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 )