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Theorem gxnn0mul 20960
Description: The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 20961 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 5882 . . . . . . . . 9  |-  ( m  =  0  ->  ( J  x.  m )  =  ( J  x.  0 ) )
21oveq2d 5890 . . . . . . . 8  |-  ( m  =  0  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  0 ) ) )
3 oveq2 5882 . . . . . . . 8  |-  ( m  =  0  ->  (
( A P J ) P m )  =  ( ( A P J ) P 0 ) )
42, 3eqeq12d 2310 . . . . . . 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 5882 . . . . . . . . 9  |-  ( m  =  k  ->  ( J  x.  m )  =  ( J  x.  k ) )
76oveq2d 5890 . . . . . . . 8  |-  ( m  =  k  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  k ) ) )
8 oveq2 5882 . . . . . . . 8  |-  ( m  =  k  ->  (
( A P J ) P m )  =  ( ( A P J ) P k ) )
97, 8eqeq12d 2310 . . . . . . 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 5882 . . . . . . . . 9  |-  ( m  =  ( k  +  1 )  ->  ( J  x.  m )  =  ( J  x.  ( k  +  1 ) ) )
1211oveq2d 5890 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  ( k  +  1 ) ) ) )
13 oveq2 5882 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  (
( A P J ) P m )  =  ( ( A P J ) P ( k  +  1 ) ) )
1412, 13eqeq12d 2310 . . . . . . 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 5882 . . . . . . . . 9  |-  ( m  =  K  ->  ( J  x.  m )  =  ( J  x.  K ) )
1716oveq2d 5890 . . . . . . . 8  |-  ( m  =  K  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  K ) ) )
18 oveq2 5882 . . . . . . . 8  |-  ( m  =  K  ->  (
( A P J ) P m )  =  ( ( A P J ) P K ) )
1917, 18eqeq12d 2310 . . . . . . 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 2296 . . . . . . . . 9  |-  (GId `  G )  =  (GId
`  G )
23 gxnn0mul.2 . . . . . . . . 9  |-  P  =  ( ^g `  G
)
2421, 22, 23gx0 20944 . . . . . . . 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 10045 . . . . . . . . . 10  |-  ( J  e.  ZZ  ->  J  e.  CC )
2726mul01d 9027 . . . . . . . . 9  |-  ( J  e.  ZZ  ->  ( J  x.  0 )  =  0 )
2827oveq2d 5890 . . . . . . . 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 20948 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P J )  e.  X )
3221, 22, 23gx0 20944 . . . . . . . 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 2338 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) )
35 oveq1 5881 . . . . . . . . . . . . . 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 9991 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 8811 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
39 adddi 8842 . . . . . . . . . . . . . . . . . . . 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 8851 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  CC  ->  ( J  x.  1 )  =  J )
4241oveq2d 5890 . . . . . . . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . . . . . . . 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 5890 . . . . . . . . . . . . . . . 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 10062 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  ZZ )
49 zmulcl 10082 . . . . . . . . . . . . . . . . . 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 20958 . . . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . . . 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 10131 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
k  e.  ZZ )
6121, 23gxsuc 20955 . . . . . . . . . . . . . . 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 2338 . . . . . . . . . . . 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 10124 . . . . 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 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   NN0cn0 9981   ZZcz 10040   GrpOpcgr 20869  GIdcgi 20870   ^gcgx 20873
This theorem is referenced by:  gxmul  20961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gx 20878
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