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Theorem scmatghm 19489
Description: There is a group homomorphism from the additive group of a ring to the additive group of the ring of scalar matrices over this ring. (Contributed by AV, 22-Dec-2019.)
Hypotheses
Ref Expression
scmatrhmval.k  |-  K  =  ( Base `  R
)
scmatrhmval.a  |-  A  =  ( N Mat  R )
scmatrhmval.o  |-  .1.  =  ( 1r `  A )
scmatrhmval.t  |-  .*  =  ( .s `  A )
scmatrhmval.f  |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  ) )
scmatrhmval.c  |-  C  =  ( N ScMat  R )
scmatghm.s  |-  S  =  ( As  C )
Assertion
Ref Expression
scmatghm  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F  e.  ( R  GrpHom  S ) )
Distinct variable groups:    x, K    x, R    x,  .1.    x,  .*    x, C    x, N
Allowed substitution hints:    A( x)    S( x)    F( x)

Proof of Theorem scmatghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scmatrhmval.k . 2  |-  K  =  ( Base `  R
)
2 eqid 2429 . 2  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2429 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2429 . 2  |-  ( +g  `  S )  =  ( +g  `  S )
5 ringgrp 17720 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
65adantl 467 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  e.  Grp )
7 scmatrhmval.a . . . 4  |-  A  =  ( N Mat  R )
8 eqid 2429 . . . 4  |-  ( Base `  A )  =  (
Base `  A )
9 eqid 2429 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
10 scmatrhmval.c . . . 4  |-  C  =  ( N ScMat  R )
117, 8, 1, 9, 10scmatsgrp 19475 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  (SubGrp `  A
) )
12 scmatghm.s . . . 4  |-  S  =  ( As  C )
1312subggrp 16771 . . 3  |-  ( C  e.  (SubGrp `  A
)  ->  S  e.  Grp )
1411, 13syl 17 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  Grp )
15 scmatrhmval.o . . . 4  |-  .1.  =  ( 1r `  A )
16 scmatrhmval.t . . . 4  |-  .*  =  ( .s `  A )
17 scmatrhmval.f . . . 4  |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  ) )
181, 7, 15, 16, 17, 10scmatf 19485 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F : K --> C )
197, 10, 12scmatstrbas 19482 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( Base `  S )  =  C )
2019feq3d 5734 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( F : K --> ( Base `  S )  <->  F : K --> C ) )
2118, 20mpbird 235 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F : K --> ( Base `  S ) )
227matsca2 19376 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  =  (Scalar `  A
) )
23 ovex 6333 . . . . . . . . . . 11  |-  ( N ScMat 
R )  e.  _V
2410, 23eqeltri 2513 . . . . . . . . . 10  |-  C  e. 
_V
25 eqid 2429 . . . . . . . . . . 11  |-  (Scalar `  A )  =  (Scalar `  A )
2612, 25resssca 15234 . . . . . . . . . 10  |-  ( C  e.  _V  ->  (Scalar `  A )  =  (Scalar `  S ) )
2724, 26mp1i 13 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
(Scalar `  A )  =  (Scalar `  S )
)
2822, 27eqtrd 2470 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  =  (Scalar `  S
) )
2928fveq2d 5885 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( +g  `  R )  =  ( +g  `  (Scalar `  S ) ) )
3029oveqd 6322 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y ( +g  `  R ) z )  =  ( y ( +g  `  (Scalar `  S ) ) z ) )
3130oveq1d 6320 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( y ( +g  `  R ) z )  .*  .1.  )  =  ( (
y ( +g  `  (Scalar `  S ) ) z )  .*  .1.  )
)
3231adantr 466 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
( y ( +g  `  R ) z )  .*  .1.  )  =  ( ( y ( +g  `  (Scalar `  S ) ) z )  .*  .1.  )
)
337matlmod 19385 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
347, 10scmatlss 19481 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  ( LSubSp `  A ) )
35 eqid 2429 . . . . . . . 8  |-  ( LSubSp `  A )  =  (
LSubSp `  A )
3612, 35lsslmod 18118 . . . . . . 7  |-  ( ( A  e.  LMod  /\  C  e.  ( LSubSp `  A )
)  ->  S  e.  LMod )
3733, 34, 36syl2anc 665 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  LMod )
3837adantr 466 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  S  e.  LMod )
3928fveq2d 5885 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( Base `  R )  =  ( Base `  (Scalar `  S ) ) )
401, 39syl5eq 2482 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  K  =  ( Base `  (Scalar `  S )
) )
4140eleq2d 2499 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y  e.  K  <->  y  e.  ( Base `  (Scalar `  S ) ) ) )
4241biimpd 210 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y  e.  K  ->  y  e.  ( Base `  (Scalar `  S )
) ) )
4342adantrd 469 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( y  e.  K  /\  z  e.  K )  ->  y  e.  ( Base `  (Scalar `  S ) ) ) )
4443imp 430 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  y  e.  ( Base `  (Scalar `  S ) ) )
4540eleq2d 2499 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( z  e.  K  <->  z  e.  ( Base `  (Scalar `  S ) ) ) )
4645biimpd 210 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( z  e.  K  ->  z  e.  ( Base `  (Scalar `  S )
) ) )
4746adantld 468 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( y  e.  K  /\  z  e.  K )  ->  z  e.  ( Base `  (Scalar `  S ) ) ) )
4847imp 430 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  z  e.  ( Base `  (Scalar `  S ) ) )
497, 8, 1, 9, 10scmatid 19470 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  A
)  e.  C )
5015a1i 11 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .1.  =  ( 1r `  A ) )
5149, 50, 193eltr4d 2532 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .1.  e.  ( Base `  S
) )
5251adantr 466 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  .1.  e.  ( Base `  S
) )
53 eqid 2429 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
5412, 16ressvsca 15235 . . . . . . 7  |-  ( C  e.  _V  ->  .*  =  ( .s `  S ) )
5524, 54ax-mp 5 . . . . . 6  |-  .*  =  ( .s `  S )
56 eqid 2429 . . . . . 6  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
57 eqid 2429 . . . . . 6  |-  ( +g  `  (Scalar `  S )
)  =  ( +g  `  (Scalar `  S )
)
582, 4, 53, 55, 56, 57lmodvsdir 18050 . . . . 5  |-  ( ( S  e.  LMod  /\  (
y  e.  ( Base `  (Scalar `  S )
)  /\  z  e.  ( Base `  (Scalar `  S
) )  /\  .1.  e.  ( Base `  S
) ) )  -> 
( ( y ( +g  `  (Scalar `  S ) ) z )  .*  .1.  )  =  ( ( y  .*  .1.  ) ( +g  `  S ) ( z  .*  .1.  ) ) )
5938, 44, 48, 52, 58syl13anc 1266 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
( y ( +g  `  (Scalar `  S )
) z )  .*  .1.  )  =  ( ( y  .*  .1.  ) ( +g  `  S
) ( z  .*  .1.  ) ) )
6032, 59eqtrd 2470 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
( y ( +g  `  R ) z )  .*  .1.  )  =  ( ( y  .*  .1.  ) ( +g  `  S ) ( z  .*  .1.  ) ) )
61 simpr 462 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  e.  Ring )
6261adantr 466 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  R  e.  Ring )
6361anim1i 570 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( R  e.  Ring  /\  (
y  e.  K  /\  z  e.  K )
) )
64 3anass 986 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  K  /\  z  e.  K )  <->  ( R  e.  Ring  /\  ( y  e.  K  /\  z  e.  K ) ) )
6563, 64sylibr 215 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( R  e.  Ring  /\  y  e.  K  /\  z  e.  K ) )
661, 3ringacl 17743 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  K  /\  z  e.  K )  ->  (
y ( +g  `  R
) z )  e.  K )
6765, 66syl 17 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
y ( +g  `  R
) z )  e.  K )
681, 7, 15, 16, 17scmatrhmval 19483 . . . 4  |-  ( ( R  e.  Ring  /\  (
y ( +g  `  R
) z )  e.  K )  ->  ( F `  ( y
( +g  `  R ) z ) )  =  ( ( y ( +g  `  R ) z )  .*  .1.  ) )
6962, 67, 68syl2anc 665 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  ( y
( +g  `  R ) z ) )  =  ( ( y ( +g  `  R ) z )  .*  .1.  ) )
701, 7, 15, 16, 17scmatrhmval 19483 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  K )  ->  ( F `  y )  =  ( y  .*  .1.  ) )
7170ad2ant2lr 752 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  y )  =  ( y  .*  .1.  ) )
721, 7, 15, 16, 17scmatrhmval 19483 . . . . 5  |-  ( ( R  e.  Ring  /\  z  e.  K )  ->  ( F `  z )  =  ( z  .*  .1.  ) )
7372ad2ant2l 750 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  z )  =  ( z  .*  .1.  ) )
7471, 73oveq12d 6323 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
( F `  y
) ( +g  `  S
) ( F `  z ) )  =  ( ( y  .*  .1.  ) ( +g  `  S ) ( z  .*  .1.  ) ) )
7560, 69, 743eqtr4d 2480 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  ( y
( +g  `  R ) z ) )  =  ( ( F `  y ) ( +g  `  S ) ( F `
 z ) ) )
761, 2, 3, 4, 6, 14, 21, 75isghmd 16843 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F  e.  ( R  GrpHom  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   _Vcvv 3087    |-> cmpt 4484   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7577   Basecbs 15084   ↾s cress 15085   +g cplusg 15152  Scalarcsca 15155   .scvsca 15156   0gc0g 15297   Grpcgrp 16620  SubGrpcsubg 16762    GrpHom cghm 16831   1rcur 17670   Ringcrg 17715   LModclmod 18026   LSubSpclss 18090   Mat cmat 19363   ScMat cscmat 19445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-0g 15299  df-gsum 15300  df-prds 15305  df-pws 15307  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-ghm 16832  df-cntz 16922  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-subrg 17941  df-lmod 18028  df-lss 18091  df-sra 18330  df-rgmod 18331  df-dsmm 19226  df-frlm 19241  df-mamu 19340  df-mat 19364  df-dmat 19446  df-scmat 19447
This theorem is referenced by:  scmatrhm  19491
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