Theorem redivcld

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

	Ref	Expression
Hypotheses	redivcld.1	$$⊢\phi \to A\in ℝ$$
	redivcld.2	$$⊢\phi \to B\in ℝ$$
	redivcld.3	$$⊢\phi \to B\ne 0$$
Assertion	redivcld	$$⊢\phi \to \frac{A}{B}\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		redivcld.1	$$⊢\phi \to A\in ℝ$$
2		redivcld.2	$$⊢\phi \to B\in ℝ$$
3		redivcld.3	$$⊢\phi \to B\ne 0$$
4		redivcl	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge B\ne 0\right)\to \frac{A}{B}\in ℝ$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \frac{A}{B}\in ℝ$$