Metamath Proof Explorer


Theorem riotaocN

Description: The orthocomplement of the unique poset element such that ps . ( riotaneg analog.) (Contributed by NM, 16-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses riotaoc.b B = Base K
riotaoc.o ˙ = oc K
riotaoc.a x = ˙ y φ ψ
Assertion riotaocN K OP ∃! x B φ ι x B | φ = ˙ ι y B | ψ

Proof

Step Hyp Ref Expression
1 riotaoc.b B = Base K
2 riotaoc.o ˙ = oc K
3 riotaoc.a x = ˙ y φ ψ
4 nfcv _ y ˙
5 nfriota1 _ y ι y B | ψ
6 4 5 nffv _ y ˙ ι y B | ψ
7 1 2 opoccl K OP y B ˙ y B
8 1 2 opoccl K OP ι y B | ψ B ˙ ι y B | ψ B
9 fveq2 y = ι y B | ψ ˙ y = ˙ ι y B | ψ
10 1 2 opoccl K OP x B ˙ x B
11 1 2 opcon2b K OP x B y B x = ˙ y y = ˙ x
12 10 11 reuhypd K OP x B ∃! y B x = ˙ y
13 6 7 8 3 9 12 riotaxfrd K OP ∃! x B φ ι x B | φ = ˙ ι y B | ψ