Theorem mulexpd

Description: Nonnegative integer exponentiation of a product. Proposition 10-4.2(c) of Gleason p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	$$⊢\phi \to A\in ℂ$$
	mulexpd.2	$$⊢\phi \to B\in ℂ$$
	mulexpd.3	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	mulexpd	$$⊢\phi \to {AB}^N=A^N{B}^N$$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		mulexpd.2	$$⊢\phi \to B\in ℂ$$
3		mulexpd.3	$$⊢\phi \to N\in {\mathbb{N}}_0$$
4		mulexp	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge N\in {\mathbb{N}}_0\right)\to {AB}^N=A^N{B}^N$$
5	1 2 3 4	syl3anc	$$⊢\phi \to {AB}^N=A^N{B}^N$$