Metamath Proof Explorer

Theorem pjnorm

Description: The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 fveq2 
 |-  ( H = if ( H e. CH , H , ~H ) -> ( projh ` H ) = ( projh ` if ( H e. CH , H , ~H ) ) )
2 1 fveq1d 
 |-  ( H = if ( H e. CH , H , ~H ) -> ( ( projh ` H ) ` A ) = ( ( projh ` if ( H e. CH , H , ~H ) ) ` A ) )
3 2 fveq2d 
 |-  ( H = if ( H e. CH , H , ~H ) -> ( normh ` ( ( projh ` H ) ` A ) ) = ( normh ` ( ( projh ` if ( H e. CH , H , ~H ) ) ` A ) ) )
4 3 breq1d 
 |-  ( H = if ( H e. CH , H , ~H ) -> ( ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) <-> ( normh ` ( ( projh ` if ( H e. CH , H , ~H ) ) ` A ) ) <_ ( normh ` A ) ) )
5 2fveq3 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( ( projh ` if ( H e. CH , H , ~H ) ) ` A ) ) = ( normh ` ( ( projh ` if ( H e. CH , H , ~H ) ) ` if ( A e. ~H , A , 0h ) ) ) )
6 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` A ) = ( normh ` if ( A e. ~H , A , 0h ) ) )
7 5 6 breq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( ( projh ` if ( H e. CH , H , ~H ) ) ` A ) ) <_ ( normh ` A ) <-> ( normh ` ( ( projh ` if ( H e. CH , H , ~H ) ) ` if ( A e. ~H , A , 0h ) ) ) <_ ( normh ` if ( A e. ~H , A , 0h ) ) ) )
8 ifchhv 
 |-  if ( H e. CH , H , ~H ) e. CH
9 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
10 8 9 pjnormi 
 |-  ( normh ` ( ( projh ` if ( H e. CH , H , ~H ) ) ` if ( A e. ~H , A , 0h ) ) ) <_ ( normh ` if ( A e. ~H , A , 0h ) )
11 4 7 10 dedth2h 
 |-  ( ( H e. CH /\ A e. ~H ) -> ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) )