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Mirrors > Home > MPE Home > Th. List > om0r | Structured version Visualization version GIF version |
Description: Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.) |
Ref | Expression |
---|---|
om0r | ⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . 3 ⊢ (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅)) | |
2 | 1 | eqeq1d 2612 | . 2 ⊢ (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅)) |
3 | oveq2 6557 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦)) | |
4 | 3 | eqeq1d 2612 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅)) |
5 | oveq2 6557 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦)) | |
6 | 5 | eqeq1d 2612 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅)) |
7 | oveq2 6557 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴)) | |
8 | 7 | eqeq1d 2612 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅)) |
9 | om0x 7486 | . 2 ⊢ (∅ ·𝑜 ∅) = ∅ | |
10 | oveq1 6556 | . . 3 ⊢ ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅)) | |
11 | 0elon 5695 | . . . . 5 ⊢ ∅ ∈ On | |
12 | omsuc 7493 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅)) | |
13 | 11, 12 | mpan 702 | . . . 4 ⊢ (𝑦 ∈ On → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅)) |
14 | oa0 7483 | . . . . . . 7 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅) | |
15 | 11, 14 | ax-mp 5 | . . . . . 6 ⊢ (∅ +𝑜 ∅) = ∅ |
16 | 15 | eqcomi 2619 | . . . . 5 ⊢ ∅ = (∅ +𝑜 ∅) |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +𝑜 ∅)) |
18 | 13, 17 | eqeq12d 2625 | . . 3 ⊢ (𝑦 ∈ On → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))) |
19 | 10, 18 | syl5ibr 235 | . 2 ⊢ (𝑦 ∈ On → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅)) |
20 | iuneq2 4473 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅) | |
21 | iun0 4512 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅ | |
22 | 20, 21 | syl6eq 2660 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅) |
23 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
24 | omlim 7500 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦)) | |
25 | 11, 24 | mpan 702 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦)) |
26 | 23, 25 | mpan 702 | . . . 4 ⊢ (Lim 𝑥 → (∅ ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦)) |
27 | 26 | eqeq1d 2612 | . . 3 ⊢ (Lim 𝑥 → ((∅ ·𝑜 𝑥) = ∅ ↔ ∪ 𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅)) |
28 | 22, 27 | syl5ibr 235 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 𝑥) = ∅)) |
29 | 2, 4, 6, 8, 9, 19, 28 | tfinds 6951 | 1 ⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∅c0 3874 ∪ ciun 4455 Oncon0 5640 Lim wlim 5641 suc csuc 5642 (class class class)co 6549 +𝑜 coa 7444 ·𝑜 comu 7445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-omul 7452 |
This theorem is referenced by: omord 7535 omwordi 7538 om00 7542 odi 7546 omass 7547 oeoa 7564 omxpenlem 7946 |
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