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 ) ∈ Sℋ )
2		shss	⊢ ( ( ⊥ ‘ 𝐴 ) ∈ Sℋ → ( ⊥ ‘ 𝐴 ) ⊆ ℋ )
3	1 2	syl	⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ⊆ ℋ )