Theorem expcllem 10297

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 8972	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2		oveq2 5775	. . . . . . 7 |- ( z = 1 -> ( A ^ z ) = ( A ^ 1 ) )
3	2	eleq1d 2206	. . . . . 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 5775	. . . . . . 7 |- ( z = w -> ( A ^ z ) = ( A ^ w ) )
6	5	eleq1d 2206	. . . . . 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 5775	. . . . . . 7 |- ( z = ( w + 1 ) -> ( A ^ z ) = ( A ^ ( w + 1 ) ) )
9	8	eleq1d 2206	. . . . . 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 5775	. . . . . . 7 |- ( z = B -> ( A ^ z ) = ( A ^ B ) )
12	11	eleq1d 2206	. . . . . 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 3088	. . . . . . . 8 |- ( A e. F -> A e. CC )
16		exp1 10292	. . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
17	15, 16	syl 14	. . . . . . 7 |- ( A e. F -> ( A ^ 1 ) = A )
18	17	eleq1d 2206	. . . . . 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 5918	. . . . . . . . . . 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 468	. . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
24		nnnn0 8977	. . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25		expp1 10293	. . . . . . . . . . . 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 2206	. . . . . . . . . 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 361	. . . . . . 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 8729	. . . 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 5775	. . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36		exp0 10290	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
37	15, 36	syl 14	. . . . 5 |- ( A e. F -> ( A ^ 0 ) = 1 )
38	35, 37	sylan9eqr 2192	. . . 4 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) = 1 )
39		expcllem.3	. . . 4 |- 1 e. F
40	38, 39	eqeltrdi 2228	. . 3 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) e. F )
41	34, 40	jaodan 786	. 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 697   = wceq 1331   e. wcel 1480   C_ wss 3066  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614   + caddc 7616   x. cmul 7618   NNcn 8713   NN0cn0 8970   ^cexp 10285

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-seqfrec 10212  df-exp 10286

This theorem is referenced by:  expcl2lemap  10298  nnexpcl  10299  nn0expcl  10300  zexpcl  10301  qexpcl  10302  reexpcl  10303  expcl  10304  expge0  10322  expge1  10323