Sets invariant to a MulAction

In this file we define SubMulAction R M; a subset of a MulAction R M which is closed with respect to scalar multiplication.

For most uses, typically Submodule R M is more powerful.

Main definitions

• SubMulAction.mulAction - the MulAction R M transferred to the subtype.
• SubMulAction.mulAction' - the MulAction S M transferred to the subtype when IsScalarTower S R M.
• SubMulAction.isScalarTower - the IsScalarTower S R M transferred to the subtype.
• SubMulAction.inclusion — the inclusion of a submulaction, as an equivariant map

Tags

submodule, mul_action

class SMulMemClass (S : Type u_1) (R : outParam (Type u_2)) (M : Type u_3) [SMul R M] [SetLike S M] : Prop

SMulMemClass S R M says S is a type of subsets s ≤ M that are closed under the scalar action of R on M.

Note that only R is marked as an outParam here, since M is supplied by the SetLike class instead.

• smul_mem {s : S} (r : R) {m : M} : m ∈ s → r • m ∈ s

  Multiplication by a scalar on an element of the set remains in the set.

class VAddMemClass (S : Type u_1) (R : outParam (Type u_2)) (M : Type u_3) [VAdd R M] [SetLike S M] : Prop

VAddMemClass S R M says S is a type of subsets s ≤ M that are closed under the additive action of R on M.

Note that only R is marked as an outParam here, since M is supplied by the SetLike class instead.

• vadd_mem {s : S} (r : R) {m : M} : m ∈ s → r +ᵥ m ∈ s

  Addition by a scalar with an element of the set remains in the set.

theorem AddSubmonoidClass.nsmulMemClass {S : Type u_1} {M : Type u_2} [AddMonoid M] [SetLike S M] [AddSubmonoidClass S M] : SMulMemClass S ℕ M

Not registered as an instance because R is an outParam in SMulMemClass S R M.

theorem AddSubgroupClass.zsmulMemClass {S : Type u_1} {M : Type u_2} [SubNegMonoid M] [SetLike S M] [AddSubgroupClass S M] : SMulMemClass S ℤ M

Not registered as an instance because R is an outParam in SMulMemClass S R M.

@[instance 50]

instance SetLike.smul {S : Type u'} {R : Type u} {M : Type v} [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] (s : S) : SMul R ↥s

A subset closed under the scalar action inherits that action.

Equations

• SetLike.smul s = { smul := fun (r : R) (x : ↥s) => ⟨r • ↑x, ⋯⟩ }

@[instance 50]

instance SetLike.vadd {S : Type u'} {R : Type u} {M : Type v} [VAdd R M] [SetLike S M] [hS : VAddMemClass S R M] (s : S) : VAdd R ↥s

A subset closed under the additive action inherits that action.

Equations

• SetLike.vadd s = { vadd := fun (r : R) (x : ↥s) => ⟨r +ᵥ ↑x, ⋯⟩ }

theorem SMulMemClass.ofIsScalarTower (S : Type u_1) (M : Type u_2) (N : Type u_3) (α : Type u_4) [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] : SMulMemClass S M α

This can't be an instance because Lean wouldn't know how to find N, but we can still use this to manually derive SMulMemClass on specific types.

instance SetLike.instIsScalarTower {S : Type u'} {R : Type u} {M : Type v} [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] [Mul M] [MulMemClass S M] [IsScalarTower R M M] (s : S) : IsScalarTower R ↥s ↥s

instance SetLike.instSMulCommClass {S : Type u'} {R : Type u} {M : Type v} [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] [Mul M] [MulMemClass S M] [SMulCommClass R M M] (s : S) : SMulCommClass R ↥s ↥s

@[simp]

theorem SetLike.val_smul {S : Type u'} {R : Type u} {M : Type v} [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] (s : S) (r : R) (x : ↥s) : ↑(r • x) = r • ↑x

@[simp]

theorem SetLike.val_vadd {S : Type u'} {R : Type u} {M : Type v} [VAdd R M] [SetLike S M] [hS : VAddMemClass S R M] (s : S) (r : R) (x : ↥s) : ↑(r +ᵥ x) = r +ᵥ ↑x

@[simp]

theorem SetLike.mk_smul_mk {S : Type u'} {R : Type u} {M : Type v} [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] (s : S) (r : R) (x : M) (hx : x ∈ s) : r • ⟨x, hx⟩ = ⟨r • x, ⋯⟩

@[simp]

theorem SetLike.mk_vadd_mk {S : Type u'} {R : Type u} {M : Type v} [VAdd R M] [SetLike S M] [hS : VAddMemClass S R M] (s : S) (r : R) (x : M) (hx : x ∈ s) : r +ᵥ ⟨x, hx⟩ = ⟨r +ᵥ x, ⋯⟩

theorem SetLike.smul_def {S : Type u'} {R : Type u} {M : Type v} [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] (s : S) (r : R) (x : ↥s) : r • x = ⟨r • ↑x, ⋯⟩

theorem SetLike.vadd_def {S : Type u'} {R : Type u} {M : Type v} [VAdd R M] [SetLike S M] [hS : VAddMemClass S R M] (s : S) (r : R) (x : ↥s) : r +ᵥ x = ⟨r +ᵥ ↑x, ⋯⟩

@[simp]

theorem SetLike.forall_smul_mem_iff {R : Type u_1} {M : Type u_2} {S : Type u_3} [Monoid R] [MulAction R M] [SetLike S M] [SMulMemClass S R M] {N : S} {x : M} : (∀ (a : R), a • x ∈ N) ↔ x ∈ N

@[instance 50]

instance SetLike.smul' {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) : SMul M ↥s

A subset closed under the scalar action inherits that action.

Equations

• SetLike.smul' s = { smul := fun (r : M) (x : ↥s) => ⟨r • ↑x, ⋯⟩ }

@[instance 50]

instance SetLike.vadd' {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [VAdd M N] [VAdd M α] [AddMonoid N] [AddAction N α] [VAddMemClass S N α] [VAddAssocClass M N α] (s : S) : VAdd M ↥s

A subset closed under the additive action inherits that action.

Equations

• SetLike.vadd' s = { vadd := fun (r : M) (x : ↥s) => ⟨r +ᵥ ↑x, ⋯⟩ }

@[simp]

theorem SetLike.val_smul_of_tower {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) (r : M) (x : ↥s) : ↑(r • x) = r • ↑x

@[simp]

theorem SetLike.val_vadd_of_tower {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [VAdd M N] [VAdd M α] [AddMonoid N] [AddAction N α] [VAddMemClass S N α] [VAddAssocClass M N α] (s : S) (r : M) (x : ↥s) : ↑(r +ᵥ x) = r +ᵥ ↑x

@[simp]

theorem SetLike.mk_smul_of_tower_mk {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) (r : M) (x : α) (hx : x ∈ s) : r • ⟨x, hx⟩ = ⟨r • x, ⋯⟩

@[simp]

theorem SetLike.mk_vadd_of_tower_mk {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [VAdd M N] [VAdd M α] [AddMonoid N] [AddAction N α] [VAddMemClass S N α] [VAddAssocClass M N α] (s : S) (r : M) (x : α) (hx : x ∈ s) : r +ᵥ ⟨x, hx⟩ = ⟨r +ᵥ x, ⋯⟩

theorem SetLike.smul_of_tower_def {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) (r : M) (x : ↥s) : r • x = ⟨r • ↑x, ⋯⟩

theorem SetLike.vadd_of_tower_def {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [VAdd M N] [VAdd M α] [AddMonoid N] [AddAction N α] [VAddMemClass S N α] [VAddAssocClass M N α] (s : S) (r : M) (x : ↥s) : r +ᵥ x = ⟨r +ᵥ ↑x, ⋯⟩

structure SubMulAction (R : Type u) (M : Type v) [SMul R M] : Type v

A SubMulAction is a set which is closed under scalar multiplication.

• carrier : Set M

  The underlying set of a SubMulAction.

• smul_mem' (c : R) {x : M} : x ∈ self.carrier → c • x ∈ self.carrier

  The carrier set is closed under scalar multiplication.

instance SubMulAction.instSetLike {R : Type u} {M : Type v} [SMul R M] : SetLike (SubMulAction R M) M

Equations

• SubMulAction.instSetLike = { coe := SubMulAction.carrier, coe_injective' := ⋯ }

instance SubMulAction.instSMulMemClass {R : Type u} {M : Type v} [SMul R M] : SMulMemClass (SubMulAction R M) R M

@[simp]

theorem SubMulAction.mem_carrier {R : Type u} {M : Type v} [SMul R M] {p : SubMulAction R M} {x : M} : x ∈ p.carrier ↔ x ∈ ↑p

theorem SubMulAction.ext {R : Type u} {M : Type v} [SMul R M] {p q : SubMulAction R M} (h : ∀ (x : M), x ∈ p ↔ x ∈ q) : p = q

def SubMulAction.copy {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) (s : Set M) (hs : s = ↑p) : SubMulAction R M

Copy of a sub_mul_action with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations

• p.copy s hs = { carrier := s, smul_mem' := ⋯ }

@[simp]

theorem SubMulAction.coe_copy {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) (s : Set M) (hs : s = ↑p) : ↑(p.copy s hs) = s

theorem SubMulAction.copy_eq {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) (s : Set M) (hs : s = ↑p) : p.copy s hs = p

instance SubMulAction.instBot {R : Type u} {M : Type v} [SMul R M] : Bot (SubMulAction R M)

Equations

• SubMulAction.instBot = { bot := { carrier := ∅, smul_mem' := ⋯ } }

instance SubMulAction.instInhabited {R : Type u} {M : Type v} [SMul R M] : Inhabited (SubMulAction R M)

Equations

• SubMulAction.instInhabited = { default := ⊥ }

theorem SubMulAction.smul_mem {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) {x : M} (r : R) (h : x ∈ p) : r • x ∈ p

instance SubMulAction.instSMulSubtypeMem {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) : SMul R ↥p

Equations

• p.instSMulSubtypeMem = { smul := fun (c : R) (x : ↥p) => ⟨c • ↑x, ⋯⟩ }

@[simp]

theorem SubMulAction.val_smul {R : Type u} {M : Type v} [SMul R M] {p : SubMulAction R M} (r : R) (x : ↥p) : ↑(r • x) = r • ↑x

def SubMulAction.subtype {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) : ↥p →ₑ[id] M

Embedding of a submodule p to the ambient space M.

Equations

• p.subtype = { toFun := Subtype.val, map_smul' := ⋯ }

@[simp]

theorem SubMulAction.subtype_apply {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) (x : ↥p) : p.subtype x = ↑x

theorem SubMulAction.subtype_eq_val {R : Type u} {M : Type v} [SMul R M] (p : SubMulAction R M) : ⇑p.subtype = Subtype.val

@[instance 75]

instance SubMulAction.SMulMemClass.toMulAction {R : Type u} {M : Type v} [Monoid R] [MulAction R M] {A : Type u_1} [SetLike A M] [hA : SMulMemClass A R M] (S' : A) : MulAction R ↥S'

A SubMulAction of a MulAction is a MulAction.

Equations

• SubMulAction.SMulMemClass.toMulAction S' = Function.Injective.mulAction Subtype.val ⋯ ⋯

def SubMulAction.SMulMemClass.subtype {R : Type u} {M : Type v} [Monoid R] [MulAction R M] {A : Type u_1} [SetLike A M] [hA : SMulMemClass A R M] (S' : A) : ↥S' →ₑ[id] M

The natural MulActionHom over R from a SubMulAction of M to M.

Equations

• SubMulAction.SMulMemClass.subtype S' = { toFun := Subtype.val, map_smul' := ⋯ }

@[simp]

theorem SubMulAction.SMulMemClass.coeSubtype {R : Type u} {M : Type v} [Monoid R] [MulAction R M] {A : Type u_1} [SetLike A M] [hA : SMulMemClass A R M] (S' : A) : ⇑(SubMulAction.SMulMemClass.subtype S') = Subtype.val

theorem SubMulAction.smul_of_tower_mem {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) (s : S) {x : M} (h : x ∈ p) : s • x ∈ p

instance SubMulAction.smul' {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) : SMul S ↥p

Equations

• p.smul' = { smul := fun (c : S) (x : ↥p) => ⟨c • ↑x, ⋯⟩ }

instance SubMulAction.isScalarTower {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) : IsScalarTower S R ↥p

instance SubMulAction.isScalarTower' {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) {S' : Type u_1} [SMul S' R] [SMul S' S] [SMul S' M] [IsScalarTower S' R M] [IsScalarTower S' S M] : IsScalarTower S' S ↥p

@[simp]

theorem SubMulAction.val_smul_of_tower {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) (s : S) (x : ↥p) : ↑(s • x) = s • ↑x

@[simp]

theorem SubMulAction.smul_mem_iff' {R : Type u} {M : Type v} [Monoid R] [MulAction R M] (p : SubMulAction R M) {G : Type u_1} [Group G] [SMul G R] [MulAction G M] [IsScalarTower G R M] (g : G) {x : M} : g • x ∈ p ↔ x ∈ p

instance SubMulAction.isCentralScalar {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) [SMul Sᵐᵒᵖ R] [SMul Sᵐᵒᵖ M] [IsScalarTower Sᵐᵒᵖ R M] [IsCentralScalar S M] : IsCentralScalar S ↥p

instance SubMulAction.mulAction' {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [Monoid S] [SMul S R] [MulAction S M] [IsScalarTower S R M] (p : SubMulAction R M) : MulAction S ↥p

If the scalar product forms a MulAction, then the subset inherits this action

Equations

• p.mulAction' = MulAction.mk ⋯ ⋯

instance SubMulAction.mulAction {R : Type u} {M : Type v} [Monoid R] [MulAction R M] (p : SubMulAction R M) : MulAction R ↥p

Equations

• p.mulAction = p.mulAction'

theorem SubMulAction.val_image_orbit {R : Type u} {M : Type v} [Monoid R] [MulAction R M] {p : SubMulAction R M} (m : ↥p) : Subtype.val '' MulAction.orbit R m = MulAction.orbit R ↑m

Orbits in a SubMulAction coincide with orbits in the ambient space.

theorem SubMulAction.val_preimage_orbit {R : Type u} {M : Type v} [Monoid R] [MulAction R M] {p : SubMulAction R M} (m : ↥p) : Subtype.val ⁻¹' MulAction.orbit R ↑m = MulAction.orbit R m

theorem SubMulAction.mem_orbit_subMul_iff {R : Type u} {M : Type v} [Monoid R] [MulAction R M] {p : SubMulAction R M} {x m : ↥p} : x ∈ MulAction.orbit R m ↔ ↑x ∈ MulAction.orbit R ↑m

theorem SubMulAction.stabilizer_of_subMul.submonoid {R : Type u} {M : Type v} [Monoid R] [MulAction R M] {p : SubMulAction R M} (m : ↥p) : MulAction.stabilizerSubmonoid R m = MulAction.stabilizerSubmonoid R ↑m

Stabilizers in monoid SubMulAction coincide with stabilizers in the ambient space

theorem SubMulAction.orbitRel_of_subMul {R : Type u} {M : Type v} [Group R] [MulAction R M] (p : SubMulAction R M) : MulAction.orbitRel R ↥p = Setoid.comap Subtype.val (MulAction.orbitRel R M)

theorem SubMulAction.stabilizer_of_subMul {R : Type u} {M : Type v} [Group R] [MulAction R M] {p : SubMulAction R M} (m : ↥p) : MulAction.stabilizer R m = MulAction.stabilizer R ↑m

Stabilizers in group SubMulAction coincide with stabilizers in the ambient space

theorem SubMulAction.zero_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : SubMulAction R M) (h : (↑p).Nonempty) : 0 ∈ p

instance SubMulAction.instZeroSubtypeMemOfNonempty {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : SubMulAction R M) [n_empty : Nonempty ↥p] : Zero ↥p

If the scalar product forms a Module, and the SubMulAction is not ⊥, then the subset inherits the zero.

Equations

• p.instZeroSubtypeMemOfNonempty = { zero := ⟨0, ⋯⟩ }

theorem SubMulAction.neg_mem {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] (p : SubMulAction R M) {x : M} (hx : x ∈ p) : -x ∈ p

@[simp]

theorem SubMulAction.neg_mem_iff {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] (p : SubMulAction R M) {x : M} : -x ∈ p ↔ x ∈ p

instance SubMulAction.instNegSubtypeMem {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] (p : SubMulAction R M) : Neg ↥p

Equations

• p.instNegSubtypeMem = { neg := fun (x : ↥p) => ⟨-↑x, ⋯⟩ }

@[simp]

theorem SubMulAction.val_neg {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] (p : SubMulAction R M) (x : ↥p) : ↑(-x) = -↑x

theorem SubMulAction.smul_mem_iff {S : Type u'} {R : Type u} {M : Type v} [GroupWithZero S] [Monoid R] [MulAction R M] [SMul S R] [MulAction S M] [IsScalarTower S R M] (p : SubMulAction R M) {s : S} {x : M} (s0 : s ≠ 0) : s • x ∈ p ↔ x ∈ p

def SubMulAction.inclusion {M : Type u_1} {α : Type u_2} [Monoid M] [MulAction M α] (s : SubMulAction M α) : ↥s →ₑ[id] α

The inclusion of a SubMulAction into the ambient set, as an equivariant map

Equations

• s.inclusion = { toFun := Subtype.val, map_smul' := ⋯ }

theorem SubMulAction.inclusion.toFun_eq_coe {M : Type u_1} {α : Type u_2} [Monoid M] [MulAction M α] (s : SubMulAction M α) : s.inclusion.toFun = Subtype.val

theorem SubMulAction.inclusion.coe_eq {M : Type u_1} {α : Type u_2} [Monoid M] [MulAction M α] (s : SubMulAction M α) : ⇑s.inclusion = Subtype.val

theorem SubMulAction.image_inclusion {M : Type u_1} {α : Type u_2} [Monoid M] [MulAction M α] (s : SubMulAction M α) : Set.range ⇑s.inclusion = s.carrier

theorem SubMulAction.inclusion_injective {M : Type u_1} {α : Type u_2} [Monoid M] [MulAction M α] (s : SubMulAction M α) : Function.Injective ⇑s.inclusion

def Units.nonZeroSubMul (R : Type u_1) (M : Type u_2) [Monoid R] [AddCommMonoid M] [DistribMulAction R M] : SubMulAction Rˣ M

The non-zero elements of M are invariant under the action by the units of R.

Equations

• Units.nonZeroSubMul R M = { carrier := {x : M | x ≠ 0}, smul_mem' := ⋯ }

instance Units.instMulActionSubtypeNeOfNat (R : Type u_1) (M : Type u_2) [Monoid R] [AddCommMonoid M] [DistribMulAction R M] : MulAction Rˣ { x : M // x ≠ 0 }

Equations

• Units.instMulActionSubtypeNeOfNat R M = (Units.nonZeroSubMul R M).mulAction'

@[simp]

theorem Units.smul_coe (R : Type u_1) (M : Type u_2) [Monoid R] [AddCommMonoid M] [DistribMulAction R M] (a : Rˣ) (x : { x : M // x ≠ 0 }) : ↑(a • x) = a • ↑x

theorem Units.orbitRel_nonZero_iff (R : Type u_1) (M : Type u_2) [Monoid R] [AddCommMonoid M] [DistribMulAction R M] (x y : { v : M // v ≠ 0 }) : (MulAction.orbitRel Rˣ { v : M // v ≠ 0 }) x y ↔ (MulAction.orbitRel Rˣ M) ↑x ↑y