| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > subcld | Structured version Visualization version GIF version | ||
| Description: Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| Ref | Expression |
|---|---|
| subcld | ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | subcl 11507 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
| Copyright terms: Public domain | W3C validator |