Theorem absdifled

Description: The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	|- ( ph -> A e. RR )
	absltd.2	|- ( ph -> B e. RR )
	absltd.3	|- ( ph -> C e. RR )
Assertion	absdifled	|- ( ph -> ( ( abs ` ( A - B ) ) <_ C <-> ( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		absltd.1	|- ( ph -> A e. RR )
2		absltd.2	|- ( ph -> B e. RR )
3		absltd.3	|- ( ph -> C e. RR )
4		absdifle	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( abs ` ( A - B ) ) <_ C <-> ( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( abs ` ( A - B ) ) <_ C <-> ( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) )