Theorems · Theorem · group theory
mem_leftAddCoset_iff
∀ {α : Type u_1} [inst : AddGroup α] {s : Set α} {x : α} (a : α), x ∈ a +ᵥ s ↔ -a + x ∈ s- Defined in
- Mathlib.GroupTheory.Coset.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses propext
- Assumes
- AddGroup
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
- AddGroupstatement and proof · cited by 4,410
- HVAdd.hVAddstatement and proof · cited by 1,820
- Set.vaddSetstatement · cited by 403
- neg_add_cancel_leftproof · cited by 84
- add_neg_cancel_leftproof · cited by 76
Cited by9
Results whose statement or proof uses this declaration.
- AddSubgroup.leftCoset_cover_const_iff_surjOnproof · cited by 3
- AddSubgroup.exists_finiteIndex_of_leftCoset_cover_auxproof · cited by 2
- eq_addCosets_of_normalproof · cited by 1
- QuotientAddGroup.eq_class_eq_leftCosetproof · cited by 1
- AddSubgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eqproof · cited by 1
- leftAddCoset_eq_iffproof · cited by 0
- leftAddCoset_mem_leftAddCosetproof · cited by 0
- NonarchimedeanAddGroup.exists_openAddSubgroup_separatingproof · cited by 0