Metamath Proof Explorer


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 ) ) )