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Theorem gxcl 21810
Description: Closure of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxnn0suc.1  |-  X  =  ran  G
gxnn0suc.2  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxcl  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )

Proof of Theorem gxcl
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6052 . . . 4  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
21eleq1d 2474 . . 3  |-  ( m  =  0  ->  (
( A P m )  e.  X  <->  ( A P 0 )  e.  X ) )
3 oveq2 6052 . . . 4  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
43eleq1d 2474 . . 3  |-  ( m  =  k  ->  (
( A P m )  e.  X  <->  ( A P k )  e.  X ) )
5 oveq2 6052 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
65eleq1d 2474 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( A P m )  e.  X  <->  ( A P ( k  +  1 ) )  e.  X ) )
7 oveq2 6052 . . . 4  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
87eleq1d 2474 . . 3  |-  ( m  =  -u k  ->  (
( A P m )  e.  X  <->  ( A P -u k )  e.  X ) )
9 oveq2 6052 . . . 4  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
109eleq1d 2474 . . 3  |-  ( m  =  K  ->  (
( A P m )  e.  X  <->  ( A P K )  e.  X
) )
11 gxnn0suc.1 . . . . 5  |-  X  =  ran  G
12 eqid 2408 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
13 gxnn0suc.2 . . . . 5  |-  P  =  ( ^g `  G
)
1411, 12, 13gx0 21806 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
1511, 12grpoidcl 21762 . . . . 5  |-  ( G  e.  GrpOp  ->  (GId `  G
)  e.  X )
1615adantr 452 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (GId `  G )  e.  X
)
1714, 16eqeltrd 2482 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  e.  X )
1811grpocl 21745 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X  /\  A  e.  X )  ->  (
( A P k ) G A )  e.  X )
19183com23 1159 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A P k )  e.  X )  ->  (
( A P k ) G A )  e.  X )
20193expia 1155 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P k )  e.  X  -> 
( ( A P k ) G A )  e.  X ) )
21203adant3 977 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P k )  e.  X  -> 
( ( A P k ) G A )  e.  X ) )
2211, 13gxnn0suc 21809 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
2322eleq1d 2474 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P ( k  +  1 ) )  e.  X  <->  ( ( A P k ) G A )  e.  X
) )
2421, 23sylibrd 226 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P k )  e.  X  -> 
( A P ( k  +  1 ) )  e.  X ) )
25243expia 1155 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN0  ->  ( ( A P k )  e.  X  -> 
( A P ( k  +  1 ) )  e.  X ) ) )
26 nnnn0 10188 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
27 eqid 2408 . . . . . . . . 9  |-  ( inv `  G )  =  ( inv `  G )
2811, 27, 13gxnn0neg 21808 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
2926, 28syl3an3 1219 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
3029adantr 452 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  /\  ( A P k )  e.  X )  ->  ( A P
-u k )  =  ( ( inv `  G
) `  ( A P k ) ) )
3111, 27grpoinvcl 21771 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X )  ->  (
( inv `  G
) `  ( A P k ) )  e.  X )
32313ad2antl1 1119 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  /\  ( A P k )  e.  X )  ->  ( ( inv `  G ) `  ( A P k ) )  e.  X )
3330, 32eqeltrd 2482 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  /\  ( A P k )  e.  X )  ->  ( A P
-u k )  e.  X )
3433ex 424 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  ->  (
( A P k )  e.  X  -> 
( A P -u k )  e.  X
) )
35343expia 1155 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN  ->  ( ( A P k )  e.  X  -> 
( A P -u k )  e.  X
) ) )
362, 4, 6, 8, 10, 17, 25, 35zindd 10331 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( A P K )  e.  X ) )
37363impia 1150 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ran crn 4842   ` cfv 5417  (class class class)co 6044   0cc0 8950   1c1 8951    + caddc 8953   -ucneg 9252   NNcn 9960   NN0cn0 10181   ZZcz 10242   GrpOpcgr 21731  GIdcgi 21732   invcgn 21733   ^gcgx 21735
This theorem is referenced by:  gxneg  21811  gxneg2  21812  gxcom  21814  gxinv  21815  gxsuc  21817  gxid  21818  gxnn0add  21819  gxadd  21820  gxnn0mul  21822  gxmul  21823  gxmodid  21824  gxdi  21841
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-seq 11283  df-grpo 21736  df-gid 21737  df-ginv 21738  df-gx 21740
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