Theorems · Theorem · group theory
Subgroup.isComplement_iff_existsUnique
∀ {G : Type u_1} [inst : Group G] {S T : Set G}, Subgroup.IsComplement S T ↔ ∀ (g : G), ∃! x, ↑x.1 * ↑x.2 = g- Defined in
- Mathlib.GroupTheory.Complement
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- Groupstatement and proof · cited by 6,238
- ExistsUniquestatement · cited by 268
- Subgroup.IsComplementstatement · cited by 94
- Function.bijective_iff_existsUniqueproof · cited by 15
Cited by4
Results whose statement or proof uses this declaration.
- Subgroup.isComplement_iff_existsUnique_mul_inv_memproof · cited by 3
- Subgroup.IsComplement.existsUniqueproof · cited by 2
- Subgroup.isComplement_iff_existsUnique_inv_mul_memproof · cited by 2
- Subgroup.isComplement'_stabilizerproof · cited by 1