Theorem expge0d

Description: Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	reexpcld.1	$$⊢\phi \to A\in ℝ$$
	reexpcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
	expge0d.3	$$⊢\phi \to 0\le A$$
Assertion	expge0d	$$⊢\phi \to 0\le A^N$$

Proof

Step	Hyp	Ref	Expression
1		reexpcld.1	$$⊢\phi \to A\in ℝ$$
2		reexpcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		expge0d.3	$$⊢\phi \to 0\le A$$
4		expge0	$$⊢\left(A\in ℝ\wedge N\in {\mathbb{N}}_0\wedge 0\le A\right)\to 0\le A^N$$
5	1 2 3 4	syl3anc	$$⊢\phi \to 0\le A^N$$