Theorems · Theorem · group theory
Subgroup.one_le_sum_inv_index_of_leftCoset_cover
∀ {G : Type u_1} [inst : Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
⋃ i ∈ s, g i • ↑(H i) = Set.univ → 1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹Let the group G be the union of finitely many left cosets g i • H i. Then the
sum of the inverses of the indexes of the subgroups H i is greater than or equal to 1.
- Defined in
- Mathlib.GroupTheory.CosetCover
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Finsetstatement and proof · cited by 13,712
- SetLike.coestatement and proof · cited by 8,199
- Groupstatement and proof · cited by 6,238
- Finset.sumstatement · cited by 5,195
- Set.univstatement and proof · cited by 3,945
- Subgroupstatement and proof · cited by 3,593
- Set.iUnionstatement and proof · cited by 2,483
- Set.smulSetstatement · cited by 608
- Subgroup.indexstatement · cited by 150
- Subgroup.FiniteIndexproof · cited by 113
- Classical.decPredproof · cited by 26
Cited by1
Results whose statement or proof uses this declaration.
- Subgroup.exists_index_le_card_of_leftCoset_coverproof · cited by 0