Group actions on and by Mˣ

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.instVAddAddUnits {M : Type u_3} {α : Type u_5} [AddMonoid M] [VAdd M α] : VAdd (AddUnits M) α

Equations

• AddUnits.instVAddAddUnits = { vadd := fun (m : AddUnits M) (a : α) => ↑m +ᵥ a }

instance Units.instSMulUnits {M : Type u_3} {α : Type u_5} [Monoid M] [SMul M α] : SMul Mˣ α

Equations

• Units.instSMulUnits = { 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) : IsUnit.unit hm • a = m • a

theorem IsUnit.inv_smul {α : Type u_5} [Monoid α] {a : α} (h : IsUnit a) : (IsUnit.unit h)⁻¹ • a = 1

instance AddUnits.instFaithfulVAddAddUnitsInstVAddAddUnits {M : Type u_3} {α : Type u_5} [AddMonoid M] [VAdd M α] [FaithfulVAdd M α] : FaithfulVAdd (AddUnits M) α

Equations

• ⋯ = ⋯

instance Units.instFaithfulSMulUnitsInstSMulUnits {M : Type u_3} {α : Type u_5} [Monoid M] [SMul M α] [FaithfulSMul M α] : FaithfulSMul Mˣ α

Equations

• ⋯ = ⋯

instance AddUnits.instAddAction {M : Type u_3} {α : Type u_5} [AddMonoid M] [AddAction M α] : AddAction (AddUnits M) α

Equations

• AddUnits.instAddAction = AddAction.mk ⋯ ⋯

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 Units.instMulAction {M : Type u_3} {α : Type u_5} [Monoid M] [MulAction M α] : MulAction Mˣ α

Equations

• Units.instMulAction = MulAction.mk ⋯ ⋯

instance Units.instSMulZeroClass {M : Type u_3} {α : Type u_5} [Monoid M] [Zero α] [SMulZeroClass M α] : SMulZeroClass Mˣ α

Equations

• Units.instSMulZeroClass = SMulZeroClass.mk ⋯

instance Units.instDistribSMulUnits {M : Type u_3} {α : Type u_5} [Monoid M] [AddZeroClass α] [DistribSMul M α] : DistribSMul Mˣ α

Equations

• Units.instDistribSMulUnits = DistribSMul.mk ⋯

instance Units.instDistribMulAction {M : Type u_3} {α : Type u_5} [Monoid M] [AddMonoid α] [DistribMulAction M α] : DistribMulAction Mˣ α

Equations

• Units.instDistribMulAction = let __spread.0 := Units.instDistribSMulUnits; DistribMulAction.mk ⋯ ⋯

instance Units.instMulDistribMulAction {M : Type u_3} {α : Type u_5} [Monoid M] [Monoid α] [MulDistribMulAction M α] : MulDistribMulAction Mˣ α

Equations

• Units.instMulDistribMulAction = MulDistribMulAction.mk ⋯ ⋯

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] : SMulCommClass Mˣ 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] : SMulCommClass M Nˣ α

Equations

• ⋯ = ⋯

instance Units.instIsScalarTowerUnitsInstSMulUnitsInstSMulUnits {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] : IsScalarTower Mˣ 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] : MulAction G Mˣ

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

Equations

• Units.mulAction' = MulAction.mk ⋯ ⋯

@[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) : (g • m)⁻¹ = g⁻¹ • m⁻¹

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] : SMulCommClass G H Mˣ

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] : IsScalarTower G H Mˣ

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] : IsScalarTower G Mˣ α

Transfer IsScalarTower G M α to IsScalarTower G Mˣ α

Equations

• ⋯ = ⋯

instance Units.mulDistribMulAction' {G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulDistribMulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] : MulDistribMulAction G Mˣ

A stronger form of Units.mul_action'.

Equations

• Units.mulDistribMulAction' = let __src := Units.mulAction'; MulDistribMulAction.mk ⋯ ⋯

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)