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| Mirrors > Home > MPE Home > Th. List > mulassd | Structured version Visualization version GIF version | ||
| Description: Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| addcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| addcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| addassd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| Ref | Expression |
|---|---|
| mulassd | ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | addcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | addcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | addassd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 4 | mulass 11243 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
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