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Metamath Proof Explorer

Theorem redivclzi

Description: Closure law for division of reals. (Contributed by NM, 9-May-1999)

Ref Expression
Hypotheses redivcl.1 ⊢ A ∈ ℝ
redivcl.2 ⊢ B ∈ ℝ
Assertion redivclzi ⊢ B ≠ 0 → A B ∈ ℝ

Proof

Step Hyp Ref Expression
1 redivcl.1 ⊢ A ∈ ℝ
2 redivcl.2 ⊢ B ∈ ℝ
3 redivcl ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ B ≠ 0 → A B ∈ ℝ
4 1 2 3 mp3an12 ⊢ B ≠ 0 → A B ∈ ℝ