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Theorem omllat 38568
Description: An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
Assertion
Ref Expression
omllat (𝐾 ∈ OML → 𝐾 ∈ Lat)

Proof of Theorem omllat
StepHypRef Expression
1 omlol 38566 . 2 (𝐾 ∈ OML → 𝐾 ∈ OL)
2 ollat 38539 . 2 (𝐾 ∈ OL → 𝐾 ∈ Lat)
31, 2syl 17 1 (𝐾 ∈ OML → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Latclat 18385  OLcol 38500  OMLcoml 38501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-ol 38504  df-oml 38505
This theorem is referenced by:  omllaw2N  38570  omllaw4  38572  omllaw5N  38573  cmtcomlemN  38574  cmt2N  38576  cmtbr2N  38579  cmtbr3N  38580  cmtbr4N  38581  lecmtN  38582  cmtidN  38583  omlfh1N  38584  omlfh3N  38585  omlmod1i2N  38586  omlspjN  38587
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