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Theorem gxcom 21849
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 6081 . . . . 5  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
21oveq1d 6088 . . . 4  |-  ( m  =  0  ->  (
( A P m ) G A )  =  ( ( A P 0 ) G A ) )
31oveq2d 6089 . . . 4  |-  ( m  =  0  ->  ( A G ( A P m ) )  =  ( A G ( A P 0 ) ) )
42, 3eqeq12d 2449 . . 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 6081 . . . . 5  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
65oveq1d 6088 . . . 4  |-  ( m  =  k  ->  (
( A P m ) G A )  =  ( ( A P k ) G A ) )
75oveq2d 6089 . . . 4  |-  ( m  =  k  ->  ( A G ( A P m ) )  =  ( A G ( A P k ) ) )
86, 7eqeq12d 2449 . . 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 6081 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
109oveq1d 6088 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( A P m ) G A )  =  ( ( A P ( k  +  1 ) ) G A ) )
119oveq2d 6089 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( A G ( A P m ) )  =  ( A G ( A P ( k  +  1 ) ) ) )
1210, 11eqeq12d 2449 . . 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 6081 . . . . 5  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
1413oveq1d 6088 . . . 4  |-  ( m  =  -u k  ->  (
( A P m ) G A )  =  ( ( A P -u k ) G A ) )
1513oveq2d 6089 . . . 4  |-  ( m  =  -u k  ->  ( A G ( A P m ) )  =  ( A G ( A P -u k
) ) )
1614, 15eqeq12d 2449 . . 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 6081 . . . . 5  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
1817oveq1d 6088 . . . 4  |-  ( m  =  K  ->  (
( A P m ) G A )  =  ( ( A P K ) G A ) )
1917oveq2d 6089 . . . 4  |-  ( m  =  K  ->  ( A G ( A P m ) )  =  ( A G ( A P K ) ) )
2018, 19eqeq12d 2449 . . 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 2435 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
2321, 22grpolid 21799 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
24 gxcom.2 . . . . . 6  |-  P  =  ( ^g `  G
)
2521, 22, 24gx0 21841 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625oveq1d 6088 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( (GId `  G ) G A ) )
2725oveq2d 6089 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  ( A G (GId
`  G ) ) )
2821, 22grporid 21800 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
2927, 28eqtrd 2467 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( A P 0 ) )  =  A )
3023, 26, 293eqtr4d 2477 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P 0 ) G A )  =  ( A G ( A P 0 ) ) )
31 nn0z 10296 . . . . . . . 8  |-  ( k  e.  NN0  ->  k  e.  ZZ )
32 simp1 957 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  G  e.  GrpOp )
33 simp2 958 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  A  e.  X )
3421, 24gxcl 21845 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P k )  e.  X )
3521grpoass 21783 . . . . . . . . 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 1186 . . . . . . . 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 1219 . . . . . . 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 452 . . . . . 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 21844 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
4039eqeq1d 2443 . . . . . . . 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 472 . . . . . . 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 6088 . . . . . 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 6089 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A G ( A P ( k  +  1 ) ) )  =  ( A G ( ( A P k ) G A ) ) )
4443adantr 452 . . . . . 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 2477 . . . . 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 424 . . . 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 1155 . . 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 10295 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
49 simpl1 960 . . . . . . . . . 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 961 . . . . . . . . . 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 10305 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  -u k  e.  ZZ )
5221, 24gxcl 21845 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  -u k  e.  ZZ )  ->  ( A P -u k )  e.  X )
5351, 52syl3an3 1219 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  e.  X )
5453adantr 452 . . . . . . . . . 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 2435 . . . . . . . . . . . 12  |-  ( inv `  G )  =  ( inv `  G )
5621, 55grpoinvcl 21806 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( inv `  G
) `  A )  e.  X )
5749, 50, 56syl2anc 643 . . . . . . . . . 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 21783 . . . . . . . . . 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 1186 . . . . . . . . 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 21846 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
6160adantr 452 . . . . . . . . . . . . 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 6088 . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 21821 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 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 6089 . . . . . . . . . . . . 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 21821 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 5720 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 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 2474 . . . . . . . . . . . 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 2473 . . . . . . . . . . 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 6089 . . . . . . . . . 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 21817 . . . . . . . . . . 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 1184 . . . . . . . . . 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 2467 . . . . . . . . 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 2467 . . . . . . . 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 6088 . . . . . . 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 21780 . . . . . . . . 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 1184 . . . . . . . 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 21818 . . . . . . . 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 1184 . . . . . . 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 2469 . . . . . 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 424 . . . . 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 1155 . . . 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 30 . . 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 10363 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( ( A P K ) G A )  =  ( A G ( A P K ) ) ) )
88873impia 1150 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    + caddc 8985   -ucneg 9284   NNcn 9992   NN0cn0 10213   ZZcz 10274   GrpOpcgr 21766  GIdcgi 21767   invcgn 21768   ^gcgx 21770
This theorem is referenced by:  gxinv  21850  gxsuc  21852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gx 21775
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