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Theorem madutpos 18951
Description: The adjuct of a transposed matrix is the transposition of the adjunct of the matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
Hypotheses
Ref Expression
maduf.a  |-  A  =  ( N Mat  R )
maduf.j  |-  J  =  ( N maAdju  R )
maduf.b  |-  B  =  ( Base `  A
)
Assertion
Ref Expression
madutpos  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  = tpos  ( J `  M
) )

Proof of Theorem madutpos
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . . . . 9  |-  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r
`  R ) ,  ( 0g `  R
) ) ,  ( d M c ) ) )  =  ( d  e.  N , 
c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) )
21tposmpt2 6993 . . . . . . . 8  |- tpos  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r
`  R ) ,  ( 0g `  R
) ) ,  ( d M c ) ) )  =  ( c  e.  N , 
d  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) )
3 orcom 387 . . . . . . . . . . 11  |-  ( ( d  =  a  \/  c  =  b )  <-> 
( c  =  b  \/  d  =  a ) )
43a1i 11 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( ( d  =  a  \/  c  =  b )  <->  ( c  =  b  \/  d  =  a ) ) )
5 ancom 450 . . . . . . . . . . . 12  |-  ( ( c  =  b  /\  d  =  a )  <->  ( d  =  a  /\  c  =  b )
)
65a1i 11 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( ( c  =  b  /\  d  =  a )  <->  ( d  =  a  /\  c  =  b ) ) )
76ifbid 3961 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  =  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) )
8 ovtpos 6971 . . . . . . . . . . . 12  |-  ( ctpos 
M d )  =  ( d M c )
98eqcomi 2480 . . . . . . . . . . 11  |-  ( d M c )  =  ( ctpos  M d )
109a1i 11 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( d M c )  =  ( ctpos 
M d ) )
114, 7, 10ifbieq12d 3966 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) )  =  if ( ( c  =  b  \/  d  =  a ) ,  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( ctpos  M d ) ) )
1211mpt2eq3dv 6348 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( c  e.  N ,  d  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( d M c ) ) )  =  ( c  e.  N ,  d  e.  N  |->  if ( ( c  =  b  \/  d  =  a ) ,  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( ctpos 
M d ) ) ) )
132, 12syl5eq 2520 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> tpos  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( d M c ) ) )  =  ( c  e.  N ,  d  e.  N  |->  if ( ( c  =  b  \/  d  =  a ) ,  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( ctpos 
M d ) ) ) )
1413fveq2d 5870 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( ( N maDet  R
) ` tpos  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) ) )  =  ( ( N maDet  R ) `
 ( c  e.  N ,  d  e.  N  |->  if ( ( c  =  b  \/  d  =  a ) ,  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( ctpos 
M d ) ) ) ) )
15 simpll 753 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  R  e.  CRing )
16 maduf.a . . . . . . . 8  |-  A  =  ( N Mat  R )
17 eqid 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
18 maduf.b . . . . . . . 8  |-  B  =  ( Base `  A
)
1916, 18matrcl 18721 . . . . . . . . . 10  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
2019simpld 459 . . . . . . . . 9  |-  ( M  e.  B  ->  N  e.  Fin )
2120ad2antlr 726 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  N  e.  Fin )
22 simp1ll 1059 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  ( a  e.  N  /\  b  e.  N ) )  /\  d  e.  N  /\  c  e.  N )  ->  R  e.  CRing )
23 crngrng 17022 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  R  e.  Ring )
24 eqid 2467 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
2517, 24rngidcl 17032 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
26 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  R
)
2717, 26rng0cl 17033 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( 0g
`  R )  e.  ( Base `  R
) )
2825, 27ifcld 3982 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r
`  R ) ,  ( 0g `  R
) )  e.  (
Base `  R )
)
2922, 23, 283syl 20 . . . . . . . . 9  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  ( a  e.  N  /\  b  e.  N ) )  /\  d  e.  N  /\  c  e.  N )  ->  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) )  e.  ( Base `  R
) )
3016, 17, 18matbas2i 18731 . . . . . . . . . . . . 13  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
31 elmapi 7441 . . . . . . . . . . . . 13  |-  ( M  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
3230, 31syl 16 . . . . . . . . . . . 12  |-  ( M  e.  B  ->  M : ( N  X.  N ) --> ( Base `  R ) )
3332ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R ) )
3433fovrnda 6431 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  ( a  e.  N  /\  b  e.  N ) )  /\  ( d  e.  N  /\  c  e.  N
) )  ->  (
d M c )  e.  ( Base `  R
) )
35343impb 1192 . . . . . . . . 9  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  ( a  e.  N  /\  b  e.  N ) )  /\  d  e.  N  /\  c  e.  N )  ->  ( d M c )  e.  ( Base `  R ) )
3629, 35ifcld 3982 . . . . . . . 8  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  ( a  e.  N  /\  b  e.  N ) )  /\  d  e.  N  /\  c  e.  N )  ->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( d M c ) )  e.  ( Base `  R
) )
3716, 17, 18, 21, 15, 36matbas2d 18732 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( d M c ) ) )  e.  B )
38 eqid 2467 . . . . . . . 8  |-  ( N maDet 
R )  =  ( N maDet  R )
3938, 16, 18mdettpos 18920 . . . . . . 7  |-  ( ( R  e.  CRing  /\  (
d  e.  N , 
c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) )  e.  B )  ->  (
( N maDet  R ) ` tpos  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( d M c ) ) ) )  =  ( ( N maDet  R ) `  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( d M c ) ) ) ) )
4015, 37, 39syl2anc 661 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( ( N maDet  R
) ` tpos  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) ) )  =  ( ( N maDet  R ) `
 ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) ) ) )
4114, 40eqtr3d 2510 . . . . 5  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( ( N maDet  R
) `  ( c  e.  N ,  d  e.  N  |->  if ( ( c  =  b  \/  d  =  a ) ,  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( ctpos 
M d ) ) ) )  =  ( ( N maDet  R ) `
 ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) ) ) )
4216, 18mattposcl 18762 . . . . . . . 8  |-  ( M  e.  B  -> tpos  M  e.  B )
4342adantl 466 . . . . . . 7  |-  ( ( R  e.  CRing  /\  M  e.  B )  -> tpos  M  e.  B )
4443adantr 465 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> tpos  M  e.  B )
45 simprl 755 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
a  e.  N )
46 simprr 756 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
b  e.  N )
47 maduf.j . . . . . . 7  |-  J  =  ( N maAdju  R )
4816, 38, 47, 18, 24, 26maducoeval2 18949 . . . . . 6  |-  ( ( ( R  e.  CRing  /\ tpos  M  e.  B )  /\  a  e.  N  /\  b  e.  N
)  ->  ( a
( J ` tpos  M ) b )  =  ( ( N maDet  R ) `
 ( c  e.  N ,  d  e.  N  |->  if ( ( c  =  b  \/  d  =  a ) ,  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( ctpos 
M d ) ) ) ) )
4915, 44, 45, 46, 48syl211anc 1234 . . . . 5  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( J `
tpos  M ) b )  =  ( ( N maDet  R ) `  ( c  e.  N ,  d  e.  N  |->  if ( ( c  =  b  \/  d  =  a ) ,  if ( ( d  =  a  /\  c  =  b ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( ctpos  M
d ) ) ) ) )
50 simplr 754 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  M  e.  B )
5116, 38, 47, 18, 24, 26maducoeval2 18949 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  b  e.  N  /\  a  e.  N
)  ->  ( b
( J `  M
) a )  =  ( ( N maDet  R
) `  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( d M c ) ) ) ) )
5215, 50, 46, 45, 51syl211anc 1234 . . . . 5  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( b ( J `
 M ) a )  =  ( ( N maDet  R ) `  ( d  e.  N ,  c  e.  N  |->  if ( ( d  =  a  \/  c  =  b ) ,  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( d M c ) ) ) ) )
5341, 49, 523eqtr4d 2518 . . . 4  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( J `
tpos  M ) b )  =  ( b ( J `  M
) a ) )
54 ovtpos 6971 . . . 4  |-  ( atpos  ( J `  M
) b )  =  ( b ( J `
 M ) a )
5553, 54syl6eqr 2526 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( J `
tpos  M ) b )  =  ( atpos  ( J `  M
) b ) )
5655ralrimivva 2885 . 2  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  A. a  e.  N  A. b  e.  N  ( a
( J ` tpos  M ) b )  =  ( atpos  ( J `  M ) b ) )
5716, 47, 18maduf 18950 . . . . . . 7  |-  ( R  e.  CRing  ->  J : B
--> B )
5857adantr 465 . . . . . 6  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  J : B --> B )
5958, 43ffvelrnd 6023 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  e.  B )
6016, 17, 18matbas2i 18731 . . . . 5  |-  ( ( J ` tpos  M )  e.  B  ->  ( J `
tpos  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
6159, 60syl 16 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
62 elmapi 7441 . . . 4  |-  ( ( J ` tpos  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) )  -> 
( J ` tpos  M ) : ( N  X.  N ) --> ( Base `  R ) )
63 ffn 5731 . . . 4  |-  ( ( J ` tpos  M ) : ( N  X.  N ) --> ( Base `  R )  ->  ( J ` tpos  M )  Fn  ( N  X.  N
) )
6461, 62, 633syl 20 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  Fn  ( N  X.  N
) )
6557ffvelrnda 6022 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J `  M )  e.  B )
6616, 18mattposcl 18762 . . . . 5  |-  ( ( J `  M )  e.  B  -> tpos  ( J `
 M )  e.  B )
6716, 17, 18matbas2i 18731 . . . . 5  |-  (tpos  ( J `  M )  e.  B  -> tpos  ( J `
 M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
6865, 66, 673syl 20 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B )  -> tpos  ( J `
 M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
69 elmapi 7441 . . . 4  |-  (tpos  ( J `  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) )  -> tpos  ( J `  M ) : ( N  X.  N ) --> ( Base `  R ) )
70 ffn 5731 . . . 4  |-  (tpos  ( J `  M ) : ( N  X.  N ) --> ( Base `  R )  -> tpos  ( J `
 M )  Fn  ( N  X.  N
) )
7168, 69, 703syl 20 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B )  -> tpos  ( J `
 M )  Fn  ( N  X.  N
) )
72 eqfnov2 6394 . . 3  |-  ( ( ( J ` tpos  M )  Fn  ( N  X.  N )  /\ tpos  ( J `  M )  Fn  ( N  X.  N
) )  ->  (
( J ` tpos  M )  = tpos  ( J `  M )  <->  A. a  e.  N  A. b  e.  N  ( a
( J ` tpos  M ) b )  =  ( atpos  ( J `  M ) b ) ) )
7364, 71, 72syl2anc 661 . 2  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  (
( J ` tpos  M )  = tpos  ( J `  M )  <->  A. a  e.  N  A. b  e.  N  ( a
( J ` tpos  M ) b )  =  ( atpos  ( J `  M ) b ) ) )
7456, 73mpbird 232 1  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  = tpos  ( J `  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   ifcif 3939    X. cxp 4997    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287  tpos ctpos 6955    ^m cmap 7421   Fincfn 7517   Basecbs 14493   0gc0g 14698   1rcur 16967   Ringcrg 17012   CRingccrg 17013   Mat cmat 18716   maDet cmdat 18893   maAdju cmadu 18941
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 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-addf 9572  ax-mulf 9573
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 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-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  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-se 4839  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-tpos 6956  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-rp 11222  df-fz 11674  df-fzo 11794  df-seq 12077  df-exp 12136  df-hash 12375  df-word 12509  df-concat 12511  df-s1 12512  df-substr 12513  df-splice 12514  df-reverse 12515  df-s2 12779  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-hom 14582  df-cco 14583  df-0g 14700  df-gsum 14701  df-prds 14706  df-pws 14708  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-mhm 15789  df-submnd 15790  df-grp 15871  df-minusg 15872  df-mulg 15874  df-subg 16012  df-ghm 16079  df-gim 16121  df-cntz 16169  df-oppg 16195  df-symg 16217  df-pmtr 16282  df-psgn 16331  df-evpm 16332  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-cring 17015  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-dvr 17145  df-rnghom 17177  df-drng 17210  df-subrg 17239  df-sra 17630  df-rgmod 17631  df-cnfld 18232  df-zring 18297  df-zrh 18348  df-dsmm 18570  df-frlm 18585  df-mat 18717  df-mdet 18894  df-madu 18943
This theorem is referenced by:  madulid  18954
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