Theorems · Theorem · group theory

Subgroup.exists_finiteIndex_of_leftCoset_cover_aux

∀ {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 →
    ∀ [inst_1 : DecidableEq (Subgroup G)],
      ∀ j ∈ s, ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex

Defined in: Mathlib.GroupTheory.CosetCover
Cited by: 2 results in Mathlib
Foundations: Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Group DecidableEq

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Cites48

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · 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
Set.univstatement and proof · cited by 3,945
Subgroupstatement and proof · cited by 3,593
Finset.univproof · cited by 3,473
Set.iUnionstatement and proof · cited by 2,483
Finset.cardproof · cited by 2,327
HasQuotient.Quotientproof · cited by 2,301
mul_assocproof · cited by 1,667
Finset.filterstatement and proof · cited by 949

Cited by2

Results whose statement or proof uses this declaration.

Subgroup.leftCoset_cover_filter_FiniteIndex_auxproof · cited by 3
Subgroup.exists_finiteIndex_of_leftCoset_coverproof · cited by 1