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Theorem hlclat 39359
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 39357 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp2d 1144 1 (𝐾 ∈ HL → 𝐾 ∈ CLat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  CLatccla 18543  OMLcoml 39176  CvLatclc 39266  HLchlt 39351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-hlat 39352
This theorem is referenced by:  hlomcmat  39366  glbconN  39378  glbconNOLD  39379  pmaple  39763  pmapglbx  39771  polsubN  39909  2polvalN  39916  2polssN  39917  3polN  39918  2pmaplubN  39928  paddunN  39929  poldmj1N  39930  pnonsingN  39935  ispsubcl2N  39949  psubclinN  39950  paddatclN  39951  polsubclN  39954  poml4N  39955  diaglbN  41057  diaintclN  41060  dibglbN  41168  dibintclN  41169  dihglblem2N  41296  dihglblem3N  41297  dihglblem4  41299  dihglbcpreN  41302  dihglblem6  41342  dihintcl  41346  dochval2  41354  dochcl  41355  dochvalr  41359  dochss  41367
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