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Theorem nnecl 7847
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 6801 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
21eleq1d 2835 . . . 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 6801 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
54eleq1d 2835 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 ∅) ∈ ω))
6 oveq2 6801 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
76eleq1d 2835 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝑦) ∈ ω))
8 oveq2 6801 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
98eleq1d 2835 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 suc 𝑦) ∈ ω))
10 nnon 7218 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 7756 . . . . . 6 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴𝑜 ∅) = 1𝑜)
13 df-1o 7713 . . . . . 6 1𝑜 = suc ∅
14 peano1 7232 . . . . . . 7 ∅ ∈ ω
15 peano2 7233 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2846 . . . . 5 1𝑜 ∈ ω
1812, 17syl6eqel 2858 . . . 4 (𝐴 ∈ ω → (𝐴𝑜 ∅) ∈ ω)
19 nnmcl 7846 . . . . . . . 8 (((𝐴𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω)
2019expcom 398 . . . . . . 7 (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
2120adantr 466 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
22 nnesuc 7842 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
2322eleq1d 2835 . . . . . 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 398 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω)))
265, 7, 9, 18, 25finds2 7241 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω))
273, 26vtoclga 3423 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω))
2827impcom 394 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  c0 4063  Oncon0 5866  suc csuc 5868  (class class class)co 6793  ωcom 7212  1𝑜c1o 7706   ·𝑜 comu 7711  𝑜 coe 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-oexp 7719
This theorem is referenced by: (None)
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