Theorem ollat 39914

Description: An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)

Assertion

Ref	Expression
ollat	⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)

Proof of Theorem ollat

Step	Hyp	Ref	Expression
1		isolat 39913	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
2	1	simplbi 501	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2149  Latclat 18489  OPcops 39873  OLcol 39875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-in 3920  df-ol 39879

This theorem is referenced by:  oldmm1  39918  oldmj1  39922  olj01  39926  olj02  39927  olm12  39929  latmassOLD  39930  latm12  39931  latm32  39932  latmrot  39933  latm4  39934  latmmdiN  39935  latmmdir  39936  olm01  39937  olm02  39938  omllat  39943  meetat  39997