Theorem expp1z

Description: Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014)

	Ref	Expression
Assertion	expp1z	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		1z	⊢ 1 ∈ ℤ
2		expaddz	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐴 ↑ 1 ) ) )
3	1 2	mpanr2	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐴 ↑ 1 ) ) )
4	3	3impa	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐴 ↑ 1 ) ) )
5		exp1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 1 ) = 𝐴 )
6	5	3ad2ant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 1 ) = 𝐴 )
7	6	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( ( 𝐴 ↑ 𝑁 ) · ( 𝐴 ↑ 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · 𝐴 ) )
8	4 7	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 + 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) · 𝐴 ) )