Theorem cxpexpz

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	cxpexpz	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		zcn	⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℂ )
2		cxpef	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
3	1 2	syl3an3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
4		explog	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ↑ 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
5	3 4	eqtr4d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) )