Theorem cxpmul2d

Description: Product of exponents law for complex exponentiation. Variation on cxpmul with more general conditions on A and B when C is an integer. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	$$⊢\phi \to A\in ℂ$$
	cxpcld.2	$$⊢\phi \to B\in ℂ$$
	cxpmul2d.4	$$⊢\phi \to C\in {\mathbb{N}}_0$$
Assertion	cxpmul2d	$$⊢\phi \to A^{BC}={\left(A^B\right)}^C$$

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	$$⊢\phi \to A\in ℂ$$
2		cxpcld.2	$$⊢\phi \to B\in ℂ$$
3		cxpmul2d.4	$$⊢\phi \to C\in {\mathbb{N}}_0$$
4		cxpmul2	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in {\mathbb{N}}_0\right)\to A^{BC}={\left(A^B\right)}^C$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A^{BC}={\left(A^B\right)}^C$$