Theorem pjid

Description: The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	pjid	|- ( ( H e. CH /\ A e. H ) -> ( ( projh ` H ) ` A ) = A )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( H e. CH /\ A e. H ) -> H e. CH )
2		chel	|- ( ( H e. CH /\ A e. H ) -> A e. ~H )
3	1 2	jca	|- ( ( H e. CH /\ A e. H ) -> ( H e. CH /\ A e. ~H ) )
4		pjch	|- ( ( H e. CH /\ A e. ~H ) -> ( A e. H <-> ( ( projh ` H ) ` A ) = A ) )
5	4	biimpa	|- ( ( ( H e. CH /\ A e. ~H ) /\ A e. H ) -> ( ( projh ` H ) ` A ) = A )
6	3 5	sylancom	|- ( ( H e. CH /\ A e. H ) -> ( ( projh ` H ) ` A ) = A )