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| Mirrors > Home > MPE Home > Th. List > fsumrecl | Structured version Visualization version GIF version | ||
| Description: Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.) |
| Ref | Expression |
|---|---|
| fsumcl.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
| fsumrecl.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
| Ref | Expression |
|---|---|
| fsumrecl | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-resscn 11212 | . . 3 ⊢ ℝ ⊆ ℂ | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℝ ⊆ ℂ) |
| 3 | readdcl 11238 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) | |
| 4 | 3 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ) |
| 5 | fsumcl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
| 6 | fsumrecl.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
| 7 | 0red 11264 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 8 | 2, 4, 5, 6, 7 | fsumcllem 15768 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
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