Theorem ltreci

Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
Assertion	ltreci	⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		ltrec	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ) )
4	2 3	mpanr1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ 0 < 𝐵 ) → ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ) )
5	1 4	mpanl1	⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) ) )