Theorem rpdivcld

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

	Ref	Expression
Hypotheses	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	rpdivcld	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		rpdivcl	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ+ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℝ+ )