Theorem gxpval 9382

Description: The result of the group power operator when the exponent is positive. (Contributed by Paul Chapman, 17-Apr-2009.)

Hypotheses

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

Assertion

Ref	Expression
gxpval	|- ((G e. Grp /\ A e. X /\ K e. NN) -> (APK) = ((G seq1 (NN X. {A}))` K))

Proof of Theorem gxpval

Step	Hyp	Ref	Expression
1		gxpval.1	. . . 4 |- X = ran G
2		eqid 1884	. . . 4 |- (Id` G) = (Id` G)
3		eqid 1884	. . . 4 |- (inv` G) = (inv` G)
4		gxpval.2	. . . 4 |- P = (^g` G)
5	1, 2, 3, 4	gxval 9381	. . 3 |- ((G e. Grp /\ A e. X /\ K e. ZZ) -> (APK) = if(K = 0, (Id` G), if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK)))))
6		nnz 7362	. . 3 |- (K e. NN -> K e. ZZ)
7	5, 6	syl3an3 1132	. 2 |- ((G e. Grp /\ A e. X /\ K e. NN) -> (APK) = if(K = 0, (Id` G), if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK)))))
8		nnne0 7132	. . . . . 6 |- (K e. NN -> K =/= 0)
9		df-ne 2019	. . . . . 6 |- (K =/= 0 <-> -. K = 0)
10	8, 9	sylib 215	. . . . 5 |- (K e. NN -> -. K = 0)
11		iffalse 2991	. . . . 5 |- (-. K = 0 -> if(K = 0, (Id` G), if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK)))) = if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK))))
12	10, 11	syl 12	. . . 4 |- (K e. NN -> if(K = 0, (Id` G), if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK)))) = if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK))))
13		nngt0 7129	. . . . 5 |- (K e. NN -> 0 < K)
14		iftrue 2989	. . . . 5 |- (0 < K -> if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK))) = ((G seq1 (NN X. {A}))` K))
15	13, 14	syl 12	. . . 4 |- (K e. NN -> if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK))) = ((G seq1 (NN X. {A}))` K))
16	12, 15	eqtrd 1925	. . 3 |- (K e. NN -> if(K = 0, (Id` G), if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK)))) = ((G seq1 (NN X. {A}))` K))
17	16	3ad2ant3 899	. 2 |- ((G e. Grp /\ A e. X /\ K e. NN) -> if(K = 0, (Id` G), if(0 < K, ((G seq1 (NN X. {A}))` K), ((inv` G)` ((G seq1 (NN X. {A}))` -uK)))) = ((G seq1 (NN X. {A}))` K))
18	7, 17	eqtrd 1925	1 |- ((G e. Grp /\ A e. X /\ K e. NN) -> (APK) = ((G seq1 (NN X. {A}))` K))

Syntax hints:  -. wn 2   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  ifcif 2982  {csn 3044   class class class wbr 3338   X. cxp 3984  ran crn 3987  ` cfv 3998  (class class class)co 4884  0cc0 6386  -ucneg 6446  NNcn 6449  ZZcz 6451   < clt 6653   seq1 cseq1 7720  Grpcgr 9311  Idcgi 9312  invcgn 9313  ^gcgx 9315

This theorem is referenced by:  gx1 9385  gxnn0neg 9386  gxnn0suc 9387

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-z 7345  df-gx 9320

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