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Theorem gxid 24939
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 6285 . . . 4  |-  ( m  =  0  ->  ( U P m )  =  ( U P 0 ) )
21eqeq1d 2464 . . 3  |-  ( m  =  0  ->  (
( U P m )  =  U  <->  ( U P 0 )  =  U ) )
3 oveq2 6285 . . . 4  |-  ( m  =  k  ->  ( U P m )  =  ( U P k ) )
43eqeq1d 2464 . . 3  |-  ( m  =  k  ->  (
( U P m )  =  U  <->  ( U P k )  =  U ) )
5 oveq2 6285 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( U P m )  =  ( U P ( k  +  1 ) ) )
65eqeq1d 2464 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( U P m )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
7 oveq2 6285 . . . 4  |-  ( m  =  -u k  ->  ( U P m )  =  ( U P -u k ) )
87eqeq1d 2464 . . 3  |-  ( m  =  -u k  ->  (
( U P m )  =  U  <->  ( U P -u k )  =  U ) )
9 oveq2 6285 . . . 4  |-  ( m  =  K  ->  ( U P m )  =  ( U P K ) )
109eqeq1d 2464 . . 3  |-  ( m  =  K  ->  (
( U P m )  =  U  <->  ( U P K )  =  U ) )
11 eqid 2462 . . . . 5  |-  ran  G  =  ran  G
12 gxid.1 . . . . 5  |-  U  =  (GId `  G )
1311, 12grpoidcl 24883 . . . 4  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
14 gxid.2 . . . . 5  |-  P  =  ( ^g `  G
)
1511, 12, 14gx0 24927 . . . 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 10878 . . . 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 24938 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2218, 19, 20, 21syl3anc 1223 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2311, 14gxcl 24931 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2418, 19, 20, 23syl3anc 1223 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2511, 12grporid 24886 . . . . . . . . 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 2504 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  =  ( U P ( k  +  1 ) ) )
2827eqeq1d 2464 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
2928biimpd 207 . . . . 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 10877 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
33 eqid 2462 . . . . . . . . . 10  |-  ( inv `  G )  =  ( inv `  G )
3411, 33, 14gxneg 24932 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
3513, 34syl3an2 1257 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
36353anidm12 1280 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
37363adant3 1011 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
38 fveq2 5859 . . . . . . . 8  |-  ( ( U P k )  =  U  ->  (
( inv `  G
) `  ( U P k ) )  =  ( ( inv `  G ) `  U
) )
3912, 33grpoinvid 24898 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( ( inv `  G ) `  U )  =  U )
4038, 39sylan9eqr 2525 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
41403adant2 1010 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
4237, 41eqtrd 2503 . . . . 5  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  U )
43423exp 1190 . . . 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 10953 . 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 968    = wceq 1374    e. wcel 1762   ran crn 4995   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484    + caddc 9486   -ucneg 9797   NNcn 10527   NN0cn0 10786   ZZcz 10855   GrpOpcgr 24852  GIdcgi 24853   invcgn 24854   ^gcgx 24856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-seq 12066  df-grpo 24857  df-gid 24858  df-ginv 24859  df-gx 24861
This theorem is referenced by:  gxmodid  24945
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