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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