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Theorem gxdi 24971
Description: Distribution of group power over group operation for abelian groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxdi.1  |-  X  =  ran  G
gxdi.2  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxdi  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X )  /\  K  e.  ZZ )  ->  ( ( A G B ) P K )  =  ( ( A P K ) G ( B P K ) ) )

Proof of Theorem gxdi
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6290 . . . . 5  |-  ( m  =  0  ->  (
( A G B ) P m )  =  ( ( A G B ) P 0 ) )
2 oveq2 6290 . . . . . 6  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
3 oveq2 6290 . . . . . 6  |-  ( m  =  0  ->  ( B P m )  =  ( B P 0 ) )
42, 3oveq12d 6300 . . . . 5  |-  ( m  =  0  ->  (
( A P m ) G ( B P m ) )  =  ( ( A P 0 ) G ( B P 0 ) ) )
51, 4eqeq12d 2489 . . . 4  |-  ( m  =  0  ->  (
( ( A G B ) P m )  =  ( ( A P m ) G ( B P m ) )  <->  ( ( A G B ) P 0 )  =  ( ( A P 0 ) G ( B P 0 ) ) ) )
6 oveq2 6290 . . . . 5  |-  ( m  =  k  ->  (
( A G B ) P m )  =  ( ( A G B ) P k ) )
7 oveq2 6290 . . . . . 6  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
8 oveq2 6290 . . . . . 6  |-  ( m  =  k  ->  ( B P m )  =  ( B P k ) )
97, 8oveq12d 6300 . . . . 5  |-  ( m  =  k  ->  (
( A P m ) G ( B P m ) )  =  ( ( A P k ) G ( B P k ) ) )
106, 9eqeq12d 2489 . . . 4  |-  ( m  =  k  ->  (
( ( A G B ) P m )  =  ( ( A P m ) G ( B P m ) )  <->  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) ) )
11 oveq2 6290 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  (
( A G B ) P m )  =  ( ( A G B ) P ( k  +  1 ) ) )
12 oveq2 6290 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
13 oveq2 6290 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  ( B P m )  =  ( B P ( k  +  1 ) ) )
1412, 13oveq12d 6300 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  (
( A P m ) G ( B P m ) )  =  ( ( A P ( k  +  1 ) ) G ( B P ( k  +  1 ) ) ) )
1511, 14eqeq12d 2489 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( ( A G B ) P m )  =  ( ( A P m ) G ( B P m ) )  <->  ( ( A G B ) P ( k  +  1 ) )  =  ( ( A P ( k  +  1 ) ) G ( B P ( k  +  1 ) ) ) ) )
16 oveq2 6290 . . . . 5  |-  ( m  =  -u k  ->  (
( A G B ) P m )  =  ( ( A G B ) P
-u k ) )
17 oveq2 6290 . . . . . 6  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
18 oveq2 6290 . . . . . 6  |-  ( m  =  -u k  ->  ( B P m )  =  ( B P -u k ) )
1917, 18oveq12d 6300 . . . . 5  |-  ( m  =  -u k  ->  (
( A P m ) G ( B P m ) )  =  ( ( A P -u k ) G ( B P
-u k ) ) )
2016, 19eqeq12d 2489 . . . 4  |-  ( m  =  -u k  ->  (
( ( A G B ) P m )  =  ( ( A P m ) G ( B P m ) )  <->  ( ( A G B ) P
-u k )  =  ( ( A P
-u k ) G ( B P -u k ) ) ) )
21 oveq2 6290 . . . . 5  |-  ( m  =  K  ->  (
( A G B ) P m )  =  ( ( A G B ) P K ) )
22 oveq2 6290 . . . . . 6  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
23 oveq2 6290 . . . . . 6  |-  ( m  =  K  ->  ( B P m )  =  ( B P K ) )
2422, 23oveq12d 6300 . . . . 5  |-  ( m  =  K  ->  (
( A P m ) G ( B P m ) )  =  ( ( A P K ) G ( B P K ) ) )
2521, 24eqeq12d 2489 . . . 4  |-  ( m  =  K  ->  (
( ( A G B ) P m )  =  ( ( A P m ) G ( B P m ) )  <->  ( ( A G B ) P K )  =  ( ( A P K ) G ( B P K ) ) ) )
26 ablogrpo 24959 . . . . . . . 8  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
27263ad2ant1 1017 . . . . . . 7  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
28 gxdi.1 . . . . . . . . 9  |-  X  =  ran  G
2928grpocl 24875 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
3026, 29syl3an1 1261 . . . . . . 7  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
31 eqid 2467 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
32 gxdi.2 . . . . . . . 8  |-  P  =  ( ^g `  G
)
3328, 31, 32gx0 24936 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A G B )  e.  X )  ->  (
( A G B ) P 0 )  =  (GId `  G
) )
3427, 30, 33syl2anc 661 . . . . . 6  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) P 0 )  =  (GId `  G
) )
3528, 31grpoidcl 24892 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  (GId `  G
)  e.  X )
3627, 35syl 16 . . . . . . 7  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (GId `  G )  e.  X
)
3728, 31grpolid 24894 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  (GId `  G )  e.  X
)  ->  ( (GId `  G ) G (GId
`  G ) )  =  (GId `  G
) )
3827, 36, 37syl2anc 661 . . . . . 6  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G (GId `  G )
)  =  (GId `  G ) )
3934, 38eqtr4d 2511 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) P 0 )  =  ( (GId `  G ) G (GId
`  G ) ) )
40 simp2 997 . . . . . . 7  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
4128, 31, 32gx0 24936 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
4227, 40, 41syl2anc 661 . . . . . 6  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
43 simp3 998 . . . . . . 7  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
4428, 31, 32gx0 24936 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B P 0 )  =  (GId `  G )
)
4527, 43, 44syl2anc 661 . . . . . 6  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B P 0 )  =  (GId `  G )
)
4642, 45oveq12d 6300 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A P 0 ) G ( B P 0 ) )  =  ( (GId `  G ) G (GId
`  G ) ) )
4739, 46eqtr4d 2511 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) P 0 )  =  ( ( A P 0 ) G ( B P 0 ) ) )
48 nn0z 10883 . . . . 5  |-  ( k  e.  NN0  ->  k  e.  ZZ )
49273ad2ant1 1017 . . . . . . . . 9  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  G  e.  GrpOp )
50303ad2ant1 1017 . . . . . . . . 9  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( A G B )  e.  X
)
51 simp2 997 . . . . . . . . 9  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  k  e.  ZZ )
5228, 32gxsuc 24947 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A G B )  e.  X  /\  k  e.  ZZ )  ->  (
( A G B ) P ( k  +  1 ) )  =  ( ( ( A G B ) P k ) G ( A G B ) ) )
5349, 50, 51, 52syl3anc 1228 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A G B ) P ( k  +  1 ) )  =  ( ( ( A G B ) P k ) G ( A G B ) ) )
54 oveq1 6289 . . . . . . . . 9  |-  ( ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) )  ->  (
( ( A G B ) P k ) G ( A G B ) )  =  ( ( ( A P k ) G ( B P k ) ) G ( A G B ) ) )
55543ad2ant3 1019 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( ( A G B ) P k ) G ( A G B ) )  =  ( ( ( A P k ) G ( B P k ) ) G ( A G B ) ) )
56 simp11 1026 . . . . . . . . 9  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  G  e.  AbelOp )
57403ad2ant1 1017 . . . . . . . . . 10  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  A  e.  X
)
5828, 32gxcl 24940 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P k )  e.  X )
5949, 57, 51, 58syl3anc 1228 . . . . . . . . 9  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( A P k )  e.  X
)
60433ad2ant1 1017 . . . . . . . . . 10  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  B  e.  X
)
6128, 32gxcl 24940 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  k  e.  ZZ )  ->  ( B P k )  e.  X )
6249, 60, 51, 61syl3anc 1228 . . . . . . . . 9  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( B P k )  e.  X
)
6328ablo4 24962 . . . . . . . . 9  |-  ( ( G  e.  AbelOp  /\  (
( A P k )  e.  X  /\  ( B P k )  e.  X )  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( A P k ) G ( B P k ) ) G ( A G B ) )  =  ( ( ( A P k ) G A ) G ( ( B P k ) G B ) ) )
6456, 59, 62, 57, 60, 63syl122anc 1237 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( ( A P k ) G ( B P k ) ) G ( A G B ) )  =  ( ( ( A P k ) G A ) G ( ( B P k ) G B ) ) )
6553, 55, 643eqtrd 2512 . . . . . . 7  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A G B ) P ( k  +  1 ) )  =  ( ( ( A P k ) G A ) G ( ( B P k ) G B ) ) )
6628, 32gxsuc 24947 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
6749, 57, 51, 66syl3anc 1228 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
6828, 32gxsuc 24947 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  k  e.  ZZ )  ->  ( B P ( k  +  1 ) )  =  ( ( B P k ) G B ) )
6949, 60, 51, 68syl3anc 1228 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( B P ( k  +  1 ) )  =  ( ( B P k ) G B ) )
7067, 69oveq12d 6300 . . . . . . 7  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A P ( k  +  1 ) ) G ( B P ( k  +  1 ) ) )  =  ( ( ( A P k ) G A ) G ( ( B P k ) G B ) ) )
7165, 70eqtr4d 2511 . . . . . 6  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A G B ) P ( k  +  1 ) )  =  ( ( A P ( k  +  1 ) ) G ( B P ( k  +  1 ) ) ) )
72713exp 1195 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
k  e.  ZZ  ->  ( ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) )  -> 
( ( A G B ) P ( k  +  1 ) )  =  ( ( A P ( k  +  1 ) ) G ( B P ( k  +  1 ) ) ) ) ) )
7348, 72syl5 32 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
k  e.  NN0  ->  ( ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) )  -> 
( ( A G B ) P ( k  +  1 ) )  =  ( ( A P ( k  +  1 ) ) G ( B P ( k  +  1 ) ) ) ) ) )
74 nnz 10882 . . . . 5  |-  ( k  e.  NN  ->  k  e.  ZZ )
75 eqid 2467 . . . . . . . . . 10  |-  ( inv `  G )  =  ( inv `  G )
7628, 75, 32gxneg 24941 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A G B )  e.  X  /\  k  e.  ZZ )  ->  (
( A G B ) P -u k
)  =  ( ( inv `  G ) `
 ( ( A G B ) P k ) ) )
7749, 50, 51, 76syl3anc 1228 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A G B ) P
-u k )  =  ( ( inv `  G
) `  ( ( A G B ) P k ) ) )
7828ablocom 24960 . . . . . . . . . . 11  |-  ( ( G  e.  AbelOp  /\  ( A P k )  e.  X  /\  ( B P k )  e.  X )  ->  (
( A P k ) G ( B P k ) )  =  ( ( B P k ) G ( A P k ) ) )
7956, 59, 62, 78syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A P k ) G ( B P k ) )  =  ( ( B P k ) G ( A P k ) ) )
80 eqeq1 2471 . . . . . . . . . . 11  |-  ( ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) )  ->  (
( ( A G B ) P k )  =  ( ( B P k ) G ( A P k ) )  <->  ( ( A P k ) G ( B P k ) )  =  ( ( B P k ) G ( A P k ) ) ) )
81803ad2ant3 1019 . . . . . . . . . 10  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( ( A G B ) P k )  =  ( ( B P k ) G ( A P k ) )  <->  ( ( A P k ) G ( B P k ) )  =  ( ( B P k ) G ( A P k ) ) ) )
8279, 81mpbird 232 . . . . . . . . 9  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A G B ) P k )  =  ( ( B P k ) G ( A P k ) ) )
8382fveq2d 5868 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( inv `  G ) `  (
( A G B ) P k ) )  =  ( ( inv `  G ) `
 ( ( B P k ) G ( A P k ) ) ) )
8428, 75grpoinvop 24916 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B P k )  e.  X  /\  ( A P k )  e.  X )  ->  (
( inv `  G
) `  ( ( B P k ) G ( A P k ) ) )  =  ( ( ( inv `  G ) `  ( A P k ) ) G ( ( inv `  G ) `  ( B P k ) ) ) )
8549, 62, 59, 84syl3anc 1228 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( inv `  G ) `  (
( B P k ) G ( A P k ) ) )  =  ( ( ( inv `  G
) `  ( A P k ) ) G ( ( inv `  G ) `  ( B P k ) ) ) )
8677, 83, 853eqtrd 2512 . . . . . . 7  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A G B ) P
-u k )  =  ( ( ( inv `  G ) `  ( A P k ) ) G ( ( inv `  G ) `  ( B P k ) ) ) )
8728, 75, 32gxneg 24941 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  ZZ )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
8849, 57, 51, 87syl3anc 1228 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( A P
-u k )  =  ( ( inv `  G
) `  ( A P k ) ) )
8928, 75, 32gxneg 24941 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  k  e.  ZZ )  ->  ( B P -u k )  =  ( ( inv `  G ) `  ( B P k ) ) )
9049, 60, 51, 89syl3anc 1228 . . . . . . . 8  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( B P
-u k )  =  ( ( inv `  G
) `  ( B P k ) ) )
9188, 90oveq12d 6300 . . . . . . 7  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A P -u k ) G ( B P
-u k ) )  =  ( ( ( inv `  G ) `
 ( A P k ) ) G ( ( inv `  G
) `  ( B P k ) ) ) )
9286, 91eqtr4d 2511 . . . . . 6  |-  ( ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  ZZ  /\  ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) ) )  ->  ( ( A G B ) P
-u k )  =  ( ( A P
-u k ) G ( B P -u k ) ) )
93923exp 1195 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
k  e.  ZZ  ->  ( ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) )  -> 
( ( A G B ) P -u k )  =  ( ( A P -u k ) G ( B P -u k
) ) ) ) )
9474, 93syl5 32 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
k  e.  NN  ->  ( ( ( A G B ) P k )  =  ( ( A P k ) G ( B P k ) )  -> 
( ( A G B ) P -u k )  =  ( ( A P -u k ) G ( B P -u k
) ) ) ) )
955, 10, 15, 20, 25, 47, 73, 94zindd 10958 . . 3  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( K  e.  ZZ  ->  ( ( A G B ) P K )  =  ( ( A P K ) G ( B P K ) ) ) )
96953expb 1197 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( K  e.  ZZ  ->  ( ( A G B ) P K )  =  ( ( A P K ) G ( B P K ) ) ) )
97963impia 1193 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X )  /\  K  e.  ZZ )  ->  ( ( A G B ) P K )  =  ( ( A P K ) G ( B P K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ran crn 5000   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491   -ucneg 9802   NNcn 10532   NN0cn0 10791   ZZcz 10860   GrpOpcgr 24861  GIdcgi 24862   invcgn 24863   ^gcgx 24865   AbelOpcablo 24956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-seq 12071  df-grpo 24866  df-gid 24867  df-ginv 24868  df-gx 24870  df-ablo 24957
This theorem is referenced by: (None)
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