fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
add_mem (x y : K) : x ∈ A → y ∈ A → x + y ∈ A := A.toSubring.add_mem | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | add_mem | null |
mul_mem (x y : K) : x ∈ A → y ∈ A → x * y ∈ A := A.toSubring.mul_mem | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mul_mem | null |
neg_mem (x : K) : x ∈ A → -x ∈ A := A.toSubring.neg_mem | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | neg_mem | null |
mem_or_inv_mem (x : K) : x ∈ A ∨ x⁻¹ ∈ A := A.mem_or_inv_mem' _ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_or_inv_mem | null |
toSubring_injective : Function.Injective (toSubring : ValuationSubring K → Subring K) :=
fun x y h => by cases x; cases y; congr | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | toSubring_injective | null |
mem_top (x : K) : x ∈ (⊤ : ValuationSubring K) :=
trivial | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_top | null |
le_top : A ≤ ⊤ := fun _a _ha => mem_top _ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | le_top | null |
isLocalRing : IsLocalRing A := inferInstance
@[simp] | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | isLocalRing | null |
algebraMap_apply (a : A) : algebraMap A K a = a := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | algebraMap_apply | null |
ValueGroup :=
ValuationRing.ValueGroup A K
deriving LinearOrderedCommGroupWithZero | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ValueGroup | The value group of the valuation associated to `A`. Note: it is actually a group with zero. |
valuation : Valuation K A.ValueGroup :=
ValuationRing.valuation A K | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation | Any valuation subring of `K` induces a natural valuation on `K`. |
inhabitedValueGroup : Inhabited A.ValueGroup := ⟨A.valuation 0⟩ | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | inhabitedValueGroup | null |
valuation_le_one (a : A) : A.valuation a ≤ 1 :=
(ValuationRing.mem_integer_iff A K _).2 ⟨a, rfl⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_le_one | null |
mem_of_valuation_le_one (x : K) (h : A.valuation x ≤ 1) : x ∈ A :=
let ⟨a, ha⟩ := (ValuationRing.mem_integer_iff A K x).1 h
ha ▸ a.2 | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_of_valuation_le_one | null |
valuation_le_one_iff (x : K) : A.valuation x ≤ 1 ↔ x ∈ A :=
⟨mem_of_valuation_le_one _ _, fun ha => A.valuation_le_one ⟨x, ha⟩⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_le_one_iff | null |
valuation_eq_iff (x y : K) : A.valuation x = A.valuation y ↔ ∃ a : Aˣ, (a : K) * y = x :=
Quotient.eq'' | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_eq_iff | null |
valuation_le_iff (x y : K) : A.valuation x ≤ A.valuation y ↔ ∃ a : A, (a : K) * y = x :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_le_iff | null |
valuation_surjective : Function.Surjective A.valuation := Quot.mk_surjective | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_surjective | null |
valuation_unit (a : Aˣ) : A.valuation a = 1 := by
rw [← A.valuation.map_one, valuation_eq_iff]; use a; simp | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_unit | null |
valuation_eq_one_iff (a : A) : IsUnit a ↔ A.valuation a = 1 :=
⟨fun h => A.valuation_unit h.unit, fun h => by
have ha : (a : K) ≠ 0 := by
intro c
rw [c, A.valuation.map_zero] at h
exact zero_ne_one h
have ha' : (a : K)⁻¹ ∈ A := by rw [← valuation_le_one_iff, map_inv₀, h, inv_one]
apply isUnit_of_mul_eq_one a ⟨a⁻¹, ha'⟩; ext; simp [field]⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_eq_one_iff | null |
valuation_lt_one_or_eq_one (a : A) : A.valuation a < 1 ∨ A.valuation a = 1 :=
lt_or_eq_of_le (A.valuation_le_one a) | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_lt_one_or_eq_one | null |
valuation_lt_one_iff (a : A) : a ∈ IsLocalRing.maximalIdeal A ↔ A.valuation a < 1 := by
rw [IsLocalRing.mem_maximalIdeal]
dsimp [nonunits]; rw [valuation_eq_one_iff]
exact (A.valuation_le_one a).lt_iff_ne.symm | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuation_lt_one_iff | null |
ofSubring (R : Subring K) (hR : ∀ x : K, x ∈ R ∨ x⁻¹ ∈ R) : ValuationSubring K :=
{ R with mem_or_inv_mem' := hR }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofSubring | A subring `R` of `K` such that for all `x : K` either `x ∈ R` or `x⁻¹ ∈ R` is
a valuation subring of `K`. |
mem_ofSubring (R : Subring K) (hR : ∀ x : K, x ∈ R ∨ x⁻¹ ∈ R) (x : K) :
x ∈ ofSubring R hR ↔ x ∈ R :=
Iff.refl _ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_ofSubring | null |
ofLE (R : ValuationSubring K) (S : Subring K) (h : R.toSubring ≤ S) : ValuationSubring K :=
{ S with mem_or_inv_mem' := fun x => (R.mem_or_inv_mem x).imp (@h x) (@h _) } | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofLE | An overring of a valuation ring is a valuation ring. |
inclusion (R S : ValuationSubring K) (h : R ≤ S) : R →+* S :=
Subring.inclusion h | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | inclusion | The ring homomorphism induced by the partial order. |
subtype (R : ValuationSubring K) : R →+* K :=
Subring.subtype R.toSubring
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | subtype | The canonical ring homomorphism from a valuation ring to its field of fractions. |
subtype_apply {R : ValuationSubring K} (x : R) :
R.subtype x = x := rfl | lemma | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | subtype_apply | null |
subtype_injective (R : ValuationSubring K) :
Function.Injective R.subtype :=
R.toSubring.subtype_injective
@[simp] | lemma | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | subtype_injective | null |
coe_subtype (R : ValuationSubring K) : ⇑(subtype R) = Subtype.val :=
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | coe_subtype | null |
mapOfLE (R S : ValuationSubring K) (h : R ≤ S) : R.ValueGroup →*₀ S.ValueGroup where
toFun := Quotient.map' id fun _ _ ⟨u, hu⟩ => ⟨Units.map (R.inclusion S h).toMonoidHom u, hu⟩
map_zero' := rfl
map_one' := rfl
map_mul' := by rintro ⟨⟩ ⟨⟩; rfl
@[mono] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mapOfLE | The canonical map on value groups induced by a coarsening of valuation rings. |
monotone_mapOfLE (R S : ValuationSubring K) (h : R ≤ S) : Monotone (R.mapOfLE S h) := by
rintro ⟨⟩ ⟨⟩ ⟨a, ha⟩; exact ⟨R.inclusion S h a, ha⟩
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | monotone_mapOfLE | null |
mapOfLE_comp_valuation (R S : ValuationSubring K) (h : R ≤ S) :
R.mapOfLE S h ∘ R.valuation = S.valuation := by ext; rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mapOfLE_comp_valuation | null |
mapOfLE_valuation_apply (R S : ValuationSubring K) (h : R ≤ S) (x : K) :
R.mapOfLE S h (R.valuation x) = S.valuation x := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mapOfLE_valuation_apply | null |
idealOfLE (R S : ValuationSubring K) (h : R ≤ S) : Ideal R :=
(IsLocalRing.maximalIdeal S).comap (R.inclusion S h) | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | idealOfLE | The ideal corresponding to a coarsening of a valuation ring. |
prime_idealOfLE (R S : ValuationSubring K) (h : R ≤ S) : (idealOfLE R S h).IsPrime :=
(IsLocalRing.maximalIdeal S).comap_isPrime _ | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | prime_idealOfLE | null |
ofPrime (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] : ValuationSubring K :=
ofLE A (Localization.subalgebra.ofField K _ P.primeCompl_le_nonZeroDivisors).toSubring
fun a ha => Subalgebra.mem_toSubring.mpr <|
Subalgebra.algebraMap_mem
(Localization.subalgebra.ofField K _ P.primeCompl_le_nonZeroDivisors) (⟨a, ha⟩ : A) | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofPrime | The coarsening of a valuation ring associated to a prime ideal. |
ofPrimeAlgebra (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] :
Algebra A (A.ofPrime P) :=
Subalgebra.algebra (Localization.subalgebra.ofField K _ P.primeCompl_le_nonZeroDivisors) | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofPrimeAlgebra | null |
ofPrime_scalar_tower (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] :
letI : SMul A (A.ofPrime P) := SMulZeroClass.toSMul
IsScalarTower A (A.ofPrime P) K :=
IsScalarTower.subalgebra' A K K
(Localization.subalgebra.ofField K _ P.primeCompl_le_nonZeroDivisors) | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofPrime_scalar_tower | null |
ofPrime_localization (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] :
IsLocalization.AtPrime (A.ofPrime P) P := by
apply
Localization.subalgebra.isLocalization_ofField K P.primeCompl
P.primeCompl_le_nonZeroDivisors | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofPrime_localization | null |
le_ofPrime (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] : A ≤ ofPrime A P :=
fun a ha => Subalgebra.mem_toSubring.mpr <| Subalgebra.algebraMap_mem _ (⟨a, ha⟩ : A) | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | le_ofPrime | null |
ofPrime_valuation_eq_one_iff_mem_primeCompl (A : ValuationSubring K) (P : Ideal A)
[P.IsPrime] (x : A) : (ofPrime A P).valuation x = 1 ↔ x ∈ P.primeCompl := by
rw [← IsLocalization.AtPrime.isUnit_to_map_iff (A.ofPrime P) P x, valuation_eq_one_iff]; rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofPrime_valuation_eq_one_iff_mem_primeCompl | null |
idealOfLE_ofPrime (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] :
idealOfLE A (ofPrime A P) (le_ofPrime A P) = P := by
refine Ideal.ext (fun x => ?_)
apply IsLocalization.AtPrime.to_map_mem_maximal_iff
exact isLocalRing (ofPrime A P)
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | idealOfLE_ofPrime | null |
ofPrime_idealOfLE (R S : ValuationSubring K) (h : R ≤ S) :
ofPrime R (idealOfLE R S h) = S := by
ext x; constructor
· rintro ⟨a, r, hr, rfl⟩; apply mul_mem; · exact h a.2
· rw [← valuation_le_one_iff, map_inv₀, ← inv_one, inv_le_inv₀]
· exact not_lt.1 ((not_iff_not.2 <| valuation_lt_one_iff S _).1 hr)
· simpa [Valuation.pos_iff] using fun hr₀ ↦ hr₀ ▸ hr <| Ideal.zero_mem (R.idealOfLE S h)
· exact zero_lt_one
· intro hx; by_cases hr : x ∈ R; · exact R.le_ofPrime _ hr
have : x ≠ 0 := fun h => hr (by rw [h]; exact R.zero_mem)
replace hr := (R.mem_or_inv_mem x).resolve_left hr
refine ⟨1, ⟨x⁻¹, hr⟩, ?_, ?_⟩
· simp only [Ideal.primeCompl, Submonoid.mem_mk, Subsemigroup.mem_mk, Set.mem_compl_iff,
SetLike.mem_coe, idealOfLE, Ideal.mem_comap, IsLocalRing.mem_maximalIdeal, mem_nonunits_iff,
not_not]
change IsUnit (⟨x⁻¹, h hr⟩ : S)
apply isUnit_of_mul_eq_one _ (⟨x, hx⟩ : S)
ext; simp [field]
· simp | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofPrime_idealOfLE | null |
ofPrime_le_of_le (P Q : Ideal A) [P.IsPrime] [Q.IsPrime] (h : P ≤ Q) :
ofPrime A Q ≤ ofPrime A P := fun _x ⟨a, s, hs, he⟩ => ⟨a, s, fun c => hs (h c), he⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ofPrime_le_of_le | null |
idealOfLE_le_of_le (R S : ValuationSubring K) (hR : A ≤ R) (hS : A ≤ S) (h : R ≤ S) :
idealOfLE A S hS ≤ idealOfLE A R hR := fun x hx =>
(valuation_lt_one_iff R _).2
(by
by_contra c; push_neg at c; replace c := monotone_mapOfLE R S h c
rw [(mapOfLE _ _ _).map_one, mapOfLE_valuation_apply] at c
apply not_le_of_gt ((valuation_lt_one_iff S _).1 hx) c) | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | idealOfLE_le_of_le | null |
@[simps apply]
primeSpectrumEquiv : PrimeSpectrum A ≃ {S // A ≤ S} where
toFun P := ⟨ofPrime A P.asIdeal, le_ofPrime _ _⟩
invFun S := ⟨idealOfLE _ S S.2, inferInstance⟩
left_inv P := by ext1; simp
right_inv S := by ext1; simp | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | primeSpectrumEquiv | The equivalence between coarsenings of a valuation ring and its prime ideals. |
@[simps!]
primeSpectrumOrderEquiv : (PrimeSpectrum A)ᵒᵈ ≃o {S // A ≤ S} :=
{ OrderDual.ofDual.trans (primeSpectrumEquiv A) with
map_rel_iff' {a b} :=
⟨a.rec <| fun a => b.rec <| fun b => fun h => by
simp only [OrderDual.toDual_le_toDual]
dsimp at h
have := idealOfLE_le_of_le A _ _ ?_ ?_ h
· rwa [idealOfLE_ofPrime, idealOfLE_ofPrime] at this
all_goals exact le_ofPrime A (PrimeSpectrum.asIdeal _),
fun h => by apply ofPrime_le_of_le; exact h⟩ } | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | primeSpectrumOrderEquiv | An ordered variant of `primeSpectrumEquiv`. |
le_total_ideal : IsTotal {S // A ≤ S} LE.le := by
classical
let _ : IsTotal (PrimeSpectrum A) (· ≤ ·) := ⟨fun ⟨x, _⟩ ⟨y, _⟩ => LE.isTotal.total x y⟩
exact ⟨(primeSpectrumOrderEquiv A).symm.toRelEmbedding.isTotal.total⟩
open scoped Classical in | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | le_total_ideal | null |
linearOrderOverring : LinearOrder {S // A ≤ S} where
le_total := (le_total_ideal A).1
max_def a b := congr_fun₂ sup_eq_maxDefault a b
toDecidableLE := _ | instance | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | linearOrderOverring | null |
valuationSubring : ValuationSubring K :=
{ v.integer with
mem_or_inv_mem' := by
intro x
rcases val_le_one_or_val_inv_le_one v x with h | h
exacts [Or.inl h, Or.inr h] }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuationSubring | The valuation subring associated to a valuation. |
mem_valuationSubring_iff (x : K) : x ∈ v.valuationSubring ↔ v x ≤ 1 := Iff.refl _ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_valuationSubring_iff | null |
isEquiv_iff_valuationSubring :
v₁.IsEquiv v₂ ↔ v₁.valuationSubring = v₂.valuationSubring := by
constructor
· intro h; ext x; specialize h x 1; simpa using h
· intro h; apply isEquiv_of_val_le_one
intro x
have : x ∈ v₁.valuationSubring ↔ x ∈ v₂.valuationSubring := by rw [h]
simpa using this | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | isEquiv_iff_valuationSubring | null |
isEquiv_valuation_valuationSubring : v.IsEquiv v.valuationSubring.valuation := by
rw [isEquiv_iff_val_le_one]
intro x
rw [ValuationSubring.valuation_le_one_iff]
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | isEquiv_valuation_valuationSubring | null |
valuationSubring.integers : v.Integers v.valuationSubring :=
Valuation.integer.integers _ | lemma | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuationSubring.integers | null |
@[simp]
valuationSubring_valuation : A.valuation.valuationSubring = A := by
ext; rw [← A.valuation_le_one_iff]; rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | valuationSubring_valuation | null |
unitGroup : Subgroup Kˣ :=
(A.valuation.toMonoidWithZeroHom.toMonoidHom.comp (Units.coeHom K)).ker
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitGroup | The unit group of a valuation subring, as a subgroup of `Kˣ`. |
mem_unitGroup_iff (x : Kˣ) : x ∈ A.unitGroup ↔ A.valuation x = 1 := Iff.rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_unitGroup_iff | null |
unitGroupMulEquiv : A.unitGroup ≃* Aˣ where
toFun x :=
{ val := ⟨(x : Kˣ), mem_of_valuation_le_one A _ x.prop.le⟩
inv := ⟨((x⁻¹ : A.unitGroup) : Kˣ), mem_of_valuation_le_one _ _ x⁻¹.prop.le⟩
val_inv := Subtype.ext (by simp)
inv_val := Subtype.ext (by simp) }
invFun x := ⟨Units.map A.subtype.toMonoidHom x, A.valuation_unit x⟩
map_mul' a b := by ext; rfl
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitGroupMulEquiv | For a valuation subring `A`, `A.unitGroup` agrees with the units of `A`. |
coe_unitGroupMulEquiv_apply (a : A.unitGroup) :
((A.unitGroupMulEquiv a : A) : K) = ((a : Kˣ) : K) := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | coe_unitGroupMulEquiv_apply | null |
coe_unitGroupMulEquiv_symm_apply (a : Aˣ) : ((A.unitGroupMulEquiv.symm a : Kˣ) : K) = a :=
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | coe_unitGroupMulEquiv_symm_apply | null |
unitGroup_le_unitGroup {A B : ValuationSubring K} : A.unitGroup ≤ B.unitGroup ↔ A ≤ B := by
constructor
· intro h x hx
rw [← A.valuation_le_one_iff x, le_iff_lt_or_eq] at hx
by_cases h_1 : x = 0; · simp only [h_1, zero_mem]
by_cases h_2 : 1 + x = 0
· simp only [← add_eq_zero_iff_neg_eq.1 h_2, neg_mem _ _ (one_mem _)]
rcases hx with hx | hx
· have := h (show Units.mk0 _ h_2 ∈ A.unitGroup from A.valuation.map_one_add_of_lt hx)
simpa using
B.add_mem _ _ (show 1 + x ∈ B from SetLike.coe_mem (B.unitGroupMulEquiv ⟨_, this⟩ : B))
(B.neg_mem _ B.one_mem)
· have := h (show Units.mk0 x h_1 ∈ A.unitGroup from hx)
exact SetLike.coe_mem (B.unitGroupMulEquiv ⟨_, this⟩ : B)
· rintro h x (hx : A.valuation x = 1)
apply_fun A.mapOfLE B h at hx
simpa using hx | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitGroup_le_unitGroup | null |
unitGroup_injective : Function.Injective (unitGroup : ValuationSubring K → Subgroup _) :=
fun A B h => by simpa only [le_antisymm_iff, unitGroup_le_unitGroup] using h | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitGroup_injective | null |
eq_iff_unitGroup {A B : ValuationSubring K} : A = B ↔ A.unitGroup = B.unitGroup :=
unitGroup_injective.eq_iff.symm | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | eq_iff_unitGroup | null |
unitGroupOrderEmbedding : ValuationSubring K ↪o Subgroup Kˣ where
toFun A := A.unitGroup
inj' := unitGroup_injective
map_rel_iff' {_A _B} := unitGroup_le_unitGroup | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitGroupOrderEmbedding | The map on valuation subrings to their unit groups is an order embedding. |
unitGroup_strictMono : StrictMono (unitGroup : ValuationSubring K → Subgroup _) :=
unitGroupOrderEmbedding.strictMono | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitGroup_strictMono | null |
nonunits : Subsemigroup K where
carrier := {x | A.valuation x < 1}
mul_mem' ha hb := (mul_lt_mul'' (Set.mem_setOf.mp ha) (Set.mem_setOf.mp hb)
zero_le' zero_le').trans_eq <| mul_one _ | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | nonunits | The nonunits of a valuation subring of `K`, as a subsemigroup of `K` |
mem_nonunits_iff {x : K} : x ∈ A.nonunits ↔ A.valuation x < 1 :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_nonunits_iff | null |
nonunits_le_nonunits {A B : ValuationSubring K} : B.nonunits ≤ A.nonunits ↔ A ≤ B := by
constructor
· intro h x hx
by_cases h_1 : x = 0; · simp only [h_1, zero_mem]
rw [← valuation_le_one_iff, ← not_lt, Valuation.one_lt_val_iff _ h_1] at hx ⊢
by_contra h_2; exact hx (h h_2)
· intro h x hx
by_contra h_1; exact not_lt.2 (monotone_mapOfLE _ _ h (not_lt.1 h_1)) hx | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | nonunits_le_nonunits | null |
nonunits_injective : Function.Injective (nonunits : ValuationSubring K → Subsemigroup _) :=
fun A B h => by simpa only [le_antisymm_iff, nonunits_le_nonunits] using h.symm | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | nonunits_injective | null |
nonunits_inj {A B : ValuationSubring K} : A.nonunits = B.nonunits ↔ A = B :=
nonunits_injective.eq_iff | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | nonunits_inj | null |
nonunitsOrderEmbedding : ValuationSubring K ↪o (Subsemigroup K)ᵒᵈ where
toFun A := A.nonunits
inj' := nonunits_injective
map_rel_iff' {_A _B} := nonunits_le_nonunits
variable {A} | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | nonunitsOrderEmbedding | The map on valuation subrings to their nonunits is a dual order embedding. |
coe_mem_nonunits_iff {a : A} : (a : K) ∈ A.nonunits ↔ a ∈ IsLocalRing.maximalIdeal A :=
(valuation_lt_one_iff _ _).symm | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | coe_mem_nonunits_iff | The elements of `A.nonunits` are those of the maximal ideal of `A` after coercion to `K`.
See also `mem_nonunits_iff_exists_mem_maximalIdeal`, which gets rid of the coercion to `K`,
at the expense of a more complicated right-hand side. |
nonunits_le : A.nonunits ≤ A.toSubring.toSubmonoid.toSubsemigroup := fun _a ha =>
(A.valuation_le_one_iff _).mp (A.mem_nonunits_iff.mp ha).le | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | nonunits_le | null |
nonunits_subset : (A.nonunits : Set K) ⊆ A :=
nonunits_le | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | nonunits_subset | null |
mem_nonunits_iff_exists_mem_maximalIdeal {a : K} :
a ∈ A.nonunits ↔ ∃ ha, (⟨a, ha⟩ : A) ∈ IsLocalRing.maximalIdeal A :=
⟨fun h => ⟨nonunits_subset h, coe_mem_nonunits_iff.mp h⟩, fun ⟨_, h⟩ =>
coe_mem_nonunits_iff.mpr h⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_nonunits_iff_exists_mem_maximalIdeal | The elements of `A.nonunits` are those of the maximal ideal of `A`.
See also `coe_mem_nonunits_iff`, which has a simpler right-hand side but requires the element
to be in `A` already. |
image_maximalIdeal : ((↑) : A → K) '' IsLocalRing.maximalIdeal A = A.nonunits := by
ext a
simp only [Set.mem_image, SetLike.mem_coe, mem_nonunits_iff_exists_mem_maximalIdeal]
rw [Subtype.exists]
simp_rw [exists_and_right, exists_eq_right] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | image_maximalIdeal | `A.nonunits` agrees with the maximal ideal of `A`, after taking its image in `K`. |
principalUnitGroup : Subgroup Kˣ where
carrier := {x | A.valuation (x - 1) < 1}
mul_mem' := by
intro a b ha hb
rw [Set.mem_setOf] at ha hb ⊢
refine lt_of_le_of_lt ?_ (max_lt hb ha)
rw [← one_mul (A.valuation (b - 1)), ← A.valuation.map_one_add_of_lt ha, add_sub_cancel,
← Valuation.map_mul, mul_sub_one, ← sub_add_sub_cancel]
exact A.valuation.map_add _ _
one_mem' := by simp
inv_mem' := by
dsimp
intro a ha
conv =>
lhs
rw [← mul_one (A.valuation _), ← A.valuation.map_one_add_of_lt ha]
rwa [add_sub_cancel, ← Valuation.map_mul, sub_mul, Units.inv_mul, ← neg_sub, one_mul,
Valuation.map_neg] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principalUnitGroup | The principal unit group of a valuation subring, as a subgroup of `Kˣ`. |
principal_units_le_units : A.principalUnitGroup ≤ A.unitGroup := fun a h => by
simpa only [add_sub_cancel] using A.valuation.map_one_add_of_lt h | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principal_units_le_units | null |
mem_principalUnitGroup_iff (x : Kˣ) :
x ∈ A.principalUnitGroup ↔ A.valuation ((x : K) - 1) < 1 :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | mem_principalUnitGroup_iff | null |
principalUnitGroup_le_principalUnitGroup {A B : ValuationSubring K} :
B.principalUnitGroup ≤ A.principalUnitGroup ↔ A ≤ B := by
constructor
· intro h x hx
by_cases h_1 : x = 0; · simp only [h_1, zero_mem]
by_cases h_2 : x⁻¹ + 1 = 0
· rw [add_eq_zero_iff_eq_neg, inv_eq_iff_eq_inv, inv_neg, inv_one] at h_2
simpa only [h_2] using B.neg_mem _ B.one_mem
· rw [← valuation_le_one_iff, ← not_lt, Valuation.one_lt_val_iff _ h_1,
← add_sub_cancel_right x⁻¹, ← Units.val_mk0 h_2, ← mem_principalUnitGroup_iff] at hx ⊢
simpa only [hx] using @h (Units.mk0 (x⁻¹ + 1) h_2)
· intro h x hx
by_contra h_1; exact not_lt.2 (monotone_mapOfLE _ _ h (not_lt.1 h_1)) hx | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principalUnitGroup_le_principalUnitGroup | null |
principalUnitGroup_injective :
Function.Injective (principalUnitGroup : ValuationSubring K → Subgroup _) := fun A B h => by
simpa [le_antisymm_iff, principalUnitGroup_le_principalUnitGroup] using h.symm | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principalUnitGroup_injective | null |
eq_iff_principalUnitGroup {A B : ValuationSubring K} :
A = B ↔ A.principalUnitGroup = B.principalUnitGroup :=
principalUnitGroup_injective.eq_iff.symm | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | eq_iff_principalUnitGroup | null |
principalUnitGroupOrderEmbedding : ValuationSubring K ↪o (Subgroup Kˣ)ᵒᵈ where
toFun A := A.principalUnitGroup
inj' := principalUnitGroup_injective
map_rel_iff' {_A _B} := principalUnitGroup_le_principalUnitGroup | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principalUnitGroupOrderEmbedding | The map on valuation subrings to their principal unit groups is an order embedding. |
coe_mem_principalUnitGroup_iff {x : A.unitGroup} :
(x : Kˣ) ∈ A.principalUnitGroup ↔
A.unitGroupMulEquiv x ∈ (Units.map (IsLocalRing.residue A).toMonoidHom).ker := by
rw [MonoidHom.mem_ker, Units.ext_iff]
let π := Ideal.Quotient.mk (IsLocalRing.maximalIdeal A); convert_to _ ↔ π _ = 1
rw [← π.map_one, ← sub_eq_zero, ← π.map_sub, Ideal.Quotient.eq_zero_iff_mem, valuation_lt_one_iff]
simp [mem_principalUnitGroup_iff] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | coe_mem_principalUnitGroup_iff | null |
principalUnitGroupEquiv :
A.principalUnitGroup ≃* (Units.map (IsLocalRing.residue A).toMonoidHom).ker where
toFun x :=
⟨A.unitGroupMulEquiv ⟨_, A.principal_units_le_units x.2⟩,
A.coe_mem_principalUnitGroup_iff.1 x.2⟩
invFun x :=
⟨A.unitGroupMulEquiv.symm x, by
rw [A.coe_mem_principalUnitGroup_iff]; simp⟩
left_inv x := by simp
right_inv x := by simp
map_mul' _ _ := rfl | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principalUnitGroupEquiv | The principal unit group agrees with the kernel of the canonical map from
the units of `A` to the units of the residue field of `A`. |
principalUnitGroupEquiv_apply (a : A.principalUnitGroup) :
(((principalUnitGroupEquiv A a : Aˣ) : A) : K) = (a : Kˣ) :=
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principalUnitGroupEquiv_apply | null |
principalUnitGroup_symm_apply (a : (Units.map (IsLocalRing.residue A).toMonoidHom).ker) :
((A.principalUnitGroupEquiv.symm a : Kˣ) : K) = ((a : Aˣ) : A) :=
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | principalUnitGroup_symm_apply | null |
unitGroupToResidueFieldUnits : A.unitGroup →* (IsLocalRing.ResidueField A)ˣ :=
MonoidHom.comp (Units.map <| (Ideal.Quotient.mk _).toMonoidHom) A.unitGroupMulEquiv.toMonoidHom
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitGroupToResidueFieldUnits | The canonical map from the unit group of `A` to the units of the residue field of `A`. |
coe_unitGroupToResidueFieldUnits_apply (x : A.unitGroup) :
(A.unitGroupToResidueFieldUnits x : IsLocalRing.ResidueField A) =
Ideal.Quotient.mk _ (A.unitGroupMulEquiv x : A) :=
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | coe_unitGroupToResidueFieldUnits_apply | null |
ker_unitGroupToResidueFieldUnits :
A.unitGroupToResidueFieldUnits.ker = A.principalUnitGroup.comap A.unitGroup.subtype := by
ext
simp_rw [Subgroup.mem_comap, Subgroup.coe_subtype, coe_mem_principalUnitGroup_iff,
unitGroupToResidueFieldUnits, IsLocalRing.residue, RingHom.toMonoidHom_eq_coe,
MulEquiv.toMonoidHom_eq_coe, MonoidHom.mem_ker, MonoidHom.coe_comp, MonoidHom.coe_coe,
Function.comp_apply] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | ker_unitGroupToResidueFieldUnits | null |
surjective_unitGroupToResidueFieldUnits :
Function.Surjective A.unitGroupToResidueFieldUnits :=
(IsLocalRing.surjective_units_map_of_local_ringHom _ Ideal.Quotient.mk_surjective
IsLocalRing.isLocalHom_residue).comp
(MulEquiv.surjective _) | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | surjective_unitGroupToResidueFieldUnits | null |
unitsModPrincipalUnitsEquivResidueFieldUnits :
A.unitGroup ⧸ A.principalUnitGroup.comap A.unitGroup.subtype ≃* (IsLocalRing.ResidueField A)ˣ :=
(QuotientGroup.quotientMulEquivOfEq A.ker_unitGroupToResidueFieldUnits.symm).trans
(QuotientGroup.quotientKerEquivOfSurjective _ A.surjective_unitGroupToResidueFieldUnits) | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitsModPrincipalUnitsEquivResidueFieldUnits | The quotient of the unit group of `A` by the principal unit group of `A` agrees with
the units of the residue field of `A`. |
unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk :
(A.unitsModPrincipalUnitsEquivResidueFieldUnits : _ ⧸ Subgroup.comap _ _ →* _).comp
(QuotientGroup.mk' (A.principalUnitGroup.subgroupOf A.unitGroup)) =
A.unitGroupToResidueFieldUnits := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk | null |
unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk_apply
(x : A.unitGroup) :
A.unitsModPrincipalUnitsEquivResidueFieldUnits.toMonoidHom (QuotientGroup.mk x) =
A.unitGroupToResidueFieldUnits x := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk_apply | null |
pointwiseHasSMul : SMul G (ValuationSubring K) where
smul g S :=-- TODO: if we add `ValuationSubring.map` at a later date, we should use it here
{ g • S.toSubring with
mem_or_inv_mem' := fun x =>
(mem_or_inv_mem S (g⁻¹ • x)).imp Subring.mem_pointwise_smul_iff_inv_smul_mem.mpr fun h =>
Subring.mem_pointwise_smul_iff_inv_smul_mem.mpr <| by rwa [smul_inv''] }
scoped[Pointwise] attribute [instance] ValuationSubring.pointwiseHasSMul
open scoped Pointwise
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | pointwiseHasSMul | The action on a valuation subring corresponding to applying the action to every element.
This is available as an instance in the `Pointwise` locale. |
coe_pointwise_smul (g : G) (S : ValuationSubring K) : ↑(g • S) = g • (S : Set K) := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | coe_pointwise_smul | null |
pointwise_smul_toSubring (g : G) (S : ValuationSubring K) :
(g • S).toSubring = g • S.toSubring := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | pointwise_smul_toSubring | null |
pointwiseMulAction : MulAction G (ValuationSubring K) :=
toSubring_injective.mulAction toSubring pointwise_smul_toSubring
scoped[Pointwise] attribute [instance] ValuationSubring.pointwiseMulAction
open scoped Pointwise | def | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | pointwiseMulAction | The action on a valuation subring corresponding to applying the action to every element.
This is available as an instance in the `Pointwise` locale.
This is a stronger version of `ValuationSubring.pointwiseSMul`. |
smul_mem_pointwise_smul (g : G) (x : K) (S : ValuationSubring K) : x ∈ S → g • x ∈ g • S :=
(Set.smul_mem_smul_set : _ → _ ∈ g • (S : Set K)) | theorem | RingTheory | [
"Mathlib.RingTheory.Valuation.ValuationRing",
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.Ring.Subring.Pointwise",
"Mathlib.Algebra.Ring.Action.Field",
"Mathlib.RingTheory.LocalRing.ResidueField.Basic"
] | Mathlib/RingTheory/Valuation/ValuationSubring.lean | smul_mem_pointwise_smul | null |
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