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Mirrors > Home > MPE Home > Th. List > latpos | Structured version Visualization version GIF version |
Description: A lattice is a poset. (Contributed by NM, 17-Sep-2011.) |
Ref | Expression |
---|---|
latpos | ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2728 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2728 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | 1, 2, 3 | islat 18419 | . 2 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))) |
5 | 4 | simplbi 497 | 1 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 × cxp 5671 dom cdm 5673 ‘cfv 6543 Basecbs 17174 Posetcpo 18293 joincjn 18297 meetcmee 18298 Latclat 18417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-xp 5679 df-dm 5683 df-iota 6495 df-fv 6551 df-lat 18418 |
This theorem is referenced by: latref 18427 latasymb 18428 lattr 18430 latjcom 18433 latjle12 18436 latleeqj1 18437 latmcom 18449 latlem12 18452 latleeqm1 18453 atlpos 38768 cvlposN 38794 hlpos 38833 |
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