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Theorem hloml 37014
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 37013 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp1d 1143 1 (𝐾 ∈ HL → 𝐾 ∈ OML)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  CLatccla 17833  OMLcoml 36832  CvLatclc 36922  HLchlt 37007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7173  df-hlat 37008
This theorem is referenced by:  hlol  37018  hlomcmat  37022  poml4N  37610  doca2N  38783  djajN  38794  dihoml4c  39033
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