Theorem cxpexpzd

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cxpefd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	cxpexpzd.3	⊢ ( 𝜑 → 𝐵 ∈ ℤ )
Assertion	cxpexpzd	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cxpefd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		cxpexpzd.3	⊢ ( 𝜑 → 𝐵 ∈ ℤ )
4		cxpexpz	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) )