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Theorem hlomcmat 38748
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 38740 . 2 (𝐾 ∈ HL → 𝐾 ∈ OML)
2 hlclat 38741 . 2 (𝐾 ∈ HL → 𝐾 ∈ CLat)
3 hlatl 38743 . 2 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
41, 2, 33jca 1125 1 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  CLatccla 18463  OMLcoml 38558  AtLatcal 38647  HLchlt 38733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  hlatmstcOLDN  38781  hlatle  38782  hlrelat1  38784  pmaple  39145  pol1N  39294  polpmapN  39296  pmaplubN  39308
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