Theorem nnecl 7693

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 6658	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐵))
2	1	eleq1d 2686	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 𝐵) ∈ ω))
3	2	imbi2d 330	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ↑𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ↑𝑜 𝐵) ∈ ω)))
4		oveq2 6658	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
5	4	eleq1d 2686	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 ∅) ∈ ω))
6		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
7	6	eleq1d 2686	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 𝑦) ∈ ω))
8		oveq2 6658	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
9	8	eleq1d 2686	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 suc 𝑦) ∈ ω))
10		nnon 7071	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
11		oe0 7602	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
12	10, 11	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ↑𝑜 ∅) = 1𝑜)
13		df-1o 7560	. . . . . 6 ⊢ 1𝑜 = suc ∅
14		peano1 7085	. . . . . . 7 ⊢ ∅ ∈ ω
15		peano2 7086	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
17	13, 16	eqeltri 2697	. . . . 5 ⊢ 1𝑜 ∈ ω
18	12, 17	syl6eqel 2709	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ↑𝑜 ∅) ∈ ω)
19		nnmcl 7692	. . . . . . . 8 ⊢ (((𝐴 ↑𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω)
20	19	expcom 451	. . . . . . 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 7688	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
23	22	eleq1d 2686	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 suc 𝑦) ∈ ω ↔ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
24	21, 23	sylibrd 249	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ∈ ω → (𝐴 ↑𝑜 suc 𝑦) ∈ ω))
25	24	expcom 451	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ↑𝑜 𝑦) ∈ ω → (𝐴 ↑𝑜 suc 𝑦) ∈ ω)))
26	5, 7, 9, 18, 25	finds2 7094	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑𝑜 𝑥) ∈ ω))
27	3, 26	vtoclga 3272	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑𝑜 𝐵) ∈ ω))
28	27	impcom 446	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑𝑜 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∅c0 3915  Oncon0 5723  suc csuc 5725  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by: (None)