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Mirrors > Home > MPE Home > Th. List > oe0 | Structured version Visualization version GIF version |
Description: Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oe0 | ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6697 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 ∅) = (∅ ↑𝑜 ∅)) | |
2 | oe0m0 7645 | . . . . 5 ⊢ (∅ ↑𝑜 ∅) = 1𝑜 | |
3 | 1, 2 | syl6eq 2701 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 ∅) = 1𝑜) |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
5 | 0elon 5816 | . . . . . 6 ⊢ ∅ ∈ On | |
6 | oevn0 7640 | . . . . . 6 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅)) | |
7 | 5, 6 | mpanl2 717 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅)) |
8 | 1on 7612 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
9 | 8 | elexi 3244 | . . . . . 6 ⊢ 1𝑜 ∈ V |
10 | 9 | rdg0 7562 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅) = 1𝑜 |
11 | 7, 10 | syl6eq 2701 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
12 | 11 | adantll 750 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
13 | 4, 12 | oe0lem 7638 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
14 | 13 | anidms 678 | 1 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∅c0 3948 ↦ cmpt 4762 Oncon0 5761 ‘cfv 5926 (class class class)co 6690 reccrdg 7550 1𝑜c1o 7598 ·𝑜 comu 7603 ↑𝑜 coe 7604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oexp 7611 |
This theorem is referenced by: oecl 7662 oe1 7669 oe1m 7670 oen0 7711 oewordri 7717 oeoalem 7721 oeoelem 7723 oeoe 7724 oeeulem 7726 nnecl 7738 oaabs2 7770 cantnff 8609 |
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