Theorem gxnn0suc 9387

Description: Induction on group power (lemma with nonnegative exponent - use gxsuc 9395 instead). (Contributed by Paul Chapman, 17-Apr-2009.)

Hypotheses

Ref	Expression
gxnn0suc.1	|- X = ran G
gxnn0suc.2	|- P = (^g` G)

Assertion

Ref	Expression
gxnn0suc	|- ((G e. Grp /\ A e. X /\ K e. NN0) -> (AP(K + 1)) = ((APK)GA))

Proof of Theorem gxnn0suc

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . . . . . . . 12 |- (g = G -> (g seq1 (NN X. {A})) = (G seq1 (NN X. {A})))
2	1	fveq1d 4683	. . . . . . . . . . 11 |- (g = G -> ((g seq1 (NN X. {A}))` (K + 1)) = ((G seq1 (NN X. {A}))` (K + 1)))
3	1	fveq1d 4683	. . . . . . . . . . . . 13 |- (g = G -> ((g seq1 (NN X. {A}))` K) = ((G seq1 (NN X. {A}))` K))
4	3	opreq1d 4897	. . . . . . . . . . . 12 |- (g = G -> (((g seq1 (NN X. {A}))` K)g((NN X. {A})` (K + 1))) = (((G seq1 (NN X. {A}))` K)g((NN X. {A})` (K + 1))))
5		opreq 4888	. . . . . . . . . . . 12 |- (g = G -> (((G seq1 (NN X. {A}))` K)g((NN X. {A})` (K + 1))) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1))))
6	4, 5	eqtrd 1925	. . . . . . . . . . 11 |- (g = G -> (((g seq1 (NN X. {A}))` K)g((NN X. {A})` (K + 1))) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1))))
7	2, 6	eqeq12d 1899	. . . . . . . . . 10 |- (g = G -> (((g seq1 (NN X. {A}))` (K + 1)) = (((g seq1 (NN X. {A}))` K)g((NN X. {A})` (K + 1))) <-> ((G seq1 (NN X. {A}))` (K + 1)) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1)))))
8	7	imbi2d 674	. . . . . . . . 9 |- (g = G -> ((K e. NN -> ((g seq1 (NN X. {A}))` (K + 1)) = (((g seq1 (NN X. {A}))` K)g((NN X. {A})` (K + 1)))) <-> (K e. NN -> ((G seq1 (NN X. {A}))` (K + 1)) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1))))))
9		visset 2295	. . . . . . . . . 10 |- g e. _V
10		nnex 7116	. . . . . . . . . . 11 |- NN e. _V
11		snex 3492	. . . . . . . . . . 11 |- {A} e. _V
12	10, 11	xpex 4096	. . . . . . . . . 10 |- (NN X. {A}) e. _V
13	9, 12	seq1p1 7731	. . . . . . . . 9 |- (K e. NN -> ((g seq1 (NN X. {A}))` (K + 1)) = (((g seq1 (NN X. {A}))` K)g((NN X. {A})` (K + 1))))
14	8, 13	vtoclg 2346	. . . . . . . 8 |- (G e. Grp -> (K e. NN -> ((G seq1 (NN X. {A}))` (K + 1)) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1)))))
15	14	imp 377	. . . . . . 7 |- ((G e. Grp /\ K e. NN) -> ((G seq1 (NN X. {A}))` (K + 1)) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1))))
16	15	3adant2 895	. . . . . 6 |- ((G e. Grp /\ A e. X /\ K e. NN) -> ((G seq1 (NN X. {A}))` (K + 1)) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1))))
17		gxnn0suc.1	. . . . . . . 8 |- X = ran G
18		gxnn0suc.2	. . . . . . . 8 |- P = (^g` G)
19	17, 18	gxpval 9382	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ (K + 1) e. NN) -> (AP(K + 1)) = ((G seq1 (NN X. {A}))` (K + 1)))
20		peano2nn 7118	. . . . . . 7 |- (K e. NN -> (K + 1) e. NN)
21	19, 20	syl3an3 1132	. . . . . 6 |- ((G e. Grp /\ A e. X /\ K e. NN) -> (AP(K + 1)) = ((G seq1 (NN X. {A}))` (K + 1)))
22	17, 18	gxpval 9382	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ K e. NN) -> (APK) = ((G seq1 (NN X. {A}))` K))
23		fvconst 4814	. . . . . . . . . 10 |- (((NN X. {A}):NN-->{A} /\ (K + 1) e. NN) -> ((NN X. {A})` (K + 1)) = A)
24		fconstg 4604	. . . . . . . . . 10 |- (A e. X -> (NN X. {A}):NN-->{A})
25	23, 24, 20	syl2an 503	. . . . . . . . 9 |- ((A e. X /\ K e. NN) -> ((NN X. {A})` (K + 1)) = A)
26	25	3adant1 894	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ K e. NN) -> ((NN X. {A})` (K + 1)) = A)
27	26	eqcomd 1889	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ K e. NN) -> A = ((NN X. {A})` (K + 1)))
28	22, 27	opreq12d 4900	. . . . . 6 |- ((G e. Grp /\ A e. X /\ K e. NN) -> ((APK)GA) = (((G seq1 (NN X. {A}))` K)G((NN X. {A})` (K + 1))))
29	16, 21, 28	3eqtr4d 1937	. . . . 5 |- ((G e. Grp /\ A e. X /\ K e. NN) -> (AP(K + 1)) = ((APK)GA))
30	29	3expia 1069	. . . 4 |- ((G e. Grp /\ A e. X) -> (K e. NN -> (AP(K + 1)) = ((APK)GA)))
31	17, 18	gx1 9385	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> (AP1) = A)
32		eqid 1884	. . . . . . . . . . 11 |- (Id` G) = (Id` G)
33	17, 32, 18	gx0 9384	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X) -> (AP0) = (Id` G))
34	33	opreq1d 4897	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> ((AP0)GA) = ((Id` G)GA))
35	17, 32	grplid 9345	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> ((Id` G)GA) = A)
36	34, 35	eqtrd 1925	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> ((AP0)GA) = A)
37	31, 36	eqtr4d 1928	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> (AP1) = ((AP0)GA))
38	37	3adant3 896	. . . . . 6 |- ((G e. Grp /\ A e. X /\ K = 0) -> (AP1) = ((AP0)GA))
39		opreq1 4889	. . . . . . . . . 10 |- (K = 0 -> (K + 1) = (0 + 1))
40		ax1cn 6422	. . . . . . . . . . 11 |- 1 e. CC
41	40	addid2i 6484	. . . . . . . . . 10 |- (0 + 1) = 1
42	39, 41	syl6eq 1944	. . . . . . . . 9 |- (K = 0 -> (K + 1) = 1)
43	42	opreq2d 4898	. . . . . . . 8 |- (K = 0 -> (AP(K + 1)) = (AP1))
44		opreq2 4890	. . . . . . . . 9 |- (K = 0 -> (APK) = (AP0))
45	44	opreq1d 4897	. . . . . . . 8 |- (K = 0 -> ((APK)GA) = ((AP0)GA))
46	43, 45	eqeq12d 1899	. . . . . . 7 |- (K = 0 -> ((AP(K + 1)) = ((APK)GA) <-> (AP1) = ((AP0)GA)))
47	46	3ad2ant3 899	. . . . . 6 |- ((G e. Grp /\ A e. X /\ K = 0) -> ((AP(K + 1)) = ((APK)GA) <-> (AP1) = ((AP0)GA)))
48	38, 47	mpbird 213	. . . . 5 |- ((G e. Grp /\ A e. X /\ K = 0) -> (AP(K + 1)) = ((APK)GA))
49	48	3expia 1069	. . . 4 |- ((G e. Grp /\ A e. X) -> (K = 0 -> (AP(K + 1)) = ((APK)GA)))
50	30, 49	jaod 469	. . 3 |- ((G e. Grp /\ A e. X) -> ((K e. NN \/ K = 0) -> (AP(K + 1)) = ((APK)GA)))
51		elnn0 7310	. . 3 |- (K e. NN0 <-> (K e. NN \/ K = 0))
52	50, 51	syl5ib 223	. 2 |- ((G e. Grp /\ A e. X) -> (K e. NN0 -> (AP(K + 1)) = ((APK)GA)))
53	52	3impia 1064	1 |- ((G e. Grp /\ A e. X /\ K e. NN0) -> (AP(K + 1)) = ((APK)GA))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {csn 3044   X. cxp 3984  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  0cc0 6386  1c1 6387   + caddc 6389  NNcn 6449  NN0cn0 6450   seq1 cseq1 7720  Grpcgr 9311  Idcgi 9312  ^gcgx 9315

This theorem is referenced by:  gxcl 9388  gxcom 9392  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-gx 9320

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