Theorem fsumzcl2

Description: A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018)

	Ref	Expression
Assertion	fsumzcl2	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		csbeq1a	⊢ ( 𝑘 = 𝑥 → 𝐵 = ⦋ 𝑥 / 𝑘 ⦌ 𝐵 )
2		nfcv	⊢ Ⅎ 𝑥 𝐵
3		nfcsb1v	⊢ Ⅎ 𝑘 ⦋ 𝑥 / 𝑘 ⦌ 𝐵
4	1 2 3	cbvsum	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑥 ∈ 𝐴 ⦋ 𝑥 / 𝑘 ⦌ 𝐵
5		simpl	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → 𝐴 ∈ Fin )
6		rspcsbela	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
7	6	expcom	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ → ( 𝑥 ∈ 𝐴 → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ ) )
8	7	adantl	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → ( 𝑥 ∈ 𝐴 → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ ) )
9	8	imp	⊢ ( ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) ∧ 𝑥 ∈ 𝐴 ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
10	5 9	fsumzcl	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑥 ∈ 𝐴 ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
11	4 10	eqeltrid	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℤ )