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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. Definition 2.6 of [Schloeder] p. 4. (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 7438 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o ∅) = (∅ ↑o ∅)) | |
2 | oe0m0 8557 | . . . . 5 ⊢ (∅ ↑o ∅) = 1o | |
3 | 1, 2 | eqtrdi 2791 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑o ∅) = 1o) |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o ∅) = 1o) |
5 | 0elon 6440 | . . . . . 6 ⊢ ∅ ∈ On | |
6 | oevn0 8552 | . . . . . 6 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅)) | |
7 | 5, 6 | mpanl2 701 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅)) |
8 | 1oex 8515 | . . . . . 6 ⊢ 1o ∈ V | |
9 | 8 | rdg0 8460 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o |
10 | 7, 9 | eqtrdi 2791 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = 1o) |
11 | 10 | adantll 714 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = 1o) |
12 | 4, 11 | oe0lem 8550 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑o ∅) = 1o) |
13 | 12 | anidms 566 | 1 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ↦ cmpt 5231 Oncon0 6386 ‘cfv 6563 (class class class)co 7431 reccrdg 8448 1oc1o 8498 ·o comu 8503 ↑o coe 8504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oexp 8511 |
This theorem is referenced by: oecl 8574 oe1 8581 oe1m 8582 oen0 8623 oewordri 8629 oeoalem 8633 oeoelem 8635 oeoe 8636 oeeulem 8638 nnecl 8650 oaabs2 8686 cantnff 9712 onexoegt 43233 oe0suclim 43267 oenassex 43308 omabs2 43322 omcl2 43323 |
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