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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 7439 | . . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅)) | |
2 | 1 | eqeq1d 2737 | . 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅)) |
3 | oveq2 7439 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦)) | |
4 | 3 | eqeq1d 2737 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅)) |
5 | oveq2 7439 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦)) | |
6 | 5 | eqeq1d 2737 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅)) |
7 | oveq2 7439 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴)) | |
8 | 7 | eqeq1d 2737 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅)) |
9 | 0elon 6440 | . . 3 ⊢ ∅ ∈ On | |
10 | om0 8554 | . . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (∅ ·o ∅) = ∅ |
12 | oveq1 7438 | . . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)) | |
13 | omsuc 8563 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) | |
14 | 9, 13 | mpan 690 | . . . 4 ⊢ (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) |
15 | oa0 8553 | . . . . . . 7 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
16 | 9, 15 | ax-mp 5 | . . . . . 6 ⊢ (∅ +o ∅) = ∅ |
17 | 16 | eqcomi 2744 | . . . . 5 ⊢ ∅ = (∅ +o ∅) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +o ∅)) |
19 | 14, 18 | eqeq12d 2751 | . . 3 ⊢ (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))) |
20 | 12, 19 | imbitrrid 246 | . 2 ⊢ (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅)) |
21 | iuneq2 5016 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅) | |
22 | iun0 5067 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅ | |
23 | 21, 22 | eqtrdi 2791 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅) |
24 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | omlim 8570 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) | |
26 | 9, 25 | mpan 690 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) |
27 | 24, 26 | mpan 690 | . . . 4 ⊢ (Lim 𝑥 → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) |
28 | 27 | eqeq1d 2737 | . . 3 ⊢ (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅)) |
29 | 23, 28 | imbitrrid 246 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅)) |
30 | 2, 4, 6, 8, 11, 20, 29 | tfinds 7881 | 1 ⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∅c0 4339 ∪ ciun 4996 Oncon0 6386 Lim wlim 6387 suc csuc 6388 (class class class)co 7431 +o coa 8502 ·o comu 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 df-omul 8510 |
This theorem is referenced by: omord 8605 omwordi 8608 om00 8612 odi 8616 omass 8617 oeoa 8634 omxpenlem 9112 onmcl 43321 omcl2 43323 omcl3g 43324 |
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