Direct sum

This file defines the direct sum of abelian groups, indexed by a discrete type.

Notation

⨁ i, β i is the n-ary direct sum DirectSum. This notation is in the DirectSum locale, accessible after open DirectSum.

References

• https://en.wikipedia.org/wiki/Direct_sum

def DirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] : Type (max w v)

DirectSum ι β is the direct sum of a family of additive commutative monoids β i.

Note: open DirectSum will enable the notation ⨁ i, β i for DirectSum ι β.

Equations

• DirectSum ι β = Π₀ (i : ι), β i

instance instInhabitedDirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] : Inhabited (DirectSum ι β)

Equations

• instInhabitedDirectSum ι β = inferInstanceAs (Inhabited (Π₀ (i : ι), β i))

instance instAddCommMonoidDirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] : AddCommMonoid (DirectSum ι β)

Equations

• instAddCommMonoidDirectSum ι β = inferInstanceAs (AddCommMonoid (Π₀ (i : ι), β i))

instance instDFunLikeDirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] : DFunLike (DirectSum ι β) ι fun (i : ι) => β i

Equations

• instDFunLikeDirectSum ι β = inferInstanceAs (DFunLike (Π₀ (i : ι), β i) ι fun (i : ι) => β i)

instance instCoeFunDirectSumForall (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] : CoeFun (DirectSum ι β) fun (x : DirectSum ι β) => (i : ι) → β i

Equations

• instCoeFunDirectSumForall ι β = inferInstanceAs (CoeFun (Π₀ (i : ι), β i) fun (x : Π₀ (i : ι), β i) => (i : ι) → β i)

def DirectSum.«term⨁_,_» : Lean.ParserDescr

⨁ i, f i is notation for DirectSum _ f and equals the direct sum of fun i ↦ f i. Taking the direct sum over multiple arguments is possible, e.g. ⨁ (i) (j), f i j.

Equations

• One or more equations did not get rendered due to their size.

def DirectSum.«term⨁_,_».delab : Lean.PrettyPrinter.Delaborator.Delab

Pretty printer defined by notation3 command.

Equations

• One or more equations did not get rendered due to their size.

instance instDecidableEqDirectSum (ι : Type v) [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → DecidableEq (β i)] : DecidableEq (DirectSum ι β)

Equations

• instDecidableEqDirectSum ι β = inferInstanceAs (DecidableEq (Π₀ (i : ι), β i))

instance DirectSum.instAddCommGroup {ι : Type v} (β : ι → Type w) [(i : ι) → AddCommGroup (β i)] : AddCommGroup (DirectSum ι β)

Equations

• DirectSum.instAddCommGroup β = inferInstanceAs (AddCommGroup (Π₀ (i : ι), β i))

@[simp]

theorem DirectSum.sub_apply {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommGroup (β i)] (g₁ : DirectSum ι fun (i : ι) => β i) (g₂ : DirectSum ι fun (i : ι) => β i) (i : ι) : (g₁ - g₂) i = g₁ i - g₂ i

@[simp]

theorem DirectSum.zero_apply {ι : Type v} (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) : 0 i = 0

@[simp]

theorem DirectSum.add_apply {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] (g₁ : DirectSum ι fun (i : ι) => β i) (g₂ : DirectSum ι fun (i : ι) => β i) (i : ι) : (g₁ + g₂) i = g₁ i + g₂ i

def DirectSum.mk {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (s : Finset ι) : ((i : ↑↑s) → β ↑i) →+ DirectSum ι fun (i : ι) => β i

mk β s x is the element of ⨁ i, β i that is zero outside s and has coefficient x i for i in s.

Equations

• DirectSum.mk β s = { toFun := DFinsupp.mk s, map_zero' := ⋯, map_add' := ⋯ }

def DirectSum.of {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) : β i →+ DirectSum ι fun (i : ι) => β i

of i is the natural inclusion map from β i to ⨁ i, β i.

Equations

• DirectSum.of β i = DFinsupp.singleAddHom β i

@[simp]

theorem DirectSum.of_eq_same {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) (x : β i) : ((DirectSum.of β i) x) i = x

theorem DirectSum.of_eq_of_ne {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) (j : ι) (x : β i) (h : i ≠ j) : ((DirectSum.of β i) x) j = 0

theorem DirectSum.of_apply {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] {i : ι} (j : ι) (x : β i) : ((DirectSum.of β i) x) j = if h : i = j then Eq.recOn h x else 0

@[simp]

theorem DirectSum.support_zero {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] : DFinsupp.support 0 = ∅

@[simp]

theorem DirectSum.support_of {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (i : ι) (x : β i) (h : x ≠ 0) : DFinsupp.support ((DirectSum.of β i) x) = {i}

theorem DirectSum.support_of_subset {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {i : ι} {b : β i} : DFinsupp.support ((DirectSum.of β i) b) ⊆ {i}

theorem DirectSum.sum_support_of {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (x : DirectSum ι fun (i : ι) => β i) : ∑ i ∈ DFinsupp.support x, (DirectSum.of β i) (x i) = x

theorem DirectSum.sum_univ_of {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [Fintype ι] (x : DirectSum ι fun (i : ι) => β i) : ∑ i : ι, (DirectSum.of β i) (x i) = x

theorem DirectSum.mk_injective {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] (s : Finset ι) : Function.Injective ⇑(DirectSum.mk β s)

theorem DirectSum.of_injective {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] (i : ι) : Function.Injective ⇑(DirectSum.of β i)

theorem DirectSum.induction_on {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {C : (DirectSum ι fun (i : ι) => β i) → Prop} (x : DirectSum ι fun (i : ι) => β i) (H_zero : C 0) (H_basic : ∀ (i : ι) (x : β i), C ((DirectSum.of β i) x)) (H_plus : ∀ (x y : DirectSum ι fun (i : ι) => β i), C x → C y → C (x + y)) : C x

theorem DirectSum.addHom_ext {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u_1} [AddMonoid γ] ⦃f : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ ⦃g : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ (H : ∀ (i : ι) (y : β i), f ((DirectSum.of (fun (i : ι) => β i) i) y) = g ((DirectSum.of (fun (i : ι) => β i) i) y)) : f = g

If two additive homomorphisms from ⨁ i, β i are equal on each of β i y, then they are equal.

theorem DirectSum.addHom_ext' {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u_1} [AddMonoid γ] ⦃f : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ ⦃g : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ (H : ∀ (i : ι), f.comp (DirectSum.of (fun (i : ι) => β i) i) = g.comp (DirectSum.of (fun (i : ι) => β i) i)) : f = g

If two additive homomorphisms from ⨁ i, β i are equal on each of β i y, then they are equal.

See note [partially-applied ext lemmas].

def DirectSum.toAddMonoid {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (φ : (i : ι) → β i →+ γ) : (DirectSum ι fun (i : ι) => β i) →+ γ

toAddMonoid φ is the natural homomorphism from ⨁ i, β i to γ induced by a family φ of homomorphisms β i → γ.

Equations

• DirectSum.toAddMonoid φ = DFinsupp.liftAddHom φ

@[simp]

theorem DirectSum.toAddMonoid_of {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (φ : (i : ι) → β i →+ γ) (i : ι) (x : β i) : (DirectSum.toAddMonoid φ) ((DirectSum.of β i) x) = (φ i) x

theorem DirectSum.toAddMonoid.unique {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (ψ : (DirectSum ι fun (i : ι) => β i) →+ γ) (f : DirectSum ι fun (i : ι) => β i) : ψ f = (DirectSum.toAddMonoid fun (i : ι) => ψ.comp (DirectSum.of β i)) f

theorem DirectSum.toAddMonoid_injective {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] : Function.Injective DirectSum.toAddMonoid

@[simp]

theorem DirectSum.toAddMonoid_inj {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] {f : (i : ι) → β i →+ γ} {g : (i : ι) → β i →+ γ} : DirectSum.toAddMonoid f = DirectSum.toAddMonoid g ↔ f = g

def DirectSum.fromAddMonoid {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] : (DirectSum ι fun (i : ι) => γ →+ β i) →+ γ →+ DirectSum ι fun (i : ι) => β i

fromAddMonoid φ is the natural homomorphism from γ to ⨁ i, β i induced by a family φ of homomorphisms γ → β i.

Note that this is not an isomorphism. Not every homomorphism γ →+ ⨁ i, β i arises in this way.

Equations

• DirectSum.fromAddMonoid = DirectSum.toAddMonoid fun (i : ι) => AddMonoidHom.compHom (DirectSum.of β i)

@[simp]

theorem DirectSum.fromAddMonoid_of {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (i : ι) (f : γ →+ β i) : DirectSum.fromAddMonoid ((DirectSum.of (fun (i : ι) => γ →+ β i) i) f) = (DirectSum.of β i).comp f

theorem DirectSum.fromAddMonoid_of_apply {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (i : ι) (f : γ →+ β i) (x : γ) : (DirectSum.fromAddMonoid ((DirectSum.of (fun (i : ι) => γ →+ β i) i) f)) x = (DirectSum.of β i) (f x)

def DirectSum.setToSet {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (S : Set ι) (T : Set ι) (H : S ⊆ T) : (DirectSum ↑S fun (i : ↑S) => β ↑i) →+ DirectSum ↑T fun (i : ↑T) => β ↑i

setToSet β S T h is the natural homomorphism ⨁ (i : S), β i → ⨁ (i : T), β i, where h : S ⊆ T.

Equations

• DirectSum.setToSet β S T H = DirectSum.toAddMonoid fun (i : ↑S) => DirectSum.of (fun (i : Subtype T) => β ↑i) ⟨↑i, ⋯⟩

instance DirectSum.unique {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] [∀ (i : ι), Subsingleton (β i)] : Unique (DirectSum ι fun (i : ι) => β i)

Equations

• DirectSum.unique = DFinsupp.unique

instance DirectSum.uniqueOfIsEmpty {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] [IsEmpty ι] : Unique (DirectSum ι fun (i : ι) => β i)

A direct sum over an empty type is trivial.

Equations

• DirectSum.uniqueOfIsEmpty = DFinsupp.uniqueOfIsEmpty

def DirectSum.id (M : Type v) (ι : optParam (Type u_1) PUnit.{u_1 + 1} ) [AddCommMonoid M] [Unique ι] : (DirectSum ι fun (x : ι) => M) ≃+ M

The natural equivalence between ⨁ _ : ι, M and M when Unique ι.

Equations

• One or more equations did not get rendered due to their size.

def DirectSum.equivCongrLeft {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {κ : Type u_1} (h : ι ≃ κ) : (DirectSum ι fun (i : ι) => β i) ≃+ DirectSum κ fun (k : κ) => β (h.symm k)

Reindexing terms of a direct sum.

Equations

• DirectSum.equivCongrLeft h = let __src := DFinsupp.equivCongrLeft h; { toEquiv := __src, map_add' := ⋯ }

@[simp]

theorem DirectSum.equivCongrLeft_apply {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {κ : Type u_1} (h : ι ≃ κ) (f : DirectSum ι fun (i : ι) => β i) (k : κ) : ((DirectSum.equivCongrLeft h) f) k = f (h.symm k)

@[simp]

theorem DirectSum.addEquivProdDirectSum_apply {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] (f : Π₀ (i : Option ι), (fun (i : Option ι) => (fun (i : Option ι) => α i) i) i) : DirectSum.addEquivProdDirectSum f = (f none, DFinsupp.comapDomain some ⋯ f)

@[simp]

theorem DirectSum.addEquivProdDirectSum_symm_apply_toFun {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] (f : (fun (i : Option ι) => (fun (i : Option ι) => α i) i) none × Π₀ (i : ι), (fun (i : Option ι) => (fun (i : Option ι) => α i) i) (some i)) (i : Option ι) : (DirectSum.addEquivProdDirectSum.symm f) i = match i with | none => f.1 | some val => f.2 val

@[simp]

theorem DirectSum.addEquivProdDirectSum_symm_apply_support' {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] (f : (fun (i : Option ι) => (fun (i : Option ι) => α i) i) none × Π₀ (i : ι), (fun (i : Option ι) => (fun (i : Option ι) => α i) i) (some i)) : (DirectSum.addEquivProdDirectSum.symm f).support' = Trunc.map (fun (s : { s : Multiset ι // ∀ (i : ι), i ∈ s ∨ f.2 i = 0 }) => ⟨none ::ₘ Multiset.map some ↑s, ⋯⟩) f.2.support'

noncomputable def DirectSum.addEquivProdDirectSum {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] : (DirectSum (Option ι) fun (i : Option ι) => α i) ≃+ α none × DirectSum ι fun (i : ι) => α (some i)

Isomorphism obtained by separating the term of index none of a direct sum over Option ι.

Equations

• DirectSum.addEquivProdDirectSum = let __src := DFinsupp.equivProdDFinsupp; { toEquiv := __src, map_add' := ⋯ }

def DirectSum.sigmaCurry {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] : (DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd) →+ DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j

The natural map between ⨁ (i : Σ i, α i), δ i.1 i.2 and ⨁ i (j : α i), δ i j.

Equations

• DirectSum.sigmaCurry = { toFun := DFinsupp.sigmaCurry, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem DirectSum.sigmaCurry_apply {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] (f : DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd) (i : ι) (j : α i) : ((DirectSum.sigmaCurry f) i) j = f ⟨i, j⟩

def DirectSum.sigmaUncurry {ι : Type v} {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → DecidableEq (α i)] [(i : ι) → (j : α i) → DecidableEq (δ i j)] : (DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j) →+ DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd

The natural map between ⨁ i (j : α i), δ i j and Π₀ (i : Σ i, α i), δ i.1 i.2, inverse of curry.

Equations

• DirectSum.sigmaUncurry = { toFun := DFinsupp.sigmaUncurry, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem DirectSum.sigmaUncurry_apply {ι : Type v} {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → DecidableEq (α i)] [(i : ι) → (j : α i) → DecidableEq (δ i j)] (f : DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j) (i : ι) (j : α i) : (DirectSum.sigmaUncurry f) ⟨i, j⟩ = (f i) j

def DirectSum.sigmaCurryEquiv {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → DecidableEq (α i)] [(i : ι) → (j : α i) → DecidableEq (δ i j)] : (DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd) ≃+ DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j

The natural map between ⨁ (i : Σ i, α i), δ i.1 i.2 and ⨁ i (j : α i), δ i j.

Equations

• One or more equations did not get rendered due to their size.

def DirectSum.coeAddMonoidHom {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ι → S) : (DirectSum ι fun (i : ι) => ↥(A i)) →+ M

The canonical embedding from ⨁ i, A i to M where A is a collection of AddSubmonoid M indexed by ι.

When S = Submodule _ M, this is available as a LinearMap, DirectSum.coe_linearMap.

Equations

• DirectSum.coeAddMonoidHom A = DirectSum.toAddMonoid fun (i : ι) => AddSubmonoidClass.subtype (A i)

@[simp]

theorem DirectSum.coeAddMonoidHom_of {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ι → S) (i : ι) (x : ↥(A i)) : (DirectSum.coeAddMonoidHom A) ((DirectSum.of (fun (i : ι) => ↥(A i)) i) x) = ↑x

theorem DirectSum.coe_of_apply {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] {A : ι → S} (i : ι) (j : ι) (x : ↥(A i)) : ↑(((DirectSum.of (fun (i : ι) => ↥(A i)) i) x) j) = ↑(if i = j then x else 0)

def DirectSum.IsInternal {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ι → S) : Prop

The DirectSum formed by a collection of additive submonoids (or subgroups, or submodules) of M is said to be internal if the canonical map (⨁ i, A i) →+ M is bijective.

For the alternate statement in terms of independence and spanning, see DirectSum.subgroup_isInternal_iff_independent_and_supr_eq_top and DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top.

Equations

• DirectSum.IsInternal A = Function.Bijective ⇑(DirectSum.coeAddMonoidHom A)

theorem DirectSum.IsInternal.addSubmonoid_iSup_eq_top {ι : Type v} {M : Type u_1} [DecidableEq ι] [AddCommMonoid M] (A : ι → AddSubmonoid M) (h : DirectSum.IsInternal A) : iSup A = ⊤