Step |
Hyp |
Ref |
Expression |
1 |
|
eleq1 |
|- ( A = if ( A e. ~H , A , 0h ) -> ( A e. ~H <-> if ( A e. ~H , A , 0h ) e. ~H ) ) |
2 |
|
fveq2 |
|- ( A = if ( A e. ~H , A , 0h ) -> ( ( projh ` ~H ) ` A ) = ( ( projh ` ~H ) ` if ( A e. ~H , A , 0h ) ) ) |
3 |
|
id |
|- ( A = if ( A e. ~H , A , 0h ) -> A = if ( A e. ~H , A , 0h ) ) |
4 |
2 3
|
eqeq12d |
|- ( A = if ( A e. ~H , A , 0h ) -> ( ( ( projh ` ~H ) ` A ) = A <-> ( ( projh ` ~H ) ` if ( A e. ~H , A , 0h ) ) = if ( A e. ~H , A , 0h ) ) ) |
5 |
1 4
|
bibi12d |
|- ( A = if ( A e. ~H , A , 0h ) -> ( ( A e. ~H <-> ( ( projh ` ~H ) ` A ) = A ) <-> ( if ( A e. ~H , A , 0h ) e. ~H <-> ( ( projh ` ~H ) ` if ( A e. ~H , A , 0h ) ) = if ( A e. ~H , A , 0h ) ) ) ) |
6 |
|
helch |
|- ~H e. CH |
7 |
|
ifhvhv0 |
|- if ( A e. ~H , A , 0h ) e. ~H |
8 |
6 7
|
pjchi |
|- ( if ( A e. ~H , A , 0h ) e. ~H <-> ( ( projh ` ~H ) ` if ( A e. ~H , A , 0h ) ) = if ( A e. ~H , A , 0h ) ) |
9 |
5 8
|
dedth |
|- ( A e. ~H -> ( A e. ~H <-> ( ( projh ` ~H ) ` A ) = A ) ) |
10 |
9
|
ibi |
|- ( A e. ~H -> ( ( projh ` ~H ) ` A ) = A ) |