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

Metamath Proof Explorer

Theorem redivcld

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

Ref Expression
Hypotheses redivcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
redivcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
redivcld.3 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion redivcld ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 redivcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 redivcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 redivcld.3 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
4 redivcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) ∈ ℝ )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℝ )