Theorem ltmuldivi

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999)

	Ref	Expression
Hypotheses	ltplus1.1	|- A e. RR
	prodgt0.2	|- B e. RR
	ltmul1.3	|- C e. RR
Assertion	ltmuldivi	|- ( 0 < C -> ( ( A x. C ) < B <-> A < ( B / C ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	|- A e. RR
2		prodgt0.2	|- B e. RR
3		ltmul1.3	|- C e. RR
4		ltmuldiv	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> A < ( B / C ) ) )
5	1 2 4	mp3an12	|- ( ( C e. RR /\ 0 < C ) -> ( ( A x. C ) < B <-> A < ( B / C ) ) )
6	3 5	mpan	|- ( 0 < C -> ( ( A x. C ) < B <-> A < ( B / C ) ) )