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Theorem hlomcmat 38699
Description: A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
hlomcmat (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Proof of Theorem hlomcmat
StepHypRef Expression
1 hloml 38691 . 2 (𝐾 ∈ HL → 𝐾 ∈ OML)
2 hlclat 38692 . 2 (𝐾 ∈ HL → 𝐾 ∈ CLat)
3 hlatl 38694 . 2 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
41, 2, 33jca 1127 1 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2105  CLatccla 18461  OMLcoml 38509  AtLatcal 38598  HLchlt 38684
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-cvlat 38656  df-hlat 38685
This theorem is referenced by:  hlatmstcOLDN  38732  hlatle  38733  hlrelat1  38735  pmaple  39096  pol1N  39245  polpmapN  39247  pmaplubN  39259
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