Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2736 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2736 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2736 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | 1, 2, 3 | isclat 17960 | . 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)))) |
5 | 4 | simplbi 501 | 1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 𝒫 cpw 4499 dom cdm 5536 ‘cfv 6358 Basecbs 16666 Posetcpo 17768 lubclub 17770 glbcglb 17771 CLatccla 17958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-dm 5546 df-iota 6316 df-fv 6366 df-clat 17959 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |