MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latpos Unicode version

Theorem latpos 14155
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2283 . . 3  |-  ( meet `  K )  =  (
meet `  K )
41, 2, 3islat 14153 . 2  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) ( ( x ( join `  K
) y )  e.  ( Base `  K
)  /\  ( x
( meet `  K )
y )  e.  (
Base `  K )
) ) )
54simplbi 446 1  |-  ( K  e.  Lat  ->  K  e.  Poset )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Posetcpo 14074   joincjn 14078   meetcmee 14079   Latclat 14151
This theorem is referenced by:  latref  14159  latasymb  14160  lattr  14162  latjcom  14165  latjle12  14168  latleeqj1  14169  latmcom  14181  latlem12  14184  latleeqm1  14185  atlpos  29491  cvlposN  29517  hlpos  29555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-lat 14152
  Copyright terms: Public domain W3C validator