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Metamath Proof Explorer

Theorem expm1

Description: Value of a nonzero 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 ) )