Theorem nnecl 8554

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 7377	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
2	1	eleq1d 2813	. . . 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 7377	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
5	4	eleq1d 2813	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o ∅) ∈ ω))
6		oveq2 7377	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
7	6	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o 𝑦) ∈ ω))
8		oveq2 7377	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	8	eleq1d 2813	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o suc 𝑦) ∈ ω))
10		nnon 7828	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
11		oe0 8463	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
12	10, 11	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) = 1o)
13		df-1o 8411	. . . . . 6 ⊢ 1o = suc ∅
14		peano1 7845	. . . . . . 7 ⊢ ∅ ∈ ω
15		peano2 7846	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
17	13, 16	eqeltri 2824	. . . . 5 ⊢ 1o ∈ ω
18	12, 17	eqeltrdi 2836	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) ∈ ω)
19		nnmcl 8553	. . . . . . . 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 8549	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
23	22	eleq1d 2813	. . . . . 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 7854	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑o 𝑥) ∈ ω))
27	3, 26	vtoclga 3540	. 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 1540   ∈ wcel 2109  ∅c0 4292  Oncon0 6320  suc csuc 6322  (class class class)co 7369  ωcom 7822  1_oc1o 8404   ·_o comu 8409   ↑_o coe 8410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-oadd 8415  df-omul 8416  df-oexp 8417

This theorem is referenced by:  nnamecl  43269  nnoeomeqom  43294