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Theorem gxcom 20936
Description: The group power operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxcom.1  |-  X  =  ran  G
gxcom.2  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxcom  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  (
( A P K ) G A )  =  ( A G ( A P K ) ) )

Proof of Theorem gxcom
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . 5  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
21oveq1d 5873 . . . 4  |-  ( m  =  0  ->  (
( A P m ) G A )  =  ( ( A P 0 ) G A ) )
31oveq2d 5874 . . . 4  |-  ( m  =  0  ->  ( A G ( A P m ) )  =  ( A G ( A P 0 ) ) )
42, 3eqeq12d 2297 . . 3  |-  ( m  =  0  ->  (
( ( A P m ) G A )  =  ( A G ( A P m ) )  <->  ( ( A P 0 ) G A )  =  ( A G ( A P 0 ) ) ) )
5 oveq2 5866 . . . . 5  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
65oveq1d 5873 . . . 4  |-  ( m  =  k  ->  (
( A P m ) G A )  =  ( ( A P k ) G A ) )
75oveq2d 5874 . . . 4  |-  ( m  =  k  ->  ( A G ( A P m ) )  =  ( A G ( A P k ) ) )
86, 7eqeq12d 2297 . . 3  |-  ( m  =  k  ->  (
( ( A P m ) G A )  =  ( A G ( A P m ) )  <->  ( ( A P k ) G A )  =  ( A G ( A P k ) ) ) )
9 oveq2 5866 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
109oveq1d 5873 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( A P m ) G A )  =  ( ( A P ( k  +  1 ) ) G A ) )
119oveq2d 5874 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( A G ( A P m ) )  =  ( A G ( A P ( k  +  1 ) ) ) )
1210, 11eqeq12d 2297 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( ( A P m ) G A )  =  ( A G ( A P m ) )  <->  ( ( A P ( k  +  1 ) ) G A )  =  ( A G ( A P ( k  +  1 ) ) ) ) )
13 oveq2 5866 . . . . 5  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
1413oveq1d 5873 . . . 4  |-  ( m  =  -u k  ->  (
( A P m ) G A )  =  ( ( A P -u k ) G A ) )
1513oveq2d 5874 . . . 4  |-  ( m  =  -u k  ->  ( A G ( A P m ) )  =  ( A G ( A P -u k
) ) )
1614, 15eqeq12d 2297 . . 3  |-  ( m  =  -u k  ->  (
( ( A P m ) G A )  =  ( A G ( A P m ) )  <->  ( ( A P -u k ) G A )  =  ( A G ( A P -u k
) ) ) )
17 oveq2 5866 . . . . 5  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
1817oveq1d 5873 . . . 4  |-  ( m  =  K  ->  (
( A P m ) G A )  =  ( ( A P K ) G A ) )
1917oveq2d 5874 . . . 4  |-  ( m  =  K  ->  ( A G ( A P m ) )  =  ( A G ( A P K ) ) )
2018, 19eqeq12d 2297 . . 3  |-  ( m  =  K  ->  (
( ( A P m ) G A )  =  ( A G ( A P m ) )  <->  ( ( A P K ) G A )  =  ( A G ( A P K ) ) ) )
21 gxcom.1 . . . . 5  |-  X  =  ran  G
22 eqid 2283 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
2321, 22grpolid 20886 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
24 gxcom.2 . . . . . 6  |-  P  =  ( ^g `  G
)
2521, 22, 24gx0 20928 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625oveq1d 5873 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( (GId `  G ) G A ) )
2725oveq2d 5874 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  ( A G (GId
`  G ) ) )
2821, 22grporid 20887 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
2927, 28eqtrd 2315 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  A )
3023, 26, 293eqtr4d 2325 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( A G ( A P 0 ) ) )
31 nn0z 10046 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  ZZ )
32 simp1 955 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  G  e.  GrpOp )
33 simp2 956 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  A  e.  X )
3421, 24gxcl 20932 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P k )  e.  X )
3521grpoass 20870 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  ( A P k )  e.  X  /\  A  e.  X ) )  -> 
( ( A G ( A P k ) ) G A )  =  ( A G ( ( A P k ) G A ) ) )
3632, 33, 34, 33, 35syl13anc 1184 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  (
( A G ( A P k ) ) G A )  =  ( A G ( ( A P k ) G A ) ) )
3731, 36syl3an3 1217 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A G ( A P k ) ) G A )  =  ( A G ( ( A P k ) G A ) ) )
3837adantr 451 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A G ( A P k ) ) G A )  =  ( A G ( ( A P k ) G A ) ) )
3921, 24gxnn0suc 20931 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
4039eqeq1d 2291 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P ( k  +  1 ) )  =  ( A G ( A P k ) )  <->  ( ( A P k ) G A )  =  ( A G ( A P k ) ) ) )
4140biimpar 471 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A P ( k  +  1 ) )  =  ( A G ( A P k ) ) )
4241oveq1d 5873 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A P ( k  +  1 ) ) G A )  =  ( ( A G ( A P k ) ) G A ) )
4339oveq2d 5874 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A G ( A P ( k  +  1 ) ) )  =  ( A G ( ( A P k ) G A ) ) )
4443adantr 451 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A G ( A P ( k  +  1 ) ) )  =  ( A G ( ( A P k ) G A ) ) )
4538, 42, 443eqtr4d 2325 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A P ( k  +  1 ) ) G A )  =  ( A G ( A P ( k  +  1 ) ) ) )
4645ex 423 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( ( A P k ) G A )  =  ( A G ( A P k ) )  -> 
( ( A P ( k  +  1 ) ) G A )  =  ( A G ( A P ( k  +  1 ) ) ) ) )
47463expia 1153 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN0  ->  ( ( ( A P k ) G A )  =  ( A G ( A P k ) )  -> 
( ( A P ( k  +  1 ) ) G A )  =  ( A G ( A P ( k  +  1 ) ) ) ) ) )
48 nnz 10045 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
49 simpl1 958 . . . . . . . . . 10  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  G  e.  GrpOp )
50 simpl2 959 . . . . . . . . . 10  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  A  e.  X
)
51 znegcl 10055 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  -u k  e.  ZZ )
5221, 24gxcl 20932 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  -u k  e.  ZZ )  ->  ( A P -u k )  e.  X )
5351, 52syl3an3 1217 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  e.  X )
5453adantr 451 . . . . . . . . . 10  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A P
-u k )  e.  X )
55 eqid 2283 . . . . . . . . . . . 12  |-  ( inv `  G )  =  ( inv `  G )
5621, 55grpoinvcl 20893 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( inv `  G
) `  A )  e.  X )
5749, 50, 56syl2anc 642 . . . . . . . . . 10  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( inv `  G ) `  A
)  e.  X )
5821grpoass 20870 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  ( A P -u k
)  e.  X  /\  ( ( inv `  G
) `  A )  e.  X ) )  -> 
( ( A G ( A P -u k ) ) G ( ( inv `  G
) `  A )
)  =  ( A G ( ( A P -u k ) G ( ( inv `  G ) `  A
) ) ) )
5949, 50, 54, 57, 58syl13anc 1184 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A G ( A P
-u k ) ) G ( ( inv `  G ) `  A
) )  =  ( A G ( ( A P -u k
) G ( ( inv `  G ) `
 A ) ) ) )
6021, 55, 24gxneg 20933 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
6160adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A P
-u k )  =  ( ( inv `  G
) `  ( A P k ) ) )
6261oveq1d 5873 . . . . . . . . . . . 12  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A P -u k ) G ( ( inv `  G ) `  A
) )  =  ( ( ( inv `  G
) `  ( A P k ) ) G ( ( inv `  G ) `  A
) ) )
6334adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A P k )  e.  X
)
6421, 55grpoinvop 20908 . . . . . . . . . . . . 13  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A P k )  e.  X )  ->  (
( inv `  G
) `  ( A G ( A P k ) ) )  =  ( ( ( inv `  G ) `
 ( A P k ) ) G ( ( inv `  G
) `  A )
) )
6549, 50, 63, 64syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( inv `  G ) `  ( A G ( A P k ) ) )  =  ( ( ( inv `  G ) `
 ( A P k ) ) G ( ( inv `  G
) `  A )
) )
66 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( ( ( A P k ) G A )  =  ( A G ( A P k ) )  ->  (
( inv `  G
) `  ( ( A P k ) G A ) )  =  ( ( inv `  G
) `  ( A G ( A P k ) ) ) )
6766adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( inv `  G ) `  (
( A P k ) G A ) )  =  ( ( inv `  G ) `
 ( A G ( A P k ) ) ) )
6821, 55grpoinvop 20908 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X  /\  A  e.  X )  ->  (
( inv `  G
) `  ( ( A P k ) G A ) )  =  ( ( ( inv `  G ) `  A
) G ( ( inv `  G ) `
 ( A P k ) ) ) )
6949, 63, 50, 68syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( inv `  G ) `  (
( A P k ) G A ) )  =  ( ( ( inv `  G
) `  A ) G ( ( inv `  G ) `  ( A P k ) ) ) )
7061oveq2d 5874 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( ( inv `  G ) `
 A ) G ( A P -u k ) )  =  ( ( ( inv `  G ) `  A
) G ( ( inv `  G ) `
 ( A P k ) ) ) )
7169, 70eqtr4d 2318 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( inv `  G ) `  (
( A P k ) G A ) )  =  ( ( ( inv `  G
) `  A ) G ( A P
-u k ) ) )
7267, 71eqtr3d 2317 . . . . . . . . . . . 12  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( inv `  G ) `  ( A G ( A P k ) ) )  =  ( ( ( inv `  G ) `
 A ) G ( A P -u k ) ) )
7362, 65, 723eqtr2d 2321 . . . . . . . . . . 11  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A P -u k ) G ( ( inv `  G ) `  A
) )  =  ( ( ( inv `  G
) `  A ) G ( A P
-u k ) ) )
7473oveq2d 5874 . . . . . . . . . 10  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A G ( ( A P
-u k ) G ( ( inv `  G
) `  A )
) )  =  ( A G ( ( ( inv `  G
) `  A ) G ( A P
-u k ) ) ) )
7521, 55grpoasscan1 20904 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A P -u k )  e.  X )  -> 
( A G ( ( ( inv `  G
) `  A ) G ( A P
-u k ) ) )  =  ( A P -u k ) )
7649, 50, 54, 75syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A G ( ( ( inv `  G ) `  A
) G ( A P -u k ) ) )  =  ( A P -u k
) )
7774, 76eqtrd 2315 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A G ( ( A P
-u k ) G ( ( inv `  G
) `  A )
) )  =  ( A P -u k
) )
7859, 77eqtrd 2315 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A G ( A P
-u k ) ) G ( ( inv `  G ) `  A
) )  =  ( A P -u k
) )
7978oveq1d 5873 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( ( A G ( A P -u k ) ) G ( ( inv `  G ) `
 A ) ) G A )  =  ( ( A P
-u k ) G A ) )
8021grpocl 20867 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A P -u k )  e.  X )  -> 
( A G ( A P -u k
) )  e.  X
)
8149, 50, 54, 80syl3anc 1182 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( A G ( A P -u k ) )  e.  X )
8221, 55grpoasscan2 20905 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A G ( A P
-u k ) )  e.  X  /\  A  e.  X )  ->  (
( ( A G ( A P -u k ) ) G ( ( inv `  G
) `  A )
) G A )  =  ( A G ( A P -u k ) ) )
8349, 81, 50, 82syl3anc 1182 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( ( A G ( A P -u k ) ) G ( ( inv `  G ) `
 A ) ) G A )  =  ( A G ( A P -u k
) ) )
8479, 83eqtr3d 2317 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  /\  ( ( A P k ) G A )  =  ( A G ( A P k ) ) )  ->  ( ( A P -u k ) G A )  =  ( A G ( A P -u k
) ) )
8584ex 423 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  (
( ( A P k ) G A )  =  ( A G ( A P k ) )  -> 
( ( A P
-u k ) G A )  =  ( A G ( A P -u k ) ) ) )
86853expia 1153 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  ZZ  ->  ( ( ( A P k ) G A )  =  ( A G ( A P k ) )  -> 
( ( A P
-u k ) G A )  =  ( A G ( A P -u k ) ) ) ) )
8748, 86syl5 28 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN  ->  ( ( ( A P k ) G A )  =  ( A G ( A P k ) )  -> 
( ( A P
-u k ) G A )  =  ( A G ( A P -u k ) ) ) ) )
884, 8, 12, 16, 20, 30, 47, 87zindd 10113 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( ( A P K ) G A )  =  ( A G ( A P K ) ) ) )
89883impia 1148 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  (
( A P K ) G A )  =  ( A G ( A P K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   ^gcgx 20857
This theorem is referenced by:  gxinv  20937  gxsuc  20939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gx 20862
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