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Theorem omlol 39222
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.
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2735 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2735 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2735 . . 3 (meet‘𝐾) = (meet‘𝐾)
5 eqid 2735 . . 3 (oc‘𝐾) = (oc‘𝐾)
61, 2, 3, 4, 5isoml 39220 . 2 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦𝑦 = (𝑥(join‘𝐾)(𝑦(meet‘𝐾)((oc‘𝐾)‘𝑥))))))
76simplbi 497 1 (𝐾 ∈ OML → 𝐾 ∈ OL)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  OLcol 39156  OMLcoml 39157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-oml 39161
This theorem is referenced by:  omlop  39223  omllat  39224  omllaw3  39227  omllaw4  39228  cmtcomlemN  39230  cmtbr2N  39235  cmtbr3N  39236  omlfh1N  39240  omlfh3N  39241  omlspjN  39243  hlol  39343
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