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Theorem latpos 17404
Description: A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
Assertion
Ref Expression
latpos (𝐾 ∈ Lat → 𝐾 ∈ Poset)

Proof of Theorem latpos
StepHypRef 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)
41, 2, 3islat 17401 . 2 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
54simplbi 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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