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 ) )