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 ) ) )