Theorem olop 39170

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

Assertion

Ref	Expression
olop	⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)

Proof of Theorem olop

Step	Hyp	Ref	Expression
1		isolat 39168	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
2	1	simprbi 496	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)

Syntax hints:   → wi 4   ∈ wcel 2108  Latclat 18501  OPcops 39128  OLcol 39130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-in 3983  df-ol 39134

This theorem is referenced by:  olposN  39171  oldmm1  39173  oldmm2  39174  oldmm3N  39175  oldmm4  39176  oldmj1  39177  oldmj2  39178  oldmj3  39179  oldmj4  39180  olj01  39181  olj02  39182  olm11  39183  olm12  39184  latmassOLD  39185  olm01  39192  olm02  39193  omlop  39197  meetat  39252  hlop  39318  polatN  39888