Theorem cxpef

Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	cxpef	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cxpval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
2	1	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) )
3		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → 𝐴 ≠ 0 )
4	3	neneqd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ¬ 𝐴 = 0 )
5	4	iffalsed	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → if ( 𝐴 = 0 , if ( 𝐵 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
6	2 5	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )