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Theorem gxcl 21858
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 6092 . . . 4  |-  ( m  =  0  ->  ( A P m )  =  ( A P 0 ) )
21eleq1d 2504 . . 3  |-  ( m  =  0  ->  (
( A P m )  e.  X  <->  ( A P 0 )  e.  X ) )
3 oveq2 6092 . . . 4  |-  ( m  =  k  ->  ( A P m )  =  ( A P k ) )
43eleq1d 2504 . . 3  |-  ( m  =  k  ->  (
( A P m )  e.  X  <->  ( A P k )  e.  X ) )
5 oveq2 6092 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( A P m )  =  ( A P ( k  +  1 ) ) )
65eleq1d 2504 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( A P m )  e.  X  <->  ( A P ( k  +  1 ) )  e.  X ) )
7 oveq2 6092 . . . 4  |-  ( m  =  -u k  ->  ( A P m )  =  ( A P -u k ) )
87eleq1d 2504 . . 3  |-  ( m  =  -u k  ->  (
( A P m )  e.  X  <->  ( A P -u k )  e.  X ) )
9 oveq2 6092 . . . 4  |-  ( m  =  K  ->  ( A P m )  =  ( A P K ) )
109eleq1d 2504 . . 3  |-  ( m  =  K  ->  (
( A P m )  e.  X  <->  ( A P K )  e.  X
) )
11 gxnn0suc.1 . . . . 5  |-  X  =  ran  G
12 eqid 2438 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
13 gxnn0suc.2 . . . . 5  |-  P  =  ( ^g `  G
)
1411, 12, 13gx0 21854 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  (GId `  G )
)
1511, 12grpoidcl 21810 . . . . 5  |-  ( G  e.  GrpOp  ->  (GId `  G
)  e.  X )
1615adantr 453 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (GId `  G )  e.  X
)
1714, 16eqeltrd 2512 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  e.  X )
1811grpocl 21793 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X  /\  A  e.  X )  ->  (
( A P k ) G A )  e.  X )
19183com23 1160 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( A P k )  e.  X )  ->  (
( A P k ) G A )  e.  X )
20193expia 1156 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A P k )  e.  X  -> 
( ( A P k ) G A )  e.  X ) )
21203adant3 978 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P k )  e.  X  -> 
( ( A P k ) G A )  e.  X ) )
2211, 13gxnn0suc 21857 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P ( k  +  1 ) )  =  ( ( A P k ) G A ) )
2322eleq1d 2504 . . . . 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 227 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  (
( A P k )  e.  X  -> 
( A P ( k  +  1 ) )  e.  X ) )
25243expia 1156 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
k  e.  NN0  ->  ( ( A P k )  e.  X  -> 
( A P ( k  +  1 ) )  e.  X ) ) )
26 nnnn0 10233 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
27 eqid 2438 . . . . . . . . 9  |-  ( inv `  G )  =  ( inv `  G )
2811, 27, 13gxnn0neg 21856 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN0 )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
2926, 28syl3an3 1220 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  ->  ( A P -u k )  =  ( ( inv `  G ) `  ( A P k ) ) )
3029adantr 453 . . . . . 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 21819 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A P k )  e.  X )  ->  (
( inv `  G
) `  ( A P k ) )  e.  X )
32313ad2antl1 1120 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  /\  ( A P k )  e.  X )  ->  ( ( inv `  G ) `  ( A P k ) )  e.  X )
3330, 32eqeltrd 2512 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  /\  ( A P k )  e.  X )  ->  ( A P
-u k )  e.  X )
3433ex 425 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  k  e.  NN )  ->  (
( A P k )  e.  X  -> 
( A P -u k )  e.  X
) )
35343expia 1156 . . 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 10376 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( K  e.  ZZ  ->  ( A P K )  e.  X ) )
37363impia 1151 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4882   ` cfv 5457  (class class class)co 6084   0cc0 8995   1c1 8996    + caddc 8998   -ucneg 9297   NNcn 10005   NN0cn0 10226   ZZcz 10287   GrpOpcgr 21779  GIdcgi 21780   invcgn 21781   ^gcgx 21783
This theorem is referenced by:  gxneg  21859  gxneg2  21860  gxcom  21862  gxinv  21863  gxsuc  21865  gxid  21866  gxnn0add  21867  gxadd  21868  gxnn0mul  21870  gxmul  21871  gxmodid  21872  gxdi  21889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gx 21788
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