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 ⌀ |
|---|---|---|---|---|---|---|
rangeRestrict_surjective (f : R →ₙ+* S) : Function.Surjective f.rangeRestrict :=
fun ⟨_y, hy⟩ =>
let ⟨x, hx⟩ := mem_range.mp hy
⟨x, Subtype.ext 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 | rangeRestrict_surjective | null |
range_eq_top {f : R →ₙ+* S} :
f.range = (⊤ : NonUnitalSubring S) ↔ Function.Surjective f :=
SetLike.ext'_iff.trans <| Iff.trans (by rw [coe_range, coe_top]) Set.range_eq_univ | 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_top | null |
@[simp]
range_eq_top_of_surjective (f : R →ₙ+* S) (hf : Function.Surjective f) :
f.range = (⊤ : NonUnitalSubring S) :=
range_eq_top.2 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 | range_eq_top_of_surjective | The range of a surjective ring homomorphism is the whole of the codomain. |
eqLocus (f g : R →ₙ+* S) : NonUnitalSubring R :=
{ (f : R →ₙ* S).eqLocus g, (f : R →+ S).eqLocus g with carrier := {x | f x = g x} }
@[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 | eqLocus | The `NonUnitalSubring` of elements `x : R` such that `f x = g x`, i.e.,
the equalizer of f and g as a `NonUnitalSubring` of R |
mem_eqLocus {f g : R →ₙ+* S} {x : R} : x ∈ f.eqLocus g ↔ f x = g x := Iff.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 | mem_eqLocus | null |
eqLocus_same (f : R →ₙ+* S) : f.eqLocus f = ⊤ :=
SetLike.ext fun _ => eq_self_iff_true _ | 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 | eqLocus_same | null |
eqOn_set_closure {f g : R →ₙ+* S} {s : Set R} (h : Set.EqOn f g s) :
Set.EqOn f g (closure s) :=
show closure s ≤ f.eqLocus g from closure_le.2 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 | eqOn_set_closure | If two ring homomorphisms are equal on a set, then they are equal on its
`NonUnitalSubring` closure. |
eq_of_eqOn_set_top {f g : R →ₙ+* S} (h : Set.EqOn f g (⊤ : NonUnitalSubring R)) : f = g :=
ext fun _x => h trivial | 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 | eq_of_eqOn_set_top | null |
eq_of_eqOn_set_dense {s : Set R} (hs : closure s = ⊤) {f g : R →ₙ+* S} (h : s.EqOn f g) :
f = g :=
eq_of_eqOn_set_top <| hs ▸ eqOn_set_closure 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 | eq_of_eqOn_set_dense | null |
closure_preimage_le (f : R →ₙ+* S) (s : Set S) : closure (f ⁻¹' s) ≤ (closure s).comap f :=
closure_le.2 fun _x hx => SetLike.mem_coe.2 <| mem_comap.2 <| subset_closure 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 | closure_preimage_le | null |
map_closure (f : R →ₙ+* S) (s : Set R) : (closure s).map f = closure (f '' s) :=
Set.image_preimage.l_comm_of_u_comm (gc_map_comap f) (NonUnitalSubring.gi S).gc
(NonUnitalSubring.gi R).gc fun _ ↦ 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 | map_closure | The image under a ring homomorphism of the `NonUnitalSubring` generated by a set equals
the `NonUnitalSubring` generated by the image of the set. |
@[simp]
range_subtype (s : NonUnitalSubring R) : (NonUnitalSubringClass.subtype s).range = s :=
SetLike.coe_injective <| (coe_srange _).trans Subtype.range_coe | 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_subtype | null |
range_fst : NonUnitalRingHom.srange (fst R S) = ⊤ :=
NonUnitalSubsemiring.range_fst | 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_fst | null |
range_snd : NonUnitalRingHom.srange (snd R S) = ⊤ :=
NonUnitalSubsemiring.range_snd | 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_snd | null |
nonUnitalSubringCongr (h : s = t) : s ≃+* t :=
{
Equiv.setCongr <| congr_arg _ h 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 | nonUnitalSubringCongr | Makes the identity isomorphism from a proof two `NonUnitalSubring`s of a multiplicative
monoid are equal. |
ofLeftInverse' {g : S → R} {f : R →ₙ+* S} (h : Function.LeftInverse g f) : R ≃+* f.range :=
{ f.rangeRestrict with
toFun := fun x => f.rangeRestrict x
invFun := fun x => (g ∘ NonUnitalSubringClass.subtype f.range) x
left_inv := h
right_inv := fun x =>
Subtype.ext <|
let ⟨x', hx'⟩ := NonUnitalRingHom.mem_range.mp x.prop
show f (g x) = x by rw [← hx', h x'] }
@[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 | ofLeftInverse' | Restrict a ring homomorphism with a left inverse to a ring isomorphism to its
`RingHom.range`. |
ofLeftInverse'_apply {g : S → R} {f : R →ₙ+* S} (h : Function.LeftInverse g f) (x : R) :
↑(ofLeftInverse' h x) = f x :=
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 | ofLeftInverse'_apply | null |
ofLeftInverse'_symm_apply {g : S → R} {f : R →ₙ+* S} (h : Function.LeftInverse g f)
(x : f.range) : (ofLeftInverse' h).symm x = g 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 | ofLeftInverse'_symm_apply | null |
closure_preimage_le (f : F) (s : Set S) :
closure ((f : R → S) ⁻¹' s) ≤ (closure s).comap f :=
closure_le.2 fun _x hx => SetLike.mem_coe.2 <| mem_comap.2 <| subset_closure 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 | closure_preimage_le | null |
NonUnitalSubringClass (S : Type*) (R : Type u) [NonUnitalNonAssocRing R] [SetLike S R] : Prop
extends NonUnitalSubsemiringClass S R, NegMemClass S R where | class | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | NonUnitalSubringClass | `NonUnitalSubringClass S R` states that `S` is a type of subsets `s ⊆ R` that
are both a multiplicative submonoid and an additive subgroup. |
subtype (s : S) : s →ₙ+* R :=
{ NonUnitalSubsemiringClass.subtype s,
AddSubgroupClass.subtype s with
toFun := Subtype.val }
variable {s} in
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | subtype | A non-unital subring of a non-unital ring inherits a non-unital ring structure -/
instance (priority := 75) toNonUnitalNonAssocRing : NonUnitalNonAssocRing s := fast_instance%
Subtype.val_injective.nonUnitalNonAssocRing _ rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl)
(fun _ _ => rfl) (fun _ _ => rfl) fun _ _ => rfl
-- Prefer subclasses of `NonUnitalRing` over subclasses of `NonUnitalSubringClass`.
/-- A non-unital subring of a non-unital ring inherits a non-unital ring structure -/
instance (priority := 75) toNonUnitalRing {R : Type*} [NonUnitalRing R] [SetLike S R]
[NonUnitalSubringClass S R] (s : S) : NonUnitalRing s := fast_instance%
Subtype.val_injective.nonUnitalRing _ rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl)
(fun _ _ => rfl) (fun _ _ => rfl) fun _ _ => rfl
-- Prefer subclasses of `NonUnitalRing` over subclasses of `NonUnitalSubringClass`.
/-- A non-unital subring of a `NonUnitalCommRing` is a `NonUnitalCommRing`. -/
instance (priority := 75) toNonUnitalCommRing {R} [NonUnitalCommRing R] [SetLike S R]
[NonUnitalSubringClass S R] : NonUnitalCommRing s := fast_instance%
Subtype.val_injective.nonUnitalCommRing _ rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl)
(fun _ _ => rfl) (fun _ _ => rfl) fun _ _ => rfl
/-- The natural non-unital ring hom from a non-unital subring of a non-unital ring `R` to `R`. |
subtype_apply (x : s) : subtype s x = x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | subtype_apply | null |
subtype_injective : Function.Injective (subtype s) :=
Subtype.coe_injective
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | subtype_injective | null |
coe_subtype : (subtype s : s → R) = Subtype.val :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_subtype | null |
NonUnitalSubring (R : Type u) [NonUnitalNonAssocRing R] extends
NonUnitalSubsemiring R, AddSubgroup R | structure | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | NonUnitalSubring | `NonUnitalSubring R` is the type of non-unital subrings of `R`. A non-unital subring of `R`
is a subset `s` that is a multiplicative subsemigroup and an additive subgroup. Note in particular
that it shares the same 0 as R. |
toSubsemigroup (s : NonUnitalSubring R) : Subsemigroup R :=
{ s.toNonUnitalSubsemiring.toSubsemigroup with carrier := s.carrier } | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toSubsemigroup | Reinterpret a `NonUnitalSubring` as a `NonUnitalSubsemiring`. -/
add_decl_doc NonUnitalSubring.toNonUnitalSubsemiring
/-- Reinterpret a `NonUnitalSubring` as an `AddSubgroup`. -/
add_decl_doc NonUnitalSubring.toAddSubgroup
namespace NonUnitalSubring
/-- The underlying submonoid of a `NonUnitalSubring`. |
@[simps]
ofClass {S R : Type*} [NonUnitalNonAssocRing R] [SetLike S R] [NonUnitalSubringClass S R]
(s : S) : NonUnitalSubring R where
carrier := s
add_mem' := add_mem
zero_mem' := zero_mem _
mul_mem' := mul_mem
neg_mem' := neg_mem | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | ofClass | The actual `NonUnitalSubring` obtained from an element of a `NonUnitalSubringClass`. |
mem_carrier {s : NonUnitalSubring R} {x : R} : x ∈ s.toNonUnitalSubsemiring ↔ x ∈ s :=
Iff.rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mem_carrier | null |
mem_mk {S : NonUnitalSubsemiring R} {x : R} (h) :
x ∈ (⟨S, h⟩ : NonUnitalSubring R) ↔ x ∈ S :=
Iff.rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mem_mk | null |
coe_set_mk (S : NonUnitalSubsemiring R) (h) :
((⟨S, h⟩ : NonUnitalSubring R) : Set R) = S :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_set_mk | null |
mk_le_mk {S S' : NonUnitalSubsemiring R} (h h') :
(⟨S, h⟩ : NonUnitalSubring R) ≤ (⟨S', h'⟩ : NonUnitalSubring R) ↔ S ≤ S' :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mk_le_mk | null |
@[ext]
ext {S T : NonUnitalSubring R} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=
SetLike.ext h | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | ext | Two non-unital subrings are equal if they have the same elements. |
protected copy (S : NonUnitalSubring R) (s : Set R) (hs : s = ↑S) : NonUnitalSubring R :=
{ S.toNonUnitalSubsemiring.copy s hs with
carrier := s
neg_mem' := hs.symm ▸ S.neg_mem' }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | copy | Copy of a non-unital subring with a new `carrier` equal to the old one. Useful to fix
definitional equalities. |
coe_copy (S : NonUnitalSubring R) (s : Set R) (hs : s = ↑S) : (S.copy s hs : Set R) = s :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_copy | null |
copy_eq (S : NonUnitalSubring R) (s : Set R) (hs : s = ↑S) : S.copy s hs = S :=
SetLike.coe_injective hs | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | copy_eq | null |
toNonUnitalSubsemiring_injective :
Function.Injective (toNonUnitalSubsemiring : NonUnitalSubring R → NonUnitalSubsemiring R)
| _r, _s, h => ext (SetLike.ext_iff.mp h :)
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toNonUnitalSubsemiring_injective | null |
toNonUnitalSubsemiring_strictMono :
StrictMono (toNonUnitalSubsemiring : NonUnitalSubring R → NonUnitalSubsemiring R) := fun _ _ =>
id
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toNonUnitalSubsemiring_strictMono | null |
toNonUnitalSubsemiring_mono :
Monotone (toNonUnitalSubsemiring : NonUnitalSubring R → NonUnitalSubsemiring R) :=
toNonUnitalSubsemiring_strictMono.monotone | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toNonUnitalSubsemiring_mono | null |
toAddSubgroup_injective :
Function.Injective (toAddSubgroup : NonUnitalSubring R → AddSubgroup R)
| _r, _s, h => ext (SetLike.ext_iff.mp h :)
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toAddSubgroup_injective | null |
toAddSubgroup_strictMono :
StrictMono (toAddSubgroup : NonUnitalSubring R → AddSubgroup R) := fun _ _ => id
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toAddSubgroup_strictMono | null |
toAddSubgroup_mono : Monotone (toAddSubgroup : NonUnitalSubring R → AddSubgroup R) :=
toAddSubgroup_strictMono.monotone | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toAddSubgroup_mono | null |
toSubsemigroup_injective :
Function.Injective (toSubsemigroup : NonUnitalSubring R → Subsemigroup R)
| _r, _s, h => ext (SetLike.ext_iff.mp h :)
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toSubsemigroup_injective | null |
toSubsemigroup_strictMono :
StrictMono (toSubsemigroup : NonUnitalSubring R → Subsemigroup R) := fun _ _ => id
@[mono] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toSubsemigroup_strictMono | null |
toSubsemigroup_mono : Monotone (toSubsemigroup : NonUnitalSubring R → Subsemigroup R) :=
toSubsemigroup_strictMono.monotone | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toSubsemigroup_mono | null |
protected mk' (s : Set R) (sm : Subsemigroup R) (sa : AddSubgroup R) (hm : ↑sm = s)
(ha : ↑sa = s) : NonUnitalSubring R :=
{ sm.copy s hm.symm, sa.copy s ha.symm with }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mk' | Construct a `NonUnitalSubring R` from a set `s`, a subsemigroup `sm`, and an additive
subgroup `sa` such that `x ∈ s ↔ x ∈ sm ↔ x ∈ sa`. |
coe_mk' {s : Set R} {sm : Subsemigroup R} (hm : ↑sm = s) {sa : AddSubgroup R}
(ha : ↑sa = s) : (NonUnitalSubring.mk' s sm sa hm ha : Set R) = s :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_mk' | null |
mem_mk' {s : Set R} {sm : Subsemigroup R} (hm : ↑sm = s) {sa : AddSubgroup R} (ha : ↑sa = s)
{x : R} : x ∈ NonUnitalSubring.mk' s sm sa hm ha ↔ x ∈ s :=
Iff.rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mem_mk' | null |
mk'_toSubsemigroup {s : Set R} {sm : Subsemigroup R} (hm : ↑sm = s) {sa : AddSubgroup R}
(ha : ↑sa = s) : (NonUnitalSubring.mk' s sm sa hm ha).toSubsemigroup = sm :=
SetLike.coe_injective hm.symm
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mk'_toSubsemigroup | null |
mk'_toAddSubgroup {s : Set R} {sm : Subsemigroup R} (hm : ↑sm = s) {sa : AddSubgroup R}
(ha : ↑sa = s) : (NonUnitalSubring.mk' s sm sa hm ha).toAddSubgroup = sa :=
SetLike.coe_injective ha.symm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mk'_toAddSubgroup | null |
protected zero_mem : (0 : R) ∈ s :=
zero_mem _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | zero_mem | A non-unital subring contains the ring's 0. |
protected mul_mem {x y : R} : x ∈ s → y ∈ s → x * y ∈ s :=
mul_mem | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mul_mem | A non-unital subring is closed under multiplication. |
protected add_mem {x y : R} : x ∈ s → y ∈ s → x + y ∈ s :=
add_mem | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | add_mem | A non-unital subring is closed under addition. |
protected neg_mem {x : R} : x ∈ s → -x ∈ s :=
neg_mem | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | neg_mem | A non-unital subring is closed under negation. |
protected sub_mem {x y : R} (hx : x ∈ s) (hy : y ∈ s) : x - y ∈ s :=
sub_mem hx hy | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | sub_mem | A non-unital subring is closed under subtraction |
toNonUnitalRing {R : Type*} [NonUnitalRing R] (s : NonUnitalSubring R) :
NonUnitalRing s := fast_instance%
Subtype.coe_injective.nonUnitalRing _ rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl)
(fun _ _ => rfl) (fun _ _ => rfl) fun _ _ => rfl | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toNonUnitalRing | A non-unital subring of a non-unital ring inherits a non-unital ring structure |
protected zsmul_mem {x : R} (hx : x ∈ s) (n : ℤ) : n • x ∈ s :=
zsmul_mem hx n
@[simp, norm_cast] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | zsmul_mem | null |
val_add (x y : s) : (↑(x + y) : R) = ↑x + ↑y :=
rfl
@[simp, norm_cast] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | val_add | null |
val_neg (x : s) : (↑(-x) : R) = -↑x :=
rfl
@[simp, norm_cast] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | val_neg | null |
val_mul (x y : s) : (↑(x * y) : R) = ↑x * ↑y :=
rfl
@[simp, norm_cast] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | val_mul | null |
val_zero : ((0 : s) : R) = 0 :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | val_zero | null |
coe_eq_zero_iff {x : s} : (x : R) = 0 ↔ x = 0 := by
simp | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_eq_zero_iff | null |
toNonUnitalCommRing {R} [NonUnitalCommRing R] (s : NonUnitalSubring R) :
NonUnitalCommRing s := fast_instance%
Subtype.coe_injective.nonUnitalCommRing _ rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl)
(fun _ _ => rfl) (fun _ _ => rfl) fun _ _ => rfl
/-! ## Partial order -/
@[simp] | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | toNonUnitalCommRing | A non-unital subring of a `NonUnitalCommRing` is a `NonUnitalCommRing`. |
mem_toSubsemigroup {s : NonUnitalSubring R} {x : R} : x ∈ s.toSubsemigroup ↔ x ∈ s :=
Iff.rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mem_toSubsemigroup | null |
coe_toSubsemigroup (s : NonUnitalSubring R) : (s.toSubsemigroup : Set R) = s :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_toSubsemigroup | null |
mem_toAddSubgroup {s : NonUnitalSubring R} {x : R} : x ∈ s.toAddSubgroup ↔ x ∈ s :=
Iff.rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mem_toAddSubgroup | null |
coe_toAddSubgroup (s : NonUnitalSubring R) : (s.toAddSubgroup : Set R) = s :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_toAddSubgroup | null |
mem_toNonUnitalSubsemiring {s : NonUnitalSubring R} {x : R} :
x ∈ s.toNonUnitalSubsemiring ↔ x ∈ s :=
Iff.rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | mem_toNonUnitalSubsemiring | null |
coe_toNonUnitalSubsemiring (s : NonUnitalSubring R) :
(s.toNonUnitalSubsemiring : Set R) = s :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | coe_toNonUnitalSubsemiring | null |
inclusion {S T : NonUnitalSubring R} (h : S ≤ T) : S →ₙ+* T :=
NonUnitalRingHom.codRestrict (NonUnitalSubringClass.subtype S) _ fun x => h x.2 | def | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Defs",
"Mathlib.Tactic.FastInstance"
] | Mathlib/RingTheory/NonUnitalSubring/Defs.lean | inclusion | The ring homomorphism associated to an inclusion of `NonUnitalSubring`s. |
@[mono]
toSubsemigroup_strictMono :
StrictMono (toSubsemigroup : NonUnitalSubsemiring R → Subsemigroup R) := fun _ _ => id
@[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 | toSubsemigroup_strictMono | null |
toSubsemigroup_mono : Monotone (toSubsemigroup : NonUnitalSubsemiring R → Subsemigroup R) :=
toSubsemigroup_strictMono.monotone
@[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 | toSubsemigroup_mono | null |
toAddSubmonoid_strictMono :
StrictMono (toAddSubmonoid : NonUnitalSubsemiring R → AddSubmonoid R) := fun _ _ => id
@[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 | toAddSubmonoid_strictMono | null |
toAddSubmonoid_mono : Monotone (toAddSubmonoid : NonUnitalSubsemiring R → AddSubmonoid R) :=
toAddSubmonoid_strictMono.monotone | 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 | toAddSubmonoid_mono | null |
@[simps!]
topEquiv : (⊤ : NonUnitalSubsemiring R) ≃+* R :=
{ Subsemigroup.topEquiv, AddSubmonoid.topEquiv 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 | topEquiv | The ring equiv between the top element of `NonUnitalSubsemiring R` and `R`. |
comap (f : F) (s : NonUnitalSubsemiring S) : NonUnitalSubsemiring R :=
{ s.toSubsemigroup.comap (f : MulHom R S), s.toAddSubmonoid.comap (f : R →+ S) with
carrier := f ⁻¹' s }
@[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 | comap | The preimage of a non-unital subsemiring along a non-unital ring homomorphism is a
non-unital subsemiring. |
coe_comap (s : NonUnitalSubsemiring S) (f : F) : (s.comap f : Set R) = f ⁻¹' s :=
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_comap | null |
mem_comap {s : NonUnitalSubsemiring S} {f : F} {x : R} : x ∈ s.comap f ↔ f x ∈ s :=
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_comap | null |
comap_comap (s : NonUnitalSubsemiring T) (g : G) (f : F) :
((s.comap g : NonUnitalSubsemiring S).comap f : NonUnitalSubsemiring R) =
s.comap ((g : S →ₙ+* T).comp (f : R →ₙ+* 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 | comap_comap | null |
map (f : F) (s : NonUnitalSubsemiring R) : NonUnitalSubsemiring S :=
{ s.toSubsemigroup.map (f : R →ₙ* S), s.toAddSubmonoid.map (f : R →+ S) with carrier := f '' s }
@[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 | map | The image of a non-unital subsemiring along a ring homomorphism is a non-unital subsemiring. |
coe_map (f : F) (s : NonUnitalSubsemiring R) : (s.map f : Set S) = f '' s :=
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_map | null |
mem_map {f : F} {s : NonUnitalSubsemiring R} {y : S} : y ∈ s.map f ↔ ∃ x ∈ s, f x = y :=
Iff.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 | mem_map | null |
map_id : s.map (NonUnitalRingHom.id R) = s :=
SetLike.coe_injective <| Set.image_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 | map_id | null |
map_map (g : G) (f : F) :
(s.map (f : R →ₙ+* S)).map (g : S →ₙ+* T) = s.map ((g : S →ₙ+* T).comp (f : R →ₙ+* S)) :=
SetLike.coe_injective <| Set.image_image _ _ _ | 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_map | null |
map_le_iff_le_comap {f : F} {s : NonUnitalSubsemiring R} {t : NonUnitalSubsemiring S} :
s.map f ≤ t ↔ s ≤ t.comap f :=
Set.image_subset_iff | 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_le_iff_le_comap | null |
gc_map_comap (f : F) :
@GaloisConnection (NonUnitalSubsemiring R) (NonUnitalSubsemiring S) _ _ (map f) (comap f) :=
fun _ _ => map_le_iff_le_comap | 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 | 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.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 | equivMapOfInjective | A non-unital subsemiring 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.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_equivMapOfInjective_apply | null |
srange : NonUnitalSubsemiring S :=
((⊤ : NonUnitalSubsemiring R).map (f : R →ₙ+* S)).copy (Set.range f) Set.image_univ.symm
@[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 | srange | The range of a non-unital ring homomorphism is a non-unital subsemiring.
See note [range copy pattern]. |
coe_srange : (srange f : Set S) = Set.range f :=
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_srange | null |
mem_srange {f : F} {y : S} : y ∈ srange f ↔ ∃ x, f x = y :=
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_srange | null |
srange_eq_map : srange f = (⊤ : NonUnitalSubsemiring R).map f := by
ext
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_eq_map | null |
mem_srange_self (f : F) (x : R) : f x ∈ srange f :=
mem_srange.mpr ⟨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 | mem_srange_self | null |
map_srange (g : S →ₙ+* T) (f : R →ₙ+* S) : map g (srange f) = srange (g.comp f) := by
simpa only [srange_eq_map] using (⊤ : NonUnitalSubsemiring R).map_map g 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_srange | null |
finite_srange [Finite R] (f : F) : Finite (srange f : NonUnitalSubsemiring S) :=
(Set.finite_range f).to_subtype | 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 | finite_srange | The range of a morphism of non-unital semirings is finite if the domain is a finite. |
@[simp, norm_cast]
coe_sInf (S : Set (NonUnitalSubsemiring R)) :
((sInf S : NonUnitalSubsemiring R) : Set R) = ⋂ s ∈ S, ↑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_sInf | null |
mem_sInf {S : Set (NonUnitalSubsemiring R)} {x : R} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p :=
Set.mem_iInter₂
@[simp, norm_cast] | 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_sInf | null |
coe_iInf {ι : Sort*} {S : ι → NonUnitalSubsemiring R} :
(↑(⨅ i, S i) : Set R) = ⋂ i, S i := by
simp only [iInf, coe_sInf, Set.biInter_range] | 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_iInf | null |
mem_iInf {ι : Sort*} {S : ι → NonUnitalSubsemiring R} {x : R} :
(x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by
simp only [iInf, mem_sInf, Set.forall_mem_range]
@[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_iInf | null |
sInf_toSubsemigroup (s : Set (NonUnitalSubsemiring R)) :
(sInf s).toSubsemigroup = ⨅ t ∈ s, NonUnitalSubsemiring.toSubsemigroup t :=
mk'_toSubsemigroup _ _
@[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 | sInf_toSubsemigroup | null |
sInf_toAddSubmonoid (s : Set (NonUnitalSubsemiring R)) :
(sInf s).toAddSubmonoid = ⨅ t ∈ s, NonUnitalSubsemiring.toAddSubmonoid t :=
mk'_toAddSubmonoid _ _ | 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 | sInf_toAddSubmonoid | null |
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