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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 7359 | . . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅)) | |
2 | 1 | eqeq1d 2739 | . 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅)) |
3 | oveq2 7359 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦)) | |
4 | 3 | eqeq1d 2739 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅)) |
5 | oveq2 7359 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦)) | |
6 | 5 | eqeq1d 2739 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅)) |
7 | oveq2 7359 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴)) | |
8 | 7 | eqeq1d 2739 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅)) |
9 | 0elon 6369 | . . 3 ⊢ ∅ ∈ On | |
10 | om0 8455 | . . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (∅ ·o ∅) = ∅ |
12 | oveq1 7358 | . . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)) | |
13 | omsuc 8464 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) | |
14 | 9, 13 | mpan 688 | . . . 4 ⊢ (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) |
15 | oa0 8454 | . . . . . . 7 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
16 | 9, 15 | ax-mp 5 | . . . . . 6 ⊢ (∅ +o ∅) = ∅ |
17 | 16 | eqcomi 2746 | . . . . 5 ⊢ ∅ = (∅ +o ∅) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +o ∅)) |
19 | 14, 18 | eqeq12d 2753 | . . 3 ⊢ (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))) |
20 | 12, 19 | syl5ibr 245 | . 2 ⊢ (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅)) |
21 | iuneq2 4971 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅) | |
22 | iun0 5020 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅ | |
23 | 21, 22 | eqtrdi 2793 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅) |
24 | vex 3447 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | omlim 8471 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) | |
26 | 9, 25 | mpan 688 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) |
27 | 24, 26 | mpan 688 | . . . 4 ⊢ (Lim 𝑥 → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) |
28 | 27 | eqeq1d 2739 | . . 3 ⊢ (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅)) |
29 | 23, 28 | syl5ibr 245 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅)) |
30 | 2, 4, 6, 8, 11, 20, 29 | tfinds 7788 | 1 ⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 Vcvv 3443 ∅c0 4280 ∪ ciun 4952 Oncon0 6315 Lim wlim 6316 suc csuc 6317 (class class class)co 7351 +o coa 8401 ·o comu 8402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-oadd 8408 df-omul 8409 |
This theorem is referenced by: omord 8507 omwordi 8510 om00 8514 odi 8518 omass 8519 oeoa 8536 omxpenlem 8975 omcl2 41573 omcl3g 41574 |
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