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Theorem hlclat 29473
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 29471 . 2  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
21simp2d 970 1  |-  ( K  e.  HL  ->  K  e.  CLat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   CLatccla 14463   OMLcoml 29290   CvLatclc 29380   HLchlt 29465
This theorem is referenced by:  hlomcmat  29479  glbconN  29491  pmaple  29875  pmapglbx  29883  polsubN  30021  2polvalN  30028  2polssN  30029  3polN  30030  2pmaplubN  30040  paddunN  30041  poldmj1N  30042  pnonsingN  30047  ispsubcl2N  30061  psubclinN  30062  paddatclN  30063  polsubclN  30066  poml4N  30067  diaglbN  31170  diaintclN  31173  dibglbN  31281  dibintclN  31282  dihglblem2N  31409  dihglblem3N  31410  dihglblem4  31412  dihglbcpreN  31415  dihglblem6  31455  dihintcl  31459  dochval2  31467  dochcl  31468  dochvalr  31472  dochss  31480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-hlat 29466
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