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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 2826 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2826 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2826 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | 1, 2, 3 | islat 17401 | . 2 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))) |
5 | 4 | simplbi 493 | 1 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 × cxp 5341 dom cdm 5343 ‘cfv 6124 Basecbs 16223 Posetcpo 17294 joincjn 17298 meetcmee 17299 Latclat 17399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-xp 5349 df-dm 5353 df-iota 6087 df-fv 6132 df-lat 17400 |
This theorem is referenced by: latref 17407 latasymb 17408 lattr 17410 latjcom 17413 latjle12 17416 latleeqj1 17417 latmcom 17429 latlem12 17432 latleeqm1 17433 atlpos 35377 cvlposN 35403 hlpos 35442 |
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