Theorems · Theorem · group theory
finsum_mem_empty
∀ {α : Type u_1} {M : Type u_5} [inst : AddCommMonoid M] {f : α → M}, ∑ᶠ (i : α) (_ : i ∈ ∅), f i = 0The sum of any function over an empty set is 0.
- Defined in
- Mathlib.Algebra.BigOperators.Finprod
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- AddCommMonoidstatement and proof · cited by 12,281
- finsumstatement and proof · cited by 286
- Set.mem_empty_iff_falseproof · cited by 32
- finsum_congr_Propproof · cited by 18
- finsum_zeroproof · cited by 18
- finsum_falseproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- finsum_mem_union'proof · cited by 3
- nonempty_of_finsum_mem_ne_zeroproof · cited by 0