| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > mul12d | Structured version Visualization version GIF version | ||
| Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| muld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| addcomd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| addcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| Ref | Expression |
|---|---|
| mul12d | ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | muld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | addcomd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | addcand.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | mul12 11426 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) |
| Copyright terms: Public domain | W3C validator |