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 ) ) ) )