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+ )