Theorem abs3difi

Description: Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	|- A e. CC
	abssub.2	|- B e. CC
	abs3dif.3	|- C e. CC
Assertion	abs3difi	|- ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	|- A e. CC
2		abssub.2	|- B e. CC
3		abs3dif.3	|- 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	mp3an	|- ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) )