(Semi)linear equivalences #

In this file we define

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

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structure LinearEquiv {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] (σ : R →+* S) {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] (M : Type u_3) (M₂ : Type u_4) [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] extends LinearMap :

Type (maxu_3u_4)

The backwards directed function underlying a linear equivalence.
invFun : M₂ → M
LinearEquiv.invFun is a left inverse to the linear equivalence's underlying function.
left_inv : Function.LeftInverse invFun toLinearMap.toAddHom.toFun
LinearEquiv.invFun is a right inverse to the linear equivalence's underlying function.
right_inv : Function.RightInverse invFun toLinearMap.toAddHom.toFun

A linear equivalence is an invertible linear map.

Instances For

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abbrev LinearEquiv.toAddEquiv {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] {M : Type u_3} {M₂ : Type u_4} [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] (self : M ≃ₛₗ[σ] M₂) :

M ≃+ M₂

The additive equivalence of types underlying a linear equivalence.

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

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def «term_≃ₛₗ[_]_» :

Lean.TrailingParserDescr

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

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

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def «term_≃ₗ[_]_» :

Lean.TrailingParserDescr

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

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

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def «term_≃ₗ⋆[_]_» :

Lean.TrailingParserDescr

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

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

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class SemilinearEquivClass (F : Type u_1) {R : outParam (Type u_2)} {S : outParam (Type u_3)} [inst : Semiring R] [inst : Semiring S] (σ : outParam (R →+* S)) {σ' : outParam (S →+* R)} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] (M : outParam (Type u_4)) (M₂ : outParam (Type u_5)) [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] extends AddEquivClass :

Type (max(maxu_1u_4)u_5)

Applying a semilinear equivalence f over σ to r • x equals σ r • f x.
map_smulₛₗ : ∀ (f : F) (r : R) (x : M), ↑f (r • x) = ↑σ r • ↑f x

SemilinearEquivClass F σ M M₂ asserts F is a type of bundled σ-semilinear equivs M → M₂→ 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→+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

Instances

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@[inline]

abbrev LinearEquivClass (F : Type u_1) (R : outParam (Type u_2)) (M : outParam (Type u_3)) (M₂ : outParam (Type u_4)) [inst : Semiring R] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module R M₂] :

Type (max(maxu_1u_3)u_4)

LinearEquivClass F R M M₂ asserts F is a type of bundled R-linear equivs M → M₂→ 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

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@[infer_tc_goals_rl]

instance SemilinearEquivClass.instSemilinearMapClass {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} (F : Type u_5) [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] [s : SemilinearEquivClass F σ M M₂] :

SemilinearMapClass F σ M M₂

Equations

SemilinearEquivClass.instSemilinearMapClass F = SemilinearMapClass.mk (_ : ∀ (f : F) (r : R) (x : M), ↑f (r • x) = ↑σ r • ↑f x)

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instance LinearEquiv.instCoeLinearEquivLinearMap {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] :

Coe (M ≃ₛₗ[σ] M₂) (M →ₛₗ[σ] M₂)

Equations

LinearEquiv.instCoeLinearEquivLinearMap = { coe := LinearEquiv.toLinearMap }

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def LinearEquiv.toEquiv {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : 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

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theorem LinearEquiv.toEquiv_injective {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] :

Function.Injective LinearEquiv.toEquiv

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theorem LinearEquiv.toEquiv_inj {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} :

LinearEquiv.toEquiv e₁ = LinearEquiv.toEquiv e₂ ↔ e₁ = e₂

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theorem LinearEquiv.toLinearMap_injective {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] :

Function.Injective LinearEquiv.toLinearMap

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theorem LinearEquiv.toLinearMap_inj {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] {e₁ : M ≃ₛₗ[σ] M₂} {e₂ : M ≃ₛₗ[σ] M₂} :

↑e₁ = ↑e₂ ↔ e₁ = e₂

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instance LinearEquiv.instSemilinearEquivClassLinearEquiv {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] :

SemilinearEquivClass (M ≃ₛₗ[σ] M₂) σ M M₂

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

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theorem LinearEquiv.coe_mk {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : 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), AddHom.toFun { toFun := to_fun, map_add' := map_add } (r • x) = ↑σ r • AddHom.toFun { toFun := to_fun, map_add' := map_add } 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

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theorem LinearEquiv.coe_injective {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] :

Function.Injective CoeFun.coe

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theorem LinearEquiv.coe_coe {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.coe_toEquiv {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.coe_toLinearMap {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.toFun_eq_coe {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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).toAddHom.toFun = ↑e

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theorem LinearEquiv.ext {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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'

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theorem LinearEquiv.ext_iff {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.congr_arg {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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'

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theorem LinearEquiv.congr_fun {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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def LinearEquiv.refl (R : Type u_1) (M : Type u_2) [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] :

M ≃ₗ[R] M

The identity map is a linear equivalence.

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

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theorem LinearEquiv.refl_apply {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] (x : M) :

↑(LinearEquiv.refl R M) x = x

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def LinearEquiv.symm {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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.

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

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def LinearEquiv.Simps.apply {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] {M : Type u_3} {M₂ : Type u_4} [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] (e : M ≃ₛₗ[σ] M₂) :

M → M₂

See Note [custom simps projection]

Equations

LinearEquiv.Simps.apply e = ↑e

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def LinearEquiv.Simps.symm_apply {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] {σ : R →+* S} {σ' : S →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] {M : Type u_3} {M₂ : Type u_4} [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module S M₂] (e : M ≃ₛₗ[σ] M₂) :

M₂ → M

See Note [custom simps projection]

Equations

LinearEquiv.Simps.symm_apply e = ↑(LinearEquiv.symm e)

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@[simp]

theorem LinearEquiv.invFun_eq_symm {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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)

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theorem LinearEquiv.coe_toEquiv_symm {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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) :

Equiv.symm (LinearEquiv.toEquiv e) = ↑(LinearEquiv.symm e)

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def LinearEquiv.trans {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [inst : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [inst : RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [inst : RingHomInvPair σ₃₁ σ₁₃] (e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂) (e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃) :

M₁ ≃ₛₗ[σ₁₃] M₃

Linear equivalences are transitive.

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

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def LinearEquiv.«term_≪≫ₗ_» :

Lean.TrailingParserDescr

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

Equations

LinearEquiv.«term_≪≫ₗ_» = Lean.ParserDescr.trailingNode `LinearEquiv.term_≪≫ₗ_ 80 80 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ≪≫ₗ ") (Lean.ParserDescr.cat `term 81))

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@[simp]

theorem LinearEquiv.coe_toAddEquiv {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.toAddMonoidHom_commutes {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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@[simp]

theorem LinearEquiv.trans_apply {R₁ : Type u_3} {R₂ : Type u_5} {R₃ : Type u_4} {M₁ : Type u_2} {M₂ : Type u_6} {M₃ : Type u_1} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [inst : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [inst : RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [inst : RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} (c : M₁) :

↑(LinearEquiv.trans e₁₂ e₂₃) c = ↑e₂₃ (↑e₁₂ c)

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theorem LinearEquiv.coe_trans {R₁ : Type u_3} {R₂ : Type u_5} {R₃ : Type u_4} {M₁ : Type u_1} {M₂ : Type u_6} {M₃ : Type u_2} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [inst : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [inst : RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [inst : RingHomInvPair σ₃₁ σ₁₃] {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} {e₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃} :

↑(LinearEquiv.trans e₁₂ e₂₃) = LinearMap.comp ↑e₂₃ ↑e₁₂

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@[simp]

theorem LinearEquiv.apply_symm_apply {R : Type u_4} {S : Type u_3} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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@[simp]

theorem LinearEquiv.symm_apply_apply {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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@[simp]

theorem LinearEquiv.trans_symm {R₁ : Type u_4} {R₂ : Type u_5} {R₃ : Type u_3} {M₁ : Type u_1} {M₂ : Type u_6} {M₃ : Type u_2} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [inst : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [inst : RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [inst : 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)

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theorem LinearEquiv.symm_trans_apply {R₁ : Type u_4} {R₂ : Type u_5} {R₃ : Type u_3} {M₁ : Type u_1} {M₂ : Type u_6} {M₃ : Type u_2} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [inst : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₃ : RingHomInvPair σ₂₃ σ₃₂} [inst : RingHomInvPair σ₁₃ σ₃₁] {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {re₃₂ : RingHomInvPair σ₃₂ σ₂₃} [inst : 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)

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@[simp]

theorem LinearEquiv.trans_refl {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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@[simp]

theorem LinearEquiv.refl_trans {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.symm_apply_eq {R : Type u_4} {S : Type u_3} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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 : (fun x => M) x} :

↑(LinearEquiv.symm e) x = y ↔ x = ↑e y

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theorem LinearEquiv.eq_symm_apply {R : Type u_4} {S : Type u_3} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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 : (fun x => M) x} :

y = ↑(LinearEquiv.symm e) x ↔ ↑e y = x

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theorem LinearEquiv.eq_comp_symm {R₁ : Type u_5} {R₂ : Type u_4} {M₁ : Type u_3} {M₂ : Type u_2} [inst : Semiring R₁] [inst : Semiring R₂] [inst : AddCommMonoid M₁] [inst : 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_1} (f : M₂ → α) (g : M₁ → α) :

f = g ∘ ↑(LinearEquiv.symm e₁₂) ↔ f ∘ ↑e₁₂ = g

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theorem LinearEquiv.comp_symm_eq {R₁ : Type u_5} {R₂ : Type u_4} {M₁ : Type u_3} {M₂ : Type u_2} [inst : Semiring R₁] [inst : Semiring R₂] [inst : AddCommMonoid M₁] [inst : 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_1} (f : M₂ → α) (g : M₁ → α) :

g ∘ ↑(LinearEquiv.symm e₁₂) = f ↔ g = f ∘ ↑e₁₂

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theorem LinearEquiv.eq_symm_comp {R₁ : Type u_5} {R₂ : Type u_4} {M₁ : Type u_2} {M₂ : Type u_3} [inst : Semiring R₁] [inst : Semiring R₂] [inst : AddCommMonoid M₁] [inst : 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_1} (f : α → M₁) (g : α → M₂) :

f = ↑(LinearEquiv.symm e₁₂) ∘ g ↔ ↑e₁₂ ∘ f = g

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theorem LinearEquiv.symm_comp_eq {R₁ : Type u_5} {R₂ : Type u_4} {M₁ : Type u_2} {M₂ : Type u_3} [inst : Semiring R₁] [inst : Semiring R₂] [inst : AddCommMonoid M₁] [inst : 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_1} (f : α → M₁) (g : α → M₂) :

↑(LinearEquiv.symm e₁₂) ∘ g = f ↔ g = ↑e₁₂ ∘ f

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theorem LinearEquiv.eq_comp_toLinearMap_symm {R₁ : Type u_5} {R₂ : Type u_1} {R₃ : Type u_2} {M₁ : Type u_6} {M₂ : Type u_3} {M₃ : Type u_4} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [inst : 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

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theorem LinearEquiv.comp_toLinearMap_symm_eq {R₁ : Type u_5} {R₂ : Type u_1} {R₃ : Type u_2} {M₁ : Type u_6} {M₂ : Type u_3} {M₃ : Type u_4} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [inst : 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

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theorem LinearEquiv.eq_toLinearMap_symm_comp {R₁ : Type u_2} {R₂ : Type u_5} {R₃ : Type u_1} {M₁ : Type u_4} {M₂ : Type u_6} {M₃ : Type u_3} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [inst : 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

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theorem LinearEquiv.toLinearMap_symm_comp_eq {R₁ : Type u_2} {R₂ : Type u_5} {R₃ : Type u_1} {M₁ : Type u_4} {M₂ : Type u_6} {M₃ : Type u_3} [inst : Semiring R₁] [inst : Semiring R₂] [inst : Semiring R₃] [inst : AddCommMonoid M₁] [inst : AddCommMonoid M₂] [inst : 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} [inst : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁] {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} {e₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂} [inst : 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

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theorem LinearEquiv.refl_symm {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] :

LinearEquiv.symm (LinearEquiv.refl R M) = LinearEquiv.refl R M

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theorem LinearEquiv.self_trans_symm {R₁ : Type u_1} {R₂ : Type u_2} {M₁ : Type u_3} {M₂ : Type u_4} [inst : Semiring R₁] [inst : Semiring R₂] [inst : AddCommMonoid M₁] [inst : 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

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theorem LinearEquiv.symm_trans_self {R₁ : Type u_1} {R₂ : Type u_2} {M₁ : Type u_3} {M₂ : Type u_4} [inst : Semiring R₁] [inst : Semiring R₂] [inst : AddCommMonoid M₁] [inst : 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

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theorem LinearEquiv.refl_toLinearMap {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst : AddCommMonoid M] [inst : Module R M] :

↑(LinearEquiv.refl R M) = LinearMap.id

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theorem LinearEquiv.comp_coe {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} {M₃ : Type u_4} [inst : Semiring R] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : AddCommMonoid M₃] [inst : Module R M] [inst : Module R M₂] [inst : Module R M₃] (f : M ≃ₗ[R] M₂) (f' : M₂ ≃ₗ[R] M₃) :

LinearMap.comp ↑f' ↑f = ↑(LinearEquiv.trans f f')

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theorem LinearEquiv.mk_coe {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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).toAddHom.toFun) (h₂ : Function.RightInverse f (↑e).toAddHom.toFun) :

{ toLinearMap := ↑e, invFun := f, left_inv := h₁, right_inv := h₂ } = e

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theorem LinearEquiv.map_add {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.map_zero {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.map_smulₛₗ {R : Type u_2} {S : Type u_4} {M : Type u_3} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.map_smul {R₁ : Type u_1} {N₁ : Type u_2} {N₂ : Type u_3} [inst : Semiring R₁] [inst : AddCommMonoid N₁] [inst : 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

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theorem LinearEquiv.map_eq_zero_iff {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.map_ne_zero_iff {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.symm_symm {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.symm_bijective {R : Type u_1} {S : Type u_3} {M : Type u_2} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] {σ : R →+* S} {σ' : S →+* R} [inst : Module R M] [inst : Module S M₂] [inst : RingHomInvPair σ' σ] [inst : RingHomInvPair σ σ'] :

Function.Bijective LinearEquiv.symm

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theorem LinearEquiv.mk_coe' {R : Type u_4} {S : Type u_3} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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), AddHom.toFun { toFun := f, map_add' := h₁ } (r • x) = ↑σ' r • AddHom.toFun { toFun := f, map_add' := h₁ } 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

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theorem LinearEquiv.symm_mk {R : Type u_3} {S : Type u_4} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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), AddHom.toFun { toFun := ↑e, map_add' := h₁ } (r • x) = ↑σ r • AddHom.toFun { toFun := ↑e, map_add' := h₁ } 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), AddHom.toFun (↑src).toAddHom (x + y) = AddHom.toFun (↑src).toAddHom x + AddHom.toFun (↑src).toAddHom y) }, map_smul' := (_ : ∀ (r : S) (x : M₂), AddHom.toFun (↑src).toAddHom (r • x) = ↑σ' r • AddHom.toFun (↑src).toAddHom x) }, invFun := ↑e, left_inv := (_ : Function.LeftInverse src.invFun (↑src).toAddHom.toFun), right_inv := (_ : Function.RightInverse src.invFun (↑src).toAddHom.toFun) }

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theorem LinearEquiv.coe_symm_mk {R : Type u_1} {M : Type u_2} {M₂ : Type u_3} [inst : Semiring R] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : 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), AddHom.toFun { toFun := to_fun, map_add' := map_add } (r • x) = ↑(RingHom.id R) r • AddHom.toFun { toFun := to_fun, map_add' := map_add } 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

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theorem LinearEquiv.bijective {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.injective {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.surjective {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.image_eq_preimage {R : Type u_3} {S : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem LinearEquiv.image_symm_eq_preimage {R : Type u_4} {S : Type u_3} {M : Type u_2} {M₂ : Type u_1} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : 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

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theorem RingEquiv.toSemilinearEquiv_symm_apply {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] (f : R ≃+* S) :

∀ (a : S), ↑(LinearEquiv.symm (RingEquiv.toSemilinearEquiv f)) a = Equiv.invFun f.toEquiv a

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theorem RingEquiv.toSemilinearEquiv_apply {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] (f : R ≃+* S) (a : R) :

↑(RingEquiv.toSemilinearEquiv f) a = ↑f a

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def RingEquiv.toSemilinearEquiv {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] (f : R ≃+* S) :

R ≃ₛₗ[↑f] S

Interpret a RingEquiv f as an f-semilinear equiv.

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

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def LinearEquiv.ofInvolutive {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst : AddCommMonoid M] {σ : R →+* R} {σ' : R →+* R} [inst : RingHomInvPair σ σ'] [inst : 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.

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

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theorem LinearEquiv.coe_ofInvolutive {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst : AddCommMonoid M] {σ : R →+* R} {σ' : R →+* R} [inst : RingHomInvPair σ σ'] [inst : RingHomInvPair σ' σ] :

∀ {x : Module R M} (f : M →ₛₗ[σ] M) (hf : Function.Involutive ↑f), ↑(LinearEquiv.ofInvolutive f hf) = ↑f

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theorem LinearEquiv.restrictScalars_symm_apply (R : Type u_1) {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module R M₂] [inst : Module S M] [inst : Module S M₂] [inst : LinearMap.CompatibleSMul M M₂ R S] (f : M ≃ₗ[S] M₂) (a : M₂) :

↑(LinearEquiv.symm (LinearEquiv.restrictScalars R f)) a = ↑(LinearEquiv.symm f) a

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theorem LinearEquiv.restrictScalars_apply (R : Type u_1) {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module R M₂] [inst : Module S M] [inst : Module S M₂] [inst : LinearMap.CompatibleSMul M M₂ R S] (f : M ≃ₗ[S] M₂) (a : M) :

↑(LinearEquiv.restrictScalars R f) a = ↑f a

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def LinearEquiv.restrictScalars (R : Type u_1) {S : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R] [inst : Semiring S] [inst : AddCommMonoid M] [inst : AddCommMonoid M₂] [inst : Module R M] [inst : Module R M₂] [inst : Module S M] [inst : Module S M₂] [inst : 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.