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Theorem nnecl 8228
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 7153 . . . . 5 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
21eleq1d 2894 . . . 4 (𝑥 = 𝐵 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o 𝐵) ∈ ω))
32imbi2d 342 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴o 𝐵) ∈ ω)))
4 oveq2 7153 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
54eleq1d 2894 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o ∅) ∈ ω))
6 oveq2 7153 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
76eleq1d 2894 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o 𝑦) ∈ ω))
8 oveq2 7153 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
98eleq1d 2894 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o suc 𝑦) ∈ ω))
10 nnon 7575 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 8136 . . . . . 6 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴o ∅) = 1o)
13 df-1o 8091 . . . . . 6 1o = suc ∅
14 peano1 7590 . . . . . . 7 ∅ ∈ ω
15 peano2 7591 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2906 . . . . 5 1o ∈ ω
1812, 17syl6eqel 2918 . . . 4 (𝐴 ∈ ω → (𝐴o ∅) ∈ ω)
19 nnmcl 8227 . . . . . . . 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 8223 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2322eleq1d 2894 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴o suc 𝑦) ∈ ω ↔ ((𝐴o 𝑦) ·o 𝐴) ∈ ω))
2421, 23sylibrd 260 . . . . 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 7599 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴o 𝑥) ∈ ω))
273, 26vtoclga 3571 . 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 1528  wcel 2105  c0 4288  Oncon0 6184  suc csuc 6186  (class class class)co 7145  ωcom 7569  1oc1o 8084   ·o comu 8089  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-oexp 8097
This theorem is referenced by: (None)
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