Theorems · Theorem · commutative algebra
Ideal.sup_iInf_eq_top
∀ {R : Type u} [inst : Semiring R] {I : Ideal R} {ι : Type u_1} {s : Finset ι} {J : ι → Ideal R}
[∀ (i : ι), (J i).IsTwoSided], (∀ i ∈ s, I ⊔ J i = ⊤) → I ⊔ ⨅ i ∈ s, J i = ⊤- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringIdeal.IsTwoSided
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Finsetstatement and proof · cited by 13,712
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- iInfstatement and proof · cited by 1,690
- le_transproof · cited by 985
- Eq.geproof · cited by 375
- iInf_congr_Propproof · cited by 218
- Ideal.IsTwoSidedstatement and proof · cited by 179
- sup_of_le_rightproof · cited by 143
- inf_commproof · cited by 139
- top_uniqueproof · cited by 102
Cited by5
Results whose statement or proof uses this declaration.
- Submodule.supIndep_torsionBySet_idealproof · cited by 4
- Ideal.isCoprime_biInfproof · cited by 2
- Ideal.iSup_iInf_eq_top_iff_pairwiseproof · cited by 1
- FractionalIdeal.isPrincipal.of_finite_maximals_of_invproof · cited by 1
- Ideal.iInf_sup_eq_topproof · cited by 0