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| Mirrors > Home > MPE Home > Th. List > lediv1dd | Structured version Visualization version GIF version | ||
| Description: Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.) | 
| Ref | Expression | 
|---|---|
| ltmul1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| ltmul1d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| ltmul1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) | 
| lediv1dd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) | 
| Ref | Expression | 
|---|---|
| lediv1dd | ⊢ (𝜑 → (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lediv1dd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 2 | ltmul1d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 3 | ltmul1d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | ltmul1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
| 5 | 2, 3, 4 | lediv1d 13124 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶))) | 
| 6 | 1, 5 | mpbid 232 | 1 ⊢ (𝜑 → (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)) | 
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