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| Mirrors > Home > MPE Home > Th. List > letrd | Structured version Visualization version GIF version | ||
| Description: Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.) |
| Ref | Expression |
|---|---|
| ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| letrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
| letrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| letrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
| Ref | Expression |
|---|---|
| letrd | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | letrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 2 | letrd.5 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
| 3 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 4 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 5 | letrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
| 6 | letr 11355 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
| 7 | 3, 4, 5, 6 | syl3anc 1373 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
| 8 | 1, 2, 7 | mp2and 699 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
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