Theorem reclt1d

Description: The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
Assertion	reclt1d	⊢ ( 𝜑 → ( 𝐴 < 1 ↔ 1 < ( 1 / 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2	1	rpregt0d	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
3		reclt1	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 < 1 ↔ 1 < ( 1 / 𝐴 ) ) )
4	2 3	syl	⊢ ( 𝜑 → ( 𝐴 < 1 ↔ 1 < ( 1 / 𝐴 ) ) )