Theorem ollat 36351

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 36350	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
2	1	simplbi 500	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2114  Latclat 17657  OPcops 36310  OLcol 36312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-in 3945  df-ol 36316

This theorem is referenced by:  oldmm1  36355  oldmj1  36359  olj01  36363  olj02  36364  olm12  36366  latmassOLD  36367  latm12  36368  latm32  36369  latmrot  36370  latm4  36371  latmmdiN  36372  latmmdir  36373  olm01  36374  olm02  36375  omllat  36380  meetat  36434