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Theorem hloml 36485
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 36484 . 2 (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
21simp1d 1137 1 (𝐾 ∈ HL → 𝐾 ∈ OML)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  CLatccla 17709  OMLcoml 36303  CvLatclc 36393  HLchlt 36478
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7151  df-hlat 36479
This theorem is referenced by:  hlol  36489  hlomcmat  36493  poml4N  37081  doca2N  38254  djajN  38265  dihoml4c  38504
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