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Theorem matinv 19046
Description: The inverse of a matrix is the adjunct of the matrix multiplied with the inverse of the determinant of the matrix if the determinant is a unit in the underlying ring. Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.)
Hypotheses
Ref Expression
matinv.a  |-  A  =  ( N Mat  R )
matinv.j  |-  J  =  ( N maAdju  R )
matinv.d  |-  D  =  ( N maDet  R )
matinv.b  |-  B  =  ( Base `  A
)
matinv.u  |-  U  =  (Unit `  A )
matinv.v  |-  V  =  (Unit `  R )
matinv.h  |-  H  =  ( invr `  R
)
matinv.i  |-  I  =  ( invr `  A
)
matinv.t  |-  .xb  =  ( .s `  A )
Assertion
Ref Expression
matinv  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( M  e.  U  /\  ( I `  M
)  =  ( ( H `  ( D `
 M ) ) 
.xb  ( J `  M ) ) ) )

Proof of Theorem matinv
StepHypRef Expression
1 matinv.b . 2  |-  B  =  ( Base `  A
)
2 eqid 2467 . 2  |-  ( .r
`  A )  =  ( .r `  A
)
3 eqid 2467 . 2  |-  ( 1r
`  A )  =  ( 1r `  A
)
4 matinv.u . 2  |-  U  =  (Unit `  A )
5 matinv.i . 2  |-  I  =  ( invr `  A
)
6 matinv.a . . . . . . 7  |-  A  =  ( N Mat  R )
76, 1matrcl 18781 . . . . . 6  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
87simpld 459 . . . . 5  |-  ( M  e.  B  ->  N  e.  Fin )
983ad2ant2 1018 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  N  e.  Fin )
10 simp1 996 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  R  e.  CRing )
116matassa 18813 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
129, 10, 11syl2anc 661 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  A  e. AssAlg )
13 assaring 17837 . . 3  |-  ( A  e. AssAlg  ->  A  e.  Ring )
1412, 13syl 16 . 2  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  A  e.  Ring )
15 simp2 997 . 2  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  M  e.  B )
16 assalmod 17836 . . . 4  |-  ( A  e. AssAlg  ->  A  e.  LMod )
1712, 16syl 16 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  A  e.  LMod )
18 crngring 17079 . . . . . 6  |-  ( R  e.  CRing  ->  R  e.  Ring )
19183ad2ant1 1017 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  R  e.  Ring )
20 simp3 998 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( D `  M )  e.  V )
21 matinv.v . . . . . 6  |-  V  =  (Unit `  R )
22 matinv.h . . . . . 6  |-  H  =  ( invr `  R
)
23 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2421, 22, 23ringinvcl 17195 . . . . 5  |-  ( ( R  e.  Ring  /\  ( D `  M )  e.  V )  ->  ( H `  ( D `  M ) )  e.  ( Base `  R
) )
2519, 20, 24syl2anc 661 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( H `  ( D `  M ) )  e.  ( Base `  R
) )
266matsca2 18789 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
279, 10, 26syl2anc 661 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  R  =  (Scalar `  A )
)
2827fveq2d 5876 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( Base `  R )  =  ( Base `  (Scalar `  A ) ) )
2925, 28eleqtrd 2557 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( H `  ( D `  M ) )  e.  ( Base `  (Scalar `  A ) ) )
30 matinv.j . . . . . 6  |-  J  =  ( N maAdju  R )
316, 30, 1maduf 19010 . . . . 5  |-  ( R  e.  CRing  ->  J : B
--> B )
32313ad2ant1 1017 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  J : B --> B )
3332, 15ffvelrnd 6033 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( J `  M )  e.  B )
34 eqid 2467 . . . 4  |-  (Scalar `  A )  =  (Scalar `  A )
35 matinv.t . . . 4  |-  .xb  =  ( .s `  A )
36 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  A )
)  =  ( Base `  (Scalar `  A )
)
371, 34, 35, 36lmodvscl 17398 . . 3  |-  ( ( A  e.  LMod  /\  ( H `  ( D `  M ) )  e.  ( Base `  (Scalar `  A ) )  /\  ( J `  M )  e.  B )  -> 
( ( H `  ( D `  M ) )  .xb  ( J `  M ) )  e.  B )
3817, 29, 33, 37syl3anc 1228 . 2  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
)  .xb  ( J `  M ) )  e.  B )
391, 34, 36, 35, 2assaassr 17835 . . . 4  |-  ( ( A  e. AssAlg  /\  (
( H `  ( D `  M )
)  e.  ( Base `  (Scalar `  A )
)  /\  M  e.  B  /\  ( J `  M )  e.  B
) )  ->  ( M ( .r `  A ) ( ( H `  ( D `
 M ) ) 
.xb  ( J `  M ) ) )  =  ( ( H `
 ( D `  M ) )  .xb  ( M ( .r `  A ) ( J `
 M ) ) ) )
4012, 29, 15, 33, 39syl13anc 1230 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( M ( .r `  A ) ( ( H `  ( D `
 M ) ) 
.xb  ( J `  M ) ) )  =  ( ( H `
 ( D `  M ) )  .xb  ( M ( .r `  A ) ( J `
 M ) ) ) )
41 matinv.d . . . . . 6  |-  D  =  ( N maDet  R )
426, 1, 30, 41, 3, 2, 35madurid 19013 . . . . 5  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( M ( .r
`  A ) ( J `  M ) )  =  ( ( D `  M ) 
.xb  ( 1r `  A ) ) )
4315, 10, 42syl2anc 661 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( M ( .r `  A ) ( J `
 M ) )  =  ( ( D `
 M )  .xb  ( 1r `  A ) ) )
4443oveq2d 6311 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
)  .xb  ( M
( .r `  A
) ( J `  M ) ) )  =  ( ( H `
 ( D `  M ) )  .xb  ( ( D `  M )  .xb  ( 1r `  A ) ) ) )
45 eqid 2467 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
46 eqid 2467 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
4721, 22, 45, 46unitlinv 17196 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
) ( .r `  R ) ( D `
 M ) )  =  ( 1r `  R ) )
4819, 20, 47syl2anc 661 . . . . . 6  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
) ( .r `  R ) ( D `
 M ) )  =  ( 1r `  R ) )
4927fveq2d 5876 . . . . . . 7  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( .r `  R )  =  ( .r `  (Scalar `  A ) ) )
5049oveqd 6312 . . . . . 6  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
) ( .r `  R ) ( D `
 M ) )  =  ( ( H `
 ( D `  M ) ) ( .r `  (Scalar `  A ) ) ( D `  M ) ) )
5127fveq2d 5876 . . . . . 6  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( 1r `  R )  =  ( 1r `  (Scalar `  A ) ) )
5248, 50, 513eqtr3d 2516 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
) ( .r `  (Scalar `  A ) ) ( D `  M
) )  =  ( 1r `  (Scalar `  A ) ) )
5352oveq1d 6310 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( ( H `  ( D `  M ) ) ( .r `  (Scalar `  A ) ) ( D `  M
) )  .xb  ( 1r `  A ) )  =  ( ( 1r
`  (Scalar `  A )
)  .xb  ( 1r `  A ) ) )
5423, 21unitcl 17178 . . . . . . 7  |-  ( ( D `  M )  e.  V  ->  ( D `  M )  e.  ( Base `  R
) )
55543ad2ant3 1019 . . . . . 6  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( D `  M )  e.  ( Base `  R
) )
5655, 28eleqtrd 2557 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( D `  M )  e.  ( Base `  (Scalar `  A ) ) )
571, 3ringidcl 17089 . . . . . 6  |-  ( A  e.  Ring  ->  ( 1r
`  A )  e.  B )
5814, 57syl 16 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( 1r `  A )  e.  B )
59 eqid 2467 . . . . . 6  |-  ( .r
`  (Scalar `  A )
)  =  ( .r
`  (Scalar `  A )
)
601, 34, 35, 36, 59lmodvsass 17406 . . . . 5  |-  ( ( A  e.  LMod  /\  (
( H `  ( D `  M )
)  e.  ( Base `  (Scalar `  A )
)  /\  ( D `  M )  e.  (
Base `  (Scalar `  A
) )  /\  ( 1r `  A )  e.  B ) )  -> 
( ( ( H `
 ( D `  M ) ) ( .r `  (Scalar `  A ) ) ( D `  M ) )  .xb  ( 1r `  A ) )  =  ( ( H `  ( D `  M ) )  .xb  ( ( D `  M )  .xb  ( 1r `  A
) ) ) )
6117, 29, 56, 58, 60syl13anc 1230 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( ( H `  ( D `  M ) ) ( .r `  (Scalar `  A ) ) ( D `  M
) )  .xb  ( 1r `  A ) )  =  ( ( H `
 ( D `  M ) )  .xb  ( ( D `  M )  .xb  ( 1r `  A ) ) ) )
62 eqid 2467 . . . . . 6  |-  ( 1r
`  (Scalar `  A )
)  =  ( 1r
`  (Scalar `  A )
)
631, 34, 35, 62lmodvs1 17409 . . . . 5  |-  ( ( A  e.  LMod  /\  ( 1r `  A )  e.  B )  ->  (
( 1r `  (Scalar `  A ) )  .xb  ( 1r `  A ) )  =  ( 1r
`  A ) )
6417, 58, 63syl2anc 661 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( 1r `  (Scalar `  A ) )  .xb  ( 1r `  A ) )  =  ( 1r
`  A ) )
6553, 61, 643eqtr3d 2516 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
)  .xb  ( ( D `  M )  .xb  ( 1r `  A
) ) )  =  ( 1r `  A
) )
6640, 44, 653eqtrd 2512 . 2  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( M ( .r `  A ) ( ( H `  ( D `
 M ) ) 
.xb  ( J `  M ) ) )  =  ( 1r `  A ) )
671, 34, 36, 35, 2assaass 17834 . . . 4  |-  ( ( A  e. AssAlg  /\  (
( H `  ( D `  M )
)  e.  ( Base `  (Scalar `  A )
)  /\  ( J `  M )  e.  B  /\  M  e.  B
) )  ->  (
( ( H `  ( D `  M ) )  .xb  ( J `  M ) ) ( .r `  A ) M )  =  ( ( H `  ( D `  M )
)  .xb  ( ( J `  M )
( .r `  A
) M ) ) )
6812, 29, 33, 15, 67syl13anc 1230 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( ( H `  ( D `  M ) )  .xb  ( J `  M ) ) ( .r `  A ) M )  =  ( ( H `  ( D `  M )
)  .xb  ( ( J `  M )
( .r `  A
) M ) ) )
696, 1, 30, 41, 3, 2, 35madulid 19014 . . . . 5  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( ( J `  M ) ( .r
`  A ) M )  =  ( ( D `  M ) 
.xb  ( 1r `  A ) ) )
7015, 10, 69syl2anc 661 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( J `  M
) ( .r `  A ) M )  =  ( ( D `
 M )  .xb  ( 1r `  A ) ) )
7170oveq2d 6311 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( H `  ( D `  M )
)  .xb  ( ( J `  M )
( .r `  A
) M ) )  =  ( ( H `
 ( D `  M ) )  .xb  ( ( D `  M )  .xb  ( 1r `  A ) ) ) )
7268, 71, 653eqtrd 2512 . 2  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  (
( ( H `  ( D `  M ) )  .xb  ( J `  M ) ) ( .r `  A ) M )  =  ( 1r `  A ) )
731, 2, 3, 4, 5, 14, 15, 38, 66, 72invrvald 19045 1  |-  ( ( R  e.  CRing  /\  M  e.  B  /\  ( D `  M )  e.  V )  ->  ( M  e.  U  /\  ( I `  M
)  =  ( ( H `  ( D `
 M ) ) 
.xb  ( J `  M ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118   -->wf 5590   ` cfv 5594  (class class class)co 6295   Fincfn 7528   Basecbs 14506   .rcmulr 14572  Scalarcsca 14574   .scvsca 14575   1rcur 17023   Ringcrg 17068   CRingccrg 17069  Unitcui 17158   invrcinvr 17190   LModclmod 17381  AssAlgcasa 17826   Mat cmat 18776   maDet cmdat 18953   maAdju cmadu 19001
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-fal 1385  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12522  df-concat 12524  df-s1 12525  df-substr 12526  df-splice 12527  df-reverse 12528  df-s2 12792  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-0g 14713  df-gsum 14714  df-prds 14719  df-pws 14721  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-mhm 15838  df-submnd 15839  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mulg 15931  df-subg 16069  df-ghm 16136  df-gim 16178  df-cntz 16226  df-oppg 16252  df-symg 16274  df-pmtr 16338  df-psgn 16387  df-evpm 16388  df-cmn 16671  df-abl 16672  df-mgp 17012  df-ur 17024  df-ring 17070  df-cring 17071  df-oppr 17142  df-dvdsr 17160  df-unit 17161  df-invr 17191  df-dvr 17202  df-rnghom 17234  df-drng 17267  df-subrg 17296  df-lmod 17383  df-lss 17448  df-sra 17687  df-rgmod 17688  df-assa 17829  df-cnfld 18289  df-zring 18357  df-zrh 18408  df-dsmm 18630  df-frlm 18645  df-mamu 18753  df-mat 18777  df-mdet 18954  df-madu 19003
This theorem is referenced by:  matunit  19047
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