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 ) ) ) |
| 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 ) ) ) |