(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_17} {S : Type u_18} [] [] (σ : R →+* S) {σ' : S →+* R} [] [] (M : Type u_19) (M₂ : Type u_20) [] [] [Module R M] [Module S M₂] extends :
Type (max u_19 u_20)

A linear equivalence is an invertible linear map.

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

The backwards directed function underlying a linear equivalence.

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

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

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

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

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

The additive equivalence of types underlying a linear equivalence.

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

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

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

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

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

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

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

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

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

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

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

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

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

Instances
theorem SemilinearEquivClass.map_smulₛₗ {F : Type u_17} {R : outParam (Type u_18)} {S : outParam (Type u_19)} [] [] {σ : outParam (R →+* S)} {σ' : outParam (S →+* R)} [] [] {M : outParam (Type u_20)} {M₂ : outParam (Type u_21)} [] [] [Module R M] [Module S M₂] [EquivLike F M M₂] [self : SemilinearEquivClass F σ M M₂] (f : F) (r : R) (x : M) :
f (r x) = σ r f x

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

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

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

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

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

Equations
• f = let __src := f; let __src_1 := f; { toFun := __src.toFun, map_add' := , map_smul' := , invFun := __src.invFun, left_inv := , right_inv := }
Instances For
instance SemilinearEquivClass.instCoeToSemilinearEquiv {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} {F : Type u_17} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] [EquivLike F M M₂] [SemilinearEquivClass F σ M M₂] :
CoeHead F (M ≃ₛₗ[σ] M₂)

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

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

The equivalence of types underlying a linear equivalence.

Equations
• f.toEquiv = f.toAddEquiv.toEquiv
Instances For
theorem LinearEquiv.toEquiv_injective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] :
Function.Injective LinearEquiv.toEquiv
@[simp]
theorem LinearEquiv.toEquiv_inj {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} :
e₁.toEquiv = e₂.toEquiv e₁ = e₂
theorem LinearEquiv.toLinearMap_injective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] :
Function.Injective LinearEquiv.toLinearMap
@[simp]
theorem LinearEquiv.toLinearMap_inj {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} :
e₁ = e₂ e₁ = e₂
instance LinearEquiv.instEquivLike {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] :
EquivLike (M ≃ₛₗ[σ] M₂) M M₂
Equations
• LinearEquiv.instEquivLike = { coe := fun (e : M ≃ₛₗ[σ] M₂) => (↑e).toFun, inv := LinearEquiv.invFun, left_inv := , right_inv := , coe_injective' := }
instance LinearEquiv.instFunLike {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] :
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.instFunLike = { coe := DFunLike.coe, coe_injective' := }
instance LinearEquiv.instSemilinearEquivClass {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] :
SemilinearEquivClass (M ≃ₛₗ[σ] M₂) σ M M₂
Equations
• =
@[simp]
theorem LinearEquiv.coe_mk {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] {to_fun : MM₂} {inv_fun : M₂M} {map_add : ∀ (x y : M), to_fun (x + y) = to_fun x + to_fun y} {map_smul : ∀ (m : R) (x : M), { toFun := to_fun, map_add' := map_add }.toFun (m x) = σ m { toFun := to_fun, map_add' := map_add }.toFun x} {left_inv : Function.LeftInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun} {right_inv : Function.RightInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun} :
{ toFun := to_fun, map_add' := map_add, map_smul' := map_smul, invFun := inv_fun, left_inv := left_inv, right_inv := right_inv } = to_fun
theorem LinearEquiv.coe_injective {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [] [] :
@[simp]
theorem LinearEquiv.coe_coe {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } (e : M ≃ₛₗ[σ] M₂) :
e = e
@[simp]
theorem LinearEquiv.coe_toEquiv {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } (e : M ≃ₛₗ[σ] M₂) :
e.toEquiv = e
@[simp]
theorem LinearEquiv.coe_toLinearMap {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } (e : M ≃ₛₗ[σ] M₂) :
e = e
theorem LinearEquiv.toFun_eq_coe {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } (e : M ≃ₛₗ[σ] M₂) :
(↑e).toFun = e
theorem LinearEquiv.ext_iff {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} :
e = e' ∀ (x : M), e x = e' x
theorem LinearEquiv.ext {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} (h : ∀ (x : M), e x = e' x) :
e = e'
theorem LinearEquiv.congr_arg {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } {e : M ≃ₛₗ[σ] M₂} {x : M} {x' : M} :
x = x'e x = e x'
theorem LinearEquiv.congr_fun {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} (h : e = e') (x : M) :
e x = e' x
def LinearEquiv.refl (R : Type u_1) (M : Type u_8) [] [] [Module R M] :

The identity map is a linear equivalence.

Equations
• = let __src := LinearMap.id; let __src_1 := ; { toLinearMap := __src, invFun := __src_1.invFun, left_inv := , right_inv := }
Instances For
@[simp]
theorem LinearEquiv.refl_apply {R : Type u_1} {M : Type u_8} [] [] [Module R M] (x : M) :
def LinearEquiv.symm {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [] [] [] [] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : } {re₂ : } (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_18} {S : Type u_19} [] [] {σ : R →+* S} {σ' : S →+* R} [] [] {M : Type u_20} {M₂ : Type u_21} [] [] [Module R M] [Module S M₂] (e : M ≃ₛₗ[σ] M₂) :
MM₂

See Note [custom simps projection]

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

See Note [custom simps projection]

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

Linear equivalences are transitive.

Equations
• e₁₂.trans e₂₃ = let __src := (↑e₂₃).comp e₁₂; let __src_1 := e₁₂.toEquiv.trans e₂₃.toEquiv; { toLinearMap := __src, invFun := __src_1.invFun, left_inv := , right_inv := }
Instances For

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

Equations
Instances For

Pretty printer defined by notation3 command.

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

The two paths coercion can take to an AddMonoidHom are equivalent

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

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

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

Interpret a RingEquiv f as an f-semilinear equiv.

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

An involutive linear map is a linear equivalence.

Equations
• = let __src := ; { toLinearMap := f, invFun := __src.invFun, left_inv := , right_inv := }
Instances For
@[simp]
theorem LinearEquiv.coe_ofInvolutive {R : Type u_1} {M : Type u_8} [] [] {σ : R →+* R} {σ' : R →+* R} [] [] :
∀ {x : Module R M} (f : M →ₛₗ[σ] M) (hf : ), = f