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Metamath Proof Explorer

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 ) ) )