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Theorem expcllem 10780

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 9379	. 2 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
2		oveq2 6015	. . . . . . 7 ⊢ (𝑧 = 1 → (𝐴↑𝑧) = (𝐴↑1))
3	2	eleq1d 2298	. . . . . 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 6015	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝐴↑𝑧) = (𝐴↑𝑤))
6	5	eleq1d 2298	. . . . . 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 6015	. . . . . . 7 ⊢ (𝑧 = (𝑤 + 1) → (𝐴↑𝑧) = (𝐴↑(𝑤 + 1)))
9	8	eleq1d 2298	. . . . . 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 6015	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴↑𝑧) = (𝐴↑𝐵))
12	11	eleq1d 2298	. . . . . 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 3220	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ ℂ)
16		exp1 10775	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴)
17	15, 16	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑1) = 𝐴)
18	17	eleq1d 2298	. . . . . 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 6166	. . . . . . . . . . 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 9384	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℕ → 𝑤 ∈ ℕ₀)
25		expp1 10776	. . . . . . . . . . . 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 2298	. . . . . . . . . 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 9134	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐴 ∈ 𝐹 → (𝐴↑𝐵) ∈ 𝐹))
34	33	impcom 125	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ) → (𝐴↑𝐵) ∈ 𝐹)
35		oveq2 6015	. . . . 5 ⊢ (𝐵 = 0 → (𝐴↑𝐵) = (𝐴↑0))
36		exp0 10773	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
37	15, 36	syl 14	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑0) = 1)
38	35, 37	sylan9eqr 2284	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) = 1)
39		expcllem.3	. . . 4 ⊢ 1 ∈ 𝐹
40	38, 39	eqeltrdi 2320	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) ∈ 𝐹)
41	34, 40	jaodan 802	. 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 713 = wceq 1395 ∈ wcel 2200 ⊆ wss 3197 (class class class)co 6007 ℂcc 8005 0cc0 8007 1c1 8008 + caddc 8010 · cmul 8012 ℕcn 9118 ℕ₀cn0 9377 ↑cexp 10768

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-n0 9378 df-z 9455 df-uz 9731 df-seqfrec 10678 df-exp 10769

This theorem is referenced by: expcl2lemap 10781 nnexpcl 10782 nn0expcl 10783 zexpcl 10784 qexpcl 10785 reexpcl 10786 expcl 10787 expge0 10805 expge1 10806 lgsfcl2 15693

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