Theorem expcllem 10693

Description: Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)

Hypotheses

Ref	Expression
expcllem.1	|- F C_ CC
expcllem.2	|- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )
expcllem.3	|- 1 e. F

Assertion

Ref	Expression
expcllem	|- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )

Distinct variable groups:   x, y, A   x, B   x, F, y

Allowed substitution hint:   B( y)

Proof of Theorem expcllem

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 9296	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2		oveq2 5951	. . . . . . 7 |- ( z = 1 -> ( A ^ z ) = ( A ^ 1 ) )
3	2	eleq1d 2273	. . . . . 6 |- ( z = 1 -> ( ( A ^ z ) e. F <-> ( A ^ 1 ) e. F ) )
4	3	imbi2d 230	. . . . 5 |- ( z = 1 -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ 1 ) e. F ) ) )
5		oveq2 5951	. . . . . . 7 |- ( z = w -> ( A ^ z ) = ( A ^ w ) )
6	5	eleq1d 2273	. . . . . 6 |- ( z = w -> ( ( A ^ z ) e. F <-> ( A ^ w ) e. F ) )
7	6	imbi2d 230	. . . . 5 |- ( z = w -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ w ) e. F ) ) )
8		oveq2 5951	. . . . . . 7 |- ( z = ( w + 1 ) -> ( A ^ z ) = ( A ^ ( w + 1 ) ) )
9	8	eleq1d 2273	. . . . . 6 |- ( z = ( w + 1 ) -> ( ( A ^ z ) e. F <-> ( A ^ ( w + 1 ) ) e. F ) )
10	9	imbi2d 230	. . . . 5 |- ( z = ( w + 1 ) -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
11		oveq2 5951	. . . . . . 7 |- ( z = B -> ( A ^ z ) = ( A ^ B ) )
12	11	eleq1d 2273	. . . . . 6 |- ( z = B -> ( ( A ^ z ) e. F <-> ( A ^ B ) e. F ) )
13	12	imbi2d 230	. . . . 5 |- ( z = B -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ B ) e. F ) ) )
14		expcllem.1	. . . . . . . . 9 |- F C_ CC
15	14	sseli 3188	. . . . . . . 8 |- ( A e. F -> A e. CC )
16		exp1 10688	. . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
17	15, 16	syl 14	. . . . . . 7 |- ( A e. F -> ( A ^ 1 ) = A )
18	17	eleq1d 2273	. . . . . 6 |- ( A e. F -> ( ( A ^ 1 ) e. F <-> A e. F ) )
19	18	ibir 177	. . . . 5 |- ( A e. F -> ( A ^ 1 ) e. F )
20		expcllem.2	. . . . . . . . . . . 12 |- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )
21	20	caovcl 6100	. . . . . . . . . . 11 |- ( ( ( A ^ w ) e. F /\ A e. F ) -> ( ( A ^ w ) x. A ) e. F )
22	21	ancoms 268	. . . . . . . . . 10 |- ( ( A e. F /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
23	22	adantlr 477	. . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
24		nnnn0 9301	. . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25		expp1 10689	. . . . . . . . . . . 12 |- ( ( A e. CC /\ w e. NN0 ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
26	15, 24, 25	syl2an 289	. . . . . . . . . . 11 |- ( ( A e. F /\ w e. NN ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
27	26	eleq1d 2273	. . . . . . . . . 10 |- ( ( A e. F /\ w e. NN ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
28	27	adantr 276	. . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
29	23, 28	mpbird 167	. . . . . . . 8 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( A ^ ( w + 1 ) ) e. F )
30	29	exp31 364	. . . . . . 7 |- ( A e. F -> ( w e. NN -> ( ( A ^ w ) e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
31	30	com12 30	. . . . . 6 |- ( w e. NN -> ( A e. F -> ( ( A ^ w ) e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
32	31	a2d 26	. . . . 5 |- ( w e. NN -> ( ( A e. F -> ( A ^ w ) e. F ) -> ( A e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
33	4, 7, 10, 13, 19, 32	nnind 9051	. . . 4 |- ( B e. NN -> ( A e. F -> ( A ^ B ) e. F ) )
34	33	impcom 125	. . 3 |- ( ( A e. F /\ B e. NN ) -> ( A ^ B ) e. F )
35		oveq2 5951	. . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36		exp0 10686	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
37	15, 36	syl 14	. . . . 5 |- ( A e. F -> ( A ^ 0 ) = 1 )
38	35, 37	sylan9eqr 2259	. . . 4 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) = 1 )
39		expcllem.3	. . . 4 |- 1 e. F
40	38, 39	eqeltrdi 2295	. . 3 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) e. F )
41	34, 40	jaodan 798	. 2 |- ( ( A e. F /\ ( B e. NN \/ B = 0 ) ) -> ( A ^ B ) e. F )
42	1, 41	sylan2b 287	1 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1372   e. wcel 2175   C_ wss 3165  (class class class)co 5943   CCcc 7922   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   NNcn 9035   NN0cn0 9294   ^cexp 10681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-seqfrec 10591  df-exp 10682

This theorem is referenced by:  expcl2lemap  10694  nnexpcl  10695  nn0expcl  10696  zexpcl  10697  qexpcl  10698  reexpcl  10699  expcl  10700  expge0  10718  expge1  10719  lgsfcl2  15454