![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clatlubcl2 | Structured version Visualization version GIF version |
Description: Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.) |
Ref | Expression |
---|---|
clatlubcl.b | ⊢ 𝐵 = (Base‘𝐾) |
clatlubcl.u | ⊢ 𝑈 = (lub‘𝐾) |
Ref | Expression |
---|---|
clatlubcl2 | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatlubcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
2 | 1 | fvexi 6910 | . . . . 5 ⊢ 𝐵 ∈ V |
3 | 2 | elpw2 5348 | . . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
4 | 3 | biimpri 227 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
5 | 4 | adantl 480 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
6 | clatlubcl.u | . . . . 5 ⊢ 𝑈 = (lub‘𝐾) | |
7 | eqid 2725 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
8 | 1, 6, 7 | isclat 18511 | . . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵))) |
9 | simprl 769 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom (glb‘𝐾) = 𝒫 𝐵)) → dom 𝑈 = 𝒫 𝐵) | |
10 | 8, 9 | sylbi 216 | . . 3 ⊢ (𝐾 ∈ CLat → dom 𝑈 = 𝒫 𝐵) |
11 | 10 | adantr 479 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝑈 = 𝒫 𝐵) |
12 | 5, 11 | eleqtrrd 2828 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 𝒫 cpw 4604 dom cdm 5678 ‘cfv 6549 Basecbs 17199 Posetcpo 18318 lubclub 18320 glbcglb 18321 CLatccla 18509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-dm 5688 df-iota 6501 df-fv 6557 df-clat 18510 |
This theorem is referenced by: lublem 18521 |
Copyright terms: Public domain | W3C validator |