Theorem ltdiv2d

Description: Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	ltdiv2d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
Assertion	ltdiv2d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 / 𝐵 ) < ( 𝐶 / 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		ltdiv2d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4	1	rpregt0d	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
5	2	rpregt0d	⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) )
6	3	rpregt0d	⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) )
7		ltdiv2	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 / 𝐵 ) < ( 𝐶 / 𝐴 ) ) )
8	4 5 6 7	syl3anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 / 𝐵 ) < ( 𝐶 / 𝐴 ) ) )