MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gxnn0suc Structured version   Unicode version

Theorem gxnn0suc 23686
Description: Induction on group power (lemma with nonnegative exponent - use gxsuc 23694 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 10577 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 peano2nn 10330 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  +  1 )  e.  NN )
3 gxnn0suc.1 . . . . . . . 8  |-  X  =  ran  G
4 gxnn0suc.2 . . . . . . . 8  |-  P  =  ( ^g `  G
)
53, 4gxpval 23681 . . . . . . 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 1248 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  ( K  +  1 ) ) )
7 fvconst2g 5928 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( K  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( K  +  1 ) )  =  A )
82, 7sylan2 471 . . . . . . . . 9  |-  ( ( A  e.  X  /\  K  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( K  +  1
) )  =  A )
983adant1 1001 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (
( NN  X.  { A } ) `  ( K  +  1 ) )  =  A )
109oveq2d 6106 . . . . . . 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 11817 . . . . . . . . 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 10892 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1311, 12eleq2s 2533 . . . . . . . 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 1006 . . . . . . 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 23681 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
1615oveq1d 6105 . . . . . . 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 2483 . . . . . 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 2473 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
19183expia 1184 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) ) )
20 eqid 2441 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
213, 20grpolid 23641 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
2221adantr 462 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( (GId `  G ) G A )  =  A )
23 simpr 458 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  K  = 
0 )
2423oveq2d 6106 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  ( A P 0 ) )
253, 20, 4gx0 23683 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625adantr 462 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 0 )  =  (GId `  G )
)
2724, 26eqtrd 2473 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  (GId
`  G ) )
2827oveq1d 6105 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( ( A P K ) G A )  =  ( (GId `  G ) G A ) )
2923oveq1d 6105 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  ( 0  +  1 ) )
30 0p1e1 10429 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
3129, 30syl6eq 2489 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  1 )
3231oveq2d 6106 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( A P 1 ) )
333, 4gx1 23684 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 1 )  =  A )
3433adantr 462 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 1 )  =  A )
3532, 34eqtrd 2473 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  A )
3622, 28, 353eqtr4rd 2484 . . . . 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 1179 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 960    = wceq 1364    e. wcel 1761   {csn 3874    X. cxp 4834   ran crn 4837   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281   NNcn 10318   NN0cn0 10575   ZZ>=cuz 10857    seqcseq 11802   GrpOpcgr 23608  GIdcgi 23609   ^gcgx 23612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-seq 11803  df-grpo 23613  df-gid 23614  df-gx 23617
This theorem is referenced by:  gxcl  23687  gxcom  23691  gxinv  23692  gxsuc  23694
  Copyright terms: Public domain W3C validator