Metamath Proof Explorer


Theorem lhpoc2N

Description: The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012) (New usage is discouraged.)

Ref Expression
Hypotheses lhpoc.b B = Base K
lhpoc.o ˙ = oc K
lhpoc.a A = Atoms K
lhpoc.h H = LHyp K
Assertion lhpoc2N K HL W B W A ˙ W H

Proof

Step Hyp Ref Expression
1 lhpoc.b B = Base K
2 lhpoc.o ˙ = oc K
3 lhpoc.a A = Atoms K
4 lhpoc.h H = LHyp K
5 hlop K HL K OP
6 1 2 opoccl K OP W B ˙ W B
7 5 6 sylan K HL W B ˙ W B
8 1 2 3 4 lhpoc K HL ˙ W B ˙ W H ˙ ˙ W A
9 7 8 syldan K HL W B ˙ W H ˙ ˙ W A
10 1 2 opococ K OP W B ˙ ˙ W = W
11 5 10 sylan K HL W B ˙ ˙ W = W
12 11 eleq1d K HL W B ˙ ˙ W A W A
13 9 12 bitr2d K HL W B W A ˙ W H