Bundled non-unital subsemirings

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

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

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

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

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

def NonUnitalSubsemiring.topEquiv {R : Type u} [NonUnitalNonAssocSemiring R] : ↥⊤ ≃+* R

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

@[simp]

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

@[simp]

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

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.

@[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) : ↑(comap f s) = ⇑f ⁻¹' ↑s

@[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 ∈ comap f s ↔ f x ∈ s

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

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.

@[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) : ↑(map f s) = ⇑f '' ↑s

@[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 ∈ map f s ↔ ∃ x ∈ s, f x = y

@[simp]

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

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

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} : map f s ≤ t ↔ s ≤ comap f t

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 (map f) (comap f)

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 ≃+* ↥(map f s)

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

@[simp]

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

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].

@[simp]

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) : ↑(srange f) = Set.range ⇑f

@[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 ∈ srange f ↔ ∃ (x : R), f x = y

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) : srange f = NonUnitalSubsemiring.map f ⊤

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 ∈ srange f

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 (srange f) = srange (g.comp f)

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 ↥(srange f)

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

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

@[simp]

theorem NonUnitalSubsemiring.coe_sInf {R : Type u} [NonUnitalNonAssocSemiring R] (S : Set (NonUnitalSubsemiring R)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

@[simp]

theorem NonUnitalSubsemiring.mem_sInf {R : Type u} [NonUnitalNonAssocSemiring R] {S : Set (NonUnitalSubsemiring R)} {x : R} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

@[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)

@[simp]

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

@[simp]

theorem NonUnitalSubsemiring.sInf_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (NonUnitalSubsemiring R)) : (sInf s).toSubsemigroup = ⨅ t ∈ s, t.toSubsemigroup

@[simp]

theorem NonUnitalSubsemiring.sInf_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (NonUnitalSubsemiring R)) : (sInf s).toAddSubmonoid = ⨅ t ∈ s, t.toAddSubmonoid

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

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

theorem NonUnitalSubsemiring.eq_top_iff' {R : Type u} [NonUnitalNonAssocSemiring R] (A : NonUnitalSubsemiring R) : A = ⊤ ↔ ∀ (x : R), x ∈ A

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

theorem NonUnitalSubsemiring.coe_center (R : Type u) [NonUnitalNonAssocSemiring R] : ↑(center R) = Set.center R

@[simp]

theorem NonUnitalSubsemiring.center_toSubsemigroup (R : Type u) [NonUnitalNonAssocSemiring R] : (center R).toSubsemigroup = Subsemigroup.center R

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

The center is commutative and associative.

theorem Set.mem_center_iff_addMonoidHom (R : Type u) [NonUnitalNonAssocSemiring R] (a : R) : a ∈ center R ↔ AddMonoidHom.mulLeft a = AddMonoidHom.mulRight a ∧ AddMonoidHom.mul.compr₂ (AddMonoidHom.mulLeft a) = AddMonoidHom.mul.comp (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.

def NonUnitalSubsemiring.centerCongr {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) : ↥(center R) ≃+* ↥(center S)

The center of isomorphic (not necessarily unital or associative) semirings are isomorphic.

@[simp]

theorem NonUnitalSubsemiring.centerCongr_symm_apply_coe {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (s : ↥(Subsemigroup.center S)) : ↑((centerCongr e).symm s) = (↑e).symm ↑s

@[simp]

theorem NonUnitalSubsemiring.centerCongr_apply_coe {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (r : ↥(Subsemigroup.center R)) : ↑((centerCongr e) r) = e ↑r

def NonUnitalSubsemiring.centerToMulOpposite {R : Type u} [NonUnitalNonAssocSemiring R] : ↥(center R) ≃+* ↥(center Rᵐᵒᵖ)

The center of a (not necessarily unital or associative) semiring is isomorphic to the center of its opposite.

@[simp]

theorem NonUnitalSubsemiring.centerToMulOpposite_apply_coe {R : Type u} [NonUnitalNonAssocSemiring R] (r : ↥(Subsemigroup.center R)) : ↑(centerToMulOpposite r) = MulOpposite.op ↑r

@[simp]

theorem NonUnitalSubsemiring.centerToMulOpposite_symm_apply_coe {R : Type u} [NonUnitalNonAssocSemiring R] (r : ↥(Subsemigroup.center Rᵐᵒᵖ)) : ↑(centerToMulOpposite.symm r) = MulOpposite.unop ↑r

theorem NonUnitalSubsemiring.mem_center_iff {R : Type u_1} [NonUnitalSemiring R] {z : R} : z ∈ center R ↔ ∀ (g : R), g * z = z * g

instance NonUnitalSubsemiring.decidableMemCenter {R : Type u_1} [NonUnitalSemiring R] [DecidableEq R] [Fintype R] : DecidablePred fun (x : R) => x ∈ center R

@[simp]

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

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

The centralizer of a set as non-unital subsemiring.

@[simp]

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

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

theorem NonUnitalSubsemiring.mem_centralizer_iff {R : Type u_1} [NonUnitalSemiring R] {s : Set R} {z : R} : z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g

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

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

@[simp]

theorem NonUnitalSubsemiring.centralizer_eq_top_iff_subset {R : Type u_1} [NonUnitalSemiring R] {s : Set R} : centralizer s = ⊤ ↔ s ⊆ ↑(center R)

@[simp]

theorem NonUnitalSubsemiring.centralizer_univ {R : Type u_1} [NonUnitalSemiring R] : centralizer Set.univ = center R

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

The NonUnitalSubsemiring generated by a set.

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

@[simp]

theorem NonUnitalSubsemiring.subset_closure {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} : s ⊆ ↑(closure s)

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

theorem NonUnitalSubsemiring.mem_closure_of_mem {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {x : R} (hx : x ∈ s) : x ∈ closure s

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

@[simp]

theorem NonUnitalSubsemiring.closure_le {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {t : NonUnitalSubsemiring R} : closure s ≤ t ↔ s ⊆ ↑t

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

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

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

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

theorem NonUnitalSubsemiring.closure_le_centralizer_centralizer {R : Type u_1} [NonUnitalSemiring R] (s : Set R) : closure s ≤ centralizer ↑(centralizer s)

@[reducible, inline]

abbrev NonUnitalSubsemiring.closureNonUnitalCommSemiringOfComm {R : Type u_1} [NonUnitalSemiring R] {s : Set R} (hcomm : ∀ x ∈ s, ∀ y ∈ s, x * y = y * x) : NonUnitalCommSemiring ↥(closure s)

If all the elements of a set s commute, then closure s is a non-unital commutative semiring.

theorem NonUnitalSubsemiring.mem_map_equiv {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {f : R ≃+* S} {K : NonUnitalSubsemiring R} {x : S} : x ∈ map (↑f) K ↔ f.symm x ∈ K

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

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

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.

theorem Subsemigroup.nonUnitalSubsemiringClosure_coe {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) : ↑M.nonUnitalSubsemiringClosure = ↑(AddSubmonoid.closure ↑M)

theorem Subsemigroup.nonUnitalSubsemiringClosure_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) : M.nonUnitalSubsemiringClosure.toAddSubmonoid = AddSubmonoid.closure ↑M

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 .

@[simp]

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

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

theorem NonUnitalSubsemiring.mem_closure_iff {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {x : R} : x ∈ closure s ↔ x ∈ AddSubmonoid.closure ↑(Subsemigroup.closure s)

@[simp]

theorem NonUnitalSubsemiring.closure_addSubmonoid_closure {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} : closure ↑(AddSubmonoid.closure s) = closure s

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

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

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

def NonUnitalSubsemiring.gi (R : Type u) [NonUnitalNonAssocSemiring R] : GaloisInsertion closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

@[simp]

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

Closure of a non-unital subsemiring S equals S.

@[simp]

theorem NonUnitalSubsemiring.closure_empty {R : Type u} [NonUnitalNonAssocSemiring R] : closure ∅ = ⊥

@[simp]

theorem NonUnitalSubsemiring.closure_univ {R : Type u} [NonUnitalNonAssocSemiring R] : closure Set.univ = ⊤

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

theorem NonUnitalSubsemiring.closure_iUnion {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_2} (s : ι → Set R) : closure (⋃ (i : ι), s i) = ⨆ (i : ι), closure (s i)

theorem NonUnitalSubsemiring.closure_sUnion {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (Set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t

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 t : NonUnitalSubsemiring R) (f : F) : map f (s ⊔ t) = map f s ⊔ map f t

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

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 t : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective ⇑f) : map f (s ⊓ t) = map f s ⊓ map f t

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

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 t : NonUnitalSubsemiring S) (f : F) : comap f (s ⊓ t) = comap f s ⊓ comap f t

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

@[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) : map f ⊥ = ⊥

@[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) : comap f ⊤ = ⊤

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.

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

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

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

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

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

theorem NonUnitalSubsemiring.prod_top {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (s : NonUnitalSubsemiring R) : s.prod ⊤ = comap (NonUnitalRingHom.fst R S) s

theorem NonUnitalSubsemiring.top_prod {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (s : NonUnitalSubsemiring S) : ⊤.prod s = comap (NonUnitalRingHom.snd R S) s

@[simp]

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

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.

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

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)

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

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

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

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+* ↥(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.

@[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) : ↑((srangeRestrict f) x) = f x

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 ⇑(srangeRestrict f)

theorem NonUnitalRingHom.srange_eq_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} : srange f = ⊤ ↔ Function.Surjective ⇑f

@[simp]

theorem NonUnitalRingHom.srange_eq_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) : srange f = ⊤

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

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 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.

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 g : F} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

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)

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.

@[simp]

theorem NonUnitalSubsemiring.srange_subtype {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) : NonUnitalRingHom.srange (NonUnitalSubsemiringClass.subtype s) = s

@[simp]

theorem NonUnitalSubsemiring.range_fst {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] : NonUnitalRingHom.srange (NonUnitalRingHom.fst R S) = ⊤

@[simp]

theorem NonUnitalSubsemiring.range_snd {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] : NonUnitalRingHom.srange (NonUnitalRingHom.snd R S) = ⊤

def RingEquiv.nonUnitalSubsemiringCongr {R : Type u} [NonUnitalNonAssocSemiring R] {s 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.

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.

@[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) : ↑((sofLeftInverse' h) x) = f x

@[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)) : (sofLeftInverse' h).symm x = g ↑x

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, nonUnitalSubsemiringMap e s is the induced equivalence between s and s.map e

@[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

@[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