Theorem rpexpcld

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

	Ref	Expression
Hypotheses	rpexpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion	rpexpcld	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		rpexpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
3		rpexpcl	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )