Metamath Proof Explorer


Theorem 0cxpd

Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses cxp0d.1
|- ( ph -> A e. CC )
cxpefd.2
|- ( ph -> A =/= 0 )
Assertion 0cxpd
|- ( ph -> ( 0 ^c A ) = 0 )

Proof

Step Hyp Ref Expression
1 cxp0d.1
 |-  ( ph -> A e. CC )
2 cxpefd.2
 |-  ( ph -> A =/= 0 )
3 0cxp
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( 0 ^c A ) = 0 )
4 1 2 3 syl2anc
 |-  ( ph -> ( 0 ^c A ) = 0 )