(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_12} {S : Type u_13} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_14) (M₂ : Type u_15) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends LinearMap : Type (max u_14 u_15)

A linear equivalence is an invertible linear map.

• toFun : M → M₂
• 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 : 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_12} {S : Type u_13} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_14} {M₂ : Type u_15} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃ₛₗ[σ] M₂) : M ≃+ M₂

The additive equivalence of types underlying a linear equivalence.

Equations

• self.toAddEquiv = { toFun := (↑self).toFun, invFun := self.invFun, left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }

theorem LinearEquiv.right_inv {R : Type u_12} {S : Type u_13} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_14} {M₂ : Type u_15} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃ₛₗ[σ] M₂) : Function.RightInverse self.invFun (↑self).toFun

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

theorem LinearEquiv.left_inv {R : Type u_12} {S : Type u_13} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_14} {M₂ : Type u_15} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃ₛₗ[σ] M₂) : Function.LeftInverse self.invFun (↑self).toFun

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

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.

class SemilinearEquivClass (F : Type u_12) {R : outParam (Type u_13)} {S : outParam (Type u_14)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam (Type u_15)) (M₂ : outParam (Type u_16)) [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.

theorem SemilinearEquivClass.map_smulₛₗ {F : Type u_12} {R : outParam (Type u_13)} {S : outParam (Type u_14)} : ∀ {inst : Semiring R} {inst_1 : Semiring S} {σ : outParam (R →+* S)} {σ' : outParam (S →+* R)} {inst_2 : RingHomInvPair σ σ'} {inst_3 : RingHomInvPair σ' σ} {M : outParam (Type u_15)} {M₂ : outParam (Type u_16)} {inst_4 : AddCommMonoid M} {inst_5 : AddCommMonoid M₂} {inst_6 : Module R M} {inst_7 : Module S M₂} {inst_8 : EquivLike F M M₂} [self : SemilinearEquivClass F σ M M₂] (f : F) (r : R) (x : M), f (r • x) = σ r • f x

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

@[reducible, inline]

abbrev LinearEquivClass (F : Type u_12) (R : outParam (Type u_13)) (M : outParam (Type u_14)) (M₂ : outParam (Type u_15)) [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 100]

instance SemilinearEquivClass.instSemilinearMapClass {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} (F : Type u_12) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [EquivLike F M M₂] [s : SemilinearEquivClass F σ M M₂] : SemilinearMapClass F σ M M₂

Equations

• ⋯ = ⋯

def SemilinearEquivClass.semilinearEquiv {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} {F : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [EquivLike F M M₂] [SemilinearEquivClass F σ M M₂] (f : F) : M ≃ₛₗ[σ] M₂

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

Equations

• ↑f = { toFun := (↑f).toFun, map_add' := ⋯, map_smul' := ⋯, invFun := (↑f).invFun, left_inv := ⋯, right_inv := ⋯ }

instance SemilinearEquivClass.instCoeToSemilinearEquiv {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} {F : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [EquivLike F M M₂] [SemilinearEquivClass F σ M M₂] : CoeHead F (M ≃ₛₗ[σ] M₂)

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

Equations

• SemilinearEquivClass.instCoeToSemilinearEquiv = { coe := fun (f : F) => ↑f }

instance LinearEquiv.instCoeLinearMap {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] : Coe (M ≃ₛₗ[σ] M₂) (M →ₛₗ[σ] M₂)

Equations

• LinearEquiv.instCoeLinearMap = { coe := LinearEquiv.toLinearMap }

def LinearEquiv.toEquiv {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] : (M ≃ₛₗ[σ] M₂) → M ≃ M₂

The equivalence of types underlying a linear equivalence.

Equations

• f.toEquiv = f.toAddEquiv.toEquiv

theorem LinearEquiv.toEquiv_injective {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} : e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂

theorem LinearEquiv.toLinearMap_injective {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} : ↑e₁ = ↑e₂ ↔ e₁ = e₂

instance LinearEquiv.instEquivLike {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] : EquivLike (M ≃ₛₗ[σ] M₂) M M₂

Equations

• LinearEquiv.instEquivLike = { coe := fun (e : M ≃ₛₗ[σ] M₂) => (↑e).toFun, inv := LinearEquiv.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }

instance LinearEquiv.instSemilinearEquivClass {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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 : ∀ (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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : ⇑e.toEquiv = ⇑e

@[simp]

theorem LinearEquiv.coe_toLinearMap {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : (↑e).toFun = ⇑e

theorem LinearEquiv.ext_iff {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} : e = e' ↔ ∀ (x : M), e x = e' x

theorem LinearEquiv.ext {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e : M ≃ₛₗ[σ] M₂} {e' : M ≃ₛₗ[σ] M₂} (h : ∀ (x : M), e x = e' x) : e = e'

theorem LinearEquiv.congr_arg {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_6) [Semiring R] [AddCommMonoid M] [Module R M] : M ≃ₗ[R] M

The identity map is a linear equivalence.

Equations

• LinearEquiv.refl R M = { toLinearMap := LinearMap.id, invFun := (Equiv.refl M).invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LinearEquiv.refl_apply {R : Type u_1} {M : Type u_6} [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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : M₂ ≃ₛₗ[σ'] M

Linear equivalences are symmetric.

Equations

• e.symm = { toFun := ⇑((↑e).inverse e.invFun ⋯ ⋯), map_add' := ⋯, map_smul' := ⋯, invFun := e.toEquiv.symm.invFun, left_inv := ⋯, right_inv := ⋯ }

def LinearEquiv.Simps.apply {R : Type u_12} {S : Type u_13} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_14} {M₂ : Type u_15} [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_12} {S : Type u_13} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_14} {M₂ : Type u_15} [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 = ⇑e.symm

@[simp]

theorem LinearEquiv.invFun_eq_symm {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : e.invFun = ⇑e.symm

@[simp]

theorem LinearEquiv.coe_toEquiv_symm {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : e.toEquiv.symm = ↑e.symm

def LinearEquiv.trans {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] (e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂) (e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃) : M₁ ≃ₛₗ[σ₁₃] M₃

Linear equivalences are transitive.

Equations

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

def LinearEquiv.transNotation : Lean.TrailingParserDescr

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

Equations

• LinearEquiv.transNotation = Lean.ParserDescr.trailingNode `LinearEquiv.transNotation 80 80 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ≪≫ₗ ") (Lean.ParserDescr.cat `term 81))

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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : e.toAddEquiv = ↑e

theorem LinearEquiv.toAddMonoidHom_commutes {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : (↑e).toAddMonoidHom = e.toAddEquiv.toAddMonoidHom

The two paths coercion can take to an AddMonoidHom are equivalent

@[simp]

theorem LinearEquiv.trans_apply {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} (c : M₁) : (e₁₂.trans e₂₃) c = e₂₃ (e₁₂ c)

theorem LinearEquiv.coe_trans {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} : ↑(e₁₂.trans e₂₃) = (↑e₂₃).comp ↑e₁₂

@[simp]

theorem LinearEquiv.apply_symm_apply {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (c : M₂) : e (e.symm c) = c

@[simp]

theorem LinearEquiv.symm_apply_apply {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (b : M) : e.symm (e b) = b

@[simp]

theorem LinearEquiv.trans_symm {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} : (e₁₂.trans e₂₃).symm = e₂₃.symm.trans e₁₂.symm

theorem LinearEquiv.symm_trans_apply {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} (c : M₃) : (e₁₂.trans e₂₃).symm c = e₁₂.symm (e₂₃.symm c)

@[simp]

theorem LinearEquiv.trans_refl {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : e.trans (LinearEquiv.refl S M₂) = e

@[simp]

theorem LinearEquiv.refl_trans {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : (LinearEquiv.refl R M).trans e = e

theorem LinearEquiv.symm_apply_eq {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) {x : M₂} {y : M} : e.symm x = y ↔ x = e y

theorem LinearEquiv.eq_symm_apply {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) {x : M₂} {y : M} : y = e.symm x ↔ e y = x

theorem LinearEquiv.eq_comp_symm {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_7} {M₂ : Type u_8} [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_12} (f : M₂ → α) (g : M₁ → α) : f = g ∘ ⇑e₁₂.symm ↔ f ∘ ⇑e₁₂ = g

theorem LinearEquiv.comp_symm_eq {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_7} {M₂ : Type u_8} [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_12} (f : M₂ → α) (g : M₁ → α) : g ∘ ⇑e₁₂.symm = f ↔ g = f ∘ ⇑e₁₂

theorem LinearEquiv.eq_symm_comp {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_7} {M₂ : Type u_8} [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_12} (f : α → M₁) (g : α → M₂) : f = ⇑e₁₂.symm ∘ g ↔ ⇑e₁₂ ∘ f = g

theorem LinearEquiv.symm_comp_eq {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_7} {M₂ : Type u_8} [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_12} (f : α → M₁) (g : α → M₂) : ⇑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_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₂₁ σ₁₃ σ₂₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃) : f = g.comp ↑e₁₂.symm ↔ f.comp ↑e₁₂ = g

theorem LinearEquiv.comp_toLinearMap_symm_eq {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {σ₂₁ : R₂ →+* R₁} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₂₁ σ₁₃ σ₂₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃) : g.comp ↑e₁₂.symm = f ↔ g = f.comp ↑e₁₂

theorem LinearEquiv.eq_toLinearMap_symm_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₃₁ σ₁₂ σ₃₂] (f : M₃ →ₛₗ[σ₃₁] M₁) (g : M₃ →ₛₗ[σ₃₂] M₂) : f = (↑e₁₂.symm).comp g ↔ (↑e₁₂).comp f = g

theorem LinearEquiv.toLinearMap_symm_comp_eq {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {σ₃₂ : R₃ →+* R₂} {σ₃₁ : R₃ →+* R₁} [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [RingHomCompTriple σ₃₁ σ₁₂ σ₃₂] (f : M₃ →ₛₗ[σ₃₁] M₁) (g : M₃ →ₛₗ[σ₃₂] M₂) : (↑e₁₂.symm).comp g = f ↔ g = (↑e₁₂).comp f

@[simp]

theorem LinearEquiv.comp_toLinearMap_eq_iff {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [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[σ₃₁] M₁) : (↑e₁₂).comp f = (↑e₁₂).comp g ↔ f = g

@[simp]

theorem LinearEquiv.eq_comp_toLinearMap_iff {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_7} {M₂ : Type u_8} {M₃ : Type u_9} [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 : M₂ →ₛₗ[σ₂₃] M₃) : f.comp ↑e₁₂ = g.comp ↑e₁₂ ↔ f = g

@[simp]

theorem LinearEquiv.refl_symm {R : Type u_1} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] : (LinearEquiv.refl R M).symm = LinearEquiv.refl R M

@[simp]

theorem LinearEquiv.self_trans_symm {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_7} {M₂ : Type u_8} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} (f : M₁ ≃ₛₗ[σ₁₂] M₂) : f.trans f.symm = LinearEquiv.refl R₁ M₁

@[simp]

theorem LinearEquiv.symm_trans_self {R₁ : Type u_2} {R₂ : Type u_3} {M₁ : Type u_7} {M₂ : Type u_8} [Semiring R₁] [Semiring R₂] [AddCommMonoid M₁] [AddCommMonoid M₂] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} (f : M₁ ≃ₛₗ[σ₁₂] M₂) : f.symm.trans f = LinearEquiv.refl R₂ M₂

@[simp]

theorem LinearEquiv.refl_toLinearMap {R : Type u_1} {M : Type u_6} [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_6} {M₂ : Type u_8} {M₃ : Type u_9} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_10} {N₂ : Type u_11} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) : e.symm.symm = e

theorem LinearEquiv.symm_bijective {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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 : ∀ (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_5} {M : Type u_6} {M₂ : Type u_8} [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 : ∀ (m : R) (x : M), { toFun := ⇑e, map_add' := h₁ }.toFun (m • x) = σ m • { toFun := ⇑e, map_add' := h₁ }.toFun x) (h₃ : Function.LeftInverse f { toFun := ⇑e, map_add' := h₁, map_smul' := h₂ }.toFun) (h₄ : Function.RightInverse f { toFun := ⇑e, map_add' := h₁, map_smul' := h₂ }.toFun) : M₂ ≃ₛₗ[σ'] M

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

Equations

• LinearEquiv.symm_mk.aux e f h₁ h₂ h₃ h₄ = { toFun := ⇑e, map_add' := h₁, map_smul' := h₂, invFun := f, left_inv := h₃, right_inv := h₄ }.symm

@[simp]

theorem LinearEquiv.symm_mk {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [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 : ∀ (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_6} {M₂ : Type u_8} [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 : ∀ (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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [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_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (s : Set M) : ⇑e '' s = ⇑e.symm ⁻¹' s

theorem LinearEquiv.image_symm_eq_preimage {R : Type u_1} {S : Type u_5} {M : Type u_6} {M₂ : Type u_8} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) (s : Set M₂) : ⇑e.symm '' s = ⇑e ⁻¹' s

@[simp]

theorem RingEquiv.toSemilinearEquiv_apply {R : Type u_1} {S : Type u_5} [Semiring R] [Semiring S] (f : R ≃+* S) (a : R) : f.toSemilinearEquiv a = f a

@[simp]

theorem RingEquiv.toSemilinearEquiv_symm_apply {R : Type u_1} {S : Type u_5} [Semiring R] [Semiring S] (f : R ≃+* S) : ∀ (a : S), f.toSemilinearEquiv.symm a = f.invFun a

def RingEquiv.toSemilinearEquiv {R : Type u_1} {S : Type u_5} [Semiring R] [Semiring S] (f : R ≃+* S) : R ≃ₛₗ[↑f] S

Interpret a RingEquiv f as an f-semilinear equiv.

Equations

• f.toSemilinearEquiv = { toFun := ⇑f, map_add' := ⋯, map_smul' := ⋯, invFun := f.invFun, left_inv := ⋯, right_inv := ⋯ }

def LinearEquiv.ofInvolutive {R : Type u_1} {M : Type u_6} [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

• LinearEquiv.ofInvolutive f hf = { toLinearMap := f, invFun := (Function.Involutive.toPerm (⇑f) hf).invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LinearEquiv.coe_ofInvolutive {R : Type u_1} {M : Type u_6} [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