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Theorem gxcom 20952
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 5882 . . . . 5  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
21oveq1d 5889 . . . 4  |-  ( m  =  0  ->  (
( A P m ) G A )  =  ( ( A P 0 ) G A ) )
31oveq2d 5890 . . . 4  |-  ( m  =  0  ->  ( A G ( A P m ) )  =  ( A G ( A P 0 ) ) )
42, 3eqeq12d 2310 . . 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 5882 . . . . 5  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
65oveq1d 5889 . . . 4  |-  ( m  =  k  ->  (
( A P m ) G A )  =  ( ( A P k ) G A ) )
75oveq2d 5890 . . . 4  |-  ( m  =  k  ->  ( A G ( A P m ) )  =  ( A G ( A P k ) ) )
86, 7eqeq12d 2310 . . 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 5882 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
109oveq1d 5889 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( A P m ) G A )  =  ( ( A P ( k  +  1 ) ) G A ) )
119oveq2d 5890 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( A G ( A P m ) )  =  ( A G ( A P ( k  +  1 ) ) ) )
1210, 11eqeq12d 2310 . . 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 5882 . . . . 5  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
1413oveq1d 5889 . . . 4  |-  ( m  =  -u k  ->  (
( A P m ) G A )  =  ( ( A P -u k ) G A ) )
1513oveq2d 5890 . . . 4  |-  ( m  =  -u k  ->  ( A G ( A P m ) )  =  ( A G ( A P -u k
) ) )
1614, 15eqeq12d 2310 . . 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 5882 . . . . 5  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
1817oveq1d 5889 . . . 4  |-  ( m  =  K  ->  (
( A P m ) G A )  =  ( ( A P K ) G A ) )
1917oveq2d 5890 . . . 4  |-  ( m  =  K  ->  ( A G ( A P m ) )  =  ( A G ( A P K ) ) )
2018, 19eqeq12d 2310 . . 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 2296 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
2321, 22grpolid 20902 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
24 gxcom.2 . . . . . 6  |-  P  =  ( ^g `  G
)
2521, 22, 24gx0 20944 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625oveq1d 5889 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( (GId `  G ) G A ) )
2725oveq2d 5890 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  ( A G (GId
`  G ) ) )
2821, 22grporid 20903 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
2927, 28eqtrd 2328 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  A )
3023, 26, 293eqtr4d 2338 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( A G ( A P 0 ) ) )
31 nn0z 10062 . . . . . . . 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 20948 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P k )  e.  X )
3521grpoass 20886 . . . . . . . . 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 20947 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
4039eqeq1d 2304 . . . . . . . 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 5889 . . . . . 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 5890 . . . . . . 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 2338 . . . . 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 10061 . . . 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 10071 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  -u k  e.  ZZ )
5221, 24gxcl 20948 . . . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( inv `  G )  =  ( inv `  G )
5621, 55grpoinvcl 20909 . . . . . . . . . . 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 20886 . . . . . . . . . 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 20949 . . . . . . . . . . . . . 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 5889 . . . . . . . . . . . 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 20924 . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . 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 20924 . . . . . . . . . . . . . . 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 5890 . . . . . . . . . . . . . 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 2331 . . . . . . . . . . . . 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 2330 . . . . . . . . . . . 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 2334 . . . . . . . . . . 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 5890 . . . . . . . . . 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 20920 . . . . . . . . . . 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 2328 . . . . . . . . 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 2328 . . . . . . . 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 5889 . . . . . . 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 20883 . . . . . . . . 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 20921 . . . . . . . 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 2330 . . . . . 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 10129 . 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 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   -ucneg 9054   NNcn 9762   NN0cn0 9981   ZZcz 10040   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871   ^gcgx 20873
This theorem is referenced by:  gxinv  20953  gxsuc  20955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gx 20878
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