Metamath Proof Explorer


Theorem cxpexpzd

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses cxp0d.1
|- ( ph -> A e. CC )
cxpefd.2
|- ( ph -> A =/= 0 )
cxpexpzd.3
|- ( ph -> B e. ZZ )
Assertion cxpexpzd
|- ( ph -> ( A ^c B ) = ( A ^ B ) )

Proof

Step Hyp Ref Expression
1 cxp0d.1
 |-  ( ph -> A e. CC )
2 cxpefd.2
 |-  ( ph -> A =/= 0 )
3 cxpexpzd.3
 |-  ( ph -> B e. ZZ )
4 cxpexpz
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( A ^c B ) = ( A ^ B ) )