Theorem sheli

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
Hypothesis	shssi.1	⊢ 𝐻 ∈ Sℋ
Assertion	sheli	⊢ ( 𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ )

Proof

Step	Hyp	Ref	Expression
1		shssi.1	⊢ 𝐻 ∈ Sℋ
2	1	shssii	⊢ 𝐻 ⊆ ℋ
3	2	sseli	⊢ ( 𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ )