Theorem cxpne0d

Description: Complex exponentiation is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	|- ( ph -> A e. CC )
	cxpefd.2	|- ( ph -> A =/= 0 )
	cxpefd.3	|- ( ph -> B e. CC )
Assertion	cxpne0d	|- ( ph -> ( A ^c B ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	|- ( ph -> A e. CC )
2		cxpefd.2	|- ( ph -> A =/= 0 )
3		cxpefd.3	|- ( ph -> B e. CC )
4		cxpne0	|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) =/= 0 )
5	1 2 3 4	syl3anc	|- ( ph -> ( A ^c B ) =/= 0 )