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Theorem hlomcmat 34477
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 34470 . 2 (𝐾 ∈ HL → 𝐾 ∈ OML)
2 hlclat 34471 . 2 (𝐾 ∈ HL → 𝐾 ∈ CLat)
3 hlatl 34473 . 2 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
41, 2, 33jca 1241 1 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1037  wcel 1989  CLatccla 17101  OMLcoml 34288  AtLatcal 34377  HLchlt 34463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-cvlat 34435  df-hlat 34464
This theorem is referenced by:  hlatmstcOLDN  34509  hlatle  34510  hlrelat1  34512  pmaple  34873  pol1N  35022  polpmapN  35024  pmaplubN  35036
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