Theorem omllat 39997

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

Step	Hyp	Ref	Expression
1		omlol 39995	. 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
2		ollat 39968	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2143  Latclat 18488  OLcol 39929  OMLcoml 39930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-ol 39933  df-oml 39934

This theorem is referenced by:  omllaw2N  39999  omllaw4  40001  omllaw5N  40002  cmtcomlemN  40003  cmt2N  40005  cmtbr2N  40008  cmtbr3N  40009  cmtbr4N  40010  lecmtN  40011  cmtidN  40012  omlfh1N  40013  omlfh3N  40014  omlmod1i2N  40015  omlspjN  40016