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Theorem hlomcmat 33669
 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 33662 . 2 (𝐾 ∈ HL → 𝐾 ∈ OML)
2 hlclat 33663 . 2 (𝐾 ∈ HL → 𝐾 ∈ CLat)
3 hlatl 33665 . 2 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
41, 2, 33jca 1235 1 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   ∈ wcel 1977  CLatccla 16930  OMLcoml 33480  AtLatcal 33569  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cvlat 33627  df-hlat 33656 This theorem is referenced by:  hlatmstcOLDN  33701  hlatle  33702  hlrelat1  33704  pmaple  34065  pol1N  34214  polpmapN  34216  pmaplubN  34228
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