Theorem pjoci

Description: Projection of orthocomplement. First part of Theorem 27.3 of Halmos p. 45. (Contributed by NM, 26-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjidmco.1	|- H e. CH
Assertion	pjoci	|- ( ( projh ` ~H ) -op ( projh ` H ) ) = ( projh ` ( _|_ ` H ) )

Proof

Step	Hyp	Ref	Expression
1		pjidmco.1	|- H e. CH
2	1	pjtoi	|- ( ( projh ` H ) +op ( projh ` ( _|_ ` H ) ) ) = ( projh ` ~H )
3		helch	|- ~H e. CH
4	3	pjfi	|- ( projh ` ~H ) : ~H --> ~H
5	1	pjfi	|- ( projh ` H ) : ~H --> ~H
6	1	choccli	|- ( _|_ ` H ) e. CH
7	6	pjfi	|- ( projh ` ( _|_ ` H ) ) : ~H --> ~H
8	4 5 7	hodsi	|- ( ( ( projh ` ~H ) -op ( projh ` H ) ) = ( projh ` ( _|_ ` H ) ) <-> ( ( projh ` H ) +op ( projh ` ( _|_ ` H ) ) ) = ( projh ` ~H ) )
9	2 8	mpbir	|- ( ( projh ` ~H ) -op ( projh ` H ) ) = ( projh ` ( _|_ ` H ) )