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Theorem gxcom 25982
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 6309 . . . . 5  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
21oveq1d 6316 . . . 4  |-  ( m  =  0  ->  (
( A P m ) G A )  =  ( ( A P 0 ) G A ) )
31oveq2d 6317 . . . 4  |-  ( m  =  0  ->  ( A G ( A P m ) )  =  ( A G ( A P 0 ) ) )
42, 3eqeq12d 2444 . . 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 6309 . . . . 5  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
65oveq1d 6316 . . . 4  |-  ( m  =  k  ->  (
( A P m ) G A )  =  ( ( A P k ) G A ) )
75oveq2d 6317 . . . 4  |-  ( m  =  k  ->  ( A G ( A P m ) )  =  ( A G ( A P k ) ) )
86, 7eqeq12d 2444 . . 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 6309 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
109oveq1d 6316 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( A P m ) G A )  =  ( ( A P ( k  +  1 ) ) G A ) )
119oveq2d 6317 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( A G ( A P m ) )  =  ( A G ( A P ( k  +  1 ) ) ) )
1210, 11eqeq12d 2444 . . 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 6309 . . . . 5  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
1413oveq1d 6316 . . . 4  |-  ( m  =  -u k  ->  (
( A P m ) G A )  =  ( ( A P -u k ) G A ) )
1513oveq2d 6317 . . . 4  |-  ( m  =  -u k  ->  ( A G ( A P m ) )  =  ( A G ( A P -u k
) ) )
1614, 15eqeq12d 2444 . . 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 6309 . . . . 5  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
1817oveq1d 6316 . . . 4  |-  ( m  =  K  ->  (
( A P m ) G A )  =  ( ( A P K ) G A ) )
1917oveq2d 6317 . . . 4  |-  ( m  =  K  ->  ( A G ( A P m ) )  =  ( A G ( A P K ) ) )
2018, 19eqeq12d 2444 . . 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 2422 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
2321, 22grpolid 25932 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
24 gxcom.2 . . . . . 6  |-  P  =  ( ^g `  G
)
2521, 22, 24gx0 25974 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625oveq1d 6316 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( (GId `  G ) G A ) )
2725oveq2d 6317 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  ( A G (GId
`  G ) ) )
2821, 22grporid 25933 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
2927, 28eqtrd 2463 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  A )
3023, 26, 293eqtr4d 2473 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( A G ( A P 0 ) ) )
31 nn0z 10960 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  ZZ )
32 simp1 1005 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  G  e.  GrpOp )
33 simp2 1006 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  A  e.  X )
3421, 24gxcl 25978 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P k )  e.  X )
3521grpoass 25916 . . . . . . . . 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 1266 . . . . . . . 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 1299 . . . . . . 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 466 . . . . . 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 25977 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
4039eqeq1d 2424 . . . . . . . 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 487 . . . . . . 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 6316 . . . . . 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 6317 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A G ( A P ( k  +  1 ) ) )  =  ( A G ( ( A P k ) G A ) ) )
4443adantr 466 . . . . . 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 2473 . . . . 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 435 . . . 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 1207 . . 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 10959 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
49 simpl1 1008 . . . . . . . . . 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 1009 . . . . . . . . . 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 10972 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  -u k  e.  ZZ )
5221, 24gxcl 25978 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  -u k  e.  ZZ )  ->  ( A P -u k )  e.  X )
5351, 52syl3an3 1299 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  e.  X )
5453adantr 466 . . . . . . . . . 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 2422 . . . . . . . . . . . 12  |-  ( inv `  G )  =  ( inv `  G )
5621, 55grpoinvcl 25939 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( inv `  G
) `  A )  e.  X )
5749, 50, 56syl2anc 665 . . . . . . . . . 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 25916 . . . . . . . . . 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 1266 . . . . . . . . 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 25979 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
6160adantr 466 . . . . . . . . . . . . 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 6316 . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 25954 . . . . . . . . . . . . 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 1264 . . . . . . . . . . . 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 )
) )
6661oveq2d 6317 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) ) )
6721, 55grpoinvop 25954 . . . . . . . . . . . . . 14  |-  ( ( 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 ) ) ) )
6849, 63, 50, 67syl3anc 1264 . . . . . . . . . . . . 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 ( ( inv `  G ) `  ( A P k ) ) ) )
69 fveq2 5877 . . . . . . . . . . . . . 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 ) ) ) )
7069adantl 467 . . . . . . . . . . . . 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 ) ) ) )
7166, 68, 703eqtr2rd 2470 . . . . . . . . . . . 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 ) ) )
7262, 65, 713eqtr2d 2469 . . . . . . . . . . 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 ) ) )
7372oveq2d 6317 . . . . . . . . . 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 ) ) ) )
7421, 55grpoasscan1 25950 . . . . . . . . . . 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 ) )
7549, 50, 54, 74syl3anc 1264 . . . . . . . . . 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
) )
7673, 75eqtrd 2463 . . . . . . . . 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
) )
7759, 76eqtrd 2463 . . . . . . . 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
) )
7877oveq1d 6316 . . . . . . 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 ) )
7921grpocl 25913 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A P -u k )  e.  X )  -> 
( A G ( A P -u k
) )  e.  X
)
8049, 50, 54, 79syl3anc 1264 . . . . . . . 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 )
8121, 55grpoasscan2 25951 . . . . . . . 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 ) ) )
8249, 80, 50, 81syl3anc 1264 . . . . . . 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
) ) )
8378, 82eqtr3d 2465 . . . . . 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
) ) )
8483ex 435 . . . . 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 ) ) ) )
85843expia 1207 . . . 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 ) ) ) ) )
8648, 85syl5 33 . . 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 ) ) ) ) )
874, 8, 12, 16, 20, 30, 47, 86zindd 11036 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( ( A P K ) G A )  =  ( A G ( A P K ) ) ) )
88873impia 1202 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  (
( A P K ) G A )  =  ( A G ( A P K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   ran crn 4850   ` cfv 5597  (class class class)co 6301   0cc0 9539   1c1 9540    + caddc 9542   -ucneg 9861   NNcn 10609   NN0cn0 10869   ZZcz 10937   GrpOpcgr 25899  GIdcgi 25900   invcgn 25901   ^gcgx 25903
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 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-seq 12213  df-grpo 25904  df-gid 25905  df-ginv 25906  df-gx 25908
This theorem is referenced by:  gxinv  25983  gxsuc  25985
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