Metamath Proof Explorer


Theorem ltmul2d

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of Apostol p. 20. (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 ltmul2d
|- ( ph -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )

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 ltmul2
 |-  ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )
6 1 2 4 5 syl3anc
 |-  ( ph -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )