Metamath Proof Explorer


Theorem recgt1

Description: The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005)

Ref Expression
Assertion recgt1 ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < 𝐴 ↔ ( 1 / 𝐴 ) < 1 ) )

Proof

Step Hyp Ref Expression
1 1re 1 ∈ ℝ
2 0lt1 0 < 1
3 ltrec ( ( ( 1 ∈ ℝ ∧ 0 < 1 ) ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) → ( 1 < 𝐴 ↔ ( 1 / 𝐴 ) < ( 1 / 1 ) ) )
4 1 2 3 mpanl12 ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < 𝐴 ↔ ( 1 / 𝐴 ) < ( 1 / 1 ) ) )
5 1div1e1 ( 1 / 1 ) = 1
6 5 breq2i ( ( 1 / 𝐴 ) < ( 1 / 1 ) ↔ ( 1 / 𝐴 ) < 1 )
7 4 6 bitrdi ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < 𝐴 ↔ ( 1 / 𝐴 ) < 1 ) )