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| Mirrors > Home > MPE Home > Th. List > npcand | Structured version Visualization version GIF version | ||
| Description: Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| Ref | Expression |
|---|---|
| npcand | ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | npcan 11517 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
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