Metamath Proof Explorer

Theorem pjsubi

Description: Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000) (New usage is discouraged.)

Ref Expression
Hypothesis pjadjt.1 
|- H e. CH
Assertion pjsubi 
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( projh ` H ) ` ( A -h B ) ) = ( ( ( projh ` H ) ` A ) -h ( ( projh ` H ) ` B ) ) )

Proof

Step Hyp Ref Expression
1 pjadjt.1 
 |-  H e. CH
2 fvoveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( projh ` H ) ` ( A -h B ) ) = ( ( projh ` H ) ` ( if ( A e. ~H , A , 0h ) -h B ) ) )
3 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( projh ` H ) ` A ) = ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) )
4 3 oveq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( projh ` H ) ` A ) -h ( ( projh ` H ) ` B ) ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` B ) ) )
5 2 4 eqeq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( projh ` H ) ` ( A -h B ) ) = ( ( ( projh ` H ) ` A ) -h ( ( projh ` H ) ` B ) ) <-> ( ( projh ` H ) ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` B ) ) ) )
6 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. ~H , A , 0h ) -h B ) = ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) )
7 6 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( projh ` H ) ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( ( projh ` H ) ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
8 fveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( projh ` H ) ` B ) = ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) )
9 8 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` B ) ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) ) )
10 7 9 eqeq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( projh ` H ) ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` B ) ) <-> ( ( projh ` H ) ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) ) ) )
11 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
12 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
13 1 11 12 pjsubii 
 |-  ( ( projh ` H ) ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) -h ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) )
14 5 10 13 dedth2h 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( projh ` H ) ` ( A -h B ) ) = ( ( ( projh ` H ) ` A ) -h ( ( projh ` H ) ` B ) ) )