Theorems · Theorem · group theory
CoxeterSystem.subgroup_closure_range_simple
∀ {B : Type u_1} {W : Type u_3} [inst : Group W] {M : CoxeterMatrix B} (cs : CoxeterSystem M W),
Subgroup.closure (Set.range cs.simple) = ⊤The simple reflections of W generate W as a group.
- Defined in
- Mathlib.GroupTheory.Coxeter.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 82 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.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setproof · cited by 53,352
- Top.topstatement and proof · cited by 9,680
- Groupstatement and proof · cited by 6,238
- Set.imageproof · cited by 5,609
- Set.rangestatement and proof · cited by 4,705
- Subgroupstatement and proof · cited by 3,593
- MulEquiv.symmproof · cited by 482
- Subgroup.mapproof · cited by 301
- Set.range_compproof · cited by 223
- Subgroup.closurestatement and proof · cited by 196
- CoxeterMatrixstatement and proof · cited by 140
Cited by1
Results whose statement or proof uses this declaration.
- CoxeterSystem.submonoid_closure_range_simpleproof · cited by 4