Theorem rpreccld

Description: Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
Assertion	rpreccld	⊢ ( 𝜑 → ( 1 / 𝐴 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpreccl	⊢ ( 𝐴 ∈ ℝ+ → ( 1 / 𝐴 ) ∈ ℝ+ )
3	1 2	syl	⊢ ( 𝜑 → ( 1 / 𝐴 ) ∈ ℝ+ )