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Theorem gxnn0suc 20947
Description: Induction on group power (lemma with nonnegative exponent - use gxsuc 20955 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 9983 . . 3  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 peano2nn 9774 . . . . . . 7  |-  ( K  e.  NN  ->  ( K  +  1 )  e.  NN )
3 gxnn0suc.1 . . . . . . . 8  |-  X  =  ran  G
4 gxnn0suc.2 . . . . . . . 8  |-  P  =  ( ^g `  G
)
53, 4gxpval 20942 . . . . . . 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 1217 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  ( K  +  1 ) ) )
7 fvconst2g 5743 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( K  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( K  +  1 ) )  =  A )
82, 7sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  X  /\  K  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( K  +  1
) )  =  A )
983adant1 973 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  (
( NN  X.  { A } ) `  ( K  +  1 ) )  =  A )
109oveq2d 5890 . . . . . . 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 11077 . . . . . . . . 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 10279 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
1311, 12eleq2s 2388 . . . . . . . 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 978 . . . . . . 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 20942 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) )
1615oveq1d 5889 . . . . . . 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 2338 . . . . . 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 2328 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
19183expia 1153 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) ) )
20 eqid 2296 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
213, 20grpolid 20902 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
2221adantr 451 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( (GId `  G ) G A )  =  A )
23 simpr 447 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  K  = 
0 )
2423oveq2d 5890 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  ( A P 0 ) )
253, 20, 4gx0 20944 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
2625adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 0 )  =  (GId `  G )
)
2724, 26eqtrd 2328 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P K )  =  (GId
`  G ) )
2827oveq1d 5889 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( ( A P K ) G A )  =  ( (GId `  G ) G A ) )
2923oveq1d 5889 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  ( 0  +  1 ) )
30 0p1e1 9855 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
3129, 30syl6eq 2344 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( K  +  1 )  =  1 )
3231oveq2d 5890 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( A P 1 ) )
333, 4gx1 20945 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 1 )  =  A )
3433adantr 451 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P 1 )  =  A )
3532, 34eqtrd 2328 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  A )
3622, 28, 353eqtr4rd 2339 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  K  =  0 )  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) )
3736ex 423 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  =  0  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) ) )
3819, 37jaod 369 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( K  e.  NN  \/  K  =  0
)  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) ) )
391, 38syl5bi 208 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  NN0  ->  ( A P ( K  + 
1 ) )  =  ( ( A P K ) G A ) ) )
40393impia 1148 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 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {csn 3653    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   NNcn 9762   NN0cn0 9981   ZZ>=cuz 10246    seq cseq 11062   GrpOpcgr 20869  GIdcgi 20870   ^gcgx 20873
This theorem is referenced by:  gxcl  20948  gxcom  20952  gxinv  20953  gxsuc  20955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-grpo 20874  df-gid 20875  df-gx 20878
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