Theorem expnegd

Description: Value of a nonzero complex number raised to the negative of an integer power. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	sqrecd.1	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	expclzd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion	expnegd	⊢ ( 𝜑 → ( 𝐴 ↑ - 𝑁 ) = ( 1 / ( 𝐴 ↑ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		sqrecd.1	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		expclzd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		expnegz	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ - 𝑁 ) = ( 1 / ( 𝐴 ↑ 𝑁 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑ - 𝑁 ) = ( 1 / ( 𝐴 ↑ 𝑁 ) ) )