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Theorem madurid 19669
Description: Multiplying a matrix with its adjunct results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 16-Jul-2018.)
Hypotheses
Ref Expression
madurid.a  |-  A  =  ( N Mat  R )
madurid.b  |-  B  =  ( Base `  A
)
madurid.j  |-  J  =  ( N maAdju  R )
madurid.d  |-  D  =  ( N maDet  R )
madurid.i  |-  .1.  =  ( 1r `  A )
madurid.t  |-  .x.  =  ( .r `  A )
madurid.s  |-  .xb  =  ( .s `  A )
Assertion
Ref Expression
madurid  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( M  .x.  ( J `  M )
)  =  ( ( D `  M ) 
.xb  .1.  ) )

Proof of Theorem madurid
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . 3  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
2 eqid 2451 . . 3  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2451 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 simpr 463 . . 3  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  R  e.  CRing )
5 madurid.a . . . . . 6  |-  A  =  ( N Mat  R )
6 madurid.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6matrcl 19437 . . . . 5  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
87simpld 461 . . . 4  |-  ( M  e.  B  ->  N  e.  Fin )
98adantr 467 . . 3  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  N  e.  Fin )
105, 2, 6matbas2i 19447 . . . 4  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
1110adantr 467 . . 3  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  M  e.  ( ( Base `  R )  ^m  ( N  X.  N
) ) )
12 madurid.j . . . . . . 7  |-  J  =  ( N maAdju  R )
135, 12, 6maduf 19666 . . . . . 6  |-  ( R  e.  CRing  ->  J : B
--> B )
1413adantl 468 . . . . 5  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  J : B --> B )
15 simpl 459 . . . . 5  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  M  e.  B )
1614, 15ffvelrnd 6023 . . . 4  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( J `  M
)  e.  B )
175, 2, 6matbas2i 19447 . . . 4  |-  ( ( J `  M )  e.  B  ->  ( J `  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
1816, 17syl 17 . . 3  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( J `  M
)  e.  ( (
Base `  R )  ^m  ( N  X.  N
) ) )
191, 2, 3, 4, 9, 9, 9, 11, 18mamuval 19411 . 2  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( M ( R maMul  <. N ,  N ,  N >. ) ( J `
 M ) )  =  ( a  e.  N ,  b  e.  N  |->  ( R  gsumg  ( c  e.  N  |->  ( ( a M c ) ( .r `  R
) ( c ( J `  M ) b ) ) ) ) ) )
205, 1matmulr 19463 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
218, 20sylan 474 . . . 4  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
22 madurid.t . . . 4  |-  .x.  =  ( .r `  A )
2321, 22syl6eqr 2503 . . 3  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  .x.  )
2423oveqd 6307 . 2  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( M ( R maMul  <. N ,  N ,  N >. ) ( J `
 M ) )  =  ( M  .x.  ( J `  M ) ) )
25 madurid.d . . . . . 6  |-  D  =  ( N maDet  R )
26 simp1l 1032 . . . . . 6  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  M  e.  B )
27 simp1r 1033 . . . . . 6  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  R  e.  CRing )
28 elmapi 7493 . . . . . . . . . 10  |-  ( M  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
2911, 28syl 17 . . . . . . . . 9  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  M : ( N  X.  N ) --> ( Base `  R ) )
30293ad2ant1 1029 . . . . . . . 8  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  M : ( N  X.  N ) --> (
Base `  R )
)
3130adantr 467 . . . . . . 7  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  c  e.  N )  ->  M : ( N  X.  N ) --> ( Base `  R ) )
32 simpl2 1012 . . . . . . 7  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  c  e.  N )  ->  a  e.  N )
33 simpr 463 . . . . . . 7  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  c  e.  N )  ->  c  e.  N )
3431, 32, 33fovrnd 6441 . . . . . 6  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  c  e.  N )  ->  (
a M c )  e.  ( Base `  R
) )
35 simp3 1010 . . . . . 6  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  b  e.  N )
365, 12, 6, 25, 3, 2, 26, 27, 34, 35madugsum 19668 . . . . 5  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  ( R  gsumg  ( c  e.  N  |->  ( ( a M c ) ( .r
`  R ) ( c ( J `  M ) b ) ) ) )  =  ( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) ) )
37 iftrue 3887 . . . . . . . . 9  |-  ( a  =  b  ->  if ( a  =  b ,  ( D `  M ) ,  ( 0g `  R ) )  =  ( D `
 M ) )
3837adantl 468 . . . . . . . 8  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  =  b )  ->  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) )  =  ( D `  M ) )
39 ffn 5728 . . . . . . . . . . . . 13  |-  ( M : ( N  X.  N ) --> ( Base `  R )  ->  M  Fn  ( N  X.  N
) )
4029, 39syl 17 . . . . . . . . . . . 12  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  M  Fn  ( N  X.  N ) )
41 fnov 6404 . . . . . . . . . . . 12  |-  ( M  Fn  ( N  X.  N )  <->  M  =  ( d  e.  N ,  c  e.  N  |->  ( d M c ) ) )
4240, 41sylib 200 . . . . . . . . . . 11  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  M  =  ( d  e.  N ,  c  e.  N  |->  ( d M c ) ) )
4342adantr 467 . . . . . . . . . 10  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  =  b )  ->  M  =  ( d  e.  N ,  c  e.  N  |->  ( d M c ) ) )
44 equtr2 1869 . . . . . . . . . . . . . . 15  |-  ( ( a  =  b  /\  d  =  b )  ->  a  =  d )
4544oveq1d 6305 . . . . . . . . . . . . . 14  |-  ( ( a  =  b  /\  d  =  b )  ->  ( a M c )  =  ( d M c ) )
4645ifeq1da 3911 . . . . . . . . . . . . 13  |-  ( a  =  b  ->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) )  =  if ( d  =  b ,  ( d M c ) ,  ( d M c ) ) )
47 ifid 3918 . . . . . . . . . . . . 13  |-  if ( d  =  b ,  ( d M c ) ,  ( d M c ) )  =  ( d M c )
4846, 47syl6eq 2501 . . . . . . . . . . . 12  |-  ( a  =  b  ->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) )  =  ( d M c ) )
4948adantl 468 . . . . . . . . . . 11  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  =  b )  ->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) )  =  ( d M c ) )
5049mpt2eq3dv 6357 . . . . . . . . . 10  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  =  b )  ->  ( d  e.  N ,  c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) )  =  ( d  e.  N ,  c  e.  N  |->  ( d M c ) ) )
5143, 50eqtr4d 2488 . . . . . . . . 9  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  =  b )  ->  M  =  ( d  e.  N ,  c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) )
5251fveq2d 5869 . . . . . . . 8  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  =  b )  ->  ( D `  M
)  =  ( D `
 ( d  e.  N ,  c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) ) )
5338, 52eqtr2d 2486 . . . . . . 7  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  =  b )  ->  ( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) )  =  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) ) )
54533ad2antl1 1170 . . . . . 6  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  a  =  b )  -> 
( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) )  =  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) ) )
55 eqid 2451 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
56 simpl1r 1060 . . . . . . . 8  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  R  e.  CRing )
5793ad2ant1 1029 . . . . . . . . 9  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  N  e.  Fin )
5857adantr 467 . . . . . . . 8  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  N  e.  Fin )
5930ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  /\  c  e.  N
)  ->  M :
( N  X.  N
) --> ( Base `  R
) )
60 simpll2 1048 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  /\  c  e.  N
)  ->  a  e.  N )
61 simpr 463 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  /\  c  e.  N
)  ->  c  e.  N )
6259, 60, 61fovrnd 6441 . . . . . . . 8  |-  ( ( ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  /\  c  e.  N
)  ->  ( a M c )  e.  ( Base `  R
) )
6330adantr 467 . . . . . . . . . 10  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  M : ( N  X.  N ) --> (
Base `  R )
)
6463fovrnda 6440 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  /\  ( d  e.  N  /\  c  e.  N
) )  ->  (
d M c )  e.  ( Base `  R
) )
65643impb 1204 . . . . . . . 8  |-  ( ( ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  /\  d  e.  N  /\  c  e.  N
)  ->  ( d M c )  e.  ( Base `  R
) )
66 simpl3 1013 . . . . . . . 8  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  b  e.  N )
67 simpl2 1012 . . . . . . . 8  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  a  e.  N )
68 df-ne 2624 . . . . . . . . . . 11  |-  ( a  =/=  b  <->  -.  a  =  b )
6968biimpri 210 . . . . . . . . . 10  |-  ( -.  a  =  b  -> 
a  =/=  b )
7069necomd 2679 . . . . . . . . 9  |-  ( -.  a  =  b  -> 
b  =/=  a )
7170adantl 468 . . . . . . . 8  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  b  =/=  a )
7225, 2, 55, 56, 58, 62, 65, 66, 67, 71mdetralt2 19634 . . . . . . 7  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  ( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  if ( d  =  a ,  ( a M c ) ,  ( d M c ) ) ) ) )  =  ( 0g `  R ) )
73 ifid 3918 . . . . . . . . . . 11  |-  if ( d  =  a ,  ( d M c ) ,  ( d M c ) )  =  ( d M c )
74 oveq1 6297 . . . . . . . . . . . . 13  |-  ( d  =  a  ->  (
d M c )  =  ( a M c ) )
7574adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  /\  d  =  a
)  ->  ( d M c )  =  ( a M c ) )
7675ifeq1da 3911 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  if ( d  =  a ,  ( d M c ) ,  ( d M c ) )  =  if ( d  =  a ,  ( a M c ) ,  ( d M c ) ) )
7773, 76syl5eqr 2499 . . . . . . . . . 10  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  ( d M c )  =  if ( d  =  a ,  ( a M c ) ,  ( d M c ) ) )
7877ifeq2d 3900 . . . . . . . . 9  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) )  =  if ( d  =  b ,  ( a M c ) ,  if ( d  =  a ,  ( a M c ) ,  ( d M c ) ) ) )
7978mpt2eq3dv 6357 . . . . . . . 8  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  ( d  e.  N ,  c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) )  =  ( d  e.  N ,  c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  if ( d  =  a ,  ( a M c ) ,  ( d M c ) ) ) ) )
8079fveq2d 5869 . . . . . . 7  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  ( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) )  =  ( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  if ( d  =  a ,  ( a M c ) ,  ( d M c ) ) ) ) ) )
81 iffalse 3890 . . . . . . . 8  |-  ( -.  a  =  b  ->  if ( a  =  b ,  ( D `  M ) ,  ( 0g `  R ) )  =  ( 0g
`  R ) )
8281adantl 468 . . . . . . 7  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) )  =  ( 0g `  R ) )
8372, 80, 823eqtr4d 2495 . . . . . 6  |-  ( ( ( ( M  e.  B  /\  R  e. 
CRing )  /\  a  e.  N  /\  b  e.  N )  /\  -.  a  =  b )  ->  ( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) )  =  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) ) )
8454, 83pm2.61dan 800 . . . . 5  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  ( D `  (
d  e.  N , 
c  e.  N  |->  if ( d  =  b ,  ( a M c ) ,  ( d M c ) ) ) )  =  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) ) )
8536, 84eqtrd 2485 . . . 4  |-  ( ( ( M  e.  B  /\  R  e.  CRing )  /\  a  e.  N  /\  b  e.  N )  ->  ( R  gsumg  ( c  e.  N  |->  ( ( a M c ) ( .r
`  R ) ( c ( J `  M ) b ) ) ) )  =  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) ) )
8685mpt2eq3dva 6355 . . 3  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( a  e.  N ,  b  e.  N  |->  ( R  gsumg  ( c  e.  N  |->  ( ( a M c ) ( .r
`  R ) ( c ( J `  M ) b ) ) ) ) )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( D `  M ) ,  ( 0g `  R ) ) ) )
87 madurid.i . . . . 5  |-  .1.  =  ( 1r `  A )
8887oveq2i 6301 . . . 4  |-  ( ( D `  M ) 
.xb  .1.  )  =  ( ( D `  M )  .xb  ( 1r `  A ) )
89 crngring 17791 . . . . . 6  |-  ( R  e.  CRing  ->  R  e.  Ring )
9089adantl 468 . . . . 5  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  R  e.  Ring )
9125, 5, 6, 2mdetf 19620 . . . . . . 7  |-  ( R  e.  CRing  ->  D : B
--> ( Base `  R
) )
9291adantl 468 . . . . . 6  |-  ( ( M  e.  B  /\  R  e.  CRing )  ->  D : B --> ( Base `  R ) )
9392, 15ffvelrnd 6023 . . . . 5  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( D `  M
)  e.  ( Base `  R ) )
94 madurid.s . . . . . 6  |-  .xb  =  ( .s `  A )
955, 2, 94, 55matsc 19475 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  ( D `  M )  e.  ( Base `  R
) )  ->  (
( D `  M
)  .xb  ( 1r `  A ) )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( D `
 M ) ,  ( 0g `  R
) ) ) )
969, 90, 93, 95syl3anc 1268 . . . 4  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( ( D `  M )  .xb  ( 1r `  A ) )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( D `  M ) ,  ( 0g `  R ) ) ) )
9788, 96syl5eq 2497 . . 3  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( ( D `  M )  .xb  .1.  )  =  ( a  e.  N ,  b  e.  N  |->  if ( a  =  b ,  ( D `  M ) ,  ( 0g `  R ) ) ) )
9886, 97eqtr4d 2488 . 2  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( a  e.  N ,  b  e.  N  |->  ( R  gsumg  ( c  e.  N  |->  ( ( a M c ) ( .r
`  R ) ( c ( J `  M ) b ) ) ) ) )  =  ( ( D `
 M )  .xb  .1.  ) )
9919, 24, 983eqtr3d 2493 1  |-  ( ( M  e.  B  /\  R  e.  CRing )  -> 
( M  .x.  ( J `  M )
)  =  ( ( D `  M ) 
.xb  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045   ifcif 3881   <.cotp 3976    |-> cmpt 4461    X. cxp 4832    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292    ^m cmap 7472   Fincfn 7569   Basecbs 15121   .rcmulr 15191   .scvsca 15194   0gc0g 15338    gsumg cgsu 15339   1rcur 17735   Ringcrg 17780   CRingccrg 17781   maMul cmmul 19408   Mat cmat 19432   maDet cmdat 19609   maAdju cmadu 19657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-xor 1406  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-splice 12669  df-reverse 12670  df-s2 12944  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-0g 15340  df-gsum 15341  df-prds 15346  df-pws 15348  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-gim 16923  df-cntz 16971  df-oppg 16997  df-symg 17019  df-pmtr 17083  df-psgn 17132  df-evpm 17133  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-rnghom 17943  df-drng 17977  df-subrg 18006  df-lmod 18093  df-lss 18156  df-sra 18395  df-rgmod 18396  df-cnfld 18971  df-zring 19040  df-zrh 19075  df-dsmm 19295  df-frlm 19310  df-mamu 19409  df-mat 19433  df-mdet 19610  df-madu 19659
This theorem is referenced by:  madulid  19670  matinv  19702  cpmadurid  19891  cpmidgsum2  19903
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