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Theorem nnecl 7864
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) → (𝐴𝑜 𝐵) ∈ ω)

Proof of Theorem nnecl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6822 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
21eleq1d 2824 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝐵) ∈ ω))
32imbi2d 329 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω)))
4 oveq2 6822 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
54eleq1d 2824 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 ∅) ∈ ω))
6 oveq2 6822 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
76eleq1d 2824 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝑦) ∈ ω))
8 oveq2 6822 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
98eleq1d 2824 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 suc 𝑦) ∈ ω))
10 nnon 7237 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 7773 . . . . . 6 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴𝑜 ∅) = 1𝑜)
13 df-1o 7730 . . . . . 6 1𝑜 = suc ∅
14 peano1 7251 . . . . . . 7 ∅ ∈ ω
15 peano2 7252 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2835 . . . . 5 1𝑜 ∈ ω
1812, 17syl6eqel 2847 . . . 4 (𝐴 ∈ ω → (𝐴𝑜 ∅) ∈ ω)
19 nnmcl 7863 . . . . . . . 8 (((𝐴𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω)
2019expcom 450 . . . . . . 7 (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
2120adantr 472 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
22 nnesuc 7859 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
2322eleq1d 2824 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 suc 𝑦) ∈ ω ↔ ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
2421, 23sylibrd 249 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω))
2524expcom 450 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω)))
265, 7, 9, 18, 25finds2 7260 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω))
273, 26vtoclga 3412 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω))
2827impcom 445 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  c0 4058  Oncon0 5884  suc csuc 5886  (class class class)co 6814  ωcom 7231  1𝑜c1o 7723   ·𝑜 comu 7728  𝑜 coe 7729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-omul 7735  df-oexp 7736
This theorem is referenced by: (None)
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