Theorem hloml 39613

Description: A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)

Assertion

Ref	Expression
hloml	⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)

Proof of Theorem hloml

Step	Hyp	Ref	Expression
1		hlomcmcv 39612	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp1d 1142	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)

Syntax hints:   → wi 4   ∈ wcel 2113  CLatccla 18421  OMLcoml 39431  CvLatclc 39521  HLchlt 39606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-hlat 39607

This theorem is referenced by:  hlol  39617  hlomcmat  39621  poml4N  40209  doca2N  41382  djajN  41393  dihoml4c  41632