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Theorem gxnn0mul 20944
Description: The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 20945 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxnn0mul.1  |-  X  =  ran  G
gxnn0mul.2  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxnn0mul  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  NN0 ) )  -> 
( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) )

Proof of Theorem gxnn0mul
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . . . . . 9  |-  ( m  =  0  ->  ( J  x.  m )  =  ( J  x.  0 ) )
21oveq2d 5874 . . . . . . . 8  |-  ( m  =  0  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  0 ) ) )
3 oveq2 5866 . . . . . . . 8  |-  ( m  =  0  ->  (
( A P J ) P m )  =  ( ( A P J ) P 0 ) )
42, 3eqeq12d 2297 . . . . . . 7  |-  ( m  =  0  ->  (
( A P ( J  x.  m ) )  =  ( ( A P J ) P m )  <->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) ) )
54imbi2d 307 . . . . . 6  |-  ( m  =  0  ->  (
( ( G  e. 
GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  m
) )  =  ( ( A P J ) P m ) )  <->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) ) ) )
6 oveq2 5866 . . . . . . . . 9  |-  ( m  =  k  ->  ( J  x.  m )  =  ( J  x.  k ) )
76oveq2d 5874 . . . . . . . 8  |-  ( m  =  k  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  k ) ) )
8 oveq2 5866 . . . . . . . 8  |-  ( m  =  k  ->  (
( A P J ) P m )  =  ( ( A P J ) P k ) )
97, 8eqeq12d 2297 . . . . . . 7  |-  ( m  =  k  ->  (
( A P ( J  x.  m ) )  =  ( ( A P J ) P m )  <->  ( A P ( J  x.  k ) )  =  ( ( A P J ) P k ) ) )
109imbi2d 307 . . . . . 6  |-  ( m  =  k  ->  (
( ( G  e. 
GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  m
) )  =  ( ( A P J ) P m ) )  <->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  k
) )  =  ( ( A P J ) P k ) ) ) )
11 oveq2 5866 . . . . . . . . 9  |-  ( m  =  ( k  +  1 )  ->  ( J  x.  m )  =  ( J  x.  ( k  +  1 ) ) )
1211oveq2d 5874 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  ( k  +  1 ) ) ) )
13 oveq2 5866 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  (
( A P J ) P m )  =  ( ( A P J ) P ( k  +  1 ) ) )
1412, 13eqeq12d 2297 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  (
( A P ( J  x.  m ) )  =  ( ( A P J ) P m )  <->  ( A P ( J  x.  ( k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) )
1514imbi2d 307 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  (
( ( G  e. 
GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  m
) )  =  ( ( A P J ) P m ) )  <->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  (
k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) ) )
16 oveq2 5866 . . . . . . . . 9  |-  ( m  =  K  ->  ( J  x.  m )  =  ( J  x.  K ) )
1716oveq2d 5874 . . . . . . . 8  |-  ( m  =  K  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  K ) ) )
18 oveq2 5866 . . . . . . . 8  |-  ( m  =  K  ->  (
( A P J ) P m )  =  ( ( A P J ) P K ) )
1917, 18eqeq12d 2297 . . . . . . 7  |-  ( m  =  K  ->  (
( A P ( J  x.  m ) )  =  ( ( A P J ) P m )  <->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
2019imbi2d 307 . . . . . 6  |-  ( m  =  K  ->  (
( ( G  e. 
GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  m
) )  =  ( ( A P J ) P m ) )  <->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  K
) )  =  ( ( A P J ) P K ) ) ) )
21 gxnn0mul.1 . . . . . . . . 9  |-  X  =  ran  G
22 eqid 2283 . . . . . . . . 9  |-  (GId `  G )  =  (GId
`  G )
23 gxnn0mul.2 . . . . . . . . 9  |-  P  =  ( ^g `  G
)
2421, 22, 23gx0 20928 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
25243adant3 975 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P 0 )  =  (GId `  G )
)
26 zcn 10029 . . . . . . . . . 10  |-  ( J  e.  ZZ  ->  J  e.  CC )
2726mul01d 9011 . . . . . . . . 9  |-  ( J  e.  ZZ  ->  ( J  x.  0 )  =  0 )
2827oveq2d 5874 . . . . . . . 8  |-  ( J  e.  ZZ  ->  ( A P ( J  x.  0 ) )  =  ( A P 0 ) )
29283ad2ant3 978 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( A P 0 ) )
30 simp1 955 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  G  e.  GrpOp )
3121, 23gxcl 20932 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P J )  e.  X )
3221, 22, 23gx0 20928 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A P J )  e.  X )  ->  (
( A P J ) P 0 )  =  (GId `  G
) )
3330, 31, 32syl2anc 642 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  (
( A P J ) P 0 )  =  (GId `  G
) )
3425, 29, 333eqtr4d 2325 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) )
35 oveq1 5865 . . . . . . . . . . . . . 14  |-  ( ( A P ( J  x.  k ) )  =  ( ( A P J ) P k )  ->  (
( A P ( J  x.  k ) ) G ( A P J ) )  =  ( ( ( A P J ) P k ) G ( A P J ) ) )
3635adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 )
)  /\  ( A P ( J  x.  k ) )  =  ( ( A P J ) P k ) )  ->  (
( A P ( J  x.  k ) ) G ( A P J ) )  =  ( ( ( A P J ) P k ) G ( A P J ) ) )
37 nn0cn 9975 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 8795 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
39 adddi 8826 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( J  x.  ( k  +  1 ) )  =  ( ( J  x.  k )  +  ( J  x.  1 ) ) )
4038, 39mp3an3 1266 . . . . . . . . . . . . . . . . . . 19  |-  ( ( J  e.  CC  /\  k  e.  CC )  ->  ( J  x.  (
k  +  1 ) )  =  ( ( J  x.  k )  +  ( J  x.  1 ) ) )
41 mulid1 8835 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  CC  ->  ( J  x.  1 )  =  J )
4241oveq2d 5874 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  CC  ->  (
( J  x.  k
)  +  ( J  x.  1 ) )  =  ( ( J  x.  k )  +  J ) )
4342adantr 451 . . . . . . . . . . . . . . . . . . 19  |-  ( ( J  e.  CC  /\  k  e.  CC )  ->  ( ( J  x.  k )  +  ( J  x.  1 ) )  =  ( ( J  x.  k )  +  J ) )
4440, 43eqtrd 2315 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  CC  /\  k  e.  CC )  ->  ( J  x.  (
k  +  1 ) )  =  ( ( J  x.  k )  +  J ) )
4526, 37, 44syl2an 463 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( J  x.  (
k  +  1 ) )  =  ( ( J  x.  k )  +  J ) )
4645oveq2d 5874 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( A P ( J  x.  ( k  +  1 ) ) )  =  ( A P ( ( J  x.  k )  +  J ) ) )
47463ad2ant3 978 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
( A P ( J  x.  ( k  +  1 ) ) )  =  ( A P ( ( J  x.  k )  +  J ) ) )
48 nn0z 10046 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  ZZ )
49 zmulcl 10066 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  ZZ  /\  k  e.  ZZ )  ->  ( J  x.  k
)  e.  ZZ )
5048, 49sylan2 460 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( J  x.  k
)  e.  ZZ )
51 simpl 443 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  ->  J  e.  ZZ )
5250, 51jca 518 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( ( J  x.  k )  e.  ZZ  /\  J  e.  ZZ ) )
5321, 23gxadd 20942 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  (
( J  x.  k
)  e.  ZZ  /\  J  e.  ZZ )
)  ->  ( A P ( ( J  x.  k )  +  J ) )  =  ( ( A P ( J  x.  k
) ) G ( A P J ) ) )
5452, 53syl3an3 1217 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
( A P ( ( J  x.  k
)  +  J ) )  =  ( ( A P ( J  x.  k ) ) G ( A P J ) ) )
5547, 54eqtrd 2315 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
( A P ( J  x.  ( k  +  1 ) ) )  =  ( ( A P ( J  x.  k ) ) G ( A P J ) ) )
5655adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 )
)  /\  ( A P ( J  x.  k ) )  =  ( ( A P J ) P k ) )  ->  ( A P ( J  x.  ( k  +  1 ) ) )  =  ( ( A P ( J  x.  k
) ) G ( A P J ) ) )
57 simp1 955 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  ->  G  e.  GrpOp )
58313adant3r 1179 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
( A P J )  e.  X )
59 simp3r 984 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
k  e.  NN0 )
6059nn0zd 10115 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
k  e.  ZZ )
6121, 23gxsuc 20939 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  ( A P J )  e.  X  /\  k  e.  ZZ )  ->  (
( A P J ) P ( k  +  1 ) )  =  ( ( ( A P J ) P k ) G ( A P J ) ) )
6257, 58, 60, 61syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
( ( A P J ) P ( k  +  1 ) )  =  ( ( ( A P J ) P k ) G ( A P J ) ) )
6362adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 )
)  /\  ( A P ( J  x.  k ) )  =  ( ( A P J ) P k ) )  ->  (
( A P J ) P ( k  +  1 ) )  =  ( ( ( A P J ) P k ) G ( A P J ) ) )
6436, 56, 633eqtr4d 2325 . . . . . . . . . . . 12  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 )
)  /\  ( A P ( J  x.  k ) )  =  ( ( A P J ) P k ) )  ->  ( A P ( J  x.  ( k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) )
6564ex 423 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
( ( A P ( J  x.  k
) )  =  ( ( A P J ) P k )  ->  ( A P ( J  x.  (
k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) )
66653expia 1153 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( J  e.  ZZ  /\  k  e.  NN0 )  ->  ( ( A P ( J  x.  k
) )  =  ( ( A P J ) P k )  ->  ( A P ( J  x.  (
k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) ) )
6766exp3a 425 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( J  e.  ZZ  ->  ( k  e.  NN0  ->  ( ( A P ( J  x.  k ) )  =  ( ( A P J ) P k )  -> 
( A P ( J  x.  ( k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) ) ) )
68673impia 1148 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  (
k  e.  NN0  ->  ( ( A P ( J  x.  k ) )  =  ( ( A P J ) P k )  -> 
( A P ( J  x.  ( k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) ) )
6968com12 27 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  (
( A P ( J  x.  k ) )  =  ( ( A P J ) P k )  -> 
( A P ( J  x.  ( k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) ) )
7069a2d 23 . . . . . 6  |-  ( k  e.  NN0  ->  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  k ) )  =  ( ( A P J ) P k ) )  ->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  (
k  +  1 ) ) )  =  ( ( A P J ) P ( k  +  1 ) ) ) ) )
715, 10, 15, 20, 34, 70nn0ind 10108 . . . . 5  |-  ( K  e.  NN0  ->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
7271com12 27 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( K  e.  NN0  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
73723expia 1153 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( J  e.  ZZ  ->  ( K  e.  NN0  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) ) )
7473imp3a 420 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( J  e.  ZZ  /\  K  e.  NN0 )  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
75743impia 1148 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  NN0 ) )  -> 
( A P ( J  x.  K ) )  =  ( ( A P J ) 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   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NN0cn0 9965   ZZcz 10024   GrpOpcgr 20853  GIdcgi 20854   ^gcgx 20857
This theorem is referenced by:  gxmul  20945
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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