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Theorem gxid 25702
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 6288 . . . 4  |-  ( m  =  0  ->  ( U P m )  =  ( U P 0 ) )
21eqeq1d 2406 . . 3  |-  ( m  =  0  ->  (
( U P m )  =  U  <->  ( U P 0 )  =  U ) )
3 oveq2 6288 . . . 4  |-  ( m  =  k  ->  ( U P m )  =  ( U P k ) )
43eqeq1d 2406 . . 3  |-  ( m  =  k  ->  (
( U P m )  =  U  <->  ( U P k )  =  U ) )
5 oveq2 6288 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( U P m )  =  ( U P ( k  +  1 ) ) )
65eqeq1d 2406 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( U P m )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
7 oveq2 6288 . . . 4  |-  ( m  =  -u k  ->  ( U P m )  =  ( U P -u k ) )
87eqeq1d 2406 . . 3  |-  ( m  =  -u k  ->  (
( U P m )  =  U  <->  ( U P -u k )  =  U ) )
9 oveq2 6288 . . . 4  |-  ( m  =  K  ->  ( U P m )  =  ( U P K ) )
109eqeq1d 2406 . . 3  |-  ( m  =  K  ->  (
( U P m )  =  U  <->  ( U P K )  =  U ) )
11 eqid 2404 . . . . 5  |-  ran  G  =  ran  G
12 gxid.1 . . . . 5  |-  U  =  (GId `  G )
1311, 12grpoidcl 25646 . . . 4  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
14 gxid.2 . . . . 5  |-  P  =  ( ^g `  G
)
1511, 12, 14gx0 25690 . . . 4  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U P 0 )  =  U )
1613, 15mpdan 668 . . 3  |-  ( G  e.  GrpOp  ->  ( U P 0 )  =  U )
17 nn0z 10930 . . . 4  |-  ( k  e.  NN0  ->  k  e.  ZZ )
18 simpl 457 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  G  e.  GrpOp )
1913adantr 465 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  U  e.  ran  G )
20 simpr 461 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  k  e.  ZZ )
2111, 14gxsuc 25701 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2218, 19, 20, 21syl3anc 1232 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2311, 14gxcl 25694 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2418, 19, 20, 23syl3anc 1232 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2511, 12grporid 25649 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( U P k )  e. 
ran  G )  -> 
( ( U P k ) G U )  =  ( U P k ) )
2624, 25syldan 470 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k ) G U )  =  ( U P k ) )
2722, 26eqtr2d 2446 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  =  ( U P ( k  +  1 ) ) )
2827eqeq1d 2406 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
2928biimpd 209 . . . . 5  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k )  =  U  -> 
( U P ( k  +  1 ) )  =  U ) )
3029ex 434 . . . 4  |-  ( G  e.  GrpOp  ->  ( k  e.  ZZ  ->  ( ( U P k )  =  U  ->  ( U P ( k  +  1 ) )  =  U ) ) )
3117, 30syl5 32 . . 3  |-  ( G  e.  GrpOp  ->  ( k  e.  NN0  ->  ( ( U P k )  =  U  ->  ( U P ( k  +  1 ) )  =  U ) ) )
32 nnz 10929 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
33 eqid 2404 . . . . . . . . . 10  |-  ( inv `  G )  =  ( inv `  G )
3411, 33, 14gxneg 25695 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
3513, 34syl3an2 1266 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
36353anidm12 1289 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
37363adant3 1019 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
38 fveq2 5851 . . . . . . . 8  |-  ( ( U P k )  =  U  ->  (
( inv `  G
) `  ( U P k ) )  =  ( ( inv `  G ) `  U
) )
3912, 33grpoinvid 25661 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( ( inv `  G ) `  U )  =  U )
4038, 39sylan9eqr 2467 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
41403adant2 1018 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
4237, 41eqtrd 2445 . . . . 5  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  U )
43423exp 1198 . . . 4  |-  ( G  e.  GrpOp  ->  ( k  e.  ZZ  ->  ( ( U P k )  =  U  ->  ( U P -u k )  =  U ) ) )
4432, 43syl5 32 . . 3  |-  ( G  e.  GrpOp  ->  ( k  e.  NN  ->  ( ( U P k )  =  U  ->  ( U P -u k )  =  U ) ) )
452, 4, 6, 8, 10, 16, 31, 44zindd 11006 . 2  |-  ( G  e.  GrpOp  ->  ( K  e.  ZZ  ->  ( U P K )  =  U ) )
4645imp 429 1  |-  ( ( G  e.  GrpOp  /\  K  e.  ZZ )  ->  ( U P K )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   ran crn 4826   ` cfv 5571  (class class class)co 6280   0cc0 9524   1c1 9525    + caddc 9527   -ucneg 9844   NNcn 10578   NN0cn0 10838   ZZcz 10907   GrpOpcgr 25615  GIdcgi 25616   invcgn 25617   ^gcgx 25619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-seq 12154  df-grpo 25620  df-gid 25621  df-ginv 25622  df-gx 25624
This theorem is referenced by:  gxmodid  25708
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