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Theorem invrvald 19156
Description: If a matrix multiplied with a given matrix (from the left as well as from the right) results in the identity matrix, this matrix is the inverse (matrix) of the given matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
Hypotheses
Ref Expression
invrvald.b  |-  B  =  ( Base `  R
)
invrvald.t  |-  .x.  =  ( .r `  R )
invrvald.o  |-  .1.  =  ( 1r `  R )
invrvald.u  |-  U  =  (Unit `  R )
invrvald.i  |-  I  =  ( invr `  R
)
invrvald.r  |-  ( ph  ->  R  e.  Ring )
invrvald.x  |-  ( ph  ->  X  e.  B )
invrvald.y  |-  ( ph  ->  Y  e.  B )
invrvald.xy  |-  ( ph  ->  ( X  .x.  Y
)  =  .1.  )
invrvald.yx  |-  ( ph  ->  ( Y  .x.  X
)  =  .1.  )
Assertion
Ref Expression
invrvald  |-  ( ph  ->  ( X  e.  U  /\  ( I `  X
)  =  Y ) )

Proof of Theorem invrvald
StepHypRef Expression
1 invrvald.x . . . . 5  |-  ( ph  ->  X  e.  B )
2 invrvald.y . . . . 5  |-  ( ph  ->  Y  e.  B )
3 invrvald.b . . . . . 6  |-  B  =  ( Base `  R
)
4 eqid 2443 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
5 invrvald.t . . . . . 6  |-  .x.  =  ( .r `  R )
63, 4, 5dvdsrmul 17276 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X ( ||r `
 R ) ( Y  .x.  X ) )
71, 2, 6syl2anc 661 . . . 4  |-  ( ph  ->  X ( ||r `
 R ) ( Y  .x.  X ) )
8 invrvald.yx . . . 4  |-  ( ph  ->  ( Y  .x.  X
)  =  .1.  )
97, 8breqtrd 4461 . . 3  |-  ( ph  ->  X ( ||r `
 R )  .1.  )
10 eqid 2443 . . . . . . 7  |-  (oppr `  R
)  =  (oppr `  R
)
1110, 3opprbas 17257 . . . . . 6  |-  B  =  ( Base `  (oppr `  R
) )
12 eqid 2443 . . . . . 6  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
13 eqid 2443 . . . . . 6  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
1411, 12, 13dvdsrmul 17276 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X ( ||r `
 (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) )
151, 2, 14syl2anc 661 . . . 4  |-  ( ph  ->  X ( ||r `
 (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) )
163, 5, 10, 13opprmul 17254 . . . . 5  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
17 invrvald.xy . . . . 5  |-  ( ph  ->  ( X  .x.  Y
)  =  .1.  )
1816, 17syl5eq 2496 . . . 4  |-  ( ph  ->  ( Y ( .r
`  (oppr
`  R ) ) X )  =  .1.  )
1915, 18breqtrd 4461 . . 3  |-  ( ph  ->  X ( ||r `
 (oppr
`  R ) )  .1.  )
20 invrvald.u . . . 4  |-  U  =  (Unit `  R )
21 invrvald.o . . . 4  |-  .1.  =  ( 1r `  R )
2220, 21, 4, 10, 12isunit 17285 . . 3  |-  ( X  e.  U  <->  ( X
( ||r `
 R )  .1. 
/\  X ( ||r `  (oppr `  R
) )  .1.  )
)
239, 19, 22sylanbrc 664 . 2  |-  ( ph  ->  X  e.  U )
24 invrvald.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
25 eqid 2443 . . . . . 6  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
2620, 25, 21unitgrpid 17297 . . . . 5  |-  ( R  e.  Ring  ->  .1.  =  ( 0g `  ( (mulGrp `  R )s  U ) ) )
2724, 26syl 16 . . . 4  |-  ( ph  ->  .1.  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2817, 27eqtrd 2484 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2920, 25unitgrp 17295 . . . . 5  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3024, 29syl 16 . . . 4  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
313, 4, 5dvdsrmul 17276 . . . . . . 7  |-  ( ( Y  e.  B  /\  X  e.  B )  ->  Y ( ||r `
 R ) ( X  .x.  Y ) )
322, 1, 31syl2anc 661 . . . . . 6  |-  ( ph  ->  Y ( ||r `
 R ) ( X  .x.  Y ) )
3332, 17breqtrd 4461 . . . . 5  |-  ( ph  ->  Y ( ||r `
 R )  .1.  )
3411, 12, 13dvdsrmul 17276 . . . . . . 7  |-  ( ( Y  e.  B  /\  X  e.  B )  ->  Y ( ||r `
 (oppr
`  R ) ) ( X ( .r
`  (oppr
`  R ) ) Y ) )
352, 1, 34syl2anc 661 . . . . . 6  |-  ( ph  ->  Y ( ||r `
 (oppr
`  R ) ) ( X ( .r
`  (oppr
`  R ) ) Y ) )
363, 5, 10, 13opprmul 17254 . . . . . . 7  |-  ( X ( .r `  (oppr `  R
) ) Y )  =  ( Y  .x.  X )
3736, 8syl5eq 2496 . . . . . 6  |-  ( ph  ->  ( X ( .r
`  (oppr
`  R ) ) Y )  =  .1.  )
3835, 37breqtrd 4461 . . . . 5  |-  ( ph  ->  Y ( ||r `
 (oppr
`  R ) )  .1.  )
3920, 21, 4, 10, 12isunit 17285 . . . . 5  |-  ( Y  e.  U  <->  ( Y
( ||r `
 R )  .1. 
/\  Y ( ||r `  (oppr `  R
) )  .1.  )
)
4033, 38, 39sylanbrc 664 . . . 4  |-  ( ph  ->  Y  e.  U )
4120, 25unitgrpbas 17294 . . . . 5  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
42 fvex 5866 . . . . . . 7  |-  (Unit `  R )  e.  _V
4320, 42eqeltri 2527 . . . . . 6  |-  U  e. 
_V
44 eqid 2443 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
4544, 5mgpplusg 17124 . . . . . . 7  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
4625, 45ressplusg 14721 . . . . . 6  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
4743, 46ax-mp 5 . . . . 5  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
48 eqid 2443 . . . . 5  |-  ( 0g
`  ( (mulGrp `  R )s  U ) )  =  ( 0g `  (
(mulGrp `  R )s  U
) )
49 invrvald.i . . . . . 6  |-  I  =  ( invr `  R
)
5020, 25, 49invrfval 17301 . . . . 5  |-  I  =  ( invg `  ( (mulGrp `  R )s  U
) )
5141, 47, 48, 50grpinvid1 16077 . . . 4  |-  ( ( ( (mulGrp `  R
)s 
U )  e.  Grp  /\  X  e.  U  /\  Y  e.  U )  ->  ( ( I `  X )  =  Y  <-> 
( X  .x.  Y
)  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) ) )
5230, 23, 40, 51syl3anc 1229 . . 3  |-  ( ph  ->  ( ( I `  X )  =  Y  <-> 
( X  .x.  Y
)  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) ) )
5328, 52mpbird 232 . 2  |-  ( ph  ->  ( I `  X
)  =  Y )
5423, 53jca 532 1  |-  ( ph  ->  ( X  e.  U  /\  ( I `  X
)  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14614   ↾s cress 14615   +g cplusg 14679   .rcmulr 14680   0gc0g 14819   Grpcgrp 16032  mulGrpcmgp 17120   1rcur 17132   Ringcrg 17177  opprcoppr 17250   ||rcdsr 17266  Unitcui 17267   invrcinvr 17299
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-uni 4235  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-om 6686  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-mgp 17121  df-ur 17133  df-ring 17179  df-oppr 17251  df-dvdsr 17269  df-unit 17270  df-invr 17300
This theorem is referenced by:  matinv  19157  matunit  19158
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