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 ⌀ |
|---|---|---|---|---|---|---|
liftCLM_mk {σ : R →+* S} (f : M →SL[σ] N) (hf : ∀ x y, Inseparable x y → f x = f y)
(x : M) : liftCLM f hf (mk x) = f x := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean | liftCLM_mk | null |
instAlgebra : Algebra R (SeparationQuotient A) where
algebraMap := mkRingHom.comp (algebraMap R A)
commutes' r := Quotient.ind fun a => congrArg _ <| Algebra.commutes r a
smul_def' r := Quotient.ind fun a => congrArg _ <| Algebra.smul_def r a
@[simp] | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean | instAlgebra | null |
mk_algebraMap (r : R) : mk (algebraMap R A r) = algebraMap R (SeparationQuotient A) r :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean | mk_algebraMap | null |
SeparationQuotient.instModuleFinite
{R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] [Module.Finite R M]
[TopologicalSpace M] [ContinuousAdd M] [ContinuousConstSMul R M] :
Module.Finite R (SeparationQuotient M) :=
Module.Finite.of_surjective (mkCLM R M).toLinearMap Quotient.mk_surjective | instance | Topology | [
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/Topology/Algebra/SeparationQuotient/FiniteDimensional.lean | SeparationQuotient.instModuleFinite | The separation quotient of a finite module is a finite module. |
@[to_additive /-- The lift of an additive monoid hom from `M` to an additive monoid hom from
`SeparationQuotient M`. -/]
noncomputable liftContinuousMonoidHom [CommMonoid M] [ContinuousMul M] [CommMonoid N]
(f : ContinuousMonoidHom M N) (hf : ∀ x y, Inseparable x y → f x = f y) :
ContinuousMonoidHom (Separation... | def | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic"
] | Mathlib/Topology/Algebra/SeparationQuotient/Hom.lean | liftContinuousMonoidHom | The lift of a monoid hom from `M` to a monoid hom from `SeparationQuotient M`. |
liftContinuousCommMonoidHom_mk [CommMonoid M] [ContinuousMul M] [CommMonoid N]
(f : ContinuousMonoidHom M N) (hf : ∀ x y, Inseparable x y → f x = f y) (x : M) :
liftContinuousMonoidHom f hf (mk x) = f x := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic"
] | Mathlib/Topology/Algebra/SeparationQuotient/Hom.lean | liftContinuousCommMonoidHom_mk | null |
exists_out_continuousLinearMap :
∃ f : SeparationQuotient E →L[K] E, mkCLM K E ∘L f = .id K (SeparationQuotient E) := by
rcases (mkCLM K E).toLinearMap.exists_rightInverse_of_surjective
(LinearMap.range_eq_top.mpr surjective_mk) with ⟨f, hf⟩
replace hf : mk ∘ f = id := congr_arg DFunLike.coe hf
exact ⟨⟨f,... | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | exists_out_continuousLinearMap | There exists a continuous `K`-linear map from `SeparationQuotient E` to `E`
such that `mk (outCLM x) = x` for all `x`.
Note that continuity of this map comes for free, because `mk` is a topology inducing map. |
noncomputable outCLM : SeparationQuotient E →L[K] E :=
(exists_out_continuousLinearMap K E).choose
@[simp] | def | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | outCLM | A continuous `K`-linear map from `SeparationQuotient E` to `E`
such that `mk (outCLM x) = x` for all `x`. |
mkCLM_comp_outCLM : mkCLM K E ∘L outCLM K E = .id K (SeparationQuotient E) :=
(exists_out_continuousLinearMap K E).choose_spec
variable {E} in
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | mkCLM_comp_outCLM | null |
mk_outCLM (x : SeparationQuotient E) : mk (outCLM K E x) = x :=
DFunLike.congr_fun (mkCLM_comp_outCLM K E) x
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | mk_outCLM | null |
mk_comp_outCLM : mk ∘ outCLM K E = id := funext (mk_outCLM K)
variable {K} in | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | mk_comp_outCLM | null |
postcomp_mkCLM_surjective {L : Type*} [Semiring L] (σ : L →+* K)
(F : Type*) [AddCommMonoid F] [Module L F] [TopologicalSpace F] :
Function.Surjective ((mkCLM K E).comp : (F →SL[σ] E) → (F →SL[σ] SeparationQuotient E)) := by
intro f
use (outCLM K E).comp f
rw [← ContinuousLinearMap.comp_assoc, mkCLM_comp_... | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | postcomp_mkCLM_surjective | null |
isEmbedding_outCLM : IsEmbedding (outCLM K E) :=
Function.LeftInverse.isEmbedding (mk_outCLM K) continuous_mk (map_continuous _) | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | isEmbedding_outCLM | The `SeparationQuotient.outCLM K E` map is a topological embedding. |
outCLM_injective : Function.Injective (outCLM K E) :=
(isEmbedding_outCLM K E).injective | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | outCLM_injective | null |
outCLM_isUniformInducing : IsUniformInducing (outCLM K E) := by
rw [← isUniformInducing_mk.isUniformInducing_comp_iff, mk_comp_outCLM]
exact .id | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | outCLM_isUniformInducing | null |
outCLM_isUniformEmbedding : IsUniformEmbedding (outCLM K E) where
injective := outCLM_injective K E
toIsUniformInducing := outCLM_isUniformInducing K E | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | outCLM_isUniformEmbedding | null |
outCLM_uniformContinuous : UniformContinuous (outCLM K E) :=
(outCLM_isUniformInducing K E).uniformContinuous | theorem | Topology | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Maps.OpenQuotient"
] | Mathlib/Topology/Algebra/SeparationQuotient/Section.lean | outCLM_uniformContinuous | null |
@[simp]
NormedField.v_eq_valuation (x : K) : Valued.v x = NormedField.valuation x := rfl | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | NormedField.v_eq_valuation | null |
mem_iff {x : K} : x ∈ 𝒪[K] ↔ ‖x‖ ≤ 1 := by
simp [Valuation.mem_integer_iff, ← NNReal.coe_le_coe] | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | mem_iff | An element is in the valuation ring if the norm is bounded by 1. This is a variant of
`Valuation.mem_integer_iff`, phrased using norms instead of the valuation. |
norm_le_one (x : 𝒪[K]) : ‖x‖ ≤ 1 := mem_iff.mp x.prop
@[simp] | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | norm_le_one | null |
norm_coe_unit (u : 𝒪[K]ˣ) : ‖((u : 𝒪[K]) : K)‖ = 1 := by
simpa [← NNReal.coe_inj] using
(Valuation.integer.integers (NormedField.valuation (K := K))).valuation_unit u | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | norm_coe_unit | null |
norm_unit (u : 𝒪[K]ˣ) : ‖(u : 𝒪[K])‖ = 1 := by
simp | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | norm_unit | null |
isUnit_iff_norm_eq_one {u : 𝒪[K]} : IsUnit u ↔ ‖u‖ = 1 := by
simpa [← NNReal.coe_inj] using
(Valuation.integer.integers (NormedField.valuation (K := K))).isUnit_iff_valuation_eq_one | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | isUnit_iff_norm_eq_one | null |
norm_irreducible_lt_one {ϖ : 𝒪[K]} (h : Irreducible ϖ) : ‖ϖ‖ < 1 :=
Valuation.integer.v_irreducible_lt_one h | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | norm_irreducible_lt_one | null |
norm_irreducible_pos {ϖ : 𝒪[K]} (h : Irreducible ϖ) : 0 < ‖ϖ‖ :=
Valuation.integer.v_irreducible_pos h | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | norm_irreducible_pos | null |
coe_span_singleton_eq_closedBall (x : 𝒪[K]) :
(Ideal.span {x} : Set 𝒪[K]) = Metric.closedBall 0 ‖x‖ := by
simp [Valuation.integer.coe_span_singleton_eq_setOf_le_v_coe, Set.ext_iff, ← NNReal.coe_le_coe] | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | coe_span_singleton_eq_closedBall | null |
_root_.Irreducible.maximalIdeal_eq_closedBall [IsDiscreteValuationRing 𝒪[K]]
{ϖ : 𝒪[K]} (h : Irreducible ϖ) :
(𝓂[K] : Set 𝒪[K]) = Metric.closedBall 0 ‖ϖ‖ := by
simp [h.maximalIdeal_eq_setOf_le_v_coe, Set.ext_iff, ← NNReal.coe_le_coe] | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | _root_.Irreducible.maximalIdeal_eq_closedBall | null |
_root_.Irreducible.maximalIdeal_pow_eq_closedBall_pow [IsDiscreteValuationRing 𝒪[K]]
{ϖ : 𝒪[K]} (h : Irreducible ϖ) (n : ℕ) :
((𝓂[K] ^ n : Ideal 𝒪[K]) : Set 𝒪[K]) = Metric.closedBall 0 (‖ϖ‖ ^ n) := by
simp [h.maximalIdeal_pow_eq_setOf_le_v_coe_pow, Set.ext_iff, ← NNReal.coe_le_coe]
variable (K) in | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | _root_.Irreducible.maximalIdeal_pow_eq_closedBall_pow | null |
exists_norm_coe_lt_one : ∃ x : 𝒪[K], 0 < ‖(x : K)‖ ∧ ‖(x : K)‖ < 1 := by
obtain ⟨x, hx, hx'⟩ := NormedField.exists_norm_lt_one K
refine ⟨⟨x, hx'.le⟩, ?_⟩
simpa [hx', Subtype.ext_iff] using hx
variable (K) in | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | exists_norm_coe_lt_one | null |
exists_norm_lt_one : ∃ x : 𝒪[K], 0 < ‖x‖ ∧ ‖x‖ < 1 :=
exists_norm_coe_lt_one K
variable (K) in | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | exists_norm_lt_one | null |
exists_nnnorm_lt_one : ∃ x : 𝒪[K], 0 < ‖x‖₊ ∧ ‖x‖₊ < 1 :=
exists_norm_coe_lt_one K | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | exists_nnnorm_lt_one | null |
finite_quotient_maximalIdeal_pow_of_finite_residueField [IsDiscreteValuationRing 𝒪[K]]
(h : Finite 𝓀[K]) (n : ℕ) :
Finite (𝒪[K] ⧸ 𝓂[K] ^ n) := by
induction n with
| zero =>
simp only [pow_zero, Ideal.one_eq_top]
exact Finite.of_fintype (↥𝒪[K] ⧸ ⊤)
| succ n ih =>
have : 𝓂[K] ^ (n + 1) ≤ �... | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | finite_quotient_maximalIdeal_pow_of_finite_residueField | null |
totallyBounded_iff_finite_residueField [(Valued.v : Valuation K Γ₀).RankOne]
[IsDiscreteValuationRing 𝒪[K]] :
TotallyBounded (Set.univ (α := 𝒪[K])) ↔ Finite 𝓀[K] := by
constructor
· intro H
obtain ⟨p, hp⟩ := IsDiscreteValuationRing.exists_irreducible 𝒪[K]
have := Metric.finite_approx_of_totallyB... | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | totallyBounded_iff_finite_residueField | null |
locallyFiniteOrder_units_mrange_of_isCompact_integer (hc : IsCompact (X := K) 𝒪[K]) :
Nonempty (LocallyFiniteOrder (MonoidHom.mrange (Valued.v : Valuation K Γ₀))ˣ):= by
constructor
refine LocallyFiniteOrder.ofFiniteIcc ?_
suffices ∀ z : (MonoidHom.mrange (Valued.v : Valuation K Γ₀))ˣ, (Set.Icc z 1).Finite by... | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | locallyFiniteOrder_units_mrange_of_isCompact_integer | null |
mulArchimedean_mrange_of_isCompact_integer (hc : IsCompact (X := K) 𝒪[K]) :
MulArchimedean (MonoidHom.mrange (Valued.v : Valuation K Γ₀)) := by
rw [← Units.mulArchimedean_iff]
obtain ⟨_⟩ := locallyFiniteOrder_units_mrange_of_isCompact_integer hc
exact MulArchimedean.of_locallyFiniteOrder | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | mulArchimedean_mrange_of_isCompact_integer | null |
isPrincipalIdealRing_of_compactSpace [hc : CompactSpace 𝒪[K]] :
IsPrincipalIdealRing 𝒪[K] := by
have hi : Valuation.Integers (R := K) Valued.v 𝒪[K] := Valuation.integer.integers v
have hc : IsCompact (X := K) 𝒪[K] := isCompact_iff_compactSpace.mpr hc
obtain ⟨_⟩ := locallyFiniteOrder_units_mrange_of_isComp... | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | isPrincipalIdealRing_of_compactSpace | null |
_root_.Valuation.isNontrivial_iff_not_a_field {K Γ : Type*} [Field K]
[LinearOrderedCommGroupWithZero Γ] (v : Valuation K Γ) :
v.IsNontrivial ↔ IsLocalRing.maximalIdeal v.integer ≠ ⊥ := by
simp_rw [ne_eq, eq_bot_iff, v.isNontrivial_iff_exists_lt_one, SetLike.le_def, Ideal.mem_bot,
not_forall, exists_prop,... | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | _root_.Valuation.isNontrivial_iff_not_a_field | null |
isDiscreteValuationRing_of_compactSpace [hn : (Valued.v : Valuation K Γ₀).IsNontrivial]
[CompactSpace 𝒪[K]] : IsDiscreteValuationRing 𝒪[K] where
__ := isPrincipalIdealRing_of_compactSpace
not_a_field' := v.isNontrivial_iff_not_a_field.mp hn | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | isDiscreteValuationRing_of_compactSpace | null |
compactSpace_iff_completeSpace_and_isDiscreteValuationRing_and_finite_residueField
[(Valued.v : Valuation K Γ₀).RankOne] :
CompactSpace 𝒪[K] ↔ CompleteSpace 𝒪[K] ∧ IsDiscreteValuationRing 𝒪[K] ∧ Finite 𝓀[K] := by
refine ⟨fun h ↦ ?_, fun ⟨_, _, h⟩ ↦ ⟨?_⟩⟩
· have : IsDiscreteValuationRing 𝒪[K] := isDiscr... | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | compactSpace_iff_completeSpace_and_isDiscreteValuationRing_and_finite_residueField | null |
properSpace_iff_compactSpace_integer [(Valued.v : Valuation K Γ₀).RankOne] :
ProperSpace K ↔ CompactSpace 𝒪[K] := by
simp only [← isCompact_univ_iff, Subtype.isCompact_iff, Set.image_univ, Subtype.range_coe_subtype,
toNormedField.setOf_mem_integer_eq_closedBall]
constructor <;> intro h
· exact i... | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | properSpace_iff_compactSpace_integer | null |
properSpace_iff_completeSpace_and_isDiscreteValuationRing_integer_and_finite_residueField
[(Valued.v : Valuation K Γ₀).RankOne] :
ProperSpace K ↔ CompleteSpace K ∧ IsDiscreteValuationRing 𝒪[K] ∧ Finite 𝓀[K] := by
simp only [properSpace_iff_compactSpace_integer,
compactSpace_iff_completeSpace_and_isDis... | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Lemmas",
"Mathlib.Analysis.Normed.Field.ProperSpace",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient",
"Mathlib.RingTheory.Valuation.Archimedean",
"Mathlib.Topology.Algebra.Valued.NormedValued",
"Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Valued/LocallyCompact.lean | properSpace_iff_completeSpace_and_isDiscreteValuationRing_integer_and_finite_residueField | null |
valuation : Valuation K ℝ≥0 where
toFun := nnnorm
map_zero' := nnnorm_zero
map_one' := nnnorm_one
map_mul' := nnnorm_mul
map_add_le_max' := IsUltrametricDist.norm_add_le_max
@[simp] | def | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | valuation | The valuation on a nonarchimedean normed field `K` defined as `nnnorm`. |
valuation_apply (x : K) : valuation x = ‖x‖₊ := rfl | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | valuation_apply | null |
toValued : Valued K ℝ≥0 :=
{ hK.toUniformSpace,
inferInstanceAs (IsUniformAddGroup K) with
v := valuation
is_topological_valuation := fun U => by
rw [Metric.mem_nhds_iff]
exact ⟨fun ⟨ε, hε, h⟩ =>
⟨Units.mk0 ⟨ε, le_of_lt hε⟩ (ne_of_gt hε), fun x hx ↦ h (mem_ball_zero_iff.mpr hx)⟩,
... | def | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | toValued | The valued field structure on a nonarchimedean normed field `K`, determined by the norm. |
norm : L → ℝ := fun x : L => hv.hom (Valued.v x) | def | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm | The norm function determined by a rank one valuation on a field `L`. |
norm_def {x : L} : Valued.norm x = hv.hom (Valued.v x) := rfl | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_def | null |
norm_nonneg (x : L) : 0 ≤ norm x := by simp only [norm, NNReal.zero_le_coe] | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_nonneg | null |
norm_add_le (x y : L) : norm (x + y) ≤ max (norm x) (norm y) := by
simp only [norm, NNReal.coe_le_coe, le_max_iff, StrictMono.le_iff_le hv.strictMono]
exact le_max_iff.mp (Valuation.map_add_le_max' val.v _ _) | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_add_le | null |
norm_eq_zero {x : L} (hx : norm x = 0) : x = 0 := by
simpa [norm, NNReal.coe_eq_zero, RankOne.hom_eq_zero_iff, zero_iff] using hx | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_eq_zero | null |
norm_pos_iff_valuation_pos {x : L} : 0 < Valued.norm x ↔ (0 : Γ₀) < v x := by
rw [norm_def, ← NNReal.coe_zero, NNReal.coe_lt_coe, ← map_zero (RankOne.hom (v (R := L))),
StrictMono.lt_iff_lt]
exact RankOne.strictMono v
variable (L) (Γ₀) | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_pos_iff_valuation_pos | null |
toNormedField : NormedField L :=
{ (inferInstance : Field L) with
norm := norm
dist := fun x y => norm (x - y)
dist_self := fun x => by
simp only [sub_self, norm, Valuation.map_zero, hv.hom.map_zero, NNReal.coe_zero]
dist_comm := fun x y => by simp only [norm]; rw [← neg_sub, Valuation.map_neg]
... | def | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | toNormedField | The normed field structure determined by a rank one valuation. |
protected isNonarchimedean_norm : IsNonarchimedean ((‖·‖) : L → ℝ) := Valued.norm_add_le | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | isNonarchimedean_norm | null |
coe_valuation_eq_rankOne_hom_comp_valuation : ⇑NormedField.valuation = hv.hom ∘ val.v := rfl | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | coe_valuation_eq_rankOne_hom_comp_valuation | null |
@[simp]
norm_le_iff : ‖x‖ ≤ ‖x'‖ ↔ val.v x ≤ val.v x' :=
(Valuation.RankOne.strictMono val.v).le_iff_le
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_le_iff | null |
norm_lt_iff : ‖x‖ < ‖x'‖ ↔ val.v x < val.v x' :=
(Valuation.RankOne.strictMono val.v).lt_iff_lt
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_lt_iff | null |
norm_le_one_iff : ‖x‖ ≤ 1 ↔ val.v x ≤ 1 := by
simpa only [map_one] using (Valuation.RankOne.strictMono val.v).le_iff_le (b := 1)
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_le_one_iff | null |
norm_lt_one_iff : ‖x‖ < 1 ↔ val.v x < 1 := by
simpa only [map_one] using (Valuation.RankOne.strictMono val.v).lt_iff_lt (b := 1)
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | norm_lt_one_iff | null |
one_le_norm_iff : 1 ≤ ‖x‖ ↔ 1 ≤ val.v x := by
simpa only [map_one] using (Valuation.RankOne.strictMono val.v).le_iff_le (a := 1)
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | one_le_norm_iff | null |
one_lt_norm_iff : 1 < ‖x‖ ↔ 1 < val.v x := by
simpa only [map_one] using (Valuation.RankOne.strictMono val.v).lt_iff_lt (a := 1) | theorem | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | one_lt_norm_iff | null |
setOf_mem_integer_eq_closedBall :
{ x : L | x ∈ Valued.v.integer } = Metric.closedBall 0 1 := by
ext x
simp [mem_integer_iff] | lemma | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | setOf_mem_integer_eq_closedBall | null |
toNontriviallyNormedField : NontriviallyNormedField L := {
val.toNormedField with
non_trivial := by
obtain ⟨x, hx⟩ := Valuation.RankOne.nontrivial val.v
rcases Valuation.val_le_one_or_val_inv_le_one val.v x with h | h
· use x⁻¹
simp only [toNormedField.one_lt_norm_iff, map_inv₀, one_lt_inv₀ (zero_... | def | Topology | [
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.Group.Ultra",
"Mathlib.RingTheory.Valuation.RankOne",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/NormedValued.lean | toNontriviallyNormedField | The nontrivially normed field structure determined by a rank one valuation. |
map_eq_one_of_forall_lt [MulArchimedean Γ₀] {v : Valuation K Γ₀} {r : Γ₀} (hr : r ≠ 0)
(h : ∀ x : K, v x ≠ 0 → r < v x) (x : K) (hx : v x ≠ 0) : v x = 1 := by
lift r to Γ₀ˣ using IsUnit.mk0 _ hr
rcases lt_trichotomy (Units.mk0 _ hx) 1 with H | H | H
· obtain ⟨k, hk⟩ := exists_pow_lt H r
specialize h (x ^ ... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | map_eq_one_of_forall_lt | null |
subgroups_basis : RingSubgroupsBasis fun γ : Γ₀ˣ => (v.ltAddSubgroup γ : AddSubgroup R) :=
{ inter := by
rintro γ₀ γ₁
use min γ₀ γ₁
simp only [ltAddSubgroup, Units.min_val, lt_inf_iff, le_inf_iff, AddSubgroup.mk_le_mk,
AddSubmonoid.mk_le_mk, AddSubsemigroup.mk_le_mk, setOf_subset_setOf]
... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | subgroups_basis | The basis of open subgroups for the topology on a ring determined by a valuation. |
Valued (R : Type u) [Ring R] (Γ₀ : outParam (Type v))
[LinearOrderedCommGroupWithZero Γ₀] extends UniformSpace R, IsUniformAddGroup R where
v : Valuation R Γ₀
is_topological_valuation : ∀ s, s ∈ 𝓝 (0 : R) ↔ ∃ γ : Γ₀ˣ, { x : R | v x < γ } ⊆ s | class | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | Valued | A valued ring is a ring that comes equipped with a distinguished valuation. The class `Valued`
is designed for the situation that there is a canonical valuation on the ring.
TODO: show that there always exists an equivalent valuation taking values in a type belonging to
the same universe as the ring.
See Note [forget... |
mk' (v : Valuation R Γ₀) : Valued R Γ₀ :=
{ v
toUniformSpace := @IsTopologicalAddGroup.toUniformSpace R _ v.subgroups_basis.topology _
toIsUniformAddGroup := @isUniformAddGroup_of_addCommGroup _ _ v.subgroups_basis.topology _
is_topological_valuation := by
letI := @IsTopologicalAddGroup.toUniformSpa... | def | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | mk' | Alternative `Valued` constructor for use when there is no preferred `UniformSpace` structure. |
hasBasis_nhds_zero :
(𝓝 (0 : R)).HasBasis (fun _ => True) fun γ : Γ₀ˣ => { x | v x < (γ : Γ₀) } := by
simp [Filter.hasBasis_iff, is_topological_valuation]
open Uniformity in | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | hasBasis_nhds_zero | null |
hasBasis_uniformity : (𝓤 R).HasBasis (fun _ => True)
fun γ : Γ₀ˣ => { p : R × R | v (p.2 - p.1) < (γ : Γ₀) } := by
rw [uniformity_eq_comap_nhds_zero]
exact (hasBasis_nhds_zero R Γ₀).comap _ | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | hasBasis_uniformity | null |
toUniformSpace_eq :
toUniformSpace = @IsTopologicalAddGroup.toUniformSpace R _ v.subgroups_basis.topology _ :=
UniformSpace.ext
((hasBasis_uniformity R Γ₀).eq_of_same_basis <| v.subgroups_basis.hasBasis_nhds_zero.comap _)
variable {R Γ₀} | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | toUniformSpace_eq | null |
mem_nhds {s : Set R} {x : R} : s ∈ 𝓝 x ↔ ∃ γ : Γ₀ˣ, { y | (v (y - x) : Γ₀) < γ } ⊆ s := by
simp only [← nhds_translation_add_neg x, ← sub_eq_add_neg, preimage_setOf_eq, true_and,
((hasBasis_nhds_zero R Γ₀).comap fun y => y - x).mem_iff] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | mem_nhds | null |
mem_nhds_zero {s : Set R} : s ∈ 𝓝 (0 : R) ↔ ∃ γ : Γ₀ˣ, { x | v x < (γ : Γ₀) } ⊆ s := by
simp only [mem_nhds, sub_zero] | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | mem_nhds_zero | null |
loc_const {x : R} (h : (v x : Γ₀) ≠ 0) : { y : R | v y = v x } ∈ 𝓝 x := by
rw [mem_nhds]
use Units.mk0 _ h
rw [Units.val_mk0]
intro y y_in
exact Valuation.map_eq_of_sub_lt _ y_in | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | loc_const | null |
discreteTopology_of_forall_map_eq_one (h : ∀ x : R, x ≠ 0 → v x = 1) :
DiscreteTopology R := by
simp only [discreteTopology_iff_isOpen_singleton_zero, isOpen_iff_mem_nhds, mem_singleton_iff,
forall_eq, mem_nhds_zero, subset_singleton_iff, mem_setOf_eq]
use 1
contrapose! h
obtain ⟨x, hx, hx'⟩ := h
exac... | lemma | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | discreteTopology_of_forall_map_eq_one | null |
discreteTopology_of_forall_lt [MulArchimedean Γ₀] [Valued K Γ₀] {r : Γ₀} (hr : r ≠ 0)
(h : ∀ x : K, v x ≠ 0 → r < v x) :
DiscreteTopology K :=
discreteTopology_of_forall_map_eq_one (by simpa using Valued.v.map_eq_one_of_forall_lt hr h) | lemma | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | discreteTopology_of_forall_lt | null |
cauchy_iff {F : Filter R} : Cauchy F ↔
F.NeBot ∧ ∀ γ : Γ₀ˣ, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), (v (y - x) : Γ₀) < γ := by
rw [toUniformSpace_eq, AddGroupFilterBasis.cauchy_iff]
apply and_congr Iff.rfl
simp_rw [Valued.v.subgroups_basis.mem_addGroupFilterBasis_iff]
constructor
· intro h γ
exact h _ (Valued.v.... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | cauchy_iff | null |
isOpen_ball (r : Γ₀) : IsOpen (X := R) {x | v x < r} := by
rw [isOpen_iff_mem_nhds]
rcases eq_or_ne r 0 with rfl | hr
· simp
intro x hx
rw [mem_nhds]
simp only [setOf_subset_setOf]
exact ⟨Units.mk0 _ hr,
fun y hy => (sub_add_cancel y x).symm ▸ (v.map_add _ x).trans_lt (max_lt hy hx)⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isOpen_ball | An open ball centred at the origin in a valued ring is open. |
isClosed_ball (r : Γ₀) : IsClosed (X := R) {x | v x < r} := by
rcases eq_or_ne r 0 with rfl | hr
· simp
exact AddSubgroup.isClosed_of_isOpen
(Valuation.ltAddSubgroup v (Units.mk0 r hr))
(isOpen_ball _ _) | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClosed_ball | An open ball centred at the origin in a valued ring is closed. |
isClopen_ball (r : Γ₀) : IsClopen (X := R) {x | v x < r} :=
⟨isClosed_ball _ _, isOpen_ball _ _⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClopen_ball | An open ball centred at the origin in a valued ring is clopen. |
isOpen_closedball {r : Γ₀} (hr : r ≠ 0) : IsOpen (X := R) {x | v x ≤ r} := by
rw [isOpen_iff_mem_nhds]
intro x hx
rw [mem_nhds]
simp only [setOf_subset_setOf]
exact ⟨Units.mk0 _ hr,
fun y hy => (sub_add_cancel y x).symm ▸ le_trans (v.map_add _ _) (max_le (le_of_lt hy) hx)⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isOpen_closedball | A closed ball centred at the origin in a valued ring is open. |
isClosed_closedBall (r : Γ₀) : IsClosed (X := R) {x | v x ≤ r} := by
rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]
intro x hx
rw [mem_nhds]
have hx' : v x ≠ 0 := ne_of_gt <| lt_of_le_of_lt zero_le' <| lt_of_not_ge hx
exact ⟨Units.mk0 _ hx', fun y hy hy' => ne_of_lt hy <| map_sub_swap v x y ▸
(Valuation.m... | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClosed_closedBall | A closed ball centred at the origin in a valued ring is closed. |
isClopen_closedBall {r : Γ₀} (hr : r ≠ 0) : IsClopen (X := R) {x | v x ≤ r} :=
⟨isClosed_closedBall _ _, isOpen_closedball _ hr⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClopen_closedBall | A closed ball centred at the origin in a valued ring is clopen. |
isClopen_sphere {r : Γ₀} (hr : r ≠ 0) : IsClopen (X := R) {x | v x = r} := by
have h : {x : R | v x = r} = {x | v x ≤ r} \ {x | v x < r} := by
ext x
simp [← le_antisymm_iff]
rw [h]
exact IsClopen.diff (isClopen_closedBall _ hr) (isClopen_ball _ _) | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClopen_sphere | A sphere centred at the origin in a valued ring is clopen. |
isOpen_sphere {r : Γ₀} (hr : r ≠ 0) : IsOpen (X := R) {x | v x = r} :=
isClopen_sphere _ hr |>.isOpen | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isOpen_sphere | A sphere centred at the origin in a valued ring is open. |
isClosed_sphere (r : Γ₀) : IsClosed (X := R) {x | v x = r} := by
rcases eq_or_ne r 0 with rfl | hr
· simpa using isClosed_closedBall R 0
exact isClopen_sphere _ hr |>.isClosed | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClosed_sphere | A sphere centred at the origin in a valued ring is closed. |
isOpen_integer : IsOpen (_i.v.integer : Set R) :=
isOpen_closedball _ one_ne_zero
@[deprecated (since := "2025-04-25")]
alias integer_isOpen := isOpen_integer | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isOpen_integer | The closed unit ball in a valued ring is open. |
isClosed_integer : IsClosed (_i.v.integer : Set R) :=
isClosed_closedBall _ _ | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClosed_integer | The closed unit ball of a valued ring is closed. |
isClopen_integer : IsClopen (_i.v.integer : Set R) :=
⟨isClosed_integer _, isOpen_integer _⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClopen_integer | The closed unit ball of a valued ring is clopen. |
isOpen_valuationSubring (K : Type u) [Field K] [hv : Valued K Γ₀] :
IsOpen (hv.v.valuationSubring : Set K) :=
isOpen_integer K
@[deprecated (since := "2025-04-25")]
alias valuationSubring_isOpen := isOpen_valuationSubring | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isOpen_valuationSubring | The valuation subring of a valued field is open. |
isClosed_valuationSubring (K : Type u) [Field K] [hv : Valued K Γ₀] :
IsClosed (hv.v.valuationSubring : Set K) :=
isClosed_integer K | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClosed_valuationSubring | The valuation subring of a valued field is closed. |
isClopen_valuationSubring (K : Type u) [Field K] [hv : Valued K Γ₀] :
IsClopen (hv.v.valuationSubring : Set K) :=
isClopen_integer K | theorem | Topology | [
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformFilterBasis",
"Mathlib.RingTheory.Valuation.ValuationSubring"
] | Mathlib/Topology/Algebra/Valued/ValuationTopology.lean | isClopen_valuationSubring | The valuation subring of a valued field is clopen. |
of_zero [ContinuousConstVAdd R R]
(h₀ : ∀ s : Set R, s ∈ 𝓝 0 ↔ ∃ γ : (ValueGroupWithZero R)ˣ, { z | v z < γ } ⊆ s) :
IsValuativeTopology R where
mem_nhds_iff {s x} := by
simpa [← vadd_mem_nhds_vadd_iff (t := s) (-x), ← image_vadd, ← image_subset_iff] using
h₀ ((x + ·) ⁻¹' s) | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | of_zero | Assuming `ContinuousConstVAdd R R`, we only need to check the neighbourhood of `0` in order to
prove `IsValuativeTopology R`. |
mem_nhds_iff' {s : Set R} {x : R} :
s ∈ 𝓝 (x : R) ↔
∃ γ : (ValueGroupWithZero R)ˣ, { z | v (z - x) < γ } ⊆ s := by
convert mem_nhds_iff (s := s) using 4
ext z
simp [neg_add_eq_sub]
@[deprecated (since := "2025-08-01")]
alias _root_.ValuativeTopology.mem_nhds := mem_nhds_iff' | lemma | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | mem_nhds_iff' | A version mentioning subtraction. |
mem_nhds_zero_iff (s : Set R) : s ∈ 𝓝 (0 : R) ↔
∃ γ : (ValueGroupWithZero R)ˣ, { x | v x < γ } ⊆ s := by
convert IsValuativeTopology.mem_nhds_iff' (x := (0 : R))
rw [sub_zero]
@[deprecated (since := "2025-08-04")]
alias _root_.ValuativeTopology.mem_nhds_iff := mem_nhds_zero_iff | lemma | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | mem_nhds_zero_iff | null |
hasBasis_nhds (x : R) :
(𝓝 x).HasBasis (fun _ => True)
fun γ : (ValueGroupWithZero R)ˣ => { z | v (z - x) < γ } := by
simp [Filter.hasBasis_iff, mem_nhds_iff']
variable (R) in | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | hasBasis_nhds | null |
hasBasis_nhds_zero :
(𝓝 (0 : R)).HasBasis (fun _ => True)
fun γ : (ValueGroupWithZero R)ˣ => { x | v x < γ } := by
convert hasBasis_nhds (0 : R); rw [sub_zero]
@[deprecated (since := "2025-08-01")]
alias _root_.ValuativeTopology.hasBasis_nhds_zero := hasBasis_nhds_zero
variable (R) in | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | hasBasis_nhds_zero | null |
isOpen_ball (r : ValueGroupWithZero R) :
IsOpen {x | v x < r} := by
rw [isOpen_iff_mem_nhds]
rcases eq_or_ne r 0 with rfl | hr
· simp
· intro x hx
rw [mem_nhds_iff']
simp only [setOf_subset_setOf]
exact ⟨Units.mk0 _ hr,
fun y hy => (sub_add_cancel y x).symm ▸ ((v).map_add _ x).trans_lt (ma... | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isOpen_ball | null |
isClosed_ball (r : ValueGroupWithZero R) :
IsClosed {x | v x < r} := by
rcases eq_or_ne r 0 with rfl | hr
· simp
· exact AddSubgroup.isClosed_of_isOpen (Valuation.ltAddSubgroup v (Units.mk0 r hr))
(isOpen_ball _)
@[deprecated (since := "2025-08-01")]
alias _root_.ValuativeTopology.isClosed_ball := isClo... | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isClosed_ball | null |
isClopen_ball (r : ValueGroupWithZero R) :
IsClopen {x | v x < r} :=
⟨isClosed_ball _, isOpen_ball _⟩
@[deprecated (since := "2025-08-01")]
alias _root_.ValuativeTopology.isClopen_ball := isClopen_ball | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isClopen_ball | null |
isOpen_closedBall {r : ValueGroupWithZero R} (hr : r ≠ 0) :
IsOpen {x | v x ≤ r} := by
rw [isOpen_iff_mem_nhds]
intro x hx
rw [mem_nhds_iff']
simp only [setOf_subset_setOf]
exact ⟨Units.mk0 _ hr, fun y hy => (sub_add_cancel y x).symm ▸
le_trans ((v).map_add _ _) (max_le (le_of_lt hy) hx)⟩
@[deprecated... | lemma | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isOpen_closedBall | null |
isClosed_closedBall (r : ValueGroupWithZero R) :
IsClosed {x | v x ≤ r} := by
rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]
intro x hx
simp only [mem_compl_iff, mem_setOf_eq, not_le] at hx
rw [mem_nhds_iff']
have hx' : v x ≠ 0 := ne_of_gt <| lt_of_le_of_lt zero_le' <| hx
exact ⟨Units.mk0 _ hx', fun y hy ... | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isClosed_closedBall | null |
isClopen_closedBall {r : ValueGroupWithZero R} (hr : r ≠ 0) :
IsClopen {x | v x ≤ r} :=
⟨isClosed_closedBall _, isOpen_closedBall hr⟩
@[deprecated (since := "2025-08-01")]
alias _root_.ValuativeTopology.isClopen_closedBall := isClopen_closedBall | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isClopen_closedBall | null |
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