Theorem cxpefd

Description: Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cxpefd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		cxpefd.3	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
4		cxpef	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )