Documentation

Mathlib.Algebra.DirectSum.Basic

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 #

def DirectSum (ι : Type v) (β : ιType 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
instance instInhabitedDirectSum (ι : Type v) (β : ιType w) [(i : ι) → AddCommMonoid (β i)] :
Equations
instance instAddCommMonoidDirectSum (ι : Type v) (β : ιType w) [(i : ι) → AddCommMonoid (β i)] :
Equations
instance instDFunLikeDirectSum (ι : Type v) (β : ιType w) [(i : ι) → AddCommMonoid (β i)] :
DFunLike (DirectSum ι β) ι fun (i : ι) => β i
Equations
instance instCoeFunDirectSumForall (ι : Type v) (β : ιType w) [(i : ι) → AddCommMonoid (β i)] :
CoeFun (DirectSum ι β) fun (x : DirectSum ι β) => (i : ι) → β i
Equations

Pretty printer defined by notation3 command.

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

⨁ 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.
instance instDecidableEqDirectSum (ι : Type v) (β : ιType w) [DecidableEq ι] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → DecidableEq (β i)] :
Equations
def DirectSum.coeFnAddMonoidHom {ι : Type v} (β : ιType w) [(i : ι) → AddCommMonoid (β i)] :
(DirectSum ι fun (i : ι) => β i) →+ (i : ι) → β i

Coercion from a DirectSum to a pi type is an AddMonoidHom.

Equations
@[simp]
theorem DirectSum.coeFnAddMonoidHom_apply {ι : Type v} (β : ιType w) [(i : ι) → AddCommMonoid (β i)] (v : DirectSum ι fun (i : ι) => β i) :
(coeFnAddMonoidHom β) v = v
instance DirectSum.instAddCommGroup {ι : Type v} (β : ιType w) [(i : ι) → AddCommGroup (β i)] :
Equations
@[simp]
theorem DirectSum.sub_apply {ι : Type v} {β : ιType w} [(i : ι) → AddCommGroup (β i)] (g₁ g₂ : DirectSum ι fun (i : ι) => β i) (i : ι) :
(g₁ - g₂) i = g₁ i - g₂ i
theorem DirectSum.ext {ι : Type v} (β : ιType w) [(i : ι) → AddCommMonoid (β i)] {x y : DirectSum ι β} (w : ∀ (i : ι), x i = y i) :
x = y
theorem DirectSum.ext_iff {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] {x y : DirectSum ι β} :
x = y ∀ (i : ι), x i = y i
@[simp]
theorem DirectSum.zero_apply {ι : Type v} (β : ιType w) [(i : ι) → AddCommMonoid (β i)] (i : ι) :
0 i = 0
@[simp]
theorem DirectSum.add_apply {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] (g₁ g₂ : DirectSum ι fun (i : ι) => β i) (i : ι) :
(g₁ + g₂) i = g₁ i + g₂ i
def DirectSum.mk {ι : Type v} (β : ιType w) [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] (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
def DirectSum.of {ι : Type v} (β : ιType w) [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] (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
@[simp]
theorem DirectSum.of_eq_same {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] (i : ι) (x : β i) :
((of β i) x) i = x
theorem DirectSum.of_eq_of_ne {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] (i j : ι) (x : β i) (h : i j) :
((of β i) x) j = 0
theorem DirectSum.of_apply {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {i : ι} (j : ι) (x : β i) :
((of β i) x) j = if h : i = j then Eq.recOn h x else 0
theorem DirectSum.mk_apply_of_mem {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {s : Finset ι} {f : (i : s) → β i} {n : ι} (hn : n s) :
((mk β s) f) n = f n, hn
theorem DirectSum.mk_apply_of_notMem {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {s : Finset ι} {f : (i : s) → β i} {n : ι} (hn : ns) :
((mk β s) f) n = 0
@[deprecated DirectSum.mk_apply_of_notMem (since := "2025-05-23")]
theorem DirectSum.mk_apply_of_not_mem {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {s : Finset ι} {f : (i : s) → β i} {n : ι} (hn : ns) :
((mk β s) f) n = 0

Alias of DirectSum.mk_apply_of_notMem.

@[simp]
theorem DirectSum.support_zero {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] [(i : ι) → (x : β i) → Decidable (x 0)] :
@[simp]
theorem DirectSum.support_of {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] [(i : ι) → (x : β i) → Decidable (x 0)] (i : ι) (x : β i) (h : x 0) :
DFinsupp.support ((of β i) x) = {i}
theorem DirectSum.support_of_subset {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] [(i : ι) → (x : β i) → Decidable (x 0)] {i : ι} {b : β i} :
theorem DirectSum.sum_support_of {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] [(i : ι) → (x : β i) → Decidable (x 0)] (x : DirectSum ι fun (i : ι) => β i) :
iDFinsupp.support x, (of β i) (x i) = x
theorem DirectSum.sum_univ_of {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] [Fintype ι] (x : DirectSum ι fun (i : ι) => β i) :
i : ι, (of β i) (x i) = x
theorem DirectSum.mk_injective {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] (s : Finset ι) :
theorem DirectSum.of_injective {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] (i : ι) :
theorem DirectSum.induction_on {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {motive : (DirectSum ι fun (i : ι) => β i)Prop} (x : DirectSum ι fun (i : ι) => β i) (zero : motive 0) (of : ∀ (i : ι) (x : β i), motive ((of β i) x)) (add : ∀ (x y : DirectSum ι fun (i : ι) => β i), motive xmotive ymotive (x + y)) :
motive x
theorem DirectSum.addHom_ext {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u_1} [AddZeroClass γ] f g : (DirectSum ι fun (i : ι) => β i) →+ γ (H : ∀ (i : ι) (y : β i), f ((of β i) y) = g ((of β 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} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u_1} [AddZeroClass γ] f g : (DirectSum ι fun (i : ι) => β i) →+ γ (H : ∀ (i : ι), f.comp (of β i) = g.comp (of β 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].

theorem DirectSum.addHom_ext'_iff {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u_1} [AddZeroClass γ] {f g : (DirectSum ι fun (i : ι) => β i) →+ γ} :
f = g ∀ (i : ι), f.comp (of β i) = g.comp (of β i)
def DirectSum.toAddMonoid {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {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
@[simp]
theorem DirectSum.toAddMonoid_of {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u₁} [AddCommMonoid γ] (φ : (i : ι) → β i →+ γ) (i : ι) (x : β i) :
(toAddMonoid φ) ((of β i) x) = (φ i) x
theorem DirectSum.toAddMonoid.unique {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u₁} [AddCommMonoid γ] (ψ : (DirectSum ι fun (i : ι) => β i) →+ γ) (f : DirectSum ι fun (i : ι) => β i) :
ψ f = (toAddMonoid fun (i : ι) => ψ.comp (of β i)) f
theorem DirectSum.toAddMonoid_injective {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u₁} [AddCommMonoid γ] :
@[simp]
theorem DirectSum.toAddMonoid_inj {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u₁} [AddCommMonoid γ] {f g : (i : ι) → β i →+ γ} :
def DirectSum.fromAddMonoid {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {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
@[simp]
theorem DirectSum.fromAddMonoid_of {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u₁} [AddCommMonoid γ] (i : ι) (f : γ →+ β i) :
fromAddMonoid ((of (fun (i : ι) => γ →+ β i) i) f) = (of β i).comp f
theorem DirectSum.fromAddMonoid_of_apply {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] {γ : Type u₁} [AddCommMonoid γ] (i : ι) (f : γ →+ β i) (x : γ) :
(fromAddMonoid ((of (fun (i : ι) => γ →+ β i) i) f)) x = (of β i) (f x)
def DirectSum.setToSet {ι : Type v} (β : ιType w) [(i : ι) → AddCommMonoid (β i)] [DecidableEq ι] (S 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
instance DirectSum.unique {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] [∀ (i : ι), Subsingleton (β i)] :
Unique (DirectSum ι fun (i : ι) => β i)
Equations
instance DirectSum.uniqueOfIsEmpty {ι : Type v} {β : ιType 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
def DirectSum.id (M : Type v) (ι : 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
def DirectSum.equivCongrLeft {ι : Type v} {β : ιType 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
@[simp]
theorem DirectSum.equivCongrLeft_apply {ι : Type v} {β : ιType w} [(i : ι) → AddCommMonoid (β i)] {κ : Type u_1} (h : ι κ) (f : DirectSum ι fun (i : ι) => β i) (k : κ) :
((equivCongrLeft h) f) k = f (h.symm k)
noncomputable def DirectSum.addEquivProdDirectSum {ι : Type v} {α : Option ιType 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
@[simp]
theorem DirectSum.addEquivProdDirectSum_symm_apply_support' {ι : Type v} {α : Option ιType w} [(i : Option ι) → AddCommMonoid (α i)] (f : (fun (i : Option ι) => α i) none × Π₀ (i : ι), (fun (i : Option ι) => α i) (some i)) :
(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'
@[simp]
theorem DirectSum.addEquivProdDirectSum_apply {ι : Type v} {α : Option ιType w} [(i : Option ι) → AddCommMonoid (α i)] (f : Π₀ (i : Option ι), (fun (i : Option ι) => α i) i) :
@[simp]
theorem DirectSum.addEquivProdDirectSum_symm_apply_toFun {ι : Type v} {α : Option ιType w} [(i : Option ι) → AddCommMonoid (α i)] (f : (fun (i : Option ι) => α i) none × Π₀ (i : ι), (fun (i : Option ι) => α i) (some i)) (i : Option ι) :
(addEquivProdDirectSum.symm f) i = match i with | none => f.1 | some val => f.2 val
def DirectSum.sigmaCurry {ι : Type v} [DecidableEq ι] {α : ιType u} {δ : (i : ι) → α iType 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
@[simp]
theorem DirectSum.sigmaCurry_apply {ι : Type v} [DecidableEq ι] {α : ιType u} {δ : (i : ι) → α iType 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) :
((sigmaCurry f) i) j = f i, j
def DirectSum.sigmaUncurry {ι : Type v} [DecidableEq ι] {α : ιType u} {δ : (i : ι) → α iType w} [(i : ι) → (j : α i) → AddCommMonoid (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
@[simp]
theorem DirectSum.sigmaUncurry_apply {ι : Type v} [DecidableEq ι] {α : ιType u} {δ : (i : ι) → α iType w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] (f : DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j) (i : ι) (j : α i) :
(sigmaUncurry f) i, j = (f i) j
def DirectSum.sigmaCurryEquiv {ι : Type v} [DecidableEq ι] {α : ιType u} {δ : (i : ι) → α iType 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
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 : ιS) :
(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
theorem DirectSum.coeAddMonoidHom_eq_dfinsuppSum {ι : Type v} [DecidableEq ι] {M : Type u_1} {S : Type u_2} [DecidableEq M] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ιS) (x : DirectSum ι fun (i : ι) => (A i)) :
(DirectSum.coeAddMonoidHom A) x = DFinsupp.sum x fun (i : ι) (x : (A i)) => x
@[deprecated DirectSum.coeAddMonoidHom_eq_dfinsuppSum (since := "2025-04-06")]
theorem DirectSum.coeAddMonoidHom_eq_dfinsupp_sum {ι : Type v} [DecidableEq ι] {M : Type u_1} {S : Type u_2} [DecidableEq M] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ιS) (x : DirectSum ι fun (i : ι) => (A i)) :
(DirectSum.coeAddMonoidHom A) x = DFinsupp.sum x fun (i : ι) (x : (A i)) => x

Alias of DirectSum.coeAddMonoidHom_eq_dfinsuppSum.

@[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 : ιS) (i : ι) (x : (A i)) :
(DirectSum.coeAddMonoidHom A) ((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 : ιS} (i j : ι) (x : (A i)) :
(((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 : ιS) :

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_iSupIndep_and_supr_eq_top and DirectSum.isInternal_submodule_iff_iSupIndep_and_iSup_eq_top.

Equations
theorem DirectSum.support_subset {ι : Type v} {M : Type u_1} {S : Type u_2} [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] [DecidableEq ι] [DecidableEq M] (A : ιS) (x : DirectSum ι fun (i : ι) => (A i)) :
(Function.support fun (i : ι) => (x i)) (DFinsupp.support x)
theorem DirectSum.finite_support {ι : Type v} {M : Type u_1} {S : Type u_2} [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ιS) (x : DirectSum ι fun (i : ι) => (A i)) :
(Function.support fun (i : ι) => (x i)).Finite
def DirectSum.map {ι : Type u_3} {α : ιType u_4} {β : ιType u_5} [(i : ι) → AddCommMonoid (α i)] [(i : ι) → AddCommMonoid (β i)] (f : (i : ι) → α i →+ β i) :
(DirectSum ι fun (i : ι) => α i) →+ DirectSum ι fun (i : ι) => β i

create a homomorphism from ⨁ i, α i to ⨁ i, β i by giving the component-wise map f.

Equations
@[simp]
theorem DirectSum.map_of {ι : Type u_3} {α : ιType u_4} {β : ιType u_5} [(i : ι) → AddCommMonoid (α i)] [(i : ι) → AddCommMonoid (β i)] (f : (i : ι) → α i →+ β i) [DecidableEq ι] (i : ι) (x : α i) :
(map f) ((of α i) x) = (of β i) ((f i) x)
@[simp]
theorem DirectSum.map_apply {ι : Type u_3} {α : ιType u_4} {β : ιType u_5} [(i : ι) → AddCommMonoid (α i)] [(i : ι) → AddCommMonoid (β i)] (f : (i : ι) → α i →+ β i) (i : ι) (x : DirectSum ι fun (i : ι) => α i) :
((map f) x) i = (f i) (x i)
@[simp]
theorem DirectSum.map_id {ι : Type u_3} {α : ιType u_4} [(i : ι) → AddCommMonoid (α i)] :
(map fun (i : ι) => AddMonoidHom.id (α i)) = AddMonoidHom.id (DirectSum ι fun (i : ι) => α i)
@[simp]
theorem DirectSum.map_comp {ι : Type u_3} {α : ιType u_4} {β : ιType u_5} [(i : ι) → AddCommMonoid (α i)] [(i : ι) → AddCommMonoid (β i)] (f : (i : ι) → α i →+ β i) {γ : ιType u_6} [(i : ι) → AddCommMonoid (γ i)] (g : (i : ι) → β i →+ γ i) :
(map fun (i : ι) => (g i).comp (f i)) = (map g).comp (map f)
theorem DirectSum.map_injective {ι : Type u_3} {α : ιType u_4} {β : ιType u_5} [(i : ι) → AddCommMonoid (α i)] [(i : ι) → AddCommMonoid (β i)] (f : (i : ι) → α i →+ β i) :
Function.Injective (map f) ∀ (i : ι), Function.Injective (f i)
theorem DirectSum.map_surjective {ι : Type u_3} {α : ιType u_4} {β : ιType u_5} [(i : ι) → AddCommMonoid (α i)] [(i : ι) → AddCommMonoid (β i)] (f : (i : ι) → α i →+ β i) :
Function.Surjective (map f) ∀ (i : ι), Function.Surjective (f i)
theorem DirectSum.map_eq_iff {ι : Type u_3} {α : ιType u_4} {β : ιType u_5} [(i : ι) → AddCommMonoid (α i)] [(i : ι) → AddCommMonoid (β i)] (f : (i : ι) → α i →+ β i) (x y : DirectSum ι fun (i : ι) => α i) :
(map f) x = (map f) y ∀ (i : ι), (f i) (x i) = (f i) (y i)
def DirectSum.addEquivProd {ι : Type u_1} [Fintype ι] (G : ιType u_2) [(i : ι) → AddCommMonoid (G i)] :
DirectSum ι G ≃+ ((i : ι) → G i)

The canonical isomorphism of a finite direct sum of additive commutative monoids and the corresponding finite product.

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