Theorem oc0

Description: The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	oc0	⊢ ( 𝐻 ∈ Sℋ → 0ℎ ∈ ( ⊥ ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		shocsh	⊢ ( 𝐻 ∈ Sℋ → ( ⊥ ‘ 𝐻 ) ∈ Sℋ )
2		sh0	⊢ ( ( ⊥ ‘ 𝐻 ) ∈ Sℋ → 0ℎ ∈ ( ⊥ ‘ 𝐻 ) )
3	1 2	syl	⊢ ( 𝐻 ∈ Sℋ → 0ℎ ∈ ( ⊥ ‘ 𝐻 ) )