Theorem elspani

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

	Ref	Expression
Hypothesis	elspan.1	$$⊢B\in \mathrm{V}$$
Assertion	elspani	$$⊢A\subseteq \mathscr{H}\to \left(B\in \mathrm{span}\left(A\right)\leftrightarrow \forall x\in {\mathbf{S}}_{\mathscr{H}}\left(A\subseteq x\to B\in x\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elspan.1	$$⊢B\in \mathrm{V}$$
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	2	eleq2d	$$⊢A\subseteq \mathscr{H}\to \left(B\in \mathrm{span}\left(A\right)\leftrightarrow B\in \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\right)$$
4	1	elintrab	$$⊢B\in \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\leftrightarrow \forall x\in {\mathbf{S}}_{\mathscr{H}}\left(A\subseteq x\to B\in x\right)$$
5	3 4	bitrdi	$$⊢A\subseteq \mathscr{H}\to \left(B\in \mathrm{span}\left(A\right)\leftrightarrow \forall x\in {\mathbf{S}}_{\mathscr{H}}\left(A\subseteq x\to B\in x\right)\right)$$