Theorem mulcxpd

Description: Complex exponentiation of a product. Proposition 10-4.2(c) of Gleason p. 135. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	recxpcld.1	$$⊢\phi \to A\in ℝ$$
	recxpcld.2	$$⊢\phi \to 0\le A$$
	recxpcld.3	$$⊢\phi \to B\in ℝ$$
	mulcxpd.4	$$⊢\phi \to 0\le B$$
	mulcxpd.5	$$⊢\phi \to C\in ℂ$$
Assertion	mulcxpd	$$⊢\phi \to {AB}^C=A^C{B}^C$$

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	$$⊢\phi \to A\in ℝ$$
2		recxpcld.2	$$⊢\phi \to 0\le A$$
3		recxpcld.3	$$⊢\phi \to B\in ℝ$$
4		mulcxpd.4	$$⊢\phi \to 0\le B$$
5		mulcxpd.5	$$⊢\phi \to C\in ℂ$$
6		mulcxp	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\wedge C\in ℂ\right)\to {AB}^C=A^C{B}^C$$
7	1 2 3 4 5 6	syl221anc	$$⊢\phi \to {AB}^C=A^C{B}^C$$