Metamath Proof Explorer


Theorem ocss

Description: An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

Ref Expression
Assertion ocss ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ⊆ ℋ )

Proof

Step Hyp Ref Expression
1 ocsh ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ∈ S )
2 shss ( ( ⊥ ‘ 𝐴 ) ∈ S → ( ⊥ ‘ 𝐴 ) ⊆ ℋ )
3 1 2 syl ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ⊆ ℋ )