Transfer algebraic structures across Equivs

In this file we prove theorems of the following form: if β has a group structure and α ≃ β then α has a group structure, and similarly for monoids, semigroups, rings, integral domains, fields and so on.

Note that most of these constructions can also be obtained using the transport tactic.

Implementation details

When adding new definitions that transfer type-classes across an equivalence, please mark them @[reducible]. See note [reducible non-instances].

Tags

equiv, group, ring, field, module, algebra

def Equiv.zero {α : Type u} {β : Type v} (e : α ≃ β) [Zero β] : Zero α

Transfer Zero across an Equiv

@[reducible]

def Equiv.one {α : Type u} {β : Type v} (e : α ≃ β) [One β] : One α

Transfer One across an Equiv

theorem Equiv.zero_def {α : Type u} {β : Type v} (e : α ≃ β) [Zero β] : 0 = ↑e.symm 0

theorem Equiv.one_def {α : Type u} {β : Type v} (e : α ≃ β) [One β] : 1 = ↑e.symm 1

def Equiv.add {α : Type u} {β : Type v} (e : α ≃ β) [Add β] : Add α

Transfer Add across an Equiv

@[reducible]

def Equiv.mul {α : Type u} {β : Type v} (e : α ≃ β) [Mul β] : Mul α

Transfer Mul across an Equiv

theorem Equiv.add_def {α : Type u} {β : Type v} (e : α ≃ β) [Add β] (x : α) (y : α) : x + y = ↑e.symm (↑e x + ↑e y)

theorem Equiv.mul_def {α : Type u} {β : Type v} (e : α ≃ β) [Mul β] (x : α) (y : α) : x * y = ↑e.symm (↑e x * ↑e y)

def Equiv.sub {α : Type u} {β : Type v} (e : α ≃ β) [Sub β] : Sub α

Transfer Sub across an Equiv

@[reducible]

def Equiv.div {α : Type u} {β : Type v} (e : α ≃ β) [Div β] : Div α

Transfer Div across an Equiv

theorem Equiv.sub_def {α : Type u} {β : Type v} (e : α ≃ β) [Sub β] (x : α) (y : α) : x - y = ↑e.symm (↑e x - ↑e y)

theorem Equiv.div_def {α : Type u} {β : Type v} (e : α ≃ β) [Div β] (x : α) (y : α) : x / y = ↑e.symm (↑e x / ↑e y)

def Equiv.Neg {α : Type u} {β : Type v} (e : α ≃ β) [Neg β] : Neg α

Transfer Neg across an Equiv

@[reducible]

def Equiv.Inv {α : Type u} {β : Type v} (e : α ≃ β) [Inv β] : Inv α

Transfer Inv across an Equiv

theorem Equiv.neg_def {α : Type u} {β : Type v} (e : α ≃ β) [Neg β] (x : α) : -x = ↑e.symm (-↑e x)

theorem Equiv.inv_def {α : Type u} {β : Type v} (e : α ≃ β) [Inv β] (x : α) : x⁻¹ = ↑e.symm (↑e x)⁻¹

@[reducible]

def Equiv.smul {α : Type u} {β : Type v} (e : α ≃ β) (R : Type u_1) [SMul R β] : SMul R α

Transfer SMul across an Equiv

theorem Equiv.smul_def {α : Type u} {β : Type v} (e : α ≃ β) {R : Type u_1} [SMul R β] (r : R) (x : α) : r • x = ↑e.symm (r • ↑e x)

@[reducible]

def Equiv.pow {α : Type u} {β : Type v} (e : α ≃ β) (N : Type u_1) [Pow β N] : Pow α N

Transfer Pow across an Equiv

theorem Equiv.pow_def {α : Type u} {β : Type v} (e : α ≃ β) {N : Type u_1} [Pow β N] (n : N) (x : α) : x ^ n = ↑e.symm (↑e x ^ n)

theorem Equiv.addEquiv.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [Add β] (x : α) (y : α) : Equiv.toFun { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) } (x + y) = Equiv.toFun { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) } x + Equiv.toFun { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) } y

def Equiv.addEquiv {α : Type u} {β : Type v} (e : α ≃ β) [Add β] : let mul := Equiv.add e; α ≃+ β

An equivalence e : α ≃ β gives an additive equivalence α ≃+ β where the additive structure on α is the one obtained by transporting an additive structure on β back along e.

def Equiv.mulEquiv {α : Type u} {β : Type v} (e : α ≃ β) [Mul β] : let mul := Equiv.mul e; α ≃* β

An equivalence e : α ≃ β gives a multiplicative equivalence α ≃* β where the multiplicative structure on α is the one obtained by transporting a multiplicative structure on β back along e.

@[simp]

theorem Equiv.addEquiv_apply {α : Type u} {β : Type v} (e : α ≃ β) [Add β] (a : α) : ↑(Equiv.addEquiv e) a = ↑e a

@[simp]

theorem Equiv.mulEquiv_apply {α : Type u} {β : Type v} (e : α ≃ β) [Mul β] (a : α) : ↑(Equiv.mulEquiv e) a = ↑e a

theorem Equiv.addEquiv_symm_apply {α : Type u} {β : Type v} (e : α ≃ β) [Add β] (b : β) : ↑(AddEquiv.symm (Equiv.addEquiv e)) b = ↑e.symm b

theorem Equiv.mulEquiv_symm_apply {α : Type u} {β : Type v} (e : α ≃ β) [Mul β] (b : β) : ↑(MulEquiv.symm (Equiv.mulEquiv e)) b = ↑e.symm b

def Equiv.ringEquiv {α : Type u} {β : Type v} (e : α ≃ β) [Add β] [Mul β] : let add := Equiv.add e; let mul := Equiv.mul e; α ≃+* β

An equivalence e : α ≃ β gives a ring equivalence α ≃+* β where the ring structure on α is the one obtained by transporting a ring structure on β back along e.

@[simp]

theorem Equiv.ringEquiv_apply {α : Type u} {β : Type v} (e : α ≃ β) [Add β] [Mul β] (a : α) : ↑(Equiv.ringEquiv e) a = ↑e a

theorem Equiv.ringEquiv_symm_apply {α : Type u} {β : Type v} (e : α ≃ β) [Add β] [Mul β] (b : β) : ↑(RingEquiv.symm (Equiv.ringEquiv e)) b = ↑e.symm b

def Equiv.addSemigroup {α : Type u} {β : Type v} (e : α ≃ β) [AddSemigroup β] : AddSemigroup α

Transfer add_semigroup across an Equiv

theorem Equiv.addSemigroup.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddSemigroup β] : ∀ (x y : α), ↑e (↑e.symm (↑e x + ↑e y)) = ↑e x + ↑e y

@[reducible]

def Equiv.semigroup {α : Type u} {β : Type v} (e : α ≃ β) [Semigroup β] : Semigroup α

Transfer Semigroup across an Equiv

@[reducible]

def Equiv.semigroupWithZero {α : Type u} {β : Type v} (e : α ≃ β) [SemigroupWithZero β] : SemigroupWithZero α

Transfer SemigroupWithZero across an Equiv

theorem Equiv.addCommSemigroup.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommSemigroup β] : ∀ (x y : α), ↑e (↑e.symm (↑e x + ↑e y)) = ↑e x + ↑e y

def Equiv.addCommSemigroup {α : Type u} {β : Type v} (e : α ≃ β) [AddCommSemigroup β] : AddCommSemigroup α

Transfer AddCommSemigroup across an Equiv

@[reducible]

def Equiv.commSemigroup {α : Type u} {β : Type v} (e : α ≃ β) [CommSemigroup β] : CommSemigroup α

Transfer CommSemigroup across an Equiv

@[reducible]

def Equiv.mulZeroClass {α : Type u} {β : Type v} (e : α ≃ β) [MulZeroClass β] : MulZeroClass α

Transfer MulZeroClass across an Equiv

theorem Equiv.addZeroClass.proof_2 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddZeroClass β] : ∀ (x y : α), ↑e (↑e.symm (↑e x + ↑e y)) = ↑e x + ↑e y

theorem Equiv.addZeroClass.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddZeroClass β] : ↑e (↑e.symm 0) = 0

def Equiv.addZeroClass {α : Type u} {β : Type v} (e : α ≃ β) [AddZeroClass β] : AddZeroClass α

Transfer AddZeroClass across an Equiv

@[reducible]

def Equiv.mulOneClass {α : Type u} {β : Type v} (e : α ≃ β) [MulOneClass β] : MulOneClass α

Transfer MulOneClass across an Equiv

@[reducible]

def Equiv.mulZeroOneClass {α : Type u} {β : Type v} (e : α ≃ β) [MulZeroOneClass β] : MulZeroOneClass α

Transfer MulZeroOneClass across an Equiv

theorem Equiv.addMonoid.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddMonoid β] : ↑e (↑e.symm 0) = 0

theorem Equiv.addMonoid.proof_3 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddMonoid β] : ∀ (x : α) (n : ℕ), ↑e (↑e.symm (n • ↑e x)) = n • ↑e x

def Equiv.addMonoid {α : Type u} {β : Type v} (e : α ≃ β) [AddMonoid β] : AddMonoid α

Transfer AddMonoid across an Equiv

theorem Equiv.addMonoid.proof_2 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddMonoid β] : ∀ (x y : α), ↑e (↑e.symm (↑e x + ↑e y)) = ↑e x + ↑e y

@[reducible]

def Equiv.monoid {α : Type u} {β : Type v} (e : α ≃ β) [Monoid β] : Monoid α

Transfer Monoid across an Equiv

theorem Equiv.addCommMonoid.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommMonoid β] : ↑e (↑e.symm 0) = 0

theorem Equiv.addCommMonoid.proof_3 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommMonoid β] : ∀ (x : α) (n : ℕ), ↑e (↑e.symm (n • ↑e x)) = n • ↑e x

def Equiv.addCommMonoid {α : Type u} {β : Type v} (e : α ≃ β) [AddCommMonoid β] : AddCommMonoid α

Transfer AddCommMonoid across an Equiv

theorem Equiv.addCommMonoid.proof_2 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommMonoid β] : ∀ (x y : α), ↑e (↑e.symm (↑e x + ↑e y)) = ↑e x + ↑e y

@[reducible]

def Equiv.commMonoid {α : Type u} {β : Type v} (e : α ≃ β) [CommMonoid β] : CommMonoid α

Transfer CommMonoid across an Equiv

theorem Equiv.addGroup.proof_3 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddGroup β] : ∀ (x : α), ↑e (↑e.symm (-↑e x)) = -↑e x

theorem Equiv.addGroup.proof_5 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddGroup β] : ∀ (x : α) (n : ℕ), ↑e (↑e.symm (n • ↑e x)) = n • ↑e x

theorem Equiv.addGroup.proof_2 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddGroup β] : ∀ (x y : α), ↑e (↑e.symm (↑e x + ↑e y)) = ↑e x + ↑e y

theorem Equiv.addGroup.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddGroup β] : ↑e (↑e.symm 0) = 0

theorem Equiv.addGroup.proof_6 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddGroup β] : ∀ (x : α) (n : ℤ), ↑e (↑e.symm (n • ↑e x)) = n • ↑e x

theorem Equiv.addGroup.proof_4 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddGroup β] : ∀ (x y : α), ↑e (↑e.symm (↑e x - ↑e y)) = ↑e x - ↑e y

def Equiv.addGroup {α : Type u} {β : Type v} (e : α ≃ β) [AddGroup β] : AddGroup α

Transfer AddGroup across an Equiv

@[reducible]

def Equiv.group {α : Type u} {β : Type v} (e : α ≃ β) [Group β] : Group α

Transfer Group across an Equiv

theorem Equiv.addCommGroup.proof_1 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommGroup β] : ↑e (↑e.symm 0) = 0

theorem Equiv.addCommGroup.proof_6 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommGroup β] : ∀ (x : α) (n : ℤ), ↑e (↑e.symm (n • ↑e x)) = n • ↑e x

theorem Equiv.addCommGroup.proof_2 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommGroup β] : ∀ (x y : α), ↑e (↑e.symm (↑e x + ↑e y)) = ↑e x + ↑e y

theorem Equiv.addCommGroup.proof_5 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommGroup β] : ∀ (x : α) (n : ℕ), ↑e (↑e.symm (n • ↑e x)) = n • ↑e x

theorem Equiv.addCommGroup.proof_4 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommGroup β] : ∀ (x y : α), ↑e (↑e.symm (↑e x - ↑e y)) = ↑e x - ↑e y

def Equiv.addCommGroup {α : Type u} {β : Type v} (e : α ≃ β) [AddCommGroup β] : AddCommGroup α

Transfer AddCommGroup across an Equiv

theorem Equiv.addCommGroup.proof_3 {α : Type u_2} {β : Type u_1} (e : α ≃ β) [AddCommGroup β] : ∀ (x : α), ↑e (↑e.symm (-↑e x)) = -↑e x

@[reducible]

def Equiv.commGroup {α : Type u} {β : Type v} (e : α ≃ β) [CommGroup β] : CommGroup α

Transfer CommGroup across an Equiv

@[reducible]

def Equiv.nonUnitalNonAssocSemiring {α : Type u} {β : Type v} (e : α ≃ β) [NonUnitalNonAssocSemiring β] : NonUnitalNonAssocSemiring α

Transfer NonUnitalNonAssocSemiring across an Equiv

@[reducible]

def Equiv.nonUnitalSemiring {α : Type u} {β : Type v} (e : α ≃ β) [NonUnitalSemiring β] : NonUnitalSemiring α

Transfer NonUnitalSemiring across an Equiv

@[reducible]

def Equiv.addMonoidWithOne {α : Type u} {β : Type v} (e : α ≃ β) [AddMonoidWithOne β] : AddMonoidWithOne α

Transfer AddMonoidWithOne across an Equiv

@[reducible]

def Equiv.addGroupWithOne {α : Type u} {β : Type v} (e : α ≃ β) [AddGroupWithOne β] : AddGroupWithOne α

Transfer AddGroupWithOne across an Equiv

@[reducible]

def Equiv.nonAssocSemiring {α : Type u} {β : Type v} (e : α ≃ β) [NonAssocSemiring β] : NonAssocSemiring α

Transfer NonAssocSemiring across an Equiv

@[reducible]

def Equiv.semiring {α : Type u} {β : Type v} (e : α ≃ β) [Semiring β] : Semiring α

Transfer Semiring across an Equiv

@[reducible]

def Equiv.nonUnitalCommSemiring {α : Type u} {β : Type v} (e : α ≃ β) [NonUnitalCommSemiring β] : NonUnitalCommSemiring α

Transfer NonUnitalCommSemiring across an Equiv

@[reducible]

def Equiv.commSemiring {α : Type u} {β : Type v} (e : α ≃ β) [CommSemiring β] : CommSemiring α

Transfer CommSemiring across an Equiv

@[reducible]

def Equiv.nonUnitalNonAssocRing {α : Type u} {β : Type v} (e : α ≃ β) [NonUnitalNonAssocRing β] : NonUnitalNonAssocRing α

Transfer NonUnitalNonAssocRing across an Equiv

@[reducible]

def Equiv.nonUnitalRing {α : Type u} {β : Type v} (e : α ≃ β) [NonUnitalRing β] : NonUnitalRing α

Transfer NonUnitalRing across an Equiv

@[reducible]

def Equiv.nonAssocRing {α : Type u} {β : Type v} (e : α ≃ β) [NonAssocRing β] : NonAssocRing α

Transfer NonAssocRing across an Equiv

@[reducible]

def Equiv.ring {α : Type u} {β : Type v} (e : α ≃ β) [Ring β] : Ring α

Transfer Ring across an Equiv

@[reducible]

def Equiv.nonUnitalCommRing {α : Type u} {β : Type v} (e : α ≃ β) [NonUnitalCommRing β] : NonUnitalCommRing α

Transfer NonUnitalCommRing across an Equiv

@[reducible]

def Equiv.commRing {α : Type u} {β : Type v} (e : α ≃ β) [CommRing β] : CommRing α

Transfer CommRing across an Equiv

@[reducible]

theorem Equiv.nontrivial {α : Type u} {β : Type v} (e : α ≃ β) [Nontrivial β] : Nontrivial α

Transfer Nontrivial across an Equiv

@[reducible]

theorem Equiv.isDomain {α : Type u} {β : Type v} [Ring α] [Ring β] [IsDomain β] (e : α ≃+* β) : IsDomain α

Transfer IsDomain across an Equiv

@[reducible]

def Equiv.RatCast {α : Type u} {β : Type v} (e : α ≃ β) [RatCast β] : RatCast α

Transfer RatCast across an Equiv

@[reducible]

def Equiv.divisionRing {α : Type u} {β : Type v} (e : α ≃ β) [DivisionRing β] : DivisionRing α

Transfer DivisionRing across an Equiv

@[reducible]

def Equiv.field {α : Type u} {β : Type v} (e : α ≃ β) [Field β] : Field α

Transfer Field across an Equiv

@[reducible]

def Equiv.mulAction {α : Type u} {β : Type v} (R : Type u_1) [Monoid R] (e : α ≃ β) [MulAction R β] : MulAction R α

Transfer MulAction across an Equiv

@[reducible]

def Equiv.distribMulAction {α : Type u} {β : Type v} (R : Type u_1) [Monoid R] (e : α ≃ β) [AddCommMonoid β] [DistribMulAction R β] : DistribMulAction R α

Transfer DistribMulAction across an Equiv

@[reducible]

def Equiv.module {α : Type u} {β : Type v} (R : Type u_1) [Semiring R] (e : α ≃ β) [AddCommMonoid β] : let addCommMonoid := Equiv.addCommMonoid e; [inst : Module R β] → Module R α

Transfer Module across an Equiv

def Equiv.linearEquiv {α : Type u} {β : Type v} (R : Type u_1) [Semiring R] (e : α ≃ β) [AddCommMonoid β] [Module R β] : let addCommMonoid := Equiv.addCommMonoid e; let module := Equiv.module R e; α ≃ₗ[R] β

An equivalence e : α ≃ β gives a linear equivalence α ≃ₗ[R] β where the R-module structure on α is the one obtained by transporting an R-module structure on β back along e.

@[reducible]

def Equiv.algebra {α : Type u} {β : Type v} (R : Type u_1) [CommSemiring R] (e : α ≃ β) [Semiring β] : let semiring := Equiv.semiring e; [inst : Algebra R β] → Algebra R α

Transfer Algebra across an Equiv

def Equiv.algEquiv {α : Type u} {β : Type v} (R : Type u_1) [CommSemiring R] (e : α ≃ β) [Semiring β] [Algebra R β] : let semiring := Equiv.semiring e; let algebra := Equiv.algebra R e; α ≃ₐ[R] β

An equivalence e : α ≃ β gives an algebra equivalence α ≃ₐ[R] β where the R-algebra structure on α is the one obtained by transporting an R-algebra structure on β back along e.