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Theorem madutpos 18460
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 2443 . . . . . . . . 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 6794 . . . . . . . 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 3823 . . . . . . . . . 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 6772 . . . . . . . . . . . 12  |-  ( ctpos 
M d )  =  ( d M c )
98eqcomi 2447 . . . . . . . . . . 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 3828 . . . . . . . . 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 6164 . . . . . . . 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 2487 . . . . . . 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 5707 . . . . . 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 2443 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
18 maduf.b . . . . . . . 8  |-  B  =  ( Base `  A
)
1916, 18matrcl 18324 . . . . . . . . . 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 1051 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  ( a  e.  N  /\  b  e.  N ) )  /\  d  e.  N  /\  c  e.  N )  ->  R  e.  CRing )
23 crngrng 16667 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  R  e.  Ring )
24 eqid 2443 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
2517, 24rngidcl 16677 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
26 eqid 2443 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  R
)
2717, 26rng0cl 16678 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( 0g
`  R )  e.  ( Base `  R
) )
2825, 27ifcld 3844 . . . . . . . . . 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 18335 . . . . . . . . . . . . 13  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
31 elmapi 7246 . . . . . . . . . . . . 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 6246 . . . . . . . . . 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 1183 . . . . . . . . 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 3844 . . . . . . . 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 18336 . . . . . . 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 2443 . . . . . . . 8  |-  ( N maDet 
R )  =  ( N maDet  R )
3938, 16, 18mdettpos 18429 . . . . . . 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 2477 . . . . 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 18349 . . . . . . . 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 18458 . . . . . 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 1224 . . . . 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 18458 . . . . . 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 1224 . . . . 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 2485 . . . 4  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( J `
tpos  M ) b )  =  ( b ( J `  M
) a ) )
54 ovtpos 6772 . . . 4  |-  ( atpos  ( J `  M
) b )  =  ( b ( J `
 M ) a )
5553, 54syl6eqr 2493 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( J `
tpos  M ) b )  =  ( atpos  ( J `  M
) b ) )
5655ralrimivva 2820 . 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 18459 . . . . . . 7  |-  ( R  e.  CRing  ->  J : B
--> B )
5857adantr 465 . . . . . 6  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  J : B --> B )
5958, 43ffvelrnd 5856 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J ` tpos  M )  e.  B )
6016, 17, 18matbas2i 18335 . . . . 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 7246 . . . 4  |-  ( ( J ` tpos  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) )  -> 
( J ` tpos  M ) : ( N  X.  N ) --> ( Base `  R ) )
63 ffn 5571 . . . 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 5855 . . . . 5  |-  ( ( R  e.  CRing  /\  M  e.  B )  ->  ( J `  M )  e.  B )
6616, 18mattposcl 18349 . . . . 5  |-  ( ( J `  M )  e.  B  -> tpos  ( J `
 M )  e.  B )
6716, 17, 18matbas2i 18335 . . . . 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 7246 . . . 4  |-  (tpos  ( J `  M )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) )  -> tpos  ( J `  M ) : ( N  X.  N ) --> ( Base `  R ) )
70 ffn 5571 . . . 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 6209 . . 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 965    = wceq 1369    e. wcel 1756   A.wral 2727   _Vcvv 2984   ifcif 3803    X. cxp 4850    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105  tpos ctpos 6756    ^m cmap 7226   Fincfn 7322   Basecbs 14186   0gc0g 14390   1rcur 16615   Ringcrg 16657   CRingccrg 16658   Mat cmat 18292   maDet cmdat 18407   maAdju cmadu 18450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-ot 3898  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-tpos 6757  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-seq 11819  df-exp 11878  df-hash 12116  df-word 12241  df-concat 12243  df-s1 12244  df-substr 12245  df-splice 12246  df-reverse 12247  df-s2 12487  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-0g 14392  df-gsum 14393  df-prds 14398  df-pws 14400  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-mulg 15560  df-subg 15690  df-ghm 15757  df-gim 15799  df-cntz 15847  df-oppg 15873  df-symg 15895  df-pmtr 15960  df-psgn 16009  df-evpm 16010  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-cring 16660  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-dvr 16787  df-rnghom 16818  df-drng 16846  df-subrg 16875  df-sra 17265  df-rgmod 17266  df-cnfld 17831  df-zring 17896  df-zrh 17947  df-dsmm 18169  df-frlm 18184  df-mat 18294  df-mdet 18408  df-madu 18452
This theorem is referenced by:  madulid  18463
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