ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ideal.map_comap_map
@[simp]
theorem map_comap_map : ((I.map f).comap f).map f = I.map f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal R)
Goal: Ideal.map f (Ideal.comap f (Ideal.map f I)) = Ideal.map f I
-- Ideal.comap_map_comap
@[simp]
theorem comap_map_comap : ((K.comap f).map f).comap f = K.comap f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (K : Ideal S)
Goal: Ideal.comap f (Ideal.map f (Ideal.comap f K)) = Ideal.comap f K
-- Ideal.map_sup
theorem map_sup : (I ⊔ J).map f = I.map f ⊔ J.map f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I J : Ideal R)
Goal: Ideal.map f (I ⊔ J) = Ideal.map f I ⊔ Ideal.map f J
-- Ideal.comap_inf
theorem comap_inf : comap f (K ⊓ L) = comap f K ⊓ comap f L 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (K L : Ideal S)
Goal: Ideal.comap f (K ⊓ L) = Ideal.comap f K ⊓ Ideal.comap f L
-- Ideal.map_iSup
theorem map_iSup (K : ι → Ideal R) : (iSup K).map f = ⨆ i, (K i).map f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (K : ι → Ideal R)
Goal: Ideal.map f (iSup K) = ⨆ i, Ideal.map f (K i)
-- Ideal.comap_iInf
theorem comap_iInf (K : ι → Ideal S) : (iInf K).comap f = ⨅ i, (K i).comap f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (K : ι → Ideal S)
Goal: Ideal.comap f (iInf K) = ⨅ i, Ideal.comap f (K i)
-- Ideal.map_sSup
theorem map_sSup (s : Set (Ideal R)) : (sSup s).map f = ⨆ I ∈ s, (I : Ideal R).map f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (s : Set (Ideal R))
Goal: Ideal.map f (sSup s) = ⨆ I ∈ s, Ideal.map f I
-- Ideal.comap_sInf
theorem comap_sInf (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ s, (I : Ideal S).comap f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (s : Set (Ideal S))
Goal: Ideal.comap f (sInf s) = ⨅ I ∈ s, Ideal.comap f I
-- Ideal.comap_sInf'
theorem comap_sInf' (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ comap f '' s, I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (s : Set (Ideal S))
Goal: Ideal.comap f (sInf s) = ⨅ I ∈ Ideal.comap f '' s, I
-- Ideal.comap_isPrime
theorem comap_isPrime [H : IsPrime K] : IsPrime (comap f K) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (K : Ideal S) [H : K.IsPrime]
Goal: (Ideal.comap f K).IsPrime
-- Ideal.map_inf_le
theorem map_inf_le : map f (I ⊓ J) ≤ map f I ⊓ map f J 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {I J : Ideal R}
Goal: Ideal.map f (I ⊓ J) ≤ Ideal.map f I ⊓ Ideal.map f J
-- Ideal.le_comap_sup
theorem le_comap_sup : comap f K ⊔ comap f L ≤ comap f (K ⊔ L) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K L : Ideal S}
Goal: Ideal.comap f K ⊔ Ideal.comap f L ≤ Ideal.comap f (K ⊔ L)
-- element_smul_restrictScalars
theorem _root_.element_smul_restrictScalars {R S M}
    [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M]
    [Module R M] [Module S M] [IsScalarTower R S M] (r : R) (N : Submodule S M) :
    (algebraMap R S r • N).restrictScalars R = r • N.restrictScalars R 
Arguments: {R : Type u_4} {S : Type u_5} {M : Type u_6} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] (r : R) (N : Submodule S M)
Goal: Submodule.restrictScalars R ((algebraMap R S) r • N) = r • Submodule.restrictScalars R N
-- Ideal.smul_restrictScalars
theorem smul_restrictScalars {R S M} [CommSemiring R] [CommSemiring S]
    [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M]
    [IsScalarTower R S M] (I : Ideal R) (N : Submodule S M) :
    (I.map (algebraMap R S) • N).restrictScalars R = I • N.restrictScalars R 
Arguments: {R : Type u_4} {S : Type u_5} {M : Type u_6} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] (I : Ideal R) (N : Submodule S M)
Goal: Submodule.restrictScalars R (Ideal.map (algebraMap R S) I • N) = I • Submodule.restrictScalars R N
-- Ideal.smul_top_eq_map
@[simp]
theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
    (I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R 
Arguments: {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] [Algebra R S] (I : Ideal R)
Goal: I • ⊤ = Submodule.restrictScalars R (Ideal.map (algebraMap R S) I)
-- Ideal.coe_restrictScalars
@[simp]
theorem coe_restrictScalars {R S : Type*} [CommSemiring R] [Semiring S] [Algebra R S]
    (I : Ideal S) : (I.restrictScalars R : Set S) = ↑I 
Arguments: {R : Type u_4} {S : Type u_5} [CommSemiring R] [Semiring S] [Algebra R S] (I : Ideal S)
Goal: ↑(Submodule.restrictScalars R I) = ↑I
-- Ideal.restrictScalars_mul
@[simp]
theorem restrictScalars_mul {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S]
    (I J : Ideal S) : (I * J).restrictScalars R = I.restrictScalars R * J.restrictScalars R 
Arguments: {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] [Algebra R S] (I J : Ideal S)
Goal: Submodule.restrictScalars R (I * J) = Submodule.restrictScalars R I * Submodule.restrictScalars R J
-- Ideal.map_comap_of_surjective
theorem map_comap_of_surjective (I : Ideal S) : map f (comap f I) = I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal S)
Goal: Ideal.map f (Ideal.comap f I) = I
-- Ideal.map_surjective_of_surjective
theorem map_surjective_of_surjective : Surjective (map f) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f)
Goal: Function.Surjective (Ideal.map f)
-- Ideal.comap_injective_of_surjective
theorem comap_injective_of_surjective : Injective (comap f) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f)
Goal: Function.Injective (Ideal.comap f)
-- Ideal.map_sup_comap_of_surjective
theorem map_sup_comap_of_surjective (I J : Ideal S) : (I.comap f ⊔ J.comap f).map f = I ⊔ J 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I J : Ideal S)
Goal: Ideal.map f (Ideal.comap f I ⊔ Ideal.comap f J) = I ⊔ J
-- Ideal.map_iSup_comap_of_surjective
theorem map_iSup_comap_of_surjective (K : ι → Ideal S) : (⨆ i, (K i).comap f).map f = iSup K 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (hf : Function.Surjective ⇑f) (K : ι → Ideal S)
Goal: Ideal.map f (⨆ i, Ideal.comap f (K i)) = iSup K
-- Ideal.map_inf_comap_of_surjective
theorem map_inf_comap_of_surjective (I J : Ideal S) : (I.comap f ⊓ J.comap f).map f = I ⊓ J 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I J : Ideal S)
Goal: Ideal.map f (Ideal.comap f I ⊓ Ideal.comap f J) = I ⊓ J
-- Ideal.map_iInf_comap_of_surjective
theorem map_iInf_comap_of_surjective (K : ι → Ideal S) : (⨅ i, (K i).comap f).map f = iInf K 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (hf : Function.Surjective ⇑f) (K : ι → Ideal S)
Goal: Ideal.map f (⨅ i, Ideal.comap f (K i)) = iInf K
-- Ideal.mem_image_of_mem_map_of_surjective
theorem mem_image_of_mem_map_of_surjective {I : Ideal R} {y} (H : y ∈ map f I) : y ∈ f '' I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) {I : Ideal R} {y : S} (H : y ∈ Ideal.map f I)
Goal: y ∈ ⇑f '' ↑I
-- Ideal.mem_map_iff_of_surjective
theorem mem_map_iff_of_surjective {I : Ideal R} {y} : y ∈ map f I ↔ ∃ x, x ∈ I ∧ f x = y 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) {I : Ideal R} {y : S}
Goal: y ∈ Ideal.map f I ↔ ∃ x ∈ I, f x = y
-- Ideal.le_map_of_comap_le_of_surjective
theorem le_map_of_comap_le_of_surjective : comap f K ≤ I → K ≤ map f I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {I : Ideal R} {K : Ideal S} (hf : Function.Surjective ⇑f)
Goal: Ideal.comap f K ≤ I → K ≤ Ideal.map f I
-- Ideal.map_eq_submodule_map
theorem map_eq_submodule_map (f : R →+* S) [h : RingHomSurjective f] (I : Ideal R) :
    I.map f = Submodule.map f.toSemilinearMap I 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R →+* S) [h : RingHomSurjective f] (I : Ideal R)
Goal: Ideal.map f I = Submodule.map f.toSemilinearMap I
-- Ideal.comap_bot_le_of_injective
theorem comap_bot_le_of_injective (hf : Function.Injective f) : comap f ⊥ ≤ I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {I : Ideal R} (hf : Function.Injective ⇑f)
Goal: Ideal.comap f ⊥ ≤ I
-- Ideal.comap_bot_of_injective
theorem comap_bot_of_injective (hf : Function.Injective f) : Ideal.comap f ⊥ = ⊥ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Injective ⇑f)
Goal: Ideal.comap f ⊥ = ⊥
-- Ideal.map_of_equiv
@[simp]
theorem map_of_equiv (I : Ideal R) (f : R ≃+* S) :
    (I.map (f : R →+* S)).map (f.symm : S →+* R) = I 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (f : R ≃+* S)
Goal: Ideal.map (↑f.symm) (Ideal.map (↑f) I) = I
-- Ideal.comap_of_equiv
@[simp]
theorem comap_of_equiv (I : Ideal R) (f : R ≃+* S) :
    (I.comap (f.symm : S →+* R)).comap (f : R →+* S) = I 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (f : R ≃+* S)
Goal: Ideal.comap (↑f) (Ideal.comap (↑f.symm) I) = I
-- Ideal.map_comap_of_equiv
theorem map_comap_of_equiv (I : Ideal R) (f : R ≃+* S) : I.map (f : R →+* S) = I.comap f.symm 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (f : R ≃+* S)
Goal: Ideal.map (↑f) I = Ideal.comap f.symm I
-- Ideal.comap_symm
@[simp]
theorem comap_symm (I : Ideal R) (f : R ≃+* S) : I.comap f.symm = I.map f 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (f : R ≃+* S)
Goal: Ideal.comap f.symm I = Ideal.map f I
-- Ideal.map_symm
@[simp]
theorem map_symm (I : Ideal S) (f : R ≃+* S) : I.map f.symm = I.comap f 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal S) (f : R ≃+* S)
Goal: Ideal.map f.symm I = Ideal.comap f I
-- Ideal.comap_map_of_surjective
theorem comap_map_of_surjective (hf : Function.Surjective f) (I : Ideal R) :
    comap f (map f I) = I ⊔ comap f ⊥ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal R)
Goal: Ideal.comap f (Ideal.map f I) = I ⊔ Ideal.comap f ⊥
-- Ideal.map_eq_top_or_isMaximal_of_surjective
theorem map_eq_top_or_isMaximal_of_surjective (hf : Function.Surjective f) {I : Ideal R}
    (H : IsMaximal I) : map f I = ⊤ ∨ IsMaximal (map f I) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) {I : Ideal R} (H : I.IsMaximal)
Goal: Ideal.map f I = ⊤ ∨ (Ideal.map f I).IsMaximal
-- Ideal.comap_isMaximal_of_surjective
theorem comap_isMaximal_of_surjective (hf : Function.Surjective f) {K : Ideal S} [H : IsMaximal K]:
    IsMaximal (comap f K) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) {K : Ideal S} [H : K.IsMaximal]
Goal: (Ideal.comap f K).IsMaximal
-- Ideal.comap_le_comap_iff_of_surjective
theorem comap_le_comap_iff_of_surjective (hf : Function.Surjective f) (I J : Ideal S) :
    comap f I ≤ comap f J ↔ I ≤ J 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I J : Ideal S)
Goal: Ideal.comap f I ≤ Ideal.comap f J ↔ I ≤ J
-- Ideal.comap_le_iff_le_map
theorem comap_le_iff_le_map (hf : Function.Bijective f) {I : Ideal R} {K : Ideal S} :
    comap f K ≤ I ↔ K ≤ map f I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Bijective ⇑f) {I : Ideal R} {K : Ideal S}
Goal: Ideal.comap f K ≤ I ↔ K ≤ Ideal.map f I
-- Ideal.map.isMaximal
theorem map.isMaximal (hf : Function.Bijective f) {I : Ideal R} (H : IsMaximal I) :
    IsMaximal (map f I) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Bijective ⇑f) {I : Ideal R} (H : I.IsMaximal)
Goal: (Ideal.map f I).IsMaximal
-- Ideal.RingEquiv.bot_maximal_iff
theorem RingEquiv.bot_maximal_iff (e : R ≃+* S) :
    (⊥ : Ideal R).IsMaximal ↔ (⊥ : Ideal S).IsMaximal 
Arguments: {R : Type u} {S : Type v} [Ring R] [Ring S] (e : R ≃+* S)
Goal: ⊥.IsMaximal ↔ ⊥.IsMaximal
-- Ideal.map_mul
theorem map_mul : map f (I * J) = map f I * map f J 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I J : Ideal R)
Goal: Ideal.map f (I * J) = Ideal.map f I * Ideal.map f J
-- Ideal.mapHom_apply
@[simps]
def mapHom : Ideal R →*₀ Ideal S 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal R)
Goal: (Ideal.mapHom f) I = Ideal.map f I
-- Ideal.map_pow
protected theorem map_pow (n : ℕ) : map f (I ^ n) = map f I ^ n 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal R) (n : ℕ)
Goal: Ideal.map f (I ^ n) = Ideal.map f I ^ n
-- Ideal.comap_radical
theorem comap_radical : comap f (radical K) = radical (comap f K) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (K : Ideal S)
Goal: Ideal.comap f K.radical = (Ideal.comap f K).radical
-- Ideal.IsRadical.comap
theorem IsRadical.comap (hK : K.IsRadical) : (comap f K).IsRadical 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (hK : K.IsRadical)
Goal: (Ideal.comap f K).IsRadical
-- Ideal.map_radical_le
theorem map_radical_le : map f (radical I) ≤ radical (map f I) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {I : Ideal R}
Goal: Ideal.map f I.radical ≤ (Ideal.map f I).radical
-- Ideal.le_comap_mul
theorem le_comap_mul : comap f K * comap f L ≤ comap f (K * L) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K L : Ideal S}
Goal: Ideal.comap f K * Ideal.comap f L ≤ Ideal.comap f (K * L)
-- Ideal.le_comap_pow
theorem le_comap_pow (n : ℕ) : K.comap f ^ n ≤ (K ^ n).comap f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (n : ℕ)
Goal: Ideal.comap f K ^ n ≤ Ideal.comap f (K ^ n)
-- RingHom.mem_ker
theorem mem_ker {r} : r ∈ ker f ↔ f r = 0 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F) {r : R}
Goal: r ∈ RingHom.ker f ↔ f r = 0
-- RingHom.ker_eq
theorem ker_eq : (ker f : Set R) = Set.preimage f {0} 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F)
Goal: ↑(RingHom.ker f) = ⇑f ⁻¹' {0}
-- RingHom.ker_eq_comap_bot
theorem ker_eq_comap_bot (f : F) : ker f = Ideal.comap f ⊥ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F)
Goal: RingHom.ker f = Ideal.comap f ⊥
-- RingHom.comap_ker
theorem comap_ker (f : S →+* R) (g : T →+* S) : f.ker.comap g = ker (f.comp g) 
Arguments: {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : S →+* R) (g : T →+* S)
Goal: Ideal.comap g (RingHom.ker f) = RingHom.ker (f.comp g)
-- RingHom.not_one_mem_ker
theorem not_one_mem_ker [Nontrivial S] (f : F) : (1 : R) ∉ ker f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] [Nontrivial S] (f : F)
Goal: 1 ∉ RingHom.ker f
-- RingHom.ker_ne_top
theorem ker_ne_top [Nontrivial S] (f : F) : ker f ≠ ⊤ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] [Nontrivial S] (f : F)
Goal: RingHom.ker f ≠ ⊤
-- Pi.ker_ringHom
lemma _root_.Pi.ker_ringHom {ι : Type*} {R : ι → Type*} [∀ i, Semiring (R i)]
    (φ : ∀ i, S →+* R i) : ker (Pi.ringHom φ) = ⨅ i, ker (φ i) 
Arguments: {S : Type v} [Semiring S] {ι : Type u_3} {R : ι → Type u_4} [(i : ι) → Semiring (R i)] (φ : (i : ι) → S →+* R i)
Goal: RingHom.ker (Pi.ringHom φ) = ⨅ i, RingHom.ker (φ i)
-- RingHom.ker_rangeSRestrict
@[simp]
theorem ker_rangeSRestrict (f : R →+* S) : ker f.rangeSRestrict = ker f 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R →+* S)
Goal: RingHom.ker f.rangeSRestrict = RingHom.ker f
-- RingHom.injective_iff_ker_eq_bot
theorem injective_iff_ker_eq_bot : Function.Injective f ↔ ker f = ⊥ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F)
Goal: Function.Injective ⇑f ↔ RingHom.ker f = ⊥
-- RingHom.ker_eq_bot_iff_eq_zero
theorem ker_eq_bot_iff_eq_zero : ker f = ⊥ ↔ ∀ x, f x = 0 → x = 0 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F)
Goal: RingHom.ker f = ⊥ ↔ ∀ (x : R), f x = 0 → x = 0
-- RingHom.ker_coe_equiv
@[simp]
theorem ker_coe_equiv (f : R ≃+* S) : ker (f : R →+* S) = ⊥ 
Arguments: {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R ≃+* S)
Goal: RingHom.ker ↑f = ⊥
-- RingHom.ker_equiv
@[simp]
theorem ker_equiv {F' : Type*} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') : ker f = ⊥ 
Arguments: {R : Type u} {S : Type v} [Ring R] [Semiring S] {F' : Type u_2} [EquivLike F' R S] [RingEquivClass F' R S] (f : F')
Goal: RingHom.ker f = ⊥
-- RingHom.sub_mem_ker_iff
theorem sub_mem_ker_iff {x y} : x - y ∈ ker f ↔ f x = f y 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {x y : R}
Goal: x - y ∈ RingHom.ker f ↔ f x = f y
-- RingHom.ker_rangeRestrict
@[simp]
theorem ker_rangeRestrict (f : R →+* S) : ker f.rangeRestrict = ker f 
Arguments: {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S)
Goal: RingHom.ker f.rangeRestrict = RingHom.ker f
-- RingHom.ker_isPrime
theorem ker_isPrime {F : Type*} [Ring R] [Ring S] [IsDomain S]
    [FunLike F R S] [RingHomClass F R S] (f : F) :
    (ker f).IsPrime 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [IsDomain S] [FunLike F R S] [RingHomClass F R S] (f : F)
Goal: (RingHom.ker f).IsPrime
-- RingHom.ker_isMaximal_of_surjective
theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K]
    [FunLike F R K] [RingHomClass F R K] (f : F)
    (hf : Function.Surjective f) : (ker f).IsMaximal 
Arguments: {R : Type u_1} {K : Type u_2} {F : Type u_3} [Ring R] [Field K] [FunLike F R K] [RingHomClass F R K] (f : F) (hf : Function.Surjective ⇑f)
Goal: (RingHom.ker f).IsMaximal
-- Ideal.map_eq_bot_iff_le_ker
theorem map_eq_bot_iff_le_ker {I : Ideal R} (f : F) : I.map f = ⊥ ↔ I ≤ RingHom.ker f 
Arguments: {R : Type u_1} {S : Type u_2} {F : Type u_3} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {I : Ideal R} (f : F)
Goal: Ideal.map f I = ⊥ ↔ I ≤ RingHom.ker f
-- Ideal.ker_le_comap
theorem ker_le_comap {K : Ideal S} (f : F) : RingHom.ker f ≤ comap f K 
Arguments: {R : Type u_1} {S : Type u_2} {F : Type u_3} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {K : Ideal S} (f : F)
Goal: RingHom.ker f ≤ Ideal.comap f K
-- Ideal.map_isPrime_of_equiv
theorem map_isPrime_of_equiv {F' : Type*} [EquivLike F' R S] [RingEquivClass F' R S]
    (f : F') {I : Ideal R} [IsPrime I] : IsPrime (map f I) 
Arguments: {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] {F' : Type u_4} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') {I : Ideal R} [I.IsPrime]
Goal: (Ideal.map f I).IsPrime
-- Ideal.map_sInf
theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
    (∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) 
Arguments: {R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] {A : Set (Ideal R)} {f : F} (hf : Function.Surjective ⇑f)
Goal: (∀ J ∈ A, RingHom.ker f ≤ J) → Ideal.map f (sInf A) = sInf (Ideal.map f '' A)
-- Ideal.map_isPrime_of_surjective
theorem map_isPrime_of_surjective {f : F} (hf : Function.Surjective f) {I : Ideal R} [H : IsPrime I]
    (hk : RingHom.ker f ≤ I) : IsPrime (map f I) 
Arguments: {R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} (hf : Function.Surjective ⇑f) {I : Ideal R} [H : I.IsPrime] (hk : RingHom.ker f ≤ I)
Goal: (Ideal.map f I).IsPrime
-- Ideal.map_eq_bot_iff_of_injective
theorem map_eq_bot_iff_of_injective {I : Ideal R} {f : F} (hf : Function.Injective f) :
    I.map f = ⊥ ↔ I = ⊥ 
Arguments: {R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] {I : Ideal R} {f : F} (hf : Function.Injective ⇑f)
Goal: Ideal.map f I = ⊥ ↔ I = ⊥
-- Ideal.map_eq_iff_sup_ker_eq_of_surjective
theorem map_eq_iff_sup_ker_eq_of_surjective {I J : Ideal R} (f : R →+* S)
    (hf : Function.Surjective f) : map f I = map f J ↔ I ⊔ RingHom.ker f = J ⊔ RingHom.ker f 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {I J : Ideal R} (f : R →+* S) (hf : Function.Surjective ⇑f)
Goal: Ideal.map f I = Ideal.map f J ↔ I ⊔ RingHom.ker f = J ⊔ RingHom.ker f
-- Ideal.map_radical_of_surjective
theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
    (h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {f : R →+* S} (hf : Function.Surjective ⇑f) {I : Ideal R} (h : RingHom.ker f ≤ I)
Goal: Ideal.map f I.radical = (Ideal.map f I).radical
-- RingHom.liftOfRightInverseAux_comp_apply
@[simp]
theorem liftOfRightInverseAux_comp_apply (hf : Function.RightInverse f_inv f) (g : A →+* C)
    (hg : RingHom.ker f ≤ RingHom.ker g) (a : A) :
    (f.liftOfRightInverseAux f_inv hf g hg) (f a) = g a 
Arguments: {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : RingHom.ker f ≤ RingHom.ker g) (a : A)
Goal: (f.liftOfRightInverseAux f_inv hf g hg) (f a) = g a