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Theorem latpos 16990
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 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2621 . . 3 (meet‘𝐾) = (meet‘𝐾)
41, 2, 3islat 16987 . 2 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)))))
54simplbi 476 1 (𝐾 ∈ Lat → 𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   × cxp 5082  dom cdm 5084  cfv 5857  Basecbs 15800  Posetcpo 16880  joincjn 16884  meetcmee 16885  Latclat 16985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-dm 5094  df-iota 5820  df-fv 5865  df-lat 16986
This theorem is referenced by:  latref  16993  latasymb  16994  lattr  16996  latjcom  16999  latjle12  17002  latleeqj1  17003  latmcom  17015  latlem12  17018  latleeqm1  17019  atlpos  34107  cvlposN  34133  hlpos  34171
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