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 | lemul2d | |- ( ph -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) ) |
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 | lemul2 | |- ( ( 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 ) ) ) |