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 2824	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o 𝐵) ∈ ω))
3	2	imbi2d 341	. . 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 2824	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o ∅) ∈ ω))
6		oveq2 7364	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
7	6	eleq1d 2824	. . . 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 2824	. . . 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 2835	. . . . 5 ⊢ 1o ∈ ω
18	12, 17	eqeltrdi 2847	. . . 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 414	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ↑o 𝑦) ∈ ω → ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω))
21	20	adantr 481	. . . . . 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 2824	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑o suc 𝑦) ∈ ω ↔ ((𝐴 ↑o 𝑦) ·o 𝐴) ∈ ω))
24	21, 23	sylibrd 260	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑o 𝑦) ∈ ω → (𝐴 ↑o suc 𝑦) ∈ ω))
25	24	expcom 414	. . . 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 3520	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑o 𝐵) ∈ ω))
28	27	impcom 408	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∅c0 4261  Oncon0 6310  suc csuc 6312  (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 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  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  43732  nnoeomeqom  43757