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Metamath Proof Explorer

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 )