Theorem spanid

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

	Ref	Expression
Assertion	spanid	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \mathrm{span}\left(A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		shss	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\subseteq \mathscr{H}$$
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	syl	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \mathrm{span}\left(A\right)=\bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}$$
4		intmin	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}=A$$
5	3 4	eqtrd	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \mathrm{span}\left(A\right)=A$$