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Theorem hlclat 39340
Description: A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
Assertion
Ref Expression
hlclat (𝐾 ∈ HL → 𝐾 ∈ CLat)

Proof of Theorem hlclat
StepHypRef Expression
1 hlomcmcv 39338 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp2d 1142 1 (𝐾 ∈ HL → 𝐾 ∈ CLat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  CLatccla 18556  OMLcoml 39157  CvLatclc 39247  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-hlat 39333
This theorem is referenced by:  hlomcmat  39347  glbconN  39359  glbconNOLD  39360  pmaple  39744  pmapglbx  39752  polsubN  39890  2polvalN  39897  2polssN  39898  3polN  39899  2pmaplubN  39909  paddunN  39910  poldmj1N  39911  pnonsingN  39916  ispsubcl2N  39930  psubclinN  39931  paddatclN  39932  polsubclN  39935  poml4N  39936  diaglbN  41038  diaintclN  41041  dibglbN  41149  dibintclN  41150  dihglblem2N  41277  dihglblem3N  41278  dihglblem4  41280  dihglbcpreN  41283  dihglblem6  41323  dihintcl  41327  dochval2  41335  dochcl  41336  dochvalr  41340  dochss  41348
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