Theorem ollat 39589

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 39588	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
2	1	simplbi 496	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2114  Latclat 18366  OPcops 39548  OLcol 39550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-ol 39554

This theorem is referenced by:  oldmm1  39593  oldmj1  39597  olj01  39601  olj02  39602  olm12  39604  latmassOLD  39605  latm12  39606  latm32  39607  latmrot  39608  latm4  39609  latmmdiN  39610  latmmdir  39611  olm01  39612  olm02  39613  omllat  39618  meetat  39672