ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsFractional.map
theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} :
    IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
  
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P →ₐ[R] P') {I : Submodule R P}
Goal: IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I)
-- FractionalIdeal.coe_map
@[simp, norm_cast]
theorem coe_map (g : P →ₐ[R] P') (I : FractionalIdeal S P) :
    ↑(map g I) = Submodule.map g.toLinearMap I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P →ₐ[R] P') (I : FractionalIdeal S P)
Goal: ↑(FractionalIdeal.map g I) = Submodule.map g.toLinearMap ↑I
-- FractionalIdeal.mem_map
@[simp]
theorem mem_map {I : FractionalIdeal S P} {g : P →ₐ[R] P'} {y : P'} :
    y ∈ I.map g ↔ ∃ x, x ∈ I ∧ g x = y 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] {I : FractionalIdeal S P} {g : P →ₐ[R] P'} {y : P'}
Goal: y ∈ FractionalIdeal.map g I ↔ ∃ x ∈ I, g x = y
-- FractionalIdeal.map_id
@[simp]
theorem map_id : I.map (AlgHom.id _ _) = I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] (I : FractionalIdeal S P)
Goal: FractionalIdeal.map (AlgHom.id R P) I = I
-- FractionalIdeal.map_comp
@[simp]
theorem map_comp (g' : P' →ₐ[R] P'') : I.map (g'.comp g) = (I.map g).map g' 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] {P'' : Type u_4} [CommRing P''] [Algebra R P''] (I : FractionalIdeal S P) (g : P →ₐ[R] P') (g' : P' →ₐ[R] P'')
Goal: FractionalIdeal.map (g'.comp g) I = FractionalIdeal.map g' (FractionalIdeal.map g I)
-- FractionalIdeal.map_coeIdeal
@[simp, norm_cast]
theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P →ₐ[R] P') (I : Ideal R)
Goal: FractionalIdeal.map g ↑I = ↑I
-- FractionalIdeal.map_one
@[simp]
theorem map_one : (1 : FractionalIdeal S P).map g = 1 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P →ₐ[R] P')
Goal: FractionalIdeal.map g 1 = 1
-- FractionalIdeal.map_zero
@[simp]
theorem map_zero : (0 : FractionalIdeal S P).map g = 0 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P →ₐ[R] P')
Goal: FractionalIdeal.map g 0 = 0
-- FractionalIdeal.map_add
@[simp]
theorem map_add : (I + J).map g = I.map g + J.map g 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (I J : FractionalIdeal S P) (g : P →ₐ[R] P')
Goal: FractionalIdeal.map g (I + J) = FractionalIdeal.map g I + FractionalIdeal.map g J
-- FractionalIdeal.map_mul
@[simp]
theorem map_mul : (I * J).map g = I.map g * J.map g 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (I J : FractionalIdeal S P) (g : P →ₐ[R] P')
Goal: FractionalIdeal.map g (I * J) = FractionalIdeal.map g I * FractionalIdeal.map g J
-- FractionalIdeal.map_map_symm
@[simp]
theorem map_map_symm (g : P ≃ₐ[R] P') : (I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (I : FractionalIdeal S P) (g : P ≃ₐ[R] P')
Goal: FractionalIdeal.map (↑g.symm) (FractionalIdeal.map (↑g) I) = I
-- FractionalIdeal.map_symm_map
@[simp]
theorem map_symm_map (I : FractionalIdeal S P') (g : P ≃ₐ[R] P') :
    (I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (I : FractionalIdeal S P') (g : P ≃ₐ[R] P')
Goal: FractionalIdeal.map (↑g) (FractionalIdeal.map (↑g.symm) I) = I
-- FractionalIdeal.map_mem_map
theorem map_mem_map {f : P →ₐ[R] P'} (h : Function.Injective f) {x : P} {I : FractionalIdeal S P} :
    f x ∈ map f I ↔ x ∈ I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] {f : P →ₐ[R] P'} (h : Function.Injective ⇑f) {x : P} {I : FractionalIdeal S P}
Goal: f x ∈ FractionalIdeal.map f I ↔ x ∈ I
-- FractionalIdeal.map_injective
theorem map_injective (f : P →ₐ[R] P') (h : Function.Injective f) :
    Function.Injective (map f : FractionalIdeal S P → FractionalIdeal S P') 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (f : P →ₐ[R] P') (h : Function.Injective ⇑f)
Goal: Function.Injective (FractionalIdeal.map f)
-- FractionalIdeal.coeFun_mapEquiv
@[simp]
theorem coeFun_mapEquiv (g : P ≃ₐ[R] P') :
    (mapEquiv g : FractionalIdeal S P → FractionalIdeal S P') = map g 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P ≃ₐ[R] P')
Goal: ⇑(FractionalIdeal.mapEquiv g) = FractionalIdeal.map ↑g
-- FractionalIdeal.mapEquiv_apply
@[simp]
theorem mapEquiv_apply (g : P ≃ₐ[R] P') (I : FractionalIdeal S P) : mapEquiv g I = map (↑g) I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P ≃ₐ[R] P') (I : FractionalIdeal S P)
Goal: (FractionalIdeal.mapEquiv g) I = FractionalIdeal.map (↑g) I
-- FractionalIdeal.mapEquiv_symm
@[simp]
theorem mapEquiv_symm (g : P ≃ₐ[R] P') :
    ((mapEquiv g).symm : FractionalIdeal S P' ≃+* _) = mapEquiv g.symm 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {P' : Type u_3} [CommRing P'] [Algebra R P'] (g : P ≃ₐ[R] P')
Goal: (FractionalIdeal.mapEquiv g).symm = FractionalIdeal.mapEquiv g.symm
-- FractionalIdeal.mapEquiv_refl
@[simp]
theorem mapEquiv_refl : mapEquiv AlgEquiv.refl = RingEquiv.refl (FractionalIdeal S P) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P]
Goal: FractionalIdeal.mapEquiv AlgEquiv.refl = RingEquiv.refl (FractionalIdeal S P)
-- FractionalIdeal.isFractional_span_iff
theorem isFractional_span_iff {s : Set P} :
    IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {s : Set P}
Goal: IsFractional S (Submodule.span R s) ↔ ∃ a ∈ S, ∀ b ∈ s, IsLocalization.IsInteger R (a • b)
-- FractionalIdeal.isFractional_of_fg
theorem isFractional_of_fg {I : Submodule R P} (hI : I.FG) : IsFractional S I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {I : Submodule R P} (hI : I.FG)
Goal: IsFractional S I
-- FractionalIdeal.mem_span_mul_finite_of_mem_mul
theorem mem_span_mul_finite_of_mem_mul {I J : FractionalIdeal S P} {x : P} (hx : x ∈ I * J) :
    ∃ T T' : Finset P, (T : Set P) ⊆ I ∧ (T' : Set P) ⊆ J ∧ x ∈ span R (T * T' : Set P) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] {I J : FractionalIdeal S P} {x : P} (hx : x ∈ I * J)
Goal: ∃ T T', ↑T ⊆ ↑I ∧ ↑T' ⊆ ↑J ∧ x ∈ Submodule.span R (↑T * ↑T')
-- FractionalIdeal.coeIdeal_fg
theorem coeIdeal_fg (inj : Function.Injective (algebraMap R P)) (I : Ideal R) :
    FG ((I : FractionalIdeal S P) : Submodule R P) ↔ I.FG 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) {P : Type u_2} [CommRing P] [Algebra R P] (inj : Function.Injective ⇑(algebraMap R P)) (I : Ideal R)
Goal: (↑↑I).FG ↔ I.FG
-- FractionalIdeal.fg_unit
theorem fg_unit (I : (FractionalIdeal S P)ˣ) : FG (I : Submodule R P) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] (I : (FractionalIdeal S P)ˣ)
Goal: (↑↑I).FG
-- FractionalIdeal.fg_of_isUnit
theorem fg_of_isUnit (I : FractionalIdeal S P) (h : IsUnit I) : FG (I : Submodule R P) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] (I : FractionalIdeal S P) (h : IsUnit I)
Goal: (↑I).FG
-- Ideal.fg_of_isUnit
theorem _root_.Ideal.fg_of_isUnit (inj : Function.Injective (algebraMap R P)) (I : Ideal R)
    (h : IsUnit (I : FractionalIdeal S P)) : I.FG 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] (inj : Function.Injective ⇑(algebraMap R P)) (I : Ideal R) (h : IsUnit ↑I)
Goal: I.FG
-- FractionalIdeal.canonicalEquiv_def
noncomputable irreducible_def canonicalEquiv : FractionalIdeal S P ≃+* FractionalIdeal S P' 
Arguments: {R : Type u_5} [CommRing R] (S : Submonoid R) (P : Type u_6) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (P' : Type u_7) [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P']
Goal: FractionalIdeal.canonicalEquiv S P P' =
  FractionalIdeal.mapEquiv
    (let __src := IsLocalization.ringEquivOfRingEquiv P P' (RingEquiv.refl R) ⋯;
    { toEquiv := __src.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ })
-- FractionalIdeal.mem_canonicalEquiv_apply
@[simp]
theorem mem_canonicalEquiv_apply {I : FractionalIdeal S P} {x : P'} :
    x ∈ canonicalEquiv S P P' I ↔
      ∃ y ∈ I,
        IsLocalization.map P' (RingHom.id R) (fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy)
            (y : P) =
          x 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (P' : Type u_3) [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P'] {I : FractionalIdeal S P} {x : P'}
Goal: x ∈ (FractionalIdeal.canonicalEquiv S P P') I ↔ ∃ y ∈ I, (IsLocalization.map P' (RingHom.id R) ⋯) y = x
-- FractionalIdeal.canonicalEquiv_symm
@[simp]
theorem canonicalEquiv_symm : (canonicalEquiv S P P').symm = canonicalEquiv S P' P 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (P' : Type u_3) [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P']
Goal: (FractionalIdeal.canonicalEquiv S P P').symm = FractionalIdeal.canonicalEquiv S P' P
-- FractionalIdeal.canonicalEquiv_flip
theorem canonicalEquiv_flip (I) : canonicalEquiv S P P' (canonicalEquiv S P' P I) = I 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (P' : Type u_3) [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P'] (I : FractionalIdeal S P')
Goal: (FractionalIdeal.canonicalEquiv S P P') ((FractionalIdeal.canonicalEquiv S P' P) I) = I
-- FractionalIdeal.canonicalEquiv_canonicalEquiv
@[simp]
theorem canonicalEquiv_canonicalEquiv (P'' : Type*) [CommRing P''] [Algebra R P'']
    [IsLocalization S P''] (I : FractionalIdeal S P) :
    canonicalEquiv S P' P'' (canonicalEquiv S P P' I) = canonicalEquiv S P P'' I 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (P' : Type u_3) [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P'] (P'' : Type u_5) [CommRing P''] [Algebra R P''] [IsLocalization S P''] (I : FractionalIdeal S P)
Goal: (FractionalIdeal.canonicalEquiv S P' P'') ((FractionalIdeal.canonicalEquiv S P P') I) =
  (FractionalIdeal.canonicalEquiv S P P'') I
-- FractionalIdeal.canonicalEquiv_trans_canonicalEquiv
theorem canonicalEquiv_trans_canonicalEquiv (P'' : Type*) [CommRing P''] [Algebra R P'']
    [IsLocalization S P''] :
    (canonicalEquiv S P P').trans (canonicalEquiv S P' P'') = canonicalEquiv S P P'' 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (P' : Type u_3) [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P'] (P'' : Type u_5) [CommRing P''] [Algebra R P''] [IsLocalization S P'']
Goal: (FractionalIdeal.canonicalEquiv S P P').trans (FractionalIdeal.canonicalEquiv S P' P'') =
  FractionalIdeal.canonicalEquiv S P P''
-- FractionalIdeal.canonicalEquiv_coeIdeal
@[simp]
theorem canonicalEquiv_coeIdeal (I : Ideal R) : canonicalEquiv S P P' I = I 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (P' : Type u_3) [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P'] (I : Ideal R)
Goal: (FractionalIdeal.canonicalEquiv S P P') ↑I = ↑I
-- FractionalIdeal.canonicalEquiv_self
@[simp]
theorem canonicalEquiv_self : canonicalEquiv S P P = RingEquiv.refl _ 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P]
Goal: FractionalIdeal.canonicalEquiv S P P = RingEquiv.refl (FractionalIdeal S P)
-- FractionalIdeal.exists_ne_zero_mem_isInteger
theorem exists_ne_zero_mem_isInteger [Nontrivial R] (hI : I ≠ 0) :
    ∃ x, x ≠ 0 ∧ algebraMap R K x ∈ I 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K] {I : FractionalIdeal R⁰ K} [Nontrivial R] (hI : I ≠ 0)
Goal: ∃ x, x ≠ 0 ∧ (algebraMap R K) x ∈ I
-- FractionalIdeal.map_ne_zero
theorem map_ne_zero [Nontrivial R] (hI : I ≠ 0) : I.map h ≠ 0 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} {K' : Type u_4} [Field K] [Field K'] [Algebra R K] [IsFractionRing R K] [Algebra R K'] [IsFractionRing R K'] {I : FractionalIdeal R⁰ K} (h : K →ₐ[R] K') [Nontrivial R] (hI : I ≠ 0)
Goal: FractionalIdeal.map h I ≠ 0
-- FractionalIdeal.map_eq_zero_iff
@[simp]
theorem map_eq_zero_iff [Nontrivial R] : I.map h = 0 ↔ I = 0 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} {K' : Type u_4} [Field K] [Field K'] [Algebra R K] [IsFractionRing R K] [Algebra R K'] [IsFractionRing R K'] {I : FractionalIdeal R⁰ K} (h : K →ₐ[R] K') [Nontrivial R]
Goal: FractionalIdeal.map h I = 0 ↔ I = 0
-- FractionalIdeal.coeIdeal_injective
theorem coeIdeal_injective : Function.Injective (fun (I : Ideal R) ↦ (I : FractionalIdeal R⁰ K)) 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K]
Goal: Function.Injective fun I => ↑I
-- FractionalIdeal.coeIdeal_inj
theorem coeIdeal_inj {I J : Ideal R} :
    (I : FractionalIdeal R⁰ K) = (J : FractionalIdeal R⁰ K) ↔ I = J 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K] {I J : Ideal R}
Goal: ↑I = ↑J ↔ I = J
-- FractionalIdeal.coeIdeal_eq_zero
@[simp]
theorem coeIdeal_eq_zero {I : Ideal R} : (I : FractionalIdeal R⁰ K) = 0 ↔ I = ⊥ 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K] {I : Ideal R}
Goal: ↑I = 0 ↔ I = ⊥
-- FractionalIdeal.coeIdeal_ne_zero
theorem coeIdeal_ne_zero {I : Ideal R} : (I : FractionalIdeal R⁰ K) ≠ 0 ↔ I ≠ ⊥ 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K] {I : Ideal R}
Goal: ↑I ≠ 0 ↔ I ≠ ⊥
-- FractionalIdeal.coeIdeal_eq_one
@[simp]
theorem coeIdeal_eq_one {I : Ideal R} : (I : FractionalIdeal R⁰ K) = 1 ↔ I = 1 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K] {I : Ideal R}
Goal: ↑I = 1 ↔ I = 1
-- FractionalIdeal.coeIdeal_ne_one
theorem coeIdeal_ne_one {I : Ideal R} : (I : FractionalIdeal R⁰ K) ≠ 1 ↔ I ≠ 1 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K] {I : Ideal R}
Goal: ↑I ≠ 1 ↔ I ≠ 1
-- FractionalIdeal.num_eq_zero_iff
theorem num_eq_zero_iff [Nontrivial R] {I : FractionalIdeal R⁰ K} : I.num = 0 ↔ I = 0 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_3} [Field K] [Algebra R K] [IsFractionRing R K] [Nontrivial R] {I : FractionalIdeal R⁰ K}
Goal: I.num = 0 ↔ I = 0
-- FractionalIdeal.ne_zero_of_mul_eq_one
theorem ne_zero_of_mul_eq_one (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1) : I ≠ 0 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1)
Goal: I ≠ 0
-- IsFractional.div_of_nonzero
theorem _root_.IsFractional.div_of_nonzero {I J : Submodule R₁ K} :
    IsFractional R₁⁰ I → IsFractional R₁⁰ J → J ≠ 0 → IsFractional R₁⁰ (I / J)
  
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I J : Submodule R₁ K}
Goal: IsFractional R₁⁰ I → IsFractional R₁⁰ J → J ≠ 0 → IsFractional R₁⁰ (I / J)
-- FractionalIdeal.fractional_div_of_nonzero
theorem fractional_div_of_nonzero {I J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
    IsFractional R₁⁰ (I / J : Submodule R₁ K) 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I J : FractionalIdeal R₁⁰ K} (h : J ≠ 0)
Goal: IsFractional R₁⁰ (↑I / ↑J)
-- FractionalIdeal.div_zero
@[simp]
theorem div_zero {I : FractionalIdeal R₁⁰ K} : I / 0 = 0 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I : FractionalIdeal R₁⁰ K}
Goal: I / 0 = 0
-- FractionalIdeal.div_nonzero
theorem div_nonzero {I J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
    I / J = ⟨I / J, fractional_div_of_nonzero h⟩ 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I J : FractionalIdeal R₁⁰ K} (h : J ≠ 0)
Goal: I / J = ⟨↑I / ↑J, ⋯⟩
-- FractionalIdeal.coe_div
@[simp]
theorem coe_div {I J : FractionalIdeal R₁⁰ K} (hJ : J ≠ 0) :
    (↑(I / J) : Submodule R₁ K) = ↑I / (↑J : Submodule R₁ K) 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I J : FractionalIdeal R₁⁰ K} (hJ : J ≠ 0)
Goal: ↑(I / J) = ↑I / ↑J
-- FractionalIdeal.mem_div_iff_of_nonzero
theorem mem_div_iff_of_nonzero {I J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) {x} :
    x ∈ I / J ↔ ∀ y ∈ J, x * y ∈ I 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) {x : K}
Goal: x ∈ I / J ↔ ∀ y ∈ J, x * y ∈ I
-- FractionalIdeal.mul_one_div_le_one
theorem mul_one_div_le_one {I : FractionalIdeal R₁⁰ K} : I * (1 / I) ≤ 1 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I : FractionalIdeal R₁⁰ K}
Goal: I * (1 / I) ≤ 1
-- FractionalIdeal.le_self_mul_one_div
theorem le_self_mul_one_div {I : FractionalIdeal R₁⁰ K} (hI : I ≤ (1 : FractionalIdeal R₁⁰ K)) :
    I ≤ I * (1 / I) 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I : FractionalIdeal R₁⁰ K} (hI : I ≤ 1)
Goal: I ≤ I * (1 / I)
-- FractionalIdeal.le_div_iff_of_nonzero
theorem le_div_iff_of_nonzero {I J J' : FractionalIdeal R₁⁰ K} (hJ' : J' ≠ 0) :
    I ≤ J / J' ↔ ∀ x ∈ I, ∀ y ∈ J', x * y ∈ J 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I J J' : FractionalIdeal R₁⁰ K} (hJ' : J' ≠ 0)
Goal: I ≤ J / J' ↔ ∀ x ∈ I, ∀ y ∈ J', x * y ∈ J
-- FractionalIdeal.le_div_iff_mul_le
theorem le_div_iff_mul_le {I J J' : FractionalIdeal R₁⁰ K} (hJ' : J' ≠ 0) :
    I ≤ J / J' ↔ I * J' ≤ J 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I J J' : FractionalIdeal R₁⁰ K} (hJ' : J' ≠ 0)
Goal: I ≤ J / J' ↔ I * J' ≤ J
-- FractionalIdeal.div_one
@[simp]
theorem div_one {I : FractionalIdeal R₁⁰ K} : I / 1 = I 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I : FractionalIdeal R₁⁰ K}
Goal: I / 1 = I
-- FractionalIdeal.eq_one_div_of_mul_eq_one_right
theorem eq_one_div_of_mul_eq_one_right (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1) :
    J = 1 / I 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1)
Goal: J = 1 / I
-- FractionalIdeal.mul_div_self_cancel_iff
theorem mul_div_self_cancel_iff {I : FractionalIdeal R₁⁰ K} : I * (1 / I) = 1 ↔ ∃ J, I * J = 1 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {I : FractionalIdeal R₁⁰ K}
Goal: I * (1 / I) = 1 ↔ ∃ J, I * J = 1
-- FractionalIdeal.map_div
@[simp]
theorem map_div (I J : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
    (I / J).map (h : K →ₐ[R₁] K') = I.map h / J.map h 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {K' : Type u_5} [Field K'] [Algebra R₁ K'] [IsFractionRing R₁ K'] (I J : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K')
Goal: FractionalIdeal.map (↑h) (I / J) = FractionalIdeal.map (↑h) I / FractionalIdeal.map (↑h) J
-- FractionalIdeal.map_one_div
theorem map_one_div (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
    (1 / I).map (h : K →ₐ[R₁] K') = 1 / I.map h 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] {K' : Type u_5} [Field K'] [Algebra R₁ K'] [IsFractionRing R₁ K'] (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K')
Goal: FractionalIdeal.map (↑h) (1 / I) = 1 / FractionalIdeal.map (↑h) I
-- FractionalIdeal.eq_zero_or_one
theorem eq_zero_or_one (I : FractionalIdeal K⁰ L) : I = 0 ∨ I = 1 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] [IsFractionRing K L] (I : FractionalIdeal K⁰ L)
Goal: I = 0 ∨ I = 1
-- FractionalIdeal.eq_zero_or_one_of_isField
theorem eq_zero_or_one_of_isField (hF : IsField R₁) (I : FractionalIdeal R₁⁰ K) : I = 0 ∨ I = 1 
Arguments: {R₁ : Type u_3} {K : Type u_4} [CommRing R₁] [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] (hF : IsField R₁) (I : FractionalIdeal R₁⁰ K)
Goal: I = 0 ∨ I = 1
-- FractionalIdeal.spanFinset_coe
@[simp] lemma spanFinset_coe {ι : Type*} (s : Finset ι) (f : ι → K) :
    (spanFinset R₁ s f : Submodule R₁ K) = Submodule.span R₁ (f '' s) 
Arguments: (R₁ : Type u_3) [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] {ι : Type u_5} (s : Finset ι) (f : ι → K)
Goal: ↑(FractionalIdeal.spanFinset R₁ s f) = Submodule.span R₁ (f '' ↑s)
-- FractionalIdeal.spanFinset_eq_zero
@[simp]
theorem spanFinset_eq_zero {ι : Type*} {s : Finset ι} {f : ι → K} :
    spanFinset R₁ s f = 0 ↔ ∀ j ∈ s, f j = 0 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] {ι : Type u_5} {s : Finset ι} {f : ι → K}
Goal: FractionalIdeal.spanFinset R₁ s f = 0 ↔ ∀ j ∈ s, f j = 0
-- FractionalIdeal.spanFinset_ne_zero
theorem spanFinset_ne_zero {ι : Type*} {s : Finset ι} {f : ι → K} :
    spanFinset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f j ≠ 0 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] {ι : Type u_5} {s : Finset ι} {f : ι → K}
Goal: FractionalIdeal.spanFinset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f j ≠ 0
-- FractionalIdeal.isFractional_span_singleton
theorem isFractional_span_singleton (x : P) : IsFractional S (span R {x} : Submodule R P) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P)
Goal: IsFractional S (Submodule.span R {x})
-- FractionalIdeal.spanSingleton_def
irreducible_def spanSingleton (x : P) : FractionalIdeal S P 
Arguments: {R : Type u_5} [CommRing R] (S : Submonoid R) {P : Type u_6} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P)
Goal: FractionalIdeal.spanSingleton S x = ⟨Submodule.span R {x}, ⋯⟩
-- FractionalIdeal.coe_spanSingleton
@[simp]
theorem coe_spanSingleton (x : P) : (spanSingleton S x : Submodule R P) = span R {x} 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P)
Goal: ↑(FractionalIdeal.spanSingleton S x) = Submodule.span R {x}