Theorem reclt1

Description: The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005)

	Ref	Expression
Assertion	reclt1	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 < 1 ↔ 1 < ( 1 / 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		0lt1	⊢ 0 < 1
3		ltrec	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 1 ∈ ℝ ∧ 0 < 1 ) ) → ( 𝐴 < 1 ↔ ( 1 / 1 ) < ( 1 / 𝐴 ) ) )
4	1 2 3	mpanr12	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 < 1 ↔ ( 1 / 1 ) < ( 1 / 𝐴 ) ) )
5		1div1e1	⊢ ( 1 / 1 ) = 1
6	5	breq1i	⊢ ( ( 1 / 1 ) < ( 1 / 𝐴 ) ↔ 1 < ( 1 / 𝐴 ) )
7	4 6	bitrdi	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 < 1 ↔ 1 < ( 1 / 𝐴 ) ) )