Theorem sqrecd

Description: Square of reciprocal. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	$$⊢\phi \to A\in ℂ$$
	sqrecd.1	$$⊢\phi \to A\ne 0$$
Assertion	sqrecd	$$⊢\phi \to {\left(\frac{1}{A}\right)}^2=\frac{1}{A^2}$$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		sqrecd.1	$$⊢\phi \to A\ne 0$$
3		2z	$$⊢2\in \mathbb{Z}$$
4	3	a1i	$$⊢\phi \to 2\in \mathbb{Z}$$
5		exprec	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge 2\in \mathbb{Z}\right)\to {\left(\frac{1}{A}\right)}^2=\frac{1}{A^2}$$
6	1 2 4 5	syl3anc	$$⊢\phi \to {\left(\frac{1}{A}\right)}^2=\frac{1}{A^2}$$