Theorem chel

Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chel	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻 ) → 𝐴 ∈ ℋ )

Proof

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