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Theorem omlol 34007
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 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2621 . . 3 (meet‘𝐾) = (meet‘𝐾)
5 eqid 2621 . . 3 (oc‘𝐾) = (oc‘𝐾)
61, 2, 3, 4, 5isoml 34005 . 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 476 1 (𝐾 ∈ OML → 𝐾 ∈ OL)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  occoc 15870  joincjn 16865  meetcmee 16866  OLcol 33941  OMLcoml 33942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-oml 33946
This theorem is referenced by:  omlop  34008  omllat  34009  omllaw3  34012  omllaw4  34013  cmtcomlemN  34015  cmtbr2N  34020  cmtbr3N  34021  omlfh1N  34025  omlfh3N  34026  omlspjN  34028  hlol  34128
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