Theorem omlol 39263

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 2736	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2736	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2736	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2736	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2736	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 39261	. 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 497	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3052   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283  occoc 17284  joincjn 18328  meetcmee 18329  OLcol 39197  OMLcoml 39198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-oml 39202

This theorem is referenced by:  omlop  39264  omllat  39265  omllaw3  39268  omllaw4  39269  cmtcomlemN  39271  cmtbr2N  39276  cmtbr3N  39277  omlfh1N  39281  omlfh3N  39282  omlspjN  39284  hlol  39384