Theorem rpexpcld

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

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

Proof

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