Theorem recgt0i

Description: The reciprocal of a positive number is positive. Exercise 4 of Apostol p. 21. (Contributed by NM, 15-May-1999)

	Ref	Expression
Hypothesis	ltplus1.1	⊢ 𝐴 ∈ ℝ
Assertion	recgt0i	⊢ ( 0 < 𝐴 → 0 < ( 1 / 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		recgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( 1 / 𝐴 ) )
3	1 2	mpan	⊢ ( 0 < 𝐴 → 0 < ( 1 / 𝐴 ) )