Theorem reexpclzd

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

	Ref	Expression
Hypotheses	rpexpclzd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rpexpclzd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	rpexpclzd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion	reexpclzd	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		rpexpclzd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rpexpclzd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		rpexpclzd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		reexpclz	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )