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| Mirrors > Home > MPE Home > Th. List > nn0addcld | Structured version Visualization version GIF version | ||
| Description: Closure of addition 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 |
|---|---|
| nn0addcld | ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nn0red.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
| 2 | nn0addcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ0) | |
| 3 | nn0addcl 12561 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℕ0) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) |
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