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Mirrors > Home > MPE Home > Th. List > clatpos | Structured version Visualization version GIF version |
Description: A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.) |
Ref | Expression |
---|---|
clatpos | ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2610 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | 1, 2, 3 | isclat 16932 | . 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)))) |
5 | 4 | simplbi 475 | 1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 𝒫 cpw 4108 dom cdm 5038 ‘cfv 5804 Basecbs 15695 Posetcpo 16763 lubclub 16765 glbcglb 16766 CLatccla 16930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 df-clat 16931 |
This theorem is referenced by: (None) |
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