Theorem omlol 38110

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 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2733	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2733	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2733	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2733	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 38108	. 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 499	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ∀wral 3062   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  occoc 17205  joincjn 18264  meetcmee 18265  OLcol 38044  OMLcoml 38045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-oml 38049

This theorem is referenced by:  omlop  38111  omllat  38112  omllaw3  38115  omllaw4  38116  cmtcomlemN  38118  cmtbr2N  38123  cmtbr3N  38124  omlfh1N  38128  omlfh3N  38129  omlspjN  38131  hlol  38231