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| Mirrors > Home > MPE Home > Th. List > divassd | Structured version Visualization version GIF version | ||
| Description: An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| divmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| divassd.4 | ⊢ (𝜑 → 𝐶 ≠ 0) |
| Ref | Expression |
|---|---|
| divassd | ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | divmuld.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | divassd.4 | . 2 ⊢ (𝜑 → 𝐶 ≠ 0) | |
| 5 | divass 11940 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) | |
| 6 | 1, 2, 3, 4, 5 | syl112anc 1376 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) |
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