Theorem reexpclz

Description: Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014) (Revised by Mario Carneiro, 9-Sep-2014)

	Ref	Expression
Assertion	reexpclz	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		ax-resscn	⊢ ℝ ⊆ ℂ
2		remulcl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
3		1re	⊢ 1 ∈ ℝ
4		rereccl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑥 ≠ 0 ) → ( 1 / 𝑥 ) ∈ ℝ )
5	1 2 3 4	expcl2lem	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )