|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6920 | . . . . 5 ⊢ 𝐵 ∈ V | 
| 3 | 2 | elpw2 5334 | . . . 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 2737 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
| 7 | clatglbcl.g | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
| 8 | 1, 6, 7 | isclat 18545 | . . . 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 2844 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 dom cdm 5685 ‘cfv 6561 Basecbs 17247 Posetcpo 18353 lubclub 18355 glbcglb 18356 CLatccla 18543 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-clat 18544 | 
| This theorem is referenced by: isglbd 18554 clatglb 18561 clatglble 18562 glbconN 39378 | 
| Copyright terms: Public domain | W3C validator |