Documentation

Mathlib.Algebra.Module.Equiv

(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_16} {S : Type u_17} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_18) (M₂ : Type u_19) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends LinearMap :
Type (max u_18 u_19)

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' : ∀ (r : R) (x : M), self.toFun (r x) = σ r 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_16} {S : Type u_17} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_18} {M₂ : Type u_19} [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
    • LinearEquiv.toAddEquiv self = { toEquiv := { toFun := self.toFun, invFun := self.invFun, left_inv := , right_inv := }, map_add' := }
    Instances For

      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

          The notation M ≃ₗ⋆[R] M₂ denotes the type of star-linear equivalences between M and M₂ over the endomorphism of the underlying starred ring R.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
            class SemilinearEquivClass (F : Type u_16) {R : outParam (Type u_17)} {S : outParam (Type u_18)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam (Type u_19)) (M₂ : outParam (Type u_20)) [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
              @[inline, reducible]
              abbrev LinearEquivClass (F : Type u_16) (R : outParam (Type u_17)) (M : outParam (Type u_18)) (M₂ : outParam (Type u_19)) [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 SemilinearEquivClass.instSemilinearMapClassToFunLike {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} (F : Type u_16) [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
                • =
                instance LinearEquiv.instCoeLinearEquivLinearMap {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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.instCoeLinearEquivLinearMap = { coe := LinearEquiv.toLinearMap }
                def LinearEquiv.toEquiv {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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
                Instances For
                  theorem LinearEquiv.toEquiv_injective {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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} :
                  theorem LinearEquiv.toLinearMap_injective {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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.instEquivLikeLinearEquiv {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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.instEquivLikeLinearEquiv = { coe := fun (e : M ≃ₛₗ[σ] M₂) => e.toFun, inv := LinearEquiv.invFun, left_inv := , right_inv := , coe_injective' := }
                  instance LinearEquiv.instFunLikeLinearEquiv {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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.instFunLikeLinearEquiv = { coe := DFunLike.coe, coe_injective' := }
                  instance LinearEquiv.instSemilinearEquivClassLinearEquivInstEquivLikeLinearEquiv {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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 : ∀ (r : R) (x : M), { toFun := to_fun, map_add' := map_add }.toFun (r x) = σ r { toFun := to_fun, map_add' := map_add }.toFun x} {left_inv : Function.LeftInverse inv_fun { toAddHom := { toFun := to_fun, map_add' := map_add }, map_smul' := map_smul }.toFun} {right_inv : Function.RightInverse inv_fun { toAddHom := { toFun := to_fun, map_add' := map_add }, map_smul' := map_smul }.toFun} :
                  { toLinearMap := { toAddHom := { 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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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) :
                  @[simp]
                  theorem LinearEquiv.coe_toLinearMap {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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 {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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.ext_iff {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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.congr_arg {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_7) [Semiring R] [AddCommMonoid M] [Module R M] :

                  The identity map is a linear equivalence.

                  Equations
                  • LinearEquiv.refl R M = let __src := LinearMap.id; let __src_1 := Equiv.refl M; { toLinearMap := __src, invFun := __src_1.invFun, left_inv := , right_inv := }
                  Instances For
                    @[simp]
                    theorem LinearEquiv.refl_apply {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] (x : M) :
                    def LinearEquiv.symm {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      def LinearEquiv.Simps.apply {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] (e : M ≃ₛₗ[σ] M₂) :
                      MM₂

                      See Note [custom simps projection]

                      Equations
                      Instances For
                        def LinearEquiv.Simps.symm_apply {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] (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_6} {M : Type u_7} {M₂ : Type u_9} [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 = (LinearEquiv.symm e)
                          @[simp]
                          theorem LinearEquiv.coe_toEquiv_symm {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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) :
                          def LinearEquiv.trans {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [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
                          • One or more equations did not get rendered due to their size.
                          Instances For

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

                            Equations
                            • One or more equations did not get rendered due to their size.
                            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_6} {M : Type u_7} {M₂ : Type u_9} [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.toAddMonoidHom_commutes {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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) :

                                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_8} {M₂ : Type u_9} {M₃ : Type u_10} [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) :
                                (LinearEquiv.trans e₁₂ e₂₃) c = e₂₃ (e₁₂ c)
                                theorem LinearEquiv.coe_trans {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [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} :
                                (LinearEquiv.trans e₁₂ e₂₃) = LinearMap.comp e₂₃ e₁₂
                                @[simp]
                                theorem LinearEquiv.apply_symm_apply {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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 ((LinearEquiv.symm e) c) = c
                                @[simp]
                                theorem LinearEquiv.symm_apply_apply {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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) :
                                (LinearEquiv.symm e) (e b) = b
                                @[simp]
                                theorem LinearEquiv.trans_symm {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [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} :
                                theorem LinearEquiv.symm_trans_apply {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [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) :
                                (LinearEquiv.symm (LinearEquiv.trans e₁₂ e₂₃)) c = (LinearEquiv.symm e₁₂) ((LinearEquiv.symm e₂₃) c)
                                @[simp]
                                theorem LinearEquiv.trans_refl {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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) :
                                @[simp]
                                theorem LinearEquiv.refl_trans {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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.symm_apply_eq {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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} :
                                (LinearEquiv.symm e) x = y x = e y
                                theorem LinearEquiv.eq_symm_apply {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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 = (LinearEquiv.symm e) x e y = x
                                theorem LinearEquiv.eq_comp_symm {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_8} {M₂ : Type u_9} [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_17} (f : M₂α) (g : M₁α) :
                                f = g (LinearEquiv.symm e₁₂) f e₁₂ = g
                                theorem LinearEquiv.comp_symm_eq {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_8} {M₂ : Type u_9} [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_17} (f : M₂α) (g : M₁α) :
                                g (LinearEquiv.symm e₁₂) = f g = f e₁₂
                                theorem LinearEquiv.eq_symm_comp {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_8} {M₂ : Type u_9} [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_17} (f : αM₁) (g : αM₂) :
                                f = (LinearEquiv.symm e₁₂) g e₁₂ f = g
                                theorem LinearEquiv.symm_comp_eq {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_8} {M₂ : Type u_9} [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_17} (f : αM₁) (g : αM₂) :
                                (LinearEquiv.symm e₁₂) 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_8} {M₂ : Type u_9} {M₃ : Type u_10} [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 = LinearMap.comp g (LinearEquiv.symm e₁₂) LinearMap.comp f e₁₂ = g
                                theorem LinearEquiv.comp_toLinearMap_symm_eq {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [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) :
                                LinearMap.comp g (LinearEquiv.symm e₁₂) = f g = LinearMap.comp f e₁₂
                                theorem LinearEquiv.eq_toLinearMap_symm_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [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 = LinearMap.comp ((LinearEquiv.symm e₁₂)) g LinearMap.comp (e₁₂) f = g
                                theorem LinearEquiv.toLinearMap_symm_comp_eq {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_8} {M₂ : Type u_9} {M₃ : Type u_10} [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) :
                                LinearMap.comp ((LinearEquiv.symm e₁₂)) g = f g = LinearMap.comp (e₁₂) f
                                @[simp]
                                theorem LinearEquiv.self_trans_symm {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_8} {M₂ : Type u_9} [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) :
                                @[simp]
                                theorem LinearEquiv.symm_trans_self {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_8} {M₂ : Type u_9} [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) :
                                @[simp]
                                theorem LinearEquiv.refl_toLinearMap {R : Type u_1} {M : Type u_7} [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_7} {M₂ : Type u_9} {M₃ : Type u_10} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_11} {N₂ : Type u_12} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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.symm_bijective {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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 : ∀ (r : S) (x : M₂), { toFun := f, map_add' := h₁ }.toFun (r x) = σ' r { toFun := f, map_add' := h₁ }.toFun x) (h₃ : Function.LeftInverse e { toAddHom := { toFun := f, map_add' := h₁ }, map_smul' := h₂ }.toFun) (h₄ : Function.RightInverse e { toAddHom := { toFun := f, map_add' := h₁ }, map_smul' := h₂ }.toFun) :
                                { toLinearMap := { toAddHom := { toFun := f, map_add' := h₁ }, map_smul' := h₂ }, invFun := e, left_inv := h₃, right_inv := h₄ } = LinearEquiv.symm e
                                @[simp]
                                theorem LinearEquiv.symm_mk {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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 : ∀ (r : R) (x : M), { toFun := e, map_add' := h₁ }.toFun (r x) = σ r { toFun := e, map_add' := h₁ }.toFun x) (h₃ : Function.LeftInverse f { toAddHom := { toFun := e, map_add' := h₁ }, map_smul' := h₂ }.toFun) (h₄ : Function.RightInverse f { toAddHom := { toFun := e, map_add' := h₁ }, map_smul' := h₂ }.toFun) :
                                LinearEquiv.symm { toLinearMap := { toAddHom := { toFun := e, map_add' := h₁ }, map_smul' := h₂ }, invFun := f, left_inv := h₃, right_inv := h₄ } = let __src := LinearEquiv.symm { toLinearMap := { toAddHom := { toFun := e, map_add' := h₁ }, map_smul' := h₂ }, invFun := f, left_inv := h₃, right_inv := h₄ }; { toLinearMap := { toAddHom := { 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_7} {M₂ : Type u_9} [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 : ∀ (r : R) (x : M), { toFun := to_fun, map_add' := map_add }.toFun (r x) = (RingHom.id R) r { toFun := to_fun, map_add' := map_add }.toFun x} {left_inv : Function.LeftInverse inv_fun { toAddHom := { toFun := to_fun, map_add' := map_add }, map_smul' := map_smul }.toFun} {right_inv : Function.RightInverse inv_fun { toAddHom := { toFun := to_fun, map_add' := map_add }, map_smul' := map_smul }.toFun} :
                                (LinearEquiv.symm { toLinearMap := { toAddHom := { toFun := to_fun, map_add' := map_add }, map_smul' := map_smul }, invFun := inv_fun, left_inv := left_inv, right_inv := right_inv }) = inv_fun
                                theorem LinearEquiv.bijective {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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_6} {M : Type u_7} {M₂ : Type u_9} [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 = (LinearEquiv.symm e) ⁻¹' s
                                theorem LinearEquiv.image_symm_eq_preimage {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [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) :
                                (LinearEquiv.symm e) '' s = e ⁻¹' s
                                @[simp]
                                theorem RingEquiv.toSemilinearEquiv_symm_apply {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) :
                                ∀ (a : S), (LinearEquiv.symm (RingEquiv.toSemilinearEquiv f)) a = f.invFun a
                                @[simp]
                                theorem RingEquiv.toSemilinearEquiv_apply {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) (a : R) :
                                def RingEquiv.toSemilinearEquiv {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) :
                                R ≃ₛₗ[f] S

                                Interpret a RingEquiv f as an f-semilinear equiv.

                                Equations
                                • RingEquiv.toSemilinearEquiv f = { toLinearMap := { toAddHom := { 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_7} [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_7} [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
                                    @[simp]
                                    theorem LinearEquiv.restrictScalars_symm_apply (R : Type u_1) {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (f : M ≃ₗ[S] M₂) (a : M₂) :
                                    @[simp]
                                    theorem LinearEquiv.restrictScalars_apply (R : Type u_1) {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (f : M ≃ₗ[S] M₂) (a : M) :
                                    def LinearEquiv.restrictScalars (R : Type u_1) {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (f : M ≃ₗ[S] M₂) :
                                    M ≃ₗ[R] M₂

                                    If M and M₂ are both R-semimodules and S-semimodules and R-semimodule structures are defined by an action of R on S (formally, we have two scalar towers), then any S-linear equivalence from M to M₂ is also an R-linear equivalence.

                                    See also LinearMap.restrictScalars.

                                    Equations
                                    • One or more equations did not get rendered due to their size.
                                    Instances For
                                      theorem LinearEquiv.restrictScalars_injective (R : Type u_1) {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] :
                                      @[simp]
                                      theorem LinearEquiv.restrictScalars_inj (R : Type u_1) {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (f : M ≃ₗ[S] M₂) (g : M ≃ₗ[S] M₂) :
                                      theorem Module.End_isUnit_iff {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] (f : Module.End R M) :
                                      instance LinearEquiv.automorphismGroup {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :
                                      Equations
                                      @[simp]
                                      theorem LinearEquiv.coe_one {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :
                                      1 = id
                                      @[simp]
                                      theorem LinearEquiv.coe_toLinearMap_one {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :
                                      1 = LinearMap.id
                                      @[simp]
                                      theorem LinearEquiv.coe_toLinearMap_mul {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] {e₁ : M ≃ₗ[R] M} {e₂ : M ≃ₗ[R] M} :
                                      (e₁ * e₂) = e₁ * e₂
                                      theorem LinearEquiv.coe_pow {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] (e : M ≃ₗ[R] M) (n : ) :
                                      (e ^ n) = (e)^[n]
                                      theorem LinearEquiv.pow_apply {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] (e : M ≃ₗ[R] M) (n : ) (m : M) :
                                      (e ^ n) m = (e)^[n] m
                                      @[simp]
                                      theorem LinearEquiv.automorphismGroup.toLinearMapMonoidHom_apply {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] (e : M ≃ₗ[R] M) :
                                      LinearEquiv.automorphismGroup.toLinearMapMonoidHom e = e

                                      Restriction from R-linear automorphisms of M to R-linear endomorphisms of M, promoted to a monoid hom.

                                      Equations
                                      • LinearEquiv.automorphismGroup.toLinearMapMonoidHom = { toOneHom := { toFun := fun (e : M ≃ₗ[R] M) => e, map_one' := }, map_mul' := }
                                      Instances For

                                        The tautological action by M ≃ₗ[R] M on M.

                                        This generalizes Function.End.applyMulAction.

                                        Equations
                                        @[simp]
                                        theorem LinearEquiv.smul_def {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] (f : M ≃ₗ[R] M) (a : M) :
                                        f a = f a
                                        instance LinearEquiv.apply_faithfulSMul {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :

                                        LinearEquiv.applyDistribMulAction is faithful.

                                        Equations
                                        • =
                                        instance LinearEquiv.apply_smulCommClass {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :
                                        Equations
                                        • =
                                        instance LinearEquiv.apply_smulCommClass' {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] :
                                        Equations
                                        • =
                                        @[simp]
                                        theorem LinearEquiv.ofSubsingleton_symm_apply {R : Type u_1} (M : Type u_7) (M₂ : Type u_9) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Subsingleton M] [Subsingleton M₂] :
                                        ∀ (x : M₂), (LinearEquiv.symm (LinearEquiv.ofSubsingleton M M₂)) x = 0
                                        @[simp]
                                        theorem LinearEquiv.ofSubsingleton_apply {R : Type u_1} (M : Type u_7) (M₂ : Type u_9) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Subsingleton M] [Subsingleton M₂] :
                                        ∀ (x : M), (LinearEquiv.ofSubsingleton M M₂) x = 0
                                        def LinearEquiv.ofSubsingleton {R : Type u_1} (M : Type u_7) (M₂ : Type u_9) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Subsingleton M] [Subsingleton M₂] :
                                        M ≃ₗ[R] M₂

                                        Any two modules that are subsingletons are isomorphic.

                                        Equations
                                        • One or more equations did not get rendered due to their size.
                                        Instances For
                                          @[simp]
                                          theorem Module.compHom.toLinearEquiv_apply {R : Type u_16} {S : Type u_17} [Semiring R] [Semiring S] (g : R ≃+* S) (a : R) :
                                          def Module.compHom.toLinearEquiv {R : Type u_16} {S : Type u_17} [Semiring R] [Semiring S] (g : R ≃+* S) :

                                          g : R ≃+* S is R-linear when the module structure on S is Module.compHom S g .

                                          Equations
                                          Instances For
                                            @[simp]
                                            theorem DistribMulAction.toLinearEquiv_apply (R : Type u_1) {S : Type u_6} (M : Type u_7) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) :
                                            ∀ (a : M), (DistribMulAction.toLinearEquiv R M s) a = s a
                                            @[simp]
                                            theorem DistribMulAction.toLinearEquiv_symm_apply (R : Type u_1) {S : Type u_6} (M : Type u_7) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) :
                                            def DistribMulAction.toLinearEquiv (R : Type u_1) {S : Type u_6} (M : Type u_7) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) :

                                            Each element of the group defines a linear equivalence.

                                            This is a stronger version of DistribMulAction.toAddEquiv.

                                            Equations
                                            • One or more equations did not get rendered due to their size.
                                            Instances For
                                              def DistribMulAction.toModuleAut (R : Type u_1) {S : Type u_6} (M : Type u_7) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] :
                                              S →* M ≃ₗ[R] M

                                              Each element of the group defines a module automorphism.

                                              This is a stronger version of DistribMulAction.toAddAut.

                                              Equations
                                              Instances For
                                                def AddEquiv.toLinearEquiv {R : Type u_1} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (e : M ≃+ M₂) (h : ∀ (c : R) (x : M), e (c x) = c e x) :
                                                M ≃ₗ[R] M₂

                                                An additive equivalence whose underlying function preserves smul is a linear equivalence.

                                                Equations
                                                • AddEquiv.toLinearEquiv e h = { toLinearMap := { toAddHom := { toFun := e.toFun, map_add' := }, map_smul' := h }, invFun := e.invFun, left_inv := , right_inv := }
                                                Instances For
                                                  @[simp]
                                                  theorem AddEquiv.coe_toLinearEquiv {R : Type u_1} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (e : M ≃+ M₂) (h : ∀ (c : R) (x : M), e (c x) = c e x) :
                                                  @[simp]
                                                  theorem AddEquiv.coe_toLinearEquiv_symm {R : Type u_1} {M : Type u_7} {M₂ : Type u_9} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (e : M ≃+ M₂) (h : ∀ (c : R) (x : M), e (c x) = c e x) :
                                                  def AddEquiv.toNatLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃+ M₂) :

                                                  An additive equivalence between commutative additive monoids is a linear equivalence between ℕ-modules

                                                  Equations
                                                  Instances For
                                                    @[simp]
                                                    theorem AddEquiv.coe_toNatLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃+ M₂) :
                                                    @[simp]
                                                    theorem AddEquiv.toNatLinearEquiv_toAddEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃+ M₂) :
                                                    @[simp]
                                                    @[simp]
                                                    theorem AddEquiv.toNatLinearEquiv_trans {M : Type u_7} {M₂ : Type u_9} {M₃ : Type u_10} [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] (e : M ≃+ M₂) (e₂ : M₂ ≃+ M₃) :
                                                    def AddEquiv.toIntLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃+ M₂) :

                                                    An additive equivalence between commutative additive groups is a linear equivalence between ℤ-modules

                                                    Equations
                                                    Instances For
                                                      @[simp]
                                                      theorem AddEquiv.coe_toIntLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃+ M₂) :
                                                      @[simp]
                                                      theorem AddEquiv.toIntLinearEquiv_toAddEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃+ M₂) :
                                                      @[simp]
                                                      @[simp]
                                                      theorem AddEquiv.toIntLinearEquiv_trans {M : Type u_7} {M₂ : Type u_9} {M₃ : Type u_10} [AddCommGroup M] [AddCommGroup M₂] [AddCommGroup M₃] (e : M ≃+ M₂) (e₂ : M₂ ≃+ M₃) :
                                                      @[simp]
                                                      theorem LinearMap.ringLmapEquivSelf_apply (R : Type u_1) (S : Type u_6) (M : Type u_7) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] (f : R →ₗ[R] M) :
                                                      @[simp]
                                                      def LinearMap.ringLmapEquivSelf (R : Type u_1) (S : Type u_6) (M : Type u_7) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] :
                                                      (R →ₗ[R] M) ≃ₗ[S] M

                                                      The equivalence between R-linear maps from R to M, and points of M itself. This says that the forgetful functor from R-modules to types is representable, by R.

                                                      This is an S-linear equivalence, under the assumption that S acts on M commuting with R. When R is commutative, we can take this to be the usual action with S = R. Otherwise, S = ℕ shows that the equivalence is additive. See note [bundled maps over different rings].

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