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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 6802 | . . . 4 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅)) | |
2 | 1 | eleq1d 2835 | . . 3 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 ∅) ∈ On)) |
3 | oveq2 6802 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦)) | |
4 | 3 | eleq1d 2835 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝑦) ∈ On)) |
5 | oveq2 6802 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦)) | |
6 | 5 | eleq1d 2835 | . . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 suc 𝑦) ∈ On)) |
7 | oveq2 6802 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵)) | |
8 | 7 | eleq1d 2835 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝐵) ∈ On)) |
9 | om0 7752 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅) | |
10 | 0elon 5922 | . . . 4 ⊢ ∅ ∈ On | |
11 | 9, 10 | syl6eqel 2858 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ∈ On) |
12 | oacl 7770 | . . . . . . 7 ⊢ (((𝐴 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On) | |
13 | 12 | expcom 398 | . . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)) |
14 | 13 | adantr 466 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)) |
15 | omsuc 7761 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)) | |
16 | 15 | eleq1d 2835 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 suc 𝑦) ∈ On ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)) |
17 | 14, 16 | sylibrd 249 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On)) |
18 | 17 | expcom 398 | . . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On))) |
19 | vex 3354 | . . . . . 6 ⊢ 𝑥 ∈ V | |
20 | iunon 7590 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ ∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On) | |
21 | 19, 20 | mpan 664 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On) |
22 | omlim 7768 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦)) | |
23 | 19, 22 | mpanr1 677 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦)) |
24 | 23 | eleq1d 2835 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝑥) ∈ On ↔ ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On)) |
25 | 21, 24 | syl5ibr 236 | . . . 4 ⊢ ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On)) |
26 | 25 | expcom 398 | . . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On))) |
27 | 2, 4, 6, 8, 11, 18, 26 | tfinds3 7212 | . 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·𝑜 𝐵) ∈ On)) |
28 | 27 | impcom 394 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 Vcvv 3351 ∅c0 4064 ∪ ciun 4655 Oncon0 5867 Lim wlim 5868 suc csuc 5869 (class class class)co 6794 +𝑜 coa 7711 ·𝑜 comu 7712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-oadd 7718 df-omul 7719 |
This theorem is referenced by: oecl 7772 omordi 7801 omord2 7802 omcan 7804 omword 7805 omwordri 7807 om00 7810 om00el 7811 omlimcl 7813 odi 7814 omass 7815 oneo 7816 omeulem1 7817 omeulem2 7818 omopth2 7819 oeoelem 7833 oeoe 7834 oeeui 7837 oaabs2 7880 omxpenlem 8218 omxpen 8219 cantnfle 8733 cantnflt 8734 cantnflem1d 8750 cantnflem1 8751 cantnflem3 8753 cantnflem4 8754 cnfcomlem 8761 xpnum 8978 infxpenc 9042 dfac12lem2 9169 |
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