Theorem hloml 39945

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 39944	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp1d 1154	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)

Syntax hints:   → wi 4   ∈ wcel 2141  CLatccla 18513  OMLcoml 39763  CvLatclc 39853  HLchlt 39938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-ov 7395  df-hlat 39939

This theorem is referenced by:  hlol  39949  hlomcmat  39953  poml4N  40541  doca2N  41714  djajN  41725  dihoml4c  41964