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	$$⊢\phi \to A\in ℂ$$
	expeq0d.2	$$⊢\phi \to N\in \mathbb{N}$$
	expeq0d.3	$$⊢\phi \to A^N=0$$
Assertion	expeq0d	$$⊢\phi \to A=0$$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$$⊢\phi \to A\in ℂ$$
2		expeq0d.2	$$⊢\phi \to N\in \mathbb{N}$$
3		expeq0d.3	$$⊢\phi \to A^N=0$$
4		expeq0	$$⊢\left(A\in ℂ\wedge N\in \mathbb{N}\right)\to \left(A^N=0\leftrightarrow A=0\right)$$
5	1 2 4	syl2anc	$$⊢\phi \to \left(A^N=0\leftrightarrow A=0\right)$$
6	3 5	mpbid	$$⊢\phi \to A=0$$