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) [inst : AddMonoidWithOne R] [inst : SetLike S R] extends AddSubmonoidClass , OneMemClass : Prop

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

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

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

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• AddSubmonoidWithOneClass.toAddMonoidWithOne s = let src := AddSubmonoidClass.toAddMonoid s; AddMonoidWithOne.mk

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

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

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

Equations

• SubsemiringClass.addSubmonoidWithOneClass = SubsemiringClass.addSubmonoidWithOneClass.proof_1

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theorem coe_nat_mem {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : SetLike S R] [hSR : SubsemiringClass S R] (s : S) [inst : Nontrivial R] : Nontrivial { x // x ∈ s }

Equations

• @SubsemiringClass.nontrivial = @SubsemiringClass.nontrivial.proof_1

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

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• @SubsemiringClass.noZeroDivisors = @SubsemiringClass.noZeroDivisors.proof_1

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def SubsemiringClass.subtype {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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.

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theorem SubsemiringClass.coe_subtype {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : Semiring R] [inst : SetLike S R] [inst : SubsemiringClass S R] : Semiring { x // x ∈ s }

A subsemiring of a Semiring is a Semiring.

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theorem SubsemiringClass.coe_pow {S : Type v} (s : S) {R : Type u_1} [inst : Semiring R] [inst : SetLike S R] [inst : 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} [inst : CommSemiring R] [inst : SetLike S R] [inst : 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} [inst : OrderedSemiring R] [inst : SetLike S R] [inst : 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} [inst : StrictOrderedSemiring R] [inst : SetLike S R] [inst : SubsemiringClass S R] : StrictOrderedSemiring { x // x ∈ s }

A subsemiring of an StrictOrderedSemiring is an StrictOrderedSemiring.

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instance SubsemiringClass.toOrderedCommSemiring {S : Type v} (s : S) {R : Type u_1} [inst : OrderedCommSemiring R] [inst : SetLike S R] [inst : 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} [inst : StrictOrderedCommSemiring R] [inst : SetLike S R] [inst : SubsemiringClass S R] : StrictOrderedCommSemiring { x // x ∈ s }

A subsemiring of an StrictOrderedCommSemiring is an StrictOrderedCommSemiring.

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instance SubsemiringClass.toLinearOrderedSemiring {S : Type v} (s : S) {R : Type u_1} [inst : LinearOrderedSemiring R] [inst : SetLike S R] [inst : 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} [inst : LinearOrderedCommSemiring R] [inst : SetLike S R] [inst : SubsemiringClass S R] : LinearOrderedCommSemiring { x // x ∈ s }

A subsemiring of a LinearOrderedCommSemiring is a LinearOrderedCommSemiring.

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

• The sum of two elements of an additive subsemigroup belongs to the subsemigroup.

  add_mem' : ∀ {a b : R}, a ∈ toSubmonoid.toSubsemigroup.carrier → b ∈ toSubmonoid.toSubsemigroup.carrier → a + b ∈ toSubmonoid.toSubsemigroup.carrier

• An additive submonoid contains 0.

  zero_mem' : 0 ∈ toSubmonoid.toSubsemigroup.carrier

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

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

Reinterpret a Subsemiring as an AddSubmonoid.

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instance Subsemiring.instSetLikeSubsemiring {R : Type u} [inst : NonAssocSemiring R] : SetLike (Subsemiring R) R

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instance Subsemiring.instSubsemiringClassSubsemiringInstSetLikeSubsemiring {R : Type u} [inst : NonAssocSemiring R] : SubsemiringClass (Subsemiring R) R

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• @Subsemiring.instSubsemiringClassSubsemiringInstSetLikeSubsemiring = @Subsemiring.instSubsemiringClassSubsemiringInstSetLikeSubsemiring.proof_1

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theorem Subsemiring.mem_toSubmonoid {R : Type u} [inst : NonAssocSemiring R] {s : Subsemiring R} {x : R} : x ∈ s.toSubmonoid ↔ x ∈ s

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theorem Subsemiring.mem_carrier {R : Type u} [inst : NonAssocSemiring R] {s : Subsemiring R} {x : R} : x ∈ s.toSubmonoid.toSubsemigroup.carrier ↔ x ∈ s

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theorem Subsemiring.ext {R : Type u} [inst : 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} [inst : 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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theorem Subsemiring.coe_copy {R : Type u} [inst : 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} [inst : 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} [inst : NonAssocSemiring R] : Function.Injective Subsemiring.toSubmonoid

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

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

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theorem Subsemiring.toAddSubmonoid_injective {R : Type u} [inst : NonAssocSemiring R] : Function.Injective Subsemiring.toAddSubmonoid

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

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

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def Subsemiring.mk' {R : Type u} [inst : 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∈ s ↔ x ∈ sm ↔ x ∈ sa↔ x ∈ sm ↔ x ∈ sa∈ sm ↔ x ∈ sa↔ x ∈ sa∈ sa.

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theorem Subsemiring.coe_mk' {R : Type u} [inst : 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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theorem Subsemiring.mem_mk' {R : Type u} [inst : 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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theorem Subsemiring.mk'_toSubmonoid {R : Type u} [inst : 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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theorem Subsemiring.mk'_toAddSubmonoid {R : Type u} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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 an Subsemiring of an Semiring indexed by a Finset is in the add_subsemiring.

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

A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

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• Subsemiring.toNonAssocSemiring s = SubsemiringClass.toNonAssocSemiring s

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theorem Subsemiring.coe_one {R : Type u} [inst : NonAssocSemiring R] (s : Subsemiring R) : ↑1 = 1

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theorem Subsemiring.coe_zero {R : Type u} [inst : NonAssocSemiring R] (s : Subsemiring R) : ↑0 = 0

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

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

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• @Subsemiring.nontrivial = @Subsemiring.nontrivial.proof_1

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theorem Subsemiring.pow_mem {R : Type u_1} [inst : 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} [inst : NonAssocSemiring R] (s : Subsemiring R) [inst : NoZeroDivisors R] : NoZeroDivisors { x // x ∈ s }

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• @Subsemiring.noZeroDivisors = @Subsemiring.noZeroDivisors.proof_1

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

A subsemiring of a Semiring is a Semiring.

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theorem Subsemiring.coe_pow {R : Type u_1} [inst : 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} [inst : CommSemiring R] (s : Subsemiring R) : CommSemiring { x // x ∈ s }

A subsemiring of a CommSemiring is a CommSemiring.

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• Subsemiring.toCommSemiring s = let src := Subsemiring.toSemiring s; CommSemiring.mk (_ : ∀ (x x_1 : { x // x ∈ s }), x * x_1 = x_1 * x)

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def Subsemiring.subtype {R : Type u} [inst : 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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theorem Subsemiring.coe_subtype {R : Type u} [inst : NonAssocSemiring R] (s : Subsemiring R) : ↑(Subsemiring.subtype s) = Subtype.val

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instance Subsemiring.toOrderedSemiring {R : Type u_1} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : 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} [inst : LinearOrderedCommSemiring R] (s : Subsemiring R) : LinearOrderedCommSemiring { x // x ∈ s }

A subsemiring of a LinearOrderedCommSemiring is a LinearOrderedCommSemiring.

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• One or more equations did not get rendered due to their size.

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

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

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theorem Subsemiring.coe_carrier_toSubmonoid {R : Type u} [inst : NonAssocSemiring R] (s : Subsemiring R) : s.toSubmonoid.toSubsemigroup.carrier = ↑s

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theorem Subsemiring.mem_toAddSubmonoid {R : Type u} [inst : NonAssocSemiring R] {s : Subsemiring R} {x : R} : x ∈ Subsemiring.toAddSubmonoid s ↔ x ∈ s

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

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instance Subsemiring.instTopSubsemiring {R : Type u} [inst : NonAssocSemiring R] : Top (Subsemiring R)

The subsemiring R of the semiring R.

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

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theorem Subsemiring.coe_top {R : Type u} [inst : NonAssocSemiring R] : ↑⊤ = Set.univ

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

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theorem Subsemiring.topEquiv_symm_apply_coe {R : Type u} [inst : NonAssocSemiring R] (r : R) : ↑(↑(RingEquiv.symm Subsemiring.topEquiv) r) = r

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def Subsemiring.topEquiv {R : Type u} [inst : 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} [inst : NonAssocSemiring R] [inst : 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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theorem Subsemiring.coe_comap {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] (s : Subsemiring S) (f : R →+* S) : ↑(Subsemiring.comap f s) = ↑f ⁻¹' ↑s

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theorem Subsemiring.mem_comap {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : 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} [inst : NonAssocSemiring R] [inst : 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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theorem Subsemiring.coe_map {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) : ↑(Subsemiring.map f s) = ↑f '' ↑s

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

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

Equations

• RingHom.rangeS f = Subsemiring.copy (Subsemiring.map f ⊤) (Set.range ↑f) (_ : Set.range ↑f = ↑f '' Set.univ)

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theorem RingHom.coe_rangeS {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] (f : R →+* S) : ↑(RingHom.rangeS f) = Set.range ↑f

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theorem RingHom.mem_rangeS {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : 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} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : Fintype R] [inst : 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.

Equations

• RingHom.fintypeRangeS f = Set.fintypeRange ↑f

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instance Subsemiring.instBotSubsemiring {R : Type u} [inst : NonAssocSemiring R] : Bot (Subsemiring R)

Equations

• Subsemiring.instBotSubsemiring = { bot := RingHom.rangeS (Nat.castRingHom R) }

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instance Subsemiring.instInhabitedSubsemiring {R : Type u} [inst : NonAssocSemiring R] : Inhabited (Subsemiring R)

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• Subsemiring.instInhabitedSubsemiring = { default := ⊥ }

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theorem Subsemiring.coe_bot {R : Type u} [inst : NonAssocSemiring R] : ↑⊥ = Set.range Nat.cast

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theorem Subsemiring.mem_bot {R : Type u} [inst : NonAssocSemiring R] {x : R} : x ∈ ⊥ ↔ ∃ n, ↑n = x

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instance Subsemiring.instInfSubsemiring {R : Type u} [inst : NonAssocSemiring R] : Inf (Subsemiring R)

The inf of two subsemirings is their intersection.

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theorem Subsemiring.coe_inf {R : Type u} [inst : NonAssocSemiring R] (p : Subsemiring R) (p' : Subsemiring R) : ↑(p ⊓ p') = ↑p ∩ ↑p'

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theorem Subsemiring.mem_inf {R : Type u} [inst : NonAssocSemiring R] {p : Subsemiring R} {p' : Subsemiring R} {x : R} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

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instance Subsemiring.instInfSetSubsemiring {R : Type u} [inst : NonAssocSemiring R] : InfSet (Subsemiring R)

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theorem Subsemiring.coe_infₛ {R : Type u} [inst : NonAssocSemiring R] (S : Set (Subsemiring R)) : ↑(infₛ S) = Set.interᵢ fun s => Set.interᵢ fun h => ↑s

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theorem Subsemiring.mem_infₛ {R : Type u} [inst : NonAssocSemiring R] {S : Set (Subsemiring R)} {x : R} : x ∈ infₛ S ↔ ∀ (p : Subsemiring R), p ∈ S → x ∈ p

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theorem Subsemiring.infₛ_toSubmonoid {R : Type u} [inst : NonAssocSemiring R] (s : Set (Subsemiring R)) : (infₛ s).toSubmonoid = ⨅ t, ⨅ h, t.toSubmonoid

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theorem Subsemiring.infₛ_toAddSubmonoid {R : Type u} [inst : NonAssocSemiring R] (s : Set (Subsemiring R)) : Subsemiring.toAddSubmonoid (infₛ s) = ⨅ t, ⨅ h, Subsemiring.toAddSubmonoid t

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instance Subsemiring.instCompleteLatticeSubsemiring {R : Type u} [inst : 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} [inst : NonAssocSemiring R] (A : Subsemiring R) : A = ⊤ ↔ ∀ (x : R), x ∈ A

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

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

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theorem Subsemiring.coe_center (R : Type u_1) [inst : Semiring R] : ↑(Subsemiring.center R) = Set.center R

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theorem Subsemiring.center_toSubmonoid (R : Type u_1) [inst : Semiring R] : (Subsemiring.center R).toSubmonoid = Submonoid.center R

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theorem Subsemiring.mem_center_iff {R : Type u_1} [inst : 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} [inst : Semiring R] [inst : DecidableEq R] [inst : Fintype R] : DecidablePred fun x => x ∈ Subsemiring.center R

Equations

• Subsemiring.decidableMemCenter x = decidable_of_iff' (∀ (g : R), g * x = x * g) (_ : x ∈ Subsemiring.center R ↔ ∀ (g : R), g * x = x * g)

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theorem Subsemiring.center_eq_top (R : Type u_1) [inst : CommSemiring R] : Subsemiring.center R = ⊤

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instance Subsemiring.commSemiring {R : Type u_1} [inst : Semiring R] : CommSemiring { x // x ∈ Subsemiring.center R }

The center is commutative.

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

The centralizer of a set as subsemiring.

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

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

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theorem Subsemiring.centralizer_univ {R : Type u_1} [inst : Semiring R] : Subsemiring.centralizer Set.univ = Subsemiring.center R

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

The Subsemiring generated by a set.

Equations

• Subsemiring.closure s = infₛ { S | s ⊆ ↑S }

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theorem Subsemiring.mem_closure {R : Type u} [inst : NonAssocSemiring R] {x : R} {s : Set R} : x ∈ Subsemiring.closure s ↔ ∀ (S : Subsemiring R), s ⊆ ↑S → x ∈ S

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

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theorem Subsemiring.closure_le {R : Type u} [inst : 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} [inst : 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⊆ t, then closure s ≤ closure t≤ closure t.

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

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theorem Subsemiring.mem_map_equiv {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] (f : R ≃+* S) (K : Subsemiring S) : Subsemiring.comap (↑f) K = Subsemiring.map (↑(RingEquiv.symm f)) K

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

The additive closure of a submonoid is a subsemiring.

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• One or more equations did not get rendered due to their size.

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theorem Submonoid.subsemiringClosure_coe {R : Type u} [inst : NonAssocSemiring R] (M : Submonoid R) : ↑(Submonoid.subsemiringClosure M) = ↑(AddSubmonoid.closure ↑M)

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theorem Submonoid.subsemiringClosure_toAddSubmonoid {R : Type u} [inst : NonAssocSemiring R] (M : Submonoid R) : Subsemiring.toAddSubmonoid (Submonoid.subsemiringClosure M) = AddSubmonoid.closure ↑M

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theorem Submonoid.subsemiringClosure_eq_closure {R : Type u} [inst : 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 .

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theorem Subsemiring.closure_submonoid_closure {R : Type u} [inst : NonAssocSemiring R] (s : Set R) : Subsemiring.closure ↑(Submonoid.closure s) = Subsemiring.closure s

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theorem Subsemiring.coe_closure_eq {R : Type u} [inst : 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} [inst : NonAssocSemiring R] {s : Set R} {x : R} : x ∈ Subsemiring.closure s ↔ x ∈ AddSubmonoid.closure ↑(Submonoid.closure s)

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theorem Subsemiring.closure_addSubmonoid_closure {R : Type u} [inst : NonAssocSemiring R] {s : Set R} : Subsemiring.closure ↑(AddSubmonoid.closure s) = Subsemiring.closure s

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theorem Subsemiring.closure_induction {R : Type u} [inst : 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_induction₂ {R : Type u} [inst : 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.

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theorem Subsemiring.mem_closure_iff_exists_list {R : Type u_1} [inst : 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

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def Subsemiring.gi (R : Type u) [inst : NonAssocSemiring R] : GaloisInsertion Subsemiring.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

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• One or more equations did not get rendered due to their size.

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

Closure of a subsemiring S equals S.

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theorem Subsemiring.closure_empty {R : Type u} [inst : NonAssocSemiring R] : Subsemiring.closure ∅ = ⊥

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theorem Subsemiring.closure_univ {R : Type u} [inst : NonAssocSemiring R] : Subsemiring.closure Set.univ = ⊤

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theorem Subsemiring.closure_union {R : Type u} [inst : 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_unionᵢ {R : Type u} [inst : NonAssocSemiring R] {ι : Sort u_1} (s : ι → Set R) : Subsemiring.closure (Set.unionᵢ fun i => s i) = ⨆ i, Subsemiring.closure (s i)

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theorem Subsemiring.closure_unionₛ {R : Type u} [inst : NonAssocSemiring R] (s : Set (Set R)) : Subsemiring.closure (⋃₀ s) = ⨆ t, ⨆ h, Subsemiring.closure t

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

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theorem Subsemiring.map_supᵢ {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subsemiring R) : Subsemiring.map f (supᵢ s) = ⨆ i, Subsemiring.map f (s i)

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

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theorem Subsemiring.comap_infᵢ {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subsemiring S) : Subsemiring.comap f (infᵢ s) = ⨅ i, Subsemiring.comap f (s i)

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

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

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def Subsemiring.prod {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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× t as a subsemiring of R × S× S.

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• One or more equations did not get rendered due to their size.

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

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theorem Subsemiring.prod_mono_right {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] (s : Subsemiring S) : Subsemiring.prod ⊤ s = Subsemiring.comap (RingHom.snd R S) s

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theorem Subsemiring.top_prod_top {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : Subsemiring.prod ⊤ ⊤ = ⊤

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def Subsemiring.prodEquiv {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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.

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• One or more equations did not get rendered due to their size.

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theorem Subsemiring.mem_supᵢ_of_directed {R : Type u} [inst : 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, S i) ↔ ∃ i, x ∈ S i

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theorem Subsemiring.coe_supᵢ_of_directed {R : Type u} [inst : NonAssocSemiring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subsemiring R} (hS : Directed (fun x x_1 => x ≤ x_1) S) : ↑(⨆ i, S i) = Set.unionᵢ fun i => ↑(S i)

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theorem Subsemiring.mem_supₛ_of_directedOn {R : Type u} [inst : NonAssocSemiring R] {S : Set (Subsemiring R)} (Sne : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) {x : R} : x ∈ supₛ S ↔ ∃ s, s ∈ S ∧ x ∈ s

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theorem Subsemiring.coe_supₛ_of_directedOn {R : Type u} [inst : NonAssocSemiring R] {S : Set (Subsemiring R)} (Sne : Set.Nonempty S) (hS : DirectedOn (fun x x_1 => x ≤ x_1) S) : ↑(supₛ S) = Set.unionᵢ fun s => Set.unionᵢ fun h => ↑s

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

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

Equations

• RingHom.domRestrict f s = RingHom.comp f (SubsemiringClass.subtype s)

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theorem RingHom.restrict_apply {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {σR : Type u_1} [inst : SetLike σR R] [inst : 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} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {σS : Type u_1} [inst : SetLike σS S] [inst : 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.

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• One or more equations did not get rendered due to their size.

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def RingHom.restrict {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [inst : SetLike σR R] [inst : SetLike σS S] [inst : SubsemiringClass σR R] [inst : 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.

Equations

• RingHom.restrict f s' s h = RingHom.codRestrict (RingHom.domRestrict f s') s (_ : ∀ (x : { x // x ∈ s' }), ↑f ↑x ∈ s)

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theorem RingHom.coe_restrict_apply {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [inst : SetLike σR R] [inst : SetLike σS S] [inst : SubsemiringClass σR R] [inst : 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

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theorem RingHom.comp_restrict {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [inst : SetLike σR R] [inst : SetLike σS S] [inst : SubsemiringClass σR R] [inst : 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')

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def RingHom.rangeSRestrict {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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.

Equations

• RingHom.rangeSRestrict f = RingHom.codRestrict f (RingHom.rangeS f) (_ : ∀ (x : R), ↑f x ∈ RingHom.rangeS f)

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theorem RingHom.coe_rangeSRestrict {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] {f : R →+* S} : RingHom.rangeS f = ⊤ ↔ Function.Surjective ↑f

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

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

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• One or more equations did not get rendered due to their size.

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

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theorem RingHom.eqOn_sclosure {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : NonAssocSemiring R] [inst : 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} [inst : 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.

Equations

• Subsemiring.inclusion h = RingHom.codRestrict (Subsemiring.subtype S) T (_ : ∀ (x : { x // x ∈ S }), ↑(Subsemiring.subtype S) x ∈ T)

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theorem Subsemiring.rangeS_subtype {R : Type u} [inst : NonAssocSemiring R] (s : Subsemiring R) : RingHom.rangeS (Subsemiring.subtype s) = s

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theorem Subsemiring.range_fst {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : RingHom.rangeS (RingHom.fst R S) = ⊤

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theorem Subsemiring.range_snd {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : RingHom.rangeS (RingHom.snd R S) = ⊤

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

Equations

• One or more equations did not get rendered due to their size.

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def RingEquiv.ofLeftInverseS {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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.

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• One or more equations did not get rendered due to their size.

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theorem RingEquiv.ofLeftInverseS_apply {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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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theorem RingEquiv.ofLeftInverseS_symm_apply {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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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theorem RingEquiv.subsemiringMap_apply_coe {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) : ∀ (a : ↑↑(Subsemiring.toAddSubmonoid s)), ↑(↑(RingEquiv.subsemiringMap e s) a) = ↑e ↑a

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theorem RingEquiv.subsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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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def RingEquiv.subsemiringMap {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst : 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≃+* S of semirings and a subsemiring s of R, subsemiring_map e s is the induced equivalence between s and s.map e

Equations

• One or more equations did not get rendered due to their size.

Actions by Subsemirings

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.of_subsemiring 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} [inst : NonAssocSemiring R'] [inst : SMul R' α] (S : Subsemiring R') : SMul { x // x ∈ S } α

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

Equations

• Subsemiring.smul S = Submonoid.smul S.toSubmonoid

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

Equations

• @Subsemiring.smulCommClass_left = @Subsemiring.smulCommClass_left.proof_1

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

Equations

• @Subsemiring.smulCommClass_right = @Subsemiring.smulCommClass_right.proof_1

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instance Subsemiring.isScalarTower {R' : Type u_1} {α : Type u_2} {β : Type u_3} [inst : NonAssocSemiring R'] [inst : SMul α β] [inst : SMul R' α] [inst : SMul R' β] [inst : 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.

Equations

• @Subsemiring.isScalarTower = @Subsemiring.isScalarTower.proof_1

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

Equations

• @Subsemiring.faithfulSMul = @Subsemiring.faithfulSMul.proof_1

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

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

Equations

• One or more equations did not get rendered due to their size.

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

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

Equations

• Subsemiring.mulAction S = Submonoid.mulAction S.toSubmonoid

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

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

Equations

• Subsemiring.distribMulAction S = Submonoid.distribMulAction S.toSubmonoid

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

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

Equations

• Subsemiring.mulDistribMulAction S = Submonoid.mulDistribMulAction S.toSubmonoid

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

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

Equations

• Subsemiring.mulActionWithZero S = MulActionWithZero.compHom α (RingHom.toMonoidWithZeroHom (Subsemiring.subtype S))

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

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

Equations

• Subsemiring.module S = Module.compHom α (Subsemiring.subtype S)

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

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

Equations

• One or more equations did not get rendered due to their size.

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instance Subsemiring.center.smulCommClass_left {R' : Type u_1} [inst : Semiring R'] : SMulCommClass { x // x ∈ Subsemiring.center R' } R' R'

The center of a semiring acts commutatively on that semiring.

Equations

• @Subsemiring.center.smulCommClass_left = @Subsemiring.center.smulCommClass_left.proof_1

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instance Subsemiring.center.smulCommClass_right {R' : Type u_1} [inst : Semiring R'] : SMulCommClass R' { x // x ∈ Subsemiring.center R' } R'

The center of a semiring acts commutatively on that semiring.

Equations

• @Subsemiring.center.smulCommClass_right = @Subsemiring.center.smulCommClass_right.proof_1

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def Subsemiring.closureCommSemiringOfComm {R' : Type u_1} [inst : 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.

Equations

• Subsemiring.closureCommSemiringOfComm hcomm = let src := Subsemiring.toSemiring (Subsemiring.closure s); CommSemiring.mk (_ : ∀ (x y : { x // x ∈ Subsemiring.closure s }), x * y = y * x)

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

Submonoid of positive elements of an ordered semiring.

Equations

• posSubmonoid R = { toSubsemigroup := { carrier := { x | 0 < x }, mul_mem' := (_ : ∀ {x y : R}, 0 < x → 0 < y → 0 < x * y) }, one_mem' := (_ : 0 < 1) }

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theorem mem_posSubmonoid {R : Type u_1} [inst : StrictOrderedSemiring R] (u : Rˣ) : ↑u ∈ posSubmonoid R ↔ 0 < ↑u