Theorem ltdiv23i

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
Assertion	ltdiv23i	⊢ ( ( 0 < 𝐵 ∧ 0 < 𝐶 ) → ( ( 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		ltdiv23	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
5	1 4	mp3an1	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
6	2 5	mpanl1	⊢ ( ( 0 < 𝐵 ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )
7	3 6	mpanr1	⊢ ( ( 0 < 𝐵 ∧ 0 < 𝐶 ) → ( ( 𝐴 / 𝐵 ) < 𝐶 ↔ ( 𝐴 / 𝐶 ) < 𝐵 ) )