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Theorem hlclat 28921
Description: A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
Assertion
Ref Expression
hlclat  |-  ( K  e.  HL  ->  K  e.  CLat )

Proof of Theorem hlclat
StepHypRef Expression
1 hlomcmcv 28919 . 2  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
21simp2d 968 1  |-  ( K  e.  HL  ->  K  e.  CLat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   CLatccla 14213   OMLcoml 28738   CvLatclc 28828   HLchlt 28913
This theorem is referenced by:  hlomcmat  28927  glbconN  28939  pmaple  29323  pmapglbx  29331  polsubN  29469  2polvalN  29476  2polssN  29477  3polN  29478  2pmaplubN  29488  paddunN  29489  poldmj1N  29490  pnonsingN  29495  ispsubcl2N  29509  psubclinN  29510  paddatclN  29511  polsubclN  29514  poml4N  29515  diaglbN  30618  diaintclN  30621  dibglbN  30729  dibintclN  30730  dihglblem2N  30857  dihglblem3N  30858  dihglblem4  30860  dihglbcpreN  30863  dihglblem6  30903  dihintcl  30907  dochval2  30915  dochcl  30916  dochvalr  30920  dochss  30928
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-hlat 28914
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