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Metamath Proof Explorer

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	$⊢ A \in ℋ \to span (\{A\}) \in S_{ℋ}$

Step	Hyp	Ref	Expression
1		spansnch	$⊢ A \in ℋ \to span (\{A\}) \in C_{ℋ}$
2		chsh	$⊢ span (\{A\}) \in C_{ℋ} \to span (\{A\}) \in S_{ℋ}$
3	1 2	syl	$⊢ A \in ℋ \to span (\{A\}) \in S_{ℋ}$