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 ) ) |
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 ) ) |