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Metamath Proof Explorer

Theorem cxpne0

Description: Complex exponentiation is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014)

Ref Expression
Assertion cxpne0 
|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) =/= 0 )

Proof

Step Hyp Ref Expression
1 cxpef 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
2 id 
 |-  ( B e. CC -> B e. CC )
3 logcl 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. CC )
4 mulcl 
 |-  ( ( B e. CC /\ ( log ` A ) e. CC ) -> ( B x. ( log ` A ) ) e. CC )
5 2 3 4 syl2anr 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ B e. CC ) -> ( B x. ( log ` A ) ) e. CC )
6 5 3impa 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( B x. ( log ` A ) ) e. CC )
7 efne0 
 |-  ( ( B x. ( log ` A ) ) e. CC -> ( exp ` ( B x. ( log ` A ) ) ) =/= 0 )
8 6 7 syl 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( exp ` ( B x. ( log ` A ) ) ) =/= 0 )
9 1 8 eqnetrd 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) =/= 0 )