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 )