Theorem reexpcld

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

	Ref	Expression
Hypotheses	reexpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	reexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	reexpcld	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		reexpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		reexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3		reexpcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )