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 ∈ ℝ )
	ltmul1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltmul1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	ltdiv1dd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	ltmul2dd	⊢ ( 𝜑 → ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) )

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	ltmul2d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → ( 𝐶 · 𝐴 ) < ( 𝐶 · 𝐵 ) )