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Theorem invrvald 19470
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 2402 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
5 invrvald.t . . . . . 6  |-  .x.  =  ( .r `  R )
63, 4, 5dvdsrmul 17617 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X ( ||r `
 R ) ( Y  .x.  X ) )
71, 2, 6syl2anc 659 . . . 4  |-  ( ph  ->  X ( ||r `
 R ) ( Y  .x.  X ) )
8 invrvald.yx . . . 4  |-  ( ph  ->  ( Y  .x.  X
)  =  .1.  )
97, 8breqtrd 4419 . . 3  |-  ( ph  ->  X ( ||r `
 R )  .1.  )
10 eqid 2402 . . . . . . 7  |-  (oppr `  R
)  =  (oppr `  R
)
1110, 3opprbas 17598 . . . . . 6  |-  B  =  ( Base `  (oppr `  R
) )
12 eqid 2402 . . . . . 6  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
13 eqid 2402 . . . . . 6  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
1411, 12, 13dvdsrmul 17617 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X ( ||r `
 (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) )
151, 2, 14syl2anc 659 . . . 4  |-  ( ph  ->  X ( ||r `
 (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) )
163, 5, 10, 13opprmul 17595 . . . . 5  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
17 invrvald.xy . . . . 5  |-  ( ph  ->  ( X  .x.  Y
)  =  .1.  )
1816, 17syl5eq 2455 . . . 4  |-  ( ph  ->  ( Y ( .r
`  (oppr
`  R ) ) X )  =  .1.  )
1915, 18breqtrd 4419 . . 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 17626 . . 3  |-  ( X  e.  U  <->  ( X
( ||r `
 R )  .1. 
/\  X ( ||r `  (oppr `  R
) )  .1.  )
)
239, 19, 22sylanbrc 662 . 2  |-  ( ph  ->  X  e.  U )
24 invrvald.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
25 eqid 2402 . . . . . 6  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
2620, 25, 21unitgrpid 17638 . . . . 5  |-  ( R  e.  Ring  ->  .1.  =  ( 0g `  ( (mulGrp `  R )s  U ) ) )
2724, 26syl 17 . . . 4  |-  ( ph  ->  .1.  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2817, 27eqtrd 2443 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2920, 25unitgrp 17636 . . . . 5  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3024, 29syl 17 . . . 4  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
313, 4, 5dvdsrmul 17617 . . . . . . 7  |-  ( ( Y  e.  B  /\  X  e.  B )  ->  Y ( ||r `
 R ) ( X  .x.  Y ) )
322, 1, 31syl2anc 659 . . . . . 6  |-  ( ph  ->  Y ( ||r `
 R ) ( X  .x.  Y ) )
3332, 17breqtrd 4419 . . . . 5  |-  ( ph  ->  Y ( ||r `
 R )  .1.  )
3411, 12, 13dvdsrmul 17617 . . . . . . 7  |-  ( ( Y  e.  B  /\  X  e.  B )  ->  Y ( ||r `
 (oppr
`  R ) ) ( X ( .r
`  (oppr
`  R ) ) Y ) )
352, 1, 34syl2anc 659 . . . . . 6  |-  ( ph  ->  Y ( ||r `
 (oppr
`  R ) ) ( X ( .r
`  (oppr
`  R ) ) Y ) )
363, 5, 10, 13opprmul 17595 . . . . . . 7  |-  ( X ( .r `  (oppr `  R
) ) Y )  =  ( Y  .x.  X )
3736, 8syl5eq 2455 . . . . . 6  |-  ( ph  ->  ( X ( .r
`  (oppr
`  R ) ) Y )  =  .1.  )
3835, 37breqtrd 4419 . . . . 5  |-  ( ph  ->  Y ( ||r `
 (oppr
`  R ) )  .1.  )
3920, 21, 4, 10, 12isunit 17626 . . . . 5  |-  ( Y  e.  U  <->  ( Y
( ||r `
 R )  .1. 
/\  Y ( ||r `  (oppr `  R
) )  .1.  )
)
4033, 38, 39sylanbrc 662 . . . 4  |-  ( ph  ->  Y  e.  U )
4120, 25unitgrpbas 17635 . . . . 5  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
42 fvex 5859 . . . . . . 7  |-  (Unit `  R )  e.  _V
4320, 42eqeltri 2486 . . . . . 6  |-  U  e. 
_V
44 eqid 2402 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
4544, 5mgpplusg 17465 . . . . . . 7  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
4625, 45ressplusg 14955 . . . . . 6  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
4743, 46ax-mp 5 . . . . 5  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
48 eqid 2402 . . . . 5  |-  ( 0g
`  ( (mulGrp `  R )s  U ) )  =  ( 0g `  (
(mulGrp `  R )s  U
) )
49 invrvald.i . . . . . 6  |-  I  =  ( invr `  R
)
5020, 25, 49invrfval 17642 . . . . 5  |-  I  =  ( invg `  ( (mulGrp `  R )s  U
) )
5141, 47, 48, 50grpinvid1 16422 . . . 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 1230 . . 3  |-  ( ph  ->  ( ( I `  X )  =  Y  <-> 
( X  .x.  Y
)  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) ) )
5328, 52mpbird 232 . 2  |-  ( ph  ->  ( I `  X
)  =  Y )
5423, 53jca 530 1  |-  ( ph  ->  ( X  e.  U  /\  ( I `  X
)  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   ↾s cress 14842   +g cplusg 14909   .rcmulr 14910   0gc0g 15054   Grpcgrp 16377  mulGrpcmgp 17461   1rcur 17473   Ringcrg 17518  opprcoppr 17591   ||rcdsr 17607  Unitcui 17608   invrcinvr 17640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641
This theorem is referenced by:  matinv  19471  matunit  19472
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