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Theorem gxcl 21364
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 5989 . . . 4  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
21eleq1d 2432 . . 3  |-  ( m  =  0  ->  (
( A P m )  e.  X  <->  ( A P 0 )  e.  X ) )
3 oveq2 5989 . . . 4  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
43eleq1d 2432 . . 3  |-  ( m  =  k  ->  (
( A P m )  e.  X  <->  ( A P k )  e.  X ) )
5 oveq2 5989 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
65eleq1d 2432 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( A P m )  e.  X  <->  ( A P ( k  +  1 ) )  e.  X ) )
7 oveq2 5989 . . . 4  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
87eleq1d 2432 . . 3  |-  ( m  =  -u k  ->  (
( A P m )  e.  X  <->  ( A P -u k )  e.  X ) )
9 oveq2 5989 . . . 4  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
109eleq1d 2432 . . 3  |-  ( m  =  K  ->  (
( A P m )  e.  X  <->  ( A P K )  e.  X
) )
11 gxnn0suc.1 . . . . 5  |-  X  =  ran  G
12 eqid 2366 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
13 gxnn0suc.2 . . . . 5  |-  P  =  ( ^g `  G
)
1411, 12, 13gx0 21360 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
1511, 12grpoidcl 21316 . . . . 5  |-  ( G  e.  GrpOp  ->  (GId `  G
)  e.  X )
1615adantr 451 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (GId `  G )  e.  X
)
1714, 16eqeltrd 2440 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  e.  X )
1811grpocl 21299 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X  /\  A  e.  X )  ->  (
( A P k ) G A )  e.  X )
19183com23 1158 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A P k )  e.  X )  ->  (
( A P k ) G A )  e.  X )
20193expia 1154 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P k )  e.  X  -> 
( ( A P k ) G A )  e.  X ) )
21203adant3 976 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P k )  e.  X  -> 
( ( A P k ) G A )  e.  X ) )
2211, 13gxnn0suc 21363 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
2322eleq1d 2432 . . . . 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 225 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P k )  e.  X  -> 
( A P ( k  +  1 ) )  e.  X ) )
25243expia 1154 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN0  ->  ( ( A P k )  e.  X  -> 
( A P ( k  +  1 ) )  e.  X ) ) )
26 nnnn0 10121 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
27 eqid 2366 . . . . . . . . 9  |-  ( inv `  G )  =  ( inv `  G )
2811, 27, 13gxnn0neg 21362 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
2926, 28syl3an3 1218 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
3029adantr 451 . . . . . 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 21325 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X )  ->  (
( inv `  G
) `  ( A P k ) )  e.  X )
32313ad2antl1 1118 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  /\  ( A P k )  e.  X )  ->  ( ( inv `  G ) `  ( A P k ) )  e.  X )
3330, 32eqeltrd 2440 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  /\  ( A P k )  e.  X )  ->  ( A P
-u k )  e.  X )
3433ex 423 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  ->  (
( A P k )  e.  X  -> 
( A P -u k )  e.  X
) )
35343expia 1154 . . 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 10264 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( A P K )  e.  X ) )
37363impia 1149 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 358    /\ w3a 935    = wceq 1647    e. wcel 1715   ran crn 4793   ` cfv 5358  (class class class)co 5981   0cc0 8884   1c1 8885    + caddc 8887   -ucneg 9185   NNcn 9893   NN0cn0 10114   ZZcz 10175   GrpOpcgr 21285  GIdcgi 21286   invcgn 21287   ^gcgx 21289
This theorem is referenced by:  gxneg  21365  gxneg2  21366  gxcom  21368  gxinv  21369  gxsuc  21371  gxid  21372  gxnn0add  21373  gxadd  21374  gxnn0mul  21376  gxmul  21377  gxmodid  21378  gxdi  21395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-seq 11211  df-grpo 21290  df-gid 21291  df-ginv 21292  df-gx 21294
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