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Theorem gxid 21866
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 6092 . . . 4  |-  ( m  =  0  ->  ( U P m )  =  ( U P 0 ) )
21eqeq1d 2446 . . 3  |-  ( m  =  0  ->  (
( U P m )  =  U  <->  ( U P 0 )  =  U ) )
3 oveq2 6092 . . . 4  |-  ( m  =  k  ->  ( U P m )  =  ( U P k ) )
43eqeq1d 2446 . . 3  |-  ( m  =  k  ->  (
( U P m )  =  U  <->  ( U P k )  =  U ) )
5 oveq2 6092 . . . 4  |-  ( m  =  ( k  +  1 )  ->  ( U P m )  =  ( U P ( k  +  1 ) ) )
65eqeq1d 2446 . . 3  |-  ( m  =  ( k  +  1 )  ->  (
( U P m )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
7 oveq2 6092 . . . 4  |-  ( m  =  -u k  ->  ( U P m )  =  ( U P -u k ) )
87eqeq1d 2446 . . 3  |-  ( m  =  -u k  ->  (
( U P m )  =  U  <->  ( U P -u k )  =  U ) )
9 oveq2 6092 . . . 4  |-  ( m  =  K  ->  ( U P m )  =  ( U P K ) )
109eqeq1d 2446 . . 3  |-  ( m  =  K  ->  (
( U P m )  =  U  <->  ( U P K )  =  U ) )
11 eqid 2438 . . . . 5  |-  ran  G  =  ran  G
12 gxid.1 . . . . 5  |-  U  =  (GId `  G )
1311, 12grpoidcl 21810 . . . 4  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
14 gxid.2 . . . . 5  |-  P  =  ( ^g `  G
)
1511, 12, 14gx0 21854 . . . 4  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U P 0 )  =  U )
1613, 15mpdan 651 . . 3  |-  ( G  e.  GrpOp  ->  ( U P 0 )  =  U )
17 nn0z 10309 . . . 4  |-  ( k  e.  NN0  ->  k  e.  ZZ )
18 simpl 445 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  G  e.  GrpOp )
1913adantr 453 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  U  e.  ran  G )
20 simpr 449 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  k  e.  ZZ )
2111, 14gxsuc 21865 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2218, 19, 20, 21syl3anc 1185 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P ( k  +  1 ) )  =  ( ( U P k ) G U ) )
2311, 14gxcl 21858 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2418, 19, 20, 23syl3anc 1185 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  e. 
ran  G )
2511, 12grporid 21813 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( U P k )  e. 
ran  G )  -> 
( ( U P k ) G U )  =  ( U P k ) )
2624, 25syldan 458 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k ) G U )  =  ( U P k ) )
2722, 26eqtr2d 2471 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P k )  =  ( U P ( k  +  1 ) ) )
2827eqeq1d 2446 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k )  =  U  <->  ( U P ( k  +  1 ) )  =  U ) )
2928biimpd 200 . . . . 5  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  (
( U P k )  =  U  -> 
( U P ( k  +  1 ) )  =  U ) )
3029ex 425 . . . 4  |-  ( G  e.  GrpOp  ->  ( k  e.  ZZ  ->  ( ( U P k )  =  U  ->  ( U P ( k  +  1 ) )  =  U ) ) )
3117, 30syl5 31 . . 3  |-  ( G  e.  GrpOp  ->  ( k  e.  NN0  ->  ( ( U P k )  =  U  ->  ( U P ( k  +  1 ) )  =  U ) ) )
32 nnz 10308 . . . 4  |-  ( k  e.  NN  ->  k  e.  ZZ )
33 eqid 2438 . . . . . . . . . 10  |-  ( inv `  G )  =  ( inv `  G )
3411, 33, 14gxneg 21859 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
3513, 34syl3an2 1219 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
36353anidm12 1242 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
37363adant3 978 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  ( ( inv `  G ) `  ( U P k ) ) )
38 fveq2 5731 . . . . . . . 8  |-  ( ( U P k )  =  U  ->  (
( inv `  G
) `  ( U P k ) )  =  ( ( inv `  G ) `  U
) )
3912, 33grpoinvid 21825 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( ( inv `  G ) `  U )  =  U )
4038, 39sylan9eqr 2492 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
41403adant2 977 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  (
( inv `  G
) `  ( U P k ) )  =  U )
4237, 41eqtrd 2470 . . . . 5  |-  ( ( G  e.  GrpOp  /\  k  e.  ZZ  /\  ( U P k )  =  U )  ->  ( U P -u k )  =  U )
43423exp 1153 . . . 4  |-  ( G  e.  GrpOp  ->  ( k  e.  ZZ  ->  ( ( U P k )  =  U  ->  ( U P -u k )  =  U ) ) )
4432, 43syl5 31 . . 3  |-  ( G  e.  GrpOp  ->  ( k  e.  NN  ->  ( ( U P k )  =  U  ->  ( U P -u k )  =  U ) ) )
452, 4, 6, 8, 10, 16, 31, 44zindd 10376 . 2  |-  ( G  e.  GrpOp  ->  ( K  e.  ZZ  ->  ( U P K )  =  U ) )
4645imp 420 1  |-  ( ( G  e.  GrpOp  /\  K  e.  ZZ )  ->  ( U P K )  =  U )
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:  gxmodid  21872
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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