Documentation

Mathlib.RingTheory.NonUnitalSubring.Basic

NonUnitalSubrings #

Let R be a non-unital ring. This file defines the "bundled" non-unital subring type NonUnitalSubring R, a type whose terms correspond to non-unital subrings of R. This is the preferred way to talk about non-unital subrings in mathlib.

We prove that non-unital subrings are a complete lattice, and that you can map (pushforward) and comap (pull back) them along ring homomorphisms.

We define the closure construction from Set R to NonUnitalSubring R, sending a subset of R to the non-unital subring it generates, and prove that it is a Galois insertion.

Main definitions #

Notation used here:

(R : Type u) [NonUnitalRing R] (S : Type u) [NonUnitalRing S] (f g : R →ₙ+* S) (A : NonUnitalSubring R) (B : NonUnitalSubring S) (s : Set R)

Implementation notes #

A non-unital subring is implemented as a NonUnitalSubsemiring which is also an additive subgroup.

Lattice inclusion (e.g. and ) is used rather than set notation ( and ), although is defined as membership of a non-unital subring's underlying set.

Tags #

non-unital subring

NonUnitalSubringClass S R states that S is a type of subsets s ⊆ R that are both a multiplicative submonoid and an additive subgroup.

    Instances
      @[instance 100]
      Equations
      • =
      @[instance 75]

      A non-unital subring of a non-unital ring inherits a non-unital ring structure

      Equations
      @[instance 75]

      A non-unital subring of a non-unital ring inherits a non-unital ring structure

      Equations
      @[instance 75]

      A non-unital subring of a NonUnitalCommRing is a NonUnitalCommRing.

      Equations
      def NonUnitalSubringClass.subtype {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [SetLike S R] [hSR : NonUnitalSubringClass S R] (s : S) :
      s →ₙ+* R

      The natural non-unital ring hom from a non-unital subring of a non-unital ring R to R.

      Equations
      Instances For
        @[simp]
        theorem NonUnitalSubringClass.coe_subtype {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [SetLike S R] [hSR : NonUnitalSubringClass S R] (s : S) :
        (NonUnitalSubringClass.subtype s) = Subtype.val

        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.

        • carrier : Set R
        • add_mem' : ∀ {a b : R}, a self.carrierb self.carriera + b self.carrier
        • zero_mem' : 0 self.carrier
        • mul_mem' : ∀ {a b : R}, a self.carrierb self.carriera * b self.carrier
        • neg_mem' : ∀ {x : R}, x self.carrier-x self.carrier

          G is closed under negation

        Instances For
          @[reducible]

          Reinterpret a NonUnitalSubring as an AddSubgroup.

          Equations
          • self.toAddSubgroup = { toAddSubmonoid := self.toAddSubmonoid, neg_mem' := }
          Instances For

            The underlying submonoid of a NonUnitalSubring.

            Equations
            • s.toSubsemigroup = { carrier := s.carrier, mul_mem' := }
            Instances For
              Equations
              • NonUnitalSubring.instSetLike = { coe := fun (s : NonUnitalSubring R) => s.carrier, coe_injective' := }
              theorem NonUnitalSubring.mem_carrier {R : Type u} [NonUnitalNonAssocRing R] {s : NonUnitalSubring R} {x : R} :
              x s.toNonUnitalSubsemiring x s
              @[simp]
              theorem NonUnitalSubring.mem_mk {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubsemiring R} {x : R} (h : ∀ {x : R}, x S.carrier-x S.carrier) :
              x { toNonUnitalSubsemiring := S, neg_mem' := h } x S
              @[simp]
              theorem NonUnitalSubring.coe_set_mk {R : Type u} [NonUnitalNonAssocRing R] (S : NonUnitalSubsemiring R) (h : ∀ {x : R}, x S.carrier-x S.carrier) :
              { toNonUnitalSubsemiring := S, neg_mem' := h } = S
              @[simp]
              theorem NonUnitalSubring.mk_le_mk {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubsemiring R} {S' : NonUnitalSubsemiring R} (h : ∀ {x : R}, x S.carrier-x S.carrier) (h' : ∀ {x : R}, x S'.carrier-x S'.carrier) :
              { toNonUnitalSubsemiring := S, neg_mem' := h } { toNonUnitalSubsemiring := S', neg_mem' := h' } S S'
              theorem NonUnitalSubring.ext_iff {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubring R} {T : NonUnitalSubring R} :
              S = T ∀ (x : R), x S x T
              theorem NonUnitalSubring.ext {R : Type u} [NonUnitalNonAssocRing R] {S : NonUnitalSubring R} {T : NonUnitalSubring R} (h : ∀ (x : R), x S x T) :
              S = T

              Two non-unital subrings are equal if they have the same elements.

              def NonUnitalSubring.copy {R : Type u} [NonUnitalNonAssocRing R] (S : NonUnitalSubring R) (s : Set R) (hs : s = S) :

              Copy of a non-unital subring with a new carrier equal to the old one. Useful to fix definitional equalities.

              Equations
              • S.copy s hs = { carrier := s, add_mem' := , zero_mem' := , mul_mem' := , neg_mem' := }
              Instances For
                @[simp]
                theorem NonUnitalSubring.coe_copy {R : Type u} [NonUnitalNonAssocRing R] (S : NonUnitalSubring R) (s : Set R) (hs : s = S) :
                (S.copy s hs) = s
                theorem NonUnitalSubring.copy_eq {R : Type u} [NonUnitalNonAssocRing R] (S : NonUnitalSubring R) (s : Set R) (hs : s = S) :
                S.copy s hs = S
                theorem NonUnitalSubring.toNonUnitalSubsemiring_strictMono {R : Type u} [NonUnitalNonAssocRing R] :
                StrictMono NonUnitalSubring.toNonUnitalSubsemiring
                theorem NonUnitalSubring.toNonUnitalSubsemiring_mono {R : Type u} [NonUnitalNonAssocRing R] :
                Monotone NonUnitalSubring.toNonUnitalSubsemiring
                theorem NonUnitalSubring.toAddSubgroup_strictMono {R : Type u} [NonUnitalNonAssocRing R] :
                StrictMono NonUnitalSubring.toAddSubgroup
                theorem NonUnitalSubring.toAddSubgroup_mono {R : Type u} [NonUnitalNonAssocRing R] :
                Monotone NonUnitalSubring.toAddSubgroup
                theorem NonUnitalSubring.toSubsemigroup_strictMono {R : Type u} [NonUnitalNonAssocRing R] :
                StrictMono NonUnitalSubring.toSubsemigroup
                theorem NonUnitalSubring.toSubsemigroup_mono {R : Type u} [NonUnitalNonAssocRing R] :
                Monotone NonUnitalSubring.toSubsemigroup
                def NonUnitalSubring.mk' {R : Type u} [NonUnitalNonAssocRing R] (s : Set R) (sm : Subsemigroup R) (sa : AddSubgroup R) (hm : sm = s) (ha : sa = s) :

                Construct a NonUnitalSubring R from a set s, a subsemigroup sm, and an additive subgroup sa such that x ∈ s ↔ x ∈ sm ↔ x ∈ sa.

                Equations
                • NonUnitalSubring.mk' s sm sa hm ha = { carrier := (sm.copy s ).carrier, add_mem' := , zero_mem' := , mul_mem' := , neg_mem' := }
                Instances For
                  @[simp]
                  theorem NonUnitalSubring.coe_mk' {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {sm : Subsemigroup R} (hm : sm = s) {sa : AddSubgroup R} (ha : sa = s) :
                  (NonUnitalSubring.mk' s sm sa hm ha) = s
                  @[simp]
                  theorem NonUnitalSubring.mem_mk' {R : Type u} [NonUnitalNonAssocRing R] {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
                  @[simp]
                  theorem NonUnitalSubring.mk'_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {sm : Subsemigroup R} (hm : sm = s) {sa : AddSubgroup R} (ha : sa = s) :
                  (NonUnitalSubring.mk' s sm sa hm ha).toSubsemigroup = sm
                  @[simp]
                  theorem NonUnitalSubring.mk'_toAddSubgroup {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {sm : Subsemigroup R} (hm : sm = s) {sa : AddSubgroup R} (ha : sa = s) :
                  (NonUnitalSubring.mk' s sm sa hm ha).toAddSubgroup = sa

                  A non-unital subring contains the ring's 0.

                  theorem NonUnitalSubring.mul_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} {y : R} :
                  x sy sx * y s

                  A non-unital subring is closed under multiplication.

                  theorem NonUnitalSubring.add_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} {y : R} :
                  x sy sx + y s

                  A non-unital subring is closed under addition.

                  theorem NonUnitalSubring.neg_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} :
                  x s-x s

                  A non-unital subring is closed under negation.

                  theorem NonUnitalSubring.sub_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} {y : R} (hx : x s) (hy : y s) :
                  x - y s

                  A non-unital subring is closed under subtraction

                  theorem NonUnitalSubring.list_sum_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {l : List R} :
                  (∀ xl, x s)l.sum s

                  Sum of a list of elements in a non-unital subring is in the non-unital subring.

                  theorem NonUnitalSubring.multiset_sum_mem {R : Type u_1} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) (m : Multiset R) :
                  (∀ am, a s)m.sum s

                  Sum of a multiset of elements in a NonUnitalSubring of a NonUnitalRing is in the NonUnitalSubring.

                  theorem NonUnitalSubring.sum_mem {R : Type u_1} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {ι : Type u_2} {t : Finset ι} {f : ιR} (h : ct, f c s) :
                  it, f i s

                  Sum of elements in a NonUnitalSubring of a NonUnitalRing indexed by a Finset is in the NonUnitalSubring.

                  A non-unital subring of a non-unital ring inherits a non-unital ring structure

                  Equations
                  theorem NonUnitalSubring.zsmul_mem {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : R} (hx : x s) (n : ) :
                  n x s
                  @[simp]
                  theorem NonUnitalSubring.val_add {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) (x : s) (y : s) :
                  (x + y) = x + y
                  @[simp]
                  theorem NonUnitalSubring.val_neg {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) (x : s) :
                  (-x) = -x
                  @[simp]
                  theorem NonUnitalSubring.val_mul {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) (x : s) (y : s) :
                  (x * y) = x * y
                  theorem NonUnitalSubring.coe_eq_zero_iff {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) {x : s} :
                  x = 0 x = 0

                  A non-unital subring of a NonUnitalCommRing is a NonUnitalCommRing.

                  Equations

                  Partial order #

                  @[simp]
                  theorem NonUnitalSubring.mem_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] {s : NonUnitalSubring R} {x : R} :
                  x s.toSubsemigroup x s
                  @[simp]
                  theorem NonUnitalSubring.coe_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) :
                  s.toSubsemigroup = s
                  @[simp]
                  theorem NonUnitalSubring.mem_toAddSubgroup {R : Type u} [NonUnitalNonAssocRing R] {s : NonUnitalSubring R} {x : R} :
                  x s.toAddSubgroup x s
                  @[simp]
                  theorem NonUnitalSubring.coe_toAddSubgroup {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) :
                  s.toAddSubgroup = s
                  @[simp]
                  theorem NonUnitalSubring.mem_toNonUnitalSubsemiring {R : Type u} [NonUnitalNonAssocRing R] {s : NonUnitalSubring R} {x : R} :
                  x s.toNonUnitalSubsemiring x s
                  @[simp]
                  theorem NonUnitalSubring.coe_toNonUnitalSubsemiring {R : Type u} [NonUnitalNonAssocRing R] (s : NonUnitalSubring R) :
                  s.toNonUnitalSubsemiring = s

                  top #

                  The non-unital subring R of the ring R.

                  Equations
                  • NonUnitalSubring.instTop = { top := let __src := ; let __src_1 := ; { carrier := __src.carrier, add_mem' := , zero_mem' := , mul_mem' := , neg_mem' := } }
                  @[simp]
                  @[simp]
                  theorem NonUnitalSubring.coe_top {R : Type u} [NonUnitalNonAssocRing R] :
                  = Set.univ
                  @[simp]
                  theorem NonUnitalSubring.topEquiv_apply {R : Type u} [NonUnitalNonAssocRing R] (x : ) :
                  NonUnitalSubring.topEquiv x = x
                  @[simp]
                  theorem NonUnitalSubring.topEquiv_symm_apply_coe {R : Type u} [NonUnitalNonAssocRing R] (x : R) :
                  (NonUnitalSubring.topEquiv.symm x) = x

                  The ring equiv between the top element of NonUnitalSubring R and R.

                  Equations
                  • NonUnitalSubring.topEquiv = NonUnitalSubsemiring.topEquiv
                  Instances For

                    comap #

                    The preimage of a NonUnitalSubring along a ring homomorphism is a NonUnitalSubring.

                    Equations
                    Instances For
                      @[simp]
                      theorem NonUnitalSubring.coe_comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring S) (f : F) :
                      (NonUnitalSubring.comap f s) = f ⁻¹' s
                      @[simp]
                      theorem NonUnitalSubring.mem_comap {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {s : NonUnitalSubring S} {f : F} {x : R} :

                      map #

                      The image of a NonUnitalSubring along a ring homomorphism is a NonUnitalSubring.

                      Equations
                      • NonUnitalSubring.map f s = { carrier := f '' s.carrier, add_mem' := , zero_mem' := , mul_mem' := , neg_mem' := }
                      Instances For
                        @[simp]
                        theorem NonUnitalSubring.coe_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.map f s) = f '' s
                        @[simp]
                        theorem NonUnitalSubring.mem_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} {y : S} :
                        y NonUnitalSubring.map f s xs, f x = y
                        noncomputable def NonUnitalSubring.equivMapOfInjective {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring R) (f : F) (hf : Function.Injective f) :

                        A NonUnitalSubring is isomorphic to its image under an injective function

                        Equations
                        • s.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑s) hf, map_mul' := , map_add' := }
                        Instances For
                          @[simp]
                          theorem NonUnitalSubring.coe_equivMapOfInjective_apply {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring R) (f : F) (hf : Function.Injective f) (x : s) :
                          ((s.equivMapOfInjective f hf) x) = f x

                          range #

                          The range of a ring homomorphism, as a NonUnitalSubring of the target. See Note [range copy pattern].

                          Equations
                          Instances For
                            @[simp]
                            theorem NonUnitalRingHom.coe_range {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) :
                            f.range = Set.range f
                            @[simp]
                            theorem NonUnitalRingHom.mem_range {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f : R →ₙ+* S} {y : S} :
                            y f.range ∃ (x : R), f x = y
                            theorem NonUnitalRingHom.mem_range_self {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) (x : R) :
                            f x f.range
                            theorem NonUnitalRingHom.map_range {R : Type u} {S : Type v} {T : Type u_1} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T] (g : S →ₙ+* T) (f : R →ₙ+* S) :
                            NonUnitalSubring.map g f.range = (g.comp f).range

                            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.

                            Equations

                            bot #

                            Equations
                            Equations
                            • NonUnitalSubring.instInhabited = { default := }

                            inf #

                            The inf of two NonUnitalSubrings is their intersection.

                            Equations
                            • One or more equations did not get rendered due to their size.
                            @[simp]
                            theorem NonUnitalSubring.coe_inf {R : Type u} [NonUnitalNonAssocRing R] (p : NonUnitalSubring R) (p' : NonUnitalSubring R) :
                            (p p') = p p'
                            @[simp]
                            theorem NonUnitalSubring.mem_inf {R : Type u} [NonUnitalNonAssocRing R] {p : NonUnitalSubring R} {p' : NonUnitalSubring R} {x : R} :
                            x p p' x p x p'
                            Equations
                            @[simp]
                            theorem NonUnitalSubring.coe_sInf {R : Type u} [NonUnitalNonAssocRing R] (S : Set (NonUnitalSubring R)) :
                            (sInf S) = sS, s
                            theorem NonUnitalSubring.mem_sInf {R : Type u} [NonUnitalNonAssocRing R] {S : Set (NonUnitalSubring R)} {x : R} :
                            x sInf S pS, x p
                            @[simp]
                            theorem NonUnitalSubring.coe_iInf {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_1} {S : ιNonUnitalSubring R} :
                            (⨅ (i : ι), S i) = ⋂ (i : ι), (S i)
                            theorem NonUnitalSubring.mem_iInf {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_1} {S : ιNonUnitalSubring R} {x : R} :
                            x ⨅ (i : ι), S i ∀ (i : ι), x S i
                            @[simp]
                            theorem NonUnitalSubring.sInf_toSubsemigroup {R : Type u} [NonUnitalNonAssocRing R] (s : Set (NonUnitalSubring R)) :
                            (sInf s).toSubsemigroup = ts, t.toSubsemigroup
                            @[simp]
                            theorem NonUnitalSubring.sInf_toAddSubgroup {R : Type u} [NonUnitalNonAssocRing R] (s : Set (NonUnitalSubring R)) :
                            (sInf s).toAddSubgroup = ts, t.toAddSubgroup

                            NonUnitalSubrings of a ring form a complete lattice.

                            Equations

                            Center of a ring #

                            The center of a ring R is the set of elements that commute with everything in R

                            Equations
                            Instances For
                              theorem NonUnitalSubring.mem_center_iff {R : Type u} [NonUnitalRing R] {z : R} :
                              z NonUnitalSubring.center R ∀ (g : R), g * z = z * g

                              NonUnitalSubring closure of a subset #

                              The NonUnitalSubring generated by a set.

                              Equations
                              Instances For
                                theorem NonUnitalSubring.mem_closure {R : Type u} [NonUnitalNonAssocRing R] {x : R} {s : Set R} :
                                x NonUnitalSubring.closure s ∀ (S : NonUnitalSubring R), s Sx S
                                @[simp]

                                The NonUnitalSubring generated by a set includes the set.

                                @[simp]

                                A NonUnitalSubring t includes closure s if and only if it includes s.

                                NonUnitalSubring closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

                                theorem NonUnitalSubring.closure_induction {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {p : (x : R) → x NonUnitalSubring.closure sProp} (mem : ∀ (x : R) (hx : x s), p x ) (zero : p 0 ) (add : ∀ (x y : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s), p x hxp y hyp (x + y) ) (neg : ∀ (x : R) (hx : x NonUnitalSubring.closure s), p x hxp (-x) ) (mul : ∀ (x y : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s), p x hxp y hyp (x * y) ) {x : R} (hx : x NonUnitalSubring.closure s) :
                                p x hx

                                An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition, negation, and multiplication, then p holds for all elements of the closure of s.

                                @[deprecated NonUnitalSubring.closure_induction]
                                theorem NonUnitalSubring.closure_induction' {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {p : (NonUnitalSubring.closure s)Prop} (a : (NonUnitalSubring.closure s)) (mem : ∀ (x : R) (hx : x s), p x, ) (zero : p 0) (add : ∀ (x y : (NonUnitalSubring.closure s)), p xp yp (x + y)) (neg : ∀ (x : (NonUnitalSubring.closure s)), p xp (-x)) (mul : ∀ (x y : (NonUnitalSubring.closure s)), p xp yp (x * y)) :
                                p a

                                The difference with NonUnitalSubring.closure_induction is that this acts on the subtype.

                                theorem NonUnitalSubring.closure_induction₂ {R : Type u} [NonUnitalNonAssocRing R] {s : Set R} {p : (x y : R) → x NonUnitalSubring.closure sy NonUnitalSubring.closure sProp} (mem_mem : ∀ (x y : R) (hx : x s) (hy : y s), p x y ) (zero_left : ∀ (x : R) (hx : x NonUnitalSubring.closure s), p 0 x hx) (zero_right : ∀ (x : R) (hx : x NonUnitalSubring.closure s), p x 0 hx ) (neg_left : ∀ (x y : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s), p x y hx hyp (-x) y hy) (neg_right : ∀ (x y : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s), p x y hx hyp x (-y) hx ) (add_left : ∀ (x y z : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s) (hz : z NonUnitalSubring.closure s), p x z hx hzp y z hy hzp (x + y) z hz) (add_right : ∀ (x y z : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s) (hz : z NonUnitalSubring.closure s), p x y hx hyp x z hx hzp x (y + z) hx ) (mul_left : ∀ (x y z : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s) (hz : z NonUnitalSubring.closure s), p x z hx hzp y z hy hzp (x * y) z hz) (mul_right : ∀ (x y z : R) (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s) (hz : z NonUnitalSubring.closure s), p x y hx hyp x z hx hzp x (y * z) hx ) {x : R} {y : R} (hx : x NonUnitalSubring.closure s) (hy : y NonUnitalSubring.closure s) :
                                p x y hx hy

                                An induction principle for closure membership, for predicates with two arguments.

                                def NonUnitalSubring.closureNonUnitalCommRingOfComm {R : Type u} [NonUnitalRing R] {s : Set R} (hcomm : as, bs, a * b = b * a) :

                                If all elements of s : Set A commute pairwise, then closure s is a commutative ring.

                                Equations
                                Instances For
                                  def NonUnitalSubring.gi (R : Type u) [NonUnitalNonAssocRing R] :
                                  GaloisInsertion NonUnitalSubring.closure SetLike.coe

                                  closure forms a Galois insertion with the coercion to set.

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                                    theorem NonUnitalSubring.closure_iUnion {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_1} (s : ιSet R) :
                                    NonUnitalSubring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), NonUnitalSubring.closure (s i)
                                    theorem NonUnitalSubring.map_iSup {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_1} (f : F) (s : ιNonUnitalSubring R) :
                                    NonUnitalSubring.map f (iSup s) = ⨆ (i : ι), NonUnitalSubring.map f (s i)
                                    theorem NonUnitalSubring.map_iInf {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_1} [Nonempty ι] (f : F) (hf : Function.Injective f) (s : ιNonUnitalSubring R) :
                                    NonUnitalSubring.map f (iInf s) = ⨅ (i : ι), NonUnitalSubring.map f (s i)
                                    theorem NonUnitalSubring.comap_iInf {F : Type w} {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_1} (f : F) (s : ιNonUnitalSubring S) :

                                    Given NonUnitalSubrings s, t of rings R, S respectively, s.prod t is s ×ˢ t as a NonUnitalSubring of R × S.

                                    Equations
                                    • s.prod t = { carrier := s ×ˢ t, add_mem' := , zero_mem' := , mul_mem' := , neg_mem' := }
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                                      theorem NonUnitalSubring.coe_prod {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (s : NonUnitalSubring R) (t : NonUnitalSubring S) :
                                      (s.prod t) = s ×ˢ t
                                      theorem NonUnitalSubring.mem_prod {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {s : NonUnitalSubring R} {t : NonUnitalSubring S} {p : R × S} :
                                      p s.prod t p.1 s p.2 t
                                      theorem NonUnitalSubring.prod_mono {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] ⦃s₁ : NonUnitalSubring R ⦃s₂ : NonUnitalSubring R (hs : s₁ s₂) ⦃t₁ : NonUnitalSubring S ⦃t₂ : NonUnitalSubring S (ht : t₁ t₂) :
                                      s₁.prod t₁ s₂.prod t₂
                                      def NonUnitalSubring.prodEquiv {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (s : NonUnitalSubring R) (t : NonUnitalSubring S) :
                                      (s.prod t) ≃+* s × t

                                      Product of NonUnitalSubrings is isomorphic to their product as rings.

                                      Equations
                                      • s.prodEquiv t = { toEquiv := Equiv.Set.prod s t, map_mul' := , map_add' := }
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                                        theorem NonUnitalSubring.mem_iSup_of_directed {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_1} [hι : Nonempty ι] {S : ιNonUnitalSubring R} (hS : Directed (fun (x1 x2 : NonUnitalSubring R) => x1 x2) S) {x : R} :
                                        x ⨆ (i : ι), S i ∃ (i : ι), x S i

                                        The underlying set of a non-empty directed Sup of NonUnitalSubrings is just a union of the NonUnitalSubrings. Note that this fails without the directedness assumption (the union of two NonUnitalSubrings is typically not a NonUnitalSubring)

                                        theorem NonUnitalSubring.coe_iSup_of_directed {R : Type u} [NonUnitalNonAssocRing R] {ι : Sort u_1} [Nonempty ι] {S : ιNonUnitalSubring R} (hS : Directed (fun (x1 x2 : NonUnitalSubring R) => x1 x2) S) :
                                        (⨆ (i : ι), S i) = ⋃ (i : ι), (S i)
                                        theorem NonUnitalSubring.mem_sSup_of_directedOn {R : Type u} [NonUnitalNonAssocRing R] {S : Set (NonUnitalSubring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : NonUnitalSubring R) => x1 x2) S) {x : R} :
                                        x sSup S sS, x s
                                        theorem NonUnitalSubring.coe_sSup_of_directedOn {R : Type u} [NonUnitalNonAssocRing R] {S : Set (NonUnitalSubring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : NonUnitalSubring R) => x1 x2) S) :
                                        (sSup S) = sS, s
                                        theorem NonUnitalSubring.mem_map_equiv {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f : R ≃+* S} {K : NonUnitalSubring R} {x : S} :
                                        x NonUnitalSubring.map (↑f) K f.symm x K

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

                                        This is the bundled version of Set.rangeFactorization.

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                                          @[simp]
                                          theorem NonUnitalRingHom.coe_rangeRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) (x : R) :
                                          (f.rangeRestrict x) = f x
                                          @[simp]

                                          The range of a surjective ring homomorphism is the whole of the codomain.

                                          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

                                          Equations
                                          • f.eqLocus g = { carrier := {x : R | f x = g x}, add_mem' := , zero_mem' := , mul_mem' := , neg_mem' := }
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                                            @[simp]
                                            theorem NonUnitalRingHom.eqLocus_same {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] (f : R →ₙ+* S) :
                                            f.eqLocus f =
                                            theorem NonUnitalRingHom.eqOn_set_closure {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f : R →ₙ+* S} {g : R →ₙ+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) :

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

                                            theorem NonUnitalRingHom.eq_of_eqOn_set_top {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f : R →ₙ+* S} {g : R →ₙ+* S} (h : Set.EqOn f g ) :
                                            f = g
                                            theorem NonUnitalRingHom.eq_of_eqOn_set_dense {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {s : Set R} (hs : NonUnitalSubring.closure s = ) {f : R →ₙ+* S} {g : R →ₙ+* S} (h : Set.EqOn (⇑f) (⇑g) s) :
                                            f = g

                                            The image under a ring homomorphism of the NonUnitalSubring generated by a set equals the NonUnitalSubring generated by the image of the set.

                                            The ring homomorphism associated to an inclusion of NonUnitalSubrings.

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                                              def RingEquiv.nonUnitalSubringCongr {R : Type u} [NonUnitalRing R] {s : NonUnitalSubring R} {t : NonUnitalSubring R} (h : s = t) :
                                              s ≃+* t

                                              Makes the identity isomorphism from a proof two NonUnitalSubrings of a multiplicative monoid are equal.

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                                                def RingEquiv.ofLeftInverse' {R : Type u} {S : Type v} [NonUnitalRing R] [NonUnitalRing S] {g : SR} {f : R →ₙ+* S} (h : Function.LeftInverse g f) :
                                                R ≃+* f.range

                                                Restrict a ring homomorphism with a left inverse to a ring isomorphism to its RingHom.range.

                                                Equations
                                                • One or more equations did not get rendered due to their size.
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                                                  @[simp]
                                                  theorem RingEquiv.ofLeftInverse'_apply {R : Type u} {S : Type v} [NonUnitalRing R] [NonUnitalRing S] {g : SR} {f : R →ₙ+* S} (h : Function.LeftInverse g f) (x : R) :
                                                  @[simp]
                                                  theorem RingEquiv.ofLeftInverse'_symm_apply {R : Type u} {S : Type v} [NonUnitalRing R] [NonUnitalRing S] {g : SR} {f : R →ₙ+* S} (h : Function.LeftInverse g f) (x : f.range) :
                                                  (RingEquiv.ofLeftInverse' h).symm x = g x