Theorem cxpmul2zd

Description: Generalize cxpmul2 to negative integers. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	$$⊢\phi \to A\in ℂ$$
	cxpefd.2	$$⊢\phi \to A\ne 0$$
	cxpefd.3	$$⊢\phi \to B\in ℂ$$
	cxpmul2zd.4	$$⊢\phi \to C\in \mathbb{Z}$$
Assertion	cxpmul2zd	$$⊢\phi \to A^{BC}={\left(A^B\right)}^C$$

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	$$⊢\phi \to A\in ℂ$$
2		cxpefd.2	$$⊢\phi \to A\ne 0$$
3		cxpefd.3	$$⊢\phi \to B\in ℂ$$
4		cxpmul2zd.4	$$⊢\phi \to C\in \mathbb{Z}$$
5		cxpmul2z	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge C\in \mathbb{Z}\right)\right)\to A^{BC}={\left(A^B\right)}^C$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to A^{BC}={\left(A^B\right)}^C$$