Theorem expm1

Description: Value of a complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008) (Revised by Mario Carneiro, 4-Jun-2014)

	Ref	Expression
Assertion	expm1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 − 1 ) ) = ( ( 𝐴 ↑ 𝑁 ) / 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		1z	⊢ 1 ∈ ℤ
2		expsub	⊢ ( ( ( 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 ) )