Theorem ltsub23

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999)

	Ref	Expression
Assertion	ltsub23	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> ( A - C ) < B ) )

Proof

Step	Hyp	Ref	Expression
1		ltsubadd	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( C + B ) ) )
2		ltsubadd2	|- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( ( A - C ) < B <-> A < ( C + B ) ) )
3	2	3com23	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - C ) < B <-> A < ( C + B ) ) )
4	1 3	bitr4d	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> ( A - C ) < B ) )