Bundled subsemirings #

We define bundled subsemirings and some standard constructions: CompleteLattice structure, Subtype and inclusion ring homomorphisms, subsemiring map, comap and range (rangeS) of a RingHom etc.

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class AddSubmonoidWithOneClass (S : Type u_1) (R : Type u_2) [AddMonoidWithOne R] [SetLike S R] extends AddSubmonoidClass , OneMemClass :

Prop

AddSubmonoidWithOneClass S R says S is a type of subsets s ≤ R that contain 0, 1, and are closed under (+)

Instances

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theorem natCast_mem {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] (s : S) [AddSubmonoidWithOneClass S R] (n : ℕ) :

↑n ∈ s

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instance AddSubmonoidWithOneClass.toAddMonoidWithOne {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] (s : S) [AddSubmonoidWithOneClass S R] :

AddMonoidWithOne { x // x ∈ s }

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class SubsemiringClass (S : Type u_1) (R : Type u) [NonAssocSemiring R] [SetLike S R] extends SubmonoidClass , AddSubmonoidClass :

Prop

SubsemiringClass S R states that S is a type of subsets s ⊆ R that are both a multiplicative and an additive submonoid.

Instances

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instance SubsemiringClass.addSubmonoidWithOneClass (S : Type u_1) (R : Type u) [NonAssocSemiring R] [SetLike S R] [h : SubsemiringClass S R] :

AddSubmonoidWithOneClass S R

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theorem coe_nat_mem {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) (n : ℕ) :

↑n ∈ s

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instance SubsemiringClass.toNonAssocSemiring {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) :

NonAssocSemiring { x // x ∈ s }

A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

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instance SubsemiringClass.nontrivial {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [Nontrivial R] :

Nontrivial { x // x ∈ s }

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instance SubsemiringClass.noZeroDivisors {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [NoZeroDivisors R] :

NoZeroDivisors { x // x ∈ s }

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def SubsemiringClass.subtype {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) :

{ x // x ∈ s } →+* R

The natural ring hom from a subsemiring of semiring R to R.

Instances For

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@[simp]

theorem SubsemiringClass.coe_subtype {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) :

↑(SubsemiringClass.subtype s) = Subtype.val

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instance SubsemiringClass.toSemiring {S : Type v} (s : S) {R : Type u_1} [Semiring R] [SetLike S R] [SubsemiringClass S R] :

Semiring { x // x ∈ s }

A subsemiring of a Semiring is a Semiring.

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@[simp]

theorem SubsemiringClass.coe_pow {S : Type v} (s : S) {R : Type u_1} [Semiring R] [SetLike S R] [SubsemiringClass S R] (x : { x // x ∈ s }) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

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instance SubsemiringClass.toCommSemiring {S : Type v} (s : S) {R : Type u_1} [CommSemiring R] [SetLike S R] [SubsemiringClass S R] :

CommSemiring { x // x ∈ s }

A subsemiring of a CommSemiring is a CommSemiring.

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instance SubsemiringClass.toOrderedSemiring {S : Type v} (s : S) {R : Type u_1} [OrderedSemiring R] [SetLike S R] [SubsemiringClass S R] :

OrderedSemiring { x // x ∈ s }

A subsemiring of an OrderedSemiring is an OrderedSemiring.

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instance SubsemiringClass.toStrictOrderedSemiring {S : Type v} (s : S) {R : Type u_1} [StrictOrderedSemiring R] [SetLike S R] [SubsemiringClass S R] :

StrictOrderedSemiring { x // x ∈ s }

A subsemiring of a StrictOrderedSemiring is a StrictOrderedSemiring.

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instance SubsemiringClass.toOrderedCommSemiring {S : Type v} (s : S) {R : Type u_1} [OrderedCommSemiring R] [SetLike S R] [SubsemiringClass S R] :

OrderedCommSemiring { x // x ∈ s }

A subsemiring of an OrderedCommSemiring is an OrderedCommSemiring.

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instance SubsemiringClass.toStrictOrderedCommSemiring {S : Type v} (s : S) {R : Type u_1} [StrictOrderedCommSemiring R] [SetLike S R] [SubsemiringClass S R] :

StrictOrderedCommSemiring { x // x ∈ s }

A subsemiring of a StrictOrderedCommSemiring is a StrictOrderedCommSemiring.

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instance SubsemiringClass.toLinearOrderedSemiring {S : Type v} (s : S) {R : Type u_1} [LinearOrderedSemiring R] [SetLike S R] [SubsemiringClass S R] :

LinearOrderedSemiring { x // x ∈ s }

A subsemiring of a LinearOrderedSemiring is a LinearOrderedSemiring.

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instance SubsemiringClass.toLinearOrderedCommSemiring {S : Type v} (s : S) {R : Type u_1} [LinearOrderedCommSemiring R] [SetLike S R] [SubsemiringClass S R] :

LinearOrderedCommSemiring { x // x ∈ s }

A subsemiring of a LinearOrderedCommSemiring is a LinearOrderedCommSemiring.

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instance SubsemiringClass.instCharZero {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [CharZero R] :

CharZero { x // x ∈ s }

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structure Subsemiring (R : Type u) [NonAssocSemiring R] extends Submonoid :

Type u

carrier : Set R
mul_mem' : ∀ {a b : R}, a ∈ s.carrier → b ∈ s.carrier → a * b ∈ s.carrier
one_mem' : 1 ∈ s.carrier
add_mem' : ∀ {a b : R}, a ∈ s.carrier → b ∈ s.carrier → a + b ∈ s.carrier
The sum of two elements of an additive subsemigroup belongs to the subsemigroup.
zero_mem' : 0 ∈ s.carrier
An additive submonoid contains 0.

A subsemiring of a semiring R is a subset s that is both a multiplicative and an additive submonoid.

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@[reducible]

abbrev Subsemiring.toAddSubmonoid {R : Type u} [NonAssocSemiring R] (self : Subsemiring R) :

AddSubmonoid R

Reinterpret a Subsemiring as an AddSubmonoid.

Instances For

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instance Subsemiring.instSetLikeSubsemiring {R : Type u} [NonAssocSemiring R] :

SetLike (Subsemiring R) R

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instance Subsemiring.instSubsemiringClassSubsemiringInstSetLikeSubsemiring {R : Type u} [NonAssocSemiring R] :

SubsemiringClass (Subsemiring R) R

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@[simp]

theorem Subsemiring.mem_toSubmonoid {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} :

x ∈ s.toSubmonoid ↔ x ∈ s

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theorem Subsemiring.mem_carrier {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} :

x ∈ s.carrier ↔ x ∈ s

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theorem Subsemiring.ext {R : Type u} [NonAssocSemiring R] {S : Subsemiring R} {T : Subsemiring R} (h : ∀ (x : R), x ∈ S ↔ x ∈ T) :

S = T

Two subsemirings are equal if they have the same elements.

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def Subsemiring.copy {R : Type u} [NonAssocSemiring R] (S : Subsemiring R) (s : Set R) (hs : s = ↑S) :

Subsemiring R

Copy of a subsemiring with a new carrier equal to the old one. Useful to fix definitional equalities.

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@[simp]

theorem Subsemiring.coe_copy {R : Type u} [NonAssocSemiring R] (S : Subsemiring R) (s : Set R) (hs : s = ↑S) :

↑(Subsemiring.copy S s hs) = s

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theorem Subsemiring.copy_eq {R : Type u} [NonAssocSemiring R] (S : Subsemiring R) (s : Set R) (hs : s = ↑S) :

Subsemiring.copy S s hs = S

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theorem Subsemiring.toSubmonoid_injective {R : Type u} [NonAssocSemiring R] :

Function.Injective Subsemiring.toSubmonoid

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theorem Subsemiring.toSubmonoid_strictMono {R : Type u} [NonAssocSemiring R] :

StrictMono Subsemiring.toSubmonoid

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theorem Subsemiring.toSubmonoid_mono {R : Type u} [NonAssocSemiring R] :

Monotone Subsemiring.toSubmonoid

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theorem Subsemiring.toAddSubmonoid_injective {R : Type u} [NonAssocSemiring R] :

Function.Injective Subsemiring.toAddSubmonoid

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theorem Subsemiring.toAddSubmonoid_strictMono {R : Type u} [NonAssocSemiring R] :

StrictMono Subsemiring.toAddSubmonoid

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theorem Subsemiring.toAddSubmonoid_mono {R : Type u} [NonAssocSemiring R] :

Monotone Subsemiring.toAddSubmonoid

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def Subsemiring.mk' {R : Type u} [NonAssocSemiring R] (s : Set R) (sm : Submonoid R) (hm : ↑sm = s) (sa : AddSubmonoid R) (ha : ↑sa = s) :

Subsemiring R

Construct a Subsemiring R from a set s, a submonoid sm, and an additive submonoid sa such that x ∈ s ↔ x ∈ sm ↔ x ∈ sa.

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@[simp]

theorem Subsemiring.coe_mk' {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) :

↑(Subsemiring.mk' s sm hm sa ha) = s

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@[simp]

theorem Subsemiring.mem_mk' {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) {x : R} :

x ∈ Subsemiring.mk' s sm hm sa ha ↔ x ∈ s

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@[simp]

theorem Subsemiring.mk'_toSubmonoid {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) :

(Subsemiring.mk' s sm hm sa ha).toSubmonoid = sm

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@[simp]

theorem Subsemiring.mk'_toAddSubmonoid {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) :

Subsemiring.toAddSubmonoid (Subsemiring.mk' s sm hm sa ha) = sa

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theorem Subsemiring.one_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

1 ∈ s

A subsemiring contains the semiring's 1.

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theorem Subsemiring.zero_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

0 ∈ s

A subsemiring contains the semiring's 0.

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theorem Subsemiring.mul_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {x : R} {y : R} :

x ∈ s → y ∈ s → x * y ∈ s

A subsemiring is closed under multiplication.

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theorem Subsemiring.add_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {x : R} {y : R} :

x ∈ s → y ∈ s → x + y ∈ s

A subsemiring is closed under addition.

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theorem Subsemiring.list_prod_mem {R : Type u_1} [Semiring R] (s : Subsemiring R) {l : List R} :

(∀ (x : R), x ∈ l → x ∈ s) → List.prod l ∈ s

Product of a list of elements in a Subsemiring is in the Subsemiring.

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theorem Subsemiring.list_sum_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {l : List R} :

(∀ (x : R), x ∈ l → x ∈ s) → List.sum l ∈ s

Sum of a list of elements in a Subsemiring is in the Subsemiring.

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theorem Subsemiring.multiset_prod_mem {R : Type u_1} [CommSemiring R] (s : Subsemiring R) (m : Multiset R) :

(∀ (a : R), a ∈ m → a ∈ s) → Multiset.prod m ∈ s

Product of a multiset of elements in a Subsemiring of a CommSemiring is in the Subsemiring.

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theorem Subsemiring.multiset_sum_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) (m : Multiset R) :

(∀ (a : R), a ∈ m → a ∈ s) → Multiset.sum m ∈ s

Sum of a multiset of elements in a Subsemiring of a Semiring is in the add_subsemiring.

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theorem Subsemiring.prod_mem {R : Type u_1} [CommSemiring R] (s : Subsemiring R) {ι : Type u_2} {t : Finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

(Finset.prod t fun i => f i) ∈ s

Product of elements of a subsemiring of a CommSemiring indexed by a Finset is in the subsemiring.

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theorem Subsemiring.sum_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {ι : Type u_1} {t : Finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

(Finset.sum t fun i => f i) ∈ s

Sum of elements in a Subsemiring of a Semiring indexed by a Finset is in the add_subsemiring.

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instance Subsemiring.toNonAssocSemiring {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

NonAssocSemiring { x // x ∈ s }

A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

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@[simp]

theorem Subsemiring.coe_one {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

↑1 = 1

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@[simp]

theorem Subsemiring.coe_zero {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

↑0 = 0

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@[simp]

theorem Subsemiring.coe_add {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) (x : { x // x ∈ s }) (y : { x // x ∈ s }) :

↑(x + y) = ↑x + ↑y

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@[simp]

theorem Subsemiring.coe_mul {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) (x : { x // x ∈ s }) (y : { x // x ∈ s }) :

↑(x * y) = ↑x * ↑y

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instance Subsemiring.nontrivial {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) [Nontrivial R] :

Nontrivial { x // x ∈ s }

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theorem Subsemiring.pow_mem {R : Type u_1} [Semiring R] (s : Subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) :

x ^ n ∈ s

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instance Subsemiring.noZeroDivisors {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) [NoZeroDivisors R] :

NoZeroDivisors { x // x ∈ s }

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instance Subsemiring.toSemiring {R : Type u_1} [Semiring R] (s : Subsemiring R) :

Semiring { x // x ∈ s }

A subsemiring of a Semiring is a Semiring.

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@[simp]

theorem Subsemiring.coe_pow {R : Type u_1} [Semiring R] (s : Subsemiring R) (x : { x // x ∈ s }) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

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instance Subsemiring.toCommSemiring {R : Type u_1} [CommSemiring R] (s : Subsemiring R) :

CommSemiring { x // x ∈ s }

A subsemiring of a CommSemiring is a CommSemiring.

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def Subsemiring.subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

{ x // x ∈ s } →+* R

The natural ring hom from a subsemiring of semiring R to R.

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@[simp]

theorem Subsemiring.coe_subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

↑(Subsemiring.subtype s) = Subtype.val

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instance Subsemiring.toOrderedSemiring {R : Type u_1} [OrderedSemiring R] (s : Subsemiring R) :

OrderedSemiring { x // x ∈ s }

A subsemiring of an OrderedSemiring is an OrderedSemiring.

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instance Subsemiring.toStrictOrderedSemiring {R : Type u_1} [StrictOrderedSemiring R] (s : Subsemiring R) :

StrictOrderedSemiring { x // x ∈ s }

A subsemiring of a StrictOrderedSemiring is a StrictOrderedSemiring.

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instance Subsemiring.toOrderedCommSemiring {R : Type u_1} [OrderedCommSemiring R] (s : Subsemiring R) :

OrderedCommSemiring { x // x ∈ s }

A subsemiring of an OrderedCommSemiring is an OrderedCommSemiring.

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instance Subsemiring.toStrictOrderedCommSemiring {R : Type u_1} [StrictOrderedCommSemiring R] (s : Subsemiring R) :

StrictOrderedCommSemiring { x // x ∈ s }

A subsemiring of a StrictOrderedCommSemiring is a StrictOrderedCommSemiring.

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instance Subsemiring.toLinearOrderedSemiring {R : Type u_1} [LinearOrderedSemiring R] (s : Subsemiring R) :

LinearOrderedSemiring { x // x ∈ s }

A subsemiring of a LinearOrderedSemiring is a LinearOrderedSemiring.

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instance Subsemiring.toLinearOrderedCommSemiring {R : Type u_1} [LinearOrderedCommSemiring R] (s : Subsemiring R) :

LinearOrderedCommSemiring { x // x ∈ s }

A subsemiring of a LinearOrderedCommSemiring is a LinearOrderedCommSemiring.

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theorem Subsemiring.nsmul_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) :

n • x ∈ s

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@[simp]

theorem Subsemiring.coe_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

↑s.toSubmonoid = ↑s

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@[simp]

theorem Subsemiring.coe_carrier_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

s.carrier = ↑s

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theorem Subsemiring.mem_toAddSubmonoid {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} :

x ∈ Subsemiring.toAddSubmonoid s ↔ x ∈ s

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theorem Subsemiring.coe_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

↑(Subsemiring.toAddSubmonoid s) = ↑s

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instance Subsemiring.instTopSubsemiring {R : Type u} [NonAssocSemiring R] :

Top (Subsemiring R)

The subsemiring R of the semiring R.

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@[simp]

theorem Subsemiring.mem_top {R : Type u} [NonAssocSemiring R] (x : R) :

x ∈ ⊤

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@[simp]

theorem Subsemiring.coe_top {R : Type u} [NonAssocSemiring R] :

↑⊤ = Set.univ

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@[simp]

theorem Subsemiring.topEquiv_symm_apply_coe {R : Type u} [NonAssocSemiring R] (r : R) :

↑(↑(RingEquiv.symm Subsemiring.topEquiv) r) = r

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@[simp]

theorem Subsemiring.topEquiv_apply {R : Type u} [NonAssocSemiring R] (r : { x // x ∈ ⊤ }) :

↑Subsemiring.topEquiv r = ↑r

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def Subsemiring.topEquiv {R : Type u} [NonAssocSemiring R] :

{ x // x ∈ ⊤ } ≃+* R

The ring equiv between the top element of Subsemiring R and R.

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def Subsemiring.comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring S) :

Subsemiring R

The preimage of a subsemiring along a ring homomorphism is a subsemiring.

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@[simp]

theorem Subsemiring.coe_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring S) (f : R →+* S) :

↑(Subsemiring.comap f s) = ↑f ⁻¹' ↑s

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@[simp]

theorem Subsemiring.mem_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Subsemiring S} {f : R →+* S} {x : R} :

x ∈ Subsemiring.comap f s ↔ ↑f x ∈ s

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theorem Subsemiring.comap_comap {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (s : Subsemiring T) (g : S →+* T) (f : R →+* S) :

Subsemiring.comap f (Subsemiring.comap g s) = Subsemiring.comap (RingHom.comp g f) s

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def Subsemiring.map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) :

Subsemiring S

The image of a subsemiring along a ring homomorphism is a subsemiring.

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@[simp]

theorem Subsemiring.coe_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) :

↑(Subsemiring.map f s) = ↑f '' ↑s

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@[simp]

theorem Subsemiring.mem_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {s : Subsemiring R} {y : S} :

y ∈ Subsemiring.map f s ↔ ∃ x, x ∈ s ∧ ↑f x = y

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@[simp]

theorem Subsemiring.map_id {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

Subsemiring.map (RingHom.id R) s = s

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theorem Subsemiring.map_map {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (s : Subsemiring R) (g : S →+* T) (f : R →+* S) :

Subsemiring.map g (Subsemiring.map f s) = Subsemiring.map (RingHom.comp g f) s

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theorem Subsemiring.map_le_iff_le_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {s : Subsemiring R} {t : Subsemiring S} :

Subsemiring.map f s ≤ t ↔ s ≤ Subsemiring.comap f t

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theorem Subsemiring.gc_map_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

GaloisConnection (Subsemiring.map f) (Subsemiring.comap f)

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noncomputable def Subsemiring.equivMapOfInjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (f : R →+* S) (hf : Function.Injective ↑f) :

{ x // x ∈ s } ≃+* { x // x ∈ Subsemiring.map f s }

A subsemiring is isomorphic to its image under an injective function

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@[simp]

theorem Subsemiring.coe_equivMapOfInjective_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (f : R →+* S) (hf : Function.Injective ↑f) (x : { x // x ∈ s }) :

↑(↑(Subsemiring.equivMapOfInjective s f hf) x) = ↑f ↑x

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def RingHom.rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

Subsemiring S

The range of a ring homomorphism is a subsemiring. See Note [range copy pattern].

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@[simp]

theorem RingHom.coe_rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

↑(RingHom.rangeS f) = Set.range ↑f

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@[simp]

theorem RingHom.mem_rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {y : S} :

y ∈ RingHom.rangeS f ↔ ∃ x, ↑f x = y

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theorem RingHom.rangeS_eq_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

RingHom.rangeS f = Subsemiring.map f ⊤

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theorem RingHom.mem_rangeS_self {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (x : R) :

↑f x ∈ RingHom.rangeS f

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theorem RingHom.map_rangeS {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (g : S →+* T) (f : R →+* S) :

Subsemiring.map g (RingHom.rangeS f) = RingHom.rangeS (RingHom.comp g f)

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instance RingHom.fintypeRangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] [Fintype R] [DecidableEq S] (f : R →+* S) :

Fintype { x // x ∈ RingHom.rangeS f }

The range of a morphism of semirings is a fintype, if the domain is a fintype. Note: this instance can form a diamond with Subtype.fintype in the presence of Fintype S.

source

instance Subsemiring.instBotSubsemiring {R : Type u} [NonAssocSemiring R] :

Bot (Subsemiring R)

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instance Subsemiring.instInhabitedSubsemiring {R : Type u} [NonAssocSemiring R] :

Inhabited (Subsemiring R)

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theorem Subsemiring.coe_bot {R : Type u} [NonAssocSemiring R] :

↑⊥ = Set.range Nat.cast

source

theorem Subsemiring.mem_bot {R : Type u} [NonAssocSemiring R] {x : R} :

x ∈ ⊥ ↔ ∃ n, ↑n = x

source

instance Subsemiring.instInfSubsemiring {R : Type u} [NonAssocSemiring R] :

Inf (Subsemiring R)

The inf of two subsemirings is their intersection.

source

@[simp]

theorem Subsemiring.coe_inf {R : Type u} [NonAssocSemiring R] (p : Subsemiring R) (p' : Subsemiring R) :

↑(p ⊓ p') = ↑p ∩ ↑p'

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@[simp]

theorem Subsemiring.mem_inf {R : Type u} [NonAssocSemiring R] {p : Subsemiring R} {p' : Subsemiring R} {x : R} :

x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

source

instance Subsemiring.instInfSetSubsemiring {R : Type u} [NonAssocSemiring R] :

InfSet (Subsemiring R)

source

@[simp]

theorem Subsemiring.coe_sInf {R : Type u} [NonAssocSemiring R] (S : Set (Subsemiring R)) :

↑(sInf S) = ⋂ (s : Subsemiring R) (_ : s ∈ S), ↑s

source

theorem Subsemiring.mem_sInf {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} {x : R} :

x ∈ sInf S ↔ ∀ (p : Subsemiring R), p ∈ S → x ∈ p

source

@[simp]

theorem Subsemiring.sInf_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Set (Subsemiring R)) :

(sInf s).toSubmonoid = ⨅ (t : Subsemiring R) (_ : t ∈ s), t.toSubmonoid

source

@[simp]

theorem Subsemiring.sInf_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (s : Set (Subsemiring R)) :

Subsemiring.toAddSubmonoid (sInf s) = ⨅ (t : Subsemiring R) (_ : t ∈ s), Subsemiring.toAddSubmonoid t

source

instance Subsemiring.instCompleteLatticeSubsemiring {R : Type u} [NonAssocSemiring R] :

CompleteLattice (Subsemiring R)

Subsemirings of a semiring form a complete lattice.

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theorem Subsemiring.eq_top_iff' {R : Type u} [NonAssocSemiring R] (A : Subsemiring R) :

A = ⊤ ↔ ∀ (x : R), x ∈ A

source

def Subsemiring.center (R : Type u_1) [Semiring R] :

Subsemiring R

The center of a semiring R is the set of elements that commute with everything in R

Instances For

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theorem Subsemiring.coe_center (R : Type u_1) [Semiring R] :

↑(Subsemiring.center R) = Set.center R

source

@[simp]

theorem Subsemiring.center_toSubmonoid (R : Type u_1) [Semiring R] :

(Subsemiring.center R).toSubmonoid = Submonoid.center R

source

theorem Subsemiring.mem_center_iff {R : Type u_1} [Semiring R] {z : R} :

z ∈ Subsemiring.center R ↔ ∀ (g : R), g * z = z * g

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instance Subsemiring.decidableMemCenter {R : Type u_1} [Semiring R] [DecidableEq R] [Fintype R] :

DecidablePred fun x => x ∈ Subsemiring.center R

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@[simp]

theorem Subsemiring.center_eq_top (R : Type u_1) [CommSemiring R] :

Subsemiring.center R = ⊤

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instance Subsemiring.commSemiring {R : Type u_1} [Semiring R] :

CommSemiring { x // x ∈ Subsemiring.center R }

The center is commutative.

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def Subsemiring.centralizer {R : Type u_1} [Semiring R] (s : Set R) :

Subsemiring R

The centralizer of a set as subsemiring.

Instances For

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@[simp]

theorem Subsemiring.coe_centralizer {R : Type u_1} [Semiring R] (s : Set R) :

↑(Subsemiring.centralizer s) = Set.centralizer s

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theorem Subsemiring.centralizer_toSubmonoid {R : Type u_1} [Semiring R] (s : Set R) :

(Subsemiring.centralizer s).toSubmonoid = Submonoid.centralizer s

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theorem Subsemiring.mem_centralizer_iff {R : Type u_1} [Semiring R] {s : Set R} {z : R} :

z ∈ Subsemiring.centralizer s ↔ ∀ (g : R), g ∈ s → g * z = z * g

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theorem Subsemiring.center_le_centralizer {R : Type u_1} [Semiring R] (s : Set R) :

Subsemiring.center R ≤ Subsemiring.centralizer s

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theorem Subsemiring.centralizer_le {R : Type u_1} [Semiring R] (s : Set R) (t : Set R) (h : s ⊆ t) :

Subsemiring.centralizer t ≤ Subsemiring.centralizer s

source

@[simp]

theorem Subsemiring.centralizer_eq_top_iff_subset {R : Type u_1} [Semiring R] {s : Set R} :

Subsemiring.centralizer s = ⊤ ↔ s ⊆ ↑(Subsemiring.center R)

source

@[simp]

theorem Subsemiring.centralizer_univ {R : Type u_1} [Semiring R] :

Subsemiring.centralizer Set.univ = Subsemiring.center R

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def Subsemiring.closure {R : Type u} [NonAssocSemiring R] (s : Set R) :

Subsemiring R

The Subsemiring generated by a set.

Instances For

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theorem Subsemiring.mem_closure {R : Type u} [NonAssocSemiring R] {x : R} {s : Set R} :

x ∈ Subsemiring.closure s ↔ ∀ (S : Subsemiring R), s ⊆ ↑S → x ∈ S

source

@[simp]

theorem Subsemiring.subset_closure {R : Type u} [NonAssocSemiring R] {s : Set R} :

s ⊆ ↑(Subsemiring.closure s)

The subsemiring generated by a set includes the set.

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theorem Subsemiring.not_mem_of_not_mem_closure {R : Type u} [NonAssocSemiring R] {s : Set R} {P : R} (hP : ¬P ∈ Subsemiring.closure s) :

¬P ∈ s

source

@[simp]

theorem Subsemiring.closure_le {R : Type u} [NonAssocSemiring R] {s : Set R} {t : Subsemiring R} :

Subsemiring.closure s ≤ t ↔ s ⊆ ↑t

A subsemiring S includes closure s if and only if it includes s.

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theorem Subsemiring.closure_mono {R : Type u} [NonAssocSemiring R] ⦃s : Set R⦄ ⦃t : Set R⦄ (h : s ⊆ t) :

Subsemiring.closure s ≤ Subsemiring.closure t

Subsemiring closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

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theorem Subsemiring.closure_eq_of_le {R : Type u} [NonAssocSemiring R] {s : Set R} {t : Subsemiring R} (h₁ : s ⊆ ↑t) (h₂ : t ≤ Subsemiring.closure s) :

Subsemiring.closure s = t

source

theorem Subsemiring.mem_map_equiv {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R ≃+* S} {K : Subsemiring R} {x : S} :

x ∈ Subsemiring.map (↑f) K ↔ ↑(RingEquiv.symm f) x ∈ K

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theorem Subsemiring.map_equiv_eq_comap_symm {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (K : Subsemiring R) :

Subsemiring.map (↑f) K = Subsemiring.comap (↑(RingEquiv.symm f)) K

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theorem Subsemiring.comap_equiv_eq_map_symm {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (K : Subsemiring S) :

Subsemiring.comap (↑f) K = Subsemiring.map (↑(RingEquiv.symm f)) K

source

def Submonoid.subsemiringClosure {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :

Subsemiring R

The additive closure of a submonoid is a subsemiring.

Instances For

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theorem Submonoid.subsemiringClosure_coe {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :

↑(Submonoid.subsemiringClosure M) = ↑(AddSubmonoid.closure ↑M)

source

theorem Submonoid.subsemiringClosure_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :

Subsemiring.toAddSubmonoid (Submonoid.subsemiringClosure M) = AddSubmonoid.closure ↑M

source

theorem Submonoid.subsemiringClosure_eq_closure {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :

Submonoid.subsemiringClosure M = Subsemiring.closure ↑M

The Subsemiring generated by a multiplicative submonoid coincides with the Subsemiring.closure of the submonoid itself .

source

@[simp]

theorem Subsemiring.closure_submonoid_closure {R : Type u} [NonAssocSemiring R] (s : Set R) :

Subsemiring.closure ↑(Submonoid.closure s) = Subsemiring.closure s

source

theorem Subsemiring.coe_closure_eq {R : Type u} [NonAssocSemiring R] (s : Set R) :

↑(Subsemiring.closure s) = ↑(AddSubmonoid.closure ↑(Submonoid.closure s))

The elements of the subsemiring closure of M are exactly the elements of the additive closure of a multiplicative submonoid M.

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theorem Subsemiring.mem_closure_iff {R : Type u} [NonAssocSemiring R] {s : Set R} {x : R} :

x ∈ Subsemiring.closure s ↔ x ∈ AddSubmonoid.closure ↑(Submonoid.closure s)

source

@[simp]

theorem Subsemiring.closure_addSubmonoid_closure {R : Type u} [NonAssocSemiring R] {s : Set R} :

Subsemiring.closure ↑(AddSubmonoid.closure s) = Subsemiring.closure s

source

theorem Subsemiring.closure_induction {R : Type u} [NonAssocSemiring R] {s : Set R} {p : R → Prop} {x : R} (h : x ∈ Subsemiring.closure s) (Hs : (x : R) → x ∈ s → p x) (H0 : p 0) (H1 : p 1) (Hadd : (x y : R) → p x → p y → p (x + y)) (Hmul : (x y : R) → p x → p y → p (x * y)) :

p x

An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition and multiplication, then p holds for all elements of the closure of s.

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theorem Subsemiring.closure_induction' {R : Type u} [NonAssocSemiring R] {s : Set R} {p : (x : R) → x ∈ Subsemiring.closure s → Prop} (Hs : (x : R) → (h : x ∈ s) → p x (_ : x ∈ ↑(Subsemiring.closure s))) (H0 : p 0 (_ : 0 ∈ Subsemiring.closure s)) (H1 : p 1 (_ : 1 ∈ Subsemiring.closure s)) (Hadd : (x : R) → (hx : x ∈ Subsemiring.closure s) → (y : R) → (hy : y ∈ Subsemiring.closure s) → p x hx → p y hy → p (x + y) (_ : x + y ∈ Subsemiring.closure s)) (Hmul : (x : R) → (hx : x ∈ Subsemiring.closure s) → (y : R) → (hy : y ∈ Subsemiring.closure s) → p x hx → p y hy → p (x * y) (_ : x * y ∈ Subsemiring.closure s)) {a : R} (ha : a ∈ Subsemiring.closure s) :

p a ha

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theorem Subsemiring.closure_induction₂ {R : Type u} [NonAssocSemiring R] {s : Set R} {p : R → R → Prop} {x : R} {y : R} (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s) (Hs : (x : R) → x ∈ s → (y : R) → y ∈ s → p x y) (H0_left : (x : R) → p 0 x) (H0_right : (x : R) → p x 0) (H1_left : (x : R) → p 1 x) (H1_right : (x : R) → p x 1) (Hadd_left : (x₁ x₂ y : R) → p x₁ y → p x₂ y → p (x₁ + x₂) y) (Hadd_right : (x y₁ y₂ : R) → p x y₁ → p x y₂ → p x (y₁ + y₂)) (Hmul_left : (x₁ x₂ y : R) → p x₁ y → p x₂ y → p (x₁ * x₂) y) (Hmul_right : (x y₁ y₂ : R) → p x y₁ → p x y₂ → p x (y₁ * y₂)) :

p x y

An induction principle for closure membership for predicates with two arguments.

source

theorem Subsemiring.mem_closure_iff_exists_list {R : Type u_1} [Semiring R] {s : Set R} {x : R} :

x ∈ Subsemiring.closure s ↔ ∃ L, (∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s) ∧ List.sum (List.map List.prod L) = x

source

def Subsemiring.gi (R : Type u) [NonAssocSemiring R] :

GaloisInsertion Subsemiring.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Instances For

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theorem Subsemiring.closure_eq {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

Subsemiring.closure ↑s = s

Closure of a subsemiring S equals S.

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@[simp]

theorem Subsemiring.closure_empty {R : Type u} [NonAssocSemiring R] :

Subsemiring.closure ∅ = ⊥

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@[simp]

theorem Subsemiring.closure_univ {R : Type u} [NonAssocSemiring R] :

Subsemiring.closure Set.univ = ⊤

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theorem Subsemiring.closure_union {R : Type u} [NonAssocSemiring R] (s : Set R) (t : Set R) :

Subsemiring.closure (s ∪ t) = Subsemiring.closure s ⊔ Subsemiring.closure t

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theorem Subsemiring.closure_iUnion {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} (s : ι → Set R) :

Subsemiring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), Subsemiring.closure (s i)

source

theorem Subsemiring.closure_sUnion {R : Type u} [NonAssocSemiring R] (s : Set (Set R)) :

Subsemiring.closure (⋃₀ s) = ⨆ (t : Set R) (_ : t ∈ s), Subsemiring.closure t

source

theorem Subsemiring.map_sup {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring R) (f : R →+* S) :

Subsemiring.map f (s ⊔ t) = Subsemiring.map f s ⊔ Subsemiring.map f t

source

theorem Subsemiring.map_iSup {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subsemiring R) :

Subsemiring.map f (iSup s) = ⨆ (i : ι), Subsemiring.map f (s i)

source

theorem Subsemiring.comap_inf {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring S) (t : Subsemiring S) (f : R →+* S) :

Subsemiring.comap f (s ⊓ t) = Subsemiring.comap f s ⊓ Subsemiring.comap f t

source

theorem Subsemiring.comap_iInf {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subsemiring S) :

Subsemiring.comap f (iInf s) = ⨅ (i : ι), Subsemiring.comap f (s i)

source

@[simp]

theorem Subsemiring.map_bot {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

Subsemiring.map f ⊥ = ⊥

source

@[simp]

theorem Subsemiring.comap_top {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

Subsemiring.comap f ⊤ = ⊤

source

def Subsemiring.prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) :

Subsemiring (R × S)

Given Subsemirings s, t of semirings R, S respectively, s.prod t is s × t as a subsemiring of R × S.

Instances For

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theorem Subsemiring.coe_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) :

↑(Subsemiring.prod s t) = ↑s ×ˢ ↑t

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theorem Subsemiring.mem_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Subsemiring R} {t : Subsemiring S} {p : R × S} :

p ∈ Subsemiring.prod s t ↔ p.fst ∈ s ∧ p.snd ∈ t

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theorem Subsemiring.prod_mono {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] ⦃s₁ : Subsemiring R⦄ ⦃s₂ : Subsemiring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ : Subsemiring S⦄ ⦃t₂ : Subsemiring S⦄ (ht : t₁ ≤ t₂) :

Subsemiring.prod s₁ t₁ ≤ Subsemiring.prod s₂ t₂

source

theorem Subsemiring.prod_mono_right {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) :

Monotone fun t => Subsemiring.prod s t

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theorem Subsemiring.prod_mono_left {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (t : Subsemiring S) :

Monotone fun s => Subsemiring.prod s t

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theorem Subsemiring.prod_top {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) :

Subsemiring.prod s ⊤ = Subsemiring.comap (RingHom.fst R S) s

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theorem Subsemiring.top_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring S) :

Subsemiring.prod ⊤ s = Subsemiring.comap (RingHom.snd R S) s

source

@[simp]

theorem Subsemiring.top_prod_top {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] :

Subsemiring.prod ⊤ ⊤ = ⊤

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def Subsemiring.prodEquiv {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) :

{ x // x ∈ Subsemiring.prod s t } ≃+* { x // x ∈ s } × { x // x ∈ t }

Product of subsemirings is isomorphic to their product as monoids.

Instances For

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theorem Subsemiring.mem_iSup_of_directed {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subsemiring R} (hS : Directed (fun x x_1 => x ≤ x_1) S) {x : R} :

x ∈ ⨆ (i : ι), S i ↔ ∃ i, x ∈ S i

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theorem Subsemiring.coe_iSup_of_directed {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subsemiring R} (hS : Directed (fun x x_1 => x ≤ x_1) S) :

↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

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theorem Subsemiring.mem_sSup_of_directedOn {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} (Sne : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) {x : R} :

x ∈ sSup S ↔ ∃ s, s ∈ S ∧ x ∈ s

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theorem Subsemiring.coe_sSup_of_directedOn {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} (Sne : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) :

↑(sSup S) = ⋃ (s : Subsemiring R) (_ : s ∈ S), ↑s

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def RingHom.domRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} [SetLike σR R] [SubsemiringClass σR R] (f : R →+* S) (s : σR) :

{ x // x ∈ s } →+* S

Restriction of a ring homomorphism to a subsemiring of the domain.

Instances For

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@[simp]

theorem RingHom.restrict_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} [SetLike σR R] [SubsemiringClass σR R] (f : R →+* S) {s : σR} (x : { x // x ∈ s }) :

↑(RingHom.domRestrict f s) x = ↑f ↑x

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def RingHom.codRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σS : Type u_2} [SetLike σS S] [SubsemiringClass σS S] (f : R →+* S) (s : σS) (h : ∀ (x : R), ↑f x ∈ s) :

R →+* { x // x ∈ s }

Restriction of a ring homomorphism to a subsemiring of the codomain.

Instances For

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def RingHom.restrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ (x : R), x ∈ s' → ↑f x ∈ s) :

{ x // x ∈ s' } →+* { x // x ∈ s }

The ring homomorphism from the preimage of s to s.

Instances For

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@[simp]

theorem RingHom.coe_restrict_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ (x : R), x ∈ s' → ↑f x ∈ s) (x : { x // x ∈ s' }) :

↑(↑(RingHom.restrict f s' s h) x) = ↑f ↑x

source

@[simp]

theorem RingHom.comp_restrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ (x : R), x ∈ s' → ↑f x ∈ s) :

RingHom.comp (SubsemiringClass.subtype s) (RingHom.restrict f s' s h) = RingHom.comp f (SubsemiringClass.subtype s')

source

def RingHom.rangeSRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

R →+* { x // x ∈ RingHom.rangeS f }

Restriction of a ring homomorphism to its range interpreted as a subsemiring.

This is the bundled version of Set.rangeFactorization.

Instances For

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@[simp]

theorem RingHom.coe_rangeSRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (x : R) :

↑(↑(RingHom.rangeSRestrict f) x) = ↑f x

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theorem RingHom.rangeSRestrict_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

Function.Surjective ↑(RingHom.rangeSRestrict f)

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theorem RingHom.rangeS_top_iff_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} :

RingHom.rangeS f = ⊤ ↔ Function.Surjective ↑f

source

@[simp]

theorem RingHom.rangeS_top_of_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (hf : Function.Surjective ↑f) :

RingHom.rangeS f = ⊤

The range of a surjective ring homomorphism is the whole of the codomain.

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def RingHom.eqLocusS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : R →+* S) :

Subsemiring R

The subsemiring of elements x : R such that f x = g x

Instances For

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@[simp]

theorem RingHom.eqLocusS_same {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

RingHom.eqLocusS f f = ⊤

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theorem RingHom.eqOn_sclosure {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {g : R →+* S} {s : Set R} (h : Set.EqOn (↑f) (↑g) s) :

Set.EqOn ↑f ↑g ↑(Subsemiring.closure s)

If two ring homomorphisms are equal on a set, then they are equal on its subsemiring closure.

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theorem RingHom.eq_of_eqOn_stop {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {g : R →+* S} (h : Set.EqOn ↑f ↑g ↑⊤) :

f = g

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theorem RingHom.eq_of_eqOn_sdense {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Set R} (hs : Subsemiring.closure s = ⊤) {f : R →+* S} {g : R →+* S} (h : Set.EqOn (↑f) (↑g) s) :

f = g

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theorem RingHom.sclosure_preimage_le {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Set S) :

Subsemiring.closure (↑f ⁻¹' s) ≤ Subsemiring.comap f (Subsemiring.closure s)

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theorem RingHom.map_closureS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Set R) :

Subsemiring.map f (Subsemiring.closure s) = Subsemiring.closure (↑f '' s)

The image under a ring homomorphism of the subsemiring generated by a set equals the subsemiring generated by the image of the set.

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def Subsemiring.inclusion {R : Type u} [NonAssocSemiring R] {S : Subsemiring R} {T : Subsemiring R} (h : S ≤ T) :

{ x // x ∈ S } →+* { x // x ∈ T }

The ring homomorphism associated to an inclusion of subsemirings.

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@[simp]

theorem Subsemiring.rangeS_subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :

RingHom.rangeS (Subsemiring.subtype s) = s

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@[simp]

theorem Subsemiring.range_fst {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] :

RingHom.rangeS (RingHom.fst R S) = ⊤

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@[simp]

theorem Subsemiring.range_snd {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] :

RingHom.rangeS (RingHom.snd R S) = ⊤

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@[simp]

theorem Subsemiring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) :

Subsemiring.prod s ⊥ ⊔ Subsemiring.prod ⊥ t = Subsemiring.prod s t

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def RingEquiv.subsemiringCongr {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {t : Subsemiring R} (h : s = t) :

{ x // x ∈ s } ≃+* { x // x ∈ t }

Makes the identity isomorphism from a proof two subsemirings of a multiplicative monoid are equal.

Instances For

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def RingEquiv.ofLeftInverseS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ↑f) :

R ≃+* { x // x ∈ RingHom.rangeS f }

Restrict a ring homomorphism with a left inverse to a ring isomorphism to its RingHom.rangeS.

Instances For

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@[simp]

theorem RingEquiv.ofLeftInverseS_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ↑f) (x : R) :

↑(↑(RingEquiv.ofLeftInverseS h) x) = ↑f x

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@[simp]

theorem RingEquiv.ofLeftInverseS_symm_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ↑f) (x : { x // x ∈ RingHom.rangeS f }) :

↑(RingEquiv.symm (RingEquiv.ofLeftInverseS h)) x = g ↑x

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@[simp]

theorem RingEquiv.subsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) :

∀ (a : ↑(↑↑(RingEquiv.toAddEquiv e) '' ↑(Subsemiring.toAddSubmonoid s))), ↑(↑(RingEquiv.symm (RingEquiv.subsemiringMap e s)) a) = ↑(AddEquiv.symm ↑e) ↑a

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@[simp]

theorem RingEquiv.subsemiringMap_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) :

∀ (a : ↑↑(Subsemiring.toAddSubmonoid s)), ↑(↑(RingEquiv.subsemiringMap e s) a) = ↑e ↑a

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def RingEquiv.subsemiringMap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) :

{ x // x ∈ s } ≃+* { x // x ∈ Subsemiring.map (RingEquiv.toRingHom e) s }

Given an equivalence e : R ≃+* S of semirings and a subsemiring s of R, subsemiring_map e s is the induced equivalence between s and s.map e

Instances For

Actions by `Subsemiring`s #

These are just copies of the definitions about Submonoid starting from submonoid.mul_action. The only new result is subsemiring.module.

When R is commutative, Algebra.ofSubsemiring provides a stronger result than those found in this file, which uses the same scalar action.

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instance Subsemiring.smul {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [SMul R' α] (S : Subsemiring R') :

SMul { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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theorem Subsemiring.smul_def {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [SMul R' α] {S : Subsemiring R'} (g : { x // x ∈ S }) (m : α) :

g • m = ↑g • m

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instance Subsemiring.smulCommClass_left {R' : Type u_1} {α : Type u_2} {β : Type u_3} [NonAssocSemiring R'] [SMul R' β] [SMul α β] [SMulCommClass R' α β] (S : Subsemiring R') :

SMulCommClass { x // x ∈ S } α β

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instance Subsemiring.smulCommClass_right {R' : Type u_1} {α : Type u_2} {β : Type u_3} [NonAssocSemiring R'] [SMul α β] [SMul R' β] [SMulCommClass α R' β] (S : Subsemiring R') :

SMulCommClass α { x // x ∈ S } β

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instance Subsemiring.isScalarTower {R' : Type u_1} {α : Type u_2} {β : Type u_3} [NonAssocSemiring R'] [SMul α β] [SMul R' α] [SMul R' β] [IsScalarTower R' α β] (S : Subsemiring R') :

IsScalarTower { x // x ∈ S } α β

Note that this provides IsScalarTower S R R which is needed by smul_mul_assoc.

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instance Subsemiring.faithfulSMul {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [SMul R' α] [FaithfulSMul R' α] (S : Subsemiring R') :

FaithfulSMul { x // x ∈ S } α

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instance Subsemiring.instSMulWithZeroSubtypeMemSubsemiringInstMembershipInstSetLikeSubsemiringZeroToZeroToMulZeroOneClassToZeroMemClassToAddZeroClassToAddMonoidToAddMonoidWithOneToAddCommMonoidWithOneToAddSubmonoidClassInstSubsemiringClassSubsemiringInstSetLikeSubsemiring {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [Zero α] [SMulWithZero R' α] (S : Subsemiring R') :

SMulWithZero { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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instance Subsemiring.mulAction {R' : Type u_1} {α : Type u_2} [Semiring R'] [MulAction R' α] (S : Subsemiring R') :

MulAction { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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instance Subsemiring.distribMulAction {R' : Type u_1} {α : Type u_2} [Semiring R'] [AddMonoid α] [DistribMulAction R' α] (S : Subsemiring R') :

DistribMulAction { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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instance Subsemiring.mulDistribMulAction {R' : Type u_1} {α : Type u_2} [Semiring R'] [Monoid α] [MulDistribMulAction R' α] (S : Subsemiring R') :

MulDistribMulAction { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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instance Subsemiring.mulActionWithZero {R' : Type u_1} {α : Type u_2} [Semiring R'] [Zero α] [MulActionWithZero R' α] (S : Subsemiring R') :

MulActionWithZero { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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instance Subsemiring.module {R' : Type u_1} {α : Type u_2} [Semiring R'] [AddCommMonoid α] [Module R' α] (S : Subsemiring R') :

Module { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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instance Subsemiring.instMulSemiringActionSubtypeMemSubsemiringToNonAssocSemiringInstMembershipInstSetLikeSubsemiringToMonoidToMonoidToMonoidWithZeroToSubmonoid {R' : Type u_1} {α : Type u_2} [Semiring R'] [Semiring α] [MulSemiringAction R' α] (S : Subsemiring R') :

MulSemiringAction { x // x ∈ S } α

The action by a subsemiring is the action by the underlying semiring.

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instance Subsemiring.center.smulCommClass_left {R' : Type u_1} [Semiring R'] :

SMulCommClass { x // x ∈ Subsemiring.center R' } R' R'

The center of a semiring acts commutatively on that semiring.

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instance Subsemiring.center.smulCommClass_right {R' : Type u_1} [Semiring R'] :

SMulCommClass R' { x // x ∈ Subsemiring.center R' } R'

The center of a semiring acts commutatively on that semiring.

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def Subsemiring.closureCommSemiringOfComm {R' : Type u_1} [Semiring R'] {s : Set R'} (hcomm : ∀ (a : R'), a ∈ s → ∀ (b : R'), b ∈ s → a * b = b * a) :

CommSemiring { x // x ∈ Subsemiring.closure s }

If all the elements of a set s commute, then closure s is a commutative monoid.

Instances For

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def posSubmonoid (R : Type u_1) [StrictOrderedSemiring R] :

Submonoid R

Submonoid of positive elements of an ordered semiring.

Instances For

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@[simp]

theorem mem_posSubmonoid {R : Type u_1} [StrictOrderedSemiring R] (u : Rˣ) :

↑u ∈ posSubmonoid R ↔ 0 < ↑u

Bundled subsemirings #

Actions by Subsemirings #

Actions by `Subsemiring`s #