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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 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝐾) ∈ V | |
3 | 1, 2 | eqeltri 2697 | . . . . 5 ⊢ 𝐵 ∈ V |
4 | 3 | elpw2 4828 | . . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
5 | 4 | biimpri 218 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
6 | 5 | adantl 482 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
7 | eqid 2622 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
8 | clatglbcl.g | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
9 | 1, 7, 8 | isclat 17109 | . . . 4 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
10 | simprr 796 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵) | |
11 | 9, 10 | sylbi 207 | . . 3 ⊢ (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵) |
12 | 11 | adantr 481 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵) |
13 | 6, 12 | eleqtrrd 2704 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 𝒫 cpw 4158 dom cdm 5114 ‘cfv 5888 Basecbs 15857 Posetcpo 16940 lubclub 16942 glbcglb 16943 CLatccla 17107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-clat 17108 |
This theorem is referenced by: isglbd 17117 clatglb 17124 clatglble 17125 |
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