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| Mirrors > Home > MPE Home > Th. List > subdid | Structured version Visualization version GIF version | ||
| Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| mulm1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| mulnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| subdid.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| Ref | Expression |
|---|---|
| subdid | ⊢ (𝜑 → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mulm1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | mulnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | subdid.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | subdi 11696 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) |
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