Metamath Proof Explorer


Theorem ltsub13

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004)

Ref Expression
Assertion ltsub13
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( B - C ) <-> C < ( B - A ) ) )

Proof

Step Hyp Ref Expression
1 ltaddsub
 |-  ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( ( A + C ) < B <-> A < ( B - C ) ) )
2 ltaddsub2
 |-  ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( ( A + C ) < B <-> C < ( B - A ) ) )
3 1 2 bitr3d
 |-  ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < ( B - C ) <-> C < ( B - A ) ) )
4 3 3com23
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( B - C ) <-> C < ( B - A ) ) )