(Semi)linear equivalences

In this file we define

• LinearEquiv σ M M₂, M ≃ₛₗ[σ] M₂: an invertible semilinear map. Here, σ is a RingHom from R to R₂ and an e : M ≃ₛₗ[σ] M₂ satisfies e (c • x) = (σ c) • (e x). The plain linear version, with σ being RingHom.id R, is denoted by M ≃ₗ[R] M₂, and the star-linear version (with σ being starRingEnd) is denoted by M ≃ₗ⋆[R] M₂.

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

• Parts of this file have not yet been generalized to semilinear maps

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 : M → M₂
• map_add' : ∀ (x y : M), self.toAddHom.toFun (x + y) = self.toAddHom.toFun x + self.toAddHom.toFun y
• map_smul' : ∀ (r : R) (x : M), self.toAddHom.toFun (r • x) = σ r • self.toAddHom.toFun x
• invFun : M₂ → M

  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.

@[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

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

def «term_≃ₛₗ[_]_» : Lean.TrailingParserDescr

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.

def «term_≃ₗ[_]_» : Lean.TrailingParserDescr

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.

def «term_≃ₗ⋆[_]_» : Lean.TrailingParserDescr

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.

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 : Prop

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.

@[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] : Prop

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

• LinearEquivClass F R M M₂ = SemilinearEquivClass F (RingHom.id R) M M₂

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

• (_ : SemilinearMapClass F σ M M₂) = (_ : SemilinearMapClass F σ M M₂)

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

• LinearEquiv.toEquiv f = (LinearEquiv.toAddEquiv f).toEquiv

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} : LinearEquiv.toEquiv e₁ = LinearEquiv.toEquiv e₂ ↔ e₁ = e₂

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

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

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 EquivLike → FunLike.

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

Equations

• LinearEquiv.instFunLikeLinearEquiv = { coe := DFunLike.coe, coe_injective' := (_ : Function.Injective fun (f : M ≃ₛₗ[σ] M₂) => ⇑f) }

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

• (_ : SemilinearEquivClass (M ≃ₛₗ[σ] M₂) σ M M₂) = (_ : SemilinearEquivClass (M ≃ₛₗ[σ] M₂) σ M M₂)

@[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 : M → M₂} {inv_fun : M₂ → M} {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 }.toAddHom.toFun} {right_inv : Function.RightInverse inv_fun { toAddHom := { toFun := to_fun, map_add' := map_add }, map_smul' := map_smul }.toAddHom.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] : Function.Injective CoeFun.coe

@[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) : ⇑(LinearEquiv.toEquiv e) = ⇑e

@[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] : M ≃ₗ[R] M

The identity map is a linear equivalence.

Equations

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

@[simp]

theorem LinearEquiv.refl_apply {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] (x : M) : (LinearEquiv.refl R M) x = x

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.

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) : M → M₂

See Note [custom simps projection]

Equations

• LinearEquiv.Simps.apply e = ⇑e

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) : M₂ → M

See Note [custom simps projection]

Equations

• LinearEquiv.Simps.symm_apply e = ⇑(LinearEquiv.symm e)

@[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) : (LinearEquiv.toEquiv e).symm = ↑(LinearEquiv.symm e)

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.

def LinearEquiv.transNotation : Lean.TrailingParserDescr

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.

def LinearEquiv.transNotation.delab : Lean.PrettyPrinter.Delaborator.Delab

Pretty printer defined by notation3 command.

Equations

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

@[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) : LinearEquiv.toAddEquiv e = ↑e

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) : LinearMap.toAddMonoidHom ↑e = AddEquiv.toAddMonoidHom (LinearEquiv.toAddEquiv e)

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} : LinearEquiv.symm (LinearEquiv.trans e₁₂ e₂₃) = LinearEquiv.trans (LinearEquiv.symm e₂₃) (LinearEquiv.symm e₁₂)

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) : LinearEquiv.trans e (LinearEquiv.refl S M₂) = e

@[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) : LinearEquiv.trans (LinearEquiv.refl R M) e = e

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 : M₂ → α) (g : M₁ → α) : 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 : M₂ → α) (g : M₁ → α) : 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 : α → M₁) (g : α → M₂) : 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 : α → M₁) (g : α → M₂) : ⇑(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.refl_symm {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : LinearEquiv.symm (LinearEquiv.refl R M) = LinearEquiv.refl R M

@[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) : LinearEquiv.trans f (LinearEquiv.symm f) = LinearEquiv.refl R₁ M₁

@[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) : LinearEquiv.trans (LinearEquiv.symm f) f = LinearEquiv.refl R₂ 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 : M₂ → M) (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) : LinearEquiv.symm (LinearEquiv.symm e) = e

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 : M₂ → M) (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 }.toAddHom.toFun) (h₄ : Function.RightInverse ⇑e { toAddHom := { toFun := f, map_add' := h₁ }, map_smul' := h₂ }.toAddHom.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 : M₂ → M) (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 }.toAddHom.toFun) (h₄ : Function.RightInverse f { toAddHom := { toFun := ⇑e, map_add' := h₁ }, map_smul' := h₂ }.toAddHom.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' := (_ : ∀ (x y : M₂), src.toAddHom.toFun (x + y) = src.toAddHom.toFun x + src.toAddHom.toFun y) }, map_smul' := (_ : ∀ (r : S) (x : M₂), src.toAddHom.toFun (r • x) = σ' r • src.toAddHom.toFun x) }, invFun := ⇑e, left_inv := (_ : Function.LeftInverse src.invFun src.toFun), right_inv := (_ : Function.RightInverse src.invFun src.toFun) }

@[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 : M → M₂} {inv_fun : M₂ → M} {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 }.toAddHom.toFun} {right_inv : Function.RightInverse inv_fun { toAddHom := { toFun := to_fun, map_add' := map_add }, map_smul' := map_smul }.toAddHom.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) : Function.Bijective ⇑e

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) : Function.Injective ⇑e

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) : Function.Surjective ⇑e

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.toEquiv.invFun a

@[simp]

theorem RingEquiv.toSemilinearEquiv_apply {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R ≃+* S) (a : R) : (RingEquiv.toSemilinearEquiv f) a = f a

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

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

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 ⇑f → M ≃ₛₗ[σ] M

An involutive linear map is a linear equivalence.

Equations

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

@[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) : (LinearEquiv.symm (LinearEquiv.restrictScalars R f)) a = (LinearEquiv.symm f) a

@[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) : (LinearEquiv.restrictScalars R f) a = f a

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.

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] : Function.Injective (LinearEquiv.restrictScalars R)

@[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) : LinearEquiv.restrictScalars R f = LinearEquiv.restrictScalars R g ↔ f = g

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) : IsUnit f ↔ Function.Bijective ⇑f

instance LinearEquiv.automorphismGroup {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : Group (M ≃ₗ[R] M)

Equations

• LinearEquiv.automorphismGroup = Group.mk (_ : ∀ (f : M ≃ₗ[R] M), f⁻¹ * f = 1)

@[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 : 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 : 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

def LinearEquiv.automorphismGroup.toLinearMapMonoidHom {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : (M ≃ₗ[R] M) →* M →ₗ[R] M

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

Equations

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

instance LinearEquiv.applyDistribMulAction {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : DistribMulAction (M ≃ₗ[R] M) M

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

This generalizes Function.End.applyMulAction.

Equations

• LinearEquiv.applyDistribMulAction = DistribMulAction.mk (_ : ∀ (e : M ≃ₗ[R] M), e 0 = 0) (_ : ∀ (e : M ≃ₗ[R] M) (a b : M), e (a + b) = e a + e b)

@[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] : FaithfulSMul (M ≃ₗ[R] M) M

LinearEquiv.applyDistribMulAction is faithful.

Equations

• (_ : FaithfulSMul (M ≃ₗ[R] M) M) = (_ : FaithfulSMul (M ≃ₗ[R] M) M)

instance LinearEquiv.apply_smulCommClass {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : SMulCommClass R (M ≃ₗ[R] M) M

Equations

• (_ : SMulCommClass R (M ≃ₗ[R] M) M) = (_ : SMulCommClass R (M ≃ₗ[R] M) M)

instance LinearEquiv.apply_smulCommClass' {R : Type u_1} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : SMulCommClass (M ≃ₗ[R] M) R M

Equations

• (_ : SMulCommClass (M ≃ₗ[R] M) R M) = (_ : SMulCommClass (M ≃ₗ[R] M) R M)

@[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.

@[simp]

theorem LinearEquiv.ofSubsingleton_self {R : Type u_1} (M : Type u_7) [Semiring R] [AddCommMonoid M] [Module R M] [Subsingleton M] : LinearEquiv.ofSubsingleton M M = LinearEquiv.refl R M

@[simp]

theorem Module.compHom.toLinearEquiv_apply {R : Type u_16} {S : Type u_17} [Semiring R] [Semiring S] (g : R ≃+* S) (a : R) : (Module.compHom.toLinearEquiv g) a = g a

@[simp]

theorem Module.compHom.toLinearEquiv_symm_apply {R : Type u_16} {S : Type u_17} [Semiring R] [Semiring S] (g : R ≃+* S) (a : S) : (LinearEquiv.symm (Module.compHom.toLinearEquiv g)) a = (RingEquiv.symm g) a

def Module.compHom.toLinearEquiv {R : Type u_16} {S : Type u_17} [Semiring R] [Semiring S] (g : R ≃+* S) : R ≃ₗ[R] S

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

Equations

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

@[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) : ∀ (a : M), (LinearEquiv.symm (DistribMulAction.toLinearEquiv R M s)) a = s⁻¹ • a

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) : M ≃ₗ[R] M

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.

@[simp]

theorem DistribMulAction.toModuleAut_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) : (DistribMulAction.toModuleAut R M) s = DistribMulAction.toLinearEquiv R M s

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

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

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

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

@[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) : ⇑(AddEquiv.toLinearEquiv e h) = ⇑e

@[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) : ⇑(LinearEquiv.symm (AddEquiv.toLinearEquiv e h)) = ⇑(AddEquiv.symm e)

def AddEquiv.toNatLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃+ M₂) : M ≃ₗ[ℕ] M₂

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

Equations

• AddEquiv.toNatLinearEquiv e = AddEquiv.toLinearEquiv e (_ : ∀ (c : ℕ) (a : M), e (c • a) = c • e a)

@[simp]

theorem AddEquiv.coe_toNatLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃+ M₂) : ⇑(AddEquiv.toNatLinearEquiv e) = ⇑e

@[simp]

theorem AddEquiv.toNatLinearEquiv_toAddEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃+ M₂) : ↑(AddEquiv.toNatLinearEquiv e) = e

@[simp]

theorem LinearEquiv.toAddEquiv_toNatLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃ₗ[ℕ] M₂) : AddEquiv.toNatLinearEquiv ↑e = e

@[simp]

theorem AddEquiv.toNatLinearEquiv_symm {M : Type u_7} {M₂ : Type u_9} [AddCommMonoid M] [AddCommMonoid M₂] (e : M ≃+ M₂) : LinearEquiv.symm (AddEquiv.toNatLinearEquiv e) = AddEquiv.toNatLinearEquiv (AddEquiv.symm e)

@[simp]

theorem AddEquiv.toNatLinearEquiv_refl {M : Type u_7} [AddCommMonoid M] : AddEquiv.toNatLinearEquiv (AddEquiv.refl M) = LinearEquiv.refl ℕ M

@[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) : AddEquiv.toNatLinearEquiv e ≪≫ₗ AddEquiv.toNatLinearEquiv e₂ = AddEquiv.toNatLinearEquiv (AddEquiv.trans e e₂)

def AddEquiv.toIntLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃+ M₂) : M ≃ₗ[ℤ] M₂

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

Equations

• AddEquiv.toIntLinearEquiv e = AddEquiv.toLinearEquiv e (_ : ∀ (c : ℤ) (a : M), (AddEquiv.toAddMonoidHom e) (c • a) = c • (AddEquiv.toAddMonoidHom e) a)

@[simp]

theorem AddEquiv.coe_toIntLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃+ M₂) : ⇑(AddEquiv.toIntLinearEquiv e) = ⇑e

@[simp]

theorem AddEquiv.toIntLinearEquiv_toAddEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃+ M₂) : ↑(AddEquiv.toIntLinearEquiv e) = e

@[simp]

theorem LinearEquiv.toAddEquiv_toIntLinearEquiv {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃ₗ[ℤ] M₂) : AddEquiv.toIntLinearEquiv ↑e = e

@[simp]

theorem AddEquiv.toIntLinearEquiv_symm {M : Type u_7} {M₂ : Type u_9} [AddCommGroup M] [AddCommGroup M₂] (e : M ≃+ M₂) : LinearEquiv.symm (AddEquiv.toIntLinearEquiv e) = AddEquiv.toIntLinearEquiv (AddEquiv.symm e)

@[simp]

theorem AddEquiv.toIntLinearEquiv_refl {M : Type u_7} [AddCommGroup M] : AddEquiv.toIntLinearEquiv (AddEquiv.refl M) = LinearEquiv.refl ℤ M

@[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) : AddEquiv.toIntLinearEquiv e ≪≫ₗ AddEquiv.toIntLinearEquiv e₂ = AddEquiv.toIntLinearEquiv (AddEquiv.trans e e₂)

@[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) : (LinearMap.ringLmapEquivSelf R S M) f = f 1

@[simp]

theorem LinearMap.ringLmapEquivSelf_symm_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] (x : M) : (LinearEquiv.symm (LinearMap.ringLmapEquivSelf R S M)) x = LinearMap.smulRight 1 x

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.