Theorem sshjval2

Description: Value of join in the set of closed subspaces of Hilbert space CH . (Contributed by NM, 1-Nov-2000) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	sshjval2	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\to A{\vee }_{\mathscr{H}}B=\bigcap \left\{x\in {\mathbf{C}}_{\mathscr{H}}|A\cup B\subseteq x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		sshjval	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\to A{\vee }_{\mathscr{H}}B=\perp \left(\perp \left(A\cup B\right)\right)$$
2		unss	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\leftrightarrow A\cup B\subseteq \mathscr{H}$$
3		ococin	$$⊢A\cup B\subseteq \mathscr{H}\to \perp \left(\perp \left(A\cup B\right)\right)=\bigcap \left\{x\in {\mathbf{C}}_{\mathscr{H}}|A\cup B\subseteq x\right\}$$
4	2 3	sylbi	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\to \perp \left(\perp \left(A\cup B\right)\right)=\bigcap \left\{x\in {\mathbf{C}}_{\mathscr{H}}|A\cup B\subseteq x\right\}$$
5	1 4	eqtrd	$$⊢\left(A\subseteq \mathscr{H}\wedge B\subseteq \mathscr{H}\right)\to A{\vee }_{\mathscr{H}}B=\bigcap \left\{x\in {\mathbf{C}}_{\mathscr{H}}|A\cup B\subseteq x\right\}$$