Theorem olop 37224

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 37222	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
2	1	simprbi 497	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)

Syntax hints:   → wi 4   ∈ wcel 2110  Latclat 18147  OPcops 37182  OLcol 37184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ol 37188

This theorem is referenced by:  olposN  37225  oldmm1  37227  oldmm2  37228  oldmm3N  37229  oldmm4  37230  oldmj1  37231  oldmj2  37232  oldmj3  37233  oldmj4  37234  olj01  37235  olj02  37236  olm11  37237  olm12  37238  latmassOLD  37239  olm01  37246  olm02  37247  omlop  37251  meetat  37306  hlop  37372  polatN  37941