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| 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 | 
|---|---|
| subdird | ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 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 | subdir 11698 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶))) | 
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