Theorem expcllem 10081

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 8773	. 2 |- ( B e. NN0 <-> ( B e. NN \/ B = 0 ) )
2		oveq2 5698	. . . . . . 7 |- ( z = 1 -> ( A ^ z ) = ( A ^ 1 ) )
3	2	eleq1d 2163	. . . . . 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 5698	. . . . . . 7 |- ( z = w -> ( A ^ z ) = ( A ^ w ) )
6	5	eleq1d 2163	. . . . . 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 5698	. . . . . . 7 |- ( z = ( w + 1 ) -> ( A ^ z ) = ( A ^ ( w + 1 ) ) )
9	8	eleq1d 2163	. . . . . 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 5698	. . . . . . 7 |- ( z = B -> ( A ^ z ) = ( A ^ B ) )
12	11	eleq1d 2163	. . . . . 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 3035	. . . . . . . 8 |- ( A e. F -> A e. CC )
16		exp1 10076	. . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
17	15, 16	syl 14	. . . . . . 7 |- ( A e. F -> ( A ^ 1 ) = A )
18	17	eleq1d 2163	. . . . . 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 5837	. . . . . . . . . . 11 |- ( ( ( A ^ w ) e. F /\ A e. F ) -> ( ( A ^ w ) x. A ) e. F )
22	21	ancoms 265	. . . . . . . . . 10 |- ( ( A e. F /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
23	22	adantlr 462	. . . . . . . . 9 |- ( ( ( A e. F /\ w e. NN ) /\ ( A ^ w ) e. F ) -> ( ( A ^ w ) x. A ) e. F )
24		nnnn0 8778	. . . . . . . . . . . 12 |- ( w e. NN -> w e. NN0 )
25		expp1 10077	. . . . . . . . . . . 12 |- ( ( A e. CC /\ w e. NN0 ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
26	15, 24, 25	syl2an 284	. . . . . . . . . . 11 |- ( ( A e. F /\ w e. NN ) -> ( A ^ ( w + 1 ) ) = ( ( A ^ w ) x. A ) )
27	26	eleq1d 2163	. . . . . . . . . 10 |- ( ( A e. F /\ w e. NN ) -> ( ( A ^ ( w + 1 ) ) e. F <-> ( ( A ^ w ) x. A ) e. F ) )
28	27	adantr 271	. . . . . . . . 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 357	. . . . . . 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 8536	. . . 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 5698	. . . . 5 |- ( B = 0 -> ( A ^ B ) = ( A ^ 0 ) )
36		exp0 10074	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
37	15, 36	syl 14	. . . . 5 |- ( A e. F -> ( A ^ 0 ) = 1 )
38	35, 37	sylan9eqr 2149	. . . 4 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) = 1 )
39		expcllem.3	. . . 4 |- 1 e. F
40	38, 39	syl6eqel 2185	. . 3 |- ( ( A e. F /\ B = 0 ) -> ( A ^ B ) e. F )
41	34, 40	jaodan 749	. 2 |- ( ( A e. F /\ ( B e. NN \/ B = 0 ) ) -> ( A ^ B ) e. F )
42	1, 41	sylan2b 282	1 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 667   = wceq 1296   e. wcel 1445   C_ wss 3013  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448   + caddc 7450   x. cmul 7452   NNcn 8520   NN0cn0 8771   ^cexp 10069

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-seqfrec 10001  df-exp 10070

This theorem is referenced by:  expcl2lemap  10082  nnexpcl  10083  nn0expcl  10084  zexpcl  10085  qexpcl  10086  reexpcl  10087  expcl  10088  expge0  10106  expge1  10107