Theorem hloml 39727

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 39726	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp1d 1143	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)

Syntax hints:   → wi 4   ∈ wcel 2114  CLatccla 18433  OMLcoml 39545  CvLatclc 39635  HLchlt 39720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-hlat 39721

This theorem is referenced by:  hlol  39731  hlomcmat  39735  poml4N  40323  doca2N  41496  djajN  41507  dihoml4c  41746