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Theorem gxnn0suc 21805
Description: Induction on group power (lemma with nonnegative exponent - use gxsuc 21813 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 10179 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 peano2nn 9968 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  +  1 )  e.  NN )
3 gxnn0suc.1 . . . . . . . 8  |-  X  =  ran  G
4 gxnn0suc.2 . . . . . . . 8  |-  P  =  ( ^g `  G
)
53, 4gxpval 21800 . . . . . . 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 5904 . . . . . . . . . 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 6056 . . . . . . 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 11293 . . . . . . . . 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 10477 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1311, 12eleq2s 2496 . . . . . . . 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 21800 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
1615oveq1d 6055 . . . . . . 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 2446 . . . . . 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 2436 . . . . 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 2404 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
213, 20grpolid 21760 . . . . . . 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 6056 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  ( A P 0 ) )
253, 20, 4gx0 21802 . . . . . . . . 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 2436 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  (GId
`  G ) )
2827oveq1d 6055 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( ( A P K ) G A )  =  ( (GId `  G ) G A ) )
2923oveq1d 6055 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  ( 0  +  1 ) )
30 0p1e1 10049 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
3129, 30syl6eq 2452 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  1 )
3231oveq2d 6056 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( A P 1 ) )
333, 4gx1 21803 . . . . . . . 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 2436 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  A )
3622, 28, 353eqtr4rd 2447 . . . . 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 1721   {csn 3774    X. cxp 4835   ran crn 4838   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947    + caddc 8949   NNcn 9956   NN0cn0 10177   ZZ>=cuz 10444    seq cseq 11278   GrpOpcgr 21727  GIdcgi 21728   ^gcgx 21731
This theorem is referenced by:  gxcl  21806  gxcom  21810  gxinv  21811  gxsuc  21813
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-seq 11279  df-grpo 21732  df-gid 21733  df-gx 21736
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