Theorem expcllem 10487

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 9137	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2		oveq2 5861	. . . . . . 7 |- ( z = 1 -> ( A ^ z ) = ( A ^ 1 ) )
3	2	eleq1d 2239	. . . . . 6 |- ( z = 1 -> ( ( A ^ z ) e. F <-> ( A ^ 1 ) e. F ) )
4	3	imbi2d 229	. . . . 5 |- ( z = 1 -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ 1 ) e. F ) ) )
5		oveq2 5861	. . . . . . 7 |- ( z = w -> ( A ^ z ) = ( A ^ w ) )
6	5	eleq1d 2239	. . . . . 6 |- ( z = w -> ( ( A ^ z ) e. F <-> ( A ^ w ) e. F ) )
7	6	imbi2d 229	. . . . 5 |- ( z = w -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ w ) e. F ) ) )
8		oveq2 5861	. . . . . . 7 |- ( z = ( w + 1 ) -> ( A ^ z ) = ( A ^ ( w + 1 ) ) )
9	8	eleq1d 2239	. . . . . 6 |- ( z = ( w + 1 ) -> ( ( A ^ z ) e. F <-> ( A ^ ( w + 1 ) ) e. F ) )
10	9	imbi2d 229	. . . . 5 |- ( z = ( w + 1 ) -> ( ( A e. F -> ( A ^ z ) e. F ) <-> ( A e. F -> ( A ^ ( w + 1 ) ) e. F ) ) )
11		oveq2 5861	. . . . . . 7 |- ( z = B -> ( A ^ z ) = ( A ^ B ) )
12	11	eleq1d 2239	. . . . . 6 |- ( z = B -> ( ( A ^ z ) e. F <-> ( A ^ B ) e. F ) )
13	12	imbi2d 229	. . . . 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 3143	. . . . . . . 8 |- ( A e. F -> A e. CC )
16		exp1 10482	. . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
17	15, 16	syl 14	. . . . . . 7 |- ( A e. F -> ( A ^ 1 ) = A )
18	17	eleq1d 2239	. . . . . 6 |- ( A e. F -> ( ( A ^ 1 ) e. F <-> A e. F ) )
19	18	ibir 176	. . . . 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 6007	. . . . . . . . . . 11 |- ( ( ( A ^ w ) e. F /\ A e. F ) -> ( ( A ^ w ) x. A ) e. F )
22	21	ancoms 266	. . . . . . . . . 10 |- ( ( A e. F /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
23	22	adantlr 474	. . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
24		nnnn0 9142	. . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25		expp1 10483	. . . . . . . . . . . 12 |- ( ( A e. CC /\ w e. NN0 ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
26	15, 24, 25	syl2an 287	. . . . . . . . . . 11 |- ( ( A e. F /\ w e. NN ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
27	26	eleq1d 2239	. . . . . . . . . 10 |- ( ( A e. F /\ w e. NN ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
28	27	adantr 274	. . . . . . . . 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 166	. . . . . . . 8 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( A ^ ( w + 1 ) ) e. F )
30	29	exp31 362	. . . . . . 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 8894	. . . 4 |- ( B e. NN -> ( A e. F -> ( A ^ B ) e. F ) )
34	33	impcom 124	. . 3 |- ( ( A e. F /\ B e. NN ) -> ( A ^ B ) e. F )
35		oveq2 5861	. . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36		exp0 10480	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
37	15, 36	syl 14	. . . . 5 |- ( A e. F -> ( A ^ 0 ) = 1 )
38	35, 37	sylan9eqr 2225	. . . 4 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) = 1 )
39		expcllem.3	. . . 4 |- 1 e. F
40	38, 39	eqeltrdi 2261	. . 3 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) e. F )
41	34, 40	jaodan 792	. 2 |- ( ( A e. F /\ ( B e. NN \/ B = 0 ) ) -> ( A ^ B ) e. F )
42	1, 41	sylan2b 285	1 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   C_ wss 3121  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   NNcn 8878   NN0cn0 9135   ^cexp 10475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476

This theorem is referenced by:  expcl2lemap  10488  nnexpcl  10489  nn0expcl  10490  zexpcl  10491  qexpcl  10492  reexpcl  10493  expcl  10494  expge0  10512  expge1  10513  lgsfcl2  13701