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Theorem gxnn0mul 21826
Description: The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 21827 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 6056 . . . . . . . . 9  |-  ( m  =  0  ->  ( J  x.  m )  =  ( J  x.  0 ) )
21oveq2d 6064 . . . . . . . 8  |-  ( m  =  0  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  0 ) ) )
3 oveq2 6056 . . . . . . . 8  |-  ( m  =  0  ->  (
( A P J ) P m )  =  ( ( A P J ) P 0 ) )
42, 3eqeq12d 2426 . . . . . . 7  |-  ( m  =  0  ->  (
( A P ( J  x.  m ) )  =  ( ( A P J ) P m )  <->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) ) )
54imbi2d 308 . . . . . 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 6056 . . . . . . . . 9  |-  ( m  =  k  ->  ( J  x.  m )  =  ( J  x.  k ) )
76oveq2d 6064 . . . . . . . 8  |-  ( m  =  k  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  k ) ) )
8 oveq2 6056 . . . . . . . 8  |-  ( m  =  k  ->  (
( A P J ) P m )  =  ( ( A P J ) P k ) )
97, 8eqeq12d 2426 . . . . . . 7  |-  ( m  =  k  ->  (
( A P ( J  x.  m ) )  =  ( ( A P J ) P m )  <->  ( A P ( J  x.  k ) )  =  ( ( A P J ) P k ) ) )
109imbi2d 308 . . . . . 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 6056 . . . . . . . . 9  |-  ( m  =  ( k  +  1 )  ->  ( J  x.  m )  =  ( J  x.  ( k  +  1 ) ) )
1211oveq2d 6064 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  ( k  +  1 ) ) ) )
13 oveq2 6056 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  (
( A P J ) P m )  =  ( ( A P J ) P ( k  +  1 ) ) )
1412, 13eqeq12d 2426 . . . . . . 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 308 . . . . . 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 6056 . . . . . . . . 9  |-  ( m  =  K  ->  ( J  x.  m )  =  ( J  x.  K ) )
1716oveq2d 6064 . . . . . . . 8  |-  ( m  =  K  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  K ) ) )
18 oveq2 6056 . . . . . . . 8  |-  ( m  =  K  ->  (
( A P J ) P m )  =  ( ( A P J ) P K ) )
1917, 18eqeq12d 2426 . . . . . . 7  |-  ( m  =  K  ->  (
( A P ( J  x.  m ) )  =  ( ( A P J ) P m )  <->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
2019imbi2d 308 . . . . . 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 2412 . . . . . . . . 9  |-  (GId `  G )  =  (GId
`  G )
23 gxnn0mul.2 . . . . . . . . 9  |-  P  =  ( ^g `  G
)
2421, 22, 23gx0 21810 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
25243adant3 977 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P 0 )  =  (GId `  G )
)
26 zcn 10251 . . . . . . . . . 10  |-  ( J  e.  ZZ  ->  J  e.  CC )
2726mul01d 9229 . . . . . . . . 9  |-  ( J  e.  ZZ  ->  ( J  x.  0 )  =  0 )
2827oveq2d 6064 . . . . . . . 8  |-  ( J  e.  ZZ  ->  ( A P ( J  x.  0 ) )  =  ( A P 0 ) )
29283ad2ant3 980 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( A P 0 ) )
30 simp1 957 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  G  e.  GrpOp )
3121, 23gxcl 21814 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P J )  e.  X )
3221, 22, 23gx0 21810 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A P J )  e.  X )  ->  (
( A P J ) P 0 )  =  (GId `  G
) )
3330, 31, 32syl2anc 643 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  (
( A P J ) P 0 )  =  (GId `  G
) )
3425, 29, 333eqtr4d 2454 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) )
35 oveq1 6055 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 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 10195 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 9012 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
39 adddi 9043 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( J  x.  ( k  +  1 ) )  =  ( ( J  x.  k )  +  ( J  x.  1 ) ) )
4038, 39mp3an3 1268 . . . . . . . . . . . . . . . . . . 19  |-  ( ( J  e.  CC  /\  k  e.  CC )  ->  ( J  x.  (
k  +  1 ) )  =  ( ( J  x.  k )  +  ( J  x.  1 ) ) )
41 mulid1 9052 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  CC  ->  ( J  x.  1 )  =  J )
4241oveq2d 6064 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  CC  ->  (
( J  x.  k
)  +  ( J  x.  1 ) )  =  ( ( J  x.  k )  +  J ) )
4342adantr 452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( J  e.  CC  /\  k  e.  CC )  ->  ( ( J  x.  k )  +  ( J  x.  1 ) )  =  ( ( J  x.  k )  +  J ) )
4440, 43eqtrd 2444 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  CC  /\  k  e.  CC )  ->  ( J  x.  (
k  +  1 ) )  =  ( ( J  x.  k )  +  J ) )
4526, 37, 44syl2an 464 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( J  x.  (
k  +  1 ) )  =  ( ( J  x.  k )  +  J ) )
4645oveq2d 6064 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( A P ( J  x.  ( k  +  1 ) ) )  =  ( A P ( ( J  x.  k )  +  J ) ) )
47463ad2ant3 980 . . . . . . . . . . . . . . 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 10268 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  ZZ )
49 zmulcl 10288 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  ZZ  /\  k  e.  ZZ )  ->  ( J  x.  k
)  e.  ZZ )
5048, 49sylan2 461 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( J  x.  k
)  e.  ZZ )
51 simpl 444 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  ->  J  e.  ZZ )
5250, 51jca 519 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  ZZ  /\  k  e.  NN0 )  -> 
( ( J  x.  k )  e.  ZZ  /\  J  e.  ZZ ) )
5321, 23gxadd 21824 . . . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . 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 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  +  1 ) ) )  =  ( ( A P ( J  x.  k
) ) G ( A P J ) ) )
57 simp1 957 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  ->  G  e.  GrpOp )
58313adant3r 1181 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
( A P J )  e.  X )
59 simp3r 986 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
k  e.  NN0 )
6059nn0zd 10337 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
k  e.  ZZ )
6121, 23gxsuc 21821 . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . 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 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 ) P ( k  +  1 ) )  =  ( ( ( A P J ) P k ) G ( A P J ) ) )
6436, 56, 633eqtr4d 2454 . . . . . . . . . . . 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 424 . . . . . . . . . . 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 1155 . . . . . . . . . 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 426 . . . . . . . . 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 1150 . . . . . . . 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 29 . . . . . . 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 24 . . . . . 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 10330 . . . . 5  |-  ( K  e.  NN0  ->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
7271com12 29 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( K  e.  NN0  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
73723expia 1155 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( J  e.  ZZ  ->  ( K  e.  NN0  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) ) )
7473imp3a 421 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( J  e.  ZZ  /\  K  e.  NN0 )  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) ) )
75743impia 1150 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ran crn 4846   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959   NN0cn0 10185   ZZcz 10246   GrpOpcgr 21735  GIdcgi 21736   ^gcgx 21739
This theorem is referenced by:  gxmul  21827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-seq 11287  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gx 21744
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