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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