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 e. CC -> ( A ^ -u 1 ) = ( 1 / A ) )

Proof

Step	Hyp	Ref	Expression
1		1nn0	|- 1 e. NN0
2		expneg	|- ( ( A e. CC /\ 1 e. NN0 ) -> ( A ^ -u 1 ) = ( 1 / ( A ^ 1 ) ) )
3	1 2	mpan2	|- ( A e. CC -> ( A ^ -u 1 ) = ( 1 / ( A ^ 1 ) ) )
4		exp1	|- ( A e. CC -> ( A ^ 1 ) = A )
5	4	oveq2d	|- ( A e. CC -> ( 1 / ( A ^ 1 ) ) = ( 1 / A ) )
6	3 5	eqtrd	|- ( A e. CC -> ( A ^ -u 1 ) = ( 1 / A ) )