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

Metamath Proof Explorer

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 )