Theorem omllat 40057

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 40055	. 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
2		ollat 40028	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Latclat 18512  OLcol 39989  OMLcoml 39990

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-ol 39993  df-oml 39994

This theorem is used by:  omllaw2N  40059  omllaw4  40061  omllaw5N  40062  cmtcomlemN  40063  cmt2N  40065  cmtbr2N  40068  cmtbr3N  40069  cmtbr4N  40070  lecmtN  40071  cmtidN  40072  omlfh1N  40073  omlfh3N  40074  omlmod1i2N  40075  omlspjN  40076