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| Mirrors > Home > MPE Home > Th. List > redivcld | Structured version Visualization version GIF version | ||
| Description: Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| redivcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| redivcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| redivcld.3 | ⊢ (𝜑 → 𝐵 ≠ 0) |
| Ref | Expression |
|---|---|
| redivcld | ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | redivcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | redivcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 3 | redivcld.3 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
| 4 | redivcl 11986 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℝ) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
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