i necomd w3a - cbvmptv a1dd cid cdm neanior elpri orcomd con3i 3adant1 pmtrmvd neleqtrrd - wo sylbi wb wfn pmtrf ffn fnelnfp necon2bbid mpbird 3exp fveq2d grprinv ) - AIDUEZEUBHUFUEZUGUEZUBUHZEUIUEZHUJUEZUKZUEZEULUEZUCHUCUHZUYOUEZVUAIUMZUNZ - UOUMZEUPUEZUMZUNZUOUMZEVUHHUQUEZURZUOUMZEVUHUYNVUJUSZURZUOUMZEUTUEZUMZJAI - CVAZUYLVUIVBQUCBCDUYNEUYQVUFUYTIHUYPUBLMNUYNVCZUYPVCZUYQVCZVUFVCZUYTVCZVD - VEAUYNEUGUEZVUJVUMVUPUBVUHEVUGVVDVCZVUPVCZAEVFVAZEVGVAAEVHVAZVVGPEVIVEZEV - JVEZAHVKVAZUYNVKVAZAVVKEVLVAZAVURVVKVVMVMQBCEHIMNVNVEVOZHUYNUYMUYMVCZVUSV - PVEZAUYOUYNVAZVMZVVGUYSVVDVAVUEVVDVAZVUGVVDVAAVVGVVQVVIVQAUYNVVDUYOUYRAUY - RUYMUYTVRUMVAZUYNVVDUYRVSAVVGVVKVVTVVIVVNHEVTWAUYNVVDUYMUYTUYRVUSVVDEUYTV - VCVVEWCZWBVEWDVVRVVDUCUYTHVUCVWAAUYTVGVAZVVQAVVHVWBPEUYTVVCWMVEVQAVVKVVQV - VNVQVVRVUCVVDVAUCHVVRVUAHVAZVMVUBVUAVVDHHIAHHWEZVVDIVSZVVQVWCAIVVDVWDWFUM - VAZVWEAVURVWFQBCEVVDIHMVVENWGVEIVVDVWDWHVEWIVVRHHVUAUYOVVRHHUYOWJZHHUYOVS - ZVVQVWGAHUYNUYOUYMVVOVUSWKWLHHUYOXDVEZWDVVRVWCWNWOWPWQZVVDEVUFUYSVUEVVEVV - BWRWSVUJVUMWTXAVBAVUJUYNXBXCUYNVUJVUMUUAZVBAVWKUYNVUJUYNXOZVWKUYNVBHUYNUY - MVVOVUSUUBZVUJUYNUUCUUQUUDXCVUHVCUUEAVUQEUBVUJVUEUNZUOUMZVWOEUUFUEZUEZVUP - UMZJAVULVWOVUOVWQVUPAVUKVWNEUOAVUKUBVUJVUGUNZVWNVWLVUKVWSVBVWMUBUYNVUJVUG - XEXFAUBVUJVUGVUEAUYOVUJVAZVMZVUGEUUGUEZVUEVUFUMZVUEVXAUYSVXBVUEVUFVXAVVGV - VKVWTUYSVXBVBAVVGVWTVVIVQZAVVKVWTVVNVQZAVWTWNEUYQVXBUYOHUYPVUTVVAVXBVCZUU - HWSXGVXAVVGVVSVXCVUEVBVXDVWTAVVQVVSVUJUYNUYOVWMUUIZVWJXHZVVDEVUFVXBVUEVVE - VVBVXFUUJWAXIXJXKXLAVUOEVWPUBVUMVUEUNZUKZUOUMEVXIUOUMZVWPUEVWQAVUNVXJEUOA - 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Corollary if ( i = I , ( X .+ ( W .x. Y ) ) , if ( i = J , Y , Z ) ) ) ) = ( D ` ( i e. N , j e. N |-> if ( i = I , X , if ( i = J , Y , Z ) ) ) ) ) $= - ( cv wceq co cif cmpt2 cfv c0g wcel 3adant2 w3a crg ccrg crngrng 3ad2ant1 - syl rngcl syl3anc ifcld mdetrlin2 mdetrsca2 eqid mdetralt2 oveq2d syl2anc - rngrz 3eqtrd cgrp rnggrp cmat cbs wf mdetf matbas2d ffvelrnd grprid ) AFG - KKFUIZHUJZMLNEUKZCUKWDIUJZNOULZULUMBUNFGKKWEMWHULZUMZBUNZFGKKWEWFWHULUMBU - NZCUKWKDUOUNZCUKZWKABCDFGHJKMWFWHPQRTUAAGUIKUPZMJUPWDKUPZUBUQZAWPWOURZDUS - UPZLJUPZNJUPZWFJUPAWPWSWOADUTUPZWSTDVAVCZVBAWPWTWOUEVBAWOXAWPUCUQZJDELNQS - VDVEWRWGNOJXDUDVFZUFVGAWLWMWKCAWLLFGKKWENWHULUMBUNZEUKLWMEUKZWMABDEFGLHJK - NWHPQSTUAXDXEUEUFVHAXFWMLEABDFGHIJKNOWMPQWMVIZTUAUCUDUFUGUHVJVKAWSWTXGWMU - JXCUEJDELWMQSXHVMVLVNVKADVOUPZWKJUPWNWKUJAWSXIXCDVPVCAKDVQUKZVRUNZJWJBAXB - XKJBVSTXJXKBDJKPXJVIZXKVIZQVTVCAFGXJXKWIDJKUTXLQXMUATWRWEMWHJWQXEVFWAWBJC - DWKWMQRXHWCVLVN $. + ( cv wceq co cif cmpt2 cfv c0g wcel 3adant2 w3a crg crngring syl 3ad2ant1 + ccrg ringcl ifcld mdetrlin2 mdetrsca2 eqid mdetralt2 oveq2d ringrz 3eqtrd + syl3anc syl2anc cgrp ringgrp cmat cbs wf mdetf matbas2d ffvelrnd grprid ) + AFGKKFUIZHUJZMLNEUKZCUKWDIUJZNOULZULUMBUNFGKKWEMWHULZUMZBUNZFGKKWEWFWHULU + MBUNZCUKWKDUOUNZCUKZWKABCDFGHJKMWFWHPQRTUAAGUIKUPZMJUPWDKUPZUBUQZAWPWOURZ + DUSUPZLJUPZNJUPZWFJUPAWPWSWOADVCUPZWSTDUTVAZVBAWPWTWOUEVBAWOXAWPUCUQZJDEL + NQSVDVMWRWGNOJXDUDVEZUFVFAWLWMWKCAWLLFGKKWENWHULUMBUNZEUKLWMEUKZWMABDEFGL + HJKNWHPQSTUAXDXEUEUFVGAXFWMLEABDFGHIJKNOWMPQWMVHZTUAUCUDUFUGUHVIVJAWSWTXG + WMUJXCUEJDELWMQSXHVKVNVLVJADVOUPZWKJUPWNWKUJAWSXIXCDVPVAAKDVQUKZVRUNZJWJB + AXBXKJBVSTXJXKBDJKPXJVHZXKVHZQVTVAAFGXJXKWIDJKVCXLQXMUATWRWEMWHJWQXEVEWAW + BJCDWKWMQRXHWCVNVL $. $( [16-Jul-2018] $) $} @@ -260755,29 +260984,29 @@ each row (matrices are given explicitly by their entries). Corollary ( ( D ` ( a e. N , b e. N |-> if ( a = E , F , H ) ) ) .+ ( D ` ( a e. N , b e. N |-> if ( a = E , G , H ) ) ) ) ) $= ( cv wceq cif cmpt2 wcel csn cxp cres cof cdif cfv crg cfn syl 3ad2ant1 - w3a simp1d simp2d rngacl syl3anc simp3d ifcld matbas2d cvv snex a1i wss - co snssd simp2 sseldd syld3an2 eqidd offval22 eqcomd wa elsni mpt2eq3ia - adantr oveq12i 3eqtr4g ssid resmpt2 sylancl oveq12d 3eqtr4d wn eldifsni - iftrue 3ad2ant2 df-ne sylib iffalse eqtr4d mpt2eq3dva difss mdetunilem3 - wne mp2an syl332anc ) BAUAUBSSUAUSZNUTZOPJWFZQVAZVBZHVCUAUBSSXTOQVAZVBZ - HVCUAUBSSXTPQVAZVBZHVCNSVCYCNVDZSVEZVFZYEYIVFZYGYIVFZJVGZWFZUTYCSYHVHZS - VEZVFZYEYPVFZUTYQYGYPVFZUTYCIVIYEIVIYGIVIJWFUTUPBUAUBGHYBKRSVJUCUEUDBAS - VKVCUPUJVLZBAKVJVCZUPUKVLZBXSSVCZUBUSSVCZVNZXTYAQRUUEUUAORVCZPRVCZYARVC - BUUCUUAUUDUUBVMUUEUUFUUGQRVCZURVOZUUEUUFUUGUUHURVPZRJKOPUEUHVQVRUUEUUFU - UGUUHURVSZVTWABUAUBGHYDKRSVJUCUEUDYTUUBUUEXTOQRUUIUUKVTWABUAUBGHYFKRSVJ - UCUEUDYTUUBUUEXTPQRUUJUUKVTWAUQBUAUBYHSYBVBZUAUBYHSYDVBZUAUBYHSYFVBZYMW - FZYJYNBUAUBYHSYAVBZUAUBYHSOVBZUAUBYHSPVBZYMWFZUULUUOBUUSUUPBUAUBYHSOPJU - UQUURWBVKRRYHWBVCBNWCWDYTBUUCXSYHVCZUUDUUFBUUTUUDVNYHSXSBUUTYHSWEZUUDBN - SUQWGZVMBUUTUUDWHWIZUUIWJBUUCUUTUUDUUGUVCUUJWJBUUQWKBUURWKWLWMUAUBYHSYB - YAUUTUUDWNZXTYBYAUTUUTXTUUDXSNWOWQZXTYAQXGVLWPUUMUUQUUNUURYMUAUBYHSYDOU - VDXTYDOUTUVEXTOQXGVLWPUAUBYHSYFPUVDXTYFPUTUVEXTPQXGVLWPWRWSBUVASSWEZYJU - ULUTUVBSWTZUAUBSSYHSYBXAXBBYKUUMYLUUNYMBUVAUVFYKUUMUTUVBUVGUAUBSSYHSYDX - AXBBUVAUVFYLUUNUTUVBUVGUAUBSSYHSYFXAXBXCXDBUAUBYOSYBVBZUAUBYOSYDVBZYQYR - BUAUBYOSYBYDBXSYOVCZUUDVNZXTXEZYBYDUTUVKXSNXPZUVLUVJBUVMUUDXSSNXFXHXSNX - IXJZUVLYBQYDXTYAQXKZXTOQXKXLVLXMYOSWEZUVFYQUVHUTSYHXNZUVGUAUBSSYOSYBXAX - QZUVPUVFYRUVIUTUVQUVGUAUBSSYOSYDXAXQWSBUVHUAUBYOSYFVBZYQYSBUAUBYOSYBYFU - VKUVLYBYFUTUVNUVLYBQYFUVOXTPQXKXLVLXMUVRUVPUVFYSUVSUTUVQUVGUAUBSSYOSYFX - AXQWSACDEFGHIJKLMYCYEYGNRSTUCUDUEUFUGUHUIUJUKULUMUNUOXOXR $. + co w3a simp1d simp2d ringacl syl3anc simp3d ifcld matbas2d cvv snex a1i + wss snssd simp2 sseldd syld3an2 offval22 eqcomd adantr iftrue mpt2eq3ia + eqidd wa elsni oveq12i 3eqtr4g ssid resmpt2 sylancl oveq12d 3eqtr4d wne + wn eldifsni 3ad2ant2 df-ne sylib iffalse eqtr4d difss mp2an mdetunilem3 + mpt2eq3dva syl332anc ) BAUAUBSSUAUSZNUTZOPJVNZQVAZVBZHVCUAUBSSXTOQVAZVB + ZHVCUAUBSSXTPQVAZVBZHVCNSVCYCNVDZSVEZVFZYEYIVFZYGYIVFZJVGZVNZUTYCSYHVHZ + SVEZVFZYEYPVFZUTYQYGYPVFZUTYCIVIYEIVIYGIVIJVNUTUPBUAUBGHYBKRSVJUCUEUDBA + SVKVCUPUJVLZBAKVJVCZUPUKVLZBXSSVCZUBUSSVCZVOZXTYAQRUUEUUAORVCZPRVCZYARV + CBUUCUUAUUDUUBVMUUEUUFUUGQRVCZURVPZUUEUUFUUGUUHURVQZRJKOPUEUHVRVSUUEUUF + UUGUUHURVTZWAWBBUAUBGHYDKRSVJUCUEUDYTUUBUUEXTOQRUUIUUKWAWBBUAUBGHYFKRSV + JUCUEUDYTUUBUUEXTPQRUUJUUKWAWBUQBUAUBYHSYBVBZUAUBYHSYDVBZUAUBYHSYFVBZYM + VNZYJYNBUAUBYHSYAVBZUAUBYHSOVBZUAUBYHSPVBZYMVNZUULUUOBUUSUUPBUAUBYHSOPJ + UUQUURWCVKRRYHWCVCBNWDWEYTBUUCXSYHVCZUUDUUFBUUTUUDVOYHSXSBUUTYHSWFZUUDB + NSUQWGZVMBUUTUUDWHWIZUUIWJBUUCUUTUUDUUGUVCUUJWJBUUQWPBUURWPWKWLUAUBYHSY + BYAUUTUUDWQZXTYBYAUTUUTXTUUDXSNWRWMZXTYAQWNVLWOUUMUUQUUNUURYMUAUBYHSYDO + UVDXTYDOUTUVEXTOQWNVLWOUAUBYHSYFPUVDXTYFPUTUVEXTPQWNVLWOWSWTBUVASSWFZYJ + UULUTUVBSXAZUAUBSSYHSYBXBXCBYKUUMYLUUNYMBUVAUVFYKUUMUTUVBUVGUAUBSSYHSYD + XBXCBUVAUVFYLUUNUTUVBUVGUAUBSSYHSYFXBXCXDXEBUAUBYOSYBVBZUAUBYOSYDVBZYQY + RBUAUBYOSYBYDBXSYOVCZUUDVOZXTXGZYBYDUTUVKXSNXFZUVLUVJBUVMUUDXSSNXHXIXSN + XJXKZUVLYBQYDXTYAQXLZXTOQXLXMVLXQYOSWFZUVFYQUVHUTSYHXNZUVGUAUBSSYOSYBXB + XOZUVPUVFYRUVIUTUVQUVGUAUBSSYOSYDXBXOWTBUVHUAUBYOSYFVBZYQYSBUAUBYOSYBYF + UVKUVLYBYFUTUVNUVLYBQYFUVOXTPQXLXMVLXQUVRUVPUVFYSUVSUTUVQUVGUAUBSSYOSYF + XBXOWTACDEFGHIJKLMYCYEYGNRSTUCUDUEUFUGUHUIUJUKULUMUNUOXPXR $. $} ${ @@ -260793,34 +261022,34 @@ each row (matrices are given explicitly by their entries). Corollary ( a e. N , b e. N |-> if ( a = E , H , if ( a = F , G , I ) ) ) ) ) ) $= ( cv wceq cif cmpt2 cfv cminusg co wcel wne simp1d w3a wa simprd simpld - 3adant2 cgrp crg rnggrp 3syl adantr grpcl ifcld mdetunilem5 mdetunilem2 - syl3anc 3jca simp2d simp3d necomd oveq1d neneqd eqtr2 3ad2ant1 ifcomnan - wn nsyl syl mpt2eq3dva fveq2d wf cfn matbas2d ffvelrnd eqeltrrd syl2anc - grplid 3eqtrd 3eqtr4d oveq2d grprid oveq12d 3eqtr3rd eqid mpbird eqcomd - wb grpinvid1 ) BUBUCTTUBVAZNVBZQXROVBZPRVCZVCZVDZIVEZKVFVEZVEZUBUCTTXSP - XTQRVCZVCZVDZIVEZBYFYJVBZYDYJJVGZUAVBZBUBUCTTXSQPJVGZXTYNRVCZVCVDIVEUBU - CTTXSQYOVCZVDZIVEZUBUCTTXSPYOVCZVDZIVEZJVGUAYLABCDEFGHIJKLMNQPYOSTUAUBU - CUDUEUFUGUHUIUJUKULUMUNUOUPUQBNTVHZOTVHZNOVIZURVJZBXRTVHZUCVATVHZVKZQSV - HZPSVHZYOSVHBUUGUUIUUFBUUGVLZUUJUUIUSVMZVOZBUUGUUJUUFUUKUUJUUIUSVNZVOZU - UHXTYNRSBUUGYNSVHZUUFUUKKVPVHZUUIUUJUUPBUUQUUGBAKVQVHUUQUQULKVRVSZVTUUL - UUNSJKQPUFUIWAWEZVOUTWBWFWCABCDEFGHIJKLMNYNORSTUAUBUCUDUEUFUGUHUIUJUKUL - UMUNUOUPUQURUUSUTWDBYRYDUUAYJJBUBUCTTXTYNXSQRVCZVCZVDZIVEZUBUCTTXTPUUTV - CZVDZIVEZYRYDBUVCUBUCTTXTQUUTVCVDIVEZUVFJVGUAUVFJVGZUVFABCDEFGHIJKLMOQP - UUTSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQBUUBUUCUUDURWGZUUHUUIUUJUUTSVHUU - MUUOUUHXSQRSUUMUTWBWFWCBUVGUAUVFJABCDEFGHIJKLMOQNRSTUAUBUCUDUEUFUGUHUIU - JUKULUMUNUOUPUQBUUCUUBONVIUVIUUEBNOBUUBUUCUUDURWHZWIWFZUULUTWDWJBUUQUVF - SVHUVHUVFVBUURBYDUVFSBYCUVEIBUBUCTTYBUVDUUHXSXTVLZWOZYBUVDVBBUUFUVMUUGB - NOVBUVLBNOUVJWKXRNOWLWPWMZXSXTQPRWNWQWRWSZBHSYCIBAHSIWTUQUMWQZBUBUCGHYB - KSTVPUDUFUEBATXAVHUQUKWQZUURUUHXSQYASUUMUUHXTPRSUUOUTWBWBXBXCZXDSJKUVFU - AUFUIUGXFXEXGBYQUVBIBUBUCTTYPUVAUUHUVMYPUVAVBUVNXSXTQYNRWNWQWRWSUVOXHBU - BUCTTXTYNXSPRVCZVCZVDZIVEZUBUCTTXTQUVSVCZVDZIVEZUUAYJBUWBUWEUBUCTTXTPUV - SVCVDIVEZJVGUWEUAJVGZUWEABCDEFGHIJKLMOQPUVSSTUAUBUCUDUEUFUGUHUIUJUKULUM - UNUOUPUQUVIUUHUUIUUJUVSSVHUUMUUOUUHXSPRSUUOUTWBWFWCBUWFUAUWEJABCDEFGHIJ - KLMOPNRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQUVKUUNUTWDXIBUUQUWESVHUWGUWE - VBUURBYJUWESBYIUWDIBUBUCTTYHUWCUUHUVMYHUWCVBUVNXSXTPQRWNWQWRWSZBHSYIIUV - PBUBUCGHYHKSTVPUDUFUEUVQUURUUHXSPYGSUUOUUHXTQRSUUMUTWBWBXBXCZXDSJKUWEUA - UFUIUGXJXEXGBYTUWAIBUBUCTTYSUVTUUHUVMYSUVTVBUVNXSXTPYNRWNWQWRWSUWHXHXKX - LBUUQYDSVHYJSVHYKYMXPUURUVRUWISJKYEYDYJUAUFUIUGYEXMXQWEXNXO $. + 3adant2 cgrp ringgrp adantr grpcl syl3anc ifcld mdetunilem5 mdetunilem2 + crg 3syl 3jca simp2d simp3d necomd oveq1d wn neneqd eqtr2 nsyl 3ad2ant1 + ifcomnan syl mpt2eq3dva fveq2d wf cfn matbas2d ffvelrnd eqeltrrd grplid + syl2anc 3eqtrd 3eqtr4d oveq2d grprid oveq12d 3eqtr3rd wb eqid grpinvid1 + mpbird eqcomd ) BUBUCTTUBVAZNVBZQXROVBZPRVCZVCZVDZIVEZKVFVEZVEZUBUCTTXS + PXTQRVCZVCZVDZIVEZBYFYJVBZYDYJJVGZUAVBZBUBUCTTXSQPJVGZXTYNRVCZVCVDIVEUB + UCTTXSQYOVCZVDZIVEZUBUCTTXSPYOVCZVDZIVEZJVGUAYLABCDEFGHIJKLMNQPYOSTUAUB + UCUDUEUFUGUHUIUJUKULUMUNUOUPUQBNTVHZOTVHZNOVIZURVJZBXRTVHZUCVATVHZVKZQS + VHZPSVHZYOSVHBUUGUUIUUFBUUGVLZUUJUUIUSVMZVOZBUUGUUJUUFUUKUUJUUIUSVNZVOZ + UUHXTYNRSBUUGYNSVHZUUFUUKKVPVHZUUIUUJUUPBUUQUUGBAKWDVHUUQUQULKVQWEZVRUU + LUUNSJKQPUFUIVSVTZVOUTWAWFWBABCDEFGHIJKLMNYNORSTUAUBUCUDUEUFUGUHUIUJUKU + LUMUNUOUPUQURUUSUTWCBYRYDUUAYJJBUBUCTTXTYNXSQRVCZVCZVDZIVEZUBUCTTXTPUUT + VCZVDZIVEZYRYDBUVCUBUCTTXTQUUTVCVDIVEZUVFJVGUAUVFJVGZUVFABCDEFGHIJKLMOQ + PUUTSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQBUUBUUCUUDURWGZUUHUUIUUJUUTSVHU + UMUUOUUHXSQRSUUMUTWAWFWBBUVGUAUVFJABCDEFGHIJKLMOQNRSTUAUBUCUDUEUFUGUHUI + UJUKULUMUNUOUPUQBUUCUUBONVIUVIUUEBNOBUUBUUCUUDURWHZWIWFZUULUTWCWJBUUQUV + FSVHUVHUVFVBUURBYDUVFSBYCUVEIBUBUCTTYBUVDUUHXSXTVLZWKZYBUVDVBBUUFUVMUUG + BNOVBUVLBNOUVJWLXRNOWMWNWOZXSXTQPRWPWQWRWSZBHSYCIBAHSIWTUQUMWQZBUBUCGHY + BKSTVPUDUFUEBATXAVHUQUKWQZUURUUHXSQYASUUMUUHXTPRSUUOUTWAWAXBXCZXDSJKUVF + UAUFUIUGXEXFXGBYQUVBIBUBUCTTYPUVAUUHUVMYPUVAVBUVNXSXTQYNRWPWQWRWSUVOXHB + UBUCTTXTYNXSPRVCZVCZVDZIVEZUBUCTTXTQUVSVCZVDZIVEZUUAYJBUWBUWEUBUCTTXTPU + VSVCVDIVEZJVGUWEUAJVGZUWEABCDEFGHIJKLMOQPUVSSTUAUBUCUDUEUFUGUHUIUJUKULU + MUNUOUPUQUVIUUHUUIUUJUVSSVHUUMUUOUUHXSPRSUUOUTWAWFWBBUWFUAUWEJABCDEFGHI + JKLMOPNRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQUVKUUNUTWCXIBUUQUWESVHUWGUW + EVBUURBYJUWESBYIUWDIBUBUCTTYHUWCUUHUVMYHUWCVBUVNXSXTPQRWPWQWRWSZBHSYIIU + VPBUBUCGHYHKSTVPUDUFUEUVQUURUUHXSPYGSUUOUUHXTQRSUUMUTWAWAXBXCZXDSJKUWEU + AUFUIUGXJXFXGBYTUWAIBUBUCTTYSUVTUUHUVMYSUVTVBUVNXSXTPYNRWPWQWRWSUWHXHXK + XLBUUQYDSVHYJSVHYKYMXMUURUVRUWISJKYEYDYJUAUFUIUGYEXNXOVTXPXQ $. $} ${ @@ -260832,89 +261061,89 @@ each row (matrices are given explicitly by their entries). Corollary ( vc vd ve vf wf1o wcel w3a cv cfv co cmpt2 czrh cpsgn ccom wceq cplusg csymg c0g cbs cpmtr crn weq fveq1 oveq1d mpt2eq3dv fveq2d fveq2 eqeq12d eqid cfn cgrp cmnd 3ad2ant1 symggrp grpmnd 3syl wss symgtrf a1i csubmnd - cmrc symggen2 syl eqcomd crg wf simp3 ffvelrnd rnglidm syl2anc cmgp cid - cmhm fveq1d simp2 eqtr3d mpt2eq3dva wfn 3ad2ant3 eqtr4d symgbasf1o f1of - ffn eqtrd wne cpr wrex cif adantl adantr ad2antrr simprlr simpr simprll - wa fovrnd pmtrprfv syl13anc sylan9eqr iftrue fveq2i cdif mpbird syl3anc - wn wb iffalse pm2.61dan zrhpsgnmhm rngidval mhm0 cres symgid fvresi cxp - cmap matbas2i elmapi fnov sylib 3eqtr4rd cminusg sseli fvco3 pmtrrn2 wi - symgov simpll1 df-3an biimpri simpllr simp1lr mdetunilem6 simpl1 simprr - jca prcom fveq1i simplrl simprd simpld simplrr necomd syl5eq adantlr wo - cdm vex elpr notbii ioran bitr2i biimpi adantll c2o cen wbr prssi pr2ne - pmtrmvd eleq2d pmtrf fnelnfp necon2bbid 3adant3 sylan pm2.61i mpt2eq3ia - notbid 3eqtr4d eqeq1d syl5ibrcom expr impd rexlimdvva syl5 syl3an2 mhmf - 3impia mgpbas simp13 rngmneg1 mgpplusg mhmlin cevpm zrhpsgnodpm rngnegr - pmtrodpm oveq2d 3eqtrrd 3eqtrd elsymgbas mrcmndind ) APPMUQZNGURZUSZRSP - PRUTZUMUTZVAZSUTZNVBZVCZHVAZUYJJVDVAZPVEVAZVFZVAZNHVAZKVBZVGRSPPUYIUNUT - ZVAZUYLNVBZVCZHVAZVUBUYRVAZUYTKVBZVGZRSPPUYIVUBUOUTZPVIVAZVHVAZVBZVAZUY - LNVBZVCZHVAZVUMUYRVAZUYTKVBZVGRSPPUYIVUKVJVAZVAZUYLNVBZVCZHVAZVUTUYRVAZ - UYTKVBZVGRSPPUYIMVAZUYLNVBZVCZHVAZMUYRVAZUYTKVBZVGUMUNUOMVUKVKVAZVULPVL - VAZVMZVUKVUTUMUNVNZUYOVUFVUAVUHVVPUYNVUEHVVPRSPPUYMVUDVVPUYKVUCUYLNUYIU - YJVUBVOVPVQVRVVPUYSVUGUYTKUYJVUBUYRVSVPVTUYJVUMVGZUYOVUQVUAVUSVVQUYNVUP - HVVQRSPPUYMVUOVVQUYKVUNUYLNUYIUYJVUMVOVPVQVRVVQUYSVURUYTKUYJVUMUYRVSVPV - TUYJVUTVGZUYOVVDVUAVVFVVRUYNVVCHVVRRSPPUYMVVBVVRUYKVVAUYLNUYIUYJVUTVOVP - VQVRVVRUYSVVEUYTKUYJVUTUYRVSVPVTUYJMVGZUYOVVJVUAVVLVVSUYNVVIHVVSRSPPUYM - VVHVVSUYKVVGUYLNUYIUYJMVOVPVQVRVVSUYSVVKUYTKUYJMUYRVSVPVTVUTWAZVULWAZVV - MWAZUYHPWBURZVUKWCURVUKWDURAUYFVWCUYGUGWEZPVUKWBVUKWAZWFVUKWGWHVVOVVMWI 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Corollary mdetunilem8 $p |- ( ( ph /\ E : N --> N ) -> ( D ` ( a e. N , b e. N |-> if ( ( E ` a ) = b , .1. , .0. ) ) ) = .0. ) $= ( vc vd wf wa wf1 cv cfv wceq cif cmpt2 wi co czrh cpsgn ccom wf1o wcel - cur simpl cen wbr cfn wb enrefg syl f1finf1o syl2anc biimpa matrng eqid - crg rngidcl adantr mdetunilem7 syl3anc w3a 3ad2ant1 simp1r f1f ffvelrnd - simp2 simp3 mat1ov mpt2eq3dva fveq2d oveq2d csymg cbs zrhpsgnmhm mgpbas - cmgp cmhm mhmf elsymgbas mpbird rngrz eqtrd 3eqtr3d ex wn wne wrex wral + cur simpl cen wbr cfn wb enrefg syl f1finf1o syl2anc biimpa crg matring + eqid ringidcl adantr mdetunilem7 syl3anc 3ad2ant1 simp1r simp2 ffvelrnd + w3a f1f simp3 mat1ov mpt2eq3dva fveq2d oveq2d csymg cbs cmgp zrhpsgnmhm + cmhm mgpbas mhmf elsymgbas mpbird ringrz eqtrd 3eqtr3d ex wne wrex wral weq ibar adantl dff13 syl6rbbr notbid rexnal df-ne anbi2i bitr2i rexbii - annim bitr3i syl6bb simprrl eqeq1d iftrue eqtr4d iffalse eqcomd pm2.61i - fveq2 ifbid syl6reqr ifcld eqcoms ifeq1d ifeq2d syl5eq mpt2eq3dv simpll - eqeq1 simprll simprlr simprrr 3jca rng0cl ad3antrrr simp1ll mdetunilem2 - expr rexlimdvva sylbid pm2.61d ) AOOMUOZUPZOOMUQZQROOQURZMUSZRURZUTZLPV - AZVBZHUSZPUTZAUVBUVJVCUUTAUVBUVJAUVBUPZQROOUVDUVEFVJUSZVDZVBZHUSZMJVEUS - OVFUSVGZUSZUVLHUSZKVDZUVIPUVKAOOMVHZUVLGVIZUVOUVSUTAUVBVKAUVBUVTAOOVLVM - ZOVNVIZUVBUVTVOAUWCUWBUFOVNVPVQUFOOMVRVSVTZAUWAUVBAFWCVIZUWAAUWCJWCVIZU - WEUFUGFJOSWAVSGFUVLTUVLWBZWDVQWEABCDEFGHIJKLMUVLNOPQRSTUAUBUCUDUEUFUGUH - UIUJUKWFWGUVKUVNUVHHUVKQROOUVMUVGUVKUVCOVIZUVEOVIZWHZFJUVLLUVDUVEOPSUCU - BUVKUWHUWCUWIAUWCUVBUFWEZWIUVKUWHUWFUWIAUWFUVBUGWEZWIUWJOOUVCMUWJUVBUUT - AUVBUWHUWIWJOOMWKVQUVKUWHUWIWMWLUVKUWHUWIWNUWGWOWPWQUVKUVSUVQPKVDZPUVKU - VRPUVQKAUVRPUTUVBULWEWRUVKUWFUVQNVIUWMPUTUWLUVKOWSUSZWTUSZNMUVPAUWONUVP - UOZUVBAUVPUWNJXCUSZXDVDVIZUWPAUWFUWCUWRUGUFOJXAVSUWONUWNUWQUVPUWOWBZNJU - WQUWQWBUAXBXEVQWEUVKMUWOVIZUVTUWDUVKUWCUWTUVTVOUWKOUWOMUWNVNUWNWBUWSXFV - QXGWLNJKUVQPUAUEUBXHVSXIXJXKWEUVAUVBXLZUMURZMUSZUNURZMUSZUTZUXBUXDXMZUP - ZUNOXNZUMOXNZUVJUVAUXAUXFUMUNXPZVCZUNOXOZUMOXOZXLZUXJUVAUVBUXNUVAUXNUUT - UXNUPZUVBUUTUXNUXPVOAUUTUXNXQXRUMUNOOMXSXTYAUXOUXMXLZUMOXNUXJUXMUMOYBUX - 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Corollary cid mpan anbi2d mpteq2dv cmpt2 mpt2mpt wfn ffn simp2 fnopfvb mpt2eq3dva bicomd syl5eq mdetunilem8 sylan2 chvarv adantrl ex ralrimivva xpfi ssid 3eqtrd elab2g cun simp3 ssun1 sstr2 3anim123i imim1i csg simpl1 simprll - id simpl2 cof 3ad2ant3 reseq1d cgrp crg rnggrp unssbd snss sylibr snssd - xpss1 sselda ffvelrnd rngidcl rng0cl grpnpcan grplid snfi c0g cur elsni - ovex 3syl mpteq2ia eqidd offval2 wne eldifsni neneqd 3eqtr4rd ffvelrnda - nsyl anidms matbas2 syl6eqr grpsubcl 3anbi123d 3anbi12d 3anbi13d xpeq1d - difss sneq difeq2d mp3and rngridm rngrz fconstmpt anbi12d xpeq2d anbi1d - ad2antrr oveq1 mp2and simpl3 simprlr simprr csts ralss ibar wrel 1st2nd + id simpl2 cof 3ad2ant3 reseq1d cgrp crg ringgrp unssbd snss snssd xpss1 + sylibr sselda ffvelrnd ringidcl ring0cl grpnpcan grplid snfi ovex elsni + c0g cur 3syl mpteq2ia eqidd offval2 eldifsni neneqd difss 3eqtr4rd nsyl + wne ffvelrnda matbas2 syl6eqr grpsubcl 3anbi123d 3anbi12d 3anbi13d sneq + anidms xpeq1d difeq2d mp3and ringridm ad2antrr ringrz fconstmpt anbi12d + xpeq2d anbi1d oveq1 mp2and simpl3 simprlr simprr csts ralss ibar 1st2nd relxp eleq1d xp2nd fsets syl211anc fvsetsid mp3an syl6eq eqcom 3bitr2rd - syl6bb ad3antrrr xpopth 3bitr3rd a1d wo con3i elun biimpi orel2 adantll - sylc setsres eleq2d df-ne biimpri eldifsn sylanbrc opres 1st2nd2 bitr4d - syl2an ifeq2 syl5ibrcom embantd equequ1 ifbieq12d fvmpt ralimdva syl5bi - sylibrd impr 3adantr1 simpr2 simpr1 ibi eleq2 mpd unssad simpr3 ralunsn - ssel2 ifid syl33anc expr cbvral2v sylib elab2 3expia con3d 3adant3 syl5 - a2d findcard2s mpcom 3exp mpi mt4d 0ex ) AULGULUQZHURZUSULGPUSZHGPUTVAZ - AULGVXOPAVXNGVBZVCZEUQZVXNURZVXTUUJNVDZVBZLPVEZVFZEVGVKZVXOPVFZVYEEUUAV - XSVXRVYBNNUUBVHZVBZVXTCUQZURZEDVIZLPVEZVFZEVGVKZVYJHURZPVFZVJZDVYHVKCGV - KZVYFVYGVJZAVXRUUCAVYIVXRAVYINNVYBVLZNNVYBUUDWUAANUUENNVYBUUFVMANVNVBZW - UBVYIWUAWCUDUDNNVYBVNVNVOVPVQVRVXSVGOVBZVYSAWUCVXRANNVAZOVBZWUCAWUEVYNE - 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Corollary mdetuni0 $p |- ( ph -> ( D ` F ) = ( ( D ` ( 1r ` A ) ) .x. ( E ` F ) ) ) $= ( va vb vc vd ve cfv cur co csg wceq cv cmpt csn cxp wel cif wral wi cmap - cab wcel cgrp crg rnggrp syl adantr ffvelrnda cfn jca matrng eqid rngidcl - wa 3syl ffvelrnd wf mdetf rngcl syl3anc grpsubcl fmptd wne w3a simpr1 weq + cab wcel cgrp crg ringgrp syl adantr ffvelrnda cfn jca eqid ringidcl 3syl + wa matring ffvelrnd wf mdetf ringcl syl3anc grpsubcl fmptd wne w3a simpr1 ccrg fveq2 oveq2d oveq12d fvmpt 3adant3 simp1 simp21 simp3r oveq2 eqeq12d - cbvralv sylib simp22 simp23 simp3l mdetunilem1 syl33anc 3ad2ant1 grpsubid - ovex syl2anc eqtrd 3eqtrd 3expia cres cof simp2ll simp2lr simp2rl simp2rr - simprll simprlr simprrl syl13anc 3eqtr4d anassrs ralrimivva cvv c0g a1i - fvmptd mdetralt grpidcl ralrimivvva cdif simp31 simp32 simp33 mdetunilem3 - rngrz syl332anc mdetrlin cabl rngabl ablsub4 syl122anc eqcomd mdetunilem4 - rngdi 3eqtr2d syl133anc mdetrsca rngsubdi cmgp ccmn mgpbas mgpplusg cmn12 - crngmgp eqidd mdet1 rngridm sylan9eqr eqeltri mdetunilem9 fveq1d fvconst2 - fvex adantl 3eqtr3d wb grpsubeq0 mpbid ) ANHUSZFUTUSZHUSZNMUSZKVAZJVBUSZV - AZQVCZUWCUWGVCZANUNGUNVDZHUSZUWEUWLMUSZKVAZUWHVAZVEZUSNGQVFVGZUSZUWIQANUW - QUWRAUOUPUQURFGUWQIJKLOPURVDZUPVDZUSURUQVHLQVIVCURUOVDZVJUXAUWQUSZQVCVKUQ - PPVLVAVJUPGVJUOVMZQRSTUAUBUCUDUEUFAUNGUWPOUWQAUWLGVNZWFZJVOVNZUWMOVNUWOOV - NZUWPOVNAUXGUXEAJVPVNZUXGUFJVQVRZVSAGOUWLHUGVTUXFUXIUWEOVNZUWNOVNUXHAUXIU - XEUFVSAUXKUXEAGOUWDHUGAPWAVNZUXIWFFVPVNUWDGVNAUXLUXIUEUFWBFJPRWCGFUWDSUWD - WDZWEWGZWHZVSAGOUWLMAJWSVNZGOMWIZULFGMJOPUKRSTWJVRZVTOJKUWEUWNTUDWKWLOJUW - HUWMUWOTUWHWDZWMWLUWQWDZWNAUXAUQVDZWOZUXAUWTUXBVAZUYAUWTUXBVAZVCZURPVJZWF - ZUXBUWQUSZQVCZVKUOUPUQGPPAUXBGVNZUXAPVNZUYAPVNZWPZUYGUYIAUYMUYGWPZUYHUXBH - USZUWEUXBMUSZKVAZUWHVAZQUWEQKVAZUWHVAZQAUYMUYHUYRVCZUYGAUYMWFUYJVUAAUYJUY - KUYLWQUNUXBUWPUYRGUWQUNUOWRZUWMUYOUWOUYQUWHUWLUXBHWTVUBUWNUYPUWEKUWLUXBMW - TXAXBUXTUYOUYQUWHXSXCZVRXDUYNUYOQUYQUYSUWHUYNAUYJUXAEVDZUXBVAZUYAVUDUXBVA - ZVCZEPVJZUYKUYLUYBUYOQVCAUYMUYGXEAUYJUYKUYLUYGXFZUYNUYFVUHAUYMUYBUYFXGZUY - 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MZYRYSUXDWCUVNUVOAUNNUWPUWIGUWQYPWUHUWLNVCZUWPUWIVCAWUPUWMUWCUWOUWGUWHUWL + NHWSWUPUWNUWFUWEKUWLNMWSWTXAUVPUMUWIYPVNAUWCUWGUWHXJYRYSANGVNUWSQVCUMGQNW + UOUVRVRUVSAUXGUWCOVNUWGOVNZUWJUWKUVTUXJAGONHUGUMWHAUXIUXKUWFOVNWUQUFUXOAG + ONMUXRUMWHOJKUWEUWFTUDWKWLOJUWHUWCUWGQTUAUXSUWAWLUWB $. mdetuni.no $e |- ( ph -> ( D ` ( 1r ` A ) ) = .1. ) $. $( According to the definition in [Weierstrass] p. 272, the determinant @@ -261257,10 +261486,10 @@ this function D for a matrix F (mdetuni.f) is the determinant of this matrix. (Contributed by Stefan O'Rear, 15-Jul-2018.) (Revised by Alexander van der Vekens, 8-Feb-2019.) $) mdetuni $p |- ( ph -> ( D ` F ) = ( E ` F ) ) $= - ( cfv cur co mdetuni0 oveq1d wcel wceq ccrg mdetcl syl2anc rnglidm 3eqtrd - crg ) ANHUOFUPUOHUOZNMUOZKUQLVIKUQZVIABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUH - UIUJUKULUMURAVHLVIKUNUSAJVGUTVIOUTZVJVIVAUFAJVBUTNGUTVKULUMFGMJONPUKRSTVC - VDOJKLVITUDUBVEVDVF $. + ( cfv cur co mdetuni0 oveq1d crg wcel wceq mdetcl syl2anc ringlidm 3eqtrd + ccrg ) ANHUOFUPUOHUOZNMUOZKUQLVIKUQZVIABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGU + HUIUJUKULUMURAVHLVIKUNUSAJUTVAVIOVAZVJVIVBUFAJVGVANGVAVKULUMFGMJONPUKRSTV + CVDOJKLVITUDUBVEVDVF $. $} ${ @@ -261279,89 +261508,89 @@ this function D for a matrix F (mdetuni.f) is the determinant of this mdetmul $p |- ( ( R e. CRing /\ F e. B /\ G e. B ) -> ( D ` ( F .xb G ) ) = ( ( D ` F ) .x. ( D ` G ) ) ) $= ( wcel co cfv wceq 3adant3 cres va vb vc vd ve ccrg w3a cv cur cplusg cbs - cmpt c0g eqid cfn cvv matrcl simpld 3ad2ant2 crngrng 3ad2ant1 wa wf mdetf - crg adantr matrng syl2anc simpr simpl3 rngcl syl3anc ffvelrnd wne wral wi - fmptd simp21 weq oveq1 fveq2d fvex syl simp11 simpr1 simp22 simp23 simp3l - fvmpt cgsu simpl3r ralimi mpteq12 sylancr oveq2d cotp cmmul simp1 matmulr - cmulr syl6eqr ad2antrr simpll1 cxp cmap simplr1 matbas2i 3ad2ant3 simplr2 - oveqd mamufv eqtr3d 3adantl3 simplr3 ralrimiva mdetralt eqtrd ralrimivvva - 3eqtr4d 3expia csn cof simprll simprlr simprrl simp2rr oveq1d a1i simprrr - snssd elmapssres reseq1d simpl1 mamures oveq12d difssd anassrs ralrimivva - cdif wss simp31 snfi mamudi simp32 simp33 mdetrlin simp2lr mamuvs1 eqtr4d - xpss1 simp3r mdetrsca simp2ll simp2rl simp2 mdetuni0 rngidcl 3syl rnglidm - simp3 crngcom 3eqtr3d ) DUFOZGBOZHBOZUGZGUABUAUHZHEPZCQZULZQZAUIQZUVJQZGC - 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determinant of this -> ( Y ` ( S ` Q ) ) = .1. ) $= ( wcel c1 cop c2 cpr wceq cfv eqid crg wa cpsgn cmg co cvv 1ex prex prid1 cn csymg symg2bas syl5eleqr mp2an eleq1 mpbiri m2detleiblem1 sylan2 fveq2 - 2nn adantl cpmtr crn psgnprfval1 syl6eq oveq1d cbs rngidcl adantr 3eqtrd + 2nn adantl cpmtr crn psgnprfval1 syl6eq oveq1d cbs ringidcl adantr 3eqtrd mulg1 syl ) CUAMZBNNOZPPOZQZRZUBZBDSGSZBFUCSZSZECUDSZUEZNEWBUEZEVQVMBAMZV SWCRVQWEVPAMZNUFMZPUJMZWFUGUTWGWHUBVPVPNPOPNOQZQAVPWIVNVOUHUIFAFUKSZNPUFU JWJTZIHULUMUNBVPAUOUPABCDEFGHIJKLUQURVRWANEWBVRWAVPVTSZNVQWAWLRVMBVPVTUSV @@ -261406,12 +261635,12 @@ this function D for a matrix F (mdetuni.f) is the determinant of this -> ( Y ` ( S ` Q ) ) = ( I ` .1. ) ) $= ( wcel c1 c2 cop cfv eqid crg cpr wceq wa cpsgn cmg cneg cvv 1ex 2nn prex co prid2 csymg symg2bas syl5eleqr mp2an eleq1 mpbiri m2detleiblem1 sylan2 - cn fveq2 adantl cpmtr crn psgnprfval2 syl6eq oveq1d rnggrp rngidcl mulgm1 - cgrp cbs syl2anc adantr 3eqtrd ) CUAOZBPQRZQPRZUBZUCZUDZBDSHSZBGUESZSZECU - FSZULZPUGZEWGULZEFSZWBVRBAOZWDWHUCWBWLWAAOZPUHOZQVBOZWMUIUJWNWOUDWAPPRQQR - UBZWAUBAWPWAVSVTUKUMGAGUNSZPQUHVBWQTZJIUOUPUQBWAAURUSABCDEGHIJKLMUTVAWCWF - WIEWGWCWFWAWESZWIWBWFWSUCVRBWAWEVCVDAGGVESVFZWQWEIWRJWTTWETVGVHVIVRWJWKUC - ZWBVRCVMOECVNSZOXACVJXBCEXBTZMVKXBWGCFEXCWGTNVLVOVPVQ $. + cn fveq2 adantl cpmtr crn psgnprfval2 syl6eq oveq1d cgrp ringgrp ringidcl + cbs mulgm1 syl2anc adantr 3eqtrd ) CUAOZBPQRZQPRZUBZUCZUDZBDSHSZBGUESZSZE + CUFSZULZPUGZEWGULZEFSZWBVRBAOZWDWHUCWBWLWAAOZPUHOZQVBOZWMUIUJWNWOUDWAPPRQ + QRUBZWAUBAWPWAVSVTUKUMGAGUNSZPQUHVBWQTZJIUOUPUQBWAAURUSABCDEGHIJKLMUTVAWC + WFWIEWGWCWFWAWESZWIWBWFWSUCVRBWAWEVCVDAGGVESVFZWQWEIWRJWTTWETVGVHVIVRWJWK + UCZWBVRCVJOECVMSZOXACVKXBCEXBTZMVLXBWGCFEXCWGTNVNVOVPVQ $. m2detleiblem1.t $e |- .x. = ( .r ` R ) $. m2detleiblem1.m $e |- .- = ( -g ` R ) $. @@ -261419,10 +261648,10 @@ this function D for a matrix F (mdetuni.f) is the determinant of this m2detleiblem7 $p |- ( ( R e. Ring /\ X e. ( Base ` R ) /\ Z e. ( Base ` R ) ) -> ( X ( +g ` R ) ( ( I ` .1. ) .x. Z ) ) = ( X .- Z ) ) $= - ( cfv crg wcel cbs w3a co cplusg wceq wa eqid simpl simpr rngnegl 3adant2 - oveq2d grpsubval 3adant1 eqtr4d ) BUAUBZIBUCTZUBZKUSUBZUDZIEFTKDUEZBUFTZU - EIKFTZVDUEZIKGUEZVBVCVEIVDURVAVCVEUGUTURVAUHUSBDEFKUSUIZRPQURVAUJURVAUKUL - UMUNUTVAVGVFUGURUSVDBFGIKVHVDUIQSUOUPUQ $. + ( cfv crg wcel cbs w3a co cplusg wceq wa eqid simpl simpr ringnegl oveq2d + 3adant2 grpsubval 3adant1 eqtr4d ) BUAUBZIBUCTZUBZKUSUBZUDZIEFTKDUEZBUFTZ + UEIKFTZVDUEZIKGUEZVBVCVEIVDURVAVCVEUGUTURVAUHUSBDEFKUSUIZRPQURVAUJURVAUKU + LUNUMUTVAVGVFUGURUSVDBFGIKVHVDUIQSUOUPUQ $. $} ${ @@ -261436,11 +261665,11 @@ this function D for a matrix F (mdetuni.f) is the determinant of this shortened by AV, 1-Jan-2019.) $) m2detleiblem2 $p |- ( ( R e. Ring /\ Q e. P /\ M e. B ) -> ( G gsum ( n e. N |-> ( ( Q ` n ) M n ) ) ) e. ( Base ` R ) ) $= - ( wcel cfv c1 c2 co cfz crg w3a cbs cv eqid mgpbas cmnd 3ad2ant1 2eluzge1 - rngmgp cuz a1i wceq caddc cz 1z fzpr ax-mp 1p1e2 preq2i eqtri df-2 oveq2i - cpr 3eqtr4ri matepmcl gsummptfzcl ) EUAOZDCOZHBOZUBZEUCPZFGIQRFUDZDPVMHSV - LEGNVLUEUFVHVIGUGOVJEGNUJUHRQUKPOVKUIULIQRTSZUMVKQQQUNSZTSZQRVDZVNIVPQVOV - DZVQQUOOVPVRUMUPQUQURVORQUSUTVARVOQTVBVCJVEULABCDEFHILMKVFVG $. + ( wcel cfv c1 c2 co cfz crg w3a cbs eqid mgpbas cmnd ringmgp 3ad2ant1 cuz + cv 2eluzge1 a1i wceq caddc cpr cz 1z fzpr ax-mp 1p1e2 preq2i eqtri oveq2i + df-2 3eqtr4ri matepmcl gsummptfzcl ) EUAOZDCOZHBOZUBZEUCPZFGIQRFUJZDPVMHS + VLEGNVLUDUEVHVIGUFOVJEGNUGUHRQUIPOVKUKULIQRTSZUMVKQQQUNSZTSZQRUOZVNIVPQVO + UOZVQQUPOVPVRUMUQQURUSVORQUTVAVBRVOQTVDVCJVEULABCDEFHILMKVFVG $. m2detleiblem3.m $e |- .x. = ( +g ` G ) $. $( Lemma 3 for ~ m2detleib . (Contributed by AV, 16-Dec-2018.) (Proof @@ -261505,49 +261734,49 @@ this function D for a matrix F (mdetuni.f) is the determinant of this m2detleib $p |- ( ( R e. Ring /\ M e. B ) -> ( D ` M ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) ) $= ( wcel cfv co c1 c2 wceq vk vn crg wa csymg cbs cpsgn czrh cmgp cmpt cgsu - cop cpr csn cplusg eqid mdetleib1 adantl ccmn rngcmn adantr cfn symgbasfi - cv prfi eqeltri ax-mp a1i simpl zrhpsgnelbas mp3an2 adantlr m2detleiblem2 - simpr syl3anc rngcl cvv wne wo cin opex pm3.2i 1ne2 olci 1ex opthne mpbir - c0 orci prneimg imp disjsn2 3syl cn symg2bas mp2an df-pr gsummptfidmsplit - cun 2nn eqtri cminusg cmnd rngmnd prex prid1 eleqtrri sylan2 fveq2 fveq2d - cur fveq1 oveq1d mpteq2dv oveq2d oveq12d gsumsn prid2 eqidd m2detleiblem5 - mgpplusg m2detleiblem3 eleq2i biimpi matecl prid2g rnglidm m2detleiblem6 - syldan eqtrd m2detleiblem4 m2detleiblem7 3eqtrd ) DUCOZFBOZUDZFCPZDUAHUEP - ZUFPZUAVDZHUGPZPZDUHPZPZDUIPZUBHUBVDZYTPZUUFFQZUJZUKQZEQZUJUKQZDUARRULZSS - ULZUMZUNZUUKUJUKQZDUARSULZSRULZUMZUNZUUKUJUKQZDUOPZQZRRFQZSSFQZEQZSRFQZRS - FQZEQZGQZYOYQUULTYNUBABCYSDUUAEUUEFHUUCUAJKLYSUPZUUCUPZUUAUPZNUUEUPZUQURY - PYSDUFPZUUPUVAUVCUADUUKUVPUPZUVCUPYNDUSOYODUTVAYSVBOZYPHVBOZUVRHRSUMZVBIR - SVEVFZHYSYRYRUPZUVLVCVGVHYPYTYSOZUDZYNUUDUVPOZUUJUVPOZUUKUVPOYPYNUWCYNYOV - IZVAZYNUWCUWEYOYNUVSUWCUWEUWAYSYTDUUAHUUCUVLUVNUVMVJVKVLUWDYNUWCYOUWFUWHY - PUWCVNYPYOUWCYNYOVNZVAABYSYTDUBUUEFHIUVLKLUVOVMVOUVPDEUUDUUJUVQNVPVOYPUUM - VQOZUUNVQOZUDZUURVQOZUUSVQOZUDZUDZUUMUURVRZUUMUUSVRZUDZUUNUURVRUUNUUSVRUD - ZVSZUDZUUOUUTVRZUUPUVAVTWHTUXBYPUWPUXAUWLUWOUWJUWKRRWASSWAWBUWMUWNRSWASRW - AWBWBUWSUWTUWQUWRUWQRRVRZRSVRZVSUXEUXDWCWDRRRSWEWEWFWGUWRUXEUXDVSUXEUXDWC - WIRRSRWEWEWFWGWBWIWBVHUWPUXAUXCUUMUUNUURUUSVQVQVQVQWJWKUUOUUTWLWMYSUUPUVA - WSZTYPYSUUOUUTUMZUXFRVQOSWNOZYSUXGTWEWTHYSYRRSVQWNUWBUVLIWOWPZUUOUUTWQXAV - HWRYPUVDUUOUUAPZUUCPZUUEUBHUUFUUOPZUUFFQZUJZUKQZEQZUUTUUAPZUUCPZUUEUBHUUF - UUTPZUUFFQZUJZUKQZEQZUVCQUVGDXKPZDXBPZPZUVJEQZUVCQZUVKYPUUQUXPUVBUYCUVCYP - DXCOZUUOVQOZUXPUVPOZUUQUXPTYNUYIYODXDVAZUYJYPUUMUUNXEZVHYPYNUXKUVPOZUXOUV - POZUYKUWGYOYNUUOYSOZUYNUYPYOUUOUXGYSUUOUUTUYMXFUXIXGZVHYNUVSUYPUYNUWAYSUU - ODUUAHUUCUVLUVNUVMVJVKXHYNUYPYOUYOUYQABYSUUODUBUUEFHIUVLKLUVOVMVKUVPDEUXK - UXOUVQNVPVOUUKUVPUXPUADUUOVQUVQYTUUOTZUUDUXKUUJUXOEUYRUUBUXJUUCYTUUOUUAXI - XJUYRUUIUXNUUEUKUYRUBHUUHUXMUYRUUGUXLUUFFUUFYTUUOXLXMXNXOXPXQVOYPUYIUUTVQ - OZUYCUVPOZUVBUYCTUYLUYSYPUURUUSXEZVHYPYNUXRUVPOZUYBUVPOZUYTUWGYOYNUUTYSOZ - VUBVUDYOUUTUXGYSUUOUUTVUAXRUXIXGZVHYNUVSVUDVUBUWAYSUUTDUUAHUUCUVLUVNUVMVJ - VKXHYNVUDYOVUCVUEABYSUUTDUBUUEFHIUVLKLUVOVMVKUVPDEUXRUYBUVQNVPVOUUKUVPUYC - UADUUTVQUVQYTUUTTZUUDUXRUUJUYBEVUFUUBUXQUUCYTUUTUUAXIXJVUFUUIUYAUUEUKVUFU - BHUUHUXTVUFUUGUXSUUFFUUFYTUUTXLXMXNXOXPXQVOXPYPUXPUVGUYCUYGUVCYPUXPUYDUVG - EQZUVGYPUXKUYDUXOUVGEYOYNUUOUUOTZUXKUYDTYOUUOXSYSUUODUUAUYDHUUCIUVLUVMUVN - UYDUPZXTXHYPYNVUHYOUXOUVGTUWGYPUUOXSUWIABYSUUODEUBUUEFHIUVLKLUVODEUUEUVON - YAZYBVOXPYNYOUVGUVPOZVUGUVGTYPYNUVEUVPOZUVFUVPOZVUKUWGYPRHOZVUNFAUFPZOZVU - LVUNYPRUVTHRSWEXFIXGVHZVUQYOVUPYNYOVUPBVUOFLYCYDURZADRRUVPFHKUVQYEVOYPSHO - ZVUSVUPVUMVUSYPSUVTHUXHSUVTOWTRSWNYFVGIXGVHZVUTVURADSSUVPFHKUVQYEVOUVPDEU - VEUVFUVQNVPVOZUVPDEUYDUVGUVQNVUIYGYIYJYPUXRUYFUYBUVJEYOYNUUTUUTTZUXRUYFTY - OUUTXSYSUUTDUUAUYDUYEHUUCIUVLUVMUVNVUIUYEUPZYHXHYPYNVVBYOUYBUVJTUWGYPUUTX - SUWIABYSUUTDEUBUUEFHIUVLKLUVOVUJYKVOXPXPYPYNVUKUVJUVPOZUYHUVKTUWGVVAYPYNU - VHUVPOZUVIUVPOZVVDUWGYPVUSVUNVUPVVEVUTVUQVURADSRUVPFHKUVQYEVOYPVUNVUSVUPV - VFVUQVUTVURADRSUVPFHKUVQYEVOUVPDEUVHUVIUVQNVPVOYSDUUAEUYDUYEGHUVGUUCUVJIU - VLUVMUVNVUIVVCNMYLVOYMYM $. + cop cpr csn cplusg eqid mdetleib1 adantl ccmn ringcmn adantr prfi eqeltri + cv cfn symgbasfi ax-mp a1i simpl zrhpsgnelbas adantlr simpr m2detleiblem2 + mp3an2 syl3anc ringcl cvv wne wo cin c0 opex pm3.2i 1ne2 1ex opthne mpbir + olci orci prneimg imp disjsn2 3syl cun cn symg2bas mp2an gsummptfidmsplit + 2nn df-pr eqtri cur cminusg cmnd ringmnd prex prid1 eleqtrri sylan2 fveq2 + fveq2d fveq1 oveq1d mpteq2dv oveq2d oveq12d gsumsn m2detleiblem5 mgpplusg + prid2 eqidd m2detleiblem3 eleq2i biimpi matecl prid2g eqtrd m2detleiblem6 + ringlidm syldan m2detleiblem4 m2detleiblem7 3eqtrd ) DUCOZFBOZUDZFCPZDUAH + UEPZUFPZUAVDZHUGPZPZDUHPZPZDUIPZUBHUBVDZYTPZUUFFQZUJZUKQZEQZUJUKQZDUARRUL + ZSSULZUMZUNZUUKUJUKQZDUARSULZSRULZUMZUNZUUKUJUKQZDUOPZQZRRFQZSSFQZEQZSRFQ + ZRSFQZEQZGQZYOYQUULTYNUBABCYSDUUAEUUEFHUUCUAJKLYSUPZUUCUPZUUAUPZNUUEUPZUQ + URYPYSDUFPZUUPUVAUVCUADUUKUVPUPZUVCUPYNDUSOYODUTVAYSVEOZYPHVEOZUVRHRSUMZV + EIRSVBVCZHYSYRYRUPZUVLVFVGVHYPYTYSOZUDZYNUUDUVPOZUUJUVPOZUUKUVPOYPYNUWCYN + YOVIZVAZYNUWCUWEYOYNUVSUWCUWEUWAYSYTDUUAHUUCUVLUVNUVMVJVNVKUWDYNUWCYOUWFU + WHYPUWCVLYPYOUWCYNYOVLZVAABYSYTDUBUUEFHIUVLKLUVOVMVOUVPDEUUDUUJUVQNVPVOYP + UUMVQOZUUNVQOZUDZUURVQOZUUSVQOZUDZUDZUUMUURVRZUUMUUSVRZUDZUUNUURVRUUNUUSV + RUDZVSZUDZUUOUUTVRZUUPUVAVTWATUXBYPUWPUXAUWLUWOUWJUWKRRWBSSWBWCUWMUWNRSWB + SRWBWCWCUWSUWTUWQUWRUWQRRVRZRSVRZVSUXEUXDWDWHRRRSWEWEWFWGUWRUXEUXDVSUXEUX + DWDWIRRSRWEWEWFWGWCWIWCVHUWPUXAUXCUUMUUNUURUUSVQVQVQVQWJWKUUOUUTWLWMYSUUP + UVAWNZTYPYSUUOUUTUMZUXFRVQOSWOOZYSUXGTWEWSHYSYRRSVQWOUWBUVLIWPWQZUUOUUTWT + XAVHWRYPUVDUUOUUAPZUUCPZUUEUBHUUFUUOPZUUFFQZUJZUKQZEQZUUTUUAPZUUCPZUUEUBH + UUFUUTPZUUFFQZUJZUKQZEQZUVCQUVGDXBPZDXCPZPZUVJEQZUVCQZUVKYPUUQUXPUVBUYCUV + CYPDXDOZUUOVQOZUXPUVPOZUUQUXPTYNUYIYODXEVAZUYJYPUUMUUNXFZVHYPYNUXKUVPOZUX + OUVPOZUYKUWGYOYNUUOYSOZUYNUYPYOUUOUXGYSUUOUUTUYMXGUXIXHZVHYNUVSUYPUYNUWAY + SUUODUUAHUUCUVLUVNUVMVJVNXIYNUYPYOUYOUYQABYSUUODUBUUEFHIUVLKLUVOVMVNUVPDE + UXKUXOUVQNVPVOUUKUVPUXPUADUUOVQUVQYTUUOTZUUDUXKUUJUXOEUYRUUBUXJUUCYTUUOUU + AXJXKUYRUUIUXNUUEUKUYRUBHUUHUXMUYRUUGUXLUUFFUUFYTUUOXLXMXNXOXPXQVOYPUYIUU + TVQOZUYCUVPOZUVBUYCTUYLUYSYPUURUUSXFZVHYPYNUXRUVPOZUYBUVPOZUYTUWGYOYNUUTY + SOZVUBVUDYOUUTUXGYSUUOUUTVUAXTUXIXHZVHYNUVSVUDVUBUWAYSUUTDUUAHUUCUVLUVNUV + MVJVNXIYNVUDYOVUCVUEABYSUUTDUBUUEFHIUVLKLUVOVMVNUVPDEUXRUYBUVQNVPVOUUKUVP + UYCUADUUTVQUVQYTUUTTZUUDUXRUUJUYBEVUFUUBUXQUUCYTUUTUUAXJXKVUFUUIUYAUUEUKV + UFUBHUUHUXTVUFUUGUXSUUFFUUFYTUUTXLXMXNXOXPXQVOXPYPUXPUVGUYCUYGUVCYPUXPUYD + UVGEQZUVGYPUXKUYDUXOUVGEYOYNUUOUUOTZUXKUYDTYOUUOYAYSUUODUUAUYDHUUCIUVLUVM + 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fveq2i syl6eqr wss cmulr cplusg cbs simpl1l cfn - simp1r cvv matrcl simpld syl adantr crg simp1l ad2antrr crngrng rng0cl wf - eqid cxp cmap simpl1r matbas2i elmapi 3syl eldifi ad2antll eldifad fovrnd - simpr ifcld rngidcl 3adant2 fovrnda 3impb simpl2 simpl3 wne mdetero ifnot - eldifsni eqcomi a1i oveq2 ifsb rngridm syl2anc adantl ifeq1da rngrz eqtrd - eqtr4d syl5eq oveq12d rngmnd iftrue syl5ibrcom imp neneqd 3ad2ant1 notbid - cmnd id eleq1 3eqtr4d pm2.61dan wi imnan mndifsplit syl3anc pm2.1 3eqtr2d - mpbi oveq1 eqeq1 eldifn mpt2eq3dva neeq2 biimparc necomd snid elun2 ax-mp - vex mpbiri 3eqtr4rd orc orel2 impbid2 orbi2i bitr2i syl6bb 3eqtr3d biimpd - elun elsn difss ssfi sylancl findcard2d iba neqned anim2i adantlr eldifsn - orcs sylibr biorf intnand eqcomd eqidd ifbieq12d 3eqtrd syl6eq ) DUDUEZJB - UEZUFZHKUEZGKUEZUGZHGJIUHUIZFMKKFUJZGTZMUJZHTZELUKZUWPKGULZUMZUEZUWSLUWPU - WRJUIZUKZUXDUKZUKZUNZCUHZFMKKUWQUWSUOZUWSUWQUFZELUKZUXDUKZUNZCUHUWNUWOFMK - 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CRing -> J : B --> B ) $= ( vm vi vj vk vl ccrg wcel weq cfv cv co eqid cur c0g cif cmpt2 cmdat cbs wa cfn cvv matrcl adantl simpld simpl w3a wf mdetf adantr 3ad2ant1 simp1l - crg simp11l crngrng rngidcl rng0cl ifcl syl2anc 3syl cxp simp11r matbas2i - cmap elmapi simp2 simp3 fovrnd matbas2d ffvelrnd madufval fmptd ) CNOZIBJ - KEELMEELKPZMJPZCUAQZCUBQZUCZLRZMRZIRZSZUCZUDZECUESZQZUDBDVTWHBOZUGZJKABWM - CCUFQZENFWPTZHWOEUHOZCUIOZWNWRWSUGVTABCEWHFHUJUKULZVTWNUMWOJREOZKREOZUNZB - WPWKWLWOXABWPWLUOZXBVTXDWNABWLCWPEWLTZFHWQUPUQURXCLMABWJCWPENFWQHWOXAWRXB - WTURVTWNXAXBUSXCWFEOZWGEOZUNZWEWPOZWIWPOWJWPOXHVTCUTOZXIVTWNXAXBXFXGVACVB - XJWCWPOWDWPOXIWPCWCWQWCTZVCWPCWDWQWDTZVDWBWCWDWPVEVFVGXHWFWGWPEEWHXHWNWHW - PEEVHZVKSOXMWPWHUOVTWNXAXBXFXGVIABCWPWHEFWQHVJWHWPXMVLVGXCXFXGVMXCXFXGVNV - OWAWEWIWPVEVFVPVQVPABWLCWCJKLIDEWDMFXEGHXKXLVRVS $. + crg simp11l crngring ringidcl ring0cl ifcl syl2anc 3syl cxp cmap matbas2i + simp11r elmapi simp2 simp3 fovrnd matbas2d ffvelrnd madufval fmptd ) CNOZ + IBJKEELMEELKPZMJPZCUAQZCUBQZUCZLRZMRZIRZSZUCZUDZECUESZQZUDBDVTWHBOZUGZJKA + BWMCCUFQZENFWPTZHWOEUHOZCUIOZWNWRWSUGVTABCEWHFHUJUKULZVTWNUMWOJREOZKREOZU + NZBWPWKWLWOXABWPWLUOZXBVTXDWNABWLCWPEWLTZFHWQUPUQURXCLMABWJCWPENFWQHWOXAW + RXBWTURVTWNXAXBUSXCWFEOZWGEOZUNZWEWPOZWIWPOWJWPOXHVTCUTOZXIVTWNXAXBXFXGVA + CVBXJWCWPOWDWPOXIWPCWCWQWCTZVCWPCWDWQWDTZVDWBWCWDWPVEVFVGXHWFWGWPEEWHXHWN + WHWPEEVHZVISOXMWPWHUOVTWNXAXBXFXGVKABCWPWHEFWQHVJWHWPXMVLVGXCXFXGVMXCXFXG + VNVOWAWEWIWPVEVFVPVQVPABWLCWCJKLIDEWDMFXEGHXKXLVRVS $. $d ph a b c d e i j $. $d D a b c d e $. $d J a b c d e i $. $d K a b c d e i j $. $d M a b c d e i j $. $d N a b c d e $. @@ -261782,25 +262011,25 @@ is equal to the minor determined in the standard way (as determinant of ( va vb vc vd wcel wa cfv wceq cv co eqid ccrg ctpos wral weq cur c0g cif wo cmpt2 cmdat tposmpt2 orcom a1i ancom ovtpos eqcomi ifbieq12d mpt2eq3dv ifbid syl5eq fveq2d simpll cbs cfn cvv matrcl simpld ad2antlr w3a simp1ll - 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See @@ -261934,16 +262163,16 @@ is equal to the minor determined in the standard way (as determinant of ( ( D ` M ) .xb .1. ) ) $= ( wcel co wceq ccrg cfv ctpos simpr maduf ffvelrnda ancoms simpl mattposm wa syl3anc madutpos oveq2d mattposcl madurid sylan 3eqtr2d tposeqd matrcl - crg cfn cvv simpld crngrng matrng syl2an rngcl mattpostpos syl eqid mdetf - cbs wf adantl adantr ffvelrnd rngidcl mattposvs syl2anc mdettpos mattpos1 - oveq12d eqtrd 3eqtr3d ) IBRZDUARZUJZIHUBZIFSZUCZUCZIUCZCUBZGESZUCZWIICUBZ - GESZWGWJWNWGWJWLWHUCZFSZWLWLHUBZFSZWNWGWFWHBRZWEWJWSTWEWFUDWFWEXBWFBBIHAB - DHJKMLUEUFUGZWEWFUHZABDFJWHIKLPUIUKWGWTWRWLFWFWEWTWRTABDHIJKMLULUGUMWEWLB - RZWFXAWNTABDIJKLUNZABCDEFGHWLJKLMNOPQUOUPUQURWGWIBRZWKWITWGAUTRZXBWEXGWEJ - VARZDUTRZXHWFWEXIDVBRABDJIKLUSVCZDVDZADJKVEVFZXCXDBAFWHILPVGUKABDWIJKLVHV - IWGWOWMGUCZESZWQWGWMDVLUBZRGBRZWOXOTWGBXPWLCWFBXPCVMWEABCDXPJNKLXPVJZVKVN - WEXEWFXFVOVPWGXHXQXMBAGLOVQVIABDEXPJWMGKLXRQVRVSWGWMWPXNGEWFWEWMWPTABCDIJ - NKLVTUGWEXIXJXNGTWFXKXLADGJKOWAVFWBWCWD $. + crg cfn cvv simpld crngring matring syl2an ringcl mattpostpos syl wf eqid + mdetf adantl adantr ffvelrnd ringidcl mattposvs syl2anc mdettpos mattpos1 + cbs oveq12d eqtrd 3eqtr3d ) IBRZDUARZUJZIHUBZIFSZUCZUCZIUCZCUBZGESZUCZWII + CUBZGESZWGWJWNWGWJWLWHUCZFSZWLWLHUBZFSZWNWGWFWHBRZWEWJWSTWEWFUDWFWEXBWFBB + IHABDHJKMLUEUFUGZWEWFUHZABDFJWHIKLPUIUKWGWTWRWLFWFWEWTWRTABDHIJKMLULUGUMW + EWLBRZWFXAWNTABDIJKLUNZABCDEFGHWLJKLMNOPQUOUPUQURWGWIBRZWKWITWGAUTRZXBWEX + GWEJVARZDUTRZXHWFWEXIDVBRABDJIKLUSVCZDVDZADJKVEVFZXCXDBAFWHILPVGUKABDWIJK + LVHVIWGWOWMGUCZESZWQWGWMDWAUBZRGBRZWOXOTWGBXPWLCWFBXPCVJWEABCDXPJNKLXPVKZ + VLVMWEXEWFXFVNVOWGXHXQXMBAGLOVPVIABDEXPJWMGKLXRQVQVRWGWMWPXNGEWFWEWMWPTAB + CDIJNKLVSUGWEXIXJXNGTWFXKXLADGJKOVTVFWBWCWD $. $} ${ @@ -262025,10 +262254,10 @@ is equal to the minor determined in the standard way (as determinant of minmar1marrep $p |- ( ( R e. Ring /\ M e. B ) -> ( ( N minMatR1 R ) ` M ) = ( M ( N matRRep R ) .1. ) ) $= ( vk vl vi vj wcel co cfv weq eqid crg wa cminmar1 c0g cmpt2 cmarrep wceq - cif minmar1val0 adantl cbs simpr rngidcl adantr marrepval0 syl2anc eqtr4d - cv ) DUAPZFBPZUBZFGDUCQZRZLMGGNOGGNLSOMSEDUDRZUHNUROURFQUHUEUEZFEGDUFQZQZ - UTVCVEUGUSABVBDENOLFGVDMHIVBTKVDTZUIUJVAUTEDUKRZPZVGVEUGUSUTULUSVJUTVIDEV - ITKUMUNABVFDENOLFGVDMHIVFTVHUOUPUQ $. + cif cv minmar1val0 adantl simpr ringidcl adantr marrepval0 syl2anc eqtr4d + cbs ) DUAPZFBPZUBZFGDUCQZRZLMGGNOGGNLSOMSEDUDRZUHNUIOUIFQUHUEUEZFEGDUFQZQ + ZUTVCVEUGUSABVBDENOLFGVDMHIVBTKVDTZUJUKVAUTEDURRZPZVGVEUGUSUTULUSVJUTVIDE + VITKUMUNABVFDENOLFGVDMHIVFTVHUOUPUQ $. $} ${ @@ -262038,11 +262267,11 @@ is equal to the minor determined in the standard way (as determinant of for a minor is a matrix. (Contributed by AV, 13-Feb-2019.) $) minmar1cl $p |- ( ( ( R e. Ring /\ M e. B ) /\ ( K e. N /\ L e. N ) ) -> ( K ( ( N minMatR1 R ) ` M ) L ) e. B ) $= - ( crg wcel wa cminmar1 co cfv cur cmarrep wceq eqid adantr oveqd cbs 3jca - minmar1marrep w3a simpl simpr rngidcl marrepcl sylan eqeltrd ) CJKZFBKZLZ - DGKEGKLZLZDEFGCMNOZNDEFCPOZGCQNZNZNZBUPUQUTDEUNUQUTRUOABUSCURFGHIUSSURSZU - DTUAUNULUMURCUBOZKZUEUOVABKUNULUMVDULUMUFULUMUGULVDUMVCCURVCSVBUHTUCABCUR - DEFGHIUIUJUK $. + ( crg wcel wa cminmar1 co cfv cur cmarrep wceq eqid adantr oveqd ringidcl + minmar1marrep cbs w3a simpl simpr 3jca marrepcl sylan eqeltrd ) CJKZFBKZL + ZDGKEGKLZLZDEFGCMNOZNDEFCPOZGCQNZNZNZBUPUQUTDEUNUQUTRUOABUSCURFGHIUSSURSZ + UCTUAUNULUMURCUDOZKZUEUOVABKUNULUMVDULUMUFULUMUGULVDUMVCCURVCSVBUBTUHABCU + RDEFGHIUIUJUK $. $} ${ @@ -262276,11 +262505,11 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, marep01ma $p |- ( M e. B -> ( k e. N , l e. N |-> if ( k = H , if ( l = I , .1. , .0. ) , ( k M l ) ) ) e. B ) $= ( wcel cv wceq cif co cbs cfv ccrg eqid cfn cvv matrcl simpld a1i w3a crg - crngrng rngidcl mp2b rng0cl keepel simp2 simp3 id syl6eleq matecl syl3anc - 3ad2ant1 ifcld matbas2d ) HBQZEKABERZFSZKRZGSZDJTZVHVJHUAZTCCUBUCZIUDLVNU - EZMVGIUFQCUGQABCIHLMUHUICUDQZVGNUJVGVHIQZVJIQZUKZVIVLVMVNVLVNQVSVKDJVNVPC - ULQZDVNQNCUMZVNCDVOPUNUOVPVTJVNQNWAVNCJVOOUPUOUQUJVSVQVRHAUBUCZQZVMVNQVGV - QVRURVGVQVRUSVGVQWCVRVGHBWBVGUTMVAVDACVHVJVNHILVOVBVCVEVF $. + crngring ringidcl ring0cl keepel simp2 simp3 id syl6eleq 3ad2ant1 syl3anc + mp2b matecl ifcld matbas2d ) HBQZEKABERZFSZKRZGSZDJTZVHVJHUAZTCCUBUCZIUDL + VNUEZMVGIUFQCUGQABCIHLMUHUICUDQZVGNUJVGVHIQZVJIQZUKZVIVLVMVNVLVNQVSVKDJVN + VPCULQZDVNQNCUMZVNCDVOPUNVCVPVTJVNQNWAVNCJVOOUOVCUPUJVSVQVRHAUBUCZQZVMVNQ + VGVQVRUQVGVQVRURVGVQWCVRVGHBWBVGUSMUTVAACVHVJVNHILVOVDVBVEVF $. $d i j m n B $. $d i j m n q K $. $d i j m n q L $. $d i j m n M $. $d i j m n N $. $d i j m n q P $. $d i j m n q Q $. $d i j m n R $. @@ -262296,16 +262525,16 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) ) ) = .0. ) ) $= ( vm wcel w3a cv cfv wceq crab cdif cif co cmpt2 cmpt cgsu wa a1i cfn cvv - ccrg matrcl simpld 3ad2ant1 adantr cbs wral wi crngrng mp1i eldifi adantl - crg marep01ma matepm2cl syl3anc weq id oveq12d eleq1d rspccv syl imp wrex - fveq2 symgmatr01 3adant1 gsummgp0 ex ) MBUEZKNUEZLNUEZUFZDCKPUGUHLUIPCUJZ - UKUEZJINIUGZWPDUHZGHNNGUGZKUIHUGZLUIFOULWRWSMUMULUNZUMZUOUPUMOUIWMWOUQZXA - UDUGZXCDUHZWTUMZEUDIJNOUCTEVAUEZXBSURWMNUSUEZWOWJWKXGWLWJXGEUTUEABENMQRVB - VCVDVEXBWPNUEZXAEVFUHZUEZXBXEXIUEZUDNVGZXHXJVHXBEVMUEZDCUEZWTBUEZXLXFXMXB - SEVIVJWOXNWMDCWNVKVLWMXOWOWJWKXOWLABEFGKLMNOHQRSTUAVNVDVEABCDEUDWTNQRUBVO - VPXKXJUDWPNUDIVQZXEXAXIXPXCWPXDWQWTXPVRXCWPDWEVSVTWAWBWCIUDVQZXAXEUIXBXQW - PXCWQXDWTXQVRWPXCDWEVSVLWMWOXEOUIUDNWDZWKWLWOXRVHWJCDEFGHUDKLMNOPUBTUAWFW - GWCWHWI $. + ccrg matrcl simpld 3ad2ant1 adantr cbs wral wi crg crngring eldifi adantl + mp1i marep01ma matepm2cl syl3anc weq fveq2 oveq12d eleq1d rspccv syl wrex + id imp symgmatr01 3adant1 gsummgp0 ex ) MBUEZKNUEZLNUEZUFZDCKPUGUHLUIPCUJ + ZUKUEZJINIUGZWPDUHZGHNNGUGZKUIHUGZLUIFOULWRWSMUMULUNZUMZUOUPUMOUIWMWOUQZX + AUDUGZXCDUHZWTUMZEUDIJNOUCTEVAUEZXBSURWMNUSUEZWOWJWKXGWLWJXGEUTUEABENMQRV + BVCVDVEXBWPNUEZXAEVFUHZUEZXBXEXIUEZUDNVGZXHXJVHXBEVIUEZDCUEZWTBUEZXLXFXMX + BSEVJVMWOXNWMDCWNVKVLWMXOWOWJWKXOWLABEFGKLMNOHQRSTUAVNVDVEABCDEUDWTNQRUBV + OVPXKXJUDWPNUDIVQZXEXAXIXPXCWPXDWQWTXPWDXCWPDVRVSVTWAWBWEIUDVQZXAXEUIXBXQ + WPXCWQXDWTXQWDWPXCDVRVSVLWMWOXEOUIUDNWCZWKWLWOXRVHWJCDEFGHUDKLMNOPUBTUAWF + WGWEWHWI $. $d n p $. madetminlem.y $e |- Y = ( ZRHom ` R ) $. @@ -262335,16 +262564,16 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( p ` n ) ) ) ) ) ) ) = .0. ) $= ( wcel w3a cv cfv wceq crab cdif ccom co cmpt2 cmpt cgsu wa smadiadetlem0 - cif imp oveq2d mpteq2dva crg cbs ccrg crngrng mp1i matrcl simpld 3ad2ant1 - cfn adantr eldifi adantl zrhcopsgnelbas syl3anc eqid rngrz syl2anc rngmnd - cvv cmnd mp2b csymg fvex eqeltri difexg gsumz sylancr 3eqtrd ) NBUJZLOUJZ - MOUJZUKZDSCLRULUMMUNRCUOZUPZSULZPEUQUMZKJOJULZXDXBUMHIOOHULZLUNIULZMUNGQV - DXEXFNURVDUSURUTVAURZFURZUTZVAURDSXAXCQFURZUTZVAURDSXAQUTZVAURZQWSXIXKDVA - WSSXAXHXJWSXBXAUJZVBZXGQXCFWSXNXGQUNABCXBDGHIJKLMNOQRTUAUBUCUDUEUFVCVEVFV - GVFWSXKXLDVAWSSXAXJQXODVHUJZXCDVIUMZUJZXJQUNDVJUJZXPXOUBDVKZVLZXOXPOVPUJZ - XBCUJZXRYAWSYBXNWPWQYBWRWPYBDWFUJABDONTUAVMVNVOVQXNYCWSXBCWTVRVSCXBDEOPUE - UHUGVTWAXQDFXCQXQWBUIUCWCWDVGVFWSDWGUJZXAWFUJZXMQUNXSXPYDUBXTDWEWHCWFUJYE - WSCOWIUMZVIUMWFUEYFVIWJWKCWTWFWLVLXASDWFQUCWMWNWO $. + cif imp oveq2d mpteq2dva crg cbs ccrg crngring mp1i cfn cvv matrcl simpld + 3ad2ant1 adantr eldifi adantl zrhcopsgnelbas syl3anc eqid syl2anc ringmnd + ringrz cmnd mp2b csymg fvex eqeltri difexg gsumz sylancr 3eqtrd ) NBUJZLO + UJZMOUJZUKZDSCLRULUMMUNRCUOZUPZSULZPEUQUMZKJOJULZXDXBUMHIOOHULZLUNIULZMUN + GQVDXEXFNURVDUSURUTVAURZFURZUTZVAURDSXAXCQFURZUTZVAURDSXAQUTZVAURZQWSXIXK + DVAWSSXAXHXJWSXBXAUJZVBZXGQXCFWSXNXGQUNABCXBDGHIJKLMNOQRTUAUBUCUDUEUFVCVE + VFVGVFWSXKXLDVAWSSXAXJQXODVHUJZXCDVIUMZUJZXJQUNDVJUJZXPXOUBDVKZVLZXOXPOVM + UJZXBCUJZXRYAWSYBXNWPWQYBWRWPYBDVNUJABDONTUAVOVPVQVRXNYCWSXBCWTVSVTCXBDEO + PUEUHUGWAWBXQDFXCQXQWCUIUCWFWDVGVFWSDWGUJZXAVNUJZXMQUNXSXPYDUBXTDWEWHCVNU + JYEWSCOWIUMZVIUMVNUEYFVIWJWKCWTVNWLVLXASDVNQUCWMWNWO $. $( Lemma 2 for ~ smadiadet : The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and @@ -262397,11 +262626,11 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, |-> ( n ( i e. ( N \ { K } ) , j e. ( N \ { K } ) |-> ( i M j ) ) ( p ` n ) ) ) ) ) ) ) ) $= ( wcel wa ccmn cv ccom cfv csn cdif co cmpt2 cmpt cmulr crn cbs wss ccntz - cgsu ccrg crg crngrng rngcmn mp2b smadiadetlem3lem0 ralrimiva rnmptss syl - wral eqid cntzcmnss sylancr ) MBULLNULUMZDUNULZSOSUOZPRUPUQKJNLURUSZJUOZW - FWDUQHIWEWEHUOIUOMUTVAUTVBVHUTDVCUQUTZVBZVDZDVEUQZVFZWIWIDVGUQZUQVFDVIULD - VJULWCUBDVKDVLVMWBWGWJULZSOVRWKWBWMSOABCWDDEFGHIJKLMNOPQRTUAUBUCUDUEUFUGU - HUIUJUKVNVOSOWGWJWHWHVSVPVQWJWIDWLWJVSWLVSVTWA $. + cgsu ccrg crg crngring ringcmn mp2b wral smadiadetlem3lem0 ralrimiva eqid + rnmptss syl cntzcmnss sylancr ) MBULLNULUMZDUNULZSOSUOZPRUPUQKJNLURUSZJUO + ZWFWDUQHIWEWEHUOIUOMUTVAUTVBVHUTDVCUQUTZVBZVDZDVEUQZVFZWIWIDVGUQZUQVFDVIU + LDVJULWCUBDVKDVLVMWBWGWJULZSOVNWKWBWMSOABCWDDEFGHIJKLMNOPQRTUAUBUCUDUEUFU + GUHUIUJUKVOVPSOWGWJWHWHVQVRVSWJWIDWLWJVQWLVQVTWA $. $d G p y $. $d K i j n y $. $d M y $. $d N y $. $d R y $. $d W y $. $d Y p y $. $d Z p y $. @@ -262416,27 +262645,27 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, |-> ( n ( i e. ( N \ { K } ) , j e. ( N \ { K } ) |-> ( i M j ) ) ( p ` n ) ) ) ) ) ) ) ) $= ( vy wcel wa cv ccom cfv csn cdif co cmpt2 cmpt cgsu cmulr wceq crab cres - cbs cvv ccntz eqid cmnd ccrg crg crngrng mp2b a1i csymg smadiadetlem3lem1 - rngmnd fvex syl5eqel smadiadetlem3lem2 cfn matrcl simpld adantr symgbasfi - diffi 3syl ovex c0g eqeltri fsuppmptdm wf1o fveq1 eqeq1d symgfixf1o sylan - cbvrabv gsumzf1o symgfixelsi adantll eqidd fveq2 mpteq2dv oveq12d cbvmptv - oveq2d fvres sylan9eq mpteq2dva fmptco wi zrhcopsgndif imp oveq1d eqtr2d - eqtrd ) MBUNZLNUNZUOZDTOTUPZPRUQZURZKJNLUSZUTZJUPZYIYDURZHIYHYHHUPIUPMVAV - BZVAZVCZVDVAZDVEURZVAZVCZVDVADYQTLSUPZURZLVFZSCVGZYDYHVHZVCZUQZVDVADTUUAY - DPEUQURZYNYOVAZVCZVDVAYCODVIURZUUAYQDUUCVJQDVKURZUUHVLUDUUIVLDVMUNZYCDVNU - NDVOUNUUJUCDVPDWAVQVRYCOYHVSURZVIURZVJUKUULVJUNYCUUKVIWBVRWCABCDEFGHIJKLM - NOPQRTUAUBUCUDUEUFUGUHUIUJUKULVTABCDEFGHIJKLMNOPQRTUAUBUCUDUEUFUGUHUIUJUK - ULWDYCTOYQVJVJYPQYQVLYCOUULWEUKYCNWEUNZYHWEUNUULWEUNYAUUMYBYAUUMDVJUNABDN - MUAUBWFWGZWHNYGWJYHUULUUKUUKVLUULVLWIWKWCYPVJUNYCYDOUNUOYFYNYOWLVRQVJUNYC - QDWMURVJUDDWMWBWNVRWOYAUUMYBUUAOUUCWPUUNCUUAOUUCLNWETUFYTLYDURZLVFSTCYRYD - VFYSUUOLLYRYDWQWRXAUKUUCVLWSWTXBYCUUDUUGDVDYCUUDTUUAUUBYEURZYNYOVAZVCUUGY - CTUMUUAOUUBUMUPZYEURZKJYHYIYIUURURZYKVAZVCZVDVAZYOVAZUUQUUCYQYBYDUUAUNZUU - BOUNYAYHCUUAOYDLNSUFUUAVLUKYHVLXCXDYCUUCXEYQUMOUVDVCVFYCTUMOYPUVDYDUURVFZ - YFUUSYNUVCYOYDUURYEXFUVFYMUVBKVDUVFJYHYLUVAUVFYJUUTYIYKYIYDUURWQXJXGXJXHX - IVRUURUUBVFZUUSUUPUVCYNYOUURUUBYEXFUVGUVBYMKVDUVGJYHUVAYLUVGYIYHUNZUOUUTY - JYIYKUVGUVHUUTYIUUBURYJYIUURUUBWQYIYHYDXKXLXJXMXJXHXNYCTUUAUUQUUFYCUVEUOU - UPUUEYNYOYCUVEUUPUUEVFZYAUUMYBUVEUVIXOUUNCYDDELNPRSUFUIULUHXPWTXQXRXMXTXJ - XS $. + cbs cvv ccntz eqid cmnd ccrg crg crngring ringmnd mp2b a1i csymg syl5eqel + fvex smadiadetlem3lem1 smadiadetlem3lem2 cfn matrcl simpld symgbasfi 3syl + adantr diffi ovex eqeltri fsuppmptdm wf1o fveq1 eqeq1d cbvrabv symgfixf1o + c0g sylan gsumzf1o symgfixelsi adantll eqidd fveq2 oveq2d oveq12d cbvmptv + mpteq2dv fvres sylan9eq mpteq2dva fmptco wi zrhcopsgndif imp oveq1d eqtrd + eqtr2d ) MBUNZLNUNZUOZDTOTUPZPRUQZURZKJNLUSZUTZJUPZYIYDURZHIYHYHHUPIUPMVA + VBZVAZVCZVDVAZDVEURZVAZVCZVDVADYQTLSUPZURZLVFZSCVGZYDYHVHZVCZUQZVDVADTUUA + YDPEUQURZYNYOVAZVCZVDVAYCODVIURZUUAYQDUUCVJQDVKURZUUHVLUDUUIVLDVMUNZYCDVN + UNDVOUNUUJUCDVPDVQVRVSYCOYHVTURZVIURZVJUKUULVJUNYCUUKVIWBVSWAABCDEFGHIJKL + MNOPQRTUAUBUCUDUEUFUGUHUIUJUKULWCABCDEFGHIJKLMNOPQRTUAUBUCUDUEUFUGUHUIUJU + KULWDYCTOYQVJVJYPQYQVLYCOUULWEUKYCNWEUNZYHWEUNUULWEUNYAUUMYBYAUUMDVJUNABD + NMUAUBWFWGZWJNYGWKYHUULUUKUUKVLUULVLWHWIWAYPVJUNYCYDOUNUOYFYNYOWLVSQVJUNY + CQDWTURVJUDDWTWBWMVSWNYAUUMYBUUAOUUCWOUUNCUUAOUUCLNWETUFYTLYDURZLVFSTCYRY + DVFYSUUOLLYRYDWPWQWRUKUUCVLWSXAXBYCUUDUUGDVDYCUUDTUUAUUBYEURZYNYOVAZVCUUG + YCTUMUUAOUUBUMUPZYEURZKJYHYIYIUURURZYKVAZVCZVDVAZYOVAZUUQUUCYQYBYDUUAUNZU + UBOUNYAYHCUUAOYDLNSUFUUAVLUKYHVLXCXDYCUUCXEYQUMOUVDVCVFYCTUMOYPUVDYDUURVF + ZYFUUSYNUVCYOYDUURYEXFUVFYMUVBKVDUVFJYHYLUVAUVFYJUUTYIYKYIYDUURWPXGXJXGXH + XIVSUURUUBVFZUUSUUPUVCYNYOUURUUBYEXFUVGUVBYMKVDUVGJYHUVAYLUVGYIYHUNZUOUUT + YJYIYKUVGUVHUUTYIUUBURYJYIUURUUBWPYIYHYDXKXLXGXMXGXHXNYCTUUAUUQUUFYCUVEUO + UUPUUEYNYOYCUVEUUPUUEVFZYAUUMYBUVEUVIXOUUNCYDDELNPRSUFUIULUHXPXAXQXRXMXSX + GXT $. $d G i j $. $( Lemma 4 for ~ smadiadet . (Contributed by AV, 31-Jan-2019.) $) @@ -262451,18 +262680,18 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, |-> ( i M j ) ) ( p ` n ) ) ) ) ) ) ) ) $= ( wcel wa cv cfv wceq crab ccom cif co cmpt2 cmpt cgsu cmulr csn cdif cfn ccmn cbs wral ccrg crngmgp mp1i matrcl simpld adantr simprl simprr eleq2i - cvv jca biimpi eqid matecl syl3anc mgpbas syl6eleq ralrimivva crg crngrng - rng0cl eleqtri jctir simpr rngidval gsummatr01 syl113anc oveq2d mpteq2dva - mp2b smadiadetlem3 eqtrd ) MBUMZLNUMZUNZDTLSUOUPLUQSCURZTUOZPEUSUPZKJNJUO - ZXJXHUPZHINNHUOZLUQIUOZLUQGQUTXLXMMVAZUTVBVAVCVDVAZDVEUPZVAZVCZVDVADTXGXI - KJNLVFVGZXJXKHIXSXSXNVBVAVCVDVAZXPVAZVCZVDVADTOXHPRUSUPXTXPVAVCVDVAXFXRYB - DVDXFTXGXQYAXFXHXGUMZUNZXOXTXIXPYDKVIUMZNVHUMZUNZXNKVJUPZUMZINVKHNVKZQYHU - MZUNXEXEYCXOXTUQXFYGYCXFYEYFDVLUMZYEXFUCDKUGVMVNXDYFXEXDYFDWAUMABDNMUAUBV - OVPVQWBVQYDYJYKXFYJYCXFYIHINNXFXLNUMZXMNUMZUNZUNZXNDVJUPZYHYPYMYNMAVJUPZU - MZXNYQUMXFYMYNVRXFYMYNVSXFYSYOXDYSXEXDYSBYRMUBVTWCVQVQADXLXMYQMNUAYQWDZWE - WFYQDKUGYTWGZWHWIVQQYQYHYLDWJUMQYQUMUCDWKYQDQYTUDWLXAUUAWMWNXFXEYCXDXEWOV - QZUUBXFYCWOMQCXHXGYHHIJKLLNGSUFXGWDDGKUGUEWPYHWDWQWRWSWTWSABCDEFGHIJKLMNO - PQRSTUAUBUCUDUEUFUGUHUIUJUKULXBXC $. + cvv jca biimpi eqid matecl syl3anc mgpbas syl6eleq ralrimivva crg ring0cl + crngring mp2b eleqtri jctir simpr rngidval gsummatr01 syl113anc mpteq2dva + oveq2d smadiadetlem3 eqtrd ) MBUMZLNUMZUNZDTLSUOUPLUQSCURZTUOZPEUSUPZKJNJ + UOZXJXHUPZHINNHUOZLUQIUOZLUQGQUTXLXMMVAZUTVBVAVCVDVAZDVEUPZVAZVCZVDVADTXG + XIKJNLVFVGZXJXKHIXSXSXNVBVAVCVDVAZXPVAZVCZVDVADTOXHPRUSUPXTXPVAVCVDVAXFXR + YBDVDXFTXGXQYAXFXHXGUMZUNZXOXTXIXPYDKVIUMZNVHUMZUNZXNKVJUPZUMZINVKHNVKZQY + HUMZUNXEXEYCXOXTUQXFYGYCXFYEYFDVLUMZYEXFUCDKUGVMVNXDYFXEXDYFDWAUMABDNMUAU + BVOVPVQWBVQYDYJYKXFYJYCXFYIHINNXFXLNUMZXMNUMZUNZUNZXNDVJUPZYHYPYMYNMAVJUP + ZUMZXNYQUMXFYMYNVRXFYMYNVSXFYSYOXDYSXEXDYSBYRMUBVTWCVQVQADXLXMYQMNUAYQWDZ + WEWFYQDKUGYTWGZWHWIVQQYQYHYLDWJUMQYQUMUCDWLYQDQYTUDWKWMUUAWNWOXFXEYCXDXEW + PVQZUUBXFYCWPMQCXHXGYHHIJKLLNGSUFXGWDDGKUGUEWQYHWDWRWSXAWTXAABCDEFGHIJKLM + NOPQRSTUAUBUCUDUEUFUGUHUIUJUKULXBXC $. $} ${ @@ -262484,29 +262713,29 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, ( vi vj vp wcel co cfv eqid vn vq wa csubma csn cdif cv cminmar1 submaval cmpt2 wceq 3anidm23 fveq2d cur c0g cif csymg cbs czrh ccom cmgp cmpt cgsu cpsgn minmar1val ccrg marep01ma mdetleib2 sylancr adantr crab cplusg ccmn - crg crngrng rngcmn mp2b a1i cfn cvv matrcl simpld symgbasfi smadiadetlem1 - cmulr syl cin disjdif cun wss ssrab2 undif gsummptfidmsplit smadiadetlem4 - sylib eqcomd smadiadetlem2 oveq12d cmnd rngmnd diffi cmat simpll submabas - c0 difssd syl2anc madetsmelbas2 syl3anc ralrimiva gsummptcl mndrid sylan2 - simpr jctil eqtr4d 3eqtrd ) GBQZFHQZUCZFFGHDUDRZSRZESNOHFUEZUFZYDNUGZOUGZ - GRZUJZESZFFGHDUHRZSRZCSZXTYBYHEXRXSYBYHUKABYADNOFFGHIYATJUIULUMXTYLNOHHYE - FUKYFFUKDUNSZDUOSZUPYGUPUJZCSZDPHUQSZURSZPUGZDUSSZHVDSZUTSDVASZUAHUAUGZUU - CYSSZYORVBVCRDWESZRZVBVCRZYIXTYKYOCXRXSYKYOUKABYJDYMNOFFGHYNIJYJTYMTZYNTZ - VEULUMXRYPUUGUKZXSXRDVFQZYOBQUUJKABDYMNFFGHYNOIJKUUIUUHVGUAABCYRDUUAUUEUU - BYOHYTPLIJYRTZYTTZUUATZUUETZUUBTZVHVIVJXTUUGDPFUBUGSFUKZUBYRVKZUUFVBVCRZD - PYRUURUFZUUFVBVCRZDVLSZRDPYDUQSZURSZYSYTYDVDSZUTSUUBUAYDUUCUUDYHRVBVCRUUE - RZVBVCRZYNUVBRZYIXTYRDURSZUURUUTUVBPDUUFUVITZUVBTZDVMQZXTUUKDVNQZUVLKDVOZ - DVPVQVRZXRYRVSQZXSXRHVSQZUVPXRUVQDVTQABDHGIJWAWBZHYRYQYQTUULWCWFVJABYRDUU - AUUEYMNOUAUUBFGHYTYNPIJKUUIUUHUULUUPUUMUUNUUOWDUURUUTWGXEUKXTUURYRWHVRXTU - URUUTWIZYRXTUURYRWJZUVSYRUKUVTXTUUQUBYRWKVRUURYRWLWOWPWMXTUUSUVGUVAYNUVBA - BYRDUUAUUEYMNOUAUUBFGHUVDYTYNUVEUBPIJKUUIUUHUULUUPUUMUUNUUOUVDTZUVETZWNAB - YRDUUAUUEYMNOUAUUBFGHYTYNUBPIJKUUIUUHUULUUPUUMUUNUUOWQWRXTUVHUVGYIXTDWSQZ - UVGUVIQUVHUVGUKUUKUVMUWCKUVNDWTVQXTUVIPDUVDUVFUVJUVOXTYDVSQZUVDVSQXRUWDXS - XRUVQUWDUVRHYCXAWFVJYDUVDUVCUVCTUWAWCWFXTUVFUVIQZPUVDXTYSUVDQZUCZUUKYHYDD - XBRZURSZQZUWFUWEUUKUWGKVRUWGXRYDHWJZUWJXRXSUWFXCUWGHYCXFABYDDNOGHIJXDZXGX - TUWFXNUWHUWIUVDYSDUVEUAUUBYHYDYTUWAUWBUUMUWHTZUWITZUUPXHXIXJXKUVIUVBDUVGY - NUVJUVKUUIXLVIXTUUKUWJUCZYIUVGUKXSXRUWKUWOXSHYCXFXRUWKUCUWJUUKUWLKXOXMUAU - WHUWIEUVDDUVEUUEUUBYHYDYTPMUWMUWNUWAUUMUWBUUOUUPVHWFXPXQXQXP $. + cmulr crg crngring ringcmn mp2b a1i matrcl simpld symgbasfi smadiadetlem1 + cfn cvv syl cin c0 disjdif cun ssrab2 undif sylib eqcomd gsummptfidmsplit + smadiadetlem4 smadiadetlem2 oveq12d cmnd ringmnd diffi cmat simpll difssd + wss submabas syl2anc simpr madetsmelbas2 ralrimiva gsummptcl mndrid jctil + syl3anc sylan2 eqtr4d 3eqtrd ) GBQZFHQZUCZFFGHDUDRZSRZESNOHFUEZUFZYDNUGZO + UGZGRZUJZESZFFGHDUHRZSRZCSZXTYBYHEXRXSYBYHUKABYADNOFFGHIYATJUIULUMXTYLNOH + HYEFUKYFFUKDUNSZDUOSZUPYGUPUJZCSZDPHUQSZURSZPUGZDUSSZHVDSZUTSDVASZUAHUAUG + ZUUCYSSZYORVBVCRDVNSZRZVBVCRZYIXTYKYOCXRXSYKYOUKABYJDYMNOFFGHYNIJYJTYMTZY + NTZVEULUMXRYPUUGUKZXSXRDVFQZYOBQUUJKABDYMNFFGHYNOIJKUUIUUHVGUAABCYRDUUAUU + EUUBYOHYTPLIJYRTZYTTZUUATZUUETZUUBTZVHVIVJXTUUGDPFUBUGSFUKZUBYRVKZUUFVBVC + RZDPYRUURUFZUUFVBVCRZDVLSZRDPYDUQSZURSZYSYTYDVDSZUTSUUBUAYDUUCUUDYHRVBVCR + UUERZVBVCRZYNUVBRZYIXTYRDURSZUURUUTUVBPDUUFUVITZUVBTZDVMQZXTUUKDVOQZUVLKD + VPZDVQVRVSZXRYRWDQZXSXRHWDQZUVPXRUVQDWEQABDHGIJVTWAZHYRYQYQTUULWBWFVJABYR + DUUAUUEYMNOUAUUBFGHYTYNPIJKUUIUUHUULUUPUUMUUNUUOWCUURUUTWGWHUKXTUURYRWIVS + XTUURUUTWJZYRXTUURYRXEZUVSYRUKUVTXTUUQUBYRWKVSUURYRWLWMWNWOXTUUSUVGUVAYNU + VBABYRDUUAUUEYMNOUAUUBFGHUVDYTYNUVEUBPIJKUUIUUHUULUUPUUMUUNUUOUVDTZUVETZW + PABYRDUUAUUEYMNOUAUUBFGHYTYNUBPIJKUUIUUHUULUUPUUMUUNUUOWQWRXTUVHUVGYIXTDW + SQZUVGUVIQUVHUVGUKUUKUVMUWCKUVNDWTVRXTUVIPDUVDUVFUVJUVOXTYDWDQZUVDWDQXRUW + DXSXRUVQUWDUVRHYCXAWFVJYDUVDUVCUVCTUWAWBWFXTUVFUVIQZPUVDXTYSUVDQZUCZUUKYH + YDDXBRZURSZQZUWFUWEUUKUWGKVSUWGXRYDHXEZUWJXRXSUWFXCUWGHYCXDABYDDNOGHIJXFZ + XGXTUWFXHUWHUWIUVDYSDUVEUAUUBYHYDYTUWAUWBUUMUWHTZUWITZUUPXIXNXJXKUVIUVBDU + VGYNUVJUVKUUIXLVIXTUUKUWJUCZYIUVGUKXSXRUWKUWOXSHYCXDXRUWKUCUWJUUKUWLKXMXO + UAUWHUWIEUVDDUVEUUEUUBYHYDYTPMUWMUWNUWAUUMUWBUUOUUPVHWFXPXQXQXP $. $d S i j $. $( Lemma 1 for ~ smadiadetg . (Contributed by AV, 13-Feb-2019.) $) @@ -262515,16 +262744,16 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, = ( ( K ( ( N minMatR1 R ) ` M ) K ) |` ( ( N \ { K } ) X. N ) ) ) $= ( vi vj wcel co cres wceq cbs cfv w3a cmarrep csn cxp cur cminmar1 cv c0g cdif cif cmpt2 mpt2difsnif eqtr4i wss wa ssid pm3.2i resmpt2 mp1i 3eqtr4a - difss simp1 simp3 eqid marrepval syl22anc reseq1d crg crngrng rngidcl syl - simp2 ccrg 3eqtr4d ax-mp minmar1marrep sylancr eqcomd oveqd eqtrd ) HBQZG - IQZEDUAUBZQZUCZGGHEIDUDRZRRZIGUEZUKZIUFZSZGGHDUGUBZWHRZRZWLSZGGHIDUHRUBZR - ZWLSWGOPIIOUIZGTZPUIZGTZEDUJUBZULZWTXBHRZULZUMZWLSZOPIIXAXCWNXDULZXFULZUM - ZWLSZWMWQWGOPWKIXGUMZOPWKIXKUMZXIXMXNOPWKIXFUMXOIIXEXFOPGUNIIXJXFOPGUNUOW - KIUPZIIUPZUQZXIXNTWGXPXQIWJVCIURUSZOPIIWKIXGUTVAXRXMXOTWGXSOPIIWKIXKUTVAV - BWGWIXHWLWGWCWFWDWDWIXHTWCWDWFVDZWCWDWFVEWCWDWFVNZYAABWHDEOPGGHIXDJKWHVFZ - XDVFZVGVHVIWGWPXLWLWGWCWNWEQZWDWDWPXLTXTDVOQZYDWGLYEDVJQZYDDVKZWEDWNWEVFW - NVFZVLVMVAYAYAABWHDWNOPGGHIXDJKYBYCVGVHVIVPWGWPWSWLWGWOWRGGWGWRWOWGYFWCWR - WOTYEYFLYGVQXTABWHDWNHIJKYBYHVRVSVTWAVIWB $. + difss simp1 simp3 simp2 eqid marrepval syl22anc reseq1d ccrg crg crngring + ringidcl syl 3eqtr4d ax-mp minmar1marrep sylancr eqcomd oveqd eqtrd ) HBQ + ZGIQZEDUAUBZQZUCZGGHEIDUDRZRRZIGUEZUKZIUFZSZGGHDUGUBZWHRZRZWLSZGGHIDUHRUB + ZRZWLSWGOPIIOUIZGTZPUIZGTZEDUJUBZULZWTXBHRZULZUMZWLSZOPIIXAXCWNXDULZXFULZ + UMZWLSZWMWQWGOPWKIXGUMZOPWKIXKUMZXIXMXNOPWKIXFUMXOIIXEXFOPGUNIIXJXFOPGUNU + OWKIUPZIIUPZUQZXIXNTWGXPXQIWJVCIURUSZOPIIWKIXGUTVAXRXMXOTWGXSOPIIWKIXKUTV + AVBWGWIXHWLWGWCWFWDWDWIXHTWCWDWFVDZWCWDWFVEWCWDWFVFZYAABWHDEOPGGHIXDJKWHV + GZXDVGZVHVIVJWGWPXLWLWGWCWNWEQZWDWDWPXLTXTDVKQZYDWGLYEDVLQZYDDVMZWEDWNWEV + GWNVGZVNVOVAYAYAABWHDWNOPGGHIXDJKYBYCVHVIVJVPWGWPWSWLWGWOWRGGWGWRWOWGYFWC + WRWOTYEYFLYGVQXTABWHDWNHIJKYBYHVRVSVTWAVJWB $. $d .x. i j $. smadiadetg.x $e |- .x. = ( .r ` R ) $. @@ -262535,25 +262764,25 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, .x. ( ( K ( ( N minMatR1 R ) ` M ) K ) |` ( { K } X. N ) ) ) ) $= ( vi vj wcel wceq cbs cfv w3a csn cxp cur c0g cif cmpt2 cof cminmar1 cres cv co cmarrep cvv snex a1i cfn matrcl elex adantr syl 3ad2ant1 simp13 crg - wa ccrg crngrng mp1i eqid rngidcl ifcld fconstmpt2 eqidd offval22 rngridm - rng0cl mpancom 3ad2ant3 ad2antrl iftrue oveq2d 3eqtr4d wn rngrz pm2.61ian - iffalse 3adant2 mpt2eq3dva eqtrd minmar1val syld3an3 reseq1d wss 3ad2ant2 - simp2 snssi ssid resmpt2 syl2anc mpt2snif 3eqtrd 3simpb syl12anc 3eqtr4rd - marrepval ) IBSZHJSZEDUAUBZSZUCZHUDZJUEZEUDUEZQRXMJRUMZHTZDUFUBZDUGUBZUHZ - UIZFUJZUNZQRXMJXQEXSUHZUIZXOHHIJDUKUNZUBUNZXNULZYBUNHHIEJDUOUNZUNUNZXNULZ - XLYCQRXMJEXTFUNZUIYEXLQRXMJEXTFXOYAUPUPXJXJXMUPSXLHUQURXHXIJUPSZXKXHJUSSZ - DUPSZVGYMABDJIKLUTYNYMYOJUSVAVBVCVDXHXIXKQUMZXMSZXPJSZVEXLYQYRUCZDVFSZXTX - JSDVHSZYTYSMDVIZVJYTXQXRXSXJXJDXRXJVKZXRVKZVLXJDXSUUCXSVKZVRVMVCXOQRXMJEU - ITXLQRXMJEVNURXLYAVOVPXLQRXMJYLYDXLYRYLYDTZYQXQXLYRVGZUUFXQUUGVGZEXRFUNZE - YLYDXLUUIETZXQYRXKXHUUJXIYTXKUUJUUAYTXKMUUBVJZXJDFXREUUCPUUDVQVSVTWAUUHXT - XREFXQXTXRTUUGXQXRXSWBVBWCXQYDETUUGXQEXSWBVBWDXQWEZUUGVGEXSFUNZXSYLYDXLUU - MXSTZUULYRXKXHUUNXIYTXKUUNUUKXJDFEXSUUCPUUEWFVSVTWAUULYLUUMTUUGUULXTXSEFX - QXRXSWHWCVBUULYDXSTUUGXQEXSWHVBWDWGWIWJWKXLYHYAXOYBXLYHQRJJYPHTZXTYPXPIUN - ZUHZUIZXNULZQRXMJUUQUIZYAXLYGUURXNXHXIXKXIYGUURTXHXIXKWQZABYFDXRQRHHIJXSK - LYFVKUUDUUEWLWMWNXLXMJWOZJJWOZUUSUUTTXIXHUVBXKHJWRWPZUVCXLJWSURZQRJJXMJUU - QWTXAUUTYATXLJXTUUPQRHXBURXCWCXLYKQRJJUUOYDUUPUHZUIZXNULZQRXMJUVFUIZYEXLY - JUVGXNXLXHXKVGXIXIYJUVGTXHXIXKXDUVAUVAABYIDEQRHHIJXSKLYIVKUUEXGXEWNXLUVBU - VCUVHUVITUVDUVEQRJJXMJUVFWTXAUVIYETXLJYDUUPQRHXBURXCXF $. + wa ccrg crngring mp1i ringidcl ring0cl ifcld fconstmpt2 offval22 ringridm + eqid eqidd mpancom 3ad2ant3 ad2antrl iftrue oveq2d 3eqtr4d ringrz iffalse + wn pm2.61ian 3adant2 mpt2eq3dva eqtrd simp2 minmar1val syld3an3 wss snssi + reseq1d 3ad2ant2 resmpt2 syl2anc mpt2snif 3eqtrd 3simpb syl12anc 3eqtr4rd + ssid marrepval ) IBSZHJSZEDUAUBZSZUCZHUDZJUEZEUDUEZQRXMJRUMZHTZDUFUBZDUGU + BZUHZUIZFUJZUNZQRXMJXQEXSUHZUIZXOHHIJDUKUNZUBUNZXNULZYBUNHHIEJDUOUNZUNUNZ + XNULZXLYCQRXMJEXTFUNZUIYEXLQRXMJEXTFXOYAUPUPXJXJXMUPSXLHUQURXHXIJUPSZXKXH + JUSSZDUPSZVGYMABDJIKLUTYNYMYOJUSVAVBVCVDXHXIXKQUMZXMSZXPJSZVEXLYQYRUCZDVF + SZXTXJSDVHSZYTYSMDVIZVJYTXQXRXSXJXJDXRXJVQZXRVQZVKXJDXSUUCXSVQZVLVMVCXOQR + XMJEUITXLQRXMJEVNURXLYAVRVOXLQRXMJYLYDXLYRYLYDTZYQXQXLYRVGZUUFXQUUGVGZEXR + FUNZEYLYDXLUUIETZXQYRXKXHUUJXIYTXKUUJUUAYTXKMUUBVJZXJDFXREUUCPUUDVPVSVTWA + UUHXTXREFXQXTXRTUUGXQXRXSWBVBWCXQYDETUUGXQEXSWBVBWDXQWGZUUGVGEXSFUNZXSYLY + DXLUUMXSTZUULYRXKXHUUNXIYTXKUUNUUKXJDFEXSUUCPUUEWEVSVTWAUULYLUUMTUUGUULXT + XSEFXQXRXSWFWCVBUULYDXSTUUGXQEXSWFVBWDWHWIWJWKXLYHYAXOYBXLYHQRJJYPHTZXTYP + XPIUNZUHZUIZXNULZQRXMJUUQUIZYAXLYGUURXNXHXIXKXIYGUURTXHXIXKWLZABYFDXRQRHH + IJXSKLYFVQUUDUUEWMWNWQXLXMJWOZJJWOZUUSUUTTXIXHUVBXKHJWPWRZUVCXLJXFURZQRJJ + XMJUUQWSWTUUTYATXLJXTUUPQRHXAURXBWCXLYKQRJJUUOYDUUPUHZUIZXNULZQRXMJUVFUIZ + YEXLYJUVGXNXLXHXKVGXIXIYJUVGTXHXIXKXCUVAUVAABYIDEQRHHIJXSKLYIVQUUEXGXDWQX + LUVBUVCUVHUVITUVDUVEQRJJXMJUVFWSWTUVIYETXLJYDUUPQRHXAURXBXE $. $( The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals @@ -262563,14 +262792,14 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, smadiadetg $p |- ( ( M e. B /\ K e. N /\ S e. ( Base ` R ) ) -> ( D ` ( K ( M ( N matRRep R ) S ) K ) ) = ( S .x. ( E ` ( K ( ( N subMat R ) ` M ) K ) ) ) ) $= - ( wcel cbs cfv co w3a cmarrep cminmar1 csubma eqid ccrg a1i crngrng simp1 - crg simp3 simp2 marrepcl syl32anc minmar1cl smadiadetglem2 smadiadetglem1 - mp1i syl22anc mdetrsca wceq smadiadet 3adant3 eqcomd oveq2d eqtrd ) IBQZH - JQZEDRSZQZUAZHHIEJDUBTTTZCSEHHIJDUCTSTZCSZFTEHHIJDUDTSTGSZFTVKABCDFHVIJVL - EVMNKLVIUEPDUFQZVKMUGVKDUJQZVGVJVHVHVLBQVPVQVKMDUHURZVGVHVJUIZVGVHVJUKZVG - VHVJULZWAABDEHHIJKLUMUNVTVKVQVGVHVHVMBQVRVSWAWAABDHHIJKLUOUSWAABCDEFGHIJK - LMNOPUPABCDEGHIJKLMNOUQUTVKVNVOEFVKVOVNVGVHVOVNVAVJABCDGHIJKLMNOVBVCVDVEV - F $. + ( wcel cbs cfv co w3a cmarrep cminmar1 csubma eqid ccrg a1i crngring mp1i + crg simp1 simp3 simp2 marrepcl syl32anc minmar1cl syl22anc smadiadetglem2 + smadiadetglem1 mdetrsca wceq smadiadet 3adant3 eqcomd oveq2d eqtrd ) IBQZ + HJQZEDRSZQZUAZHHIEJDUBTTTZCSEHHIJDUCTSTZCSZFTEHHIJDUDTSTGSZFTVKABCDFHVIJV + LEVMNKLVIUEPDUFQZVKMUGVKDUJQZVGVJVHVHVLBQVPVQVKMDUHUIZVGVHVJUKZVGVHVJULZV + GVHVJUMZWAABDEHHIJKLUNUOVTVKVQVGVHVHVMBQVRVSWAWAABDHHIJKLUPUQWAABCDEFGHIJ + KLMNOPURABCDEGHIJKLMNOUSUTVKVNVOEFVKVOVNVGVHVOVNVAVJABCDGHIJKLMNOVBVCVDVE + VF $. $} ${ @@ -262665,24 +262894,24 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, matinv $p |- ( ( R e. CRing /\ M e. B /\ ( D ` M ) e. V ) -> ( M e. U /\ ( I ` M ) = ( ( H ` ( D ` M ) ) .xb ( J ` M ) ) ) ) $= ( ccrg wcel cfv w3a cmulr cur co eqid casa crg cfn matrcl simpld 3ad2ant2 - cvv simp1 matassa syl2anc assarng syl simp2 cbs assalmod crngrng 3ad2ant1 - clmod csca simp3 rnginvcl wceq matsca2 eleqtrd wf maduf ffvelrnd lmodvscl - fveq2d syl3anc assaassr syl13anc madurid oveq2d unitlinv 3eqtr3d 3ad2ant3 - oveq1d unitcl rngidcl lmodvsass lmodvs1 3eqtrd assaass madulid invrvald - oveqd ) DUBUCZJBUCZJCUDZLUCZUEZBAAUFUDZFAUGUDZHJWSGUDZJIUDZEUHZPXBUIZXCUI - ZQTXAAUJUCZAUKUCZXAKULUCZWQXIWRWQXKWTWRXKDUPUCABDKJMPUMUNUOZWQWRWTUQZADKM - URUSZAUTVAZWQWRWTVBZXAAVGUCZXDAVHUDZVCUDZUCZXEBUCZXFBUCXAXIXQXNAVDVAZXAXD - DVCUDZXSXADUKUCZWTXDYCUCWQWRYDWTDVEVFZWQWRWTVIZYCDLGWSRSYCUIZVJUSXADXRVCX - AXKWQDXRVKXLXMADKUBMVLUSZVRZVMZXABBJIWQWRBBIVNWTABDIKMNPVOVFXPVPZXDEXRXSB - AXEPXRUIZUAXSUIZVQVSXAJXFXBUHZXDJXEXBUHZEUHZXDWSXCEUHZEUHZXCXAXIXTWRYAYNY - PVKXNYJXPYKXDXSEXBXRBAJXEPYLYMUAXGVTWAXAYOYQXDEXAWRWQYOYQVKXPXMABCDEXBXCI - JKMPNOXHXGUAWBUSWCXAXDWSXRUFUDZUHZXCEUHZXRUGUDZXCEUHZYRXCXAYTUUBXCEXAXDWS - DUFUDZUHZDUGUDZYTUUBXAYDWTUUEUUFVKYEYFDUUDLUUFGWSRSUUDUIUUFUIWDUSXAUUDYSX - DWSXADXRUFYHVRWPXADXRUGYHVRWEWGXAXQXTWSXSUCXCBUCZUUAYRVKYBYJXAWSYCXSWTWQW - SYCUCWRYCDLWSYGRWHWFYIVMXAXJUUGXOBAXCPXHWIVAZXDWSEYSXRXSBAXCPYLUAYMYSUIWJ - WAXAXQUUGUUCXCVKYBUUHEUUBXRBAXCPYLUAUUBUIWKUSWEZWLXAXFJXBUHZXDXEJXBUHZEUH - ZYRXCXAXIXTYAWRUUJUULVKXNYJYKXPXDXSEXBXRBAXEJPYLYMUAXGWMWAXAUUKYQXDEXAWRW - QUUKYQVKXPXMABCDEXBXCIJKMPNOXHXGUAWNUSWCUUIWLWO $. + cvv simp1 matassa syl2anc assaring syl simp2 clmod csca assalmod crngring + 3ad2ant1 simp3 ringinvcl wceq matsca2 fveq2d eleqtrd wf ffvelrnd lmodvscl + cbs maduf syl3anc assaassr syl13anc madurid oveq2d unitlinv oveqd 3eqtr3d + oveq1d unitcl 3ad2ant3 ringidcl lmodvsass lmodvs1 3eqtrd assaass invrvald + madulid ) DUBUCZJBUCZJCUDZLUCZUEZBAAUFUDZFAUGUDZHJWSGUDZJIUDZEUHZPXBUIZXC + UIZQTXAAUJUCZAUKUCZXAKULUCZWQXIWRWQXKWTWRXKDUPUCABDKJMPUMUNUOZWQWRWTUQZAD + KMURUSZAUTVAZWQWRWTVBZXAAVCUCZXDAVDUDZVQUDZUCZXEBUCZXFBUCXAXIXQXNAVEVAZXA + XDDVQUDZXSXADUKUCZWTXDYCUCWQWRYDWTDVFVGZWQWRWTVHZYCDLGWSRSYCUIZVIUSXADXRV + QXAXKWQDXRVJXLXMADKUBMVKUSZVLZVMZXABBJIWQWRBBIVNWTABDIKMNPVRVGXPVOZXDEXRX + SBAXEPXRUIZUAXSUIZVPVSXAJXFXBUHZXDJXEXBUHZEUHZXDWSXCEUHZEUHZXCXAXIXTWRYAY + NYPVJXNYJXPYKXDXSEXBXRBAJXEPYLYMUAXGVTWAXAYOYQXDEXAWRWQYOYQVJXPXMABCDEXBX + CIJKMPNOXHXGUAWBUSWCXAXDWSXRUFUDZUHZXCEUHZXRUGUDZXCEUHZYRXCXAYTUUBXCEXAXD + WSDUFUDZUHZDUGUDZYTUUBXAYDWTUUEUUFVJYEYFDUUDLUUFGWSRSUUDUIUUFUIWDUSXAUUDY + SXDWSXADXRUFYHVLWEXADXRUGYHVLWFWGXAXQXTWSXSUCXCBUCZUUAYRVJYBYJXAWSYCXSWTW + QWSYCUCWRYCDLWSYGRWHWIYIVMXAXJUUGXOBAXCPXHWJVAZXDWSEYSXRXSBAXCPYLUAYMYSUI + WKWAXAXQUUGUUCXCVJYBUUHEUUBXRBAXCPYLUAUUBUIWLUSWFZWMXAXFJXBUHZXDXEJXBUHZE + UHZYRXCXAXIXTYAWRUUJUULVJXNYJYKXPXDXSEXBXRBAXEJPYLYMUAXGWNWAXAUUKYQXDEXAW + RWQUUKYQVJXPXMABCDEXBXCIJKMPNOXHXGUAWPUSWCUUIWMWO $. $} ${ @@ -262695,19 +262924,19 @@ According to Wikipedia ("Laplace expansion", 08-Mar-2019, in the underlying ring. (Contributed by Stefan O'Rear, 17-Jul-2018.) $) matunit $p |- ( ( R e. CRing /\ M e. B ) -> ( M e. U <-> ( D ` M ) e. V ) ) $= - ( wcel cfv wceq eqid simpld sylancom co ccrg wa cinvr cbs cur crg crngrng - cmulr ad2antrr ffvelrnda adantr wf cfn cvv matrcl ad2antlr matrng syl2anc - rnginvcl ffvelrnd unitrinv fveq2d simpll mdetmul syl3anc 3eqtr3d unitlinv - mdetf simplr mdet1 invrvald w3a cmadu cvsca matinv 3expa impbida ) DUANZF - BNZUBZFENZFCOZHNZVTWAUBZWCWBDUCOZOZFAUCOZOZCOZPWDDUDOZDDUHOZHDUEOZWEWBWIW - JQZWKQZWLQZMWEQZVRDUFNZVSWADUGUIZVTWBWJNWAVRBWJFCABCDWJGJIKWMVHZUJUKWDBWJ - WHCVRBWJCULVSWAWSUIVTWAAUFNZWHBNZWDGUMNZWQWTVSXBVRWAVSXBDUNNABDGFIKUORUPZ - WRADGIUQURZBAEWGFLWGQZKUSSZUTWDFWHAUHOZTZCOZAUEOZCOZWBWIWKTZWLWDXHXJCVTWA - WTXHXJPXDAXGEXJWGFLXEXGQZXJQZVASVBWDVRVSXAXIXLPVRVSWAVCZVRVSWAVIZXFABCDXG - WKFWHGIKJWNXMVDVEWDVRXBXKWLPXOXCACDWLXJGJIXNWOVJURZVFWDWHFXGTZCOZXKWIWBWK - TZWLWDXRXJCVTWAWTXRXJPXDAXGEXJWGFLXEXMXNVGSVBWDVRXAVSXSXTPXOXFXPABCDXGWKW - HFGIKJWNXMVDVEXQVFVKRVRVSWCWAVRVSWCVLWAWHWFFGDVMTZOAVNOZTPABCDYBEWEWGYAFG - HIYAQJKLMWPXEYBQVORVPVQ $. + ( wcel cfv wceq eqid simpld sylancom co ccrg cinvr cbs cmulr cur crngring + wa crg ad2antrr mdetf ffvelrnda adantr wf cfn cvv matrcl ad2antlr matring + syl2anc ringinvcl ffvelrnd unitrinv fveq2d simpll mdetmul syl3anc 3eqtr3d + simplr mdet1 unitlinv invrvald w3a cmadu cvsca matinv 3expa impbida ) DUA + NZFBNZUGZFENZFCOZHNZVTWAUGZWCWBDUBOZOZFAUBOZOZCOZPWDDUCOZDDUDOZHDUEOZWEWB + WIWJQZWKQZWLQZMWEQZVRDUHNZVSWADUFUIZVTWBWJNWAVRBWJFCABCDWJGJIKWMUJZUKULWD + BWJWHCVRBWJCUMVSWAWSUIVTWAAUHNZWHBNZWDGUNNZWQWTVSXBVRWAVSXBDUONABDGFIKUPR + UQZWRADGIURUSZBAEWGFLWGQZKUTSZVAWDFWHAUDOZTZCOZAUEOZCOZWBWIWKTZWLWDXHXJCV + TWAWTXHXJPXDAXGEXJWGFLXEXGQZXJQZVBSVCWDVRVSXAXIXLPVRVSWAVDZVRVSWAVHZXFABC + DXGWKFWHGIKJWNXMVEVFWDVRXBXKWLPXOXCACDWLXJGJIXNWOVIUSZVGWDWHFXGTZCOZXKWIW + BWKTZWLWDXRXJCVTWAWTXRXJPXDAXGEXJWGFLXEXMXNVJSVCWDVRXAVSXSXTPXOXFXPABCDXG + WKWHFGIKJWNXMVEVFXQVGVKRVRVSWCWAVRVSWCVLWAWHWFFGDVMTZOAVNOZTPABCDYBEWEWGY + AFGHIYAQJKLMWPXEYBQVORVPVQ $. $} $( @@ -262787,21 +263016,21 @@ left to right implication is valid (see ~ cramer0 ), because assuming the slesolinv $p |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( ( D ` X ) e. ( Unit ` R ) /\ ( X .x. Z ) = Y ) ) -> Z = ( ( I ` X ) .x. Y ) ) $= - ( wcel wa co c0 wne ccrg cfv cui wceq w3a cotp cmmul cbs crngrng adantl - eqid crg 3ad2ant1 cfn matrcl simpld adantr 3ad2ant2 cmap anim2i 3adant3 - cvv anim1i 3ad2ant3 slesolvec sylc syl6eleq anim12ci matrng syl matunit - simpr wb bicomd ad2ant2lr biimpd adantrd 3impia rnginvcl syl2anc eleq2i - biimpi mavmulass cur cmulr matmulr oveqd unitlinv eqtrd 1mavmul 3eqtr3d - oveq1d oveq2 ) GUAUBZDUCRZSZIBRZJHRZSZICUDDUEUDZRZIKETZJUFZSZUGZIFUDZID - GGGUHUITZTZKETZXHXDETZKXHJETZXGADUJUDZDEXIGXHKILXNUMZOWRXADUNRZXFWQXPWP - DUKZULZUOZXAWRGUPRZXFWSXTWTWSXTDVDRABDGILMUQURUSZUTZXGKHXNGVATXGWPXPSZX - ASZXEKHRWRXAYDXFWRYCXAWQXPWPXQVBVEVCXFWRXEXAXCXEVNVFABDEGHIJKLMNOVGVHNV - IZXIUMZXGAUNRZIAUEUDZRZXHAUJUDZRXGXTXPSZYGWRXAYKXFWRXPXAXTXRYAVJVCADGLV - KVLZWRXAXFYIWRXASZXCYIXEYMXCYIWQWSXCYIVOWPWTWQWSSYIXCABCDYHIGXBLPMYHUMZ - XBUMVMVPVQVRVSVTZYJAYHFIYNQYJUMWAWBXAWRIYJRZXFWSYPWTWSYPBYJIMWCWDUSUTWE - XGXKAWFUDZKETKXGXJYQKEXGXJXHIAWGUDZTZYQXGXIYRXHIXGXTWQSZXIYRUFWRXAYTXFW - RWQXAXTWPWQVNYAVJVCADXIGUCLYFWHVLWIXGYGYIYSYQUFYLYOAYRYHYQFIYNQYRUMYQUM - WJWBWKWNXGAXNDEGKLXOOXSYBYEWLWKXFWRXLXMUFZXAXEUUAXCXDJXHEWOULVFWM $. + ( wcel wa co wne ccrg cfv cui wceq w3a cotp cmmul cbs eqid crg crngring + c0 adantl 3ad2ant1 cfn matrcl simpld adantr 3ad2ant2 cmap anim2i anim1i + cvv 3adant3 simpr 3ad2ant3 slesolvec sylc syl6eleq anim12ci matring syl + matunit bicomd ad2ant2lr biimpd adantrd 3impia ringinvcl syl2anc eleq2i + wb biimpi mavmulass cur cmulr matmulr oveqd unitlinv eqtrd oveq1d oveq2 + 1mavmul 3eqtr3d ) GUMUAZDUBRZSZIBRZJHRZSZICUCDUDUCZRZIKETZJUEZSZUFZIFUC + ZIDGGGUGUHTZTZKETZXHXDETZKXHJETZXGADUIUCZDEXIGXHKILXNUJZOWRXADUKRZXFWQX + PWPDULZUNZUOZXAWRGUPRZXFWSXTWTWSXTDVDRABDGILMUQURUSZUTZXGKHXNGVATXGWPXP + SZXASZXEKHRWRXAYDXFWRYCXAWQXPWPXQVBVCVEXFWRXEXAXCXEVFVGABDEGHIJKLMNOVHV + INVJZXIUJZXGAUKRZIAUDUCZRZXHAUIUCZRXGXTXPSZYGWRXAYKXFWRXPXAXTXRYAVKVEAD + GLVLVMZWRXAXFYIWRXASZXCYIXEYMXCYIWQWSXCYIWCWPWTWQWSSYIXCABCDYHIGXBLPMYH + UJZXBUJVNVOVPVQVRVSZYJAYHFIYNQYJUJVTWAXAWRIYJRZXFWSYPWTWSYPBYJIMWBWDUSU + TWEXGXKAWFUCZKETKXGXJYQKEXGXJXHIAWGUCZTZYQXGXIYRXHIXGXTWQSZXIYRUEWRXAYT + XFWRWQXAXTWPWQVFYAVKVEADXIGUBLYFWHVMWIXGYGYIYSYQUEYLYOAYRYHYQFIYNQYRUJY + QUJWJWAWKWLXGAXNDEGKLXOOXSYBYEWNWKXFWRXLXMUEZXAXEUUAXCXDJXHEWMUNVGWO $. $( The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the @@ -262812,19 +263041,19 @@ left to right implication is valid (see ~ cramer0 ), because assuming the -> ( ( X .x. Z ) = Y <-> Z = ( ( I ` X ) .x. Y ) ) ) $= ( wcel cfv co c0 wne ccrg wa cui w3a wceq simpl1 simpl2 simp3 slesolinv anim1i syl3anc oveq2 cmmul cur cmulr cfn simpr cvv matrcl simpld adantr - cotp anim12ci 3adant3 eqid matmulr syl oveqd crg crngrng adantl matunit - matrng wb ad2ant2lr biimp3ar unitrinv syl2anc eqtrd oveq1d cbs 3ad2ant1 - 3ad2ant2 cmap eleq2i biimpi mavmulass 1mavmul 3eqtr3d sylan9eqr impbida - rnginvcl ) GUAUBZDUCRZUDZIBRZJHRZUDZICSDUESZRZUFZIKETZJUGZKIFSZJETZUGZX - CXEUDWQWTXBXEUDXHWQWTXBXEUHWQWTXBXEUIXCXBXEWQWTXBUJULABCDEFGHIJKLMNOPQU - KUMXHXCXDIXGETZJKXGIEUNXCIXFDGGGVDUOTZTZJETAUPSZJETXIJXCXKXLJEXCXKIXFAU - QSZTZXLXCXJXMIXFXCGURRZWPUDZXJXMUGWQWTXPXBWQWPWTXOWOWPUSWRXOWSWRXODUTRA - BDGILMVAVBVCZVEVFADXJGUCLXJVGZVHVIVJXCAVKRZIAUESZRZXNXLUGXCXODVKRZUDZXS - WQWTYCXBWQYBWTXOWPYBWODVLVMZXQVEVFADGLVOVIZWQWTYAXBWPWRYAXBVPWOWSABCDXT - IGXALPMXTVGZXAVGVNVQVRZAXMXTXLFIYFQXMVGXLVGVSVTWAWBXCADWCSZDEXJGIJXFLYH - VGZOWQWTYBXBYDWDZWTWQXOXBXQWEZWTWQJYHGWFTZRZXBWSYMWRWSYMHYLJNWGWHVMWEZX - RWTWQIAWCSZRZXBWRYPWSWRYPBYOIMWGWHVCWEXCXSYAXFYORYEYGYOAXTFIYFQYOVGWNVT - WIXCAYHDEGJLYIOYJYKYNWJWKWLWM $. + cotp anim12ci 3adant3 eqid matmulr syl oveqd crngring adantl matring wb + crg matunit ad2ant2lr biimp3ar unitrinv syl2anc eqtrd 3ad2ant1 3ad2ant2 + oveq1d cmap eleq2i biimpi ringinvcl mavmulass 1mavmul 3eqtr3d sylan9eqr + cbs impbida ) GUAUBZDUCRZUDZIBRZJHRZUDZICSDUESZRZUFZIKETZJUGZKIFSZJETZU + GZXCXEUDWQWTXBXEUDXHWQWTXBXEUHWQWTXBXEUIXCXBXEWQWTXBUJULABCDEFGHIJKLMNO + PQUKUMXHXCXDIXGETZJKXGIEUNXCIXFDGGGVDUOTZTZJETAUPSZJETXIJXCXKXLJEXCXKIX + FAUQSZTZXLXCXJXMIXFXCGURRZWPUDZXJXMUGWQWTXPXBWQWPWTXOWOWPUSWRXOWSWRXODU + TRABDGILMVAVBVCZVEVFADXJGUCLXJVGZVHVIVJXCAVORZIAUESZRZXNXLUGXCXODVORZUD + ZXSWQWTYCXBWQYBWTXOWPYBWODVKVLZXQVEVFADGLVMVIZWQWTYAXBWPWRYAXBVNWOWSABC + DXTIGXALPMXTVGZXAVGVPVQVRZAXMXTXLFIYFQXMVGXLVGVSVTWAWDXCADWMSZDEXJGIJXF + LYHVGZOWQWTYBXBYDWBZWTWQXOXBXQWCZWTWQJYHGWETZRZXBWSYMWRWSYMHYLJNWFWGVLW + CZXRWTWQIAWMSZRZXBWRYPWSWRYPBYOIMWFWGVCWCXCXSYAXFYORYEYGYOAXTFIYFQYOVGW + HVTWIXCAYHDEGJLYIOYJYKYNWJWKWLWN $. $} $d A z $. $d B z $. $d D z $. $d N z $. $d R z $. $d V z $. $d X z $. @@ -262836,15 +263065,16 @@ left to right implication is valid (see ~ cramer0 ), because assuming the /\ ( D ` X ) e. ( Unit ` R ) ) -> E. z e. V ( X .x. z ) = Y ) $= ( wcel wa cfv co eqid c0 wne ccrg cui w3a cv wceq wrex cinvr cbs cmap crg - cmulr crngrng adantl 3ad2ant1 cfn matrcl simpld 3ad2ant2 anim12ci 3adant3 - cvv adantr matrng syl wb matunit bicomd ad2ant2lr biimp3a rnginvcl eleq2i - syl2anc mavmulcl syl6eleqr slesolinvbi biimprd impancom rspcimedv pm2.43i - biimpi ) GUAUBZEUCPZQZICPZJHPZQZIDREUDRZPZUEZIAUFZFSJUGZAHUHWKWMWKAIBUIRZ - RZJFSZHWKWPEUJRZGUKSZHWKBWQEEUMRZFGWOJKNWQTWSTWEWHEULPZWJWDWTWCEUNUOZUPWH - WEGUQPZWJWFXBWGWFXBEVCPBCEGIKLURUSVDZUTWKBULPZIBUDRZPZWOBUJRZPWKXBWTQZXDW - EWHXHWJWEWTWHXBXAXCVAVBBEGKVEVFWEWHWJXFWDWFWJXFVGWCWGWDWFQXFWJBCDEXEIGWIK - OLXETZWITVHVIVJVKXGBXEWNIXIWNTZXGTVLVNWHWEJWRPZWJWGXKWFWGXKHWRJMVMWBUOUTV - OMVPWKWKWLWPUGZWMWKWKQWMXLWKWMXLVGWKBCDEFWNGHIJWLKLMNOXJVQVDVRVSVTWA $. + cmulr crngring adantl 3ad2ant1 cfn matrcl simpld adantr 3ad2ant2 anim12ci + cvv 3adant3 matring wb matunit bicomd ad2ant2lr biimp3a ringinvcl syl2anc + syl eleq2i biimpi mavmulcl syl6eleqr slesolinvbi biimprd impancom pm2.43i + rspcimedv ) GUAUBZEUCPZQZICPZJHPZQZIDREUDRZPZUEZIAUFZFSJUGZAHUHWKWMWKAIBU + IRZRZJFSZHWKWPEUJRZGUKSZHWKBWQEEUMRZFGWOJKNWQTWSTWEWHEULPZWJWDWTWCEUNUOZU + PWHWEGUQPZWJWFXBWGWFXBEVCPBCEGIKLURUSUTZVAWKBULPZIBUDRZPZWOBUJRZPWKXBWTQZ + XDWEWHXHWJWEWTWHXBXAXCVBVDBEGKVEVMWEWHWJXFWDWFWJXFVFWCWGWDWFQXFWJBCDEXEIG + WIKOLXETZWITVGVHVIVJXGBXEWNIXIWNTZXGTVKVLWHWEJWRPZWJWGXKWFWGXKHWRJMVNVOUO + VAVPMVQWKWKWLWPUGZWMWKWKQWMXLWKWMXLVFWKBCDEFWNGHIJWLKLMNOXJVRUTVSVTWBWA + $. $} ${ @@ -262860,20 +263090,20 @@ the ith column replaced by a (column) vector equals the ith component of cramerimplem1 $p |- ( ( ( N e. Fin /\ R e. CRing /\ I e. N ) /\ Z e. V ) -> ( D ` E ) = ( Z ` I ) ) $= ( wcel wa cfv co wceq eqid cfn ccrg w3a cmarrep csubma csn cdif cmdat crg - cmulr crngrng anim2i ancomd 3adant3 adantr simp3 cur cmat 1marepvmarrepid - anim1i fveq2i syl2anc eqcomd fveq2d a1i fveq1d cbs simpl2 cmatrepV eqcomi - ma1repvcl syl5eqel wi cmap wf elmapi ffvelrn ex syl eleq2s com12 3ad2ant3 - smadiadetr syl22anc eqtrd 1marepvsma1 oveq2d diffi mdet1 3ad2ant2 rngridm - imp 3eqtrd ) GUAOZDUBOZFGOZUCZIHOZPZECQFFEFIQZGDUDRRRZCQZWTFFEGDUERQRZGFU - FZUGZDUHRZQZDUJQZRZWTWSEXACWSXAEWSDUIOZWNPZWPWRPZXAESWQXKWRWNWOXKWPWNWOPZ - WNXJWOXJWNDUKZULUMUNUOZWQWPWRWNWOWPUPZUTZDAUQQZFGHEILAGDURRZUQJVAZMUSVBVC - VDWSXBXAGDUHRZQZXIWSXACYACYASWSNVEVFWSWOEXSVGQZOWPWTDVGQZOZYBXISWNWOWPWRV - HWSEFXRIGDVIRRQZYCMWSXKWRWPPYFYCOXOWSWPWRXQUMAYCIDXRFGHJXSAVGAXSJVJVALXRT - VKVBVLWQWPWRXPUOWQWRYEWPWNWRYEVMWOWRWPYEWPYEVMZIYDGVNRZHIYHOGYDIVOZYGIYDG - VPYIWPYEGYDFIVQVRVSLVTWAWBWLZDWTFEGWCWDWEWSXIWTXEDURRZUQQZXFQZXHRWTDUQQZX - HRZWTWSXGYMWTXHWSXCYLXFWSXKXLXCYLSXOXQDXRFGHEILXTMWFVBVDWGWSYMYNWTXHWSWOX - EUAOZPZYMYNSWQYQWRWNWOYQWPXMYPWOWNYPWOGXDWHUTUMUNUOYKXFDYNYLXEXFTYKTYLTYN - TZWIVSWGWSXJYEYOWTSWQXJWRWOWNXJWPXNWJUOYJYDDXHYNWTYDTXHTYRWKVBWMWM $. + cmulr crngring anim2i ancomd adantr simp3 anim1i cur cmat 1marepvmarrepid + 3adant3 fveq2i syl2anc eqcomd fveq2d a1i fveq1d simpl2 cmatrepV ma1repvcl + cbs eqcomi syl5eqel wi wf elmapi ffvelrn ex syl eleq2s com12 3ad2ant3 imp + cmap smadiadetr syl22anc eqtrd 1marepvsma1 oveq2d diffi 3ad2ant2 ringridm + mdet1 3eqtrd ) GUAOZDUBOZFGOZUCZIHOZPZECQFFEFIQZGDUDRRRZCQZWTFFEGDUERQRZG + FUFZUGZDUHRZQZDUJQZRZWTWSEXACWSXAEWSDUIOZWNPZWPWRPZXAESWQXKWRWNWOXKWPWNWO + PZWNXJWOXJWNDUKZULUMUTUNZWQWPWRWNWOWPUOZUPZDAUQQZFGHEILAGDURRZUQJVAZMUSVB + VCVDWSXBXAGDUHRZQZXIWSXACYACYASWSNVEVFWSWOEXSVJQZOWPWTDVJQZOZYBXISWNWOWPW + RVGWSEFXRIGDVHRRQZYCMWSXKWRWPPYFYCOXOWSWPWRXQUMAYCIDXRFGHJXSAVJAXSJVKVALX + RTVIVBVLWQWPWRXPUNWQWRYEWPWNWRYEVMWOWRWPYEWPYEVMZIYDGWCRZHIYHOGYDIVNZYGIY + DGVOYIWPYEGYDFIVPVQVRLVSVTWAWBZDWTFEGWDWEWFWSXIWTXEDURRZUQQZXFQZXHRWTDUQQ + ZXHRZWTWSXGYMWTXHWSXCYLXFWSXKXLXCYLSXOXQDXRFGHEILXTMWGVBVDWHWSYMYNWTXHWSW + OXEUAOZPZYMYNSWQYQWRWNWOYQWPXMYPWOWNYPWOGXDWIUPUMUTUNYKXFDYNYLXEXFTYKTYLT + YNTZWLVRWHWSXJYEYOWTSWQXJWRWOWNXJWPXNWJUNYJYDDXHYNWTYDTXHTYRWKVBWMWM $. $} ${ @@ -262900,25 +263130,25 @@ the ith column replaced by a (column) vector equals the ith component of ( vi vj vl ccrg wcel wa co wceq w3a cv cmulr cfv cmpt cgsu cif cbs eqid cmpt2 3ad2ant1 cfn cvv matrcl simpld adantr 3ad2ant2 cxp cmap wi anim2i simpl ancomd matbas2 syl syl6reqr eleq2d biimpd ex com12 pm2.43a impcom - 3adant3 crg crngrng anim12i c0 wne ne0i adantl 3jca eleq2i biimpi simp3 - mavmulsolcl imp syl21anc simpr cur cmatrepV ma1repvcl syl5eqel syl12anc - ad2ant2r eleqtrrd mamuval simp2 c0g mulmarep1gsum2 syl113anc mpt2eq3dva - eqcomd marepvval syl3anc syl5req 3eqtrd ) CUDUEZHIUEZUFZKBUEZLJUEZUFZKM - DUGLUHZUIZKFEUGUAUBIICUCIUAUJZUCUJZKUGYDUBUJZFUGCUKULZUGUMUNUGZURUAUBII - YEHUHYCLULYCYEKUGUOZURZGYBCUPULZICYFUAUCUBEIIUDKFTYJUQZYFUQXQXTXOYAXOXP - VJZUSXTXQIUTUEZYAXRYMXSXRYMCVAUEABCIKNOVBVCZVDZVEZYPYPXQXTKYJIIVFVGUGZU - EZYAXTXQYRXRXQYRVHXSXQXRYRXQXRXRYRVHZXOXRYSVHXPXOXRYSXOXRUFZXRYRYTBYQKY - TYQAUPULZBYTYMXOUFYQUUAUHYTXOYMXRYMXOYNVIVKACYJIUDNYKVLVMOVNZVOVPVQVDVR - VSVDVTWAYBFBYQYBCWBUEZYMUFZMJUEZXPFBUEXQXTUUDYAXQUUCXTYMXOUUCXPCWCVDZYO - WDWAYBYMYMIWEWFZUIZXOLYJIVGUGZUEZUFZYAUUEYBYMYMUUGYPYPXQXTUUGYAXPUUGXOI - HWGWHUSWIXQXTUUKYAXQXOXTUUJYLXSUUJXRXSUUJJUUILPWJWKWHWDWAXQXTYAWLZUUHUU - KUFYAUUEKYJYQJCDUUIIIUDMLYKYQUQPSUUIUQWMWNWOZXQXTXPYAXOXPWPUSZUUDUUEXPU - FUFFHAWQULZMICWRUGZUGULBQABMCUUOHIJNOPUUOUQZWSWTXAXQXTYQBUHZYAXOXRUURXP - XSYTBYQUUBXJXBWAXCXDYBUAUBIIYGYHYBYCIUEZYEIUEZUIUUCXRUUEXPUIZUUSUUTYAYG - YHUHYBUUSUUCUUTXQXTUUCYAUUFUSUSYBUUSUVAUUTYBXRUUEXPXTXQXRYAXRXSVJVEZUUM - UUNWIUSYBUUSUUTXEYBUUSUUTWLYBUUSYAUUTUULUSABMCDUUOFYCYEHIJKCXFULZLUCNOP - UUQUVCUQQSXGXHXIYBGHKLUUPUGULZYIRYBXRXSXPUVDYIUHUVBXTXQXSYAXRXSWPVEUUNA - BLUUPCUAUBHKIJNOUUPUQPXKXLXMXN $. + 3adant3 crg crngring anim12i c0 wne ne0i adantl 3jca eleq2i mavmulsolcl + biimpi simp3 syl21anc simpr cmatrepV ma1repvcl syl5eqel syl12anc eqcomd + imp ad2ant2r eleqtrrd mamuval simp2 mulmarep1gsum2 syl113anc mpt2eq3dva + cur c0g marepvval syl3anc syl5req 3eqtrd ) CUDUEZHIUEZUFZKBUEZLJUEZUFZK + MDUGLUHZUIZKFEUGUAUBIICUCIUAUJZUCUJZKUGYDUBUJZFUGCUKULZUGUMUNUGZURUAUBI + IYEHUHYCLULYCYEKUGUOZURZGYBCUPULZICYFUAUCUBEIIUDKFTYJUQZYFUQXQXTXOYAXOX + PVJZUSXTXQIUTUEZYAXRYMXSXRYMCVAUEABCIKNOVBVCZVDZVEZYPYPXQXTKYJIIVFVGUGZ + UEZYAXTXQYRXRXQYRVHXSXQXRYRXQXRXRYRVHZXOXRYSVHXPXOXRYSXOXRUFZXRYRYTBYQK + YTYQAUPULZBYTYMXOUFYQUUAUHYTXOYMXRYMXOYNVIVKACYJIUDNYKVLVMOVNZVOVPVQVDV + RVSVDVTWAYBFBYQYBCWBUEZYMUFZMJUEZXPFBUEXQXTUUDYAXQUUCXTYMXOUUCXPCWCVDZY + OWDWAYBYMYMIWEWFZUIZXOLYJIVGUGZUEZUFZYAUUEYBYMYMUUGYPYPXQXTUUGYAXPUUGXO + IHWGWHUSWIXQXTUUKYAXQXOXTUUJYLXSUUJXRXSUUJJUUILPWJWLWHWDWAXQXTYAWMZUUHU + UKUFYAUUEKYJYQJCDUUIIIUDMLYKYQUQPSUUIUQWKXAWNZXQXTXPYAXOXPWOUSZUUDUUEXP + UFUFFHAXIULZMICWPUGZUGULBQABMCUUOHIJNOPUUOUQZWQWRWSXQXTYQBUHZYAXOXRUURX + PXSYTBYQUUBWTXBWAXCXDYBUAUBIIYGYHYBYCIUEZYEIUEZUIUUCXRUUEXPUIZUUSUUTYAY + GYHUHYBUUSUUCUUTXQXTUUCYAUUFUSUSYBUUSUVAUUTYBXRUUEXPXTXQXRYAXRXSVJVEZUU + MUUNWIUSYBUUSUUTXEYBUUSUUTWMYBUUSYAUUTUULUSABMCDUUOFYCYEHIJKCXJULZLUCNO + PUUQUVCUQQSXFXGXHYBGHKLUUPUGULZYIRYBXRXSXPUVDYIUHUVBXTXQXSYAXRXSWOVEUUN + ABLUUPCUAUBHKIJNOUUPUQPXKXLXMXN $. $} cramerimp.d $e |- D = ( N maDet R ) $. @@ -262936,15 +263166,15 @@ the ith column replaced by a (column) vector equals the ith component of -> ( ( D ` X ) .(x) ( D ` E ) ) = ( D ` H ) ) $= ( ccrg wcel wa co wceq w3a cotp cmmul cfv cmulr cfn simpl matrcl simpld cvv adantr anim12ci 3adant3 eqid matmulr syl oveqd fveq2d cramerimplem2 - simp1l simp2l cur cmatrepV crg crngrng anim12i c0 wne wi ne0i slesolvec - 3impia simp1r ma1repvcl syl12anc syl5eqel mdetmul syl3anc 3eqtr3rd - sylan ) DUCUDZIJUDZUEZLBUDZMKUDZUEZLNEUFMUGZUHZLGDJJJUIUJUFZUFZCUKLGAUL - UKZUFZCUKZHCUKLCUKGCUKFUFZWOWQWSCWOWPWRLGWOJUMUDZWHUEZWPWRUGWJWMXCWNWJW - HWMXBWHWIUNWKXBWLWKXBDUQUDABDJLOPUOUPURZUSUTADWPJUCOWPVAZVBVCVDVEWOWQHC - ABDEWPGHIJKLMNOPQRSTXEVFVEWOWHWKGBUDWTXAUGWHWIWMWNVGWJWKWLWNVHWOGIAVIUK - ZNJDVJUFUFUKZBRWODVKUDZXBUEZNKUDZWIXGBUDWJWMXIWNWJXHWMXBWHXHWIDVLZURXDV - MUTWJWMWNXJWJJVNVOZXHUEWMWNXJVPWHXHWIXLXKJIVQUSABDEJKLMNOPQTVRWGVSWHWIW - MWNVTABNDXFIJKOPQXFVAWAWBWCABCDWRFLGJOPUAUBWRVAWDWEWF $. + simp1l simp2l cur cmatrepV crg crngring anim12i c0 ne0i slesolvec sylan + wne 3impia simp1r ma1repvcl syl12anc syl5eqel mdetmul syl3anc 3eqtr3rd + wi ) DUCUDZIJUDZUEZLBUDZMKUDZUEZLNEUFMUGZUHZLGDJJJUIUJUFZUFZCUKLGAULUKZ + UFZCUKZHCUKLCUKGCUKFUFZWOWQWSCWOWPWRLGWOJUMUDZWHUEZWPWRUGWJWMXCWNWJWHWM + XBWHWIUNWKXBWLWKXBDUQUDABDJLOPUOUPURZUSUTADWPJUCOWPVAZVBVCVDVEWOWQHCABD + EWPGHIJKLMNOPQRSTXEVFVEWOWHWKGBUDWTXAUGWHWIWMWNVGWJWKWLWNVHWOGIAVIUKZNJ + DVJUFUFUKZBRWODVKUDZXBUEZNKUDZWIXGBUDWJWMXIWNWJXHWMXBWHXHWIDVLZURXDVMUT + WJWMWNXJWJJVNVRZXHUEWMWNXJWGWHXHWIXLXKJIVOUSABDEJKLMNOPQTVPVQVSWHWIWMWN + VTABNDXFIJKOPQXFVAWAWBWCABCDWRFLGJOPUAUBWRVAWDWEWF $. $} cramerimp.q $e |- ./ = ( /r ` R ) $. @@ -262963,21 +263193,21 @@ the ith column replaced by a (column) vector equals the ith component of cramerimp $p |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( ( X .x. Z ) = Y /\ ( D ` X ) e. ( Unit ` R ) ) ) -> ( Z ` I ) = ( ( D ` H ) ./ ( D ` X ) ) ) $= - ( ccrg wcel wa co wceq cfv cui w3a cmulr crg crngrng adantr 3ad2ant1 eqid - cbs mdetf cur cmatrepV cfn cvv matrcl simpld anim12i 3adant3 wne anim12ci - wf ne0i anim1i simpl 3ad2ant3 slesolvec simpr ma1repvcl syl12anc syl5eqel - c0 sylc ffvelrnd dvrcan3 syl3anc unitcl adantl crngcom 3jca cramerimplem1 - oveq1d syl2anc 3eqtr3rd cramerimplem3 3adant3r eqtrd ) EUCUDZIJUDZUEZLBUD - ZMKUDZUEZLNFUFMUGZLCUHZEUIUHZUDZUEZUJZINUHZXBGCUHZEUKUHZUFZXBDUFZHCUHZXBD - UFXFXHXBXIUFZXBDUFZXHXKXGXFEULUDZXHEUQUHZUDZXDXNXHUGWQWTXOXEWOXOWPEUMZUNZ - UOXFBXPGCWQWTBXPCVIZXEWOXTWPABCEXPJUAOPXPUPZURUNUOXFGIAUSUHZNJEUTUFUFUHZB - RXFXOJVAUDZUEZNKUDZWPYCBUDWQWTYEXEWQXOWTYDXSWRYDWSWRYDEVBUDABEJLOPVCVDUNZ - VEVFXFJVSVGZXOUEZWTUEZXAYFWQWTYJXEWQYIWTWOXOWPYHXRJIVJVHVKVFXEWQXAWTXAXDV - LVMABEFJKLMNOPQTVNVTZWQWTWPXEWOWPVOZUOABNEYBIJKOPQYBUPVPVQVRWAZXEWQXDWTXA - XDVOVMXPDEXIXCXHXBYAXCUPZUBXIUPZWBWCXFXMXJXBDXFWOXQXBXPUDZXMXJUGWQWTWOXEW - OWPVLZUOYMXEWQYPWTXDYPXAXPEXCXBYAYNWDWEVMXPEXIXHXBYAYOWFWCWIXFYDWOWPUJZYF - XHXGUGWQWTYRXEWQWTUEYDWOWPWTYDWQYGWEWQWOWTYQUNWQWPWTYLUNWGVFYKABCEGIJKNOP - QRUAWHWJWKXFXJXLXBDWQWTXAXJXLUGXDABCEFXIGHIJKLMNOPQRSTUAYOWLWMWIWN $. + ( ccrg wcel wa co wceq cfv cui w3a cmulr crg cbs crngring adantr 3ad2ant1 + wf eqid mdetf cur cmatrepV cfn cvv matrcl simpld anim12i 3adant3 wne ne0i + c0 anim12ci anim1i simpl 3ad2ant3 slesolvec sylc simpr ma1repvcl syl12anc + syl5eqel ffvelrnd dvrcan3 syl3anc unitcl adantl oveq1d 3jca cramerimplem1 + crngcom syl2anc 3eqtr3rd cramerimplem3 3adant3r eqtrd ) EUCUDZIJUDZUEZLBU + DZMKUDZUEZLNFUFMUGZLCUHZEUIUHZUDZUEZUJZINUHZXBGCUHZEUKUHZUFZXBDUFZHCUHZXB + DUFXFXHXBXIUFZXBDUFZXHXKXGXFEULUDZXHEUMUHZUDZXDXNXHUGWQWTXOXEWOXOWPEUNZUO + ZUPXFBXPGCWQWTBXPCUQZXEWOXTWPABCEXPJUAOPXPURZUSUOUPXFGIAUTUHZNJEVAUFUFUHZ + BRXFXOJVBUDZUEZNKUDZWPYCBUDWQWTYEXEWQXOWTYDXSWRYDWSWRYDEVCUDABEJLOPVDVEUO + ZVFVGXFJVJVHZXOUEZWTUEZXAYFWQWTYJXEWQYIWTWOXOWPYHXRJIVIVKVLVGXEWQXAWTXAXD + VMVNABEFJKLMNOPQTVOVPZWQWTWPXEWOWPVQZUPABNEYBIJKOPQYBURVRVSVTWAZXEWQXDWTX + AXDVQVNXPDEXIXCXHXBYAXCURZUBXIURZWBWCXFXMXJXBDXFWOXQXBXPUDZXMXJUGWQWTWOXE + WOWPVMZUPYMXEWQYPWTXDYPXAXPEXCXBYAYNWDWEVNXPEXIXHXBYAYOWIWCWFXFYDWOWPUJZY + FXHXGUGWQWTYRXEWQWTUEYDWOWPWTYDWQYGWEWQWOWTYQUOWQWPWTYLUOWGVGYKABCEGIJKNO + PQRUAWHWJWKXFXJXLXBDWQWTXAXJXLUGXDABCEFXIGHIJKLMNOPQRSTUAYOWLWMWFWN $. $} ${ @@ -263077,12 +263307,12 @@ only systems of linear equations whose matrix has a unit (of the |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) <-> ( X .x. Z ) = Y ) ) $= ( wcel wa ccrg c0 wne cfv cui w3a cv cmatrepV cmpt wceq pm3.22 cramerlem3 - syl3an1 simpl1l simpl2 simpl3 crg crngrng anim1i ancomd 3adant3 slesolvec - co wi imp sylan simpr cramerlem1 syl113anc ex impbid ) EUASZHUBUCZTZJBSKI - STZJCUDZEUEUDSZUFZLGHGUGJKHEUHVCVCUDCUDVPDVCUIUJZJLFVCKUJZVNVMVLTVOVQVSVT - VDVLVMUKABCDEFGHIJKLMNOPQRULUMVRVTVSVRVTTVLVOVQLISZVTVSVLVMVOVQVTUNVNVOVQ - VTUOVNVOVQVTUPVRVMEUQSZTZVOTZVTWAVNVOWDVQVNWCVOVNWBVMVLWBVMEURUSUTUSVAWDV - TWAABEFHIJKLMNOQVBVEVFVRVTVGABCDEFGHIJKLMNOPQRVHVIVJVK $. + co wi syl3an1 simpl1l simpl2 simpl3 crg crngring anim1i 3adant3 slesolvec + ancomd imp sylan simpr cramerlem1 syl113anc ex impbid ) EUASZHUBUCZTZJBSK + ISTZJCUDZEUEUDSZUFZLGHGUGJKHEUHUMUMUDCUDVPDUMUIUJZJLFUMKUJZVNVMVLTVOVQVSV + TUNVLVMUKABCDEFGHIJKLMNOPQRULUOVRVTVSVRVTTVLVOVQLISZVTVSVLVMVOVQVTUPVNVOV + QVTUQVNVOVQVTURVRVMEUSSZTZVOTZVTWAVNVOWDVQVNWCVOVNWBVMVLWBVMEUTVAVDVAVBWD + VTWAABEFHIJKLMNOQVCVEVFVRVTVGABCDEFGHIJKLMNOPQRVHVIVJVK $. $} $( @@ -263109,17 +263339,17 @@ only systems of linear equations whose matrix has a unit (of the $) ${ - pmatrng.p $e |- P = ( Poly1 ` R ) $. - pmatrng.c $e |- C = ( N Mat P ) $. + pmatring.p $e |- P = ( Poly1 ` R ) $. + pmatring.c $e |- C = ( N Mat P ) $. $( The set of polynomial matrices over a ring is a ring. (Contributed by AV, 6-Nov-2019.) $) - pmatrng $p |- ( ( N e. Fin /\ R e. Ring ) -> C e. Ring ) $= - ( crg wcel cfn ply1rng matrng sylan2 ) CGHDIHBGHAGHBCEJABDFKL $. + pmatring $p |- ( ( N e. Fin /\ R e. Ring ) -> C e. Ring ) $= + ( crg wcel cfn ply1ring matring sylan2 ) CGHDIHBGHAGHBCEJABDFKL $. $( The set of polynomial matrices over a ring is a left module. (Contributed by AV, 6-Nov-2019.) $) pmatlmod $p |- ( ( N e. Fin /\ R e. Ring ) -> C e. LMod ) $= - ( crg wcel cfn clmod ply1rng matlmod sylan2 ) CGHDIHBGHAJHBCEKABDFLM $. + ( crg wcel cfn clmod ply1ring matlmod sylan2 ) CGHDIHBGHAJHBCEKABDFLM $. $d N i j $. $d P i j $. pmat0op.z $e |- .0. = ( 0g ` P ) $. @@ -263127,8 +263357,8 @@ only systems of linear equations whose matrix has a unit (of the (Contributed by AV, 16-Nov-2019.) $) pmat0op $p |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` C ) = ( i e. N , j e. N |-> .0. ) ) $= - ( crg wcel cfn c0g cfv cmpt2 wceq ply1rng mat0op sylan2 ) CKLFMLBKLANODEF - FGPQBCHRABDEFGIJST $. + ( crg wcel cfn c0g cfv cmpt2 wceq ply1ring mat0op sylan2 ) CKLFMLBKLANODE + FFGPQBCHRABDEFGIJST $. $d C i j $. $d .0. i j $. $d .1. i j $. pmat1op.o $e |- .1. = ( 1r ` P ) $. @@ -263136,8 +263366,8 @@ only systems of linear equations whose matrix has a unit (of the (Contributed by AV, 16-Nov-2019.) $) pmat1op $p |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` C ) = ( i e. N , j e. N |-> if ( i = j , .1. , .0. ) ) ) $= - ( crg wcel cfn cur cfv weq cif cmpt2 wceq ply1rng mat1 sylan2 ) CMNGONBMN - APQEFGGEFRDHSTUABCIUBABDEFGHJLKUCUD $. + ( crg wcel cfn cur cfv weq cif cmpt2 wceq ply1ring mat1 sylan2 ) CMNGONBM + NAPQEFGGEFRDHSTUABCIUBABDEFGHJLKUCUD $. pmat1ovd.n $e |- ( ph -> N e. Fin ) $. pmat1ovd.r $e |- ( ph -> R e. Ring ) $. @@ -263147,7 +263377,7 @@ only systems of linear equations whose matrix has a unit (of the $( Entries of the identity polynomial matrix over a ring, deduction form. (Contributed by AV, 16-Nov-2019.) $) pmat1ovd $p |- ( ph -> ( I U J ) = if ( I = J , .1. , .0. ) ) $= - ( crg wcel ply1rng syl mat1ov ) ABCEFGHIJLNMOADTUACTUAPCDKUBUCQRSUD $. + ( crg wcel ply1ring syl mat1ov ) ABCEFGHIJLNMOADTUACTUAPCDKUBUCQRSUD $. $} ${ @@ -263251,9 +263481,9 @@ only systems of linear equations whose matrix has a unit (of the AV, 4-Dec-2019.) $) 1pmatscmul $p |- ( ( N e. Fin /\ R e. Ring /\ Q e. E ) -> ( Q .* .1. ) e. B ) $= - ( cfn wcel crg w3a 3adant3 wa co ply1rng anim2i simp3 pmatrng rngidcl syl - matvscl syl12anc ) IPQZERQZDGQZSZUKCRQZUAZUMFAQZDFHUBAQUKULUPUMULUOUKCEJU - CUDTUKULUMUEUNBRQZUQUKULURUMBCEIJKUFTABFLOUGUHBADCHGIFMKLNUIUJ $. + ( cfn wcel crg w3a 3adant3 wa ply1ring anim2i simp3 pmatring ringidcl syl + co matvscl syl12anc ) IPQZERQZDGQZSZUKCRQZUAZUMFAQZDFHUHAQUKULUPUMULUOUKC + EJUBUCTUKULUMUDUNBRQZUQUKULURUMBCEIJKUETABFLOUFUGBADCHGIFMKLNUIUJ $. $} $( @@ -263420,19 +263650,19 @@ whose entries are constant polynomials (or "scalar polynomials", see is a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.) $) 1elcpmat $p |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` C ) e. S ) $= ( vn vi vj wcel wa cfv cco1 wceq cn wral eqid mpbird cfn crg cur cv cascl - co c0g cif cbs rngidcl adantl ad2antrl cply1coe0 syl iftrue fveq2d fveq1d - ancli wb eqeq1d ralbidv adantr rng0cl iffalse pm2.61ian ralrimivva simpll - wn simplr simprl simprr pmat1ovscd 2ralbidva pmatrng cpmatel mpd3an3 ) EU - ALZCUBLZMZAUCNZDLZIUDZJUDZKUDZVTUFZONZNZCUGNZPZIQRZKERJERZVSWKWBWCWDPZCUC - NZBUENZNZWHWNNZUHZONZNZWHPZIQRZKERJERVSXAJKEEWLVSWCELZWDELZMZMZXAWLXEMZXA - WBWOONZNZWHPZIQRZXFVRWMCUINZLZMZXJVSXMWLXDVRXMVQVRXLXKCWMXKSZWMSZUJURUKUL - WNBUINZBCWMIXKWHXNWHSZGXPSZWNSZUMUNWLXAXJUSXEWLWTXIIQWLWSXHWHWLWBWRXGWLWQ - WOOWLWOWPUOUPUQUTVAVBTWLVHZXEMZXAWBWPONZNZWHPZIQRZVSYEXTXDVSVRWHXKLZMZYEV - RYGVQVRYFXKCWHXNXQVCURUKWNXPBCWHIXKWHXNXQGXRXSUMUNULYAWTYDIQYAWSYCWHYAWBW - RYBYAWQWPOXTWQWPPXEWLWOWPVDVBUPUQUTVATVEVFVSWJXAJKEEXEWIWTIQXEWGWSWHXEWBW - FWRXEWEWQOXEWNABCVTWMWCWDEWHGHXSXQXOVQVRXDVGVQVRXDVIVSXBXCVJVSXBXCVKVTSZV - LUPUQUTVAVMTVQVRVTAUINZLZWAWKUSVSAUBLYJABCEGHVNYIAVTYISZYHUJUNYIABCDJKIVT - EUBFGHYKVOVPT $. + co c0g cif cbs ringidcl ancli adantl ad2antrl cply1coe0 syl iftrue fveq2d + wb fveq1d eqeq1d ralbidv adantr wn ring0cl pm2.61ian simpll simplr simprl + iffalse ralrimivva simprr pmat1ovscd 2ralbidva pmatring cpmatel mpd3an3 ) + EUALZCUBLZMZAUCNZDLZIUDZJUDZKUDZVTUFZONZNZCUGNZPZIQRZKERJERZVSWKWBWCWDPZC + UCNZBUENZNZWHWNNZUHZONZNZWHPZIQRZKERJERVSXAJKEEWLVSWCELZWDELZMZMZXAWLXEMZ + XAWBWOONZNZWHPZIQRZXFVRWMCUINZLZMZXJVSXMWLXDVRXMVQVRXLXKCWMXKSZWMSZUJUKUL + UMWNBUINZBCWMIXKWHXNWHSZGXPSZWNSZUNUOWLXAXJURXEWLWTXIIQWLWSXHWHWLWBWRXGWL + WQWOOWLWOWPUPUQUSUTVAVBTWLVCZXEMZXAWBWPONZNZWHPZIQRZVSYEXTXDVSVRWHXKLZMZY + EVRYGVQVRYFXKCWHXNXQVDUKULWNXPBCWHIXKWHXNXQGXRXSUNUOUMYAWTYDIQYAWSYCWHYAW + BWRYBYAWQWPOXTWQWPPXEWLWOWPVIVBUQUSUTVATVEVJVSWJXAJKEEXEWIWTIQXEWGWSWHXEW + BWFWRXEWEWQOXEWNABCVTWMWCWDEWHGHXSXQXOVQVRXDVFVQVRXDVGVSXBXCVHVSXBXCVKVTS + ZVLUQUSUTVAVMTVQVRVTAUINZLZWAWKURVSAUBLYJABCEGHVNYIAVTYISZYHUJUOYIABCDJKI + VTEUBFGHYKVOVPT $. $d C a b c $. $d N a b c i j x y $. $d P a b c $. $d R a b c x y $. $d S y $. @@ -263442,35 +263672,35 @@ whose entries are constant polynomials (or "scalar polynomials", see cpmatacl $p |- ( ( N e. Fin /\ R e. Ring ) -> A. x e. S A. y e. S ( x ( +g ` C ) y ) e. S ) $= ( vi vj wcel wa cfv co wceq wral wi eqid vc va vb cfn crg cv cplusg cascl - cbs wrex cpmatelimp2 r19.26-2 rngacl 3expb wb ply1sca eqcomd fveq2d oveqd - eleq1d adantr mpbird ad3antlr imp fveq2 eqeq2d adantl simpr ancomd anim1i - csca ex ad2antrr matplusgcell syl oveq12 ancoms ply1rng ad4antlr ply1lmod - cghm clmod asclghm eleq2d biimpd adantrd adantld ghmlin syl3anc sylan9eqr - eqtrd expd anassrs rexlimdva com23 impd ralimdvva syl5bir expr com34 syld - rspcedvd imp32 simpl pmatrng w3a anim2i df-3an sylibr cpmatel2 ralrimivva - cpmatpmat ) GUDMZEUEMZNZAUFZBUFZCUGOZPZFMZABFFXOXPFMZXQFMZNZNZXTKUFZLUFZX - SPZUAUFZDUHOZOZQZUAEUIOZUJZLGRKGRZXOYAYBYNXOYAXPCUIOZMZYEYFXPPZUBUFZYIOZQ - ZUBYLUJZLGRKGRZNZYBYNSYIYOCDEFKLUBYLXPGHIJYOTZYLTZYITZUKXOYBUUCYNXOYBXQYO - MZYEYFXQPZUCUFZYIOZQZUCYLUJZLGRKGRZNUUCYNSZYIYOCDEFKLUCYLXQGHIJUUDUUEUUFU - KXOUUGUUMUUNXOUUGUUCUUMYNXOUUGUUCUUMYNSZSXOUUGNYPUUBUUOXOUUGYPUUBUUOSXOUU - GYPNZNZUUBUUMYNUUBUUMNUUAUULNZLGRKGRUUQYNUUAUULKLGGULUUQUURYMKLGGUUQYEGMY - FGMNZNZUUAUULYMUUTYTUULYMSUBYLUUTYRYLMZNZUULYTYMUVBUUKYTYMSZUCYLUUTUVAUUI - YLMZUUKUVCSUUTUVAUVDNZNZUUKYTYMUVFUUKYTNZYMUVFUVGNZYKYGYRUUIDVKOZUGOZPZYI - OZQZUAUVKYLUVFUVKYLMZUVGUUTUVEUVNXNUVEUVNSXMUUPUUSXNUVEUVNXNUVENUVNYRUUIE - UGOZPZYLMZXNUVAUVDUVQYLUVOEYRUUIUUEUVOTUMUNXNUVNUVQUOUVEXNUVKUVPYLXNUVJUV - OYRUUIXNUVIEUGXNEUVIDEUEIUPZUQURUSUTVAVBVLVCVDVAYHUVKQZYKUVMUOUVHUVSYJUVL - YGYHUVKYIVEVFVGUVHYGYQUUHDUGOZPZUVLUVHYPUUGNZUUSNZYGUWAQUUTUWCUVEUVGUUQUW - BUUSUUQUUGYPXOUUPVHVIVJVMCYOUVTXRDYEYFGXPXQJUUDXRTZUVTTZVNVOUVGUVFUWAYSUU - JUVTPZUVLYTUUKUWAUWFQYQYSUUHUUJUVTVPVQUVFUVLUWFUVFYIUVIDWAPMYRUVIUIOZMZUU - IUWGMZUVLUWFQUVFYIUVIDUUFUVITXNDUEMXMUUPUUSUVEDEIVRVSXNDWBMXMUUPUUSUVEDEI - VTVSWCUUTUVEUWHUUTUVAUWHUVDXOUVAUWHSUUPUUSXOUVAUWHXOYLUWGYRXOEUVIUIXNEUVI - QZXMUVRVGURWDWEVMWFVDUUTUVEUWIUUTUVDUWIUVAUUTUVDUWIUUTYLUWGUUIUUTEUVIUIXN - UWJXMUUPUUSUVRVCURWDWEWGVDUVJUVTUVIDYRYIUUIUWGUWGTUVJTUWEWHWIUQWJWKXBVLWL - WMWNWOWNWPWQWRWLWSWPVLWTWPXAWOXAXCYDXMXNXSYOMZXTYNUOXOXMYCXMXNXDVAXOXNYCX - MXNVHVAYDCUEMZYPUUGUWKXOUWLYCCDEGIJXEVAYDXMXNYAXFZYPYDXOYANUWMYCYAXOYAYBX - DXGXMXNYAXHXIYOCDEFXPGUEHIJUUDXLVOYDXMXNYBXFZUUGYDXOYBNUWNYCYBXOYAYBVHXGX - MXNYBXHXIYOCDEFXQGUEHIJUUDXLVOYOXRCXPXQUUDUWDUMWIYIYOCDEFKLUAYLXSGHIJUUDU - UEUUFXJWIVBXK $. + cbs wrex cpmatelimp2 r19.26-2 csca ringacl 3expb wb ply1sca eqcomd fveq2d + oveqd eleq1d adantr mpbird ad3antlr imp fveq2 eqeq2d adantl ancomd anim1i + ex simpr ad2antrr matplusgcell oveq12 ancoms cghm ply1ring ad4antlr clmod + syl ply1lmod asclghm eleq2d biimpd adantrd adantld ghmlin sylan9eqr eqtrd + syl3anc rspcedvd expd anassrs rexlimdva com23 impd ralimdvva syl5bir expr + com34 syld imp32 simpl pmatring anim2i df-3an sylibr cpmatpmat ralrimivva + w3a cpmatel2 ) GUDMZEUEMZNZAUFZBUFZCUGOZPZFMZABFFXOXPFMZXQFMZNZNZXTKUFZLU + FZXSPZUAUFZDUHOZOZQZUAEUIOZUJZLGRKGRZXOYAYBYNXOYAXPCUIOZMZYEYFXPPZUBUFZYI + OZQZUBYLUJZLGRKGRZNZYBYNSYIYOCDEFKLUBYLXPGHIJYOTZYLTZYITZUKXOYBUUCYNXOYBX + QYOMZYEYFXQPZUCUFZYIOZQZUCYLUJZLGRKGRZNUUCYNSZYIYOCDEFKLUCYLXQGHIJUUDUUEU + UFUKXOUUGUUMUUNXOUUGUUCUUMYNXOUUGUUCUUMYNSZSXOUUGNYPUUBUUOXOUUGYPUUBUUOSX + OUUGYPNZNZUUBUUMYNUUBUUMNUUAUULNZLGRKGRUUQYNUUAUULKLGGULUUQUURYMKLGGUUQYE + GMYFGMNZNZUUAUULYMUUTYTUULYMSUBYLUUTYRYLMZNZUULYTYMUVBUUKYTYMSZUCYLUUTUVA + UUIYLMZUUKUVCSUUTUVAUVDNZNZUUKYTYMUVFUUKYTNZYMUVFUVGNZYKYGYRUUIDUMOZUGOZP + ZYIOZQZUAUVKYLUVFUVKYLMZUVGUUTUVEUVNXNUVEUVNSXMUUPUUSXNUVEUVNXNUVENUVNYRU + UIEUGOZPZYLMZXNUVAUVDUVQYLUVOEYRUUIUUEUVOTUNUOXNUVNUVQUPUVEXNUVKUVPYLXNUV + JUVOYRUUIXNUVIEUGXNEUVIDEUEIUQZURUSUTVAVBVCVKVDVEVBYHUVKQZYKUVMUPUVHUVSYJ + UVLYGYHUVKYIVFVGVHUVHYGYQUUHDUGOZPZUVLUVHYPUUGNZUUSNZYGUWAQUUTUWCUVEUVGUU + QUWBUUSUUQUUGYPXOUUPVLVIVJVMCYOUVTXRDYEYFGXPXQJUUDXRTZUVTTZVNWAUVGUVFUWAY + SUUJUVTPZUVLYTUUKUWAUWFQYQYSUUHUUJUVTVOVPUVFUVLUWFUVFYIUVIDVQPMYRUVIUIOZM + ZUUIUWGMZUVLUWFQUVFYIUVIDUUFUVITXNDUEMXMUUPUUSUVEDEIVRVSXNDVTMXMUUPUUSUVE + DEIWBVSWCUUTUVEUWHUUTUVAUWHUVDXOUVAUWHSUUPUUSXOUVAUWHXOYLUWGYRXOEUVIUIXNE + UVIQZXMUVRVHUSWDWEVMWFVEUUTUVEUWIUUTUVDUWIUVAUUTUVDUWIUUTYLUWGUUIUUTEUVIU + IXNUWJXMUUPUUSUVRVDUSWDWEWGVEUVJUVTUVIDYRYIUUIUWGUWGTUVJTUWEWHWKURWIWJWLV + KWMWNWOWPWOWQWRWSWMWTWQVKXAWQXBWPXBXCYDXMXNXSYOMZXTYNUPXOXMYCXMXNXDVBXOXN + YCXMXNVLVBYDCUEMZYPUUGUWKXOUWLYCCDEGIJXEVBYDXMXNYAXKZYPYDXOYANUWMYCYAXOYA + YBXDXFXMXNYAXGXHYOCDEFXPGUEHIJUUDXIWAYDXMXNYBXKZUUGYDXOYBNUWNYCYBXOYAYBVL + XFXMXNYBXGXHYOCDEFXQGUEHIJUUDXIWAYOXRCXPXQUUDUWDUNWKYIYOCDEFKLUAYLXSGHIJU + UDUUEUUFXLWKVCXJ $. $( The set of all constant polynomial matrices over a ring ` R ` is closed under inversion. (Contributed by AV, 17-Nov-2019.) (Proof shortened by @@ -263478,25 +263708,25 @@ whose entries are constant polynomials (or "scalar polynomials", see cpmatinvcl $p |- ( ( N e. Fin /\ R e. Ring ) -> A. x e. S ( ( invg ` C ) ` x ) e. S ) $= ( vi vj wcel crg wa cv cminusg cfv wceq eqid adantr vc va cfn co cbs wrex - cascl wral cpmatelimp2 csca wi ply1sca adantl eqcomd fveq2d fveq1d rnggrp - cgrp grpinvcl sylan eqeltrd ex ad2antrr imp fveq2 eqeq2d ply1rng ad3antlr - wb w3a simplr simpr 3jca matinvgcell syl cghm clmod asclghm eleq2d biimpd - ply1lmod ghminv syl2anc sylan9eqr eqtrd rspcedvd rexlimdva ralimdvva syld - expimpd simpll pmatrng cpmatpmat 3expa cpmatel2 syl3anc mpbird ralrimiva - ) FUCLZDMLZNZAOZBPQZQZELZAEXAXBELZNZXEJOZKOZXDUDZUAOZCUGQZQZRZUADUEQZUFZK - FUHJFUHZXAXFXQXAXFXBBUEQZLZXHXIXBUDZUBOZXLQZRZUBXOUFZKFUHJFUHZNXQXLXRBCDE - JKUBXOXBFGHIXRSZXOSZXLSZUIXAXSYEXQXAXSNZYDXPJKFFYIXHFLXIFLNZNZYCXPUBXOYKY - AXOLZNZYCXPYMYCNZXNXJYACUJQZPQZQZXLQZRZUAYQXOYMYQXOLZYCYKYLYTXAYLYTUKXSYJ - XAYLYTXAYLNZYQYADPQZQZXOUUAYAYPUUBUUAYODPUUADYOXADYORZYLWTUUDWSCDMHULUMZT - UNUOUPXADURLZYLUUCXOLWTUUFWSDUQUMXODUUBYAYGUUBSUSUTVAVBVCVDTXKYQRZXNYSVIY - NUUGXMYRXJXKYQXLVEVFUMYNXJXTCPQZQZYRYNCMLZXSYJVJZXJUUIRYKUUKYLYCYKUUJXSYJ - WTUUJWSXSYJCDHVGVHZXAXSYJVKYIYJVLVMVCBXRCXHXIFUUHXCXBIYFUUHSZXCSZVNVOYCYM - UUIYBUUHQZYRXTYBUUHVEYMYRUUOYMXLYOCVPUDLYAYOUEQZLZYRUUORYMXLYOCYHYOSYKUUJ - YLUULTYKCVQLZYLWTUURWSXSYJCDHWAVHTVRYKYLUUQXAYLUUQUKXSYJXAYLUUQXAXOUUPYAX - ADYOUEUUEUOVSVTVCVDUUPYOCXLYPUUHYAUUPSYPSUUMWBWCUNWDWEWFVBWGWHWJWIVDXGWSW - TXDXRLZXEXQVIWSWTXFWKWSWTXFVKXGBURLZXSUUSXAUUTXFXABMLUUTBCDFHIWLBUQVOTWSW - TXFXSXRBCDEXBFMGHIYFWMWNXRBXCXBYFUUNUSWCXLXRBCDEJKUAXOXDFGHIYFYGYHWOWPWQW - R $. + wral cpmatelimp2 csca wi ply1sca adantl eqcomd fveq2d fveq1d cgrp ringgrp + cascl grpinvcl sylan eqeltrd ex ad2antrr imp wb fveq2 eqeq2d w3a ply1ring + ad3antlr simplr simpr 3jca matinvgcell cghm clmod ply1lmod asclghm eleq2d + syl biimpd ghminv syl2anc sylan9eqr eqtrd rspcedvd rexlimdva expimpd syld + ralimdvva simpll pmatring cpmatpmat cpmatel2 syl3anc mpbird ralrimiva + 3expa ) FUCLZDMLZNZAOZBPQZQZELZAEXAXBELZNZXEJOZKOZXDUDZUAOZCURQZQZRZUADUE + QZUFZKFUGJFUGZXAXFXQXAXFXBBUEQZLZXHXIXBUDZUBOZXLQZRZUBXOUFZKFUGJFUGZNXQXL + XRBCDEJKUBXOXBFGHIXRSZXOSZXLSZUHXAXSYEXQXAXSNZYDXPJKFFYIXHFLXIFLNZNZYCXPU + BXOYKYAXOLZNZYCXPYMYCNZXNXJYACUIQZPQZQZXLQZRZUAYQXOYMYQXOLZYCYKYLYTXAYLYT + UJXSYJXAYLYTXAYLNZYQYADPQZQZXOUUAYAYPUUBUUAYODPUUADYOXADYORZYLWTUUDWSCDMH + UKULZTUMUNUOXADUPLZYLUUCXOLWTUUFWSDUQULXODUUBYAYGUUBSUSUTVAVBVCVDTXKYQRZX + NYSVEYNUUGXMYRXJXKYQXLVFVGULYNXJXTCPQZQZYRYNCMLZXSYJVHZXJUUIRYKUUKYLYCYKU + UJXSYJWTUUJWSXSYJCDHVIVJZXAXSYJVKYIYJVLVMVCBXRCXHXIFUUHXCXBIYFUUHSZXCSZVN + VTYCYMUUIYBUUHQZYRXTYBUUHVFYMYRUUOYMXLYOCVOUDLYAYOUEQZLZYRUUORYMXLYOCYHYO + SYKUUJYLUULTYKCVPLZYLWTUURWSXSYJCDHVQVJTVRYKYLUUQXAYLUUQUJXSYJXAYLUUQXAXO + UUPYAXADYOUEUUEUNVSWAVCVDUUPYOCXLYPUUHYAUUPSYPSUUMWBWCUMWDWEWFVBWGWJWHWIV + DXGWSWTXDXRLZXEXQVEWSWTXFWKWSWTXFVKXGBUPLZXSUUSXAUUTXFXABMLUUTBCDFHIWLBUQ + VTTWSWTXFXSXRBCDEXBFMGHIYFWMWRXRBXCXBYFUUNUSWCXLXRBCDEJKUAXOXDFGHIYFYGYHW + NWOWPWQ $. $d C k $. $d N c i j k l x y $. $d P k $. $d R k l $. $( Lemma for ~ cpmatmcl . (Contributed by AV, 18-Nov-2019.) $) @@ -263508,32 +263738,32 @@ whose entries are constant polynomials (or "scalar polynomials", see cbs wi cpmatelimp adantr ralcom r19.26-2 bitr3i nfv nfra1 simp-4r simplrl nfan simpr matecld simplrr jca32 adantlr oveq2 fveq2d fveq1d eqeq1d oveq1 anbi12d rspcva exp4b com23 imp31 ralimdva impancom cply1mul sylc r19.21bi - a1i imp an32s mpteq2dva oveq2d cmnd rngmnd anim2i gsumz syl ad4antr eqtrd - ancomd ex ralrimi cn0 nnnn0 adantl ply1rng ad4antlr rngcl syl3anc simp-4l - wb ralrimiva coe1fzgsumd ralbidva mpbird syl5bi expd expr impd syld imp32 - ) JUAPZEUBPZQZAUCZFPZBUCZFPZKUCZDIJGUCZIUCZYAUDZYGHUCZYCUDZDUERZUDZUFUGUD - UHRRZEUIRZUJZKSTZHJTZGJTZXTYBYACULRZPZYEYFOUCZYAUDZUHRZRZYNUJZKSTOJTZGJTZ - QYDYRUMZYSCDEFGOKYAJLMNYSUKZUNXTYTUUGUUHXTYTUUGUUHUMXTYTQZYDUUGYRUUJYDYCY - SPZYEUUAYIYCUDZUHRZRZYNUJZKSTZHJTOJTZQZUUGYRUMZXTYDUURUMYTYSCDEFOHKYCJLMN - UUIUNUOUUJUUKUUQUUSXTYTUUKUUQUUSUMXTYTUUKQZQZUUQUUSUVAUUQQZUUFYQGJUVBYFJP - ZUUFYQUMZUVAUVCUUQUVDUVAUVCQZUUFUUQYQUVEUUFUUQYQUMUUQUUPOJTZHJTUVEUUFQZYQ - UUPOHJJUPUVGUVFYPHJUVEUUFYIJPZUVFYPUMZUVEUVHUUFUVIUVAUVCUVHUUFUVIUMUVAUVC - UVHQZQZUUFUVFYPUUFUVFQZUUEUUOQZOJTZKSTZUVKYPUVLUVMKSTOJTUVOUUEUUOOKJSUQUV - MOKJSUPURUVKUVOYPUVKUVOQZYPEIJYEYLUHRRZUFZUGUDZYNUJZKSTZUVPUVTKSUVKUVOKUV - KKUSUVNKSUTVCUVPYESPZUVTUVPUWBQZUVSEIJYNUFZUGUDZYNUWCUVRUWDEUGUWCIJUVQYNU - VPYGJPZUWBUVQYNUJZUVPUWFQZUWGKSUWHXSYHDULRZPZYJUWIPZQQZYEYHUHRZRZYNUJZYEY - JUHRZRZYNUJZQZKSTZUWGKSTUVKUWFUWLUVOUVKUWFQZXSUWJUWKXRXSUUTUVJUWFVAUXACYS - DYFYGUWIYAJNUWIUKZUUIUVAUVCUVHUWFVBUVKUWFVDZUVKYTUWFXTYTUUKUVJVBUOVEZUXAC - YSDYGYIUWIYCJNUXBUUIUXCUVAUVCUVHUWFVFUVKUUKUWFXTYTUUKUVJVFUOVEZVGVHUVPUWF - UWTUVKUWFUVOUWTUXAUVNUWSKSUVKUWFUWBUVNUWSUMZUVKUWBUWFUXFUVKUWBUWFUVNUWSUW - FUVNQUWSUMUVKUWBQZUVMUWSOYGJUUAYGUJZUUEUWOUUOUWRUXHUUDUWNYNUXHYEUUCUWMUXH - UUBYHUHUUAYGYFYAVIVJVKVLUXHUUNUWQYNUXHYEUUMUWPUXHUULYJUHUUAYGYIYCVMVJVKVL - VNVOWDVPVQVRVSVTWEUWIDEYKYHYJYNKMUXBYNUKZYKUKZWAWBWCWFWGWHXTUWEYNUJZUUTUV - JUVOUWBXTEWIPZXRQUXKXTXRUXLXSUXLXREWJWKWPJIEUAYNUXIWLWMWNWOWQWRUVKYPUWAXG - UVOUVKYOUVTKSUXGYMUVSYNUXGIUWIDEYEYLJMUXBXRXSUUTUVJUWBVAUWBYEWSPUVKYEWTXA - UVKYLUWIPZIJTUWBUVKUXMIJUXADUBPZUWJUWKUXMXSUXNXRUUTUVJUWFDEMXBXCUXDUXEUWI - DYKYHYJUXBUXJXDXEXHUOXRXSUUTUVJUWBXFXIVLXJUOXKWQXLXMXNVQVRVSXLWQVQVTWEVSW - QXNXOXPVQWQXOXPXQ $. + a1i imp an32s mpteq2dva oveq2d cmnd ringmnd anim2i ancomd gsumz syl eqtrd + ad4antr ex ralrimi wb cn0 nnnn0 adantl ply1ring ad4antlr ringcl ralrimiva + syl3anc simp-4l coe1fzgsumd ralbidva mpbird syl5bi expd expr impd imp32 + syld ) JUAPZEUBPZQZAUCZFPZBUCZFPZKUCZDIJGUCZIUCZYAUDZYGHUCZYCUDZDUERZUDZU + FUGUDUHRRZEUIRZUJZKSTZHJTZGJTZXTYBYACULRZPZYEYFOUCZYAUDZUHRZRZYNUJZKSTOJT + ZGJTZQYDYRUMZYSCDEFGOKYAJLMNYSUKZUNXTYTUUGUUHXTYTUUGUUHUMXTYTQZYDUUGYRUUJ + YDYCYSPZYEUUAYIYCUDZUHRZRZYNUJZKSTZHJTOJTZQZUUGYRUMZXTYDUURUMYTYSCDEFOHKY + CJLMNUUIUNUOUUJUUKUUQUUSXTYTUUKUUQUUSUMXTYTUUKQZQZUUQUUSUVAUUQQZUUFYQGJUV + BYFJPZUUFYQUMZUVAUVCUUQUVDUVAUVCQZUUFUUQYQUVEUUFUUQYQUMUUQUUPOJTZHJTUVEUU + FQZYQUUPOHJJUPUVGUVFYPHJUVEUUFYIJPZUVFYPUMZUVEUVHUUFUVIUVAUVCUVHUUFUVIUMU + VAUVCUVHQZQZUUFUVFYPUUFUVFQZUUEUUOQZOJTZKSTZUVKYPUVLUVMKSTOJTUVOUUEUUOOKJ + SUQUVMOKJSUPURUVKUVOYPUVKUVOQZYPEIJYEYLUHRRZUFZUGUDZYNUJZKSTZUVPUVTKSUVKU + VOKUVKKUSUVNKSUTVCUVPYESPZUVTUVPUWBQZUVSEIJYNUFZUGUDZYNUWCUVRUWDEUGUWCIJU + VQYNUVPYGJPZUWBUVQYNUJZUVPUWFQZUWGKSUWHXSYHDULRZPZYJUWIPZQQZYEYHUHRZRZYNU + JZYEYJUHRZRZYNUJZQZKSTZUWGKSTUVKUWFUWLUVOUVKUWFQZXSUWJUWKXRXSUUTUVJUWFVAU + XACYSDYFYGUWIYAJNUWIUKZUUIUVAUVCUVHUWFVBUVKUWFVDZUVKYTUWFXTYTUUKUVJVBUOVE + ZUXACYSDYGYIUWIYCJNUXBUUIUXCUVAUVCUVHUWFVFUVKUUKUWFXTYTUUKUVJVFUOVEZVGVHU + VPUWFUWTUVKUWFUVOUWTUXAUVNUWSKSUVKUWFUWBUVNUWSUMZUVKUWBUWFUXFUVKUWBUWFUVN + UWSUWFUVNQUWSUMUVKUWBQZUVMUWSOYGJUUAYGUJZUUEUWOUUOUWRUXHUUDUWNYNUXHYEUUCU + WMUXHUUBYHUHUUAYGYFYAVIVJVKVLUXHUUNUWQYNUXHYEUUMUWPUXHUULYJUHUUAYGYIYCVMV + JVKVLVNVOWDVPVQVRVSVTWEUWIDEYKYHYJYNKMUXBYNUKZYKUKZWAWBWCWFWGWHXTUWEYNUJZ + UUTUVJUVOUWBXTEWIPZXRQUXKXTXRUXLXSUXLXREWJWKWLJIEUAYNUXIWMWNWPWOWQWRUVKYP + UWAWSUVOUVKYOUVTKSUXGYMUVSYNUXGIUWIDEYEYLJMUXBXRXSUUTUVJUWBVAUWBYEWTPUVKY + EXAXBUVKYLUWIPZIJTUWBUVKUXMIJUXADUBPZUWJUWKUXMXSUXNXRUUTUVJUWFDEMXCXDUXDU + XEUWIDYKYHYJUXBUXJXEXGXFUOXRXSUUTUVJUWBXHXIVLXJUOXKWQXLXMXNVQVRVSXLWQVQVT + WEVSWQXNXOXQVQWQXOXQXP $. $d S c i j $. $( The set of all constant polynomial matrices over a ring ` R ` is closed @@ -263541,19 +263771,19 @@ whose entries are constant polynomials (or "scalar polynomials", see cpmatmcl $p |- ( ( N e. Fin /\ R e. Ring ) -> A. x e. S A. y e. S ( x ( .r ` C ) y ) e. S ) $= ( vc vi vj wcel crg wa cv cfv co wral vk cfn cmulr cco1 wceq cn cmpt cgsu - c0g cpmatmcllem cbs ply1rng ad4antlr eqid cpmatpmat 3expa anim12dan simpr - adantr anim1i matmulcell syl3anc fveq2d fveq1d eqeq1d ralbidva wb pmatrng - mpbird simpl w3a anim2i df-3an sylibr syl rngcl cpmatel ralrimivva ) GUBN - ZEONZPZAQZBQZCUCRZSZFNZABFFWAWBFNZWCFNZPZPZWFKQZLQZMQZWESZUDRZRZEUIRZUEZK - UFTZMGTZLGTZWJXAWKDUAGWLUAQZWBSXBWMWCSDUCRZSUGUHSZUDRZRZWQUEZKUFTZMGTZLGT - ABCDEFLMUAGKHIJUJWJWTXILGWJWLGNZPZWSXHMGXKWMGNZPZWRXGKUFXMWKUFNZPZWPXFWQX - OWKWOXEXMWOXEUEXNXMWNXDUDXMDONZWBCUKRZNZWCXQNZPZXJXLPWNXDUEVTXPVSWIXJXLDE - IULUMXKXTXLWJXTXJWAWGXRWHXSVSVTWGXRXQCDEFWBGOHIJXQUNZUOZUPVSVTWHXSXQCDEFW - CGOHIJYAUOZUPUQUSUSXKXJXLWJXJURUTCXQDXCWDUAWLWMGWBWCJYAXCUNWDUNZVAVBVCUSV - DVEVFVFVFVIWJVSVTWEXQNZWFXAVGWAVSWIVSVTVJUSWAVTWIVSVTURUSWJCONZXRXSYEWAYF - WICDEGIJVHUSWJVSVTWGVKZXRWJWAWGPYGWIWGWAWGWHVJVLVSVTWGVMVNYBVOWJVSVTWHVKZ - XSWJWAWHPYHWIWHWAWGWHURVLVSVTWHVMVNYCVOXQCWDWBWCYAYDVPVBXQCDEFLMKWEGOHIJY - AVQVBVIVR $. + cpmatmcllem ply1ring ad4antlr eqid cpmatpmat 3expa anim12dan adantr simpr + c0g cbs anim1i matmulcell syl3anc fveq2d fveq1d eqeq1d ralbidva mpbird wb + simpl pmatring w3a anim2i df-3an sylibr syl ringcl cpmatel ralrimivva ) G + UBNZEONZPZAQZBQZCUCRZSZFNZABFFWAWBFNZWCFNZPZPZWFKQZLQZMQZWESZUDRZRZEURRZU + EZKUFTZMGTZLGTZWJXAWKDUAGWLUAQZWBSXBWMWCSDUCRZSUGUHSZUDRZRZWQUEZKUFTZMGTZ + LGTABCDEFLMUAGKHIJUIWJWTXILGWJWLGNZPZWSXHMGXKWMGNZPZWRXGKUFXMWKUFNZPZWPXF + WQXOWKWOXEXMWOXEUEXNXMWNXDUDXMDONZWBCUSRZNZWCXQNZPZXJXLPWNXDUEVTXPVSWIXJX + LDEIUJUKXKXTXLWJXTXJWAWGXRWHXSVSVTWGXRXQCDEFWBGOHIJXQULZUMZUNVSVTWHXSXQCD + EFWCGOHIJYAUMZUNUOUPUPXKXJXLWJXJUQUTCXQDXCWDUAWLWMGWBWCJYAXCULWDULZVAVBVC + UPVDVEVFVFVFVGWJVSVTWEXQNZWFXAVHWAVSWIVSVTVIUPWAVTWIVSVTUQUPWJCONZXRXSYEW + AYFWICDEGIJVJUPWJVSVTWGVKZXRWJWAWGPYGWIWGWAWGWHVIVLVSVTWGVMVNYBVOWJVSVTWH + VKZXSWJWAWHPYHWIWHWAWGWHUQVLVSVTWHVMVNYCVOXQCWDWBWCYAYDVPVBXQCDEFLMKWEGOH + IJYAVQVBVGVR $. $d C m $. $d C x y $. $d N i j k m $. $d R k m $. $d S x $. $( The set of all constant polynomial matrices over a ring ` R ` is an @@ -263563,12 +263793,12 @@ whose entries are constant polynomials (or "scalar polynomials", see -> S e. ( SubGrp ` C ) ) $= ( vx vy vk vi vj vm wcel crg cfv cv wral eqid cfn wa csubg cbs wss c0 wne cplusg co cminusg cco1 c0g wceq cn crab ssrab2 syl6eqss cur 1elcpmat ne0i - syl cpmatacl cpmatinvcl r19.26 sylanbrc cgrp w3a wb ply1rng matrng sylan2 - cpmat rnggrp issubg2 3syl mpbir3and ) EUAOZCPOZUBZDAUCQOZDAUDQZUEZDUFUGZI - RZJRAUHQZUIDOJDSZWDAUJQZQDOZUBIDSZVSDKRLRMRNRUIUKQQCULQUMKUNSMESLESZNWAUO - WAWAABCDLMKNEPFGHWATZVLWJNWAUPUQVSAURQZDOWCABCDEFGHUSDWLUTVAVSWFIDSWHIDSW - IIJABCDEFGHVBIABCDEFGHVCWFWHIDVDVEVSAPOZAVFOVTWBWCWIVGVHVRVQBPOWMBCGVIABE - HVJVKAVMIJWAWEDAWGWKWETWGTVNVOVP $. + cpmat syl cpmatacl cpmatinvcl r19.26 sylanbrc cgrp w3a wb ply1ring sylan2 + matring ringgrp issubg2 3syl mpbir3and ) EUAOZCPOZUBZDAUCQOZDAUDQZUEZDUFU + GZIRZJRAUHQZUIDOJDSZWDAUJQZQDOZUBIDSZVSDKRLRMRNRUIUKQQCULQUMKUNSMESLESZNW + AUOWAWAABCDLMKNEPFGHWATZVAWJNWAUPUQVSAURQZDOWCABCDEFGHUSDWLUTVBVSWFIDSWHI + DSWIIJABCDEFGHVCIABCDEFGHVDWFWHIDVEVFVSAPOZAVGOVTWBWCWIVHVIVRVQBPOWMBCGVJ + ABEHVLVKAVMIJWAWEDAWGWKWETWGTVNVOVP $. $( The set of all constant polynomial matrices over a ring ` R ` is a subring of the ring of all polynomial matrices over the ring ` R ` . @@ -263576,10 +263806,10 @@ whose entries are constant polynomials (or "scalar polynomials", see cpmatsrgpmat $p |- ( ( N e. Fin /\ R e. Ring ) -> S e. ( SubRing ` C ) ) $= ( vx vy cfn wcel crg wa csubrg cfv csubg cv wral eqid cmulr cpmatsubgpmat - cur co 1elcpmat cpmatmcl w3a ply1rng matrng sylan2 cbs issubrg2 mpbir3and - wb syl ) EKLZCMLZNZDAOPLZDAQPLZAUCPZDLZIRJRAUAPZUDDLJDSIDSZABCDEFGHUBABCD - EFGHUEIJABCDEFGHUFURAMLZUSUTVBVDUGUNUQUPBMLVEBCGUHABEHUIUJIJDAUKPZAVCVAVF - TVATVCTULUOUM $. + cur co 1elcpmat cpmatmcl w3a wb ply1ring matring sylan2 cbs syl mpbir3and + issubrg2 ) EKLZCMLZNZDAOPLZDAQPLZAUCPZDLZIRJRAUAPZUDDLJDSIDSZABCDEFGHUBAB + CDEFGHUEIJABCDEFGHUFURAMLZUSUTVBVDUGUHUQUPBMLVEBCGUIABEHUJUKIJDAULPZAVCVA + VFTVATVCTUOUMUN $. $} ${ @@ -263651,15 +263881,15 @@ whose entries are constant polynomials (or "scalar polynomials", see mat2pmatbas $p |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( T ` M ) e. ( Base ` C ) ) $= ( vx vy wcel crg cfv cbs eqid cfn w3a cv co cascl cmpt2 mat2pmatval simp1 - cvv cpl1 fvex eqeltri csca ply1rng 3ad2ant2 3ad2ant1 clmod ply1lmod asclf - a1i wf wceq fveq2d feq2d mpbird simp2 simp3 eleq2i biimpi 3ad2ant3 matecl - ply1sca syl3anc ffvelrnd matbas2d eqeltrd ) HUAPZEQPZGBPZUBZGFRNOHHNUCZOU - CZGUDZDUERZRZUFCSRZNOABDEWDFGHQIJKLWDTZUGVTNOCWFWEDDSRZHUIMWHTZWFTVQVRVSU - HDUIPVTDEUJRUILEUJUKULUTVTWAHPZWBHPZUBZESRZWHWCWDWLWMWHWDVADUMRZSRZWHWDVA - WLWDWHWNWODWGWNTVTWJDQPZWKVRVQWPVSDELUNUOUPVTWJDUQPZWKVRVQWQVSDELURUOUPWO - TWIUSWLWMWOWHWDVTWJWMWOVBZWKVRVQWRVSVREWNSDEQLVLVCUOUPVDVEWLWJWKGASRZPZWC - WMPVTWJWKVFVTWJWKVGVTWJWTWKVSVQWTVRVSWTBWSGKVHVIVJUPAEWAWBWMGHJWMTVKVMVNV - OVP $. + cvv cpl1 fvex eqeltri a1i ply1ring 3ad2ant2 3ad2ant1 clmod ply1lmod asclf + wf csca wceq ply1sca fveq2d feq2d mpbird simp2 simp3 eleq2i biimpi matecl + 3ad2ant3 syl3anc ffvelrnd matbas2d eqeltrd ) HUAPZEQPZGBPZUBZGFRNOHHNUCZO + UCZGUDZDUERZRZUFCSRZNOABDEWDFGHQIJKLWDTZUGVTNOCWFWEDDSRZHUIMWHTZWFTVQVRVS + UHDUIPVTDEUJRUILEUJUKULUMVTWAHPZWBHPZUBZESRZWHWCWDWLWMWHWDUTDVARZSRZWHWDU + TWLWDWHWNWODWGWNTVTWJDQPZWKVRVQWPVSDELUNUOUPVTWJDUQPZWKVRVQWQVSDELURUOUPW + OTWIUSWLWMWOWHWDVTWJWMWOVBZWKVRVQWRVSVREWNSDEQLVCVDUOUPVEVFWLWJWKGASRZPZW + CWMPVTWJWKVGVTWJWKVHVTWJWTWKVSVQWTVRVSWTBWSGKVIVJVLUPAEWAWBWMGHJWMTVKVMVN + VOVP $. mat2pmatbas0.h $e |- H = ( Base ` C ) $. $( The result of a matrix transformation is a polynomial matrix. @@ -263704,35 +263934,35 @@ whose entries are constant polynomials (or "scalar polynomials", see group homomorphism. (Contributed by AV, 28-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.) $) mat2pmatghm $p |- ( ( N e. Fin /\ R e. Ring ) -> T e. ( A GrpHom C ) ) $= - ( vi vj wcel crg cfv co vx cfn cplusg eqid cgrp matrng rnggrp syl pmatrng - vy wa mat2pmatf cv cascl cmpt2 cof wceq cbs adantr ply1rng adantl simp1lr - simpl w3a simp2 simp3 simp1rl matecld ply1sclcl syl2anc simp1rr matplusg2 - matbas2d cvv fvex eqidd offval22 csca simpr 3ad2ant1 matplusgcell ply1sca - a1i 3simpc fveq2d oveqd eqtrd clmod ply1lmod asclghm eqcomd eleq2d mpbird - cghm wb ghmlin syl3anc eqtr2d mpt2eq3dva cmnd rngmnd anim1i 3anass sylibr - mndcl df-3an sylanbrc mat2pmatval anim2i oveq12d 3eqtr4d isghmd ) HUBQZER - QZUKZUAUJAUCSZCUCSZACFBGKNXPUDZXQUDZXOARQZAUEQAEHJUFZAUGUHXOCRQCUEQCDEHLM - UICUGUHABCDEFGHIJKLMNULXOUAUMZBQZUJUMZBQZUKZUKZOPHHOUMZPUMZYBYDXPTZTZDUNS - ZSZUOZOPHHYHYIYBTZYLSZUOZOPHHYHYIYDTZYLSZUOZXQTZYJFSZYBFSZYDFSZXQTYGUUAYQ - YTDUCSZUPTZYNYGYQGQYTGQUUAUUFUQYGOPCGYPDDURSZHRMUUGUDZNXOXMYFXMXNVCUSZXOD - RQZYFXNUUJXMDELUTVAUSZYGYHHQZYIHQZVDZXNYOEURSZQZYPUUGQXMXNYFUULUUMVBZUUNA - BEYHYIUUOYBHJUUOUDZKYGUULUUMVEZYGUULUUMVFZYCYEXOUULUUMVGVHZYLUUGDEYOUUOLY - LUDZUURUUHVIVJVMYGOPCGYSDUUGHRMUUHNUUIUUKUUNXNYRUUOQZYSUUGQUUQUUNABEYHYIU - UOYDHJUURKUUSUUTYCYEXOUULUUMVKVHZYLUUGDEYRUUOLUVBUURUUHVIVJVMCGUUEXQDHYQY - TMNXSUUEUDZVLVJYGUUFOPHHYPYSUUETZUOYNYGOPHHYPYSUUEYQYTUBUBVNVNUUIUUIYPVNQ - UUNYOYLVOWCYSVNQUUNYRYLVOWCYGYQVPYGYTVPVQYGOPHHUVFYMUUNYMYOYRDVRSZUCSZTZY - LSZUVFUUNYKUVIYLUUNYKYOYREUCSZTZUVIUUNYFUULUUMUKYKUVLUQYGUULYFUUMXOYFVSVT - YGUULUUMWDABUVKXPEYHYIHYBYDJKXRUVKUDWAVJYGUULUVLUVIUQZUUMXOUVMYFXOUVKUVHY - OYRXOEUVGUCXNEUVGUQXMDERLWBVAZWEWFUSVTWGWEUUNYLUVGDWNTQYOUVGURSZQZYRUVOQZ - UVJUVFUQUUNYLUVGDUVBUVGUDYGUULUUJUUMUUKVTYGUULDWHQZUUMXOUVRYFXNUVRXMDELWI - VAUSVTWJUUNUVPUUPUVAYGUULUVPUUPWOZUUMXOUVSYFXOUVOUUOYOXOUVGEURXOEUVGUVNWK - WEZWLUSVTWMUUNUVQUVCUVDYGUULUVQUVCWOZUUMXOUWAYFXOUVOUUOYRUVTWLUSVTWMUVHUU - EUVGDYOYLYRUVOUVOUDUVHUDUVEWPWQWRWSWGWRYGXMXNYJBQZVDZUUBYNUQYGXOUWBUWCXOY - FVCYGAWTQZYCYEVDZUWBYGUWDYFUKUWEXOUWDYFXOXTUWDYAAXAUHXBUWDYCYEXCXDBXPAYBY - DKXRXEUHXMXNUWBXFXGOPABDEYLFYJHRIJKLUVBXHUHYGUUCYQUUDYTXQYGXMXNYCVDZUUCYQ - UQYGXOYCUKUWFYFYCXOYCYEVCXIXMXNYCXFXDOPABDEYLFYBHRIJKLUVBXHUHYGXMXNYEVDZU - UDYTUQYGXOYEUKUWGYFYEXOYCYEVSXIXMXNYEXFXDOPABDEYLFYDHRIJKLUVBXHUHXJXKXL - $. + ( vi vj wcel crg cfv co vx vy cfn wa cplusg eqid cgrp matring ringgrp syl + pmatring mat2pmatf cv cascl cmpt2 cof wceq cbs adantr ply1ring adantl w3a + simpl simp1lr simp2 simp3 simp1rl matecld ply1sclcl syl2anc matplusg2 cvv + matbas2d simp1rr fvex a1i eqidd offval22 csca simpr 3ad2ant1 matplusgcell + 3simpc ply1sca fveq2d oveqd eqtrd clmod ply1lmod asclghm wb eqcomd eleq2d + cghm mpbird ghmlin syl3anc eqtr2d mpt2eq3dva ringmnd anim1i 3anass sylibr + cmnd mndcl df-3an sylanbrc mat2pmatval anim2i oveq12d 3eqtr4d isghmd ) HU + CQZERQZUDZUAUBAUESZCUESZACFBGKNXPUFZXQUFZXOARQZAUGQAEHJUHZAUIUJXOCRQCUGQC + DEHLMUKCUIUJABCDEFGHIJKLMNULXOUAUMZBQZUBUMZBQZUDZUDZOPHHOUMZPUMZYBYDXPTZT + ZDUNSZSZUOZOPHHYHYIYBTZYLSZUOZOPHHYHYIYDTZYLSZUOZXQTZYJFSZYBFSZYDFSZXQTYG + UUAYQYTDUESZUPTZYNYGYQGQYTGQUUAUUFUQYGOPCGYPDDURSZHRMUUGUFZNXOXMYFXMXNVCU + SZXODRQZYFXNUUJXMDELUTVAUSZYGYHHQZYIHQZVBZXNYOEURSZQZYPUUGQXMXNYFUULUUMVD + ZUUNABEYHYIUUOYBHJUUOUFZKYGUULUUMVEZYGUULUUMVFZYCYEXOUULUUMVGVHZYLUUGDEYO + UUOLYLUFZUURUUHVIVJVMYGOPCGYSDUUGHRMUUHNUUIUUKUUNXNYRUUOQZYSUUGQUUQUUNABE + YHYIUUOYDHJUURKUUSUUTYCYEXOUULUUMVNVHZYLUUGDEYRUUOLUVBUURUUHVIVJVMCGUUEXQ + DHYQYTMNXSUUEUFZVKVJYGUUFOPHHYPYSUUETZUOYNYGOPHHYPYSUUEYQYTUCUCVLVLUUIUUI + YPVLQUUNYOYLVOVPYSVLQUUNYRYLVOVPYGYQVQYGYTVQVRYGOPHHUVFYMUUNYMYOYRDVSSZUE + SZTZYLSZUVFUUNYKUVIYLUUNYKYOYREUESZTZUVIUUNYFUULUUMUDYKUVLUQYGUULYFUUMXOY + FVTWAYGUULUUMWCABUVKXPEYHYIHYBYDJKXRUVKUFWBVJYGUULUVLUVIUQZUUMXOUVMYFXOUV + KUVHYOYRXOEUVGUEXNEUVGUQXMDERLWDVAZWEWFUSWAWGWEUUNYLUVGDWNTQYOUVGURSZQZYR + UVOQZUVJUVFUQUUNYLUVGDUVBUVGUFYGUULUUJUUMUUKWAYGUULDWHQZUUMXOUVRYFXNUVRXM + DELWIVAUSWAWJUUNUVPUUPUVAYGUULUVPUUPWKZUUMXOUVSYFXOUVOUUOYOXOUVGEURXOEUVG + UVNWLWEZWMUSWAWOUUNUVQUVCUVDYGUULUVQUVCWKZUUMXOUWAYFXOUVOUUOYRUVTWMUSWAWO + UVHUUEUVGDYOYLYRUVOUVOUFUVHUFUVEWPWQWRWSWGWRYGXMXNYJBQZVBZUUBYNUQYGXOUWBU + WCXOYFVCYGAXDQZYCYEVBZUWBYGUWDYFUDUWEXOUWDYFXOXTUWDYAAWTUJXAUWDYCYEXBXCBX + PAYBYDKXRXEUJXMXNUWBXFXGOPABDEYLFYJHRIJKLUVBXHUJYGUUCYQUUDYTXQYGXMXNYCVBZ + UUCYQUQYGXOYCUDUWFYFYCXOYCYEVCXIXMXNYCXFXCOPABDEYLFYBHRIJKLUVBXHUJYGXMXNY + EVBZUUDYTUQYGXOYEUDUWGYFYEXOYCYEVTXIXMXNYEXFXCOPABDEYLFYDHRIJKLUVBXHUJXJX + KXL $. $d A k l $. $d B k l $. $d N k l $. $d P i j k l m x y $. $d R k l $. $( The transformation of matrices into polynomial matrices preserves the @@ -263742,74 +263972,74 @@ whose entries are constant polynomials (or "scalar polynomials", see -> A. x e. B A. y e. 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(Contributed by AV, 29-Oct-2019.) $) mat2pmat1 $p |- ( ( N e. Fin /\ R e. Ring ) -> ( T ` ( 1r ` A ) ) = ( 1r ` C ) ) $= - ( vi vj wcel crg cfv eqid cfn wa wceq cv co wral cascl simpl simpr matrng - cur w3a rngidcl syl 3jca mat2pmatvalel weq c0g cif fvif ply1scl1 ad2antlr - sylan ply1scl0 ifeq12d syl5eq adantr adantl mat1ov fveq2d ply1rng 3eqtr4d - eqtrd ralrimivva wb mat2pmatbas0 sylan2 eqmat syl2anc mpbird ) HUAQZERQZU - BZAUKSZFSZCUKSZUCZOUDZPUDZWEUEZWHWIWFUEZUCZPHUFOHUFZWCWLOPHHWCWHHQZWIHQZU - BZUBZWJWHWIWDUEZDUGSZSZWKWCWAWBWDBQZULZWPWJWTUCWCWAWBXAWAWBUHZWAWBUIZWCAR - QXAAEHJUJBAWDKWDTZUMUNUOZABDEWSFWDHRWHWIIJKLWSTZUPVCWQOPUQZEUKSZEURSZUSZW - SSZXHDUKSZDURSZUSZWTWKWQXLXHXIWSSZXJWSSZUSXOXHXIXJWSUTWQXHXPXMXQXNWBXPXMU - CWAWPWSDEXIXMLXGXITZXMTZVAVBWBXQXNUCWAWPWSDEXNXJLXGXJTZXNTZVDVBVEVFWQWRXK - WSWQAEWDXIWHWIHXJJXRXTWCWAWPXCVGZWCWBWPXDVGWPWNWCWNWOUHVHZWPWOWCWNWOUIVHZ - XEVIVJWQCDWFXMWHWIHXNMXSYAYBWBDRQZWAWPDELVKZVBYCYDWFTZVIVLVMVNWCWEGQZWFGQ - ZWGWMVOWCXBYHXFABCDEFGWDHIJKLMNVPUNWCCRQZYIWBWAYEYJYFCDHMUJVQGCWFNYGUMUNC - GDOPHWEWFMNVRVSVT $. + ( vi vj wcel crg cfv eqid cfn wa cur wceq cv wral cascl w3a simpl matring + simpr ringidcl syl 3jca mat2pmatvalel sylan weq c0g cif ply1scl1 ad2antlr + fvif ply1scl0 ifeq12d syl5eq adantr adantl mat1ov fveq2d ply1ring 3eqtr4d + co eqtrd ralrimivva wb mat2pmatbas0 sylan2 eqmat syl2anc mpbird ) HUAQZER + QZUBZAUCSZFSZCUCSZUDZOUEZPUEZWEVLZWHWIWFVLZUDZPHUFOHUFZWCWLOPHHWCWHHQZWIH + QZUBZUBZWJWHWIWDVLZDUGSZSZWKWCWAWBWDBQZUHZWPWJWTUDWCWAWBXAWAWBUIZWAWBUKZW + CARQXAAEHJUJBAWDKWDTZULUMUNZABDEWSFWDHRWHWIIJKLWSTZUOUPWQOPUQZEUCSZEURSZU + SZWSSZXHDUCSZDURSZUSZWTWKWQXLXHXIWSSZXJWSSZUSXOXHXIXJWSVBWQXHXPXMXQXNWBXP + XMUDWAWPWSDEXIXMLXGXITZXMTZUTVAWBXQXNUDWAWPWSDEXNXJLXGXJTZXNTZVCVAVDVEWQW + RXKWSWQAEWDXIWHWIHXJJXRXTWCWAWPXCVFZWCWBWPXDVFWPWNWCWNWOUIVGZWPWOWCWNWOUK + VGZXEVHVIWQCDWFXMWHWIHXNMXSYAYBWBDRQZWAWPDELVJZVAYCYDWFTZVHVKVMVNWCWEGQZW + FGQZWGWMVOWCXBYHXFABCDEFGWDHIJKLMNVPUMWCCRQZYIWBWAYEYJYFCDHMUJVQGCWFNYGUL + UMCGDOPHWEWFMNVRVSVT $. $( The transformation of matrices into polynomial matrices is a homomorphism of multiplicative monoids. (Contributed by AV, @@ -263817,23 +264047,23 @@ whose entries are constant polynomials (or "scalar polynomials", see mat2pmatmhm $p |- ( ( N e. Fin /\ R e. CRing ) -> T e. ( ( mulGrp ` A ) MndHom ( mulGrp ` C ) ) ) $= ( vx vy wcel cfv crg eqid cfn ccrg wa cmgp cmnd wf cv cmulr wceq wral cur - co w3a crngrng matrng sylan2 rngmgp syl ply1rng jca mat2pmatf mat2pmatmul - cmhm mat2pmat1 3jca mgpbas mgpplusg rngidval ismhm sylanbrc ) HUAQZEUBQZU - CZAUDRZUEQZCUDRZUEQZUCBGFUFZOUGZPUGZAUHRZULFRVSFRVTFRCUHRZULUIPBUJOBUJZAU - KRZFRCUKRZUIZUMFVNVPVCULQVMVOVQVMASQZVOVLVKESQZWGEUNZAEHJUOUPAVNVNTZUQURV - MCSQZVQVLVKDSQZWKVLWHWLWIDELUSURCDHMUOUPCVPVPTZUQURUTVMVRWCWFVLVKWHVRWIAB - CDEFGHIJKLMNVAUPOPABCDEFGHIJKLMNVBVLVKWHWFWIABCDEFGHIJKLMNVDUPVEOPBGWAWBV - NVPFWEWDBAVNWJKVFGCVPWMNVFAWAVNWJWATVGCWBVPWMWBTVGAWDVNWJWDTVHCWEVPWMWETV - HVIVJ $. + co w3a crngring matring sylan2 ringmgp syl ply1ring mat2pmatf mat2pmatmul + cmhm jca mat2pmat1 3jca mgpbas mgpplusg rngidval ismhm sylanbrc ) HUAQZEU + BQZUCZAUDRZUEQZCUDRZUEQZUCBGFUFZOUGZPUGZAUHRZULFRVSFRVTFRCUHRZULUIPBUJOBU + JZAUKRZFRCUKRZUIZUMFVNVPVBULQVMVOVQVMASQZVOVLVKESQZWGEUNZAEHJUOUPAVNVNTZU + QURVMCSQZVQVLVKDSQZWKVLWHWLWIDELUSURCDHMUOUPCVPVPTZUQURVCVMVRWCWFVLVKWHVR + WIABCDEFGHIJKLMNUTUPOPABCDEFGHIJKLMNVAVLVKWHWFWIABCDEFGHIJKLMNVDUPVEOPBGW + AWBVNVPFWEWDBAVNWJKVFGCVPWMNVFAWAVNWJWATVGCWBVPWMWBTVGAWDVNWJWDTVHCWEVPWM + WETVHVIVJ $. $( The transformation of matrices into polynomial matrices is a ring homomorphism. (Contributed by AV, 29-Oct-2019.) $) mat2pmatrhm $p |- ( ( N e. Fin /\ R e. CRing ) -> T e. ( A RingHom C ) ) $= - ( wcel wa crg co syl jca cfn ccrg cghm cmgp cfv crh crngrng anim2i matrng - cmhm ply1rng mat2pmatghm mat2pmatmhm eqid isrhm sylibr ) HUAOZEUBOZPZAQOZ - CQOZPZFACUCROZFAUDUEZCUDUEZUJROZPZPFACUFROUSVBVGUSUTVAUSUQEQOZPZUTURVHUQE - UGZUHZAEHJUISUSUQDQOZPVAURVLUQURVHVLVJDELUKSUHCDHMUISTUSVCVFUSVIVCVKABCDE - FGHIJKLMNULSABCDEFGHIJKLMNUMTTACFVDVEVDUNVEUNUOUP $. + ( wcel wa crg co syl jca cfn ccrg cghm cmgp cfv cmhm crh crngring matring + anim2i ply1ring mat2pmatghm mat2pmatmhm eqid isrhm sylibr ) HUAOZEUBOZPZA + QOZCQOZPZFACUCROZFAUDUEZCUDUEZUFROZPZPFACUGROUSVBVGUSUTVAUSUQEQOZPZUTURVH + UQEUHZUJZAEHJUISUSUQDQOZPVAURVLUQURVHVLVJDELUKSUJCDHMUISTUSVCVFUSVIVCVKAB + CDEFGHIJKLMNULSABCDEFGHIJKLMNUMTTACFVDVEVDUNVEUNUOUP $. $d K i j $. $d S i j $. $d X i j $. $d Y i j $. $d .X. i j $. $d .x. i j $. @@ -263850,26 +264080,26 @@ whose entries are constant polynomials (or "scalar polynomials", see -> ( T ` ( X .x. Y ) ) = ( ( S ` X ) .X. ( T ` Y ) ) ) $= ( vi vj cfn wcel ccrg wa co cfv wceq cv wral cmulr crh csca casa ply1assa cbs eqid asclrhm 3syl ply1sca adantl oveq1d eleqtrrd adantr eleq2i biimpi - ad2antlr simprl simplrr matecld rhmmul syl3anc crngrng matvscacell fveq2d - simpr crg anim2i anim12i df-3an sylibr mat2pmatvalel sylan oveq2d 3eqtr4d - w3a simpll matvscl syl31anc ply1rng syl ply1sclcl syl2an mat2pmatbas0 jca - simpl ralrimivva wb syl21anc eqmat syl2anc mpbird ) LUGUHZEUIUHZUJZMKUHZN - BUHZUJZUJZMNHUKZGULZMFULZNGULZIUKZUMZUEUNZUFUNZXPUKZYAYBXSUKZUMZUFLUOUELU - OZXNYEUEUFLLXNYALUHZYBLUHZUJZUJZYAYBXOUKZFULZXQYAYBXRUKZDUPULZUKZYCYDYJMY - AYBNUKZEUPULZUKZFULZXQYPFULZYNUKZYLYOYJFEDUQUKZUHZMEVAULZUHZYPUUDUHYSUUAU - MXNUUCYIXJUUCXMXJFDURULZDUQUKZUUBXJXIDUSUHFUUGUHXHXIWADERUTFUUFDUBUUFVBVC - VDXJEUUFDUQXIEUUFUMXHDEUIRVEVFVGVHVIVIXMUUEXJYIXKUUEXLXKUUEKUUDMUAVJVKVIV - LYJABEYAYBUUDNLPUUDVBZQXNYGYHVMYIYHXNYGYHWAVFXJXKXLYIVNVOMYPEDYQYNFUUDUUH - YQVBZYNVBZVPVQYJYKYRFYJEWBUHZXMYIYKYRUMXNUUKYIXIUUKXHXMEVRZVLZVIZXNXMYIXJ - XMWAVIXNYIWAZABEHYQYAYBKLMNPQUAUCUUIVSVQVTYJYMYTXQYNXNXHUUKXLWKZYIYMYTUMX - NXHUUKUJZXLUJUUPXJUUQXMXLXIUUKXHUULWCZXKXLWAWDXHUUKXLWEWFZABDEFGNLWBYAYBO - PQRUBWGWHWIWJYJXHUUKXOBUHZYIYCYLUMXNXHYIXHXIXMWLZVIUUNXNUUTYIXJUUQXMUUTUU - RABMEHKLNUAPQUCWMWHZVIUUOABDEFGXOLWBYAYBOPQRUBWGWNYJDWBUHZXQDVAULZUHZXRJU - HZUJZYIYDYOUMXNUVCYIXIUVCXHXMXIUUKUVCUULDERWOWPVLZVIXNUVGYIXNUVEUVFXJUUKX - KUVEXMXIUUKXHUULVFXKXLXAFUVDDEMKRUBUAUVDVBZWQWRXNUUPUVFUUSABCDEGJNLOPQRST - WSWPWTZVIUUOCJDIYNYAYBUVDLXQXRSTUVIUDUUJVSVQWJXBXNXPJUHZXSJUHZXTYFXCXNXHU - UKUUTUVKUVAUUMUVBABCDEGJXOLOPQRSTWSVQXNXHUVCUVGUVLUVAUVHUVJCJXQDIUVDLXRUV - ISTUDWMXDCJDUEUFLXPXSSTXEXFXG $. + simpr ad2antlr simprl simplrr matecld rhmmul syl3anc crngring matvscacell + crg fveq2d anim2i anim12i df-3an sylibr mat2pmatvalel sylan oveq2d simpll + w3a 3eqtr4d matvscl syl31anc ply1ring simpl ply1sclcl syl2an mat2pmatbas0 + syl jca ralrimivva wb syl21anc eqmat syl2anc mpbird ) LUGUHZEUIUHZUJZMKUH + ZNBUHZUJZUJZMNHUKZGULZMFULZNGULZIUKZUMZUEUNZUFUNZXPUKZYAYBXSUKZUMZUFLUOUE + LUOZXNYEUEUFLLXNYALUHZYBLUHZUJZUJZYAYBXOUKZFULZXQYAYBXRUKZDUPULZUKZYCYDYJ + MYAYBNUKZEUPULZUKZFULZXQYPFULZYNUKZYLYOYJFEDUQUKZUHZMEVAULZUHZYPUUDUHYSUU + AUMXNUUCYIXJUUCXMXJFDURULZDUQUKZUUBXJXIDUSUHFUUGUHXHXIVLDERUTFUUFDUBUUFVB + VCVDXJEUUFDUQXIEUUFUMXHDEUIRVEVFVGVHVIVIXMUUEXJYIXKUUEXLXKUUEKUUDMUAVJVKV + IVMYJABEYAYBUUDNLPUUDVBZQXNYGYHVNYIYHXNYGYHVLVFXJXKXLYIVOVPMYPEDYQYNFUUDU + UHYQVBZYNVBZVQVRYJYKYRFYJEWAUHZXMYIYKYRUMXNUUKYIXIUUKXHXMEVSZVMZVIZXNXMYI + XJXMVLVIXNYIVLZABEHYQYAYBKLMNPQUAUCUUIVTVRWBYJYMYTXQYNXNXHUUKXLWKZYIYMYTU + MXNXHUUKUJZXLUJUUPXJUUQXMXLXIUUKXHUULWCZXKXLVLWDXHUUKXLWEWFZABDEFGNLWAYAY + BOPQRUBWGWHWIWLYJXHUUKXOBUHZYIYCYLUMXNXHYIXHXIXMWJZVIUUNXNUUTYIXJUUQXMUUT + UURABMEHKLNUAPQUCWMWHZVIUUOABDEFGXOLWAYAYBOPQRUBWGWNYJDWAUHZXQDVAULZUHZXR + JUHZUJZYIYDYOUMXNUVCYIXIUVCXHXMXIUUKUVCUULDERWOWTVMZVIXNUVGYIXNUVEUVFXJUU + KXKUVEXMXIUUKXHUULVFXKXLWPFUVDDEMKRUBUAUVDVBZWQWRXNUUPUVFUUSABCDEGJNLOPQR + STWSWTXAZVIUUOCJDIYNYAYBUVDLXQXRSTUVIUDUUJVTVRWLXBXNXPJUHZXSJUHZXTYFXCXNX + HUUKUUTUVKUVAUUMUVBABCDEGJXOLOPQRSTWSVRXNXHUVCUVGUVLUVAUVHUVJCJXQDIUVDLXR + UVISTUDWMXDCJDUEUFLXPXSSTXEXFXG $. $} ${ @@ -263949,12 +264179,12 @@ whose entries are constant polynomials (or "scalar polynomials", see mat2pmatscmxcl $p |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( M e. K /\ L e. NN0 ) ) -> ( ( L .^ X ) .* ( T ` M ) ) e. B ) $= - ( cfn wcel crg wa cn0 cbs cfv simpll ply1rng ad2antlr cmgp eqid ply1moncl - ad2ant2l w3a simpl anim2i df-3an sylibr mat2pmatbas0 syl matvscl syl22anc - co ) LUCUDZEUEUDZUFZKIUDZJUGUDZUFZUFZVGDUEUDZJMGVFZDUHUIZUDZKFUIZBUDZVOVR - HVFBUDVGVHVLUJVHVNVGVLDEQUKULVHVKVQVGVJVPJDEGDUMUIZMQUBVTUNUAVPUNZUOUPVMV - GVHVJUQZVSVMVIVJUFWBVLVJVIVJVKURUSVGVHVJUTVAAICDEFBKLPNOQRSVBVCCBVODHVPLV - RWARSTVDVE $. + ( cfn wcel crg wa cn0 co cbs simpll ply1ring ad2antlr cmgp eqid ply1moncl + cfv ad2ant2l w3a simpl anim2i df-3an sylibr mat2pmatbas0 matvscl syl22anc + syl ) LUCUDZEUEUDZUFZKIUDZJUGUDZUFZUFZVGDUEUDZJMGUHZDUIUPZUDZKFUPZBUDZVOV + RHUHBUDVGVHVLUJVHVNVGVLDEQUKULVHVKVQVGVJVPJDEGDUMUPZMQUBVTUNUAVPUNZUOUQVM + VGVHVJURZVSVMVIVJUFWBVLVJVIVJVKUSUTVGVHVJVAVBAICDEFBKLPNOQRSVCVFCBVODHVPL + VRWARSTVDVE $. $} ${ @@ -264019,22 +264249,22 @@ whose entries are constant polynomials (or "scalar polynomials", see m2cpmmhm $p |- ( ( N e. Fin /\ R e. CRing ) -> T e. ( ( mulGrp ` A ) MndHom ( mulGrp ` U ) ) ) $= ( wcel cfv co cvv cfn ccrg cmgp cmhm cbs eqid mat2pmatmhm csubmnd crn wss - wa wb crg csubrg crngrng anim2i cpmatsrgpmat subrgsubm 3syl wf m2cpmf frn - cress wceq cmat ovex eqeltri ccpmat mgpress mp2an eqcomi resmhm2b syl2anc - mpbid ) IUAQZEUBQZUKZGAUCRZCUCRZUDSQZGVRHUCRZUDSQZABCDEGCUERZIKLMNOWCUFUG - VQFVSUHRQZGUIFUJZVTWBULVQVOEUMQZUKZFCUNRQWDVPWFVOEUOUPZCDEFIJNOUQFCVSVSUF - ZURUSVQWGBFGUTWEWHABEFGIJKLMVABFGVBUSVRVSWAGFVSFVCSZWACTQFTQWJWAVDCIDVEST - OIDVEVFVGFIEVHSTJIEVHVFVGFCHVSTTPWIVIVJVKVLVMVN $. + wa wb crg csubrg crngring anim2i cpmatsrgpmat subrgsubm 3syl m2cpmf cress + wf frn wceq cmat ovex eqeltri ccpmat mgpress mp2an resmhm2b syl2anc mpbid + eqcomi ) IUAQZEUBQZUKZGAUCRZCUCRZUDSQZGVRHUCRZUDSQZABCDEGCUERZIKLMNOWCUFU + GVQFVSUHRQZGUIFUJZVTWBULVQVOEUMQZUKZFCUNRQWDVPWFVOEUOUPZCDEFIJNOUQFCVSVSU + FZURUSVQWGBFGVBWEWHABEFGIJKLMUTBFGVCUSVRVSWAGFVSFVASZWACTQFTQWJWAVDCIDVES + TOIDVEVFVGFIEVHSTJIEVHVFVGFCHVSTTPWIVIVJVNVKVLVM $. $( The transformation of matrices into constant polynomial matrices is a ring homomorphism. (Contributed by AV, 18-Nov-2019.) $) m2cpmrhm $p |- ( ( N e. Fin /\ R e. CRing ) -> T e. ( A RingHom U ) ) $= - ( wcel wa crg co cfn ccrg cghm cmgp cfv cmhm crngrng matrng sylan2 csubrg - crh cpmatsrgpmat subrgrng syl jca m2cpmghm m2cpmmhm eqid isrhm sylanbrc ) - IUAQZEUBQZRZASQZHSQZRGAHUCTQZGAUDUEZHUDUEZUFTQZRGAHUKTQVCVDVEVBVAESQZVDEU - GZAEILUHUIVCFCUJUEQZVEVBVAVJVLVKCDEFIJNOULUIFCHPUMUNUOVCVFVIVBVAVJVFVKABC - DEFGHIJKLMNOPUPUIABCDEFGHIJKLMNOPUQUOAHGVGVHVGURVHURUSUT $. + ( wcel wa crg co cfn ccrg cghm cmgp cfv crh crngring matring cpmatsrgpmat + cmhm sylan2 csubrg subrgring syl m2cpmghm m2cpmmhm eqid isrhm sylanbrc + jca ) IUAQZEUBQZRZASQZHSQZRGAHUCTQZGAUDUEZHUDUEZUJTQZRGAHUFTQVCVDVEVBVAES + QZVDEUGZAEILUHUKVCFCULUEQZVEVBVAVJVLVKCDEFIJNOUIUKFCHPUMUNUTVCVFVIVBVAVJV + FVKABCDEFGHIJKLMNOPUOUKABCDEFGHIJKLMNOPUPUTAHGVGVHVGUQVHUQURUS $. $} ${ @@ -264064,17 +264294,17 @@ whose entries are constant polynomials (or "scalar polynomials", see /\ ( s e. NN0 /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( Y gsum ( i e. ( 0 ... s ) |-> ( ( i .^ X ) .x. ( T ` ( b ` i ) ) ) ) ) e. ( Base ` Y ) ) $= - ( cfn wcel ccrg w3a cv cn0 cc0 cfz cmap cbs cfv eqid ccmn crngrng ply1rng - co wa crg syl matrng sylan2 rngcmn 3adant3 adantr fzfid 3ad2ant2 ad2antrr - simpll1 elfznn0 cmgp ply1moncl syl2an anim2i anim12i df-3an sylibr simprr - simpl anim1i m2pmfzmap syl2anc matvscl syl22anc ralrimiva gsummptcl ) JUC - UDZDUEUDZIBUDZUFZMUGZUHUDZNUGZBUIWLUJURZUKURUDZUSZUSZLULUMZGLWOGUGZKHURZW - TWNUMEUMZFURZWSUNZWKLUOUDZWQWHWIXEWJWHWIUSLUTUDZXEWIWHCUTUDZXFWIDUTUDZXGD - UPZCDQUQVAZLCJRVBVCLVDVAVEVFWRUIWLVGWRXCWSUDZGWOWRWTWOUDZUSZWHXGXACULUMZU - DZXBWSUDZXKWHWIWJWQXLVJWKXGWQXLWIWHXGWJXJVHVIWRXHWTUHUDXOXLWKXHWQWIWHXHWJ - XIVHVFWTWLVKXNWTCDHCVLUMZKQTXQUNUAXNUNZVMVNXMWHXHWMUFZWPXLUSXPWRXSXLWRWHX - HUSZWMUSXSWKXTWQWMWHWIXTWJWIXHWHXIVOVEWMWPVTVPWHXHWMVQVRVFWRWPXLWKWMWPVSW - AABCDWLEWTJLNOPQRSWBWCLWSXACFXNJXBXRRXDUBWDWEWFWG $. + ( cfn wcel ccrg w3a cv cn0 cc0 cfz co cmap cbs cfv eqid ccmn crg crngring + ply1ring syl matring sylan2 ringcmn 3adant3 adantr fzfid simpll1 3ad2ant2 + ad2antrr elfznn0 cmgp ply1moncl syl2an anim2i simpl anim12i df-3an sylibr + wa simprr anim1i m2pmfzmap syl2anc matvscl syl22anc ralrimiva gsummptcl ) + JUCUDZDUEUDZIBUDZUFZMUGZUHUDZNUGZBUIWLUJUKZULUKUDZVSZVSZLUMUNZGLWOGUGZKHU + KZWTWNUNEUNZFUKZWSUOZWKLUPUDZWQWHWIXEWJWHWIVSLUQUDZXEWIWHCUQUDZXFWIDUQUDZ + XGDURZCDQUSUTZLCJRVAVBLVCUTVDVEWRUIWLVFWRXCWSUDZGWOWRWTWOUDZVSZWHXGXACUMU + NZUDZXBWSUDZXKWHWIWJWQXLVGWKXGWQXLWIWHXGWJXJVHVIWRXHWTUHUDXOXLWKXHWQWIWHX + HWJXIVHVEWTWLVJXNWTCDHCVKUNZKQTXQUOUAXNUOZVLVMXMWHXHWMUFZWPXLVSXPWRXSXLWR + WHXHVSZWMVSXSWKXTWQWMWHWIXTWJWIXHWHXIVNVDWMWPVOVPWHXHWMVQVRVEWRWPXLWKWMWP + VTWAABCDWLEWTJLNOPQRSWBWCLWSXACFXNJXBXRRXDUBWDWEWFWG $. $} ${ @@ -264280,14 +264510,14 @@ whose entries are constant polynomials (or "scalar polynomials", see ring isomorphism. (Contributed by AV, 19-Nov-2019.) $) m2cpmrngiso $p |- ( ( N e. Fin /\ R e. CRing ) -> T e. ( A RingIso U ) ) $= - ( wcel wa crg syl vm cfn ccrg crs co crh cbs wf1o m2cpmrhm crngrng anim2i - cfv m2cpmf1o wceq wb wss cv cpmatpmat 3expia ssrdv ressbas2 eqcomd f1oeq3 - eqid mpbird cgrp csubg cpmatsubgpmat subggrp 3syl isrim syl2anc mpbir2and - matrng ) IUBQZDUCQZRZFAGUDUEQZFAGUFUEQZHGUGULZFUHZAHBCDEFGIJKLMNOPUIVQVOD - SQZRZWAVPWBVODUJUKZWCWAHEFUHZADEFHIJKLMUMWCVTEUNWAWEUOWCEVTWCEBUGULZUPEVT - UNWCUAEWFVOWBUAUQZEQWGWFQWFBCDEWGISJNOWFVDZURUSUTEWFGBPWHVATVBVTEHFVCTVET - VQASQZGVFQZVRVSWARUOVQWCWIWDADILVNTVQWCEBVGULQWJWDBCDEIJNOVHEBGPVIVJHVTAG - FSVFMVTVDVKVLVM $. + ( wcel wa crg syl vm cfn ccrg crs co crh cbs cfv m2cpmrhm crngring anim2i + wf1o m2cpmf1o wceq wb wss cv eqid cpmatpmat 3expia ressbas2 eqcomd f1oeq3 + ssrdv mpbird cgrp matring csubg cpmatsubgpmat subggrp 3syl isrim syl2anc + mpbir2and ) IUBQZDUCQZRZFAGUDUEQZFAGUFUEQZHGUGUHZFULZAHBCDEFGIJKLMNOPUIVQ + VODSQZRZWAVPWBVODUJUKZWCWAHEFULZADEFHIJKLMUMWCVTEUNWAWEUOWCEVTWCEBUGUHZUP + EVTUNWCUAEWFVOWBUAUQZEQWGWFQWFBCDEWGISJNOWFURZUSUTVDEWFGBPWHVATVBVTEHFVCT + VETVQASQZGVFQZVRVSWARUOVQWCWIWDADILVGTVQWCEBVHUHQWJWDBCDEIJNOVIEBGPVJVKHV + TAGFSVFMVTURVLVMVN $. $} ${ @@ -264338,10 +264568,10 @@ whose entries are constant polynomials (or "scalar polynomials", see 15-Dec-2019.) $) m2cpminv0 $p |- ( ( N e. Fin /\ R e. Ring ) -> ( I ` Z ) = .0. ) $= ( wcel crg cfv co wceq c0g cfn cmat2pmat eqid cmat fveq2i eqtri 0mat2pmat - wa ancoms eqcomd fveq2d cbs matrng rng0cl syl m2cpminvid mpd3an3 eqtrd ) - FUAOZDPOZUHZHEQGFDUBRZQZEQZGVAHVCEVAVCHUTUSVCHSCDVBFGHVBUCZKGATQFDUDRZTQM - AVFTIUEUFHBTQFCUDRZTQNBVGTLUEUFUGUIUJUKUSUTGAULQZOZVDGSVAAPOVIADFIUMVHAGV - HUCZMUNUOADVBEVHGFJIVJVEUPUQUR $. + wa ancoms eqcomd fveq2d cbs matring ring0cl syl m2cpminvid mpd3an3 eqtrd + ) FUAOZDPOZUHZHEQGFDUBRZQZEQZGVAHVCEVAVCHUTUSVCHSCDVBFGHVBUCZKGATQFDUDRZT + QMAVFTIUEUFHBTQFCUDRZTQNBVGTLUEUFUGUIUJUKUSUTGAULQZOZVDGSVAAPOVIADFIUMVHA + GVHUCZMUNUOADVBEVHGFJIVJVEUPUQUR $. $} $( @@ -264461,18 +264691,18 @@ multiples of constant (polynomial) matrices is prepared by collecting the -> E. s e. NN0 A. x e. NN0 ( s < x -> ( M decompPMat x ) = .0. ) ) $= ( vi vj wcel wa cn0 va vb crg cv clt wbr cdecpmat wceq wral wrex cco1 cfv co wi c0g cfn matrcl simpld adantl simpl simpr eqid pmatcoe1fsupp syl3anc - cvv cbs wb decpmatcl 3expa matrng rng0cl 3syl adantr eqmat syl2anc df-3an - jca w3a decpmate sylanbr cmpt2 mat0op syl5eq syl weq fvex ovmpt2d eqeq12d - eqidd a1i 2ralbidva bitrd imbi2d ralbidva rexbidv mpbird ) FUCRZGCRZSZJUD - AUDZUEUFZGWTUGUMZIUHZUNZATUIZJTUJXAWTPUDZQUDZGUMUKULULZFUOULZUHZQHUIPHUIZ - UNZATUIZJTUJZWSHUPRZWQWRXNWRXOWQWRXOEVERDCEHGLMUQURUSZWQWRUTZWQWRVAACDEFP - QGHXIJKLMXIVBZVCVDWSXEXMJTWSXDXLATWSWTTRZSZXCXKXAXTXCXFXGXBUMZXFXGIUMZUHZ - QHUIPHUIZXKXTXBBVFULZRZIYERZXCYDVGWQWRXSYFBCDYEEFWTGHUCKLMNYEVBZVHVIWSYGX - SWSXOWQSZBUCRYGWSXOWQXPXQVQZBFHNVJYEBIYHOVKVLVMBYEFPQHXBINYHVNVOXTYCXJPQH - HXTXFHRZXGHRZSZSZYAXHYBXIXTWQWRXSVRYMYAXHUHWQWRXSVPCDEFXFXGWTGHUCKLMVSVTY - NUAUBXFXGHHXIXIIVEYNYIIUAUBHHXIWAZUHXTYIYMWSYIXSYJVMVMYIIBUOULYOOBFUAUBHX - INXRWBWCWDYNUAPWEUBQWESSXIWIYMYKXTYKYLUTUSYMYLXTYKYLVAUSXIVERYNFUOWFWJWGW - HWKWLWMWNWOWP $. + cvv cbs decpmatcl 3expa jca matring ring0cl 3syl adantr eqmat syl2anc w3a + wb df-3an decpmate sylanbr cmpt2 mat0op syl5eq syl weq eqidd fvex ovmpt2d + a1i eqeq12d 2ralbidva bitrd imbi2d ralbidva rexbidv mpbird ) FUCRZGCRZSZJ + UDAUDZUEUFZGWTUGUMZIUHZUNZATUIZJTUJXAWTPUDZQUDZGUMUKULULZFUOULZUHZQHUIPHU + IZUNZATUIZJTUJZWSHUPRZWQWRXNWRXOWQWRXOEVERDCEHGLMUQURUSZWQWRUTZWQWRVAACDE + FPQGHXIJKLMXIVBZVCVDWSXEXMJTWSXDXLATWSWTTRZSZXCXKXAXTXCXFXGXBUMZXFXGIUMZU + HZQHUIPHUIZXKXTXBBVFULZRZIYERZXCYDVQWQWRXSYFBCDYEEFWTGHUCKLMNYEVBZVGVHWSY + GXSWSXOWQSZBUCRYGWSXOWQXPXQVIZBFHNVJYEBIYHOVKVLVMBYEFPQHXBINYHVNVOXTYCXJP + QHHXTXFHRZXGHRZSZSZYAXHYBXIXTWQWRXSVPYMYAXHUHWQWRXSVRCDEFXFXGWTGHUCKLMVSV + TYNUAUBXFXGHHXIXIIVEYNYIIUAUBHHXIWAZUHXTYIYMWSYIXSYJVMVMYIIBUOULYOOBFUAUB + HXINXRWBWCWDYNUAPWEUBQWESSXIWFYMYKXTYKYLUTUSYMYLXTYKYLVAUSXIVERYNFUOWGWIW + HWJWKWLWMWNWOWP $. $( The mapping to the matrices consisting of the coefficients in the polynomial entries of a given matrix for the same power is finitely @@ -264500,36 +264730,37 @@ multiples of constant (polynomial) matrices is prepared by collecting the decpmatid $p |- ( ( N e. Fin /\ R e. Ring /\ K e. NN0 ) -> ( I decompPMat K ) = if ( K = 0 , .1. , .0. ) ) $= ( vi vj wcel cfv wceq vk cfn crg cn0 w3a cdecpmat co cv cco1 weq cc0 csca - cmpt2 cur c0g cif cbs pmatrng 3adant3 eqid rngidcl syl decpmatval syl2anc - simp3 simp11 simp12 simp2 pmat1ovd fveq2d fveq1d fvif iffv eqtri cv1 cmgp - fveq1i cmg cvsca ply1idvr1 3ad2ant2 eqcomd clmod ply1lmod ply1moncl mpan2 - 0nn0 lmodvs1 cmpt ply1sca eqeltrd coe1tm syl3anc eqidd eqeq1 ifbid adantl - a1i cvv fvex fvmptd eqtrd 3eqtrd csn cxp coe1z fvconst2g ifeq12d 3ad2ant1 - ifex syl5eq mpt2eq3dva wa ifeq1d mpt2eq3dv iftrue mat1 3eqtr4d wn iffalse - adantr ifid 3simpa mat0op pm2.61ian ) HUBRZDUCRZGUDRZUEZFGUFUGZPQHHGPUHZQ - UHZFUGZUISZSZUMZPQHHPQUJZGUKTZCULSZUNSZDUOSZUPZUUAUPZUMZYREIUPZYIFBUQSZRZ - YHYJYPTYIBUCRZUUGYFYGUUHYHBCDHJKURUSUUFBFUUFUTZLVAVBYFYGYHVEZBUUFCPQGFHKU - UIVCVDYIPQHHYOUUCYIYKHRZYLHRZUEZYOGYQCUNSZCUOSZUPZUISZSZUUCUUMGYNUUQUUMYM - UUPUIUUMBCDFUUNYKYLHUUOJKUUOUTZUUNUTYFYGYHUUKUULVFYFYGYHUUKUULVGYIUUKUULV - HYIUUKUULVELVIVJVKUUMUURYQGUUNUISZSZGUUOUISZSZUPZUUCUURGYQUUTUVBUPZSUVDGU - 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l ) ) ) ) ) ) ) ) $= ( wcel crg cfv co cgsu vk cfn wa cn0 w3a cmulr cdecpmat cco1 cmpt cc0 cfz - cmin wceq simpr 3ad2ant1 pmatrng adantr simpl adantl eqid syl3anc 3adant3 - rngcl simp33 3simpa 3ad2ant3 decpmate syl31anc cotp cmmul ply1rng matmulr - cv eqcomd sylan2 oveqd cbs cmap wi matbas2 syl6reqr eleq2d biimpcd impcom - cxp simp31 simp32 mamufv eqtrd fveq2d simpl2l matecld simpl2r coe1fzgsumd - fveq1d ralrimiva simpl1r coe1mul oveq2 oveq1 oveq2d mpteq12dv ovex fvmptd - cvv a1i mpteq2dva 3eqtrd ) JUBPZEQPZUCZFBPZKBPZUCZGJPZHJPZIUDPZUEZUEZGHFK - CUFRZSZIUGSSZIGHYASZUHRZRZIDAJGAVMZFSZYFHKSZDUFRZSZUITSZUHRZRZEAJELUJIUKS - ZLVMZYGUHRRZIYOULSZYHUHRZRZEUFRZSZUIZTSZUIZTSZXSXJYABPZXQXOXPUCZYBYEUMXKX - NXJXRXIXJUNUOZXKXNUUFXRXKXNUCCQPZXLXMUUFXKUUIXNCDEJMNUPUQXNXLXKXLXMURUSXN - XMXKXLXMUNUSBCXTFKOXTUTVCVAVBXKXNXOXPXQVDZXRXKUUGXNXOXPXQVEVFBCDEGHIYAJQM - NOVGVHXSIYDYLXSYCYKUHXSYCGHFKDJJJVIVJSZSZSYKXSYAUULGHXSXTUUKFKXKXNXTUUKUM - ZXRXJXIDQPZUUMDEMVKZXIUUNUCZUUKXTCDUUKJQNUUKUTZVLVNVOUOVPVPXSDVQRZJDYIAUU - 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oveq1 wi wral wrex cmulr cmpt2 simpll simplr pmatrng anim1i 3anass sylibr - cco1 w3a eqid rngcl syl pmatcoe1fsupp syl3anc fveq2d fveq1d eqeq1d expcom - rspc2va adantl 3impib mpt2eq3dva mat0op syl5eq ad3antrrr eqtr4d ex imim2d - ralimdva reximdv oveqi mpteq2dv simpllr simpr decpmatmul decpmatval sylan - mpd 3eqtr2d imbi2d ralbidva rexbidv mpbird mptnn0fsuppd ) JUGSZGUHSZUIZAU - JZDSZBUJZDSZUIZUIZUAUKCIULLUJZUQTZYCIUJZUMTZYEYIYKUNTZUMTZHTZUOZUPTZCIULU - AUJZUQTZYLYEYRYKUNTZUMTZHTZUOZUPTZLUKKUBKUKSYHKCURUSZUKRCURUTVAVBYQUKSYHY - IVCSUICYPUPVDVBYIYRVEZYPUUCCUPUUFIYJYOYSUUBYIYRULUQVFUUFYNUUAYLHUUFYMYTYE - UMYIYRYKUNVKVGVGVHVGYHUBUJYRVIVJZUUDKVEZVLZUAVCVMZUBVCVNUUGUCUDJJYRUCUJZU - DUJZYCYEEVOUSZTZTZWCUSZUSZVPZKVEZVLZUAVCVMZUBVCVNZYHUUGYRUEUJZUFUJZUUNTZW - CUSZUSZGURUSZVEZUFJVMUEJVMZVLZUAVCVMZUBVCVNZUVBYHXTYAUUNDSZUVMXTYAYGVQXTY - AYGVRYHEUHSZYDYFWDZUVNYHUVOYGUIUVPYBUVOYGEFGJMNVSVTUVOYDYFWAWBDEUUMYCYEOU - UMWEWFWGZUADEFGUEUFUUNJUVHUBMNOUVHWEZWHWIYHUVLUVAUBVCYHUVKUUTUAVCYHYRVCSZ - UIZUVJUUSUUGUVTUVJUUSUVTUVJUIZUURUCUDJJUVHVPZKUWAUCUDJJUUQUVHUWAUUKJSZUUL - 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( n .^ X ) ) ) ) ) ) $= ( wcel co va vb cfn crg w3a cdecpmat cmpt cgsu cmpt2 wceq pmatcollpw1lem2 cn0 cv wral wa cbs cfv oveq12 oveq1d mpteq2dv oveq2d adantl simprl simprr - eqidd cvv c0g eqid ccmn ply1rng rngcmn syl 3ad2ant2 adantr nn0ex a1i cmat - simpl2 simplrl simpl3 simpr decpmatcl syl3anc matecld cmgp ply1tmcl fmptd - cfsupp pmatcollpw1lem1 3expb gsumcl ovmpt2d eqtr4d ralrimivva simp3 simp1 - wbr wb 3ad2ant1 simpl12 matbas2d eqmat syl2anc mpbird ) KUCSZDUDSZJASZUEZ - JFGKKCHULFUMZGUMZJHUMZUFTZTZXKLITZETZUGZUHTZUIZUJZUAUMZUBUMZJTZXTYAXRTZUJ - ZUBKUNUAKUNZXHYDUAUBKKXHXTKSZYAKSZUOZUOZYBCHULXTYAXLTZXNETZUGZUHTZYCABCDE - HIJKLUAUBMNOPQRUKYIFGXTYAKKXQYMXRCUPUQZYIXRVEXIXTUJXJYAUJUOZXQYMUJYIYOXPY - LCUHYOHULXOYKYOXMYJXNEXIXTXJYAXLURUSUTVAVBXHYFYGVCXHYFYGVDZYIULYNYLCVFCVG - UQZYNVHZYQVHZXHCVISZYHXFXEYTXGXFCUDSZYTCDMVJZCVKVLVMZVNULVFSZYIVOVPYIHULY - KYNYLYIXKULSZUOZXFYJDUPUQZSUUEYKYNSYIXFUUEXEXFXGYHVRVNZUUFKDVQTZUUIUPUQZD - XTYAUUGXLKUUIVHZUUGVHZUUJVHZXHYFYGUUEVSYIYGUUEYPVNUUFXFXGUUEXLUUJSZUUHYIX - 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rexralbidv oveq12d adantl id ancri 3ad2ant1 fvmptd ply1rng sylibr 3adant3 - anim2i mat0op eqeq12d nne imbi2i rexbii rabssnn0fi wfun funmpt funisfsupp - eqeltrd mptex ) KUBUCZDUDUCZJAUCZUEZHTFGKKFUFZGUFZJHUFZUGUHZUHZUUTLIUHZEU - HZUIZUJZBUKULZUMUNZUVFUVGUOUHZUBUCZUUQUVISUFZUVFULZUVGUPZSTUQZUBUUQUVFTUR - ZTVGUCZUVGVGUCZUVIUVNUSUUQUVEVGUCZHTUTUVOUUQUVRHTUUQUUNUUNUVRUUNUUOUUPVAZ - UVSFGKKUVDUBUBVBVCVDHTUVEUVFVGUVFVHVEVIUVPUUQVFVJUVQUUQBUKVKVJZSUVFVGVGTU - VGVLVMUUQUAUFUVKVNUNZUVMVOZVPZSTUTZUATVQZUVNUBUCUUQUWAUVLUVGUSZVPZSTUTZUA - TVQZUWEUUQUWIUWAFGKKUURUUSJUVKUGUHZUHZUVKLIUHZEUHZUIZFGKKCUKULZUIZUSZVPZS - TUTZUATVQZUUQUWAKKUSZUXAUWMUWOUSZGKUTZVTZFKUTZVTZVPZSTUTZUATVQZUWTUUQUXIU - WAUXCFKUTZVPZSTUTZUATVQZUUQUXMUWAUVKUURUUSJUHVRULULZUWLEUHZUWOUSZGKUTFKUT - ZVPZSTUTZUATVQZUUQUWAUXNDUKULZUSZGKUTFKUTZVPZSTUTZUATVQUXTSABCDFGJKUYAUAM - NOUYAVHZVSUUQUYEUXSUATUUQUYDUXRSTUUQUVKTUCZVTZUYCUXQUWAUYHUYBUXPFGKKUYHUU - RKUCZUUSKUCZUYBUXPVPUYHUYIVTZUYJVTZUYBUXPUYBUYLUXOUYAUWLEUHZUWOUXNUYAUWLE - WAUYLUYMCWBULZUKULZUWLCWCULZUHZUYAUWLUYPUHZUWOUUQUYMUYQUSUYGUYIUYJUUQUYAU - 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NN0 |-> ( i e. N , j e. N |-> ( ( i ( M decompPMat n ) j ) .X. ( n .^ X ) ) ) ) ) ) $= ( wcel co cfn crg w3a cn0 cv cdecpmat cmpt cgsu cmpt2 pmatcollpw1 cvv c0g - cfv eqid simp1 nn0ex a1i ply1rng 3ad2ant2 wa cbs simp1l2 cmat simp2 simp3 - adantr simpr 3jca 3ad2ant1 decpmatcl matecld simp1r cmgp ply1tmcl syl3anc - syl matbas2d pmatcollpw2lem matgsum eqtr4d ) KUASZDUBSZJASZUCZJFGKKCHUDFU - EZGUEZJHUEZUFTZTZWGLITETZUGUHTUIBHUDFGKKWJUIUGUHTABCDEFGHIJKLMNOPQRUJWDHB - ACWJFGUDKUKBULUMZNOWKUNWAWBWCUOZUDUKSWDUPUQWBWACUBSZWCCDMURUSZWDWGUDSZUTZ - FGBAWJCCVAUMZKUBNWQUNZOWDWAWOWLVFWDWMWOWNVFWPWEKSZWFKSZUCZWBWIDVAUMZSWOWJ - WQSWAWBWCWOWSWTVBXAKDVCTZXCVAUMZDWEWFXBWHKXCUNZXBUNZXDUNZWPWSWTVDWPWSWTVE - XAWBWCWOUCZWHXDSWPWSXHWTWPWBWCWOWDWBWOWAWBWCVDVFWDWCWOWAWBWCVEVFWDWOVGVHV - IXCABXDCDWGJKUBMNOXEXGVJVPVKWDWOWSWTVLWQWIWGCDEIXBCVMUMZLXFMRPXIUNQWRVNVO - VQABCDEFGHIJKLMNOPQRVRVSVT $. + cfv eqid simp1 nn0ex a1i ply1ring 3ad2ant2 cbs adantr simp1l2 simp2 simp3 + wa cmat simpr 3jca 3ad2ant1 decpmatcl syl matecld simp1r ply1tmcl syl3anc + cmgp matbas2d pmatcollpw2lem matgsum eqtr4d ) KUASZDUBSZJASZUCZJFGKKCHUDF + UEZGUEZJHUEZUFTZTZWGLITETZUGUHTUIBHUDFGKKWJUIUGUHTABCDEFGHIJKLMNOPQRUJWDH + BACWJFGUDKUKBULUMZNOWKUNWAWBWCUOZUDUKSWDUPUQWBWACUBSZWCCDMURUSZWDWGUDSZVE + ZFGBAWJCCUTUMZKUBNWQUNZOWDWAWOWLVAWDWMWOWNVAWPWEKSZWFKSZUCZWBWIDUTUMZSWOW + JWQSWAWBWCWOWSWTVBXAKDVFTZXCUTUMZDWEWFXBWHKXCUNZXBUNZXDUNZWPWSWTVCWPWSWTV + DXAWBWCWOUCZWHXDSWPWSXHWTWPWBWCWOWDWBWOWAWBWCVCVAWDWCWOWAWBWCVDVAWDWOVGVH + VIXCABXDCDWGJKUBMNOXEXGVJVKVLWDWOWSWTVMWQWIWGCDEIXBCVPUMZLXFMRPXIUNQWRVNV + OVQABCDEFGHIJKLMNOPQRVRVSVT $. $} ${ @@ -264848,49 +265079,49 @@ polynomial matrices for the same power (Contributed by AV, -> ( ( ( L .^ X ) .x. ( T ` M ) ) decompPMat I ) = if ( I = L , M , .0. ) ) $= ( vi vj vl vx vy vz vw cfn wcel ccrg wa cn0 w3a co cdecpmat cv cco1 cmpt2 - cfv wceq c0g cif cbs crg simpll crngrng ply1rng syl ad2antlr adantl simp2 - cmpt cmgp eqid ply1moncl syl2an anim2i anim12i df-3an sylibr mat2pmatbas0 - simp1 jca matvscl syl21anc simpr3 decpmatval syl2anc cmulr cvsca 3ad2ant1 - 3simpc matvscacell fveq2d fveq1d cascl mat2pmatvalel oveq2d casa ply1assa - syl3anc csca simp3 eleq2i biimpi matecld ply1sca eqcomd eleqtrrd asclmul2 - adantr eqtrd simp1r2 coe1tm 3eqtrd mpt2eq3dva cvv mat0op syl5eq weq eqidd - wral simprl simprr fvex ovmpt2d ifeq2d oveq12 ifeq1d mpteq2dv eqeq1 ifbid - a1i ovex fvmptd rng0cl ifcld ifex sylan9eqr 3eqtr4d ralrimivva wb eqeltrd - ifov simplr matbas2d matrng 3syl eqmat mpbird ) LUKULZDUMULZUNZKIULZJUOUL - ZHUOULZUPZUNZJMGUQZKEVBZFUQZHURUQZUDUELLHUDUSZUEUSZUVDUQZUTVBZVBZVAZUDUEL - LHUFUOUFUSZJVCZUVFUVGKUQZDVDVBZVEZVOZVBZVAZHJVCZKNVEZUVAUVDBVFVBZULZUUSUV - EUVKVCUVAUUNCVGULZUVBCVFVBZULZUVCUWBULZUNZUWCUUNUUOUUTVHZUUOUWDUUNUUTUUOD - 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co decpmatcl syl3anc matecld crg crngrng ply1sca syl eqcomd fveq2d eleq2d - simpr wb mpbird cmgp ply1moncl sylan asclmul2 cvv eqidd oveq12 adantl a1i - fvex ovmpt2d oveq2d eqtr3d ply1rng ply1sclcl syl2anc matbas2d matvscacell - simpl1 jca mat2pmatval oveqd 3eqtr2d ) JUCUDZDUEUDZIAUDZUFZFUGZUHUDZUIZLU - GZJUDZMUGZJUDZUFZXLXNIXIUJVHZVHZXIKGVHZCUKUPZVHZXSXLXNUAUBJJUAUGZUBUGZXQV - HZCULUPZUPZUMZVHZCUNUPZVHZXLXNXSYGHVHZVHZXLXNXSXQEUPZHVHZVHZXPXSXRYEUPZYI - VHZYAYJXPCUOUDZXRCUQUPZURUPZUDZXSCURUPZUDZYQYAUSXKXMYRXOXHYRXJXFXEYRXGCDN - UTVAVBVCXPUUAXRDURUPZUDZXPJDVDVHZUUFURUPZDXLXNUUDXQJUUFVEZUUDVEZUUGVEZXKX - MXOVFZXKXMXOVGZXKXMXQUUGUDZXOXKXFXGXJUUMXHXFXJXEXFXGVFVBZXHXGXJXEXFXGVGVB - XHXJVSUUFABUUGCDXIIJUENOPUUHUUJVIVJZVCVKXKXMUUAUUEVTZXOXHUUPXJXHYTUUDXRXH - YSDURXHDYSXHDVLUDZDYSUSXFXEUUQXGDVMZVAZCDVLNVNVOVPVQVRVBVCWAXKXMUUCXOXHUU - QXJUUCUUSUUBXICDGCWBUPZKNSUUTVERUUBVEZWCWDZVCYEXRXTYIYSYTUUBCXSYEVEZYSVEY - TVEUVAYIVEZXTVEWEVJXPYPYHXSYIXPYHYPXPUAUBXLXNJJYFYPYGWFXPYGWGYBXLUSYCXNUS - UIZYFYPUSXPUVEYDXRYEYBXLYCXNXQWHVQWIUUKUULYPWFUDXPXRYEWKWJWLVPWMWNXPCVLUD - 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NN0 |-> ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) ) ) ) $= ( wcel co vi vj va vb cfn ccrg w3a cn0 cdecpmat cvsca cfv cmpt2 cmpt cgsu - cv crg wceq crngrng eqid pmatcollpw2 syl3an2 wral cbs eqidd oveq12 oveq1d - adantl simpl simpr simp2 adantr syl cmat decpmatcl syl3anc matecld simplr - wa simp3 ply1tmcl ovmpt2d pmatcollpwlem 3expb eqtrd ralrimivva wb ply1rng - simpl1 3ad2ant2 3ad2ant1 matbas2d ply1moncl mat2pmatbas syl6eleqr matvscl - cmgp sylan syl22anc eqmat syl2anc mpbird mpteq2dva oveq2d ) JUESZDUFSZIAS - ZUGZIBFUHUAUBJJUAUOZUBUOZIFUOZUITZTZXJKGTZCUJUKZTZULZUMZUNTZBFUHXMXKEUKZH - TZUMZUNTXEXDDUPSZXFIXRUQDURZABCDXNUAUBFGIJKLMNXNUSZPQUTVAXGXQYABUNXGFUHXP - XTXGXJUHSZVRZXPXTUQZUCUOZUDUOZXPTZYHYIXTTZUQZUDJVBUCJVBZYFYLUCUDJJYFYHJSZ - YIJSZVRZVRZYJYHYIXKTZXMXNTZYKYQUAUBYHYIJJXOYSXPCVCUKZYQXPVDXHYHUQXIYIUQVR - ZXOYSUQYQUUAXLYRXMXNXHYHXIYIXKVEVFVGYPYNYFYNYOVHVGZYPYOYFYNYOVIVGZYQYBYRD - VCUKZSYEYSYTSYFYBYPYFXEYBXGXEYEXDXEXFVJVKZYCVLZVKYQJDVMTZUUGVCUKZDYHYIUUD - XKJUUGUSZUUDUSZUUHUSZUUBUUCYFXKUUHSZYPYFXEXFYEUULUUEXGXFYEXDXEXFVSVKXGYEV - IZUUGABUUHCDXJIJUFLMNUUIUUKVNVOZVKVPXGYEYPVQYTYRXJCDXNGUUDCWPUKZKUUJLQYDU - UOUSZPYTUSZVTVOWAYFYNYOYSYKUQABCDEFGHIJKUCUDLMNOPQRWBWCWDWEYFXPASXTASZYGY - MWFYFUAUBBAXOCYTJUPMUUQNXDXEXFYEWHZXGCUPSZYEXEXDUUTXFXEYBUUTYCCDLWGVLWIVK - ZYFXHJSZXIJSZUGZYBXLUUDSYEXOYTSYFUVBYBUVCUUFWJUVDUUGUUHDXHXIUUDXKJUUIUUJU - UKYFUVBUVCVJYFUVBUVCVSYFUVBUULUVCUUNWJVPYFUVBYEUVCUUMWJYTXLXJCDXNGUUDUUOK - UUJLQYDUUPPUUQVTVOWKYFXDUUTXMYTSZXSASUURUUSUVAXGYBYEUVEXEXDYBXFYCWIZYTXJC - DGUUOKLQUUPPUUQWLWQYFXSBVCUKZAYFXDYBUULXSUVGSUUSXGYBYEUVFVKUUNUUGUUHBCDEX - KJRUUIUUKLMWMVONWNBAXMCHYTJXSUUQMNOWOWRBACUCUDJXPXTMNWSWTXAXBXCWD $. + cv crg wceq crngring eqid pmatcollpw2 syl3an2 wa wral eqidd oveq12 oveq1d + cbs adantl simpl simpr simp2 adantr syl cmat simp3 syl3anc matecld simplr + decpmatcl ply1tmcl ovmpt2d pmatcollpwlem 3expb eqtrd ralrimivva wb simpl1 + ply1ring 3ad2ant2 3ad2ant1 matbas2d ply1moncl sylan mat2pmatbas syl6eleqr + cmgp matvscl syl22anc eqmat syl2anc mpbird mpteq2dva oveq2d ) JUESZDUFSZI + ASZUGZIBFUHUAUBJJUAUOZUBUOZIFUOZUITZTZXJKGTZCUJUKZTZULZUMZUNTZBFUHXMXKEUK + ZHTZUMZUNTXEXDDUPSZXFIXRUQDURZABCDXNUAUBFGIJKLMNXNUSZPQUTVAXGXQYABUNXGFUH + XPXTXGXJUHSZVBZXPXTUQZUCUOZUDUOZXPTZYHYIXTTZUQZUDJVCUCJVCZYFYLUCUDJJYFYHJ + SZYIJSZVBZVBZYJYHYIXKTZXMXNTZYKYQUAUBYHYIJJXOYSXPCVGUKZYQXPVDXHYHUQXIYIUQ + VBZXOYSUQYQUUAXLYRXMXNXHYHXIYIXKVEVFVHYPYNYFYNYOVIVHZYPYOYFYNYOVJVHZYQYBY + RDVGUKZSYEYSYTSYFYBYPYFXEYBXGXEYEXDXEXFVKVLZYCVMZVLYQJDVNTZUUGVGUKZDYHYIU + UDXKJUUGUSZUUDUSZUUHUSZUUBUUCYFXKUUHSZYPYFXEXFYEUULUUEXGXFYEXDXEXFVOVLXGY + EVJZUUGABUUHCDXJIJUFLMNUUIUUKVSVPZVLVQXGYEYPVRYTYRXJCDXNGUUDCWPUKZKUUJLQY + DUUOUSZPYTUSZVTVPWAYFYNYOYSYKUQABCDEFGHIJKUCUDLMNOPQRWBWCWDWEYFXPASXTASZY + GYMWFYFUAUBBAXOCYTJUPMUUQNXDXEXFYEWGZXGCUPSZYEXEXDUUTXFXEYBUUTYCCDLWHVMWI + VLZYFXHJSZXIJSZUGZYBXLUUDSYEXOYTSYFUVBYBUVCUUFWJUVDUUGUUHDXHXIUUDXKJUUIUU + JUUKYFUVBUVCVKYFUVBUVCVOYFUVBUULUVCUUNWJVQYFUVBYEUVCUUMWJYTXLXJCDXNGUUDUU + OKUUJLQYDUUPPUUQVTVPWKYFXDUUTXMYTSZXSASUURUUSUVAXGYBYEUVEXEXDYBXFYCWIZYTX + JCDGUUOKLQUUPPUUQWLWMYFXSBVGUKZAYFXDYBUULXSUVGSUUSXGYBYEUVFVLUUNUUGUUHBCD + EXKJRUUIUUKLMWNVPNWOBAXMCHYTJXSUUQMNOWQWRBACUCUDJXPXTMNWSWTXAXBXCWD $. $d B n s $. $d C n $. $d M s $. $d N s $. $d R s $. $( Write a polynomial matrix (over a commutative ring) as a finite sum of @@ -264971,29 +265202,29 @@ polynomial matrices for the same power (Contributed by AV, -> E. s e. NN0 M = ( C gsum ( n e. ( 0 ... s ) |-> ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) ) ) ) $= ( wcel cfn ccrg w3a cv clt wbr cdecpmat co cmat c0g cfv wceq wi wral wrex - cn0 cc0 cfz cmpt crg crngrng 3ad2ant2 simp3 decpmataa0 syl2anc pmatcollpw - cgsu eqid wa ad2antrr nfv nfra1 nfan ccmn simp1 pmatrng rngcmn syl adantr - ply1rng anim1i cmgp ply1moncl simpl2 simpr decpmatcl syl3anc mat2pmatbas0 - cbs matvscl syl22anc ralrimiva fveq2 jca 0mat2pmat sylan9eqr oveq2d clmod - csca pmatlmod adantlr ply1crng anim2i 3adant3 eqcomd fveq2d eqcomi fveq2i - matsca2 eleqtrrd oveq2i lmodvs0 syl5eq eqtrd imim2d ralimdva gsummptnn0fz - ex imp reximdva mpd ) JUATZDUBTZIATZUCZLUDZFUDZUEUFZIYGUGUHZJDUIUHZUJUKZU - LZUMZFUPUNZLUPUOZIBFUQYFURUHYGKGUHZYIEUKZHUHZUSVGUHZULZLUPUOYEDUTTZYDYOYC - YBUUAYDDVAVBZYBYCYDVCZFYJABCDIJYKLMNOYJVHZYKVHZVDVEYEYNYTLUPYEYFUPTZVIZYN - YTUUGYNVIZIBFUPYRUSVGUHZYSYEIUUIULUUFYNABCDEFGHIJKMNOPQRSVFVJUUHAYRYFFBBU - JUKZUUGYNFUUGFVKYMFUPVLVMOUUJVHZYEBVNTZUUFYNYEBUTTZUULYEYBUUAUUMYBYCYDVOZ - UUBBCDJMNVPVEBVQVRVJYEYRATZFUPUNUUFYNYEUUOFUPYEYGUPTZVIZYBCUTTZYPCWIUKZTZ - YQATZUUOYEYBUUPUUNVSZUUQUUAUURYEUUAUUPUUBVSZCDMVTVRUUQUUAUUPVIZUUTYEUUAUU - PUUBWAZUUSYGCDGCWBUKZKMRUVFVHQUUSVHZWCZVRUUQYBUUAYIYJWIUKZTZUVAUVBUVCUUQY - CYDUUPUVJYBYCYDUUPWDYEYDUUPUUCVSYEUUPWEYJABUVICDYGIJUBMNOUUDUVIVHZWFWGYJU - VIBCDEAYIJSUUDUVKMNOWHWGBAYPCHUUSJYQUVGNOPWJWKWLVJUUGUUFYNYEUUFWEVSUUGYNY - HYRUUJULZUMZFUPUNUUGYMUVMFUPUUGUUPVIZYLUVLYHUVNYLUVLUVNYLVIZYRYPJCUIUHZUJ - UKZHUHZUUJUVOYQUVQYPHYLUVNYQYKEUKZUVQYIYKEWMUVNUUAYBVIZUVSUVQULYEUVTUUFUU - PYEUUAYBUUBUUNWNVJCDEJYKUVQSMUUEUVQVHWOVRWPWQUVNUVRUUJULZYLUVNBWRTZYPBWSU - KZWIUKZTZUWAYEUWBUUFUUPYEYBUUAUWBUUNUUBBCDJMNWTVEVJUVNYPUUSUWDUVNUVDUUTYE - UUPUVDUUFUVEXAUVHVRUVNUWCCWIYEUWCCULUUFUUPYECUWCYEYBCUBTZVIZCUWCULYBYCUWG - YDYCUWFYBCDMXBXCXDBCJUBNXIVRXEVJXFXJUWBUWEVIUVRYPUUJHUHUUJUVQUUJYPHUVPBUJ - BUVPNXGXHXKHUWCUWDBYPUUJUWCVHPUWDVHUUKXLXMVEVSXNXRXOXPXSXQXNXRXTYA $. + cn0 cc0 cfz cmpt cgsu crg crngring 3ad2ant2 simp3 eqid decpmataa0 syl2anc + wa pmatcollpw ad2antrr nfv nfra1 nfan ccmn simp1 pmatring ringcmn syl cbs + adantr ply1ring anim1i cmgp ply1moncl simpl2 simpr decpmatcl mat2pmatbas0 + syl3anc matvscl syl22anc ralrimiva fveq2 0mat2pmat sylan9eqr oveq2d clmod + jca csca pmatlmod adantlr ply1crng anim2i 3adant3 matsca2 eqcomd eleqtrrd + fveq2d eqcomi fveq2i oveq2i lmodvs0 syl5eq eqtrd ex ralimdva gsummptnn0fz + imim2d imp reximdva mpd ) JUATZDUBTZIATZUCZLUDZFUDZUEUFZIYGUGUHZJDUIUHZUJ + UKZULZUMZFUPUNZLUPUOZIBFUQYFURUHYGKGUHZYIEUKZHUHZUSUTUHZULZLUPUOYEDVATZYD + YOYCYBUUAYDDVBVCZYBYCYDVDZFYJABCDIJYKLMNOYJVEZYKVEZVFVGYEYNYTLUPYEYFUPTZV + HZYNYTUUGYNVHZIBFUPYRUSUTUHZYSYEIUUIULUUFYNABCDEFGHIJKMNOPQRSVIVJUUHAYRYF + FBBUJUKZUUGYNFUUGFVKYMFUPVLVMOUUJVEZYEBVNTZUUFYNYEBVATZUULYEYBUUAUUMYBYCY + DVOZUUBBCDJMNVPVGBVQVRVJYEYRATZFUPUNUUFYNYEUUOFUPYEYGUPTZVHZYBCVATZYPCVSU + KZTZYQATZUUOYEYBUUPUUNVTZUUQUUAUURYEUUAUUPUUBVTZCDMWAVRUUQUUAUUPVHZUUTYEU + UAUUPUUBWBZUUSYGCDGCWCUKZKMRUVFVEQUUSVEZWDZVRUUQYBUUAYIYJVSUKZTZUVAUVBUVC + UUQYCYDUUPUVJYBYCYDUUPWEYEYDUUPUUCVTYEUUPWFYJABUVICDYGIJUBMNOUUDUVIVEZWGW + IYJUVIBCDEAYIJSUUDUVKMNOWHWIBAYPCHUUSJYQUVGNOPWJWKWLVJUUGUUFYNYEUUFWFVTUU + GYNYHYRUUJULZUMZFUPUNUUGYMUVMFUPUUGUUPVHZYLUVLYHUVNYLUVLUVNYLVHZYRYPJCUIU + HZUJUKZHUHZUUJUVOYQUVQYPHYLUVNYQYKEUKZUVQYIYKEWMUVNUUAYBVHZUVSUVQULYEUVTU + UFUUPYEUUAYBUUBUUNWRVJCDEJYKUVQSMUUEUVQVEWNVRWOWPUVNUVRUUJULZYLUVNBWQTZYP + BWSUKZVSUKZTZUWAYEUWBUUFUUPYEYBUUAUWBUUNUUBBCDJMNWTVGVJUVNYPUUSUWDUVNUVDU + UTYEUUPUVDUUFUVEXAUVHVRUVNUWCCVSYEUWCCULUUFUUPYECUWCYEYBCUBTZVHZCUWCULYBY + CUWGYDYCUWFYBCDMXBXCXDBCJUBNXEVRXFVJXHXGUWBUWEVHUVRYPUUJHUHUUJUVQUUJYPHUV + PBUJBUVPNXIXJXKHUWCUWDBYPUUJUWCVEPUWDVEUUKXLXMVGVTXNXOXRXPXSXQXNXOXTYA $. $d B f i j $. $d B k l x y $. $d B m $. $d C f $. $d B f n x $. $d D f k $. $d I i j k l x y $. $d I f n k x $. $d M k l x y $. @@ -265102,51 +265333,51 @@ polynomial matrices for the same power (Contributed by AV, -> M = ( C gsum ( n e. ( 0 ... 1 ) |-> ( ( n .^ X ) .* ( T ` ( H ` n ) ) ) ) ) ) $= ( cfn wcel crg wa cc0 csn cmap co cv cfv cmpt cgsu wceq c1 caddc cplusg - w3a cfz c0g cmnd pmatrng rngmnd syl adantr ccmn rngcmn snfi a1i simplll - simpr cn0 simpllr elmapi adantl ffvelrnda elsni syl6eqel mat2pmatscmxcl - wf 0nn0 syl22anc ralrimiva gsummptcl eqid mndrid syl2anc cz fzsn eqcomi - 0z ax-mp cif ad2antlr eqtrd iftrue fveq2 eqcomd cuz 1nn0 nn0uz syl6eleq - eluzfz1 eleq1 mpbird wi ffvelrn imp fvmptd fveq2d oveq2d mpteq12dva cvv - ex ovex mndidcl wn 0p1e1 eqeq2i ax-1ne0 neii eqeq1 mtbiri sylbi iffalse - wb notbid syl6eq cmat fveq2i ad2antrr cbs oveq2i eluzfz2 fvex 0mat2pmat - ancoms clmod csca pmatlmod cmgp ply1moncl ply1rng matsca2 sylan2 eleq2d - lmodvs0 gsumsnd oveq12d eqtr3d 3impa id c0ex snid ffvelrnd matrng ifcld - rng0cl fmptd feq2i sylibr elfznn0 gsummptfzsplit 3adant3 eqtr4d mpteq1 - ) NUIUJZFUKUJZULZJDUMUNZUOUPUJZMCHUVQHUQZOIUPZUVSJURZGURZLUPZUSZUTUPZVA - ZVEZMCHUMUMVBVCUPZVFUPZUVTUVSKURZGURZLUPZUSZUTUPZCHUMVBVFUPZUWLUSZUTUPU - 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( 0 ... s ) |-> ( ( n .^ X ) .* ( T ` ( f ` n ) ) ) ) ) ) ) $= ( vg vl cc0 csn cv co cfv cmpt cgsu wceq cmap wrex cfn wcel w3a cfz fveq1 - cn fveq2d oveq2d mpteq2dv eqeq2d cbvrexv wa c1 c0g cif crg crngrng anim2i - ccrg 3adant3 ad2antrr simplr simpr eqid pmatcollpw3fi1lem1 syl3anc elmapi - wf wi c0ex a1i ffvelrn sylan2 ex syl adantl imp matrng rng0cl ifcld fmptd - snid cvv wb cbs fvex eqeltri ovex pm3.2i elmapg mpbird adantr rspcedv 1nn - mp1i oveq2 mpteq1d rexeqbidv mpd mpdan rexlimdva syl5bi ) LCIUGUHZIUIZNJU - JZXTHUIZUKZGUKZKUJZULZUMUJZUNZHDXSUOUJZUPLCIXSYAXTUEUIZUKZGUKZKUJZULZUMUJ - ZUNZUEYIUPMUQURZFVOURZLBURZUSZLCIUGOUIZUTUJZYEULZUMUJZUNZHDUUBUOUJZUPZOVB - UPZYHYPHUEYIYBYJUNZYGYOLUUIYFYNCUMUUIIXSYEYMUUIYDYLYAKUUIYCYKGXTYBYJVAVCV - DVEVDVFVGYTYPUUHUEYIYTYJYIURZVHZYPUUHUUKYPVHZLCIUGVIUTUJZYAXTUFUUMUFUIZUG - UNZUGYJUKZAVJUKZVKZULZUKZGUKZKUJZULZUMUJZUNZUUHUULYQFVLURZVHZUUJYPUVEYTUV - GUUJYPYQYRUVGYSYRUVFYQFVMZVNVPVQYTUUJYPVRUUKYPVSABCDEFGIJYJUUSKLMNUUQUFPQ - RSTUAUBUCUDUUQVTZUUSVTZWAWBUULUVEVHZLCIUUMYEULZUMUJZUNZHDUUMUOUJZUPZUUHUU - 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wceq w3a ply1rng anim2i simpr df-3an sylibr eqid scmatscmide sylan syl5eq - anim12i fveq2d fveq1d fvif fveq1i iffv eqtri syl6eq oveq1d ovif csn coe1z - cxp ad2antlr cvv fvex a1i simpl fvconst2g syl eqtrd clmod ply1lmod rngmgp - csca cmnd 0nn0 mgpbas mulgnn0cl syl3anc lmod0vs syl2anc wb ply1sca adantl - vr1cl eqeq1d adantr mpbird ifeq2d cur cbs ancomd coe1fvalcl eqcomd eleq2d - syl6eqr asclval ply1idvr1 oveq2d eqtr2d ifeq1d 3eqtrd ) RUQURZGUSURZUTZPV - AURZFLURZUTZUTZTVBZRURUAVBZRURUTZUTZPUUEUUFQVCZVDVEZVEZVFGVGVEZEVHVEZVIVE - ZVCZEVJVEZVCUUEUUFVNZPFVDVEZVEZPEVKVEZVDVEZVEZVLZUUOUUPVCZUUQUUSUUOUUPVCZ - UUTVLZUUQUUSJVEZUUTVLZUUHUUKUVCUUOUUPUUHUUKPUUQFUUTVLZVDVEZVEZUVCUUHPUUJU - VJUUHUUIUVIVDUUHUUIUUEUUFFKNVCZVCZUVIQUVLUUEUUFUPVMUUDYREUSURZUUBVOZUUGUV - MUVIVNUUDYRUVNUTZUUBUTUVOYTUVPUUCUUBYSUVNYREGUBVPZVQUUAUUBVRWEYRUVNUUBVSV - TCLFEKUUENUUFRUUTUCUMUUTWAZUOUEWBWCWDWFWGUVKPUUQUURUVAVLZVEUVCPUVJUVSUUQF - UUTVDWHWIUUQPUURUVAWJWKWLWMUUHUVDUUQUVEUVBUUOUUPVCZVLZUVFUUQUUSUVBUUOUUPW - NUUDUWAUVFVNUUGUUDUUQUVTUUTUVEUUDUVTGVKVEZUUOUUPVCZUUTUUDUVBUWBUUOUUPUUDU - 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wral cvv oveq1d adantl simprr a1i ovmpt2d simpll ply1rng pm3.22 ply1sclcl - ovex scmatscmide sylan 3eqtr4d ralrimivva wb 0nn0 ply1tmcl matbas2d eqmat - mpbird eqtrd 3eqtrd ) RUSUTZGVAUTZVBZPVCUTZFLUTZVBZVBZQPVDVEZIVFZUOUPRRUO - VGZUPVGZUUCVEZEVHVFZVFZVIZUOUPRRPUUEUUFQVEZVJVFZVFZUUHVFZVIZPFVJVFZVFZJVF - ZKNVEZUUBYPYQUUCDUTZVKZUUDUUJWBUUBYRUUTUVAYRUUAVLUUBYQQBUTZYSUUTYRYQUUAYP - YQVMVNZUUBYPYQYTVKZUVBUUBYRYTVBUVDUUAYTYRYSYTVMVOYPYQYTVPVQUVDQFKNVEBUNBC - EFGKLNRTUAUBUKUCUMVRVSVTZYRYSYTWCZABCDEGPQRVATUAUBUGUHWAWDYPYQUUTVPWEUOUP - ADEGUUHIUUCRVAUFUGUHTUUHWKZWFVTUUBUOUPRRUUIUUNUUBUUERUTZUUFRUTZVKZUUGUUMU - UHUVJYQUVBYSVKZUVHUVIVBUUGUUMWBUUBUVHUVKUVIUUBYQUVBYSUVCUVEUVFWLWGUUBUVHU - VIWHBCEGUUEUUFPQRVATUAUBWIWJWMWNUUBUUOUOUPRRUUMWOGWPVFZEXCVFZWQVFZVEZEWRV - FZVEZVIZUUSUUBUOUPRRUUNUVQUVJYQUUMOUTZUUNUVQWBYPYQUUAUVHUVIWSZUVJUUKLUTYS - UVSUVJCBEUUEUUFLQRUAUKUBUUBUVHUVIWTUUBUVHUVIXAUUBUVHUVBUVIUVEWGXBUUBUVHYS - UVIUVFWGUULLEGUUKOPUULWKUKTUJXDWJZUUHEGUVPUVNUUMOUVMUVLUJTUVLWKZUVPWKZUVM - WKZUVNWKZUVGXEWJWNUUBUVRUUSWBZUQVGZURVGZUVRVEZUWGUWHUUSVEZWBZURRXLUQRXLZU - 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NN0 |-> ( ( n .^ X ) .* ( ( U ` ( ( coe1 ` Q ) ` n ) ) .* .1. ) ) ) ) ) $= ( cfn wcel ccrg w3a cn0 cv co cdecpmat cfv cmpt cgsu cco1 wceq 1pmatscmul - crg crngrng syl5eqel syl3an2 pmatcollpw syld3an3 wa anim2i 3adant3 adantr - simp3 anim1i ancomd pmatcollpwscmatlem2 syl2anc oveq2d mpteq2dva eqtrd ) + crg crngring syl5eqel syl3an2 pmatcollpw syld3an3 wa anim2i 3adant3 simp3 + adantr anim1i ancomd pmatcollpwscmatlem2 syl2anc oveq2d mpteq2dva eqtrd ) RUOUPZGUQUPZFMUPZURZQCLUSLUTZSNVAZQWKVBVAIVCZOVAZVDZVEVAZCLUSWLWKFVFVCVCJ VCKOVAZOVAZVDZVEVAWGWHWIQBUPZQWPVGWHWGGVIUPZWIWTGVJZWGXAWIURQFKOVABUNBCEF GKMORTUAUBUKUCUMVHVKVLBCEGILNOQRSTUAUBUCUDUEUFVMVNWJWOWSCVEWJLUSWNWRWJWKU - SUPZVOZWMWQWLOXDWGXAVOZXCWIVOWMWQVGWJXEXCWGWHXEWIWHXAWGXBVPVQVRXDWIXCWJWI - XCWGWHWIVSVTWAABCDEFGHIJKMNOPWKQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNWBWCWDWEWD + SUPZVOZWMWQWLOXDWGXAVOZXCWIVOWMWQVGWJXEXCWGWHXEWIWHXAWGXBVPVQVSXDWIXCWJWI + XCWGWHWIVRVTWAABCDEFGHIJKMNOPWKQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNWBWCWDWEWD WF $. $} @@ -265352,7 +265583,7 @@ polynomial matrices for the same power (Contributed by AV, In this section, the following conventions are mostly used: