Theorem ollat 39179

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 39178	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
2	1	simplbi 497	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2109  Latclat 18366  OPcops 39138  OLcol 39140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-in 3918  df-ol 39144

This theorem is referenced by:  oldmm1  39183  oldmj1  39187  olj01  39191  olj02  39192  olm12  39194  latmassOLD  39195  latm12  39196  latm32  39197  latmrot  39198  latm4  39199  latmmdiN  39200  latmmdir  39201  olm01  39202  olm02  39203  omllat  39208  meetat  39262