Metamath Proof Explorer


Theorem ltdiv1dd

Description: Division of both sides of 'less than' by a positive number. (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 ltdiv1dd
|- ( ph -> ( A / C ) < ( 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 ltdiv1dd.4
 |-  ( ph -> A < B )
5 1 2 3 ltdiv1d
 |-  ( ph -> ( A < B <-> ( A / C ) < ( B / C ) ) )
6 4 5 mpbid
 |-  ( ph -> ( A / C ) < ( B / C ) )