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Theorem expcllem 10919

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

**Hypotheses**
Ref	Expression
expcllem.1	⊢ 𝐹 ⊆ ℂ
expcllem.2	⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹)
expcllem.3	⊢ 1 ∈ 𝐹

**Assertion**
Ref	Expression
expcllem	⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ₀) → (𝐴↑𝐵) ∈ 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 9503	. 2 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
2		oveq2 6060	. . . . . . 7 ⊢ (𝑧 = 1 → (𝐴↑𝑧) = (𝐴↑1))
3	2	eleq1d 2303	. . . . . 6 ⊢ (𝑧 = 1 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑1) ∈ 𝐹))
4	3	imbi2d 230	. . . . 5 ⊢ (𝑧 = 1 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑1) ∈ 𝐹)))
5		oveq2 6060	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝐴↑𝑧) = (𝐴↑𝑤))
6	5	eleq1d 2303	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑𝑤) ∈ 𝐹))
7	6	imbi2d 230	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑𝑤) ∈ 𝐹)))
8		oveq2 6060	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (𝐴↑𝑧) = (𝐴↑(𝑤 + 1)))
9	8	eleq1d 2303	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑(𝑤 + 1)) ∈ 𝐹))
10	9	imbi2d 230	. . . . 5 ⊢ (𝑧 = (𝑤 + 1) → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹)))
11		oveq2 6060	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴↑𝑧) = (𝐴↑𝐵))
12	11	eleq1d 2303	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑𝐵) ∈ 𝐹))
13	12	imbi2d 230	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑𝐵) ∈ 𝐹)))
14		expcllem.1	. . . . . . . . 9 ⊢ 𝐹 ⊆ ℂ
15	14	sseli 3236	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ ℂ)
16		exp1 10914	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴)
17	15, 16	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑1) = 𝐴)
18	17	eleq1d 2303	. . . . . 6 ⊢ (𝐴 ∈ 𝐹 → ((𝐴↑1) ∈ 𝐹 ↔ 𝐴 ∈ 𝐹))
19	18	ibir 177	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑1) ∈ 𝐹)
20		expcllem.2	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹)
21	20	caovcl 6211	. . . . . . . . . . 11 ⊢ (((𝐴↑𝑤) ∈ 𝐹 ∧ 𝐴 ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)
22	21	ancoms 268	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐹 ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)
23	22	adantlr 477	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)
24		nnnn0 9508	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℕ → 𝑤 ∈ ℕ₀)
25		expp1 10915	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℕ₀) → (𝐴↑(𝑤 + 1)) = ((𝐴↑𝑤) · 𝐴))
26	15, 24, 25	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) → (𝐴↑(𝑤 + 1)) = ((𝐴↑𝑤) · 𝐴))
27	26	eleq1d 2303	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) → ((𝐴↑(𝑤 + 1)) ∈ 𝐹 ↔ ((𝐴↑𝑤) · 𝐴) ∈ 𝐹))
28	27	adantr 276	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑(𝑤 + 1)) ∈ 𝐹 ↔ ((𝐴↑𝑤) · 𝐴) ∈ 𝐹))
29	23, 28	mpbird 167	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → (𝐴↑(𝑤 + 1)) ∈ 𝐹)
30	29	exp31 364	. . . . . . 7 ⊢ (𝐴 ∈ 𝐹 → (𝑤 ∈ ℕ → ((𝐴↑𝑤) ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹)))
31	30	com12 30	. . . . . 6 ⊢ (𝑤 ∈ ℕ → (𝐴 ∈ 𝐹 → ((𝐴↑𝑤) ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹)))
32	31	a2d 26	. . . . 5 ⊢ (𝑤 ∈ ℕ → ((𝐴 ∈ 𝐹 → (𝐴↑𝑤) ∈ 𝐹) → (𝐴 ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹)))
33	4, 7, 10, 13, 19, 32	nnind 9258	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ 𝐹 → (𝐴↑𝐵) ∈ 𝐹))
34	33	impcom 125	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ) → (𝐴↑𝐵) ∈ 𝐹)
35		oveq2 6060	. . . . 5 ⊢ (𝐵 = 0 → (𝐴↑𝐵) = (𝐴↑0))
36		exp0 10912	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
37	15, 36	syl 14	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑0) = 1)
38	35, 37	sylan9eqr 2289	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) = 1)
39		expcllem.3	. . . 4 ⊢ 1 ∈ 𝐹
40	38, 39	eqeltrdi 2325	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) ∈ 𝐹)
41	34, 40	jaodan 805	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝐴↑𝐵) ∈ 𝐹)
42	1, 41	sylan2b 287	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ₀) → (𝐴↑𝐵) ∈ 𝐹)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 = wceq 1398 ∈ wcel 2205 ⊆ wss 3213 (class class class)co 6052 ℂcc 8130 0cc0 8132 1c1 8133 + caddc 8135 · cmul 8137 ℕcn 9242 ℕ₀cn0 9501 ↑cexp 10907

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-frec 6624 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-n0 9502 df-z 9583 df-uz 9860 df-seqfrec 10817 df-exp 10908

This theorem is referenced by: expcl2lemap 10920 nnexpcl 10921 nn0expcl 10922 zexpcl 10923 qexpcl 10924 reexpcl 10925 expcl 10926 expge0 10944 expge1 10945 lgsfcl2 15928

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