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| Mirrors > Home > MPE Home > Th. List > subge0d | Structured version Visualization version GIF version | ||
| Description: Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| Ref | Expression |
|---|---|
| subge0d | ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 3 | subge0 11776 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) |
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