Step |
Hyp |
Ref |
Expression |
1 |
|
fvoveq1 |
|- ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h B ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) ) |
2 |
|
oveq2 |
|- ( A = if ( A e. ~H , A , 0h ) -> ( B -h A ) = ( B -h if ( A e. ~H , A , 0h ) ) ) |
3 |
2
|
fveq2d |
|- ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( B -h A ) ) = ( normh ` ( B -h if ( A e. ~H , A , 0h ) ) ) ) |
4 |
1 3
|
eqeq12d |
|- ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h B ) ) = ( normh ` ( B -h A ) ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( normh ` ( B -h if ( A e. ~H , A , 0h ) ) ) ) ) |
5 |
|
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 ) ) ) |
6 |
5
|
fveq2d |
|- ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) ) |
7 |
|
fvoveq1 |
|- ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( B -h if ( A e. ~H , A , 0h ) ) ) = ( normh ` ( if ( B e. ~H , B , 0h ) -h if ( A e. ~H , A , 0h ) ) ) ) |
8 |
6 7
|
eqeq12d |
|- ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( normh ` ( B -h if ( A e. ~H , A , 0h ) ) ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( normh ` ( if ( B e. ~H , B , 0h ) -h if ( A e. ~H , A , 0h ) ) ) ) ) |
9 |
|
ifhvhv0 |
|- if ( A e. ~H , A , 0h ) e. ~H |
10 |
|
ifhvhv0 |
|- if ( B e. ~H , B , 0h ) e. ~H |
11 |
9 10
|
normsubi |
|- ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( normh ` ( if ( B e. ~H , B , 0h ) -h if ( A e. ~H , A , 0h ) ) ) |
12 |
4 8 11
|
dedth2h |
|- ( ( A e. ~H /\ B e. ~H ) -> ( normh ` ( A -h B ) ) = ( normh ` ( B -h A ) ) ) |