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Theorem gxnn0suc 25089
Description: Induction on group power (lemma with nonnegative exponent - use gxsuc 25097 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxnn0suc.1  |-  X  =  ran  G
gxnn0suc.2  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxnn0suc  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e. 
NN0 )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )

Proof of Theorem gxnn0suc
StepHypRef Expression
1 elnn0 10809 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 peano2nn 10560 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  +  1 )  e.  NN )
3 gxnn0suc.1 . . . . . . . 8  |-  X  =  ran  G
4 gxnn0suc.2 . . . . . . . 8  |-  P  =  ( ^g `  G
)
53, 4gxpval 25084 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  +  1 )  e.  NN )  -> 
( A P ( K  +  1 ) )  =  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  ( K  +  1
) ) )
62, 5syl3an3 1263 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  ( K  +  1 ) ) )
7 fvconst2g 6125 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( K  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( K  +  1 ) )  =  A )
82, 7sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  X  /\  K  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( K  +  1
) )  =  A )
983adant1 1014 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (
( NN  X.  { A } ) `  ( K  +  1 ) )  =  A )
109oveq2d 6311 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (
(  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 K ) G ( ( NN  X.  { A } ) `  ( K  +  1
) ) )  =  ( (  seq 1
( G ,  ( NN  X.  { A } ) ) `  K ) G A ) )
11 seqp1 12102 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  1
)  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  ( K  +  1
) )  =  ( (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  K
) G ( ( NN  X.  { A } ) `  ( K  +  1 ) ) ) )
12 nnuz 11129 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1311, 12eleq2s 2575 . . . . . . . 8  |-  ( K  e.  NN  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  ( K  +  1
) )  =  ( (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  K
) G ( ( NN  X.  { A } ) `  ( K  +  1 ) ) ) )
14133ad2ant3 1019 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  ( K  +  1
) )  =  ( (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  K
) G ( ( NN  X.  { A } ) `  ( K  +  1 ) ) ) )
153, 4gxpval 25084 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
1615oveq1d 6310 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (
( A P K ) G A )  =  ( (  seq 1 ( G , 
( NN  X.  { A } ) ) `  K ) G A ) )
1710, 14, 163eqtr4d 2518 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  ( K  +  1
) )  =  ( ( A P K ) G A ) )
186, 17eqtrd 2508 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
19183expia 1198 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) ) )
20 eqid 2467 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
213, 20grpolid 25044 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
2221adantr 465 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( (GId `  G ) G A )  =  A )
23 simpr 461 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  K  = 
0 )
2423oveq2d 6311 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  ( A P 0 ) )
253, 20, 4gx0 25086 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625adantr 465 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 0 )  =  (GId `  G )
)
2724, 26eqtrd 2508 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  (GId
`  G ) )
2827oveq1d 6310 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( ( A P K ) G A )  =  ( (GId `  G ) G A ) )
2923oveq1d 6310 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  ( 0  +  1 ) )
30 0p1e1 10659 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
3129, 30syl6eq 2524 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  1 )
3231oveq2d 6311 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( A P 1 ) )
333, 4gx1 25087 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 1 )  =  A )
3433adantr 465 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 1 )  =  A )
3532, 34eqtrd 2508 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  A )
3622, 28, 353eqtr4rd 2519 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
3736ex 434 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  =  0  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) ) )
3819, 37jaod 380 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  NN  \/  K  =  0
)  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) ) )
391, 38syl5bi 217 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN0  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) ) )
40393impia 1193 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e. 
NN0 )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {csn 4033    X. cxp 5003   ran crn 5006   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507   NNcn 10548   NN0cn0 10807   ZZ>=cuz 11094    seqcseq 12087   GrpOpcgr 25011  GIdcgi 25012   ^gcgx 25015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-seq 12088  df-grpo 25016  df-gid 25017  df-gx 25020
This theorem is referenced by:  gxcl  25090  gxcom  25094  gxinv  25095  gxsuc  25097
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