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Theorem hlomcmat 39346
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 39338 . 2 (𝐾 ∈ HL → 𝐾 ∈ OML)
2 hlclat 39339 . 2 (𝐾 ∈ HL → 𝐾 ∈ CLat)
3 hlatl 39341 . 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 18555  OMLcoml 39156  AtLatcal 39245  HLchlt 39331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-cvlat 39303  df-hlat 39332
This theorem is referenced by:  hlatmstcOLDN  39379  hlatle  39380  hlrelat1  39382  pmaple  39743  pol1N  39892  polpmapN  39894  pmaplubN  39906
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