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Mirrors > Home > ILE Home > Th. List > oeiv | GIF version |
Description: Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.) |
Ref | Expression |
---|---|
oeiv | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 6031 | . . 3 ⊢ 1𝑜 ∈ On | |
2 | vex 2604 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
3 | omexg 6054 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·𝑜 𝐴) ∈ V) | |
4 | 2, 3 | mpan 414 | . . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ·𝑜 𝐴) ∈ V) |
5 | 4 | ralrimivw 2435 | . . . . 5 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V) |
6 | eqid 2081 | . . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) | |
7 | 6 | fnmpt 5045 | . . . . 5 ⊢ (∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V) |
8 | 5, 7 | syl 14 | . . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V) |
9 | rdgexggg 5987 | . . . 4 ⊢ (((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V ∧ 1𝑜 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V) | |
10 | 8, 9 | syl3an1 1202 | . . 3 ⊢ ((𝐴 ∈ On ∧ 1𝑜 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V) |
11 | 1, 10 | mp3an2 1256 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V) |
12 | oveq2 5540 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·𝑜 𝑦) = (𝑥 ·𝑜 𝐴)) | |
13 | 12 | mpteq2dv 3869 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))) |
14 | rdgeq1 5981 | . . . . 5 ⊢ ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜) = rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)) | |
15 | 13, 14 | syl 14 | . . . 4 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜) = rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)) |
16 | 15 | fveq1d 5200 | . . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧)) |
17 | fveq2 5198 | . . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) | |
18 | df-oexpi 6030 | . . 3 ⊢ ↑𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧)) | |
19 | 16, 17, 18 | ovmpt2g 5655 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) |
20 | 11, 19 | mpd3an3 1269 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∀wral 2348 Vcvv 2601 ↦ cmpt 3839 Oncon0 4118 Fn wfn 4917 ‘cfv 4922 (class class class)co 5532 reccrdg 5979 1𝑜c1o 6017 ·𝑜 comu 6022 ↑𝑜 coei 6023 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-oexpi 6030 |
This theorem is referenced by: oei0 6062 oeicl 6065 |
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