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 ⌀ |
|---|---|---|---|---|---|---|
isClopen_sphere {r : ValueGroupWithZero R} (hr : r ≠ 0) :
IsClopen {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 _)
@[deprecated (since := "2025-08-01")]
alias _root... | theorem | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isClopen_sphere | null |
isOpen_sphere {r : ValueGroupWithZero R} (hr : r ≠ 0) :
IsOpen {x | v x = r} :=
isClopen_sphere hr |>.isOpen
@[deprecated (since := "2025-08-01")]
alias _root_.ValuativeTopology.isOpen_sphere := isOpen_sphere
open WithZeroTopology in | lemma | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | isOpen_sphere | null |
continuous_valuation : Continuous v := by
simp only [continuous_iff_continuousAt, ContinuousAt]
rintro x
by_cases hx : v x = 0
· simpa [hx, (hasBasis_nhds _).tendsto_iff WithZeroTopology.hasBasis_nhds_zero,
Valuation.map_sub_of_right_eq_zero _ hx] using fun i hi ↦ ⟨.mk0 i hi, fun y ↦ id⟩
· simpa [(hasBa... | lemma | Topology | [
"Mathlib.RingTheory.Valuation.ValuativeRel.Basic",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology"
] | Mathlib/Topology/Algebra/Valued/ValuativeRel.lean | continuous_valuation | null |
Valuation.inversion_estimate {x y : K} {γ : Γ₀ˣ} (y_ne : y ≠ 0)
(h : v (x - y) < min (γ * (v y * v y)) (v y)) : v (x⁻¹ - y⁻¹) < γ := by
have hyp1 : v (x - y) < γ * (v y * v y) := lt_of_lt_of_le h (min_le_left _ _)
have hyp1' : v (x - y) * (v y * v y)⁻¹ < γ := mul_inv_lt_of_lt_mul₀ hyp1
have hyp2 : v (x - y) <... | theorem | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | Valuation.inversion_estimate | null |
Valuation.inversion_estimate' {x y r s : K} (y_ne : y ≠ 0) (hr : r ≠ 0) (hs : s ≠ 0)
(h : v (x - y) < min ((v s / v r) * (v y * v y)) (v y)) : v (x⁻¹ - y⁻¹) * v r < v s := by
have hr' : 0 < v r := by simp [zero_lt_iff, hr]
let γ : Γ₀ˣ := .mk0 (v s / v r) (by simp [hs, hr])
calc
v (x⁻¹ - y⁻¹) * v r < γ * v... | theorem | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | Valuation.inversion_estimate' | null |
noncomputable extension : hat K → Γ₀ :=
Completion.isDenseInducing_coe.extend (v : K → Γ₀) | def | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | extension | The topology coming from a valuation on a division ring makes it a topological division ring
[BouAC, VI.5.1 middle of Proposition 1] -/
instance (priority := 100) Valued.isTopologicalDivisionRing [Valued K Γ₀] :
IsTopologicalDivisionRing K :=
{ (by infer_instance : IsTopologicalRing K) with
continuousAt_inv₀ ... |
continuous_extension : Continuous (Valued.extension : hat K → Γ₀) := by
refine Completion.isDenseInducing_coe.continuous_extend ?_
intro x₀
rcases eq_or_ne x₀ 0 with (rfl | h)
· refine ⟨0, ?_⟩
erw [← Completion.isDenseInducing_coe.isInducing.nhds_eq_comap]
exact Valued.continuous_valuation.tendsto' 0 0 ... | theorem | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | continuous_extension | null |
extension_extends (x : K) : extension (x : hat K) = v x := by
refine Completion.isDenseInducing_coe.extend_eq_of_tendsto ?_
rw [← Completion.isDenseInducing_coe.nhds_eq_comap]
exact Valued.continuous_valuation.continuousAt | theorem | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | extension_extends | null |
noncomputable extensionValuation : Valuation (hat K) Γ₀ where
toFun := Valued.extension
map_zero' := by
rw [← v.map_zero (R := K), ← Valued.extension_extends (0 : K)]
rfl
map_one' := by
rw [← Completion.coe_one, Valued.extension_extends (1 : K)]
exact Valuation.map_one _
map_mul' x y := by
a... | def | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | extensionValuation | the extension of a valuation on a division ring to its completion. |
extensionValuation_apply_coe (x : K) :
Valued.extensionValuation (x : hat K) = v x :=
extension_extends x
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | extensionValuation_apply_coe | null |
extension_eq_zero_iff {x : hat K} :
extension x = 0 ↔ x = 0 := by
suffices extensionValuation x = 0 ↔ x = 0 from this
simp | lemma | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | extension_eq_zero_iff | null |
continuous_extensionValuation : Continuous (Valued.extensionValuation : hat K → Γ₀) :=
continuous_extension | lemma | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | continuous_extensionValuation | null |
exists_coe_eq_v (x : hat K) : ∃ r : K, extensionValuation x = v r := by
rcases eq_or_ne x 0 with (rfl | h)
· use 0
exact extensionValuation_apply_coe 0
· refine Completion.denseRange_coe.induction_on x ?_ (by simp)
simpa [eq_comm] using
valuation_isClosedMap.isClosed_range.preimage continuous_extens... | lemma | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | exists_coe_eq_v | null |
closure_coe_completion_v_lt {γ : Γ₀ˣ} :
closure ((↑) '' { x : K | v x < (γ : Γ₀) }) =
{ x : hat K | extensionValuation x < (γ : Γ₀) } := by
ext x
let γ₀ := extensionValuation x
suffices γ₀ ≠ 0 → (x ∈ closure ((↑) '' { x : K | v x < (γ : Γ₀) }) ↔ γ₀ < (γ : Γ₀)) by
rcases eq_or_ne γ₀ 0 with h | h
· ... | theorem | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | closure_coe_completion_v_lt | null |
closure_coe_completion_v_mul_v_lt {r s : K} (hr : r ≠ 0) (hs : s ≠ 0) :
closure ((↑) '' { x : K | v x * v r < v s }) =
{ x : hat K | extensionValuation x * v r < v s } := by
have hrs : v s / v r ≠ 0 := by simp [hr, hs]
convert closure_coe_completion_v_lt (γ := .mk0 _ hrs) using 3
all_goals simp [← lt_div_... | theorem | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | closure_coe_completion_v_mul_v_lt | null |
noncomputable valuedCompletion : Valued (hat K) Γ₀ where
v := extensionValuation
is_topological_valuation s := by
suffices
HasBasis (𝓝 (0 : hat K)) (fun _ => True) fun γ : Γ₀ˣ => { x | extensionValuation x < γ } by
rw [this.mem_iff]
exact exists_congr fun γ => by simp
simp_rw [← closure_c... | instance | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | valuedCompletion | null |
valuedCompletion_apply (x : K) : Valued.v (x : hat K) = v x :=
extension_extends x | theorem | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | valuedCompletion_apply | null |
valuedCompletion_surjective_iff :
Function.Surjective (v : hat K → Γ₀) ↔ Function.Surjective (v : K → Γ₀) := by
constructor <;> intro h γ <;> obtain ⟨a, ha⟩ := h γ
· induction a using Completion.induction_on
· by_cases H : ∃ x : K, (v : K → Γ₀) x = γ
· simp [H]
· simp only [H, imp_false]
... | lemma | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | valuedCompletion_surjective_iff | null |
@[reducible]
integer : Subring K := (vK.v).integer
@[inherit_doc]
scoped notation "𝒪[" K "]" => Valued.integer K | def | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | integer | A `Valued` version of `Valuation.integer`, enabling the notation `𝒪[K]` for the
valuation integers of a valued field `K`. |
@[reducible]
maximalIdeal : Ideal 𝒪[K] := IsLocalRing.maximalIdeal 𝒪[K]
@[inherit_doc]
scoped notation "𝓂[" K "]" => maximalIdeal K | def | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | maximalIdeal | An abbreviation for `IsLocalRing.maximalIdeal 𝒪[K]` of a valued field `K`, enabling the notation
`𝓂[K]` for the maximal ideal in `𝒪[K]` of a valued field `K`. |
@[reducible]
ResidueField := IsLocalRing.ResidueField (𝒪[K])
@[inherit_doc]
scoped notation "𝓀[" K "]" => ResidueField K | def | Topology | [
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.Topology.Algebra.WithZeroTopology",
"Mathlib.Topology.Algebra.UniformField"
] | Mathlib/Topology/Algebra/Valued/ValuedField.lean | ResidueField | An abbreviation for `IsLocalRing.ResidueField 𝒪[K]` of a `Valued` instance, enabling the
notation `𝓀[K]` for the residue field of a valued field `K`. |
@[nolint unusedArguments]
WithVal [Ring R] : Valuation R Γ₀ → Type _ := fun _ => R | def | Topology | [
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.NumberTheory.NumberField.Basic"
] | Mathlib/Topology/Algebra/Valued/WithVal.lean | WithVal | Type synonym for a ring equipped with the topology coming from a valuation. |
equiv : WithVal v ≃+* R := RingEquiv.refl _ | def | Topology | [
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.NumberTheory.NumberField.Basic"
] | Mathlib/Topology/Algebra/Valued/WithVal.lean | equiv | Canonical ring equivalence between `WithVal v` and `R`. |
apply_equiv (r : WithVal v) : v (equiv v r) = Valued.v r := rfl
@[simp] theorem apply_symm_equiv (r : R) : Valued.v ((equiv v).symm r) = v r := rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.NumberTheory.NumberField.Basic"
] | Mathlib/Topology/Algebra/Valued/WithVal.lean | apply_equiv | null |
Completion := UniformSpace.Completion (WithVal v) | abbrev | Topology | [
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.NumberTheory.NumberField.Basic"
] | Mathlib/Topology/Algebra/Valued/WithVal.lean | Completion | The completion of a field with respect to a valuation. |
@[simps!]
withValEquiv (R : Type*) [CommRing R] [Algebra R K] [IsIntegralClosure R ℤ K] :
𝓞 (WithVal v) ≃+* R := NumberField.RingOfIntegers.equiv R | def | Topology | [
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.NumberTheory.NumberField.Basic"
] | Mathlib/Topology/Algebra/Valued/WithVal.lean | withValEquiv | The ring equivalence between `𝓞 (WithVal v)` and an integral closure of
`ℤ` in `K`. |
@[simps! apply]
Rat.ringOfIntegersWithValEquiv (v : Valuation ℚ Γ₀) : 𝓞 (WithVal v) ≃+* ℤ :=
NumberField.RingOfIntegers.withValEquiv v ℤ | def | Topology | [
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.Valued.ValuationTopology",
"Mathlib.NumberTheory.NumberField.Basic"
] | Mathlib/Topology/Algebra/Valued/WithVal.lean | Rat.ringOfIntegersWithValEquiv | The ring of integers of `WithVal v`, when `v` is a valuation on `ℚ`, is
equivalent to `ℤ`. |
tendsto_zero_pow_of_v_lt_one [MulArchimedean Γ₀] [Valued R Γ₀] {x : R} (hx : v x < 1) :
Tendsto (fun n : ℕ ↦ x ^ n) atTop (𝓝 0) := by
simp only [(hasBasis_nhds_zero _ _).tendsto_right_iff, mem_setOf_eq, map_pow, eventually_atTop,
forall_const]
intro y
obtain ⟨n, hn⟩ := exists_pow_lt₀ hx y
refine ⟨n, fu... | lemma | Topology | [
"Mathlib.GroupTheory.ArchimedeanDensely",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/WithZeroMulInt.lean | tendsto_zero_pow_of_v_lt_one | null |
tendsto_zero_pow_of_le_exp_neg_one [Valued R ℤᵐ⁰] {x : R} (hx : v x ≤ exp (-1)) :
Tendsto (fun n : ℕ ↦ x ^ n) atTop (𝓝 0) := by
refine tendsto_zero_pow_of_v_lt_one (hx.trans_lt ?_)
rw [← exp_zero, exp_lt_exp]
simp | lemma | Topology | [
"Mathlib.GroupTheory.ArchimedeanDensely",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/WithZeroMulInt.lean | tendsto_zero_pow_of_le_exp_neg_one | In a `ℤᵐ⁰`-valued ring, powers of `x` tend to zero if `v x ≤ exp (-1)`. |
exists_pow_lt_of_le_exp_neg_one [Valued R ℤᵐ⁰] {x : R} (hx : v x ≤ exp (-1)) (γ : ℤᵐ⁰ˣ) :
∃ n, v x ^ n < γ := by
refine exists_pow_lt₀ (hx.trans_lt ?_) _
rw [← exp_zero, exp_lt_exp]
simp | lemma | Topology | [
"Mathlib.GroupTheory.ArchimedeanDensely",
"Mathlib.Topology.Algebra.Valued.ValuationTopology"
] | Mathlib/Topology/Algebra/Valued/WithZeroMulInt.lean | exists_pow_lt_of_le_exp_neg_one | null |
@[pp_with_univ]
ProfiniteGrp where
/-- The underlying profinite topological space. -/
toProfinite : Profinite
/-- The group structure. -/
[group : Group toProfinite]
/-- The above data together form a topological group. -/
[topologicalGroup : IsTopologicalGroup toProfinite] | structure | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteGrp | The category of profinite groups. A term of this type consists of a profinite
set with a topological group structure. |
@[pp_with_univ]
ProfiniteAddGrp where
/-- The underlying profinite topological space. -/
toProfinite : Profinite
/-- The additive group structure. -/
[addGroup : AddGroup toProfinite]
/-- The above data together form a topological additive group. -/
[topologicalAddGroup : IsTopologicalAddGroup toProfinite]
... | structure | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteAddGrp | The category of profinite additive groups. A term of this type consists of a profinite
set with a topological additive group structure. |
@[to_additive /-- Construct a term of `ProfiniteAddGrp` from a type endowed with the structure of a
compact and totally disconnected topological additive group.
(The condition of being Hausdorff can be omitted here because totally disconnected implies that {0}
is a closed set, thus implying Hausdorff in a topological a... | abbrev | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteGrp.of | Construct a term of `ProfiniteGrp` from a type endowed with the structure of a
compact and totally disconnected topological group.
(The condition of being Hausdorff can be omitted here because totally disconnected implies that {1}
is a closed set, thus implying Hausdorff in a topological group.) |
ProfiniteGrp.coe_of (G : Type u) [Group G] [TopologicalSpace G] [IsTopologicalGroup G]
[CompactSpace G] [TotallyDisconnectedSpace G] : (ProfiniteGrp.of G : Type u) = G :=
rfl | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteGrp.coe_of | null |
@[ext]
ProfiniteAddGrp.Hom (A B : ProfiniteAddGrp.{u}) where
private mk ::
/-- The underlying `ContinuousAddMonoidHom`. -/
hom' : A →ₜ+ B | structure | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteAddGrp.Hom | The type of morphisms in `ProfiniteAddGrp`. |
@[to_additive existing (attr := ext)]
ProfiniteGrp.Hom (A B : ProfiniteGrp.{u}) where
private mk ::
/-- The underlying `ContinuousMonoidHom`. -/
hom' : A →ₜ* B
@[to_additive] | structure | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteGrp.Hom | The type of morphisms in `ProfiniteGrp`. |
@[to_additive /-- The underlying `ContinuousAddMonoidHom`. -/]
ProfiniteGrp.Hom.hom {M N : ProfiniteGrp.{u}} (f : ProfiniteGrp.Hom M N) :
M →ₜ* N :=
ConcreteCategory.hom (C := ProfiniteGrp) f | abbrev | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteGrp.Hom.hom | The underlying `ContinuousMonoidHom`. |
@[to_additive /-- Typecheck a `ContinuousAddMonoidHom` as a morphism in `ProfiniteAddGrp`. -/]
ProfiniteGrp.ofHom {X Y : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]
[CompactSpace X] [TotallyDisconnectedSpace X] [Group Y] [TopologicalSpace Y]
[IsTopologicalGroup Y] [CompactSpace Y] [TotallyDisc... | abbrev | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ProfiniteGrp.ofHom | Typecheck a `ContinuousMonoidHom` as a morphism in `ProfiniteGrp`. |
@[to_additive (attr := simp)]
hom_id {A : ProfiniteGrp.{u}} : (𝟙 A : A ⟶ A).hom = ContinuousMonoidHom.id A := rfl
/- Provided for rewriting. -/
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | hom_id | null |
id_apply (A : ProfiniteGrp.{u}) (a : A) :
(𝟙 A : A ⟶ A) a = a := by simp
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | id_apply | null |
hom_comp {A B C : ProfiniteGrp.{u}} (f : A ⟶ B) (g : B ⟶ C) :
(f ≫ g).hom = g.hom.comp f.hom := rfl
/- Provided for rewriting. -/
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | hom_comp | null |
comp_apply {A B C : ProfiniteGrp.{u}} (f : A ⟶ B) (g : B ⟶ C) (a : A) :
(f ≫ g) a = g (f a) := by
simp only [hom_comp, ContinuousMonoidHom.comp_toFun]
@[to_additive (attr := ext)] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | comp_apply | null |
hom_ext {A B : ProfiniteGrp.{u}} {f g : A ⟶ B} (hf : f.hom = g.hom) : f = g :=
Hom.ext hf
variable {X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X]
[CompactSpace X] [TotallyDisconnectedSpace X] [Group Y] [TopologicalSpace Y]
[IsTopologicalGroup Y] [CompactSpace Y] [TotallyDisconnectedSpa... | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | hom_ext | null |
hom_ofHom (f : X →ₜ* Y) : (ofHom f).hom = f := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | hom_ofHom | null |
ofHom_hom {A B : ProfiniteGrp.{u}} (f : A ⟶ B) :
ofHom (Hom.hom f) = f := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofHom_hom | null |
ofHom_id : ofHom (ContinuousMonoidHom.id X) = 𝟙 (of X) := rfl
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofHom_id | null |
ofHom_comp (f : X →ₜ* Y) (g : Y →ₜ* Z) :
ofHom (g.comp f) = ofHom f ≫ ofHom g :=
rfl
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofHom_comp | null |
ofHom_apply (f : X →ₜ* Y) (x : X) : ofHom f x = f x := rfl
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofHom_apply | null |
inv_hom_apply {A B : ProfiniteGrp.{u}} (e : A ≅ B) (x : A) : e.inv (e.hom x) = x := by
simp
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | inv_hom_apply | null |
hom_inv_apply {A B : ProfiniteGrp.{u}} (e : A ≅ B) (x : B) : e.hom (e.inv x) = x := by
simp
@[to_additive (attr := simp)] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | hom_inv_apply | null |
coe_id (X : ProfiniteGrp) : (𝟙 X : X → X) = id :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | coe_id | null |
coe_comp {X Y Z : ProfiniteGrp} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g : X → Z) = g ∘ f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | coe_comp | null |
@[to_additive /-- Construct a term of `ProfiniteAddGrp` from a type endowed with the structure of a
profinite topological additive group. -/]
ofProfinite (G : Profinite) [Group G] [IsTopologicalGroup G] :
ProfiniteGrp := of G | abbrev | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofProfinite | Construct a term of `ProfiniteGrp` from a type endowed with the structure of a
profinite topological group. |
@[to_additive /-- The pi-type of profinite additive groups is a
profinite additive group. -/]
pi {α : Type u} (β : α → ProfiniteGrp) : ProfiniteGrp :=
let pitype := Profinite.pi fun (a : α) => (β a).toProfinite
letI (a : α): Group (β a).toProfinite := (β a).group
letI : Group pitype := Pi.group
letI : IsTopolog... | def | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | pi | The pi-type of profinite groups is a profinite group. |
@[to_additive /-- A `FiniteAddGrp` when given the discrete topology can be considered as a
profinite additive group. -/]
ofFiniteGrp (G : FiniteGrp) : ProfiniteGrp :=
letI : TopologicalSpace G := ⊥
letI : DiscreteTopology G := ⟨rfl⟩
letI : IsTopologicalGroup G := {}
of G
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofFiniteGrp | A `FiniteGrp` when given the discrete topology can be considered as a profinite group. |
@[to_additive /-- A closed additive subgroup of a profinite additive group is profinite. -/]
ofClosedSubgroup {G : ProfiniteGrp} (H : ClosedSubgroup G) : ProfiniteGrp :=
letI : CompactSpace H := inferInstance
of H.1 | def | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofClosedSubgroup | A closed subgroup of a profinite group is profinite. |
@[to_additive /-- A topological additive group that has a `ContinuousAddEquiv` to a
profinite additive group is profinite. -/]
ofContinuousMulEquiv {G : ProfiniteGrp.{u}} {H : Type v} [TopologicalSpace H]
[Group H] [IsTopologicalGroup H] (e : G ≃ₜ* H) : ProfiniteGrp.{v} :=
let _ : CompactSpace H := Homeomorph.com... | def | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ofContinuousMulEquiv | A topological group that has a `ContinuousMulEquiv` to a profinite group is profinite. |
ContinuousMulEquiv.toProfiniteGrpIso {X Y : ProfiniteGrp} (e : X ≃ₜ* Y) : X ≅ Y where
hom := ofHom e
inv := ofHom e.symm | def | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | ContinuousMulEquiv.toProfiniteGrpIso | Build an isomorphism in the category `ProfiniteGrp` from
a `ContinuousMulEquiv` between `ProfiniteGrp`s. |
@[to_additive /-- Auxiliary construction to obtain the additive group structure on the limit of
profinite additive groups. -/]
limitConePtAux : Subgroup (Π j : J, F.obj j) where
carrier := {x | ∀ ⦃i j : J⦄ (π : i ⟶ j), F.map π (x i) = x j}
mul_mem' hx hy _ _ π := by simp only [Pi.mul_apply, map_mul, hx π, hy π]
o... | def | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | limitConePtAux | The functor mapping a profinite group to its underlying profinite space. -/
@[to_additive]
instance : HasForget₂ ProfiniteGrp Profinite where
forget₂ := {
obj G := G.toProfinite
map f := CompHausLike.ofHom _ ⟨f, by continuity⟩}
@[to_additive]
instance : (forget₂ ProfiniteGrp Profinite).Faithful := {
map_in... |
limitCone : Limits.Cone F where
pt := ofProfinite (Profinite.limitCone (F ⋙ (forget₂ ProfiniteGrp Profinite))).pt
π :=
{ app := fun j => ⟨{
toFun := fun x => x.1 j
map_one' := rfl
map_mul' := fun x y => rfl
continuous_toFun := by
exact (continuous_apply j).comp (continuous_iff_le_i... | abbrev | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | limitCone | The explicit limit cone in `ProfiniteGrp`. |
limitConeIsLimit : Limits.IsLimit (limitCone F) where
lift cone := ofHom
{ ((Profinite.limitConeIsLimit (F ⋙ (forget₂ ProfiniteGrp Profinite))).lift
((forget₂ ProfiniteGrp Profinite).mapCone cone)).hom with
map_one' := Subtype.ext (funext fun j ↦ map_one (cone.π.app j).hom)
map_mul' := fun _ _... | def | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | limitConeIsLimit | `ProfiniteGrp.limitCone` is a limit cone. |
limit : ProfiniteGrp := ProfiniteGrp.of (ProfiniteGrp.limitConePtAux F)
@[ext] | abbrev | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | limit | The abbreviation for the limit of `ProfiniteGrp`s. |
limit_ext (x y : limit F) (hxy : ∀ j, x.val j = y.val j) : x = y :=
Subtype.ext (funext hxy)
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | limit_ext | null |
limit_one_val (j : J) : (1 : limit F).val j = 1 :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | limit_one_val | null |
limit_mul_val (x y : limit F) (j : J) : (x * y).val j = x.val j * y.val j :=
rfl | lemma | Topology | [
"Mathlib.Algebra.Category.Grp.FiniteGrp",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Separation.Connected"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean | limit_mul_val | null |
toFiniteQuotientFunctor (P : ProfiniteGrp) : OpenNormalSubgroup P ⥤ FiniteGrp where
obj := fun H => FiniteGrp.of (P ⧸ H.toSubgroup)
map := fun fHK => FiniteGrp.ofHom (QuotientGroup.map _ _ (.id _) (leOfHom fHK))
map_id _ := ConcreteCategory.ext <| QuotientGroup.map_id _
map_comp f g := ConcreteCategory.ext <| (... | def | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | toFiniteQuotientFunctor | The functor from `OpenNormalSubgroup P` to `FiniteGrp` sending `U` to `P ⧸ U`,
where `P : ProfiniteGrp`. |
toLimit_fun (P : ProfiniteGrp.{u}) : P →*
limit (toFiniteQuotientFunctor P ⋙ forget₂ FiniteGrp ProfiniteGrp) where
toFun p := ⟨fun _ => QuotientGroup.mk p, fun _ ↦ fun _ _ ↦ rfl⟩
map_one' := Subtype.val_inj.mp rfl
map_mul' _ _ := Subtype.val_inj.mp rfl | def | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | toLimit_fun | The `MonoidHom` from a profinite group `P` to the projective limit of its quotients by
open normal subgroups ordered by inclusion |
toLimit_fun_continuous (P : ProfiniteGrp.{u}) : Continuous (toLimit_fun P) := by
apply continuous_induced_rng.mpr (continuous_pi _)
intro H
dsimp only [Functor.comp_obj, CompHausLike.coe_of, Functor.comp_map,
CompHausLike.toCompHausLike_map, CompHausLike.compHausLikeToTop_map, Set.mem_setOf_eq,
toLimit_fu... | lemma | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | toLimit_fun_continuous | null |
toLimit (P : ProfiniteGrp.{u}) : P ⟶
limit (toFiniteQuotientFunctor P ⋙ forget₂ FiniteGrp ProfiniteGrp) :=
ofHom { toLimit_fun P with
continuous_toFun := toLimit_fun_continuous P } | def | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | toLimit | The morphism in the category of `ProfiniteGrp` from a profinite group `P` to
the projective limit of its quotients by open normal subgroups ordered by inclusion |
denseRange_toLimit (P : ProfiniteGrp.{u}) : DenseRange (toLimit P) := by
apply dense_iff_inter_open.mpr
rintro U ⟨s, hsO, hsv⟩ ⟨⟨spc, hspc⟩, uDefaultSpec⟩
simp_rw [← hsv, Set.mem_preimage] at uDefaultSpec
rcases (isOpen_pi_iff.mp hsO) _ uDefaultSpec with ⟨J, fJ, hJ1, hJ2⟩
let M := iInf (fun (j : J) => j.1.1.1... | theorem | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | denseRange_toLimit | An auxiliary result, superseded by `toLimit_surjective` |
toLimit_surjective (P : ProfiniteGrp.{u}) : Function.Surjective (toLimit P) := by
have : IsClosed (Set.range P.toLimit) :=
P.toLimit.hom.continuous_toFun.isClosedMap.isClosed_range
rw [← Set.range_eq_univ, ← closure_eq_iff_isClosed.mpr this,
Dense.closure_eq (denseRange_toLimit P)] | theorem | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | toLimit_surjective | null |
toLimit_injective (P : ProfiniteGrp.{u}) : Function.Injective (toLimit P) := by
change Function.Injective (toLimit P).hom.toMonoidHom
rw [← MonoidHom.ker_eq_bot_iff, Subgroup.eq_bot_iff_forall]
intro x h
by_contra xne1
rcases exist_openNormalSubgroup_sub_open_nhds_of_one (isOpen_compl_singleton)
(Set.mem_... | theorem | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | toLimit_injective | null |
noncomputable continuousMulEquivLimittoFiniteQuotientFunctor (P : ProfiniteGrp.{u}) :
P ≃ₜ* (limit (toFiniteQuotientFunctor P ⋙ forget₂ FiniteGrp ProfiniteGrp)) := {
(Continuous.homeoOfEquivCompactToT2
(f := Equiv.ofBijective _ ⟨toLimit_injective P, toLimit_surjective P⟩)
P.toLimit.hom.continuous_toFun) w... | def | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | continuousMulEquivLimittoFiniteQuotientFunctor | The topological group isomorphism between a profinite group and the projective limit of
its quotients by open normal subgroups |
isIso_toLimit (P : ProfiniteGrp.{u}) : IsIso (toLimit P) := by
rw [CategoryTheory.ConcreteCategory.isIso_iff_bijective]
exact ⟨toLimit_injective P, toLimit_surjective P⟩ | instance | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | isIso_toLimit | null |
noncomputable isoLimittoFiniteQuotientFunctor (P : ProfiniteGrp.{u}) :
P ≅ (limit (toFiniteQuotientFunctor P ⋙ forget₂ FiniteGrp ProfiniteGrp)) :=
ContinuousMulEquiv.toProfiniteGrpIso (continuousMulEquivLimittoFiniteQuotientFunctor P) | def | Topology | [
"Mathlib.CategoryTheory.ConcreteCategory.EpiMono",
"Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic",
"Mathlib.Topology.Algebra.ClopenNhdofOne"
] | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | isoLimittoFiniteQuotientFunctor | The isomorphism in the category of profinite group between a profinite group and
the projective limit of its quotients by open normal subgroups |
ContinuousAlternatingMap (R M N ι : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
[TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] extends
ContinuousMultilinearMap R (fun _ : ι => M) N, M [⋀^ι]→ₗ[R] N where | structure | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | ContinuousAlternatingMap | A continuous alternating map from `ι → M` to `N`, denoted `M [⋀^ι]→L[R] N`,
is a continuous map that is
- multilinear : `f (update m i (c • x)) = c • f (update m i x)` and
`f (update m i (x + y)) = f (update m i x) + f (update m i y)`;
- alternating : `f v = 0` whenever `v` has two equal coordinates. |
@[simps!]
codRestrict (f : M [⋀^ι]→L[R] N) (p : Submodule R N) (h : ∀ v, f v ∈ p) : M [⋀^ι]→L[R] p :=
{ f.toAlternatingMap.codRestrict p h with toContinuousMultilinearMap := f.1.codRestrict p h } | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | codRestrict | Projection to `ContinuousMultilinearMap`s. -/
add_decl_doc ContinuousAlternatingMap.toContinuousMultilinearMap
/-- Projection to `AlternatingMap`s. -/
add_decl_doc ContinuousAlternatingMap.toAlternatingMap
@[inherit_doc]
notation M " [⋀^" ι "]→L[" R "] " N:100 => ContinuousAlternatingMap R M N ι
namespace Continuous... |
@[simp]
coe_zero : ⇑(0 : M [⋀^ι]→L[R] N) = 0 :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | coe_zero | null |
toContinuousMultilinearMap_zero : (0 : M [⋀^ι]→L[R] N).toContinuousMultilinearMap = 0 :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toContinuousMultilinearMap_zero | null |
toAlternatingMap_zero : (0 : M [⋀^ι]→L[R] N).toAlternatingMap = 0 :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toAlternatingMap_zero | null |
@[simp]
coe_smul (f : M [⋀^ι]→L[A] N) (c : R') : ⇑(c • f) = c • ⇑f :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | coe_smul | null |
smul_apply (f : M [⋀^ι]→L[A] N) (c : R') (v : ι → M) : (c • f) v = c • f v :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | smul_apply | null |
toContinuousMultilinearMap_smul (c : R') (f : M [⋀^ι]→L[A] N) :
(c • f).toContinuousMultilinearMap = c • f.toContinuousMultilinearMap :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toContinuousMultilinearMap_smul | null |
toAlternatingMap_smul (c : R') (f : M [⋀^ι]→L[A] N) :
(c • f).toAlternatingMap = c • f.toAlternatingMap :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toAlternatingMap_smul | null |
@[simp]
coe_add : ⇑(f + g) = ⇑f + ⇑g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | coe_add | null |
add_apply (v : ι → M) : (f + g) v = f v + g v :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | add_apply | null |
toContinuousMultilinearMap_add (f g : M [⋀^ι]→L[R] N) : (f + g).1 = f.1 + g.1 :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toContinuousMultilinearMap_add | null |
toAlternatingMap_add (f g : M [⋀^ι]→L[R] N) :
(f + g).toAlternatingMap = f.toAlternatingMap + g.toAlternatingMap :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toAlternatingMap_add | null |
addCommMonoid : AddCommMonoid (M [⋀^ι]→L[R] N) :=
toContinuousMultilinearMap_injective.addCommMonoid _ rfl (fun _ _ => rfl) fun _ _ => rfl | instance | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | addCommMonoid | null |
applyAddHom (v : ι → M) : M [⋀^ι]→L[R] N →+ N :=
⟨⟨fun f => f v, rfl⟩, fun _ _ => rfl⟩
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | applyAddHom | Evaluation of a `ContinuousAlternatingMap` at a vector as an `AddMonoidHom`. |
sum_apply {α : Type*} (f : α → M [⋀^ι]→L[R] N) (m : ι → M) {s : Finset α} :
(∑ a ∈ s, f a) m = ∑ a ∈ s, f a m :=
map_sum (applyAddHom m) f s | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | sum_apply | null |
@[simps]
toMultilinearAddHom : M [⋀^ι]→L[R] N →+ ContinuousMultilinearMap R (fun _ : ι => M) N :=
⟨⟨fun f => f.1, rfl⟩, fun _ _ => rfl⟩ | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toMultilinearAddHom | Projection to `ContinuousMultilinearMap`s as a bundled `AddMonoidHom`. |
@[simps! apply]
toContinuousLinearMap [DecidableEq ι] (m : ι → M) (i : ι) : M →L[R] N :=
f.1.toContinuousLinearMap m i | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | toContinuousLinearMap | If `f` is a continuous alternating map, then `f.toContinuousLinearMap m i` is the continuous
linear map obtained by fixing all coordinates but `i` equal to those of `m`, and varying the
`i`-th coordinate. |
@[simps!]
prod (f : M [⋀^ι]→L[R] N) (g : M [⋀^ι]→L[R] N') : M [⋀^ι]→L[R] (N × N') :=
⟨f.1.prod g.1, (f.toAlternatingMap.prod g.toAlternatingMap).map_eq_zero_of_eq⟩ | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | prod | The Cartesian product of two continuous alternating maps, as a continuous alternating map. |
pi {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)] [∀ i, TopologicalSpace (M' i)]
[∀ i, Module R (M' i)] (f : ∀ i, M [⋀^ι]→L[R] M' i) : M [⋀^ι]→L[R] ∀ i, M' i :=
⟨ContinuousMultilinearMap.pi fun i => (f i).1,
(AlternatingMap.pi fun i => (f i).toAlternatingMap).map_eq_zero_of_eq⟩
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | pi | Combine a family of continuous alternating maps with the same domain and codomains `M' i` into a
continuous alternating map taking values in the space of functions `Π i, M' i`. |
coe_pi {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
[∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)] (f : ∀ i, M [⋀^ι]→L[R] M' i) :
⇑(pi f) = fun m j => f j m :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | coe_pi | null |
pi_apply {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
[∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)] (f : ∀ i, M [⋀^ι]→L[R] M' i) (m : ι → M)
(j : ι') : pi f m j = f j m :=
rfl | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | pi_apply | null |
@[simps! apply_apply symm_apply_apply apply_toContinuousMultilinearMap]
ofSubsingleton [Subsingleton ι] (i : ι) :
(M →L[R] N) ≃ M [⋀^ι]→L[R] N where
toFun f :=
{ AlternatingMap.ofSubsingleton R M N i f with
toContinuousMultilinearMap := ContinuousMultilinearMap.ofSubsingleton R M N i f }
invFun f := (... | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | ofSubsingleton | The natural equivalence between continuous linear maps from `M` to `N`
and continuous 1-multilinear alternating maps from `M` to `N`. |
ofSubsingleton_toAlternatingMap [Subsingleton ι] (i : ι) (f : M →L[R] N) :
(ofSubsingleton R M N i f).toAlternatingMap = AlternatingMap.ofSubsingleton R M N i f :=
rfl
variable (ι) {N} | theorem | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | ofSubsingleton_toAlternatingMap | null |
@[simps! toContinuousMultilinearMap apply]
constOfIsEmpty [IsEmpty ι] (m : N) : M [⋀^ι]→L[R] N :=
{ AlternatingMap.constOfIsEmpty R M ι m with
toContinuousMultilinearMap := ContinuousMultilinearMap.constOfIsEmpty R (fun _ => M) m }
@[simp] | def | Topology | [
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.Multilinear.Basic"
] | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | constOfIsEmpty | The constant map is alternating when `ι` is empty. |
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