Theorem pjnorm2

Description: A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch yield Theorem 26.3 of Halmos p. 44. (Contributed by NM, 7-Apr-2001) (New usage is discouraged.)

	Ref	Expression
Assertion	pjnorm2	|- ( ( H e. CH /\ A e. ~H ) -> ( A e. H <-> ( normh ` ( ( projh ` H ) ` A ) ) = ( normh ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjhcl	|- ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` H ) ` A ) e. ~H )
2		normcl	|- ( ( ( projh ` H ) ` A ) e. ~H -> ( normh ` ( ( projh ` H ) ` A ) ) e. RR )
3	1 2	syl	|- ( ( H e. CH /\ A e. ~H ) -> ( normh ` ( ( projh ` H ) ` A ) ) e. RR )
4		normcl	|- ( A e. ~H -> ( normh ` A ) e. RR )
5	4	adantl	|- ( ( H e. CH /\ A e. ~H ) -> ( normh ` A ) e. RR )
6	3 5	eqleltd	|- ( ( H e. CH /\ A e. ~H ) -> ( ( normh ` ( ( projh ` H ) ` A ) ) = ( normh ` A ) <-> ( ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) /\ -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) ) )
7		pjnorm	|- ( ( H e. CH /\ A e. ~H ) -> ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) )
8	7	biantrurd	|- ( ( H e. CH /\ A e. ~H ) -> ( -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) <-> ( ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) /\ -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) ) )
9		pjnel	|- ( ( H e. CH /\ A e. ~H ) -> ( -. A e. H <-> ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) )
10	9	con1bid	|- ( ( H e. CH /\ A e. ~H ) -> ( -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) <-> A e. H ) )
11	6 8 10	3bitr2rd	|- ( ( H e. CH /\ A e. ~H ) -> ( A e. H <-> ( normh ` ( ( projh ` H ) ` A ) ) = ( normh ` A ) ) )