Theorem sshjval3

Description: Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in Beran p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice CH . (Contributed by NM, 2-Mar-2004) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	sshjval3	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( \/H ` { A , B } ) )

Proof

Step	Hyp	Ref	Expression
1		ax-hilex	|- ~H e. _V
2	1	elpw2	|- ( A e. ~P ~H <-> A C_ ~H )
3	1	elpw2	|- ( B e. ~P ~H <-> B C_ ~H )
4		uniprg	|- ( ( A e. ~P ~H /\ B e. ~P ~H ) -> U. { A , B } = ( A u. B ) )
5	2 3 4	syl2anbr	|- ( ( A C_ ~H /\ B C_ ~H ) -> U. { A , B } = ( A u. B ) )
6	5	fveq2d	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( _|_ ` U. { A , B } ) = ( _|_ ` ( A u. B ) ) )
7	6	fveq2d	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( _|_ ` ( _|_ ` U. { A , B } ) ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
8		prssi	|- ( ( A e. ~P ~H /\ B e. ~P ~H ) -> { A , B } C_ ~P ~H )
9	2 3 8	syl2anbr	|- ( ( A C_ ~H /\ B C_ ~H ) -> { A , B } C_ ~P ~H )
10		hsupval	|- ( { A , B } C_ ~P ~H -> ( \/H ` { A , B } ) = ( _|_ ` ( _|_ ` U. { A , B } ) ) )
11	9 10	syl	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( \/H ` { A , B } ) = ( _|_ ` ( _|_ ` U. { A , B } ) ) )
12		sshjval	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
13	7 11 12	3eqtr4rd	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( \/H ` { A , B } ) )