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| Mirrors > Home > MPE Home > Th. List > fsumcl | Structured version Visualization version GIF version | ||
| Description: Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.) | 
| Ref | Expression | 
|---|---|
| fsumcl.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) | 
| fsumcl.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| Ref | Expression | 
|---|---|
| fsumcl | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssidd 4007 | . 2 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
| 2 | addcl 11237 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ) | 
| 4 | fsumcl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
| 5 | fsumcl.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
| 6 | 0cnd 11254 | . 2 ⊢ (𝜑 → 0 ∈ ℂ) | |
| 7 | 1, 3, 4, 5, 6 | fsumcllem 15768 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) | 
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