Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omlol Structured version   Visualization version   GIF version

Theorem omlol 35049
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 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2771 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2771 . . 3 (join‘𝐾) = (join‘𝐾)
4 eqid 2771 . . 3 (meet‘𝐾) = (meet‘𝐾)
5 eqid 2771 . . 3 (oc‘𝐾) = (oc‘𝐾)
61, 2, 3, 4, 5isoml 35047 . 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 485 1 (𝐾 ∈ OML → 𝐾 ∈ OL)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  occoc 16157  joincjn 17152  meetcmee 17153  OLcol 34983  OMLcoml 34984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-oml 34988
This theorem is referenced by:  omlop  35050  omllat  35051  omllaw3  35054  omllaw4  35055  cmtcomlemN  35057  cmtbr2N  35062  cmtbr3N  35063  omlfh1N  35067  omlfh3N  35068  omlspjN  35070  hlol  35170
  Copyright terms: Public domain W3C validator