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Metamath Proof Explorer

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 → ∑ 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 → ∑ k ∈ A B ∈ ℤ