Metamath Proof Explorer

Theorem pjmuli

Description: Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 pjadjt.1 
 |-  H e. CH
2 fvoveq1 
 |-  ( A = if ( A e. CC , A , 0 ) -> ( ( projh ` H ) ` ( A .h B ) ) = ( ( projh ` H ) ` ( if ( A e. CC , A , 0 ) .h B ) ) )
3 oveq1 
 |-  ( A = if ( A e. CC , A , 0 ) -> ( A .h ( ( projh ` H ) ` B ) ) = ( if ( A e. CC , A , 0 ) .h ( ( projh ` H ) ` B ) ) )
4 2 3 eqeq12d 
 |-  ( A = if ( A e. CC , A , 0 ) -> ( ( ( projh ` H ) ` ( A .h B ) ) = ( A .h ( ( projh ` H ) ` B ) ) <-> ( ( projh ` H ) ` ( if ( A e. CC , A , 0 ) .h B ) ) = ( if ( A e. CC , A , 0 ) .h ( ( projh ` H ) ` B ) ) ) )
5 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. CC , A , 0 ) .h B ) = ( if ( A e. CC , A , 0 ) .h if ( B e. ~H , B , 0h ) ) )
6 5 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( projh ` H ) ` ( if ( A e. CC , A , 0 ) .h B ) ) = ( ( projh ` H ) ` ( if ( A e. CC , A , 0 ) .h if ( B e. ~H , B , 0h ) ) ) )
7 fveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( projh ` H ) ` B ) = ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) )
8 7 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. CC , A , 0 ) .h ( ( projh ` H ) ` B ) ) = ( if ( A e. CC , A , 0 ) .h ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) ) )
9 6 8 eqeq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( projh ` H ) ` ( if ( A e. CC , A , 0 ) .h B ) ) = ( if ( A e. CC , A , 0 ) .h ( ( projh ` H ) ` B ) ) <-> ( ( projh ` H ) ` ( if ( A e. CC , A , 0 ) .h if ( B e. ~H , B , 0h ) ) ) = ( if ( A e. CC , A , 0 ) .h ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) ) ) )
10 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
11 0cn 
 |-  0 e. CC
12 11 elimel 
 |-  if ( A e. CC , A , 0 ) e. CC
13 1 10 12 pjmulii 
 |-  ( ( projh ` H ) ` ( if ( A e. CC , A , 0 ) .h if ( B e. ~H , B , 0h ) ) ) = ( if ( A e. CC , A , 0 ) .h ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) )
14 4 9 13 dedth2h 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( ( projh ` H ) ` ( A .h B ) ) = ( A .h ( ( projh ` H ) ` B ) ) )