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Theorem nnecl 7653
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 6623 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
21eleq1d 2683 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝐵) ∈ ω))
32imbi2d 330 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω)))
4 oveq2 6623 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
54eleq1d 2683 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 ∅) ∈ ω))
6 oveq2 6623 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
76eleq1d 2683 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝑦) ∈ ω))
8 oveq2 6623 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
98eleq1d 2683 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 suc 𝑦) ∈ ω))
10 nnon 7033 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 7562 . . . . . 6 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴𝑜 ∅) = 1𝑜)
13 df-1o 7520 . . . . . 6 1𝑜 = suc ∅
14 peano1 7047 . . . . . . 7 ∅ ∈ ω
15 peano2 7048 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2694 . . . . 5 1𝑜 ∈ ω
1812, 17syl6eqel 2706 . . . 4 (𝐴 ∈ ω → (𝐴𝑜 ∅) ∈ ω)
19 nnmcl 7652 . . . . . . . 8 (((𝐴𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω)
2019expcom 451 . . . . . . 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 7648 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
2322eleq1d 2683 . . . . . 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 451 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω)))
265, 7, 9, 18, 25finds2 7056 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω))
273, 26vtoclga 3262 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω))
2827impcom 446 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  c0 3897  Oncon0 5692  suc csuc 5694  (class class class)co 6615  ωcom 7027  1𝑜c1o 7513   ·𝑜 comu 7518  𝑜 coe 7519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-oexp 7526
This theorem is referenced by: (None)
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