This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem normsq

Description: The square of a norm. (Contributed by NM, 12-May-2005) (New usage is discouraged.)

Ref Expression
Assertion normsq 
|- ( A e. ~H -> ( ( normh ` A ) ^ 2 ) = ( A .ih A ) )

Proof

Step Hyp Ref Expression
1 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` A ) = ( normh ` if ( A e. ~H , A , 0h ) ) )
2 1 oveq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` A ) ^ 2 ) = ( ( normh ` if ( A e. ~H , A , 0h ) ) ^ 2 ) )
3 id 
 |-  ( A = if ( A e. ~H , A , 0h ) -> A = if ( A e. ~H , A , 0h ) )
4 3 3 oveq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A .ih A ) = ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) )
5 2 4 eqeq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( normh ` A ) ^ 2 ) = ( A .ih A ) <-> ( ( normh ` if ( A e. ~H , A , 0h ) ) ^ 2 ) = ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) ) )
6 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
7 6 normsqi 
 |-  ( ( normh ` if ( A e. ~H , A , 0h ) ) ^ 2 ) = ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) )
8 5 7 dedth 
 |-  ( A e. ~H -> ( ( normh ` A ) ^ 2 ) = ( A .ih A ) )