Theorem expcld

Description: Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		expcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		expcl	$$⊢\left(A\in ℂ\wedge N\in {\mathbb{N}}_0\right)\to A^N\in ℂ$$
4	1 2 3	syl2anc	$$⊢\phi \to A^N\in ℂ$$