Metamath Proof Explorer


Theorem ltmul2d

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 ( 𝜑𝐴 ∈ ℝ )
ltmul1d.2 ( 𝜑𝐵 ∈ ℝ )
ltmul1d.3 ( 𝜑𝐶 ∈ ℝ+ )
Assertion ltmul2d ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 ltmul1d.1 ( 𝜑𝐴 ∈ ℝ )
2 ltmul1d.2 ( 𝜑𝐵 ∈ ℝ )
3 ltmul1d.3 ( 𝜑𝐶 ∈ ℝ+ )
4 3 rpregt0d ( 𝜑 → ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) )
5 ltmul2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )
6 1 2 4 5 syl3anc ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )