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Mirrors > Home > ILE Home > Th. List > rpexpcld | GIF version |
Description: Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpexpcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
rpexpcld.2 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
rpexpcld | ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpexpcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rpexpcld.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | rpexpcl 10482 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) |
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