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Theorem nnecl 8234
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 7158 . . . . 5 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
21eleq1d 2902 . . . 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 7158 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
54eleq1d 2902 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o ∅) ∈ ω))
6 oveq2 7158 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
76eleq1d 2902 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o 𝑦) ∈ ω))
8 oveq2 7158 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
98eleq1d 2902 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o suc 𝑦) ∈ ω))
10 nnon 7579 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 8143 . . . . . 6 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴o ∅) = 1o)
13 df-1o 8098 . . . . . 6 1o = suc ∅
14 peano1 7594 . . . . . . 7 ∅ ∈ ω
15 peano2 7595 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2914 . . . . 5 1o ∈ ω
1812, 17syl6eqel 2926 . . . 4 (𝐴 ∈ ω → (𝐴o ∅) ∈ ω)
19 nnmcl 8233 . . . . . . . 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 8229 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2322eleq1d 2902 . . . . . 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 7603 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴o 𝑥) ∈ ω))
273, 26vtoclga 3579 . 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 1530  wcel 2107  c0 4295  Oncon0 6190  suc csuc 6192  (class class class)co 7150  ωcom 7573  1oc1o 8091   ·o comu 8096  o coe 8097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-omul 8103  df-oexp 8104
This theorem is referenced by: (None)
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