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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 2823 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2823 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | eqid 2823 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | 1, 2, 3 | isclat 17721 | . 2 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 (Base‘𝐾) ∧ dom (glb‘𝐾) = 𝒫 (Base‘𝐾)))) |
5 | 4 | simplbi 500 | 1 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 𝒫 cpw 4541 dom cdm 5557 ‘cfv 6357 Basecbs 16485 Posetcpo 17552 lubclub 17554 glbcglb 17555 CLatccla 17719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-dm 5567 df-iota 6316 df-fv 6365 df-clat 17720 |
This theorem is referenced by: (None) |
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