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