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Theorem gxnn0suc 21702
Description: Induction on group power (lemma with nonnegative exponent - use gxsuc 21710 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 10157 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 peano2nn 9946 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  +  1 )  e.  NN )
3 gxnn0suc.1 . . . . . . . 8  |-  X  =  ran  G
4 gxnn0suc.2 . . . . . . . 8  |-  P  =  ( ^g `  G
)
53, 4gxpval 21697 . . . . . . 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 1219 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  ( K  +  1 ) ) )
7 fvconst2g 5886 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( K  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( K  +  1 ) )  =  A )
82, 7sylan2 461 . . . . . . . . 9  |-  ( ( A  e.  X  /\  K  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( K  +  1
) )  =  A )
983adant1 975 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (
( NN  X.  { A } ) `  ( K  +  1 ) )  =  A )
109oveq2d 6038 . . . . . . 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 11267 . . . . . . . . 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 10455 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1311, 12eleq2s 2481 . . . . . . . 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 980 . . . . . . 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 21697 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
1615oveq1d 6037 . . . . . . 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 2431 . . . . . 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 2421 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
19183expia 1155 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) ) )
20 eqid 2389 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
213, 20grpolid 21657 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
2221adantr 452 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( (GId `  G ) G A )  =  A )
23 simpr 448 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  K  = 
0 )
2423oveq2d 6038 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  ( A P 0 ) )
253, 20, 4gx0 21699 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625adantr 452 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 0 )  =  (GId `  G )
)
2724, 26eqtrd 2421 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  (GId
`  G ) )
2827oveq1d 6037 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( ( A P K ) G A )  =  ( (GId `  G ) G A ) )
2923oveq1d 6037 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  ( 0  +  1 ) )
30 0p1e1 10027 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
3129, 30syl6eq 2437 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  1 )
3231oveq2d 6038 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( A P 1 ) )
333, 4gx1 21700 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 1 )  =  A )
3433adantr 452 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 1 )  =  A )
3532, 34eqtrd 2421 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  A )
3622, 28, 353eqtr4rd 2432 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
3736ex 424 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  =  0  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) ) )
3819, 37jaod 370 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  NN  \/  K  =  0
)  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) ) )
391, 38syl5bi 209 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN0  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) ) )
40393impia 1150 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e. 
NN0 )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {csn 3759    X. cxp 4818   ran crn 4821   ` cfv 5396  (class class class)co 6022   0cc0 8925   1c1 8926    + caddc 8928   NNcn 9934   NN0cn0 10155   ZZ>=cuz 10422    seq cseq 11252   GrpOpcgr 21624  GIdcgi 21625   ^gcgx 21628
This theorem is referenced by:  gxcl  21703  gxcom  21707  gxinv  21708  gxsuc  21710
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-gx 21633
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