Theorems · Theorem · commutative algebra
exists_sum_eq_one_iff_pairwise_coprime
∀ {R : Type u} {I : Type v} [inst : CommSemiring R] {s : I → R} {t : Finset I} [inst_1 : DecidableEq I],
t.Nonempty → ((∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (Function.onFun IsCoprime fun i => s ↑i))- Defined in
- Mathlib.RingTheory.Coprime.Lemmas
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites42
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
- CommSemiringstatement and proof · cited by 10,911
- Finset.sumstatement and proof · cited by 5,195
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- Finset.prodstatement and proof · cited by 2,356
- Finset.sum_congrproof · cited by 2,323
- CommMonoidproof · cited by 2,264
- mul_commproof · cited by 2,262
- mul_assocproof · cited by 1,667
- MulZeroClass.zero_mulproof · cited by 1,625
- Finset.Nonemptystatement and proof · cited by 1,001
Cited by1
Results whose statement or proof uses this declaration.
- exists_sum_eq_one_iff_pairwise_coprime'proof · cited by 0