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Theorem hloml 38829
Description: A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
Assertion
Ref Expression
hloml (𝐾 ∈ HL → 𝐾 ∈ OML)

Proof of Theorem hloml
StepHypRef Expression
1 hlomcmcv 38828 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp1d 1140 1 (𝐾 ∈ HL → 𝐾 ∈ OML)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  CLatccla 18489  OMLcoml 38647  CvLatclc 38737  HLchlt 38822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-hlat 38823
This theorem is referenced by:  hlol  38833  hlomcmat  38837  poml4N  39426  doca2N  40599  djajN  40610  dihoml4c  40849
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