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Theorem hloml 37371
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 37370 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp1d 1141 1 (𝐾 ∈ HL → 𝐾 ∈ OML)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  CLatccla 18216  OMLcoml 37189  CvLatclc 37279  HLchlt 37364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-hlat 37365
This theorem is referenced by:  hlol  37375  hlomcmat  37379  poml4N  37967  doca2N  39140  djajN  39151  dihoml4c  39390
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