Theorem ltdiv23ii

Description: Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
	ltmul1.3	⊢ 𝐶 ∈ ℝ
	ltdiv23i.4	⊢ 0 < 𝐵
	ltdiv23i.5	⊢ 0 < 𝐶
Assertion	ltdiv23ii	⊢ ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		ltmul1.3	⊢ 𝐶 ∈ ℝ
4		ltdiv23i.4	⊢ 0 < 𝐵
5		ltdiv23i.5	⊢ 0 < 𝐶
6	1 2 3	ltdiv23i	⊢ ( ( 0 < 𝐵 ∧ 0 < 𝐶 ) → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
7	4 5 6	mp2an	⊢ ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 )