data.equiv.ring - mathlib docs

structure ring_equiv (R : Type u_4) (S : Type u_5) [has_mul R] [has_add R] [has_mul S] [has_add S] :

Type (max u_4 u_5)

to_fun : R → S
inv_fun : S → R
left_inv : function.left_inverse c.inv_fun c.to_fun
right_inv : function.right_inverse c.inv_fun c.to_fun
map_mul' : ∀ (x y : R), c.to_fun (x * y) = (c.to_fun x) * c.to_fun y
map_add' : ∀ (x y : R), c.to_fun (x + y) = c.to_fun x + c.to_fun y

An equivalence between two (semi)rings that preserves the algebraic structure.

def ring_equiv.to_mul_equiv {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (self : R ≃+* S) :

R ≃* S

The equivalence of multiplicative monoids underlying an equivalence of (semi)rings.

source

def ring_equiv.to_add_equiv {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (self : R ≃+* S) :

R ≃+ S

The equivalence of additive monoids underlying an equivalence of (semi)rings.

source

def ring_equiv.to_equiv {R : Type u_4} {S : Type u_5} [has_mul R] [has_add R] [has_mul S] [has_add S] (self : R ≃+* S) :

R ≃ S

The "plain" equivalence of types underlying an equivalence of (semi)rings.

source

@[instance]

def ring_equiv.has_coe_to_fun {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] :

has_coe_to_fun (R ≃+* S)

Equations

ring_equiv.has_coe_to_fun = {F := λ (x : R ≃+* S), R → S, coe := ring_equiv.to_fun _inst_4}

source

@[simp]

theorem ring_equiv.to_fun_eq_coe {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :

f.to_fun = ⇑f

source

@[simp]

theorem ring_equiv.map_mul {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x y : R) :

⇑e (x * y) = (⇑e x) * ⇑e y

A ring isomorphism preserves multiplication.

source

@[simp]

theorem ring_equiv.map_add {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x y : R) :

⇑e (x + y) = ⇑e x + ⇑e y

A ring isomorphism preserves addition.

source

@[ext]

theorem ring_equiv.ext {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] {f g : R ≃+* S} (h : ∀ (x : R), ⇑f x = ⇑g x) :

f = g

Two ring isomorphisms agree if they are defined by the same underlying function.

source

@[simp]

theorem ring_equiv.coe_mk {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R → S) (e' : S → R) (h₁ : function.left_inverse e' e) (h₂ : function.right_inverse e' e) (h₃ : ∀ (x y : R), e (x * y) = (e x) * e y) (h₄ : ∀ (x y : R), e (x + y) = e x + e y) :

⇑{to_fun := e, inv_fun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄} = e

source

@[simp]

theorem ring_equiv.mk_coe {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (e' : S → R) (h₁ : function.left_inverse e' ⇑e) (h₂ : function.right_inverse e' ⇑e) (h₃ : ∀ (x y : R), ⇑e (x * y) = (⇑e x) * ⇑e y) (h₄ : ∀ (x y : R), ⇑e (x + y) = ⇑e x + ⇑e y) :

{to_fun := ⇑e, inv_fun := e', left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄} = e

source

theorem ring_equiv.congr_arg {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] {f : R ≃+* S} {x x' : R} :

x = x' → ⇑f x = ⇑f x'

source

theorem ring_equiv.congr_fun {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] {f g : R ≃+* S} (h : f = g) (x : R) :

⇑f x = ⇑g x

source

theorem ring_equiv.ext_iff {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] {f g : R ≃+* S} :

f = g ↔ ∀ (x : R), ⇑f x = ⇑g x

source

@[instance]

def ring_equiv.has_coe_to_mul_equiv {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] :

has_coe (R ≃+* S) (R ≃* S)

Equations

ring_equiv.has_coe_to_mul_equiv = {coe := ring_equiv.to_mul_equiv _inst_4}

source

@[instance]

def ring_equiv.has_coe_to_add_equiv {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] :

has_coe (R ≃+* S) (R ≃+ S)

Equations

ring_equiv.has_coe_to_add_equiv = {coe := ring_equiv.to_add_equiv _inst_4}

source

theorem ring_equiv.to_add_equiv_eq_coe {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :

f.to_add_equiv = ↑f

source

theorem ring_equiv.to_mul_equiv_eq_coe {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :

f.to_mul_equiv = ↑f

source

@[simp]

theorem ring_equiv.coe_to_mul_equiv {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :

⇑↑f = ⇑f

source

@[simp]

theorem ring_equiv.coe_to_add_equiv {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R ≃+* S) :

⇑↑f = ⇑f

source

def ring_equiv.ring_equiv_of_unique_of_unique {M : Type u_1} {N : Type u_2} [unique M] [unique N] [has_add M] [has_mul M] [has_add N] [has_mul N] :

M ≃+* N

The ring_equiv between two semirings with a unique element.

Equations

ring_equiv.ring_equiv_of_unique_of_unique = {to_fun := add_equiv.add_equiv_of_unique_of_unique.to_fun, inv_fun := add_equiv.add_equiv_of_unique_of_unique.inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[instance]

def ring_equiv.unique {M : Type u_1} {N : Type u_2} [unique M] [unique N] [has_add M] [has_mul M] [has_add N] [has_mul N] :

unique (M ≃+* N)

Equations

ring_equiv.unique = {to_inhabited := {default := ring_equiv.ring_equiv_of_unique_of_unique _inst_12}, uniq := _}

source

def ring_equiv.refl (R : Type u_1) [has_mul R] [has_add R] :

R ≃+* R

The identity map is a ring isomorphism.

Equations

ring_equiv.refl R = {to_fun := (mul_equiv.refl R).to_fun, inv_fun := (mul_equiv.refl R).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_equiv.refl_apply (R : Type u_1) [has_mul R] [has_add R] (x : R) :

⇑(ring_equiv.refl R) x = x

source

@[simp]

theorem ring_equiv.coe_add_equiv_refl (R : Type u_1) [has_mul R] [has_add R] :

↑(ring_equiv.refl R) = add_equiv.refl R

source

@[simp]

theorem ring_equiv.coe_mul_equiv_refl (R : Type u_1) [has_mul R] [has_add R] :

↑(ring_equiv.refl R) = mul_equiv.refl R

source

@[instance]

def ring_equiv.inhabited (R : Type u_1) [has_mul R] [has_add R] :

inhabited (R ≃+* R)

Equations

ring_equiv.inhabited R = {default := ring_equiv.refl R _inst_2}

source

def ring_equiv.symm {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :

S ≃+* R

The inverse of a ring isomorphism is a ring isomorphism.

Equations

e.symm = {to_fun := e.to_mul_equiv.symm.to_fun, inv_fun := e.to_mul_equiv.symm.inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

def ring_equiv.simps.symm_apply {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :

S → R

See Note [custom simps projection]

Equations

ring_equiv.simps.symm_apply e = ⇑(e.symm)

source

@[simp]

theorem ring_equiv.symm_symm {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :

e.symm.symm = e

source

theorem ring_equiv.symm_bijective {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] :

function.bijective ring_equiv.symm

source

@[simp]

theorem ring_equiv.mk_coe' {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (f : S → R) (h₁ : function.left_inverse ⇑e f) (h₂ : function.right_inverse ⇑e f) (h₃ : ∀ (x y : S), f (x * y) = (f x) * f y) (h₄ : ∀ (x y : S), f (x + y) = f x + f y) :

{to_fun := f, inv_fun := ⇑e, left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄} = e.symm

source

@[simp]

theorem ring_equiv.symm_mk {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (f : R → S) (g : S → R) (h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) (h₃ : ∀ (x y : R), f (x * y) = (f x) * f y) (h₄ : ∀ (x y : R), f (x + y) = f x + f y) :

{to_fun := f, inv_fun := g, left_inv := h₁, right_inv := h₂, map_mul' := h₃, map_add' := h₄}.symm = {to_fun := g, inv_fun := f, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

def ring_equiv.trans {R : Type u_1} {S : Type u_2} {S' : Type u_3} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :

R ≃+* S'

Transitivity of ring_equiv.

Equations

e₁.trans e₂ = {to_fun := (e₁.to_mul_equiv.trans e₂.to_mul_equiv).to_fun, inv_fun := (e₁.to_mul_equiv.trans e₂.to_mul_equiv).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_equiv.trans_apply {R : Type u_1} {S : Type u_2} {S' : Type u_3} [has_mul R] [has_add R] [has_mul S] [has_add S] [has_mul S'] [has_add S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : R) :

⇑(e₁.trans e₂) a = ⇑e₂ (⇑e₁ a)

source

theorem ring_equiv.bijective {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :

function.bijective ⇑e

source

theorem ring_equiv.injective {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :

function.injective ⇑e

source

theorem ring_equiv.surjective {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) :

function.surjective ⇑e

source

@[simp]

theorem ring_equiv.apply_symm_apply {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x : S) :

⇑e (⇑(e.symm) x) = x

source

@[simp]

theorem ring_equiv.symm_apply_apply {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (x : R) :

⇑(e.symm) (⇑e x) = x

source

theorem ring_equiv.image_eq_preimage {R : Type u_1} {S : Type u_2} [has_mul R] [has_add R] [has_mul S] [has_add S] (e : R ≃+* S) (s : set R) :

⇑e '' s = ⇑(e.symm) ⁻¹' s

source

def ring_equiv.to_opposite (R : Type u_1) [comm_semiring R] :

R ≃+* Rᵒᵖ

A commutative ring is isomorphic to its opposite.

Equations

ring_equiv.to_opposite R = {to_fun := opposite.equiv_to_opposite.to_fun, inv_fun := opposite.equiv_to_opposite.inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_equiv.to_opposite_apply (R : Type u_1) [comm_semiring R] (r : R) :

⇑(ring_equiv.to_opposite R) r = opposite.op r

source

@[simp]

theorem ring_equiv.to_opposite_symm_apply (R : Type u_1) [comm_semiring R] (r : Rᵒᵖ) :

⇑((ring_equiv.to_opposite R).symm) r = opposite.unop r

source

@[simp]

theorem ring_equiv.map_zero {R : Type u_1} {S : Type u_2} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) :

⇑f 0 = 0

A ring isomorphism sends zero to zero.

source

@[simp]

theorem ring_equiv.map_eq_zero_iff {R : Type u_1} {S : Type u_2} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) {x : R} :

⇑f x = 0 ↔ x = 0

source

theorem ring_equiv.map_ne_zero_iff {R : Type u_1} {S : Type u_2} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S] (f : R ≃+* S) {x : R} :

⇑f x ≠ 0 ↔ x ≠ 0

source

@[simp]

theorem ring_equiv.map_one {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :

⇑f 1 = 1

A ring isomorphism sends one to one.

source

@[simp]

theorem ring_equiv.map_eq_one_iff {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) {x : R} :

⇑f x = 1 ↔ x = 1

source

theorem ring_equiv.map_ne_one_iff {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) {x : R} :

⇑f x ≠ 1 ↔ x ≠ 1

source

def ring_equiv.of_bijective {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (hf : function.bijective ⇑f) :

R ≃+* S

Produce a ring isomorphism from a bijective ring homomorphism.

Equations

ring_equiv.of_bijective f hf = {to_fun := (equiv.of_bijective ⇑f hf).to_fun, inv_fun := (equiv.of_bijective ⇑f hf).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_equiv.map_neg {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R ≃+* S) (x : R) :

⇑f (-x) = -⇑f x

source

@[simp]

theorem ring_equiv.map_sub {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R ≃+* S) (x y : R) :

⇑f (x - y) = ⇑f x - ⇑f y

source

@[simp]

theorem ring_equiv.map_neg_one {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R ≃+* S) :

⇑f (-1) = -1

source

def ring_equiv.to_ring_hom {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :

R →+* S

Reinterpret a ring equivalence as a ring homomorphism.

Equations

e.to_ring_hom = {to_fun := e.to_mul_equiv.to_monoid_hom.to_fun, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

source

theorem ring_equiv.to_ring_hom_injective {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] :

function.injective ring_equiv.to_ring_hom

source

@[instance]

def ring_equiv.has_coe_to_ring_hom {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] :

has_coe (R ≃+* S) (R →+* S)

Equations

ring_equiv.has_coe_to_ring_hom = {coe := ring_equiv.to_ring_hom _inst_2}

source

theorem ring_equiv.to_ring_hom_eq_coe {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :

f.to_ring_hom = ↑f

source

@[simp]

theorem ring_equiv.coe_to_ring_hom {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :

⇑↑f = ⇑f

source

theorem ring_equiv.coe_ring_hom_inj_iff {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f g : R ≃+* S) :

f = g ↔ ↑f = ↑g

source

def ring_equiv.to_monoid_hom {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :

R →* S

Reinterpret a ring equivalence as a monoid homomorphism.

source

def ring_equiv.to_add_monoid_hom {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :

R →+ S

Reinterpret a ring equivalence as an add_monoid homomorphism.

source

theorem ring_equiv.to_add_monoid_hom_commutes {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :

↑f.to_add_monoid_hom = ↑f.to_add_monoid_hom

The two paths coercion can take to an add_monoid_hom are equivalent

source

theorem ring_equiv.to_monoid_hom_commutes {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :

↑f.to_monoid_hom = ↑f.to_monoid_hom

The two paths coercion can take to an monoid_hom are equivalent

source

theorem ring_equiv.to_equiv_commutes {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) :

↑f.to_equiv = ↑f.to_equiv

The two paths coercion can take to an equiv are equivalent

source

@[simp]

theorem ring_equiv.to_ring_hom_refl {R : Type u_1} [non_assoc_semiring R] :

(ring_equiv.refl R).to_ring_hom = ring_hom.id R

source

@[simp]

theorem ring_equiv.to_monoid_hom_refl {R : Type u_1} [non_assoc_semiring R] :

(ring_equiv.refl R).to_monoid_hom = monoid_hom.id R

source

@[simp]

theorem ring_equiv.to_add_monoid_hom_refl {R : Type u_1} [non_assoc_semiring R] :

(ring_equiv.refl R).to_add_monoid_hom = add_monoid_hom.id R

source

@[simp]

theorem ring_equiv.to_ring_hom_apply_symm_to_ring_hom_apply {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) (y : S) :

⇑(e.to_ring_hom) (⇑(e.symm.to_ring_hom) y) = y

source

@[simp]

theorem ring_equiv.symm_to_ring_hom_apply_to_ring_hom_apply {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) (x : R) :

⇑(e.symm.to_ring_hom) (⇑(e.to_ring_hom) x) = x

source

@[simp]

theorem ring_equiv.to_ring_hom_trans {R : Type u_1} {S : Type u_2} {S' : Type u_3} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') :

(e₁.trans e₂).to_ring_hom = e₂.to_ring_hom.comp e₁.to_ring_hom

source

@[simp]

theorem ring_equiv.to_ring_hom_comp_symm_to_ring_hom {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :

e.to_ring_hom.comp e.symm.to_ring_hom = ring_hom.id S

source

@[simp]

theorem ring_equiv.symm_to_ring_hom_comp_to_ring_hom {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) :

e.symm.to_ring_hom.comp e.to_ring_hom = ring_hom.id R

source

def ring_equiv.of_hom_inv {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (hom : R →+* S) (inv : S →+* R) (hom_inv_id : inv.comp hom = ring_hom.id R) (inv_hom_id : hom.comp inv = ring_hom.id S) :

R ≃+* S

Construct an equivalence of rings from homomorphisms in both directions, which are inverses.

Equations

ring_equiv.of_hom_inv hom inv hom_inv_id inv_hom_id = {to_fun := hom.to_fun, inv_fun := ⇑inv, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_equiv.of_hom_inv_apply {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (hom : R →+* S) (inv : S →+* R) (hom_inv_id : inv.comp hom = ring_hom.id R) (inv_hom_id : hom.comp inv = ring_hom.id S) (r : R) :

⇑(ring_equiv.of_hom_inv hom inv hom_inv_id inv_hom_id) r = ⇑hom r

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@[simp]

theorem ring_equiv.of_hom_inv_symm_apply {R : Type u_1} {S : Type u_2} [non_assoc_semiring R] [non_assoc_semiring S] (hom : R →+* S) (inv : S →+* R) (hom_inv_id : inv.comp hom = ring_hom.id R) (inv_hom_id : hom.comp inv = ring_hom.id S) (s : S) :

⇑((ring_equiv.of_hom_inv hom inv hom_inv_id inv_hom_id).symm) s = ⇑inv s

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def mul_equiv.to_ring_equiv {R : Type u_1} {S : Type u_2} [has_add R] [has_add S] [has_mul R] [has_mul S] (h : R ≃* S) (H : ∀ (x y : R), ⇑h (x + y) = ⇑h x + ⇑h y) :

R ≃+* S

Gives a ring_equiv from a mul_equiv preserving addition.

Equations

h.to_ring_equiv H = {to_fun := h.to_equiv.to_fun, inv_fun := h.to_equiv.inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

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@[simp]

theorem ring_equiv.trans_symm {R : Type u_1} {S : Type u_2} [has_add R] [has_add S] [has_mul R] [has_mul S] (e : R ≃+* S) :

e.trans e.symm = ring_equiv.refl R

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@[simp]

theorem ring_equiv.symm_trans {R : Type u_1} {S : Type u_2} [has_add R] [has_add S] [has_mul R] [has_mul S] (e : R ≃+* S) :

e.symm.trans e = ring_equiv.refl S

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theorem ring_equiv.is_integral_domain {A : Type u_1} (B : Type u_2) [ring A] [ring B] (hB : is_integral_domain B) (e : A ≃+* B) :

is_integral_domain A

If two rings are isomorphic, and the second is an integral domain, then so is the first.

source

def ring_equiv.integral_domain {A : Type u_1} (B : Type u_2) [ring A] [integral_domain B] (e : A ≃+* B) :

integral_domain A

If two rings are isomorphic, and the second is an integral domain, then so is the first.

Equations

ring_equiv.integral_domain B e = {add := ring.add _inst_5, add_assoc := _, zero := ring.zero _inst_5, zero_add := _, add_zero := _, nsmul := ring.nsmul _inst_5, nsmul_zero' := _, nsmul_succ' := _, neg := ring.neg _inst_5, sub := ring.sub _inst_5, sub_eq_add_neg := _, gsmul := ring.gsmul _inst_5, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, mul := ring.mul _inst_5, mul_assoc := _, one := ring.one _inst_5, one_mul := _, mul_one := _, npow := ring.npow _inst_5, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, mul_comm := _, exists_pair_ne := _, eq_zero_or_eq_zero_of_mul_eq_zero := _}

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def equiv.units_equiv_ne_zero (K : Type u_4) [division_ring K] :

units K ≃ ↥{a : K | a ≠ 0}

In a division ring K, the unit group units K is equivalent to the subtype of nonzero elements.

Equations

equiv.units_equiv_ne_zero K = {to_fun := λ (a : units K), ⟨a.val, _⟩, inv_fun := λ (a : ↥{a : K | a ≠ 0}), units.mk0 a.val _, left_inv := _, right_inv := _}

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@[simp]

theorem equiv.coe_units_equiv_ne_zero {K : Type u_4} [division_ring K] (a : units K) :

↑(⇑(equiv.units_equiv_ne_zero K) a) = ↑a

mathlib documentation

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