Theorem matinvgcell 20241

Description: Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)

Hypotheses

Ref	Expression
matplusgcell.a	|- A = ( N Mat R )
matplusgcell.b	|- B = ( Base ` A )
matinvgcell.v	|- V = ( invg ` R )
matinvgcell.w	|- W = ( invg ` A )

Assertion

Ref	Expression
matinvgcell	|- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I ( W ` X ) J ) = ( V ` ( I X J ) ) )

Proof of Theorem matinvgcell

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		matplusgcell.a	. . . . . . . . . 10 |- A = ( N Mat R )
2		matplusgcell.b	. . . . . . . . . 10 |- B = ( Base ` A )
3	1, 2	matrcl 20218	. . . . . . . . 9 |- ( X e. B -> ( N e. Fin /\ R e. _V ) )
4	3	simpld 475	. . . . . . . 8 |- ( X e. B -> N e. Fin )
5	4	adantl 482	. . . . . . 7 |- ( ( R e. Ring /\ X e. B ) -> N e. Fin )
6		simpl 473	. . . . . . 7 |- ( ( R e. Ring /\ X e. B ) -> R e. Ring )
7	1	matgrp 20236	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> A e. Grp )
8	5, 6, 7	syl2anc 693	. . . . . 6 |- ( ( R e. Ring /\ X e. B ) -> A e. Grp )
9		eqid 2622	. . . . . . 7 |- ( 0g ` A ) = ( 0g ` A )
10	2, 9	grpidcl 17450	. . . . . 6 |- ( A e. Grp -> ( 0g ` A ) e. B )
11	8, 10	syl 17	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> ( 0g ` A ) e. B )
12		simpr 477	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> X e. B )
13	11, 12	jca 554	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( ( 0g ` A ) e. B /\ X e. B ) )
14	13	3adant3 1081	. . 3 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( ( 0g ` A ) e. B /\ X e. B ) )
15		eqid 2622	. . . 4 |- ( -g ` A ) = ( -g ` A )
16		eqid 2622	. . . 4 |- ( -g ` R ) = ( -g ` R )
17	1, 2, 15, 16	matsubgcell 20240	. . 3 |- ( ( R e. Ring /\ ( ( 0g ` A ) e. B /\ X e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( ( 0g ` A ) ( -g ` A ) X ) J ) = ( ( I ( 0g ` A ) J ) ( -g ` R ) ( I X J ) ) )
18	14, 17	syld3an2 1373	. 2 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I ( ( 0g ` A ) ( -g ` A ) X ) J ) = ( ( I ( 0g ` A ) J ) ( -g ` R ) ( I X J ) ) )
19		matinvgcell.w	. . . . . 6 |- W = ( invg ` A )
20	2, 15, 19, 9	grpinvval2 17498	. . . . 5 |- ( ( A e. Grp /\ X e. B ) -> ( W ` X ) = ( ( 0g ` A ) ( -g ` A ) X ) )
21	8, 12, 20	syl2anc 693	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( W ` X ) = ( ( 0g ` A ) ( -g ` A ) X ) )
22	21	3adant3 1081	. . 3 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( W ` X ) = ( ( 0g ` A ) ( -g ` A ) X ) )
23	22	oveqd 6667	. 2 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I ( W ` X ) J ) = ( I ( ( 0g ` A ) ( -g ` A ) X ) J ) )
24		ringgrp 18552	. . . . 5 |- ( R e. Ring -> R e. Grp )
25	24	3ad2ant1 1082	. . . 4 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> R e. Grp )
26		simp3 1063	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I e. N /\ J e. N ) )
27	2	eleq2i 2693	. . . . . . . 8 |- ( X e. B <-> X e. ( Base ` A ) )
28	27	biimpi 206	. . . . . . 7 |- ( X e. B -> X e. ( Base ` A ) )
29	28	3ad2ant2 1083	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> X e. ( Base ` A ) )
30		df-3an 1039	. . . . . 6 |- ( ( I e. N /\ J e. N /\ X e. ( Base ` A ) ) <-> ( ( I e. N /\ J e. N ) /\ X e. ( Base ` A ) ) )
31	26, 29, 30	sylanbrc 698	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I e. N /\ J e. N /\ X e. ( Base ` A ) ) )
32		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
33	1, 32	matecl 20231	. . . . 5 |- ( ( I e. N /\ J e. N /\ X e. ( Base ` A ) ) -> ( I X J ) e. ( Base ` R ) )
34	31, 33	syl 17	. . . 4 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I X J ) e. ( Base ` R ) )
35		matinvgcell.v	. . . . 5 |- V = ( invg ` R )
36		eqid 2622	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
37	32, 16, 35, 36	grpinvval2 17498	. . . 4 |- ( ( R e. Grp /\ ( I X J ) e. ( Base ` R ) ) -> ( V ` ( I X J ) ) = ( ( 0g ` R ) ( -g ` R ) ( I X J ) ) )
38	25, 34, 37	syl2anc 693	. . 3 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( V ` ( I X J ) ) = ( ( 0g ` R ) ( -g ` R ) ( I X J ) ) )
39	4	anim1i 592	. . . . . . . . 9 |- ( ( X e. B /\ R e. Ring ) -> ( N e. Fin /\ R e. Ring ) )
40	39	ancoms 469	. . . . . . . 8 |- ( ( R e. Ring /\ X e. B ) -> ( N e. Fin /\ R e. Ring ) )
41	1, 36	mat0op 20225	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` A ) = ( x e. N , y e. N |-> ( 0g ` R ) ) )
42	40, 41	syl 17	. . . . . . 7 |- ( ( R e. Ring /\ X e. B ) -> ( 0g ` A ) = ( x e. N , y e. N |-> ( 0g ` R ) ) )
43	42	3adant3 1081	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( 0g ` A ) = ( x e. N , y e. N |-> ( 0g ` R ) ) )
44		eqidd 2623	. . . . . 6 |- ( ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) /\ ( x = I /\ y = J ) ) -> ( 0g ` R ) = ( 0g ` R ) )
45	26	simpld 475	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> I e. N )
46		simp3r 1090	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> J e. N )
47		fvexd 6203	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( 0g ` R ) e. _V )
48	43, 44, 45, 46, 47	ovmpt2d 6788	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I ( 0g ` A ) J ) = ( 0g ` R ) )
49	48	eqcomd 2628	. . . 4 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( 0g ` R ) = ( I ( 0g ` A ) J ) )
50	49	oveq1d 6665	. . 3 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( ( 0g ` R ) ( -g ` R ) ( I X J ) ) = ( ( I ( 0g ` A ) J ) ( -g ` R ) ( I X J ) ) )
51	38, 50	eqtrd 2656	. 2 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( V ` ( I X J ) ) = ( ( I ( 0g ` A ) J ) ( -g ` R ) ( I X J ) ) )
52	18, 23, 51	3eqtr4d 2666	1 |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I ( W ` X ) J ) = ( V ` ( I X J ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   Ringcrg 18547   Mat cmat 20213

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-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-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-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-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-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-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-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  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-mat 20214

This theorem is referenced by:  cpmatinvcl  20522