Theorem sshjval2

Description: Value of join in the set of closed subspaces of Hilbert space CH . (Contributed by NM, 1-Nov-2000) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	sshjval2	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = |^| { x e. CH | ( A u. B ) C_ x } )

Proof

Step	Hyp	Ref	Expression
1		sshjval	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
2		unss	|- ( ( A C_ ~H /\ B C_ ~H ) <-> ( A u. B ) C_ ~H )
3		ococin	|- ( ( A u. B ) C_ ~H -> ( _|_ ` ( _|_ ` ( A u. B ) ) ) = |^| { x e. CH | ( A u. B ) C_ x } )
4	2 3	sylbi	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( _|_ ` ( _|_ ` ( A u. B ) ) ) = |^| { x e. CH | ( A u. B ) C_ x } )
5	1 4	eqtrd	|- ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = |^| { x e. CH | ( A u. B ) C_ x } )