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Theorem madutpos 19591
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 2420 . . . . . . . . 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 7009 . . . . . . . 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 388 . . . . . . . . . . 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 451 . . . . . . . . . . . 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 3928 . . . . . . . . . 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 6987 . . . . . . . . . . . 12  |-  ( ctpos 
M d )  =  ( d M c )
98eqcomi 2433 . . . . . . . . . . 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 3933 . . . . . . . . 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 6362 . . . . . . . 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 2473 . . . . . . 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 5876 . . . . . 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 758 . . . . . . 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 2420 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
18 maduf.b . . . . . . . 8  |-  B  =  ( Base `  A
)
1916, 18matrcl 19361 . . . . . . . . . 10  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
2019simpld 460 . . . . . . . . 9  |-  ( M  e.  B  ->  N  e.  Fin )
2120ad2antlr 731 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  N  e.  Fin )
22 simp1ll 1068 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  ( a  e.  N  /\  b  e.  N ) )  /\  d  e.  N  /\  c  e.  N )  ->  R  e.  CRing )
23 crngring 17719 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  R  e.  Ring )
24 eqid 2420 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
2517, 24ringidcl 17729 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
26 eqid 2420 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  R
)
2717, 26ring0cl 17730 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( 0g
`  R )  e.  ( Base `  R
) )
2825, 27ifcld 3949 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  if ( ( c  =  b  /\  d  =  a ) ,  ( 1r
`  R ) ,  ( 0g `  R
) )  e.  (
Base `  R )
)
2922, 23, 283syl 18 . . . . . . . . 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 19371 . . . . . . . . . . . . 13  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
31 elmapi 7492 . . . . . . . . . . . . 13  |-  ( M  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
3230, 31syl 17 . . . . . . . . . . . 12  |-  ( M  e.  B  ->  M : ( N  X.  N ) --> ( Base `  R ) )
3332ad2antlr 731 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R ) )
3433fovrnda 6445 . . . . . . . . . 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 1201 . . . . . . . . 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 3949 . . . . . . . 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 19372 . . . . . . 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 2420 . . . . . . . 8  |-  ( N maDet 
R )  =  ( N maDet  R )
3938, 16, 18mdettpos 19560 . . . . . . 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 665 . . . . . 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 2463 . . . . 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 19402 . . . . . . . 8  |-  ( M  e.  B  -> tpos  M  e.  B )
4342adantl 467 . . . . . . 7  |-  ( ( R  e.  CRing  /\  M  e.  B )  -> tpos  M  e.  B )
4443adantr 466 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> tpos  M  e.  B )
45 simprl 762 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
a  e.  N )
46 simprr 764 . . . . . 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 19589 . . . . . 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 1270 . . . . 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 760 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  ->  M  e.  B )
5116, 38, 47, 18, 24, 26maducoeval2 19589 . . . . . 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 1270 . . . . 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 2471 . . . 4  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( J `
tpos  M ) b )  =  ( b ( J `  M
) a ) )
54 ovtpos 6987 . . . 4  |-  ( atpos  ( J `  M
) b )  =  ( b ( J `
 M ) a )
5553, 54syl6eqr 2479 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( J `
tpos  M ) b )  =  ( atpos  ( J `  M
) b ) )
5655ralrimivva 2844 . 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 19590 . . . . . . 7  |-  ( R  e.  CRing  ->  J : B
--> B )
5857adantr 466 . . . . . 6  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  J : B --> B )
5958, 43ffvelrnd 6029 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  e.  B )
6016, 17, 18matbas2i 19371 . . . . 5  |-  ( ( J ` tpos  M )  e.  B  ->  ( J `
tpos  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
6159, 60syl 17 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
62 elmapi 7492 . . . 4  |-  ( ( J ` tpos  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) )  -> 
( J ` tpos  M ) : ( N  X.  N ) --> ( Base `  R ) )
63 ffn 5737 . . . 4  |-  ( ( J ` tpos  M ) : ( N  X.  N ) --> ( Base `  R )  ->  ( J ` tpos  M )  Fn  ( N  X.  N
) )
6461, 62, 633syl 18 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  Fn  ( N  X.  N
) )
6557ffvelrnda 6028 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J `  M )  e.  B )
6616, 18mattposcl 19402 . . . . 5  |-  ( ( J `  M )  e.  B  -> tpos  ( J `
 M )  e.  B )
6716, 17, 18matbas2i 19371 . . . . 5  |-  (tpos  ( J `  M )  e.  B  -> tpos  ( J `
 M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
6865, 66, 673syl 18 . . . 4  |-  ( ( R  e.  CRing  /\  M  e.  B )  -> tpos  ( J `
 M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
69 elmapi 7492 . . . 4  |-  (tpos  ( J `  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) )  -> tpos  ( J `  M ) : ( N  X.  N ) --> ( Base `  R ) )
70 ffn 5737 . . . 4  |-  (tpos  ( J `  M ) : ( N  X.  N ) --> ( Base `  R )  -> tpos  ( J `
 M )  Fn  ( N  X.  N
) )
7168, 69, 703syl 18 . . 3  |-  ( ( R  e.  CRing  /\  M  e.  B )  -> tpos  ( J `
 M )  Fn  ( N  X.  N
) )
72 eqfnov2 6408 . . 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 665 . 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 235 1  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  = tpos  ( J `  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078   ifcif 3906    X. cxp 4843    Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298  tpos ctpos 6971    ^m cmap 7471   Fincfn 7568   Basecbs 15073   0gc0g 15290   1rcur 17663   Ringcrg 17708   CRingccrg 17709   Mat cmat 19356   maDet cmdat 19533   maAdju cmadu 19581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-xor 1401  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-ot 4002  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-sup 7953  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-word 12640  df-lsw 12641  df-concat 12642  df-s1 12643  df-substr 12644  df-splice 12645  df-reverse 12646  df-s2 12918  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-sca 15158  df-vsca 15159  df-ip 15160  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-hom 15166  df-cco 15167  df-0g 15292  df-gsum 15293  df-prds 15298  df-pws 15300  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-mhm 16526  df-submnd 16527  df-grp 16617  df-minusg 16618  df-mulg 16620  df-subg 16758  df-ghm 16825  df-gim 16867  df-cntz 16915  df-oppg 16941  df-symg 16963  df-pmtr 17027  df-psgn 17076  df-evpm 17077  df-cmn 17360  df-abl 17361  df-mgp 17652  df-ur 17664  df-ring 17710  df-cring 17711  df-oppr 17779  df-dvdsr 17797  df-unit 17798  df-invr 17828  df-dvr 17839  df-rnghom 17871  df-drng 17905  df-subrg 17934  df-sra 18323  df-rgmod 18324  df-cnfld 18899  df-zring 18967  df-zrh 18999  df-dsmm 19219  df-frlm 19234  df-mat 19357  df-mdet 19534  df-madu 19583
This theorem is referenced by:  madulid  19594
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