fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
isNoetherian_submodule' [IsNoetherian R M] (N : Submodule R M) : IsNoetherian R N :=
isNoetherian_submodule.2 fun _ _ => IsNoetherian.noetherian _ | instance | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | isNoetherian_submodule' | null |
isNoetherian_of_le {s t : Submodule R M} [ht : IsNoetherian R t] (h : s ≤ t) :
IsNoetherian R s :=
isNoetherian_submodule.mpr fun _ hs' => isNoetherian_submodule.mp ht _ (le_trans hs' h) | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | isNoetherian_of_le | null |
isNoetherian_iff' : IsNoetherian R M ↔ WellFoundedGT (Submodule R M) := by
refine .trans ?_ ((CompleteLattice.wellFoundedGT_characterisations <| Submodule R M).out 0 3).symm
exact
⟨fun ⟨h⟩ => fun k => (fg_iff_compact k).mp (h k), fun h =>
⟨fun k => (fg_iff_compact k).mpr (h k)⟩⟩ | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | isNoetherian_iff' | null |
isNoetherian_iff :
IsNoetherian R M ↔ WellFounded ((· > ·) : Submodule R M → Submodule R M → Prop) := by
rw [isNoetherian_iff', ← isWellFounded_iff]
alias ⟨IsNoetherian.wf, _⟩ := isNoetherian_iff
alias ⟨IsNoetherian.wellFoundedGT, isNoetherian_mk⟩ := isNoetherian_iff' | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | isNoetherian_iff | null |
wellFoundedGT [h : IsNoetherian R M] : WellFoundedGT (Submodule R M) :=
h.wellFoundedGT | instance | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | wellFoundedGT | null |
isNoetherian_iff_fg_wellFounded :
IsNoetherian R M ↔ WellFoundedGT { N : Submodule R M // N.FG } := by
let α := { N : Submodule R M // N.FG }
constructor
· intro H
let f : α ↪o Submodule R M := OrderEmbedding.subtype _
exact OrderEmbedding.wellFoundedLT f.dual
· intro H
constructor
intro N
obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=
WellFounded.has_min H.wf { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩
convert h₁
refine (e.antisymm ?_).symm
by_contra h₃
obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃
apply hx₂
rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂
⟨_, Submodule.FG.sup ⟨{x}, by rw [Finset.coe_singleton]⟩ h₁⟩ <|
sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) e)]
exact (le_sup_left : (R ∙ x) ≤ _) (Submodule.mem_span_singleton_self _) | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | isNoetherian_iff_fg_wellFounded | null |
set_has_maximal_iff_noetherian :
(∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M := by
rw [isNoetherian_iff, WellFounded.wellFounded_iff_has_min] | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | set_has_maximal_iff_noetherian | A module is Noetherian iff every nonempty set of submodules has a maximal submodule among them. |
monotone_stabilizes_iff_noetherian :
(∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by
rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]
variable [IsNoetherian R M]
open Filter | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | monotone_stabilizes_iff_noetherian | A module is Noetherian iff every increasing chain of submodules stabilizes. |
Module.End.eventually_disjoint_ker_pow_range_pow (f : End R M) :
∀ᶠ n in atTop, Disjoint (LinearMap.ker (f ^ n)) (LinearMap.range (f ^ n)) := by
obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.ker (f ^ n) = LinearMap.ker (f ^ m)⟩ :=
monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer
refine eventually_atTop.mpr ⟨n, fun m hm ↦ disjoint_iff.mpr ?_⟩
rw [← hn _ hm, Submodule.eq_bot_iff]
rintro - ⟨hx, ⟨x, rfl⟩⟩
apply pow_map_zero_of_le hm
replace hx : x ∈ LinearMap.ker (f ^ (n + m)) := by
simpa [f.pow_apply n, f.pow_apply m, ← f.pow_apply (n + m), ← iterate_add_apply] using hx
rwa [← hn _ (n.le_add_right m)] at hx | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | Module.End.eventually_disjoint_ker_pow_range_pow | For an endomorphism of a Noetherian module, any sufficiently large iterate has disjoint kernel
and range. |
LinearMap.eventually_iSup_ker_pow_eq (f : M →ₗ[R] M) :
∀ᶠ n in atTop, ⨆ m, LinearMap.ker (f ^ m) = LinearMap.ker (f ^ n) := by
obtain ⟨n, hn : ∀ m, n ≤ m → ker (f ^ n) = ker (f ^ m)⟩ :=
monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer
refine eventually_atTop.mpr ⟨n, fun m hm ↦ ?_⟩
refine le_antisymm (iSup_le fun l ↦ ?_) (le_iSup (fun i ↦ LinearMap.ker (f ^ i)) m)
rcases le_or_gt m l with h | h
· rw [← hn _ (hm.trans h), hn _ hm]
· exact f.iterateKer.monotone h.le | lemma | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | LinearMap.eventually_iSup_ker_pow_eq | null |
IsNoetherianRing (R) [Semiring R] :=
IsNoetherian R R | abbrev | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | IsNoetherianRing | A (semi)ring is Noetherian if it is Noetherian as a module over itself,
i.e. all its ideals are finitely generated. |
isNoetherianRing_iff {R} [Semiring R] : IsNoetherianRing R ↔ IsNoetherian R R :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | isNoetherianRing_iff | null |
isNoetherianRing_iff_ideal_fg (R : Type*) [Semiring R] :
IsNoetherianRing R ↔ ∀ I : Ideal R, I.FG :=
isNoetherianRing_iff.trans isNoetherian_def | theorem | RingTheory | [
"Mathlib.Order.Filter.AtTopBot.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Noetherian/Defs.lean | isNoetherianRing_iff_ideal_fg | A ring is Noetherian if and only if all its ideals are finitely-generated. |
eventuallyConst_of_isNoetherian [IsNoetherian R M] (f : ℕ →o Submodule R M) :
atTop.EventuallyConst f := by
simp_rw [eventuallyConst_atTop, eq_comm]
exact (monotone_stabilizes_iff_noetherian.mpr inferInstance) f | theorem | RingTheory | [
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Noetherian.Defs"
] | Mathlib/RingTheory/Noetherian/Filter.lean | eventuallyConst_of_isNoetherian | null |
IsNoetherianRing.isNilpotent_nilradical (R : Type*) [CommSemiring R] [IsNoetherianRing R] :
IsNilpotent (nilradical R) := by
obtain ⟨n, hn⟩ := Ideal.exists_radical_pow_le_of_fg (⊥ : Ideal R) (IsNoetherian.noetherian _)
exact ⟨n, eq_bot_iff.mpr hn⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Defs"
] | Mathlib/RingTheory/Noetherian/Nilpotent.lean | IsNoetherianRing.isNilpotent_nilradical | null |
Ideal.FG.isNilpotent_iff_le_nilradical {R : Type*} [CommSemiring R] {I : Ideal R}
(hI : I.FG) : IsNilpotent I ↔ I ≤ nilradical R :=
⟨fun ⟨n, hn⟩ _ hx ↦ ⟨n, hn ▸ Ideal.pow_mem_pow hx n⟩,
fun h ↦ let ⟨n, hn⟩ := exists_pow_le_of_le_radical_of_fg h hI; ⟨n, le_bot_iff.mp hn⟩⟩ | lemma | RingTheory | [
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Defs"
] | Mathlib/RingTheory/Noetherian/Nilpotent.lean | Ideal.FG.isNilpotent_iff_le_nilradical | null |
IsNoetherian.injective_of_surjective_of_injective (i f : N →ₗ[R] M)
(hi : Injective i) (hf : Surjective f) : Injective f := by
haveI := isNoetherian_of_injective i hi
obtain ⟨n, H⟩ := monotone_stabilizes_iff_noetherian.2 ‹_›
⟨_, monotone_nat_of_le_succ <| f.iterateMapComap_le_succ i ⊥ (by simp)⟩
exact LinearMap.ker_eq_bot.1 <| bot_unique <|
f.ker_le_of_iterateMapComap_eq_succ i ⊥ n (H _ (Nat.le_succ _)) hf hi | theorem | RingTheory | [
"Mathlib.Algebra.Module.Submodule.IterateMapComap",
"Mathlib.Order.PartialSups",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.OrzechProperty"
] | Mathlib/RingTheory/Noetherian/Orzech.lean | IsNoetherian.injective_of_surjective_of_injective | **Orzech's theorem** for Noetherian modules: if `R` is a ring (not necessarily commutative),
`M` and `N` are `R`-modules, `M` is Noetherian, `i : N →ₗ[R] M` is injective,
`f : N →ₗ[R] M` is surjective, then `f` is also injective. The proof here is adapted from
Djoković's paper *Epimorphisms of modules which must be isomorphisms* [djokovic1973],
utilizing `LinearMap.iterateMapComap`.
See also Orzech's original paper: *Onto endomorphisms are isomorphisms* [orzech1971]. |
IsNoetherian.injective_of_surjective_of_submodule
{N : Submodule R M} (f : N →ₗ[R] M) (hf : Surjective f) : Injective f :=
IsNoetherian.injective_of_surjective_of_injective N.subtype f N.injective_subtype hf | theorem | RingTheory | [
"Mathlib.Algebra.Module.Submodule.IterateMapComap",
"Mathlib.Order.PartialSups",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.OrzechProperty"
] | Mathlib/RingTheory/Noetherian/Orzech.lean | IsNoetherian.injective_of_surjective_of_submodule | **Orzech's theorem** for Noetherian modules: if `R` is a ring (not necessarily commutative),
`M` is a Noetherian `R`-module, `N` is a submodule, `f : N →ₗ[R] M` is surjective, then `f` is also
injective. |
IsNoetherian.injective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Injective f :=
IsNoetherian.injective_of_surjective_of_injective _ f (LinearEquiv.refl _ _).injective s | theorem | RingTheory | [
"Mathlib.Algebra.Module.Submodule.IterateMapComap",
"Mathlib.Order.PartialSups",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.OrzechProperty"
] | Mathlib/RingTheory/Noetherian/Orzech.lean | IsNoetherian.injective_of_surjective_endomorphism | Any surjective endomorphism of a Noetherian module is injective. |
IsNoetherian.bijective_of_surjective_endomorphism (f : M →ₗ[R] M)
(s : Surjective f) : Bijective f :=
⟨IsNoetherian.injective_of_surjective_endomorphism f s, s⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Module.Submodule.IterateMapComap",
"Mathlib.Order.PartialSups",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.OrzechProperty"
] | Mathlib/RingTheory/Noetherian/Orzech.lean | IsNoetherian.bijective_of_surjective_endomorphism | Any surjective endomorphism of a Noetherian module is bijective. |
IsNoetherian.subsingleton_of_prod_injective (f : M × N →ₗ[R] M)
(i : Injective f) : Subsingleton N := .intro fun x y ↦ by
have h := IsNoetherian.injective_of_surjective_of_injective f _ i LinearMap.fst_surjective
simpa using h (show LinearMap.fst R M N (0, x) = LinearMap.fst R M N (0, y) from rfl) | theorem | RingTheory | [
"Mathlib.Algebra.Module.Submodule.IterateMapComap",
"Mathlib.Order.PartialSups",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.OrzechProperty"
] | Mathlib/RingTheory/Noetherian/Orzech.lean | IsNoetherian.subsingleton_of_prod_injective | If `M ⊕ N` embeds into `M`, for `M` Noetherian over `R`, then `N` is trivial. |
@[simps!]
IsNoetherian.equivPUnitOfProdInjective (f : M × N →ₗ[R] M)
(i : Injective f) : N ≃ₗ[R] PUnit.{w + 1} :=
haveI := IsNoetherian.subsingleton_of_prod_injective f i
.ofSubsingleton _ _ | def | RingTheory | [
"Mathlib.Algebra.Module.Submodule.IterateMapComap",
"Mathlib.Order.PartialSups",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.OrzechProperty"
] | Mathlib/RingTheory/Noetherian/Orzech.lean | IsNoetherian.equivPUnitOfProdInjective | If `M ⊕ N` embeds into `M`, for `M` Noetherian over `R`, then `N` is trivial. |
protected list_sum_mem {l : List R} : (∀ x ∈ l, x ∈ s) → l.sum ∈ s :=
list_sum_mem | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | list_sum_mem | Sum of a list of elements in a non-unital subring is in the non-unital subring. |
protected multiset_sum_mem {R} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R)
(m : Multiset R) : (∀ a ∈ m, a ∈ s) → m.sum ∈ s :=
multiset_sum_mem _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | multiset_sum_mem | Sum of a multiset of elements in a `NonUnitalSubring` of a `NonUnitalRing` is
in the `NonUnitalSubring`. |
protected sum_mem {R : Type*} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R)
{ι : Type*} {t : Finset ι} {f : ι → R} (h : ∀ c ∈ t, f c ∈ s) : (∑ i ∈ t, f i) ∈ s :=
sum_mem h
/-! ## top -/ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | sum_mem | Sum of elements in a `NonUnitalSubring` of a `NonUnitalRing` indexed by a `Finset`
is in the `NonUnitalSubring`. |
@[simps!]
topEquiv : (⊤ : NonUnitalSubring R) ≃+* R := NonUnitalSubsemiring.topEquiv | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | topEquiv | The non-unital subring `R` of the ring `R`. -/
instance : Top (NonUnitalSubring R) :=
⟨{ (⊤ : Subsemigroup R), (⊤ : AddSubgroup R) with }⟩
@[simp]
theorem mem_top (x : R) : x ∈ (⊤ : NonUnitalSubring R) :=
Set.mem_univ x
@[simp]
theorem coe_top : ((⊤ : NonUnitalSubring R) : Set R) = Set.univ :=
rfl
/-- The ring equiv between the top element of `NonUnitalSubring R` and `R`. |
comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
[FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubring S) :
NonUnitalSubring R :=
{ s.toSubsemigroup.comap (f : R →ₙ* S), s.toAddSubgroup.comap (f : R →+ S) with
carrier := f ⁻¹' s.carrier }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | comap | The preimage of a `NonUnitalSubring` along a ring homomorphism is a `NonUnitalSubring`. |
coe_comap (s : NonUnitalSubring S) (f : F) : (s.comap f : Set R) = f ⁻¹' s :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_comap | null |
mem_comap {s : NonUnitalSubring S} {f : F} {x : R} : x ∈ s.comap f ↔ f x ∈ s :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_comap | null |
comap_comap (s : NonUnitalSubring T) (g : S →ₙ+* T) (f : R →ₙ+* S) :
(s.comap g).comap f = s.comap (g.comp f) :=
rfl
/-! ## map -/ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | comap_comap | null |
map {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
[FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubring R) :
NonUnitalSubring S :=
{ s.toSubsemigroup.map (f : R →ₙ* S), s.toAddSubgroup.map (f : R →+ S) with
carrier := f '' s.carrier }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map | The image of a `NonUnitalSubring` along a ring homomorphism is a `NonUnitalSubring`. |
coe_map (f : F) (s : NonUnitalSubring R) : (s.map f : Set S) = f '' s :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_map | null |
mem_map {f : F} {s : NonUnitalSubring R} {y : S} : y ∈ s.map f ↔ ∃ x ∈ s, f x = y :=
Set.mem_image _ _ _
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_map | null |
map_id : s.map (NonUnitalRingHom.id R) = s :=
SetLike.coe_injective <| Set.image_id _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_id | null |
map_map (g : S →ₙ+* T) (f : R →ₙ+* S) : (s.map f).map g = s.map (g.comp f) :=
SetLike.coe_injective <| Set.image_image _ _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_map | null |
map_le_iff_le_comap {f : F} {s : NonUnitalSubring R} {t : NonUnitalSubring S} :
s.map f ≤ t ↔ s ≤ t.comap f :=
Set.image_subset_iff | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_le_iff_le_comap | null |
gc_map_comap (f : F) :
GaloisConnection (map f : NonUnitalSubring R → NonUnitalSubring S) (comap f) := fun _S _T =>
map_le_iff_le_comap | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | gc_map_comap | null |
noncomputable equivMapOfInjective (f : F) (hf : Function.Injective (f : R → S)) :
s ≃+* s.map f :=
{
Equiv.Set.image f s
hf with
map_mul' := fun _ _ => Subtype.ext (map_mul f _ _)
map_add' := fun _ _ => Subtype.ext (map_add f _ _) }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | equivMapOfInjective | A `NonUnitalSubring` is isomorphic to its image under an injective function |
coe_equivMapOfInjective_apply (f : F) (hf : Function.Injective f) (x : s) :
(equivMapOfInjective s f hf x : S) = f x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_equivMapOfInjective_apply | null |
range {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
(f : R →ₙ+* S) : NonUnitalSubring S :=
((⊤ : NonUnitalSubring R).map f).copy (Set.range f) Set.image_univ.symm
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | range | The range of a ring homomorphism, as a `NonUnitalSubring` of the target.
See Note [range copy pattern]. |
coe_range : (f.range : Set S) = Set.range f :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_range | null |
mem_range {f : R →ₙ+* S} {y : S} : y ∈ f.range ↔ ∃ x, f x = y :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_range | null |
range_eq_map (f : R →ₙ+* S) : f.range = NonUnitalSubring.map f ⊤ := by ext; simp | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | range_eq_map | null |
mem_range_self (f : R →ₙ+* S) (x : R) : f x ∈ f.range :=
mem_range.mpr ⟨x, rfl⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_range_self | null |
map_range : f.range.map g = (g.comp f).range := by
simpa only [range_eq_map] using (⊤ : NonUnitalSubring R).map_map g f | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_range | null |
fintypeRange [Fintype R] [DecidableEq S] (f : R →ₙ+* S) : Fintype (range f) :=
Set.fintypeRange f | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | fintypeRange | The range of a ring homomorphism 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`. |
coe_bot : ((⊥ : NonUnitalSubring R) : Set R) = {0} :=
(NonUnitalRingHom.coe_range (0 : R →ₙ+* R)).trans (@Set.range_const R R _ 0) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_bot | null |
mem_bot {x : R} : x ∈ (⊥ : NonUnitalSubring R) ↔ x = 0 :=
show x ∈ ((⊥ : NonUnitalSubring R) : Set R) ↔ x = 0 by rw [coe_bot, Set.mem_singleton_iff]
/-! ## inf -/ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_bot | null |
center : NonUnitalSubring R :=
{ NonUnitalSubsemiring.center R with
neg_mem' := Set.neg_mem_center } | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | center | The inf of two `NonUnitalSubring`s is their intersection. -/
instance : Min (NonUnitalSubring R) :=
⟨fun s t =>
{ s.toSubsemigroup ⊓ t.toSubsemigroup, s.toAddSubgroup ⊓ t.toAddSubgroup with
carrier := s ∩ t }⟩
@[simp]
theorem coe_inf (p p' : NonUnitalSubring R) :
((p ⊓ p' : NonUnitalSubring R) : Set R) = (p : Set R) ∩ p' :=
rfl
@[simp]
theorem mem_inf {p p' : NonUnitalSubring R} {x : R} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' :=
Iff.rfl
instance : InfSet (NonUnitalSubring R) :=
⟨fun s =>
NonUnitalSubring.mk' (⋂ t ∈ s, ↑t) (⨅ t ∈ s, NonUnitalSubring.toSubsemigroup t)
(⨅ t ∈ s, NonUnitalSubring.toAddSubgroup t) (by simp) (by simp)⟩
@[simp, norm_cast]
theorem coe_sInf (S : Set (NonUnitalSubring R)) :
((sInf S : NonUnitalSubring R) : Set R) = ⋂ s ∈ S, ↑s :=
rfl
theorem mem_sInf {S : Set (NonUnitalSubring R)} {x : R} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p :=
Set.mem_iInter₂
@[simp, norm_cast]
theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubring R} : (↑(⨅ i, S i) : Set R) = ⋂ i, S i := by
simp only [iInf, coe_sInf, Set.biInter_range]
theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubring R} {x : R} :
(x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_mem_range]
@[simp]
theorem sInf_toSubsemigroup (s : Set (NonUnitalSubring R)) :
(sInf s).toSubsemigroup = ⨅ t ∈ s, NonUnitalSubring.toSubsemigroup t :=
mk'_toSubsemigroup _ _
@[simp]
theorem sInf_toAddSubgroup (s : Set (NonUnitalSubring R)) :
(sInf s).toAddSubgroup = ⨅ t ∈ s, NonUnitalSubring.toAddSubgroup t :=
mk'_toAddSubgroup _ _
/-- `NonUnitalSubring`s of a ring form a complete lattice. -/
instance : CompleteLattice (NonUnitalSubring R) :=
{ completeLatticeOfInf (NonUnitalSubring R) fun _s =>
IsGLB.of_image (@fun _ _ : NonUnitalSubring R => SetLike.coe_subset_coe)
isGLB_biInf with
bot := ⊥
bot_le := fun s _x hx => (mem_bot.mp hx).symm ▸ zero_mem s
top := ⊤
le_top := fun _ _ _ => trivial
inf := (· ⊓ ·)
inf_le_left := fun _ _ _ => And.left
inf_le_right := fun _ _ _ => And.right
le_inf := fun _s _t₁ _t₂ h₁ h₂ _x hx => ⟨h₁ hx, h₂ hx⟩ }
theorem eq_top_iff' (A : NonUnitalSubring R) : A = ⊤ ↔ ∀ x : R, x ∈ A :=
eq_top_iff.trans ⟨fun h m => h <| mem_top m, fun h m _ => h m⟩
end Order
/-! ## Center of a ring -/
section Center
variable {R : Type u}
section NonUnitalNonAssocRing
variable (R) [NonUnitalNonAssocRing R]
/-- The center of a ring `R` is the set of elements that commute with everything in `R` |
coe_center : ↑(center R) = Set.center R :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_center | null |
center_toNonUnitalSubsemiring :
(center R).toNonUnitalSubsemiring = NonUnitalSubsemiring.center R :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | center_toNonUnitalSubsemiring | null |
center.instNonUnitalCommRing : NonUnitalCommRing (center R) :=
{ NonUnitalSubsemiring.center.instNonUnitalCommSemiring R,
inferInstanceAs <| NonUnitalNonAssocRing (center R) with }
variable {R} | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | center.instNonUnitalCommRing | The center is commutative and associative. |
@[simps!] centerCongr {S} [NonUnitalNonAssocRing S] (e : R ≃+* S) : center R ≃+* center S :=
NonUnitalSubsemiring.centerCongr e | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | centerCongr | The center of isomorphic (not necessarily unital or associative) rings are isomorphic. |
@[simps!] centerToMulOpposite : center R ≃+* center Rᵐᵒᵖ :=
NonUnitalSubsemiring.centerToMulOpposite | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | centerToMulOpposite | The center of a (not necessarily unital or associative) ring
is isomorphic to the center of its opposite. |
mem_center_iff {z : R} : z ∈ center R ↔ ∀ g, g * z = z * g := Subsemigroup.mem_center_iff | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_center_iff | null |
decidableMemCenter [DecidableEq R] [Fintype R] : DecidablePred (· ∈ center R) := fun _ =>
decidable_of_iff' _ mem_center_iff
@[simp] | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | decidableMemCenter | null |
center_eq_top (R) [NonUnitalCommRing R] : center R = ⊤ :=
SetLike.coe_injective (Set.center_eq_univ R) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | center_eq_top | null |
closure (s : Set R) : NonUnitalSubring R :=
sInf {S | s ⊆ S} | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure | The `NonUnitalSubring` generated by a set. |
mem_closure {x : R} {s : Set R} : x ∈ closure s ↔ ∀ S : NonUnitalSubring R, s ⊆ S → x ∈ S :=
mem_sInf | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_closure | null |
@[simp]
closure_le {s : Set R} {t : NonUnitalSubring R} : closure s ≤ t ↔ s ⊆ t :=
⟨Set.Subset.trans subset_closure, fun h => sInf_le h⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_le | The `NonUnitalSubring` generated by a set includes the set. -/
@[simp, aesop safe 20 (rule_sets := [SetLike])]
theorem subset_closure {s : Set R} : s ⊆ closure s := fun _x hx => mem_closure.2 fun _S hS => hS hx
@[aesop 80% (rule_sets := [SetLike])]
theorem mem_closure_of_mem {s : Set R} {x : R} (hx : x ∈ s) : x ∈ closure s := subset_closure hx
theorem notMem_of_notMem_closure {s : Set R} {P : R} (hP : P ∉ closure s) : P ∉ s := fun h =>
hP (subset_closure h)
@[deprecated (since := "2025-05-23")] alias not_mem_of_not_mem_closure := notMem_of_notMem_closure
/-- A `NonUnitalSubring` `t` includes `closure s` if and only if it includes `s`. |
@[gcongr]
closure_mono ⦃s t : Set R⦄ (h : s ⊆ t) : closure s ≤ closure t :=
closure_le.2 <| Set.Subset.trans h subset_closure | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_mono | `NonUnitalSubring` closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`. |
closure_eq_of_le {s : Set R} {t : NonUnitalSubring R} (h₁ : s ⊆ t) (h₂ : t ≤ closure s) :
closure s = t :=
le_antisymm (closure_le.2 h₁) h₂ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_eq_of_le | null |
@[elab_as_elim]
closure_induction {s : Set R} {p : (x : R) → x ∈ closure s → Prop}
(mem : ∀ (x) (hx : x ∈ s), p x (subset_closure hx)) (zero : p 0 (zero_mem _))
(add : ∀ x y hx hy, p x hx → p y hy → p (x + y) (add_mem hx hy))
(neg : ∀ x hx, p x hx → p (-x) (neg_mem hx))
(mul : ∀ x y hx hy, p x hx → p y hy → p (x * y) (mul_mem hx hy))
{x} (hx : x ∈ closure s) : p x hx :=
let K : NonUnitalSubring R :=
{ carrier := { x | ∃ hx, p x hx }
mul_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, mul _ _ _ _ hpx hpy⟩
add_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, add _ _ _ _ hpx hpy⟩
neg_mem' := fun ⟨_, hpx⟩ ↦ ⟨_, neg _ _ hpx⟩
zero_mem' := ⟨_, zero⟩ }
closure_le (t := K) |>.mpr (fun y hy ↦ ⟨subset_closure hy, mem y hy⟩) hx |>.elim fun _ ↦ id | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_induction | An induction principle for closure membership. If `p` holds for `0`, `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. |
@[elab_as_elim]
closure_induction₂ {s : Set R} {p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop}
(mem_mem : ∀ (x) (y) (hx : x ∈ s) (hy : y ∈ s), p x y (subset_closure hx) (subset_closure hy))
(zero_left : ∀ x hx, p 0 x (zero_mem _) hx) (zero_right : ∀ x hx, p x 0 hx (zero_mem _))
(neg_left : ∀ x y hx hy, p x y hx hy → p (-x) y (neg_mem hx) hy)
(neg_right : ∀ x y hx hy, p x y hx hy → p x (-y) hx (neg_mem hy))
(add_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x + y) z (add_mem hx hy) hz)
(add_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y + z) hx (add_mem hy hz))
(mul_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x * y) z (mul_mem hx hy) hz)
(mul_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y * z) hx (mul_mem hy hz))
{x y : R} (hx : x ∈ closure s) (hy : y ∈ closure s) :
p x y hx hy := by
induction hy using closure_induction with
| mem z hz => induction hx using closure_induction with
| mem _ h => exact mem_mem _ _ h hz
| zero => exact zero_left _ _
| mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
| add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
| neg _ _ h => exact neg_left _ _ _ _ h
| zero => exact zero_right x hx
| mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
| add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
| neg _ _ h => exact neg_right _ _ _ _ h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_induction₂ | An induction principle for closure membership, for predicates with two arguments. |
mem_closure_iff {s : Set R} {x} :
x ∈ closure s ↔ x ∈ AddSubgroup.closure (Subsemigroup.closure s : Set R) :=
⟨fun h => by
induction h using closure_induction with
| mem _ hx => exact AddSubgroup.subset_closure (Subsemigroup.subset_closure hx)
| zero => exact zero_mem _
| add _ _ _ _ hx hy => exact add_mem hx hy
| neg x _ hx => exact neg_mem hx
| mul _ _ _hx _hy hx hy =>
clear _hx _hy
induction hx, hy using AddSubgroup.closure_induction₂ with
| mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
| zero_left => simp
| zero_right => simp
| add_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
| add_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
| neg_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
| neg_right _ _ _ _ h => simpa [mul_neg] using neg_mem h,
fun h => by
induction h using AddSubgroup.closure_induction with
| mem _ hx => induction hx using Subsemigroup.closure_induction with
| mem _ h => exact subset_closure h
| mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
| zero => exact zero_mem _
| add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
| neg _ _ h => exact neg_mem h⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_closure_iff | null |
closureNonUnitalCommRingOfComm {R : Type u} [NonUnitalRing R] {s : Set R}
(hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) : NonUnitalCommRing (closure s) :=
{ (closure s).toNonUnitalRing with
mul_comm := fun ⟨x, hx⟩ ⟨y, hy⟩ => by
ext
simp only [MulMemClass.mk_mul_mk]
induction hx, hy using closure_induction₂ with
| mem_mem x y hx hy => exact hcomm x hx y hy
| zero_left x _ => exact Commute.zero_left x
| zero_right x _ => exact Commute.zero_right x
| mul_left _ _ _ _ _ _ h₁ h₂ => exact Commute.mul_left h₁ h₂
| mul_right _ _ _ _ _ _ h₁ h₂ => exact Commute.mul_right h₁ h₂
| add_left _ _ _ _ _ _ h₁ h₂ => exact Commute.add_left h₁ h₂
| add_right _ _ _ _ _ _ h₁ h₂ => exact Commute.add_right h₁ h₂
| neg_left _ _ _ _ h => exact Commute.neg_left h
| neg_right _ _ _ _ h => exact Commute.neg_right h }
variable (R) in | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closureNonUnitalCommRingOfComm | If all elements of `s : Set A` commute pairwise, then `closure s` is a commutative ring. |
protected gi : GaloisInsertion (@closure R _) SetLike.coe where
choice s _ := closure s
gc _s _t := closure_le
le_l_u _s := subset_closure
choice_eq _s _h := rfl | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | gi | `closure` forms a Galois insertion with the coercion to set. |
@[simp]
closure_eq (s : NonUnitalSubring R) : closure (s : Set R) = s :=
(NonUnitalSubring.gi R).l_u_eq s
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_eq | Closure of a `NonUnitalSubring` `S` equals `S`. |
closure_empty : closure (∅ : Set R) = ⊥ :=
(NonUnitalSubring.gi R).gc.l_bot
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_empty | null |
closure_univ : closure (Set.univ : Set R) = ⊤ :=
@coe_top R _ ▸ closure_eq ⊤ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_univ | null |
closure_union (s t : Set R) : closure (s ∪ t) = closure s ⊔ closure t :=
(NonUnitalSubring.gi R).gc.l_sup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_union | null |
closure_iUnion {ι} (s : ι → Set R) : closure (⋃ i, s i) = ⨆ i, closure (s i) :=
(NonUnitalSubring.gi R).gc.l_iSup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_iUnion | null |
closure_sUnion (s : Set (Set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t :=
(NonUnitalSubring.gi R).gc.l_sSup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | closure_sUnion | null |
map_sup (s t : NonUnitalSubring R) (f : F) : (s ⊔ t).map f = s.map f ⊔ t.map f :=
(gc_map_comap f).l_sup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_sup | null |
map_iSup {ι : Sort*} (f : F) (s : ι → NonUnitalSubring R) :
(iSup s).map f = ⨆ i, (s i).map f :=
(gc_map_comap f).l_iSup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_iSup | null |
map_inf (s t : NonUnitalSubring R) (f : F) (hf : Function.Injective f) :
(s ⊓ t).map f = s.map f ⊓ t.map f := SetLike.coe_injective (Set.image_inter hf) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_inf | null |
map_iInf {ι : Sort*} [Nonempty ι] (f : F) (hf : Function.Injective f)
(s : ι → NonUnitalSubring R) : (iInf s).map f = ⨅ i, (s i).map f := by
apply SetLike.coe_injective
simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_iInf | null |
comap_inf (s t : NonUnitalSubring S) (f : F) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
(gc_map_comap f).u_inf | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | comap_inf | null |
comap_iInf {ι : Sort*} (f : F) (s : ι → NonUnitalSubring S) :
(iInf s).comap f = ⨅ i, (s i).comap f :=
(gc_map_comap f).u_iInf
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | comap_iInf | null |
map_bot (f : R →ₙ+* S) : (⊥ : NonUnitalSubring R).map f = ⊥ :=
(gc_map_comap f).l_bot
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_bot | null |
comap_top (f : R →ₙ+* S) : (⊤ : NonUnitalSubring S).comap f = ⊤ :=
(gc_map_comap f).u_top | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | comap_top | null |
prod (s : NonUnitalSubring R) (t : NonUnitalSubring S) : NonUnitalSubring (R × S) :=
{ s.toSubsemigroup.prod t.toSubsemigroup, s.toAddSubgroup.prod t.toAddSubgroup with
carrier := s ×ˢ t }
@[norm_cast] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | prod | Given `NonUnitalSubring`s `s`, `t` of rings `R`, `S` respectively, `s.prod t` is `s ×ˢ t`
as a `NonUnitalSubring` of `R × S`. |
coe_prod (s : NonUnitalSubring R) (t : NonUnitalSubring S) :
(s.prod t : Set (R × S)) = (s : Set R) ×ˢ t :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_prod | null |
mem_prod {s : NonUnitalSubring R} {t : NonUnitalSubring S} {p : R × S} :
p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t :=
Iff.rfl
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_prod | null |
prod_mono ⦃s₁ s₂ : NonUnitalSubring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : NonUnitalSubring S⦄
(ht : t₁ ≤ t₂) : s₁.prod t₁ ≤ s₂.prod t₂ :=
Set.prod_mono hs ht | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | prod_mono | null |
prod_mono_right (s : NonUnitalSubring R) :
Monotone fun t : NonUnitalSubring S => s.prod t :=
prod_mono (le_refl s) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | prod_mono_right | null |
prod_mono_left (t : NonUnitalSubring S) : Monotone fun s : NonUnitalSubring R => s.prod t :=
fun _s₁ _s₂ hs => prod_mono hs (le_refl t) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | prod_mono_left | null |
prod_top (s : NonUnitalSubring R) :
s.prod (⊤ : NonUnitalSubring S) = s.comap (NonUnitalRingHom.fst R S) :=
ext fun x => by simp [mem_prod] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | prod_top | null |
top_prod (s : NonUnitalSubring S) :
(⊤ : NonUnitalSubring R).prod s = s.comap (NonUnitalRingHom.snd R S) :=
ext fun x => by simp [mem_prod]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | top_prod | null |
top_prod_top : (⊤ : NonUnitalSubring R).prod (⊤ : NonUnitalSubring S) = ⊤ :=
(top_prod _).trans <| comap_top _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | top_prod_top | null |
prodEquiv (s : NonUnitalSubring R) (t : NonUnitalSubring S) : s.prod t ≃+* s × t :=
{ Equiv.Set.prod (s : Set R) (t : Set S) with
map_mul' := fun _ _ => rfl
map_add' := fun _ _ => rfl } | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | prodEquiv | Product of `NonUnitalSubring`s is isomorphic to their product as rings. |
mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → NonUnitalSubring R}
(hS : Directed (· ≤ ·) S) {x : R} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i := by
refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩
let U : NonUnitalSubring R :=
NonUnitalSubring.mk' (⋃ i, (S i : Set R)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup)
(Subsemigroup.coe_iSup_of_directed hS) (AddSubgroup.coe_iSup_of_directed hS)
suffices ⨆ i, S i ≤ U by simpa [U] using @this x
exact iSup_le fun i x hx ↦ Set.mem_iUnion.2 ⟨i, hx⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_iSup_of_directed | The underlying set of a non-empty directed Sup of `NonUnitalSubring`s is just a union of the
`NonUnitalSubring`s. Note that this fails without the directedness assumption (the union of two
`NonUnitalSubring`s is typically not a `NonUnitalSubring`) |
coe_iSup_of_directed {ι} [Nonempty ι] {S : ι → NonUnitalSubring R}
(hS : Directed (· ≤ ·) S) : ((⨆ i, S i : NonUnitalSubring R) : Set R) = ⋃ i, S i :=
Set.ext fun x ↦ by simp [mem_iSup_of_directed hS] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_iSup_of_directed | null |
mem_sSup_of_directedOn {S : Set (NonUnitalSubring R)} (Sne : S.Nonempty)
(hS : DirectedOn (· ≤ ·) S) {x : R} : x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by
haveI : Nonempty S := Sne.to_subtype
simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists,
exists_prop] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_sSup_of_directedOn | null |
coe_sSup_of_directedOn {S : Set (NonUnitalSubring R)} (Sne : S.Nonempty)
(hS : DirectedOn (· ≤ ·) S) : (↑(sSup S) : Set R) = ⋃ s ∈ S, ↑s :=
Set.ext fun x => by simp [mem_sSup_of_directedOn Sne hS] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_sSup_of_directedOn | null |
mem_map_equiv {f : R ≃+* S} {K : NonUnitalSubring R} {x : S} :
x ∈ K.map (f : R →ₙ+* S) ↔ f.symm x ∈ K :=
@Set.mem_image_equiv _ _ (K : Set R) f.toEquiv x | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | mem_map_equiv | null |
map_equiv_eq_comap_symm (f : R ≃+* S) (K : NonUnitalSubring R) :
K.map (f : R →ₙ+* S) = K.comap f.symm :=
SetLike.coe_injective (f.toEquiv.image_eq_preimage K) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | map_equiv_eq_comap_symm | null |
comap_equiv_eq_map_symm (f : R ≃+* S) (K : NonUnitalSubring S) :
K.comap (f : R →ₙ+* S) = K.map f.symm :=
(map_equiv_eq_comap_symm f.symm K).symm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | comap_equiv_eq_map_symm | null |
rangeRestrict (f : R →ₙ+* S) : R →ₙ+* f.range :=
NonUnitalRingHom.codRestrict f f.range fun x => ⟨x, rfl⟩
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | rangeRestrict | Restriction of a ring homomorphism to its range interpreted as a `NonUnitalSubring`.
This is the bundled version of `Set.rangeFactorization`. |
coe_rangeRestrict (f : R →ₙ+* S) (x : R) : (f.rangeRestrict x : S) = f x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Algebra.Group.Submonoid.BigOperators",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.RingTheory.NonUnitalSubring.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | coe_rangeRestrict | null |
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