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Metamath Proof Explorer

Theorem reexpclz

Description: Closure of integer 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 )