Theorem 0cxp

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

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

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		cxpval	⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 0 ↑𝑐 𝐴 ) = if ( 0 = 0 , if ( 𝐴 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐴 · ( log ‘ 0 ) ) ) ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℂ → ( 0 ↑𝑐 𝐴 ) = if ( 0 = 0 , if ( 𝐴 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐴 · ( log ‘ 0 ) ) ) ) )
4		eqid	⊢ 0 = 0
5	4	iftruei	⊢ if ( 0 = 0 , if ( 𝐴 = 0 , 1 , 0 ) , ( exp ‘ ( 𝐴 · ( log ‘ 0 ) ) ) ) = if ( 𝐴 = 0 , 1 , 0 )
6	3 5	eqtrdi	⊢ ( 𝐴 ∈ ℂ → ( 0 ↑𝑐 𝐴 ) = if ( 𝐴 = 0 , 1 , 0 ) )
7		ifnefalse	⊢ ( 𝐴 ≠ 0 → if ( 𝐴 = 0 , 1 , 0 ) = 0 )
8	6 7	sylan9eq	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 ↑𝑐 𝐴 ) = 0 )