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.

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 IsAddUnit.neg_vadd {α : Type u_5} [AddMonoid α] {a : α} (h : IsAddUnit a) : -h.addUnit +ᵥ a = 0

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] : FaithfulVAdd (AddUnits M) α

Equations

• ⋯ = ⋯

instance Units.instFaithfulSMul {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 AddUnits.vaddCommClass_left {M : Type u_3} {N : Type u_4} {α : Type u_5} [AddMonoid M] [VAdd M α] [VAdd N α] [VAddCommClass M N α] : VAddCommClass (AddUnits M) N α

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] : SMulCommClass Mˣ N α

Equations

• ⋯ = ⋯

instance AddUnits.vaddCommClass_right {M : Type u_3} {N : Type u_4} {α : Type u_5} [AddMonoid N] [VAdd M α] [VAdd N α] [VAddCommClass M N α] : VAddCommClass M (AddUnits 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 AddUnits.instIsScalarTower {M : Type u_3} {N : Type u_4} {α : Type u_5} [AddMonoid M] [VAdd M N] [VAdd M α] [VAdd N α] [VAddAssocClass M N α] : VAddAssocClass (AddUnits 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] : IsScalarTower Mˣ N α

Equations

• ⋯ = ⋯

Action of a group G on units of M

theorem AddUnits.addAction'.proof_4 {G : Type u_2} {M : Type u_1} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] : ∀ (x x_1 : G) (x_2 : AddUnits M), x + x_1 +ᵥ x_2 = x +ᵥ (x_1 +ᵥ x_2)

theorem AddUnits.addAction'.proof_3 {G : Type u_2} {M : Type u_1} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] : ∀ (x : AddUnits M), 0 +ᵥ x = x

instance AddUnits.addAction' {G : Type u_1} {M : Type u_3} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] : AddAction G (AddUnits M)

Equations

• AddUnits.addAction' = AddAction.mk ⋯ ⋯

theorem AddUnits.addAction'.proof_2 {G : Type u_2} {M : Type u_1} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] (g : G) (m : AddUnits M) : -g +ᵥ ↑(-m) + (g +ᵥ ↑m) = 0

theorem AddUnits.addAction'.proof_1 {G : Type u_2} {M : Type u_1} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] (g : G) (m : AddUnits M) : g +ᵥ ↑m + (-g +ᵥ ↑(-m)) = 0

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 AddUnits.val_vadd {G : Type u_1} {M : Type u_3} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] (g : G) (m : AddUnits M) : ↑(g +ᵥ m) = g +ᵥ ↑m

@[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 AddUnits.vadd_neg {G : Type u_1} {M : Type u_3} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] (g : G) (m : AddUnits M) : -(g +ᵥ m) = -g +ᵥ -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 AddUnits.vaddCommClass' {G : Type u_1} {H : Type u_2} {M : Type u_3} [AddGroup G] [AddGroup H] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [AddAction H M] [VAddCommClass H M M] [VAddAssocClass G M M] [VAddAssocClass H M M] [VAddCommClass G H M] : VAddCommClass G H (AddUnits M)

Transfer VAddCommClass G H M to VAddCommClass G H (AddUnits M).

Equations

• ⋯ = ⋯

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 AddUnits.isScalarTower' {G : Type u_1} {H : Type u_2} {M : Type u_3} [VAdd G H] [AddGroup G] [AddGroup H] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [AddAction H M] [VAddCommClass H M M] [VAddAssocClass G M M] [VAddAssocClass H M M] [VAddAssocClass G H M] : VAddAssocClass G H (AddUnits M)

Transfer VAddAssocClass G H M to VAddAssocClass G H (AddUnits M).

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 AddUnits.isScalarTower'_left {G : Type u_1} {M : Type u_3} {α : Type u_5} [AddGroup G] [AddMonoid M] [AddAction G M] [VAdd M α] [VAdd G α] [VAddCommClass G M M] [VAddAssocClass G M M] [VAddAssocClass G M α] : VAddAssocClass G (AddUnits M) α

Transfer VAddAssocClass G M α to VAddAssocClass G (AddUnits M) α.

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

• ⋯ = ⋯

theorem IsAddUnit.vadd {G : Type u_1} {M : Type u_3} [AddGroup G] [AddMonoid M] [AddAction G M] [VAddCommClass G M M] [VAddAssocClass G M M] {m : M} (g : G) (h : IsAddUnit m) : IsAddUnit (g +ᵥ m)

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)