Bundled non-unital subsemirings #

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.

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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.

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

GaloisInsertion closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

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@[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.

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@[simp]

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

closure ∅ = ⊥

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@[simp]

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

closure Set.univ = ⊤

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

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

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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)

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

closure (⋃₀ s) = ⨆ t ∈ s, 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 t : NonUnitalSubsemiring R) (f : F) :

map f (s ⊔ t) = map f s ⊔ map f t

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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)

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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

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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)

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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

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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)

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@[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 ⊥ = ⊥

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@[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 ⊤ = ⊤

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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.

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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 s₂ : NonUnitalSubsemiring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : NonUnitalSubsemiring S⦄ (ht : t₁ ≤ t₂) :

s₁.prod t₁ ≤ s₂.prod t₂

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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

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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

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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

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@[simp]

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

⊤.prod ⊤ = ⊤

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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.

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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

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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)

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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

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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

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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

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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.

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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) :

↑((srangeRestrict f) x) = f x

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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)

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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

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@[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.

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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.

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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 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)

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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.

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@[simp]

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

NonUnitalRingHom.srange (NonUnitalSubsemiringClass.subtype s) = s

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@[simp]

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

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

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@[simp]

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

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

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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.

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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.

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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) :

↑((sofLeftInverse' h) x) = f x

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@[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

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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

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@[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

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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

Documentation

Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Bundled non-unital subsemirings #