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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 → Σ 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 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ )