Metamath Proof Explorer


Theorem ltdiv1d

Description: Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses ltmul1d.1 φ A
ltmul1d.2 φ B
ltmul1d.3 φ C +
Assertion ltdiv1d φ A < B A C < B C

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 ltdiv1 A B C 0 < C A < B A C < B C
6 1 2 4 5 syl3anc φ A < B A C < B C