Metamath Proof Explorer


Theorem ltmul1d

Description: The ratio of nonnegative and positive numbers is nonnegative. (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 ltmul1d
|- ( 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 ltmul1
 |-  ( ( 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 ) ) )