Theorem expp1d

Description: Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of Gleason p. 134. (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	expp1d	$$⊢\phi \to A^{N+1}=A^{N}A$$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		expcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		expp1	$$⊢\left(A\in ℂ\wedge N\in {\mathbb{N}}_0\right)\to A^{N+1}=A^{N}A$$
4	1 2 3	syl2anc	$$⊢\phi \to A^{N+1}=A^{N}A$$