Theorem spancl

Description: The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		spanval	$$⊢A\subseteq \mathscr{H}\to \mathrm{span}\left(A\right)=\bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}$$
2		ssrab2	$$⊢\left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\subseteq {\mathbf{S}}_{\mathscr{H}}$$
3		helsh	$$⊢\mathscr{H}\in {\mathbf{S}}_{\mathscr{H}}$$
4		sseq2	$$⊢x=\mathscr{H}\to \left(A\subseteq x\leftrightarrow A\subseteq \mathscr{H}\right)$$
5	4	rspcev	$$⊢\left(\mathscr{H}\in {\mathbf{S}}_{\mathscr{H}}\wedge A\subseteq \mathscr{H}\right)\to \exists x\in {\mathbf{S}}_{\mathscr{H}}A\subseteq x$$
6	3 5	mpan	$$⊢A\subseteq \mathscr{H}\to \exists x\in {\mathbf{S}}_{\mathscr{H}}A\subseteq x$$
7		rabn0	$$⊢\left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\ne \varnothing \leftrightarrow \exists x\in {\mathbf{S}}_{\mathscr{H}}A\subseteq x$$
8	6 7	sylibr	$$⊢A\subseteq \mathscr{H}\to \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\ne \varnothing$$
9		shintcl	$$⊢\left(\left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\subseteq {\mathbf{S}}_{\mathscr{H}}\wedge \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\ne \varnothing \right)\to \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\in {\mathbf{S}}_{\mathscr{H}}$$
10	2 8 9	sylancr	$$⊢A\subseteq \mathscr{H}\to \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\in {\mathbf{S}}_{\mathscr{H}}$$
11	1 10	eqeltrd	$$⊢A\subseteq \mathscr{H}\to \mathrm{span}\left(A\right)\in {\mathbf{S}}_{\mathscr{H}}$$