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Metamath Proof Explorer

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+ )