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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 6035 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)) | |
2 | omfnex 6029 | . . 3 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V) | |
3 | 0elon 4129 | . . . 4 ⊢ ∅ ∈ On | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝐴 ∈ On → ∅ ∈ On) |
5 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | oacl 6040 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 +𝑜 𝐴) ∈ On) | |
7 | oveq1 5519 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴)) | |
8 | eqid 2040 | . . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) | |
9 | 7, 8 | fvmptg 5248 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ (𝑦 +𝑜 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) = (𝑦 +𝑜 𝐴)) |
10 | 5, 6, 9 | sylancr 393 | . . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) = (𝑦 +𝑜 𝐴)) |
11 | 10, 6 | eqeltrd 2114 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On) |
12 | 11 | ancoms 255 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On) |
13 | 12 | ralrimiva 2392 | . . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On) |
14 | 2, 4, 13 | rdgon 5973 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ On) |
15 | 1, 14 | eqeltrd 2114 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∅c0 3224 ↦ cmpt 3818 Oncon0 4100 ‘cfv 4902 (class class class)co 5512 reccrdg 5956 +𝑜 coa 5998 ·𝑜 comu 5999 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 |
This theorem is referenced by: oeicl 6042 omv2 6045 omsuc 6051 |
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