Theorem expdivd

Description: Nonnegative integer exponentiation of a quotient. (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	$$⊢\phi \to B\ne 0$$
	expdivd.3	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	expdivd	$$⊢\phi \to {\left(\frac{A}{B}\right)}^N=\frac{A^N}{B^N}$$

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		expdivd.3	$$⊢\phi \to N\in {\mathbb{N}}_0$$
5		expdiv	$$⊢\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\wedge N\in {\mathbb{N}}_0\right)\to {\left(\frac{A}{B}\right)}^N=\frac{A^N}{B^N}$$
6	1 2 3 4 5	syl121anc	$$⊢\phi \to {\left(\frac{A}{B}\right)}^N=\frac{A^N}{B^N}$$