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| Mirrors > Home > MPE Home > Th. List > mulge0d | Structured version Visualization version GIF version | ||
| Description: The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| addge0d.3 | ⊢ (𝜑 → 0 ≤ 𝐴) |
| addge0d.4 | ⊢ (𝜑 → 0 ≤ 𝐵) |
| Ref | Expression |
|---|---|
| mulge0d | ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | addge0d.3 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
| 3 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | addge0d.4 | . 2 ⊢ (𝜑 → 0 ≤ 𝐵) | |
| 5 | mulge0 11781 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 · 𝐵)) | |
| 6 | 1, 2, 3, 4, 5 | syl22anc 839 | 1 ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) |
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