Theorem chsupval2

Description: The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of Kalmbach p. 65. (Contributed by NM, 13-Aug-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	chsupval2	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to {\bigvee }_{\mathscr{H}}\left(A\right)=\bigcap \left\{x\in {\mathbf{C}}_{\mathscr{H}}|\bigcup A\subseteq x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		chsspwh	$$⊢{\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}$$
2		sstr2	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to \left({\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}\to A\subseteq 𝒫\mathscr{H}\right)$$
3	1 2	mpi	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to A\subseteq 𝒫\mathscr{H}$$
4		hsupval2	$$⊢A\subseteq 𝒫\mathscr{H}\to {\bigvee }_{\mathscr{H}}\left(A\right)=\bigcap \left\{x\in {\mathbf{C}}_{\mathscr{H}}|\bigcup A\subseteq x\right\}$$
5	3 4	syl	$$⊢A\subseteq {\mathbf{C}}_{\mathscr{H}}\to {\bigvee }_{\mathscr{H}}\left(A\right)=\bigcap \left\{x\in {\mathbf{C}}_{\mathscr{H}}|\bigcup A\subseteq x\right\}$$