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Theorem gxnn0mul 21865
Description: The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 21866 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 6089 . . . . . . . . 9  |-  ( m  =  0  ->  ( J  x.  m )  =  ( J  x.  0 ) )
21oveq2d 6097 . . . . . . . 8  |-  ( m  =  0  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  0 ) ) )
3 oveq2 6089 . . . . . . . 8  |-  ( m  =  0  ->  (
( A P J ) P m )  =  ( ( A P J ) P 0 ) )
42, 3eqeq12d 2450 . . . . . . 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 6089 . . . . . . . . 9  |-  ( m  =  k  ->  ( J  x.  m )  =  ( J  x.  k ) )
76oveq2d 6097 . . . . . . . 8  |-  ( m  =  k  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  k ) ) )
8 oveq2 6089 . . . . . . . 8  |-  ( m  =  k  ->  (
( A P J ) P m )  =  ( ( A P J ) P k ) )
97, 8eqeq12d 2450 . . . . . . 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 6089 . . . . . . . . 9  |-  ( m  =  ( k  +  1 )  ->  ( J  x.  m )  =  ( J  x.  ( k  +  1 ) ) )
1211oveq2d 6097 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  ( k  +  1 ) ) ) )
13 oveq2 6089 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  (
( A P J ) P m )  =  ( ( A P J ) P ( k  +  1 ) ) )
1412, 13eqeq12d 2450 . . . . . . 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 6089 . . . . . . . . 9  |-  ( m  =  K  ->  ( J  x.  m )  =  ( J  x.  K ) )
1716oveq2d 6097 . . . . . . . 8  |-  ( m  =  K  ->  ( A P ( J  x.  m ) )  =  ( A P ( J  x.  K ) ) )
18 oveq2 6089 . . . . . . . 8  |-  ( m  =  K  ->  (
( A P J ) P m )  =  ( ( A P J ) P K ) )
1917, 18eqeq12d 2450 . . . . . . 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 2436 . . . . . . . . 9  |-  (GId `  G )  =  (GId
`  G )
23 gxnn0mul.2 . . . . . . . . 9  |-  P  =  ( ^g `  G
)
2421, 22, 23gx0 21849 . . . . . . . 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 10287 . . . . . . . . . 10  |-  ( J  e.  ZZ  ->  J  e.  CC )
2726mul01d 9265 . . . . . . . . 9  |-  ( J  e.  ZZ  ->  ( J  x.  0 )  =  0 )
2827oveq2d 6097 . . . . . . . 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 21853 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P J )  e.  X )
3221, 22, 23gx0 21849 . . . . . . . 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 2478 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  J  e.  ZZ )  ->  ( A P ( J  x.  0 ) )  =  ( ( A P J ) P 0 ) )
35 oveq1 6088 . . . . . . . . . . . . . 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 10231 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  CC )
38 ax-1cn 9048 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
39 adddi 9079 . . . . . . . . . . . . . . . . . . . 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 9088 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J  e.  CC  ->  ( J  x.  1 )  =  J )
4241oveq2d 6097 . . . . . . . . . . . . . . . . . . . 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 2468 . . . . . . . . . . . . . . . . . 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 6097 . . . . . . . . . . . . . . . 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 10304 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN0  ->  k  e.  ZZ )
49 zmulcl 10324 . . . . . . . . . . . . . . . . . 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 21863 . . . . . . . . . . . . . . . 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 2468 . . . . . . . . . . . . . 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 10373 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  k  e.  NN0 ) )  -> 
k  e.  ZZ )
6121, 23gxsuc 21860 . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 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 10366 . . . . 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 1652    e. wcel 1725   ran crn 4879   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   NN0cn0 10221   ZZcz 10282   GrpOpcgr 21774  GIdcgi 21775   ^gcgx 21778
This theorem is referenced by:  gxmul  21866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gx 21783
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