Theorem gxinv 9393

Description: The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.)

Hypotheses

Ref	Expression
gxinv.1	|- X = ran G
gxinv.2	|- N = (inv` G)
gxinv.3	|- P = (^g` G)

Assertion

Ref	Expression
gxinv	|- ((G e. Grp /\ A e. X /\ K e. ZZ) -> ((N` A)PK) = (N` (APK)))

Proof of Theorem gxinv

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . 4 |- (m = 0 -> ((N` A)Pm) = ((N` A)P0))
2		opreq2 4890	. . . . 5 |- (m = 0 -> (APm) = (AP0))
3	2	fveq2d 4685	. . . 4 |- (m = 0 -> (N` (APm)) = (N` (AP0)))
4	1, 3	eqeq12d 1899	. . 3 |- (m = 0 -> (((N` A)Pm) = (N` (APm)) <-> ((N` A)P0) = (N` (AP0))))
5		opreq2 4890	. . . 4 |- (m = k -> ((N` A)Pm) = ((N` A)Pk))
6		opreq2 4890	. . . . 5 |- (m = k -> (APm) = (APk))
7	6	fveq2d 4685	. . . 4 |- (m = k -> (N` (APm)) = (N` (APk)))
8	5, 7	eqeq12d 1899	. . 3 |- (m = k -> (((N` A)Pm) = (N` (APm)) <-> ((N` A)Pk) = (N` (APk))))
9		opreq2 4890	. . . 4 |- (m = (k + 1) -> ((N` A)Pm) = ((N` A)P(k + 1)))
10		opreq2 4890	. . . . 5 |- (m = (k + 1) -> (APm) = (AP(k + 1)))
11	10	fveq2d 4685	. . . 4 |- (m = (k + 1) -> (N` (APm)) = (N` (AP(k + 1))))
12	9, 11	eqeq12d 1899	. . 3 |- (m = (k + 1) -> (((N` A)Pm) = (N` (APm)) <-> ((N` A)P(k + 1)) = (N` (AP(k + 1)))))
13		opreq2 4890	. . . 4 |- (m = -uk -> ((N` A)Pm) = ((N` A)P-uk))
14		opreq2 4890	. . . . 5 |- (m = -uk -> (APm) = (AP-uk))
15	14	fveq2d 4685	. . . 4 |- (m = -uk -> (N` (APm)) = (N` (AP-uk)))
16	13, 15	eqeq12d 1899	. . 3 |- (m = -uk -> (((N` A)Pm) = (N` (APm)) <-> ((N` A)P-uk) = (N` (AP-uk))))
17		opreq2 4890	. . . 4 |- (m = K -> ((N` A)Pm) = ((N` A)PK))
18		opreq2 4890	. . . . 5 |- (m = K -> (APm) = (APK))
19	18	fveq2d 4685	. . . 4 |- (m = K -> (N` (APm)) = (N` (APK)))
20	17, 19	eqeq12d 1899	. . 3 |- (m = K -> (((N` A)Pm) = (N` (APm)) <-> ((N` A)PK) = (N` (APK))))
21		eqid 1884	. . . . . 6 |- (Id` G) = (Id` G)
22		gxinv.2	. . . . . 6 |- N = (inv` G)
23	21, 22	grpinvid 9358	. . . . 5 |- (G e. Grp -> (N` (Id` G)) = (Id` G))
24	23	adantr 425	. . . 4 |- ((G e. Grp /\ A e. X) -> (N` (Id` G)) = (Id` G))
25		gxinv.1	. . . . . 6 |- X = ran G
26		gxinv.3	. . . . . 6 |- P = (^g` G)
27	25, 21, 26	gx0 9384	. . . . 5 |- ((G e. Grp /\ A e. X) -> (AP0) = (Id` G))
28	27	fveq2d 4685	. . . 4 |- ((G e. Grp /\ A e. X) -> (N` (AP0)) = (N` (Id` G)))
29	25, 22	grpinvcl 9352	. . . . 5 |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
30	25, 21, 26	gx0 9384	. . . . 5 |- ((G e. Grp /\ (N` A) e. X) -> ((N` A)P0) = (Id` G))
31	29, 30	syldan 516	. . . 4 |- ((G e. Grp /\ A e. X) -> ((N` A)P0) = (Id` G))
32	24, 28, 31	3eqtr4rd 1939	. . 3 |- ((G e. Grp /\ A e. X) -> ((N` A)P0) = (N` (AP0)))
33		opreq2 4890	. . . . . . 7 |- (((N` A)Pk) = (N` (APk)) -> ((N` A)G((N` A)Pk)) = ((N` A)G(N` (APk))))
34	33	adantl 424	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((N` A)Pk) = (N` (APk))) -> ((N` A)G((N` A)Pk)) = ((N` A)G(N` (APk))))
35		simp1 876	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> G e. Grp)
36	29	3adant3 896	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (N` A) e. X)
37		simp3 878	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> k e. NN0)
38	25, 26	gxnn0suc 9387	. . . . . . . . 9 |- ((G e. Grp /\ (N` A) e. X /\ k e. NN0) -> ((N` A)P(k + 1)) = (((N` A)Pk)G(N` A)))
39	35, 36, 37, 38	syl111anc 1100	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> ((N` A)P(k + 1)) = (((N` A)Pk)G(N` A)))
40	25, 26	gxcom 9392	. . . . . . . . . 10 |- ((G e. Grp /\ (N` A) e. X /\ k e. ZZ) -> (((N` A)Pk)G(N` A)) = ((N` A)G((N` A)Pk)))
41		nn0z 7363	. . . . . . . . . 10 |- (k e. NN0 -> k e. ZZ)
42	40, 41	syl3an3 1132	. . . . . . . . 9 |- ((G e. Grp /\ (N` A) e. X /\ k e. NN0) -> (((N` A)Pk)G(N` A)) = ((N` A)G((N` A)Pk)))
43	35, 36, 37, 42	syl111anc 1100	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (((N` A)Pk)G(N` A)) = ((N` A)G((N` A)Pk)))
44	39, 43	eqtrd 1925	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> ((N` A)P(k + 1)) = ((N` A)G((N` A)Pk)))
45	44	adantr 425	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((N` A)Pk) = (N` (APk))) -> ((N` A)P(k + 1)) = ((N` A)G((N` A)Pk)))
46	25, 26	gxnn0suc 9387	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (AP(k + 1)) = ((APk)GA))
47	46	fveq2d 4685	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (N` (AP(k + 1))) = (N` ((APk)GA)))
48	25, 26	gxcl 9388	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (APk) e. X)
49	48, 41	syl3an3 1132	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (APk) e. X)
50		simp2 877	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> A e. X)
51	25, 22	grpinvop 9365	. . . . . . . . 9 |- ((G e. Grp /\ (APk) e. X /\ A e. X) -> (N` ((APk)GA)) = ((N` A)G(N` (APk))))
52	35, 49, 50, 51	syl111anc 1100	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (N` ((APk)GA)) = ((N` A)G(N` (APk))))
53	47, 52	eqtrd 1925	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (N` (AP(k + 1))) = ((N` A)G(N` (APk))))
54	53	adantr 425	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((N` A)Pk) = (N` (APk))) -> (N` (AP(k + 1))) = ((N` A)G(N` (APk))))
55	34, 45, 54	3eqtr4d 1937	. . . . 5 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((N` A)Pk) = (N` (APk))) -> ((N` A)P(k + 1)) = (N` (AP(k + 1))))
56	55	ex 402	. . . 4 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (((N` A)Pk) = (N` (APk)) -> ((N` A)P(k + 1)) = (N` (AP(k + 1)))))
57	56	3expia 1069	. . 3 |- ((G e. Grp /\ A e. X) -> (k e. NN0 -> (((N` A)Pk) = (N` (APk)) -> ((N` A)P(k + 1)) = (N` (AP(k + 1))))))
58		simp1 876	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> G e. Grp)
59	29	3adant3 896	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (N` A) e. X)
60		simp3 878	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> k e. ZZ)
61	25, 22, 26	gxneg 9389	. . . . . . . . 9 |- ((G e. Grp /\ (N` A) e. X /\ k e. ZZ) -> ((N` A)P-uk) = (N` ((N` A)Pk)))
62	58, 59, 60, 61	syl111anc 1100	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> ((N` A)P-uk) = (N` ((N` A)Pk)))
63	62	adantr 425	. . . . . . 7 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((N` A)Pk) = (N` (APk))) -> ((N` A)P-uk) = (N` ((N` A)Pk)))
64	25, 22, 26	gxneg 9389	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (AP-uk) = (N` (APk)))
65	64	adantr 425	. . . . . . . . 9 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((N` A)Pk) = (N` (APk))) -> (AP-uk) = (N` (APk)))
66		simpr 350	. . . . . . . . 9 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((N` A)Pk) = (N` (APk))) -> ((N` A)Pk) = (N` (APk)))
67	65, 66	eqtr4d 1928	. . . . . . . 8 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((N` A)Pk) = (N` (APk))) -> (AP-uk) = ((N` A)Pk))
68	67	fveq2d 4685	. . . . . . 7 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((N` A)Pk) = (N` (APk))) -> (N` (AP-uk)) = (N` ((N` A)Pk)))
69	63, 68	eqtr4d 1928	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((N` A)Pk) = (N` (APk))) -> ((N` A)P-uk) = (N` (AP-uk)))
70	69	ex 402	. . . . 5 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (((N` A)Pk) = (N` (APk)) -> ((N` A)P-uk) = (N` (AP-uk))))
71	70	3expia 1069	. . . 4 |- ((G e. Grp /\ A e. X) -> (k e. ZZ -> (((N` A)Pk) = (N` (APk)) -> ((N` A)P-uk) = (N` (AP-uk)))))
72		nnz 7362	. . . 4 |- (k e. NN -> k e. ZZ)
73	71, 72	syl5 20	. . 3 |- ((G e. Grp /\ A e. X) -> (k e. NN -> (((N` A)Pk) = (N` (APk)) -> ((N` A)P-uk) = (N` (AP-uk)))))
74	4, 8, 12, 16, 20, 32, 57, 73	zindd 7427	. 2 |- ((G e. Grp /\ A e. X) -> (K e. ZZ -> ((N` A)PK) = (N` (APK))))
75	74	3impia 1064	1 |- ((G e. Grp /\ A e. X /\ K e. ZZ) -> ((N` A)PK) = (N` (APK)))

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:  gxinv2 9394  gxmul 9401

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

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