Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > omcl | GIF version |
Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.) |
Ref | Expression |
---|---|
omcl | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omv 6058 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)) | |
2 | omfnex 6052 | . . 3 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V) | |
3 | 0elon 4147 | . . . 4 ⊢ ∅ ∈ On | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝐴 ∈ On → ∅ ∈ On) |
5 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | oacl 6063 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 +𝑜 𝐴) ∈ On) | |
7 | oveq1 5539 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴)) | |
8 | eqid 2081 | . . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) | |
9 | 7, 8 | fvmptg 5269 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ (𝑦 +𝑜 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) = (𝑦 +𝑜 𝐴)) |
10 | 5, 6, 9 | sylancr 405 | . . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) = (𝑦 +𝑜 𝐴)) |
11 | 10, 6 | eqeltrd 2155 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On) |
12 | 11 | ancoms 264 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On) |
13 | 12 | ralrimiva 2434 | . . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On) |
14 | 2, 4, 13 | rdgon 5996 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ On) |
15 | 1, 14 | eqeltrd 2155 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ∅c0 3251 ↦ cmpt 3839 Oncon0 4118 ‘cfv 4922 (class class class)co 5532 reccrdg 5979 +𝑜 coa 6021 ·𝑜 comu 6022 |
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-oadd 6028 df-omul 6029 |
This theorem is referenced by: oeicl 6065 omv2 6068 omsuc 6074 |
Copyright terms: Public domain | W3C validator |