Theorem ollat 40536

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 40535	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
2	1	simplbi 502	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Latclat 18655  OPcops 40495  OLcol 40497

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-ol 40501

This theorem is used by:  oldmm1  40540  oldmj1  40544  olj01  40548  olj02  40549  olm12  40551  latmassOLD  40552  latm12  40553  latm32  40554  latmrot  40555  latm4  40556  latmmdiN  40557  latmmdir  40558  olm01  40559  olm02  40560  omllat  40565  meetat  40619