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Mirrors > Home > MPE Home > Th. List > clatglbcl2 | Structured version Visualization version GIF version |
Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018.) |
Ref | Expression |
---|---|
clatglbcl.b | ⊢ 𝐵 = (Base‘𝐾) |
clatglbcl.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
clatglbcl2 | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatglbcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
2 | 1 | fvexi 6906 | . . . . 5 ⊢ 𝐵 ∈ V |
3 | 2 | elpw2 5346 | . . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
4 | 3 | biimpri 227 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
5 | 4 | adantl 483 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
6 | eqid 2733 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | clatglbcl.g | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
8 | 1, 6, 7 | isclat 18453 | . . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
9 | simprr 772 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵) | |
10 | 8, 9 | sylbi 216 | . . 3 ⊢ (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵) |
11 | 10 | adantr 482 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵) |
12 | 5, 11 | eleqtrrd 2837 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 𝒫 cpw 4603 dom cdm 5677 ‘cfv 6544 Basecbs 17144 Posetcpo 18260 lubclub 18262 glbcglb 18263 CLatccla 18451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-dm 5687 df-iota 6496 df-fv 6552 df-clat 18452 |
This theorem is referenced by: isglbd 18462 clatglb 18469 clatglble 18470 glbconN 38247 |
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