Theorem lemul1d

Description: Multiplication of both sides of 'less than or equal to' by a positive number. (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	lemul1d	|- ( ph -> ( A <_ B <-> ( 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	3	rpregt0d	|- ( ph -> ( C e. RR /\ 0 < C ) )
5		lemul1	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) )
6	1 2 4 5	syl3anc	|- ( ph -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) )