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

Metamath Proof Explorer

Theorem rpmulcld

Description: Closure law for multiplication of positive reals. Part of Axiom 7 of Apostol p. 20. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

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