Theorem divcxpd

Description: Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	recxpcld.1	$$⊢\phi \to A\in ℝ$$
	recxpcld.2	$$⊢\phi \to 0\le A$$
	divcxpd.4	$$⊢\phi \to B\in {ℝ}^{+}$$
	divcxpd.5	$$⊢\phi \to C\in ℂ$$
Assertion	divcxpd	$$⊢\phi \to {\left(\frac{A}{B}\right)}^C=\frac{A^C}{B^C}$$

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	$$⊢\phi \to A\in ℝ$$
2		recxpcld.2	$$⊢\phi \to 0\le A$$
3		divcxpd.4	$$⊢\phi \to B\in {ℝ}^{+}$$
4		divcxpd.5	$$⊢\phi \to C\in ℂ$$
5		divcxp	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge B\in {ℝ}^{+}\wedge C\in ℂ\right)\to {\left(\frac{A}{B}\right)}^C=\frac{A^C}{B^C}$$
6	1 2 3 4 5	syl211anc	$$⊢\phi \to {\left(\frac{A}{B}\right)}^C=\frac{A^C}{B^C}$$