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

Theorem nnecl 7898
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 6850 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
21eleq1d 2829 . . . 4 (𝑥 = 𝐵 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝐵) ∈ ω))
32imbi2d 331 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω)))
4 oveq2 6850 . . . . 5 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
54eleq1d 2829 . . . 4 (𝑥 = ∅ → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 ∅) ∈ ω))
6 oveq2 6850 . . . . 5 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
76eleq1d 2829 . . . 4 (𝑥 = 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 𝑦) ∈ ω))
8 oveq2 6850 . . . . 5 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
98eleq1d 2829 . . . 4 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝑥) ∈ ω ↔ (𝐴𝑜 suc 𝑦) ∈ ω))
10 nnon 7269 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ On)
11 oe0 7807 . . . . . 6 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
1210, 11syl 17 . . . . 5 (𝐴 ∈ ω → (𝐴𝑜 ∅) = 1𝑜)
13 df-1o 7764 . . . . . 6 1𝑜 = suc ∅
14 peano1 7283 . . . . . . 7 ∅ ∈ ω
15 peano2 7284 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
1614, 15ax-mp 5 . . . . . 6 suc ∅ ∈ ω
1713, 16eqeltri 2840 . . . . 5 1𝑜 ∈ ω
1812, 17syl6eqel 2852 . . . 4 (𝐴 ∈ ω → (𝐴𝑜 ∅) ∈ ω)
19 nnmcl 7897 . . . . . . . 8 (((𝐴𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω)
2019expcom 402 . . . . . . 7 (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
2120adantr 472 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 𝑦) ∈ ω → ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
22 nnesuc 7893 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
2322eleq1d 2829 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 suc 𝑦) ∈ ω ↔ ((𝐴𝑜 𝑦) ·𝑜 𝐴) ∈ ω))
2421, 23sylibrd 250 . . . . 5 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω))
2524expcom 402 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑜 𝑦) ∈ ω → (𝐴𝑜 suc 𝑦) ∈ ω)))
265, 7, 9, 18, 25finds2 7292 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝑥) ∈ ω))
273, 26vtoclga 3424 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝑜 𝐵) ∈ ω))
2827impcom 396 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝑜 𝐵) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  c0 4079  Oncon0 5908  suc csuc 5910  (class class class)co 6842  ωcom 7263  1𝑜c1o 7757   ·𝑜 comu 7762  𝑜 coe 7763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-omul 7769  df-oexp 7770
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator