Theorem nnecl 7580

Description: Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 24-Mar-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
nnecl	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑𝑜 𝐵) ∈ ω)

Proof of Theorem nnecl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐵))
2	1	eleq1d 2672	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 𝐵) ∈ ω))
3	2	imbi2d 329	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ↑𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ↑𝑜 𝐵) ∈ ω)))
4		oveq2 6557	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
5	4	eleq1d 2672	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 ∅) ∈ ω))
6		oveq2 6557	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
7	6	eleq1d 2672	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 𝑦) ∈ ω))
8		oveq2 6557	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
9	8	eleq1d 2672	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 suc 𝑦) ∈ ω))
10		nnon 6963	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
11		oe0 7489	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
12	10, 11	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ↑𝑜 ∅) = 1𝑜)
13		df-1o 7447	. . . . . 6 ⊢ 1𝑜 = suc ∅
14		peano1 6977	. . . . . . 7 ⊢ ∅ ∈ ω
15		peano2 6978	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
17	13, 16	eqeltri 2684	. . . . 5 ⊢ 1𝑜 ∈ ω
18	12, 17	syl6eqel 2696	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ↑𝑜 ∅) ∈ ω)
19		nnmcl 7579	. . . . . . . 8 ⊢ (((𝐴 ↑𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω)
20	19	expcom 450	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ↑𝑜 𝑦) ∈ ω → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
21	20	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ∈ ω → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
22		nnesuc 7575	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
23	22	eleq1d 2672	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 suc 𝑦) ∈ ω ↔ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
24	21, 23	sylibrd 248	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ∈ ω → (𝐴 ↑𝑜 suc 𝑦) ∈ ω))
25	24	expcom 450	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ↑𝑜 𝑦) ∈ ω → (𝐴 ↑𝑜 suc 𝑦) ∈ ω)))
26	5, 7, 9, 18, 25	finds2 6986	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑𝑜 𝑥) ∈ ω))
27	3, 26	vtoclga 3245	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑𝑜 𝐵) ∈ ω))
28	27	impcom 445	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑𝑜 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874  Oncon0 5640  suc csuc 5642  (class class class)co 6549  ωcom 6957  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by: (None)