Theorem cxpexpzd

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	$$⊢\phi \to A\in ℂ$$
	cxpefd.2	$$⊢\phi \to A\ne 0$$
	cxpexpzd.3	$$⊢\phi \to B\in \mathbb{Z}$$
Assertion	cxpexpzd	$$⊢\phi \to A^B=A^B$$

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	$$⊢\phi \to A\in ℂ$$
2		cxpefd.2	$$⊢\phi \to A\ne 0$$
3		cxpexpzd.3	$$⊢\phi \to B\in \mathbb{Z}$$
4		cxpexpz	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in \mathbb{Z}\right)\to A^B=A^B$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A^B=A^B$$