Theorem ltmul1dd

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	ltmul1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltmul1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltmul1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	ltdiv1dd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	ltmul1dd	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ltmul1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltmul1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltmul1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		ltdiv1dd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
5	1 2 3	ltmul1d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) )