Theorem shelii

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shssi.1	⊢ 𝐻 ∈ Sℋ
	sheli.1	⊢ 𝐴 ∈ 𝐻
Assertion	shelii	⊢ 𝐴 ∈ ℋ

Proof

Step	Hyp	Ref	Expression
1		shssi.1	⊢ 𝐻 ∈ Sℋ
2		sheli.1	⊢ 𝐴 ∈ 𝐻
3	1	shssii	⊢ 𝐻 ⊆ ℋ
4	3 2	sselii	⊢ 𝐴 ∈ ℋ