Theorem spanss2

Description: A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spanss2	$$⊢A\subseteq \mathscr{H}\to A\subseteq \mathrm{span}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssintub	$$⊢A\subseteq \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}$$
2		spanval	$$⊢A\subseteq \mathscr{H}\to \mathrm{span}\left(A\right)=\bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}$$
3	1 2	sseqtrrid	$$⊢A\subseteq \mathscr{H}\to A\subseteq \mathrm{span}\left(A\right)$$