Theorem omlol 40265

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 2761	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2761	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2761	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2761	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2761	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 40263	. 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 2145  ∀wral 3077   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  lecple 17415  occoc 17416  joincjn 18465  meetcmee 18466  OLcol 40199  OMLcoml 40200

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-oml 40204

This theorem is used by:  omlop  40266  omllat  40267  omllaw3  40270  omllaw4  40271  cmtcomlemN  40273  cmtbr2N  40278  cmtbr3N  40279  omlfh1N  40283  omlfh3N  40284  omlspjN  40286  hlol  40386