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 ) ) ) |