Metamath Proof Explorer


Theorem abs3difd

Description: Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses abscld.1
|- ( ph -> A e. CC )
abssubd.2
|- ( ph -> B e. CC )
abs3difd.3
|- ( ph -> C e. CC )
Assertion abs3difd
|- ( ph -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )

Proof

Step Hyp Ref Expression
1 abscld.1
 |-  ( ph -> A e. CC )
2 abssubd.2
 |-  ( ph -> B e. CC )
3 abs3difd.3
 |-  ( ph -> C e. CC )
4 abs3dif
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )