Theorems · Theorem · group theory
MulAction.fixedBy_mem_fixedBy_of_commute
∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] {g h : G},
Commute g h → MulAction.fixedBy α g ∈ MulAction.fixedBy (Set α) hIf g and h commute, then g fixes h • x iff g fixes x.
This is equivalent to say that the set fixedBy α g is fixed by h.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Groupstatement and proof · cited by 6,238
- Set.extproof · cited by 2,266
- MulActionstatement and proof · cited by 1,294
- Commutestatement and proof · cited by 639
- SemigroupAction.mul_smulproof · cited by 291
- Set.mulActionSetstatement · cited by 95
- MulAction.fixedBystatement and proof · cited by 36
- Set.mem_smul_set_iff_inv_smul_memproof · cited by 18
- smul_left_cancel_iffproof · cited by 10
- MulAction.mem_fixedByproof · cited by 6
- Commute.inv_rightproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- MulAction.movedBy_mem_fixedBy_of_commuteproof · cited by 2
- MulAction.smul_zpow_fixedBy_eq_of_commuteproof · cited by 0