Metamath Proof Explorer

Theorem normsub0

Description: Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999) (New usage is discouraged.)

Ref Expression
Assertion normsub0 
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( normh ` ( A -h B ) ) = 0 <-> A = B ) )

Proof

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 1 eqeq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h B ) ) = 0 <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = 0 ) )
3 eqeq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A = B <-> if ( A e. ~H , A , 0h ) = B ) )
4 2 3 bibi12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( normh ` ( A -h B ) ) = 0 <-> A = B ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = 0 <-> if ( A e. ~H , A , 0h ) = B ) ) )
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 fveqeq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = 0 <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = 0 ) )
7 eqeq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. ~H , A , 0h ) = B <-> if ( A e. ~H , A , 0h ) = if ( B e. ~H , B , 0h ) ) )
8 6 7 bibi12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = 0 <-> if ( A e. ~H , A , 0h ) = B ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = 0 <-> if ( A e. ~H , A , 0h ) = if ( B e. ~H , B , 0h ) ) ) )
9 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
10 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
11 9 10 normsub0i 
 |-  ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = 0 <-> if ( A e. ~H , A , 0h ) = if ( B e. ~H , B , 0h ) )
12 4 8 11 dedth2h 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( normh ` ( A -h B ) ) = 0 <-> A = B ) )