Theorems · Theorem · group theory
Subgroup.normalClosure_le_normal
∀ {G : Type u_1} [inst : Group G] {s : Set G} {N : Subgroup G} [N.Normal], s ⊆ ↑N → Subgroup.normalClosure s ≤ NThe normal closure of s is the smallest normal subgroup containing s.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Basic
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- GroupSubgroup.Normal
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 and proof · cited by 53,352
- SetLike.coestatement and proof · cited by 8,199
- Groupstatement and proof · cited by 6,238
- Subgroupstatement and proof · cited by 3,593
- Subgroup.Normalstatement and proof · cited by 334
- Subgroup.closureproof · cited by 196
- MulMemClass.mul_memproof · cited by 173
- OneMemClass.one_memproof · cited by 87
- InvMemClass.inv_memproof · cited by 52
- Subgroup.normalClosurestatement and proof · cited by 35
- Subgroup.closure_inductionproof · cited by 14
- Group.conjugatesOfSetproof · cited by 9
Cited by10
Results whose statement or proof uses this declaration.
- Subgroup.map_normalClosureproof · cited by 3
- Subgroup.normalClosure_eq_selfproof · cited by 2
- Subgroup.normalClosure_subset_iffproof · cited by 2
- Subgroup.normalClosure_closure_eq_normalClosureproof · cited by 1
- PresentedGroup.closure_rels_subset_kerproof · cited by 1
- Subgroup.IsFinitelyNormallyGenerated.of_FGproof · cited by 1
- Subgroup.normalClosure_monoproof · cited by 1
- Subgroup.commutator_def'proof · cited by 1
- IsConj.normalClosure_eq_top_ofproof · cited by 0
- Subgroup.normalClosure_eq_iInfproof · cited by 0