Theorem ltmul1dd

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	ltmul1d.1	|- ( ph -> A e. RR )
	ltmul1d.2	|- ( ph -> B e. RR )
	ltmul1d.3	|- ( ph -> C e. RR+ )
	ltdiv1dd.4	|- ( ph -> A < B )
Assertion	ltmul1dd	|- ( ph -> ( A x. C ) < ( B x. 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		ltdiv1dd.4	|- ( ph -> A < B )
5	1 2 3	ltmul1d	|- ( ph -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
6	4 5	mpbid	|- ( ph -> ( A x. C ) < ( B x. C ) )