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Metamath Proof Explorer

Theorem recgt1i

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

Ref Expression
Assertion recgt1i ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( 0 < ( 1 / 𝐴 ) ∧ ( 1 / 𝐴 ) < 1 ) )

Proof

Step Hyp Ref Expression
1 0lt1 ⊢ 0 < 1
2 0re ⊢ 0 ∈ ℝ
3 1re ⊢ 1 ∈ ℝ
4 lttr ⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 0 < 1 ∧ 1 < 𝐴 ) → 0 < 𝐴 ) )
5 2 3 4 mp3an12 ⊢ ( 𝐴 ∈ ℝ → ( ( 0 < 1 ∧ 1 < 𝐴 ) → 0 < 𝐴 ) )
6 1 5 mpani ⊢ ( 𝐴 ∈ ℝ → ( 1 < 𝐴 → 0 < 𝐴 ) )
7 6 imdistani ⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
8 recgt0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( 1 / 𝐴 ) )
9 7 8 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → 0 < ( 1 / 𝐴 ) )
10 recgt1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < 𝐴 ↔ ( 1 / 𝐴 ) < 1 ) )
11 10 biimpa ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ 1 < 𝐴 ) → ( 1 / 𝐴 ) < 1 )
12 7 11 sylancom ⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( 1 / 𝐴 ) < 1 )
13 9 12 jca ⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( 0 < ( 1 / 𝐴 ) ∧ ( 1 / 𝐴 ) < 1 ) )