Theorem hloml 36495

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 36494	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
2	1	simp1d 1138	1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)

Syntax hints:   → wi 4   ∈ wcel 2114  CLatccla 17719  OMLcoml 36313  CvLatclc 36403  HLchlt 36488

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-hlat 36489

This theorem is referenced by:  hlol  36499  hlomcmat  36503  poml4N  37091  doca2N  38264  djajN  38275  dihoml4c  38514