Theorem cxpexpz

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	cxpexpz	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in \mathbb{Z}\right)\to A^B=A^B$$

Proof

Step	Hyp	Ref	Expression
1		zcn	$$⊢B\in \mathbb{Z}\to B\in ℂ$$
2		cxpef	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to A^B={\mathrm{e}}^{B\log A}$$
3	1 2	syl3an3	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in \mathbb{Z}\right)\to A^B={\mathrm{e}}^{B\log A}$$
4		explog	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in \mathbb{Z}\right)\to A^B={\mathrm{e}}^{B\log A}$$
5	3 4	eqtr4d	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in \mathbb{Z}\right)\to A^B=A^B$$