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.

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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
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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
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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.

Instances For
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    @[simp]
    theorem MulOpposite.coe_opLinearEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑(opLinearEquiv R) = op
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_symm (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑(opLinearEquiv R).symm = unop
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_toLinearMap (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑↑(opLinearEquiv R) = op
    source
    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_symm_toLinearMap (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑↑(opLinearEquiv R).symm = unop
    source
    theorem MulOpposite.opLinearEquiv_toAddEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    (opLinearEquiv R).toAddEquiv = opAddEquiv
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    @[simp]
    theorem MulOpposite.coe_opLinearEquiv_addEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    ↑(opLinearEquiv R) = opAddEquiv
    source
    theorem MulOpposite.opLinearEquiv_symm_toAddEquiv (R : Type u) {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] :
    (opLinearEquiv R).symm.toAddEquiv = opAddEquiv.symm
    source
    @[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