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Metamath Proof Explorer

Theorem hvnegdi

Description: Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000) (New usage is discouraged.)

Ref Expression
Assertion hvnegdi 
|- ( ( A e. ~H /\ B e. ~H ) -> ( -u 1 .h ( A -h B ) ) = ( B -h A ) )

Proof

Step Hyp Ref Expression
1 oveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A -h B ) = ( if ( A e. ~H , A , 0h ) -h B ) )
2 1 oveq2d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( -u 1 .h ( A -h B ) ) = ( -u 1 .h ( if ( A e. ~H , A , 0h ) -h B ) ) )
3 oveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( B -h A ) = ( B -h if ( A e. ~H , A , 0h ) ) )
4 2 3 eqeq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( -u 1 .h ( A -h B ) ) = ( B -h A ) <-> ( -u 1 .h ( if ( A e. ~H , A , 0h ) -h B ) ) = ( 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 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( -u 1 .h ( if ( A e. ~H , A , 0h ) -h B ) ) = ( -u 1 .h ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
7 oveq1 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( B -h if ( A e. ~H , A , 0h ) ) = ( if ( B e. ~H , B , 0h ) -h if ( A e. ~H , A , 0h ) ) )
8 6 7 eqeq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( -u 1 .h ( if ( A e. ~H , A , 0h ) -h B ) ) = ( B -h if ( A e. ~H , A , 0h ) ) <-> ( -u 1 .h ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( 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 hvnegdii 
 |-  ( -u 1 .h ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( if ( B e. ~H , B , 0h ) -h if ( A e. ~H , A , 0h ) )
12 4 8 11 dedth2h 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( -u 1 .h ( A -h B ) ) = ( B -h A ) )