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Theorem matinvgcell 18810
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.
StepHypRef Expression
1 matplusgcell.a . . . . . . . . . 10  |-  A  =  ( N Mat  R )
2 matplusgcell.b . . . . . . . . . 10  |-  B  =  ( Base `  A
)
31, 2matrcl 18787 . . . . . . . . 9  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 459 . . . . . . . 8  |-  ( X  e.  B  ->  N  e.  Fin )
54adantl 466 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  N  e.  Fin )
6 simpl 457 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Ring )
71matgrp 18805 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Grp )
85, 6, 7syl2anc 661 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  A  e.  Grp )
9 eqid 2443 . . . . . . 7  |-  ( 0g
`  A )  =  ( 0g `  A
)
102, 9grpidcl 15952 . . . . . 6  |-  ( A  e.  Grp  ->  ( 0g `  A )  e.  B )
118, 10syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( 0g `  A )  e.  B )
12 simpr 461 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  B )
1311, 12jca 532 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 0g `  A
)  e.  B  /\  X  e.  B )
)
14133adant3 1017 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( ( 0g `  A )  e.  B  /\  X  e.  B ) )
15 eqid 2443 . . . 4  |-  ( -g `  A )  =  (
-g `  A )
16 eqid 2443 . . . 4  |-  ( -g `  R )  =  (
-g `  R )
171, 2, 15, 16matsubgcell 18809 . . 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 ) ) )
1814, 17syld3an2 1276 . 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 )
202, 15, 19, 9grpinvval2 15995 . . . . 5  |-  ( ( A  e.  Grp  /\  X  e.  B )  ->  ( W `  X
)  =  ( ( 0g `  A ) ( -g `  A
) X ) )
218, 12, 20syl2anc 661 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( W `  X )  =  ( ( 0g
`  A ) (
-g `  A ) X ) )
22213adant3 1017 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( W `  X )  =  ( ( 0g `  A
) ( -g `  A
) X ) )
2322oveqd 6298 . 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 17077 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
25243ad2ant1 1018 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  R  e.  Grp )
26 simp3 999 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( I  e.  N  /\  J  e.  N ) )
272eleq2i 2521 . . . . . . . 8  |-  ( X  e.  B  <->  X  e.  ( Base `  A )
)
2827biimpi 194 . . . . . . 7  |-  ( X  e.  B  ->  X  e.  ( Base `  A
) )
29283ad2ant2 1019 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  X  e.  ( Base `  A )
)
30 df-3an 976 . . . . . 6  |-  ( ( I  e.  N  /\  J  e.  N  /\  X  e.  ( Base `  A ) )  <->  ( (
I  e.  N  /\  J  e.  N )  /\  X  e.  ( Base `  A ) ) )
3126, 29, 30sylanbrc 664 . . . . 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 2443 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
331, 32matecl 18800 . . . . 5  |-  ( ( I  e.  N  /\  J  e.  N  /\  X  e.  ( Base `  A ) )  -> 
( I X J )  e.  ( Base `  R ) )
3431, 33syl 16 . . . 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 2443 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
3732, 16, 35, 36grpinvval2 15995 . . . 4  |-  ( ( R  e.  Grp  /\  ( I X J )  e.  ( Base `  R ) )  -> 
( V `  (
I X J ) )  =  ( ( 0g `  R ) ( -g `  R
) ( I X J ) ) )
3825, 34, 37syl2anc 661 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( V `  ( I X J ) )  =  ( ( 0g `  R
) ( -g `  R
) ( I X J ) ) )
394anim1i 568 . . . . . . . . 9  |-  ( ( X  e.  B  /\  R  e.  Ring )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
4039ancoms 453 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
411, 36mat0op 18794 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  A
)  =  ( x  e.  N ,  y  e.  N  |->  ( 0g
`  R ) ) )
4240, 41syl 16 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( 0g `  A )  =  ( x  e.  N ,  y  e.  N  |->  ( 0g `  R
) ) )
43423adant3 1017 . . . . . 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 2444 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  ( I  e.  N  /\  J  e.  N
) )  /\  (
x  =  I  /\  y  =  J )
)  ->  ( 0g `  R )  =  ( 0g `  R ) )
4526simpld 459 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  I  e.  N )
46 simp3r 1026 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  J  e.  N )
47 fvex 5866 . . . . . . 7  |-  ( 0g
`  R )  e. 
_V
4847a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( 0g `  R )  e.  _V )
4943, 44, 45, 46, 48ovmpt2d 6415 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( I
( 0g `  A
) J )  =  ( 0g `  R
) )
5049eqcomd 2451 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( 0g `  R )  =  ( I ( 0g `  A ) J ) )
5150oveq1d 6296 . . 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 ) ) )
5238, 51eqtrd 2484 . 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 ) ) )
5318, 23, 523eqtr4d 2494 1  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( I
( W `  X
) J )  =  ( V `  (
I X J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Fincfn 7518   Basecbs 14509   0gc0g 14714   Grpcgrp 15927   invgcminusg 15928   -gcsg 15929   Ringcrg 17072   Mat cmat 18782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-hom 14598  df-cco 14599  df-0g 14716  df-prds 14722  df-pws 14724  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-sbg 15933  df-subg 16072  df-mgp 17016  df-ur 17028  df-ring 17074  df-subrg 17301  df-lmod 17388  df-lss 17453  df-sra 17692  df-rgmod 17693  df-dsmm 18636  df-frlm 18651  df-mat 18783
This theorem is referenced by:  cpmatinvcl  19091
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