Theorem 0cxpd

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

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

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cxpefd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		0cxp	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 ↑𝑐 𝐴 ) = 0 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 0 ↑𝑐 𝐴 ) = 0 )