Theorem chsupcl

Description: Closure of supremum of subset of CH . Definition of supremum in Proposition 1 of Kalmbach p. 65. Shows that CH is a complete lattice. Also part of Definition 3.4-1 in MegPav2000 p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999) (New usage is discouraged.)

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

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		hsupcl	|- ( A C_ ~P ~H -> ( \/H ` A ) e. CH )
5	3 4	syl	|- ( A C_ CH -> ( \/H ` A ) e. CH )