Metamath Proof Explorer

Theorem ltmul2dd

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of Apostol p. 20. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses ltmul1d.1 φ A
ltmul1d.2 φ B
ltmul1d.3 φ C +
ltdiv1dd.4 φ A < B
Assertion ltmul2dd φ C A < C B


Step Hyp Ref Expression
1 ltmul1d.1 φ A
2 ltmul1d.2 φ B
3 ltmul1d.3 φ C +
4 ltdiv1dd.4 φ A < B
5 1 2 3 ltmul2d φ A < B C A < C B
6 4 5 mpbid φ C A < C B