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Mathlib.Algebra.Module.Equiv.Defs

(Semi)linear equivalences #

In this file we define

Implementation notes #

To ensure that composition works smoothly for semilinear equivalences, we use the typeclasses RingHomCompTriple, RingHomInvPair and RingHomSurjective from Algebra/Ring/CompTypeclasses.

The group structure on automorphisms, LinearEquiv.automorphismGroup, is provided elsewhere.

TODO #

Tags #

linear equiv, linear equivalences, linear isomorphism, linear isomorphic

structure LinearEquiv {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_19) (M₂ : Type u_20) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends LinearMap :
Type (max u_19 u_20)

A linear equivalence is an invertible linear map.

  • toFun : MM₂
  • map_add' : ∀ (x y : M), (↑self).toFun (x + y) = (↑self).toFun x + (↑self).toFun y
  • map_smul' : ∀ (m : R) (x : M), (↑self).toFun (m x) = σ m (↑self).toFun x
  • invFun : M₂M

    The backwards directed function underlying a linear equivalence.

  • left_inv : Function.LeftInverse self.invFun (↑self).toFun

    LinearEquiv.invFun is a left inverse to the linear equivalence's underlying function.

  • right_inv : Function.RightInverse self.invFun (↑self).toFun

    LinearEquiv.invFun is a right inverse to the linear equivalence's underlying function.

Instances For
    @[reducible]
    abbrev LinearEquiv.toAddEquiv {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_19} {M₂ : Type u_20} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃ₛₗ[σ] M₂) :
    M ≃+ M₂

    The additive equivalence of types underlying a linear equivalence.

    Equations
    • self.toAddEquiv = { toFun := (↑self).toFun, invFun := self.invFun, left_inv := , right_inv := , map_add' := }
    Instances For
      theorem LinearEquiv.right_inv {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_19} {M₂ : Type u_20} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃ₛₗ[σ] M₂) :
      Function.RightInverse self.invFun (↑self).toFun

      LinearEquiv.invFun is a right inverse to the linear equivalence's underlying function.

      theorem LinearEquiv.left_inv {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_19} {M₂ : Type u_20} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃ₛₗ[σ] M₂) :
      Function.LeftInverse self.invFun (↑self).toFun

      LinearEquiv.invFun is a left inverse to the linear equivalence's underlying function.

      The notation M ≃ₛₗ[σ] M₂ denotes the type of linear equivalences between M and M₂ over a ring homomorphism σ.

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

        The notation M ≃ₗ [R] M₂ denotes the type of linear equivalences between M and M₂ over a plain linear map M →ₗ M₂.

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For
          class SemilinearEquivClass (F : Type u_17) {R : outParam (Type u_18)} {S : outParam (Type u_19)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam (Type u_20)) (M₂ : outParam (Type u_21)) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] [EquivLike F M M₂] extends AddEquivClass :

          SemilinearEquivClass F σ M M₂ asserts F is a type of bundled σ-semilinear equivs M → M₂.

          See also LinearEquivClass F R M M₂ for the case where σ is the identity map on R.

          A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

          • map_add : ∀ (f : F) (a b : M), f (a + b) = f a + f b
          • map_smulₛₗ : ∀ (f : F) (r : R) (x : M), f (r x) = σ r f x

            Applying a semilinear equivalence f over σ to r • x equals σ r • f x.

          Instances
            theorem SemilinearEquivClass.map_smulₛₗ {F : Type u_17} {R : outParam (Type u_18)} {S : outParam (Type u_19)} :
            ∀ {inst : Semiring R} {inst_1 : Semiring S} {σ : outParam (R →+* S)} {σ' : outParam (S →+* R)} {inst_2 : RingHomInvPair σ σ'} {inst_3 : RingHomInvPair σ' σ} {M : outParam (Type u_20)} {M₂ : outParam (Type u_21)} {inst_4 : AddCommMonoid M} {inst_5 : AddCommMonoid M₂} {inst_6 : Module R M} {inst_7 : Module S M₂} {inst_8 : EquivLike F M M₂} [self : SemilinearEquivClass F σ M M₂] (f : F) (r : R) (x : M), f (r x) = σ r f x

            Applying a semilinear equivalence f over σ to r • x equals σ r • f x.

            @[reducible, inline]
            abbrev LinearEquivClass (F : Type u_17) (R : outParam (Type u_18)) (M : outParam (Type u_19)) (M₂ : outParam (Type u_20)) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [EquivLike F M M₂] :

            LinearEquivClass F R M M₂ asserts F is a type of bundled R-linear equivs M → M₂. This is an abbreviation for SemilinearEquivClass F (RingHom.id R) M M₂.

            Equations
            Instances For
              @[instance 100]
              instance SemilinearEquivClass.instSemilinearMapClass {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} (F : Type u_17) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [EquivLike F M M₂] [s : SemilinearEquivClass F σ M M₂] :
              SemilinearMapClass F σ M M₂
              Equations
              • =
              def SemilinearEquivClass.semilinearEquiv {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} {F : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [EquivLike F M M₂] [SemilinearEquivClass F σ M M₂] (f : F) :
              M ≃ₛₗ[σ] M₂

              Reinterpret an element of a type of semilinear equivalences as a semilinear equivalence.

              Equations
              • f = { toFun := (↑f).toFun, map_add' := , map_smul' := , invFun := (↑f).invFun, left_inv := , right_inv := }
              Instances For
                instance SemilinearEquivClass.instCoeToSemilinearEquiv {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} {F : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [EquivLike F M M₂] [SemilinearEquivClass F σ M M₂] :
                CoeHead F (M ≃ₛₗ[σ] M₂)

                Reinterpret an element of a type of semilinear equivalences as a semilinear equivalence.

                Equations
                • SemilinearEquivClass.instCoeToSemilinearEquiv = { coe := fun (f : F) => f }
                instance LinearEquiv.instCoeLinearMap {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                Coe (M ≃ₛₗ[σ] M₂) (M →ₛₗ[σ] M₂)
                Equations
                • LinearEquiv.instCoeLinearMap = { coe := LinearEquiv.toLinearMap }
                def LinearEquiv.toEquiv {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                (M ≃ₛₗ[σ] M₂)M M₂

                The equivalence of types underlying a linear equivalence.

                Equations
                • f.toEquiv = f.toAddEquiv.toEquiv
                Instances For
                  theorem LinearEquiv.toEquiv_injective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                  Function.Injective LinearEquiv.toEquiv
                  @[simp]
                  theorem LinearEquiv.toEquiv_inj {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} :
                  e₁.toEquiv = e₂.toEquiv e₁ = e₂
                  theorem LinearEquiv.toLinearMap_injective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                  Function.Injective LinearEquiv.toLinearMap
                  @[simp]
                  theorem LinearEquiv.toLinearMap_inj {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} :
                  e₁ = e₂ e₁ = e₂
                  instance LinearEquiv.instEquivLike {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                  EquivLike (M ≃ₛₗ[σ] M₂) M M₂
                  Equations
                  • LinearEquiv.instEquivLike = { coe := fun (e : M ≃ₛₗ[σ] M₂) => (↑e).toFun, inv := LinearEquiv.invFun, left_inv := , right_inv := , coe_injective' := }
                  instance LinearEquiv.instFunLike {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                  FunLike (M ≃ₛₗ[σ] M₂) M M₂

                  Helper instance for when inference gets stuck on following the normal chain EquivLikeFunLike.

                  TODO: this instance doesn't appear to be necessary: remove it (after benchmarking?)

                  Equations
                  • LinearEquiv.instFunLike = { coe := DFunLike.coe, coe_injective' := }
                  instance LinearEquiv.instSemilinearEquivClass {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                  SemilinearEquivClass (M ≃ₛₗ[σ] M₂) σ M M₂
                  Equations
                  • =
                  @[simp]
                  theorem LinearEquiv.coe_mk {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {to_fun : MM₂} {inv_fun : M₂M} {map_add : ∀ (x y : M), to_fun (x + y) = to_fun x + to_fun y} {map_smul : ∀ (m : R) (x : M), { toFun := to_fun, map_add' := map_add }.toFun (m x) = σ m { toFun := to_fun, map_add' := map_add }.toFun x} {left_inv : Function.LeftInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun} {right_inv : Function.RightInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun} :
                  { toFun := to_fun, map_add' := map_add, map_smul' := map_smul, invFun := inv_fun, left_inv := left_inv, right_inv := right_inv } = to_fun
                  theorem LinearEquiv.coe_injective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                  @[simp]
                  theorem LinearEquiv.coe_coe {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                  e = e
                  @[simp]
                  theorem LinearEquiv.coe_toEquiv {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                  e.toEquiv = e
                  @[simp]
                  theorem LinearEquiv.coe_toLinearMap {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                  e = e
                  theorem LinearEquiv.toFun_eq_coe {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                  (↑e).toFun = e
                  theorem LinearEquiv.ext_iff {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} :
                  e = e' ∀ (x : M), e x = e' x
                  theorem LinearEquiv.ext {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} (h : ∀ (x : M), e x = e' x) :
                  e = e'
                  theorem LinearEquiv.congr_arg {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e : M ≃ₛₗ[σ] M₂} {x : M} {x' : M} :
                  x = x'e x = e x'
                  theorem LinearEquiv.congr_fun {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} (h : e = e') (x : M) :
                  e x = e' x
                  def LinearEquiv.refl (R : Type u_1) (M : Type u_8) [Semiring R] [AddCommMonoid M] [Module R M] :

                  The identity map is a linear equivalence.

                  Equations
                  Instances For
                    @[simp]
                    theorem LinearEquiv.refl_apply {R : Type u_1} {M : Type u_8} [Semiring R] [AddCommMonoid M] [Module R M] (x : M) :
                    def LinearEquiv.symm {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                    M₂ ≃ₛₗ[σ'] M

                    Linear equivalences are symmetric.

                    Equations
                    • e.symm = { toFun := ((↑e).inverse e.invFun ), map_add' := , map_smul' := , invFun := e.toEquiv.symm.invFun, left_inv := , right_inv := }
                    Instances For
                      def LinearEquiv.Simps.apply {R : Type u_18} {S : Type u_19} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_20} {M₂ : Type u_21} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (e : M ≃ₛₗ[σ] M₂) :
                      MM₂

                      See Note [custom simps projection]

                      Equations
                      Instances For
                        def LinearEquiv.Simps.symm_apply {R : Type u_18} {S : Type u_19} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_20} {M₂ : Type u_21} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (e : M ≃ₛₗ[σ] M₂) :
                        M₂M

                        See Note [custom simps projection]

                        Equations
                        Instances For
                          @[simp]
                          theorem LinearEquiv.invFun_eq_symm {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                          e.invFun = e.symm
                          @[simp]
                          theorem LinearEquiv.coe_toEquiv_symm {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                          e.toEquiv.symm = e.symm
                          def LinearEquiv.trans {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] (e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂) (e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃) :
                          M₁ ≃ₛₗ[σ₁₃] M₃

                          Linear equivalences are transitive.

                          Equations
                          • e₁₂.trans e₂₃ = { toLinearMap := (↑e₂₃).comp e₁₂, invFun := (e₁₂.toEquiv.trans e₂₃.toEquiv).invFun, left_inv := , right_inv := }
                          Instances For

                            The notation e₁ ≪≫ₗ e₂ denotes the composition of the linear equivalences e₁ and e₂.

                            Equations
                            Instances For

                              Pretty printer defined by notation3 command.

                              Equations
                              • One or more equations did not get rendered due to their size.
                              Instances For
                                @[simp]
                                theorem LinearEquiv.coe_toAddEquiv {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                e.toAddEquiv = e
                                theorem LinearEquiv.toAddMonoidHom_commutes {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                (↑e).toAddMonoidHom = e.toAddEquiv.toAddMonoidHom

                                The two paths coercion can take to an AddMonoidHom are equivalent

                                @[simp]
                                theorem LinearEquiv.trans_apply {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} (c : M₁) :
                                (e₁₂.trans e₂₃) c = e₂₃ (e₁₂ c)
                                theorem LinearEquiv.coe_trans {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} :
                                (e₁₂.trans e₂₃) = (↑e₂₃).comp e₁₂
                                @[simp]
                                theorem LinearEquiv.apply_symm_apply {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (c : M₂) :
                                e (e.symm c) = c
                                @[simp]
                                theorem LinearEquiv.symm_apply_apply {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (b : M) :
                                e.symm (e b) = b
                                @[simp]
                                theorem LinearEquiv.trans_symm {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} :
                                (e₁₂.trans e₂₃).symm = e₂₃.symm.trans e₁₂.symm
                                theorem LinearEquiv.symm_trans_apply {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} (c : M₃) :
                                (e₁₂.trans e₂₃).symm c = e₁₂.symm (e₂₃.symm c)
                                @[simp]
                                theorem LinearEquiv.trans_refl {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                e.trans (LinearEquiv.refl S M₂) = e
                                @[simp]
                                theorem LinearEquiv.refl_trans {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                (LinearEquiv.refl R M).trans e = e
                                theorem LinearEquiv.symm_apply_eq {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) {x : M₂} {y : M} :
                                e.symm x = y x = e y
                                theorem LinearEquiv.eq_symm_apply {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) {x : M₂} {y : M} :
                                y = e.symm x e y = x
                                theorem LinearEquiv.eq_comp_symm {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_9} {M₂ : Type u_10} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {α : Type u_18} (f : M₂α) (g : M₁α) :
                                f = g e₁₂.symm f e₁₂ = g
                                theorem LinearEquiv.comp_symm_eq {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_9} {M₂ : Type u_10} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {α : Type u_18} (f : M₂α) (g : M₁α) :
                                g e₁₂.symm = f g = f e₁₂
                                theorem LinearEquiv.eq_symm_comp {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_9} {M₂ : Type u_10} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {α : Type u_18} (f : αM₁) (g : αM₂) :
                                f = e₁₂.symm g e₁₂ f = g
                                theorem LinearEquiv.symm_comp_eq {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_9} {M₂ : Type u_10} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {α : Type u_18} (f : αM₁) (g : αM₂) :
                                e₁₂.symm g = f g = e₁₂ f
                                theorem LinearEquiv.eq_comp_toLinearMap_symm {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₂₁ σ₁₃ σ₂₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃) :
                                f = g.comp e₁₂.symm f.comp e₁₂ = g
                                theorem LinearEquiv.comp_toLinearMap_symm_eq {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₂₁ σ₁₃ σ₂₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃) :
                                g.comp e₁₂.symm = f g = f.comp e₁₂
                                theorem LinearEquiv.eq_toLinearMap_symm_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₃₁ σ₁₂ σ₃₂] (f : M₃ →ₛₗ[σ₃₁] M₁) (g : M₃ →ₛₗ[σ₃₂] M₂) :
                                f = (↑e₁₂.symm).comp g (↑e₁₂).comp f = g
                                theorem LinearEquiv.toLinearMap_symm_comp_eq {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_9} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₃₁ σ₁₂ σ₃₂] (f : M₃ →ₛₗ[σ₃₁] M₁) (g : M₃ →ₛₗ[σ₃₂] M₂) :
                                (↑e₁₂.symm).comp g = f g = (↑e₁₂).comp f
                                @[simp]
                                theorem LinearEquiv.refl_symm {R : Type u_1} {M : Type u_8} [Semiring R] [AddCommMonoid M] [Module R M] :
                                @[simp]
                                theorem LinearEquiv.self_trans_symm {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_9} {M₂ : Type u_10} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} (f : M₁ ≃ₛₗ[σ₁₂] M₂) :
                                f.trans f.symm = LinearEquiv.refl R₁ M₁
                                @[simp]
                                theorem LinearEquiv.symm_trans_self {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_9} {M₂ : Type u_10} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} (f : M₁ ≃ₛₗ[σ₁₂] M₂) :
                                f.symm.trans f = LinearEquiv.refl R₂ M₂
                                @[simp]
                                theorem LinearEquiv.refl_toLinearMap {R : Type u_1} {M : Type u_8} [Semiring R] [AddCommMonoid M] [Module R M] :
                                (LinearEquiv.refl R M) = LinearMap.id
                                @[simp]
                                theorem LinearEquiv.comp_coe {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M ≃ₗ[R] M₂) (f' : M₂ ≃ₗ[R] M₃) :
                                f' ∘ₗ f = (f ≪≫ₗ f')
                                @[simp]
                                theorem LinearEquiv.mk_coe {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (f : M₂M) (h₁ : Function.LeftInverse f (↑e).toFun) (h₂ : Function.RightInverse f (↑e).toFun) :
                                { toLinearMap := e, invFun := f, left_inv := h₁, right_inv := h₂ } = e
                                theorem LinearEquiv.map_add {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (a : M) (b : M) :
                                e (a + b) = e a + e b
                                theorem LinearEquiv.map_zero {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                e 0 = 0
                                theorem LinearEquiv.map_smulₛₗ {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (c : R) (x : M) :
                                e (c x) = σ c e x
                                theorem LinearEquiv.map_smul {R₁ : Type u_2} {N₁ : Type u_12} {N₂ : Type u_13} [Semiring R₁] [AddCommMonoid N₁] [AddCommMonoid N₂] {module_N₁ : Module R₁ N₁} {module_N₂ : Module R₁ N₂} (e : N₁ ≃ₗ[R₁] N₂) (c : R₁) (x : N₁) :
                                e (c x) = c e x
                                theorem LinearEquiv.map_eq_zero_iff {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) {x : M} :
                                e x = 0 x = 0
                                theorem LinearEquiv.map_ne_zero_iff {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) {x : M} :
                                e x 0 x 0
                                @[simp]
                                theorem LinearEquiv.symm_symm {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                e.symm.symm = e
                                theorem LinearEquiv.symm_bijective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {σ : R →+* S} {σ' : S →+* R} [Module R M] [Module S M₂] [RingHomInvPair σ' σ] [RingHomInvPair σ σ'] :
                                Function.Bijective LinearEquiv.symm
                                @[simp]
                                theorem LinearEquiv.mk_coe' {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (f : M₂M) (h₁ : ∀ (x y : M₂), f (x + y) = f x + f y) (h₂ : ∀ (m : S) (x : M₂), { toFun := f, map_add' := h₁ }.toFun (m x) = σ' m { toFun := f, map_add' := h₁ }.toFun x) (h₃ : Function.LeftInverse e { toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun) (h₄ : Function.RightInverse e { toFun := f, map_add' := h₁, map_smul' := h₂ }.toFun) :
                                { toFun := f, map_add' := h₁, map_smul' := h₂, invFun := e, left_inv := h₃, right_inv := h₄ } = e.symm
                                def LinearEquiv.symm_mk.aux {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (f : M₂M) (h₁ : ∀ (x y : M), e (x + y) = e x + e y) (h₂ : ∀ (m : R) (x : M), { toFun := e, map_add' := h₁ }.toFun (m x) = σ m { toFun := e, map_add' := h₁ }.toFun x) (h₃ : Function.LeftInverse f { toFun := e, map_add' := h₁, map_smul' := h₂ }.toFun) (h₄ : Function.RightInverse f { toFun := e, map_add' := h₁, map_smul' := h₂ }.toFun) :
                                M₂ ≃ₛₗ[σ'] M

                                Auxiliary definition to avoid looping in dsimp with LinearEquiv.symm_mk.

                                Equations
                                • LinearEquiv.symm_mk.aux e f h₁ h₂ h₃ h₄ = { toFun := e, map_add' := h₁, map_smul' := h₂, invFun := f, left_inv := h₃, right_inv := h₄ }.symm
                                Instances For
                                  @[simp]
                                  theorem LinearEquiv.symm_mk {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (f : M₂M) (h₁ : ∀ (x y : M), e (x + y) = e x + e y) (h₂ : ∀ (m : R) (x : M), { toFun := e, map_add' := h₁ }.toFun (m x) = σ m { toFun := e, map_add' := h₁ }.toFun x) (h₃ : Function.LeftInverse f { toFun := e, map_add' := h₁, map_smul' := h₂ }.toFun) (h₄ : Function.RightInverse f { toFun := e, map_add' := h₁, map_smul' := h₂ }.toFun) :
                                  { toFun := e, map_add' := h₁, map_smul' := h₂, invFun := f, left_inv := h₃, right_inv := h₄ }.symm = let __src := LinearEquiv.symm_mk.aux e f h₁ h₂ h₃ h₄; { toFun := f, map_add' := , map_smul' := , invFun := e, left_inv := , right_inv := }
                                  @[simp]
                                  theorem LinearEquiv.coe_symm_mk {R : Type u_1} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {to_fun : MM₂} {inv_fun : M₂M} {map_add : ∀ (x y : M), to_fun (x + y) = to_fun x + to_fun y} {map_smul : ∀ (m : R) (x : M), { toFun := to_fun, map_add' := map_add }.toFun (m x) = (RingHom.id R) m { toFun := to_fun, map_add' := map_add }.toFun x} {left_inv : Function.LeftInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun} {right_inv : Function.RightInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun} :
                                  { toFun := to_fun, map_add' := map_add, map_smul' := map_smul, invFun := inv_fun, left_inv := left_inv, right_inv := right_inv }.symm = inv_fun
                                  theorem LinearEquiv.bijective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                  theorem LinearEquiv.injective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                  theorem LinearEquiv.surjective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) :
                                  theorem LinearEquiv.image_eq_preimage {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (s : Set M) :
                                  e '' s = e.symm ⁻¹' s
                                  theorem LinearEquiv.image_symm_eq_preimage {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (s : Set M₂) :
                                  e.symm '' s = e ⁻¹' s
                                  @[simp]
                                  theorem RingEquiv.toSemilinearEquiv_symm_apply {R : Type u_1} {S : Type u_7} [Semiring R] [Semiring S] (f : R ≃+* S) :
                                  ∀ (a : S), f.toSemilinearEquiv.symm a = f.invFun a
                                  @[simp]
                                  theorem RingEquiv.toSemilinearEquiv_apply {R : Type u_1} {S : Type u_7} [Semiring R] [Semiring S] (f : R ≃+* S) (a : R) :
                                  f.toSemilinearEquiv a = f a
                                  def RingEquiv.toSemilinearEquiv {R : Type u_1} {S : Type u_7} [Semiring R] [Semiring S] (f : R ≃+* S) :
                                  R ≃ₛₗ[f] S

                                  Interpret a RingEquiv f as an f-semilinear equiv.

                                  Equations
                                  • f.toSemilinearEquiv = { toFun := f, map_add' := , map_smul' := , invFun := f.invFun, left_inv := , right_inv := }
                                  Instances For
                                    def LinearEquiv.ofInvolutive {R : Type u_1} {M : Type u_8} [Semiring R] [AddCommMonoid M] {σ : R →+* R} {σ' : R →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                                    {x : Module R M} → (f : M →ₛₗ[σ] M) → Function.Involutive fM ≃ₛₗ[σ] M

                                    An involutive linear map is a linear equivalence.

                                    Equations
                                    Instances For
                                      @[simp]
                                      theorem LinearEquiv.coe_ofInvolutive {R : Type u_1} {M : Type u_8} [Semiring R] [AddCommMonoid M] {σ : R →+* R} {σ' : R →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] :
                                      ∀ {x : Module R M} (f : M →ₛₗ[σ] M) (hf : Function.Involutive f), (LinearEquiv.ofInvolutive f hf) = f