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	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltmul1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltmul1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
Assertion	ltmul2d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 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	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 ) ) )