Documentation

Mathlib.Algebra.Module.Equiv.Opposite

Module operations on `Mᵐᵒᵖ` #

This file contains definitions that build on top of the group action definitions in Mathlib/Algebra/GroupWithZero/Action/Opposite.lean.

theorem LinearMap.ext_ring_op {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [AddCommMonoid M] [Module S M] {σ : Rᵐᵒᵖ →+* S} {f g : R →ₛₗ[σ] M} (h : f 1 = g 1) :

f = g

theorem LinearMap.ext_ring_op_iff {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [AddCommMonoid M] [Module S M] {σ : Rᵐᵒᵖ →+* S} {f g : R →ₛₗ[σ] M} :

f = g ↔ f 1 = g 1

def MulOpposite.opLinearEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

M ≃ₗ[R] Mᵐᵒᵖ

The function op is a linear equivalence.

Equations

MulOpposite.opLinearEquiv R = { toFun := MulOpposite.opAddEquiv.toFun, map_add' := ⋯, map_smul' := ⋯, invFun := MulOpposite.opAddEquiv.invFun, left_inv := ⋯, right_inv := ⋯ }

Instances For

@[simp]

theorem MulOpposite.coe_opLinearEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

⇑(opLinearEquiv R) = op

@[simp]

theorem MulOpposite.coe_opLinearEquiv_symm (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

⇑(opLinearEquiv R).symm = unop

@[simp]

theorem MulOpposite.coe_opLinearEquiv_toLinearMap (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

⇑↑(opLinearEquiv R) = op

@[simp]

theorem MulOpposite.coe_opLinearEquiv_symm_toLinearMap (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

⇑↑(opLinearEquiv R).symm = unop

theorem MulOpposite.opLinearEquiv_toAddEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

(opLinearEquiv R).toAddEquiv = opAddEquiv

@[simp]

theorem MulOpposite.coe_opLinearEquiv_addEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

↑(opLinearEquiv R) = opAddEquiv

theorem MulOpposite.opLinearEquiv_symm_toAddEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

(opLinearEquiv R).symm.toAddEquiv = opAddEquiv.symm

@[simp]

theorem MulOpposite.coe_opLinearEquiv_symm_addEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :

↑(opLinearEquiv R).symm = opAddEquiv.symm