Theorem cxpp1

Description: Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	cxpp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		cxpadd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) )
3	1 2	mp3an3	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) )
4	3	3impa	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) )
5		cxp1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑𝑐 1 ) = 𝐴 )
6	5	3ad2ant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 1 ) = 𝐴 )
7	6	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )
8	4 7	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 1 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · 𝐴 ) )