Theorem omlol 40500

Description: An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)

Assertion

Ref	Expression
omlol	⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

Proof of Theorem omlol

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2763	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2763	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2763	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 40498	. 2 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 → 𝑦 = (𝑥(join‘𝐾)(𝑦(meet‘𝐾)((oc‘𝐾)‘𝑥))))))
7	6	simplbi 502	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  lecple 17467  occoc 17468  joincjn 18517  meetcmee 18518  OLcol 40434  OMLcoml 40435

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-oml 40439

This theorem is used by:  omlop  40501  omllat  40502  omllaw3  40505  omllaw4  40506  cmtcomlemN  40508  cmtbr2N  40513  cmtbr3N  40514  omlfh1N  40518  omlfh3N  40519  omlspjN  40521  hlol  40621