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| Mirrors > Home > MPE Home > Th. List > rerpdivcld | Structured version Visualization version GIF version | ||
| Description: Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
| Ref | Expression |
|---|---|
| rpgecld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| rpgecld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
| Ref | Expression |
|---|---|
| rerpdivcld | ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rpgecld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | rpgecld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
| 3 | rerpdivcl 13065 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
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