Theorem hlclat 40021

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 40019	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp2d 1159	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)

Syntax hints:   → wi 4   ∈ wcel 2149  CLatccla 18553  OMLcoml 39838  CvLatclc 39928  HLchlt 40013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-hlat 40014

This theorem is referenced by:  hlomcmat  40028  glbconN  40040  pmaple  40424  pmapglbx  40432  polsubN  40570  2polvalN  40577  2polssN  40578  3polN  40579  2pmaplubN  40589  paddunN  40590  poldmj1N  40591  pnonsingN  40596  ispsubcl2N  40610  psubclinN  40611  paddatclN  40612  polsubclN  40615  poml4N  40616  diaglbN  41718  diaintclN  41721  dibglbN  41829  dibintclN  41830  dihglblem2N  41957  dihglblem3N  41958  dihglblem4  41960  dihglbcpreN  41963  dihglblem6  42003  dihintcl  42007  dochval2  42015  dochcl  42016  dochvalr  42020  dochss  42028