Theorems · Theorem · group theory
AddSubgroup.isComplement_iff_existsUnique_add_neg_mem
∀ {G : Type u_1} [inst : AddGroup G] {S T : Set G}, AddSubgroup.IsComplement S T ↔ ∀ (g : G), ∃! t, g + -↑t ∈ S- Defined in
- Mathlib.GroupTheory.Complement
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses propext, Quot.sound
- Assumes
- AddGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Set.Elemstatement and proof · cited by 7,166
- AddGroupstatement and proof · cited by 4,410
- ExistsUniquestatement and proof · cited by 268
- neg_add_cancel_rightproof · cited by 75
- add_neg_cancel_rightproof · cited by 65
- AddSubgroup.IsComplementstatement and proof · cited by 57
- Subtype.coe_propproof · cited by 42
- AddSubgroup.isComplement_iff_existsUniqueproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- AddSubgroup.closure_add_image_add_eq_topproof · cited by 1
- AddSubgroup.closure_add_image_eqproof · cited by 1
- AddSubgroup.isComplement_addSubgroup_left_iff_existsUnique_quotientMk''proof · cited by 1