Theorem spansnsh

Description: The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansnsh	⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Sℋ )

Proof

Step	Hyp	Ref	Expression
1		spansnch	⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Cℋ )
2		chsh	⊢ ( ( span ‘ { 𝐴 } ) ∈ Cℋ → ( span ‘ { 𝐴 } ) ∈ Sℋ )
3	1 2	syl	⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Sℋ )