Theorem fsumzcl

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

	Ref	Expression
Hypotheses	fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumzcl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℤ )
Assertion	fsumzcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumzcl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℤ )
3		zsscn	⊢ ℤ ⊆ ℂ
4	3	a1i	⊢ ( 𝜑 → ℤ ⊆ ℂ )
5		zaddcl	⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) → ( 𝑥 + 𝑦 ) ∈ ℤ )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( 𝑥 + 𝑦 ) ∈ ℤ )
7		0zd	⊢ ( 𝜑 → 0 ∈ ℤ )
8	4 6 1 2 7	fsumcllem	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ )