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Theorem latpos 17652
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 2819 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2819 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2819 . . 3 (meet‘𝐾) = (meet‘𝐾)
41, 2, 3islat 17649 . 2 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
54simplbi 500 1 (𝐾 ∈ Lat → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108   × cxp 5546  dom cdm 5548  cfv 6348  Basecbs 16475  Posetcpo 17542  joincjn 17546  meetcmee 17547  Latclat 17647
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-iota 6307  df-fv 6356  df-lat 17648
This theorem is referenced by:  latref  17655  latasymb  17656  lattr  17658  latjcom  17661  latjle12  17664  latleeqj1  17665  latmcom  17677  latlem12  17680  latleeqm1  17681  atlpos  36429  cvlposN  36455  hlpos  36494
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