Metamath Proof Explorer

Theorem cxp0

Description: Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014)

Ref Expression
Assertion cxp0 
|- ( A e. CC -> ( A ^c 0 ) = 1 )

Proof

Step Hyp Ref Expression
1 0nn0 
 |-  0 e. NN0
2 cxpexp 
 |-  ( ( A e. CC /\ 0 e. NN0 ) -> ( A ^c 0 ) = ( A ^ 0 ) )
3 1 2 mpan2 
 |-  ( A e. CC -> ( A ^c 0 ) = ( A ^ 0 ) )
4 exp0 
 |-  ( A e. CC -> ( A ^ 0 ) = 1 )
5 3 4 eqtrd 
 |-  ( A e. CC -> ( A ^c 0 ) = 1 )