Bundled non-unital subsemirings #

We define the CompleteLattice structure, and non-unital subsemiring map, comap and range (srange) of a NonUnitalRingHom etc.

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theorem NonUnitalSubsemiring.toSubsemigroup_strictMono {R : Type u} [NonUnitalNonAssocSemiring R] :

StrictMono NonUnitalSubsemiring.toSubsemigroup

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theorem NonUnitalSubsemiring.toSubsemigroup_mono {R : Type u} [NonUnitalNonAssocSemiring R] :

Monotone NonUnitalSubsemiring.toSubsemigroup

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theorem NonUnitalSubsemiring.toAddSubmonoid_strictMono {R : Type u} [NonUnitalNonAssocSemiring R] :

StrictMono NonUnitalSubsemiring.toAddSubmonoid

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theorem NonUnitalSubsemiring.toAddSubmonoid_mono {R : Type u} [NonUnitalNonAssocSemiring R] :

Monotone NonUnitalSubsemiring.toAddSubmonoid

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def NonUnitalSubsemiring.topEquiv {R : Type u} [NonUnitalNonAssocSemiring R] :

↥⊤ ≃+* R

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

Equations

NonUnitalSubsemiring.topEquiv = { toEquiv := Subsemigroup.topEquiv.toEquiv, map_mul' := ⋯, map_add' := ⋯ }

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@[simp]

theorem NonUnitalSubsemiring.topEquiv_symm_apply_coe {R : Type u} [NonUnitalNonAssocSemiring R] (x : R) :

↑(NonUnitalSubsemiring.topEquiv.symm x) = x

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@[simp]

theorem NonUnitalSubsemiring.topEquiv_apply {R : Type u} [NonUnitalNonAssocSemiring R] (x : ↥⊤) :

NonUnitalSubsemiring.topEquiv x = ↑x

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def NonUnitalSubsemiring.comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubsemiring S) :

NonUnitalSubsemiring R

The preimage of a non-unital subsemiring along a non-unital ring homomorphism is a non-unital subsemiring.

Equations

NonUnitalSubsemiring.comap f s = { carrier := ⇑f ⁻¹' ↑s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

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@[simp]

theorem NonUnitalSubsemiring.coe_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring S) (f : F) :

↑(NonUnitalSubsemiring.comap f s) = ⇑f ⁻¹' ↑s

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@[simp]

theorem NonUnitalSubsemiring.mem_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {s : NonUnitalSubsemiring S} {f : F} {x : R} :

x ∈ NonUnitalSubsemiring.comap f s ↔ f x ∈ s

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theorem NonUnitalSubsemiring.comap_comap {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] {F : Type u_1} {G : Type u_2} [FunLike F R S] [NonUnitalRingHomClass F R S] [FunLike G S T] [NonUnitalRingHomClass G S T] (s : NonUnitalSubsemiring T) (g : G) (f : F) :

NonUnitalSubsemiring.comap f (NonUnitalSubsemiring.comap g s) = NonUnitalSubsemiring.comap ((↑g).comp ↑f) s

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def NonUnitalSubsemiring.map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubsemiring R) :

NonUnitalSubsemiring S

The image of a non-unital subsemiring along a ring homomorphism is a non-unital subsemiring.

Equations

NonUnitalSubsemiring.map f s = { carrier := ⇑f '' ↑s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

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@[simp]

theorem NonUnitalSubsemiring.coe_map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubsemiring R) :

↑(NonUnitalSubsemiring.map f s) = ⇑f '' ↑s

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@[simp]

theorem NonUnitalSubsemiring.mem_map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {s : NonUnitalSubsemiring R} {y : S} :

y ∈ NonUnitalSubsemiring.map f s ↔ ∃ x ∈ s, f x = y

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@[simp]

theorem NonUnitalSubsemiring.map_id {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) :

NonUnitalSubsemiring.map (NonUnitalRingHom.id R) s = s

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theorem NonUnitalSubsemiring.map_map {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] {F : Type u_1} {G : Type u_2} [FunLike F R S] [NonUnitalRingHomClass F R S] [FunLike G S T] [NonUnitalRingHomClass G S T] (s : NonUnitalSubsemiring R) (g : G) (f : F) :

NonUnitalSubsemiring.map (↑g) (NonUnitalSubsemiring.map (↑f) s) = NonUnitalSubsemiring.map ((↑g).comp ↑f) s

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theorem NonUnitalSubsemiring.map_le_iff_le_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {s : NonUnitalSubsemiring R} {t : NonUnitalSubsemiring S} :

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

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theorem NonUnitalSubsemiring.gc_map_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) :

GaloisConnection (NonUnitalSubsemiring.map f) (NonUnitalSubsemiring.comap f)

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noncomputable def NonUnitalSubsemiring.equivMapOfInjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective ⇑f) :

↥s ≃+* ↥(NonUnitalSubsemiring.map f s)

A non-unital subsemiring is isomorphic to its image under an injective function

Equations

s.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑s) hf, map_mul' := ⋯, map_add' := ⋯ }

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theorem NonUnitalSubsemiring.coe_equivMapOfInjective_apply {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective ⇑f) (x : ↥s) :

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

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def NonUnitalRingHom.srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) :

NonUnitalSubsemiring S

The range of a non-unital ring homomorphism is a non-unital subsemiring. See note [range copy pattern].

Equations

NonUnitalRingHom.srange f = (NonUnitalSubsemiring.map ↑f ⊤).copy (Set.range ⇑f) ⋯

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theorem NonUnitalRingHom.coe_srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) :

↑(NonUnitalRingHom.srange f) = Set.range ⇑f

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@[simp]

theorem NonUnitalRingHom.mem_srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {y : S} :

y ∈ NonUnitalRingHom.srange f ↔ ∃ (x : R), f x = y

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theorem NonUnitalRingHom.srange_eq_map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) :

NonUnitalRingHom.srange f = NonUnitalSubsemiring.map f ⊤

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theorem NonUnitalRingHom.mem_srange_self {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (x : R) :

f x ∈ NonUnitalRingHom.srange f

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theorem NonUnitalRingHom.map_srange {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (g : S →ₙ+* T) (f : R →ₙ+* S) :

NonUnitalSubsemiring.map g (NonUnitalRingHom.srange f) = NonUnitalRingHom.srange (g.comp f)

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instance NonUnitalRingHom.finite_srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] [Finite R] (f : F) :

Finite ↥(NonUnitalRingHom.srange f)

The range of a morphism of non-unital semirings is finite if the domain is a finite.

Equations

⋯ = ⋯

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instance NonUnitalSubsemiring.instInfSet {R : Type u} [NonUnitalNonAssocSemiring R] :

InfSet (NonUnitalSubsemiring R)

Equations

NonUnitalSubsemiring.instInfSet = { sInf := fun (s : Set (NonUnitalSubsemiring R)) => NonUnitalSubsemiring.mk' (⋂ t ∈ s, ↑t) (⨅ t ∈ s, t.toSubsemigroup) ⋯ (⨅ t ∈ s, t.toAddSubmonoid) ⋯ }

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@[simp]

theorem NonUnitalSubsemiring.coe_sInf {R : Type u} [NonUnitalNonAssocSemiring R] (S : Set (NonUnitalSubsemiring R)) :

↑(sInf S) = ⋂ s ∈ S, ↑s

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

x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

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@[simp]

theorem NonUnitalSubsemiring.coe_iInf {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_1} {S : ι → NonUnitalSubsemiring R} :

↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

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theorem NonUnitalSubsemiring.mem_iInf {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_1} {S : ι → NonUnitalSubsemiring R} {x : R} :

x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

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@[simp]

theorem NonUnitalSubsemiring.sInf_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (NonUnitalSubsemiring R)) :

(sInf s).toSubsemigroup = ⨅ t ∈ s, t.toSubsemigroup

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@[simp]

theorem NonUnitalSubsemiring.sInf_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (NonUnitalSubsemiring R)) :

(sInf s).toAddSubmonoid = ⨅ t ∈ s, t.toAddSubmonoid

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instance NonUnitalSubsemiring.instCompleteLattice {R : Type u} [NonUnitalNonAssocSemiring R] :

CompleteLattice (NonUnitalSubsemiring R)

Non-unital subsemirings of a non-unital semiring form a complete lattice.

Equations

NonUnitalSubsemiring.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

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theorem NonUnitalSubsemiring.eq_top_iff' {R : Type u} [NonUnitalNonAssocSemiring R] (A : NonUnitalSubsemiring R) :

A = ⊤ ↔ ∀ (x : R), x ∈ A

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def NonUnitalSubsemiring.center (R : Type u) [NonUnitalNonAssocSemiring R] :

NonUnitalSubsemiring R

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

Equations

NonUnitalSubsemiring.center R = { carrier := (Subsemigroup.center R).carrier, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

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theorem NonUnitalSubsemiring.coe_center (R : Type u) [NonUnitalNonAssocSemiring R] :

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

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theorem NonUnitalSubsemiring.center_toSubsemigroup (R : Type u) [NonUnitalNonAssocSemiring R] :

(NonUnitalSubsemiring.center R).toSubsemigroup = Subsemigroup.center R

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instance NonUnitalSubsemiring.center.instNonUnitalCommSemiring (R : Type u) [NonUnitalNonAssocSemiring R] :

NonUnitalCommSemiring ↥(NonUnitalSubsemiring.center R)

The center is commutative and associative.

Equations

NonUnitalSubsemiring.center.instNonUnitalCommSemiring R = NonUnitalCommSemiring.mk ⋯

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theorem Set.mem_center_iff_addMonoidHom (R : Type u) [NonUnitalNonAssocSemiring R] (a : R) :

a ∈ Set.center R ↔ AddMonoidHom.mulLeft a = AddMonoidHom.mulRight a ∧ AddMonoidHom.mul.compr₂ (AddMonoidHom.mulLeft a) = AddMonoidHom.mul.comp (AddMonoidHom.mulLeft a) ∧ AddMonoidHom.mul.comp (AddMonoidHom.mulRight a) = AddMonoidHom.mul.compl₂ (AddMonoidHom.mulLeft a) ∧ AddMonoidHom.mul.compr₂ (AddMonoidHom.mulRight a) = AddMonoidHom.mul.compl₂ (AddMonoidHom.mulRight a)

A point-free means of proving membership in the center, for a non-associative ring.

This can be helpful when working with types that have ext lemmas for R →+ R.

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theorem NonUnitalSubsemiring.mem_center_iff {R : Type u_1} [NonUnitalSemiring R] {z : R} :

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

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instance NonUnitalSubsemiring.decidableMemCenter {R : Type u_1} [NonUnitalSemiring R] [DecidableEq R] [Fintype R] :

DecidablePred fun (x : R) => x ∈ NonUnitalSubsemiring.center R

Equations

NonUnitalSubsemiring.decidableMemCenter x = decidable_of_iff' (∀ (g : R), g * x = x * g) ⋯

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@[simp]

theorem NonUnitalSubsemiring.center_eq_top (R : Type u_1) [NonUnitalCommSemiring R] :

NonUnitalSubsemiring.center R = ⊤

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def NonUnitalSubsemiring.centralizer {R : Type u_1} [NonUnitalSemiring R] (s : Set R) :

NonUnitalSubsemiring R

The centralizer of a set as non-unital subsemiring.

Equations

NonUnitalSubsemiring.centralizer s = { carrier := s.centralizer, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

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theorem NonUnitalSubsemiring.coe_centralizer {R : Type u_1} [NonUnitalSemiring R] (s : Set R) :

↑(NonUnitalSubsemiring.centralizer s) = s.centralizer

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theorem NonUnitalSubsemiring.centralizer_toSubsemigroup {R : Type u_1} [NonUnitalSemiring R] (s : Set R) :

(NonUnitalSubsemiring.centralizer s).toSubsemigroup = Subsemigroup.centralizer s

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theorem NonUnitalSubsemiring.mem_centralizer_iff {R : Type u_1} [NonUnitalSemiring R] {s : Set R} {z : R} :

z ∈ NonUnitalSubsemiring.centralizer s ↔ ∀ g ∈ s, g * z = z * g

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theorem NonUnitalSubsemiring.center_le_centralizer {R : Type u_1} [NonUnitalSemiring R] (s : Set R) :

NonUnitalSubsemiring.center R ≤ NonUnitalSubsemiring.centralizer s

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

NonUnitalSubsemiring.centralizer t ≤ NonUnitalSubsemiring.centralizer s

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@[simp]

theorem NonUnitalSubsemiring.centralizer_eq_top_iff_subset {R : Type u_1} [NonUnitalSemiring R] {s : Set R} :

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

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@[simp]

theorem NonUnitalSubsemiring.centralizer_univ {R : Type u_1} [NonUnitalSemiring R] :

NonUnitalSubsemiring.centralizer Set.univ = NonUnitalSubsemiring.center R

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def NonUnitalSubsemiring.closure {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) :

NonUnitalSubsemiring R

The NonUnitalSubsemiring generated by a set.

Equations

NonUnitalSubsemiring.closure s = sInf {S : NonUnitalSubsemiring R | s ⊆ ↑S}

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theorem NonUnitalSubsemiring.mem_closure {R : Type u} [NonUnitalNonAssocSemiring R] {x : R} {s : Set R} :

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

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@[simp]

theorem NonUnitalSubsemiring.subset_closure {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} :

s ⊆ ↑(NonUnitalSubsemiring.closure s)

The non-unital subsemiring generated by a set includes the set.

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theorem NonUnitalSubsemiring.not_mem_of_not_mem_closure {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {P : R} (hP : P ∉ NonUnitalSubsemiring.closure s) :

P ∉ s

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@[simp]

theorem NonUnitalSubsemiring.closure_le {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {t : NonUnitalSubsemiring R} :

NonUnitalSubsemiring.closure s ≤ t ↔ s ⊆ ↑t

A non-unital subsemiring S includes closure s if and only if it includes s.

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

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

NonUnitalSubsemiring.closure s = t

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

x ∈ NonUnitalSubsemiring.map (↑f) K ↔ f.symm x ∈ K

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

NonUnitalSubsemiring.map (↑f) K = NonUnitalSubsemiring.comap f.symm K

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

NonUnitalSubsemiring.comap (↑f) K = NonUnitalSubsemiring.map f.symm K

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def Subsemigroup.nonUnitalSubsemiringClosure {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) :

NonUnitalSubsemiring R

The additive closure of a non-unital subsemigroup is a non-unital subsemiring.

Equations

M.nonUnitalSubsemiringClosure = { toAddSubmonoid := AddSubmonoid.closure ↑M, mul_mem' := ⋯ }

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theorem Subsemigroup.nonUnitalSubsemiringClosure_coe {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) :

↑M.nonUnitalSubsemiringClosure = ↑(AddSubmonoid.closure ↑M)

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theorem Subsemigroup.nonUnitalSubsemiringClosure_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) :

M.nonUnitalSubsemiringClosure.toAddSubmonoid = AddSubmonoid.closure ↑M

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theorem Subsemigroup.nonUnitalSubsemiringClosure_eq_closure {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) :

M.nonUnitalSubsemiringClosure = NonUnitalSubsemiring.closure ↑M

The NonUnitalSubsemiring generated by a multiplicative subsemigroup coincides with the NonUnitalSubsemiring.closure of the subsemigroup itself .

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@[simp]

theorem NonUnitalSubsemiring.closure_subsemigroup_closure {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) :

NonUnitalSubsemiring.closure ↑(Subsemigroup.closure s) = NonUnitalSubsemiring.closure s

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theorem NonUnitalSubsemiring.coe_closure_eq {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) :

↑(NonUnitalSubsemiring.closure s) = ↑(AddSubmonoid.closure ↑(Subsemigroup.closure s))

The elements of the non-unital subsemiring closure of M are exactly the elements of the additive closure of a multiplicative subsemigroup M.

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theorem NonUnitalSubsemiring.mem_closure_iff {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {x : R} :

x ∈ NonUnitalSubsemiring.closure s ↔ x ∈ AddSubmonoid.closure ↑(Subsemigroup.closure s)

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@[simp]

theorem NonUnitalSubsemiring.closure_addSubmonoid_closure {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} :

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

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theorem NonUnitalSubsemiring.closure_induction {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {p : (x : R) → x ∈ NonUnitalSubsemiring.closure s → Prop} (mem : ∀ (x : R) (hx : x ∈ s), p x ⋯) (zero : p 0 ⋯) (add : ∀ (x y : R) (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s), p x hx → p y hy → p (x + y) ⋯) (mul : ∀ (x y : R) (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s), p x hx → p y hy → p (x * y) ⋯) {x : R} (hx : x ∈ NonUnitalSubsemiring.closure s) :

p x hx

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 NonUnitalSubsemiring.closure_induction₂ {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {p : (x y : R) → x ∈ NonUnitalSubsemiring.closure s → y ∈ NonUnitalSubsemiring.closure s → Prop} (mem_mem : ∀ (x : R) (hx : x ∈ s) (y : R) (hy : y ∈ s), p x y ⋯ ⋯) (zero_left : ∀ (x : R) (hx : x ∈ NonUnitalSubsemiring.closure s), p 0 x ⋯ hx) (zero_right : ∀ (x : R) (hx : x ∈ NonUnitalSubsemiring.closure s), p x 0 hx ⋯) (add_left : ∀ (x y z : R) (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s) (hz : z ∈ NonUnitalSubsemiring.closure s), p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz) (add_right : ∀ (x y z : R) (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s) (hz : z ∈ NonUnitalSubsemiring.closure s), p x y hx hy → p x z hx hz → p x (y + z) hx ⋯) (mul_left : ∀ (x y z : R) (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s) (hz : z ∈ NonUnitalSubsemiring.closure s), p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz) (mul_right : ∀ (x y z : R) (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s) (hz : z ∈ NonUnitalSubsemiring.closure s), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯) {x : R} {y : R} (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s) :

p x y hx hy

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

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def NonUnitalSubsemiring.gi (R : Type u) [NonUnitalNonAssocSemiring R] :

GaloisInsertion NonUnitalSubsemiring.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

NonUnitalSubsemiring.gi R = { choice := fun (s : Set R) (x : ↑(NonUnitalSubsemiring.closure s) ≤ s) => NonUnitalSubsemiring.closure s, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

Instances For

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theorem NonUnitalSubsemiring.closure_eq {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) :

NonUnitalSubsemiring.closure ↑s = s

Closure of a non-unital subsemiring S equals S.

source

@[simp]

theorem NonUnitalSubsemiring.closure_empty {R : Type u} [NonUnitalNonAssocSemiring R] :

NonUnitalSubsemiring.closure ∅ = ⊥

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@[simp]

theorem NonUnitalSubsemiring.closure_univ {R : Type u} [NonUnitalNonAssocSemiring R] :

NonUnitalSubsemiring.closure Set.univ = ⊤

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

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

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theorem NonUnitalSubsemiring.closure_iUnion {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_2} (s : ι → Set R) :

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

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theorem NonUnitalSubsemiring.closure_sUnion {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (Set R)) :

NonUnitalSubsemiring.closure (⋃₀ s) = ⨆ t ∈ s, NonUnitalSubsemiring.closure t

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theorem NonUnitalSubsemiring.map_sup {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring R) (f : F) :

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

source

theorem NonUnitalSubsemiring.map_iSup {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} (f : F) (s : ι → NonUnitalSubsemiring R) :

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

source

theorem NonUnitalSubsemiring.map_inf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective ⇑f) :

NonUnitalSubsemiring.map f (s ⊓ t) = NonUnitalSubsemiring.map f s ⊓ NonUnitalSubsemiring.map f t

source

theorem NonUnitalSubsemiring.map_iInf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} [Nonempty ι] (f : F) (hf : Function.Injective ⇑f) (s : ι → NonUnitalSubsemiring R) :

NonUnitalSubsemiring.map f (iInf s) = ⨅ (i : ι), NonUnitalSubsemiring.map f (s i)

source

theorem NonUnitalSubsemiring.comap_inf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring S) (t : NonUnitalSubsemiring S) (f : F) :

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

source

theorem NonUnitalSubsemiring.comap_iInf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} (f : F) (s : ι → NonUnitalSubsemiring S) :

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

source

@[simp]

theorem NonUnitalSubsemiring.map_bot {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) :

NonUnitalSubsemiring.map f ⊥ = ⊥

source

@[simp]

theorem NonUnitalSubsemiring.comap_top {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) :

NonUnitalSubsemiring.comap f ⊤ = ⊤

source

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

NonUnitalSubsemiring (R × S)

Given NonUnitalSubsemirings s, t of semirings R, S respectively, s.prod t is s × t as a non-unital subsemiring of R × S.

Equations

s.prod t = { carrier := ↑s ×ˢ ↑t, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

Instances For

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

↑(s.prod t) = ↑s ×ˢ ↑t

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

p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t

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

s₁.prod t₁ ≤ s₂.prod t₂

source

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

Monotone fun (t : NonUnitalSubsemiring S) => s.prod t

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

Monotone fun (s : NonUnitalSubsemiring R) => s.prod t

source

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

s.prod ⊤ = NonUnitalSubsemiring.comap (NonUnitalRingHom.fst R S) s

source

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

⊤.prod s = NonUnitalSubsemiring.comap (NonUnitalRingHom.snd R S) s

source

@[simp]

theorem NonUnitalSubsemiring.top_prod_top {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] :

⊤.prod ⊤ = ⊤

source

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

↥(s.prod t) ≃+* ↥s × ↥t

Product of non-unital subsemirings is isomorphic to their product as semigroups.

Equations

s.prodEquiv t = { toEquiv := Equiv.Set.prod ↑s ↑t, map_mul' := ⋯, map_add' := ⋯ }

Instances For

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theorem NonUnitalSubsemiring.mem_iSup_of_directed {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_2} [hι : Nonempty ι] {S : ι → NonUnitalSubsemiring R} (hS : Directed (fun (x1 x2 : NonUnitalSubsemiring R) => x1 ≤ x2) S) {x : R} :

x ∈ ⨆ (i : ι), S i ↔ ∃ (i : ι), x ∈ S i

source

theorem NonUnitalSubsemiring.coe_iSup_of_directed {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_2} [hι : Nonempty ι] {S : ι → NonUnitalSubsemiring R} (hS : Directed (fun (x1 x2 : NonUnitalSubsemiring R) => x1 ≤ x2) S) :

↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

source

theorem NonUnitalSubsemiring.mem_sSup_of_directedOn {R : Type u} [NonUnitalNonAssocSemiring R] {S : Set (NonUnitalSubsemiring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : NonUnitalSubsemiring R) => x1 ≤ x2) S) {x : R} :

x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s

source

theorem NonUnitalSubsemiring.coe_sSup_of_directedOn {R : Type u} [NonUnitalNonAssocSemiring R] {S : Set (NonUnitalSubsemiring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : NonUnitalSubsemiring R) => x1 ≤ x2) S) :

↑(sSup S) = ⋃ s ∈ S, ↑s

source

theorem NonUnitalRingHom.eq_of_eqOn_stop {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] {f : F} {g : F} (h : Set.EqOn ⇑f ⇑g ↑⊤) :

f = g

source

def NonUnitalRingHom.srangeRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) :

R →ₙ+* ↥(NonUnitalRingHom.srange f)

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

This is the bundled version of Set.rangeFactorization.

Equations

NonUnitalRingHom.srangeRestrict f = NonUnitalRingHom.codRestrict f (NonUnitalRingHom.srange f) ⋯

Instances For

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@[simp]

theorem NonUnitalRingHom.coe_srangeRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (x : R) :

↑((NonUnitalRingHom.srangeRestrict f) x) = f x

source

theorem NonUnitalRingHom.srangeRestrict_surjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) :

Function.Surjective ⇑(NonUnitalRingHom.srangeRestrict f)

source

theorem NonUnitalRingHom.srange_top_iff_surjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {f : F} :

NonUnitalRingHom.srange f = ⊤ ↔ Function.Surjective ⇑f

source

@[simp]

theorem NonUnitalRingHom.srange_top_of_surjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) :

NonUnitalRingHom.srange f = ⊤

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

source

theorem NonUnitalRingHom.eqOn_sclosure {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {f : F} {g : F} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) :

Set.EqOn ⇑f ⇑g ↑(NonUnitalSubsemiring.closure s)

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

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theorem NonUnitalRingHom.eq_of_eqOn_sdense {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {s : Set R} (hs : NonUnitalSubsemiring.closure s = ⊤) {f : F} {g : F} (h : Set.EqOn (⇑f) (⇑g) s) :

f = g

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theorem NonUnitalRingHom.sclosure_preimage_le {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (s : Set S) :

NonUnitalSubsemiring.closure (⇑f ⁻¹' s) ≤ NonUnitalSubsemiring.comap f (NonUnitalSubsemiring.closure s)

source

theorem NonUnitalRingHom.map_sclosure {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (s : Set R) :

NonUnitalSubsemiring.map f (NonUnitalSubsemiring.closure s) = NonUnitalSubsemiring.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.

source

@[simp]

theorem NonUnitalSubsemiring.srange_subtype {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) :

NonUnitalRingHom.srange (NonUnitalSubsemiringClass.subtype s) = s

source

@[simp]

theorem NonUnitalSubsemiring.range_fst {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] :

NonUnitalRingHom.srange (NonUnitalRingHom.fst R S) = ⊤

source

@[simp]

theorem NonUnitalSubsemiring.range_snd {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] :

NonUnitalRingHom.srange (NonUnitalRingHom.snd R S) = ⊤

source

def RingEquiv.nonUnitalSubsemiringCongr {R : Type u} [NonUnitalNonAssocSemiring R] {s : NonUnitalSubsemiring R} {t : NonUnitalSubsemiring R} (h : s = t) :

↥s ≃+* ↥t

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

Equations

RingEquiv.nonUnitalSubsemiringCongr h = { toEquiv := Equiv.setCongr ⋯, map_mul' := ⋯, map_add' := ⋯ }

Instances For

source

def RingEquiv.sofLeftInverse' {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {g : S → R} {f : F} (h : Function.LeftInverse g ⇑f) :

R ≃+* ↥(NonUnitalRingHom.srange f)

Restrict a non-unital ring homomorphism with a left inverse to a ring isomorphism to its NonUnitalRingHom.srange.

Equations

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

Instances For

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@[simp]

theorem RingEquiv.sofLeftInverse'_apply {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {g : S → R} {f : F} (h : Function.LeftInverse g ⇑f) (x : R) :

↑((RingEquiv.sofLeftInverse' h) x) = f x

source

@[simp]

theorem RingEquiv.sofLeftInverse'_symm_apply {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {g : S → R} {f : F} (h : Function.LeftInverse g ⇑f) (x : ↥(NonUnitalRingHom.srange f)) :

(RingEquiv.sofLeftInverse' h).symm x = g ↑x

source

def RingEquiv.nonUnitalSubsemiringMap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (s : NonUnitalSubsemiring R) :

↥s ≃+* ↥(NonUnitalSubsemiring.map e.toNonUnitalRingHom s)

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

Equations

e.nonUnitalSubsemiringMap s = { toEquiv := (e.toAddEquiv.addSubmonoidMap s.toAddSubmonoid).toEquiv, map_mul' := ⋯, map_add' := ⋯ }

Instances For

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@[simp]

theorem RingEquiv.nonUnitalSubsemiringMap_apply_coe {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (s : NonUnitalSubsemiring R) (x : ↑↑s.toAddSubmonoid) :

↑((e.nonUnitalSubsemiringMap s) x) = e ↑x

source

@[simp]

theorem RingEquiv.nonUnitalSubsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (s : NonUnitalSubsemiring R) (y : ↑(⇑↑e.toAddEquiv '' ↑s.toAddSubmonoid)) :

↑((e.nonUnitalSubsemiringMap s).symm y) = (↑e).symm ↑y

Documentation

Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Bundled non-unital subsemirings #