Theorems · Theorem · group theory
Subgroup.isComplement_iff_existsUnique_mul_inv_mem
∀ {G : Type u_1} [inst : Group G] {S T : Set G}, Subgroup.IsComplement S T ↔ ∀ (g : G), ∃! t, g * (↑t)⁻¹ ∈ S- Defined in
- Mathlib.GroupTheory.Complement
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses propext, Quot.sound
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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 and proof · cited by 268
- Subgroup.IsComplementstatement and proof · cited by 94
- inv_mul_cancel_rightproof · cited by 70
- mul_inv_cancel_rightproof · cited by 53
- Subgroup.isComplement_iff_existsUniqueproof · cited by 4
Cited by3
Results whose statement or proof uses this declaration.
- Subgroup.closure_mul_image_eqproof · cited by 1
- Subgroup.IsComplement.equiv_snd_eq_iff_rightCosetEquivalenceproof · cited by 1
- Subgroup.closure_mul_image_mul_eq_topproof · cited by 1