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Mirrors > Home > MPE Home > Th. List > nnecl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nnecl | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑𝑜 𝐵) ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐵)) | |
2 | 1 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 𝐵) ∈ ω)) |
3 | 2 | imbi2d 329 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ↑𝑜 𝑥) ∈ ω) ↔ (𝐴 ∈ ω → (𝐴 ↑𝑜 𝐵) ∈ ω))) |
4 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅)) | |
5 | 4 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = ∅ → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 ∅) ∈ ω)) |
6 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦)) | |
7 | 6 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 𝑦) ∈ ω)) |
8 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦)) | |
9 | 8 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ↑𝑜 𝑥) ∈ ω ↔ (𝐴 ↑𝑜 suc 𝑦) ∈ ω)) |
10 | nnon 6963 | . . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
11 | oe0 7489 | . . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ↑𝑜 ∅) = 1𝑜) |
13 | df-1o 7447 | . . . . . 6 ⊢ 1𝑜 = suc ∅ | |
14 | peano1 6977 | . . . . . . 7 ⊢ ∅ ∈ ω | |
15 | peano2 6978 | . . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ suc ∅ ∈ ω |
17 | 13, 16 | eqeltri 2684 | . . . . 5 ⊢ 1𝑜 ∈ ω |
18 | 12, 17 | syl6eqel 2696 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ↑𝑜 ∅) ∈ ω) |
19 | nnmcl 7579 | . . . . . . . 8 ⊢ (((𝐴 ↑𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω) | |
20 | 19 | expcom 450 | . . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝐴 ↑𝑜 𝑦) ∈ ω → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω)) |
21 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ∈ ω → ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω)) |
22 | nnesuc 7575 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)) | |
23 | 22 | eleq1d 2672 | . . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 suc 𝑦) ∈ ω ↔ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ∈ ω)) |
24 | 21, 23 | sylibrd 248 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ↑𝑜 𝑦) ∈ ω → (𝐴 ↑𝑜 suc 𝑦) ∈ ω)) |
25 | 24 | expcom 450 | . . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ↑𝑜 𝑦) ∈ ω → (𝐴 ↑𝑜 suc 𝑦) ∈ ω))) |
26 | 5, 7, 9, 18, 25 | finds2 6986 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑𝑜 𝑥) ∈ ω)) |
27 | 3, 26 | vtoclga 3245 | . 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ↑𝑜 𝐵) ∈ ω)) |
28 | 27 | impcom 445 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑𝑜 𝐵) ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 Oncon0 5640 suc csuc 5642 (class class class)co 6549 ωcom 6957 1𝑜c1o 7440 ·𝑜 comu 7445 ↑𝑜 coe 7446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-oexp 7453 |
This theorem is referenced by: (None) |
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