Theorems · Theorem · group theory
AddAction.movedBy_mem_fixedBy_of_addCommute
∀ {α : Type u_1} {G : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G α] {g h : G},
AddCommute g h → (AddAction.fixedBy α g)ᶜ ∈ AddAction.fixedBy (Set α) hIf g and h commute, then g moves h +ᵥ x iff g moves x.
This is equivalent to say that the set (fixedBy α g)ᶜ is fixed by h.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Compl.complstatement and proof · cited by 2,925
- AddActionstatement and proof · cited by 820
- AddCommutestatement and proof · cited by 185
- Set.addActionSetstatement · cited by 39
- AddAction.fixedBystatement and proof · cited by 26
- AddAction.mem_fixedByproof · cited by 12
- Set.vadd_set_complproof · cited by 2
- AddAction.fixedBy_mem_fixedBy_of_addCommuteproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- AddAction.not_addCommute_of_disjoint_movedBy_preimageproof · cited by 0
- AddAction.vadd_zsmul_movedBy_eq_of_addCommuteproof · cited by 0