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Theorem latpos 15343
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 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2454 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2454 . . 3  |-  ( meet `  K )  =  (
meet `  K )
41, 2, 3islat 15340 . 2  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  ( join `  K )  =  ( ( Base `  K )  X.  ( Base `  K ) )  /\  dom  ( meet `  K )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
54simplbi 460 1  |-  ( K  e.  Lat  ->  K  e.  Poset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4949   dom cdm 4951   ` cfv 5529   Basecbs 14296   Posetcpo 15233   joincjn 15237   meetcmee 15238   Latclat 15338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-dm 4961  df-iota 5492  df-fv 5537  df-lat 15339
This theorem is referenced by:  latref  15346  latasymb  15347  lattr  15349  latjcom  15352  latjle12  15355  latleeqj1  15356  latmcom  15368  latlem12  15371  latleeqm1  15372  atlpos  33309  cvlposN  33335  hlpos  33373
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