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 )