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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 ) )