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Theorems · Theorem · group theory

MulAction.mem_fixedBy

∀ {M : Type u_1} {α : Type u_3} [inst : Monoid M] [inst_1 : MulAction M α] {m : M} {a : α},
  a ∈ MulAction.fixedBy α m ↔ m • a = a
Defined in
Mathlib.GroupTheory.GroupAction.Defs
Cited by
6 results in Mathlib
Foundations
Depth 5 from the axioms · uses no axioms
Assumes
MonoidMulAction
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

MulAction.fixedBy_inv · cited by 4MulAction.fixedBy_invMulAction.movedBy_mem_fixedBy_of_commute · cited by 2MulAction.movedBy_mem_fix…MulAction.smul_mem_fixedBy_iff_mem_fixedBy · cited by 2MulAction.smul_mem_fixedB…MulAction.mem_fixedBy_zpow · cited by 2MulAction.mem_fixedBy_zpowMulAction.fixedBy_mem_fixedBy_of_commute · cited by 2MulAction.fixedBy_mem_fix…MulAction.fixedBy_mul · cited by 0MulAction.fixedBy_mulSet · cited by 53352SetMonoid · cited by 3887MonoidMulAction · cited by 1294MulActionMulAction.fixedBy · cited by 36MulAction.fixedByMulAction.mem_fixedByCITED BYCITES

Cites4

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by6

Results whose statement or proof uses this declaration.