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Theorem invrvald 18942
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 2467 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
5 invrvald.t . . . . . 6  |-  .x.  =  ( .r `  R )
63, 4, 5dvdsrmul 17078 . . . . 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 4471 . . 3  |-  ( ph  ->  X ( ||r `
 R )  .1.  )
10 eqid 2467 . . . . . . 7  |-  (oppr `  R
)  =  (oppr `  R
)
1110, 3opprbas 17059 . . . . . 6  |-  B  =  ( Base `  (oppr `  R
) )
12 eqid 2467 . . . . . 6  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
13 eqid 2467 . . . . . 6  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
1411, 12, 13dvdsrmul 17078 . . . . 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 17056 . . . . 5  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
17 invrvald.xy . . . . 5  |-  ( ph  ->  ( X  .x.  Y
)  =  .1.  )
1816, 17syl5eq 2520 . . . 4  |-  ( ph  ->  ( Y ( .r
`  (oppr
`  R ) ) X )  =  .1.  )
1915, 18breqtrd 4471 . . 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 17087 . . 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 2467 . . . . . 6  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
2620, 25, 21unitgrpid 17099 . . . . 5  |-  ( R  e.  Ring  ->  .1.  =  ( 0g `  ( (mulGrp `  R )s  U ) ) )
2724, 26syl 16 . . . 4  |-  ( ph  ->  .1.  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2817, 27eqtrd 2508 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  =  ( 0g
`  ( (mulGrp `  R )s  U ) ) )
2920, 25unitgrp 17097 . . . . 5  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3024, 29syl 16 . . . 4  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
313, 4, 5dvdsrmul 17078 . . . . . . 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 4471 . . . . 5  |-  ( ph  ->  Y ( ||r `
 R )  .1.  )
3411, 12, 13dvdsrmul 17078 . . . . . . 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 17056 . . . . . . 7  |-  ( X ( .r `  (oppr `  R
) ) Y )  =  ( Y  .x.  X )
3736, 8syl5eq 2520 . . . . . 6  |-  ( ph  ->  ( X ( .r
`  (oppr
`  R ) ) Y )  =  .1.  )
3835, 37breqtrd 4471 . . . . 5  |-  ( ph  ->  Y ( ||r `
 (oppr
`  R ) )  .1.  )
3920, 21, 4, 10, 12isunit 17087 . . . . 5  |-  ( Y  e.  U  <->  ( Y
( ||r `
 R )  .1. 
/\  Y ( ||r `  (oppr `  R
) )  .1.  )
)
4033, 38, 39sylanbrc 664 . . . 4  |-  ( ph  ->  Y  e.  U )
4120, 25unitgrpbas 17096 . . . . 5  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
42 fvex 5874 . . . . . . 7  |-  (Unit `  R )  e.  _V
4320, 42eqeltri 2551 . . . . . 6  |-  U  e. 
_V
44 eqid 2467 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
4544, 5mgpplusg 16932 . . . . . . 7  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
4625, 45ressplusg 14590 . . . . . 6  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
4743, 46ax-mp 5 . . . . 5  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
48 eqid 2467 . . . . 5  |-  ( 0g
`  ( (mulGrp `  R )s  U ) )  =  ( 0g `  (
(mulGrp `  R )s  U
) )
49 invrvald.i . . . . . 6  |-  I  =  ( invr `  R
)
5020, 25, 49invrfval 17103 . . . . 5  |-  I  =  ( invg `  ( (mulGrp `  R )s  U
) )
5141, 47, 48, 50grpinvid1 15896 . . . 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 1228 . . 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 1379    e. wcel 1767   _Vcvv 3113   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   ↾s cress 14484   +g cplusg 14548   .rcmulr 14549   0gc0g 14688   Grpcgrp 15720  mulGrpcmgp 16928   1rcur 16940   Ringcrg 16983  opprcoppr 17052   ||rcdsr 17068  Unitcui 17069   invrcinvr 17101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-mgp 16929  df-ur 16941  df-rng 16985  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102
This theorem is referenced by:  matinv  18943  matunit  18944
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