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Theorem matinvgcell 19104
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 19081 . . . . . . . . 9  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 457 . . . . . . . 8  |-  ( X  e.  B  ->  N  e.  Fin )
54adantl 464 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  N  e.  Fin )
6 simpl 455 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Ring )
71matgrp 19099 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Grp )
85, 6, 7syl2anc 659 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  A  e.  Grp )
9 eqid 2454 . . . . . . 7  |-  ( 0g
`  A )  =  ( 0g `  A
)
102, 9grpidcl 16277 . . . . . 6  |-  ( A  e.  Grp  ->  ( 0g `  A )  e.  B )
118, 10syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( 0g `  A )  e.  B )
12 simpr 459 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  B )
1311, 12jca 530 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 0g `  A
)  e.  B  /\  X  e.  B )
)
14133adant3 1014 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( ( 0g `  A )  e.  B  /\  X  e.  B ) )
15 eqid 2454 . . . 4  |-  ( -g `  A )  =  (
-g `  A )
16 eqid 2454 . . . 4  |-  ( -g `  R )  =  (
-g `  R )
171, 2, 15, 16matsubgcell 19103 . . 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 1273 . 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 16320 . . . . 5  |-  ( ( A  e.  Grp  /\  X  e.  B )  ->  ( W `  X
)  =  ( ( 0g `  A ) ( -g `  A
) X ) )
218, 12, 20syl2anc 659 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( W `  X )  =  ( ( 0g
`  A ) (
-g `  A ) X ) )
22213adant3 1014 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( W `  X )  =  ( ( 0g `  A
) ( -g `  A
) X ) )
2322oveqd 6287 . 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 17398 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
25243ad2ant1 1015 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  R  e.  Grp )
26 simp3 996 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( I  e.  N  /\  J  e.  N ) )
272eleq2i 2532 . . . . . . . 8  |-  ( X  e.  B  <->  X  e.  ( Base `  A )
)
2827biimpi 194 . . . . . . 7  |-  ( X  e.  B  ->  X  e.  ( Base `  A
) )
29283ad2ant2 1016 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  X  e.  ( Base `  A )
)
30 df-3an 973 . . . . . 6  |-  ( ( I  e.  N  /\  J  e.  N  /\  X  e.  ( Base `  A ) )  <->  ( (
I  e.  N  /\  J  e.  N )  /\  X  e.  ( Base `  A ) ) )
3126, 29, 30sylanbrc 662 . . . . 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 2454 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
331, 32matecl 19094 . . . . 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 2454 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
3732, 16, 35, 36grpinvval2 16320 . . . 4  |-  ( ( R  e.  Grp  /\  ( I X J )  e.  ( Base `  R ) )  -> 
( V `  (
I X J ) )  =  ( ( 0g `  R ) ( -g `  R
) ( I X J ) ) )
3825, 34, 37syl2anc 659 . . 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 566 . . . . . . . . 9  |-  ( ( X  e.  B  /\  R  e.  Ring )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
4039ancoms 451 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
411, 36mat0op 19088 . . . . . . . 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 1014 . . . . . 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 2455 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  ( I  e.  N  /\  J  e.  N
) )  /\  (
x  =  I  /\  y  =  J )
)  ->  ( 0g `  R )  =  ( 0g `  R ) )
4526simpld 457 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  I  e.  N )
46 simp3r 1023 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  J  e.  N )
47 fvex 5858 . . . . . . 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 6403 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( I
( 0g `  A
) J )  =  ( 0g `  R
) )
5049eqcomd 2462 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  (
I  e.  N  /\  J  e.  N )
)  ->  ( 0g `  R )  =  ( I ( 0g `  A ) J ) )
5150oveq1d 6285 . . 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 2495 . 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 2505 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Fincfn 7509   Basecbs 14716   0gc0g 14929   Grpcgrp 16252   invgcminusg 16253   -gcsg 16254   Ringcrg 17393   Mat cmat 19076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-hom 14808  df-cco 14809  df-0g 14931  df-prds 14937  df-pws 14939  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-mgp 17337  df-ur 17349  df-ring 17395  df-subrg 17622  df-lmod 17709  df-lss 17774  df-sra 18013  df-rgmod 18014  df-dsmm 18936  df-frlm 18951  df-mat 19077
This theorem is referenced by:  cpmatinvcl  19385
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