Theorem ltmuldivd

Description: 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltmul1d.1	|- ( ph -> A e. RR )
	ltmul1d.2	|- ( ph -> B e. RR )
	ltmul1d.3	|- ( ph -> C e. RR+ )
Assertion	ltmuldivd	|- ( ph -> ( ( A x. C ) < B <-> A < ( B / C ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltmul1d.1	|- ( ph -> A e. RR )
2		ltmul1d.2	|- ( ph -> B e. RR )
3		ltmul1d.3	|- ( ph -> C e. RR+ )
4	3	rpregt0d	|- ( ph -> ( C e. RR /\ 0 < C ) )
5		ltmuldiv	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> A < ( B / C ) ) )
6	1 2 4 5	syl3anc	|- ( ph -> ( ( A x. C ) < B <-> A < ( B / C ) ) )