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 C_ CH -> ( \/H ` A ) = |^| { x e. CH | U. A C_ x } )

Proof

Step	Hyp	Ref	Expression
1		chsspwh	|- CH C_ ~P ~H
2		sstr2	|- ( A C_ CH -> ( CH C_ ~P ~H -> A C_ ~P ~H ) )
3	1 2	mpi	|- ( A C_ CH -> A C_ ~P ~H )
4		hsupval2	|- ( A C_ ~P ~H -> ( \/H ` A ) = |^| { x e. CH | U. A C_ x } )
5	3 4	syl	|- ( A C_ CH -> ( \/H ` A ) = |^| { x e. CH | U. A C_ x } )