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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 7027 | . . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅)) | |
2 | 1 | eqeq1d 2796 | . 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅)) |
3 | oveq2 7027 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦)) | |
4 | 3 | eqeq1d 2796 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅)) |
5 | oveq2 7027 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦)) | |
6 | 5 | eqeq1d 2796 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅)) |
7 | oveq2 7027 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴)) | |
8 | 7 | eqeq1d 2796 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅)) |
9 | 0elon 6122 | . . 3 ⊢ ∅ ∈ On | |
10 | om0 7996 | . . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (∅ ·o ∅) = ∅ |
12 | oveq1 7026 | . . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)) | |
13 | omsuc 8005 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) | |
14 | 9, 13 | mpan 686 | . . . 4 ⊢ (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) |
15 | oa0 7995 | . . . . . . 7 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
16 | 9, 15 | ax-mp 5 | . . . . . 6 ⊢ (∅ +o ∅) = ∅ |
17 | 16 | eqcomi 2803 | . . . . 5 ⊢ ∅ = (∅ +o ∅) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝑦 ∈ On → ∅ = (∅ +o ∅)) |
19 | 14, 18 | eqeq12d 2809 | . . 3 ⊢ (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))) |
20 | 12, 19 | syl5ibr 247 | . 2 ⊢ (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅)) |
21 | iuneq2 4845 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∪ 𝑦 ∈ 𝑥 ∅) | |
22 | iun0 4886 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝑥 ∅ = ∅ | |
23 | 21, 22 | syl6eq 2846 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅) |
24 | vex 3439 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | omlim 8012 | . . . . . 6 ⊢ ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) | |
26 | 9, 25 | mpan 686 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) |
27 | 24, 26 | mpan 686 | . . . 4 ⊢ (Lim 𝑥 → (∅ ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦)) |
28 | 27 | eqeq1d 2796 | . . 3 ⊢ (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ ∪ 𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅)) |
29 | 23, 28 | syl5ibr 247 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅)) |
30 | 2, 4, 6, 8, 11, 20, 29 | tfinds 7433 | 1 ⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ∀wral 3104 Vcvv 3436 ∅c0 4213 ∪ ciun 4827 Oncon0 6069 Lim wlim 6070 suc csuc 6071 (class class class)co 7019 +o coa 7953 ·o comu 7954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-oadd 7960 df-omul 7961 |
This theorem is referenced by: omord 8047 omwordi 8050 om00 8054 odi 8058 omass 8059 oeoa 8076 omxpenlem 8468 |
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