Metamath Proof Explorer

Theorem abs3dif

Description: Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006)

Ref Expression
Assertion abs3dif 
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )

Proof

Step Hyp Ref Expression
1 npncan 
 |-  ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A - C ) + ( C - B ) ) = ( A - B ) )
2 1 3com23 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) + ( C - B ) ) = ( A - B ) )
3 2 fveq2d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( ( A - C ) + ( C - B ) ) ) = ( abs ` ( A - B ) ) )
4 subcl 
 |-  ( ( A e. CC /\ C e. CC ) -> ( A - C ) e. CC )
5 4 3adant2 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - C ) e. CC )
6 subcl 
 |-  ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
7 6 ancoms 
 |-  ( ( B e. CC /\ C e. CC ) -> ( C - B ) e. CC )
8 7 3adant1 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - B ) e. CC )
9 abstri 
 |-  ( ( ( A - C ) e. CC /\ ( C - B ) e. CC ) -> ( abs ` ( ( A - C ) + ( C - B ) ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )
10 5 8 9 syl2anc 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( ( A - C ) + ( C - B ) ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )
11 3 10 eqbrtrrd 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )