Documentation

Mathlib.Algebra.Module.Submodule.Basic

Submodules of a module #

In this file we define

Tags #

submodule, subspace, linear map

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

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.

  • carrier : Set M
  • add_mem' : ∀ {a b : M}, a self.carrierb self.carriera + b self.carrier
  • zero_mem' : 0 self.carrier
  • smul_mem' : ∀ (c : R) {x : M}, x self.carrierc x self.carrier

    The carrier set is closed under scalar multiplication.

Instances For
    @[reducible]
    abbrev Submodule.toSubMulAction {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (self : Submodule R M) :

    Reinterpret a Submodule as a SubMulAction.

    Equations
    • self.toSubMulAction = { carrier := self.carrier, smul_mem' := }
    Instances For
      instance Submodule.setLike {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
      Equations
      • Submodule.setLike = { coe := fun (s : Submodule R M) => s.carrier, coe_injective' := }
      Equations
      • =
      instance Submodule.smulMemClass {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
      Equations
      • =
      @[simp]
      theorem Submodule.mem_toAddSubmonoid {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (x : M) :
      x p.toAddSubmonoid x p
      @[simp]
      theorem Submodule.mem_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {S : AddSubmonoid M} {x : M} (h : ∀ (c : R) {x : M}, x S.carrierc x S.carrier) :
      x { toAddSubmonoid := S, smul_mem' := h } x S
      @[simp]
      theorem Submodule.coe_set_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (S : AddSubmonoid M) (h : ∀ (c : R) {x : M}, x S.carrierc x S.carrier) :
      { toAddSubmonoid := S, smul_mem' := h } = S
      @[simp]
      theorem Submodule.eta {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} (h : ∀ (c : R) {x : M}, x p.carrierc x p.carrier) :
      { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := h } = p
      @[simp]
      theorem Submodule.mk_le_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {S : AddSubmonoid M} {S' : AddSubmonoid M} (h : ∀ (c : R) {x : M}, x S.carrierc x S.carrier) (h' : ∀ (c : R) {x : M}, x S'.carrierc x S'.carrier) :
      { toAddSubmonoid := S, smul_mem' := h } { toAddSubmonoid := S', smul_mem' := h' } S S'
      theorem Submodule.ext_iff {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R M} :
      p = q ∀ (x : M), x p x q
      theorem Submodule.ext {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R M} (h : ∀ (x : M), x p x q) :
      p = q
      @[simp]
      theorem Submodule.carrier_inj {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R M} :
      p.carrier = q.carrier p = q
      def Submodule.copy {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (s : Set M) (hs : s = p) :

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

      Equations
      • p.copy s hs = { carrier := s, add_mem' := , zero_mem' := , smul_mem' := }
      Instances For
        @[simp]
        theorem Submodule.coe_copy {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (S : Submodule R M) (s : Set M) (hs : s = S) :
        (S.copy s hs) = s
        theorem Submodule.copy_eq {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (S : Submodule R M) (s : Set M) (hs : s = S) :
        S.copy s hs = S
        theorem Submodule.toAddSubmonoid_injective {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
        Function.Injective Submodule.toAddSubmonoid
        @[simp]
        theorem Submodule.toAddSubmonoid_eq {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R M} :
        p.toAddSubmonoid = q.toAddSubmonoid p = q
        theorem Submodule.toAddSubmonoid_strictMono {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
        StrictMono Submodule.toAddSubmonoid
        theorem Submodule.toAddSubmonoid_le {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R M} :
        p.toAddSubmonoid q.toAddSubmonoid p q
        theorem Submodule.toAddSubmonoid_mono {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
        Monotone Submodule.toAddSubmonoid
        @[simp]
        theorem Submodule.coe_toAddSubmonoid {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) :
        p.toAddSubmonoid = p
        theorem Submodule.toSubMulAction_injective {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
        Function.Injective Submodule.toSubMulAction
        theorem Submodule.toSubMulAction_eq {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R M} :
        p.toSubMulAction = q.toSubMulAction p = q
        theorem Submodule.toSubMulAction_strictMono {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
        StrictMono Submodule.toSubMulAction
        theorem Submodule.toSubMulAction_mono {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
        Monotone Submodule.toSubMulAction
        @[simp]
        theorem Submodule.coe_toSubMulAction {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) :
        p.toSubMulAction = p
        @[instance 75]
        instance SMulMemClass.toModule {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {A : Type u_1} [SetLike A M] [AddSubmonoidClass A M] [SMulMemClass A R M] (S' : A) :
        Module R S'

        A submodule of a Module is a Module.

        Equations
        def SMulMemClass.toModule' (S : Type u_2) (R' : Type u_3) (R : Type u_4) (A : Type u_5) [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] [SetLike S A] [AddSubmonoidClass S A] [SMulMemClass S R A] (s : S) :
        Module R' s

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

        Equations
        Instances For
          theorem Submodule.mem_carrier {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} :
          x p.carrier x p
          @[simp]
          theorem Submodule.zero_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) :
          0 p
          theorem Submodule.add_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} (h₁ : x p) (h₂ : y p) :
          x + y p
          theorem Submodule.smul_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} (r : R) (h : x p) :
          r x p
          theorem Submodule.smul_of_tower_mem {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [SMul S R] [SMul S M] [IsScalarTower S R M] (r : S) (h : x p) :
          r x p
          theorem Submodule.sum_mem {R : Type u} {M : Type v} {ι : Type w} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {t : Finset ι} {f : ιM} :
          (∀ ct, f c p)it, f i p
          theorem Submodule.sum_smul_mem {R : Type u} {M : Type v} {ι : Type w} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {t : Finset ι} {f : ιM} (r : ιR) (hyp : ct, f c p) :
          it, r i f i p
          @[simp]
          theorem Submodule.smul_mem_iff' {G : Type u''} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [Group G] [MulAction G M] [SMul G R] [IsScalarTower G R M] (g : G) :
          g x p x p
          instance Submodule.add {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) :
          Add p
          Equations
          • p.add = { add := fun (x y : p) => x + y, }
          instance Submodule.zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) :
          Zero p
          Equations
          • p.zero = { zero := 0, }
          instance Submodule.inhabited {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) :
          Equations
          • p.inhabited = { default := 0 }
          instance Submodule.smul {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [SMul S R] [SMul S M] [IsScalarTower S R M] :
          SMul S p
          Equations
          • p.smul = { smul := fun (c : S) (x : p) => c x, }
          instance Submodule.isScalarTower {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [SMul S R] [SMul S M] [IsScalarTower S R M] :
          IsScalarTower S R p
          Equations
          • =
          instance Submodule.isScalarTower' {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {S' : Type u_1} [SMul S R] [SMul S M] [SMul S' R] [SMul S' M] [SMul S S'] [IsScalarTower S' R M] [IsScalarTower S S' M] [IsScalarTower S R M] :
          IsScalarTower S S' p
          Equations
          • =
          instance Submodule.isCentralScalar {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [SMul S R] [SMul S M] [IsScalarTower S R M] [SMul Sᵐᵒᵖ R] [SMul Sᵐᵒᵖ M] [IsScalarTower Sᵐᵒᵖ R M] [IsCentralScalar S M] :
          Equations
          • =
          theorem Submodule.nonempty {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) :
          (↑p).Nonempty
          @[simp]
          theorem Submodule.mk_eq_zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} (h : x p) :
          x, h = 0 x = 0
          theorem Submodule.coe_eq_zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} {x : p} :
          x = 0 x = 0
          @[simp]
          theorem Submodule.coe_add {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : p) (y : p) :
          (x + y) = x + y
          @[simp]
          theorem Submodule.coe_zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} :
          0 = 0
          theorem Submodule.coe_smul {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (r : R) (x : p) :
          (r x) = r x
          @[simp]
          theorem Submodule.coe_smul_of_tower {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} [SMul S R] [SMul S M] [IsScalarTower S R M] (r : S) (x : p) :
          (r x) = r x
          theorem Submodule.coe_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : M) (hx : x p) :
          x, hx = x
          theorem Submodule.coe_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : p) :
          x p
          instance Submodule.addCommMonoid {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) :
          Equations
          instance Submodule.module' {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [Semiring S] [SMul S R] [Module S M] [IsScalarTower S R M] :
          Module S p
          Equations
          instance Submodule.module {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) :
          Module R p
          Equations
          • p.module = p.module'
          instance Submodule.noZeroSMulDivisors {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [NoZeroSMulDivisors R M] :
          Equations
          • =

          Additive actions by Submodules #

          These instances transfer the action by an element m : M of an R-module M written as m +ᵥ a onto the action by an element s : S of a submodule S : Submodule R M such that s +ᵥ a = (s : M) +ᵥ a. These instances work particularly well in conjunction with AddGroup.toAddAction, enabling s +ᵥ m as an alias for ↑s + m.

          instance Submodule.instVAddSubtypeMem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {α : Type u_1} [VAdd M α] :
          VAdd (↥p) α
          Equations
          • p.instVAddSubtypeMem = p.vadd
          instance Submodule.vaddCommClass {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {α : Type u_1} {β : Type u_2} [VAdd M β] [VAdd α β] [VAddCommClass M α β] :
          VAddCommClass (↥p) α β
          Equations
          • =
          instance Submodule.instFaithfulVAddSubtypeMem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {α : Type u_1} [VAdd M α] [FaithfulVAdd M α] :
          FaithfulVAdd (↥p) α
          Equations
          • =
          theorem Submodule.vadd_def {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} {α : Type u_1} [VAdd M α] (g : p) (m : α) :
          g +ᵥ m = g +ᵥ m
          instance Submodule.addSubgroupClass {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] :
          Equations
          • =
          theorem Submodule.neg_mem {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} (hx : x p) :
          -x p
          def Submodule.toAddSubgroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) :

          Reinterpret a submodule as an additive subgroup.

          Equations
          • p.toAddSubgroup = { toAddSubmonoid := p.toAddSubmonoid, neg_mem' := }
          Instances For
            @[simp]
            theorem Submodule.coe_toAddSubgroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) :
            p.toAddSubgroup = p
            @[simp]
            theorem Submodule.mem_toAddSubgroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} :
            x p.toAddSubgroup x p
            theorem Submodule.toAddSubgroup_injective {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} :
            Function.Injective Submodule.toAddSubgroup
            @[simp]
            theorem Submodule.toAddSubgroup_eq {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (p' : Submodule R M) :
            p.toAddSubgroup = p'.toAddSubgroup p = p'
            theorem Submodule.toAddSubgroup_strictMono {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} :
            StrictMono Submodule.toAddSubgroup
            theorem Submodule.toAddSubgroup_le {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (p' : Submodule R M) :
            p.toAddSubgroup p'.toAddSubgroup p p'
            theorem Submodule.toAddSubgroup_mono {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} :
            Monotone Submodule.toAddSubgroup
            theorem Submodule.sub_mem {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} :
            x py px - y p
            theorem Submodule.neg_mem_iff {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} :
            -x p x p
            theorem Submodule.add_mem_iff_left {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} :
            y p(x + y p x p)
            theorem Submodule.add_mem_iff_right {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} :
            x p(x + y p y p)
            theorem Submodule.coe_neg {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (x : p) :
            (-x) = -x
            theorem Submodule.coe_sub {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (x : p) (y : p) :
            (x - y) = x - y
            theorem Submodule.sub_mem_iff_left {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} (hy : y p) :
            x - y p x p
            theorem Submodule.sub_mem_iff_right {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} {y : M} (hx : x p) :
            x - y p y p
            instance Submodule.addCommGroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) :
            Equations
            theorem Submodule.neg_coe {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) :
            -p = p
            theorem Submodule.not_mem_of_ortho {R : Type u} {M : Type v} [Ring R] [IsDomain R] [AddCommGroup M] [Module R M] {x : M} {N : Submodule R M} (ortho : ∀ (c : R), yN, c x + y = 0c = 0) :
            xN
            theorem Submodule.ne_zero_of_ortho {R : Type u} {M : Type v} [Ring R] [IsDomain R] [AddCommGroup M] [Module R M] {x : M} {N : Submodule R M} (ortho : ∀ (c : R), yN, c x + y = 0c = 0) :
            x 0
            @[instance 75]
            instance SubmoduleClass.module' {S : Type u'} {R : Type u} {M : Type v} {T : Type u_1} [Semiring R] [AddCommMonoid M] [Semiring S] [Module R M] [SMul S R] [Module S M] [IsScalarTower S R M] [SetLike T M] [AddSubmonoidClass T M] [SMulMemClass T R M] (t : T) :
            Module S t
            Equations
            @[instance 75]
            instance SubmoduleClass.module {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [SetLike S M] [AddSubmonoidClass S M] [SMulMemClass S R M] (s : S) :
            Module R s
            Equations
            theorem Submodule.smul_mem_iff {S : Type u'} {R : Type u} {M : Type v} [DivisionSemiring S] [Semiring R] [AddCommMonoid M] [Module R M] [SMul S R] [Module S M] [IsScalarTower S R M] (p : Submodule R M) {s : S} {x : M} (s0 : s 0) :
            s x p x p
            @[reducible, inline]
            abbrev Subspace (R : Type u) (M : Type v) [DivisionRing R] [AddCommGroup M] [Module R M] :

            Subspace of a vector space. Defined to equal Submodule.

            Equations
            Instances For