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| Mirrors > Home > MPE Home > Th. List > reccld | Structured version Visualization version GIF version | ||
| Description: Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| reccld.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
| Ref | Expression |
|---|---|
| reccld | ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | reccld.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
| 3 | reccl 11929 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
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