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Theorem gxnn0mul 21715
Description: The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 21716 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 6030 . . . . . . . . 9  |-  ( m  =  0  ->  ( J  x.  m )  =  ( J  x.  0 ) )
21oveq2d 6038 . . . . . . . 8  |-  ( m  =  0  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  0 ) ) )
3 oveq2 6030 . . . . . . . 8  |-  ( m  =  0  ->  (
( A P J ) P m )  =  ( ( A P J ) P 0 ) )
42, 3eqeq12d 2403 . . . . . . 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 6030 . . . . . . . . 9  |-  ( m  =  k  ->  ( J  x.  m )  =  ( J  x.  k ) )
76oveq2d 6038 . . . . . . . 8  |-  ( m  =  k  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  k ) ) )
8 oveq2 6030 . . . . . . . 8  |-  ( m  =  k  ->  (
( A P J ) P m )  =  ( ( A P J ) P k ) )
97, 8eqeq12d 2403 . . . . . . 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 6030 . . . . . . . . 9  |-  ( m  =  ( k  +  1 )  ->  ( J  x.  m )  =  ( J  x.  ( k  +  1 ) ) )
1211oveq2d 6038 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  ( k  +  1 ) ) ) )
13 oveq2 6030 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  (
( A P J ) P m )  =  ( ( A P J ) P ( k  +  1 ) ) )
1412, 13eqeq12d 2403 . . . . . . 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 6030 . . . . . . . . 9  |-  ( m  =  K  ->  ( J  x.  m )  =  ( J  x.  K ) )
1716oveq2d 6038 . . . . . . . 8  |-  ( m  =  K  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  K ) ) )
18 oveq2 6030 . . . . . . . 8  |-  ( m  =  K  ->  (
( A P J ) P m )  =  ( ( A P J ) P K ) )
1917, 18eqeq12d 2403 . . . . . . 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 2389 . . . . . . . . 9  |-  (GId `  G )  =  (GId
`  G )
23 gxnn0mul.2 . . . . . . . . 9  |-  P  =  ( ^g `  G
)
2421, 22, 23gx0 21699 . . . . . . . 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 10221 . . . . . . . . . 10  |-  ( J  e.  ZZ  ->  J  e.  CC )
2726mul01d 9199 . . . . . . . . 9  |-  ( J  e.  ZZ  ->  ( J  x.  0 )  =  0 )
2827oveq2d 6038 . . . . . . . 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 21703 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P J )  e.  X )
3221, 22, 23gx0 21699 . . . . . . . 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 2431 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) )
35 oveq1 6029 . . . . . . . . . . . . . 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 10165 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 8983 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
39 adddi 9014 . . . . . . . . . . . . . . . . . . . 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 9023 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  CC  ->  ( J  x.  1 )  =  J )
4241oveq2d 6038 . . . . . . . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . . . . 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 6038 . . . . . . . . . . . . . . . 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 10238 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  ZZ )
49 zmulcl 10258 . . . . . . . . . . . . . . . . . 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 21713 . . . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . 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 10307 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
k  e.  ZZ )
6121, 23gxsuc 21710 . . . . . . . . . . . . . . 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 2431 . . . . . . . . . . . 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 10300 . . . . 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 1717   ran crn 4821   ` cfv 5396  (class class class)co 6022   CCcc 8923   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930   NN0cn0 10155   ZZcz 10216   GrpOpcgr 21624  GIdcgi 21625   ^gcgx 21628
This theorem is referenced by:  gxmul  21716
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-seq 11253  df-grpo 21629  df-gid 21630  df-ginv 21631  df-gx 21633
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