Theorem cxpmuld

Description: Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of Gleason p. 135. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	rpcxpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	rpcxpcld.2	$$⊢\phi \to B\in ℝ$$
	cxpmuld.4	$$⊢\phi \to C\in ℂ$$
Assertion	cxpmuld	$$⊢\phi \to A^{BC}={\left(A^B\right)}^C$$

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		rpcxpcld.2	$$⊢\phi \to B\in ℝ$$
3		cxpmuld.4	$$⊢\phi \to C\in ℂ$$
4		cxpmul	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\wedge C\in ℂ\right)\to A^{BC}={\left(A^B\right)}^C$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A^{BC}={\left(A^B\right)}^C$$