Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnecl Structured version   Visualization version   GIF version

Theorem nnecl 8249
 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 2836 . . . 4 (𝑥 = 𝐵 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o 𝐵) ∈ ω))
32imbi2d 344 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴o 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴o 𝐵) ∈ ω)))
4 oveq2 7158 . . . . 5 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
54eleq1d 2836 . . . 4 (𝑥 = ∅ → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o ∅) ∈ ω))
6 oveq2 7158 . . . . 5 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
76eleq1d 2836 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o 𝑦) ∈ ω))
8 oveq2 7158 . . . . 5 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
98eleq1d 2836 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝑥) ∈ ω ↔ (𝐴o suc 𝑦) ∈ ω))
10 nnon 7585 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 8157 . . . . . 6 (𝐴 ∈ On → (𝐴o ∅) = 1o)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴o ∅) = 1o)
13 df-1o 8112 . . . . . 6 1o = suc ∅
14 peano1 7600 . . . . . . 7 ∅ ∈ ω
15 peano2 7601 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2848 . . . . 5 1o ∈ ω
1812, 17eqeltrdi 2860 . . . 4 (𝐴 ∈ ω → (𝐴o ∅) ∈ ω)
19 nnmcl 8248 . . . . . . . 8 (((𝐴o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴o 𝑦) ·o 𝐴) ∈ ω)
2019expcom 417 . . . . . . 7 (𝐴 ∈ ω → ((𝐴o 𝑦) ∈ ω → ((𝐴o 𝑦) ·o 𝐴) ∈ ω))
2120adantr 484 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴o 𝑦) ∈ ω → ((𝐴o 𝑦) ·o 𝐴) ∈ ω))
22 nnesuc 8244 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2322eleq1d 2836 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴o suc 𝑦) ∈ ω ↔ ((𝐴o 𝑦) ·o 𝐴) ∈ ω))
2421, 23sylibrd 262 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴o 𝑦) ∈ ω → (𝐴o suc 𝑦) ∈ ω))
2524expcom 417 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴o 𝑦) ∈ ω → (𝐴o suc 𝑦) ∈ ω)))
265, 7, 9, 18, 25finds2 7610 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴o 𝑥) ∈ ω))
273, 26vtoclga 3492 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴o 𝐵) ∈ ω))
2827impcom 411 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴o 𝐵) ∈ ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4225  Oncon0 6169  suc csuc 6171  (class class class)co 7150  ωcom 7579  1oc1o 8105   ·o comu 8110   ↑o coe 8111 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-oadd 8116  df-omul 8117  df-oexp 8118 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator