Theorem shel

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shel	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ) → 𝐴 ∈ ℋ )

Proof

Step	Hyp	Ref	Expression
1		shss	⊢ ( 𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ )
2	1	sselda	⊢ ( ( 𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ) → 𝐴 ∈ ℋ )