Theorem expeq0d

Description: Positive integer exponentiation is 0 iff its base is 0. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	|- ( ph -> A e. CC )
	expeq0d.2	|- ( ph -> N e. NN )
	expeq0d.3	|- ( ph -> ( A ^ N ) = 0 )
Assertion	expeq0d	|- ( ph -> A = 0 )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	|- ( ph -> A e. CC )
2		expeq0d.2	|- ( ph -> N e. NN )
3		expeq0d.3	|- ( ph -> ( A ^ N ) = 0 )
4		expeq0	|- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) = 0 <-> A = 0 ) )
5	1 2 4	syl2anc	|- ( ph -> ( ( A ^ N ) = 0 <-> A = 0 ) )
6	3 5	mpbid	|- ( ph -> A = 0 )