Metamath Proof Explorer


Theorem absdifltd

Description: The absolute value of a difference and 'less than' 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 absdifltd
|- ( 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 absdiflt
 |-  ( ( 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 ) ) ) )