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Theorem latpos 16004
Description: A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
Assertion
Ref Expression
latpos  |-  ( K  e.  Lat  ->  K  e.  Poset )

Proof of Theorem latpos
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2402 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2402 . . 3  |-  ( meet `  K )  =  (
meet `  K )
41, 2, 3islat 16001 . 2  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  ( join `  K )  =  ( ( Base `  K )  X.  ( Base `  K ) )  /\  dom  ( meet `  K )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
54simplbi 458 1  |-  ( K  e.  Lat  ->  K  e.  Poset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    X. cxp 4821   dom cdm 4823   ` cfv 5569   Basecbs 14841   Posetcpo 15893   joincjn 15897   meetcmee 15898   Latclat 15999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-dm 4833  df-iota 5533  df-fv 5577  df-lat 16000
This theorem is referenced by:  latref  16007  latasymb  16008  lattr  16010  latjcom  16013  latjle12  16016  latleeqj1  16017  latmcom  16029  latlem12  16032  latleeqm1  16033  atlpos  32319  cvlposN  32345  hlpos  32383
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