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Metamath Proof Explorer

Theorem nnrecgt0

Description: The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999)

Ref Expression
Assertion nnrecgt0 ⊢ A ∈ ℕ → 0 < 1 A

Proof

Step Hyp Ref Expression
1 nnge1 ⊢ A ∈ ℕ → 1 ≤ A
2 0lt1 ⊢ 0 < 1
3 nnre ⊢ A ∈ ℕ → A ∈ ℝ
4 0re ⊢ 0 ∈ ℝ
5 1re ⊢ 1 ∈ ℝ
6 ltletr ⊢ 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ A ∈ ℝ → 0 < 1 ∧ 1 ≤ A → 0 < A
7 4 5 6 mp3an12 ⊢ A ∈ ℝ → 0 < 1 ∧ 1 ≤ A → 0 < A
8 recgt0 ⊢ A ∈ ℝ ∧ 0 < A → 0 < 1 A
9 8 ex ⊢ A ∈ ℝ → 0 < A → 0 < 1 A
10 7 9 syld ⊢ A ∈ ℝ → 0 < 1 ∧ 1 ≤ A → 0 < 1 A
11 3 10 syl ⊢ A ∈ ℕ → 0 < 1 ∧ 1 ≤ A → 0 < 1 A
12 2 11 mpani ⊢ A ∈ ℕ → 1 ≤ A → 0 < 1 A
13 1 12 mpd ⊢ A ∈ ℕ → 0 < 1 A