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Theorem scmatghm 18904
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 2467 . 2  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2467 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2467 . 2  |-  ( +g  `  S )  =  ( +g  `  S )
5 ringgrp 17075 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
65adantl 466 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  e.  Grp )
7 scmatrhmval.a . . . 4  |-  A  =  ( N Mat  R )
8 eqid 2467 . . . 4  |-  ( Base `  A )  =  (
Base `  A )
9 eqid 2467 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
10 scmatrhmval.c . . . 4  |-  C  =  ( N ScMat  R )
117, 8, 1, 9, 10scmatsgrp 18890 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  (SubGrp `  A
) )
12 scmatghm.s . . . 4  |-  S  =  ( As  C )
1312subggrp 16076 . . 3  |-  ( C  e.  (SubGrp `  A
)  ->  S  e.  Grp )
1411, 13syl 16 . 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 18900 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F : K --> C )
197, 10, 12scmatstrbas 18897 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( Base `  S )  =  C )
20 feq3 5721 . . . 4  |-  ( (
Base `  S )  =  C  ->  ( F : K --> ( Base `  S )  <->  F : K
--> C ) )
2119, 20syl 16 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( F : K --> ( Base `  S )  <->  F : K --> C ) )
2218, 21mpbird 232 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F : K --> ( Base `  S ) )
237matsca2 18791 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  =  (Scalar `  A
) )
24 ovex 6320 . . . . . . . . . . 11  |-  ( N ScMat 
R )  e.  _V
2510, 24eqeltri 2551 . . . . . . . . . 10  |-  C  e. 
_V
26 eqid 2467 . . . . . . . . . . 11  |-  (Scalar `  A )  =  (Scalar `  A )
2712, 26resssca 14650 . . . . . . . . . 10  |-  ( C  e.  _V  ->  (Scalar `  A )  =  (Scalar `  S ) )
2825, 27mp1i 12 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
(Scalar `  A )  =  (Scalar `  S )
)
2923, 28eqtrd 2508 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  =  (Scalar `  S
) )
3029fveq2d 5876 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( +g  `  R )  =  ( +g  `  (Scalar `  S ) ) )
3130oveqd 6312 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y ( +g  `  R ) z )  =  ( y ( +g  `  (Scalar `  S ) ) z ) )
3231oveq1d 6310 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( y ( +g  `  R ) z )  .*  .1.  )  =  ( (
y ( +g  `  (Scalar `  S ) ) z )  .*  .1.  )
)
3332adantr 465 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
( y ( +g  `  R ) z )  .*  .1.  )  =  ( ( y ( +g  `  (Scalar `  S ) ) z )  .*  .1.  )
)
347matlmod 18800 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
357, 10scmatlss 18896 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  ( LSubSp `  A ) )
36 eqid 2467 . . . . . . . 8  |-  ( LSubSp `  A )  =  (
LSubSp `  A )
3712, 36lsslmod 17477 . . . . . . 7  |-  ( ( A  e.  LMod  /\  C  e.  ( LSubSp `  A )
)  ->  S  e.  LMod )
3834, 35, 37syl2anc 661 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  LMod )
3938adantr 465 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  S  e.  LMod )
4029fveq2d 5876 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( Base `  R )  =  ( Base `  (Scalar `  S ) ) )
411, 40syl5eq 2520 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  K  =  ( Base `  (Scalar `  S )
) )
4241eleq2d 2537 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y  e.  K  <->  y  e.  ( Base `  (Scalar `  S ) ) ) )
4342biimpd 207 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( y  e.  K  ->  y  e.  ( Base `  (Scalar `  S )
) ) )
4443adantrd 468 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( y  e.  K  /\  z  e.  K )  ->  y  e.  ( Base `  (Scalar `  S ) ) ) )
4544imp 429 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  y  e.  ( Base `  (Scalar `  S ) ) )
4641eleq2d 2537 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( z  e.  K  <->  z  e.  ( Base `  (Scalar `  S ) ) ) )
4746biimpd 207 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( z  e.  K  ->  z  e.  ( Base `  (Scalar `  S )
) ) )
4847adantld 467 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( y  e.  K  /\  z  e.  K )  ->  z  e.  ( Base `  (Scalar `  S ) ) ) )
4948imp 429 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  z  e.  ( Base `  (Scalar `  S ) ) )
507, 8, 1, 9, 10scmatid 18885 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  A
)  e.  C )
5115a1i 11 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .1.  =  ( 1r `  A ) )
5250, 51, 193eltr4d 2570 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .1.  e.  ( Base `  S
) )
5352adantr 465 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  .1.  e.  ( Base `  S
) )
54 eqid 2467 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
5512, 16ressvsca 14651 . . . . . . 7  |-  ( C  e.  _V  ->  .*  =  ( .s `  S ) )
5625, 55ax-mp 5 . . . . . 6  |-  .*  =  ( .s `  S )
57 eqid 2467 . . . . . 6  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
58 eqid 2467 . . . . . 6  |-  ( +g  `  (Scalar `  S )
)  =  ( +g  `  (Scalar `  S )
)
592, 4, 54, 56, 57, 58lmodvsdir 17407 . . . . 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.  ) ) )
6039, 45, 49, 53, 59syl13anc 1230 . . . 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.  ) ) )
6133, 60eqtrd 2508 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
( y ( +g  `  R ) z )  .*  .1.  )  =  ( ( y  .*  .1.  ) ( +g  `  S ) ( z  .*  .1.  ) ) )
62 simpr 461 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  e.  Ring )
6362adantr 465 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  R  e.  Ring )
6462anim1i 568 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( R  e.  Ring  /\  (
y  e.  K  /\  z  e.  K )
) )
65 3anass 977 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  K  /\  z  e.  K )  <->  ( R  e.  Ring  /\  ( y  e.  K  /\  z  e.  K ) ) )
6664, 65sylibr 212 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( R  e.  Ring  /\  y  e.  K  /\  z  e.  K ) )
671, 3ringacl 17098 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  K  /\  z  e.  K )  ->  (
y ( +g  `  R
) z )  e.  K )
6866, 67syl 16 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  (
y ( +g  `  R
) z )  e.  K )
691, 7, 15, 16, 17scmatrhmval 18898 . . . 4  |-  ( ( R  e.  Ring  /\  (
y ( +g  `  R
) z )  e.  K )  ->  ( F `  ( y
( +g  `  R ) z ) )  =  ( ( y ( +g  `  R ) z )  .*  .1.  ) )
7063, 68, 69syl2anc 661 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  ( y
( +g  `  R ) z ) )  =  ( ( y ( +g  `  R ) z )  .*  .1.  ) )
711, 7, 15, 16, 17scmatrhmval 18898 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  K )  ->  ( F `  y )  =  ( y  .*  .1.  ) )
7271ad2ant2lr 747 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  y )  =  ( y  .*  .1.  ) )
731, 7, 15, 16, 17scmatrhmval 18898 . . . . 5  |-  ( ( R  e.  Ring  /\  z  e.  K )  ->  ( F `  z )  =  ( z  .*  .1.  ) )
7473ad2ant2l 745 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  z )  =  ( z  .*  .1.  ) )
7572, 74oveq12d 6313 . . 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.  ) ) )
7661, 70, 753eqtr4d 2518 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( y  e.  K  /\  z  e.  K
) )  ->  ( F `  ( y
( +g  `  R ) z ) )  =  ( ( F `  y ) ( +g  `  S ) ( F `
 z ) ) )
771, 2, 3, 4, 6, 14, 22, 76isghmd 16148 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F  e.  ( R  GrpHom  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    |-> cmpt 4511   -->wf 5590   ` cfv 5594  (class class class)co 6295   Fincfn 7528   Basecbs 14507   ↾s cress 14508   +g cplusg 14572  Scalarcsca 14575   .scvsca 14576   0gc0g 14712   Grpcgrp 15925  SubGrpcsubg 16067    GrpHom cghm 16136   1rcur 17025   Ringcrg 17070   LModclmod 17383   LSubSpclss 17449   Mat cmat 18778   ScMat cscmat 18860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-gsum 14715  df-prds 14720  df-pws 14722  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-subrg 17298  df-lmod 17385  df-lss 17450  df-sra 17689  df-rgmod 17690  df-dsmm 18632  df-frlm 18647  df-mamu 18755  df-mat 18779  df-dmat 18861  df-scmat 18862
This theorem is referenced by:  scmatrhm  18906
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