Theorem remulcld

Description: Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	recnd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	readdcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	remulcld	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		recnd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		readdcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		remulcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℝ )