Theorem nnecl 8539

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 7364	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
2	1	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o 𝐵) ∈ ω))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ↑o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ↑o 𝐵) ∈ ω)))
4		oveq2 7364	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
5	4	eleq1d 2819	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o ∅) ∈ ω))
6		oveq2 7364	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
7	6	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o 𝑦) ∈ ω))
8		oveq2 7364	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	8	eleq1d 2819	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o suc 𝑦) ∈ ω))
10		nnon 7812	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
11		oe0 8447	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
12	10, 11	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) = 1o)
13		df-1o 8395	. . . . . 6 ⊢ 1o = suc ∅
14		peano1 7829	. . . . . . 7 ⊢ ∅ ∈ ω
15		peano2 7830	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
17	13, 16	eqeltri 2830	. . . . 5 ⊢ 1o ∈ ω
18	12, 17	eqeltrdi 2842	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) ∈ ω)
19		nnmcl 8538	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω)
20	19	expcom 413	. . . . . . 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 8534	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
23	22	eleq1d 2819	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑o suc 𝑦) ∈ ω ↔ ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω))
24	21, 23	sylibrd 259	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑o 𝑦) ∈ ω → (𝐴 ↑o suc 𝑦) ∈ ω))
25	24	expcom 413	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ↑o 𝑦) ∈ ω → (𝐴 ↑o suc 𝑦) ∈ ω)))
26	5, 7, 9, 18, 25	finds2 7838	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑o 𝑥) ∈ ω))
27	3, 26	vtoclga 3530	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑o 𝐵) ∈ ω))
28	27	impcom 407	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∅c0 4283  Oncon0 6315  suc csuc 6317  (class class class)co 7356  ωcom 7806  1_oc1o 8388   ·_o comu 8393   ↑_o coe 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-oexp 8401

This theorem is referenced by:  nnamecl  43471  nnoeomeqom  43496