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Metamath Proof Explorer

Theorem norm3dif

Description: Norm of differences around common element. Part of Lemma 3.6 of Beran p. 101. (Contributed by NM, 20-Apr-2006) (New usage is discouraged.)

Ref Expression
Assertion norm3dif 
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( normh ` ( A -h B ) ) <_ ( ( normh ` ( A -h C ) ) + ( normh ` ( C -h 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 fvoveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h C ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) )
3 2 oveq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h C ) ) + ( normh ` ( C -h B ) ) ) = ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h B ) ) ) )
4 1 3 breq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h B ) ) <_ ( ( normh ` ( A -h C ) ) + ( normh ` ( C -h B ) ) ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) <_ ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h 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 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 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( C -h B ) = ( C -h if ( B e. ~H , B , 0h ) ) )
8 7 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( C -h B ) ) = ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) )
9 8 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h B ) ) ) = ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) ) )
10 6 9 breq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) <_ ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h B ) ) ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) <_ ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) ) ) )
11 oveq2 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( if ( A e. ~H , A , 0h ) -h C ) = ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) )
12 11 fveq2d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) )
13 fvoveq1 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) = ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
14 12 13 oveq12d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) ) = ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) + ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) ) )
15 14 breq2d 
 |-  ( C = if ( C e. ~H , C , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) <_ ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) + ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) <_ ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) + ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) ) ) )
16 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
17 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
18 ifhvhv0 
 |-  if ( C e. ~H , C , 0h ) e. ~H
19 16 17 18 norm3difi 
 |-  ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) <_ ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) + ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
20 4 10 15 19 dedth3h 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( normh ` ( A -h B ) ) <_ ( ( normh ` ( A -h C ) ) + ( normh ` ( C -h B ) ) ) )