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Theorem latpos 18424
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 2728 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2728 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2728 . . 3 (meet‘𝐾) = (meet‘𝐾)
41, 2, 3islat 18419 . 2 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
54simplbi 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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