Theorem cjexpd

Description: Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	$$⊢\phi \to A\in ℂ$$
	cjexpd.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	cjexpd	$$⊢\phi \to \overline{A^N}={\bar{A}}^N$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		cjexpd.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		cjexp	$$⊢\left(A\in ℂ\wedge N\in {\mathbb{N}}_0\right)\to \overline{A^N}={\bar{A}}^N$$
4	1 2 3	syl2anc	$$⊢\phi \to \overline{A^N}={\bar{A}}^N$$