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Mirrors > Home > MPE Home > Th. List > lemul2ad | Structured version Visualization version GIF version |
Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltp1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
divgt0d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lemul1ad.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lemul1ad.4 | ⊢ (𝜑 → 0 ≤ 𝐶) |
lemul1ad.5 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
lemul2ad | ⊢ (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltp1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | divgt0d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | lemul1ad.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | lemul1ad.4 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐶) | |
5 | 3, 4 | jca 512 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
6 | lemul1ad.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
7 | lemul2a 11828 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)) | |
8 | 1, 2, 5, 6, 7 | syl31anc 1372 | 1 ⊢ (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)) |
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