This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )