Theorem nnecl 8605

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 7427	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
2	1	eleq1d 2850	. . . 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 7427	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
5	4	eleq1d 2850	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o ∅) ∈ ω))
6		oveq2 7427	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
7	6	eleq1d 2850	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o 𝑦) ∈ ω))
8		oveq2 7427	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	8	eleq1d 2850	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑o 𝑥) ∈ ω ↔ (𝐴 ↑o suc 𝑦) ∈ ω))
10		nnon 7874	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
11		oe0 8513	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
12	10, 11	syl 18	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) = 1o)
13		df-1o 8459	. . . . . 6 ⊢ 1o = suc ∅
14		peano1 7891	. . . . . . 7 ⊢ ∅ ∈ ω
15		peano2 7892	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
17	13, 16	eqeltri 2861	. . . . 5 ⊢ 1o ∈ ω
18	12, 17	eqeltrdi 2873	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ↑o ∅) ∈ ω)
19		nnmcl 8604	. . . . . . . 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 8600	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
23	22	eleq1d 2850	. . . . . 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 7901	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑o 𝑥) ∈ ω))
27	3, 26	vtoclga 3543	. 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 1570   ∈ wcel 2146  ∅c0 4286  Oncon0 6364  suc csuc 6366  (class class class)co 7419  ωcom 7868  1_oc1o 8452   ·_o comu 8457   ↑_o coe 8458

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-omul 8464  df-oexp 8465

This theorem is used by:  nnamecl  44074  nnoeomeqom  44099