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Theorem nnecl 8598
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) → (𝐴o 𝐵) ∈ ω)

Proof of Theorem nnecl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7402 . . . . 5 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
21eleq1d 2818 . . . 4 (𝑥 = 𝐵 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o 𝐵) ∈ ω))
32imbi2d 340 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴o 𝐵) ∈ ω)))
4 oveq2 7402 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
54eleq1d 2818 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o ∅) ∈ ω))
6 oveq2 7402 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
76eleq1d 2818 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o 𝑦) ∈ ω))
8 oveq2 7402 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
98eleq1d 2818 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o suc 𝑦) ∈ ω))
10 nnon 7845 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 8506 . . . . . 6 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴o ∅) = 1o)
13 df-1o 8450 . . . . . 6 1o = suc ∅
14 peano1 7863 . . . . . . 7 ∅ ∈ ω
15 peano2 7865 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2829 . . . . 5 1o ∈ ω
1812, 17eqeltrdi 2841 . . . 4 (𝐴 ∈ ω → (𝐴o ∅) ∈ ω)
19 nnmcl 8597 . . . . . . . 8 (((𝐴o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴o 𝑦) ·o 𝐴) ∈ ω)
2019expcom 414 . . . . . . 7 (𝐴 ∈ ω → ((𝐴o 𝑦) ∈ ω → ((𝐴o 𝑦) ·o 𝐴) ∈ ω))
2120adantr 481 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴o 𝑦) ∈ ω → ((𝐴o 𝑦) ·o 𝐴) ∈ ω))
22 nnesuc 8593 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2322eleq1d 2818 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴o suc 𝑦) ∈ ω ↔ ((𝐴o 𝑦) ·o 𝐴) ∈ ω))
2421, 23sylibrd 258 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴o 𝑦) ∈ ω → (𝐴o suc 𝑦) ∈ ω))
2524expcom 414 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴o 𝑦) ∈ ω → (𝐴o suc 𝑦) ∈ ω)))
265, 7, 9, 18, 25finds2 7875 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴o 𝑥) ∈ ω))
273, 26vtoclga 3563 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴o 𝐵) ∈ ω))
2827impcom 408 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴o 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  c0 4319  Oncon0 6354  suc csuc 6356  (class class class)co 7394  ωcom 7839  1oc1o 8443   ·o comu 8448  o coe 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-oadd 8454  df-omul 8455  df-oexp 8456
This theorem is referenced by:  nnamecl  41872  nnoeomeqom  41897
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