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Theorem invrvald 18487
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 16745 . . . . 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 4321 . . 3  |-  ( ph  ->  X ( ||r `
 R )  .1.  )
10 eqid 2443 . . . . . . 7  |-  (oppr `  R
)  =  (oppr `  R
)
1110, 3opprbas 16726 . . . . . 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 16745 . . . . 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 16723 . . . . 5  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
17 invrvald.xy . . . . 5  |-  ( ph  ->  ( X  .x.  Y
)  =  .1.  )
1816, 17syl5eq 2487 . . . 4  |-  ( ph  ->  ( Y ( .r
`  (oppr
`  R ) ) X )  =  .1.  )
1915, 18breqtrd 4321 . . 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 16754 . . 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 16766 . . . . 5  |-  ( R  e.  Ring  ->  .1.  =  ( 0g `  ( (mulGrp `  R )s  U ) ) )
2724, 26syl 16 . . . 4  |-  ( ph  ->  .1.  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2817, 27eqtrd 2475 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2920, 25unitgrp 16764 . . . . 5  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3024, 29syl 16 . . . 4  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
313, 4, 5dvdsrmul 16745 . . . . . . 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 4321 . . . . 5  |-  ( ph  ->  Y ( ||r `
 R )  .1.  )
3411, 12, 13dvdsrmul 16745 . . . . . . 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 16723 . . . . . . 7  |-  ( X ( .r `  (oppr `  R
) ) Y )  =  ( Y  .x.  X )
3736, 8syl5eq 2487 . . . . . 6  |-  ( ph  ->  ( X ( .r
`  (oppr
`  R ) ) Y )  =  .1.  )
3835, 37breqtrd 4321 . . . . 5  |-  ( ph  ->  Y ( ||r `
 (oppr
`  R ) )  .1.  )
3920, 21, 4, 10, 12isunit 16754 . . . . 5  |-  ( Y  e.  U  <->  ( Y
( ||r `
 R )  .1. 
/\  Y ( ||r `  (oppr `  R
) )  .1.  )
)
4033, 38, 39sylanbrc 664 . . . 4  |-  ( ph  ->  Y  e.  U )
4120, 25unitgrpbas 16763 . . . . 5  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
42 fvex 5706 . . . . . . 7  |-  (Unit `  R )  e.  _V
4320, 42eqeltri 2513 . . . . . 6  |-  U  e. 
_V
44 eqid 2443 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
4544, 5mgpplusg 16600 . . . . . . 7  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
4625, 45ressplusg 14285 . . . . . 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 16770 . . . . 5  |-  I  =  ( invg `  ( (mulGrp `  R )s  U
) )
5141, 47, 48, 50grpinvid1 15591 . . . 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 1218 . . 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 1369    e. wcel 1756   _Vcvv 2977   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   ↾s cress 14180   +g cplusg 14243   .rcmulr 14244   0gc0g 14383   Grpcgrp 15415  mulGrpcmgp 16596   1rcur 16608   Ringcrg 16650  opprcoppr 16719   ||rcdsr 16735  Unitcui 16736   invrcinvr 16768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-tpos 6750  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769
This theorem is referenced by:  matinv  18488  matunit  18489
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