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Mirrors > Home > MPE Home > Th. List > nn0mulcld | Structured version Visualization version GIF version |
Description: Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nn0red.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
nn0addcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0mulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0red.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | nn0addcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ0) | |
3 | nn0mulcl 12005 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) ∈ ℕ0) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ0) |
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