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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 6921 | . . . . 5 ⊢ 𝐵 ∈ V |
3 | 2 | elpw2 5340 | . . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
4 | 3 | biimpri 228 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
5 | 4 | adantl 481 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
6 | eqid 2735 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | clatglbcl.g | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
8 | 1, 6, 7 | isclat 18558 | . . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
9 | simprr 773 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵) | |
10 | 8, 9 | sylbi 217 | . . 3 ⊢ (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵) |
11 | 10 | adantr 480 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵) |
12 | 5, 11 | eleqtrrd 2842 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 𝒫 cpw 4605 dom cdm 5689 ‘cfv 6563 Basecbs 17245 Posetcpo 18365 lubclub 18367 glbcglb 18368 CLatccla 18556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 df-clat 18557 |
This theorem is referenced by: isglbd 18567 clatglb 18574 clatglble 18575 glbconN 39359 |
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