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	$$⊢A\in ℂ\to A^{-1}=\frac{1}{A}$$

Proof

Step	Hyp	Ref	Expression
1		1nn0	$$⊢1\in {\mathbb{N}}_0$$
2		expneg	$$⊢\left(A\in ℂ\wedge 1\in {\mathbb{N}}_0\right)\to A^{-1}=\frac{1}{A^1}$$
3	1 2	mpan2	$$⊢A\in ℂ\to A^{-1}=\frac{1}{A^1}$$
4		exp1	$$⊢A\in ℂ\to A^1=A$$
5	4	oveq2d	$$⊢A\in ℂ\to \frac{1}{A^1}=\frac{1}{A}$$
6	3 5	eqtrd	$$⊢A\in ℂ\to A^{-1}=\frac{1}{A}$$