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Mirrors > Home > ILE Home > Th. List > om0 | GIF version |
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
om0 | ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 4177 | . . 3 ⊢ ∅ ∈ On | |
2 | omv 6126 | . . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅)) | |
3 | 1, 2 | mpan2 416 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅)) |
4 | 0ex 3926 | . . 3 ⊢ ∅ ∈ V | |
5 | 4 | rdg0 6062 | . 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘∅) = ∅ |
6 | 3, 5 | syl6eq 2131 | 1 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 Vcvv 2611 ∅c0 3268 ↦ cmpt 3860 Oncon0 4148 ‘cfv 4955 (class class class)co 5569 reccrdg 6044 +𝑜 coa 6088 ·𝑜 comu 6089 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3994 ax-un 4218 ax-setind 4310 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-id 4078 df-iord 4151 df-on 4153 df-suc 4156 df-xp 4400 df-rel 4401 df-cnv 4402 df-co 4403 df-dm 4404 df-rn 4405 df-res 4406 df-ima 4407 df-iota 4920 df-fun 4957 df-fn 4958 df-f 4959 df-f1 4960 df-fo 4961 df-f1o 4962 df-fv 4963 df-ov 5572 df-oprab 5573 df-mpt2 5574 df-1st 5824 df-2nd 5825 df-recs 5980 df-irdg 6045 df-oadd 6095 df-omul 6096 |
This theorem is referenced by: nnm0 6146 nnm0r 6150 |
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