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Mirrors > Home > MPE Home > Th. List > omcl | Structured version Visualization version GIF version |
Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
omcl | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . 4 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅)) | |
2 | 1 | eleq1d 2672 | . . 3 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 ∅) ∈ On)) |
3 | oveq2 6557 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦)) | |
4 | 3 | eleq1d 2672 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝑦) ∈ On)) |
5 | oveq2 6557 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦)) | |
6 | 5 | eleq1d 2672 | . . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 suc 𝑦) ∈ On)) |
7 | oveq2 6557 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵)) | |
8 | 7 | eleq1d 2672 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝐵) ∈ On)) |
9 | om0 7484 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅) | |
10 | 0elon 5695 | . . . 4 ⊢ ∅ ∈ On | |
11 | 9, 10 | syl6eqel 2696 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ∈ On) |
12 | oacl 7502 | . . . . . . 7 ⊢ (((𝐴 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On) | |
13 | 12 | expcom 450 | . . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)) |
14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)) |
15 | omsuc 7493 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)) | |
16 | 15 | eleq1d 2672 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 suc 𝑦) ∈ On ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)) |
17 | 14, 16 | sylibrd 248 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On)) |
18 | 17 | expcom 450 | . . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On))) |
19 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
20 | iunon 7323 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On) | |
21 | 19, 20 | mpan 702 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On) |
22 | omlim 7500 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦)) | |
23 | 19, 22 | mpanr1 715 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦)) |
24 | 23 | eleq1d 2672 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On)) |
25 | 21, 24 | syl5ibr 235 | . . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On)) |
26 | 25 | expcom 450 | . . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On))) |
27 | 2, 4, 6, 8, 11, 18, 26 | tfinds3 6956 | . 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·𝑜 𝐵) ∈ On)) |
28 | 27 | impcom 445 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On) |
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-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-omul 7452 |
This theorem is referenced by: oecl 7504 omordi 7533 omord2 7534 omcan 7536 omword 7537 omwordri 7539 om00 7542 om00el 7543 omlimcl 7545 odi 7546 omass 7547 oneo 7548 omeulem1 7549 omeulem2 7550 omopth2 7551 oeoelem 7565 oeoe 7566 oeeui 7569 oaabs2 7612 omxpenlem 7946 omxpen 7947 cantnfle 8451 cantnflt 8452 cantnflem1d 8468 cantnflem1 8469 cantnflem3 8471 cantnflem4 8472 cnfcomlem 8479 xpnum 8660 infxpenc 8724 dfac12lem2 8849 |
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