Theorem gxid 9396

Description: The identity element of a group to any power remains unchanged. (Contributed by Paul Chapman, 17-Apr-2009.)

Hypotheses

Ref	Expression
gxid.1	|- U = (Id` G)
gxid.2	|- P = (^g` G)

Assertion

Ref	Expression
gxid	|- ((G e. Grp /\ K e. ZZ) -> (UPK) = U)

Proof of Theorem gxid

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . 4 |- (m = 0 -> (UPm) = (UP0))
2	1	eqeq1d 1892	. . 3 |- (m = 0 -> ((UPm) = U <-> (UP0) = U))
3		opreq2 4890	. . . 4 |- (m = k -> (UPm) = (UPk))
4	3	eqeq1d 1892	. . 3 |- (m = k -> ((UPm) = U <-> (UPk) = U))
5		opreq2 4890	. . . 4 |- (m = (k + 1) -> (UPm) = (UP(k + 1)))
6	5	eqeq1d 1892	. . 3 |- (m = (k + 1) -> ((UPm) = U <-> (UP(k + 1)) = U))
7		opreq2 4890	. . . 4 |- (m = -uk -> (UPm) = (UP-uk))
8	7	eqeq1d 1892	. . 3 |- (m = -uk -> ((UPm) = U <-> (UP-uk) = U))
9		opreq2 4890	. . . 4 |- (m = K -> (UPm) = (UPK))
10	9	eqeq1d 1892	. . 3 |- (m = K -> ((UPm) = U <-> (UPK) = U))
11		eqid 1884	. . . . 5 |- ran G = ran G
12		gxid.1	. . . . 5 |- U = (Id` G)
13	11, 12	grpidcl 9343	. . . 4 |- (G e. Grp -> U e. ran G)
14		gxid.2	. . . . 5 |- P = (^g` G)
15	11, 12, 14	gx0 9384	. . . 4 |- ((G e. Grp /\ U e. ran G) -> (UP0) = U)
16	13, 15	mpdan 768	. . 3 |- (G e. Grp -> (UP0) = U)
17		simpl 346	. . . . . . . . 9 |- ((G e. Grp /\ k e. ZZ) -> G e. Grp)
18	13	adantr 425	. . . . . . . . 9 |- ((G e. Grp /\ k e. ZZ) -> U e. ran G)
19		simpr 350	. . . . . . . . 9 |- ((G e. Grp /\ k e. ZZ) -> k e. ZZ)
20	11, 14	gxsuc 9395	. . . . . . . . 9 |- ((G e. Grp /\ U e. ran G /\ k e. ZZ) -> (UP(k + 1)) = ((UPk)GU))
21	17, 18, 19, 20	syl111anc 1100	. . . . . . . 8 |- ((G e. Grp /\ k e. ZZ) -> (UP(k + 1)) = ((UPk)GU))
22	11, 14	gxcl 9388	. . . . . . . . . 10 |- ((G e. Grp /\ U e. ran G /\ k e. ZZ) -> (UPk) e. ran G)
23	17, 18, 19, 22	syl111anc 1100	. . . . . . . . 9 |- ((G e. Grp /\ k e. ZZ) -> (UPk) e. ran G)
24	11, 12	grprid 9346	. . . . . . . . 9 |- ((G e. Grp /\ (UPk) e. ran G) -> ((UPk)GU) = (UPk))
25	17, 23, 24	syl11anc 524	. . . . . . . 8 |- ((G e. Grp /\ k e. ZZ) -> ((UPk)GU) = (UPk))
26	21, 25	eqtr2d 1926	. . . . . . 7 |- ((G e. Grp /\ k e. ZZ) -> (UPk) = (UP(k + 1)))
27	26	eqeq1d 1892	. . . . . 6 |- ((G e. Grp /\ k e. ZZ) -> ((UPk) = U <-> (UP(k + 1)) = U))
28	27	biimpd 170	. . . . 5 |- ((G e. Grp /\ k e. ZZ) -> ((UPk) = U -> (UP(k + 1)) = U))
29	28	ex 402	. . . 4 |- (G e. Grp -> (k e. ZZ -> ((UPk) = U -> (UP(k + 1)) = U)))
30		nn0z 7363	. . . 4 |- (k e. NN0 -> k e. ZZ)
31	29, 30	syl5 20	. . 3 |- (G e. Grp -> (k e. NN0 -> ((UPk) = U -> (UP(k + 1)) = U)))
32		eqid 1884	. . . . . . . . . . 11 |- (inv` G) = (inv` G)
33	11, 32, 14	gxneg 9389	. . . . . . . . . 10 |- ((G e. Grp /\ U e. ran G /\ k e. ZZ) -> (UP-uk) = ((inv` G)` (UPk)))
34	33, 13	syl3an2 1131	. . . . . . . . 9 |- ((G e. Grp /\ G e. Grp /\ k e. ZZ) -> (UP-uk) = ((inv` G)` (UPk)))
35	34	3anidm12 1154	. . . . . . . 8 |- ((G e. Grp /\ k e. ZZ) -> (UP-uk) = ((inv` G)` (UPk)))
36	35	3adant3 896	. . . . . . 7 |- ((G e. Grp /\ k e. ZZ /\ (UPk) = U) -> (UP-uk) = ((inv` G)` (UPk)))
37		fveq2 4681	. . . . . . . . 9 |- ((UPk) = U -> ((inv` G)` (UPk)) = ((inv` G)` U))
38	12, 32	grpinvid 9358	. . . . . . . . 9 |- (G e. Grp -> ((inv` G)` U) = U)
39	37, 38	sylan9eqr 1951	. . . . . . . 8 |- ((G e. Grp /\ (UPk) = U) -> ((inv` G)` (UPk)) = U)
40	39	3adant2 895	. . . . . . 7 |- ((G e. Grp /\ k e. ZZ /\ (UPk) = U) -> ((inv` G)` (UPk)) = U)
41	36, 40	eqtrd 1925	. . . . . 6 |- ((G e. Grp /\ k e. ZZ /\ (UPk) = U) -> (UP-uk) = U)
42	41	3expia 1069	. . . . 5 |- ((G e. Grp /\ k e. ZZ) -> ((UPk) = U -> (UP-uk) = U))
43	42	ex 402	. . . 4 |- (G e. Grp -> (k e. ZZ -> ((UPk) = U -> (UP-uk) = U)))
44		nnz 7362	. . . 4 |- (k e. NN -> k e. ZZ)
45	43, 44	syl5 20	. . 3 |- (G e. Grp -> (k e. NN -> ((UPk) = U -> (UP-uk) = U)))
46	2, 4, 6, 8, 10, 16, 31, 45	zindd 7427	. 2 |- (G e. Grp -> (K e. ZZ -> (UPK) = U))
47	46	imp 377	1 |- ((G e. Grp /\ K e. ZZ) -> (UPK) = U)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ran crn 3987  ` cfv 3998  (class class class)co 4884  0cc0 6386  1c1 6387   + caddc 6389  -ucneg 6446  NNcn 6449  NN0cn0 6450  ZZcz 6451  Grpcgr 9311  Idcgi 9312  invcgn 9313  ^gcgx 9315

This theorem is referenced by:  gxmodid 9402

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-seq1 7721  df-grp 9316  df-gid 9317  df-ginv 9318  df-gx 9320

Copyright terms: Public domain