Theorems · Theorem · group theory
AddSubgroup.normalClosure_le_normal
∀ {G : Type u_1} [inst : AddGroup G] {s : Set G} {N : AddSubgroup G} [N.Normal],
s ⊆ ↑N → AddSubgroup.normalClosure s ≤ NThe normal closure of s is the smallest normal additive subgroup containings.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroupAddSubgroup.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
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement and proof · cited by 3,232
- AddMemClass.add_memproof · cited by 229
- AddSubgroup.Normalstatement and proof · cited by 183
- ZeroMemClass.zero_memproof · cited by 162
- AddSubgroup.closureproof · cited by 156
- NegMemClass.neg_memproof · cited by 63
- AddSubgroup.normalClosurestatement and proof · cited by 21
- AddSubgroup.closure_inductionproof · cited by 14
- AddGroup.addConjugatesOfSetproof · cited by 9
Cited by9
Results whose statement or proof uses this declaration.
- AddSubgroup.map_normalClosureproof · cited by 3
- AddSubgroup.normalClosure_eq_selfproof · cited by 2
- AddSubgroup.comap_normalClosure_image_geproof · cited by 1
- AddSubgroup.addCommutator_def'proof · cited by 1
- AddSubgroup.IsFinitelyNormallyGenerated.of_FGproof · cited by 1
- AddSubgroup.normalClosure_closure_eq_normalClosureproof · cited by 1
- AddSubgroup.normalClosure_monoproof · cited by 1
- AddSubgroup.normalClosure_subset_iffproof · cited by 1
- AddSubgroup.normalClosure_eq_iInfproof · cited by 0