Theorem cxpaddd

Description: Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of Gleason p. 135. (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	$$⊢\phi \to B\in ℂ$$
	cxpaddd.4	$$⊢\phi \to C\in ℂ$$
Assertion	cxpaddd	$$⊢\phi \to A^{B+C}=A^B{A}^C$$

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		cxpaddd.4	$$⊢\phi \to C\in ℂ$$
5		cxpadd	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge B\in ℂ\wedge C\in ℂ\right)\to A^{B+C}=A^B{A}^C$$
6	1 2 3 4 5	syl211anc	$$⊢\phi \to A^{B+C}=A^B{A}^C$$