Theorem hlclat 39477

Description: A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)

Assertion

Ref	Expression
hlclat	⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)

Proof of Theorem hlclat

Step	Hyp	Ref	Expression
1		hlomcmcv 39475	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp2d 1143	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)

Syntax hints:   → wi 4   ∈ wcel 2113  CLatccla 18406  OMLcoml 39294  CvLatclc 39384  HLchlt 39469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-hlat 39470

This theorem is referenced by:  hlomcmat  39484  glbconN  39496  pmaple  39880  pmapglbx  39888  polsubN  40026  2polvalN  40033  2polssN  40034  3polN  40035  2pmaplubN  40045  paddunN  40046  poldmj1N  40047  pnonsingN  40052  ispsubcl2N  40066  psubclinN  40067  paddatclN  40068  polsubclN  40071  poml4N  40072  diaglbN  41174  diaintclN  41177  dibglbN  41285  dibintclN  41286  dihglblem2N  41413  dihglblem3N  41414  dihglblem4  41416  dihglbcpreN  41419  dihglblem6  41459  dihintcl  41463  dochval2  41471  dochcl  41472  dochvalr  41476  dochss  41484