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Theorem gxid 21822
Description: The identity element of a group to any power remains unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxid.1  |-  U  =  (GId `  G )
gxid.2  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxid  |-  ( ( G  e.  GrpOp  /\  K  e.  ZZ )  ->  ( U P K )  =  U )

Proof of Theorem gxid
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6056 . . . 4  |-  ( m  =  0  ->  ( U P m )  =  ( U P 0 ) )
21eqeq1d 2420 . . 3  |-  ( m  =  0  ->  (
( U P m )  =  U  <->  ( U P 0 )  =  U ) )
3 oveq2 6056 . . . 4  |-  ( m  =  k  ->  ( U P m )  =  ( U P k ) )
43eqeq1d 2420 . . 3  |-  ( m  =  k  ->  (
( U P m )  =  U  <->  ( U P k )  =  U ) )
5 oveq2 6056 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( U P m )  =  ( U P ( k  +  1 ) ) )
65eqeq1d 2420 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( U P m )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
7 oveq2 6056 . . . 4  |-  ( m  =  -u k  ->  ( U P m )  =  ( U P -u k ) )
87eqeq1d 2420 . . 3  |-  ( m  =  -u k  ->  (
( U P m )  =  U  <->  ( U P -u k )  =  U ) )
9 oveq2 6056 . . . 4  |-  ( m  =  K  ->  ( U P m )  =  ( U P K ) )
109eqeq1d 2420 . . 3  |-  ( m  =  K  ->  (
( U P m )  =  U  <->  ( U P K )  =  U ) )
11 eqid 2412 . . . . 5  |-  ran  G  =  ran  G
12 gxid.1 . . . . 5  |-  U  =  (GId `  G )
1311, 12grpoidcl 21766 . . . 4  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
14 gxid.2 . . . . 5  |-  P  =  ( ^g `  G
)
1511, 12, 14gx0 21810 . . . 4  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U P 0 )  =  U )
1613, 15mpdan 650 . . 3  |-  ( G  e.  GrpOp  ->  ( U P 0 )  =  U )
17 nn0z 10268 . . . 4  |-  ( k  e.  NN0  ->  k  e.  ZZ )
18 simpl 444 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  G  e.  GrpOp )
1913adantr 452 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  U  e.  ran  G )
20 simpr 448 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  k  e.  ZZ )
2111, 14gxsuc 21821 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2218, 19, 20, 21syl3anc 1184 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2311, 14gxcl 21814 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2418, 19, 20, 23syl3anc 1184 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2511, 12grporid 21769 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( U P k )  e. 
ran  G )  -> 
( ( U P k ) G U )  =  ( U P k ) )
2624, 25syldan 457 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k ) G U )  =  ( U P k ) )
2722, 26eqtr2d 2445 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  =  ( U P ( k  +  1 ) ) )
2827eqeq1d 2420 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
2928biimpd 199 . . . . 5  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k )  =  U  -> 
( U P ( k  +  1 ) )  =  U ) )
3029ex 424 . . . 4  |-  ( G  e.  GrpOp  ->  ( k  e.  ZZ  ->  ( ( U P k )  =  U  ->  ( U P ( k  +  1 ) )  =  U ) ) )
3117, 30syl5 30 . . 3  |-  ( G  e.  GrpOp  ->  ( k  e.  NN0  ->  ( ( U P k )  =  U  ->  ( U P ( k  +  1 ) )  =  U ) ) )
32 nnz 10267 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
33 eqid 2412 . . . . . . . . . 10  |-  ( inv `  G )  =  ( inv `  G )
3411, 33, 14gxneg 21815 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
3513, 34syl3an2 1218 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
36353anidm12 1241 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
37363adant3 977 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
38 fveq2 5695 . . . . . . . 8  |-  ( ( U P k )  =  U  ->  (
( inv `  G
) `  ( U P k ) )  =  ( ( inv `  G ) `  U
) )
3912, 33grpoinvid 21781 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( ( inv `  G ) `  U )  =  U )
4038, 39sylan9eqr 2466 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
41403adant2 976 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
4237, 41eqtrd 2444 . . . . 5  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  U )
43423exp 1152 . . . 4  |-  ( G  e.  GrpOp  ->  ( k  e.  ZZ  ->  ( ( U P k )  =  U  ->  ( U P -u k )  =  U ) ) )
4432, 43syl5 30 . . 3  |-  ( G  e.  GrpOp  ->  ( k  e.  NN  ->  ( ( U P k )  =  U  ->  ( U P -u k )  =  U ) ) )
452, 4, 6, 8, 10, 16, 31, 44zindd 10335 . 2  |-  ( G  e.  GrpOp  ->  ( K  e.  ZZ  ->  ( U P K )  =  U ) )
4645imp 419 1  |-  ( ( G  e.  GrpOp  /\  K  e.  ZZ )  ->  ( U P K )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ran crn 4846   ` cfv 5421  (class class class)co 6048   0cc0 8954   1c1 8955    + caddc 8957   -ucneg 9256   NNcn 9964   NN0cn0 10185   ZZcz 10246   GrpOpcgr 21735  GIdcgi 21736   invcgn 21737   ^gcgx 21739
This theorem is referenced by:  gxmodid  21828
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-seq 11287  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gx 21744
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