Theorems · Definition · group theory
Subgroup.IsComplement.equiv
{G : Type u_1} → [inst : Group G] → {S T : Set G} → Subgroup.IsComplement S T → G ≃ ↑S × ↑TThe equivalence G ≃ S × T, such that the inverse is (*) : S × T → G
- Defined in
- Mathlib.GroupTheory.Complement
- Cited by
- 34 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses Classical.choice
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Equivstatement · cited by 8,337
- Set.Elemstatement and proof · cited by 7,166
- Groupstatement and proof · cited by 6,238
- Equiv.symmproof · cited by 3,681
- Subgroup.IsComplementstatement and proof · cited by 94
- Equiv.ofBijectiveproof · cited by 70
Cited by37
Results whose statement or proof uses this declaration.
- Subgroup.IsComplement.equiv_fst_eq_mul_invstatement · cited by 8
- Subgroup.IsComplement.equiv_snd_eq_inv_mulstatement · cited by 8
- Monoid.PushoutI.NormalWord.consproof · cited by 7
- HNNExtension.NormalWord.unitsSMulGroupproof · cited by 6
- Subgroup.IsComplement.equiv_fst_mul_equiv_sndstatement · cited by 5
- Subgroup.IsComplement.equiv_mul_leftstatement and proof · cited by 3
- Monoid.PushoutI.NormalWord.rconsproof · cited by 3
- Subgroup.IsComplement.equiv_fst_eq_one_of_mem_of_one_memstatement · cited by 2
- Subgroup.IsComplement.equiv_fst_eq_self_of_mem_of_one_memstatement and proof · cited by 2
- Subgroup.IsComplement.equiv_snd_eq_self_iff_memstatement and proof · cited by 2
- Subgroup.IsComplement.equiv_snd_eq_self_of_mem_of_one_memstatement and proof · cited by 2
- Monoid.PushoutI.NormalWord.prod_consproof · cited by 2