Metamath Proof Explorer


Theorem ltdivmuld

Description: 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses ltmul1d.1 φ A
ltmul1d.2 φ B
ltmul1d.3 φ C +
Assertion ltdivmuld φ A C < B A < C B

Proof

Step Hyp Ref Expression
1 ltmul1d.1 φ A
2 ltmul1d.2 φ B
3 ltmul1d.3 φ C +
4 3 rpregt0d φ C 0 < C
5 ltdivmul A B C 0 < C A C < B A < C B
6 1 2 4 5 syl3anc φ A C < B A < C B