Theorem rereccld

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

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

Proof

Step	Hyp	Ref	Expression
1		redivcld.1	$$⊢\phi \to A\in ℝ$$
2		rereccld.2	$$⊢\phi \to A\ne 0$$
3		rereccl	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to \frac{1}{A}\in ℝ$$
4	1 2 3	syl2anc	$$⊢\phi \to \frac{1}{A}\in ℝ$$