Theorem gxcom 9392

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

Hypotheses

Ref	Expression
gxcom.1	|- X = ran G
gxcom.2	|- P = (^g` G)

Assertion

Ref	Expression
gxcom	|- ((G e. Grp /\ A e. X /\ K e. ZZ) -> ((APK)GA) = (AG(APK)))

Proof of Theorem gxcom

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (m = 0 -> (APm) = (AP0))
2	1	opreq1d 4897	. . . 4 |- (m = 0 -> ((APm)GA) = ((AP0)GA))
3	1	opreq2d 4898	. . . 4 |- (m = 0 -> (AG(APm)) = (AG(AP0)))
4	2, 3	eqeq12d 1899	. . 3 |- (m = 0 -> (((APm)GA) = (AG(APm)) <-> ((AP0)GA) = (AG(AP0))))
5		opreq2 4890	. . . . 5 |- (m = k -> (APm) = (APk))
6	5	opreq1d 4897	. . . 4 |- (m = k -> ((APm)GA) = ((APk)GA))
7	5	opreq2d 4898	. . . 4 |- (m = k -> (AG(APm)) = (AG(APk)))
8	6, 7	eqeq12d 1899	. . 3 |- (m = k -> (((APm)GA) = (AG(APm)) <-> ((APk)GA) = (AG(APk))))
9		opreq2 4890	. . . . 5 |- (m = (k + 1) -> (APm) = (AP(k + 1)))
10	9	opreq1d 4897	. . . 4 |- (m = (k + 1) -> ((APm)GA) = ((AP(k + 1))GA))
11	9	opreq2d 4898	. . . 4 |- (m = (k + 1) -> (AG(APm)) = (AG(AP(k + 1))))
12	10, 11	eqeq12d 1899	. . 3 |- (m = (k + 1) -> (((APm)GA) = (AG(APm)) <-> ((AP(k + 1))GA) = (AG(AP(k + 1)))))
13		opreq2 4890	. . . . 5 |- (m = -uk -> (APm) = (AP-uk))
14	13	opreq1d 4897	. . . 4 |- (m = -uk -> ((APm)GA) = ((AP-uk)GA))
15	13	opreq2d 4898	. . . 4 |- (m = -uk -> (AG(APm)) = (AG(AP-uk)))
16	14, 15	eqeq12d 1899	. . 3 |- (m = -uk -> (((APm)GA) = (AG(APm)) <-> ((AP-uk)GA) = (AG(AP-uk))))
17		opreq2 4890	. . . . 5 |- (m = K -> (APm) = (APK))
18	17	opreq1d 4897	. . . 4 |- (m = K -> ((APm)GA) = ((APK)GA))
19	17	opreq2d 4898	. . . 4 |- (m = K -> (AG(APm)) = (AG(APK)))
20	18, 19	eqeq12d 1899	. . 3 |- (m = K -> (((APm)GA) = (AG(APm)) <-> ((APK)GA) = (AG(APK))))
21		gxcom.1	. . . . . . 7 |- X = ran G
22		eqid 1884	. . . . . . 7 |- (Id` G) = (Id` G)
23		gxcom.2	. . . . . . 7 |- P = (^g` G)
24	21, 22, 23	gx0 9384	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (AP0) = (Id` G))
25	24	opreq1d 4897	. . . . 5 |- ((G e. Grp /\ A e. X) -> ((AP0)GA) = ((Id` G)GA))
26	21, 22	grplid 9345	. . . . 5 |- ((G e. Grp /\ A e. X) -> ((Id` G)GA) = A)
27	25, 26	eqtrd 1925	. . . 4 |- ((G e. Grp /\ A e. X) -> ((AP0)GA) = A)
28	24	opreq2d 4898	. . . . 5 |- ((G e. Grp /\ A e. X) -> (AG(AP0)) = (AG(Id` G)))
29	21, 22	grprid 9346	. . . . 5 |- ((G e. Grp /\ A e. X) -> (AG(Id` G)) = A)
30	28, 29	eqtrd 1925	. . . 4 |- ((G e. Grp /\ A e. X) -> (AG(AP0)) = A)
31	27, 30	eqtr4d 1928	. . 3 |- ((G e. Grp /\ A e. X) -> ((AP0)GA) = (AG(AP0)))
32		simp1 876	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> G e. Grp)
33		simp2 877	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> A e. X)
34	21, 23	gxcl 9388	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (APk) e. X)
35	21	grpass 9327	. . . . . . . . 9 |- ((G e. Grp /\ (A e. X /\ (APk) e. X /\ A e. X)) -> ((AG(APk))GA) = (AG((APk)GA)))
36	32, 33, 34, 33, 35	syl13anc 1102	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> ((AG(APk))GA) = (AG((APk)GA)))
37		nn0z 7363	. . . . . . . 8 |- (k e. NN0 -> k e. ZZ)
38	36, 37	syl3an3 1132	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> ((AG(APk))GA) = (AG((APk)GA)))
39	38	adantr 425	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((APk)GA) = (AG(APk))) -> ((AG(APk))GA) = (AG((APk)GA)))
40	21, 23	gxnn0suc 9387	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (AP(k + 1)) = ((APk)GA))
41	40	eqeq1d 1892	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> ((AP(k + 1)) = (AG(APk)) <-> ((APk)GA) = (AG(APk))))
42	41	biimpar 461	. . . . . . 7 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((APk)GA) = (AG(APk))) -> (AP(k + 1)) = (AG(APk)))
43	42	opreq1d 4897	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((APk)GA) = (AG(APk))) -> ((AP(k + 1))GA) = ((AG(APk))GA))
44	40	opreq2d 4898	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (AG(AP(k + 1))) = (AG((APk)GA)))
45	44	adantr 425	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((APk)GA) = (AG(APk))) -> (AG(AP(k + 1))) = (AG((APk)GA)))
46	39, 43, 45	3eqtr4d 1937	. . . . 5 |- (((G e. Grp /\ A e. X /\ k e. NN0) /\ ((APk)GA) = (AG(APk))) -> ((AP(k + 1))GA) = (AG(AP(k + 1))))
47	46	ex 402	. . . 4 |- ((G e. Grp /\ A e. X /\ k e. NN0) -> (((APk)GA) = (AG(APk)) -> ((AP(k + 1))GA) = (AG(AP(k + 1)))))
48	47	3expia 1069	. . 3 |- ((G e. Grp /\ A e. X) -> (k e. NN0 -> (((APk)GA) = (AG(APk)) -> ((AP(k + 1))GA) = (AG(AP(k + 1))))))
49		simpl1 879	. . . . . . . . . 10 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> G e. Grp)
50		simpl2 880	. . . . . . . . . 10 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> A e. X)
51		simpl3 881	. . . . . . . . . . 11 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> k e. ZZ)
52	21, 23	gxcl 9388	. . . . . . . . . . . 12 |- ((G e. Grp /\ A e. X /\ -uk e. ZZ) -> (AP-uk) e. X)
53		znegcl 7372	. . . . . . . . . . . 12 |- (k e. ZZ -> -uk e. ZZ)
54	52, 53	syl3an3 1132	. . . . . . . . . . 11 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (AP-uk) e. X)
55	49, 50, 51, 54	syl111anc 1100	. . . . . . . . . 10 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (AP-uk) e. X)
56		eqid 1884	. . . . . . . . . . . 12 |- (inv` G) = (inv` G)
57	21, 56	grpinvcl 9352	. . . . . . . . . . 11 |- ((G e. Grp /\ A e. X) -> ((inv` G)` A) e. X)
58	49, 50, 57	syl11anc 524	. . . . . . . . . 10 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((inv` G)` A) e. X)
59	21	grpass 9327	. . . . . . . . . 10 |- ((G e. Grp /\ (A e. X /\ (AP-uk) e. X /\ ((inv` G)` A) e. X)) -> ((AG(AP-uk))G((inv` G)` A)) = (AG((AP-uk)G((inv` G)` A))))
60	49, 50, 55, 58, 59	syl13anc 1102	. . . . . . . . 9 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((AG(AP-uk))G((inv` G)` A)) = (AG((AP-uk)G((inv` G)` A))))
61		fveq2 4681	. . . . . . . . . . . 12 |- (((APk)GA) = (AG(APk)) -> ((inv` G)` ((APk)GA)) = ((inv` G)` (AG(APk))))
62	61	adantl 424	. . . . . . . . . . 11 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((inv` G)` ((APk)GA)) = ((inv` G)` (AG(APk))))
63	49, 50, 51, 34	syl111anc 1100	. . . . . . . . . . . . 13 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (APk) e. X)
64	21, 56	grpinvop 9365	. . . . . . . . . . . . 13 |- ((G e. Grp /\ (APk) e. X /\ A e. X) -> ((inv` G)` ((APk)GA)) = (((inv` G)` A)G((inv` G)` (APk))))
65	49, 63, 50, 64	syl111anc 1100	. . . . . . . . . . . 12 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((inv` G)` ((APk)GA)) = (((inv` G)` A)G((inv` G)` (APk))))
66	21, 56, 23	gxneg 9389	. . . . . . . . . . . . . 14 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (AP-uk) = ((inv` G)` (APk)))
67	49, 50, 51, 66	syl111anc 1100	. . . . . . . . . . . . 13 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (AP-uk) = ((inv` G)` (APk)))
68	67	opreq2d 4898	. . . . . . . . . . . 12 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (((inv` G)` A)G(AP-uk)) = (((inv` G)` A)G((inv` G)` (APk))))
69	65, 68	eqtr4d 1928	. . . . . . . . . . 11 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((inv` G)` ((APk)GA)) = (((inv` G)` A)G(AP-uk)))
70	21, 56	grpinvop 9365	. . . . . . . . . . . . 13 |- ((G e. Grp /\ A e. X /\ (APk) e. X) -> ((inv` G)` (AG(APk))) = (((inv` G)` (APk))G((inv` G)` A)))
71	49, 50, 63, 70	syl111anc 1100	. . . . . . . . . . . 12 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((inv` G)` (AG(APk))) = (((inv` G)` (APk))G((inv` G)` A)))
72	67	opreq1d 4897	. . . . . . . . . . . 12 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((AP-uk)G((inv` G)` A)) = (((inv` G)` (APk))G((inv` G)` A)))
73	71, 72	eqtr4d 1928	. . . . . . . . . . 11 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((inv` G)` (AG(APk))) = ((AP-uk)G((inv` G)` A)))
74	62, 69, 73	3eqtr3rd 1936	. . . . . . . . . 10 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((AP-uk)G((inv` G)` A)) = (((inv` G)` A)G(AP-uk)))
75	74	opreq2d 4898	. . . . . . . . 9 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (AG((AP-uk)G((inv` G)` A))) = (AG(((inv` G)` A)G(AP-uk))))
76	21, 56	grpasscan1 9361	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X /\ (AP-uk) e. X) -> (AG(((inv` G)` A)G(AP-uk))) = (AP-uk))
77	49, 50, 55, 76	syl111anc 1100	. . . . . . . . 9 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (AG(((inv` G)` A)G(AP-uk))) = (AP-uk))
78	60, 75, 77	3eqtrd 1929	. . . . . . . 8 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((AG(AP-uk))G((inv` G)` A)) = (AP-uk))
79	78	opreq1d 4897	. . . . . . 7 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (((AG(AP-uk))G((inv` G)` A))GA) = ((AP-uk)GA))
80	21	grpcl 9324	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ (AP-uk) e. X) -> (AG(AP-uk)) e. X)
81	49, 50, 55, 80	syl111anc 1100	. . . . . . . 8 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (AG(AP-uk)) e. X)
82	21, 56	grpasscan2 9362	. . . . . . . 8 |- ((G e. Grp /\ (AG(AP-uk)) e. X /\ A e. X) -> (((AG(AP-uk))G((inv` G)` A))GA) = (AG(AP-uk)))
83	49, 81, 50, 82	syl111anc 1100	. . . . . . 7 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> (((AG(AP-uk))G((inv` G)` A))GA) = (AG(AP-uk)))
84	79, 83	eqtr3d 1927	. . . . . 6 |- (((G e. Grp /\ A e. X /\ k e. ZZ) /\ ((APk)GA) = (AG(APk))) -> ((AP-uk)GA) = (AG(AP-uk)))
85	84	ex 402	. . . . 5 |- ((G e. Grp /\ A e. X /\ k e. ZZ) -> (((APk)GA) = (AG(APk)) -> ((AP-uk)GA) = (AG(AP-uk))))
86	85	3expia 1069	. . . 4 |- ((G e. Grp /\ A e. X) -> (k e. ZZ -> (((APk)GA) = (AG(APk)) -> ((AP-uk)GA) = (AG(AP-uk)))))
87		nnz 7362	. . . 4 |- (k e. NN -> k e. ZZ)
88	86, 87	syl5 20	. . 3 |- ((G e. Grp /\ A e. X) -> (k e. NN -> (((APk)GA) = (AG(APk)) -> ((AP-uk)GA) = (AG(AP-uk)))))
89	4, 8, 12, 16, 20, 31, 48, 88	zindd 7427	. 2 |- ((G e. Grp /\ A e. X) -> (K e. ZZ -> ((APK)GA) = (AG(APK))))
90	89	3impia 1064	1 |- ((G e. Grp /\ A e. X /\ K e. ZZ) -> ((APK)GA) = (AG(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:  gxinv 9393  gxsuc 9395

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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