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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 6451 | . . . . 5 ⊢ 𝐵 ∈ V |
3 | 2 | elpw2 5052 | . . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
4 | 3 | biimpri 220 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
5 | 4 | adantl 475 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
6 | eqid 2825 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | clatglbcl.g | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
8 | 1, 6, 7 | isclat 17469 | . . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
9 | simprr 789 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵) | |
10 | 8, 9 | sylbi 209 | . . 3 ⊢ (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵) |
11 | 10 | adantr 474 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵) |
12 | 5, 11 | eleqtrrd 2909 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 𝒫 cpw 4380 dom cdm 5346 ‘cfv 6127 Basecbs 16229 Posetcpo 17300 lubclub 17302 glbcglb 17303 CLatccla 17467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-dm 5356 df-iota 6090 df-fv 6135 df-clat 17468 |
This theorem is referenced by: isglbd 17477 clatglb 17484 clatglble 17485 |
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