Theorems · Theorem · group theory
Finset.sum_induction_nonempty
∀ {ι : Type u_1} {s : Finset ι} {M : Type u_7} [inst : AddCommMonoid M] (f : ι → M) (p : M → Prop),
(∀ (a b : M), p a → p b → p (a + b)) → s.Nonempty → (∀ x ∈ s, p (f x)) → p (∑ x ∈ s, f x)To prove a property of a sum, it suffices to prove that the property is additive and holds on summands.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 60 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.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- AddCommMonoidstatement and proof · cited by 12,281
- Finset.sumstatement · cited by 5,195
- Finset.Nonemptystatement and proof · cited by 1,001
- Finset.nonempty_iff_ne_emptyproof · cited by 34
- Finset.val_eq_zeroproof · cited by 12
- Multiset.forall_mem_map_iffproof · cited by 8
- Multiset.map_eq_zeroproof · cited by 3
- Multiset.sum_induction_nonemptyproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Finset.le_expect_nonempty_of_subadditive_on_predproof · cited by 2
- AkraBazziRecurrence.strictAnti_sumCoeffsExpproof · cited by 1