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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 6657 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 ∅) = (∅ ↑𝑜 ∅)) | |
2 | oe0m0 7600 | . . . . 5 ⊢ (∅ ↑𝑜 ∅) = 1𝑜 | |
3 | 1, 2 | syl6eq 2672 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 ∅) = 1𝑜) |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
5 | 0elon 5778 | . . . . . 6 ⊢ ∅ ∈ On | |
6 | oevn0 7595 | . . . . . 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 7567 | . . . . . . 7 ⊢ 1𝑜 ∈ On | |
9 | 8 | elexi 3213 | . . . . . 6 ⊢ 1𝑜 ∈ V |
10 | 9 | rdg0 7517 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅) = 1𝑜 |
11 | 7, 10 | syl6eq 2672 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
12 | 11 | adantll 750 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
13 | 4, 12 | oe0lem 7593 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑𝑜 ∅) = 1𝑜) |
14 | 13 | anidms 677 | 1 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ↦ cmpt 4729 Oncon0 5723 ‘cfv 5888 (class class class)co 6650 reccrdg 7505 1𝑜c1o 7553 ·𝑜 comu 7558 ↑𝑜 coe 7559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oexp 7566 |
This theorem is referenced by: oecl 7617 oe1 7624 oe1m 7625 oen0 7666 oewordri 7672 oeoalem 7676 oeoelem 7678 oeoe 7679 oeeulem 7681 nnecl 7693 oaabs2 7725 cantnff 8571 |
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