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Theorem expcllem 10466

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 9116	. 2 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
2		oveq2 5850	. . . . . . 7 ⊢ (𝑧 = 1 → (𝐴↑𝑧) = (𝐴↑1))
3	2	eleq1d 2235	. . . . . 6 ⊢ (𝑧 = 1 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑1) ∈ 𝐹))
4	3	imbi2d 229	. . . . 5 ⊢ (𝑧 = 1 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑1) ∈ 𝐹)))
5		oveq2 5850	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝐴↑𝑧) = (𝐴↑𝑤))
6	5	eleq1d 2235	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑𝑤) ∈ 𝐹))
7	6	imbi2d 229	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑𝑤) ∈ 𝐹)))
8		oveq2 5850	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (𝐴↑𝑧) = (𝐴↑(𝑤 + 1)))
9	8	eleq1d 2235	. . . . . 6 ⊢ (𝑧 = (𝑤 + 1) → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑(𝑤 + 1)) ∈ 𝐹))
10	9	imbi2d 229	. . . . 5 ⊢ (𝑧 = (𝑤 + 1) → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹)))
11		oveq2 5850	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴↑𝑧) = (𝐴↑𝐵))
12	11	eleq1d 2235	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑𝐵) ∈ 𝐹))
13	12	imbi2d 229	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑𝐵) ∈ 𝐹)))
14		expcllem.1	. . . . . . . . 9 ⊢ 𝐹 ⊆ ℂ
15	14	sseli 3138	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ ℂ)
16		exp1 10461	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴)
17	15, 16	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑1) = 𝐴)
18	17	eleq1d 2235	. . . . . 6 ⊢ (𝐴 ∈ 𝐹 → ((𝐴↑1) ∈ 𝐹 ↔ 𝐴 ∈ 𝐹))
19	18	ibir 176	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑1) ∈ 𝐹)
20		expcllem.2	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹)
21	20	caovcl 5996	. . . . . . . . . . 11 ⊢ (((𝐴↑𝑤) ∈ 𝐹 ∧ 𝐴 ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)
22	21	ancoms 266	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐹 ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)
23	22	adantlr 469	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)
24		nnnn0 9121	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℕ → 𝑤 ∈ ℕ₀)
25		expp1 10462	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℕ₀) → (𝐴↑(𝑤 + 1)) = ((𝐴↑𝑤) · 𝐴))
26	15, 24, 25	syl2an 287	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) → (𝐴↑(𝑤 + 1)) = ((𝐴↑𝑤) · 𝐴))
27	26	eleq1d 2235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) → ((𝐴↑(𝑤 + 1)) ∈ 𝐹 ↔ ((𝐴↑𝑤) · 𝐴) ∈ 𝐹))
28	27	adantr 274	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑(𝑤 + 1)) ∈ 𝐹 ↔ ((𝐴↑𝑤) · 𝐴) ∈ 𝐹))
29	23, 28	mpbird 166	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → (𝐴↑(𝑤 + 1)) ∈ 𝐹)
30	29	exp31 362	. . . . . . 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 8873	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ 𝐹 → (𝐴↑𝐵) ∈ 𝐹))
34	33	impcom 124	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ) → (𝐴↑𝐵) ∈ 𝐹)
35		oveq2 5850	. . . . 5 ⊢ (𝐵 = 0 → (𝐴↑𝐵) = (𝐴↑0))
36		exp0 10459	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
37	15, 36	syl 14	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑0) = 1)
38	35, 37	sylan9eqr 2221	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) = 1)
39		expcllem.3	. . . 4 ⊢ 1 ∈ 𝐹
40	38, 39	eqeltrdi 2257	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) ∈ 𝐹)
41	34, 40	jaodan 787	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝐴↑𝐵) ∈ 𝐹)
42	1, 41	sylan2b 285	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ₀) → (𝐴↑𝐵) ∈ 𝐹)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1343 ∈ wcel 2136 ⊆ wss 3116 (class class class)co 5842 ℂcc 7751 0cc0 7753 1c1 7754 + caddc 7756 · cmul 7758 ℕcn 8857 ℕ₀cn0 9114 ↑cexp 10454

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-n0 9115 df-z 9192 df-uz 9467 df-seqfrec 10381 df-exp 10455

This theorem is referenced by: expcl2lemap 10467 nnexpcl 10468 nn0expcl 10469 zexpcl 10470 qexpcl 10471 reexpcl 10472 expcl 10473 expge0 10491 expge1 10492 lgsfcl2 13547

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