Theorem cxpp1d

Description: Value of a nonzero complex number raised to a complex power plus one. (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	cxpp1d	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 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		cxpp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )