HepLean 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.

source
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 : R →ₛₗ[σ] M} {g : R →ₛₗ[σ] M} :
f = g ↔ f 1 = g 1
source
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 : R →ₛₗ[σ] M} {g : R →ₛₗ[σ] M} (h : f 1 = g 1) :
f = g
source
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
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑(MulOpposite.opLinearEquiv R) = MulOpposite.op
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_symm (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑(MulOpposite.opLinearEquiv R).symm = MulOpposite.unop
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_toLinearMap (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑↑(MulOpposite.opLinearEquiv R) = MulOpposite.op
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_symm_toLinearMap (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑↑(MulOpposite.opLinearEquiv R).symm = MulOpposite.unop
    source
    theorem MulOpposite.opLinearEquiv_toAddEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    (MulOpposite.opLinearEquiv R).toAddEquiv = MulOpposite.opAddEquiv
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_addEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ↑(MulOpposite.opLinearEquiv R) = MulOpposite.opAddEquiv
    source
    theorem MulOpposite.opLinearEquiv_symm_toAddEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    (MulOpposite.opLinearEquiv R).symm.toAddEquiv = MulOpposite.opAddEquiv.symm
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_symm_addEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ↑(MulOpposite.opLinearEquiv R).symm = MulOpposite.opAddEquiv.symm