Theorem scmatghm 20339

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 = ( A ↾s 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.

Step	Hyp	Ref	Expression
1		scmatrhmval.k	. 2 |- K = ( Base ` R )
2		eqid 2622	. 2 |- ( Base ` S ) = ( Base ` S )
3		eqid 2622	. 2 |- ( +g ` R ) = ( +g ` R )
4		eqid 2622	. 2 |- ( +g ` S ) = ( +g ` S )
5		ringgrp 18552	. . 3 |- ( R e. Ring -> R e. Grp )
6	5	adantl 482	. 2 |- ( ( N e. Fin /\ R e. Ring ) -> R e. Grp )
7		scmatrhmval.a	. . . 4 |- A = ( N Mat R )
8		eqid 2622	. . . 4 |- ( Base ` A ) = ( Base ` A )
9		eqid 2622	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
10		scmatrhmval.c	. . . 4 |- C = ( N ScMat R )
11	7, 8, 1, 9, 10	scmatsgrp 20325	. . 3 |- ( ( N e. Fin /\ R e. Ring ) -> C e. (SubGrp ` A ) )
12		scmatghm.s	. . . 4 |- S = ( A ↾s C )
13	12	subggrp 17597	. . 3 |- ( C e. (SubGrp ` A ) -> S e. Grp )
14	11, 13	syl 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. ) )
18	1, 7, 15, 16, 17, 10	scmatf 20335	. . 3 |- ( ( N e. Fin /\ R e. Ring ) -> F : K --> C )
19	7, 10, 12	scmatstrbas 20332	. . . 4 |- ( ( N e. Fin /\ R e. Ring ) -> ( Base ` S ) = C )
20	19	feq3d 6032	. . 3 |- ( ( N e. Fin /\ R e. Ring ) -> ( F : K --> ( Base ` S ) <-> F : K --> C ) )
21	18, 20	mpbird 247	. 2 |- ( ( N e. Fin /\ R e. Ring ) -> F : K --> ( Base ` S ) )
22	7	matsca2 20226	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> R = (Scalar ` A ) )
23		ovex 6678	. . . . . . . . . . 11 |- ( N ScMat R ) e. _V
24	10, 23	eqeltri 2697	. . . . . . . . . 10 |- C e. _V
25		eqid 2622	. . . . . . . . . . 11 |- (Scalar ` A ) = (Scalar ` A )
26	12, 25	resssca 16031	. . . . . . . . . 10 |- ( C e. _V -> (Scalar ` A ) = (Scalar ` S ) )
27	24, 26	mp1i 13	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> (Scalar ` A ) = (Scalar ` S ) )
28	22, 27	eqtrd 2656	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> R = (Scalar ` S ) )
29	28	fveq2d 6195	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( +g ` R ) = ( +g ` (Scalar ` S ) ) )
30	29	oveqd 6667	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> ( y ( +g ` R ) z ) = ( y ( +g ` (Scalar ` S ) ) z ) )
31	30	oveq1d 6665	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> ( ( y ( +g ` R ) z ) .* .1. ) = ( ( y ( +g ` (Scalar ` S ) ) z ) .* .1. ) )
32	31	adantr 481	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( ( y ( +g ` R ) z ) .* .1. ) = ( ( y ( +g ` (Scalar ` S ) ) z ) .* .1. ) )
33	7	matlmod 20235	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> A e. LMod )
34	7, 10	scmatlss 20331	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> C e. ( LSubSp ` A ) )
35		eqid 2622	. . . . . . . 8 |- ( LSubSp ` A ) = ( LSubSp ` A )
36	12, 35	lsslmod 18960	. . . . . . 7 |- ( ( A e. LMod /\ C e. ( LSubSp ` A ) ) -> S e. LMod )
37	33, 34, 36	syl2anc 693	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> S e. LMod )
38	37	adantr 481	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> S e. LMod )
39	28	fveq2d 6195	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> ( Base ` R ) = ( Base ` (Scalar ` S ) ) )
40	1, 39	syl5eq 2668	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> K = ( Base ` (Scalar ` S ) ) )
41	40	eleq2d 2687	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> ( y e. K <-> y e. ( Base ` (Scalar ` S ) ) ) )
42	41	biimpd 219	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( y e. K -> y e. ( Base ` (Scalar ` S ) ) ) )
43	42	adantrd 484	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> ( ( y e. K /\ z e. K ) -> y e. ( Base ` (Scalar ` S ) ) ) )
44	43	imp 445	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> y e. ( Base ` (Scalar ` S ) ) )
45	40	eleq2d 2687	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> ( z e. K <-> z e. ( Base ` (Scalar ` S ) ) ) )
46	45	biimpd 219	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( z e. K -> z e. ( Base ` (Scalar ` S ) ) ) )
47	46	adantld 483	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> ( ( y e. K /\ z e. K ) -> z e. ( Base ` (Scalar ` S ) ) ) )
48	47	imp 445	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> z e. ( Base ` (Scalar ` S ) ) )
49	7, 8, 1, 9, 10	scmatid 20320	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` A ) e. C )
50	15	a1i 11	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> .1. = ( 1r ` A ) )
51	49, 50, 19	3eltr4d 2716	. . . . . 6 |- ( ( N e. Fin /\ R e. Ring ) -> .1. e. ( Base ` S ) )
52	51	adantr 481	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> .1. e. ( Base ` S ) )
53		eqid 2622	. . . . . 6 |- (Scalar ` S ) = (Scalar ` S )
54	12, 16	ressvsca 16032	. . . . . . 7 |- ( C e. _V -> .* = ( .s ` S ) )
55	24, 54	ax-mp 5	. . . . . 6 |- .* = ( .s ` S )
56		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` S ) )
57		eqid 2622	. . . . . 6 |- ( +g ` (Scalar ` S ) ) = ( +g ` (Scalar ` S ) )
58	2, 4, 53, 55, 56, 57	lmodvsdir 18887	. . . . 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. ) ) )
59	38, 44, 48, 52, 58	syl13anc 1328	. . . 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. ) ) )
60	32, 59	eqtrd 2656	. . 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 477	. . . . 5 |- ( ( N e. Fin /\ R e. Ring ) -> R e. Ring )
62	61	adantr 481	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> R e. Ring )
63	61	anim1i 592	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( R e. Ring /\ ( y e. K /\ z e. K ) ) )
64		3anass 1042	. . . . . 6 |- ( ( R e. Ring /\ y e. K /\ z e. K ) <-> ( R e. Ring /\ ( y e. K /\ z e. K ) ) )
65	63, 64	sylibr 224	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( R e. Ring /\ y e. K /\ z e. K ) )
66	1, 3	ringacl 18578	. . . . 5 |- ( ( R e. Ring /\ y e. K /\ z e. K ) -> ( y ( +g ` R ) z ) e. K )
67	65, 66	syl 17	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( y ( +g ` R ) z ) e. K )
68	1, 7, 15, 16, 17	scmatrhmval 20333	. . . 4 |- ( ( R e. Ring /\ ( y ( +g ` R ) z ) e. K ) -> ( F ` ( y ( +g ` R ) z ) ) = ( ( y ( +g ` R ) z ) .* .1. ) )
69	62, 67, 68	syl2anc 693	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( F ` ( y ( +g ` R ) z ) ) = ( ( y ( +g ` R ) z ) .* .1. ) )
70	1, 7, 15, 16, 17	scmatrhmval 20333	. . . . 5 |- ( ( R e. Ring /\ y e. K ) -> ( F ` y ) = ( y .* .1. ) )
71	70	ad2ant2lr 784	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( F ` y ) = ( y .* .1. ) )
72	1, 7, 15, 16, 17	scmatrhmval 20333	. . . . 5 |- ( ( R e. Ring /\ z e. K ) -> ( F ` z ) = ( z .* .1. ) )
73	72	ad2ant2l 782	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( F ` z ) = ( z .* .1. ) )
74	71, 73	oveq12d 6668	. . 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. ) ) )
75	60, 69, 74	3eqtr4d 2666	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( y e. K /\ z e. K ) ) -> ( F ` ( y ( +g ` R ) z ) ) = ( ( F ` y ) ( +g ` S ) ( F ` z ) ) )
76	1, 2, 3, 4, 6, 14, 21, 75	isghmd 17669	1 |- ( ( N e. Fin /\ R e. Ring ) -> F e. ( R GrpHom S ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588   GrpHom cghm 17657   1rcur 18501   Ringcrg 18547   LModclmod 18863   LSubSpclss 18932   Mat cmat 20213   ScMat cscmat 20295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-dmat 20296  df-scmat 20297

This theorem is referenced by:  scmatrhm  20341