Theorem hsupcl

Description: Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to CH even if the subsets do not. (Contributed by NM, 10-Nov-1999) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	hsupcl	|- ( A C_ ~P ~H -> ( \/H ` A ) e. CH )

Proof

Step	Hyp	Ref	Expression
1		hsupval	|- ( A C_ ~P ~H -> ( \/H ` A ) = ( _|_ ` ( _|_ ` U. A ) ) )
2		sspwuni	|- ( A C_ ~P ~H <-> U. A C_ ~H )
3		ocss	|- ( U. A C_ ~H -> ( _|_ ` U. A ) C_ ~H )
4		occl	|- ( ( _|_ ` U. A ) C_ ~H -> ( _|_ ` ( _|_ ` U. A ) ) e. CH )
5	3 4	syl	|- ( U. A C_ ~H -> ( _|_ ` ( _|_ ` U. A ) ) e. CH )
6	2 5	sylbi	|- ( A C_ ~P ~H -> ( _|_ ` ( _|_ ` U. A ) ) e. CH )
7	1 6	eqeltrd	|- ( A C_ ~P ~H -> ( \/H ` A ) e. CH )