Theorem sqdivd

Description: Distribution of square over division. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	$$⊢\phi \to A\in ℂ$$
	mulexpd.2	$$⊢\phi \to B\in ℂ$$
	sqdivd.3	$$⊢\phi \to B\ne 0$$
Assertion	sqdivd	$$⊢\phi \to {\left(\frac{A}{B}\right)}^2=\frac{A^2}{B^2}$$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		mulexpd.2	$$⊢\phi \to B\in ℂ$$
3		sqdivd.3	$$⊢\phi \to B\ne 0$$
4		sqdiv	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to {\left(\frac{A}{B}\right)}^2=\frac{A^2}{B^2}$$
5	1 2 3 4	syl3anc	$$⊢\phi \to {\left(\frac{A}{B}\right)}^2=\frac{A^2}{B^2}$$