File: Mathlib/Algebra/Module/LocalizedModuleIntegers.lean

full_name: IsLocalizedModule.IsInteger
code: /-- Given `x : M'`, `M'` a localization of `M` via `f`, `IsInteger f x` iff `x` is in the image of
the localization map `f`. -/
def IsInteger (x : M') : Prop 
goal: Prop

full_name: IsLocalizedModule.isInteger_zero
code: lemma isInteger_zero : IsInteger f (0 : M') 
goal: IsLocalizedModule.IsInteger f 0

full_name: IsLocalizedModule.isInteger_add
code: theorem isInteger_add {x y : M'} (hx : IsInteger f x) (hy : IsInteger f y) : IsInteger f (x + y) 
goal: IsLocalizedModule.IsInteger f (x + y)

full_name: IsLocalizedModule.isInteger_smul
code: theorem isInteger_smul {a : R} {x : M'} (hx : IsInteger f x) : IsInteger f (a • x) 
goal: IsLocalizedModule.IsInteger f (a • x)

full_name: IsLocalizedModule.exists_integer_multiple
code: /-- Each element `x : M'` has an `S`-multiple which is an integer. -/
theorem exists_integer_multiple (x : M') : ∃ a : S, IsInteger f (a.val • x) 
goal: ∃ a, IsLocalizedModule.IsInteger f (↑a • x)

full_name: IsLocalizedModule.exist_integer_multiples
code: /-- We can clear the denominators of a `Finset`-indexed family of fractions. -/
theorem exist_integer_multiples {ι : Type*} (s : Finset ι) (g : ι → M') :
    ∃ b : S, ∀ i ∈ s, IsInteger f (b.val • g i) 
goal: ∃ b, ∀ i ∈ s, IsLocalizedModule.IsInteger f (↑b • g i)

full_name: IsLocalizedModule.exist_integer_multiples_of_finite
code: /-- We can clear the denominators of a finite indexed family of fractions. -/
theorem exist_integer_multiples_of_finite {ι : Type*} [Finite ι] (g : ι → M') :
    ∃ b : S, ∀ i, IsInteger f ((b : R) • g i) 
goal: ∃ b, ∀ (i : ι), IsLocalizedModule.IsInteger f (↑b • g i)

full_name: IsLocalizedModule.exist_integer_multiples_of_finset
code: /-- We can clear the denominators of a finite set of fractions. -/
theorem exist_integer_multiples_of_finset (s : Finset M') :
    ∃ b : S, ∀ a ∈ s, IsInteger f ((b : R) • a) 
goal: ∃ b, ∀ a ∈ s, IsLocalizedModule.IsInteger f (↑b • a)

full_name: IsLocalizedModule.commonDenom
code: /-- A choice of a common multiple of the denominators of a `Finset`-indexed family of fractions. -/
noncomputable def commonDenom {ι : Type*} (s : Finset ι) (g : ι → M') : S 
goal: ↥S

full_name: IsLocalizedModule.integerMultiple
code: /-- The numerator of a fraction after clearing the denominators
of a `Finset`-indexed family of fractions. -/
noncomputable def integerMultiple {ι : Type*} (s : Finset ι) (g : ι → M') (i : s) : M 
goal: M

full_name: IsLocalizedModule.map_integerMultiple
code: @[simp]
theorem map_integerMultiple {ι : Type*} (s : Finset ι) (g : ι → M') (i : s) :
    f (integerMultiple S f s g i) = commonDenom S f s g • g i 
goal: f (IsLocalizedModule.integerMultiple S f s g i) = IsLocalizedModule.commonDenom S f s g • g ↑i

full_name: IsLocalizedModule.commonDenomOfFinset
code: /-- A choice of a common multiple of the denominators of a finite set of fractions. -/
noncomputable def commonDenomOfFinset (s : Finset M') : S 
goal: ↥S

full_name: IsLocalizedModule.finsetIntegerMultiple
code: /-- The finset of numerators after clearing the denominators of a finite set of fractions. -/
noncomputable def finsetIntegerMultiple [DecidableEq M] (s : Finset M') : Finset M 
goal: Finset M

full_name: IsLocalizedModule.finsetIntegerMultiple_image
code: theorem finsetIntegerMultiple_image [DecidableEq M] (s : Finset M') :
    f '' finsetIntegerMultiple S f s = commonDenomOfFinset S f s • (s : Set M') 
goal: ⇑f '' ↑(IsLocalizedModule.finsetIntegerMultiple S f s) = IsLocalizedModule.commonDenomOfFinset S f s • ↑s

full_name: IsLocalizedModule.smul_mem_finsetIntegerMultiple_span
code: theorem smul_mem_finsetIntegerMultiple_span [DecidableEq M] (x : M) (s : Finset M')
    (hx : f x ∈ Submodule.span R s) :
    ∃ (m : S), m • x ∈ Submodule.span R (IsLocalizedModule.finsetIntegerMultiple S f s) 
goal: ∃ m, m • x ∈ Submodule.span R ↑(IsLocalizedModule.finsetIntegerMultiple S f s)