Theorem omlol 39345

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 2731	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2731	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2731	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2731	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2731	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 39343	. 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 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5093  ‘cfv 6487  (class class class)co 7352  Basecbs 17126  lecple 17174  occoc 17175  joincjn 18223  meetcmee 18224  OLcol 39279  OMLcoml 39280

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6443  df-fv 6495  df-ov 7355  df-oml 39284

This theorem is referenced by:  omlop  39346  omllat  39347  omllaw3  39350  omllaw4  39351  cmtcomlemN  39353  cmtbr2N  39358  cmtbr3N  39359  omlfh1N  39363  omlfh3N  39364  omlspjN  39366  hlol  39466