Submodules of a module

In this file we define

• Submodule R M : a subset of a Module M that contains zero and is closed with respect to addition and scalar multiplication.

• Subspace k M : an abbreviation for Submodule assuming that k is a Field.

Tags

submodule, subspace, linear map

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class SubmoduleClass (S : Type u_1) (R : outParam (Type u_2)) (M : Type u_3) [inst : AddZeroClass M] [inst : SMul R M] [inst : SetLike S M] [inst : AddSubmonoidClass S M] extends SMulMemClass : Type

SubmoduleClass S R M says S is a type of submodules s ≤ M.

Note that only R is marked as outParam since M is already supplied by the SetLike class.

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structure Submodule (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] extends AddSubmonoid : Type v

• The carrier set is closed under scalar multiplication.

  smul_mem' : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid.toAddSubsemigroup.carrier → c • x ∈ toAddSubmonoid.toAddSubsemigroup.carrier

A submodule of a module is one which is closed under vector operations. This is a sufficient condition for the subset of vectors in the submodule to themselves form a module.

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abbrev Submodule.toSubMulAction {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (self : Submodule R M) : SubMulAction R M

Reinterpret a Submodule as an SubMulAction.

Equations

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

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instance Submodule.setLike {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : SetLike (Submodule R M) M

Equations

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

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instance Submodule.addSubmonoidClass {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : AddSubmonoidClass (Submodule R M) M

Equations

• @Submodule.addSubmonoidClass = @Submodule.addSubmonoidClass.proof_1

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instance Submodule.submoduleClass {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : SubmoduleClass (Submodule R M) R M

Equations

• Submodule.submoduleClass = SubmoduleClass.mk

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theorem Submodule.mem_toAddSubmonoid {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (p : Submodule R M) (x : M) : x ∈ p.toAddSubmonoid ↔ x ∈ p

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theorem Submodule.mem_mk {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {S : AddSubmonoid M} {x : M} (h : ∀ (c : R) {x : M}, x ∈ S.toAddSubsemigroup.carrier → c • x ∈ S.toAddSubsemigroup.carrier) : x ∈ { toAddSubmonoid := S, smul_mem' := h } ↔ x ∈ S

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theorem Submodule.coe_set_mk {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (S : AddSubmonoid M) (h : ∀ (c : R) {x : M}, x ∈ S.toAddSubsemigroup.carrier → c • x ∈ S.toAddSubsemigroup.carrier) : ↑{ toAddSubmonoid := S, smul_mem' := h } = ↑S

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theorem Submodule.mk_le_mk {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {S : AddSubmonoid M} {S' : AddSubmonoid M} (h : ∀ (c : R) {x : M}, x ∈ S.toAddSubsemigroup.carrier → c • x ∈ S.toAddSubsemigroup.carrier) (h' : ∀ (c : R) {x : M}, x ∈ S'.toAddSubsemigroup.carrier → c • x ∈ S'.toAddSubsemigroup.carrier) : { toAddSubmonoid := S, smul_mem' := h } ≤ { toAddSubmonoid := S', smul_mem' := h' } ↔ S ≤ S'

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theorem Submodule.ext {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {p : Submodule R M} {q : Submodule R M} (h : ∀ (x : M), x ∈ p ↔ x ∈ q) : p = q

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theorem Submodule.carrier_inj {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {p : Submodule R M} {q : Submodule R M} : p.toAddSubmonoid.toAddSubsemigroup.carrier = q.toAddSubmonoid.toAddSubsemigroup.carrier ↔ p = q

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def Submodule.copy {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (p : Submodule R M) (s : Set M) (hs : s = ↑p) : Submodule R M

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

Equations

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

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theorem Submodule.coe_copy {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (S : Submodule R M) (s : Set M) (hs : s = ↑S) : ↑(Submodule.copy S s hs) = s

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theorem Submodule.copy_eq {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (S : Submodule R M) (s : Set M) (hs : s = ↑S) : Submodule.copy S s hs = S

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theorem Submodule.toAddSubmonoid_injective {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : Function.Injective Submodule.toAddSubmonoid

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theorem Submodule.toAddSubmonoid_eq {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {p : Submodule R M} {q : Submodule R M} : p.toAddSubmonoid = q.toAddSubmonoid ↔ p = q

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theorem Submodule.toAddSubmonoid_strictMono {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : StrictMono Submodule.toAddSubmonoid

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theorem Submodule.toAddSubmonoid_le {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {p : Submodule R M} {q : Submodule R M} : p.toAddSubmonoid ≤ q.toAddSubmonoid ↔ p ≤ q

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theorem Submodule.toAddSubmonoid_mono {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : Monotone Submodule.toAddSubmonoid

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theorem Submodule.coe_toAddSubmonoid {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (p : Submodule R M) : ↑p.toAddSubmonoid = ↑p

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theorem Submodule.toSubMulAction_injective {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : Function.Injective Submodule.toSubMulAction

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theorem Submodule.toSubMulAction_eq {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {p : Submodule R M} {q : Submodule R M} : Submodule.toSubMulAction p = Submodule.toSubMulAction q ↔ p = q

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theorem Submodule.toSubMulAction_strictMono {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : StrictMono Submodule.toSubMulAction

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theorem Submodule.toSubMulAction_mono {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] : Monotone Submodule.toSubMulAction

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theorem Submodule.coe_toSubMulAction {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (p : Submodule R M) : ↑(Submodule.toSubMulAction p) = ↑p

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instance SubmoduleClass.toModule {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {A : Type u_1} [inst : SetLike A M] [inst : AddSubmonoidClass A M] [inst : SubmoduleClass A R M] (S' : A) : Module R { x // x ∈ S' }

A submodule of a Module is a Module.

Equations

• SubmoduleClass.toModule S' = Function.Injective.module R (AddSubmonoidClass.Subtype S') (_ : Function.Injective fun a => ↑a) (_ : ∀ (r : R) (x : { x // x ∈ S' }), ↑(r • x) = r • ↑x)

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def SubmoduleClass.subtype {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {A : Type u_1} [inst : SetLike A M] [inst : AddSubmonoidClass A M] [inst : SubmoduleClass A R M] (S' : A) : { x // x ∈ S' } →ₗ[R] M

The natural R-linear map from a submodule of an R-module M to M.

Equations

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

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theorem SubmoduleClass.coeSubtype {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] {A : Type u_1} [inst : SetLike A M] [inst : AddSubmonoidClass A M] [inst : SubmoduleClass A R M] (S' : A) : ↑(SubmoduleClass.subtype S') = Subtype.val

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theorem Submodule.mem_carrier {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} : x ∈ p.toAddSubmonoid.toAddSubsemigroup.carrier ↔ x ∈ ↑p

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theorem Submodule.zero_mem {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : 0 ∈ p

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theorem Submodule.add_mem {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} (h₁ : x ∈ p) (h₂ : y ∈ p) : x + y ∈ p

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theorem Submodule.smul_mem {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} (r : R) (h : x ∈ p) : r • x ∈ p

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theorem Submodule.smul_of_tower_mem {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [inst : SMul S R] [inst : SMul S M] [inst : IsScalarTower S R M] (r : S) (h : x ∈ p) : r • x ∈ p

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theorem Submodule.sum_mem {R : Type u} {M : Type v} {ι : Type w} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {t : Finset ι} {f : ι → M} : (∀ (c : ι), c ∈ t → f c ∈ p) → (Finset.sum t fun i => f i) ∈ p

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theorem Submodule.sum_smul_mem {R : Type u} {M : Type v} {ι : Type w} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {t : Finset ι} {f : ι → M} (r : ι → R) (hyp : ∀ (c : ι), c ∈ t → f c ∈ p) : (Finset.sum t fun i => r i • f i) ∈ p

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theorem Submodule.smul_mem_iff' {G : Type u''} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [inst : Group G] [inst : MulAction G M] [inst : SMul G R] [inst : IsScalarTower G R M] (g : G) : g • x ∈ p ↔ x ∈ p

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instance Submodule.add {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Add { x // x ∈ p }

Equations

• Submodule.add p = { add := fun x y => { val := ↑x + ↑y, property := (_ : ↑x + ↑y ∈ p) } }

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instance Submodule.zero {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Zero { x // x ∈ p }

Equations

• Submodule.zero p = { zero := { val := 0, property := (_ : 0 ∈ p) } }

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instance Submodule.inhabited {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Inhabited { x // x ∈ p }

Equations

• Submodule.inhabited p = { default := 0 }

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instance Submodule.smul {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [inst : SMul S R] [inst : SMul S M] [inst : IsScalarTower S R M] : SMul S { x // x ∈ p }

Equations

• Submodule.smul p = { smul := fun c x => { val := c • ↑x, property := (_ : c • ↑x ∈ p) } }

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instance Submodule.isScalarTower {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [inst : SMul S R] [inst : SMul S M] [inst : IsScalarTower S R M] : IsScalarTower S R { x // x ∈ p }

Equations

• @Submodule.isScalarTower = @Submodule.isScalarTower.proof_1

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instance Submodule.isScalarTower' {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {S' : Type u_1} [inst : SMul S R] [inst : SMul S M] [inst : SMul S' R] [inst : SMul S' M] [inst : SMul S S'] [inst : IsScalarTower S' R M] [inst : IsScalarTower S S' M] [inst : IsScalarTower S R M] : IsScalarTower S S' { x // x ∈ p }

Equations

• @Submodule.isScalarTower' = @Submodule.isScalarTower'.proof_1

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instance Submodule.isCentralScalar {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [inst : SMul S R] [inst : SMul S M] [inst : IsScalarTower S R M] [inst : SMul Sᵐᵒᵖ R] [inst : SMul Sᵐᵒᵖ M] [inst : IsScalarTower Sᵐᵒᵖ R M] [inst : IsCentralScalar S M] : IsCentralScalar S { x // x ∈ p }

Equations

• @Submodule.isCentralScalar = @Submodule.isCentralScalar.proof_1

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theorem Submodule.nonempty {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Set.Nonempty ↑p

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theorem Submodule.mk_eq_zero {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} (h : x ∈ p) : { val := x, property := h } = 0 ↔ x = 0

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theorem Submodule.coe_eq_zero {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} {x : { x // x ∈ p }} : ↑x = 0 ↔ x = 0

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theorem Submodule.coe_add {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : { x // x ∈ p }) (y : { x // x ∈ p }) : ↑(x + y) = ↑x + ↑y

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theorem Submodule.coe_zero {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} : ↑0 = 0

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theorem Submodule.coe_smul {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (r : R) (x : { x // x ∈ p }) : ↑(r • x) = r • ↑x

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theorem Submodule.coe_smul_of_tower {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} [inst : SMul S R] [inst : SMul S M] [inst : IsScalarTower S R M] (r : S) (x : { x // x ∈ p }) : ↑(r • x) = r • ↑x

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theorem Submodule.coe_mk {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : M) (hx : x ∈ p) : ↑{ val := x, property := hx } = x

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theorem Submodule.coe_mem {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : { x // x ∈ p }) : ↑x ∈ p

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instance Submodule.addCommMonoid {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : AddCommMonoid { x // x ∈ p }

Equations

• Submodule.addCommMonoid p = let src := AddSubmonoid.toAddCommMonoid p.toAddSubmonoid; AddCommMonoid.mk (_ : ∀ (a b : { x // x ∈ p.toAddSubmonoid }), a + b = b + a)

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instance Submodule.module' {S : Type u'} {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [inst : Semiring S] [inst : SMul S R] [inst : Module S M] [inst : IsScalarTower S R M] : Module S { x // x ∈ p }

Equations

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

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instance Submodule.module {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Module R { x // x ∈ p }

Equations

• Submodule.module p = Submodule.module' p

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instance Submodule.noZeroSMulDivisors {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [inst : NoZeroSMulDivisors R M] : NoZeroSMulDivisors R { x // x ∈ p }

Equations

• @Submodule.noZeroSMulDivisors = @Submodule.noZeroSMulDivisors.proof_1

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def Submodule.subtype {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : { x // x ∈ p } →ₗ[R] M

Embedding of a submodule p to the ambient space M.

Equations

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

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theorem Submodule.subtype_apply {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) (x : { x // x ∈ p }) : ↑(Submodule.subtype p) x = ↑x

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theorem Submodule.coeSubtype {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : ↑(Submodule.subtype p) = Subtype.val

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theorem Submodule.injective_subtype {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Function.Injective ↑(Submodule.subtype p)

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theorem Submodule.coe_sum {R : Type u} {M : Type v} {ι : Type w} [inst : Semiring R] [inst : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) (x : ι → { x // x ∈ p }) (s : Finset ι) : ↑(Finset.sum s fun i => x i) = Finset.sum s fun i => ↑(x i)

Note the AddSubmonoid version of this lemma is called AddSubmonoid.coe_finset_sum.

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def Submodule.restrictScalars (S : Type u') {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (V : Submodule R M) : Submodule S M

V.restrict_scalars S is the S-submodule of the S-module given by restriction of scalars, corresponding to V, an R-submodule of the original R-module.

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

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theorem Submodule.coe_restrictScalars (S : Type u') {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (V : Submodule R M) : ↑(Submodule.restrictScalars S V) = ↑V

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theorem Submodule.restrictScalars_mem (S : Type u') {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (V : Submodule R M) (m : M) : m ∈ Submodule.restrictScalars S V ↔ m ∈ V

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theorem Submodule.restrictScalars_self {R : Type u} {M : Type v} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (V : Submodule R M) : Submodule.restrictScalars R V = V

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theorem Submodule.restrictScalars_injective (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] : Function.Injective (Submodule.restrictScalars S)

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theorem Submodule.restrictScalars_inj (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] {V₁ : Submodule R M} {V₂ : Submodule R M} : Submodule.restrictScalars S V₁ = Submodule.restrictScalars S V₂ ↔ V₁ = V₂

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instance Submodule.restrictScalars.origModule (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (p : Submodule R M) : Module R { x // x ∈ Submodule.restrictScalars S p }

Even though p.restrictScalars S has type Submodule S M, it is still an R-module.

Equations

• Submodule.restrictScalars.origModule S R M p = inferInstance

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instance Submodule.restrictScalars.isScalarTower (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (p : Submodule R M) : IsScalarTower S R { x // x ∈ Submodule.restrictScalars S p }

Equations

• Submodule.restrictScalars.isScalarTower = Submodule.restrictScalars.isScalarTower.proof_1

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theorem Submodule.restrictScalarsEmbedding_apply (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (V : Submodule R M) : ↑(Submodule.restrictScalarsEmbedding S R M).toEmbedding V = Submodule.restrictScalars S V

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def Submodule.restrictScalarsEmbedding (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] : Submodule R M ↪o Submodule S M

restrictScalars S is an embedding of the lattice of R-submodules into the lattice of S-submodules.

Equations

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

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theorem Submodule.restrictScalarsEquiv_apply (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (p : Submodule R M) (a : { x // x ∈ Submodule.restrictScalars S p }) : ↑(Submodule.restrictScalarsEquiv S R M p) a = a

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@[simp]

theorem Submodule.restrictScalarsEquiv_symm_apply (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (p : Submodule R M) (a : { x // x ∈ p }) : ↑(LinearEquiv.symm (Submodule.restrictScalarsEquiv S R M p)) a = a

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def Submodule.restrictScalarsEquiv (S : Type u') (R : Type u) (M : Type v) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Semiring S] [inst : Module S M] [inst : Module R M] [inst : SMul S R] [inst : IsScalarTower S R M] (p : Submodule R M) : { x // x ∈ Submodule.restrictScalars S p } ≃ₗ[R] { x // x ∈ p }

Turning p : Submodule R M into an S-submodule gives the same module structure as turning it into a type and adding a module structure.

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instance Submodule.addSubgroupClass {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] [inst : Module R M] : AddSubgroupClass (Submodule R M) M

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• @Submodule.addSubgroupClass = @Submodule.addSubgroupClass.proof_1

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

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def Submodule.toAddSubgroup {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) : AddSubgroup M

Reinterpret a submodule as an additive subgroup.

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theorem Submodule.coe_toAddSubgroup {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) : ↑(Submodule.toAddSubgroup p) = ↑p

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theorem Submodule.mem_toAddSubgroup {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} : x ∈ Submodule.toAddSubgroup p ↔ x ∈ p

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theorem Submodule.toAddSubgroup_injective {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} : Function.Injective Submodule.toAddSubgroup

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theorem Submodule.toAddSubgroup_eq {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (p' : Submodule R M) : Submodule.toAddSubgroup p = Submodule.toAddSubgroup p' ↔ p = p'

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theorem Submodule.toAddSubgroup_strictMono {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} : StrictMono Submodule.toAddSubgroup

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theorem Submodule.toAddSubgroup_le {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (p' : Submodule R M) : Submodule.toAddSubgroup p ≤ Submodule.toAddSubgroup p' ↔ p ≤ p'

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theorem Submodule.toAddSubgroup_mono {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} : Monotone Submodule.toAddSubgroup

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theorem Submodule.sub_mem {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} : x ∈ p → y ∈ p → x - y ∈ p

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

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theorem Submodule.add_mem_iff_left {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} : y ∈ p → (x + y ∈ p ↔ x ∈ p)

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theorem Submodule.add_mem_iff_right {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} : x ∈ p → (x + y ∈ p ↔ y ∈ p)

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theorem Submodule.coe_neg {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (x : { x // x ∈ p }) : ↑(-x) = -↑x

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theorem Submodule.coe_sub {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (x : { x // x ∈ p }) (y : { x // x ∈ p }) : ↑(x - y) = ↑x - ↑y

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theorem Submodule.sub_mem_iff_left {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} (hy : y ∈ p) : x - y ∈ p ↔ x ∈ p

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theorem Submodule.sub_mem_iff_right {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} (hx : x ∈ p) : x - y ∈ p ↔ y ∈ p

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instance Submodule.addCommGroup {R : Type u} {M : Type v} [inst : Ring R] [inst : AddCommGroup M] {module_M : Module R M} (p : Submodule R M) : AddCommGroup { x // x ∈ p }

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• Submodule.addCommGroup p = let src := AddSubgroup.toAddCommGroup (Submodule.toAddSubgroup p); AddCommGroup.mk (_ : ∀ (a b : { x // x ∈ Submodule.toAddSubgroup p }), a + b = b + a)

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theorem Submodule.not_mem_of_ortho {R : Type u} {M : Type v} [inst : Ring R] [inst : IsDomain R] [inst : AddCommGroup M] [inst : Module R M] {x : M} {N : Submodule R M} (ortho : ∀ (c : R) (y : M), y ∈ N → c • x + y = 0 → c = 0) : ¬x ∈ N

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theorem Submodule.ne_zero_of_ortho {R : Type u} {M : Type v} [inst : Ring R] [inst : IsDomain R] [inst : AddCommGroup M] [inst : Module R M] {x : M} {N : Submodule R M} (ortho : ∀ (c : R) (y : M), y ∈ N → c • x + y = 0 → c = 0) : x ≠ 0

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instance Submodule.toOrderedAddCommMonoid {R : Type u} [inst : Semiring R] {M : Type u_1} [inst : OrderedAddCommMonoid M] [inst : Module R M] (S : Submodule R M) : OrderedAddCommMonoid { x // x ∈ S }

A submodule of an OrderedAddCommMonoid is an OrderedAddCommMonoid.

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instance Submodule.toLinearOrderedAddCommMonoid {R : Type u} [inst : Semiring R] {M : Type u_1} [inst : LinearOrderedAddCommMonoid M] [inst : Module R M] (S : Submodule R M) : LinearOrderedAddCommMonoid { x // x ∈ S }

A submodule of a LinearOrderedAddCommMonoid is a LinearOrderedAddCommMonoid.

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instance Submodule.toOrderedCancelAddCommMonoid {R : Type u} [inst : Semiring R] {M : Type u_1} [inst : OrderedCancelAddCommMonoid M] [inst : Module R M] (S : Submodule R M) : OrderedCancelAddCommMonoid { x // x ∈ S }

A submodule of an OrderedCancelAddCommMonoid is an OrderedCancelAddCommMonoid.

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instance Submodule.toLinearOrderedCancelAddCommMonoid {R : Type u} [inst : Semiring R] {M : Type u_1} [inst : LinearOrderedCancelAddCommMonoid M] [inst : Module R M] (S : Submodule R M) : LinearOrderedCancelAddCommMonoid { x // x ∈ S }

A submodule of a LinearOrderedCancelAddCommMonoid is a LinearOrderedCancelAddCommMonoid.

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instance Submodule.toOrderedAddCommGroup {R : Type u} [inst : Ring R] {M : Type u_1} [inst : OrderedAddCommGroup M] [inst : Module R M] (S : Submodule R M) : OrderedAddCommGroup { x // x ∈ S }

A submodule of an OrderedAddCommGroup is an OrderedAddCommGroup.

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instance Submodule.toLinearOrderedAddCommGroup {R : Type u} [inst : Ring R] {M : Type u_1} [inst : LinearOrderedAddCommGroup M] [inst : Module R M] (S : Submodule R M) : LinearOrderedAddCommGroup { x // x ∈ S }

A submodule of a LinearOrderedAddCommGroup is a LinearOrderedAddCommGroup.

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theorem Submodule.smul_mem_iff {S : Type u'} {R : Type u} {M : Type v} [inst : DivisionRing S] [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] [inst : SMul S R] [inst : Module S M] [inst : IsScalarTower S R M] (p : Submodule R M) {s : S} {x : M} (s0 : s ≠ 0) : s • x ∈ p ↔ x ∈ p

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@[inline]

abbrev Subspace (R : Type u) (M : Type v) [inst : DivisionRing R] [inst : AddCommGroup M] [inst : Module R M] : Type v

Subspace of a vector space. Defined to equal Submodule.

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• Subspace R M = Submodule R M