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 ⌀ |
|---|---|---|---|---|---|---|
center : NonUnitalSubsemiring R :=
{ Subsemigroup.center R with
zero_mem' := Set.zero_mem_center
add_mem' := Set.add_mem_center } | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | center | Non-unital subsemirings of a non-unital semiring form a complete lattice. -/
instance : CompleteLattice (NonUnitalSubsemiring R) :=
{ completeLatticeOfInf (NonUnitalSubsemiring R)
fun _ => IsGLB.of_image SetLike.coe_subset_coe isGLB_biInf with
bot := ⊥
bot_le := fun s _ 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 _ _ _ h₁ h₂ _ hx => ⟨h₁ hx, h₂ hx⟩ }
theorem eq_top_iff' (A : NonUnitalSubsemiring R) : A = ⊤ ↔ ∀ x : R, x ∈ A :=
eq_top_iff.trans ⟨fun h m => h <| mem_top m, fun h m _ => h m⟩
section NonUnitalNonAssocSemiring
variable (R)
/-- The center of a semiring `R` is the set of elements that commute and associate with everything
in `R` |
coe_center : ↑(center R) = Set.center R :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | coe_center | null |
center_toSubsemigroup :
(center R).toSubsemigroup = Subsemigroup.center R :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | center_toSubsemigroup | null |
center.instNonUnitalCommSemiring : NonUnitalCommSemiring (center R) :=
{ Subsemigroup.center.commSemigroup,
NonUnitalSubsemiringClass.toNonUnitalNonAssocSemiring (center R) with } | instance | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | center.instNonUnitalCommSemiring | The center is commutative and associative. |
_root_.Set.mem_center_iff_addMonoidHom (a : R) :
a ∈ Set.center R ↔
AddMonoidHom.mulLeft a = .mulRight a ∧
AddMonoidHom.compr₂ .mul (.mulLeft a) = .comp .mul (.mulLeft a) ∧
AddMonoidHom.compr₂ .mul (.mulRight a) = .compl₂ .mul (.mulRight a) := by
rw [Set.mem_center_iff, isMulCentral_iff]
simp [DFunLike.ext_iff, commute_iff_eq]
variable {R} | lemma | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | _root_.Set.mem_center_iff_addMonoidHom | 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`. |
@[simps!] centerCongr [NonUnitalNonAssocSemiring S] (e : R ≃+* S) : center R ≃+* center S where
__ := Subsemigroup.centerCongr e
map_add' _ _ := Subtype.ext <| by exact map_add e .. | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | centerCongr | The center of isomorphic (not necessarily unital or associative) semirings are isomorphic. |
@[simps!] centerToMulOpposite : center R ≃+* center Rᵐᵒᵖ where
__ := Subsemigroup.centerToMulOpposite
map_add' _ _ := rfl | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | centerToMulOpposite | The center of a (not necessarily unital or associative) semiring
is isomorphic to the center of its opposite. |
mem_center_iff {R} [NonUnitalSemiring R] {z : R} : z ∈ center R ↔ ∀ g, g * z = z * g := by
rw [← Semigroup.mem_center_iff]
exact Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_center_iff | null |
decidableMemCenter {R} [NonUnitalSemiring R] [DecidableEq R] [Fintype R] :
DecidablePred (· ∈ center R) := fun _ => decidable_of_iff' _ mem_center_iff
@[simp] | instance | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | decidableMemCenter | null |
center_eq_top (R) [NonUnitalCommSemiring R] : center R = ⊤ :=
SetLike.coe_injective (Set.center_eq_univ R) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | center_eq_top | null |
centralizer {R} [NonUnitalSemiring R] (s : Set R) : NonUnitalSubsemiring R :=
{ Subsemigroup.centralizer s with
carrier := s.centralizer
zero_mem' := Set.zero_mem_centralizer
add_mem' := Set.add_mem_centralizer }
@[simp, norm_cast] | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | centralizer | The centralizer of a set as non-unital subsemiring. |
coe_centralizer {R} [NonUnitalSemiring R] (s : Set R) :
(centralizer s : Set R) = s.centralizer :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | coe_centralizer | null |
centralizer_toSubsemigroup {R} [NonUnitalSemiring R] (s : Set R) :
(centralizer s).toSubsemigroup = Subsemigroup.centralizer s :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | centralizer_toSubsemigroup | null |
mem_centralizer_iff {R} [NonUnitalSemiring R] {s : Set R} {z : R} :
z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_centralizer_iff | null |
center_le_centralizer {R} [NonUnitalSemiring R] (s) : center R ≤ centralizer s :=
s.center_subset_centralizer | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | center_le_centralizer | null |
centralizer_le {R} [NonUnitalSemiring R] (s t : Set R) (h : s ⊆ t) :
centralizer t ≤ centralizer s :=
Set.centralizer_subset h
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | centralizer_le | null |
centralizer_eq_top_iff_subset {R} [NonUnitalSemiring R] {s : Set R} :
centralizer s = ⊤ ↔ s ⊆ center R :=
SetLike.ext'_iff.trans Set.centralizer_eq_top_iff_subset
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | centralizer_eq_top_iff_subset | null |
centralizer_univ {R} [NonUnitalSemiring R] : centralizer Set.univ = center R :=
SetLike.ext' (Set.centralizer_univ R) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | centralizer_univ | null |
closure (s : Set R) : NonUnitalSubsemiring R :=
sInf { S | s ⊆ S } | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure | The `NonUnitalSubsemiring` generated by a set. |
mem_closure {x : R} {s : Set R} :
x ∈ closure s ↔ ∀ S : NonUnitalSubsemiring R, s ⊆ S → x ∈ S :=
mem_sInf | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_closure | null |
@[simp]
closure_le {s : Set R} {t : NonUnitalSubsemiring R} : closure s ≤ t ↔ s ⊆ t :=
⟨Set.Subset.trans subset_closure, fun h => sInf_le h⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_le | The non-unital subsemiring generated by a set includes the set. -/
@[simp, aesop safe 20 (rule_sets := [SetLike])]
theorem subset_closure {s : Set R} : s ⊆ closure s := fun _ hx => mem_closure.2 fun _ 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 non-unital subsemiring `S` 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.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_mono | Subsemiring 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 : NonUnitalSubsemiring R} (h₁ : s ⊆ t)
(h₂ : t ≤ closure s) : closure s = t :=
le_antisymm (closure_le.2 h₁) h₂ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_eq_of_le | null |
closure_le_centralizer_centralizer {R : Type*} [NonUnitalSemiring R] (s : Set R) :
closure s ≤ centralizer (centralizer s) :=
closure_le.mpr Set.subset_centralizer_centralizer | lemma | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_le_centralizer_centralizer | null |
closureNonUnitalCommSemiringOfComm {R : Type*} [NonUnitalSemiring R] {s : Set R}
(hcomm : ∀ x ∈ s, ∀ y ∈ s, x * y = y * x) : NonUnitalCommSemiring (closure s) :=
{ NonUnitalSubsemiringClass.toNonUnitalSemiring (closure s) with
mul_comm := fun ⟨_, h₁⟩ ⟨_, h₂⟩ ↦
have := closure_le_centralizer_centralizer s
Subtype.ext <| Set.centralizer_centralizer_comm_of_comm hcomm _ (this h₁) _ (this h₂) }
variable [NonUnitalNonAssocSemiring S] | abbrev | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closureNonUnitalCommSemiringOfComm | If all the elements of a set `s` commute, then `closure s` is a non-unital commutative
semiring. |
mem_map_equiv {f : R ≃+* S} {K : NonUnitalSubsemiring R} {x : S} :
x ∈ K.map (f : R →ₙ+* S) ↔ f.symm x ∈ K := by
convert @Set.mem_image_equiv _ _ (↑K) f.toEquiv x | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_map_equiv | null |
map_equiv_eq_comap_symm (f : R ≃+* S) (K : NonUnitalSubsemiring 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.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | map_equiv_eq_comap_symm | null |
comap_equiv_eq_map_symm (f : R ≃+* S) (K : NonUnitalSubsemiring 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.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | comap_equiv_eq_map_symm | null |
nonUnitalSubsemiringClosure (M : Subsemigroup R) : NonUnitalSubsemiring R :=
{ AddSubmonoid.closure (M : Set R) with mul_mem' := MulMemClass.mul_mem_add_closure } | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | nonUnitalSubsemiringClosure | The additive closure of a non-unital subsemigroup is a non-unital subsemiring. |
nonUnitalSubsemiringClosure_coe :
(M.nonUnitalSubsemiringClosure : Set R) = AddSubmonoid.closure (M : Set R) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | nonUnitalSubsemiringClosure_coe | null |
nonUnitalSubsemiringClosure_toAddSubmonoid :
M.nonUnitalSubsemiringClosure.toAddSubmonoid = AddSubmonoid.closure (M : Set R) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | nonUnitalSubsemiringClosure_toAddSubmonoid | null |
nonUnitalSubsemiringClosure_eq_closure :
M.nonUnitalSubsemiringClosure = NonUnitalSubsemiring.closure (M : Set R) := by
ext
refine ⟨fun hx => ?_,
fun hx => (NonUnitalSubsemiring.mem_closure.mp hx) M.nonUnitalSubsemiringClosure fun s sM => ?_⟩
<;> rintro - ⟨H1, rfl⟩
<;> rintro - ⟨H2, rfl⟩
· exact AddSubmonoid.mem_closure.mp hx H1.toAddSubmonoid H2
· exact H2 sM | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | nonUnitalSubsemiringClosure_eq_closure | The `NonUnitalSubsemiring` generated by a multiplicative subsemigroup coincides with the
`NonUnitalSubsemiring.closure` of the subsemigroup itself . |
@[simp]
closure_subsemigroup_closure (s : Set R) : closure ↑(Subsemigroup.closure s) = closure s :=
le_antisymm
(closure_le.mpr fun _ hy =>
(Subsemigroup.mem_closure.mp hy) (closure s).toSubsemigroup subset_closure)
(closure_mono Subsemigroup.subset_closure) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_subsemigroup_closure | null |
coe_closure_eq (s : Set R) :
(closure s : Set R) = AddSubmonoid.closure (Subsemigroup.closure s : Set R) := by
simp [← Subsemigroup.nonUnitalSubsemiringClosure_toAddSubmonoid,
Subsemigroup.nonUnitalSubsemiringClosure_eq_closure] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | coe_closure_eq | The elements of the non-unital subsemiring closure of `M` are exactly the elements of the
additive closure of a multiplicative subsemigroup `M`. |
mem_closure_iff {s : Set R} {x} :
x ∈ closure s ↔ x ∈ AddSubmonoid.closure (Subsemigroup.closure s : Set R) :=
Set.ext_iff.mp (coe_closure_eq s) x
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_closure_iff | null |
closure_addSubmonoid_closure {s : Set R} :
closure ↑(AddSubmonoid.closure s) = closure s := by
ext x
refine ⟨fun hx => ?_, fun hx => closure_mono AddSubmonoid.subset_closure hx⟩
rintro - ⟨H, rfl⟩
rintro - ⟨J, rfl⟩
refine (AddSubmonoid.mem_closure.mp (mem_closure_iff.mp hx)) H.toAddSubmonoid fun y hy => ?_
refine (Subsemigroup.mem_closure.mp hy) H.toSubsemigroup fun z hz => ?_
exact (AddSubmonoid.mem_closure.mp hz) H.toAddSubmonoid fun w hw => J hw | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_addSubmonoid_closure | 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))
(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 : NonUnitalSubsemiring R :=
{ carrier := { x | ∃ hx, p x hx }
mul_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, mul _ _ _ _ hpx hpy⟩
add_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, add _ _ _ _ hpx hpy⟩
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.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/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 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) (hx : x ∈ s) (y) (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 _))
(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
| zero => exact zero_right x hx
| mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
| add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
variable (R) in | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_induction₂ | An induction principle for closure membership for predicates with two arguments. |
protected gi : GaloisInsertion (@closure R _) (↑) where
choice s _ := closure s
gc _ _ := closure_le
le_l_u _ := subset_closure
choice_eq _ _ := rfl
variable [NonUnitalNonAssocSemiring S]
variable {F : Type*} [FunLike F R S] [NonUnitalRingHomClass F R S] | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | gi | `closure` forms a Galois insertion with the coercion to set. |
@[simp]
closure_eq (s : NonUnitalSubsemiring R) : closure (s : Set R) = s :=
(NonUnitalSubsemiring.gi R).l_u_eq s
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_eq | Closure of a non-unital subsemiring `S` equals `S`. |
closure_empty : closure (∅ : Set R) = ⊥ :=
(NonUnitalSubsemiring.gi R).gc.l_bot
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_empty | null |
closure_univ : closure (Set.univ : Set R) = ⊤ :=
@coe_top R _ ▸ closure_eq ⊤ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_univ | null |
closure_union (s t : Set R) : closure (s ∪ t) = closure s ⊔ closure t :=
(NonUnitalSubsemiring.gi R).gc.l_sup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_union | null |
closure_iUnion {ι} (s : ι → Set R) : closure (⋃ i, s i) = ⨆ i, closure (s i) :=
(NonUnitalSubsemiring.gi R).gc.l_iSup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_iUnion | null |
closure_sUnion (s : Set (Set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t :=
(NonUnitalSubsemiring.gi R).gc.l_sSup | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | closure_sUnion | null |
map_sup (s t : NonUnitalSubsemiring R) (f : F) :
(map f (s ⊔ t) : NonUnitalSubsemiring S) = map f s ⊔ map f t :=
@GaloisConnection.l_sup _ _ s t _ _ _ _ (gc_map_comap f) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | map_sup | null |
map_iSup {ι : Sort*} (f : F) (s : ι → NonUnitalSubsemiring R) :
(map f (iSup s) : NonUnitalSubsemiring S) = ⨆ i, map f (s i) :=
@GaloisConnection.l_iSup _ _ _ _ _ _ _ (gc_map_comap f) s | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | map_iSup | null |
map_inf (s t : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective f) :
(map f (s ⊓ t) : NonUnitalSubsemiring S) = map f s ⊓ map f t :=
SetLike.coe_injective (Set.image_inter hf) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | map_inf | null |
map_iInf {ι : Sort*} [Nonempty ι] (f : F) (hf : Function.Injective f)
(s : ι → NonUnitalSubsemiring R) :
(map f (iInf s) : NonUnitalSubsemiring S) = ⨅ i, map f (s i) := by
apply SetLike.coe_injective
simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | map_iInf | null |
comap_inf (s t : NonUnitalSubsemiring S) (f : F) :
(comap f (s ⊓ t) : NonUnitalSubsemiring R) = comap f s ⊓ comap f t :=
@GaloisConnection.u_inf _ _ s t _ _ _ _ (gc_map_comap f) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | comap_inf | null |
comap_iInf {ι : Sort*} (f : F) (s : ι → NonUnitalSubsemiring S) :
(comap f (iInf s) : NonUnitalSubsemiring R) = ⨅ i, comap f (s i) :=
@GaloisConnection.u_iInf _ _ _ _ _ _ _ (gc_map_comap f) s
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | comap_iInf | null |
map_bot (f : F) : map f (⊥ : NonUnitalSubsemiring R) = (⊥ : NonUnitalSubsemiring S) :=
(gc_map_comap f).l_bot
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | map_bot | null |
comap_top (f : F) : comap f (⊤ : NonUnitalSubsemiring S) = (⊤ : NonUnitalSubsemiring R) :=
(gc_map_comap f).u_top | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | comap_top | null |
prod (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring S) : NonUnitalSubsemiring (R × S) :=
{ s.toSubsemigroup.prod t.toSubsemigroup, s.toAddSubmonoid.prod t.toAddSubmonoid with
carrier := (s : Set R) ×ˢ (t : Set S) }
@[norm_cast] | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | prod | Given `NonUnitalSubsemiring`s `s`, `t` of semirings `R`, `S` respectively, `s.prod t` is
`s × t` as a non-unital subsemiring of `R × S`. |
coe_prod (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring S) :
(s.prod t : Set (R × S)) = (s : Set R) ×ˢ (t : Set S) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | coe_prod | null |
mem_prod {s : NonUnitalSubsemiring R} {t : NonUnitalSubsemiring S} {p : R × S} :
p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t :=
Iff.rfl
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_prod | null |
prod_mono ⦃s₁ s₂ : NonUnitalSubsemiring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : NonUnitalSubsemiring S⦄
(ht : t₁ ≤ t₂) : s₁.prod t₁ ≤ s₂.prod t₂ :=
Set.prod_mono hs ht | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | prod_mono | null |
prod_mono_right (s : NonUnitalSubsemiring R) :
Monotone fun t : NonUnitalSubsemiring S => s.prod t :=
prod_mono (le_refl s) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | prod_mono_right | null |
prod_mono_left (t : NonUnitalSubsemiring S) :
Monotone fun s : NonUnitalSubsemiring R => s.prod t := fun _ _ hs => prod_mono hs (le_refl t) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | prod_mono_left | null |
prod_top (s : NonUnitalSubsemiring R) :
s.prod (⊤ : NonUnitalSubsemiring S) = s.comap (NonUnitalRingHom.fst R S) :=
ext fun x => by simp [mem_prod] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | prod_top | null |
top_prod (s : NonUnitalSubsemiring S) :
(⊤ : NonUnitalSubsemiring R).prod s = s.comap (NonUnitalRingHom.snd R S) :=
ext fun x => by simp [mem_prod]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | top_prod | null |
top_prod_top : (⊤ : NonUnitalSubsemiring R).prod (⊤ : NonUnitalSubsemiring S) = ⊤ :=
(top_prod _).trans <| comap_top _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | top_prod_top | null |
prodEquiv (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring 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.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | prodEquiv | Product of non-unital subsemirings is isomorphic to their product as semigroups. |
mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → NonUnitalSubsemiring 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 : NonUnitalSubsemiring R :=
NonUnitalSubsemiring.mk' (⋃ i, (S i : Set R))
(⨆ i, (S i).toSubsemigroup) (Subsemigroup.coe_iSup_of_directed hS)
(⨆ i, (S i).toAddSubmonoid) (AddSubmonoid.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.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_iSup_of_directed | null |
coe_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → NonUnitalSubsemiring R}
(hS : Directed (· ≤ ·) S) : ((⨆ i, S i : NonUnitalSubsemiring R) : Set R) = ⋃ i, S i :=
Set.ext fun x ↦ by simp [mem_iSup_of_directed hS] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | coe_iSup_of_directed | null |
mem_sSup_of_directedOn {S : Set (NonUnitalSubsemiring 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, Subtype.exists, exists_prop] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | mem_sSup_of_directedOn | null |
coe_sSup_of_directedOn {S : Set (NonUnitalSubsemiring 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.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | coe_sSup_of_directedOn | null |
eq_of_eqOn_stop {f g : F}
(h : Set.EqOn (f : R → S) (g : R → S) (⊤ : NonUnitalSubsemiring R)) : f = g :=
DFunLike.ext _ _ fun _ => h trivial
variable [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T]
[NonUnitalRingHomClass F R S]
{S' : Type*} [SetLike S' S] [NonUnitalSubsemiringClass S' S]
{s : NonUnitalSubsemiring R}
open NonUnitalSubsemiringClass NonUnitalSubsemiring | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | eq_of_eqOn_stop | null |
srangeRestrict (f : F) : R →ₙ+* (srange f : NonUnitalSubsemiring S) :=
codRestrict f (srange f) (mem_srange_self f)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | srangeRestrict | Restriction of a non-unital ring homomorphism to its range interpreted as a
non-unital subsemiring.
This is the bundled version of `Set.rangeFactorization`. |
coe_srangeRestrict (f : F) (x : R) : (srangeRestrict f x : S) = f x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | coe_srangeRestrict | null |
srangeRestrict_surjective (f : F) :
Function.Surjective (srangeRestrict f : R → (srange f : NonUnitalSubsemiring S)) :=
fun ⟨_, hy⟩ =>
let ⟨x, hx⟩ := mem_srange.mp hy
⟨x, Subtype.ext hx⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | srangeRestrict_surjective | null |
srange_eq_top_iff_surjective {f : F} :
srange f = (⊤ : NonUnitalSubsemiring S) ↔ Function.Surjective (f : R → S) :=
SetLike.ext'_iff.trans <| Iff.trans (by rw [coe_srange, coe_top]) Set.range_eq_univ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | srange_eq_top_iff_surjective | null |
@[simp]
srange_eq_top_of_surjective (f : F) (hf : Function.Surjective (f : R → S)) :
srange f = (⊤ : NonUnitalSubsemiring S) :=
srange_eq_top_iff_surjective.2 hf | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | srange_eq_top_of_surjective | The range of a surjective non-unital ring homomorphism is the whole of the codomain. |
eqOn_sclosure {f g : F} {s : Set R} (h : Set.EqOn (f : R → S) (g : R → S) s) :
Set.EqOn f g (closure s) :=
show closure s ≤ eqSlocus f g from closure_le.2 h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | eqOn_sclosure | If two non-unital ring homomorphisms are equal on a set, then they are equal on its
non-unital subsemiring closure. |
eq_of_eqOn_sdense {s : Set R} (hs : closure s = ⊤) {f g : F}
(h : s.EqOn (f : R → S) (g : R → S)) : f = g :=
eq_of_eqOn_stop <| hs ▸ eqOn_sclosure h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | eq_of_eqOn_sdense | null |
sclosure_preimage_le (f : F) (s : Set S) :
closure ((f : R → S) ⁻¹' s) ≤ (closure s).comap f :=
closure_le.2 fun _ hx => SetLike.mem_coe.2 <| mem_comap.2 <| subset_closure hx | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | sclosure_preimage_le | null |
map_sclosure (f : F) (s : Set R) : (closure s).map f = closure ((f : R → S) '' s) :=
Set.image_preimage.l_comm_of_u_comm (gc_map_comap f) (NonUnitalSubsemiring.gi S).gc
(NonUnitalSubsemiring.gi R).gc fun _ ↦ rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | map_sclosure | The image under a ring homomorphism of the subsemiring generated by a set equals
the subsemiring generated by the image of the set. |
@[simp]
srange_subtype (s : NonUnitalSubsemiring R) : NonUnitalRingHom.srange (subtype s) = s :=
SetLike.coe_injective <| (coe_srange _).trans Subtype.range_coe
variable [NonUnitalNonAssocSemiring S]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | srange_subtype | null |
range_fst : NonUnitalRingHom.srange (fst R S) = ⊤ :=
NonUnitalRingHom.srange_eq_top_of_surjective (fst R S) Prod.fst_surjective
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | range_fst | null |
range_snd : NonUnitalRingHom.srange (snd R S) = ⊤ :=
NonUnitalRingHom.srange_eq_top_of_surjective (snd R S) <| Prod.snd_surjective | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | range_snd | null |
nonUnitalSubsemiringCongr (h : s = t) : s ≃+* t :=
{ Equiv.setCongr <| congr_arg _ h with
map_mul' := fun _ _ => rfl
map_add' := fun _ _ => rfl } | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | nonUnitalSubsemiringCongr | Makes the identity isomorphism from a proof two non-unital subsemirings of a multiplicative
monoid are equal. |
sofLeftInverse' {g : S → R} {f : F} (h : Function.LeftInverse g f) : R ≃+* srange f :=
{ srangeRestrict f with
toFun := srangeRestrict f
invFun := fun x => g (subtype (srange f) x)
left_inv := h
right_inv := fun x =>
Subtype.ext <|
let ⟨x', hx'⟩ := NonUnitalRingHom.mem_srange.mp x.prop
show f (g x) = x by rw [← hx', h x'] }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | sofLeftInverse' | Restrict a non-unital ring homomorphism with a left inverse to a ring isomorphism to its
`NonUnitalRingHom.srange`. |
sofLeftInverse'_apply {g : S → R} {f : F} (h : Function.LeftInverse g f) (x : R) :
↑(sofLeftInverse' h x) = f x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | sofLeftInverse'_apply | null |
sofLeftInverse'_symm_apply {g : S → R} {f : F} (h : Function.LeftInverse g f)
(x : srange f) : (sofLeftInverse' h).symm x = g x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | sofLeftInverse'_symm_apply | null |
@[simps!]
nonUnitalSubsemiringMap (e : R ≃+* S) (s : NonUnitalSubsemiring R) :
s ≃+* NonUnitalSubsemiring.map e.toNonUnitalRingHom s :=
{ e.toAddEquiv.addSubmonoidMap s.toAddSubmonoid,
e.toMulEquiv.subsemigroupMap s.toSubsemigroup with } | def | RingTheory | [
"Mathlib.Algebra.Group.Submonoid.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Membership",
"Mathlib.Algebra.Group.Subsemigroup.Operations",
"Mathlib.Algebra.GroupWithZero.Center",
"Mathlib.Algebra.Ring.Center",
"Mathlib.Algebra.Ring.Centralizer",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.R... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | nonUnitalSubsemiringMap | Given an equivalence `e : R ≃+* S` of non-unital semirings and a non-unital subsemiring
`s` of `R`, `non_unital_subsemiring_map e s` is the induced equivalence between `s` and
`s.map e` |
NonUnitalSubsemiringClass (S : Type*) (R : outParam (Type u)) [NonUnitalNonAssocSemiring R]
[SetLike S R] : Prop
extends AddSubmonoidClass S R where
mul_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a * b ∈ s | class | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | NonUnitalSubsemiringClass | This lemma exists for `aesop`, as `aesop` simplifies `-x * y` to `-(x * y)` before applying
unsafe rules like `mul_mem`, leading to a dead end in cases where `neg_mem` does not hold. -/
@[aesop unsafe 80% (rule_sets := [SetLike])]
theorem neg_mul_mem {x y : R} (hx : -x ∈ s) (hy : y ∈ s) : -(x * y) ∈ s := by
simpa using mul_mem hx hy
/-- This lemma exists for `aesop`, as `aesop` simplifies `x * -y` to `-(x * y)` before applying
unsafe rules like `mul_mem`, leading to a dead end in cases where `neg_mem` does not hold. -/
@[aesop unsafe 80% (rule_sets := [SetLike])]
theorem mul_neg_mem {x y : R} (hx : x ∈ s) (hy : -y ∈ s) : -(x * y) ∈ s := by
simpa using mul_mem hx hy
-- doesn't work without the above `aesop` lemmas
example {x y z : R} (hx : x ∈ s) (hy : -y ∈ s) (hz : z ∈ s) :
x * (-y) * z ∈ s := by aesop
end neg_mul
variable {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R]
/-- `NonUnitalSubsemiringClass S R` states that `S` is a type of subsets `s ⊆ R` that
are both an additive submonoid and also a multiplicative subsemigroup. |
subtype : s →ₙ+* R :=
{ AddSubmonoidClass.subtype s, MulMemClass.subtype s with toFun := (↑) }
variable {s} in
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | subtype | A non-unital subsemiring of a `NonUnitalNonAssocSemiring` inherits a
`NonUnitalNonAssocSemiring` structure -/
instance (priority := 75) toNonUnitalNonAssocSemiring :
NonUnitalNonAssocSemiring s := fast_instance%
Subtype.coe_injective.nonUnitalNonAssocSemiring Subtype.val rfl (by simp) (fun _ _ => rfl)
fun _ _ => rfl
instance noZeroDivisors [NoZeroDivisors R] : NoZeroDivisors s :=
Subtype.coe_injective.noZeroDivisors Subtype.val rfl fun _ _ => rfl
/-- The natural non-unital ring hom from a non-unital subsemiring of a non-unital semiring `R` to
`R`. |
subtype_apply (x : s) : subtype s x = x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | subtype_apply | null |
subtype_injective : Function.Injective (subtype s) :=
Subtype.coe_injective
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | subtype_injective | null |
coe_subtype : (subtype s : s → R) = ((↑) : s → R) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | coe_subtype | null |
toNonUnitalSemiring {R} [NonUnitalSemiring R] [SetLike S R]
[NonUnitalSubsemiringClass S R] : NonUnitalSemiring s := fast_instance%
Subtype.coe_injective.nonUnitalSemiring Subtype.val rfl (by simp) (fun _ _ => rfl) fun _ _ => rfl | instance | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | toNonUnitalSemiring | A non-unital subsemiring of a `NonUnitalSemiring` is a `NonUnitalSemiring`. |
toNonUnitalCommSemiring {R} [NonUnitalCommSemiring R] [SetLike S R]
[NonUnitalSubsemiringClass S R] : NonUnitalCommSemiring s := fast_instance%
Subtype.coe_injective.nonUnitalCommSemiring Subtype.val rfl (by simp) (fun _ _ => rfl)
fun _ _ => rfl
/-! Note: currently, there are no ordered versions of non-unital rings. -/ | instance | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | toNonUnitalCommSemiring | A non-unital subsemiring of a `NonUnitalCommSemiring` is a `NonUnitalCommSemiring`. |
NonUnitalSubsemiring (R : Type u) [NonUnitalNonAssocSemiring R] extends AddSubmonoid R,
Subsemigroup R | structure | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | NonUnitalSubsemiring | A non-unital subsemiring of a non-unital semiring `R` is a subset `s` that is both an additive
submonoid and a semigroup. |
@[simps]
ofClass {S R : Type*} [NonUnitalNonAssocSemiring R] [SetLike S R]
[NonUnitalSubsemiringClass S R] (s : S) : NonUnitalSubsemiring R where
carrier := s
add_mem' := add_mem
zero_mem' := zero_mem _
mul_mem' := mul_mem | def | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | ofClass | Reinterpret a `NonUnitalSubsemiring` as a `Subsemigroup`. -/
add_decl_doc NonUnitalSubsemiring.toSubsemigroup
/-- Reinterpret a `NonUnitalSubsemiring` as an `AddSubmonoid`. -/
add_decl_doc NonUnitalSubsemiring.toAddSubmonoid
namespace NonUnitalSubsemiring
instance : SetLike (NonUnitalSubsemiring R) R where
coe s := s.carrier
coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective' h
/-- The actual `NonUnitalSubsemiring` obtained from an element of a `NonUnitalSubsemiringClass`. |
mem_carrier {s : NonUnitalSubsemiring R} {x : R} : x ∈ s.carrier ↔ x ∈ s :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | mem_carrier | null |
@[ext]
ext {S T : NonUnitalSubsemiring R} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=
SetLike.ext h | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | ext | Two non-unital subsemirings are equal if they have the same elements. |
protected copy (S : NonUnitalSubsemiring R) (s : Set R) (hs : s = ↑S) :
NonUnitalSubsemiring R :=
{ S.toAddSubmonoid.copy s hs, S.toSubsemigroup.copy s hs with carrier := s }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | copy | Copy of a non-unital subsemiring with a new `carrier` equal to the old one. Useful to fix
definitional equalities. |
coe_copy (S : NonUnitalSubsemiring R) (s : Set R) (hs : s = ↑S) :
(S.copy s hs : Set R) = s :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | coe_copy | null |
copy_eq (S : NonUnitalSubsemiring R) (s : Set R) (hs : s = ↑S) : S.copy s hs = S :=
SetLike.coe_injective hs | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | copy_eq | null |
toSubsemigroup_injective :
Function.Injective (toSubsemigroup : NonUnitalSubsemiring R → Subsemigroup R)
| _, _, h => ext (SetLike.ext_iff.mp h :) | theorem | RingTheory | [
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Group.Submonoid.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubsemiring/Defs.lean | toSubsemigroup_injective | null |
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