Theorem expn1

Description: A complex number raised to the negative one power is its reciprocal. When A = 0 , both sides have the "value" ( 1 / 0 ) ; relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014)

	Ref	Expression
Assertion	expn1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ - 1 ) = ( 1 / 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		1nn0	⊢ 1 ∈ ℕ0
2		expneg	⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℕ0 ) → ( 𝐴 ↑ - 1 ) = ( 1 / ( 𝐴 ↑ 1 ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ - 1 ) = ( 1 / ( 𝐴 ↑ 1 ) ) )
4		exp1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 1 ) = 𝐴 )
5	4	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( 1 / ( 𝐴 ↑ 1 ) ) = ( 1 / 𝐴 ) )
6	3 5	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ - 1 ) = ( 1 / 𝐴 ) )