Documentation

Mathlib.GroupTheory.GroupAction.Units

Group actions on and by #

This file provides the action of a unit on a type α, SMul Mˣ α, in the presence of SMul M α, with the obvious definition stated in Units.smul_def. This definition preserves MulAction and DistribMulAction structures too.

Additionally, a MulAction G M for some group G satisfying some additional properties admits a MulAction G Mˣ structure, again with the obvious definition stated in Units.coe_smul. These instances use a primed name.

The results are repeated for AddUnits and VAdd where relevant.

Action of the units of M on a type α #

instance AddUnits.instVAdd {M : Type u_3} {α : Type u_5} [AddMonoid M] [VAdd M α] :
VAdd (AddUnits M) α
Equations
  • AddUnits.instVAdd = { vadd := fun (m : AddUnits M) (a : α) => m +ᵥ a }
instance Units.instSMul {M : Type u_3} {α : Type u_5} [Monoid M] [SMul M α] :
SMul Mˣ α
Equations
  • Units.instSMul = { smul := fun (m : Mˣ) (a : α) => m a }
theorem AddUnits.vadd_def {M : Type u_3} {α : Type u_5} [AddMonoid M] [VAdd M α] (m : AddUnits M) (a : α) :
m +ᵥ a = m +ᵥ a
theorem Units.smul_def {M : Type u_3} {α : Type u_5} [Monoid M] [SMul M α] (m : Mˣ) (a : α) :
m a = m a
theorem AddUnits.vadd_mk_apply {M : Type u_6} {α : Type u_7} [AddMonoid M] [VAdd M α] (m : M) (n : M) (h₁ : m + n = 0) (h₂ : n + m = 0) (a : α) :
{ val := m, neg := n, val_neg := h₁, neg_val := h₂ } +ᵥ a = m +ᵥ a
@[simp]
theorem Units.smul_mk_apply {M : Type u_6} {α : Type u_7} [Monoid M] [SMul M α] (m : M) (n : M) (h₁ : m * n = 1) (h₂ : n * m = 1) (a : α) :
{ val := m, inv := n, val_inv := h₁, inv_val := h₂ } a = m a
@[simp]
theorem Units.smul_isUnit {M : Type u_3} {α : Type u_5} [Monoid M] [SMul M α] {m : M} (hm : IsUnit m) (a : α) :
hm.unit a = m a
theorem IsUnit.inv_smul {α : Type u_5} [Monoid α] {a : α} (h : IsUnit a) :
h.unit⁻¹ a = 1
instance AddUnits.instFaithfulVAdd {M : Type u_3} {α : Type u_5} [AddMonoid M] [VAdd M α] [FaithfulVAdd M α] :
Equations
  • =
instance Units.instFaithfulSMul {M : Type u_3} {α : Type u_5} [Monoid M] [SMul M α] [FaithfulSMul M α] :
Equations
  • =
theorem AddUnits.instAddAction.proof_1 {M : Type u_1} {α : Type u_2} [AddMonoid M] [AddAction M α] (m : AddUnits M) (n : AddUnits M) (b : α) :
m + n +ᵥ b = m +ᵥ (n +ᵥ b)
instance AddUnits.instAddAction {M : Type u_3} {α : Type u_5} [AddMonoid M] [AddAction M α] :
Equations
instance Units.instMulAction {M : Type u_3} {α : Type u_5} [Monoid M] [MulAction M α] :
Equations
instance Units.instSMulZeroClass {M : Type u_3} {α : Type u_5} [Monoid M] [Zero α] [SMulZeroClass M α] :
Equations
instance Units.instDistribSMulUnits {M : Type u_3} {α : Type u_5} [Monoid M] [AddZeroClass α] [DistribSMul M α] :
Equations
instance Units.instDistribMulAction {M : Type u_3} {α : Type u_5} [Monoid M] [AddMonoid α] [DistribMulAction M α] :
Equations
  • Units.instDistribMulAction = let __spread.0 := Units.instDistribSMulUnits; DistribMulAction.mk
Equations
instance Units.smulCommClass_left {M : Type u_3} {N : Type u_4} {α : Type u_5} [Monoid M] [SMul M α] [SMul N α] [SMulCommClass M N α] :
Equations
  • =
instance Units.smulCommClass_right {M : Type u_3} {N : Type u_4} {α : Type u_5} [Monoid N] [SMul M α] [SMul N α] [SMulCommClass M N α] :
Equations
  • =
instance Units.instIsScalarTower {M : Type u_3} {N : Type u_4} {α : Type u_5} [Monoid M] [SMul M N] [SMul M α] [SMul N α] [IsScalarTower M N α] :
Equations
  • =

Action of a group G on units of M #

instance Units.mulAction' {G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] :

If an action G associates and commutes with multiplication on M, then it lifts to an action on . Notably, this provides MulAction Mˣ Nˣ under suitable conditions.

Equations
@[simp]
theorem Units.val_smul {G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] (g : G) (m : Mˣ) :
(g m) = g m
@[simp]
theorem Units.smul_inv {G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] (g : G) (m : Mˣ) :

Note that this lemma exists more generally as the global smul_inv

instance Units.smulCommClass' {G : Type u_1} {H : Type u_2} {M : Type u_3} [Group G] [Group H] [Monoid M] [MulAction G M] [SMulCommClass G M M] [MulAction H M] [SMulCommClass H M M] [IsScalarTower G M M] [IsScalarTower H M M] [SMulCommClass G H M] :

Transfer SMulCommClass G H M to SMulCommClass G H Mˣ

Equations
  • =
instance Units.isScalarTower' {G : Type u_1} {H : Type u_2} {M : Type u_3} [SMul G H] [Group G] [Group H] [Monoid M] [MulAction G M] [SMulCommClass G M M] [MulAction H M] [SMulCommClass H M M] [IsScalarTower G M M] [IsScalarTower H M M] [IsScalarTower G H M] :

Transfer IsScalarTower G H M to IsScalarTower G H Mˣ

Equations
  • =
instance Units.isScalarTower'_left {G : Type u_1} {M : Type u_3} {α : Type u_5} [Group G] [Monoid M] [MulAction G M] [SMul M α] [SMul G α] [SMulCommClass G M M] [IsScalarTower G M M] [IsScalarTower G M α] :

Transfer IsScalarTower G M α to IsScalarTower G Mˣ α

Equations
  • =

A stronger form of Units.mul_action'.

Equations
theorem IsUnit.smul {G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] {m : M} (g : G) (h : IsUnit m) :
IsUnit (g m)