Theorem recgt0d

Description: The reciprocal of a positive number is positive. Exercise 4 of Apostol p. 21. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	recgt0d.2	⊢ ( 𝜑 → 0 < 𝐴 )
Assertion	recgt0d	⊢ ( 𝜑 → 0 < ( 1 / 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		recgt0d.2	⊢ ( 𝜑 → 0 < 𝐴 )
3		recgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( 1 / 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 0 < ( 1 / 𝐴 ) )