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Theorem nnecl 7558
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 6535 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
21eleq1d 2671 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝐵) ∈ ω))
32imbi2d 328 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω)))
4 oveq2 6535 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
54eleq1d 2671 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 ∅) ∈ ω))
6 oveq2 6535 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
76eleq1d 2671 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝑦) ∈ ω))
8 oveq2 6535 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
98eleq1d 2671 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 suc 𝑦) ∈ ω))
10 nnon 6941 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 7467 . . . . . 6 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴𝑜 ∅) = 1𝑜)
13 df-1o 7425 . . . . . 6 1𝑜 = suc ∅
14 peano1 6955 . . . . . . 7 ∅ ∈ ω
15 peano2 6956 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2683 . . . . 5 1𝑜 ∈ ω
1812, 17syl6eqel 2695 . . . 4 (𝐴 ∈ ω → (𝐴𝑜 ∅) ∈ ω)
19 nnmcl 7557 . . . . . . . 8 (((𝐴𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω)
2019expcom 449 . . . . . . 7 (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
2120adantr 479 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
22 nnesuc 7553 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
2322eleq1d 2671 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 suc 𝑦) ∈ ω ↔ ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
2421, 23sylibrd 247 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω))
2524expcom 449 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω)))
265, 7, 9, 18, 25finds2 6964 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω))
273, 26vtoclga 3244 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω))
2827impcom 444 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  c0 3873  Oncon0 5626  suc csuc 5628  (class class class)co 6527  ωcom 6935  1𝑜c1o 7418   ·𝑜 comu 7423  𝑜 coe 7424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-omul 7430  df-oexp 7431
This theorem is referenced by: (None)
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