Theorem reexpcld

Description: Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		reexpcld.1	$$⊢\phi \to A\in ℝ$$
2		reexpcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		reexpcl	$$⊢\left(A\in ℝ\wedge N\in {\mathbb{N}}_0\right)\to A^N\in ℝ$$
4	1 2 3	syl2anc	$$⊢\phi \to A^N\in ℝ$$