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| Mirrors > Home > MPE Home > Th. List > zmulcld | Structured version Visualization version GIF version | ||
| Description: Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
| Ref | Expression |
|---|---|
| zred.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
| zaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
| Ref | Expression |
|---|---|
| zmulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
| 2 | zaddcld.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
| 3 | zmulcl 12666 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) ∈ ℤ) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
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