ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NonUnitalAlgebra.adjoin_induction_subtype
@[elab_as_elim]
lemma adjoin_induction_subtype {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)
    (mem : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)
    (add : ∀ x y, p x → p y → p (x + y)) (zero : p 0)
    (mul : ∀ x y, p x → p y → p (x * y)) (smul : ∀ (r : R) x, p x → p (r • x)) : p a 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {s : Set A} {p : ↥(NonUnitalAlgebra.adjoin R s) → Prop} (a : ↥(NonUnitalAlgebra.adjoin R s)) (mem : ∀ (x : A) (h : x ∈ s), p ⟨x, ⋯⟩) (add : ∀ (x y : ↥(NonUnitalAlgebra.adjoin R s)), p x → p y → p (x + y)) (zero : p 0) (mul : ∀ (x y : ↥(NonUnitalAlgebra.adjoin R s)), p x → p y → p (x * y)) (smul : ∀ (r : R) (x : ↥(NonUnitalAlgebra.adjoin R s)), p x → p (r • x))
Goal: p a
-- NonUnitalAlgebra.adjoin_induction'
theorem adjoin_induction' {s : Set A} {p : ∀ x, x ∈ adjoin R s → Prop}
    (mem : ∀ (x) (h : x ∈ s), p x (subset_adjoin R h))
    (add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (add_mem ‹_› ‹_›))
    (zero : p 0 (zero_mem _))
    (mul : ∀ x hx y hy, p x hx → p y hy → p (x * y) (mul_mem ‹_› ‹_›))
    (smul : ∀ (r : R) (x hx), p x hx → p (r • x) (SMulMemClass.smul_mem _ ‹_›))
    {a} (ha : a ∈ adjoin R s) : p a ha 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {s : Set A} {p : (x : A) → x ∈ NonUnitalAlgebra.adjoin R s → Prop} (mem : ∀ (x : A) (h : x ∈ s), p x ⋯) (add :
  ∀ (x : A) (hx : x ∈ NonUnitalAlgebra.adjoin R s) (y : A) (hy : y ∈ NonUnitalAlgebra.adjoin R s),
    p x hx → p y hy → p (x + y) ⋯) (zero : p 0 ⋯) (mul :
  ∀ (x : A) (hx : x ∈ NonUnitalAlgebra.adjoin R s) (y : A) (hy : y ∈ NonUnitalAlgebra.adjoin R s),
    p x hx → p y hy → p (x * y) ⋯) (smul : ∀ (r : R) (x : A) (hx : x ∈ NonUnitalAlgebra.adjoin R s), p x hx → p (r • x) ⋯) {a : A} (ha : a ∈ NonUnitalAlgebra.adjoin R s)
Goal: p a ha
-- NonUnitalAlgebra.gc
protected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: GaloisConnection (NonUnitalAlgebra.adjoin R) SetLike.coe
-- NonUnitalAlgebra.adjoin_le
theorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ ↑S)
Goal: NonUnitalAlgebra.adjoin R s ≤ S
-- NonUnitalAlgebra.adjoin_le_iff
theorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S : NonUnitalSubalgebra R A} {s : Set A}
Goal: NonUnitalAlgebra.adjoin R s ≤ S ↔ s ⊆ ↑S
-- NonUnitalAlgebra.adjoin_union
theorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (s t : Set A)
Goal: NonUnitalAlgebra.adjoin R (s ∪ t) = NonUnitalAlgebra.adjoin R s ⊔ NonUnitalAlgebra.adjoin R t
-- NonUnitalAlgebra.adjoin_eq
lemma adjoin_eq (s : NonUnitalSubalgebra R A) : adjoin R (s : Set A) = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (s : NonUnitalSubalgebra R A)
Goal: NonUnitalAlgebra.adjoin R ↑s = s
-- NonUnitalAlgebra.adjoin_eq_span
lemma adjoin_eq_span (s : Set A) : (adjoin R s).toSubmodule = span R (Subsemigroup.closure s) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (s : Set A)
Goal: (NonUnitalAlgebra.adjoin R s).toSubmodule = Submodule.span R ↑(Subsemigroup.closure s)
-- NonUnitalAlgebra.span_eq_toSubmodule
@[simp]
lemma span_eq_toSubmodule (s : NonUnitalSubalgebra R A) :
    Submodule.span R (s : Set A) = s.toSubmodule 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] (s : NonUnitalSubalgebra R A)
Goal: Submodule.span R ↑s = s.toSubmodule
-- NonUnitalAlgebra.adjoin_empty
@[simp]
theorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: NonUnitalAlgebra.adjoin R ∅ = ⊥
-- NonUnitalAlgebra.adjoin_univ
@[simp]
theorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: NonUnitalAlgebra.adjoin R Set.univ = ⊤
-- NonUnitalAlgHom.map_adjoin
lemma _root_.NonUnitalAlgHom.map_adjoin (f : F) (s : Set A) :
    map f (adjoin R s) = adjoin R (f '' s) 
Arguments: {F : Type u_1} (R : Type u) (A : Type v) {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] (f : F) (s : Set A)
Goal: NonUnitalSubalgebra.map f (NonUnitalAlgebra.adjoin R s) = NonUnitalAlgebra.adjoin R (⇑f '' s)
-- NonUnitalAlgHom.map_adjoin_singleton
@[simp]
lemma _root_.NonUnitalAlgHom.map_adjoin_singleton (f : F) (x : A) :
    map f (adjoin R {x}) = adjoin R {f x} 
Arguments: {F : Type u_1} (R : Type u) (A : Type v) {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] (f : F) (x : A)
Goal: NonUnitalSubalgebra.map f (NonUnitalAlgebra.adjoin R {x}) = NonUnitalAlgebra.adjoin R {f x}
-- NonUnitalAlgebra.coe_top
@[simp]
theorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ↑⊤ = Set.univ
-- NonUnitalAlgebra.mem_top
@[simp]
theorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {x : A}
Goal: x ∈ ⊤
-- NonUnitalAlgebra.top_toSubmodule
@[simp]
theorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ⊤.toSubmodule = ⊤
-- NonUnitalAlgebra.top_toNonUnitalSubsemiring
@[simp]
theorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ⊤.toNonUnitalSubsemiring = ⊤
-- NonUnitalAlgebra.top_toSubring
@[simp]
theorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]
    [IsScalarTower R A A] [SMulCommClass R A A] :
    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ 
Arguments: {R : Type u_2} {A : Type u_3} [CommRing R] [NonUnitalNonAssocRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ⊤.toNonUnitalSubring = ⊤
-- NonUnitalAlgebra.toSubmodule_eq_top
@[simp]
theorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S : NonUnitalSubalgebra R A}
Goal: S.toSubmodule = ⊤ ↔ S = ⊤
-- NonUnitalAlgebra.toNonUnitalSubsemiring_eq_top
@[simp]
theorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :
    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S : NonUnitalSubalgebra R A}
Goal: S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤
-- NonUnitalAlgebra.to_subring_eq_top
@[simp]
theorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]
    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ 
Arguments: {R : Type u_2} {A : Type u_3} [CommRing R] [Ring A] [Algebra R A] {S : NonUnitalSubalgebra R A}
Goal: S.toNonUnitalSubring = ⊤ ↔ S = ⊤
-- NonUnitalAlgebra.mem_sup_left
theorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} {x : A}
Goal: x ∈ S → x ∈ S ⊔ T
-- NonUnitalAlgebra.mem_sup_right
theorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} {x : A}
Goal: x ∈ T → x ∈ S ⊔ T
-- NonUnitalAlgebra.mul_mem_sup
theorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :
    x * y ∈ S ⊔ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T)
Goal: x * y ∈ S ⊔ T
-- NonUnitalAlgebra.map_sup
theorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :
    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f 
Arguments: {F : Type u_1} {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] [FunLike F A B] [NonUnitalAlgHomClass F R A B] (f : F) (S T : NonUnitalSubalgebra R A)
Goal: NonUnitalSubalgebra.map f (S ⊔ T) = NonUnitalSubalgebra.map f S ⊔ NonUnitalSubalgebra.map f T
-- NonUnitalAlgebra.coe_inf
@[simp, norm_cast]
theorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (S T : NonUnitalSubalgebra R A)
Goal: ↑(S ⊓ T) = ↑S ∩ ↑T
-- NonUnitalAlgebra.mem_inf
@[simp]
theorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} {x : A}
Goal: x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T
-- NonUnitalAlgebra.inf_toSubmodule
@[simp]
theorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :
    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (S T : NonUnitalSubalgebra R A)
Goal: (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule
-- NonUnitalAlgebra.inf_toNonUnitalSubsemiring
@[simp]
theorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :
    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (S T : NonUnitalSubalgebra R A)
Goal: (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring
-- NonUnitalAlgebra.coe_sInf
@[simp, norm_cast]
theorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (S : Set (NonUnitalSubalgebra R A))
Goal: ↑(sInf S) = ⋂ s ∈ S, ↑s
-- NonUnitalAlgebra.mem_sInf
theorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S : Set (NonUnitalSubalgebra R A)} {x : A}
Goal: x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p
-- NonUnitalAlgebra.sInf_toSubmodule
@[simp]
theorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :
    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (S : Set (NonUnitalSubalgebra R A))
Goal: (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S)
-- NonUnitalAlgebra.sInf_toNonUnitalSubsemiring
@[simp]
theorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :
    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (S : Set (NonUnitalSubalgebra R A))
Goal: (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S)
-- NonUnitalAlgebra.coe_iInf
@[simp, norm_cast]
theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :
    (↑(⨅ i, S i) : Set A) = ⋂ i, S i 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {ι : Sort u_2} {S : ι → NonUnitalSubalgebra R A}
Goal: ↑(⨅ i, S i) = ⋂ i, ↑(S i)
-- NonUnitalAlgebra.mem_iInf
theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :
    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {ι : Sort u_2} {S : ι → NonUnitalSubalgebra R A} {x : A}
Goal: x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i
-- NonUnitalAlgebra.iInf_toSubmodule
@[simp]
theorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :
    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {ι : Sort u_2} (S : ι → NonUnitalSubalgebra R A)
Goal: (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule
-- NonUnitalAlgebra.mem_bot
theorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {x : A}
Goal: x ∈ ⊥ ↔ x = 0
-- NonUnitalAlgebra.toSubmodule_bot
theorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ⊥.toSubmodule = ⊥
-- NonUnitalAlgebra.coe_bot
@[simp]
theorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ↑⊥ = {0}
-- NonUnitalAlgebra.eq_top_iff
theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S : NonUnitalSubalgebra R A}
Goal: S = ⊤ ↔ ∀ (x : A), x ∈ S
-- NonUnitalAlgebra.range_top_iff_surjective
theorem range_top_iff_surjective (f : A →ₙₐ[R] B) :
    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] (f : A →ₙₐ[R] B)
Goal: NonUnitalAlgHom.range f = ⊤ ↔ Function.Surjective ⇑f
-- NonUnitalAlgebra.range_id
@[simp]
theorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤
-- NonUnitalAlgebra.map_top
@[simp]
theorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] (f : A →ₙₐ[R] B)
Goal: NonUnitalSubalgebra.map f ⊤ = NonUnitalAlgHom.range f
-- NonUnitalAlgebra.map_bot
@[simp]
theorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] (f : A →ₙₐ[R] B)
Goal: NonUnitalSubalgebra.map f ⊥ = ⊥
-- NonUnitalAlgebra.comap_top
@[simp]
theorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] (f : A →ₙₐ[R] B)
Goal: NonUnitalSubalgebra.comap f ⊤ = ⊤
-- NonUnitalSubalgebra.range_val
theorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] (S : NonUnitalSubalgebra R A)
Goal: NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S
-- NonUnitalSubalgebra.inclusion_injective
theorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :
    Function.Injective (inclusion h) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} (h : S ≤ T)
Goal: Function.Injective ⇑(NonUnitalSubalgebra.inclusion h)
-- NonUnitalSubalgebra.inclusion_self
@[simp]
theorem inclusion_self {S : NonUnitalSubalgebra R A} :
    inclusion (le_refl S) = NonUnitalAlgHom.id R S 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S : NonUnitalSubalgebra R A}
Goal: NonUnitalSubalgebra.inclusion ⋯ = NonUnitalAlgHom.id R ↥S
-- NonUnitalSubalgebra.inclusion_mk
@[simp]
theorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :
    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S)
Goal: (NonUnitalSubalgebra.inclusion h) ⟨x, hx⟩ = ⟨x, ⋯⟩
-- NonUnitalSubalgebra.inclusion_right
theorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :
    inclusion h ⟨x, m⟩ = x 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : ↥T) (m : ↑x ∈ S)
Goal: (NonUnitalSubalgebra.inclusion h) ⟨↑x, m⟩ = x
-- NonUnitalSubalgebra.inclusion_inclusion
@[simp]
theorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :
    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : ↥S)
Goal: (NonUnitalSubalgebra.inclusion htu) ((NonUnitalSubalgebra.inclusion hst) x) = (NonUnitalSubalgebra.inclusion ⋯) x
-- NonUnitalSubalgebra.coe_inclusion
@[simp]
theorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :
    (inclusion h s : A) = s 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : ↥S)
Goal: ↑((NonUnitalSubalgebra.inclusion h) s) = ↑s
-- NonUnitalSubalgebra.coe_prod
@[simp]
theorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B] (S : NonUnitalSubalgebra R A) (S₁ : NonUnitalSubalgebra R B)
Goal: ↑(S.prod S₁) = ↑S ×ˢ ↑S₁
-- NonUnitalSubalgebra.prod_toSubmodule
theorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B] (S : NonUnitalSubalgebra R A) (S₁ : NonUnitalSubalgebra R B)
Goal: (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule
-- NonUnitalSubalgebra.mem_prod
@[simp]
theorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :
    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B] {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B}
Goal: x ∈ S.prod S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁
-- NonUnitalSubalgebra.prod_top
@[simp]
theorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]
Goal: ⊤.prod ⊤ = ⊤
-- NonUnitalSubalgebra.prod_mono
theorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :
    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B}
Goal: S ≤ T → S₁ ≤ T₁ → S.prod S₁ ≤ T.prod T₁
-- NonUnitalSubalgebra.prod_inf_prod
@[simp]
theorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :
    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B] {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B}
Goal: S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁)
-- NonUnitalSubalgebra.coe_iSup_of_directed
theorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}
    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {ι : Type u_1} [Nonempty ι] {S : ι → NonUnitalSubalgebra R A} (dir : Directed (fun x x_1 => x ≤ x_1) S)
Goal: ↑(iSup S) = ⋃ i, ↑(S i)