Theorem ltrecii

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
	ltreci.3	⊢ 0 < 𝐴
	ltreci.4	⊢ 0 < 𝐵
Assertion	ltrecii	⊢ ( 𝐴 < 𝐵 ↔ ( 1 / 𝐵 ) < ( 1 / 𝐴 ) )

Proof

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