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

Metamath Proof Explorer

Theorem norm3adifi

Description: Norm of differences around common element. Part of Lemma 3.6 of Beran p. 101. (Contributed by NM, 3-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypothesis norm3adift.1 
|- C e. ~H
Assertion norm3adifi 
|- ( ( A e. ~H /\ B e. ~H ) -> ( abs ` ( ( normh ` ( A -h C ) ) - ( normh ` ( B -h C ) ) ) ) <_ ( normh ` ( A -h B ) ) )

Proof

Step Hyp Ref Expression
1 norm3adift.1 
 |-  C e. ~H
2 fvoveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h C ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) )
3 2 fvoveq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( abs ` ( ( normh ` ( A -h C ) ) - ( normh ` ( B -h C ) ) ) ) = ( abs ` ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( B -h C ) ) ) ) )
4 fvoveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h B ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) )
5 3 4 breq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( abs ` ( ( normh ` ( A -h C ) ) - ( normh ` ( B -h C ) ) ) ) <_ ( normh ` ( A -h B ) ) <-> ( abs ` ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( B -h C ) ) ) ) <_ ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) ) )
6 fvoveq1 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( B -h C ) ) = ( normh ` ( if ( B e. ~H , B , 0h ) -h C ) ) )
7 6 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( B -h C ) ) ) = ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( if ( B e. ~H , B , 0h ) -h C ) ) ) )
8 7 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( abs ` ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( B -h C ) ) ) ) = ( abs ` ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( if ( B e. ~H , B , 0h ) -h C ) ) ) ) )
9 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 ) ) )
10 9 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 ) ) ) )
11 8 10 breq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( abs ` ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( B -h C ) ) ) ) <_ ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) <-> ( abs ` ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( if ( B e. ~H , B , 0h ) -h C ) ) ) ) <_ ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) ) )
12 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
13 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
14 12 13 1 norm3adifii 
 |-  ( abs ` ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) - ( normh ` ( if ( B e. ~H , B , 0h ) -h C ) ) ) ) <_ ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) )
15 5 11 14 dedth2h 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( abs ` ( ( normh ` ( A -h C ) ) - ( normh ` ( B -h C ) ) ) ) <_ ( normh ` ( A -h B ) ) )