Theorem cheli

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

	Ref	Expression
Hypothesis	chssi.1	⊢ 𝐻 ∈ Cℋ
Assertion	cheli	⊢ ( 𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ )

Proof

Step	Hyp	Ref	Expression
1		chssi.1	⊢ 𝐻 ∈ Cℋ
2	1	chssii	⊢ 𝐻 ⊆ ℋ
3	2	sseli	⊢ ( 𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ )