fact
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isOpenMap_toWeakSpace_symm : IsOpenMap (toWeakSpace 𝕜 E).symm :=
IsOpenMap.of_inverse (toWeakSpaceCLM 𝕜 E).cont
(toWeakSpace 𝕜 E).left_inv (toWeakSpace 𝕜 E).right_inv
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
isOpenMap_toWeakSpace_symm
|
The canonical map from `WeakSpace 𝕜 E` to `E` is an open map.
|
WeakSpace.isOpen_of_isOpen (V : Set E)
(hV : IsOpen ((toWeakSpaceCLM 𝕜 E) '' V : Set (WeakSpace 𝕜 E))) : IsOpen V := by
simpa [Set.image_image] using isOpenMap_toWeakSpace_symm _ hV
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
WeakSpace.isOpen_of_isOpen
|
A set in `E` which is open in the weak topology is open.
|
tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter α} {f : α → WeakDual 𝕜 E}
{x : WeakDual 𝕜 E} :
Tendsto f l (𝓝 x) ↔
∀ y, Tendsto (fun i => topDualPairing 𝕜 E (f i) y) l (𝓝 (topDualPairing 𝕜 E x y)) :=
WeakBilin.tendsto_iff_forall_eval_tendsto _ ContinuousLinearMap.coe_injective
|
theorem
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
tendsto_iff_forall_eval_tendsto_topDualPairing
| null |
instAddCommGroup : AddCommGroup (WeakSpace 𝕜 E) :=
WeakBilin.instAddCommGroup (topDualPairing 𝕜 E).flip
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instAddCommGroup
| null |
instIsTopologicalAddGroup : IsTopologicalAddGroup (WeakSpace 𝕜 E) :=
WeakBilin.instIsTopologicalAddGroup (topDualPairing 𝕜 E).flip
|
instance
|
Topology
|
[
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Module.WeakBilin"
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
instIsTopologicalAddGroup
| null |
DenseRange.addChar_eq_of_eval_one_eq {A M : Type*} [TopologicalSpace A] [AddMonoidWithOne A]
[Monoid M] [TopologicalSpace M] [T2Space M] (hdr : DenseRange ((↑) : ℕ → A))
{κ₁ κ₂ : AddChar A M} (hκ₁ : Continuous κ₁) (hκ₂ : Continuous κ₂) (h : κ₁ 1 = κ₂ 1) :
κ₁ = κ₂ := by
refine DFunLike.coe_injective <| hdr.equalizer hκ₁ hκ₂ (funext fun n ↦ ?_)
simp only [Function.comp_apply, ← nsmul_one, h, AddChar.map_nsmul_eq_pow]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Group.AddChar",
"Mathlib.Topology.DenseEmbedding"
] |
Mathlib/Topology/Algebra/Monoid/AddChar.lean
|
DenseRange.addChar_eq_of_eval_one_eq
| null |
ContinuousAdd (M : Type*) [TopologicalSpace M] [Add M] : Prop where
continuous_add : Continuous fun p : M × M => p.1 + p.2
|
class
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
ContinuousAdd
|
Basic hypothesis to talk about a topological additive monoid or a topological additive
semigroup. A topological additive monoid over `M`, for example, is obtained by requiring both the
instances `AddMonoid M` and `ContinuousAdd M`.
Continuity in only the left/right argument can be stated using
`ContinuousConstVAdd α α`/`ContinuousConstVAdd αᵐᵒᵖ α`.
|
@[to_additive]
ContinuousMul (M : Type*) [TopologicalSpace M] [Mul M] : Prop where
continuous_mul : Continuous fun p : M × M => p.1 * p.2
|
class
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
ContinuousMul
|
Basic hypothesis to talk about a topological monoid or a topological semigroup.
A topological monoid over `M`, for example, is obtained by requiring both the instances `Monoid M`
and `ContinuousMul M`.
Continuity in only the left/right argument can be stated using
`ContinuousConstSMul α α`/`ContinuousConstSMul αᵐᵒᵖ α`.
|
@[to_additive (attr := continuity, fun_prop)]
continuous_mul : Continuous fun p : M × M => p.1 * p.2 :=
ContinuousMul.continuous_mul
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
continuous_mul
| null |
Filter.Tendsto.mul {α : Type*} {f g : α → M} {x : Filter α} {a b : M}
(hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x ↦ f x * g x) x (𝓝 (a * b)) :=
(continuous_mul.tendsto _).comp (hf.prodMk_nhds hg)
variable {X : Type*} [TopologicalSpace X] {f g : X → M} {s : Set X} {x : X}
@[to_additive (attr := continuity, fun_prop)]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
Filter.Tendsto.mul
| null |
Continuous.mul (hf : Continuous f) (hg : Continuous g) :
Continuous fun x => f x * g x :=
continuous_mul.comp₂ hf hg
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
Continuous.mul
| null |
ContinuousWithinAt.mul (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun x => f x * g x) s x :=
Filter.Tendsto.mul hf hg
@[to_additive (attr := fun_prop)]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
ContinuousWithinAt.mul
| null |
ContinuousAt.mul (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (fun x => f x * g x) x :=
Filter.Tendsto.mul hf hg
@[to_additive (attr := fun_prop)]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
ContinuousAt.mul
| null |
ContinuousOn.mul (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun x => f x * g x) s := fun x hx ↦
(hf x hx).mul (hg x hx)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Constructions.SumProd"
] |
Mathlib/Topology/Algebra/Monoid/Defs.lean
|
ContinuousOn.mul
| null |
continuous_map
(M : Type*) [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M]
{X Y : Type*} [Finite X] [Finite Y] (f : X → Y) :
Continuous (FunOnFinite.map (M := M) f) := by
classical
have := Fintype.ofFinite X
refine continuous_pi (fun y ↦ ?_)
simp only [FunOnFinite.map_apply_apply]
exact continuous_finset_sum _ (fun _ _ ↦ continuous_apply _)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.LinearAlgebra.Finsupp.Pi"
] |
Mathlib/Topology/Algebra/Monoid/FunOnFinite.lean
|
continuous_map
| null |
continuous_linearMap
(R M : Type*) [Semiring R] [AddCommMonoid M]
[Module R M] [TopologicalSpace M] [ContinuousAdd M]
{X Y : Type*} [Finite X] [Finite Y] (f : X → Y) :
Continuous (FunOnFinite.linearMap R M f) :=
FunOnFinite.continuous_map _ _
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.LinearAlgebra.Finsupp.Pi"
] |
Mathlib/Topology/Algebra/Monoid/FunOnFinite.lean
|
continuous_linearMap
| null |
adic_basis (I : Ideal R) : SubmodulesRingBasis fun n : ℕ => (I ^ n • ⊤ : Ideal R) :=
{ inter := by
suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by
simpa only [smul_eq_mul, mul_top, Algebra.algebraMap_self, map_id, le_inf_iff] using this
intro i j
exact ⟨max i j, pow_le_pow_right (le_max_left i j), pow_le_pow_right (le_max_right i j)⟩
leftMul := by
suffices ∀ (a : R) (i : ℕ), ∃ j : ℕ, a • I ^ j ≤ I ^ i by
simpa only [smul_top_eq_map, Algebra.algebraMap_self, map_id] using this
intro r n
use n
rintro a ⟨x, hx, rfl⟩
exact (I ^ n).smul_mem r hx
mul := by
suffices ∀ i : ℕ, ∃ j : ℕ, (↑(I ^ j) * ↑(I ^ j) : Set R) ⊆ (↑(I ^ i) : Set R) by
simpa only [smul_top_eq_map, Algebra.algebraMap_self, map_id] using this
intro n
use n
rintro a ⟨x, _hx, b, hb, rfl⟩
exact (I ^ n).smul_mem x hb }
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
adic_basis
| null |
ringFilterBasis (I : Ideal R) :=
I.adic_basis.toRing_subgroups_basis.toRingFilterBasis
|
def
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
ringFilterBasis
|
The adic ring filter basis associated to an ideal `I` is made of powers of `I`.
|
adicTopology (I : Ideal R) : TopologicalSpace R :=
(adic_basis I).topology
|
def
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
adicTopology
|
The adic topology associated to an ideal `I`. This topology admits powers of `I` as a basis of
neighborhoods of zero. It is compatible with the ring structure and is non-archimedean.
|
nonarchimedean (I : Ideal R) : @NonarchimedeanRing R _ I.adicTopology :=
I.adic_basis.toRing_subgroups_basis.nonarchimedean
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
nonarchimedean
| null |
hasBasis_nhds_zero_adic (I : Ideal R) :
HasBasis (@nhds R I.adicTopology (0 : R)) (fun _n : ℕ => True) fun n =>
((I ^ n : Ideal R) : Set R) :=
⟨by
intro U
rw [I.ringFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
constructor
· rintro ⟨-, ⟨i, rfl⟩, h⟩
replace h : ↑(I ^ i) ⊆ U := by simpa using h
exact ⟨i, trivial, h⟩
· rintro ⟨i, -, h⟩
exact ⟨(I ^ i : Ideal R), ⟨i, by simp⟩, h⟩⟩
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
hasBasis_nhds_zero_adic
|
For the `I`-adic topology, the neighborhoods of zero has basis given by the powers of `I`.
|
hasBasis_nhds_adic (I : Ideal R) (x : R) :
HasBasis (@nhds R I.adicTopology x) (fun _n : ℕ => True) fun n =>
(fun y => x + y) '' (I ^ n : Ideal R) := by
letI := I.adicTopology
have := I.hasBasis_nhds_zero_adic.map fun y => x + y
rwa [map_add_left_nhds_zero x] at this
variable (I : Ideal R) (M : Type*) [AddCommGroup M] [Module R M]
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
hasBasis_nhds_adic
| null |
adic_module_basis :
I.ringFilterBasis.SubmodulesBasis fun n : ℕ => I ^ n • (⊤ : Submodule R M) :=
{ inter := fun i j =>
⟨max i j,
le_inf_iff.mpr
⟨smul_mono_left <| pow_le_pow_right (le_max_left i j),
smul_mono_left <| pow_le_pow_right (le_max_right i j)⟩⟩
smul := fun m i =>
⟨(I ^ i • ⊤ : Ideal R), ⟨i, by simp⟩, fun a a_in => by
replace a_in : a ∈ I ^ i := by simpa [(I ^ i).mul_top] using a_in
exact smul_mem_smul a_in mem_top⟩ }
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
adic_module_basis
| null |
adicModuleTopology : TopologicalSpace M :=
@ModuleFilterBasis.topology R M _ I.adic_basis.topology _ _
(I.ringFilterBasis.moduleFilterBasis (I.adic_module_basis M))
|
def
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
adicModuleTopology
|
The topology on an `R`-module `M` associated to an ideal `M`. Submodules $I^n M$,
written `I^n • ⊤` form a basis of neighborhoods of zero.
|
openAddSubgroup (n : ℕ) : @OpenAddSubgroup R _ I.adicTopology := by
letI := I.adicTopology
refine ⟨(I ^ n).toAddSubgroup, ?_⟩
convert (I.adic_basis.toRing_subgroups_basis.openAddSubgroup n).isOpen
change (↑(I ^ n) : Set R) = ↑(I ^ n • (⊤ : Ideal R))
simp
|
def
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
openAddSubgroup
|
The elements of the basis of neighborhoods of zero for the `I`-adic topology
on an `R`-module `M`, seen as open additive subgroups of `M`.
|
IsAdic [H : TopologicalSpace R] (J : Ideal R) : Prop :=
H = J.adicTopology
|
def
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
IsAdic
|
Given a topology on a ring `R` and an ideal `J`, `IsAdic J` means the topology is the
`J`-adic one.
|
isAdic_iff [top : TopologicalSpace R] [IsTopologicalRing R] {J : Ideal R} :
IsAdic J ↔
(∀ n : ℕ, IsOpen ((J ^ n : Ideal R) : Set R)) ∧
∀ s ∈ 𝓝 (0 : R), ∃ n : ℕ, ((J ^ n : Ideal R) : Set R) ⊆ s := by
constructor
· intro H
change _ = _ at H
rw [H]
letI := J.adicTopology
constructor
· intro n
exact (J.openAddSubgroup n).isOpen'
· intro s hs
simpa using J.hasBasis_nhds_zero_adic.mem_iff.mp hs
· rintro ⟨H₁, H₂⟩
apply IsTopologicalAddGroup.ext
· apply @IsTopologicalRing.to_topologicalAddGroup
· apply (RingSubgroupsBasis.toRingFilterBasis _).toAddGroupFilterBasis.isTopologicalAddGroup
· ext s
letI := Ideal.adic_basis J
rw [J.hasBasis_nhds_zero_adic.mem_iff]
constructor <;> intro H
· rcases H₂ s H with ⟨n, h⟩
exact ⟨n, trivial, h⟩
· rcases H with ⟨n, -, hn⟩
rw [mem_nhds_iff]
exact ⟨_, hn, H₁ n, (J ^ n).zero_mem⟩
variable [TopologicalSpace R] [IsTopologicalRing R]
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
isAdic_iff
|
A topological ring is `J`-adic if and only if it admits the powers of `J` as a basis of
open neighborhoods of zero.
|
is_ideal_adic_pow {J : Ideal R} (h : IsAdic J) {n : ℕ} (hn : 0 < n) : IsAdic (J ^ n) := by
rw [isAdic_iff] at h ⊢
constructor
· intro m
rw [← pow_mul]
apply h.left
· intro V hV
obtain ⟨m, hm⟩ := h.right V hV
use m
refine Set.Subset.trans ?_ hm
cases n
· exfalso
exact Nat.not_succ_le_zero 0 hn
rw [← pow_mul, Nat.succ_mul]
apply Ideal.pow_le_pow_right
apply Nat.le_add_left
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
is_ideal_adic_pow
| null |
is_bot_adic_iff {A : Type*} [CommRing A] [TopologicalSpace A] [IsTopologicalRing A] :
IsAdic (⊥ : Ideal A) ↔ DiscreteTopology A := by
rw [isAdic_iff]
constructor
· rintro ⟨h, _h'⟩
rw [discreteTopology_iff_isOpen_singleton_zero]
simpa using h 1
· intros
constructor
· simp
· intro U U_nhds
use 1
simp [mem_of_mem_nhds U_nhds]
omit [IsTopologicalRing R] in
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
is_bot_adic_iff
| null |
IsAdic.hasBasis_nhds_zero {I : Ideal R} (hI : IsAdic I) :
(𝓝 (0 : R)).HasBasis (fun _ ↦ True) fun n ↦ ↑(I ^ n) :=
hI ▸ Ideal.hasBasis_nhds_zero_adic I
omit [IsTopologicalRing R] in
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
IsAdic.hasBasis_nhds_zero
| null |
IsAdic.hasBasis_nhds {I : Ideal R} (hI : IsAdic I) (x : R) :
(𝓝 x).HasBasis (fun _ ↦ True) fun n ↦ (x + ·) '' ↑(I ^ n) :=
hI ▸ Ideal.hasBasis_nhds_adic I x
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
IsAdic.hasBasis_nhds
| null |
WithIdeal (R : Type*) [CommRing R] where
i : Ideal R
|
class
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
WithIdeal
|
The ring `R` is equipped with a preferred ideal.
|
topologicalSpaceModule (M : Type*) [AddCommGroup M] [Module R M] : TopologicalSpace M :=
(i : Ideal R).adicModuleTopology M
/-
The next examples are kept to make sure potential future refactors won't break the instance
chaining.
-/
|
def
|
Topology
|
[
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.Topology.Algebra.Nonarchimedean.Bases",
"Mathlib.Topology.Algebra.UniformRing"
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
topologicalSpaceModule
|
The adic topology on an `R` module coming from the ideal `WithIdeal.I`.
This cannot be an instance because `R` cannot be inferred from `M`.
|
RingSubgroupsBasis {A ι : Type*} [Ring A] (B : ι → AddSubgroup A) : Prop where
/-- Condition for `B` to be a filter basis on `A`. -/
inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
/-- For each set `B` in the submodule basis on `A`, there is another basis element `B'` such
that the set-theoretic product `B' * B'` is in `B`. -/
mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
/-- For any element `x : A` and any set `B` in the submodule basis on `A`,
there is another basis element `B'` such that `B' * x` is in `B`. -/
leftMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (x * ·) ⁻¹' B i
/-- For any element `x : A` and any set `B` in the submodule basis on `A`,
there is another basis element `B'` such that `x * B'` is in `B`. -/
rightMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (· * x) ⁻¹' B i
|
structure
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
RingSubgroupsBasis
|
A family of additive subgroups on a ring `A` is a subgroups basis if it satisfies some
axioms ensuring there is a topology on `A` which is compatible with the ring structure and
admits this family as a basis of neighborhoods of zero.
|
of_comm {A ι : Type*} [CommRing A] (B : ι → AddSubgroup A)
(inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j) (mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i)
(leftMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (fun y : A => x * y) ⁻¹' B i) :
RingSubgroupsBasis B :=
{ inter
mul
leftMul
rightMul := fun x i ↦ (leftMul x i).imp fun j hj ↦ by simpa only [mul_comm] using hj }
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
of_comm
| null |
toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
RingFilterBasis A where
sets := { U | ∃ i, U = B i }
nonempty := by
inhabit ι
exact ⟨B default, default, rfl⟩
inter_sets := by
rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
obtain ⟨k, hk⟩ := hB.inter i j
use B k
constructor
· use k
· exact hk
zero' := by
rintro _ ⟨i, rfl⟩
exact (B i).zero_mem
add' := by
rintro _ ⟨i, rfl⟩
use B i
constructor
· use i
· rintro x ⟨y, y_in, z, z_in, rfl⟩
exact (B i).add_mem y_in z_in
neg' := by
rintro _ ⟨i, rfl⟩
use B i
constructor
· use i
· intro x x_in
exact (B i).neg_mem x_in
conj' := by
rintro x₀ _ ⟨i, rfl⟩
use B i
constructor
· use i
· simp
mul' := by
rintro _ ⟨i, rfl⟩
obtain ⟨k, hk⟩ := hB.mul i
use B k
constructor
· use k
· exact hk
mul_left' := by
rintro x₀ _ ⟨i, rfl⟩
obtain ⟨k, hk⟩ := hB.leftMul x₀ i
use B k
constructor
· use k
· exact hk
mul_right' := by
...
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
toRingFilterBasis
|
Every subgroups basis on a ring leads to a ring filter basis.
|
mem_addGroupFilterBasis_iff {V : Set A} :
V ∈ hB.toRingFilterBasis.toAddGroupFilterBasis ↔ ∃ i, V = B i :=
Iff.rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
mem_addGroupFilterBasis_iff
| null |
mem_addGroupFilterBasis (i) : (B i : Set A) ∈ hB.toRingFilterBasis.toAddGroupFilterBasis :=
⟨i, rfl⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
mem_addGroupFilterBasis
| null |
topology : TopologicalSpace A :=
hB.toRingFilterBasis.toAddGroupFilterBasis.topology
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
topology
|
The topology defined from a subgroups basis, admitting the given subgroups as a basis
of neighborhoods of zero.
|
hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
⟨by
intro s
rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
constructor
· rintro ⟨-, ⟨i, rfl⟩, hi⟩
exact ⟨i, trivial, hi⟩
· rintro ⟨i, -, hi⟩
exact ⟨B i, ⟨i, rfl⟩, hi⟩⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
hasBasis_nhds_zero
| null |
hasBasis_nhds (a : A) :
HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
⟨by
intro s
rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
simp only [true_and]
constructor
· rintro ⟨-, ⟨i, rfl⟩, hi⟩
use i
suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i) by
rw [h]
assumption
simp only [image_add_left, neg_add_eq_sub]
ext b
simp
· rintro ⟨i, hi⟩
use B i
constructor
· use i
· rw [image_subset_iff]
rintro b b_in
apply hi
simpa using b_in⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
hasBasis_nhds
| null |
openAddSubgroup (i : ι) : @OpenAddSubgroup A _ hB.topology :=
let _ := hB.topology
{ B i with
isOpen' := by
rw [isOpen_iff_mem_nhds]
intro a a_in
rw [(hB.hasBasis_nhds a).mem_iff]
use i, trivial
rintro b b_in
simpa using (B i).add_mem a_in b_in }
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
openAddSubgroup
|
Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology.
|
nonarchimedean : @NonarchimedeanRing A _ hB.topology := by
letI := hB.topology
constructor
intro U hU
obtain ⟨i, -, hi : (B i : Set A) ⊆ U⟩ := hB.hasBasis_nhds_zero.mem_iff.mp hU
exact ⟨hB.openAddSubgroup i, hi⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
nonarchimedean
| null |
SubmodulesRingBasis (B : ι → Submodule R A) : Prop where
/-- Condition for `B` to be a filter basis on `A`. -/
inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
/-- For any element `a : A` and any set `B` in the submodule basis on `A`,
there is another basis element `B'` such that `a • B'` is in `B`. -/
leftMul : ∀ (a : A) (i), ∃ j, a • B j ≤ B i
/-- For each set `B` in the submodule basis on `A`, there is another basis element `B'` such
that the set-theoretic product `B' * B'` is in `B`. -/
mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
|
structure
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
SubmodulesRingBasis
|
A family of submodules in a commutative `R`-algebra `A` is a submodules basis if it satisfies
some axioms ensuring there is a topology on `A` which is compatible with the ring structure and
admits this family as a basis of neighborhoods of zero.
|
toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
RingSubgroupsBasis fun i => (B i).toAddSubgroup := by
apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul
intro a i
rcases hB.leftMul a i with ⟨j, hj⟩
use j
rintro b (b_in : b ∈ B j)
exact hj ⟨b, b_in, rfl⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
toRing_subgroups_basis
| null |
topology [Nonempty ι] (hB : SubmodulesRingBasis B) : TopologicalSpace A :=
hB.toRing_subgroups_basis.topology
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
topology
|
The topology associated to a basis of submodules in an algebra.
|
SubmodulesBasis [TopologicalSpace R] (B : ι → Submodule R M) : Prop where
/-- Condition for `B` to be a filter basis on `M`. -/
inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
/-- For any element `m : M` and any set `B` in the basis, `a • m` lies in `B` for all
`a` sufficiently close to `0`. -/
smul : ∀ (m : M) (i : ι), ∀ᶠ a in 𝓝 (0 : R), a • m ∈ B i
|
structure
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
SubmodulesBasis
|
A family of submodules in an `R`-module `M` is a submodules basis if it satisfies
some axioms ensuring there is a topology on `M` which is compatible with the module structure and
admits this family as a basis of neighborhoods of zero.
|
toModuleFilterBasis : ModuleFilterBasis R M where
sets := { U | ∃ i, U = B i }
nonempty := by
inhabit ι
exact ⟨B default, default, rfl⟩
inter_sets := by
rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
obtain ⟨k, hk⟩ := hB.inter i j
use B k
constructor
· use k
· exact hk
zero' := by
rintro _ ⟨i, rfl⟩
exact (B i).zero_mem
add' := by
rintro _ ⟨i, rfl⟩
use B i
constructor
· use i
· rintro x ⟨y, y_in, z, z_in, rfl⟩
exact (B i).add_mem y_in z_in
neg' := by
rintro _ ⟨i, rfl⟩
use B i
constructor
· use i
· intro x x_in
exact (B i).neg_mem x_in
conj' := by
rintro x₀ _ ⟨i, rfl⟩
use B i
constructor
· use i
· simp
smul' := by
rintro _ ⟨i, rfl⟩
use univ
constructor
· exact univ_mem
· use B i
constructor
· use i
· rintro _ ⟨a, -, m, hm, rfl⟩
exact (B i).smul_mem _ hm
smul_left' := by
rintro x₀ _ ⟨i, rfl⟩
use B i
constructor
· use i
· intro m
...
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
toModuleFilterBasis
|
The image of a submodules basis is a module filter basis.
|
topology : TopologicalSpace M :=
hB.toModuleFilterBasis.toAddGroupFilterBasis.topology
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
topology
|
The topology associated to a basis of submodules in a module.
|
openAddSubgroup (i : ι) : @OpenAddSubgroup M _ hB.topology :=
let _ := hB.topology
{ (B i).toAddSubgroup with
isOpen' := by
letI := hB.topology
rw [isOpen_iff_mem_nhds]
intro a a_in
rw [(hB.toModuleFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
use B i
constructor
· use i
· rintro - ⟨b, b_in, rfl⟩
exact (B i).add_mem a_in b_in }
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
openAddSubgroup
|
Given a submodules basis, the basis elements as open additive subgroups in the associated
topology.
|
nonarchimedean (hB : SubmodulesBasis B) : @NonarchimedeanAddGroup M _ hB.topology := by
letI := hB.topology
constructor
intro U hU
obtain ⟨-, ⟨i, rfl⟩, hi : (B i : Set M) ⊆ U⟩ :=
hB.toModuleFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff.mp hU
exact ⟨hB.openAddSubgroup i, hi⟩
library_note "non-Archimedean non-instances"/--
The non-Archimedean subgroup basis lemmas cannot be instances because some instances
(such as `MeasureTheory.AEEqFun.instAddMonoid` or `IsTopologicalAddGroup.toContinuousAdd`)
cause the search for `@IsTopologicalAddGroup β ?m1 ?m2`, i.e. a search for a topological group where
the topology/group structure are unknown. -/
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
nonarchimedean
| null |
SubmodulesRingBasis.toSubmodulesBasis : SubmodulesBasis B :=
{ inter := hB.inter
smul := hsmul }
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
SubmodulesRingBasis.toSubmodulesBasis
| null |
RingFilterBasis.SubmodulesBasis (BR : RingFilterBasis R) (B : ι → Submodule R M) :
Prop where
/-- Condition for `B` to be a filter basis on `M`. -/
inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
/-- For any element `m : M` and any set `B i` in the submodule basis on `M`,
there is a `U` in the ring filter basis on `R` such that `U * m` is in `B i`. -/
smul : ∀ (m : M) (i : ι), ∃ U ∈ BR, U ⊆ (· • m) ⁻¹' B i
|
structure
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
RingFilterBasis.SubmodulesBasis
|
Given a ring filter basis on a commutative ring `R`, define a compatibility condition
on a family of submodules of an `R`-module `M`. This compatibility condition allows to get
a topological module structure.
|
RingFilterBasis.submodulesBasisIsBasis (BR : RingFilterBasis R) {B : ι → Submodule R M}
(hB : BR.SubmodulesBasis B) : @_root_.SubmodulesBasis ι R _ M _ _ BR.topology B :=
let _ := BR.topology
{ inter := hB.inter
smul := by
letI := BR.topology
intro m i
rcases hB.smul m i with ⟨V, V_in, hV⟩
exact mem_of_superset (BR.toAddGroupFilterBasis.mem_nhds_zero V_in) hV }
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
RingFilterBasis.submodulesBasisIsBasis
| null |
RingFilterBasis.moduleFilterBasis [Nonempty ι] (BR : RingFilterBasis R) {B : ι → Submodule R M}
(hB : BR.SubmodulesBasis B) : @ModuleFilterBasis R M _ BR.topology _ _ :=
@SubmodulesBasis.toModuleFilterBasis ι R _ M _ _ BR.topology _ _ (BR.submodulesBasisIsBasis hB)
|
def
|
Topology
|
[
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
RingFilterBasis.moduleFilterBasis
|
The module filter basis associated to a ring filter basis and a compatible submodule basis.
This allows to build a topological module structure compatible with the given module structure
and the topology associated to the given ring filter basis.
|
NonarchimedeanAddGroup (G : Type*) [AddGroup G] [TopologicalSpace G] : Prop
extends IsTopologicalAddGroup G where
is_nonarchimedean : ∀ U ∈ 𝓝 (0 : G), ∃ V : OpenAddSubgroup G, (V : Set G) ⊆ U
|
class
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
NonarchimedeanAddGroup
|
A topological additive group is nonarchimedean if every neighborhood of 0
contains an open subgroup.
|
@[to_additive]
NonarchimedeanGroup (G : Type*) [Group G] [TopologicalSpace G] : Prop
extends IsTopologicalGroup G where
is_nonarchimedean : ∀ U ∈ 𝓝 (1 : G), ∃ V : OpenSubgroup G, (V : Set G) ⊆ U
|
class
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
NonarchimedeanGroup
|
A topological group is nonarchimedean if every neighborhood of 1 contains an open subgroup.
|
NonarchimedeanRing (R : Type*) [Ring R] [TopologicalSpace R] : Prop
extends IsTopologicalRing R where
is_nonarchimedean : ∀ U ∈ 𝓝 (0 : R), ∃ V : OpenAddSubgroup R, (V : Set R) ⊆ U
|
class
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
NonarchimedeanRing
|
A topological ring is nonarchimedean if its underlying topological additive
group is nonarchimedean.
|
@[to_additive]
nonarchimedean_of_emb (f : G →* H) (emb : IsOpenEmbedding f) : NonarchimedeanGroup H :=
{ is_nonarchimedean := fun U hU =>
have h₁ : f ⁻¹' U ∈ 𝓝 (1 : G) := by
apply emb.continuous.tendsto
rwa [f.map_one]
let ⟨V, hV⟩ := is_nonarchimedean (f ⁻¹' U) h₁
⟨{ Subgroup.map f V with isOpen' := emb.isOpenMap _ V.isOpen }, Set.image_subset_iff.2 hV⟩ }
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
nonarchimedean_of_emb
|
Every nonarchimedean ring is naturally a nonarchimedean additive group. -/
instance (priority := 100) NonarchimedeanRing.to_nonarchimedeanAddGroup (R : Type*) [Ring R]
[TopologicalSpace R] [t : NonarchimedeanRing R] : NonarchimedeanAddGroup R :=
{ t with }
namespace NonarchimedeanGroup
variable {G : Type*} [Group G] [TopologicalSpace G] [NonarchimedeanGroup G]
variable {H : Type*} [Group H] [TopologicalSpace H] [IsTopologicalGroup H]
variable {K : Type*} [Group K] [TopologicalSpace K] [NonarchimedeanGroup K]
/-- If a topological group embeds into a nonarchimedean group, then it is nonarchimedean.
|
@[to_additive NonarchimedeanAddGroup.prod_subset /-- An open neighborhood of the identity in
the Cartesian product of two nonarchimedean groups contains the Cartesian product of
an open neighborhood in each group. -/]
prod_subset {U} (hU : U ∈ 𝓝 (1 : G × K)) :
∃ (V : OpenSubgroup G) (W : OpenSubgroup K), (V : Set G) ×ˢ (W : Set K) ⊆ U := by
rw [nhds_prod_eq, Filter.mem_prod_iff] at hU
rcases hU with ⟨U₁, hU₁, U₂, hU₂, h⟩
obtain ⟨V, hV⟩ := is_nonarchimedean _ hU₁
obtain ⟨W, hW⟩ := is_nonarchimedean _ hU₂
use V
grind
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
prod_subset
|
An open neighborhood of the identity in the Cartesian product of two nonarchimedean groups
contains the Cartesian product of an open neighborhood in each group.
|
@[to_additive NonarchimedeanAddGroup.prod_self_subset /-- An open neighborhood of the identity in
the Cartesian square of a nonarchimedean group contains the Cartesian square of
an open neighborhood in the group. -/]
prod_self_subset {U} (hU : U ∈ 𝓝 (1 : G × G)) :
∃ V : OpenSubgroup G, (V : Set G) ×ˢ (V : Set G) ⊆ U :=
let ⟨V, W, h⟩ := prod_subset hU
⟨V ⊓ W, by refine Set.Subset.trans (Set.prod_mono ?_ ?_) ‹_› <;> simp⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
prod_self_subset
|
An open neighborhood of the identity in the Cartesian square of a nonarchimedean group
contains the Cartesian square of an open neighborhood in the group.
|
@[to_additive /-- The Cartesian product of two nonarchimedean groups is nonarchimedean. -/]
Prod.instNonarchimedeanGroup : NonarchimedeanGroup (G × K) where
is_nonarchimedean _ hU :=
let ⟨V, W, h⟩ := prod_subset hU
⟨V.prod W, ‹_›⟩
|
instance
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
Prod.instNonarchimedeanGroup
|
The Cartesian product of two nonarchimedean groups is nonarchimedean.
|
left_mul_subset (U : OpenAddSubgroup R) (r : R) :
∃ V : OpenAddSubgroup R, r • (V : Set R) ⊆ U :=
⟨U.comap (AddMonoidHom.mulLeft r) (continuous_mul_left r), (U : Set R).image_preimage_subset _⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
left_mul_subset
|
The Cartesian product of two nonarchimedean rings is nonarchimedean. -/
instance : NonarchimedeanRing (R × S) where
is_nonarchimedean := NonarchimedeanAddGroup.is_nonarchimedean
/-- Given an open subgroup `U` and an element `r` of a nonarchimedean ring, there is an open
subgroup `V` such that `r • V` is contained in `U`.
|
mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U := by
let ⟨V, H⟩ := prod_self_subset <| (U.isOpen.preimage continuous_mul).mem_nhds <| by
simpa only [Set.mem_preimage, Prod.snd_zero, mul_zero] using U.zero_mem
use V
rintro v ⟨a, ha, b, hb, hv⟩
have hy := H (Set.mk_mem_prod ha hb)
simp only [Set.mem_preimage, SetLike.mem_coe, hv] at hy
rw [SetLike.mem_coe]
exact hy
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean
|
mul_subset
|
An open subgroup of a nonarchimedean ring contains the square of another one.
|
@[to_additive]
exists_openSubgroup_separating {a b : G} (h : a ≠ b) :
∃ V : OpenSubgroup G, Disjoint (a • (V : Set G)) (b • V) := by
obtain ⟨u, v, _, open_v, mem_u, mem_v, dis⟩ := t2_separation (h ∘ inv_mul_eq_one.mp)
obtain ⟨V, hV⟩ := is_nonarchimedean v (open_v.mem_nhds mem_v)
use V
simp only [Disjoint, Set.le_eq_subset, Set.bot_eq_empty, Set.subset_empty_iff]
intro x mem_aV mem_bV
by_contra! con
obtain ⟨s, hs⟩ := con
have hsa : s ∈ a • (V : Set G) := mem_aV hs
have hsb : s ∈ b • (V : Set G) := mem_bV hs
rw [mem_leftCoset_iff] at hsa hsb
refine dis.subset_compl_right mem_u (hV ?_)
simpa [mul_assoc] using mul_mem hsa (inv_mem hsb)
@[to_additive]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Nonarchimedean.Basic"
] |
Mathlib/Topology/Algebra/Nonarchimedean/TotallyDisconnected.lean
|
exists_openSubgroup_separating
| null |
Rat.denseRange_cast {𝕜} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
[TopologicalSpace 𝕜] [OrderTopology 𝕜]
[Archimedean 𝕜] : DenseRange ((↑) : ℚ → 𝕜) :=
dense_of_exists_between fun _ _ h => Set.exists_range_iff.2 <| exists_rat_btwn h
|
theorem
|
Topology
|
[
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Algebra.Group.Subgroup.ZPowers.Basic",
"Mathlib.Topology.Order.Basic"
] |
Mathlib/Topology/Algebra/Order/Archimedean.lean
|
Rat.denseRange_cast
|
Rational numbers are dense in a linear ordered archimedean field.
|
@[to_additive /-- An additive subgroup of an archimedean linear ordered additive commutative group
with order topology is dense provided that for all positive `ε` there exists a positive element of
the subgroup that is less than `ε`. -/]
dense_of_not_isolated_one (S : Subgroup G) (hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε) :
Dense (S : Set G) := by
cases subsingleton_or_nontrivial G
· refine fun x => _root_.subset_closure ?_
rw [Subsingleton.elim x 1]
exact one_mem S
refine dense_of_exists_between fun a b hlt => ?_
rcases hS (b / a) (one_lt_div'.2 hlt) with ⟨g, hgS, hg0, hg⟩
rcases (existsUnique_add_zpow_mem_Ioc hg0 1 a).exists with ⟨m, hm⟩
rw [one_mul] at hm
refine ⟨g ^ m, zpow_mem hgS _, hm.1, hm.2.trans_lt ?_⟩
rwa [lt_div_iff_mul_lt'] at hg
|
theorem
|
Topology
|
[
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Algebra.Group.Subgroup.ZPowers.Basic",
"Mathlib.Topology.Order.Basic"
] |
Mathlib/Topology/Algebra/Order/Archimedean.lean
|
dense_of_not_isolated_one
|
A subgroup of an archimedean linear ordered multiplicative commutative group with order
topology is dense provided that for all `ε > 1` there exists an element of the subgroup
that belongs to `(1, ε)`.
|
@[to_additive /-- Let `S` be a nontrivial additive subgroup in an archimedean linear ordered
additive commutative group `G` with order topology. If the set of positive elements of `S` does not
have a minimal element, then `S` is dense `G`. -/]
dense_of_no_min (S : Subgroup G) (hbot : S ≠ ⊥)
(H : ¬∃ a : G, IsLeast { g : G | g ∈ S ∧ 1 < g } a) : Dense (S : Set G) := by
refine S.dense_of_not_isolated_one fun ε ε1 => ?_
contrapose! H
exact exists_isLeast_one_lt hbot ε1 (disjoint_left.2 H)
|
theorem
|
Topology
|
[
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Algebra.Group.Subgroup.ZPowers.Basic",
"Mathlib.Topology.Order.Basic"
] |
Mathlib/Topology/Algebra/Order/Archimedean.lean
|
dense_of_no_min
|
Let `S` be a nontrivial subgroup in an archimedean linear ordered multiplicative commutative
group `G` with order topology. If the set of elements of `S` that are greater than one
does not have a minimal element, then `S` is dense `G`.
|
@[to_additive dense_or_cyclic
/-- An additive subgroup of an archimedean linear ordered additive commutative group `G`
with order topology either is dense in `G` or is a cyclic subgroup. -/]
dense_or_cyclic (S : Subgroup G) : Dense (S : Set G) ∨ ∃ a : G, S = closure {a} := by
refine (em _).imp (dense_of_not_isolated_one S) fun h => ?_
push_neg at h
rcases h with ⟨ε, ε1, hε⟩
exact cyclic_of_isolated_one ε1 (disjoint_left.2 hε)
variable [Nontrivial G] [DenselyOrdered G]
|
theorem
|
Topology
|
[
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Algebra.Group.Subgroup.ZPowers.Basic",
"Mathlib.Topology.Order.Basic"
] |
Mathlib/Topology/Algebra/Order/Archimedean.lean
|
dense_or_cyclic
|
A subgroup of an archimedean linear ordered multiplicative commutative group `G` with order
topology either is dense in `G` or is a cyclic subgroup.
|
@[to_additive dense_xor'_cyclic
/-- In a nontrivial densely linear ordered archimedean topological additive group,
a subgroup is either dense or is cyclic, but not both.
For a non-exclusive `Or` version with weaker assumptions, see `AddSubgroup.dense_or_cyclic` above.
-/]
dense_xor'_cyclic (s : Subgroup G) :
Xor' (Dense (s : Set G)) (∃ a, s = .zpowers a) := by
if hd : Dense (s : Set G) then
simp only [hd, xor_true]
rintro ⟨a, rfl⟩
exact not_denseRange_zpow hd
else
simp only [hd, xor_false, id, zpowers_eq_closure]
exact s.dense_or_cyclic.resolve_left hd
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Algebra.Group.Subgroup.ZPowers.Basic",
"Mathlib.Topology.Order.Basic"
] |
Mathlib/Topology/Algebra/Order/Archimedean.lean
|
dense_xor'_cyclic
|
In a nontrivial densely linear ordered archimedean topological multiplicative group,
a subgroup is either dense or is cyclic, but not both.
For a non-exclusive `Or` version with weaker assumptions, see `Subgroup.dense_or_cyclic` above.
|
dense_iff_ne_zpowers {s : Subgroup G} :
Dense (s : Set G) ↔ ∀ a, s ≠ .zpowers a := by
simp [xor_iff_iff_not.1 s.dense_xor'_cyclic]
|
theorem
|
Topology
|
[
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Algebra.Group.Subgroup.ZPowers.Basic",
"Mathlib.Topology.Order.Basic"
] |
Mathlib/Topology/Algebra/Order/Archimedean.lean
|
dense_iff_ne_zpowers
| null |
Filter.Tendsto.atTop_mul_pos {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atTop)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by
refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC))
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)), hf.eventually_ge_atTop 0] with x hg
hf using mul_le_mul_of_nonneg_left hg.le hf
@[deprecated (since := "2025-03-18")]
alias Filter.Tendsto.atTop_mul := Filter.Tendsto.atTop_mul_pos
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.atTop_mul_pos
|
In a linearly ordered semifield with the order topology, if `f` tends to `Filter.atTop` and `g`
tends to a positive constant `C` then `f * g` tends to `Filter.atTop`.
|
Filter.Tendsto.pos_mul_atTop {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atTop := by
simpa only [mul_comm] using hg.atTop_mul_pos hC hf
@[deprecated (since := "2025-03-18")]
alias Filter.Tendsto.mul_atTop := Filter.Tendsto.pos_mul_atTop
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.pos_mul_atTop
|
In a linearly ordered semifield with the order topology, if `f` tends to a positive constant `C`
and `g` tends to `Filter.atTop` then `f * g` tends to `Filter.atTop`.
|
inv_atTop₀ : (atTop : Filter 𝕜)⁻¹ = 𝓝[>] 0 :=
(((atTop_basis_Ioi' (0 : 𝕜)).map _).comp_surjective inv_surjective).eq_of_same_basis <|
(nhdsGT_basis _).congr (by simp) fun a ha ↦ by simp [inv_Ioi₀ (inv_pos.2 ha)]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
inv_atTop₀
| null |
inv_nhdsGT_zero : (𝓝[>] (0 : 𝕜))⁻¹ = atTop := by rw [← inv_atTop₀, inv_inv]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
inv_nhdsGT_zero
| null |
tendsto_inv_nhdsGT_zero : Tendsto (fun x : 𝕜 => x⁻¹) (𝓝[>] (0 : 𝕜)) atTop :=
inv_nhdsGT_zero.le
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_inv_nhdsGT_zero
|
The function `x ↦ x⁻¹` tends to `+∞` on the right of `0`.
|
tendsto_inv_atTop_nhdsGT_zero : Tendsto (fun r : 𝕜 => r⁻¹) atTop (𝓝[>] (0 : 𝕜)) :=
inv_atTop₀.le
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_inv_atTop_nhdsGT_zero
|
The function `r ↦ r⁻¹` tends to `0` on the right as `r → +∞`.
|
tendsto_inv_atTop_zero : Tendsto (fun r : 𝕜 => r⁻¹) atTop (𝓝 0) :=
tendsto_inv_atTop_nhdsGT_zero.mono_right inf_le_left
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_inv_atTop_zero
| null |
Filter.Tendsto.inv_tendsto_atTop (h : Tendsto f l atTop) : Tendsto f⁻¹ l (𝓝 0) :=
tendsto_inv_atTop_zero.comp h
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.inv_tendsto_atTop
| null |
Filter.Tendsto.inv_tendsto_nhdsGT_zero (h : Tendsto f l (𝓝[>] 0)) : Tendsto f⁻¹ l atTop :=
tendsto_inv_nhdsGT_zero.comp h
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.inv_tendsto_nhdsGT_zero
| null |
tendsto_pow_neg_atTop {n : ℕ} (hn : n ≠ 0) :
Tendsto (fun x : 𝕜 => x ^ (-(n : ℤ))) atTop (𝓝 0) := by
simpa only [zpow_neg, zpow_natCast] using (tendsto_pow_atTop (α := 𝕜) hn).inv_tendsto_atTop
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_pow_neg_atTop
|
The function `x^(-n)` tends to `0` at `+∞` for any positive natural `n`.
A version for positive real powers exists as `tendsto_rpow_neg_atTop`.
|
tendsto_zpow_atTop_zero {n : ℤ} (hn : n < 0) :
Tendsto (fun x : 𝕜 => x ^ n) atTop (𝓝 0) := by
lift -n to ℕ using le_of_lt (neg_pos.mpr hn) with N h
rw [← neg_pos, ← h, Nat.cast_pos] at hn
simpa only [h, neg_neg] using tendsto_pow_neg_atTop hn.ne'
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_zpow_atTop_zero
| null |
IsTopologicalRing.of_norm {R 𝕜 : Type*} [NonUnitalNonAssocRing R]
[Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
[TopologicalSpace R] [IsTopologicalAddGroup R] (norm : R → 𝕜)
(norm_nonneg : ∀ x, 0 ≤ norm x) (norm_mul_le : ∀ x y, norm (x * y) ≤ norm x * norm y)
(nhds_basis : (𝓝 (0 : R)).HasBasis ((0 : 𝕜) < ·) (fun ε ↦ { x | norm x < ε })) :
IsTopologicalRing R := by
have h0 : ∀ f : R → R, ∀ c ≥ (0 : 𝕜), (∀ x, norm (f x) ≤ c * norm x) →
Tendsto f (𝓝 0) (𝓝 0) := by
refine fun f c c0 hf ↦ (nhds_basis.tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_
rcases exists_pos_mul_lt ε0 c with ⟨δ, δ0, hδ⟩
refine ⟨δ, δ0, fun x hx ↦ (hf _).trans_lt ?_⟩
exact (mul_le_mul_of_nonneg_left (le_of_lt hx) c0).trans_lt hδ
apply IsTopologicalRing.of_addGroup_of_nhds_zero
case hmul =>
refine ((nhds_basis.prod nhds_basis).tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_
refine ⟨(1, ε), ⟨one_pos, ε0⟩, fun (x, y) ⟨hx, hy⟩ => ?_⟩
simp only at *
calc norm (x * y) ≤ norm x * norm y := norm_mul_le _ _
_ < ε := (mul_le_of_le_one_left (norm_nonneg _) hx.le).trans_lt hy
case hmul_left => exact fun x => h0 _ (norm x) (norm_nonneg _) (norm_mul_le x)
case hmul_right =>
exact fun y => h0 (· * y) (norm y) (norm_nonneg y) fun x =>
(norm_mul_le x y).trans_eq (mul_comm _ _)
variable {𝕜 α : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
[TopologicalSpace 𝕜] [OrderTopology 𝕜]
{l : Filter α} {f g : α → 𝕜}
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
IsTopologicalRing.of_norm
|
If a (possibly non-unital and/or non-associative) ring `R` admits a submultiplicative
nonnegative norm `norm : R → 𝕜`, where `𝕜` is a linear ordered field, and the open balls
`{ x | norm x < ε }`, `ε > 0`, form a basis of neighborhoods of zero, then `R` is a topological
ring.
|
Filter.Tendsto.atTop_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atTop)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by
have := hf.atTop_mul_pos (neg_pos.2 hC) hg.neg
simpa only [Function.comp_def, neg_mul_eq_mul_neg, neg_neg] using
tendsto_neg_atTop_atBot.comp this
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.atTop_mul_neg
|
In a linearly ordered field with the order topology, if `f` tends to `Filter.atTop` and `g`
tends to a negative constant `C` then `f * g` tends to `Filter.atBot`.
|
Filter.Tendsto.neg_mul_atTop {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atBot := by
simpa only [mul_comm] using hg.atTop_mul_neg hC hf
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.neg_mul_atTop
|
In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and
`g` tends to `Filter.atTop` then `f * g` tends to `Filter.atBot`.
|
Filter.Tendsto.atBot_mul_pos {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atBot)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_pos hC hg
simpa [Function.comp_def] using tendsto_neg_atTop_atBot.comp this
@[deprecated (since := "2025-03-18")]
alias Filter.Tendsto.atBot_mul := Filter.Tendsto.atBot_mul_pos
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.atBot_mul_pos
|
In a linearly ordered field with the order topology, if `f` tends to `Filter.atBot` and `g`
tends to a positive constant `C` then `f * g` tends to `Filter.atBot`.
|
Filter.Tendsto.atBot_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atBot)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_neg hC hg
simpa [Function.comp_def] using tendsto_neg_atBot_atTop.comp this
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.atBot_mul_neg
|
In a linearly ordered field with the order topology, if `f` tends to `Filter.atBot` and `g`
tends to a negative constant `C` then `f * g` tends to `Filter.atTop`.
|
Filter.Tendsto.pos_mul_atBot {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atBot := by
simpa only [mul_comm] using hg.atBot_mul_pos hC hf
@[deprecated (since := "2025-03-18")]
alias Filter.Tendsto.mul_atBot := Filter.Tendsto.pos_mul_atBot
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.pos_mul_atBot
|
In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and
`g` tends to `Filter.atBot` then `f * g` tends to `Filter.atBot`.
|
Filter.Tendsto.neg_mul_atBot {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atTop := by
simpa only [mul_comm] using hg.atBot_mul_neg hC hf
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.neg_mul_atBot
|
In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and
`g` tends to `Filter.atBot` then `f * g` tends to `Filter.atTop`.
|
inv_atBot₀ : (atBot : Filter 𝕜)⁻¹ = 𝓝[<] 0 :=
(((atBot_basis_Iio' (0 : 𝕜)).map _).comp_surjective inv_surjective).eq_of_same_basis <|
(nhdsLT_basis _).congr (by simp) fun a ha ↦ by simp [inv_Iio₀ (inv_neg''.2 ha)]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
inv_atBot₀
| null |
inv_nhdsLT_zero : (𝓝[<] (0 : 𝕜))⁻¹ = atBot := by
rw [← inv_atBot₀, inv_inv]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
inv_nhdsLT_zero
| null |
tendsto_inv_nhdsLT_zero : Tendsto (fun x : 𝕜 => x⁻¹) (𝓝[<] (0 : 𝕜)) atBot :=
inv_nhdsLT_zero.le
@[deprecated (since := "2025-04-23")]
alias tendsto_inv_zero_atBot := tendsto_inv_nhdsLT_zero
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_inv_nhdsLT_zero
|
The function `x ↦ x⁻¹` tends to `-∞` on the left of `0`.
|
tendsto_inv_atBot_nhdsLT_zero : Tendsto (fun r : 𝕜 => r⁻¹) atBot (𝓝[<] (0 : 𝕜)) :=
inv_atBot₀.le
@[deprecated (since := "2025-04-23")]
alias tendsto_inv_atBot_zero' := tendsto_inv_atBot_nhdsLT_zero
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_inv_atBot_nhdsLT_zero
|
The function `r ↦ r⁻¹` tends to `0` on the left as `r → -∞`.
|
tendsto_inv_atBot_zero : Tendsto (fun r : 𝕜 => r⁻¹) atBot (𝓝 0) :=
tendsto_inv_atBot_nhdsLT_zero.mono_right inf_le_left
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_inv_atBot_zero
| null |
Filter.Tendsto.div_atTop {a : 𝕜} (h : Tendsto f l (𝓝 a)) (hg : Tendsto g l atTop) :
Tendsto (fun x => f x / g x) l (𝓝 0) := by
simp only [div_eq_mul_inv]
exact mul_zero a ▸ h.mul (tendsto_inv_atTop_zero.comp hg)
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.div_atTop
| null |
Filter.Tendsto.div_atBot {a : 𝕜} (h : Tendsto f l (𝓝 a)) (hg : Tendsto g l atBot) :
Tendsto (fun x => f x / g x) l (𝓝 0) := by
simp only [div_eq_mul_inv]
exact mul_zero a ▸ h.mul (tendsto_inv_atBot_zero.comp hg)
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.div_atBot
| null |
Filter.Tendsto.const_div_atTop (hg : Tendsto g l atTop) (r : 𝕜) :
Tendsto (fun n ↦ r / g n) l (𝓝 0) :=
tendsto_const_nhds.div_atTop hg
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.const_div_atTop
| null |
Filter.Tendsto.const_div_atBot (hg : Tendsto g l atBot) (r : 𝕜) :
Tendsto (fun n ↦ r / g n) l (𝓝 0) :=
tendsto_const_nhds.div_atBot hg
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.const_div_atBot
| null |
Filter.Tendsto.inv_tendsto_atBot (h : Tendsto f l atBot) : Tendsto f⁻¹ l (𝓝 0) :=
tendsto_inv_atBot_zero.comp h
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.inv_tendsto_atBot
| null |
Filter.Tendsto.inv_tendsto_nhdsLT_zero (h : Tendsto f l (𝓝[<] 0)) : Tendsto f⁻¹ l atBot :=
tendsto_inv_nhdsLT_zero.comp h
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Order.Filter.AtTopBot.Field",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Order.Group"
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.inv_tendsto_nhdsLT_zero
| null |
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