Theorem nnecl 8633

Description: Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. Theorem 2.20 of [Schloeder] p. 6. (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 7435	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
2	1	eleq1d 2848	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o 𝐵) ∈ ω))
3	2	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ↑o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ↑o 𝐵) ∈ ω)))
4		oveq2 7435	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
5	4	eleq1d 2848	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o ∅) ∈ ω))
6		oveq2 7435	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
7	6	eleq1d 2848	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o 𝑦) ∈ ω))
8		oveq2 7435	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	8	eleq1d 2848	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o suc 𝑦) ∈ ω))
10		nnon 7890	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
11		oe0 8541	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
12	10, 11	syl 18	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) = 1o)
13		df-1o 8487	. . . . . 6 ⊢ 1o = suc ∅
14		peano1 7907	. . . . . . 7 ⊢ ∅ ∈ ω
15		peano2 7908	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
17	13, 16	eqeltri 2859	. . . . 5 ⊢ 1o ∈ ω
18	12, 17	eqeltrdi 2871	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) ∈ ω)
19		nnmcl 8632	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω)
20	19	expcom 419	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ↑o 𝑦) ∈ ω → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω))
21	20	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑o 𝑦) ∈ ω → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω))
22		nnesuc 8628	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
23	22	eleq1d 2848	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑o suc 𝑦) ∈ ω ↔ ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω))
24	21, 23	sylibrd 262	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑o 𝑦) ∈ ω → (𝐴 ↑o suc 𝑦) ∈ ω))
25	24	expcom 419	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ↑o 𝑦) ∈ ω → (𝐴 ↑o suc 𝑦) ∈ ω)))
26	5, 7, 9, 18, 25	finds2 7917	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑o 𝑥) ∈ ω))
27	3, 26	vtoclga 3539	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑o 𝐵) ∈ ω))
28	27	impcom 413	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o 𝐵) ∈ ω)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∅c0 4280  Oncon0 6368  suc csuc 6370  (class class class)co 7427  ωcom 7884  1_oc1o 8480   ·_o comu 8485   ↑_o coe 8486

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-oadd 8491  df-omul 8492  df-oexp 8493

This theorem is used by:  nnamecl  44429  nnoeomeqom  44454