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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 → 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 )