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| Description: Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.) | 
| Ref | Expression | 
|---|---|
| reexpcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| reexpcld.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) | 
| Ref | Expression | 
|---|---|
| reexpcld | ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reexpcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | reexpcld.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
| 3 | reexpcl 14120 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) | 
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