Theorem fsummsnunz

Description: A finite sum all of whose summands are integers is itself an integer (case where the summation set is the union of a finite set and a singleton). (Contributed by Alexander van der Vekens, 1-Sep-2018) (Revised by AV, 17-Dec-2021)

	Ref	Expression
Assertion	fsummsnunz	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 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 ∪ { 𝑍 } ) 𝐵 = Σ 𝑥 ∈ ( 𝐴 ∪ { 𝑍 } ) ⦋ 𝑥 / 𝑘 ⦌ 𝐵
5		snfi	⊢ { 𝑍 } ∈ Fin
6	5	a1i	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) → { 𝑍 } ∈ Fin )
7		unfi	⊢ ( ( 𝐴 ∈ Fin ∧ { 𝑍 } ∈ Fin ) → ( 𝐴 ∪ { 𝑍 } ) ∈ Fin )
8	6 7	syldan	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) → ( 𝐴 ∪ { 𝑍 } ) ∈ Fin )
9		rspcsbela	⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ { 𝑍 } ) ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
10	9	expcom	⊢ ( ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ → ( 𝑥 ∈ ( 𝐴 ∪ { 𝑍 } ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ ) )
11	10	adantl	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) → ( 𝑥 ∈ ( 𝐴 ∪ { 𝑍 } ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ ) )
12	11	imp	⊢ ( ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) ∧ 𝑥 ∈ ( 𝐴 ∪ { 𝑍 } ) ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
13	8 12	fsumzcl	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) → Σ 𝑥 ∈ ( 𝐴 ∪ { 𝑍 } ) ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ ℤ )
14	4 13	eqeltrid	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ ) → Σ 𝑘 ∈ ( 𝐴 ∪ { 𝑍 } ) 𝐵 ∈ ℤ )