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 ) ) |
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 ) ) |