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Metamath Proof Explorer

Theorem fsumrecl

Description: Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005) (Revised by Mario Carneiro, 22-Apr-2014)

Ref Expression
Hypotheses fsumcl.1 ⊢ φ → A ∈ Fin
fsumrecl.2 ⊢ φ ∧ k ∈ A → B ∈ ℝ
Assertion fsumrecl ⊢ φ → ∑ k ∈ A B ∈ ℝ

Proof

Step Hyp Ref Expression
1 fsumcl.1 ⊢ φ → A ∈ Fin
2 fsumrecl.2 ⊢ φ ∧ k ∈ A → B ∈ ℝ
3 ax-resscn ⊢ ℝ ⊆ ℂ
4 3 a1i ⊢ φ → ℝ ⊆ ℂ
5 readdcl ⊢ x ∈ ℝ ∧ y ∈ ℝ → x + y ∈ ℝ
6 5 adantl ⊢ φ ∧ x ∈ ℝ ∧ y ∈ ℝ → x + y ∈ ℝ
7 0red ⊢ φ → 0 ∈ ℝ
8 4 6 1 2 7 fsumcllem ⊢ φ → ∑ k ∈ A B ∈ ℝ