Metamath Proof Explorer


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 ) ) = ( ( 𝐴𝑁 ) / 𝐴 ) )

Proof

Step Hyp Ref Expression
1 1z 1 ∈ ℤ
2 expsub ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝑁 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝐴 ↑ ( 𝑁 − 1 ) ) = ( ( 𝐴𝑁 ) / ( 𝐴 ↑ 1 ) ) )
3 1 2 mpanr2 ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 − 1 ) ) = ( ( 𝐴𝑁 ) / ( 𝐴 ↑ 1 ) ) )
4 3 3impa ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 − 1 ) ) = ( ( 𝐴𝑁 ) / ( 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 ) → ( ( 𝐴𝑁 ) / ( 𝐴 ↑ 1 ) ) = ( ( 𝐴𝑁 ) / 𝐴 ) )
8 4 7 eqtrd ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ ( 𝑁 − 1 ) ) = ( ( 𝐴𝑁 ) / 𝐴 ) )