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 C_ ~H -> ( _|_ ` A ) C_ ~H )

Proof

Step Hyp Ref Expression
1 ocsh 
 |-  ( A C_ ~H -> ( _|_ ` A ) e. SH )
2 shss 
 |-  ( ( _|_ ` A ) e. SH -> ( _|_ ` A ) C_ ~H )
3 1 2 syl 
 |-  ( A C_ ~H -> ( _|_ ` A ) C_ ~H )