Metamath Proof Explorer

Theorem pjadji

Description: A projection is self-adjoint. Property (i) of Beran p. 109. (Contributed by NM, 6-Oct-2000) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 pjadjt.1 
 |-  H e. CH
2 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( projh ` H ) ` A ) = ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) )
3 2 oveq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( projh ` H ) ` A ) .ih B ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) .ih B ) )
4 oveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A .ih ( ( projh ` H ) ` B ) ) = ( if ( A e. ~H , A , 0h ) .ih ( ( projh ` H ) ` B ) ) )
5 3 4 eqeq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( ( projh ` H ) ` A ) .ih B ) = ( A .ih ( ( projh ` H ) ` B ) ) <-> ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) .ih B ) = ( if ( A e. ~H , A , 0h ) .ih ( ( projh ` H ) ` B ) ) ) )
6 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) .ih B ) = ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) .ih 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. ~H , A , 0h ) .ih ( ( projh ` H ) ` B ) ) = ( if ( A e. ~H , A , 0h ) .ih ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) ) )
9 6 8 eqeq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) .ih B ) = ( if ( A e. ~H , A , 0h ) .ih ( ( projh ` H ) ` B ) ) <-> ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) .ih if ( B e. ~H , B , 0h ) ) = ( if ( A e. ~H , A , 0h ) .ih ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) ) ) )
10 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
11 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
12 1 10 11 pjadjii 
 |-  ( ( ( projh ` H ) ` if ( A e. ~H , A , 0h ) ) .ih if ( B e. ~H , B , 0h ) ) = ( if ( A e. ~H , A , 0h ) .ih ( ( projh ` H ) ` if ( B e. ~H , B , 0h ) ) )
13 5 9 12 dedth2h 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( ( projh ` H ) ` A ) .ih B ) = ( A .ih ( ( projh ` H ) ` B ) ) )