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Metamath Proof Explorer

Theorem reclt1

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

Ref Expression
Assertion reclt1 ⊢ A ∈ ℝ ∧ 0 < A → A < 1 ↔ 1 < 1 A

Proof

Step Hyp Ref Expression
1 1re ⊢ 1 ∈ ℝ
2 0lt1 ⊢ 0 < 1
3 ltrec ⊢ A ∈ ℝ ∧ 0 < A ∧ 1 ∈ ℝ ∧ 0 < 1 → A < 1 ↔ 1 1 < 1 A
4 1 2 3 mpanr12 ⊢ A ∈ ℝ ∧ 0 < A → A < 1 ↔ 1 1 < 1 A
5 1div1e1 ⊢ 1 1 = 1
6 5 breq1i ⊢ 1 1 < 1 A ↔ 1 < 1 A
7 4 6 bitrdi ⊢ A ∈ ℝ ∧ 0 < A → A < 1 ↔ 1 < 1 A