Mathlib.RingTheory.NonUnitalSubring.Basic

/-- Sum of a list of elements in a non-unital subring is in the non-unital subring. -/
protected theorem list_sum_mem {l : List R} : (∀ x ∈ l, x ∈ s) → l.sum ∈ s :=
  list_sum_mem

/-- Sum of a multiset of elements in a `NonUnitalSubring` of a `NonUnitalRing` is
in the `NonUnitalSubring`. -/
protected theorem multiset_sum_mem {R} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R)
    (m : Multiset R) : (∀ a ∈ m, a ∈ s) → m.sum ∈ s :=
  multiset_sum_mem _

/-- Sum of elements in a `NonUnitalSubring` of a `NonUnitalRing` indexed by a `Finset`
is in the `NonUnitalSubring`. -/
protected theorem sum_mem {R : Type*} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R)
    {ι : Type*} {t : Finset ι} {f : ι → R} (h : ∀ c ∈ t, f c ∈ s) : (∑ i ∈ t, f i) ∈ s :=
  sum_mem h

/-- The non-unital subring `R` of the ring `R`. -/
instance : Top (NonUnitalSubring R) :=
  ⟨{ (⊤ : Subsemigroup R), (⊤ : AddSubgroup R) with }⟩

@[simp]
theorem mem_top (x : R) : x ∈ (⊤ : NonUnitalSubring R) :=
  Set.mem_univ x

@[simp]
theorem coe_top : ((⊤ : NonUnitalSubring R) : Set R) = Set.univ :=
  rfl

/-- The ring equiv between the top element of `NonUnitalSubring R` and `R`. -/
@[simps!]
def topEquiv : (⊤ : NonUnitalSubring R) ≃+* R := NonUnitalSubsemiring.topEquiv

/-- The preimage of a `NonUnitalSubring` along a ring homomorphism is a `NonUnitalSubring`. -/
def comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
    [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubring S) :
    NonUnitalSubring R :=
  { s.toSubsemigroup.comap (f : R →ₙ* S), s.toAddSubgroup.comap (f : R →+ S) with
    carrier := f ⁻¹' s.carrier }

@[simp]
theorem coe_comap (s : NonUnitalSubring S) (f : F) : (s.comap f : Set R) = f ⁻¹' s :=
  rfl

@[simp]
theorem mem_comap {s : NonUnitalSubring S} {f : F} {x : R} : x ∈ s.comap fx ∈ s.comap f ↔ f x ∈ s :=
  Iff.rfl

theorem comap_comap (s : NonUnitalSubring T) (g : S →ₙ+* T) (f : R →ₙ+* S) :
    (s.comap g).comap f = s.comap (g.comp f) :=
  rfl

/-- The image of a `NonUnitalSubring` along a ring homomorphism is a `NonUnitalSubring`. -/
def map {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
    [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubring R) :
    NonUnitalSubring S :=
  { s.toSubsemigroup.map (f : R →ₙ* S), s.toAddSubgroup.map (f : R →+ S) with
    carrier := f '' s.carrier }

@[simp]
theorem coe_map (f : F) (s : NonUnitalSubring R) : (s.map f : Set S) = f '' s :=
  rfl

@[simp]
theorem mem_map {f : F} {s : NonUnitalSubring R} {y : S} : y ∈ s.map fy ∈ s.map f ↔ ∃ x ∈ s, f x = y :=
  Set.mem_image _ _ _

@[simp]
theorem map_id : s.map (NonUnitalRingHom.id R) = s :=
  SetLike.coe_injective <| Set.image_id _

theorem map_map (g : S →ₙ+* T) (f : R →ₙ+* S) : (s.map f).map g = s.map (g.comp f) :=
  SetLike.coe_injective <| Set.image_image _ _ _

theorem map_le_iff_le_comap {f : F} {s : NonUnitalSubring R} {t : NonUnitalSubring S} :
    s.map f ≤ t ↔ s ≤ t.comap f :=
  Set.image_subset_iff

theorem gc_map_comap (f : F) :
    GaloisConnection (map f : NonUnitalSubring R → NonUnitalSubring S) (comap f) := fun _S _T =>
  map_le_iff_le_comap

/-- A `NonUnitalSubring` is isomorphic to its image under an injective function -/
noncomputable def 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]
theorem coe_equivMapOfInjective_apply (f : F) (hf : Function.Injective f) (x : s) :
    (equivMapOfInjective s f hf x : S) = f x :=
  rfl

/-- The range of a ring homomorphism, as a `NonUnitalSubring` of the target.
See Note [range copy pattern]. -/
def range {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
    (f : R →ₙ+* S) : NonUnitalSubring S :=
  ((⊤ : NonUnitalSubring R).map f).copy (Set.range f) Set.image_univ.symm

@[simp]
theorem coe_range : (f.range : Set S) = Set.range f :=
  rfl

@[simp]
theorem mem_range {f : R →ₙ+* S} {y : S} : y ∈ f.rangey ∈ f.range ↔ ∃ x, f x = y :=
  Iff.rfl

theorem range_eq_map (f : R →ₙ+* S) : f.range = NonUnitalSubring.map f ⊤ := by ext; simp

theorem mem_range_self (f : R →ₙ+* S) (x : R) : f x ∈ f.range :=
  mem_range.mpr ⟨x, rfl⟩

theorem map_range : f.range.map g = (g.comp f).range := by
  simpa only [range_eq_map] using (⊤ : NonUnitalSubring R).map_map g f

/-- The range of a ring homomorphism is a fintype, if the domain is a fintype.
Note: this instance can form a diamond with `Subtype.fintype` in the
  presence of `Fintype S`. -/
instance fintypeRange [Fintype R] [DecidableEq S] (f : R →ₙ+* S) : Fintype (range f) :=
  Set.fintypeRange f

instance : Bot (NonUnitalSubring R) :=
  ⟨(0 : R →ₙ+* R).range⟩

instance : Inhabited (NonUnitalSubring R) :=
  ⟨⊥⟩

theorem coe_bot : ((⊥ : NonUnitalSubring R) : Set R) = {0} :=
  (NonUnitalRingHom.coe_range (0 : R →ₙ+* R)).trans (@Set.range_const R R _ 0)

theorem mem_bot {x : R} : x ∈ (⊥ : NonUnitalSubring R)x ∈ (⊥ : NonUnitalSubring R) ↔ x = 0 :=
  show x ∈ ((⊥ : NonUnitalSubring R) : Set R)x ∈ ((⊥ : NonUnitalSubring R) : Set R) ↔ x = 0 by rw [coe_bot, Set.mem_singleton_iff]

/-- The inf of two `NonUnitalSubring`s is their intersection. -/
instance : Min (NonUnitalSubring R) :=
  ⟨fun s t =>
    { s.toSubsemigroup ⊓ t.toSubsemigroup, s.toAddSubgroup ⊓ t.toAddSubgroup with
      carrier := s ∩ t }⟩

@[simp]
theorem coe_inf (p p' : NonUnitalSubring R) :
    ((p ⊓ p' : NonUnitalSubring R) : Set R) = (p : Set R) ∩ p' :=
  rfl

@[simp]
theorem mem_inf {p p' : NonUnitalSubring R} {x : R} : x ∈ p ⊓ p'x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' :=
  Iff.rfl

instance : InfSet (NonUnitalSubring R) :=
  ⟨fun s =>
    NonUnitalSubring.mk' (⋂ t ∈ s, ↑t) (⨅ t ∈ s, NonUnitalSubring.toSubsemigroup t)
      (⨅ t ∈ s, NonUnitalSubring.toAddSubgroup t) (by simp) (by simp)⟩

@[simp, norm_cast]
theorem coe_sInf (S : Set (NonUnitalSubring R)) :
    ((sInf S : NonUnitalSubring R) : Set R) = ⋂ s ∈ S, ↑s :=
  rfl

theorem mem_sInf {S : Set (NonUnitalSubring R)} {x : R} : x ∈ sInf Sx ∈ sInf S ↔ ∀ p ∈ S, x ∈ p :=
  Set.mem_iInter₂

@[simp, norm_cast]
theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubring R} : (↑(⨅ i, S i) : Set R) = ⋂ i, S i := by
  simp only [iInf, coe_sInf, Set.biInter_range]

theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubring R} {x : R} :
    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_mem_range]

@[simp]
theorem sInf_toSubsemigroup (s : Set (NonUnitalSubring R)) :
    (sInf s).toSubsemigroup = ⨅ t ∈ s, NonUnitalSubring.toSubsemigroup t :=
  mk'_toSubsemigroup _ _

@[simp]
theorem sInf_toAddSubgroup (s : Set (NonUnitalSubring R)) :
    (sInf s).toAddSubgroup = ⨅ t ∈ s, NonUnitalSubring.toAddSubgroup t :=
  mk'_toAddSubgroup _ _

/-- `NonUnitalSubring`s of a ring form a complete lattice. -/
instance : CompleteLattice (NonUnitalSubring R) :=
  { completeLatticeOfInf (NonUnitalSubring R) fun _s =>
      IsGLB.of_image (@fun _ _ : NonUnitalSubring R => SetLike.coe_subset_coe)
        isGLB_biInf with
    bot := ⊥
    bot_le := fun s _x hx => (mem_bot.mp hx).symm ▸ zero_mem s
    top := ⊤
    le_top := fun _ _ _ => trivial
    inf := (· ⊓ ·)
    inf_le_left := fun _ _ _ => And.left
    inf_le_right := fun _ _ _ => And.right
    le_inf := fun _s _t₁ _t₂ h₁ h₂ _x hx => ⟨h₁ hx, h₂ hx⟩ }

/-- The center of a ring `R` is the set of elements that commute with everything in `R` -/
def center : NonUnitalSubring R :=
  { NonUnitalSubsemiring.center R with
    neg_mem' := Set.neg_mem_center }

theorem coe_center : ↑(center R) = Set.center R :=
  rfl

@[simp]
theorem center_toNonUnitalSubsemiring :
    (center R).toNonUnitalSubsemiring = NonUnitalSubsemiring.center R :=
  rfl

/-- The center is commutative and associative. -/
instance center.instNonUnitalCommRing : NonUnitalCommRing (center R) :=
  { NonUnitalSubsemiring.center.instNonUnitalCommSemiring R,
    inferInstanceAs <| NonUnitalNonAssocRing (center R) with }

/-- The center of isomorphic (not necessarily unital or associative) rings are isomorphic. -/
@[simps!] def centerCongr {S} [NonUnitalNonAssocRing S] (e : R ≃+* S) : centercenter R ≃+* center S :=
  NonUnitalSubsemiring.centerCongr e

/-- The center of a (not necessarily uintal or associative) ring
is isomorphic to the center of its opposite. -/
@[simps!] def centerToMulOpposite : centercenter R ≃+* center Rᵐᵒᵖ :=
  NonUnitalSubsemiring.centerToMulOpposite

theorem mem_center_iff {z : R} : z ∈ center Rz ∈ center R ↔ ∀ g, g * z = z * g := Subsemigroup.mem_center_iff

instance decidableMemCenter [DecidableEq R] [Fintype R] : DecidablePred (· ∈ center R) := fun _ =>
  decidable_of_iff' _ mem_center_iff

@[simp]
theorem center_eq_top (R) [NonUnitalCommRing R] : center R = ⊤ :=
  SetLike.coe_injective (Set.center_eq_univ R)

/-- The `NonUnitalSubring` generated by a set. -/
def closure (s : Set R) : NonUnitalSubring R :=
  sInf {S | s ⊆ S}

theorem mem_closure {x : R} {s : Set R} : x ∈ closure sx ∈ closure s ↔ ∀ S : NonUnitalSubring R, s ⊆ S → x ∈ S :=
  mem_sInf

/-- The `NonUnitalSubring` generated by a set includes the set. -/
@[simp, aesop safe 20 apply (rule_sets := [SetLike])]
theorem subset_closure {s : Set R} : s ⊆ closure s := fun _x hx => mem_closure.2 fun _S hS => hS hx

theorem not_mem_of_not_mem_closure {s : Set R} {P : R} (hP : P ∉ closure s) : P ∉ s := fun h =>
  hP (subset_closure h)

/-- A `NonUnitalSubring` `t` includes `closure s` if and only if it includes `s`. -/
@[simp]
theorem closure_le {s : Set R} {t : NonUnitalSubring R} : closureclosure s ≤ t ↔ s ⊆ t :=
  ⟨Set.Subset.trans subset_closure, fun h => sInf_le h⟩

/-- `NonUnitalSubring` closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`. -/
@[gcongr]
theorem closure_mono ⦃s t : Set R⦄ (h : s ⊆ t) : closure s ≤ closure t :=
  closure_le.2 <| Set.Subset.trans h subset_closure

theorem closure_eq_of_le {s : Set R} {t : NonUnitalSubring R} (h₁ : s ⊆ t) (h₂ : t ≤ closure s) :
    closure s = t :=
  le_antisymm (closure_le.2 h₁) h₂

/-- An induction principle for closure membership. If `p` holds for `0`, `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {s : Set R} {p : (x : R) → x ∈ closure s → Prop}
    (mem : ∀ (x) (hx : x ∈ s), p x (subset_closure hx)) (zero : p 0 (zero_mem _))
    (add : ∀ x y hx hy, p x hx → p y hy → p (x + y) (add_mem hx hy))
    (neg : ∀ x hx, p x hx → p (-x) (neg_mem hx))
    (mul : ∀ x y hx hy, p x hx → p y hy → p (x * y) (mul_mem hx hy))
    {x} (hx : x ∈ closure s) : p x hx :=
  let K : NonUnitalSubring R :=
    { carrier := { x | ∃ hx, p x hx }
      mul_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, mul _ _ _ _ hpx hpy⟩
      add_mem' := fun ⟨_, hpx⟩ ⟨_, hpy⟩ ↦ ⟨_, add _ _ _ _ hpx hpy⟩
      neg_mem' := fun ⟨_, hpx⟩ ↦ ⟨_, neg _ _ hpx⟩
      zero_mem' := ⟨_, zero⟩ }
  closure_le (t := K) |>.mpr (fun y hy ↦ ⟨subset_closure hy, mem y hy⟩) hx |>.elim fun _ ↦ id

/-- An induction principle for closure membership, for predicates with two arguments. -/
@[elab_as_elim]
theorem closure_induction₂ {s : Set R} {p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop}
    (mem_mem : ∀ (x) (y) (hx : x ∈ s) (hy : y ∈ s), p x y (subset_closure hx) (subset_closure hy))
    (zero_left : ∀ x hx, p 0 x (zero_mem _) hx) (zero_right : ∀ x hx, p x 0 hx (zero_mem _))
    (neg_left : ∀ x y hx hy, p x y hx hy → p (-x) y (neg_mem hx) hy)
    (neg_right : ∀ x y hx hy, p x y hx hy → p x (-y) hx (neg_mem hy))
    (add_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x + y) z (add_mem hx hy) hz)
    (add_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y + z) hx (add_mem hy hz))
    (mul_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x * y) z (mul_mem hx hy) hz)
    (mul_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y * z) hx (mul_mem hy hz))
    {x y : R} (hx : x ∈ closure s) (hy : y ∈ closure s) :
    p x y hx hy := by
  induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h
  | zero => exact zero_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
  | neg _ _ h => exact neg_right _ _ _ _ h

theorem mem_closure_iff {s : Set R} {x} :
    x ∈ closure sx ∈ closure s ↔ x ∈ AddSubgroup.closure (Subsemigroup.closure s : Set R) :=
  ⟨fun h => by
    induction h using closure_induction with
    | mem _ hx => exact AddSubgroup.subset_closure (Subsemigroup.subset_closure hx)
    | zero => exact zero_mem _
    | add _ _ _ _ hx hy => exact add_mem hx hy
    | neg x _ hx => exact neg_mem hx
    | mul _ _ _hx _hy hx hy =>
      clear _hx _hy
      induction hx, hy using AddSubgroup.closure_induction₂ with
      | mem _ _ hx hy => exact AddSubgroup.subset_closure (mul_mem hx hy)
      | one_left => simpa using zero_mem _
      | one_right => simpa using zero_mem _
      | mul_left _ _ _ _ _ _ h₁ h₂ => simpa [add_mul] using add_mem h₁ h₂
      | mul_right _ _ _ _ _ _ h₁ h₂ => simpa [mul_add] using add_mem h₁ h₂
      | inv_left _ _ _ _ h => simpa [neg_mul] using neg_mem h
      | inv_right _ _ _ _ h => simpa [mul_neg] using neg_mem h,
  fun h => by
    induction h using AddSubgroup.closure_induction with
    | mem _ hx => induction hx using Subsemigroup.closure_induction with
      | mem _ h => exact subset_closure h
      | mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
    | one => exact zero_mem _
    | mul _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
    | inv _ _ h => exact neg_mem h⟩

/-- If all elements of `s : Set A` commute pairwise, then `closure s` is a commutative ring. -/
def closureNonUnitalCommRingOfComm {R : Type u} [NonUnitalRing R] {s : Set R}
    (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) : NonUnitalCommRing (closure s) :=
  { (closure s).toNonUnitalRing with
    mul_comm := fun ⟨x, hx⟩ ⟨y, hy⟩ => by
      ext
      simp only [MulMemClass.mk_mul_mk]
      induction hx, hy using closure_induction₂ with
      | mem_mem x y hx hy => exact hcomm x hx y hy
      | zero_left x _ => exact Commute.zero_left x
      | zero_right x _ => exact Commute.zero_right x
      | mul_left _ _ _ _ _ _ h₁ h₂ => exact Commute.mul_left h₁ h₂
      | mul_right _ _ _ _ _ _ h₁ h₂ => exact Commute.mul_right h₁ h₂
      | add_left _ _ _ _ _ _ h₁ h₂ => exact Commute.add_left h₁ h₂
      | add_right _ _ _ _ _ _ h₁ h₂ => exact Commute.add_right h₁ h₂
      | neg_left _ _ _ _ h => exact Commute.neg_left h
      | neg_right _ _ _ _ h => exact Commute.neg_right h }

/-- `closure` forms a Galois insertion with the coercion to set. -/
protected def gi : GaloisInsertion (@closure R _) SetLike.coe where
  choice s _ := closure s
  gc _s _t := closure_le
  le_l_u _s := subset_closure
  choice_eq _s _h := rfl

/-- Closure of a `NonUnitalSubring` `S` equals `S`. -/
@[simp]
theorem closure_eq (s : NonUnitalSubring R) : closure (s : Set R) = s :=
  (NonUnitalSubring.gi R).l_u_eq s

@[simp]
theorem closure_empty : closure (∅ : Set R) = ⊥ :=
  (NonUnitalSubring.gi R).gc.l_bot

@[simp]
theorem closure_univ : closure (Set.univ : Set R) = ⊤ :=
  @coe_top R _ ▸ closure_eq ⊤

theorem closure_union (s t : Set R) : closure (s ∪ t) = closureclosure s ⊔ closure t :=
  (NonUnitalSubring.gi R).gc.l_sup

theorem closure_iUnion {ι} (s : ι → Set R) : closure (⋃ i, s i) = ⨆ i, closure (s i) :=
  (NonUnitalSubring.gi R).gc.l_iSup

theorem closure_sUnion (s : Set (Set R)) : closure (⋃₀ s) = ⨆ t ∈ s, closure t :=
  (NonUnitalSubring.gi R).gc.l_sSup

theorem map_sup (s t : NonUnitalSubring R) (f : F) : (s ⊔ t).map f = s.map f ⊔ t.map f :=
  (gc_map_comap f).l_sup

theorem map_iSup {ι : Sort*} (f : F) (s : ι → NonUnitalSubring R) :
    (iSup s).map f = ⨆ i, (s i).map f :=
  (gc_map_comap f).l_iSup

theorem map_inf (s t : NonUnitalSubring R) (f : F) (hf : Function.Injective f) :
    (s ⊓ t).map f = s.map f ⊓ t.map f := SetLike.coe_injective (Set.image_inter hf)

theorem map_iInf {ι : Sort*} [Nonempty ι] (f : F) (hf : Function.Injective f)
    (s : ι → NonUnitalSubring R) : (iInf s).map f = ⨅ i, (s i).map f := by
  apply SetLike.coe_injective
  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coeSetLike.coe ∘ s)

theorem comap_inf (s t : NonUnitalSubring S) (f : F) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
  (gc_map_comap f).u_inf

theorem comap_iInf {ι : Sort*} (f : F) (s : ι → NonUnitalSubring S) :
    (iInf s).comap f = ⨅ i, (s i).comap f :=
  (gc_map_comap f).u_iInf

@[simp]
theorem map_bot (f : R →ₙ+* S) : (⊥ : NonUnitalSubring R).map f = ⊥ :=
  (gc_map_comap f).l_bot

@[simp]
theorem comap_top (f : R →ₙ+* S) : (⊤ : NonUnitalSubring S).comap f = ⊤ :=
  (gc_map_comap f).u_top

/-- Given `NonUnitalSubring`s `s`, `t` of rings `R`, `S` respectively, `s.prod t` is `s ×ˢ t`
as a `NonUnitalSubring` of `R × S`. -/
def prod (s : NonUnitalSubring R) (t : NonUnitalSubring S) : NonUnitalSubring (R × S) :=
  { s.toSubsemigroup.prod t.toSubsemigroup, s.toAddSubgroup.prod t.toAddSubgroup with
    carrier := s ×ˢ t }

@[norm_cast]
theorem coe_prod (s : NonUnitalSubring R) (t : NonUnitalSubring S) :
    (s.prod t : Set (R × S)) = (s : Set R) ×ˢ t :=
  rfl

theorem mem_prod {s : NonUnitalSubring R} {t : NonUnitalSubring S} {p : R × S} :
    p ∈ s.prod tp ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t :=
  Iff.rfl

@[mono]
theorem prod_mono ⦃s₁ s₂ : NonUnitalSubring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : NonUnitalSubring S⦄
    (ht : t₁ ≤ t₂) : s₁.prod t₁ ≤ s₂.prod t₂ :=
  Set.prod_mono hs ht

theorem prod_mono_right (s : NonUnitalSubring R) :
    Monotone fun t : NonUnitalSubring S => s.prod t :=
  prod_mono (le_refl s)

theorem prod_mono_left (t : NonUnitalSubring S) : Monotone fun s : NonUnitalSubring R => s.prod t :=
  fun _s₁ _s₂ hs => prod_mono hs (le_refl t)

theorem prod_top (s : NonUnitalSubring R) :
    s.prod (⊤ : NonUnitalSubring S) = s.comap (NonUnitalRingHom.fst R S) :=
  ext fun x => by simp [mem_prod, MonoidHom.coe_fst]

theorem top_prod (s : NonUnitalSubring S) :
    (⊤ : NonUnitalSubring R).prod s = s.comap (NonUnitalRingHom.snd R S) :=
  ext fun x => by simp [mem_prod, MonoidHom.coe_snd]

@[simp]
theorem top_prod_top : (⊤ : NonUnitalSubring R).prod (⊤ : NonUnitalSubring S) = ⊤ :=
  (top_prod _).trans <| comap_top _

/-- Product of `NonUnitalSubring`s is isomorphic to their product as rings. -/
def prodEquiv (s : NonUnitalSubring R) (t : NonUnitalSubring S) : s.prod t ≃+* s × t :=
  { Equiv.Set.prod (s : Set R) (t : Set S) with
    map_mul' := fun _ _ => rfl
    map_add' := fun _ _ => rfl }

/-- The underlying set of a non-empty directed Sup of `NonUnitalSubring`s is just a union of the
`NonUnitalSubring`s. Note that this fails without the directedness assumption (the union of two
`NonUnitalSubring`s is typically not a `NonUnitalSubring`) -/
theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → NonUnitalSubring R}
    (hS : Directed (· ≤ ·) S) {x : R} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i := by
  refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩
  let U : NonUnitalSubring R :=
    NonUnitalSubring.mk' (⋃ i, (S i : Set R)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup)
      (Subsemigroup.coe_iSup_of_directed hS) (AddSubgroup.coe_iSup_of_directed hS)
  suffices ⨆ i, S i ≤ U by simpa [U] using @this x
  exact iSup_le fun i x hx ↦ Set.mem_iUnion.2 ⟨i, hx⟩

theorem coe_iSup_of_directed {ι} [Nonempty ι] {S : ι → NonUnitalSubring R}
    (hS : Directed (· ≤ ·) S) : ((⨆ i, S i : NonUnitalSubring R) : Set R) = ⋃ i, S i :=
  Set.ext fun x ↦ by simp [mem_iSup_of_directed hS]

theorem mem_sSup_of_directedOn {S : Set (NonUnitalSubring R)} (Sne : S.Nonempty)
    (hS : DirectedOn (· ≤ ·) S) {x : R} : x ∈ sSup Sx ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by
  haveI : Nonempty S := Sne.to_subtype
  simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,
    exists_prop]

theorem coe_sSup_of_directedOn {S : Set (NonUnitalSubring R)} (Sne : S.Nonempty)
    (hS : DirectedOn (· ≤ ·) S) : (↑(sSup S) : Set R) = ⋃ s ∈ S, ↑s :=
  Set.ext fun x => by simp [mem_sSup_of_directedOn Sne hS]

theorem mem_map_equiv {f : R ≃+* S} {K : NonUnitalSubring R} {x : S} :
    x ∈ K.map (f : R →ₙ+* S)x ∈ K.map (f : R →ₙ+* S) ↔ f.symm x ∈ K :=
  @Set.mem_image_equiv _ _ (K : Set R) f.toEquiv x

theorem map_equiv_eq_comap_symm (f : R ≃+* S) (K : NonUnitalSubring R) :
    K.map (f : R →ₙ+* S) = K.comap f.symm :=
  SetLike.coe_injective (f.toEquiv.image_eq_preimage K)

theorem comap_equiv_eq_map_symm (f : R ≃+* S) (K : NonUnitalSubring S) :
    K.comap (f : R →ₙ+* S) = K.map f.symm :=
  (map_equiv_eq_comap_symm f.symm K).symm

/-- Restriction of a ring homomorphism to its range interpreted as a `NonUnitalSubring`.

This is the bundled version of `Set.rangeFactorization`. -/
def rangeRestrict (f : R →ₙ+* S) : R →ₙ+* f.range :=
  NonUnitalRingHom.codRestrict f f.range fun x => ⟨x, rfl⟩

@[simp]
theorem coe_rangeRestrict (f : R →ₙ+* S) (x : R) : (f.rangeRestrict x : S) = f x :=
  rfl

theorem 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 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

/-- The range of a surjective ring homomorphism is the whole of the codomain. -/
@[simp]
theorem range_eq_top_of_surjective (f : R →ₙ+* S) (hf : Function.Surjective f) :
    f.range = (⊤ : NonUnitalSubring S) :=
  range_eq_top.2 hf

/-- 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 -/
def 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]
theorem eqLocus_same (f : R →ₙ+* S) : f.eqLocus f = ⊤ :=
  SetLike.ext fun _ => eq_self_iff_true _

/-- If two ring homomorphisms are equal on a set, then they are equal on its
`NonUnitalSubring` closure. -/
theorem 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 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

/-- The image under a ring homomorphism of the `NonUnitalSubring` generated by a set equals
the `NonUnitalSubring` generated by the image of the set. -/
theorem 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

@[simp]
theorem range_subtype (s : NonUnitalSubring R) : (NonUnitalSubringClass.subtype s).range = s :=
  SetLike.coe_injective <| (coe_srange _).trans Subtype.range_coe

theorem range_fst : NonUnitalRingHom.srange (fst R S) = ⊤ :=
  NonUnitalSubsemiring.range_fst

theorem range_snd : NonUnitalRingHom.srange (snd R S) = ⊤ :=
  NonUnitalSubsemiring.range_snd

/-- Makes the identity isomorphism from a proof two `NonUnitalSubring`s of a multiplicative
    monoid are equal. -/
def nonUnitalSubringCongr (h : s = t) : s ≃+* t :=
  {
    Equiv.setCongr <| congr_arg _ h with
    map_mul' := fun _ _ => rfl
    map_add' := fun _ _ => rfl }

/-- Restrict a ring homomorphism with a left inverse to a ring isomorphism to its
`RingHom.range`. -/
def 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]
theorem ofLeftInverse'_apply {g : S → R} {f : R →ₙ+* S} (h : Function.LeftInverse g f) (x : R) :
    ↑(ofLeftInverse' h x) = f x :=
  rfl

@[simp]
theorem 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 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