Theorem cjexpd

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

	Ref	Expression
Hypotheses	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cjexpd.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	cjexpd	⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( ∗ ‘ 𝐴 ) ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cjexpd.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3		cjexp	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( ∗ ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( ∗ ‘ 𝐴 ) ↑ 𝑁 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( ∗ ‘ 𝐴 ) ↑ 𝑁 ) )