Theorem spanss

Description: Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sstr2	$$⊢A\subseteq B\to \left(B\subseteq x\to A\subseteq x\right)$$
2	1	ralrimivw	$$⊢A\subseteq B\to \forall x\in {\mathbf{S}}_{\mathscr{H}}\left(B\subseteq x\to A\subseteq x\right)$$
3		ss2rab	$$⊢\left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}\subseteq \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\leftrightarrow \forall x\in {\mathbf{S}}_{\mathscr{H}}\left(B\subseteq x\to A\subseteq x\right)$$
4	2 3	sylibr	$$⊢A\subseteq B\to \left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}\subseteq \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}$$
5		intss	$$⊢\left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}\subseteq \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\to \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\subseteq \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}$$
6	4 5	syl	$$⊢A\subseteq B\to \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\subseteq \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}$$
7	6	adantl	$$⊢\left(B\subseteq \mathscr{H}\wedge A\subseteq B\right)\to \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}\subseteq \bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}$$
8		sstr	$$⊢\left(A\subseteq B\wedge B\subseteq \mathscr{H}\right)\to A\subseteq \mathscr{H}$$
9	8	ancoms	$$⊢\left(B\subseteq \mathscr{H}\wedge A\subseteq B\right)\to A\subseteq \mathscr{H}$$
10		spanval	$$⊢A\subseteq \mathscr{H}\to \mathrm{span}\left(A\right)=\bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}$$
11	9 10	syl	$$⊢\left(B\subseteq \mathscr{H}\wedge A\subseteq B\right)\to \mathrm{span}\left(A\right)=\bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|A\subseteq x\right\}$$
12		spanval	$$⊢B\subseteq \mathscr{H}\to \mathrm{span}\left(B\right)=\bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}$$
13	12	adantr	$$⊢\left(B\subseteq \mathscr{H}\wedge A\subseteq B\right)\to \mathrm{span}\left(B\right)=\bigcap \left\{x\in {\mathbf{S}}_{\mathscr{H}}|B\subseteq x\right\}$$
14	7 11 13	3sstr4d	$$⊢\left(B\subseteq \mathscr{H}\wedge A\subseteq B\right)\to \mathrm{span}\left(A\right)\subseteq \mathrm{span}\left(B\right)$$