Theorem ltdiv2dd

Description: Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	ltdiv2dd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	ltdiv2dd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	ltdiv2dd.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	ltdiv2dd.altb	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	ltdiv2dd	⊢ ( 𝜑 → ( 𝐶 / 𝐵 ) < ( 𝐶 / 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ltdiv2dd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		ltdiv2dd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		ltdiv2dd.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		ltdiv2dd.altb	⊢ ( 𝜑 → 𝐴 < 𝐵 )
5	1 2 3	ltdiv2d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 / 𝐵 ) < ( 𝐶 / 𝐴 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → ( 𝐶 / 𝐵 ) < ( 𝐶 / 𝐴 ) )